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New Special Instructions
| author | Klaus Thoden <kthoden@mpiwg-berlin.mpg.de> |
|---|---|
| date | Wed, 30 Jul 2014 15:58:21 +0200 |
| parents | 22d6a63640c6 |
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<?xml version="1.0"?> <!DOCTYPE TEI PUBLIC "-//TEI P5//DTD Main DTD Driver File//EN" "../dtd/tei2.dtd" [ <!ENTITY % Iso1 SYSTEM "../dtd/ISOlat1.pen"> <!ENTITY % Iso2 SYSTEM "../dtd/ISOlat2.pen"> <!ENTITY % Archent SYSTEM "../dtd/archimedes.pen"> %Iso1; %Iso2; %Archent; ]> <TEI> <teiHeader status="new" type="text"> <fileDesc><titleStmt><title>A Mathematical and Philosophical Dictionary</title><author>Charles Hutton</author></titleStmt> <publicationStmt> <p>Berlin: Max-Planck-Institut für Wissenschaftsgeschichte, 2000 <!-- archimedes_locator:078.xml: --> </p></publicationStmt><sourceDesc default="false"> <p>London, 1796. </p></sourceDesc></fileDesc> <encodingDesc> <refsDecl doctype="tei.2"> <state n="chunk" unit="entry"/> </refsDecl> <refsDecl doctype="tei.2"> <state unit="alphabetic letter"/> <state unit="entry"/> </refsDecl> <refsDecl doctype="tei.2"> <state unit="page"/> </refsDecl> </encodingDesc> <profileDesc> <langUsage default="false"> <language ident="en" xml:lang="en">English </language> <language ident="la" xml:lang="la">Latin </language> <language ident="grk" xml:lang="grk">Greek </language> </langUsage> </profileDesc> </teiHeader> <text> <front> <pb/> <titlePage> <docTitle> <titlePart rend="center" type="main">A MATHEMATICAL <hi rend="smallcaps">AND</hi> PHILOSOPHICAL DICTIONARY:</titlePart> <titlePart rend="center" type="main">CONTAINING <hi rend="italics">AN EXPLANATION OF THE TERMS, AND AN ACCOUNT OF THE SEVERAL SUBJECTS,</hi></titlePart> <titlePart rend="center" type="main">COMPRIZED UNDER THE HEADS MATHEMATICS, ASTRONOMY, <hi rend="smallcaps">AND</hi> PHILOSOPHY BOTH NATURAL AND EXPERIMENTAL: WITH AN HISTORICAL ACCOUNT OF THE RISE, PROGRESS, AND PRESENT STATE OF THESE SCIENCES: ALSO MEMOIRS OF THE LIVES AND WRITINGS OF THE MOST EMINENT AUTHORS, BOTH ANCIENT AND MODERN, <hi rend="italics">WHO BY THEIR DISCOVERIES OR IMPROVEMENTS HAVE CONTRIBUTED TO THE ADVANCEMENT OF THEM.</hi> IN TWO VOLUMES. WITH MANY CUTS AND COPPER-PLATES.</titlePart> </docTitle> <byline rend="center"><hi rend="smallcaps">By</hi> <docAuthor>CHARLES HUTTON</docAuthor>, LL.D. F. R. SS. OF LONDON AND EDINBURGH, AND OF THE PHILOSOPHICAL SOCIETIES OF HAARLEM AND AMERICA; AND PROFESSOR OF MATHEMATICS IN THE ROYAL MILITARY ACADEMY, WOOLWICK.</byline> <docTitle> <titlePart rend="center" type="main">VOL. I.</titlePart> </docTitle> <docImprint rend="center"><pubPlace rend="italics">LONDON:</pubPlace> PRINTED BY J. DAVIS, FOR J. JOHNSON, IN ST. PAUL'S CHURCH-YARD; AND G. G. AND J. ROBINSON, IN PATERNOSTER-ROW. <docDate>M.DCC.XCVI.</docDate></docImprint> </titlePage> <div1 part="N" org="uniform" sample="complete" type="preface"><pb/><pb/><head>PREFACE.</head><p>AMONG the Dictionaries of Arts and Sciences which have been published, of late years, in various parts of Europe, it is matter of surprise that Philosophy and Mathematics should have been so far overlooked as not to be thought worthy of a separate Treatise, in this form. These Sciences constitute a large portion of the present stock of human knowledge, and have been usually considered as possessing a degree of importance to which few others are entitled; and yet we have hitherto had no distinct Lexicon, in which their constituent parts and technical terms have been explained, with that amplitude and precision, which the great improvements of the Moderns, as well as the rising dignity of the Subject, seem to demand.</p><p><hi rend="smallcaps">The</hi> only works of this kind in the English language, deserving of notice, are Harris's Lexicon Technicum, and Stone's Mathematical Dictionary; the former of which, though a valuable performance at the time it was written, is now become too dry and obsolete to be referred to with pleasure or satisfaction: and the latter, consisting only of one volume in 8vo, must be regarded merely as an unfinished sketch, or brief compendium, extremely limited in its plan, and necessarily desieient in useful in formation.</p><p><hi rend="smallcaps">It</hi> became, therefore, the only resource of the Reader, in many cases where explanation was wanted, to have recourse to Chambers's Dictionary, in four large Volumes folio, or to the Encyclopædia Britannica, now in eighteen large volumes 4to, or the still more, stupendous performance of the French Encyclopedists; and even here his expectations might be frequently disappointed. These great and useful works, aiming at a general comprehension of the whole circle of the Sciences, are sometimes very delicient in their descriptions of particular branches; it being almost impossible, in such <pb n="vi"/> extensive undertakings, to appreciate, with exactness, the due value of every article: They are, besides, so voluminous and heterogeneous in their nature, as to render a frequent reference to them extremely inconvenient; and even if this were not the case, their high price puts them out of the reach of the generality of readers.</p> <p><hi rend="smallcaps">WITH</hi> a view to obviate these defects, the Public are here presented with a Dic- tionary of a moderate size and price, which is devoted solely to Philosophical and Ma- thematical subjects. It is a work for which materials have been collecting through a course of many years ; and is the result of great labour and reading. Not only most of the Encyclopedias already extant, and the various publications of the Learned Societies throughout Europe, have been carefully consulted, but also all the original works, of any reputation, which have hitherto appeared upon these subjects, from the earliest writers down to the present times.</p> <p><hi rend="smallcaps">FROM</hi> the latter of there sources, in particular, a considerable portion of information has been obtained, which the curious reader will find, in many cases, to be highly in- teresting and important. The History of Algebra, for instance, which is detailed at considerable length in the First Volume, under the head of that Article, will afford sufficient evidence to shew in what a superficial and partial way the inquiry has been hitherto investigated, even by professed writers on the subject; the principal of whom are M. Montucla, our countryman the celebrated Dr. Wallis, and the Abbé De Gua, a late French author, who has pretended to correct the Doctor's errors and misrepresen- tations.</p> <p><hi rend="smallcaps">REGULAR</hi> historical details are in like manner given of the origin and progress of each of these Sciences, as well as of the inventions and improvements by which they have been gradually brought from their first rude beginnings to their present ad- vanced state.</p> <p><hi rend="smallcaps">IT</hi> is also to be observed, that besides the articles common to the generality of Dictionaries of this kind, an interesting Biographical Account is here introduced of the most celebrated Philosophers and Mathematicians, both ancient and modern; among which will be found the Lives of many eminent characters, who have hi- therto been either wholly overlooked, or very imperfectly recorded. Complete lists of their works are also subjoined to each Article, where they could be procured; which cannot but prove highly acceptable to that class of readers, who are desirous of ob- taining the most satisfactory information upon the subjects of their particular enqui- ries and pursuits. On the head of Biography however the Author has still to <pb n="vii"/> to his list of authors, not having been able to procure any circumstantial accounts of their lives. He could have wished to have comprised in his list, the lives of all such public literary characters as the University Professors of Astronomy, Philosophy and Mathematics, as well as those of the other more respectable class of Authors on those Sciences. He will therefore thankfully receive the communication of any such memoirs from the hands of gentlemen possessed of them; as well as hints and information on such useful improvements in the sciences as may have been overlooked in this Dictionary, or any articles that may here have been imperfectly or incorrectly treated; that he may at some future time, by adding them to this work, render it still more complete and deserving the public notice.</p><p><hi rend="smallcaps">As</hi> this work is an attempt to separate the words in the sciences of Astronomy, Mathematics, and Philosophy, from those of other arts or sciences, in several of which there are already separate Dictionaries; as in Chemistry, Geography, Music, Marine and Naval affairs, &c; words sometimes occurred which it was rather doubtful whether they could be considered as properly belonging to the present work or not; in which case many of such words have been here inserted. But such as appeared clearly and peculiarly to belong to any of those other subjects, have been either wholly omitted, or else have had a very short account only given of them. The readers of this work therefore, recollecting that it is not a General Dictionary of all the Arts and Sciences, will not expect to find all sorts of words and subjects here treated of; but such only as peculiarly appertain to the proper matter of the work. And therefore, although some few words may inadvertently have been omitted; yet when the Reader does not immediately find every word which he wishes to consult, he will not always consider them as omissions of the Author, but for the most part as relating to some other science foreign to this Dictionary.</p><p><hi rend="smallcaps">In</hi> all cases where it could be conveniently done, the necessary figures and diagrams are inserted in the same page with the subjects which they are designed to elucidate; a method which will be found much more commodious than that of putting them in separate plates at the end of each volume, but, which has added very considerably to the expence of the undertaking: where the subjects are of such a nature that they could not be otherwise well represented, they are engraved on Copperplates.</p><p><hi rend="smallcaps">As</hi> the whole of this work was written before it was put to the press, the Reader will find it of an equal and uniform nature and construction throughout; in which respect many publications of this kind are very defective, from the subjects being diffusely treated under the first letters of the alphabet, while articles <pb n="viii"/> of equal importance in the latter part are so much abridged as to be rendered al- most useless, in order that the whole might be comprised in a limited number of sheets, according to proposals made before the works were composed. The pre- sent Dictionary having been completed without any of there unfavourable circum- stances, will be found in most cases equally instructive and useful, and may be consulted with no less advantage by the Man of Science than the Student. </p></div1> </front> <body><pb/> <!-- <C>A PHILOSOPHICAL and MATHEMATICAL DICTIONARY.</C> --> <div0 part="N" n="A" org="uniform" sample="complete" type="alphabetic letter"><head>A</head><cb/><div1 part="N" n="ABACIST" org="uniform" sample="complete" type="entry"><head>ABACIST</head><p>, an Arithmetician. In this sense we find the word used by William of Malmesbury, in his History <hi rend="italics">de Gestis Anglorum,</hi> written about the year 1150; where he shews that one Gerbert, a learned monk of France, who was afterwards made pope of Rome in the year 998 or 999, by the name of Silvester the 2d, was the first who got from the Saracens the abacus, and that he taught such rules concerning it, as the Abacists themselves could hardly understand.</p></div1><div1 part="N" n="ABACUS" org="uniform" sample="complete" type="entry"><head>ABACUS</head><p>, <hi rend="italics">in Arithmetic,</hi> an ancient instrument used by most nations for casting up accounts, or performing arithmetical calculations: it is by some derived from the Greek <foreign xml:lang="greek">a<*>ac</foreign>, which signifies a cupboard or beaufet, perhaps from the similarity of the form of this instrument; and by others it is derived from the Phœnician <hi rend="italics">abak,</hi> which signifies dust or powder, because it was said that this instrument was sometimes made of a square board or tablet, which was powdered over with fine sand or dust, in which were traced the figures or characters used in making calculations, which could thence be easily defaced, and the abacus refitted for use. But Lucas Paciolus, in the sirst part of his second distinction, thinks it is a corruption of Arabicus, by which he meant their Algorism, or the method of numeral computation received from them.</p><p>We find this instrument for computation in use, under some variations, with most nations, as the Greeks, Romans, Germans, French, Chinese, &c.</p><p>The Grecian abacus was an oblong frame, over which were stretched several brass wires, strung with little ivory balls, like the beads of a necklace; by the various arrangements of which all kinds of computa- <cb/> tions were easily made. Mahudel, in Hist. Acad. R. Inscr. t. 3. p. 390.</p><p>The Roman Abacus was a little varied from the Grecian, having pins sliding in grooves, instead of strings or wires and beads. Philos. Trans. No. 180.</p><p>The Chinese Abacus, or Shwan-pan, like the Grecian, consists of several series of beads strung on brass wires, stretched from the top to the bottom of the instrument, and divided in the middle by a cross piece from side to side. In the upper space every string has two beads, which are each counted for 5; and in the lower space every string has five beads, of different values, the first being counted as 1, the second as 10, the third as 100, and so on, as with us. See S<hi rend="smallcaps">HWANPAN.</hi></p><p>The Abacus chiefly used in European countries, is nearly upon the same principles, though the use of it is here more limited, because of the arbitrary and unequal divisions of money, weights, and measures, which, in China, are all divided in a tenfold proportion, like our scale of common numbers. This is made by drawing any number of parallel lines, like paper ruled for music, at such a distance as may be at least equal to twice the diameter of a calculus, or counter. Then the value of these lines, and of the spaces between them, increases, from the lowest to the highest, in a tenfold proportion. Thus, counters placed upon the first line, signify so many units or ones; on the second line 10's, on the third line 100's, on the fourth line 1000's, and so on: in like manner a counter placed in the first space, between the first and second line, denotes 5, in the second space 50, in the third space 500, in the fourth space 5000, <pb n="2"/><cb/> and so on. So that there are never more than four counters placed on any line, nor more than one placed in any space, this being of the same value as five counters on the next line below. So the counters on the Abacus, in the figure here below, express the number or sum 47382. <figure/></p><p>Besides the above instruments of computation, there have been several others invented by different persons; as <hi rend="italics">Napier's rods</hi> or <hi rend="italics">bones,</hi> deseribed in his Rabdologia, which see under the word <hi rend="smallcaps">Napier;</hi> also the <hi rend="italics">Abacus Rhabdologicus,</hi> a variation of Napier's, which is described in the first vol. of <hi rend="italics">Machines et Inventions approuvées par l'Academie Royale des Sciences.</hi> An ingenious and general one was also invented by Mr. Gamaliel Smethurst, and is described in the Philosophical Transactions, vol. 46; where the inventor remarks that computations by it are much quicker and easier than by the pen, are less burthensome to the memory, and can be performed by blind persons, or in the dark as well as in the light. A very comprehensive instrument of this kind was also contrived by the late learned Dr. Nicholas Saunderson, by which he performed very intricate calculations: an account of it is prefixed to the first volume of his Algebra, and it is there by the editor called <hi rend="italics">Palpable Arithmetic:</hi> which see.</p><div2 part="N" n="Abacus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Abacus</hi></head><p>, <hi rend="italics">Pythagorean,</hi> so denominated from its inventor, Pythagoras; a table of numbers, contrived for readily learning the principles of arithmetic; and was probably what we now call the multiplication-table.</p></div2><div2 part="N" n="Abacus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Abacus</hi></head><p>, or <hi rend="smallcaps">Abaciscus</hi>, in <hi rend="italics">Architecture,</hi> the upper part or member of the capital of a column; serving as a crowning both to the capital and to the whole column. Vitruvius informs us that the <hi rend="italics">Abacus</hi> was originally intended to represent a square flat tile laid over an urn, or a basket; and the invention is ascribed to Calimachus, an ingenious statuary of Athens, who, it is said, adopted it on observing a small basket, covered with a tile, over the root of an Acanthus plant, which grew on the grave of a young lady; the plant shooting up, encompassed the basket all around, till meeting with the tile, it curled back in the form of scrolls: Calimachus passing by, took the hint, and immediately executed a capital on this plan; representing the tile by the Abacus, the leaves of the acanthus by the volutes or scrolls, and the basket by the vase or body of the capital. See <hi rend="smallcaps">Acanthus.</hi></p><p><hi rend="italics">Abacus</hi> is also used by Scamozzi for a concave moulding in the capital of the Tuscan pedestal. And the word is used by Palladio for other members which he describes. Also, in the ancient architecture, the same term is used to denote certain compartments in the incrustation or lining of the walls of state-rooms, mosaicpavements, and the like. There were <hi rend="italics">Abaci</hi> of marble, <cb/> porphyry, jasper, alabaster, and even glass; variously shaped, as square, triangular, and such-like.</p><p><hi rend="smallcaps">Abacus</hi> <hi rend="italics">Logisticus</hi> is a right angled triangle, whose sides, about the right angle, contain all the numbers from 1 to 60; and its area the products of each two of the opposite numbers. This is also called a <hi rend="italics">canon of sexagesimals,</hi> and is no other than a multiplication-table carried to 60 both ways.</p><p><hi rend="smallcaps">Abacus</hi> <hi rend="italics">& Palmulæ,</hi> in the Ancient Music, denote the machinery by which the strings of the polyplectra, or instruments of many strings, were struck, with a plectrum made of quills.</p><p><hi rend="smallcaps">Abacus</hi> <hi rend="italics">Harmonicus</hi> is used by Kircher for the structure and disposition of the keys of a musical instrument, either to be touched with the hands or feet.</p></div2><div2 part="N" n="Abacus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Abacus</hi>, in <hi rend="italics">Geometry</hi></head><p>, a table or slate upon which schemes or diagrams are drawn.</p></div2></div1><div1 part="N" n="ABAS" org="uniform" sample="complete" type="entry"><head>ABAS</head><p>, a weight used in Persia for weighing pearls; and is an eighth part lighter than the European carat.</p></div1><div1 part="N" n="ABASSI" org="uniform" sample="complete" type="entry"><head>ABASSI</head><p>, a silver coin current in Perlia, deriving its name from Schaw Abbas II. King of Persia, and is worth near eighteen pence English money.</p></div1><div1 part="N" n="ABATIS" org="uniform" sample="complete" type="entry"><head>ABATIS</head><p>, or <hi rend="smallcaps">Abattis</hi>, from the French <hi rend="italics">abattre,</hi> to throw down, or beat down, in the Military Art, denotes a kind of retrenchment made by a quantity of whole trees cut down, and laid lengthways beside each other, the closer the better, having all their branches pointed towards the enemy, which prevents his approach, at the same time that the trunks serve as a breast-work before the men. The Abattis is a very useful work on most occasions, especially on sudden emergencies, when trees are near at hand; and has always been practised with considerable success, by the ablest commanders in all ages and nations.</p><p>ABBREVIATE; to abbreviate fractions in arithmetic and algebra, is to lessen proportionally their terms, or the numerator and denominator; which is performed by dividing those terms by any number or quantity, which will divide them without leaving a remainder. And when the terms cannot be any farther so divided, the fraction is said to be in its least terms.</p><p>So , by dividing the terms continually by 2.</p><p>And , by dividing by 2, 3, and 7.</p><p>Also , by dividing by 3 and by 2.</p><p>And , by dividing by 4 <hi rend="sup"><hi rend="italics">ax.</hi></hi></p><p>And , by dividing by <hi rend="italics">a</hi>+<hi rend="italics">x.</hi></p></div1><div1 part="N" n="ABBREVIATION" org="uniform" sample="complete" type="entry"><head>ABBREVIATION</head><p>, of fractions, in Arithmetic and Algebra, the reducing them to lower terms.</p></div1><div1 part="N" n="ABERRATION" org="uniform" sample="complete" type="entry"><head>ABERRATION</head><p>, in <hi rend="italics">Astronomy,</hi> an apparent motion of the celestial bodies, occasioned by the progressive motion of light, and the earth's annual motion in her orbit.</p><p>This effect may be explained and familiarized by the <pb n="3"/><cb/> motion of a line parallel to itself, much after the manner that the composition and resolution of forces are explained. If light have a progressive motion, let the <figure/> proportion of its velocity to that of the earth in her orbit, be as the line BC to the line AC; then, by the composition of these two motions, the particle of light will seem to describe the line BA or DC, instead of its real course BC; and will appear in the direction AB or CD, instead of its true direction CB. So that if AB represent a tube, carried with a parallel motion by an observer along the line AC, in the time that a particle of light would move over the space BC, the different places of the tube being AB, <hi rend="italics">ab, cd,</hi> CD; and when the eye, or end of the tube, is at A, let a particle of light enter the other end at B; then when the tube is at <hi rend="italics">ab,</hi> the particle of light will be at <hi rend="italics">e,</hi> exactly in the axis of the tube; and when the tube is at <hi rend="italics">cd,</hi> the particle of light will arrive at <hi rend="italics">f,</hi> still in the axis of the tube; and lastly, when the tube arrives at CD, the particle of light will arrive at the eye or point C, and consequently will appear to come in the direction DC of the tube, instead of the true direction BC. And so on, one particle succeeding another, and forming a continued stream or ray of light in the apparent direction DC. So that the apparent angle made by the ray of light with the line AE, is the angle DCE, instead of the true angle BCE; and the difference, BCD or ABC, is the quantity of the aberration.</p><p>M. de Maupertuis, in his Elements of Geography, gives also a familiar and ingenious idea of the aberration, in this manner: “It is thus,” says he, “concerning the direction in which a gun must be pointed to strike a bird in its flight; instead of pointing it straight to the bird, the fowler will point a little before it, in the path of its flight, and that so much the more as the flight of the bird is more rapid, with respect to the flight of the shot.” In this way of considering the matter, the flight of the bird represents the motion of the earth, or the line AC, in our scheme above, and the flight of the shot represents the motion of the ray of light, or the line BC.</p><p>Mr. Clairaut too, in the Memoires of the Academy of Sciences for the year 1746, illustrates this effect in a familiar way, by supposing drops of rain to fall rapidly and quickly after each other from a cloud, under which a person moves with a very narrow tube; in which case it is evident that the tube must have a certain inclination, in order that a drop which enters at the top, may fall freely through the axis of the tube, without touching the sides of it; which inclination must be more or less according to the velocity of the drops in respect to that of the tube: then the angle made by the direction of the tube and of the falling drops, is the aberration arising from the combination of those two motions.</p><p>This discovery, which is one of the brightest that have been made in the present age, we owe to the accuracy and ingenuity of the late Dr. Bradley, Astronomer Royal; to which he was occasionally led by the result <cb/> of some accurate observations which he had made with another view, namely, to determine the annual parallax of the fixed stars, or that which arises from the motion of the earth in its annual orbit about the sun.</p><p>The annual motion of the earth about the sun had been much doubted, and warmly contested. The defenders of that motion, among other proofs of the reality of it, conceived the idea of adducing an incontestable one from the annual parallax of the fixed stars, if the stars should be within such a distance, or if instruments and observations could be made with such accuracy, as to render that parallax sensible. And with this view various attempts have been made. Before the observations of M. Picard, made in 1672, it was the general opinion, that the stars did not change their position during the course of a year. Tycho Brahe and Ricciolus fancied that they had assured themselves of it from their observations; and from thence they concluded that the earth did not move round the sun, and that there was no annual parallax in the fixed stars. M. Picard, in the account of his <hi rend="italics">Voyage d'Uranibourg,</hi> made in 1672, says that the pole star, at different times of the year, has certain variations which he had observed for about 10 years, and which amounted to about 40″ a year: from whence some who favoured the annual motion of the earth were led to conclude that these variations were the effect of the parallax of the earth's orbit. But it was impossible to explain it by that parallax; because this motion was in a manner contrary to what ought to follow only from the motion of the earth in her orbit.</p><p>In 1674 Dr. Hook published an account of observations which he said he had made in 1669, and by which he had found that the star <foreign xml:lang="greek">g</foreign> Draconis was 23 more northerly in July than in October: observations which, for the present, seemed to favour the opinion of the earth's motion, although it be now known that there could not be any truth or accuracy in them.</p><p>Flamsteed having observed the pole star with his mural quadrant, in 1689 and the following years, found that its declination was 40″ less in July than in December; which observations, although very just, were yet however improper for proving the annual parallax: and he recommended the making of an instrument of 15 or 20 feet radius, to be firmly fixed on a strong foundation, for deciding a doubt which was otherwise not soon likely to be brought to a conclusion.</p><p>In this state of uncertainty and doubt, then, Dr. Bradley, in conjunction with Mr. Samuel Molineux, in the year 1725, formed the project of verifying, by a series of new observations, those which Dr. Hook had communicated to the public almost 50 years before. And as it was his attempt that chiefly gave rise to this, so it was his method in making the observations, in some measure, that they followed; for they made choice of the same star, and their instrument was constructed upon nearly the same principles: but had it not greatly exceeded the former in exactness, they might still have continued in great uncertainty as to the parallax of the fixed stars. And this was chiefly owing to the accuracy of the ingenious Mr. George Graham, to whom the lovers of astronomy are also indebted for several other exact and convenient instruments. <pb n="4"/><cb/></p><p>The success then of the intended experiment, evidently depending very much on the accuracy of the instrument, that leading object was first to be well secured. Mr. Molineux's apparatus then having been completed, and fitted for observing, about the end of November 1725, on the third day of December following, the bright star in the head of Draco, marked <foreign xml:lang="greek">g</foreign> by Bayer, was for the first time observed, as it passed near the zenith, and its situation carefully taken with the instrument. The like observations were made on the fifth, eleventh, and twelfth days of the same month; and there appearing no material difference in the place of the star, a farther repetition of them, at that season, seemed needless, it being a time of the year in which no sensible alteration of parallax, in this star, could soon be expected. It was therefore curiosity that chiefly urged Dr. Bradley, being then at Kew, where the in strument was fixed, to prepare for observing the star again on the 17th of the same month; when, having adjusted the instrument as usual, he perceived that it passed a little more southerly this day than it had done before. Not suspecting any other cause of this appearance, they first concluded that it was owing to the uncertainty of the observations, and that either this, or the foregoing, was not so exact as they had before supposed. For which reason they proposed to repeat the observation again, to determine from what cause this difference might proceed: and upon doing it, on the 20th of December, the doctor found that the star passed still more southerly than at the preceding observation. This sensible alteration surprised them the more, as it was the contrary way from what it would have been, had it proceeded from an annual parallax of the star. But being now pretty well satisfied, that it could not be entirely owing to the want of exactness in the observations, and having no notion of any thing else that could cause such an apparent motion as this in the star; they began to suspect that some change in the materials, or fabric of the instrument itself, might have occasioned it. Under these uncertainties they remained for some time; but being at length fully convinced, by several trials, of the great exactness of the instrument; and finding, by the gradual increase of the star's distance from the pole, that there must be some regular cause that produced it; they took care to examine very micely, at the time of each observation, how much the variation was; till about the beginning of March 1726, the star was found to be 20″ more southerly than at the time of the first observation: it now indeed seemed to have arrived at its utmost limit southward, as in several trials, made about this time, no sensible difference was observed in its situation. By the middle of April it appeared to be returning back again towards the north; and about the beginning of June, it passed at the same distance from the zenith, as it had done in December, when it was first observed.</p><p>From the quick alteration in the declination of the star about this time, increasing about one second in three days, it was conjectured that it would now proceed northward, as it had before gone southward, of its present situation; and it happened accordingly; for the star continued to move northward till September following, when it became stationary again; being then near 20″ more northerly than in June, and upwards of <cb/> 39″ more northerly than it had been in March. From September the star again returned towards the south, till, in December, it arrived at the same situation in which it had been observed twelve months before, allowing for the difference of declination on account of the precession of the equinox.</p><p>This was a sufficient proof that the instrument had not been the cause of this apparent motion of the star; and yet it seemed difficult to devise one that should be adequate to such an unusual effect. A nutation of the earth's axis was one of the first things that offered itself on this occasion; but it was soon found to be insufficient; for though it might have accounted for the change of declination in <foreign xml:lang="greek">g</foreign> Draconis, yet it would not at the same time accord with the phenomena observed in the other stars, particularly in a small one almost opposite in right ascension to <foreign xml:lang="greek">g</foreign> Draconis; and at about the same distance from the north pole of the equator: for though this star seemed to move the same way, as a nutation of the earth's axis would have made it; yet changing its declination but about half as much as <foreign xml:lang="greek">g</foreign> Draconis in the same time, as a peared on comparing the observations of both made on the same days, at different seasons of the year, this plainly proved that the apparent motion of the star was not occasioned by a real nutation; since, had that been the case, the alteration in both stars would have been nearly equal.</p><p>The great regularity of the observations left no room to doubt, but that there was some uniform cause by which this unexpected motion was produced, and which did not depend on the uncertainty or variety of the seasons of the year. Upon comparing the observations with each other, it was discovered that, in both the stars above mentioned, the apparent difference of declination from the <hi rend="italics">maxima,</hi> was always nearly proportional to the versed sine of the sun's distance from the equinoctial points. This was an inducement to think that the cause, whatever it was, had some relation to the sun's situation with respect to those points. But not being able to frame any hypothesis, sufficient to account for all the phenomena, and being very desivous to search a little farther into this matter, Dr. Bradley began to think of erecting an instrument for himself at Wanstead; that, having it always at hand, he might with the more ease and certainty enquire into the laws of this new motion. The consideration likewise of being able, by another instrument, to confirm the truth of the observations hitherto made with that of Mr. Molineux, was no small inducement to the undertaking; but the chief of all was, the opportunity he should thereby have of trying in what manner other stars should be affected by the same cause, whatever it might be. For Mr. Molineux's instrument being originally designed for observing <foreign xml:lang="greek">g</foreign> Draconis, to try whether it had any sensible parallax, it was so contrived, as to be capable of but little alteration in its direction; not above seven or eight minutes of a degree: and there being but few stars, within half that distance from the zenith of Kew, bright enough to be well observed, he could not, with his instrument, thoroughly examine how this cause affected stars that were differently situated, with respect to the equinoctial and solsticial points of the ecliptic.</p><p>These considerations determined him; and by the contrivance and direction of the same ingenious person, <pb n="5"/><cb/> Mr. Graham, his instrument was fixed up the 19th of August 1727. As he had no convenient place where he could make use of so long a telescope as Mr. Molineux's, he contented himself with one of but little more than half the length, namely of 12 feet and a half, the other being 24 feet and a half long, judging from the experience he had already had, that this radius would be long enough to adjust the instrument to a sufficient degree of exactness: and he had no reason afterwards to change his opinion; for by all his trials he was very well satisfied, that when it was caresully rectisied, its situation might be securely depended on to half a second. As the place where his instrument was hung, in some measure determined its radius; so did it also the length of the arc or limb, on which the divisions were made, to adjust it: for the arc could not conveniently be extended farther, than to reach to about 6 1/4 degrees on each side of his zenith. This however was sufficient, as it gave him an opportunity of making choice of several stars, very different both in magnitude and situation; there being more than two hundred, inserted in the British Catalogue, that might be observed with it. He needed not indeed to have extended the limb so far, but that he was willing to take in <hi rend="italics">Capella,</hi> the only star of the first magnitude that came so near his zenith.</p><p>His instrument being fixed, he immediately began to observe such stars as he judged most proper to give him any light into the cause of the motion already mentioned. There was a sufficient variety of small ones, and not less than twelve that he could observe through all seasons of the year, as they were bright enough to be seen in the day-time, when nearest the sun. He had not been long observing, before he perceived that the notion they had before entertained, that the stars were farthest north and south when the sun was near the equinoxes, was only true of those stars which are near the solsticial colure. And after continuing his observations a few months, he discovered what he then apprehended to be a general law observed by all the stars, namely, that each of them became stationary, or was farthest north or south, when it passed over his zenith at six of the clock, either in the evening or morning. He perceived also that whatever situation the stars were in, with respect to the cardinal points of the ecliptic, the apparent motion of every one of them tended the same way, when they passed his instrument about the same hour of the day or night; for they all moved southward when they passed in the day, and northward when in the night; so that each of them was farthest north, when it came in the evening about six of the clock, and farthest south when it came about six in the morning.</p><p>Though he afterwards discovered that the maxima, in most of these stars, do not happen exactly when they pass at those hours; yet, not being able at that time to prove the contrary, and supposing that they did, he endeavoured to find out what proportion the greatest alterations of declination, in different stars, bore to each other; it being very evident that they did not all change their declination equally. It has been before noticed, that it appeared from Mr. Molineux's observations, that <foreign xml:lang="greek">g</foreign> <hi rend="italics">Draconis</hi> changed its declination above twice as much as the before-mentioned small star that was nearly op- <cb/> posite to it; but examining the matter more nicely, he found that the greatest change in the declination of these stars, was as the sine of the latitude of each star respectively. This led him to suspect that there might be the like proportion between the <hi rend="italics">maxima</hi> of other stars; but finding that the observations of some of them would not perfectly correspond with such an hypothesis, and not knowing whether the small difference he met with might not be owing to the uncertainty and error of the observations, he deferred the farther examination into the truth of this hypothesis, till he should be sarther furnished with a series of observations made in all parts of the year; which would enable him not only to determine what errors the observations might be liable to, or how far they might safely be depended on; but also to judge, whether there had been any sensible change in the parts of the instrument itself.</p><p>When the year was completed, he began to examine and compare his observations; and having pretty well satisfied himself as to the general laws of the phenomena, he then endeavoured to sind out the cause of them. He was already convinced that the apparent motion of the stars was not owing to a nutation of the earth's axis. The next that occurred to him, was an alteration in the direction of the plumb-line, by which the instrument was constantly adjusted; but this, upon trial, proved insufficient. Then he considered what refraction might do; but here also he met with no satisfaction. At last, through an amazing sagacity, he conjectured that all the phenomena hitherto mentioned, proceeded from the progressive motion of light, and the earth's annual motion in her orbit: for he perceived, that if light were propagated in time, the apparent place of a fixed object would not be the same when the eye is at rest, as when it is moving in any other direction but that of the line passing through the object and the eye; and that when the eye is moving in different directions, the apparent place of the object would be different.</p><p>He considered this matter in the following manner. He imagined CA to be a ray of <figure/> light, falling perpendicularly upon the line BD: then, if the eye be at rest at A, the object must appear in the direction AC, whether light be propagated in time, or in an instant. But if the eye be moving from B towards A, and light be propagated in time, with a velocity that is to the velocity of the eye, as AC to AB; then, light moving from C to A, whilst the eye moves from B to A, that particle of it by which the object will be discerned, when the eye in its motion comes to A, is at C when the eye is at B. Joining the points B, C, he supposed the line BC to be a tube, inclined to the line BD in the angle DBC, and of such a diameter as to admit of but one particle of light: then it was easy to conceive, that the particle of light at C, by which the object must be seen when the eye arrives at A, would pass through the tube BC, so inclined to the line BD, and accompanying the eye in its motion from B to A; and that it would not come <pb n="6"/><cb/> to the eye, placed behind such a tube, if it had any other inclination to the line BD. If, instead of supposing BC so small a tube, we conceive it to be the axis of a larger; then, for the same reason, the particle of light at C cannot pass through that axis, unless it be inclined to BD in the same angle DBC.</p><p>In the like manner, if the eye move the contrary way, from D towards A, with the same velocity; then the tube must be inclined in the angle BDC. Although therefore the true or real place of an object, be perpendicular to the line in which the eye is moving, yet the visible place will not be so; since that must doubtless be in the direction of the tube. But the difference between the true and apparent place, will be, <hi rend="italics">cæteris paribus,</hi> greater or less, according to the different proportions between the velocity of light and that of the eye: so that if we could suppose light to be propagated in an instant, then there would be no difference between the real and visible place of an object, although the eye were in motion; for in that case, AC being infinite with respect to AB, the angle ACB, which is the difference between the true and visible place, vanishes. But if light be propagated in time, which was then allowed by most philosophers, then it is evident from the foregoing considerations, that there will always be a difference between the true and visible place of an object, except when the eye is moving either directly towards, or from the object. And in all cases, the sine of the difference between the true and visible place of the object, will be to the sine of the visible inclination of the object to the line in which the eye is moving, as the velocity of the eye, is to the velocity of light.</p><p>If light moved only 1000 times faster than the eye, and an object, supposed to be at an infinite distance, were really placed perpendicularly over the plane in which the eye is moving; it follows, from what has been saíd, that the apparent place of such object will always be inclined to that plane, in an angle of 89° 56′ 1/2; so that it will constantly appear 3′ 1/2 from its true place, and will seem so much less inclined to the plane, that way towards which the eye tends. That is, if AC be to AB or AD, as 1000 to 1, the angle ABC will be 89° 56′ 1/2, and the angle ACB 3′ 1/2, and BCD or 2ACB will be 7′, if the direction of the motion of the eye be contrary at one time to what it is at another.</p><p>If the earth revolve about the sun annually, and the velocity of light were to the velocity of the earth's motion in its orbit, as 1000 is to 1; then it is easy to conceive, that a star really placed in the pole of the ecliptic, would to an eye carried along with the earth, seem to change its place continually; and, neglecting the small difference on account of the earth's diurnal revolution on its axis, it would seem to describe a circle about that pole, every where distant from it by 3′ 1/2. So that its longitude would be varied through all the points of the ecliptic every year, but its latitude would always remain the same. Its right ascension would also change, and its declination, according to the different situation of the sun in respect of the equinoctial points; and its apparent distance from the north pole of the equator, would be 7′ less at the autumnal, than at the vernal equinox. <cb/></p><p>The greatest alteration of the place of a star, in the pole of the ecliptic, or, which in effect amounts to the same, the proportion between the velocity of light and the earth's motion in its orbit, being known, it will not be difficult to find what would be the difference, on this account, between the true and apparent place of any other star at any time; and, on the contrary, the difference between the true and apparent place being given, the proportion between the velocity of light, and the earth's motion in her orbit, may be found.</p><p>After the history of this curious discovery, related by the author nearly in the terms above, he gives the results of a multitude of accurate observations, made on a great number of stars, at all seasons of the year. From all which observations, and the theory as related above, he found that every star, in consequence of the earth's motion in her orbit and the progressive motion of light, appears to describe a small ellipse in the heavens, the transverse axis of which is equal to the same quantity for every star, namely 40″, nearly; and that the conjugate axis of the ellipse, for different stars, varies in this proportion, namely, as the right sine of the star's latitude; that is, radius is to the sine of the star's latitude, as the transverse axis to the conjugate axis: and consequently a star in the pole of the ecliptic, its latitude being there 90°, whose sine is equal to the radius, will appear to describe a small circle about that pole as a centre, whose radius is equal to 20″. He also gives the following law of the variation of the star's declination: if A denote the angle of position, or the angle at the star made by two great circles drawn from it through the poles of the ecliptic and equator, and B another angle, whose tangent is to the tangent of A, as radius is to the sine of the star's latitude; then B will be equal to the difference of longitude between the sun and the star, when the true and apparent declination of the star are the same. And if the sun's longitude in the ecliptic be reckoned from that point in which it is when this happens; then the difference between the true and apparent declination of the star, will be always as the sine of the sun's longitude from that point. It will also be found that the greatest difference of declination that can be between the true and apparent place of the star, will be to 20″, the semitransverse axis of the ellipse, as the sine of A to the sine of B.</p><p>The author then shews, by the comparison of a number of observations made on different stars, that they exactly agree with the theory deduced from the progressive motion of light, and that consequently it is highly probable that such motion is the cause of those variations in the situation of the stars. From which he infers, that the parallax of the fixed stars is much smaller, than hath been hitherto supposed by those, who have pretended to deduce it from their observations. He thinks he may venture to say, that in the stars he had observed, the parallax does not amount to 2″; nay, that if it had amounted to 1″, he should certainly have perceived it, in the great number of observations that he made, especially of <foreign xml:lang="greek">g</foreign> Draconis; which agreeing with the hypothesis, without allowing any thing for parallax, nearly as well when the sun was in conjunction with, as in opposition to, this star, it seems very pro- <pb n="7"/><cb/> bable that the parallax of it is not so much as one single second; and consequently that it is above 400000 times farther from us than the sun.</p><p>From the greatest variation in the place of the stars, namely 40″, Dr. Bradley deduces the ratio of the velocity of light in comparison with that of the earth in her orbit. In the preceding figure, AC is to AB, as the velocity of light to that of the earth in her or bit, the angle ACB being equal to 20″; so that the ratio of those velocities is that of radius to the tangent of 20″, or of radius to 20″, since the tangent has no sensible difference from so small an are: but the radius of a circle is equal to the arc of 57° 3/10 nearly, or equal to 206260″; therefore the velocity of light is to the velocity of the earth, as 206260 to 20, or as 10313 to 1.</p><p>And hence also the time in which light passes over the space from the sun to the earth, is easily deduced; for this time will be to one year, as AB or 20″ to 360° or the whole circle; that is, 360°: 20″ :: 365 1/4 days: 8<hi rend="sup">m</hi> 7<hi rend="sup">s</hi>, namely, light will pass from the sun to the earth in the time of 8 minutes, 7 seconds; and this will be the same, whatever the distance of the sun is.</p><p>Dr. Bradley having annexed to his theory the rules or formulæ for computing the aberration of the fixed stars in declination and right ascension; these rules have been variously demonstrated, and reduced to other practical forms, by Mr. Clairaut in the Memoirs of the Academy of Sciences for 1737; by Mr. Simpson in his Essays in 1740; by M. Fontaine des Crutes in 1744; and several other persons. The results of these rules are as follow: Every star appears to describe in the course of a year, by means of the aberration, a small ellipse, whose greater axis is 40″, and the less axis, perpendicular to the ecliptic, is equal to 40″ multiplied by the sine of the star's latitude, the radius being 1. The eastern extremity of the longer axis, marks the apparent place of the star, the day of the opposition; and the extremity of the less axe, which is farthest from the ecliptic, marks its situation three months after.</p><p>The greatest aberration in longitude, is equal to 20″ divided by the cosine of its latitude. And the aberration for any time, is equal to 20″ multiplied by the cosine of the elongation of the star found for the same time, and divided by the cosine of its latitude. This aberration is subtractive in the first and last quadrant of the argument, or of the difference between the longitudes of the sun and star; and additive in the second and third quadrants. The greatest aberration in latitude, is equal to 20″ multiplied by the sine of the star's latitude. And the aberration in latitude for any time, is equal to 20″ multiplied by the sine of the star's latitude, and multiplied also by the sine of the elongation. The aberration is subtractive before the opposition, and additive after it.</p><p>The greatest aberration in declination, is equal to 20″ multiplied by the sine of the angle of position A, and divided by the sine of B the difference of longitude between the sun and star when the aberration in declination is nothing. And the aberration in declination at any other time, will be equal to the greatest aberration multiplied by the sine of the difference between the sun's place at the given time and his place when the <cb/> aberration is nothing. Also the sine of the latitude of the star is to radius, as the tangent of A the angle of position at the star, is to the tangent of B, the difference of longitude between the sun and star when the aberration in declination is nothing. The greatest aberration in right-ascension, is equal to 20″ multiplied by the cosine of A the angle of position, and divided by the sine of C the difference in longitude between the sun and star when the aberration in right ascension is nothing. And the aberration in right-ascension at any other time, is equal to the greatest aberration multiplied by the sine of the difference between the sun's place at the given time, and his place when the aberration is nothing. Also the sine of the latitude of the star is to radius, as the cotangent of A the angle of position, to the tangent of C.</p><p><hi rend="smallcaps">Aberration</hi> <hi rend="italics">of the Planets,</hi> is equal to the geocentric motion of the planet, the space it appears to move as seen from the earth, during the time that light employs in passing from the planet to the earth. Thus, in the sun, the aberration in longitude is constantly 20″, that being the space moved by the sun, or, which is the same thing, by the earth, in the time of 8<hi rend="sup">m</hi> 7<hi rend="sup">s</hi>, which is the time in which light passes from the sun to the earth, as we have seen in the foregoing article. In like manner, knowing the distance of any planet from the earth, by proportion it will be, as the distance of the sun is to the distance of the planet, so is 8<hi rend="sup">m</hi> 7<hi rend="sup">s</hi> to the time of light passing from the planet to the earth: then computing the planet's geocentric motion in this time, that will be the aberration of the planet, whether it be in longitude, latitude, right-ascension, or declination.</p><p>It is evident that the aberration will be greatest in the longitude, and very small in latitude, because the planets deviate very little from the plane of the ecliptic, or path of the earth; so that the aberration in the latitudes of the planets, is commonly neglected, as insensible; the greatest in Mercury being only 4″ 1/3, and much less in the other planets. As to the aberrations in declination and right-ascension, they must depend on the situation of the planet in the zodiac. The aberration in longitude, being equal to the geocentric motion, will be more or less according as that motion is; it will therefore be least, or nothing at all, when the planet is stationary; and greatest in the superior planets Mars, Jupiter, Saturn, &c, when they are in opposition to the sun; but in the inferior planets Venus and Mercury, the aberration is greatest at the time of their superior conjunction. These maxima of aberration for the several planets, when their distance from the sun is least, are as below: viz, for <table><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data">27″</cell><cell cols="1" rows="1" role="data">.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">.2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Moon</cell><cell cols="1" rows="1" role="data"> 2/3</cell><cell cols="1" rows="1" role="data"/></row></table> And between these numbers and nothing the aberrations of the planets, in longitude, vary according to their situations. But that of the sun varies not, being constantly 20″, as has been before observed. And this may alter his declination by a quantity, which varies from 0 to near 8″; being greatest or 8″ about the equinoxes, and vanishing in the solstices. <pb n="8"/><cb/></p><p>The methods of computing these, and the formulas for all cases, are given by M. Clairaut in the Memoirs of the Academy of Sciences for the year 1746, and by M. Euler in the Berlin Memoirs, vol. 2, for 1746.</p><p><hi rend="italics">Optic</hi> <hi rend="smallcaps">Aberration</hi>, the deviation or dispersion of the rays of light, when reflected by a speculum, or refracted by a lens, by which they are prevented from meeting or uniting in the same point, called the geometrical focus, but are spread over a small space, and produce a confusion of images. Aberration is either lateral or longitudinal: the lateral aberration is measured by a perpendicular to the axis of the speculum or lens, drawn from the focus to meet the refracted or reflected ray: the longitudinal aberration is the distance, on the axis, between the focus and the point where the ray meets the axis. The aberrations are very amply treated in Smith's Complete System of Opties, in 2 volumes 4to.</p><p>There are two species of aberration, distinguished according to their different causes: the one arises from the figure of the speculum or lens, producing a geometrical dispersion of the rays, when these are perfectly equal in all respects; the other arises from the unequal refrangibility of the rays of light themselves; a discovery that was made by Sir Isaac Newton, and for this reason it is often called the Newtonian aberration. As to the former species of aberration, or that arising from the sigure, it is well known that if rays issue from a point at a given distance; then they will be reflected into the other focus of an ellipse having the given luminous point for one focus, or directly from the other focus of an hyperbola; and will be variously dispersed by all other figures. But if the luminous point be infinitely distant, or, which is the same, the incident rays be parallel, then they will be reflected by a parabola into its focus, and variously dispersed by all other figures. But those figures are very difficult to make, and therefore curved specula are commonly made spherical, the figure of which is generated by the revolution of a circular arc, which produces an aberration of all rays, whether they are parallel or not, and therefore it has no accurate geometrical focus which is common to all the rays. Let BVF represent a concave spherical speculum, whose centre is C; and let AB, EF be incident rays parallel to the axis CV. Because the angle of incidence is equal to the angle of reflection in <figure/> all cases, therefore if the radii CB, CF be drawn to the points of incidence, and thence BD making the angle CBD equal to the angle CBA, and FG making the angle CFG equal to the angle CFE; then BD, FG will be the reflected rays, and D, G, the points where <cb/> they meet the axis. Hence it appears that the point of coincidence with the axis is equally distant from the point of incidence and the centre: for because the angle CBD is equal to the angle CBA, which is equal to the alternate angle BCD, therefore their opposite sides CD, DB are equal: and in like manner, in any other, GF is equal to GC. And hence it is evident that when B is indefinitely near the vertex V, then D is in the middle of the radius CV; and the nearer the incident ray is to the axis CV, the nearer will the reflected ray come to the middle point D; and the contrary. So that the aberration DG of any ray EFG, is always more and more, as the incident ray is farther from the axis, or the incident point F from the vertex V; till when the distance VI is 60 degrees, then the reflected ray falls in the vertex V, making the aberration equal to the whole length DV. And this shews the reason why specula are made of a very small segment of a sphere, namely, that all their reflected rays may arrive very near the middle point or focus D, to produce an image the most distinct, by the least aberration of the rays. And in like manner for rays refracted through lenses.</p><p>In spherical lenses, Mr. Huygens has demonstrated that the aberration from the figure, in different lenses, is as follows.</p><p>1. In all plano-convex lenses, having their plane surface exposed to parallel rays, the longitudinal aberration of the extreme ray, or that remotest from the axis, is equal to 9/2 of the thickness of the lens.</p><p>2. In all plano-convex lenses, having their convex surface exposed to parallel rays, the longitudinal aberration of the extreme ray, is equal to 7/6 of the thickness of the lens. So that in this position of the same planoconvex lens, the aberration is but about one-fourth of that in the former; being to it only as 7 to 27.</p><p>3. In all double convex lenses of equal spheres, the aberration of the extreme ray, is equal to 5/3 of the thickness of the lens.</p><p>4. In a double convex lens, the radii of whose spheres are as 1 to 6, if the more convex surface be exposed to parallel rays, the aberration from the figure is less than in any other spherical lens; being no more than 15/14 of its thickness.</p><p>But the foregoing species of aberration, arising from the figure, is very small, and easily remedied, in comparison with the other, arising from the unequal refrangibility of the rays of light; insomuch that Sir Isaac Newton shews in his Optics, pa. 84 of the 8vo. edition, that if the object-glass of a telescope be planoconvex, the plane side being turned towards the object, and the diameter of the sphere, to which the convex side is ground, be 100 seet, the diameter of the aperture being 4 inches, and the ratio of the sine of incidence out of glass into air, be to that of refraction, as 20 to 31; then the diameter of the circle of aberrations will in this case be only 961/72000000 parts of an inch: while the diameter of the little circle, through which the same rays are scattered by unequal refrangibility, will be about the 55th part of the aperture of the object-glass, which here is 4 inches. And therefore the error arising from the spherical figure of the glass, is to the error arising from the different refrangibility of the rays, as 961/72000000 to 4/55, that is as 1 to 5449. <pb n="9"/><cb/></p><p>So that it may seem strange that objects appear through telescopes so distinct as they do, considering that the error arising from the different refrangibility, is almost incomparably larger than that of the figure. Newton however solves the difficulty by observing that the rays, under their various aberrations, are not scattered uniformly over all the circular space, but collected insinitely more dense in the centre than in any other part of the circle; and that, in the way from the centre to the circumference, they grow more and more rare, so as at the circumference to become infinitely rare; and, by reason of their rarity, they are not strong enough to be visible, unless in the centre, and very near it.</p><p>In consequence of the discovery of the unequal refrangibility of light, and the apprehension that equal refractions must produce equal divergencies in every sort of medium, it was supposed that all spherical objectglasses of telescopes would be equally affected by the different refrangibility of light, in proportion to their aperture, of whatever materials they might be constructed: and therefore that the only improvement that could be made in refracting telescopes, was that of increasing their length. So that Sir Isaac Newton, and other persons after him, despairing of success in the use and fabric of lenses, directed their chief attention to the construction of reflecting telescopes.</p><p>However, about the year 1747, M. Euler applied himself to the subject of refraction; and pursued a hint suggested by Newton, for the design of making object-glasses with two lenses of glass inclosing water between them; hoping that, by constructing them of different materials, the refractions would balance one another, and so the usual aberration be prevented. Mr. John Dollond, an ingenious optician in London, minutely examined this scheme, and found that Mr. Euler's principles were not satisfactory. M. Clairaut likewise, whose attention had been excited to the same subject, concurred in opinion that Euler's speculations were more ingenious than useful. This controversy, which seemed to be of great importance in the science of optics, engaged also the attention of M. Klingenstierna of Sweden, who was led to make a careful examination of the 8th experiment in the second part of Newton's Optics, with the conclusions there drawn from it. The consequence was, that he found that the rays of light, in the circumstances there mentioned, did not lose their colour, as Sir Isaac had imagined. This hint of the Swedish philosopher led Mr. Dollond to re-examine the same experiment: and after several trials it appeared, that different substances caused the light to diverge very differently, in proportion to their general refractive powers. In the year 1757 therefore he procured wedges of different kinds of glass, and applied them together so that the refractions might be made in contrary directions, that he might discover whether the refraction and divergency of colour would vanish together. The result of his sirst trials encouraged him to persevere; for he discovered a difference far beyond his hopes in the qualities of different kinds of glass, with respect to their divergency of colours. The Venice glass and English crown glass were found to be nearly allied in this respect: the common English plate glass made the rays diverge more; and the English flint <cb/> glass most of all. But without enquiring into the cause of this difference, he proceeded to adapt wedges of crown glass, and of white flint glass, ground to different angles, to each other, so as to refract in different directions; till the refracted light was entirely free from colours. Having measured the refractions of each wedge, he found that the refraction of the white glass was to that of the crown glass, nearly as 2 to 3: and he hence concluded in general, that any two wedges made in this proportion, and applied together so as to refract in contrary directions, would refract the light without any aberration of the rays.</p><p>Mr. Dollond's next object was to make similar trials with spherical glasses of different materials, with the view of applying his discovery to the improvement of telescopes: and here he perceived that, to obtain a refraction of light in contrary directions, the one glass must be concave, and the other convex; and the latter, which was to refract the most, that the rays might converge to a real focus, he made of crown glass, the other of white flint glass. And as the refractions of spherical glasses are inversely as their focal distances, it was necessary that the focal distances of the two glasses should be inversely as the ratios of the refractions of the wedges; because that, being thus proportioned, every ray of light that passes through this compound glass, at any distance from its axis, will constantly be refracted, by the difference between two contrary refractions, in the proportion required; and therefore the different refrangibility of the light will be entirely removed.</p><p>But in the applications of this ingenious discovery to practice, Mr. Dollond met with many and great difficulties. At length, however, after many repeated trials, by a resolute perseverance, he succeeded so far as to construct refracting telescopes much superior to any that had hitherto been made; representing objects with great distinctness, and in their true colours.</p><p>Mr. Clairaut, who had interested himself from the beginning in this discovery, now endeavoured to ascertain the principles of Mr. Dollond's theory, and to lay down rules to facilitate the construction of these new telescopes. With this view he made several experiments, to determine the resractive power of different kinds of glass, and the proportions in which they separated the rays of light: and from these experiments he deduced several theorems of general use. M. D'Alembert made likewise a great variety of calculations to the same purpose; and he shewed how to correct the errors to which these telescopes are subject, sometimes by placing the object-glasses at a small distance from each other, and sometimes by using eye-glasses of different refractive powers. But though foreigners were hereby supplied with the most accurate calculations, they were very defective in practice. And the English telescopes, made, as they imagined, without any precise rule, were greatly superior to the best of their construction.</p><p>M. Euler, whose speculations had sirst given occasion to this important and useful enquiry, was very reluctant in admitting Mr. Dollond's improvements, because they militated against a pre-conceived theory of his own. At last however, after several altercations, being convinced of their reality and importance by M. Clair- <pb n="10"/><cb/> aut, he assented; and he soon after received farther satisfaction from the experiments of M. Zeiher, of Petersburgh.</p><p>M. Zeiher shewed by experiments that it is the lead, in the composition of glass, which gives it this remarkable property, namely, that while the refraction of the mean rays is nearly the same, that of the extreme rays considerably differs. And, by increasing the lead, he produced a kind of glass, which occasioned a much greater separation of the extreme rays than that of the flint glass used by Mr. Dollond, and at the same time considerably increased the mean refraction. M. Zeiher, in the course of his experiments, made glass of minium and lead, with a mixture also of alkaline salts; and he found that this mixture greatly diminished the mean refraction, and yet made hardly any change in the dispersion: and he at length obtained a kind of glass greatly superior to the flint glass of Mr. Dollond for the construction of telescopes; as it occasioned three times as great a dispersion of the rays as the common glass, whilst the mean refraction was only as 1.61 to 1.</p><p>Other improvements were also made on the new or achromatic telescopes by the inventor Mr. John Dollond, and by his son Peter Dollond; which may be seen under the proper words. For various dissertations on the subject of the aberration of light, colours, and the figure of the glass, see Philos. Trans. vols. 35, 48, 50, 51, 52, 55, 60; Memoirs of the Academy of Sciences of Paris, for the years 1737, 1746, 1752, 1755, 1756, 1757, 1762, 1764, 1765, 1767, 1770; the Berlin Ac. 1746, 1762, 1766; Swed. Mem. vol. 16; Com. Nov. Petripol. 1762; M. Euler's Dioptrics; M. d'Alembert's Opuscules Math.; M. de Rochon Opuscules; &c, &c.</p></div1><div1 part="N" n="ABRIDGING" org="uniform" sample="complete" type="entry"><head>ABRIDGING</head><p>, <hi rend="italics">in Algebra,</hi> is the reducing a compound equation, or quantity, to a more simple form of expression. This is done either to save room, or the trouble of writing a number of symbols; or to simplisy the expression, either to ease the memory, or to render the formula more easy and general.</p><p>So the equation , by putting <hi rend="italics">p</hi> = <hi rend="italics">a, q</hi> = <hi rend="italics">ab,</hi> and <hi rend="italics">r</hi> = <hi rend="italics">abc,</hi> becomes </p><p>And the equation , by putting , and , becomes .</p></div1><div1 part="N" n="ABSCISS" org="uniform" sample="complete" type="entry"><head>ABSCISS</head><p>, <hi rend="smallcaps">Abscisse</hi>, or <hi rend="smallcaps">Abscissa</hi>, is a part or segment cut off a line, terminated at some certain point, by an ordinate to a curve; as AP or BP. <figure/></p><p>The absciss may either commence at the vertex of the curve, or at any other fixed point. And it may be taken either upon the axis or diameter of the curve, or upon any other line drawn in a given position. <cb/></p><p>Hence there are an infinite number of variable abscisses, terminated at the same fixed point at one end, the other end of them being at any point of the given line or diameter.</p><p>In the common parabola, each ordinate PQ has but <figure/> one absciss AP; in the ellipse or circle, the ordinate has two abscisses AP, BP lying on the opposite sides of it; and in the hyperbola the ordinate PQ has also two abscisses, but they lie both on the same side of it. That is, in general, a line of the second kind, or a curve of the first kind, may have two abscisses to each ordinate. But a line of the third order may have three abscisses to each ordinate; a line of the fourth order may have four; and so on.</p><p>The use of the abscisses is, in conjunction with the ordinates, to express the nature of the curves, either by some proportion or equation including the abfcifs and its ordinate, with some other fixed invariable line or lines. Every different curve has its own peculiar equation or property by which it is expressed, and different from all others: and that equation or expression is the same for every ordinate and its abscisses, whatever point of the curve be taken. So, in the circle, the square of any ordinate is equal to the rectangle of its two abscisses, or AP.PB = PQ<hi rend="sup">2</hi>; in the parabola, the square of the ordinate is equal to the rectangle of the absciss and a certain given line called the parameter; in the ellipse and hyperbola, the square of the ordinate is always in a certain constant proportion to the rectangle of the two abscisses, namely, as the square of the conjugate to the square of the transverse, or as the parameter is to the transverse axis; and so other properties in other curves.</p><p>When the natures or properties of curves are expressed by algebraic equations, any general absciss, as AP, is commonly denoted by the letter <hi rend="italics">x,</hi> and the ordinate PQ by the letter <hi rend="italics">y;</hi> the other or constant lines being represented by other letters. Then the equations expressing the nature of these curves are as follow; namely, for the circle , where <hi rend="italics">d</hi> is the diameter AB; parabola - <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi> , where <hi rend="italics">p</hi> is the parameter; <hi rend="brace"><note anchored="true" place="unspecified">ellipse - <hi rend="italics">t</hi><hi rend="sup">2</hi> : <hi rend="italics">c</hi><hi rend="sup">2</hi> :: <hi rend="italics">tx</hi> - <hi rend="italics">x</hi><hi rend="sup">2</hi> : <hi rend="italics">y</hi><hi rend="sup">2</hi>, hyperbola <hi rend="italics">t</hi><hi rend="sup">2</hi> : <hi rend="italics">c</hi><hi rend="sup">2</hi> :: <hi rend="italics">tx</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi> : <hi rend="italics">y</hi><hi rend="sup">2</hi>,</note> where <hi rend="italics">t</hi> is the transverse, & <hi rend="italics">c</hi> the conjugate axis.</hi></p></div1><div1 part="N" n="ABSIS" org="uniform" sample="complete" type="entry"><head>ABSIS</head><p>, ABSIDES. See <hi rend="smallcaps">Apsis, Apsides.</hi> <pb n="11"/><cb/></p><p>ABSOLUTE <hi rend="smallcaps">Equation</hi>, <hi rend="italics">in Aftronomy,</hi> is the sum of the optic and excentric equations. The apparent inequality of a planet's motion, arising from its not being equally dislant from the earth at all times, is called its optic equation; and this would subsist even if the planet's real motion were uniform. The excentric inequality is caused by the planet's motion being not uniform. To illustrate this, conceive the sun to move, or to appear to move, in the circumference of a circle, in whose centre the earth is placed. It is manifest, that if the sun move uniformly in this circle, then he must appear to move uniformly to a spectator at the earth; and in this case there will be no optic nor excentric equation. But suppose the earth to be placed out of the centre of the circle; and then, though the sun's motion should be really uniform, it would not appear to be so, being seen from the earth; and in this case there would be an optic equation, without an excentric one. Imagine farther, the sun's orbit to be, not circular, but elliptical, and the earth in its focus: it will be full as evident that the sun cannot appear to have an uniform motion in such ellipse; so that his motion will then be subject to two equations; that is, the optic equation, and the excentric equation. <hi rend="italics">See</hi> <hi rend="smallcaps">Equation</hi>, and <hi rend="smallcaps">Optic Inequality.</hi></p><p><hi rend="smallcaps">Absolute</hi> <hi rend="italics">Number,</hi> in Algebra, is that term or member of an equation that is completely known, and which is equal to all the other, or unknown terms, taken together; and is the same as what Vieta calls the <hi rend="italics">homogeneum comparationis.</hi> So, of the equation , or , the absolute number, or known term, is 36.</p><p><hi rend="smallcaps">Absolute</hi> <hi rend="italics">Gravity, Motion, Space, Time, &c.</hi> See the respective substantives.</p><p>ABSTRACT <hi rend="smallcaps">Mathematics</hi>, otherwise called pure mathematics, is that which treats of the properties of magnitude, figure, or quantity, absolutely and generally confidered, without restriction to any species in particular: such as Arithmetic and Geometry. In this sense, abstract or pure mathematics, is opposed to mixed mathematics, in which simple and abstract properties, and the relations of quantities, primitively considered in pure mathematics, are applied to sensible objects; as in astronomy, hydrostatics, optics, &c.</p><p><hi rend="smallcaps">Abstract</hi> <hi rend="italics">Number,</hi> is a number, or collection of units, considered in itself, without being applied to denote a collection of any particular and determinate things. So, for example, 3 is an abstract number, so far as it is not applied to something: but when we say 3 feet, or 3 persons, the 3 is no longer an abstract, but a concrete number.</p></div1><div1 part="N" n="ABSURD" org="uniform" sample="complete" type="entry"><head>ABSURD</head><p>, or <hi rend="smallcaps">Absurdum</hi>, a term commonly used in demonstrating converse propositions; a mode of demonstration, in which the proposition intended is not proved in a direct manner, by principles before laid down; but it proves that the contrary is absurd or impossible; and so indirectly as it were proves the proposition itself. The 4th proposition in the first book of Euclid, is the first in which he makes use of this mode of proof; where he shews that if the extremities of two lines coincide, those lines will coincide in all their parts, otherwise they would inclose a space, which is absurd or contrary to the 10th axiom. Most converse <cb/> propositions are proved in this way, which mode of proof is called <hi rend="italics">reductio ad absurdum.</hi></p><p>ABUNDANT <hi rend="smallcaps">Number</hi>, in <hi rend="italics">Arithmetic,</hi> is a number whose aliquot parts, added all together, make a sum which is greater than the number itself. Thus 12 is an abundant number, because its aliquot parts, namely 1, 2, 3, 4, 6, when added together, make 16, which is greater than the number 12 itself.</p><p>An abundant number is opposed to a deficient one, which is less than the sum of its aliquot parts taken together, as the number 14, whose aliquot parts 1, 2, 7, make no more than 10; and to a perfect number, which is exactly equal to the sum of all its aliquot parts, as the number 6, which is equal to the sum of 1, 2, 3, which are its aliquot parts.</p></div1><div1 part="N" n="ACADEMICIAN" org="uniform" sample="complete" type="entry"><head>ACADEMICIAN</head><p>, a member of a society called an academy, instituted for the promotion of arts, sciences, or natural knowledge in general.</p></div1><div1 part="N" n="ACADEMICS" org="uniform" sample="complete" type="entry"><head>ACADEMICS</head><p>, an ancient sect of philosophers, who followed the doctrine of Socrates and Plato, as to the uncertainty of knowledge, and the incomprehensibility of truth.</p><p><hi rend="italics">Academic,</hi> in this sense, amounts to much the same with Platonist; the difference between them being only in point of time. Those who embraced the system of Plato, among the ancients, were called <hi rend="italics">academici,</hi> academician or academic; whereas those who did the same since the restoration of learning, have assumed the denomination of Platonists.</p><p>We usually reckon three sects of academics; though some make five. The ancient academy was that of which Plato was the chief.</p><p>Arcessilas, one of Plato's successors, introducing some alterations into the philosophy of this sect, founded what they call the second academy.</p><p>The establishment of the third, called also the new academy, is attributed to Lacydes, or rather to Carneades.</p><p>Some authors add a fourth, founded by Philo; and a fifth, by Antiochus, called the Antiochan, which tempered the ancient academy with Stoicism.</p><p>The ancient academy doubted of every thing; and carried this principle so far as to make it a doubt, whether or no they ought to doubt. It was a kind of a principle with them, never to be certain or satisfied of any thing; never to affirm or to deny any thing, either for true or false.</p><p>The new academy was somewhat more reasonable; they acknowledged several things for truths, but without attaching themselves to any with entire assurance. These philosophers had found that the ordinary commerce of life and society was inconsistent with the absolute and universal doubtfulness of the ancient academy: and yet it is evident that they looked upon things rather as probable, than as true and certain: by this amendment thinking to secure themselves from those absurdities into which the ancient academy had fallen.</p></div1><div1 part="N" n="ACADEMIST" org="uniform" sample="complete" type="entry"><head>ACADEMIST</head><p>, the same as Academician.</p></div1><div1 part="N" n="ACADEMY" org="uniform" sample="complete" type="entry"><head>ACADEMY</head><p>, <hi rend="smallcaps">Academia</hi>, in <hi rend="italics">Antiquity,</hi> a fine villa or pleasure house, in one of the submbs of Athens, about a mile from the city; where Plato, and the wise men who followed him, held assemblies for disputes <pb n="12"/><cb/> and philosophical conference; which gave the name to the sect of Academics.</p><p>The house took its name, <hi rend="italics">Academy,</hi> from one Academus, or Ecademus, a citizen of Athens, to whom it originally belonged: he lived in the time of Theseus; and here he used to have gymnastic sports or exercises.</p><p>The academy was farther improved by Cimon, and adorned with fountains, trees, shady walks, &c, for the convenience of the philosophers and men of learning, who here met to confer and dispute for their mutual improvement. It was surrounded with a wall by Hipparchus, the son of Pisistratus; and it was also used as the burying-place for illustrious persons, who had deserved well of the republic.</p><p>It was here that Plato taught his philosophy; and hence it was that all public places, destined for the assemblies of the learned and ingenious, have been since called <hi rend="italics">Academies.</hi></p><p>Sylla facrificed the delicious walks and groves of the academy, which had been planted by Cimon, to the ravages of war; and employed those very trees in constructing machines to batter the walls of the city which they had adorned.</p><p>Cicero too had a villa, or country retirement, near Puzzuoli, which he called by the same name, <hi rend="italics">Academia.</hi> Here he used to entertain his philosophical friends; and here it was that he composed his Academical Questions, and his books <hi rend="italics">De Naturâ Deorum.</hi></p><div2 part="N" n="Academy" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Academy</hi></head><p>, among the moderns, denotes a regular society or company of learned persons, instituted under the protection of some prince, or other public authority, for the cultivation and improvement of arts or sciences.</p><p>Some authors confound Academy with University; but though much the same in Latin, they are very different things in English. An university is properly a body composed of graduates in the feveral faculties; of professors, who teach in the public schools; of regents or tutors, and students who learn under them, and aspire likewise to degrees. Whereas an academy is not intended to teach, or profess any art or science, but to improve it: it is not for novices to be instructed in, but for those that are more knowing; for persons of learning to confer in, and communicate their lights and discoveries to each other, for their mutual benefit and improvement.</p><p>The first modern academy we read of, was established by Charlemagne, by the advice of Alcuin, an English monk: it was composed of the chief geniuses of the court, the emperor himself being a member. In their academical conferences, every person was to give some account of the ancient authors he had read; and each one assumed the name of some ancient author, that pleased him most, or some celebrated person of antiquity. Alcuin, from whose letters we learn these particulars, took that of Flaccus, the surname of Horace; a young lord, named Augilbert, took that of Homer; Adelard, bishop of Corbie, was called Augustin; Recluse, bishop of Mentz, was Dametas; and the king himself, David.</p><p>Since the revival of learning in Europe, academies have multiplied greatly, most nations being furnished with several, and from their communications the chief <cb/> improvements have been made in the arts and sciences, and in cultivating natural knowledge. There are now academies for almost every art, or species of knowledge; but I shall give a short account only of those institutions of this kind, which regard the cultivation of subjects mathematical or philosophical, which are the proper and peculiar objects of our undertaking.</p><p>Italy abounds more in academies than all the world besides; there being enumerated by Jarckius not less than sive hundred and fifty in all; and even to the amount of twenty-five in Milan itself. These are however mostly of a private and inferior nature; the consequence of their too great number.</p><p>The first academy of a philosophical kind was established at Naples, in the house of Baptista Porta, about the year 1560, under the name of <hi rend="italics">Academy Secretorum Naturæ;</hi> being formed for the improvement of natural and mathematical knowledge. This was succeeded by the</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">of Lyncei,</hi> founded at Rome by prince Frederick Cesi, towards the end of the same century. It was rendered famous by the notable discoveries made by several of its members; among whom was the celebrated Galileo Galilei.</p><p>Several other academies contributed also to the advancement of the sciences; but it was by speculations rather than by repeated experiments on the phenomena of nature: such were the academy of Bessarian at Rome, and that of Laurence de Medicis at Florence, in the 15th century; and in the 16th were that of Infiammati at Padua, of Vegna Juoli at Rome, of Ortolani at Placentia, and of Umidi at Florence. The first of these studied fire and pyrotechnia, the second wine and vineyards, the third pot-herbs and gardens, the fourth water and hydraulics. To these may be added that of Venice, called La Veneta, and sounded by Frederick Badoara, a noble Venetian; another in the same city, of which Campegio, bishop of Feltro, appears to have been the chief; also that of Cosenza, or La Consentina, of which Bernadin Telesio, Sertorio Quatromanni, Paulus Aquinas, Julio Cavalcanti, and Fabio Cicali, celebrated philosophers, were the chief members. The compositions of all these academies, of the 16th century, were good in their kind; but none of them comparable to those of the Lyncei.</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">del Cimento,</hi> that is, of Experiments, arose at Florence, some years after the death of Torricelli, namely in the year 1657, under the protection of prince Leopold of Tuscany, afterwards cardinal de Medicis, and brother to the Grand Duke Ferdinand the Second. Galileo, Toricelli, Aggiunti, and Viviani had prepared the way sor it: and some of its chief members were Paul del Buono, who in 1657 invented the instrument for trying the incompressibility of water, namely a thick globular shell of gold, having its cavity filled with water; then the globe being compressed by a strong screw, the water came through the pores of the gold rather than yield to the compression: also, Alphonsus Borelli, well known for his ingenious treatise <hi rend="italics">De Motu Animalium,</hi> and other works; Candide del Buono, brother of Paul; Alexander Marsili, Vincent Viviani, Francis Rhedi, and the Count Laurence Magalotti, secretary of this academy, who pub- <pb n="13"/><cb/> lished a volume of their curious experiments in 1667, under the title of <hi rend="italics">Saggi di Naturali Esperienze;</hi> a copy of which being presented to the Royal Society, it was translated into English by Mr. Waller, and published at London, in 4to, 1684: A curious collection of tracts, containing ingenious experiments on the pressure of the air, on the compressing of water, on cold, heat, ice, magnets, electricity, odours, the motion of sound, projectiles, light, &c, &c. But we have heard little or nothing more of the academy since that time. It may not be improper to observe here, that the Grand Duke Ferdinand, above mentioned, was no mean philosopher and chemist, and that he invented thermometers, of which the construction and use may be seen in the collection of the academy del Cimento.</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">degl' Inquieti</hi> at Bologna, incorporated afterwards into that <hi rend="italics">della traccia</hi> in the same city, followed the example of that <hi rend="italics">del Cimento.</hi> The members met at the house of the abbot Antonio Sampieri; and here Geminiano Montanari, one of the chief members, made excellent discourses on mathematical and philosophical subjects, some parts of which were published in 1667, under the title of Pensieri Fisico-Mathematici. This academy afterwards met in an apartment of Eustachio Manfredi; and then in that of Jacob Sandri; but it arrived at its chief lustre while its assemblies were held in the palace Marsilli.</p><p><hi rend="smallcaps">Academy</hi> <hi rend="italics">of Rossano,</hi> in the kingdom of Naples, called <hi rend="italics">La Societa Scientifica Rossanese degl' Incuriosi,</hi> was founded about the year 1540, under the name of <hi rend="italics">Naviganti;</hi> and was renewed under that of <hi rend="italics">Spensierati</hi> by Camillo Tuscano, about the year 1600. It was then an academy of belles-lettres, but was afterwards transformed into an academy of sciences, on the solicitation of the learned abbot Don Giacinto Gimma; who, being made president under the title of promotergeneral, in 1695, gave it a new set of regulations. He divided the academists into several classes, namely, grammarians, rhetoricians, poets, historians, philosophers, physicians, mathematicians, lawyers, and divines; with a separate class for cardinals and persons of quality. To be admitted a member, it was necessary that the candidate have degrees in some faculty. Members, in the beginning of their books, are not allowed to take the title of <hi rend="italics">academist</hi> without a written permission from the president, which is not granted till the work has been examined by the censors of the academy. This permission is the highest honour the academy can confer; since they hereby, as it were, adopt the work, and engage to answer for it against any criticisms that may be made upon it. The president himself is not exempt from this law: and it is not permitted that any academist publish any thing against the writings of another, without leave obtained from the society.</p><p>There have been several other academies of sciences in Italy, but which have not subsisted long, for want of being supported by the princes. Such were at Naples that of the <hi rend="italics">Investiganti,</hi> founded about the year 1679, by the marquis d'Arena, Don Andrea Concubletto; and that which, about the year 1698, met in the palace of Don Lewis della Cerda, the duke de Medina, and viceroy of Naples: at Rome, that of <hi rend="italics">Fisico-Matematici,</hi> which in 1686 met in the house of <cb/> Signior Ciampini: at Verona, that of <hi rend="italics">Aletosili,</hi> founded the same year by Signior Joseph Gazola, and which met in the house of the count Serenghi della Cucca: at Brescia, that of <hi rend="italics">Filesotici,</hi> founded the same year for the cultivation of philosophy and mathematics, and terminated the year following: that of F. Francisco Lana, a jesuit of great skill in these sciences: and lastly that of Fisico-Critici at Sienna, founded in 1691, by Signior Peter Maria Gabrielli.</p><p>Some other academies, still subsisting in Italy, repair with advantage the loss of the former. One of the principal is the academy of Filarmonici at Verona, supported by the marquis Scipio Maffei, one of the most learned men in Italy; the members of which academy, though they cultivate the belles lettres, do not neglect the sciences. The academy of <hi rend="italics">Ricovrati</hi> at Padua still subsists with reputation; in which; from time to time, learned discourses are held on philosophical subjects. The like may be said of the academy of the <hi rend="italics">Muti di Reggio,</hi> at Modena. At Bologna is an academy of sciences, in a flourishing condition, known by the name of <hi rend="italics">The Institute of Bologna;</hi> which was founded in 1712 by count Marsigli, for cultivating physics, mathematics, medicine, chemistry, and natural history. The history of it is written by M. de Limiers, from memoirs furnished by the founder himself. Among the new academies, the first place, after the Institute of Bologna, is given to that of the Countess Donna Clelio Grillo Boromeo, one of the most learned ladies of the age, to whom Signior Gimma dedicates his literary history of Italy. She had lately established an academy of experimental philosophy in her palace at Milan; of which Signior Vallisnieri was nominated president, and had already drawn up the regulations for it, though we do not find it has yet taken place. In the number of these academies may also be ranked the assembly of the learned, who of late years met at Venice in the house of Signior Cristino Martinelli, a noble Venetian, and a great patron of learning.</p><p><hi rend="smallcaps">Academia</hi> <hi rend="italics">Cosmografica,</hi> or that of the Argonauts, was instituted at Venice, at the instance of F. Coronelli, for the improvement of geography; the design being to procure exact maps, geographical, topographical, hydrographical, and ichnographical, of the celestial as well as terrestrial globe, and their several regions or parts, together with geographical, historical, and astronomical descriptions accommodated to them: to promote which purposes, the several members oblige themselves, by their subscription, to take one copy or more of each piece published under the direction of the academy; and to advance the money, or part of it, to defray the charge of publication. To this end there are three societies settled, namely at Venice, Paris, and Rome; the first under F. Moro, provincial of the Minorites of Hungary; the second under the abbot Laurence au Rue Payenne au Marais; the third under F. Ant. Baldigiani, jesuit, professor of mathematics in the Roman college; to whom those address themselves who are willing to engage in this design. The Argonauts number near 200 members in the different countries of Europe; and their device is the terraqueous globe, with the motto <hi rend="italics">Plus ultra.</hi> All the globes, maps, and geographical writings of F. Coronelli have been published at the expence of this academy. <pb n="14"/><cb/></p><p><hi rend="smallcaps">The Academy</hi> <hi rend="italics">of Apatists,</hi> or Impartial Academy, deserves to be mentioned on account of the extent of its plan, including universally all arts and sciences. It holds from time to time public meetings at Florence, where any person, whether academist or not, may read his works, in whatever form, language, or subject; the academy receiving all with the greatest impartiality.</p><p>In France there are many academies for the improvement of arts and sciences. F. Mersenne, it is said, gave the first idea of a philosophical academy in France, about the beginning of the seventeenth century, by the conferences of mathematicians and naturalists, held occasionally at his lodgings; at which Des Cartes, Gassendus, Hobbes, Roberval, Pascal, Blondel, and others, assisted. F. Mersenne proposed to each of them certain problems to examine, or certain experiments to be made. These private assemblies were succeeded by more public ones, formed by M. Monmort, and M. Thevenot, the celebrated traveller. The French example animated several Englishmen of rank and learning to erect a kind of philosophical academy at Oxford, towards the close of Cromwell's administration; which after the restoration was erected, by public authority, into a Royal Society: an account of which see under the word. The English example, in its turn, animated the French. In 1666 Louis XIV, assisted by the counsels of M. Colbert, founded an academy of fciences at Paris, called the</p><p><hi rend="smallcaps">Academie</hi> <hi rend="italics">Royale des Sciences, or Royal Academy of Sciences,</hi> for the improvement of philosophy, mathematics, chemistry, medicine, belles-lettres, &c. Among the principal members, at the commencement in 1666, were the respectable names of Carcavi, Huygens, Roberval, Frenicle, Auzout, Picard, Buot, Du Hamel the Secretary, and Mariotte. There was a perfect equality among all the members, and many of them received salaries from the king, as at present. By the rules of the academy, every class was to meet twice a week; the philosophers and geometricians were to meet, separately, every Wednesday, and then both together on the Saturday, in a room of the king's library, where the philosophical and mathematical books were kept: the history class was to meet on the Monday and Thursday in the room of the historical books; and the class of belles-lettres on the Tuesday and Friday: and on the first Thursday of every month all the classes met together, and by their secretaries made a mutual report of what had been transacted by each class during the preceding month.</p><p>In 1699, on the application of the president, the abbé Bignon, the academy received, under royal authority and protection, a new form and constitution; by the articles of which, the academy was to consist of four sorts of members, namely honorary, pensionary, associates, and eleves. The honorary class to consist of ten persons, and the other three classes of twenty persons each. The president to be chosen annually out of the honorary class, and the secretary and treasurer to be perpetual, and of the pensionary class. The meetings to be twice a week, on the Wednesday and Saturday; besides two public meetings in the year.</p><p>Of the pensionaries, or those who receive salaries, three to be geometricians, three astronomers, three mechanists, three anatomists, three botanists, and three <cb/> chemists, the other two being the secretary and treasurer. Of the twenty associates, of which twelve to be French, and eight might be foreigners, two were to cultivate geometry, two astronomy, two mechanics, two anatomy, two botany, and two chemistry. Of the twenty eleves, one to be attached to each pensionary, and to cultivate his peculiar branch of science. The pensionaries and their eleves to reside at Paris. No regulars nor religious to be admitted, except into the honorary class: nor any person to be admitted a pensioner who was not known by some considerable work, or some remarkable discovery.</p><p>In 1716 the Duke of Orleans, then regent of France, by the king's authority made some alteration in their constitution. The class of eleves was suppressed; and instead of them were instituted twelve adjuncts, two to each of the six classes of pensioners. The honorary members were increased to twelve: and a class of fix free associates was made, who were not under the obligation of cultivating any particular branch of science, and in this class only could the regulars or religious be admitted. A president and vice-president to be appointed annually from the honorary class, and a director and sub-director annually from that of the pensioners. And no person to be allowed to make use of his quality of academician, in the title of any of his books that he published, unless such book were first approved by the academy.</p><p>The academy has for a device or motto, <hi rend="italics">Invenit & perficit.</hi> And the meetings, which were formerly held in the king's library, have since the year 1699 been held in a fine hall of the old Louvre.</p><p>Finally, in the year 1785 the king confirmed, by letters patent, dated April 23, the establishment of the academy of sciences, making the sollowing alterations, and adding classes of agriculture, natural history, mineralogy, and physics; incorporating the associates and adjuncts, and limiting to six the members of each class, namely three pensioners and three associates; by which the former receive an increase of salary, and the latter approach nearer to becoming pensioners.</p><p>By the articles of this instrument it is ordained, that the academy shall consist of eight classes, namely, that of geometry, 2d astronomy, 3d mechanics, 4th general physics, 5th anatomy, 6th chemistry and metallurgy, 7th botany and agriculture, and 8th natural history and mineralogy. That each class shall remain irrevocably sixed at six members; namely, three pensioners and three associates, independent however of <*> perpetual secretary and treasurer, of twelve free-associates and of eight associate strangers or foreigners, the same as before, except that the adjunct-geographer for the future be called the associate-geographer.</p><p>The classes at first to be filled by the following persons, namely, that of geometry by Messieurs de Borda, Jeaurat, Vandermonde, as pensioners; and Messieurs Cousin, Meusnier, and Charles, as associates: that of astronomy by Messieurs le Monnier, de la Lande, and le Gentil, as pensioners; and Messieurs Messier, de Cassini, and Dagelat, as associates: that of mechanics by Messieurs l'abbe Bossut, Pabbe Rochon, and de la Place, as pensioners; and Messieurs Coulomb, le Gendre, and Perrier, as associates: that of general physics by Messieurs Leroy, Brisson, and <pb n="15"/><cb/> Bailly, as pensioners; and Messieurs Monge, Mechain, and Quatremere, as associates: that of anatomy by Messieurs Daubluton, Tenon, and Portal, as pensioners; and Messieurs Sabatier, Vicq-d'azir, and Broussonet, as associates: that of chemistry and metallurgy by Messieurs Cadet, Lavoisier, and Beaume, as pensioners; and Messieurs Cornette, Bertholet, aud Fourcroy, as associates: that of botany and agriculture by Messieurs Guettard, Fougeroux, and Adanson, as pensioners; and Messieurs de Jussieu, de la Marck, and Desfontaines, as associates: and that of natural history and mineralogy by Messieurs Desmaretz, Saye, and l'abbe de Gua, as pensioners; and Messieurs Darcet, l'abbe Haui, and l'abbe Tessier, as associates. All names respectable in the common-wealth of letters; and from whom the world might expect still farther improvements in the several branches of science.</p><p>The late M. Rouille de Meslay, counsellor of the parliament of Paris, founded two prizes, the one of 2500 livres, the other of 2000 livres, which the academy distributed alternately every year: the subjects of the former prize respecting physical astronomy, and of the latter, navigation and commerce.</p><p>The world is highly indebted to this academy for the many valuable works they have executed, or published, both individually and as a body collectively, especially by their memoirs, making upwards of a hundred volumes in 4to, with the machines, indexes, &c. in which may be found most excellent compositions in every branch of science. They publish a volume of these memoirs every year, with the history of the academy, and eloges of remarkable men lately deceased: also a general index to the volumes every ten years. An alteration was introduced into the volume for 1783, which it seems is to be continued in future, by omitting, in the history, the minutes or extracts from the registers, containing some preliminary account of the subjects of the memoires; but still however retaining the eloges of distinguished men, lately deceased.</p><p>M. l'abbe Rozier also has published in four 4to volumes, an excellent index of the contents of all the volumes, and the writings of all the members, from the beginning of their publications to the year 1770; with convenient blank spaces for continuing the articles in writing.</p><p>Their history also, to the year 1697, was written by M. Du Hamel; and after that time continued from year to year by M. Fontenelle, under the following titles, Du Hamel Historiæ Regiæ Academiæ Scientiarum, Paris, 4to. Histoire de l'Academie Royale des Sciences, avec les Memoires de Mathematique & de Physique, tirez des Registres de l'Academie, Paris, 4to. Histoire de l'Academie Royale des Sciences depuis son etablissement en 1666, jusqu'en 1699, en 13 tomes, 4to. A new history, from the institution of the academy, to the period from whence M. de Fontenelle commences, has been formed; with a series of the works published under the name of this academy, during the first interval.</p><p>Since the foregoing account was written, it is said the Academy has been suppressed and abolished, by the present convention of France.</p><p>Besides the academies in the capital, there are a great many in other parts of France. The <hi rend="smallcaps">Academie</hi> <hi rend="italics">Royale,</hi> at Caen, was established by letters <cb/> patent in the year 1705; but it had its rise fifty years earlier in private conferences, held first in the house of M. de Brieux. M. de Segrais retiring to this city, to spend the rest of his days, restored and gave new lustre to their meetings. In 1707 M. Foucault, intendant of the generality of Caen, procured the king's letters patent for erecting them into a perpetual academy, of which M. Foucault was to be protector for the time, and the choice afterwards left to the members, the number of whom was fixed to thirty, chosen for this time by M. Foucault. But besides the thirty original members, leave<*>was given to add six supernumerary members, from the ecclesiastical communities in that city.</p><p>At Toulouse is the <hi rend="italics">Academie des jeux floraux,</hi> composed of forty persons, the oldest of the kingdom: besides an academy of sciences and belles-lettres, founded in 1750.</p><p>At Montpelier is the royal society of sciences, which since 1708 makes but one body with the royal academy of sciences at Paris.</p><p>There are also other academies at Bourdeaux, founded in 1703, at Soissons in 1674, at Marseilles in 1726, at Lyons in 1700, at Pau in Bearn in 1721, at Montauban in 1744, at Angers in 1685, at Amiens in 1750, at Villefranche in 1679, at Dijon in 1740, at Nimes in 1682, at Besançon in 1752, at Chalons in 1775, at Rochelle in 1734, at Beziers in 1723, at Rouen in 1744, at Metz in 1760, at Arras in 1773, &c. The number of these academies is continually augmenting; and even in such towns as have no academies, the literati form themselves into literary societies, having nearly the same objects and pursuits.</p><p>In Germany and other parts of Europe, there are various academies of sciences, &c. The</p><p><hi rend="smallcaps">Academie</hi> <hi rend="italics">Royale des Sciences & des Belles Lettres</hi> of Prussia, was founded at Berlin, in the year 1700, by Frederic I. king of Prussia, of which the famous M. Leibnitz was the first president, and its great promoter. The academy received a new form, and a new set of statutes in 1710; by which it was ordained, that the president shall be one of the counsellors of state; and that the members be divided into four classes; the first to cultivate physics, medicine, and chemistry; the second, mathematics, astronomy, and mechanics; the third, the German language, and the history of the country; and the fourth, oriental learning, particularly as it may concern the propagation of the gospel among infidels. That each class elect a director for themselves, who shall hold his post for life. That they meet in the castle called the New Marshal, the classes to meet in their turns, one each week. And that the members of any of the classes have free access into the assemblies of the rest. Several volumes of their transactions have been published in Latin, from time to time, under the title of Miscellanea Berolinensia.</p><p>In 1743 the late famous Frederic II. king of Prussia, made great alterations and improvements in the academy. Instead of a great lord or minister of state, who had usually presided over the academy, he wisely judged that office would be better filled by a man of letters; and he honoured the French academy of sciences by fixing upon one of its members for a president, namely M. Maupertuis, a distinguished character in the literary world, and whose conduct in improving the academy was a proof of the sound judgment of the <pb n="16"/><cb/> king, who at the same time made new regulations for the academy, and took the title of its Protector. From that time the transactions of the academy have been published, under the title of Histoire de l'Academie Royale des Sciences et Belles Lettres à Berlin, much in the manner of those of the French academy of sciences, and in the French language; and the volumes are now commonly published annually. Besides the ordinary private meetings of the academy, it has two public ones in the year, in January and May, at the latter of which is given a prize gold medal, of the value of 50 ducats, or about 20 guineas. The subject of the prize is successively physics, mathematics, metaphysics, and general literature. For the academy has this peculiar circumstance, that it embraces also metaphysics, logic, and morality; having one class particularly appropriated to these objects, called the class of Speculative Philosophy.</p><p><hi rend="italics">Imperial</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Petersburgh.</hi> This academy was projected by the Czar Peter I, commonly called Peter the Great, who in so many other instances also was instrumental in raising Russia from the state of barbarity in which it had been immerged for so many ages. Having visited France in 1717, and among other things informed himself of the advantages of an academy of arts and sciences, he resolved to establish one in his new capital, whither he had drawn by noble encouragements several learned strangers, and made other preparations, when his death prevented him from fully accomplishing that object, in the beginning of the year 1725. Those preparations and intentions however were carried into execution the same year, by the establishment of the academy, by his consort the czarina Catherine, who succeeded him. And soon after the academy composed the first volume of their works, published in 1728, under the title of Commentarii Academiæ Scientiarum Imperialis Petropolitanæ; which they continued almost annually till 1746, the whole amounting to 14 volumes, which were published in Latin, and the subjects divided and classed under the following heads, namely mathematics, physics, history, and astronomy. Their device a tree bearing fruit not ripe, with the modest motto <hi rend="italics">paullatim.</hi></p><p>Most part of the strangers who composed this academy being dead, or having retired, it was rather in a languishing state at the beginning of the reign of the empress Elizabeth, when the count Rasomowski was happily appointed president, who was instrumental in recovering its vigour and labours. This empress renewed and altered its constitution, by letters patent dated July 24, 1747, giving it a new form and regulations. It consists of two chief parts, an academy, and a university, having regular professors in the several saculties, who read lectures as in our colleges. The ordinary assemblies are held twice a week, and public or solemn ones thrice in the year; in which an account is given of what has been done in the private ones. The academy has a noble building for their meetings, &c, with a good apparatus of instruments, a sine library, observatory, &c. Their first volume, after this renovation, was published for the years 1747 and 1748, and they have been fince continued from year to year, now to the amount of near thirty volumes, under the title of Novi Commentarii Academiæ Scientiarum Imperialis <cb/> Petropolitanæ. They are printed in the Latin language, and contain many excellent compositions in all the sciences, especially the mathematical papers of the late excellent M. L. Euler, which always made a considerable portion of every volume. The subjects are classed under heads in the following order, mathematics, physico-mathematics, physics, which include botany, anatomy, &c, and astronomy; the whole prefaced by historical extracts, or minutes, relating to each paper or memoir, after the manner of the volumes of the French academy; but wanting however the eloges of deceased eminent men. Their device is a heap of ripe fruits piled on a table, with the motto <hi rend="italics">En addit fructus ætate recentes.</hi></p><p><hi rend="italics">Imperial and Royal</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Sciences and Belles Lettres, at Brussels.</hi> This academy was founded in the year 1773; and several volumes of their memoirs have been published.</p><p><hi rend="italics">Royal</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Sciences,</hi> at Stockholm, was instituted in 1739, and since that time it has published about sixty volumes of transactions, quarterly, in 8vo, in the Swedish language.</p><p>For an account of the Royal Society of London, and several other similar institutions, see the words Journal, Society, &c.</p><p><hi rend="italics">American</hi> <hi rend="smallcaps">Academy</hi> <hi rend="italics">of Arts and Sciences,</hi> was established in 1780 by the council and house of representatives in the province of Massachuset's Bay, for promoting the knowledge of the antiquities of America, and of the natural history of the country; for determining the uses to which its various natural productions may be applied; for encouraging medicinal discoveries, mathematical disquisitions, philosophical enquiries and experiments, astronomical, meteorological, and geographical observations, and improvements in agriculture, manufactures, and commerce; and, in short, for cultivating every art and science, which may tend to advance the interest, honour, dignity, and happiness, of a free, independent, and virtuous people. The members of this academy are never to be less than forty, nor more than two hundred.</p><p><hi rend="smallcaps">Academy</hi> is also used among us for a kind of collegiate school, or seminary; where youth are instructed in the liberal arts and sciences in a private way: now indeed it is used for all kinds of schools.</p><p>Frederic 1, king of Prussia, established an academy at Berlin in 1703, for educating the young nobility of the court, suitable to their extraction. The expence of the students was very moderate, the king having undertaken to pay the extraordinaries. This illustrious school, which was then called the academy of princes, has now lost much of its first splendour.</p><p>The Romans had a kind of military academies established in all the cities of Italy, under the name of <hi rend="italics">Campi Martis.</hi> Here the youth were admitted to be trained sor war at the public expence. And the Greeks, besides academies of this kind, had military professors, called <hi rend="italics">Tactici,</hi> who taught all the higher offices of war, &c.</p><p>We have two royal academies of this kind in England, the expences of which are defrayed by the government; the one at Woolwich, for the artillery and military engineers; and the other at Portsmouth, for the navy. The former was established by his late <pb n="17"/><cb/> majesty king George II, by warrants dated April the 30th and November the 18th, 1741, for instructing persons belonging to the military part of the ordnance, in the several branches of mathematics, fortification, &c, proper to qualify them for the service of artillery and the office of engineers. This institution is under the direction of the master-general and board of ordnance for the time being: and at first the lectures of the masters in the academy were attended by the practitioner-engineers, with the officers, serjeants, corporals, and private men of the artillery, besides the eadets. At present however none are educated there but the gentlemen cadets, to the number of 90 or 100, where they receive an education perhaps not to be obtained or purchased for money in any part of the world. The master-general of the ordnance is always captain of the cadets' company, and governor of the academy; under him are a lieutenant-governor, and an inspector of studies. The masters have been gradually increased, from two or three at first, now to the number of twelve, namely, a professor of mathematics, and two other mathematical masters, a professor of fortification, and an assistant, two drawing masters, two French masters, with masters for fencing, dancing, and chemistry. This institution is of the greatest consequence to the state, and it is hardly credible that so important an object should be accomplished at so trifling an expence. It is to be lamented however that the academy is fixed in so unhealthy a situation; that the lecture rooms and cadets' barracks are so small as to be insufsicient for the purposes of the institution; and that the salaries of the professors and masters should be so inadequate to their labours, and the benefit of their services.</p><p>The Royal Naval Academy at Portsmouth was founded by George I, in 1722, for instructing young gentlemen in the sciences useful for navigation, to breed officers for the royal navy. The establishment is under the direction of the board of admiralty, who give salaries to two masters, by one of whom the students are boarded and lodged, the expence of which is defrayed by their own friends, nothing being supplied by the government but their education.</p></div2></div1><div1 part="N" n="ACANTHUS" org="uniform" sample="complete" type="entry"><head>ACANTHUS</head><p>, <hi rend="italics">in Architecture,</hi> the leaves of a plant which forms the ornament of the capital of the Corinthian order. Vitruvius ascribes the use of it to the following accident. A young girl dying, her nurse was desirous of consecrating to her manes certain toys which she was fond of in her life-time; which the good woman carried in a little basket, covered with a square tile, and placed it among some green plants which grew on her grave. One of these, which happened to be the acanthus, as it grew up, invironed and in a manner embraced the basket; which Callimachus, a noted Greek sculptor, casting his eyes upon, from thence took the hint of this elegant ornament. See <hi rend="smallcaps">Abacus.</hi></p><p>ACCELERATED <hi rend="italics">Motion,</hi> is that which receives fresh accessions of velocity; and the acceleration may be either equably or unequably: if the accessions of velocity be always equal in equal times, the motion is said to be equably or uniformly accelerated; but if the <cb/> accessions, in equal times, either increase or decrease, then the motion is unequably or variably accelerated.</p><p>Acceleration is directly opposite to retardation, which denotes a diminution of velocity.</p><p><hi rend="smallcaps">Acceleration</hi> comes chiefly under consideration in physics, in the descent of heavy bodies, tending or falling towards the centre of the earth, by the force of gravity.</p><p>That bodies are accelerated in their natural descent, is evident both to the sight, and from observing that the greater height they fall from, the greater force they strike with, and the deeper impressions they make in soft substances.</p><p>The acceleration of falling bodies has been ascribed to various causes, by different philosophers. Some have attributed it to the pressure of the air downwards: the more a body descends, the longer and heavier, say they, must be the column of atmosphere incumbent upon it; to which they add, that the whole mass of fluid pressing by an infinity of right-lines all ultimately meeting in the earth's centre, such central point must support, as it were, the pressure of the whole mass; and that consequently the nearer a body approaches to it, the more must it receive of the pressure of a multitude of lines tending to unite in the central point.</p><p>Mr. Hobbes endeavours to account for this acceleration from a new impression of the cause which makes bodies fall; in which he is so far right. But then he as far mistakes, as to the cause of the fall, which he thinks is the air: at the same time, says he, that one particle of air ascends, another descends; for in consequence of the earth's motion being two-fold, that is circular and progressive, the air must at once both ascend and circulate; whence it follows, that a body falling in this medium, and receiving a new pressure every instant, must have its motion accelerated.</p><p>But to both these systems it may be answered, that the air is quite out of the question; for it is very evident that bodies fall, and in falling have their motion accelerated, in vacuo, as in open air, and even more than in the air, in as much as this opposes and somewhat retards their fall.</p><p>The Gassendists assign another reason for the acceleration: they pretend that there are continually issuing out of the earth certain attractive corpuscles, directed in an infinite number of rays; those, say they, afcend and then descend, in such sort that the nearer a body approaches to the earth's centre, the more of these attractive rays press upon it, in consequence of which its motion becomes more accelerated.</p><p>The peripatetics endeavour to explain the matter thus: the motion of heavy bodies downward, arises, say they, out of an intrinsic principle that causes a tendency in them to the centre, as the place appropriated to their element; where, when they can once arrive, they will be at perfect rest; and therefore, continue they, the nearer bodies approach to it, the more the velocity of their motion is increased: a notion too idle to merit confutation.</p><p>The Cartesians account for acceleration, by reiterated impulses of their <hi rend="italics">materia subtilis,</hi> acting continually <pb n="18"/><cb/> on falling bodies, and propelling them downwards: a conceit equally unintelligible and absurd with the former.</p><p>But, leaving all such visionary causes of acceleration, and only admitting the existence of such a force as gravity, so evidently inherent in all bodies, without regard to what may be the cause of it, the whole mystery of acceleration will be cleared up. Consider gravity then, with Galileo, only as a cause or force which acts continually on heavy bodies; and it will be easy to conceive that the principle of gravitation, which determines bodies to descend, must by a necessary consequence accelerate them in falling.</p><p>A body then having once begun to descend, through the impulse of gravity; the state of descending is now, by Newton's first law of nature, become as it were natural to it; insomuch that, were it left to itself, it would for ever continue to descend, even though the first cause of its descent should cease. But besides this determination to descend, impressed upon it by the first cause of motion, which would be sufficient to continue to infinity the degree of motion already begun, new impulses are continually superadded by the same cause; which continues to act upon the body already in motion, in the same manner as if it had remained at rest. There being then two causes of motion, acting both in the same direction; it necessarily follows, that the motion which they unitedly produce, must be more considerable than what either could produce separately. And as long as the velocity is thus continued, the same cause still subsisting to increase it more, the descent must of necessity be continually accelerated.</p><p>Supposing then that gravity, from whatever principle it arises, acts uniformly upon all bodies at the same distance from the centre of the earth: dividing the time which the heavy body takes up in falling to the earth, into indefinitely small equal parts; gravity will impel the body toward the centre of the earth, in the first indefinitely short instant of the descent. If after this we suppose the action of gravity to cease, the body will continue perpetually to advance uniformly toward the earth's centre, with an indefinitely small velocity, equal to that which resulted from the first impulse.</p><p>But then if we suppose that the action of gravity still continues the same after the first impulse; in the second instant, the body will receive a new impulse toward the earth, equal to that which it received in the first instant; and consequently its velocity will be doubled; in the third instant, it will be tripled; in the fourth, quadrupled; in the fifth, quintupled; and so on continually: for the impulse made in any preceding instant, is no ways altered by that which is made in the following one; but they are, on the contrary, always accumulated on each other.</p><p>So that the instants of time being supposed indefinitely small, and all equal, the velocity acquired by the falling body, will be, in every instant, proportional to the times from the beginning of the descent; and consequently the velocity will be proportional to the time in which it is produced. So that if a body, by this constant force, acquire a velocity of 16 1/12 feet suppose in one second of time; it will acquire a velocity <cb/> of 32 1/6 feet in two seconds, 48 1/4 feet in 3 seconds, 64 1/3 in 4 seconds, and so on. Nor ought it to seem strange that all bodies, small or large, acquire, by the force of gravity, the same velocity in the same time. For every equal particle of matter being endued with an equal impelling force, namely its gravity or weight, the sum of all the forces, in any compound mass of matter, will be proportional to the sum of all the weights, or quantities of matter to be moved; consequently, the forces and masses moved, being thus constantly increased in the same proportion, the velocities generated will be the same in all bodies, great or small. That is, a double force moves a double mass of matter, with the same velocity that the single force moves the single mass; and so on. Or otherwise, the whole compound mass falls all together with the same velocity, and in the same manner, as if its particles were not united, but as if each fell by itself, separated all from one another. And thus all being let go at once, they would fall together, just as if they were united into one mass.</p><p>The foregoing law of the descent of falling bodies, namely that the velocities are always proportional to the times of descent, as well as the following laws concerning the spaces passed over, &c, were first discovered and taught by the great Galileo, and that nearly in the following manner.</p><p>Because the constant velocity with which any body moves, or the space it passes over in a given time, as suppose one second, being multiplied by the time, or number of seconds it is in motion, expresses the space passed over in that time; and the area or space of a rectangular figure being denoted by the length multiplied by the breadth; therefore the space so run over, may be considered as a rectangle compounded of the time and velocity, that is a rectangle of which the time denotes the length, and the velocity the breadth. Suppose then A to be the heavy body which descends, and AB to denote the whole time of any descent; which <figure/> let be divided into a certain number of equal parts, denoting intervals or portions of the given time, as AC, CD, DE, &c. Imagine the body to descend, during the time expressed by the first of the divisions AC, with a certain uniform velocity arising from the force of gravity acting on it, which let be denoted by AF, the breadth of the rectangle CF; then the space run through during the time denoted by AC, with the velocity denoted by AF, will be expressed by the rectangular space CF.</p><p>Now the action of gravity having produced, in the first moment, the velocity AF, in the body, before at rest; in the first two moments it will produce the velocity CG, the double of the former; in the third moment, to the velocity CG will be added one degree <pb n="19"/><cb/> more, by which means will be produced the velocity DH, triple of the first; and so of the rest; so that during the whole time AB, the body will have acquired the velocity BK. Hence, taking the divisions of the line AB at pleasure; for example, the divisions AC, CD, &c, for the times; the spaces run through during those times, will be as the areas or rectangles CF, DG, &c; and so the space described by the moving body during the whole time AB, will be equal to all the rectangles, that is, equal to the whole indented space ABKIHGF. And thus it will happen if the increments of velocity be produced, as we may say, all at once, at the end of certain portions of finite time; for instance at C, at D, &c; so that the degree of motion remains the same to the instant that a new acceleration takes place.</p><p>By conceiving the divisions of time to be shorter, for example but half as long as the former, the indentures of the figure will be proportionably more contracted, and it will approach nearer to a triangle; and so much the nearer as the divisions of time are shorter: and if these be supposed infinitely small; that is, if increments of the velocity be supposed to be acquired continually, and at each indivisible particle of time, which is really the case, the rectangles so successively produced, will form a true triangle, as ABC; the whole time AB consisting of minute portions A 1, <figure/> 12, 23, &c; and the area of the triangle ABC, of all the minute surfaces, or minute trapeziums, which answer to the divisions of the times; the area of the whole triangle ABC, denoting the space run through during the whole time AB; and the area of any smaller triangle A 7 <hi rend="italics">g,</hi> denoting the space run through during the corresponding time A 7. Bnt the triangles A 1 <hi rend="italics">a,</hi> A 7 <hi rend="italics">g,</hi> &c, being similar, have their areas to each other as the squares of their like sides A 1, A 7, &c; and consequently the spaces gone through, in any times counted from the beginning, are to each other as the squares of the times.</p><p>Hence, in any right-angled triangle, as ABC, the one side AB represents the time, the other side BC the velocity acquired in that time, and the area of the triangle the space described by the falling body.</p><p>From the preceding demonstration is also drawn this other general theorem in motions that are uniformly accelerated; namely, that a body descending with a uniformly accelerated motion, describes in the whole time of its descent, a space, which is exactly the half of that which it would describe uniformly in the same time, with the velocity it has acquired at the end of its accelerated fall. For it has been shewn that the whole space which the falling body has run through in the time AB, is represented by the triangle ABC, the last velocity being BC; and the space which the <cb/> body would run through uniformly in the same time AB, constantly with the said greatest velocity BC, is represented by the rectangle ABCD: but it is well known that the rectangle ABCD is double the triangle ABC; and therefore the latter space run through, is double the former; that is, the space run through by the accelerated motion, is just half of that which the body would describe in the same time, moving uniformly with the velocity acquired at the end of its accelerated fall.</p><p>Hence then, from the foregoing considerations are deduced the following general laws of uniformly accelerated motions, namely,</p><p>1st. That the velocities acquired, are constantly proportional to the times; in a double time a double velocity, &c.</p><p>2d. That the spaces described in the whole times, each counted from the commencement of the motion, are proportional to the squares of the times, or to the squares of the velocities; that is, in twice the time, the body will describe 4 times the space; in thrice the time, it will describe 9 times the space; in quadruple the time, 16 times the space; and so on. In short, if the times are proportional to the numbers 1, 2, 3, 4, 5, &c, the spaces will be as 1, 4, 9, 16, 25, &c, which are the squares of the former. So that if a body, by the natural force of gravity, fall through the space of 16 1/12 feet in the first second of time; then in the first two seconds of time it will fall through four times as much, or 64 1/3 feet; in the first three seconds it will fall nine times as much, or 144 3/4 feet; and so on. And as the spaces fallen through are as the squares of the times, or of the velocities; therefore the times, or the velocities, are proportional to the square roots of the spaces.</p><p>3d. The spaces described by falling bodies, in a series of equal instants or intervals of time, will be as the odd numbers 1, 3, 5, 7, 9, &c, <hi rend="brace"><note anchored="true" place="unspecified">1, 4, 9, 16, 25, &c,</note> which are the differences of the squares or whole spaces</hi> that is, the body which has run through 16 1/12 feet in the firft second, will in the next second run through 48 1/3 feet, in the third second 80 3/12, and so on.</p><p>4th. If the body fall through any space in any time, it acquires a velocity equal to double that space; that is, in an equal time, with the last velocity acquired, if uniformly continued, it would pass over just double the space. So if a body fall through 16 1/12 feet in the first second of time, then it has acquired a velocity of 32 1/6 feet in a second; that is, if the body move uniformly for one second, with the velocity acquired, it will pass over 32 1/6 feet in this one second: and if in any time the body fall through 100 feet; then in another equal time, if it move uniformly with the velocity last acquired, it will pass over 200 feet. And so on.</p><p>But, as the method of demonstration used by Galileo, by means of infinitely small parts forming a regular triangle, is not approved of by many persons, the same laws may be otherwise demonstrated thus: let the whole time of a body's free descent be divided into any number of parts, calling each of these parts 1; and let <hi rend="italics">a</hi> denote the velocity acquired at the end of the first <pb n="20"/><cb/> part of time; then will 2<hi rend="italics">a,</hi> 3<hi rend="italics">a,</hi> 4<hi rend="italics">a,</hi> &c, represent the velocities at the end of the 2d, 3d, 4th, &c, part of time, because the velocities are as the times; and for the same reason 1/2<hi rend="italics">a,</hi> 3/2<hi rend="italics">a,</hi> 5/2<hi rend="italics">a,</hi> &c, will be the velocities at the middle point of the first, second, third, &c, part of time. But now as the velocities increase uniformly, the space described in any one of these parts of time, may be considered as uniformly deseribed with its middle velocity, or the velocity in the middle of that part of time; and therefore multiplying those mean velocities each by their common time 1, we have the same fractions 1/2<hi rend="italics">a,</hi> 3/2<hi rend="italics">a,</hi> 5/2<hi rend="italics">a,</hi> &c, for the spaces passed over in the successive parts of the time; that is, the space 1/2<hi rend="italics">a</hi> in the first time, 3/2<hi rend="italics">a</hi> in the second, 5/2<hi rend="italics">a</hi> in the third, and so on: then add these spaces successively to one another, and we obtain 1/2<hi rend="italics">a,</hi> 4/2<hi rend="italics">a,</hi> 9/2<hi rend="italics">a,</hi> 16/2<hi rend="italics">a,</hi> &c, for the whole spaces described from the beginning of the motion to the end of the first, second, third, &c, portion of time; namely 1/2<hi rend="italics">a</hi> space in one time, 4/2<hi rend="italics">a</hi> in 2 times, 9/2<hi rend="italics">a</hi> in 3 times, and so on: and it is evident that these spaces are as the numbers 1, 4, 9, 16, &c, which are as the squares of the times.</p><p>And from this mode of demonstration, all the properties above mentioned evidently flow: such as that the whole spaces 1/2<hi rend="italics">a,</hi> 4/2<hi rend="italics">a,</hi> 9/2<hi rend="italics">a,</hi> &c, are as the squares of the times 1, 2, 3, &c, that the separate spaces 1/2<hi rend="italics">a,</hi> 3/2<hi rend="italics">a,</hi> 5/2<hi rend="italics">a,</hi> &c, <hi rend="brace"><note anchored="true" place="unspecified">1, 3, 5, &c,</note> described in the successive times, are as the odd numbers</hi> and that the velocity <hi rend="italics">a</hi> acquired in any time 1, is double the space 1/2<hi rend="italics">a</hi> described in the same time.</p><p>As the laws of acceleration are very important, I shall here insert the two following propositions, sent me by my learned friend Mr. Abram Robertson, of Christ Church College Oxford, in which those laws are demonstrated in a manner somewhat different. <hi rend="center">“<hi rend="smallcaps">Ppoposition</hi> 1.</hi></p><p>If from the point P in the straight line AB, the points M, N begin to move at the same time, namely, M towards A with a motion, uniformly retarded, and N from rest towards B with a motion uniformly accelerated; and if the velocity of M decreases as much as the velocity of N increases in the same time; then the space MN is generated by an uniform motion, equal to the velocity with which M begins to move. <figure/></p><p>For, by hypothesis, whatever is lost in the velocity of M by retardation, is added to the velocity of N by acceleration: the joint velocities, therefore, of M and N must always be equal. But it is by the joint velocities of M and N that the space MN is generated. Consequently MN is generated by an uniform motion, which is evidently equal to the velocity with which M begins to move. <hi rend="center">“<hi rend="smallcaps">Proposition</hi> II.</hi></p><p>If a point begins to move in the direction of a straight line, and continues to move in the same di- <cb/> rection with a velocity uniformly aocelerated; the space passed over in any given time, will be equal to half the space passed over in the same time with the velocity with which the acceleration ends.</p><p>Let the point D begin to move from A towards B, along the straight line AB, with a motion unisormly accelerated; the space AD passed over, is equal to half the space which the point would pass over, in the same time with the acquired velocity at D. <figure/></p><p>Let the points M, N begin to move in the straight line GH, at the same time, with equal velocities uniformly accelerated; M beginning to move from G, and N from P; and at the same time that M comes to the point P, let N come to H. Then as M and N <figure/> move with equal velocities, uniformly accelerated, it is evident that the spaces, which they pass over in the same time, are equal to one another; consequently the space GP is equal to the space PH. Now as M begins to move from G with a velocity uniformly accelerated, it will arrive at P with an acquired velocity. Hence it is evident, if it be supposed to begin to move from P with this acquired velocity, and proceed toward G with a velocity uniformly retarded in the same degree that it was accelerated when it began to move from G, that it will pass over the same space GP in the same time. Wherefore, supposing the two points M, N to begin to move from P at the same time, namely the point M beginning to move with the acquired velocity mentioned above, and proceeding towards G with the velocity uniformly retarded, described above; and the point N as before with the velocity uniformly accelerated: then as the acceleration and retardation are uniform, they will be equal in equal spaces of time. Again, as M is retarded in the same degree that it was accelerated when it began to move from G, that is, in the same degree that N is accelerated, by the former prop. MN is generated by an uniform velocity. But when the point M arrives at G, its velocity becomes equal to o or nothing; and at the time that M arrives at G, N arrives at H with the acquired velocity. Wherefore, as the velocities of M and N taken jointly are equal, and consequently uniform, the space GH is passed over with the velocity of N at H, in the same time that PH is passed over by N beginning to move from P with a velocity uniformly accelerated to H. But PH is half of GH. “Hence the prop. is manifest.”</p><p>And hence the other laws of the spaces, before<*> mentioned, easily follow.</p><p>Since the spaces descended are as the squares of the times, and the abscisses of a parabola are as the squares of the ordinates, therefore the relation of the times and spaces descended may be very well represented by the ordinates and abscisses of that figure. Thus if AB be the axis of the parabola A<hi rend="italics">bdfh,</hi> and AC a tangent <pb n="21"/><cb/> <figure/> at the vertex perpendicular to the axis, divided into any number of equal parts A<hi rend="italics">a, ac, ce,</hi> &c, for the times; and if there be drawn <hi rend="italics">ab, cd, ef,</hi> &c, parallel to the axis: hence if <hi rend="italics">ab</hi> be the space descended in the time A<hi rend="italics">a,</hi> then <hi rend="italics">cd</hi> will be the space descended in the time A<hi rend="italics">c,</hi> and <hi rend="italics">ef</hi> the space defcended in the time A<hi rend="italics">e,</hi> and so on continually.</p><p>From the properties above-demonstrated, are derived the following practical formulas or theorems for use. Namely, if <hi rend="italics">g</hi> denote the space passed over in the first second of time, by a body urged by any constant force, denoted by 1, and <hi rend="italics">t</hi> denote the time or number of seconds in which the body passes over any other space <hi rend="italics">s,</hi> and <hi rend="italics">v</hi> the velocity acquired at the end of that time; then from the foregoing laws we have <hi rend="italics">v</hi> = 2<hi rend="italics">gt,</hi> and <hi rend="italics">s</hi> = <hi rend="italics">gt</hi><hi rend="sup">2</hi>; and from these two equations result the following general formulas:</p><p>And here, when the constant force 1, is the natural force of gravity, then the distance <hi rend="italics">g</hi> descended in the first second, in the latitude of London, is 16 1/12 feet: but if it be any other constant force, the value of <hi rend="italics">g</hi> will be different, in proportion as the force is more or less.</p><p>The motion of an ascending body, or of one that is impelled upwards, is diminished or retarded by the same principle of gravity, acting in a contrary direction, after the same manner that a falling body is accelerated.</p><p>A body projected upwards, ascends until it has lost all its motion; which it does in the same space of time, that the body would have taken up in acquiring, by falling, a velocity equal to that with which the falling body began to be projected upwards. And consequently the heights to which bodies ascend, when projected upwards with different velocities, are to each other as the squares of those velocities.</p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Bodies on Inclined Planes.</hi> The same general laws obtain here, as in bodies falling freely, or perpendicularly; namely, that the velocities are as the times, and the spaces descended down the planes as the squares of the times, or of the velocities. But those velocities are less, according to the sine of the plane's inclination; and the spaces less, according to the square of the sine. See <hi rend="smallcaps">Inclined</hi> <hi rend="italics">Plane.</hi></p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Pendulums.</hi> See P<hi rend="smallcaps">ENDULUM.</hi></p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Projectiles.</hi> See P<hi rend="smallcaps">ROJECTILE.</hi></p><p><hi rend="smallcaps">Accelerated</hi> <hi rend="italics">Motion of Compressed Bodies,</hi> in ex- <cb/> panding or restoring themselves. See <hi rend="smallcaps">Dilatation, Compression</hi>, and <hi rend="smallcaps">Elasticity.</hi></p><p><hi rend="smallcaps">Accelerating Force</hi>, in Physics, is the force that accelerates the motion or velocity of bodies; and it is equal to, or expressed by, the quotientarising from the motive or absolute force, divided by the mass or weight of the body that is moved. In treating of physical considerations respecting forces, velocities, times, and spaces gone over, the first inquiry is the accelerating or accelerative force. This force is greater or less in proportion to the velocity it generates in the same time, and by this velocity it is measured. All accelerating forces are equal, and generate equal velocities, that have the motive forces directly proportional to the quantities of matter: so a double motive force will move a double quantity of matter with the same velocity, as also a triple motive force a triple quantity, a quadruple force a quadruple quantity, &c, all with the same velocity. And this is the reason why all bodies fall equally swift by the force of gravity; for the motive force is exactly proportional to their weight or mass. In general, the accelerating force is in the direct ratio of the motive force, and inverse ratio of the quantity of matter. When a body is let fall freely, to descend by the force of its natural gravity, it has been found by experiment that it falls through 16 1/12 feet in one second of time, and requires a velocity of 32 1/6 feet in that time: but if the quantity of matter be doubled, and the motive force remain the same as before, by connecting the falling body to another of equal weight by means of a thread, this other body being laid on a horizontal plane, and the falling body hanging down off the plane, and drawing the other equal body along the plane after it; then the accelerating force will be only half of what it was before, and the space fallen in one second will be only 8 1/24 feet, and the velocity acquired 16 1/12: and if the quantity of matter be tripled, or the body drawn along the plane doubled; then the accelerating force will be only one-third of what it was at first, and the space descended in one second, and velocity acquired, each one-third of the sirst: and so on.</p><p>But accelerating forces are sometimes variable, as well as sometimes constant; and the variation may be either increasing or decreasing.</p><p>The nature of constant and variable accelerating forces, may be illuftrated in the following manner. Let two weights W, <hi rend="italics">w,</hi> be connected by a thread <figure/> passing over a pully at A, B, or C; and let the weight W descend perpendicularly down, while it draws the smaller weight <hi rend="italics">w</hi> up the line AD, or BE, or CF, the <pb n="22"/><cb/> first being a straight inclined plane, and the other two curves, the one convex and the other concave to the perpendicular. Then the small weight <hi rend="italics">w</hi> will always make some certain resistance to the free descent of the large weight W, and that resistance will be constantly the same in every part of the plane AD, the difficulty to draw it up being the same in every point of it, because every part of it has the same inclination to the horizon, or to the perpendicular; and consequently the accessions to the velocity of the descending weight W, will be always equal in equal times; that is, in this case W descends by a uniformly accelerating force. But in the two curves BE, CF, the resistance or opposition of the small weight <hi rend="italics">w</hi> will be constantly altering as it is drawn up the curves, because every part of them has a different inclination to the horizon, or to the perpendicular: in the former curve, the direction becomes more and more upright, or nearer perpendicular, as the small weight <hi rend="italics">w</hi> ascends, and the opposition it makes to the descent of W, becomes more and more; and consequently the accessions to the velocity of W will be always less and less in equal times; that is, W descends by a decreasing accelerating force: but in the latter curve CF, as <hi rend="italics">w</hi> ascends, the direction of the curve becomes less and less upright, and the opposition it makes to the descent of W, becomes always less and less; and consequently the accessions to the velocity of W will be always more and more in equal times; that is, W descends by an increasing accelerating force. So that although the velocity continually increases in all these cases, yet whilst it increases in a constant ratio to the times of motion, in the plane AD; the velocity increases in a less ratio than the time it ascended up BE, and in a greater ratio than the time increases in the other curve CF.</p><p>Now the relations between the times and velocities in all these cases, may be very well represented by the relations between the abscisses and ordinates of certain lines. Thus let AB and AC be two straight lines, <figure/> making any angle BAC; and AD, AE two curves, the former concave, and the latter convex towards AB: divide AB into any parts A<hi rend="italics">a,</hi> A<hi rend="italics">b,</hi> &c, representing the times of motion; and draw the perpendiculars <hi rend="italics">acde, bfgh,</hi> &c, representing the velocities. Then in the right line AC, the ordinates <hi rend="italics">ad, bg,</hi> being as the abscisses A<hi rend="italics">a,</hi> A<hi rend="italics">b,</hi> this represents the case of uniformly accelerated motion, in which the velocities are always as the times: but in the curve AD, the ordinates <hi rend="italics">ac, bf</hi> increase in a less ratio than the abscisses A<hi rend="italics">a,</hi> A<hi rend="italics">b;</hi> and therefore this represents the case of decreasing acceleration, in which the velocities increase <cb/> in a less ratio than the times: and in the other curve AE, the ordinates <hi rend="italics">ae, bh</hi> increase in a greater ratio than the abscisses; and therefore this represents the case of increasing acceleration, in which the velocities increase in a greater ratio than the times.</p><p>The several algebraic formulas or theorems, respecting the time, velocity, space, for constant accelerating forces, are delivered above, at the article <hi rend="italics">Accelerated Motion,</hi> where the value of each circumstance is expressed in finite determinate quantities. But in the cases of variably accelerated motions, the formulas will require the help of the method of fluxions to express, not those general relations themselves, but the fluxions of them; and consequently, taking the fluents of those expressions, in particular cases, the relations of time, space, velocity, &c, are obtained.</p><p>Now if <hi rend="italics">t</hi> denote the time in motion, <hi rend="italics">v</hi> the velocity generated by any force, <hi rend="italics">s</hi> the space passed over, and 2<hi rend="italics">g</hi> the variable force at any part of the motion, or the velocity the force would generate in one second of time, if it should continue invariable, like the force of gravity, during that one second; and therefore the value of this velocity 2<hi rend="italics">g,</hi> will be in proportion to 32 1/6 feet, as that variable force, is to 1 the force of gravity. Then because the force may be supposed constant during the indefinitely small time <hi rend="italics">t,</hi> and that in uniform motions the spaces and velocities are proportional to the times, we from thence obtain these two general fundamental porportions,</p><p>From which are derived the four formulas below, in which the value of each quantity is expressed in terms of the rest.</p><p>And these theorems equally hold good for the destruction of motion and velocity, by means of retarding forces, as for the generation of the same by means of accelerating forces.</p><div2 part="N" n="Acceleration" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Acceleration</hi>, in <hi rend="italics">Mechanics</hi></head><p>, the increase of velocity in a moving body.</p><p><hi rend="smallcaps">Acceleration.</hi> <hi rend="italics">Astron.</hi> The Diurnal Acceleration of the fixed stars, is the time which the stars, in one diurnal revolution, anticipate the mean diurnal revolution of the sun; which is 3<hi rend="sup">m</hi> 55<hi rend="sup">s</hi> 9/10 of mean time, or nearly 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi>: that is, a star rises, or sets, or passes the meridian, about 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi> sooner each day. This acceleration of the stars, which is only apparent in them, arises from the real retardation of the sun, owing to his appa- <pb n="23"/><cb/> rent motion in his orbit towards the east, which is about 59′ 8″ 2/10 of a degree every day. So that the star which passed the meridian yesterday at the same moment with the sun, is to-day about 59′ 8″ past the meridian to the west, when the sun arrives at it; which will take him up about 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi> of time to pass over; and therefore the star passes by 3<hi rend="sup">m</hi> 56<hi rend="sup">s</hi> sooner than the sun each day, or anticipates his motion at that rate. The true quantity of this anticipation, or acceleration, is found by this proportion, 360° 59′ 8″ 1/5 :: 24 hours: 3<hi rend="sup">m</hi> 55<hi rend="sup">s</hi> 9/10, the fourth term of which is the acceleration.</p><p>The diurnal acceleration serves to regulate the lengths or vibration of pendulums. If I observe a fixed star set or pass behind a hill, steeple, or such like, when the pendulum marks for instance 8<hi rend="sup">h</hi> 10<hi rend="sup">m</hi>; and the next day, the eye being in the same place as before, the passage be at 8<hi rend="sup">h</hi> 6<hi rend="sup">m</hi> 4<hi rend="sup">s</hi>; I thence conclude that the pendulum is well regulated, or truly measures mean time.</p><p><hi rend="smallcaps">Acceleration</hi> <hi rend="italics">of a Planet.</hi> A planet is said to be accelerated in its motion, when its real diurnal motion exceeds its mean diurnal motion. And, on the other hand, the planet is said to be retarded in its motion, when the mean exceeds the real diurnal motion. This inequality arises from the change in the distance of the planet from the sun, which is continually varying; the planet moving always quicker in its orbit when nearer the sun, and slower when farther off.</p><p><hi rend="smallcaps">Acceleration</hi> <hi rend="italics">of the Moon,</hi> is a term used to express the increase of the moon's mean motion from the sun, compared with the diurnal motion of the earth; by which it appears that, from some uncertain cause, it is now a little quicker than it was formerly. Dr. Halley was led to the discovery, or suspicion, of this acceleration, by comparing the ancient eclipses observed at Babylon, &c, and those observed by Albategnius in the ninth century, with some of his own time; as may be seen in N 218 of the Philosophical Transactions. He could not however ascertain the quantity of the acceleration, because the longitudes of Bagdat, Alexandria, and Aleppo, where the observations were made, had not been accurately determined. But since his time the longitude of Alexandria has been ascertained by Chazelles; and Babylon, according to Ptolemy's account, lies 50′ east of Alexandria. From these data, Mr. Dunthorne, vol. 46 Philos. Transactions, compared the recorded times of several ancient and modern eclipses, with the calculations of them by his own tables, and thereby verified the suspicion that had been started by Dr. Halley; for he found that the same tables gave the moon's place more backward than her true place in ancient eclipses, and more forward than her true place in later eclipses; and thence he justly inferred that her motion in ancient times was slower, and in later times quicker, than the tables give it.</p><p>Not content however with barely ascertaining the fact, he proceeded to determine, as well as the observations would allow, the quantity of the acceleration; and by means of the most authentic eclipse, of which any good account remains, observed at Babylon in the year 721 before Christ, he found that the observed beginning of this eclipse was about an hour and three quarters sooner than the beginning by the tables; and that therefore the moon's true place preceded her place by computation by about 50′ of a degree at that time. <cb/> Then admitting the acceleration to be uniform, and the aggregate of it as the square of the time, it will be at the rate of about 10″ in 100 years.</p><p>Dr. Long, vol. ii. p. 436 of his Astronomy, enumerates the following causes from some one or more of which the acceleration may arise. Either 1st, the annual and diurnal motion of the earth continuing the same, the moon is really carried about the earth with a greater velocity than formerly: or, 2dly, the diurnal motion of the earth, and the periodical revolution of the moon, continuing the same, the annual motion of the earth about the sun is retarded; which makes the sun's apparent motion in the ecliptic a little slower than formerly; and consequently the moon, in passing from any conjunction with the sun, takes up a less time before she again overtakes the sun, and forms a subsequent conjunction: in both these cases, the motion of the moon from the sun is really accelerated, and the synodical month actually shortened: or, 3dly, the annual motion of the earth, and the periodical revolution of the moon, continuing the same, the rotation of the earth upon its axis is a little retarded; in this case, days, hours, minutes, &c, by which all periods of time must be measured, appear of a longer duration; and consequently the synodical month will appear to be shortened, though it really contain the same quantity of absolute time as it always did. If the quantity of matter in the body of the sun be lessened, by the particles of light continually streaming from it, the motion of the earth about the sun may become slower: if the earth increases in bulk, the motion of the moon about the earth may thereby be quickened.</p><p>ACCELERATIVE <hi rend="smallcaps">Force</hi>, <hi rend="italics">&c,</hi> the same as A<hi rend="smallcaps">CCELERATING.</hi></p></div2></div1><div1 part="N" n="ACCESSIBLE" org="uniform" sample="complete" type="entry"><head>ACCESSIBLE</head><p>, something that may be approached, or to which we can come. In Surveying, it is such a place as will admit of having a distance or length of ground measured from it; or such a height or depth as can be measured by actually applying a proper instrument to it. For the means of doing which, see A<hi rend="smallcaps">LTIMETRY, Longimetry</hi>, or <hi rend="smallcaps">Heights-and-Distances.</hi></p></div1><div1 part="N" n="ACCIDENS" org="uniform" sample="complete" type="entry"><head>ACCIDENS</head><p>, <hi rend="smallcaps">Accident</hi>, <hi rend="italics">Philos.</hi></p><p><hi rend="italics">Per</hi> <hi rend="smallcaps">Accidens</hi> is a term often used among philosophers, to denote what does not follow from the nature of a thing, but from some accidental quality of it: in this sense it stands opposed to <hi rend="italics">per se,</hi> which denotes the nature and essence of a thing. Thus, fire is said to burn <hi rend="italics">per se,</hi> or considered as sire, and not <hi rend="italics">per accidens;</hi> but a piece of iron, though red-hot, only burns <hi rend="italics">per accidens,</hi> by a quality accidental to it, and not considered as iron.</p><div2 part="N" n="Accidents" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Accidents</hi>, in <hi rend="italics">Astrology</hi></head><p>, denote the most extraordinary occurrences in the course of a person's life, either good or bad: such as a remarkable instance of good fortune, a signal deliverance, a great sickness, &c.</p></div2></div1><div1 part="N" n="ACCIDENTAL" org="uniform" sample="complete" type="entry"><head>ACCIDENTAL</head><p>, something that partakes of the nature of an accident; or that is indifferent, or not essential to its subject.—Thus whiteness is accidental to marble, and sensible heat to iron.</p><p><hi rend="smallcaps">Accidental</hi> <hi rend="italics">Colours,</hi> so called by M. Buffon, are those which depend on the affections of the eye, in contradistinction to such as belong to light itself.</p><p>The impressions made upon the eye, by looking stedfastly on objects of a particular colour, are various <pb n="24"/><cb/> according to the single colour, or assemblage of colours, in the object; and they continue for some time after the eye is withdrawn, and give a false colouring to other objects that are viewed during their continuance. M. Buffon has endeavoured to trace the connection between these accidental colours, and those that are natural, in a variety of instances. M. d' Arcy contrived a machine for measuring the duration of those impressions on the eye; and from the result of several trials he inserred, that the effect of the action of light on the eye continued about eight thirds of a minute.</p><p>The subject has also been considered by M. de la Hire, and M. Aepinus, &c. See Mem. Acad. Paris 1743, and 1765; Nov. Com. Petrop. vol. 10; also Dr. Priestley's Hist. of Discoveries relating to Vision, pa. 631.</p><p><hi rend="smallcaps">Accidental</hi> <hi rend="italics">Point,</hi> in Perspective, is the point in which a right line drawn from the eye, parallel to another right line, cuts the picture or perspective plane. <figure/></p><p>Let AB be the line given to be put into perspective, CFD the picture or perspective plane, and E the eye: draw EF parallel to AB; so shall F be the accidental point of the line AB, and indeed of all lines parallel to it, since only one parallel to them, namely EF, can be drawn from the same point E: and in the accidental point concur or meet the representations of all the parallels to AB, when produced.</p><p>It is called the accidental point, to distinguish it from the principal point, or point of view, where a line drawn from the eye perpendicular to the perspective plane, meets this plane, and which is the accidental point to all lines that are perpendicular to the same plane.</p><p><hi rend="smallcaps">Accidental</hi> <hi rend="italics">Dignities,</hi> and <hi rend="italics">Debilities,</hi> in <hi rend="italics">Astrology,</hi> are certain casual dispositions, and affections, of the planets, by which they are supposed to be either strengthened, or weakened, by being in such a house of the figure.</p></div1><div1 part="N" n="ACCLIVITY" org="uniform" sample="complete" type="entry"><head>ACCLIVITY</head><p>, the slope or steepness of a line or plane inclined to the horizon, taken upwards; in contradistinction to <hi rend="italics">declivity,</hi> which is taken downwards. So the ascent of a hill, is an <hi rend="italics">acclivity:</hi> the descent of the same, a <hi rend="italics">declivity.</hi></p><p>Some writers on fortification use acclivity for <hi rend="italics">talus:</hi> though more commonly the word talus is used to denote the slope, whether in ascending or descending.</p></div1><div1 part="N" n="ACCOMPANYMENT" org="uniform" sample="complete" type="entry"><head>ACCOMPANYMENT</head><p>, in <hi rend="italics">Music,</hi> denotes either the different parts of a piece of music for the different instruments, or the instruments themselves which accompany a voice, to sustain it, as well as to make the music more full.</p><p>The Accompanyment is used in recitative, as well as in song; on the stage, as well as in the choir, &c. <cb/></p><p>The ancients had likewise their accompanyments on the theatre; and they had even different kinds of instruments to accompany the chorus, from those which accompanied the actors in the recitation.</p><p>The accompanyment among the moderns, is often a different part, or melody, from the song it accompanies. But it is disputed whether it was so among the ancients.</p><p>Organists sometimes apply the word to several pipes which they occasionally touch to accompany the treble; as the drone, the flute, &c.</p><p>ACCOMPT. See <hi rend="smallcaps">Account.</hi></p></div1><div1 part="N" n="ACCORD" org="uniform" sample="complete" type="entry"><head>ACCORD</head><p>, according to the modern French music, is the union of two or more sounds heard at the same time, and forming together a regular harmony.</p><p>They divide Accords into <hi rend="italics">persect</hi> and <hi rend="italics">imperfect;</hi> and again into <hi rend="italics">consonances</hi> and <hi rend="italics">dissonances.</hi></p><p>Accord is more commonly called <hi rend="smallcaps">Concord</hi>, which see.</p><p><hi rend="smallcaps">Accord</hi> is also spoken of the state of an instrument, when its fixed sounds have among themselves all the justness that they ought to have.</p></div1><div1 part="N" n="ACCOUNT" org="uniform" sample="complete" type="entry"><head>ACCOUNT</head><p>, or <hi rend="smallcaps">Accompt</hi>, in <hi rend="italics">Arithmetic,</hi> &c, a calculation or computation of the number or order of certain things; as the computation of time, &c.</p><p>There are various ways of accounting; as, by enumeration, or telling one by one; or by the rules of arithmetic, addition, subtraction, &c.</p><div2 part="N" n="Account" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Account</hi>, in <hi rend="italics">Chronology</hi></head><p>, is nearly synonymous with style. Thus, we say the English, the foreign, the Julian, the Gregorian, the Old, or the New account, or style.</p><p>We account time by years, months, &c; the Greeks accounted it by olympiads; the Romans, by indictions, lustres, &c.</p></div2><div2 part="N" n="Acherner" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Acherner</hi></head><p>, or <hi rend="smallcaps">Acharner</hi>, in <hi rend="italics">Astronomy,</hi> a star of the first magnitude in the southern extremity of the constellation Eridanus, marked <foreign xml:lang="greek">a</foreign> by Bayer. Its longitude for 1761, <figure/> 11° 55′ 1″; and latitude south 59° 22′ 4″.</p></div2></div1><div1 part="N" n="ACHILLES" org="uniform" sample="complete" type="entry"><head>ACHILLES</head><p>, a name given by the schools to the principal argument alleged by each sect of philosophers in behalf of their system. In this sense we say this is his Achilles; that is, his master-proof: alluding to the strength and importance of the hero Achilles among the Greeks.</p><p>Zeno's argument against motion is peculiarly termed Achilles. That philosopher made a comparison between the swiftness of Achilles, and the slowness of a tortoise, pretending that a very swift animal could never overtake a slow one that was before it, and that therefore there is no such thing as motion: for, said he, if the tortoise were one mile before Achilles, and the motion of Achilles 100 times swifter than that of the tortoise, yet he would never overtake it; and for this reason, namely, that while Achilles runs over the mile, the tortoise will creep over one hundredth part of a mile, and will be so much the foremost; again while Achilles runs over this 1/100th part, the tortoise will creep over the 100th part of that 1/100th part, and will still be this last part the foremost; and so on continually, according to an infinite series of 100th parts: from which he concluded that the swifter could never overtake the slower in any finite time, but that they must go on ap- <pb n="25"/><cb/> proaching to infinity. But this sophism lay in their considering as an infinite time, the sum of the infinite series of small times in which Achilles could run over the infinite series of spaces, 1 + 1/100 + 1/10000 + 1/1000000 &c, not knowing that the sum of this infinite series is equal to the quantity 1 1/99 of a mile, and that therefore Achilles will overtake the tortoise when the latter has crawled over 1/99th of a mile.</p></div1><div1 part="N" n="ACHROMATIC" org="uniform" sample="complete" type="entry"><head>ACHROMATIC</head><p>, in <hi rend="italics">Optics,</hi> without colour; a term which, it seems, was first used by M. de la Lande, in his astronomy, to denote telescopes of a new invention, contrived to remedy aberrations and colours. See <hi rend="italics">Aberration</hi> and <hi rend="italics">Telescope.</hi></p><p><hi rend="smallcaps">Achromatic Telescope</hi>, a singular species of refracting telescope, said to be invented by the late Mr. John Dollond, optician to the king, and since improved by his son Mr. Peter Dollond, and others.</p><p>Every ray of light passing obliquely from a rarer into a denser medium, changes its direction towards the perpendicular; and every ray passing obliquely from a denser into a rarer medium, changes its direction from the perpendieular. This bending of the ray, caused by the change of its direction, is called its refraction; and the quality of light which subjects it to this refraction, is called its refrangibility. Every ray of light, before it is refracted, is white, though it consists of a number of component rays, each of which is of a different colour. As soon as it is refracted, it is separated into its component rays, which, from that time, proceed diverging from each other, like rays from a centre: and this divergency is caused by the different refrangibility of the component rays, in such sort, that the more the original or component ray is refracted, the more will the compound rays diverge when the light is refracted by one given medium only.</p><p>From hence it has been concluded, that any two different mediums that can be made to produce equal refractions, will necessarily produce equal divergencies: whence it should also follow, that equal and contrary refractions should not only destroy each other, but that the divergency of the colours caused by one refraction, should be corrected by the other; and that to produce refraction that would not be affected by the different refrangibility of light, is impossible.</p><p>But Mr. Dollond has proved, by many experiments, that these conclusions are not well founded; from which experiments it appeared, that a ray of light, after equal and contrary refractions, was still spread into component rays differently coloured: in other words, that two different mediums may cause equal refraction, but different divergency; and equal divergency, with different refraction. It follows therefore that refraction may be produced, which is not affected by the different refrangibility of light. In other words, that, if the mediums be different, different refractions may be produced, though at the same time the divergency caused by one refraction shall be exactly counteracted by the divergency caused by the other; and so an object may be seen through mediums which, together, cause the rays to converge, without appearing of different colours.</p><p>This is the foundation of Mr. Dollond's improvement of refracting telescopes. By subsequent experiments he found, that different sorts of glass differed greatly in their refractive qualities, with respect to the divergency of colours. He found that crown glass causes the least diver- <cb/> gency, and white flint the most, when they are wrought into forms that produce equal refractions. He ground a piece of white flint glass into a wedge, whose angle was about 25 degrees; and a piece of crown glass to another, whose angle was about 29 degrees; and these he found refracted nearly alike, but that their divergency of colours was very different.</p><p>He then ground several other pieces of crown glass to wedges of different angles, till he got one that was equal, in the divergency it produced, to that of a wedge of flint glass of 25 degrees; so that when they were put together, in such a manner as to refract in contrary directions, the refracted light was perfectly free from colour. Then measuring the fractions of each wedge, he found that that of the white flint glass, was to that of the crown glass, nearly as two to three. And hence any two wedges, made of these two substances, and in this proportion, would, when applied together so as to refract in contrary directions, refract the light without any effect ariling from the different refrangibility of the component rays.</p><p>Therefore, to make two spherical glasses that refract the light in contrary directions, one must be concave, and the other convex; and as the rays, after passing through both, must meet in a focus, the excess of the refraction must be in the convex one: and as the convex is to refract most, it appears from the experiment that it must be made of crown glass; and as the concave is to refract least, it must be made of white flint.</p><p>And farther, as the refractions of spherical glasses are in an inverse ratio of their focal distances, it follows that the focal distances of the two glasses should be in the ratio of the refractions of the wedges; for, being thus proportioned, every ray of light that passes through this combined glass, at whatever distance from its axis, will constantly be refracted by the difference between two contrary refractions, in the proportion required; and therefore the effect of the different refrangibility of light will be prevented.</p><p>The removal of this impediment, however, produced another: for the two glasses, which were thus combined, being segments of very deep spheres, the aberrations from the spherical surfaces became so considerable, as greatly to disturb the distinctness of the image. Yet considering that the surfaces of spherical glasses admit of great variations, though the focal distance be limited, and that by these variations their aberration might be made more or less at pleasure; Mr. Dollond plainly saw that it was possible to make the aberrating of any two glasses equal; and that, as in this case the refractions of the two glasses were contrary to each other, and their aberrations being equal, these would destroy each other.</p><p>Thus he obtained a persect theory of making object glasses, to the apertures of which he could hardly perceive any limits: for if the practice could come up to the theory, they must admit of apertures of great extent, and consequently bear great magnifying powers.</p><p>The difficulties of the practice are, however, still very considerable. For first, the focal distances, as well as the particular surfaces, must be proportioned with the utmost accuracy to the densities and refracting powers of the glasses, which vary even in the same sort of glass, when made at different times. Secondly, there are four surfaces to be wrought persectly spherical. <pb n="26"/><cb/> However, Mr. Dollond could construct refracting telescopes upon these principles, with fuch apertures and magnifying powers, under limited lengths, as greatly exceed any that were produced before, in forming the images of objects bright, distinct, and uninfected with colours about the edges, through the whole extent of a very large field or compass of view; of which he has given abundant and undeniable testimony. See T<hi rend="smallcaps">ELESCOPE.</hi></p><p>There has lately appeared in the Gentleman's Magazine (1790, pa. 890) a paper on the refracting telescope, by an author who signs <hi rend="italics">Veritus,</hi> in which the invention is ascribed to another person, not heretofore mentioned; in these words: “As the invention has been claimed by M. Euler, M. Klingenstierna, and some other foreigners, we ought, for the honour of England, to assert our right, and give the merit of the discovery to whom it is due; and therefore, without farther preface, I shall observe, that the inventor was Chester More Hall, Esq. of More-hall, in Essex, who, about 1729, as appears by his papers, considering the different humours of the eye, imagined they were placed so as to correct the different refrangibility of light. He then conceived, that if he could find substances having such properties as he supposed these humours might possess, he should be enabled to construct an object glass that would shew objects colourless. After many experiments he had the good fortune to find those properties in two different sorts of glass, and making them disperse the rays of light in different directions, he succeeded. About 1733 he completed several achromatic object glasses (though he did not give them this name), that bore an aperture of more than 2 1/2 inches, though the focal length did not exceed 20 inches; one of which is now in the possession of the Rev. Mr. Smith, of Charlotte Street, Rathbone Place.</p><p>This glass has been examined by several gentlemen of eminence and scientific abilities, and found to possess the properties of the present achromatic glasses.</p><p>Mr. Hall used to employ the working opticians to grind his lenses; at the same time he finished them with the radii of the surfaces, not only to correct the different refrangibility of rays, but also the aberration arising from the spherical figure of the lenses. Old Mr. Bass, who at that time lived in Bridewell precinct, was one of these working opticians, from whom Mr. Hall's invention seems to have been obtained.</p><p>In the trial at Westminster hall about the patent for making achromatio telescopes, Mr. Hall was allowed to be the inventor; but Lord Mansfield observed, that “It was not the person that locked up his invention in his scrutoire that ought to profit by a patent for such an invention, but he who brought it forth for the benefit of the public.” This, perhaps, might be said with some degree of justice, as Mr. Hall was a gentleman of property, and did not look to any pecuniary advantage from his discovery; and, consequently, it is very probable that he might not have an intention to make it generally known at that time.</p><p>That Mr. Ayscough, optician on Ludgate Hill, was in possession of one of Mr. Hall's achromatic telescopes in 1754, is a fact which at this time will not be disputed.”</p></div1><div1 part="N" n="ACHRONICAL" org="uniform" sample="complete" type="entry"><head>ACHRONICAL</head><p>, or <hi rend="italics">Achronycal.</hi> See <hi rend="smallcaps">Acronychal.</hi> <cb/></p><p>ACOUSTICS. This term, in physico-mathematical meaning, signifies the doctrine of hearing, and the art of assisting that sense by means of speaking trumpets, hearing trumpets, whispering galleries, and such like. See <hi rend="smallcaps">Stentrophonic Tube.</hi></p><p>Sturmius, in his Elements of Universal Mechanics, treating of Acoustics, after examining into the nature of sounds, describes the several parts of the external and internal ear, and their several uses and connexions with each other; and from thence deduces the mechanism of hearing: and lastly, he treats of the means of adding an intensity of force to the voice and other sounds; and explains the nature of echoes, otacoustic tubes, and speaking trumpets. See <hi rend="smallcaps">Sound, Ear, Voice</hi>, and <hi rend="smallcaps">Echo.</hi></p><p>Dr. Hook, in the preface to his Micrography, asserts that the lowest whisper, by certain means, may be heard at the distance of a furlong; and that he knew a way by which it is easy to hear any one speak through a wall of three feet thick; also that by means of an extended wire, sound may be conveyed to a very great distance, almost in an instant.</p></div1><div1 part="N" n="ACRE" org="uniform" sample="complete" type="entry"><head>ACRE</head><p>, from the Saxon <hi rend="italics">æcre,</hi> or German <hi rend="italics">acker,</hi> a <hi rend="italics">field,</hi> of the Latin <hi rend="italics">ager.</hi> It is a measure of land, containing, by the ordinance for measuring land, made in the 33d and 34th of Edward I, 160 perches or square poles of land; that is, 16 in length and 10 in breadth, or in that proportion: and as the statute length of a pole is 5 1/2 yards, or 16 1/2 feet, therefore the acre will contain 4840 square yards, or 43560 square feet. The chain with which land is commonly measured, and which was invented by Gunter, is 4 poles or 22 yards in length; and therefore the acre is just 10 square chains; that is, 10 chains in length and one in breadth, or in that proportion. Farther, as a mile contains 1760 yards, or 80 chains in length, therefore the square mile contains 640 acres.</p><p>The acre, in surveying, is divided into 4 roods, and the rood is 40 perches.</p><p>The French acre, <hi rend="italics">arpent,</hi> is equal to 1 1/4 English acre;</p><p>The Strasburg contains about 1/2 an English acre;</p><p>The Welch acre contains about 2 English acres;</p><p>The Irish acre contains 1 ac. 2 r. 19 27/121 p. English.</p><p>Sir William Petty, in his Political Arithmetic, reckons that England contains 39 million acres: but Dr. Greve shews, in the Philos. Trans. N° 330, that England contains not less than 46 million acres. Whence he infers that England is above 46 times as large as the province of Holland, which it is said contains but about one million of acres.</p><p>By a statute of the 31st of Elizabeth, it is ordained, that if any man erect a cottage, he shall annex four acres of land to it.</p></div1><div1 part="N" n="ACRONYCHAL" org="uniform" sample="complete" type="entry"><head>ACRONYCHAL</head><p>, or <hi rend="smallcaps">Acronycal</hi>, in <hi rend="italics">Astronomy,</hi> is said of a star or planet, when it is opposite to the sun. It is from the Greek <foreign xml:lang="greek">axronuxos</foreign>, the point or extremity of night, because the star rose at sun-set, or the beginning of night, and set at sun-rise, or the end of night; and so it shone all the night.</p><p>The acronychal is one of the three Greek poetic risings and settings of the stars; and stands distinguished from Cosmical and Heliacal. And by means of which, for want of accurate instruments, and other observations, they might regulate the length of their year. <pb n="27"/><cb/></p></div1><div1 part="N" n="ACROTERIA" org="uniform" sample="complete" type="entry"><head>ACROTERIA</head><p>, or <hi rend="smallcaps">Acroters</hi>, in <hi rend="italics">Architecture,</hi> small pedestals, usually without bases, placed on pediments, and serving to support statues.</p><p>Those at the extremities ought to be half the height of the tympanum; and that in the middle, according to Vitruvius, one eighth part more.</p><p><hi rend="smallcaps">Acroteria</hi> also are sometimes used to signify figures, whether of stone or metal, placed as ornaments or crownings, on the tops of temples, or other buildings.</p><p>It is also sometimes used to denote those sharp pinacles or spiry battlements, that stand in ranges about flat buildings, with rails and balustres.</p></div1><div1 part="N" n="ACTION" org="uniform" sample="complete" type="entry"><head>ACTION</head><p>, in <hi rend="italics">Mechanics</hi> or <hi rend="italics">Physics,</hi> a term used to denote, sometimes the effort which some body or power exerts against another body or power, and sometimes it denotes the effects resulting from such esfort.</p><p>The Cartesians resolve all physical action into metaphysical. Bodies, according to them, do not act on one another; the action comes all immediately from the Deity; the motions of bodies, which seem to be the cause, being only the occasions of it.</p><p>It is one of the laws of nature, that action and reaction are always equal, and contrary to each other in their directions.</p><p>Action is either instantaneous or continued; that is, either by collition or perc<*>ssion, or by pressure. These two sorts of action are heterogeneous quantities, and are not comparable, the smallest action by percussion exceeding the greatest action of pressure, as the smallest surface exceeds the longest line, or as the smallest solid exceeds the largest surface: thus, a man by a small blow with a hammer, will drive a wedge below the greatest ship on the stocks, or under any other weight; that is, the smallest percussion overcomes the pressure of the greatest weight. These actions then cannot be measured the one by the other, but each must have a measure of its own kind, like as solids must be measured by solids, and surfaces by surfaces: time being concerned in the one, but not in the other.</p><p>If a body be urged at the same time by equal and contrary actions, it will remain at rest. But if one of these actions be greater than its opposite, motion will ensue towards the part least urged.</p><p>The actions of bodies upon each other, in a space that is carried uniformly forward, are the same as if the space were at rest; and any powers or forces that act upon all bodies, so as to produce equal velocities in them in the same, or in parallel right lines, have no effect on their mutual actions, or relative motions. Thus the motion of bodies on board of a ship that is carried uniformly forward, are performed in the same manner as if the ship was at rest. And the motion of the earth about its axis has no effect on the actions of bodies and agents at its surface, except in so far as it is not uniform and rectilineal. In general, the actions of bodies upon each other, depend not on their absolute, but relative motion.</p><p><hi rend="italics">Quantity of</hi> <hi rend="smallcaps">Action</hi>, in Mechanics, a name given by M. de Maupertuis, in the Memoirs of the Academy of Sciences of Paris for 1744, and in those of Berlin for 1746, to the continual product of the mass of a body, by the space which it runs through, and by its celerity. He lays it down as a general law, that in the <cb/> changes made in the state of a body, the quantity of action necessary to produce such change is the least possible. This principle he applies to the investigation of the laws of refraction, and even the laws of rest, as he calls them; that is, of the equilibrium or equipollency of pressures; and even to the modes of acting of the Supreme Being. In this way Maupertuis attempts to connect the metaphysics of sinal causes with the fundamental truths of mechanics; to shew the dependence of the collision of both elastic and hard bodies, upon one and the same law, which before had always been referred to separate laws; and to reduce the laws of motion, and those of equilibrium, to one and the same principle.</p><p>But this quantity of motion, of Maupertuis, which is defined to be the product of the mass, the space passed over, and the celerity, comes to the same thing as the mass multiplied by the square of the velocity, when the space passed over is equal to that by which the velocity is measured; and so the quantity of force will be proportional to the mass multiplied by the square of the velocity; since the space is measured by the velocity continued for a certain time.</p><p>In the same year that Maupertuis communicated the idea of his principle, professor Euler, in the supplement to his book, intitled <hi rend="italics">Methodus inveniendi lineas curvas maximi vel minimi proprietate gaudentes,</hi> demonstrates, that in the trajectories which bodies describe by central forces, the velocity multiplied by what the foreign mathematicians call the element of the curve, always makes a minimum; which Maupertuis considered as an application of his principle to the motion of the planets.</p><p>It appears from Maupertuis's Memoir of 1744, that it was his reflections on the laws of refractions, that led him to the theorem above mentioned. The principle which Fermat, and after him Leibnitz, made use of, in accounting for the laws of refraction, is sufficiently known. Those mathematicians pretended, that a particle of light, in its passage from one point to another, through two mediums, in each of which it moves with a different velocity, must do it in the shortest time possible: and from this principle they have demonstrated geometrically, that the particle cannot go from the one point to the other in a right line; but being arrived at the surface that separates the two mediums, it must alter its direction in such a manner, that the sine of its incidence shall be to the sine of its refraction, as its velocity in the first medium is to its velocity in the second: whence they deduced the well known law of the constant ratio of those sines.</p><p>This explanation, though very ingenious, is liable to this pressing difficulty, namely, that the particle must approach towards the perpendicular, in that medium where its velocity is the least, and which consequently resists it the most: which seems contrary to all the mechanical explanations of the refraction of bodies, that have hitherto been advanced, and of the refraction of light in particular.</p><p>Sir Isaac Newton's way of accounting for it, is the most satisfactory of any that has hitherto been offered, and gives a clear reason for the constant ratio of the sines, by ascribing the refraction to the attractive force of the mediums; from which it follows, that the densest <pb n="28"/><cb/> mediums, whose attraction is the strongest, should cause the ray to approach the perpendicular; a fact confirmed by experiment. But the attraction of the medium could not caúse the ray to approach towards the perpendicular, without increasing its velocity; as may easily be demonstrated. Thus then, according to Newton, the refraction must be towards the perpendicular, when the velocity is increased: contrary to the law of Fermat and Leibnitz.</p><p>Maupertuis has attempted to reconcile Newton's explanation with metaphysical principles. Instead of supposing, as the aforesaid gentlemen do, that a particle of light proceeds from one point to another in the shortest time possible; he contends that a particle of light passes from one point to another in such a manner, that the quantity of action shall be the least possible. This quantity of action, says he, is a real expence, in which nature is always frugal. In virtue of this philosophical principle he discovers, that not only the sines are in a constant ratio, but also that they are in the inverse ratio of the velocities, according to Newton's explanation, and not in the direct ratio, as had been pretended by Fermat and Leibnitz.</p><p>It is remarkable that, of the many philosophers who have written on refraction, none should have fallen upon so simple a manner of reconciling metaphysics with mechanics; since no more is necessary to that, than making a small alteration in the ealculus founded upon Fermat's principle. Now according to that principle, the time, that is, the space divided by the velocity, should be a minimum; so that calling the space run through in the first medium S, with the velocity V, and the space run through in the second medium <hi rend="italics">s,</hi> with the velocity <hi rend="italics">v,</hi> we shall have minimum; that is to say, . Now it is easy to perceive, that the sines of incidence and refraction are to each other, as S<hi rend="sup">.</hi> to-<hi rend="italics">s<hi rend="sup">.</hi></hi>; whence it follows, that those sines are in the direct ratio of the velocities V, <hi rend="italics">v</hi>; which is exactly what Fermat makes it to be. But in order to have those sines to be in the inverse ratio of the velocities, it is only supposing ; which gives a minimum: which is Maupertuis's principle.</p><p>In the Memoirs of the Academy of Berlin, above cited, may be seen all the other applications which Maupertuis has made of this principle. And whatever may be determined as to his metaphysical basis of it, as also to the idea he has annexed to the quantity of action, it will still hold good, that the product of the space by the velocity is a minimum in some of the most general laws of nature.</p></div1><div1 part="N" n="ACTIVE" org="uniform" sample="complete" type="entry"><head>ACTIVE</head><p>, the quality of an agent, or of communicating motion or action to some body. In this sense the word stands opposed to passive: thus we say an active cause, active principle, &c.</p><p>Sir Isaac Newton shews that the quantity of motion in the world must be always deereasing, in consequence of the vis inertiæ, &c. So that there is a necessity for <cb/> certain active principles to recruit it: such he takes the cause of gravity to be, and the cause of fermentation; adding, that we see but little motion in the universe, except what is owing to these active principles.</p></div1><div1 part="N" n="ACTIVITY" org="uniform" sample="complete" type="entry"><head>ACTIVITY</head><p>, the virtue or faculty of acting. As the activity of an acid, a poison, &c: the activity of fire exceeds all imagination.</p><p>According to Sir Isaac Newton, bodies derive their activity from the principle of attraction.</p><p><hi rend="italics">Sphere of</hi> <hi rend="smallcaps">Activity</hi>, is the space which surrounds a body, as far as its efficacy or virtue extends to produce any sensible effect. Thus we say, the sphere of activity of a loadstone, of an electric body, &c.</p></div1><div1 part="N" n="ACUBENE" org="uniform" sample="complete" type="entry"><head>ACUBENE</head><p>, in <hi rend="italics">Astronomy,</hi> the Arabic name of a star of the fourth magnitude, in the southern claw of Cancer, marked <foreign xml:lang="greek">a</foreign> by Bayer. Its longitude for 1761, <figure/> 10° 18′ 9″, south latitude 5° 5′ 56″.</p></div1><div1 part="N" n="ACUTE" org="uniform" sample="complete" type="entry"><head>ACUTE</head><p>, or sharp; a term opposed to obtuse. Thus, <hi rend="smallcaps">Acute</hi> <hi rend="italics">Angle,</hi> in <hi rend="italics">Geometry,</hi> is that which is less than a right angle; and is measured by less than 90°, or by less than a quadrant of a circle. As the angle ABC. <figure/></p><p><hi rend="smallcaps">Acute</hi> <hi rend="italics">angled Triangle,</hi> is that whose three angles are all acute; and is otherwise called an oxygenous triangle. As the triangle DEF.</p><p><hi rend="smallcaps">Acute</hi>-<hi rend="italics">angled Cone,</hi> is that whose opposite sides make an acute angle at the vertex, or whose axis, in a right cone, makes less than half a right angle with the side As the cone GHI.</p><p>Pappus, in his Mathematical Collections, says, this name was given to such a cone by Euclid and the ancients, before the time of Apollonius. And they called an</p><p><hi rend="smallcaps">Acute</hi>-<hi rend="italics">angled Section of a Cone,</hi> an Ellipsis, which was made by a plane cutting both sides of an acuteangled cone: not knowing that such a section could be generated from any cone whatever, till it was shewn by Apollonius.</p><div2 part="N" n="Acute" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Acute</hi>, in <hi rend="italics">Music</hi></head><p>, is understood of a tone, or sound, which is high, sharp, or shrill, in respect of some other: in which sense the word stands opposed to grave. And both these sounds are independent of loudness or force: so that the tone may be acute or high, without being loud; and loud without being high or acute. For both the affections of acute and grave, depend intirely on the quickness or slowness of the vibrations by which they are produced.</p><p>Sounds considered as grave and acute, that is, in the relation of gravity and acuteness, constitute what is called tune, the soundation of all harmony.</p></div2></div1><div1 part="N" n="ADAGIO" org="uniform" sample="complete" type="entry"><head>ADAGIO</head><p>, in <hi rend="italics">Music,</hi> one of the terms used by the Italians to express a degree or distinction of time.</p><p>Adagio denotes the slowest time except grave.</p><p>Sometimes the word is repeated, as <hi rend="italics">adagio, adagio,</hi> to denote a still slower time than the former. <pb n="29"/><cb/></p><p>Adagio also signifies a slow movement, when used substantively.</p></div1><div1 part="N" n="ADAMAS" org="uniform" sample="complete" type="entry"><head>ADAMAS</head><p>, in <hi rend="italics">Astrology,</hi> a name given to the moon.</p></div1><div1 part="N" n="ADAR" org="uniform" sample="complete" type="entry"><head>ADAR</head><p>, in the Hebrew <hi rend="italics">Chronology,</hi> is the 6th month of their civil year, but the 12th of their ecclesiastical year. It contains only 29 days; and it answers to our February; but sometimes entering into the month of March, according to the course of the moon.</p></div1><div1 part="N" n="ADDITION" org="uniform" sample="complete" type="entry"><head>ADDITION</head><p>, the uniting or joining of two or more things together; or the finding of one quantity equal to two or more others taken together.</p><div2 part="N" n="Addition" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Addition</hi>, in <hi rend="italics">Arithmetic</hi></head><p>, is the first of the four fundamental rules or operations of that science; and it consists in finding a number equal to several others taken together, or in finding the most simple expression of a number according to the established notation. The quantity so found equal to several others taken together, is named their sum.</p><p>The sign or character of addition is +, and is called <hi rend="italics">plus.</hi> This character is set between the quantities to be added, to denote their sum: thus, , that is, 3 plus 6 are equal to 9; and , that is, 2 plus 4 plus 6 are equal to 12.</p><p>Simple numbers are either added as above; or else by placing them under one another, as in the margin, and adding them together, one after another, beginning at the bottom: thus 2 and 4 make 6, and 6 make 12. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row></table></p><p>Compound numbers, or numbers consisting of more figures than one, are added, by first ranging the numbers in columns under each other, placing always the numbers of the same denomination under each other, that is, units under units, tens under tens, and so on; and then adding up each column separately, beginning at the right hand, setting down the sum of each column below it, unless it amount to ten or some number of tens, and in that case setting down only the overplus, and carrying one for each ten to the next column. Thus, to add 451 and 326, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">451</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">that is</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">400 + 50 + 1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">326</cell><cell cols="1" rows="1" rend="align=right" role="data">300 + 20 + 6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Sum 777</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">=</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">700 + 70 + 7</cell></row></table></p><p>Also to add the numbers ; set them down as in the margin, and beginning at the lowest number on the right hand, say 8 and 7 make 15, and 2 make 17, and 9 make 26; set down 6, and carry 2 to the next column, saying 2 and 4 make 6, and 4 make 10, and 6 make 16, and 2 make 18; set down 8, and carry 1, saying 1 and 3 make 4, and 5 make 9, and 3 make 12; set down 2, and carry 1, saying 1 and 2 make 3, and 1 make 4, which set down; then 1 and 2 make 3; and 7 is 7 to set down: so the sum of all together is 734286. Or it is the same as the sums of the columns set under one another, as in the margin, and then these added up in the same manner. <cb/></p><p>When a great number of separate sums or numbers are to be added, as in long accounts, it is easier to break or separate them into two or more parcels, which are added up severally, and then their sums added together for the total sum. And thus also the truth of the addition may be proved, by dividing the numbers into parcels different ways, as the totals must be the same in both cases when the operation is right.</p><p>Another method of proving addition was given by Dr. Wallis, in his Arithmetic, published 1657, by casting out the nines, which method of proof extends also to the other rules of arithmetic. The method is this: add the figures of each line of numbers together severally, casting out always 9 from the sums as they arise in so adding, adding the overplus to the next figure, and setting at the end of each line what is over the nine or nines; then do the same by the sum-total, as also by the former excesses of 9, so shall the last excesses be equal when the work is right. So the former example will be proved as below: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">329</cell><cell cols="1" rows="1" rend="rowspan=6" role="data">Excess of 9's</cell><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1562</cell><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20347</cell><cell cols="1" rows="1" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">712048</cell><cell cols="1" rows="1" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">734286</cell><cell cols="1" rows="1" role="data">3</cell></row></table></p><p>When the numbers are of different denominations; as pounds, shillings, and pence; or yards, seet, and inches; place the numbers of the same kind under one another, as pence under pence, shillings under shillings, &c; then add each column separately, and carry the overplus as before, from one column to another. As in the following examples: <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">l.</cell><cell cols="1" rows="1" rend="align=center" role="data">s.</cell><cell cols="1" rows="1" rend="align=center" role="data">d.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Yards.</cell><cell cols="1" rows="1" rend="align=center" role="data">Feet.</cell><cell cols="1" rows="1" rend="align=center" role="data">Inches.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">271</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">271</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">94</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">408</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">sums</cell><cell cols="1" rows="1" rend="align=right" role="data">323</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row></table></p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Decimals,</hi> is performed in the same manner as that of whole numbers, placing the numbers of the same denomination under each other, in which case the decimal separating points will range straight in one column; as in this example, to add together these numbers . <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">371.0496</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25.213 </cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1.704 </cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">924.61  </cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.0962</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">The sum</cell><cell cols="1" rows="1" rend="align=right" role="data">1322.6728</cell></row></table></p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Vulgar Fractions,</hi> is performed by bringing all the proposed fractions to a common denominator, if they have different ones, which is an indispensable preparation; then adding all the numerators <pb n="30"/><cb/> together, and placing their sum over the common denominator for the sum total required.</p><p>So .</p><p>ADDITION <hi rend="italics">in Algebra,</hi> or the addition of indeterminate quantities, denoted by letters of the alphabet, is performed by connecting the quantities together by their proper signs, and uniting or reducing such as are susceptible of it, namely similar quantities, by adding their co-efficients together if the signs are the same, but subtracting them when different. Thus the quantity <hi rend="italics">a</hi> added to the quantity <hi rend="italics">b,</hi> makes <hi rend="italics">a</hi> + <hi rend="italics">b</hi>; and <hi rend="italics">a</hi> joined with-<hi rend="italics">b,</hi> makes <hi rend="italics">a</hi>-<hi rend="italics">b</hi>; also-<hi rend="italics">a</hi> and-<hi rend="italics">b</hi> make-<hi rend="italics">a</hi>-<hi rend="italics">b</hi>; and 3<hi rend="italics">a</hi> and 5<hi rend="italics">a</hi> make 3<hi rend="italics">a</hi> + 5<hi rend="italics">a</hi> or 8<hi rend="italics">a,</hi> by uniting the similar numbers 3 and 5 to make 8. Thus also .</p><p>In the addition of surd or irrational quantities, they must be reduced to the same denomination, or to the same radical, if that can be done; then add or unite the rational parts, and subjoin the common surd. Otherwise connect them with their own signs.</p><p>So ; but of √5 and √6 the sum is set down √5 + √6, because the terms are incommensurable, and not reducible to a common surd.</p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Logarithms.</hi> See Logarithms. <cb/></p><p><hi rend="smallcaps">Addition</hi> <hi rend="italics">of Ratios,</hi> the same as composition of ratios; which see.</p></div2></div1><div1 part="N" n="ADDITIVE" org="uniform" sample="complete" type="entry"><head>ADDITIVE</head><p>, denotes something to be added to another, in contradistinction to something to be taken away or subtracted. So astronomers speak of additive equations, and geometricians of additive rations.</p></div1><div1 part="N" n="ADELARD" org="uniform" sample="complete" type="entry"><head>ADELARD</head><p>, or <hi rend="smallcaps">Athelard</hi>, was a learned monk of Bath, in England, who flourished about the year 1130, as appears by some manuscripts of his in Corpus Christi, and Trinity Colleges, Oxford. Vossius says he was universally learned in all the sciences of his time; and that, to acquire all sorts of knowledge, he travelled into France, Germany, Italy, Spain, Egypt, and Arabia. He wrote many books himself, and translated others from different languages: thus, he translated, from Arabic into Latin, Euclid's Elements, at a time before any Greek copies had been discovered; also Erichiafarim, upon the seven planets. He wrote a book on the seven liberal arts, another on the astrolabe, another on the causes of natural compositions, besides several on physics and on medicine.</p><p>Although Vossius refers to Oxford for some of these manuscripts, it would yet seem they were not to be found there in Wallis's time; for the Doctor, speaking of this author, and other English authors and travellers about the fame age, says, “A particular account of these travels of Sholley and Morley was a while since to be seen in two prefaces to two manuscript books of theirs in the library of Corpus-Christi College in Oxford, but hath lately (by some unknown hand) been cut out, and carried away; which prefaces (one or both of them) did also make mention of the travels of Athelardus Bathoniensis, and are, to that purpose, cited by Vossius out of that manuscript copy. Whoever hath them, would do a kindness (by some way or other) to restore them, or at leaft a copy of them.” Wallis's Algebra, pa. 6.</p></div1><div1 part="N" n="ADELM" org="uniform" sample="complete" type="entry"><head>ADELM</head><p>, <hi rend="smallcaps">Aldhelmus</hi>, or <hi rend="smallcaps">Althelmus</hi>, a learned Englishman, who flourished about the year 680. He was sirst abbot of Malmsbury, and afterward bishop of Shirburn. He died in the year 709, in the monastery of Malmsbury.</p><p>Adelm was the son of Kenred or Kenten, who was the brother of Ina, king of the West Saxons in England. Beside certain books in theology, he composed several on the mathematical sciences &c; as Arithmetic, and Astrology, and librum de philosophorum disciplinis. See Bede's History, lib. 5. cap. 19. He is also mentioned by Bale and William of Malmsbury.</p></div1><div1 part="N" n="ADERAIMIN" org="uniform" sample="complete" type="entry"><head>ADERAIMIN</head><p>, or <hi rend="smallcaps">Alderaimin</hi>, the Arabic name of a star of the third magnitude, in the left shoulder of Cepheus, marked <foreign xml:lang="greek">a</foreign> by Bayer. Its longitude for 1761, <foreign xml:lang="greek">g</foreign> 9° 30′ 8″ north latitude 68° 56′ 20″.</p></div1><div1 part="N" n="ADFECTED" org="uniform" sample="complete" type="entry"><head>ADFECTED</head><p>, see <hi rend="smallcaps">Affected.</hi></p></div1><div1 part="N" n="ADHESION" org="uniform" sample="complete" type="entry"><head>ADHESION</head><p>, <hi rend="smallcaps">Adherence</hi>, in <hi rend="italics">Physics,</hi> is the state of two bodies, joined or fastened together, whether by mutual attraction, the interposition of their own parts, or the impulse or pressure of external bodies. See <hi rend="smallcaps">Cohesion.</hi></p><p>Thus two hollow hemispheres, exhausted of air, are made to adhere firmly together by the pressure of the atmosphere on their convex or external surfaces; for if they are introduced into an exhausted receiver, they presently fall asunder. Also two very well polished <pb n="31"/><cb/> planes adhere firmly together, partly by the external pressure of the atmosphere, and partly by the attraction of their parts.</p><p>In No. 389 of the Philos. Trans. Dr. Desaguliers has given experiments of the adhesion of leaden bullets to each other: the cause of which he resolves into the principle of attraction.</p><p>M. Musschenbroeck, in his Essai de Physique, has given a great many remarks on the adhesion of bodies, and relates various experiments which he had made upon this matter, but chiefly relative to the resistance made by bodies to fracture, in virtue of the adhesion of their parts; which adhesion he ascribes principally to their mutual attraction. Common experiments prove the mutual adhesion of the parts of water to each other, as well as to the bodies they touch. The same may be said of the particles of air, on which M. Petit has a memoir among those of the Paris Academy of Sciences for the year 1731.</p><p>Some authors however are not willing to admit that the adhesion of the parts of water, or indeed of bodies in general, is to be attributed to the attraction of their parts, and they reason thus: suppose, say they, that attraction acts at any small distance, as for example to the distance of one-tenth of an inch, from a particle of water: and about this particle describe a circle whose radius is one-tenth of an inch: then the particle of water will be attracted only by the particles included within the circle; but as these particles act in contrary directions, their mutual effects must destroy one another, and there can be no attraction of the particle, since it will have no more tendency one way than another.</p></div1><div1 part="N" n="ADHIL" org="uniform" sample="complete" type="entry"><head>ADHIL</head><p>, in <hi rend="italics">Astronomy,</hi> a star, of the sixth magnitude, upon the garment of Andromeda, under the last star in her foot.</p></div1><div1 part="N" n="ADJACENT" org="uniform" sample="complete" type="entry"><head>ADJACENT</head><p>, whatever lies immediately by the side of another.</p><p><hi rend="smallcaps">Adjacent</hi> <hi rend="italics">Angle,</hi> in <hi rend="italics">Geometry,</hi> is said of an angle when it is immediately contiguous to another, so that they have both one common side. And the term is more particularly used when the two angles have not only one common side, but also when the other two sides form one continued right line.</p><p><hi rend="smallcaps">Adjacent</hi> <hi rend="italics">bodies, in Physics,</hi> are understood of those that are near, or next to, some other body.</p></div1><div1 part="N" n="ADJUTAGE" org="uniform" sample="complete" type="entry"><head>ADJUTAGE</head><p>, or rather AJUTAGE; which see.</p></div1><div1 part="N" n="ADSCRIPTS" org="uniform" sample="complete" type="entry"><head>ADSCRIPTS</head><p>, in <hi rend="italics">Trigonometry,</hi> is used by some mathematicians, for the tangents of arcs. Vieta calls them also prosines.</p><p>ADVANCE-<hi rend="smallcaps">Fosse</hi>, in Fortification, a ditch thrown round the esplanade or glacis of a place, to prevent its being surprised by the besiegers.</p><p>The ditch sometimes made in that part of the lines or retrenchments nearest the enemy, to prevent him from attacking them, is also called the advance-fosse.</p><p>The advance-fosse should always be full of water, otherwise it will serve to cover the enemy from the fire of the place, if he should become master of the fosse. Beyond the advance-fosse it is usual to construct lunettes, redouts, &c.</p></div1><div1 part="N" n="ADVENT" org="uniform" sample="complete" type="entry"><head>ADVENT</head><p>, <hi rend="italics">Adventus,</hi> in the Calendar, the time immediately preceding Christmas; and was anciently employed in pious preparation for the <hi rend="italics">adventus,</hi> or coming on, of the feast of the Nativity. <cb/></p><p>Advent includes four Sundays, or weeks; commencing either with the Sunday which falls on St. Andrew's day, namely the 30th day of November, or the nearest Sunday to that day, either before or after.</p><p>ÆOLIPILE, <hi rend="italics">Æolipile,</hi> in Hydraulics, a hollow ball of metal, with a very small hole or opening; chiefly used to shew the convertibility of water into elastic steam. The best way of fitting up this instrument, is with a very slender neck or pipe, to screw on and off, for the convenience of introducing the water into the inside; for by unscrewing the pipe, and immerging the ball in water, it readily fills, the hole being pretty large; and then the pipe is screwed on. But if the pipe do not screw off, its orifice is too small to force its way in against the included air; and therefore to expel most of the air, the ball is heated red hot, and suddenly plunged with its orifice into water, which will then rush in till the ball is about two-thirds filled with the water. The water having been introduced, the ball is set upon the fire, which gradually heats the contained water, and converts it into elastic steam, which rushes out by the pipe with great violence and noise; and thus continues till all the water is so discharged; though not with a constant and uniform blast, but by sits: and the stronger the fire is, the more elastic will the steam be, and the force of the blast. Care should be taken that the ball be not set upon a violent fire with very little water in it, and that the small pipe be not stopped with any thing; for in such case, the included elastic steam will suddenly burst the ball with a very dangerous explosion.</p><p>This instrument was known to the ancients, being mentioned by Vitruvius, lib. 1. cap. 6. It is also treated of, or mentioned, by several modern authors, as Descartes, in his Meteor. cap. 4; and Father Mersennus, in prop. 29 Phædom. Pneumat. uses it to weigh the air, by first weighing the instrument when red hot, and having no water in it; and afterwards weighing the same when it becomes cold. But the conclusion gained by this means, cannot be quite accurate, as there is supposed to be no air in the ball when it is red hot; whereas it is shewn by Varenius, in his Geography, cap. 19, sect. 6, prop. 10, that the air is raresied but about 70 times; and consequently the weight obtained by the above process, will be about one-70th too small, or more or less according to the intensity of the heat.</p><p>In Italy it is said that the Æolipile is often used to cure smoaky chimneys: for being hung over the fire, the blast arising from it carries up the loitering smoke along with it.</p><p>And some have imagined that the æolipile might be employed as bellows to blow up a fire, having the blast from the pipe directed into the fire: but experience would soon convince them of their mistake; for it would rather blow the sire <hi rend="italics">out</hi> than <hi rend="italics">up,</hi> as it is not air, but rarefied water, that is thus violently blown through the pipe.</p><p>ÆOLUS, in Mechanics, a small portable machine, not long since invented by Mr. Tidd, for refreshing and changing the air in rooms which are made too close.</p><p>The machine is adapted to supply the place of a <pb n="32"/><cb/> square of glass in a sash-window, where it works with little or no noise, on the principle of the sails of a mill, or a smoke-jack; and thus admitting an agreeable quantity of air, at a convenient part of the room.</p><p>Æ<hi rend="smallcaps">OLUS</hi>'<hi rend="italics">s Harp,</hi> or <hi rend="italics">Æolian Harp,</hi> an instrument so named, from its producing an agreeable melody, merely by the action of the wind.</p><p>Neither the age nor inventor of this instrument are very well known. It is not mentioned by Mersennus in his Harmonics, where he describes most sorts of musical instruments: and yet the description and use of it was given soon after, by Kircher, in his book, Magia Phenotactica & Phonurgia.</p><p>The construction of this instrument is thus; let a box be made of as thin deal as possible, its length answering exactly to the width of the window in which it is to be placed; five or six inches deep, and seven or eight inches wide. Across the top, and near each end, glue on a bit of wainscot, about half an inch high, and a quarter of an inch thick, to serve as two bridges for the strings to be stretched over, by means of pins inserted into holes a little behind the bridges, nearer the ends, half the number being at one end, and half at the other end: these pins are like those of a harpsichord; and for their better support in the thin deal, a piece of beech of about an inch square, and length equal to the breadth of the box, is glewed on the inside of the lid, immediately under the place of the pins, the holes for receiving them being bored through this piece. It is strung with small catgut, or blue first fiddle strings, more or less at pleasure, on the outside and lengthways of the lid, fixing one end to one of the small pins, and twisting the other end about the opposite or stretching pin. A couple of sound-holes are cut in the lid; and the thinner this is, the better will be the performance.</p><p>When the strings are tuned unison, and the instrument placed, with the top or stringed side outwards, in the window to which it is fitted, the air blowing upon that window, the instrument will give a sound like a distant choir, increasing or decreasing according to the strength of the wind.</p><p>ÆRA, in Chronology, is the same as epoch, or epocha, and means a sixed point of time, from which to begin a computation of the years ensuing.</p><p>The word is sometimes also written <hi rend="italics">cra</hi> in ancient authors. Its origin is contested, though it is generally supposed that it had its rise in Spain. Some imagine that it is formed from <hi rend="italics">a. er. a.</hi> the abbreviations of the words, <hi rend="italics">annus erat Augusti,</hi> or from <hi rend="italics">a. e. r. a.</hi> the initials of the words <hi rend="italics">annus erat regni Augusti,</hi> because the Spaniards began their computation from the time that their country came under the dominion of Augustus. Others derive it from <hi rend="italics">æs, brass,</hi> the tribute money with which Augustus taxed the world. It is also said that <hi rend="italics">æra</hi> originally signified a number stamped on money to determine its current value. And that the ancients used <hi rend="italics">æs</hi> or <hi rend="italics">æra</hi> as an article, as we do the word <hi rend="italics">item,</hi> to each particular of an account; and hence it came to stand for a sum or number itself.</p><p>Æ<hi rend="smallcaps">RA</hi> also means the way or mode of accounting time. Thus we say such a year of the Christian æra, &c.</p><p><hi rend="italics">Spanish</hi> Æ<hi rend="smallcaps">RA</hi>, otherwise called the year of Cæsar, <cb/> was introduced after the second division of the Roman provinces, between Augustus, Anthony, and Lepidus, in the 714th year of Rome, the 4676th year of the Julian period, and the 38th year before Christ. In the 447th year of this æra, the Alani, the Vandals, Suevi, &c, entered Spain. It is frequently mentioned in the Spanish affairs; their councils, and other public acts, being all dated according to it. Some say it was abolished under Peter IV, king of Arragon, in the year of Christ 1358, and the Christian æra introduced instead of it. But Mariana observes that it ceased in the year of Christ 1383, under John I, king of Castile. The like was afterwards done in Portugal.</p><p><hi rend="italics">Christian</hi> Æ<hi rend="smallcaps">RA.</hi> It is generally allowed by Chronologers, that the computation of time from the birth of Christ, was only introduced in the sixth century in the reign of Justinian; and it is commonly ascribed to Dionysius Exiguus. This æra came then into use in deeds, and such like; before which time either the olympiads, the year of Rome, or that of the reign of the emperors, was used for such purposes.</p><p>See an account of the other principal æras under the word Epoch.</p><p>AERIAL <hi rend="italics">Perspective,</hi> is that which represents bodies diminished and weakened, in proportion to their distance from the eye.</p><p>Aërial Perspective chiefly respects the colours of objects, whose force and lustre it diminishes more or less, to make them appear as if more or less remote.</p><p>It is founded upon this, that the longer the column of air an object is seen through, the more feebly do the visual rays emitted from it affect the eye.</p></div1><div1 part="N" n="AEROGRAPHY" org="uniform" sample="complete" type="entry"><head>AEROGRAPHY</head><p>, a description of the air, or atmosphere, its limits, dimensions, properties, &c.</p></div1><div1 part="N" n="AEROLOGY" org="uniform" sample="complete" type="entry"><head>AEROLOGY</head><p>, the doctrine or science of the air, and its phænomena, its properties, good and bad qualities, &c. It is much the same with the soregoing word, Aerography.</p></div1><div1 part="N" n="AEROMETRY" org="uniform" sample="complete" type="entry"><head>AEROMETRY</head><p>, <hi rend="italics">Aerometria,</hi> the science of measuring the air, its powers and properties; comprehending not only the quantity of the air itself, as a fluid body, but also its pressure or weight, its elasticity, rarefaction, condensation, &c.</p><p>The term is not much used at present; this branch of natural philosophy being usually called pneumatics, which see. Wolfius, late professor of mathematics at Hall, having reduced several properties of the air to geometrical demonstrations, sirst published at Leipsic his Elements of Aerometry, in the German language, and afterwards more enlarged in Latin, which have since been inserted in his <hi rend="italics">Cursus Mathematicus,</hi> in five volumes in 4to.</p></div1><div1 part="N" n="AERONAUTICA" org="uniform" sample="complete" type="entry"><head>AERONAUTICA</head><p>, the pretended art of sailing through the air, or atmosphere, in a vessel, sustained as a ship in the sea.</p></div1><div1 part="N" n="AEROSTATICA" org="uniform" sample="complete" type="entry"><head>AEROSTATICA</head><p>, is properly the doctrine of the weight, pressure, and balance of the air and atmosphere.</p></div1><div1 part="N" n="AEROSTATION" org="uniform" sample="complete" type="entry"><head>AEROSTATION</head><p>, in its proper and primary sense, denotes the science of weights suspended in the air; but in the modern application of the term, it siguifies the art of navigating or floating in the air, both as to the practice and principles of it. Hence also the machines which are employed for this purpose, are called <pb/><pb/><pb n="33"/><cb/> <hi rend="italics">aerostats,</hi> or <hi rend="italics">aerostatic</hi> machines; and which, on account of their round and bell-like shape, are otherwise called <hi rend="italics">air ballcons.</hi> Also <hi rend="italics">aeronaut</hi> is the name given to the person who navigates or sloats in the air by means of such machines.</p><p><hi rend="italics">Principles of</hi> <hi rend="smallcaps">Aerostation.</hi> The fundamental principles of this art bave been long and generally known, as well as speculations on the theory of it; but the successful application of them to practice seems to be altogether a modern discovery. These principles chiefly respect the weight or pressure, and elasticity of the air, with its specific gravity, and that of the other bodies to be raised or floated in it: the particular detail of which principles may be seen under the respective words in this dictionary. Suffice it therefore in this place to observe, that any body which is specifically, or bulk for bulk, lighter than the atmosphere, or air encompassing the earth, will be buoyed up by it, and ascend, like as wood, or a cork, or a blown bladder, ascends in water. And thus the body would continue to ascend to the top of the atmosphere, if the air were every where of the same density as at the surface of the earth. But as the air is compressible and elastic, its density decreases continually in ascending, on account of the diminished pressure of the superincumbent air, at the higher elevations above the earth; and therefore the body will ascend only to such height where the air is of the same specific gravity with itself; where the body will float, and move along with the wind or current of air, which it may meet with at that height. This body then is an aerostatic machine, of whatever form or nature it may be. And an air-balloon is a body of this kind, the whole mass of which, including its covering and contents, and the weights annexed to it, is of less weight than the same bulk of air in which it rises.</p><p>We know of no solid bodies however that are light enough thus to ascend and float in the atmosphere; and therefore recourse must be had to some fluid or aeriform substance.</p><p>Among these, that which is called inslammable air is the most proper of any that have hitherto been discovered. It is very elastic, and from six to ten or eleven times lighter than common atmospheric air at the surface of the earth, according to the different methods of preparing it. If therefore a sufficient quantity of this kind of air be inclosed in any thin bag or covering, the weight of the two together will be less than the weight of the same bulk of common air; and, consequently this compound mass will rise in the atmosphere, and continue to ascend till it attain a height at which the atmosphere is of the same specific gravity as itself; where it will remain or float with the current of air, as long as the inflammable air does not escape through the pores of its covering. And this is an inflammable air-balloon.</p><p>Another way is to make use of common air, rendered lighter by warming it, instead of the inflammable air. Heat, it is well known, rarefies and expands common air, and consequently lessens its specific gravity; and the diminution of its weight is proportional to the heat applied. If therefore the air, inclosed in any kind of a bag or covering, be heated, and consequently dilated, to such a degree, that the excess of the weight of an equal bulk of common air, above the <cb/> weight of the heated air, be greater than the weight of the covering and its appendages, the whole compound mass will ascend in the atmosphere, till, by the diminished density of the surrounding air, the whole become of the same specific gravity with the air in which it sloats; where it will remain, till, by the cooling and condensation of the included air, it shall gradually contract and descend again, unless the heat is renewed or kept up. And such is a heated air-balloon, otherwise called a Montgolfier, from its inventor.</p><p>Now it has been discovered, by various experiments, that one degree of heat, according to the seale of Fahrenheit's thermometer, expands the air about one five-hundredth part; and therefore that it will require about 500 degrees, or nearer 484 degrees of hent, to expand the air to just double its bulk. Which is a degree of heat far above what it is practicable to give it on such occasions. And therefore, in this respect, common air heated, is much inferior to inflammable air, in point of levity and usefulness for aerostatic machines.</p><p>Upon such principles then depends the construction of the two sorts of air-balloons. But before treating of this branch more particularly, it will be proper to give a short historical account of this late-discovered art.</p><p><hi rend="italics">History of</hi> <hi rend="smallcaps">Aerostation.</hi> Various schemes for rising in the air, and passing through it, have been devised and attempted, both by the ancients and moderns, and that upon different principles, and with various success. Of these, some attempts have been upon mechanical principles, or by virtue of the powers of mechanism: and such are conceived to be the instances related of the flying pigeon made by Archytas, the flying eagle and fly by Regiomontanus, and various others. Again, other projects have been formed for attaching wings to some part of the body, which were to be moved either by the hands or feet, by the help of mechanical powers; so that striking the air with them, aster the manner of the wings of a bird, the person might raise himself in the air, and transport himself through it, in imitation of that animal. But of these and various other devices of the like nature, a particular account will be given under the article <hi rend="italics">artisicial flying,</hi> as belonging rather to that species or principle of motion, than to our present subject of aerostation, which is properly the sailing or floating in the air by means of a machine rendered specifically lighter than that element, in imitation of aqueous navigation, or the sailing upon the water in a ship, or vessel, which is specisically lighter than the water.</p><p>The first rational account that we have upon record, for this sort of sailing, is perhaps that of our countryman Roger Bacon, who died in the year 1292. He not only affirms that the art is feasible, but assures us that he himself knew how to make an engine, in which a man sitting might be able to carry himself through the air like a bird; and he farther affirms that there was another person who had tried it with success. And the secret it seems consisted in a couple of large thin shells, or hollow globes, of copper, exhausted of air; so that the whole being thus rendered lighter than air, they would support a chair, on which a person might sit.</p><p>Bishop Wilkins too, who died in 1672, in several <pb n="34"/><cb/> of his works, makes mention of similar ideas being entertained by divers persons. “It is a pretty notion to this purpose, says he (in his Discovery of a New World, prop. 14), mentioned by Albertus de Saxonia, and out of him by Francis Mendoza, that the air is in some part of it navigable. And that upon this statick principle, any brass or iron vessel (suppose a kettle), whose substance is much heavier than that of the water; yet being filled with the lighter air, it will swim upon it, and not sink.” And again, in his Dedalus, chap. 6, “Scaliger conceives the framing of such volant automata to be very easy. <hi rend="italics">Volantis columbæ machinulam, cujus autorem Archytam tradunt,</hi> vel facillime <hi rend="italics">profiteri audeo.</hi> Those ancient motions were thought to be contrived by the force of some included air: So <hi rend="italics">Gellius, Ita erat scilicet libramentis suspensum, & aura spiritus inclusa, atque occulta consitum, &c.</hi> As if there had been some lamp, or other fire within it, which might produce such a forcible rarefaction, as should give a motion to the whole frame.” From which it would seem that Bishop Wilkins had some confused notion of such a thing as a heated air-balloon.</p><p>Again F. Francisco Lana, in his Prodroma, printed in 1670, proposes the same method with that of Roger Bacon, as his own thought.</p><p>He considered that a hollow vessel, exhausted of air, would weigh less than when filled with that fluid; he also reasoned that, as the capacity of spherical vessels increases much faster than their surface, if there were two spherical vessels, of which the diameter of one is double the diameter of the other; then the capacity of the former will be equal to 8 times the capacity of the latter, but the surface of that only equal to 4 times the surface of this: and the one sphere have its diameter equal to triple the diameter of the other; then the capacity of the greater will be equal to 27 times the capacity of the less, while its surface is only 9 times greater: and so on, the capacities increasing as the cubes of the diameters, while the surfaces increase only as the squares of the same diameters. And from this mathematical principle, father Lana deduces, that it is possible to make a spherical vessel of any given matter, and thickness, and of such a size as, when emptied of air, it will be lighter than an equal bulk of that air, and consequently that it will ascend in that element, together with some additional weight attached to it. After stating these principles, father Lana computes that a round vessel of plate brass, 14 feet in diameter, weighing 3 ounces the square foot, will only weight 1848 ounces; whereas a quantity of air of the same bulk will weigh 2155 2/3 ounces, allowing only one ounce to the cubic foot; so that the globe will not only be sustained in the air, but will also carry up a weight of 307 2/3 ounces: and by increasing the bulk of the globe, without increasing the thickness of the metal, he adds, a vessel might be made to carry a much greater weight.</p><p>Such then were the ingenious speculations of learned men, and the gradual approaches towards this art. But one thing more was yet wanting: although acquainted in some degree with the weight of any quantity of air, considered as a detached substance, it seems they were not aware of its great elasticity, and the universal pressure of the atmosphere; by which pressure, a globe of the dimensions above-described, and exhausted of its <cb/> air, would immediately be crushed inwards, for want of the equivalent internal counter pressure, to be sought for in some element, much lighter than common air, and yet nearly of equal pressure or elasticity with it; a property or circumstance attending common air when considerably heated. It is evident then that the schemes of ingenious men hitherto must have terminated in mere speculation; otherwise they could never have recorded schemes, which, on the first attempt to put in practice, must have manifested their own insufficiency, by an immediate failure of success: For instead of exhausting the vessel of air, it must either be filled with common air heated, or with some other equally elastic and lighter air. So that upon the whole it appears, that the art of traversing the air, is an invention of our own time; and the whole history of it is comprehended within a very short period.</p><p>The rarefaction and expansion of air by heat, is a property of it that has long been known, not only to philosophers, but even to the vulgar: by this means it is, that the smoke is continually carried up our chimneys; and the effect of heat upon air, is made very sensible by bringing a bladder, only partly full of air, near a fire; when the air presently expands with the heat, and distends the bladder so as almost to burst it: and so well are the common people acquainted with this effect, that it is the common practice of those who kick blown bladders about for foot-balls, to bring them from time to time to the fire, to restore the spring of the air, and distension of the ball, lost by the continual waste of that fluid through the sides of it.</p><p>But the great levity of inflammable air, is a very modern discovery. As to the inflammable property of this air itself, it had been long known to miners, and especially in coal mines, by the dreadful effects it sometimes produces by its explosions. Among them it is sometimes vulgarly called sulphur, but more properly the fire damp, or inflammable damp, to distinguish it from the choak damp, and other damps, a species of air sometimes found in deep wells and mines, and which does not explode nor take fire, but presently extinguishes candles, and suffocates the persons who may happen to go into it. But it seems that it was Mr. Cavendish who first discovered with exactness the specific gravity of inflammable air; and his experiments and observations upon it, are published in the 56th volume of the Philosophical Transactions for the year 1766. Soon after this discovery of Mr. Cavendish, it occurred to the ingenious Dr. Black of Edinburgh, that if a bladder, or other vessel, sufficiently light and thin, were filled with this air, it would form altogether a mass lighter than the same bulk of atmospheric air, and consequently that it would ascend in it.</p><p>This idea he mentioned in his chemical lectures in the year 1767 or 1768; and he farther proposed to exhibit the experiment, by filling the allantois of a calf with such air. The allantois however was not prepared just at the time when he was at that part of his lectures, and other avocations afterwards prevented his design: so that, considering it only as an amusing experiment, and being fully satisfied of the truth of so evident an effect, he contented himself with barely mentioning the experiment from time to time in his lectures. About the year 1777 or 1778 too it occurred to Mr. Cavallo, <pb n="35"/><cb/> that it might be possible to construct a vessel, which, when filled with inflammable air, would ascend in the atmosphere: and there is no doubt but that similar ideas would occur to many other persons, of so evident a consequence of Mr. Cavendish's discovery.</p><p>But it seems to have been Mr. Cavallo who first actually attempted the experiment, in which however he succeeded no farther than in being able to raise soap bubbles of two or three inches diameter: a thing which had been done by children for their amusement time immemorial. These experiments Mr. Cavallo made in the beginning of the year 1782, and an account of them was read at a public meeting of the Royal Society on the 20th day of June of that year. From which it appears that he tried bladders and paper of various sorts. But the bladders, however thin they were made by scraping, &c, were still found too heavy to ascend in the atmosphere, when fully inflated with the inflammable air: and in using China paper, he found that this air passed through its pores, like water through a sieve. And having failed of success by blowing the same air into a thick solution of gum, thick varnishes, and oil paint, he was obliged to rest satisfied with soap-balls or bubbles, which, being filled with inflammable air, by dipping the end of a small glass tube, connected with a bladder containing the air, into a thick solution of soap, and gently compressing the bladder, ascended rapidly in the atmosphere, and broke against the ceiling of the room.</p><p>Here however it seems the matter might have rested, had it not been for experiments made in France soon after, by the two brothers Stephen and Joseph Montgolfier, upon principles suggested, not by the levity of inflammable air, which probably they had never heard of, but by that of smoke and clouds ascending in the atmosphere. These two brothers it seems were natives of Annonay, a town in the Vivarais, about 36 miles distant from Lyons; and that in their youth, Stephen, the elder, had assiduously studied the mathematics, but the other had applied himself more particularly to natural philosophy and chemistry. They were not intended for any particular way of business, but the death of a brother obliged them to put themselves at the head of a considerable paper manufactory at Annonay. In the intervals of time allowed by their business they amused themselves in several philosophical pursuits, and particularly with the experiments in aerostation, of which we are now to give some account. It would be perhaps impossible to know all the particular steps and ideas which finally produced this discovery: but it has been said that the real principle, upon which the effect of the aerostatic machine depends, was unknown even for a considerable time after its discovery: that M. Montgolfier attributed the effect of the machine, not to the rarefaction of the air, which is the true cause, but to a certain gas, specifically lighter than common air, which was supposed to be developed srom burning substances, and which was commonly called Montgolfier's gas. Be this however as it may, it is well known that the two brothers began to think of the experiment of the aerostatic machine about the middle or the larter part of the year 1782. The natural ascension of the smoke and the clouds in the atmosphere, suggested the first idea; and to imitate those bodies, <cb/> or to inclose a cloud in a bag, and let the latter be lifted up by the buoyancy of the former, was the first project of those celebrated gentlemen.</p><p>Accordingly the first experiment was made at Avignon by Stephen, the elder brother, about the middle of November 1782. Having prepared a bag of sine silk, in the shape of a parallelopipedon, and of about 40 cubic feet in capacity, he applied burning paper to an aperture in the bottom, which rarefied the air, and thus formed a kind of cloud in the bag; and when it became sufficiently expanded, it ascended rapidly to the ceiling.</p><p>Soon afterwards the experiment was repeated with the same machine at Annonay, by the two brothers, in the open air; when the bag ascended to the height of about 70 feet. Encouraged by this success, they constructed another machine, of about 650 cubic feet capacity; which, when inflated as before, broke the cords which confined it, and after ascending rapidly to the height of about 600 feet, descended and fell on the adjoining ground. With another larger machine, of 37 feet diameter, they repeated the experiment on the 25th day of April, which answered exceedingly well: the machine had such force of ascension, that, breaking abruptly from its confinement of ropes, it rose to the height of more than 1000 feet, and then, being carried by the wind, descended and fell at a place about three quarters of a mile from the place of its ascension. The capacity of this machine was equal to above 23 thousand cubic feet, and, being nearly globular, when inflated, it measured 117 English feet in circumference. The covering was formed of linen, lined with paper; and its aperture at the bottom was fixed to a wooden frame, of about 4 feet square, or 16 feet in surface. When filled with vapour, which it was conjectured might be about half as heavy as common air, it was capable of lifting up about 490 pounds, besides its own weight, which, together with that of the wooden frame, was equal to about 500 pounds. With this same machine the next experiment was publicly performed at Annonay, on the 5th of June 1783, before a great multitude of spectators. The flaccid bag was suspended on a pole 35 feet high; straw and chopped wool were burned under the opening at the bottom; the vapour, or rather smoke, soon inflated the bag, so as to distend it in all its parts; and this enormous mas<*> ascended in the air with such velocity, that in less than ten minutes it reached the height of above 6 thousand feet; when a breeze carried it in an horizontal direction to the distance of 7668 feet, or near a mile and a half, where it descended gently to the ground.</p><p>As soon as the news of this experiment reached Paris, the philosophers of that city, conceiving that a new species of gas, of about half the weight of common air, had been discovered by Messrs. Montgolfier; and knowing that the weight of inflammable air was but about the eighth or tenth part of the weight of common air, they justly concluded that inflammable air would answer the purpose of this experiment better than the gas of Montgolfier, and accordingly they resolved to make trial of it.</p><p>A subscription was opened by M. Faujas de St. Fond, towards defraying the expence of the experiment. A sufficient sum of money having soon been raised, <pb n="36"/><cb/> Messrs. Roberts were appointed to construct the machine, and M. Charles, professor of experimental philosophy, to superintend the work. After a considerable time spent, and surmounting many difficulties in obtaining a sufficient quantity of inflammable air, and searching out a substance light enough for the covering, they at length constructed a globe of the silk called lutestring, which was rendered impervious to the inclosed air by a varnish of elastic gum or caeutchouc, dissolved in some kind of spirit or essential oil. The diameter of this globe was about 13 feet; and it had only one aperture, like a bladder, to which a stop-cock was adapted: and the weight of this covering, when empty, together with that of the stop-cock, was 25 pounds.</p><p>On the 23d of August 1783, they began to fill the globe with inflammable air; but this, being their first attempt, was attended with many obstructions and disappointments, which took up two or three days to overcome.</p><p>At length however it was prepared for exhibition, and on the 27th it was carried from the Place des Victoires, where it had been prepared, to the Champs de Mars, a spacious open ground in the front of the Military School, where, after introducing some more inflammable air, and disengaging it from the cords by which it was held down, it rose, in less than two minutes, to the height of 3123 feet: the specific gravity of the balloon, when it went up, being 35 pounds less than that of common air. At that height the balloon entered a cloud, but soon appeared again; and at last it was lost among other clouds. After floating about in the air for about three quarters of an hour, it fell in a field about 15 miles from the place of its ascent; where, as we may easily imagine, it occasioned great amazement to the peasants who found it. Its fall was owing to a rent in the covering, probably occasioned by the superior elasticity of the inflammable air, over that of the rare part of the atmosphere to which it had ascended.</p><p>In consequence of this brilliant experiment, numberless small balloons were made, mostly of goldbeater's skin, from 6 and 9 to 18 or 20 inches diameter; their cheapness putting it in the power of almost every family to satisfy its curiosity relative to the new experiment; and in a few days time balloons were seen flying all about Paris, from whence they were soon after sent abroad.</p><p>Mr. Joseph Montgolfier repeted an experiment with a machine of his construction before the commissaries of the Academy of Sciences, on the 11th and 12th of September. The machine was about 74 feet high, and 43 feet in diameter; it was made of canvass, covered with paper both within and without, and weighed 1000 pounds. It was filled with rarefied air in 9 minutes, and in one trial the weight of eight men was not sufficient to keep it down. It was not suffered to go up, as it had been intended for exhibition before the Royal Family, a few days after. By the violence of the rain, however, which fell about this time, it was so much spoiled, that he thought proper to construct another for that purpose, in which he used so great dispatch, that it was completed in the short space of four days time. This machine was constructed of cloth made of linnen and cotton <cb/> thread, and painted with water colours both within and without. Its height was 60 feet, and diameter 43 feet. Having made the necessary preparation for inflating it, the operation was bègun about one o'elock on the 19th of the same month; before the king and queen, the court, and the inhabitants of the place, as well as all the Parisians who could procure a conveyance to Versailles. The balloon was soon filled, and in eleven minutes after the commencement of the operation, the ropes being cut, it ascended, bearing up with it a wicker cage, containing a cock, a duck, and a sheep, the first animals that ever ascended into the atmosphere with an aerostatic machine. Its power of ascension, or the weight by which it was lighter than an equal bulk of common air, allowing for the animals and their cage, was 696 pounds. The balloon rose to the height of 1440 feet; and being driven by the wind for the space of eight minutes, it gradually descended in consequence of two large rents made in the covering by the wind, and fell in a wood at the distance of 10,200 feet, or about two miles from Versailles. The animals landed again as safe as when they went up, and the sheep was found feeding.</p><p>The success of this experiment induced M. Pilatre de Rozier, with a philosophical intrepidity which will be recorded with applause in the history of aerostation, to offer himself as the first adventurer in this aerial navigation. For this purpose M. Montgolfier constructed a new machine, of an oval shape, in a garden of the fauxbourg St. Antoine; its diameter being about 48 feet, and height 74 feet. To the aperture in the lower part was annexed a wicker gallery about three feet broad, with a ballustrade of three feet high. From the middle of the aperture an iron grate, or brazier, was suspended by chains, descending from the sides of the machine, in which a fire was lighted for inflating the machine; and towards the aperture port-holes were opened in the gallery, through which any person, who might venture to ascend, might feed the fire on the grate with fuel, and regulate at pleasure the dilatation of the air inclosed in the machine: the weight of the whole being upwards of 1600 pounds. On the 15th of October 1783. the fire being lighted, and the balloon inflated, M. P. de Rozier placed himself in the gallery, and, to the astonishment of a multitude of spectators, ascended as high as the length of the restraining cords would permit, which was about 84 feet from the ground, and there kept the machine afloat about four minutes and a half, by repeatedly throwing straw and wool upon the fire: the machine then descended gradually and gently, through a medium of increasing density, to the ground; and the intrepid adventurer assured the admiring spectators that he had not experienced the least inconvensence in this aerial excursion. This experiment was repeted on the 17th with nearly the same success; and again several times on the 19th, when M. P. de Rozier, by a partial ascent and descent, several times repeted, evinced to the multitude of observers that the machine may be made to ascend and descend at the pleasure of the aeronaut, by merely increasing or diminishing the fire in the grate. The balloon having been hauled down, by the ropes which always confined it, M. Gironde de Villette placed himself in the gallery <pb n="37"/><cb/> opposite to M. de Rozier, and the machine being suffered to ascend, it hovered for about 9 minutes over Paris, in the sight of all its inhabitants, at the height of 330 feet. And on their descending, the marquis of Arlandes ascended with M. de Rozier much in the same manner.</p><p>In consequence of the report of these experiments, signed by the commissaries of the Academy of Sciences, it was ordered that the annual prize of 600 livres should be given to Messrs. Montgolfier for the year 1783.</p><p>In the experiments above-recited, the machine was always secured by long ropes, to prevent its entire escape: but they were soon succeeded by unconfined aerial navigation. For this purpose the same balloon of 74 feet in height was conveyed to La Muette, a royal palace in the Bois de Boulogne: and all things being got ready, on the 21st of November 1783, M. P. de Rozier and the marquis d'Arlandes took their post in opposite sides of the gallery, and at 54 minutes after one the machine was absolutely abandoned to the element, and it ascended calmly and majestically in the atmosphere. On reaching the height of about 280 feet the intrepid aeronauts waved their hats to the astonished multitude: but they soon after rose too high to be distinguished, and it is supposed they rose to more than 3000 feet in height. At first they were driven, by a north-west wind, horizontally over the river Seine and part of Paris, taking care to clear the steeples and high buildings by increasing the fire; and in rising they met with a current of air which carried them southward. Having thus passed the Boulevard, and finally desisting from supplying the fire with fuel, they descended very gently in a field beyond the new Boulevard, about 9000 yards, or a little more than 5 miles distant from the palace de La Muette, having been between 20 and 25 minutes in the air. The weight of the whole apparatus including that of the two travellers, was between 1600 and 1700 pounds.</p><p>Notwithstanding the rapid progress of aerostation in France, it is remarkable that we have no authentic account of any experiments of this kind being attempted in other countries. Even in our own island, where all arts and sciences sind an indulgent nursery, and many their birth, no aerostatic machine was seen before the month of November 1783. Various speculations have been made on the reasons of this strange neglect of so novel and brilliant an experiment. But none seemed to carry any shew of probability except that it was said to be discouraged by the leader of a philosophical society, expressly instituted for the improvement of natural knowledge, for the reason, as it was said, that it was the discovery of a neighbouring nation. Be this however as it may, it is a fact that the first aerostatic experiment was exhibited in England by a foreigner unconnected and unsupported. This was a count Zambeccari, an ingenious Italian, who happened to be in London about that time. He made a balloon of oiled-silk, 10 feet in diameter, weighing only 11 pounds: it was gilt, both for ornament, and to render it more impermeable to the inflammable air with which it was to be filled. The balloon, after being publicly shewn for several days in London, was <cb/> carried to the Artillery Ground, and there being filled about three-quarters with inflammable air, and having a direction inclosed in a tin box for any person by whom it should afterwards be found, it was launched about one o'clock on the 25th of November 1783. At half past three it was taken up near Petworth in Sussex, 48 miles distant from London; so that it travelled at the rate of near 20 miles an hour. Its descent was occasioned by a rent in the silk, which must have been the effect of the rarefaction of the inflammable air when the balloon ascended to a rarer part of the atmosphere.</p><p>The French philosophers having executed the first aerial voyage with a balloon inflated by heated air, resolved to attempt a similar voyage with a balloon filled with inflammable air, which seemed to be preserable to dilated air in every respect, the expence of preparing it only excepted. A subscription was opened however to defray that expence, which was estimated at about ten thousand livres; and the balloon was constructed by Messrs. Roberts, of gores of silk, varnished with a solution of elastic gum. Its form was spherical, and it measured 27 1/2 feet in diameter. The upper hemisphere was covered by a net, which was fastened to a hoop encircling its middle, and called its equator. To this equator was suspended by ropes a car or boat, covered with painted linen, and beautifully ornamented, which swung a few feet below the balloon. To prevent the bursting of the machine by the expansion of the inflammable air in a rarer medium, or to cause the balloon to descend, it was furnished with a valve, which might be opened by means of a string descending from it, for discharging a part of the internal air, without admitting the external to enter: And the car was ballasted with bags of sand, for the purpose of lightening it occasionally, and causing it to ascend: so that by letting some of the air escape through the valve, they might descend; and by discharging some of their sand ballast, ascend. To this balloon was likewise annexed a long pipe by which it was filled. The apparatus for filling it consisted of several casks placed round a large tub of water, each having a long tin tube, that terminated under a vessel or funnel which was inverted into the water of the tub, and communicated with the long pipe annexed to the lower part of the balloon. Iron filings and diluted vitriolic acid being put into the casks, the inflammable air which was produced from these materials, passed through the tin tubes, thence through the water of the tubs to the inverted funnel, and so through the pipe into the balloon. When inflated, the weight of the common air which was equal in bulk to the balloon, was 771 1/2 pounds; also the power of ascension, or weight just necessary to keep it from ascending, was 20 pounds, and the weight of the balloon, with its car, passengers, and all its appendages, was 604 1/2 pounds, which two together make 624 1/2 pounds: and this taken from 771 1/2 pounds, the weight of common air displaced, leaves 147 pounds for the weight of the inflammable air contained in the balloon, and which is to 771 1/2 pounds, the weight of the same bulk of common air, nearly as 1 to 5 1/4; that is, the inflammable air used in this experiment was 5 1/4 times lighter than common air.</p><p>The first of December was fixed on for the display of this grand experiment; and every preparation was <pb n="38"/><cb/> made for conducting it with advantage. The garden of the Thuilleries at Paris was the scene of operation; which was soon crowded and encompassed with a prodigious multitude of observers. Signals were given, from time to time, by the siring of cannon, waving of flags, &c: and a small montgolsier was launched, for shewing the direction of the wind, and for the amusement of the people previous to the general display. At three quarters after one o'clock, M. Charles and one of the Roberts, having seated themselves in the boat attached to the balloon, and being furnished with proper instruments, cloathing, and provisions, left the ground, and ascended with a moderately accelerated velocity to the height of about 600 yards; the surrounding multitude standing silent with fear and amazement; while the aerial navigators at this height made signals of their safety. When they left the ground, the thermometer, according to Fahrenheit's scale, stood at 59 degrees; and the barometer, at 30.18 inches: and at the utmost height to which they ascended, the barometer fell to 27 inches; from which they deduced their height as above to be 600 yards, or one third part of a mile. During the rest of the vovage the quicksilver in the barometer was generally between 27 and 27.65 inches, rising and falling, as part of the ballast was thrown out, or some of the inflammable air escaped from the balloon. The thermometer generally stood between 53 and 57 degrees. Soon after their ascent, they remained stationary for some time: they then moved horizontally in the direction north-north-west: and having crossed the Seine, and passed over several towns and villages, to the great amazement of the inhabitants, they descended in a field, about 27 miles distant from Paris, at three-quarters past 3 o'clock; so that they had travelled at the rate of near 15 miles an hour, without feeling the least inconvenience.</p><p>The balloon still containing a considerable quantity of inflammable air, M. Charles re-ascended alone, and it was computed he went to the height of 3100 yards, or almost 2 miles, the barometer being then at 20 English inches: having amused himself in the air about 33 minutes, he pulled the string of the valve, and descended at 3 miles distance from the place of his ascent. All the inconvenience he experienced in his great elevation, was a dry sharp cold, with a pain in one of his ears and a part of his face, which he ascribed to the dilatation of the internal air: a circumstance that usually happens to persons who suddenly change the density of their atmosphere, either by ascending into a rarer, or descending into a denser one. The small balloon, launched at the beginning by M. Montgolfier, was found to have moved in a direction opposite to that of the aeronauts; from which it is inferred that there were two currents of air at different heights above the earth.</p><p>In the month of December this year, several experiments were made at Philadelphia in America with air balloons, by Messrs. Rittenhouse and Hopkins. They constructed and filled a great many small balloons, and connected them together; in which a man went up several times, and was drawn down again; and finally, the ropes being cut, he ascended to the height of 100 feet, and floated to a considerable distance; but, <cb/> being afraid, he cut open the balloons with a knife, and so descended.</p><p>About the close of this year small balloons were sent up in many places, and were become very common in some parts of France and England. And in the beginning of the year following, their number and magnitude increased considerably; and some of the more remarkable ones were as follow:—On the 19th of January M. Joseph Montgolfier, accompanied by six other persons, ascended from Lyons with a rarefied air balloon, to the height of 1000 yards. This was the largest machine that had been hitherto made, being 131 feet high, and 104 feet in diameter: it was formed of a double covering of linen, with three layers of paper between them; and it weighed, when it went up, 1600 pounds, including the gallery, passengers, &c. It was at first intended for six passengers; but before it went up, it was not judged safe to freight it with more than three: however no authority nor solicitations could prevail upon any of the six to quit their place, nor even to cast lots which three should resign their pretensions: so that the spectators saw them all ascend with terror and anxiety; and to add to their distress, when the ropes were cut, and the machine had ascended a foot or two from the ground, a seventh person suddenly leaped into the gallery, and the fire being increased, the whole ascended together. To add to the terror of the scene, after being in the air about 15 minutes, a large rent of about 50 feet in length was made by the balloon taking fire, in consequence of which it descended very rapidly to the ground, though fortunately without injury to any of the aeronauts.</p><p>On the 22d of February an inflammable air balloon was launched from Sandwich in Kent. It was but a small one, being only 5 feet in diameter; but it was rendered remarkable by being the first machine that crossed the sea from England to France. It was found in a field at Warneton, about 9 miles from Lisle in French Flanders, two hours and a half after it left Sandwich, the distance being about 74 miles; so that it floated at the rate of about 30 miles an hour.</p><p>The chevalier Paul Andreani, of Milan, was the first aerial traveller in Italy. The chevalier was at the sole expence of this machine, but was assisted in the construction by two brothers of the name of Gerli. They all three ascended together near Milan on the 25th of February, and remained in the atmosphere about 20 minutes, when they descended, all their fuel being exhausted. This machine was a montgolfier, of a spherical shape, and about 68 feet in diameter. From calculations made on the power of this, and other machines of the same sort, it appears that the included air is raresied commonly but about one-third, or that the included warm air weighs about two-thirds of the same bulk of the external or common air.</p><p>The next aerial voyage was performed on the 2d of March 1784, by M. Jean Pierre Blanchard, a man who has since that time made more voyages than any other person, and who has rendered himself famous by being the first who has floated in the air over the channel from England to France. M. Blanchard it scems had sor many years been in pursuit of mechanical means for flying through the air; but on hearing of the late invented air balloons, he dropped <pb n="39"/><cb/> his former pursuits, and turned his attention to them. He accordingly constructed one of 27 feet diameter, to which a boat was suspended with two wings, and a rudder to steer it by, as also a large parachute spread horizontally between the boat and the balloon, designed to check the fall in case the balloon should burst. The machine being filled with inflammable air, he ascended, from the Champs de Mars at Paris, to the height of near ten thousand feet, or almost 2 miles; and after floating in the air for an hour and a quarter, he descended at Billancourt near Seve, having experienced by turns heat, cold, hunger, and an excessive drowsiness. It appears from his own account, and as might have been expected, that the wings and rudder of his boat had little or no power in turning the balloon from the direction of the wind.</p><p>In the course of this year, 1784, aerostatic experiments and aerial voyages became so frequent, that the limits of this article will not allow of any thing farther than mentioning those which were attended with any remarkable circumstances. On the 25th of April, Messrs. de Morveau and Bertrand ascended from Dijon, with an inflammable-air balloon, to the height of thirteen thousand feet, or near 2 miles and a half, where the thermometer marked 25 degrees. They were in the air 1 hour and 25 minutes, in which time they floated 18 miles.</p><p>On the 20th of May, four ladies and a gentleman ascended from Paris, in a large montgolfier, above the highest buildings, and remained suspended there a considerable time, the balloon being confined by ropes from flying away.</p><p>On the 23d of May, M. Blanchard, with the same balloon as before, ascended from Rouen, to such height that the mercury in the barometer stood at 20.57 inches, which on the earth had been at 30.16. It was observed that in this voyage M. Blanchard's wings or oars could not turn him aside from the direction of the wind.</p><p>On the 4th of June M. Fleurant and Madame Thible, the first lady who made an aerial voyage, ascended at Lyons in a machine of 70 feet diameter. They went to the height of 8500 feet, and floated about 2 miles in 45 minutes.</p><p>On the 14th of June, M. Coustard de Massi and M. Mouchet ascended at Nantes to a great height, with a balloon of 32 1/2 feet diameter, filled with inflammable air extracted from zink; and they floated to the distance of 27 miles in 58 minutes.</p><p>On the 23d of June, the first aerial traveller M. Pilatre de Rozier, accompanied with M. Prouts, ascended at Versailles, in the presence of the royal family and the king of Sweden, with a large montgolfier, whose diameter was 79 feet, and its height 91 seet and a half. They floated to the distance of 36 miles in three-quarters of an hour, when they descended, which is at the rate of 48 miles an hour. In consequence of this experiment the king granted to M. de Rozier a pension of 2000 livres.</p><p>On the 15th of July the duke of Chartres, the two brothers Roberts, and a fourth person ascended from the park of St. Cloud, with an inflammable-air machine, of an oblong form, its diameter being 34 feet, and its length, which went in a direction parallel to the horizon, was 55 1/2 feet; and they remained in the atmosphere <cb/> for 45 minutes in the greatest fear and danger. The machine contained an interior small balloon, filled with common air, by means of which it was proposed to cause the machine to ascend or descend without the loss of any inflammable air or ballast: and the boat was furnished with a helm and oars, which were intended to guide it. Three minutes after ascending, the machine was lost in the clouds, and involved in a dense vapour. A violent agitation of the air, resembling a whirlwind, greatly alarmed the aeronauts, turned the machine three times round in a moment, and gave it such shocks as prevented them from using any of their instruments for managing the machine. After many struggles, with great difficulty they tore about 7 or 8 feet of the lower part of the covering, by which the inflamable air escaped, and they descended to the ground with great rapidity, though without any hurt. At the place of departure the barometer slood at 30.12 inches, and at their greatest elevation it stood at 24.36 inches; so that their ascent was about 5100 feet, or near one mile.</p><p>On the 18th of July, M. Blanchard, with a Mr. Boley, made his third voyage with the same balloon as he had before, and rose so high as to sink the barometer from 30.1 to 25.34 inches, answering to a height of about 4600 feet. In 2 hours and a quarter they floated 45 miles, which is at the rate of 20 miles an hour. In this voyage M. Blanchard pretended that he was able to turn the machine with his wings, and to make it ascend and descend at pleasure. After descending, it is said the balloon remained all the night at anchor full of air; and that the next day several ladies amused themselves by ascending successively to the height of 80 feet, the length of the ropes by which it was anchored.</p><p>In the course of this summer two persons had nearly lost their lives by ascending with machines of warmed air. The one in Spain, on the 5th of June, by the machine taking fire, was much burnt, and so hurt by the fall that his life was long despaired of. The latter having ascended a few feet, the machinery entangled under the eves of a house, which broke the ropes, and the man fell about twenty feet: the machine presently took fire, and was consumed. Other montgolfiers were also burned about London, by taking fire, through the defects of their construction.</p><p>The first aerial voyage performed in England was by one Vincent Lunardi, a native of Italy, who ascended from the Artillery Ground, London, with an inflammable-air balloon on the 15th of September. His machine was made of oiled silk, painted in alternate stripes of blue and red; and its diameter was 33 feet. From a net, which covered about two-thirds of the balloon, 45 cords descended to a hoop hanging below the balloon, to which the gallery was attached. The machine had no valve; and its neck, which terminated in the form of a pear, was the aperture through which the inflammable air was introduced, and through which it might be let out. The balloon was filled with air produced from zink by means of diluted vitriolic acid. And when the aeronaut departed, at 2 o'clock, he took up with him a dog, a cat, and a pigeon. After throwing out some sand to clear the houses, he ascended to a considerable height; and the direction of his motion at first was north-west by west; <pb n="40"/><cb/> but as the balloon rose higher, it came into another current of air, which carried it nearly north. In the course of his voyage the thermometer was as low as 29 degrees, and the drops of water which had collected round the balloon were frozen. About half after three he descended very near the ground, and landed the cat, which was almost dead with cold: then rising, he prosecuted his voyage, till at 10 minutes past 4 o'clock he landed near Ware in Hertfordshire. He pretends that he descended by means of his oars or wings; but other circumstances related by him, strongly contradict the fact.</p><p>The longest and most interesting voyage performed about this time, was that of Messrs. Roberts and M. Colin Hullin, who ascended at Paris, at noon on the 19th of September, with an aerostat, filled with inflammable air, which was 27 <*> feet in diameter, and 46 3/4 feet long, the machine being made to float with its longest part parallel to the horizon, and having a boat of near 17 feet long attached to it. The boat was fitted up with several wings or oars, shaped like an umbrella, and they ascended at 12 o'clock with 450 pounds of sand ballast, and after various manœuvres finally descended, at 40 minutes past 6 o'clock, near Arras in Artois, 150 miles from Paris, having still 200 pounds of ballast remaining in the boat. In one part they found the current of air uniform from 600 to 4200 feet high, which it seems was their greatest height, and the sall of the barometer had been near 5.6 degrees. They found that by means of their oars they could accelerate their course a little in the direction of the wind, when it moved slowly, which may be true; but there is great reason to doubt of the accuracy of their experiments by which they pretended to cause their path to deviate about 22 degrees from the wind, going with a considerable velocity.</p><p>The second aerial voyage in England, was performed by Mr. Blanchard, and Mr. Sheldon professor of anatomy to the Royal Academy, being the first Englishman who ascended with an aerostatic machine. They ascended at Chelsea the 16th of October, at 9 minutes past 12 o'clock. Mr. Blanchard having landed Mr. Sheldon at about 14 miles from Chelsea, re-ascended alone, and finally landed near Rumsey in Hampshire, about 75 miles distant from London, having gone nearly at the rate of 20 miles an hour. The wings used on this occasion it seems produced no deviation from the direction of the wind. Mr. Blanchard said that he ascended so high as to seel a great difficulty of breathing: and that a pigeon, which flew away from the boat, labou<*>ed for some time to sustain itself with its wings in the rarefied air, but after wandering a good while, returned, and rested on the side of the boat.</p><p>On the 4th of October, Mr. Sadler, an ingenious tradesman at Oxsord, ascended at that place with an inflammable-air balloon of his own construction and filling. And again on the 12th of the same month he ascended at Oxford, and floated to the distance of 14 miles in 17 minutes, which is at the rate of near 50 miles an hour.</p><p>The 30th of November this year Mr. Blanchard's fifth aerial voyage, still with his old machine, was performed in company with Dr. J. Jeffries, a native of America. Their voyage was about 21 miles; and it <cb/> does not appear that the greatest action of their oars produced any effect in directing the course of the balloon.</p><p>On the 4th of January, 1785, a Mr. Harper ascended at Bitmingham with an inflammable-air balloon, and went to the distance of 50 miles in an hour and a quarter, and sussered no other inconvenience than a temporary deafness, and what might be expected from the changes of wet and cold. The thermometer descended from 40 to 28 degrees.</p><p>On the 7th of January, Mr. Blanchard, accompanied with Dr. Jeffries, performed his sixth aerial voyage, by actually crossing the British channel from Dover to Calais, with the same balloon which had sive times before carried him successfully through the air. They ascended with only 30 pounds of sand ballast, besides their provisions, some books, instruments, and other necessaries. The machine parted with the gas very rapidly, and their ballast was soon all exhausted; after which, from time to time they threw out every thing else in the boat, to prevent themselves from dropping into the sea. In this way they disposed of all their provisions, their books and instruments, and finally the most part of their very clothes themselves. This however bringing them near the French coast, they gradually ascended, cleared the cliffs and houses, and landed in the forest of Guiennes. It is remarkable that a bottle, being thrown out when they were in danger of falling into the sea, struck the water with sach force, that they heard and felt the shock very sensibly on the car and balloon. In consequence of this voyage the king of France presented M. Blanchard with a gift of 12000 livres, and granted him a pension of 1200 lives a year.</p><p>On the 19th of January, Mr. Crosbie ascended at Dublin in Ireland, with an inflammable-air balloon to a great height. He rose so rapidly that he was out of sight in 3 minutes and a half. By suddenly opening the valve he descended just at the edge of the sea, as he was driving towards the channel, being unprovided for properly passing over to England.</p><p>On the 23d of March, Count Zambeccari and Admiral Sir Edward Vernon ascended at London, and sailed to Horsham in Sussex, at the distance of 35 miles in less than an hour. The voyage proved very dangerous, owing to some of the machinery about the valve being damaged, which obliged them to cut open some part of the balloon when they were about two miles perpendicular height above the earth, the barometer having fallen from 30.4 to 20.8 inches. In descending they passed through a dense cloud, which felt very cold, and covered them with snow. The observatione they made were, that the balloon kept perpetually turning round its vertical axis, sometimes so rapidly as to make each revolution in 4 or 5 seconds; that a peculiar noise, like rustling, was heard among the clouds, and that the balloon was greatly agitated in the descent.</p><p>On May the 5th, Mr. Sadler, and William Windham, esq. member of parliament for Norwich, ascended at Moulsey-hurst. The machine took a south-east course, and the current of air was so strong that they were in great danger of being driven to sea. They had the good fortune however to descend near the con- <pb n="41"/><cb/> flux of the Thames and Medway, not a mile from the water's edge. By an accident they lost their balloon: for while the aeronauts were busied in securing their instruments, the country people, whom they had employed in holding down the machine, suddenly let go the cords, when the balloon instantly ascended, and was driven many miles out to sea, where it fell, and was taken up by a trading vessel. It was afterwards restored again, and another voyage made with it from Manchester to Pontefract, in which Mr. Sadler was still more unfortunate; for no person being near when it descended, and not being able to confine it by his own strength, he was dragged by it over trees and hedges; and at last was forced to quit it at the utmost peril of his life; after which it rose, and was out of sight in a few minutes. It was afterwards found near Gainsborough.</p><p>On the 12th of May Mr. Crosbie ascended, at Dublin, as high as the tops of the houses; but soon descended again with a velocity that alarmed all the spectators for his safety. On his stepping out of the car, in an instant Mr. M'Guire, a college youth, sprung into it, and the balloon ascended with him to the astonishment of the beholders, and presently he was carried with great velocity towards the channel in the direction of Holyhead. This being observed, a crowd of horsemen pursued full speed the course he seemed to take, and could plainly perceive the balloon descending into the sea. Lord H. Fitzgerald, who was among the foremost, instantly dispatched a swift-sailing vessel mounted with oars, and all the boats that could be got, to the relief of the gallant youth; whom they found almost spent with swimming, just time enough to save his life.</p><p>The fate of M. Pilatre de Rozier, the first aerial traveller, and his companion M. Romain, has been much lamented. They ascended at Boulogne the 15th of June, with intent to cross the channel to England: for the first 20 minutes they seemed to take the proper direction; when presently the whole apparatus was seen in flames, and the unfortunate adventurers fell to the ground from the height of more than a thousand yards, and were killed on the spot, their bones being broken, and their bodies crushed in a shocking manner. The machine in which they ascended, consisted of a spherical balloon, 37 feet in diameter, filled with inflammable air; and under this balloon was suspended a small montgolfier, or fire balloon, of 10 feet diameter; the gallery which suspended the aeronauts, was attached to the net of the upper balloon by cords, which were fastened to a hoop rather larger than the montgolsier, and descended perpendicularly to the gallery. The montgolfier was intended to promote and prolong the ascension, by rarefying the atmospheric air, and by that means gaining levity. It is not certainly known whether the balloon was actually set on fire by the montgolfier, or, being over-rarefied by the heat beneath, burst, and by that means the inflammable air was set in a blaze.</p><p>On the 19th of July, at 20 minutes past 2 o'clock, Mr. Crosbie ascended at Dublin, with intent to cross the channel to Holyhead in England. The usual form of the boat had been changed, for a capacious wicker basket, of a circular form, round the upper <cb/> edges of which were fastened a great many bladders, which were intended to render his gallery buoyant, in case of a disaster at sea. About 300 pounds of ballast were put into the basket, but the aeronaut discharged half a hundred on his first rise. At first the current of air carried him due west; but it soon changed his course to nearly north-east, pointing nearly towards Whitehaven. At upwards of 40 miles from the Irish shore, he found himself within clear sight of both lands, and he said it was impossible to give any adequate idea of the unspeakable beauties which the scenery of the sea, bounded by both lands, presented. He rose at one time so high, that by the intense cold his ink was frozen, and the mercury sunk quite into the ball of the thermometer. He was sick, and felt a strong prepulsion on the tympanum of the ears. At his utmost height he thought himself stationary; but on liberating some gas, he descended to a current of air blowing north, and extremely rough. He now entered a thick cloud, and encountered strong blasts of wind, with thunder and lightning, which brought him rapidly towards the surface of the water. Here the balloon made a circuit, but falling lower, the water entered his car, and he lost his notes of observation. All his endeavours to throw out ballast were of no avail; the force of the wind plunged him into the ocean; and with much difficulty he put on his cork jacket. The propriety of his idea was now very manifest in the construction of his boat; as by the admission of the water into the lower part of it, and the suspension of his bladders, which were arranged at the top, the water, added to his own weight, became proper ballast; and the balloon maintaining its poise, it became a powerful sail, by means of which, and a snatch block to his car, he went before the wind as regularly as a sailing vessel. In this situation he found himself inclined to eat, and he took a little fowl. At the distance of a league he discovered some vessels crowding after him; but as his progress outstripped all their endeavours, he lengthened the space of the balloon from the car, which gave a check to the rapidity of his sailing, and he was at length overtaken and saved by the Dunleary barge, which took him on board, and steered to Dunleary, towing the balloon after them.</p><p>A similar accident happened to Major Money, who ascended at Norwich, on the 22d of July, at 20 minutes past 4 in the afternoon; when meeting with an improper current, and not being able to let himself down, on account of the smallness of the valve, he was driven out to sea, where, after blowing about for near two hours, he dropped into the water. Here the struggles were astonishing which he made to keep the balloon up, which was torn, and hung only like an umbrella over his head. A ship was once within a mile, but, he adds, whether from want of humanity, or by mistaking the balloon for a sea monster, they sheered off, and left him to his fate: but a boat chased him for two hours, till just dark, and then bore away. He now gave up all hopes, and began to wish that providence had given him the fate of Pilatre de Rozier, rather than such a lingering death. Exerting himself however to preserve life as long as possible, by keeping the balloon floating over his head, to keep himself out of the water, into which nevertheless he sunk gradually inch <pb n="42"/><cb/> by inch, as it lost its power, till he was at length breast deep in water, when he was providentially taken up by a revenue cutter, at half past eleven at night, but so weak that he was obliged to be lifted out of the car into the ship.</p><p>About the latter end of August, the longest aerial voyage hitherto made, was performed by Mr. Blanchard, who ascended at Lisle, accompanied by the Chevalier de L'Epinard, and travelled 300 miles in their balloon before it descended. On this occasion, as on some former ones, Mr. Blanchard made trial of a parachute, like a large umbrella, invented to break the fall in case of an accident happening the balloon: with this machine he dropped a dog from the car soon after his ascension, which descended gently and unhurt.</p><p>On September the 8th, Thomas Baldwin, Esq. ascended from the city of Chester, at 40 minutes past one o'clock, and descended at Rixton-Moss, at 25 miles distance, after a voyage of 2 hours and a quarter. The greatest perpendicular altitude ascended was about a mile and a half, and the aeronaut computed that in some parts of the voyage he moved at the rate of 30 miles an hour. Mr. Baldwin published a very circumstantial account of his voyage, with many ingenious philosophical remarks relating to aerostation, of which subject his book may be considered as one of the best treatises yet given to the public.</p><p>October the 5th, Mr. Lunardi made the first aerial voyage in Scotland. He ascended at Edinburgh, and after various turnings, landed near Cupar in Fife, having described a track of 40 miles over the sea, and 10 over the land, in an hour and a half. He said the mercury in the barometer sunk as low as 18.3 inches at his greatest elevation.</p><p>November the 19th the celebrated Blanchard ascended at Ghent to a great height, and after many dangers descended at Delft without his car, which he cut away to lighten the machine when he was descending too rapidly, and slung himself by the cords to the balloon, which served him then in the nature of a parachute. On his first ascent, when he was almost out of sight, he let down a dog, by means of a parachute, which came easily to the ground.</p><p>November the 25th Mr. Lunardi ascended at Glasgow, and in two hours it is said he described a track of 125 miles. It is further remarkable that, being overcome with drowsiness, he says he slept for about 20 minutes in the bottom of the car, during this voyage.</p><p>Many other voyages were made in different countries, and with various success. But since the year 1785, the rage for balloons has considerably abated, and we have gradually had less and less of these aerial excuisions, so that it is now become rather an uncommon thing to hear of one of them performed in any country whatever: which speedy decline in this new art is perhaps to be ascribed chiefly to the following causes; namely, a less degree of eagerness in people to pursue such experiments, from their curiosity having been satisfied; secondly, the trouble, danger, and great expence, attending them; and lastly, the want of the means of conducting them, and the small degree of utility to which they have hitherto been applied. The failure in the many attempts that have been made to direct balloons at pleasure through the air, cannot but <cb/> be felt as a very discouraging circumstance: and it ito be feared that it will ever be felt as such, notwithstanding the pretensions of some persons on this head; for they never have caused, nor is it to be expected they ever can cause the machine to deviate sensibly from the course of the wind, except only in the case when this moves with a very small celerity. For when the current blows only at the rate of 10 miles an hour, which is but a very gentle wind, it may be shewn that a balloon of 50 feet in diameter will require a force equal to the pressure of 72 pounds weight, to cause it to deviate 30 degrees from the course of the wind; and a force equal to double or triple that weight, when the wind blows with a double or triple velocity, that is, at the rate of 20 or 30 miles an hour; and so in proportion. To obviate the danger of a fall, arising from any accident happening to the balloon, some experiments have been made with a parachute, chiefly by Mr. Blanchard, whose endeavours and perseverance it seems have continued longer than in any other person: we still hear of his excursions in different parts of Europe, and improvements of the parachute, wings, &c; and have just read accounts of two voyages lately performed by him; with which, being very curious, we shall conclude our narration of these aerial excursions. They will be best related in Mr. Blanchard's own words, taken from his letter, dated Leipsick, October the 9th, 1787, to the editors of the Paris Journal. “I did not mention,” says he, “in your interesting paper, my ascension at Strasburg on the 26th of last August: the weather was so horrible that I mounted only for the sake of contenting the astonishing crowd of strangers assembled there from all parts of the country. Every body seemed satisfied at the attempt, but I assure you, gentlemen, that I was far from being pleased with so common an experiment. The only remarkable thing that occurred at that time, was the following circumstance: At the height of about 2000 yards, or a mile and half a quarter, I let down a dog tied to the parachute, who, instead of descending gently, was forcibly carried, by a whirlwind, above the clouds. I met him soon after, bending his course directly downwards, and, as on recollecting his master, he began to bark a little, I was going to take hold of the parachute, when another whirlwind lifted him again to a great height. I lost him for the space of six minutes, and perceived him afterwards, with my telescope, as if sleeping in the cradle or basket belonging to the machine. Continually agitated, and impetuously tossed through every point of the compass, by the violence of the different currents of air, I determined to end my voyage on the other side of the Rhine, after having passed vertically over Zell. I descended at a small village, with an intention to be assisted a little, and about thirty men soon came within reach of the balloon very a-propos, and fixed me to the ground. The wind was so violent that anchors or ropes would have been of no service. I had however added to the large aerostatic globe a smaller one, of 60 pounds ascensional force, which would have contributed to fix me, when once I let it loose; but notwithstanding this precaution, the men's assistance was very necessary to me. The parachute was still wavering in the air, and did not come down till 12 minutes after.” <pb n="43"/><cb/></p><p>“I performed my 27th ascension at Leipsick the 29th of September, in the midst of an incredible number of spectators, forming one of the most brilliant assemblies I ever beheld. The sky was as clear and serene as possible, and the air so calm that many of my friends, and multitudes of others, could follow me on horseback, and even on foot. I was sometimes so near them that they thought they could reach me, but I could soon find the means of rising; and once, when they had actually taken hold of the cords, to see me float with the strings in their hands, I suddenly cut them, and mounted again in the air. All these amusing evolutions were in sight of the town and its environs. At length I yielded to the earnest solicirations of the company, and entered the town triumphantly in my car, followed by a concourse of people transported with joy, and amidst the acclamations of thousands. The next day I emptied the inflammable air into another globe, with which I intended to try some experiments; and I let it off with a cradle, in which a dog was fixed. The balloon, having reached a considerable height, made an explosion in its under part, as I had imagined it would, having previously disposed it in a proper manner for that purpose; by which means the little animal fell gently to the ground.”</p><p>“The day before yesterday having repeated this experiment, at the town's request, I prepared the globe in such a manner as to cause an explosion in its upper part, and added a parachute with two small dogs fixed to it. They went so high that, notwithstanding the serenity of the sky, the balloon was lost in its immense expanse. Telescopes of the best sort became useless, and I began to be apprehensive for the death of the little animals, on account of the severity of the cold. They descended however about two hours after, quite safe and well, in the town of Delitzsch, three miles from Leipsick. I went yesterday to claim them, and found them again over the town in the air with the parachute. Such experiments had been already tried many times in the course of the day, and some officers had thrown them from the top of a steeple, in the sight of all the inhabitants of Delitzsch, from whence they descended safe.”</p><p>We have lately heard of Mr. Blanchard's 32d ascension at Brunswick in the month of August 1788, in which he much assisted his ascent by means of his wings. <hi rend="center">For several figures of balloons, see plate 1.</hi></p><p><hi rend="italics">Practice of</hi> <hi rend="smallcaps">Aerostation.</hi> The first consideration in the practice of aerostation, is the form and the size of the machine. Various shapes have been tried and proposed, but the globular, or the egg-like figure, is the most proper and convenient, for all purposes; and this form also will require less cloth or silk than any other shape of the same capacity; so that it will both come cheaper, and have a greater power of ascension. The bag or cover of an inflammable-air balloon, is best made of the silk stuff called lustring, varnished over. But for a montgolfier, or heated-air balloon, on account of its great size, linen cloth has been used, lined within or without with paper, and varnished. Small balloons are made either of varnished paper, or simply of paper unvarnished, or of gold-beater's skin, or such-like light substances. <cb/></p><p>With respect to the form of a balloon, it will be necessary that the operator remember the common proportions between the diameters, circumferences, surfaces, and solidities of spheres; for instance, that of different spheres, the circumferences are as the diameters; that the surfaces are as the squares of the diameters; and the solidities as the cubes of the same diameters: that any diameter is to its circumference as 7 to 22, or as 1 to 3 1/7; and therefore 3 times and 1/7 of any diameter will be its circumserence; so that if the diameter of a balloon be 35 feet, its circumference will be 110 feet. And if the diameter be multiplied by the circumference, the product will be the surface of the sphere; thus 35 multiplied by 110 gives 3850, which is the sursace of the same sphere in square feet. and if this surface be divided by the breadth of the stuff, in feet, which the balloon is to be made of, the quotient will be the number of feet in length necessary to construct the balloon; so if the stuff be 3 feet wide, then 3850 divided by 3, gives 1283 1/3 feet, or 428 yards nearly, the requisite quantity of stuff of 3 feet or one yard wide, to form the balloon of 35 feet diameter. Hence also, by knowing the weight of a given piece of the stuff, as of a square foot, or square yard, it is easy to find the weight of the whole bag, namely by multiplying the surface, in square feet or yards, by the weight of a square foot or yard: so if each square yard weigh 16 ounces or 1 pound, then the whole bag will weigh 428 pounds. Again, the capacity, or solid contents, of the sphere, will be found by multiplying 1/6 of the surface by the diameter, or by taking 11/22 of the cube of the diameter; which gives 22458 cubie feet for the capacity of the said balloon, that is, it will contain, or displace, 22458 cubic feet of air. From the content and surface of the balloon, so found, is to be derived its power or levity, thus: on an average, a cubic foot of common air weighs 1 1/5 ounce, and therefore to the number 22458, which is the content of our balloon, adding its (1/5)th part, we have 26950 ounces, or 1684 pounds, for the weight of the common air displaced or occupied by the balloon. From this weight must be deducted the weight of the bag, namely 428 pounds, and then there remains 1256 pounds levity of the balloon, without however considering the contained air, whether it be heated air, or of the inflammable kind. If inflammable air be used, as it is of different weights, from 1/4 to 1/10 or 1/12 the weight of common air, according to the modes of preparing it, let us suppose for instance that it is 1/6 of the weight of common air; then 1/6 of 1684 is 261 pounds, which is the weight of the bag full of that air; which being taken from 1256, leaves 995 pounds for the levity of the balloon when so filled with that inflammable air, or the weight which it will carry up, consisting of the car, the ropes, the passengers, the necessaries, and ballast. But if heated air be used; then as it is known from experiment that, by heating, the contained air is diminished in density about one-third only, therefore from 1684, take 1/3 of itself, and there remains 1123 for the weight of the contained warm air; and this being subtracted from 1256, leaves only 133 pounds for the levity of the balloon in this case; which being too small to carry up the car, passengers, &c, it shews that for those purposes a larger balloon is necessary, on <pb n="44"/><cb/> Montgolfier's principles. But if now, from the preceding computation, it be required to find how much the size of the balloon must be increased, that its levity, or power of ascension, may be equal to any given weight, as suppose 1000 pounds; then because the levities are nearly as the cubes of the diameters, therefore the diameters will be nearly as the cube roots of the levities; but the levities 133 and 1000 are nearly as 1 to 8, the cube roots of which are as 1 to 2, and consequently 1 : 2 :: 35 : 70 feet, the diameter of a montgolfier, made of the same thickness of stuff as the former, capable of lifting 1000 pounds.</p><p>On the same principles we can easily find the size of a balloon that shall just float in air when made of stuff of a given thickness or weight, and filled with air of a given density; the rule for which is this: from the weight of a cubic foot of common air, subtract that of a cubic foot of the lighter or contained air; then divide 6 times the weight of a square foot of the stuff, by the remainder; and the quotient will be the diameter, in feet, of the balloon that will just float at the surface of the earth. Suppose, for instance, that the materials are as before, namely, the stuff 1 pound to the square yard, or 16/9 ounces to the square foot, which taken 6 times is 32/3; then the cubic foot of common air weighing 1 1/5 ounce, and of heated air 2/3 of the same, whose difference is 2/5; therefore 32/3 divided by 2/5, gives 26 2/3 feet, which is the diameter of a montgolfier that will just float: but if inflammable air be used of 1/6 the weight of common air, the difference between 1 1/5 and 1/6 of it, is 1; by which dividing 32/3 or 10 2/3, the quotient is the same 10 2/3 feet, which therefore is the diameter of an inflammable-air balloon that will just float. And if the diameter be more than these dimensions, the balloons will rise up into the atmosphere.</p><p>The height nearly to which a given balloon will rise in the atmosphere, may be thus found, having given only the diameter of the balloon, and the weight which just balances it, or that is just necessary to keep it from rising: compute the capacity or content of the globe in cubic feet, and divide its restraining weight in ounces by that content, and the <table rend="border"><row role="data"><cell cols="1" rows="1" role="data">Height in miles.</cell><cell cols="1" rows="1" role="data">Density.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" role="data">1.200</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1/4</cell><cell cols="1" rows="1" role="data">1.141</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1/2</cell><cell cols="1" rows="1" role="data">1.085</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3/4</cell><cell cols="1" rows="1" role="data">1.031</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">0.980</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1 1/4</cell><cell cols="1" rows="1" role="data">0.932</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1 1/2</cell><cell cols="1" rows="1" role="data">0.886</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1 3/4</cell><cell cols="1" rows="1" role="data">0.842</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">0.800</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2 1/4</cell><cell cols="1" rows="1" role="data">0.761</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2 1/2</cell><cell cols="1" rows="1" role="data">0.723</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2 3/4</cell><cell cols="1" rows="1" role="data">0.687</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">0.653</cell></row></table> quotient will be the difference between the density or specific gravity of the atmosphere at the earth's surface and that at the height to which the balloon will rise; therefore subtract that difference or quotient from 1 1/5 or 1.2, the density at the earth, and the remainder will be the density at that height: then the height answering to that density will be found sufficiently near in the annexed table. Thus, in the foregoing examples, in which the diameter of the balloon is 35 feet, its capacity 22458, and the levity of the first one 995 pounds, or 15920 ounces, the quotient of the latter number divided by the former, is .709, which is the density at the utmost height, and to which in the tableanswers a little more than 2 1/2 miles, or 2 5/8 miles nearly, which therefore is the height to which the balloon will ascend. And when the same balloon was filled with heated air, its levity was found equal to only 133 pounds, or 2128 <cb/> ounces, then dividing this by 22458 the capacity, the quotient .095 taken from 1.200, leaves 1.105 for the density; to which in the table corresponds almost half a mile, or nearer 3/8 of a mile. And so high nearly would these balloons ascend, if they keep the same figure, and lose none of the contained air: or rather, those are the heights they would settle at; for their acquired velocity would first carry them above that height, so far as till all their motion should be destroyed; then they would descend and pass below that height, but not so much as they had gone above; after which they would re-ascend, and pass that height again, but not so far as they had gone below it; and so on for many times, vibrating alternately above and below that point, but always less and less every time. The foregoing rule, for finding the height to which the balloon will ascend, is independent of the different states of the thermometer at that highest point, and at the surface of the earth; but for greater accuracy, including the allowances depending on the different states of the thermometer, see under the word <hi rend="smallcaps">Atmosphere</hi>, where the more accurate rules are given at large.</p><p>The best way to make up the whole coating of the balloon, is by different pieces or slips joined lengthways from end to end, like the pieces composing the surface of a geographical globe, and contained between one meridian and another, or like the slices into which a melon is usually cut, and supposed to be spread flat out. Now the edges of such pieces cannot be exactly described by a pair of compasses, not being circular, but flatter or less round than circular arches; but if the slips are sufficiently narrow, or numerous, they will differ the less from circles, and may be described as such. But more accurately, the breadths of the slip, at the several distances from the <figure/> point to the middle, where it is broadest, are directly as the sines of those distances, radius being the sine of the half length of the slip, or of the distance of either point from the middle of the slip: that is, If ACBD be one of the slips, AB being half the circumference, or AE a quadrant conceived to be equal to AC or AD; then will CD be to <hi rend="italics">a b,</hi> as radius or the sine of AC, to the sine of A<hi rend="italics">a.</hi> So that if the quadrant AE or AC be divided into any number of equal parts, as here suppose 9, then divide the quadrant or 90 degrees by the number of parts 9, and the quotient 10 is the number of degrees in each part; and hence the arcs AC, A<hi rend="italics">a,</hi> A<hi rend="italics">c,</hi> &c, will be respectively 90°, 80°, 70°, &c; and CD being radius, the several breadths <hi rend="italics">ab, cd, ef,</hi> &c, will be respectively the sines of 80°, 70°, 60°, &c, which are here placed opposite them, the radius being 1. Therefore when it is proposed to cut out slips for a globe of a given diameter; compute the circumference, and make AE or AC a quarter of that circumference, and CD of any breadth, as 3 feet, or 2 feet, or any other <pb n="45"/><cb/> quantity; then multiply each of the decimal numbers, set opposite the figure, by that quantity, or breadth of CD, so shall the products be the several breadths <hi rend="italics">ab, cd, ef,</hi> &c.</p><p>Various schemes have been devised for conducting balloons in any direction, whether vertical or sideways. As to the vertical directions, namely upwards or downwards, the means are obvious, viz. in order to ascend, the aeronaut throws out some ballast; and that he may descend, he opens a valve in the top of his machine by means of a string, to let some of the gas escape; or if it be a montgolfier, he increases or diminishes the fire, as he would ascend or descend. But to direct the machine in a side or horizontal course, is a very difficult operation, and what has hitherto not been accomplished, except in a small degree, and when the current of air is very gentle indeed. The dissiculty of moving the balloon sideways, arises from the want of wind blowing upon it; for as it floats along with the current of air, it is relatively in a calm, and the aeronaut feels no more wind than if the machine were at rest in a perfect calm. For this reason, any thing in the nature of sails can be of no use; and all that can be hoped for, is to be attempted by means of oars; and how small the effect of these must be, may easily be conceived from the rarity of the medium against which they must act, and the great magnitude of the machine to be forced through it. We can easily assign what force is necessary to move a given machine in the air with any proposed velocity. From very accurate experiments I have determined, that a globe of 6 3/8 inches in diameter, and moving with a velocity of 20 feet per second of time, suffers a resistance from the air which is just equal to the weight or pressure of one ounce Averdupois; and farther that with different surfaces, and the same velocity, the resistances are directly proportional to the surface nearly, a double surface having a double resistance, a triple surface a triple resistance, and so on; and also that with different velocities, the resistances are proportional to the squares of the velocities nearly, so that a double velocity produced a quadruple resistance, and three times the velocity nine times the resistance, and so on. And hence we can assign the resistance to move a given balloon, with any velocity. Thus, take the balloon as before of 35 feet diameter; then by comparifon as above it is found that this globe, if moved with the velocity of 20 feet per second, or almost 14 miles per hour, will suffer a resistance equal to 271 pounds; to move it at the rate of 7 miles an hour, the resistance will be 68 pounds; and to move it 3 1/2 miles an hour, the resistance will be 17 pounds; and so on: and with such force must the aeronauts act on the air in a contrary direction, to communicate such a motion to the machine. And if the balloon move through a rarer part of the atmosphere, than that at the surface of the earth, as 1/3, or 1/4, &c, rarer, and consequently the resistance be less in the same proportion; yet the force of the oars will be diminished as much; and therefore the same difficulty still remains. In general, the aeronaut must strike the air, by means of his oars, with a force just equal to the resistance of the air on the balloon, and therefore he must strike that air with a velocity which must be greater as the surface of the oar is less <cb/> than the resisted surface of the globe, but not in the same proportion, because the force is as the square of the velocity.</p><p>Now suppose the aeronaut act with an oar equal to 100 square feet of surface, to move the balloon above mentioned at the rate of 20 feet per second, or 14 miles an hour; then must he move this oar with the great velocity of 62 feet per second, or near 43 miles per hour: and so in proportion for other velocities of the balloon. From whence it is highly probable, that it will never be in the power of man to guide such machine with any tolerable degree of success, especially when any considerable wind blows, which is almost always the case.</p><p>As some aeronauts have thought of using parachutes, made something like umbrellas, to break their fall, in case of any accident happening to the balloon, we shall here consider the principles and power of such a machine. Let us suppose a person wants to know what the size of a parachute must be, that he may descend with it at the uniform rate of 10 feet in a second, which is nearly equal to the velocity he acquires by falling or leaping from the height only of 17 inches, and which it is presumed he may do with safety. Now in order to descend with any uniform velocity, the resistance of the air must be equal to the whole weight that descends: then suppose the weight of the aeronaut to be 150 pounds, and that the parachute is flat, and circular, and made of such materials as that every square foot of its surface weighs 2 ounces, and farther that the weight increases in the same proportion as the surface; then the diameter of the parachute necessary to descend with the moderate velocity of 10 feet per second, must be upwards of 78 feet in diameter: but if the parachute be not a flat surface, but concave on the lower side, its power will be rather the greater, and the diameter may be somewhat less. If it be required to know the power of a flat circular parachute, or what resistance it meets with from air of a mean density, when descending with a given velocity; say as the number 800 is to the square of the velocity in feet, so is the square of the diameter in feet, to a fourth number, which will be the resistance in pounds. And hence, if it be required to know with what velocity a parachute will descend with a given weight; say as the given diameter is to the square root of the weight, so is the number 28 1/3 to a fourth term, which will be the velocity when the descent is in air of a mean density. So if the diameter of a balloon be 50, and its weight together with that of a man be 530 pounds, the square root of which is 23 very nearly; then as 50 : 23 :: 28 1/3 : 13, so that the man and parachute will descend with the velocity of 13 feet per second; which it is presumed he may safely do, as he would meet with a shock only equal to that of leaping freely from a height only of 2 feet 2 inches.</p><p>The methods of extracting inflammable air from various substances, for filling balloons, and for other purposes, may be seen under the words Air and Gas. And as to the methods of filling and constructing balloons, being matters merely mechanical, they are omitted in this place.</p><p>Ample information however on these, and many other particulars, may be met with in several books expressly written on the subject; as in Cavallo's History <pb n="46"/><cb/> and Practice of Aerostation, 8vo, 1785; in Baldwin's Airopaidia, 8vo, 1786; &c.</p><p>It has often been discussed, says the former of these gentlemen, whether the preference should be given to machines raised by inflammable air, or to those raised by heated air. Each of them has its peculiar advantages and disadvantages; a just consideration of which seems to decide in favour of those made with inflammable air. The principal comparative advantages of the other sort are, that they do not require to be made of so expensive materials; that they are filled with little or no expence; and that the combustibles necessary to fill them are found almost every where; so that when the stock of fuel is exhausted, the aeronaut may descend and recruit it again, in order to proceed on his voyage. But then this sort of machines must be made larger than the other, to take up the same weight; and the presence of a fire is a continual trouble, and a continual danger: in fact, among the many aerial voyages that have been made and attempted with such machines, very few have succeeded without an inconvenience, or an accident; and some indeed have been attended with dangerous and even fatal consequences; from which the other sort is in a great measure exempt. But, on the other hand, the inflammable air balloon must be made of a substance impermeable to the subtle gas: the gas itself cannot be produced without a considerable expence; and it is not easy to find the materials and apparatus necessary for the production of it in every place. However, it has been found that an inflammable-air balloon, of 30 feet in diameter, may be made so close as to sustain two persons, and a considerable quantity of ballast, in the air for more than 24 hours, when properly managed; and possibly one man might be supported by the same machine for three days: and it is probable that the stuff for these balloons may be so far improved, as to be quite impermeable to the gas, or very nearly so; in which case, the machine, once silled, would continue to float for a long space of time. At Paris they have already attained to a great degree of perfection in this point; and small balloons have been kept floating in a room for many weeks, without losing any considerable quantity of their levity: but the difficulty lies in the large machines: for in these, the weight of the stuff itself, with the weight and stress of the ropes and boat, and the folding them up, may easily crack and rub off the varnish, and make them leaky.</p><p>In regard to philosophical observations, derived from the new subject of aerostation, there have been very few made; the novelty of the discovery, and of the prospect enjoyed from the car of an aerostatic machine, have commonly distracted the attention of the aeronauts; not to remark that many of the adventurers were inadequate to the purpose of making improvements in philosophy, being mostly influenced either by pecuniary motives, or the vanity of adding their names to the list of aerial travellers.—The agreeable stillness and tranquillity experienced aloft in the atmosphere, have been matter of general observation. Some machines have ascended to a great height, as far, it has been said, as two miles; and they have commonly passed through fogs and clouds, above which they have enjoyed the clear light and heat of the sun, whilst the earth beneath <cb/> was actually covered by dense clouds, which poured down abundance of rain. In ascending very high, the aeronauts have often experienced a pain in their ears, arising, it is supposed, from the internal air being not of the same density as the air without; but the pain usually went off in a short time: and it seems that this effect is similar to what is experienced by persons who descend by a diving-bell to considerable depths in the sea: I remember often to have heard the late unfortunate Mr. Spalding, the celebrated diver, speak of this effect, with a marked and philosophical accuracy: after descending two or three fathoms below the surface, he began to feel a pain in his ears, which gradually increased to a very great degree if the descent was too quick; his method was therefore to descend slowly, and to make a stop for some minutes at the depth of 5 fathom, which is equal nearly to the pressure of the atmosphere, and where consequently the air in his bell was of double the density of common air at the surface; after resting here awhile, his ears, as he expressed it, gave a crack, and he was suddenly relieved of the pain. He then descended 5 fathoms more, with the same symptoms, and the same effect: and so on continually, from one five fathoms to another, descending leisurely, and stopping a little at each stage, to give time for his constitution to adapt itself to the degree of condensation of the air; after which he felt no more inconvenience, till he came to ascend again, which was performed with the same caution and circumstances. One experiment is recorded, in which the air of a high region, being brought down, and examined by means of nitrous air, was found to be purer than the air below. The temperature of the upper regions too, it has been found, is much colder than that of the air near the earth; the thermometer, in some aerostatic machines, having descended many degrees below the freezing point of water, while it was considerably higher than that degree at the earth's surface.</p><p>ÆSTIVAL, see <hi rend="smallcaps">Estival.</hi></p><p>ÆSTUARY, or <hi rend="smallcaps">Estuary</hi>, in <hi rend="italics">Geography,</hi> an arm of the sea, running up a good way into the land. Such as Bristol channel, many of the friths in Sootland, and such like.</p><p>ÆTHER, see <hi rend="smallcaps">Ether.</hi></p></div1><div1 part="N" n="AFFECTED" org="uniform" sample="complete" type="entry"><head>AFFECTED</head><p>, or <hi rend="smallcaps">Adfected, Equation</hi>, in <hi rend="italics">Algebra,</hi> is an equation in which the unknown quantity rises to two or more several powers or degrees. Such, for example, is the equation , in which there are three different powers of <hi rend="italics">x,</hi> namely, <hi rend="italics">x</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">2</hi>, and <hi rend="italics">x.</hi></p><p>The term, affected, is also used sometimes in algebra, when speaking of quantities that have co-efficients.</p><p>Thus in the quantity 2<hi rend="italics">a, a</hi> is said to be affected with the co-efficient 2.</p><p>It is also said, that an algebraic quantity is affected with the sign + or -, or with a radical sign; meaning no more than that it has the sign + or -, or that it includes a radical sign.</p><p>The term adfected, or affected, I think, was introduced by Vieta.</p></div1><div1 part="N" n="AFFECTION" org="uniform" sample="complete" type="entry"><head>AFFECTION</head><p>, in <hi rend="italics">Geometry,</hi> a term used by some ancient writers, signifying the same as property.</p><p><hi rend="smallcaps">Affection.</hi> <hi rend="italics">Phys.</hi> The affections of a body are certain modifications occasioned or induced by <pb n="47"/><cb/> motion; in virtue of which the body is disposed after such, or such a manner.</p><p>The affections of bodies, are sometimes divided into <hi rend="italics">primary</hi> and <hi rend="italics">secondary.</hi></p><p><hi rend="italics">Primary Affections,</hi> are those which arise either out of the idea of matter, as magnitude, quantity, and figure; or out of the idea of form, as quality and power; or out of both, as motion, place, and time.</p><p><hi rend="italics">Secondary,</hi> or <hi rend="italics">derivative Affections,</hi> are such as arise out of primary ones, as divisibility, continuity, contiguity, &c, which arise out of quantity; regularity, irregularity, &c, which arise out of figure, &c.</p><p>AFFIRMATIVE <hi rend="smallcaps">Quantity</hi>, or <hi rend="smallcaps">Positive Quantity</hi>, one which is to be added, or taken effectively; in contradistinction to one that is to be subtracted, or taken defectively.—The term affirmative was introduced by Vieta.</p><p><hi rend="smallcaps">Affirmative Sign</hi>, or <hi rend="smallcaps">Positive Sign</hi>, in <hi rend="italics">Algebra,</hi> the sign of addition, thus marked +, and is called <hi rend="italics">plus,</hi> or <hi rend="italics">more,</hi> or <hi rend="italics">added to.</hi> When set before any single quantity, it serves to denote that it is an affirmative or a positive quantity; when set between two or more quantities, it denotes their sum, shewing that the latter are to be added to the former. So + 6, and + <hi rend="italics">a,</hi> and + AB, are affirmative quantities; also + 6 + 8 + 10 denote the sum of 6, 8, and 10, which is 24, and are read thus, 6 plus 8 plus 10. Also <hi rend="italics">a</hi> + <hi rend="italics">b</hi> + <hi rend="italics">c</hi> denote the sum of the quantities represented by <hi rend="italics">a, b</hi> and <hi rend="italics">c,</hi> when added together. It seems now not easy to ascertain with certainty, when, or by whom, this sign was first introduced; but it was probably by the Germans, as I find it first used by Stifelius in his Arithmetic, printed in 1544.</p><p>The early writers on Algebra used the word <hi rend="italics">plus</hi> in Latin, or <hi rend="italics">piu</hi> in Italian, for addition, and afterwards the initial <hi rend="italics">p</hi> only, as a contraction; like as they used <hi rend="italics">minus,</hi> or <hi rend="italics">meno,</hi> or the initial <hi rend="italics">m</hi> only, for subtraction: and thus these operations were denoted in Italy by Lucas de Burgo, Tartalea, and Cardan, while the signs + and - were employed much about the same time in Germany by Stifelius, Scheubelius, and others, for the same operations.</p></div1><div1 part="N" n="AGE" org="uniform" sample="complete" type="entry"><head>AGE</head><p>, in <hi rend="italics">Chronology,</hi> is used for a century, being a system or a period of a hundred years.</p><p>Chronologists also divide the time since the creation of the world into three ages: The first, from Adam till Moses, which they call the age of nature; the second from Moses to Jesus Christ, called the age of the law; and the third, or age of grace, from Jesus Christ till the end of the world.</p><p><hi rend="smallcaps">Age</hi> <hi rend="italics">of the Moon,</hi> in Astronomy, is the number of days elapsed since the last new moon. To find the moon's age, for any time nearly, for ordinary uses; add together the epact, the day of the current month, and the number of months from March to the present month inclusive; the sum is the moon's age: but if the sum exceed 29, deduct 29 from it in months that have 30 days, or 30 in those that have 31; and the remainder will be the age.—At the end of 19 years the moon's age returns upon the same day of the month, but falls a little short of the same hour of the day.</p></div1><div1 part="N" n="AGENT" org="uniform" sample="complete" type="entry"><head>AGENT</head><p>, <hi rend="italics">Agens,</hi> in <hi rend="italics">Physics,</hi> that by which a thing is done or effected; or any thing having a power by <cb/> which it acts on another, called the patient, or by its action induces some change in it.</p></div1><div1 part="N" n="AGGREGATE" org="uniform" sample="complete" type="entry"><head>AGGREGATE</head><p>, the sum or result of several things added together. See <hi rend="italics">Sum.</hi></p></div1><div1 part="N" n="AGITATION" org="uniform" sample="complete" type="entry"><head>AGITATION</head><p>, in <hi rend="italics">Physics,</hi> a brisk intestine motion, excited among the particles of a body. Thus fire agitates the subtlest particles of bodies. Fermentations, and effervescences, are produced by a brisk agitation of the particles of the fermenting body.</p></div1><div1 part="N" n="AGUILON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">AGUILON</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, or <hi rend="smallcaps">Aquilon</hi>, was a jesuit of Brussels, and professor of philosophy at Doway, and of theology at Antwerp. He was one of the first that introduced mathematical studies into Flanders. He wrote a large work on <hi rend="italics">Optics,</hi> in 6 books, which was published in folio, at Antwerp, in 1613; and a treatise of <hi rend="italics">Projections of the Sphere.</hi> He promised also to treat upon <hi rend="italics">Catoptrics</hi> and <hi rend="italics">Dioptrics,</hi> but this was prevented by his death, which happened at Seville, in the year 1617.</p></div1><div1 part="N" n="AIR" org="uniform" sample="complete" type="entry"><head>AIR</head><p>, in <hi rend="italics">Physics,</hi> a thin, fluid, transparent, elastic, compressible, and dilatable body, which surrounds this terraqueous globe, and covers it to a considerable height.</p><p>Some of the ancients considered air as an element, namely, one of the four elements, air, earth, water, and fire, of which they conceived all bodies to be composed; and though it be certain that air, taken in the common acceptation, be far from the simplicity of an elementary substance, yet some of its parts may properly be so called. So that air may be distinguished into proper or elementary, and vulgar or heterogeneous.</p><p><hi rend="italics">Elementary</hi> <hi rend="smallcaps">Air</hi>, or <hi rend="italics">Air</hi> properly so called, is a subtle, homogeneous, elastic fluid; being the basis, or fundamental ingredient, of the whole air of the atmosphere, from which it takes its name. And in this sense Dr. Hales, and other modern philosophers, consider it as entering into the composition of most, or perhaps all bodies; existing in them in a solid state, devoid of its elasticity, and most of its distinguishing properties, and serving as their cement; but, by certain processes, capable of being disengaged from them, recovering its elasticity, and resembling the air of our atmosphere.</p><p>The particular nature of this aerial matter we know but little about: what authors have said concerning it being chiefly conjectural. There is no way of examining air pure and defecated from the several matters with which it is mixed; and consequently we cannot pronounce what are its peculiar properties, abstractedly from other bodies.</p><p>Dr. Hook, and some others, maintain that it is the same with the <hi rend="italics">ether,</hi> or that imaginary fine, fluid, active matter, conceived to be diffused through the whole expanse of the celestial regions: which comes to much the same thing as Newton's subtle medium, or spirit. In this sense it is supposed to be a body <hi rend="italics">sui generis,</hi> incorruptible, immutable, incapable of being generated, but present in all places, and in all bodies.</p><p>Other philosophers place its essence in elasticity, making that its distinctive character. These suppose that it may be generated, and that it is nothing else but the matter of other bodies, rendered by the <pb n="48"/><cb/> changes it has undergone, susceptible of a permanent elasticity. Mr. Boyle produces a number of experiments, which he made on the production of air, that is, according to him, the extraction of a sensible quantity of air from a body in which there appeared to be little or none at all, by whatever means this may be effected. He observes that among the different methods for this purpose, the chief are fermentation, corrosion, dissolution, decomposition, ebullition of water and other fluids, the reciprocal action of bodies, especially saline ones, upon one another; he adds, that different solid and mineral bodies, in the parts of which no elasticity could be conceived to exist, being plunged into corrosive mediums, which also are quite unelastic, will, by the attenuation of their parts from their mutual collision, produce a considerable quantity of elastic air.</p><p>Sir Isaac Newton is of the same opinion, according to whom the particles of a dense compact fixed substance, adhering to each other by a powerful attractive force, cannot be separated but by a violent heat, and perhaps never without fermentation; and these bodies, raresied by heat and fermentation, are finally transformed into a truly permanent elastic air. On these principles, he adds, gunpowder produces air on explosion. Optics, Qu. 31, &c.</p><p><hi rend="italics">Common,</hi> or <hi rend="italics">heterogeneous</hi> <hi rend="smallcaps">Air</hi>, is an assemblage of corpuscles of various kinds, which together constitute one fluid mass, in which we live and move, and which we constantly breathe; which compound mass altogether, is called the atmosphere.</p><p>In this popular and extensive meaning of the term, Mr. Boyle acknowledges that air is the most heterogeneous body in the universe; and Boerhaave proves that it is an universal chaos, a mere jumble of all species of created things. Besides the matter of light or fire, which continually flows into it from the celestial bodies, and perhaps the magnetic effluvia of the earth, whatever fire can volatilize must be found in the air.</p><p>Hence, for instance, 1. All sorts of vegetable matter must be contained in the air; being either exhaled from plants growing all over the face of the earth, or rendered volatile by putrefaction, not excepting even the more solid and vascular parts of them.</p><p>2. It is no less certain that the air must contain particles of every substance belonging to the animal kingdom. For the copious emanations which are perpetually issuing from the bodies of animals, in the perspiration constantly kept up by the vital heat, are absorbed by the air; and in such quantities too, during the course of an animal life, that, could they be recollected, they would be sufficient to compose a good round number of the like animals. And besides, when a dead animal continues exposed to the air, all its particles evaporate, and are quickly dissipated; so that the substance which composed the animal, is almost wholly incorporated with the air.</p><p>3. The whole fossil kingdom must necessarily be found in the atmosphere; for all of that kind, as salts, sulphurs, stones, metals, &c, are convertible into fume, and must consequently take place among aerial substances. Gold itself, the most fixed of all natural bodies; is found among ores, closely adhering to sulphurs in mines, and so is raised along with the mineral. <cb/></p><p>Of all the emanations which float in the vast ocean of the atmosphere, perhaps the principal are such as consist of saline particles. Many writers suppose that they are of a nitrous kind; but it is probable that they are of all sorts, as vitriol, alum, mariue salt, and many others. And Mr. Boyle thinks that there may be great quantities of compound salts, not to be met with on or in the bowels of the earth, formed by the fortuitous concourse and mixture of different saline spirits.</p><p>We often find the window-glass of old buildings corroded, as if eaten by worms; though we know of no particular salt that is capable of producing such an effect.</p><p>Sulphurs too must make a considerable portion of this compound mass, on account of the many volcanos, grotts, caverns, and mines, dispersed over the face of the globe.</p><p>Finally, the various attritions, separations, dissolutions, and other mutual operations of matter of different sorts upon one another, may be regarded as the sources of many other neutral, or anony mous bodies, unknown to us, which rise and float in the air.</p><p>Air, taken in this extensive sense, is one of the most general and considerable agents in nature; being concerned in the preservation of animal and vegetable life, and in the production of most of the phenomena that take place in the material world.</p><p>Its properties and effects, having been the principal objects of the researches and discoveries of modern philosophers, have been reduced to precise laws and demonstrations, forming no inconsiderable branch of mixed mathematics, under the titles of Pneumatics, Aerometry, &c.</p><p><hi rend="italics">Mechanical Properties and Effects of</hi> <hi rend="smallcaps">Air.</hi> Of these the most considerable are its fluidity, its weight, and its elasticity.</p><p>1. <hi rend="italics">Its Fluidity.</hi>—The great fluidity of the air is manifest from the great facility with which bodies traverse it; as in the propagation of, and easy conveyance it affords to, sounds, odours and other effluvia and emanations that escape from bodies: for these effects prove that it is a body whose parts give way to any force, and in yielding are easily moved amongst themselves; which is the definition of a fluid. That the air is a fluid is also proved from this circumstance, that it is found to exert an equal pressure in all directions; an effect which could not take place otherwise than from its extreme fluidity. Neither has it been found that the air can be deprived of this property, whether it be kept for many years together consined in glass vessels, or be exposed to the greatest natural or artificial cold, or condensed by the most powerful pressure; for in none of these circumstances has it ever been reduced to a solid state. It is true indeed that real permanent air may be extracted from solid bodies, and may also be absorbed by them; and we also know that in this case it must be exceedingly condensed, and reduced to a bulk many hundred times less than in its natural state: but in what form it exists in those bodies, or how their particles are combined together, is a mystery which remains hitherto inexplicable.</p><p>Those philosophers who, with the Cartesians, make fluidity to consist in a perpetual intestine motion of the parts, think they can prove that this character belongs to air: thus, in a darkened room, where the represen- <pb n="49"/><cb/> tations of external objects are introduced by a single ray, the corpuscles with which the air is replete, are seen to be in a continual fluctuation. Some moderns attribute the fluidity of the air, to the fire which is intermixed with it; without which, say they, the whole atmosphere would harden into a solid impenetrable mass: and indeed it must be allowed that the more fire it contains, the greater will its fluidity, mobility, and permeability be; and according as the different positions of the sun augment or diminish the degree of fire, the air always receives a proportional temperature, and is kept in a continual reciprocation.</p><p>2. <hi rend="italics">Its Weight or Gravity.</hi>—The weight or gravity of the air, is a property belonging to it as a body; for gravity is a property essential to matter, or at least a property found in all bodies. But independent of this, we have many direct proofs of its gravity from sense and experiment: thus, the hand laid close upon the end of a vessel, out of which the air is drawn at the other end, soon feels the load of the incumbent atmosphere: thus also, thin glass vessels, exhausted of their air, are easily crushed to pieces by the weight of the external air: and so two hollow segments of a sphere, 4 inches in diameter, exactly fitting each other, being emptied of air, are, by the weight of the ambient air, pressed together with a force which requires the weight of 188 pounds to separate them; and that they are thus forcibly held together by the pressure of the air, is made evident by suspending them in an exhausted receiver, for then they quickly separate of themselves, and fall asunder. Again, if a tube, close at one end, be filled with quicksilver, and the open end be immerged in a bason of the same fluid, and so held upright, the quicksilver in the tube will be kept raised up in it to the height of about 30 inches above the surface of that in the bason, being supported and balanced by the pressure of the external air upon that surface: and that this is the cause of the suspension of the quicksilver in the tube, is made evident by placing the whole apparatus under the receiver of an air-pump; for then the fluid will descend in the tube in proportion as the receiver is exhausted of its air; and then on gradually letting in the air again, the quicksilver reascends to its former beight in the tube: and this is what is called, from its inventor, the Terricellian experiment. Nay farther, air can actually be weighed like any other body: for a rigid vessel, full even of common air, by a nice balance is found to weigh more than when the air is exhausted from it; and the essect is proportionally more sensible, if the vessel be weighed full of condensed air, and more still if it be weighed in a receiver void of air.</p><p>But although we have innumerable proofs of the gravitating property of the air, yet the full discovery of the laws and circumstances of it are certainly due to the moderns. It cannot indeed be denied, that several of the ancients had some confused notions about this property: thus Aristotle says that all the elements have gravity, and even air itself; and as a proof of it, says that a bladder inflated with air, weighs more than the same when empty; and Plutarch and Stobæus quote him as teaching that the air in its weight is between that of fire and of earth; and farther, he himself, treating of respiration, reports it as the opinion of <cb/> Empedocles, that he ascribes the cause of it to the weight of the air, which by its pressure forces itself into the lungs; and much in the same way are the sentiments of Asclepiades expressed by Plutarch, who represents him as saying, among other things, that the external air, by its weight, forcibly opened its way into the breast. But nevertheless it is certain, however unreasonable it may seem, that Aristotle's followers departed in this instance from their master, by asserting the contrary for many ages together. Indeed many of the phenomena arising from this property, have been remarked from the highest antiquity. Many centuries since, it was known that by sucking the air from an open pipe, having its extremity immersed in water, this fluid rises above its level, and occupies the place of the air. In consequence of such observations, sucking pumps were contrived, and various other hydraulic machines; as Heron's syphons, described in his Spiritalia or Pneumatics, and the watering pots known in Aristotle's time under the name of <hi rend="italics">clepsydræ,</hi> which alternately stop or run as the singer closes or opens their upper orifice. Indeed the reason assigned, by philosophers many ages after, for this phenomenon, was a pretended horror that nature conceives for a vacuum, which, rather then endure it, makes a body ascend contrary to the powerful solicitation of its gravity. Even Galileo, with all his sagacity, could not for some time hit upon any thing more satisfactory; for he only assigned a limit to this dread of vacuity: having observed that sucking pumps would not raise water higher than 16 brasses, or 34 English feet, he limited this abhorring force of nature, to one that was equivalent to the weight of a column of water 34 feet high, on the same base as the void space. Consequently he pointed out a way of making a vacuum, by means of a hollow cylinder, whose piston is charged with a weight sufficient to detach it from the close bottom turned upwards: this effort he called the measure of the force of vacuity, and made use of it for explaining the cohesion of the parts of bodies.</p><p>Galileo however was well apprised of the weight of the air as a body: in his Dialogues he shews two ways of demonstrating it, by weighing it in bottles: the transition was easy from one discovery to another: yet still Galileo's knowledge of the matter was imperfect, that is, as to the particular instance of the suspension of a fluid above its level, by the pressure of the external air.</p><p>At length Torricelli fell upon the lucky guess, that the counterpoise which keeps fluids above their level, when nothing presses upon their internal surface, is the mass of air resting upon the external one. He discovered it in the following manner: In the year 1643 this disciple of Galileo, on occasion of executing an experiment on the vacuum formed in pumps, above the column of water, when it exceeds 34 feet, thought of using some heavier fluid, such as quicksilver. He conceived that whatever might be the cause by which a column of water of 34 feet high is sustained above its level, the same force would sustain a column of any other fluid, which weighed as much as that column of water, on the same base; whence he concluded that quicksilver, being about 14 times as heavy as water, would not be sustained higher than 29 or 30 inches. <pb n="50"/><cb/> He therefore took a glass tube of several feet in length, sealed it hermetieally at one end, and filled it with quicksilver; then inverting it, and holding it upright, by pressing his finger against the lower or open orifice, he immersed that end in a vessel of quicksilver; then removing his singer, and suffering the sluid to run out, the event verisied his conjecture; the quicksilver, faithful to the laws of hydrostatics, descended till the column of it was about 30 inches high above the surface of that in the vessel below. And hence Torricelli concluded that it was no other than the weight of the air incumbent on the surface of the external quicksilver, which counterbalanced the fluid contained in the tube.</p><p>By this experiment Torricelli not only proved, what Galileo had done before, that the air had weight, but also that it was its weight which kept water and quicksilver raised in pumps and tubes, and that the weight of the whole column of it was equal to that of a like column of quicksilver of 30 inches high, or of water 34 or 35 feet high; but he did not ascertain the weight of any particular quantity of it, as a gallon, or a cubic foot of it, nor its specific gravity to water, which had been done by Galileo, though to be sure with no great accuracy, for he only proved that water was more than 400 times heavier than air.</p><p>Torricelli's experiment became famous in a short time. Father Mersenne, who kept up a correspondence with most of the literati in Italy, was informed of it in 1644, and communicated it to those of France, who presently repeated the experiment: Messrs. Pascal and Petit made it first, and varied it several ways; which gave occasion to the ingenious treatise which Pascal published at 23 years of age, intitled <hi rend="italics">Experiences Nouvelles touchant la Vuide.</hi> In this treatise indeed he makes use of the old principle of <hi rend="italics">suga vacui;</hi> but afterwards getting some notion of the weight of the air, he soon adopted Torricelli's idea, and devised several experiments to consirm it. One of these was to procure a vacuum above the reservoir of quicksilver; in which case he found the column sink down to the common level: but this appearing to him not sufficiently powerful to dissipate the prejudices of the ancient philosophy, he prevailed on M. Perier, his brother-in-law, to execute the famous experiment of Puy-de-Domme, who found that the height of the quicksilver half-way up the mountain was less, by some inches, than at the foot of it, and still less at the top: so that it was now put out of doubt that it was the weight of the atmosphere which counterpoised the quicksilver.</p><p>Des Cartes too had a right notion of this effect of the air, to sustain fluids above their level, as appears by some of his letters about this time, and some years before; and in one of those he lays claim to the idea of the Puy-de-Domme experiment: After having desired M. de Carcavi to inform him of the success of that experiment, which public rumour had advertised him had been made by M. Pascal himself, he adds, “I had reason to expect this from him, rather than from you, because I first proposed it to him two years since, assuring him at the same time, that although I had not tried it, yet I could not doubt of the consequence; but as he is a friend of M. Roberval, who professes himself no friend to me, I suppose he is guided <cb/> by that gentleman's passions.” See more of this history under <hi rend="smallcaps">Barometer.</hi></p><p>As to the actual weight of any given portion of common air, it seems that Galileo was the first who determined it experimentally; and he gives two different methods, in his Dialogues, for weighing it in bottles: he did not however perform the experiment very accurately, as he stated from the result that the gravity of water was to that of air rather above 400 to 1.</p><p>A quantity of air was next weighed by Mersenne in a very ingenious manner. His idea was to weigh a vessel both when full of air, and when emptied of it a to make the vacuum for this purpose, he knew no better way than by expelling the air out of an colipile by heating it red hot: by weighing it both when cold and hot, he sound a certain difference; which however was not the exact weight of that capacity of air, because the vacuum was not perfect. But by plunging the eolipile, when red hot, into water, just so much water entered as was equal in bulk to the air that had been expelled; then he took it out and weighed it with the water, which gave the weight of the same bulk of water; and on comparing this with the former difference, or weight of air expelled, he found their proportion to be as 1300 to 1. Which is as wide of the truth as Galileo's proportion, namely 400 to 1, but the contrary way. And it is remarkable that the mean between the two, namely 850 to 1, is very near the true proportion as settled by other more accurate experiments.</p><p>Mr. Boyle, by a more accurate experiment, found the proportion to be that of 938 to 1. And Mr. Hauksbee found it as 850 to 1, proceeding on the same principles as Mersenne, with a three-gallon glass bottle, but extracting the air out of it with the air pump, instead of expelling it by fire; the height of the barometer being at that time 29.7 inches. Also by other accurate experiments made before the Royal Society by Mr. Hauksbee, Dr. Halley, Mr. Cotes, and others, the proportion was always between 800 and 900 to 1, but rather nearer the latter, namely, being first found as 840 to 1, then as 852 to 1, and a third time as 860 to 1; the barometer then standing at 29 3/4 inches, and the weather warm. Mr. Cavendish determines the ratio 800 to 1, the barometer being 29 3/4, and the thermometer at 50°; and Sir George Shuckburgh, by a very accurate experiment, finds it 836 to 1, the barometer being at that time at 29.27, and the thermometer at 51°. And the medium of all these is about 832 or or 833 to 1, when reduced to the pressure of 30 inches of the barometer, and the mean temperature 55° of the thermometer. Upon the whole therefore it may be safely concluded that, when the barometer is at 30 inches, and the thermometer at the mean temperature 55°, the density or gravity of water is to that of air, as 833 1/3 to 1, that is as 2500/3 to 1, or as 2500 to 3; and that for any changes in the height of the barometer, the ratio varies proportionally; and also that the density of the air is altered by the (1/440)th part for every degree of the thermometer above or below temperate.</p><p>This number, which is a very good medium among them all, I have chosen with the fraction 1/37, because it gives exactly 1 1/5 ounce for the mean weight of a cubis <pb n="51"/><cb/> soot of air, the weight of the cubic foot of water being just 1000 ounces averdupois, and that of quicksilver equal to 13600 ounces.</p><p><hi rend="italics">Air,</hi> then, having been shewn to be a heavy fluid substance, the laws of its gravitation and pressure must be the same as those of water and other fluids; and consequently its pressure must be proportional to its perpendicular altitude. Which is exactly conformable to experiment; for on removing the Terricellian tube to different heights, where the column of air is shorter, the column of quicksilver which it sustains is shorter also, and that nearly at the rate of 100 feet for 1/10 of an inch of quicksilver. And on these principles depend the structure and use of the barometer.</p><p>And from the same principle it likewise follows that air, like other fluids, presses equally in all directions. And hence it happens that soft bodies endure this pressure without change of figure, and hard or brittle bodies without breaking; being equally pressed on all parts; but if the pressure be taken off, or diminished, on one side, the effect of it is immediately perceived on the other. See <hi rend="smallcaps">Atmosphere</hi>, for the total quantity of effects and pressure, and the laws of different altitudes, &c.</p><p>From the weight and fluidity of the air, jointly considered, many effects and uses of it may easily be deduced. By the combination of these two qualities, it closely invests the earth, with all the bodies upon it, constringing and binding them down with a great force, namely a pressure equal to about 15 pounds upon every square inch. Hence, for example, it prevents the arterial vessels of plants and animals from being too much distended by the impetus of the circulating juices, or by the elastic force of the air so copiously abounding in them. For hence it happens, that on a diminution of the pressure of the air, in the operation of cupping, we see the parts of the body grow tumid, which causes an alteration in the circulation of the fluids in the capillary vessels. And the same cause hinders the fluids from transpiring through the pores of their containing vessels, which would otherwise cause the greatest debility, and often destroy the animal. To the same two qualities of the air, weight and fluidity, is owing the mixture of bodies contiguous to one another, especially fluids; for several liquids, as oils and salts, which readily mix of themselves in air, will not mix at all in vacuo. With many other natural phenomena.</p><p>3. <hi rend="italics">Elasticity.</hi> Another quality of the air, from whence arise a multitude of effects, is its elasticity; a quality by which it yields to the pression of any other bodies, by contracting its volume; and dilates and expands itself again on the removal or diminution of the pressure. This quality is the chief distinctive property of air, the other two being common to other fluids also.</p><p>Of this property we have innumerable instances. Thus, for example, a blown bladder being squeezed in the hand, we find a sensible resistance from the included air; and upon taking off the piessure, the compressed parts immediately restore themselves to their former round sigure. And on this property of elasticity depend the structure and uses of the air-pump.</p><p>Every particle of air makes a continual effort to dilate itself, and so it acts forcibly against all the neigh- <cb/> bouring particles, which also exert the like force in return; but if their resistance happen to cease, or be weakened, the particle immediately expands to an immense extent. Hence it is that thin glass bubbles, or bladders, filled with air, and placed under the receiver of an air-pump, do, upon pumping out the air, burst asunder by the force of the air which they contain. So likewise a close flaccid bladder, containing only a small quantity of air, being put under the receiver, swells as the receiver is exhausted, and at length appears quite full. And the same thing happens by carrying the flaccid bladder to the top of a very high mountain.</p><p>The same experiment shews that this elastic property of the air is very different from the elasticity of solid bodies, and that these are dilated after a different manner from the air. For when air ceases to be compressed, it not only dilates, but then occupies a far greater space, and exists under a volume immensely greater than before; whereas solid elastic bodies only resume the figure they had before they were compressed.</p><p>It is plain that the weight or pressure of the air does not at all depend on its elasticity, and that it is neither more nor less heavy than if it were not at all elastic. But from its being elastic, it follows that it is susceptible of a pressure, which reduces it to such a space, that the force of its elasticity, which re-acts against the pressing weight, is exactly equal to that weight. Now the law os the elasticity is such, that it increases in proportion to the density of the air, and that its density increases in proportion to the forces or weights which compress it. But there is a necessary equality between action and re-action; that is, the gravity of the air, which effects its compression, and the elasticity of it, which gives it its tendency to expansion, are equal.</p><p>So that, the elasticity increasing or diminishing, in the same proportion as the density increases or diminishes, that is, as the distance between its particles decrease or increase; it is no matter whether the air be compressed, and retained in any space, by the weight of the atmosphere, or by any other cause; as in either case it must endeavour to expand with the same force. And therefore, if such air as is near the earth be inclosed in a vessel, so as to have no communication with the external air, the pressure of such inclosed air will be exactly equal to that of the whole external atmosphere. And accordingly we find that quicksilver is sustained to the same height, by the elastic force of air inclosed in a glass vessel, as by the whole pressure of the atmosphere.—And on this principle of the condensation and elasticity of the air, depends the structure and use of the air-gun.</p><p>That the density of the air is always directly proportional to the force or weight which compresses it, was proved by Boyle and Mariotte, at least as far as their experiments go on this head: and Mr. Mariotte has shewn that the same rule takes place in condensed air. However, this rule is not to be admitted as scrupulously exact; for when air is very forcibly compressed, so as to be reduced to (1/4)th of its ordinary bulk, the effect does not answer precisely to the rule; for in this case the air begins to make a greater resistance, and requires a stronger compression, than according to the rule. And hence it would seem, that the particles of air cannot, by means of any possible weight or pressure, <pb n="52"/><cb/> how great soever, be brought into perfect contact, or that it cannot thus be reduced to a solid mass; and consequently that there must be a limit to which this con densation of the air can never arrive. And the same remark is true with regard to the rarefaction of air, namely, that in very high degrees of rarefaction, the elasticity is decreased rather more than in proportion to the weight or density of the air: and hence there must also be a limit to the rarefaction and expansion of the air, by which it is prevented from expanding to infinity.</p><p>We know not however how to assign those limits to the elasticity of the air, nor to destroy or alter it, without changing the very nature of air, which is effected by chemical processes. To what degree air is susceptible of condensation, by compression, is not certainly known. Mr. Boyle condensed it 13 times more than in its natural state, by this means: others have compressed it into (1/70)th part of its ordinary volume; Dr. Hales made it 38 times more dense, by means of a press; but by freezing water in a hollow cast-iron ball or shell, he reduced it to 1838 times less space than it naturally occupies; in which state it must have been of more than twice the density or specific gravity of water: And as water is not compressible, except in a very small degree, it follows from this experiment, that the particles of air must be of a nature very different from those of water; since it would otherwise be impossible to reduce air to a volume above 800 times less than in its common state; an inference however which militates directly against an assertion made by Dr. Halley, from some experiments performed in London, and others at Florence by the Academy del Cimento, namely, that it may be safely concluded that no force whatever is capable to reduce air into a space 800 times less than that which it naturally occupies near the surface of the earth.</p><p>The elasticity of the air exerts its force equally in all directions; and when it is at liberty, and freed from the cause which compressed it, it expands equally in all directions, and in consequence always assumes a spherical figure in the interstices of the fluids in which it is lodged. This is evident in liquors placed in the receiver of an air pump, by exhausting the air; at first there appears a multitude of exceeding small bubbles, like grains of fine sand, dispersed through the fluid mass, and rising upwards; and as more air is pumped out, they enlarge in size; but still they continue round. Also if a plate of metal be immerged in the liquor, on pumping, its surface will be seen covered over with small round bubbles, composed of the air which adhered to it, now expanding itself. And for the same reason it is that large glass globes are always blown up of a spherical shape, by blowing air through an iron tube into a piece of melted glass at the end of the pipe.</p><p>The expansion of the air, by virtue of its elastic property, when only the compressing force is taken off, or diminished, is found to be surprisingly great; and yet we are far from knowing the utmost dilatation of which it is capable. In several experiments made by Mr. Boyle, it expanded first into 9 times its former space; then into 31 times; then into 60, and then into 150 times. Afterwards, it was brought to dilate into <cb/> 8000 times its first space; then into 10000, and at last even into 13679 times its space; and this solely by its own natural expansive force, by only removing the pressure, but without the help of fire. And on this principle depends the construction and use of the M<hi rend="smallcaps">ANOMETER.</hi></p><p>The elasticity of the air, under one and the same pressure, is still farther increased by heat, and diminished by cold, and that, by some late accurate experiments made by Sir George Shuckburgh, at the rate of the 440th part of its volume nearly, for each degree of the variation of heat, from that of temperate, in Fahrenheit's thermometer.</p><p>Mr. Hauksbee observed that a portion of air inclosed in a glass tube, when the temperature was at the freezing point, formed a volume which was to that of the same quantity of air in the greatest heat of summer here in England, as 6 to 7. And it has been found by several experiments, that air is expanded 1/3 of its natural bulk by applying the heat of boiling water to it.</p><p>Dr. Hales found that the air in a retort, when the bottom of the vessel just became red hot, was dilated into twice its former space; and that in a white, or almost melting heat, it filled thrice its former space: but Mr. Robins found that air was expanded, by means of the white or fusing heat of iron, to 4 times its former bulk.</p><p>See several ingenious experiments on the elasticity of the air, in the Philos. Trans. for the year 1777, by Sir George Shuckburgh and Colonel Roy.</p><p>This properly explains the common effect observed on bringing a close flaccid bladder near the fire to warm it; when it is presently found to swell as if more air were blown into it. And upon this principle depends the structure and office of the thermometer; as also the air balloons, lately invented by Mr. Montgolfier, for floating in the atmosphere.</p><p>M. Amontons first discovered that, with the same degree of heat, air will expand in a degree proportioned to its density. And on this foundation the ingenious author has formed a discourse, to prove “that the spring and weight of the air, with a moderate degree of warmth, may enable it to produce even earthquakes, and others of the most vehement commotions of nature.” He computes that at the depth of the 74th part of the earth's radius below the surface, the natural pressure of the air would reduce to the density of gold; and thence infers that all matter below that depth, is probably heavier than the heaviest metal that we know of. And hence again, as it is proved that the more the air is compressed, the more does the same degree of fire increase the force of its elasticity; we may infer that a degree of heat, which in our orb can produee only a moderate effect, may have a very violent one in such lower orb; and that, as there are many degrees of heat in nature, beyond that of boiling water, it is probable there may be some whose violence, thus assisted by the weight of the air, may be sufficiently powerful to tear asunder the solid globe. Mem. de l'Acad. 1703.</p><p>Many philosophers have supposed that the elastic property of the air depends on the figure of its corpuscles, which they take to be ramous: some maintain <pb n="53"/><cb/> that they are so many minute <hi rend="italics">flocculi,</hi> resembling fleeces of wool: others conceive them rolled up like hoops, and curled like wires, or shavings of wood, or coiled like the springs of watches, and endeavouring to expand themselves by virtue of their texture.</p><p>But Sir Isaac Newton (Optics, Qu. 31, &c.) explains the matter in a different way; such a contexture of parts he thinks by no means sufficient to account for that amazing power of elasticity observed in air, which is capable of dilating itself into above a million of times more space than it occupied before: but, he observes, as it is known that all bodies have an attractive and a repelling power; and as both these are stronger in bodies, the denser, more compact, and solid they are; hence it follows that when, by heat, or any other powerful agent, the attractive force is overcome, and the particles of the body separated so far as to be out of the sphere of attraction; the repelling power, then commencing, makes them recede from each other with a strong force, proportionable to that with which they before cohered; and thus they become permanent air.</p><p>And hence, he says, it is, that as the particles of air are grosser, and rise from denser bodies, than those of transient air, or vapour, true air is more ponderous than vapour, and a moist atmosphere lighter than a dry one.</p><p>And M. Amontons makes the elasticity of air to arise from the fire it contains; so that by augmenting the degree of heat, the rarefaction will be increased to a far greater degree than by a mere spontaneous dilatation.</p><p>The elastic power of the air becomes the second great source of the remarkable effects of this important fluid. By this property it insinuates itself into the pores of bodies, where, by means of this virtue of expanding, which is so easily excited, it must put the particles of those bodies into perpetual vibrations, and maintain a continual motion of dilatation and contraction in all bodies, by the incessant changes in its gravity and density, and consequently its elasticity and expansion.</p><p>This reciprocation is observable in several instances, particularly in plants, in which the tracheæ or air-vessels perform the office of lungs; for as the heat increases or diminishes, the air alternately dilates and contracts, and so by turns compresses the vessels, and eases them again; thus promoting a circulation of their juices. And hence it is found that no vegetation or germination is carried on in vacuo.</p><p>It is from the same cause too, that ice is burst by the continual action of the air contained in its bubbles. Thus, too, glasses and other vessels are frequently cracked, when their contained liquors are frozen; and thus also large blocks of stone, and entire columns of marble, sometimes split in the winter season, from some little bubble of included air acquiring an increased elasticity: and for the same reason it is that so few stones will bear to be heated by a fire, without cracking into many pieces, by the increased expansive force of some air confined within their pores. From the same source arise also all putrefaction and fermentation; neither of which can be carried on in vacuo, even in the best disposed subjects. And even respiration, and animal-life itself, are supposed, by many authors, to be <cb/> conducted, in a great measure, by the same principle of the air. And as we find such great quantities of air generated by the solution of animal and vegetable substances, a good deal must constantly be raised from the dissolution of these clements in the stomach and bowels.</p><p>In fact, all natural corruption and alteration seem to depend on air; and even metals, particularly gold, only seem to be durable and incorruptible, in so far as they are impervious to air.</p><p>As to the different kinds of air, with its generation, and the effects of different ingredients of it, &c, they are omitted here, as properly belonging to a Chemical Dictionary, or to a General Dictionary of Arts, &c.</p><p>For the resistance of the air, see <hi rend="smallcaps">Resistance.</hi></p><p><hi rend="smallcaps">Air-Gun</hi>, in <hi rend="smallcaps">Pneumatics</hi>, is a machine for propelling bulleto with great violence, by the sole means of condensed air.</p><p>The first account we meet with of an air-gun, is in the <hi rend="italics">Elemens d'Artillerie</hi> of David Rivaut, who was preceptor to Louis XIII. of France. He ascribes the invention to one Marin, a burgher of Lisieux, who presented one to Henry IV.</p><p>To construct a machine of this kind, it is only necessary to take a strong vessel of any sort, into which the air is to be thrown or condensed by means of a syringe, or otherwise, the more the better; then a valve is suddenly opened, which lets the air escape by a small tube in which a bullet is placed, and which is thus violently forced out before the air.</p><p>It is evident then that the effect is produced by virtue of the elastic property of the air; the force of which, as has been shewn in the last article, is directly proportional to its condensation; and therefore the greater quantity that can be forced into the engine, the greater will be the effect. Now this effect will be exactly similar to that of a gun charged with powder, and therefore we can easily form a comparison between them: for inflamed gun-powder is nothing more than very condensed elastic air; so that the two forces are exactly similar. Now it is shewn by Mr. Robins, in his New Principles of Gunnery, that the fluid of inflamed gun-powder, has, at the first moment, a force of elasticity equal to about a 1000 times that of common air; and therefore it is necessary that air should be condensed a 1000 times more than in its natural state, to produce the same effect as gun-powder. But then it is to be considered, that the velocities with which equal balls are impelled, are directly proportional to the square roots of the forces; so that if the air in an air-gun be condensed only 10 times, then the velocity it will project a ball with, will be, by that rule, (1/10)th of that arising from gun-powder; and if the air were condensed 20 times, it would communicate a velocity of 1/7 of that of gun-powder. But in reality the air-gun shoots its ball with a much greater proportion of velocity than as above, and for this reason, namely, that as the reservoir, or magazine of condensed air, is commonly very large in proportion to the tube which contains the ball, its density is very little altered by expanding through that narrow tube, and consequently the ball is urged all the way by nearly the same uniform force as at the first instant; whereas the elastic fluid arising from inflamed gun-powder is but very small in <pb n="54"/><cb/> proportion to the tube or barrel of the gun, occupying at first indeed but a very small portion of it next the but-end: an dtherefore by dilating into a comparatively large space, as it urges the ball along the barrel, its elastic force is proportionally weakened, and it acts always less and less on the ball in the tube. From which cause it happens, that air condensed into a good large machine only 10 times, will shoot its ball with a velocity but little inferior to that given by the gunpowder. And if the valve of communication be suddenly shut again by a spring, after opening it to let some air escape, then the same collection of it may serve to impel many balls, one after another.</p><p>In all cases in which a considerable force is required, and consequently a great condensation of air, it will be requisite to have the condensing syringe of a small bore, perhaps not more than half an inch in diameter: otherwise the force to produce the compression will become so great, that the operator cannot work the machine: for, as the pressure against every square inch is about 15 pounds, and against every circular inch about 12 pounds, if the syringe be one inch in diameter, when one atmosphere is injected, there will be a resistance of 12 pounds against the piston; when 2, of 24 pounds; and when 10 are injected, there will be a force of 120 pounds to overcome; whereas 10 atmospheres act against the half-inch piston, whose area is but 1/4 of the former, with 1/4 of the force only, namely, 30 pounds; and 40 at mospheres may be injected with such a syringe, as well as ten with the larger.</p><p>There are air-guns of various constructions; an easy and portable one is represented in Plate II, fig. 1. which is a section lengthways through the axis, to shew the inside. It is made ofbrass, and has two barrels; the inner barrel D A of a small bore, from which the bullets are shot; and a larger barrel ESCDR, on the outside of it. In the stock of the gun there is a syringe MNPS, whose rod M draws out to take in air; and by pushing it in again, the piston SN drives the air before it, through the valve PE into the cavity between the two barrels. The ball K is put down into its place in the small barrel, with the rammer, as in another gun. There is another valve at SL, which, being opened by the trigger O, permits the air to come behind the ball, so as to drive it out with great force. If this valve be opened and shut suddenly, one charge of condensed air may make several discharges of bullets; because only part of the injected air will then go out at a time, and another bullet may be put into the place K: but if the whole air be discharged on a single bullet, it will impel it more forcibly. This discharge is effected by means of a lock (fig. 2) when fixed to its place as usual in other guns; for the trigger being pulled, the cock will go down and drive a lever which opens the valve.</p><p>Dr. Macbride (Exper. Ess. p. 81) mentions an improvement of the air-gun, made by Dr. Ellis; in which the chamber sor containing the condensed air is not in the stock, which renders the machine heavy and unweildy, but has five or six hollow spheres belonging to it, of about 3 inches diameter, sitted to a screw on the lock of the gun. These spheres are contrived with valves, to consine the air which is forced into their cavities, so that a servant may carry them ready charged with condensed air: and thus the gun of this <cb/> construction is rendered as light and portable as one of the smallest fowling.pieces.</p><p>Fig. 3 represents one made by the late Mr. B. Martin of London, and now by several of the mathematical instrument and gun-makers of the metropolis; which, for simplicity and perfection, perhaps exceeds any other that has been contrived. A is the gun-barrel, of the size and weight of a common fowling-piece, with the lock, stock, and ramrod. Under the lock, at <hi rend="italics">b,</hi> is a round steel tube, having a small moveable pin in the inside, which is pushed out when the trigger <hi rend="italics">a</hi> is pulled, by the springwork within the lock; to this tube <hi rend="italics">b</hi> is serewed a hollow copper ball, perfectly airtight. This copper ball is fully charged with condensed air by means of a syringe, previous to its being applied to the tube <hi rend="italics">b.</hi> Hence, if a bullet be rammed down in the barrel, the copper ball screwed fast at <hi rend="italics">b,</hi> and the trigger <hi rend="italics">a</hi> be pulled; then the pin in <hi rend="italics">b</hi> will forcibly push open a valve within the copper ball, and let out a portion of the condensed air; which air will rush up through the aperture of the lock, and forcibly act against the bullet, driving it to the distance of 60 or 70 yards, or farther. If the air be strongly condensed at every discharge, only a portion of the air escapes from the ball; therefore, by re-cocking the piece, another discharge may be made; and this repeated 15 or 16 times. An additional barrel is sometimes made, and applied for the discharge of shot, instead of the ball above described.</p><p>Sometimes the syringe is applied to the end of the barrel C (fig. 4); the lock and trigger shut up in a brass case <hi rend="italics">d</hi>; and the trigger pulled, or the discharge made, by pulling the chain <hi rend="italics">b.</hi> In this contrivance there is a round chamber for the condensed air at the end of the spring at <hi rend="italics">e,</hi> and it has a valve acting in a similar manner to that of the copper ball. When this instrument is not in use, the brass case <hi rend="italics">d</hi> is made to slide off, and the instrument then becomes a walking stick: from which circumstance, and the barrel being made of cane, or brass, &c, it has been called the <hi rend="italics">Air-cane.</hi> The head of the cane unscrews and takes off at <hi rend="italics">a,</hi> where the extremity of the piston-rod in the barrel is shewn. An iron rod is placed in a ring at the end of this, and the air is condensed in the barrel in a manner similar to that of the gun as above; but its force and action is not near so strong as in the gun.</p><p><hi rend="smallcaps">Magazine Air</hi>-<hi rend="italics">Gun.</hi> This is an improvement of the common air-gun, made by an ingenious artist, called L. Colbe. By his contrivance, ten bullets are so lodged in a cavity, near the place of discharge, that they may be successively drawn into the barrel, and shot so quickly as to be nearly of the same use as so many different guns; the only motion required, after the air has been injected, being that of shutting and opening the hammer, and cocking and pulling the trigger. Fig. 3 is a longitudinal section of this gun, as large in every part as the gun itself; and as much of its length is shewn as is peculiar to this construction; the rest of it being like the ordinary air-gun. EE is part of the stock; G is the end of the injecting syringe, with its valve H, opening into the cavity FFFF between the barrels, KK is the small or shooting barrel, which receives the bullets, one at a time, from the magazine DE, being a serpentine cavity, in which the bullets <hi rend="italics">b, b, b,</hi> &c, are <pb/><pb/><pb/><pb/><pb n="55"/><cb/> lodged, and closed at the end D; from whence, by one motion of the hammer, they are brought into the barrel at I, and thence are shot out by the opening of the valve V, which lets in the condensed air from the cavity FFF into the channel VKI, and so along the inner barrel KKK, whence the bullet is discharged. <hi rend="italics">s</hi> I <hi rend="italics">si</hi> M <hi rend="italics">k</hi> is the key of a cook, having a hole through it; which hole, in the present situation, makes part of the barrel KK, being just of the same bore: so that the air, which is let in at every opening of the valve V, comes behind this cock, and taking the ball out of it, carries it forward, and so out of the mouth of the piece.</p><p>To bring in another bullet to succeed I, which is done in an instant, bring the cylindrical cavity of the key of the cock, which made part of the barrel KKK, into the situation <hi rend="italics">ik,</hi> so that the part I may be at K; then turning the gun upside-down, one bullet next the cock will fall into it out of the magazine, but will go no farther into this cylindrical cavity, than the two little pieces <hi rend="italics">ss</hi> will permit it; by which means only one bullet at a time will be taken in to the place I, to be discharged again as before.</p><p>A more particular description of the several parts may be seen in Desaguliers' Exper. Philos. vol. ii. pa. 399 et seq.</p><p><hi rend="smallcaps">Air-Pump</hi>, in <hi rend="italics">Pneumatics,</hi> is a machine for exhausting the air out of a proper vessel, and so to make what is commonly called a vacuum; though in reality the air in the receiver is only rarefied to a great degree, so as to take off the ordinary effects of the atmosphere. So that by this machine we learn, in some measure, what our earth would be without air; and how much all vital, generative, nutritive, and alterative powers depend upon it.</p><p>The principle on which the air-pump is constructed, is the spring or elasticity of the air; as that on which the common, or water pump is formed, is the gravity of the same air: the one gradually exhausting the air from a vessel by means of a piston, with a proper valve, working in a cylindrical barrel or tube; and the other exhausting water in a similar manner.</p><p>The air-pump has proved one of the principal means of performing philosophical discoveries, that has been invented by the moderns. The idea of such a machine occurred to several persons, nearly about the same time. But the first it seems was completed by Otto Guericke, the celebrated consul of Magdeburg, who exhibited his first public experiments with it, before the emperor and the states of Germany, at the breaking up of the imperial diet at Ratisbon, in the year 1654. But it was not till the year 1672 that Guericke published a description of the instrument, with an account of his experiments, in his Experimenta Nova Magdeburgica de Vacuo Spacia: though an account of them had been published by Schottus in 1657, in his Mechanica Hydraulico Pneumatica.</p><p>Dr. Hook and M. Duhamel ascribe the invention of the air-pump to Mr. Boyle. But that great man frankly confesses that Guericke was beforehand with him in the execution. Some attempts, he assures us, he had indeed made upon the same foundation, before he knew any thing of what had been done abroad: but the information he afterwards received from the account given <cb/> by Schottus, enabled him, with the assistance of Dr. Hook, aster two or three unsuccessful trials, to bring his design to maturity. The product of their labours was a new air-pump, much more easy; convenient, and manageable, than the German one. And hence, or rather from the great variety of experiments to which this illustrious author applied the machine, it was afterwards called <hi rend="italics">Machina Boyliana,</hi> and the vacuum produced by it, <hi rend="italics">Vacuum Boylianum.</hi></p><p><hi rend="italics">Structure of the Air-Pump.</hi> Most of the air-pumps that were first made, consisted of only one barrel, or hollow cylinder of brass, with a valve at the bottom, opening inwards; and a moveable embolus or piston, having likewise a valve opening upwards, and so exactly fitted to the barrel, that when it is drawn up from the bottom, by means of an indented iron rod or rack, and a handle turning a small indented wheel, playing in the teeth of that rod, all the air will be drawn up from the cavity of the barrel: there is also a small pipe opening into the bottom of the barrel, by means of which it communicates with any proper vessel to be exhausted of air, which is called a receiver, from its office in receiving the subjects upon which experiments are to be made in vacuo: the whole being fixed in a convenient frame of wood-work, where the end of the pipe turns up into a horizontal plate, upon which the receiver is placed, just over that end of the pipe.</p><p>The other parts of the machine, being only accidental circumstances, chiefly respecting conveniency, have been diversified and improved from time to time, according to the address and several views of the makers. That of Otto Guericke was very rude and inconvenient, requiring the labour of two strong men, for more than two hours, to extract the air from a glass, which was also placed under water; and yet allowed of no change of subjects for experiments.</p><p>Mr. Boyle, from time to time, removed several of these inconveniences, and lessened others: but still the working of his pump, which had but one barrel, was laborious, by reason of the pressure of the atmosphere, a great part of which was to be removed at every lift of the piston, when the exhaustion was nearly completed. Various improvements were successively made in the machine by the philosophers about that time, and foon after, who cultivated this new and important branch of pneumatics; as Papin, Mersenne, Mariotte, and others; but still they laboured under a difficulty of working them, from the circumstance of the single barrel, till Papin, in his farther improvements of the air-pump, removed that inconvenience, by the use of a second barrel and piston, contrived to rise, as the other fell, and to sall as that rose; by which, and the great improvements made by Mr. Hauksbee, the pressure of the atmosphere on the descending piston, always nearly balanced that of the ascending one; so that the winch, which worked them up and down, was easily moved by a very gentle force with one hand; and besides, the exhaustion was hereby made in less than half the time.</p><p>Some of the Germans, and others likewise, made improvements in the air-pump, and contrived it to perform the counter office of a condenser, in order to examine the properties of the air depending on its condensation. <pb n="56"/><cb/></p><p>Mr. Boyle contrived a mercurial gauge or index to the air-pump, which is described in his first and second Physico-Mechanical Continuations, for measuring the degrees of the air's rarefaction in the receiver. This gauge is similar to the barometer, being a long glass tube, having its lower end immersed in an open bason of quicksilver, but its other end, which was open also, communicating with the receiver: which being exhausted, this tube is equally exhausted of air at the same time, and the external air presses the quicksilver up into the tube, to a height proportioned to the degree of exhaustion.</p><p>Mr. Vream, an ingenious pneumatic operator, made an improvement in Hauksbee's air-pump, by reducing the alternate up-and-down motion of the hand and winch to a circular one. In his method, the winch is turned quite round, and yet the pistons are alternately raised and depressed: by which the trouble of shifting the hand backwards and forwards, as well as the loss of time, and the shaking of the pump, are prevented.</p><p>The air-pump, thus improved, is represented in plate III. fig. 1; where <hi rend="italics">oo</hi> is the receiver to be exhausted, ground truly level at the bottom, set over a hole in the plate, from which descends the bent pipe <hi rend="italics">hh</hi> to the cistern <hi rend="italics">dd,</hi> with which the two barrels <hi rend="italics">aa</hi> communicate, in which the pistons are worked by a toothed wheel, by turning the handle <hi rend="italics">bb</hi>; by which the racks <hi rend="italics">cc,</hi> with the pistons, are worked alternately up and down. <hi rend="italics">ll</hi> is the gauge tube, immersed in a bason of quicksilver <hi rend="italics">m</hi> at bottom, and communicating with the receiver at top; from which however it may be occasionally disengaged, by turning a cock. And <hi rend="italics">n</hi> is another cock, by turning of which, the air is again let in to the exhausted receiver; into which it is heard to rush with a considerable hissing noise.</p><p>Notwithstanding the great excellency of Mr. Hauksbee's air-pump, it was still subject to inconveniences, from which it was in a great measure relieved by some contrivances of Mr. Smeaton, which are described at large in the Philos. Trans. for the year 1752. The principal improvements suggested by Mr. Smeaton, relate to the gauge, the valves of the piston, and the piston going closer down to the bottom of the barrel; for his pump has only one. By the last of these, the air was extracted more perfectly at each stroke. By the second, he remedied an inconvenience arising from the valve hole of the piston being too wide properly to support the bladder valve which covered it: instead of the usual circular orisice, Mr. Smeaton perforated the piston with seven small and equal hexagonal holes, one in the centre, and the other six around, forming together the appearance of a transverse section of a honeycomb; the bars or divisions between which, served to support the pressure of the air on the valve. His gage consists of a bulb of glass, of a pear-like shape, and capable of holding about half a pound of quicksilver: it is open at the lower end, the other terminating in a tube hermetically sealed; and it has annexed to it a seale, divided into parts of about 1/10 of an inch, and answering to the 1000th part of the whole capacity. During the exhaustion of the receiver, the gage is suspended in it by a wire; but when the pump has been worked as much as necessary, the gage is pushed down, till the open end be immersed in a bason of quicksilver <cb/> placed underneath. The air is then let into the receiver again, and the quicksilver driven by it from the bason, up into the gauge, till the air remaining in it become of the same density as the air without; and as the air always takes the highest place, the tube being uppermost, the expansion will be determined by the number of divisions occupied by the air at the top. This airpump is made to act also as a condensing engine, as some German machines had done before, by the very simple apparatus of turning a cock.</p><p>By means of this gauge, Mr. Smeaton judged that his machine was incomparably better than any former ones, as it seemed to rarefy the air in the receiver 1000, or even 2000 times, while the best of the former construction only rarefied about 140 times: and so the case has since been always understood, an implicit considence being placed in Mr. Smeaton's accuracy, till the fallacy was accidentally detected in the manner related at large by Mr. Nairne in the Philos. Trans. for the year 1777. This accurate and ingenious artist wanting to make trial of Mr. Smeaton's pear-gauge, executed an air-pump of his improved construction, in the best manner possible; which, in various experiments made with it, appeared, by the pear-gauge, to rarefy the air to an amazing degree indeed, being at times from 4000 to 10000, or 50000, or even 100000 times rarefied. But upon measuring the same expansion by the usual long and short tube gauges, which both accurately agreed together, he found that these never shewed a rarefaction of more than 600 times: widely different from the same as measured by the pear or internal gauge, by experiments often repeated. ‘Finding, says Mr. Nairne, still this disagreement between the pear-gauge and the other gauges, I tried a variety of experiments; but none of them appeared to me satisfactory, till one day in April 1776, shewing an experiment with one of these pumps to the honourable Henry Cavendish, Mr. Smeaton, and several other gentlemen of the Royal Society, when the two gauges differed some thousand times from one another, Mr. Cavendish accounted for it in the following manner. “It appeared, he said, from some experiments of his father's, Lord Cavendish, that water, whenever the pressure of the atmosphere on it is diminished to a certain degree, is immediately turned into vapour, and is as immediately turned back again into water on reftoring the pressure. This degree of pressure is different according to the heat of the water: when the heat is 72° of Fahrenhest's scale, it turns into vapour as soon as the pressure is no greater than that of three quarters of an inch of quicksilver, or about 1-40th of the usual pressure of the atmosphere; but when the heat is only 41°, the pressure must be reduced to that of a quarter of an inch of quicksilver before the water turns into vapour. It is true, that water exposed to the open air, will evaporate at any heat, and with any pressure of the atmosphere; but that evaporation is intirely owing to the action of the air upon it; whereas the evaporation here spoken of, is performed without any assistance from the air. Hence it follows, that when the receiver is exhausted to the above-mentioned degree, the moisture adhering to the different parts of the machine will turn into vapour, and supply the place of the air, which is continually drawn away by the working of <pb n="57"/><cb/> the pump; so that the fluid in the pear-gauge, as well as that in the receiver, will consist in a good measure of vapour. Now letting the air into the receiver, all the vapour within the pear-gauge will be reduced to water, and only the real air will remain uncondensed; consequently the pear-gauge shews only how much real air is left in the receiver, and not how much the pressure or spring of the included fluid is diminished; whereas the common gauges shew how much the pressure of the included fluid is diminished, and that equally, whether it consist of air or of vapour.” Mr. Cavendish having explained so satisfactorily the cause of the disagreement between the two gauges, Mr. Nairne considered that, if he were to avoid moisture as much as possible, the two gauges should nearly agree. And in fact they were found so to do, each shewing a rarefaction of about 600, when all moisture was perfectly cleared away from the pump, and the plate and the edges of the receiver were secured by a cement instead of setting it upon a soaked leather, as in the usual way. But by future experiments, Mr. Nairne found that the same excellent machine would not exhaust more than 50 or 60 times, when the receiver was set upon leather soaked in water, the heat of the room being about 57°. And from the whole, Mr. Nairne concludes that the air-pump of Otto Guericke, and those contrived by Mr. Gratorix, and Dr. Hook, and the improved one by Mr. Papin, both used by Mr. Boyle, as also Hauksbee's, s'Gravesande's, Muschenbroeck's, and those of all who have used water in the barrels of their pumps, could never have exhausted to more than between 40 and 50, if the heat of the place was about 57; and although Mr. Smeaton, with his pump, where no water was in the barrel, but where leather soaked in a mixture of water and spirit of wine was used on the pump-plate, to set the receiver upon, may have exhausted all but a thousandth, or even a tenthousandth part of the common air, according to the testimony of his pear-gauge; yet so much vapour must have arisen from the wet leather, that the contents of the receiver could never be less than a 70th or 80th part of the density of the atmosphere. But when nothing of moisture is used about this machine, it will, when in its greatest perfection, rarefy its contents of air about 600 times.</p><p>It is evident that by means of these two gauges we can ascertain the several quantities of vapour and permanent air which make up the contents of the receiver, after the exhaustion is made as perfect as can be; for the usual external gauge determines the whole contents, made up of the vapour and air, whilst the pear-gauge shews the quantity of real permanent air; consequently the difference is the quantity of vapour.</p><p>The principal cause which prevents this pump from exhausting beyond the limit above-mentioned, is the weakened elasticity of the air within the receiver, which, decreasing in proportion as the quantity of the air within is diminished, becomes at last incapable of lifting up the valve of communication between the receiver and the barrel; and consequently no more air can then pass from the former to the latter.</p><p>Several ingenious persons have used their endeavours to remove this imperfection in the best air-pumps. Amongst these it seems that one Mr. Haas has succeeded tolerably well; having, by means of a contri- <cb/> vance to open the communication valve in the bottom of the barrel, made his machine so perfect, that when every thing is in the greatest perfection, it rarefies the contents of the receiver as far as 1000 times, even when measured by the exterior gauge. The description of this machine, and an account of some experiments performed with it, are given by Mr. Cavallo in the Philos. Trans. for the year 1783.</p><p>But the imperfections it seems have more recently been removed by an ingenious contrivance of Mr. Cuthbertson, a mathematical instrument maker at Amsterdam, now of London, whose air-pump has neither cocks nor valves, and is so constructed, that what supplies their place has the advantages of both, without the inconveniences of either. He has also made improvements in the gauges, by means of which he determines the height of the mercury in the tube, by which the degree of exhaustion is indicated, to the hundredth part of an inch. And to obviate the inconvenience of the elastic vapour arising from the wet leather, upon which the receiver is placed, for common experiments, he recommends the use of leather dressed with allum, and soaked in hog's lard, which he found to yield very little of this vapour; but when the utmost degree of exhaustion is required, his advice is, to dry the receiver well, and set it upon the plate without any leather, only smearing its outer edges with hog's lard, or with a mixture of three parts of hog's lard and one of oil. But the use of the leather has long been laid aside by our English instrument-makers, a circumstance which probably had not come to Mr. Cuthbertson's knowledge. An account of this instrument, and of some experiments performed with it, was published at Amsterdam in the year 1787; from which experiments it appears that, by a coincidence of the several gauges, a rarefaction of 1200 times was shewn; but when the atmosphere was very dry, the exhaustion has been so complete, that the gauges have shewn the air in the receiver to be rarefied above 2400 times.</p><p>There are made also by different persons, portable, or small air-pumps, of various constructions, to set upon a table, to perform experiments with. In these, the gauge is varied according to the fancy of the maker, but commonly it consists of a bent glass tube, like a syphon, open only at one end. The gauge is placed under a small receiver communicating, by a pipe, with the principal pipe leading from the general receiver to the barrels. The close end of the gauge, of 3 or 4 inches long, before the exhaustion, has the quicksilver forced close up to the top by the pressure of the air on the open end; but when the exhaustion is considerably advanced, it begins to descend, and then the difference of the heights of the quicksilver in the two legs, compared with the height in the barometrical tube, determines the degree of exhaustion: so if the difference between the two be one inch, when the barometer stands at 30, the air is rarefied 30 times; but if the difference be only half an inch, the rarefaction is 60 times, and so on. See Plate 111. fig. 2.</p><p><hi rend="italics">The Use of the Air-Pump.</hi> In whatever manner or form this machine be made, the use and operation of it are always the same. The handle, which works the piston, is moved up and down in the barrel, by which <pb n="58"/><cb/> means a barrel of the contained air is drawn out at every stroke of the piston, in the following manner: by pushing the piston down to the bottom of the barrel, where the air is prevented from escaping downwards, by its elasticity it opens the valve of the piston, and escapes upwards above it into the open air; then raising the piston up, the external atmosphere shuts down its valve, and a vacuum would be made below it, but for the air in the receiver, pipe, &c, which now raises the valve in the bottom of the barrel, and rushes in and fills it again, till the whole air in the receiver and barrel be of one uniform density, but less than it was before the stroke, in proportion as the sum of all the capacities of the receiver, pipe, and barrel together, is to the same sum wanting the barrel. And thus is the air in the receiver diminished at each stroke of the piston, by the quantity of the barrel or cylinder full, and therefore always in the same proportion: so that by thus repeating the operation again and again, the air is rarefied to any proposed degree, or till it has not elasticity enough to open the valve of the piston or of the barrel, after which the exhaustion cannot be any farther carried on: the gauge, in comparison with the barometer, shewing at any time what the degree of exhaustion is, according to the particular nature and construction of it.</p><p>But, supposing no vapour from moisture, &c, to rise in the receiver, the degree of exhaustion, after any number of strokes of the piston, may be determined by knowing the respective capacities of the barrel and the receiver, including the pipe, &c. For as we have seen above that every stroke diminishes the density in a constant proportion, namely as much as the whole content exceeds that of the cylinder or barrel; and consequently the sum of as many diminutions as there are strokes of the piston, will shew the whole diminution by all the strokes. So, if the capacity of the barrel be equal to that of the receiver, in which the communication pipe is always to be included; then, the barrel being half the sum of the whole contents, half the air will be drawn out at one stroke; and consequently the remaining half, being dilated through the whole or first capacity, will be of only half the density of the first: in like manner, after the second stroke, the density of the remaining contents will be only half of that after the sirst stroke, that is only 1/4 of the original density: continuing this operation, it follows that the density of the remaining air will be 1/8 after 3 strokes of the piston, 1/16 after 4 strokes, 1/32 after 5 strokes, and so on, according to the powers of the ratio 1/2; that is, such power of the ratio as is denoted by the number of the strokes. In like manner, if the barrel be 1/3 of the whole contents, that is, the receiver double of the barrel, or 2/3 of the whole contents; then the ratio of diminution of density being 2/3, the density of the contents, after any number of strokes of the piston, will be denoted by such power of 2/3 whose exponent is that number; namely, the density will be 2/3 after one stroke, (2/3)<hi rend="sup">2</hi> or 4/9 after two strokes, (2/3)<hi rend="sup">3</hi> or 8/27 after 3 strokes, and in general it will be (2/3)<hi rend="sup">n</hi> after <hi rend="italics">n</hi> strokes: the original density of the air being 1. Hence then, universally, if <hi rend="italics">s</hi> denote the sum of the contents of the receiver and barrel, and <hi rend="italics">r</hi> that of the receiver only without the barrel, and <hi rend="italics">n</hi> any number of strokes of the piston; then, the original density of the air being 1, <cb/> the density after <hi rend="italics">n</hi> strokes will be (<hi rend="italics">r/s</hi>)<hi rend="sup">n</hi> o <hi rend="italics">r</hi><hi rend="sup">n</hi>/<hi rend="italics">s</hi><hi rend="sup">n</hi>, namely the <hi rend="italics">n</hi> power of the ratio <hi rend="italics">r/s</hi>. So, for example, if the capacity of the receiver be equal to 4 times that of the barrel; then their sum <hi rend="italics">s</hi> is 5, and <hi rend="italics">r</hi> is 4; and the density of the contents after 30 strokes, will be (4/5)<hi rend="sup">30</hi>, or the 30th power of 4/5, which is 1/808 nearly; so that the air in the receiver is raresied 808 times.</p><p>See also the Memoires de l'Acad. Royale des Sciences for the years 1693 and 1705.</p><p>From the same formula, namely (<hi rend="italics">r/s</hi>)<hi rend="sup">n</hi> = <hi rend="italics">d</hi> the density, we easily derive a rule for finding the number of strokes of the piston, necessary to rarefy the air any number of times, or to reduce it to a given density <hi rend="italics">d,</hi> that of the natural air being 1. For since ((<hi rend="italics">r/s</hi>)<hi rend="sup">n</hi> = <hi rend="italics">d,</hi> by taking the logarithm of this equation, it is <hi rend="italics">n</hi> X log. <hi rend="italics">r/s</hi> = log. of <hi rend="italics">d;</hi> and hence ; that is, divide the log. of the proposed density by the log. of the ratio of the receiver to the sum of the receiver and barrel together, and the quotient will shew the number of strokes of the piston requisite to produce the degree of exhaustion required. So, for example, if the receiver be equal to 5 times the barrel, and it be proposed to find how many strokes of the piston will rarefy the air 100 times; then <hi rend="italics">r</hi> = 5, <hi rend="italics">s</hi> = 6, <hi rend="italics">d</hi> = 1/100, whose log. is - 2, and <hi rend="italics">r/s</hi>=5/6, whose log. is - .07918; therefore 2/.07918 = 25 1/4 nearly, which is the number of strokes required.</p><p>And, farther, the same formula reduced, would give us the proportion between the receiver and barrel, when the air is rarefied to any degree by an assigned number of strokes of the piston. For since the density, therefore, extracting the <hi rend="italics">n</hi> root of both sides, it is : that is, the <hi rend="italics">n</hi> root of the density is equal to the ratio of the receiver to the sum of the receiver and barrel. So, if the density <hi rend="italics">d</hi> be 1/128, and the number of strokes <hi rend="italics">n</hi> = 7; then the 7th root of 1/128 is 1/2; which shews that the receiver is equal to half the receiver and barrel together, or that the capacity of the barrel is just equal to that of the receiver.</p><p>Some of the principal effects and phenomena of the air-pump, are the following: That, in the exhausted receiver, heavy and light bodies fall equally swift; so, a guinea and feather fall from the top of a tall receiver to the bottom exactly together. That most animals die in a minute or two: but however, That vipers and frogs, though they swell much, live an hour or two; and after being seemingly quite dead, come to life again in the open air: That snails survive about ten hours; efts, or slow-worms, two or three days; and leeches five or six. That oysters live for 24 hours. That the heart of an eel taken out of the body, continues to <pb n="59"/><cb/> beat for good part of an hour, and that more briskly than in the air. That warm blood, milk, gall, &c, undergo a considerable intumescence and ebullition. That a mouse or other animal may be brought, by degrees, to survive longer in a rarefied air, than naturally it does. That air may retain its usual pressure, after it is become unfit for respiration. That the eggs of silk-worms hatch in vacuo. That vegetation stops. That fire extinguishes; the flame of a candle usually going out in one minute; and a charcoal in about five minutes. That red-hot iron, however, seems not to be affected; and yet sulphur or gun-powder are not lighted by it, but only fused. That a match, after lying seemingly extinct a long time, revives again on re-admitting the air. That a flint and steel strike sparks of fire as copiously, and in all directions, as in air. That magnets, and magnetic needles, act the same as in air. That the smoke of an extinguished luminary gradually settles to the bottom in a darkish body, leaving the upper part of the receiver clear and transparent; and that on inclining the vessel sometimes to one side, and sometimes to another, the fume preserves its surface horizontal, after the nature of other fluids. That heat may be produced by attrition. That camphire will not take fire; and that gun-powder, though some of the grains of a heap of it be kindled by a burning glass, will not give fire to the contiguous grains. That glow-worms lose their light in proportion as the air is exhausted, and at length become totally obscure; but on re-admitting the air, they presently recover it all. That a bell, on being struck, is not heard to ring, or very faintly. That water freezes. But that a syphon will not run. That electricity appears like the aurora borealis. With multitudes of other curious and important particulars, to be met with in the numerous writings on this machine, namely, besides the Philos. Transactions of most academies and societies, in the writings of Torricelli, Pascal, Mersenne, Guericke, Schottus, Boyle, Hook, Duhamel, Mariotte, Hauksbee, Hales, Muschenbroeck, Gravesande, Desaguliers, Franklin, Cotes, Helsham, and a great many other authors.</p><p><hi rend="smallcaps">Air-Vessel</hi>, in <hi rend="italics">Hydraulics,</hi> is a vessel of air within some water engines, which being compressed, by forcing in a considerable quantity of water, by its uniform spring, forces it out at the pipe in a constant uninterrupted stream, to a great height.</p><p>Air-vessel too, in the improved fire engines, is a metallic cylinder, placed between the two forcing pumps, by the action of whose pistons the water is forced into this vessel, through two pipes, with valves; then the air, previously contained in it, is compressed by the water, in proportion to the quantity admitted, and this air, by its spring, forces the water through a pipe by a constant and equal stream; whereas in the common squirting engine, the stream is discontinued between the several strokes.</p><p>AIRY <hi rend="smallcaps">Triplicity</hi>, in <hi rend="italics">Astrology,</hi> the signs of Gemini, Libra, and Aquarius.</p></div1><div1 part="N" n="AJUTAGE" org="uniform" sample="complete" type="entry"><head>AJUTAGE</head><p>, or <hi rend="smallcaps">Adjutage</hi>, in <hi rend="italics">Hydraulics,</hi> part of the apparatus of a <hi rend="italics">jet d'eau,</hi> or artisicial fountain; being a kind of tube fitted to the aperture or mouth of the cistern, or the pipe; through which the water is to be played in any direction, and in any shape or figure. <cb/></p><p>It is chiefly the diversity in the ajutage, that makes the different kinds of fountains. So that, by having several ajutages, to be applied occasionally, one fountain is made to have the effect of many.</p><p>Mariotte, Gravesande, and Desaguliers have written pretty fully on the nature of ajutages, or spouts for jets d'eau, and especially the former. He affirms, from experiment, that an even polished round hole, made in the thin end of a pipe, gives a higher jet than either a cylindrical or a conical ajutage; but that, of these two latter however, the conical is better than the cylindrical figure. See his Traite du Mouvement des Eaux, part 4.</p><p>The quantity of water discharged by ajutages of equal area, but of different figures, is the same. And for like figures, but of different sizes, the quantity discharged, is directly proportional to the area of the ajutage, or to the square of its diameter, or of any side or other linear dimension: so, an ajutage of a double diameter, or side, will discharge 4 times the quantity of water; of a triple diameter, 9 times the quantity; and so on; supposing them at an equal depth below the surface or head of water. But if the ajutage be at different depths below the head, then the celerity with which the water issues, and consequently the quantity of it run out in any given-time, is directly proportional to the square-root of the altitude of the head, or depth of the hole: so at 4 times the depth, the celerity and quantity is double; at 9 times the depth, triple; and so on.</p><p>It has been found that jets do not rise quite so high as the head of water; owing chiefly to the resistance of the air against it, and the pressure of the upper parts of the jet upon the lower: and for this reason it is, that if the direction of the ajutage be turned a very little from the perpendicular, it is found to spout rather higher than when the jet is exactly upright.</p><p>It is sound by experiment too, that the jet is higher or lower, according to the size of the ajutage: that a circular hole of about an inch and a quarter in diameter, jets highest; and that the farther from that size, the worse. Experience also shews that the pipe leading to the ajutage, should be much larger than it; and if the pipe be along one, that it should be wider the farther it is from the ajutage.</p><p>For the other circumstances relating to jets and the issuing of water under various circumstances, see E<hi rend="smallcaps">XHAUSTION, Flux, Fountain, Jet d'Eau</hi>, &c, to which they more properly belong.</p></div1><div1 part="N" n="ALBATEGNI" org="uniform" sample="complete" type="entry"><head>ALBATEGNI</head><p>, an Arabic prince of Batan in Mesopotamia, who was a celebrated astronomer, about the year of Christ 880, as appears by his observations. He is also called <hi rend="italics">Muhammed ben Geber Albatani, Mahomet the son of Geber,</hi> and <hi rend="italics">Muhamedes Aractensis.</hi> He made astronomical observations at Antioch, and at Racah or Aracta, a town os Chaldea, which some authors call a town of Syria or of Mesopotamia. He is highly spoken of by Dr. Halley, as a <hi rend="italics">vir admirandi acuminis, ac in administrandis observationibus exercitatissimus.</hi></p><p>Finding that the tables of Ptolomy were imperfect, he computed new ones, which were long used as the best among the Arabs: these were adapted to the meridian of Aracta or Racah. Albategni composed in <pb n="60"/><cb/> Arabic a work under the title of <hi rend="italics">The Science of the Stars,</hi> comprising all parts of astronomy, according to his own observations and those of Ptolomy. This work, translated into Latin by Plato of Tibur, was published at Nuremberg in 1537, with some additions and demonstrations of Regiomontanus; and the same was reprinted at Bologna in 1645, with this author's notes. Dr. Halley detected many faults in these editions: Philos. Trans. for 1693, N° 204.</p><p>In this work, Albategni gives the motion of the sun's apogee since Ptolomy's time, as well as the motion of the stars, which he makes 1 degree in 70 years. He made the longitude of the sirst star of Aries to be 18° 2′; and the obliquity of the ecliptic 23° 35′. And upon Albategni's observations were founded the Alphonsine tables of the moon's motions; as is observed by Nic. Muler, in the Tab. Frisicæ, pa. 248.</p><p>ALBERTUS <hi rend="smallcaps">Magnus</hi>, a very learned man in the 13th century, who, among a multitude of books, wrote several upon the various mathematical sciences, as Arithmetic, Geometry, Perspective or Optics, Music, Astrology and Astronomy, particularly under the titles, <hi rend="italics">de sphæra, de astris, de astronomia, item speculum astronomicum.</hi></p><p>Albertus Magnus was born at Lawingen on the Danube, in Suabia, in 1205, or according to some in 1193; and he died at a great age, at Cologn, November 15, 1280. Vossius and other authors speak of him as a great genius, and deeply skilled in all the learning of the age. His writings were so numerous, that they make 21 volumes in folio, in the Lyons edition of 1615. He has passed also for the author of some writings relating to midwifery, &c, under the title of <hi rend="italics">De natura rerum,</hi> and <hi rend="italics">De secretis mulierum,</hi> in which there are many phrases and expressions unavoidable on such a subject, which gave great offence, and raised a clamour against him as the supposed author, and inconsistent with his character, being a Dominican friar, and sometime bishop of Ratisbon; which dignity however he soon resigned, through his love for solitude, to enter again into the monastic life. But the advocates of Albert assert, that he was not the author of either of these two works. It must be acknowledged however, that there are, in his Comment upon the Master of Sentences, some questions concerning the practice of conjugal duty, in which he has used some words rather too gross for chaste and delicate ears: but they allege what he himself used to say in his own vindication, that he came to the knowledge of so many monstrous things at confession, that it was impossible to avoid touching upon such questions. Albert was certainly a man of a most curious and inquisitive turn of mind, which gave rise to other accusations against him; such as, that he laboured to find out the philosopher's stone; that he was a magician; and that he made a machine in the shape of a man, which was an oracle to him, and explained all the difficulties he proposed: the common cant accusations of those times of ignorance and superstition. But having great knowledge in the mathematics and mechanics, by his skill in these sciences he probably formed a head, with springs capable of articulate sounds; like the machines of Boetius and others.</p><p>John Matthæus de Luna, in his treatise De Rerum Inventor<*>bus, has attributed the invention of fire-arms <cb/> to Albert; but in this he is refuted by Naude, in his Apologie des grands hommes.</p></div1><div1 part="N" n="ALBUMAZAR" org="uniform" sample="complete" type="entry"><head>ALBUMAZAR</head><p>, otherwise called <hi rend="smallcaps">Abuassar</hi>, and <hi rend="smallcaps">Japhar</hi>, was a celebrated Arabian philosopher and astrologer, of the 9th or 10th century, or according to some authors much earlier. Blancanus, Vossius, &c, speak of him as one of the most learned astronomers of his time, or astrologers, which was then the same thing. He wrote a work <hi rend="italics">De Magnis Conjunctionibus Annorum Revolutionibus, ac eorum Perfectionibus,</hi> printed at Venice in 1515, at the expence of Melchior Sessa, a work chiefly astrological.</p><p>He wrote also <hi rend="italics">Introductio in Astronomiam,</hi> printed in the year 1489. And it is reported that he observed a comet in his time, above the orb of Venus.</p></div1><div1 part="N" n="ALCOHOL" org="uniform" sample="complete" type="entry"><head>ALCOHOL</head><p>, in the Arabian Astrology, is when a heavy slow-moving planet receives another lighter one within its orb, so as to come in conjunction with it.</p></div1><div1 part="N" n="ALDEBARAN" org="uniform" sample="complete" type="entry"><head>ALDEBARAN</head><p>, an Arabian name of a fixed star, of the first magnitude, just in the eye of the sign or constellation Taurus, or the bull, and hence it is popularly called the bull's eye. For the beginning of the year 1800, its <table><row role="data"><cell cols="1" rows="1" role="data">Right Ascension is</cell><cell cols="1" rows="1" role="data">—</cell><cell cols="1" rows="1" role="data">66°</cell><cell cols="1" rows="1" role="data">6′</cell><cell cols="1" rows="1" role="data">51″</cell><cell cols="1" rows="1" role="data">.10</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Annual variation in AR</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">.31</cell></row><row role="data"><cell cols="1" rows="1" role="data">Declination</cell><cell cols="1" rows="1" role="data">—</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">.00 N.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">And Annual variat. in Decl.</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">.30</cell></row></table></p></div1><div1 part="N" n="ALDERAIMIN" org="uniform" sample="complete" type="entry"><head>ALDERAIMIN</head><p>, a star of the third magnitude in the right shoulder of the constellation Cepheus.</p></div1><div1 part="N" n="ALDHAFERA" org="uniform" sample="complete" type="entry"><head>ALDHAFERA</head><p>, or Aldhaphra, in the Arabian Astronomy, denotes a fixed star of the third magnitude, in the mane of the sign or constellation Leo, the lion.</p></div1><div1 part="N" n="ALEMBERT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ALEMBERT</surname> (<foreName full="yes"><hi rend="smallcaps">John le Rond</hi></foreName> D')</persName></head><p>, an eminent French mathematician and philosopher, and one of the brightest ornaments of the 18th century. He was perpetual secretary to the French Academy of Sciences, and a member of most of the philosophical academies and societies of Europe.</p><p>D'Alembert was born at Paris, the 16th of November 1717. He derived the name of John le Rond from that of the church near which, after his birth, he was exposed as a foundling. But his father, informed of this circumstance, listening to the voice of nature and duty, took measures for the proper education of his child, and for his future subsistence in a state of ease and independence. His mother, it is said, was a lady of of rank, the celebrated Mademoiselle Tencin, sister to cardinal Tencin, archbishop of Lyons.</p><p>He received his first education among the Jansenists, in the College of the Four Nations, where he gave early signs of genius and capacity. In the first year of his philosophical studies, he composed a Commentary on the Epistle of St. Paul to the Romans. The Jansenists considered this production as an omen, that portended to the party of Port-Royal a restoration to some part of their former splendor, and hoped to find one day in d'Alembert a second Pascal. To render this resemblance more complete, they engaged their pupil in the study of the mathematics; but they soon perceived that his growing attachment to this science was likely to disappoint the hopes they had formed with respect to his future destination: they therefore endeavoured to divert him from this line; but their endeavours were fruitless. <pb n="61"/><cb/></p><p>On his quitting the college, finding himself alone, and unconnected in the world, he sought an asylum in the house of his nurse. He hoped that his fortune, though not ample, would enlarge the subsistence, and better the condition of her family, which was the only one that he could consider as his own. It was here therefore that he fixed his residence, resolving to apply himself entirely to the study of geometry.—And here he lived, during the space of 40 years, with the greatest fimplicity, discovering the augmentation of his means only by increasing displays of his beneficence, concealing his growing reputation and celebrity from these honest people, and making their plain and uncouth manners the subject of good-natured pleasantry and philosophical observation. His good nurse perceived his ardent activity; heard him mentioned as the writer of many books; but never took it into her head that he was a great man, and rather beheld him with a kind of compassion. “You will never, said she to him one day, be any thing but a philosopher—and what is a philosopher?—a fool, who toils and plagues himself all his life, that people may talk of him when he is dead.”</p><p>As d'Alembert's fortune did not far exceed the demands of necessity, his friends advised him to think of some profession that might enable him to increase it. He accordingly turned his views to the law, and took his degrees in that faculty: but soon after, abandoning this line, he applied himself to the study of medicine. Geometry however was always drawing him back to his former pursuits; so that after many ineffectual struggles to resist its attractions, he renounced all views of a lucrative profession, and gave himself up entirely to mathematics and poverty.</p><p>In the year 1741 he was admitted a member of the Academy of Sciences; for which distinguished literary promotion, at so early an age (24), he had prepared the way by correcting the errors of a celebrated work (The <hi rend="italics">Analyse Demontrée</hi> of Reyneau), which was esteemed classical in France in the line of analytics. He afterwards set himself to examine, with close attention and assiduity, what must be the motion and path of a body, which passes from one fluid into another denser fluid, in a direction oblique to the surface between the two fluids. Every one knows the phenomenon which happens in this case, and amuses children, under the denomination of <hi rend="italics">Ducks and Drakes;</hi> but it was d'Alembert who first explained it in a satisfactory and philosophical manner.</p><p>Two years after his election to a place in the academy, he published his <hi rend="italics">Treatise on Dynamics.</hi> The new principle developed in this treatise, consisted in establishing an equality, at each instant, between the changes that the motion of a body has undergone, and the forces or powers which have been employed to produce them: or, to express the same thing otherwise, in separating into <hi rend="italics">two parts</hi> the action of the moving powers, and considering the <hi rend="italics">one</hi> as producing alone the motion of the body, in the second instant, and the <hi rend="italics">other</hi> as employed to destroy that which it had in the first.</p><p>So early as the year 1744, d'Alembert had applied this principle to the theory of the equilibrium, and the motion of fluids: and all the problems before resolved <cb/> in physics, became in some measure its corollaries. The discovery of this new principle was followed by that of a new calculus, the first essays of which were published in a <hi rend="italics">Discourse on the General Theory of the Winds,</hi> to which the prize-medal was adjudged by the Academy of Berlin in the year 1746, which proved a new and brilliant addition to the fame of d'Alembert. This new calculus of <hi rend="italics">Partial Differences</hi> he applied, the year following, to the problem of vibrating chords, the resolution of which, as well as the theory of the oscillations of the air and the propagation of sound, had been but imperfectly given by the mathematicians who preceded him; and these were his masters or his rivals.</p><p>In the year 1749 he furnished a method of applying his principle to the motion of any body of a given figure. He also resolved the problem of the precession of the equinoxes; determining its quantity, and explaining the phenomenon of the nutation of the terrestrial axis discovered by Dr. Bradley.</p><p>In 1752, d'Alembert published a treatise on the <hi rend="italics">Resistance of Fluids.</hi> to which he gave the modest title of an <hi rend="italics">Essay;</hi> though it contains a multitude of original ideas and new observations. About the same time he published, in the Memoirs of the Academy of Berlin, <hi rend="italics">Researches concerning the Integral Calculus,</hi> which is greatly indebted to him for the rapid progress it has made in the present century.</p><p>While the studies of d'Alembert were confined to mere mathematics, he was little known or celebrated in his native country. His connections were limited to a small society of select friends. But his cheerful conversation, his smart and lively sallies, a happy knack at telling a story, a singular mixture of malice of speech with goodness of heart, and of delicacy of wit with simplicity of manners, rendering him a pleasing and interesting companion, his company began to be much sought after in the fashionable circles. His reputation at length made its way to the throne, and rendered him the object of royal attention and beneficence. The consequence was a pension from government, which he owed to the friendship of count d'Argenson.</p><p>But the tranquillity of d'Alembert was abated when his same grew more extensive, and when it was known beyond the circle of his friends, that a fine and enlightened taste for literature and philosophy accompanied his mathematical genius. Our author's eulogist ascribes to envy, detraction, &c, all the opposition and censure that d'Alembert met with on account of the famous Encyclopédie, or Dictionary of Arts and Sciences, in conjunction with Diderot. None surely will refuse the well-deserved tribute of applause to the eminent displays of genius, judgment, and true literary taste, with which d'Alembert has enriched that great work. Among others, the Preliminary Discourse he has prefixed to it, concerning the rise, progress, connections, and affinities of all the branches of human knowledge, is perhaps one of the most capital productions the philosophy of the age can boast of.</p><p>Some time after this, d'Alembert published his <hi rend="italics">Philosophical, Historical,</hi> and <hi rend="italics">Philological Miscellanies.</hi> These were followed by the <hi rend="italics">Memoirs of Christina queen of Sweden;</hi> in which d'Alembert shewed that he was acquainted with the natural rights of mankind, <pb n="62"/><cb/> and was bold enough to assert them. His <hi rend="italics">Essay on the Intercourse of Men of Letters with Persons high in Rank and Ofsice,</hi> wounded the former to the quick, as it exposed to the eyes of the public the ignominy of those servile chains, which they feared to shake off, or were proud to wear. A lady of the court hearing one day the author accused of having exaggerated the despotism of the great, and the submission they require, answered slyly, “If he had consulted me, I would have told him still more of the matter.”</p><p>D'Alembert gave elegant specimens of his literary abilities in his translations of some select pieces of Tacitus. But these occupations did not divert him from his mathematical studies: for about the same time he enriched the Encyclopédie with a multitude of excellent articles in that line, and composed his <hi rend="italics">Researches on several Important Points of the System of the World,</hi> in which he carried to a higher degree of perfection the solution of the problem concerning the perturbations of the planets, that had several years before been presented to the Academy.</p><p>In 1759 he published his <hi rend="italics">Elements of Philosophy:</hi> a work much extolled as remarkable for its precision and perspicuity.</p><p>The resentment that was kindled (and the disputes that followed it) by the article <hi rend="italics">Geneva,</hi> inserted in the Encyclopédie, are well known. D'Alembert did not leave this sield of controversy with flying colours. Voltaire was an auxiliary in the contest: but as he had no reputation to lose, in point of candour and decency; and as he weakened the blows of his enemies, by throwing both them and the spectators into fits of laughter, the issue of the war gave him little uneasiness. It fell more heavily on d'Alembert; and exposed him, even at home, to much contradiction and opposition.</p><p>It was on this occasion that the late king of Prussia offered him an honourable asylum at his court, and the office of president of his academy: and the king was not offended at d'Alembert's refusal of these distinctions, but cultivated an intimate friendship with him during the rest of his life. He had refused, some time before this, a proposal made by the empress of Russia to entrust him with the education of the Grand Duke; —a proposal accompanied with all the flattering offers that could tempt a man, ambitious of titles, or desirous of making an ample fortune: but the objects of his ambition were tranquillity and study.</p><p>In the year 1765, he published his <hi rend="italics">Dissertation on the Destruction of the Jesuits.</hi> This piece drew upon him a swarm of adversaries, who only confirmed the merit and credit of his work by their manner of attacking it.</p><p>Beside the works already mentioned, he published nine volumes of memoirs and treatises, under the title of <hi rend="italics">Opuscules;</hi> in which he has resolved a multitude of problems relating to astronomy, mathematics, and natural philosophy; of which his panegyrist, Condorcet, gives a particular account, more especially of those which exhibit new subjects, or new methods of investigation.</p><p>He published also <hi rend="italics">Elements of Music</hi>; and rendered, at length, the system of Rameau intelligible: but he did not think the mathematical theory of the sonorous body sufficient to account for the rules of that art. <cb/></p><p>In the year 1772 he was chosen secretary to the French Academy of Sciences. He formed, soon after this preferment, the design of writing the lives of all the deceased academicians, from 1700 to 1772; and in the space of three years he executed this design, by composing 70 eulogies.</p><p>D'Alembert died on the 29th of October 1783, being nearly 66 years of age. In his moral character there were many amiable lines of candour, modesty, disinterestedness, and beneficence; which are described, with a diffusive detail, in his eulogium, by Condorcet, in the <hi rend="italics">Hist. de l'Acad. Royale des Sciences,</hi> 1783.</p><p>As it may be curious and useful to have in one view an entire list of d'Alembert's writings, I shall here insert a catalogue of them, from Rozier's <hi rend="italics">Nouvelle Table des Articles contenus dans les volumes de l'Academie Royale des Sciences de Paris,</hi> &c, as follows:</p><p><hi rend="italics">Traité de Dynamique,</hi> in 4to, Paris, 1743. The 2d ed. in 1758.</p><p><hi rend="italics">Traité de l'Equilibre et du Mouvement des Fluides.</hi> Paris, 1744; and the 2d edition in 1770.</p><p><hi rend="italics">Reflexions sur la Cause Générale des Vents;</hi> which gained the prize at Berlin in 1746; and was printed at Paris in 1747, in 4to.</p><p><hi rend="italics">Recherches sur la Précession des Équinoxes, & sur la Nutation de l'Axe de la Terre dans le Système Newtonien.</hi> Paris, 1749, in 4to.</p><p><hi rend="italics">Essais d'une Nouvelle Théorie du Mouvement des Fluides.</hi> Paris, 1752, in 4to.</p><p><hi rend="italics">Recherches sur differens Points importans du Système du Monde.</hi> Paris, 1754 and 1756, 3 vol. in 4to.</p><p><hi rend="italics">Elemens de Philosophie,</hi> 1759.</p><p><hi rend="italics">Opuscules Mathématiques,</hi> ou Memoires sur différens Sujets de Géométrie, de Méchaniques, d'Optiques, d'Astronomie. Paris, 9 vol. in 4to; 1761 to 1773.</p><p><hi rend="italics">Elémens de Musique,</hi> théorique & pratique, suivant les Principes de M. Rameau, eclairés, développés, & simplifiés. 1 vol. in 8vo. à Lyon.</p><p><hi rend="italics">De la Destruction des Jesuites,</hi> 1765.</p><p>In the Memoirs of the Academy of Paris are the following pieces, by d'Alembert: viz,</p><p><hi rend="italics">Précis</hi> de Dynamique, 1743, Hist. 164.</p><p><hi rend="italics">Précis</hi> de l'Equilibre & de Mouvement des Fluides, 1744, Hist. 55.</p><p><hi rend="italics">Methode</hi> générale pour déterminer les Orbites & les Mouvements de toutes les Planètes, en ayant égard à leur action mutuelle, 1745, p. 365.</p><p><hi rend="italics">Précis</hi> des Réflexions sur la Cause Générale des Vents, 1750, Hist. 41.</p><p><hi rend="italics">Précis</hi> des Recherches sur la Précession des Équinoxes, et sur la Nutation de l'Axe de la Terre dans le Système Newtonien, 1750, Hist. 134.</p><p><hi rend="italics">Essai</hi> d'une Nouvelle Théorie sur la Résistance des Fluides, 1752, Hist. 116.</p><p><hi rend="italics">Précis</hi> des Essais d'une Nouvelle Théorie de la Résistance des Fluides, 1753, Hist. 289.</p><p><hi rend="italics">Précis</hi> des Recherches sur les differens Points importans du Système du Monde, 1754, Hist. 125.</p><p><hi rend="italics">Recherches</hi> sur la Précession des Equinoxes, & sur la Nutation de l'Axe de la Terre, dans l'Hypothese de la Dissimilitude des Méridiens, 1754, p. 413, Hist. 116.</p><p><hi rend="italics">Reponse</hi> à un Article du Mémoire de M. l'Abbé de <pb n="63"/><cb/> la Caille, su<*> la Théorie du Soleil, 1757, p. 145, Hist. 118.</p><p><hi rend="italics">Addition</hi> à ce Mémoire, 1757, p. 567, Hist. 118.</p><p><hi rend="italics">Précis</hi> des Opuscules Mathématiques, 1761, Hist. 86.</p><p><hi rend="italics">Précis</hi> du troisième volume des Opuscules Mathématiques, 1764, Hist. 92.</p><p><hi rend="italics">Nouvelles</hi> Recherches sur les Verres Optiques, pour servir de suite à la théorie qui en à été donnée dans le volume 3<hi rend="sup">e</hi> des Opuscules Mathématiques. Premier Mémoire, 1764, p. 75, Hist. 175.</p><p><hi rend="italics">Nouvelles</hi> Recherches sur les Verres Optiques, pour fervir de suite à la théorie qui en a été donnée dans le troisième volume des Opuscules Mathématiques. Second Mémoire, 1765, p. 53.</p><p><hi rend="italics">Observations</hi> sur les Lunettes Achromatiques, 1765, p. 53, Hist. 119.</p><p><hi rend="italics">Suite</hi> des Recherches sur les Verres Optiques. Troisième Mémoire, 1767, p. 43, Hist. 153.</p><p><hi rend="italics">Recherches</hi> sur le Calcul Intégral, 1767, p. 573.</p><p><hi rend="italics">Accident</hi> arrivé par l'Explosion d'une Meule d'Emouleur, 1768, Hist. 31.</p><p><hi rend="italics">Précis</hi> des Opuscules de Mathématiques, 4<hi rend="sup">e</hi> & 5<hi rend="sup">e</hi> volumes. Leur Analyse, 1768, Hist. 83.</p><p><hi rend="italics">Recherches</hi> sur les Mouvemens de l'Axe d'une Planete quelconque dans l'hypothese de la Dissimilitude des Méridienes, 1768 p. 1, Hist. 95.</p><p><hi rend="italics">Suite</hi> des Recherches sur les Mouvemens, &c, 1768 p. 332, Hist. 95.</p><p><hi rend="italics">Recherches</hi> sur le Calcul Intégral, 1769, p. 73.</p><p><hi rend="italics">Mémoire</hi> sur les Principes de la Mech. 1769, p. 278.</p><p>And in the Memoirs of the Academy of Berlin, are the following pieces, by our author: viz,</p><p><hi rend="italics">Recherches</hi> sur le Calcul Intégral, premiere partie, 1746.</p><p><hi rend="italics">Solution</hi> de quelques problemes d'astronomie, 1747.</p><p><hi rend="italics">Recherches</hi> sur la cour be que forme une Corde Tendue, mise en Vibration, 1747.</p><p><hi rend="italics">Suite</hi> des recherches sur le Calcul Intégral, 1748.</p><p><hi rend="italics">Lettre</hi> à M. de Maupertuis, 1749.</p><p><hi rend="italics">Addition</hi> aux recherches sur la courbe que forme une Corde Tendue mise en Vibration, 1750.</p><p><hi rend="italics">Addition</hi> aux recherches sur le Calcul Intégral, 1750.</p><p><hi rend="italics">Lettre</hi> à M. le professeur Formey, 1755.</p><p><hi rend="italics">Extr.</hi> de differ. lettres à M. de la Grange, 1763.</p><p><hi rend="italics">Sur</hi> les Tautochrones, 1765.</p><p><hi rend="italics">Extr.</hi> de differ. lettres à M. de la Grange, 1769.</p><p>Also in the Memoirs of Turin are,</p><p><hi rend="italics">Differentes</hi> Lettres à M. de la Grange, en 1764 & 1765, tom. 3 of these Memoris.</p><p><hi rend="italics">Recherches</hi> sur differens sujets de Math. t. 4.</p></div1><div1 part="N" n="ALFECCA" org="uniform" sample="complete" type="entry"><head>ALFECCA</head><p>, or Alfeta, a name given to the star commonly called Lucida Coronæ.</p></div1><div1 part="N" n="ALFRAGAN" org="uniform" sample="complete" type="entry"><head>ALFRAGAN</head><p>, <hi rend="smallcaps">Alfergani</hi>, or <hi rend="smallcaps">Fargani</hi>, a celebrated Arabic astronomer, who flourished about the year 800. He was so called from the place of his nativity, Fergan, in Sogdiana, now called Maracanda, or Samarcand, anciently a part of Bactria. He is also called Ahmed (or Muhammed) ben-Cothair, or Katir. He wrote the Elements of Astronomy, in 30 chapters or sections. In this work the author chiefly follows Ptolomy, using the same hypotheses, and the same terms, and frequently citing him.</p><p>There are three Latin translations of Alfragan's <cb/> work. The first was made in the 12th century, by Joannes Hispalensis; and was published at Ferrara in 1493, and at Nuremberg in 1537, with a preface by Melancthon. The second was by James Christman, from the Hebrew version of James Antoli, and appeared at Frankfort in 1590. Christman added to the first chapter of the work an ample commentary, in which he compares together the calendars of the Romans, the Egyptians, the Arabians, the Persians, the Syrians, and the Hebrews, and shews the correspondence of their years.</p><p>The third and best translation was made by Golius, professor of mathematics and Oriental languages at Leyden: this work, which came out in 1669, after the death of Golius, is accompanied with the Arabic text, and many learned notes upon the first nine chapters; for this author was not spared to carry them farther.</p></div1><div1 part="N" n="ALGAROTI" org="uniform" sample="complete" type="entry"><head>ALGAROTI</head><p>, commonly called <hi rend="italics">Count Algaroti,</hi> a celebrated Italian of the present century, well skilled in Architecture and the Newtonian philosophy, &c. Algaroti was born at Padua, but in what year has not been mentioned. Led by curiosity, as well as a desire of improvement, he travelled early into foreign countries; and was very young when he arrived in France in 1736. It was here that he composed his <hi rend="italics">Newtonian Philosophy for the Ladies,</hi> as Fontenelle had done his Cartesian Astronomy, in the work intitled <hi rend="italics">The Plurality of Worlds.</hi> He was much noticed by the king of Prussia, who conferred on him many marks of his esteem. He died at Pisa the 23d of May, 1764, and gave orders for his own mausoleum, with this inscription upon it; <hi rend="italics">Hic jacet Algarotus, sed non omnis.</hi> He was esteemed to be well skilled in painting, sculpture, and architecture. His works, which are numerous, and upon a variety of subjects, abound with vivacity, elegance, and wit: a collection of them has lately been made, and printed at Leghorn; but that for which he is chiefly intitled to a place in this work is his <hi rend="italics">Newtonian Philosophy for the Ladies,</hi> a sprightly, ingenious, and popular work.</p></div1><div1 part="N" n="ALGEBRA" org="uniform" sample="complete" type="entry"><head>ALGEBRA</head><p>, a general method of resolving mathematical problems by means of equations. Or, it is a method of performing the calculations of all sorts of quantities by means of general signs or characters. At first, numbers and things were expressed by their names at full length; but afterwards these were abridged, and the initials of the words used instead of them; and, as the art advanced farther, the letters of the alphabet came to be employed as general representations of all sorts of quantities; and other marks were gradually introduced, to express all sorts of operations and combinations; so as to entitle it to different appellations— universal arithmetic, and literal arithmetic, and the arithmetic of signs.</p><p>The etymology of the name, <hi rend="italics">Algebra,</hi> is given in various ways. It is pretty certain, however, that the word is Arabian, and that from those people we had the name, as well as the art itself, as is testisied by Lucas le Burgo, the first European author whose treatise was printed on this art, and who also refers to former authors and masters, from whose writings he had learned it. The Arabic name he gives it, is <hi rend="italics">Alghebra e Almucabala,</hi> which is explained to signify the art of <pb n="64"/><cb/> <hi rend="italics">restitution and comparison,</hi> or <hi rend="italics">opposition and comparison,</hi> or <hi rend="italics">resolution and equation,</hi> all which agree well enough with the nature of this art. Some however derive it from various other arabic words; as from Geber, a celebrated philosopher, chemist, and mathematician, to whom also they ascribe the invention of this science: some likewise derive it from the word Geber, which with the particle <hi rend="italics">al,</hi> makes Algeber, which is purely Arabic, and signifies the reduction of broken numbers or fractions to integers.</p><p>But Peter Ramus, in the beginning of his Algebra, says “the name Algebra is Syriac, signifying the art and doctrine of an excellent man. For <hi rend="italics">Geber,</hi> in Syriac, is a name applied to men, and is sometimes a term of honour, as master or doctor among us. That there was a certain learned mathematician, who sent his Algebra, written in the Syriac language, to Alexander the Great, and he named it <hi rend="italics">Almucabala,</hi> that is, the book of dark or mysterious things, which others would rather call the doctrine of Algebra. And to this day the same book is in great estimation among the learned in the oriental nations, and by the Indians who cultivate this art it is called <hi rend="italics">Aljabra,</hi> and <hi rend="italics">Alboret;</hi> though the name of the author himself is not known.” But Ramus gives no authority for this singular paragraph. It has however on various occasions been distinguished by other names. Lucas Paciolus, or de Burgo, in Italy, called it <hi rend="italics">l'Arte Magiore: ditta dal vulgo la Regola de la Cosa over Alghebra e Almucabala;</hi> calling it <hi rend="italics">l'Arte Magiore,</hi> or the greater art, to distinguish it from common arithmetic, which is called <hi rend="italics">l'Arte Minore,</hi> or the lesser art. It seems too that it had been long and commonly known in his country by the name <hi rend="italics">Regola de la Cosa,</hi> or <hi rend="italics">Rule of the Thing;</hi> from whence came our rule of coss, cosic numbers, and such like terms. Some of his countrymen followed his denomination of the art; but other Italian and Latin writers called it <hi rend="italics">Regula rei & census,</hi> the rule of the thing and the product, or the root and the square, as the unknown quantity in their equations commonly ascended no higher than the square or second power. From this Italian word <hi rend="italics">census,</hi> pronounced <hi rend="italics">chensus,</hi> came the barbarous word <hi rend="italics">zenzus,</hi> used by the Germans and others, for quadratics; with the several zenzic or square roots. And hence [scruple], [dram], <figure/>, which are derived from the letters <hi rend="italics">r, z, c,</hi> the initials of <hi rend="italics">res, zenzus, cubus,</hi> or root, square, cube, came to be the signs or characters of these words: like as ℞ and √, derived from the letters <hi rend="italics">R, r,</hi> became the signs of radicality.</p><p>Later authors, and other nations, used some the one of those names, and some another. It was also called <hi rend="italics">Specious Arithmetic</hi> by Vieta, on account of the species, or letters of the alphabet, which he brought into general use; and by Newton it was called <hi rend="italics">Universal Arithmetic,</hi> from the manner in which it performs all arithmetical operations by general symbols, or indeterminate quantities.</p><p>Some authors define algebra to be <hi rend="italics">the art of resolving mathematical problems:</hi> but this is the idea of analysis, or the analytic art in general, rather than of algebra, which is only one particular species of it.</p><p>Indeed algebra properly consists of two parts: first, the method of calculating magnitudes or quantities, as represented by letters or other characters: and secondly <cb/> the manner of applying these calculations in the solution of problems.</p><p>In algebra, as applied to the resolution of problems, the first business is to translate the problem out of the common into the algebraic language, by expressing all the conditions and quantities, both known and unknown, by their proper characters, arranged in an equation, or several equations if necessary, and treating the unknown quantity, whether it be number, or line, or any other thing, in the same way as if it were a known one: this forms the composition. Then the resolution, or analytic part, is the disentangling the unknown quantity from the several others with which it is connected, so as to retain it alone on one side of the equation, while all the other, or known, quantities, are collected on the other side, and so giving the value of the unknown one. And as this disentangling of the quantity sought, is performed by the converse of the operations by which it is connected with the others, taking them always backwards in the contrary order, it hence becomes a species of the analytic art, and is called the modern analysis, in contradistinction to the ancient analysis, which chiefly respected geometry, and its applications.</p><p>There have arisen great controversies and sharp disputes among authors, concerning the history of the progress and improvements of Algebra; arifing partly from the partiality and prejudices which are natural to all nations, and partly from the want of a closer examination of the works of the older authors on this subject. From these causes it has happened, that the improvements made by the writers of one nation, have been ascribed to those of another; and the discoveries of an earlier author, to some one of much later date. Add to this also, that the peculiar methods of many authors have been described so little in detail, that our information derived from such histories, is but very imperfect, and amounting only to some general and vague ideas of the true state of the arts. To remedy this inconvenience therefore, and to reform this article, I have taken the pains carefully to read over in succession all the older authors on this subject, which I have been able to meet with, and to write down distinctly a particular account and description of their several compositions, as to their contents, notation, improvements, and peculiarities; from the comparison of all which, I have acquired an idea more precise and accurate than it was possible to obtain from other histories, and in a great many instances very different from them. The full detail of these descriptions would employ a volume of itself, and would be far too extensive for this place: I must therefore limit this article to a very brief abridgment of my notes, remarking only the most material circumstances in each author; from which a general idea of the chain of improvements may be perceived, from the first rude beginnings, down to the more perfect state; from which it will appear that the discoveries and improvements made by any one single author, are scarcely ever either very great or numerous; but that, on the contrary, the improvements are almost always very slow and gradual, from former writers, successively made, not by great leaps, and after long intervals of time, but by gradations which, viewed in succession, become almost imperceptible. <pb n="65"/><cb/></p><p>As to the origin of the analytic art, of which Algebra is a species, it is doubtless as old as any science in the world, being the natural method by which the mind investigates truths, causes, and theories, from their observed effects and properties. Accordingly, traces of it are observable in the works of the earliest philosophers and mathematicians, the subject of whose enquiries most of any require the aid of such an art. And this process constituted their Analytics. Of that part of analytics however which is properly called Algebra, the oldest treatise which has come down to us, is that of Diophantus of Alexandria, who flourished about the year 350 after Christ, and who wrote, in the Greek language, 13 books of Algebra or Arithmetic, as mentioned by himself at the end of his address to Dionysius, though only 6 of them have hitherto been printed; and an imperfect book on multangular numbers, namely in a Latin translation only, by Xilander, in the year 1575, and afterwards in 1621 and 1670 in Greek and Latin by Gaspar Bachet. These books however do not contain a treatise on the elementary parts of Algebra, but only collections of difficult questions relating to square and cube numbers, and other curious properties of numbers, with their solutions. And Diophantus only prefaces the books by an address to one Dionysius, for whose use it was probably written, in which he just mentions certain precognita, as it were to prepare him for the problems themselves. In these remarks he shews the names and generation of the powers, the square, cube, 4th, 5th, 6th, &c, which he calls dynamis, cubus, dynamodinamis, dynamocubus, cubocubus, according to the sum of the indices of the powers; and he marks these powers with the initials thus <foreign xml:lang="greek">d<hi rend="sup">n_</hi>, k<hi rend="sup">n_</hi>, dd<hi rend="sup">n_</hi>, dk<hi rend="sup">n_</hi>, kk<hi rend="sup">n_</hi></foreign>, &c: the unknown quantity he calls simply <foreign xml:lang="greek">ariqmos</foreign>, <hi rend="italics">numerus,</hi> the number; and in the solutions he commonly marks it by the final thus <foreign xml:lang="greek">s_</foreign>; also he denotes the monades, or indefinite unit, by <foreign xml:lang="greek">m<hi rend="sup">o_</hi></foreign>. Diophantus there remarks on the multiplication and division of simple species together, shewing what powers or species they produce; declares that minus (<foreign xml:lang="greek">leiyis</foreign>) multiplied by minus produces plus (<foreign xml:lang="greek">nparcin</foreign>); but that minus multiplied by plus, produces minus; and that the mark used for minus is <*> namely the <foreign xml:lang="greek">y</foreign> inverted and curtailed, but he uses no mark for plus, but a word or conjunction copulative. As to the operations, viz. of addition, subtraction, multiplication, and division of compound species, or those connected by plus and minus, Diophantus does not teach, but supposes his reader to know them. He then remarks on the preparation or simplifying of the equations that are derived from the questions, which we call reduction of equations, by collecting like quantities together, adding quantities that are minus, and subtracting such as are plus, called by the moderns Transposition, so as to bring the equation to simple terms, and then depressing it to a lower degree by equal division when the powers of the unknown quantity are in every term: which preparation, or reduction of the complex equation, being now made, or reduced to what we call a final equation, Diophantus goes no farther, but barely says what the root or <hi rend="italics">res ignota</hi> is, without giving any rules for finding it, or for the resolution of equations; thereby intimating that such rules were to be found in some other work, done either by himself or others. Of the body of the <cb/> work, <hi rend="italics">Lib.</hi> 1 contains 43 questions, concerning one, two, three, or four unknown numbers, having certain relations to each other, viz. concerning their sums, differences, ratios, products, squares, sums and differences of squares, &c, &c; but none of them concerning either square or cubic numbers. <hi rend="italics">Lib.</hi> 2 contains 36 questions. The first five questions are concerning two numbers, though only one condition is given in each question; but he supplies another by assuming the numbers in a given ratio, viz, as 2 to 1. The 6th and 7th contain each two conditions: then in the 8th question he first comes to trcat of square numbers, which is this, to divide a given square number into two other squares; and the 9th is the same, but performed in a different way: the rest, to the end, are, almost all, about one, two, or three squares. <hi rend="italics">Lib.</hi> 3 contains 24 questions concerning squares, chiefly including three or four numbers. <hi rend="italics">Lib.</hi> 4 begins with cubes; the first of which is this, to divide a given number into two cubes whose sides shall have a given sum: here he has occasion to cube the two binomials 5+<hi rend="italics">n</hi> and 5-<hi rend="italics">n;</hi> the manner of doing which shews that he knew the composition of the cube of a binomial; and many other places manifest the same thing. Only part of the questions in this book are concerning cubes; the rest are relating to squares. Two or three questions in this book have general solutions, and the theorems deduced are general, and for any numbers indefinitely; but all the other questions, in all the four books, find only particular numbers. <hi rend="italics">Lib.</hi> 5 is also concerning square and cube numbers, but of a more difficult kind, beginning with some that relate to numbers in geometrical progression. <hi rend="italics">Lib.</hi> 6 contains 26 propositions, concerning right-angled triangles; such as to make their sides, areas, perimeters, &c, &c, squares or cubes, or rational, &c. In some parts of this book it appears, that he was acquainted with the composition of the 4th power of the binomial root, as he sets down all the terms of it; and, from his great skill in such matters, it seems probable that he was acquainted with the composition of other higher powers, and with other parts of Algebra, besides what are here treated of. At the end is part of a book, in 10 propositions, concerning arithmetical progressions, and multangular or polygonal numbers. Diophantus once mentions a compound quadratic equation; but the resolution of his questions is by simple equations, and by means of only one unknown letter or character, which he chooses so ingeniously, that all the other unknown quantities in the question are easily expressed by it, and the final equation reduced to the simplest form which it seems the question can admit of. Sometimes he substitutes for a number sought immediately, and then expresses the other numbers or conditions by it: at other times he substitutes for the sum or difference, &c, and thence derives the rest, so as always to obtain the expressions in the simplest form. Thus, if the sum of two numbers be given, he substitutes for their difference; and if the difference be given, he substitutes for their sum: and in both cases he has the two numbers easily expressed by adding and subtracting the half sum and half difference; and so in other cases he uses other similar ingenious notations. In short, the chief excellence in this collection of questions, which seems to be only a set of exercises to some rules which had been given elsewhere, is the neat mode of substitution or notation; which being once made, the reduc- <pb n="66"/><cb/> tion to the final equation is easy and evident: and there he leaves the solution, only mentioning that the root or <foreign xml:lang="greek">ariqmos</foreign> is so much. Upon the whole, this work is treated in a very able and masterly manner, manifesting the utmost address and knowledge in the solutions, and forcing a persuasion that the author was deeply skilled in the science of Algebra, to some of the most abstruse parts of which these questions or exercises relate. However, as he contrives his assumptions and notations so as to reduce all his conditions to a simple equation, or at least a simple quadratic, it does not appear what his knowledge was in the resolution of compound or affected equations.</p><p>But although Diophantus was the first author on Algebra that we now know of, it was not from him, but from the Moors or Arabians that we received the knowledge of Algebra in Europe, as well as that of most other sciences. And it is matter of dispute who were the first inventors of it; some ascribing the invention to the Greeks, while others say that the Arabians had it from the Persians, and these from the Indians, as well as the arithmetical method of computing by ten characters, or digits; but the Arabians themselves say it was invented amongst them by one <hi rend="italics">Mahomet ben Musa,</hi> or son of Moses, who it seems flourished about the 8th or 9th century. It is more probable, however, that Mahomet was not the inventor, but only a person well skilled in the art; and it is farther probable, that the Arabians drew their first knowledge of it from Diophantus or other Greek writers, as they did that of Geometry and other sciences, which they improved and translated into their own language; and from them it was that we received these sciences, before the Greek authors were known to us, after the Moors settled in Spain, and after the Europeans began to hold communications with them, and that our countrymen began to travel amongst them to learn the sciences. And according to the testimony of Abulpharagius, the Arithmetic of Diophantus was translated in Arabic by Mahomet ben-yahya Ba<*>iani. But whoever were the inventors and first cultivators of Algebra, it is certain that the Europeans first received the knowledge, as well as the name, from the Arabians or Moors, in consequence of the close intercourse which subsisted between them for several centuries. And it appears that the art was pretty generally known, and much cultivated, at least in Italy, if not in other parts of Europe also, long before the invention of printing, as many writers upon the art are still extant in the libraries of manuscripts; and the first authors, presently after the invention of printing, speak of many former writers on this subject, from whom they learned the art.</p><p>It was chiefly among the Italians that this art was first cultivated in Europe. And the first author whose works we have in print, was Lucas Paciolus, or Lucas de Burgo, a Cordelier, or Minorite Friar. He wrote several treatises of Arithmetic, Algebra, and Geometry, which were printed in the years 1470, 1476, 1481, 1487, and in 1494 his principal work, intitled <hi rend="italics">Summa de Arithmetica, Geometria, Proportioni, et Proportionalita,</hi> is a very masterly and complete treatise on those sciences, as they then stood. In this work he mentions various former writers, as Euclid, St. Augustine, Sacrobosco or Halifax, Boetius, Prodocimo, Giordano, <cb/> Biagio da Parma, and Leonardus Pisanus, from whom he learned those sciences. The order of the work is, 1st Arithmetic, 2d Algebra, and 3d Geometry. Of the Arithmetic the contents, and the order of them, are nearly as follow. First, of numbers figurate, odd and even, perfect, prime and composite, and many others. Then of Common Arithmetic in 7 parts, namely numeration or notation, addition, subtraction, multiplication, division, progression, and extraction of roots. Before him, he says, duplation and mediation, or doubling and halving, were accounted two rules in Arithmetic; but that he omits them, as being included in multiplication and division. He ascribes the present notation and method of Arithmetic to the Arabs; and says that according to some the word <hi rend="italics">Abaco</hi> is a corruption of <hi rend="italics">Modo Arabico,</hi> but that according to others it was from a Greek word. All those primary operations he both performs and demonstrates in various ways, many of which are not in use at present, proving them not only by what is called casting out the nines, but also by casting out the sevens, and otherwise. In the extraction of roots he uses the initial ℞ for a root; and when the roots can be extracted, he calls them discrete or rational; otherwise surd, or indiscrete, or irrational. The square root is extracted much the same way as at present, namely, dividing always the last remainder by double the root found; and so he continues the surd roots continually nearer and nearer in vulgar fractions. Thus, for the root of 6, he firsts finds the nearest whole number 2, and the remainder 2 also; then 2/4 or 1/2 is the first correction, and 2 1/2 the second root: its square is 6 1/4, therefore 1/4 divided by 5, or 1/20 is the next correction, and 2 1/2 minus 1/20, or 2 9/20 is the 3d root: its square is 6 1/400, therefore 1/400 divided by 4 9/10, or 1/1960, is the 3d correction, which gives 2 881/1960 for the 4th root, whose square exceeds 6 by only 1/3841600: and so on continually: and this process he calls approximation. He observes that fractions, which he sets down the same way as we do at present, are extracted, by taking the root of the denominator, and of the denominated, for so he calls the numerator: and when mixed numbers occur, he directs to reduce the whole to a fraction, and then extract the roots of its two terms as above: as if it be 12 1/4; this he reduces to 49/4, and then the roots give 7/2 or 3 1/2: in like manner he finds that 4 1/2 is the root of 20 1/4; 5 1/2 the root of 30 1/4; “and so on (he adds) in infinitum;” which shews that he knew how to form the series of squares by addition. He then extracts the cube root, by a rule much the same as that which is used at present; from which it appears that he was well acquainted with the co-efficients of the binomial cubed, namely 1, 3, 3, 1; and he directs how the operation may be continued “in infinitum” in fractions, like as in the square root. After this, he describes geometrical methods for extracting the square and cube roots instrumentally: he then treats professedly of vulgar fractions, their reductions, addition, subtraction, and other operations, much the fame as at present: then of the rule-of-three, gain-and-loss, and other rules used by merchants.</p><p>Paciolus next enters on the algebraical part of this work, which he calls “<hi rend="italics">L'Arte Magiore; ditta dal vulgo la Regola de la Cosa, over Alghebra e Ahnucabala:</hi>” which last name he explains by <hi rend="italics">restauratio & oppositio,</hi> and <pb n="67"/><cb/> assigns as a reason for the first name, because it treats of things above the common affairs in business, which make the <hi rend="italics">Arte Minore.</hi> Here he ascribes the invention of Algebra to the Arabians, and denominates the series of powers, with their marks or abbreviations, as <hi rend="italics">n</hi>°, or <hi rend="italics">numero,</hi> the absolute or known number; <hi rend="italics">co.</hi> or <hi rend="italics">cosa,</hi> the thing or 1st power of the unknown quantity; <hi rend="italics">ce.</hi> or <hi rend="italics">censo,</hi> the product or square; <hi rend="italics">cu.</hi> or <hi rend="italics">cubo,</hi> the cube, or 3d power; <hi rend="italics">ce. ce.</hi> or <hi rend="italics">censo de censo,</hi> the square-squared, or 4th power; <hi rend="italics">p</hi>°. <hi rend="italics">r</hi>°. or <hi rend="italics">primo relato,</hi> or 5th power; <hi rend="italics">ce. cu.</hi> or <hi rend="italics">censo de cubo,</hi> the square of the cube, or 6th power; and so on, compounding the names or indices according to the multiplication of the numbers 2, 3, &c, and not according to their sum or addition, as used by Diophantus. He describes also the other characters made use of in this part, which are for the most part no more than the initials or other abbreviations of the words themselves; as ℞ for <hi rend="italics">radici,</hi> the root; ℞. ℞. <hi rend="italics">radici de radici,</hi> the root of the root; ℞ u. <hi rend="italics">radici universale,</hi> or <hi rend="italics">radici legata,</hi> or <hi rend="italics">radici unita;</hi> ℞ <hi rend="italics">cu. radici cuba;</hi> and ―q[dram]<hi rend="sup"><*></hi> <hi rend="italics">quantita,</hi> quantity; <hi rend="italics">p</hi> for <hi rend="italics">piu</hi> or plus, and <hi rend="italics">m</hi> for <hi rend="italics">meno</hi> or minus; and he remarks that the necessity and use of these two last characters are for connecting, by addition or subtraction, different powers together; as 3 <hi rend="italics">co.</hi> p. 4 <hi rend="italics">ce.</hi> m. 5 <hi rend="italics">cu.</hi> p. 2 <hi rend="italics">ce. ce.</hi> m. 6 <hi rend="italics">n</hi><hi rend="sup">i</hi>. that is, 3 <hi rend="italics">cosa</hi> piu 4 <hi rend="italics">censa</hi> meno 5 <hi rend="italics">cubo</hi> piu 2 <hi rend="italics">censa-censa</hi> meno 6 <hi rend="italics">numeri,</hi> or, as we now write the same thing, 3<hi rend="italics">x</hi>+4<hi rend="italics">x</hi><hi rend="sup">2</hi> - 5<hi rend="italics">x</hi><hi rend="sup">3</hi> + 2<hi rend="italics">x</hi><hi rend="sup">4</hi> - 6. He first treats very fully of proportions and proportionalities, both arithmetical and geometrical, accompanied with a large collection of questions concerning numbers in continued proportion, resolved by a kind of Algebra. He then treats of <hi rend="italics">el Cataym,</hi> which he says, according to some, is an Arabic or Phenician word, and signifies the Double Rule of False Position: but he here treats of both single and double position, as we do at present, dividing the <hi rend="italics">el Cataym</hi> into single and double. He gives also a geometrical demonstration of both the cases of the errors in the double rule, namely when the errors are both plus or both minus, and when the one error is plus and the other minus; and adds a large collection of questions, as usual. He then goes through the common operations of Algebra, with all the variety of signs, as to plus and minus; proving that, in multiplication and division, like signs give plus, and unlike signs give minus. He next treats of different roots <hi rend="italics">in infinitum,</hi> and the extraction of roots; giving also a copious treatise on radicals or surds, as to their addition, subtraction, multiplication and division, and that both in square roots and cube roots, and in the two together, much the same as at present. He makes here a digression concerning the 15 lines in the 10th book of Euclid, treating them as surd numbers, and teaching the extraction of the roots of the same, or of compound surds or binomials, such as of 23 <hi rend="italics">p</hi> ℞ 448, or of ℞ 18 <hi rend="italics">p</hi> ℞ 10; and gives this rule, among several others, namely: Divide the first term of the binomial into two such parts that their product may be 1/4 of the number in the second term; them the roots of those two parts, connected by their proper sign <hi rend="italics">p</hi> or <hi rend="italics">m,</hi> is the root of the binomial; as in this 23 <hi rend="italics">p</hi> ℞ 448, the two parts of 23 are 7 and 16, whose product, 112, is 1/4 of 448, therefore their roots give 4 <hi rend="italics">p</hi> ℞ 7 for the root ℞ u. 23 <hi rend="italics">p</hi> ℞ 448. He next treats of equations both simple and quadratic, or simple and compound, as he <cb/> calls it; and this latter he performs by completing the square, and then extracting the root, just as we do at present. He also resolves equations of the simple 4th power, and of the 4th combined with the 2d power, which he treats the same way as quadratics; expressing his rules in a kind of bad verse, and giving geometrical demonstrations of all the cases. He uses both the roots or values of the unknown quantity, in that case of the quadratics which has two positive roots; but he takes no notice of the negative roots in the other two cases. But as to any other compound equations, such as the cube and any other power, or the 4th and 1st, or 4th and 3d, &c, he gives them up as impossible, or at least says that no general rule has yet been found for them, any more, he adds, than for the quadrature of the circle. —The remainder of this part is employed on rules in trade and merchandise, such as Fellowship, Barter, Exchange, Interest, Composition or Alligation, with various other cases in trade. And in the third part of the work, he treats of Geometry, both theoretical and practical.</p><p>From this account of Lucas de Burgo's book, we may perceive what was the state of Algebra about the year 1500, in Europe; and probably it was much the same in Africa and Asia, from whence the Europeans had it. It appears that their knowledge extended only to quadratic equations, of which they used only the positive roots; that they used only one unknown quantity; that they had no marks or signs for either quantities or operations, excepting only some few abbreviations of the words or names themselves; and that the art was only employed in resolving certain numeral problems. So that either the Africans had not carried Algebra beyond quadratic equations, or else the Europeans had not learned the whole of the art, as it was then known to the former. And indeed it is not improbable but this might be the case: for whether the art was brought to us by an European, who, travelling in Africa, there learned it; or whether it was brought to us by an African; in either case we might receive the art only in an imperfect state, and perhaps far short of the degree of perfection to which it had been carried by their best authors. And this suspicion is rendered rather probable by the circumstance of an Arabic manuscript, said to be on cubic equations, deposited in the Library of the university of Leyden by the celebrated Warner, bearing a title which in Latin signifies <hi rend="italics">Omar Ben Ibrahim al'Ghajamæi Algebra cubicarum æquationum, sive de problematum solidorum resolutione;</hi> and of which book I am in some hopes of procuring either a copy or a translation, by means of my worthy friend Dr. Damen, the learned Professor of Mathematics in that university, and by that means to throw some light on this doubtful subject.</p><p>Since this was written, death has prematurely put an end to the useful labours of this ingenious and worthy successor of Gravesande.</p><p>After the publication of the books of Lucas de Burgo, the science of Algebra became more generally known, and improved, especially by many persons in Italy; and about this time, or soon after, namely about the year 1505, the first rule was there found out by Scipio Ferreus, for resolving one case of a compound cubic equation. But this science, as well as other branches of <pb n="68"/><cb/> Mathematics, was most of all cultivated and improved there by Hieronymus Cardan of Bononia, a very learned man, whose arithmetical writings were the next that appeared in print, namely in the year 1539, in 9 books, in the Latin language, at Milan, where he practised physic, and read public lectures on Mathematics; and in the year 1545 came out a 10th book, containing the whole doctrine of cubic equations, which had been in part revealed to him about the time of the publication of his first 9 books. And as it is only this 10th book which contains the new discoveries in Algebra, I shall here confine myself to it alone, as it will also afford sufficient occasion to speak of his manner of treating Algebra in general. This book is divided into 40 chapters, in which the whole science of cubic equations is most amply and ably treated. Chap. 1 treats of the nature, number and properties of the roots of equations, and particularly of single equations that have double roots. He begins with a few remarks on the invention and name of the art: calls it <hi rend="italics">Ars Magna,</hi> or <hi rend="italics">Cosa,</hi> or <hi rend="italics">Rules of Algebra,</hi> after Lucas de Burgo and others: says it was invented by Mahomet, the son of one Moses an Arabian, as is testified by Leonardus Pisanus; and that he left four rules or cases, which perhaps only included quadratic equations: that afterwards three derivatives were added by an unknown author, though some think by Lucas Paciolus; and after that again three other derivatives, for the cube and 6th power, by another unknown author; all which were resolved like quadratics: that then Scipio Ferreus, Professor of Mathematics at Bononia, about 1505, found out the rule for the case <hi rend="italics">cubum & rerum numero æqualium,</hi> or, as we now write it, , which he speaks of as a thing admirable: that the same thing was next afterwards found out, in 1535, by Tartalea, who revealed it to him, Cardan, after the most earnest intreaties: that, finally, by himself and his quondam pupil Lewis Ferrari, the cases are greatly augmented and extended, namely, by all that is not here expressly ascribed to others; and that all the demonstrations of the rules are his own, except only three adopted from Mahomet for the quadratics, and two of Ferrari for cubics.</p><p>He then delivers some remarks, shewing that all square numbers have two roots, the one positive, and the other negative, or, as he calls them, <hi rend="italics">vera & ficta,</hi> true and fictitious or false; so the <hi rend="italics">æstimatio rei,</hi> or root, of 9, is either 3 or - 3; of 16 it is 4 or - 4; the 4th root of 81 is 3 or - 3; and so on for all even <hi rend="italics">denominations</hi> or powers. And the same is remarked on compound cases of even powers that are added together; as if , then the æstimatio <hi rend="italics">x</hi> is=2 or-2; but that the form has four answers or roots, in real numbers, two plus and two minus, viz. 2 or - 2, and √ 3 or - √ 3; while the case has no real roots; and the case has two, namely 2 and - 2: and in like manner for other even powers. So that he includes both the positive and negative roots; but rejects what we now call imaginary ones. I here express the cases in our modern notation, for brevity sake, as he commonly expresses the terms by words at full length, calling the ablolute or known term the <hi rend="italics">numero,</hi> the 1st power the <hi rend="italics">res,</hi> the 2d the <hi rend="italics">quadratum,</hi> the 3d the <hi rend="italics">cubum,</hi> and so on, using no mark for the unknown <cb/> quantity, and only the initials <hi rend="italics">p</hi> and <hi rend="italics">m</hi> for plus and minus, and ℞ for radix or root. The <hi rend="italics">res</hi> he sometimes calls <hi rend="italics">positio,</hi> and <hi rend="italics">quantitas ignota;</hi> and in stating or setting down his equations, he, as well as Lucas de Burgo before him, sets down the terms on that side where they will be plus, and not minus.</p><p>On the other hand, he remarks that the odd denominations, or powers, have only one æstimatio, or root, and that true or positive, but none sictitious or negative, and for this reason, that no negative number raised to an odd power, will give a positive number; so of 2<hi rend="italics">x</hi>=16, the root is 8 only; and if 2<hi rend="italics">x</hi><hi rend="sup">3</hi>=16, the root is 2 only: and if there be ever so many odd denominations, added together, equal to a number, there will be only one æstimatio or root; as if , the only root is 2. But that when the signs of some of the terms are different as to plus and minus, they may have more roots; and he shews certain relations of the co-efficients, when they have two or more roots: so the equation has two æstimatios, the one true or 2, and the other fictitious or - 4, which he observes is the same as the true æstimatio of the case , having only the sign of the absolute number changed from the former, the 3d root 2 being the same as the first, which therefore he does not count. He next shews what are the relations of the co-efficients when a cubic equation has three roots, of which two are true, and the 3d fictitious, which is always equal to the sum of the other two, and also equal to the true root of the same equation with the sign of the absolute number changed: thus, in the equation , the two true roots are 3 and √5 1/4 - 1 1/2, and the fictitious one is - √ 5 1/4 - 1 1/2, which last is the same as the true root of , viz. √ 5 1/4 + 1 1/2; and he here infers generally that the fictitious æstimatio of the case , always answers to the true root of . Cardan also shews what the relation of the co-efficients is, when the case has no true roots, but only one fictitious root, which is the same as the true root of the reciprocal case, formed by changing the sign of the absolute number. Thus, the case has no true root, and only one false root, viz. - 3, which is the same as the true root of : and he shews in general, that changing the sign of the absolute number in such cases as want the 2d term, or changing the signs of the even terms when it is not wanting, changes the signs of all the three roots, which he also illustrates by many examples; thus, the roots of , are + √40 - 4, and - 3, and - √40 - 4; and the roots of , are - √40 + 4, and + 3, and + √40 + 4.</p><p>And he further observes, that the sum of the three roots, or the difference between the true and sictitious roots, is equal to 11, the co-efficient of the 2d term. He also shews how certain cubic cases have one, or two, or three roots, according to circumstances: that the case has sometimes four roots, and sometimes none at all, that is, no real ones: that the case may have three true æquatios, or positive roots, but no fictitious or negative ones; and for this reason, that the odd powers of minus being minus, and the even powers plus, the two terms <hi rend="italics">x</hi><hi rend="sup">3</hi>+<hi rend="italics">bx</hi> would be negative, and equal to a positive sum <hi rend="italics">ax</hi><hi rend="sup">2</hi> + <hi rend="italics">c,</hi> <pb n="69"/><cb/> which is absurd: and farther, that the case has three roots, one plus and two minus, which are the same, with the signs changed, as the roots of the case . He also shews the relation of the co-efficients when the equation has only one real root, in a variety of cases: but that the case has always one negative root, and either two positive roots, or none at all; the number of roots failing by pairs, or the impossible roots, as we now call them, being always in pairs. Of all these circumstances Cardan gives a great many particular examples in numeral co-efficients, and subjoins geometrical demonstrations of the properties here enumerated; such as, that the two corresponding or reciprocal cases have the same root or roots, but with different signs or affections; and how many true or positive roots each case has.</p><p>Upon the whole, it appears from this short chapter, that Cardan had discovered most of the principal properties of the roots of equations, and could point out the number and nature of the roots, partly from the signs of the terms, and partly from the magnitude and relations of the co-efficients. He shews in effect, that when the case has all its roots, or when none are impossible, the number of its positive roots is the same as the number of changes in the signs of the terms, when they are all brought to one side: that the co-efficient of the 2d term is equal to the sum of all the roots positive and negative collected together, and conseqnently that when the 2d term is wanting, the positive roots are equal to the negative ones: and that the signs of all the roots are changed, by changing only the signs of the even terms: with many other remarks concerning the nature of equations.</p><p>In chap. 2, Cardan enumerates all the cases of compound equations of the 2d and 3d order, namely, 3 quadratics, and 19 cubics; with 44 derivatives of these two, that is, of the same kind, with higher denominations.</p><p>In chap. 3 are treated the roots of simple cases, or simple equations, or at least that will reduce to such, having only two terms, the one equal to the other. He directs to depress the denominations equally, as much as they will, according to the height of the least; then divide by the number or co-efficient of the greatest; and lastly extract the root on both sides. So if 20<hi rend="italics">x</hi><hi rend="sup">3</hi> = 180<hi rend="italics">x</hi><hi rend="sup">5</hi>, then 20 = 180<hi rend="italics">x</hi><hi rend="sup">2</hi>, and 1/9=<hi rend="italics">x</hi><hi rend="sup">2</hi>, and <hi rend="italics">x</hi> = 1/3.</p><p>Chap. 4 treats of both general and particular roots, and contains various definitions and observations concerning them. It is here shewn that the several cases of quadratics and cubics have their roots of the following forms or kinds, namely that the case where the three parts √<hi rend="sup">3</hi>16, 2, √<hi rend="sup">3</hi>4, are in continual proportion.</p><p>Chap. 5 treats of the æstimatio of the lowest degree of compound cases, that is, affected quadratic equations; giving the rule for each of the three cases, which con- <cb/> sists in completing the square, &c, as at present, and which it seems was the method given by the Arabians; and proving them by geometrical demonstrations from Eucl. I. 43, and II. 4 and 5, in which he makes some improvement of the demonstrations of Mahomet. And hence it appears that the work of this Arabian author was in being, and well known in Cardan's time.</p><p>Chap. 6, on the methods of finding new rules, contains some curious speculations concerning the squares and cubes of binomial and residual quantities, and the proportions of the terms of which they consist, shewn from geometrical demonstrations, with many curious remarks and properties, forming a foundation of principles for investigating the rules for cubic equations.</p><p>Chap. 7 is on the transmutation of equations, shewing how to change them from one form to another, by taking away certain terms out of them; as , to , &c. The rules are demonstrated geometrically; and a table is added, of the forms into which any given cases will reduce; which transformations are extended to equations of the 4th and 5th order. And hence it appears that Cardan knew how to take away any term out of an equation.</p><p>Chap. 8 shews generally how to find the root of any such equation as this , where <hi rend="italics">m</hi> and <hi rend="italics">n</hi> are any exponents whatever, but <hi rend="italics">n</hi> the greater; and the rule is, to separate or divide the co-efficient <hi rend="italics">a</hi> into two such parts <hi rend="italics">z</hi> and <hi rend="italics">a</hi> - <hi rend="italics">z,</hi> as that the absolute number <hi rend="italics">b</hi> shall be equal to ―(<hi rend="italics">a</hi> - <hi rend="italics">z</hi>).<hi rend="italics">z</hi><hi rend="sup">m/(n-m)</hi>, the product of the one part <hi rend="italics">a</hi> - <hi rend="italics">z,</hi> and the <hi rend="italics">m</hi>/(<hi rend="italics">n</hi> - <hi rend="italics">m</hi>) power of the other part: then the root <hi rend="italics">x</hi> is = <hi rend="italics">z</hi><hi rend="sup">1/(n-m)</hi>. The rule is general for quadratics, cubics, and all the higher powers; and could not have been formed without the knowledge of the composition of the terms from the roots of the equation.</p><p>Chap. 9 and 10 contain the resolution of various questions producing equations not higher than quadratics.</p><p>Chap. 11 is of the case or form . Cardan now comes to the actual resolution of the first case of cubic equations. He begins with relating a short history of the invention of it, observing that it was first found out, about 30 years before, by Scipio Ferreus of Bononia, and by him taught to Antonio Maria Florido of Venice, who having a contest afterwards with Nicolas Tartalea of Brescia, it gave occasion to Tartalea to find it out himself, who after great entreaties taught it to Cardan, but suppressed the demonstration. By help of the rule alone, however, Cardan of himself discovered the source or geometrical investigation, which he gives here at large, from Eucl. II. 4. In this process he makes use of the Greek letters <foreign xml:lang="greek">a, <*>, g, d</foreign>, &c, to denote certain indefinite numbers or quantities, to render the investigation general; which may be considered as the first instance of such literal notation in Algebra. He then gives the rule in words at length, which comes to this, ; illustrating it in a variety of examples; in the resolution <pb n="70"/><cb/> of which, he extracts the cubic roots of such of the binomials as will admit of it, by some rule which he had for that purpose; such as , which .</p><p>Chap. 12, of the case . This he treats exactly as the last, and finds the rule ; which he illustrates by many examples, as usual. But when <hi rend="italics">b</hi><hi rend="sup">3</hi> exceeds <hi rend="italics">c</hi><hi rend="sup">2</hi>, which has since been called the irreducible case, he refers to another following book, called <hi rend="italics">Aliza,</hi> for other rules of solution, to overcome this difficulty, about which he took insinite pains.</p><p>Chap. 13, of the case . This case, by a geometrical process, he reduces to the case in the last chapter: thus, find the æstimatio <hi rend="italics">y</hi> of the case , having the same co-efficients as the given case ; then is , giving two roots. He shews also how to find the second root, when the first is known, independent of the foregoing case. From this relation of these two cases he deduces several corollaries, one of which is, that the æstimatio or root of the case , is equal to the sum of the roots of the case . As in the example , whose æstimatio is √(9 1/4 + 1 1/2), which is equal to the sum of 3 and √(9 1/4 - 1 1/2), the two roots of the case .</p><p>In chapters 14, 15, and 16, he treats of the three cases which contain the 2d and 3d powers, but wanting the first power, according to all the varieties of the signs; which he performs by exterminating the 2d term, or that which contains the 2d power of the unknown quantity <hi rend="italics">x,</hi> by substituting <hi rend="italics">y</hi> ± 1/3 the co-efficient of that term for <hi rend="italics">x,</hi> and so reducing these cases to one of the former. In these chapters Cardan sometimes also gives other rules; thus, for the case , find first the æstimatio <hi rend="italics">y</hi> of the case , then is : also for the case , first find the two roots of , then is <hi rend="italics">x</hi> = (√<hi rend="sup">3</hi>4<hi rend="italics">c</hi><hi rend="sup">2</hi>)/<hi rend="italics">y</hi> the two values of <hi rend="italics">x</hi> according to the two values of <hi rend="italics">y.</hi> He here also gives another rule, by which a second æstimatio or root is found, when the first is known, namely, if <hi rend="italics">e</hi> be the first estimatio or value of <hi rend="italics">x</hi> in the case , then is the other value of .</p><p>In chapters 17, 18, 19, 20, 21, 22, 23, Cardan treats of the cases in which all the four terms of the equation are present; and this he always effects by taking away the 2d term out of the equation, and so reducing it to one of the foregoing cases which want that term, giving always geometrical investigations, and adding a great many examples of every case of the equations.</p><p>Chap. 24, of the 44 derivative cases; which are only higher powers of the forms of quadratics and cubics.</p><p>Chap. 25, of imperfect and special cases; containing many particular examples when the co-efficients have certain relations amongst them, with easy rules for <cb/> finding the roots; also 8 other rules for the irreducible case .</p><p>Chap. 26, in like manner, contains easy rules for biquadratics, when the co-efficients have certain special relations.</p><p>Then the following chapters, from chap. 27 to chap. 38, contain a great number of questions and applications of various kinds, the titles of which are these: <hi rend="italics">De transitu capituli specialis in capitulum speciale; De operationibus radicum pronicarum seu mixtarum & Allellarum; De regula modi; De regula Aurea; De regula Magna,</hi> or the method of finding out solutions to certain questions; <hi rend="italics">De regula æqualis positionis,</hi> being a method of substituting for the half sum and half difference of two quantities, instead of the quantities themselves; <hi rend="italics">De regula inæqualiter ponendi, seu proportionis; De regula medii; De regula aggregati; De regula liberæ positionis; De regula falsum ponendi,</hi> in which some quantities come out negative; <hi rend="italics">Quomodo excidant partes & denominationes multiplicando.</hi> Among the foregoing collection of questions, which are chiefly about numbers, there are some geometrical ones, being the application of Algebra to Geometry, such as, In a rightangled triangle, given the sum of each leg and the adjacent segment of the hypotenuse, made by a perpendicular from the right angle, to determine the area &c; with other such geometrical questions, resolved algebraically.</p><p>Chap. 39, <hi rend="italics">De regula qua pluribus positionibus invenimus ignotam quantitatem</hi>; which is employed on biquadratie equations. After some examples of his own, Cardan gives a rule of Lewis Ferrari's, for resolving all biquadratics, namely by means of a cubic equation, which Ferrari investigated at his request, and which Cardan here demonstrates, and applies in all its cases. The method is very general, and consists in forming three squares, thus: first, complete one side of the equation up to a square, by adding or subtracting some multiples or parts of some of its own terms on both sides, which it is always easy to do: 2d, supposing now the three terms of this square to be but one quantity, viz, the first term of another square to which this same side is to be completed, by annexing the square of a new and assumed indeterminate quantity, with double the product of the roots of both; which evidently forms the square of a binomial, consisting of the assumed indeterminate quantity and the root of the first square: 3d, the other side of the equation is then made to become the square of a binomial also, by supposing the product of its ist and 3d terms to be equal to the square of half its 2d term; for it consists of only three terms, or three different denominations of the original unknown quantity: then this equality will determine the value of the assumed indeterminate quantity, by means of a cubic equation, and from it, that of the original ignota, by the equal roots of the 2d and 3d squares. Here we have a notable example of the use of assuming a new indeterminate quantity to introduce into an equation, long before Des Cartes was born, who made use of a like assumption for a similar purpose. And this method is very general, and is here applied to all forms of biquadratics, either having all their terms, or wanting some of them. To illustrate this rule I shall here set down the process of one of his examples, which is this, . Now first sub- <pb n="71"/><cb/> tract 2<hi rend="italics">x</hi><hi rend="sup">2</hi> + 4<hi rend="italics">x</hi> + 7 from both sides, then the first becomes a square, viz, . Next assume the indeterminate <hi rend="italics">y,</hi> and subtract 2<hi rend="italics">y</hi> (<hi rend="italics">x</hi><hi rend="sup">2</hi> - 1) - <hi rend="italics">y</hi><hi rend="sup">2</hi> from both sides, making the first side again a square, viz, . Of this latter side, make the product of the 1st and 3d terms equal to the square of half the 2d term, that is, , which reduces to ; the positive roots of which are <hi rend="italics">y</hi> = 2 or √15; and hence, using 2 for <hi rend="italics">y,</hi> the equation of equal squares becomes , the roots of which give ; and hence ; the two positive roots of which are √(3 + 1) and √(5 - 1), which are two of the values of <hi rend="italics">x</hi> in the given equation . The other roots he leaves to be tried by the reader.</p><p>The 40th, or last, chap. is entitled, Of modes of general supposition relating to this art; with some rules of an unusual kind; and æstimatios or roots of a nature different from the foregoing ones. Some of these are as follow: If , and , and <hi rend="italics">x : y :: c : d</hi>; then is .</p><p>Secondly, if , and , then is <hi rend="italics">x</hi> + <hi rend="italics">a : y</hi> - <hi rend="italics">a :: y</hi><hi rend="sup">2</hi> : <hi rend="italics">x</hi><hi rend="sup">2</hi>.</p><p>Thirdly, when , the square will be taken away, by putting ; and then the equation becomes .</p><p>Cardan adds some other remarks concerning the solutions of certain cases and questions, all evincing the accuracy of his skill, and the extent of his practice; and then he concludes the book with a remark concerning a certain transformation of equations, which quite astonishes us to find that the same person who, through the whole work, has shewn such a profound and critical skill in the nature of equations, and the solution of problems, should yet be ignorant of one of the most obvious transmutations attending them, namely increasing or diminishing the roots in any proportion. Cardan having observed that the form may be changed into another similar one, viz, , of which the co-efficient of the term <hi rend="italics">y</hi> is the quotient arising from the co-efficient of <hi rend="italics">x</hi> divided by the absolute number of the first equation: and that the absolute number of the 2d equation is the root of the quotient of 1 divided by the said absolute number of the first; he then adds, that finding the æstimatio or root of the one equation from that of the other is very difficult, <hi rend="italics">valde difficilis.</hi></p><p>It is matter of wonder that Cardan, among so many transmutations, should never think of substituting instead of <hi rend="italics">x</hi> in such equations, another positio or root, greater or less than the former in any indefinite proportion, that is, multiplied or divided by a given number; for this would have led him immediately to the same transformation as he makes above, and that by a way which would have shewn the constant proportion be- <cb/> tween the two roots. Thus, instead of <hi rend="italics">x</hi> in the given form , substitute <hi rend="italics">dy,</hi> and it becomes ; and this divided by <hi rend="italics">d</hi><hi rend="sup">3</hi> becomes ; and here if <hi rend="italics">d</hi> be taken = √<hi rend="italics">c,</hi> it becomes ; which is the transformation in question, and in which it is evident that <hi rend="italics">x</hi> is = <hi rend="italics">y√c,</hi> and <hi rend="italics">y</hi> = <hi rend="italics">x/√c.</hi> Instead of this, Cardan gives the following strange way of finding the one root <hi rend="italics">x</hi> from the other <hi rend="italics">y,</hi> when this latter is by any means known; viz, Multiply the first given equation by <hi rend="italics">y</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> + 1, then add <hi rend="italics">x</hi><hi rend="sup">2</hi>/4<hi rend="italics">y</hi><hi rend="sup">2</hi> to both sides, and lastly extract the roots of both, which can always be done, as they will always be both of them squares; and the roots will give the value of <hi rend="italics">x</hi> by a quadratic equation.</p><p>Thus, multiplied by <hi rend="italics">y</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> + 1 gives ; and theroots are ; and this 2d side of the equation he says will always have a root also. It is indeed true that it will have an exact root; but the reason of it is not obvious, which is, because <hi rend="italics">y</hi> is the root of the equation . Cardan has not shewn the reason why this happens; but I apprehend he made it out in this manner, viz, similar to the way in which he forms the last square in the case of biquadratic equations, namely, by making the product of the 1st and 3d terms equal to the square of half the 2d term: thus, in the present case, it is , which reduces to the equation in question. Therefore taking <hi rend="italics">y</hi> the root of the equation , and substituting its value in the quantity , this will become a complete square. <hi rend="center"><hi rend="italics">Of</hi> Cardan's <hi rend="italics">Libellus de Aliza Regula.</hi></hi></p><p>Subjoined to the above Treatise on cubic equations, is this <hi rend="italics">Libellus de Aliza regula,</hi> or the algebraic logistics, in which the author treats of some of the abstruser parts of Arithmetic and Algebra, especially cubic-equations, with many more attempts on the irreducible case . This book is divided into 60 chapters; <pb n="72"/><cb/> but I shall only set down the titles of some few of them, whose contents require more particular notice.</p><p>Chap. 4. <hi rend="italics">De modo redigendi quantitates omnes, quæ dicuntur latera prima ex decimo Euclidis in compendium.</hi> He treats here of all Euclid's irrational lines, as surd numbers, and persorms various operations with them.</p><p>Chap. 5. <hi rend="italics">De consideratione binomiorum & recisorum, &c; ubi de æstimatione capitulorum.</hi> Contains various operations of multiplying compound numbers and surds.</p><p>Chap. 6. <hi rend="italics">De operationibus</hi> p: & m: (i. e. + and -) <hi rend="italics">secundum communem usum.</hi> Here it is shewn that, in multiplication and division, <hi rend="italics">plus</hi> always gives the same signs, and <hi rend="italics">minus</hi> gives the contrary signs. So also in addition, every quantity retains its own sign; but in subtraction they change the signs. That the √ +, or the square root of plus, is +; but the √ -, or the square root of minus, is nothing as to common use: (but of this below.) That √<hi rend="sup">3</hi> - is -; as √ - 8 is - 2. That a residual, composed of + and - may have a root also composed of + and -: So √(5-√24) is = √3-√2. The rules for the signs in multiplication and division are illustrated by this example; to divide 8 by 2 + √6 or √6 + 2. Take the two corresponding residuals 2 - √6 and √6 - 2, and by these multiply both the divisor and dividend; then the products are + and - respectively, and the quotients still both alike. Thus, <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Divid.</cell><cell cols="1" rows="1" rend="align=center" role="data">Divis.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">|</cell><cell cols="1" rows="1" rend="align=right" role="data">Divid.</cell><cell cols="1" rows="1" rend="align=center" role="data">Divis.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">√6 + 2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">2 + √6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">√6 - 2</cell><cell cols="1" rows="1" rend="align=center" role="data">√6 - 2</cell><cell cols="1" rows="1" rend="align=right" role="data">2 - √6</cell><cell cols="1" rows="1" rend="align=center" role="data">2 - √6</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">√384 - 16 divide + 4</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">16 - √384 div. - 2</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Quot. √96-8.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Quot. √96 - 8.</cell></row></table> And this method of performing division of compound surds, was fully taught before him, by Lucas de Burgo, namely, reducing the compound divisor to a simple quantity, by multiplying by the corresponding quantity, having the sign changed.</p><p>In chap. 11 and 18, and elsewhere, Cardan makes a general notation of <hi rend="italics">a, b, c, d, e, f,</hi> for any indefinite quantities, and treats of them in a general way.</p><p>Cap. 2. <hi rend="italics">De contemplatione</hi> p: & m: (or + and -), <hi rend="italics">& quod</hi> m: <hi rend="italics">in</hi> m: <hi rend="italics">facit</hi> p: <hi rend="italics">& de causis horum juxta veritatem.</hi> Cardan here demonstrates geometrically that, in multiplication and division, like signs give plus, and unlike signs give minus. And he illustrates this numerically, by squaring the quantity 8, or 6 + 2, or 10 - 2, which must all produce the same thing, namely 64.</p><p>Among many of the chapters which treat of the irreducible case , there is a peculiar kind of way given in chap. 31, which is entitled <hi rend="italics">De æstimatione generali</hi> <hi rend="italics">solida vocata, & operationibus ejus;</hi> in which he shews how to approximate to the root of that case, in a manner similar to approximating the square root and cube root of a number. The rule he uses for this purpose, is the 3d in chap. 25 of the last book, and it is this: Divide <hi rend="italics">b</hi> into two parts, such that the sum of the products of each, multiplied by the square of the other, may be equal to (1/2)<hi rend="italics">c</hi>; then the sum of the roots of these parts is the æstimatio or value <figure/> of <hi rend="italics">x</hi> required. So, of this equation ; the two parts are 9 and 1, and their roots 3 and 1, and their sum 4=<hi rend="italics">x,</hi> as in the margin. <cb/> Again, take . Here he invents a new notation to express the root or <hi rend="italics">radix,</hi> which he calls <hi rend="italics">solida,</hi> viz, <hi rend="italics">x</hi>=√ <hi rend="italics">solida</hi> 6 <hi rend="italics">in</hi> 1/2, that is, the roots of the two parts of 6, so that each part multiplied by the root of the other, the two products may be 1/2 or (1/2)<hi rend="italics">c.</hi> Then to free this from fractions, and make the operation easier, multiply that root by some number as suppose 4, that is the square part 6 by the square of 4, and the solid part 1/2 by the cube of 4; then <hi rend="italics">x</hi>=1/4√ <hi rend="italics">solida</hi> 96 <hi rend="italics">in</hi> 32. Now, by a few trials, it is found that the parts are nearly 95 8/9 and 1/9, which give too much, or 95 9/10 and 1/10, which give too little, and thereof 95 17/19 and 2/19 are still nearer. Divide both by 4<hi rend="sup">2</hi> or 16, then 5 151/152 and 1/152 are the quot. And the sum of their roots, or is nearly the value of the root <hi rend="italics">x.</hi></p><p><hi rend="italics">Cap.</hi> 42. <hi rend="italics">De duplici æquatione comparanda in capitulo cubi & numeri æqualium rebus.</hi> Treats of the two positive roots of that case, neglecting the negative one; and shewing, not only that that case has two such roots, but that the same number may be the common root of innumerable equations.</p><p><hi rend="italics">Cap.</hi> 57. <hi rend="italics">Detractatione æstimationis generalis capituli</hi> . Cardan here again resumes the consideration of the irreducible case, making ingenious observations upon it, but still without obtaining the root by a general rule. In this place also, as well as elsewhere, he shews how to form an equation in this case, that shall have a given binomial root, as suppose √<hi rend="italics">m</hi> + <hi rend="italics">n,</hi> where the equation will be , having √<hi rend="italics">m</hi> + <hi rend="italics">n</hi> for one root, namely the positive root. From which it appears that he was well acquainted with the composition of cubic equations from given roots.</p><p><hi rend="italics">Cap.</hi> 59. <hi rend="italics">De ordine & exemplis in binomiis secudo & quinto.</hi> Contains a great many numeral forms of the same irreducible case , with their roots; from which are derived these following cases, with many curious remarks. When</p><p><hi rend="italics">Cap.</hi> 60. <hi rend="italics">Demonstratio generalis capituli cubi æqualis rebus & numero.</hi> This demonstration of the irreducible case is geometrical, like all the rest. Some more ingenious remarks are again added, as if he reluctantly finished the book without perfectly overcoming the difficulty of the irreducible case. Cardan here also uses the letters <hi rend="italics">a</hi> and <hi rend="italics">b</hi> for any two indefinite numbers, in order to shew the form and manner of the arithmetical operations: thus <hi rend="italics">a/b</hi> is the fraction for their quotient, also √<hi rend="italics">a/b</hi> or √<hi rend="italics">a/√b</hi> the square root of that quotient, and √<hi rend="sup">3</hi><hi rend="italics">a/b</hi> or √<hi rend="sup">3</hi><hi rend="italics">a</hi>/√<hi rend="sup">3</hi><hi rend="italics">b</hi> the cube root of it, &c. <pb n="73"/><cb/></p><p>Having considered the chief contents of Cardan's algebra, it will now be proper to sum them up, and set down a list of the improvements made by him, as collected from his writings:</p><p>And 1st, Tartalea having only communicated to him the rules for resolving these three cases of cubic equations, viz, having all their terms, or wanting any of them, and having all possible varieties of signs; demonstrating all these rules geometrically; and treating very fully of almost all sorts of transformations of equations, in a manner heretofore unknown.</p><p>2nd, It appears that he was well acquainted with all the roots of equations that are real, both positive and negative; or, as he calls them, true and fictitious; and that he made use of them both occasionally. He also shewed, that the even roots of positive quantities, are either positive or negative; that the odd roots of negative quantities, are real and negative; but that the even roots of them are impossible, or nothing as to common use. He was also acquainted with,</p><p>3d, The number and nature of the roots of an equation, and that partly from the signs of the terms, and partly from the magnitude and relation of the coefficients. He also knew,</p><p>4th, That the number of positive roots is equal to the number of changes of the signs of the terms.</p><p>5th, That the coefficient of the second term of the equation, is the difference between the positive and negative roots.</p><p>6th, That when the second term is wanting, the sum of the negative roots is equal to the sum of the positive roots.</p><p>7th, How to compose equations that shall have given roots.</p><p>8th, That, changing the signs of the even terms, changes the signs of all the roots.</p><p>9th, That the number of roots failed in pairs; or what we now call impossible roots were always in pairs.</p><p>10th, To change the equation from one form to another, by taking away any term out of it.</p><p>11th, To increase or diminish the roots by a given quantity. It appears also,</p><p>12th, That he had a rule for extracting the cube root of such binomials as admit of extraction.</p><p>13th, That he often used the literal notation <hi rend="italics">a, b, c, d,</hi> &c.</p><p>14th, That he gave a rule for biquadratic equations, suiting all their cases; and that, in the investigation of that rule, he made use of an assumed indeterminate quantity, and afterwards found its value by the arbitrary assumption of a relation between the terms.</p><p>15th, That he applied Algebra to the resolution of geometrical problems. And</p><p>16th, That he was well acquainted with the difficulty of what is called the irreducible case, viz, , upon which he spent a great deal of time, in attempting to overcome it. And though he did not fully succeed in this case, any more than other persons have done since, he nevertheless made many ingenious observations about it, laying down rules for many particular <cb/> forms of it, and shewing how to approximate very nearly to the root in all cases whatever. <hi rend="center"><hi rend="smallcaps">OF TARTALEA.</hi></hi></p><p>Nicholas Tartalea, or Tartaglia, of Brescia, was contemporary with Cardan, and was probably older than he was, but I do not know of any book of Algebra published by him till the year 1546, the year after the date of Cardan's work on Cubic Equations, when he printed his <hi rend="italics">Quesiti & Inventioni diverse,</hi> at Venice, where he resided as a public lecturer on mathematics. This work is dedicated to our king Henry the VIIIth of England, and consists of 9 books, containing answers to various questions which had been proposed to him at different times, concerning mechanics, statics, hydrostatics, &c.; but it is only the 9th, or last, that we shall have occasion to take notice of in this place, as it contains all those questions which relate to arithmetic and algebra. These are all set down in chronological order, forming a pretty collection of questions and solutions on those subjects, with a short account of the occasion of each of them. Among these, the correspondence between him and Cardan forms a remarkable part, as we have here the history of the invention of the rules for cubic equations, which he communicated to Cardan. under the promise, and indeed oath, to keep them secret, on the 25th of March 1539. But, notwithstanding his oath, finding that Cardan published them in 1545, as above related, it seems Tartalea published the correspondence between them in revenge for his breach of faith; and it elsewhere appears, that many other sharp bickerings passed between them on the same account, which only ended with the death of Tartalea, in the year 1557. It seems it was a common practice among the mathematicians, and others, of that time, to send to each other nice and difficult questions, as trials of skill, and to this cause it is that we owe the principal questions and discoveries in this collection, as well as many of the best discoveries of other authors. The collection now before us contains questions and solutions, with their dates, in a regular order, from the year 1521, and ending in .1541, in 42 dialogues, the last of which is with an English gentleman, namely, Mr. Richard Wentworth, who it seems was no mean mathematician, and who learned some algebra, &c, of Tartalea, while he resided at Venice. The questions at first are mostly very easy ones in arithmetic, but gradually become more difficult, and exercising simple and quadratic equations, with complex calculations of radical quantities: all shewing that he was well skilled in the art of Algebra as it then stood, and that he was very ingenious in applying it to the solutions of questions. Tartalea made no alteration in the notation or forms of expression used by Lucas de Burgo, calling the first power of the unknown quantity, in his language, <hi rend="italics">cosa,</hi> the second power <hi rend="italics">censa,</hi> the third <hi rend="italics">cubo,</hi> &c, and writing the names of all the operations in words at length, without using any contractions, except the initial R for root or radicality. So that the only thing remarkable in this collection, is the discovery of the rules for cubic equations, with the curious circumstances attending the same. <pb n="74"/><cb/></p><p>The first two of these were discovered by Tartalea in the year 1530, namely for the two cases , and , as appears by Quest. 14 and 25 of this collection, on occasion of a question then proposed to him by one Zuanni de Tonini da Coi or Colle, John Hill, who kept a school at Brescia. And from the 25th letter we learn, that he discovered the rules for the other two cases , and , on the 12th and 13th of February 1535, at Venice, where he had come to reside the year before. And the occasion of it was this: There was then at Venice one Antonio Maria Fiore or Florido, who, by his own account, had received from his preceptor Scipio Farreo, about thirty years before, a general rule for resolving the case . Being a captious man, and presuming on this discovery, Florido used to brave his contemporaries, and by his insults provoked Tartalea to enter into a wager with him, that each should propose to the other thirty different questions; and that he who soonest resolved those of his adversary, should win from him as many treats for himself and friends. These questions were to be proposed on a certain day at some weeks distance; and Tartalea made such good use of his time, that eight days before the time appointed for delivering the propositions, he discovered the rules both for the case , and the case . He therefore proposed several of his questions so as to fall either on this latter case, or on the cases of the cube and square, expecting that his adversary would propose his in the former. And what he suspected fell out accordingly; the consequence of which was, that on the day of meeting Tartalea resolved all his adversary's questions in the space of two hours, without receiving one answer from Florido in return; to whom, however, Tartalea generously remitted the forfeit of the thirty treats won of him.</p><p>Question 31 first brings us acquainted with the correspondence between Tartalea and Cardan. This correspondence is very curious, and would well deserve to be given at full length in their own words, if it were not too long for this place. I may enlarge farther upon it under the article <hi rend="italics">Cubic Equations</hi>; but must here be content with a brief abstract only. Cardan was then a respectable physician, and lecturer in mathematics at Milan; and having nearly finished the printing of a large work on Arithmetic, Algebra, and Geometry, and having heard of Tartalea's discoveries in cubic equations, he was very desirous of drawing those rules from him, that he might add them to his book before it was finished. For this purpose he first applied to Tartalea, by means of a third person, a bookseller, whom he sent to him, in the beginning of the year 1539, with many flattering compliments, and offers of his services and friendship, &c, accompanied with some critical questions for him to resolve, according to the custom of the times. Tartalea however refused to disclose his rules to any one, as the knowledge of them gained him great reputation among all people, and gave him a great advantage over his competitors for fame, who were commonly afraid of him on account of those very rules. He only sent Cardan therefore, at his request, a copy of the thirty questions which had been proposed to him in the contest with Florido. Not to be rebuffed so easily, Cardan next applied, in the most urgent manner, <cb/> by letter to Tartalea; which however procured from him only the solution of some other questions proposed by Cardan, with a few of the questions that had been proposed to Florido, but none of their solutions. Finding he could not thus prevail, with all his fair promises, Cardan then fell upon another scheme. There was at Milan a certain Marquis dal Valsto, a great patron of Cardan, and, it was said, of learned men in general. Cardan conceived the idea of making use of the influence of this nobleman to draw Tartalea to Milan, hoping that then, by personal intreaties, he should succeed in drawing the long-concealed rules from him. Accordingly he wrote a second letter to Tartalea, much in the same strain with the former, strongly inviting him to come and spend a few days in his house at Milan, and representing that, having often commended him in the highest terms to the marquis, this nobleman desired much to see him; for which reason Cardan advised him, as a friend, to come to visit them at Milan, as it might be greatly to his interest, the marquis being very liberal and bountiful; and he besides gave Tartalea to understand, that it might be dangerous to offend such a man by refusing to come, who might, in that case, take offence, and do him some injury. This manœuvre had the desired effect: Tartalea on this occasion laments to himself in these words, “By this I am reduced to a great dilemma; for if I go not to Milan, the marquis may take it amiss, and some evil may befal me on that account; I shall therefore go, although very unwillingly.” When he arrived at Milan however, the marquis was gone to Vigeveno, and Tartalea was prevailed on to stay three days with Cardan, in expectation of the marquis returning, at the end of which he set out from Milan, with a letter from Cardan, to go to Vigeveno to that nobleman. While Tartalea was at Milan the three days, Cardan plied him by all possible means to draw from him the rules for the cubic equations; and at length, just as Tartalea was about to depart from Milan, on the 25th of March 1539, he was overcome by the most solemn protestations of secrecy that could be made. Cardan says, “I shall swear to you on the holy evangelists, and by the honour of a gentleman, not only never to publish your inventions, if you reveal them to me; but I also promise to you, and pledge my faith as a true christian, to note them down in cyphers, so that after my death no other person may be able to understand them.” To this Tartalea replies, “If I refuse to give credit to these assurances, I should deservedly be accounted utterly void of belief. But as I intend to ride to Vigeveno, to see his excellency the marquis, as I have been here now these three days, and am weary of waiting so long; whenever I return therefore, I promise to shew you the whole.” Cardan answers, “Since you determine at any rate to go to Vigeveno, to the marquis, I shall give you a letter for his excellency, that he may know who you are. But now before you depart, I intreat you to shew me the rule for the equations, as you have promised.” “I am content,” says Tartalea: “But you must know, that to be able on all occasions to remember such operations, I have brought the rule into rhyme; for if I had not used that precaution, I should often have forgot it; and although my rhymes are not very good, I do not value that, as it is <pb n="75"/><cb/> sufficient that they serve to bring the rule to mind as often as I repeat them. I shall here write the rule with my own hand, that you may be sure I give you the discovery exactly.” These rude verses contain, in rather dark and enigmatical language, the rule for these three cases, viz. that their difference in the first case, and their sums in the 2d and 3d, may be equal to <hi rend="italics">c</hi> the absolute number, and their product equal to the cube of 1/3 of <hi rend="italics">b</hi> the coefficient of the less power; then the difference of their cube roots will be equal to <hi rend="italics">x</hi> in the first case, and the sum of their cube roots equal to <hi rend="italics">x</hi> in the 2d and 3d cases: that is, taking in the 1st case, or in the 2d and 3d, and ; then in the first case, and in the other two. At parting, T, fails not again to remind C. of his obligation: “Now your excellency will remember not to break your promised faith, for if unhappily you should insert these rules either in the work you are now printing, or in any other, although you should even give them under my name, and as of my invention, I promise and swear that I shall immediately print another work that will not be very pleasing to you.” “Doubt not, says C. but that I shall observe what I have promised: Go, and rest secure as to that point: and give this letter of mine to the marquis.” It should seem however that T. was much displeased at having suffered himself to be worried as it were out of his rules, for as soon as he quitted Milan, instead of going to wait upon the marquis, he turned his horse's head, and rode straight home to Venice, saying to himself, “By my faith I shall not go to Vigeveno, but shall return to Venice, come of it what will.”</p><p>After T's departure it seems C. applied himself immediately to resolving some examples in the cubic equations by the new rules, but not succeeding in them, for indeed he had mistaken the words, as it was very easy to do in such bad verses, having mistaken ((1/3)<hi rend="italics">b</hi>)<hi rend="sup">3</hi> for (1/3)<hi rend="italics">b</hi><hi rend="sup">3</hi>, or the cube of 1/3 of the coefficient, for 1/3 of the cube of the coefficient; accordingly we find him writing to T. in fourteen days after the above, blaming him much for his abrupt departure without seeing the marquis, who was so liberal a prince he said, and requesting T. to resolve him the example . This T. did to his satisfaction, rightly guessing at the nature of his mistake; and concludes his answer with these emphatical words, “Remember your promise.” On the 12th of May following C. returns him a letter of thanks, together with a copy of his book, saying, “As to my work, just finished, to remove your suspicion, I send you a copy, but unbound, as it is yet too fresh to be beaten. But as to the doubt you express lest I may print your inventions, my faith which I gave you with an oath should satisfy you; for as to the finishing of my book, that could be no security, as I could always add to it whenever I please. But on account of the dignity of the thing, I excuse you for not relying on that which you ought to have done, namely on the faith of a gentleman, instead of the finishing of a book, which might at any time be enlarged by the addition of new chapters; and there are besides a thousand other ways. But the security consists in this, that there is no <cb/> greater treachery than to break one's faith, and to ag<*> grieve those who have given us pleasure. And when you shall try me, you will find whether I be your friend or not, and whether I shall make an ungrateful return for your friendship, and the satisfaction you have given me.”</p><p>It was within less than two months after this, however, that T. received the alarming news of Cardan's shewing some symptoms of breaking the faith he had so lately pledged to him; this was in a letter from a quondam pupil of his, in which he writes, “A friend of mine at Milan has written to me, that Dr. Cardano is composing another algebraical work, concerning some latelydiscovered rules; hence I imagine they may be those same rules which you told me you had taught him; so that I fear he will deceive you.” To which T. replies, “I am heartily grieved at the news you inform me of, concerning Dr. Cardano of Milan; for if it be true, they can be no other rules but those I gave him; and therefore the proverb truly says, ‘That which you wish not to be known, tell to nobody.’ Pray endeavour to learn more of this matter, and inform me of it.”</p><p>Tartalea, after this, kept on the reserve with Cardan, not answering several letters he sent him, till one written on the 4th of August the same year, 1539, complaining greatly of T's neglect of him, and farther requesting his assistance to clear up the difficulty of the irreducible case , which C. had thus early been embarrassed with: he says that when ((1/3)<hi rend="italics">b</hi>)<hi rend="sup">3</hi> exceeds ((1/2)<hi rend="italics">c</hi>)<hi rend="sup">2</hi>, the rule cannot be applied to the equation in hand, because of the square root of the negative quantities. On this occasion T. turns the tables on C. and plays his own game back upon him; for being aware of the above difficulty, and unable to overcome it himself, he wanted to try if C. could be encouraged to accomplish it, by pretending that the case might be done, though in another way. He says thus to himself, “I have a good mind to give no answer to this letter, no more than to the other two. However I will answer it, if it be but to let him know what I have been told of him. And as I perceive that a suspicion has arisen concerning the difficulty or obstacle in the rule for the case . I have a mind to try if he can alter the data in hand, so as to remove the said obstacle, and to change the rule into another form, although I believe indeed that it cannot be done; however there is no harm in trying.”—“M. Hieronime, I have received your letter, in which you write that you understand the rule for the case , but that when ((1/3)<hi rend="italics">b</hi>)<hi rend="sup">3</hi> exceeds ((1/2)<hi rend="italics">c</hi>)<hi rend="sup">2</hi>, you cannot resolve the equation by following the rule, and therefore you request me to give you the solution of this equation . To which I reply, that you have not used a good method in that case, and that your whole process is intirely false. And as to resolving you the equation you have sent, I must say that I am very sorry that I have already given you so much as I have done, for I have been informed, by a credible person, that you are about to publish another Algebraical work, and that you have been boasting through Milan of having discovered some new rules in algebra. But, take notice, that if you break your faith with me, I shall certainly keep my word with you, nay, I even assure you to do more than I promised.” In Cardan's answer to this he says, “You have been misinformed as to my intention to <pb n="76"/><cb/> publish more on Algebra. But I suppose you have heard something about my work <hi rend="italics">de mysteriis æternitatis,</hi> which you take for some Algebra I intend to publish. As to your repenting of having given me your rules, I am not to be moved from the faith I promised you for any thing you say.” To this, and many other things contained in the same letter, T. returned no answer, being still suspicious of Cardan's intentions, and declining any more correspondence with him. This however did not discourage C. for we find him writing again to T. on the 5th of January, 1540, to clear up another difficulty which had occurred in this business, namely to extract the cube root of the binomials, of which the two parts of the rule always consisted, and for which purpose it seems C. had not yet found out a rule. On this occasion he informs T. that his quondam competitor Zuanne Colle had come to Milan, where, in some contests between them, Colle gave Cardan to understand that he had found out the rules for the two cases , and , and farther that he had discovered a general rule for extracting the cube roots of all such binomials as can be extracted; and that, in particular, the cube root of √(108 + 10) is √(3 + 1), and that of √(108 - 10) is √(3 - 1), and consequently that . He then earnestly entreats T. to try to find out the rule, and the solution of certain other questions which had been proposed to him by Colle. By this letter T. is still more confirmed in his resolution of silence; so that, without returning any answer, he only sets down among his own memorandums some curious remarks on the contents of the letter, and then concludes to himself, “Wherefore I do not choose to answer him again, as I have no more affection for him than for M. Zuanne, and therefore I shall leave the matter between them.” Among those remarks he sets down a rule for extracting the cube root of such binomials as can be extracted, and that is done from either member of the binomial alone, thus: Take either term of the binomial, and divide it into two such parts that one of them may be a complete cube, and the other part exactly divisible by 3; then the cube root of the said cubic part will be one term of the required root, and the square root of the quotient arising from the division of 1/3 of the 2d part by the cube root of the first, will be the other member of the root sought. This rule will be better understood in characters thus: let <hi rend="italics">m</hi> be one member of the given binomial, whose cube root is sought, and let it be divided into the two parts <hi rend="italics">a</hi><hi rend="sup">3</hi> and 3<hi rend="italics">b,</hi> so that <hi rend="italics">a</hi><hi rend="sup">3</hi> + 3<hi rend="italics">b</hi> be = <hi rend="italics">m</hi>; then is <hi rend="italics">a</hi> + √<hi rend="italics">b/a</hi> the cube root required, if it have one. Thus in the quantity √(108+10), taking the term 10 for <hi rend="italics">m,</hi> then 10 divides into 1 and 9, where <hi rend="italics">a</hi><hi rend="sup">3</hi>=1 or <hi rend="italics">a</hi>=1, and 3<hi rend="italics">b</hi>=9 or <hi rend="italics">b</hi>=3: therefore <hi rend="italics">a</hi> + √<hi rend="italics">b/a</hi> becomes 1 + √3 for the cube root of √(108 + 10). And taking the other member √108, this divides into the two equal parts √27 and √27, making <hi rend="italics">a</hi><hi rend="sup">3</hi> = √27, and 3<hi rend="italics">b</hi>=√27; hence <hi rend="italics">a</hi>=√3, and <hi rend="italics">b</hi>=√3 also; consequently <hi rend="italics">a</hi> + √<hi rend="italics">b/a</hi> is = √3 + √3/3 or √(3 + 1) for the cube root of the binomial sought, the same as before. “And thus, he adds, we may know whether any proposed binomial <cb/> or residual be a cube or a noncube; for if it be a cube, the same two terms for the root must arise from both the given terms separately; and if the two terms of the root cannot thus be brought to agree both ways, such binomial or residual will not be a cube.” And thus ends the correspondence between them, at least for this time. But it seems they had still more violent disputes when C. in violation of his faith so often pledged to the contrary, published his work on cubic equations 4 years afterwards, viz, in the year 1545, of which we have before given an account, which disputes, it is said, continued till the death of Tartalea in the year 1557.</p><p>The last article in the volume contains a dialogue on some other forms of the cubic equations, in the year 1541, between T. and a Mr. Richard Wentworth, an English gentleman, who it seems had resided some time at Venice, on some public service from England, as T. in the dedication of the volume to Henry VIII. king of England, makes mention of him as “a gentleman of his sacred majesty.” Mr. Wentworth had learned some mathematics of T. and being about to depart for England, requests T. to shew him his newly discovered rules for cubic equations, as a farewell-lesson; and it is worth while to note a few particulars in this conference, as they shew pretty nicely the limited knowledge of T. at that time, as to the nature and roots of such equations. T. had before, it seems, shewed Mr. W. the rules for the cases of the 3d and 1st powers, and now the latter desires him to do the same as to the three cases in which the 3d and 2d powers only are concerned. On this T. professes great gratitude to Mr. W. for many obligations, but desires to be excused from giving him the rules for these, because he says he intends soon to compose a new work on Arithmetic, Geometry, and Algebra, which he intends to dedicate to him, and in which he means to insert all his new discoveries. On Mr. W. urging him further, T. gives him the roots of some equations of that kind, as for instance: but not the rules for finding them.</p><p>In the course of the conversation T. tells him that “all such equations admit of two different answers, and perhaps more; and hence it follows that they have, or admit of, two different rules, and perhaps more, the one more difficult than the other.” And on Mr. W. expressing his wonder at this circumstance of a plurality of roots, T. replies, “It is however very true, though hardly to be believed, and indeed if experience had not confirmed it, I should scarcely have believed it myself.” He then commits a strange blunder in an example which he takes to illustrate this by, namely the equation , which, he says, it is evident has the number 2 for one of its roots; and yet, he adds, “whoever shall resolve the same equation by my rule, will find the value of <hi rend="italics">x</hi> to be √<hi rend="sup">3</hi>(7 + √50) + √<hi rend="sup">3</hi>(7 - √50), which is proved to be a true root by sub- <pb n="77"/><cb/> stituting it in the equation for <hi rend="italics">x.</hi> And therefore, continues he, it is manifest that the case <hi rend="italics">x</hi><hi rend="sup">3</hi> + <hi rend="italics">bx</hi> = <hi rend="italics">c</hi> admits of two rules, namely, one (as in the above example) which ought to give the value of <hi rend="italics">x</hi> rational, viz 2, and the other is my rule, which gives the value of <hi rend="italics">x</hi> irrational, as appears above; and there is reason to think that there may be such a rule as will give the value of <hi rend="italics">x</hi> = 2, although our ancestors may not have found it out.”——“And these two different answers will be found not only in every equation of this form , when the value of <hi rend="italics">x</hi> happens to be rational, as in the example above, but the same will also happen in all the other five forms of cubic equations: and therefore there is reason to think that they also admit of two different rules; and by certain circumstances attending some of them, I am almost certain that they admit of more than two rules, as, God willing, I shall soon demonstrate.” Now all this discourse shews a strange mixture of knowledge and ignorance: it is very probable that he had met with some equations which admit of a plurality of roots; indeed it was hardly possible for him to avoid it; but it seems he had no suspicion what the number of roots might be, nor that his reasoning in this instance was founded on an error of his own, mistaking the root , of the equation , for a different root from the number or root 2, when in reality it is the very same, as he might easily have found, if he had extracted the cube roots of the binomials by the rule which he himself had just given above for that purpose: for by that rule he would have found , and , and therefore their sum is 2 = <hi rend="italics">x,</hi> the same root as the other, which T. thought had been different. And besides this root 2, the equation in hand, , admits of no other real roots. Nor does any equation of the same form, , admit of more than one real root.</p><p>It seems also they had not yet discovered that all cases belong to the rules and forms for quadratic equations, which have only two powers in them, in which the exponent of the one is just double of the exponent of the other, as ; but some particular cases only of this sort they had as yet ventured to refer to quadratics, as the case . But in the conclusion of this dialogue T. informs W. of another case of this sort which he had accomplished, as a notable discovery, in these words: “I well remember, says he, that in the year 1536, on the night of St. Martin, which was on a Saturday, meditating in bed when I could not sleep, I discovered the general rule for the case , and also for the other two, its accompanying cases, in the same night.” And then he directs that they are to be resolved like quadratics, by completing the square, &c. And in these resolutions it is remarkable that he uses only the positive roots, without taking any notice of the negative ones.</p><p>Tartalea also published at Venice, in 1556, &c, a very large work, in folio, on Arithmetic, Geometry, and Algebra. This is a very complete and curious work upon the first two branches; but that of Algebra is carried no farther than quadratic equations, <cb/> called <hi rend="italics">book the first,</hi> with which the work terminates. It is evidently incomplete, owing to the death of the author, which happened before this latter part of the work was printed, as appears by the dates, and by the prefaces. It appears also, from several parts of this work, that the author had many severe conflicts with Cardan and his friend Lewis Ferrari: and particularly, there was a public trial of skill between them, in the year 1547; in which it would seem that Tartalea had greatly the advantage, his questions mostly remaining unanswered by his antagonists. <hi rend="center"><hi rend="smallcaps">OF MICHAEL STIFELIUS.</hi></hi></p><p>After the foregoing analysis of the works of the first algebraic writers in Italy, it will now be proper to consider those of their contemporaries in Germany; where, excepting for the discoveries in cubic equations, the art was in a more advanced state, and of a form approaching nearer to that of our modern Algebra; the state and circumstances indeed being so different, that one would almost be led to suppose they had derived their knowledge of it from a different origin.</p><p>Here Stifelius and Scheubelius were writers of the same time with Cardan and Tartalea, and even before their discoveries, or publication, concerning the rules for cubic equations, Stifelius's <hi rend="italics">Arithmetica Integra</hi> was published at Norimberg in 1544, being the year before Cardan's work on cubic equations, and is an excellent treatise, both on Arithmetic and Algebra. The work is divided into three books, and is prefaced with an Introduction by the famous Melanchthon. The first book contains a complete and ample Treatise on Arithmetic, the second an Exposition of the 10th book of Euclid's Elements, and the third a Treatise of Algebra, and it is therefore properly the part with which we are at present concerned. In the dedication of this part, he ascribes the invention of Algebra to Geber, an Arabic Astronomer; and mentions besides, the authors Campanus, Christ. Rudolph, and Adam Ris, Risen, or Gigas, whose rules and examples he has chiefly given. In other parts of the book he speaks, and makes use also, of the works of Bretius, Campanus, Cardan (i. e. his Arithmetic published in 1539, before the work on cubic equations appeared), de Cusa, Euclid, Jordan, Milichius, Schonerus, and Stapulensis.</p><p><hi rend="italics">Chap.</hi> 1. <hi rend="italics">Of the Rule of Algebra, and its parts.</hi> Stifelius here describes the notation and marks of powers, or denominations as he calls them, which marks for the several powers are thus: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1st,</cell><cell cols="1" rows="1" rend="align=right" role="data">2d,</cell><cell cols="1" rows="1" rend="align=right" role="data">3d,</cell><cell cols="1" rows="1" rend="align=right" role="data">4th,</cell><cell cols="1" rows="1" rend="align=center" role="data">5th,</cell><cell cols="1" rows="1" rend="align=right" role="data">6th,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><figure/>,</cell><cell cols="1" rows="1" rend="align=right" role="data">[dram],</cell><cell cols="1" rows="1" rend="align=right" role="data"><figure/>,</cell><cell cols="1" rows="1" rend="align=right" role="data">[dram][dram],</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">∫s,</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">[dram] <figure/>,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> which are formed from the initials of the barbarous way in which the Germans pronounced and wrote the Latin and Italic names of the powers, namely, res or cosa, zensus, cubo, zensi-zensus, sursolid, zenfi-cubo, &c. And the coss or first power <figure/>, he calls the radix or root, which is the first time that we meet with this word in the printed authors. He also here uses the signs or characters, + and -, for addition and subtraction, and the first of any that I know of: for in Italy they used none of these characters for a long time after. He has no mark however for equality, but makes use of the word itself. <pb n="78"/><cb/></p><p><hi rend="italics">Chap.</hi> 2. Of the Parts of the Rule of Geber or Algebra: teaching the various reductions by addition, subtraction, multiplication, division, involution, and evolution, &c.</p><p><hi rend="italics">Chap.</hi> 3. Of the Algorithm of Cossic Numbers: teaching the usual operations of addition, subtraction, multiplication, division, involution, and extraction of roots, much the same as they are at present. Single terms, or powers, he calls simple quantities; but such as 1[dram] + 1<figure/> a composite or compound, and 2<figure/> - 8 a defective one. In multiplication and division, he proves that like signs give +, and unlike signs -. He shews that the powers 1, <figure/>, [dram], <figure/>, &c, form a geometrical progression from unity; and that the natural series of numbers 0, 1, 2, 3, &c, from 0, are the exponents of the cossic powers; and he, for the first time, expressly calls them exponents: thus, <table><row role="data"><cell cols="1" rows="1" role="data">Exponents,</cell><cell cols="1" rows="1" role="data">0,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" rend="align=center" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">5,</cell><cell cols="1" rows="1" rend="align=center" role="data">6,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Powers,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data">[dram],</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" rend="align=center" role="data">[dram] [dram],</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">∫s,</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">[dram] <figure/>,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> And he shews the use of the exponents, in multiplication, division, powers, and roots, as we do at present; viz, adding the exponents in multiplication, and subtracting them in division, &c. And these operations he demonstrates from the nature of arithmetical and geometrical progressions. It is remarkable that these compound denominations of the powers are formed from the simple ones according to the <hi rend="italics">products</hi> of the exponents, while those of Diophantus are formed according to the <hi rend="italics">sums</hi> of them; thus the 6th power here is [dram] <figure/> or quadrato-cubi, but with Diophantus it is cubo-cubi; and so of others. Which is presumptive evidence that the Europeans had not taken their Algebra immediately from him, independent of other proofs.</p><p><hi rend="italics">Chap.</hi> 4. Of the extraction of the roots of cossic numbers. He here treats of quadratic equations, which he resolves by completing the square, from Euclid II. 4 &c. Also quadratics of the higher orders, shewing how to resolve them in all cases, whatever the height may be, provided the exponents be but in arithmetical progression, as <hi rend="brace"><note anchored="true" place="unspecified">&c; where it is plain that he always counts 0 for the exponent of the unknown quantity in the absolute term.</note> 2, 1, 0 4, 2, 0 6, 3, 0 8, 4, 0</hi></p><p><hi rend="italics">Chap.</hi> 5. Of irrational cossic numbers, and of surd or negative numbers. In this treatise of radicals, or irrationals, he first uses the character √ to denote a root, and sets after it the mark of the power whose root is intended; as √[dram] 20 for the square root of 20, and √ <figure/> 20 for the cube root of the same, and so on. He treats here also of negative numbers, or what he calls surd or fictitious, or numbers less than 0. On which he takes occasion to observe, that when a geometrical progression is continued downwards below 1, then the exponents of the terms, or the arithmetical progression, will go below 0 into negative numhers, and will yet be the true exponents of the former; as in these, <table rend="border"><row role="data"><cell cols="1" rows="1" role="data">Expon.</cell><cell cols="1" rows="1" rend="align=right" role="data">-3</cell><cell cols="1" rows="1" rend="align=right" role="data">-2</cell><cell cols="1" rows="1" rend="align=right" role="data">-1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pow.</cell><cell cols="1" rows="1" rend="align=right" role="data">1/8</cell><cell cols="1" rows="1" rend="align=right" role="data">1/4</cell><cell cols="1" rows="1" rend="align=right" role="data">1/2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell></row></table> And he gives examples to shew that these negative exponents perform their office the same as the positive ones, in all the opesations. <cb/></p><p><hi rend="italics">Chap.</hi> 6. Of the perfection of the Rule of Algebra, and of Secondary Roots. In the reduction of equations he uses a more general rule than those who had preceded him, who detailed the rule in a multitude of cases; instead of which, he directs to multiply or divide the two sides equally, to transpose the terms with + or -, and lastly to extract such root as may be denoted by the exponent of the highest power.</p><p>As to secondary roots, Cardan treated of a 2d <hi rend="italics">ignota</hi> or unknown, which he called <hi rend="italics">quantitas,</hi> and denoted it by the initial <hi rend="italics">q,</hi> to distinguish it from the first. But here Stifelius, for distinction sake, and to prevent one root from being mistaken for others, assigns literal marks to all of them, as A, B, C, D, &c, and then performs all the usual operations with them, joining them together as we do now, except that he subjoins the initial of the power, instead of its numeral exponent: thus, 3A into 9B makes 27AB,<lb/> 3[dram] into 4B makes 12[dram]B,<lb/> 2<figure/> into 4A[dram] makes 8<figure/> A[dram],<lb/> 1A squared makes 1A[dram],<lb/> 6 into 3C makes 18C,<lb/> 2A[dram] into 5A <figure/> makes 10A<hi rend="italics">∫s</hi>, &c, &c.<lb/> 8<figure/>A[dram] divided by 4 <figure/> makes 2A[dram], &c.<lb/> The square root of 25A[dram] is 5A, &c.<lb/> Also 2A added to 2<figure/> makes 2<figure/> + 2A,<lb/> and 2A subtr. from 2<figure/> makes 2<figure/> - 2A.<lb/> And he shews how to use the same, in questions concerning several unknown numbers; where he puts a different character for each of them, as<figure/>, A, B, C, &c; he then makes out, from the conditions of the question, as many equations as there are characters; from these he finds the value of each letter, in terms of some one of the rest; and so, expelling them all but that one, reduces the whole to a final equation, as we do at present.</p><p>The remainder of the book is employed with the solutions of a great number of questions to exercise all the rules and methods; some of which are geometrical ones.</p><p>From this account of the state of Algebra in Stifelius, it appears that the improvements made by himself, or other Germans, beyond those of the Italians, as contained in Cardan's book of 1539, were as follow:</p><p>1st. He introduced the characters +, -, √, for plus, minus, and root, or <hi rend="italics">radix,</hi> as he calls it.</p><p>2d. The initials <figure/>, [dram], <figure/>, &c. for the powers.</p><p>3d. He treated all the higher orders of quadratics by the same general rule.</p><p>4th. He introduced the numeral exponents of the powers, -3, -2, -1, 0, 1, 2, 3, &c, both positive and negative, so far as integral numbers, but not fractional ones; calling them by the name <hi rend="italics">exponens,</hi> exponent: and he taught the general uses of the exponents, in the several operations of powers, as we now use them, or the logarithms.</p><p>5th. And lastly, he used the general literal notation A, B, C, D, &c, for so many different unknown or general quantities. <hi rend="center"><hi rend="smallcaps">OF SCHEUBELIUS.</hi></hi></p><p>John Scheubelius published several books upon Arithmetic and Algebra. The one now before me, is intitled <hi rend="italics">Algebræ Compendiosa Facilisque Descriptio, quâ depro-</hi> <pb n="79"/><cb/> <hi rend="italics">muntur magna Arithmetices miracula. Authore Johanne Scheubelio Mathematicarum Professore in Academia Tubingensi. Parisiis</hi> 1552. But at the end of the book it is dated 1551. The work is most beautifully printed, and is a very clear and succinct treatise; and both in the form and matter much resembles a modern printed book. He says that the writers ascribe this art to Diophantus, which is the first time that I find this Greek author mentioned by the modern algebraists: he farther observes, that the Latins call it <hi rend="italics">Regula Rei & Census,</hi> the rule of the thing and the square (or of the 1st and 2d power); and the Arabs, Algebra. His characters and operations are much the same as those of Stifelius, using the signs and characters +, -, √, and the powers <figure/>, <figure/>, [dram], <figure/>, &c, where the character <figure/> is used for 1 or unity, or a number, or the o power; prefixing also the numeral coefficients; thus 44∫[dram] + 11[dram] + 31<figure/> - 53<figure/>. He uses also the exponents 0, 1, 2, 3, &c, of the powers, the same way as Stifelius, before him. He performs the algebraical calculations, first in integers, and then in fractions, much the same as we do at present. Then of equations, which he says may be of infinite degrees, though he treats only of two, namely the first and second orders, or what we call simple and quadratic equations, in the usual way, taking however only the positive roots of these; and adverting to all the higher orders of quadratics, namely, <hi rend="italics">x</hi><hi rend="sup">4</hi>, <hi rend="italics">ax</hi><hi rend="sup">2</hi>, <hi rend="italics">b</hi>; <hi rend="center"><hi rend="italics">x</hi><hi rend="sup">6</hi>, <hi rend="italics">ax</hi><hi rend="sup">3</hi>, <hi rend="italics">b</hi>;</hi> <hi rend="center"><hi rend="italics">x</hi><hi rend="sup">8</hi>, <hi rend="italics">ax</hi><hi rend="sup">4</hi>, <hi rend="italics">b</hi>; &c.</hi></p><p>Next follows a tract on surds, both simple and compound, quadratic, cubic, binomial, and residual. Here he first marks the notation, observing that the root is either denoted by the initial of the word, or, after some authors, by the mark √:, viz. the sq. root √:, the cube root w√:, and the 4th root, or root of the root thus v√:, which latter method he mostly uses. He then gives the Arithmetic of surds, in multiplication, division, addition, and subtraction. In these last two rules he squares the sum or difference of the surds, and then sets the root to the whole compound, which he calls <hi rend="italics">radix collecti,</hi> what Cardan calls <hi rend="italics">radix universalis.</hi> Thus √12 ± √20 is ra. col. 32 ± √960. But when the terms will reduce to a common surd, he then unites them into one number; as √27 + √12 is equal √75. Also of cubic surds, and 4th roots. In binomial and residual surds, he remarks the different kinds of them which answer to the several irrational lines in the 10th book of Euclid's elements; and then gives this general rule for extracting the root of any binomial or residual <hi rend="italics">a</hi> ± <hi rend="italics">b,</hi> where one or both parts are surds, and <hi rend="italics">a</hi> the greater quantity, namely, that the square root of it is ; which he illustrates by many examples. This rule will only succeed however, so as to come out in simple terms, in certain cases, namely, either when <hi rend="italics">a</hi><hi rend="sup">2</hi>-<hi rend="italics">b</hi><hi rend="sup">2</hi> is a square, or when <hi rend="italics">a</hi> and √(<hi rend="italics">a</hi><hi rend="sup">2</hi> - <hi rend="italics">b</hi><hi rend="sup">2</hi>) will reduce to a common surd, and unite: in all other cases the root is in two compound surds, instead of one. He gives also another rule, which comes however to the same thing as the former, though by the words of them they seem to be different. <cb/></p><p>Scheubelius wrote much about the time of Cardan and Stifelius. And as he takes no notice of cubic equations, it is probable he had neither seen nor heard any thing about them; which might very well happen, the one living in Italy, and the other in Germany. And, besides, I know not if this be the first edition of Scheubel's book: it is rather likely it is not, as it is printed at Paris, and he himself was professor of mathematics at Tubingen in Germany. <hi rend="center"><hi rend="smallcaps">ROBERT RECORDE.</hi></hi></p><p>The first part of his Arithmetic was published in 1552; and the second part in 1557, under the title of, “The Whetstone of Witte, which is the seconde parte of Arithmetike: containing the Extraction of Rootes: The Cossike Practise, with the Rule of Equation: and the Workes of Surde Nombers.” The work is in dialogue between the master and scholar; and is nearly after the manner of the Germans, Stifelius and Scheubelius, but especially the latter, whom he often quotes, and takes examples from. The chief parts of the work are, 1st. The properties of abstract and figurate numbers. 2nd. The extraction of the square and cube roots, much the same as at present. Here, when the number is not an exact power, but having some remainder over, he either continues the root into decimals as far as he pleases, by adding to the remainders always periods of cyphers; or else makes a vulgar fraction for the remaining part of the root, by taking the remainder for the numerator, and double the root for the denominator, in the square root; but in the cube root he takes for the nominator either the triple square of the root, which is Cardan's rule, or the triple square and triple root, with one more, which is Scheubel's rule. 3d. Of Algebra, or “Cossike Nombers.” He uses the notation of powers with their exponents the same as Stifel, with all the operations in simple and compound quantities, or integers and fractions. And he gives also many examples of extracting the roots of compound algebraic quantities, even when the roots are from two to six terms, in imitation of the same process in numbers, just as we do at present; which is the first instance of this kind that I have observed. As of this quantity: 25[dram] <figure/> + 80∫[dram]<hi rend="sup">Square</hi> - 26[dram][dram] - 63[dram] (5<figure/> +<hi rend="sup">Root.</hi> 8[dram] - 9<figure/>.</p><p>4th. “The Rule of Equation, commonly called Algeber's Rule.” He here, first of any, introduces the character =, for brevity sake. His words are, “And to avoide the tediouse repetition of these woordes: is equalle to: I will sette as I doe often in woorke use, a paire of paralleles, or gemowe lines of one lengthe, thus:=, bicause noe 2 thynges can be moare equalle.” He gives the rules for simple and quadratic equations, with many examples. He gives also some examples in higher compound equations, with a root for each of them, but gives no rule how to find it. 5th. “Of Surde Nombers.” This is a very ample treatise on surds, both simple and compound, and surds of various degrees, as square, cubic, and biquadratic, marking the roots in Scheubel's manner, thus: √, w√, v√. He here uses the names bimedial, binomial, and residual; but says they have been used by others before him, though this is the first place where I have observed the two latter.— Hence it appears that the things which chiefly are new in this author, are these three, viz. <pb n="80"/><cb/></p><p>1. The extraction of the roots of compound algebraic quantities.</p><p>2. The use of the terms binomial and residual.</p><p>3. The use of the sign of equality, or =. <hi rend="center"><hi rend="smallcaps">OF PELETARIUS.</hi></hi></p><p>The first edition of this author's algebra was printed in 4to at Paris, in 1558, under this title, <hi rend="italics">Jacobi Peletarii Cenomani, de occulta parte Numerorum, quam Algebram vocant. Lib. duo.</hi></p><p>In the preface he speaks of the supposed authors of Algebra, namely Geber, Mahomet the son of Moses, an Arabian, and Diophantus. But he thinks the art older, and mentions some of his contemporary writers, or a very little before him, as Cardan, Stifel, Scheubel, Chr. Januarius; and a little earlier again, Lucas Paciolus of Florence, and Stephen Villafrancus a Gaul.</p><p>Of the two books, into which the work is divided, the first is on rational, and the second on irrational or surd quantities; each being divided into many chapters. It will be sufficient to mention only the principal articles.</p><p>He calls the series of powers <hi rend="italics">numeri creati,</hi> or derived numbers, or also radicals, because they are all raised from one root or <hi rend="italics">radix.</hi> He names them thus, radix, quadratus cubus, quadrato-quadratus, or biquadratus, supersolidus, quadrato-cubus, &c; and marks them thus ℞, <hi rend="italics">q, <figure/>, qq, ∫s, q<figure/>, b∫s, &c.</hi> Of these he gives the following series in numbers, having the common ratio 2, with their marks set over them, and the exponents set over these again, in an arithmctical series, beginning at 0, thus: <table><row role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">℞</cell><cell cols="1" rows="1" role="data"><hi rend="italics">q</hi></cell><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data"><hi rend="italics">qq</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">q</hi><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">b∫s</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">qqq</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" rend="align=center" role="data">64</cell><cell cols="1" rows="1" rend="align=center" role="data">128</cell><cell cols="1" rows="1" rend="align=center" role="data">256</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> And he shews the use of the exponents, the same as Stifelius and Scheubelius; like whom also he prefixes coefficients to quantities of all kinds, as also the radical √. But he does not follow them in the use of the signs + and -, but employs the initials <hi rend="italics">p</hi> and <hi rend="italics">m</hi> for the same purpose. After the operations of addition, &c, he performs involution and cvolution also much the same way as at present: thus, in powers, raise the coefficient to the power required, and multiply the exponent, or sign, as he calls it, by 2, or 3, or 4, &c, for the 2nd, 3d, 4th, &c, power; and the reverse for extraction: and hence he observes, if the number or coefficient will not exactly extract, or the sign do not exactly divide, the quantity is a surd.</p><p>After the operations of compound quantities, and fractions, and reduction of equations, namely, simple and quadratic equations, as usual, in chap. 16, <hi rend="italics">De Inveniendis generatim Radicibus Denominatorum,</hi> he gives a method of finding the roots of equations among the divisors of the absolute number, when the root is rational, whether it be integral or fractional; for then, he observes, the root always lies hid in that number, and is some one of its divisors. This is exemplified in several instances, both of quadratic and cubic equations, and both for integral and fractional roots. And he here observes, that he knows not of any person who has yet given general rules for the solution of cubic equations; which shews that when he wrote this book, either Cardan's last book was not published, or else it had not yet come to his knowledge.</p><p><hi rend="italics">Chap.</hi> 17 contains, in a few words, directions for <cb/> bringing questions to equations, and for reducing these. He here observes, that some authors call the unknown number <hi rend="italics">res,</hi> and others the <hi rend="italics">positio</hi>; but that he calls it <hi rend="italics">radix,</hi> or root, and marks it thus ℞: hence the term, root of an equation. But it was before called radix by Stifelius.</p><p><hi rend="italics">Chap.</hi> 21 <hi rend="italics">& seq.</hi> treat of secondary roots, or a plurality of roots, denoted by A, B, C, &c, after Stifelius.</p><p>The 2d book contains the like operations in surds, or irrational numbers, and is a very complete work on this subject indeed. He treats first of simple or single surds, then of binomial surds, and lastly of trinomial surds. He gives here the same rule for extracting the root of a binomial and residual as Scheubelius, viz, . Individing by a binomial or residual, he proceeds as all others before him had done, namely, reducing the divisor to a simple quantity, by multiplying it by the same two terms with the sign of one of them changed, that is by the binomial if it be a residual, or by the same residual if it be a binomial; and multiplying the dividend by the same thing: thus . And, in imitation of this method, in division by trinomial surds, he directs to reduce the trinomial divisor first to a binomial or residual, by multiplying it by the same trinomial with the sign of one term changed, and then to reduce this binomial or residual to a simple nomial as above; observing to multiply the dividend by the same quantities as the divisor. Thus, if the divisor be 4 + √2 - √3; multiplying this by 4 + √2 + √3, the product is 15 + 8√2; then this binomial multiplied by the residual 15 - 8√2, gives 225 - 128 or 97 for the simple divisor: and the dividend, whatever it is, must also be multiplied by the two 4 + √2 + √3 and 15 - 8√2. Or in general, if the divisor be <hi rend="italics">a</hi> + √<hi rend="italics">b</hi> - √<hi rend="italics">c</hi>; multiply it by <hi rend="italics">a</hi> + √<hi rend="italics">b</hi> + √<hi rend="italics">c,</hi> which gives ; then multiply this by <hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi> - <hi rend="italics">c</hi> - 2<hi rend="italics">a√b,</hi> and it gives (<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi> - <hi rend="italics">c</hi>)<hi rend="sup">2</hi> - 4<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b,</hi> which will be rational, and will all collect into one single term. But Tartalea must have been in possession of some such rule as this, as one of the questions he proposed to Florido was of this nature, namely to find such a quantity as multiplied by a given trinomial surd, shall make it rational: and it appears, from what is done above, that, the given trinomial being , the answer will be .</p><p><hi rend="italics">Chap.</hi> 24 shews the composition of the cube of a binomial or residual, and thence remarks on the root of the case or equation 1 <figure/> <hi rend="italics">p</hi> 3℞ eqnal to 10, which he seems to know something about, though he had not Cardan's rules.</p><p><hi rend="italics">Chap.</hi> 30, which is the last, treats of certain precepts relating to square and cubic numbers, with a table of such squares and cubes for all numbers to 140; also shewing how to compute them both, by adding always their differences. <pb n="81"/><cb/></p><p>He then concludes with remarking that there are many curious properties of these numbers, one of which is this, that the sum of any number of the cubes, taken from the beginning, always makes a square number, the root of which is the sum of the roots of the cubes; so that the series of squares so formed, have for their roots — 1, 3, 6, 10, 15, 21, &c. whose diff. are the natural n<hi rend="sup">os</hi> 1, 2, 3, 4, 5, 6, &c. Namely, , &c. Or in general, .</p><p>This work of Peletarius is a very ingenious and masterly composition, treating in an able manner of the several parts of the subject then known, excepting the cubic equations. But his real discoveries, or improvements, may be reduced to these three, viz.</p><p>1st. That the root of an eqnation, is one of the divisors of the absolute term.</p><p>2d. He taught how to reduce trinomials to simple terms, by multiplying them by compound factors.</p><p>3d. He taught eurious precepts and properties concerning square and cube numbers, and the method of constructing a series of each by addition only, namely by adding successively their several orders of differences. <hi rend="center"><hi rend="smallcaps">RAMUS.</hi></hi></p><p>Peter Ramus wrote his arithmetic and algebra about the year 1560. His notation of the powers is thus, <hi rend="italics">l, q, c, bq,</hi> being the initials of latus, quadratus, cubus, biquadratus. He treats only of simple and quadratic equations. And the only thing remarkable in his work, is the first article, on the names and invention of Algebra, which we have noticed at the beginning of this history. <hi rend="center"><hi rend="smallcaps">BOMBELLI.</hi></hi></p><p>Raphael Bombelli's Algebra was published at Bologna in the year 1579, in the Italian language. It seems however it was written some time before, as the dedication is dated 1572. In a short, but neat, introduction, he first adverts, in a few words, to the great excellence and usefulness of arithmetic and algebra. He then laments that it had hitherto been treated in so imperfect and irregular a way; and declares it is his intention to remedy all defects, and to make the science and practice of it as easy and perfect as may be. And for this purpose he first resolved to procure and study all the former authors. He then mentions several of these, with a short history or character of them; as Mahomet the son of Moses, an Arabian; Leonard Pisano; Lucas de Burgo, the first printed author in Europe; Oroncius; Seribelius; Boglione Francesi; Stifelius in Germany; a certain Spaniard, doubtless meaning Nunez or Nonius; and lastly Cardan, Ferrari, and Tartalea; with some others since, whose names he omits. He then adds a curious paragraph concerning Diophantus: he says that some years since there had been found, in the Vatican library, a Greek work on this art, composed by a certain Diophantus, of Alexandria, a Greek author, who lived in the time of Antoninus Pius; which work having been shewn to him by Mr. Antonio Maria Pazzi Reggiano, public lecturer on mathematics at Rome; and sinding it to be a good work, these two formed the re- <cb/> solution of giving it to the world, and he says that they had already translated five books, of the six which were then extant, being as yet hindered by other avocations from completing the work. He then adds the following strange circumstance, viz. <hi rend="italics">that they had found that in the said work the Indian authors are often cited; by which they learned that this science was known among the Indians before the Arabians had it:</hi> a paragraph the more remarkable as I have never understood that asty other person could ever find, in Diophantus, any reference to Indian writers: and I have examined his work with some attention, for that purpose.</p><p>Bombelli's work is divided into three books. In the first, are laid down the definitions and operations of powers and roots, with various sorts of radicals, simple and compound, binomial, residual, &c; mostly aster the rules and manner of former writers, excepting in some few instances, which I shall here take notice of. And first of his rule for the cube root of binomials or residuals, which for the sake of brevity, may be expressed in modern notation as follows: let √<hi rend="italics">b</hi> + <hi rend="italics">a</hi> be the binomial, the term √<hi rend="italics">b</hi> being greater than <hi rend="italics">a</hi>; then the rule for the cube root of √(<hi rend="italics">b</hi> + <hi rend="italics">a</hi>) comes to this, . Which is a rule that can be of little or no use; for, in the first place, is the same as ―(P + Q)<hi rend="sup">2</hi>; and , the original quantity first proposed. The next thing remarkable in this 1st book, is his method for the square roots of negative quantities, and his rule for the cube roots of such imaginary binomials as arise from the irreducible case in cubic equations. His words, translated, are these: “I have found another sort of cubic root, very different from the former, which arises from the case of the cube equal to the first power and a number, when the cube of the 1/3d part of the (coef. of the) 1st power, is greater than the square of half the absolute number, which sort of square root hath in its algorism, names and operations different from the others; for in that case, the excess cannot be called either plus or minus; I therefore call it <hi rend="italics">plus of minus</hi> when it is to be added, and <hi rend="italics">minus of minus</hi> when it is to be subtracted.” He then gives a set of rules for the signs when such roots are multiplied, and illustrates them by a great many examples. His rule for the cube roots of such binomials, viz. such as <hi rend="italics">a</hi> + √- <hi rend="italics">b,</hi> is this: First sind √<hi rend="sup">3</hi>(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi>); then, by trials search out a number <hi rend="italics">c,</hi> and a sq. root √<hi rend="italics">d,</hi> such, that the sum of their squares <hi rend="italics">c</hi><hi rend="sup">2</hi> + <hi rend="italics">d</hi> may be <pb n="82"/><cb/> and also ; then shall sought. Thus, to extract the cube root of 2 + √ - 121: here ; then taking <hi rend="italics">c</hi> = 2, and <hi rend="italics">d</hi> = 1, it is , and , as it ought; and therefore 2+√-1 is = the cube root of 2+√-121, as required.</p><p>The notation in this book, is the initial <hi rend="italics">R</hi> for root, with <hi rend="italics">q</hi> or <hi rend="italics">c</hi> &c after it, for quadrate or cubic, &c root. Also <hi rend="italics">p</hi> for <hi rend="italics">plus,</hi> and <hi rend="italics">m</hi> for <hi rend="italics">minus.</hi></p><p>In the 2d book, Bombelli treats of the algorism with unknown quantities, and the resolution of equations. He first gives the definitions and characters of the unknown quantity and its powers, in which he deviates from the former authors, but professes to imitate Diophantus. He calls the unknown quantity <hi rend="italics">tanto,</hi> and marks it thus <figure/>, Its square or 2d power <hi rend="italics">potenza,</hi> <figure/>, Its cube <hi rend="italics">cubo,</hi> <figure/>, and the higher names are compounded of these, and marked <figure/>, &c, so that he denotes all the powers by their exponents set over the common character <figure/>. And all these powers he calls by the general name <hi rend="italics">dignita,</hi> dignity. He then performs all the algorism of these powers, by means of their exponents, as we do at present, viz, adding them in multiplication, subtracting in division, multiplying them by the index in involution, and dividing by the same in evolution.</p><p>In equations he goes regularly through all the cases, and varieties of the signs and terms; first all the simple or single powers, and then all the compound cases; demonstrating the rules geometrically, and illustrating them by many examples.</p><p>In compound quadratics, he gives two rules: the first is by freeing the <hi rend="italics">potenza</hi> or square from its coefficient by division, and then completing the square, &c, in the usual way: and the 2d rule, when the first term has its coefficient, may be thus expressed; if , then . He takes only the positive root or roots; and in the case , which has two, he observes that the nature of the problem must shew which of the two is the proper one.</p><p>In the cubic equations, he gives the rules and transformations, &c, after the manner of Cardan; remarking that some of the cases have only one root, but others two or three, of which some are true, and others false or negative. And in one place he says that by means of the case he <hi rend="italics">trisects</hi> or <hi rend="italics">divides an angle into three equal parts.</hi></p><p>When he arrives at biquadratic equations, and particularly to this case <hi rend="italics">x</hi><hi rend="sup">4</hi> + <hi rend="italics">ax</hi> - <hi rend="italics">b,</hi> he says, “Since I have seen Diophantus's work, I have always been of opinion that his chief intention was to come to this equation, because I observe he labours at finding always square numbers, and such, that adding some number to them, may make squares; and I believe that the six books, which are lost, may treat of this equation, &c.” —“But Lewis Ferrari,” he adds, “of this city, also laboured in this way, and found out a rule for such cases, which was a very fine invention, and therefore I <cb/> shall here treat of it the best I can.” This he accordingly does, in all the cases of biquadratics, both with respect to the number of terms in the equation, and the signs of the terms, except I think this most general case only ; fully applying Ferrari's method in all cases. Which concludes the 2d book.</p><p>The 3d book consists only of the resolution of near 300 practical questions, as exercises in all the rules and equations, some of which are taken from Diophantus and other authors.</p><p>Upon the whole it appears that this is a plain, explicit, and very orderly treatise on algebra, in which are very well explained the rules and methods of former writers. But Bombelli does not produce much of improvement or invention of his own, except his notation, which varies from others, and is by means of one general character, with the numeral indices of Stifelius. He also first remarks that angles are trisected by a cubie equation. But I know not how to account for his assertion, that Diophantus often cites the Indian authors; which I think must be a mistake in Bombelli. <hi rend="center"><hi rend="smallcaps">CLAVIUS.</hi></hi></p><p>Christopher Clavius wrote his Algebra about the year 1580, though it was not published till 1608, at Orleans. He mostly follows Stifelius and Scheubelius in his notation and method, &c, having scarcely any variations from them; nor does he treat of cubic equations. He mentions the names given to the art, and the opinions about its origin, in which he inclines to ascribe it to Diophantus, from what Diophantus says in his preface to Dyonisius. <hi rend="center"><hi rend="smallcaps">STEVINUS.</hi></hi></p><p>The Arithmetic of Simon Stevin of Bruges, was published in 1585, and his Algebra a little afterwards. They were also printed in an edition of his works at Leyden in 1634, with some notes and additions of Albert Girard, who it seems died the year before, this edition being published for the benefit of Girrard's widow and children. The Algebra is an ingenious and original work. He denotes the <hi rend="italics">res,</hi> or unknown quantity, in a way of his own, namely by a small circle ○, within which he places the numeral exponent of the power, as ○0, ○1, ○2, ○3, &c, which are the 0, 1, 2, 3, &c power of the quantity ○; where ○0, or the 0 power, is the beginning of quantity, or arithmetical unit. He also extends this notation to roots or fractional exponents, and even to radical ones.</p><p>Thus ○1/2, ○1/3, ○1/4, &c, are the sq. root, cube root, 4th root, &c;</p><p>and ○4/3 is the cube root of the square;</p><p>and ○3/2 is the sq. root of the cube. And so of others.</p><p>The first three powers, ○1, ○2, ○3, he also calls <hi rend="italics">coste</hi> (side), <hi rend="italics">quarre</hi> (square), <hi rend="italics">cube</hi> (cube); and the first of them, ○1, the prime quantity, which he observes is also <hi rend="italics">metaphorically</hi> called the racine or root, (the mark of which is also √), because it represents the root or origin from whence all other quantities spring or arise, called the <hi rend="italics">potences</hi> or powers of it. He condemns the terms sursolids, and numbers absurd, irrational, irregular, inexplicable, or surd, and shews that all numbers are denoted the same way, and are all equally proper ex- <pb n="83"/><cb/> pressions of some length or magnitude, or some power of the same root. He also rejects all the compound expressions of square-squared, cube-squared, cube-cubed, &c, and shews that it is best to name them all by their exponents, as the 1st, 2d, 3d, 4th, 5th, 6th, &c power or quantity in the series. And on his extension of the new notation he justly observes that what was before obscure, laborious, and tiresome, will by these marks be clear, easy, and pleasant. He also makes the notation of algebraic quantities more general in their coefficients, including in them not only integers, as 3○1, but also fractions and radicals, as (3/4)○2, and √2○3, &c. He has various other peculiarities in his notations; all shewing an original and inventive mind. A quantity of several terms, <hi rend="italics">he</hi> calls a multinomial, and also binomial, trinomial, &c, according to the number of the terms. He uses the signs + and -, and sometimes: for equality; also X for division of fractions, or to multiply crosswise thus, 5/7X2/3 : 15/14.</p><p>He teaches the generation of powers <figure/> by means of the annexed table of numbers, which are the coefficients of all the terms except the first and last. And he makes use of the same numbers also for extracting all roots whatever: both which things had first been done by Stifelius. In extracting the roots of non-quadrate or non-cubic numbers, he has the same approximations as at present, viz, either to continue the extraction indefinitely in decimals, by adding periods of ciphers, or by making a fraction of the remainder in this manner, viz, nearly, and nearly; where <hi rend="italics">n</hi> is the nearest exact root of N; which is Peletarius's rule, and which differs from Tartalea's rule, as this wants the 1 in the denominator. And in like manner he goes on to the roots of higher powers.</p><p>He then treats of equations, and their inventors, which according to him are thus: <hi rend="brace"><note anchored="true" place="unspecified">Mahomet, son of Moses, an Arabian, invented these</note> ○1 egale <hi rend="italics">à</hi> ○0, its derivatives, ○2 egale <hi rend="italics">à</hi> ○1, ○0,</hi> And some unknown author, the derivatives of this. <hi rend="brace"><note anchored="true" place="unspecified">Some unknown author invented these</note> ○3 egale <hi rend="italics">à</hi> ○1 ○0, ○3 egale <hi rend="italics">à</hi> ○2 ○0,</hi></p><p>But afterwards he mentions Ferreus, Tartalea, Cardan, &c, as being also concerned in the invetion of them.</p><p>Lewis Ferrari invented ○4 egale <hi rend="italics">à</hi> ○3 ○2 ○1 ○0.</p><p>He says also that Diophantus once resolves the case ○2 egale <hi rend="italics">à</hi> ○1 Θ. In his reduction of equations, which is full and masterly, he always puts the highest power on one side alone, equal to all the other terms, set in their order, on the other side, whether they be + or -. And he demonstrates all the rules both arithmetically and geometrically. In cubics, he gives up the irreducible case, as hopeless: but says that Bombelli resolves it by <hi rend="italics">plus of minus,</hi> and <hi rend="italics">minus of minus</hi>; thus, if , then , that is, . He resolves biquadratics by <cb/> means of cubics and quadratics. In quadratics, he takes both the two roots, but looks for no more than two in cubics or biquadratics. He gives also a general method of approaching indefinitely near, in decimals, to the root of any equation whatever: but it is very laborious, being little more than trying all numbers, one after another, finding thus the 1st figure, then the 2d, then the 3d, &c, among these ten characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. And finally he applies the rules in the resolution of a great many practical questions.</p><p>Although a general air of originality and improvement runs through the whole of Stevinus's work, yet his more remarkable or peculiar inventions, may be reduced to these few following: viz,</p><p>1st. He invented not only a new character for the unknown quantity, but greatly improved the notation of powers, by numeral indices, sirst given by Stifelius as to integral exponents; which Stevinus extended to fractional and all other sorts of exponents, thereby denoting all sorts of roots the same way as powers, by numeral exponents. A circumstance hitherto thought to be of much later invention.</p><p>2d. He improved and extended the use and notation of coefficients, including in them fractions and radicals, and all sorts of numbers in general.</p><p>3d. A quantity of several terms, he called generally a multinomial; and he denoted all nomials whatever by particular names expressing the number of their terms, binomial, trinomial, quadrinomial, &c.</p><p>4th. A numeral resolution of all equations whatever by one general method.</p><p>Besides which, he hints at some unknown author as the first inventor of the rules for cubic equations; by whom may probably be intended the author of the Arabic manuscript treatise on cubic equations, given to the library at Leyden by the celebrated Warner. <hi rend="center"><hi rend="smallcaps">VIETA.</hi></hi></p><p>Most of Vieta's algebraical works were written about or a little before, the year 1600, but some of them were not published till after his death, which happened in the year 1603. And his whole mathematical works were collected together by Francis Schooten, and elegantly printed in a folio volume in 1646. Of these, the algebraical parts are as follow:</p><p>1. Isagoge in Artem Analyticam.</p><p>2. Ad Logisticen Speciosam Notæ priores.</p><p>3. Zeteticorum libri quinque.</p><p>4. De Æquationum Recognitione, & Emendatione.</p><p>5. De Numerosâ Potestatum ad Exegesin Resolutione.</p><p>Of all these I shall give a very minute account, especially in such parts as contain any discoveries, as we here meet with more improvements and inventions in the nature of equations, than in almost any former author. And first of the <hi rend="italics">Isagoge</hi> or Introduction to the Analytic Art. In this short introduction Vieta lays down certain præcognita in this art, as definitions, axioms, notations, common precepts or operations of addition, subtraction, multiplication, and division, with rules for questions, &c. From which we find, 1st. That the names of his powers are latus, quadratum, cubus, quadrato-quadratum, quadrato-cubus, cubocubus, &c; in which he follows the method of Diophantus, and not that derived from the Arabians. <pb n="84"/><cb/> 2d. That he calls powers pure or adfected, and first here uses the terms coefficient, affirmative, negative, specious logistics or calculations, homogeneum comparationis, or the absolute known term of an equation, homogeneum adfectionis, or the 2d or other term which makes the equation adfected, &c. 3d. That he uses the capital letters to denote the known as well as unknown quantities, to render his rules and calculations general, namely, the vowels A, E, I, O, U, Y for the unknown quantities, and the consonants B, C, D, &c, for the known ones. 4th. That he uses the sign + between two terms for addition; — for subtraction, placing the grcater before the less; and when it is not known which term is the greater, he places = between them for the difference, as we now use <01>; thus A = B is the same as A <01> B; that he expresses division by placing the terms like a fraction, as at present; though he was not first in this. But that he uses no characters for multiplication or equality, but writes the words themselves, as well as the names of all the powers, as he uses no exponents, which causes much trouble and prolixity in the progress of his work; and the numeral coefficients set after the literal quantities, have a disagreeable effect.</p><p>II. <hi rend="italics">Ad Logisticen Speciosam, Notæ Priores.</hi> These consist of various theorems concerning sums, differences, products, powers, proportionals, &c, with the genesis of powers from binomial and residual roots, and certain properties of rational right-angled triangles.</p><p>III. <hi rend="italics">Zeteticorum libri quinque.</hi> The zetetics or questions in these 5 books are chiefly from Diophantus, but resolved more generally by literal arithmetic. And in these questions are also investigated rules for the resolution of quadratic and cubic equations. In these also Vieta first uses a line drawn over compound quantities, as a vinculum.</p><p>IV. <hi rend="italics">De Æquationum Recognitione, & Emendatione.</hi> These two books, which contain Vieta's chief improvements in Algebra, were not published till the year 1615, by Alexander Anderson, a learned and ingenious Scotchman, with various corrections and additions. The 1st of these two books consists of 20 chapters. In the first six chapters, rules are drawn from the zetetics for the resolution of quadratic and cubic equations. These rules are by means of certain quantities in continued proportion, but in the resolution they come to the same thing as Cardan's rules. In the cubics, Vieta sometimes changes the negative roots into affirmative, as Cardan had done, but he finds only the affirmative roots. And he here refers the irreducible cafe to angular sections for a solution, a method which had been mentioned by Bombelli.</p><p><hi rend="italics">Chap.</hi> 7 treats of the general method of transforming equations, which is done either by changing the root in various ways, namely by substituting another instead of it which is either increased or diminished, or multiplied or divided by fome known number, or raised or depressed in some known proportion; or by retaining the same root, and equally multiplying all the terms. Which sorts of transformation, it is evident, are intended to make the equation become simpler, or more convenient for solution. And all or most of these reductions and transformations were also practised by Cardan.</p><p><hi rend="italics">Chap.</hi> 8 shews what purposes are answered by the <cb/> foregoing transformations; such as taking away some of the terms out of an equation, and particularly the 2d term, which is done by increasing or diminishing the root by the coefficient of the 2d term divided by the index of the first: by which means also the affected quadratic is reduced to a simple one. And various other effects are produced.</p><p><hi rend="italics">Chap.</hi> 9 shews how to deduce compound quadratic equations from pure ones, which is done by increasing or diminishing the root by a given quantity, being one application of the foregoing reductions.</p><p><hi rend="italics">Chap.</hi> 10, the reduction of cubic equations affected with the 1st power, to such as are affected with the 2d power; by the same means.</p><p>In <hi rend="italics">chap.</hi> 11, by the same means also, the 2d term is restored to such cubic equations as want it.</p><p>In <hi rend="italics">chap.</hi> 12, quadratic and cubic equations are raised to higher degrees by substituting for the root, the square or cube of another root divided by a given quantity.</p><p>In <hi rend="italics">chap.</hi> 13 affected biquadratic equations are deduced from affected quadratics in this manner, when expressed in the modern notation: If , then shall . For since , therefore , and its square is : but , therefore , or . And in like manner for the biquadratic affected with its other terms. And in a similar manner also, in chap. 14, affected cubic equations are deduced from the affected quadratics.</p><p>In <hi rend="italics">chap.</hi> 15 it is shewn that the quadratic has two values of the root A, or has ambiguous roots, as he calls them; and also that the cubics, biquadratics, &c, which are raised or deduced from that quadratic, have also double roots.</p><p>Having, in the foregoing chapters, shewn how the coefficients of equations of the 3d and 4th degree are formed from those of the 2d degree, of the same root; and that certain quadratics, and others raised from them, have double roots; then in the 16th chap. Vieta shews what relation those two roots bear to the coefficients of the two lowest terms of an equation consisting of only three terms. Thus, And so on for the same terms with their signs variously changed. <pb n="85"/><cb/></p><p><hi rend="italics">Chap.</hi> 17 contains several theorems concerning quantities in continued geometrical progression. Which are preparatory to what follows, concerning the double roots of equations, the nature of which he expounds by means of such properties of proportional quantities.</p><p><hi rend="italics">Chap.</hi> 18, <hi rend="italics">Æquationum ancipitum constitutiva</hi>; treating of the nature of the double roots of equations. Thus, if <hi rend="italics">a, b, c, d,</hi> &c, be quantities in continual progression; then, 1st, of equations affected with the first power, If ; then , Z=<hi rend="italics">ab,</hi> & A=<hi rend="italics">a</hi> or <hi rend="italics">b.</hi> If ; then , , & A =<hi rend="italics">a</hi> or <hi rend="italics">c.</hi> And in general, if BA-A<hi rend="sup">n+1</hi>; then , , and A=<hi rend="italics">a</hi> or <hi rend="italics">k</hi> the first or last term. Where the number of terms<hi rend="italics">a</hi><hi rend="sup">n</hi>, <hi rend="italics">b</hi><hi rend="sup">n</hi>, &c, in B is <hi rend="italics">n</hi> + 1, and the number of terms in Z is <hi rend="italics">n.</hi></p><p>2d, For equations containing only the highest two powers. If ; then , the sum of all except the last, or sum of all except the first; where the number of terms in B is <hi rend="italics">n</hi>+1, and the number of terms in Z is <hi rend="italics">n.</hi></p><p>3d. Of equations affected by the intermediate powers. If .</p><p>4th. Of the remaining cases. If ; then , and , and . If ; then , and ; and .</p><p><hi rend="italics">Chap.</hi> 19. <hi rend="italics">Æqualitatum contradicentium constitutiva.</hi> Of the relation of equations of like terms, but the sign of one term different; containing these 5 theorems, viz, <cb/></p><p><hi rend="italics">Chap.</hi> 20. <hi rend="italics">Æqualitatum inversarum constitutiva.</hi> Containing these six theorems, viz,</p><p><hi rend="italics">Chap.</hi> 21. <hi rend="italics">Alia rursus æqualitatem inversarum constitutiva.</hi> In these two theorems:</p><p>Next follows the 2d of the pieces published by Alexander Anderson, namely,</p><p><hi rend="italics">De Emendatione Æquationum,</hi> in 14 chapters.</p><p><hi rend="italics">Chap.</hi> 1. Of preparing equations for their resolution in numbers, by taking away the 2d term; by which affected quadratics are reduced to pure ones, and cubia equations affected with the 2d term are reduced to such as are affected with the 3d only. Several examples of both sorts of equations are given. He here too remarks upon the method of taking away any other term out of an equation, when the highest power is combined with that other term only; and this Vieta effects by means of the coefficients, or, as he calls them, the unciæ of the power of a binomial. All which was also performed by Cardan for the same purpose.</p><p><hi rend="italics">Chap.</hi> 2. <hi rend="italics">De transmutatione <foreign xml:lang="greek">*prw_ton—e)\katon</foreign>, quæ remedium est adversus vitium negationis.</hi> Concerning the transformations by changing the given root A for another root E, which is equal to the homogeneum <pb n="86"/><cb/> comparationis divided by the first root A; by which means negative terms are changed to affirmative, and radicals are taken out of the equation when they are contained in the homogeneum comparationis.</p><p><hi rend="italics">Chap.</hi> 3, <hi rend="italics">De Anastrophe,</hi> shewing the relation between the roots of correlate equations; from whence, having given the root of the one equation, that of the other becomes known; and it consists of these following 8 theorems, mostly deduced from the last 4 chapters of the foregoing <hi rend="italics">recognitio æquationum.</hi></p><p><hi rend="italics">Chap.</hi> 4, <hi rend="italics">De Isomæria, adversus vitium fractionis.</hi> To take away fractions out of an equation. Thus, if . Put A = E/D; then .</p><p><hi rend="italics">Chap.</hi> 5, <hi rend="italics">De Symmetrica Climactismo adversus vitium asymmetriæ.</hi> To take away radicals or surds out of equations, by squaring &c the other side of the equation.</p><p><hi rend="italics">Chap.</hi> 6. To reduce biquadratic equations by means of cubics and quadratics, by methods which are small variations from those of Ferrari and Cardan.</p><p><hi rend="italics">Chap.</hi> 7. The resolution of cubic equations by rules which are the same with Cardan's.</p><p><hi rend="italics">Chap.</hi> 8. <hi rend="italics">De Canonica æquationum transmutatione, ut coefficientes subgraduales sint quæ præscribuntur.</hi> To transmute the equation so that the coefficient of the lower term, or power, may be any given number, he changes the root in the given proportion, thus: Let A be the root of the equation given, E that of the transmuted equation, B the given coefficient, and X the required one; then take A = BE/X, which substitute in the given equation, and it is done.—He commonly changes it so, that X may be 1; which he does, that the numeral root of the equation may be the easier found; and this he here performs by trials, by taking the nearest root of the highest power alone; and if that does not turn out to be the root of the whole equation, he concludes that it has no rational root.</p><p><hi rend="italics">Chap.</hi> 9. To reduce certain peculiar forms of cubics to quadratics, or to simpler forms, much the same as Cardan had done. Thus, <cb/> 1. If ; then is . 2. If ; then is . 3. If ; then is A=2B. 4. If ; then is A=B. 5. If ; then is A=B. 6. If ; then is A=D. 7. If ; then is A=B or=D. 8. If ; then is 9. If ; then is 10. If ; then is . 11. If ; then is .</p><p><hi rend="italics">Chap.</hi> 10. <hi rend="italics">Similium reductionum continuatio.</hi> Being some more similar theorems, when the equation is affected with all the powers of the unknown quantity A.</p><p><hi rend="italics">Chap.</hi> 11, 12, 13 relate also to certain peculiar forms of equations, in which the root is one of the terms of a certain series of continued proportionals.</p><p><hi rend="italics">Chap.</hi> 14, which is the last in this tract, contains, in four theorems, the general relation between the roots of an equation and the coefficients of its terms, when all its roots are positive. Namely, 1. If ; then is A=B or D. 2. If ; then is A=B or D or G. 3. If ; then is A=B or D or G or H. 4. If ; then is A=B or D or G or H or K.</p><p>And from these last 4 theorems it appears that Vieta was acquainted with the composition of these equations, that is, when all their roots are positive, for he never adverts to negative roots; and from other parts of the work it appears that he was not aware that the same properties will obtain in all sorts of roots whatever. But it is not certain in what manner he obtained these theorems, as he has not given any account of the investigations, though that was usually his way on other occasions; but he here contents himself with barely announcing the theorems as above, and for this strange reason, that he might at length bring his work to a conclusion.</p><p>To this piece is added, by Alexander Anderson, an Appendix, containing the construction of the cubic equations by the trisection of an angle, and a demonstration of the property referred to by Vieta for this purpose.</p><p><hi rend="italics">De Numerosa Potestatum Purarum Resolutione.</hi> Vieta here gives some examples of extracting the roots of pure powers, in the way that had been long before practised, by pointing the number into periods of figures according to the index of the root to be extracted, and then proceeding from one period to another, in the usual way. <pb n="87"/><cb/></p><p><hi rend="italics">De Numerosa Potestatum adsectarum Resolutione.</hi> And here, in close imitation of the above method for the roots of pure powers, Vieta extracts those of adfected ones; or finding the roots of affected equations, placing always the homogeneum comparationis, or absolute term, on one side, and all the terms affected with the unknown quantity, and their proper signs, on the other side. The method is very laborious, and is but little more than what was before done by Stevinus on this subject, depending not a little upon trials. The examples he uses are such as have either one or two roots, and indeed such as are affected commonly with only two powers of the unknown quantity, and which therefore admit only of those two varieties as to the number of roots, namely according as the higher of the two powers is affirmative or negative, the homogeneum comparationis, on the other side of the equation, being always affirmative; and he remarks this general rule, if the higher power be negative, the equation has two roots; otherwise, only one; that is, affirmative roots; for as to negative and imaginary ones, Vieta knew nothing about them, or at least he takes no notice of them. By the foregoing extraction, Vieta finds both the greater and less root of the two that are contained in the equation, and either of them that he pleases; having first, for this purpose, laid down some observations concerning the limits within which the two roots are contained. Also, having found one of the roots, he shews how the other root may be found by means of another equation, which is a degree lower than the given one; though not by depressing the given equation, by dividing it as is now done; but from the nature of proportionals, and the theorems relating to equations, as given in the former tracts, he finds the terms of another equation, different from that last mentioned, from the root &c of which, the 2d root of the original equation may be obtained.</p><p>In the course of this work, Vieta makes also some observations on equations that are ambiguous, or have three roots; namely, that the equation , or as we write it is ambiguous, when the 2d term is negative, and the 3d term affirmative, and when 1/3 of the square of 6 the coefficient of the 2d term, exceeds 11, the coefficient of the 3d term, and has then three roots. Or in general, if , and (1/3)<hi rend="italics">a</hi><hi rend="sup">2</hi>><hi rend="italics">b,</hi> the equation is ambiguous, and has three roots. He shews also, from the relation of the coefficients, how to sind whether the roots are in arithmetical progression or not, and how far the middle root differs from the extremes, by means of a cubic equation of this form . In all or most of which remarks he was preceded by Cardan.—Vieta also remarks that the case , has three roots by the same rule, viz, 2, 2, 5, but that two of them are equal. And farther, that when (1/3)<hi rend="italics">a</hi><hi rend="sup">2</hi> is=<hi rend="italics">b,</hi> then all the three roots are equal, as in the case , the three roots of which are 2, 2, 2. But when (1/3)<hi rend="italics">a</hi><hi rend="sup">2</hi> is less than <hi rend="italics">b,</hi> the case is not ambiguous, having but one root. And when <hi rend="italics">ab</hi>=<hi rend="italics">c,</hi> then <hi rend="italics">a</hi>=<hi rend="italics">x</hi> is one root itself.</p><p>Many curious notes are added at the end, with remarks on the method of finding the approximate roots, when they are not rational, which is done in two ways, in imitation of the same thing in the extraction <cb/> of pure powers, viz, the one by forming a fraction of the remainder after all the figures of the homogeneum comparationis are exhausted; the other by increasing the root of the equation in a 10 fold, or 100 fold, &c, proportion, and then dividing the root which results by 10, or 100, &c: and this is a decimal approximation. AndVieta observes that the roots will be increased 10 or 100 fold, &c, by adding the corresponding number of ciphers to the coefficient of the 2d term, double that number to the 3d, triple the same number to the 4th, and so on. So if the equation were , then will have its root 10 fold, and will have it 100 fold.</p><p>Besides the foregoing algebraical works, Vieta gave various constructions of equations by means of circles and right lines, and angular sections, which may be considered as an algebraical tract, or a method of exhibiting the roots of certain equations having all their roots affirmative, and by means of which he resolved the celebrated equation of 45 powers, proposed to all the world by Adrianus Romanus.</p><p>Having now delivered a particular analysis of Vieta's algebraical writings, it will be proper, as with other authors, to collect into one view the particulars of his more remarkable peculiarities, inventions, and improvements.</p><p>And first it may be observed, that his writings shew great originality of genius and invention, and that he made alterations and improvements in most parts of algebra; though in other parts and respects his method is inferior to some of his predecessors; as, for instance, where he neglects to avail himself of the negative roots of Cardan; the numeral exponents of Stifelius, instead of which he uses the names of the powers themselves; or the fractional exponents of Stevinus; or the commodious way of presixing the coefficient before the quantity or factor; and such like circumstances; the want of which gives his Algebra the appearance of an age much earlier than its ownBut his real inventions of things before not known, may be reduced to the following particulars.</p><p>1st. Vieta introduced the general use of the letters of the alphabet to denote indefinite given quantities; which had only been done on some particular occasions before his time. But the general use of letters for the unknown quantities was before pretty common with Stifelius and his successors. Vieta uses the vowels A, E, I, O, U, Y for the unknown quantities, and the consonants B, C, D, &c, for known ones.</p><p>2d. He invented, and introduced many expressions or terms, several of which are in use to this day: such as coefficient, affirmative and negative, pure and adsected or affected, unciæ, homogeneum adfectionis, homogeneum comparationis, the line or vinculum over compound quantities thus ―(A + B). And his method of setting down his equations, is to place the homogeneum comparationis, or absolute known term, on the righthand side alone, and on the other side all the terms which contain the unknown quantity, with their proper signs.</p><p>3d. In most of the rules and reductions for cubic and <pb n="88"/><cb/> other equations, he made some improvements, and variations in the modes.</p><p>4th. He shewed how to change the root of an equation in a given proportion.</p><p>5. He derived or raised the cubic and biquadratic, &c equations, from quadratics; but not by composition in Harriot's way, but by squaring and otherwise multiplying certain parts of the quadratic. And as some quadratic equations have two roots, therefore the cubics and others raised from them, have also the same two roots, and no more. And hence he comes to know what relation these two roots bear to the coefficients of the two lowest terms of cubic and other equations, when they have only 3 terms, namely, by comparing them with similar equations so raised from quadratics. And, on the contrary, what the roots are, in terms of such coefficients.</p><p>6. He made some observations on the limits of the two roots of certain equations.</p><p>7. He stated the general relation between the roots of certain equations and the coefficients of its terms, when the terms are alternately plus and minus, and none of them are wanting, or the roots all positive.</p><p>8. He extracted the roots of affected equations, by a method of approximation similar to that for pure powers.</p><p>9. He gave the construction of certain equations, and exhibited their roots by means of angular sections; before adverted to by Bombelli. <hi rend="center"><hi rend="smallcaps">OF ALBERT GIRARD.</hi></hi></p><p>Albert Girard was an ingenious Dutch or Flemish mathematician, who died about the year 1633. He published an edition of Stevinus's Arithmetic in 1625, augmented with many notes; and the year after his death was published by his widow, an edition of the whole works of Stevinus, in the same manner, which Girard had left ready for the press. But the work which entitles him to a particular notice in this history, is his “<hi rend="italics">Invention Nouvelle en l' Algebre, tant pour la sobution des equations, que pour recognoistre le nombre des solutions qu'elles reçoivent, avec plusieurs choses qui sont necessaires a la perfection de ceste divine science</hi>;” which was printed at Amsterdam 1629, in small quarto in 63 pages, viz, 49 pages on Arithmetic and Algebra, and the rest on the measure of the supersicies of spherical triangles and polygons, by him then lately discovered.</p><p>In this work Girard first premises a short tract on Arithmetic; in the notation of which he has something peculiar, viz, dividing the numbers into the ranks of millions, billions, trillions, &c.</p><p>He next delivers the common rules of Algebra, both in integers, fractions and radicals; with the notation of the quantities and signs. In this part he uses sometimes the letters A, B, C, &c, after the manner of Vieta, but more commonly the characters of Stevinus, viz, ○0, ○1, ○2, ○3, &c, for the powers of the unknown quantity, with their roots ○5/2, ○1/2, ○1/4, ○2/3, ○<*>/4, &c, used by Stevinus; and sometimes the more usual marks of the roots as, √ or √<hi rend="sup">2</hi>, √<hi rend="sup">3</hi>, √<hi rend="sup">4</hi>, &c; presixing the coefficients, as 6○2, or 3√<hi rend="sup">5</hi>32, or 2○1/2. In the signs he follows his predecessors so far as to have + for plus, - or ÷ for minus, = for general or indefinite difference, A + B for the sum, A - B or A = B for <cb/> the difference, AB the product, and A/B for the quotient of A and B. He uses the parentheses ( ) for the vinculum or bond of compound quantities, as is now commonly practised on the continent; as A(AB+Bq), or √<hi rend="sup">3</hi> (A cub. - 3AqB); and he introduces the new characters ff for <hi rend="italics">greater than,</hi> and § for <hi rend="italics">less than</hi>; but he uses no character for equality, only the word itself.</p><p>Girard gives a new rule for extracting the cube root of binomials, which however is in a good measure tentative, and which he explains thus: To extract the cube root of 72 + √5120. <table><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">The squares of the terms</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">5184</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">5120</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=right" role="data">their difference</cell><cell cols="1" rows="1" rend="align=right" role="data">64,</cell><cell cols="1" rows="1" role="data">and its</cell></row></table> cube root 4. Which shews that the difference between the squares of the terms required is 4; and the rational part 72 being the greater, the greater term of the root will be rational also; and farther, that the greater terms of the power and root are commensurable, as also the two less terms. Then having made a table as in the <table><row role="data"><cell cols="1" rows="1" role="data">2 + √0</cell></row><row role="data"><cell cols="1" rows="1" role="data">3 + √5</cell></row><row role="data"><cell cols="1" rows="1" role="data">4 + √12</cell></row><row role="data"><cell cols="1" rows="1" role="data">5 + √21</cell></row></table> margin, where the square of the rational term always exceeds that of the other, by the number 4 above mentioned, one of these binomials must be the cubic root sought, if the given quantity have such a root, and it must be one of these four forms, for it is known to be carried far enough by observing that the cube root of 72 is less than 5, and the cube root of 5120 less than 21; indeed, this being the case, the last binomial is excluded, as evidently too great; and the first is excluded because one of its terms is 0; therefore the root must be either 3+√5 or 4+√12. And to know whether of these two it must be, try which of them has its two terms exact divisors of the corresponding terms of the given quantity; then it is found that 3 and 4 are both divisors of 72, but that only 5, and not 12, is a divisor of 5120; therefore 3+√5 is the root sought, which upon trial is found to answer. It is remarkable here that Girard uses 4+√20 instead of 4+√12, and 5+√29 instead of 5+√20, contrary to his own rule.</p><p>Girard then gives distinct and plain rules for bringing questions to equations, and for the reduction of those equations to their simplest form, for solution, by the usual modes, and also by the way called by Vieta <hi rend="italics">Isomeria,</hi> multiplying the terms of the equation by the terms of a geometrical progression, by which means the roots are altered in the proportion of 1 to the ratio of the progression. He then treats of the methods of finding the roots of the several sorts of equations, quadratic, cubic, &c; and adds remarks on the proper number of conditions or equations for limiting questions. The quadratics are resolved by completing the square, and both the positive and negative roots are taken; and he observes that sometimes the equation is impossible, as 2<*> equ. 6○1 - 25, whose roots, he adds, are 3 + √-16 and 3 - √-16.</p><p>The cubic equations he resolves by Cardan's rule, except the irreducible case, which he the first of any resolves by a table of sines; the other cases also he resolves by tables of sines and tangents; and adds geometrical constructions by means of the hyperbola or the <pb n="89"/><cb/> trisection of angles. He next adds a particular mode of resolving all sorts of equations, that have rational roots, upon the principle of the roots being divisors of the last or absolute term, as before mentioned by Peletarius; and then gives the method of approximating to other roots that are not rational, much the same way as Stevinus.</p><p>Having found one root of an equation, by any of the former methods, by means of it he depresses the equation one degree lower, then finds another root, and so on till they are all found; for he shews that every algebraic equation admits of as many solutions or roots, as there are units in the index of the highest power, which roots may be either positive or negative, or imaginary, or, as he calls them, greater than nothing, or less than nothing, or involved; so the roots of the equation 1○3 equ. 7○1 - 6, are 2, 1, and - 3; and the roots of the equation 1○4 equ. 4○1 - 3 are <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">-1 + √-2,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">-1 - √-2.</cell></row></table></p><p>In depressing an equation to lower degrees, he does not use the method of resolution of Harriot, but that which is derived from the general relation of the roots and coefficients of the terms, which he here fully and universally states, viz, that the coefficient of the 2d term is equal to the sum of all the roots; that of the 3d term equal to the sum of all the products of the roots, taken two by two; that of the 4th term, the sum of the products, taken three by three; and so on, to the last or absolute term, which is the continual product of all the roots; a property which was before stated by Vieta, as to the equations that have all their roots positive; and here extended by Girard to all sorts of roots whatever: but how either Vieta or he came by this property, no where appears that I know of. From this general property, among other deductions, Girard shews how to find the sums of the powers of the roots of an equation; thus, let A, B, C, D, &c, be the 1st, 2d, 3d, 4th, &c, coefficient, after the first term, or the sums of the products taken one by one, two by two, three by three, &c; then, in all sorts of equations, <table><row role="data"><cell cols="1" rows="1" role="data">A</cell><cell cols="1" rows="1" rend="rowspan=4" role="data"><hi rend="size(7)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=4" role="data">will be the sum of the</cell><cell cols="1" rows="1" rend="rowspan=4" role="data"><hi rend="size(7)">{</hi></cell><cell cols="1" rows="1" role="data">roots,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aq-2B</cell><cell cols="1" rows="1" role="data">squares,</cell></row><row role="data"><cell cols="1" rows="1" role="data">A cub.-3AB+3C</cell><cell cols="1" rows="1" role="data">cubes,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aqq-4AqB+4AC+2Bq-4D</cell><cell cols="1" rows="1" role="data">biquadrates.</cell></row></table></p><p>Girard next explains the use of negative roots in Geometry, shewing that they represent lines only drawn in a direction contrary to those representing the positive roots; and he remarks that this is a thing hitherto unknown. He then terminates the Algebra by some questions having two or more unknown quantities; and subjoins to the whole a tract on the mensuration of the surfaces of spherical triangles and polygons, by him lately discovered.</p><p>From the foregoing account it appears that,</p><p>1st, He was the first person who understood the general doctrine of the formation of the coefficients of the powers, from the sums of their roots, and their products, &c.</p><p>2d, He was the first who understood the use of negative roots in the solution of geometrical problems.</p><p>3d, He was the first who spoke of the imaginary roots, and understood that every equation might have as many roots real and imaginary, and no more, as there are <cb/> units in the index of the highest power. And he was the first who gave the whimsical name of <hi rend="italics">quantities less than nothing</hi> to the negative. And,</p><p>4th, He was the first who discovered the rules for summing the powers of the roots of any equation. <hi rend="center"><hi rend="smallcaps">OF HARRIOT.</hi></hi></p><p>Thomas Harriot, a celebrated astronomer, philosopher, and mathematician, flourished about the year 1610, about which time it is probable he wrote his Algebra, as he was then, and had been for many years before, celebrated for his mathematical and astronomical labours. In that year he made observations on the spots in the sun, and on Jupiter's satellites, the same year also in which Galileo first observed them: he left many other curious astronomical observations, and amongst them, some on the remarkable comets of the years 1607 and 1618. His Algebra was left behind him unpublished, as well as those other papers, at his death, which happened in the year 1621, being then 60 years of age, and but six years after the first publication of the principal parts of Vieta's Algebra by Alexander Anderson; so that it is probable that Harriot's Algebra was written before this time, and indeed that he had never seen these pieces. Harriot's Algebra was published by his friend Walter Warner, in the year 1631: and it would doubtless be highly grateful to the learned in these sciences, if his other curious algebraical and astronomical works were published from his original papers in the possession of the Earl of Egremont, to whom they have descended from Henry Percy, the Earl of Northumberland, that noble Mæcenas of his day. The book is in folio, and intitled <hi rend="italics">Artis Analyticæ Praxis, ad Æquationes Algebraicas nova, expedita, & generali methodo, resolvendas</hi>; a work in all parts of it shewing marks of great genius and originality, and is the first instance of the modern form of Algebra in which it has ever since appeared. It is prefaced by 18 definitions, which are these: 1st, Logistica Speciosa; 2d, Equation; 3d, Synthesis; 4, Analysis; 5, Composition and Resolution; 6, Forming an Equation; 7, Reduction of an Equation; 8, Verification; 9, Numerosa & Speciosa; 10, Excogitata; 11, Resolution; 12, Roots; 13 and 14, The kinds and generation of equations by multiplication, from binomial roots or factors, called original equations, <table><row role="data"><cell cols="1" rows="1" role="data">as</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">=<hi rend="italics">aa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">ba</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> - <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">ca</hi> - <hi rend="italics">bc,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">=<hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">+<hi rend="italics">baa</hi>+<hi rend="italics">bca</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+<hi rend="italics">caa</hi>-<hi rend="italics">bda</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> - <hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-<hi rend="italics">daa</hi>-<hi rend="italics">cda</hi>-<hi rend="italics">bcd,</hi></cell></row></table> where he puts <hi rend="italics">a</hi> for the unknown quantity, and the small consonants, <hi rend="italics">b, c, d,</hi> &c, for its literal values or roots; 15, The first form of canonical equations, which are derived from the above originals, by transposing the homogeneum, or absolute term, <table><row role="data"><cell cols="1" rows="1" role="data">thus <hi rend="italics">aa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">ba</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">ca</hi></cell><cell cols="1" rows="1" role="data">= + <hi rend="italics">bc,</hi> &c;</cell></row></table> 16, The secondary canonicals, formed from the primary by expelling the 2d term, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">thus <hi rend="italics">aa</hi></cell><cell cols="1" rows="1" role="data">= + <hi rend="italics">bb,</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or <hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">- <hi rend="italics">bba</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">bca</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">cca</hi> =</cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">bbc</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ <hi rend="italics">bcc</hi>;</cell></row></table> <pb n="90"/><cb/> 17, That these are called canonicals, because they are adapted to canons or rules for finding the numeral roots, &c. 18, Reciprocal equations, in which the homogeneum is the product of the coefficients of the other terms, and the first term, or highest power of the root, is equal to the product of the powers in the other terms, as .</p><p>After these definitions, the work is divided into two principal parts; 1st, of various generations, reductions, and preparations of equations for their resolution in the 2d part. The former is divided into 6 sections as follows.</p><p><hi rend="italics">Sect.</hi> 1. <hi rend="italics">Logistices Speciosæ,</hi> exemplified in the 4 operations of addition, subtraction, multiplication, and division; as also the reduction of algebraic fractions, and the ordinary reduction of irregular equations to the form proper for the resolution of them, namely, so that all the unknown terms be on one side of the equation, and the known term on the other, the powers in the terms ranged in order, the greatest first, and the first or highest power made positive, and freed from its coefficient; as , <hi rend="center">or .</hi> In this part he explains some unusual characters which he introduces, namely <hi rend="center">= for equality, as <hi rend="italics">a</hi> = <hi rend="italics">b.</hi></hi> <hi rend="center">> for majority, as <hi rend="italics">a</hi> > <hi rend="italics">b,</hi></hi> <hi rend="center">< for minority, as <hi rend="italics">a</hi> < <hi rend="italics">b</hi></hi>; but the first had been before introduced by Robert Recorde.</p><p><hi rend="italics">Sect.</hi> 2. The generation of original equations from binomial factors or roots, and the deducing of canonicals from the originals. He supposes that every equation has as many roots as dimensions in its highest power; then supposing the values of the unknown letter <hi rend="italics">a</hi> in any equation to be <hi rend="italics">b, c, d, f,</hi> &c, that is <hi rend="italics">a</hi>=<hi rend="italics">b,</hi> and <hi rend="italics">a</hi>=<hi rend="italics">c,</hi> and <hi rend="italics">a</hi>=<hi rend="italics">d,</hi> &c; by transposition, or equal subtraction, these become , and , and , &c, or the same letters with contrary signs, for negative values or roots; then two of these binomial factors multiplied together, gives a quadratic equation, three of them a cubic, four of them a biquadratic, and so on, with all the terms on one side of the equation, and 0 on the other side, since, every binomial factor being = 0, the continual product of all of them must also be = 0. Thus, <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">= <hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">baa</hi> + <hi rend="italics">bca</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> + <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ <hi rend="italics">caa</hi> - <hi rend="italics">bda</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">a</hi> - <hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">daa</hi> - <hi rend="italics">cda</hi>-<hi rend="italics">bcd</hi> = 0</cell></row></table> an original equation, and <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">aaa</hi></cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">baa</hi> + <hi rend="italics">bca</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ <hi rend="italics">caa</hi> - <hi rend="italics">bda</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- <hi rend="italics">daa</hi> - <hi rend="italics">cda</hi></cell><cell cols="1" rows="1" role="data">= + <hi rend="italics">bcd</hi></cell></row></table> its canonical, deduced from it. And these operations are carried through all the cases of the 2d, 3d and 4th powers, as to the varieties of the signs + and -, and the proportions of the roots as to equal and unequal, with the reciprocals, &c. From which are made evident, at one glance of the eye, all the relations and properties between the roots of equations, and the coefficients of the terms.</p><p><hi rend="italics">Sect.</hi> 3. <hi rend="italics">Æquationum canonicarum secundariarum a primariis reductio per gradus alicujus parodici sublatiouem</hi> <cb/> <hi rend="italics">radice supposititia invariata manente.</hi> Containing a great many examples of preparing equations by taking away the 2d, 3d, or any other of the intermediate terms, which is done by making the positive coefficients in that term, equal to the negative ones, by which means the whole term vanishes, or becomes equal to nothing.</p><p>They are extended as far as equations of the 5th degree; and at the end are collected, and placed in regular order, all the secondary canonicals, so reduced, so that by the uniform law which is visible through them all, the series may be continued to the higher degrees as far as we please.</p><p><hi rend="italics">Sect.</hi> 4. <hi rend="italics">Æquationum canonicarum tam primariarum, quam secundariarum, radicum designatio.</hi> A great many literal equations are here set down, and their roots assigned from the form of the equation, that is all their positive roots; for their negative roots are not noticed here; and it is every where proved that they cannot have any more positive roots than these, and consequently the rest are negative. That those are roots, he proves by substituting them instead of the unknown letter <hi rend="italics">a</hi> in the equation, when they make all the terms on one side come to the same thing as the homogeneum on the other side.</p><p><hi rend="italics">Sect.</hi> 5, <hi rend="italics">In qua æquationum communium per canonicarum æquipollentiam, radicum numerus determinatur.</hi> On the number of the roots of common equations, that is the positive roots. This Harriot determines by comparing them with the like cases found among his canonical forms, which two equations, having the same number of terms with the same signs, and the relations of the coefficients and homogeneum correspondent, he calls equipollents. And whatever was the number of positive roots used in the composition of the canonical, the same, he infers, is the number in the proposed common equation. It is remarkable that in all the examples here used, the number of positive roots is just equal to the number of the changes in the signs from + to and from - to +, which is a circumstance, though not here expressly mentioned, that could not escape the observation, or the eye, of any one, much less of so clear and comprehensive a sight as that of Harriot. In this section are contained many ingenious disquisitions concerning the limits and magnitudes of quantities, with several curious lemmas laid down to demonstrate the propositions by, which lemmas are themselves demonstrated in a pure mathematical way, from the magnitudes themselves, independent of geometrical sigures; such as, 1, If a quantity be divided into any two unequal parts, the square of half the line will be greater than the product of the two unequal parts. 2, In three continued proportionals, the sum of the extremes is greater than double the mean. 3, In four continued proportionals, the sum of the extremes is greater than the sum of the two means. 4, In any two quantities, one-fourth the square of the sum of the cubes, is greater than the cube of the product of the two quantities. 5, Of any two quantities <hi rend="italics">q</hi> and <hi rend="italics">r,</hi> then (1/27)(<hi rend="italics">qq</hi> + <hi rend="italics">qr</hi> + <hi rend="italics">rr</hi>)<hi rend="sup">3</hi> > 1/4 (<hi rend="italics">qqr</hi>+<hi rend="italics">qrr</hi>)<hi rend="sup">2</hi>. 6, If any quantity be divided into three unequal parts, the square of 1/3 of the whole quantity is greater than 1/3 of the sum of the three products made of the three unequal parts. 7, Also the cube of the 1/3 part of the whole, is greater than the solid or continual product of the three unequal parts. <pb n="91"/><cb/></p><p><hi rend="italics">Sect.</hi> 6. <hi rend="italics">Æquationum communium reductio per gradus alicujus parodici exclusionem & radicis supposititiæ mutationeni.</hi> Here are a great many examples of reducing and transforming equations of the 2d, 3d, and 4th degrees; chiefly either by multiplying the roots of equations in any proportion, as was done by Vieta, or increasing or diminishing the root by a given quantity, after the manner of Cardan. The former of these reductions is performed by multiplying the terms of the equation by the corresponding terms of a geometrical progression, the 1st term being 1, and the 2d term the quantity by which the root is to be multiplied. And the other reduction, or transforming to another root, which may be greater or less than the given root by a given quantity, is performed commonly by substituting <hi rend="italics">e</hi> + or - <hi rend="italics">b</hi> for the given root <hi rend="italics">a,</hi> by which the equation is reduced to a simpler form. Other modes of substitution are also used; one of which is this, viz, substituting (<hi rend="italics">ee</hi> ± <hi rend="italics">bb</hi>)/<hi rend="italics">e</hi> or <hi rend="italics">e</hi> ± <hi rend="italics">bb/e</hi> for the root <hi rend="italics">a</hi> in the given equation by which it reduces to this quadratic form , from whence Cardan's forms are immediately deduced; namely , and therefore ; where he denotes the cube or 3d root thus √3), but without any vinculum over the compound quantities.</p><p>In this section, Harriot makes various remarks as they occur: thus he remarks, and demonstrates, that <hi rend="italics">eee</hi> - 3.<hi rend="italics">bbe</hi> = -<hi rend="italics">ccc</hi> -2.<hi rend="italics">bbb</hi> is an impossible equation, or has no affirmative root. He remarks also that the three cases of the equation <hi rend="italics">aaa</hi> - 3.<hi rend="italics">bba</hi> = + 2.<hi rend="italics">ccc</hi> are similar to the three conic sections; namely to the hyperbola when <hi rend="italics">c</hi> > <hi rend="italics">b,</hi> to the parabola when <hi rend="italics">c</hi> = <hi rend="italics">b,</hi> or to the ellipsis when <hi rend="italics">c</hi> < <hi rend="italics">b,</hi> and for which reason this case is not generally resoluble in species.</p><p>Having thus shewn how to simplify equations, and prepare them for solution, Harriot enters next upon the second part of his work, being the <hi rend="center"><hi rend="italics">Exegetice Numerosa,</hi></hi></p><p>or the numeral resolution of all sorts of equations by a general method, which is exemplified in a great number of equations, both simple and affected as far as the 5th power inclusive; and they are commonly prepared, by the foregoing parts, by freeing them from their 2d term, &c. These extractions are explained and performed in a way different from that of Vieta; and the examples are first in perfect or terminate roots, and afterwards for irrational or interminate ones, to which Harriot approximates by adding always periods of ciphers to the given number or resolvend, as far as necessary in decimals, which are continued and set down as such, but with their proper denominator 10, or 100, or 1000, &c.</p><p>He then concludes the work with <hi rend="center"><hi rend="italics">Canones Directorii,</hi></hi></p><p>which form a collection of the cases or theorems for making the foregoing numeral extractions, ready arranged for use, under the various forms of equations, with the factors necessary to form the several resolvends and subtrahends. <cb/></p><p>And from a review of the whole work, it appears that Harriot's inventions, peculiarities, and improvements in algebra, may be comprehended in the following particulars.</p><p>1st. He introduced the uniform use of the small letters <hi rend="italics">a, b, c, d,</hi> &c, viz, the vowels <hi rend="italics">a, e,</hi> &c for unknown quantities, and the consonants <hi rend="italics">b, c, d, f,</hi> &c for the known ones; which he joins together like the letters of a word, to represent the multiplication or product of any number of these literal quantities, and prefixing the numeral coefficient as we do at present, except only separated by a point, thus 5.<hi rend="italics">bbc.</hi> For a root he set the index of the root after the mark √; as √3) for the cube root. He also introduced the characters > and < for greater and less; and in the reduction of equations, he arranged the operations in separate steps or lines, setting the explanations in the margin on the left hand, for each line. By which, and other means, he may be considered as the introducer of the modern state of Algebra, which quite changed its form under his hands.</p><p>2d. He shewed the universal generation of all the compound or affected equations, by the continual multiplication of so many simple ones, or binomial roots; thereby plainly exhibiting to the eye the whole circumstances of the nature, mystery and number of the roots of equations; with the composition and relations of the coefficients of the terms; and from which many of the most important properties have since been deduced.</p><p>3d. He greatly improved the numeral exegesis, or extraction of the roots of all equations, by clear and explicit rules and methods, drawn from the foregoing generation or composition of affected equations of all degrees. <hi rend="center"><hi rend="smallcaps">OF OUGHTRED'S CLAVIS.</hi></hi></p><p>Oughtred was contemporary with Harriot, but lived a long time after him. His Clavis was first published in 1631, the same year in which Harriot's Algebra was published by his friend Warner. In this work, Oughtred chiefly follows Vieta, in the notation by the capitals A, B, C, D, &c, in the designation of products, powers, and roots, though with some few variations. His work may be comprehended under the following particulars.</p><p>1. <hi rend="italics">Notation.</hi> This extends to both Algebra and Arithmetic, vulgar and decimal. The Algebra chiefly after the manner of Vieta, as abovesaid. And he separates the decimals from the integers thus, 21<03>56, which is the first time I have observed such a separation, and the decimals set down without their denominator.</p><p>2. The common rules or operations of Arithmetic and Algebra. In algebraic multiplication, he either joins the letters together like a word, or connects them by the mark X, which is the first introduction of this character of multiplication: thus A X A or AA or A<hi rend="italics">q.</hi> But omitting the vinculum over compound factors, used by Vieta. He introduces here many neat and useful contractions in multiplication and division of decimals: as that common one of inverting the multiplier, to have fewer decimals, and abridge the work; that of omitting always one figure at a time, of the divisor, for the same purpose; dividing by the component factors of a number instead of the number itself; as 4 and 6 for 24; and many other neat contractions. He <pb n="92"/><cb/> states his proportions thus 7.9 :: 28.36, and denotes continued proportion thus <04>; which is the first time I have observed these characters.</p><p>3. Invents and describes various symbolical marks or abbreviations, which are not now used.</p><p>4. <hi rend="italics">The genesis and analysis of powers.</hi> Denotes powers like Vieta, and also roots, thus √<hi rend="italics">q</hi>6, √<hi rend="italics">c</hi>20, √<hi rend="italics">qq</hi>24, &c; and much in his manner too performs the numeral extraction of roots. He here gives a table of the powers of the binomial A + E as far as the 10th power, with all their terms and coefficients, or unciæ as he calls them, after Vieta.</p><p>5. <hi rend="italics">Equations.</hi> He here gives express and particular directions for the several sorts of reductions, according as the form of the equation may require. And he uses the letter <hi rend="italics">u</hi> after √, for universal, instead of the vinculum of Vieta. And observes that the signs of all the terms of the powers of A + E are positive, but those of A - E are alternately positive and negative.</p><p>6. Next follow many properties of triangles and other geometrical sigures; and the first instance of applying Algebra to Geometry, so as to investigate new geometrical properties; and after the algebraical resolution of each problem, he commonly deduces and gives a geometrical construction adapted to it. He gives also a good tract on angular sections.</p><p>7. The work concludes with the numeral resolution of affected equations, in which he follows the manner of Vieta, but he is more explicit. <hi rend="center"><hi rend="smallcaps">OF DESCARTES.</hi></hi></p><p>Descartes's Geometry was first published in 1637, being six years after the publication of Harriot's Algebra. That work was rather an application of Algebra to Geometry, than the science either of Algebra or Geometry itself, purely and properly so called. And yet he made improvements in both. We must observe however, that all the properties of equations, &c, which he sets down, are not to be considered as even meant by himself for new inventions or discoveries; but as statements and enumerations of properties, before known and taught by other authors, which he is about to make some use or application of, and for which reason it is that he mentions those properties.</p><p>Descartes's Geometry consists of three books. The sirst of these is, <hi rend="italics">De Problematibus, quæ construi possunt, adhibendo tantum rectas lineas & circulos.</hi> He here accommodates or performs arithmetical operations by Geometry, supposing some line to represent unity, and then, by means of proportionals, shewing how to multiply, divide, and extract roots by lines. He next describes the notation he uses, but not because it is a new one, for it is the same as had been used by former authors, viz, <hi rend="italics">a</hi> + <hi rend="italics">b</hi> for the addition of <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> also <hi rend="italics">a</hi> - <hi rend="italics">b</hi> for their subtraction, <hi rend="italics">ab</hi> multiplication, <hi rend="italics">a/b</hi> division, <hi rend="italics">aa</hi> or <hi rend="italics">a</hi><hi rend="sup">2</hi> the square of <hi rend="italics">a, a</hi><hi rend="sup">3</hi> its cube, &c: also √(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi>) for the square root of <hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi>, and for the cube root, &c. He then observes, after Stifelius, that there must be as many equations as there are unknown lines or quantities; and that they must be reduced all to one final equation, by exterminating all the unknown letters except one; when the final equation will appear like these, <cb/> Where he uses <figure/> for = or equality, setting the highest term or power alone on one side of the equation, and all the other terms on the other side, with their proper signs.</p><p>Descartes next defines plane problems, namely, such as can be resolved by right lines and circles, described on a plane superficies; and then the final equation rises only to the 2d power of the unknown letter. He then constructs such equations, viz. quadratics, by the circle, thus finding geometrically the root or roots, that is, the positive ones. But when the lines, by which the roots are determined, neither cut nor touch, he observes that the equation has then no possible root, or that the problem is impossible. He then concludes this book with the algebraical solution of the celebrated problem, before treated of by the antients, namely, to sind a point, or the locus of all the points, from whence a line being drawn to meet any number of given lines in given angles, the product of the segments of some of them shall have a given ratio to that of the rest.</p><p><hi rend="italics">Lib.</hi> 2. <hi rend="italics">De Natura Linearum Curvarum.</hi> This is a good algebraical treatise on curve lines in general, and the first of the kind that has been produced by the moderns. Here the nature of the curve is expressed by an equation containing two unknown or variable lines, and others that are known or constant, as <hi rend="italics">y</hi><hi rend="sup">z</hi> <figure/> <hi rend="italics">cy</hi> - <hi rend="italics">cxy/b</hi> + <hi rend="italics">ay</hi> - <hi rend="italics">ac.</hi> But, not relating to pure Algebra, the particulars will be most properly placed under the article of curve lines, and other terms relating to them. Only one discovery, among many ingenious applications of Algebra to Geometry, may here be particularly noticed, as it may be considered as the first step towards the arithmetic of infinites; and that is the method of tangents, here given, or, which comes to the same thing, of drawing a line perpendicular to a curve at any point, which is an ingenious application of the general form of an equation, generated in Harriot's way, that has two equal roots, to the equation of the curve. Of which a particular account will be given at the article <hi rend="smallcaps">Tangents.</hi></p><p><hi rend="italics">Lib.</hi> 3. <hi rend="italics">De Constructione Problematum Solidorum, et Solida excedentium.</hi> Descartes begins this book with remarks on the nature and roots of equations, observing that they have as many roots as dimensions, which he shews, after Harriot, by multiplying a certain number of simple binomial equations together, as <hi rend="italics">x</hi> - 2 <figure/> 0, and <hi rend="italics">x</hi> - 3 <figure/> 0, and <hi rend="italics">x</hi> - 4 <figure/> 0, producing <hi rend="italics">x</hi><hi rend="sup">3</hi> - 9<hi rend="italics">xx</hi> + 26<hi rend="italics">x</hi> - 24 <figure/> 0. He here remarks that equations may sometimes have their roots <hi rend="italics">false,</hi> or what we call negative, which he opposes to those that are positive, or as he calls them <hi rend="italics">true,</hi> as Cardan had done before. As a natural deduction from the generation or composition of equations, by multiplication, he infers their resolution, or depression, or decomposition, namely, dividing them by the binomial factors which were multiplied to produce the equation: and he observes that by this operation it is known that this divisor is one of the binomial roots, and that there can be no more roots than dimensions, or than those which form with the unknown letter <pb n="93"/><cb/> <hi rend="italics">x,</hi> binomials that will exactly divide the equation, as Harriot had shewn before. Descartes adverts to several other properties, mostly known before, which he has occasion to make use of in the progress of his work; such as, that equations may have as many true roots as the terms have changes of the signs + and -, and as many false ones as successions of the same signs: which number and nature of the roots had before been partly shewn by Cardan and Vieta, from the relation of the coefficients, and their signs, and more fully by Harriot in his 5th section. And hence Descartes infers the method of changing the true roots to false, and the false to true, namely by changing the signs of the even terms only, as Cardan had taught before. Descartes then adverts to other reductions and transmutations which had been taught by Cardan, Vieta, and Harriot, such as, To increase or diminish the roots by any quantity; To take away the 2d term: To alter the roots in any proportion, and thence to free the equation from fractions and radicals.</p><p>Descartes next remarks that the roots of equations, whether true or false, may be either real or imaginary; as in the equation <hi rend="italics">x</hi><hi rend="sup">3</hi> - 6<hi rend="italics">xx</hi> + 13<hi rend="italics">x</hi> - 10 <figure/> 0, which has only one real root, namely 2. The imaginary roots were first noticed by Albert Girard, as before mentioned. He then treats of the depression of a cubic equation to a quadratic, or plane problem, that it may be constructed by the circle, by dividing it by some one of the binomial factors, which, in Harriot's way, compose the equation. Peletarius having shewn that the simple root is one of the divisors of the known term of the equation, and Harriot that that term is the continual product of all the roots, Descartes therefore tries all the simple divisors of that term, till he finds one of them which, connected with the unknown letter <hi rend="italics">x,</hi> by + or -, will exactly divide the equation. And the process is the same for higher powers than the cube. But when a divisor cannot be thus found, for depressing a biquadratic equation to a cubic, he gives another rule, which is a new one, for dissolving it into two quadratics, by means of a cubic equation, in this manner: Let the given biqu. be + <hi rend="italics">x</hi><hi rend="sup">4</hi>* . <hi rend="italics">pxx. qx. r</hi> <figure/> 0; where the sign of (1/2)<hi rend="italics">p</hi> in the two quadratics must be the same as the sign of <hi rend="italics">p</hi> in the given equation, and in the 1st quadratic the sign of <hi rend="italics">q</hi>/2<hi rend="italics">y</hi> must be the same as the the sign of <hi rend="italics">q,</hi> but in the 2d quadratic the contrary. Then if there be found the root <hi rend="italics">yy</hi> of this cubic equation <hi rend="italics">y</hi><hi rend="sup">6</hi> . 2<hi rend="italics">py</hi><hi rend="sup">4</hi> + (+<hi rend="italics">pp</hi>)/(.4<hi rend="italics">r</hi>)<hi rend="italics">yy</hi>-<hi rend="italics">qq</hi> <figure/> 0, where the sign of 2<hi rend="italics">p</hi> is the same as of <hi rend="italics">p</hi> in the given biquadratic, but the sign of 4<hi rend="italics">r</hi> contrary to that of <hi rend="italics">r</hi> in the same: Then the value of <hi rend="italics">y,</hi> hence deduced, being substituted for it in the two quadratic equations, and their two pairs of roots taken, they will be the four roots of the proposed biquadratic. And thus also, he hints, may equations of the 6th power be reduced to those of the 5th, and those of the 8th power to those of the 7th, <cb/> and so on. Descartes does not give the investigation of this rule; but it has evidently been done, by assuming indeterminate quantities, after the manner of Ferrari and Cardan, as coefficients of the terms of the two quadratic equations, and, after multiplying the two together, determining their values by comparing the resulting terms with those of the proposed biquadratic equation.</p><p>After these reductions, which are only mentioned for the sake of the geometrical constructions which follow, by simplifying and depressing the equations as much as they will admit, Descartes then gives the construction of solid and other higher problems, or of cubic and higher equations, by means of parabolas and circles; where he observes that the false roots are denoted by the ordinates to the parabola lying on the contrary side of the axis to the true roots. Finally, these constructions are illustrated by various problems concerning the trisecting of an angle, and the finding of two or four mean proportionals; which concludes this ingenious work.</p><p>From the foregoing analysis may easily be collected the real inventions and improvements made in algebra by Descartes. His work, as has been observed before, is not algebra itself, but the application of algebra to geometry, and the algebraical doctrine of curve lines, expressing and explaining their nature by algebraical equations, and on the contrary, constructing and explaining equations by means of the curve lines. What respects the geometrical parts of this tract we shall have occasion to advert to elsewhere; and therefore shall here only enumerate the circumstances which belong more peculiarly to the science of Algebra, which I shall distinguish into the two heads of improvements and inventions. And</p><p>1st. Of his improvements. That he might fit equations the better for their application in the construction of problems, Descartes mentions, as it were by-the-bye, many things concerning the nature and reduction of equations, without troubling himself about the first inventors of them, stating them in his own terms and manner, which is commonly more clear and explicit, and osten with improvements of his own. And under this head we sind that he chiefly followed Cardan, Vieta, and Harriot, but especially the last, and explains some of their rules and discoveries more distinctly, and varies but a little in the notation, putting the first letters of the alphabet for the known, and the latter letters for the unknown quantities; also <hi rend="italics">x</hi><hi rend="sup">3</hi> for <hi rend="italics">aaa,</hi> &c; and <figure/> for =. But Herigone used the numeral exponents in the same manner two years before. Descartes explained or improved most parts of the reductions of equations, in their various transmutations, the number and nature of their roots, true and false, real and what he calls imaginary, called involved by Girard; and the depression of equations to lower degrees.</p><p>2d. As to his inventions and discoveries in algebra, they may be comprehended in these particulars, namely, the application of algebra to the geometry of curve lines, the constructing equations of the higher orders, and a rule for resolving biquadratic equations by means of a cubic and two quadratics.</p><p>Having now traced the science of Algebra from its origin and rude state, down to its modern and more polished form, in which it has ever since continued, with very little variation; having analysed all or most <pb n="94"/><cb/> of the principal authors, in a chronological order, and deduced the inventions and improvements made by each of them; from this time the authors both become too numerous, and their improvements too inconsiderable, to merit a detail in the same minute and circumstantial way: and besides, these will be better explained in a particular manner under the word or article to which each of them severally belongs. It may therefore now suffice to enumerate, or announce only in a cursory manner, the chief improvements and authors on algebra down to the present time.</p><p>After the publication of the Geometry of Descartes, a great many other ingenious men followed the same course, applying themselves to algebra and the new geometry, to the mutual improvement of them both; which was done chiefly by reasoning on the nature and forms of equations, as generated and composed by Harriot. Before proceeding upon these however, it is but proper to take notice here of Fermat, a learned and ingenious mathematician, who was contemporary and a competitor of Descartes for his brightest discoveries, which he was in possession of before the geometry of Descartes appeared. Namely, the application of algebra to curve lines, which he expressed by an algebraical equation, and by them constructing equations of the 3d and 4th orders; also a method of tangents, and a method de maximis et minimis, which approach very near to the method of Fluxions or Increments, which they strikingly resemble both in the manner of treating the problems, and in the algebraic notation and process. The particulars of which, see under their proper heads. Besides these, Fermat was deeply learned in the Diophantine problems, and the best edition of Diophantus's Arithmetic, is that which contains the notes of Fermat on that ingenious work.</p><p>But to return to the successors of Descartes. His geometry having been published in Holland, several learned and ingenious mathematicians of that country, presently applied themselves to cultivate and improve it; as Schooten, Hudde, Van-Heuraet, De Witte, Slusius, Huygens, &c; besides M. de Beaune, and perhaps some others in France.</p><p>Francis Schooten, professor of mathematics in the university of Leyden, was one of the first cultivators of the new geometry. He translated Descartes's Geometry out of French into Latin, and published it in 1649, with his commentary upon it, as also Brief Notes of M. de Beaune; both of them containing many ingenious and useful things. And in 1659 he gave a new edition of the same in two volumes, with the addition of several other ingenious pieces: as two posthumous tracts of de Beaune, the one on the nature and constitution, the other on the limits of equations, shewing how to assign the limits between which are contained the greatest and least roots of equations, extended and completed by Erasmus Bartholine: two letters of M. Hudde on the reduction of equations, and on the maxima and minima of quantities, containing many ingenious rules; among which are some concerning the drawing of tangents, and on the equal roots of equations, which he determines by multiplying the terms of the equation by the terms of any arithmetical progression, <*> being one of the terms, the equation is commonly depressed one degree lower: also a tract of Van Heuraet on the rectifi- <cb/> cation of curve lines; the elements of curves by De Witte; Schooten's principles of universal mathematics, or introduction to Descartes's geometry, which had before been published by itself in 1651; and to the end of the work is added a posthumous piece of Schooten's (for he died while the 2d vol. was printing) intitled <hi rend="italics">Tractatus de concinnandis demonstrationibus geometricis ex calculo algebraico.</hi> Schooten also published, in 1657, <hi rend="italics">Exercitationes Mathematicæ,</hi> in which are contained many curious algebraical and analytical pieces, amongst others of a geometrical nature.</p><p>An elaborate commentary on Descartes's Geometry was also published by F. Rabuel, a Jesuit; and James Bernoulli, enriched with notes, an edition of the same, printed at Basil in 169—.</p><p>The celebrated Huygens also, among his great discoveries, very much cultivated the algebraical analysis: and he is often cited by Schooten, who relates divers inventions of his, while he was his pupil.</p><p><hi rend="italics">Slusius,</hi> a canon of Liege, published in 1659, <hi rend="italics">Mesolabum, seu deæ mediæ propor. per circulum & ellips. vel hyperb. infinitis modis exhibitæ;</hi> by which, any solid problem may be constructed by infinite different ways. And in 1668 he gave a second edition of the same, with the addition of the analysis, and a miscellaneous collection of curious and important problems, relating to spirals, centres of gravity, maxima and minima, points of inflexion, and some Diophantine problems; all shewing him deeply skilled in Algebra and Geometry.</p><p>There have been a great number of other writers and improvers of Algebra, of which it may suffice slightly to mention the chief part, as in the following catalogue.</p><p>Peter Nonius, or Nunez, a Spaniard, wrote about the time of Cardan, or soon after.</p><p>In 1619 several pieces of Van Collen, or Ceulen, were translated out of Dutch into Latin, and published at Leyden by W. Snell; among which are contained a particular treatise on surds, and his proportion of the circumference of a circle, to its diameter.</p><p>In 1621 Bachet published, in Greek and Latin, an edition of Diophantus, with many notes. And another edition of the same was published in 1670, with additions by Fermat.</p><p>In 1624 Bachet's <hi rend="italics">Problemes Plaisans et Delectables,</hi> being curious problems in mathematical recreations.</p><p>In 1634 Herigone published, at Paris, the first course of mathematics, in 5 vols. 8vo; in the 2d of which is contained a good treatise on Algebra; in which he uses the notation by small letters, introduced by the Algebra of Harriot, which was published three years before, though the rest of it does not resemble that work, and one would suspect that Herigone had not seen it. The whole of this piece bears evident marks of originality and ingenuity. Besides + for <hi rend="italics">plus,</hi> he uses <01> for <hi rend="italics">minus,</hi> and | for <hi rend="italics">equality,</hi> with several other usful abbrevations and marks of his own. In the notation of powers, he does not repeat the letters like Harriot, but subjoins the numeral exponents, to the letter, as Descartes did two years afterwards. And Herigone uses the same numeral exponents for roots, as √3 for the cube root.</p><p>In 1635 Cavalerius published his <hi rend="italics">Indivisibles;</hi> which proved a new æra in analytics, and gave rise to other new modes of computation in analytics. <pb n="95"/><cb/></p><p>About 1640, et seq. Roberval made several notable improvements in analytics, which are published in the early volumes of the Memoirs of the Academy of Sciences; as, 1. A tract on the composition of motion, and a method of tangents. 2, <hi rend="italics">De recognitione æquationum.</hi> 3, <hi rend="italics">De geometrica planarum & cubicarum æquationum resolutione.</hi> 4, A treatise on indivisibles, &c.</p><p>In 1643 De Billy published <hi rend="italics">Nova Geometriæ Clavis Algebra.</hi> And in 1670 <hi rend="italics">Diophantus Redivivus.</hi> He was an author particularly well skilled in Diophantine problems.</p><p>In 1644 Renaldine published, in 4to, <hi rend="italics">Opus Algebraicum,</hi> both ancient and modern, with mathematical resolution and composition. And in 1665, in folio, the same, greatly enlarged, or rather a new work, which is very heavy and tedious. In this work Renaldine uses the parentheses (<hi rend="italics">a</hi><hi rend="sup">2</hi>+<hi rend="italics">b</hi><hi rend="sup">2</hi>) as a vinculum, instead of the line over, as ―(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi>).</p><p>In 1655 was published Wallis's <hi rend="italics">Arithmetica Infinitorum,</hi> being a new method of reasoning on quantities, or a great improvement on the Indivisibles of Cavalerius, and which in a great measure led the way to infinite series, the binomial theorem, and the method of fluxions. Wallis here treats ingeniously of quadratures and many other problems, and gives the sirst expression for the quadrature of the circle by an insinite series. Another series is here added for the same purpose, by the Lord Brouncker.</p><p>In 1659 was published <hi rend="italics">Algebra Rhonii Germanice;</hi> which was in 1668, translated into English by Mr. Thomas Brancker, with additions and alterations by Dr. John Pell.</p><p>In 1661 was published in Dutch, a neat piece of Algebra by Mr. Kinckhuysen; which Sir I. Newton, while he was professor of mathematics at Cambridge, made use of and improved, and he meant to republish it, with the addition of his method of fluxions and infinite series; but he was prevented by the accidental burning of some of his papers.</p><p>In 1665 or 1666 Sir Isaac Newton made several of his brightest discoveries, though they were not published till afterwards: such as the binomial theorem; the method of fluxions and infinite series; the quadrature, rectification, &c of curves; to find the roots of all sorts of equations, both numeral and literal, in infinite converging series; the reversion of series, &c. Of each of which a particular account may be seen in their proper articles.</p><p>In 1666 M. Frenicle gave several curious tracts concerning combinations, magic squares, triangular numbers, &c; which were printed in the early volumes of Memoirs of the Academy of Sciences.</p><p>In 1668 Thomas Brancker published a translation of Rhonius's Algebra, with many additions by Dr. John Pell, who used a peculiar method of registering the steps in any algebraical process, by means of marks and abbreviations in a small column drawn down the margin, by which each line, or step, is clearly explained, as was before done by Harriot in words at length.</p><p>In 1668 Mercator published his <hi rend="italics">Logarithmotechnia,</hi> or method of constructing logarithms; in which he gives the quadrature of the hyperbola, by means of an infinite series of algebraical terms, found by dividing a simple algebraic quantity by a compound one, and for the first time that this operation was given to the <cb/> public, though Newton had before that expanded all sorte of compound algebraical quantities into infinite series.</p><p>In the same year was published James Gregory's <hi rend="italics">Exercitationes Geometricæ,</hi> containing, among other things, a demonstration of Mercator's quadrature of the hyperbola, by the same series.</p><p>And in the same year was published, in the Philosophical Transactions, Lord Brouncker's quadrature of the hyperbola by another infinite series of simple rational terms, which he had been in possession of since the year 1657, when it was announced to the public by Dr. Wallis. Lord Brouncker's series for the quadrature of the circle, had been published by Wallis in his Arithmetic of Infinites.</p><p>In 1669 Dr. Isaac Barrow published his Optical and Geometrical Lectures, abounding with profound researches on the dimensions and properties of curve lines; but particularly to be noticed here for his method of tangents, by a mode of calculation similar to that of Fluxions, or Increments, from which these differ but little, except in the notation.</p><p>In 1673 was published, in 2 vols. folio, <hi rend="italics">Elements of Algebra,</hi> by John Kersey; a very ample and complete work, in which Diophantus's problems are fully explained.</p><p>In 1675 were published Nouveaux Elemens des Mathematiques, par J. Prestet, prêtre: a prolix and tedious work, which he presumptuously dedicated to God Almighty.</p><p>About 1677 Leibnitz discovered his <hi rend="italics">Methodus Differrentialis,</hi> or else made a variation in Newton's Fluxions, or an extension of Barrow's method, for it is not certain which. He gave the first instance of it in the Leipsic Acts for the year 1684. He also improved infinite series, and gave a simple one for the quadrature os the circle, in the same acts for 1682.</p><p>In 1682 Ismael Bulliald published, in folio, his <hi rend="italics">Opus Novum ad Arithmeticam Infinitorum,</hi> being a large amplification of Wallis's Arithmetic of Insinites.</p><p>In 1683 Tschirnausen gave a memoir, in the Leipsic Acts, concerning the extraction of the roots of all equations in a general way; in which he promised too much, as the method did not succeed.</p><p>In 1684 came out, in English and Latin, 4to, Thomas Baker's <hi rend="italics">Geometrical Key, or Gate of Equations Unlock'd;</hi> being an improvement of Descartes's construction of all equations under the 5th degree, by means of a circle and only one and the same parabola for all equations, using any diameter instead of the axis of the parabola.</p><p>In 1685 was published, in folio, Wallis's <hi rend="italics">Treatise of Algebra, both Historical and Practical,</hi> with the addition of several other pieces; shewing the origin, progress, and advancement of that science, from time to time. It cannot be denied that, in this work, Wallis has shewn too much partiality to the Algebra of Harriot. Yet, on the other hand, it is as true, that M. de Gua, in his account of it, in the Memoirs of the Academy of Sciences for 1741, has run at least as far into the same extreme on the contrary side, with respect to the discoveries of Vieta; and both these I believe from the same cause, namely, the want of examining the works of all former writers on Algebra, and specifying their several discoveries; as has been done in the course of this article. <pb n="96"/><cb/></p><p>In 1687 Dr. Halley gave, in the Philos. Trans. the construction of cubic and biquadratic equations, by a parabola and circle; with improvements on what had been done by Descartes, Baker, &c. Also, in the same Transactions, a memoir on the number of the roots of equations, with their limits and signs.</p><p>In 1690 was published, in 4to, by M. Rolle, <hi rend="italics">Traité d' Algébre;</hi> in 1699 <hi rend="italics">Une Methode pour la Resolution des Problemes indeterminés;</hi> and in 1704 <hi rend="italics">Memoires sur Pinverse des tangents;</hi> and other pieces.</p><p>In 1690 Joseph Raphson published <hi rend="italics">Analysis Æquationum Universalis;</hi> being a general method of approximating to the roots of equations in numbers. And in 1715 he published the <hi rend="italics">History of Fluxious,</hi> both in English and Latin.</p><p>In 1690 was also published, in 4 vols 4to, Dechale's <hi rend="italics">Cursus seu mundus mathematicus;</hi> in which is a piece of algebra, of a very old-fashioned sort, considering the time when it was written.</p><p>About 1692, and at different times afterwards, De Lagny published many pieces on the resolution of equations in numbers, with many theorems and rules for that purpose.</p><p>In 1693 was published, in a neat little volume, <hi rend="italics">Synopsis Algebraica, opus posthumum Johannis Alexandri.</hi></p><p>In 1694, Dr. Halley gave, in the Philos. Trans. an ingenious tract on the numeral extraction of all roots, without any previous reduction. And this tract is also added to some editions of Newton's Universal Arithmetic.</p><p>In 1695 Mr. John Ward, of Chester, published, in 8vo, <hi rend="italics">A Compendium of Algebra,</hi> containing plain, easy, and concise rules, with examples in an easy and clear way. And in 1706 he published the first edition of his <hi rend="italics">Young Mathematician's Guide,</hi> or a plain and easy introduction to the mathematics: a book which is still in great request, especially with beginners, and which has been ever since the ordinary introduction of the greatest part of the mathematicians of this country.</p><p>In 1696 the Marquis de l'Hòpital published his <hi rend="italics">Analyse des insiniment petits.</hi> And gave several papers to the Leipsic Acts and the Memoires of the Academy of Sciences. He left behind him also an ingenious treatise, which was published in 1707, intitled <hi rend="italics">Traité analytique des Sections Coniques, et de la construction des lieux geometriques.</hi></p><p>In 1697, and several other years, Mr. Ab. Demoivre gave various papers, in the Philos. Trans. containing improvements in Algebra: viz. in 1697, A method of raising an infinite multinomial to any power, or extracting any root of the same. In 1698, The extraction of the root of an infinite equation. In 1707, Analytical solution of certain equations of the 3d, 5th, 7th, &c degree. In 1722, Of algebraic fractions and recurring series. In 1738, The reduction of radicals into more simple forms. Also in 1730, he published <hi rend="italics">Miscellanea analytica de seriebus & quadraturis,</hi> containing great improvements in series, &c.</p><p>In 169, Mr. Richard Sault published, in 4to, <hi rend="italics">A New Treatise of Algebra, apply'd to numeral questions and geometry. With a converging series for all manner of Adfected Equations.</hi> The series here alluded to, is Mr. Raphson's method of approximation, which had been lately published. <cb/></p><p>In 1699 Hyac. Christopher published at Naples, in 4to, <hi rend="italics">De constructione æquationum.</hi></p><p>In 1702 was published Ozanam's Algebra; which is chiefly remarkable for the Diophantine analysis. He had published his mathematical dictionary in 1691, and in 1693 his course of mathematics, in 5 vols 8vo, containing also a piece of algebra.</p><p>In 1704, Dr. John Harris published his <hi rend="italics">Lexicon Technicum,</hi> the first dictionary of arts and sciences: a very plain and useful book, especially in the mathematical articles. And in 1705 a neat little piece on algebra and fluxions.</p><p>In 1705 M. Guisnée published, in 4to, his <hi rend="italics">Application de l'algebre a la geometrie:</hi> a useful book.</p><p>In 1706 Mr. William Jones published his <hi rend="italics">Synopsis Palmariorum Matheseos,</hi> or a new introduction to the mathematics: a very useful compendium in the mathematical sciences. And in 1711 he published, in 4to, a collection of Sir Isaac Newton's papers, intitled <hi rend="italics">Analysis per quantitatum series, fluxiones, ac differentias: cum enumeratione linearum tertii ordinis.</hi></p><p>In 1707 was published by Mr. Whiston, the first edition of Sir Isaac Newton's <hi rend="italics">Arithmetica Universalis: sive de compositione et resolutione arithmetica liber:</hi> and many editions have been published since. This work was the text book used by our great author in his lectures, while he was professor of mathematics in the university of Cambridge. And although it was never intended for publication, it contains many and great improvements in analytics; particularly in the nature and transmutation of equations; the limits of the roots of equations; the number of impossible roots; the invention of divisors, both surd and rational; the resolution of problems, arithmetical and geometrical; the linear construction of equations; approximating to the roots of all equations, &c. To the later editions of the book is commonly subjoined Dr. Halley's method of finding the roots of equations. As the principal parts of this work are not adapted to the circumstance os beginners, there have been published commentaries upon it by several persons, as s'Gravesande, Castilion, Wilder, &c.</p><p>In 1708 M. Reyneau published his <hi rend="italics">Analyse Demontrée,</hi> in 2 vols 4to. And in 1714 <hi rend="italics">La Science du Calcul, &c.</hi></p><p>In 1709 was published an English translation of Alexander's algebra. With an ingenious appendix by Humphry Ditton.</p><p>In 1715 Dr. Brooke Taylor published his <hi rend="italics">Methodus Incrementorum:</hi> an ingenious and learned work. And in the Philos. Trans. for 1718, An improvement of the method of approximating to the roots of equations in numbers.</p><p>In 1717 M. Nicole gave, in the memoirs of the academy of sciences, a tract on the calculation of finite differences. And in several following years, he gave various other tracts on the same subject, and on the resolution of equations of the 3d degree, and particularly on the irreducible case in cubic equations.</p><p>Also in 1717 was published a treatise on Algebra by Philip Ronayne.</p><p>Also in 1717 Mr. James Sterling published <hi rend="italics">Lineæ tertii Ordinis</hi>; an ingenious work, containing good improvements in analytics. Also in 1730 <hi rend="italics">Methodus Differentialis: sive tractatus de summatione et interpolatione serierum insinitarum:</hi> with great improvements on infinite series. <pb n="97"/><cb/></p><p>In 1726 and 1729 Maclaurin gave, in the Philos. Trans. tracts on the imaginary roots of equations. And afterwards was published, from his posthumous papers, his treatise on Algebra, with its application to curve lines.</p><p>In 1727 came out s'Gravesande's Algebra, with a specimen of a commentary on Newton's universal arithmetic.</p><p>In 1728 Mr. Campbell gave, in the Philos. Trans. an ingenious paper on the number of impossible roots of equations.</p><p>In 1732 was published Wolsius's Algebra, in his course of mathematics, in 5 vols. 4to.</p><p>In 1735 Mr. John Kirkby published his arithmetic and algebra. And in 1748 his doctrine of ultimators.</p><p>In 1740 were published Mr. Thomas Simpson's Essays; in 1743 his Dissertations, and in 1757 his Tracts; in all which are contained several improvements in series and other parts of Algebra. As also in his algebra, first printed in 1745, and in his Select Exercises, in 1752.</p><p>Also in 1740 was published professor Saunderson's Elements of Algebra, in 2 vols. 4to.</p><p>In 1741 M. de la Caille published <hi rend="italics">leçons de mathematiques; ou elemens d'algebre & de geometrie.</hi></p><p>Also in 1741, in the memoirs of the academy of sciences, were given two articles by M. de Gua, on the number of positive, negative, and imaginary roots of equations. With an historical account of the improvements in Algebra; in which he severely censures Wallis for his partiality; a circumstance in which he himself is not less faulty.</p><p>In 1746 M. Clairaut published his Elemens d'algebre, in which are contained several improvements, especially on the irreducible case in cubic equations. He has also several good papers on different parts of analytics, in the memoirs of the academy of sciences.</p><p>In 1747 M. Fontaine gave, in the memoirs of the academy of sciences, a paper on the resolution of equations. Besides some analytical papers in the memoirs of other years.</p><p>In 1761 M. Castillion published, in 2 vols 4to, Newton's universal arithmetic, with a large commentary.</p><p>In 1763 Mr. Emerson published his Increments. In 1764 his Algebra, &c.</p><p>In 1764 Mr. Landen published his Residual Analysis. In 1765 his Mathematical Lucubrations. And in 1780 his Mathematical Memoirs. All containing good improvements in infinite series, &c.</p><p>In 1770 was published, in the German language, Elements of Algebra by M. Euler. And in 1774 a French translation of the same. The memoirs of the Berlin and Petersburgh academies also abound with various improvements on series and other branches of analysis by this great man.</p><p>In 1775 was published at Bologna, in 2 vols 4to, <hi rend="italics">Compendio d'Analisi di</hi> Girolamo Saladini.</p><p>Besides the soregoing, there have been many other authors who have given treatises on Algebra, or who have made improvements on series and other parts of Algebra; as Schonerus, Coignet, Salignac, Laloubere, Hemischius, Degraave, Mescher, Henischins, Roberval, the Bernoullis, Malbranche, Agnesi, Wells, Dodson, Manfredi, Regnault, Rowning, Maseres, Waring, Lorg- <cb/> na, de la Grange, de la Place, Bertrand, Kuhnius' Hales, and many others.</p><div2 part="N" n="Algebra" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Algebra</hi></head><p>, <hi rend="italics">numeral,</hi> is that which is chiefly concerned in the solution of numeral problems, and in which all the given quantities are expressed by numbers only. As used by the more early authors, Diophantus, Paciolus, Stifelius, &c.</p></div2><div2 part="N" n="Algebra" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Algebra</hi></head><p>, <hi rend="italics">specious,</hi> or <hi rend="italics">literal,</hi> is that commonly used by the moderns, in which all the quantities, both known and unknown, are represented or expressed by species or general characters, as the letters of the alphabet, &c; in consequence of which general designation, all the conclusions become universal theorems for performing every operation of the like kind. There are specimens of this method from Cardan and others about his time, but it was more generally employed and introduced by Vieta.</p></div2><div2 part="N" n="Algebraical" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Algebraical</hi></head><p>, something relating to <hi rend="italics">algebra.</hi></p><p>Thus we say <hi rend="italics">algebraical</hi> solutions, curves, characters or symbols, &c.</p><p><hi rend="smallcaps">Algebraical</hi> <hi rend="italics">Curve,</hi> is a curve in which the general relation between the abscisses and ordinates can be expressed by a common algebraical equation.</p><p>These are also called <hi rend="italics">geometrical lines</hi> or <hi rend="italics">curves,</hi> in contradistinction to <hi rend="italics">mechanical</hi> or <hi rend="italics">transcendental</hi> ones.</p></div2></div1><div1 part="N" n="ALGEBRAIST" org="uniform" sample="complete" type="entry"><head>ALGEBRAIST</head><p>, a person skilled in <hi rend="italics">algebra.</hi></p></div1><div1 part="N" n="ALGENEB" org="uniform" sample="complete" type="entry"><head>ALGENEB</head><p>, or <hi rend="smallcaps">Algenib</hi>, a fixed star of the second magnitude, on the right side of Perseus.</p></div1><div1 part="N" n="ALGOL" org="uniform" sample="complete" type="entry"><head>ALGOL</head><p>, or <hi rend="italics">Medusa's Head,</hi> a fixed star of the third magnitude, in the constellation Perseus.</p></div1><div1 part="N" n="ALGORAB" org="uniform" sample="complete" type="entry"><head>ALGORAB</head><p>, a fixed star of the third magnitude, in the right wing of the constellation Corvus.</p></div1><div1 part="N" n="ALGORISM" org="uniform" sample="complete" type="entry"><head>ALGORISM</head><p>, or <hi rend="smallcaps">Algorithm</hi>, is similar to logistics, signifying the art of computing in any particular way, or about some particular subject; or the common rules of computing in any art. As the <hi rend="italics">algorithm</hi> of numbers, of algebra, of integers, of fractions, of surds, &c; meaning the common rules for performing the operations of arithmetic, or algebra, or fractions, &c.</p></div1><div1 part="N" n="ALHAZEN" org="uniform" sample="complete" type="entry"><head>ALHAZEN</head><p>, <hi rend="smallcaps">Allacen</hi>, or <hi rend="smallcaps">Abdilazum</hi>, was a learned Arabian, who lived in Spain about the year 1100, according to his editor Risner, and Weidler. He wrote upon Astrology; and his work upon Optics was printed, in Latin, at Basil, in 1572, under the title of <hi rend="italics">Opticæ Thesaurus,</hi> by Risner. Alhazen was the first who shewed the importance of refractions in astronomy, so little known to the ancients. He is also the first author who has treated on the twilight, upon which he wrote a work, in which he also speaks of the height of the clouds.</p></div1><div1 part="N" n="ALIDADE" org="uniform" sample="complete" type="entry"><head>ALIDADE</head><p>, an Arabic name for the label, index, or ruler, which is moveable about the centre of an astrolabe, quadrant, &c, and carrying the sights or telescope, and by which are shewn the degrees cut off the limb or arch of the instrument.</p><p>ALIQUANT <hi rend="italics">part,</hi> is that part which will not exactly measure or divide the whole, without leaving some remainder. Or the <hi rend="italics">aliquant</hi> part is such, as being taken or repeated any number of times, does not make up the whole exactly, but is either greater or less than it. Thus 4 is an aliquant part of 10; for 4 twice taken makes 8 which is less than 10, and three times taken makes 12 which is greater than 10. <pb n="98"/><cb/></p><p>ALIQUOT <hi rend="italics">part,</hi> is such a part of any whole, as will exactly measure it without any remainder. Or the <hi rend="italics">aliquot</hi> part is such, as being taken or repeated a certain number of times, exactly makes up, or is equal to the whole. So 1 is an aliquot part of 6, or of any other whole number; 2 is also an aliquot part of 6, being contained just 3 times in 6; and 3 is also an aliquot part of 6, being contained just 2 times: so that all the aliquot parts of 6 are 1, 2, 3.</p><p>All the <hi rend="italics">aliquot</hi> parts of any number may be thus found: Divide the given number by its least divisor; then divide the quotient also by its least divisor; and so on always dividing the last quotient by its least divisor, till the quotient 1 is obtained; and all the divisors, thus taken, are the prime aliquot parts of the given number. Then multiply continually together these prime divisors, viz. every two of them, every three of them, every four of them, &c; and the products will be the other or compound <hi rend="italics">aliquot</hi> parts of the given number. So if the <hi rend="italics">aliquot</hi> parts of 60 be required; first divide it by 2, and the quotient is 30: then 30 divided by 2 also, gives 15, and 15 divided by 3 gives 5, and 5 divided by 5 gives 1: so that all the prime divisors or aliquot parts are 1, 2, 2, 3, 5. Then the compound ones, by multiplying every two, are 4, 6, 10, 15; and every three 10, 20, 30. So that all the <hi rend="italics">aliquot</hi> parts of the given number 60, are 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, 30.—In like manner it will be found that all the <hi rend="italics">aliquot</hi> parts of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 180.</p><p>ALLEN (<hi rend="smallcaps">Thomas</hi>) a celebrated mathematician of the 16th century. He was born at Uttoxeter in Staffordshire, in 1542; was admitted a scholar of Trinity college, Oxford, in 1561; where he took his degree of master of arts in 1567. In 1570 he quitted his college and fellowship, and retired to Glocester hall, where he studied very closely, and became famous for his knowledge in antiquity, philosophy and mathematics. He received an invitation from Henry earl of Northumberland, a great friend and patron of the mathematicians, and he spent some time at the earl's house; where he became acquainted with those celebrated mathematicians Thomas Harriot, John Dee, Walter Warner, and Nathaniel Torporley. Robert earl of Leicester, too, had a great esteem for Allen, and would have conferred a bishopric upon him; but his love for solitude and retirement made him decline the offer. His great skill in the mathematics gave occasion to the ignorant and vulgar to look upon him as a magician or conjurer. Allen was very curious and indefatigable in collecting scattered manuscripts relating to history, antiquity, astronomy, philosophy, and mathematics: which collections have been quoted by several learned authors, and mentioned as in the Bibliotheca Alleniana. He published in Latin the second and third books of Ptolemy, <hi rend="italics">Concerning the Judgment of the Stars,</hi> or, as it is usually called, of the <hi rend="italics">quadripartite construction,</hi> with an exposition. He wrote also notes on many of Lilly's books, and some on John Bale's work, <hi rend="italics">De scriptoribus Maj. Britanniæ.</hi> He died at Glocester hall in 1632, being 90 years of age.</p><p>Mr. Burton, the author of his funeral oration, calls him the very soul and sun of all the mathematicians of his age. And Selden mentions him as a person of the most extensive learning and consummate judgment, the bright- <cb/> est ornament of the university of Oxford. Also Camden says he was skilled in most of the best arts and sciences. A. Wood has also transcribed part of his character from a manuscript in the library of Trinity college, in these words: “He studied polite literature with great application; he was strictly tenacious of academic discipline, always highly esteemed both by foreigners and those of the university, and by all of the highest stations in the church of England, and the university of Oxford. He was a sagacious observer, an agreeable companion, &c.”</p></div1><div1 part="N" n="ALLIGATION" org="uniform" sample="complete" type="entry"><head>ALLIGATION</head><p>, one of the rules in arithmetic, by which are resolved questions which relate to the compounding or mixing together of divers simples or ingredients, being so called from <hi rend="italics">alligare,</hi> to tie or connect together, probably from certain vincula, or crooked ligatures, commonly used to connect or bind the numbers together.</p><p>It is probable that this rule came to us from the Moorish or Arabic writers, as we find it, with all the other rules of arithmetic, in <hi rend="italics">Lucas de Burgo,</hi> and the other early authors in Europe.</p><p><hi rend="italics">Alligation</hi> is of two kinds, <hi rend="italics">medial</hi> and <hi rend="italics">alternate.</hi></p><p><hi rend="smallcaps">Alligation</hi> <hi rend="italics">medial</hi> is the method of finding the rate or quality of the composition, from having given the rates and quantities of the simples or ingredients.</p><p>The rule of operation is this: multiply each quantity by its rate, and add all the products together; then divide the sum of the products by the sum of the quantities, or whole compound, and the quotient will be the rate sought.</p><p><hi rend="italics">For example,</hi> Suppose it were required to mix together 6 gallons of wine, worth 5s. a gallon; 8 gallons, worth 6s. the gallon; and 4 gallons, worth 8s. the gallon; and to find the worth or value, per gallon, of the whole mixture. <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Gal.</cell><cell cols="1" rows="1" rend="align=center" role="data">s.</cell><cell cols="1" rows="1" rend="align=center" role="data">products.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Here</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">mult. by 5 gives</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">" by 6 "</cell><cell cols="1" rows="1" rend="align=center" role="data">48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">" by 8 "</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">whole comp.</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">110</cell><cell cols="1" rows="1" role="data">sum of prod.</cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data">Then 18)</cell><cell cols="1" rows="1" role="data">110 (6 2/18 or (6 1/9)s, is the rate sought.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">108</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">   2</cell></row></table></p><p><hi rend="smallcaps">Alligation</hi> <hi rend="italics">alternate</hi> is the method of finding the quantities of ingredients or simples, necessary to form a compound of a given rate.</p><p>The <hi rend="italics">rule</hi> of operation is this: 1st, Place the given rates of the simples in a column, under each other; noting which rates are less, and which are greater than the proposed compound. 2d, Connect or link with a crooked line, each rate which is less than the proposed compound rate, with one or any number of those which are greater than the same; and every greater rate with one or any number of the less ones. 3d, Take the difference between the given compound rate and that of each simple rate, and set this difference opposite every rate with which that one is linked. 4th, Then if only one difference stand opposite any rate, it will be the quantity belonging to that rate; but when there are several differences to any one, take their sum for its quantity.</p><p><hi rend="italics">For example,</hi> Suppose it were required to mix together gold of various degrees of fineness, viz. of 19, <pb n="99"/><cb/> of 21, and of 23 caracts fine, so that the mixture shall be of 20 caracts fine. Hence, <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Rates</cell><cell cols="1" rows="1" rend="align=center" role="data">Diffs.</cell><cell cols="1" rows="1" rend="align=center" role="data">Sum of Diffs.</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=3" role="data">Comp. rate 20</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(7)">{</hi></cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(7)">{</hi></cell><cell cols="1" rows="1" role="data">1 of 21 caracts sine,</cell></row><row role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1+3</cell><cell cols="1" rows="1" role="data">4 of 19 caracts sine,</cell></row><row role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1 of 23 caracts sine.</cell></row></table> That is, there must be an equal quantity of 21 and 23 caracts fine, and 4 times as much of 19 caracts fine.</p><p>Various limitations, both of the compound and the ingredients, may be conceived; and in such cases, the differences are to be altered proportionally.</p><p>Questions of this sort are however commonly best and easiest resolved by common Algebra, of which they form a species of indeterminate problems, as they admit of many, or an indesinite number of answers.</p><p>There is recorded a remarkable instance of a discovery made by Archimedes, both by <hi rend="italics">alligation</hi> and specific gravity at the same time, namely, concerning the crown of Hiero, king of Syracuse. The king had ordered a crown to be made of pure gold, but when brought to him, a suspicion arose that it was mixed with alloy of either silver or copper, and the king recommended it to Archimedes to discover the cheat without defacing the crown. Archimedes, after long thinking on the matter, without lighting on the means of doing it, being one day in the bath, and observing how his body raised the water higher, conceived the idea that different metals of the same weight would occupy different spaces, and so raise or expel different quantities of water. Upon which he procured two other masses, each of the same weight with the crown, the one of pure gold, and the other of alloy; then immersing them all three, separately, in water, and observing the space each occupied, by the quantity it raised the water, he from thence computed the quantities of gold and alloy contained in the crown.</p></div1><div1 part="N" n="ALLIOTH" org="uniform" sample="complete" type="entry"><head>ALLIOTH</head><p>, a star in the tail of the great bear. The word in Arabic denotes a horse; and they gave this name to each of the three stars, in the tail of the great bear, as they are placed like three horses, thus arranged for the purpose of drawing the waggon commonly called Charles's wain, represented by the four stars on the body of the same constellation.</p><p>ALMACANTAR. See <hi rend="smallcaps">Almucantar.</hi></p></div1><div1 part="N" n="ALMAGEST" org="uniform" sample="complete" type="entry"><head>ALMAGEST</head><p>, the name of a celebrated book composed by Ptolemy; being a collection of a great number of the observations and problems of the ancients, relating to geometry and astronomy, but especially the latter. And being the sirst work of this kind which has come down to us, and containing a catalogue of the fixed stars, with their places, beside numerous records and observations of eclipses, the motions of the planets, &c, this work will ever be held dear and valuable to the cultivators of astronomy.</p><p>In the original Greek it is called <foreign xml:lang="greek">suntacis m<*>gish</foreign>, the <hi rend="italics">great composition</hi> or <hi rend="italics">collection.</hi> And to the word <foreign xml:lang="greek">megish</foreign>, <hi rend="italics">megiste,</hi> the Arabians joined the particle <hi rend="italics">al,</hi> and thence called it <hi rend="italics">Almaghesti,</hi> or, as we call it, from them, the Almagest.</p><p>Ptolemy was born about the year of Christ 69, and died in 147, and wrote this work, consisting of 13 books, at Alexandria in Egypt, where the Arabians found it on the capture of that kingdom. It was by them <cb/> translated out of Greek, into Arabic, by order of the caliph Almaimon, about the year 827; and sirst into Latin about 1230, by favour of the emperor Frederic II. The Greek text however was not known in Europe till about the beginning of the 15th century, when it was brought from Constantinople, then taken by the Turks, by <hi rend="italics">George,</hi> a monk of Trabezond, who translated it into Latin, which translation has several times been published.</p><p>Riccioli, an Italian jesuit, also published, in 1651, a body of Astronomy, which, in imitation of Ptolemy, he called <hi rend="italics">Almagestum Novum,</hi> the <hi rend="italics">New Almagest</hi>; being a large collection of ancient and modern observations and discoveries, in the science of Astronomy.</p></div1><div1 part="N" n="ALMAMON" org="uniform" sample="complete" type="entry"><head>ALMAMON</head><p>, caliph of Bagdat, a philosopher and astronomer in the beginning of the 9th century, he having ascended the throne in the year 814. He was son of Harun Al-Rashid, and grand son of Almansor. His name is otherwise written <hi rend="italics">Mamon, Almaon, Almamun, Alamoun,</hi> or <hi rend="italics">Al-Maimon.</hi> Having been educated with great care and with a love for the liberal sciences, he applied himself to cultivate and encourage them in his own country. For this purpose he requested the Greek emperors to supply him with such books on philosophy as they had among them; and he collected skilful interpreters to translate them into the Arabic language. He also encouraged his subjects to study them; frequenting the meetings of the learned, and assisting at their exercises and deliberations. He caused Ptolemy's Almagest to be translated in 827, by Isaac Ben-honain, and Thabet Ben-korah, according to Herbelot, but according to others by Sergius, and Alhazen, the son of Joseph. In his reign, and doubtless by his encouragement, an astronomer of Bagdat, named Habash, composed three sets of astronomical tables.</p><p>Almamon himself made many astronomical observations, and determined the obliquity of the ecliptic to be then 23° 35′ (or 23° 33′ in some manuscripts), but Vossius says 23° 51′ or 23° 34′. He also caused skilful observers to procure proper instruments to be made, and to exercise themselves in astronomical observations; which they did accordingly at Shemasi in the province os Bagdat, and upon Mount Casius near Damas.</p><p>Under the auspices of Mamon also a degree of the meridian was measured on the plains of Sinjar or Sindgiar (or according to some Fingar), upon the borders of the Red Sea; by which the degree was found to contain 56 2/3 miles, of 4000 coudees each, the coudee being a foot and a half: but it is not known what foot is here meant, whether the Roman, the Alexandrian, or some other. Riccioli makes this measure of the degree amount to 81 ancient Roman miles, which value answers to 62046 French toises; a quantity more than the true value of the degree by almost one-third.</p><p>Finally, Mamon revived the sciences in the East to such a degree, that many learned men were found, not only in his own time, but after him, in a country where the study of the sciences had been long forgotten. This learned king died near Tarsus in Cilicia, by having eaten too freely of some dates, on his return from a military expedition, in the year 833.</p></div1><div1 part="N" n="ALMANAC" org="uniform" sample="complete" type="entry"><head>ALMANAC</head><p>, a calendar or table, in which are set down and marked the days and feasts of the year, the common ecclesiastical notes, the course and phases of <pb n="100"/><cb/> the moon, &c, sor each month: and answers to the <hi rend="italics">fasti</hi> of the ancient Romans.</p><p>The etymology of the word is much controverted among grammarians.—Some derive it from the Arabic, viz, from the particle <hi rend="italics">al,</hi> and <hi rend="italics">manah,</hi> to count. While Scaliger, and others, derive it from the same <hi rend="italics">al,</hi> and the Greek <foreign xml:lang="greek">manakos</foreign>, the course of the months. But Golius controverts these opinions, and ascribes the word to another origin, though he still makes it of Arabic extract, which it more evidently is. He says that, in the East it is the custom for the people, at the beginning of the year, to make presents to their princes; and that, among the rest, the astrologers present them with their almanacs, or ephemerides, for the year ensuing; whence these came to be called <hi rend="italics">almanha,</hi> that is, new-year's gifts. But this derivation seems rather strained and improbable; for, by the same rule, the gifts or productions of other artists, or classes of men, might also be called almanacs. There are other guesses at the etymology; and Verstegan writes the word <hi rend="italics">almonac,</hi> and makes it of German original. Our ancestors, he observes, used to carve the courses of the moon, for the whole year, upon a square piece of wood, which they called <hi rend="italics">al-monaght,</hi> which is as much as to say, in old English or Saxon, <hi rend="italics">all-moon-heed.</hi></p><p>Almanacs are of various kinds and composition, some books, others sheets, &c, some annual, others perpetual. The essential part is the calendar of months, weeks, and days; the motions, changes, and phases of the moon; with the rising and setting of the sun and moon. To these are commonly added various matters, astronomical, astrological, chronological, meteorological, and even political, rural, medical, &c; as also eclipses, solar ingresses, aspects, and configurations of the heavenly bodies, lunations, heliocentric and geocentric motions of the planets, prognostications of the weather, and predictions of other events, the tides, twilight, equation of time, &c.</p><p>Till about the 4th century, almanacs bore the marks of heatheniim only; from thence to the 7th century, they were a mixture of heathenism and christianity; and ever since they have been altogether christian: but at all times, astrological and other predictions have been considered as an essential part, and still are so to this day with several of them, notwithstanding that most people <hi rend="italics">assect</hi> to disbelieve in such predictions.</p><p><hi rend="italics">Nautical</hi> <hi rend="smallcaps">Almanac</hi>, and <hi rend="italics">Astronomical Ephemeris,</hi> is a kind of national almanac, chiefly for nautical purposes, which was begun in the year 1767 under the direction of the Board of Longitude, on the recommendation of the present worthy Astronomer Royal, who has the immediate conducting of it. It is still published by anticipation for several years before hand, for the convenience of ships going out upon long voyages, for which it is highly useful, and was found eminently so in the course of the late voyages round the world for making discoveries. Besides most things essential to general use, that are to be found in other almanacs, it contains many new and important particulars; more especially, the distances of the moon from the sun and fixed stars, which are computed for the meridian of the Royal Observatory of Greenwich, and set down to every three hours of time, expressly designed for computing the longitude at sea, by comparing these with the like distances observed there. <cb/></p></div1><div1 part="N" n="ALMANAR" org="uniform" sample="complete" type="entry"><head>ALMANAR</head><p>, in the Arabian astrology, denotes the pre-eminence or prevalence of one planet over another.</p></div1><div1 part="N" n="ALMUCANTARS" org="uniform" sample="complete" type="entry"><head>ALMUCANTARS</head><p>, <hi rend="smallcaps">Almacantars</hi>, or A<hi rend="smallcaps">LMICANTARS</hi>, from the Arabic <hi rend="italics">almocantharat,</hi> are circles parallel to the horizon, conceived to pass through every degree of the meridian; serving to shew the height of the sun, moon, or stars, &c; and are the same as the parallels of altitude.</p><p><hi rend="smallcaps">Almucantar</hi>-<hi rend="italics">Staff,</hi> was an instrument formerly used at sea to observe the sun's amplitude at rising or setting, and thence to determine the variation of the compass, &c. The instrument had an arch of 15 degrees, made of some smooth wood.</p><p>ALPHONSINE <hi rend="italics">Tables,</hi> are astronomical tables compiled by order of Alphonsus, king of Castile. In the compiling of these it is thought that prince himself assisted. See <hi rend="italics">Astronomical tables.</hi></p><p>ALPHONSUS the 10th, king of Leon and Castile, who has been surnamed The Wise, on account of his attachment to literature, and is now more celebrated for having been an astronomer than a king. He was born in 1203; succeeded his father Ferdinand the 3d, in 1252; and died in 1284, consequently at the age of 81.</p><p>The affairs of the reign of Alphonsus were very extraordinary and unfortunate for him. But we shall here only consider him in that part of his character, on account of which he has a place in this work, namely, as an astronomer and man of letters. He understood astronomy, philosophy, and history, as if he had been only a man of letters; and composed books upon the motions of the heavens, and on the history of Spain, which are highly commended. “What can be more surprising,” says Mariana, “than that a prince, educated in a camp, and handling- arms from his childhood, should have such a knowledge of the stars, of philosophy, and the transactions of the world, as men of leisure can scarcely acquire in their retirements? There are extant some books of Alphonsus on the motions of the stars, and the history of Spain, written with great skill and incredible care.” In his astronomical pursuits he discovered that the tables of Ptolemy were full of errors; and thence he conceived the first of any the resolution of correcting them. For this purpose, about the year 1240, and during the life of his father, he assembled at Toledo the most skilful astronomers of his time, Christians, Moors, or Jews, when a plan was formed for constructing new tables. This task was accomplished about 1252, the first year of his reign; the tables being drawn up chiefly by the skill and pains of Rabbi Isaac Hazan a learned Jew, and the work called the <hi rend="italics">Alphonsine Tables,</hi> in honour of the prince, who was at vast expences concerning them. He fixed the epoch of the tables to the 30th of May 1252, being the day of his accession to the throne. They were printed for the first time in 1483, at Venice, by Radtolt, who excelled in printing at that time; an edition extremely rare: there are others of 1492, 1521, 1545, &c. (Weidler, p. 280).</p><p>We must not omit a memorable saying of Alphonfus, which has been recorded for its boldness and pretended impiety; namely, “that if he had been of God's privy council when he made the world, he could have advised him better.” Mariana however says only in general, that Alphonsus was so bold as to blame the works of <pb n="101"/><cb/> Providence, and the construction of our bodies; and he says that this story concerning him rested only upon a vulgar tradition. The Jesuit's words are curious: “Emanuel, the uncle of Sanchez (the son of Alphonsus), in his own name, and in the name of other nobles, deprived Alphonsus of his kingdom by a public sentence: which that prince merited, for daring severely and boldly to censure the works of divine Providence, and the construction of the human body, as tradition says he did. Heaven most justly punished the folly of his tongue.” Though the silence of such an historian as Mariana, in regard to Ptolemy's system, ought to be of some weight, yet we cannot think it improbable, that if Alphonsus did pass so bold a censure on any part of the univerfe, it was on the celestial sphere, and meant to glance upon the contrivers and supporters of that system. For, besides that he studied nothing more, it is certain that at that time astronomers explained the motions of the heavens by intricate and confused hypotheses, which did no honour to God, nor anywise answered the idea of an able workman. So that, from considering the multitude of spheres composing the system of Ptolemy, and those numerous eccentric cycles and epicycles with which it is embarrassed, if we suppose Alphonsus to have said, “That if God had asked his advice when he made the world, he would have given him better council,” the boldness and impiety of the censure will be greatly diminished.</p></div1><div1 part="N" n="ALSTED" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ALSTED</surname> (<foreName full="yes"><hi rend="smallcaps">John-Henry</hi></foreName>)</persName></head><p>, a German protestant divine, and one of the most indefatigable writers of the 17th century. He was some time professor of philosophy and divinity at Herborn in the county of Nassau: from thence he went into Transilvania, to be professor at Alba Julia; where he continued till his death, which happened in 1638, being then 50 years of age. He applied himself chiefly to compose methods, and to reduce the several branches of arts and sciences into systems. His <hi rend="italics">Encyclopædia</hi> has been much esteemed even by Roman Catholics; it was printed at Lyons, and sold very well throughout all France. Vossius mentions the Encyclopædia in general, but speaks of his treatise of <hi rend="italics">Arithmetic</hi> more particularly, and allows the author to have been a man of great reading and universal erudition. His <hi rend="italics">Thesaurus Chronologicus</hi> is by some esteemed one of his best works, and has gone through several editions, though others speak of it with contempt. In his <hi rend="italics">Triumphus Biblicus</hi> Alsted endeavours to prove that the materials and principles of all the arts and sciences may be found in the scriptures; but he gained very few to his opinion. John Himmelius wrote a piece against his <hi rend="italics">Theologia Polemica,</hi> which was one of Alsted's best performances. It seems he was a millenarian, having published, in 1672, a treatise <hi rend="italics">De Mille Annis,</hi> in which he asserts that the faithful shall reign with Jesus Christ upon earth a thousand years; after which will be the general resurrection, and the last judgment; and he pretended that this reign would commence in the year 1694.</p><p>ALTERNATE <hi rend="italics">angles,</hi> are the internal angles, A and B, or <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> <figure/> made by a line cutting two parallel lines, and lying on opposite sides of the cutting line. It is the property of these angles to be always equal to each other, namely the angle A = the <cb/> angle B, and the angle <hi rend="italics">a</hi> = the angle <hi rend="italics">b.</hi> And the exterior alternate angles are also equal.</p><p><hi rend="smallcaps">Alternate</hi> <hi rend="italics">Ratio</hi> or <hi rend="italics">Proportion,</hi> is the ratio of the one antecedent to the other, or of one consequent to the other, in any proportion, in which the quantities are of the same kind. So if A : B :: C : D, then alternately, or by alternation A : C :: B : D.</p></div1><div1 part="N" n="ALTERNATION" org="uniform" sample="complete" type="entry"><head>ALTERNATION</head><p>, or <hi rend="italics">Permutation,</hi> of quantities or things, is the varying or changing the order or position of them.</p><p>As suppose two things <hi rend="italics">a</hi> and <hi rend="italics">b</hi>; these may be placed either thus <hi rend="italics">ab</hi> or <hi rend="italics">ba</hi> that is two ways, or 1 X 2. If there be three things, <hi rend="italics">a, b, c,</hi> then the 3d thing <hi rend="italics">c,</hi> may be placed three different ways with respect to each of the two positions <hi rend="italics">ab</hi> and <hi rend="italics">ba</hi> of the other two things, it may stand either before them, or between them, or after them both, that is, it may stand either 1st, 2d, or 3d; and therefore with three things there will be three times as many changes as with two, that is 1X2X3 or six changes with three things. Again, if there be four things <hi rend="italics">a, b, c, d</hi>; then the fourth thing <hi rend="italics">d</hi> may be placed in four different ways with respect to each of the six positions of the other three; for it may be set either 1st or 2d or 3d or 4th in the order of each position; consequently from four things there will be four times as many alternations as there are from three things; and therefore 1 X 2 X 3 X 4 = 24 is the number of changes with four things. And so on, always multiplying the last found number of alternations by the next number of things; or to find the number of changes for any number of things, as <hi rend="italics">n,</hi> multiply the series of natural numbers 1, 2, 3, 4, 5, &c, to <hi rend="italics">n,</hi> continually together, and the last product will be the number of alternations sought; so 1X2X3X4X5 - - - - <hi rend="italics">n</hi> is the number of changes in <hi rend="italics">n</hi> things.</p><p>So if, for example, it were required to find how many changes may be rung on 12 bells; it would be 1 X 2 X 3 X 4 X 5 X 6X7X8X9X10X11X12= 479001600, the number of changes. Now supposing there might be rung 10 changes in one minute, that is 10X12 or 120 strokes in a minute, or 2 strokes in each second of time; then, according to this rate, it would take upwards of 91 years to ring over all these changes on the 12 bells only. Also, if but two more bells were added, making 14 bells; then, at the same rate of ringing, it would require about 16575 years to ring all the changes on 14 bells but once over. And if the number of bells were 24, it would require more than 117000000000000000, years to ring all the different changes upon them!</p></div1><div1 part="N" n="ALTIMETRY" org="uniform" sample="complete" type="entry"><head>ALTIMETRY</head><p>, <hi rend="smallcaps">Altimetria</hi>, the art of taking or measuring altitudes or heights, whether accessible or inaccessible. Or</p><p>Altimetria is the part of practical Geometry which respects the theory and practice of measuring both heights and depths, and both in respect of perpendicular and oblique lines.</p></div1><div1 part="N" n="ALTING" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ALTING</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, was born at Heidelberg in 1618. He travelled into England in 1640, where he was ordained by the learned Dr. Prideaux, bishop of Worcester. He afterwards succeeded Gomarus in the professorship of Groninghen. He died in 697; and recommended the edition of his works to Menso Alting (author of Notitia German. Infer. Antiquæ); but they were published in 5 vols folio, with his life, by <pb n="102"/><cb/> Bekker of Amsterdam. They contain various analytical, exegetical, practical, problematical, and philosophical tracts, which shew his great industry and knowledge.</p></div1><div1 part="N" n="ALTITUDE" org="uniform" sample="complete" type="entry"><head>ALTITUDE</head><p>, in Geometry is the third dimension of body, considered with respect to its elevation above the ground: and is otherwise called its height when measured from bottom to top, or its depth when measured from top to bottom.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of a figure,</hi> is the distance of its vertex from the base, or the length of a perpendicular let fall from its vertex to the base. The altitudes of sigures are useful in computing their areas or solidities.</p><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">Height</hi> of any point of a terrestrial object, is the perpendicular let fall from that point to the plane of the horizon. <hi rend="italics">Altitudes</hi> are distinguished into <hi rend="italics">accessible</hi> and <hi rend="italics">inaccessible.</hi></p><p><hi rend="italics">Accessible</hi> <hi rend="smallcaps">Altitude</hi> of an object, is that whose base there is access to, to measure the nearest distance to it on the ground, from any place.</p><p><hi rend="italics">Inaccessible</hi> <hi rend="smallcaps">Altitude</hi>, of an object, is that whose base there is not free access to, by which a distance may be measured to it, by reason of some impediment, such as water, wood, or the like.</p><p><hi rend="italics">To measure</hi> or <hi rend="italics">take</hi> <hi rend="smallcaps">Altitudes.</hi> If an altitude cannot be measured by stretching a string from top to bottom, which is the direct and most accurate way, then some indirect way is used, by actually measuring some other line or distance which may serve as a basis, in conjunction with some angles, or other proportional lines, either to compute, or geometrically determine, the altitude of the object sought.</p><p>There are various ways of measuring altitudes, or depths, by means of different instruments, and by shadows or reflected images, on optical principles. There are also various ways of computing the altitude in numbers, from the measurements taken as above, either by geometrical construction, or trigonometrical calculation, or by simple numeral computation from the property of parallel lines, &c.</p><p>The instrumcnts mostly used in measuring altitudes, are the quadrant, theodolite, geometrical square, line of shadows, &c; the descriptions of each of which may be seen under their respective names.</p><p><hi rend="italics">To measure an Accessible Altitude Geometrically.</hi> Thus, suppose the height of the accessible tower AB be required. First, by means of two rode, the one longer than the other: plant the longer upright at C; then move the shorter back from it, till by trials you find such a place, D, that the eye placed at the top of it at E, may see the top of the other, F, and the top of the object B straight in a line: next measure the distances DA or EG and DC or EH, also HF the difference between the heights of the rods: then, by similar triangles, as EH : EG :: HF : the 4th proportional GB; to which add AG or DE, and the sum will be the whole altitude AB sought. <figure/> <cb/></p><p>Or, with one rod CF only: plant it at such a place C, that the eye at the ground, or near it, at I, may see the tops F and B in a right line: then, having measured IC, IA, CF, the 4th proportional to these will be the altitude AB sought.</p><p>Or thus, by means of Shadows. Plant a rod <hi rend="italics">ab</hi> at <hi rend="italics">a,</hi> and measure its shadow <hi rend="italics">ac,</hi> as also the shadow AC of the object AB; then the 4th proportional to <hi rend="italics">ac, ab,</hi> AC will be the altitude AB sought. <figure/></p><p>Or thus, by means of Optical Reflection. Place a vessel of water, or a mirror or other reflecting smface, horizontal at C; and move off from it to such a distance, D, that the eye E may see the image of the top of the object in the mirror at C: then, by similar figures, CD : DE :: CA : AB the altitude sought. <figure/></p><p>Or thus, by the Geometrical Square. At any place, C, fix the stand, and turn the square about the centre of motion, D, till the eye there see the top of the object through the sights or telescope on the side DE of the quadrant, and note the number of divisions cut off the other side by the plumb line EG: then as EF : FG :: DH : HB; to which add AH or CD, for the whole height AB. <figure/></p><p><hi rend="italics">To measure an Accessible Altitude Trigonometrically.</hi> At any convenient station, C, with a quadrant, theodolite, or other graduated instrument, observe the angle of elevation ACB above the horizontal line AC; and measure the distance AC. Then, A being a right angle, it will be, as radius is to the tangent of the angle A, so is AC to AB sought.</p><p>If AC be not horizontal, but an inclined plane; then the angle above it must be observed at two stations C and D in a right line, and the distances AC, CD both measured. Then, from the angle C take the angle D, and there remains the angle CBD; hence in the triangle BCD, are given the angles and the side DC, to sind the side CB; and then in the triangle ABC, are given the <pb n="103"/><cb/> two sides CA and CB, with the included angle C, to find the third side AB. <figure/></p><p>Or thus, measure only the distance AC, and the angles A and C: then, in the triangle ABC, are given all the angles and the side AC, to find the side AB.</p><p><hi rend="italics">To measure an Inaccessible Altitude,</hi> as a hill, cloud, or other object. This is commonly done, by observing the angle of its altitude at two stations, and measuring the distance between them. Thus, for the height AB of a hill, measure the distance CD at the foot of it, and observe the quantity of the two angles C and D. Then, from the angle C taking the angle D, leaves the augle CBD; hence As sine [angle]CBD: sine [angle]D :: CD : CB; and As rad.: sine [angle]ACB :: CB : AB the altitude. <figure/></p><p>And for a balloon, or cloud, or other moveable object C, let two observers at A and B, in a plane with C, take at the same time the angles A and B, and measure the distance between them AB; then calculate the altitude CD exactly as in the last example. <figure/></p><p>To find the height of an object, by knowing the utmost distance at which its top can be just seen in the horizon. As suppose the top H of a tower FH can be just seen from E when the distance EF is 25 miles, supposing the circumference of the earth to be 25000 miles, or the radius 3979 miles or 21009120 feet. First, as 25000 : 25 :: 360° : 21′ 36″ equal to the angle G; then as radius : sec. [angle]G :: EG : GH, which will be found to be 21009536 feet; from which take EG or GF, and there remains 416 feet, for FH the height of the tower sought.— Or rather thus, as 10000000 radius: 198=sec. [angle]G—radius :: 21009120=EG : 416 = FH, as before.</p><p>Or the same may be found easier thus: The horizon dips nearly 8 inches or 2/3 of a foot at the distance of 1 mile, and according to the square of the distance for <cb/> other distances; therefore as 1<hi rend="sup">2</hi> or 1 : 25<hi rend="sup">2</hi> or 625 :: 2/3 : 2/3 of 625 or 416 feet, the same as before.</p><p>There is a very easy method of taking great terrestrial altitudes, such as mountains &c, by means of the difference between the heights of the barometer observed at the bottom and top of the same. Which see under the article <hi rend="smallcaps">Barometer.</hi></p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the Eye,</hi> in <hi rend="italics">Perspective,</hi> is a right line let fall from the eye, perpendicular to the geometrical plane.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi>, in <hi rend="italics">Astronomy</hi></head><p>, is the arch of a vertical circle, measuring the height of the sun, moon, star, or other celestial object, above the horizon.</p><p>This altitude may be either <hi rend="italics">true</hi> or <hi rend="italics">apparent.</hi> The apparent altitude is that which appears by sensible observations made at any place on the surface of the earth. And the true altitude is that which results by correcting the apparent, on account of refraction and parallax.</p><p>The quantity of the refraction is different at different altitudes; and the quantity of the parallax is different according to the distance of the different luminaries: in the fixed stars this is too small to be observed; in the sun it is but about 8 3/4 seconds; but in the moon it is about 52 minutes.</p><p>Altitudes are observed by a quadrant, or sextant, or by the shadow of a gnomon or high pole, and by various other ways, as may be seen in most books of astronomy.</p><p><hi rend="italics">Meridian</hi> <hi rend="smallcaps">Altitude</hi>, is an arch of the meridian intercepted between any point in it and the horizon. So if HO be the horizon, and HEZO the meridian; then the arch HE, or the angle HCE, is the meridian altitude of an object in the meridian at the point E. <figure/></p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">elevation, of the Pole,</hi> is the angle OCP, or arch OP of the meridian, intercepted between the horizon and pole P.</p><p>This is equal to the latitude of the place; and it may be found by observing the meridian altitude of the pole star, when it is both above and below the pole, and taking half the sum, when corrected on account of refraction. Or the same may be found by the declination and meridian altitude of the sun.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">elevation,</hi> of the <hi rend="italics">equator,</hi> is the angle HCE, or arch HE of the meridian, between the horizon and the equator at E; and it is equal to ZP the colatitude of the place.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the Tropics,</hi> the same as what is otherwise called the <hi rend="italics">solstitial altitude</hi> of the sun, or his meridian altitude when in the solstitial points.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, <hi rend="italics">or height, of the horizon,</hi> or of stars &c seen in it, is the quantity by which it is raised by refraction.</p><p><hi rend="italics">Refraction of</hi> <hi rend="smallcaps">Altitude</hi>, is an arch of a vertical circle, by which the true altitude of the moon, or a star, or other object, is increased by means of the refraction; <pb n="104"/><cb/> and is different at different altitudes, being nothing in the zenith, and greatest at the horizon, where it is about 33′.</p><p><hi rend="italics">Parallax of</hi> <hi rend="smallcaps">Altitude</hi>, is an arch of a vertical circle, by which the true altitude, observed at the centre of the earth, exceeds that which is observed on the surface; or the difference between the angles <figure/> LM and <figure/> IK of altitude there; and is equal to the angle I <figure/> L formed at the moon or other body, and subtended by the radius IL of the earth.</p><p>It is evident that this angle is less, as the luminary is farther distant from the earth; and also less, for any one luminary, as it is higher above the horizon; being greatest there, and nothing in the zenith.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the Nonagesimal,</hi> is the altitude of the 90th degree of the ecliptic, counted upon it from where it cuts the horizon, or of the middle or highest point of it which is above the horizon, at any time; and is equal to the angle made by the ecliptic and horizon where they intersect at that time.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the cone of the earth's or moon's shadow,</hi> the height of the shadow of the body, made by the sun, and measured from the centre of the body. To find it, say, As the tangent of the angle of the sun's apparent semidiameter is to radius, so is 1 to a 4th proportional, which will be the height of the shadow, in semidiameters of the body.</p><p>So, the greatest height of the earth's shadow, is 217.8 semidiameters of the earth, when the sun is at his greatest distance, or his semidiameter subtends an angle of about 15′ 47″; and the height of the same is 210.7 semidiameters of the earth, when the sun is nearest the earth, or when his semidiameter is about 16′ 19″: And proportionally between these limits for the intermediate distances or semidiameters of the sun.</p><p>The altitudes of the shadows of the earth and moon, are nearly as 11 to 3, the proportion of their diameters.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, or <hi rend="italics">exaltation,</hi> in astrology, denotes the second of the five essential dignities, which the planets acquire by virtue of the signs they are found in.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of motion,</hi> is a term used by Dr. Wallis, for the measure of any motion, estimated in the line of direction of the moving force.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, in speaking of fluids, is more frequently expressed by the term <hi rend="italics">depth.</hi> The pressure of fluids, in every direction, is in proportion to their altitude or depth.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the mercury,</hi> in the <hi rend="italics">barometer</hi> and <hi rend="italics">thermometer,</hi> is marked by degrees, or equal divisions, placed by the side of the tube of those instruments.</p><p>The altitude of the barometer, or of the mercury in its tube, at London, is usually comprised between the limits of 28 and 31 inches; and the mean height, for every day in several years, is nearly 29.87 inches.</p><p><hi rend="smallcaps">Altitude</hi> <hi rend="italics">of the pyramids in Egypt,</hi> was measured so long since as the time of Thales, which he effected by means of their shadow, and that of a pole set upright beside them, making the altitudes of the pole and pyramid proportional to the lengths of their shadows. Plutarch has given an account of the manner of this operation, which is one of the first geometrical observations we have an exact account of.</p></div2><div2 part="N" n="Altitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Altitude</hi></head><p>, <hi rend="italics">circles of, parallels of, quadrant of, &c.</hi> See the respective words. <cb/></p><p><hi rend="italics">Equal</hi> <hi rend="smallcaps">Altitude</hi> <hi rend="italics">Instrument,</hi> is an instrument used to observe a celestial object, when it has the same or an equal altitude, on both sides of the meridian, or before and after it passes the meridian: an instrument very useful in adjusting clocks &c, and for comparing equal and apparent time.</p></div2></div1><div1 part="N" n="AMBIENT" org="uniform" sample="complete" type="entry"><head>AMBIENT</head><p>, encompassing round about; as the bodies which are placed about any other body, are called ambient bodies, and sometimcs <hi rend="italics">circum-ambient</hi> bodies; and the whole mass of the air or atmosphere, because it encompasses all things on the face of the earth, is called the <hi rend="italics">ambient</hi> air.</p><p>AMBIGENAL <hi rend="italics">Hyperbola,</hi> a name given by Newton, in his <hi rend="italics">Enumeratio linearum tertii ordinis,</hi> to one of the triple hyperbolas EGF of the second order, having one of its infinite legs, as EG, falling within the angle ACD, formed by the asymptotes AC, CD, and the other leg GF falling without that angle. <figure/></p></div1><div1 part="N" n="AMBIT" org="uniform" sample="complete" type="entry"><head>AMBIT</head><p>, <hi rend="italics">of a figure,</hi> in Geometry, is the perimeter, or line, or sum of the lines, by which the figure is bounded.</p></div1><div1 part="N" n="AMBLIGON" org="uniform" sample="complete" type="entry"><head>AMBLIGON</head><p>, or <hi rend="smallcaps">Ambligonal</hi>, in Geometry, signisies obtuse-angular, as a triangle which has one of its angles obtuse, or consisting of more than 90 degrees.</p><p>AMICABLE <hi rend="italics">numbers,</hi> denote pairs of numbers, of which each of them is mutually equal to the sum of all the aliquot parts of the other. So the first or least pair of amicable numbers are 220 and 284; all the aliquot parts of which, with their sums, are as follow, viz, of 220, they are 1, 2, 4, 5, 10, 11, 20, 22, 44, 55, 110, their sum — — 284; of 284, they are 1, 2, 4, 71, 142, and their sum is 220.</p><p>The 2d pair of amicable numbers are 17296 and 18416, which have also the same property as above.</p><p>And the 3d pair of amicable numbers are 9363584 and 9437056.</p><p>These three pairs of amicable numbers were found out by F. Schooten, sect. 9 of his <hi rend="italics">Exercitationes Mathematicæ,</hi> who I believe first gave the name of <hi rend="italics">amicable</hi> to such numbers, though such properties of numbers it seems had before been treated of by Rudolphus, Descartes, and others.</p><p>To find the sirst pair, Schooten puts 4<hi rend="italics">x</hi> and 4<hi rend="italics">yz,</hi> or <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> and <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">yz</hi> for the two numbers where <hi rend="italics">a</hi> = 2; then making each of these equal to the sum of the aliquot parts of the other, gives two equations, from which are found the values of <hi rend="italics">x</hi> and <hi rend="italics">z,</hi> and consequently, assuming a proper value for <hi rend="italics">y,</hi> the two amicable numbers themselves 4<hi rend="italics">x</hi> and 4<hi rend="italics">yz.</hi></p><p>In like manner for the other pairs of such numbers; in which he finds it necessary to assume 16<hi rend="italics">x</hi> and 16<hi rend="italics">yz</hi> or <hi rend="italics">a</hi><hi rend="sup">4</hi><hi rend="italics">x</hi> and <hi rend="italics">a</hi><hi rend="sup">4</hi><hi rend="italics">yz</hi> for the 2d pair, and 128<hi rend="italics">x</hi> and 128<hi rend="italics">yz</hi> or <hi rend="italics">a</hi><hi rend="sup">7</hi><hi rend="italics">x</hi> and <hi rend="italics">a</hi><hi rend="sup">7</hi><hi rend="italics">yz</hi> for the 3d pair. <pb n="105"/><cb/></p><p>Schooten then gives this practical rule, from Descartes, for finding amicable numbers, viz, Assume the number 2, or some power of the number 2, such that if unity or 1 be subtracted from each of these three following quantities, viz; from 3 times the assumed number,<lb/> also from 6 times the assumed number,<lb/> and from 18 times the square of the assumed number,<lb/> the three remainders may be all prime numbers; then the last prime number being multiplied by double the assumed number, the product will be one of the amicable numbers sought, and the sum of its aliquot parts will be the other.</p><p>That is, if <hi rend="italics">a</hi> be put = the number 2, and <hi rend="italics">n</hi> some integer number, such that 3<hi rend="italics">a</hi><hi rend="sup">n</hi>-1, and 6<hi rend="italics">a</hi><hi rend="sup">n</hi>-1, and 18<hi rend="italics">a</hi><hi rend="sup">2n</hi>-1 be all three prime numbers; then is ―(18<hi rend="italics">a</hi><hi rend="sup">2n</hi>-1) X2<hi rend="italics">a</hi><hi rend="sup">n</hi> one of the amicable numbers; and the sum of its aliquot parts is the other.</p></div1><div1 part="N" n="AMONTONS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">AMONTONS</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an ingenious French experimental philosopher, was born in Normandy the 31st of August 1663. While at the grammar school, he by sickness contracted a deafness that almost excluded him from the conversation of mankind. In this situation he applied himself to the study of geometry and mechanics; with which he was so delighted that it is said he refused to try any remedy for his disorder, either because he deemed it incurable, or because it increased his attention to his studies. Among other objects of his study, were the arts of drawing, of land-surveying, and of building; and shortly after he acquired some knowledge of those more sublime laws by which the universe is regulated. He studied with great care the nature of barometers and thermometers; and wrote his treatise of <hi rend="italics">Observations and Experiments concerning a new Hour-glass, and concerning Barometers, Thermometers, and Hygroscopes;</hi> as also some pieces in the Journal des Savans. In 1687, he presented a new hygroscope to the Academy of Sciences, which was much approved. He found out a method of conveying intelligence to a great distance in a short space of time: this was by making signals from one person to another, placed at as great distances from each other as they could see the signals by means of telescopes. When the Royal Academy was new regulated in 1699, Amontons was chosen a member of it, as an eleve under the third Astronomer; and he read there his <hi rend="italics">New Theory of Friction,</hi> in which he happily cleared up an important object in mechanics. In fact he had a particular genius for making experiments: his notions were just and delicate: and he knew how to prevent the inconveniences of his new inventions, and had a wonderful skill in executing them. He died of an inflammation in his bowels, the 11th of October 1705, being only 42 years of age.</p><p>The eloge of Amontons may be seen in the volume of the Memoirs of the Academy of Sciences for the year 1705, Hist. pa. 150. And his pieces contained in the different volumes of that work, which are pretty numerous, and upon various subjects, as the air, action of fire, barometers, thermometers, hygrometers, friction, machines, heat, cold, rarefactions, pumps, &c, may be seen in the volumes for the years 1696, 1699, 1702, 1703, 1704, and 1705.</p></div1><div1 part="N" n="AMPHISCII" org="uniform" sample="complete" type="entry"><head>AMPHISCII</head><p>, or <hi rend="smallcaps">Amphiscians</hi>, are the people who inhabit the torrid zone; which are so called, because <cb/> they have their shadow at noon turned sometimes one way, and sometimes another, namely, at one time of the year towards the north, and at the other towards the south.</p></div1><div1 part="N" n="AMPLITUDE" org="uniform" sample="complete" type="entry"><head>AMPLITUDE</head><p>, in <hi rend="italics">gunnery,</hi> the range of the projectile, or the right line upon the ground subtending the curvilinear path in which it moves.</p><div2 part="N" n="Amplitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Amplitude</hi>, in <hi rend="italics">astronomy</hi></head><p>, is an arch of the horizon, intercepted between the true east or west point, and the centre of the sun or a star at its rising or setting: so that the amplitude is of two kinds; <hi rend="italics">ortive</hi> or eastern, and <hi rend="italics">occiduous</hi> or western. Each of these amplitudes is also either northern or southern, according as the point of rising or setting is in the northern or southern part of the horizon: and the complement of the amplitude, or the arch of distance of the point of rising or setting, from the north or south point of the horizon, is the azimuth.</p><p>The amplitude is of use in navigation, to find the variation of the compass or magnetic needle. And the rule to find it is this: As the cosine of the latitude is to radius, so is the sine of the sun's or star's declination, to the sine of the amplitude. So in the latitude of London, viz, 51° 31′, when the sun's declination is 23° 28′; then <table><row role="data"><cell cols="1" rows="1" role="data">cos. 51° 31′ the lat.</cell><cell cols="1" rows="1" rend="align=right" role="data">-9.7939907</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 23 28 the decl.</cell><cell cols="1" rows="1" rend="align=right" role="data">+9.6001181</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 39 47 the ampl.</cell><cell cols="1" rows="1" rend="align=right" role="data">9.8061274</cell></row></table> That is, the sun then rises or sets 39° 47′ from the east or west point, to the north or south according as the declination is north or south.</p><p><hi rend="italics">Magnetical</hi> <hi rend="smallcaps">Amplitude</hi>, is an arch of the horizon, contained between the sun or star, at the rising or setting, and the magnetical east or west point of the horizon, pointed out by the magnetical compass, or the amplitude or azimuth compass. And the difference between this magnetical amplitude, so observed, and the true amplitude, as computed in the last article, is the variation of the compass.</p><p>So if, for instance, the magnetical amplitude be observed, by the compass, to be 61° 47′, at the time when <table><row role="data"><cell cols="1" rows="1" role="data">it is computed to be</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">47,</cell></row><row role="data"><cell cols="1" rows="1" role="data">then the difference</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">0 is the variation west.</cell></row></table></p></div2></div1><div1 part="N" n="ANABIBAZON" org="uniform" sample="complete" type="entry"><head>ANABIBAZON</head><p>, a name sometimes given to the dragon's tail, or northern node of the moon.</p></div1><div1 part="N" n="ANACAMPTICS" org="uniform" sample="complete" type="entry"><head>ANACAMPTICS</head><p>, or the science of the reflections of sounds, frequently used in reference to echoes, which are said to be sounds produced <hi rend="italics">anacamptically,</hi> or by reflection. And in this sense it was used by the ancients for that part of optics which is otherwise called Catoptrics.</p></div1><div1 part="N" n="ANACHRONISM" org="uniform" sample="complete" type="entry"><head>ANACHRONISM</head><p>, in Chronology, an error in computation of time, by which an event is placed earlier than it really happened. Such is that of Virgil, who makes Dido to reign at Carthage in the time of Æneas, though, in reality, she did not arrive in Africa till 300 years after the taking of Troy.</p><p>An error on the other side, by which a fact is placed later, or lower than it should be, is called a <hi rend="italics">parachronism.</hi> But in common use, this distinction, though proper, is not attended to; and the word <hi rend="italics">anachronism</hi> is used indifferently for the mistake on both sides.</p></div1><div1 part="N" n="ANACLASTICS" org="uniform" sample="complete" type="entry"><head>ANACLASTICS</head><p>, <hi rend="italics">or</hi> <hi rend="smallcaps">Anaclatics</hi>, an ancient name for that part of Optics which considers refracted light; <pb n="106"/><cb/> being the same as what is more usually called <hi rend="italics">dioptrics.</hi> See the Compendium of <hi rend="italics">Ambrosius Rhodius, lib.</hi> 3. <hi rend="italics">Opticæ, pa.</hi> 384 <hi rend="italics">& seq.</hi></p><p><hi rend="smallcaps">Anaclastic</hi> <hi rend="italics">Curves,</hi> a name given by M. de Mairan to certain apparent curves formed at the bottom of a vessel full of water, to an eye placed in the air; or the vault of the heavens, seen by refraction through the atmosphere.</p><p>M. de Mairan determines these curves by a principle not admitted by all authors; but Dr. Barrow, at the end of his Optics, determines the same curves by other principles.</p></div1><div1 part="N" n="ANALEMMA" org="uniform" sample="complete" type="entry"><head>ANALEMMA</head><p>, a planisphere, or projection of the sphere, orthographically made on the plane of the meridian, by perpendiculars from every point of that plane, the eye supposed to be at an infinite distance, and in the east or west point of the horizon. In this projection, the solstitial colure, and all its parallels, are projected into concentric circles, equal to the real circles in the sphere; and all circles whose planes pass through the eye, as the horizon and its parallels, are projected into right lines equal to their diameters; but all oblique circles are projected into ellipses, having the diameter of the circle for the transverse axis.</p><p>This instrument, having the furniture drawn on a plate of wood or brass, with an horizon fitted to it, is used for resolving many astronomical problems; as the time of the sun's rising and setting, the length and hour of the day, &c. It is also useful in dialling, for laying down the signs of the zodiac, with the lengths of days, and other matters of furniture, upon dials.</p><p>The oldest treatise we have on the analemma, was written by Ptolemy, which was printed at Rome in 1562, with a commentary by F. Commandine. Pappus also treated of the same. Since that time, many other authors have treated very well of the analemma; as Aguilonius, Taquet, Dechales, Witty, &c.</p></div1><div1 part="N" n="ANALOGY" org="uniform" sample="complete" type="entry"><head>ANALOGY</head><p>, the same as proportion, or equality, or similitude of ratios. Which see.</p></div1><div1 part="N" n="ANALYSIS" org="uniform" sample="complete" type="entry"><head>ANALYSIS</head><p>, is, generally, the resolution of any thing into its component parts, to discover the thing or the composition. And in mathematics it is properly the method of resolving problems, by reducing them to equations. <hi rend="italics">Analysis</hi> may be distinguished into the <hi rend="italics">ancient</hi> and the <hi rend="italics">modern.</hi></p><p>The <hi rend="italics">ancient analysis,</hi> as described by Pappus, is the method of proceeding from the thing sought as <hi rend="italics">taken</hi> for granted, through its consequences, to something that is <hi rend="italics">really</hi> granted or known; in which sense it is the reverse of synthesis or composition, in which we lay that down <hi rend="italics">first</hi> which was the <hi rend="italics">last</hi> step of the analysis, and tracing the steps of the analysis back, making that antecedent here which was consequent there, till we arrive at the thing sought, which was taken or assumed as granted in the first step of the analysis. This chiefly respected geometrical enquiries.</p><p>The principal authors on the ancient analysis, as recounted by Pappus, in the 7th book of his <hi rend="italics">Mathematical Collections,</hi> are Euclid in his <hi rend="italics">Data, Porismata, & de Locis ad Superficiem;</hi> Apollonius <hi rend="italics">de Sectione Rationis, de Sectione Spatii, de Taclionibus, de Inclinationibus, de Locis Planis, & de Sectionibus Conicis;</hi> Aristæus, <hi rend="italics">de Locis Solidis;</hi> and Eratosthenes, <hi rend="italics">de Mediis Proportionalibus;</hi> from which <cb/> Pappus gives many examples in the same book. T these authors we may add Pappus himself. The same sort of analysis has also been well cultivated by many of the moderns; as Fermat, Viviani, Getaldus, Snellius, Huygens, Simson, Stewart, Lawson, &c, and more especially Hugo d'Omerique, in his <hi rend="italics">Analysis Geometrica,</hi> in which he has endeavoured to restore the Analysis of the ancients. And, on this head, Dr. Pemberton tells us “that Sir Ifaac Newton used to censure himself for not following the ancients more closely than he did; and spoke with regret of his mistake, at the beginning of his mathematical studies, in applying himself to the works of Descartes, and other algebraical writers, before he had considered the Elements of Euclid with that attention so excellent a writer deserves: that he highly approved the laudable attempt of Hugo d'Omerique to restore the ancient analysis.”</p><p>In the application of the ancient analysis in geometrical problems, every thing cannot be brought within strict rules; nor any invariable directions given, by which we may succeed in all cases; but some previous preparation is necessary, a kind of mental contrivance and construction, to form a connexion between the <hi rend="italics">data</hi> and <hi rend="italics">quæsita,</hi> which must be left to every one's fancy to find out; being various, according to the various nature of the problems proposed: Right lines must be drawn in particular directions, or of particular magnitudes; bisecting perhaps a given angle, or perpendicular to a given line; or perhaps tangents must be drawn to a given curve, from a given point; or circles described from a given centre, with a given radius, or touching given lines, or other given circles; or such-like other operations. Whoever is conversant with the works of Archimedes, Apollonius, or Pappus, well knows that they founded their analysis upon some such previous operations; and the great skill of the analyst consists in discovering the most proper affections on which to found his analysis: for the same problem may often be effected in many different ways: of which it may be proper to give here an example or two. Let there be taken, for instance, this problem, which is the 155th prop. of the 7th book of Pappus.</p><p>From the extremities of the base A, B, of a given segment of a circle, it is required to draw two lines AC, BC, meeting at a point C in the circumference, so that they shall have a given ratio to each other, suppose that of F to G.</p><p>The solution of this problem, as given by Pappus, is thus. <hi rend="center"><hi rend="smallcaps">Analysis.</hi></hi></p><p>Suppose the thing done, and that the point C is found: then suppose CD is drawn a tangent to the circle at C, and meeting the line AB produced in the point D. Now by the hypothesis AC : BC :: F : G, and also AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi> :: DA : DB, as may be thus proved. <figure/></p><p>Since DC touches the circle, and BC cuts it, the angle BCD is equal to BAC by Euc. iii. 32; also the angle <pb n="107"/><cb/> D is common to both the triangles DCA, DCB; these are therefore similar, and so, by vi 4, DA : DC :: DC : DB, and hence DA<hi rend="sup">2</hi> : DC<hi rend="sup">2</hi> :: DA : DB by cor. vi 20. But also, by vi 4, DA : AC :: DC : CB, and by permutation DA : DC :: AC : BC, or DA<hi rend="sup">2</hi> : DC<hi rend="sup">2</hi> :: AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi>; and hence, by equality, AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi> :: DA : DB.</p><p>But the ratio of AC<hi rend="sup">2</hi> to BC<hi rend="sup">2</hi> is given by prop. LVII of Simson's edition of the <hi rend="italics">Data,</hi> because the ratio of AC to BC is given, and consequently that of DA to DB is given. Now since the ratio of DA to DB is given, therefore also, by Data vi, that of DA to AB, and hence, by Data ii, DA is given in magnitude.</p><p>And here the analysis properly ends. For it having been shewn that DA is given, or that a point D may be found in AB produced, such, the a tangent being drawn from it to the circumserence, the point of contact will be the point sought; we may now begin the composition, or synthetical demonstration; which must be done by finding the point D, or laying down the line AD, which, it was affirmed, was given, in the last step of the analysis. <hi rend="center"><hi rend="smallcaps">Synthesis.</hi></hi></p><p><hi rend="italics">Construction.</hi> Make as F<hi rend="sup">3</hi> : G<hi rend="sup">2</hi> :: AD : DB, (which may be done, since AB is given, by making it as F<hi rend="sup">2</hi>G<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> :: AB : DB, and then by composition it will be as F<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> :: AD : DB); and then from the point D, thus found, draw a tangent to the circle, and from the point of contact C drawing CA and CB, the thing is done.</p><p><hi rend="italics">Demonstration.</hi> Since, by the constr. F<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> :: AD : DB, and also AD : DB :: AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi>, which has been already demonstrated in the analysis, and might be here proved in the same manner. Therefore F<hi rend="sup">2</hi> : G<hi rend="sup">2</hi> :: AC<hi rend="sup">2</hi> : BC<hi rend="sup">2</hi>, and consequently F : G :: AC : BC. <hi rend="italics">Q.E.D.</hi></p><p>Here we see an instance of the method of <hi rend="italics">resolution</hi> and <hi rend="italics">composition,</hi> as it was practised by the ancients, the solution here given being that of Pappus himself. But as the method of referring and reducing every thing to the <hi rend="italics">Data,</hi> and constantly quoting the same, may appear now to be tedious and troublesome: and indeed it is unnecessary to those who have already made themselves masters of the substance of that valuable book of Euclid, and have by practice and experience acquired a facility of reasoning in such matters: I shall therefore now shew how we may abate something of the rigour and strict from of the ancient method of solution, without diminishing any part of its admirable elegance and perspicuity. And this may be done by the instance of another solution, of the many more which might be given, of the same problem, as follows. <hi rend="center"><hi rend="smallcaps">Analysis.</hi></hi></p><p>Let us again suppose that the thing is done, viz AC : BC :: F : G, and let there be drawn BH making the angle ABH equal to the angle ACB, and meeting AC produced in H. Then, the angle A being also common, the two triangles ABC and ABH are equiangular, and therefore, by vi 4, AC : BC :: AB : BH, in a given ratio; and, AB being given, therefore BH is given in position and magnitude. <cb/> <hi rend="center"><hi rend="smallcaps">Synthesis.</hi></hi></p><p><hi rend="italics">Construction.</hi> Draw BH making the angle ABH equal to that which may be contained in the given segment, and take AB to BH in the given ratio of F to G. Draw ACH, and BC.</p><p><hi rend="italics">Demonstration.</hi> The triangles ABC, ABH are equiangular, therefore, vi 4, AC : CB :: AB : BH, which is the given ratio by construction.</p><p><hi rend="italics">Modern</hi> <hi rend="smallcaps">Analysis</hi>, consists chiefly of algebra, arithmetic of insinites, insinite series, increments, fluxions, &c; of each of which a particular account may be seen under their respective articles.</p><p>These form a kind of arithmetical and symbolical analysis, depending partly on modes of arithmetical computation, partly on rules peculiar to the symbols made use of, and partly on rules drawn from the nature and species of the quantities they represent, or from the modes of their existence or generation.</p><p>The modern analysis is a general instrument by which the sinest inventions and the greatest improvements have been made in mathematics and philosophy, for near two centuries past. It furnishes the most perfect examples of the manner in which the art of reasoning should be employed; it gives to the mind a wonderful skill for discovering things unknown, by means of a small number that are given; and by employing short and easy symbols for expressing ideas, it presents to the understanding things which otherwise would seem to lie above its sphere. By this means geometrical demonstrations may be greatly abridged: a long train of arguments, in which the mind cannot, without the greatest effort of attention, discover the connection of ideas, is converted into visible symbols; and the various operations which they require, are simply effected by the combination of those symbols. And, what is still more extraordinary, by this artifice, a great number of truths are often expressed in one line only: instead of which, by following the ordinary way of explanation and demonstration, the same truths would occupy whole pages or volumes. And thus, by the bare contemplation of one line of calculation, we may undersland in a short time whole sciences, which otherwise could hardly be comprehended in several years.</p><p>It is true that Newton, who best knew all the advantages of analysis in geometry and other sciences, laments, in several parts of his works, that the study of the ancient geometry is abandoned or neglected. And indeed the method employed by the ancients in their geometrical writings, is commonly regarded as more rigorous, than that of the modern analysis: and though it be greatly inferior to that of the moderns, in point of dispatch and facility of invention; it is nevertheless highly useful in strengthening the mind, improving the reasoning faculties, and in accustoming the young mathematician to a pure, clear, and accurate mode of investigation and demonstration, though by a long and laboured process, which he would with difficulty have submitted to if his taste had before been vitiated, as it were, by the more piquant sweets of the modern analysis. And it is principally on this that the complaints of Newton are founded, who feared lest by the too early and frequent use of the modern analysis, the science of geometry should lose that rigour and purity which characterise its investigations, and the mind become debilitated by the <pb n="108"/><cb/> facility of our analysis. This great man was therefore well founded, in recommending, to a certain extent, the study of the ancient geometricians: for, their demonstrations being more difficult, give more exercise to the mind, accustom it to a closer application, give it a greater scope, and habituate it to patience and resolution, so necessary for making discoveries. But this is the only or principal advantage from it; for if we should look no farther than the method of the ancients, it is probable that, even with the best genius, we should have made but few or small discoveries, in comparison of those obtained by means of the modern analysis. And even with regard to the advantage given to investigations made in the manner of the ancients, namely of being more rigorous, it may perhaps be doubted whether this pretension be well founded. For to instance in those of Newton himself, although his demonstrations be managed in the manner of the ancients; yet at the same time it is evident that he investigates his theorems by a method different from that employed in the demonstrations, which are commonly analytical calculations, disguised by substituting the name of lines for their algebraical value: and though it be true that his demonstrations are rigorous, it is no less so that they would be the same when translated and delivered in algebraic language; and what difference can it make in this respect, whether we call a line AB, or denote it by the algebraic character <hi rend="italics">a?</hi> Indeed this last designation has this peculiarity, that when all the lines are denoted by algebraic characters, many operations can be performed upon them, without thinking of the lines or the figure. And this circumstance proves of no small advantage: the mind is relieved, and spared as much as possible, that its whole force may be employed in overcoming the natural difficulty of the problem alone.</p><p>Upon the whole therefore the state of the comparison seems to be this; That the method of the ancients is fittest to begin our studies with, to form the mind and to establish proper habits; and that of the moderns to succeed, by extending our views beyond the present limits, and enabling us to make new discoveries and improvements.</p><p><hi rend="italics">Analysis</hi> is divided, with respect to its object, into that of <hi rend="italics">finites,</hi> and that of <hi rend="italics">infinites.</hi></p><p><hi rend="italics">Analysis of finite quantities,</hi> is what is otherwise called <hi rend="italics">algebra,</hi> or <hi rend="italics">specious arithmetic.</hi></p><p><hi rend="italics">Analysis of infinites,</hi> called also the <hi rend="italics">new analysis,</hi> is that which is concerned in calculating the relations of quantities which are considered as infinite, or infinitely little; one of its chief branches being the <hi rend="italics">method of fluxions,</hi> or the <hi rend="italics">differential calculus.</hi> And the great advantage of the modern mathematicians over the ancients, arises chiefly from the use of this modern analysis.</p><p><hi rend="smallcaps">Analysis</hi> <hi rend="italics">of powers,</hi> is the same as resolving them into their roots, and is otherwise called <hi rend="italics">evolution.</hi></p><p><hi rend="smallcaps">Analysis</hi> <hi rend="italics">of curve lines,</hi> shews their constitution, nature and properties, their points of inflexion, station, retrogradation, variation, &c.</p></div1><div1 part="N" n="ANALYST" org="uniform" sample="complete" type="entry"><head>ANALYST</head><p>, a person who analyses something, or makes use of the analytical method. In mathematics, it is a person skilled in algebra, or in the mathematical analysis in general. <cb/></p><div2 part="N" n="Analyst" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Analyst</hi></head><p>, the title of an ingenious, though sophistical book, written by the celebrated Dr. Berkeley, against the doctrine of fluxions.</p></div2></div1><div1 part="N" n="ANALYTIC" org="uniform" sample="complete" type="entry"><head>ANALYTIC</head><p>, or <hi rend="smallcaps">Analytical</hi>, something belonging to, or partaking of, the nature of analysis; or performed by the method of analysis.</p><p>Thus we say <hi rend="italics">analytical</hi> demonstration, <hi rend="italics">analytical</hi> enquiry, <hi rend="italics">analytical</hi> table or scheme, <hi rend="italics">analytical</hi> method, &c. The <hi rend="italics">analytical</hi> stands opposed to the <hi rend="italics">synthetical,</hi> or that which proceeds by the way of <hi rend="italics">synthesis.</hi></p></div1><div1 part="N" n="ANALYTICS" org="uniform" sample="complete" type="entry"><head>ANALYTICS</head><p>, the science, or doctrine, and use of analysis.</p></div1><div1 part="N" n="ANAMORPHOSIS" org="uniform" sample="complete" type="entry"><head>ANAMORPHOSIS</head><p>, in perspective and painting, a monstrous projection; or a representation of some image, either on a plane or curve surface, deformed or distorted; but which in a certain point of view shall appear regular, and drawn in just proportion.</p><p><hi rend="italics">To construct an Anamorphosis, or monstrous projection, on a plane.</hi>—Draw the square ABCD (fig. 1), of any size at pleasure, and divide it by crossing lines into a number of areolæ or smaller squares: and then in this square, or reticle, called also the <hi rend="italics">cratioular prototype,</hi> draw the regular image which is to be distorted.—Or, about any image, proposed to be distorted, draw a reticle of small squares. <figure/></p><p>Then draw the line <hi rend="italics">ab</hi> (fig. 2.) equal to AB, dividing it into the same number of equal parts, as the side of the prototype AB; and on its middle point E erect the perpendicular EV, and also VS perpendicular to EV, making EV so much the longer, and VS so much the shorter, as it is intended the image shall be more distorted. From each of the points of division draw right lines to the point V, and draw the right line <hi rend="italics">a</hi>S. Lastly through the points <hi rend="italics">c, e, f, g,</hi> &c, draw lines parallel to <hi rend="italics">ab:</hi> So shall<hi rend="italics">abcd</hi> be the space upon which the monstrous projection is to be drawn; and is called the <hi rend="italics">craticular ectype.</hi></p><p>Then, in every areola, or small trapezium, of the space <hi rend="italics">abcd,</hi> draw what appears contained in the corresponding areola of the original space ABCD: so shall there be produced a deformed image in the spacc <hi rend="italics">abcd,</hi> which yet will appear in just proportion to an eye distant <pb n="109"/><cb/> from it the length of EV, and raised above it by a height equal to VS.</p><p>It will be amusing to contrive it so, that the deformed image may not represent a mere chaos, but some certain figure: thus, a river with soldiers, waggons, and other objects on the side of it, have been so drawn and distorted, that when viewed by an eye at S, it appeared like the face of a satyr.</p><p>An image may also be distorted mechanically, by perforating through in several places with a fine pin; then, placing it against a candle or lamp, observe where the rays, which pass through these small holes, fall on any plane or curve superficies; for they will give the correspondent points of the image deformed, and by means of which the deformation may be completed.</p><p><hi rend="italics">To draw an Anamorphosis upon the convex surface of a cone.</hi> It appears from the construction above, that we have only to make a craticular ectype upon the surface of the cone, which may appear equal to the craticular prototype, to an eye placed at a proper height above the vertex of the cone. Hence,</p><p>Let the base, or circumference, ABCD, of the cone (fig. 3) be divided by radii into any number of equal parts; and let some one radius be likewise divided into equal parts; then through each point of division draw concentric circles: so shall the craticular prototype be formed. <figure/></p><p>With double the diameter AB, as a radius, describe the quadrant EFG (fig. 4) so as the arch EG be equal to the whole periphery; then this quadrant, being plied or bent round, will form the superficies of a cone, whose base is the circle.</p><p>Next divide the arch EG into the same number of equal parts as the craticular prototype is divided into; and draw radii from all the points of division. Produce GF to I, so that FI be equal to FG; and from the centre I, with the radius IF, describe the quadrant FKH; and draw the right line IE. Then divide the arch KF into the same number of equal parts as the radius of the eraticular prototype is divided into; and from the centre I draw radii through all the points of division, meeting EF in 1, 2, 3, &c. Lastly, from the centre F, with the radii F1, F2, F3, &c, describe concentric circles. So will the craticular ectype be formed, whose areolas will appear equal to each other.</p><p>Hence, what is delineated in every areola of the craticular prototype, being transferred into the areolas of the craticular ectype, the images will be distorted or deformed; and yet they will appear in just proportion <cb/> to an eye elevated above the vertex at a height equal to the height of the cone itself.</p><p>If the chords of the quadrants be drawn in the craticular prototype, and chords of each of the 4th parts in the craticular ectype, every thing else remaining the same, there will be obtained the craticular ectype in a quadrangular pyramid.</p><p>And hence it will be easy to deform an image, in any other pyramid, whose base is any regular polygon.</p><p>Because the illusion is more perfect when the eye, by the contiguous objects, cannot estimate the distance of the parts of the deformed image, it is therefore proper to view it through a small hole.</p><p>Anamorphoses, or monstrous images, may also be made to appear in their natural shape and just proportions, by means of mirrors of certain shapes, from which those images are reflected again; and then they are said to be reformed.</p><p>For farther particulars, see Wolfius's <hi rend="italics">Catoptrics and Dioptrics,</hi> and some other optical authois.</p></div1><div1 part="N" n="ANAPHORA" org="uniform" sample="complete" type="entry"><head>ANAPHORA</head><p>, in <hi rend="italics">Astrology,</hi> the second house, or that part of the heavens which is 30 degrees from the horoscope.</p><p>The term <hi rend="italics">anaphora</hi> is also sometimes applied promiscuously to some of the succeeding houses, as the 5th, the 8th, and the 11th. In this sense <hi rend="italics">anaphora</hi> is the same as <hi rend="italics">epanaphora,</hi> and stands opposed to <hi rend="italics">cataphora.</hi></p><p>ANASTROUS <hi rend="italics">signs,</hi> in <hi rend="italics">Astronomy,</hi> a name given to the <hi rend="italics">duodecatemoria,</hi> or the 12 portions of the ecliptic, which the signs possessed anciently, but have since deserted by the precession of the equinox.</p></div1><div1 part="N" n="ANAXAGORAS" org="uniform" sample="complete" type="entry"><head>ANAXAGORAS</head><p>, one of the most celebrated philosophers among the ancients. He was born at Clazomene in Ionia, about the 70th Olympiad. He was a disciple of Anaximenes; and he gave up his patrimony, to be more at leisure for the study of philosophy, giving lectures in that science at Athens. Being persecuted in this place, and at last banished from it, he opened a school at Lampsacum, where he was greatly honoured during his life, and still more after his death, statues having been erected to his memory. It is said he made some predictions relative to the phenomena of nature, as earthquakes &c, upon which he wrote some treatises. His principal tenets may be reduced to the following:— All things were in the beginning confusedly mixed together, without order and without motion. The principle of things is at the same time one and multiplex, which had the name of <hi rend="italics">homæmeries,</hi> or similar particles, deprived of life. But there is beside this, from all eternity, another principle, an infinite and incorporeal spirit, who gave motion to these particles; in virtue of which, such as are homogeneal united, and such as were heterogeneal separated according to their different kinds. All things being thus put into motion by the spirit, and every thing being united to such as are similar, those that had a circular motion produced heavenly bodies, the lighter particles ascending, while those that were heavier descended. The rocks of the earth, being drawn up by the whirling force of the air, took fire, and became stars, beneath which the sun and moon took their stations. —It was said he also wrote upon the <hi rend="italics">Quadrature of the Circle;</hi> the treatise upon which, Plutarch says, he composed during his imprisonment at Athens. <pb n="110"/><cb/></p></div1><div1 part="N" n="ANAXIMANDER" org="uniform" sample="complete" type="entry"><head>ANAXIMANDER</head><p>, a very celebrated Greek philosopher, was born at Miletus in the 42d olympiad; for, according to Apollodorus, he was 64 years of age in the 2d year of the 58th olympiad. He was one of the first who publicly taught philosophy, and wrote upon philosophical subjects. He was the kinsman, companion, and disciple of Thales. He wrote also upon the sphere and geometry, &c. And he carried his researches into nature very far, for the time in which he lived. It is said that he discovered the obliquity of the zodiac; that he first published a geographical table; that he invented the gnomon, and set up the first sun-dial in an open place at Lacedæmon. He taught, that infinity of things was the principal and universal element; that this infinite always preserved its unity, but that its parts underwent changes; that all things came from it; and that all were about to return to it. By this obscure and indeterminate principle he probably meant the chaos of other philosophers. He held that the worlds are insinite; that the stars are composed of air and sire, which are carried about in their spheres, and that these spheres are gods; and that the earth is placed in the midst of the universe, as in a common centre. Farther, that insinite worlds were the produce of infinity; and that corruption proceeded from separation.</p></div1><div1 part="N" n="ANAXIMENES" org="uniform" sample="complete" type="entry"><head>ANAXIMENES</head><p>, an eminent Greek philosopher, born at Miletus, the friend, scholar, and successor of Anaximander. He diffused some degree of light upon the obscurity of his master's system. He made the first principle of things to consist in the air, which he considered as infinite or immense, and to which he ascribed a perpetual motion; that this air was the same as spirit or God, since the divine power resided in it, and agitated it. The stars were as fiery nails in the heavens; the sun a flat plate of fire; the earth an extended flat surface, &c.</p></div1><div1 part="N" n="ANDERSON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ANDERSON</surname> (<foreName full="yes"><hi rend="smallcaps">Alexander</hi></foreName>)</persName></head><p>, one of the brightest ornaments of the mathematical world, who flourished about 200 years ago. He was born at Aberdeen in Scotland, it would seem towards the latter part of the 16th century, as he was professor of mathematics at Paris in the early part of the 17th, where he published several ingenious works in geometry and algebra, both of his own, and of his friend Vieta's. Thus he published his “Supplementum Apollonii Redivivi; (of Ghetaldus) sive analysis problematis hactenus desiderati ad Apollonii Pergæi doctrinam <foreign xml:lang="greek">weri neusewn</foreign>, a Marino Ghetaldo Patritio Ragusino hujusque, non ita pridem restitutam. In qua exhibetur mechanice æqualitatum tertii gradus sive solidarum, in quibus magnitudo omnino data, æquatur homogeneæ sub altero tantum coëfficiente ignoto. Huic subnexa est variorum problematum practice.” Paris, 1612, in 4to.</p><p>“<foreign xml:lang="greek">*aitiologia</foreign>: Pro Zetetico Apolloniani problematis a se jam pridem edito in supplemento Apollonii Redivivi. Ad clarissimum & ornatissimum virum Marinum Ghetaldum Patritium Ragusinum. In qua ad ea quae obiter mihi perstrinxit Ghetaldus respondetur, & analytices clarius detegitur.” Paris, 1615, in 4to. <hi rend="center">He published also,</hi></p><p>“Francisci Vietæ Fontenacensis de Aequationum Recognitione & Emendatione Tractatus duo.” Paris, 1615, in 4to; with a Dedication, Prefaee, and an Appendix, by Anderson. <cb/></p><p>And Vieta's Angulares Sectiones, with the Demonstrations by Anderson.</p><p>Alexander was cousin german to a Mr. David Anderson, of Finshaugh, a gentleman who also possessed a singular turn for mathematical and mechanical knowledge. This mathematical genius was hereditary in the family of the Andersons, and from them it seems to have been transmitted to their descendants of the name of Gregory in the same country: the daughter of the said David Anderson having been the mother of the celebrated mathematician James Gregory, and who herself first instructed her son James in the elements of the Mathematics, upon her observing in him, while yet a child, a strong propensity to those sciences.</p><p>The time either of the birth or death of our author Alexander, has not come to my knowledge.</p></div1><div1 part="N" n="ANDROGYNOUS" org="uniform" sample="complete" type="entry"><head>ANDROGYNOUS</head><p>, an appellation given, by astrologers, to such of the planets as are sometimes hot, and sometimes cold; as mercury, which is accounted hot and dry when near the sun, and cold and moist when near the moon.</p></div1><div1 part="N" n="ANDROMEDA" org="uniform" sample="complete" type="entry"><head>ANDROMEDA</head><p>, in <hi rend="italics">Astronomy,</hi> a constellation of the northern hemisphere, representing the sigure of a woman almost naked, her seet at a distance from each other, and her arms extended and chained; being one of the original 48 asterisms, or sigures under which the ancients comprehended the stars, as derived to us from the Greeks, who probably had them from the Egyptians or Indians, and who, it is suspected, altered their names, and accompanied them with fabulous stories of their own. According to them, Cepheus, the father of Andromeda, was obliged to give her up to be devoured by a monster, to preserve his kingdom from the plague; but that she was delivered by Perseus, who slew the monster, and espoused her. And the family were all translated by Minerva to heaven, the mother being the constellation Cassiopeia.</p><p>She is sometimes called, in Latin, <hi rend="italics">Persea, Mulier catenata, Virgo devota,</hi> &c. The Arabians, whose religion did not permit them to draw the figure of the human body on any occasion whatever, have changed this constellation into the figure of a sea-calf. Schickard has changed the name for that of the scripture name <hi rend="italics">Abigail.</hi> And Schiller has also changed the figure of the constellation, for that of a sepulchre, and calls it the <hi rend="italics">Holy Sepulchre.</hi></p><p>This constellation contains about 27 stars that are visible to the naked eye; of which the principal are, <foreign xml:lang="greek">a</foreign> <hi rend="italics">Andromeda's head;</hi> <foreign xml:lang="greek">b</foreign> in the girdle, and called <hi rend="italics">mirach</hi> or <hi rend="italics">mizar;</hi> <foreign xml:lang="greek">g</foreign> on the south foot, and named <hi rend="italics">alamak,</hi> and sometimes <hi rend="italics">alhames.</hi></p><p>The number of stars placed in this constellation by the catalogue of Ptolemy is 23, by that of Tycho Brahe also 23, by that of Hevelius 47, and by that of Flamsteed 66.</p></div1><div1 part="N" n="ANEMOMETER" org="uniform" sample="complete" type="entry"><head>ANEMOMETER</head><p>, an instrument for measuring the force of the wind.</p><p>An instrument of this sort, it seems, was first invented by Wolfius in the year 1708, and first published in his <hi rend="italics">Areometry</hi> in 1709, also in the <hi rend="italics">Acta Eruditorum</hi> of the same year; afterwards in his <hi rend="italics">Mathematical Dictionary,</hi> and in his <hi rend="italics">Elem. Matheseos.</hi> He says he tried the goodness of it, and observes that the internal struc- <pb n="111"/><cb/> ture may be preserved, so as to measure the force of running water, or that of men or horses when they draw or pull. The machine consists of sails, A, B, C, like those of a wind-mill, against which the wind blows, and by turning them about, raises an arm K with a weight L upon it, to different angles of elevation, shewn by the index M, according to the force of the wind. (Plate III. fig. 3)</p><p>In the Philos. Trans. another anemometer is described, in which the wind being supposed to blow directly against a flat side, or board, which moves along the graduated arch of a quadrant, the number of degrees it advances shews the comparative forceof the wind.</p><p>In the same Transactions, for the year 1766, Mr. Alex. Brice describes a method, successfully practised by him, of measuring the velocity of the wind, by means of that of the shadow of clouds passing over a plane upon the earth.</p><p>Also in the same Transactions, for the year 1775, Dr. Lind gives a description of a very ingenious portable <hi rend="italics">Wind-Gauge,</hi> by which the force of the wind is easily measured; a brief description of the principal parts of which here follows. This simple instrument consists of two glass tubes, AB, CD, (Plate III. fig. 4.) which should not be less than 8 or 9 inches long, the bore of each being about 4/10 of an inch diameter, and connected together by a small bent glass tube <hi rend="italics">ab,</hi> only of about 1/10 of an inch diameter, to check the undulations of the water caused by a sudden gust of wind. On the upper end of the leg AB is fitted a thin metal tube, which is bent perpendicularly outwards, and having its mouth open to receive the wind blowing horizoutally into it. The two tubes, or rather the two branches of the tube, are connected to a steel spindle KL by slips of brass near the top and bottom, by the sockets of which at <hi rend="italics">e</hi> and <hi rend="italics">f</hi> the whole instrument turns easily about the spindle, which is fixed into a block by a screw in its bottom, by the wind blowing in at the orifice at F. When the instrument is used, a quantity of water is poured in, till the tubes are about half full; then exposing the instrument to the wind, by blowing in at the orifice F, it forces the water down lower in the tube AB, and raises it so much higher in the other tube; and the distance between the surfaces of the water in the two tubes, estimated by a scale of inches and parts HI, placed by the sides of the tubes, will be the height of a column of water whose weight is equal to the force or momentum of the wind blowing or striking against an equal base. And as a cubic foot of water weighs 1000 ounces, or 62 1/2 pounds, the 12th part of which is 5 5/24 or 5 1/5 pounds nearly, therefore for every inch the surface of the water is raised, the force of the wind will be equal to so many times 5 1/5 pounds on a square foot. Thus, suppose the water stand 3 inches higher in the one tube, than in the other; then 3 times 5 1/5 or 15 3/5 pounds is equal to the pressure or force of the wind on the surface of a foot square.</p><p>This instrument of Dr. Lind's, measures only the force or momentum of the wind, but not its velocity. However the velocity of the wind may be deduced from its force so obtained, by help of some experiments performed by me at the Royal Military Academy, in the years 1786, 1787, and 1788; from which experiments it appears that a plane sursace of a square foot suffers a <cb/> resistance of 12 ounces from the wind, when blowing with a velocity of 20 feet per second; and that the sorce is nearly as the square of the velocity. Hence then, taking the force of 15 2/5 pounds, above found for the force of the wind when it sustains 3 inches of water, and taking the square roots of the forces, it will be, as √12 : √15 3/5 :: 20 : 22 4/5 the 4th proportional, that is a velocity of 22 4/5 feet per second, or 15 1/2 miles per hour, is the rate or velocity at which the wind blows, when it raises the water 3 inches higher in the one tube than the other. And farther, as the said height is as the force, and the force as the square of the velocity, we shall have the force and velocity, corresponding to several heights of the water in the one tube, above that in the other, as in the following table. <hi rend="center"><hi rend="italics">Table of the corresponding height of water, force on a square foot, and velocity of wind.</hi></hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Height of water.</cell><cell cols="1" rows="1" rend="align=center" role="data">Force of wind.</cell><cell cols="1" rows="1" rend="align=center" role="data">Velocity of wind per hour.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Inches.</cell><cell cols="1" rows="1" rend="align=center" role="data">Pounds.</cell><cell cols="1" rows="1" rend="align=center" role="data">Miles.</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 0 1/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4.5</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 0 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6.4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data"> 5.2</cell><cell cols="1" rows="1" rend="align=right" role="data">9.0</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">10.4</cell><cell cols="1" rows="1" rend="align=right" role="data">12.7</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" rend="align=right" role="data">15.5</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">20.8</cell><cell cols="1" rows="1" rend="align=right" role="data">18.0</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">26.0</cell><cell cols="1" rows="1" rend="align=right" role="data">20.1</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">31.25</cell><cell cols="1" rows="1" rend="align=right" role="data">22.0</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">36.5</cell><cell cols="1" rows="1" rend="align=right" role="data">23.8</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">41.7</cell><cell cols="1" rows="1" rend="align=right" role="data">25.4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">46.9</cell><cell cols="1" rows="1" rend="align=right" role="data">27.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">52.1</cell><cell cols="1" rows="1" rend="align=right" role="data">28.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">57.3</cell><cell cols="1" rows="1" rend="align=right" role="data">29.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">62.5</cell><cell cols="1" rows="1" rend="align=right" role="data">31.0</cell></row></table></p><p>In one instance Dr. Lind found that the force of the wind was such as to be equal 34 9/10 pounds, on a square foot; and this by proportion, in the following table, will be found to answer to a velocity of 23 1/4 miles per hour.</p><p>Mr. Leutmann improved upon Wolfius's anemometer, by placing the sails horizontal, instead of vertical, which are easier to move, and turn what way soever the wind blows.</p><p>Mr. Benjamin Martin also (Plate III. fig. 5) improved upon the same. He made the axis like the fusee of a watch, having a cord winding upon it, with two weights at the ends which make always a balance to the force of the wind on the sails. See his Philos. Britan.</p><p>And M. D'Ons-en-Bray invented a new anemometer, which of itself expresses on paper, not only the several winds that have blown during the space of 24 hours, and at what hour each began and ended, but also the different strength or velocity of each. See Mem. Acad. Scienc. an. 1734. See also the article <hi rend="italics">Wind-Gauge.</hi></p></div1><div1 part="N" n="ANEMOSCOPE" org="uniform" sample="complete" type="entry"><head>ANEMOSCOPE</head><p>, is sometimes used to denote a machine invented to foretell the changes of the wind, or weather; and sometimes for an instrument shewing by an index what the present direction of the wind is. Of this latter sort, it seems, was that used by the ancients, and described by Vitruvius; and we have many of them at present in large or public buildings, where an index withinside a room or hall, points to the name of the quarter from whence the wind blows without; which is simply effected by connecting an index to the lower end of the spindle of a weather-cock. <pb n="112"/><cb/></p><p>It has been observed that hygroscopes made of catgut, or such like, prove very good anemoscopes; seldom failing, by the turning of the index, to foretell the shifting of the wind. See accounts of two different anemoscopes; one by Mr. Pickering, vol. 43 Philos. Trans. the other by Mr. B. Martin, vol. 2 of his Philos. Britan.</p><p>Otto Gueric also gave the title anemoscope to a machine invented by him to foretell the change of the weather, as to rain and fair. It consisted of the small wooden figure of a man, which rose and fell in a glass tube, as the atmosphere was more or less heavy. Which was only an application of the common barometer, as shewn by M. Couriers in the <hi rend="italics">Acta Eruditorum</hi> for 1684.</p></div1><div1 part="N" n="ANGLE" org="uniform" sample="complete" type="entry"><head>ANGLE</head><p>, <hi rend="italics">Angulus,</hi> in <hi rend="italics">Geometry,</hi> the opening or mutual inclination of two lines, or two planes, or three planes, meeting in a point called the vertex or angular point. Such as the angle formed by, or between, the two lines AB and AC, at the vertex or angular point A.—Also the two lines AB and AC, are called the <hi rend="italics">legs</hi> or the <hi rend="italics">sides</hi> of the angle. <figure/></p><p>Angles are sometimes denoted, or named, by the single letter placed at the angular point, as the angle A; and sometimes by three letters, placing always that of the vertex in the middle. The former method is used when only one angle has the same vertex; and the latter method it is necessary to use when several angles have the same vertex, to distinguish them from one another.</p><p>The measure of an angle, by which its quantity or magnitude is expressed, is an arch, as DE described from the centre A, with any radius at pleasure, and contained between its legs AB and AC.—Hence angles are compared and distinguished by the ratio of the arcs which subtend them, to the whole circumference of the circle; or by the number of degrees contained in the arc DE by which they are measured, to 360, the number of degrees in the whole circumference of the circle. And thus an angle is said to be of so many degrees, viz, as are contained in the arc DE.</p><p>Hence it matters not, with what radius the arc is described, by which an angle is measured, when great or small, as AD, or A<hi rend="italics">d,</hi> or any other: for the arcs DE, <hi rend="italics">de,</hi> being similar, have the same ratio as their respective radii or circumferences, and therefore they contain the same number of degrees.—Hence it follows, that the quantity or magnitude of the angle remains still the same, though the legs be ever so much increased or diminished.—And thus, in similar figures, the like or corresponding angles are equal.</p><p>The <hi rend="italics">taking</hi> or <hi rend="italics">measuring</hi> of <hi rend="italics">angles,</hi> is an operation of great use and extent in surveying, navigation, geography, astronomy, &c. And the instruments chiefly used for this purpose, are quadrants, sextants, octants, <cb/> theodolites, circumferentors, &c. Mr. Hadley invented an excellent instrument for taking the <hi rend="italics">larger</hi> sort of angles, where much accuracy is required, or where the motion of the object, or any circumstance causing an unsteadiness in the common instruments, renders the observations difficult, or uncertain. And Mr. Dollond contrived an instrument for measuring <hi rend="italics">small</hi> angles. See <hi rend="italics">Hadley's Quadrant, Micrometer,</hi> and the Philos. Trans. Numbers 420, 425, and vol. 48. <hi rend="center"><hi rend="italics">To measure the Quantity of an Angle.</hi></hi></p><p>1. <hi rend="italics">On paper.</hi> Apply the centre of a protractor to the vertex A of the angle, so that the radius may coincide with one leg, as AB; then the degree on the arch that is cut by the other leg AC, will give the measure of the angle required.</p><p>Or thus, by a line of chords. Take off the chord of 60 with a pair of compasses; and with that radius, from the centre A, describe an arc as DE. Then take this arc DE between the compasses, and apply the extent to the scale of chords, which will give the degrees in the angle as before.</p><p>M. De Lagny gave, in several memoirs of the Royal Academy of Sciences, a new method of measuring angles, which he called <hi rend="italics">Goniometry.</hi> The method consists in measuring, with a pair of compasses, the are which subtends the proposed angle, not by applying its extent to a pre-constructed scale, like chords, but in the following manner: From the angular point as a centre, with a pretty large radius, describe a circle, producing one leg of the angle backwards to cut off a semicircle; then search out what part of the semicircle the arc is which measures the given angle, in this manner; viz, take the extent of this arc with a very fine pair of compasses, and apply it several times to the arc of the semicircle, to find how often it is contained, with a small part remaining over; in the same manner take the extent of this small part, and apply it to the first arc, to find how often it is contained in it; and what remains this 2d time, apply in like manner to the first remainder; then the 3d remainder apply to the 2d, and so on, always counting how often the last remainder is contained in the next foregoing, till nothing remain, or till the remainder is insensible, and too small to be measured: Then, beginning at the last, and returning backwards, make a series of fractions of which the numerators are always 1, and the denominators are the number of times each remainder is contained in its next remainder, with the fractional part more, as derived from the following remainder; then the last fraction, thus obtained, will shew what part the given angle is of 180° or the semicircle; and being turned into degrees &c, will be the measure of the angle, and nearer, it is asserted, than it can be obtained by any other means; whether it be measuring, or calculating by trigonometrical tables.— Thus, if it be required to measure the angle GFH: With a large radius describe the semicircle GHI, meeting the leg FG produced in I; then take the extent of the arc GH in the compasses, and applying it from G upon the semicircle, suppose it contains 4 times to the point 4, and the part 4 I over; take 4 I and apply it from H to 1, so that HG contains 4 I once, and 1 G over; also apply this remainder to the former 4 I, and it contains 5 times, from 4 to 5, and 5 I over; <pb n="113"/><cb/> and lastly the remainder 5 I is just two times contained in the former remainder 1 G or 12, without any remaindor. Here then, the series of quotients, or numbers of times contained, are 4, 1, 5, 2; therefore, beginning at the last, the first fraction is 1/2, or the last remainder is half the preceding one; and the 2d fraction is 1/(5 1/2) or 2/11; the 3d is 1/(1 2/11) or 11/13; and the fourth is 1/(4 11/13) or 13/63; that is, the arc GH is 13/63 of a semicircle, or the angle GFH is 13/63 of two right angles, or of 180°, which is equivalent to 37 1/7 degrees, or 37° 8′ 34″ 2/7.</p><p>2. <hi rend="italics">On the ground.</hi> Place a surveying instrument with its centre over the angular point to be measured, turning the instrument about till 0, the beginning of its arch, fall in the line or direction of one leg of the angle; then turn the index about to the direction of the other leg, and it will cut off from the arch the degrees answering to the given angle.</p><p>To plot or lay down any given angle, either on paper or on the ground, may be performed in the same manner; and the method is farther explained under the articles <hi rend="smallcaps">Plotting</hi> and <hi rend="smallcaps">Protracting</hi>, and under the names of the several instruments.</p><p><hi rend="italics">To bisect a given angle,</hi> as suppose the angle LKM. From the centre K, with any radius, describe the arc LM; then with the centres L and M, describe two arcs intersecting in N; and draw the line KN, which will bisect the given angle LKM, dividing it into the two equal angles LKN, MKN. <figure/></p><p><hi rend="italics">To trisect an angle,</hi> see <hi rend="smallcaps">Trisection.</hi></p><p>Pappus, in his Mathematical Collections, book 4, treats of angular sections, but particularly and more largely, of trisections. He also treats of any section in general, in the 36th and following propositions.</p><p><hi rend="smallcaps">Angles</hi> are of various kinds and denominations. With regard to the form of their legs, they are divided into <hi rend="italics">rectilinear, curvilinear,</hi> and <hi rend="italics">mixed.</hi></p><p><hi rend="italics">Rectilinear,</hi> or <hi rend="italics">right-lined</hi> <hi rend="smallcaps">Angle</hi>, is that whose legs are both right lines; as the foregoing angle CAB.</p><p><hi rend="italics">Curvilinear</hi> <hi rend="smallcaps">Angle</hi>, is that whose legs are both of them curves.</p><p><hi rend="italics">Mixt,</hi> or <hi rend="italics">mixtilinear</hi> <hi rend="smallcaps">Angle</hi>, is that of which one leg is a right line, and the other a curve.</p><p>With regard to their <hi rend="italics">magnitude,</hi> angles are again divided into <hi rend="italics">right</hi> and <hi rend="italics">oblique, acute</hi> and <hi rend="italics">obtuse.</hi></p><p><hi rend="italics">Right</hi> <hi rend="smallcaps">Angle</hi>, is that which is formed by one line perpendicular to another; or that which is subtended by a quadrant of a circle. As the angle BAC.—Therefore the measure of a right angle is a quadrant of a circle, or 90°; and consequently all right angles are equal to each other.</p><p><hi rend="italics">Oblique</hi> <hi rend="smallcaps">Angle</hi>, is a common name for any angle that is not a right one; and it is either <hi rend="italics">acute</hi> or <hi rend="italics">obtuse.</hi> <cb/></p><p><hi rend="italics">Acute</hi> <hi rend="smallcaps">Angle</hi>, is that which is less than a righ<*> angle, or less than 90 degrees; as the angle BAD And</p><p><hi rend="italics">Obtuse</hi> <hi rend="smallcaps">Angle</hi>, is greater than a right angle, or whos<*> measure exceeds 90 degrees; as the angle BAE.</p><p>With regard to their situation in respect of eac<*> other, angles are distinguished into <hi rend="italics">contiguous, adjacen<*> vertical, opposite,</hi> and <hi rend="italics">alternate.</hi></p><p><hi rend="italics">Contiguous</hi> <hi rend="smallcaps">Angles</hi>, are such as have the same vertex<*> and one leg common to both. As the angles BAD CAD, which have AD common.</p><p><hi rend="italics">Adjacent</hi> <hi rend="smallcaps">Angles</hi>, are those of which a leg of th<*> one produced forms a leg of the other: as the angle<*> GFH and IFH, which have the legs IF and FG in <*> straight line.—Hence adjacent angles are supplement<*> to each other, making together 180 degrees. An<*> therefore if one of these be given, the other will b<*> known by subtracting the given one from 180 degrees. Which property is useful in surveying, to find the quantity of an inaccessible angle; viz, measure its adjacent accessible one, and subtract this from 180 degrees. <figure/></p><p><hi rend="italics">Vertical</hi> or <hi rend="italics">opposite</hi> <hi rend="smallcaps">Angles</hi>, are such as have their legs mutually continuations of each other; as the two angles <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> or <hi rend="italics">c</hi> and <hi rend="italics">d.</hi>—The property of these is, that the vertical or opposite angles are always equal to each other, viz, [angle] <hi rend="italics">a</hi> = [angle] <hi rend="italics">b,</hi> and [angle] <hi rend="italics">c</hi> = [angle] <hi rend="italics">d.</hi> And hence the quantity of an inaccessible angle of a field, &c, may be found, by measuring its accessible opposite angle.</p><p><hi rend="italics">Alternate</hi> <hi rend="smallcaps">Angles</hi>, are those made on the opposite sides of a line cutting two parallel lines; so, the angles <hi rend="italics">e</hi> and <hi rend="italics">f,</hi> or <hi rend="italics">g</hi> and <hi rend="italics">h,</hi> are alternates. And these are always equal to each other; viz, the [angle] <hi rend="italics">c</hi> = [angle] <hi rend="italics">f,</hi> or [angle] <hi rend="italics">g</hi> = [angle] <hi rend="italics">h.</hi> <figure/></p><p><hi rend="italics">External</hi> <hi rend="smallcaps">Angles</hi>, are the angles of a figure made without it, by producing its sides outwards; as the angles <hi rend="italics">i, k, l, m.</hi> All the external angles of any rightlined figure, taken together, are equal to 4 right angles; and the external angle of a triangle is equal to both the internal opposite ones taken together; also any external angle of a trapezium inseribed in a circle, is equal to the internal opposite angle.</p><p><hi rend="italics">Internal</hi> <hi rend="smallcaps">Angles</hi>, are the angles within any figure, made by the sides of it; as the angles <hi rend="italics">n, o, p, q.</hi>—In any right-lined figure, an internal angle as <hi rend="italics">n,</hi> and its adjacent external angle <hi rend="italics">k,</hi> together make two right angles, or 180 degrees; and all the internal angles <hi rend="italics">n, o, p, q,</hi> <pb n="114"/><cb/> taken together, make twice as many right angles, wanting 4 right angles; also any two opposite internal angles of a trapezium inscribed in a circle, taken together, make two right angles, or 180 degrees.</p><p><hi rend="italics">Homologous,</hi> or <hi rend="italics">like</hi> <hi rend="smallcaps">Angles</hi>, are such angles in two figures, as retain the same order from the first, in both figures.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">out of the centre,</hi> as G, is one whose vertex is not in the centre of the circle.—And its measure is half the sum (<hi rend="italics">a</hi>+<hi rend="italics">b</hi>)/2 of the arcs intercepted by its legs when it is within the circle, or half the difference (<hi rend="italics">a</hi>-<hi rend="italics">b</hi>)/2 when it is without. <figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">at the centre,</hi> is an angle whose vertex is in the centre; as the angle AFC, formed by two radii AF, FC, and measured by the arc ADC.—An angle at the centre, as AFC, is always double of the angle ABC at the circumference, standing upon the same arc ADC; and all angles at the centre are equal that stand upon the same or equal arcs: also all angles at the centre, are proportional to the arcs they stand upon; and so also are all angles at the circumference. <figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">at the circumference,</hi> is an angle whose vertex is somewhere in the circumference of a circle; as the angle ABC.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">in a segment,</hi> is an angle whose legs meet the extremities of the base of the segment, and its vertex is anywhere in its arch; as the angle B is in the segment ABC, or standing upon the supplemental segment ADC; and is comprehended between two chords AB and BC.—An angle at the circumference is measured by half the arc ADC upon which it stands; and all the angles ABC, AEC, in the same segment, are equal to each other.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">in a semicircle</hi> is an angle at the circumference contained in a semicircle, or standing upon a semicircle, or on a diameter.—An angle in a semicircle, is always a right angle; in a greater segment, the angle is less, and in a less segment the angle is greater than a right angle.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of a segment,</hi> is that made by a chord with a tangent, at the point of contact. So IHK is the <cb/> angle of the less segment IMH, and IHL, the angle of the greater segment INH.—And the measure of each of these angles, is half the alternate or supplemental segment, or equal to the angle in it; viz, the [angle] IHK = [angle] INH, and the [angle] IHL = [angle] IMH.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of a semicircle,</hi> is the angle which the diameter of a circle makes with the circumference. And Euclid demonstrates that this is less than a right angle, but greater than any acute angle. <figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of contact,</hi> is that made by a curve line and a tangent to it, at the point of contact; as the angle IHK. It is proved by Euclid, that the angle of contact between a right line and a circle, is less than any right-lined angle whatever; though it does not therefore follow that it is of no magnitude or quantity. This has been the subject of great disputes amongst geometricians, in which Peletarius, Clavius, Taquet, Wallis, &c, bore a considerable share; Peletarius and Wallis contending that it is no angle at all, against Clavius, who rightly asserts that it is not absolutely nothing in itself, but only of no magnitude in comparison with a right-lined angle, being a quantity of a different kind or nature; like as a line in respect to a surface, or a surface in respect to a solid, &c. And since his time, it has been proved by Sir I. Newton, and others, that angles of contact can be compared to each other, though not to right-lined angles, and what are the proportions which they bear to each other. Thus, the circular angles of contact IHK, IHL, are to each other in the reciprocal subduplicate ratio of the diameters HM, HN. And hence the circular angle of contact may be divided, by describing intermediate circles, into any number of parts, and in any proportion. And if, instead of circles, the curves be parabolas, and the point of contact H the common vertex of their axes; the angles of contact would then be reciprocally in the subduplicate ratio of their parameters. But in such elliptical and hyperbolical angles of contact, these will be reciprocally in the subduplicate of the ratio compounded of the ratios of the parameters, and the transverse axes. Moreover, if TOQ be a common parabola, to the axis OP, and tangent VOW, and whose equation is , or <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">2</hi>, where <hi rend="italics">x</hi> is the absciss OP, and <hi rend="italics">y</hi> the ordinate PQ, the parameter being 1<*> and if OR, OS, &c, be other parabolas to the same axis, tangent, and parameter, their ordinate <hi rend="italics">y</hi> being PR, or PS, &c, and their equations <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">4</hi>, <hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">5</hi>, &c: then the series of angles of contact will be in succession infinitely greater than each other, viz, the angle of contact WOQ infinitely greater than WOR, and this infinitely greater than WOS, and so on infinitely.</p><p>And farther, between the angles of contact of any two of this kind, may other angles of contact be found <hi rend="italics">ad infinitum,</hi> which shall infinitely exceed each other, <pb n="115"/><cb/> and yet the greatest of them be infinitely less than the smallest right-lined angle. So also <hi rend="italics">x</hi><hi rend="sup">2</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>=<hi rend="italics">y</hi><hi rend="sup">4</hi>, <hi rend="italics">x</hi><hi rend="sup">4</hi>=<hi rend="italics">y</hi><hi rend="sup">5</hi>, &c, denote a series of curves, of which every succeeding one makes an angle with its tangent, infinitely greater than the preceding one; and the least of these, viz, that whose equation is <hi rend="italics">x</hi><hi rend="sup">2</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>, or the semicubical parabola, is infinitely greater than any circular angle of contact.</p><p><hi rend="smallcaps">Angles</hi> are again divided into <hi rend="italics">plane, spherical,</hi> and <hi rend="italics">solid.</hi></p><p><hi rend="italics">Plane</hi> <hi rend="smallcaps">Angles</hi>, are all those above treated of; which are defined by the inclination of two lines in a plane, meeting in a point.</p><p><hi rend="italics">Spherical</hi> <hi rend="smallcaps">Angle</hi>, is an angle formed on the surface of a sphere by the intersection of two great circles; or, it is the inclination of the planes of the two great circles.</p><p>The measure of a spherical angle, is the arc of a great circle of the sphere, intercepted between the two planes which form the angle, and which cuts the said planes at right angles. For their properties, &c, see <hi rend="smallcaps">Sphere, Spherical</hi>, and <hi rend="smallcaps">Spherical Trigonometry.</hi></p><p><hi rend="italics">Solid</hi> <hi rend="smallcaps">Angle</hi>, is the mutual inclination of more than two planes, or plane angles, meeting in a point, and not contained in the same plane; like the angles or corners of solid bodies. For their measure, properties, &c, see <hi rend="smallcaps">Solid</hi> <hi rend="italics">Angle.</hi></p><p><hi rend="italics">Angles</hi> of other less usual kinds and denominations, are also to be found in some books of Geometry. As,</p><p><hi rend="italics">Horned</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus cornutus,</hi> that which is made by a right line, whether a tangent or secant, with the circumserence of a circle.</p><p><hi rend="italics">Lunular</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus lunularis,</hi> is that which is formed by the intersection of two curve lines, the one concave, and the other convex.</p><p><hi rend="italics">Cissoid</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus cissoides,</hi> the inner angle made by two spherical convex lines intersecting each other.</p><p><hi rend="italics">Sistroid</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus sistroides,</hi> is that which is in form of a sistrum.</p><p><hi rend="italics">Pelecoid</hi> <hi rend="smallcaps">Angle</hi>, <hi rend="italics">angulus pelecoides,</hi> is that in form of a hatchet.</p><div2 part="N" n="Angle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angle</hi></head><p>, in <hi rend="italics">Trigonometry.</hi> See <hi rend="smallcaps">Triangle</hi>, T<hi rend="smallcaps">RICONOMETRY, Sine, Tangent</hi>, <hi rend="italics">&c.</hi></p></div2><div2 part="N" n="Angle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angle</hi>, in <hi rend="italics">Mechanics.—Angle of Direction</hi></head><p>, is that which is comprehended between the lines of direction of two conspiring forces.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of Elevation,</hi> is that which is comprehended between the line of direction and any plane upon which the projection is made, whether horizontal or oblique.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of Incidence,</hi> is that made by the line of direction of an impinging body, at the point of impact. As the angle ABC.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of Reflection,</hi> is that made by the line of direction of the reflected body, at the point of impact. As the angle DBE.</p><p>Instead of the angles of incidence and reflection being estimated from the plane on which the body impinges, sometimes the complements of these are understood, viz, as estimated from a perpendicular to the reflecting plane; as the two angles ABF and DBF. <figure/> <cb/></p><p><hi rend="smallcaps">Angle</hi> in <hi rend="italics">Optics.—Visual</hi> or <hi rend="italics">Optic Angle,</hi> is the angle included between the two rays drawn from the two extreme points of an object to the centre of the pupil of the eye: as the angle HGI. The apparent magnitude of objects is greater or less, according to the angle under which they appear.—Objects seen under the same or an equal angle, always appear equal—.The least <hi rend="italics">visible</hi> angle, or least angle under which a body can be seen, according to Dr. Hook, is one minute; but Dr. Jurin shews, that at the time of his debate with Hevelius on this subject, the latter could probably discover a single star under so small an angle as 20″. But bodies are visible under smaller angles as they are more bright or luminous. Dr. Jurin states the grounds of this controversy, and discusses the question at large, in his Essay upon distinct and indistinct Vision, published in Smith's Optics, pa. 148, <hi rend="italics">& seq.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the interval,</hi> of two places, is the angle subtended by two lines directed from the eye to those places.</p><p><hi rend="smallcaps">Angle</hi> of <hi rend="italics">incidence,</hi> or <hi rend="italics">reflection,</hi> or <hi rend="italics">refraction,</hi> &c. See the respective words <hi rend="smallcaps">Incidence, Reflection, Refraction</hi>, &c.</p><p><hi rend="smallcaps">Angle</hi> in <hi rend="italics">Astronomy.—Angle of Commutation.</hi> See <hi rend="smallcaps">Commutation.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of elongation,</hi> or <hi rend="italics">Angle at the Earth.</hi> See <hi rend="smallcaps">Elongation.</hi></p><p><hi rend="italics">Parallactic</hi> <hi rend="smallcaps">Angle</hi>, or the <hi rend="italics">parallax,</hi> is the angle made at the centre of a star, the sun, &c, by two lines drawn, the one to the centre of the earth, and the other to its surface. See <hi rend="smallcaps">Parallactic</hi>, and <hi rend="smallcaps">Parallai.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the position of the sun, of the sun's apparent semi-diameter,</hi> &c. See the respective words.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">at the sun,</hi> is the angle under which the distance of a planet from the ecliptic, is seen from the sun.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the East.</hi> See <hi rend="smallcaps">Nonagesimal.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of obliquity,</hi> of the ecliptic, or the angle of inclination of the axis of the earth, to the axis of the ecliptic, is now nearly 23° 28′. See <hi rend="smallcaps">Obliquity</hi>, and <hi rend="smallcaps">Ecliptic.</hi></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of longitude,</hi> is the angle which the circle of a star's longitude makes with the meridian, at the pole of the ecliptic.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of right ascension,</hi> is the angle which the circle of a star's right ascension makes with the meridian at the pole of the equator.</p><p><hi rend="smallcaps">Angle</hi> in <hi rend="italics">Navigation.</hi> <hi rend="smallcaps">Angle</hi> <hi rend="italics">of the rhumb,</hi> or <hi rend="italics">loxodromic angle.</hi> See <hi rend="smallcaps">Rhumb</hi> and <hi rend="smallcaps">Loxodromic.</hi></p></div2><div2 part="N" n="Angles" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angles</hi>, in <hi rend="italics">Fortification</hi></head><p>, are understood of those formed by the several lines used in fortifying, or making a place defensible.</p><p>These are of two sorts; <hi rend="italics">real</hi> and <hi rend="italics">imaginary.—Real angles</hi> are those which actually exist and appear in the works. Such as the <hi rend="italics">flanked angle,</hi> the <hi rend="italics">angle of the epaule, angle of the flank,</hi> and the <hi rend="italics">re-entering angle of the counterscarp.</hi> Imaginary, or <hi rend="italics">occult angles,</hi> are those which are only subservient to the construction, and which exist no more after the fortification is drawn. Such as the <hi rend="italics">angle of the centre, angle of the polygan, flanking angle, sallant angle of the counterscarp,</hi> &c.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of,</hi> or <hi rend="italics">at, the centre,</hi> is the angle formed at the centre of the polygon, by two radii drawn from the centre to two adjacent angles, and subtended by a side <pb n="116"/><cb/> of it, as the angle ACB. This is found by dividing 360 degrees by the number of sides in the regular polygon. <figure/></p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the Polygon,</hi> is the angle intercepted between two sides of the polygon; as DAB, or ABE. This is the supplement of the angle at the centre, and is therefore found by subtracting the angle C from 380 degrees.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the Triangle,</hi> is half the angle of the polygon; as CAB or CBA; and is therefore half the supplement of the angle C at the centre.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the Bastion,</hi> is the angle FAG made by the two faces of the bastion. And is otherwise called the flanked angle.</p><p><hi rend="italics">Diminisbed</hi> <hi rend="smallcaps">Angle</hi>, is the angle BAG made by the meeting of the exterior side of the polygon with the face AG of the bastion.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the curtin,</hi> or <hi rend="italics">of the flank,</hi> is the angle GHI made between the curtin and the flank.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the epaule,</hi> or <hi rend="italics">shoulder,</hi> is the angle AGH made by the flank and the face of the bastion.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the tenaille,</hi> or <hi rend="italics">exterior flanking angle,</hi> is the angle AKB made by the two rasant lines of defence, or the faces of two bastions produced.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the counterscarp,</hi> is the angle made by the two sides of the counterscarp, meeting before the middle of the curtin.</p></div2><div2 part="N" n="Angle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angle</hi></head><p>, <hi rend="italics">flanking inward,</hi> is the angle made by the flanking line with the curtin.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">forming the flank,</hi> is that consisting of one flank and one demigorge.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">forming the face,</hi> is that composed of one flank and one face.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of the moat,</hi> is that made before the curtin, where it is intersected.</p><p><hi rend="italics">Re-entering,</hi> or <hi rend="italics">re-entrant</hi> <hi rend="smallcaps">Angle</hi>, is that whose vertex is turned inwards, towards the place; as H or I.</p><p><hi rend="italics">Saliant,</hi> or <hi rend="italics">sortant</hi> <hi rend="smallcaps">Angle</hi>, is that turned outwards, advancing its point towards the field; as A or G.</p><p><hi rend="italics">Dead</hi> <hi rend="smallcaps">Angle</hi>, is a re-entering angle, which is not flanked or defended.</p><p><hi rend="smallcaps">Angle</hi> <hi rend="italics">of a wall,</hi> in <hi rend="italics">Architecture,</hi> is the point or corner where the two sides or faces of a wall meet.</p></div2><div2 part="N" n="Angles" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Angles</hi>, in <hi rend="italics">Astrology</hi></head><p>, denote certain houses of a figure, or scheme of the heavens. So the horoscope of the first house, is termed the <hi rend="italics">angle of the east.</hi></p><p>ANGUINEAL <hi rend="italics">Hyperbola,</hi> a name given by Sir <cb/> I. Newton to four of his curves of the second order, viz, species 33, 34, 35, 36, expressed by the equation ; being hyperbolas of a serpentine figure. See <hi rend="smallcaps">Curves.</hi></p></div2></div1><div1 part="N" n="ANGULAR" org="uniform" sample="complete" type="entry"><head>ANGULAR</head><p>, something relating to, or that hath angles.</p><p>At a distance, angular bodies appear round; the angles and small inequalities disappearing at a much less distance than the bulk of the body.</p><p>ANGULAR <hi rend="italics">Motion,</hi> is the motion of a body which moves circularly about a point; or the variation in the angle described by a line, or radius, connecting a body with the centre about which it moves.—Thus, a pendulum has an angular motion about its centre of motion; and the planets have an angular motion about the sun.—Two moveable points M and O, of which the one describes <figure/> the arc MN, and the other the arc OP, in the same time, have an equal, or the same angularmotion, although the real motion of the point O be much greater than that of the point M, viz, as the arc OP is greater than the arc MN. The angular motions of revolving bodies, as of the planets about the sun, are reciproeally proportional to their periodic times. And they are also, as their real or absolute motions directly, and as their radii of motion inverfely.</p><p><hi rend="smallcaps">Angular</hi> <hi rend="italics">motion</hi> is also a kind of compound motion composed of a circular and a rectilinear motion; like the wheel of a coach, or other vehicle.</p><p>ANIMATED <hi rend="italics">needle,</hi> a needle touched with a magnet or load-stone.</p></div1><div1 part="N" n="ANNUAL" org="uniform" sample="complete" type="entry"><head>ANNUAL</head><p>, in <hi rend="italics">Astronomy,</hi> something that returns every year, or which terminates with the year.</p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">motion of the earth.</hi> See <hi rend="smallcaps">Earth.</hi></p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">argument of langitude.</hi> See <hi rend="smallcaps">Argument.</hi></p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">epacts.</hi> See <hi rend="smallcaps">Epact.</hi></p><p><hi rend="smallcaps">Annual</hi> <hi rend="italics">equation</hi> of the mean motion of the sun and moon, and of the moon's apogee and nodes. See E<hi rend="smallcaps">QUATION.</hi></p></div1><div1 part="N" n="ANNUITIES" org="uniform" sample="complete" type="entry"><head>ANNUITIES</head><p>, a term for any periodical income, arising from money lent, or from houses, lands, salaries, pensions, &c; payable from time to time; either annually, or at other intervals of time.</p><p>Annuities may be divided into such as are <hi rend="italics">certain,</hi> and such as depend on some <hi rend="italics">contingency,</hi> as the continuance of a life, &c.</p><p>Annuities are also divided into annuities in <hi rend="italics">possession,</hi> and annuities in <hi rend="italics">reversion;</hi> the former meaning such as have commenced; and the latter such as will not commence till some particular event has happened, or till some given period of time has elapsed.</p><p><hi rend="italics">Annuities</hi> may be farther considered as payable either <hi rend="italics">yearly,</hi> or <hi rend="italics">half yearly,</hi> or <hi rend="italics">quarterly,</hi> &c.</p><p>The <hi rend="italics">present value</hi> of an annuity, is that sum, which, being improved at interest, will be sufficient to pay the annuity.</p><p>The <hi rend="italics">present value</hi> of an <hi rend="italics">annuity certain,</hi> payable yearly, is calculated in the following manner.—Let the annuity be 1, and let <hi rend="italics">r</hi> denote the amount of 1<hi rend="italics">l.</hi> for a year, or 1<hi rend="italics">l.</hi> increased by its interest for one year. Then, 1 being the present value of the sum <hi rend="italics">r,</hi> and having to find the present value of the sum 1, it will be, by proportion <pb n="117"/><cb/> thus, <hi rend="italics">r</hi> : 1 :: 1 : 1/<hi rend="italics">r</hi> the present value of 1<hi rend="italics">l.</hi> due a year hence. In like manner 1/<hi rend="italics">r</hi><hi rend="sup">2</hi> will be the present value of 1<hi rend="italics">l.</hi> due 2 years hence; for <hi rend="italics">r</hi> : 1 :: 1/<hi rend="italics">r</hi> : 1/<hi rend="italics">r</hi><hi rend="sup">2</hi>. In like manner 1/<hi rend="italics">r</hi><hi rend="sup">3</hi>, 1/<hi rend="italics">r</hi><hi rend="sup">4</hi>, 1/<hi rend="italics">r</hi><hi rend="sup">5</hi>, &c, will be the present value of 1<hi rend="italics">l.</hi> due at the end of 3, 4, 5, &c, years respectively; and in general, 1/<hi rend="italics">r</hi><hi rend="sup">n</hi> will be the value of 1<hi rend="italics">l.</hi> to be received after the expiration of <hi rend="italics">n</hi> years. Consequently the sum of all these, or (1/<hi rend="italics">r</hi>)+(1/<hi rend="italics">r</hi><hi rend="sup">2</hi>)+(1/<hi rend="italics">r</hi><hi rend="sup">3</hi>)+(1/<hi rend="italics">r</hi><hi rend="sup">4</hi>)+ &c, contined to <hi rend="italics">n</hi> terms, will be the present value of all the <hi rend="italics">n</hi> years annuities. And the value of the perpetuity, is the sum of the series continued <hi rend="italics">ad infinitum.</hi></p><p>But this series, it is evident, is a geometrical progression, whose first term and common ratio are each 1/<hi rend="italics">r,</hi> and the number of its terms <hi rend="italics">n</hi>; and therefore the sum <hi rend="italics">s</hi> of all the terms, or the present value of all the annual payments, will be .</p><p>When the annuity is a perpetuity, it is plain that the last term 1/<hi rend="italics">r</hi><hi rend="sup">n</hi> vanishes, and therefore (1/(<hi rend="italics">r</hi>-1))X(1/<hi rend="italics">r</hi><hi rend="sup">n</hi>) also vanishes; and consequently the expression becomes barely <hi rend="italics">s</hi>=1/(<hi rend="italics">r</hi>-1); that is, any annuity divided by its interest for one year, is the value of the perpetuity. So, if the rate of interest be 5 per cent; then (100/5)=20 is the value of the perpetuity at 5 per cent. Also 100/4 =25 is the value of the perpetuity at 4 per cent. And 100/3 = 33 1/3 is the value of the perpetuity at 3 per cent. interest. And so on.</p><p>If the annuity is not to be entered on immediately, but after a certain number of years, as <hi rend="italics">m</hi> years; then the present value of the reversion is equal to the difference between two present values, the one for the first term of <hi rend="italics">m</hi> years, and the other for the end of the last term <hi rend="italics">n</hi>: that is, equal to the difference between .</p><p>Annuities certain differ in value, as they are made payable <hi rend="italics">yearly, half-yearly,</hi> or <hi rend="italics">quarterly.</hi> And by proceeding as above, using the interest or amount of a half year, or a quarter, as those for the whole year were <cb/> used, the following set of theorems will arise; where <*> denotes, as before, the amount of 1<hi rend="italics">l.</hi> and its interest for a year, and <hi rend="italics">n</hi> the number of years, during which, any annuity is to be paid; also P denotes the perpetnity 1/(<hi rend="italics">r</hi>-1), Y denotes (1/(<hi rend="italics">r</hi>-1))-(1/(<hi rend="italics">r</hi>-1))X(1/<hi rend="italics">r</hi><hi rend="sup">n</hi>) the value of the annuity supposed payable yearly, H the value of the same when it is payable half-yearly, and Q the value when payable quarterly; or universally, M the value when it is payable every <hi rend="italics">m</hi> part of a year.</p><p><hi rend="smallcaps">Theor.</hi> 1. .</p><p><hi rend="smallcaps">Theor.</hi> 2. .</p><p><hi rend="smallcaps">Theor.</hi> 3. .</p><p><hi rend="smallcaps">Theor.</hi> 4. <hi rend="center"><hi rend="italics">Example</hi> 1.</hi></p><p>Let the rate of interest be 4. per cent, and the term 5 years; and consequently <hi rend="italics">r</hi> = 1.04, <hi rend="italics">n</hi> = 5, P = 25; also let <hi rend="italics">m</hi> = 12, or the interest payable monthly in theorem 4: then the present value of such annuity of 1<hi rend="italics">l.</hi> a year, for 5 years, according as it is supposed payable 1<hi rend="italics">l.</hi> yearly, or (1/2)<hi rend="italics">l.</hi> every half year, or (1/4)<hi rend="italics">l.</hi> every quarter, or (1/12)<hi rend="italics">l.</hi> every month or (1/12)th part of a year, will be as follows:</p><p><hi rend="smallcaps">Example</hi> 2. Supposing the annuity to continue 25 years, the rate of interest and every thing else being as before; then the values of the annuities for 25 years will be</p><p><hi rend="smallcaps">Example</hi> 3. And if the term be 50 years, the values will be</p><p><hi rend="smallcaps">Example</hi> 4. Also if the term be 100 years, the values will be</p><p>Hence the difference in the value by making periods of payments smaller, for any given term of years, is the more as the intervals are smaller, or the periods more frequent. The same difference is also variable, both as the rate of interest varies, and also as the whole term of years <hi rend="italics">n</hi> varies; and, for any given rate of interest, it <pb n="118"/><cb/> is evident that the difference, for any periods <hi rend="italics">m</hi> of payments, first increases from nothing as the term <hi rend="italics">n</hi> increases, when <hi rend="italics">n</hi> is 0, to some certain finite term or value of <hi rend="italics">n,</hi> when the difference D is the greatest or a maximum; and that afterwards, as <hi rend="italics">n</hi> increases more, that difference will continually decrease to nothing again, and vanish when <hi rend="italics">n</hi> is infinite: also the term or value of <hi rend="italics">n,</hi> for the maximum of the difference, will be different according to the periods of payment, or value of <hi rend="italics">m.</hi> And the general value of <hi rend="italics">n,</hi> when the difference is a maximum between the yearly payments and the payments of <hi rend="italics">m</hi> times in a year, is expressed by this formula, viz, , where <hi rend="italics">l.</hi> denotes the logarithm of the quantity following it. Hence, taking the different values of <hi rend="italics">m,</hi> viz, 2 for half years, 4 for quarters, 12 for monthly payments, &c, and substituting in the general formula, the term or value of <hi rend="italics">n</hi> for each case, when the difference in the present worths of the annuities, will be as follows, reckoning interest at 4 per cent, viz, for half-yearly payments, for quarterly payments, for monthly payments.</p><p><hi rend="italics">Annuities</hi> may also be considered as in arrears, or as forborn, for any number of years; in which case each payment is to be considered as a sum put out to interest for the remainder of the term after the time it becomes due. And as 1<hi rend="italics">l.</hi> due at the end of 1 year, amounts to <hi rend="italics">r</hi> at the end of another year, and to <hi rend="italics">r</hi><hi rend="sup">2</hi> at the end of the 3d year, and to <hi rend="italics">r</hi><hi rend="sup">3</hi> at the end of the 4th year, and so on; therefore by adding always the last year's annuity, or 1, to the amounts of all the former years, the sum of all the annuities and their interests, will be the sum of the following geometrical series, 1 + <hi rend="italics">r</hi> + <hi rend="italics">r</hi><hi rend="sup">2</hi> + <hi rend="italics">r</hi><hi rend="sup">3</hi> + <hi rend="italics">r</hi><hi rend="sup">4</hi> to <hi rend="italics">r</hi><hi rend="sup">n-x</hi>, continued till the last term be <hi rend="italics">r</hi><hi rend="sup">n-x</hi>, or till the number of terms be <hi rend="italics">n,</hi> the number of years the annuity is forborn. But the sum of this geometrical progression is (<hi rend="italics">r</hi><hi rend="sup">n</hi>-1)/(<hi rend="italics">r</hi>-1), <cb/> which therefore is the amount of 1<hi rend="italics">l.</hi> annuity forborn for <hi rend="italics">n</hi> years. And this quantity being multiplied by any other annuity <hi rend="italics">a,</hi> instead of 1, will produce the amount for that other annuity.</p><p>But the amounts of annuities, or their present values, are easiest found by the two following tables of numbers for the annuity of 1<hi rend="italics">l.</hi> ready computed from the foregoing principles. <table rend="border"><head><hi rend="smallcaps">Table</hi> I.</head><head><hi rend="italics">The Amount of an Annuity of</hi> 1<hi rend="italics">l. at Comp. Interest.</hi></head><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Yrs.</cell><cell cols="1" rows="1" rend="align=center" role="data">at 3 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">3 1/2 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 1/2 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">5 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">6 per cent.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">2.03000</cell><cell cols="1" rows="1" rend="align=right" role="data">2.03500</cell><cell cols="1" rows="1" rend="align=right" role="data">2.04000</cell><cell cols="1" rows="1" rend="align=right" role="data">2.04500</cell><cell cols="1" rows="1" rend="align=right" role="data">2.05000</cell><cell cols="1" rows="1" rend="align=right" role="data">2.06000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">3.09090</cell><cell cols="1" rows="1" rend="align=right" role="data">3.10623</cell><cell cols="1" rows="1" rend="align=right" role="data">3.12160</cell><cell cols="1" rows="1" rend="align=right" role="data">3.13703</cell><cell cols="1" rows="1" rend="align=right" role="data">3.15250</cell><cell cols="1" rows="1" rend="align=right" role="data">3.18360</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4.18363</cell><cell cols="1" rows="1" rend="align=right" role="data">4.21494</cell><cell cols="1" rows="1" rend="align=right" role="data">4.24646</cell><cell cols="1" rows="1" rend="align=right" role="data">4.27819</cell><cell cols="1" rows="1" rend="align=right" role="data">4.31013</cell><cell cols="1" rows="1" rend="align=right" role="data">4.37462</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">5.30914</cell><cell cols="1" rows="1" rend="align=right" role="data">5.36247</cell><cell cols="1" rows="1" rend="align=right" role="data">5.41632</cell><cell cols="1" rows="1" rend="align=right" role="data">5.47071</cell><cell cols="1" rows="1" rend="align=right" role="data">5.52563</cell><cell cols="1" rows="1" rend="align=right" role="data">5.63709</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">6.46841</cell><cell cols="1" rows="1" rend="align=right" role="data">6.55015</cell><cell cols="1" rows="1" rend="align=right" role="data">6.63298</cell><cell cols="1" rows="1" rend="align=right" role="data">6.71689</cell><cell cols="1" rows="1" rend="align=right" role="data">6.80191</cell><cell cols="1" rows="1" rend="align=right" role="data">6.97532</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7.66246</cell><cell cols="1" rows="1" rend="align=right" role="data">7.77941</cell><cell cols="1" rows="1" rend="align=right" role="data">7.89829</cell><cell cols="1" rows="1" rend="align=right" role="data">8.01915</cell><cell cols="1" rows="1" rend="align=right" role="data">8.14201</cell><cell cols="1" rows="1" rend="align=right" role="data">8.39384</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">8.89234</cell><cell cols="1" rows="1" rend="align=right" role="data">9.05169</cell><cell cols="1" rows="1" rend="align=right" role="data">9.21423</cell><cell cols="1" rows="1" rend="align=right" role="data">9.38001</cell><cell cols="1" rows="1" rend="align=right" role="data">9.54911</cell><cell cols="1" rows="1" rend="align=right" role="data">9.89747</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">10.15911</cell><cell cols="1" rows="1" rend="align=right" role="data">10.36850</cell><cell cols="1" rows="1" rend="align=right" role="data">10.58280</cell><cell cols="1" rows="1" rend="align=right" role="data">10.80211</cell><cell cols="1" rows="1" rend="align=right" role="data">11.02656</cell><cell cols="1" rows="1" rend="align=right" role="data">11.49132</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">11.46388</cell><cell cols="1" rows="1" rend="align=right" role="data">11.73139</cell><cell cols="1" rows="1" rend="align=right" role="data">12.00611</cell><cell cols="1" rows="1" rend="align=right" role="data">12.28821</cell><cell cols="1" rows="1" rend="align=right" role="data">12.57789</cell><cell cols="1" rows="1" rend="align=right" role="data">13.18079</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">12.80780</cell><cell cols="1" rows="1" rend="align=right" role="data">13.14199</cell><cell cols="1" rows="1" rend="align=right" role="data">13.48635</cell><cell cols="1" rows="1" rend="align=right" role="data">13.84118</cell><cell cols="1" rows="1" rend="align=right" role="data">14.20679</cell><cell cols="1" rows="1" rend="align=right" role="data">14.97164</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">14.19203</cell><cell cols="1" rows="1" rend="align=right" role="data">14.60196</cell><cell cols="1" rows="1" rend="align=right" role="data">15.02581</cell><cell cols="1" rows="1" rend="align=right" role="data">15.46403</cell><cell cols="1" rows="1" rend="align=right" role="data">15.91713</cell><cell cols="1" rows="1" rend="align=right" role="data">16.86994</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">15.61779</cell><cell cols="1" rows="1" rend="align=right" role="data">16.11303</cell><cell cols="1" rows="1" rend="align=right" role="data">16.62684</cell><cell cols="1" rows="1" rend="align=right" role="data">17.15991</cell><cell cols="1" rows="1" rend="align=right" role="data">17.71298</cell><cell cols="1" rows="1" rend="align=right" role="data">18.88214</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">17.08632</cell><cell cols="1" rows="1" rend="align=right" role="data">17.67699</cell><cell cols="1" rows="1" rend="align=right" role="data">18.29191</cell><cell cols="1" rows="1" rend="align=right" role="data">18.93211</cell><cell cols="1" rows="1" rend="align=right" role="data">19.59863</cell><cell cols="1" rows="1" rend="align=right" role="data">21.01507</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">18.59891</cell><cell cols="1" rows="1" rend="align=right" role="data">19.29568</cell><cell cols="1" rows="1" rend="align=right" role="data">20.32359</cell><cell cols="1" rows="1" rend="align=right" role="data">20.78405</cell><cell cols="1" rows="1" rend="align=right" role="data">21.57856</cell><cell cols="1" rows="1" rend="align=right" role="data">23.27597</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">20.15688</cell><cell cols="1" rows="1" rend="align=right" role="data">20.97103</cell><cell cols="1" rows="1" rend="align=right" role="data">21.82453</cell><cell cols="1" rows="1" rend="align=right" role="data">22.71934</cell><cell cols="1" rows="1" rend="align=right" role="data">23.65749</cell><cell cols="1" rows="1" rend="align=right" role="data">25.67253</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">21.76159</cell><cell cols="1" rows="1" rend="align=right" role="data">22.70502</cell><cell cols="1" rows="1" rend="align=right" role="data">23.69751</cell><cell cols="1" rows="1" rend="align=right" role="data">24.74171</cell><cell cols="1" rows="1" rend="align=right" role="data">25.84037</cell><cell cols="1" rows="1" rend="align=right" role="data">28.21288</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">23.41444</cell><cell cols="1" rows="1" rend="align=right" role="data">24.49969</cell><cell cols="1" rows="1" rend="align=right" role="data">25.64541</cell><cell cols="1" rows="1" rend="align=right" role="data">26.85508</cell><cell cols="1" rows="1" rend="align=right" role="data">28.13238</cell><cell cols="1" rows="1" rend="align=right" role="data">30.90565</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">25.11687</cell><cell cols="1" rows="1" rend="align=right" role="data">26.35718</cell><cell cols="1" rows="1" rend="align=right" role="data">27.67123</cell><cell cols="1" rows="1" rend="align=right" role="data">29.06356</cell><cell cols="1" rows="1" rend="align=right" role="data">30.53900</cell><cell cols="1" rows="1" rend="align=right" role="data">33.75999</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">26.87037</cell><cell cols="1" rows="1" rend="align=right" role="data">28.27968</cell><cell cols="1" rows="1" rend="align=right" role="data">29.77808</cell><cell cols="1" rows="1" rend="align=right" role="data">31.37142</cell><cell cols="1" rows="1" rend="align=right" role="data">33.06595</cell><cell cols="1" rows="1" rend="align=right" role="data">36.78559</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">28.67649</cell><cell cols="1" rows="1" rend="align=right" role="data">30.26947</cell><cell cols="1" rows="1" rend="align=right" role="data">31.96920</cell><cell cols="1" rows="1" rend="align=right" role="data">33.78314</cell><cell cols="1" rows="1" rend="align=right" role="data">35.71925</cell><cell cols="1" rows="1" rend="align=right" role="data">39.99273</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">30.53678</cell><cell cols="1" rows="1" rend="align=right" role="data">32.32890</cell><cell cols="1" rows="1" rend="align=right" role="data">34.24797</cell><cell cols="1" rows="1" rend="align=right" role="data">36.30338</cell><cell cols="1" rows="1" rend="align=right" role="data">38.50521</cell><cell cols="1" rows="1" rend="align=right" role="data">43.39229</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">32.45288</cell><cell cols="1" rows="1" rend="align=right" role="data">34.46041</cell><cell cols="1" rows="1" rend="align=right" role="data">36.61789</cell><cell cols="1" rows="1" rend="align=right" role="data">38.93703</cell><cell cols="1" rows="1" rend="align=right" role="data">41.43048</cell><cell cols="1" rows="1" rend="align=right" role="data">46.99583</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">34.42647</cell><cell cols="1" rows="1" rend="align=right" role="data">36.66653</cell><cell cols="1" rows="1" rend="align=right" role="data">39.08260</cell><cell cols="1" rows="1" rend="align=right" role="data">41.68920</cell><cell cols="1" rows="1" rend="align=right" role="data">44.50200</cell><cell cols="1" rows="1" rend="align=right" role="data">50.81558</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">36.45926</cell><cell cols="1" rows="1" rend="align=right" role="data">38.94986</cell><cell cols="1" rows="1" rend="align=right" role="data">41.64591</cell><cell cols="1" rows="1" rend="align=right" role="data">44.56521</cell><cell cols="1" rows="1" rend="align=right" role="data">47.72710</cell><cell cols="1" rows="1" rend="align=right" role="data">54.86451</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">38.55304</cell><cell cols="1" rows="1" rend="align=right" role="data">41.31310</cell><cell cols="1" rows="1" rend="align=right" role="data">44.31174</cell><cell cols="1" rows="1" rend="align=right" role="data">47.57064</cell><cell cols="1" rows="1" rend="align=right" role="data">51.11345</cell><cell cols="1" rows="1" rend="align=right" role="data">59.15638</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">40.70963</cell><cell cols="1" rows="1" rend="align=right" role="data">43.75906</cell><cell cols="1" rows="1" rend="align=right" role="data">47.08421</cell><cell cols="1" rows="1" rend="align=right" role="data">50.71132</cell><cell cols="1" rows="1" rend="align=right" role="data">54.66913</cell><cell cols="1" rows="1" rend="align=right" role="data">63.70577</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">42.93092</cell><cell cols="1" rows="1" rend="align=right" role="data">46.29063</cell><cell cols="1" rows="1" rend="align=right" role="data">49.96758</cell><cell cols="1" rows="1" rend="align=right" role="data">53.99333</cell><cell cols="1" rows="1" rend="align=right" role="data">58.40258</cell><cell cols="1" rows="1" rend="align=right" role="data">68.52811</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">45.21885</cell><cell cols="1" rows="1" rend="align=right" role="data">48.91080</cell><cell cols="1" rows="1" rend="align=right" role="data">52.96629</cell><cell cols="1" rows="1" rend="align=right" role="data">57.42303</cell><cell cols="1" rows="1" rend="align=right" role="data">62.32271</cell><cell cols="1" rows="1" rend="align=right" role="data">73.63980</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">47.57542</cell><cell cols="1" rows="1" rend="align=right" role="data">51.62268</cell><cell cols="1" rows="1" rend="align=right" role="data">56.08494</cell><cell cols="1" rows="1" rend="align=right" role="data">61.00707</cell><cell cols="1" rows="1" rend="align=right" role="data">66.43885</cell><cell cols="1" rows="1" rend="align=right" role="data">79.05819</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">50.00268</cell><cell cols="1" rows="1" rend="align=right" role="data">54.42947</cell><cell cols="1" rows="1" rend="align=right" role="data">59.32834</cell><cell cols="1" rows="1" rend="align=right" role="data">64.75239</cell><cell cols="1" rows="1" rend="align=right" role="data">70.76079</cell><cell cols="1" rows="1" rend="align=right" role="data">84.80168</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">52.50276</cell><cell cols="1" rows="1" rend="align=right" role="data">57.33450</cell><cell cols="1" rows="1" rend="align=right" role="data">62.70147</cell><cell cols="1" rows="1" rend="align=right" role="data">68.66625</cell><cell cols="1" rows="1" rend="align=right" role="data">75.29883</cell><cell cols="1" rows="1" rend="align=right" role="data">90.88978</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">55.07784</cell><cell cols="1" rows="1" rend="align=right" role="data">60.34121</cell><cell cols="1" rows="1" rend="align=right" role="data">66.20953</cell><cell cols="1" rows="1" rend="align=right" role="data">72.75623</cell><cell cols="1" rows="1" rend="align=right" role="data">80.06377</cell><cell cols="1" rows="1" rend="align=right" role="data">97.34316</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">57.73018</cell><cell cols="1" rows="1" rend="align=right" role="data">63.45315</cell><cell cols="1" rows="1" rend="align=right" role="data">69.85791</cell><cell cols="1" rows="1" rend="align=right" role="data">77.03026</cell><cell cols="1" rows="1" rend="align=right" role="data">85.06696</cell><cell cols="1" rows="1" rend="align=right" role="data">104.18375</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">60.46208</cell><cell cols="1" rows="1" rend="align=right" role="data">66.67401</cell><cell cols="1" rows="1" rend="align=right" role="data">73.65222</cell><cell cols="1" rows="1" rend="align=right" role="data">81.49662</cell><cell cols="1" rows="1" rend="align=right" role="data">90.32031</cell><cell cols="1" rows="1" rend="align=right" role="data">111.43478</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">63.27594</cell><cell cols="1" rows="1" rend="align=right" role="data">70.00760</cell><cell cols="1" rows="1" rend="align=right" role="data">77.59831</cell><cell cols="1" rows="1" rend="align=right" role="data">86.16397</cell><cell cols="1" rows="1" rend="align=right" role="data">95.83632</cell><cell cols="1" rows="1" rend="align=right" role="data">119.12087</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">66.17422</cell><cell cols="1" rows="1" rend="align=right" role="data">73.45787</cell><cell cols="1" rows="1" rend="align=right" role="data">81.70225</cell><cell cols="1" rows="1" rend="align=right" role="data">91.04134</cell><cell cols="1" rows="1" rend="align=right" role="data">101.62814</cell><cell cols="1" rows="1" rend="align=right" role="data">127.26812</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">69.15945</cell><cell cols="1" rows="1" rend="align=right" role="data">77.02889</cell><cell cols="1" rows="1" rend="align=right" role="data">85.97034</cell><cell cols="1" rows="1" rend="align=right" role="data">96.13820</cell><cell cols="1" rows="1" rend="align=right" role="data">107.70955</cell><cell cols="1" rows="1" rend="align=right" role="data">135.90421</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">72.23423</cell><cell cols="1" rows="1" rend="align=right" role="data">80.72491</cell><cell cols="1" rows="1" rend="align=right" role="data">90.40915</cell><cell cols="1" rows="1" rend="align=right" role="data">101.46442</cell><cell cols="1" rows="1" rend="align=right" role="data">114.09502</cell><cell cols="1" rows="1" rend="align=right" role="data">145.05846</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">75.40126</cell><cell cols="1" rows="1" rend="align=right" role="data">84.55028</cell><cell cols="1" rows="1" rend="align=right" role="data">95.02552</cell><cell cols="1" rows="1" rend="align=right" role="data">107.03032</cell><cell cols="1" rows="1" rend="align=right" role="data">120.79977</cell><cell cols="1" rows="1" rend="align=right" role="data">154.76197</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">78.66330</cell><cell cols="1" rows="1" rend="align=right" role="data">88.50954</cell><cell cols="1" rows="1" rend="align=right" role="data">99.82654</cell><cell cols="1" rows="1" rend="align=right" role="data">112.84669</cell><cell cols="1" rows="1" rend="align=right" role="data">127.83976</cell><cell cols="1" rows="1" rend="align=right" role="data">165.04768</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">82.02320</cell><cell cols="1" rows="1" rend="align=right" role="data">92.60737</cell><cell cols="1" rows="1" rend="align=right" role="data">104.81960</cell><cell cols="1" rows="1" rend="align=right" role="data">118.92479</cell><cell cols="1" rows="1" rend="align=right" role="data">135.23175</cell><cell cols="1" rows="1" rend="align=right" role="data">175.95054</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">85.48389</cell><cell cols="1" rows="1" rend="align=right" role="data">96.84863</cell><cell cols="1" rows="1" rend="align=right" role="data">110.01238</cell><cell cols="1" rows="1" rend="align=right" role="data">125.27640</cell><cell cols="1" rows="1" rend="align=right" role="data">142.99334</cell><cell cols="1" rows="1" rend="align=right" role="data">187.50758</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">89.04841</cell><cell cols="1" rows="1" rend="align=right" role="data">101.23833</cell><cell cols="1" rows="1" rend="align=right" role="data">115.41288</cell><cell cols="1" rows="1" rend="align=right" role="data">131.91384</cell><cell cols="1" rows="1" rend="align=right" role="data">151.14301</cell><cell cols="1" rows="1" rend="align=right" role="data">199.75803</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">92.71986</cell><cell cols="1" rows="1" rend="align=right" role="data">105.78167</cell><cell cols="1" rows="1" rend="align=right" role="data">121.02939</cell><cell cols="1" rows="1" rend="align=right" role="data">138.84997</cell><cell cols="1" rows="1" rend="align=right" role="data">159.70016</cell><cell cols="1" rows="1" rend="align=right" role="data">212.74351</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">96.50146</cell><cell cols="1" rows="1" rend="align=right" role="data">110.48403</cell><cell cols="1" rows="1" rend="align=right" role="data">126.87057</cell><cell cols="1" rows="1" rend="align=right" role="data">146.09821</cell><cell cols="1" rows="1" rend="align=right" role="data">168.68516</cell><cell cols="1" rows="1" rend="align=right" role="data">226.50812</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">100.39650</cell><cell cols="1" rows="1" rend="align=right" role="data">115.35097</cell><cell cols="1" rows="1" rend="align=right" role="data">132.94539</cell><cell cols="1" rows="1" rend="align=right" role="data">153.67263</cell><cell cols="1" rows="1" rend="align=right" role="data">178.11942</cell><cell cols="1" rows="1" rend="align=right" role="data">241.09861</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">104.40840</cell><cell cols="1" rows="1" rend="align=right" role="data">120.38826</cell><cell cols="1" rows="1" rend="align=right" role="data">139.26321</cell><cell cols="1" rows="1" rend="align=right" role="data">161.58790</cell><cell cols="1" rows="1" rend="align=right" role="data">188.02539</cell><cell cols="1" rows="1" rend="align=right" role="data">256.56453</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">108.54065</cell><cell cols="1" rows="1" rend="align=right" role="data">125.60185</cell><cell cols="1" rows="1" rend="align=right" role="data">145.83373</cell><cell cols="1" rows="1" rend="align=right" role="data">169.85936</cell><cell cols="1" rows="1" rend="align=right" role="data">198.42666</cell><cell cols="1" rows="1" rend="align=right" role="data">272.95840</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">112.79687</cell><cell cols="1" rows="1" rend="align=right" role="data">130.99791</cell><cell cols="1" rows="1" rend="align=right" role="data">152.66708</cell><cell cols="1" rows="1" rend="align=right" role="data">178.50303</cell><cell cols="1" rows="1" rend="align=right" role="data">209.34800</cell><cell cols="1" rows="1" rend="align=right" role="data">290.33590</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">117.18077</cell><cell cols="1" rows="1" rend="align=right" role="data">136.58284</cell><cell cols="1" rows="1" rend="align=right" role="data">159.77377</cell><cell cols="1" rows="1" rend="align=right" role="data">187.53566</cell><cell cols="1" rows="1" rend="align=right" role="data">220.81540</cell><cell cols="1" rows="1" rend="align=right" role="data">308.75606</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">121.69620</cell><cell cols="1" rows="1" rend="align=right" role="data">142.36324</cell><cell cols="1" rows="1" rend="align=right" role="data">167.16472</cell><cell cols="1" rows="1" rend="align=right" role="data">196.97477</cell><cell cols="1" rows="1" rend="align=right" role="data">232.85617</cell><cell cols="1" rows="1" rend="align=right" role="data">328.28142</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">126.34708</cell><cell cols="1" rows="1" rend="align=right" role="data">148.34595</cell><cell cols="1" rows="1" rend="align=right" role="data">174.85131</cell><cell cols="1" rows="1" rend="align=right" role="data">206.83863</cell><cell cols="1" rows="1" rend="align=right" role="data">245.49897</cell><cell cols="1" rows="1" rend="align=right" role="data">348.97831</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">131.13750</cell><cell cols="1" rows="1" rend="align=right" role="data">154.53806</cell><cell cols="1" rows="1" rend="align=right" role="data">182.84536</cell><cell cols="1" rows="1" rend="align=right" role="data">217.14637</cell><cell cols="1" rows="1" rend="align=right" role="data">258.77392</cell><cell cols="1" rows="1" rend="align=right" role="data">370.91701</cell></row></table> <pb n="119"/><cb/> <table rend="border"><head><hi rend="smallcaps">Table</hi> II.</head><head><hi rend="italics">The present Value of an Annuity of</hi> 1<hi rend="italics">l.</hi></head><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Yrs.</cell><cell cols="1" rows="1" rend="align=center" role="data">at 3 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">3 1/2 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 1/2 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">5 per cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">6 per cent.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0.97087</cell><cell cols="1" rows="1" rend="align=right" role="data">0.96618</cell><cell cols="1" rows="1" rend="align=right" role="data">0.96154</cell><cell cols="1" rows="1" rend="align=right" role="data">0.95694</cell><cell cols="1" rows="1" rend="align=right" role="data">0.95238</cell><cell cols="1" rows="1" rend="align=right" role="data">0.94340</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">1.91347</cell><cell cols="1" rows="1" rend="align=right" role="data">1.89969</cell><cell cols="1" rows="1" rend="align=right" role="data">1.88610</cell><cell cols="1" rows="1" rend="align=right" role="data">1.87267</cell><cell cols="1" rows="1" rend="align=right" role="data">1.85941</cell><cell cols="1" rows="1" rend="align=right" role="data">1.83339</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">2.82861</cell><cell cols="1" rows="1" rend="align=right" role="data">2.80164</cell><cell cols="1" rows="1" rend="align=right" role="data">2.77509</cell><cell cols="1" rows="1" rend="align=right" role="data">2.74896</cell><cell cols="1" rows="1" rend="align=right" role="data">2.72325</cell><cell cols="1" rows="1" rend="align=right" role="data">2.67301</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3.71710</cell><cell cols="1" rows="1" rend="align=right" role="data">3.67308</cell><cell cols="1" rows="1" rend="align=right" role="data">3.62990</cell><cell cols="1" rows="1" rend="align=right" role="data">3.58753</cell><cell cols="1" rows="1" rend="align=right" role="data">3.54595</cell><cell cols="1" rows="1" rend="align=right" role="data">3.46511</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">4.57971</cell><cell cols="1" rows="1" rend="align=right" role="data">4.51505</cell><cell cols="1" rows="1" rend="align=right" role="data">4.45182</cell><cell cols="1" rows="1" rend="align=right" role="data">4.38998</cell><cell cols="1" rows="1" rend="align=right" role="data">4.32948</cell><cell cols="1" rows="1" rend="align=right" role="data">4.21236</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5.41719</cell><cell cols="1" rows="1" rend="align=right" role="data">5.32855</cell><cell cols="1" rows="1" rend="align=right" role="data">5.24214</cell><cell cols="1" rows="1" rend="align=right" role="data">5.15787</cell><cell cols="1" rows="1" rend="align=right" role="data">5.07569</cell><cell cols="1" rows="1" rend="align=right" role="data">4.91732</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">6.23028</cell><cell cols="1" rows="1" rend="align=right" role="data">6.11454</cell><cell cols="1" rows="1" rend="align=right" role="data">6.00205</cell><cell cols="1" rows="1" rend="align=right" role="data">5.89270</cell><cell cols="1" rows="1" rend="align=right" role="data">5.78637</cell><cell cols="1" rows="1" rend="align=right" role="data">5.58238</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">7.01969</cell><cell cols="1" rows="1" rend="align=right" role="data">6.87396</cell><cell cols="1" rows="1" rend="align=right" role="data">6.73274</cell><cell cols="1" rows="1" rend="align=right" role="data">6.59589</cell><cell cols="1" rows="1" rend="align=right" role="data">6.46321</cell><cell cols="1" rows="1" rend="align=right" role="data">6.20979</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">7.78611</cell><cell cols="1" rows="1" rend="align=right" role="data">7.60769</cell><cell cols="1" rows="1" rend="align=right" role="data">7.43533</cell><cell cols="1" rows="1" rend="align=right" role="data">7.26879</cell><cell cols="1" rows="1" rend="align=right" role="data">7.10782</cell><cell cols="1" rows="1" rend="align=right" role="data">6.80169</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">8.53020</cell><cell cols="1" rows="1" rend="align=right" role="data">8.31661</cell><cell cols="1" rows="1" rend="align=right" role="data">8.11090</cell><cell cols="1" rows="1" rend="align=right" role="data">7.91272</cell><cell cols="1" rows="1" rend="align=right" role="data">7.72173</cell><cell cols="1" rows="1" rend="align=right" role="data">7.36009</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">9.25262</cell><cell cols="1" rows="1" rend="align=right" role="data">9.00155</cell><cell cols="1" rows="1" rend="align=right" role="data">8.76048</cell><cell cols="1" rows="1" rend="align=right" role="data">8.52892</cell><cell cols="1" rows="1" rend="align=right" role="data">8.30541</cell><cell cols="1" rows="1" rend="align=right" role="data">7.88687</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">9.95400</cell><cell cols="1" rows="1" rend="align=right" role="data">9.66333</cell><cell cols="1" rows="1" rend="align=right" role="data">9.38507</cell><cell cols="1" rows="1" rend="align=right" role="data">9.11858</cell><cell cols="1" rows="1" rend="align=right" role="data">8.86325</cell><cell cols="1" rows="1" rend="align=right" role="data">8.38384</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">10.63496</cell><cell cols="1" rows="1" rend="align=right" role="data">10.30274</cell><cell cols="1" rows="1" rend="align=right" role="data">9.98565</cell><cell cols="1" rows="1" rend="align=right" role="data">9.68285</cell><cell cols="1" rows="1" rend="align=right" role="data">9.39357</cell><cell cols="1" rows="1" rend="align=right" role="data">8.85268</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">11.29607</cell><cell cols="1" rows="1" rend="align=right" role="data">10.92052</cell><cell cols="1" rows="1" rend="align=right" role="data">10.56312</cell><cell cols="1" rows="1" rend="align=right" role="data">10.22283</cell><cell cols="1" rows="1" rend="align=right" role="data">9.89864</cell><cell cols="1" rows="1" rend="align=right" role="data">9.29498</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">11.93794</cell><cell cols="1" rows="1" rend="align=right" role="data">11.51741</cell><cell cols="1" rows="1" rend="align=right" role="data">11.11839</cell><cell cols="1" rows="1" rend="align=right" role="data">10.73955</cell><cell cols="1" rows="1" rend="align=right" role="data">10.37966</cell><cell cols="1" rows="1" rend="align=right" role="data">9.71225</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">12.56110</cell><cell cols="1" rows="1" rend="align=right" role="data">12.09412</cell><cell cols="1" rows="1" rend="align=right" role="data">11.65230</cell><cell cols="1" rows="1" rend="align=right" role="data">11.23402</cell><cell cols="1" rows="1" rend="align=right" role="data">10.83777</cell><cell cols="1" rows="1" rend="align=right" role="data">10.10590</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">13.16612</cell><cell cols="1" rows="1" rend="align=right" role="data">12.65132</cell><cell cols="1" rows="1" rend="align=right" role="data">12.16567</cell><cell cols="1" rows="1" rend="align=right" role="data">11.70719</cell><cell cols="1" rows="1" rend="align=right" role="data">11.27407</cell><cell cols="1" rows="1" rend="align=right" role="data">10.47726</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">13.75351</cell><cell cols="1" rows="1" rend="align=right" role="data">13.18968</cell><cell cols="1" rows="1" rend="align=right" role="data">12.65930</cell><cell cols="1" rows="1" rend="align=right" role="data">12.15999</cell><cell cols="1" rows="1" rend="align=right" role="data">11.68959</cell><cell cols="1" rows="1" rend="align=right" role="data">10.82760</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">14.32380</cell><cell cols="1" rows="1" rend="align=right" role="data">13.70984</cell><cell cols="1" rows="1" rend="align=right" role="data">13.13394</cell><cell cols="1" rows="1" rend="align=right" role="data">12.59329</cell><cell cols="1" rows="1" rend="align=right" role="data">12.08532</cell><cell cols="1" rows="1" rend="align=right" role="data">11.15812</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">14.87747</cell><cell cols="1" rows="1" rend="align=right" role="data">14.21240</cell><cell cols="1" rows="1" rend="align=right" role="data">13.59033</cell><cell cols="1" rows="1" rend="align=right" role="data">13.00794</cell><cell cols="1" rows="1" rend="align=right" role="data">12.46221</cell><cell cols="1" rows="1" rend="align=right" role="data">11.46992</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">15.41502</cell><cell cols="1" rows="1" rend="align=right" role="data">14.69797</cell><cell cols="1" rows="1" rend="align=right" role="data">14.02916</cell><cell cols="1" rows="1" rend="align=right" role="data">13.40472</cell><cell cols="1" rows="1" rend="align=right" role="data">12.82115</cell><cell cols="1" rows="1" rend="align=right" role="data">11.76408</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">15.93692</cell><cell cols="1" rows="1" rend="align=right" role="data">15.16712</cell><cell cols="1" rows="1" rend="align=right" role="data">14.45112</cell><cell cols="1" rows="1" rend="align=right" role="data">13.78442</cell><cell cols="1" rows="1" rend="align=right" role="data">13.16300</cell><cell cols="1" rows="1" rend="align=right" role="data">12.04158</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">16.44361</cell><cell cols="1" rows="1" rend="align=right" role="data">15.62041</cell><cell cols="1" rows="1" rend="align=right" role="data">14.85684</cell><cell cols="1" rows="1" rend="align=right" role="data">14.14777</cell><cell cols="1" rows="1" rend="align=right" role="data">13.48857</cell><cell cols="1" rows="1" rend="align=right" role="data">12.30338</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">16.93554</cell><cell cols="1" rows="1" rend="align=right" role="data">16.05837</cell><cell cols="1" rows="1" rend="align=right" role="data">15.24696</cell><cell cols="1" rows="1" rend="align=right" role="data">14.49548</cell><cell cols="1" rows="1" rend="align=right" role="data">13.79864</cell><cell cols="1" rows="1" rend="align=right" role="data">12.55036</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">17.41315</cell><cell cols="1" rows="1" rend="align=right" role="data">16.48151</cell><cell cols="1" rows="1" rend="align=right" role="data">15.62208</cell><cell cols="1" rows="1" rend="align=right" role="data">14.82821</cell><cell cols="1" rows="1" rend="align=right" role="data">14.09394</cell><cell cols="1" rows="1" rend="align=right" role="data">12.78336</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">17.87684</cell><cell cols="1" rows="1" rend="align=right" role="data">16.89035</cell><cell cols="1" rows="1" rend="align=right" role="data">15.98277</cell><cell cols="1" rows="1" rend="align=right" role="data">15.14661</cell><cell cols="1" rows="1" rend="align=right" role="data">14.37519</cell><cell cols="1" rows="1" rend="align=right" role="data">13.00317</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">18.32703</cell><cell cols="1" rows="1" rend="align=right" role="data">17.28536</cell><cell cols="1" rows="1" rend="align=right" role="data">16.32959</cell><cell cols="1" rows="1" rend="align=right" role="data">15.45130</cell><cell cols="1" rows="1" rend="align=right" role="data">14.64303</cell><cell cols="1" rows="1" rend="align=right" role="data">13.21053</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">18.76411</cell><cell cols="1" rows="1" rend="align=right" role="data">17.66702</cell><cell cols="1" rows="1" rend="align=right" role="data">16.66306</cell><cell cols="1" rows="1" rend="align=right" role="data">15.74287</cell><cell cols="1" rows="1" rend="align=right" role="data">14.89813</cell><cell cols="1" rows="1" rend="align=right" role="data">13.40616</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">19.18845</cell><cell cols="1" rows="1" rend="align=right" role="data">18.03577</cell><cell cols="1" rows="1" rend="align=right" role="data">16.98371</cell><cell cols="1" rows="1" rend="align=right" role="data">16.02189</cell><cell cols="1" rows="1" rend="align=right" role="data">15.14107</cell><cell cols="1" rows="1" rend="align=right" role="data">13.59072</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">19.60044</cell><cell cols="1" rows="1" rend="align=right" role="data">18.39205</cell><cell cols="1" rows="1" rend="align=right" role="data">17.29203</cell><cell cols="1" rows="1" rend="align=right" role="data">16.28889</cell><cell cols="1" rows="1" rend="align=right" role="data">15.37245</cell><cell cols="1" rows="1" rend="align=right" role="data">13.76483</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">20.00043</cell><cell cols="1" rows="1" rend="align=right" role="data">18.73628</cell><cell cols="1" rows="1" rend="align=right" role="data">17.58849</cell><cell cols="1" rows="1" rend="align=right" role="data">16.54439</cell><cell cols="1" rows="1" rend="align=right" role="data">15.59281</cell><cell cols="1" rows="1" rend="align=right" role="data">13.92909</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">20.38877</cell><cell cols="1" rows="1" rend="align=right" role="data">19.06887</cell><cell cols="1" rows="1" rend="align=right" role="data">17.87355</cell><cell cols="1" rows="1" rend="align=right" role="data">16.78889</cell><cell cols="1" rows="1" rend="align=right" role="data">15.80268</cell><cell cols="1" rows="1" rend="align=right" role="data">14.08404</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">20.76579</cell><cell cols="1" rows="1" rend="align=right" role="data">19.39021</cell><cell cols="1" rows="1" rend="align=right" role="data">18.14765</cell><cell cols="1" rows="1" rend="align=right" role="data">17.02286</cell><cell cols="1" rows="1" rend="align=right" role="data">16.00255</cell><cell cols="1" rows="1" rend="align=right" role="data">14.23023</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">21.13184</cell><cell cols="1" rows="1" rend="align=right" role="data">19.70068</cell><cell cols="1" rows="1" rend="align=right" role="data">18.41120</cell><cell cols="1" rows="1" rend="align=right" role="data">17.24676</cell><cell cols="1" rows="1" rend="align=right" role="data">16.19290</cell><cell cols="1" rows="1" rend="align=right" role="data">14.36814</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">21.48722</cell><cell cols="1" rows="1" rend="align=right" role="data">20.00066</cell><cell cols="1" rows="1" rend="align=right" role="data">18.66461</cell><cell cols="1" rows="1" rend="align=right" role="data">17.46101</cell><cell cols="1" rows="1" rend="align=right" role="data">16.37419</cell><cell cols="1" rows="1" rend="align=right" role="data">14.49825</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">21.83225</cell><cell cols="1" rows="1" rend="align=right" role="data">20.29049</cell><cell cols="1" rows="1" rend="align=right" role="data">18.90828</cell><cell cols="1" rows="1" rend="align=right" role="data">17.66604</cell><cell cols="1" rows="1" rend="align=right" role="data">16.54685</cell><cell cols="1" rows="1" rend="align=right" role="data">14.62099</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">22.16724</cell><cell cols="1" rows="1" rend="align=right" role="data">20.<*>7053</cell><cell cols="1" rows="1" rend="align=right" role="data">19.14258</cell><cell cols="1" rows="1" rend="align=right" role="data">17.86224</cell><cell cols="1" rows="1" rend="align=right" role="data">16.71129</cell><cell cols="1" rows="1" rend="align=right" role="data">14.73678</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">22.49246</cell><cell cols="1" rows="1" rend="align=right" role="data">20.84109</cell><cell cols="1" rows="1" rend="align=right" role="data">19.36786</cell><cell cols="1" rows="1" rend="align=right" role="data">18.04999</cell><cell cols="1" rows="1" rend="align=right" role="data">16.86789</cell><cell cols="1" rows="1" rend="align=right" role="data">14.84602</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">22.80822</cell><cell cols="1" rows="1" rend="align=right" role="data">21.10250</cell><cell cols="1" rows="1" rend="align=right" role="data">19.58448</cell><cell cols="1" rows="1" rend="align=right" role="data">18.22966</cell><cell cols="1" rows="1" rend="align=right" role="data">17.01704</cell><cell cols="1" rows="1" rend="align=right" role="data">14.94907</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">23.11477</cell><cell cols="1" rows="1" rend="align=right" role="data">21.35507</cell><cell cols="1" rows="1" rend="align=right" role="data">19.79277</cell><cell cols="1" rows="1" rend="align=right" role="data">18.40158</cell><cell cols="1" rows="1" rend="align=right" role="data">17.15909</cell><cell cols="1" rows="1" rend="align=right" role="data">15.04630</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">23.41240</cell><cell cols="1" rows="1" rend="align=right" role="data">21.59910</cell><cell cols="1" rows="1" rend="align=right" role="data">19.99305</cell><cell cols="1" rows="1" rend="align=right" role="data">18.56611</cell><cell cols="1" rows="1" rend="align=right" role="data">17.29437</cell><cell cols="1" rows="1" rend="align=right" role="data">15.13802</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">23.70136</cell><cell cols="1" rows="1" rend="align=right" role="data">21.83488</cell><cell cols="1" rows="1" rend="align=right" role="data">20.18563</cell><cell cols="1" rows="1" rend="align=right" role="data">18.72355</cell><cell cols="1" rows="1" rend="align=right" role="data">17.42321</cell><cell cols="1" rows="1" rend="align=right" role="data">15.22454</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">23.98190</cell><cell cols="1" rows="1" rend="align=right" role="data">22.06269</cell><cell cols="1" rows="1" rend="align=right" role="data">20.37079</cell><cell cols="1" rows="1" rend="align=right" role="data">18.87421</cell><cell cols="1" rows="1" rend="align=right" role="data">17.54591</cell><cell cols="1" rows="1" rend="align=right" role="data">15.30617</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">24.25427</cell><cell cols="1" rows="1" rend="align=right" role="data">22.28279</cell><cell cols="1" rows="1" rend="align=right" role="data">20.54884</cell><cell cols="1" rows="1" rend="align=right" role="data">19.01838</cell><cell cols="1" rows="1" rend="align=right" role="data">17.66277</cell><cell cols="1" rows="1" rend="align=right" role="data">15.38318</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">24.51871</cell><cell cols="1" rows="1" rend="align=right" role="data">22.49545</cell><cell cols="1" rows="1" rend="align=right" role="data">20.72004</cell><cell cols="1" rows="1" rend="align=right" role="data">19.15635</cell><cell cols="1" rows="1" rend="align=right" role="data">17.77407</cell><cell cols="1" rows="1" rend="align=right" role="data">15.45583</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">24.77545</cell><cell cols="1" rows="1" rend="align=right" role="data">22.70092</cell><cell cols="1" rows="1" rend="align=right" role="data">20.88465</cell><cell cols="1" rows="1" rend="align=right" role="data">19.28837</cell><cell cols="1" rows="1" rend="align=right" role="data">17.88007</cell><cell cols="1" rows="1" rend="align=right" role="data">15.52437</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">25.02471</cell><cell cols="1" rows="1" rend="align=right" role="data">22.89944</cell><cell cols="1" rows="1" rend="align=right" role="data">21.04294</cell><cell cols="1" rows="1" rend="align=right" role="data">19.41471</cell><cell cols="1" rows="1" rend="align=right" role="data">17.98102</cell><cell cols="1" rows="1" rend="align=right" role="data">15.58903</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">25.26671</cell><cell cols="1" rows="1" rend="align=right" role="data">23.09124</cell><cell cols="1" rows="1" rend="align=right" role="data">21.19513</cell><cell cols="1" rows="1" rend="align=right" role="data">19.53561</cell><cell cols="1" rows="1" rend="align=right" role="data">18.07716</cell><cell cols="1" rows="1" rend="align=right" role="data">15.65003</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">25.50166</cell><cell cols="1" rows="1" rend="align=right" role="data">23.27656</cell><cell cols="1" rows="1" rend="align=right" role="data">21.34147</cell><cell cols="1" rows="1" rend="align=right" role="data">19.65130</cell><cell cols="1" rows="1" rend="align=right" role="data">18.16872</cell><cell cols="1" rows="1" rend="align=right" role="data">15.70757</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">25.72976</cell><cell cols="1" rows="1" rend="align=right" role="data">23.45562</cell><cell cols="1" rows="1" rend="align=right" role="data">21.48218</cell><cell cols="1" rows="1" rend="align=right" role="data">19.76201</cell><cell cols="1" rows="1" rend="align=right" role="data">18.25593</cell><cell cols="1" rows="1" rend="align=right" role="data">15.76186</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">25.95123</cell><cell cols="1" rows="1" rend="align=right" role="data">23.62862</cell><cell cols="1" rows="1" rend="align=right" role="data">21.61749</cell><cell cols="1" rows="1" rend="align=right" role="data">19.86795</cell><cell cols="1" rows="1" rend="align=right" role="data">18.33898</cell><cell cols="1" rows="1" rend="align=right" role="data">15.81308</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">26.16624</cell><cell cols="1" rows="1" rend="align=right" role="data">23.79576</cell><cell cols="1" rows="1" rend="align=right" role="data">21.74758</cell><cell cols="1" rows="1" rend="align=right" role="data">19.96933</cell><cell cols="1" rows="1" rend="align=right" role="data">18.41807</cell><cell cols="1" rows="1" rend="align=right" role="data">15.86139</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">26.37499</cell><cell cols="1" rows="1" rend="align=right" role="data">23.95726</cell><cell cols="1" rows="1" rend="align=right" role="data">21.87267</cell><cell cols="1" rows="1" rend="align=right" role="data">20.06634</cell><cell cols="1" rows="1" rend="align=right" role="data">18.49340</cell><cell cols="1" rows="1" rend="align=right" role="data">15.90697</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">26.57766</cell><cell cols="1" rows="1" rend="align=right" role="data">24.11330</cell><cell cols="1" rows="1" rend="align=right" role="data">21.99296</cell><cell cols="1" rows="1" rend="align=right" role="data">20.15918</cell><cell cols="1" rows="1" rend="align=right" role="data">18.56515</cell><cell cols="1" rows="1" rend="align=right" role="data">15.94998</cell></row></table> <hi rend="center"><hi rend="smallcaps">The Use of Table</hi> I.</hi></p><p><hi rend="italics">To find the Amount of an annuity forborn any number of years.</hi> Take out the amount from the 1st table, for the proposed years and rate of interest; then multiply it by the annuity in question; and the product will be its amount for the same number of years, and rate of interest.</p><p>And the converse to find the rate or time. <cb/></p><p><hi rend="italics">Exam.</hi> 1. To find how much an annuity of 50l. will amount to in 20 years at 3 1/2 per cent. compound interest.—On the line of 20 years, and in the column of 3 1/2 per cent, stands 28.27968, which is the amount of an annuity of 1l. for the 20 years; and therefore 28.27968 multiplied by 50, gives 1413.9841. or 1413l. 19s. 8d. for the answer.</p><p><hi rend="italics">Exam.</hi> 2. In what time will an annuity of 20l. amount to 1000l. at 4 per cent. compound interest?—Here the amount of 1000l. divided by 20l. the annuity, gives 50, the amount of 1l. annuity for the same time and rate. Then, the nearest tabular number in the column of 4 per cent. is 49.96758, which standing on the line of 28, shews that 28 years is the answer.</p><p><hi rend="italics">Exam.</hi> 3. If it be required to find at what rate of interest an annuity of 20l. will amount to 1000l. forborn for 28 years.—Here 1000 divided by 20 gives 50 as before. Then looking along the line of 28 years, for the nearest to this number 50, I find 49.96758 in the column of 4 per cent. which is therefore the rate of interest required. <hi rend="center"><hi rend="smallcaps">The Use of Table</hi> II.</hi></p><p><hi rend="italics">Exam.</hi> 1. To find the present value of an annuity of 50l. which is to continue 20 years, at 3 1/2 per cent.— By the table, the present value of 1l. for the same rate and time, is 14.21240; therefore 14.2124 X 50 = 710.62l. or 710l. 128. 4d. is the present value sought.</p><p><hi rend="italics">Exam.</hi> 2. To find the present value of an annuity of 20l. to commence 10 years hence, and then to continue for 40 years, or to terminate 50 years hence, at 4 per cent. interest.—In such cases as this, it is plain we have to sind the difference between the present values of two equal annuities, for the two given times; which therefore will be effected by subtracting the tabular value of the one term from that of the other, and multiplying by the annuity. Thus, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">tabular value for 50 years</cell><cell cols="1" rows="1" rend="align=right" role="data">21.48218</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">tabular value for 10 years</cell><cell cols="1" rows="1" rend="align=right" role="data">8.11090</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">the difference</cell><cell cols="1" rows="1" rend="align=right" role="data">13.37128</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">mult. by</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">gives</cell><cell cols="1" rows="1" rend="align=right" role="data">267.4256</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">2671. 8s. 6d.</cell><cell cols="1" rows="1" role="data">the answer.</cell></row></table></p><p>The foregoing observations, rules, and tables, contain all that is important in the doctrine of <hi rend="italics">annuities certain.</hi> And for farther information, reference may be had to arithmetical writings, particularly Malcolm's Arithmetic, page 595; Simpson's Algebra, sect. 16; Dodson's Mathematical Repository, page 298, &c; Jones's Synopsis, ch. 10; Philos. Trans. vol. lxvi, page 109.</p><p>For what relates to the doctrine of <hi rend="italics">annuities on lives,</hi> see <hi rend="smallcaps">Assurance, Complement, Expectation, Life Annuities, Reversions</hi>, &c.</p></div1><div1 part="N" n="ANNULETS" org="uniform" sample="complete" type="entry"><head>ANNULETS</head><p>, in <hi rend="italics">Architecture,</hi> are small square members, in the Doric capital, placed under the quarter round.</p><p><hi rend="italics">Annulet</hi> is also used for a narrow flat moulding, common to other parts of a column, as well as to the capital; and so called, because it encompasses the column <pb n="120"/><cb/> around. In which sense annulet is frequently used for <hi rend="italics">baguette,</hi> or little <hi rend="italics">astragal.</hi></p></div1><div1 part="N" n="ANNULUS" org="uniform" sample="complete" type="entry"><head>ANNULUS</head><p>, a species of <hi rend="smallcaps">Voluta.</hi> See also <hi rend="smallcaps">Ring.</hi></p><p>ANOMALISTICAL <hi rend="italics">Year,</hi> in <hi rend="italics">Astronomy,</hi> called <*>lso <hi rend="italics">periodical year,</hi> is the space of time in which the <*>arth, or a planet, passes through its orbit. The ano- <*>nalistical, or common year, is somewhat longer than <*>he tropical year; by reason of the precession of the <*>quinox.</p><p>And the apses of all the planets have a like progresive motion; by which it happens that a longer time is <*>ecessary to anive at the aplielion, which has advanced <*> little, than to arrive at the same fixed star. For eximple, the tropical revolution of the sun, with respect <table><row role="data"><cell cols="1" rows="1" role="data">to the equinox, is</cell><cell cols="1" rows="1" role="data">365<hi rend="sup">da</hi></cell><cell cols="1" rows="1" role="data">5<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">48<hi rend="sup">m</hi></cell><cell cols="1" rows="1" role="data">45<hi rend="sup">s</hi>;</cell></row><row role="data"><cell cols="1" rows="1" role="data">but the sidereal, or return to the same star,</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">11;</cell></row><row role="data"><cell cols="1" rows="1" role="data">and the anomalistic revolution is</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">20,</cell></row></table> because the sun's apogee advances each year 65″ 1/2 with respect to the equinoxes, and the sun cannot arrive at the apogee till he has passed over the 65″ 1/2 more than the revolution of the year answering to the equinoxes.</p><p>To find the anomalistic revolution, say, As the whole secular motion of a planet <hi rend="italics">minus</hi> the motion of its aphelion, is to 100 years or 3155760000 seconds, so is 360°, to the duration of the anomalistic revolution.</p></div1><div1 part="N" n="ANOMALOUS" org="uniform" sample="complete" type="entry"><head>ANOMALOUS</head><p>, is something irregular, or that deviates from the ordinary rule and method of other things of the same kind.</p></div1><div1 part="N" n="ANOMALY" org="uniform" sample="complete" type="entry"><head>ANOMALY</head><p>, in <hi rend="italics">Astronomy,</hi> is an irregularity in the motion of a planet, by which it deviates from the aphelion or apogee; or it is the angular distance of the planet from the aphelion or apogee; that is, the angle formed by the line of the apses, and another line drawn through the planet.</p><p>Kepler distinguishes three kinds of anomaly; <hi rend="italics">mean, eccentric,</hi> and <hi rend="italics">true.</hi></p><p><hi rend="italics">Mean</hi> or <hi rend="italics">Simple</hi> <hi rend="smallcaps">Anomaly</hi>, in the ancient astronomy, is the distance of a planet's mean place from the apogee. Which Ptolomy calls the angle of the mean motion.</p><p>But in the modern astronomy, in which a planet P is considered as describing an ellipse APB about the sun S, placed in one focus, it is the time in which the planet moves from its aphelion A, to the mean place or point of its orbit P. <figure/></p><p>Hence, as the elliptic area ASP is proportional to the time in which the planet describes the arc AP, that area may represent the mean anomaly.—Or, if PD be drawn perpendicular to the transverse axis AB, and meet the circle in D described on the same axis; then the mean anomaly may also be represented by the cir- <cb/> cular trilineal ASD, which is always proportional to the elliptic one ASP, as is proved in my Mensuration, pr. 3, page 296, second edition.—Or, drawing SG perpendicular to the radius DC produced; then the mean anomaly is alse proportional to SG + the circular arc AD, as is demonstrated by Keil in his <hi rend="italics">Lect. Astron.</hi>—Hence, taking DH = SG, the arc AH, or angle ACH will be the mean anomaly in practice, as expressed in degrees of a circle, the number of those degrees being to 360°, as the elliptic trilineal area ASP, is to the whole area of the ellipse; the degrees of mean anomaly, being those in the arc AH, or angle ACH.</p><p><hi rend="italics">Eccentric</hi> <hi rend="smallcaps">Anomaly</hi>, or <hi rend="italics">of the centre,</hi> in the modern astronomy, is the arc AD of the circle ADB intercepted between the apsis A and the point D determined by the perpendicular DPE to the line of the apses, drawn through the place P of the planet. Or it is the angle ACD at the centre of the circle.—Hence the eccentric anomaly is to the mean anomaly, as AD to AD + SG, or as AD to AH, or as the angle ACD to the angle ACH.</p><p><hi rend="italics">True</hi> or <hi rend="italics">Equated</hi> <hi rend="smallcaps">Anomaly</hi>, is the angle ASP at the sun, which the planet's distance AP from the aphelion, appears under; or the angle formed by the radius vector or line SP drawn from the sun to the planet, with the line of the apses.</p><p>The true anomaly being given, it is easy from thence to find the mean anomaly. For the angle ASP, which is the true anomaly, being given, the point P in the ellipse is given, and thence the proportion of the area ASP to the whole ellipse, or of the mean anomaly to 360 degrees. And for this purpose, the following easy rules for practice are deduced from the properties of the ellipse, by M. de la Caille in his Elements of Astronomy, and M. de la Lande, art. 1240 &c of his astronomy: 1st, As the square root of SB the perihelion distance, is to the square root of SA the aphelion distance, so is the tangent of half the true anomaly ASP, to the tangent of half the eccentric anomaly ACD. 2nd, The difference DH or SG between the eccentric and mean anomaly, is equal to the product of the eccentricity CS, by the sine of SCG the eccentric anomaly just found. And in this case, it is proper to express the eccentricity in seconds of a degree, which will be found by this proportion, as the mean distance 1: the eccentricity :: 206264.8 seconds, or 57° 17′ 44″.8, in the arch whose length is equal to the radius, to the seconds in the are which is equal to the eccentricity CS; which being multiplied by the sine of the eccentric anomaly, to radius 1, as above, gives the seconds in SG, or in the are DH, being the difference between the mean and eccentric anomalies. 3d, To find the radius vector SP, or distance of the planet from the sun, say either, as the sine of the true anomaly is to the sine of the eccentric anomaly, so is half the less axis of the orbit, to the radius vector SP; or as the sine of half the true anomaly is to the sine of half the eccentric anomaly, so is the square root of the perihelion distance SB, to the square root of the radius vector or planet's distance SP.</p><p>But the mean anomaly being given, it is not so easy to find the true anomaly, at least by a direct process<*> Kepler, who first proposed this problem, could not find <pb n="121"/><cb/> a direct way of resolving it, and therefore made use of an indirect one, by the rule of false position, as may be seen page 695 of Kepler's <hi rend="italics">Epitom. Astron. Copernic.</hi> See also §628 Wolfius <hi rend="italics">Elem. Astron.</hi> Now the easiest method of performing this operation, would be to work first for the eccentric anomaly, viz, assume it nearly, and from it so assumed compute what would be its mean anomaly by the rule above given, and find the difference between this result and the mean anomaly given; then assume another eccentric anomaly, and proceed in the same way with it, finding another computed mean anomaly, and its difference from the given one; and treating these differences as in the rule of position for a nearer value of the eccentric anomaly: repeating the operation till the result comes out exact. Then, from the eccentric anomaly, thus found, compute the true anomaly by the 1st rule above laid down.</p><p>Of this problem, Dr. Wallis first gave the geometrical solution by means of the protracted cycloid; and Sir Isaac Newton did the same at prop. 31 <hi rend="italics">lib.</hi> 1 <hi rend="italics">Princip.</hi> But these methods being unsit for the purpose of the practical astronomer, various series for approximation have been given, viz, several by Sir Isaac Newton in his <hi rend="italics">Fragmenta Epistolarum,</hi> page 26, as also in the Schol. to the prop. above-mentioned, which is his best, being not only fit for the planets, but also for the comets, whose orbits are very eccentric. Dr. Gregory, in his <hi rend="italics">Astron.</hi> lib. 3, has also given the solution by a series, as well as M. Reyneau, in his <hi rend="italics">Analyse Demontrée,</hi> page 713, &c. And a better still for converging is given by Keil in his <hi rend="italics">Prælect. Astron.</hi> page 375; he says, if the are AH be the mean anomaly, calling its sine <hi rend="italics">e,</hi> consine <hi rend="italics">f,</hi> the eccentricity <hi rend="italics">g,</hi> also putting <hi rend="italics">z</hi> = <hi rend="italics">ge,</hi> and ; then the eccentric anomaly AD will be , supposing <hi rend="italics">r</hi> = 57.29578 degrees; of which the first term <hi rend="italics">rz/a</hi> is sufficient for all the planets, even for Mars itself, where the error will not exceed the 200th part of a degree; and in the orbit of the earth, the error is less than the 10000th part of a degree.</p><p>Dr. Seth Ward, in his <hi rend="italics">Astronomia Geometrica,</hi> takes the angle AFP at the other focus, where the sun is not, for the mean anomaly, and thence gives an elegant solution. But this method is not sufficiently accurate when the orbit is very eccentric, as in that of the planet Mars, as is shewn by Bullialdus, in his defence of the <hi rend="italics">Philolaic. Astron.</hi> against Dr. Ward. However, when Newton's correction is made, as in the Schol. abovementioned, and the problem resolved according to Ward's hypothesis, Sir Isaac affirms that, even in the orbit of Mars, there will scarce ever be an error of more than one second.</p><p>ANSÆ, <hi rend="smallcaps">Anses</hi>, in <hi rend="italics">Astronomy,</hi> those seemingly prominent parts of the ring of the planet Saturn, discovered in its opening, and appearing like handles to the body of the planet; srom which appearance the name <hi rend="italics">ansæ</hi> is taken.</p></div1><div1 part="N" n="ANSER" org="uniform" sample="complete" type="entry"><head>ANSER</head><p>, in <hi rend="italics">Astronomy,</hi> a small star, of the 5th or 6th magnitude, in the milky-way, between the eagle and swan, first brought into order by Hevelius. <cb/></p><p>ANTARCTIC <hi rend="italics">pole,</hi> denotes the southern pole, or southern end of the earth's axis.—The stars near the antarctic pole never appear above our horizon in these latitudes.</p><p>ANTARCTIC <hi rend="italics">circle,</hi> is a small circle parallel to the equator, at the distance of 23° 28′ from the antarctic or south pole.—At one time of the year the sun never rises above the horizon of any part within this circle; and at other times he never sets.</p></div1><div1 part="N" n="ANTARES" org="uniform" sample="complete" type="entry"><head>ANTARES</head><p>, in Astronomy, the scorpion's heart; a fixed star of the first magnitude, in the constellation <hi rend="italics">Scorpio.</hi></p><p>ANTECANIS is used by some astronomers, to denote the constellation otherwise called <hi rend="italics">canis minor,</hi> or the star <hi rend="italics">procyon.</hi> It is so called, as preceding, or being the forerunner of the <hi rend="italics">canis major,</hi> and rising a little before it.</p></div1><div1 part="N" n="ANTECEDENT" org="uniform" sample="complete" type="entry"><head>ANTECEDENT</head><p>, <hi rend="italics">of a ratio,</hi> denotes the first of the two terms of the ratio, or that term which is compared with the other. Thus, if the ratio be 2 to 3. or <hi rend="italics">a</hi> to <hi rend="italics">b;</hi> then 2 or <hi rend="italics">a</hi> is the antecedent.</p><p>ANTECEDENTAL <hi rend="italics">Method,</hi> is a branch of general geometrical proportion, or universal comparison, and is derived from an examination of the Antecedents of ratios, having given consequents, and a given standard of comparison, in the various degrees of augmentation and diminution, which they undergo by composition and decomposition. This is a method invented by Mr. James Glenie, and published by him in 1793; a method which he says he always used instead of the fluxional and differential methods, and which is totally unconnected with the ideas of motion and time. See the author's treatise above-mentioned, and also his Doctrine of Universal Comparison, or General Proportion, 1789, upon which it is founded.</p></div1><div1 part="N" n="ANTECEDENTIA" org="uniform" sample="complete" type="entry"><head>ANTECEDENTIA</head><p>, a term used by astronomers when a planet &c moves westward, or contrary to the order of the signs aries, taurus, &c.—Like as when it moves eastward, or according to the order of the signs aries, taurus, &c, it is then said to move <hi rend="italics">in consequentia.</hi></p></div1><div1 part="N" n="ANTECIANS" org="uniform" sample="complete" type="entry"><head>ANTECIANS</head><p>, or <hi rend="smallcaps">Antoeci</hi>, in <hi rend="italics">Geography,</hi> the inhabitants of the earth which occupy the same semicircle of the same meridian, but equally distant from the equator, the one north and the other south; as Peloponnesus and the Cape of Good Hope.</p><p>These have their noon, or midnight, or any other hour at the same time; but their seasons are contrary, being spring to the one, when it is autumn with the other; and summer with the one, when it is winter with the other; also the length of the day to the one, is equal to the length of night to the other.</p></div1><div1 part="N" n="ANTES" org="uniform" sample="complete" type="entry"><head>ANTES</head><p>, in Architecture, are small pilastres placed at the corners of buildings.</p></div1><div1 part="N" n="ANTICS" org="uniform" sample="complete" type="entry"><head>ANTICS</head><p>, in Architecture, figures of men, beasts, &c, placed as ornaments to buildings.</p></div1><div1 part="N" n="ANTICUM" org="uniform" sample="complete" type="entry"><head>ANTICUM</head><p>, in Architecture, a porch before a door; also that part of a temple, which is called the outer temple, and lies between the body of the temple and the portico.</p></div1><div1 part="N" n="ANTILOGARITHM" org="uniform" sample="complete" type="entry"><head>ANTILOGARITHM</head><p>, the complement of the logarithm of a sine, tangent, secant, &c, to that of the radius. This is sound by beginning at the lest hand, subtracting each sigure from 9, and the last figure from 10. <pb n="122"/><cb/></p></div1><div1 part="N" n="ANTINOUS" org="uniform" sample="complete" type="entry"><head>ANTINOUS</head><p>, in <hi rend="italics">Astronomy,</hi> a part of the constellation <hi rend="italics">aquila,</hi> or the eagle.</p><p>ANTIOCHIAN <hi rend="italics">Sect,</hi> or <hi rend="italics">Academy,</hi> a name given to the sifth academy or branch of academics. It took its name from being founded by Antiochus, a philosopher contemporary with Cicero; and it succeeded the Philonian academy. Though Antiochus was really a stoic, and only nominally an academic.</p><p><hi rend="smallcaps">Antiochian</hi> <hi rend="italics">epocha,</hi> a method of computing time from the proclamation of liberty granted to the city of Antioch, about the time of the battle of Pharsalia.</p></div1><div1 part="N" n="ANTIPARALLELS" org="uniform" sample="complete" type="entry"><head>ANTIPARALLELS</head><p>, in <hi rend="italics">Geometry,</hi> are those lines which make equal angles with two other lines, but contrary ways; that is, calling the former pair the first and 2d lines, and the latter pair the 3d and 4th lines, if the angle made by the 1st and 3d liues be equal to the angle made by the 2d and 4th, and contrariwise the angle made by the 1st and 4th equal to the angle made by the 2d and 3d; then each pair of lines are antiparallels with respect to each other, viz, the first and 2d, and the 3d and 4th. So, if AB and AC be any two lines, and FC and FE be two others, cutting themso, that the angle B is equal to the angle E, and the angle C is equal to the angle D; then BC and DE are antiparallels with respect to AB and AC; also these latter are antiparallels with regard to the two former.—See also <hi rend="smallcaps">Subcontrary.</hi></p><p>It is a property of these lines, that each pair cuts the other into proportional segments, taking them alternately, viz AB : AC :: AE : AD :: DB : EC, and FE : FC :: FB : FD :: DE : BC. <figure/></p></div1><div1 part="N" n="ANTIPODES" org="uniform" sample="complete" type="entry"><head>ANTIPODES</head><p>, in Geography, are the inhabitants of two places on the earth which lie diametrically opposite to each other, or that walk feet to feet; that is, if a line be continued down from our feet, quite through the centre of the earth, till it arrive at the surface on the other side, it will fall on the feet of our Antipodes, and <hi rend="italics">vice versa.</hi>——Antipodes are 180 degrees distant from each other every way on the surface of the globe; they have equal latitudes, the one north and the other south, but they differ by 180 degrees of longitude: they have therefore the same climates or degrees of heat and cold, with the same seasons and length of days and nights; but all of these at contrary times, it being day to the one, when it is night to the other, summer to the one when it is winter to the other, &c: they have also the same horizon, the one being as far distant on the one side, as the other on the other side, and therefore when the sun, &c, rises to the one, it sets to the other. The Antipodes to London are a part a little south of New Zealand.</p><p>It has been said that Plato first started the notion of Antipodes, and gave them the name; which is likely enough, as he conceived that the earth was of a glo- <cb/> bular figure. But there have been many disputes upon this point, and the fathers of the church have greatly opposed it, especially Lactantius and Augustine, who laughed at it, and were greatly perplexed to think how men and trees should hang pendulous in the air with their feet uppermost, as he thought they must do, in the other hemisphere.</p></div1><div1 part="N" n="ANTISCIANS" org="uniform" sample="complete" type="entry"><head>ANTISCIANS</head><p>, or ANTISCII, in Geography, are people who dwell in the opposite hemispheres of the earth, as to north and south, and whose shadows at noon fall in contrary directions. This term is more general than <hi rend="italics">antæci,</hi> with which it is often confounded. The <hi rend="italics">Antiscians</hi> stand contradistinguished from <hi rend="italics">Periscians.</hi></p><p><hi rend="smallcaps">Antiscii</hi> is also used sometimes, among <hi rend="italics">Astrologers,</hi> for two points of the heavens equally distant from the tropics. Thus the signs Leo and Taurus are accounted <hi rend="italics">antiscii</hi> to each other.</p><p>ANTŒCI, see <hi rend="smallcaps">Antecians.</hi></p></div1><div1 part="N" n="APERTURE" org="uniform" sample="complete" type="entry"><head>APERTURE</head><p>, in <hi rend="italics">Geometry,</hi> is used for the space left between two lines which mutually incline towards each other, to form an angle.</p><div2 part="N" n="Aperture" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Aperture</hi></head><p>, in Optics, is the hole next the objectglass of a telescope or microscope, through which the light and the image of the object come into the tube, and are thence conveyed to the eye.</p><p>Aperture is also understood of that part of the object-glass itself which covers the former, and which is left pervious to the rays.</p><p>A great deal depends upon having a just aperture.— To find it experimentally: apply several circles of dark paper, of various sizes, upon the face of the glass, from the breadth of a straw, to such as leave only a small hole in the glass; and with each of these, separately, view some distinct objects, as the moon, stars, &c; then that aperture is to be chosen through which they appear the most distinctly.</p><p>Huygens first found the use of apertures to conduce much to the perfection of telescopes; and he found by experience (<hi rend="italics">Dioptr.</hi> prop. 56.) that the best aperture for an object-glass, for example of 30 feet, is to be de termined by this proportion, as 30 to 3, so is the square root of 30 times the distance of the focus of any lens, to its proper aperture: and that the focal distances of the eye-glasses are proportional to the apertures. And M. Auzout says he found, by experience, that the apertures of telescopes ought to be nearly in the subduplicate ratio of their lengths. It has also been found by experience that object-glasses will admit of greater apertures, if the tubes be blacked within side, and their passage furnished with wooden rings.</p><p>It is to be noted, that the greater or less aperture of an object-glass, does not increase or diminish the visible area of the object; all that is effected by this, is the admittance of more or fewer rays, and consequently the more or less bright the appearance of the object. But the largeness of the aperture or focal distance, causes the irregularity of its refractions. Hence, in viewing Venus through a telescope, a much less aperture is to be used than for the moon, or Jupiter, or Saturn, because her light is so bright and glaring. And this circumstance somewhat invalidates and disturbs Azout's proportion, as is shewn by Dr. Hook, Philos. Trans. No. 4.</p></div2></div1><div1 part="N" n="APHELION" org="uniform" sample="complete" type="entry"><head>APHELION</head><p>, or <hi rend="smallcaps">Aphelium</hi>, in Astronomy, that <pb n="123"/><cb/> point in the orbit of the earth, or a planet, in which it is at the greatest distance from the sun. Which is the point A (in the fig. to the art. <hi rend="smallcaps">Anomaly</hi>) or extremity of the transverse axis, of the elliptic orbit, farthest from the focus S, where the sun is placed; and diametrically opposite to the perihelion B, or nearer extremity of the same axis. In the Ptolemaic system, or in the supposition that the sun moves about the earth, the aphelion becomes the <hi rend="italics">apogee.</hi></p><p>The times of the aphelia of the primary planets, may be known by their apparent diameter appearing the smallest, and also by their moving slowest in a given time. Calculations and methods of finding them have been given by many astronomers, as Ricciolli, <hi rend="italics">Almag. Nov.</hi> lib. 7, sect. 2 and 3; Wolfius, <hi rend="italics">Elem. Astron.</hi> § 659; Dr. Halley, <hi rend="italics">Philos. Trans.</hi> No. 128; Sir I. Newton, <hi rend="italics">Princip.</hi> lib. 3, prop. 14; Dr. Gregory, <hi rend="italics">Astron.</hi> lib. 3, prop. 14; Keil, <hi rend="italics">Astron. Lect.</hi>; De la Lande, <hi rend="italics">Memoires de l'Acad.</hi> 1755, 1757, 1766, and in his <hi rend="italics">Astron.</hi> liv. 22; also in the writings of MM. Euler, D'Alembert, Clairaut, &c, upon attraction.</p><p>The aphelia of the planets are not fixed; for their mutual actions upon one another keep those points of their orbits in a continual motion, which is greater or less in the different planets. This motion is made in consequentia, or according to the order of the signs; and Sir I. Newton shews that it is in the sesquiplicate ratio of the distance of the planet from the sun, that is, as the square root of the cube of the distance.</p><p>The quantities of this motion, as well as the place of the aphelion for a given time, are variously given by different authors. Kepler states them, for the year 1700 as in the following table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Planets.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Aphelion.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Annual Motion.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data"><figure/>  8°</cell><cell cols="1" rows="1" role="data">25′</cell><cell cols="1" rows="1" role="data">30″</cell><cell cols="1" rows="1" role="data">1′</cell><cell cols="1" rows="1" role="data">45″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data"><figure/>  3</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data"><figure/>  0</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"> 7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data"><figure/>  8</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">47</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data"><figure/> 28</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Earth</cell><cell cols="1" rows="1" role="data"><figure/>  8</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> <hi rend="center">By De la Hire they are given as follows, for the same year 1700.</hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Planets.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Aphelion.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Annual Motion.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data"><figure/> 13°</cell><cell cols="1" rows="1" role="data"> 3′</cell><cell cols="1" rows="1" role="data">40″</cell><cell cols="1" rows="1" role="data">1′</cell><cell cols="1" rows="1" role="data">39″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data"><figure/>  6</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data"><figure/>  0</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"> 7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data"><figure/> 10</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">34</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data"><figure/> 29</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">22</cell></row></table> <hi rend="center">And De la Lande states them as follows, for the year 1750.</hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Planets.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Aphelion.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Secular Motion.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">13°</cell><cell cols="1" rows="1" role="data">33′</cell><cell cols="1" rows="1" role="data">1°</cell><cell cols="1" rows="1" role="data">57′</cell><cell cols="1" rows="1" role="data">40″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"> 0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Earth</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">10</cell></row></table> <cb/></p><p>Of the new planet, Herschel, or Georgium Sidus, the aphelion for 1790 was 11<*>23°29′42″, and its annual motion 50″ 3/8. See Connoissance des Temps, 1786 and 1787.</p></div1><div1 part="N" n="APHRODISIUS" org="uniform" sample="complete" type="entry"><head>APHRODISIUS</head><p>, in Chronology, denotes the eleventh month in the Bythinian year, commencing on the 25th of July in ours.</p><p>APIAN or <hi rend="smallcaps">Appian (Peter</hi>), called in German <hi rend="italics">Bienewitz,</hi> a celebrated astronomer and mathematician, was born at Leisnig or Leipsick in Misnia, 1495, and made professor of mathematics at Ingolstadt, in 1524, where he died in the year 1552, at 57 years of age.</p><p>Apian wrote treatises upon many of the mathematical sciences, and greatly improved them; more especially astronomy and astrology, which in that age were much the same thing; also geometry, geography, arithmetic, &c. He particularly enriched astronomy with many instruments, and observations of eclipses, comets, &c. His principal work was the <hi rend="italics">Astronomicum Cæsareum,</hi> published in folio at Ingolstadt in 1540, and which contains a number of interesting observations, with the descriptions and divisions of instruments. In this work he predicts eclipses, and constructs the figures of them in plano. In the 2d part of the work, or the <hi rend="italics">Meteoroscopium Planum,</hi> he gives the description of the most accurate astronomical quadrant, and its uses. To it are added observations of five different comets, viz, in the years 1531, 1532, 1533, 1538, and 1539; where he first shews that the tails of comets are always projected in a direction from the sun.</p><p>Apian also wrote a treatise on <hi rend="italics">Cosmography,</hi> or <hi rend="italics">Geographical Instruction,</hi> with various mathematical instruments. This work Vossius says he published in 1524, and that Gemma Frisius republished it in 1540. But Weidler says he wrote it only in 1530, and that Gemma Frisius published it at Antwerp in 1550 and 1584, with observations of many eclipses. The truth may be, that perhaps all these editions were published.</p><p>In 1533 he made, at Norimberg, a curious instrument, which from its sigure he called <hi rend="italics">Folium Populi;</hi> which, by the sun's rays, shewed the hour in all parts of the earth, and even the unequal hours of the Jews.</p><p>In 1534 he published his <hi rend="italics">Inscriptiones Orbis.</hi></p><p>In 1540, his <hi rend="italics">Instrumentum Sinuum, sive Primi Mobilis,</hi> with 100 problems.</p><p>Beside these, Apian was the author of many other works: among which may be mentioned the <hi rend="italics">Ephemerides</hi> from the year 1534 to 1570: Books upon <hi rend="italics">Shadows: Arithmetical Centilogues:</hi> Books upon <hi rend="italics">Arithmetic,</hi> with the <hi rend="italics">Rule of Coss</hi> (Algebra) demonstrated: Upon <hi rend="italics">Gauging: Almanacs,</hi> with Astrological directions: A book upon <hi rend="italics">Conjunctions:</hi> Ptolemy, with very correct figures, drawn in a quadrangular form: Ptolemy's works in Greek: Books of Eclipses: the works of Azoph, a very ancient astrologer: the works of Gebre: the Perspective of Vitello: of Critical Days, and of the Rainbow: a new Astronomical and Geometrical Radius, with various uses of Sines and Chords: Universal Astrolabe of Numbers: Maps of the World, and of particular countries: &c, &c.</p><p>Apian left a son, who many years afterwards taught mathematics at Ingolstadt, and at Tubinga. Tycho has preserved (Progymn. p. 643) his letter to the Landgrave of Hesse, in which he gives an opinion on the new star in Cassiopeia, of the year 1572. <pb n="124"/><cb/></p><p>One of the comets observed by Apian, viz, that of 1532, had its elements nearly the same as of one observed 128 1/4 years after, viz, in 1661, by Hevelius and other Astronomers; from hence Dr. Halley judged that they were the same comet, and that therefore it might be expected to appear again in the beginning of the year 1789. But it was not found that it returned at this period, although the astronomers then looked anxiously for it; and it is doubtful whether the disappointment might be owing to its passing unobserved, or to any errors in the observations of Apian, or to its period being disturbed and greatly altered by the actions of the superior planets, &c.</p></div1><div1 part="N" n="APIS" org="uniform" sample="complete" type="entry"><head>APIS</head><p>, <hi rend="italics">musca,</hi> the <hi rend="italics">Bee,</hi> or <hi rend="italics">Fly,</hi> in Astronomy, one of the southern constellations, containing 4 stars.</p></div1><div1 part="N" n="APOCATASTASIS" org="uniform" sample="complete" type="entry"><head>APOCATASTASIS</head><p>, in Astronomy, is the period of a planet, or the time employed in returning to the same point of the zodiac from whence it set out.</p></div1><div1 part="N" n="APOGEE" org="uniform" sample="complete" type="entry"><head>APOGEE</head><p>, <hi rend="italics">Apogæum,</hi> in Astronomy, that point in the orbit of the sun, moon, &c, which is sarthest distant from the earth. It is at the extremity of the line of the apsides; and the point opposite to it is called the <hi rend="italics">perigee,</hi> where the distance from the earth is the least.</p><p>The ancient astronomers, considering the earth as the centre of the system, chiefly regarded the apogee and perigee: but the moderns, placing the sun in the centre, change these terms for the <hi rend="italics">aphelion</hi> and <hi rend="italics">perihelion.</hi> —The apogee of the sun, is the same thing as the aphelion of the earth; and the perigee of the sun is the same as the perihelion of the earth.</p><p>The manner of finding the apogee of the sun or moon, is shewn by Ricciolus, <hi rend="italics">Almag. Nov.</hi> lib. 3, cap. 24; by Wolfius in <hi rend="italics">Elem. Astr.</hi> § 618; by Cassini, De la Hire, and many others: see also <hi rend="italics">Memoires de l'Academie,</hi> the <hi rend="italics">Philos. Trans.</hi> vol. 5, 47, &c.</p><p>The quantity of motion in the apogee may be found by comparing two observations of it made at a great distance of time; converting the difference into minutes, and dividing them by the number of years elapsed between the two observations; the quotient gives the annual motion of the apogee. Thus, srom an observation made by Hipparchus in the year before Christ 140, by which the sun's apogee was found 5° 30′ of <figure/> and another made by Ricciolus, in the year of Christ 1646, by which it was found 7° 26′ of <figure/>; the annual motion of the apogee is found to be 1′ 2″ And the annual motion of the moon's apogee is about 1<*> 10° 39′ 52″.</p><p>But the moon's apogee moves unequably. When she is in the syzygy with the sun, it moves forwards; but in the quadratures, backwards; and these progressions and regressions are not equable, but it goes forward slower when the moon is in the quadratures, or perhaps goes retrograde; and when the moon is in the syzygy, it goes forward the fastest of all.—See also Newton's Theory of the Moon for more upon this subject.</p></div1><div1 part="N" n="APOLLODORUS" org="uniform" sample="complete" type="entry"><head>APOLLODORUS</head><p>, a celebrated architect, under Trajan and Adrian, was born at Damascus, and flourished about the year of Christ 100. He had the direction of the stone bridge which Trajan ordered to <*>e built over the Danube in the year 104, which was asteemed the most magnifieent of all the works of that <cb/> emperor. Adrian, one day as Trajan was discoursing with this architect upon the buildings he had raised at Rome, would needs give his judgment, in which he shewed that he knew nothing of the matter. Apollodorus turned upon him bluntly, and said to him, Go paint Citruls, for you are very ignorant of the subject we are talking upon. Adrian at this time boasted of his painting Citruls well. This was the first step towards the ruin of Apollodorus; a slip which he was so far from attempting to retrieve, that he even added a new offence, and that too after Adrian was advanced to the empire, upon the following occasion: Adrian sent to him the plan of a temple of Venus; and though he asked his opinion, yet to shew that he had no need of him, and that he did not mean to be directed by it, the temple was already built. Apollodorus wrote his opinion very freely, and remarked such essential faults in it, as the emperor could neither deny nor remedy. He shewed that it was neither high nor large enough; that the statues in it were disproportioned to its bulk: for, said he, if the goddesses should have a mind to rise and go out, they could not do it. This put Adrian into a great passion, and prompted him to the destruction of Apollodorus. He banished him at first; then under the pretext of certain supposed crimes, of which he had him accused, he at last put him to death.</p></div1><div1 part="N" n="APOLLONIUS" org="uniform" sample="complete" type="entry"><head>APOLLONIUS</head><p>, of Perga, a city in Pamphilia, was a celebrated geometrician who flourished in the reign of Ptolemy Euergetes, about 240 years before Christ; being about 60 years after Euclid, and 30 years later than Archimedes. He studied a long time in Alexandria under the disciples of Euclid; and afterwards he composed several curious and ingenious geometrical works, of which only his books of Conic Sections are now extant, and even these not perfect. For it appears from the author's dedicatory epistle to Eudemus, a geometrician in Pergamus, that this work consisted of 8 books; only 7 of which however have come down to us.</p><p>From the Collections of Pappus, and the Commentaries of Eutocius, it appears that Apollonius was the author of various pieces in geometry, on account of which he acquired the title of the Great Geometrician. His <hi rend="italics">Conics</hi> was the principal of them. Some have thought that Apollonius appropriated the writings and discoveries of Archimedes; Heraclius, who wrote the life of Archimedes, affirms it; though Eutocius endeavours to refute him. Although it should be allowed a groundless supposition, that Archimedes was the first who wrote upon Conics, notwithstanding his treatise on Conics was greatly esteemed; yet it is highly probable that Apollonius would avail himself of the writings of that author, as well as others who had gone before him; and, upon the whole, he is allowed the honour of explaining a difficult subject better than had been done before; having made several improvements both in Archimedes's problems, and in Euclid. His work upon Conics was doubtless the most perfect of the kind among the ancients, and in some respects among the moderns also. Before Apollonius, it had been customary, as we are informed by Eutocius, for the writers on Conics to require three disserent sorts of cones to cut the three different sections from, viz, the parabola from a right angled cone, the ellipse from an acute, and <pb n="125"/><cb/> the hyperbola from an obtuse cone; because they always supposed the sections made by a plane cutting the cones to be perpendicular to the side of them: but Apollonius cut his sections all from any one cone, by only varying the inclination or position of the cutting plane; an improvement that has been followed by all other authors since his time. But that Archimedes was acquainted with the same manner of cutting any cone, is sufficiently proved, against Eutocius, Pappus, and others, by Guido Ubaldus, in the beginning of his Commentary on the 2d book of Archimedes's Equiponderantes, published at Pisa in 1588.</p><p>The first four books of Apollonius's Conics only have come down to us in their original Greek language; but the next three, the 5th, 6th, and 7th, in an Arabic version; and the 8th not at all. These have been commented upon, translated, and published by various authors. Pappus, in his Mathematical Collections, has lest some account of his various works, with notes and lemmas upon them, and particularly on the Conics. And Eutocius wrote a regular elaborate commentary on the propositions of several of the books of the Conics.</p><p>The first four books were badly translated by Joan. Baptista Memmius. But a better translation of these in Latin was made by Commandine, and published at Bononia in 1566.—Vossius mentions an edition of the Conics in 1650; the 5th, 6th, and 7th books being recovered by Golius.—Claude Richard, Professor of mathematics in the imperial college of his order at Madrid, in the year 1632, explained, in his public lectures, the first four books of Apollonius, which were printed at Antwerp in 1655, in folio.—And the Grand Duke Ferdinand the 2d, and his brother Prince Leopold de Medicis, employed a professor of the Oriental languages at Rome to translate the 5th, 6th, and 7th books into Latin. These were published at Florence in 1661, by Borelli, with his own notes, who also maintains that these books are the genuine production of Apollonius, by many strong authorities, against Mydorgius and others, who suspected that these three books were not the real production of Apollonius.</p><p>As to the 8th book, some mention is made of it in a book of Golius's, where he had written that it had not been translated into Arabic; because it was wanting in the Greek copies, from whence the Arabians translated the others. But the learned Mersenne, in the preface to Apollonius's Conics, printed in his Synopsis of the Mathematics, quotes the Arabic philosopher Aben Nedin for a work of his about the year 400 of Mahomet, in which is part of that 8th book, and who asserts that all the books of Apollonius are extant in his language, and even more than are enumerated by Pappus; and Vossius says he has read the same; <hi rend="italics">De Scientiis Mathematicis,</hi> pa. 55.—A neat edition of the first four books in Latin was published by Dr. Barrow, in 4to, at London in 1675.—A magnificent edition of all the 8 books, was published in folio, by Dr. Halley, at Oxford in 1710; together with the Lemmas of Pappus, and the Commentaries of Eutoeius. The first four in Greek and Latin, but the latter four in Latin only, the 8th book being restored by himself.</p><p>The other writings of Apollonius, mentioned by Pappus, are, <cb/></p><p>1. The Section of a Ratio, or Proportional Sections, two books.</p><p>2. The Section of a Space, in two books.</p><p>3. Determinate Section, in two books.</p><p>4. The Tangencies, in two books.</p><p>5. The Inclinations, in two books.</p><p>6. The Plane Loci, in two books.</p><p>The contents of all these are mentioned by Pappus, and many lemmas are delivered relative to them; but none, or very little of these books themselves have descended down to the moderns. From the account however that has been given of their contents, many restorations have been made of these works, by the modern mathematicians, as follow: viz,</p><p><hi rend="italics">Victa,</hi> Apollonius Gallus. The Tangencies. Paris, 1600, in 4to.</p><p><hi rend="italics">Snellius,</hi> Apollonius Batavus. Determinate Section. Lugd. 1601, 4to.</p><p><hi rend="italics">Snellius,</hi> Sectio Rationis & Spatii. 1607.</p><p><hi rend="italics">Ghetaldus,</hi> Apollonius Redivivus. The Inclinations. Venice, 1607, 4to.</p><p><hi rend="italics">Ghetaldus,</hi> Supplement to the Apollonius Redivivus. Tangencies. 1607.</p><p><hi rend="italics">Ghetaldus,</hi> Apollonius Redivivus, lib. 2. 1613.</p><p><hi rend="italics">Alex. Anderson,</hi> Supplem. Apol. Redivivi. Inclin. Paris, 1612, 4to.</p><p><hi rend="italics">Alex. Anderson.</hi> Pro Zetetico Apolloniani problematis a se jam pridem edito in Supplemento Apollonil Redivivi. Paris, 1615, 4to.</p><p><hi rend="italics">Schooten,</hi> Loca Plana restituta. Lug. Bat. 1656.</p><p><hi rend="italics">Fermat,</hi> Loca Plana, 2 lib. Tolos. 1679, solio.</p><p><hi rend="italics">Halley,</hi> Apol. de Sectione Rationis libri duo ex Arabico MS. Latine versi duo restituti. Oxon. 1706, 8vo.</p><p><hi rend="italics">Simson,</hi> Loca Plana, libri duo. Glasg. 1749, 4to.</p><p><hi rend="italics">Simson,</hi> Sectio Determinat. Glasg. 1776, 4to.</p><p><hi rend="italics">Horsley,</hi> Apol. Inclinat. libri duo. Oxon. 1770, 4to.</p><p><hi rend="italics">Lawson,</hi> The Tangencies, in two books. Lond. 1771, 4to.</p><p><hi rend="italics">Lawson,</hi> Determinate Section, two books. Lond. 1772, 4to.</p><p><hi rend="italics">Wales,</hi> Determinate Section, two books. Lond. 1772, 4to.</p><p><hi rend="italics">Burrow,</hi> The Inclinations. Lond. 1779, 4to.</p></div1><div1 part="N" n="APONO" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">APONO</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName> <hi rend="italics">de</hi>)</persName></head><p>, a learned astronomer and philosopher, was born at Apono near Padua, about the year 1250. He described the <hi rend="italics">Astrolabium Planum,</hi> by which were shewn the equations of the celestial houses for any hour and minute, and sor any part of the world: it was published at Venice in 1502. He acquired the name of the <hi rend="italics">Conciliator,</hi> on account of a book of his, in which he reconciies the writings of the ancient philosophers and physicians: the book was published at Venice in 1483. He resided at Padua, where, from his practising medicine, and his skill in astronomy, he fell under the suspicion of magic. He died in 1316, at 66 years of age.</p></div1><div1 part="N" n="APOPHYGE" org="uniform" sample="complete" type="entry"><head>APOPHYGE</head><p>, in Architecture, is a concave part or ring of a column, lying above or below the flat member; and it owes its origin to the ring by which the ends of wooden columns were hooped, to prevent them from splitting.</p></div1><div1 part="N" n="APOTOME" org="uniform" sample="complete" type="entry"><head>APOTOME</head><p>, the remainder or difference between two lines or quantities which are only commensurable in power. Such is the difference between 1 and √2, <pb n="126"/><cb/> or the difference between the side of the square and its diagonal.</p><p>The term is used by Euclid; and a pretty full explanation of such quantities is given in the tenth book of his Elements, where he distinguishes six kinds of apotomes, and shews how to find them all geometrically.</p><p><hi rend="smallcaps">Apotome</hi> <hi rend="italics">Prima,</hi> is when the greater term is rational, and the difference of the squares of the two is a square number; as the difference 3-√5.</p><p><hi rend="smallcaps">Apotome</hi> <hi rend="italics">Secunda,</hi> is when the less number is rational, and the square root of the difference of the squares of the two terms, has to the greater term, a ratio expressible in numbers; such is √18-4, because the difference of the fquares 18 and 16 is 2, and √2 is to √18 as √1 to √9 or as 1 to 3.</p><p><hi rend="smallcaps">Apotome</hi> <hi rend="italics">Tertia,</hi> is when both the terms are irrational, and, as in the second, the square root of the difference of their squares, has to the greater term, a rational ratio: as √24-√18; for the difference of their squares 24 and 18 is 6, and √6 is to √24 as √1 to √4 or as 1 to 2.</p><p><hi rend="smallcaps">Apotome</hi> <hi rend="italics">Quarta,</hi> is when the greater term is a rational number, and the square root of the difference of the squares of the two terms, has not a rational ratio to it: as 4-√3, where the difference of the squares 16 and 3 is 13, and √13 has not a ratio in numbers to 4.</p><p><hi rend="smallcaps">Apotome</hi> <hi rend="italics">Quinta,</hi> is when the less term is a rational number, and the square root of the difference of the squares of the two, has not a rational ratio to the greater: as √6-2, where the difference of the squares 6 and 4 is 2, and √2 to √6 or √1 to √3 or 1 to √3 is not a rational ratio.</p><p><hi rend="smallcaps">Apotome</hi> <hi rend="italics">Sexta,</hi> is where both terms are irrational, and the square root of the difference of their squares has not a rational ratio to the greater: as √6-√2; where the difference of the squares 6 and 2 is 4, and √4 to √6 or 2 to √6, is not a rational ratio.</p><p>The doctrine of apotomes, in lines, as delivered by Euclid in the tenth book, is a very curious subject, and has always been much admired and cultivated by all mathematicians who have rightly understood this part of the elements; and therefore Peter Ramus has greatly exposed his judgment by censuring that book. And the first algebraical writers in Europe commonly employed a considerable portion of their works on an algebraical exposition of that book, which led them to the doctrine of surd quantities; as Lucas de Burgo, Cardan, Tartalea, Stifelius, Peletarius, &c, &c. See also Pappus, lib. 4, prop. 3, and the introduc. to lib. 7. And Dr. Wallis's Algebra, pa. 109.</p><div2 part="N" n="Apotome" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Apotome</hi>, in <hi rend="italics">Music</hi></head><p>, is the difference between a greater and less semitone, being expressed by the ratio of 128 to 125.</p></div2></div1><div1 part="N" n="APPARENT" org="uniform" sample="complete" type="entry"><head>APPARENT</head><p>, that which is visible, or evident to the eye, or the understanding.</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">conjunction</hi> of the planets, is when a right line, supposed to be drawn through their centres, passes through the eye of the spectator, and not through the centre of the earth.—And, in general, the apparent conjunction of any objects, is when they appear or are placed in the same right line with the eye.</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">Altitude, Diameter, Distance, Horizon, Magnitude, Motion, Place, Time,</hi> &c. See the respective substantives, for the quantity and measure of it. <cb/></p><p>The apparent state of things, is commonly very different from their real state, either as to distance, figure, magnitude, position, &c, &c. Thus,</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">Diameter,</hi> or <hi rend="italics">Magnitude,</hi> as for example of the heavenly bodies, is not the real length of the diameter, but the angle which they subtend at the eye, or under which they appear. And hence, the angle, or apparent extent, diminishing with the distance of the object, a very small object, as AB, may have the same apparent diameter as a very large one FG; and indeed the objects have all the same apparent diameter, that are contained in the same angle FEG. And if these are parallel, the real magnitudes are directly proportional to their distances. <figure/></p><p>But the apparent magnitude varies not only by the distance, but also by the position of it. So, if the object CD be changed from the direct position to the oblique one C<hi rend="italics">d,</hi> its apparent magnitude would then be only the angle CE<hi rend="italics">d,</hi> instead of the angle CED. <figure/></p><p>If the eye E be placed between two parallels AB, CD, these parallels will appear to converge or come nearer and nearer to each other the farther they are continued out, and at last they will appear to coincide in that point where the sight terminates, which will happen when the optic angle BED becomes equal to about one minute of a degree, the smallest angle under which an object is visible.—Also the apparent magnitudes of the same object FG or BD, seen at different distances, that is the angles FEG, BED, are in a less ratio than the reciprocal ratio of the distances, or the distance increases in a greater ratio than the angle or apparent magnitude diminishes. But when the object is very remote, or the optic angle is very small, as one degree or thereabouts, the angle then varies nearly as the distance reciprocally.</p><p>But although the optic angle be the usual or sensible measure of the apparent magnitude of an object, yet habit, and the frequent experience of looking at distant objects, by which we know that they are larger than they appear, has so far prevailed upon the imagination and judgment, as to cause this too to have some share in our estimation of apparent magnitudes; so that these will be judged to be more than in the ratio of the optic angles.</p><p>The apparent magnitude of the same object, at the same distance, is different to different persons, and different animals, and even to the same person, when viewed in different lights, all which may be occasioned <pb n="127"/><cb/> by the different magnitudes of the eye, causing the optic angle to differ as that is greater or less: and since, in the same person, the more light there comes from an object, the less is the pupil of the eye, looking at that object; therefore the optic angle will also be less, and consequently the apparent magnitude of the object. Every one must have experienced the truth of this, by looking at another person in a room, and afterwards abroad in the sunshine, when he always appears smaller than in a room where the light is less. So also, objects up in the air, having more light coming from them than when they are upon the ground, or near it, may appear less in the former case than in the latter; like as the ball of the cross on the top of St. Paul's church, which is 6 feet in diameter, appears less than an object of the same diameter seen at the same distance below, near the ground. And this may be the chief reason why the sun and moon appear so much larger when seen in the horizon, where their beams are weak, then when they are raised higher, and their light is more bright and glaring.</p><p>Again, if the eye be placed in a rare medium, and view an object through a denser, as glass or water, having plane surfaces; the object will appear larger than it is: and contrariwise, smaller. And hence it is that fishes, and other objects, seen in the water, by an eye in the air, always appear larger than in the air.—In like manner, an object will appear larger when viewed through a globe of glass or water, or any convex spherical segments of these; and, on the contrary, it will appear smaller when viewed through a concave of glass or water.</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">Distance,</hi> is that distance which we judge an object to be from us, when seen afar off. This is commonly very different from the true distance; because we are apt to think that all very remote objects, whose parts cannot well be distinguished, and which have no other visible objects near them, are at the same distance from us; though perhaps they may bethousands or millions of miles off; as in the case of the sun and moon. The apparent distances of objects are also greatly altered by the refraction of the medium through which they are seen.</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">Figure,</hi> is the figure or shape which an object appears under when viewed at a distance; and is osten very different from the true figure. For a straight line, viewed at a distance, may appear but as a point; a surface, as a line; and a solid, as a surface. Also these may appear of different magnitudes, and the surface and solid of different figures, according to their situation with respect to the eye: thus, the arch of a circle may appear a straight line; a square, a trapezium, or even a triangle; a circle, an ellipsis; angular magnitudes, round; and a sphere, a circle. Also all objects have a tendency to roundness and smoothness, or appear less angular, as their distance is greater: for, as the distance is increased, the smaller angles and asperities first disappear, by subtending a less angle than one minute; after these, the next larger disappear, for the same reason; and so on continually, as the distance is more and more increased; the object seeming still more and more round and smooth. So, a triangle, or square, at a great distance, shews only as a round speck; and the edge of the moon appears round to the eye, <cb/> notwithstanding the hills and valleys on her surface. And hence it is also, that near objects, as a range of lamps, or such like, seen at a great distance, appear to be contiguous, and to form one uniform continued magnitude, by the intervals between them disappearing, from the smallness of the angles subtended by them.</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">Motion,</hi> is either that motion which we perceive in a distant body that moves, the eye at the same time being either in motion or at rest; or that motion which an object at rest seems to have, while the eye itself only is in motion.</p><p>The motions of bodies at a great distance, though really moving equally, or passing over equal spaces in equal times, may appear to be very unequal and irregular to the eye, which can only judge of them by the mutation of the angle at the eye. And motions, to be equally visible, or appear equal, must be directly proportional to the distances of the objects moving. Again, very swift motions, as those of the luminaries, may not appear to be any motions at all, but like that of the hour hand of a clock, on account of the great distance of the objects: and this will always happen, when the space actually passed over in one second of time, is less than about the 14000th part of its distance from the eye; for the hour hand of a clock, and the stars about the earth, move at the rate of 15 seconds of a degree in one second of time, which is only the 13751 part of the radius or distance from the eye. On the other hand, it is possible for the motion of a body to be so swift, as not to appear any motion at all; as when through the whole space it describes there constantly appears a continued surface or solid as it were generated by the motion of the object, like as when any thing is whirled very swiftly round, describing a ring, &c.</p><p>Also, the more oblique the eye is to the line which a distant body moves in, the more will the apparent motion differ from the true one. So, if a body revolve with an equable motion in the circumference of the circle ABCD &c, and the eye be <figure/> at E in the plane of the circle; as the body moves from A to B and C, it seems to move slower and slower along the line ALK, till when the body arrives at C, it appears at rest at K; then while it really moves from C by D to F, it appears to move quicker and quicker from K by L to A, where its motion is quickest of all; after this it appears to move slower and slower from A to N while the body moves from F to H: there becoming stationary again, it appears to return from N to A in the straight line, while it really moves from H by I to A in the circle. And thus it appears to move in the line KN by a motion continually varying between the least, or nothing, at the extremes K and N, and the greatest of all at the middle point A. Or, if the motion be referred to the concave side of the circle, instead of the line KN, the appearances will be the same.</p><p>If an eye move directly forwards from E to O, &c; any remote object at rest at P, will appear to move the <pb n="128"/><cb/> contrary way, or from P to Q, <figure/> with the same velocity. But if the object P move the same way, and with the same velocity as the eye; it will seem to stand still. If the object have a less velocity than the eye, it will appear to move back towards Q with the difference of the velocities; and if it move faster than the eye, it will appear to move forwards from Q, with the same difference of the velocities. And so likewise when the object P moves contrary to the motion of the eye, it appears to move backwards with the sum of the motions of the two. And the truth of all this is experienced by persons sitting in a boat moving on a river, or in a wheel-carriage when running fast, and viewing houses or trees, &c, on the shore or side of the road, or other boats or wheel-carriages also in motion.</p><p><hi rend="smallcaps">Apparent Place</hi> <hi rend="italics">of an object,</hi> in Optics, is that in which it appears, when seen in or through glass, water, or other refracting mediums; which is commonly different from the true place. So, if an object be seen in or through glass, or water, either plane or concave, it will appear nearer to the eye than its true place; but when seen through a convex glass, it appears more remote from the eye than the real place of it.</p><p><hi rend="smallcaps">Apparent Place</hi> <hi rend="italics">of the Image of an object,</hi> in Catoptrics, is that where the image of an object made by the reflexion of a speculum appears to be; and the optical writers, from Euclid downwards, give it as a general rule that this is where the reflected rays meet the perpendicular to the speculum drawn from the object: so that if the speculum be a plane, the apparent place of the image will be at the same distance behind the speculum as the eye is besore it; if convex, it will appear behind the glass nearer to the same; but if concave, it will appear before the speculum. And yet in some cases there are some exceptions to this rule, as is shewn by Kepler in his <hi rend="italics">Paralipomena in Vitellionem, prop.</hi> 18. See also Wolfius <hi rend="italics">Catoptr.</hi> § 51, 188, 233, 234.</p><p><hi rend="smallcaps">Apparent</hi> <hi rend="italics">Place of a Planet,</hi> &c, in Astronomy, is that point in the sursace of the sphere of the world, where the centre of the luminary appears from the surface of the earth.</p></div1><div1 part="N" n="APPARITION" org="uniform" sample="complete" type="entry"><head>APPARITION</head><p>, in Astronomy, denotes a star's or other luminary's becoming visible, which before was hid. So, the heliacal rising, is rather an apparition than a proper rising.</p><p><hi rend="italics">Circle of perpetual</hi> <hi rend="smallcaps">Apparition.</hi> See <hi rend="smallcaps">Circle</hi> <hi rend="italics">of perpetual apparition.</hi></p></div1><div1 part="N" n="APPEARANCE" org="uniform" sample="complete" type="entry"><head>APPEARANCE</head><p>, in Perspective, is the representation or projection of a figure, body, or the like object, on the perspective plane.—The appearance of an objective right line, is always a right line. See P<hi rend="smallcaps">ERSPECTIVE.</hi>—Having given the appearance of an opake body, and of a luminary, to sind the appearance of the shadow; see <hi rend="smallcaps">Shadow.</hi></p><p><hi rend="smallcaps">Appearance</hi> <hi rend="italics">of a star or planet.</hi> See <hi rend="smallcaps">Apparition.</hi></p><div2 part="N" n="Appearances" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Appearances</hi></head><p>, in Astronomy, &c, are more usually called <hi rend="italics">phænomena</hi> and <hi rend="italics">phases.</hi>—In <hi rend="italics">Optics,</hi> the term <hi rend="italics">direct appearance</hi> is used for the view or sight of any object by direct rays; without either refraction or reflexion.</p><p>APPIAN. See <hi rend="smallcaps">Apian.</hi> <cb/></p></div2></div1><div1 part="N" n="APPLICATE" org="uniform" sample="complete" type="entry"><head>APPLICATE</head><p>, <hi rend="smallcaps">Applicata</hi>, <hi rend="italics">Ordinate</hi> <hi rend="smallcaps">Applicate</hi>, in Geometry, is a right line drawn to a curve, and bisected by its diameter. This is otherwise called an <hi rend="smallcaps">Ordinate</hi>, which see.</p><p><hi rend="smallcaps">Applicate</hi> <hi rend="italics">Number.</hi> See <hi rend="smallcaps">Concrete.</hi></p><div2 part="N" n="Application" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Application</hi></head><p>, the act of applying one thing to another, by approaching or bringing them nearer together. So a longer space as measured by the continual <hi rend="italics">application</hi> of a less, as a foot or yard by an inch, &c. And motion is determined by a successive application of any thing to different parts of space.</p><p><hi rend="smallcaps">Application</hi> is sometimes used, both in Arithmetic and Geometry, for the rule or operation of division, or what is similar to it in geometry. Thus 20 applied to, or divided by 4, gives 5. And a rectangle <hi rend="italics">ab,</hi> applied to a line <hi rend="italics">c,</hi> gives the 4th proportional <hi rend="italics">ab/c</hi>, or another line which, with the given line <hi rend="italics">c,</hi> will contain another rectangle which shall be equal to the given rectangle <hi rend="italics">ab.</hi> And this is the sense in which Euclid uses the term, <hi rend="italics">lib.</hi> 6, <hi rend="italics">pr.</hi> 28.</p></div2><div2 part="N" n="Application" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Application</hi></head><p>, in Geometry, is also used for the act or supposition of putting or placing one figure upon another, to find whether they be equal or unequal; which seems to be the primary way in which the mind first acquires both the idea and proof of equality. And in this way Euclid, and other geometricians, demonstrate some of the first or leading properties in geometry. Thus, if two triangles have two sides in the one triangle equal to two sides in the other, and also the angle included by the same sides equal to each other; then are the two triangles equal in all respects: for by conceiving the one triangle placed on the other, it is proved that they coincide or exactly agree in all their parts. And the same happens if, of two triangles, one side and the two adjacent angles of the one triangle, are equal, respectively, to one side and the two corresponding angles of the other. Thus also it may be proved that the diameter of a circle divides it into two equal parts, as also that the diagonal of a square or parallelogram bisects or divides it into two equal parts.</p><p><hi rend="smallcaps">Application</hi> of one science to another, as of Algebra to Geometry, is said of the use made of the principles and properties of the one for augmenting and perfecting the other. Indeed all arts and sciences mutually receive aid from each other. But the application here meant, is of a more express and immediate nature; as will appear by what follows.</p><p><hi rend="smallcaps">Application</hi> <hi rend="italics">of Algebra</hi> or <hi rend="italics">of Analysis to Geometry.</hi> The first and principal applications of algebra, were to arithmetical questions and computations, as being the first and most useful science in all the concerns of human life. Afterwards algebra was applied to geometry and all the other sciences in their turn. The application of algebra to geometry, is of two kinds; that which regards the plane or common geometry, and that which respects the higher geometry, or the nature of curve lines.</p><p>The first of these, or the application of algebra to common geometry, is concerned in the algebraical solution of geometrical problems, and finding out theorems in geometrical figures, by means of algebraical investigations or demonstrations. This kind of appli- <pb n="129"/><cb/> cation has been made from the time of the most early writers on algebra, as Diophantus, Lucas de Burgo, Cardan, Tartalea, &c, &c, down to the present times. Some of the best precepts and exercises of this kind of application, are to be met with in Newton's <hi rend="italics">Universal Arithmetic,</hi> and in Thomas Simpson's <hi rend="italics">Algebra</hi> and <hi rend="italics">Select Exercises.</hi> Geometrical Problems are commonly resolved more directly and easily by algebra, than by the geometrical analysis, especially by young beginners; but then the synthesis, or construction and demonstration, is most elegant as deduced from the latter method. Now it commonly happens that the algebraical solution succeeds best in such problems as respect the sides and other lines in geometrical figures, and on the contrary, those problems in which angles are concerned, are best effected by the geometrical analysis. Newton gives these, among many other remarks on this branch. Having any problem proposed; compare together the quantities concerned in it; and, making no difference between the known and unknown quantities, consider how they depend upon, or are related to, one another; that we may perceive what quantities, if they are assumed, will, by proceeding synthetically, give the rest, and that in the simplest manner. And in this comparison, the geometrical figure is to be feigned and constructed at random, as if all the parts were actually known or given, and any other lines drawn that may appear to conduce to the easier and simpler solution of the problem. Having considered the method of computation, and drawn out the scheme, names are then to be given to the quantities entering into the computation, that is, to some few of them, both known and unknown, from which the rest may most naturally and simply be derived or expressed, by means of the geometrical properties of figures, till an equation be obtained, by which the value of the unknown quantity may be derived by the ordinary methods of reduction of equations, when only one unknown quantity is in the notation; or till as many equations are obtained as there are unknown letters in the notation.</p><p>For example, suppose it were <figure/> required to inscribe a square in a given triangle. Let ABC be the given triangle; and feign DEFG to be the required square; also draw the perpendicular BP of the triangle, which will be given, as well as all the sides of it. Then, considering that the triangles BAC, BEF are similar, it will be proper to make the notation as follows, viz, making the base AC=<hi rend="italics">b,</hi> the perpendicular BP=<hi rend="italics">p,</hi> and the side of the square DE or EF=<hi rend="italics">x.</hi> Hence then ; consequently, by the proportionality of the parts of those two similar triangles, viz, BP : AC :: BQ : EF, it is <hi rend="italics">p : b :: p</hi>-<hi rend="italics">x: x;</hi> then, multiply extremes and means &c, there arises , or , and the side of the square sought; that is, a fourth proportional to the base and perpendicular, and the sum of the two, taking this sum for the first term, or AC + BP : BP :: AC : EF.</p><p>The other branch of the application of algebra to geo- <cb/> metry, was introduced by Descartes, in his Geometry, which is the new or higher geometry, and respects the nature and properties of curve lines. In this branch, the nature of the curve is expressed or denoted by an algebraic equation, which is thus derived: A line is conceived to be drawn, as the diameter or some other principal line about the curve; and upon any indesinite points of this line other lines are erected perpendicularly, which are called ordinates, whilst the parts of the first line cut off by them, are called abscisses. Then, calling any absciss <hi rend="italics">x,</hi> and its corresponding ordinate <hi rend="italics">y,</hi> by means of the known nature, or relations of the other lines in the curve, an equation is derived, involving <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> with other given quantities in it. Hence, as <hi rend="italics">x</hi> and <hi rend="italics">y</hi> are common to every point in the primary line, that equation, so derived, will belong to every position or value of the absciss and ordinate, and so is properly considered as expressing the nature of the curve in all points of it; and is commonly called the equation of the curve.</p><p>In this way it is found that any curve line has a peculiar form of equation belonging to it, and which is different from that of every other curve, either as to the number of the terms, the powers of the unknown letters <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> or the signs or co- <figure/> efficients of the terms of the equation. Thus, if the curve line HK be a circle, of which HI is part of the diameter, and IK a perpendicular ordinate: then put HI=<hi rend="italics">x,</hi> IK=<hi rend="italics">y,</hi> and <hi rend="italics">p</hi>=the diameter of the circle, the equation of the circle will be . But if HK be an ellipse, an hyperbola, or parabola, the equation of the curve will be different, and for all the four curves, will be respectively as follows, viz, where <hi rend="italics">t</hi> is the transverse axis, and <hi rend="italics">p</hi> its parameter. And, in like manner for other curves.</p><p>This way of expressing the nature of curve lines, by algebraic equations, has given occasion to the greatest improvement and extension of the geometry of curve lines; for thus, all the properties of algebraic equations, and their roots, are transferred and added to the curve lines, whose abscisses and ordinates have similar properties. Indeed the benefit of this sort of application is mutual and reciprocal, the known properties of equations being transferred to the curves they represent; and, on the contrary, the known properties of curves transferred to their representative equations. See <hi rend="smallcaps">Curves.</hi></p><p><hi rend="smallcaps">Application</hi> <hi rend="italics">of Geometry to Algebra.</hi> Besides the use and application of the higher geometry, namely, of curve lines, to detecting the nature and roots of equations, and to the finding the values of those roots by the geometrical construction of curve lines, even common geometry may be made subservient to the purposes of algebra. Thus, to take a very plain and simple instance, if it were required to square the binomial <hi rend="italics">a</hi>+<hi rend="italics">b</hi>; <pb n="130"/><cb/> by forming a square, as in the annexed figure, whose side is equal to <hi rend="italics">a</hi>+<hi rend="italics">b,</hi> or the two lines or parts added together denoted by the letters <hi rend="italics">a</hi> and <hi rend="italics">b</hi>; and then drawing two lines parallel to the sides, from the points where the two parts join, it will be im- <figure/> mediately evident that the whole square of the compound quantity <hi rend="italics">a</hi>+<hi rend="italics">b,</hi> is equal to the squares of both the parts, together with two rectangles under the two parts, or <hi rend="italics">a</hi><hi rend="sup">2</hi> and <hi rend="italics">b</hi><hi rend="sup">2</hi> and 2<hi rend="italics">ab,</hi> that is the square of <hi rend="italics">a</hi>+<hi rend="italics">b</hi> is equal to <hi rend="italics">a</hi><hi rend="sup">2</hi>+<hi rend="italics">b</hi><hi rend="sup">2</hi>+2<hi rend="italics">ab,</hi> as derived from a geometrical figure or construction. And in this very manner it was, that the Arabians, and the early European writers on algebra, derived and demonstrated the common rule for resolving compound quadratic equations. And thus also, in a similar way, it was, that Tartalea and Cardan derived and demonstrated all the rules for the resolution of cubic equations, using cubes and parallelopipedons instead of squares and rectangles. And many other instances might be given of the use and application of geometry in algebra.</p><p><hi rend="smallcaps">Application</hi> <hi rend="italics">of Algebra and Geometry to Mechanics.</hi> This is founded on the same principles as the application of algebra to geometry. It consists principally in representing by equations the curves described by bodies in motion, by determining the equation between the spaces which the bodies describe, when actuated by any forces, and the times employed in describing them, &c. A familiar instance also of the application of geometry to mechanies, may be seen at the article A<hi rend="smallcaps">CCELERATION</hi>, where the perpendiculars of triangles represent the times, the bases the velocities, and the areas the spaces described by bodies in motion; a method first given by Galileo. In short, as velocities, times, forces, spaces, &c, may be represented by lines and geometrical figures; and as these again may be treated algebraically; it is evident how the principles and properties, of both algebra and geometry, may be applied to mechanics, and indeed to all the other branches of mixt mathematics.</p><p><hi rend="smallcaps">Application</hi> <hi rend="italics">of Mechanies to Geometry.</hi> This consists chiefly in the use that is sometimes made of the centre of gravity of figures, for determining the contents of solids described by those figures.</p><p><hi rend="smallcaps">Application</hi> <hi rend="italics">of Geometry and Astronomy to Geography.</hi> This consists chiefly in three articles. 1st, In determining the figure of the globe we inhabit, by means of geometrical and astronomical operations. 2d, In determining the positions of places, by observations of latitudes and longitudes. 3d, In determining, by geometrical operations, the positions of such places as are not far distant from one another.</p><p>Geometry and Astronomy are also of great use in Navigation.</p><p><hi rend="smallcaps">Application</hi> <hi rend="italics">of Geometry and Algebra to Physics or Natural Philosophy.</hi> This application we owe to Newton, whose philosophy may therefore be called the geometrical or mathematical philosophy; and upon this application are founded all the physico-mathematical sciences. Here a single observation or experiment will often produce a whole science: so when we know, as we do by experience, that the rays of light, in reflect- <cb/> ing, make the angle of incidence equal to the angle of reflexion; we thence deduce the whole science of catoptrics: for that experiment once admitted, catoptrics become a science purely geometrical, since it is reduced to the comparison of angles and lines given in position. And the same in many other sciences.</p><p>APPLICATION <hi rend="italics">of one thing to another,</hi> in general, is employed to denote the use that is made of the former, to understand or to perfect the latter. Thus, the application of the cycloid to pendulums, means the use made of the cycloidal curve for improving the doctrine and use of pendulums.</p><p>APPLY. This term is used two different ways, in geometry.</p><p>1st, It signifies to transfer or place a given line, either in a circle or some other figure, so that the extremities of the line shall be in the perimeter of the figure.</p><p>2d, It is also used to express division in geometry, or to find one dimension of a rectangle, when the area and the other dimension are given. As the area <hi rend="italics">ab</hi> applied to the line <hi rend="italics">c,</hi> is <hi rend="italics">ab/c.</hi></p></div2></div1><div1 part="N" n="APPROACH" org="uniform" sample="complete" type="entry"><head>APPROACH</head><p>, the <hi rend="italics">curve of equable approach.</hi> It was first proposed by Leibnitz, namely, to find a curve, down which a body descending by the force of gravity, shall make equal approaches to the horizon in equal portions of time. It has been found by Bernoulli and others, that the curve is the second cubical parabola, placed with its vertex uppermost, and which the descending body must enter with a certain determinate velocity.—Varignon rendered the question general for any law of gravity, by which a body may approach towards a given point by equal spaces in equal times. And Maupertuis also resolved the problem in the cafe of a body descending in a medium which resists as the square of the velocity. <hi rend="italics">See Hist. de l' Acad. des Sciences</hi> for 1699 and 1730.</p><p><hi rend="italics">Method of</hi> <hi rend="smallcaps">Approaches</hi>, a name given by Dr. Wallis, in his Algebra, to a method of resolving certain problems relating to square numbers, &c. This is done by first assigning certain limits to the quantities required, and then approaching nearer and nearer till a coincidence is obtained.—In this sense, the method of Trial-and-error, or double rule of False Position, may be considered as a method of approaches.</p></div1><div1 part="N" n="APPROACHES" org="uniform" sample="complete" type="entry"><head>APPROACHES</head><p>, in <hi rend="italics">Fortification,</hi> the several works made by the besiegers, for advancing or getting nearer to a fortress or place besieged. Such as the trenches, mines, saps, lodgments, batteries, galleries, epaulments, &c.</p><div2 part="N" n="Approaches" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Approaches</hi></head><p>, or <hi rend="italics">Lines of</hi> <hi rend="smallcaps">Approach</hi>, are particularly used for trenches dug in the ground, and the earth thrown up on the side next the place besieged; under the defence or shelter of which, the besiegers may approach without loss, as near as possible to the place, to raise batteries and plant guns &c, to batter it.—The lines of approach are commonly carried on, in a zig-zag way, parallel to the opposite faces of the besieged work, or nearly so, that they may not be enfiladed by the guns from the enemy's works. And they are also connected by parallels or lines of communication.—The besieged commonly make counter-approaches, to interrupt and defeat the approaches of the besiegers. <pb n="131"/><cb/></p><p>The ancients made their approaches towards the place besieged, much after the same manner as the moderns. Folard shews, that they had their trenches, their parallels, saps, &c.; which, though usually thought of modern invention, it appears, have been practised long before, by the Greeks, Romans, Asiatics, &c.</p></div2></div1><div1 part="N" n="APPROXIMATION" org="uniform" sample="complete" type="entry"><head>APPROXIMATION</head><p>, a continual approach, still nearer and nearer, to a root or any quantity sought.— Methods of continual approximation for the square roots and cube roots of numbers, have been employed by algebraists and arithmeticians, from Lucas de Burgo down to the present time. And the later writers have given various approximations, not only for the roots of higher powers, or all simple equations, but for the roots of all sorts of compound equations whatever: especially Newton, Wallis, Raphson, Halley, De Lagny, &c, &c; all of them forming a kind of insinite series, either expressed or understood, converging nearer and nearer to the quantity sought, according to the nature of the process.</p><p>It is evident that if a number proposed be not a true square, then no exact square root of it can be found, explicable by rational numbers, whether integers or fractions: therefore, in such cases, we must be content with approximations, or coming continually nearer and nearer to the truth. In like manner, for the cube and other roots, when the proposed quantities are not exact cubes, or other powers.</p><p>The most easy and general method of approximation, is perhaps by the rule of Double Position, or, what is sometimes called, the Method of Trial-and-error; which method see under its own name. And among all the methods for the roots of pure powers, of which there are many, I believe the best is that which was discovered by myself, and given in the first volume of my Mathematical Tracts, in point of ease, both of execution and for remembering it. The method is this: if N denote any number, out of which is to be extracted the root whose index is denoted by <hi rend="italics">r,</hi> and if <hi rend="italics">n</hi> be the nearest root first taken; then shall the required root of N very nearly; or as <hi rend="italics">r</hi> - 1 times the given number added to <hi rend="italics">r</hi> + 1 times the nearest power, is to <hi rend="italics">r</hi> + 1 times the given number added to <hi rend="italics">r</hi> - 1 times the nearest power, so is the assumed root <hi rend="italics">n,</hi> to the required root, very nearly. Then this last value of the root, so found, if one still nearer is wanted, is to be used for <hi rend="italics">n</hi> in the same theorem, to repeat the operation with it. And so on, repeating the operation as often as necessary. Which theorem includes all the rational formulæ of Halley and De Lagny.</p><p>For example, suppose it were required to double the cube, or to find the cube root of the number 2. Here <hi rend="italics">r</hi>=3; consequently , and ; and therefore the general theorem becomes for the cube root of N; or as N + 2<hi rend="italics">n</hi><hi rend="sup">3</hi> : 2N + <hi rend="italics">n</hi><hi rend="sup">3</hi> :: <hi rend="italics">n</hi> : the root sought nearly. Now, in this case, N = 2, and therefore the nearest root <hi rend="italics">n</hi> is 1, and its cube <hi rend="italics">n</hi><hi rend="sup">3</hi> = 1 also: hence , and ; therefore, as 4 : 5 :: 1 : 5/4 or 1 1/4=1.25 the first approximation. <cb/> Again, taking <hi rend="italics">r</hi>=(5/4), and consequently <hi rend="italics">r</hi><hi rend="sup">3</hi>=(125/64); hence ; therefore as 378 : 381, or as 126 : 127 :: 5/4 : (635/504)= 1.259921, which is the cube root of 2, true in all the sigures. And by taking 635/504 for a new value of <hi rend="italics">n,</hi> and repeating the process again, a great many more figures may be found.</p><p><hi rend="italics">Of the Roots of Equations by</hi> <hi rend="smallcaps">Approximation.</hi>— Stevinus and Vieta gave methods for sinding values, always nearer and nearer, of the roots of equations. And Oughtred and others pursued and improved the same. These however were very tedious and imperfect, and required a different process for every degree of equations. But Newton introduced, not only general methods for expressing radical quantities by approximating infinite series, but also for the roots of all sorts of compound equations whatever, which are both easy and expeditious: which will be more particularly described under each respective word or article. His method for approximating of roots, is in substance this: First take a value of the root as near as may be, by trials, either greater or less; then assuming another letter to denote the unknown difference between this and the true value, substitute into the equation the sum or difference of the approximate root and this assumed letter, instead of the unknown letter or root of the equation, which will produce a new equation having only the assumed small difference for its root or unknown letter; and, by any means, find, from this equation, a near value of this small assumed quantity. Assume then another letter for the small difference between this last value and the true one, and substitute the sum or difference of them into the last equation, by which will arise a third equation, involving the second assumed quantity; whose near value is found as before. Proceeding thus as far as we please, all the near values, connected together by their proper signs, will form a series approaching still nearer and nearer to the true value of the root of the first or proposed equation. The approximate values of the several small assumed differences, may be found in different ways: Newton's method is this: As the quan tity sought is small, its higher powers decrease more and more, and therefore neglecting them will not lead to any great error, Newton therefore neglects all the terms having in them the 2d and higher powers, leaving only the 1st power and the absolute known term; from which simple equation he always sinds the value of the assumed unknown letter nearly, in a very simple and easy manner. Halley's method of doing the samething, was to neglect all the terms above the square or 2d power, and then to sind the root of the remaining quadratic equation; which would indeed be a nearer value of the assumed letter than Newton's was, but then it is much more troublesome to perform.—Raphson has another way, which is a little varied from that of New- <pb n="132"/><cb/> ton's again, which is this: having found a near value of the first assumed small quantity or difference, by this he corrects the first approximation to the root of the proposed equation; and then, assuming another letter for the next, or smaller difference, he introduces it into the original equation in the same way as before. And thus he proceeds, from one correction to another, employing always the first proposed equation to find them, instead of the fuccessive new equations used by Newton.</p><p>For example, let it be required to find the root of the equation , or :—Here the root <hi rend="italics">x,</hi> it is evident, is nearly = 8; for <hi rend="italics">x</hi> therefore take 8 + <hi rend="italics">z,</hi> and substitute 8 + <hi rend="italics">z</hi> for <hi rend="italics">x</hi> in the given equation, and the terms will be thus; Hence, then, collecting all the assumed differences, with their signs, it is found that the root of the equation required, by Newton's method. <hi rend="center"><hi rend="italics">The same by Raphson's way.</hi></hi> . <cb/></p><p><hi rend="smallcaps">Example</hi> 2. Again, taking the cubic equation the root of the equation . And in the same manner Newton performs the approximation for the roots of literal equations, that is, equations having literal coefficients; so the root of this equation</p><p>See also a memoir on this method by the Marquis de Courtivron, in the <hi rend="italics">Memoires de l'Academie</hi> for 1744.</p><p><hi rend="italics">Other Methods of</hi> <hi rend="smallcaps">Approximation.</hi> Besides the foregoing general methods, other particular ways of approximating, for various purposes, have been given by many other persons.—As for example, methods of approximating, by series, to the roots of cubic equations belonging to the irreducible case, by Nicole in the same <hi rend="italics">Memoirs,</hi> by M. Clairaut in his Algebra, and by myself in the <hi rend="italics">Philos. Trans.</hi> for 1780. See also several parts of Simpson's works, and my Tracts vol. 1. Also the methods of infinite series by Wallis, Newton, Gregory, Mercator, &c, may be considered as approximations, in quadratures, and other branches of the mathematics, many instances of which may be seen in Wallis's Algebra, and other books:—Likewise the method of exhaustions of the ancients, by which Archimedes and others have approximated to the quadrature and rectification of the circle, &c, which was performed by continually bisecting the sides of polygons, both inscribed in a circle and circumscribed about it; by which means the sum of the sides of the like polygons approach continually nearer and nearer together, and the circumference of the circle is nearly a mean between the two sums. See also <hi rend="smallcaps">Equations.</hi></p></div1><div1 part="N" n="APPULSE" org="uniform" sample="complete" type="entry"><head>APPULSE</head><p>, in Astronomy, means the actual contact of two luminaries, according to some authors; but others describe it as their near approach to each other, so as to be seen, for instance, within the same telescope.</p><p>The appulses of the planets to the fixed stars have <pb n="133"/><cb/> always been very useful to astronomers, as serving to fix and determine the places of the former. The ancients, wanting an easy method of comparing the planets with the ecliptic, which is not visible, had scarce any other way of fixing their situations, but by observing their track among the fixed stars, and marking their appulses to some of those visible points. See Hist. Acad. Scienc. for 1710, pa. 417. And Philos. Trans. No. 369, where Dr. Halley has given a method of determining the places of the planets, by observing their near appulses to the fixed stars. See also Philos. Trans. No. 76, pa. 361, and Mem. Acad. Scienc. for 1708, where Flamsteed and De la Hire have given observations of the moon's appulses to the Pleiades. See also Flamsteed's Historia Cœlestis, where a multitude of observations of appulses, or small distances, of the moon and planets, from the fixed stars, are recorded. And Dr. Halley has published a map or planisphere of the starry zodiac, in which are accurately laid down all the stars to which the moon's appulse has ever been observed in any part of the world. See Philos. Trans. No. 369; or Abridg. vol. vi. pa. 170.</p></div1><div1 part="N" n="APRIL" org="uniform" sample="complete" type="entry"><head>APRIL</head><p>, the 4th month of the year according to the common computation, and the 2d from the vernal equinox.—The word is derived from <hi rend="italics">Aprilis,</hi> of <hi rend="italics">aperio, I open</hi>; because the earth, in this month, begins to open her bosom for the production of vegetables.—In this month the sun travels through part of the signs Aries and Taurus.</p></div1><div1 part="N" n="APRON" org="uniform" sample="complete" type="entry"><head>APRON</head><p>, in Gunnery, a piece of thin or sheet lead, used to cover the vent or touch-hole of a cannon.</p></div1><div1 part="N" n="APSES" org="uniform" sample="complete" type="entry"><head>APSES</head><p>, in Astronomy, are the two points in the orbits of planets, where they are at their greatest and least distance, from the sun or the earth. The point at the greatest distance being called the <hi rend="italics">higher apsis,</hi> and that at the nearest distance the <hi rend="italics">lower apsis.</hi> And the two apses are also called <hi rend="italics">auges.</hi> Also the higher apsis is more particularly called the <hi rend="italics">aphelion,</hi> or the <hi rend="italics">apogee</hi>; and the lower apsis, the <hi rend="italics">perihelion,</hi> or the <hi rend="italics">perigee.</hi> The diameter which joins these two points, is called the <hi rend="italics">line of the apses</hi> or of the <hi rend="italics">apsides</hi>; and it paffes through the centre of the orbit of the planet, and the centre of the sun or the earth; and in the modern astronomy this line makes the longer or transverse axis of the elliptical orbit of the planet. In this line is counted the excentricity of the orbit; being the distance between the centre of the orbit and the focus, where is placed the sun or the earth.</p><p>The foregoing definitions suppose the lines of the greatest and least distances to lie in the same straight line; which is not always precisely the case; as they are sometimes out of a right line, making an angle greater or less than 180 degrees, and the difference from 180 degrees is called the motion of the line of the apses: when the angle is less than 180 degrees, the motion of the apses is said to be contrary to the order of the signs; on the other hand, when the angle exceeds 180 degrees, the motion is according to the order of the signs.</p><p>Different means have been employed to determine the motion of the <hi rend="italics">apses.</hi> Dr. Keil explains, in his Astronomy, the method used by the ancients, who supposed the orbits of the planets to be perfectly circular, and the sun out of the centre. But since it has been <cb/> discovered that they describe elliptical orbits, various other methods have been devised for determining it. Halley has given one, which supposes to be known only the time of the planet's revolution, or periodic time. Seth Ward has also given a determination from three different observations of a planet, in any three places of its orbit: but his method being founded on an hypothesis not strictly true, Euler has given one much more exact in vol. 7. of the <hi rend="italics">Petersburgh Commentaries.</hi> See various ways explained in the Astronomy of Keil and Mounier.</p><p>Newton has also given, in the <hi rend="italics">Principia,</hi> an excellent method of determining the same motion, on the supposition that the orbits of the planets differ but little from circles, which is the case nearly. That great philosopher shews, that if the sun be immoveable, and all the planets gravitate towards him in the inverse ratio of the squares of their distances, then the apses will be fixed, or their motion nothing; that is, the lines of greatest and least distance will form one right line, and the apses will be directly opposite, or at 180 degrees distance from each other. But, because of the mutual tendency of the planets towards each other, their gravitation towards the sun is not precisely in that ratio; and hence it happens, that the apses are not always exactly in a right line with the sun. And Newton has given a very elegant method of determining the motion of the <hi rend="italics">apses,</hi> on the supposition that we know the force which is thereby added to the gravitation of the planet towards the sun, and that this additional force is always in that direction.</p><p>APUS or <hi rend="smallcaps">Apous</hi>, <hi rend="italics">Avis Indica,</hi> in Astronomy, a constellation of the southern hemisphere, situated near the south pole, between the <hi rend="italics">triangulum australe</hi> and the chameleon, and supposed to represent the bird of paradise. Also supposed to be one of the birds named <hi rend="italics">Apodes,</hi> as having no feet.</p><p>The number of stars contained in this constellation, are 11 in the British Catalogue, in Bayer's Maps 12, and a still greater number in La Caille's Catalogue; the principal star being but of the 4th or 5th order of magnitude. See <hi rend="italics">Cœlum Australe Stelliferum,</hi> and the <hi rend="italics">Memoires de l'Acad.</hi> for 1752, pa. 569.</p></div1><div1 part="N" n="AQUARIUS" org="uniform" sample="complete" type="entry"><head>AQUARIUS</head><p>, in Astronomy, one of the celestial constellations, being the eleventh sign in the zodiac, reckoning from Aries, and is marked by the character <figure/>, representing part of a stream of water, issuing from the vessel of Aquarius, or the water-pourer. This sign also gives name to the eleventh part of the ecliptic, through which the sun moves in part of the months of January and February.</p><p>The poets feign that Aquarius was Ganymede, whom Jupiter ravished under the shape of an eagle, and carried away into Heaven to serve as a cup-bearer, instead of Hebe and Vulcan; whence the name. Others hold, that the sign was thus called, because that when it appears in the horizon, the weather commonly proves rainy.</p><p>The stars in the constellation Aquarius, are, in Ptolemy's Catalogue, 45; in Tycho's 41; in Hevelius's 47; and in Flamsteed's 108. See the article C<hi rend="smallcaps">ONSTELLATION;</hi> also <hi rend="smallcaps">Catalogue.</hi></p></div1><div1 part="N" n="AQUEDUCT" org="uniform" sample="complete" type="entry"><head>AQUEDUCT</head><p>, or <hi rend="smallcaps">Aquæduct</hi>, as much as to say <hi rend="italics">ductus aquæ, a conduit of water,</hi> is a construction of <pb n="134"/><cb/> stone or timber built on uneven ground, to preserve the level of water, and convey it, by a canal, from one place to another.—Some aqueducts are under ground, being conducted through hills, &c; and others are raised above ground, and supported on arches, to conduct the water over vallies, &c.</p><p>The Romans were very magnificent in their aqueducts; having some that extended a hundred miles, or more. Frontinus, a man of consular dignity, who had the direction of the aqueducts under the emperor Nerva, speaks of nine that emptied themselves through 13594 pipes, of an inch diameter. And it is observed by Vigenere, that in the space of 24 hours, Rome received from these aqueducts not less than 500000 hogsheads of water. The chief aqueducts now in being, are these: 1st, that of the Aqua Virginia, repaired by pope Paul IV; 2d, the Aqua Felice, conitructed by pope Sixtus V, and is called from the name he assumed before he was exalted to the papal throne; 3d, the Aqua Paulina, repaired by pope Paul V, in the year 1611; and 4thly, the aqueduct built by Lewis XIV, near Maintenon, to convey the river Bure to Versailles, which is perhaps the largest in the world; being 7000 fathoms long, elevated 2560 fathoms in height, and containing 242 arcades. See Philos. Trans. for 1685, No. 171; or Abridg. vol. 1. pa. 594.</p><p>AQUEOUS <hi rend="smallcaps">Humour</hi>, or the <hi rend="italics">walry humour</hi> of the eye, is the first or outermost, and the rarest of the three humours of the eye. It is transparent and colourless, like water; and it fills up the space that lies between the cornea tunica, and the crystalline humour.</p></div1><div1 part="N" n="AQUILA" org="uniform" sample="complete" type="entry"><head>AQUILA</head><p>, <hi rend="italics">the Eagle,</hi> or the <hi rend="italics">Vulture</hi> as it is sometimes called, is a constellation of the northern hemisphere, usually joined with Antinous. It is one of the 48 old constellations, according to the division of which Hipparchus made his Catalogue of the Fixed Stars, and which are described by Ptolemy. The number of stars in Aquila, and those near it, now in the later-formed constellation Antinous, amount to 15 in Ptolemy's Catalogue, to 19 in Tycho's, to 42 in that of Hevelius, and to 71 in Flamsteed's. But in Aquila alone, Tycho counts only 12 stars, and Hevelius 23; the principal star being Lucida Aquila, and is between the 1st and 2d magnitude. The Greeks, as usual, relate various fables of this constellation, to make the science appear as of their own invention.</p></div1><div1 part="N" n="ARA" org="uniform" sample="complete" type="entry"><head>ARA</head><p>, the <hi rend="italics">Altar,</hi> one of the 48 old constellations, mentioned by the ancient astronomers, and is situated in the southern hemisphere; containing only 7 stars in Ptolemy's Catalogue, and 9 in that of Flamsteed; none of which exceed the 4th magnitude.</p></div1><div1 part="N" n="ARATUS" org="uniform" sample="complete" type="entry"><head>ARATUS</head><p>, celebrated for his Greek poem intitled <foreign xml:lang="greek">faino/mena</foreign>, the <hi rend="italics">Phenomena,</hi> flourished about the 127th Olympiad, or near 300 years besore Christ, while Ptolomy Philadelphus reigned in Egypt. Being educated under Dionysius Heracleotes, a Stoic philosopher, he espoused the principles of that sect, and became physician to Antigonus Gonatus, the son of Demetrius Poliorcetes, King of Macedon. The <hi rend="italics">Phenomena</hi> of Aratus gives him a title to the character of an astronomer, as well as a poet. In this work he describes the nature and motion of the stars, and shews their various dispositions and relations; he describes the figures of <cb/> the constellations, their situations in the sphere, the origin of the names which they bear in Greece and in Egypt, the fables which have given rise to them, the rising and setting of the stars, and he indicates the manner of knowing the constellations by their respective situations.</p><p>The poem of Aratus was commented upon and translated by many authors: of whom among the ancients were Cicero, Germanicus Cæsar, and Festus Avienus, who made Latin translations of it; a part of the former of which is still extant. Aratus must have been much esteemed by the ancients, since we sind so great a number of scholiasts and commentators upon him; among whom are Aristarchus of Samos, the Arystylli the geometricians, Apollonius, the Evæneti, Crates, Numenius the grammarian, Pyrrhus of Magnesia, Thales, Zeno, and many others, as may be seen in Vossius, p. 156. Suidas ascribes several other works to Aratus. Virgil, in his Georgies, has translated or imitated many passages from this author: Ovid speaks of him with admiration, as well as many others of the poets: And St. Paul has quoted a passage from him; which is in his speech to the Athenians (Acts xvii. 28) where he tells them that some of their own poets have said, <hi rend="italics">For we are also his offspring,</hi> these words being the beginning of the 5th line of the Phenomena of Aratus.</p><p>His modern editors are as follow: Henry Stephens published his poem at Paris in 1566, in his collection of the poets, in folio. Grotius published an edition of the Phenomena at Leyden in quarto, 1600, in Greek and Latin, with the fragments of Cicero's version, and the translations of Germanicus and Avienus; all which the editor has illustrated with curious notes. Also a neat and correct edition of Aratus was published at Oxford, 1672, in 8vo. with the Scholia.</p><p>ARÆOMETER, see <hi rend="smallcaps">Areometer.</hi></p></div1><div1 part="N" n="ARC" org="uniform" sample="complete" type="entry"><head>ARC</head><p>, or <hi rend="smallcaps">Arch;</hi> which see.</p></div1><div1 part="N" n="ARCADE" org="uniform" sample="complete" type="entry"><head>ARCADE</head><p>, in Architecture, denotes an opening in the wall of a building formed by an arch.</p><p>ARC-BOUTANT, is a kind of arched buttress, formed of a flat arch, or part of an arch, abutting against the feet or sides of another arch or vault, to support them and prevent them from bursting or giving way.</p></div1><div1 part="N" n="ARCAS" org="uniform" sample="complete" type="entry"><head>ARCAS</head><p>, a name by which some of the old writers call the star Arcturus; a single and very bright star of the first magnitude, between the legs of the constellation Bootes. They say Areas, the son of Calisto by Jupiter, when he was about to have killed his mother in the shape of a bear, was, together with her, snatched up into Heaven; where she was converted into the constellation of the Great Bear, near the north pole, and the youth into this single star.</p></div1><div1 part="N" n="ARCH" org="uniform" sample="complete" type="entry"><head>ARCH</head><p>, <hi rend="smallcaps">Arc</hi>, <hi rend="italics">Arcus,</hi> in Geometry, a part of any curve line; as, of a circle, or ellipsis, or the like.</p><p>It is by means of circular arcs, or arches, that all angles are measured; the arc being deseribed fiom the angular point as a centre. For this purpose, every circle is supposed to be divided into 360 degrees, or equal parts; and an arch, or the angle it subtends and measures, is estimated according to the number of those degrees it contains: thus, an are, or angle, is said to be of 30 or 80, or 100 degrees.—Circular arcs are also of great use in finding of fluents. <pb n="135"/><cb/></p><p><hi rend="italics">Concentric</hi> <hi rend="smallcaps">Arcs</hi>, are such as have the same centre.</p><p><hi rend="italics">Equal</hi> <hi rend="smallcaps">Arcs</hi>, are such arcs, of the same circle, or of equal circles, as contain the same number of degrees. These have also equal chords, sines, tangents, &c.</p><p><hi rend="italics">Similar</hi> <hi rend="smallcaps">Arcs</hi>, of unequal circles, &c, are such as contain the same number of de- <figure/> grees, or that are the like part or parts of their respective whole circles. Hence, in concentric circles, any two radii cut off, or intercept, similar arcs MN and OP.—Similar arcs are proportional to the radii LM, LO, or to the whole circumferences.—Similar arcs of other like curves, are also like parts of the wholes, or determined by like parts alike posited.</p><p><hi rend="italics">Of the Length of Circular</hi> <hi rend="smallcaps">Arcs.</hi> The lengths of circular arcs, as found and expressed in various ways, may be seen in my large Treatise on Mensuration, pa. 118, & seq. 2d edition: some of which are as follow. The radius of a circle being 1; and of any arc <hi rend="italics">a,</hi> if the tangent be <hi rend="italics">t,</hi> the sine <hi rend="italics">s,</hi> the cosine <hi rend="italics">c,</hi> and the versed sine <hi rend="italics">v</hi>: then the arc <hi rend="italics">a</hi> will be truly expressed by several series, as follow, viz, the arc ; where <hi rend="italics">d</hi> denotes the number of degrees in the given arc. Also nearly; where C is the chord of the arc, and <hi rend="italics">c</hi> the chord of half the arc; whatever the radius is.</p><p><hi rend="italics">To investigate the length of the arc of any curve.</hi> Put <hi rend="italics">x</hi>=the absciss, <hi rend="italics">y</hi>=the ordinate, of the arc <hi rend="italics">z,</hi> of any eurve whatever. Put ; then, by means of the equation of the curve, find the value of <hi rend="italics">x</hi><hi rend="sup">.</hi> in terms of <hi rend="italics">y</hi><hi rend="sup">.</hi>, or of <hi rend="italics">y</hi><hi rend="sup">.</hi> in terms of <hi rend="italics">x</hi><hi rend="sup">.</hi>, and substitute that value instead of it in the above expression ; hence, taking the fluents, they will give the length of the arc <hi rend="italics">z,</hi> in terms of <hi rend="italics">x</hi> or <hi rend="italics">y.</hi></p><div2 part="N" n="Arch" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Arch</hi></head><p>, in Astronomy. Of this, there are various kinds. Thus, the latitude, elevation of the pole, and the declination, are measured by an arch of the meridian; and the longitude, by an arch of a parallel circle, &c.</p><p><hi rend="italics">Diurnal</hi> <hi rend="smallcaps">Arch</hi> of the sun, is part of a circle parallel to the equator, described by the sun in his course from his rising to the setting. And his <hi rend="italics">Nocturnal</hi> <hi rend="smallcaps">Arch</hi> is of the same kind; excepting that it is described from setting to rising.</p><p><hi rend="smallcaps">Arch</hi> <hi rend="italics">of Progression,</hi> or <hi rend="italics">Direction,</hi> is an arch of the ecliptic, which a planet seems to pass over, when its motion is direct, or according to the order of the signs.</p><p><hi rend="smallcaps">Arch</hi> of <hi rend="italics">Retrogradation,</hi> is an arch of the ecliptic, <cb/> described while a planet is retrograde, or moves contrary to the order of the signs.</p><p><hi rend="smallcaps">Arch</hi> <hi rend="italics">between the Centres,</hi> in eclipses, is an arch passing from the centre of the earth's shadow, perpendicular to the moon's orbit, meeting her centre at the middle of an eclipse.—If the aggregate of this arch and the apparent semi-diameter of the moon, be equal to the semi-diameter of the shadow, the eclipse will be total for an instant, or without any duration; and if that sum be less than the radius of the shadow, the eclipse will be total, with some duration; but if greater, the eclipse will be only partial.</p><p><hi rend="smallcaps">Arch</hi> <hi rend="italics">of Vision,</hi> is that which measures the sun's depth below the horizon, when a star, before hid by his rays, begins to appear again.—The quantity of this arch is not always the same, but varies with the latitude, declination, right ascension, or descension, and distance, of any planet or star. Ricciol. Almag. v. 1, pa. 42. However, the following numbers will serve nearly for the stars and planets. <hi rend="center">TABLE exhibiting the <hi rend="italics">Arch of Vision</hi> of the <hi rend="smallcaps">Planets</hi> and <hi rend="smallcaps">Fixed Stars.</hi></hi> <table><row role="data"><cell cols="1" rows="1" rend="colspan=3 align=center" role="data"><hi rend="smallcaps">PLANETS.</hi></cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="smallcaps">FIXED STARS.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Magnitude.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" rend="align=right" role="data">10°</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12°</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">13</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">17</cell></row></table></p></div2><div2 part="N" n="Arch" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Arch</hi></head><p>, in Architecture, is a concave structure, raised or turned upon a mould, called the centering, in form of the arch of a curve, and serving as the inward support of some superstructure. Sir Henry Wotton says, An arch is nothing but a narrow or contracted vault; and a vault is a dilated arch.</p><p>Arches are used in large intercolumnations of spacious buildings; in porticoes, both within and without temples; in public halls, as ceilings, the courts of palaces, cloisters, theatres, and amphitheatres. They are also used to cover the cellars in the foundations of houses, and powder magazines; also as buttresses and counter-forts, to support large walls laid deep in the earth; for triumphal arches, gates, windows, &c; and, above all, for the foundations of bridges and aqueducts. And they are supported by piers, butments, imposts, &c.</p><p>Arches are of several kinds, and are commonly denominated from the figure or curve of them; as circular, elliptical, cycloidal, catenarian, &c, according as their curve is in the form of a circle, ellipse, cycloid, catenary, &c.</p><p>There are also other denominations of circular arches, according to the different parts of a circle, or manner of placing them. Thus,</p><p><hi rend="italics">Semicircular</hi> <hi rend="smallcaps">Arches</hi>, which are those that make an exact semicircle, having their centre in the middle of the span or chord of the arch; called also by the French builders, perfect arches, and <hi rend="italics">arches en plein centre.</hi> The arches of Westminster Bridge are semicircular.</p><p><hi rend="italics">Scheme</hi> <hi rend="smallcaps">Arches</hi>, or <hi rend="italics">skene,</hi> are those which are less than semicircles, and are consequently flatter arches; <pb n="136"/><cb/> containing 120, or 90, or 60, degrees, &c. They are also called <hi rend="italics">imperfect</hi> and <hi rend="italics">diminisbed arches.</hi></p><p><hi rend="smallcaps">Arches</hi> <hi rend="italics">of the third and fourth point,</hi> or <hi rend="italics">Gothic arches</hi>; or, as the Italians call them, <hi rend="italics">di terzo</hi> and <hi rend="italics">quarto acuto,</hi> because they always meet in an acute angle at top. These consist of two excentric circular arches, meeting in an angle above, and are drawn from the division of the chord into three or four or more parts at pleasure. Of this kind are many of the arches in churches and other old Gothic buildings.</p><p><hi rend="italics">Elliptical</hi> <hi rend="smallcaps">Arches</hi>, usually consist of semi-ellipses; and were formerly much used instead of mantle-trees in chimnies; and are now much used, from their bold and beautiful appearance, for many purposes, and particularly for the arches of a bridge, like that at Black-Friars, both for their strength, beauty, convenience, and cheapness.</p><p><hi rend="italics">Straight</hi> <hi rend="smallcaps">Arches</hi>, are those which have their upper and under edges parallel straight lines, instead of curves. These are chiefly used over doors and windows; and have their ends and joints all pointing towards one common centre.</p><p><hi rend="smallcaps">Arch</hi> is particularly used for the space between the two piers of a bridge, intended for the passage of the water, boats, &c.</p><p><hi rend="smallcaps">Arch</hi> <hi rend="italics">of equilibration,</hi> is that which is in equilibrium in all its parts, having no tendency to break in one part more than in another, and which is therefore safer and slronger than any other figure. Every particular figure of the extrados, or upper side of the wall above an arch, requires a peculiar curve for the under side of the arch itself, to form an arch of equilibration, so that the incumbent pressure on every part may be proportional to the strength or resistance there. When the arch is equally thick throughout, a case that can hardly ever happen, then the catenarian curve is the arch of equilibration; but in no other case: and therefore it is a great mistake in some authors to suppose that this curve is the best figure for arches in all cases; when in reality it is commonly the worst. This subject is fully treated in my <hi rend="italics">Principles of Bridges,</hi> pr. 5, where the proper intrados is investigated for every extrados, so as to form an arch of equilibration in all cases whatever. It there appears that, when the upper side of the wall is a straight horizontal line, as in the annexed figure, the equation <figure/> of the curve is thus expressed, <cb/> where <hi rend="italics">x</hi>=DP, <hi rend="italics">y</hi>=PC, <hi rend="italics">r</hi>=DQ, <hi rend="italics">h</hi>=AQ, and <hi rend="italics">a</hi>= DK. And hence, when <hi rend="italics">a, h, r,</hi> are any given numbers, a table is formed for the corresponding values of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> by which the curve is constructed for any particular occasion. Thus supposing <hi rend="italics">a</hi> or DK=6, <hi rend="italics">h</hi> or AQ= 50, and <hi rend="italics">r</hi> or DQ = 40; then the corresponding values of KI and IC, or horizontal and vertical lines, will be as in this table. <hi rend="center"><hi rend="italics">Table for constructing the Curve of Equilibration.</hi></hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Value of KI.</cell><cell cols="1" rows="1" rend="align=center" role="data">Value of IC.</cell><cell cols="1" rows="1" rend="align=center" role="data">Value of KI.</cell><cell cols="1" rows="1" rend="align=center" role="data">Value of IC.</cell><cell cols="1" rows="1" rend="align=center" role="data">Value of KI.</cell><cell cols="1" rows="1" rend="align=center" role="data">Value of IC.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">6.000</cell><cell cols="1" rows="1" rend="align=center" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">10.381</cell><cell cols="1" rows="1" rend="align=center" role="data">36</cell><cell cols="1" rows="1" rend="align=center" role="data">21.774</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">6.035</cell><cell cols="1" rows="1" rend="align=center" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">10.858</cell><cell cols="1" rows="1" rend="align=center" role="data">37</cell><cell cols="1" rows="1" rend="align=center" role="data">22.948</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">6.144</cell><cell cols="1" rows="1" rend="align=center" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">11.368</cell><cell cols="1" rows="1" rend="align=center" role="data">38</cell><cell cols="1" rows="1" rend="align=center" role="data">24.190</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">6.324</cell><cell cols="1" rows="1" rend="align=center" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">11.911</cell><cell cols="1" rows="1" rend="align=center" role="data">39</cell><cell cols="1" rows="1" rend="align=center" role="data">25.505</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">6.580</cell><cell cols="1" rows="1" rend="align=center" role="data">25</cell><cell cols="1" rows="1" rend="align=center" role="data">12.489</cell><cell cols="1" rows="1" rend="align=center" role="data">40</cell><cell cols="1" rows="1" rend="align=center" role="data">26.894</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">6.914</cell><cell cols="1" rows="1" rend="align=center" role="data">26</cell><cell cols="1" rows="1" rend="align=center" role="data">13.106</cell><cell cols="1" rows="1" rend="align=center" role="data">41</cell><cell cols="1" rows="1" rend="align=center" role="data">28.364</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">7.330</cell><cell cols="1" rows="1" rend="align=center" role="data">27</cell><cell cols="1" rows="1" rend="align=center" role="data">13.761</cell><cell cols="1" rows="1" rend="align=center" role="data">42</cell><cell cols="1" rows="1" rend="align=center" role="data">29.919</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">7.571</cell><cell cols="1" rows="1" rend="align=center" role="data">28</cell><cell cols="1" rows="1" rend="align=center" role="data">14.457</cell><cell cols="1" rows="1" rend="align=center" role="data">43</cell><cell cols="1" rows="1" rend="align=center" role="data">31.563</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">7.834</cell><cell cols="1" rows="1" rend="align=center" role="data">29</cell><cell cols="1" rows="1" rend="align=center" role="data">15.196</cell><cell cols="1" rows="1" rend="align=center" role="data">44</cell><cell cols="1" rows="1" rend="align=center" role="data">33.299</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">8.120</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">15.980</cell><cell cols="1" rows="1" rend="align=center" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">35.135</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">8.430</cell><cell cols="1" rows="1" rend="align=center" role="data">31</cell><cell cols="1" rows="1" rend="align=center" role="data">16.811</cell><cell cols="1" rows="1" rend="align=center" role="data">46</cell><cell cols="1" rows="1" rend="align=center" role="data">37.075</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=center" role="data">8.766</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" rend="align=center" role="data">17.693</cell><cell cols="1" rows="1" rend="align=center" role="data">47</cell><cell cols="1" rows="1" rend="align=center" role="data">39.126</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">9.168</cell><cell cols="1" rows="1" rend="align=center" role="data">33</cell><cell cols="1" rows="1" rend="align=center" role="data">18.627</cell><cell cols="1" rows="1" rend="align=center" role="data">48</cell><cell cols="1" rows="1" rend="align=center" role="data">41.293</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">9.517</cell><cell cols="1" rows="1" rend="align=center" role="data">34</cell><cell cols="1" rows="1" rend="align=center" role="data">19.617</cell><cell cols="1" rows="1" rend="align=center" role="data">49</cell><cell cols="1" rows="1" rend="align=center" role="data">43.581</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">9.934</cell><cell cols="1" rows="1" rend="align=center" role="data">35</cell><cell cols="1" rows="1" rend="align=center" role="data">20.665</cell><cell cols="1" rows="1" rend="align=center" role="data">50</cell><cell cols="1" rows="1" rend="align=center" role="data">46.000</cell></row></table></p><p>The doctrine and use of arches are neatly delivered by Sir Henry Wotton, though he is not always mathematically accurate in the principles. He says; <hi rend="italics">First,</hi> All matter, unless impeded, tends to the centre of the earth in a perpendicular line. <hi rend="italics">Secondly</hi>; All solid materials, as bricks, stones, &c, in their ordinary rectangular form, if laid in numbers, one by the side of another, in a level row, and their extreme ones sustained between two supporters; those in the middle will necessarily sink, even by their own gravity, much more if forced down by any superincumbent weight. To make them stand, therefore, either their figure or their position must be altered.—<hi rend="italics">Thirdly</hi>; Stones, or other materials, being figured <hi rend="italics">cuneatim,</hi> or wedge-like, broader above than below, and laid in a level row, with their two extremes supported as in the last article, and pointing all to the same centre; none of them can sink, till the supporters or butments give way, because they want room in that situation to descend perpendicularly. But this is a weak structure; because the supporters are subject to too much impulsion, especially where the line is long; for which reason the form of straight arches is seldom used, excepting over doors and windows, where the line is short and the side walls strong. In order to fortify the work, therefore, we must change not only the figure of the materials, but also their position.—<hi rend="italics">Fourthly</hi>; If the materials be shaped wedgewise, and be disposed in form of an arch, and pointing to some centre; in this case, neither the pieces of the said arch can sink downwards, for want of room to descend perpendicularly; nor can the supporters or butments suffer much violence, as in the preceding flat form: for the convexity will always make the incumbent rather rest upon the supporters, than thrust or <pb n="137"/><cb/> push them outwards. His reasoning, however, afterwards, on the effect of circular and other arches, is not accurate, as he attends only to the side pressure, without considering the effect of different vertical pressures.</p><p>The chief properties of arches of different curves, may be seen in the 2d sect. of my <hi rend="italics">Principles of Bridges,</hi> above quoted. It there appears that none, except the mechanical curve of the arch of equilibration, can admit of a horizontal line at top: that this arch is of a form both graceful and convenient, as it may be made higher or lower at pleasure, with the same span or opening: that all other arches require extrados that are curved, more or less, either upwards or downwards: of these, the elliptical arch approaches the nearest to that of equilibration for equality of strength and convenience; and it is also the best form for most bridges, as it can be made of any height to the same span, its hanches being at the same time sufficiently elevated above the water, even when it is very flat at top: elliptical arches also look bolder and lighter, are more uniformly strong, and much cheaper than most others, as they require less materials and labour. Of the other curves, the cycloidal arch is next in quality to the elliptical one, for those properties, and, lastly, the circle. As to the others, the parabola, hyperbola, and catenary, they are quite inadmissible in bridges that consist of several arches; but may, in some cases, be employed for a bridge of one single arch which may be intended to rise very high, as in such cases as they are not much loaded at the hanches.</p><p><hi rend="smallcaps">Arch</hi> <hi rend="italics">Mural.</hi> See <hi rend="smallcaps">Mural</hi> <hi rend="italics">arch.</hi></p></div2></div1><div1 part="N" n="ARCHER" org="uniform" sample="complete" type="entry"><head>ARCHER</head><p>, or <hi rend="italics">Sagittarius,</hi> one of the constellations of the northern hemisphere, and one of the twelve signs of the zodiac, placed between the Scorpion and Capricorn. See <hi rend="smallcaps">Sagittarius.</hi></p></div1><div1 part="N" n="ARCHIMEDES" org="uniform" sample="complete" type="entry"><head>ARCHIMEDES</head><p>, one of the most celebrated mathematicians among the ancients, who flourished about 250 years before Christ, being about 50 years later than Euclid. He was born at Syracuse in Sicily, and was related to Hiero, who was then king of that city. The mathematical genius of Archimedes set him with such distinguished excellence in the view of the world, as rendered him both the honour of his own age, and the admiration of posterity. He was indeed the prince of the ancient mathematicians, being to them what Newton is to the moderns, to whom in his genius and character he bears a very near resemblance. He was frequently lost in a kind of reverie, so as to appear hardly sensible; he would study for days and nights together, neglecting his food; and Plutarch tells us that he used to be carried to the baths by force. Many particulars of his life, and works, mathematical and mechanical, are recorded by several of the ancients, as Polybius, Livy, Plutarch, Pappus, &c. He was equally skilled in all the sciences, astronomy, geometry, mechanics, hydrostatics, optics, &c, in all of which he excelled, and made many and great inventions.</p><p>Archimedes, it is said, made a sphere of glass, of a most furprising contrivance and workmanship, exhibiting the motions of the heavenly bodies in a very pleasing manner. Claudian has an epigram upon this invention, which has been thus translated: When in a glass's narrow space confin'd,<lb/> Iove saw the fabric of th' almighty mind,<lb/> <cb/> He smil'd, and said, Can mortals' art alone,<lb/> Our heavenly labours mimic with their own?<lb/> The Syracusian's brittle work contains<lb/> Th' eternal law, that through all nature reigns.<lb/> Fram'd by his art, see stars unnumber'd burn,<lb/> And in their courses rolling orbs return:<lb/> His sun through various signs describes the year;<lb/> And every month his mimic moons appear.<lb/> Our rival's laws his little planets bind,<lb/> And rule their motions with a human mind.<lb/> Salmoneus could our thunder imitate,<lb/> But Archimedes can a world create.<lb/></p><p>Many wonderful stories are told of his discoveries, and of his very powerful and curious machines, &c. Hiero once admiring them, Archimedes replied, these effects are nothing, “But give me, said he, some other place to fix a machine on, and I shall move the earth.” He fell upon a curious device for discovering the deceit which had been practiced by a workman, employed by the said king Hiero to make a golden crown. Hiero, having a mind to make an offering to the gods of a golden crown, agreed for one of great value, and weighed out the gold to the artificer. After some time he brought the crown home of the full weight; but it was afterwards discovered or suspected that a part of the gold had been stolen, and the like weight of silver substituted in its stead. Hiero, being angry at this imposition, desired Archimedes to take it into consideration, how such a fraud might be certainly discovered. While engaged in the solution of this difficulty, he happened to go into the bath; where observing that a quantity of water overflowed, equal to the bulk of his body, it presently occurred to him, that Hiero's question might be answered by a like method: upon which he leaped out, and ran homeward, crying out <foreign xml:lang="greek">eu(\rhka! eu(\rhka</foreign>! I have found it! I have found it! He then made two masses, each of the same weight as the crown, one of gold and the other of silver: this done, he filled a vessel to the brim with water, and put the silver mass into it, upon which a quantity of water overflowed equal to the bulk of the mass; then taking the mass of silver out he filled up the vessel again, measuring the water exactly, which he put in; this shewed him what measure of water answered to a certain quantity of silver. Then he tried the gold in like manner, and sound that it caused a less quantity of water to overflow, the gold being less in bulk than the silver, though of the same weight. He then filled the vessel a third time, and putting in the crown itself, he found that it caused more water to overflow than the golden mass of the same weight, but less than the silver one; so that, finding its bulk between the two masses of gold and silver, and that in certain known proportions, he hence computed the real quantities of gold and silver in the crown, and so manifestly discovered the fraud.</p><p>Archimedes also contrived many machines for useful and beneficial purposes: among these, engines for launching large ships; screw pumps, for exhausting the water out of ships, marshes or overflowed lands, as Egypt, &c, which they would do from any depth.</p><p>But he became most famous by his curious contrivances, by which the city of Syracuse was so long defended, when besieged by the Roman consul Marcellus; showering upon the enemy sometimes long darts, and stones of vast weight and in great quantities; at other times lifting their ships up into the air, that had come near <pb n="138"/><cb/> the walls, and dashing them to pieces by letting them fall down again; nor could they find their safety in removing out of the reach of his cranes and levers, for there he contrived to fire them with the rays of the sun reflected srom burning glasses.</p><p>However, notwithstanding all his art, Syracuse was at length taken by storm, and Archimedes was so very intent upon some geometrical problem, that he neither heard the noise, nor minded any thing else, till a soldier that found him tracing of lines, asked him his name, and upon his request to begone, and not disorder his figures, slew him. “What gave Marcellus the greatest concern, says Plutarch, was the unhappy fate of Archimedes, who was at that time in his museum; and his mind, as well as his eyes, so sixed and intent upon some geometrical sigures, that he neither heard the noise and hurry of the Romans, nor perceived the city to be taken. In this depth of study and contemplation, a soldier came suddenly upon him, and commanded him to follow him to Marcellus; which he refusing to do, till he had finished his problem, the soldier, in a rage, drew his sword, and ran him through.” Livy says he was slain by a soldier, not knowing who he was, while he was drawing schemes in the dust: that Marcellus was grieved at his death, and took care of his funeral; and made his name a protection and honour to those who could claim a relationship to him. His death it seems happened about the 142 or 143 Olympiad, or 210 years before the birth of Christ.</p><p>When Cicero was questor for Sicily, he discovered the tomb of Archimedes, all overgrown with bushes and brambles; which he caused to be cleared, and the place set in order. There was a sphere and cylinder cut upon it, with an inscription, but the latter part of the verses quite worn out.</p><p>Many of the works of this great man are still extant, though the greatest part of them are lost. The pieces remaining are as follow: 1. Two books on the Sphere and Cylinder.—2. The Dimension of the Circle, or proportion between the diameter and the circumference.— 3. Of Spiral lines.—4. Of Conoids and Spheroids.— 5. Of Equiponderants, or Centres of Gravity.—6. The Quadrature of the Parabola.—7. Of Bodies floating on Fluids.—8. Lemmata.—9. Of the Number of the Sand.</p><p>Among the works of Archimedes which are lost, may be reckoned the descriptions of the following inventions, which may be gathered from himself and other ancient authors. 1. His account of the method which he employed to discover the mixture of gold and silver in the crown, mentioned by Vitruvius.—2. His description of the Cochleon, or engine to draw water out of places where it is stagnated, still in use under the name of Archimedes's Screw. Athenæus, speaking of the prodigious ship built by the order of Hiero, says, that Archimedes invented the cochleon, by means of which the hold, notwithstanding its depth, could be drained by one man. And Diodorus Siculus says, that he contrived this machine to drain Egypt, and that by a wonderful mechanism it would exhaust the water from any depth.—3. The Helix, by means of which, Athenæus informs us, he launched Hiero's great ship.—4. The Trispaston, which, according to Tzetzes and Oribasius, could draw the most stupendous weights.—5. The machines, which, according to Polybius, Livy, and Plutarch, <cb/> he used in the defence of Syracuse against Marcellus, consisting of Tormenta, Balistæ, Catapults, Sagittarii, Scorpions, Cranes, &c.—6. His Burning Glasses, with which he set fire to the Roman gallies.—7. His Pneumamatic and Hydrostatic engines, concerning which subjects he wrote some books, according to Tzetzes, Pappus, and Tertullian.—8. His Sphere, which exhibited the celestial motions. And probably many others.</p><p>A whole volume might be written upon the curious methods and inventions of Archimedes, that appear in his mathematical writings now extant only. He was the first who squared a curvilineal space; unless Hypocrates must be excepted on account of his lunes. In his time the conic sections were admitted into geometry, and he applied himself closely to the measuring of them, as well as other figures. Accordingly he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and the relations of parabolas to rectilineal planes whose quadratures had long before been determined by Euclid. He has left us also his attempts upon the circle: he proved that a circle is equal to a right-angled triangle, whose base is equal to the circumference, and its altitude equal to the radius; and consequently, that its area is equal to the rectangle of half the diameter and half the circumference; thus reducing the quadrature of the circle to the determination of the ratio between the diameter and circumference; which determination however has never yet been done. Being disappointed of the <hi rend="italics">exact</hi> quadrature of the circle, for want of the rectification of its circumference, which all his methods would not effect, he proceeded to assign an useful approximation to it: this he effected by the numeral calculation of the perimeters of the inscribed and circumscribed polygons: from which calculation it appears that the perimeter of the circumscribed regular polygon of 192 sides, is to the diameter, in a less ratio than that of 3 1/7 or 3 10/70 to 1; and that the perimeter of the inscribed polygon of 96 sides, is to the diameter, in a greater ratio than that of 3 10/71 to 1; and consequently that the ratio of the circumference to the diameter, lies between these two ratios. Now the first ratio, of 3 1/7 to 1, reduced to whole numbers, gives that of 22 to 7, for 3 1/7 : 1 :: 22 : 7; which therefore is nearly the ratio of the circumference to the diameter. From this ratio between the circumference and the diameter, Archimedes computed the approximate area of the circle, and he found that it is to the square of the diameter, as 11 is to 14. He determined also the relation between the circle and ellipse, with that of their similar parts. And it is probable that he likewise attempted the hyperbola; but it is not to be expected that he met with any success, since approximations to its area are all that can be given by the various methods that have since been invented.</p><p>Beside these sigures, he determined the measures of the spiral, described by a point moving unisormly along a right line, the line at the same time revolving with a uniform angular motion; determining the proportion of its area to that of the circumscribed circle, as also the proportion of their sectors.</p><p>Throughout the whole works of this great man, we every where perceive the deepest design, and the finest invention. He seems to have been, with Euclid, exceedingly careful of admitting into his demonstrations <pb n="139"/><cb/> nothing but principles perfectly geometrical and unexceptionable: and although his most general method of demonstrating the relations of curved figures to straight ones, be by inscribing polygons in them; yet to determine those relations, he does not increase the number, and diminish the magnitude, of the sides of the polygon <hi rend="italics">ad infinitum;</hi> but from this plain fundamental principle, allowed in Euclid's Elements, (viz, that any quantity may be so often multiplied, or added to itself, as that the result shall exceed any proposed finite quantity of the same kind,) he proves that to deny his figures to have the proposed relations, would involve an absurdity. And when he demonstrated many geometrical properties, particularly in the parabola, by means of certain progressions of numbers, whose terms are similar to the inscribed figures; this was still done without considering such series as continued <hi rend="italics">ad infinitum,</hi> and then collecting or summing up the terms of such infinite series.</p><p>There have been various editions of the existing writings of Archimedes. The whole of these works, together with the commentary of Eutocius, were found in their original Greek language, on the taking of Constantinople, from whence they were brought into Italy; and here they were found by that excellent mathematician John Muller, otherwise called Regiomontanus, who brought them into Germany: where they were, with that Commentary, published long afterwards, viz, in 1544, at Basil, being most beautifully printed in folio, both in Greek and Latin, by Hervagius, under the care of Thomas Gechauff Venatorius.—A Latin translation was published at Paris 1557, by Pascalius Hamellius.—Another edition of the whole, in Greek and Latin, was published at Paris 1615, in folio, by David Rivaltus, illustrated with new demonstrations and commentaries: a life of the author is presixed; and at the end of the volume is added some account, by way of restoration, of our author's other works, which have been lost; viz, The Crown of Hiero; the Cochleon or Water Screw; the Helicon, a kind of endless screw; the Trispaston, consisting of a combination of wheels and axles; the Machines employed in the defence of Syracuse; the Burning Speculum; the Machines moved by Air and Water; and the Material Sphere.—In 1675, Dr. Isaac Barrow published a neat edition of the works, in Latin, at London, in 4to; illustrated, and succinctly demonstrated in a new method.—But the most complete of any, is the magnisicent edition, in folio, lately printed at the Clarendon press, Oxford, 1792. This edition was prepared ready for the press by the learned Joseph Torelli, of Verona, and in that state presented to the University of Oxford. The Latin translation is a new one. Torelli also wrote a preface, a commentary on some of the pieces, and notes on the whole. An account of the life and writings of Torelli is prefixed, by Clemens Sibiliati. And at the end a a large appendix is added, in two parts; the first being a Commentary on Archimedes's paper upon Bodies that float on Fluids, by the Rev. Abram Robertson of Christ Church College; and the latter is a large collection of various readings in the Manuscript works of Archimedes, found in the library of the late king of France, and of another at Florence, as collated with the Basil edition above mentioned.</p><p>There are also extant other editions of certain parts <cb/> of the works of Archimedes. Thus, Commandine published, in 4to, at Bologna 1565, the two books concerning Bodies that Float upon Fluids, with a Commentary. Commandine published also a translation of the Arenarius. And Borelli published, in folio, at Florence 1661, Archimedes's <hi rend="italics">Liber Assumplorum,</hi> translated into Latin from an Arabic manuscript copy. This is accompanied with the like translation, from the Arabic, of the 5th, 6th, and 7th books of Apollonius's Conics. Mr. G. Anderson published (in 8vo. Lond. 1784) an English translation of the Arenarius of Archimedes, with learned and ingenious notes and illustrations. Dr. Wallis published a translation of the Arenarius. And there may be other editions beside the above, but these are all that I have got, or know of.</p><p><hi rend="smallcaps">Archimedes's</hi> <hi rend="italics">Screw.</hi> See <hi rend="smallcaps">Screw</hi> <hi rend="italics">of Archimedes.</hi></p><p><hi rend="smallcaps">Archimedes's</hi> <hi rend="italics">Burning-glass.</hi> See <hi rend="smallcaps">Burning</hi>-<hi rend="italics">glass.</hi></p></div1><div1 part="N" n="ARCHITECT" org="uniform" sample="complete" type="entry"><head>ARCHITECT</head><p>, a person skilled in architecture, or the art of building; who forms plans and designs for edifices, conducts the work, and directs the various artificers employed in it.</p><p>The most celebrated architects are, Vitruvius, Palladio, Scamozzi, Serlio, Vignola, Barbaro, Cataneo, Alberti, Viola, Inigo Jones, De Lorme, Perrault, S. Le Clerc, Sir Christopher Wren, and the Earl of Burlington.</p></div1><div1 part="N" n="ARCHITECTURE" org="uniform" sample="complete" type="entry"><head>ARCHITECTURE</head><p>, <hi rend="italics">Architectura,</hi> the art of planning and building or erecting any edifice, so as properly to answer the end proposed, for solidity, conveniency, and beauty; whether houses, temples, churches, bridges, halls, theatres, &c, &c.—Architecture is divided into civil, military, and haval or marine.</p><p><hi rend="italics">Civil</hi> <hi rend="smallcaps">Architecture</hi>, is the art of designing and erecting edifices of every kind for the uses of civil lise in every capacity; as churches, palaces, private houses, &c; and it has been divided into five orders or manners of building, under the names of the <hi rend="italics">Tuscan, Doric, Ionic, Corinthian,</hi> and <hi rend="italics">Composite.</hi></p><p>There were many authors on architecture among the Greeks and Romans, before Vitruvius; but he is the sirst whose work is entire and extant. He lived in the reigns of Julius Cæsar and Augustus, and composed a complete system of architecture, in ten books, which he dedicated to this prince. The principal authors on architecture since Vitruvius, are Philander, Barbarus, Salmasius, Baldus, Alberti, Gauricus, Demoniosius, Perrault, De l'Orme, Rivius, Wotton, Serlio, Palladio, Strada, Vignola, Scamozzi, Dieussart, Catanei, Freard, De Cambray, Blondel, Goldman, Sturmy, Wolfius, De Rosi, Desgodetz, Baratteri, Mayer, Gulielmus, Ware, &c, &c. See also <hi rend="smallcaps">Architect.</hi></p><p><hi rend="italics">Military</hi> <hi rend="smallcaps">Architecture</hi>, otherwise more usually called <hi rend="italics">Fortification,</hi> is the art of strengthening and fortifying places, to screen them from the insults or attacks of enemies, and the violence of arms; by erecting forts, castles, and other fortresses, with ramparts, bastions, &c. —The authors who have chiefly excelled in this art, are Coehorn, Pagan, Vauban, Scheiter, Blondel, and Montalembert.</p><p><hi rend="italics">Naval</hi> <hi rend="smallcaps">Architecture</hi>, or <hi rend="italics">ship-building,</hi> is the art of constructing ships, galleys, and other vessels proper to float on the water.</p></div1><div1 part="N" n="ARCHITRAVE" org="uniform" sample="complete" type="entry"><head>ARCHITRAVE</head><p>, is that part of a column which bears immediately upon the capital. It is the lowest <pb n="140"/><cb/> member of the entablature, and is supposed to represent the principal beam in timber buildings, in which it is sometimes called the <hi rend="italics">reason piece,</hi> or <hi rend="italics">master-piece.</hi> Also, in chimneys it is called the <hi rend="italics">mantle-piece;</hi> and the <hi rend="italics">hyperthyron</hi> over the jaumbs of doors, or lintels of windows.</p><p><hi rend="smallcaps">Architrave</hi> <hi rend="italics">Corniche.</hi> See <hi rend="smallcaps">Corniche.</hi></p><p><hi rend="smallcaps">Architrave</hi> <hi rend="italics">doors,</hi> are those which have an architrave on the jaumbs, and over the door; upon the cap piece if straight; or on the arch, if the top be curved.</p><p><hi rend="smallcaps">Architrave</hi> <hi rend="italics">windows,</hi> of timber, are usually an ogee raised out of the solid timber, with a list over it: though sometimes the mouldings are struck, and laid on; and sometimes they are cut in brick.</p></div1><div1 part="N" n="ARCHIVOLT" org="uniform" sample="complete" type="entry"><head>ARCHIVOLT</head><p>, the contour of an arch; or a band or frame adorned with mouldings, running over the faces of the voussoirs or arch-stones, and bearing upon the imposts.</p></div1><div1 part="N" n="ARCHYTAS" org="uniform" sample="complete" type="entry"><head>ARCHYTAS</head><p>, of Tarentum, a celebrated mathematician, cosmographer, and Pythagorean philosopher, whom Horace calls ——Maris ac Terræ, numeroque carentis Arenæ<lb/> Mensorem.<lb/> He flourished about 400 years before Christ; and was the master of Plato, Eudoxus, and Philolaus. He gave a method of finding two mean proportionals between two given lines, and thence the Duplication of the Cube, by means of the conic sections. His skill in Mechanics was such, that he was said to be the inventor of the crane and the screw; and he made a wooden pigeon that could fly about, when it was once set off, but it could not rise again of itself, after it rested. He wrote several works, though none of them are now extant, particularly a treatise <foreign xml:lang="greek">peri\ t<*> *pan<*>o\s</foreign>, <hi rend="italics">de Universo,</hi> cited by Simplicius in Aristot. Categ. It is said he invented the ten categories. He acquired great reputation both in his legislative and military capacity; having commanded an army seven times without ever being defeated. He was at last shipwrecked, and drowned in the Adriatic sea.</p><p>ARCTIC <hi rend="italics">Circle,</hi> is a lesser circle of the sphere, parallel to the equator, and passing through the north pole of the ecliptic, or distant from the north or arctic pole, by a quantity equal to the obliquity of the ecliptic, which was formerly estimated at 23° 30′, but its mean quantity is now 23° 28′ nearly. This, and its opposite, the antarctic circle, are also called the polar circles, where the longest day and longest night are 24 hours, and within all the space of these circles, at one time of the year, the sun never sets, and at the opposite season he never rises for some days, more or less according as the place is nearer the pole.</p><p><hi rend="smallcaps">Arctic</hi> <hi rend="italics">Pole,</hi> the north pole of the world, and so called from <foreign xml:lang="greek">ark<*>os</foreign>, <hi rend="italics">ursa,</hi> the <hi rend="italics">bear,</hi> from its proximity to the constellation of that name.</p></div1><div1 part="N" n="ARCTOPHYLAX" org="uniform" sample="complete" type="entry"><head>ARCTOPHYLAX</head><p>, a constellation otherwise called <hi rend="smallcaps">Bootes.</hi> Which see.</p></div1><div1 part="N" n="ARCTURUS" org="uniform" sample="complete" type="entry"><head>ARCTURUS</head><p>, a fixed star of the first magnitude, between the thighs of the constellation Bootes. So called from <foreign xml:lang="greek">ark<*>os</foreign>, <hi rend="italics">bear,</hi> and <foreign xml:lang="greek"><*>ra</foreign>, <hi rend="italics">tail;</hi> as being near the <hi rend="italics">bear's tail.</hi></p><p>This star is twice mentioned in the book of Job, viz, ix. 9, and xxxviii. 32, by the name Aish if the translation <cb/> be right; and by many of the ancients under its Greek name <hi rend="italics">Arcturus.</hi> The Greeks gave the fabulous history of this star, or constellation, to this purport: Calisto, who was afterwards, in form of the great bear, raised up into a constellation, they tell us, brought forth a son to Jupiter, whom they called Arcas. That Lyacon, when Jupiter afterwards came to visit him, cut the boy in pieces, and served him up at table. Jupiter, in revenge, as well as by way of punishment, called down lightning to consume the palace, and turned the monarch into a wolf. The limbs of the boy were gathered up, to which the god gave life again, and he was taken and educated by some of the people. His mother, who was all this time a bear in the woods, fell in his way: he chased her, ignorant of the fact, and, to avoid him, she threw herself into the temple of Jupiter: he followed her thither to destroy her; and this being death by the laws of the country, Jupiter took them both up into heaven, to prevent the punishment, making her the constellation of the great bear, and converting the youth into this single star behind her.</p><p>Dr. Hornsby, the Savilian Professor of Astronomy, concludes that Arcturus is the nearest star to our system visible in the northern hemisphere, because the variation of its place, in consequence of a proper motion of its own, is more remarkable than that of any other of the stars; and by comparing a variety of observations respecting both the quantity and direction of the motion of this star, he infers, that the obliquity of the ecliptic decreases at the rate of 58″ in 100 years; a quantity which nearly corresponds to the mean of the computations framed by Euler and De la Lande, upon the principles of attraction. <hi rend="italics">Philos. Trans.</hi> v. 63.</p></div1><div1 part="N" n="ARCTUS" org="uniform" sample="complete" type="entry"><head>ARCTUS</head><p>, a name given by the Greeks to two constellations of the northern hemisphere; by the Latins called <hi rend="smallcaps">Ursa</hi> <hi rend="italics">major</hi> and <hi rend="italics">minor,</hi> and by us the <hi rend="italics">greater</hi> and <hi rend="italics">lesser</hi> <hi rend="smallcaps">BEAR.</hi></p></div1><div1 part="N" n="AREA" org="uniform" sample="complete" type="entry"><head>AREA</head><p>, in general, denotes any plain surface to walk upon; and derived from <hi rend="italics">arere, to be dry.</hi></p><div2 part="N" n="Area" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Area</hi>, in <hi rend="italics">Architecture</hi></head><p>, denotes the space or scite of ground on which an edisice stands. It is also used for inner courts, and such like portions of ground.</p></div2><div2 part="N" n="Area" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Area</hi>, in <hi rend="italics">Geometry</hi></head><p>, denotes the superficial content of any figure. <figure/> The areas of figures are estimated in squares and parts of squares. Thus, suppose a rectangle EFGH have its length EH equal to 4 inches, or feet, or yards, &c, and its breadth EF equal to 3; its area will then be 3 times 4, or 12 squares, each side of which is respectively one inch, or foot, or yard, &c. The areas of other particular figures may be seen under their respective names.</p><p>The areas of all similar figures, are in the duplicate ratio, or as the squares of their like sides, or of any like linear dimensions.—Also the law by which the planets move round the sun, is regulated by the areas described by a line connecting the sun and planet; that is, the time in which the planet describes, or passes over, any are of its elliptic orbit, is proportional to the elliptic area described in that time by the said line, or the sector contained by the said are and two radii <pb n="141"/><cb/> drawn from its extremities to the focus in which the sun is placed.</p></div2><div2 part="N" n="Area" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Area</hi></head><p>, in <hi rend="italics">Optics.</hi> See <hi rend="smallcaps">Field.</hi></p></div2></div1><div1 part="N" n="ARENARIUS" org="uniform" sample="complete" type="entry"><head>ARENARIUS</head><p>, the name of a book of Archimedes, in which he demonstrated, that not only the sands of the earth, but even a greater quantity of particles than could be contained in the immense sphere of the fixed stars, might be expressed by numbers, in a way by him invented and described. This notation proceeds by certain geometrical progressions; and in denoting and producing certain very distant terms of the progression, he here first of any one makes use of a property similar to that of logarithms, viz, adding the indices of the terms, to find the index of the product of them. See <hi rend="smallcaps">Archimedes.</hi></p></div1><div1 part="N" n="AREOMETER" org="uniform" sample="complete" type="entry"><head>AREOMETER</head><p>, <hi rend="smallcaps">Aræometrum</hi>, an instrument to measure the density or gravity of fluids.</p><p>The areometer, or water-poise, is commonly made of glass; consisting of a round hollow ball, which terminates in a long slender neck, hermetically sealed at top; having first as much running mercury put into it, as will serve to balance or keep it swimming in an erect position. The stem, or neck, is divided into degrees or parts which are numbered, to shew, by the depth of its descent into any liquor, the lightness or density of it: for that fluid is heaviest in which it sinks least, and lightest in which it sinks deepest.</p><p>Another instrument of this kind is described by Homberg of Paris, in the Memoirs of the Acad. of Sciences for the year 1699; also in the Philos. Trans. N° 262, where a table of numbers is given, expressing the density of various fluids, as determined by this inflrument both in summer and winter. By this table it appears that the density, or specific gravity of quicksilver and distilled water, in the two seasons, were as follow, viz, <table><row role="data"><cell cols="1" rows="1" role="data">in summer as</cell><cell cols="1" rows="1" role="data">13.61 to 1,</cell></row><row role="data"><cell cols="1" rows="1" role="data">in winter as</cell><cell cols="1" rows="1" role="data">13.53 to 1;</cell></row><row role="data"><cell cols="1" rows="1" role="data">and the medium of these two is as</cell><cell cols="1" rows="1" role="data">13.57 to 1.</cell></row></table></p><p>See also the Philos. Trans. vol. 36, or Abridg. vol. 6, for the description and use of another new areometer.</p></div1><div1 part="N" n="AREOMETRY" org="uniform" sample="complete" type="entry"><head>AREOMETRY</head><p>, the science of measuring the lightness and density of fluids.—See the Philos. Trans. vol. 68, for an essay on areometry, &c.</p></div1><div1 part="N" n="AREOSTYLE" org="uniform" sample="complete" type="entry"><head>AREOSTYLE</head><p>, in Architecture, a sort of intercolumnation in which the columns were placed at a great distance from one another.</p></div1><div1 part="N" n="ARGENTICOMUS" org="uniform" sample="complete" type="entry"><head>ARGENTICOMUS</head><p>, among Ancient Astrologers, denotes a kind of silver-haired comet, of uncommon lustre, supposed to be the cause of great changes in the planetary system.</p></div1><div1 part="N" n="ARGETENAR" org="uniform" sample="complete" type="entry"><head>ARGETENAR</head><p>, a star of the fourth magnitude, in the flexure of the constellation Eridanus.</p><p>ARGO NAVIS, or <hi rend="italics">the ship,</hi> is a constellation of fixed stars, in the southern hemisphere, being one of the 48 old constellations. The number of stars in this constellation, are, in Ptolemy's catalogue 45, in Tycho Brahe's 11, in Flamsteed's 64.</p><p>The Greeks tell us, that this was the famous ship in which the Argonauts performed that celebrated expedition, which has been so famous in all their history.</p></div1><div1 part="N" n="ARGUMENT" org="uniform" sample="complete" type="entry"><head>ARGUMENT</head><p>, in Astronomy, is an arch given, by which another arch is found in some proportion to it. Hence, <cb/></p><p><hi rend="smallcaps">Argument</hi> <hi rend="italics">of Inclination,</hi> or <hi rend="smallcaps">Argument</hi> <hi rend="italics">of Latitude,</hi> of any planet, is an arch of a planet's orbit, intercepted between the ascending node, and the place of the planet from the sun, numbered according to the succession of the signs.</p><p><hi rend="italics">Menstrual</hi> <hi rend="smallcaps">Argument</hi> <hi rend="italics">of Latitude,</hi> is the distance of the moon's true place from the sun's true place.— By this is found the quantity of the real obscuration in eclipses, or how many digits are darkened in any place.</p><p><hi rend="italics">Anuual</hi> <hi rend="smallcaps">Argument</hi> <hi rend="italics">of the moon's apogee,</hi> or simply, <hi rend="italics">Annual Argument,</hi> is the distance of the sun's place from the place of the moon's apogee; that is, the arc of the ecliptic comprised between those two places.</p></div1><div1 part="N" n="ARIES" org="uniform" sample="complete" type="entry"><head>ARIES</head><p>, or the <hi rend="italics">Ram,</hi> in <hi rend="italics">Astronomy,</hi> one of the constellations of the northern hemisphere, and the first of the old twelve signs of the zodiac, and marked <foreign xml:lang="greek">g</foreign> in imitation of a ram's head. It gives name to a twelfth part of the ecliptic, which the sun enters commonly about the 20th of March.—The stars of this constellation in Ptolemy's catalogue are 18, in Tycho Brahe's 21, in Hevelius's 27, and in Flamsteed's 66: but they are mostly very small, only one being of the 2d magnitude, two of the 3d magnitude, and all the rest smaller.</p><p>The fabulous account of this constellation, as given by the Greeks, is to this effect. That Nephele gave Phryxus, her son, a ram, which bore a golden fleece, as a guard against the greatest dangers. Juno, the stepmother both of him and Helle, laid designs against their lives; But Phryxus, remembering the admonition of his mother, took his sister with him, and getting upon the back of the ram, they were carried to the sea. The ram plunged in, and the youth was carried over; but Helle dropped off, and was drowned, and so gave name to the Hellespont. When he arrived in Colchis, Æeta, the king, received him kindly; and, sacrificing the ram to Jupiter, dedicated the fleece to the god; which was afterwards carried off by Jason. The animal itself, they say, Jupiter snatched up into the heavens, and made of it the constellation Aries. They have other fables also to account for its origin. But it is most probable that the inventors of this sign, placed it there as the father of those animals which are brought forth about the time the sun approaches to that part of the heavens, and so marking the beginning of spring.</p><p><hi rend="smallcaps">Aries</hi> also denotes a <hi rend="italics">battering ram;</hi> being a military engine with an iron head, much used by the ancients, to batter and beat down the walls of places besieged. See <hi rend="smallcaps">Ram</hi>, and <hi rend="smallcaps">Battering</hi> <hi rend="italics">ram.</hi></p></div1><div1 part="N" n="ARISTARCHUS" org="uniform" sample="complete" type="entry"><head>ARISTARCHUS</head><p>, a celebrated Greek philosopher and astronomer, and a native of the city of Samos; but of what date is not exactly known; it must have been however before the time of Archimedes, as some parts of his writings and opinions are cited by that author, viz, in his Arenarius; he probably flourished about 420 years before Christ. He held the opinion of Pythagoras as to the system of the world, but whether before or after him, is uncertain, teaching that the sun and stars were fixed in the heavens, and that the earth moved in a circle about the sun, at the same time that it revolved about its own centre or axis. He taught also, that the annual orbit of the earth, compared with the distance of the fixed stars, is but as a point. On this head Ar- <pb n="142"/><cb/> chimedes says, “Aristarchus, the Samian, confuting the notions of astrologers, laid down certain positions, from whence it follows, that the world is much larger than is generally imagined; for he lays it down, that the fixed stars and the sun are immoveable; and that the earth is carried round the sun in the circumference of a circle.” On which account, although he might not suffer persecution and imprisonment like Galileo, yet he did not escape censure for his supposed impiety; for it is said Cleanthus was of opinion, that Greece ought to have tried Aristarchus for irreligion, for endeavouring to preserve the regular appearance of the heavenly bodies, by supposing that the heavens themselves stood still; but that the earth revolved in an oblique circle, and at the same time turned round its own axis.</p><p>Aristarchus invented a peculiar kind of sun-dials, mentioned by Vitruvius. There is extant of his works only a treatise upon the magnitude and distance of the sun and moon: this was translated into Latin, and commented upon by Commandine, who first published it with Pappus's Explanations, in 1572. Dr. Wallis afterwards published it in Greek, with Commandine's Latin version, in 1688, and which he inserted again in the 3d volume of his Mathematical Works, printed in folio at Oxford, 1699. The piece was animadverted upon by Mr. Foster, in his Miscellanies. There is another piece which has gone under the name of Aristarchus, of the <hi rend="italics">Mundane System,</hi> its parts, and motions, published in Latin by Robervale, and by Mersenne, in his <hi rend="italics">Mathematical Synopsis.</hi> But this piece is censured, by Menagius (in Diog. Laert.), and Descartes, in his Epistles, as a fictitious piece of Robervale's, and not the genuine work of Aristarchus.</p></div1><div1 part="N" n="ARISTOTELIAN" org="uniform" sample="complete" type="entry"><head>ARISTOTELIAN</head><p>, something that relates to the philosopher <hi rend="italics">Aristotle.</hi> Thus we say, an <hi rend="italics">Aristotelian</hi> dogma, the <hi rend="italics">Aristotelian</hi> school, &c.</p><p><hi rend="smallcaps">Aristotelian</hi> <hi rend="italics">Philosophy,</hi> the philosophy taught by <hi rend="italics">Aristotle,</hi> and maintained by his followers. It is otherwise called the <hi rend="italics">peripatetic</hi> philosophy, from their practice of teaching while they were walking.—The principles of Aristotle's philosophy, the learned agree, are chiefly laid down in the four books <hi rend="italics">de Cælo.</hi> Instead of the more ancient systems, he introduced <hi rend="italics">matter, form,</hi> and <hi rend="italics">privation,</hi> as the principles of all things; but it does not seem that he derived much benefit from them in natural philosophy. And his doctrines are, for the most part, so obscurely expressed, that it has not yet been satisfactorily ascertained, what were his sentiments on some of the most important subjects. He attempted to refute the Pythagorean doctrine concerning the two-fold motion of the earth; and pretended to demonstrate, that the matter of the heavens is ungenerated, incorruptible, and not subject to any alteration: and he supposed that the stars were carried round the earth in solid orbs.</p></div1><div1 part="N" n="ARISTOTELIANS" org="uniform" sample="complete" type="entry"><head>ARISTOTELIANS</head><p>, a sect of philosophers, so called, from their leader <hi rend="italics">Aristotle,</hi> and are otherwise called <hi rend="italics">Peripatetics.</hi>—The Aristotelians and their dogmata prevailed for a long while, in the schools and universities; even in spite of all the efforts of the Cartesians, Newtonians, and other corpuscularians. But the systems of the latter have at length gained the ascendency; and the Newtonian philosophy in particular is now very generally received. <cb/></p></div1><div1 part="N" n="ARISTOTLE" org="uniform" sample="complete" type="entry"><head>ARISTOTLE</head><p>, a Grecian philosopher, the son of Nicomachus, physician to Amyntas king of Macedonia, was born 384 years before Christ, at Stagira, a town of Macedonia, or, as others say, of Thrace; from which he is also called the Stagirite. Not succeeding in the profession of arms, to which it seems he first applied himself, he turned his views to philosophy, and at 17 years of age entered himself a disciple of Plato, and attended in the academy till the death of that philosopher. Aristotle then retired to Atarna, where the prince Hermias gave him his daughter to wife. Repairing afterwards to the court of king Philip, he became preceptor to his son, Alexander the Great, whose education he attended for the space of 8 years; and by the magnificent encouragement of this prince he was afterwards enabled to procure all sorts of animals, from the inspection of which to write the history of them. On his quitting Macedon, he settled at Athens, where he established his school, having the Lyceum assigned him, by the magistrates, for the place of his instruction or disputation; where he became the head and founder of the sect called after his name, as also Peripatetics, from the circumstance of his giving instructions while walking. But being here accused of impiety by Eurymedon, priest of Ceres, and fearing the fate of Socrates, he retired to Chalcis, where he died at 63 years of age, and 322 years before Christ. Some say that he poisoned himself, others that he died of a cholic, and others again pretend that he threw himself into the sea for grief that he could not discover the cause of the flux and reflux of the waters. Laertius, in his life of Aristotle, estimates his books at the number of 4000; of which however scarce 20 have come down to us: these may be comprised under five heads; the first, relating to poetry and rhetoric; the second, to logics; the third, to ethics and politics; the fourth, to physics; and the fifth, to metaphysics. In the schools, Aristotle has been called the philosopher, and the prince of philosophers. And such was the veneration paid to him, that his opinion was allowed to stand on a level with reason itself: nor was any appeal from it admitted, the parties, in every dispute, being obliged to shew, that their conclusions were no less conformable to the doctrine of Aristotle than to truth.</p></div1><div1 part="N" n="ARITHMETIC" org="uniform" sample="complete" type="entry"><head>ARITHMETIC</head><p>, the art and science of numbers; or, that part of mathematics which confiders their powers and properties, and teaches how to compute or calculate truly, and with ease and expedition. It is by some authors also defined the science of discrete quantity. Arithmetic consists chiefly in the four principal rules or operations of Addition, Subtraction, Multiplication, and Division; to which may perhaps be added involution and evolution, or raising of powers and extraction of roots. But besides these, for the facilitating and expediting of computations, mercantile, astronomical, &c, many other useful rules have been contrived, which are applications of the former, such as, the rules of proportion, progression, alligation, false position, fellowship, interest, barter, rebate, equation of payments, reduction, tare and tret, &c. Besides the doctrine of the curious and abstract properties of numbers.</p><p>Very little is known of the origin and invention of arithmetic. In fact it must have commenced with mankind, or as soon as they began to hold any sort of com- <pb n="143"/><cb/> merce together; and must have undergone continual improvements, as occasion was given by the extension of commerce, and by the discovery and cultivation of other sciences. It is therefore very probable that the art has been greatly indebted to the Phœnicians or Tyrians; and indeed Proclus, in his commentary on the first book of Euclid, says, that the Phœnicians, by reason of their traffic and commerce, were accounted the first inventors of Arithmetic. From Asia the art passed into Egypt, whither it was carried by Abraham, according to the opinion of Josephus. Here it was greatly cultivated and improved; insomuch that a considerable part of the Egyptian philosophy and theology seems to have turned altogether upon numbers. Hence those wonders related by them about unity, trinity, with the numbers 4, 7, 9, &c. In effect, Kircher, in his Oedip. Ægypt. shews, that the Egyptians explained every thing by numbers; Pythagoras himself affirming, that the nature of numbers pervades the whole universe; and that the knowledge of numbers is the knowledge of the deity.</p><p>From Egypt arithmetic was transmitted to the Greeks, by means of Pythagoras and other travellers; amongst whom it was greatly cultivated and improved, as appears by the writings of Euclid, Archimedes, and others: with these improvements it passed to the Romans, and from them it has descended to us.</p><p>The nature of the arithmetic however that is now in use, is very different from that above alluded to; this art having undergone a total alteration by the introduction of the Arabic notation, about 800 years since, into Europe: so that nothing now remains of use from the Greeks, but the theory and abstract properties of numbers, which have no dependence on the peculiar nature of any particular scale or mode of notation. That used by the Hebrews, Greeks, and Romans, was chiefly by means of the letters of their alphabets. The Greeks, particularly, had two different methods; the first of these was much the same with the Roman notation, which is sufficiently well known, being still in common use with us, to denote dates, chapters and sections of books, &c. Afterwards they had a better method, in which the first nine letters of their alphabet represented the first numbers, from one to nine, and the next nine letters represented any number of tens, from one to nine, that is, 10, 20, 30, &c, to 90. Any number of hundreds they expressed by other letters, supplying what they wanted with some other marks or characters: and in this order they went on, using the same letters again, with some different marks, to express thousands, tens of thousands, hundreds of thousands, &c: In which it is evident that they approached very near to the more perfect decuple scale of progression used by the Arabians, and who acknowledge that they had received it from the Indians. Archimedes also invented another peculiar scale and notation of his own, which he employed in his Arenarius, to compute the number of the sands. In the 2d century of christianity lived Cl. Ptolemy, who, it is supposed, invented the sexagesimal division of numbers, with its peculiar notation and operations: a mode of computation still used in astronomy &c, for the subdivisions of the degrees of circles. Those notations however were ill adapted to the practical operations of arithmetic: and hence it is that the art ad- <cb/> vanced but very little in this part; for, setting aside Euclid, who has given many plain and useful properties of numbers in his Elements, and Archimedes, in his Arenarius, they mostly consist in dry and tedious distinctions and divisions of numbers; as appears from the treatises of Nicomachus, supposed to be written in the 3d century of Rome, and published at Paris in 1538; as also that of Boethius, written at Rome in the 6th century of Christ. A compendium of the ancient arithmetic, written in Greek, by Psellus, in the 9th century, was published in Latin by Xylander, in 1556. A similar work was written soon after in Greek by Jodocus Willichius; and a more ample work of the same kind was written by Jordanus, in the year 1200, and published with a comment by Faber Stapulensis in 1480.</p><p>Since the introduction of the Indian notation into Europe, about the 10th century, arithmetic has greatly changed its form, the whole algorithm, or practical operations with numbers, being quite altered, as the notation required; and the authors of arithmetic have gradually become more and more numerous. This method was brought into Spain by the Moors or Saracens; whither the learned men from all parts of Europe repaired, to learn the arts and sciences of them. This, Dr. Wallis proves, began about the year 1000; particularly that a monk, called Gilbert, afterwards pope, by the name of Sylvester II, who died in the year 1003, brought this art from Spain into France, long before the date of his death: and that it was known in Britain before the year 1150, where it was brought into common use before 1250, as appears by the treatise of arithmetic of Johannes de Sacro Bosco, or Halifax, who died about 1256. Since that time, the principal writers on this art have been, Barlaam, Lucas de Burgo, Tonstall, Aventinus, Purbach, Cardan, Scheubelius, Tartalia, Faber, Stifelius, Recorde, Ramus, Maurolycus, Hemischius, Peletarius, Stevinus, Xylander, Kersey, Snellius, Tacquet, Clavius, Metius, Gemma Frisius, Buteo, Ursinus, Romanus, Napier, Ceulen, Wingate, Kepler, Briggs, Ulacq, Oughtred, Cruger, Van Schooten, Wallis, Dee, Newton, Morland, Moore, Jeake, Ward, Hatton, Malcolm, &c, &c; the particular inventions or excellencies of whom, will be noticed under the articles of the several species or kinds of arithmetic here following, which may be included under these heads, viz, <hi rend="italics">theoretical, practical, instrumental, logarithmical, numerous, specious, universal, common or decadal, fractional, radical or of surds, decimal, duodecimal, sexagesimal, dynamical or binary, tetractycal, political,</hi> &c.</p><p><hi rend="italics">Theoretical</hi> <hi rend="smallcaps">Arithmetic</hi>, is the science of the properties, relations, &c, of numbers, abstractedly considered; with the reasons and demonstrations of the several rules. Such is that contained in the 7th, 8th, and 9th books of Euclid's Elements; the <hi rend="italics">Logisties</hi> of Barlaam the monk, published in Latin by J. Chambers, in 1600; the <hi rend="italics">Summa Arithmetica</hi> of Lucas de Burgo, printed 1494, who gives the several divisions of numbers from Nicomachus, and their properties from Euclid, with the algorithm, both in integers, fractions, extraction of roots, &c; Malcolm's <hi rend="italics">New System of Arithmetic, theoretical and practical,</hi> in 1730, in which the subject is very completely treated, in all its branches, &c. <pb n="144"/><cb/></p><p><hi rend="italics">Practical</hi> <hi rend="smallcaps">Arithmetic</hi>, is the art or practice of numbering or computing; that is, from certain numbers given, to find others which shall have any proposed relation to the former. As, having the two numbers 4 and 6 given; to find their sum, which is 10; or their difference, which is 2; or their product, 24; or their quotient, 1 1/2; or a third proportional to them, which is 9; &c.—Lucas de Burgo's works contain the whole practice of arithmetic, then used, as well as the theory. Tunstall gave a neat practical treatise of Arithmetic in 1526; as did Stifelius, in 1544, both on the practical and other parts. Tartalea gave an entire body of practical arithmetic, which was printed at Venice in 1556, consisting of two parts; the former, the application of arithmetic to civil uses; the latter, the grounds of Algebra. And most of the authors in the list before enumerated, joined the practice of arithmetic with the theory.</p><p><hi rend="italics">Binary</hi> or <hi rend="italics">Dyadic</hi> <hi rend="smallcaps">Arithmetic</hi>, is that in which only two figures are used, viz 1 and 0. See <hi rend="smallcaps">Binary.</hi> —Leibnitz and De Lagny both invented an arithmetic of this sort, about the same time: and Dangicourt, in the Miscel. Berol. gives a specimen of the use of it in arithmetical progressions; where he shews, that the laws of progression may be more easily discovered by it, than by any other method where more characters are used.</p><p><hi rend="italics">Common</hi> or <hi rend="italics">Vulgar</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which is concerning integers and vulgar fractions.</p><p><hi rend="italics">Decimal</hi> or <hi rend="italics">Decadal</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which is performed by a series of ten characters or figures, the progression being ten-fold, or from 1 to 10's, 100's, &c; which includes both integers and decimal fractions, in the common scale of numbers; and the characters used are the ten Arabic or Indian figures 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. This method of arithmetic was not known to the Greeks and Romans; but was borrowed from the Moors while they possessed a great part of Spain, and who acknowledge that it came to them from the Indians. It is probable that this method took its origin from the ten fingers of the hands, which were used in computations before arithmetic was brought into an art. The Eastern missionaries assure us, that to this day the Indians are very expert at computing on their fingers, without any use of pen and ink. And it is asserted, that the Peruvians, who perform all computations by the different arrangements of grains of maize outdo any European, both for certainty and dispatch, with all his rules.</p><p><hi rend="italics">Duodecimal</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which proceeds from 12 to 12, or by a continual subdivision according to 12. This is greatly used by most artificers, in calculating the quantity of their work; as Bricklayers, Carpenters, Painters, Tilers, &c.</p><p><hi rend="italics">Fractional</hi> <hi rend="smallcaps">Arithmetic</hi>, or <hi rend="italics">of fractions,</hi> is that which treats of fractions, both vulgar and decimal.</p><p><hi rend="italics">Harmonical</hi> <hi rend="smallcaps">Arithmetic</hi>, is so much of the doctrine of numbers, as relates to the making the comparisons, reductions, &c of musical intervals.</p><p><hi rend="smallcaps">Arithmetic</hi> <hi rend="italics">of Infinites,</hi> is the method of summing up a series of numbers, of which the numbers of terms is infinite. This method was first invented by Dr. Wallis, as appears by his treatise on that subject; where he shews its uses in geometry, in finding the areas of <cb/> superficies, the contents of solids, &c. But the method of fluxions, which is a kind of universal arithmetie of infinites, performs all these more easily; as well as a great many other things, which the former will not reach.</p><p><hi rend="italics">Instrumental</hi> <hi rend="smallcaps">Arithmetic</hi>, is that in which the common rules are performed by instruments, or some sort of tangible or palpable substance. Such are the methods of computing by the ten fingers and the grains of maize, by the East Indians and Peruvians, above-mentioned; by the Abacus-or Shwanpan of the Chinese; the several sorts of scales and sliding rules; Napier's bones or rods; the arithmetical machine of Pascal, and others; Sir Samuel Morland's instrument, described in 1666; that of Leibnitz, described in the Miscell. Berol.; that of Polenus, published in the Venetian Miscellany, 1709; and that of Dr. Saunderson, of Cambridge, described in the introduction to his algebra.</p><p><hi rend="italics">Integral</hi> <hi rend="smallcaps">Arithmetic</hi>, or <hi rend="italics">of integers,</hi> is that which respects integers, or whole numbers.</p><p><hi rend="italics">Literal</hi> or <hi rend="italics">Algebra</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which is performed by letters, which represent any numbers indefinitely.</p><p><hi rend="italics">Logarithmical</hi> <hi rend="smallcaps">Arithmetic</hi> is performed by the tables of logarithms. These were invented by baron Napier; and the best treatise on the subject, is Briggs's Arithmetica Logarithmica, 1624.</p><p><hi rend="italics">Logistical</hi> <hi rend="smallcaps">Arithmetic.</hi> See <hi rend="smallcaps">Logistical.</hi></p><p><hi rend="italics">Numerous</hi> or <hi rend="italics">Numeral</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which teaches the calculus of numbers, or of abstract quantities; and is performed by the common numeral or Arabic characters.</p><p><hi rend="italics">Political</hi> <hi rend="smallcaps">Arithmetic</hi>, is the application of arithmetic to political subjects; such as, the strength and revenues of nations, the number of people, births, burials, &c. See <hi rend="smallcaps">Political</hi> Arithmetic. To this head may also be referred the doctrine of Chances, Gaming, &c.</p><p><hi rend="smallcaps">Arithmetic</hi> of <hi rend="italics">Radicals, Rationals,</hi> and <hi rend="italics">Irrationals.</hi> See <hi rend="smallcaps">Radical</hi>, &c.</p><p><hi rend="italics">Sexagesimal</hi> or <hi rend="italics">Sexagenary</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which proceeds by sixties; or the doctrine of sexagesimal fractions: a method which, it is supposed, was invented by Ptolemy, in the 2d century; at least they were used by him. In this notation, the integral numbers from 1 to 59 were expressed in the common way, by the alphabetical letters: then sixty was called a <hi rend="italics">sexagena prima,</hi> and marked with a dash to the character, thus I′; twice sixty, or 120, thus II′; and so on to 59 times 60, or 3540, which is LIX′. Again, 60 times 60, or 3600, was called <hi rend="italics">sexagena secunda,</hi> and marked with two dashes, thus I″; twice 3600, thus II″; and ten times 3600, thus X″; &c. And in this way the notation was continued to any length. But when a number less than sixty was to be joined with any of the sexagesimal integers, their proper expression was annexed without the dash: thus 4 times 60 and 25, is IV′XXV; the sum of twice 60 and 10 times 3600 and 15, is X″II′XV. So near did the inventor of this method approach to the Arabic notation: instead of the sexagesimal progression, he had only to substitute decimal; and to make the signs of numbers, from 1 to 9, simple characters, and to introduce another character, which should signify nothing by itself, but serving only to fill up places. —The <hi rend="italics">sexagenæ integrorum</hi> were soon laid aside, in or- <pb n="145"/><cb/> dinary calculations, after the introduction of the Arabic notation; but the sexagesimal fractions continued till the invention of decimals, and indeed are still used in the subdivisions of the degrees of circular arcs and angles.</p><p>Sam. Reyher has invented a kind of sexagenal rods, in imitation of Napier's bones, by means of which the sexagesimal arithmetic is easily performed.</p><p><hi rend="italics">Specious</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which gives the calculus of quantities as designed by the letters of the alphabet: a method which was more generally introduced into algebra, by Vieta; being the same as literal arithmetic, or algebra.—Dr. Wallis has joined the numeral with the literal calculus; by which means he has demonstrated the rules for fractions, proportions, extraction of roots, &c; of which a compendium is given by himself, under the title of Elementa Arithmeticæ, in the year 1698.</p><p><hi rend="italics">Tabular</hi> <hi rend="smallcaps">Arithmetic</hi>, is that in which the operations of multiplication, division, &c, are performed by means of tables calculated for that purpose: such as those of Herwart, in 1610; and my tables of powers and products, published by order of the Commissioners of Longitude, in 1781.</p><p><hi rend="italics">Tetractic</hi> <hi rend="smallcaps">Arithmetic</hi>, is that in which only the four characters 0, 1, 2, 3 are used. A treatise of this kind of arithmetic is extant, by Ethard or Echard Weigel. But both this, and binary arithmetic, are little better than curiosities, especially with regard to practice; as all numbers are much more compendiously and conveniently expressed by the common decuple scale.</p><p><hi rend="italics">Vulgar,</hi> or <hi rend="italics">Common</hi> <hi rend="smallcaps">Arithmetic</hi>, is that which is conversant about integers and vulgar fractions.</p><p><hi rend="italics">Universal</hi> <hi rend="smallcaps">Arithmetic</hi>, is the name given by Newton to the science of algebra; of which he left at Cambridge an excellent treatise, being the text-book drawn up for the use of his lectures, while he was professor of Mathematics in that University.</p></div1><div1 part="N" n="ARITHMETICAL" org="uniform" sample="complete" type="entry"><head>ARITHMETICAL</head><p>, something relating to or after the manner of arithmetic.</p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Complement,</hi> of a logarithm, is what the logarithm wants of 10.00000 &c; and the easiest way to find it is, beginning at the left hand, to subtract every figure from 9, and the last from 10. So, the arithmetical complement of 8.2501396 is 1.7498604.—It is commonly used in trigonometrical calculations, when the first term of a proportion is not radius; in that case, adding all together, the logarithms of the 3d, 2d, and arithmetical complement of the 1st term.</p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Instruments,</hi> or <hi rend="italics">Machines,</hi> are instruments for performing arithmetical computations; such as Napier's bones, seales, sliding rules, Pascal's machine, &c.</p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Mean,</hi> or <hi rend="italics">Medium,</hi> is the middle term of three quantities in arithmetical progression; and is always equal to half the sum of the extremes. So, an arithmetical mean between 3 and 7, is 5; and between <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> is (<hi rend="italics">a</hi>+<hi rend="italics">b</hi>)/2, or (1/2)<hi rend="italics">a</hi>+(1/2)<hi rend="italics">b.</hi></p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Progression,</hi> is a series of three or more quantities that have all the same common difference: as 3, 5, 7, &c, which have the common difference 2; and <hi rend="italics">a, a</hi>+<hi rend="italics">d, a</hi>+2<hi rend="italics">d,</hi> &c, which have all the same difference <hi rend="italics">d.</hi> <cb/></p><p>In an arithmetical progression, the chief properties are these: 1st, The sum of any two terms, is equal to the sum of every other two that are taken at equal distances from the two former, and equal to double the middle term when there is one equally distant between those two: so, in the series 0, 1, 2, 3, 4, 5, 6, &c, twice 3 or 6.—2d, The sum of all the terms of any arithmetical progression, is equal to the sum of as many terms of which each is the arithmetical mean between the extremes; or equal to half the sum of the extremes multiplied by the number of terms: so, the sum of these ten terms 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, is (0+9)/2 X 10, or 9 X 5, which is 45: and the reason of this will appear by inverting the terms, setting them under the former terms, and adding each two together, which will make double the same series; <table><row role="data"><cell cols="1" rows="1" role="data">thus</cell><cell cols="1" rows="1" role="data">0, 1, 2, 3, 4, 5, 6, 7, 8, 9,</cell></row><row role="data"><cell cols="1" rows="1" role="data">inverted</cell><cell cols="1" rows="1" role="data">9, 8, 7, 6, 5, 4, 3, 2, 1, 0,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">sums</cell><cell cols="1" rows="1" role="data">9, 9, 9, 9, 9, 9, 9, 9, 9 9;</cell></row></table> where the double series being the same number of 9's, or sum of the extremes, the single series must be the half of that sum.—3d, The last, or any term, of such a series, is equal to the first term, with the product added of the common difference multiplied by 1 less than the number of terms, when the series ascends or increases; or the same product subtracted when the series descends or decreases: so, of the series 1, 2, 3, 4, &c, whose common difference is 1, the 50th term is 1+1X49, or 1+49, that is 50; and of the series 50, 49, 48, &c, the 50th term is 50-1X49, or 50-49, which is 1. Also, if <hi rend="italics">a</hi> denote the least term, <hi rend="center"><hi rend="italics">z</hi> the greatest term,</hi> <hi rend="center"><hi rend="italics">d</hi> the common difference,</hi> <hi rend="center"><hi rend="italics">n</hi> the number of the terms,</hi> <hi rend="center">and <hi rend="italics">s</hi> the sum of them all;</hi></p><p>then the principal properties are expressed by these equations, viz, Moreover, when the first term <hi rend="italics">a,</hi> is 0 or nothing; the theorems become <hi rend="center">and .</hi></p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Proportion,</hi> is when the difference between two terms, is equal to the difference between other two terms. So, the four terms, 2, 4, 10, 12, are in arithmetical proportion, because the difference between 2 and 4, which is 2, is equal to the difference between 10 and 12.—The principal property, besides the above, and which indeed depends upon it, is this, that the sum of the first and last, is equal to the sum of the two means: so 2+12, or the sum of 2 and 12, is equal 4+10, or the sum of 4 and 10, which is 14. <pb n="146"/><cb/></p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Ratio,</hi> is the same as the difference of any two terms: so, the arithmetical ratio of the series 2, 4, 6, 8, is 2; and the arithmetical ratio of <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> is <hi rend="italics">a</hi>-<hi rend="italics">b.</hi></p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Scales,</hi> a name given by M. de Buffon, in the <hi rend="italics">Memoirs of the Acad.</hi> for 1741, to different progressions of numbers. according to which, arithmetical computations might be made. It has already been remarked above, that our common decuple scale of numbers was probably derived from the number of fingers on the two hands, by means of which the earliest and most natural mode of computation was performed; and that other scales of numbers, formed in a similar way, but of a different number of characters, have been devised; such as the binary and tetractic scales of arithmetic. In the memoir above cited, Buffon gives a short and simple method to find, at once, the manner of writing down a number given in any scale of numbers whatever; with remarks on different scales. The general effect of any number of characters, different from ten, is, that by a smaller number of oharacters, any given number would require more places of sigures to express or denote it by, but then arithmetical calculations, by multiplication and division, would be easier, as the small numbers 2, 3, 4, &c, are easier to use than the larger 7, 8, 9; and by employing more than ten characters, although any given number would be expressed by fewer of them, yet the calculations in arithmetic would be more difficult, as by the larger numbers 11, 12, 13, &c. It is therefore concluded, upon the whole, that the ordinary decuple scale is a good convenient medium amongst them all, the numbers expressed being tolerably short and compendious, and no single character representing too large a number. The same might also be said, and perhaps more, of a duodecimal scale, by twelve characters, which would express all numbers in a more compendious way than the decuple one, and yet no single character would represent a number too large to compute by; as is consirmed by the now common practice of extending the multiplication table, in school books, to 12 numbers or dimensions, each way, instead of 10; and every person is taught, with sufficient ease, to multiply and divide by 11 and 12 as easily as by 8 or 9 or 10. Another convenience might be added, namely, that the number 12 admitting of more submultiples than the number 10, there would be fewer expressions of interminate fractions in that way than in decimals. So that on all accounts, it is very probable that the duodecimal would be the best of any scale of numbers whatever.</p><p><hi rend="smallcaps">Arithmetical</hi> <hi rend="italics">Triangle.</hi> See <hi rend="italics">Arithmetical</hi> T<hi rend="smallcaps">RIANGLE.</hi></p><p>ARMED. A magnet or loadstone is said to be armed, when it is capped, cased, or set in iron or steel; to make it take up a greater weight; and also readily to distinguish its poles.</p><p>It is surprising, that a little iron fastened to the poles of a magnet, should so greatly improve its power, as to make it even 150 times stronger, or more, than it is naturally, or when unarmed. The effect however, it seems, is not uniform; but that some magnets, by arming, gain much more, and others much less, than one would expect; and that some magnets even lose some of their efficacy by arming. In general, however, <cb/> the thickness of the iron armour ought to be nearly proportioned to the natural strength of the magnet; giving thick irons to a strong magnet, and to the weaker ones thinner: so that a magnet may easily be over-loaded.</p><p>The usual armour of a load-stone, in form of a rightangled parallelopipedon, consists of two thin pieces of iron or steel, of a square figure, and of a thickness proportioned to the goodness of the stone; the proper thickness being found by trials; always filing it thinner and thinner, till the effect be found to be the greatest possible.—The armour of a spherical load-stone, consists of two steel shells, fastened together by a joint, and covering a good part of the convexity of the stone. This also is to be filed away, till the effect is found to be the greatest.</p><p>Kircher, in his book <hi rend="italics">de Magnete,</hi> says, that the best way to arm a load-stone, is to drill a hole through the stone, from pole to pole, in which is to be placed a steel rod of a moderate length: this rod, he asserts, will take up more weight at the end, than the stone itself when armed in the common way. And Gassendus and Cabæus prescribe the same method of arming. But Muschenbroek found, by repeated trials, that the usual armour already mentioned, is preferable to Kircher's; and he gives the following directions for preparing it. When, by means of steel filings and a small needle, the poles of a magnet have been discovered, he directs that the adjacent parts should be rubbed or ground into parallel planes, without shortening the polar axis; and the magnet may be afterwards shaped into the figure of a cube or parallelopipedon, or any other figure that may be more convenient. Plates of the softest iron are then prepared, of the same length and breadth with the whole polar sides of the magnet: the thickness of which plates, so as that they may admit and convey the greatest quantity of the magnetic virtue, is to be previously determined by experiment, in a manner which he prescribes for the purpose. A thicker piece of iron is to be annexed at right angles to these plates, which is called <hi rend="italics">pes armaturæ,</hi> the foot or base of the armour: then the plates, nicely smoothed and polished, are to be firmly attached to each of the polar sides, whilst the thicker part or base is brought into close contact with the lower part of the magnet. In this way, he says, almost all the magnetic virtue issuing from the poles, enters into the armour, is directed to the base, and condensed by means of its roundness, so as to sustain the greatest weight of iron. Phys. Exper. and Geom. Dissert. 1729, pa. 131.</p><p>ARMILLARY <hi rend="italics">Sphere,</hi> a name given to the artificial sphere, composed of a number of circles of metal, wood, or paper, which represent the several circles of the system of the world, put together in their natural order. It serves to assist the imagination to conceive the disposition of the heavens, and the motion of the celestial bodies.</p><p>This sphere is represented at Plate II, Fig. 6, where P and Q represent the poles of the world, AD the equator, EL the ecliptic and zodiac, PAGD the meridian, or the solstitial colure, T the earth, FG the tropic of cancer, HT the tropic of capricorn, MN the arctic circle, OV the antarctic, N and O the poles of the ecliptic, and RS the horizon.</p><p>The Armillary sphere constructed not long since by Dr. Long, in Pembroke-hall, Cambridge, is 18 feet in <pb n="147"/><cb/> diameter; and will contain more than 30 persons sitting within it, to view, as from a centre, the representation of the celestial spheres. The lower part of the sphere, which is not visible to England, is cut off; and the whole apparatus is so contrived, that it may be turned round with as little labour as is employed to wind up a common jack.</p><p>See also Mr. Ferguson's sphere in his Lectures, p. 194.</p><p><hi rend="smallcaps">Armillary</hi> <hi rend="italics">Trigonometer,</hi> an instrument first contrived by Mr. Mungo Murray, and improved by Mr. Ferguson, consisting of five semicircles; viz, meridian, vertical circle, horizon, hour circle, and equator; so adapted to each other by joints and hinges, and so divided and graduated, as to serve for expeditiously resolving many problems in astronomy, dialling, and spherical trigonometry. The drawing, description, and method of using it, may be seen in Ferguson's Tracts, pa. 80, &c.</p><p>ARTIFICIAL <hi rend="italics">Numbers, Sines, Tangents,</hi> &c, are the same as the Logarithms of the natural numbers, sines, tangents, &c.</p></div1><div1 part="N" n="ARTILLERY" org="uniform" sample="complete" type="entry"><head>ARTILLERY</head><p>, the heavy equipage of war; comprehending all sorts of large fire-arms, with their appurtenances; as cannon, mortars, howitzers, balls, shells, petards, musquets, carbines, &c; being what is otherwise called <hi rend="italics">Ordnance.</hi> The term is also applied to the larger instruments of war used by the ancients, as the catapult, balista, battering ram, &c.</p><p>The term <hi rend="italics">Artillery,</hi> or <hi rend="italics">Royal Artillery,</hi> is also applied to the persons employed in that service; and likewise to the art or science itself; and formerly it was used for what is otherwise called <hi rend="italics">pyrotechnia,</hi> or the art of fireworks, with the apparatus and instruments belonging to the same.</p><p>There have been many authors on the subject of artillery; the principal of which are, Buoherius, Braunius, Tartalea, Collado, Sardi, Ufano, Hanzelet, Digges, Moretti, Simienowitz, Mieth, d'Avelour, Manesson, Mallet, St. Julien; and the later authors, of still more consequence, are Belidor, St. Remy, le Blond, Valiere, Morogue, Puget, Coudray, Robins, Muller, Euler, Antoni, Tignola, Scheele; to which may be added the extensive and accurate experiments published in my 1st vol. of Tracts, and in the Philos. Trans. for 1778.</p><p><hi rend="italics">Park of</hi> <hi rend="smallcaps">Artillery</hi>, is that place in a camp which is set apart for the Artillery, or large fire arms.</p><p><hi rend="italics">Traile</hi> or <hi rend="italics">Train</hi> of <hi rend="smallcaps">Artillery</hi>, a number of pieces of ordnance, mounted on carriages, with all their furniture fit for marching. To this commonly belong mortars, cannon, balls, shells, &c.—There are trains of Artillery in most of the royal magazines; as in the Tower, at Portsmouth, Plymonth, &c, but, above all, at Woolwich, from whence the ships commonly receive their ordnance, and where they are all completely proved before they are received into the public service.</p><p>The officers and men of the artillery were formerly called also the Train of Artillery, but are now called the Royal Regiment of Artillery; consisting at present of four battalions, besides a battalion of invalids, and four troops of Horse or Cavalry Artillery.</p></div1><div1 part="N" n="ASCENDANT" org="uniform" sample="complete" type="entry"><head>ASCENDANT</head><p>, in <hi rend="italics">Astrology,</hi> denotes the horoscope; or the degree of the ecliptic which rises upon the horizon, at the time of the birth of any one.—This, <cb/> it is supposed, has an influence on the person's life and fortune, by giving him a bent and propensity to one thing more than other.—In the science of Astrology, this is called the sirst house, the Oriental angle, or angle of the East, and the significator of life: and the astrologer says, such a planet ruled in his Ascendant, or Jupiter was in his Ascendant, &c.</p></div1><div1 part="N" n="ASCENDING" org="uniform" sample="complete" type="entry"><head>ASCENDING</head><p>, in Astronomy, a term used to denote any star, or degree, or other point of the heavens, rising above the horizon.</p><p><hi rend="smallcaps">Ascending</hi> <hi rend="italics">Latitude,</hi> is the latitude of a planet when going towards the north.</p><p><hi rend="smallcaps">Ascending</hi> <hi rend="italics">Node,</hi> is that point of a planet's orbit where it crosses the ecliptic, in proceeding northward. It is otherwise called the Northern Node, and is denoted by this character <figure/>, representing a node, or knot, with the larger part upwards; like as the same character reversed is used to denote the opposite, or Descending Node.</p><p><hi rend="smallcaps">Ascending</hi> <hi rend="italics">Signs,</hi> are such as are upon their ascent, or rise, from the nadir or lowest point of the heavens, towards the zenith, or highest point.</p><p>ASCENSION-<hi rend="smallcaps">Day</hi>, otherwise called <hi rend="italics">Holy Thursday,</hi> is a festival of the church, held 10 days before Whitsunday, in memory of our Saviour's Ascension.</p></div1><div1 part="N" n="ASCENSION" org="uniform" sample="complete" type="entry"><head>ASCENSION</head><p>, in Astronomy, is either Right or Oblique.</p><p><hi rend="italics">Right</hi> <hi rend="smallcaps">Ascension</hi> of the sun, or of a star, is that degree of the equinoctial, accounted from the beginning of Aries, which rises with them, in a right sphere.— Or, <hi rend="italics">Right Ascension,</hi> is that point of the equinoctial, counted as before, which comes to the meridian with the sun or star, or other point of the heavens. And the reason of thus referring it to the meridian, is, because this is always at right angles to the equinoctial; whereas the horizon is so only in a right or direct sphere.—The right ascension, stands opposed to the right descension; and is similar to the longitude of places on the earth. All the fixed stars, &c, which have the same right-ascension, that is, which are at the same distance from the first point of Aries, or, which comes to the same thing, which are in the same meridian, rise at the same time in a right sphere, namely to the people who live at the equator. And if they be not in the same meridian, the difference between their times of rising, or of coming to the meridian of any place, is the precise difference of their right ascension.—But, in an oblique sphere, where the horizon cuts all the meridians obliquely, different points of the same meridian never rise or set together: so that two or more stars on the same meridian, or having the same right ascension, never rise or set at the same time in an oblique sphere; and the more oblique the sphere is, the greater is the interval of time between them.</p><p>To find the right ascension of the sun, stars, &c, by Trigonometry, say, As radius is to the cosine of the sun's greatest declination, or obliquity of the ecliptic, so is the tangent of the sun's or star's longitude, to the tangent of the right ascension.</p><p><hi rend="smallcaps">Right Ascension</hi> <hi rend="italics">of the Mid-heaven,</hi> often used by astronomers, especially in calculating eclipses by means of the nonagesimal degree, is the right ascension of that point of the equator which is in the meridian; and it is equal to the sum of the sun's right ascension and <pb n="148"/><cb/> the horary angle or true time reduced to degrees, or to the sum of the mean longitude and mean time.</p><p><hi rend="italics">Oblique</hi> <hi rend="smallcaps">Ascension</hi>, is an arch of the equator intercepted between the first point of Aries, and that point of the equator which rises together with the star, &c, in an oblique sphere.—The Oblique Ascension is counted from west to east; and is greater or less, according to the various obliquity of the sphere.—To sind the Oblique Ascension of the sun, see <hi rend="smallcaps">Ascensional</hi> and <hi rend="smallcaps">Globe.</hi></p><p>The <hi rend="italics">Arch of Oblique Ascension,</hi> is an arch of the horizon intercepted between the beginning of Aries, and the point of the equator which rises with a star or planet, in an oblique sphere; and it varies with the latitude of the place.</p><p><hi rend="italics">Refraction of</hi> <hi rend="smallcaps">Ascension</hi> <hi rend="italics">and Descension.</hi> See R<hi rend="smallcaps">EFRACTION.</hi></p><p>ASCENSIONAL <hi rend="smallcaps">Difference</hi>, is the difference between the right and oblique ascension of the same point on the surface of the sphere. Or it is the time the sun rises or sets before or after 6 o'clock.</p><p>To find the Ascensional Difference, having given the sun's declination and the latitude of the place, say, As radius is to the tangent of the latitude, so is the tangent of the sun's declination to the sine of the Ascensional Difference sought. The sun's Ascensional Difference converted into time, shews how much he rises before, or sets after, 6 o'clock. When the sun has north declination, the right ascension is greater than the oblique; but the contrary, when the sun has south declination; and the difference, in either case, is the Ascensional Difference.</p></div1><div1 part="N" n="ASCENT" org="uniform" sample="complete" type="entry"><head>ASCENT</head><p>, the motion of a body from below tending upwards; or the continual recess of a body from the earth, or from some other centre of force. And it is opposed to <hi rend="italics">descent,</hi> or motion downwards.</p><p>The Peripatetics attributed the spontaneous ascent of bodies to a principle of levity, inherent in them. But the moderns deny that there is any such thing as spontaneous levity; and they shew, that whatever ascends, does so by virtue of some external impulse or extrusion. Thus it is that smoke, and other rare bodies, ascend in the atmosphere; and oil, light woods, &c, in water: not by any inherent principle of levity; but by the superior gravity, or tendency downwards of the medium in which they ascend and float.</p><p>The ascent of light bodies in heavy mediums, is produced after the same manner as the ascent of the lighter scale of a balance. It is not that such scale has an internal principle, by which it immediately tends upwards; but it is impelled upwards by the preponderancy of the other scale; the excess of the weight in the one having the same effect, by augmenting its impetus downwards, as so much real levity in the other: because the tendencies mutually oppose each other, and action and reaction are always equal.—See this farther illustrated under the articles <hi rend="smallcaps">Fluid</hi>, and <hi rend="smallcaps">Specific Gravity.</hi></p><p><hi rend="smallcaps">Ascent</hi> <hi rend="italics">of Bodies on Inclined Planes.</hi> See the doctrine and laws of them under <hi rend="smallcaps">Inclined Plane.</hi></p><p><hi rend="smallcaps">Ascent</hi> <hi rend="italics">of Fluids,</hi> is particularly understood of their rising above their own level, between the surfaces of nearly contiguous bodies, or in slender capillary glass tubes, or in vessels filled with sand, ashes, or the like porous substances. Which is an effect that takes place as well <hi rend="italics">in vacuo,</hi> as in the open air, and in crooked, <cb/> as well as straight tubes. Indeed some fluids ascend swifter than others, as spirit of wine, and oil of turpentine; and some rise after a different manner from others. The phenomenon, with its causes, &c, in the instance of capillary tubes, will be treated more at large under <hi rend="smallcaps">Capillary</hi> <hi rend="italics">Tube.</hi></p><p>As to planes: Two smooth polished plates of glass, metal, stone, or other matter, being placed almost contiguous, have the effect of several capillary tubes, and the fluid rises in them accordingly: the like may be said of a vessel filled with sand, &c; the various small interstices of which form, as it were, a kind of capillary tubes. So that the same principle accounts for the appearance in them all. And to the same cause may probably be ascribed the ascent of the sap in vegetables. And on this subject Sir I. Newton says, “If a large pipe of glass be filled with sifted ashes, well pressed together, and one end dipped into stagnant water, the fluid will ascend slowly in the ashes, so as in the space of a week or fortnight, to reach the height of 30 or 40 inches above the stagnant water. This ascent is wholly owing to the action of those particles of the ashes which are upon the surface of the elevated water; those within the water attracting as much downwards as upwards: it follows, that the action of such particles is very strong; though being less dense and close than those of glass, their action is not equal to that of glass, which keeps quicksilver suspended to the height of 60 or 70 inches, and therefore acts with a force which would keep water suspended to the height of above 60 feet. By the same principle, a spunge sucks in water, and the glands in the bodies of animals, according to their several natures and dispositions, imbibe various juices from the blood.” Optics, pa. 367.</p><p>Again, if a drop of water, oil, or other fluid, be dropped upon a glass plane, perpendicular to the horizon, so as to stand without breaking, or running off; and another plane touching it at one end, be gradually inclined towards the former, till it touch the drop; then will the drop break and move along towards the touching end of the planes; and it will move the faster in proportion as it proceeds farther, because the distance between the planes is constantly diminishing. And after the same manner, the drop may be brought to any part of the planes, either upward or downward, or sideways, by altering the angle of inclination.</p><p>Lastly, if the same perpendicular planes be so placed, as that two of their sides meet, and form a small angle, the other two being only kept apart by the interposition of some thin body; and thus immerged in a fluid, tinged with some colour to render it visible; the fluid will ascend between the planes, and that the highest where the planes are nearest; so as to form a curve line which is found to be a true hyperbola, of which one of the asymptotes is the line of the fluid, the other being a line drawn along the touching sides.</p><p>And the physical cause of all these phenomena, is the same power of attraction.</p><p><hi rend="smallcaps">Ascent</hi> <hi rend="italics">of Vapour.</hi> See <hi rend="smallcaps">Cloud</hi> and <hi rend="smallcaps">Vapour.</hi></p><div2 part="N" n="Ascent" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ascent</hi></head><p>, in Astronomy, &c. See <hi rend="smallcaps">Ascension.</hi></p></div2><div2 part="N" n="Ascii" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ascii</hi></head><p>, are those inhabitants of the globe, who, at certain times of the year, have no shadow. Such are the inhabitants of the torrid zone, who twice a year having the sun at noon in their zenith, have then no shadow. <pb n="149"/><cb/></p></div2></div1><div1 part="N" n="ASELLI" org="uniform" sample="complete" type="entry"><head>ASELLI</head><p>, two fixed stars of the fourth magnitude, in the constellation Cancer.</p><p>ASH-<hi rend="smallcaps">Wednesday</hi>, the first day of Lent, supposed to have been so called from a custom in the church, of sprinkling ashes that day on the heads of penitents then admitted to penance.</p></div1><div1 part="N" n="ASPECT" org="uniform" sample="complete" type="entry"><head>ASPECT</head><p>, is the situation of the stars and planets in respect of each other. Or, in Astrology, it denotes a certain configuration and mutual relation between the planets, arising from their situations in the zodiac, by which it is supposed that their powers are mutually either increased or diminished, as they happen to agree or disagree in their active or passive qualities. Though such eonfigurations may be varied and combined a thousand ways, yet only a few of them are considered. Hence, Wolfius more accurately defines aspect to be, the meeting of luminous rays emitted from two planets to the earth, either posited in the same right line, or including an angle which is an aliquot part, or some number of aliquot parts, of four right angles, or of 360 degrees.</p><p>The doctrine of aspects was introduced by the astrologers, as the foundation of their predictions. And hence Kepler defines aspect to be, an angle formed by the rays of two planets meeting on the earth, capable of exciting some natural power or influence.</p><p>The ancients reckoned five aspects, viz, <hi rend="italics">conjunction, sextile, quartile, trine,</hi> and <hi rend="italics">opposition.</hi></p><p><hi rend="italics">Conjunction</hi> is denoted by this character <figure/>, and is when the planets are in the same sign and degree, or have the same longitude.</p><p><hi rend="italics">Sextile</hi> is denoted by <figure/>, and is when the planets are distant by the 6th part of a circle, or 2 signs, or 60 degrees.</p><p><hi rend="italics">Quartile</hi> is denoted by <figure/>, and is when the planets are distant 1/4 of the circle, or 90 degrees, or 3 signs.</p><p><hi rend="italics">Trine</hi> is denoted by ▵, and is when the planets are distant by 1/3 of the circle, or 4 signs, or 120 degrees. And</p><p><hi rend="italics">Opposition</hi> is denoted by <figure/>, and is when the planets are in opposite points of the circle, or differ by 1/2 the circle, or 6 signs, or 180 degrees of longitude.</p><p>Or their characters and distances are as in this following tablet. <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">NAME.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">CHARACTER.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">DISTANCE.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Conjunction</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=right" role="data">0°</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sextile</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell></row><row role="data"><cell cols="1" rows="1" role="data">Quartile</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell></row><row role="data"><cell cols="1" rows="1" role="data">Trine</cell><cell cols="1" rows="1" rend="align=center" role="data">▵</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell></row><row role="data"><cell cols="1" rows="1" role="data">Opposition</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=right" role="data">180</cell></row></table></p><p>These intervals are reckoned according to the longitudes of the planets; so that the aspects are the same, whether the planet be in the ecliptic or out of it.</p><p>To the five ancient aspects, modern writers have added several more: as <hi rend="italics">decile,</hi> for the 10th part of a circle; <hi rend="italics">tridecile,</hi> or 3/10; and biquintile, or 2/5 of a circle. And Kepler adds others, from meteorological observations, as he tells us: as the <hi rend="italics">semisextile,</hi> being 1/12, or 30°; and <hi rend="italics">quincunx,</hi> or 5/12, or 150°. Lastly, to the astrological physicians, &c, we owe <hi rend="italics">octile,</hi> or 1/8; <hi rend="italics">trioctile,</hi> or 3/8; and <hi rend="italics">quintile,</hi> 1/5 of the circle.</p><p>The aspects are divided with regard to their supposed influences, into <hi rend="italics">benign, malign,</hi> and <hi rend="italics">indifferent.</hi> <cb/> The trine and sextile aspects being esteemed benign or friendly; the quartile and opposition, malign or unfriendly; and the conjunction an indifferent aspect.</p><p><hi rend="italics">Aspects</hi> are also distinguished into <hi rend="italics">partile</hi> and <hi rend="italics">platic.</hi></p><p><hi rend="italics">Partile</hi> <hi rend="smallcaps">Aspect</hi>, is when the planets are just so many degrees distant, as are expressed above. And these only are the proper aspects.</p><p><hi rend="italics">Platic</hi> <hi rend="smallcaps">Aspect</hi>, is when the planets do not regard each other exactly from these very degrees of distance; but the one exceeding as much as the other falls short. So that the one does not cast its rays immediately on the body of the other, but only on its orb or sphere of light.</p></div1><div1 part="N" n="ASPERITY" org="uniform" sample="complete" type="entry"><head>ASPERITY</head><p>, signifies the inequality or roughness of the surface of any body; by which some parts of it are more prominent than the rest, so as to hinder the hand, &c, from passing over it with ease and freedom; and thus producing what is called friction.—Asperity, or roughness, stands opposed to smoothness, evenness, politure, &c.</p><p>According to the relations of Vermausen, the blind man so celebrated for distinguishing colours by the touch, it seems that every colour has its peculiar degree and kind of asperity. He makes black the roughest, as it is the darkest of colours; but the others are not smoother in proportion as they are lighter; that is, the roughest do not always reflect the least light: for, according to him, yellow is two degrees rougher than blue, and as much smoother than green. See <hi rend="smallcaps">Colours.</hi></p><p>ASSIGNABLE <hi rend="italics">Magnitude,</hi> is used for any finite magnitude that can be expressed or denoted. And,</p><p><hi rend="smallcaps">Assignable</hi> <hi rend="italics">Ratio,</hi> for any expressible ratio.</p></div1><div1 part="N" n="ASSUMPTION" org="uniform" sample="complete" type="entry"><head>ASSUMPTION</head><p>, a feast celebrated in the Romish church, in honour of the miraculous ascent of the Holy Virgin, as they describe it, body and soul, into heaven. It is kept on the 15th of August.</p><p>ASSURANCE <hi rend="italics">on Lives,</hi> a compact by which security is granted for the payment of a certain sum of money on the expiration of the life on which the policy is granted, in consideration of such a previous payment made to the assurer as is accounted a sufficient compensation for the loss and hazard to which he exposes himself.</p><p>The sum at which this compensation should be valued, depends principally on these two circumstances, viz, 1 st, On the rate of interest given for the use of money; and 2d, On the probability of the duration of the life assured, and the values of annuities. For, 1st, If the interest of money be high, the value of the assurance will be proportionally low, <hi rend="italics">& è contra</hi>; because the higher the rate of interest, the less will be the present value which amounts to a certain proposed sum in any given time. Also, if the probability of the duration of life be high, the value of the assurance will again be proportionably low, <hi rend="italics">& è contra</hi>; because the longer the time is, the less will be the principal which will amount to any assigned sum. Thus, if it be required to know the premium or present value, to be given for 100 pounds to be received at the end of any time, as suppose 10 years; then, if the interest of money be at the rate of 5 per cent. the answer, or present premium, would be 61l. 7s. rod; but at four per cent. it would be 67l. 11s. 1d; and at 3 per cent. it would amount to 74l. 8s. 2d. Again, suppose it were <pb n="150"/><cb/> required to assure 100l. on a life, for any time, for instance 1 year; that is, let 100l. be supposed to be payable a year hence, provided a life of a given age fails in that time: here it is evident that, whatever be the rate of interest, the less the probability of the life failing within the year, the less the risk is, and the less the premium ought to be. In effect, the rate of interest being 5 per cent, if it were sure that the life would fail in that year, the value of the assurance would be the same as the present value of 100l. payable at the end of the year, which is 95l. 4s. 9d. But, if it be an equal chance whether the life does or does not fail in the year, in which case the probability of failing is 1/2; then the value of the assurance will be but half the former value, or 47l. 12s. (4 1/2)d. Or if the odds against its failing be as 2 to 1, that is, if one person out of every 3 die at the age of the proposed life, the probability of dying being only 1/3, the value of the assurance will be 1/3 of the first value, or 31l. 14s. 11d. And if the odds be 19 to 1, or one person die out of 20, of that age, the probability of dying will be 1/20, and the value of the assurance will also be 1/20 of the sirst value, or 4l. 15s. 3d. nearly. Lastly, if only one person die out of 50 at the given age, the probability of dying will be 1/50, and the value of the assurance will be accordingly only 1/50 of the sirst sum, or 1l. 18s. 1d: the interest of money being all along considered as after the rate of 5 per cent.—Now, according to Dr. Halley's table of observations, one person dies out of 3, at the age of 87; one in 20 at the age of 64; and one in 50 at the age of 39: It follows, therefore, that the value of the assurance of 100l. for one year, on a life aged 87, is 31l. 14s. 11d; on a life aged 64, it is 4l. 15s. 3d; and on a life aged 39, it is 1l. 18s. 1d: reckoning interest at 5 per cent. But if interest were rated at 3 per cent. these values would be 32l. 7s. 3d, and 4l. 17s. 1d, and 1l. 18s. 10d.</p><p>The assurances most commonly practised, are such as these, on single lives, and for single years. But many private assurers, and even some large assuring offices, either from ignorance or imposition, pay no regard to any difference of age, but demand 5l. from all ages indiseriminately, for the assurance of 100l. for one year: a practice very absurd and inequitable; for it appears that this is more than the value of the assurance of a life of 64 years of age, and even more than double the value of the assurance of a life of 39 years of age; allowing the assured to make 5 per cent. of the money he advances.</p><p>When a life is assured for any number of years; the premium or value may be paid, either in one single present payment; in consequence of which the sum assured will become payable without any farther compensation, whenever, within the given term, the life shall happen to drop: or the value may be paid in annual payments, to be continued till the failure of the life, should that happen within the term; or, if not, till the determination of the term. And the determination of the value of assurance, in all cases, is to be made out from the rules for computing annuities on lives; the principal writers on which are Halley, De Moivre, Simpson, Smart, Kersseboom, De Parcieux, Price, Morgan, and Maseres. See also <hi rend="smallcaps">Life</hi> A<hi rend="smallcaps">NNUITIES, Reversion</hi>, &c. <cb/></p><p>Assurances may be made either on <hi rend="italics">single lives;</hi> as above explained; or they may be made on any number of <hi rend="italics">joint</hi> lives, or on the <hi rend="italics">longest</hi> of any lives; that is, an assurer may bind himself to pay any sums at the extinction of any <hi rend="italics">joint</hi> lives, or the <hi rend="italics">longest</hi> of any lives, or at the extinction of any one or two of any number of lives. There are further assurances on survivorships; by which is meant an obligation, for the value received, to pay a given sum or annuity, provided a given life shall survive any other given life or lives. For which see S<hi rend="smallcaps">URVIVORSHIP.</hi></p><p>The principal offices for making these insurances, in England, are the “London and the Royal Exchange Assurance Offices;” “the Amicable Society, incorporated for a perpetual Assurance Office;” “the Society for equitable Assurances on Lives and Survivorships;” and “the Westminster Society for granting Annuities and insuring Money on Lives.”</p><p>The first two of these offices, having chiefly in view assurances on ships and houses, deal but little in the way of assurances on lives; and all the business they transact in th<*> way, is at 5l. for every 100l. assured on a single life for a single year, without paying any regard to the ages of the lives assured.</p><p>The next, or Amicable Society at Serjeant's Inn, requires an annual payment of 5l. from every member during life, payable quarterly. The whole annual income, hence arising, is equally divided among the nominees, or heirs, of such members as die every year. But this society engages that the dividends shall not be less than 150l. to each claimant, though they may be more. No members are admitted whose ages are greater than 45, or less than 12; nor is any difference of contribution allowed on account of difference of age. The society has subsisted ever since the year 1706, and its credit and usefulness are well established.</p><p>The Equitable Society for Assurances on Lives and Survivorships, which meets at Black-Friars' Bridge, was established in the year 1762, in consequence of proposals which had been made, and lectures, recommending such a design, which had been read by Mr. Dodson, author of the Mathematical Repository. It assures any sums or reversionary annuities, on any life or lives for any number of years, as well as for the whole continuance of the lives; and in any manner that may be best adapted to the views of the persons assured: that is, either by making the assured sums payable certainly at the failure of any given lives; or on condition of survivorship; and also, either by taking the price of the assurance in one present payment, or in annual payments, during any single or joint lives, or any terms, less than the whole possible duration of the lives. In short, there are no kinds of assurances on lives and survivorships, which this society does not make.</p><p>In doing this, the Society follows the rules which have been given by the best mathematical writers on the doctrine of Life Annuities and Reversions, particularly Mr. Thomas Simpson, professor of Mathematics in the Royal Military Academy. It is to be observed however that the Society takes the advantage of making its calculations on the supposition that the interest of money is at so low a rate as 3 per cent, instead of the usual interest of 4 per cent; which consequently raises the insurance proportionally higher; and it also founds its calculations <pb n="151"/><cb/> on the tables of the probabilities and values of lives in London; another circumstance which secures a very advantageous profit to the Society, as experience has proved that the deaths are really in a much lower proportion than according to those tables, and even lower than those of Dr. Halley, which are founded on the bills of mortality of Breslaw. By these means the Society finding itself, by experience, well secured against future hazards, and being unwilling to take from the public an extravagant profit, have determined to reduce all the future payments for assurances, one tenth, and also generously to return, to the persons now assured, one tenth of all the payments they have made: and it seems there is reason to expect that this will be only a preparation to farther reduction.</p><p>From the foregoing account of this society, it is manifest that its business is such, that none but skilful mathematicians are qualified to conduct it. The interest of the society therefore requires, that it should make the places of those who manage its business sufficiently advantageous, to induce the ablest mathematicians to accept them: and this will render it the more necessary for the society to take care, in filling up any future vacancies, to pay no regard to any other considerations than the ability and integrity of the candidates. The consequence of granting good pay, will be a multitude of solicitations on every vacancy, from persons who, however unqualisied, will hope for success from their connexions, and the interest they are able to make. And should the society, in any future time, be led by such causes to trust its business in the hands of persons not possessed of sufficient ability, as mathematicians and calculators, such mistakes may be committed, as may prove, in the highest degree, detrimental and dangerous. There is reason to believe, that at present the society is in no danger of this kind; and one of the great public advantages attending it, is, that it has established an office, where not only the business above described, is transacted with faithfulness and skill; but where also all persons, who want solutions of any questions relating to life annuities and reversions, may apply, and be sure of receiving just answers. The following is a <hi rend="center"><hi rend="italics">Table of the rates of assurance on single lives in the Society for Equitable Assurances. The Sum assured</hi> 100<hi rend="italics">l.</hi></hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Age.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">For one year.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">For seven years at an annual payment of</cell><cell cols="1" rows="1" rend="colspan=3" role="data">For the whole life at an annual payment of</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">l.</cell><cell cols="1" rows="1" rend="align=right" role="data">s.</cell><cell cols="1" rows="1" rend="align=right" role="data">d.</cell><cell cols="1" rows="1" rend="align=right" role="data">l.</cell><cell cols="1" rows="1" rend="align=right" role="data">s.</cell><cell cols="1" rows="1" rend="align=right" role="data">d.</cell><cell cols="1" rows="1" rend="align=right" role="data">l.</cell><cell cols="1" rows="1" rend="align=right" role="data">s.</cell><cell cols="1" rows="1" rend="align=right" role="data">d.</cell></row><row role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell></row><row role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell></row><row role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell></row><row role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell></row><row role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell></row></table> <cb/></p><p>These rates are 10 per cent. lower than the true values, according to the decrements of life in London, reckoning interest at 3 per cent; but at the same time, it is to be observed that, for all ages under 50, they are near one third higher than all the true values, according to Dr. Halley's table of the decrements of life at Breslaw, and Dr. l'rice's tables of the decrements of life at Northampton and Norwich. But as the society has lately found that the decrements of life among its members have hitherto been lower than even those given in these last tables, it may reasonably be expected, that they will in time reduce their rates of assurance to the true values, as determined by these tables.</p><p>As to the <hi rend="italics">Westminster Society for granting Annuities, and insuring Money on Lives,</hi> lately established, viz, in the year 1789, from the number and respectability of its members, the equitable terms upon which it proposes to deal, and the known ability and accuracy of the mathematicians and calculators employed in conducting it, there is every reason to expect an honourable and equitable treatment of the public, and a permanent continuance of its usefulness.</p></div1><div1 part="N" n="ASTERISM" org="uniform" sample="complete" type="entry"><head>ASTERISM</head><p>, the same with constellation, or a collection of many stars, which are usually represented on globes by some particular image or sigure, to distinguish the stars which compose this constellation from those of others.</p><p>ASTRÆA, a name given by some to the sign Virgo, by others called Erigone, and sometimes Isis. The poets feign that Justice quitted heaven to reside on earth, in the golden age; but, growing weary of the iniquities of mankind, she left the earth, and returned to heaven, placing herself in that part of the zodiac called Virgo, where she became a constellation of stars, and from her orb still looks down on the ways of men. Ovid. Metam. lib. i. ver. 149.</p></div1><div1 part="N" n="ASTRAGAL" org="uniform" sample="complete" type="entry"><head>ASTRAGAL</head><p>, in Architecture, a small round moulding, which encompasses the top of the fust or shaft of a column, like a ring or bracelet. The shaft always terminates at top with an astragal, and at bottom with a fillet, which in this place is called <hi rend="italics">ozia.</hi></p><div2 part="N" n="Astragal" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Astragal</hi></head><p>, in Gunuery, is a kind of ring or moulding on a piece of ordnance, at about half a foot distance from the muzzle or mouth; serving as an ornament to the gun. as the former does to a column.</p></div2></div1><div1 part="N" n="ASTRAL" org="uniform" sample="complete" type="entry"><head>ASTRAL</head><p>, something belonging to or depending on the stars.</p><p><hi rend="smallcaps">Astral</hi> <hi rend="italics">Year,</hi> or <hi rend="italics">Sidereal Year.</hi> See <hi rend="smallcaps">Year.</hi></p></div1><div1 part="N" n="ASTRODICTICUM" org="uniform" sample="complete" type="entry"><head>ASTRODICTICUM</head><p>, an astronomical instrument invented by M. Weighel, by means of which many persons shall be able to view the same star at the same time.</p></div1><div1 part="N" n="ASTROGNOSIA" org="uniform" sample="complete" type="entry"><head>ASTROGNOSIA</head><p>, the art of knowing the fixed stars, their names, ranks, situations in the constellations, and the like.</p></div1><div1 part="N" n="ASTROLABE" org="uniform" sample="complete" type="entry"><head>ASTROLABE</head><p>, from <foreign xml:lang="greek">ashr</foreign>, star, and <foreign xml:lang="greek">lam<*>anw</foreign>, I take; alluding to its use in taking, or observing, the stars. The Arabians call it in their tongue <hi rend="italics">astharlab</hi>; a word formed by corruption from the common Greek name.</p><p>This name was originally used for a system or assemblage of the several circles of the sphere, in their proper order and situation with respect to each other. And the ancient instruments were much the same as our armillary spheres. <pb n="152"/><cb/></p><p>The first, and most celebrated of this kind, was that of Hipparchus, which he made at Alexandria, the capital of Egypt, and lodged in a secure place, where it served for divers astronomical operations. Ptolemy made the same use of it: but as the instrument had several inconveniences, he contrived to change its figure, though perfectly natural, and agreeable to the doctrine of the sphere; and to reduce the whole Astrolabe to a plane surface, to which he gave the name of the <hi rend="italics">Planisphere.</hi> Hence,</p><p><hi rend="smallcaps">Atsrolabe</hi> is used among the moderns for a <hi rend="italics">Planisphere,</hi> or a stereographic projection of the circles of the sphere upon the plane of one of the great circles; which is usually either the plane of the equinoctial, the eye being then placed in the pole of the world; or that of the meridian, the eye being supposed in the point of intersection of the equinoctial and horizon; or on that of the horizon.</p><p>The Astrolabe has been treated at large by Stoffler, Gemma Frisius, Clavius, &c. And for a farther account of the nature and kinds of it, see the article <hi rend="smallcaps">Planisphere.</hi></p><div2 part="N" n="Astrolabe" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Astrolabe</hi></head><p>, or <hi rend="smallcaps">Sea Astrolabe</hi>, more particularly denotes an instrument chiefly used for taking altitudes at sea; as the altitude of the pole, the sun, or the stars.</p><p>The common Astrolabe, represented Plate II, <hi rend="italics">fig.</hi> 7, consists of a large brass ring, about 15 inches in diameter, whose limb, or a convenient part of it, is divided into degrees and minutes. It is fitted with a moveable label or index, which turns upon the centre, and carries two sights; and having a small ring, at A, to hang it by in time of observation.</p><p>To make use of the Astrolabe in taking altitudes; suspend it by the ring A, and turn it to the sun, &c, so as that the rays may pass freely through both the sights F and G; then will the label cut or point out the altitude on the divided limb. There are many other uses of the Astrolabe; of which Clavius, Henrion, and others have written very largely.</p><p>The Astrolabe, though now grown into disuse, is by many esteemed equal to any other instrument for taking the altitude at sea; especially between the tropics, where the sun comes near the zenith.</p></div2></div1><div1 part="N" n="ASTROLOGER" org="uniform" sample="complete" type="entry"><head>ASTROLOGER</head><p>, a person professing or practising the art of Astrology.</p></div1><div1 part="N" n="ASTROLOGICAL" org="uniform" sample="complete" type="entry"><head>ASTROLOGICAL</head><p>, something relating to Astrology.</p></div1><div1 part="N" n="ASTROLOGY" org="uniform" sample="complete" type="entry"><head>ASTROLOGY</head><p>, the art of foretelling future events, from the positions, aspects, and influences of the heavenly bodies.</p><p>The word is compounded of <foreign xml:lang="greek">ashr</foreign>, <hi rend="italics">star,</hi> and <foreign xml:lang="greek">logos</foreign>, <hi rend="italics">discourse;</hi> whence, in the literal sense of the term, Astrology should signify no more than the <hi rend="italics">doctrine</hi> or <hi rend="italics">science of the stars;</hi> which indeed was its original acceptation, and constituted the ancient Astrology; which consisted formerly of both the branches now called Astronomy and Astrology, under the name of the latter only; and for the sake of making judiciary predictions it was that astronomical observations, properly so called, were chiefly made by the ancients. And though the two branches be now perfectly separated, and that of Astrology almost universally rejected by men of real learning, this has but lately been the case, as their union subsisted, in <cb/> some degree, from Ptolemy till Kepler, who had a strong bias towards the ancient astrology.</p><p>Astrology may be divided into two branches, <hi rend="italics">natural</hi> and <hi rend="italics">judiciary.</hi></p><p><hi rend="smallcaps">To Natural Astrology</hi> belongs the predicting of natural effects; such as the changes of weather, winds, storms, hurricanes, thunder, floods, earthquakes, &c. But this art properly belongs to <hi rend="italics">Physiology,</hi> or <hi rend="italics">Natural Philosophy;</hi> and is only to be deduced, <hi rend="italics">à posteriori,</hi> from phenomena and observations. And for this sort of Astrology it is that Mr. Boyle makes an apology in his History of the Air. Its foundation and merits may be gathered from what is said under the articles <hi rend="smallcaps">Air</hi>, A<hi rend="smallcaps">TMOSPHERE</hi>, and <hi rend="smallcaps">Weather.</hi></p><p><hi rend="italics">Judicial</hi> or <hi rend="italics">Judiciary</hi> <hi rend="smallcaps">Astrology</hi>, which is what is commonly and properly called <hi rend="italics">Astrology,</hi> is that which professes to foretel moral events, or such as have a dependence on the free will and agency of man; as if they were produced or directed by the stars.</p><p>The professors of this kind of Astrology maintain, “That the heavens are one great volume or book, wherein God has written the history of the world; and in which every man may read his own fortune, and the transactions of his time. The art, say they, had its rise from the same hands as Astronomy itself: while the ancient Assyrians, whose serene unclouded sky favoured their celestial observations, were intent on tracing the paths and periods of the heavenly bodies, they discovered a constant, settled relation of analogy, between them and things below; and hence were led to conclude these to be the <hi rend="italics">Parcæ,</hi> the Destinies so much talked of, which preside at our births, and dispose of our future fate.”</p><p>“The laws therefore of this relation being ascertained by a series of observations, and the share each planet has therein; by knowing the precise time of any person's nativity, they were enabled, from their knowledge in astronomy, to erect a scheme or horoscope of the situation of the planets, at that point of time; and hence, by considering their degrees of power and influence, and how each was either strengthened or tempered by some other, to compute what must be the result thereof.”</p><p>Judicial Astrology, it is commonly said, was invented in Chaldæa, and from thence transmitted to the Egyptians, Greeks, and Romans; though some insist that it was of Egyptian origin, and ascribe the invention to Cham. But it is to the Arabs that we owe it. At Rome the people were so infatuated with it, that the astrologers, or, as they were then called, the mathematicians, maintained their ground in spite of all the edicts of the emperors to expel them out of the city. See <hi rend="smallcaps">Genethliaci.</hi></p><p>Among the Indians, the Bramins, who introduced and practised this art in the East, have hereby made themselves the arbiters of good and evil hours, which, gives them great authority: they are consulted as oracles; and they have taken care always to sell their answers at good rates.</p><p>The same superstition has prevailed in more modern ages and nations. The French historians remark, that in the time of queen Catharine de Medicis, Astrology was so much in vogue, that the most inconsiderable thing was not to be done without consulting the stars. And in the reigns of king Henry III. and IV. of France, the <pb n="153"/><cb/> predictions of Astrologers were the common theme of the court conversation. And this predominant humour in that court was well rallied by Barclay, in his Argenis, <hi rend="italics">lib.</hi> 2, on occasion of an Astrologer, who had undertaken to instruct king Henry in the event of a war which was then threatened by the faction of the Guises.</p></div1><div1 part="N" n="ASTROMETEOROLOGIA" org="uniform" sample="complete" type="entry"><head>ASTROMETEOROLOGIA</head><p>, the art of foretelling the weather, and its changes, from the aspects and configurations of the moon and planets: a species of Astrology disting uished by some under the denomination of <hi rend="italics">meteorological astrology.</hi></p></div1><div1 part="N" n="ASTRONOMICAL" org="uniform" sample="complete" type="entry"><head>ASTRONOMICAL</head><p>, something relating to Astronomy.</p><p><hi rend="smallcaps">Astronomical</hi> <hi rend="italics">Calendar, Characters, Column, Horizon, Hours, Month, Quadrant, Ring-Dial, Sector, Tables, Telescope, Time, Year.</hi> See the several substantives.</p><p><hi rend="smallcaps">Astronomical</hi> <hi rend="italics">Observations.</hi> Of these there are records, or mention, in almost all ages. It is said that the Chinese have observations for a course of many thousand years. But of these, as well as those of the Indians, we have never yet had any benefit. But the observations of most of the other ancients, as Babylonians, Greeks, &c, amongst which those of Hipparchus make a principal figure, are carefully preserved by Ptolemy, in his Almagest.</p><p>About the year 880, Albategni, a Saracen, applied himself to the making of observations; in which he was followed by others of the same nation, as well as Persians and Tartars; among whom were Nassir-EddinEttusi, Arzachel, who also constructed a table of sines, and Ulug Beigh. In 1457 Regiomontanus undertook the province at Norimberg; and his disciples, J. Werner and Ber. Walther, continued the same from 1475 to 1504. Their observations were published together in 1544.—In 1509, Copernicus, and after him the landgrave of Hesse, with his assistants Rothman and Byrge, observed; and after them Tycho Brahe, assisted by the celebrated Kepler, from 1582 to 1601.—All the foregoing observations, together with Tycho's apparatus of instruments, are contained in the Historia Cœlestis, published in 1672, by order of the emperor Ferdinand.—In 1651, was published at Bononia, by Ricciolus, Almagestum Novum, being a complete body of ancient and modern observations, which he so named after the work of the same nature by Ptolemy.—Soon after, Hevelius, with a magnificent and well-contrived apparatus of instruments, described in his Machina Cœlestis, began a course of observations. It has been objected to him, that he only used plain sights, and could never be brought to take the advantage of telescopic ones; which occasioned Dr. Hook to write animadversions on Hevelius's instruments, printed in 1674, in which he too rashly despises them, on account of their inaccuracy: but Dr. Halley, who at the instance of the Royal Society went over to Dantzick in the year 1679, to inspect his instruments, approved of their justness, as well as of the observations made with them. See <hi rend="smallcaps">Sights.</hi>—Our two countrymen Jer. Horrox and Will. Crabtree, are celebrated for their observations from the year 1635 to 1645, who first observed the transit of Venus over the sun in the year 1639.—They were followed by Flamsteed, Cassini the father and son, Halley, de la Hire, Roemer, and K irchius.—The observations of the celebrated Dr. Bradley have not yet been published, though long expected. We have also now published, <cb/> from time to time, the accurate observations of the present British Astronomer Royal: as also those of the French and other observatories, with the observations of many ingenious private astronomers, which are to be found in the Transactions and Memoirs of the various Philosophical Societies.—There have been also observations of many other eminent astronomers; as, Galileo, Huygens, and our countryman Harriot, whose very interesting observations have lately been brought to light by the earl of Egremont, and count Bruhl, by whose means they may come to be published. Other publications of celestial observations, are those of Cassini, La Caille, Monnier, &c.—See farther under <hi rend="smallcaps">Celestial</hi> <hi rend="italics">Observations,</hi> <hi rend="smallcaps">Catalogue, Observatory</hi>, &c.</p><p><hi rend="smallcaps">Astronomical</hi> <hi rend="italics">Place</hi> of a star or planet, is its longitude, or place in the ecliptic, reckoued from the beginning of Aries, <hi rend="italics">in consequentia,</hi> or according to the order of the signs.</p></div1><div1 part="N" n="ASTRONOMICALS" org="uniform" sample="complete" type="entry"><head>ASTRONOMICALS</head><p>, a name used by some writers for sexagesimal fractions; on account of their use in astronomical calculations.</p><p>ASTRONOMICUS <hi rend="italics">Radius.</hi> See <hi rend="smallcaps">Radius.</hi></p></div1><div1 part="N" n="ASTRONOMY" org="uniform" sample="complete" type="entry"><head>ASTRONOMY</head><p>, the doctrine of the heavens, and their phenomena.</p><p>Astronomy is properly a mixed mathematical science, by which we become acquainted with the celestial bodies, their motions, periods, eclipses, magnitudes, distances, and other phenomena. Some, however, understand the term astronomy in a more extensive sense, as comprising in it the theory of the universe, with the primary laws of nature: in which sense it seems to be rather a branch of physics than of mathematics. <hi rend="center"><hi rend="italics">History of Astronomy.</hi></hi></p><p>The invention of astronomy has been variously given, and ascribed to several persons, several nations, and several ages. Indeed it is probable that mankind never existed without some knowledge of astronomy amongst them. For, besides the motives of mere curiosity, which are sufficient of themselves to have excited men to a contemplation of the glorious and varying celestial canopy, it is obvious that some parts of the science answer such essential purposes to mankind, as to make the cultivation of it a matter of indispensable necessity. Accordingly we find traces of it, in different degrees of improvement, among all nations.</p><p>Adam, in his state of innocence, it is supposed by some of the Jewish rabbins, was endowed with a knowledge of the nature, influence, and uses of the heavenly bodies; and Josephus ascribes to Seth and his posterity a considerable knowledge of astronomy: he speaks of two pillars, the one of stone and the other of brick, called the pillars of Seth, upon which they engraved the principles of the science; and he says that the former was still entire in his time. But be this as it may, it is evident that the great length of the antediluvian lives would afford such excellent opportunities for observing the heavenly bodies, that we cannot but suppose that the science of astronomy was considerably advanced before the flood. Indeed Josephus says that longevity was bestowed upon them for the very purpose of cultivating the sciences of geometry and astronomy; observing that the latter could not be learned in less than 600 years; “for that period, he adds, is the <hi rend="italics">grand year.</hi>” <pb n="154"/><cb/> An expression remarkable enough; and by which it may be supposed is meant the period in which the sun and moon come again into the same situation in which they were at the beginning of it, with regard to the nodes, apogee of the moon, &c. “This period, says Cassini, of which we find no intimation in any monument of any other nation, is the finest period that ever was invented: for it brings out the solar year more exactly than that of Hipparchus and Ptolemy; and the lunar month within about one second of what is determined by modern astronomers.” If the Antediluvians had such a period of 600 years, they must have known the motions of the sun and moon more exactly than their descendants knew them some ages after the slood.</p><p>On the building of the Tower of Babel, it is supposed that Noah retired with his children, born after the flood, to the north-eastern part of Asia, where his descendants peopled the vast empire of China. And this, says Dr. Long, “may perhaps account for the Chinese having so early cultivated the study of astronomy, &c.” It is said that the Jesuit missionaries have found traditional accounts among the Chinese, of their having been taught this science by their first emperor Fo-hi, who is supposed to be the same with Noah; and Kempfer asserts that Fo hi discovered the motions of the heavens, divided time into years and months, and invented the 12 signs into which they divide the zodiac, and which they distinguish by these names following; 1, the mouse; 2, the ox or cow; 3, the tiger; 4, the hare; 5, the dragon; 6, the serpent; 7, the horse; 8, the sheep; 9, the monkey; 10, the cock or hen; 11, the dog; and, 12, the boar. They divide the heavens into 28 constellations, or classes of stars, allotting 4 to each of the 7 planets; so that the year always begins with the same planet; and their constellations answer to the 28 lunar mansions used by the Arabian astronomers. These constellations however they do not mark with the figures of animals, like most other nations, but by connecting the stars by straight lines, and denoting the stars themselves by small circles: so, for instance, the great bear would be marked thus, <figure/> The Chinese themselves have many records and traditions of the high antiquity of their astronomy; though not without suspicion of great mistakes. But, on more certain authority, it is asserted by F. Gaubil, that at least 120 years before Christ, the Chinese had determined by observation the number and extent of their constellations as they now stand; the situation of the fixed stars with respect to the equinoctial and solstitial points; and the obliquity of the ecliptic; with the theory of eclipses: and that they were, long before that, acquainted with the true length of the solar year, the method of observing meridian altitudes of the sun by the shadow of a gnomon, and of deducing from thence his declination, and the height of the pole. The same missionary also says, that the Chinese have yet remaining some books of astronomy, which were written about 200 years before Christ; from which it appears, <cb/> that the Chinese had known the daily motion of the sun and moon, and the times of the revolutions of the planets, many years before that period.</p><p>Du Halde informs us, that Tcheou cong, the most skilful astronomer that ever China produced, lived more than a thousand years before Christ; that he passed whole nights in observing the celestial bodies, and arranging them into constellations, &c. At present however, the state of astronomy is but very low in that country, although it be cultivated at Peking, by public authority, in like manner as in most of the capital cities of Europe.</p><p>The inhabitants of Japan, of Siam, and of the Mogul's empire, have also been acquainted with astronomy from time immemorial; and the celebrated observatory at Benares, is a monument both of the ingenuity of the people, and of their skill in that science.</p><p>According to Porphyry, astronomy must have been of very ancient standing in the East. He informs us that, when Babylon was taken by Alexander, there were brought from thence celestial observations for the space of 1903 years; which therefore must have commenced within 115 years after the flood, or within 15 years after the building of Babel.—Epigenes, according to Pliny, affirmed that the Babylonians had observations of 720 years engraven on bricks.—Again, Achilles Tatius ascribes the invention of astronomy to the Egyptians; and adds, that their knowledge of that science was engraven on pillars, and by that means transmitted to posterity.</p><p>M. Bailly, in his elaborate History of ancient and modern astronomy, endeavours to trace the origin of this science among the Chaldeans, Egyptians, Persians, Indians and Chinese, to a very early period. And thence he maintains, that it was cultivated in Egypt and Chaldea 2800 years before Christ; in Persia, 3209; in India, 3101; and in China, 2952 years before that æra. He also apprehends, that astronomy had been studied even long before this distant period, and that we are only to date its revival from thence.</p><p>In investigating the antiquity and progress of astronomy among the Indians, M. Bailly examines and compares four different sets of astronomical tables of the Indian philosophers, namely that of the Siamese, explained by M. Cassini in 1689; that brought from India by M. le Gentil of the Academy of Sciences; and two other manuscript tables, found among the papers of the late M. de Lisle; all of which he found to accord together, and all referring to the meridian of Benares, above-mentioned. It appears that the fundamental epoch of the Indian astronomy, is a conjunction of the sun and moon, which took place at the amazing distance of 3102 years before Christ: and M. Bailly informs us that, by our most accurate astronomical ta bles, such a conjunction did really happen at that time. He further observes that, at present, the Indians calculate eclipses by the mean motions of the sun and moon observed 5000 years since; and that their accuracy, with regard to the solar motion, far exceeds that of the best Grecian astronomers. They had also settled the lunar motions by computing the space through which that luminary had passed in 1,600,984 days, or a little more than 4383 years. M. Bailly also informs us, that they make use of the cycle of 19 years, the same as that <pb n="155"/><cb/> ascribed by the Greeks to Meton; that their theory of the planets is much better than Ptolemy's, as they do not suppose the earth in the centre of the celestial motions, and believe that Venus and Mercury move round the sun; and that their astronomy agrees with the most modern discoveries as to the decrease of the obliquity of the ecliptic, the acceleration of the motion of the equinoctial points, &c.</p><p>In the 2d vol. of the Transactions of the Royal Society of Edinburgh is also a learned and ingenious dissertation on the astronomy of the Brahmins of India, by Mr. Professor Playfair; in which the great accuracy and high antiquity of the science, among them, is reduced to the greatest probability. It hence appears that their tables and rules of computation, have peculiar reference to an epoch, and to observations, 3 or 4 thousand years before Christ; and many other instances are there adduced, of their critical knowledge in the other mathematical sciences, employed in their precepts and calculations.</p><p>Astronomy, it seems, too, was not unknown to the Americans; though in their division of time, they made use only of the solar, and not of the lunar motions. And that the Mexicans, in particular, had a strange predilection for the number 13, by means of which they regulated almost every thing: their shortest periods consisted of 13 days; their cycle of 13 months, each containing 20 days; and their century of 4 periods, of 13 years each: and this excessive veneration for the number 13, arose, according to Siguenza, from its being the number of their greater gods. And it is very remarkable, that the Abbé Clavigero asserts it as a fact, that, having discovered the excess of a few hours in the solar above the lunar year, they made use of intercalary days, to bring them to an equality, as established by Julius Cæsar in the Roman Calendar; but with this difference, that, instead of one day every 4 years, they interposed 13 days every 52 years, which produces the same effect.</p><p>Most authors however fix the origin of astronomy and astrology, either in Chaldea or in Egypt; and accordingly among the ancients we find the word Chaldean often used for astronomer, or, which was the same thing, astrologer. Indeed both of these nations pretended to a very high antiquity, and claimed the honour of producing the first cultivators of this science. The Chaldeans boasted of their temple or tower of Belus, and of Zoroaster, whom they placed 5000 years before the destruction of Troy; while the Egyptians boasted of their colleges of priests, where astronomy was taught, and of the monument of Osymandyas, in which, it is said, there was a golden circle of 365 cubits in circumference, and one cubit thick, divided into 365 equal parts according to the days of the year, &c.</p><p>It is, indeed, evident, that both Chaldea and Egypt were countries very proper for astronomical observations, on account of the extended flatness of the country, and the purity and serenity of the air. The tower of Belus, or of Babel itself, of a great height, was probably an astronomical observatory; and the lofty pyramids of Egypt, whatever they were originally designed for, might perhaps answer the same purpose; and at least they shew the skill of this people in practical astronomy, as they are all placed with their four fronts exactly <cb/> facing the cardinal points of the compass. The Chal deans certainly began to make observations soon after the confusion of languages, as appears from the observations found there on the taking of Babylon by Alexander; and it is probable they began much earlier. It hence appears that they had determined, with tolerable exactness, the length both of a periodical and synodical month. They had also discovered, that the motion of the moon was not uniform; and they even attempted to assign those parts of the orbit in which the motion is quicker or slower. We are also assured by Ptolemy that they were not unacquainted with the motion of the moon's apogee and nodes, the latter of which they supposed made a complete revolation in 6585 1/3 days, or a little more than 18 years, and contained 223 complete lunations, which period is called the Chaldean <hi rend="italics">Saros.</hi> From Hipparchus, the same author also gives us several observations of lunar eclipses made at Babylon above 720 years before Christ. And Aristotle informs us, that they had many occultations of the planets and sixed stars by the moon; a circumstance which led them to conceive that eclipses of the sun were to be attributed to the same cause. They had also no inconsiderable share in arranging the stars into constellations. Nor had even those eccentric bodies the comets escaped their observation: for both Diodorus Siculus and Appollinus Myndicus, Seneca informs us, accounted these to be permanent bodies, having stated revolutions as well as the planets, but in much more extensive orbits: although others of them were of opinion, that the comets were only meteors raised very high in the air, which, blazing for a while, disappear when the matter of which they consist is consumed or dispersed. The branch of dialling was also practised among them long before the Greeks were acquainted with that science.</p><p>The Egyptians, it appears from various circumstances, were much of the same standing in Astronomy as the Chaldeans. Herodotus ascribes their knowledge in the science to Sesostris; probably not the same whom Newton makes contemporary with Solomon, as they were acquainted with astronomy at least many hundred years before that æra. We learn, from the testimony of some ancient authors, many particulars relative to the state of their knowledge in astronomy; such as, that they believed the figure of the earth was spherical; that the moon was eclipsed by passing through the earth's shadow, though it does not certainly appear that they had any knowledge of the true system of the universe; that they attempted to measure the magnitude of the earth and sun, though their methods of ascertaining the latter were very erroneous; and that they even pretended to foretel the appearance of comets, as well as earthquakes and inundations; and the same is also asscribed to the Chaldeans; though these must probably have been rather a kind of astrological predictions, than observations drawn from astronomy, properly so called.</p><p>This science however fell into great decay with the Egyptians, and in the time of the emperor Augustus, it was entirely extinct among them.</p><p>From Chaldea and Egypt the science of astronomy passed into Phenicia, which this people applied to the purposes of navigation, steering their course by the north polar star; and hence they became masters of the sea, and of almost all the commerce in the world. <pb n="156"/><cb/></p><p>The Greeks, it is probable, derived their astronomical knowledge chiefly from the Egyptians and Phenicians, by means of several of their countrymen whovisited these nations, for the purpose of learning the different sciences. Newton supposes that most of the constellations were invented about the time of the Argonautic expedition; but it is more probable that they were, at least most part of them, of a much older date, and derived from other nations, though cloathed in fables of their own invention or application. Several of the constellations are mentioned by Hesiod and Homer, the two most ancient writers among the Greeks, and who lived about 870 years before Christ. Their knowledge in this science however was greatly improved by Thales the Milesian, and other Greeks, who travelled into Egypt, and brought from thence the chief principles of the science. Thales was born about 640 years before Christ; and he, first of all among the Greeks, observed the stars, the solstices, the eclipses of the sun and moon, and predicted the same. And the same was farther cultivated and extended by his successors Anaximander, Anaximanes, and Anaxagoras; but most especially by Pythagoras, who was born 577 years before Christ, and having resided for several years in Egypt, &c, brought from thence the learning of these people, taught the same in Greece and Italy, and founded the sect of the Pythagoreans. He taught that the sun was in the centre of the universe; that the earth was round, and people had antipodes; that the moon reflected the rays of the sun, and was inhabited like the earth; that comets were a kind of wandering stars, disappearing in the further parts of their orbits; that the white colour of the milky-way was owing to the united brightness of a great multitude of small stars; and he supposed that the distances of the moon and planets from the earth, were in certain harmonic proportions to one another.</p><p>Philolaus, a Pythagorean, who flourished about 450 years before Christ, asserted the annual motion of the earth about the sun; and not long after, the diurnal motion of the earth on her own axis, was taught by Hicetas, a Syracusan. About the same time flourished at Athens, Meton and Euctemon, where they observed the summer solstice 432 years before Christ, and observed the risings and settings of the stars, and what seasons they answered to. Meton also invented the cycle of 19 years, which still bears his name.</p><p>Eudoxus the Cnidian lived about 370 years before Christ, and was accounted one of the most skilful astronomers and geometricians of antiquity, being accounted the inventor of many of the propositions in Euclid's Elements, and having introduced geometry into the science of astronomy. He travelled into Asia, Africa, Sicily, and Italy, for improvements in astronomy; and we are informed by Pliny, that he determined the annual year to contain 365 days 6 hours, that he determined also the periodical times of the planets, and made other important observations and discoveries.</p><p>Calippus flourished foon after Eudoxus, and his celestial sphere is mentioned by Aristotle; but he is better known by a period of 76 which he invented, containing 4 corrected Metonic periods, and which commenced at the summer solstice in the year 330 before Christ. About his time the knowledge of the Pythagorean system was <cb/> carried into Italy, Gaul, and Egypt, by certain colonies of Greeks.</p><p>However, the introduction of Astronomy into Greece is represented by Vitruvius in a manner somewhat different. He maintains, that Berosus, a Babylonian, brought it immediately from Babylon itself, and opened an astronomical school in the isle of Cos. And Pliny says, that in consideration of his wonderful predictions, the Athenians erected him a statue in the gymnasium, with a gilded tongue. But if this Berosus be the same with the author of the Chaldaic histories, he must have lived before Alexander.</p><p>After the death of this conqueror, the sciences flourished chiesly in Egypt, under the auspices of Ptolemy Philadelphus and his successors. He founded a school there, which continued to be the grand feminary of learning, till the invasion of the Saracens in the year of Christ 650. From the founding of that school, the science of astronomy advanced considerably. Aristarchus, about 270 years before Christ, strenuously asserted the Pythagorean system, and gave a method of determining the sun's distance by the dichotomy of the moon.—Eratosthenes, who was born at Cyrene in the year 27<*> before Christ, measured the circumference of the earth by means of a gnomon; and being invited to Alexandria, from Athens, by Ptolemy Euergetes, and made keeper of the royal library there, he set up for that Prince those armillary spheres, which Hipparchus and Ptolemy the astronomer afterwards employed so successfully in observing the heavens. He also determined the distance between the tropicsto be 11/83 of the whole meridian circle, which makes the obliquity of the ecliptic in his time to be 23° 51′ 1/3.— The celebrated Archimedes, too, cultivated astronomy, as well as geometry and mechanics: he determined the distances of the planets from one another, and constructed a kind of planetarium or orrery, to represent the phenomena and motions of the heavenly bodies.</p><p>To pass by several others of the ancients, who practised or cultivated astronomy, more or less, we find that Hipparchus, who flourished about 140 years before Christ, was the first who applied himself to the study of every part of astronomy, and, as we are informed by Ptolemy, made great improvements in it: he discovered that the orbits of the planets are eccentric, that the moon moved slower in the apogee than in her perigee, and, that there was a motion of anticipation of the moon's nodes: he constructed tables of the motions of the sun and moon, collected accounts of such eclipses, &c, as had been made by the Egyptians and Chaldeans, and calculated all that were to happen for 600 years to come: he discovered that the fixed stars changed their places, having a slow motion of their own from west to east: he corrected the Calippic period, and pointed out some errors in the method of Eratosthenes for measuring the circumference of the earth: he computed the sun's distance more accurately than any of his predecessors: but his chief work is a catalogue which he made of the fixed stars, to the number of 1022, with their longitudes, latitudes, and apparent magnitudes; which, with most of his other observations, are preserved by Ptolemy in his Almagest.</p><p>There was but little progress made in astronomy from the time of Hipparchus to that of Ptolemy, who was <pb n="157"/><cb/> born at Pelusium in Egypt, in the first century of christianity, and who made the greatest part of his observations at the celebrated school of Alexandria in that country. Prositing of those of Hipparchus and other ancient astronomers, he formed a system of his own, which, though erroneous, was followed for many ages by all nations. He compiled a great work, called the Almagest, which contained the observations and collections of Hipparchus and others his predecessors in astronomy, on which account it will ever be valuable to the professors of that science. This work was preserved from the grievous conflagration of the Alexandrine library by the Saracens, and translated out of Greek into Arabic in the year 827, and from thence into Latin in 1230. The Greek original was not known in Europe till the beginning of the 15th century, when it was brought from Constantinople, then taken by the Turks, by George, a monk of Trapezond, by whom it was translated into Latin; and various other editions have been since made.</p><p>During the long period from the year 800 till the beginning of the 14th century, the western parts of Europe were immersed in gross ignorance and barbarity, while the Arabians, profiting by the books they had preserved from the wreck of the Alexandrine library, cultivated and improved all the sciences, and particularly that of astronomy, in which they had many able professors and authors. The caliph Al Mansur first introduced a taste for the sciences into his empire. His grandson Al Mamun, who ascended the throne in 814, was a great encourager and improver of the sciences, and especially of astronomy. Having constructed proper ininstruments, he made many observations; determined the obliquity of the ecliptic to be 23° 35′; and under his auspices a degree of the circle of the earth was measured a second time in the plain of Singar, on the border of the Red Sea. About the same time Alferganus wrote elements of astronomy; and the science was from hence greatly cultivated by the Arabians, but principally by Albategnius, who flourished about the year 880, and who greatly reformed astronomy, by comparing his own observations with those of Ptolemy: hence he computed the motion of the sun's apogee from Ptolemy's time to his own; settled the precession of the equinoxes at one degree in 70 years; and fixed the obliquity of the ecliptic at 23° 35′. The tables which he composed, for the meridian of Aracta, were long esteemed by the Arabians. After his time, though the Saracens had many eminent astronomers, several centuries elapsed without producing any very valuable observations, excepting those of some eclipses observed by Ebn Younis, astronomer to the caliph of Egypt, by means of which the quantity of the moon's acceleration since that time may be determined.</p><p>Other eminent Arabic astronomers, were, Arzachel a Moor of Spain, who observed the obliquity of the ecliptic: he also improved Trigonometry by constructing tables of sines, instead of chords of arches, dividing the diameter into 300 equal parts. And Alhazen, his contemporary, who wrote upon the twilight, the height of the clouds, the phenomenon of the horizontal moon, and who first shewed the importance of the theory of refractions in astronomy.</p><p>Ulug Beg, grandson of the celebrated Tartar prince Tamerlane, was a great proficient in practical astronomy; <cb/> he had very large instruments, particularly a quadrant of about 180 feet high, with which he made good observations. From these he determined the latitude of Samercand, his capital, to be 39° 37′ 23″; and com posed astronomical tables for the meridian of the same so exact, that they differ very little from those constructed afterwards by Tycho Brahe; but his principal work was his catalogue of the fixed stars, made also from his own observations in the year 1437.</p><p>During this period, almost all Europe was immersed in gross ignorance. But the settlement of the Moors in Spain introduced the sciences into Europe; from which time they have continued to improve, and to be communicated from one people to another, to the present time, when astronomy and all the sciences, have arrived at a very eminent degree of perfection. The emperor Frederick II, about 1230, first began to encourage learning; restoring some decayed universities, and founding a new one in Vienna: he also caused the works of Aristotle, and Ptolemy's Almagest, to be translated into Latin; and from the translation of this work we may date the revival of astronomy in Europe. Two years after this, John de Sacro Bosco, that is, of Halifax, compiled, from Ptolemy, Albategnius, Alferganus, and other Arabic astronomers, his work <hi rend="italics">De Sphæra,</hi> which was held in the greatest estimation for 300 years after, and was honoured with commentaries by Clavius and other learned men. In 1240, Alphonso, king of Castile, not only cultivated astronomy himself, but greatly encouraged others; and by the assistance of several learned men he corrected the tables of Ptolemy, and composed those which were denominated from him the Alphonsine Tables. About the same time also, Roger Bacon, an English monk, wrote several tracts relative to astronomy, particularly of the lunar aspects, the solar rays, and the places of the fixed stars. And, about the year 1270, Vitello, a Polander, composed a treatise on optics, in which he shewed the use of refractions in astronomy.</p><p>Little other improvement was made in astronomy till the time of Purbach, who was born in 1423. He composed new tables of sines for every 10 minutes, making the radius 60, with four ciphers annexed. He constructed spheres and globes, and wrote several astronomical tracts; as, a commentary on Ptolemy's Almagest; some treatises on Arithmetic and Dialling, with tables for various climates; new tables of the fixed stars reduced to the middle of that century; and he corrected the tables of the planets, making new equations to them where the Alphonsine tables were erroneous. In his solar tables, he placed the sun's apogee in the beginning of Cancer; but retained the obliquity of the ecliptic 23° 33 1/2′, as determined by the latest observations. He also observed some eclipses, made new tables for computing them, and had just finished a theory of the planets, when he died in 1462, being only 39 years of age.</p><p>Purbach was succeeded in his astronomical and mathematical labours by his pupil and friend, John Muller, commonly called Regiomontanus, from Monteregio, or Koningsberg, a town of Franconia, where he was born. He completed the epitome of Ptolemy's Almagest, which Purbach had begun; and after the death of his friend, was invited to Rome, where he made many <pb n="158"/><cb/> astronomical observations. Being returned to Nuremberg in 1471, by the encouragement of a wealthy citizen named Bernard Walther, he made several instruments for astronomical observations, among which was an armillary astrolabe, like that used at Alexandria by Hipparchus and Ptolemy, with which he made many observations, using also a good clock, which was then but a late invention. He made ephemerides for 30 years to come, shewing the lunations, eclipses, &c; and, the art of printing having then been lately invented, he printed the works of many of the most celebrated ancient astronomers. He wrote the Theory of the Planets and Comets, and a treatise on triangles, still in repute for several good theorems; computing the table of sines for every single minute, to the radius 1000000, and introducing the use of tangents also into trigonometry. After his death, which happened at Rome in 1476, being only 40 years of age, Walther collected his papers, and continued the astronomical observations till his own death also. The observations of both were collected by order of the senate of Nuremberg, and published there in 1544 by John Schoner: they were also afterwards published in 1618 by Snellius, at the end of the observations made by the Landgrave of Hesse; and lastly with those of Tycho Brahe in 1666.</p><p>Walther was succeeded, as astronomer at Nuremberg, by John Werner, a clergyman. He observed the motion of the comet in 1500; and wrote several tracts on geometry, astronomy, and geography, in a masterly manner; the most remarkable of which, are those concerning the motion of the 8th sphere, or of the fixed stars; in this tract, by comparing his own observations, made in 1514, with those of Ptolemy, Alphonsus, and others, he shewed that the motion of the fixed stars, since called the precession of the equinoxes, is 1° 10′ in 100 years. He made also the first star of Aries 26° distant from the equinoctial point, and the obliquity of the ecliptic only 23° 28′. He constructed a planetarium, representing the celestial motions according to the Ptolemaic hypothesis; and he published a translation of Ptolemy's Geography, with a commentary, in which he first proposed the method of finding the longitude at sea by observing the moon's distance from the fixed stars; now so successfully practised for that purpose. Werner died in 1528, at 60 years of age.</p><p>Nicolaus Copernicus was the next who made any considerable figure in astronomy, by whom indeed the old Pythagorean system of the world was restored, which had been till now set aside from the time of Ptolemy. About the year 1507 Copernicus conceived doubts of this system, and entertained notions about the true one, which he gradually improved by a series of astronomical observations, and the contemplation of former authors. By these he formed new tables, and completed his work in the year 1530, containing these, and his renovation of the true system of the universe, in which all the planets are considered as revolving about the sun, placed in the centre. But the work was only printed in 1543, underthe care of Schoner and Osiander, by the title of <hi rend="italics">Revolutiones Orbium Cœlestium</hi>; and the author just received a copy of the work a few hours before his death, which happened on the 23d of May 1543, at 70 years of age. <cb/></p><p>After the death of Copernicus, the science and practice of astronomy were greatly improved by many other persons, as Schoner, Nonius, Appian, Gemma Frisius, Rothman, Byrgius, the Landgrave of Hesse, &c.— Schoner reformed and explained the calendar, improved the methods of making celestial observations, and published a treatise on cosmography; but he died 4 years after Copernicus.—Nonius wrote several works on mathematics, astronomy and navigation, and invented some useful and more accurate instruments than formerly; one of these was the astronomical quadrant, on which he divided the degrees into minutes by a number of concentric circles; the first of which was divided in 90 equal parts or degrees, the second into 89, the thirdinto 88, and so on, to 46; so that, the index of the quadrant always falling upon or near one of the divisions, the minutes would be known by an easy computation.—The chief work of Appian, <hi rend="italics">The Cæsarean Astronomy,</hi> was published at Ingoldstat in 1540; in which he shews, how to observethe places of the stars and planets by the astrolabe; to resolve astronomical problems by certain instruments; to predict eclipses, and to describe the figures of them; and the method of dividing and using an astronomical quadrant: at the end are added observations of 5 comets, one of which has been supposed the same with that observed by Hevelius, and if so, it ought to have returned again in the year 1789;—but it was not observed then. Gemma Frisius wrote a commentary on Appian's <hi rend="italics">Cosmography,</hi> accompanied with many observations of eclipses: he also invented the astronomical ring, and several other instruments, useful in taking observations at sea; and he was the first who recommended a timekeeper for determining the longitude at sea.—Rheticus gave up his professorship of mathematics at Wittemberg, that he might attend the astronomical lectures of Copernicus; and, for improving astronomical calculations, he began a very extensive work, being a table of sines, tangents and secants, to a very large radius, and to every 10 seconds, or 1/6 of a minute; which was completed by his pupil Valentine Otho, and published in 1594.</p><p>About the year 1561, William IV, Landgrave of Hesse Cassel, applied himself to the study of astronomy, having furnished himself with the best instruments that could then be made: with these he made a great number of observations, which were published by Snellius in 1618, and which were preferred by Hevelius to those of Tycho Brahe. From these observations he formed a catalogue of 400 stars, with their latitudes and longitudes, adapted to the beginning of the year 1593.</p><p>Tycho Brahe, a noble Dane, began his observations about the same time with the Landgrave of Hesse, abovementioned, and he observed the great conjunction of Jupiter and Saturn: but finding the usual instruments very inaccurate, he constructed many others much larger and exacter, with which he applied himself diligently to observe the celestial phenomena. In 1571 he discovered a new star in the chair of Cassiopeia; which induced him, like Hipparchus on a similar occasion, to make a new catalogue of the stars; which he composed to the number of 777, and adapted their places to the year 1600. In the year 1576, by favour of the king of Denmark, he built his new observatory, called Uraniburg, on the small island Huenna, opposite <pb n="159"/><cb/> to Copenhagen, and which he very amply furnished with many large instruments, some of them so divided as to shew single minutes, and in others the arch might be read off to 10 seconds. One quadrant was divided according to the method invented by Nonius, that is, by 47 concentric circles; but most of them were divided by diagonals; a method of division invented by a Mr. Richard Chanceler, an Englishman. Tycho employed his time at Uraniburg to the best advantage, till the death of the king, when, falling into discredit, he was obliged to remove to Holstein; and he afterwards found means of introducing himself to the Emperor Rodolph, with whom he continued at Prague till the time of his death in 1601.—It is well known that Tycho was the inventor of a system of astronomy, a kind of Semi-PtoIemaic, which he vainly endeavoured to establish instead of the Copernican or true system. His works, however, which are very numerous, shew that he was a man of great abilities; and his discoveries, together with those of Purbach and Regiomontanus, were collected and published together in 1621, by Longomontanus, the favourite disciple of Tycho.</p><p>While Tycho resided at Prague with the emperor, he prevailed on Kepler to leave the university of Glatz, and to come to him, which he did with his family and library in 1600: but Tycho dying in 1601, Kepler enjoyed all his life the title of mathematician to the Emperor, who ordered him to sinish the tables of Tycho Brahe, which he did accordingly, and published them in 1627, under the title of Rodolphine. He died about the year 1630 at Ratisbon, where he was soliciting the arrears of his pension. From his own observations, and those of Tycho, Kepler discovered several of the true laws of nature, by which the motions of the celestial bodies are regulated. He discovered that all the planets revolved about the sun, not in circular, but in elliptical orbits, having the sun in one of the foci of the ellipse; that their motions were not equable, but varying, quicker or slower as they were near to the sun or farther from him; but that this motion was so regulated, that the areas described by the variable line drawn from the planet to the sun, are equal in equal times, and always proportional to the times of describing them. He also discovered, by trials, that the cubes of the distances of the planets from the sun, were in the same proportion as the squares of their periodical times of revolution. By observations also on comets, he concluded that they are freely carried about among the orbits of the planets, in paths that are nearly rectilinear, but which he could not then determine. Besides many other discoveries, which are to be found in his writings.</p><p>In Kepler's time there were many other good prosicients in astronomy; as Edward Wright, baron Napier, John Bayer, &c. Wright made several good meridional observations of the sun, with a quadrant of 6 feet radius, in the years 1594, 1595, and 1596; from which he greatly improved the theory of the sun's motion, and computed more accurately his declination, than any person had done besore. In 1599 he published also an excellent work entitled “Certain Errors in Navigation discovered and detected,” containing a method which has commonly, though erroneously, been ascribed to Mercator.—To Napier we owe some excellent theorems and improvements in spherics, besides the ever memo- <cb/> rable invention of logarithms, one of the most usefut ever made in the art of numbering, and of the greatest use in all the other mathematical sciences.—Bayer, a German, published his <hi rend="italics">Uranometria,</hi> being a complete celestial atlas, or the figures of all the constellations visible in Europe, with the stars marked on them, and the stars also accompanied by names, or the letters of the Greek alphabet; a contrivance by which the stars may easily be referred to with distinctness and precision. —About the same time too, astronomy was cultivated by many other persons; namely, abroad by Mercator, Maurolycus, Maginus, Homelius, Schultet, Stevin, Galileo, &c; and in England by Thomas and Leonard Digges, John Dee, Robert Flood, Harriot, &c.</p><p>The beginning of the 17th century was particularly distinguished by the invention of telescopes, and the application of them to astronomical observations; an invention to which we owe the most brilliant discoveries, and all the accuracy to which the science is now brought. The more distinguished early observations with the telescope, were made by Galileo, Harriot, Huygens, Hook, Cassini, &c. It is said that from report only, and before he had seen one, Galileo made for himself telescopes, by which he discovered inequalities in the moon's surface, Jupiter's satellites, and the ring of Saturn; also spots on the sun, by which he found out the revolution of that luminary on his axis; and he discovered that the nebulæ and milky way were full of small stars. Harriot also, who has hitherto been known only as an algebraist, made much the same discoveries as Galileo, and as early, if not more so, as appears by his papers not yet printed, in the possession of the earl of Egremont.</p><p>Mr. Horrox, a young astronomer of great talents, made considerable discoveries and improvements. In 1633 he found out that the planet Venus would pass over the sun's disc on the 24th of November, 1639; an event which he announced only to his friend Mr. Crabtree; and these two were the only persons in the world that observed this transit, which was also the first time it had ever been seen by mortal eyes. Mr. Horrox made also many other useful observations, and had even formed a new theory of the moon, taken notice of by Newton; but his early death, in the beginning of the year 1640, put a slop to his useful and valuable labours.</p><p>About the same time flourished Hevelius, Burgomaster of Dantzic, who furnished an excellent observatory in his own house, where he observed the spots and phases of the moon, from which observations he compiled his <hi rend="italics">Selenographia;</hi> and an account of his apparatus is contained in his work entitled <hi rend="italics">Machina Cœlestis,</hi> a book now very scarce, as most of the copies were accidentally burnt, with the whole house and apparatus, in 1679. Hevelius died in 1688, aged 76.</p><p>Dr. Hook, a contemporary of Hevelius, invented instruments with telescopic sights, and censured those of the latter; which occasioned a sharp dispute between them, and to settle which the celebrated Dr. Halley was sent over to Hevelius to examine his instruments. The two astronomers made several observations together, very much to their satisfaction, and amongst them was one of an occultation of Jupiter by the moon, when they determined the diameter of the latter to be 30′ 33.″ <pb n="160"/><cb/></p><p>Before the middle of the 17th century the construction of telescopes had been greatly improved, particularly by Huygens and Fontana. The former constructed one of 123 feet, with which he long observed the moon and planets, and discovered that Saturn was encompassed with a ring. With telescopes too, of 200 and 300 feet focus, Cassini saw five satellites of Saturn, with his zones or belts, and the shadows of Jupiter's satellites passing over his body. In 1666 Azout applied a micrometer to telescopes, to measure the diameters of the planets, and other small distances in the heavens: but an instrument of this kind had been invented before, by Mr. Gascoigne, though it was but little known abroad.</p><p>To obviate the difficulties of the great lengths of refracting telescopes and the aberration of the rays, it is said that Mersennus first started the idea of making telescopes of reflectors, instead of lenses, in a letter to Descartes; and in 1663 James Gregory of Aberdeen shewed how such a telescope might be constructed. After some time spent also by Newton, on the construction of both sorts of telescopes, he found out the great inconvenience which arises to refractors from the different refrangibility of the rays of light, for which he could not then find a remedy; and therefore, pursuing the other kind, in the year 1672 he presented to the Royal Society two reflectors, which were constructed with spherical speculums, as he could not procure other figures. The inconveniences however arising from the different refrangibility of the rays of light, have since been fully obviated by the ingenious Mr. Dollond. Towards the latter part of the 17th, and beginning of the 18th century, practical altronomy it seems rather languished. But at the same time the speculative part was carried to the highest perfection by the immortal Newton in his Principia, and by the Astronomy of David Gregory.</p><p>Soon after this however, great improvements of astronomical instruments began to take place, particularly in Britain. Mr. Graham, a celebrated mechanic and watchmaker, not only improved clocks and watch work, but also carried the accuracy of astronomical instruments to a surprising degree. He constructed the old 8 feet mural arch at the Royal Observatory Greenwich, and a small equatorial sector for making observations out of the meridian; but he is chiefly remarkable for contriving the zenith sector of 24 feet radius, and afterwards one of 12 1/2 feet, with which Dr. Bradley discovered the aberration of the fixed stars. The reflecting telescope of Gregory and Newton, was greatly improved by Mr. Hadley, who presented a very powerful instrument of that kind to the Royal Society in 1719. The same gentleman has also immortalized his memory by the invention of the reflecting quadrant or sector, now called by his name, which he presented to the society in 1731, and which is now so universally useful at sea, especially where nice observations are required. It appears however that an instrument similar to this in its principles, had been invented by Newton; and a description with a drawing of it given by him to Dr. Halley, when he was preparing for his voyage in 1701, to discover the variation of the needle: it has also been asserted that a Mr. Godfrey of Philadelphia in America, made the same discovery, and the first instrument of this kind. About the middle of this century, the constructing and dividing of large astronomical instruments were carried to great <cb/> perfection by Mr. John Bird; and reflecting telescopes were not less improved by Mr. Short, who also first executed the divided object-glass micrometer, which had been proposed and described by M. Louville and others, Mr. Dollond also brought refracting telescopes to the greatest perfection, by means of his acromatic glasses; and lately the discoveries of Herschel are owing to the amazing powers of reflectors of his own construction.</p><p>Thus the astronomical improvements in the present century, have been chiefly owing to the foregoing inventions and improvements in the instruments, and to the establishment of regular observatories in England, France, and other parts of Europe. Roemer, a celebrated Danish Astronomer, first made use of a meridional telescope; and, by observing the eclipses of Jupiter's satellites, he first discovered the progressive motion of light, concerning which he read a differtation before the Academy of Sciences at Paris in 1675.—Mr. Flamsteed was appointed the first Astronomer Royal at Greenwich in 1675. He observed, for 44 years, all the celestial phenomena, the sun, moon, planets, and fixed stars, of all which he gave an improved theory and tables, viz, a catalogue of 3000 stars with their places, to the year 1689; also new solar tables, and a theory of the moon according to Horrox; likewise, in Sir Jonas Moore's System of Mathematics, he gave a curious tract on the doctrine of the sphere, shewing how, geometrically to construct eclipses of the sun and moon, as well as occultations of the fixed stars by the moon. And it was upon his tables that were constructed, both Halley's tables, and Newton's theory of the moon.—Cassini also, the first French Astronomer Royal, very much distinguished himself, making many observations on the sun, moon, planets and comets, and greatly improved the elements of their motions. He also erected the gnomon, and drew the celebrated meridian line in the church of Petronia at Bologna.</p><p>In 1719 Mr. Flamsteed was succeeded by Dr. Halley, as Astronomer Royal at Greenwich. The Doctor had been sent at the early age of 21 to the island of St. Helena, to observe the southern stars, and make a catalogue of them, which was published in 1679. In 1705 he published his <hi rend="italics">Synopsis Astronomiæ Cometicæ,</hi> in which he ventured to predict the return of a comet in 1758 or 1759. He was the first who discovered the acceleration of the moon, and he gave a very ingenious method for finding her parallax by three observed phases of a solar eclipse. He published, in the Philosophical Transactions, many learned papers, and amongst them some that were concerning the use that might be made of the next transit of Venus in determining the distance of the sun from the earth. He composed tables of the sun, moon, and all the planets, which are still in great repute; with which he compared the observations he made of the moon at Greenwich, amounting to near 1500, and noted the differences. He recommended the method of determining the longitude by the moon's distances from the sun and certain fixed stars; a method which had before been noticed, and which has since been carried into execution, more particularly at the instance of the present Astronomer Royal.</p><p>About this time a dispute arose concerning the figure of the earth. Sir Isaac Newton had determined, from a consideration of the laws of gravity, and the diurnal motion of the earth, that the figure of it was an oblate <pb n="161"/><cb/> spheroid, and flatted at the poles: but Cassini had determined, from the measures of Picart, that the figure was an oblong spheroid, or lengthened at the poles. To settle this dispute, it was resolved, under Lewis XV, to measure two degrees of the meridian; one near the equator, and the other as near the pole as possible. For this purpose, the Royal Academy of Sciences sent to Lapland, Mess. Maupertuis, Clairault, Camus, and Le Monier; being also accompanied by the Abbé Outhier, and by M. Celsus, professor of anatomy at Upsal. And on the southern expedition were sent Mess. Godin, Condamine, and Bouguer, to whom the king of Spain joined Don George Juan and Don Antonio de Ulloa. These set out in 1735, and returned at different times in 1744, 1745, and 1746; but the former party, who set out only in 1736, returned the year following; having both fulfilled their commissions. Picart's measure was also revised by Cassini and De la Caille, which after his errors were corrected, was found to agree very well with the other two; and the result of the whole served to confirm the determination of the figure before laid down by Newton.—On the southern expedition, it was found that the attraction of the great mountains of Peru had a sensible effect on the plumb-line of one of their largest instruments, deflecting it 7 or 8 seconds from the true perpendicular.</p><p>On the death of Dr. Halley, in 1742, he was succeeded by Dr. Bradley, as Astronomer Royal at Greenwich. The accuracy of his observations enabled him to detect the smaller inequalities in the motions of the planets and fixed stars. The consequence of this accuracy was, the discovery of the aberration of light, the nutation of the earth's axis, and a much greater degree of perfection in the lunar tables. He also observed the places, and computed the elements of the comets which appeared in the years 1723, 1736, 1743, and 1757. He made new and accurate tables of the motions of Jupiter's satellites; and from a multitude of observations of the luminaries, he constructed the most accurate table of refractions yet extant. Also, with a very large transit instrument, and a new mural quadrant of 8 feet radius, constructed by Mr. Bird in 1750, he made an immense number of observations for settling the places of all the stars in the British catalogue, together with near 1500 places of the moon, the greater part of which he compared with Mayer's tables. Dr. Bradley died in 1762.</p><p>In the mean time the mathematicians and astronomers elsewhere were assidnous in their endeavours to promote the science of astronomy. The theory of the moon was particularly considered by Messrs Clairault, D'Alembert, Euler, Meyer, Simpson, and Walmsly, and especially Clairault, Euler, and Mayer, who computed complete sets of lunar tables; those of the last of these authors, for their superior accuracy, were rewarded with a premium of 3000 pounds by the Board of Longitude, who brought them into use in the computation of the Nautical Ephemeris, published by that Board.—The most accurate tables of the satellites of Jupiter were composed, from observations by Mr. Wargentin, an excellent Swedish astronomer.—Among the many French astronomers who contrib<*>ted to the advancement of the science, it was particalarly indebted to M. De la Caille, for an excellent set of solar tables. And in 1750 he went to the Cape of Good Hope to make observations in concert <cb/> with the most celebrated astronomers in Europe, for determining the parallax of Mars and the moon, and thence, that of the sun, which it was concluded did not much exceed 10 seconds. Here he re-examined and adjusted, with great accuracy, the stars about the southern pole; and also measured a degree of the meridian. —In Italy the science was assiduously cultivated by Bianchini, Boscovich, Frisi, Manfredi, Zanotti, and many others; in Sweden by Wargentin already mentioned, Blingenstern, Mallet, and Planman; and in Germany by the Eulers, Mayer, Lambert, Grischow, and others.</p><p>In the year 1760 all the learned Societies in Europe made preparations for observing the transit of Venus over the sun, which had been predicted by Dr. Halley more than 80 years before, and the use that might be made of it in determining the sun's parallax, and the distances of the planets from the sun. And the same exertions were repeated, to observe the transit in 1769, by sending observers to different parts of the world, for the more convenience in observing. And from the whole, Mr. Short computed that the sun's parallax was nearly 8 3/5 seconds, and consequently the distance of the sun from the earth about 24114 of the earth's diameters, or 96 millions of miles.</p><p>Dr. Bradley was succeeded, in 1762, in his office of Astronomer Royal, by Mr. Bliss, Savilian professor of astronomy; who being in a declining state of health, did not long enjoy it. But, dying in 1765, was sueceeded by the learned Nevil Maskelyne, D. D. the present Astronomer Royal, who has discharged the duties of that office with the greatest honour to himself, and benesit to the science. In January 1761 this gentleman was sent by the Royal Society, at a very early age, to the Island of St. Helena, to observe the transit of Venus over the sun, and the parallax of the star Sirius. The sirst of these objects partly failed, by clouds preventing the sight of the 2d internal contact; and the 2d also, owing to Mr. Short having suspended the plumbline by a <hi rend="italics">loop</hi> from the neck of the central pin. However, our astronomer indemnified himself by many other valuable observations: Thus, he observed at St. Helena, the tides; the horary parallaxes of the moon; and the going of a clock, to sind, by comparison with its previous going which had been observed in England, the difference of gravity at the two places: also, in going out and returning, he practised the method of finding the longitude by the lunar distances taken with a Hadley's Quadrant, making out rules for the use of seamen, and taught the method to the ossicers on board the ship; which he afterwards explained in a l<*>tter to the Secretary of the Royal Society, inserted in the Philos. Trans. sor the year 1762, and still more fully afterwards, in the British Mariner's Guide, which he published in the year 1763. He returned from St. Helena in the spring of 1762, after a stay there of 10 months; and in September 1763 sailed for the island of Barbadoes, to settle the longitude of the place, and to compare Mr. Harrison's watch with the time there when this gentleman should bring it out: another object was also to try Mr. Irwin's marine chair, which he did in his way out. While at Barbadoes, he also made many other observations, and amongst them, many relating to the moon's horary parallaxes, not yet published. Returning to England in the latter part of the year 1764, he was <pb n="162"/><cb/> appointed in 1765 to succeed Mr. Bliss as Astronomer Royal, and immediately recommended to the Board of Longitude the lunar method of finding the longitude, and proposed to them the project of a Nautical Almanac, to be calculated and published to facilitate that method; this they agreed to, and the first vol. was published for 1767, and it has continued ever since under his direction, to the great benefit of navigation and universal commerce.</p><p>A multitude of other useful writings by this gentleman are inserted in the volumes of the Philos. Trans. and particularly in consequence of a proposal, made by him to the Royal Society, the noble project was formed of measuring very accurately the effect of some mountain on the plumb line, in deflecting it from the perpendicular; and the mountain of Schehallien, in Scotland, having been found the most convenient in this island for the purpose, at the request of the Society, he went into Scotland to conduct the business, which he performed in the most accurate manner, shewing that the sum of the deflections on the two opposite sides was about 11 3/5 seconds of a degree; and proving, to the satisfaction of the whole world, the universal attraction of all matter. From the data resulting from these measures I have computed the mean density of the whole matter in the earth, which I have found to be about 4 1/2 times that of common water. Besides many learned and valuable papers in the Philosophical Transactions, and the most assiduous exertions in the duties of the observatory, as abundantly appears by the curious and voluminous observations which he has published, the world is particularly obliged to his endeavours with the Board of Longitude, for the publication of the Nautical Ephemeris, and the method of observing the longitude, by the distances of the moon and stars, now adopted by all nations, and by which the practice of Navigation has been brought to the greatest perfection.</p><p>Finally, the discoveries of Dr. Herschel form a new æra in astronomy. He first, in 1781, began with observations on the periodical star <hi rend="italics">in Collo Ceti,</hi> and a new method of measuring the lunar mountains, none of which he made more than half <*> mile in height: and, having constructed telescopes vastly more powerful than any former ones, he proceeded to other observations, concerning which he has had several papers printed in the Philosophical Transactions; as, on the rotation of the planets round their axes; On the parallax of the fixed stars; Catalogues of double, triple, &c stars; On the proper motion of the sun and solar system; On the remarkable appearances of the solar regions of the planet Mars, &c. And, above all, his discovery of a new primary planet, on the 13th of March 1781, which he calls the <hi rend="italics">Georgian Planet,</hi> but it is named the <hi rend="italics">Planet Herschel</hi> by the French and other foreign astronomers; by which, and its two satellites, which he has also discovered since that time, he has greatly enlarged the bounds of the solar system, this new planet being more than twice as far from the sun as the planet Saturn.</p><p>Lists and historical accounts of the principal writings and authors on astronomy are contained in Weidler's History of Astronomy, which is brought down to the year 1737. There is also Bailly's History of astronomy, ancient and modern. For this purpose, consult also the following authors, viz, Adam, Vossius, Bayle, Chauffepié, Niceron, Perraut, the chronological table of <cb/> Riccioli, and that of Sherburn, at the end of his edition of Manilius; also the first volume of De la Lande's astronomy. The more modern, and popular books on astronomy, are very numerous, and well known: as those of Ferguson, Long, Emerson, Vince, De la Lande, Leadbetter, Brent, Keil, Whiston, Wing, Street, &c, &c.</p></div1><div1 part="N" n="ASTROSCOPE" org="uniform" sample="complete" type="entry"><head>ASTROSCOPE</head><p>, a kind of astronomical instrument, composed of two cones, on the surface of which are delineated the constellations, with their stars, by means of which these may easily be known in the heavens. The astroscope was invented by William Shukhard, professor of mathematics at Tubingen, upon which he published a treatise in 1698.</p></div1><div1 part="N" n="ASTROSCOPIA" org="uniform" sample="complete" type="entry"><head>ASTROSCOPIA</head><p>, the art of observing and examining the stars, by means of telescopes, to discover their nature and properties.</p></div1><div1 part="N" n="ASTROTHEMATA" org="uniform" sample="complete" type="entry"><head>ASTROTHEMATA</head><p>, the places or positions of the stars, in an astrological scheme of the heavens.</p></div1><div1 part="N" n="ASTROTHESIA" org="uniform" sample="complete" type="entry"><head>ASTROTHESIA</head><p>, is used by some for a constellation or collection of stars in the heavens.</p></div1><div1 part="N" n="ASTRUM" org="uniform" sample="complete" type="entry"><head>ASTRUM</head><p>, or <hi rend="smallcaps">Astron</hi>, a constellation, or assemblage of stars: in which sense it is distinguished from <hi rend="italics">Aster,</hi> which denotes a single star. Some apply the term, in a more particular sense, to the Great Dog, or rather to the large bright star in his mouth.</p></div1><div1 part="N" n="ASYMMETRY" org="uniform" sample="complete" type="entry"><head>ASYMMETRY</head><p>, the want of proportion, otherwise called <hi rend="italics">incommensur ability,</hi> or the relation of two quantities which have no common measure, as between 1 and √2, or the side and diagonal of a square.</p></div1><div1 part="N" n="ASYMPTOTE" org="uniform" sample="complete" type="entry"><head>ASYMPTOTE</head><p>, is properly a right line, which approaches continually nearer and nearer to some curve, whose asympote it is said to be, in such sort, that when they are both indefinitely produced, they are nearer together than by any assignable finite distance; or it may be considered as a tangent to the curve when conceived to be produced to an infinite distance. Two curves are also said to be asymptotical, when they thus continually approach indefinitely to a coincidence: thus, two parabolas, placed with their axes in the same right line, are asymptotes to one another.</p><p>Of lines of the second kind, or curves of the sirst kind, that is the conic sections, only the hyperbola has asymptotes, which are two in number. All curves of the second kind have at least one asymptote; but they may have three. And all curves of the third kind may have four asymptotes. The conchoid, cissoid, and logarithmic curve, though not reputed geometrical curves, have each one asymptote. And the branch or leg of a curve that has an asymptote, is said to be of the hyperbolic kind.</p><p>The nature of asymptotes will be easily conceived from the instance of the asymptote to the conchoid. Thus, if, ABC &c be part of a conchoid, and the line MN be so drawn that the parts FB, GC, HD, IE, &c, of right lines, drawn from the pole P, be equal to each other; then will the line MN be the asymptote of the curve: because the perpendicular C<hi rend="italics">c</hi> is shorter than FB, and D<hi rend="italics">d</hi> than C<hi rend="italics">c,</hi> &c; so that the two lines continually approach; yet the points E<hi rend="italics">e</hi> &c can never coincide. <figure/> <pb n="163"/><cb/></p><p><hi rend="smallcaps">Asymptotes</hi> <hi rend="italics">of the</hi> <hi rend="smallcaps">Hyperbola</hi> are thus described. Suppose a right line DE drawn to touch the curve in any point A, and equal to the conjugate <hi rend="italics">de</hi> of the diamete ACB drawn to that point A, viz, AD or AE equal to the semiconjugate C<hi rend="italics">d</hi> or C<hi rend="italics">e</hi>; then the two lines CDF, CEH, drawn from the centre C, through the points D and E, are the two asymptotes of the curve. <figure/></p><p>The parts of any right line, lying between the curve of the common hyperbola and its asymptotes, are equal to one another on both sides, that is, <hi rend="italics">g</hi>G=<hi rend="italics">h</hi>H. In like manner, in hyperbolas of the second kind, if there be drawn any right line cutting both the curve and its three asymptotes in three points, the sum of the two parts of that right line extended in the same direction from any two of the asymptotes to two points of the curve, is equal to the third part which extends in the contrary direction from the third asymptote to the third point of the curve.</p><p>If AGK be an hyperbola of any kind, whose nature, with regard to the curve and asymptote, is expressed by this general equation <hi rend="italics">x</hi><hi rend="sup">m</hi><hi rend="italics">y</hi><hi rend="sup">n</hi> = <hi rend="italics">a</hi><hi rend="sup">m4n</hi>, where <hi rend="italics">x</hi> is=CF, and <hi rend="italics">y</hi>=FG drawn any where parallel to the other asymptote CH; and the parallelogram CFGI be completed: Then <hi rend="italics">m—n</hi> is to <hi rend="italics">n,</hi> as this parallelogram CFGI is to the hyperbolic space FGK, contained under the determinate line FG, with the asymptote FK and the curve GK, both indefinitely continued towards K. So that, if <hi rend="italics">m</hi> be greater than <hi rend="italics">n,</hi> the said asymptotic space is finite and quadrable: but when <hi rend="italics">m</hi>=<hi rend="italics">n,</hi> as in the common or conic hyperbola, then <hi rend="italics">m</hi>-<hi rend="italics">n</hi>=o, the ratio of that space to the said parallelogram, is as <hi rend="italics">n</hi> to o: that is, the hyperbolic space is insinitely great, in respect of the finite parallelogram: and when <hi rend="italics">m</hi> is less than <hi rend="italics">n,</hi> then, <hi rend="italics">m</hi>-<hi rend="italics">n</hi> being negative, the asymptotic space is to the determinate parallelogram, as a positive number is to a negative one, and is what Dr. Wallis calls more than infinite.</p><p><hi rend="smallcaps">Asymptote</hi> <hi rend="italics">of the</hi> <hi rend="smallcaps">Logarithmic Curve.</hi> If LMN be the logarithmic curve, QON an asymptote, LQ and MP ordinates, MO a tangent, and PO the subtangent, which in this curve is a constant quantity. Then the indeterminate space LMNQ is equal to LQ X PO, the rectangle under the ordinate LQ and the constant subtangent PO; and the solid generated by the rotation of that curve space about the asymptote NQ, is equal to half the cylinder, whose altitude is the said constant subtangent PO, and the radius of its base is LQ.</p><div2 part="N" n="Asymptotes" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Asymptotes</hi></head><p>, by some, are distinguished into various orders. The asymptote is said to be of the first order, when it coincides with the base of the curvilinear figure: of the 2d order, when it is a right line parallel to the base: of the 3d order, when it is a right line ob- <cb/> lique to the base: of the 4th order, when it is the common parabola, having its axis perpendicular to the base: and, in general, of the <hi rend="italics">n</hi> + 2 order, when it is a parabola whose ordinate is always as the <hi rend="italics">n</hi> power of the base. The asymptote is oblique to the base, when the ratio of the first fluxion of the ordinate to the fluxion of the base, approaches to an assignable ratio, as its limit; but it is parallel to the base, or coincides with it, when this limit is not assignable.</p><p>The doctrine and determination of the asymptotes of curves, is a curious part of the higher geometry. Fontenelle has given several theorems relating to this subject, in his <hi rend="italics">Geometrie de l'Infini.</hi> See also Stirling's <hi rend="italics">Lineæ Tertii Ordinis,</hi> prop. vi, where the subject of Asymptotes is learnedly treated; and Cramer's <hi rend="italics">Introduction à l'analyse des lignes courbes,</hi> art. 147 <hi rend="italics">& seq.</hi> for an excellent theory of asymptotes of geometrical curves and their branches. Likewise Maclaurin's Algebra, and his Fluxions, book i, chap. 10, where he has carefully avoided the modern paradoxes concerning infinites and infinitesimals. But the easiest way of determining asymptotes, it seems, is by considering them as tangents to the curves at an infinite distance from the beginning of the absciss, that is when the absciss <hi rend="italics">x</hi> is infinite in the equation of the curve, and in the proportion of <hi rend="italics">x<hi rend="sup">.</hi></hi> to <hi rend="italics">y<hi rend="sup">.</hi>,</hi> or in that of the subtangent to the ordinate.</p><p>The areas bounded by curves and their asymptotes, though indefinitely extended, have sometimes limits to which they may approach indefinitely near: and this happens in hyperbolas of all kinds, except the sirst or Apollonian, and in the logarithmic curve; as was observed above. But in the common hyperbola, and many other curves, the asymptotical area has no such limit, but is infinitely great.—Solids, too, generated by hyperbolic areas, revolving about their asymptotes, have sometimes their limits; and sometimes they may be produced till they exceed any given solid.—Also the surface of such solid, when supposed to be infinitely produced, is either finite or infinite, according as the area of the generating figure is finite or infinite.</p></div2></div1><div1 part="N" n="ATLANTIDES" org="uniform" sample="complete" type="entry"><head>ATLANTIDES</head><p>, a name given to the Pleiades, or seven stars, sometimes also called Vergiliæ. They were thus called, as being supposed by the poets to have been the daughters either of Atlas or his brother Hesperus, who were translated to heaven.</p></div1><div1 part="N" n="ATMOSPHERE" org="uniform" sample="complete" type="entry"><head>ATMOSPHERE</head><p>, a term used to signify the whole of the fluid mass, consisting of air, aqueous and other vapours, electric fluids, &c, which surrounds the earth to a considerable height, and partaking of all its motions, both annual and diurnal.</p><p>The composition of that part of our atmosphere properly called air, was till lately but very little known. Formerly it was supposed to be a simple, homogeneous, and elementary fluid. But the experiments of Dr. Priestley and others, have discovered, that even the purest kind of air, which they call dephlogisticated, is in reality a compound, and might be artificially produced in various ways. This dephlogisticated air, however, is but a small part of the composition of our atmosphere. By accurate experiments, the air we usually breathe, is composed of only one-fourth part of this dephlogisticated air, or perhaps less; the other three parts, or more, consisting of what Dr. Priestley calls <hi rend="italics">phlogisticated,</hi> and M. Lavoisier <hi rend="italics">mephitic air.</hi> <pb n="164"/><cb/></p><p>Beside these sorts of air, it is obvious that the whole mass of the atmosphere contains a great deal of water, together with a vast heterogeneous collection of particles raised from all bodies of matter on the surface of the earth, by effluvia, exhalations, &c; so that it may be considered as a chaos of the particles of all sorts of matter confusedly mingled together. And hence the atmosphere has been considered as a large chemical vessel, in which the matter of all kinds of sublunary bodies is copiously floating; and thus exposed to the continual action of that immense surface the sun; from whence proceed innumerable operations, sublimations, separations, compositions, digestions, fermentations, putrefractions, &c.</p><p>There is, however, one substance, namely the electrical fluid, which is very distinguishable in the mass of the atmosphere. To measure the absolute quantity of this fluid, either in the atmosphere or any other substance, is perhaps impossible: and all that we know on this subject is, that the electric fluid pervades the atmosphere; that it appears to be more abundant in the superior than the inferior regions; that it seems to be the immediate bond of connection between the atmosphere and the water which is suspended in it; and that by its various operations, the phenomena of hail, rain, snow, lightning, and the other kinds of meteors are occasioned. See those respective articles; and see also Beccaria's Essay on Atmospheric Electricity, annexed to the English translation of his Artisicial Electricity.</p><p><hi rend="italics">Uses of the</hi> <hi rend="smallcaps">Atmosphere.</hi>—Theuses of the atmosphere are so many and great, that it seems indeed absolutely necessary, not only to the comfort and convenience of men, but even to the existence of all animal and vegetable life, and to the very constitution of all kinds of matter whatever, and without which they would not be what they are: for by it we live, breathe, and have our being; and by insinuating itself into all the vacuities of bodies, it becomes the great spring of most of the mutations here below; as generation, corruption, dissolution, &c; and without which none of these operations could be carried on. Without the atmosphere, no animal could exist, or indeed be produced; neither any plant, all vegetation ceasing without its aid; there would be neither rain nor dews to moisten the face of the ground; and though we might perceive the sun and stars like bright specks, we should be in utter darkness, having none of what we call day light or even twilight: nor would either fire or heat exist without it. In short, the nature and constitution of all matter would be changed and cease; wanting this universal bond and constituting principle.</p><p>By the mechanical force of the atmosphere too, as well as its chemical virtues, many necessary purposes are answered. We employ it as a moving power, in the motion of ships, to turn mills, and for other such uses. And it is one of the great discoveries of the modern philosophers, that the several motions attributed by the ancients to a <hi rend="italics">fuga vaceui,</hi> are really owing to the pressure of the atmosphere. Galileo, having observed that there was a certain standard altitude, beyond which no water could be elevated by pumping, took an occasion from thence to call in question the doctrine of the schools, which ascribed the ascent of water in pumps, to the <cb/> <hi rend="italics">fuga vacui,</hi> and instead of it he happily substituted the hypothesis of the weight and pressure of the air. It was with him, indeed, little better than an hypothesis, since it had not then those confirmations from experiment, afterwards found out by his pupil Torricelli, and other succeeding philosophers, particularly Mr Boyle.</p><p>Nor have the attempts to fly or float through the air been altogether without success. F. De Lana thought he had contrived an aeronautic machine for navigating the atmosphere: and Sturmius, who examined it, asserted that it was not impracticable: though Dr Hook was of a different opinion, and detected the fallacy of the contrivance. Roger Bacon, long before, proposed something of the same kind. The great secret of this art is to contrive a machine so much lighter than the air, that it will rise up and float in the atmosphere, and together with itself, buoy up and carry men along with it. The principle on which it is to be effected is, by means of an air pump, to exhaust the air from a very thin and light, yet firm, metalline vessel. But the hopes of success in such an enterprize will appear very small if it be considered, that if a globe were formed of brass of the thickness only of 1/12 of an inch, such a globe would require to be about 277 feet in diameter to float in the air: and if, as De Lana supposes, the diameter of the globe were but 25 feet, the thickness of the metal could not exceed 1/33 of an inch. See Herman's Phoronomia pa. 158. However, what is not to be expected from metalline globes or shells, has now been successfully accomplished by the balloons of cloth, silk, or skin, of Montgolfier and others. See the article <hi rend="italics">Aerostation,</hi> &c.</p><p><hi rend="italics">Salubrity of the</hi> <hi rend="smallcaps">Atmosphere.</hi> On the tops of mountains the air is generally more salubrious than in pits or very deep places. Indeed dense air is always more proper for respiration, as to the mere quality of density only, than that which is rarer. But then the air on mountains, though rarer, is freer from phlogistic vapours than that of pits; and hence it has been found that people can live very well on the tops of mountains, even when the air is but about half the density of that below. But it would seem that at some intermediate height between the two extremes, the air is the most salubrious and proper for animal life; and this height, according to M. de Saussure, is about 500 or 600 yards above the level of the sea.</p><p>Besides the difference arising from the mere difference of altitude, the salubrity of the atmosphere is greatly affected by many other circumstances. The air, when confined or stagnant, is commonly more impure than when agitated and shifted: thus, all close places are unhealthy, and even the air in a bed chamber is less salubrious in a morning, after it has been slept in, than in the evening. Dr White, in vol. 68 Philos. Trans. gives an account of experiments on this quality of the air, and remarks one instance when the air was particularly impure, viz September 13, 1777; when the barometer stood at 30.30, the thermometer at 69°; the air being then dry and sultry, and no rain having fallen for more than two weeks. A slight shock of an earthquake was perceived that day. In vol. 70 of the same Transactions Dr. Ingenhousz gives an account of some experiments on this head, made in various places and <pb n="165"/><cb/> situations: he finds,” That the air at sea, and close to it, is in general purer, and fitter for animal life, than the air on the land:” but the Doctor did not find much difference between the air of the towns and of the country, nor between one town and another. The Abbé Fontana, made nearly the same conclusious, from accurate experiments, asserting, “that the difference between the air of one country and that of another, at different times, is much less than what is commonly believed; and yet that this difference in the purity of the air at different times, is much greater than the difference between the air of the different places observed by him.” Finally M. Fontana concludes, that “Nature is not so partial as we commonly believe. She has not only given us an air almost equally good every where at every time, but has allowed us a certain latitude, or a power of living and being in health in qualities of air which differ to a certain degree. By this I do not mean to deny the existence of certain kinds of noxious air in some particular places; but only say, that in general the air is good every where, and that the small differences are not to be feared so much as some people would make us believe. Nor do I mean to speak here of those vapours and other bodies which are accidentally joined to the common air in particular places, but do not change its nature and intrinsical property. This state of the air cannot be known by the test of nitrous air; and those vapours are to be considered in the same manner as we should consider so many particles of arsenic swimming in the atmosphere. In this case it is the arsenic, and not the degenerated air, that would kill the animals who ventured to breathe it.”</p><p><hi rend="italics">Figure of the</hi> <hi rend="smallcaps">Atmosphere.</hi>—As the atmosphere envelops all parts of the surface of our globe, if they both continued at rest, and were not endowed with a diurnal motion about their common axis, then the atmosphere would be exactly globular, according to the laws of gravity; for all the parts of the surface of a fluid in a state of rest, must be equally removed from its centre. But as the earth and the ambient parts of the atmosphere revolve uniformly together about their axis, the different parts of both have a centrifugal force, the tendency of which is more considerable, and that of the centripetal less, as the parts are more remote from the axis; and hence the figure of the atmosphere must become an oblate spheroid; since the parts that correspond to the equator are father removed from the axis, than the parts which correspond to the poles. Besides, the figure of the atmosphere must, on another account, represent a flattened spheroid, namely because the sun strikes more directly the air which encompasses the equator, and is comprehended between the two tropics, than that which pertains to the polar regions: for, from hence it follows, that the mass of air, or part of the atmosphere, adjoining to the poles, being less heated, cannot expand so much, nor reach so high. And yet, notwithstanding, as the same force which contributes to elevate the air, diminishes its gravity and pressure on the surface of the earth, higher columns of it about the equatorial parts, all other circumstances being the same, may not be heavier than those about the poles.</p><p>In the Transactions of the Royal Irish Academy for 1788 Mr Kirwin has an ingenious dissertation on the <cb/> figure, height, weight, &c, of the atmosphere. He observes that, in the natural state of the atmosphere, that is, when the barometer would every where, at the level of the sea, stand at 30 inches, the weight of the atmosphere, at the surface of the sea, must be equal all over the globe; and in order to produce this equality, as the weight proceeds from its density and height, it must be lowest where the denfity is greatest, and highest where the density is least; that is, highest at the equator and lowest at the poles, with several intermediate gradations.</p><p>Though the equatorial air however be less dense to a certain height than the polar, yet at some greater heights it must be more dense: for since an equatorial and polar column are equal in total weight or mass, the lower part of the equatorial column, being more expanded by heat &c than that of the polar, must have less mass, and therefore a proportionably greater part of its mass must be found in its superior section; so that the lower extremity of the superior section of the equatorial column is more compressed, and consequently denser, than the corresponding part of the polar column. The same thing is to be understood also of the extra-tropical columns with respect to each other, where differences of heat prevail.</p><p>Hence, in the highest regions of the atmosphere, the denser equatorial air, not being supported by the collateral extra-tropical columns, gradually flows over, and rolls down to the north and south.</p><p>These superior tides consist chiefly of inflammable air, as it is much lighter than any other, and is generated in great plenty between the tropics; it furnishes the matter of the auroræ borealis and australis, by whose combustion it is destroyed, else its quantity would in time become too great, and the weight of the atmosphere annually increased; but its combustion is the primary source of the greatest perturbations of the atmosphere.</p><p><hi rend="italics">Weight or Pressure of the</hi> <hi rend="smallcaps">Atmosphere.</hi>—It is evident that the mass of the atmosphere, in common with all other matter, must be endowed with weight and pressure; and this principle was asserted by almost all philosophers, both ancient and modern. But it was only by means of the experiments made with pumps and the barometrical tube, by Galileo and Torricelli, that we came to the proof, not only that the atmosphere is endued with a pressure, but also what the measure and quantity of that pressure is. Thus, it is sound that the pressure of the atmosphere sustains a column of quicksilver, in the tube of the barometer, of about 30 inches in height; it therefore follows, that the whole pressure of the atmosphere is equal to the weight of a column of quicksilver, of an equal base, and 30 inches height: and because a cubical inch of quicksilver is found to weigh nearly half a pound averdupois, therefore the whole 30 inches, or the weight of the atmosphere on every square inch of surface, is equal to 15 pounds. Again, it has been found that the pressure of the atmosphere balances, in the case of pumps &c, a column of water of about 34 1/2 feet high; and, the cubical foot of water weighing just 1000 ounces, or 62 1/2 pounds, 34 1/2 times 62 1/2, or 2158lb, will be the weight of the column of water, or of the atmosphere on a base of a square foot; and consequently the 144th part of this, or 15lb, is the weight of the atmosphere on a square <pb n="166"/><cb/> inch; the same as before. Hence Mr Cotes computed that the pressure of this ambient fluid on the whole surface of the earth, is equivalent to that of a globe of lead of 60 miles in diameter. And hence also it appears, that the pressure upon the human body must be very considerable; for as every square inch of surface sustains a pressure of 15 pounds, every square foot will sustaín 144 times as much, or 2160 pounds; then, if the whole surface of a man's body he supposed to contain 15 square feet, which is pretty near the truth, he must sustain 15 times 2160, or 32400 pounds, that is nearly 14 1/2 tons weight, for his ordinary load. By this enormous pressure we should undoubtedly be crushed in a moment, if all parts of our bodies were not filled either with air or some other elastic fluid, the spring of which is just sufficient to counterbalance the weight of the atmosphere. But whatever this fluid may be, it is certain that it is just able to counteract the weight of the atmosphere, and no more: for, if any considerable pressure be superadded to that of the air, as by going into deep water, or the like, it is always severely felt let it be ever so equable, at least when the change is made suddenly; and if, on the other hand, the pressure of the atmosphere be taken off from any part of the human body, as the hand for instance, when put over an open receiver, from whence the air is afterwards extracted, the weight of the external atmosphere then prevails, and we imagine the hand strongly sucked down into the glass.</p><p>The difference in the weight of the air which our bodies sustain at one time more than another, is also very considerable, from the natural changes in the state of the atmosphere. This change takes place chiefly in countries at some distance from the equator; and as the barometer varies at times from 28 to 31 inches, or about one tenth of the whole quantity, it follows that this difference amounts to about a ton and a half on the whole body of a man, which he therefore sustains at one time more than at another. On the increase of this natural weight, the weather is commonly fine, and we feel ourselves what we call braced and more alert and active; but, on the contrary, when the weight of the air diminishes, the weather is bad, and people feel a listlessness and inactivity about them. And hence it is no wonder that persons sufser very much in their health, from such changes in the atmofphere, especially when they take place very suddenly, for it is to this circumstance chiefly that a sensation of uneasiness and indisposition is to be attributed; thus, when the variations of the barometer and atmosphere are sudden and great, we feel the alteration and effect on our bodies and spirits very much; but when the change takes place by very slow degrees, and by a long continuance, we are scarcely sensible of it, owing, undoubtedly, to the power with which the body is naturally endowed, of accommodating itself to this change in the state of the air, as well as to the change of many other circumstances of life, the body requiring a certain interval of time to effect the alteration in its state, proper to that of the air &c. Thus, in going up to the tops of mountains, where the pressure of the atmosphere is diminished two or three times more than on the plain below, little or no inconvenience is felt from the rarity of the air, if it is not mixed with other noxious vapours &c; because <cb/> that, in the ascent the body has had sufficient time to accommodate itself gradually to the slow variation in the state of the atmosphere: but, when a person ascends with a balloon, very rapidly to a great height in the atmosphere, he feels a difficulty in breathing and an uneasiness of body; and the same is soon felt by an animal when inclosed in a receiver, and the air suddenly drawn or pumped out of it. So also, on the condensation of the air, we feel little or no alteration in ourselves, except when the change happens suddenly, as in very rapid changes in the weather, and in descending to great depths in a diving bell, &c. I have often heard the late unfortunate Mr. Spalding speak of his experience on this point: he always found it absolutely necessary to descend with the bell very slowly, and that only from one depth to another, resting a while at each depth before he began to descend farther: he first descended slowly for about 5 or 6 fathom, and then stopped a while; he felt an uneasiness in his head and ears, which increased more and more as he descended, till he was obliged to stop at the depth above mentioned, where the density of the air was nearly doubled; having remained there a while, he felt his ears give a sudden crack, and after that he was soon relieved from any uneasiness in that part, and it seemed as if the density of the air was not altered. He then descended other 5 fathoms or 30 feet more, with the same precaution and the same sensations as before, being again relieved, in the same manner, after remaining awhile stationary at the next stage of his descent, where the density of the air was tripled. And thus he continued proceeding to a great depth, always with the same circumstances, repeated at every 5 or 6 fathoms, and adding the pressure of one more atmosphere at every period of the progress.</p><p>It is not easy to assign the true reason for the variations that happen in the gravity of the atmosphere in the same place. One cause of it however, either immediate or otherwise, it seems, is the heat of the sun; for where this is uniform, the changes are small and regular; thus between the tropics it seems the change depends on the heat of the sun, as the barometer constantly sinks about half an inch every day, and rises again to its former station in the night time. But in the temperate zones the barometer ranges from 28 to near 31 inches, shewing, by its various altitudes, the changes that are about to take place in the weather. If we could know therefore, the causes by which the weather is influenced; we should also know those by which the gravity of the atmosphere is affected. These may perhaps be reduced to immediate ones, viz, an emission of latent heat from the vapour contained in the atmosphere, or of electric fluid from the same, or from the earth; as it is observed that they both produce the same effect with the solar heat in the tropical climates, viz, to rarefy the air, by mixing with it, or setting loose a lighter fluid, which did not before act in such large proportion in any particular place.</p><p>With regard to the alteration of heat and cold in the atmosphere, many reasons and hypotheses have been given, and many experiments made; as may be seen by consulting the authors upon this subject, viz, M. Bouguer's observations in Peru, Lambert, De Luc, Saussure's journeys on the Alps, Sex's and Darwin's <pb n="167"/><cb/> experiments in vol. 78 Philos. Trans. This last gentleman hence infers, “There is good reason to conclude that in all circumstances where air is mechanically expanded, it becomes capable of attracting the fluid matter of heat from other bodies in contact with it. Now, as the vast region of air which surrounds our globe is perpetually moving along its surface, climbing up the sides of mountains, and descending into the valleys; as it passes along it must be perpetually varying the degree of heat according to the elevation of the country it traverses: for, in rising to the summits of mountains, it becomes expanded, having so much of the pressure of the superincumbent atmosphere taken away; and when thus expanded, it attracts or absorbs heat from the mountains in contiguity with it; and, when it descends into the valleys and is compressed into less compass, it again gives out the heat it has acquired to the bodies it comes in contact with. The same thing must happen to the higher regions of the atmosphere, which are regions of perpetual frost, as has lately been discovered by the aerial navigators. When large districts of air, from the lower parts of the atmosphere, are raised two or three miles high, they become so much expanded by the great diminution of the pressure over them, and thence become so cold, that hail or snow is produced by the precipitation of the vapour: and as there is, in these high regions of the atmosphere, nothing else for the expanded air to acquire heat from after it has parted with its vapour, the same degree of cold continues till the air, on descending to the earth, acquires its former state of condensation and of warmth. The Andes, almost under the line, rests its base on burning sands: about its middle height is a most pleasant and temperate climate covering an extensive plain, on which is built the city of Quito; while its forehead is encircled with eternal snow, perhaps coeval with the mountain. Yet, according to the accounts of Don Ulloa, these three discordant climates seldom encroach much on each other's territories. The hot winds below, if they ascend, become cooled by their expansion; and hence they cannot affect the snow upon the summit; and the cold winds that sweep the summit, become condensed as they descend, and of temperate warmth before they reach the fertile plains of Quito.”</p><p><hi rend="italics">Height and Density of the</hi> <hi rend="smallcaps">Atmosphere.</hi> Various attempts have been made to ascertain the height to which the atmosphere is extended all round the earth. These commenced soon after it was discovered by means of the Torricellian tube, that air is endued with weight and pressure. And had not the air an elastic power, but were it every where of the same density, from the surface of the earth to the extreme limit of the atmosphere, like water, which is equally dense at all depths it would be a very easy matter to determine its height from its density and the column of mercury which it would counterbalance in the barometer tube: for, it having been observed that the weight of the atmosphere is equivalent to a column of 30 inches or 2 1/2 feet of quicksilver, and the density of the former to that of the latter, as 1 to 11040; therefore the height of the uniform atmosphere would be 11040 times 2 1/2 feet, that is 27600 feet, or little more than 5 miles and a quarter. But the air, by its elastic quality, expands and contracts; and it being found by repeated experiments in <cb/> most nations of Europe, that the spaces it occupies, when compressed by different weights, are reciprocally proportional to those weights themselves; or, that the more the air is pressed, so much the less space it takes up; it follows that the air in the upper regions of the atmosphere must grow continually more and more rare, as it ascends higher; and indeed that, according to that law, it must necessarily be extended to an indefinite height. Now, if we suppose the height of the whole divided into innumerable equal parts; the quantity of each part will be as its density; and the weight of the whole incumbent atmosphere being also as its density; it follows, that the weight of the incumbent air, is every where as the quantity contained in the subjacent part; which causes a difference between the weights of each two contiguous parts of air. But, by a theorem in arithmetic, when a magnitude is continually diminished by the like part of itself, and the remainders the same, these will be a series of continued quantities decreasing in geometrical progression: therefore if, according to the supposition, the altitude of the air, by the addition of new parts into which it is divided, do continually increase in arithmetical progression, its density will be diminished, or, which is the same thing, its gravity decreased, in continued geometrical proportion. And hence, again, it appears that, according to the hypothesis of the density being always proportional to the compressing force, the height of the atmosphere must necessarily be extended indefinitely. And, farther, as an arithmetical series adapted to a geometrical one, is analogous to the logarithms of the said geometrical one; it follows therefore that the altitudes are proportional to the logarithms of the densities, or weights of air; and that any height taken from the earth's surface, which is the difference of two altitudes to the top of the atmosphere, is proportional to the difference of the logarithms of the two densities there, or to the logarithm of the ratio of those densities, or their corresponding compressing forces, as measured by the two heights of the barometer there. This law was first observed and demonstrated by Dr. Halley, from the nature of the hyperbola; and afterwards by Dr. Gregory, by means of the logarithmic curve. See Philos. Trans. N°. 181, or Abridg. vol. 2, p. 13, and Greg. Astron. lib. v, prop. 3.</p><p>It is now easy, from the foregoing property, and two or three experiments, or barometrical observations, made at known altitudes, to deduce a general rule to determine the absolute height answering to any density, or the density answering to any given altitude above the earth. And accordingly, calculations were made upon this plan by many philosophers, particularly by the French; but it having been found that the barometrical observations did not correspond with the altitudes as measured in a geometrical manner, it was suspected that the upper parts of the atmospherical regions were not subject to the same laws with the lower ones, in regard to the density and elasticity. And indeed, when it is considered that the atmosphere is a heterogeneous mass of particles of all sorts of matter, some elastic, and others not, it is not improbable but this may be the case, at least in the regions very high in the atmosphere, which it is likely may more copiously abound with the electrical fluid. Be this however as it may, it has lately been discovered that the law <pb n="168"/><cb/> above given, holds very well for all such altitudes as are within our reach, or as far as to the tops of the highest mountains on the earth, when a correction is made for the difference of the heat or temperature of the air only, as was fully evinced by M. De Luc, in a long series of observations, in which he determined the altitudes of hills both by the barometer, and by geometrical measurement, from which he deduced a practical rule to allow for the difference of temperature. See his Treatise on the Modifications of the Atmosphere. Similar rules have also been deduced from accurate experiments, by Sir George Shuckburgh and General Roy, both concurring to shew, that such a rule for the altitudes and densities, holds true for all heights that are accessible to us, when the elasticity of the air is corrected on account of its density: and the result of their experiments shewed, that the difference of the logarithms of the heights of the mercury in the barometer, at two stations, when multiplied by 10000, is equal to the altitude in English fathoms, of the one place above the other; that is, when the temperature of the air is about 31 or 32 degrees of Fahrenheit's thermometer; and a certain quantity more or less, according as the actual temperature is different from that degree.</p><p>But it may here be shewn, that the same rule may be deduced independent of such a train of experiments as those above, merely by the density of the air at the surface of the earth alone. Thus, let D denote the density of the air at one place, and <hi rend="italics">d</hi> the density at the other; both measured by the column of mercury in the barometrical tube: then the difference of altitude between the two places, will be proportional to the log. of D - the log. of <hi rend="italics">d,</hi> or to the log. of D/<hi rend="italics">d.</hi> But as this formula expresses only the relation between different altitudes, and not the absolute quantity of them, assume some indeterminate, but constant quantity <hi rend="italics">h,</hi> which multiplying the expression log. D/<hi rend="italics">d,</hi> may be equal to the real difference of altitude <hi rend="italics">a,</hi> that is, . Then, to determine the value of the general quantity <hi rend="italics">h,</hi> let us take a case in which we know the altitude <hi rend="italics">a</hi> which corresponds to a known density <hi rend="italics">d</hi>; as for instance, taking <hi rend="italics">a</hi> = 1 foot, or 1 inch, or some such small altitude: then because the density D may be measured by the pressure of the whole atmosphere, or the uniform column of 27600 feet, when the temperature is 55°; therefore 27600 feet will denote the density D at the lower place, and 27599 the less density <hi rend="italics">d</hi> at 1 foot above it; consequently , which, by the nature of logarithms, is nearly nearly; and hence we find <hi rend="italics">h</hi> = 63551 feet; which gives us this formula for any altitude <hi rend="italics">a</hi> in general, viz, fathoms; where M denotes the column of mercury <cb/> in the tube at the lower place, and <hi rend="italics">m</hi> that at the upper This formula is adapted to the mean temperature of the air 55°: but it has been found, by the experiments of Sir Geo. Shuckburgh and General Roy, that for every degree of the thermometer, different from 55°, the altitude <hi rend="italics">a</hi> will vary by its 435th part; hence, if we would change the factor <hi rend="italics">h</hi> from 10592 to 10000, because the difference 592 is the 18th part of the whole factor 10592, and because 18 is the 24th part of 435; therefore the change of temperature, answering to the change of the factor <hi rend="italics">h,</hi> is 24°, which reduces the 55° to 31°. So that, fathoms, is the easiest expression for the altitude, and answers to the temperature of 31°, or very nearly the freezing point: and for every degree above that, the result must be increased by so many times its 435th part, and diminished when below it.</p><p>From this theorem it follows, that, at the height of 3 1/2 miles, the density of the atmosphere is nearly 2 times rarer than it is at the surface of the earth; at the height of 7 miles, 4 times rarer; and so on, according to the following table: <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Height in miles.</cell><cell cols="1" rows="1" rend="align=center" role="data">Number of times rarer.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">256</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">1624</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">4096</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">16384</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">65536</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">63</cell><cell cols="1" rows="1" rend="align=right" role="data">262144</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">1048576</cell></row></table> And, by pursuing the calculations in this table, it might be easily shewn, that a cubic inch of the air we breathe would be so much raresied at the height of 500 miles, that it would fill a sphere equal in diameter to the orbit of Saturn.</p><p>Hence we may perceive how very soon the air becomes so extremely rare and light, as to be utterly imperceptible to all experience; and that hence, if all the planets have such atmospheres as our earth, they will, at the distances of the planets from one another, be so extremely attenuated, as to give no sensible resistance to the planets in their motion round the sun for many, perhaps hundreds or thousands of ages to come. Even at the height of about 50 miles, it is so rare as to have no sensible effect on the rays of light: for it was found by Kepler, and De la Hire after him, who computed the height of the sensible atmosphere from the duration of twilight, and from the magnitude of the terrestrial shadow in lunar eclipses, that the effect of the atmosphere to reflect and intercept the light of the sun, is only sensible to the altitude of between 40 and 50 miles: and at that altitude we may collect, from what has been already said, that the air is above 10000 times rarer than at the surface of the earth. It is well known that the twilight begins and ends when the centre of the sun is about 18 degrees below the horizon, or only 17° 27′, by subtracting 33′ for <pb n="169"/><cb/> refraction, which raises the sun so much higher than he would be. And a ray coming from the sun in that position, and entering the earth's atmosphere, is refracted and bent into a curve line in passing through it to the eye. M. de la Hire took great pains to demonstrate, that, supposing the density of the atmosphere proportional to its weight, this curve is a cycloid: he also says, that if the ray be a tangent to the atmosphere, the diameter of its generating circle will be the height of the atmosphere; and that this diameter increases, till at last, when the rays are perpendicular, it becomes infinite, or the circle degenerates into a right line. This reasoning supposes that the refracting surface of the atmosphere is a plane; but since it is in reality a curve, he observes that these cycloids become in fact epicycloids. But Herman detected the error of M. de la Hire, and shewed that this curve is infinitely extended, and has an asymptote. And it is observed by Dr. Brook Taylor, in his Methodus Increm. pa. 168, &c, that this curve is one of the most intricate and perplexed that can well be proposed. The same ingenious author computes, that the refractive power of the air is to the force of gravity at the surface of the earth, as 320 millions to 1.</p><p>Considering the extreme rarity of the atmosphere at only 40 or 50 miles in height, it seems to be surprizing that some meteors should be enflamed at such great heights as they have been observed at. A very remarkable one of this kind was observed by Dr. Halley in the month of March 1719, the altitude of which he computed at between 69 and 73 1/2 English miles; its diameter 2800 yards, or more than a mile and a half; and its velocity about 350 miles in a minute. Others, apparently of the same kind, but whose altitude and velocity were still greater, have been observed; particularly that very remarkable one, of August 18th 1783, whose distance from the earth could not be less than 90 miles, its diameter at least as large as the former, while its velocity was certainly not less than 1000 miles in a minute. Now, from analogy of reasoning, it seems very probable, that the meteors which appear at such great heights in the air, are not essentially different from those which are seen on or near the surface of the earth. The difficulty with regard to the former is, that at the great heights above-mentioned, the atmosphere ought not to have any density sufficient to support flame, or to propagate sound; and yet such meteors are commonly succeeded by one explosion or more, and it is said are even sometimes accompanied with a hissing noise as they pass over our heads. The meteor of 1719 was not only very bright, seeming for a short time to turn night into day, but was attended with an explosion heard over all the island of Britain, causing a violent concussion in the atmosphere, and seeming to shake the earth itself: And yet, in the regions in which this meteor moved, the air ought to have been 300 thousand times rarer than the air we breathe, or 1000 times rarer than the vacuum commonly made by a good air-pump. Dr. Halley offers a conjecture, indeed, that the vast magnitude of such bodies might compensate for the thinness of the medium in which they moved. But <cb/> appearances of this kind are, by some others, attributed to electricity; though the circumstances of them cannot be reconciled to that cause; for the meteors move with all different degrees of velocity; and though the electrical fire easily pervades the vacuum of an airpump, yet it does not in that case appear in bright well-desined sparks, as in the open air, but rather in long streams resembling the aurora borealis; and from some late experiments it has been concluded that the electric fluid cannot even penetrate a perfect vacuum.</p><p><hi rend="italics">Of the Refractive and Reflective Power</hi> of the A<hi rend="smallcaps">TMOSPHERE.</hi> It has been observed above, that the atmosphere has a refractive power, by which the rays of light are bent from the right lined direction, as in the case of the twilight; and many other experiments manifest the same virtue, which is the cause of many phenomena. Alhazen, the Arabian, who lived about the year 1100, it seems was more inquisitive into the nature of refraction than former writers. But neither Alhazen, nor his follower Vitello, knew any thing of its just quantity, which was not known, to any tolerable degree of exactness, till Tycho Brahe, with great diligence, settled it. But neither did Tycho nor Kepler discover in what manner the rays of light were refracted by the atmosphere. Tycho thought the refraction was chiefly caused by dense vapours, very near the earth's surface: while Kepler placed the cause wholly at the top of the atmosphere, which he thought was uniformly dense; and thence he determined its altitude to be little more than that of the highest mountains. But the true constitution of the density of the atmosphere, deduced afterwards from the Torricellian experiment, afforded a juster idea of these refractions, especially after it appeared, by a repetition of Mr. Lowthorp's experiment, that the refractive power of the air is proportional to its density. By this variation in the density and refractive power of the air, a ray of light, in passing through the atmosphere, is continually refracted at every point, and thereby made to describe a curve, and not a straight line, as it would have done were there no atmosphere, or were its density uniform.</p><p>The atmosphere, or air, has also a reflective power; and this power is the means by which objects are enlightened so uniformly on all sides. The want of this power would occasion a strange alteration in the appearance of things; the shadows of which would be so very dark, and their sides enlightened by the sun so very bright, that probably we could see no more of them than their bright halves; so that for a view of the other halves, we must turn them half round, or if immoveable, must wait till the sun could come round upon them. Such a pellucid unreflective atmosphere would indeed have been very commodious for astronomical observations on the course of the sun and planets among the fixed stars, visible by day as well as by night; but then such a sudden transition from darkness to light, and from light to darkuess, immediately upon the rising and setting of the sun, without any twilight, and even upon turning to or from the sun at noon day, would have been very inconvenient and offensive to our eyes. However, though the atmosphere <pb n="170"/><cb/> be greatly assistant in the illumination of objects, yet it must also be observed that it stops a great deal of light. By M. Bouguer's experiments, it seems that the light of the moon is often 2000 times weaker in the horizon, than at the altitude of 66 degrees; and that the proportion of her light at the altitudes of 66 and 19 degrees, is about 3 to 2; and the lights of the sun must bear the same proportion to each other at those heights; which Bouguer made choice of, as being the meridian heights of the sun, at the summer and winter solstices, in the latitude of Croisic in France. Smith's optics. Rem. 95.</p><p>For the Atmosphere of the sun, moon, and planets, see the respective articles.</p><p><hi rend="smallcaps">Atmosphere</hi> <hi rend="italics">of Solid or Consistent Bodies,</hi> is a kind of sphere formed by the effluvia, or minute corpuscles, emitted from them. Mr. Boyle endeavours to shew, that all bodies, even the hardest and most coherent, as gems, &c, have their atmospheres.</p><div2 part="N" n="Atmosphere" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Atmosphere</hi></head><p>, <hi rend="italics">in Electricity,</hi> denotes that medium which is conceived to be diffused over the surface of electrified bodies, and to some distance around them, and consisting of effluvia issuing from them; by which, other bodies immerged in it become endued with an electricity contrary to that of the body to which the atmosphere belongs. This was first noticed at a very early period in the history of this science by Otto Guericke, and afterwards by the academicians <hi rend="italics">del Cimento,</hi> who contrived to render the electric atmosphere visible, by means of smoke attracted by a piece of amber, and gently rising from it, but vanishing as the amber cooled. Dr. Franklin exhibited this electric atmosphere with greater advantage, by dropping rosin on hot iron plates held under electrified bodies, from which the smoke arose and encompassed the bodies, giving them a very beautiful appearance. But the theory of electric atmospheres was not well explained and understood for a considerable time; and the investigation led to many curious experiments and observations. The experiments of Mr. Canton and Dr. Franklin prepared the way for the conclusion that was afterwards drawn from them by Mess. Wilcke and Epinus, though they retained the common opinion of electric atmospheres, and endeavoured to explain the phenomena by it. The conclusion was, that the electric fluid, when there is a redundancy of it in any body, repels the electric fluid in any other body, when they are brought within the sphere of each other's influence, and drives it into the remote parts of the body, or quite out of it, if there be any outlet for that purpose.</p><p>By Atmosphere, M. Epinus says, no more is to be understood than the sphere of action belonging to any body, or the neighbouring air electrified by it. Sig. Beccaria agrees in the same opinion, that electrified bodies have no other atmosphere than the electricity communicated to the neighbouring air, and which goes with the air, and not with the electrified bodies. Mr. Canton also, having relinquished the opinion that electrical atmospheres were composed of effluvia from excited or electrified bodies, maintained that they only result from an alteration in the state of the electric fluid contained in it, or belonging to the air surrounding these bodies to a certain distance; for instance, that <cb/> excited glass repels the electric fluid from it, and consequently beyond that distance makes it more dense; whereas excited wax attracts the electric fluid existing in the air nearer to it, making it rarer than it was before. In the course of experiments that were performed on this occasion, Mess. Wilcke and Epinus succeeded in charging a plate of air, by suspending large boards of wood covered with tin, with the flat sides parallel to one another, and at some inches asunder: for they found, upon electrifying one of the boards positively, that the other was always negative; and a shock was produced by forming a communication between the upper and lower plates. Beccaria has largely considered the subject of electric atmospheres, in his Artificial Electricity, pa. 179 &c, Eng. edit. See also Dr Priestley's Hist. of Electricity, vol. ii. sect. 5. and Cavallo's Electricity, pa. 241.</p></div2><div2 part="N" n="Atmosphere" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Atmosphere</hi></head><p>, <hi rend="italics">Magnetic, &c,</hi> is understood of the sphere within which the virtue of the magnet, &c, acts.</p></div2></div1><div1 part="N" n="ATOM" org="uniform" sample="complete" type="entry"><head>ATOM</head><p>, a particle of matter indivisible on account of its solidity, hardness, and impenetrability; which preclude all division, and leave no vacancy for the admission of any foreign force to separate or disunite its parts. As atoms are the sirst matter, it is necessary they should be indissolvable, that they may be incorruptible. Newton adds, it is also required that they be immutable, that the world may continue in the same state, and bodies be of the same nature now as formerly.</p><p>ATOMICAL <hi rend="smallcaps">Philosophy</hi>, or the doctrine of atoms, a system which accounted for the origin and formation of things, from the hypothesis that atoms are endued with weight and motion. This philosophy was first taught by Moschus, some time before the Trojan war: but it was most cultivated by Epicurus; whence it is called the Epicurean philosophy.</p></div1><div1 part="N" n="ATTRACTION" org="uniform" sample="complete" type="entry"><head>ATTRACTION</head><p>, or <hi rend="smallcaps">Attactive Power</hi>, a general term used to denote the cause, power, or principle, by which all bodies mutually tend towards each other, and cohere, till separated by some other power. The laws, phenomena, &c, of attraction, form the chief subject of Newton's philosophy, being the principal agent of nature, in almost all her wonderful operations.</p><p>The principle of attraction, in the Newtonian sense of it, it seems was first surmised by Copernicus. “As for gravity, says he, I consider it as nothing more than a certain natural appetence <hi rend="italics">(appetentia)</hi> that the Creator has impressed upon all the parts of matter, in order to their uniting or coalescing into a globular form, for their better preservation; and it is probable that the same power is also inherent in the sun and moon, and planets, that those bodies may constantly retain that round form in which we see them.” <hi rend="italics">De Revol. Orb. Cælest.</hi> lib. i, cap. 9. Kepler calls gravity a corporeal and mutual affection between similar bodies, in order to their union. <hi rend="italics">Ast. Nov. in Introd.</hi> And he pronounced more positively that no bodies whatever were absolutely light, but only relatively so; and consequently that all matter was subjected to the power and law of gravitation. <hi rend="italics">Ibid.</hi></p><p>The first in this country who adopted the notion of attraction, was Dr. Gilbert, in his book <hi rend="italics">De Magnete</hi>; and the next was the celebrated Lord Bacon, in his <hi rend="italics">Nov. Organ.</hi> lib. ii, aphor. 36, 45, 48. <hi rend="italics">Sylv.</hi> cent. i, <pb n="171"/><cb/> exp. 33; also in his treatise <hi rend="italics">De Motu,</hi> particularly under the articles of the 9th and the 13th sorts of Motion. In France it was received by Fermat and Roberval; and in Italy by Galileo and Borelli. But till Newton appeared, this principle was very imperfectly defined and applied.</p><p>It must be observed, that though this great author makes use of the word attraction, in common with the school philosophers, yet he very studiously distinguishes between the ideas. The ancient attraction was conceived to be a kind of quality inherent in certain bodies themselves, and arising from their particular or specific forms. But the Newtonian attraction is a more indefinite principle; denoting not any particular kind or mode of action; nor the physical cause of such action; but only a general tendency, a <hi rend="italics">conatus accedendi,</hi> to whatever cause, physical or metaphysical, such effect be owing; whether to a power inherent in the bodies themselves, or to the impulse of an external agent. Accordingly, that author remarks, in his <hi rend="italics">Philos. Nat. Prin. Math.</hi> “that he uses the words <hi rend="italics">attraction, impulse,</hi> and <hi rend="italics">propension</hi> to the centre, indifferently; and cautions the reader not to imagine that by attraction he expresses the modus of the action, or its efficient cause, as if there were any proper powers in the centres, which in reality are only mathematical points; or as if centres could attract.” Lib. 1, pa. 5. So, he “considers centripetal powers as attractions, though, physically speaking, it were perhaps more just to call them impulses. Ib. pa. 147. He adds, “that what he calls attraction may possibly be effected by impulse, though not a common or corporeal impulse, or after some other manner unknown to us.” <hi rend="italics">Optic.</hi> p. 322.</p><p>Attraction, if considered as a quality arising from the specific forms of bodies, ought, together with sympathy, antipathy, and the whole tribe of occult qualities, to be exploded. But when these are set aside, there will remain innumerable phenomena of nature, and particularly the gravity or weight of bodies, or their tendency to a centre, that argue a principle of action seemingly distinct from impulse; where, at least, there is no sensible impulsion concerned. Nay, what is more, this action, in some respects, differs from all impulsion we know of; impulse being always found to act in proportion to the surfaces of bodies; whereas gravity acts according to their solid content, and consequently it must arise from some cause that penetrates or pervades the whole Tubstance of it. This unknown principle, unknown we mean in respect of its cause, for its phenomena and effects are most obvious, with all its species and modifications, is called attraction; being a general name, under which may be ranged all mutual tendencies, where no physical impulse appears, and which consequently cannot be accounted for upon any known laws of nature.</p><p>And hence arise divers particular kinds of attraction; as <hi rend="italics">Gravity, Magnetism, Electricity, &c,</hi> which are so many different principles, acting by different laws; and only agreeing in this, that we do not perceive any physical causes of them: but that, as to our senses, they may really arise from some power or efficacy in such bodies, by which they are enabled to act even upon distant bodies; though our reason absolutely disallows of any such action. <cb/></p><p>Attraction may be divided, with respect to the law it observes, into two kinds.</p><p>1. That which extends to a sensible distance. As the attraction of gravity, which is found in all bodies; and the attractions of magnetism and electricity, found only in particular bodies. The several laws and phenomena of each, see under their respective articles.</p><p>The attraction of gravity, called also among mathematicians the <hi rend="italics">centripetal force,</hi> is one of the greatest and most universal principles of all nature. We see and feel it operate on bodies near the earth, and find by observation that the same power (i. e. a power which acts in the same manner, and by the same rules, viz, aways proportionally to the quantities of matter, and inversely as the squares of the distances) does also obtain in the moon, and the other planets, both primary and secondary, as well as in the comets; and even that this is the very power by which they are all retained in their orbits, &c. And hence, as gravity is found in all the bodies which come under our observation, it is easily inferred, by one of the established rules of philosophizing, that it obtains in all others. And since it is found to be proportional to the quantity of matter in any body, it must exist in every particle of it: and hence it is proved that every particle in nature attracts every other particle.</p><p>From this attraction arises all the motion, and consequently all the mutation, in the great world. By this heavy bodies descend, and light ones are made to ascend: by this projectiles are directed, vapours and exhalations rise, and rains &c fall: by this rivers glide, the ocean swells, the air presses, &c. In short, the motions and forces arising from this principle, constitute the subject of that extensive branch of mathematics, called <hi rend="italics">mechanics</hi> or <hi rend="italics">statics,</hi> with the parts or appendages of it, as hydrostatics, pneumatics, hydraulics, &c.</p><p>2. That which does not extend to sensible distances. Such is found to obtain in the minute particles of which bodies are composed, attracting each other at or extremely near the point of contact, with forces often much superior to that of gravity, but which at any distance decrease much faster than the power of gravity. This power a late ingenious author calls the <hi rend="italics">attraction of cohesion,</hi> as being that by which the atoms or insensible particles of bodies are united into sensible masses.</p><p>This kind of attraction owns Sir Isaac Newton for its discoverer; as the former does for its improver. The laws of motion, percussion, &c, in sensible bodies, under various circumstances, as falling, projected, &c, ascertained by the later philosophers, do not reach those more recluse, intestine motions in the component particles of the same bodies, on which depend the changes in the texture, colour, properties, &c, of bodies. So that our philosophy, if it were only founded on the principle of gravitation, and even carried as far as this would lead us, would still be very deficient.</p><p>But besides the common laws of sensible masses, the minute parts they are composed of are found subject to some others, which have but lately been noticed, and are even yet imperfectly known. Newton himself, to whose happy penetration we owe the hint, limits himself with establishing that there are such motions in the <pb n="172"/><cb/> <hi rend="italics">minima naturæ,</hi> and that they flow from certain powers or forces, not reducible to any of those in the great world. He shews that, by virtue of these powers, “the small particles act on one another even at a distance; and that many of the phenomena of nature result from it. Sensible bodies, we have already observed, act on one another divers ways; and as we thus perceive the tenor and course of nature, it appears highly probable that there may be other powers of the like kind; nature being very uniform and consistent with herself. Those just mentioned, reach to sensible distances, and so have been observed by vulgar eyes; but there may be others which reach to such small distances as have hitherto escaped observation; and it is probable electricity may reach to such distances, even without being excited by friction.”</p><p>The great author just mentioned proceeds to confirm the reality of these suspicions from a great number of phenomena and experiments, which plainly argue such powers and actions between the particles, for example of salts and water, oil of vitriol and water, aquafortis and iron, spirit of vitriol and saltpetre. He also shews, that these powers, &c, are unequally strong between different bodies; stronger, for instance, between the particles of salt of tartar and those of aquafortis than those of silver, between aquafortis and lapis calaminaris than iron, between iron than copper, and copper than silver or mercury. So spirit of vitriol acts on water, but more on iron or copper, &c. And the other experiments are innumerable which countenance the existence of such principle of attraction in the particles of matter.</p><p>These actions, by virtue of which the particles of the bodies above-mentioned tend towards each other, the author calls by a general indefinite name, <hi rend="italics">attraction</hi>; a name equally applicable to all actions by which bodies tend towards one another, whether by impulse, or by any other more latent power: and from hence he accounts for an infinity of phenomena, otherwise inexplicable, to which the principle of gravity is inadequate.</p><p>Thus, adds our author, “will nature be found very conformable to herself, and very simple; performing all the great motions of the heavenly bodies by the attraction of gravity, which intercedes those bodies, and almost all the small ones of their parts, by some other attractive power diffused through their particles. Without such principles, there never would have been any motion in the world; and without the continuance of it, motion would soon perish, there being otherwise a great decrease or diminution of it, which is only supplied by these active principles.”</p><p>It need not be said how unjust it is in the generality of foreign philosophers to declare against a principle which furnishes so beautiful a view, for no other reason but because they cannot conceive how one body should act on another at a distance. It is indeed true, that philosophy allows of no action but what is by immediate contact and impulsion; for how can a body exert any active power where it does not exist? yet we see effects, without perceiving any such impulse; and where effects are observed, there must exist causes whether we see them or not. But we may contemplate such effects, without entering into the considera- <cb/> tion of the causes, as indeed it seems the business of a philosopher to do: for to exclude a number of phenomena which we do see, would be to leave a great chasm in the history of nature; and to argue about actions which we do not see, would be to raise castles in the air. It follows therefore, that the phenomena of attraction are matter of physical consideration, and as such intitled to a share in the system of physics; but that their causes will only become so when they become sensible, that is when they appear to be the effect of some other higher causes; for a cause is no otherwise seen than as itself is an effect, so that the first cause must needs be always invisible: we are therefore at liberty to suppose the causes of attractions what we please, without any injury to the effects. The illustrious author himself seems to be a little indetermined as to the causes; inclining sometimes to attribute gravity to the action of an immaterial cause (<hi rend="italics">Optics,</hi> pa. 343 &c), and sometimes to that of a material one, <hi rend="italics">Ib.</hi> pa. 325.</p><p>In his philosophy, the research into causes is the last thing, and never comes under consideration till the laws and phenomena of the effect be settled; it being to these phenomena that the cause is to be accommodated. The cause even of any, the grossest and most sensible action, is not adequately known. How impulse or percussion itself produces its effects, that is how motion is communicated from body to body, confounds the deepest philosophers; yet is impulse received not only into philosophy, but into mathematics: and accordingly the laws and phenomena of its effects make the chief part of common mechanics.</p><p>The other species of attraction, therefore, in which no impulse is observable, when their phenomena are sufficiently ascertained, have the same title to be promoted from physical to mathematical consideration; and this without any previous inquiry into their causes, to which our conceptions may not be proportionate.</p><p>Our great philosopher, then, far from adulterating science with any thing foreign or metaphysical, as many have reproached him with doing, has the glory of having thrown every thing of this kind out of his system, and of having opened a new source of sublimer mechanics, which, duly cultivated, might be of far greater extent than all the mechanics yet known. Hence it is alone that we must expect to learn the manner of the changes, productions, generations, corruptions, &c, of natural things; with all that scene of wonders opened to us by the operations of chemistry.</p><p>Some of our own countrymen have prosecuted the discovery with laudable zeal. Dr. Keil particularly has endeavoured to deduce some of the laws of this new action, and applied them in resolving several of the more general phenomena of matter, as cohesion, fluidity, elasticity, softness, fermentation, coagulation, &c: and Dr. Freind, seconding his endeavours, has made a farther application of the same principles, at once to account for almost all the phenomena that chemistry presents. So that some philosophers are inclined to think that the new mechanics should seem already raised to a complete science, and that nothing now can occur but what we have an immediate solution of, from the principles of attractive forces. <pb n="173"/><cb/></p><p>But this seems a little too precipitate: a principle so fertile should have been further explained; its particular laws, limits, &c, more industriously detected and laid down, before we had proceeded to the application. Attraction in the gross is so complex a thing, that it may solve a thousand different phenomena alike. The notion is but one degree more simple and precise than action itself; and, till its properties are more fully ascertained, it were better to apply it less, and study it more. It may be added, that some of Newton's followers have been charged with falling into that error which he industriously avoided, viz, of considering attraction as a cause or active property in bodies, not merely as a phenomenon or effect.</p><p>For the laws, properties, &c, of the different sorts of Attraction, see their particular articles <hi rend="smallcaps">Cohesion, Gravity, Magnetism</hi>, &c.</p><div2 part="N" n="Attraction" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Attraction</hi></head><p>, <hi rend="italics">Centre of,</hi> See <hi rend="smallcaps">Centre</hi> <hi rend="italics">of Attraction.</hi></p><p><hi rend="smallcaps">Attraction</hi> <hi rend="italics">of Mountains,</hi> See <hi rend="smallcaps">Mountains.</hi></p></div2><div2 part="N" n="Attrition" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Attrition</hi></head><p>, the striking or rubbing of bodies against one another, so as to throw off some of their superficial particles: such as the grinding and polishing of bodies. Or simply the act of rubbing: as when amber and other electric bodies are rubbed, to make them attract, or emit their electric force.</p><p><hi rend="smallcaps">Avant-Foss</hi>, or Ditch of the Counterscarp, in Fortification, is a wet ditch surrounding the counterscarp, on the outer side, next to the country, at the foot of the glacis. It would not be proper to have such a ditch if it could be laid dry, as it would then serve as a lodgment for the enemy.</p></div2><div2 part="N" n="Averroes" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Averroes</hi></head><p>, or <hi rend="smallcaps">Aben-Roes</hi>, a very subtile Arabian philosopher, who flourished about the end of the 11th century, when the Moors had possession of part of Spain. He was the son of the high priest and chief judge of Corduba or Cordova in Spain: but he was educated in the university of Morocco, where he was professor, and where he died in 1206, having there studied natural philosophy, medicine, mathematics, law, and divinity. After the death of his father, he enjoyed his posts in Spain, to which was afterward added that of judge of Morocco and Mauritania, where having settled deputies, he returned to his duty in Spain. Notwithstanding he was very rich, and had a very great income, his liberality to men of letters in necessity, whether they were his friends or his enemies, kept him always in debt. He was afterwards stripped of all his posts, and thrown into prison, for heresy, by the instigations of bad men, his enemies; but the oppressions of the judge who succeeded him, caused him to be restored to his former employments.</p><p>He was excessively fat, though he eat but once a day, and spent most part of the night in the study of philosophy, when he was fatigued, amusing himself with reading poetry or history. He was never seen to play at any game, or to partake in any diversion. He was extremely fond of Aristotle's works, and wrote commentaries upon them; whence he was styled <hi rend="italics">the Commentator,</hi> by way of eminence. He wrote many other pieces; among them a work on the Whole Art of Physic; an Epitome of Ptolomy's Almagest, which Vossius dates about the year 1149; also a Treatise of Astrology, which was translated into Hebrew by R. <cb/> Jacob Ben Samson, and said to be extant in the French king's library. He wrote also several poems, and many amorous verses, but these last he threw into the fire when he grew old. His other poems are lost, except a small piece, in which he says, “that when he was young, he acted against his reason; but that when he was in years, he sollowed its dictates;” upon which he utters this wish, “Would to God I had been born old, and that in my youth I had been in a state of perfection!” As to religion, his opinions were, that Christianity is absurd; Judaism, the religion of children; Mahometanism, the religion of swine.</p></div2></div1><div1 part="N" n="AVICENA" org="uniform" sample="complete" type="entry"><head>AVICENA</head><p>, <hi rend="smallcaps">Avicenne</hi>, or <hi rend="smallcaps">Avicenes</hi>, has been accounted the prince of Arabian philosophers and physicians. He was born at Assena, near Bokhara, in 978; and died at Hamadan in 1036, being 58 years of age.</p><p>The first years of Avicena were employed on the study of the Belles Lettres, and the Koran, and at 10 years of age he was perfect master of the hidden senses of that book. Then applying to the study of logic, philosophy and mathematics, he quickly made a rapid progress. After studying under a master the first principles of logic, and the first 5 or 6 propositions of Euclid's elements, he became disgusted with the slow manner of the schools, applied himself alone, and soon accomplished all the rest by the help of the commentators only.</p><p>Possessed with an extreme avidity to be acquainted with all the sciences, he studied medicine also. Persuaded that this art consists as much in practice as in theory, he sought all opportunities of seeing the sick; and asterwards confessed that he had learned more from such experience than from all the books he had read. Being now in his 16th year, and already celebrated for being the light of his age, he determined to resume his studies of philosophy, which medicine, &c, had made him for some time neglect: and he spent a year and a half in this painful labour, without ever sleeping all this time a whole night together. At the age of 21, he conceived the bold design of incorporating, in one work, all the objects of human knowledge; and he carried it into execution in an Encyclopedia of 20 volumes, to which he gave the title of the <hi rend="italics">Utility of Utilities.</hi></p><p>Many wonderful stories are related of his skill in medicine, and the cures which he performed. Several princes had been taken dangerously ill, and Avicenes was the only one that could know their ailments, and cure them. His reputation increased daily, and all the princes of the east desired to retain him in their families, and in fact he passed through several of them. But the irregularities of his conduct sometimes lost him their favour, and threw him into great distresses. His excesses in pleasures, and his infirmities, made a poet say, who wrote his epitaph, that the profound study of philosophy had not taught him good morals, nor that of medicine the art of preserving his own health.</p><p>After his death however, he enjoyed so great a reputation, that till the 12th century he was preferred for the study of philosophy and medicine to all his predecessors. Even in Europe his works were the only writings in vogue in the schools. They were very numerous, and various, the titles of which are as follow: <pb n="174"/><cb/> 1. Of the Utility and Advantage of the Sciences, in 20 books.—2. Of Innocence and Criminality, 2 books. —3. Of Health and Remedies, 18 books.—4. On the Means of preserving Health, 3 books.—5. Canons of Physic, 14 books.—6. On Astronomical Observations, 1 book.—7. On Mathematical Sciences.—8. Of Theorems, or Mathematical and Theological Demonstrations, 1 book.—9. On the Arabic Language, and its Properties, 10 books.—10. On the Last Judgment.— 11. On the Origin of the Soul, and the Resurrection of Bodies.—12. On the end we should propose to ourselves in Harangues and Philosophical Argumentations. —18. Demonstration of the Collateral Lines in the Sphere.—14. Abridgment of Euclid.—15. On Finity and Infinity.—16. On Physics and Metaphysics.— 17. On Animals and Vegetables, &c.—18. Encyclopedie, 20 volumes.</p></div1><div1 part="N" n="AUGUST" org="uniform" sample="complete" type="entry"><head>AUGUST</head><p>, the 8th month of the year, containing 31 days. In the antient Roman calendar this was called <hi rend="italics">sextilis,</hi> as being the 6th month from March, with which their year began; but changed to its present name by the emperor Augustus, calling it aster his own name on account of his having obtained many victories and honours in that month.</p><p>AVOIRDUPOIS <hi rend="italics">Weight,</hi> a weight used in England for weighing all the larger and coarser sorts of goods; as groceries, cheese, butter, flesh, wool, salt, hops, &c, and all metals except gold and silver. Avoirdupois weight is thus divided, viz. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">dr. or drams make</cell><cell cols="1" rows="1" role="data">1 ounce, marked <hi rend="italics">oz.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">oz.</cell><cell cols="1" rows="1" role="data">1 pound, <hi rend="italics">lb.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">112</cell><cell cols="1" rows="1" role="data">lb.</cell><cell cols="1" rows="1" role="data">1 hundred weight, <hi rend="italics">cwt.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">cwt.</cell><cell cols="1" rows="1" role="data">1 ton <hi rend="italics">ton.</hi></cell></row></table> The Avoirdupois ounce is less than the Troy ounce, in the proportion of 700 to 768, but the Avoirdupois pound greater than the Troy pound in the proportion of 700 to 576; <table><row role="data"><cell cols="1" rows="1" role="data">for 1lb Avoird. is</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">7000</cell><cell cols="1" rows="1" role="data">grains Troy,</cell></row><row role="data"><cell cols="1" rows="1" role="data">but 1lb Troy is</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">5760</cell><cell cols="1" rows="1" role="data">grains Troy,</cell></row><row role="data"><cell cols="1" rows="1" role="data">also 1 oz Avoird. is</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">437 1/2</cell><cell cols="1" rows="1" role="data">grains Troy,</cell></row><row role="data"><cell cols="1" rows="1" role="data">and 1 oz Troy is</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">480</cell><cell cols="1" rows="1" role="data">grains Troy.</cell></row></table></p></div1><div1 part="N" n="AURIGA" org="uniform" sample="complete" type="entry"><head>AURIGA</head><p>, the <hi rend="italics">Waggoner,</hi> a constellation in the northern hemisphere, consisting of 14 stars in Ptolemy's catalogue; but in Tycho's, 27; in Hevelius's, 40; and in the Britannic catalogue, 66.</p><p>This is one of the 48 old asterisms, mentioned by all the most ancient astronomers. It is represented by the figure of an old man, in a posture somewhat like sitting, with a goat and her kids in his left hand, and a bridle in his right.</p><p>The Greeks probably received this and all their other constellations from the Egyptians; but, wanting to appear the inventors of them themselves, and not understanding the meaning of the figures, they have cloathed them with some of their own fabulous dresses, to favour the deceit. They accordingly tell us that this figure of a waggoner was an honourable character, and Erichthonius, the inventor of coaches. Vulcan, say they, once fell in love with Minerva, and when he could not prevail with her to marry him, he would have obtained her upon less honourable terms. There was a struggle between them, and some way or other Erichthonius was <cb/> begotten, though it does not seem that Minerva had much share in it: she took care of the offspring how ever. Some have supposed it was only a serpent; but the graver authors say, Erichthonius was a man with legs only like the body of a serpent, and that to hide this monstrous part of his sigure he invented coaches to carry him about. They add, that Jupiter, doing him honour for an invention that was, in some degree, imitating the sun's carriage on the earth, raised him up among the stars.</p><p>But others, ill satisfied with a story which so badly agreed with the figure, have said that it belonged to Myrtillus, a son of Mercury and Clytie, and charioteer to Aenomanus; they say, that at his death, his father Mercury, by permission of his superiors, raised him up into the skies. All this however does not at all account for the goat and her two kids in the hands of Auriga. To set this right, they afterward made Auriga to be Olenus, a son of Vulcan, and the father of Aega and Helice, two of the Cretan nymphs that nursed the infant Jupiter. They talk of a goat that was used for giving milk to the young deity, and they suppose that this creature, and its two young ones, were placed in the hands of the father of the virgins, to commemorate the creature they took into their service on that occasion.</p><p>Besides the Hœdi, this constellation contains also another of those stars which the ancients honoured with peculiar names, the goat Capra, and Amalthæa Capra: this is the bright one near the shoulder, and supposed to be the mother of the Hœdi, and the nurse of Jupiter.</p><p>Although the whole constellation of Auriga is not mentioned among those from which the ancients formed presages of the succeeding weather, the two stars in his arm were of the foremost in that rank. It is these they called by the name Hœdi, and dreaded so extremely on account of the storms and tempests that succeeded their rising, that it is said they shut up the navigation of the sea for their season. And the day of their influence being over, we find, was celebrated as a festival with sports and games, under the name of Natalis Navigationis. Germanicus calls them unfriendly stars to mariners; and Virgil couples them with Arcturus, mentioning their setting and its rising as things of the most important presage. Horace also puts them together as the most formidable of all the stars to those who follow the traffic of the sea. And to the same purpose speak all the ancient writers, thus making a part of the conftellation Auriga, if not the whole constellation, a thing to be observed with the utmost attention, and to be feared as much as the blazing Arcturus.</p></div1><div1 part="N" n="AURORA" org="uniform" sample="complete" type="entry"><head>AURORA</head><p>, the morning twilight; or that faint light which appears in the morning when the sun is within 18 degrees of the horizon.</p><p>AURORA BOREALIS, <hi rend="smallcaps">Northern Light</hi>, or <hi rend="italics">Streamers;</hi> a kind of meteor appearing in the northern part of the heavens, mostly in the winter season, and in frosty weather. It is usually of a reddish colour, inclining to yellow, and sends out frequent coruscations of pale light, which seem to rise from the horizon in a pyramidal undulating form, and shooting with great velocity up to the zenith. It appears often in form of an arch, <pb n="175"/><cb/> which is partly bright, and partly dark, but generally transparent. And the matter of it is not found to have any effect on the rays of light, which pass freely through it. Dr. Hamilton observes, that he could plainly discern the smallest speck in the Pleiades through the density of those clouds which formed part of the Aurora borealis in 1763, without the least diminution of its splendour, or increase of twinkling. Philos. Essays, pa. 106.</p><p>Sometimes it produces an Iris. Hence M. Godin judges, that most of the extraordinary meteors and phenomena in the skies, related as prodigies by historians, as battles, and the like, may probably euough be reduced to the class of Auroræ boreales. Hist. Acad. R. Scienc. for 1762, pa. 405.</p><p>This kind of meteor never appears near the equator; but, it seems, is frequent enough towards the south pole, like as towards the north, having been observed there by voyagers. See Philos. Trans. N° 461, and vol. 54; also Forster's account of his voyage round the world with Captain Cook, where he describes their appearance as observed for several nights together, in sharp frosty weather, which was much the same as those observed in the north, excepting that they were of a lighter colour.</p><p>It seems that meteors of this kind have appeared sometimes more frequently than others. They were so rare in England, or else so little regarded, that none are recorded in our annals since that remarkable one of Nov. 14, 1574, till the surprising Aurora borealis of March 6, 1716, which appeared for three nights successively, but by far more strongly on the first: except that five small ones were observed in the year 1707 and 1708. Hence it would seem, that the air, or earth, or both, are not at all times disposed to produce this phenomenon.</p><p>The extent of these appearances is also amazingly great. That in March 1716 was visible from the west of Ireland, to the confines of Russia and the east of Poland; extending at least near 30 degrees of longitude, and from about the 50th degree in latitude, over almost all the north of Europe; and in all places, at the same time, it exhibited the like wondrous appearances. Father Boscovich has determined the height of an aurora borealis, which was observed by the Marquis of Polini the 16th of December, 1737, and found it was 825 miles high; and Mr. Bergman, from a mean of 30 computations, makes the average height of the aurora borealis amount to 70 Swedish, or 469 English miles. But Euler supposes the height to be several thousands of miles; and Mairan also assigns to them a very elevated region.</p><p>Many attempts have been made to determine the cause of this phenomenon. Dr. Halley imagines that the watery vapours, or effluvia, exceedingly rarefied by subterraneous fire, and tinged with sulphureous streams, which many naturalists have supposed to be the cause of earthquakes, may also be the cause of this appearance: or that it is produced by a kind of subtile matter, freely pervading the pores of the earth, and which, entering into it nearer the southern pole, passes out again with some force into the æther, at the same distance from the northern. This subtile matter, by becoming more dense, or having its velocity increased, may perhaps be <cb/> capable of producing a small degree of light, after the manner of effluvia from electric bodies, which, by a strong and quick friction, emit light in the dark; to which sort of light this seems to have a great affinity. Philos. Trans. N° 347. See also Mr. Cotes's description of this phenomenon, and his method of explaining it, by streams emitted from the heterogeneous and fermenting vapours of the atmosphere, in Smith's Optics, pa. 69; or Philos. Trans. abr. vol. 6, part 2.</p><p>The celebrated M. de Mairan, in an express treatise on the Aurora Borealis, published in 1731, supposes its cause to be the zodiacal light, which, according to him, is no other than the sun's atmosphere: this light happening, on some occasions, to meet the upper parts or our atmosphere about the limits where universal gravity begins to act more forcibly towards the earth than towards the sun, falls into our air to a greater or less depth, as its specific gravity is greater or less, compared with the air through which it passes. See Tract. Phys. et Hist. de l'Aurore Boreale. Suite des Memoires de l'Acad. R. des Scien. 1731. Also Philos. Trans. N° 433, or Abridg. vol. 8, pa. 540.</p><p>However M. Euler thinks the cause of the aurora borealis not owing to the zodiacal light, as M. de Mairan supposes; but to particles of our atmosphere, driven beyond its limits by the impulse of the solar light. And on this supposition he endeavours to account for the phenomena observed concerning this light. He supposes the zodiacal light, and the tails of comets, to be owing to a similar cause.</p><p>But ever since the identity of lightning and the electric matter has been determined, philosophers have been naturally led to seek for the explication of aerial meteors in the principles of electricity; and there is now no doubt but most of them, and especially the aurora borealis, are electrical phenomena. Besides the more obvious and known appearances which constitute a resemblance between this meteor and the electric matter by which lightning is produced, it has been observed that the aurora occasions a very sensible fluctuation in the magnetic needle; and that when it has extended lower than usual in the atmosphere, the flashes have been attended with various sounds of rumbling and hissing, especially in Russia and the other more northern parts of Europe; as noticed by Sig. Beccaria and M. Messier. Mr. Canton, soon after he had obtained electricity from the clouds, offered a conjecture, that the aurora is occafioned by the dashing of electric fire positive towards negative clouds at a great distance, through the upper part of the atmosphere, where the resistance is least: and he supposes that the aurora which happens at the time when the magnetic needle is disturbed by the heat of the earth, is the electricity of the heated air above it. and this appears chiefly in the northern regions, as the alteration in the heat of the air in those parts is the greatest. Nor is this hypothesis improbable, when it is considered, that electricity is the cause of thunder and lightning; that it has been extracted from the air at the time of the aurora borealis; that the inhabitants of the northern countries observe it remarkably strong when a sudden thaw succeeds very cold severe weather; and that the tourmalin is known to emit and absorb the electric fluid only by the increase or diminution of its <pb n="176"/><cb/> heat. Positive and negative electricity in the air, with a p<*>oper quantity of moisture to serve as a conductor, will account for this and other meteors, sometimes seen in a serene sky. Mr. Canton has since contrived to exhibit this meteor by means of the Torricellian vacuum, in a glass tube about 3 feet long, and sealed hermetically. When one end of the tube is held in the hand, and the other applied to the conductor, the whole tube will be illuminated from end to end, and will continue luminous without interruption for a considerable time after it has been removed from the conductor. If, after this, it be drawn through the hand either way, the light will be remarkably intense through the whole length of the tube. And though a great part of the electricity be discharged by this operation, it will still flash at intervals, when held only at one extremity, and kept quite still; but if, at the same time, it be grasped by the other hand in a different place, strong flashes of light will dart from one end to the other; and these will continue 24 hours or more, without a fresh excitation. Sig. Beccaria conjectures that there is a constant and regular circulation of the electric fluid from north to south; and he thinks that the aurora borealis may be this electric matter performing its circulation in such a state of the atmosphere as renders it visible, or approaching nearer than usual to the earth. Though probably this is not the mode of its operation, as the meteor is observed in the southern hemisphere, with the same appearances as in the northern. Dr. Franklin supposes, that the electric fire discharged into the polar regions, from many leagues of vaporised air raised from the ocean between the tropics, accounts for the aurora borealis; and that it appears first, where it is first in motion, namely in the most northern part; and the appearance proceeds southward, though the fire really moves northward. Franklin's Exper. and Obs. 1769, pa. 49. Philos. Trans. vol. 48, pa. 358, 784; <hi rend="italics">Ib.</hi> vol. 51, pa. 403; Lettere dell' Ellettricismo, pa. 269; or Priestley's Hist. of Electricity. See also an ingenious solution of this phenomenon, on the same principles, by Dr. Hamilton, in his Philos. Essays. Mr. Kirwan (in the Transactions of the Royal Irish Academy, ann. 1788) has some ingenious remarks on the <hi rend="italics">auroræ borealis & australis.</hi> He gives his reasons for supposing the rarefaction of the atmosphere in the polar regions to proceed from them, and these from a combustion of inflammable air caused by electricity. He observes, that after an aurora borealis the barometer commonly falls, and high winds from the south generally follow.</p><p>AURUM <hi rend="smallcaps">Fulminans</hi>, a preparation from gold, which being thrown into the fire, it explodes with a violent noise, like thunder. The matter is produced by dissolving gold in aqua regia, and precipitating the solution by oil of tartar <hi rend="italics">per deliquium,</hi> or volatile spirit of sal ammoniac. The powder being washed in warm water, and dried to the consistence of a paste, is afterwards formed into small grains of the size of hempseed.</p><p>It is inflammable, not only by fire, but also by a gentle warmth; and gives a report much louder than that of gunpowder. A single grain laid on the point <cb/> of a knife, and lighted at a candle, explodes with a greater report than a musquet: and a scruple of this powder, it is said, acts more loudly than half a pound of gunpowder; and yet it is said that, by mixture, it does not increase the elastic force of sired gunpowder. Dr. Black attributes the increase of weight, and also the explosive property of this powder, to adhering fixable air.—This is a very dangerous preparation, and should be used with great caution.</p></div1><div1 part="N" n="AUSTRAL" org="uniform" sample="complete" type="entry"><head>AUSTRAL</head><p>, the same with <hi rend="italics">southern.</hi> Thus, Austral signs, are the last 6 signs of the zodiac; and are so called because they are on the south side of the equinoctial.</p><p>AUSTRALIS <hi rend="italics">Corona</hi>; see <hi rend="smallcaps">Corona</hi> <hi rend="italics">Australis.</hi></p><p><hi rend="smallcaps">Australis Piscis</hi>, the <hi rend="italics">Southern Fish,</hi> is a constellation of the southern hemisphere. See <hi rend="smallcaps">Piscis</hi> <hi rend="italics">Australis.</hi></p></div1><div1 part="N" n="AUTOMATON" org="uniform" sample="complete" type="entry"><head>AUTOMATON</head><p>, a seemingly self-moving machine; or one so constructed, by means of weights, levers, pullies, springs, &c, as to move for a considerable time, as if it were endued with animal life. And according to this description, clocks, watches, and all machines of that kind, are automata.</p><p>It is said, that Archytas of Tarentum, 400 years before Christ, made a wooden pigeon that could fly; that Archimedes also made such-like automatons; that Regiomontanus made a wooden eagle that flew forth from the city, met the emperor, saluted him, and returned; also that he made an iron fly, which flew out of his hand at a feast, and returned again after flying about the room; that Dr. Hook made the model of a flying chariot, capable of supporting itself in the air. Many other surprizing automatons we have been eye-witnesses of, in the present age: thus, we have seen figures that could write, and perform many other actions in imitation of animals: M. Vaucanson made a figure that played on the flute; the same gentleman also made a duck, which was capable of eating, drinking, and imitating exactly the voice of a natural one; and, what is still more surprizing, the food it swallowed was evacuated in a digested state, or considerably altered on the principles of solution; also the wings, viscera, and bones were formed so as strongly to resemble those of a living duck; and the actions of eating and drinking shewed the strongest resemblance, even to the muddling the water with its bill. M. Le Droz of la Chaux de Fonds, in the province of Neuschatel, has also executed some very curious pieces of mechanism: one was a clock, presented to the king of Spain; which had, among other curiosities, a sheep that imitated the bleating of a natural one, and a dog watching a basket of fruit, that barked and snarled when any one offered to take it away; besides a variety of human figures, exhibiting motions truly surprising. But all these seem to be inferior to M. Kempell's chess-player, which may truly be considered as the greatest master-piece in mechanics that ever appeared in the world. See also Baptista Porta's <hi rend="italics">Magia Nat.</hi> c. 19, and Scaliger's <hi rend="italics">Subtil.</hi> 326.</p></div1><div1 part="N" n="AUTUMN" org="uniform" sample="complete" type="entry"><head>AUTUMN</head><p>, the third season, when the harvest and fruits are gathered in. This begins at the descending equinox, which, in the northern hemisphere, <pb n="177"/><cb/> is when the sun enters the sign Libra, or about the 22d day of August; and it ends, when winter commences, about the same day in December.</p><div2 part="N" n="Autumnal" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Autumnal</hi></head><p>, something belonging to autumn. Thus,</p><p><hi rend="smallcaps">Autumnal</hi> <hi rend="italics">Equinox,</hi> the time when the sun enters the descending point of the ecliptic, where it crosses the equinoctial; and is so called, because the nights and days are then equal.</p><p><hi rend="smallcaps">Autumnal</hi> <hi rend="italics">Point,</hi> the point of the ecliptic answering to the autumnal equinox.</p><p><hi rend="smallcaps">Autumnal</hi> <hi rend="italics">Signs,</hi> are the signs Libra, Scorpio, Sagittary, through which the sun passes during the autumn.</p></div2></div1><div1 part="N" n="AXIOM" org="uniform" sample="complete" type="entry"><head>AXIOM</head><p>, a self-evident truth, or a proposition assented to by every person at first sight. Such as, that the whole is greater than its part; that a thing cannot both be and not be at the same time; and that from nothing, nothing can arise.</p><p>Some Axioms are in effect, strictly speaking, no other than identical propositions. Thus, to say that all right angles are equal to each other, is as much as to say, all right angles are right angles; such equality being implied in the very definition, or the very name or term itself.</p><p><hi rend="smallcaps">Axiom</hi> is also an established principle in some art or science. Thus, it is an axiom in physics, that nature does nothing in vain; that effects are proportional to their causes; &c. It is an axiom in geometry, that two things equal to the same third thing, are also equal to each other; that if to equal things equals be added, the sums will be equal. And it is an axiom in optics, that the angle of incidence is equal to the angle of reflection. In this sense also the general laws of motion are called axioms; as, that all motion is rectilinear, that action and reaction are equal, &c.</p><p>AXE or AXIS, in <hi rend="italics">Geometry,</hi> the straight line in a plane figure, about which it revolves, to produce or generate a solid. Thus, if a semicircle be moved round its diameter at rest, it will generate a sphere, whose axis is that diameter. And if a right-angled triangle be turned about its perpendicular at rest, it will describe a cone, whose axis is that perpendicular.</p><p><hi rend="smallcaps">Axis</hi> is yet more generally used for a right line conceived to be drawn from the vertex of a figure to the middle of the base. So the</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a circle or sphere,</hi> is any line drawn through the centre, and terminated at the circumference, on both sides.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a cone,</hi> is the line from the vertex to the centre of the base.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a cylinder,</hi> is the line from the centre of the one end to that of the other.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a conic section,</hi> is the line from the principal vertex, or vertices, perpendicular to the tangent at that point. The ellipse and hyperbola have each two axes, which are sinite and perpendicular to each other; but the parabola has only one, and that infinite in length.</p><p><hi rend="italics">Transverse</hi> <hi rend="smallcaps">Axis</hi>, in the Ellipse and Hyperbola, is the diameter passing through the two foci, and the two principal vertices of the figure. In the hyperbola it is the shortest diameter, but in the ellipse it is the longest. <cb/></p><p><hi rend="italics">Conjugate</hi> <hi rend="smallcaps">Axis</hi>, or <hi rend="italics">Second Axis,</hi> in the Ellipse and Hyperbola, is the diameter passing through the centre, and perpendicular to the transverse axis; and is the shortest of all the conjugate diameters.</p><div2 part="N" n="Axis" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Axis</hi></head><p>, of a curve line, is still more generally used for that diameter which has its ordinates at right angles to it, when that is possible. For, like as in the conic sections, any diameter bisects all its parallel ordinates, making the two parts of them on both sides of it equal; and that diameter which has such ordinates perpendicular to it, is an Axis: So, in curves of the second order, if any two parallel lines each meeting the curve in three points; the right line which cuts these two parallels so, that the sum of the two parts on one side of the cutting line, between it and the curve, is equal to the third part terminated by the curve on the other side, then the said line will in like manner cut all other parallels to the former two lines, viz, so that, of every one of them, the sum of the two parts, or ordinates, on one side, will be equal to the third part or ordinate on the other side. Such cutting line then is a diameter; and that diameter whose parallel ordinates are at right angles to it, when possible, is an Axis. And the same for other curves of still higher orders. Newton, Enumeratio Linearum Tertii Ordinis, sect. 2, art. 1.</p></div2><div2 part="N" n="Axis" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Axis</hi></head><p>, <hi rend="italics">in Astronomy.</hi> As, the <hi rend="smallcaps">Axis</hi> <hi rend="italics">of the world,</hi> is an imaginary right line conceived to pass through the centre of the earth, and terminating at each end in the sursace of the mundane sphere. About this line, as an axis, the sphere, in the Ptolomaic system, is supposed daily to revolve.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of the Earth,</hi> is the line connecting its two poles, and about which the earth performs its diurnal rotation, from west to east. This is a part of the axis of the world, and always remains parallel to itself during the motion of the earth in its orbit about the sun, and perpendicular to the plane of the equator.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a Planet,</hi> is the line passing through its centre, and about which the planet revolves.—The Sun, Earth, Moon, Jupiter, Mars, and Venus, it is known from observation, move about their several axes; and the like motion is easily inferred of the other three, Mercuty, Saturn, and Georgian planet.</p><p><hi rend="smallcaps">Axis</hi> of the <hi rend="italics">Horizon, Equator, Ecliptic, Zodiac,</hi> &c, are right lines passing through the centres of those circles, perpendicular to their planes.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a Magnet,</hi> or <hi rend="italics">magnetical</hi> Axis, is a line passing through the middle of a magnet, lengthwise; in such manner, that however the magnet be divided, provided the division be made according to a plane passing through that line, the magnet will then be cut into two loadstones. And the extremities of such lines are called the poles of the stone.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">in Mechanics.</hi>—The axis of a balance, is the line upon which it moves or turns.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of Oscillation,</hi> is a line parallel to the horizon, passing through the centre about which a pendulum vibrates, and perpendicular to the plane in which it oscillates.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">in Peritrochio,</hi> or <hi rend="italics">wheel and axle,</hi> is one of the five mechanical powers, or simple machines; contrived chiefly for the raising of weights to a consider- <pb n="178"/><cb/> able height, as water from a well, &c. This machine <figure/> consists of a circle AB, concentric with the base of a cylinder, and moveable together with it about its axis CD. This cylinder is called the <hi rend="italics">axis</hi>; and the circle, the <hi rend="italics">peritrochium;</hi> and the radii, or spokes, which are sometimes fitted immediately into the cylinder, without any circle, the <hi rend="italics">scytalæ.</hi> About the axis winds a rope, or chain, by means of which great weights are raised by turning the wheel.—The axis in peritrochio takes place in the motion of every machine, in which a circle may be conceived as described about a fixed axis, concentric with the plane of a cylinder about which it is placed; as in Crane wheels, Mill wheels, Capstans, &c.</p><p>The chief properties of the Axe-in-peritrochio, are as follow:</p><p>1. If the power F applied in the direction AF a tangent to the circumference, or perpendicular to the spoke, be to a weight E, as the radius of the axis C<hi rend="italics">e</hi> is to the radius of the wheel AD, or the length of the spoke; the power will just sustain the weight; that is, the power and the weight will be in equilibrio, when they are in the reciprocal proportion of their distances from the centre.</p><p>2. When the wheel moves, with the power and weight; the velocities of their motion, and the spaces passed over by them, will be both in the same proportion as above, namely, directly proportional to their distances from the centre, and reciprocally proportional to their own weights when they are in equilibrio.</p><p>3. A power, and a weight, being given to construct an axis-in-peritrochio, by which it shall be sustained and raised. Let the axis be taken large enough to support the weight and power without breaking: then, as the weight is to the power, so make the radius of the wheel to the radius of the axis. Hence, if the power be very small in respect of the weight, the radius of the wheel will be vastly great. For example, suppose the weight 4050, and the power only 50; then the radius of the wheel will be 81 times that of the axis; which would be a very inconvenient fize. But this inconvenience is provided against by increasing the number of the wheels and axes; making one to turn another, by means of teeth or pinions. And to find the effect of a number of wheels and axes, thus turning one another, multiply together, all the radii of the axes, and all the radii of the wheels, and then it will be, as the product of the former is to the product of the latter, so is the power to the weight. So, if there be 4 wheels and axes, the radius of each axis being 1 foot, and the <cb/> radius of each wheel 3 feet; then the continual product of all the wheels is 3X3X3X3 or 81 feet, and that of the axis only 1; therefore the effect is as 81 to 1, or the weight is 81 times the power. And, on the contrary, if it be required to find the diameter of each of four equal wheels, by which a weight of 4050lb shall be balanced by a power of 50lb, the diameter of each axis being 1 foot: dividing 4050 by 50, the quotient is 81; extract the 4th root of 81, or twice the square root, and it will give 3, for the diameter of the four wheels sought.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a vessel,</hi> is that quiescent right line passing through the middle of it, perpendicular to its base, and equally distant from its sides.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">in Optics.—Optic Axis,</hi> or <hi rend="italics">visual axis,</hi> is a ray passing through the centre of the eye, or falling perpendicularly on the eye.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of a lens,</hi> or <hi rend="italics">glass,</hi> is the axis of the solid of which the lens is a segment. Or the axis of a glass, is the line joining the two vertices or middle points of the two opposite surfaces of the glass.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of Incidence,</hi> in Dioptrics, is the line passing through the point of incidence, perpendicularly to the refracting surface.</p><p><hi rend="smallcaps">Axis</hi> <hi rend="italics">of Refraction,</hi> is the line continued from the point of incidence or refraction, perpendicularly to the refracting surface, along the further medium.</p></div2></div1><div1 part="N" n="AZIMUTH" org="uniform" sample="complete" type="entry"><head>AZIMUTH</head><p>, of the sun, or star, &c, is an arch of the horizon, intercepted between the meridian of the place, and the azimuth or vertical circle passing through the sun or star; and is equal to the angle at the zenith formed by the said meridian and vertical circle. Or it is the complement to the eastern or western amplitude.—The azimuth is thus found by trigonometry; As radius is to the tangent of the latitude,<lb/> So is the tangent of the altitude of the sun or star,<lb/> To the cosine of the azimuth from the south, at<lb/> the time of the equinox.<lb/></p><div2 part="N" n="Azimuth" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Azimuth</hi></head><p>, <hi rend="italics">magnetical,</hi> an arch of the horizon contained between the magnetical meridian, and the azimuth or vertical circle of the object; or its apparent distance from the north or south point of the compass. This is found by observing the sun, or star, &c, with an azimuth compass, when it is 10 or 15 degrees high, either before or after noon.</p><p><hi rend="smallcaps">Azimuth Compass</hi>, an instrument for finding either the magnetical azimuth or amplitude of a celestial object. The description and use of this instrument, see under the article <hi rend="smallcaps">Compass.</hi></p><p><hi rend="smallcaps">Azimuth Dial</hi>, a dial whose stile or gnomon is perpendicular to the plane of the horizon.</p></div2><div2 part="N" n="Azimuths" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Azimuths</hi></head><p>, or <hi rend="italics">Vertical Circles,</hi> are great circles of the sphere intersecting each other in the zenith and nadir, and cutting the horizon at right angles.—These azimuths are represented by the rhumbs on common sea charts; and on the globe by the quadrant of altitude, when screwed in the zenith. On these azimuths is counted the height of the sun or stars, &c, when out of the meridian. <pb n="179"/></p></div2></div1></div0><div0 part="N" n="B" org="uniform" sample="complete" type="alphabetic letter"><head>B</head><cb/><p>BACK-<hi rend="smallcaps">Staff</hi>, an instrument formerly used for taking the sun's altitude at sea; being so called because the back of the observer is turned towards the sun when he makes the observation. It was sometimes called Davis's quadrant, from its inventor captain John Davis, a Welchman, and a celebrated navigator, who produced it about the year 1590.</p><p>This instrument consists of two concentric arches of box-wood, and three vanes: the arch of the longer radius is of 30 degrees, and the other 60 degrees, making between them 90 degrees, or a quadrant: also the vane A at the centre is called the <hi rend="italics">horizon-vane,</hi> that on the arch of 60° at B the <hi rend="italics">shade-vane,</hi> and that on the other arch at C the <hi rend="italics">sight-vane.</hi> <figure/></p><p><hi rend="italics">To use the Back-Staff.</hi> The shade-vane is to be set upon the 60 arch, at an even degree of some latitude, less by 10 or 15 degrees than you judge the complement of the sun's altitude will be; also the horizon-vane being put on at A, and the sight-vane on the 30 arch FG, the observer turns his back to the sun, lifts up the instrument, and looks through the sight-vane, raising or falling the quadrant, till the shadow of the upper edge of the shade-vane fall on the upper edge of the slit in the horizon-vane; and then if he can see the horizon though the said slit, the observation is exact, and the vanes are right set: But if the sea appear instead of the horizon, the sight-vane must be moved downward towards F; or if the sky appear, it must be moved upward towards G; thus trying till it comes right: the observer then examines how many degrees and minutes are cut by that edge of the sight-vane that answers to the sight hole, and to them he adds the degrees cut by the upper edge of the shade-vane; then the sum is the sun's distance from the zenith, or the <cb/> complement of the altitude; that is, of his upper limb when the upper end of the shade-vane is used in the observation, or of his lower limb when the lower part of that vane is used; therefore in the former case add 16 minutes, the sun's semidiameter, and subtract 16 minutes in the latter case, to give the zenith distance or co-altitude of the sun's centre.</p><p>Mr. Flamsteed contrived a glass lens, or double convex, to be placed in the middle of the shade-vane, which throws a small bright spot on the slit of the horizon-vane, instead of the shade; which is a great improvement, if the glass be truly made; for by this means, the instrument may be used in hazy weather, and a much more accurate observation made at all times.</p><div1 part="N" n="BACON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BACON</surname> (<foreName full="yes"><hi rend="smallcaps">Roger</hi></foreName>)</persName></head><p>, an English monk of the Franciscan order, an amazing instance of genius and learning, was born near Ilchester in Somersetshire, in the year 1214. He commenced his studies at Oxford; from whence he removed to the university of Paris, which at that time was esteemed the centre of literature; and where it seems he made such progress in the sciences, that he was esteemed the glory of that university, and was there greatly caressed by several of his countrymen, particularly by Robert Groothead or Grouthead, afterwards bishop of Lincoln, his great friend and patron. Having taken the degree of doctor, he returned to England in 1240, and took the habit of the Franciscan order, being but about 26 years of age; but according to some he became a monk before he left France. He now pursued his favourite study of experimental philosophy with unremitting ardour and assiduity. In this pursuit, in experiments, instruments, and in scarce books, he informs us he spent, in the course of 20 years, no less than 2000<hi rend="italics">l,</hi> an amazing sum in those days, and which sum it seems was generously furnished to him by some of the heads of the university, to enable him the better to pursue his noble researches. By such extraordinary talents, and amazing progress in the sciences, which in that ignorant age were so little known to the rest of mankind, while they raised the admiration of the more intelligent few, could not fail to excite the envy of his illiterate fraternity, whose malice he farther drew upon him by the freedom with which he treated the clergy in his writings, in which he spared neither their ignorance nor their want of morals: these therefore found no difficulty in possessing the vulgar with the notion of Bacon's dealing with the devil. Under this pretence he was restrained from reading lectures; his writings were confined to his convent; and at length, in 1278, he himself was imprisoned in his cell, at 64 years of age. However, being allowed the use of his books, he still proceeded in the rational pursuit of knowledge, correcting his former labours, and writing several curious pieces. <pb n="180"/><cb/></p><p>When Bacon had been 10 years in confinement, Jerom de Ascoli, general of his order, who had condemned his doctrine, was chosen pope by the name of Nicholas IV; and being reputed a person of great abilities, and one who had turned his thoughts to philosophical studies, Bacon resolved to apply to him for his discharge; and to shew both the innocence and the usefulness of his studies, addressed to him a treatise <hi rend="italics">On the means of avoiding the infirmities of old age.</hi> What effect this had on the pope does not appear; it did not at least produce an immediate discharge: however, towards the latter end of his reign, by the interposition of some noblemen, Bacon obtained his liberty; after which he spent the remainder of his life in the college of his order, where he died in the year 1294, at 80 years of age, and was buried in the Franciscan church. Such are the few particulars which the most diligent researches have been able to discover concerning the life of this very extraordinary man.</p><p>Bacon's printed works are, 1. <hi rend="italics">Epistola Fratris Rogeri Baconis de Secretis Operibus Artis et Naturæ, et de Nullitate Magiæ:</hi> Paris, 1542, in 4to. Basil, 1593, in 8vo. 2. <hi rend="italics">Opus Majus</hi>: London, 1733, in fol. published by Dr. Jebb. 3. <hi rend="italics">Thesaurus Chemicus</hi>: Francf. 1603 and 1620. These printed works of Bacon contain a considerable number of essays, which have been considered as distinct books in the catalogue of his writings by Bale, Pitts, &c; but there remain also in different libraries several manuscripts not yet published.</p><p>By an attentive perusal of his works, the reader is astonished to find that this great luminary of the 13th century was deeply skilled in all the arts and sciences, and in many of them made the most important inventions and discoveries. He was, says Dr. Peter Shaw, beyond all comparison the greatest man of his time, and he might perhaps stand in competition with the greatest that have appeared since. It is wonderful, considering the ignorant age in which he lived, how he came by such a depth of knowledge on all subjects. His writings are composed with that elegance, conciseness and strength, and adorned with such just and exquisite observations on nature, that, among all the chemists, we do not know his equal. In his chemical writings, he attempts to shew how imperfect metals may be ripened into perfect ones; making, with Geber, mercury the common basis of all metals, and sulphur the cement.</p><p>His other physical writings shew no less genius and force of mind. In his treatise <hi rend="italics">Of the Secret Works of Art and Nature,</hi> he shews that a person perfectly acquainted with the manner observed by nature in her operations, would be able to rival, and even to surpass her. In another piece, <hi rend="italics">Of the Nullity of Magic,</hi> he shews with great sagacity and penetration, whence the notion of it sprung, and how weak all pretences to it are. From a perusal of his works, adds the same author, we find Bacon was no stranger to many of the capital discoveries of the present and past ages. Gunpowder he certainly knew: thunder and lightning, he tells us, may be produced by art; for that sulphur, nitre and charcoal, which when separate have no sensible effect, yet when mixed together in due proportion, and closely confined, and fired, they yield a loud report. A more precise description of gunpowder cannot be <cb/> given in words. He also mentions a sort of unextinguishable fire prepared by art: which shews he was not unacquainted with phosphorus: and that he had a notion of the rarefaction of the air, and the structure of an air-pump, is past contradiction. He was the miracle, says Dr. Freind, of the age he lived in, and the greatest genius, perhaps, for mechanical knowledge, that ever appeared in the world since Archimedes. He appears likewise to have been master of the whole science of optics: he has aecurately described the uses of reading-glasses, and shewn the way of making them. Dr. Freind adds, that he also describes the camera obscnra, and all sorts of glasses, which magnify or diminish any object, or bring it nearer to the eye, or remove it farther off. Bacon says himself, that he had great numbers of burning-glasses: and that there were none ever in use among the Latins, till his friend Peter de Mahara Curia applied himself to the making of them. That the telescope was not unknown to him, appears from a passage where he says, that he was able to form glasses in such a manner, with respect to our sight and the objects, that the rays shall be refracted and reflected wherever we please, so that we may see a thing under what angle we think proper, either near or at a distance, and be able to read the smallest letters at an incredible distance, and to count the dust and sand, on account of the greatness of the angle under which we see the objects; and also that we shall scarce see the greatest bodies near us, on account of the smallness of the angle under which we view them. His skill in astronomy was amazing: he discovered that error which occasioned the reformation of the calendar; one of the greatest efforts, according to Dr. Jebb, of human industry: and his plan for correcting it was followed by pope Gregory the 13th, with this variation, that Bacon would have had the correction to begin from the birth of our Saviour, whereas Gregory's amendment reaches no higher than the Nicene council.</p></div1><div1 part="N" n="BACON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BACON</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, baron of Verulam, viscount of St. Albans, and lord high chancellor of England under king James I. He was born in 1560, being son of Sir Nicholas Bacon lord keeper of the great seal in the reign of queen Elizabeth, by Anne daughter of Sir Anthony Cook, eminent for her skill in the Latin and Greek languages. He gave even in his infancy tokens of what he would one day become; and queen Elizabeth had mauy times occasion to admire his wit and talents, and used to call him her young lord keeper. He studied the philosophy of Aristotle at Cambridge; where he made such progress in his studies, that at 16 years of age he had run through the whole circle of the liberal arts as they were then taught, and even began to perceive those imperfections in the reigning philosophy, which he afterwards so effectually exposed, and thence not only overturned that tyranny which prevented the progress of true knowledge, but laid the foundation of that free and useful philosophy which has since opened a way to so many glorious discoveries. On his leaving the university, his father sent him to France; where, before he was 19 years of age, he wrote a general view of the state of Europe: but his father dying, he was obliged suddenly to return to England; where he applied himself to the study of the common law, at Gray's-inn. His merit at length raised him to the <pb n="181"/><cb/> highest dignities in his profession, attorney-general, and lord high chancellor. But being of an easy and liberal disposition, his servants took advantage of that temper, and their situation under him, by accepting presents in the line of his profession. Being abandoned by the king, he was tried by the house of lords, for bribery and corruption, and by them sentenced to pay a fine of 40,000<hi rend="italics">l.</hi> and to remain prisoner in the Tower during the king's pleasure. The king however soon after remitted the fine and punishment: but his misfortunes had given him a distaste for public affairs. and he afterwards mostly lived a retired life, closely pursuing his philosophical studies and amusements, in which time he composed the greatest part of his English and Latin works. Though even in the midst of his honours and employments he forgot not his philosophy, but in 1620 published his great work <hi rend="italics">Novum Organum.</hi> After some years spent in his philosophical retirement, he died in 1626, being 66 years of age.</p><p>The chancellor Bacon is one of those extraordinary geniuses who have contributed the most to the advancement of the sciences. He clearly perceived the imperfection of the school philosophy, and he pointed out the only means of reforming it, by proceeding in the opposite way, from experiments to the discovery of the laws of nature. Addison has said of him, That he had the sound, distinct, comprehensive knowledge of Aristotle, with all the beautiful light graces and embellishments of Cicero. Mr. Walpole calls him the <hi rend="italics">Prophet of Arts,</hi> which Newton was afterwards to reveal; and adds, that his genius and his works will be universally admired as long as science exists. He did not yet, said another great man, understand nature, but he knew and pointed out all the ways that lead to her. He very early despised all that the universities called philosophy; and he did every thing in his power that they should not disgrace her by their quiddities, their horrors of a vacuum, their substantial forms, and such like impertinencies.</p><p>He composed two works for perfecting the sciences. The former <hi rend="italics">On the Dignity and Augmentation of the Sciences.</hi> He here shews the state in which they then were, and points out what remains to be discovered for perfecting them; condemning the unnatural way of Aristotle, in reversing the natural order of things. He here also proposes his celebrated division of the sciences.</p><p>To remedy the faults of the common logic, Bacon composed his second work, the <hi rend="italics">New Organ of Sciences,</hi> above-mentioned. He here teaches a new logic, the chief end of which is to shew how to make a good inference, as that of Aristotle's is to make a syllogism. Bacon was 18 years in composing this work, and he always estemed it as the chief of his compositions.</p><p>The pains which Bacon bestowed upon all the sciences in general, prevented him from making any considerable applications to any one in particular: and as he knew that natural philosophy is the foundation of all the other sciences, he chiefly endeavoured to give perfection to it. He therefore proposed to establish a new system of physics, rejecting the doubtful principles of the ancients. For this parpose he took the resolution of composing every month a treatise on some <cb/> branch of physics; he accordingly began with that of the winds; then he gave that of heat; next that of motion; and lastly that of life and death. But as it was impossible that one man alone could so compose the whole circle of sciences with the same precision, after having given these patterns, to serve as a model to those who might choose to labour upon his principles, he contented himself with tracing in a few words the design of four other tracts, and with furnishing the materials, in his Silva Silvarum, where he has amassed a vast number of experiments, to serve as a foundation for his new physics. In fact, no one before Bacon understood any thing of the experimental philosophy; and of all the physical experiments which have been made since his time, there is scarcely one that is not pointed out in his works.</p><p>This great precursor of philosophy was also an elegant writer, an historian, and a wit. His moral essays are valuable, but are formed more to instruct than to please. There are excellent things too in his work <hi rend="italics">On the Wisdom of the Ancients,</hi> in which he has moralized the fables which formed the theology of the Greeks and Romans. He wrote also <hi rend="italics">The History of Henry the VIIth king of England,</hi> by which it appears that he was not less a great politician than a great philosopher.</p><p>Bacon had also some other writings, published at different times; the whole of which were collected together, and published at Frankfort, in the year 1665, in a large folio volume, with an introduction concerning his life and writings. Another edition of his works was published at London in 1740; the enumeration of which is as below:</p><p>1. De Dignitate et Augmentis Scientiarum.</p><p>2. Novum Organum Scientiarum, sive Judicia vera de Interpretatione Naturæ; cum Parasceve ad Historiam Naturalem & Experimentalem.</p><p>3. Phænomeua Universi, sive Historia Naturalis & Experimentalis de Ventis; Historia Densi & Rari; Historia Gravis & Levis; Historia Sympathiæ & Antipathiæ Rerum; Historia Sulphuris, Mercurii, & Salis; Historia Vitæ & Mortis; Historia Naturalis & Experimentalis de Forma Calidi; De Motus, sive Virtutis activæ variis speciebus; Ratio inveniendi causas Fluxus & Refluxus Maris; &c, &c.</p><p>4. Silva Silvarum, sive Historia Naturalis.</p><p>5. Novus Atlas.</p><p>6. Historia Regni Henrici vii Angliæ Regis.</p><p>7. Sermones Fideles, Ethici, Politici, Oeconomici.</p><p>8. De Sapientia Veterum.</p></div1><div1 part="N" n="BACULE" org="uniform" sample="complete" type="entry"><head>BACULE</head><p>, in <hi rend="italics">Fortification,</hi> a kind of portcullis, or gate, made like a pit-fall, with a counterpoise, and supported with two great stakes. It is usually made before the <hi rend="italics">corps de garde,</hi> not far from the gate of a place.</p></div1><div1 part="N" n="BACULOMETRY" org="uniform" sample="complete" type="entry"><head>BACULOMETRY</head><p>, the art of measuring either accessible or inaccessible lines, by the help of <hi rend="italics">baculi,</hi> staves, or rods. Schwenter has explained this art in his <hi rend="italics">Geometria Practica;</hi> and the rules of it are delivered by Wolfius, in his Elements: Ozanam also gives an illustration of the principles of Baculometry.</p></div1><div1 part="N" n="BAILLY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BAILLY</surname> (<foreName full="yes"><hi rend="smallcaps">Jean Sylvain</hi></foreName>)</persName></head><p>, a celebrated French astronomer, historiographer, and politician, was born at Paris the 15th of September 1736, and has figured as <pb n="182"/><cb/> one of the greatest men of the age, being a member of several academies, and an excellent scholar and writer. He enjoyed for several years the office of keeper of the king's pictures at Paris. He published, in 1766, a volume in 4to, <hi rend="italics">An Essay on the Theory of Jupiter's Satellites,</hi> preceded by a History of the Astronomy of these Satellites. In the Journal Encyclopédique for May and July 1773, he addressed a letter to M. Bernoulli, Astromer Royal at Berlin, upon some discoveries relative to these satellites, which he had disputed. In 1768 he published the Eulogy of Leibnitz, which obtained the prize at the Academy of Berlin, where it was printed. In 1770 he printed at Paris, in 8vo, the Eulogies of Charles the Vth, of de la Caille, of Leibnitz, and of Corneille. This last had the second prize at the Academy of Rouen, and that of Moliere had the same honour at the French Academy.</p><p>M. Bailly was admitted into the Academy as Adjunct the 29th of January 1763, and as Associate the 14th of July 1770.—In 1775 came out at Paris, in 4to, his <hi rend="italics">History of the Ancient Astronomy,</hi> in 1 volume: In 1779 the <hi rend="italics">History of Modern Astronomy</hi> in 2 volumes: and in 1787 the <hi rend="italics">History of the Indian and Oriental Astronomy,</hi> being the 2d vol. of the Ancient Astronomy.</p><p>M. Bailly's memoris published in the volumes of the Academy, are as follow:</p><p>Memoir upon the Theory of the Comet of 1759.</p><p>Memoir upon the Epoques of the Moon's motions at the end of the last century.</p><p>First, second, and third Memoris on the Theory of Jupiter's Satellites, 1763.</p><p>Memoir on the Comet of 1762: vol. for 1763.</p><p>Astronomical Observations, made at Noslon: 1764.</p><p>On the Sum's Eclipse of the 1st of April, 1764.</p><p>On the Longitude of Polling; 1764.</p><p>Observations made at the Louvre from 1760 to 1764: 1765.</p><p>On the cause of the Variation of the Inclination of the Orbit of Jupiter's second Satellite; 1765.</p><p>On the Motion of the Nodes, and on the Variation of the Inclination of Jupiter's Satellites; 1766.</p><p>On the Theory of Jupiter's Satellites, published by M. Bailly, and according to the Tables of their Motions and of those of Jupiter, published by M. Jeaurat; 1766.</p><p>Observations on the Opposition of the Sun and Jupiter; 1768.</p><p>On the Equation of Jupiter's Centre, and on some other Elements of the Theory of that Planet; 1768.</p><p>On the Transit of Venus over the Sun, the 3d of June 1769; and on the Solar Eclipse the 4th of June the same year; 1769.</p><p>In the beginning of the revolution in France, in 1789, M. Bailly took an active part in that business, and was so popular and generally esteemed, that he was chosen the first president of the States General, and of the National Assembly, and was afterwards for two years together the Mayor of Paris; in both which offices he conducted himself with great spirit, and gave general satisfaction.</p><p>He soon afterward however experienced a sad reverse of fortune; being accused by the ruling party of favouring the king, he was condemned for incivism and wanting to overturn the Republic, and died by <cb/> the Guillotine at Paris on the eleventh day of November, 1793, at 57 years of age.</p></div1><div1 part="N" n="BAINBRIDGE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BAINBRIDGE</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent physician, astronomer, and mathematician. He was born in 1582, at Ashby de la Zouch, Leicestershire. He studied at Cambridge, where having taken his degrees of Bachelor and Master of Arts, he returned to Leicestershire, where for some years he kept a grammar-school, and at the same time practised physic; employing his leisure hours in studying mathematics, especially astronomy, which had been his favourite science from his earliest years. By the advice of his friends, he removed to London, to better his condition, and improve himself with the conversation of learned men there; and here he was admitted a fellow of the college of physicians. His description of the comet, which appeared in 1618, greatly raised his character, and procured him the acquaintance of Sir Henry Savile, who, in 1619, appointed him his first professor of astronomy at Oxford. On his removal to this university, he entered a master commoner of Merton college; the master and fellows of which appointed him junior reader of Linacer's lecture in 1631, and superior reader in 1635. As he resolved to publish correct editions of the ancient astronomers, agreeably to the statutes of the founder of his professorship, that he might acquaint himself with the discoveries of the Arabian astronomers, he began the study of the Arabic language when he was above 40 years of age. Before completing that work however he died, in the year 1643, at 61 years of age.</p><p>Dr. Bainbridge wrote many works, but most of them have never been published; those that were published, were the three following, viz:</p><p>1. An Astronomical Description of the late Comet, from the 18th of November 1618, to the 16th of December following; 4to, London, 1619.—This piece was only a specimen of a larger work, which the author intended to publish in Latin, under the title of Cometographia.</p><p>2. Procli Sphæra, Ptolomæi de Hypothesibus Planetarum liber singularis. To which he added Ptolomy's Canon Regnorum. He collated these pieces with ancient manuscripts, and gave a Latin version of them, illustrated with figures: printed in 4to, 1620.</p><p>3. Canicularia. A treatise concerning the Dog-star, and the Canicular Days: published at Oxford in 1648, by Mr. Greaves, together with a demonstration of the heliacal rising of Sirius, the dog-star, for the parallel of Lower Egypt. Dr. Bainbridge undertook this work at the request of archbishop Usher, but he left it imperfect; being prevented by the breaking out of the civil war, or by death.</p><p>There were also several dissertations of his prepared for and committed to the press the year after his death, but the edition of them was never completed. The titles of them are as follow:</p><p>1st, Antiprognosticon, in quo <foreign xml:lang="greek">*man<*>ikh_s</foreign> Astrologicæ, Cœlestium Domorum, et Triplicitatum Commentis, magnisque Saturni et Jovis (cujusmodi anno 1623, et 1643, contigerunt, et vicesimo fere quoque deinceps anno, ratis naturæ legibus, recurrent) Conjunctionibus innixæ, Vanitas breviter detegitur.</p><p>2nd, De Meridionorum sive Longitudinum Differentiis inveniendis Dissertatio. <pb n="183"/><cb/></p><p>3d, De Stella Veneris Diatriba.</p><p>Beside the foregoing, there were several other tracts, never printed, but left by his will to archbishop Usher; among whose manuscripts they are preserved in the library of the college of Dublin. Among which are the following: 1. A Theory of the Sun. 2. A Theory of the Moon. 3. A Discourse concerning the Quantity of the Year. 4. Two volumes of Astronomical Observations. 5. Nine or ten volumes of Miscellaneous Papers relating to Mathematical subjects.</p></div1><div1 part="N" n="BAKER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BAKER</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>)</persName></head><p>, a mathematician of some eminence, was born at Ilton in Somersetshire, in 1625. He entered upon his studies at Oxford in 1640, where he remained seven years. He was afterwards appointed vicar of Bishop's-Nymmet in Devonshire, where he lived a studious and retired life for many years, chiefly pursuing the mathematical sciences; of which he gave a proof of his critical knowledge, in the book he published, concerning the general construction of biquadratic equations, by a parabola and a circle; the title of which book at full length is, “The Geometrical Key; or the Gate of Equations unlocked: or a new discovery of the Construction of all Equations, howsoever affected, not exceeding the 4th degree, viz, of Linears, Quadratics, Cubics, Biquadratics, and the finding of all their Roots, as well False as True, without the use of Mesolabe, Trisection of Angles, without Reduction, Depression, or any other previous preparations of equations by a circle, and any (and that one only) Parabole, &c.”: 1684, 4to, in English and Latin.</p><p>There is some account of this work in the Philos. Trans. an. 1684. And a little before his death, the Royal Society sent him some mathematical queries; to which he returned such satisfactory answers, as procured the present of a medal, with an inscription full of honour and respect.—Mr. Baker died at Bishop'sNymmet, 1690, in the 65th year of his age.</p></div1><div1 part="N" n="BAKER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BAKER</surname> (<foreName full="yes"><hi rend="smallcaps">Henry</hi></foreName>)</persName></head><p>, an ingenious and diligent naturalist, was born in London about the beginning of the 18th century. He was brought up under an eminent bookseller, but being of a philosophical turn of mind, he quitted that line of business soon after the expiration of his apprenticeship, and took to the employment of teaching deaf and dumb persons to speak and write &c, in which occupation, in the course of his life he acquired a handsome fortune. For his amusement he cultivated various natural and philosophical sciences, particularly botany, natural history, and microscopical subjects, in which he especially excelled, having, in the year 1744, obtained the Royal Society's gold medal, for his microscopical experiments on the crystallizations and consigurations of saline particles. He had various papers published in the Philos. Trans. of the Royal Society, of which he was a worthy member, as well as of the Societies of Antiquaries, and of Arts. He was author of many pieces, on various subjects, the principal of which were, his Treatise on the Water Polype, and two Treatises on the Microscope, viz, <hi rend="italics">The Microscope made easy,</hi> and <hi rend="italics">Employment for the Microscope,</hi> which have gone through several editions.</p><p>Mr. Baker married Sophia, youngest daughter of <cb/> the celebrated Daniel Defoe, by whom he had two sons, who both died before him. He terminated an honourable and useful life, at his apartments in the Strand on the 25th of November 1774, being then upwards of 70 years of age.</p><p>BAKER's <hi rend="smallcaps">Central Rule</hi>, <hi rend="italics">for the Construction of Equations,</hi> is a method of constructing all equations, not exceeding the 4th degree, by means of a given parabola and a circle, without any previous reduction of them, or first taking away their second term. See <hi rend="smallcaps">Central</hi> <hi rend="italics">Rule.</hi></p></div1><div1 part="N" n="BALANCE" org="uniform" sample="complete" type="entry"><head>BALANCE</head><p>, one of the six simple powers in mechanics, chiefly used in determining the equality or difference of weight in heavy bodies, and consequently their masses or quantities of matter.</p><p>The balance is of two kinds, the <hi rend="italics">ancient</hi> and <hi rend="italics">modern.</hi> The ancient or Roman, called also <hi rend="italics">Statera Romana,</hi> or <hi rend="italics">Steelyard,</hi> consists of a lever or beam, moveable on a centre, and suspended near one of its extremities. The bodies to be weighed are suspended from the shorter end, and their weight is shewn by the division marked on the beam, where the power or constant weight, which is moveable along the lever, keeps the steelyard in equilibrio. This balance is still in common use for weighing heavy bodies.</p><p>The modern balance, now commonly used, consists of a lever or beam suspended exactly in the middle, and having scales suspended from the two extremities, to receive the weights to be weighed.</p><p>In either case the lever is called the <hi rend="italics">jugum</hi> or the <hi rend="italics">beam,</hi> and its two halves on each side the axis, the <hi rend="italics">brachia</hi> or <hi rend="italics">arms;</hi> also the line on which the beam turns, or which divides it in two, is called the <hi rend="italics">axis;</hi> and when considered with regard to the length of the brachia, is esteemed only a point, and called the <hi rend="italics">centre of the balance,</hi> or <hi rend="italics">centre of motion:</hi> the extremities where the weights are applied, are the <hi rend="italics">points of application</hi> or <hi rend="italics">suspension;</hi> the handle by which the balance is held, or by which the whole apparatus is suspended, is called <hi rend="italics">trutina;</hi> and the slender part perpendicular to the beam, by which is determined either the equilibrium or preponderancy of bodies, is called the <hi rend="italics">tongue of the balance.</hi></p><p>From these descriptions we easily gather the characteristic distinction between the Roman balance and the common one, viz, that in the Roman balance, there is one constant weight used as a counterpoise, the point where it is suspended being varied; but, on the contrary, in the common balance or scales, the points of suspension remain the same, and the counterpoise is varied. The principle of both of them may be easily understood from the general properties of the lever, and the following observations.</p><p>The beam ABC, the principal part of the balance, is a lever of the first kind; but instead of resting on a fulcrum, it is suspended by a handle, &c, fastened to its centre of motion B: and hence the mechanism of the balance depends on the same theorems as that of the lever. Consequently as the distance between the centre of motion and the place of the unknown weight, is to the distance between the same centre and the place of the known weight, so is the latter weight, to the former. So that the unknown weight is discovered by means <pb n="184"/><cb/> of the known one, and their distances from the common centre of motion; viz, if the distances from the centre be equal, then the two weights will be equal also, as in the common balance; but if the distances be unequal, then the weights will also be unequal, and in the very same proportion, alternately, the less weight having so much the greater distance, as in the steelyard. <figure><head><hi rend="italics">The Common Balance or Scales.</hi></head></figure> The two brachia AB, BC, should be exactly equal in length, and in weight also when their scales D and E are fixed on their ends; the beam should hang exactly level or horizontal in the case of an equipoise; and for this purpose the centre of gravity of the whole should fall a little below the centre of motion, and but a little, that the balance be sufficiently sensible to the least variation of weight: the friction on the centre should also be as small as possible. <figure><head><hi rend="italics">The Steel Yard.</hi></head></figure></p><p>Having made a proper bar of steel AB, tapering at the longer end, and very strong at the other, suspend it by a centre C near the shorter or thicker end, so that it may exactly balance itself in equilibrio, and prepare a constant weight I to weigh with: then hang on any weight, as one pound for instance, at the shorter arm, and slide the constant weight backwards and forwards upon the longer arm, till it be just in equilibrio with the former; and there make a notch and number 1, for the place of 1 pound: take off the 1lb, and hang a two pound weight in its stead at the shorter arm; then slide the constant weight back on the longer arm, till the whole come again into equilibrio, making a notch at the place of the constant weight and the number 2, for the place of 2lb. Proceed in the same manner for all other weights 3, 4, 5, &c; as also for the intermediate halves and quarters, &c, if it be necessary; always sus- <cb/> pending the variable weights at the end of the shorter arm, shisting the constant weight so as to balance them, and marking and numbering the places on the longer arm where the constant weight always makes a counterpoise. The use of the Steelyard is hence very evident: the thing whose weight is required being suspended by a hook at the short end, move the constant weight backwards and forwards on the longer arm, till the beam is balanced horizontally: then look what notch the constant weight is placed at, and its number will shew the weight of the body that was required. DC is the handle and tongue; F the centre of motion; EG a scale sometimes hung on at the end by the hook H. <figure><head><hi rend="italics">The Bent-Lever Balance.</hi></head></figure></p><p>This instrument operates by a fixed weight, C, increasing in power as it ascends along the are FG of a circle, and pointing by an index to the number or division of the are which denotes the weight of any body put into the scale at E. And thus one constant weight serves to weigh all others, by only varying the position of the arms of the balance, instead of varying the places or points of suspension in the arms themselves.</p><p><hi rend="italics">The Deceitful Balance.</hi> This operates in the same manner as the steelyard, and cheats or deceives by having one arm a little longer than the other; though the deception is not perceived, because the shorter arm is made somewhat heavier, so as to compensate for its shortness, by which means the beam of the balance, when no weights are in the scales, hangs horizontal in equilibrio. The consequence of this construction is, that any commodity put in the scale of the longer arm, requires a greater weight in the other scale to balance it; and so the body is fallaciously accounted heavier than it really is. But the trick will easily be detected by making the body and the weight change places, removing them to the opposite scales, when the weight will immediately be seen to preponderate.</p><p><hi rend="italics">Assay-Balance.</hi> This is a very nice balance, used in determining the exact weights of very small bodies. Its structure is but little different from the common sort; <pb/><pb/><pb n="185"/><cb/> except that it is made of the best and hardest steel, and made to turn with the smallest weight.</p><p><hi rend="italics">Hydrostatical Balance.</hi> This is un instrument for determining the specific gravity of bodies. See H<hi rend="smallcaps">YDROSTATICAL</hi>, and <hi rend="smallcaps">Specific</hi> <hi rend="italics">Gravity.</hi></p><div2 part="N" n="Balance" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Balance</hi></head><p>, in Astronomy, the same as <hi rend="italics">Libra.</hi></p><p><hi rend="smallcaps">Balance</hi> <hi rend="italics">of a Clock or Watch,</hi> is that part which, by its motion, regulates and determines the beats. The circular part of it is called the <hi rend="italics">rim,</hi> and its spindle the <hi rend="italics">verge;</hi> there belong to it also two pallets or nuts, that play in the fangs of the crown-wheel: in pocket watches, that strong stud in which the lower pivot of the verge plays, and in the middle of which one pivot of the crown-wheel runs, is called the <hi rend="italics">potence:</hi> the wrought piece which covers the balance, and in which the upper pivot of the balance plays, is the <hi rend="italics">cock;</hi> and the small spring in the new pocket watches, is called the <hi rend="italics">regulator.</hi></p></div2></div1><div1 part="N" n="BALCONY" org="uniform" sample="complete" type="entry"><head>BALCONY</head><p>, a projecture in the front of a house, or other building, commonly supported by pillars or consoles, and encompassed by a ballustrade.</p></div1><div1 part="N" n="BALL" org="uniform" sample="complete" type="entry"><head>BALL</head><p>, any spherical, globular, or round body.</p><div2 part="N" n="Ball" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ball</hi></head><p>, in the military art, signifies all sorts of bullets for fire arms, from the pistol up to the largest cannon. Cannon balls are made of cast iron; but the musket and pistol balls of lead, as these are both heavier under the same bulk, and do not furrow the barrels of the pieces.</p><p><hi rend="smallcaps">Ball</hi> <hi rend="italics">of a Pendulum,</hi> is the weight at the bottom of it; and is sometimes, especially in sh<*>rter pendulums, called the <hi rend="italics">bob.</hi></p><p><hi rend="smallcaps">Balls</hi> <hi rend="italics">of Fire,</hi> in Meteorology. See <hi rend="smallcaps">Fire</hi> <hi rend="italics">balls.</hi></p></div2><div2 part="N" n="Balls" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Balls</hi></head><p>, in Electricity, invented by Mr. Canton, are two pieces of cork, or pith of elder tree, nicely turned in a lathe to the size of a small pea, and suspended by fine linen threads. They are used as electrometers, and are of excellent use to discover small degrees of electricity; and to observe its changes from positive to negative, or the reverse; as also to estimate the force of a shock before the discharge, so that the operator shall always be able to tell very nearly before hand what the explosion will be, by knowing how high he has charged his jars.</p></div2></div1><div1 part="N" n="BALLISTA" org="uniform" sample="complete" type="entry"><head>BALLISTA</head><p>, a military engine much used by the ancients for throwing stones, darts, and javelins; and somewhat resembling our cross-bows, but much larger and stronger. The word is Latin, signifying a cross-bow; but is derived from the Greek <foreign xml:lang="greek">ballw</foreign>, to <hi rend="italics">shoot,</hi> or <hi rend="italics">throw.</hi></p><p>Vegetius informs us, that the ballista discharged darts with such violence and rapidity, that nothing could resist their force: and Athenæus adds, that Agistratus made one of little more than 2 feet in length, that shot darts 500 paces. Authors have often confounded the ballista with the catapulta, attributing to the one what belongs to the other. According to Vitruvius, the ballista was made after divers manners, though all were used to the same purpose: one sort was framed with levers and bars; another with pullies; some with a crane; and others again with a toothed wheel. Marcellinus describes the ballista thus; A round iron cylinder is fastened between two planks, from which reaches a hollow square beam placed cross-wise, fastened with cords, to which are added screws. At one end of this stands the engineer, who puts a wooden shaft, or arrow, with a large head, into the cavity of the beam; this done, two men bend the engine, by drawing some <cb/> wheels; when the top of the head is drawn to the utmost end of the cords, the shaft is driven out of the ballista, &c.</p><p>The ballista is ranked by the ancients in the sling kind, and its structure and effect reduced to the principles of that instrument; whence it is called by Hero and others, <hi rend="italics">funda,</hi> and <hi rend="italics">fundibulus.</hi> Gunther calls it <hi rend="italics">Balearica machina,</hi> as a sling peculiar to the Balearic islands.—Perrault, in his notes on Vitruvius, gives a new contrivance of a like engine for throwing bombs without powder.</p><p>Fig. 1, Plate v, represents the ballista used in sieges, according to Folard: where 2, 2, denote the base of the ballista; 3, 4, upright beams; 5, 6, transverse beams; 7, 7, the two capitals in the upper transverse beam, (the lower transverse beam has also two similar capitals, which cannot be seen in this transverse figure); 9, 9, two posts or supports for strengthening the transverse beams; 10, 10, two skains of cords fastened to the capitals; 11, 11, two arms inserted between the two strands, or parts of the skains; 12, a cord fastened to the two arms; 13, darts which are shot by the ballista; 14, 14, curves in the upright beams, and in the concavity of which cushions are fastened, in order to break the force of the arms, which strike against them with great force when the dart is discharged; 16, the arbor of the machine, in which a straight groove or canal is formed to receive the darts, in order to their being shot by the ballista; 17, the nut of the trigger; 18, the roll or windlass, about which the cord is wound; 19, a hook, by which the cord is drawn towards the centre, and the ballista cocked; 20, a stage or table on which the arbor is in part sustained.</p><div2 part="N" n="Ballista" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ballista</hi></head><p>, in Practical Geometry, the same as the geometrical cross, called also <hi rend="italics">Jacob's Staff.</hi> See <hi rend="smallcaps">Cross</hi>-<hi rend="italics">Staff.</hi></p><p>BALLISTIC <hi rend="smallcaps">Pendulum</hi>, an ingenious machine invented by Benjamin Robins, for ascertaining the velocity of military projectiles, and consequently the force of fired gun-powder. It consists of a large block of wood, annexed to the end of a strong iron stem, having a cross steel axis at the other end, placed horizontally, about which the whole vibrates together like the pendulum of a clock. The machine being at rest, a piece of ordnance is pointed straight towards the wooden block, or ball of this pendulum, and then discharged: the consequence is this; the ball discharged from the gun strikes and enters the block, and causes the pendulum to vibrate more or less according to the velocity of the projectile, or the force of the blow; and by observing the extent of the vibration, the force of that blow becomes known, or the greatest velocity with which the block is moved out of its place, and consequently the velocity of the projectile itself which struck the blow and urged the pendulum.</p><p>The more minute and particular description may be seen in my Tracts, vol. 1, where are given all the rules for using it, and for computing the velocities, with a multitude of accurate experiments performed with cannon balls, by means of which the most useful and important conclufions have been deduced in military projectiles and the nature of physics. I have also since that publication, made many other experiments of the same kind, by discharging cannon balls at various distances from the block; from which have resulted the <pb n="186"/><cb/> discovery of a complete series of the resistances of the air to balls passing through it with all degrees of velocity, from o up to 2000 feet in a second of time.</p><p>Other writers on this subject are Euler, Antoni, Le Roy, Darcy, &c. See also Robins's Mathematical Tracts.</p></div2></div1><div1 part="N" n="BALLISTICS" org="uniform" sample="complete" type="entry"><head>BALLISTICS</head><p>, is used by some for projectiles, or the art of throwing heavy bodies Mersennus has published a treatise on the projection of bodies, under this title.</p></div1><div1 part="N" n="BALLOON" org="uniform" sample="complete" type="entry"><head>BALLOON</head><p>, or <hi rend="smallcaps">Ballon</hi>, in a general sense, signifies any spherical hollow body. Thus, with chemists, it denotes a round short-necked vessel, used to receive what is distilled by means of fire: in architecture, a ball or globe on the top of a pillar, &c: and among engineers, a kind of bomb made of pasteboard, and played off in fireworks, in imitation of a real iron bomb-shell.</p><p><hi rend="italics">Air</hi>-<hi rend="smallcaps">Balloon.</hi> See <hi rend="smallcaps">Aerostation</hi> and <hi rend="smallcaps">Air</hi>-<hi rend="italics">Balloon.</hi></p></div1><div1 part="N" n="BALLUSTER" org="uniform" sample="complete" type="entry"><head>BALLUSTER</head><p>, a small kind of column or pillar, used for ballustrades.</p></div1><div1 part="N" n="BALLUSTRADE" org="uniform" sample="complete" type="entry"><head>BALLUSTRADE</head><p>, a series or row of ballusters, joined by a rail; serving for a rest to the arms, or as a fence or inclosure to balconies, altars, staircases, &c.</p></div1><div1 part="N" n="BAND" org="uniform" sample="complete" type="entry"><head>BAND</head><p>, in Architecture, denotes any flat low member, or moulding, that is broad, but not very deep. The word <hi rend="italics">lace</hi> sometimes means the same thing.</p></div1><div1 part="N" n="BANQUET" org="uniform" sample="complete" type="entry"><head>BANQUET</head><p>, or <hi rend="smallcaps">Banquette</hi>, in Fortification, a little foot-bank, or elevation of earth, forming a path along the inside of a parapet, for the soldiers to stand upon to discover the counterscarp, or to fire on the enemy, in the moat, or in the covert way. It is commonly about 3 feet wide, and a foot and a half high.</p></div1><div1 part="N" n="BARLOWE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BARLOWE</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an eminent mathematician and divine, in the 16th century. He was born in Pembrokeshire, his father (William Barlowe) being then bishop of St. David's. In 1560 he was entered commoner of Baliol college in Oxford; and in 1564, having taken a degree in arts, he left the university, and went to sea; but in what capacity is uncertain: however he thence acquired considerable knowledge in the art of navigation, as his writings afterwards shewed. About the year 1573, he entered into orders, and became prebendary of Winchester, and rector of Easton, near that city. In 1588 he was made prebendary of Litchfield, which he exchanged for the office of treasurer of that church. He afterwards was appointed chaplain to prince Henry, eldest son of king James the first; and, in 1614, archdeacon of Salisbury. Barlowe was remarkable, especially for having been the first writer on the nature and properties of the loadstone, 20 years before Gilbert published his book on that subject. He was the first who made the inclinatory instrument transparent, and to be used with a glass on both sides. It was he also who suspended it in a compass-box, where, with 2 ounces weight, it was made fit for use at sea. He also found out the difference between iron and steel, and their tempers for magnetical uses. He likewise discovered the proper way of touching magnetical needles; and of piecing and cementing of loadstones; and also why a loadstone, being double-capped, must take up so great a weight.</p><p>Barlowe died in the year 1625.—His works are as follow:</p><p>1. <hi rend="italics">The Navigator's Supply, containing many things of principal importance belonging to Navigation, and use of</hi> <cb/> <hi rend="italics">diverse Instruments framed chiefly for that purpose.</hi> Lond. 1597, 4to; dedicated to Robert earl of Essex.</p><p>2. <hi rend="italics">Magnetical Advertisement, or Diverse Pertinent Observations and improved Experiments concerning the nature and properties of the Loadstone.</hi> Lond. 1616, 4to.</p><p>3. <hi rend="italics">A Brief Discovery of the idle Animadversions of Mark Ridley, M. D. upon a treatise entitled Magnetical Advertisements.</hi> Lond. 1618, 4to.</p><p>In the first of these pieces, Barlowe gave a demonstration of Wright's or Mercator's division of the meridian line, as communicated by a friend; observing that “This manner of carde has been publiquely extant in print thes thirtie yeares at least [he should have said 28 only<*>, but a cloude (as it were) and thicke miste of ignorance doth keepe it hitherto concealed: And so much the more, because some who were reckoned for men of good knowledge, have by glauncing speeches (but never by any one reason of moment) gone about what they could to disgrace it.”</p><p>This work of Barlowe's contains descriptions of several instruments for the use of Navigation, the principal of which is an Azimuth Compass, with two upright sights; and as the author was very curious in making experiments on the loadstone, he treats well and fully upon the SeaCompass. And he treated still farther on the same instrument in his second work, the Magnetical Advertisement.</p></div1><div1 part="N" n="BAROMETER" org="uniform" sample="complete" type="entry"><head>BAROMETER</head><p>, an instrument for measuring the weight or pressure of the at mosphere; and by that means the variations in the state of the air, foretelling the changes in the weather, and measuring heights or depths, &c.</p><p>This instrument is founded on what is called the Torricellian experiment, related below, and commonly consists of a glass tube, open at one end; which being first filled with quicksilver, and then inverted with the open end downwards into a bason of the same, the mercury descends in the tube till it remains at about the height of 29 or 30 inches, according to the weight or pressure of the atmosphere at the time, which is just equal to the weight of that column of the quicksilver. Hence it follows that, if by any means the pressure of the air be altered, it will be indicated by the rising or falling of the mercury in the tube; or if the barometer be carried to a higher station, the quicksilver will descend lower in the tube, but when carried to a lower place, it will rise higher in the tube, according to the difference in elevation between the two places.</p><p><hi rend="italics">History of the Barometer.</hi>—About the beginning of the last century, when the doctrine of a plenum was in vogue, it was a common opinion among philosophers, that the ascent of water in pumps was owing to what they called nature's abhorrence of a vacuum; and that thus fluids might be raised by suction to any height whatever. But an accident having just discovered, that water could not be raised in a pump unless the sucker reached to within 33 feet of the water in the well, it was conjectured by Galileo, who flourished about that time, that there might be some other cause of the ascent of water in pumps, or at least that this abhorrence was limited to the finite height of 33 feet. Being unable to satisfy himself on this head, he recommended the consideration of the difficulty to Torricelli, who had been his disciple. After some time Torricelli fell upon the suspicion that the pressure of the atmosphere was the cause of the ascent of water in pumps; that a column <pb/><pb/><pb n="187"/><cb/> of water 33 feet high was a just counterpoise to a column of air, of the same base, and which extended up to the top of the atmosphere; and that this was the true reason why the water did not follow the sucker any farther. And this suspicion was soon after confirmed by various experiments. Torricelli considered, that if a column of water 33 feet high were a counterpoise to a whole column of the atmosphere, then a column of mereury of about 2 feet and a half high would also be a counterpoise to it, since quicksilver is near 14 times heavier than water, and so the 14th part of the height, or near 2 feet and a half, would be as heavy as the column of water. This reasoning was soon verisied; for having filled a glass tube with quicksilver, and inverted it into a bason of the same, the mercury presently descended till its height, above that in the bason, was about two feet and a half, just as he expected. And this is what has, from him, been called the Torricellian experiment.</p><p>The new opinion, with this confirmation of it, was readily acquiesced in by most of the philosophers, who repeated the experiment in various ways. Others however still adhered to the old doctrine, and raised several pretended objections against the new one; such as that there was a film or imperceptible <hi rend="italics">rope of mercury,</hi> extended through the upper part of the tube, which suspended the column of mercury, and kept it from falling into that in the bason. This and other objections were however soon overcome by additional confirmations of the true doctrine, particularly by varying the elevation of the place. It was hinted by Descartes and Pascal, that if the mercury be sustained in the tube by the pressure of the atmosphere, by carrying it to a higher situation, it would descend lower in the tube, having a shorter column of the atmosphere to sustain it, and vice versa. And Pascal engaged his brother-in-law, M. Perier, to try that experiment for him, being more conveniently situated for that purpose than he was at Paris. This he accordingly executed, by observing the height of the quicksilver in the tube, first at the bottom of a mountain in Auvergne, and then at several stations, or different altitudes, in ascending, by which it was found that the mercury fell lower and lower all the way to the top of the mountain; and so confirming the truth of the doctrine relating to the universal pressure of the atmosphere, and the consequent suspension of the mercury in the tube of the barometer. Thus, by the united endeavours of Torricelli, Descartes, Pascal, Mersenne, Huygens, and others, the cause of the suspension of the quicksilver in the tube of the barometer, became pretty generally established.</p><p>It was some time however after this general consent, before it was known that the pressure of the air was various at different times, in the same place. This could not however remain long unknown, as the frequent measuring of the column of mercury, must soon shew its variations in altitude; and experience and observation would presently shew that those variations in the mercurial column, were always succeeded by certain changes in the weather, as to rain, wind, frosts, &c. Hence this instrument soon came into use as the means of foretelling the changes of the weather; and on this account it obtained the name of the <hi rend="italics">weather-glass,</hi> as it did that of <hi rend="italics">barometer</hi> from its being the measure of the weight or <cb/> pressure of the air. We may now proceed to take a view of its various forms and uses.</p><p><hi rend="italics">The Common Barometer.</hi> This is represented at fig. 1, plate iv, such as it was invented by Torricelli. AB is a glass tube, of 1/4, or 1/3, or 1/2 inch wide, the more the better, and about 34 inches long, being close at the top A, and the open end B immersed in a bason of quicksilver CD, which is the better the wider it is. To fill this, or any other barometer; take a clean new glass tube, of the dimensions as above, and pour into it well purified quicksilver, with a small funnel either of glass or paper, in a fine continued stream, till it wants about half an inch or an inch of being full; then stopping it close with the finger, invert it slowly, and the air in the empty part will ascend gradually to the other end, collecting into itself such other small air bubbles as unavoidably get into the tube among the mercury, in filling it with the funnel: and thus continue to invert it several times, turning the two ends alternately upwards, till all the air bubbles are collected, and brought up to the open end of the tube, and when the part filled shall appear, without speck, like a fine polished steel rod. This done, pour in a little more quicksilver, to sill the empty part quite full, and so exclude all air from the tube: then, stopping the orifice again with the finger, invert the tube, and immerse the finger and end, thus stopped, into a bason of like purified quicksilver; in this position withdraw the finger, so shall the mercury descend in the tube to some place as E, between 28 and 31 inches above that in the bason at F, as these are the limits between which it always stands in this country on the common surface of the earth. Then measure, from the surface of the quicksilver in the bason at F, 28 inches to G, and 31 inches to H, dividing the space between them into inches and tenths, which are marked on a scale placed against the side of the tube; and the tenths are subdivided into hundredth parts of an inch by a sliding index carrying a vernier or nonius. These 3 inches, between 28 and 31, so divided, will answer for all the ordinary purposes of a stationary or chamber barometer; but for experiments on altitudes and depths, it is proper to have the divisions carried on a little higher up, and a great deal lower down. In the proper filling and otherwise fitting up of the barometer, several circumstances are to be carefully noted; as, that the bore of the tube be pretty wide, to allow the freer motion of the quicksilver, without being impeded by an adhesion to the sides; that the bason below it be also pretty large, in order that the surface of the mercury at F may not sensibly rise or fall with that in the tube; that the bottom of the tube be cut off rather obliquely, that when it rests on the bottom of the bason there may be a free passage for the quicksilver; and that, to have the quicksilver very pure, it is best to boil it in the tube, which will expel all the air from it. This barometer is commonly sitted up in a neat mahogany case, together with a thermometer and hygrometer, as represented in plate 4, fig. 13.</p><p>As the scale of variation is but small, being included within 3 inches in the common barometer, several contrivances have been devised to enlarge the scale, or to render the motion of the quicksilver more sensible.</p><p>Descartes first suggested a method of increasing the sensibility, which was executed by Huygens. This <pb n="188"/><cb/> was effected by making the barometrical tube end in a pretty large cylindrical vessel at top, into which was inserted also the lower or open end of a much finer tube than the former, which was partly filled with water, to give little obstruction by its weight to the motion of the mercury, while it moved through a pretty long space of the very sine tube by a small variation of the mercury below it, and so rendered the small changes in the state of the air very sensible. But the inconvenience was this, that the air contained in the water gradually disengaged itself, and escaped through into the vacuum in the top of the small tube, till it was collected in a body there, and by its elasticity preventing the sree rise of the fluids in the tubes, spoiled the instrument as a barometer. And this, it may be observed by-the-bye, is the reason why a water barometer cannot succeed. This barometer is here represented in fig. 2, where CD is the vessel, in which are united the upper or small water tube AC, with the lower or mercurial one CB.</p><p>To remedy this inconvenience, Huygens thought of placing the mercury at top, and the water at bottom, which he thus contrived. ADG (fig. 3) is a bent tube hermetically sealed at A, but open at G, of about one line in diameter, and passing through the two equal cylindrical vessels BC, EF, which are about 20 inches apart, and of 15 lines diameter, their length being 10. The mercury being put into the tube, will stand between the middle of the vessels EF and BC, the remaining space to A being void both of air and mercury. Lastly, common water, tinged with a 6th part of aqua regis, to prevent its freezing, is poured into the tube FG, till it rises a foot above the mercury in DF. To prevent the water from evaporating, a drop of oil of sweet almonds floats on the top of it. But the column of water will be sensibly affected by heat and cold, which spoils the accuracy of the instrument. For which reason other contrivances have been made, as below.</p><p><hi rend="italics">The Horizontal or Rectangular Barometer,</hi> fig. 4, was invented by J. Bernoulli and Cassini; where AB is a pretty wide cylindrical part at the top of the tube, which tube is bent at right angles at C, the lower part of it CD being turned into the horizontal direction, and close above at A, but open at the lower end D, where however the mercury cannot run out, being there opposed by the pressure of the atmosphere. This and the foregoing contrivance of Huygens are founded on the theorem in hydrostatics, that fluids of the same base press according to their perpendicular altitude, not according to the quantity of their matter; so that the same pressure of the atmosphere sustains the quicksilver that fills the tube ACD, and the cistern B, as would support the mercury in the tube alone. Hence, having fixed upon the size of the scale, as suppose the extent of 12 inches, instead of the 3, in the common barometer from 28 to 31, that is 4 times as long; then the area of a section of the cylinder AB must be 4 times that of the tube, and consequently its diameter double, since the areas of circles are as the squares of their diameters: then for every natural variation of an inch in the cylinder AB, there will be a variation of four inches in the tube CD.—But on account of the attrition of the mercury against the sides of the glass, and the great momentum from the quick motion in CD, the quicksilver is apt to break, and the rife and fall is no longer equa- <cb/> ble; and besides, the mercury is apt to be thrown out of the orifice at D by sudden motions of the machine.</p><p><hi rend="italics">The Diagonal Barometer</hi> of Sir Samuel Moreland, fig. 5, is another method of enlarging the natural scale of three inches perpendicular, or CD, by extending it to any leught BC in an oblique direction. This is liable in some degree to the same inconvenience, from friction and breaking, as the horizontal one; and hence it is found that the diagonal part BC cannot properly be bent from the perpendicular more than in an angle of 45°, which only increases the scale nearly in the proportion of 7 to 5.</p><p><hi rend="italics">Doctor Hook's Wheel Barometer,</hi> fig. 6. This was invented about 1668, and is meant to render the alterations in the air more sensible. Here the barometer tube has a large ball AB at top, and is bent up at the lower or open end, where an iron ball G floats on the top of the mercury in the tube, to which is connected another ball H by a cord, hanging freely over a pulley, turning an index KL about its centre. When the mercury rises in the part FG, it raises the ball, and the other ball descends and turns the pulley with the index round a graduated circle from N towards M and P; and the contrary way when the quicksilver and the ball sink in the bent part of the tube. Hence the scale is easily enlarged 10 or 12 fold; being increased in proportion of the axis of the pulley to the length of the index KL. But then the friction of the pulley and axis is some obstruction to the free motion of the quicksilver. Contrivances to lessen the friction &c, may also be seen in the <hi rend="italics">Philos. Trans.</hi> vol. 52, art. 29, and vol. 60, art. 10.</p><p><hi rend="italics">The Steelyard Barometer,</hi> for so that may be called which is represented by fig. 7, which enlarges the scale in the proportion of the shorter to the longer arm of a steelyard. AB is the barometer tube, close at A and open at B, immersed in a cylindrical glass cistern CD, which is but very little wider than the tube AB is. The barometer tube is suspended to the shorter arm of an index like a steelyard, moving on the fulcrum E, and the extremity of its longer arm pointing to the divisions of a graduated arch, with which index the tube is nearly in equilibrio. When the pressure of the atmosphere is lessened, the mercury descends out of the tube into the cistern which raises the tube and the shorter arm of the index, and consequently the extremity of the longer moves downwards, and passes over a part of the graduated arch. And on the contrary this moves upwards when the pressure of the atmosphere increases.</p><p><hi rend="italics">The Pendant Barometer,</hi> fig. 8, was invented by M. Amontons, in 1695. It consists of a single conical tube AB, hung up by a thread, the larger or open end downwards, and having no vessel or cistern, because the conical sigure supplies that, and the column of mercury suftained is always equal to that in the common barometer tube; which is effected thus; when the pressure of the air is less, the mercury sinks down to a lower and wider part of the tube, and consequently the altitude of its column will be less; and on the contrary, by a greater pressure of the atmosphere the mercury is forced up to a higher and narrower part, till the length of the column CD be equal to that in the tube of the common barometer.—The inconvenience of this barometer is, that as the bore must be made very small, to prevent the mercury from falling out by an accidental shake, the <pb n="189"/><cb/> friction and adhesion to the sides of the tube prevent the free motion of the mercury.</p><p><hi rend="italics">Mr. Rowning's Compound Barometers.</hi> This gentleman has several contrivances for enlarging the scale, and that in any proportion whatever. One of these is described in the <hi rend="italics">Philos. Trans.</hi> N° 427, and also in his <hi rend="italics">Nat. Philos.</hi> part 2; and another in the same part, which is here represented at fig. 9. ABC is a compound tube, hermetically sealed at A, and open at C; empty from A to D, filled with mercury from thence to B, and from hence to E with water. Hence by varying the proportions of the two tubes AF and FC, the scale of variation may be changed in any degree.</p><p><hi rend="italics">The Marine Barometer.</hi> This was first invented by Dr. Hook, to be used on board of ship, being contrived so as not to be affected or injured by the motion of the ship. His contrivance consisted of a double thermometer, or a couple of tubes half filled with spirit of wine; the one sealed at both ends, with a quantity of air included; the other sealed at one end only. The former of these is affected only by the warmth of the air; but the other is affected both by the external warmth and by the variable pressure of the atmosphere. Hence, considering the spirit thermometer as a standard, the excess of the rise or fall of the other above it will shew the increase or decrease of the pressure of the atmosphere. This instrument is deseribed by Dr. Halley, in the <hi rend="italics">Philos. Trans.</hi> N° 269, where he says of it, “I had one of these barometers with me in my late southern voyage, and it never failed to prognosticate and give early notice of all the bad weather we had, so that I depended thereon, and made provision accordingly; and from my own experience I conclude, that a more useful contrivance hath not for this long time been offered for the benefit of navigation.”</p><p><hi rend="italics">Mr. Nairne,</hi> an ingenious artist in London, has lately invented a new kind of <hi rend="italics">Marine Barometer;</hi> which differs from the common barometer by having the lower part of the tube, for about 2 feet long, made very small, to check the vibrations of the mercury, which would otherwise arise from the motions of the ship. This is also assisted by being hung in gimbals, by a part which subjects it to be the least affected by such motions.</p><p>Another sort of Marine Barometer has also been invented by M. Passemente, an ingenious artist at Paris. This contrivance consists only in twisting the middle of the tube into a spiral of two revolutions; by which contrivance the impulses which the mercury receives from the motions of the ship are destroyed, by being transmitted in contrary directions.</p><p><hi rend="italics">The Statical Baroscope,</hi> or <hi rend="italics">Barometer,</hi> of Mr. Boyle, &c. This consists of a large glass bubble, blown very thin, and then balanced by a small brass weight. Hence these two bodies being of unequal bulk, the larger will be very much affected by a change of the density of the medium, but the less not at all as to sense: So that, when the atmosphere becomes denser, the ball loses more of its weight, and the brass weight preponderates; and contrariwise when the air grows lighter.</p><p><hi rend="italics">Mr. Caswell's Baroscope,</hi> or <hi rend="italics">Barometer.</hi> This is described in the <hi rend="italics">Philos. Trans.</hi> vol. 24, and seems to be the most sensible and exact of any. It is thus described: Suppose ABCD, (fig. 10) is a bucket of water, in <cb/> which is the baroscope <hi rend="italics">xrezyosm,</hi> which consists of a body <hi rend="italics">xrsm,</hi> and a tube <hi rend="italics">ezyo,</hi> which are both concave cylinders, made of tin, or rather glass, and communicating with each other. The bottom of the tube <hi rend="italics">zy</hi> has a leaden weight to sink it, so that the top of the body may just swim even with the suiface of the water by the addition of some grain weights on the top. When the instrument is forced with its mouth downwards, the water ascends into the tube to the height <hi rend="italics">yu.</hi> To the top is added a small concave cylinder, or pipe, to keep the instrument from sinking down to the bottom: <hi rend="italics">md</hi> is a wire: and <hi rend="italics">mS, de</hi> are two threads oblique to the surface of the water, which perform the office of diagonals: for while the instrument sinks more or less by an alteration in the gravity of the air, where the surface of the water cuts the thread is formed a small bubble, which ascends up the thread while the mercury of the common baroscope ascends, and vice versa.</p><p>It appears from a calculation which the author makes, that this instrument shews the alterations in the air 1200 times more accurately than the common barometer. He observes, that the bubble is seldom known to stand still even for a minute; that a small blast of wind, which cannot be heard in a chamber, will sensibly make it sink; and that a cloud passing over it always makes it descend, &c.</p><p>While some have been increasing the sensibility of the barometer by enlarging the variations, others have endeavoured to make it more convenient by reducing the length of the tube. <hi rend="italics">M. Amontons,</hi> in 1688, first proposed this alteration in the structure of barometers, by joining several tubes to one another, alternately filled with mercury and with air, or some other fluid; and the number of these tubes may be increased at pleasure: but the contrivance is perhaps more ingenious than useful.</p><p><hi rend="italics">M. Mairan's reduced Barometer,</hi> which is only 3 inches long, serves the purpose of a manometer, in shewing the dilatations of the air in the receiver of an airpump; and instruments of this kind are now commonly applied to this use.</p><p><hi rend="italics">The Portable Barometer,</hi> is so contrived that it may be carried from one place to another without being disordered. The end of the tube is tied up in a leathern bag not quite full of mercury; which being pressed by the air, forces the mercury into the tube, and keeps it suspended at its proper height. This bag is usually inclosed in a box, through the bottom of which passes a screw, by means of which the mercury may be forced up to the top of the tube, and prevented from breaking it by dashing against the top when the instrument is removed from one station to another. It seems Mr. Patrick first made a contrivance of this kind: but the portable barometer has received various improvements since; and the most complete of this kind has been described by M. De Luc, in his Recherches, vol. 2, pa. 5 &c, together with the apparatus belonging to it, the method of construction and use, and the advantages attending it. Improvements have also been suggested by Sir George Shuckburgh, and Col. Roy, which have been carried into execution, with farther improvements also, by Mr. Ramsden, and other ingenious artists in London. <pb n="190"/><cb/></p><p>Fig. 11 represents this instrument, as inclosed in its mahogany case by means of three metallic rings <hi rend="italics">aaa.</hi> This case is a hollow cone, so shaped within as to contain steadily the body of the barometer, and is divided into three branches from <hi rend="italics">b</hi> to <hi rend="italics">c,</hi> forming three legs or supports for the instrument when observations are making, and sustaining it at the part <hi rend="italics">d</hi> of the case, as it appears in Fig. 12, by an improved kind of gimbals, in which its own weight renders it sufficiently steady at any time. In the part of the frame <hi rend="italics">fg</hi> where the barometer tube appears, is made a long slit or opening, that the column of mercury may be seen against the light, and the vernier piece <hi rend="italics">f</hi> brought down to coincide very nicely with the edge of the mercury. When the instrument is fixed in its stand, the screw <hi rend="italics">t</hi> is to be turned to let the mercury down to its proper station, and a peg at <hi rend="italics">i</hi> must be loosened, to admit the external air to act upon the mercury contained in the box <hi rend="italics">k.</hi> The proper adjustment, or mode of observing what is called the <hi rend="italics">zero</hi> or o division of the column of mercury, is by observing it in the transparent part of the box <hi rend="italics">k,</hi> which has a glass tube or reservoir for the quicksilver, and an edged piece of metal attached to the external part of it; with the edge of which the mercury is to be brought into contact by turning the screw <hi rend="italics">l</hi> to the right or left as occasion requires. The vernier piece at <hi rend="italics">f,</hi> which determines the altitude of the mercurial column, is first brought down by the hand to a near contact, and then accurately adjusted by turning the screw <hi rend="italics">e</hi> at the top. The divisions annexed to the tube of this instrument may be of any sort, or of any degree of smallness, according to the purposes it is intended to serve. To accommodate it to the use of foreigners as well as natives, there are commonly added scales of both French and English inches, with their subdivisions to any extent required. It is usual to place the French scale of inches on the right side at <hi rend="italics">fg,</hi> from 19 to 31 inches, measured from the zero or surface of the mercury in the box <hi rend="italics">k</hi> below; each inch being divided into lines or 12th parts, and each line subdivided by the vernier into 10th parts, or 120th parts of inches; by means of which therefore the length of the mercurial column may be determined to the 120th part of a French inch. The other scale, which is placed on the left side of the instrument, is divided into English inches, and each inch into 20th parts, which by a vernier are subdivided into 25th parts, or 500th parts of inches; by this means shewing the height of the mercury to the 500th part of an English inch. But this vernier is figured double or each division is accounted 2, which reduces the measures to 1000ths of an inch for the conveniency of calculation, in measuring altitudes of hills &c.</p><p>A thermometer is always attached to the instrument, as a necessary appendage to it, being fastened to the body at <hi rend="italics">h,</hi> and sunk into the surface of the frame, to preserve it from injury: the degrees of this thermometer are marked on two scales, one on each side of it, viz, the scale of Fahrenheit, and that of Reaumur; the freezing point of the former being at 32, and of the latter at o. Also on the right hand side of these two scales there is a third, called a scale of <hi rend="italics">correction,</hi> <cb/> placed oppositely to that of Fahrenheit, with the word <hi rend="italics">add</hi> and <hi rend="italics">subtract</hi> marked; which shews the necessary correction of the observed altitude of the mercury at any given temperature of the air, indicated by the thermometer.</p><p>There are several other pieces of mechanism about the instrument, which will be evident by inspection; and the manner of making the observations, with the necessary calculations, are fully explained in M. de Luc's <hi rend="italics">Recherches fur les Modifications de l'Atmosphere,</hi> and the <hi rend="italics">Philos. Trans.</hi> vol. 67 and 68, before cited.</p><p><hi rend="italics">The Common Chamber Weatherglass,</hi> is also usually fitted up in a neat mahogany frame, and other embellishments, to make it an ornamental piece of furniture. It confists of the common tube barometer, with a thermometer by the side of it, and an hygrometer at the top, as exhibited in fig. 13.</p><p>To the foregoing may be added a new sort of <hi rend="italics">Barometer,</hi> or <hi rend="italics">Weather Instrument by the Sound of a Wire.</hi> This is mentioned by M. Lazowski in his Tour through Switzerland: it is as yet but in an imperfect state, and was lately diseovered there by accident. It seems that a clergyman, though near-sighted, often amused himself with firing at a mark, and contrived to stretch a wire so as to draw the mark to him to see how he had aimed. He observed that the wire sometimes sounded as if it vibrated like a musical cord; and that after such soundings, a change always ensued in the state of the atmosphere; from whence he came to predict rain or sine weather. On making farther experiments, it was found that the sounds were most distinct when extended in the plane of the meridian. And according to the weather which was to follow, it was found that the sounds were more or less soft, or more or less continued; also fine weather, it is said, was announced by the tones of counter-tenor, and rain by those of bass. It has been said that M. Volta mounted 15 chords in this way at Pavia, to bring this method to some precision, but no accounts have yet appeared of the success of his observations.</p><p><hi rend="italics">The Phænomena and Observations of the Barometer</hi><note anchored="true" place="unspecified">An ingenious author observes that, by means of barometers we may regain the knowledge that still resides, in brutes, and which we forfeited by not continuing in the open air, as they mostly do; and, by our intemperance, corrupting the <hi rend="italics">crasis</hi> of our organs of sense.</note> The phænomena of the barometer are various; but authors are not yet agreed upon the causes of them; nor is the use of it, as a weather-glass, yet perfectly ascertained, though daily observations and experience lead us still nearer to precision. Mr. Boyle observes that the phænomena of the barometer are so precarious, that it is exceedingly difficult to form any certain general rules concerning the rise and fall of the mercury. Even that rule fails which seems to hold the most generally, viz, that the mercury is low in high winds. The best rules however that have been deduced by several authors are as follow.</p><p><hi rend="italics">Dr Halley's Rules for judging of the Weather.</hi></p><p>1. In calm weather, when the air is inclined to rain, the mercury is commonly low.</p><p>2. In serene, good, and settled weather, the mercury is generally high.</p><p>3. Upon very great winds, though they be not accompanied with rain, the mercury sinks lowest of all, according to the point of the compass the wind blows from. <pb n="191"/><cb/></p><p>4. The greatest heights of the mercury are found upon easterly or north-easterly winds, other circumstances alike.</p><p>5. In calm frosty weather, the mercury commonly stands high.</p><p>6. After very great storms of wind, when the mercury has been very low, it generally rises again very fast.</p><p>7. The more northerly places have greater alterations of the barometer than the more southerly, near the equator.</p><p>8. Within the tropics, and near them, there is little or no variation of the barometer, in all weathers. For instance, at St. Helena it is little or nothing, at Jamaica 3-10ths of an inch, and at Naples the variation hardly ever exceeds an inch; whereas in England it amounts to 2 inches and a half, and at Petersburgh to 3 1/3 nearly.</p><p><hi rend="italics">Dr Beal,</hi> who followed the opinion of M. Pascal, observes that, <hi rend="italics">cæteris paribus,</hi> the mercury is higher in cold weather than in warm: and in the morning and evening usually higher than at mid-day.—That in settled and fair weather, the mercury is higher than either a little before or after, or in the rain; and that it commonly descends lower after rain than it was before it. And he ascribes these effects to the vapours with which the air is charged in the former case, and which are dispersed by the falling rain in the latter. If it chance to rise higher after rain, it is usually followed by a settled serenity. And that there are often great changes in the air, without any perceptible alteration in the barometer.</p><p><hi rend="italics">Mr Patrick's Rules for judging of the Weather.</hi> These are esteemed the best of any general rules hitherto made:</p><p>1. The rising of the mercury presages, in general, fair weather; and its falling, foul weather, as rain, snow, high winds, and storms.</p><p>2. In very hot weather, the falling of the mercury indicates thunder.</p><p>3. In winter, the rising presages frost: and in frosty weather, if the mercury falls 3 or 4 divisions, there will certainly follow a thaw. But in a continued frost, if the mercury rises, it will certainly snow.</p><p>4. When foul weather happens soon after the falling of the mercury, expect but little of it; and on the contrary, expect but little fair weather when it proves fair shortly after the mercury has risen.</p><p>5. In foul weather, when the mercury rises much and high, and fo continues for 2 or 3 days before the foul weather is quite over, then expect a continuance of fair weather to follow.</p><p>6. In fair weather, when the mercury falls much and low, and thus continues for 2 or 3 days before the rain comes; then expect a great deal of wet, and probably high winds.</p><p>7. The unsettled motion of the mercury, denotes uncertain and changeable weather.</p><p>8. You are not so strictly to observe the words engraved on the plates, as the mercury's rising and falling; though in general it will agree with them. For if it stands at <hi rend="italics">much rain,</hi> and then rises up to <hi rend="italics">changeable,</hi> it presages fair weather; though not to continue so long as if the mercury had risen higher. And so, on <cb/> the contrary, if the mercury stood at <hi rend="italics">fair,</hi> and falls to <hi rend="italics">changeable,</hi> it presages foul weather; though not so much of it as if it had sunk lower.</p><p>Upon these rules of Mr Patrick, the following <hi rend="italics">Remarks</hi> are made by <hi rend="italics">Mr Rowning.</hi> That it is not so much the absolute height of the mercury in the tube that indicates the weather, as its motion up and down: wherefore, to pass a right judgment of what weather is to be expected, we ought to know whether the mercury is actually rising or salling; to which end the following rules are of use.</p><p>1. If the surface of the mercury is convex, standing higher in the middle of the tube than at the sides, it is a sign that the mercury is then rising.</p><p>2. But if the surface be concave, or hollow in the middle, it is then sinking. And,</p><p>3. If it be plain, or rather a very little convex, the mercury is stationary: for mercury being put into a glass tube, especially a small one, naturally has its surface a little convex, because the particles of mercury attract one another more forcibly than they are attracted by glass. Farther,</p><p>4. If the glass be small, shake the tube; then if the air be grown heavier, the mercury will rise about half a 10th of an inch higher than it stood before; but if it be grown lighter, it will sink as much. And, it may added, in the wheel or circular barometer, tap the instrument gently with the singer, and the index will visibly start forwards or backwards according to the tendency to rise or fall at that time. This proceeds from the mercury's sticking to the sides of the tube, which prevents the free motion of it till it be disengaged by the shock: and therefore when an observation is to be made with such a tube, it ought to be first shaken; for sometimes the mercury will not vary of its own accord, till the weather is present which it ought to have indicated.</p><p>And to the foregoing may be added the following additional rules, more accurately drawn from later and more clofe observation of the motions of the barometer, and the consequent changes in the air in this country.</p><p>1. In winter, spring, and autumn, the sudden falling of the mercury, and that for a large space, denotes high winds and storms; but in summer it denotes heavy showers, and often thunder: and it always sinks lowest of all for great winds, though not accompanied with rain; though it falls more for wind and rain together than for either of them alone. Also, if, after rain, the wind change into any part of the north, with a clear and dry sky, and the mercury rise, it is a certain sign of fair weather.</p><p>2. After very great storms of wind, when the mercury has bein low, it commonly rises again very fast. In settled sair and dry weather, except the barometer fink much, expect but little rain; for its small sinking then, is only for a little wind, or a few drops of rain; and the mercury soon rises again to its former station. In a wet season, suppose in hay-time and harvest, the smallest sinking of the mercury must be minded; for when the constitution of the air is much inclined to showers, a little sinking in the barometer then denotes more rain, as it never then stands very high. And, if <pb n="192"/><cb/> in such a season, it rise suddenly, very fast, and high, expect not fair weather more than a day or two, but rather that the mercury will fall again very soon, and rain immediately to follow: the slow gradual rising, and keeping on for 2 or 3 days, being most to be depended on for a week's fair weather. And the unsettled state of the quicksilver always denoting uncertain and changeable weather, especially when the mercury stands any where about the word <hi rend="italics">changeable</hi> on the scale.</p><p>3. The greatest heights of the mercury, in this country, are found upon easterly and north-easterly winds; and it may often rain or snow, the wind being in these points, and the barometer sink little or none, or it may even be in a rising state, the effect of those winds counteracting. But the mercury sinks for wind, as well as rain, in all the other points of the compass; but rises as the wind shifts about to the north or east, or between those points: but if the barometer should sink with the wind in that quarter, expect it soon to change from thence; or else, should the fall of the mercury be much, a heavy rain is then likely to ensue, as it sometimes happens. <hi rend="center"><hi rend="italics">Cause of the Phænomena of the Barometer.</hi></hi></p><p>To account for the foregoing phænomena of the barometer, many hypotheses have been framed, which may be reduced to two general heads, viz, <hi rend="italics">mechanical</hi> and <hi rend="italics">chemical.</hi> The chief writers upon these causes, are Pascal, Beal, Wallis, Garcin, Garden, Lister, Halley, Garsten, De la Hire, Mariotte, Le Cat, Woodward, Leibnitz, De Mairan, Hamberger, D. Bernoulli, Muschenbrock, Chambers, De Luc, Black, &c; and an account of most of their hypotheses may be seen at large in M. De Luc's <hi rend="italics">Recherches sur les Modifications de l'Atmosphere,</hi> vol. 1. chap. 3; see also the <hi rend="italics">Philos. Trans.</hi> and various other works on this subject. It may suffice to notice here slightly a few of the principal of them.</p><p><hi rend="italics">Dr. Lister</hi> accounts for the changes of the barometer from the alterations by heat and cold in the mercury itself; contracting by cold, and expanding by heat. But this, it is now well known, is quite insufficient to account for the whole of the effect.</p><p>The changes in the weight or pressure of the atmosphere must theresore be regarded as the principal cause of those in the barometer. But then, the difficulty will be to assign the cause of that cause, or whence arise those alterations that take place in the atmosphere, which are sometimes so great as to alter its pressure by the 10th part of the whole quantity. It is probable that the winds, as driven about in different directions, have a great share in them; vapours and exhalations, rising from the earth, may also have some share; and some perhaps the flux and reflux occasioned in the air by the moon; as well as some chemical causes operating between the different particles of matter.</p><p>Dr. Halley thinks the winds and exhalations sufficient; and on this principle gives a theory, the substance of which may be comprised in what follows:</p><p>1st, That the winds must alter the weight of the air in any particular country; and this, either by bringing together a greater quantity of air, and so load- <cb/> ing the atmosphere of any place; which will be the case as often as two winds blow from opposite parts, at the same time, towards the same point: or by sweeping away some part of the air, and giving room for the atmosphere to expand itself; which will happen when two winds blow opposite ways from the same point at the same time: or lastly by cutting off the perpendicular pressure of the air; which is the case when a single wind blows briskly any way; it being found by experience, that a strong blast of wind, even made by art, will render the atmosphere lighter; and hence the mercury in a tube below it, as well as in others more distant, will considerably subside. See <hi rend="italics">Philos. Trans.</hi> N° 292.</p><p>2dly, That the cold nitrous particles, and even the air itself condensed in the northern regions, and driven elsewhere, must load the atmosphere, and increase its pressure.</p><p>3dly, That heavy dry exhalations from the earth must increase the weight of the atmosphere, as well as its elastic force; as we find the specific gravity of menstruums increased by dissolved salts and metals.</p><p>4thly, That the air being rendered heavier by these and the like causes, is thence better able to support the vapours; which being likewise intimately mixed with it, make the weather serene and fair. Again, the air being made lighter from the contrary causes, it becomes unable to support the vapours with which it is replete; these therefore precipitating, are collected into clouds, the particles of which in their progress unite into drops of rain.</p><p>Hence he infers, it is evident that the same causes which increase the weight of the air, and render it more able to support the mercury in the barometer, do likewise produce a serene sky, and a dry season; and that the same causes which render the air lighter, and less able to support the mercury, do likewise generate clouds and rain.</p><p>But these principles, though well adapted to many of the particular cases of the barometer, seem however to fall short of some of the principal and most obvious ones, besides being liable to several objections.</p><p><hi rend="italics">Leibnitz</hi> accounted for the fall of the mercury before rain by another principle, viz, That as a body specifically lighter than a fluid, while it is sustained by it, adds more weight to that fluid than when, by being reduced in bulk, it becomes specifically heavier, and descends; so the vapour, after it is reduced into the form of clouds, and descends, adds less weight to the air than it did before; and hence the mercury sinks in the tube.—But here, granting that the drops of rain formed from the vapours always increasing in size as they fall lower, were continually accelerated also in their motion, and so the air suffer a continued loss of their weight as they descend; it may however be objected, that by the descent of the mercury the rain is foretold a much longer time before it comes, than the vapour can be supposed to take up in falling: that many times, and in different places, there falls a great deal of rain, without any sinking of the mercury at all; as also that there often happens a fall of the mercury without any rain ensuing: and that sometimes the mercury will suddenly sink, in a short space <pb n="193"/><cb/> of time, half an inch or more, which answers to 7 inches of rain, or about one third of the whole quantity falling in the whole year.</p><p>Mr. <hi rend="italics">De Luc</hi> supposes that the changes observed in the pressure of the atmosphere, are chiefly produced by the greater or less quantity of vapours floating in it: as others have attributed them to the same cause, but have given a different explanation of it. His opinion is, that vapours diminish the specific gravity, and consequently the absolute weight, of those columns of the atmosphere into which they are received, and which, notwithstanding this admixture, still remain of the same height with adjoining columns that consist of pure or dry air. He afterwards vindicates and more fully explains this theory, and applies it to the solution of the principal phenomena of the barometer, as depending on the varying denfity and weight of the atmosphere.</p><p>Dr. <hi rend="italics">James Hutton,</hi> in his <hi rend="italics">Theory of Rain,</hi> printed in the Transactions of the Royal Society of Edinburgh, vol. 1, gives ingenious and plausible reasons for thinking that the lessening the weight of the atmosphere by the fall of rain, is not the cause of the fall of the barometer; but that the principal, if not the only cause, arises from the commotions in the atmosphere, which are chiefly produced by sudden changes of heat and cold in the air. “The barometer, says he, is an instrument necessarily connected with motions in the atmosphere; but it is not equally affected with every motion in that fluid body. The barometer is chiefly affected by those motions by which there are produced accumulations and abstractions of this fluid, in places or regions of sufficient extent to affect the pressure of the atmosphere upon the surface of the earth. But as every commotion in the atmosphere may, under proper conditions, be a cause for rain, and as the want of commotion in the atmosphere is naturally a cause of fair weather, this instrument may be made of great importance for the purpose of meteorological observations, although not in the certain and more simple manner in which it has been, with the increase of science, so successfully applied to the measuring of heights.” See <hi rend="smallcaps">Rain.</hi></p><p>In the <hi rend="italics">Encyclopædia Britannica</hi> there is another theory of the changes in the barometer, as depending on the <hi rend="italics">heat</hi> in the atmosphere, not as producing commotions there, but as altering the specific gravity of the air by the changes of heat and cold. The principles of this theory are, 1st, That vapour is formed by an intimate union between the elements of fire and water, by which the fire or heat is so totally enveloped, and its action so perfectly suspended by the aqueous particles, that it not only loses its properties of burning and of giving light, but becomes incapable of affecting the most sensible thermometer, in which case it is said to be in a latent state: and 2d, That if the atmosphere be affected by any unusual degree of heat, it thence becomes incapable of supporting so long a column of mercury as before; for which reason it is that the barometer sinks.</p><p>From these axioms it would follow, that as vapour is formed by an union of fire with water, whether by attraction or a solution of the water in the fire, the vapour cannot be condensed till this union, attraction, <cb/> or solution, is at an end. Hence the beginning of the condensation of the vapour, or the first signs of approaching rain, must be the separation of the fire which is latent in the vapour. In the beginning, this may be either slow and partial, or it may be sudden and violent: in the first case, the rain will come on slowly, and after a considerable time; but in the other, it will come very quickly, and in a great quantity. But Dr. Black has proved, that when fire quits its latent state, however long it may have lain dormant and insensible, it always reassumes its proper qualities, and affects the thermometer just the same as if it had never been absorbed. The consequence of this is, that in proportion as the latent heat is discharged from the vapour, those parts of the atmosphere into which it is discharged must be sensibly affected by it; and in proportion to the heat communicated to those parts, they will become specifically lighter, and the mercury will sink of course.</p><p>In the <hi rend="italics">Memoirs of the Literary Society of Manchester,</hi> vol. 4, is also a curious paper on this subject, viz, <hi rend="italics">Metevrological Observations made on different Parts of the Western Coast of Great Britain: arranged by T. Garnett, M. D.</hi> This paper is composed of materials furnished by several observers; those of Mr. Copland, surgeon at Dumfries, are of special importance. This gentleman is of opinion that the changes of the barometer indicate approaching hot and cold weather, with much more certainty than dry and wet. “Every remarkable elevation of the barometer, says he, where it is of any duration, is followed by very warm or dry weather, and moderate as to wind, or by all of them; but heat seems to have most influence and connexion; and when it is deficient, the continuance of the other two will be longer and more remarkable; therefore the calculation must be in a compound ratio of the excess and deficiency of the heat, and of the dryness of the weather in comparison of the medium of the sea son; and with regard to the want of strong wind, it appears to be intimately connected with the last, as they shew that no precipitation is going on in any of the neighbouring regions.” <hi rend="center">In his 14th and 15th remarks, he had said,</hi></p><p>‘14th, That the barometer being lower, and continuing so longer than what can be accounted for by immediate falls, or stormy weather, indicates the approach of very cold weather for the season; and also, cold weather, though dry, is always accompanied by a low barometer, till near its termination.’</p><p>‘15th, That warm weather is always preceded and mostly accompanied by a high barometer; and the rising of the barometer in the time of broken or cold weather, is a sign of the approach of warmer weather: and also if the wind is in any of the cold points, a sudden rise of the barometer indicates the approach of a southerly wind, which in winter generally brings rain with it.’</p><p>In the two following remarks, Mr. Copland had explained certain phenomena from a principle similar to that on which Dr. Darwin has so much insisted: (Betanic Garden, I. notes p. 79, &c.)</p><p>‘That the falling of the barometer may proceed from a decomposition of the atmosphere occurring a- <pb n="194"/><cb/> round or near that part of the globe where we are placed, which will occasion the electricity of the atmosphere to be repelled upwards in fine lambent portions; or driven downwards or upwards in more compacted balls of fire; or lastly, to be carried along with the rain, &c, in an imperceptible manner to the surface of the earth: the precipitation of the watery parts generally very soon takes place, which diminishes the real gravity of the atmosphere, and also by the decomposition of some of the more active parts, the air loses part of that elastic and repulsive power which it so eminently possessed, and will therefore press with less force on the mercury of the barometer than before, by which means a fall ensues.</p><p>‘That the cause of the currents of air, or winds, may also be this way accounted for: and in very severe storms, where great decompositions of the atmosphere take place, this is particularly evident, such as generally occur in one or more of the West India islands at one time, a great loss of real gravity, together with a considerable diminution of the spring of the air immediately ensues; hence a current commences, first in that direction whence the air has most gravity, or is most disposed to undergo such a change; but it being soon relieved of its superior weight or spring on that side, by the decomposition going on as fast as the wind arrives on the island, it immediately veers to another point, which then rushes in mostly with an increase of force; thus it goes on till it has blown more than half way round the points of the compass during the continuation of the hurricane. For in this manner the West India phenomena, as well as the alteration of the wind during heavy rains in this country, can only be properly accounted for.’ See remark No. 4.</p><p>Mr. C.'s 4th aphorism is, ‘That the heaviest rains, when of long continuance, generally beign with the wind blowing easterly, when it gradually veers round to the south; and that the rain does not then begin to cease till the wind has got to the west, or rather a little to the northward of it, when, it may be added, it commonly blows with some violence.’</p><p>Many other observations on the barometer, the weather, &c. may be seen in various parts of the Philos. Trans. And for other curious papers on the same, and other subjects connected with the barometer, see the Gentleman's Magazine for 1789, p. 317; also Greu's Journal of Nat. Philos. printed at Leipzig 1792, for the influence of the sun and moon upon the barometer. <hi rend="center"><hi rend="italics">The Barometer applied to the measuring of Altitudes.</hi></hi></p><p>The secondary character of the barometer, namely as an instrument for measuring accessible heights or depths, was first proposed by Pascal, and Descartes, as has been before observed; and succeeding philosophers have been at great pains to ascertain the proportion between the fall of the barometer and the height to which it is carried; as Halley, Mariotte, Maraldi, Scheuchzer, J. Cassini, D. Bernoulli, Horrebow, Bouguer, Shuckburgh, Roy, and more especially by De Luc, who has given a critical and historical detail of most of the attempts that have at different times been made for applying the motion of the mercury in the barometer to the measurement of accessible heights. <cb/> And for this purpose serves the portable barometer, before described, (fig. 11 and 12, plate 4,) which should be made with all the accuracy possible. Various rules have been given by the writers on this subject, for computing the height ascended from the given fall of the mercury in the tube of the barometer, the most accurate of which was that of Dr. Halley, till it was rendered much more accurate by the indefatigable researches of De Luc, by introducing into it the corrections of the columns of mercury and air, on account of heat. And other corrections and modifications of the same may be seen inserted under the article A<hi rend="smallcaps">TMOSPHERE</hi>, where the most correct rule is deduced from one single experiment only. This rule is as follows: <hi rend="center"><hi rend="italics">The Rule for Computing Altitudes,</hi> is this,</hi></p><p>Viz, 10000 X log. of M/<hi rend="italics">m</hi> is the altitude in fathoms, in the mean temperature of 31°; and for every degree of the thermometer above that, the result must be increased by so many times its 435th part, and diminished when below it: in which theorem M denotes the length of the column of mercury in the barometer tube at the bottom, and <hi rend="italics">m</hi> that at the top of the hill, or other eminence; which lengths may be expressed in any one and the same sort of measures, whether feet, or inches, or tenths, &c, and either English, or French, or of any other nation; but the refult is always in fathoms, of 6 English feet each.</p><p>And the <hi rend="italics">Precepts,</hi> in words, for the practice of measurements by the barometer, are these following:</p><p>1st, Observe the height of the barometer at the bottom of any height or depth, proposed to be measured; together with the temperature of the mercury by means of the thermometer attached to the barometer, and also the temperature of the air in the shade by another thermometer which is detached from the barometer.</p><p>2dly, Let the same thing be done also at the top of the said height or depth, and as near to the same time with the former as may be. And let those altitudes of mercury be reduced to the same temperature, if it be thought necessary, by correcting either the one or the other, viz, augmenting the height of the mercury in the colder temperature, or diminishing that in the warmer, by its 9600th part for every degree of difference between the two; and the altitudes of mercury so corrected, are what are denoted by M and <hi rend="italics">m,</hi> in the algebraic formula above.</p><p>3dly, Take out the common logarithms of the two heights of mercury, so corrected, and subtract the less from the greater, cutting off from the right hand side of the remainder three places for decimals; so shall those on the left be fathoms in whole numbers, the tables of logarithms being understood to be such as have 7 places of decimals.</p><p>4thly, Correct the number last found, for the difference of the temperature of the air, as follows: viz, Take half the sum of the two temperatures of the air, shewn by the detached thermometers, for the mean one; and sor every degree which this differs from the standard temperature of 31°, take so many times the 435th part of the fathoms above found, and add them if the mean <pb n="195"/><cb/> temperature be more than 31°, but subtract them if it be below 31°; so shall the sum or difference be the true altitude in fathoms, or being multiplied by 6, it will give the true altitude in English feet.</p><p><hi rend="italics">Example</hi> 1. Let the state of the barometers and thermometers be as follows, to find the altitude: viz. <figure/></p><p><hi rend="italics">Example</hi> 2. To sind the altitude of a hill, when the state of the barometer and thermometer, as observed at the bottom and top of it, is as follows; viz, <figure/></p><p>See this rule investigated under the article P<hi rend="smallcaps">NEUMATICS</hi>, at the end.</p><p>N. B. The mean height of the barometer in London, upon an average of two observations in every day of the year, kept at the house of the Royal Society, for many years past, is 29.88; the medium temperature, or height of the thermometer, according to the same, being 58°. But the medium height at the surface of the sea, according to Sir Geo. Shuckburgh (Philos. Trans. 1777, p. 586) is 30.04 inches, the heat of the barometer being 55°, and of the air 62°.</p></div1><div1 part="N" n="BAROSCOPE" org="uniform" sample="complete" type="entry"><head>BAROSCOPE</head><p>, a machine for shewing the alterations in the weight or pressure of the atmosphere. See <hi rend="smallcaps">Barometer.</hi></p></div1><div1 part="N" n="BARREL" org="uniform" sample="complete" type="entry"><head>BARREL</head><p>, an English vessel or cask, containing 36 gallons of beer measure, or 32 gallons of ale measure. The barrel of beer, vinegar, or of liquor preparing for vinegar, ought to contain 34 gallons, according to the standard of the ale quart. <cb/></p><div2 part="N" n="Barrel" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Barrel</hi></head><p>, in Clock-work, is the cylinder about which the spring is wrapped.</p></div2></div1><div1 part="N" n="BARRICADE" org="uniform" sample="complete" type="entry"><head>BARRICADE</head><p>, or <hi rend="smallcaps">Barricado</hi>, a military term for a fence, or retrenchment, hastily made with vessels, or baskets of earth, carts, trees, stakes, or the like, to preserve an army from the shot or assault of an enemy.</p></div1><div1 part="N" n="BARRIER" org="uniform" sample="complete" type="entry"><head>BARRIER</head><p>, a kind of fence made at a passage, retrenchment, gate, or such like, to stop it up against an enemy.</p></div1><div1 part="N" n="BARROW" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BARROW</surname> (<foreName full="yes"><hi rend="smallcaps">Isaac</hi></foreName>)</persName></head><p>, a very eminent mathematician and divine of the 17th century, was born at London in October, 1630, being the son of Thomas Barrow, then a linen-draper of that city, but descended from an ancient family in Sussolk. He was at first placed at the Charter-house school for two or three years; where his behaviour afforded but little hopes of success in the profession of a scholar, being fond of fighting, and promoting it among his school-sellows: but being removed to Felsted in Essex, his disposition took a different turn; and having soon made a great progress in learning, he was first admitted a pensioner of Peter House in Cambridge; but when he came to join the university, in Feb. 1645, he was entered at Trinity college. He now a<*>plied himself with great diligence to the study of all parts of literature, especially natural philosophy. He afterward turned his attention to the profession of physic, and made a considerable progress in anatomy, botany, and chemistry: he next studied divinity; then chronology, astronomy, geometry, and the other branches of the mathematics; with what success, his writings afterwards most eminently shewed.</p><p>When Dr. Duport resigned the chair of Greek professor, he recommended his pupil Mr. Barrow for his successor, who, in his probation exercise, shewed himself equal to the character that had been given him by this gentleman; but being suspected of favouring Arminianism, he was not preferred. This disappointment it seems determined him to quit the college, and visit foreign countries; but his finances were so low, that he was obliged to dispose of his books, to enable him to execute that design.</p><p>He left England in June 1655, and visited France, Italy, Turkey, &c. At several places, in the course of this tour, he met with kindness and liberal assistance from the English ambassadors, &c, which enabled him to benefit the more from it, by protracting his stay, and prolonging his journey. He spent more than a year in Turkey, and returned to England by way of Venice, Germany, and Holland, in 1659. At Constantinople he read over the works of St. Chrysostom, once bishop of that see, whom he preferred to all the other fathers.</p><p>On his return home Barrow was episcopally ordained by bishop Brownrig; and in 1660, he was chosen to the Greek professorship at Cambridge. In July 1662, he was elected professor of geometry in Gresham college: in which station, he not only discharged his own duty, but supplied likewise the absence of Dr. Pope the astronomy professor. Among his lectures, some were upon the projection of the sphere and perspective, which are lost; but his Latin oration, previous to his lectures, is still extant. About this time Mr. Barrow was offered a good living; but the condition annexed, of teaching the patron's son, made him refuse it, as thinking it too <pb n="196"/><cb/> like a simonial contract. Upon the 20th of May 1663 he was elected a fellow of the Royal Society, in the first choice made by the council after their charter. The same year the executors of Mr. Lucas having, according to his appointment, founded a mathematical lecture at Cambridge, they selected Mr. Barrow for the first professor; and though his two professorships were not incompatible with each other, he chose to resign that of Gresham-college, which he did May the 20th, 1664. In 1669 he resigned the mathematical chair to his learned friend Mr. Isaac Newton, being now determined to quit the study of mathematics for that of divinity. On quitting his professorship, he had only his fellowship of Trinity-college, till his uncle gave him a small sinecure in Wales, and Dr. Seth Ward bishop of Salisbury conferred upon him a prebend in his church. In the year 1670 he was created doctor in divinity by mandate; and, upon the promotion of Dr. Pearson master of Trinity college to the see of Chester, he was appointed to succeed him by the king's patent bearing date the 13th of February 1672: upon which occasion the king was pleased to say, “he had given it to the best scholar in England.” In this, his majesty did not speak from report, but from his own knowledge; the doctor being then his chaplain, he used often to converse with him, and, in his humourous way, to call him an “unfair preacher,” because he exhausted every subject, and left no room for others to come after him. In 1675 he was chosen vice-chancellor of the university; and he omitted no endeavours for the good of that society, nor in the line of his profession as a divine, for the promotion of piety and virtue; but his useful labours were abruptly terminated by a fever on the 4th of May 1677, in the 47th year of his age. He was interred in Westminster abbey, where a monument, adorned with his bust, was soon after erected, by the contribution of his friends.</p><p>Dr. Barrow's works are very numerous, and indeed various, mathematical, theological, poetical, &c, and such as do honour to the English nation. They are principally as follow:</p><p>1. Euclidis Elementa. Cantab. 1655, in 8vo.</p><p>2. Euclidis Data. Cantab. 1657, in 8vo.</p><p>3. Lectiones Opticæ xviii, Lond. 1669, 4to.</p><p>4. Lectiones Geometricæ xiii, Lond. 1670, 4to.</p><p>5. Arehimedis Opera, Apollonii Conicorum libri iv, Theodosii Sphericorum lib. iii; nova methodo illustrata, et succincte demonstrata. Lond. 1675, in 4to.</p><p>The following were published after his decease, viz:</p><p>6. Lectio, in qua theoremata Archimedis de sphæra et cylindro per methodum indivisibilium investigata, ac breviter investigata, exhibentur. Lond. 1678, 12mo.</p><p>7. Mathematicæ Lectiones habitæ in scholis publicis academiæ Cantabrigiensis, an. 1664, 5, 6, &c. Lond. 1683.</p><p>8. All his English works in 3 volumes, Lond. 1683, folio.—These are all theological, and were published by Dr. John Tillotson.</p><p>9. Isaaci Barrow Opuscula, viz, Determinationes, Conciones ad Clerum, Orationes, Poemata, &c. volumen quartum. Lond. 1687, folio.</p><p>Dr. Barrow left also several curious papers on mathematieal subjects, written in his own hand, which were <cb/> communicated by Mr. Jones to the author of “The Lives of the Gresham Professors,” a particular account of which may be seen in that book, in the Life of Barrow.</p><p>Several of his works have been translated into English, and published; as the Elements and Data of Euclid; the Geometrical Lectures, the Mathematical Lectures. And accounts of some of them were also given in several volumes of the Philos. Trans.</p><p>Dr. Barrow must ever be esteemed, in all the subjects which exercised his pen, a person of the clearest perception, the finest fancy, the soundest judgment, the profoundest thought, and the closest and most nervous reasoning. “The name of Dr. Barrow (says the learned Mr. Granger) will ever be illustrious for a strength of mind and a compass of knowledge that did honour to his country. He was unrivalled in mathematical learning, and especially in the sublime geometry; in which he has been excelled only by his successor Newton. The same genius that seemed to be born only to bring hidden truths to light, and to rise to the heights or descend to the depths of science, would sometimes amuse itself in the flowery paths of poetry, and he composed verses both in Greek and Latin. He at length gave himself up entirely to divinity; and particularly to the most useful part of it, that which has a tendency to make men wiser and better.”</p><p>Several good anecdotes are told of Barrow, as well of his great integrity, as of his wit, and bold intrepid spirit and strength of body. His early attachment to fighting when a boy is some indication of the latter; to which may be added the two following anecdotes: In his voyage between Leghorn and Smyrna the ship was attacked by an Algerine pirate, which after a stout resistance they compelled to sheer off, Barrow keeping his post at the gun assigned him to the last. And when Dr. Pope in their conversation asked him, “Why he did not go down into the hold, and leave the defence of the ship to those, to whom it did belong? He replied, It concerned no man more than myself: I would rather have lost my life, than to have fallen into the hands of those merciless infidels.”</p><p>There is another anecdote told of him, which shewed not only his intrepidity, but an uncommon goodness of disposition, in circumstances where an ordinary share of it would have been probably extinguished. Being once on a visit at a gentleman's house in the country, where the necessary was at the end of a long garden, and consequently at a great distance from the room where he lodged; as he was going to it before day, for he was a very early riser, a fierce mastiff, that used to be chained up all day, and let loose at night for the security of the house, perceiving a strange person in the garden at that unusual time, set upon him with great fury. The doctor caught him by the throat, grappled with him, and, throwing him down, lay upon him: once he had a mind to kill him; but he altered his resolution, on recollecting that this would be unjust, since the dog did only his duty, and he himself was in fault for rambling out of his room before it was light. At length he called out so loud, that he was heard by some of the family, who came presently out, and freed the doctor and the dog from the danger they both had been in. <pb n="197"/><cb/></p><p>Among other instances of his wit and vivacity, they relate the following rencontre between him and that wicked wit lord Rochester. These two meeting one day at the court, while the doctor was king's chaplain in ordinary, Rochester, thinking to banter him, with a flippant air, and a low formal bow, accosted him with, “Doctor, I am yours to my shoe-tie:” Barrow perceiving his drift, and determined upon defending himself, returned the salute, with, “My lord, I am yours to the ground.” Rochester, on this, improving his blow, quickly returned it, with, “Doctor, I am yours to the centre;” which was as smartly followed up by Barrow, with, “My lord, I am yours to the antipodes.” Upon which, Rochester, disdaining to be foiled by a musty old piece of divinity, as he used to call him, exclaimed, “Doctor, I am yours to the lowest pit of hell;” upon which Barrow, turning upon his heel, with a farcastic fmile, archly replied, “There, my lord, I leave you.”</p></div1><div1 part="N" n="BARS" org="uniform" sample="complete" type="entry"><head>BARS</head><p>, in Music, are the spaces quite through any composition, separated by upright lines drawn across the five horizontal lines, each of which either contains the fame number of notes of the same kind, or so many other notes as will make up a like interval of time; for all the bars, in any piece, must be of the same length, and played in the same time.</p></div1><div1 part="N" n="BARTER" org="uniform" sample="complete" type="entry"><head>BARTER</head><p>, or <hi rend="smallcaps">Truck</hi>, is the exchanging of one commodity for another; and forms a rule in the commercial part of arithmetic, by which the commodities are properly calculated and equalled, by computing first the value of the commodity which is given, and then the quantity of the other which will amount to the same sum.</p></div1><div1 part="N" n="BASE" org="uniform" sample="complete" type="entry"><head>BASE</head><p>, <hi rend="smallcaps">Basis</hi>, in Architecture, denotes the lower part of a column or pedestal.</p><div2 part="N" n="Base" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Base</hi></head><p>, in Geometry, the lowest side of any figure. Any side of a figure may be considered as its base, according to the position in which it may be conceived as standing; but commonly it is understood of the lowest side: as the base of a triangle, of a cone, cylinder, &c.</p><p><hi rend="smallcaps">Base Line</hi>, in Perspective, denotes the common section of the picture and the geometrical plane.</p><p><hi rend="smallcaps">Base Ring</hi>, of a Cannon, is the great ring next behind the vent or touch-hole.</p></div2><div2 part="N" n="Base" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Base</hi></head><p>, <hi rend="italics">alternate.</hi> See <hi rend="smallcaps">Alternate.</hi></p></div2></div1><div1 part="N" n="BASEMENT" org="uniform" sample="complete" type="entry"><head>BASEMENT</head><p>, in Architecture, a continued base, extended a considerable length, as about a house, a room, or other piece of building.</p></div1><div1 part="N" n="BASILIC" org="uniform" sample="complete" type="entry"><head>BASILIC</head><p>, in the ancient Architecture, was a large hall, or court of judicature, where the magistrates sat to administer justice.</p></div1><div1 part="N" n="BASILICA" org="uniform" sample="complete" type="entry"><head>BASILICA</head><p>, or <hi rend="smallcaps">Basilicus</hi>, the same as <hi rend="italics">Regulus,</hi> or <hi rend="italics">Cor Leonis,</hi> being a fixed star of the first magnitude in the constellation <hi rend="italics">Leo.</hi></p></div1><div1 part="N" n="BASILISK" org="uniform" sample="complete" type="entry"><head>BASILISK</head><p>, in the older Artillery, was a large piece of ordnance so called from its resemblance to the supposed serpent of that name. It threw an iron ball of 200 pounds weight; and was in great repute in the time of Solyman emperor of the Turks, in the wars of Hungary; but it is now grown out of use in most parts of Europe. Paulus Jovius relates the terrible slaughter made in a Spanish ship by a single ball from one of these basilisks; after passing through the beams and planks in the ship's head, it killed upwards of 30 men. And Maffeus speaks of basilisks made of brass, each os which <cb/> required 100 yoke of oxen to draw them.—More modern writers also give the name basilisk to a much smaller and sizeable piece of ordnance, made of 15 feet long by the Dutch, but of only 10 by the French, and carrying a ball of 48 pounds. The largest size of cannon now used by the English, are the 32 pounders.</p></div1><div1 part="N" n="BASIS" org="uniform" sample="complete" type="entry"><head>BASIS</head><p>, in Geometry, the same as <hi rend="smallcaps">Base.</hi></p></div1><div1 part="N" n="BASS" org="uniform" sample="complete" type="entry"><head>BASS</head><p>, the lowest in the four parts of music; by some esteemed the basis and principal part of all, and by others as scarcely necessary in some tunes.</p></div1><div1 part="N" n="BASSANTIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BASSANTIN</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, a Scotch astronomer of the 16th century, born in the reign of James the 4th of Scotland. He was a son of the Laird of Bassantin in the Merse. After finifhing his education at the university of Glasgow, he travelled through Germany and Italy, and then settled in the university of Paris, where he taught mathematics with great applause. Having acquired some property in this employment, he returned to Scotland in 1562, where he died 6 years after.</p><p>From his writings it appears he was no inconsiderable astronomer, for the age he lived in; but, according to the fashion of the times, he was not a little addicted to judicial astrology. It was doubtless to our author that Sir James Melvil alludes in his Memoirs, when he says that his brother Sir Robert, when he was using his endeavours to reconcile the two queens Elizabeth and Mary, met with one Bassantin a man learned in the high sciences, who told him, “that all his travel would be in vain; for, said he, they will never meet together; and next, there will never be any thing but dissembling and secret hatred for a while, and at length captivity and utter wreck to our queen from England.” He added, “that the kingdom of England at length shall fall, of right, to the crown of Scotland: but it shall cost many bloody battles; and the Spaniards shall be helpers, and take a part to themselves for their labour.” A prediction in which Bassantin partly guessed right, which it is likely he was enabled to do from a judicious consideration of probable circumstances and appearances.</p><p>Bassantin's works are,</p><p>1. <hi rend="italics">Astronomia Jacubi Bassantini Scoti, opus absolutissimum, &c; ter. edit. Latine et Gallice.</hi> Genev. 1599, fol. This is the title given it by Tornœsius, who translated it into Latin from the French, in which language it was first published.</p><p>2. <hi rend="italics">Paraphrase de l'Astrolabe, avec une amplification de l'usage de l'Astrolabe.</hi> Lyons 1555. Paris 1617, 8vo.</p><p>3. <hi rend="italics">Mathematica Genethliaca.</hi></p><p>4. <hi rend="italics">Arithmetica.</hi></p><p>5. <hi rend="italics">Musica secundum Platonem.</hi></p><p>6. <hi rend="italics">De Mathesi in Genere.</hi></p></div1><div1 part="N" n="BASSOON" org="uniform" sample="complete" type="entry"><head>BASSOON</head><p>, a musical instrument of the wind kind, ferving for a bass to the haut-boy. It is blown with a reed, and furnished with eleven holes.</p><p>BASS-VIOL, a bass to the viol.</p></div1><div1 part="N" n="BASTION" org="uniform" sample="complete" type="entry"><head>BASTION</head><p>, in the modern fortification, a large mass of earth at the angles of a work, connecting the curtains to each other; and answers to the bulwark of the ancients. It is formed by two faces, two flanks, and two demigorges. The two faces form the saliant angle, or angle of the bastion; the two flanks form with the faces, the <hi rend="italics">epaules</hi> or shoulders; and the union of <pb n="198"/><cb/> the other two ends of the flanks with the curtains forms the two angles of the flanks. <figure/></p><p><hi rend="italics">Solid</hi> <hi rend="smallcaps">Bastion</hi>, are those that are entirely filled up with earth to the height of the rampart, without any void space towards the centre.</p><p><hi rend="italics">Void</hi> or <hi rend="italics">Hollow</hi> <hi rend="smallcaps">Bastion</hi>, has the rampart and parapet ranging only round the flanks and spaces, so that a void space is left within towards the centre, where the ground is so low that if the rampart be taken, no retrenchment can be made in the centre, but what will lie under the fire of the besieged.</p><p><hi rend="italics">Regular</hi> <hi rend="smallcaps">Bastion</hi>, is that which has its due proportion of faces, flanks, and gorges.</p><p><hi rend="italics">Deformed</hi> or <hi rend="italics">Irregular</hi> <hi rend="smallcaps">Bastion</hi>, is when the irregularity of the lines and angles throws the bastion out of shape: as when it wants one of the demigorges, one side of the interior polygon being too short, &c.</p><p><hi rend="italics">Demi</hi> <hi rend="smallcaps">Bastion</hi>, or <hi rend="italics">Half bastion,</hi> also otherwise called an <hi rend="italics">Epaulment,</hi> has but one face and flanlt.</p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Bastion</hi>, is when one bastion is raised within, and upon the plane of another baslion.</p><p><hi rend="italics">Flat</hi> <hi rend="smallcaps">Bastion</hi>, is one built in the middle of the curtain, when it is too long to be defended by the usual bastions at the extremities.</p><p><hi rend="italics">Composed</hi> <hi rend="smallcaps">Bastion</hi>, is when the two sides of the interior polygon are very unequal, which makes the gorges also unequal.</p><p><hi rend="italics">Cut</hi> <hi rend="smallcaps">Bastion</hi>, is that which has a re-entering angle at the point, and is sometimes called a <hi rend="smallcaps">Bastion</hi> <hi rend="italics">with a Tenaille,</hi> whose point is cut off, making an angle inwards, and two points outwards. This is used when the saliant angle would be too sharp, or when water or some other impediment prevents it from being carried out to its full extent.</p></div1><div1 part="N" n="BASTON" org="uniform" sample="complete" type="entry"><head>BASTON</head><p>, or <hi rend="smallcaps">Batoon</hi>, in Architecture, a moulding in the base of a column, called also a <hi rend="italics">Tore</hi> or <hi rend="italics">Torus.</hi></p></div1><div1 part="N" n="BATTEN" org="uniform" sample="complete" type="entry"><head>BATTEN</head><p>, a name given by workmen to a scantling or piece of wooden stuff, about an inch thick, and from 2 to 4 inches broad; of a considerable but indeterminate length.</p></div1><div1 part="N" n="BATTERING" org="uniform" sample="complete" type="entry"><head>BATTERING</head><p>, the attacking a place, work, or the like, with heavy artillery.</p><p><hi rend="smallcaps">Battering-Ram</hi>, a military engine used for beating down walls, before the invention of gunpowder and the modern artillery. It was no other than a long heavy beam of timber, armed with an iron head, something like the head of a ram. This being pushed violently with constant successive blows against a wall, gradually shakes it with a vibratory motion, till the stones are <cb/> disjointed and the wall falls down. There were several kinds of the battering-ram, the first rude and plain, which the soldiers carried in their arms by main force, and so struck the head of it against the wall. The second was slung by a rope about the middle to another beam lying across upon a couple of posts; which was the kind described by Josephus as used at the siege of Jerusalem. A third sort was covered over with a shell or screen of boards, to defend the men from the stones and darts of the besieged upon the walls, and thence called <hi rend="italics">testudo arietaria.</hi> And Felibien describes a fourth sort of battering-ram, which ran upon wheels; and was the most perfect and effectual of any.</p><p>Vitruvius affirms, that the battering-ram was sirst invented by the Carthaginians, while they laid siege to Cadiz: yet Pliny assures us, that the ram was invented or used at the siege of Troy; and that it was this that gave occasion to the fable of the wooden horse. In fact there can be no doubt but that the use of some sort of a battering-ram is as old as the art of war itself. And it has even been suspected that the walls of Jericho, mentioned in the book of Joshua, were beaten down by this instrument, the rams horns there mentioned, by means of which they were overthrown, being no other than the horns of the battering-rams. Pephasmenos, a Tyrian, afterwards contrived to suspend it with ropes; and lastly, Polydus, the Thessalian, mounted it on wheels, at the siege of Byzantium, under Philip of Macedon.</p><p>Plutarch relates, that Marc Anthony, in the Parthian war, made use of a ram 80 feet long: and Vitruvius affirms that they were sometimes 106, and even 120 feet long; which must have given an amazing force to this engine. The ram required 100 soldiers to work and manage it at one time; who being exhausted, another century relieved them; by which means in was kept playing continually without intermission. See fig. 2, plate V, which represents the battering-ram suspended in its open frame; in which 3 denotes the form of the head, fastened to the enormous beam 2, by three or four bands (4) of iron, of about four feet in breadth. At the extremity of each of these bands was an iron chain (5), one end of which was fastened to a hook (6), and to the last link at the other extremity was firmly bound a cable, which ran the whole length of the beam to the end of the ram 7, where these cables were bound all together as fast as possible with small ropes. To the end of these cables was fastened another, that consisted of several strong cords platted together to a certain length, and then running single (8), at each of which were placed several men, to balance and work the machine. 10 Is the chain or cable by which the ram was hung to the cross beam (11), fixed on the top of the frame; and 12 is the base of the machine.</p><p>The unsuspended ram differed from this only in the manner of working it; as it moved on small wheels upon another large beam, instead of being slung by a chain or cable.</p></div1><div1 part="N" n="BATTERY" org="uniform" sample="complete" type="entry"><head>BATTERY</head><p>, in the Military Art, a place raised to plant cannon upon, to play with more advantage upon the enemy. It consists of an epaulment or a breastwork, of about 8 feet high, and 18 or 20 feet thick.</p><p>In all batteries, the open spaces through which the <pb n="199"/><cb/> muzzles of the cannon are pointed, are called <hi rend="italics">Embrasures,</hi> and the distances between the embrasures, <hi rend="italics">merlons.</hi> The guns are placed upon a platform of planks &c, ascending a little from the parapet, to check the recoil, and that the gun may be the easier brought back again to the parapet: they are placed from 12 to 16 feet distant from one another, that the parapet may be strong, and the gunners have room to work.</p><p><hi rend="italics">Mortar</hi> <hi rend="smallcaps">Batteries</hi> differ from the others, in that the slope of the parapet is inwards, and it is without embrasures, the shells being fired quite over the parapet, commonly at an angle of 45 degrees elevation.</p><p><hi rend="italics">Open</hi> <hi rend="smallcaps">Battery</hi>, is nothing more than a number of cannon, commonly field-pieces, ranged in a row abreast of one another, perhaps on some small natural elevation of the ground, or an artificial bank a little raised for the purpose.</p><p><hi rend="italics">Covered</hi> or <hi rend="italics">Masked</hi> <hi rend="smallcaps">Battery</hi>, is when the cannon and gunners are covered by a bank or breast-work, commonly made of brush-wood, faggots, and earth, called a fascine battery.</p><p><hi rend="italics">Sunk</hi> or <hi rend="italics">Buried</hi> <hi rend="smallcaps">Battery</hi>, is when its platform is sunk, or let down into the ground, so that trenches must be cut in the earth opposite the muzzles of the guns, to serve as embrasures to fire through. This is mostly used on the first making of approaches in besieging and battering a place.</p><p><hi rend="italics">Cross</hi> <hi rend="smallcaps">Batteries</hi>, are two batteries playing athwart each other upon the same object, forming an angle there, and battering to more effect, because what one battery shakes, the other beats down.</p><p><hi rend="smallcaps">Battery</hi> <hi rend="italics">d'Enfilade,</hi> is one that scours or sweeps the whole length of a straight line.</p><p><hi rend="smallcaps">Battery</hi> <hi rend="italics">en Echarpe,</hi> is one that plays obliquely.</p><p><hi rend="smallcaps">Battery</hi> <hi rend="italics">de Reverse,</hi> or <hi rend="italics">Murdering Battery,</hi> is one that plays upon the enemy's back.</p><p><hi rend="italics">Camerade</hi> or <hi rend="italics">Joint</hi> <hi rend="smallcaps">Battery</hi>, is when several guns play upon one place at the same time.</p><div2 part="N" n="Battery" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Battery</hi></head><p>, <hi rend="italics">in Electricity,</hi> is a combination of coated surfaces of glass, commonly jars, so connected together that they may be charged at once, and discharged by a common conductor. Mr. Gralath, a German electrician, first contrived to increase the shock by charging several phials at the same time.—Dr Franklin, having analysed the Leyden phial, and found that it lost at one surface the electric fire received at the other, constructed a battery of eleven large panes of sash window glass, coated on both sides, and so connected that the whole might be charged together, and with the same labour as one single pane; then by bringing all the <hi rend="italics">giving</hi> sides into contact with one wire, and all the <hi rend="italics">receiving</hi> sides with another, he contrived to unite the force of all the plates, and to discharge them at once.—Dr. Priestley describes a still more complete battery. This consists of 64 jars, each 10 inches long, and 2 1/2 inches in diameter, all coated within an inch and a half of the top, forming in the whole about 32 square feet of coated surface. A piece of very fine wire is twisted about the lower end of the wire of each jar, to touch the inside coating in several places; and it is put through a pretty large piece of cork, within the jar, to prevent any part of it from touching the side, by which a spontaneous discharge might be made. Each wire is turned round so as to make a loop at the <cb/> upper end; and through these loops passes a pretty thick brass rod with knobs, each rod serving for one row of the jars; and these rods are made to communicate together by a chick chain laid over them, or as many of them as may be wanted. The jars stand in a box, the bottom of which is covered with a tin plate; and a bent wire touching the plate passes through the box, and appears on the outside. To this wire is fastened any conductor designed to communicate with the outside of the battery; and the discharge is made by bringing the brass knob to any of the knobs of the battery. When a very great force is required, the size or number of the jars may be increased, or two or more batteries may be used.—But the largest and most powerful battery of all, is that employed by Dr. Van Marum, to the amazing large electrical machine, lately constructed for Teyler's museum at Haarlem. This grand battery consists of a great number of jars coated as above, to the amount of about 130 square feet; and the effects of it, which are truly astonishing, are related by Dr. Van Marum in his description of this machine, and of the experiments made with it, at Haarlem 1785, &c. See also Franklin's Exper. and Observ. and Priestley's History of Electricity.</p></div2></div1><div1 part="N" n="BATTLEMENTS" org="uniform" sample="complete" type="entry"><head>BATTLEMENTS</head><p>, in Architecture, are notches or indentures in the top of a wall or other building, like embrasures, to look through.</p></div1><div1 part="N" n="BAY" org="uniform" sample="complete" type="entry"><head>BAY</head><p>, in Geography, denotes a small gulph, or an arm of the sea stretching up into the land; being larger in the middle within, than at its entrance, which is called the mouth of the bay.</p></div1><div1 part="N" n="BAYER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BAYER</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a German lawyer and astronomer of the latter part of the 16th and beginning of the 17th century, but in what particular year or place he was born, is not certainly known: however, his name will be ever memorable in the annals of astronomy, on account of that great and excellent work which he first published in the year 1603, under the title of <hi rend="italics">Uranometria,</hi> being a complete celestial atlas, or large folio charts of all the constellations, with a nomenclature collected from all the tables of astronomy, ancient and modern, with the ufeful invention of denoting the stars in every constellation by the letters of the Greek alphabet, in their order, and according 10 the order of magnitude of the stars in each constellation. By means of these marks, the stars of the heavens may, with as great facility, be distinguished and referred to, as the several places of the earth are by means of geographical tables; and as a proof of the usefulness of this method, our celestial globes and atlasses have ever since retained it; and hence it is become of general use through all the literary world; astronomers, in speaking of any star in the constellation, denoting it by saying it is marked by Bayer, <foreign xml:lang="greek">a</foreign>, or <foreign xml:lang="greek">b</foreign>, or <foreign xml:lang="greek">g</foreign>, &c.</p><p>Bayer lived many years after the first publication of this work, which he greatly improved and augmented by his constant attention to the study of the stars. At length, in the year 1627, it was republished under a new title, viz, <hi rend="italics">Coelum Stellatum Christianum,</hi> that is, the <hi rend="italics">Christian Stellated Heaven,</hi> or the <hi rend="italics">Starry Heavens Christianized:</hi> for in this work, the heathen names and characters, or figures of the constellations, were rejected, and others, taken from the scriptures, were in- <pb n="200"/><cb/> serted in their stead, to circumscribe the respective constellations. This was the project of one Julius Schiller, a civilian of the same place. But this attempt was too great an innovation, to sind success, or a general reception, which might occasion great confusion. And, we even sind in the later editions of this work, that the ancient sigures and names were restored again; at least so I find them in two editions, of the years 1654, and 1661, which are now before me.</p></div1><div1 part="N" n="BEAD" org="uniform" sample="complete" type="entry"><head>BEAD</head><p>, in Architecture, is a round moulding, carved in short embossments, like beads in necklaces: and sometimes an astragal is thus carved. There is also a sort of plain bead often set on the edge of each facia of an architrave; as also sometimes on the lining hoard of a door case, the upper edge of skirting boards, &c.</p><div2 part="N" n="Bead" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Bead</hi></head><p>, in assaying, the small ball or mass of pure metal separated from the <hi rend="italics">scoria,</hi> and seen distinct and pure in the middle of the coppel while in the fire.</p></div2></div1><div1 part="N" n="BEAM" org="uniform" sample="complete" type="entry"><head>BEAM</head><p>, in Architecture, a large timber laid across a building, into which the principal rafters are framed. Several ingenious authors have considered the force or strength of beams, as supporting their own weight and any other additional weight; particularly Varignon, and Parent in the Memoir. Acad. R. Scien. an. 1708, and Mr. Emerson, on the Strength and Stress of Timber, in his Mechanics. Mr. Parent makes the proportion of the depth to the breadth of a beam to be as 7 to 5 when it is strongest.</p><p><hi rend="smallcaps">Beams</hi> <hi rend="italics">of a ship,</hi> are the large, main, cross timbers, stretched from side to side, to support the decks, and keep the sides of the ship from falling together.</p><p><hi rend="smallcaps">Beam</hi> <hi rend="italics">of a balance,</hi> is the horizontal piece of wood or iron supported on a pivot in the middle, and at the extremities of which the two scales are suspended, for weighing any thing.</p><p><hi rend="smallcaps">Beam</hi>-<hi rend="italics">Compass,</hi> an instrument consisting of a wooden or brass square beam, having sliding sockets carrying steel or pencil points; and are used for describing large circles, the radii of which are beyond the extent of the common compasses.</p></div1><div1 part="N" n="BEAR" org="uniform" sample="complete" type="entry"><head>BEAR</head><p>, in Astronomy, a name given to two constellations, called the <hi rend="italics">greater</hi> and the <hi rend="italics">lesser bear,</hi> or <hi rend="smallcaps">Ursa</hi> <hi rend="italics">major</hi> and <hi rend="italics">minor.</hi> The pole star is in the tail of the little bear, and is within less than 2 degrees of the north pole. See <hi rend="smallcaps">Ursa</hi>, <hi rend="italics">major</hi> and <hi rend="italics">minor.</hi></p></div1><div1 part="N" n="BEARD" org="uniform" sample="complete" type="entry"><head>BEARD</head><p>, <hi rend="italics">of a Comet,</hi> the rays which it emits in the direction in which it moves; as distinguished from the tail, or the rays emitted or left behind it as it moves along, being always in a direction from the sun.</p></div1><div1 part="N" n="BEARER" org="uniform" sample="complete" type="entry"><head>BEARER</head><p>, in Arehitecture, a post or brick wall, trimmed up between the two ends of a piece of timber, to shorten its bearing, or to prevent its bearing with the whole weight at the ends only.</p></div1><div1 part="N" n="BEARING" org="uniform" sample="complete" type="entry"><head>BEARING</head><p>, in Geography and Navigation, the situation of one place from another, with regard to the points of the compass; or an arch of the horizon between the meridian of a place and a line drawn through this and another place, or the angle formed by a line drawn through the two places and their meridians.— The bearings of places on the ground are usually determined by the magnetic needle. <cb/></p></div1><div1 part="N" n="BEATS" org="uniform" sample="complete" type="entry"><head>BEATS</head><p>, in a Clock or Watch, are the strokes made by the fangs or pallets of the spindle of the balance; or of the pads in a royal pendulum. For the number and use of the beats, see Derham's Artificial Clock Maker, pa. 14 and seq.</p></div1><div1 part="N" n="BED" org="uniform" sample="complete" type="entry"><head>BED</head><p>, <hi rend="italics">of a Great Gun,</hi> a plank laid between the cheeks of the carriage, on the middle transum, for the breech of the gun to rest upon.</p><div2 part="N" n="Bed" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Bed</hi></head><p>, or <hi rend="italics">Stool,</hi> of a mortar, a thick and strong planking on which a mortar is placed, hollowed a little to receive the breech and trunions.</p><p>BED-MOULDING, in Architecture, a term used by workmen for those members in a cornice which are placed below the coronet, or crown. It usually consists of these four members, an ogee, a list, a large boultine, and another list under the coronet.</p></div2></div1><div1 part="N" n="BELIDOR" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BELIDOR</surname> (<foreName full="yes"><hi rend="smallcaps">Bernard Forest de</hi></foreName>)</persName></head><p>, an engineer in the service of France, but born in Catalonia in 1698. He was professor in the new school of artillery at la Fere, where he published his course of mathematics for the use of the artillery and engineers. He was the first who seriously considered the quantity of gunpowder proper for charges, and reduced it to 2-3ds the quantity. He was named Associate in the Academy of Sciences in 1751; and died Sept. 8, 1761, at 63 years of age. <hi rend="center">His works that have been published, are:</hi></p><p>1. Sommaire d'un Cours d'Architecture militaire, civile & hydraulique, in 12mo, 1720.</p><p>2. Nouveau Cours de Mathematiques, &c. in 4to, 1725.</p><p>3. La Science des Ingénieurs, in 4to, 1729.</p><p>4. Le Bombardier Francois, in 4to, 1734.</p><p>5. Architecture Hydraulique, 4 vols. in 4to, 1737.</p><p>6. Dictionnaire portatif de l'Ingénieur, in 8vo.</p><p>7. Traité des Fortifications, 4 vols. in 4to.</p><p>Besides several pieces inserted in the volumes of the Memoirs of the Academy of Sciences, for the years 1737, 1750, 1753, and 1756.</p></div1><div1 part="N" n="BELLATRIX" org="uniform" sample="complete" type="entry"><head>BELLATRIX</head><p>, in Astronomy, a ruddy, glittering star of the 2d magnitude, in the left shoulder of Orion. Its name is from the Latin <hi rend="italics">bellum,</hi> as being anciently supposed to have great influence in kindling wars, and forming warriors.</p></div1><div1 part="N" n="BELTS" org="uniform" sample="complete" type="entry"><head>BELTS</head><p>, <hi rend="italics">Fasciæ,</hi> in Astronomy, two zones or girdles surrounding the planet Jupiter, brighter than the rest of his body, and terminated by parallel lines. They are observed however to be sometimes broader and sometimes narrower, and not always occupying exactly the same part of the dise. Jupiter's belts were first observed and described by Huygens, in his Syst. Saturn. Dark spots have often been observed on the belts of Jupiter; and M. Cassini observed a permanent one on the northern side of the most southern belt, by which he determined the length of Jupiter's days, or the time in which the planet revolves upon its axis, which is 9h. 56m. Some astronomers suppose that these belts are seas, which alternately cover and leave bare large tracts of the planet's surface: and that the spots are gulphs in those seas, which are sometimes dry, and sometimes full. But Azout conceived that the spots are protuberances of the belts; and others <pb n="201"/><cb/> again are of opinion that the transparent and moveable spots are the shadows of Jupiter's satellites.</p><p>Cassini also speaks of the belts of Saturn; being three dark, straight, parallel bands, or <hi rend="italics">fasciæ,</hi> on the disc of that planet. But it does not appear that Saturn's belts adhere to his body, as those of Jupiter do; but rather that they are large dark rings surrounding the planet at a distance. Some imagine that they are clouds in the atmosphere of Saturn, though it would seem that the middlemost is the shadow of his ring.</p></div1><div1 part="N" n="BENDING" org="uniform" sample="complete" type="entry"><head>BENDING</head><p>, the reducing a body to a curved or crooked form. The bending of boards, planks, &c, is effected by means of heat, whether by boiling or otherwise, by which their fibres are so relaxed that they may be bent into any figure. Bernoulli has a discourse on the bending of springs, or elastic bodies. And Amontons gives several experiments concerning the bending of ropes. The friction of a rope bent or wound about an immoveable cylinder, is sufficient, with a very small power, to sustain very great weights.</p><p>BERENICE's Hair; see <hi rend="smallcaps">Coma</hi> <hi rend="italics">Berenices.</hi></p></div1><div1 part="N" n="BERKELEY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERKELEY</surname> (<foreName full="yes"><hi rend="smallcaps">George</hi></foreName>)</persName></head><p>, the virtuous and learned bishop of Cloyne in Ireland, was born in that kingdom, at Kilcrin, the 12th of March 1684. After receiving the first part of his education at Kilkenny school, he was admitted a pensioner of Trinity College, Dublin, at 15 years old; and chosen fellow of that college in 1707.</p><p>The first public proof he gave of his literary abilities was, <hi rend="italics">Arithmetica absque Algebra aut Euclide demonstrata;</hi> which from the preface it appears he wrote before he was 20 years old, though he did not publish it till 1707. It is followed by a Mathematical Miscellany, containing observations and theorems inscribed to his pupil Samuel Molineux.</p><p>In 1709 came out the <hi rend="italics">Theory of Vision;</hi> which of all his works it seems does the greatest honour to his sagacity; being, it has been observed, the first attempt that ever was made to distinguish the immediate and natural objects of sight, from the conclusions we have been accustomed from infancy to draw from them. The boundary is here traced out between the ideas of sight and touch; and it is shewn, that though habit hath so connected these two classes of ideas in the mind, that they are not without a strong effort to be separated from each other, yet originally they have no such connection; insomuch, that a person born blind, and suddenly made to see, would at first be utterly unable to tell how any object that affected his sight would affect his touch; and particularly would not from sight receive any idea of distance, or external space, but would imagine all objects to be in his eye, or rather in his mind.</p><p>In 1710 appeared <hi rend="italics">The Principles of Human Knowledge;</hi> and in 1713 <hi rend="italics">Dialogues hetween Hylas and Philonous:</hi> the object of both which pieces is, to prove that the commonly received notion of the existence of matter, is false; that sensible material objects, as they are called, are not external to the mind, but exist in it, and are nothing more than impressions made upon it by the immediate act of God, according to certain rules termed laws of nature. <cb/></p><p>Acuteness of parts and beauty of imagination were so conspicuous in Berkeley's writings, that his reputation was now established, and his company courted; men of opposite parties concurred in recommending him. For Steele he wrote several papers in the Guardian, and at his house became acquainted with Pope, with whom he always lived in friendship. Swift recommended him to the celebrated earl of Peterborough, who being appointed ambassador to the king of Sicily and the Italian States, took Berkeley with him as chaplain and secretary in 1713, with whom he returned to England the year following.</p><p>His hopes of preferment expiring with the fall of queen Anue's ministry, he some time after embraced an offer made him by Ashe, bishop of Clogher, of accompanying his son in a tour through Europe. In this he employed four years; and besides those places which fall within the grand tour, he visited some that are less frequented, and with great industry collected materials for a natural history of those parts, but which were unfortunately lost in-the passage to Naples. He arrived at London in 1721; and being much affected with the miseries of the nation, occasioned by the South-sea scheme in 1720, he published the same year <hi rend="italics">An Essay towards preventing the ruin of Great Britain:</hi> reprinted in his <hi rend="italics">Miscellaneous Tracts.</hi></p><p>His way was now open into the very first company. Pope introduced him to lord Burlington, by whom he was recommended to the duke of Grafton, then appointed lord-lieutenant of Ireland, who took Berkeley over as one of his chaplains in 1721. The latter part of this year he accumulated the degrees of bachelor and doctor in divinity: and the year following he had a very unexpected increase of fortune from the death of Mrs. Vanhomrigh, the celebrated Vanessa, to whom he had been introduced by Swift: this lady had intended Swift for her heir; but perceiving herself to be slighted by him, she left her fortune, of 8000l. between her two executors, of whom Berkeley was one. In 1724 he was promoted to the deanery of Derry, worth 1100l. a year.</p><p>In 1725 he published, and it has since been reprinted in his Miscellaneous Tracts, <hi rend="italics">A Proposal for converting the savage Americans to Christianity, by a college to be erected in the Summer Isles, otherwise called the Isles of Bermuda.</hi> The proposal was well received, at least by the king; and he obtained a charter for founding the college, with a parliamentary grant of 20,000l. toward carrying it into execution: but he could never get the money, it being otherwise employed by the minister; so that after two years stay in America on this business, he was obliged to return, and the scheme dropped.</p><p>In 1732 he published <hi rend="italics">The Minute Philosopher,</hi> in 2 volumes 8vo, against Freethinkers. In 1733 he was made bishop of Cloyne; and might have been removed in 1745, by lord Chesterfield, to Clogher; but declined it. He resided constantly at Cloyne, where he faithfully discharged all the offices of a good bishop, yet continued his studies with unabated attention.</p><p>About this time he engaged in a controversy with the mathematicians, which made a good deal of noise in the literary world; and the occasion of it was this: <pb n="202"/><cb/> Addison had given the bishop an account of the behaviour of their common friend Dr. Garth in his last illness, which was equally unpleasing to both these advocates of revealed religion. For when Addison went to see the doctor, and began to discourse with him seriously about another world, “Surely, Addison, replied he, I have good reason not to believe those trifles, since my friend Dr. Halley, who has dealt so much in demonstration, has assured me, that the doctrines of christianity are incomprehensible, and the religion itself an imposture.” The bishop therefore took up arms against Halley, and addressed to him, as to an Infidel Mathematician, a discourse called <hi rend="italics">The Analyst</hi>; with a view of shewing that mysteries in faith were unjustly objected to by mathematicians, who he thought admitted much greater mysteries, and even falshoods in science, of which he endeavoured to prove that the doctrine of Fluxions furnished a clear example. This attack gave occasion to <hi rend="italics">Robins's Discourse concerning the Method of Fluxions,</hi> to <hi rend="italics">Maclaurin's Fluxions,</hi> and to other smaller works upon the same subject; but the direct answers to <hi rend="italics">The Analyst</hi> were made by a person under the name of Philalethes Cantabrigiensis, but commonly supposed to be Dr. Jurin, whose first piece was, <hi rend="italics">Geometry no Friend to Infidelity,</hi> 1734. To this the bishop replied in <hi rend="italics">A Defence of Freethinking in Mathematics; with an Appendix concerning Mr. Walton's Vindication,</hi> 1735; which drew a second answer the same year from Philalethes, under the title of <hi rend="italics">The Minute Mathematician, or the Freethinker no just Thinker,</hi> 1735. Other writings in this controversy, beside those before mentioned, were</p><p>1. A Vindication of Newton's Principles of Fluxions against the objections contained in the Analyst, by J. Walton, Dublin, 1735.</p><p>2. The Catechism of the Author of the Minute Philosopher fully answered, by J. Walton, Dublin, 1735.</p><p>3. Reasons for not replying to Mr. Walton's Full Answer, in a letter to P. T. P. by the author of the Minute Philosopher, Dublin, 1735.</p><p>4. An Introduction to the Doctrine of Fluxions, and Defence of the Mathematicians against the objections of the author of the Analyst, &c. Lond. 1736.</p><p>5. A new Treatise of Fluxions; with answers to the principal objections in the Analyst, by James Smith, A. M. Lond. 1737.</p><p>6. Mr. Robins's Discourse of Newton's Methods of Fluxions, and of Prime and Ultimate Ratios, 1735.</p><p>7. Mr. Robins's Account of the preceding Discourse, in the Repub. of Letters, for October 1735.</p><p>8. Philalethes's Considerations upon some passages contained in two letters to the author of the Analyst &c, in Repub. of Letters, Novemb. 1735.</p><p>9. Mr. Robins's Review of some of the principal objections that have been made to the doctrine of Fluxions &c. Repub. of Letters for Decem. 1735.</p><p>10. Philalethes's Reply to ditto, in the Repub. of Letters, Jan. 1736.</p><p>11. Mr. Robins's Dissertation, shewing that the account of the doctrines of Fluxions &c, is agreeable to the real sense and meaning of their great Inventor, &c, Repub. of Letters, April 1736. <cb/></p><p>12. Philalethes's Considerations upon ditto, in Repub. of Letters, July 1736.</p><p>13. Mr. Robins's Remarks on ditto, in Repub. of Letters, Aug. 1736.</p><p>14. Mr. Robins's Remainder of ditto, in an Appendix to the Repub. of Letters, Sept. 1736.</p><p>15. Philalethes's Observations upon ditto, in an Appendix to the Repub. of Letters, Nov. 1736.</p><p>16. Mr. Robins's Advertisement in Repub. of Letters, Decemb. 1736.</p><p>17. Philalethes's Reply to ditto, in an Appendix to the Repub. of Letters for Decem. 1736.</p><p>18. Some Observations on the Appendix to the Repub. of Letters for Decem. 1736, by Dr. Pemberton, in the Works of the Learned for Feb. 1737. With some smaller pieces in the same.</p><p>In 1736 bishop Berkeley published <hi rend="italics">The Querist,</hi> “a discourse addressed to magistrates, occasioned by the enormous licence and irreligion of the times;” and many other things afterward of a smaller kind. In 1744 came out his celebrated and curious book, “<hi rend="italics">Siris</hi>; a Chain of Philosophical Reflections and Inquiries concerning the virtues of Tar-water.” It had a second impression, with additions and emendations, in 1747; and was followed by <hi rend="italics">Farther Thoughts on Tar-water,</hi> in 1752. In July the same year he removed, with his lady and family, to Oxford, partly to superintend the education of a son, but chiefly to indulge the passion for learned retirement, which had always strongly possessed him. He would have resigned his bishoprick for a canonry or headship at Oxford; but it was not permitted him. Here he lived highly respected, and collected and printed the same year all his smaller pieces in 8vo. But this happiness did not long continue, being suddenly cut off by a palsy of the heart Jan. 14, 1753, in the 69th year of his age, while listening to a sermon that his lady was reading to him.</p><p>The excellence of Berkeley's moral character is conspicuous in his writings: he was certainly a very amiable as well as a very great man; and it is thought that Pope searcely said too much, when he ascribed <hi rend="center">“To Berkeley every virtue under heaven.”</hi></p></div1><div1 part="N" n="BERME" org="uniform" sample="complete" type="entry"><head>BERME</head><p>, in Fortification, a small space of ground, 4 or 5 feet wide, left without the rampart, between it and the side of the moat, to receive the earth that rolls down from the rampart, and prevent its falling into the ditch and filling it up.—Sometimes, for greater security, the berme is pallisadoed.</p></div1><div1 part="N" n="BERNARD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERNARD</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">Edward</hi></foreName>)</persName></head><p>, a learned ast ronomer, critic and linguist, was born at Perry St. Paul, near Towcester, the 2d of May 1638, and educated at Merchant-Taylor's school, and at St. John's college Oxford. Having laid in a good fund of classical learning at school, in the Greek and Latin languages, he applied himself very diligently at the university to the study of history, the eastern languages, and mathematics under the celebrated Dr. Wallis. In 1668 he went to Leyden to consult some Oriental manuscripts left to that university by Joseph Scaliger and Levin Warner, and especially the 5th, 6th, and 7th books of Apollonius's Conics, the Greek text of which is lost, and this Arabic version having been brought from the east by the celebrated <pb n="203"/><cb/> Golius, a transcript of which was thence taken by Bernard, and brought with him to Oxford, with intent to publish it there with a Latin translation; but he was obliged to drop that design for want of encouragement. This however was afterwards carried into effect by Dr. Halley in 1710, with the addition of the 8th book, which he supplied by his own ingenuity and industry.</p><p>At his return to Oxford, Bernard examined and collated the most valuable manuscripts in the Bodleian library. In 1669, the celebrated Christopher Wren, Savilian professor of astronomy at Oxford, having been appointed surveyor-general of his majesty's works, and being much detained at London by this employment, obtained leave to name a deputy at Oxford, and pitched upon Mr. Bernard, which engaged the latter in a more particular application to the study of astronomy. But in 1673 he was appointed to the professorship himself, which Wren was obliged to resign, as, by the statutes of the founder, Sir Henry Saville, the professors are not allowed to hold any other office either ecclesiastical or civil.</p><p>About this time a scheme was set on foot at Oxford, of collecting and publishing the ancient mathematicians. Mr. Bernard, who had first formed the project, collected all the old books published on that subject since the invention of printing, and all the manuscripts he could discover in the Bodleian and Savilian libraries, which he arranged in order of time, and according to the matter they contained; of this he drew up a synopsis or view; and as a specimen he published a few sheets of Euclid, containing the Greek text, and a Latin version, with Proclus's commentary in Greek and Latin, and learned scholia and corollaries. The synopsis itself was published by Dr. Smith, at the end of his life of our author, under the title of <hi rend="italics">Veterum Mathematicorum Græcorum, Latinorum, et Arabum, Synopsis.</hi> And at the end of it there is a catalogue of some Greek writers, whose works are supposed to be lost in their own language, but are preserved in the Syriac or Arabic translations of them.</p><p>Mr. Bernard undertook also an edition of the <hi rend="italics">Parva Syntaxis Alexandrina;</hi> in which, besides Euclid, are contained the small treatises of Theodosius, Menelaus, A<*>istarchus, and Hipsicles; but it never was published.</p><p>In 1676 he was sent to France, as tutor to the dukes of Grafton and Northumberland, sons to king Charles the 2d by the dutchess of Cleveland, who then lived with their mother at Paris: but the simplicity of his manners not suiting the gaiety of the dutchess's family, he returned about a year after to Oxford, and pursued his studies; in which he made great proficiency, as appears by his many learned and critical works. In 1691, being presented to the rectory of Brightwell in Berkshire, he quitted his professorship at Oxford, in which he was succeeded by David Gregory, professor of mathematics at Edinburgh.</p><p>Toward the latter end of his life he was much afflicted with the stone; yet notwithstanding this, and other infirmities, he undertook a voyage to Holland, to attend the sale of Golius's manuscripts, as he had once before done at the sale of Heinsius's library. On his return to England, he fell into a languishing consumption, which put an end to his life the 12th of January 1696, in the 58th year of his age.</p><p>Beside the works of his before mentioned, he was <cb/> author of many other compositions. He composed tables of the longitudes, latitudes, right-ascensions, &c, of the fixed stars: he wrote Observations on the Obliquity of the Ecliptic; and other pieces inserted in the Philosophical Transactious. He wrote also,</p><p>1. <hi rend="italics">A Treatise of the Ancient Weights and Measures.</hi></p><p>2. <hi rend="italics">Chronologiæ Samaritanæ Synopsis,</hi> in two tables.</p><p>3. <hi rend="italics">Testimonies of the Ancients concerning the Greek Version of the Old Testament by the Seventy.</hi></p><p>And several other learned works. Besides a great number of valuable manuscripts left at his death.</p></div1><div1 part="N" n="BERNARD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERNARD</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, professor of philosophy and mathematics, and minister of the Walloon church at Ley den, was born September the 1st 1658, at Nions in Dauphine. Having studied at Geneva, he returned to France in 1679, and was chosen minister of Venterol, a village in Dauphine; but some time after he was removed to the church of Vinsobres in the same province. To avoid the persecutions against the protestants in France, he went into Holland, where he was appointed one of the pensionary ministers of Ganda. He here published several political and historical works. And in 1699 he began the <hi rend="italics">Nouvelles de la Republique des Lettres,</hi> which continued till December 1710. In 1705 he was chosen minister of the Walloon church at Leyden; and about the same time, Mr. de Volder, professor of philosophy and mathematics at Leyden, having resigned, Mr. Bernard was appointed his successor; upon which occasion the university also presented him with the degrees of doctor of philosophy and master of arts. In 1716 he published a supplement to Moreri's dictionary in 2 volumes folio. The same year he resumed his <hi rend="italics">Nouvelles de la Republique des Lettres;</hi> which he continued till his death, which happened the 27th of April 1718, in the 60th year of his age.</p></div1><div1 part="N" n="BERNOULLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERNOULLI</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, a celebrated mathematician, born at Basil the 27th of December 1654. Having taken his degrees in that university, he applied himself to divinity at the entreaties of his father, but against his own inclination, which led him to astronomy and mathematics. He gave very early proofs of his genius for these sciences, and soon became a geometrician, without a preceptor, and almost without books; for if one by chance fell into his hands, he was obliged to conceal it, to avoid the displeasure of his father, who designed him for other studies. This situation induced him to choose for his device, Phaeton driving the chariot of the sun, with these words, <hi rend="italics">Invito patre sidera verso,</hi> “I traverse the stars against my father's will:” alluding particularly to astronomy, to which he then chiefly applied himself.</p><p>In 1676 he began his travels. When he was at Geneva, he fell upon a method to teach a young girl to write who had been blind from two months old. At Bourdeaux he composed universal gnomonic tables; but they were never published. He returned from France to his own country in 1680. About this time there appeared a comet, the return of which he foretold; and wrote a small treatise upon it. Soon after this he went into Holland, where he applied himself to the study of the new philosophy. Having visited Flanders and Brabant, he passed over to Fngland; where he formed an acquaintance with the most eminent men in the sciences, and was frequent at their philosophical meetings. He <pb n="204"/><cb/> returned to his native country in 1682; and exhibited at Basil a course of experiments in natural philosophy and mechanics, which consisted of a variety of new discoveries. The same year he published his Essay on a new System of Comets; and the year following, his Dissertation on the weight of the air. About this time Leibnitz having published, in the Acta Eruditorum at Leipsic, some essays on his new <hi rend="italics">Calculus Disscrentialis,</hi> but concealing the art and method of it, Mr. Bernoulli and his brother John discovered, by the little which they saw, the beauty and extent of it: this induced them to endeavour to unravel the secret; which they did with such success, that Leibnitz declared that the invention belonged to them as much as to himself.</p><p>In 1687 James Bernoulli succeeded to the professorship of mathematics at Basil; a trust which he discharged with great applause; and his reputation drew a great number of foreigners from all parts to attend his lectures. In 1699 he was admitted a foreign member of the Academy of Sciences of Paris; and in 1701 the same honour was conferred upon him by the Academy of Berlin: in both of which he published several ingenious compositions, about the years 1702, 3, and 4. He wrote also several pieces in the <hi rend="italics">Acta Eruditorum</hi> of Leipsic, and in the <hi rend="italics">Journal des Sçavans.</hi> His intense application to study brought upon him the gout, and by degrees a slow fever, which put a period to his life the 16th of August 1705, in the 51st year of his age.— Archimedes having found out the proportion of a sphere and its circumscribing cylinder, ordered them to be engraven on his monument: In imitation of him, Bernoulli appointed that a logarithmic spiral curve should be inscribed on his tomb, with these words, <hi rend="italics">Eadem mutata resurgo;</hi> in allusion to the hopes of the resurrection, which are in some measure represented by the properties of that curve, which he had the honour of discovering.</p><p>James Bernoulli had an excellent genius for invention and elegant simplicity, as well as a close application. He was eminently skilled in all the branches of the mathematics, and contributed much to the promoting the new analysis, infinite series, &c. He carried to a great height the theory of the quadrature of the parabola; the geometry of curve lines, of spirals, of cycloids and epicycloids.</p><p>His works, that had been-published, were collected, and printed in 2 volumes 4to, at Geneva in 1744. At the time of his death he was occupied on a great work entitled <hi rend="italics">De Arte Conjectandi,</hi> which was published in 4to, in 1713. It contains one of the best and most elegant introductions to Infinite Series, &c. This posthumous work is omitted in the collection of his works above mentioned.</p></div1><div1 part="N" n="BERNOULLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERNOULLI</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, the brother of James, last mentioned, and a celebrated mathematician, was born at Basil the 7th of August 1667. His father intended him for trade; but his own inclination was at first for the Belles-Lettres, which however, like his brother, he left for mathematics. He laboured with his brother to discover the method used by Leibnitz, in his essays on the Differential Calculus, and gave the first principles of the Integral Calculus. Our author, with Messieurs Huygens and Leibnitz, was the first who gave the solution of the problem proposed by James <cb/> Bernoulli, concerning the catenary, or curve formed by a chain suspended by its two extremities.</p><p>John Bernoulli had the degree of doctor of physic at Basil, and two years afterward was named professor of mathematics in the university of Groningen. It was here that he discovered the mercurial phosphorus or luminous barometer; and where he resolved the problem proposed by his brother concerning Isoperimetricals.</p><p>On the death of his brother James, the professor at Basil, our author returned to his native country, against the pressing invitations of the magistrates of Utrecht to come to that city, and of the university of Groningen, who wished to retain him. The Academic Senate of Basil soon appointed him to fucceed his brother, without assembling competitors, and contrary to the established practice: an appointment which he held during his whole life.</p><p>In 1714 was published his treatise on the management of ships; and in 1730, his memoir on the elliptical figure of the planets gained the prize of the Academy of Sciences. The same academy also divided the prize, for their question concerning the inclination of the planetary orbits, between our author and his son Daniel.</p><p>John Bernoulli was a member of most of the academies of Europe, and reecived as a foreign associate of that of Paris in 1699. After a long life spent in constant study and improvement of all the branches of the mathematics, he died full of honours the 1st of January 1748, in the 81st year of his age. Of five sons which he had, three pursued the same sciences with himself. One of these died before him; the two others, Nicolas and Daniel, he lived to see become eminent and much respected in the same sciences.</p><p>The writings of this great man were dispersed through the periodical memoirs of several academies, as well as in many separate treatises. And the whole of them were carefully collected and published at Lausanne and Geneva, 1742, in 4 volumes, 4to.</p></div1><div1 part="N" n="BERNOULLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERNOULLI</surname> (<foreName full="yes"><hi rend="smallcaps">Daniel</hi></foreName>)</persName></head><p>, a celebrated physician and philosopher, and son of John Bernoulli last mentioned, was born at Groningen Feb. the 9th, 1700, where his father was then professor of mathematics. He was intended by his father for trade, but his genius led him to other pursuits. He passed some time in Italy; and at 24 years of age he declined the honour offered him of becoming president of an academy intended to have been established at Genoa. He spent several years with great credit at Petersburgh; and in 1733 returned to Basil, where his father was then professor of mathematics; and here our author successively filled the chair of physic, of natural and of speculative philosophy.</p><p>In his work <hi rend="italics">Exercitationes Mathematicæ,</hi> 1724, he took the only title he then had, viz, “Son of John Bernoulli,” and never would suffer any other to be added to it. This work was published in Italy, while he was there on his travels; and it classed him in the rank of inventors. In his work, <hi rend="italics">Hydrodynamica,</hi> published in 4to at Argentoratum or Strasbourg, in 1738, to the same title was also added that of <hi rend="italics">Med. Prof. Basil.</hi></p><p>Daniel Bernoulli wrote a multitude of other pieces, which have been published in the Mem. Acad. of Sciences at Paris, and in those of other Academies. He gained and divided ten prizes from the Academy of <pb n="205"/><cb/> Sciences, which were contended for by the most illustrious mathematicians in Europe. The only person who has had similar success of the same kind, is Euler, his countryman, disciple, rival, and friend. His first prize he gained at 24 years of age. In 1734 he divided one with his father; which hurt the family union; for the father considered the contest itself as a want of respect; and the son did not sufficiently conceal that he thought (what was really the case) his own piece better than his father's. And besides, he declared for Newton, against whom his father had contended all his life. In 1740 our author divided the prize, “On the Tides of the Sea,” with Euler and Maclaurin. The Academy at the same time crowned a fourth piece, whose chief merit was that of being Cartesian; but this was the last public act of adoration paid by the Academy to the authority of the author of the Vortices, which it had obeyed but too long. In 1748 Daniel Bernoulli succeeded his father John in the Academy of Sciences, who had succeeded his brother James; this place, since its first erection in 1699, having never been without a Bernoulli to fill it.</p><p>Our author was extremely respected at Basil; and to bow to Daniel Bernoulli, when they met him in the streets, was one of the first lessons which every father gave every child. He was a man of great simplicity and modesty of manners. He used to tell two little adventures, which he said had given him more pleasure than all the other honours he had received. Travelling with a learned stranger, who, being pleased with his conversation, asked his name; “I am Daniel Bernoulli,” answered he with great modesty; “And I,” said the stranger (who thought he meant to laugh at him), “am Isaac Newton.” Another time having to dinner with him the celebrated Koenig the mathematician, who boasted, with some degree of self-complacency, of a difficult problem he had resolved with much trouble, Bernoulli went on doing the honours of his table, and when they went to drink coffee he presented Koenig with a solution of the problem more elegant than his own.</p><p>After a long, useful, and honourable life, Daniel Bernoulli died the 17th of March 1782, in the 83d year of his age.</p></div1><div1 part="N" n="BETELGEUSE" org="uniform" sample="complete" type="entry"><head>BETELGEUSE</head><p>, a fixed star of the first magnitude in the right shoulder of Orion.</p></div1><div1 part="N" n="BEZOUT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BEZOUT</surname> (<foreName full="yes"><hi rend="smallcaps">Stephen</hi></foreName>)</persName></head><p>, a celebrated French mathematician, Member of the Academies of Sciences and the Marine, and Examiner of the Guards of the Marine and of the Eleves of Artillery, was born at Nemours the 31st of March 1730. In the course of his studies he met with some books of geometry, which gave him a taste for that science; and the Eloges of Fontenelle, which shewed him the honours attendant on talents and the love of the sciences. His father in vain opposed the strong attachment of young Bezout to the mathematical sciences. April 8, 1758, he was named adjoint-mechanician in the French academy of sciences; having before that sent them two ingenious memoirs on the integral calculus, and given other proofs of his proficiency in the sciences. In 1763, he was named to the new office of Examiner to the Marine, and appointed to compose a Course of Mathematics for their use; and in 1768, on the death of <cb/> M. Camus, he succeeded as Examiner of the Artillery Eleves.</p><p>Bezout fixed his attention more particularly to the resolution of algebraic equations; and he first found out the solution of a particular class of equations of all degrees. This method, different from all former ones, was general for the cubic and biquadratic equations, and just became particular only at those of the 5th degree. Upon this work of finding the roots of equations, our author laboured from 1762 till 1779, when he published it. He composed two courses of mathematics; the one for the Marine, the other for the Artillery: The foundation of these two works was the same; the applications only being different, according to the two different objects: these courses have every where been held in great estimation. In his office of examiner he discharged the duties with great attention, care, and tenderness; a trait of his justice and zeal is remarkable in the following instance: During an examination which he held at Toulon, he was told that two of the pupils could not be present, being confined by the small-pox: he himself had never had that disease, and he was greatly afraid of it; but as he knew that if he did not see these two young men, it would much impede their improvement; he ventured therefore to their bed-sides, to examine them, and was happy to find them so deserving of the hazard he put himself into for their benefit.</p><p>Mr. Bezout lived thus several years beloved of his family and friends, and respected by all, enjoying the fruits and the credit of his labours. But the trouble and fatigues of his offices, with some personal chagrines, had reduced his strength and constitution; he was attacked by a malignant fever, of which he died Sept. 27, 1783, in the 54th year of his age, regretted by his samily, his friends, the young students, and by all his acquaintance in general.</p><p>The books published by him, were:</p><p>1. Course of Mathematics for the use of the Marine, with a Treatise on Navigation; 6 vols. in 8vo, Paris, 1764.</p><p>2. Course of Mathematics for the Corps of Artillery; 4 vols. in 8vo, 1770.</p><p>3. General Theory of Algebraic Equations; 1779.</p><p>His papers printed in the volumes of the Memoirs of the Academy of Sciences, are:</p><p>1. On curves whose rectification depends on a given quantity; in the vol. for 1758.</p><p>2. On several classes of equations that admit of an algebraic solution; 1762.</p><p>3. First vol. of a course of mathematics, 1764.</p><p>4. On certain equations, &c; 1764.</p><p>5. General resolution of all equations; 1765.</p><p>6. Second vol. of a course of mathematics; 1765.</p><p>7. Third vol. of the same; 1766.</p><p>8. Fourth vol. of the same; 1767.</p><p>9. Integration of differentials, &c. vol. 3, Sav. Etr.</p><p>10. Experiments on cold; 1777.</p></div1><div1 part="N" n="BIANCHINI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BIANCHINI</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, a very learned Italian philosopher and mathematician of the 17th century, was born at Verona the 13th of December 1662. He was much esteemed by the learned, and was a member of several academies; and was even the founder of that at Verona, called the Academy of Aletofili, or <pb n="206"/><cb/> Lovers of Truth. He went to Rome in 1684; and was made librarian to cardinal Ottoboni, who was afterwards Pope by the name of Alexander the 8th. He entered into the church, and became canon of St. Mary de la Rotondo, and afterward of St. Lawrence in Damaso.</p><p>Bianchini was author of several learned and ingenious dissertations. In 1697 he published <hi rend="italics">La Istoria universale provata con monumenti, & figurata con simboli de gli antichi.</hi> In 1701 pope Clement the 11th named him secretary of the conferences for the reformation of the calendar; and he published in 1703, <hi rend="italics">De Calendario & Cyclo Cæsaris, ac de Canone Paschali fancti Hyppoliti, Martyris, Dissertationes duæ.</hi> Bianchini was employed on the construction of the large gnomon in the church of the Chartreux at Rome, upon which he published an ample dissertation intitled, <hi rend="italics">De Nummo & Gnomone Clementino.</hi> The research concerning the parallax and the spots of Venus occupied him a long time; but his most remarkable discovery is that of the parallelism of the axis of Venus in her orbit. He proposed to trace a meridian line through the whole extent of Italy. He was admitted a foreign Associate in the Paris Academy of Sciences, in 1706; and he had many astronomical dissertations inserted in their Memoirs, particularly in those of the years 1702, 1703, 1704, 1706, 1707, 1708, 1713, and 1718.—Bianchiui died the 2d of March 1729, in the 67th year of his age.</p><p>BIMEDIAL <hi rend="italics">Line,</hi> is the sum of two Medials. Euclid reckons two of these bimedials, in pr. 38 and 39 lib. X; the first is when the rectangle is rational, which is contained by the two medial lines whose sum makes the bimedial; and the second when that rectangle is a medial, or contained under two lines that are commensurable only in power.</p><p>BINARY Number, that which is composed of two units.</p><p><hi rend="smallcaps">Binary</hi> <hi rend="italics">Arithmetic,</hi> that in which two figures or characters, viz, 1 and 0, only are used; the cipher multiplying every thing by 2, as in the common arithmetic by ten: thus, 1 is one, 10 is 2, 11 is 3, 100 is 4, 101 is 5, 110 is 6, 111 is 7, 1000 is 8, 1001 is 9, 1010 is ten; being founded on the same principles as common arithmetic.—This sort of arithmetic was invented by Leibnitz, who pretended that it is better adapted than the common arithmetic, for discovering certain properties of numbers, and for constructing tables; but he does not venture to recommend it, for ordinary use, on account of the great number of places of figures requisite to express all numbers, even very small ones. Jos. Pelican of Prague has more largely explained the principles and practice of the binary arithmetic, in a book entitled <hi rend="italics">Arithmeticus Perfectus, qui tria numerare nescit;</hi> 1712. And De Lagni proposed a new system of logarithms, on the plan of the binary arithmetic; which he finds shorter, and more easy and natural than the common ones.</p></div1><div1 part="N" n="BINOCLE" org="uniform" sample="complete" type="entry"><head>BINOCLE</head><p>, or <hi rend="smallcaps">Binocular Telescope</hi>, is one by which an object is viewed with both eyes at the same time. It consists of two tubes, each furnished with glasses of the same power, by which means it has been said to shew objects larger and more clearly than a mo- <cb/> nocular or single telescope; though this is probably only an illusion, occasioned by the stronger impression which two equal images, alike illuminated, make upon the eyes; and they have been found more embarrassing than useful in practice. This telescope has been chiefly treated of by the fathers Reita and Cherubin of Orleans.—There are also microscopes of the same kind, though but little used; being subject to the same inconveniences as the telescopes.</p></div1><div1 part="N" n="BINOMIAL" org="uniform" sample="complete" type="entry"><head>BINOMIAL</head><p>, a quantity consisting of two terms or members connected by the sign + or -, viz, plus or minus; as <hi rend="italics">a</hi>+<hi rend="italics">b,</hi> or 3<hi rend="italics">a</hi>-2<hi rend="italics">c,</hi> or <hi rend="italics">a</hi><hi rend="sup">2</hi>+<hi rend="italics">b,</hi> or <hi rend="italics">x</hi><hi rend="sup">2</hi>-2√<hi rend="italics">c,</hi> &c; denoting the sum or the difference of the two terms. But the difference is also sometimes named a <hi rend="italics">residual,</hi> and by Euclid an <hi rend="italics">apotome.</hi> The term binomial was first introduced by Robert Recorde; see his algebra, pa. 46<hi rend="sup">2</hi>.</p><p><hi rend="smallcaps">Binomial</hi> <hi rend="italics">Line,</hi> or <hi rend="italics">Surd,</hi> is that in which at least one of the parts is a surd. Euclid enumerates six kinds of binomial lines or surds, in the 10th book of his Elements, which are exactly similar to the six residuals or apotomes there treated of also, and of which an account is given under the art. <hi rend="smallcaps">Apotome</hi>, which see. Those apotomes become binomials by only changing the sign of the latter term from minus to plus, which therefore are as below. <hi rend="center"><hi rend="italics">Euclid's</hi> 6 <hi rend="italics">Binomial Lines.</hi></hi> <hi rend="center">First binomial 3+√5,</hi> <hi rend="center">2d binomial √18+4,</hi> <hi rend="center">3d binomial √24+√18,</hi> <hi rend="center">4th binomial 4+√3,</hi> <hi rend="center">5th binomial √6+2,</hi> <hi rend="center">6th binomial √6+√2.</hi></p><p><hi rend="italics">To extract the Square Root of a Binomial,</hi> as of <hi rend="italics">a</hi>+√<hi rend="italics">b,</hi> or √<hi rend="italics">c</hi>+√<hi rend="italics">b.</hi> Various rules have been given for this purpose. The first is that of Lucas De Burgo, in his <hi rend="italics">Summa de Arith.</hi> &c, which is this: When one part, as <hi rend="italics">a,</hi> is rational, divide it into two parts such, that their product may be equal to 1/4th of the number under the radical <hi rend="italics">b</hi>; then shall the sum of the roots of those parts be the root of the binomial sought: or their difference is the root when the quantity is residual. That is, if <hi rend="italics">c</hi>+<hi rend="italics">e</hi>=<hi rend="italics">a,</hi> and <hi rend="italics">ce</hi>=(1/4)<hi rend="italics">b</hi>; then is the root sought. As if the binomial be 23+√448; then the parts of 23 are 16 and 7, and their product is 112, which is 1/4th of 448; therefore the sum of their roots 4+√7 is the root sought of 23+√448.</p><p>De Burgo gives also another rule for the same extractions, which is this: The given binomial being, for example, √<hi rend="italics">c</hi>+√<hi rend="italics">b,</hi> its root will be ; conseq. 4+√7 is the root sought, as before. Again, if the binomial be √18+√10; here <hi rend="italics">c</hi>=18, <pb n="207"/><cb/> and <hi rend="italics">b</hi>=10; theref. is the root of √18+√10 sought. And this latter rule has been used by all other authors, down to the present time. <hi rend="italics">To extract the Cubic and other higher Roots of a Binomial.</hi> This is useful in resolving cubic and higher equations, and was introduced with the resolution of those equations by Tartalea and Cardan. The rules for such extractions are in great measure tentative; and some of the principal ones are the following.</p><p><hi rend="italics">Tartalea's Rule for the Cube Root of a Binomial p</hi>+<hi rend="italics">q.</hi> This rule is given in his 9th book of Miscellaneous Questions, quest. 40; and it is made out from either of the terms, <hi rend="italics">p</hi> or <hi rend="italics">q,</hi> of the binomial, taken fingly, in this manner: Separate either term, as <hi rend="italics">p,</hi> into two such parts that the one of them may be a cubic number, and the other part divisible by 3 without a remainder; then the cube root of the said cubic part will be one term of the root, and the other term will be the square root of the quotient arising from dividing the aforesaid third part by the first term just found. So if <hi rend="italics">p</hi> be divided into <hi rend="italics">r</hi><hi rend="sup">3</hi>+3<hi rend="italics">s,</hi> then the root is <hi rend="italics">r</hi>+√<hi rend="italics">s/r</hi>. For example, to extract the cube root of √108+10. Suppose the part 10 be taken: this separates into the parts 1 and 9, the former of which is a cube, and the latter divisible by 3; that is <hi rend="italics">r</hi><hi rend="sup">3</hi>=1, and 3<hi rend="italics">s</hi>=9; hence <hi rend="italics">r</hi>=1, and <hi rend="italics">s</hi>=3; consequently is the cubic root of √108+10 sought. Again, to use the other term √108: this divides into √27+√27, of which the former is a cube, and the latter divisible by 3; that is, the cube root, the same as before.</p><p><hi rend="italics">Bombelli's Rule for the Cubic Root of the Binomial a</hi>+√-<hi rend="italics">b.</hi> First find ; then, by trials, search out a number <hi rend="italics">c,</hi> and a square root √<hi rend="italics">d,</hi> such that the sum of their squares ; then shall <hi rend="italics">c</hi>+√-<hi rend="italics">d</hi> be the cube root of <hi rend="italics">a</hi>+√-<hi rend="italics">b</hi> sought. For example, to find the cube root of 2+√-121: here ; then taking <hi rend="italics">c</hi>=2, and <hi rend="italics">d</hi>=1, it is , and , as it ought; therefore 2+√-1 is the cube root of 2+√-121 sought.—Bombelli gave also a rule for the cube root of the binomial <hi rend="italics">a</hi>+√<hi rend="italics">b,</hi> but it is good for nothing.</p><p><hi rend="italics">Albert Girard's Rule for the Cube Root of a Binomial.</hi> This is given in his <hi rend="italics">Invention Nouvelle en l'Algebre,</hi> and is explained by him thus: Let 72+√5120 be the given binomial whose cubic root is sought. <cb/> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">The square of 72 the greater term is</cell><cell cols="1" rows="1" rend="align=right" role="data">5184</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and of the less term is</cell><cell cols="1" rows="1" rend="align=right" role="data">5120</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">their difference</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">its cube root</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell></row></table> which 4 must be the difference between the squares of the two terms of the root sought; and as the rational part 72 of the given binomial is the greater term, therefore the rational part of the required root <table><row role="data"><cell cols="1" rows="1" role="data">2+√0</cell></row><row role="data"><cell cols="1" rows="1" role="data">3+√5</cell></row><row role="data"><cell cols="1" rows="1" role="data">4+√12</cell></row><row role="data"><cell cols="1" rows="1" role="data">5+√20</cell></row></table> will be the greater part also; consequently the cubic root sought must be one of the binominals here set in the margin, where the difference of the squares of the terms is always 4, as required; and to find out which of them it must be, proceed thus: The first, 2+√0 must be rejected, because one term of it is 0 or nothing; also because 5 exceeds the cube root of 72, or √20 exceeds the cube root of √5120, therefore 5+√20, and all after it must be rejected too; so that the root must be either 3+√5 or 4+√12, if the given quantity has a binomial root: to know which of these is to be taken, it must be considered that the rational term of the root must measure the rational term given; and also the irrational term of the root must measure the irrational term given; then, on examination it is found that both 3 and 4 measure or divide the 72 without a remainder, but that only the √5, and not √12, measures √5120; consequently none but 3+√5 can be the cube root of the given quantity 72+√5120; which is found to answer, by cubing the said root 3+√5.</p><p><hi rend="italics">Dr. Wallis's Rule for the Cube Root of a Binomial a</hi>±<hi rend="italics">m</hi>√<hi rend="italics">b or a</hi>±<hi rend="italics">m</hi>√-<hi rend="italics">b.</hi> In these forms the greatest rational part <hi rend="italics">m</hi> is extracted out of the radical part, leaving only <hi rend="italics">b</hi> the least radical part possible under the radical sign. He then observes that if the given quantity have a binomial root, it must be of this form <hi rend="italics">c</hi>±<hi rend="italics">n</hi>√<hi rend="italics">b,</hi> with the same radical <hi rend="italics">b.</hi> Then to find the value of <hi rend="italics">c</hi> and <hi rend="italics">n,</hi> he raises this root to the 3d power, which gives , which must be=<hi rend="italics">a</hi>±<hi rend="italics">m</hi>√<hi rend="italics">b</hi> the given quantity; hence putting the rational part of the one quantity equal to that of the other, and also the radical part of the one equal to that of the other, gives . Then assuming several values of <hi rend="italics">n,</hi> from the last equation he finds the value of <hi rend="italics">c</hi>; hence if these values of <hi rend="italics">c</hi> and <hi rend="italics">n,</hi> substituted in the first equation, make it just, they are right; but if not, another value of <hi rend="italics">n</hi> is assumed, and so on, till the first equation hold true. And it is to be noted that <hi rend="italics">n</hi> is always an integer or else the half of an integer. For example, if the cube root of 135 ± √1825 be required, or 135±78√3; here <hi rend="italics">a</hi>=135, <hi rend="italics">m</hi>=78, and <hi rend="italics">b</hi>=3; hence ; then assuming <hi rend="italics">n</hi>=1, this last equation becomes , from whence <hi rend="italics">c</hi> is found=5; which values of <hi rend="italics">c</hi> and <hi rend="italics">n</hi> being substituted in the first equation , makes , but ought to be 135, shewing that <hi rend="italics">c</hi> is too great, and consequently <hi rend="italics">n</hi> taken too little. Let <hi rend="italics">n</hi> therefore be assumed=2, so shall , and <hi rend="italics">c</hi>=3; and the first equation becomes as it ought, which shews that the true value of <hi rend="italics">n</hi> is 2, and that of <hi rend="italics">c</hi> is 3; hence then the cube root of 135±78√3 or <hi rend="italics">c</hi>±<hi rend="italics">n</hi>√<hi rend="italics">b</hi> is <pb n="208"/><cb/> 3±2√3 or 3±√12. And in like manner is the process instituted when the number in the radical is negative, as the cube root of 81±30√-3, which is (9/2)±(1/2)√-3.</p><p>Another rule for extracting the cube root of an imaginary binomial was also given by Demoivre, at the end of Saunderson's Algebra, by means of the trisection of an arc or angle.</p><p><hi rend="italics">Sir I. Newton's Rule for any Root of a Binomial a</hi>±<hi rend="italics">b.</hi> In his Universal Arith. is given a rule for the square root of a binomial, which is the same as the 2d of Lucas de Burgo, before given; and also a general rule for any root of a binomial, which I have not met with elsewhere; and it is this: Of the given quantity <hi rend="italics">a</hi>±<hi rend="italics">b,</hi> let <hi rend="italics">a</hi> be the greater term, and <hi rend="italics">c</hi> the index of the root to be extracted. Seek the least number <hi rend="italics">n</hi> whose power <hi rend="italics">n</hi><hi rend="sup"><hi rend="italics">c</hi></hi> can be divided by <hi rend="italics">aa</hi>-<hi rend="italics">bb</hi> without a remainder, and let the quotient be <hi rend="italics">q</hi>; Compute in the nearest integer number, which call <hi rend="italics">r</hi>; divide <hi rend="italics">a</hi>√<hi rend="italics">q</hi> by its greatest rational divisor, calling the quotient <hi rend="italics">s</hi>; and let the nearest integer number above be the root sought, if the root can be extracted. And this rule is demonstrated by <hi rend="italics">s'Gravesande</hi> in his commentary on Newton's Arithmetic. And many numeral examples, illustrating this rule, are given in s'Gravesande's Algebra, abovementioned, pa. 160, as also in Newton's Univers. Arith. pa. 53 2d edit. and in Maclaurin's Algebra pa. 118. Other rules may be found in Schooten's Commentary on the Geometry of Descartes, and elsewhere.</p><p><hi rend="italics">Impossible</hi> or <hi rend="italics">Imaginary</hi> <hi rend="smallcaps">Binomial</hi>, is a binomial which has one of its terms an impossible or an imaginary quantity; as <hi rend="italics">a</hi>+√-<hi rend="italics">b.</hi></p><p><hi rend="smallcaps">Binomial</hi> <hi rend="italics">Curve,</hi> is a curve whose ordinate is expressed by a binomial quantity; as the curve whose ordinate is . Stirling, Method. Diff. pa. 58.</p><p><hi rend="smallcaps">Binomial</hi> <hi rend="italics">Theorem,</hi> is used to denote the celebrated theorem given by Sir I. Newton for raising a binomial to any power, or for extracting any root of it by an approximating infinite series. It was known by Stifelius, and others, about the beginning of the 16th century, how to raise the integral powers, not barely by a continued multiplication of the binomial given, but Stifelius formed also a table of numbers by a continued addition, which shewed by inspection the coefficients of the terms of any power of the binomial, contained within the limits of the table; but still they could not independent of a table, and of any of the lower powers, raise any power of a binomial at once, by determining its terms one from another only, viz, the 2d term from the 1st, the 3d from the 2d, and so on as far as we please, by a general rule; and much less could they extract general algebraic roots in infinite series by any rule whatever.</p><p>For although the nature and construction of that table, which is a table of figurate numbers, was so early known, and employed in the raising of powers, and the extracting the roots of pure numbers; yet it was only by raising the numbers one from another by con tinual additions, and then taking them from the table <cb/> when wanted, till Mr. Briggs first pointed out the way of raising any line in the table by itself, without any of the preceding lines; and thus teaching to raise the terms of any power of a binomial, independent of any of the other powers; and so gave the substance of the binomial theorem in words, wanting only the algebraic notation in symbols; as is shewn at large at pa. 75 of the historical introduction to my Mathematical Tables. Whatever was known however of this matter, related only to pure or integral powers, no one before Newton having thought of extracting roots by infinite series. He happily discovered that, by considering powers and roots in a continued series, roots being as powers having fractional exponents, the same binomial series would equally serve for them all, whether the index should be fractional or integral, or whether the series be finite or infinite. The truth of this method however was long known only by trial in particular cases, and by induction from analogy; nor does it appear that even Newton himself ever attempted any direct proof of it: however, various demonstrations of the theorem have since been given by the more modern mathematicians, some of which are by means of the doctrine of fluxions, and others, more legally, from the pure principles of algebra only: for a full account of which, see pa. 71 &c, of my Mathematical Tracts, vol. 1.</p><p>This theorem was first discovered by Sir I. Newton in 1669, and sent in a letter of June 13, 1676, to Mr. Oldenburgh, Secretary to the Royal Society, to be by him communicated to Mr. Leibnitz; and it was in this form: : where <hi rend="italics">p</hi> + <hi rend="italics">pq</hi> signifies the quantity whose root, or power, or root of any power, is to be found; <hi rend="italics">p</hi> being the first term of that quantity; <hi rend="italics">q</hi> the quotient of all the rest of the terms divided by that first term; and <hi rend="italics">m/n</hi> the numeral index of the power or root of the quantity <hi rend="italics">p</hi>+<hi rend="italics">pq,</hi> whether it be integral or fractional, positive or negative; and lastly <hi rend="italics">a, b, c, d,</hi> &c, are assumed to denote the several terms in their order as they are found, viz, <hi rend="italics">a</hi>=the first term <hi rend="italics">p</hi>(<hi rend="italics">m</hi>/n), <hi rend="italics">b</hi>=the 2d term (<hi rend="italics">m/n</hi>)<hi rend="italics">aq, c</hi>=the 3d term (<hi rend="italics">m</hi>-<hi rend="italics">n</hi>)/2<hi rend="italics">n bq,</hi> and so on. As Newton's general notation of indices was not commonly known, he takes this occasion to explain it; and then he gives many examples of the application of this theorem, one of which is the following.</p><p><hi rend="italics">Ex.</hi> 1. To find the value of , that is, to extract the square root of <hi rend="italics">c</hi><hi rend="sup">2</hi>+<hi rend="italics">x</hi><hi rend="sup">2</hi> in an infinite series. Here , &c; and therefore the root sought is . <pb n="209"/><cb/></p><p>A variety of other examples are also given in the same place, by which it is shewn that the theorem is of universal application to all sorts of quantities whatever.</p><p>This theorem is sometimes represented in other forms, as ; which comes to the same thing. Or also thus .</p><p>In another letter to Mr. Oldenburgh, of Oct. 24, 1676, Newton explains the train of reasoning by which he obtained the said theorem, as follows: “In the beginning of my mathematical studies, when I was perusing the works of the celebrated Dr. Wallis (see his Arith. of Insinites, prop. 118, and 121, also his Algebra chap. 82), and considering the series by the interpolation of which he exhibits the area of the circle and hyperbola; for instance, in this series of curves, whose common base or axis is <hi rend="italics">x,</hi> and the ordinates respectively ; I perceived that if the areas of the alternate curves, which are <hi rend="italics">x,</hi> , &c; could be interpolated, we should obtain the areas of the intermediate ones; the first of which, or , is the area of the circle: now in order to this, it appeared that in all the series the first term was <hi rend="italics">x;</hi> that the 2d terms (0/3)<hi rend="italics">x</hi><hi rend="sup">3</hi>, (1/3)<hi rend="italics">x</hi><hi rend="sup">3</hi>, (2/3)<hi rend="italics">x</hi><hi rend="sup">3</hi>, (3/3)<hi rend="italics">x</hi><hi rend="sup">3</hi>, &c, were in arithmetical progression; and consequently that the first two terms of all the series to be interpolated would be , &c.</p><p>“Now for the interpolation of the rest, I considered that the denominators 1, 3, 5, 7, &c, were in arithmetical progression; and that therefore only the numeral coefficients of the numerators were to be investigated. But these in the alternate areas, which are given, were the same with the figures of which the several powers of 11 consist, viz, of 11<hi rend="sup">0</hi>, 11<hi rend="sup">1</hi>, 11<hi rend="sup">2</hi>, 11<hi rend="sup">3</hi> &c; that is, the first 1, <hi rend="center">the second 1, 1</hi> <hi rend="center">the third 1, 2, 1</hi> <hi rend="center">the fourth 1, 3, 3, 1</hi> <hi rend="center">the fifth 1, 4, 6, 4, 1</hi> <hi rend="center">&c.</hi></p><p>“I enquired therefore how, in these series, the rest of the terms may be derived from the first two being given; and I found that by putting <hi rend="italics">m</hi> for the 2d figure or term, <cb/> the rest would be produced by the continued multiplication of the terms of this series, .</p><p>“For instance, if the 2d term <hi rend="italics">m</hi>=4; then shall , or 6, be the 3d term; and , or 4, the 4th term; and , or 1, the 5th term; and , or 0, the 6th; which shew<*> that in this case the series terminates.</p><p>“This rule therefore I applied to the series to be interpolated. And since, in the series for the circle, the 2d term was (1/2)<hi rend="italics">x</hi><hi rend="sup">3</hi>/3 I put <hi rend="italics">m</hi>=1/2, which produced the terms ; and so on <hi rend="italics">ad infinitum.</hi> And hence I found that the required area of the circular segment is .</p><p>“And in the same manner might be produced the interpolated areas of the other curves: as also the area of the hyperbola and the other alternates in this series . And in the same way also may other series be interpolated, and that too if they should be taken at the distance of two or more terms.</p><p>“This was the way then in which I first entered upon these speculations; which I should not have remembered, but that in turning over my papers a few weeks since, I chanced to cast my eyes on those relating to this matter.</p><p>“Having proceeded so far, I considered that the terms , &c, that is, 1 , &c, might be interpolated in the same manner as the areas generated by them: and for this, nothing more was required but to omit the denominators 1, 3, 5, 7, &c, in the terms expressing the areas; that is, the coefficients of the terms of the quantity to be interpolated, , will be produced by the continued multiplication of the terms of this series .</p><p>“Thus, for example, there would be found <pb n="210"/><cb/></p><p>“Thus then I discovered a general method of reducing radical quantities into infinite series, by the theorem which I sent in the beginning of the former letter, before I knew the same by the extraction of roots.</p><p>“But having discovered that way, this other could not long remain unknown: for, to prove the truth of those operations, I multiplied &c, by itself, and the product is 1 - <hi rend="italics">x</hi><hi rend="sup">2</hi>, all the rest of the terms vanishing after these, <hi rend="italics">in infinitum.</hi> In like manner, &c, twice multiplied by itself, produced 1 - <hi rend="italics">x</hi><hi rend="sup">2</hi>. But as this was a certain proof of those conclusions, so I was naturally led to try conversely whether these series, which were thus known to be the roots of the quantity 1-<hi rend="italics">x</hi><hi rend="sup">2</hi>, could not be extracted out of it after the manner of arithmetic; and upon trial I found it to succeed. The process for the square root is here set down</p><p>“These methods being found, I laid aside the other way by interpolation of series, and used these operations only as a more genuine foundation. Neither was I ignorant of the reduction by division, which is so much easier.” See Collins's <hi rend="italics">Commercium Epistolicum.</hi></p><p>And this is all the account that Newton gives of the invention of this theorem, which is engraved on his monument in Westminster Abbey, as one of his greatest discoveries.</p></div1><div1 part="N" n="BIPARTIENT" org="uniform" sample="complete" type="entry"><head>BIPARTIENT</head><p>, is a number that divides another into two equal parts without a remainder. So 2 is a bipartient to 4, and 5 a bipartient to 10.</p></div1><div1 part="N" n="BIPARTITION" org="uniform" sample="complete" type="entry"><head>BIPARTITION</head><p>, is a division into two equal parts.</p></div1><div1 part="N" n="BIQUADRATE" org="uniform" sample="complete" type="entry"><head>BIQUADRATE</head><p>, or <hi rend="smallcaps">Biquadratic</hi> <hi rend="italics">Power,</hi> is the squared square, or 4th power of any number or quantity. Thus 16 is the biquadrate or 4th power of 2, or it is the square of 4 which is the 2d power of 2.</p><p><hi rend="smallcaps">Biouadratic</hi> <hi rend="italics">Root,</hi> of any quantity, is the square root of the square root, or the 4th root of that quantity. So the biquadratic root of 16 is 2, and the biquadratic root of 81 is 3.</p><p><hi rend="smallcaps">Biquadratic</hi> <hi rend="italics">Equation,</hi> is that which rises to 4 dimensions, or in which the unknown quantity rises to the 4th power; as .</p><p>Any biquadratic equation may be conceived to be generated or produced from the continual multiplication of four simple equations, as ; or from that of two quadratic equations, as ; or, lastly, from that of a cubic and a simple equation, as : which was the invention of Harriot. And, on the contrary, a biquadratic equation may be resolved into four simple equations, or into two quadratics, or into a cubic and a simple equation, having all the same roots with it. <cb/> <hi rend="center">1. <hi rend="italics">Ferrari's Method for Biquadratic Equations:</hi></hi></p><p>The first resolution of a biquadratic equation was given in Cardan's Algebra, chap. 39, being the invention of his pupil and friend Lewis Ferrari, about the year 1540. This is effected by means of a cubic equation, and is indeed a method of depressing the biquadratic equation to a cubic, which Cardan demonstrates, and applies in a great variety of examples. The principle is very general, and consists in completing one side of the equation up to a square, by the help of some multiples or parts of its own terms and an assumed unknown quantity; which it is always easy to do; and then the other side is made to be a square also, by assuming the product of its 1st and 3d terms equal to the square of half the 2d term; for it consists only of three terms, or three different denominations of the original letter; then this equality will determine the value of the assumed quantity by a cubic equation: other circumstances depend on the artist's judgment. But the method will be farther explained by the following examples, extracted from Cardan's book.</p><p><hi rend="italics">Ex.</hi> 1. Given , to be resolved. Add 6<hi rend="italics">x</hi><hi rend="sup">2</hi> to both sides of the equation, so shall . Assume <hi rend="italics">y,</hi> and add to both sides, then is . Make now the , this gives , or ; and hence . From which <hi rend="italics">x</hi> may be found by a quadratic equation.</p><p><hi rend="italics">Ex.</hi> 2. Given . Before applying Ferrari's method to this example, Cardan resolves it by another way as follows: subtract 1, then is ; divide by <hi rend="italics">x</hi> + 1, then is ; and hence .</p><p>But to resolve it by Ferrari's rule: Because . theresore ; hence ; and the root is : by means of which <hi rend="italics">x</hi> is found by a quadratic equation.</p><p><hi rend="italics">Ex.</hi> 3. Given .—Add 240, then ; complete square again, then ; make the last side a sq. by the rule, which gives . Put now , and the last transforms to ; then the value of <hi rend="italics">z</hi> found from this, gives the value of <hi rend="italics">y,</hi> and hence the value of <hi rend="italics">x,</hi> as before. <pb n="211"/><cb/> <hi rend="center">2. <hi rend="italics">Descartes's Rule for Biquadratic Equations.</hi></hi></p><p>Another solution was given of biquadratic equations by Descartes, in the 3d book of his Geometry. In this solution he resolved the given biquadratic equation into two quadratics, by means of a cubic equation, in this manner: First, let the 2d term or 3d power be taken away out of the equation, after which it will stand thus, . Find <hi rend="italics">y</hi> in this , and these values of <hi rend="italics">x</hi> will be the roots of the given biquadratic equation.</p><p><hi rend="italics">Ex.</hi> Let the equ. be , Hence <hi rend="italics">p</hi> = - 17, <hi rend="italics">q</hi> = - 20, & <hi rend="italics">r</hi> = - 6; and the cubic equ. is , the root of which is <hi rend="italics">y</hi><hi rend="sup">2</hi> = 16, or <hi rend="italics">y</hi> = 4; , the four roots of which are 2 ± √7 and - 2 ± √2. <hi rend="center">3. <hi rend="italics">Euler's Method for Biquadratic Equations.</hi></hi></p><p>The celebrated Leonard Euler gave, in the 6th volume of the Petersburgh Ancient Commentaries, for the year 1738, an ingenious and general method of resolving equations of all degrees, by means of the equation of the next lower degree, and among them of the biquadratic equation by means of the cubic; and this last was also given more at large in his treatise of Algebra, translated from the German into French in 1774, in 2 volumes 8vo. The method is this: Let , be the given biquadratic equation, wanting the 2d term. Take with which values of <hi rend="italics">f, g, h,</hi> form the cubic equation . find the three roots of this cubic equation, and let them be called <hi rend="italics">p, q, r.</hi> Then shall the four roots of the proposed biquadratic be these following, viz, <table><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">When (1/8)<hi rend="italics">b</hi> is positive:</cell><cell cols="1" rows="1" rend="align=right" role="data">When (1/8)<hi rend="italics">b</hi> is negative:</cell></row><row role="data"><cell cols="1" rows="1" role="data">1st.</cell><cell cols="1" rows="1" role="data">√<hi rend="italics">p</hi> + √<hi rend="italics">q</hi> + √<hi rend="italics">r</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">√<hi rend="italics">p</hi> + √<hi rend="italics">q</hi> - √<hi rend="italics">r</hi>,</cell></row><row role="data"><cell cols="1" rows="1" role="data">2d.</cell><cell cols="1" rows="1" role="data">√<hi rend="italics">p</hi> + √<hi rend="italics">q</hi> - √<hi rend="italics">r</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">√<hi rend="italics">p</hi> - √<hi rend="italics">q</hi> + √<hi rend="italics">r</hi>,</cell></row><row role="data"><cell cols="1" rows="1" role="data">3d.</cell><cell cols="1" rows="1" role="data">√<hi rend="italics">p</hi> - √<hi rend="italics">q</hi> + √<hi rend="italics">r</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">-√<hi rend="italics">p</hi> + √<hi rend="italics">q</hi> + √<hi rend="italics">r</hi>,</cell></row><row role="data"><cell cols="1" rows="1" role="data">4th.</cell><cell cols="1" rows="1" role="data">√<hi rend="italics">p</hi> - √<hi rend="italics">q</hi> - √<hi rend="italics">r</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">-√<hi rend="italics">p</hi> - √<hi rend="italics">q</hi> - √<hi rend="italics">r</hi>,</cell></row></table></p><p><hi rend="italics">Ex.</hi> Let the eq. be . Here <hi rend="italics">a</hi> = 25, <hi rend="italics">b</hi> = - 60, and <hi rend="italics">c</hi> = 36; theref. <hi rend="italics">f</hi> = 25/2, , and <hi rend="italics">h</hi> = 225/4. Conseq. the cubic equation will be . The three roots of which are <hi rend="italics">z</hi> = 9/4 = <hi rend="italics">p,</hi> and <hi rend="italics">z</hi> = 4 = <hi rend="italics">q,</hi> and <hi rend="italics">z</hi> = 25/4 = <hi rend="italics">r</hi>; the roots of which are √<hi rend="italics">p</hi> = 3/2, √<hi rend="italics">q</hi> = 2 or 4/2, √<hi rend="italics">r</hi> = 5/2. Hence, as the value of (1/8)<hi rend="italics">b</hi> is negative, the four roots are <cb/> . <hi rend="center">4. <hi rend="italics">Simpson's Rule for Biquadratic Equations.</hi></hi></p><p>Mr. Simpson gave also a general rule for the solution of biquadratic equations, in the 2d edit. of his Algebra, pa. 150, in which the given equation is also resolved by means of a cubic equation, as well as the two former ways; and it is investigated on the principle, that the given equation is equal to the difference between two squares; being indeed a kind of generalization of Ferrari's method.</p><p>Thus, he supposes the given equation, viz, ; then from a comparison of the like terms, the values of the assumed letters are found, and the final equation becomes , where . The value of A being found in this cubic equation, from it will be had the values of B and C, which have these general values, viz, . Hence, finally, the root <hi rend="italics">x</hi> will be obtained from the assumed equation , in four several values.</p><p><hi rend="italics">Ex.</hi> Given the equ. . Here <hi rend="italics">p</hi> = -6, <hi rend="italics">q</hi> = - 58, <hi rend="italics">r</hi> = -114, and <hi rend="italics">s</hi> = - 11, whence ; and therefore the cubic equation becomes , the root of which is A = 4. Hence then B or : and the quadratic equation becomes , the four roots of which are .</p><p>Mr. Simpson here subjoins an observation which it has since been found is erroneous, viz, that “The value of A, in this equation, will be <hi rend="italics">commensurate</hi> and <hi rend="italics">rational</hi> (and therefore the easier to be discovered), not only when all the roots of the given equation are <hi rend="italics">commensurate,</hi> but when they are <hi rend="italics">irrational</hi> and even <hi rend="italics">impossible;</hi> as will appear from the examples subjoined.” This is a strange reason for Simpson to give for the proof of a proposition; and it is wonderful that he fell upon no examples that disprove it, as the instances in which it holds true, are very few indeed, in comparison with the number of those in which it fails.</p><p><hi rend="italics">Note.</hi> In any biquadratic equation having all its terms, if 3/8 of the square of the coefficient of the 2d term be greater than the product of the coefficients of the 1st and 3d terms, or 3/8 of the square of the coefficient <pb n="212"/><cb/> of the 4th term be greater than the product of the coefficients of the 3d and 5th terms, or 4/9 of the square of the coefficient of the 3d term greater than the product of the coefficients of the 2d and 4th terms; then all the roots of that equation will be real and unequal; but if either of the said parts of those squares be less than either of those products, the equation will have imaginary roots.</p><p>For the construction of biquadratic equations, see <hi rend="italics">Construction.</hi> See also <hi rend="italics">Descartes's Geometry,</hi> with the <hi rend="italics">Commentaries of Schooten</hi> and others; <hi rend="italics">Baker's Geometrical Key; Slusius's Mesolabium; l'Hospital's Conic Sections; Wolfius's Elementa Matheseos; &c.</hi></p><p><hi rend="smallcaps">Biouadratic</hi> <hi rend="italics">Parabola,</hi> a curve of the 3d order, having two infinite legs, and expressed by one of these three equations, viz, <hi rend="italics">a</hi><hi rend="sup">3</hi><hi rend="italics">x</hi> = <hi rend="italics">y</hi><hi rend="sup">4</hi> , as in fig. 1, , as in fig. 2, , as in fig. 3; where <hi rend="italics">x</hi> = AP the absciss, <hi rend="italics">y</hi> = PQ the ordinate, <hi rend="italics">b</hi> = AB, <hi rend="italics">c</hi> = AC, and <hi rend="italics">a</hi> = a certain given quantity. <figure><head><hi rend="italics">Fig1</hi></head></figure> <figure><head><hi rend="italics">Fig2</hi></head></figure> <figure><head><hi rend="italics">Fig3</hi></head></figure></p><p>But the most general equation of this curve is the sollowing, which belongs to fig. 4, viz, <figure><head><hi rend="italics">Fig4</hi></head></figure> ; where <hi rend="italics">x</hi> = A<hi rend="italics">p</hi> or AP the absciss, and - <hi rend="italics">y</hi> or + <hi rend="italics">y</hi> is the ordinate <hi rend="italics">pm</hi> or PM, also <hi rend="italics">a, b, c, d, e,</hi> are constant quantities; the beginning of the absciss being at any point A in the indefinite line AP.</p><p>But if the beginning of the absciss A be where this line intersects the curve, as in fig. 5, then the nature of the curve will be defined by this equation , wherever the point <hi rend="italics">p</hi> is taken in the infinite line RS. <figure><head><hi rend="italics">Fig5</hi></head></figure> <figure><head><hi rend="italics">Fig6</hi></head></figure></p><p>When the curve has no serpentine part, as fig. 6, the equation is more simple, being in this case barely <cb/> . See <hi rend="smallcaps">Curve</hi> <hi rend="italics">Lines,</hi> and <hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Lines.</hi></p></div1><div1 part="N" n="BIQUINTILE" org="uniform" sample="complete" type="entry"><head>BIQUINTILE</head><p>, an aspect of the planets when the distance between them is 144 degrees, or twice the 5th part of 360 degrees.</p></div1><div1 part="N" n="BISECTION" org="uniform" sample="complete" type="entry"><head>BISECTION</head><p>, or <hi rend="smallcaps">Bissection</hi>, the division of a quantity into two equal parts, otherwise called <hi rend="italics">bipartition.</hi></p></div1><div1 part="N" n="BISSEXTILE" org="uniform" sample="complete" type="entry"><head>BISSEXTILE</head><p>, or <hi rend="italics">Leap-year,</hi> a year consisting of 366 days, and happening every 4th year, by the addition of a day in the month of February, which that year consists of 29 days. And this is done to recover the 6 hours which the sun takes up nearly in his course, more than the 365 days commonly allowed for it in other years.</p><p>The day thus added was by Julius Cæsar appointed to be the day before the 24th of February, which among the Romans was the 6th of the calends, and which on this occasion was reckoned twice; whence it was called the <hi rend="italics">bissextile.</hi></p><p>By the statute <hi rend="italics">De anno bissextile,</hi> 21 Hen. III, to prevent misunderstandings, the intercalary day and that next before it are to be accounted as one day.</p><p>To find what year of the period any given year is; divide the given year by 4; then if o remains, it is leap year; but if any thing remain, the given year is so many after leap year.</p><p>But the astronomers concerned in reforming the calendar in 1582, by order of pope Gregory XIII, observing that in 4 years the bissextile added 44 minutes more than the sun spent in returning to the same point of the ecliptic; and computing that in 133 years these fupernumerary minutes would form a day; to prevent any changes being thus insensibly introduced into the seasons, directed, that in the course of 400 years there should be three sextiles retrenched; so that every centesimal year, which is a leap year according to the Julian account, is a common year in the Gregorian account, unless the number of centuries can be divided by 4 without a remainder. So 1600 and 2000 are bissextile; but 1700, 1800, and 1900 are common years.</p><p>The Gregorian computation has been received in most foreign countries ever since the reformation of the calendar in 1582; excepting some northern countries, as sussia, &c. And by act of parliament, passed in 1751, it commenced in all the dominions under the crown of Great Britain in the year following; it being ordered by that act that the natural day next following the 2d of September, should be accounted the 14th; omitting the intermediate 11 days of the common calendar. The supernumerary day, in leap years, being added at the end of the month February, and called the 29th of that month.</p></div1><div1 part="N" n="BLACK" org="uniform" sample="complete" type="entry"><head>BLACK</head><p>, a colour so called, or rather a privation of all colour. This, it seems, arises from such a peculiar texture and situation of the superficial parts of a black body, that they absorb all or most of the rays of light, reflecting little or none to the eye: and hence it happens that black bodies, thus imbibing the rays, are always found to be hotter than those of a lighter colour. Dr. Franklin observes that black cloaths heat more, and dry sooner in the sun than white cloaths; that therefore black is a bad colour for cloaths in hot climates; but a fit colour for the linings of ladies' summer hats; <pb n="213"/><cb/> and that a chimney painted black, when exposed to the sun, will draw more strongly. Franklin's Experim. &c.—Dr. Watson, the present bishop of Landaff, covered the bulb of a thermometer with a black coating of Indian ink, and the consequence was that the mercury rose 10 degrees higher. Philos. Trans. vol. 63, pa. 40.—And a virtuoso of unsuspected credit assured Mr. Boyle, that in a hot climate by blacking the shells of eggs, and exposing them to the sun, he had seen them thus well roasted in a short time.</p></div1><div1 part="N" n="BLACKNESS" org="uniform" sample="complete" type="entry"><head>BLACKNESS</head><p>, the quality of a black body, as to colour; arising from its stifling or absorbing the rays of light, and reflecting little or none. In which sense it stands directly opposed to whiteness; which consists in such a texture of parts, as indifferently reflects all the rays thrown upon it, of whatever colour they may be,</p><p>Descartes, it seems, first rightly distinguished these causes of black and white, though he might be mistaken with respect to the general nature of light and colours. —Sir lsaac Newton shews, in his Optics, that to produce black colours, the corpuscles must be smaller than for exhibiting the other colours; because, where the sizes of the component particles of a body are greater, the light reflected is too much for constituting this colour: but when they are a little smaller than is requisite to reflect the white, and very faint blue of the first order, they will reflect so little light, as to appear intensely black; and yet they may perhaps reflect it variously to and fro within them so long, till it be stifled and lost.</p><p>And hence, it appears, why fire, or putrefaction, by dividing the particles of substances, turn them black: why small quantities of black substances impart their colours very freely, and intensely, to other substances, to which they are applied; the minute particles of these, on account of their very great number, easily overspreading the gross particles of others. Hence it also appears, why glass ground very elaborately on a copper-plate with sand, till it be well polished, makes the sand, with what is rubbed off from the copper and glass, become very black: also why blacks commonly incline a little towards a blueish colour; as may be seen by illuminating white paper with light reflected from black substances, when the paper usually appears of a blueish white; the reason of which is, that black borders on the obscure blue of the first order of colours, and therefore reflects more rays of that colour than of any other: and lastly why black substances do sooner than others become hot in the sun's light, and burn; an effect which may proceed partly from the multitude of refractions in a little space, and partly from the easy commotion among such minute particles.</p></div1><div1 part="N" n="BLAGRAVE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BLAGRAVE</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent mathematician, who flourished in the 16th and 17th centuries. He was the second son of John Blagrave, of Bulmarshcourt near Sunning in Berkshire, descended from an ancient family in that country. From a grammar school at Reading he was sent to St. John's college in Oxford, where he applied himself chiefly to the study of mathematics. From hence he retired to his patrimonial seat of Southcote-lodge near Reading, where he spent the rest of his life, in a retired manner, without marrying, that he might have more leisure to pursue his favourite studies; which he did with great <cb/> application and success. After a life thus spent in study, and in acts of benevolence to all around him, he died in the year 1611; and was buried at Reading in the church of St. Lawrence, where a sumptuous monument was erected to his memory.</p><p>He left the bulk of his fortune to the posterity of his three brothers, which were very numerous. There have been mentioned various acts of his benesicence in private life, for the encouragement of learning, the reward of merit, and the relief of distress. Some of these were the result of a quaint, humorous disposition, discovered chiefly in his legacies: One of these was 10 pounds left to be annually disposed of in the following manner: On Good-friday, the churchwardens of each of the three parishes of Reading send to the town-hall “one virtuous maid who has lived five years with her master:” there, in the presence of the magistrates these three virtuous maids throw dice for the ten pounds. The year following the two losers are returned with a fresh one, and again the third year, till each has had three chances. He also left an annuity to 80 poor widows, who should attend annually on Good-friday also, and hear a sermon, for the preaching of which he left ten shillings to the minister. He took care also for the maintenance of his servants, rewarding their diligence and fidelity, and providing amply for their support. Thus it appears he was not more remarkable for his scientific knowledge, than for his generosity and philanthropy. His works are,</p><p>1. A Mathematical Jewel. Lond. 1585, folio.</p><p>2. Of the Making and Use of the Familiar Staff. Lond. 1590, 4to.</p><p>3. Astrolabium Uranicum Generale. Lond. 1596, 4to.</p><p>4. The Art of Dyalling. Lond. 1609, 4to.</p></div1><div1 part="N" n="BLAIR" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BLAIR</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent chronologist, was educated at Edinburgh. Afterward, coming to London, he was for some time usher of a school in Hedge-lane. In 1754 he first published “The Chronology and History of the World, from the Creation to the year of Christ 1753; illustrated in 56 tables. In 1755 he was elected a fellow of the Royal Society, and in 1761 of the Society of Antiquaries. In 1756 he published a 2d edition of his Chronological Tables; and in 1768 an improved edition of the same with the addition of 14 maps of Ancient and Modern Geography, for illustrating the Tables of Chronology and History; to which is prefixed a Dissertation on the Progress of Geography. In 1757 he was appointed chaplain to the Princess Dowager of Wales, and Mathematical Tutor to the Duke of York; whom he attended in 1763 in a tour to the continent, from which they returned the year after. Dr. Blair had successively several good church livings: as, a prebendal stall at Westminster, the vicarage of Hinckley, and the rectory of Burton Coggles in Lincolnshire, all in 1761; the vicarage of St. Bride's in London, in 1771, in exchange for that of Hinckley; the rectory of St. John the Evangelist in Westminster, in 1776, in exchange for the vicarage of St. Bride's; in the same year the rectory of Horton near Colebrooke, Bucks. Dr. Blair died the 24th of June, 1782.</p></div1><div1 part="N" n="BLIND" org="uniform" sample="complete" type="entry"><head>BLIND</head><p>, an epithet applied to an animal deprived of the use of eyes; or one from whom light, colours, <pb n="214"/><cb/> and all the glorious objects of the visible creation, are intercepted by some natural or accidental cause.</p></div1><div1 part="N" n="BLINDNESS" org="uniform" sample="complete" type="entry"><head>BLINDNESS</head><p>, a privation of the sense of sight. The ordinary causes of blindness are, some external violence, vicious conformation, growth of a cataract, gutta serena, small-pox, &c; or a decay of the optic nerve; an instance of which we have in the Academy of Sciences, where, upon opening the eye of a person long blind, it was found that the optic nerve was extremely shrunk and decayed, and without any medulla in it. The more extraordinary causes of blindness are malignant stenches, poisonous juices dropped into the eye, baneful vermin, long confinement in the dark, or the like.</p><p>We find various recompenses for blindness, or substitutes for the use of eyes, in the wonderful sagacity of many blind persons, related by Zahnius in his <hi rend="italics">Oculus Artificialis,</hi> and others. In some, the defect has been supplied by a most excellent gift of remembering what they had seen before; others by a delicate nose or the sense of smelling; others by a very nice ear; and others again by an exquisite touch, or sense of feeling, which they have had in such perfection, that as it has been said of some, they learned to hear with their eyes, it may be said of these that they taught themselves to see with their hands.</p><p>Some have been able to perform all sorts of curious works in the nicest and most dexterous manner. Aldrovandus speaks of a sculptor, who had become blind at 20 years of age, and yet 10 years afterwards he made a perfect marble statue of Cosmo II de Medicis, and another of clay like pope Urban VIII. Bartholin speaks of a blind sculptor in Denmark, who, by mere touch, distinguished perfectly well all sorts of wood, and even colours; and father Grimaldi relates an instance of the same kind; besides the blind organist, lately living in Paris, who it was said did the same thing. What seems more extraordinary still, we are told, by authors of good report, of a blind guide, who used to conduct the merchants through the sands and desarts of Arabia: and a not less marvellous instance is now existing in this country, in one John Metcalf near Manchester, who became quite blind at a very early age; and yet passed many years of his life as a waggoner, and occasionally, as a guide in different roads during the night, or when the paths were covered with snow; and, what is stranger still, his present occupation is that of surveyor and projector of highways in difsicult and mountainous parts, particularly about Buxton, and the Peak in Derbyshire.</p><p>There are also many instances of blind men who have been highly distinguished for their mental and literary talents, not to speak of the poets Homer, Milton, Ossian, &c; of which we have a remarkable instance in the late Dr. Sanderson, professor of mathematics in the university of Cambridge, and in the present Dr. Henry Moyes, public lecturer in philosophy, who both of them lost their sight by the small pox at an age before they had any recollection; these men were well skilled in all branches of the mathematics, philosophy, and optics, &c, which they taught with the greatest reputation; besides the monument of fame which the former has left behind him in his mathematical and philosophical works. <cb/></p><p>The effects of a sudden recovery of sight in such as have been born blind, are also very remarkable: devoid of the experience of distance and figure arising from sight, they are liable to the greatest mistakes in this respect, in so much it has been said that they could not distinguish by the mere sight which was a cube and which a globe, without first touching them. Mr. Boyle mentions a gentleman of this sort, who having been restored to sight at eighteen years of age, was near going distracted with the joy: see Boyle's works abridg. vol. 1. pa. 4. See also a remarkable case of this kind in the Tatler, N° 55, vol. 1. And the gentleman couched by Mr. Cheselden had no ideas of colour, shape, or distance: though he knew the colours asunder in a good light during his blind state; yet when he saw them after he had been couched, the faint ideas he had of them before, were not sufficient for him to know them by afterwards: as to distance, his ideas were so deficient, that he thought all the objects he saw touched his eyes, as what he felt did his skin; and it was a considerable time before he could remember which was the dog and which the cat, though often informed, without feeling them.</p></div1><div1 part="N" n="BLINDS" org="uniform" sample="complete" type="entry"><head>BLINDS</head><p>, or <hi rend="smallcaps">Blindes</hi>, in Fortification, a kind of defence usually made of oziers or branches interwoven, and laid across between two rows of stakes, about a man's height, and 4 or 5 feet asunder. They are used particularly at the heads of trenches, when these are extended in front towards the glacis; serving to defend the workmen, and prevent the enemy from overlooking them.</p></div1><div1 part="N" n="BLOCKADE" org="uniform" sample="complete" type="entry"><head>BLOCKADE</head><p>, is the blocking up a place, by posting troops all about it at the avenues, to prevent supplies of men and provisions from getting into it; and thus starving it out, without forming any regular siege or attacks.</p></div1><div1 part="N" n="BLONDEL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BLONDEL</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, a celebrated French mathematician and military engineer. He was born at Ribemond in Picardy in 1617. While he was yet but young, he was chosen Regius Professor of Mathematies and Architecture at Paris. Not long after he was appointed governor to Lewis-Henry de Lomenix, Count de Brienne, whom he accompanied in his travels from 1652 to 1655, of which he published an account. He enjoyed many honourable employments, both in the navy and army; and was entrusted with the management of several negociations with foreign princes. He arrived at the dignity of marshal de camp, and counsellor of state, and had the honour to be appointed mathematical preceptor to the Dauphin. He was a member of the Royal Academy of Sciences, director of the Academy of Architecture, and lecturer to the Royal College: in all which he supported his character with dignity and applause. Blondel was no less versed in the knowledge of the Belles Lettres than in the mathematical sciences, as appears by the comparison he published between Pindar and Horace. He died at Paris the 22d of February 1686, in the 69th year of his age. His chief mathematical works were,</p><p>1. Cours d'Architecture. Paris, 1675, in folio.</p><p>2. Resolution des quatre principaux problemes d'Architecture. Paris, 1676, in folio.</p><p>3. Histoire du Calendrier Romain. Paris, 1682, in 4to. <pb n="215"/><cb/></p><p>4. Cours de Mathematiques. Paris, 1683, in 4to.</p><p>5. L'Art de jetter des Bombes. La Haye, 1685, in 4to.</p><p>Besides a New Method of fortifying Places, and other works.</p><p>Blondel had also many ingenious pieces inserted in the Memoirs of the French Academy of Sciences, particularly in the year 1666.</p></div1><div1 part="N" n="BLOW" org="uniform" sample="complete" type="entry"><head>BLOW</head><p>, in a general sense, denotes a stroke given either with the hand, a weapon, or an instrument. The effect of a blow is estimated like the force of percussion, and so is expressed by the velocity of the body multiplied by its weight.</p><p>BLOWING <hi rend="italics">of a Fire-arm,</hi> is when the vent or touch-hole is run or gullied, and becomes wide, so that the powder will flame out.</p></div1><div1 part="N" n="BLUE" org="uniform" sample="complete" type="entry"><head>BLUE</head><p>, one of the seven primitive colours of the rays of light, into which they are divided when refracted through a glass prism. See Newton's Optics &c. See also <hi rend="smallcaps">Chromatics.</hi></p></div1><div1 part="N" n="BLUENESS" org="uniform" sample="complete" type="entry"><head>BLUENESS</head><p>, that quality of a body, as to colour, from whence it is called blue; depending on such a size and texture of the parts that compose the surface of a body, as disposes them to reflect only the blue or azure rays of light to the eye.</p><p>The blueness of the sky is thus accounted for by De la Hire, after Da Vinci; viz, that a black body viewed through a thin white one, gives the sensation of blue, like the immense expanse viewed through the air illuminated and whitened by the sun. For the same reason he says it is, that soot mixed with white, makes a blue; for that white bodies, being always a little transparent, when mixing with a black behind, give the perception of blue. From the same principle too he accounts for the blueness of the veins on the surface of the skin, though the blood they are filled with be a deep red.</p><p>In the same manner was the blueness of the sky accounted for by many other of the earlier writers, as Fromondus, Funceius, Otto Guericke, and many others, together with several of the more modern writers, as Wolfius, Muschenbroek, &c. But in the explication of this phenomenon, Newton observes that all the vapours, when they begin to condense and coalesce into natural particles, become first of such a magnitude as to reflect the azure rays, before they can constitute clouds of any other colour. This being therefore the first colour they begin to reflect, must be that of the finest and most transparent skies, in which the vapours are not yet arrived at a grossness sufficient to reflect other colours.</p><p>Bouguer however ascribes this blueness of the sky to the constitution of the air itself, being of such a nature that these fainter-coloured rays are incapable of making their way through any considerable tract of it. And as to the blue shadows which were first observed by Buffon in the year 1742, he accounts for them by the aerial colour of the atmosphere, which enlightens these shadows, and in which the blue rays prevail; whilst the red rays are not reflected so soon, but pass on to the remoter regions of the atmosphere. And the Abbé Mazeas accounts for the phenomenon of blue shadows by the diminution of light; observing that, <cb/> of two shadows which were cast upon <*> white wall from an opaque body, illuminated by the moon and by a candle at the same time, that from the candle was reddish, while the other from the moon was blue. See Newton's Optics pa. 228, Bouguer Traité d'Optique pa. 368, Edinb. Ess. vol. 2 pa. 75, or Priestley's Hist. of Vision &c, pa. 436.</p><p>BOB <hi rend="italics">of a Pendulum,</hi> the same as the ball, which see.</p></div1><div1 part="N" n="BODY" org="uniform" sample="complete" type="entry"><head>BODY</head><p>, in Geometry, is a figure conceived to be extended in all directions, or what is usually said to consist of length, breadth, and thickness; being otherwise called a <hi rend="italics">Solid.</hi> A body is conceived to be formed or generated by the motion of a surface; like as a surface by the motion of a line, and a line by the motion of a point.—Similar bodies, or solids, are in proportion to each other, as the cubes of their like sides, or linear dimensions.</p><div2 part="N" n="Body" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Body</hi></head><p>, in Physics, or Natural Philosophy, a solid, extended, palpable substance; of itself merely passive, being indifferent either to motion or rest, and capable of any sort of motion or figure.</p><p>Body is composed, according to the Peripatetics, of matter, form, and privation; according to the Epicureans and Corpuscularians, of an assemblage of hooked, heavy atoms; according to the Cartesians, of a certain extension; and according to the Newtonians, of a system or association of solid, massy, hard, impenetrable, moveable particles, ranged or disposed in this or that manner; from which arise bodies of this or that form, and distinguished by this or that name. These elementary or component particles of bodies, they assert, must be perfectly hard, so as never to wear or break in pieces; which, Newton observes, is necessary, in order to the world's persisting in the same state, and bodies continuing of the same nature and texture in several ages.</p><p><hi rend="smallcaps">Body</hi> <hi rend="italics">of a piece of Ordnance,</hi> the part contained between the centre of the trunnions and the caseabel. This should always be more fortified or stronger than the rest. See <hi rend="smallcaps">Cannon.</hi></p><p><hi rend="smallcaps">Body</hi> <hi rend="italics">of the Place,</hi> in Fortification, denotes either the buildings inclosed, or more generally the inclosure itself. Thus, to construct the body of the place, is to fortify or inclose the place with bastions and curtains.</p><p><hi rend="smallcaps">Body</hi> <hi rend="italics">of a Pump,</hi> the thickest part of the barrel or pipe of a pump, within which the piston moves.</p></div2><div2 part="N" n="Bodies" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Bodies</hi></head><p>, <hi rend="italics">Regular</hi> or <hi rend="italics">Platonic,</hi> are those which have all their sides, angles, and planes, similar and equal. Of these there are only 5; viz, the <hi rend="italics">tetraedron,</hi> contained by 4 equilateral triangles; the <hi rend="italics">hexaedron</hi> or <hi rend="italics">cube,</hi> by 6 squares; the <hi rend="italics">octaedron,</hi> by 8 triangles; the <hi rend="italics">dodecaedron,</hi> by 12 pentagons; and the <hi rend="italics">icosaedron,</hi> by 20 triangles. <hi rend="center"><hi rend="italics">To form the five Regular Bodies.</hi></hi></p><p>Let the annexed sigures be exactly drawn on pasteboard, or stiff paper, and cut out from it by the extreme or bounding lines: then cut the others, or internal lines, only half through, so that the parts may be turned up by them, and then glued or otherwise fastened together with paste, sealing-wax, &c; so shall they form the respective body marked with the corresponding <pb n="216"/><cb/> number; viz, N° 1 the tetraedron, N° 2 the hexaedron or cube, N° 3 the octaedron, N° 4 the dodecaedron, and N° 5 the icosaedron. <figure/> <hi rend="center"><hi rend="italics">To find the Superficies or Solidity of the Regular Bodies.</hi></hi></p><p>1. Multiply the proper tabular area (taken from the following table) by the square of the linear edge of the solid, for the superficies.</p><p>2. Multiply the tabular solidity by the cube of the linear edge, for the solid content. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">Table of the Surfaces and Solidities of the five Regular Bodies, the linear edge being 1.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">No. of Faces</cell><cell cols="1" rows="1" rend="align=center" role="data">Names</cell><cell cols="1" rows="1" rend="align=center" role="data">Surfaces</cell><cell cols="1" rows="1" rend="align=center" role="data">Solidities</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">Tetraedron</cell><cell cols="1" rows="1" rend="align=right" role="data">1.73205</cell><cell cols="1" rows="1" rend="align=right" role="data">0.11785</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">Hexaedron</cell><cell cols="1" rows="1" rend="align=right" role="data">6.00000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.00000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">Octaedron</cell><cell cols="1" rows="1" rend="align=right" role="data">3.46410</cell><cell cols="1" rows="1" rend="align=right" role="data">0.47140</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">Dodecaedron</cell><cell cols="1" rows="1" rend="align=right" role="data">20.64573</cell><cell cols="1" rows="1" rend="align=right" role="data">7.66312</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">Icosaedron</cell><cell cols="1" rows="1" rend="align=right" role="data">8.66025</cell><cell cols="1" rows="1" rend="align=right" role="data">2.18169</cell></row></table> <cb/></p><p>For more particular properties, see each respective word. See also my large <hi rend="italics">Mensuration,</hi> pa. 248, edit. 2.</p><p>These bodies were called <hi rend="italics">platonic,</hi> because they were said to have been invented, or first treated of, by <hi rend="italics">Plato,</hi> who conceived certain mysteries annexed to them.</p></div2></div1><div1 part="N" n="BOFFRAND" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BOFFRAND</surname> (<foreName full="yes"><hi rend="smallcaps">Germain</hi></foreName>)</persName></head><p>, a celebrated French architect and engineer, was born at Nantes in Bretagne in 1667. He was brought up under Harduin Mansarad, who trusted him with conducting his greatest works. Boffrand was admitted into the French Academy of Architecture in 1709. Many German princes chose him for their architect, and raised considerable edifices on his plans. His manner of building approached that of Palladio; and there was much of grandeur in all his designs. As engineer and inspectorgeneral of bridges and highways, he directed and constructed a number of canals, sluices, bridges, and other mechanical works. He published a curious and useful book, containing the general principles of his art; with an account of the plans, profiles, and elevations of the principal works, which he executed in France and other countries. Boffrand died at Paris in 1755, dean of the Academy of Architecture, first engineer and inspectorgeneral of the bridges and highways, architect and administrator of the general hospital.</p></div1><div1 part="N" n="BOILING" org="uniform" sample="complete" type="entry"><head>BOILING</head><p>, or <hi rend="smallcaps">Ebullition</hi>, the bubbling up of any fluid, by the application of heat. This is, in general, occasioned by the discharge of an elastic vapour through the fluid that boils; whether that be common air, fixed air, or steam, &c. It is proved by Dr. Hamilton of Dublin, in his Essay on the ascent of vapour, that the boiling of water is occasioned by the lowermost particles of it being heated and rarefied into vapour, or steam; in consequence of this diminution of their specific gravity, they ascend through the surrounding heavier fluid with great velocity, lacerating and throwing up the body of water in the ascent, and so giving it the tumultuous motion called <hi rend="italics">boiling.</hi></p><p>That this is occasioned by elastic steam, and not by particles of fire or air, as some have imagined, is easily proved by the following fimple experiment: Take a common drinking glass filled with hot water, and invert it into a vessel of the same: then, as soon as the water in the vessel begins to boil, large bubbles will be seen to ascend in the glass, by which the water in it will be displaced, and there will soon be a continued bubbling from under its edge: but if the glass be then drawn up, so that its mouth may just touch the water, and a cloth wetted in cold water be applied to the outside, the elastic steam within it will be instantly condensed, upon which the water will ascend so as nearly to sill it again. Some small parts of air &c, that may happen to be lodged in the fluid, may also perhaps be expelled, as well as the rarefied steam. And this is particularly recommended as a method of purifying quicksilver, for making more accurately barometers and thermometers.</p><p>We commonly annex the idea of a certain very great degree of heat to the boiling of liquids, though often without reason; for different liquids boil with different degrees of heat; and any one given liquid also, under different pressures of the atmosphere. Thus, a vessel of tar being set over the fire till it boils, it is said a person may then put his hand into it without <pb n="217"/><cb/> injury: and by putting water under the receiver of an air-pump, and applying the flame of a candle or lamp under it, by gradually exhausting, the water is made to boil with always less and less degrees of heat; and without applying any heat at all, the water, or even the moisture about the bottom or edges of the receiver, will rise in an elastic vapour up into it, when the exhaustion is near completed.</p><p>Spirit of wine boils still sooner in vacuo than water. And Dr. Freind gives a table of the different times required to make several fluids boil by the same heat. See also Philos. Trans. N° 122.</p></div1><div1 part="N" n="BOMB" org="uniform" sample="complete" type="entry"><head>BOMB</head><p>, in Artillery, a shell, or hollow ball of castiron, having a large vent, by which it is filled with gun-powder, and which is fitted with a fuze or hollow plug to give fire by, when thrown out of a mortar, &c: about the time when the shell arrives at the intended place, the composition in the pipe of the fuze sets fire to the powder in the shell, which blows it all in pieces, to the great annoyance of the enemy, by killing the people, or firing the houses, &c. They are now commonly called <hi rend="italics">shells</hi> simply, in the English artillery.</p><p>These shells, or bombs, are of various sizes, from that of 17 or 18 inches diameter downwards. The very large ones are not used by the English, that of 13 inches diameter being the highest size now employed by them; the weight, dimensions, and other circumstances of them, and the others downwards, are as in the following table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Diameter of the Shell.</cell><cell cols="1" rows="1" rend="align=center" role="data">Weight of the Shell.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Powder to fill them.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Powder to burst them into most pieces.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">lbs.</cell><cell cols="1" rows="1" rend="align=right" role="data">lb.</cell><cell cols="1" rows="1" rend="align=center" role="data">oz.</cell><cell cols="1" rows="1" rend="align=right" role="data">lb.</cell><cell cols="1" rows="1" rend="align=right" role="data">oz.</cell></row><row role="data"><cell cols="1" rows="1" role="data">13 inch</cell><cell cols="1" rows="1" role="data">195</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">4 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">14 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">3 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 5 4/5 Royal</cell><cell cols="1" rows="1" role="data">14 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 4 3/5 Cohorn</cell><cell cols="1" rows="1" role="data">7 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">8 </cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell></row></table></p><p>Mr. Muller gives the following proportion for all shells. Dividing the diameter of the mortar into 30 equal parts, then the other dimensions, in 30ths of that diameter, will be thus: <table><row role="data"><cell cols="1" rows="1" role="data">Diameter of the bore, or mortar</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diameter of the shell,</cell><cell cols="1" rows="1" role="data">29 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diameter of the hollow sphere</cell><cell cols="1" rows="1" role="data">21</cell></row><row role="data"><cell cols="1" rows="1" role="data">Thickness of metal at the fuze hole</cell><cell cols="1" rows="1" role="data">3 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Thickness at the opposite part</cell><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diameter of the fuze hole</cell><cell cols="1" rows="1" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Weight of shell empty</cell><cell cols="1" rows="1" role="data">10/117<hi rend="italics">d</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Weight of powder to fill it</cell><cell cols="1" rows="1" role="data">2/473<hi rend="italics">d</hi></cell></row></table> where <hi rend="italics">d</hi> denotes the cube of the diameter of the bore in inches.—But shells have also lately been made with the metal all of the same thickness quite around.</p><p>In general, the windage, or difference between the diameter of the shell and mortar, is 1/60 of the latter; also the diameter of the hollow part of the shell is 7/10 of the same.</p><p>Bombs are thrown out of mortars, or howitzers; but they may also be thrown out of cannon; and a very small sort are thrown by the hand, which are called granados: and the Venetians at the siege of Candia, <cb/> when the Turks had possessed themselves of the ditch used large bombs without any piece of ordnance, but barely rolled them down upon the enemy along a plank set aslope, with ledges on the sides to keep the bomb right forwards.</p><p>Mr. Blondel, in his <hi rend="italics">Art de jetter des Bombes,</hi> says the first bombs were those thrown into the city of Watchtendonch in Guelderland, in 1588; and they are described by our countryman Lucar, in his book on Artillery published this same year 1588; though it is pretended by others that they were in use near a century before, namely at the siege of Naples in 1495. They only came into common use, however, in 1634, and then only in the Dutch and Spanish armies. It is said that one Malthus, an English engineer, was sent for from Holland by Lewis the 14th, who used them for him with much success, particularly at the siege of Cohoure in 1642.</p><p>The art of throwing bombs, or shells, forms a principal branch of Gunnery, founded on the theory of projectiles, and the quantities and laws of force of gunpowder. And the principal writers on this art are Mess. Blondel, Guisnée, De Ressons, De La Hire, &c.</p><p><hi rend="smallcaps">Bomb-Chest</hi>, is a kind of chest usually filled with bombs, and sometimes only with gunpowder, placed under ground, to blow it up into the air with those who stand upon it; being set on fire by means of a saucissee fastened at one end. But they are now much out of use.</p></div1><div1 part="N" n="BOMBARD" org="uniform" sample="complete" type="entry"><head>BOMBARD</head><p>, an ancient piece of ordnance, now out of use. It was very short and thick, with a large mouth; some of which it is said threw balls of 300 pounds weight, requiring the use of cranes to load them. The Bombard is by some called <hi rend="italics">basilisk,</hi> and by the Dutch <hi rend="italics">donderbus.</hi></p><p><hi rend="italics">To</hi> <hi rend="smallcaps">BOMBARD</hi>, is to attack by throwing of bombs, or shells.</p></div1><div1 part="N" n="BOMBARDIER" org="uniform" sample="complete" type="entry"><head>BOMBARDIER</head><p>, a person employed about throwing bombs or shells. He adjusts the fuze, and loads and fires the mortar.</p></div1><div1 part="N" n="BONES" org="uniform" sample="complete" type="entry"><head>BONES</head><p>, <hi rend="italics">Napier's.</hi> See <hi rend="smallcaps">Napier.</hi></p></div1><div1 part="N" n="BONING" org="uniform" sample="complete" type="entry"><head>BONING</head><p>, in Surveying and Levelling, &c, is the placing three or more rods or poles, all of the same length, in or upon the ground, in such a manner that the tops of them be all in one continued straight line, whether it be horizontal or inclined, so that the eye can look along the tops of them all, from one end of the line to the other.</p></div1><div1 part="N" n="BONNET" org="uniform" sample="complete" type="entry"><head>BONNET</head><p>, in Fortification, a small work of two faces, having only a parapet, with two rows of palisadoes at about 10 or 12 feet distance. It is commonly placed before the saliant angle of the counterscarp, and having a communication with the covered way, by means of a trench cut through the glacis, and palisadoes on each side.</p><p><hi rend="smallcaps">Bonnet</hi> <hi rend="italics">à Prêtre,</hi> or <hi rend="italics">Priest's Cap,</hi> is an outwork, having three saliant angles at the head, besides two inwards. It differs from the double tenaille only in this, that its sides, instead of being parallel, grow narrower, or closer, at the gorge, and opening at the front; from whence it is called <hi rend="italics">queue d'aronde,</hi> or swallow's tail.</p></div1><div1 part="N" n="BOOTES" org="uniform" sample="complete" type="entry"><head>BOOTES</head><p>, a constellation of the northern hemisphere, and one of the 48 old ones; having 23. stars in <pb n="218"/><cb/> Ptolemy's catalogue, 28 in Tycho's, 34 in Bayer's, 52 in Hevilus's, and 54 in Flamsteed's; of which one, in the skirt of his coat, is of the first magnitude, and called <hi rend="italics">Arcturus.</hi></p><p>Bootes is represented as a man in the posture of walking; his right hand grasping a club, and his left extended upwards, and holding the cord of the two dogs which seem barking at the Great Bear.</p><p>The Greeks, contrary to their usual custom, do not give any certain account of the origin of this constellation. Those who in very early days made the stars which were afterwards formed into the great bear represent a waggon drawn by oxen, made this Bootes the driver of them, from whence he was called the waggoner: others continued the office when the waggon was destroyed, and made a celestial bearward of Bootes, making it his office to drive the two bears round about the pole: and some, when the greater waggon was turned into the greater bear, were still for preserving the form of that machine in those stars which constitute Bootes.</p><p>This constellation is called by various other names; as <hi rend="italics">Arcas, Arctophylax, Arcturus-Minor, Bubulcus, Bubulus, Canis-Latrans, Clamator, Icarus, Lycaon, Philometus, Plaustri-Custos, Plorans, Thegnis,</hi> and <hi rend="italics">Vociferator;</hi> by Hesychius it is called <hi rend="italics">Orion,</hi> and by the Arabs <hi rend="italics">Aramech,</hi> or <hi rend="italics">Archamech.</hi> Schiller, instead of Bootes, makes the figure of St. Sylvester; Schickhard, that of Nimrod; and Weigelius, the three Swedish crowns. See Wolf. Lex Math. p. 266.</p></div1><div1 part="N" n="BORE" org="uniform" sample="complete" type="entry"><head>BORE</head><p>, of a gun, or other piece of ordnance, is the chase, cylinder, or hollow part of the piece.</p><p>BOREAL <hi rend="smallcaps">Signs</hi>, are the first six signs of the Zodiac, or those on the northern side of the equinoctial; viz, the signs <figure/> aries, <figure/> taurus, <figure/> gemini, <figure/> cancer, <figure/> leo, <figure/> virgo.</p></div1><div1 part="N" n="BOREALIS" org="uniform" sample="complete" type="entry"><head>BOREALIS</head><p>, <hi rend="smallcaps">Aurora.</hi> See <hi rend="smallcaps">Aurora</hi> <hi rend="italics">Borealis.</hi></p></div1><div1 part="N" n="BORELLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BORELLI</surname> (<foreName full="yes"><hi rend="smallcaps">John Alphonso</hi></foreName>)</persName></head><p>, a celebrated philosopher and mathematician, born at Naples the 28th of January 1608. He was professor of philosophy and mathematics in some of the most celebrated universities of Italy, particularly at Florence and Pisa, where he became highly in favour with the princes of the house of Medicis. But having been concerned in the revolt of Messina, he was obliged to retire to Rome, where he spent the remainder of his life under the protection of Christina queen of Sweden, who honoured him with her friendship, and by her liberality towards him softened the rigour of his hard fortune. He continued two years in the convent of the regular clergy of St. Pantaleon, called the <hi rend="italics">Pious Schools,</hi> where he instructed the youth in mathematical studies. And this study he prosecuted with great diligence for many years afterward, as appears by his correspondence with several ingenious mathematicians of his time, and the frequent mention that has been made of him by others, who have endeavoured to do justice to his memory. He wrote a letter to Mr. John Collins, in which he discovers his great desire and endeavours to promote the improvement of those sciences: he also speaks of his correspondence with, and great affection for, Mr. Henry Oldenburgh, Secretary of the Royal Society; of Dr. Wallis; of the then late learned Mr. Boyle, and lamented the loss sustained by his <cb/> death to the commonwealth of learning. Mr. Baxter, in his <hi rend="italics">Enquiry into the Nature of the Human Soul,</hi> makes frequent use of our author's book <hi rend="italics">De Motu Animalium,</hi> and tells us, that he was the first who discovered that the force exerted within the body prodigiously exceeds the weight to be moved without, or that nature employs an immense power to move a small weight. But he acknowledges that Dr. James Keil had shewn that Borelli was mistaken in his calculation of the force of the muscle of the heart; but that he nevertheless ranks him with the most authentic writers, and says he is seldom mistaken: and, having remarked that it is so far from being true, that great things are brought about by small powers, that, on the contrary, a stupendous power is manifest in the most ordinary operations of nature, he observes that the ingenious Borelli first observed this in animal motion; and that Dr. Stephen Hales, by a course of experiments in his <hi rend="italics">Vegetable Statics,</hi> had shewn the same in the force of the ascending sap in vegetables.</p><p>After a course of unceasing labours, Borelli died at Pantaleon of a pleurisy, the 31st of December 1679, at 72 years of age.</p><p>Beside several books on physical subjects, Borelli published the following mathematical ones: viz.</p><p>1. Apollonii Pergæi Conicorum Lib. 5, 6, & 7. Floren. 1661, fol.</p><p>2. Theoriæ Medicorum Planetarum ex causis physicis deductæ. Flor. 1666, 4to.</p><p>3. De Vi Percussionis. Bologna, 1667, 4to.—This piece was reprinted, with his celebrated treatise De Motu Animalium, and that other De Motionibus Naturalibus, in 1686.</p><p>4. Euclides Restitutus, &c. Pisa, 1668, 4to.</p><p>5. Osservatione intorno alla vistu ineguali degli Occi. —This piece was inserted in the Journal of Rome, for the year 1669.</p><p>6. De Motionibus Naturalibus de Gravitate pendentibus. Regio Julio, 1670, 4to.</p><p>7. Meteorologia Aetnea, &c. Regio Julio, 1670, 4to.</p><p>8. Osservatione dell' Ecclissi Lunare, 11 Gennaro 1675.—Inserted in the Journal of Rome 1675, p. 34.</p><p>9. Elementa Conica Appollonii Pergæi, & Archimedis Opera, nova & breviori methodo demonstrata.— Printed at Rome in 1679, in 12mo, at the end of the 3d edition of his Euclides Restitutus.</p><p>10. De Motu Animalium. Pars prima in 1680, and Pars altera in 1681, 4to.—These were reprinted at Leyden 1685, revised and purged from many errors; with the addition of John Bernoulli's Mathematical Meditations concerning the Motion of the Muscles.</p><p>11. At Leyden, 1686, in 4to, a more correct and accurate edition, revised by J. Broen, M. D. of Leyden, of his two pieces, De Vi Percussionis, & De Motionibus de Gravitate pendentibus.</p></div1><div1 part="N" n="BOUGUER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BOUGUER</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, a celebrated French mathematician, was born at Croisic, in Lower Bretagne, the 10th of February 1698. He was the son of John Bouguer, Professor Royal of Hydrography, a tolerably good mathematician, and author of A complete Treatise on Navigation. Young Bouguer was accustomed to learn mathematics from his father, from the <pb n="219"/><cb/> time he was able to speak, and thus became a proficient in those sciences while he was yet a child. He was sent very early to the Jesuits' college at Vannes, where he had the honour to instruct his regent in the mathematics, at eleven years of age.</p><p>Two years after this he had a public contest with a professor of mathematics, upon a proposition which the latter had advanced erroneously; and he triumphed over him; upon which the professor, unable to bear the disgrace, left the country.</p><p>Two years after this, when young Bouguer had not yet finished his studies, he lost his father; whom he was appointed to succeed in his office of hydrographer, after a public examination of his qualifications; being then only 15 years of age; an occupation which he discharged with great respect and dignity at that early age.</p><p>In 1727, at the age of 29, he obtained the prize proposed by the Academy of Sciences, for the best way of masting of ships. This first success of Bouguer was soon after followed by two others of the same kind; he successively gained the prizes of 1729 and 1731; the former, for the best manner of observing at sea the height of the stars, and the latter, for the most advantageous way of observing the declination of the magnetic needle, or the variation of the compass.</p><p>In 1729, he gave an <hi rend="italics">Optical Essay upon the Gradation of Light;</hi> a subject quite new, in which he examined the intensity of light, and determined its degrees of diminution in passing through different pellucid mediums, and particularly that of the sun in traversing the earth's atmosphere. Mairan gave an extract of this first essay in the Journal des Savans, in 1730.</p><p>In this same year, 1730, he was removed from the port of Croisic to that of Havre, which brought him into a nearer connection with the Academy of Sciences, in which he obtained, in 1731, the place of associate geometrician, vacant by the promotion of Maupertuis to that of pensioner; and in 1735 he was promoted to the office of pensioner-astronomer. The same year he was sent on the commission to South America, along with Messieurs Godin, Condamine, and Jeussieu, to determine the measure of the degrees of the meridian, and the figure of the earth. In this painful and troublesome business, of 10 years duration, chiefly among the lofty Cordelier mountains, our author determined many other new circumstances, beside the main object of the voyage; such as the expansion and contraction of metals and other substances, by the sudden and alternate changes of heat and cold among those mountains; observations on the refraction of the atmosphere from the tops of the same, with the singular phenomenon of the sudden increase of the refraction, when the star can be observed below the line of the level; the laws of the density of the air at different heights, from observations made at different points of these enormous mountains; a determination that the mountains have an effect upon a plummet, though he did not assign the exact quantity of it; a method of estimating the errors committed by navigators in determining their route; a new construction of the log for measuring a ship's way; with several other useful improvements. <cb/></p><p>Other inventions of Bouguer, made upon different occasions, were as follow: The heliometer, being a telescope with two object glasses, affording a good method of measuring the diameters of the larger planets with ease and exactness: his researches on the figure in which two lines or two long ranges of parallel trees appear: his experiments on the famous reciprocation of the pendulum: and those upon the manner of measuring the force of the light: &c, &c.</p><p>The close application which Bouguer gave to study, undermined his health, and terminated his life the 15th of August 1758, at 60 years of age.—His chief works, that have been published, are,</p><p>1. The Figure of the Earth, determined by the observations made in South America; 1749, in 4to.</p><p>2. Treatise on Navigation and Pilotage; Paris, 1752, in 4to. This work has been abridged by M. La Caille, in 1 volume, 8vo, 1768.</p><p>3. Treatise on Ships, their Construction and Motions; in 4to, 1756.</p><p>4. Optical Treatise on the Gradation of Light; first in 1729; then a new edition in 1760, in 4to.</p><p>His papers that were inserted in the Memoirs of the Academy, are very numerous and important: as, in the Memoirs for 1726, Comparison of the force of the solar and lunar light with that of candles.—1731, Observations on the curvilinear motion of bodies in mediums.—1732, Upon the new curves called the <hi rend="italics">lines of pursuit.</hi>—1733, To determine the species of conoid, to be constructed upon a given base which is exposed to the shock of a fluid, so that the impulse may be the least possible.—Determination of the orbit of comets.—1734, Comparison of the two laws which the earth and the other planets must observe in the figure which gravity causes them to take.—On the curve lines proper to form the arches in domes.—1735, Observations on the equinoxes.—On the length of the pendulum.—1736, On the length of the pendulum in the torrid zone.—On the manner of determining the figure of the earth by the measure of the degrees of latitude and longitude.—1739, On the astronomical refractions in the torrid zone.—Observations on the lunar eclipse, of the 8th of September 1737, made at Quito. —1744, Short account of the voyage to Peru, by the members of the Royal Academy of Sciences, to measure the degrees of the meridian near the equator, and from thence to determine the figure of the earth. —1745, Experiments made at Quito and divers other places in the torrid zone, on the expansion and contraction of metals by heat and cold.—On the problem of the masting of ships.—1746, Treatise on ships, their structure and motions.—On the impulse of fluids upon the fore parts of pyramidoids having their base a trapezium.—Continuation of the short account given in 1744, of the voyage to Peru for measuring the earth.—1747, On a new construction of the log, and other instruments for measuring the run of a ship.— 1748, Of the diameters of the larger planets. The new instrument called a <hi rend="italics">heliometer,</hi> proper for determining them; with observations of the sun.—Observation of the eclipse of the moon the 8th of August 1748.—1749, Second memoir on astronomical refractions, observed in the torrid zone, with remarks on the manner of constructing the tables of them.— <pb n="220"/><cb/> Figure of the earth determined by MM. Bouguer and Condamine, with an abridgment of the expedition to Peru.—1750, Observation of the lunar eclipse of the 13th of December 1750.—1751, On the form of bodies most proper to turn about themselves, when they are pushed by one of their extremities, or any other point.—On the moon's parallax, with the estimation of the changes caused in the parallaxes by the figure of the earth.—Observation of the lunar eclipse, the 2d of December 1751.—1752, On the operations made by seamen, called <hi rend="italics">Corrections.</hi>—1753, Observation of the passage of Mercury over the sun, the 6th of May 1753.—On the dilatations of the air in the atmosphere.—New treatise of navigation, containing the theory and practice of pilotage, or working of ships.—1754, Operations, &c, for distinguishing, among the different determinations of the degree of the meridian near Paris, that which ought to be preferred.—On the direction which the string of a plummet takes.—Solution of the chief problems in the working of ships.—1755, On the apparent magnitude of objects.—Second memoir on the chief problems in the working of ships.—1757, Account of the treatise on the working of ships.—On the means of measuring the light.—1758, His Eulogy.</p><p>In the volumes of the prizes given by the academy, are the following pieces by Bouguer:</p><p>In vol. 1, on the masting of ships.—Vol. 2, On the method of exactly observing at sea the height of the stars; and the variation of the compass. Also on the cause of the inclination of the planets' orbits.</p></div1><div1 part="N" n="BOULTINE" org="uniform" sample="complete" type="entry"><head>BOULTINE</head><p>, in Architecture, a convex moulding, of a quarter of a circle, and placed next below the plinth in the Tuscan and Dorick capital.</p></div1><div1 part="N" n="BOW" org="uniform" sample="complete" type="entry"><head>BOW</head><p>, an offensive weapon made of wood, horn, steel, or other elastic matter, by which arrows are thrown with great force. This instrument was of very ancient and general use, and is still found among all savage nations who have not the use of fire arms, by which it has been superseded among us. There are two species of the Bow, the <hi rend="italics">Long,</hi> and the <hi rend="italics">Cross</hi> Bow.</p><p>The <hi rend="italics">Long Bow</hi> is simply a bow, or a rod, with a string fastened to each end of it, to the middle of which the end of an arrow being applied, and then drawn by the hand, on suddenly quitting the hold, the bow returns by means of its elasticity, and impels the arrow from the string with great violence. The old English archers were famous for the long bow, by means of which they gained many victories in France and elsewhere.</p><p>The <hi rend="italics">Cross Bow,</hi> called also <hi rend="italics">arbalest</hi> or <hi rend="italics">arbalet,</hi> is a bow strung and set in a shaft of wood, and furnished with a trigger; serving to throw bullets, darts, and large arrows, &c. The ancients had large machines for throwing many arrows at once, called <hi rend="italics">arbalets,</hi> or <hi rend="italics">balistæ.</hi></p><p>The force of a bow may be calculated on this principle, that its spring, <hi rend="italics">i. e.</hi> the power by which it restores itself to its natural position, is always proportional to the space or distance it is bent or removed from it.</p><p>Bow, a mathematical instrument formerly used at sea for taking the sun's altitude. It consisted of a large arch divided into 90 degrees, fixed on a staff, <cb/> and furnished with three vanes, viz, a side vane, a sight vane, and a horizon vane.</p><p>Bow-<hi rend="italics">Compass,</hi> an instrument for drawing arches of very large circles, for which the common compasses are too small. It consists of a beam of wood or brass, with three long screws that govern or bend a lath of wood or steel to any arch.</p><p>BOX <hi rend="smallcaps">AND Needle</hi>, the small compass of a theodolite, circumferentor, or plain-table.</p></div1><div1 part="N" n="BOYAU" org="uniform" sample="complete" type="entry"><head>BOYAU</head><p>, in Fortification, a ditch covered by a parapet, and serving as a communication between two trenches. It runs parallel to the works of the body of the place; and serves as a line of contravallation, both to hinder the sallies of the besieged, and to secure the miners. When it is a particular cut running from the trenches, to cover some spot of ground, it is drawn so as not to be ensiladed or scoured by the enemy's shot.</p></div1><div1 part="N" n="BOYLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BOYLE</surname> (<foreName full="yes"><hi rend="smallcaps">Robert</hi></foreName>)</persName></head><p>, one of the grea est philosophers, as well as best men, that any country has ever produced, was the 7th son and the 14th child of Richard earl of Cork, and was born at Lismore in the province of Munster in Ireland, the 25th of January, 1626-7; the very year of the death of the learned Lord Bacon, whose plans of experimental philosophy our author afterwards so ably seconded. While very young, he was instructed in his father's house to read and write, and to speak French and Latin. In 1635, when only 8 years old, he was sent over to England, to be educated at Eton school. Here he soon discovered an extraordinary force of understanding, with a disposition to cultivate and improve it to the utmost.</p><p>After remaining at Eton between 3 and 4 years, his father sent our author and his brother Francis, in 1638, on their travels upon the continent. They passed through France to Geneva, where they settled for some time to pursue their studies: here our author resumed his acquaintance with the elements of the mathematics, which he had commenced at Eton when 10 years old, on occasion of an illness which prevented his other usual studies.</p><p>In the autumn of 1641, he quitted Geneva, and travelled through Switzerland and Italy to Venice, from whence he returned again to Florence, where he spent the winter, studying the Italian language and history, and the works of the celebrated astronomer Galileo, who died in a village near this city during Mr. Boyle's residence here.</p><p>About the end of March 1642, he set out from Florence, visited Rome and other places in Italy, then returned to the south of France. At Marseilles, in May 1642, Mr. Boyle received letters from his father, which informed him that the rebellion had broken out in Ireland, and with how much difficulty he had procured 250l. then remitted to help him and his brother home. This remittance however never reached them, and they were obliged to return to Geneva with their governor Mr. Marcombes, who contrived on his own credit, and by selling some jewels, to raise money enough to send them to England, where they arrived in 1644. On their arrival they found that their father was dead, and had left our author the manor of Stalbridge in England, with some other considerable estates in Ireland. <pb n="221"/><cb/></p><p>From this time Mr. Boyle's chief residence, for some years at least, was at his manor of Stalbridge, from whence he made occasional excursions to Oxford, London, &c; applying himself with great industry to various kinds of studies, but especially to philosophy and chemistry; and seizing every opportunity of cultivating the acquaintance of the most learned men of his time. He was one of the members of that small but learned body of men who, when all academical studies were interrupted by the civil wars, secreted themselves about the year 1645; and held private meetings, first in London, afterwards at Oxford, to cultivate subjects of natural knowledge upon that plan of experiment which Lord Bacon had delineated. They styled themselves then <hi rend="italics">The Philosophic College;</hi> but after the restoration, when they were incorporated, and distinguished openly, they took the name of the <hi rend="italics">Royal Society.</hi></p><p>In the summer of 1654 he retired to settle at Oxford, the Philosophical Society being removed from London to that place, that he might enjoy the conversation of the other learned members, his friends, who had retired thither, such as Wilkins, Wallis, Ward, Willis, Wren, &c. It was during his residence here that he improved that admirable engine the airpump; and by numerous experiments was enabled to discover several qualities of the air, so as to lay a foundation for a complete theory. He declared against the philosophy of Aristotle, as having in it more of words than things; promising much, and performing little; and giving the inventions of men for indubitable proofs, instead of building upon observation and experiment. He was so zealous for this true method of learning by experiment, and so careful about it, that though the Cartesian philosophy then made a great noise in the world, yet he could never be persuaded to read the works of Descartes, for fear he should be amused and led away by plausible accounts of things founded on conjecture, and merely hypothetical. But philosophy, and enquiries into nature, though they engaged his attention deeply, did not occupy him entirely; as he still continued to pursue critical and theological studies. He had offers of preferment to enter into holy orders, by the government, after the restoration. But he declined the offer, choosing rather to pursue his studies as a layman, in such a manner as might be most effectual for the support of religion; and began to communicate to the world the fruits of these studies. These fruits were very numerous and important, as well as various: the principal of which, as well as of some other memorable occurrences of his life, were nearly in the following order.</p><p>In 1660 came out, 1. New experiments, physicomechanical, touching the spring of the air and its effects.—2. Seraphic love; or some motives and incentives to the love of God, pathetically discoursed of in a letter to a friend. A work which it has been said was owing to his courtship of a lady, the daughter of Cary earl of Monmouth; though our author was never married.—3. Certain physiological essays and other tracts, in 1661.—4. Sceptical chemist, 1662; reprinted about the year 1679, with the addition of divers experiments and notes on the producibleness of chemical principles. <cb/></p><p>In the year 1663, the Royal Society being incorporated by king Charles the 2d, Mr. Boyle was named one of the council; and as he might justly be reckoned among the founders of that learned body, so he continued one of the most useful and industrious of its members during the whole course of his life. His next publications were, 5. Considerations touching the usefulness of experimental natural philosophy, 1663.— 6. Experiments and considerations upon colours; to which was added a letter, containing Observations on a diamond that shines in the dark, 1663. This treatise is full of curious and useful remarks on the hitherto unexplained doctrine of light and colours; in which he shews great judgment, accuracy, and penetration; and which may be said to have led the way to Newton, who made such great discoveries in that branch of physics.— 7. Considerations on the style of the holy scriptures, 1663. This was an extract from a larger work, intitled An essay on scripture; which was afterwards published by Sir Peter Pett, a friend of Mr. Boyle's.</p><p>In 1664 he was elected into the company of the royal mines; and was all this year occupied in prosecuting various good designs, which was probably the reason that he did not publish any works in this year. Soon after came out, 8. Occasional reflections upon several subjects, 1665. This piece exposed our author to the censure of the celebrated Dean Swift, who, to ridicule these discourses, wrote <hi rend="italics">A pious meditation upon a broomstick, in the style of the honourable Mr. Boyle.</hi>— 9. New experiments and observations upon cold, 1665. —10. Hydrostatical paradoxes made out by new experiments, for the most part physical and easy, 1666.— 11. The origin of forms and qualities, according to the corpuscular philosophy, 1666.—Both in this and the former year, our author communicated to his friend Mr. Oldenburgh, then secretary to the Royal Society, several curious and excellent short pieces of his own, upon a great variety of subjects, and others transmitted to him by his learned friends, which are printed in the Philos. Trans.</p><p>In the year 1668 Mr. Boyle resolved to settle in London for life; and for that purpose he removed to the house of his sister, the lady Ranelagh, in Pall-Mall. This removal was to the great benefit of the learned in general, and particularly of the Royal Society, to whom he gave great and continual assistance, as abundantly appears by the several pieces communicated to them from time to time, and printed in their Transactions. To avoid improper waste of time, he had set hours in the day appointed for receiving such persons as wanted to consult him, either for their own assistance, or to communicate new discoveries to him: And he besides kept up an extensive correspondence with the most learned men in Europe; so that it is wonderful how he could bring out so many new works as he did. His next publications were, 12. A continuation of new experiments touching the weight and spring of the air; to which is added, A discourse of the atmosphere of consistent bodies, 1669.—13. Tracts about the cosmical qualities of things; cosmical suspicions; the temperature of the subterraneous regions; the bottom of the sea; to which is prefixed an introduction to the history of particular qualities, 1669.—14. Considerations on the usefulness of experimental and natural phi- <pb n="222"/><cb/> losophy, the 2d part, 1671.—15. A collection of tracts upon several useful and important points of practical philosophy, 1671.—16. An essay upon the origin and virtues of gems, 1672.—17. A collection of tracts upon the relation between flame and air; and several other useful and curious subjects, 1672. Besides furnishing, in this and the former year, a number of short dissertations upon a great variety of topics, addressed to the Royal Society, and inserted in their Transactions.— 18. Essays on the strange subtilty, great efficacy, and determinate nature, of effluvia; with a variety of experiments on other subjects, 1673.—19. The excellency of theology compared with philosophy, 1673. This discourse was written in the year 1665, while our author, to avoid the great plague which then raged in London, was forced to go from place to place in the country, having little or no opportunity of consulting his books.—20. A collection of tracts upon the saltness of the sea, the moisture of the air, the natural and preternatural state of bodies; to which is prefixed a dialogue concerning cold, 1674.—21. A collection of tracts containing suspicions about hidden qualities of the air; with an appendix touching celestial magnets; animadversions upon Mr. Hobbes's problem about a vacuum; a discourse of the cause of attraction and suction, 1674. —22. Some considerations about the reasonableness of reason and religion; by T. E. (the final letters of his names). To which is annexed a discourse about the possibility of the resurrection; by Mr. Boyle, 1675. The same year several papers communicated to the Royal Society, among which were two upon quicksilver growing hot with gold.—23. Experiments and notes about the mechanical origin or production of particular qualities, in several discourses on a great variety of subjects, and among the rest on electricity, 1676.—He then communicated to Mr. Hook a short memorial of some observations made upon an artisicial substance that shines without any preceding illustration; published by Hook in his <hi rend="italics">Lectiones Cutlerianæ.</hi>—24. Historical account of a degradation of gold made by an anti-elixir.—25. Aerial noctiluca; or some new phænomena, and a process of a factitious self-shining substance, 1680. This year the Royal Society, as a proof of the just sense of his great worth, and of the constant and particular services which through a course of many years he had done them, made choice of him for their president; but he being extremely, and, as he says, peculiarly tender in point of oaths, declined that honour.—26. Discourse of things above reason; inquiring, whether a philosopher should admit any such, 1681.—27. New experiments and observations upon the icy noctiluca; to which is added a chemical paradox, grounded upon new experiments, making it probable that chemical principles are transmutable, so that out of one of them others may be produced, 1682.—28. A continuation of new experiments, physico-mechanical, touching the spring and weight of the air, and their effects, 1682.—29. A short letter to Dr. Beale, in relation to the making of fresh water out of salt, 1683.—30. Memoris for the natural history of human blood, especially the spirit of that liquor, 1684.— 31. Experiments and considerations about the porosity of bodies, 1684.—32. Short memoris for the natural experimental history of mineral waters, &c, 1685.— <cb/> 33. An essay on the great effects of even languid and unheeded motion, &c, 1685.—34. Of the reconcileableness of specific medicines to the corpuscular philosophy, &c, 1685.—35. Of the high veneration man's intellect owes to God, peculiarly for his wisdom and power, 1685. —36. Free inquiry into the vulgarly received notion of nature, 1686.—37. The martyrdom of Theodora and Didymia, 1687. A work he had drawn up in his youth.—38. A disquisition about the final causes of natural things, and about vitiated light, 1688.</p><p>Mr. Boyle's health declining very much, he abridged greatly his time given to conversations and communications with other persons, to have more time to prepare for the press some others of his papers, before his death, which were as follow:—39. <hi rend="italics">Medicina Hydrostatica,</hi> &c, 1690.—40. The Christian virtuoso, &c, 1690. 41. <hi rend="italics">Experimenta et Observationes Physicæ,</hi> &c, 1691; which is the last work that he published.</p><p>Mr. Boyle died on the last day of December of the same year 1691, in the 65th year of his age, and was buried in St. Martin's church in the Fields, Westminster; his funeral sermon being preached by Dr. Gilbert Burnet bishop of Salisbury; in which he displayed the excellent qualities of our author, with many circumstances of his life, &c. But as the limits of this work will not allow us to follow the bishop in the copious and eloquent account he has given of this great man's abilities, we must content ourselves with adding the following short eulogium by the celebrated physician, philosopher, and chemist, Dr. Boerhaave; who, after having declared lord Bacon to be the father of experimental philosophy, asserts, that “Mr. Boyle, the ornament of his age and country, succeeded to the genius and inquiries of the great chancellor Verulam. Which, says he, of Mr. Boyle's writings shall I recommend? All of them. To him we owe the secrets of fire, air, water, animals, vegetables, fossils: so that from his works may be deduced the whole system of natural knowledge.”</p><p>Mr. Boyle left also several papers behind him, which have been published since his death. Beautiful editions of all his works have been printed at London, in 5 volumes folio, and 6 volumes 4to. Dr. Shaw also published in 3 volumes 4to, the same works “abridged, methodized, and disposed under the general heads of Physics, Statics, Pneumatics, Natural History, Chymistry, and Medicine;” to which he has prefixed a short catalogue of the philosophical writings, according to the order of time when they were first published, &c, as follows: <table><row role="data"><cell cols="1" rows="1" rend="valign=bottom" role="data">Physico-mechanical experiments on the spring and weight of the air</cell><cell cols="1" rows="1" role="data">1661</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Sceptical Chymist</cell><cell cols="1" rows="1" role="data">1661</cell></row><row role="data"><cell cols="1" rows="1" role="data">Physiological Essays</cell><cell cols="1" rows="1" role="data">1662</cell></row><row role="data"><cell cols="1" rows="1" role="data">History of Colours</cell><cell cols="1" rows="1" role="data">1663</cell></row><row role="data"><cell cols="1" rows="1" role="data">Usefulness of Experimental Philosophy</cell><cell cols="1" rows="1" role="data">1663</cell></row><row role="data"><cell cols="1" rows="1" role="data">History of Cold</cell><cell cols="1" rows="1" role="data">1665</cell></row><row role="data"><cell cols="1" rows="1" role="data">Historical Paradoxes</cell><cell cols="1" rows="1" role="data">1666</cell></row><row role="data"><cell cols="1" rows="1" role="data">Origin of Forms and Qualities</cell><cell cols="1" rows="1" role="data">1666</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cosmical Qualities</cell><cell cols="1" rows="1" role="data">1670</cell></row><row role="data"><cell cols="1" rows="1" role="data">The admirable Rarefaction of the air</cell><cell cols="1" rows="1" role="data">1670</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Origin and Virtues of Gems</cell><cell cols="1" rows="1" role="data">1672</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Relation betwixt Flame and Air</cell><cell cols="1" rows="1" role="data">1672</cell></row></table> <pb n="223"/><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">Effluviums</cell><cell cols="1" rows="1" role="data">1673</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saltness of the Sea</cell><cell cols="1" rows="1" role="data">1674</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hidden Qualities in the Air</cell><cell cols="1" rows="1" role="data">1674</cell></row><row role="data"><cell cols="1" rows="1" rend="valign=bottom" role="data">The Excellence &c of the Mechanical Hypothesis</cell><cell cols="1" rows="1" role="data">1674</cell></row><row role="data"><cell cols="1" rows="1" role="data">Considerations on the Resurrection</cell><cell cols="1" rows="1" role="data">1675</cell></row><row role="data"><cell cols="1" rows="1" role="data">Particular Qualities</cell><cell cols="1" rows="1" role="data">1676</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aerial Noctiluca</cell><cell cols="1" rows="1" role="data">1680</cell></row><row role="data"><cell cols="1" rows="1" role="data">Icy Noctiluca</cell><cell cols="1" rows="1" role="data">1680</cell></row><row role="data"><cell cols="1" rows="1" role="data">Things above Reason</cell><cell cols="1" rows="1" role="data">1681</cell></row><row role="data"><cell cols="1" rows="1" role="data">Natural History of Human Blood</cell><cell cols="1" rows="1" role="data">1684</cell></row><row role="data"><cell cols="1" rows="1" role="data">Porosity of Bodies</cell><cell cols="1" rows="1" role="data">1684</cell></row><row role="data"><cell cols="1" rows="1" role="data">Natural History of Mineral Waters</cell><cell cols="1" rows="1" role="data">1684</cell></row><row role="data"><cell cols="1" rows="1" role="data">Specific Medicines</cell><cell cols="1" rows="1" role="data">1685</cell></row><row role="data"><cell cols="1" rows="1" role="data">The High Veneration due to God</cell><cell cols="1" rows="1" role="data">1685</cell></row><row role="data"><cell cols="1" rows="1" role="data">Languid Motion</cell><cell cols="1" rows="1" role="data">1685</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Notion of Nature</cell><cell cols="1" rows="1" role="data">1685</cell></row><row role="data"><cell cols="1" rows="1" role="data">Final Causes</cell><cell cols="1" rows="1" role="data">1688</cell></row><row role="data"><cell cols="1" rows="1" role="data">Medicina Hydrostatica</cell><cell cols="1" rows="1" role="data">1690</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Christian Virtuoso</cell><cell cols="1" rows="1" role="data">1690</cell></row><row role="data"><cell cols="1" rows="1" role="data">Experimenta & Observationes Physicæ</cell><cell cols="1" rows="1" role="data">1691</cell></row><row role="data"><cell cols="1" rows="1" role="data">Natural History of the Air</cell><cell cols="1" rows="1" role="data">1692</cell></row><row role="data"><cell cols="1" rows="1" role="data">Medicinal Experiments</cell><cell cols="1" rows="1" role="data">1718</cell></row></table></p></div1><div1 part="N" n="BRADLEY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BRADLEY</surname> (<foreName full="yes"><hi rend="smallcaps">Dr. James</hi></foreName>)</persName></head><p>, a celebrated English astronomer, the third son of William Bradley, was born at Sherborne in Gloucestershire in the year 1692. He was fitted for the university at Northleach in the same county, at the boarding school of Mr. Egles and Mr. Brice. From thence he was sent to Oxford, and admitted a commoner of Baliol college March 15, 1710; where he took the degree of bachelor the 14th of October 1714, and of master of arts the 21st of January 1716. His friends intending him for the church, his studies were regulated with that view; and as soon as he was of a proper age to receive holy orders, the bishop of Hereford, who had conceived a great esteem for him, gave him the living of Bridstow, and soon after he was inducted to that of Landewy Welfry in Pembrokeshire.</p><p>He was nephew to Mr. Pound, a gentleman well known in the learned world, by many excellent astronomical and other observations, and who would have enriched it much more, if the journals of his voyages had not been burnt at Pulo Condor, when the place was set on fire, and the English who were settled there cruelly massacred, Mr. Pound himself very narrowly escaping with his life. With this gentleman, at Wanstead, Mr. Bradley passed all the time that he could spare from the duties of his function; being then sufficiently acquainted with the mathematics to improve by Mr. Pound's conversation. It may easily be imagined that the example and conversation of this gentleman did not render Bradley more fond of his profession, to which he had before no great attachment: he continued however as yet to fulfil the duties of it, though at this time he had made such observations as laid the foundation of those discoveries which afterward distinguished him as one of the greatest astronomers of his age. These observations gained him the notice and friendship of the lord chancellor Macclesfield, Mr. Newton afterward Sir Isaac, Mr. Halley, and many other members of the Royal Society, into which he was soon after elected a member.</p><p>Soon after, the chair of Savilian professor of astro- <cb/> nomy at Oxford became vacant, by the death of the celebrated Dr. John Keil; and Mr. Bradley was elected to succeed him on the 31st of October 1721, at 29 years of age: his colleague being Mr. Halley, who was professor of geometry on the same foundation. Upon this appointment, Mr. Bradley resigned his church livings, and applied himself wholly to the study of his favourite science. In the course of his observations, which were innumerable, he discovered and settled the laws of the alterations of the fixed stars, from the progressive motion of light, combined with the earth's annual motion about the sun, and the nutation of the earth's axis, arising from the unequal attraction of the sun and moon on the different parts of the earth. The former of these effects is called the <hi rend="italics">aberration</hi> of the fixed stars, the theory of which he published in 1727; and the latter the <hi rend="italics">nutation</hi> of the earth's axis, the theory of which appeared in 1737: so that in the space of about 10 years, he communicated to the world two of the finest discoveries in modern astronomy; which will for ever make a memorable epoch in the history of that science. See <hi rend="smallcaps">Aberration</hi> and <hi rend="smallcaps">Nutation.</hi></p><p>In 1730 our author succeeded Mr. Whiteside, as lecturer in astronomy and experimental philosophy in the Museum at Oxford: which was a considerable emolument to him, and which he held till within a year or two of his death; when the ill state of his health induced him to resign it.</p><p>Our author always preserved the esteem and friendship of Dr. Halley; who, being worn out by age and infirmities, thought he could not do better for the service of astronomy, than procure for Mr. Bradley the place of regius professor of astronomy at Greenwich, which he himself had many years possessed with the greatest reputation. With this view he wrote many letters, desiring Mr. Bradley's permission to apply for a grant of the reversion of it to him, and even offered to resign it in his favour, if it should be thought necessary: but Dr. Halley died before he could accomplish this kind object. Our author however obtained the place, by the interest of lord Macclesfield, who was afterward president of the Royal Society; and upon this appointment the university of Oxford sent him a diploma of doctor of divinity.</p><p>The appointment of astronomer royal at Greenwich, which was dated the 3d of February 1741-2, placed our author in his proper element; and he pursued his observations with unwearied diligence. However numerous the collection of astronomical instruments at that observatory, it was impossible that such an observer as Dr. Bradley should not desire to increase them, as well to answer those particular views, as in general to make observations with greater exactness. In the year 1748 therefore, he took the opportunity of the visit of the Royal Society to the observatory, annually made to examine the instruments and receive the professor's observations for the year, to represent so strongly the necessity of repairing the old instruments, and providing new ones, that the society thought proper to make application to the king, who was pleased to order 1000 pounds for that purpose. This sum was laid out under the direction of our author, who, with the assistance of the late celebrated Mr. Graham and Mr. Bird, furnished the observatory with as complete a col- <pb n="224"/><cb/> lection of astronomical instruments, as the most skilful and diligent observer could desire. Dr. Bradley, thus furnished with such assistance, pursued his observations with great assiduity during the rest of his life; an immense number of which was found after his death, in 13 folio volumes, and were presented to the university of Oxford in the year 1776, on condition of their printing and publishing them; but which however, unfortunately for the improvement of astronomy, now after a lapse of almost 20 years, has never yet been done.</p><p>During Dr. Bradley's residence at the Royal Observatory, the living of the church at Greenwich became vacant, and was offered to him: upon his refusing to accept it, from a conscientious scruple, “that the duty of a pastor was incompatible with his other studies and necessary engagements,” the king was pleased to grant him a pension of 250l. over and above the astronomer's original salary from the Board of Ordnance, “in consideration (as the sign manual, dated the 15th Feb. 1752, expresses it) of his great skill and knowledge in the several branches of astronomy and other parts of the mathematics, which have proved so useful to the trade and navigation of this kingdom.” A pension which has been regularly continued to the astronomers royal ever since.</p><p>About 1748, our author became entitled to bishop Crew's benefaction of 30l. a year, to the lecture reader in experimental philosophy at Oxford. He was elected a member of the Academy of Sciences at Berlin, in 1747; of that at Paris, in 1748; of that at Petersburgh, in 1754; and of that at Bologna, in 1757. He was married in the year 1744, but never had more than one child, a daughter.</p><p>By too close application to study and observations, Dr. Bradley became afflicted, for near two years before his death, with a grievous oppression on his spirits; which interrupted his useful labours. This distress arose chiefly from an apprehension that he should outlive his rational faculties: but this so much dreaded evil never came upon him. In June 1762 he was seized with a suppression of urine, occasioned by an inflammation in the reins, which terminated his existence the 13th of July following. His death happened at Chalfont in Gloucestershire, in the 70th year of his age; and he was interred at Minchinhampton in the same county.</p><p>As to his character, Dr. Bradley was remarkable for a placid and gentle modesty, very uncommon in persons of an active temper and robust constitution. Although he was a good speaker, and possessed the rare but happy art of expressing his ideas with the utmost precision and clearness, yet no man was a greater lover of silence, for he never spoke but when he thought it absolutely necessary. Nor was he more inclined to write than to speak, as he has published very little: he had a natural diffidence, which made him always afraid that his works might injure his character; so that he suppressed many which might have been worthy of publication.</p><p>His papers, which have been inserted in the Philos. Trans. are,</p><p>1. Observations on the comet of 1703. Vol. 33, p. 41. <cb/></p><p>2. The longitude of Lisbon and of the fort of New York from Wansted and London determined by the eclipse of the first satellite of Jupiter. Vol. 34, p. 85.</p><p>3. An account of a new discovered Motion of the Fixed Stars. Vol. 35, p. 637.</p><p>4. On the Going of Clocks with Isochronal Pendulums. Vol. 38, p. 302.</p><p>5. Observations on the Comet of 1736-7. Vol. 40, p. 111.</p><p>6. On the apparent Motion of the fixed Stars. Vol. 45, p. 1.</p><p>7. On the Occultation of Venus by the Moon, the 15th of April 1751. Vol. 46, p. 201.</p><p>8. On the Comet of 1757. Vol. 50, p. 408.</p><p>9. Directions for using the Common Micrometer. Vol. 62, p. 46.</p></div1><div1 part="N" n="BRADWARDIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BRADWARDIN</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>)</persName></head><p>, archbishop of Canterbury, was born at Hartfield in Sussex, about the close of the 13th century. He was educated at Merton College, Oxford, where he took the degree of doctor of divinity; and acquired the reputation of a profound scholar, a skilful mathematician, and consummate divine. It has been said he was professor of divinity at Oxford; that he was chancellor of the diocese of London, and confessor to Edward the 3d, whom he constantly attended during his war with France. After his return from the war, he was made prebendary of Lincoln, and afterward archbishop of Canterbury. He died at Lambeth in the year 1349, forty days after his consecration. His works are, 1. <hi rend="italics">De Causa Dei,</hi> printed London 1618, published by J. H. Savil.—2. <hi rend="italics">De Geometria speculativa, &c.</hi> Paris, 1495, 1512, 1530.— 3. <hi rend="italics">De Arithmetica practica,</hi> Paris, 1502, 1512.—4. <hi rend="italics">De Proportionibus,</hi> Paris, 1495. Venice, 1505, folio.— 5. <hi rend="italics">De Quadratura Circuli,</hi> Paris, 1495, folio.</p></div1><div1 part="N" n="BRAHE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BRAHE</surname> (<foreName full="yes"><hi rend="smallcaps">Tycho</hi></foreName>)</persName></head><p>, a celebrated astronomer, descended from a noble family originally of Sweden but settled in Denmark, was born the 14th of December 1546, at Knudstorp in the county of Schonen, near Helsimbourg. He was taught Latin when 7 years old, and studied 5 years under private tutors. His father dying while our author was very young, his uncle, George Brahe, having no children, adopted him, and sent him, in 1559, to study philosophy and rhetoric at Copenhagen. The great eclipse of the sun, on the 21st of August 1560, happening at the precise time the astronomers had foretold, he began to consider astronomy as something divine; and purchasing the tables of Stadius, he gained some notion of the theory of the planets. In 1562 he was sent by his uncle to Leipsic to study the law, where his acquirements gave manifest indications of extraordinary abilities. His natural inclination however was to the study of the heavens, to which he applied himself so assiduously, that, notwithstanding the care of his tutor to keep him close to the study of the law, he made use of every means in his power for improving his knowledge of astronomy; he purchased with his pocket money whatever books he could meet with on the subject, and read them with great attention, procuring assistance in difficult cases from Bartholomew Scultens his private tutor; and having procured a small celestial globe, he took opportunities, when his tutor was in bed, and when the weather was clear, to examine the constellations in the heavens, to learn their <pb n="225"/><cb/> names from the globe, and their motions from observation.</p><p>After a course of 3 years study at Leipsic, his uncle dying, he returned home in 1565. In this year a difference arising between Brahe and a Danish nobleman, they fought, and our author had part of his nose cut off by a blow; a defect which he so artfully supplied with one made of gold and silver, that it was not perceivable. About this time he began to apply himself to chemistry, proposing nothing less than to obtain the philosopher's stone. But becoming greatly disgusted to see the liberal arts despised, and finding his own relations and friends uneasy that he applied himself to astronomy, as thinking it a study unsuitable to a person of his quality, he went to Wirtemberg in 1566, from whence the breaking out of the plague soon occasioned his removal to Rostock, and in 1569 to Augsburg, where he was visited by Peter Ramus, then professor of astronomy at Paris, and who greatly admired his uncommon skill in this science.</p><p>In 1571 he returned to Denmark; and was favoured by his maternal uncle, Steno Billes, a lover of learning, with a convenient place at his castle of Herritzvad near Knudstorp, for making his observations, and building a laboratory. And here it was he discovered, in 1573, a new star in the constellation Cassiopeia. But soon after, his marrying a country girl, beneath his rank, occasioned so violent a quarrel between him and his relations, that the king was obliged to interpose to reconcile them.</p><p>In 1574, by the king's command, he read lectures at Copenhagen on the theory of the planets. The year following he began his travels through Germany, and proceeded as far as Venice. He then resolved to remove his family, and settle at Basil; but Frederic the 2d, king of Denmark, being informed of his design, and unwilling to lose a man who was capable of doing so much honour to his country, he promised to enable him to pursue his studies, and bestowed upon him for life the island of Huen in the Sound, and promised that an observatory and laboratory should be built for him, with a supply of money for carrying on his designs: and accordingly the first stone of the observatory was laid the 8th of August 1576, under the name of Uranibourg: The king also gave him a pension of 2000 crowns out of his treasury, a fee in Norway, and a canonry of Roshild, which brought him in 1000 more. This situation he enjoyed for the space of about 20 years, pursuing his observations and studies with great industry: here he kept always in his house ten or twelve young men, who assisted him in his observations, and whom he instructed in astronomy and mathematics. Here it was that he received a visit from James the 6th, king of Scotland, afterward James the 1st of England, having come to Denmark to espouse Anne, daughter of Frederick the 2d: James made our author some noble presents, and wrote a copy of Latin verses in his praise.</p><p>Brahe's tranquillity however in this happy situation was at length fatally interrupted. Soon after the death of king Frederick, by the aspersions of envious and malevolent ministers, he was deprived of his pension, fee, and canonry, in 1596. Being thus rendered incapable of supporting the expences of his establish- <cb/> ment, he quitted his favourite Uranibourg, and withdrew to Copenhagen, with some of his instruments, and continued his astronomical observations and chemical experiments in that city, till the same malevolence procured from the new king, Charles the 4th, an order for him to discontinue them. This induced him to fall upon means of being introduced to the emperor Rodolphus, who was fond of mechanism and chemical experiments: and to smooth the way to an interview, Tycho now published his book, <hi rend="italics">Astronomia instaurata Mechanica,</hi> adorned with figures, and dedicated it to the emperor. That prince received him at Prague with great civility and respect; gave him a magnificent house, till he could procure one for him more fit for astronomical observations; he also assigned him a pension of 3000 crowns; and promised him a fee for himself and his descendants. Here then he settled in the latter part of 1598, with his sons and scholars, and among them the celebrated Kepler, who had joined him. But he did not long enjoy this happy situation; for, about 3 years after, he died, on the 24th of October 1601, of a retention of urine, in the 55th year of his age, and was interred in a very magnisicent manner in the principal church at Prague, where a noble monument was erected to him; leaving, beside his wife, two sons and four daughters. On the approach of death, he enjoined his sons to take care that none of his works should be lost; exhorted the students to attend closely to their exercises; and recommended to Kepler the finishing of the Rudolphine tables he had constructed for regulating the motion of the planets.</p><p>Brahe's skill in astronomy is universally known; and he is famed for being the inventor of a new system of the planets, which he endeavoured, though without success, to establish on the ruins of that of Copernicus. He was very credulous with regard to judicial astrology and presages: If he met an old woman when he went out of doors, or a hare upon the road on a journey, he would turn back immediately, being persuaded that it was a bad omen: Also, when he lived at Uranibourg, he kept at his house a madman, whom he placed at his feet at table, and fed himself; for as he imagined that every thing spoken by mad persons presaged something, he carefully observed all that this man said; and because it sometimes proved true, he fancied it might always be depended on. He was of a very irritable disposition: a mere trifle put him in a passion; and against persons of the first rank, whom he thought his enemies, he openly discovered his resentment. He was very apt to rally others, but highly provoked when the same liberty was taken with himself.—The principal part of his writings, according to Gassendus, are,</p><p>1. An account of the New Star, which appeared Nov. 11th 1572, in Cassiopeia; Copenh. 1573, in 4to. —2. An Oration concerning the Mathematical Sciences, pronounced in the university of Copenhagen, in the year 1574: published by Conrad Aslac, of Bergen in Norway.—3. A treatise on the Comet of the year 1577, immediately after it disappeared. Nine years afterward, he revised it, and added a 10th chapter. Printed at Uranibourg, 1589.—4. Another treatise on the New Phenomena of the heavens. In the first part of which he treats of the Restitution, as he calls it, of the sun, and of the sixed stars. And in the 2d part, of <pb n="226"/><cb/> a New Star, which then had made its appearance.— 5. A collection of Astronomical Epistles: printed in 4to, at Uranibourg in 1596; Nuremberg in 1602; and at Franckfort in 1610. It was dedicated to Maurice landgrave of Hesse; because there are in it a considerable number of letters of the landgrave William his father, and of Christopher Rothmann, the mathematician of that prince, to Tycho, and of Tycho to them. —6. The Mechanical Principles of Astronomy restored: Wandesburg, 1598, in folio.—7. An Answer to the Letter of a certain Scotchman, concerning the comet, in the year 1577.—8. On the composition of an Elixir for the Plague; addressed to the emperor Rodolphus.—9. An elegy upon his Exile: Rostock, 1614, 4to.—10. The Rudolphine Tables; which he had not finished when he died; but were revised, and published by Kepler, as Tycho had desired.—11. An accurate Enumeration of the Fixed Stars: addressed to the emperor Rodolphus.—12. A complete Catalogue of 1000 of the Fixed Stars; which Kepler has inserted in the Rudolphine Tables.—13. <hi rend="italics">Historia Cælestis;</hi> or a History of the Heavens; in two parts: The 1st contains the Observations he had made at Uranibourg, in 16 books: The latter contains the Observations made at Wandesburg, Wittenberg, Prague, &c; in 4 books.— 14. Is an Epistle to Caster Pucer; printed at Copenhagen 1668.</p></div1><div1 part="N" n="BRANCKER" org="uniform" sample="complete" type="entry"><head>BRANCKER</head><p>, or <hi rend="smallcaps">Branker (Thomas</hi>), an eminent mathematician of the 17th century, son of Thomas Brancker some time bachelor of arts in Exeter College, Oxford, was born in Devonshire in 1636, and was admitted butler of the said college Nov. 8, 1652, in the 17th year of his age. In 1655, June the 15th, he took the degree of bachelor of arts, and was elected probationary fellow the 30th of the same month. In 1658, April the 22d, he took the degree of master of arts, and became a preacher; but after the restoration, refusing to conform to the ceremonies of the church of England, he quitted his fellowship in 1662, and retired to Chester: But not long after, he became reconciled to the service of the church, took orders from a bishop, and was made a minister of Whitegate. He had however, for some time, enjoyed great opportunity and leisure for pursuing the bent of his genius in the mathematical sciences; and his skill both in the mathematics and chemistry procured him the favour of lord Brereton, who gave him the rectory of Tilston. He was afterward chosen master of the well-endowed school at Macclesfield, in that county, where he spent the remaining years of his life, which was terminated by a short illness in 1676, at 40 years of age; and he was interred in the church at Macclesfield.</p><p>Brancker wrote a piece on the Doctrine of the Sphere, in Latin, which was published at Oxford in 1662; and in 1668, he published at London, in 4to, a translation of Rhonius's Algebra, with the title of <hi rend="italics">An Introduction to Algebra;</hi> which treatise having communicated to Dr. John Pell, he received from him some assistance towards improving it; which he generously acknowledges in a letter to Mr. John Collins; with whom, and some other gentlemen, prosicients in this science, he continued a correspondence during his life.</p></div1><div1 part="N" n="BREACH" org="uniform" sample="complete" type="entry"><head>BREACH</head><p>, in Fortification, a gap or opening made in any part of the works of a town, by the cannon or <cb/> mines of the besiegers, with intent to storm or attack the place.</p><p>BREAKING <hi rend="italics">Ground,</hi> in Military Affairs, is beginning the works for carrying on the siege of a place; more especially the beginning to dig trenches, or approaches.</p><p>BREECH <hi rend="italics">of a Gun,</hi> the hinder part, from the cascabel to the lower part of the bore.</p></div1><div1 part="N" n="BREREWOOD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BREREWOOD</surname> (<foreName full="yes"><hi rend="smallcaps">Edward</hi></foreName>)</persName></head><p>, a learned mathematician and antiquary, was the son of Robert Brerewood, a reputable tradesman, who was three times mayor of Chester. Our author was born in that city in 1565, where he was educated in grammar learning at the free school; and was afterward admitted, in 1581, of Brazen-nose College, Oxford; where he soon acquired the character of a hard student; as he has shewn by the commentaries he wrote upon Aristotle's Ethics, which were written by him about the age of 21.</p><p>In the year 1596 he was chosen the first Professor of Astronomy in Gresham College, being one of the two who, at the desire of the electors, were recommended to them by the university of Oxford. He loved retirement, and wholly devoted himself to the pursuit of knowledge. And though he never published any thing himself, yet he was very communicative, and ready to impart what he knew, to others, either in conversation or in writing. His retired situation at Gresham College being agreeable, it did not appear that he had any other views, but continued there the remainder of his life, which was terminated by a fever the 4th of November 1613, at 48 years of age, in the midst of his pursuits, and before he had taken proper care to collect and digest his learned labours; which however were not lost; being reduced to order, and published after his death. These were little or nothing mathematical, being of a miscellaneous nature, upon the several subjects of Weights, Money, Languages, Religion, Logic, the Sabbath, Meteors, the Eye, Ethics, &c.</p></div1><div1 part="N" n="BRIDGE" org="uniform" sample="complete" type="entry"><head>BRIDGE</head><p>, a work of carpentry or masonry, built over a river, canal, or the like, for the convenience of passing from one side to the other; and may be considered as a road over water, supported by one or more arches, and these again supported by proper piers or buttments. Besides these essential parts, may be added the paving at top, the banquet, or raised footway, on each side, leaving a sufficient breadth in the middle for horses and carriages, also the parapet wall either with or without a balustrade, or other ornamental and useful parts. The breadth of a bridge for a great city should be such, as to allow an easy passage for three carriages and two horsemen abreast in the middle way, and for 3 foot passengers in the same manner on each banquet: but for other smaller bridges, a less breadth.</p><p>Bridges are commonly very difficult to execute, on account of the inconvenience of laying foundations and walling under water. The earliest rules and instructions for building of bridges are given by Alberti, in his <hi rend="italics">Archit.</hi> 1. 8. Other rules were afterwards laid down by Palladio, Serlio, and Scamozzi, which are collected by Blondel, in his <hi rend="italics">Cours d'Archit.</hi> pa. 629 &c. The best of these rules were also given by Goldman, Baukhurst, and in Hawkesmoor's History of <pb n="227"/><cb/> London Bridge. M. Gautier has a considerable volume expressly on bridges, antient and modern. See also Riou's Short Principles for the Architecture of Stone Bridges; as also Emerson, Muller, Labelye, and my own Principles of Bridges.</p><p>The conditions required in a bridge are, That it be well designed, commodious, durable, and suitably decorated. It should be of such a height as to be quite convenient for the passage over it, and yet easily admitting through its arches the vessels that navigate upon it, and all the water, even at high tides and floods: the neglect of this precept has been the ruin of many bridges. Bridges are commonly continued in a straight direction perpendicular to the stream; though some think they should be made convex towards the stream, the better to resist floods, &c. And bridges of this sort have been executed in some places, as Pont St. Esprit near Lyons. Again, a bridge should not be made in too narrow a part of a navigable river, or one subject to tides or floods: because the breadth being still more contracted by the piers, this will increase the depth, velocity, and fall of the water under the arches, and endanger the whole bridge and navigation. There ought to be an uneven number of arches, or an even number of piers; both that the middle of the stream or chief current may flow freely without the interruption of a pier; and that the two halves of the bridge, by gradually rising from the ends to the middle, may there meet in the highest and largest arch; and also, that by being open in the middle, the eye in viewing it may look directly through there. When the middle and ends are of different heights, their difference however ought not to be great in proportion to the length, that the ascent and descent may be easy; and in that case also it is more beautiful to make the top in one continued curve, than two straight lines forming an angle in the middle. Bridges should rather be of few and large arches, than of many smaller ones, if the height and situation will possibly allow of it; for this will leave more free passage for the water and navigation, and be a great faving in materials and labour, as there will be fewer piers and centres, and the arches &c will require less materials; a remarkable instance of which appears in the difference between the bridges of Westminster and Blackfriars, the expence of the former being more than double the latter.</p><p>For the proper execution of a bridge, and making an estimate of the expence, &c, it is necessary to have three plans, three sections, and an elevation. The three plans are so many horizontal sections, viz, first a plan of the foundation under the piers, with the particular circumstances attending it, whether of gratings, planks, piles, &c; the 2d is the plan of the piers and arches; and the 3d is the plan of the superstructure, with the paved road and banquet. The three sections are vertical ones; the first of them a longitudinal section from end to end of the bridge, and through the middle of the breadth; the 2d a transverse one, or across it, and through the summit of an arch; and the 3d also across, but taken upon a pier. The elevation is an orthographic projection of one side or face of the bridge, or its appearance as viewed at a distance, shewing the exterior aspect of the materials, with the manner in which they are disposed &c. <cb/></p><p>For the figure of the arches, some prefer the semicircle, though perhaps without knowing any good reason why; others the elliptical form, as having many advantages over the semi-circular; and some talk of the catenarian arch, though its pretended advantages are only chimerical; but the arch of equilibration is the only perfect one, so as to be equally strong in every part: see my Principles of Bridges. The piers are of diverse thickness, according to the sigure, span, and height of the arches; as may be seen in the work above mentioned.</p><p>With the Romans, the repairing and building of bridges were committed to the priests, thence named <hi rend="italics">pontifices;</hi> next to the censors, or curators of the roads; but at last the emperors took the care of the bridges into their own hands. Thus, the Pons Janiculensis was built of marble by Antoninus Pius; the Pons Cestius was restored by Gordian; and Arian built a new one which was called after his own name. In the middle age, bridge-building was counted among the acts of religion; and, toward the end of the 12th century, St. Benezet founded a regular order of hospitallers, under the name of <hi rend="italics">pontifices,</hi> or bridge-builders, whose office was to assist travellers, by making bridges, settling ferries, and receiving strangers into hospitals, or houses, built on the banks of rivers. We read of an hospital of this kind at Avignon, where the hospitallers resided under the direction of their first superior St. Benezet: and the Jesuit Raynaldus has a treatise on St. John the bridge-builder.</p><p>Among the bridges of antiquity, that built by Trajan over the Danube, it is allowed, is the most magnificent. It was demolished by his next successor Adrian, and the ruins are still to be seen in the middle of the Danube, near the city Warhel in Hungary. It had 20 piers, of square stone, each of which was 150 feet high above the foundation, 60 feet in breadth, and 170 feet distant from one another, which is the span or width of the arches; so that the whole length of the bridge was more than 1530 yards, or one mile nearly.</p><p>In France, the Pont de Garde is a very bold structure; the piers being only 13 feet thick, yet serving to support an immense weight of a triplicate arcade, and joining two mountains. It consists of three bridges, one over another; the uppermost of which is an aqueduct.</p><p>The bridge of Avignon, which was finished in the year 1188, consists of 18 arches, and measures 1340 paces, or about 1000 yards in length.</p><p>The famous bridge at Venice, called the Rialto, passes for a master piece of art, consisting of only one very flat and bold arch, near 100 feet span, and only 23 feet high above the water: it was built in 1591.— Poulet also mentions a bridge of a single arch, in the city of Munster in Bothnia, much bolder than that of the Rialto at Venice. Yet these are nothing to a bridge in China, built from one mountain to another, consisting only of a single arch, 400 cubits long, and 500 cubits high, whence it is called the flying bridge; and a figure of it is given in the Philos. Trans. Kircher also speaks of a bridge in the same country 360 perches long without any arch, but supported by 300 pillars.</p><p>There are many bridges of considerable note in our <pb n="228"/><cb/> own country. The triangular bridge at Crowland in Lincolnshire, it is said, is the most ancient Gothic structure remaining intire in the kingdom; and was erected about the year 860.</p><p>London bridge is on the old Gothick structure, with 20 small locks or arches, each of only 20 feet wide; but there are now only 18 open, two having lately been thrown into one in the centre, and another next one side is concealed or covered up. It is 900 feet long, 60 high, and 74 wide; the piers are from 25 to 34 feet broad, with starlings projecting at the ends; so that the greatest water-way, when the tide is above the starlings, was 450 feet, scarce half the breadth of the river; and below the starlings, the water-way was reduced to 194 feet, before the late opening of the centre. London bridge was first built with timber between the years 993 and 1016; and it was repaired, or rather new built with timber, 1163. The stone bridge was begun in 1176, and finished in 1209. It is probable there were no houses on this bridge for upwards of 200 years; since we read of a tilt and tournament held on it in 1395. Houses it seems were erected on it afterwards; but being found of great inconvenience and nuisance, they were removed in 1758, and the avenues to it enlarged and the whole made more commodious; the two middle arches were then thrown into one, by removing the pier from between them; the whole repairs amounting to above 80,000l.</p><p>There are still some more bridges in England built in the old manner of London bridge; as the bridge at Rochester, and some others; also the late bridge at Newcastle upon Tyne, which was broken down by a great flood in the year 1771, for want of a sufficient quantity of water-way through the arches.</p><p>The longest bridge in England is that over the Trent at Burton, built in the 12th century, of squared free stone, and is strong and lofty; it contains 34 arches, and the whole length is 1545 feet. But this falls far short of the wooden bridge over the Drave, which according to Dr. Brown is at least 5 miles long.</p><p>But one of the most singular bridges in Europe is that built over the Taaf in Glamorganshire, by William Edward, a poor country mason, in the year 1756. This remarkable bridge consists of only one stupendous arch, which, though only 8 feet broad, and 35 feet high, is no less than 140 span, being part of a circle of 175 feet diameter.</p><p>There is also a remarkable bridge of one arch, built at Colebrook Dale in 1779, of cast iron: and another still larger of the same metal, is now raising over the river Wear, at Sunderland, the arch being of 240 feet span.</p><p>Of modern bridges, perhaps the two finest in Europe, are the Westminster and Blackfriars bridges over the river Thames at London. The former is 1220 feet long, and 44 feet wide, having a commodious broad foot path on each side for passengers. It consists of 13 large and two small arches, all semicircular, with 14 intermediate piers. The arches all spring from about 2 feet above low-water mark; the middle arch is 76 feet wide, and the others on each side decrease always by 4 feet at a time. The two middle piers are each 17 feet thick at the springing of the arches; and the others decrease equally on each side by one foot at <cb/> a time; every pier terminating with a saliant right angle against either stream. This bridge is built of the best materials, and in a neat and elegant taste, but the arches are too small for the quantity of masonry contained in it. This bridge was begun in 1738, and opened in 1750; and the whole sum of money granted and paid for the erection of this bridge, with the purchase of houses to take down, and widening the avenues, &c, amounted to 389,500l.</p><p>Blackfriars bridge, nearly opposite the centre of the city of London, was begun in 1760, and was completed in 10 years and three quarters; and is an exceeding light and elegant structure, but the materials unfortunately do not seem to be of the best, as many of the arch stones are decaying. It consists of 9 large, elegant, elliptical arches; the centre arch being 100 feet wide, and those on each side decreasing in a regular gradation, to the smallest, at each extremity, which is 70 feet wide. The breadth of the bridge is 42 feet, and the length from wharf to wharf 995. The upper surface is a portion of a very large circle, which forms an elegant figure, and is of convenient passage over it. The whole expence was 150,840l.</p><p>There are various sorts of bridges, of stone, wood, or metal, of boats or floats, pendant or hanging bridges, draw bridges, flying bridges, &c, &c, and even natural bridges, or such as are found formed by nature, of which kind a most wonderful one is described by Mr. Jefferson, in his <hi rend="italics">State of Virginia;</hi> and another, but smaller, is described by Don Ulloa, in the province of Angaraez in South America.</p></div1><div1 part="N" n="BRIGGS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BRIGGS</surname> (<foreName full="yes"><hi rend="smallcaps">Henry</hi></foreName>)</persName></head><p>, one of the greatest mathematicians in the 16th and 17th centuries, was born at Warleywood, near Halifax, in Yorkshire, in 1556. From a grammar school in that country he was sent to St. John's College, Cambridge, 1579; where after taking both the degrees in arts, he was chosen fellow of his college in 1588. He applied himself chiefly to the study of the mathematics, in which he greatly excelled; in consequence in 1592 he was made examiner and lecturer in that faculty; and soon after, reader of the physic lecture, founded by Dr. Linacer.</p><p>Upon the settlement of Gresham College, in London, he was chosen the first professor of geometry there, in 1596. Soon after this, he constructed a table, for finding the latitude, from the variation of the magnetic needle being given. In the year 1609 he contracted an acquaintance with the learned Mr. James Usher, afterwards archbishop of Armagh, which continued many years after by letters, two of Mr. Briggs being still extant in the collection of Usher's letters that were published: in the former of these, dated August 1610, he writes among other things, that he was engaged in the subject of eclipses; and in the latter, dated the 10th of March 1615, that he was wholly taken up and employed about the noble invention of logarithms, which had come out the year before, and in the improvement of which he had afterwards so great a concern. For Briggs immediately set himself to the study and improvement of them; expounding them also to his auditors in his lectures at Gresham college. In these lectures he proposed the alteration of the scale of logarithms, from the hyperbolic form which Napier <pb n="229"/><cb/> had given them, to that in which 1 should be the logarithm of the ratio of 10 to 1; and soon after he wrote to Napier to make the same proposal to himself. In the year 1616 Briggs made a visit to Napier at Edinburgh, to confer with him upon this change; and the next year he did the same also. In these conferences, the alteration was agreed upon accordingly, and upon Briggs's return from his second visit, in 1617, he published the first chiliad, or 1000 of his logarithms. See the Introduction to my Logarithms.</p><p>In 1619 he was made the first Savilian professor of geometry; and resigned the professorship of Gresham college the 25th of July 1620. At Oxford he settled himself at Merton college, where he continued a most laborious and studious life, employed partly in the duties of his office as geometry lecturer, and partly in the computation of the logarithms, and in other useful works. In the year 1622 he published a small tract on the “North-west passage to the South Seas, through the continent of Virginia and Hudson's Bay;” the reason of which was probably, that he was then a member of the company trading to Virginia. His next performance was his great and elaborate work, the <hi rend="italics">Arithmetica Logarithmica</hi> in folio, printed at London in 1624; a stupendous work for so short a time! containing the logarithms of 30 thousand natural numbers, to 14 places of figures beside the index. Briggs lived also to complete a table of logarithmic sines and tangents for the 100th part of every degree, to 14 places of figures beside the index; with a table of natural sines for the same 100th parts to 15 places, and the tangents and secants for the same to ten places; with the construction of the whole. These tables were printed at Gouda in 1631, under the care of Adrian Vlacq, and published in 1633, with the title of <hi rend="italics">Trigonometria Britannica.</hi> In the construction of these two works, on the logarithms of numbers, and of sines and tangents, our author, beside extreme labour and application, manifests the highest powers of genius and invention; as we here for the first time meet with several of the most important discoveries in the mathematics, and what have hitherto been considered as of much later invention; such as the Binomial Theorem; the Differential Method and Construction of Tables by Differences; the Interpolation by Differences; with Angular Sections, and several other ingenious compositions: a particular account of which may be seen in the Introduction to my Mathematical Tables.</p><p>This truly great man terminated his useful life the 26 of January 1630, and was buried in the choir of the chapel of Merton College. As to his character, he was not less esteemed for his great probity and other eminent virtues, than for his excellent skill in mathematics. Doctor Smith gives him the character of a man of great probity; easy of access to all; free from arrogance, moroseness, envy, ambition and avarice; a contemner of riches, and contented in his own situation; preferring a studious retirement to all the splendid circumstances of life. The learned Mr. Thomas Gataker, who attended his lectures when he was reader of mathematics at Cambridge, represents him as highly esteemed by all persons skilled in mathematics, both at home and abroad; and says, that desiring him once to give his judgment concerning judicial astrology, his <cb/> answer was, “that he conceived it to be a mere system of groundless conceits.” Oughtred calls him the mirror of the age, for his excellent skill in geometry. And one of his successors at Gresham college, the learned Dr. Isaac Barrow, in his oration there upon his admission, has drawn his character more fully; celebrating his great abilities, skill, and industry, particularly in perfecting the invention of logarithms, which, without his care and pains, might have continued an imperfect and useless design.</p><p>His writings were more important than numerous: some of them were published by other persons: the list of the principal part of them as follows.</p><p>1. <hi rend="italics">A Table to find the Height of the Pole; the Magnetical Declination being given.</hi> This was published in Mr. Thomas Blundevile's Theoriques of the Seven Planets: London 1602, 4to.</p><p>2. <hi rend="italics">Tables for the improvement of Navigation.</hi> These consist of, A table of declination of every minute of the ecliptic, in degrees, minutes and seconds: A table of the sun's prosthaphaereses: A table of equations of the sun's ephemerides: A table of the sun's declination: Tables to find the height of the pole in any latitude, from the height of the pole star. These tables are printed in the 2d edition of Edward Wright's treatise, intitled, Certain Errors in Navigation detected and corrected; London 1610, 4to.</p><p>3. <hi rend="italics">A description of an Instrumental Table to find the Part Proportional, devised by Mr. Edward Wright.</hi> This is subjoined to Napier's table of logarithms, translated into English by Mr. Wright, and after his death published by Briggs, with a preface of his own: Lond. 1616 and 1618, 12mo.</p><p>4. <hi rend="italics">Logarithmorum chilias prima.</hi> Lond. 1617, 8vo.</p><p>5. <hi rend="italics">Lucubrationes & Annotationes in opera posthuma J. Neperi:</hi> Edinb. 1619, 4to.</p><p>6. <hi rend="italics">Euclidis Elementorum</hi> vi <hi rend="italics">libri priores &c.</hi> Lond. 1620, folio. This was printed without his name to it.</p><p>7. <hi rend="italics">A treatise of the North-west passage to the South Sea &c.</hi> By H. B. Lond. 1622, 4to. This was reprinted in Purchas's Pilgrims, vol. 3, p. 852.</p><p>8. <hi rend="italics">Arithmetica Logarithmica, &c.</hi> Lond. 1624, folio.</p><p>9. <hi rend="italics">Trigonometria Britannica, &c.</hi> Goudæ 1633, folio.</p><p>10. <hi rend="italics">Two letters to archbishop Usher.</hi></p><p>11. <hi rend="italics">Mathematica ab antiquis minus cognita.</hi>—This is a summary account of the most observable inventions of modern mathematicians, communicated by Mr. Briggs to Dr. George Hakewill, and published by him in his <hi rend="italics">Apologie;</hi> Lond. folio.</p><p>Beside these publications, Briggs wrote some other pieces, that have not been printed: as,</p><p>(1). <hi rend="italics">Commentaries on the Geometry of Peter Ramus.</hi></p><p>(2). <hi rend="italics">Duæ Epistolæ ad celeberrimum virum, Chr. Sever. Longomontanum.</hi> One of these letters contained some remarks on a treatise of Longomontanus, about squaring the circle; and the other a defence of arithmetical geometry.</p><p>(3). <hi rend="italics">Animadversiones Geometricæ:</hi> 4to.</p><p>(4). <hi rend="italics">De eodem Argumento:</hi> 4to.—These two were in the possession of the late Mr. Jones. They both contain a great variety of geometrical propositions, concerning the properties of many figures, with several <pb n="230"/><cb/> arithmetical computations, relating to the circle, angular sections, &c.—The two following were also in possession of Mr. Jones.</p><p>(5). <hi rend="italics">A treatise of Common Arithmetic;</hi> folio.</p><p>(6). <hi rend="italics">A letter to Mr. Clarke of Gravesend,</hi> dated 25 Feb. 1606; with which he sends him the description of a ruler, called Bedwell's ruler, with directions how to use it.</p></div1><div1 part="N" n="BRIGGS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BRIGGS</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an eminent physician in the latter part of the 17th century, was born at Norwich, for which town his father was four times member of parliament. He studied at the university of Cambridge. He afterwards travelled into France, where he attended the lectures of the famous anatomist Vieussens, at Montpelier. Upon his return he published his <hi rend="italics">Ophthalmographia,</hi> in 1676. The year following he was made doctor of medicine at Cambridge, and soon after fellow of the college of physicians at London. In 1682 he resigned his fellowship to his brother; and the same year his <hi rend="italics">Theory of Vision</hi> was published by Hook. The ensuing year he sent to the Royal Society a continuation of that discourse, which was published in their Transactions; and the same year he was appointed physician to St. Thomas's hospital. In 1684 he communicated to the Royal Society two remarkable cases relating to vision, which were likewise printed in their Transactions; and in 1685 he published a Latin version of his <hi rend="italics">Theory of Vision,</hi> at the desire of Mr. Newton, afterwards Sir Isaac, then professor of mathematics at Cambridge, with a recommendatory epistle from him prefixed to it. He was afterwards made physician in ordinary to king William, and continued in great esteem for his skill in his profession till he died the 4th of September 1704.</p><p><hi rend="smallcaps">Briggs's</hi> <hi rend="italics">Logarithms,</hi> that species of them in which 1 is the logarithm of the ratio of 10 to 1, or the logarithm of 10. See <hi rend="smallcaps">Logarithms.</hi></p><p>BROKEN <hi rend="italics">Number,</hi> the same as <hi rend="italics">Fraction;</hi> which see.</p><p><hi rend="smallcaps">Broken</hi> <hi rend="italics">Ray,</hi> or <hi rend="italics">Ray of Refraction,</hi> in Dioptrics, is the line into which an incident ray is refracted or broken, in crossing the second medium.</p></div1><div1 part="N" n="BROUNCKER" org="uniform" sample="complete" type="entry"><head>BROUNCKER</head><p>, or BROUNKER, (<hi rend="smallcaps">William</hi>), lord viscount of Castle Lyons in Ireland, son of Sir William Brounker, afterwards made viscount in 1645, was born about the year 1620. He very early discovered a genius for mathematics, in which he afterwards became very eminent. He was made doctor of physic at Oxford June 23, 1646. In 1657 and 1658, he was engaged in a correspondence by letters on mathematical subjects with Dr. John Wallis, who published them in his <hi rend="italics">Commercium Epistolicum,</hi> printed 1658, at Oxford. He was one of the nobility and gentry who signed the remarkable declaration concerning king Charles the 2d, published in April 1660.</p><p>After the restoration, lord Brounker was made chancellor and keeper of the great seal to the queen consort, one of the commissioners of the navy, and inaster of St. Katherine's hospital near the tower of London. He was one of those great men who first formed the Royal Society, of which he was by the charter appointed the first president in 1662: which office he held, with great advantage to the Society, and honour to himself, till the anniversary election, Nov. 30, 1677. He died at his house in St. James's street, Westmin- <cb/> ster, the 5th of April 1684; and was succeeded in his title by his younger brother Harry, who died in Jan. 1687.</p><p>Lord Brounker had several papers inserted in the Philosophical Transactions, the chief of which were, 1. Experiments concerning the Recoiling of Guns.— 2. — Series for the Quadrature of the Hyperbola; which was the first series of the kind upon that subject. —3. Several of his letters to archbishop Usher were also printed in Usher's letters; as well as some to Dr. Wallis, in his Commercium Epistolicum, above mentioned.</p></div1><div1 part="N" n="BROWN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BROWN</surname> (Sir <foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, a noted physician and miscellaneous writer, of the 18th century. He was settled originally at Lynn in Norfolk, where he published a translation of Dr. Gregory's Elements of Catoptrics and Dioptrics; to which he added, 1. A Method for finding the Foci of all Specula and Lenses universally; as also Magnifying or Lessening a given object by a given Speculum or lens, in any assigned proportion.—2. A Solution of those Problems which Dr. Gregory has left undemonstrated.—3. A particular account of Microscopes and Telescopes, from Mr. Huygens; with the discoveries made by Catoptrics and Dioptrics.</p><p>Having acquired a competence by his profession, he removed to Queen's Square, Ormond Street, London, where he resided till his death, in 1774, at 82 years of age; leaving by his will two prize-medals to be annually contended for by the Cambridge poets.</p><p>Sir William Brown was a very facetious man; and a great number of his lively essays, both in prose and verse, were printed and circulated among his friends. The active part taken by him in the contest with the licentiates, in 1768, occasioned his being introduced by Mr. Foote in his <hi rend="italics">Devil upon Two Sticks.</hi>— Upon Foote's exact representation of him with his identical wig and coat, tall sigure, and glass stiffly applied to his eye, he sent him a card complimenting him on having so happily represented him; but as he had forgot his muff, he had sent him his own.— This good-natured way of resenting disarmed Foote.— He used to frequent the annual ball at the ladies boarding-school, Queen Square, merely as a neighbour, a good-natured man, and sond of the company of sprightly young folks. A dignitary of the church being there one day to see his daughter dance, and finding this upright sigure stationed there, told him he believed he was Hermippus redivivus who lived <hi rend="italics">anhelitu puellarum.</hi> —When he lived at Lynn, a pamphlet was written against him; which he nailed up against his house-door. —At the age of 80, on St. Luke's day 1771, he came to Batson's coffee-house in his laced coat and band, and fringed white gloves, to shew himself to Mr. Crosby, then lord mayor. A gentleman present observing that he looked very well, he replied, he had neither wise nor debts.</p></div1><div1 part="N" n="BULLIALD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BULLIALD</surname> (<foreName full="yes"><hi rend="smallcaps">Ismael</hi></foreName>)</persName></head><p>, an eminent astronomer and mathematician, was born at Laon in the Isle of France in 1605. He travelled in his youth, for the sake of improvement, and gave very early prooss of his astronomical genius; and his riper years rendered him beloved and admired. Riccioli styled him, <hi rend="italics">Astronomus prosimdæ indaginis.</hi> He first published his disser- <pb n="231"/><cb/> tation intitled, <hi rend="italics">Philolaus, sive de vero Systemate Mundi;</hi> or his true system of the world, according to Philolaus, an ancient philosopher and astronomer. Afterward, in the year 1645, he set forth his <hi rend="italics">Astronomia Philolaica,</hi> grounded upon the hypothesis of the earth's motion, and the elliptical orbit described by the planet's motion about a cone. To which he added tables intitled, <hi rend="italics">Tabulæ Philolaicæ:</hi> a work which Riccioli says ought to be attentively read by all students of astronomy.—He considered the hypothesis, or approximation of bishop Ward, and found it not to agree with the planet Mars; and shewed in his defence of the Philolaic astronomy against the bishop, that from four observations made by Tycho on the planet Mars, that planet in the first and third quarters of the mean anomaly, was more forward than it ought to be according to Ward's hypothesis; but in the 2d and 4th quadrant of the same, the planet was not so far advanced as that hypothesis required. He therefore set about a correction of the bishop's hypothesis, and made it to answer more exactly to the orbits of the planets, which were most eccentric, and introduced what is called, by Street in his <hi rend="italics">Caroline Tables,</hi> the Variation: for these tables were calculated from this correction of Bulliald's, and exceeded all in exactness that went before. This correction is, in the judgment of Dr. Gregory, a very happy one, if it be not set above its due place; and be accounted no more than a correction of an approximation to the true system: For by this means we are enabled to gather the coequate anomaly <hi rend="italics">a priori</hi> and directly from the mean, and the observations are well enough answered at the same time; which, in Mercator's opinion, no one had effected before.—It is remarkable that the ellipsis which he has chosen for a planet's motion, is such a one as, if cut out of a cone, will have the axis of the cone passing through one of its foci, viz, that next the aphelion.</p><p>In 1657 was published his treatise <hi rend="italics">De Lineis Spiralibus, Exerc. Geom. & Astron.</hi> Paris, 4to.—In 1682 came out at Paris, in folio, his large work intitled, <hi rend="italics">Opus novum ad Arithmeticam Infinitorum:</hi> A work which is a diffuse amplification of Dr. Wallis's Arithmetic of Infinites, and which Wallis treats of particularly in the 80th chapter of his historical treatise of Algebra.—He wrote also two Admonitions to Astronomers. The sirst, concerning a new star in the neck of the Whale, appearing at some times, and disappearing at others. The 2d, concerning a nebulous star in the northern part of Andromeda's girdle, not discovered by any of the ancients. This star also appeared and disappeared by turns. And as these phenomena appeared new and surprizing, he strongly recommended the observing them to all that might be curious in astronomy.</p></div1><div1 part="N" n="BURNING" org="uniform" sample="complete" type="entry"><head>BURNING</head><p>, the action of fire on some pabulum or fuel, by which its minute parts are put into a violent motion, and some of them, assuming the nature of fire, fly off <hi rend="italics">in orbem,</hi> while the rest are dissipated in vapour or reduced to ashes..</p><p><hi rend="smallcaps">Burning</hi>-<hi rend="italics">Glass,</hi> or <hi rend="italics">Burning-Mirror,</hi> a machine by which the sun's rays are collected into a point; and by that means their force and effect are extremely heightened, so as to burn objects placed in it. <cb/></p><p>Burning glasses are of two kinds, <hi rend="italics">convex</hi> and <hi rend="italics">concave.</hi> The convex ones are lenses, which acting according to the laws of refraction, incline the rays of light towards the axis, and unite them in a point or focus. The concave ones are mirrors or reflectors, whether made of polished metal or silvered glass, and which acting by the laws of reflection, throw the rays back into a point or focus before the glass.</p><p>The use of burning glasses it appears is very ancient, many of the old authors relating some effects of them. Diodorus Siculus, Lucian, Dion, Zonaras, Galen, Anthemius, Eustatius, Tzetzes, and others, relate that by means of them Archimedes set fire to the Roman fleet at the siege of Syracuse. Tzetzes is so particular in his account of this matter, that his description suggested to Kircher the method by which it was probably accomplished. That author says that “Archimedes set fire to Marcellus's navy by means of a burning glass composed of small square mirrors, moving every way upon hinges; which when placed in the sun's rays, directed them upon the Roman fleet, so as to reduce it to ashes at the distance of a bowshot.” And the burning power of reflectors is mentioned in Euclid's Optics, theor. 31. Again, Aristophanes, in his comedy of The Clouds, introduces Socrates as examining Strepsiades about a method he had discovered of getting clear of his debts. He replies, that “he thought of making use of a burning-glass which he had hitherto used in kindling his fire; for should they bring a writ against me, I'll immediately place my glass in the sun at some little distance from it, and set it on fire.” Pliny and Lactantius have also spoken of glasses that burn by refraction. The former calls them <hi rend="italics">balls</hi> or <hi rend="italics">globes</hi> of <hi rend="italics">crystal</hi> or <hi rend="italics">glass,</hi> which being exposed to the sun, transmit a heat sufficient to set sire to cloth, or corrode the dead flesh of those patients who stand in need of caustics; and the latter, after Clemens Alexandrinus, observes that fire may be kinkled by interposing glasses filled with water between the sun and the object, so as to transmit the rays to it.</p><p>Among the ancients the most celebrated burning mirrors were those of Archimedes and Proclus; by the former was burnt the fleet of Marcellus, as above mentioned; and by the latter, the navy of Vitellius, besieging Byzantium, according to Zonaras was burnt to ashes.</p><p>Among the moderns, the most remarkable burningglasses, are those of Magine of 20 inches diameter: of Sepatala of Milan, near 42 inches diameter, and which burnt at the distance of 15 feet; of Settala of Villette, of Tschirnhausen, of Buffon, of Trudaine, and of Parker.</p><p>Villette, a French artist at Lyons, made a large mirror, which was bought by Tavernier, and presented to the king of Prussia; a second, bought by the king of Denmark; a third, presented to the Royal Academy by the king of France; and a 4th came to England, and was publicly shewn. This mirror is 47 inches wide, being a segment of a sphere of 76 inches radius; so that its focus is about 38 inches from the vertex; and its substance is a composition of tin, copper, and tin-glass. Some of its effects were as follow: <pb n="232"/><cb/> <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">sec.</cell></row><row role="data"><cell cols="1" rows="1" role="data">A silver sixpence melted in</cell><cell cols="1" rows="1" role="data"> 7 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">A George the 1st's halfpenny in</cell><cell cols="1" rows="1" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" role="data">and runs with a hole in</cell><cell cols="1" rows="1" role="data">34</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tin melts in</cell><cell cols="1" rows="1" role="data"> 3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cast iron in</cell><cell cols="1" rows="1" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" role="data">Slate in</cell><cell cols="1" rows="1" role="data"> 3</cell></row><row role="data"><cell cols="1" rows="1" role="data">A fossil shell calcines in</cell><cell cols="1" rows="1" role="data"> 7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Piece of Pompey's pillar vitrifies, the black part in</cell><cell cols="1" rows="1" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">the white part in</cell><cell cols="1" rows="1" role="data">54</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper ore in</cell><cell cols="1" rows="1" role="data"> 8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bone calcines in 4, and vitrifies in</cell><cell cols="1" rows="1" role="data">33</cell></row></table> An emerald melts into a substance like a torquois stone; a diamond weighing 4 grains loses 7/8 of its weight: the asbestos vitrifies; as all other bodies will do if kept long enough in the focus; but when once vitrified, the mirror can go no farther with them. Philos. Trans. vol. iv. pa. 198.</p><p>Tschirnhausen's reflecting mirrors produced equally surprizing effects; as they may be seen described in the Acta Erudit. for 1687, pa. 52. And other persons have made very good ones of wood, straw, paper, ice, and other substances capable of taking a proper form and polish.</p><p>Every lens, whether convex, plano-convex, or convexo-convex, collects the sun's rays, dispersed over its convexity, into a point by refraction; and it is therefore a burning-glass. The most considerable of this kind is that made by Tschirnhausen, and described in the same Acta Erudit. The diameters of his lenses are from 3 to 4 feet, having the focus at the distance of 12 feet, and its diameter an inch and a half. To make the focus more vivid, the rays are collected a second time, by a second lens parallel to the first, and placed at such a distance that the diameter of the cone of rays formed by the first lens is equal to the diameter of the second; so that it receives them all; and the focus is reduced from an inch and a half to half the quantity, and consequently its force is quadrupled. This glass vitrifies tiles, slates, pumice-stones &c. in a moment. It melts sulphur, pitch, and all rosins, under water; the ashes of vegetables, woods and other matters, are transmuted into glass; and every thing applied to its focus is either melted, changed into a calx, or into fumes. The author observes that it succeeds best when the matter applied is laid on a hard charcoal well burnt. —But though the force of the solar rays be thus found so surprizing, yet the rays of the full moon, collected by the same burning-glass, do not shew the least increase of heat.</p><p>Sir Isaac Newton presented a burning-glass to the Royal Society, consisting of 7 concave glasses, so placed that all their foci join in one physical point. Each glass is about 11 1/2 inches diameter: six of them are placed contiguous to, and round the seventh, forming a kind of spherical segment, whose subtense is about 34 1/2 inches: the common focus is about 22 1/2 inches distant, and about an inch in diameter. This glass vitrifies brick or tile in 1 second, and melts gold in 30 seconds.</p><p>M. Buffon also made a variety of very powerfulburningglasses, both as mirrors and as lenses; but at length concluded with one which is probably of the same nature with that of Archimedes, and consisted of 400 mirrors <cb/> reflecting their rays all to one point, and with which he could melt lead and tin at the distance of 140 feet; and with others he consumed substances at the distance of 210 feet. See Philos. Trans. vol. 44; or Buffon's Histoire Naturelle, Suppl. vol. 1; or Montucla's Histoire des Math. vol. i. pa. 246.</p><p>It would seem there is no substance capable of resisting the efficacy of modern burning-glasses; though water &c. are not affected by them at all. Thus, Messrs Macquer and Baumé have succeeded in melting small portions of platina by means of a concave glass, 22 inches diameter, and 28 inches focus; though this metal is not fusible by the strongest fires that can be excited in furnaces, or sustainèd by any chemical apparatus. Yet it was long since observed, by the Academicians del Cimento, that spirit of. wine could not be sired by any burning-glass which they used; and notwithstanding the great improvements these instruments have since received, M. Nollet has not been able, by the most powerful burning mirrors, to set fire to any inflammable liquors whatever.</p><p>However, a large burning lens, for fusing and vitrifying such substances as resist the fires of furnaces, and especially for the application of heat in vacuo, and in certain other circumstances in which heat cannot be applied by other means, has long been a desideratum with persons concerned in philosophical experiments: and this it appears is now in a great measure accomplished by Mr. Parker, an ingenious glass manufacturer in Fleet-street, London. His lens is made of flint glass, and is 3 feet in diameter, but when fixed in its frame exposes a surface of 32 inches in the clear; the length of the focus is 6 feet 8 inches, and its diameter one inch. The rays from this large lens are received and transmitted through a smaller, of 13 inches diameter in the clear within the frame, its focal length 29 inches, and diameter of its focus 3-8ths of an inch: so that this second lens increases the power of the former more than 7 times, or as the square of 8 to the square of 3.</p><p>From a great number of experiments made with this lens, the following are selected to serve as specimens of its powers: <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Substances fused; with their weight, and time of fusion.</cell><cell cols="1" rows="1" rend="align=center" role="data">Time in sec.</cell><cell cols="1" rows="1" rend="align=center" role="data">Weight in grs.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Scoria of wrought iron</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common slate</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Silver, pure</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Platina, pure</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Nickell</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cast Iron, a cube</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Kearsh</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gold, pure</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Crystal pebble</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cauk, or terra ponderosa</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lava</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Asbestos</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bar Iron, a cube</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Steel, a cube</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Garnet</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper, pure</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell></row><row role="data"><cell cols="1" rows="1" role="data">Onyx</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row></table> <pb n="233"/><cb/> <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">SUBSTANCES FUSED</hi>, &c.</cell><cell cols="1" rows="1" rend="align=center" role="data">Time in sec.</cell><cell cols="1" rows="1" rend="align=center" role="data">Wgt. in grs.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Zeolites</cell><cell cols="1" rows="1" rend="align=center" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pumice Stone</cell><cell cols="1" rows="1" rend="align=center" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oriental Emerald</cell><cell cols="1" rows="1" rend="align=center" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jasper</cell><cell cols="1" rows="1" rend="align=center" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">White Agate</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Flint, oriental</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Topaz, or chrysolite</cell><cell cols="1" rows="1" rend="align=center" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common Limestone</cell><cell cols="1" rows="1" rend="align=center" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">White <*>homboidal Spar</cell><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Volcanic Clay</cell><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cornish Moorstone</cell><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rough Cornelian</cell><cell cols="1" rows="1" rend="align=center" role="data">75</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rotten Stone</cell><cell cols="1" rows="1" rend="align=center" role="data">80</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row></table></p><p><hi rend="smallcaps">Burning</hi> <hi rend="italics">Zone,</hi> or <hi rend="italics">Torrid Zone,</hi> the space within 23 1/2 degrees of the equator, both north and south. <cb/></p></div1><div1 part="N" n="BUSHEL" org="uniform" sample="complete" type="entry"><head>BUSHEL</head><p>, a measure of capacity for dry goods; as grain, pulse, fruits, &c; containing 4 pecks, or 8 gallons, or 1/8 of a quarter. By act of Parliament, made in 1697, it was ordained that “Every round bushel with a plain and even bottom, being made 18 1/2 inches wide throughout, and 8 inches deep, shall be esteemed a Legal Winchester Bushel, according to the standard in his majesty's exchequer.” Now a bushel being thus made will contain 2150.42 cubic inches, and consequently the corn gallon contains only 268 4/5 cubic inches.</p></div1><div1 part="N" n="BUTMENTS" org="uniform" sample="complete" type="entry"><head>BUTMENTS</head><p>, are those supporters, or props, by which the feet of arches, or the extremities of bridges are supported; and should be made very strong and firm.</p></div1><div1 part="N" n="BUTTRESS" org="uniform" sample="complete" type="entry"><head>BUTTRESS</head><p>, is an arch, or a mass of masonry, serving to support the sides of a building, wall, or the like, on the outside. See <hi rend="smallcaps">Arch</hi>, and A<hi rend="smallcaps">RCH-BOUTANT.</hi> </p></div1></div0><div0 part="N" n="C" org="uniform" sample="complete" type="alphabetic letter"><head>C</head><cb/><div1 part="N" n="CAILLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CAILLE</surname> (<foreName full="yes"><hi rend="smallcaps">Nicholas Lewis de la</hi></foreName>)</persName></head><p>, an eminent French mathematician and astronomer, was born at Rumigny in the diocese of Rheims in 1713. His fa, ther having quitted the army, in which he had served, amused himself in his retirement with studying mathematics and mechanics, in which he proved the happy author of several inventions of considerable use to the public. From this example of his father, our author almost in his infancy took a fancy to mechanics, which proved of signal service to him in his maturer years. At school he discovered early tokens of genius. He next came to Paris in 1729; where he studied the classics, philosophy and mathematics. He afterwards studied divinity in the college de Navarre, with the view of embracing the ecclesiastical life: however he never entered into priest's orders, apprehending that his astronomical studies, to which he had become much devoted, might too much interfere with his religious duties. His turn for astronomy soon connected him with the celebrated Cassini, who procured him an apartment in the observatory; where, assisted by the counsels of this master, he soon acquired a name among the astronomers. In 1739 he was joined with M. Cassini de Thury, son to M. Cassini, in verifying the meridian through the whole extent of France: and in the same year he was named professor of mathematics in the college of Mazarine. In 1741 our author was admitted into the Academy of Sciences as an adjoint member for astronomy; and had many excellent papers inserted in their memoirs; beside which he published several useful treatises, viz, Elements of Geometry, Astronomy, Mechanics, and Optics. He also carefully computed all the eclipses of the sun and moon that had happened <cb/> since the christian era, which were printed in the work entitled <hi rend="italics">l'Art de verifier les dates,</hi> &c, Paris, 1750, in 4to. He also compiled a volume of astronomical ephemerides for the years 1745 to 1755; another for the years 1755 to 1765; and a third for the years 1765 to 1775: as also the most correct solar tables of any; and an excellent work entitled <hi rend="italics">Astronomiæ fundamenta novissimis solis & stellarum observationibus stabilita.</hi></p><p>Having gone through a seven years series of astronomical observations in his own observatory in the Mazarine college, he formed the project of going to observe the southern stars at the Cape of Good Hope: being countenanced by the court, he set out upon this expedition in 1750, and in the space of two years he observed there the places of about 10 thousand stars in the southern hemisphere that are not visible in our latitudes, as well as many other important elements, viz, the parallaxes of the sun, moon, and some of the planets, the obliquity of the ecliptic, the refractions, &c. Having thus executed the purpose of his voyage, and no present opportunity offering for his return, he thought of employing the vacant time in another arduous attempt; no less than that of taking the measure of the earth, as he had already done that of the heavens. This indeed had been done before by different sets of learned men both in Europe and America; some determining the quantity of a degree at the equator, and others at the arctic circle: but it had not as yet been decided whether in the southern parallels of latitude the same dimensions obtained as in the northern. His labours were rewarded with the satisfaction he wished for; having determined a distance of 410814 feet from a place called <hi rend="italics">Klip-Fontyn</hi> to the <pb n="234"/><cb/> Cape, by means of a base of 38802 feet, three times actually measured; whence he discovered a new secret of nature, namely, that the radii of the parallels in south latitude are not the same length as those of the corresponding parallels in north latitude. About the 23d degree of south latitude he found a degree on the meridian to contain 342222 Paris feet. The court of Versailles also sent him an order to go and fix the situation of the Isles of France and of Bourbon. While at the Cape too he observed a wonderful effect of the atmosphere in some states of it: Although the sky at the Cape be generally pure and serene, yet when the southeast wind blows, which is pretty often, it is attended with some strange and even terrible effects: the stars look larger, and seem to dance; the moon has an undulating tremor; and the planets have a sort of beard like comets.</p><p>M. de la Caille returned to France in the autumn of 1754, after an absence of about 4 years; loaded, not with the spoils of the east, but with those of the southern heavens, before then almost unknown to astronomers. Upon his return, he first drew up a reply to some strictures which the celebrated Euler had published relative to the meridian: after which he settled the results of the comparison of his observations for the parallaxes, with those of other astronomers: that of the sun he fixed at 9 1/2″; of the moon at 56′ 56″; of Mars in his opposition, 36″; of Venus 38″. He also settled the laws by which astronomical refractions are varied by the different density or rarity of the air, by heat or cold, and by dryness or moisture. And lastly he shewed an easy and practicable method of finding the longitude at sea by means of the moon. His fame being now celebrated every where, M. de la Caille was soon elected a member of most of the academies and societies of Europe, as London, Bologna, Petersburgh, Berlin, Stockholm, and Gottingen.</p><p>In 1760 our author was attacked with a severe fit of the gout; which however did not interrupt the course of his studies; for he then planned out a new and large work, no less than a history of astronomy through all ages, with a comparison of the aucient and modern observations, and the construction and use of the instruments employed in making them. Towards the latter part of 1761, his constitution became greatly reduced; though his mind remained unaffected, and he resolutely persisted in his studies to the last; death only putting an end to his labours the 21st of March 1762, at 49 years of age; after having committed his manuscripts to the care and discretion of his esteemed friend M. Maraldi.</p><p>Beside the publications before mentioned, and perhaps some others also, he had a vast number inserted in the volumes of the Memoirs of the French Academy of Sciences, much too numerous indeed, though very important, to be here all mentioned particularly; suffice it therefore just to distinguish the years of those volumes in which his pieces are to be found, by the following list of them, viz, 1741, 1742, 1743, 1744, 1745, 1746, 1747, 1748, 1749, 1750, 1751, 1752, 1753, 1754, 1755, 1756, 1757, 1758, 1759, 1760, 1761, 1763; in all or most of which years there are two or three or more of his papers.</p></div1><div1 part="N" n="CAISSON" org="uniform" sample="complete" type="entry"><head>CAISSON</head><p>, in Architecture, a kind of chest or flat- <cb/> bottomed boat, in which the pier of a bridge is built, then sunk to the bottom of the water, and the sides loosened and taken ofs from the bottom, by a contrivance for that purpose; the bottom of the caisson being left under the pier as a foundation to it. The caisson is kept afloat till the pier is built to above the height of low-water mark; and for that purpose, its sides are either made of more than that height at first, or else gradually raised to it as it sinks by the weight of the work, so as always to keep its top above water. Mr. Labelye tells us, that the caissons in which he built some of the piers of Westminster bridge, contained above 150 load of fir timber, of 40 cubic feet each, and that it was of more tonnage or capacity than a 40 gun ship of war.</p><div2 part="N" n="Caisson" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Caisson</hi></head><p>, in Military Affairs, is sometimes used for a chest; and in particular for a bomb or shell chest, and is used as a supersicial mine, or fourneau. This is done by filling a chest either with gun powder and loaded shells, or else with shells alone, and burying it in a spot where an enemy, besieging a place, is expected to come, and then firing it by a train to blow the men up.</p></div2></div1><div1 part="N" n="CALCULATION" org="uniform" sample="complete" type="entry"><head>CALCULATION</head><p>, the act of computing several sums, by adding, subtracting, multiplying, dividing, &c. From <hi rend="italics">calculus,</hi> in allusion to the practice of the ancients, who used <hi rend="italics">calculi,</hi> or little stoues, in making computations, in taking suffrages, and in keeping accounts, &c; as we now use counters, figures, &c. Calculation is more particularly used to signify the computations in astronomy, trigonometry, &c, for making tables of astronomy, of logarithms, ephemerides, finding the times of eclipses, and such like.</p></div1><div1 part="N" n="CALCULATCR" org="uniform" sample="complete" type="entry"><head>CALCULATCR</head><p>, a person who makes or performs calculations.—It is also the name given by Mr. Ferguson to a machine in the shape of an orrery, which he constructed for exhibiting the motions of the earth and moon, and resolving a variety of astronomical problems. See his Astron. 4to pa. 265, or 8vo pa. 393.</p></div1><div1 part="N" n="CALCULATORES" org="uniform" sample="complete" type="entry"><head>CALCULATORES</head><p>, were anciently accountants who reckoned their sums by <hi rend="italics">calculi,</hi> or little stones, or counters.—In ancient canons too we find a sort of diviners or enchanters, censured under the denomination of <hi rend="italics">calculatores.</hi></p><p>CALCULUS denotes primarily a small stone, pebble, or counter, used by the ancients in making calculations or computations, taking of suffrages, playing at tables, and the like.</p><p><hi rend="smallcaps">Calculus</hi> denotes now a certain way of performing mathematical investigations and resolutions. Thus, we say the Arithmetical or Numeral Calculus, the Algebraical Calculus, the Differential Calculus, the Exponential Calculus, the Fluxional Calculus, the Integral Calculus, the Literal or Symbolical Calculus, &c; for which, see each respective word.</p><p><hi rend="italics">Arithmetical</hi> or <hi rend="italics">Numeral</hi> <hi rend="smallcaps">Calculus</hi>, is the method of performing arithmetical computations by numbers. See <hi rend="smallcaps">Arithmetic</hi>, and <hi rend="smallcaps">Number.</hi></p><p><hi rend="italics">Algebraical, Literal,</hi> or <hi rend="italics">Symbolical</hi> <hi rend="smallcaps">Calculus</hi>, is the method of performing algebraical calculations by letters or other symbols. See <hi rend="smallcaps">Algebra.</hi></p><p><hi rend="italics">Differential</hi> <hi rend="smallcaps">Calculus</hi>, is the arithmetic of the indefinitely small differences of variable quantities; a mode of computation much used by foreign mathematicians, and introduced by Leibnitz, as similar to the <pb n="235"/><cb/> method of Fluxions of Newton. See <hi rend="smallcaps">Differential</hi> &c.</p><p><hi rend="italics">Exponential</hi> <hi rend="smallcaps">Calculus</hi>, is the applying the fluxional or differential methods to exponential quantities; such as <hi rend="italics">a<hi rend="sup">x</hi>,</hi> or <hi rend="italics">x<hi rend="sup">x</hi>,</hi> or <hi rend="italics">ay<hi rend="sup">x</hi>,</hi> &c. See <hi rend="smallcaps">Exponential.</hi></p><p><hi rend="italics">Fluxional</hi> <hi rend="smallcaps">Calculus</hi>, is the method of fluxions, invented by Newton. See <hi rend="smallcaps">Fluxions.</hi></p><p><hi rend="italics">Integral</hi> <hi rend="smallcaps">Calculus</hi>, or <hi rend="italics">Summatorius,</hi> is a method of integrating, or summing up differential quantities; and is similar to the finding of fluents. See <hi rend="smallcaps">Integral</hi> and <hi rend="smallcaps">Fluent.</hi></p><p><hi rend="smallcaps">Calculus</hi> <hi rend="italics">Literalis,</hi> or <hi rend="italics">Literal Calculus,</hi> is the same with algebra, or specious arithmetic, so called from its using the letters of the alphabet; in contradistinction to numeral arithmetic, in which figures are used.</p></div1><div1 part="N" n="CALENDAR" org="uniform" sample="complete" type="entry"><head>CALENDAR</head><p>, or <hi rend="smallcaps">Kalendar</hi>, a distribution of time as accommodated to the uses of life; or an Almanac, or table, containing the order of days, weeks, months, feasts, &c, occurring in the course of the year: being so called from the word <hi rend="italics">Calendæ,</hi> which among the Romans denoted the first days of every month, and anciently was written in large characters at the head of each month. See <hi rend="smallcaps">Almanac, Calends, Month, Time, Year</hi>, &c.</p><p>In Calendars the days were originally divided into octoades, or eights; but afterwards, in imitation of the Jews, they were divided into hebdomades, or sevens, for what we now call a week: which custom, Scaliger observes, was not in use among the Romans till after the time of Theodosius.</p><p>Divers calendars are established in different countries, according to the different forms of the year, and distributions of time: As the Persian, the Roman, the Jewish, the Julian, the Gregorian, &c, calendars.—The ancient Roman Calendar is given by Ricciolus, Struvius, Danet, and others; in which we perceive the order and number of the Roman holy-days and work-days.—The Jewish calendar was fixed by Rabbi Hillel, about the year 360; from which time the days of their year may be reduced to those of the Julian calendar.—The three Christian calendars are given by Wolfius in his Elements of Chronology; as also the Jewish and Mohamedan calendars. Other writers on the calendars are Vieta, Clavius, Scaliger, Blondel, &c.</p><p><hi rend="italics">The Roman</hi> <hi rend="smallcaps">Calendar</hi> was first formed by Romulus, who distributed time into several periods for the use of his followers and people. He divided the year into 10 months, of 304 days; beginning on the first of March, and ending with December.</p><p>Numa reformed the calendar of Romulus. He added the months of January and February, making it to commence on the first of January, and to consist of 355 days. But as this was evidently deficient of the true year, he ordered an intercalation of 45 days to be made every 4 years, in this manner, viz, Every 2 years an additional month of 22 days, between February and March; and at the end of each two years more, another month of 23 days; the month thus interposed, being called Marcedonius, or the intercalary February.</p><p>Julius Cæsar, with the aid of Sosigenes, a celebrated astronomer of those times, farther reformed the Roman calendar, from whence arose the Julian calendar, and the Julian or old style. Finding that the sun per- <cb/> formed his annual course in 365 days and a quarter nearly, he divided the year into 365 days, but every 4th year 366 days, adding a day that year before the 24th of February, which being the 6th of the calends, and being thus reckoned twice, gave occasion to the name <hi rend="italics">bissextile,</hi> or what we also call leap-year.</p><p>This calendar was farther reformed by order of the pope Gregory XIII, from whence arose the term Gregorian calendar and style, or what we also call the new style, which is now observed by almost all European nations. The year of Julius was too long by nearly 11 minutes, which amounts to about 3 days in 400 years; the pope therefore, by the advice of Clavius and Ciaconius, ordained that there should be omitted a day in every 3 centuries out of 4; so that every century, which would otherwise be a bissextile year, is made to be only a common year, excepting only such centuries as are exactly divisible by 4, which happens once in 4 centuries. See <hi rend="smallcaps">Bissextile.</hi> This reformation of the calendar, or the new style, as we call it, commenced in the countries under the popish influence, on the 4th of October 1582, when 10 days were omitted at once, which had been over-run since the time of the council of Nice, in the year 325, by the surplus of 11 minutes each year. But in England it only commenced in 1752, when 11 days were omitted at once, the 3d of September being accounted the 14th that year; as the surplus minutes had then amounted to 11 days.</p><p><hi rend="italics">Julian Christian</hi> <hi rend="smallcaps">Calendar</hi>, is that in which the days of the week are determined by the letters A, B, C, D, E, F, G, by means of the solar cycle; and the new and full moons, particularly the paschal full moon, with the feast of Easter, and the other moveable feasts depending upon it, by means of golden numbers, or lunar cycles, rightly disposed through the Julian year. See <hi rend="smallcaps">Cycle</hi>, and <hi rend="smallcaps">Golden Number.</hi></p><p>In this calendar, it is supposed that the vernal equinox is sixed to the 21st day of March; and that the golden numbers, or cycles of 19 years, constantly indicate the places of the new and full moons; though both are erroneous; and from hence arose a great irregularity in the time of Easter.</p><p><hi rend="italics">Gregorian</hi> <hi rend="smallcaps">Calendar</hi>, is that which, by means of Epacts, rightly disposed through the several months, determines the new and full moons, with the time of Easter, and the moveable feasts depending upon it, in the Gregorian year. This differs therefore from the Julian calendar, both in the form of the year, and in as much as epacts are substituted instead of golden numbers. See <hi rend="smallcaps">Epact.</hi></p><p>Though the Gregorian calendar be more accurate than the Julian, yet it is not without imperfections, as Scaliger and Calvisius have fully shewn; nor is it perhaps possible to devise any one that shall be quite perfect. Yet the Reformed Calendar, and that which is ordered to be observed in England, by act of Parliament made the 24th of George II, come very near to the point of accuracy: For, by that act it is ordered that “Easter-day, on which the rest depend, is always the first Sunday after the full moon, which happens upon, or next after the 21st day of March; and if the full-moon happens upon a Sunday, Easter-day is the Sunday after.”</p><p><hi rend="italics">Reformed,</hi> or <hi rend="italics">Corrected,</hi> <hi rend="smallcaps">Calendar</hi> is that which, <pb n="236"/><cb/> rejecting all the apparatus of golden numbers, epacts, and dominical letters, determines the equinox, and the paschal full-moon, with the moveable feasts depending upon it, by computation from astronomical tables. This calendar was introduced among the protestant states of Germany in the year 1700, when 11 days were omitted in the month of February, to make the corrected style agree with the Gregorian. This alteration in the form of the year, they admitted for a time; in expectation that, the true quantity of the tropical year being at length more accurately determined by observation, the Romanists would agree with them on some more convenient intercalation.</p><p><hi rend="italics">French New</hi> <hi rend="smallcaps">Calendar</hi>, is a quite new form of calendar that commenced in France on the 22d of September 1792. At the time of printing this (viz, in July 1794), it does not certainly appear whether this new calendar will be made permanent or not; but merely as a curiosity in the science of chronology, a very brief notice of it may here be added, as follows.</p><p>The year, in this calendar, commences at midnight the beginning of that day in which falls the true autumnal equinox for the observatory of Paris. The year is divided into 12 equal months, of 30 days each; after which 5 supplementary days are added, to complete the 365 days of the ordinary year: these 5 days do not belong to any month. Each month is divided into three decades of 10 days each; distinguished by 1st, 2d, and 3d decade. All these are named according to the order of the natural numbers, viz, the 1st, 2d, 3d, &c, month, or day of the decade, or of the supplementary days. The years which receive an intercalary day, when the position of the equinox requires it, which we call embolismic or bissextile, they call olimpic; and the period of four years, ending with an olimpic year, is called an olimpiade; the intercalary day being placed after the ordinary five supplementary days, and making the last day of the olimpic year. Each day, from midnight to midnight, is divided into 10 parts, each part into 10 others, and so on to the last measurable portion of time.</p><p>In this calendar too the months and days of them have new names. The first three months of the year, of which the autumn is composed, take their etymology, the first from the vintage which takes place from September to October, and is called <hi rend="italics">vendemaire;</hi> the second, <hi rend="italics">brumaire,</hi> from the mists and low fogs, which shew as it were the transudation of nature from October to November: the third, <hi rend="italics">frimaire,</hi> from the cold, sometimes dry and sometimes moist, which is felt from November to December. The three winter months take their etymology, the first, <hi rend="italics">nivose,</hi> from the snow which whitens the earth from December to January; the second, <hi rend="italics">pluviose,</hi> from the rains which usually fall in greater abundance from January to February; the third, <hi rend="italics">ventose,</hi> from the wind which dries the earth from February to March. The three spring months take their etymology, the first, <hi rend="italics">germinal,</hi> from the fermentation and development of the sap from March to April; the second, <hi rend="italics">floreal,</hi> from the blowing of the flowers from April to May: the third, <hi rend="italics">prairial,</hi> from the smiling secundity of the meadow crops from May to June. Lastly, the three summer months take their <cb/> etymology, the first, <hi rend="italics">messidor,</hi> from the appearance of the waving ears of corn and the golden harvests which cover the fields from June to July; the second, <hi rend="italics">thermidor,</hi> from the heat, at once solar and terrestrial, which inflames the air from July to August; the third, <hi rend="italics">fructidor,</hi> from the fruits gilt and ripened by the sun from August to September. Thus, the whole 12 months are, <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">Autumn.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">Spring.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Vendemaire</cell><cell cols="1" rows="1" role="data">Germinal</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brumaire</cell><cell cols="1" rows="1" role="data">Floreal</cell></row><row role="data"><cell cols="1" rows="1" role="data">Frimaire.</cell><cell cols="1" rows="1" role="data">Prairial.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">Winter.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="smallcaps">Summer.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Nivose</cell><cell cols="1" rows="1" role="data">Messidor</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pluviose</cell><cell cols="1" rows="1" role="data">Thermidor</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ventose.</cell><cell cols="1" rows="1" role="data">Fructidor.</cell></row></table></p><p>From these denominations it follows, that by the mere pronunciation of the name of the month, every one readily perceives three things and all their relations, viz, the kind of season, the temperature, and the state of vegetation: for instance, in the word <hi rend="italics">germinal,</hi> his imagination will easily conceive, by the termination of the word, that the spring commences; by the construction of the word, that the elementary agents are busied; and by the signification of the word, that the buds unfold themselves.</p><p>As to the names of the days of the week, or decade of 10 days each, which they have adopted instead of seven, as these bear the stamp of judicial astrology and heathen mythology, they are simply called from the first ten numbers; thus, <table><row role="data"><cell cols="1" rows="1" role="data">Primdi</cell><cell cols="1" rows="1" role="data">Sextidi</cell></row><row role="data"><cell cols="1" rows="1" role="data">Duodi</cell><cell cols="1" rows="1" role="data">Septidi</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tridi</cell><cell cols="1" rows="1" role="data">Octidi</cell></row><row role="data"><cell cols="1" rows="1" role="data">Quartidi</cell><cell cols="1" rows="1" role="data">Nonidi</cell></row><row role="data"><cell cols="1" rows="1" role="data">Quintidi</cell><cell cols="1" rows="1" role="data">Decadi.</cell></row></table></p><p>In the almanac, or annual calendar, instead of the multitude of saints, one for each day of the year, as in the popish calendars, they annex to every day the name of some animal, or utensil, or work, or fruit, or flower, or vegetable, &c, appropriate and most proper to the times.</p><p><hi rend="italics">Astronomical</hi> <hi rend="smallcaps">Calendar</hi>, an instrument engraven upon copper-plates, printed on paper, and pasted on board, with a brass slider which carries a hair, and shews by inspection, the sun's meridian, altitude, right ascension, declination, rising, setting, amplitude, &c, to a greater exactness than ean be shewn by the common globes.</p></div1><div1 part="N" n="CALENDS" org="uniform" sample="complete" type="entry"><head>CALENDS</head><p>, <hi rend="italics">Calendæ,</hi> in the Roman Chronology, denoted the first days of each month; being so named from <foreign xml:lang="greek">kalew</foreign>, <hi rend="italics">calo, I call,</hi> or <hi rend="italics">proclaim</hi>; because that, before the publication of the Roman <hi rend="italics">Fasti,</hi> and counting their months by the motion of the moon, a priest was appointed to observe the first appearance of the new moon; who, having seen her, gave notice to the president of the sacrifices to offer one; and cailing the people together, he proclaimed unto them how they should reckon the days until the nones; pronouncing the word <hi rend="italics">Caleo</hi> 5 times if the nones should happen on the 5th day, or seven times if they happened on the 7th day of the month.</p><p>The calends were reckoned backwards, or in a re- <pb n="237"/><cb/> trograde order; thus, for example, the first of May being the calends of May; the last or 30th day of April, was the <hi rend="italics">pridie calendarum,</hi> or 2d of the calends of May; the 29th of April, the 3d of the calends, or before the calends: and so back to the 13th, where the ides commence; which are likewise numbered backwards to the 5th, where the nones begin; which are also reckoned after the same manner to the sirst day of the month, which is the calends of April.</p><p>Hence comes this rule to find the day of the calends answering to any day of the month, viz, Consider how many days of the month are yet remaining after the day proposed, and to that number add 2, for the number of or from the calends. For example, suppose it were the 23d day of April, it would then be the 9th of the caledns of May: for April containing 30 days, from which 23 being taken, there remains 7; to which 2 being added, makes the sum 9. And the reason for this addition of the constant number 2, is because the last day of the month is called the 2d of the calends of the month following.</p></div1><div1 part="N" n="CALIBER" org="uniform" sample="complete" type="entry"><head>CALIBER</head><p>, or <hi rend="smallcaps">Caliper</hi>, is the thickness or diameter of a round body, particularly the bore or width of a piece of ordnance, or that of its ball.</p><p><hi rend="smallcaps">Caliber</hi>-<hi rend="italics">Compasses,</hi> or <hi rend="smallcaps">Caliper</hi>-<hi rend="italics">Compasses,</hi> or simple <hi rend="smallcaps">Calipers</hi>, a sort of compasses made with bowed or arched legs, the better to take the diameter of any round body; as the diameters of balls, or the bores of guns; or the diameter, and even the length of casks, and such like. The best sort of calipers usually contain the following articles, viz, 1st, the measure of convex diameters in inches &c; 2d, of concave diameters: 3d, the weight of iron shot of given diameters; 4th, the weight of iron shot for given gun bores; 5th, the degrees of a semicircle; 6th, the proportion of troy and averdupois weight; 7th, the proportion of English and French feet and pounds weight; 8th, factors used in circular and spherical figures; 9th, tables of the specific gravities and weights of bodies; 10th, tables of the quantity of powder necessary for the proof and service of brass and iron guns; 11th, rules for computing the number of shot or shells in a complete pile; 12th, rules for the fall or descent of heavy bodies; 13th, rules for the raising of water; 14th, rules for firing artillery and mortars; 15th, a line of inches; 16th, logarithmic scales of numbers, sines, versed sines, and tangents; 17th, a sectoral line of equal parts, or the line of lines; 18th, a sectoral line of planes and superficies; and 19th, a sectoral line of solids.</p><p><hi rend="smallcaps">Calippic</hi> <hi rend="italics">Period,</hi> in Chronology, a period of 76 years, continually recurring; at every repetition of which, it was supposed, by its inventor Calippus, an Athenian astronomer, that the mean new and full moons would always return to the same day and hour.</p><p>About a century before, the golden number, or cycle of 19 years, had been invented by Meton, which Calippus finding to contain 19 of Nabonassar's year, 4 days and 331/459, to avoid fractions he quadrupled it, and so produced his period of 76 years, or 4 times 19; after which he supposed all the lunations &c would regularly return to the same hour. But neither is this exact, as it brings them too late by a whole day in 225 years. <cb/></p><p>CAMBER-<hi rend="smallcaps">Beam</hi>, a piece of timber cut arch-wise, or with an obtuse angle in the middle. They are commonly used in platforms, as for church-roofs, and other occasions where long timbers are wanted to lie at a small slope. A camber-beam is much stronger than another of the same dimensions; for being laid with the hollow side downwards, and having good butments at the ends, they serve for a kind of arch.</p></div1><div1 part="N" n="CAMELEON" org="uniform" sample="complete" type="entry"><head>CAMELEON</head><p>, one of the constellations of the southern hemisphere, near the south pole, and invisible in our latitude. There are 10 stars marked in this constellation in Sharp's catalogue.</p></div1><div1 part="N" n="CAMELOPARDALUS" org="uniform" sample="complete" type="entry"><head>CAMELOPARDALUS</head><p>, a new constellation of the northern hemisphere, formed by Hevelius, consisting of 32 stars first observed by him. It is situated between Cepheus, Cassiopeia, Perseus, the Two Bears, and Draco; and it contains 58 stars in the British catalogue.</p><p>CAMERA <hi rend="italics">Æolia,</hi> a name given by Kircher to a contrivance for blowing the fire, for the fusion of ores, without bellows. This is effected by means of water falling through a sunnel into a close vessel, which sends from it so much air or vapour, as continually blows the fire. See Hook's Philof. Coll. n° 3, pa. 80.</p><p><hi rend="smallcaps">Camera</hi> <hi rend="italics">Lucida,</hi> a contrivance of Dr. Hook to make the image of any thing appear on a wall in a light room, either by day or night. See Philos. Trans. n° 38, pa. 741.</p><p><hi rend="smallcaps">Camera</hi> <hi rend="italics">Obscura,</hi> or <hi rend="italics">Dark Chamber,</hi> an optical machine or apparatus, representing an artificial eye, by which the images of external objects, received through a double convex glass, are shewn distinctly, and in their native colours, on a white ground placed within the machine, in the focus of the glass. The first invention of the camera obscura is ascribed to John Baptista Porta. See his <hi rend="italics">Magia Naturalis, lib.</hi> 17, <hi rend="italics">cap.</hi> 6, where he largely describes the effects of it. See also the end of s'Gravesande's Perspective, and other optical writers, for the construction and uses of various sorts of camera obscuras.</p><p>This machine serves for many useful and entertaining purposes. For example, it is very useful in explaining the nature of vision, representing a kind of artificial eye: it exhibits very diverting sights or spectacles; shewing images perfectly like their objects, clothed in their natural colours, but more intense and vivid, and at the same time accompanied with all their motions; an advantage which no art can imitate: and by this instrument, a person unacquainted with painting, or drawing, may delineate objects with the greatest accuracy of drawing and colouring. <figure/></p><p><hi rend="italics">Theory of the Camera Obscura.</hi> The theory and <pb n="238"/><cb/> principle of this instrument may be thus explained. If any object AB radiate through a small aperture L, upon a white ground opposite to it, within a darkened room, or box, &c; the image of the object will be painted on that ground in an inverted situation. For, by the smallness of the aperture, the rays from the object will cross each other there, the image of the point A being at <hi rend="italics">a,</hi> and that of B at <hi rend="italics">b</hi>; so that the whole object AB will appear inverted, as at <hi rend="italics">ab.</hi> And as the corresponding rays make equal angles on both sides of the aperture, if the ground be parallel to the object, their heights will be to each other directly as their distances from the aperture.</p><p><hi rend="italics">Construction of a Camera Obscura,</hi> by which the images of external objects shall be represented distinctly, and in their genuine colours. 1st, Darken a chamber that has one of its windows looking towards a place containing various objects to be viewed; leaving only a small aperture open in one shutter. 2d, In this aperture fit a proper lens, either plano-convex, or convex on both sides; the convexity forming a small portion of a large sphere. But note, that if the aperture be made very small, as of the size of a pea, the objects will be represented even without any lens at all. 3d, At a proper distance to be determined by trials, stretch a paper or white cloth, unless there be a white wall at that distance, to receive the images of the objects: or the best way is to have some plaister of Paris cast on a convex mould, so as to form a concave smooth surface, and of a curvature and size adapted to the lens, to be placed occasionally at the proper distance. 4th, If it be rather desired to have the objects appear erect, instead of inverted, this may be done either by placing a concave lens between the centre and the focus of the sirst lens; or by reflecting the image from a plane speculum inclined to the horizon in an angle of 45 degrees; or by having two lenses included in a draw-tube, instead of one.</p><p>That the images be clear and distinct, it is necessary that the objects be illuminated by the sun's light shining upon them from the opposite quarter: so that, in a western prospect the images will be best seen in a forenoon, an eastern prospect the afternoon, and a northern prospect about noon; a southern aspect is the least eligible of any. But the best way of any is, if the lens be fixed in a proper frame, on the top of a building, and made to move easily round in all directions, by a handle extended to the person who manages the instrument; the images being then thrown down into a dark room immediately below it, upon a horizontal round plaister of Paris ground: for thus a view of all the objects quite around may easily be taken in the space of a few minutes; as is the case of the excellent camera obscura placed on the top of the Royal Observatory at Greenwich.</p><p>The objects will be seen brighter, if the spectator sirst wait a few minutes in the dark. Care should also be taken, that no light escape through any chinks; and that the ground be not too much illuminated. It may further be observed, that the greater distance there is between the aperture and the ground, the larger the images will be; but then at the same time the brightness is weakened more and more with the increase of distance. <cb/></p><p><hi rend="italics">To construct a Portable Camera Obscura.</hi> 1st, Provide a small box or chest of dry wood, and of about 10 inches broad, and 2 feet long or more, according to the size of the lenses. 2d, In one side of it, as BD, fit a sliding tube EF with two lenses; or, to have the image at a less distance from the tube, with three lenses, convex on both sides; the diameter of the two outer ones to be about 7 inches, but that of the inner to be less, as 4 3/4 or 5 inches. 3d, At a proper distance, within the box, set up perpendicularly an oiled paper GH, so that images thrown upon it may be seen through. 4th, In the opposite side, at I, make a round hole, for a person to look conveniently through with both eyes. Then if the tube be turned towards the objects, and the lenses be placed by trials at the proper distance, by sliding the tube in and out, the objects will be seen delineated on the paper, erect as before.</p><p>The machine may be better accommodated for drawing, by placing a mirror to pass from G to C; for this will reflect the image upon a rough glass plane, or an oiled paper, placed horizontally at AB; and a copy of it may there be sketched out with a black-lead pencil. <figure/></p><p><hi rend="italics">Another Portable Camera Obscura</hi> is thus made. 1st. On the top of a box or chest raise a little turret HI, open towards the object AB. 2d, Behind the aperture, incline a small mirror <hi rend="italics">ab</hi> at an angle of 45 degrees, to reflect the rays A<hi rend="italics">a</hi> and B<hi rend="italics">b</hi> upon a lens G convex on both sides, and included in a tube GL. Or the lens may be fixed in the aperture. 3d, At the dis- <figure/> tance of the focus place a table, or board EF, covered <pb n="239"/><cb/> with a white paper, to receive the image <hi rend="italics">ab.</hi> Lastly, In MN make an oblong aperture to look through; and an opening may also be made in the side of the box, for the convenience of drawing.</p><p>This sort of camera is easily changed into a show-box, for viewing prints, &c: placing the print at the bottom of the box, with its upper part inwards, where it is enlightened through the front, left open for this purpose, either by day or candle-light; and the print may be viewed through the aperture in HI.—A variety of contrivances for this purpose may be seen described in Harris's Optics, b. ii. sect. 4.—Mr. Storer has also procured a patent for an instrument of this sort, which he calls a delineator; being formed of two double convex lenses and a plane mirror, fitted into a proper box. One lens is placed close to the mirror, making with it an angle of 45 degrees; the other being placed at right augles to the former, and fixed in a moveable tube. If the moveable lens be directed towards the object, which is to be viewed or copied, and moved nearer to or farther from the mirror, till the image is distinctly formed on a greyed glass, laid upon that surface of the upper lens which is next the eye, it will be found more sharp and vivid than those formed in the common instruments; because the image is taken up so near the upper lens. And by increasing the diameter and curvature of the lenses, the effect will be much heightened.</p></div1><div1 part="N" n="CAMUS" org="uniform" sample="complete" type="entry"><head>CAMUS</head><p>, <hi rend="italics">(Charles-Stephen-Lewis),</hi> a celebrated French mathematician, Examiner of the royal Schools of Artillery and Engineers, Secretary and Professor of the Royal Academy of Architecture, Honorary member of that of the Marine, and fellow of the Royal Society of London, was born at Cressy en Brie, the 25th of August 1699. His early ingenuity in mechanics and his own intreaties induced his parents to send him to study at a college in Paris, at 10 years of age; where in the space of two years his progress was so great, that he was able to give lessons in mathematics, and thus to defray his own expences at the college, without any farther charge to his parents. By the assistance of the celebrated Varignon, young Camus soon ran through the course of the higher mathematics, and acquired a name among the learned. He made himself more particularly known to the Academy of Sciences in 1727, by his memoir upon the subject of the prize which they had proposed for that year, viz, ‘To determine the most advantageous way of masting ships;’ in consequence of which he was named that year Adjoint-Mechanician to the Academy; and in 1730 he was appointed professor of Architecture. In less than three years after, he was honoured with the secretaryship of the same; and the 18th of April 1733, he obtained the degree of Associate in the Academy, where he distinguished himself greatly by his memoirs upon living forces, or bodies in motion acted upon by forces, on the figure of the teeth of wheels and pinions, on pump work, and several other ingenious memoirs.</p><p>In 1736 he was sent, in company with Messieurs Clairaut, Maupertuis, and Monnier, upon the celebrated expedition to measure a degree at the north polar circle; in which he rendered himself highly useful, not only as a mathematician, but also as a mechanician and an artist, branches for which he had a remarkable talent. <cb/></p><p>In 1741 Camus had the honour to be appointed Pensioner-Geometrician in the Academy; and the same year he invented a gauging-rod and sliding-rule proper at once to gauge all sorts of casks, and to calculate their contents. About the year 1747 he was named Examiner of the Sehools of Artillery and Engineers: and, in 1756, one of the eight mathematicians appointed to examine by a new measurement, the base which had formerly been measured by Picard, between Villejuifve and Juvisi; an operation in which his ingenuity and exactness were of great utility. In 1765 M. Camus was elected a fellow of the Royal Society of London; and died the 4th of May 1768, in the 69th year of his age; being succeeded by the celebrated d'Alembert in his office of Geometrician in the French Academy; and leaving behind him a great number of manuscript treatises on various branches of the mathematics.</p><p>The works published by M. Camus, are:</p><p>1. Course of Mathematics for the use of the Engineers, 4 vols. in 8vo.</p><p>2. Elements of Mechanics.</p><p>3. Elements of Arithmetic.</p><p>And his memoirs printed in the volumes of the Academy, are:</p><p>1. Of Accelerated Motions by living forces: vol. for 1728.</p><p>2. Solution of a Geometrical Problem of M. Cramer: 1732.</p><p>3. On the figure of the teeth and pinions in clocks: 1733.</p><p>4. On the action of a Musket ball, piercing a pretty thick piece of wood, without communicating any considerable velocity to it: 1738.</p><p>5. On the best manner of employing buckets for raising water: 1739.</p><p>6. A Problem in Statics: 1740.</p><p>7. On an Instrument for gauging of vessels: 1741.</p><p>8. On the Standard of the Ell Measure: 1746.</p><p>9. On the Tangents of Points common to several branches of the same curve: 1747.</p><p>10. On the Operations in measuring the distance between the centres of the pyramids of Villejuive and Juvisy, to discover the best measure of the degree about Paris: 1754.</p><p>11. On the Masting of Ships: Prize Tom. 2.</p><p>12. The Manner of working Oars: Mach. tom. 2.</p><p>13. A Machine for moving many Colters at once: Mach. tom. 2.</p></div1><div1 part="N" n="CANCER" org="uniform" sample="complete" type="entry"><head>CANCER</head><p>, the <hi rend="italics">Crab,</hi> one of the twelve signs of the zodiac, usually drawn on the globe in the form of a crab, and in books of Astronomy denoted by a character resembling the number sixty-nine, turned sideways, thus <figure/>.</p><p>This is one of the 48 old constellations; and, from the hieroglyphic mode of writing among the Egyptians &c, it is probable that they gave the name and figure to this constellation from the following circumstance, viz, that as the crab is an animal that goes sideling backwards, so the sun, in his annual course through the zodiac, when he arrives at this part of the ecliptic, having reached his utmost limit northwards, begins there to return back again towards the south. But the Greeks, who adapted some fable of their own to every thing of this kind, pretend that when Hercules was fighting with <pb n="240"/><cb/> the Lernæan hydra, there was a crab upon the marsh which seized his foot. The hero crushed the reptile to pieces under his heel; but Juno, in gratitude for the offered service, little as it was, raised the creature into the heavens.</p><p>The number of stars in the sign cancer, Ptolemy makes 13, Tycho 15, Bayer and Hevelius 29, and Flamsteed 83.</p><p><hi rend="italics">Tropic of</hi> <hi rend="smallcaps">Cancer</hi>, a little circle of the sphere parallel to the equinoctial, and passing through the beginning of the sign cancer.</p></div1><div1 part="N" n="CANDLEMAS" org="uniform" sample="complete" type="entry"><head>CANDLEMAS</head><p>, or the <hi rend="italics">Purification,</hi> a feast of the church, held on the 2d of February, in memory of the purification of the Virgin; taking its name of Candlemas, either from the number of lighted candles used by the Romish church, in the processions of this day, or because that the church then consecrated candles for the whole year.</p><p>CANES <hi rend="italics">Venatici,</hi> the <hi rend="italics">Hounds,</hi> or the <hi rend="italics">Greyhounds,</hi> one of the new constellations of the northern hemisphere, which Hevelius has formed out of the unformed stars of the old catalogues. These two dogs are farther distinguished by the names of <hi rend="italics">asterion</hi> and <hi rend="italics">chara.</hi> They contain 23 stars according to Hevelius, but 25 in the British catalogue.</p></div1><div1 part="N" n="CANICULA" org="uniform" sample="complete" type="entry"><head>CANICULA</head><p>, a name given by many of the earlier astronomers to the constellation which we call the Lesser Dog, and Canis Minor, but some Procyon and Antecanis. See <hi rend="smallcaps">Canis minor.</hi></p><p>It is also used for one of the stars of the constellation Canis Major; called also simply the Dog-star; and by the Greeks <foreign xml:lang="greek">*seirios</foreign>, Sirius. It is situated in the mouth of the constellation, and is the largest and brightest of all the stars in the heavens. From the heliacal rising of this star, that is, its emersion from the sun's rays, which now happens with us about the 11th of August, the ancients reckoned their <hi rend="italics">dies caniculares,</hi> or dog-days.</p><p>The Egyptians and Ethiopians began their year at the heliacal rising of Canicula; reckoning to its rise again the next year, which is called the <hi rend="italics">Annus Canarius.</hi></p><p>CANICULAR <hi rend="italics">Days,</hi> or <hi rend="italics">Eog-days,</hi> denote a certain number of days, before and after the heliacal rising of canicula, or the dog-star, in the morning. The ancients imagined that this star, so rising, occasioned the sultry weather usually felt in the latter part of the summer, or dog-days; with all the distempers of that sickly season: Homer's Il. lib. 5, v. 10, and Virgil's Æn. lib. 10, v. 270. Some authors say, from Hippocrates and Pliny, that the day this star first rises in the morning, the sea boils, wine turns sour, dogs begin to grow mad, the bile increases and irritates, and all animals grow languid; also that the diseases it usually occasions in men, are burning fevers, dysenteries, and phrensies. The Romans too sacrisiced a brown dog every year to Canicula at his first rising, to appease its rage. All this however arose from a groundless idea that the dog-star, so rising, was the occasion of the extreme heat and the diseases of that season; for the star not only varies in its rising, in any one year, as the latitude varies, but it is always later and later every year in all latitudes; so that in time the star may, by the same rule, come to be charged with bringing frost and snow, when he comes to rise in winter. <cb/></p><p>The dog-days were commonly counted for about 40 days, viz, 20 days before and 20 days after the heliacal rising; and almanac-makers have usually set down the dog-days in their almanacs to the changing time of the star's rising, by which means they had at length fallen considerably after the hottest time of the year, till of late we have observed an alteration of them in the almanacs, and very properly, from July 3 to August 11. For, by the dog-days, the ancients meant to express the hottest time of the year, which is commonly during the month of July, about which month the dog-star rose heliacally in the time of the most ancient astronomers that we know of: but the precession of the equinoxes has carried this heliacal rising into a much later and cooler part of the year; and because Hesiod tells us that the hot time of the year ends on the 50th day after the summer solstice, which brings us to about August 10 or 11, therefore the above alteration seems to be very proper.</p><p><hi rend="smallcaps">Canicular</hi> <hi rend="italics">Year,</hi> denotes the Egyptian natural year, which was computed from one heliacal rising of canicula, to the next. This year was also called <hi rend="italics">annus canarius,</hi> and <hi rend="italics">annus cynicus;</hi> and by the Egyptians themselves the <hi rend="italics">Sethic year,</hi> from <hi rend="italics">Seth,</hi> by which name they called Sirius. Some call it also the <hi rend="italics">heliacal year.</hi> This year consisted ordinarily of 365 days, and every 4th year of 366; by which means it was accommodated to the civil year, like the Julian account. And the reason why they chose this star, in preference to others, to compute their time by, was not only the superior brightness of that star, but because that in Egypt its heliacal rising was a time of very singular note, as coinciding with the greatest augmentation of the Nile, the reputed father of Egypt. Ephestion adds, that from the aspect of canicula, its colour &c, the Egyptians drew prognostics concerning the rise of the Nile; and, according to Florus, predicted the future state of the year. So that it is no wonder the first rising of this star was observed with great attention. Bainbrigge, <hi rend="italics">Canicul.</hi> cap. 4. p. 26.</p><p>CANIS <hi rend="italics">Major,</hi> the <hi rend="italics">Great Dog,</hi> a constellation of the southern hemisphere, below the feet of Orion, and one of the old 48 constellarions. The Greeks, as usual, have many fables of their own about the exaltation of the dog into the skies; but the origin of this constellation, as well as its other name Sirius, lies more probably among the Egyptians, who carefully watched the rising of this star, and by it judged of the swelling of the Nile, calling the star the sentinel and watch of the year; and hence, according to their manner of hieroglyphic writing, represented it under the figure of a dog. They also called the Nile <hi rend="italics">Siris;</hi> and hence their <hi rend="italics">Osiris.</hi></p><p>The stars in this constellation, Ptolemy makes 29; Tycho however observed only 13, and Hevelius 21; but in Flamsteed's catalogue they are 31.</p><p><hi rend="smallcaps">Canis</hi> <hi rend="italics">Minor,</hi> a constellation of the northern hemisphere, just below Gemini, and is one of the 48 old constellations. The Greeks fabled that this is one of Orion's hounds; but the Egyptians were most probably the inventors of this constellation, and they may have given it this figure to express a little dog, or watchful creature, going before as leading in the larger, or rising before it: and hence the Latins have called it Antecanis, the star before the dog. <pb n="241"/><cb/></p><p>The stars in this constellation are, in Ptolomy's catalogue 2, the principal of which is the star Procyon; in Tycho's 5, in Hevelius's 13; and in Flamsteed's 14.</p></div1><div1 part="N" n="CANNON" org="uniform" sample="complete" type="entry"><head>CANNON</head><p>, in Military Affairs, a long round hollow engine, made of iron or brass, &c, for throwing balls, &c, by means of gunpowder. The length is distinguished into three parts; the first reinforce, the second reinforce, and the chase: the inside hollow where the charge is lodged, being also called the chase, or bore. But for the several parts and members of a cannon, see <hi rend="smallcaps">Astragal, Base-ring, Bore, Breech, Cascabel, Chase, Muzzle, Ogre, Reinforce-ring</hi>, T<hi rend="smallcaps">RUNNIONS</hi>, &c. See also <hi rend="smallcaps">Gun</hi>, and <hi rend="smallcaps">Gunnery.</hi></p><p>Cannon were first made of several bars of iron adapted to each other lengthways, and hooped together with strong iron rings. They were employed in throwing stones and metal of several hundred weight. Others were made of thin sheets of iron rolled up, and hooped: and on emergencies they have been even made of leather, with plates of iron or copper. They are now made of cast iron or brass; being cast solid, and the tube bored out of the middle of the solid metal.</p><p>Larrey makes brass cannon the invention of J. Owen; and asserts that the first known in England, were in 1535; and farther that iron cannon were first cast here in 1547. He acknowledged that cannon were known before; and remarks that at the battle of Cressi, in 1346, there were 5 pieces of cannon in the English army, which were the first ever seen in France. Mezeray also observes that king Edward struck terror into the French army, by 5 or 6 pieces of cannon; it being the first time they had met such thundering machines.</p><p>In the list of aids raised for the redemption of king John of France, in 1368, mention is made of an officer in the French army called <hi rend="italics">master of the king's cannon,</hi> and of his providing 4 large cannon for the garrison of Harfleur. But father Daniel, in his life of Philip of Valois, produces a proof from the records of the chamber of accounts at Paris, that cannon and gunpowder were used in the year 1338. And Du-Cange even finds mention of the same engines in Froissart, and other French historians, some time earlier.</p><p>The Germans carry the invention of cannon farther back, and ascribe it to Albertus Magnus, a Dominican monk, about the year 1250. But Isaac Vossius finds cannon in China upwards of 1700 years ago; being used by the emperor Kitey, in the year of Christ 85. The ancients too, of Europe and Asia, had their fiery tubes, or <hi rend="italics">cannæ,</hi> which being loaden with pitch, stones, and iron balls, were exploded with a vehement noise, smoke, and great effect.</p><p>Cannon were formerly made of a very great length, which rendered them exceedingly heavy, and their use very troublesome and confined. But it has lately been found by experiment that there is very little added to the force of the ball by a great length of the cannon, and therefore they have very properly been much reduced both in their length and weight, and rendered easily manageable upon all occasions. They were formerly distinguished by many hard and, terrible names, but are now only named from the weight of their ball; as a 6 pounder, a 12 pounder, a 24 pounder, or a 42 pounder, which as the largest size now used by the English for battering.</p></div1><div1 part="N" n="CANON" org="uniform" sample="complete" type="entry"><head>CANON</head><p>, in Algebra, Arithmetic, Geometry, &c, <cb/> is a general rule for resolving all cases of a like nature with the present enquiry. Thus the last step of every equation is such a canon, and if turned into words, becomes a rule to resolve all cases or questions of the same kind with that proposed.</p><p>Tables of sines, tangents, &c, whether natural or artificial, are also called canons.</p><div2 part="N" n="Canon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Canon</hi></head><p>, in Ancient Music, is a rule or way of determining the intervals of musical notes. Ptolomy, rejecting the Aristoxenian way of measuring the intervals in music by the magnitude of a tone, formed by the difference between a diapente and a diatessaron, thought that they should be distinguished by the ratios which the sounds terminating those intervals bear to one another, when considered according to their degree of acuteness or gravity; which, before Aristoxenus, was the old Pythagorean way. He therefore made the diapason consist in a double ratio; the diapente consist in a sesquialterate; the diatessaron, in a sesquitertian; and the tone itself, in a sesquioctave; and all the other intervals, according to the proportion of the sounds that terminate them: wherefore, taking the canon, as it is called, for a determinate line of any length, he shews how this is to be cut, that it may represent the respective intervals: and this method answers exactly to experiments in the different lengths of musical chords. From this canon, Ptolomy and his followers have been called <hi rend="italics">Canonici;</hi> as those of Aristoxenus were called <hi rend="italics">Musici.</hi></p><p><hi rend="italics">Pascal</hi> <hi rend="smallcaps">Canon</hi>, a table of the moveable feasts, shewing the day of Easter, and the other feasts depending upon it, for a cycle or period of 19 years. It is said that the Pascal Canon was the calculation of Eusebius of Cæsarea, and that it was made by order of the council of Nice.</p></div2></div1><div1 part="N" n="CANOPUS" org="uniform" sample="complete" type="entry"><head>CANOPUS</head><p>, a name given by some of the old astronomers to a star under the 2d bend of Eridanus. These writers say that the river in the heavens is not the Eridanus, but the Nile, and that this star commemorates an island made by that river, which was called by the same name.</p><p><hi rend="smallcaps">Canopus</hi> is also the name of a bright star of the first magnitude in the rudder of the ship Argo, one of the southern constellations. Its situation, as given by several authors, at different times, is as follows: <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Authors</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Dates</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Longit.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Lat.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Rt. Ascen.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Declin. So.</cell></row><row role="data"><cell cols="1" rows="1" role="data">F. Thomas</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" role="data">1682</cell><cell cols="1" rows="1" role="data">8° <figure/></cell><cell cols="1" rows="1" role="data">52′</cell><cell cols="1" rows="1" role="data">75°</cell><cell cols="1" rows="1" role="data">15′</cell><cell cols="1" rows="1" role="data">93°</cell><cell cols="1" rows="1" role="data">32′</cell><cell cols="1" rows="1" role="data">20″</cell><cell cols="1" rows="1" role="data">52°</cell><cell cols="1" rows="1" role="data">31′</cell><cell cols="1" rows="1" role="data">33″</cell></row><row role="data"><cell cols="1" rows="1" role="data">F. Noel</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1697</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Dr. Halley</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">170<*></cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">F. Feuille</cell><cell cols="1" rows="1" role="data">Mar.</cell><cell cols="1" rows="1" role="data">1709</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">4</cell></row></table></p></div1><div1 part="N" n="CANTALIVERS" org="uniform" sample="complete" type="entry"><head>CANTALIVERS</head><p>, in Architecture, are the same with modillions, except that the former are plain, and the latter carved. They are both a kind of cartouses, set at equal distances under the corona of the cornice of a building.</p></div1><div1 part="N" n="CANTON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CANTON</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an ingenious natural philosopher, was born at Stroud, in Gloucestershire, in 1718; and was placed, when young, under the care of Mr Davis, an able mathematician of that place, with whom he had learned both vulgar and decimal arithmetic before he was quite 9 years of age. He next proceeded to higher parts of the mathematics, and particularly to algebra and astronomy, in which he had made a considerable progress when his father took him from school, and set him to learn his own business, which was that <pb n="242"/><cb/> of a broad cloth weaver. This circumstance was not able to damp his zeal for acquiring knowledge. All his leisure time was devoted to the assiduous cultivation of astronomical science; by which he was soon able to calculate lunar eclipses and other phenomena, and to construct various kinds of sun-dials, even at times when he ought to have slept, being done without the knowledge and consent of his father, who feared that such studies might injure his health. It was during this prohibition, and at these hours, that he computed, and cut upon stone, with no better an instrument than a common knife, the lines of a large upright sun-dial, on which, beside the hour of the day, were shewn the sun's rising, his place in the ecliptic, and some other particulars. When this was finished and made known to his father, he permitted it to be placed against the front of his house, where it excited the admiration of several neighbouring gentlemen, and introduced young Canton to their acquaintance, which was followed by the offer of the use of their libraries. In the library of one of these gentlemen he found Martin's Philosophical Grammar, which was the first book that gave him a taste for natural philosophy. In the possession of another gentleman he first saw a pair of globes; a circumstance that afforded him great pleasure, from the great ease with which he could resolve those problems he had hitherto been accustomed to compute.</p><p>Among other persons with whom he became acquainted in early life, was Dr. Henry Miles of Tooting; who perceiving that young Canton possessed abilities too promising to be confined within the narrow limits of a country town, prevailed on his father to permit him to come up to London. Accordingly he arrived at the metropolis the 4th of March 1737, and resided with Dr. Miles at Tooting till the 6th of May following; when he articled himself, for the term of 5 years, as a clerk to Mr. Samuel Watkins, master of the academy in Spital Square. In this situation his ingenuity, diligence, and prudence, were so distinguished, that on the expiration of his clerkship in May 1742, he was taken into partnership with Mr. Watkins for 3 years; which gentleman he afterward succeeded in the school, and there continued during his whole life.</p><p>Towards the end of 1745, electricity received a great improvement by the discovery of the famous Leyden phial. This event turned the thoughts of most of the philosophers of Europe to that branch of natural philosophy; and our author, who was one of the first to repeat and to pursue the experiment, found his endeavours rewarded by many notable discoveries.—Towards the end of 1749, he was engaged with his friend, the late ingenious Benjamin Robins, in making experiments to determine the height to which rockets may be made to ascend, and at what distance their light may be seen. —In 1750 was read at the Royal Society, Mr. Canton's “Method of making Artificial Magnets, without the use of, and yet far superior to, any natural ones.” This paper procured him the honour of being elected a member of the Society, and the present of their gold medal. The same year he was complimented with the degree of M. A. by the university of Aberdeen. And in 1751 he was chosen one of the council of the Royal Society; an honour which was twice repeated afterwards. <cb/></p><p>In 1752, our philosopher was so fortunate as to be the first person in England who, by attracting the electric fire from the clouds during a thunder-storm, verified Dr. Franklin's hypothesis of the similarity of lightning and electricity. Next year his paper intitled “Electrical Experiments, with an attempt to account for their several phenomena,” was read at the Royal Society. In the same paper Mr. Canton mentioned his having discovered, by many experiments, that some clouds were in a positive, and some in a negative state of electricity: a discovery which was also made by Dr. Franklin in America much about the same time. This circumstance, together with our author's constant desence of the doctor's hypothesis, induced that excellent philosopher, on his arrival in England, to pay Mr. Canton a visit, and gave rise to a friendship which ever after continued between them.—In the Ladies' Diary for 1756, our author answered the prize query that had been proposed in the preceding year, concerning the meteor called shooting stars. The solution, though only signed A. M. was so satisfactory to his friend, the excellent mathematician Mr. Thomas Simpson, who then conducted that ingenious and useful little work, that he sent Mr. Canton the prize, accompanied with a note, in which he said he was sure that he was not mistaken in the author of it, as no one besides, that he knew of, could have given that answer.— Our philosopher's next communication to the public, was a letter in the Gentleman's Magazine for September 1759, on the electrical properties of the tourmalin, in which the laws of that wonderful stone are laid down in a very concise and elegant manner. On the 13th of December in the same year was read at the Royal Society, “An attempt to account for the Regular Diurnal Variation of the Horizontal Magnetic Needle; and also for its Irregular Variation at the time of an Aurora Borealis.” A complete year's observations of the diurnal variations of the needle are annexed to the paper. —Nov. 5, 1761, our author communicated to the Royal Society an account of the Transit of Venus of the 6th of June that year, observed in Spital Square. His next communication to the Society, was a Letter, read the 4th of Feb. 1762, containing some remarks on Mr. Delaval's electrical experiments. On the 16th of Dec. the same year, another curious addition was made by him to philosophical knowledge, in a paper, intitled, “Experiments to prove that Water is not Incompressible.” And on Nov. 8, the year following, were read before the Society, his farther “Experiments and Observations on the Compressibility of Water, and some other fluids.” These experiments are a complete refutation of the famous Florentine experiment, which so many philosophers have mentioned as a proof of the incompressibility of water. For this communication he had a second time the Society's prize gold medal.</p><p>Another communication was made by our author to the Society, on Dec. 22, 1768, being “An eafy method of making a phosphorus that will imbibe and emit light like the Bolognian Stone; with experiments and observations.” When he sirst shewed to Dr. Franklin the instantaneous light acquired by some of this phosphorus from the near discharge of an electrified bottle, the doctor immediately exclaimed, “And God said let there be light, and there was light.” <pb n="243"/><cb/></p><p>The Dean and Chapter of St. Paul's having, in a letter, dated March 6, 1769, requested the opinion of the Royal Society relative to the best method of fixing electrical conductors to preserve that cathedral from damage by lightning, Mr. Canton was one of the committee appointed to take the letter into consideration, and to report their opinion upon it. The gentlemen joined with him in this business were, Mr. Delaval, Dr. Franklin, Dr. Watson, and Mr. Wilson. Their report was made on the 8th of June following: and the mode recommended by them has been carried into execution. —Our author's last communication to the Royal Society, was a paper read Dec. 21, 1769, containing “Experiments to prove that the Luminousness of the Sea arises from the Putrefaction of its animal Substances.”</p><p>Besides the papers above mentioned, Mr. Canton wrote a number of others, both in the earlier and the later parts of his life, which appeared in several publications, and particularly in the Gentleman's Magazine.— He died of a dropsy, the 22d of March 1772, in the 54th year of his age.</p></div1><div1 part="N" n="CAPACITY" org="uniform" sample="complete" type="entry"><head>CAPACITY</head><p>, is the solid content of any body. Also our hollow measures for corn, beer, wine, &c, are called measures of capacity.</p></div1><div1 part="N" n="CAPE" org="uniform" sample="complete" type="entry"><head>CAPE</head><p>, or <hi rend="italics">Promontory,</hi> is any high land, running out with a point into the sea; as Cape Verde, Cape Horn, the Cape of Good Hope, &c.</p></div1><div1 part="N" n="CAPELLA" org="uniform" sample="complete" type="entry"><head>CAPELLA</head><p>, a bright star of the first magnitude, in the left shoulder of Auriga.</p><p>CAPILLARY <hi rend="italics">Tubes,</hi> in Physics, are very small pipes, whose canals are exceedingly narrow; being so called from their resemblance to a hair in smallness. Their usual diameter may be from 1/20 to 1/50 of an inch: though Dr. Hook assures us that he drew tubes in the flame of a lamp much smaller, and resembling a spider's thread.</p><p>The <hi rend="italics">Ascent of Water &c,</hi> in capillary tubes, is a noted phenomenon in philosophy. Take several small glass tubes, of different diameters, and open at both ends; immerse them a little way into water, and the fluid will be seen to stand higher in the tubes than the surface of the water without, and higher as the tube is smaller, almost in the reciprocal ratio of the diameter of the tube; and that both in open air, and in vacuo. The greatest height to which Dr. Hook ever observed the water to stand, in the smallest tubes, was 21 inches above the surface in the vessel.</p><p>This does not however happen uniformly the same in all fluids; some standing higher than others; and in quicksilver the contrary takes place, as that fluid stands lower within the tube than its surface in the vessel, and the lower as the tube is smaller. See Philof. Trans. N° 355, or Abr. vol. 4, pa. 423, &c, or Cotes's Hydr. and Pneum. Lect. pa. 265.</p><p>Another phenomenon of these tubes is, that such of them as would only naturally discharge water by drops, when electrisied, yield a continued and accelerated stream; and the acceleration is proportional to the smallness of the tube: indeed the effect of electricity is so considerable, that it produces a continued stream from a very small tube, out of which the water had not before been able to drop. Priestley's Hist. Electr. 8vo. vol. 1, pa. 171, ed. 3d. <cb/></p><p>This ascent and suspension of the water in the tube, is by Dr. Jurin, Mr. Hauksbee, and other philosophers, ascribed to the attraction of the periphery of the concave surface of the tube, to which the upper surface of the water is contiguous and adheres.</p></div1><div1 part="N" n="CAPITAL" org="uniform" sample="complete" type="entry"><head>CAPITAL</head><p>, in Architecture, the uppermost part of a column or pilaster, serving as a head or crowning to it; being placed immediately over the shaft, and under the entablature. It is made differently in the different orders, and is that indeed which chiefly distinguishes the orders themselves.</p><p><hi rend="smallcaps">Capital</hi> <hi rend="italics">of a Bastion,</hi> is an imaginary line dividing any work into two equal and similar parts; or a line drawn from the angle of the polygon to the point of the bastion, or from the point of the bastion to the middle of the gorge.</p></div1><div1 part="N" n="CAPONIERE" org="uniform" sample="complete" type="entry"><head>CAPONIERE</head><p>, or <hi rend="smallcaps">Caponniere</hi>, in Fortification, is a passage made from one work to another, of 10 or 12 feet wide, and about 5 feet deep, covered on each side by a parapet, terminating in a glacis or slope. Sometimes it is covered with planks and earth.</p></div1><div1 part="N" n="CAPRA" org="uniform" sample="complete" type="entry"><head>CAPRA</head><p>, or the <hi rend="italics">She-goat,</hi> a name given to the star Capella, on the left shoulder of Auriga; and sometimes to the constellation Capricorn. Some again represent Capra as a constellation in the northern hemisphere, consisting of 3 stars, comprised between the 45th and 55th degree of latitude.</p><p>The poets fable her to be Amalthea's goat, which suckled Jupiter in his infancy.</p></div1><div1 part="N" n="CAPRICORN" org="uniform" sample="complete" type="entry"><head>CAPRICORN</head><p>, the <hi rend="italics">Goat,</hi> a southern constellation, and the 10th sign of the zodiac, as also one of the 48 original constellations received by the Greeks from the Egyptians. The figure of this sign is drawn as having the fore part of a goat, but the hinder part of a fish; and sometimes simply under the form of a goat. In writing, it is denoted by a character representing the crooked horns of a goat's head, thus <figure/>.</p><p>As to the figure of this constellation, the Greeks pretend that Pan, to avoid the terrible giant Typhon, threw himself into the Nile, and was changed into the figure here drawn; in commemoration of which exploit, Jupiter took it up to heaven. But it is probable, as Macrobius observes, that the Egyptians marked the point of the ecliptic appropriated to this sign, where the sun begins again to ascend up towards the north, with the figure of a goat, an animal which is always climbing the sides of mountains.</p><p>The stars in this constellation, in Ptolomy's and Tycho's catalogue, are 28; in that of Hevelius 29; though it is to be remarked that one of those in the tail, of the 6th magnitude, marked the 27th in Tycho's book, was lost in Hevelius's time. Flamsteed gives 51 stars to this sign.</p><p><hi rend="italics">Tropic of</hi> <hi rend="smallcaps">Capricorn</hi>, a little circle of the sphere, parallel to the equator, passing through the beginning of Capricorn, or the winter solstice, or the point of the sun's greatest south declination.</p></div1><div1 part="N" n="CAPSTAN" org="uniform" sample="complete" type="entry"><head>CAPSTAN</head><p>, a large massy column shaped like a truncated cone; being set upright on the deck of a ship, and turned by levers or bars, passing through holes in its upper extremity. The capstan is a kind of perpetual lever, or an axis-in-peritrochio, which, by means of a strong rope or cable passed round, serves to raise very great weights; such as to hoist sails, to weigh <pb n="244"/><cb/> the anchors, to draw the vessels on shore, and hoist them up to be refitted, &c.</p><p>CAPUT <hi rend="smallcaps">Draconis</hi>, or dragon's head, a name given by some to a fixed star of the sirst magnitude, in the head of the constellation Draco.</p></div1><div1 part="N" n="CARACT" org="uniform" sample="complete" type="entry"><head>CARACT</head><p>, or <hi rend="smallcaps">Carat</hi>, a name given to the weight which expresses the degree of goodness or fineness of gold. The whole quantity of metal is considered as consisting of 24 parts, which are the carats, so that the carat is the 24th part of the whole; this carat is divided into 4 equal parts, called grains of a carat, and the grain into halves and quarters.</p><p>When gold is purified to the utmost degree possible, so that it loses no more by farther trials, it is considered as quite pure, and said to be 24 carats fine; if it lose 1 carat, or 1—24th in purifying, it was of 23 carats fine; and if it lose 2 carats, it was 22 carats fine; and so on.</p></div1><div1 part="N" n="CARCASS" org="uniform" sample="complete" type="entry"><head>CARCASS</head><p>, is a hollow cafe formed of ribs of iron, and covered over with pitched cloth &c, about the size of bomb-shells; or sometimes made all of iron except two or three holes for the fire to blaze through. These are filled with various matters and combustibles, to five houses, when thrown out of mortar pieces into besieged places.</p></div1><div1 part="N" n="CARCAVI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CARCAVI</surname> (<foreName full="yes"><hi rend="smallcaps">Peter de</hi></foreName>)</persName></head><p>, was born at Lyons, but in what year is not known. He was Counsellor to the Parliament of Toulouse, afterward Counsellor to the Grand Council, and Keeper of the King's Library. He was appointed Geometrician to the French Academy of Sciences in 1666; and died at Paris in 1684. There are extant some letters of his, printed among those of Descartes.</p></div1><div1 part="N" n="CARDAN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CARDAN</surname> (<foreName full="yes"><hi rend="smallcaps">Hieronymus</hi></foreName>, or <hi rend="smallcaps">Jerom</hi>)</persName></head><p>, one of the most extraordinary geniuses of his age, was born at Pavia, in Italy, Sept. 24, 1501. At 4 years old he was carried to Milan, his father being an advocate and physician in that city: at the age of 20 he went to study in the university of the same city; and two years afterward he explained Euclid. In 1524, he went to Padua: the same year he was admitted to the degree of master of arts; and the year following, that of doctor of physic. He married about the year 1531; and became professor of mathematics, and practised medicine at Milan about 1533. In 1539 he was admitted a member of the college of physicians at Milan: in 1543 he read public lectures in medicine there; and the same at Pavia the year following; but he discontinued them because he could not get payment of his salary, and returned to Milan.</p><p>In 1552 he went into Scotland, having been sent for by the archbishop of St. Andrews, to cure him of a grievous disorder, after trying the physicians of the king of France and of the emperor of Germany, without benefit. He began to recover from the day that Cardan prescribed for him: our author took his leave of him at the end of six weeks and three days, leaving him prescriptions which in two years wrought a complete cure. Upon this visit Cardan passed through London, and calculated king Edward's nativity; for he was famous for his knowledge in astrology, as well as those of mathematics and medicine. Returning to Milan, after four months absence, he remained there till the beginning of Oct. 1552; and then went to <cb/> Pavia, from whence he was invited to Bologna in 1562. He taught in this last city till the year 1570; at which time he was thrown into prison; but some months after he was sent home to his own house. He quitted Bologna in 1571; and went to Rome, where he lived for some time without any public employment. He was however admitted a member of the college of physicians, and received a pension from the Pope, till the time of his death, which happened at Rome on the 21st of September 1575.</p><p>Cardan, at the same time that he was one of the greatest geniuses and most learned men of his age, in all the sciences, was one of the most eccentric and fickle in conduct of all men that ever lived: despising all good principles and opinions, and without one friend in the world. The same capriciousness that was remarkable in his outward conduct, is also observable in the composition of his numerous and elaborate works. In many of his treatises the reader is stopped almost every moment by the obscurity of his text, or by digressions from the point in hand. In his arithmetical writings there are several discourses on the motions of the planets, on the creation, on the Tower of Babel, and such like. And the apology which he made for these frequent digressions is, that he might by that means enlarge and fill up his book, his bargain with the bookseller being at so much per sheet; and that he worked as much for his daily support as for fame. The Lyons edition of his works, printed in 1663, consists of no less than 10 volumes in folio.</p><p>In fact, when we consider the transcendent qualities of Cardan's mind, it cannot be denied that he cultivated it with every species of knowledge, and made a greater progress in philosophy, in the medical art, in astronomy, in mathematics, and the other sciences, than the most part of his contemporaries who had applied themselves but to one only of those sciences. In particular, he was perhaps the very best algebraist of his time, a science in which he made great improvements; and his labours in cubic equations especially have rendered his name immortal, the rules for resolving them having ever since borne his name, and are likely to do so as long as the science shall exist, although he received the first knowledge of them from another person; the account of which, and his disputes with Tartalea, have been given at large under the article <hi rend="smallcaps">Algebra.</hi></p><p>Scaliger affirms, that Cardan, having by astrology predicted and fixed the time of his death, abstained from all food, that his prediction might be fulfilled, and that his continuance to live might not discredit his art. It is farther remarkable, that Cardan's father also died in this manner, in the year 1524, having abstained from sustenance for nine days. Our author too informs us that his father had white eyes, and could see in the night-time.</p><p>CARDINAL <hi rend="italics">Points,</hi> in Geography, are the east, west, north, and south points of the horizon.</p><p><hi rend="smallcaps">Cardinal</hi> <hi rend="italics">Points of the Heavens,</hi> or <hi rend="italics">of a Nativity,</hi> are the rising and setting of the sun, the zenith and nadir.</p><p><hi rend="smallcaps">Cardinal</hi> <hi rend="italics">Signs,</hi> are those at the four quarters, or the equinoxes and solstices, viz, the signs Aries, Libra, Cancer, and Capricorn. <pb n="245"/><cb/></p><p><hi rend="smallcaps">Cardinal</hi> <hi rend="italics">Winds,</hi> are those that blow from the four cardinal points, viz, the east, west, north, and south winds.</p></div1><div1 part="N" n="CARDIOIDE" org="uniform" sample="complete" type="entry"><head>CARDIOIDE</head><p>, the name of a curve so called by Castilliani.—But it was first treated of by Koersma, and by Carré. See Philos. Trans. 1741, and Memoires de l'Acad. 1705.</p><p><hi rend="italics">The Cardioide is thus generated.</hi> APB is a circle, and AB its diameter. Through one extremity A of the diameter draw a number of lines APQ, cutting the circle in P; upon these set off always PQ equal to the diameter AB; so shall the points Q be always in the curve of the cardioide. <figure/></p><p>From this generation of the curve, its chief properties are evident, viz, that,</p><p>everywhere PQ = AB,</p></div1><div1 part="N" n="CQ" org="uniform" sample="complete" type="entry"><head>CQ</head><p>, or QQ is = A<hi rend="italics">a</hi> or 2 AB,</p><p>,</p><p>P always bisects QQ.</p><p>The cardioide is an algebraical curve, and the equation expressing its nature is thus:</p><p>Put <hi rend="italics">a</hi> = AB the diameter, <hi rend="italics">x</hi> = <hi rend="italics">a</hi>D perp. AB, <hi rend="italics">y</hi> = DQ perp. AD; then is which is the equation of the curve.</p><p>Many properties of the cardioide may be seen in the places above cited.</p><p>CARRÉ (<hi rend="smallcaps">Lewis</hi>), was born in the year 1663, in the province of Brie in France. His father, a substantial farmer, intended him for the church. But young Carré, after going through the usual course of education for that purpose, having an utter aversion to it, he refused to enter upon that function; by which he incurred his father's displeasure. His resources being thus cut off, he was obliged to quit the university, and look out into the world for some employment. In this exigency he had the good fortune to be engaged as an amanuensis by the celebrated father Malebranche; by which he found himself transported all at once from the mazes of scholastic darkness, to the source of the most brilliant and enlightened philosophy. Under this great master he studied mathematics and the most sublime metaphysics. After seven years spent in this excellent school, M. Carré found it necessary, in order to procure himself some less precarious establishment, to teach mathematics and philosophy in Paris; but especially that philosophy which, on account of its tendency to improve our morals, he valued more than all the mathematics in the world. And accordingly his greatest care was to make geome- <cb/> try serve as an introduction to his well beloved metaphysics.</p><p>Most of M. Carré's pupils were of the fair sex. The first of these, who soon perceived that his language was rather the reverse of elegant and correct, told him pleasantly that, as an acknowledgement for the pains he took to teach her philosophy, she would teach him French; and he ever after owned that her lessons were of great service to him. In general he seemed to set more value upon the genius of women than that of men.</p><p>M. Carré, although he gave the preference to metaphysics, did not neglect mathematics; and while he taught both, he took care to make himself acquainted with all the new discoveries in the latter. This was all that his constant attendance on his pupils would allow him to do, till the year 1697, when M. Varignon, so remarkable for his extreme scrupulousness in the choice of his eleves, took M. Carré to him in that station. Soon after, viz. in the year 1700, our author thinking himself bound to do something that might render him worthy of that title, published the first complete work on the Integral Calculus, under the title of “A method of measuring Surfaces and Solids, and finding their Centres of Gravity, Percussion, and Oscillation.” He afterwards discovered some errors in the work, and was candid enough to own and correct them in a subsequent edition.</p><p>In a little time M. Carré became Associate, and at length one of the Pensioners of the Academy. And as this was a sufficient establishment for one, who knew so well how to keep his desires within just bounds, he gave himself up entirely to study; and as he enjoyed the appointment of Mechanician, he applied himself more particularly to mechanics. He took also a survey of every branch relating to music; such as the doctrine of sounds, the description of musical instruments; though he despised the practice of music, as a mere sensual pleasure. Some sketches of his ingenuity and industry in this way may be seen in the Memoirs of the French Academy of Sciences. M. Carré also composed some treatises on other branches of natural philosophy, and some on mathematical subjects; all which he bequeathed to that illustrious body; though it does not appear that any of them have yet been published. It is not unlikely that he was hindered from putting the last hand to them by a train of disorders proceeding from a bad digestion, which, after harassing him during the space of five or six years, at length brought him to the grave in 1711, at 48 years of age.</p><p>His memoirs printed in the volumes of the Academy, with the years of the volumes, are as below.</p><p>1. The Rectification of Curve Lines by Tangents: 1701.</p><p>2. Solution of a problem proposed to Geometricians, &c. 1701.</p><p>3. Reflections on the Table of Equations: 1701.</p><p>4. On the Cause of the Refraction of Light: 1702.</p><p>5. Why the Tides are always augmenting from Brest to St. Malo, and diminishing along the coasts of Normandy: 1702.</p><p>6. The Number and the Names of Musical Instruments: 1702. <pb n="246"/><cb/></p><p>7. On the Vinegar which causes small stones to roll upon an inclined plane: 1703.</p><p>8. On the Rectification &c. of the Caustics by reflection: 1703.</p><p>9. Method for the Rectification of Curves: 1704.</p><p>10. Observations on the Production of Sound: 1704.</p><p>11. On a Curve formed from a Circle: 1705.</p><p>12. On the Refraction of Musket-balls in water, and on the Resistance of that fluid: 1705.</p><p>13. Experiments on Capillary Tubes: 1705.</p><p>14. On the Proportion of Pipes to have a determinate quantity of water: 1705.</p><p>15. On the Laws of Motion: 1706.</p><p>16. On the Properties of Pendulums; with some new properties of the Parabola: 1707.</p><p>17. On the Proportion of Cylinders that their sounds may form the musical chords: 1709.</p><p>18. On the Elasticity of the Air: 1710.</p><p>19. On Catoptrics: 1710.</p><p>20. On the Monochord: in the Machines, tom. 1. with some other pieces, not mathematical.</p></div1><div1 part="N" n="CARRIAGE" org="uniform" sample="complete" type="entry"><head>CARRIAGE</head><p>, <hi rend="italics">of a Cannon,</hi> is the machine upon which it is mounted; serving to point or direct it for shooting, and to convey it from place to place.</p><p><hi rend="italics">Wheel</hi> <hi rend="smallcaps">Carriage</hi>, one that is mounted and moved about upon wheels. Horses draw in general, to most advantage, when the direction of their draft is parallel to the ground, or rather a little upwards. A carriage also goes easiest when the centre of gravity is placed very high; since then, when once put in motion, it continues it with very little labour to the horses.</p></div1><div1 part="N" n="CARTES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CARTES</surname> (<foreName full="yes"><hi rend="smallcaps">Rene des</hi></foreName>)</persName></head><p>, one of the most eminent philosophers and mathematicians of the 17th century, or indeed of any age whatever. He was descended of an ancient noble family in Touraine in France, being a younger son of a counsellor in the parliament of Rennes, and was born March 31, 1596. His father gave him a liberal education, and the more so as he observed in him the appearance of a promising genius, using to call him the philosopher, on account of his insatiable curiosity in asking the reasons of every thing that he did not understand.</p><p>Des Cartes was sent to the Jesuits college at La Fleche in 1604, and put under the tuition of Father Charlet. Here he made a great progress in the learned languages and polite literature; but having passed through his course of philosophy without any great satisfaction to himself, he left the college in 1612, and began to learn military arts, to ride and fence, and other such like exercises. But notwithstanding his inclination to military achievements, the weakness of his constitution not permitting him early to expose himself to the fatigues of war, he was sent to Paris in 1613. Here he formed an acquaintance with several learned persons, who helped to reclaim him from his intention of declining his studies, particularly Father Mersenne, whose conversation revived in him a love for truth, and induced him to retire from the world to pursue his studies without interruption; which he did for two years: but in May 1616, at the repeated solicitations of his relations, he set out for Holland, and entered as a volunteer under the Prince of Orange.</p><p>Whilst he lay in garrison at Breda, during the truce between the Spanish and Dutch, an unknown person <cb/> caused a problem in mathematics, in the Dutch language, to be fixed up in the streets: when Des Cartes, seeing a concourse of people stop to read it, desired one who stood near him to explain it to him in Latin or French. The person promised to satisfy him, upon condition that he would engage to resolve the problem; and Des Cartes agreed to the condition with such an air, that though he little expected such a thing from a young military cadet, he gave him his address, desiring he would bring him the solution. Des Cartes next day visited Beekman, principal of the college of Dort, who was the person that had translated the problem to him. Beekman was surprised at his having resolved it in such a short time; but his wonder was much increased, to find, in the course of conversation, that the young man's knowledge was much superior to his own in those sciences, in which he had employed his whole time for several years. During his stay at Breda, Des Cartes wrote in Latin, a treatise on Music, and laid the foundation of several of his other works.</p><p>In 1619, he entered himself a volunteer in the army of the duke of Bavaria. In 1621, he made the campaign in Hungary, under the count de Bucquoy; but the loss of his general, who was killed at a siege that year, determined him to quit the army. He soon after began his travels into the north, and visited Silesia, Poland, Pomerania, the coasts of the Baltic, Brandenburgh, Holstein, East Friesland, West Friesland; in his passage to which last place he was in danger of being murdered. The sailors fancied he was a merchant, who had a large sum of money about him; and perceiving that he was a foreigner who had little acquaintance in the country, and a man of a mild disposition, they resolved to kill him, and throw his body into the sea. They even discoursed of their design before his face, thinking he understood no language but French, in which he always spoke to his servant. Des Cartes suddenly started up; and drawing his sword, spoke to them in their own language, in such a tone as struck terror into them: upon which they behaved very civilly. The year following he went to Paris, where he cleared himself from the imputation of having been received among the Rosicrusians, whom he confidered as a company of visionaries and impostors.</p><p>Dropping the study of mathematics, he now applied himself again to ethics and natural philosophy. The same year he took a journey through Switzerland to Italy. Upon his return he settled at Paris; but his studies being interrupted by frequent visits, he went in 1628 to the siege of Rochelle. He returned to Paris in November; but in the following spring he repaired to Amsterdam; and from thence to a place near Franeker in Friesland, where he began his Metaphysical Meditations, and spent some time in Dioptrics; about this time too he wrote his thoughts upon Meteors. After about six months he returned to Amsterdam.</p><p>Des Cartes imagined that nothing could more promote the temporal felicity of mankind, than the union of natural philosophy with mathematics. But before he should set himself to relieve men's labours, or multiply the conveniencies of life by mechanics, he thought it necessary to discover some means of securing the hu- <pb n="247"/><cb/> man body from disease and debility: this led him to the study of anatomy and chemistry, in which he employed the winter at Amsterdam.</p><p>He now, viz, about 1630 or 1631, took a short tour to England, and made some observations near London on the variation of the compass. In the spring of 1633 he removed to Deventer, where he completed several works that were left unfinished the year before, and resumed his studies in astronomy. In the summer he put the last hand to his “Treatise of the World.” The next year he returned to Amsterdam; but soon after took a journey into Denmark, and the lower parts of Germany. In autumn 1635 he went to Lewarden in Friesland, where he remained till 1637, and wrote his “Treatise of Mechanics.” The same year he published his four treatises concerning Method, Dioptrics, Meteors, and Geometry. About this time he received an invitation to settle in England from Sir Charles Cavendish, brother to the earl of Newcastle, with which he did not appear backward to comply, especially upon being assured that the king was a catholic in his heart: but the breaking out of the civil wars in this country prevented his journey.</p><p>At the end of 1641, Lewis the 13th, of France, invited him to his court, upon very honourable terms; but he could not be persuaded to quit his retirement. This year he published his Meditations concerning the Existence of God, and the Immortality of the Soul. In 1645 he again applied to anatomy; but was a little diverted from this study, by the question concerning the Quadrature of the Circle, which was at that time agitated. During the winter of the same year he composed a small tract against Gassendus's Institutes; and another on the Nature of the Passions. About this time he carried on an epistolary correspondence with the princess Elizabeth, daughter to Frederick the 5th, elector palatine, and king of Bohemia, who had been his pupil in Holland.</p><p>A dispute arising between Christina, queen of Sweden, and M. Chanut, the resident of France, concerning the following question; When a man carries love or hatred to excess, which of these two irregularities is the worst? The resident sent the question to Des Cartes, who upon that occasion drew up the dissertation upon Love, that is published in the first volume of his letters, which proved highly satisfactory to the queen. In June 1647 he took a journey to France, where the king settled on him a pension of 3000 livres; but he returned to Holland about the end of September. In November he received a letter from M. Chanut, in queen Christina's name, desiring his opinion of the sovereign good; which he accordingly sent her, with some letters upon the same subject formerly written to the princess Elizabeth, and his treatise on the Passions. The queen was so highly pleased with them, that she wrote him a letter of thanks with her own hand, and invited him to come to Sweden. He arrived at Stockholm in Oct. 1648. The queen engaged him to attend her every morning at five o'clock, to instruct her in his philosophy; and desired him to revise and digest all his unpublished writings, and to draw up from them a complete body of philosophy. She purposed also to fix him in Sweden, by allowing him a revenue of 3000 crowns a year, with an estate which should <cb/> descend to his heirs and assigns for ever; and to establish an academy, of which he was to be the director. But these designs were frustrated by his death, which happened the 11th of Feb. 1650, in the 54th year of his age. His body was interred at Stockholm: but 17 years after it was removed to Paris, where a magnificent monument was erected to him in the church of Genevieve du Mont. <hi rend="center">As to the character of our author:</hi></p><p>Dr. Barrow in his <hi rend="italics">Opuscula</hi> tells us, that Des Cartes was doubtless a very ingenious man, and a real philosopher, and one who seems to have brought those assistances to that part of philosophy relating to matter and motion, which perhaps no one had done before: namely, a great skill in mathematics; a mind habituated, both by nature and custom, to profound meditation; a judgment exempt from all prejudices and popular errors, and furnished with a good number of certain and select experiments; a great deal of leisure; an entire disengagement, by his own choice, from the reading of useless books, and the avocations of life; with an incomparable acuteness of wit, and an excellent talent of thinking clearly and distinctly, and of expressing his thoughts with the utmost perspicuity.</p><p>Dr. Halley, in a paper concerning Optics, communicated to Mr. Wotton, and published by the latter in his “Reflections upon Ancient and Modern Learning,” writes as follows: As to dioptrics, though some of the ancients mention refraction, as a natural effect of transparent media; yet Des Cartes was the first, who in this age has discovered the laws of refraction, and brought dioptrics to a science.</p><p>Dr. Keil, in the introduction to his “Examination of Dr. Burnet's Theory of the Earth,” tells us, that Des Cartes was so far from applying geometry and observations to natural philosophy, that his whole system is but one continued blunder on account of his negligence in that point; which he could easily prove, by shewing that his theory of the vortices, upon which his system is founded, is absolutely false, for that Newton has shewn that the periodical times of all bodies, that swim in vortices, must be directly as the squares of their distances from the centre of them: but it is evident from observations, that the planets, in moving round the sun, observe a law quite different from this; for the squares of their periodical times are always as the cubes of their distances: and therefore, since they do not observe that law, which of necessity they must if they swim in a vortex, it is a demonstration that there are no vortices in which the planets are carried round the sun.</p><p>“Nature, fays Voltaire, had favoured Des Cartes with a strong and clear imagination, whence he became a very singular person, both in private life, and in his manner of reasoning. This imagination could not be concealed even in his philosophical writings, which are every where adorned with very brilliant ingenious metaphors. Nature had almost made him a poet; and indeed he wrote a piece of poetry for the entertainment of Christina queen of Sweden, which however was suppressed in honour of his memory. He extended the limits of geometry as far beyond the place where he found them, as Newton did after him; and first taught the method of expressing curves by equations. He ap- <pb n="248"/><cb/> plied this geometrical and inventive genius to dioptrics, which when treated by him became a new art; and if he was mistaken in some things, the reason is, that a man who discovers a new tract of land, cannot at once know all the properties of the soil. Those who come after him, and make these lands fruitful, are at least obliged to him for the discovery.” Voltaire acknowledges, that there are innumerable errors in the rest of Des Cartes' works; but adds, that geometry was a guide which he himself had in some measure formed, and which would have safely conducted him through the several paths of natural philosophy: nevertheless he had at last abandoned this guide, and gave entirely into the humour of framing hypotheses; and then philosophy was no more than an ingenious romance, fit only to amuse the ignorant.</p><p>It has been pretty generally acknowledged, that he borrowed his improvements in Algebra from Harriot's <hi rend="italics">Artis Analyticæ Praxis</hi>; which is highly probable, as he was in England about the time when Harriot's book was published, and as he follows the manner of Harriot, except in the method of noting the powers. Upon this head the following anecdote is told by Dr. Pell, in Wallis's Algebra, pa. 198. Sir Charles Cavendish, then resident at Paris, discoursing there with M. Roberval, concerning Des Cartes's Geometry, then lately published: I admire, said Roberval, that method in Des Cartes, of placing all the terms of the equation on one side, making the whole equal to nothing, and how he lighted upon it. The reason why you admire it, said Sir Charles, is because you are a Frenchman; for if you were an Englishman, you would not admire it. Why so? asked Roberval. Because, replied Sir Charles, we in England know whence he had it; namely from Harriot's Algebra. What book is that? says Roberval, I never saw it. Next time you come to my chamber, saith Sir Charles, I will shew it to you. Which a while after he did; and upon perusal of it, Roberval exclaimed with admiration, <hi rend="italics">Il l'a vu! il l'a vu! He had seen it! he had seen it!</hi> finding all that in Harriot which he had before admired in Des Cartes, and not doubting but that Des Cartes had it from thence. See also Montucla's History of Mathematics.</p><p>The real improvements of Des Cartes in Algebra and Geometry, I have particularly treated of under the article <hi rend="smallcaps">Algebra;</hi> and his philosophical doctrines are displayed in the article <hi rend="smallcaps">Cartesian</hi> <hi rend="italics">Philosophy,</hi> here following. He was never married, but had one natural daughter, who died when she was but five years old. There have been several editions of his works, and commentaries upon them; particularly those of Schooten on his Geometry.</p><p>CARTESIAN <hi rend="italics">Philosophy,</hi> or <hi rend="italics">Cartesianism,</hi> the system of philosophy advanced by Des Cartes, and maintained by his followers, the Cartesians.</p><p>The Cartesian philosophy is founded on two great principles, the one metaphysical, the other physical. The metaphysical one is this: <hi rend="italics">I think,</hi> therefore <hi rend="italics">I am,</hi> or <hi rend="italics">I exist:</hi> the physical principle is, that <hi rend="italics">nothing exists but substances.</hi> Substance he makes of two kinds; the one a substance that thinks, the other a substance extended: so that actual thought and actual extension make the essence of substance.</p><p>The essence of matter being thus fixed in extension, <cb/> Des Cartes concludes that there is no vacuum, nor any possibility of it in nature; but that the universe is absolutely full: by this principle, mere space is quite excluded; for extension being implied in the idea of space, matter is so too.</p><p>Des Cartes defines motion to be the translation of a body from the neighbourhood of others that are in contact with it, and considered as at rest, to the neighbourhood of other bodies: by which he destroys the distinction between motion that is absolute or real, and that which is relative or apparent. He maintains that the same quantity of motion is always preserved in the universe, because God must be supposed to act in the most constant and immutable manner. And hence also he deduces his three laws of motion. See <hi rend="smallcaps">Motion.</hi></p><p>Upon these principles Des Cartes explains mechanically how the world was formed, and how the present phenomena of nature came to arise. He supposes that God created matter of an indefinite extension, which he separated into small square portions or masses, full of angles <*> that he impressed too motions on this matter; the one, by which each part revolved about its own centre; and another, by which an assemblage, or system of them, turned round a common centre. From whence arose as many different vortices, or eddies, as there were different masses of matter, thus moving about common centres.</p><p>The consequence of these motions in each vortex, according to Des Cartes, is as follows: The parts of matter could not thus move and revolve amongst one another, without having their angles gradually broken; and this continual friction of parts and angles must produce three elements: the first of these, an infinitely fine dust, formed of the angles broken off; the second, the spheres remaining, after all the angular parts are thus removed; and those particles not yet rendered smooth and spherical, but still retaining some of their angles, and hamous parts, from the third element.</p><p>Now the first or subtilest element, according to the laws of motion, must occupy the centre of each system, or vortex, by reason of the smallness of its parts: and this is the matter which constitutes the sun, and the fixed stars above, and the fire below. The second element, made up of spheres, forms the atmosphere, and all the matter between the earth and the fixed stars; in such sort, that the largest spheres are always next the circumserence of the vortex, and the smallest next its centre. The third element, formed of the irregular particles, is the matter that composes the earth, and all terrestrial bodies, together with comets, spots in the sun, &c.</p><p>He accounts for the gravity of terrestrial bodies from the centrifugal force of the ether revolving round the earth: and upon the same general principles he pretends to explain the phenomena of the magnet, and to account for all the other operations in nature.</p></div1><div1 part="N" n="CARY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CARY</surname> (<foreName full="yes"><hi rend="smallcaps">Robert</hi></foreName>)</persName></head><p>, a learned English chronologer and divine, was born at Cockington, in the county of Devon, about the year 1615. He took his degrees in arts, and LL.D. in Oxford. After returning from his travels he was presented to the rectory of Portlemouth, near Kingsbridge in Devonshire: but not long after he was drawn over by the presbyterian ministers to their party, and chosen moderator of that part of the second division of the county of Devon, which was ap- <pb n="249"/><cb/> pointed to meet at Kingsbridge. And yet, upon the restoration of Charles the 2d, he was one of the first to congratulate that prince upon his return, and soon after was preferred to the archdeaconry of Exeter; but from which he was however some time afterward ejected. He spent the rest of his days at his rectory at Portlemouth, and died in 1688, at 73 years of age.— He published <hi rend="italics">Palælogia Chronica,</hi> a chronological account of ancient time, in three parts. 1, Didactical; 2, Apodidactical; 3, Canonical: in 1677.</p></div1><div1 part="N" n="CASATI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CASATI</surname> (<foreName full="yes"><hi rend="smallcaps">Paul</hi></foreName>)</persName></head><p>, a learned Jesuit, born at Placentia in 1617. He entered early among the Jesuits; and after having taught mathematics and divinity at Rome, he was sent into Sweden to queen Christina, whom he prevailed on to embrace the popish religion. His writings are as follow:</p><p>1. <hi rend="italics">Vacuum Proscriptum.</hi>—2. <hi rend="italics">Terra Machinis mota.</hi>— 3. <hi rend="italics">Mechanicorum, libri octo.</hi>—4. <hi rend="italics">De Igne Dissertationes.</hi> —5. <hi rend="italics">De Angelis Disputatio Theolog.</hi>—6. <hi rend="italics">Hydrostaticæ Dissertationes.</hi>—7. <hi rend="italics">Opticæ Disputationes.</hi> It is remarkable that he wrote this treatse on optics at 88 years of age, and after he was blind. He was also author of several books in the Italian language.</p></div1><div1 part="N" n="CASCABEL" org="uniform" sample="complete" type="entry"><head>CASCABEL</head><p>, the knob or button of metal behind the breech of a cannon, as a kind of handle by which to elevate and direct the piece; to which some add the fillet and ogees as far as the base-ring.</p></div1><div1 part="N" n="CASEMATE" org="uniform" sample="complete" type="entry"><head>CASEMATE</head><p>, or <hi rend="smallcaps">Cazemate</hi>, in Fortification, a kind of vault or arch, of stone-work, in that part of the flank of a bastion next the curtain; serving as a battery, to defend the face of the opposite bastion, and the moat or ditch.</p><p>The casemate sometimes consists of three platforms, one above another; the highest being on the rampart; though it is common to withdraw this within the bastion.</p><p>The casemate is also called the low place, and low flank, as being at the bottom of the wall next the ditch; and sometimes the retired flank, as being the part of the flank nearest the curtain, and the centre of the bastion. It was formerly covered by an apaulement, or a massive body, either round or square, which prevented the enemy from seeing within the batteries; whence it was also called <hi rend="italics">covered flank.</hi></p><p>It is now seldom used, because the batteries of the enemy are apt to bury the artillery of the casemate in the ruins of the vault: beside, the great smoke made by the discharge of the cannon, renders it intolerable to the men. So that, instead of the ancient covered casemates, later engineers have contrived open ones, only guarded by a parapet, &c.</p><p><hi rend="smallcaps">Casemate</hi> is also used for a well with several subterraneous branches, dug in the passage of the bastion, till the miner is heard at work, and air given to the mine.</p></div1><div1 part="N" n="CASERNS" org="uniform" sample="complete" type="entry"><head>CASERNS</head><p>, or <hi rend="smallcaps">Cazerns</hi>, in Fortification, small rooms, or huts, erected between the ramparts and the houses of fortified towns, or even on the ramparts themselves; to serve as lodgings for the soldiers on immediate duty, to ease the garrison.</p><p>CASE-<hi rend="smallcaps">Shot</hi>, or <hi rend="smallcaps">Cannister-Shot</hi>, are a number of small balls put into a round tin cannister, and so shot out of great guns. These have superseded, and been substituted instead of the grape-shot, which have been laid aside.</p></div1><div1 part="N" n="CASSINI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CASSINI</surname> (<foreName full="yes"><hi rend="smallcaps">John Dominic</hi></foreName>)</persName></head><p>, an eminent astrono- <cb/> mer, was born of noble parents, at a town in Piedmont in Italy, June 8, 1625. After laying a proper foundation in his studies at home, he was sent to continue them in a college of Jesuits at Genoa. He had an uncommon turn for Latin poetry, which he exercised so very early, that some of his poems were published when he was but 11 years old. At length he met with books of astronomy, which he read, with great eagerness. Pursuing the bent of his inclinations in this way, in a short time he made so amazing a progress, that in 1650 the senate of Bologna invited him to be their public mathematical professor. Cassini was but 25 years of age when he went to Bologna, where he taught mathematics, and made observations upon the heavens, with great care and assiduity. In 1652 a comet appeared, which he observed with great accuracy; and he discovered that comets were not bodies accidentally generated in the atmosphere, as had been supposed, but of the same nature, and probably governed by the same laws, as the planets. The same year he resolved an astronomical problem, which Kepler and Bulliald had given up as insolvable; viz, to determine geometrically the apogee and eccentricity of a planet, from its true and mean place.—In 1653, when a church in Bologna was repaired and enlarged, he obtained leave of the senate to correct and settle a meridian line, which had been drawn by an astronomer in 1575.—In 1657 he attended, as an assistant, a nobleman, who was sent to Rome to compose some differences, which had arisen between Bologna and Ferrara, from the inundations of the Po; and he shewed so much skill and judgment in the management of the affair, that in 1663 the pope's brother appointed him inspector general of the fortifications of the castle of Urbino: and he had afterward committed to him the care of all the rivers in the ecclesiastical state.</p><p>Mean while he did not neglect his astronomical studies, but cultivated them with great care. He made several discoveries relating to the planets Mars and Venus, particularly the revolution of Mars upon his own axis: but the point he had chiefly in view, was to settle an accurate theory of Jupiter's satellites; which, after much labour and observation, he happily effected, and published it at Rome, among other astronomical pieces, in 1666.</p><p>Picard, the French astronomer, getting Cassini's tables of Jupiter's satellites, found them so very exact, that he conceived the highest opinion of his skill; and from that time his fame increased so fast in France, that the government desired to have him a member of the academy. Cassini however could not leave his station without leave of his superiors; and therefore the king, Lewis the 14th, requested of the pope and the senate of Bologna, that Cassini might be permitted to come into France. Leave was granted for 6 years; and he came to Paris in the beginning of 1669, where he was immediately made the king's astronomer. When this term of 6 years was near expiring, the pope and the senate of Bologna insisted upon his return, on pain of forfeiting his revenues and emoluments, which had hitherto been remitted to him: but the minister Colbert prevailed on him to stay, and he was naturalized in 1673; the same year also in which he was married.</p><p>The Royal Observatory of Paris had been finished <pb n="250"/><cb/> some time. The occasion of its being built was this: In 1638, the celebrated Mersenne was the chief institutor and promoter of a society, where several ingenious and learned men met together to talk upon physical and astronomical subjects; among whom were Gassend, Defcartes, Monmort, Thevenot, Bulliald, our countryman Hobbes, &c: and this society was kept up by a succession of learned men for many years. At length the government considering that a number of such men, acting in a body, would succeed much better in the promotion of science, than if they acted separately, each in his particular art or province, established under the direction of Colbert, in 1666, the Royal Academy of Sciences: and for the advancement of astronomy in particular, erected the Royal Observatory at Paris, and furnished it with all kinds of instruments that were necessary to make observations. The foundation of this noble pile was laid in 1667, and the building completed in 1670. Of this observatory, Cassini was appointed to be the first inhabiter; which he took possession of in Sept. 1671, when he set himself with fresh alacrity to attend the duties of his profession. In 1672 he endeavoured to determine the parallax of Mars and the sun: and in 1677 he proved that the diurnal rotation of Jupiter round his axis was performed in 9 hours 58 minutes, from the motion of a spot in one of his larger belts: also in 1684 he discovered four satellites of Saturn, besides that which Huygens had found out. In 1693 he published a new edition of his “Tables of Jupiter's Satellites,” corrected by later observations. In 1695 he took a journey to Bologna, to examine the meridian line, which he had fixed there in 1655; and he shewed, in the presence of eminent mathematicians, that it had not varied in the least, during that 40 years. In 1700 he continued the meridian line through France, which Picard had begun, to the very southern limits of that country.</p><p>After our author had resided at the royal observatory for more than 40 years, making many excellent and useful discoveries, which he published from time to time, he died September the 14th, 1712, at 87 years of age; and was succeeded by his only son James Cassini. His publications were very numerous, far too much so, even to be enumerated in this place.</p></div1><div1 part="N" n="CASSINI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CASSINI</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, a celebrated French astronomer, and member of the several Academies of Sciences of France, England, Prussia, and Bologna, was born at Paris Feb. 18, 1677, being the younger son of John-Dominic Cassini, above mentioned, whom he succeeded as astronomer at the royal observatory, the elder son having lost his life at the battle of La Hogue.</p><p>After his first studies in his father's house, in which it is not to be supposed that mathematics and astronomy were neglected, he was sent to study philosophy at the Mazarine college, where the celebrated Varignon was then professor of mathematics; from whose assistance young Cassini profited so well, that at 15 years of age he supported a mathematical thesis with great honour. At the age of 17 he was admitted a member of the Academy of Sciences; and the same year he accompanied his father in his journey to Italy, where he assisted him in the verification of the meridian at Bologna, and other measurements. On his return he made other similar operations in a journey into Holland, where he <cb/> discovered some errors in the measure of the earth by Snell, the result of which was communicated to the Academy in 1702. He made also a visit to England in 1696, where he was made a member of the Royal Society.—In 1712 he succeeded his father as astronomer royal at the observatory.—In 1717 he gave to the Academy his researches on the distance of the fixed stars, in which he shewed that the whole annual orbit, of near 200 million of miles diameter, is but as a point in comparison of that distance. The same year he communicated also his discoveries concerning the inclination of the orbits of the satellites in general, and especially of those of Saturn's satellites and ring.—In 1725 he undertook to determine the cause of the moon's libration, by which she shews sometimes a little towards one side, and sometimes a little on the other, of that half which is commonly behind or hid from our view.</p><p>In 1732 an important question in astronomy exercised the ingenuity of our author. His father had determined, by his observations, that the planet Venus revolved about her axis in the space of 23 hours: and M. Bianchini had published a work in 1729, in which he settled the period of the same revolution at 24 days 8 hours. From an examination of Bianchini's observations, which were upon the spots in Venus, he discovered that he had intermitted his observations for the space of 3 hours, from which cause he had probably mistaken new spots for the old ones, and so had been led into the mistake. He soon afterwards determined the nature and quantity of the acceleration of the motion of Jupiter, at half a second per year, and of that of the retardation of Saturn at two minutes per year; that these quantities would go on increasing for 2000 years, and then would decrease again.—In 1740 he published his Astronomical Tables; and his Elements of Astronomy; very extensive and accurate works.</p><p>Although astronomy was the principal object of our author's consideration, he did not confine himself absolutely to that branch, but made occasional excursions into other fields. We owe also to him, for example, Experiments on Electricity, or the light produced by bodies by friction. Experiments on the recoil of fire arms; Researches on the rise of the mercury in the barometer at different heights above the level of the sea; Reflections on the perfecting of burning-glasses; and other memoirs.</p><p>The French Academy had properly judged that one of its most important objects, was the measurement of the earth. In 1669 Picard measured a little more than a degree of latitude to the north of Paris; but as that extent appeared too small from which to conclude the whole circumference with sufficient accuracy, it was resolved to continue that measurement on the meridian of Paris to the north and the south, through the whole extent of the country. Accordingly, in 1683, the late M. de la Hire continued that on the north side of Paris, and the older Cassini that on the south side. The latter was assisted in 1700 in the continuation of this operation by his son our author. The same work was farther continued by the same Academicians; and finally the part left unfinished by de la Hire in the north, was finished in 1718 by our author, with the late Maraldi, and de la Hire the younger.</p><p>These operations produced a considerable degree of <pb n="251"/><cb/> precision. It appeared also, from this measured extent of 6 degrees, that the degrees were of different lengths in different parts of the meridian; and in such sort that our author concluded, in the volume published for 1718, that they decreased more and more towards the pole, and that therefore the figure of the earth was that of an oblong spheroid, or having its axe longer than the equatorial diameter. He also measured the perpendicular to the same meridian, and compared the measured distance with the differences of longitude as before determined by the eclipses of Jupiter's satellites; from whence he concluded that the length of the degrees of longitude was smaller than it would be on a sphere, and that therefore again the figure of the earth was an oblong spheroid; contrary to the determination of Newton by the theory of gravity. In consequence of these assertions of our author, the French government sent two different sets of measurers, the one to measure a degree at the equator, the other at the polar circle; and the comparison of the whole determined the figure to be an oblate spheroid, contrary to Cassini's determination.</p><p>After a long and laborious life, our author James Cassini lost his life by a fall in April 1756, in the 80th year of his age, and was succeeded in the Academy and Observatory by his second son Cesar-François de Thury. He published, A Treatise on the Magnitude and Figure of the Earth; as also The Elements or Theory of the Planets, with Tables; beside an infinite number of papers in the Memoirs of the Academy, from the year 1699 to 1755.</p><p>CASSINI <hi rend="smallcaps">DE Thury (Cesar-Francois</hi>), a celebrated French astronomer, director of the observatory, pensioner astronomer, and member of most of the learned societies of Europe, was born at Paris June 17, 1714, being the second son of James Cassini, whose occupations and talents our author inherited and supported, with great honour. He received his first lessons in astronomy and mathematics from MM. Maraldi and Camus. He was hardly 10 years of age when he calculated the phases of the total eclipse of the sun of 1727. At the age of 18 he accompanied his father in his two journies undertaken for drawing the perpendicular to the observatory meridian from Strasbourg to Brest. From that time a general chart of France was devised; for which purpose it was necessary to traverse the country by several lines parallel and perpendicular to the meridian of Paris, and our author was charged with the conduct of this business. He did not content himself with the measure of a degree by Picard: suspecting even that the measures which had been taken by his father and grandfather were not exempt from some errors, which the imperfections of their instruments at least would be liable to, he again undertook to measure the meridian of Paris, by means of a new series of triangles, of a smaller number, and more advantageously disposed. This great work was published in 1740, with a chart shewing the new meridian of Paris, by two different series of triangles, passing along the sea coasts, to Bayonne, traversing the frontiers of Spain to the Mediterranean and Antibes, and thence along the eastern limits of France to Dunkirk, with parallel and perpendicular lines described at the distance of 6000 toises from one another, from side to side of the country.—In <cb/> 1735, he had been received into the academy as adjoint supernumerary at 21 years of age.</p><p>A tour which our author made in Flanders, in company with the king, about 1741, gave rise to the particular chart of France, at the instance of the king. Cassini published different works relative to these charts, and a great number of the sheets of the charts themselves.</p><p>In 1761, Cassini undertook an expedition into Germany; for the purpose of continuing to Vienna the perpendicular of the Paris meridian; to unite the triangles of the chart of France with the points taken in Germany; to prepare the means of extending into this country the same plan as in France; and thus to establish successively for all Europe a most useful uniformity. Our author was at Vienna the 6th of June 1761, the day of the transit of the planet Venus over the sun, of which he observed as much as the state of the weather would permit him to do, and published the account of it in his <hi rend="italics">Voyage en Allemague.</hi></p><p>Finally, M. Cassini, always meditating the perfection of his grand design, prosited of the late peace to propose the joining of certain points taken upon the English coast with those which had been determined on the coast of France, and thus to connect the general chart of the latter with that of the British isles, like as he had before united it with those of Flanders and Germany. The proposal was favourably received by the English government, and presently carried into effect, under the direction of the Royal Society, the execution being committed to the late General Roy; after whose death the business was for some time suspended; but it has lately been revived under the auspices of the duke of Richmond, Master General of the Ordnance, and the execution committed to the care of Col. Edward Williams and Capt. William Mudge, both respectable officers of the Artillery, and Mr. Isaac Dalby, who had before accompanied and assisted General Roy; from whose united skill and zeal the happiest prosecution of this business may be expected.</p><p>M. Cassini published in the volumes of Memoirs of the French Academy a prodigious number of pieces, chiefly astronomical, too numerous to particularize in this place, between the years 1735 and 1770; consisting of astronomical observations and questions; among which are observable, Researches concerning the Parallax of the sun, the moon, Mars, and Venus; On astronomical refractions, and the effect caused in their quantity and laws by the weather; Numerous observations on the obliquity of the ecliptic, and on the law of its variations. In short, he cultivated astronomy for 50 years, of the most important for that science that ever elapsed, for the magnitude and variety of objects, in which he commonly sustained a principal share.</p><p>M. Cassini was of a very strong and vigorous constitution, which carried him through the many laborious operations in geography and astronomy which he conducted. An habitual retention of urine however rendered the last 12 years of his life very painful and distressing, till it was at length terminated by the small-pox the 4th of September 1784, in the 71st year of his age; being succeeded in the academy, and as director of the observatory, by his only son the present count JohnDominic Cassini; who is the 4th in order by direct descent in that honourable station. <pb n="252"/><cb/></p></div1><div1 part="N" n="CASSIOPEIA" org="uniform" sample="complete" type="entry"><head>CASSIOPEIA</head><p>, one of the 48 old constellations, placed near Cepheus, not far from the north pole. The Greeks probably received this figure, as they did the rest, from the Egyptians, and in their fables added it to the family in the neighbouring part of the heavens, making her the wife of Cepheus, and mother of Andromeda. They pretend she was placed in this situation, to behold the destruction of her favourite daughter Andromeda, who is chained just by her on the shore, to be devoured; and that as a punishment for her pride and vanity in presuming to stand the comparison of beauty with the Nereids.</p><p>In the year 1572 there burst out all at once in this constellation a new star, which at first surpassed Jupiter himself in magnitude and brightness; but it diminished by degrees, till it quite disappeared at the end of 18 months. This star alarmed all the astronomers of that age, many of whom wrote dissertations upon it; among the rest Tycho Brahe, Kepler, Maurolycus, Lycetus, Gramineus, and others. Beza, the Landgrave of Hesse, Rosa, and others, wrote to prove it a comet, and the same that appeared to the Magi at the birth of Christ, and that it came to declare his second coming: these were answered by Tycho.</p><p>The stars in the constellation Cassiopeia, are in Ptolomy's catalogue 13, in Hevelius's 37, in Tycho's 46, and in Flamsteed's 55.</p></div1><div1 part="N" n="CASTOR" org="uniform" sample="complete" type="entry"><head>CASTOR</head><p>, a moiety of the constellation Gemini; called also Apollo. Also a star in this constellation, whose latitude, for the year 1700, according to Hevelius, was 10° 4′ 20″ north; and its longitude <figure/> 16° 4′ 14″.</p><p>CASTOR <hi rend="italics">and</hi> <hi rend="smallcaps">Pollux.</hi> See <hi rend="smallcaps">Gemini.</hi></p><p>CASTOR <hi rend="italics">and</hi> <hi rend="smallcaps">Pollux</hi>, in Meteorology, is a fiery meteor, which at sea appears sometimes adhering to a part of the ship, in the form of a ball, or even several balls. When one is seen alone, it is properly called Helena; but two are called Castor and Pollux, and sometimes Tyndaridæ.</p><p>By the Spaniards, Castor and Pollux are called San Elmo; by the French, St. Elme, St. Nicholas, St. Clare, St. Helene; by the Italians, Hermo; and by the Dutch, Vree Vuuren.</p><p>The meteor Castor and Pollux, it is commonly thought, denotes a cessation of the storm, and a future calm; as it is rarely seen till the tempest is nigh spent. But Helena alone portends ill weather, and denotes the severest part of the storm yet behind.</p><p>When the metcor adheres to the masts, yards, &c, it is concluded, from the air not having motion enough to dissipate this flame, that a profound calm is at hand; but if it flutter about, that it denotes a storm.</p></div1><div1 part="N" n="CASTRAMETATION" org="uniform" sample="complete" type="entry"><head>CASTRAMETATION</head><p>, the art, or act, of encamping an army.</p></div1><div1 part="N" n="CATACAUSTICS" org="uniform" sample="complete" type="entry"><head>CATACAUSTICS</head><p>, or <hi rend="italics">Catacaustic Curves,</hi> in the Higher Geometry, are the species of caustic curves formed by reflection.</p><p>These curves are generated after the following manner: If there be an infinite number of rays AB, AC, AD, &c, proceeding from the radiating point A, and reflected at any given curve BCDH, so that the angles of incidence be still equal to the angles of reflection; then the curve BEG, to which the reflect- <cb/> ed rays BI, CE, DF, &c, are always tangents, as at the points I, E, F, &c, is the catacaustic, or causticby-reflection. Or it is the same thing as to say, that a caustic curve is that formed by joining the points of concurrence of the several reflected rays. <figure/></p><p>Some properties of these curves are as follow. If the reflected ray IB be produced to K, so that AB = BK, and the curve KL be the evolute of the caustic BEG, beginning at the point K; then the portion of the caustic BE is , that is, the difference of the two incident rays added to the difference of the two reflected rays.</p><p>When the given curve BCD is a geometrical one, the caustic will be so too, and will always be rectifiable. The caustic of the circle, is a cycloid, or epicycloid, formed by the revolution of a circle upon a circle. <figure/></p><p>Thus, ABD being a semicircle exposed to parallel rays; then those rays which fall near the axis CB will be reflected to F, the middle point of BC; and those which fall at A, as they touch the curve only, will not be reflected at all; but any intermediate ray HI will be reflected to a point K, somewhere between A and F. And since every different incident ray will have a different focal point, therefore those various focal points will form a curve line AEF in one quadrant, and FGD in the other, being the cycloid above-mentioned. And this figure may be beautifully exhibited experimentally by exposing the inside of a smooth bowl, or glass, to the sun beams, or strong candle light; for then this curve AEFGD will appear plainly delineated on any white surface placed horizontally within the same, or on the surface of milk contained in the bowl.</p><p>The caustic of the common cycloid, when the rays are parallel to its axis, is also a common cycloid, described by the revolution of a circle upon the same base. The caustic of the logarithmic spiral, is the same curve.</p><p>The principal writers on the caustics, are l'Hôpital, Carré, &c. See Memoires de l'Acad, an. 1666. & 1703. <pb n="253"/><cb/></p></div1><div1 part="N" n="CATACOUSTICS" org="uniform" sample="complete" type="entry"><head>CATACOUSTICS</head><p>, or <hi rend="italics">Cataphonics,</hi> is the science of reflected sounds; or that part of acoustics which treats of the properties of echoes.</p><p>CATADIOPTRICAL <hi rend="italics">Telescope,</hi> the same as Reflecting telescope; which see.</p><p>CATALOGUE <hi rend="italics">of the Stars,</hi> is a list of the fixed stars, disposed according to some order; in their several constellations; with the longitudes, latitudes, right-ascensions, &c, of each.</p><p>Catalogues of the stars have usually been disposed, either as collected into certain figures called constellations, or according to their right ascensions, that is the order of their passing over the meridian. All the catalogues, from the most ancient down to Flamsteed's inclusively, were of the first of these forms, or in constellations: but most of the others since that have been of the latter form, as being much more convenient for most purposes. Indeed one has lately been disposed in classes according to zones or degrees of polar distance.</p><p>Hipparchus of Rhodes first undertook to make a catalogue of the stars, about 128 years before Christ; in which he made use of the observations of Timocharis and Aristyllus, for about 140 years before him. Ptolomy retained Hipparchus's catalogue, containing 1026 fixed stars in 48 constellations, though he himself made abundance of observations, with a view to a new catalogue, an. dom. 140. Albategni, a Syrian, brought the same down to his own time, viz, about the year of Christ 880. Anno 1437, Ulugh Beigh, or Beg, king of Parthia and India, made a new catalogue of 1022 fixed stars, or according to some 1016; since translated out of Persian into Latin by Dr. Hyde, in 1665. The third person who made a catalogue of stars from his own observations was Tycho Brahe, who determined the places of 777 stars for the end of the year 1600; which Kepler, from other observations of Tycho, afterwards increased to the number of 1000 in the Rudolphine tables; adding those of Ptolomy and other authors, omitted by Tycho; so that his catalogue amounts to above 1160. About the same time, William, landgrave of Hesse, with his mathematicians Byrgius and Rothman, determined the places of 400 stars from new observations, rectifying them for the year 1593; which Hevelius prefers to those of Tycho. Ricciolus, in his Astronomia Reformata, determined the places of 101 stars for the year 1700, from his own observations: for the rest he followed Tycho's catalogue; altering it where he thought fit. Anno 1667 Dr. Halley, in the island of St. Helena, observed 350 of the southern stars, not visible in our horizon. The same labour was repeated by Father Noel in 1710, who published a new catalogue of the same stars constructed for the year 1687. Also De la Caille, at the Cape of Good Hope, made accurate observations of about 10 thousand stars near the south pole, in the years 1751 and 1752; the catalogue of which was published in the Memoirs of the French Academy of Sciehces for the year 1752, and in some of his own works, as more particularly noticed below.</p><p>Bayer, in his Uranometria, published in 1603 a catalogue of 1160 stars, compiled chiefly from Ptolomy and Tycho, in which every star is marked with some letter of the Greek alphabet; the brightest or princi- <cb/> pal star in any constellation being denoted by the first letter of the alphabet, the next star in order by the 2d letter, and so on; and when the number of stars exceeds the Greek alphabet, the remaining stars are marked by the letters of the Roman alphabet; which letters are preserved by Flamsteed in his catalogue, and by Senex on his globes, and indeed by most astronomers since that time.</p><p>In 1673, the celebrated John Hevelius, of Dantzick, published, in his <hi rend="italics">Machina Cœlestis,</hi> a catalogue of 1888 stars, of which 1553 were observed by himself; and their places set down for the end of the year 1660. But this catalogue, as it stands in Flamsteed's <hi rend="italics">Historia Cœlestis</hi> of 1725, contains only 1520 stars.</p><p>The most complete catalogue ever given from the labours of one man, was the Britannic catalogue, compiled from the observations of the accurate and indefatigable Mr. Flamsteed, the first Royal Astronomer at Greenwich; who for a long series of years devoted himself wholly to that business. As there was nothing wanting either in the observer or apparatus, his may be considered as a perfect work, so far as it goes. It is however to be regretted that the edition had not passed through his own hands: that now extant was published by authority, but without the author's consent, and contains 2734 stars. Another edition was published in 1725, pursuant to his testament, containing 3000 stars, with their places adapted to the beginning of the year 1689; to which is added Mr. Sharp's catalogue of the southern stars not visible in our hemisphere, set down for the year 1726. See vol. 3 of his <hi rend="italics">Historia Cœlestis,</hi> in which are printed the catalogues of Ptolomy, Ulugh Beigh, Tycho, the Prince of Hessc, and Hevelius; with an account of each of them in the Prolegomena.</p><p>The first catalogue we believe that was printed in the new or second form, according to the order of the right ascensions, is that of De la Caille, given in his Ephemerides for the 10 years between 1755 and 1765, and printed in 1755. It contains the right ascensions and declinations of 307 stars, adapted to the beginning of the year 1750.—In 1757, De la Caille published his <hi rend="italics">Astronomiæ Fundamenta,</hi> containing a catalogue of the right ascensions and declinations of 398 stars, likewise adapted to the beginning of 1750.—And in 1763, the year after his death, was published the <hi rend="italics">Cœlum Australe Stelliferum</hi> of the same author; containing a catalogue of the places of 1942 stars, all situated to the southward of the tropic of Capricorn, and observed by him while he was at the Cape of Good Hope, in 1751 and 1752; their places being also adapted to the beginning of 1750.—In the same year was published his Ephemerides for the 10 years between 1765 and 1775; in the introduction to which are given the places of 515 zodiacal stars, all deduced from the observations of the same author; the places adapted to the beginning of the year 1765.</p><p>In the Nautical Almanac for 1773, is given a catalogue of 387 stars, in right ascension, declination, longitude, and latitude, derived from the observations of the late celebrated Dr. Bradley, and adjusted to the beginning of the year 1760. This small catalogue, and the results of about 1200 observations of the moon, are all that the public have yet seen of the multiplied labours <pb n="254"/><cb/> of this most accurate and indefatigable observer, although he has now (1794) been dead upwards of 32 years.</p><p>In 1775 was published a thin volume entitled <hi rend="italics">Opera Inedita,</hi> containing several papers of the late Tobias Mayer, and among them a catalogue of the right ascensions and declinations of 998 stars, which may be occulted by the moon and planets; the places being adapted to the beginning of the year 1756.</p><p>At the end of the first volume of ‘Astronomical Observations made at the Royal Observatory at Greenwich,’ published in 1776, Dr. Maskelyne, the present Astronomer Royal, has given a catalogue of the places of 34 principal stars, in right-ascension and north-polar distance, adapted to the beginning of the year 1770. These, being the result of several years' repeated observations, made with the utmost care, and the best instruments, it may be presumed are exceedingly accurate.</p><p>In 1782, M. Bode, of Berlin, published a very extensive catalogue of 5058 of the fixed stars, collected from the observations of Flamsteed, Bradley, Hevelius, Mayer, De la Caille, Messier, Monnier, D'Arquier, and other astronomers; all adapted to the beginning of the year 1780; and accompanied with a Celestial Atlas, or set of maps of the constellations, engraved in a most delicate and beautiful manner.</p><p>To these may be added, Dr. Herschel's catalogue of double stars, printed in the Philos. Trans. for 1782 and 1783; Messier's nebulæ and clusters of stars, published in the Connoissance des Temps for 1784; and Herschel's catalogue of the same kind, given in the Philos. Trans. for 1786.</p><p>In 1789 Mr. Francis Wollaston published ‘A Specimen of a General Astronomical Catalogue, in Zones of North-polar Distance, and adapted to Jan. 1, 1790.’ These stars are collected from all the catalogues beforementioned, from that of Hevelius downwards. This work contains five distinct catalogues; viz, Dr. Maskelyne's new catalogue of 36 principal stars; a general catalogue of all the stars, in zones of north-polar distance; an index to the general catalogue; a catalogue of all the stars, in the order in which they pass the meridian; and a catalogue of zodiacal stars, in longitude and latitude.</p><p>Finally, in 1792, Dr. Zach published at Gotha, <hi rend="italics">Tabulæ Motuum Solis,</hi> to which is annexed a new catalogue of the principal fixed stars, from his own observations made in the years 1787, 1788, 1789, 1790. This catalogue contains the right ascensions and declinations of 381 principal stars, adapted to the beginning of the year 1800.</p></div1><div1 part="N" n="CATAPULT" org="uniform" sample="complete" type="entry"><head>CATAPULT</head><p>, <hi rend="italics">Catapulta,</hi> a military engine, much used by the ancients for throwing huge stones, and sometimes large darts and javelins, 12, 15, or even 18 feet long, on the enemy. It is sometimes confounded with the Ballista, which is more peculiarly adapted for throwing stones; some authors making them the same, and others different.</p><p>The catapulta, which it is said was invented by the Syrians, consisted of two huge timbers, like masts of ships, placed against each other, and bent by an engine for the purpose; these being suddenly unbent again by the stroke of a hammer, threw the javelins with prodigious force. Its structure and the manner of <cb/> working it are described by Vitruvius; and a figure of it is also given by Perrault. M. Folard asserts that the catapult made infinitely more disorder in the ranks than our cannon charged with iron balls. See Vitruv. Archit. lib. x. cap. 15 and 18; and Perr. notes on the same; also Rivius, pa. 597.</p><p>Josephus takes notice of the surprising effects of these engines, and says, that the stones thrown out of them, of a hundred weight or more, beat down the battlements, knocked the angles off the towers, and would level a whole file of men from one end to the other, were the phalanx ever so deep.</p><p>See plate V, fig. 3 and 4, for two forms of the catapult, the one for throwing darts and javelins, the other for stones.</p></div1><div1 part="N" n="CATENARY" org="uniform" sample="complete" type="entry"><head>CATENARY</head><p>, a curve line which a chain, cord, or such like, forms itself into, by hanging freely from two points of suspension, whether these be in the same horizontal line or not; as the curve ACB, formed by a heavy flexible line suspended by any two points A and B. <figure/></p><p>The nature of this curve was sought after by Galileo, who thought it was the same with the parabola; but though Jungius detected this mistake, its true nature was not discovered till the year 1691, in consequence of M. John Bernoulli having published it as a problem in the Acta Eruditorum, to the mathematicians of Europe. In 1697 Dr. David Gregory published an investigation of the properties before discovered by Bernoulli and Leibnitz; in which he pretends that an inverted catenary is the best figure for an arch of a bridge &c. See Philos. Trans. abr. vol. 1. pa. 39; also Bernoulli Opera, vol. 1. pa. 48, and vol. 3. pa. 491; and Cotes's Harmon. Mensur. pa. 108.</p><p>The catenary is a curve of the mechanical kind, and cannot be expressed by a sinite algebraical equation, in simple terms of its absciss and ordinate; but is easily expressed by means of fluxions; thus if AQ be its axis perpendicular to the horizon, and PQ an ordinate parallel to the same, or perp. to AQ; also <hi rend="italics">pq</hi> another ordinate indefinitely near the former, and <hi rend="italics">po</hi> parallel to AQ; then, <hi rend="italics">a</hi> being some given or constant quantity, the fundamental property of the curve is this, viz, P<hi rend="italics">o : op ::</hi> AP : <hi rend="italics">a,</hi> or <hi rend="italics">x<hi rend="sup">.</hi> : y<hi rend="sup">.</hi> :: z : a,</hi> that is, the fluxion of the axis, is to the fluxion of the ordinate, as the length of the curve is to the given quantity <hi rend="italics">a;</hi> where <hi rend="italics">x</hi> = AQ, <hi rend="italics">y</hi> = PQ, and <hi rend="italics">z</hi> = AP. This, and the other properties of the curve, will easily appear from the following considerations: First, supposing the curve hung up by its two points B and C against a perpendicular or upright wall: then, every lower part of the curve being kept in its position by the tension of that which is immediately above it, the lower parts of the curve will retain the same position unvaried, by whatever points it is sus- <pb n="255"/><cb/> pended above; thus, if it were fixed to the wall by the point F, or G, &c, the whole curve CAB will remain just as it was; for the tensions at F and G have the same effect upon the other parts of the curve as when it is fixed by those points: and hence it follows that the tension of the curve at the point A, in the horizontal direction, is a constant quantity, whether the two legs or branches of the curve, on both sides of it, be longer or shorter: which constant tension at A let be denoted by the quantity <hi rend="italics">a.</hi></p><p>Now because any portion of the curve, as AP, is kept in its position by three forces, viz, the tensions at its extremities A and P, and its own weight, of which the tension at A acts in the direction AH or <hi rend="italics">po,</hi> and the tension at P acts in the direction P<hi rend="italics">p,</hi> and the wt. of the line acts in the perpendicular direction <hi rend="italics">o</hi>P; that is, the three forces which retain the curve AP in its position, act in the directions of the sides of the elementary triangle <hi rend="italics">op</hi>P; but, by the principles of mechanics, any three forces, keeping a body in equilibrio, are proportional to the three sides of a triangle drawn in the directions in which those forces act; therefore it follows that the forces keeping AP in its position, viz, the tension at A, the tension at P, and the wt. of AP, are respectively as <hi rend="italics">op, p</hi>P, and <hi rend="italics">o</hi>P, that is, as <hi rend="italics">y<hi rend="sup">.</hi>, z<hi rend="sup">.</hi>,</hi> and <hi rend="italics">x<hi rend="sup">.</hi>.</hi> But the tension at A is the constant quantity <hi rend="italics">a,</hi> and the wt. of the uniform curve AP may be expounded by its length <hi rend="italics">z</hi>; therefore it follows that <hi rend="italics">x<hi rend="sup">.</hi> : y<hi rend="sup">.</hi> :: z : a</hi>; which was to be proved.</p><p>Also from this last proportion, by proper analogy, or similar combinations of the terms, there arises this other property, ; and the fluents of these give . But at the vertex of the curve, where <hi rend="italics">x</hi> = o, and <hi rend="italics">z</hi> = o, this becomes ; and therefore by correction the true equation of the fluents is : and hence also . Any of which is the equation of the curve in terms of the arch and its absciss; in which it appears that <hi rend="italics">a</hi> + <hi rend="italics">x</hi> is the hypothenuse of a right-angled triangle whose two legs are <hi rend="italics">a</hi> and <hi rend="italics">z.</hi> So that, if in QA and HA produced, there be taken AD = <hi rend="italics">a,</hi> and AE = the curve <hi rend="italics">z</hi> or AP; then will the hypothenuse DE be = <hi rend="italics">a</hi> + <hi rend="italics">x</hi> or DQ. And hence, any two of these three, <hi rend="italics">a, x, z,</hi> being given, the third is given also.</p><p>Again, from the first simple property, viz, <hi rend="italics">x<hi rend="sup">.</hi> : y<hi rend="sup">.</hi> :: z : a,</hi> or <hi rend="italics">ax<hi rend="sup">.</hi></hi> = <hi rend="italics">zy,</hi> by substituting the value of <hi rend="italics">z</hi> above found, it becomes ; and the fluent of this equation is <hi rend="italics">y</hi> = 2<hi rend="italics">a</hi> X hyp. log. of . But at the vertex of the curve, where <hi rend="italics">x</hi> = 0 and <hi rend="italics">y</hi> = 0, this becomes 0 = 2<hi rend="italics">a</hi> X hyp. log. of √2<hi rend="italics">a</hi>; therefore the correct equation of the fluents is <hi rend="italics">y</hi>=2<hi rend="italics">a</hi> X hyp. log. of <cb/> ; an equation to the curve also, in terms of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> but not in simple algebraic terms. This last equation however may be brought to much simpler terms in different ways; as first by squaring the logarithmic quantity and dividing its coef. by 2, then <hi rend="italics">y</hi> = <hi rend="italics">a</hi> X hyp. log. of hyp. log. ; and 2d by multiplying both numerator and denominator by , then squaring the product, and dividing the coef. by 2, which gives <hi rend="italics">y</hi> = <hi rend="italics">a</hi> X hyp. log. hyp. log. hyp. log. .</p></div1><div1 part="N" n="CATHETUS" org="uniform" sample="complete" type="entry"><head>CATHETUS</head><p>, in Geometry, a name by which the perpendicular leg of a right-angled triangle is sometimes called. Or it is in general any line or radius falling perpendicularly on another line, or surface.</p><p><hi rend="smallcaps">Cathetus</hi> <hi rend="italics">of Incidence;</hi> in Catoptrics, is a right line drawn from a radiant point, or point of incidence, perpendicular to the reflecting line, or plane of the speculum.</p><p><hi rend="smallcaps">Cathetus</hi> <hi rend="italics">of Reflection,</hi> or <hi rend="italics">of the Eye,</hi> a right line drawn from the eye, perpendicular to the plane of reflection.</p><div2 part="N" n="Cathetus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cathetus</hi></head><p>, in Architecture, denotes the axis of a column &c. In the Ionic Capital, it denotes a line passing perpendicularly through the eye or centre of the volute.</p></div2></div1><div1 part="N" n="CATOPTRICS" org="uniform" sample="complete" type="entry"><head>CATOPTRICS</head><p>, the science of reflex vision; or that part of opties which explains the laws and properties of light reflected srom mirrors, or specula.</p><p>The first treatise extant on catoptrics, is that which was compofed by Euclid: this was published in Latin in 1604 by John Pena; it is also contained in Herigon's Course of Mathematics, and in Gregory's edition of the works of Euclid: though it is suspected by some that this piece was not the work of that great geometrician, notwithstanding that it is ascribed to him by Proclus in lib. 2, and by Marinus in his Preface to Euclid's Data. Alhazen, an Arabian author, composed a large volume of optics about the year 1100, in which he treats pretty fully of catoptrics: and after him Vitello, a Polish writer, composed another about the year 1270. Tacquet, in his Optics, has very well demonstrated the chief propositions of plane and spherical speculums. And the same is very ably done by Dr. Barrow in his Optical Lectures. There are also Trabe's Catoptrics; David Gregory's Elements of Catoptrics; Wolfius's Elements of Catoptrics; and those of Dr. Smith, contained in his learned and very elaborate Treatise on Optics; and many others of less note.</p><p>As this subject is treated under the general term Optics, the less need be said of it here. The whole doctrine of Catoptrics depends upon this simple principle, that the angle of incidence is equal to the angle of reflection, that is, that the angle in which a ray of light falls upon any surface, called the angle of incidence, <pb n="256"/><cb/> is equal to the angle in which it quits it when reflected from it, called the angle of reflection; though it is sometimes defined that the angles of incidence and reflection, are those which the incident and reflected rays make, not with the reflecting surface itself, but with a perpendicular to that surface, at the point of contact, which are the complements to the others: but it matters not by what name these angles are called, as to the truth and principles of the science; since, if the angles are equal, their complements are also equal. This principle of the equality between the angles of incidence and reflection, is mere matter of experience, being a phenomenon that has always been observed to take place, in every case that has fallen under observation, as near at least as mechanical measurements can ascertain; and hence it is inferred that it is a universal law of nature, and to be considered as matter of fact in all cases. Thus, let AC be an incident ray falling upon the reflecting surface DE, and CB the reflected ray, also CO perpendicular to DE; then is the angle ACD = BCE, or the angle ACO = BCO. <figure/></p><p>Of this law in nature, viz, the equality between the angles of incidence and reflection, it is remarkable, that in this way, the length or rout AC + CB, in a ray passing from any point A to another given point B, by being reflected from any surface DE, is the shortest possible, namely AC + CB is shorter than the sum AG + GB of any other two lines inflected at the line DE; and hence also the passage of the ray from A to B is performed in a shorter time than if it had passed by any other way.</p><p>From this simple principle, and the common properties of lineal geometry, the chief phenomena of catoptrics are easily deduced, and are as here following. <figure/></p><p>1. Rays of light reflected from a plane surface, have the same inclination to each other after reflection as they had before it. Thus, the rays AC, AI, AK, issuing from the radiant point A, and reflected by the surface DE into the lines CF, IL, KM; these latter lines will have the same inclination to each other as the former AC, AI, AK have. For draw ABG perpendicular to DE, and produce FC, LI, MK backwards to meet this perpendicular, so shall they all meet in the same point G, and AB will in every case be equal to BD: for the incident [angle] ACB is equal to the re- <cb/> flected [angle] FCE, which is equal to the opposite angle BCG; so that the two triangles ABC, GBC, have the angles at C equal, as also the right angles at B equal, and consequently the 3d angles at A and G equal; and having also the side BC common, they are equal in all respects, and so AB = BG. And the same for the other rays. Consequently the angles BGC, BGI, BGK are respectively equal to the angles BAC, BAI, BAK; that is, the reflected rays have the same inclinations as the incident ones have.</p><p>2. Hence it is that the image of an object, seen by reflection from a plain mirror, seems to proceed from a place G as far beyond, or on the other side of the reflecting plane DE, as the object A itself is before the plane. This is when the incident rays diverge from some point as A.</p><p>But if the case be reversed, and FC, LI, MK be considered as incident rays, issuing from points F, L, M, and converging to some point G beyond the reflecting plane; then CA, IA, KA will become the reflected rays, and they will converge to the point A as far before the plane, as the point G is beyond it.</p><p>So that universally, when the incident rays diverge from a point A, the reflected rays will also diverge from a point G; and when the incident rays converge towards a point G, the reflected ones will also converge to a point A; and in both cases these two points are at equal distances on the opposite sides of the reflecting plane DE.</p><p>3. Parallel rays reflected from a concave spherical surface, converge after reflection. For, let AF, CD, EB be three parallel rays falling upon the concave surface FB, whose centre is C. To the centre draw the perpendiculars FC and BC; also draw FM making the reflected angle CFM equal to the incident angle CFA; and in like manner BM to make the angle CBM = the angle CBE; so shall the rays AF and EB be reflected into the converging rays FM and BM. As to the ray CD, being perpendicular to the surface, it is reflected back again in the same line DC. <figure/></p><p>4. Converging rays falling upon the concave surface are made to converge more. Thus, let GB and HF be the incident rays: then because the incident angle HFC is larger than the angle AFC, therefore the equal <pb n="257"/><cb/> reflected angle NFC is greater than the reflected angle MFC, and so the point N is below the point M, or the line FN below the line FM; and in like manner BN is below BM; that is, the reflected rays FN and BN are more converging in this case, than FM and BM in the other.</p><p>5. The focus to which all parallel rays, falling near the vertex D, are reflected, is in the middle of the radius M. For, because the [angle] MFC = [angle] AFC which is = the alternate [angle] FCM, therefore the sides opposite these angles are also equal, namely the side FM = CM; consequently when the point F is very near the vertex D, then the sum CM + MF is nearly = CD, and so CM nearly = MD, or the focus of the parallel rays is nearly in the middle of the radius.—But the focus of other reflected rays is either above or below that of the parallel rays; namely, below when the incident rays are converging, and above when they are diverging; as is evident by inspection; thus, N the reflected focus of the converging rays GB and HF, is below M; I that of the diverging rays YB and YF, is above M.</p><p>6. Incident and reflected rays are reciprocal, or so that if the reflected rays be returned back, or considered as incident ones, they will be reflected back into what were before their incident rays. And hence it follows that diverging rays, after reflection from a concave spherical surface, become either parallel or less diverging than before. Thus the incident rays MF and MB are reflected into the parallel rays FA and BE, and the rays NF and NB are reflected into FH and BG, which are less diverging; also the rays IF and IB are reflected into FK and BL, which converge.—And hence all the phenomena of concave mirrors will be evident.</p><p>7. Rays reflected from a convex speculum, become quite contrary to those reflected from a concave one; so that the parallel rays become diverging, and the diverging rays become still more diverging; also converging rays will become either diverging, or parallel, or else less converging. Thus BDF being a spherical surface, whose centre is C, produce the radii CBV and CFT which are perpendicular to the surface; then it is evident that the parallel rays AF and EB will be reflected into the diverging ones FK and BL; and the diverging rays YB and YF become BO and FP which are more diverging; also the converging rays HF and GB become FR and BS which diverge, or else KF and LB become FA and BE which are parallel, or else lastly PF and OB become FY and BY which are converging. <figure/> <cb/></p><p>8. Hence, as in the concave speculum, so also in the convex one, of parallel incident rays AF and EB, the imaginary focus M of their reflected rays FL and BK, is in the middle of the radius when the speculum is a small segment of a sphere: but the reflected imaginary focus of other rays is either above or below the middle point M, viz N being that of the converging rays GB and HF, below M; but I, that of the diverging rays YB and YF, above M.</p><p>9. When the speculum is the small segment of a sphere, either convex or concave, and the incident rays either converging or diverging, the distances of the foci, or points of concurrence, of the incident rays, and of the reslected rays, from the vertex of the speculum, are directly proportional to the distances of the same from the centre of it; that is YD : ID :: YC : IC, and QD : ND :: QC : NC. For because the radius CF, or the same produced, bisects the angle YFI in the concave speculum, or the external angle YFP in the convex one, therefore YF : IF :: YC : IC; but when F is very near to D, then YF and IF become nearly YD and ID; consequently YD : ID :: YC : IC.</p><p>In like manner, because CF bisects the angle QFN in the convex, or its external angle NFH in the concave speculum, therefore QF : FN :: QC : NC; but when F is very near to D, then QF and FN become nearly QD and ND; and therefore QD : ND :: QC : NC.</p><p>For example, suppose it were required to sind the focal distance of diverging rays incident upon a convex surface, the radius of the sphere being 5 inches, and the distance of the radiant point from the surface 20 inches. Here then are given YD = 20, and CD = 5, to find ID : then the theorem YD : ID :: YC : IC, in numbers is 20 : ID :: 25 : 5 - ID, or by permutation 20 : 25 :: ID : 5 - ID, and by composition 45 : 20 :: 5 : ID = 100/45=<*>0/9= 2 2/9 the focal distance sought.</p><p>And if it should happen in any case that the value of ID in the calculation should come out a negative quantity, the focal distance must then be taken on the contrary side of the surface.</p><p>From the foregoing principles may be deduced and collected the following practical maxims, for plane and spherical mirrors, viz, <hi rend="center">I. <hi rend="italics">In a Plane Mirror,</hi></hi></p><p>(1). The image will appear as far behind the mirror, as the object is before it.</p><p>(2). The image will appear of the same size, and in the same position as the object.</p><p>(3). Any plain mirror will reflect the image of an object of twice its own length and breadth. <hi rend="center">II. <hi rend="italics">In a Spherical Convex Mirror,</hi></hi></p><p>(1). The image will always appear behind the mirror, or within the sphere.</p><p>(2). The image will be in the same position, but less than the object.</p><p>(3). The image will be curved, but not spherical, like the mirror.</p><p>(4). Parallel rays falling on this mirror, will have <pb n="258"/><cb/> the image at half the distance of the centre from the mirror.</p><p>(5). In converging rays, the distance of the object must be equal to half the distance of the centre, to make the image appear behind the mirror.</p><p>(6). Diverging rays will have their image at less than half the distance of the centre. <hi rend="center">III. <hi rend="italics">In a Spherical Concave Mirror,</hi></hi></p><p>(1). Parallel rays have their focus, or the image, at half the distance of the centre.</p><p>(2). In the centre of the sphere the image appears of the same dimensions as the object.</p><p>(3). Converging rays form an image before the mirror.</p><p>(4). In diverging rays, if the object be at less than half the distance of the centre, the image will be behind the mirror, erect, curved, and magnified; but if the distance of the object be greater, the image will be before the mirror, inverted and diminished.</p><p>(5). The solar rays, being parallel, will be collected in a focus at half the distance of its centre, where their heat will be augmented in proportion as the surface of the mirror exceeds that of the focal spot.</p><p>(6). If a luminous body be placed in the focus of a concave mirror, its rays, being reflected in parallel lines, will strongly enlighten a space of the same dimensions with the mirror, at a great distance. If the luminous object be placed nearer than the focus, its rays will diverge, and so enlighten a larger space, but not so strongly. And upon this principle it is that reverberators are constructed.</p><p><hi rend="smallcaps">Catoptric</hi> <hi rend="italics">Dial,</hi> a dial that exhibits objects by neflected rays. See <hi rend="italics">Reflecting</hi> <hi rend="smallcaps">Dial.</hi></p><p><hi rend="smallcaps">Catoptric</hi> <hi rend="italics">Telescope,</hi> a telescope that exhibits objects by reflection. See <hi rend="italics">Reflecting</hi> <hi rend="smallcaps">Telescope.</hi></p><p><hi rend="smallcaps">Catoptric</hi> <hi rend="italics">Cistula,</hi> a machine, or apparatus, by which small bodies are represented extremely large, and near ones extremely wide, and diffused through a vast space; with other very pleasing phenomena, by means of mirrors, disposed by the laws of catoptrics, in the concavity of a kind of chest.</p><p>There are various kinds of these machines, accommodated to the various intentions of the artificer: some multiply the objects, some magnify, some deform them, &c. The structure of one or two of them will suffice to shew how many more may be made.</p><p><hi rend="italics">To make a Catoptric Cistula to represent several difserent scenes of objects, when viewed by different holes.</hi> <figure/> <cb/></p><p>Provide a polygonal cistula, or box, like the multangular prism ABCDEF, and divide its cavity by diagonal planes AD, BE, CF, intersecting in the centre, into as many triangular cells as the chest has sides. Line those diagonal partitions with plain mirrors; and in the sides of the box make round holes, through which the eye may peep within the cells of it. These holes are to be covered with plain glasses, ground within-side, but not polished, to prevent the objects in the cells from appearing too distinctly. In each cell are to be placed the different objects whose images are to be exhibited; then covering up the top of the chest with a thin transparent membrane, or parchment, to admit the light, the machine is complete.</p><p>For, from the laws of reflection, it follows, that the images of objects, placed within the angles of mirrors, are multiplied, and appear some more remote than others; by which the objects in one cell will appear to take up more room than is contained in the whole box. Therefore by looking through one hole only, the objects in one cell will be seen, but those multiplied, and diffused through a space much larger than the whole box. Thus every hole will afford a new scene; and according to the different angles the mirrors make with each other, the representations will be different: if they be at an angle greater than a right one, the images will be monstrous, &c.</p><p><hi rend="italics">To make a Catoptric Cistula to represent the objects within it prodigiously multiplied, and diffused through a vast space.</hi></p><p>Make a polygonous cistula or box, as before, but without dividing the inner cavity into any apartments, or cells; line the insides CBHI, BHLA, ALMF, &c, with plane mirrors, and at the holes pare off the tin and quicksilver, to look through; place any object in the bottom MI, as a bird in a cage, &c.</p><p>Now by looking through the aperture <hi rend="italics">hi,</hi> each object placed at the bottom will be seen vastly multiplied, and the images removed at equal distances from one another, like a great multitude of birds, or a large aviary.</p></div1><div1 part="N" n="CAVALIER" org="uniform" sample="complete" type="entry"><head>CAVALIER</head><p>, in Fortification, a mount of earth raised in a fortress higher than the other works, on which to place cannon &c for scouring the field, or opposing a commanding work. Cavaliers are of different shapes; and are bordered with a parapet, to cover the cannon mounted upon them; their situation is also various, either in the curtain, bastion, or gorge. The cavalier is sometimes called a double bastion, and its use is to overlook the enemy's batteries, and to scour their trenches.</p></div1><div1 part="N" n="CAVALIERI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CAVALIERI</surname> (<foreName full="yes"><hi rend="smallcaps">Bonaventura</hi></foreName>)</persName></head><p>, an eminent Italian mathematician in the 17th century. He was a native of Milan, and a friar of the order of the Jesuati of St. Jerome. Cavalieri was a disciple of Galileo, and the friend of Torricelli. He was a very eminent mathematician, and was professor of that science at Bologna; where several of his books were published, and where he died in the year 1647. His works that have been published, as far as I can find, are as follow:</p><p>1. <hi rend="italics">Directorium Generale Uranometricum;</hi> 4<hi rend="italics">to, Bononiæ,</hi> 1632.—In this work the author treats of Trigonome try; and Logarithms, their construction, uses, and applications. The work includes also tables of logarithms <pb n="259"/><cb/> of common numbers; with trigonometrical tables, of natural sines, and logarithmic sines, tangents, secants and versed sines.</p><p>2. <hi rend="italics">Lo Spechio Ustorio overo Trattato delle Settioni Coniche:</hi> 4<hi rend="italics">to, Bologna,</hi> 1632.—An ingenious treatise of conic sections.</p><p>3. <hi rend="italics">Geometria Indivisibilibus continuorum nova quadam ratione promota:</hi> 4<hi rend="italics">to, Bononiæ,</hi> 1635; and a 2d edition in 1653.—This is a curious original work in geometry, in which the author conceives the geometrical figures as resolved into their very small elements, or as made up of an infinite number of infinitely small parts, and on account of which he passes in Italy for the inventor of the infinitesimal calculns.</p><p>4. <hi rend="italics">Trigonometria Plana & Sphærica, Linearis, & Logarithmica:</hi> 4<hi rend="italics">to, Bononiæ,</hi> 1643.—A very neat and ingenioun. treatise on Trigonometry; with the tables of sines, tangents, and secants, both natural and logarithmical.</p><p>5. <hi rend="italics">Exercitationes Geometricæ Sex:</hi> 4<hi rend="italics">to, Bononiæ,</hi> 1647. This work contains Exercises on the method of Indivisibles; Answers to the objections of Guldini; The use of Indivisibles in cossic powers or algebra, and in considerations about gravity; with a miscellaneous collection of problems.</p><p>CAUDA <hi rend="italics">Capricorni,</hi> a fixed star of the 4th magnitude, in the tail of Capricorn; called also by the Arabs, Dineb Algedi; and marked <foreign xml:lang="greek">g</foreign> by Bayer.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Ceti,</hi> a fixed star of the 3d magnitude; called also by the Arabs, Dineb Kaetos; marked <foreign xml:lang="greek">b</foreign> by Bayer.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Cygni,</hi> a fixed star of the 2d magnitude in the Swan's tail; called by the Arabs, Dineb Adigege, or Eldegiagich; and marked <hi rend="italics">a</hi> by Bayer.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Delphini,</hi> a fixed star of the 3d magnitude, in the tail of the Dolphin; marked <foreign xml:lang="greek">e</foreign> by Bayer.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Draconis,</hi> or Dragon's tail, the moon's southern or descending node.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Leonis,</hi> a fixed star of the sirst magnitude in the Lion's tail; called also by the Arabs, Dineb Eleced; and marked <foreign xml:lang="greek">b</foreign> by Bayer. It is called also Lucida Cauda.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Ursæ Majoris,</hi> a sixed star of the 3d magnitude, in the tip of the Great Bear's tail; called also by the Arabs, Alalioth, and Benenath; and marked <foreign xml:lang="greek">h</foreign> by Bayer.</p><p><hi rend="smallcaps">Cauda</hi> <hi rend="italics">Ursæ Minoris,</hi> a fixed star of the 3d magnitude, at the end of the Lesser Bear's tail; called also the Pole Star, and by the Arabs, Alrukabah; and marked <foreign xml:lang="greek">a</foreign> by Bayer.</p></div1><div1 part="N" n="CAVETTO" org="uniform" sample="complete" type="entry"><head>CAVETTO</head><p>, a hollow member or moulding, containing a quadrant of a circle, and having an effect just contrary to that of a quarter round. It is used as an ornament in cornices.</p><p>CAUSTIC <hi rend="smallcaps">Curves.</hi> See <hi rend="italics">Catacaustics,</hi> and <hi rend="italics">Diacaustics.</hi></p><p>CAZEMATE. See <hi rend="smallcaps">Casemate.</hi></p><p>CAZERN. See <hi rend="smallcaps">Casern.</hi></p></div1><div1 part="N" n="CEGINUS" org="uniform" sample="complete" type="entry"><head>CEGINUS</head><p>, a sixed star of the 3d magnitude, in the left shoulder of Bootes; and marked <foreign xml:lang="greek">g</foreign> by Bayer.</p></div1><div1 part="N" n="CELERITY" org="uniform" sample="complete" type="entry"><head>CELERITY</head><p>, is the velocity or swiftness of a body in motion; or that affection of a body in motion by which it can pass over a certain space in a certain time. <cb/></p><p>CELESTIAL <hi rend="smallcaps">Globe</hi>, &c. See <hi rend="smallcaps">Globe</hi>, &c.</p></div1><div1 part="N" n="CELLARIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CELLARIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Christopher</hi></foreName>)</persName></head><p>, a learned geographer and historiographer of the 17th century. He was born in 1638, at Smalcalde in Franconia, where his father was minister. Our author was successively rector of the colleges at Weymar, Zeits, and Mersbourg, and professor of eloquence and history in the university founded by the king of Prussia at Hall in 1693, where he composed the greatest part of his works.</p><p>His great application to study hastened the infirmities of old age; for it has been said, he would spend whole days and nights together over his books, without any attention to his health, or even the calls of nature. He died in 1707, at 69 years of age.</p><p>Cellar was author of an amazing number of books, upon various subjects; but those on account of which he has a place here, are his geographical works, which are as follow:</p><p>1. <hi rend="italics">Notitia Orbis Antiqui,</hi> 2 vols in 4to; and is esteemed the best work extant on the ancient geography.</p><p>2. <hi rend="italics">Atlas Cœlestis;</hi> in folio.</p><p>3. <hi rend="italics">Historia Antiqua,</hi> 2 vols in 12mo; being an abridgement of universal history.</p></div1><div1 part="N" n="CENTAURUS" org="uniform" sample="complete" type="entry"><head>CENTAURUS</head><p>, the <hi rend="italics">Centaur,</hi> one of the 48 old constellations, being a southern one, and is in form half man and half horse. It is fabled by the Greeks that it was Chiron the Centaur, who was the tutor of Achilles and Esculapius. The stars of this constellation are, in Ptolomy's catalogue 37, in Tycho's 4, and in the Britannic catalogue, with Sharp's appendix, 35.</p><p>CENTER. See <hi rend="smallcaps">Centre.</hi></p></div1><div1 part="N" n="CENTESM" org="uniform" sample="complete" type="entry"><head>CENTESM</head><p>, the 100th part of any thing.</p></div1><div1 part="N" n="CENTRAL" org="uniform" sample="complete" type="entry"><head>CENTRAL</head><p>, something relating to a centre. Thus we say central eclipse, central forces, central rule, &c.</p><p><hi rend="smallcaps">Central</hi> <hi rend="italics">Eclipse,</hi> is when the centres of the luminaries exactly coincide, and come in a line with the eye.</p><p><hi rend="smallcaps">Central</hi> <hi rend="italics">Forces,</hi> are forces having a tendency directly towards or from some point or centre; or forces which cause a moving body to tend towards, or recede from, the centre of motion. And accordingly they are divided into two kinds, in respect to their different relations to the centre, and hence are called centripetal, and centrifugal.</p><p>The doctrine of central forces makes a considerable branch of the Newtonian philosophy, and has been greatly cultivated by mathematicians, on account of its extensive use in the theory of gravity, and other physico-mathematical sciences.</p><p>In this doctrine, it is supposed that matter is equally indifferent to motion or rest; or that a body at rest never moves itself, and that a body in motion never of itself changes either the velocity or the direction of its motion; but that every motion would continue uniformly, and its direction rectilinear, unless some external force or resistance should affect it, or act upon it. Hence, when a body at rest always tends to move, or when the velocity of any rectilinear motion is continnally accelerated or retarded, or when the direction of a motion is continually changed, and a curve line is thereby described, it is supposed that these circumstances proceed from the influence of some power that acts incessantly; which power may be measured, in the first case, by the pressure of the quiescent body against the obstacle which prevents it from moving; or by the ve- <pb n="260"/><cb/> locity gained or lost in the second case, or by the flexure of the curve described in the 3d ease: due regard being had to the time in which these effects are produced, and other circumstances, according to the principles of mechanics. Now the power or force of gravity produces effects of each of these kinds, which fall under our constant observation near the surface of the earth; for the same power which renders bodies heavy, while they are at rest, accelerates their motion when they descend perpendicularly; and bends the track of the motion into a curve line, when they are projected in a direction oblique to that of their gravity. But we can judge of the forces or powers that act on the celestial bodies by effects of the last kind only. And hence it is, that the doctrine of central forces is of so much use in the theory of the planetary motions.</p><p>Sir I. Newton has treated of central forces in lib. 1. sec. 2 of his Principia, and has demonstrated this fundamental theorem of central forces, viz, that the areas which revolving bodies describe by radii drawn to an immoveable centre, lie in the same immoveable planes, and are proportional to the times in which they are described. Prop. 1.</p><p>It is remarked by a late eminent mathematician, that this law, which was originally observed by Kepler, is the only general principle in the doctrine of centripetal forces; but since this law, as Newton himself has proved, cannot hold in cases where a body has a tendency to any other than one and the same point, there seems to be wanting some law that may serve to explain the motions of the moon and satellites which gravitate towards two different centres: the law he lays down for this purpose is, That when a body is urged by two forces tending constantly to two fixed points, it will describe, by lines drawn from the two fixed points, equal solids in equal times, about the line joining those fixed points, See Machin, on the Laws of the Moon's Motion, in the Postscript. See also a demonstration of this law by Mr. William Jones, in the Philos. Trans. vol. 59. Very learned tracts have also been since given, when the motion respects, not two only, but several centres, by many ingenious authors, and practical rules deduced from them for computing the places &c of planets and satellites; as by La Grange, De la Place, Waring, &c, &c. See Berlin Memoirs; those of the Academy of Sciences at Paris; and the Philof. Trans. of London.</p><p>M. De Moivre gave elegant general theorems relating to central forces, in the Philos. Trans. and in his Miscel. Analyt. pa. 231.—Let MPQ be any given curve, in which a body moves: let P be the place of the body at any time; S the centre of force, or the point to which the central force acting on the body is always directed; PG the radius of curvature at the point P; and ST perpendicular to the tangent PT; then will the centripetal force be everywhere proportional to the quantity SP/(GP X ST<hi rend="sup">3</hi>). Vid. ut supra. <figure/> <cb/></p><p>M. Varignon has also given two general theorems on this subject in the Memoirs of the Acad. an. 1700, 1701; and has shewn their application to the motions of the planets. See also the same Memoirs, an. 1706, 1710.</p><p>Mr. MacLaurin has also treated the subject of central forces very ably and fully, in his Treatise on Fluxions, art. 416 to 493; where he gives a great variety of expressions for these forces, and several elegant methods of investigating them.</p><p><hi rend="italics">Laws of</hi> <hi rend="smallcaps">Central Forces.</hi></p><p>1. The following is a very clear and comprehensive rule, for which we are obliged to the marquis de l'Hôpital: Suppose a body of any determinate weight to revolve uniformly about a centre, with any given velocity; find from what height it must have fallen, by the force of gravity, to acquire that velocity; then, as the radius of the circle it describes is to double that height, so is its weight to its centrifugal force. So that, if <hi rend="italics">b</hi> be the body, or its weight or quantity of matter, <hi rend="italics">v</hi> its velocity, and <hi rend="italics">r</hi> the radius of the circle described, also <hi rend="italics">g</hi> = 16 <*>/12 feet; then, first 4<hi rend="italics">g</hi><hi rend="sup">2</hi> : <hi rend="italics">v</hi><hi rend="sup">2</hi> :: <hi rend="italics">g</hi> : <hi rend="italics">v</hi><hi rend="sup">2</hi>/4<hi rend="italics">g</hi> the height due to the velocity <hi rend="italics">v</hi>; and as the centrifugal force. And hence, if the centrifugal force be equal to the gravity, the velocity is equal to that acquired by falling through half the radius.</p><p>2. The central force of a body moving in the periphery of a circle, is as the versed sine AM of the indefinitely small arc AE; or it is as the square of that arc AE directly, and as the diameter AB inversely. For AM is the space through which the body is drawn from the tangent in the given time, and 2AM is the proper measure of the central force. But, AE being very small, and therefore nearly equal to its chord, by the nature of the circle <figure/></p><p>3. If two bodies revolve uniformly in different circles; their central forces are in the duplicate ratio of their velocities directly, and the diameters or radii of the circles inversely; that is F : <hi rend="italics">f</hi> :: V<hi rend="sup">2</hi>/D : <hi rend="italics">v</hi><hi rend="sup">2</hi>/<hi rend="italics">d</hi> :: V<hi rend="sup">2</hi>/R : <hi rend="italics">v</hi><hi rend="sup">2</hi>/<hi rend="italics">r</hi> For the force, by the last article, is as <pb n="261"/><cb/> AE<hi rend="sup">2</hi>/AB or AE<hi rend="sup">2</hi>/D; and the velocity <hi rend="italics">v</hi> is as the space AE uniformly described.</p><p>4. And hence, if the radii or diameters be reciprocally in the duplicate ratio of the velocities, the central forces will be reciprocally in the duplicate ratio of the radii, or directly as the 4th power of the velocities; that is, if V<hi rend="sup">2</hi> : <hi rend="italics">v</hi><hi rend="sup">2</hi> :: <hi rend="italics">r</hi> : R, then F : <hi rend="italics">f :: r</hi><hi rend="sup">2</hi> : R<hi rend="sup">2</hi> :: V<hi rend="sup">4</hi> : <hi rend="italics">v</hi><hi rend="sup">4</hi>.</p><p>5. The central forces are as the diameters of the circles directly, and squares of the periodic times inversely. For if <hi rend="italics">c</hi> be the circumference described in the time <hi rend="italics">t,</hi> with the velocity <hi rend="italics">v</hi>; then the space <hi rend="italics">c</hi> = <hi rend="italics">tv,</hi> or <hi rend="italics">v</hi> = <hi rend="italics">c/t</hi>; hence, using this value of <hi rend="italics">v</hi> in the 3d rule, it becomes F : <hi rend="italics">f</hi> :: C<hi rend="sup">2</hi>/DF<hi rend="sup">2</hi> : <hi rend="italics">c</hi><hi rend="sup">2</hi>/<hi rend="italics">dt</hi><hi rend="sup">2</hi> :: D/T<hi rend="sup">2</hi> : <hi rend="italics">d/t</hi><hi rend="sup">2</hi> :: R/T<hi rend="sup">2</hi> : <hi rend="italics">r/t</hi><hi rend="sup">2</hi>; since the diameter is as the circumference.</p><p>6. If two bodies, revolving in different circles, be acted on by the same central force; the periodic times are in the subduplicate ratio of the diameters or radii of the circles; for when F = <hi rend="italics">f,</hi> then D/T<hi rend="sup">2</hi> = <hi rend="italics">d/t</hi><hi rend="sup">2</hi>, and D : <hi rend="italics">d</hi> :: T<hi rend="sup">2</hi> : <hi rend="italics">t</hi><hi rend="sup">2</hi>, or T : <hi rend="italics">t</hi> :: √D : √<hi rend="italics">d</hi> :: √R : √<hi rend="italics">r.</hi></p><p>7. If the velocities be reciprocally as the distances from the centre, the central forces will be reciprocally as the cubes of the same distances, or directly as the cubes of the velocities. That is, if V : <hi rend="italics">v :: r</hi> : R, then is F : <hi rend="italics">f :: r</hi><hi rend="sup">3</hi> : R<hi rend="sup">3</hi> :: V<hi rend="sup">3</hi> : <hi rend="italics">v</hi><hi rend="sup">3</hi>.</p><p>8. If the velocities be reciprocally in the subduplicate ratio of the central distances, the squares of the times will be as the cubes of the distances: for if V<hi rend="sup">2</hi> : <hi rend="italics">v</hi><hi rend="sup">2</hi> :: <hi rend="italics">r</hi> : R, then is T<hi rend="sup">2</hi> : <hi rend="italics">t</hi><hi rend="sup">2</hi> :: R<hi rend="sup">3</hi> : <hi rend="italics">r</hi><hi rend="sup">3</hi>.</p><p>9. Wherefore, if the forces be reciprocally as the squares of the central distances, the squares of the periodic times will be as the cubes of the distances; or when F : <hi rend="italics">f :: r</hi><hi rend="sup">2</hi> : R<hi rend="sup">2</hi>, then is T<hi rend="sup">2</hi> : <hi rend="italics">t</hi><hi rend="sup">2</hi> :: R<hi rend="sup">3</hi> : <hi rend="italics">r</hi><hi rend="sup">3</hi>.</p><p><hi rend="italics">Exam.</hi> From this, and some of the foregoing theorems, may be deduced the velocity and periodic time of a body revolving in a circle, at any given distance from the earth's centre, by means of its own gravity. Put <hi rend="italics">g</hi> = 16 1/12 feet, the space described by gravity, at the surface, in the first second of time, viz = AM in the foregoing fig. and by rule 2; then, putting <hi rend="italics">r</hi> = the radius AC; it is the velocity in a circle at its surface, in one second of time; and hence, putting <hi rend="italics">c</hi> = 3.14159 &c, the circumference of the earth being 2<hi rend="italics">cr</hi> = 25,000 miles, or 132,000,000. feet, it will be √(2<hi rend="italics">gr</hi>) : 2<hi rend="italics">cr</hi> :: 1″ : <hi rend="italics">c</hi>√ 2<hi rend="italics">r/g</hi> = 5078 seconds nearly, or 1<hi rend="sup">h</hi> 24<hi rend="sup">m</hi> 38, the periodic time at the circumference: Also the velocity there, or √(2<hi rend="italics">gr</hi>) is = 26000 feet per second nearly. Then, since the force of gravity varies in the inverse duplicate ratio of the distance, by rules 8 and 9, it is √R : √<hi rend="italics">r :: v</hi> or the velocity of a body revolving about the earth at the distance R; and √<hi rend="italics">r</hi><hi rend="sup">3</hi> : √R<hi rend="sup">3</hi> :: <hi rend="italics">t</hi> or 5078″ : 5078 √(R<hi rend="sup">3</hi>/<hi rend="italics">r</hi><hi rend="sup">3</hi>) = T the time of revolution in the same. So if, for instance, it be the moon <cb/> revolving about the earth at the distance of 60 semidiameters; then R = 60<hi rend="italics">r,</hi> and the above expressions become V = 26000√(1/60) = 3357 feet per second, or 38 1/7 miles per minute, for the velocity of the moon in her orbit; and T = 5078√(R<hi rend="sup">3</hi>/<hi rend="italics">r</hi><hi rend="sup">3</hi>) = 2360051 seconds or 27 3/10 days nearly, for the periodic time of the moon in her orbit at that distance.</p><p>Thus also the ratio of the forces of gravitation of the moon towards the sun and earth may be estimated. For, 1 year or 365 1/4 days being the periodic time of the earth and moon about the sun, and 27 3/10 days the periodic time of the moon about the earth, also 60 being the distance of the moon from the earth, and 23920 the distance from the sun, in semidiameters of the earth, by art. 5 it is ; that is, the proportion of the moon's gravitation towards the sun, is to that towards the earth, as 2 2/9 to 1 nearly.</p><p>Again, we may hence compute the centrifugal force of a body at the equator, arising from the earth's rotation. For, the periodic time when the centrifugal force is equal to the force of gravity, it has been shewn above, is 5078 seconds, and 23 hours, 56 minutes, or 86160 seconds, is the period of the earth's rotation on its axis; therefore, by art. 5, as 86160<hi rend="sup">2</hi> : 5078<hi rend="sup">2</hi> :: 1 : 1/289, the centrifugal force required, which therefore is the 289th part of gravity at the earth's surface. Simpson's Flux. pa. 240, &c.</p><p>Also for another example, suppose A to be a ball of 1 ounce, which is whirled about the centre C, so as to describe the circle ABE, each revolution being made in half a second; and the length of the cord AC equal to 2 feet. Here then <hi rend="italics">t</hi> = 1/2, <hi rend="italics">r</hi> = 2, and it having been sound above that <hi rend="italics">c</hi>√(2R/<hi rend="italics">g</hi>) = T is the periodic time at the circumference of the earth when the centrifugal force is equal to gravity; hence then, by art. 5, as R/T<hi rend="sup">2</hi> : <hi rend="italics">r/t</hi><hi rend="sup">2</hi> :: F or 1 : <hi rend="italics">f,</hi> which proportion becomes = the centrifugal force, or that by which the string is stretched, viz, nearly to ounces, or 10 times the weight of the ball.</p><p>Lastly, suppose the string and ball be suspended from a point D, and describes in its motion a conical surface ADB; then putting DC = <hi rend="italics">a,</hi> AC = <hi rend="italics">r,</hi> and AD = <hi rend="italics">h</hi>; and putting F = 1 the force of gravity as before; then will the body A be affected by three forces, viz, gravity acting parallel to DC, a centrifugal force in the direction CA, and the tension of the string, or force by which it is stretched, in the direction DA; hence these three powers will be as the three sides of the triangle ADC respectively, and therefo re as CD or <hi rend="italics">a</hi> : AD or <hi rend="italics">h</hi> :: 1 : <hi rend="italics">h/a</hi> the tension of the string as compared with the weight of the body. Also AC or <hi rend="italics">a</hi> : AC or <hi rend="italics">r</hi> :: 1 : (2<hi rend="italics">c</hi><hi rend="sup">2</hi><hi rend="italics">r</hi>)/(<hi rend="italics">gt</hi><hi rend="sup">2</hi>) the general expression for the centrifugal force above-found; <pb n="262"/><cb/> hence, <hi rend="italics">gt</hi><hi rend="sup">2</hi> = 2<hi rend="italics">ac</hi><hi rend="sup">2</hi> and so <hi rend="italics">t</hi> = <hi rend="italics">c</hi>√(2<hi rend="italics">a</hi>)/<hi rend="italics">g</hi> = 1.108√<hi rend="italics">a</hi> = the periodic time. And <figure/></p><p>10. When the force by which a body is urged towards a point is not always the same, but is either increased or decreased as some power of the distance; several curves will thence arise according to that power. If the force decrease as the squares of the distances increase, the body will describe an ellipsis, and the force is directed towards one of its foci; so that in every revolution the body once approaches towards it, and once recedes from it; also the eccentricity of the ellipse is greater or less, according to the projectile force; and the curve may sometimes become a circle, when the eccentricity is nothing; the body may also describe the other two conic sections, the parabola and hyperbola, which do not return into themselves, by supposing the velocity greater in certain proportions. Also if the force increase in the simple ratio as the distance increases, the body will still describe an ellipse; but the force will in this case be directed to the centre of the ellipse; and the body, in each revolution, will twice approach towards it, and again twice recede from that point.</p><p><hi rend="smallcaps">Central Rule</hi>, is a rule or method discovered by Mr. Thomas Baker, rector of Nympton in Devonshire, which he published in his Geometrical Key, in the year 1684, for determining the centre of a circle which shall cut a given parabola in as many points as a given equation, to be constructed, has real roots; which he has applied with good success in the construction of all equations as far as the 4th power inclusive.</p><p>The <hi rend="italics">Central Rule</hi> is chiefly founded on this property of the parabola; that if a line be inscribed in the curve perpendicular to any diameter, the rectangle of the segments of this line, is equal to the rectangle of the intercepted part of the diameter and the parameter of the axis.</p><p>The Central Rule has the advantage over the methods of constructing equations by Des Cartes and De Latteres, which are liable to the trouble of preparing the equations by taking away the second term; whereas Baker's method effects the same thing without any previous preparation whatever. See also Philos. Trans. N° 157.</p></div1><div1 part="N" n="CENTRE" org="uniform" sample="complete" type="entry"><head>CENTRE</head><p>, or <hi rend="smallcaps">Center</hi>, in a general sense, signifies a point equally remote from the extremes of a line, plane, or solid; or a middle point dividing them so that some certain effects are equal on all fides of it.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Astraction,</hi> or <hi rend="italics">Gravitation,</hi> is the point to which bodies tend by gravity; or that point to which <cb/> a revolving planet or comet is impelled or attracted, by the force or impetus of gravity.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Bastion,</hi> is a point in the middle of the gorge, where the capital line commences, and which is usually at the angle of the inner polygon of the figure. Or it is the point where the two adjacent curtains produced intersect each other.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Circle,</hi> is the point in the middle of a circle, or circular figure, from which all lines drawn to the circumfereuce are equal.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Conic Section,</hi> is the middle point of any diameter, or the point in which all the diameters intersect and bisect one another.</p><p>In the ellipse the centre is within the figure; but in the hyperbola it is without, or between the conjugate hyperbolas; and in the parabola it is at an infinite distance from the vertex.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Conversion,</hi> in Mechanics, a term first used by M. Parent, and may be thus conceived: Suppose a stick laid on stagnant water, and then drawn by a thread sastened to it, so that the thread always makes the same angle with the stick, either a right angle or any other; then it will be found that the stick will turn about one point of it, which will be immoveable; and this point is termed the centre of conversion.</p><p>This effect arises from the resistance of the fluid to the stick partly immersed in it. And if, instead of the body thus floating on a fluid, the same be conceived to be laid on the surface of another body; then the resistance of this plane to the stick will always have the same effect, and will determine the same centre of conversion. And this resistance is precisely what is called friction, so prejudicial to the effects of machines.</p><p>M. Parent has determined this centre in some certain cases, with much laborious calculation. When the thread is sastened to the extremity of the stick, he found that the distance of the centre from this extremity would be nearly 13/20 of the whole length. But when it is a surface or a solid, there will be some change in the place of this centre, according to the nature of the figure. See Mem. of the Acad. of Sciences, vol. 1, pa. 191.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Curve,</hi> of the higher kind, is the point where two diameters meet.—When all the diameters meet in the same point, it is called, by Sir Isaac Newton, the <hi rend="italics">general centre.</hi></p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Dial,</hi> is the point where its gnomon or stile, which is placed parallel to the axis of the earth, meets the plane of the dial; and from hence all the hour-lines are drawn, in such dials as have centres, viz, all except that whose plane is parallel to the axis of the world; all the hour-lines of which are parallel to the stile, and to one another, the centre being as it were at an infinite distance.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of an Ellipse,</hi> is the middle of any diameter, or the point where all the diameters intersect.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of the Equant,</hi> in the Old Astronomy, is a point in the line of the aphelion; being as far distant from the centre os the eccentric, towards the aphelion, as the sun is from the same centre of the eccentric towards the perihelion.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Equilibrium,</hi> is the same with respect to bodies immersed in a fluid, as the centre of gravity is to bodies in free space; being a certain point, upon <pb n="263"/><cb/> which if the body or bodies be suspended, they will rest in any position. To determine this centre, see Emmerson's Mechanics, prop. 92, pa. 134.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Friction,</hi> is that point in the base of a body on which it revolves, into which if the whole surface of the base, and the mass of the body were collected, and made to revolve about the centre of the base of the given body, the angular velocity destroyed by its friction would be equal to the angular velocity destroyed in the given body by its friction in the same time.—See Vince on the Motion of Bodies assected by friction, in the Philos. Trans. 1785.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Gravity,</hi> is that point about which all the parts of a body do in any situation exactly balance each other. Hence, by means of this property, if the body be supported or suspended by this point, the body will rest in any position into which it is put; as also that if a plane pass through the same point, the segments on each side will equiponderate, neither of them being able to move the other.</p><p>The whole gravity, or the whole matter, of a body may be conceived united in the centre of gravity; and in demonstrations it is usual to conceive all the matter as really collected in that point.</p><p>Through the centre of gravity passes a right line, called the <hi rend="italics">diameter of gravity;</hi> and therefore the intersection of two such diameters determines the centre. Also the plane upon which the centre of gravity is placed, is called the <hi rend="italics">plane of gravity;</hi> so that the common intersection of two such planes determines the diameter of gravity.</p><p>In homogeneal bodies, which may be divided lengthways into similar and equal parts, the centre of gravity is the same with the centre of magnitude. Hence therefore the centre of gravity of a line is in the middle point of it, or that point which bisects the line. Also the centre of gravity of a parallelogram, or cylinder, or any prisin whatever, is in the middle point of the axis. And the centre of gravity of a circle or any regular sigure, is the same as the centre of magnitude.</p><p>Also, if a line can be so drawn as to divide a plane into equal and similar parts, that line will be a diameter of gravity, or will pass through the centre of gravity; and it is the same as the axis of the plane. Thus the line drawn from the vertex and perpendicular to the base of the isosceles triangle, is a diameter of gravity; and thus also the axis of an ellipse, or a parabola, &c, is a diameter of gravity. The centre of gravity of a segment or arc of a circle, is in the radius or line perpendicularly bisecting its chord or base.</p><p>Likewise, if a plane divide a solid in the same manner, making the parts on both sides of it perfectly equal and similar in all respects, it will be a plane of gravity, or will pass through the centre of gravity. Thus, as the intersection of two such planes determines the diameter of gravity, the centre of gravity of a right cone, or spherical segment, or conoid, &c, will be in the axis of the same.</p><p><hi rend="italics">Common Centre of Gravity</hi> of two or more bodies, or the different parts of the same body, is such a point as that, if it be suspended or supported, the system of bodies will equiponderate, and rest in any position. Thus, the point of suspension in a common balance beam, or sseelyard, is the centre of gravity of the same. <cb/> <hi rend="center"><hi rend="italics">Laws and Determination of the Centre of Gravity.</hi></hi></p><p>1. In two equal bodies, or masses, the centre of gravity is equally distant from their two respective centres. For these are as two equal weights suspended at equal distances from the point of suspension; in which case they will equiponderate, and rest in any position. <figure/></p><p>2. If the centres of gravity of two bodies A and B be connected by the right line AB, the distances AC and BC from the common centre of gravity C, are reciprocally as the weights or bodies A and B; that is, AC : BC :: B : A.</p><p>See this demonstrated under the article <hi rend="smallcaps">Balance.</hi></p><p>Hence, if the weights of the bodies A and B be equal, their common centre of gravity C will be in the middle of the right line AB, as in the foregoing article. Also since A : B :: BC : AC, therefore ; whence it appears that the powers of equiponderating bodies are to be estimated by the product of the mass multiplied by the distance from the centre of gravity; which product is usually called the <hi rend="italics">momentum</hi> of the weights.</p><p>Further, from the foregoing proportion, by composition it will be A + B : A :: AB : BC, or A + B : B :: AB : AC. So that the common centre of gravity C of two bodies will be found, if the product of one weight by the whole distance between the two, be divided by the sum of the two weights. Suppose, for example, that A = 12 pounds, B = 4lb, and AB = 36 inches; then 16 : 12 :: 36 : 27 = BC, and consequently AC = 9, the two distances from the common centre of gravity. <figure/></p><p>3. <hi rend="italics">The Common Centre of Gravity of three or more given bodies or points</hi> A, B, C, D, &c, will be thus determined.—If the given bodies lie all in the same straight line AD; by the last article, find P the centre of gravity of the two A and B, and Q the centre of gravity of C and D; then, considering P as the place of a body equal to the sum of A and B, and Q as the place of another body equal to both C and D, sind S the common centre of gravity of these two sums, viz A + B collected in P, and C + D united in Q; so shall S be the common centre of gravity of all the four bodies A, B, C, D. And the same for any other number of bodies, always considering the sum of any number of them as united or placed in their common centre of gravity, when found.</p><p><hi rend="italics">Otherwise, thus.</hi> Take the distances of the given bodies from some fixed point as V, calling the distance VA = <hi rend="italics">a,</hi> VB = <hi rend="italics">b,</hi> VC = <hi rend="italics">c,</hi> VD = <hi rend="italics">d,</hi> and the distance of the centre of gravity VS = <hi rend="italics">x</hi>; then SA = <hi rend="italics">x</hi>-<hi rend="italics">a,</hi> SB = <hi rend="italics">x</hi>-<hi rend="italics">b,</hi> SC = <hi rend="italics">c</hi>-<hi rend="italics">x,</hi> SD = <hi rend="italics">d</hi>-<hi rend="italics">x,</hi> and by the nature of the lever A.―(<hi rend="italics">x</hi>-<hi rend="italics">a</hi>) + B.―(<hi rend="italics">x</hi>-<hi rend="italics">b</hi>) = C.―(<hi rend="italics">c</hi>-<hi rend="italics">x</hi>) + D.―(<hi rend="italics">d</hi>-<hi rend="italics">x</hi>); hence A<hi rend="italics">x</hi>+B<hi rend="italics">x</hi>+C<hi rend="italics">x</hi>+D<hi rend="italics">x</hi>=A<hi rend="italics">a</hi>+ B<hi rend="italics">b</hi>+C<hi rend="italics">c</hi>+D<hi rend="italics">d,</hi> and the distance sought; which therefore is equal to the <pb n="264"/><cb/> sum of all the momenta, divided by the sum of all the we ghts or bodies. <figure/></p><p><hi rend="italics">Or thus.</hi> When the bodies are not in the same straight line, connect them with the lines AB, CD; then, as before, find P the common centre of A and B, and Q the common centre of C and D; then, conceiving A and B united in P, and C and D united in Q, find S the common centre of P and Q, which will again be the centre of gravity of the whole.</p><p>Or the bodies may be all reduced to any line V<hi rend="italics">AB</hi> &c, drawn in any direction whatever, by perpendiculars B<hi rend="italics">B,</hi> C<hi rend="italics">C,</hi> &c, and then the common centre <hi rend="italics">S</hi> in this ine, found as before, will be at the same distance from V as the true centre S is; and consequently the perpendicular from <hi rend="italics">S</hi> will pass through S the real centre.</p><p>4. From the foregoing general expression, viz, , for the centre of gravity of any system of bodies, may be derived a general method for finding that centre; for A, B, C, &c, may be considered as the elementary parts of any body, whose sum or mass is M = A + B + C &c, and A<hi rend="italics">a,</hi> B<hi rend="italics">b,</hi> C<hi rend="italics">c,</hi> &c, are the several momenta of all these parts, viz, the product of each part multiplied by its distance from the fixed point V. Hence then, in any body, find a general expression for the sum of the momenta, and divide it by the content of the body, so shall the quotient be the distance of the centre of gravity from the vertex, or from any other fixed point, from which the momenta are estimated. <figure/></p><p>5. <hi rend="italics">Thus, in a right line</hi> AB, all the particles which compose it may be considered as so many very small weights, each equal to <hi rend="italics">x<hi rend="sup">.</hi>,</hi> which is therefore the fluxion of the weights, or of the line denoted by <hi rend="italics">x.</hi> So that the small weight <hi rend="italics">x<hi rend="sup">.</hi></hi> multiplied by its distance from A, viz <hi rend="italics">x,</hi> is <hi rend="italics">xx<hi rend="sup">.</hi></hi> the momentum of that weight <hi rend="italics">x<hi rend="sup">.</hi></hi>; that is, <hi rend="italics">xx<hi rend="sup">.</hi></hi> is the fluxion of all the momenta in the line AB or <hi rend="italics">x</hi>; and therefore its fluent (1/2)<hi rend="italics">x</hi><hi rend="sup">2</hi> is the sum of all those momenta; which being divided by <hi rend="italics">x</hi> the sum of all the weights, gives (1/2)<hi rend="italics">x</hi> or (1/2)AB for the distance of the centre of gravity C from the point A; that is, the centre is in the middle of the line. <figure/></p><p>6. <hi rend="italics">Also in the parallelogram,</hi> whose axis or length AB = <hi rend="italics">x,</hi> and its breadth DE = <hi rend="italics">b</hi>; drawing <hi rend="italics">de</hi> parallel and indefinitely near DE, the areola <hi rend="italics">d</hi>DE<hi rend="italics">e</hi> = <hi rend="italics">bx<hi rend="sup">.</hi></hi> <cb/> will be the fluxion of all the weights, which multipl ed by its distance <hi rend="italics">x</hi> from the point A, gives <hi rend="italics">bxx<hi rend="sup">.</hi></hi> for the fluxion of all the momenta, and consequently the fluent (1/2)<hi rend="italics">bx</hi><hi rend="sup">2</hi> is the sum of all those momenta themselves; which being divided by <hi rend="italics">bx</hi> the sum of all the weights, gives (1/2)<hi rend="italics">x</hi> = (1/2)AB for the distance of the centre C from the extremity at A, and is therefore in the middle of the axis, as is known from other principles.</p><p>And the process and conclusion will be exactly the same for a cylinder, or any prism whatever, making <hi rend="italics">b</hi> to denote the area of the end or of a transverse section of the body. <figure/></p><p>7. <hi rend="italics">In a Triangle</hi> ABC; the line AD drawn from one angle to bisect the opposite side, will be a diameter of gravity, or will pass through the centre of gravity; for if that line be supported, or conceived to be laid upon the edge of something, the two halves of the triangle on both sides of that line will just balance one another, since all the parallels EF &c to the base will be bisected, as well as the base itself, and so the two halves of each line will just balance each other. Therefore, putting the base BC = <hi rend="italics">b,</hi> and the axis or bisecting line AD = <hi rend="italics">a,</hi> the variable part AS = <hi rend="italics">x</hi>; then, by similar triangles AD : BC :: AS : EF, that is <hi rend="italics">a : b :: x : bx/a</hi> = EF; which, as a weight, multiplied by <hi rend="italics">x<hi rend="sup">.</hi>,</hi> gives <hi rend="italics">bxx<hi rend="sup">.</hi>/a</hi> for the fluxion of the weights; and this again multiplied by <hi rend="italics">x</hi> = AS, the distance from A, gives <hi rend="italics">bx</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi>/a</hi> for the fluxion of the momenta; the fluent of which, or <hi rend="italics">bx</hi><hi rend="sup">3</hi>/3<hi rend="italics">a</hi> divided by <hi rend="italics">bx</hi><hi rend="sup">2</hi>/<hi rend="italics">a</hi> the fluent for the weights, gives (2/3)<hi rend="italics">x</hi> = (2/3)AS for the distance of the centre of gravity from the vertex A in the triangle AEF; and when <hi rend="italics">x</hi> = AD, then (2/3)AD is the distance of the centre of gravity of the triangle ABC.</p><p><hi rend="italics">The Same Otherwise, without Fluxions.</hi>—Since a line drawn from any angle to the middle of the opposite side passes through the centre of gravity, therefore the intersection of any two of such lines, will be that centre: thus then the centre of gravity is in the line AD; and it is also in the line CG bisecting AB; it is therefore in their intersection S. Now to determine the distance of S from any angle, as A, produce CG to meet BH parallel to AS in H; then the two triangles AGS, BGH are mutually equal and similar; for the opposite angles at G are equal, as are the alternate angles at H and S, and at A and B, also the side AG = BG; therefore the other sides BH, AS are equal. But the triangles CDS, CBH are similar, and the side CB = 2CD, therefore BH or its equal AS = 2DS, that is AS = (2/3)AD, the same as was found before. And in like manner CS = (2/3)CG. <pb n="265"/><cb/></p><p>8. <hi rend="italics">In a Trapezium.</hi> Divide the figure into two triangles by the diagonal AC, and find the centres of gravity E and F of these triangles; join EF, and find the common centre G of these two by this proportion, ABC : ADC :: FG : EG, or ABCD : ADC :: EF : EG.</p><p>In like manner, for any other sigure, whatever be the number of sides, divide it into several triangles, and find the centre of gravity of each; then connect two centres together, and find their common centre as above; then connect this and the centre of a third, and find the common centre of these; and so on, always connecting the last found common centre to another centre, till the whole are included in this process; so shall the last common centre be that which is required. <figure/></p><p>9. <hi rend="italics">In the Parabola</hi> BAC. Put AD = <hi rend="italics">x,</hi> BD = <hi rend="italics">y,</hi> and the parameter = <hi rend="italics">p.</hi> Then, by the nature of the figure, <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, and 2<hi rend="italics">y</hi> = 2√<hi rend="italics">px</hi>; hence 2<hi rend="italics">x<hi rend="sup">.</hi>√px</hi> is the fluxion of the weights, and 2<hi rend="italics">xx<hi rend="sup">.</hi>√px</hi> is the fluxion of the momenta; then the fluent of the latter divided by that of the former, or (4/5)<hi rend="italics">x</hi><hi rend="sup">5/2</hi>√<hi rend="italics">p</hi> divided by (4/3)<hi rend="italics">x</hi><hi rend="sup">3/2</hi>√<hi rend="italics">p,</hi> gives (3/5)<hi rend="italics">x</hi> = (3/5)AD, for AG, the distance of the centre of gravity G from the vertex A of the parabola. <figure/></p><p>10. <hi rend="italics">In the Circular Arc</hi> ABD, considered as a physical line having gravity. It is manisest that the centre of gravity G of the arc, will be somewhere in the axis, or middle radius BC, C being the centre of the circle, which is considered as the point of suspension. Suppose F indefinitely near to A, and FH parallel to BC. Put the radius BC or AC = <hi rend="italics">r,</hi> the semiarc AB = <hi rend="italics">z,</hi> and the semichord AE = <hi rend="italics">x</hi>; then is AH = <hi rend="italics">x<hi rend="sup">.</hi>,</hi> and AF = <hi rend="italics">z<hi rend="sup">.</hi></hi> the fluxion of the weights, and therefore CE X <hi rend="italics">z<hi rend="sup">.</hi></hi> is the fluxion of the momenta. But, by similar triangles, AC or <hi rend="italics">r</hi> : CE :: AF or <hi rend="italics">z<hi rend="sup">.</hi></hi> : AH or <hi rend="italics">x<hi rend="sup">.</hi>,</hi> therefore <hi rend="italics">rx<hi rend="sup">.</hi></hi> = CE X <hi rend="italics">z<hi rend="sup">.</hi>,</hi> and so <hi rend="italics">rx<hi rend="sup">.</hi></hi> is also the fluxion of the momenta; the fluent of which is <hi rend="italics">rx,</hi> and this divided by <hi rend="italics">z</hi> the weight, gives the distance of the centre of gravity from the centre C of the circle; being a 4th proportional to the given arc, its chord, and the radius of the circle. <cb/></p><p>Hence, when the arc becomes the semicircle ABK, the above expression becomes IC<hi rend="sup">2</hi>/IB or <hi rend="italics">r</hi><hi rend="sup">2</hi>/(1.5708<hi rend="italics">r</hi>)=<hi rend="italics">r</hi>/(1.5708) = .6366<hi rend="italics">r,</hi> viz a third proportional to a quadrant and the radius.</p><p>11. <hi rend="italics">In the Circular Sector</hi> ABDC. Here also the centre of gravity will be in the axis or middle radius BC. Now with any smaller radius describe the concentric arc LMN, and put the radius AC or BC = <hi rend="italics">r,</hi> the arc ABD = <hi rend="italics">a,</hi> its chord AED = <hi rend="italics">c,</hi> and the variable radius CL or CM = <hi rend="italics">y</hi>; then as <hi rend="italics">r : y :: a</hi> : (<hi rend="italics">ay</hi>)/<hi rend="italics">r</hi> = the arc LMN, and <hi rend="italics">r : y :: c</hi> : <hi rend="italics">cy/r</hi> = the chord LON; also, by the last article, the distance of the centre of gravity of the arc LMN is ; hence the arc LMN or <hi rend="italics">ay/r</hi> multiplied by <hi rend="italics">y<hi rend="sup">.</hi></hi> gives (<hi rend="italics">ayy<hi rend="sup">.</hi></hi>)/<hi rend="italics">r</hi> the fluxion of the weights, and this mul tipliedby <hi rend="italics">cy/a</hi> the distance of the common centre of gravity, gives (<hi rend="italics">cy</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">.</hi></hi>)/<hi rend="italics">r</hi> the fluxion of the momenta; the fluent of which, viz (<hi rend="italics">cy</hi><hi rend="sup">3</hi>)/(3<hi rend="italics">r</hi>), divided by <hi rend="italics">ay</hi><hi rend="sup">2</hi>/2<hi rend="italics">r</hi>, the fluent of the weights, gives (2<hi rend="italics">cy</hi>)/(3<hi rend="italics">a</hi>) for the distance of the centre of gravity of the sector CLEN from the centre C; and when <hi rend="italics">y</hi> = <hi rend="italics">r,</hi> it becomes 2<hi rend="italics">cr</hi>/3<hi rend="italics">a</hi> = CG for that of the sector CABD proposed; being 2/3 of a 4th proportional to the arc of the sector, its chord, and the radius of the circle.</p><p>Hence, when the sector becomes a semicircle, the last expression becomes 4<hi rend="italics">r</hi><hi rend="sup">2</hi>/3<hi rend="italics">a</hi> = 2IC<hi rend="sup">2</hi>/3IB or 2/3 of a 3d proportional to a quadrantal arc and the radius. Or it is equal to 4<hi rend="italics">r</hi>/3<hi rend="italics">p</hi> = .4244<hi rend="italics">r</hi> from the centre C; where <hi rend="italics">p</hi>=3.1416. <figure/></p><p>12. <hi rend="italics">In the Cone</hi> ADB. Putting <hi rend="italics">a</hi> = DC, <hi rend="italics">b</hi> = area of the base AEB, and <hi rend="italics">x</hi> = D<hi rend="italics">c</hi> any variable altitude; then as <hi rend="italics">a</hi><hi rend="sup">2</hi> : <hi rend="italics">x</hi><hi rend="sup">2</hi> :: <hi rend="italics">b</hi> : <hi rend="italics">bx</hi><hi rend="sup">2</hi>/<*><hi rend="sup">2</hi> = area <hi rend="italics">aeb</hi>; hence the flux- <pb n="266"/><cb/> ion of the weights is <hi rend="italics">bx</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>/<hi rend="italics">a</hi><hi rend="sup">2</hi>, whose fluent, or the solid, is <hi rend="italics">bx</hi><hi rend="sup">3</hi>/3<hi rend="italics">a</hi><hi rend="sup">2</hi>; and the fluxion of the momenta is <hi rend="italics">bx</hi><hi rend="sup">3</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>/<hi rend="italics">a</hi><hi rend="sup">2</hi>, whose fluent is <hi rend="italics">bx</hi><hi rend="sup">4</hi>/4<hi rend="italics">a</hi><hi rend="sup">2</hi>; then this fluent divided by the former fluent gives (3/4)<hi rend="italics">x</hi> or (3/4)D<hi rend="italics">c</hi> for the distance of the centre of gravity of the cone D<hi rend="italics">ab,</hi> or (3/4)DC for that of the cone DAB below the vertex D.</p><p>And the same is the distance in any other pyramid. So that all pyramids of the same altitude, have the same centre of gravity.</p><p>13. In like manner are we to proceed for the centre of gravity in other bodies. Thus, the altitude of the segment of a sphere, or spheroid, or conoid, being <hi rend="italics">x, a</hi> being the whole of that axis itself; then the distance of the centre of gravity in each of these bodies, from the vertex, will be as follows, viz, (4<hi rend="italics">a</hi>-3<hi rend="italics">x</hi>)/(6<hi rend="italics">a</hi>-4<hi rend="italics">x</hi>)<hi rend="italics">x</hi> in the sphere or spheroid,<lb/> (5/8)<hi rend="italics">x</hi> in the semisphere or semispheroid,<lb/> (2/3)<hi rend="italics">x</hi> in the parabolic conoid,<lb/> (4<hi rend="italics">a</hi> + 3<hi rend="italics">x</hi>)/(6<hi rend="italics">a</hi> + 3<hi rend="italics">x</hi>)<hi rend="italics">x</hi> in the hyperbolic conoid.<lb/></p><p>14. <hi rend="italics">To determine the Centre of Gravity in any Body Mechanically.</hi> Lay the body on the edge of any thing, as a triangular prism, or such like, moving it backward and forward till the parts on both sides are in equilibrio; then is that line just in, or under the centre of gravity. Balance it again in another position, to find another line passing through the centre of gravity; then the intersection of these two lines will give the place of that centre itself.</p><p>The same may be done by laying the body on an horizontal table, as near the edge as possible without its falling, and that in two positions, as lengthwise and breadthwise: then the common interfection of the two lines contiguous to the edge, will be its centre of gravity. Or it may be done by placing the body on the point of a style, &c, till it rest in equilibrio. It was by this method that Borelli found that the centre of gravity in a human body, is between the nates and pubis; so that the whole gravity of the body is collected into the place of the genitals; an instance of the wisdom of the creator, in placing the membrum virile in the part, which is the most convenient for copulation.</p><p><hi rend="italics">The same otherwise thus.</hi> Hang the body up by any point; then a plumb-line hung over the same point, will pass through the centre of gravity; because that centre will always descend to the lowest point when the body comes to rest, which it cannot do except when it falls in the plumb line. Therefore, marking that line upon it, and suspending the body by another point, with the plummet, to find another such line, the intersection of the two will give the centre of gravity.</p><p><hi rend="italics">Or thus.</hi> Hang the body by two strings from the same tack, but fixed to different points of the body; then a plummet, hung by the same tack, will fall on the centre of gravity.</p><p>In the 4th volume of the New Acts of the Academy of Petersburgh, is the demonstration of a very general theorem concerning centres of gravity, by M. Lhuilier; <cb/> a particular example only of the general proposition, will be as follows: Let A, B, C, be the centres of gravity of three bodies; <hi rend="italics">a, b, c</hi> their respective masses, and Q their common centre of gravity. Let right lines QA, QB, QC, be drawn from the common centre to that of each body, and the latter be connected by right lines AB, AC, and BC; then</p><p>.</p><p><hi rend="italics">Uses of the Centre of Gravity.</hi> This point is of the greatest use in mechanics, and many important concerns in life, because the place of that centre is to be considered as the place of the body itself in computing mechanical effects; as in the oblique pressures of bodies, banks of earth, arches of bridges, and such like.</p><p>The same centre is even useful in finding the superficial and solid contents of bodies; for it is a general rule, that the superficies or solid generated by the rotation of a line or plane about any axis, is always equal to the product of the said line or plane drawn into the circumference or path described by the centre of gravity. For example, it was found above at art 11, that in a semicircle, the distance of the centre of gravity from the centre of the circle, is 4<hi rend="italics">r</hi>/3<hi rend="italics">p</hi>; and therefore the path of that centre, or circumference described by it whilst the femicircle revolves about its diameter, is (2/3)<hi rend="italics">r</hi>; also the area of the semicircle is (1/2)<hi rend="italics">pr</hi><hi rend="sup">2</hi>; hence the product of the two is (4/3)<hi rend="italics">pr</hi><hi rend="sup">3</hi>, which, it is well known, is equal to the solidity of the sphere generated by the revolution of the semicircle.</p><p>And hence also is obtained another method of finding mathematically the centre of gravity of a line or plane, from the contents of the superficies or solid gene<*> rated by it. For if the generated superficies or solid be divided by the generating line or plane, the quotient will be the circumference described by the centre of gravity; and consequently this divided by 2<hi rend="italics">p</hi> gives the radius, or distance of that centre from the axis of rotation. So, in the semicircle, whose area is (1/2)<hi rend="italics">pr</hi><hi rend="sup">2</hi>, and the content of the sphere generated by it (4/3)<hi rend="italics">pr</hi><hi rend="sup">3</hi>; here the latter divided by the former is (8/3)<hi rend="italics">r,</hi> and this divided by 2<hi rend="italics">p</hi> gives 4<hi rend="italics">r</hi>/3<hi rend="italics">p</hi> for the distance of the centre of gravity from the axis, or from the centre of the semicircle. The property last mentioned, relative to the relation between the centre of gravity and the figure generated by the revolution of any line or plane, is mentioned by Pappus, in the preface to his 7th book; and father Guldin has more fully demonstrated it in his 2d and 3d books on the Centre of Gravity.</p><p>The principal writers on the centre of gravity are Archimedes, Pappus, Guldini, Wallis, Casatus, Carré, Hays, Wolfius, &c.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Gyration,</hi> is that point in which if the whole mass be collected, the same angular velocity will be generated in the same time, by a given force acting at any place, as in the body or system itself. This point differs from the centre of oscillation, in as much as in this latter case the motion of the body is produced by the gravity of its own particles, but in the case of the <pb n="267"/><cb/> centre of gyration the body is put in motion by some other force acting at one place only.</p><p><hi rend="italics">To determine the Centre of Gyration,</hi> in any body, or system of bodies composed of the parts A, B, C, &c, moving about the point S, when urged by a force <hi rend="italics">f</hi> acting at any point P. Let R be that centre: then, by mechanics, the angular velocity generated in the system by the force <hi rend="italics">f,</hi> is as , and, by the same, the angular velocity of the matter placed all in the point R, is ; then since these two are to be equal, their equation will give , for the distance of the centre of gyration sought, below the axis of motion.</p><p>Now because the quantity , where G is the centre of gravity, O the centre of oscillation, and <hi rend="italics">b</hi> the whole body or sum of A, B, C, &c; therefore it follows that ; that is, the distance of the centre of gyration, is a mean proportional between those of gravity and oscillation.</p><p>And hence also, if <hi rend="italics">p</hi> denote any particle of a body, placed at the distance <hi rend="italics">d</hi> from the axis of motion; then is ; from whence the point R may be determined in bodies by means of Fluxions.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of an Hyperbola,</hi> is the middle of the axis, or of any other diameter, being the point without the figure in which all the diameters intersect one another; and it is common to all the four conjugate hyperbolas.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Magnitude,</hi> is the point which is equally distant from all the similar external parts of a body. This is the same as the centre of gravity in homogeneal bodies that can be cut into like and equal parts according to their length, as in a cylinder or any other prism.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Motion,</hi> is the point about which any body, or system of bodies, moves, in a revolving motion.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">os Oscillation,</hi> is that point in the axis or line of suspension of a vibrating body, or system of bodies, in which if the whole matter or weight be collected, the vibrations will still be performed in the same time, and with the same angular velocity, as before. Hence, in a compound pendulum, its distance from the point of suspension is equal to the length of a simple pendulum whose oscillations are isochronal with those of the compound one.</p><p>Mr. Huygens, in his Horologium Oscillatorium, first shewed how to find the centre of oscillation. At the beginning of his discourse on this subject, he says, that Mersennus first proposed the problem to him while he was yet very young, requiring him to resolve it in the cases of sectors of circles suspended by their angles, and by the middle of their bases, both when they oscillate sideways and flatways; as also for triangles and the segments of circles, either suspended from their vertex or the middle of their bases. But, says he, not <cb/> having immediately discovered any thing that would open a passage into this business, I was repulsed at first setting out, and stopped from a further prosecution of the thing; till being farther incited to it by adjusting the motion of the pendulums of my clock, I surmounted all difficulties, going far beyond Descartes, Fabry, and others, who had done the thing in a few of the most easy cases only, without any sufficient demonstration; and solving not only the problems proposed by Mersennus, but many others that were much more difficult, and shewing a general way of determining this centre, in lines, superficies, and solids.</p><p>In the Leipsic Acts for 1691 and 1714, this doctrine is handled by the two Bernoullis: and the same is also done by Herman, in his treatise De Motu Corporum Solidorum et Fluidorum.</p><p>It may also be seen in treatises on the Inverse Method of Fluxions, where it is introduced as one of the examples of that method. See Hayes, Carré, Wolfius, &c.</p><p><hi rend="italics">To determine the Centre of Oscillation,</hi> in any Compound Mass or Body MN, or of any System of Bodies A, B, C, &c. <figure/></p><p>Let MN be the plane of vibration, to which plane conceive all the matter to be reduced by letting fall perpendiculars to this plane from every particle in the body; a supposition which will not alter the vibration of the body, because the particles are still at the same distance from the axis of motion. Let O be the centre of oscillation, and G the centre of gravity; through the axis S draw SGO, and the horizontal line ST; then from every particle A, B, C, &c, let fall perpendiculars A<hi rend="italics">a</hi> and A<hi rend="italics">p,</hi> B<hi rend="italics">b</hi> and B<hi rend="italics">q,</hi> C<hi rend="italics">c</hi> and C<hi rend="italics">r,</hi> &c, to these two lines; and join SA, SB, SC; also draw G<hi rend="italics">m</hi> and O<hi rend="italics">n</hi> perpendicular to ST.</p><p>Now the forces of the weights A, B, C, to turn the body about the axis, are A . S<hi rend="italics">p,</hi> B . S<hi rend="italics">q,</hi>-C . S<hi rend="italics">r</hi>; and, by mechanics, the forces opposing that motion are A . SA<hi rend="sup">2</hi>, B . SB<hi rend="sup">2</hi>, C . SC<hi rend="sup">2</hi>; therefore the angular motion generated in the system is . In like manner, the angular velocity which any body or particle <hi rend="italics">p,</hi> situated in O, generates in the system, by its weight, is because of <pb n="268"/><cb/> the similar triangles SG<hi rend="italics">m,</hi> SO<hi rend="italics">n.</hi> But, by the conditions of the problem, the vibrations are performed alike in both these cases; therefore these two expressions must be equal to each other, that is , and consequently . But, by mechanics again, the sum of the forces A . S<hi rend="italics">p</hi> + B . S<hi rend="italics">q</hi>-C . S<hi rend="italics">r</hi> is equal the force of the same matter collected all into its centre of gravity G; and therefore , which is the distance of the centre of oscillation O below the axis of suspension.</p><p>Farther, because it was found under the article <hi rend="italics">Centre of Gravity,</hi> that , therefore is the same distance of the centre of oscillation; where any of the products A . S<hi rend="italics">a,</hi> B . S<hi rend="italics">b,</hi> &c are to be taken negatively when the points <hi rend="italics">a, b,</hi> &c lie above the point S, or where the axis passes through. Again, because, by Eucl. II 12 and 13, it is ; and because by Mechanics, the sum of the last terms is nothing, ; therefore the sum of the others, or ; where <hi rend="italics">b</hi> denotes the body, or sum A+B+C &c of all the parts: this value then being substituted in the numerator of the 2d value of SO above-found, it becomes</p><p>From which it appears that the centre of oscillation is always below the centre of gravity, and that the difference or distance between them is .</p><p>It farther follows from hence, that ; that is, the rectangle SG.GO is always the same constant quantity, wherever the point of suspension S is placed, since the point G and the bodies A, B, &c, are constant. Or GO is always reciprocally as SG, that is GO is less as SG is greater; and the points G and O coincide when SG is infinite; but when S coincides with G, then GO is infinite, or O is at an infinite distance.</p><p><hi rend="italics">To find the Centre of Oscillation by means of Fluxions.</hi> From the premises is derived this general method for the centre of oscillation, viz, let <hi rend="italics">x</hi> be the abscissa of an oscillating body, and <hi rend="italics">y</hi> its corresponding ordinate or section; then will the distance SO of the centre of oscil- <cb/> lation below the axis of suspension S, be equal to the fluent of <hi rend="italics">yx</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> divided by the fluent of <hi rend="italics">yxx<hi rend="sup">.</hi>.</hi> So that, if from the nature or equation of any given figure, the value of <hi rend="italics">y</hi> be expressed in terms of <hi rend="italics">x,</hi> or otherwise, and substituted in these two fluxions; then the fluents being duly found, and the one divided by the other, the quotient will be the distance to the centre of oscillation in terms of the absciss <hi rend="italics">x.</hi></p><p>But when the body is suspended by a very fine thread of a given length <hi rend="italics">a,</hi> then the fluent of ―(<hi rend="italics">a</hi> + <hi rend="italics">x</hi>))<hi rend="sup">2</hi> . <hi rend="italics">yx<hi rend="sup">.</hi></hi> divided by the fluent of ―(<hi rend="italics">a</hi> + <hi rend="italics">x</hi>) . <hi rend="italics">yx<hi rend="sup">.</hi></hi> gives the distance of the same centre of oscillation below the point of suspension.</p><p><hi rend="italics">Ex.</hi> For example, in a right line, or rectangle or cylinder or any other prism, whose constant section is <hi rend="italics">y,</hi> or the constant quantity <hi rend="italics">a</hi>; then <hi rend="italics">yx</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi> is <hi rend="italics">ax</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi>,</hi> whose fluent is (1/3)<hi rend="italics">ax</hi><hi rend="sup">3</hi>; also <hi rend="italics">yxx<hi rend="sup">.</hi></hi> is <hi rend="italics">axx<hi rend="sup">.</hi>,</hi> whose fluent is (1/2)<hi rend="italics">ax</hi><hi rend="sup">2</hi>; and the quotient of the former (1/3)<hi rend="italics">ax</hi><hi rend="sup">3</hi> divided by the latter (1/2)<hi rend="italics">ax</hi><hi rend="sup">2</hi>, is (2/3)<hi rend="italics">x</hi> for the distance of the centre of oscillation below the vertex in any such figure, namely having every where the same breadth or section, that is, at twothirds of its length.</p><p>In like manner the centre of oscillation is found for various figures, vibrating flatways, and are as they are expressed below, viz, <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Nature of the Figure.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">When suspended by Vertex.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Isosceles triangle</cell><cell cols="1" rows="1" role="data">3/4 of its altitude</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common Parabola</cell><cell cols="1" rows="1" role="data">5/7 of its altitude</cell></row><row role="data"><cell cols="1" rows="1" role="data">Any Parabola</cell><cell cols="1" rows="1" role="data">(2<hi rend="italics">m</hi>+1)/(3<hi rend="italics">m</hi>+1) X its altitude.</cell></row></table></p><p>As to figures moved laterally or sideways, or edgeways, that is about an axis perpendicular to the plane of the figure, the finding the centre of oscillation is somewhat difficult; because all the parts of the weight in the same horizontal plane, on account of their unequal distances from the point of suspension, do not move with the same velocity; as is shewn by Huygens, in his Horol. Oscil. He found, in this case, the distance of the centre of oscillation below the axis, viz, <table><row role="data"><cell cols="1" rows="1" role="data">In a circle,</cell><cell cols="1" rows="1" role="data">3/4 of the diameter:</cell></row><row role="data"><cell cols="1" rows="1" role="data">In a rectangle, susp. by one angle,</cell><cell cols="1" rows="1" role="data">2/3 of the diagonal:</cell></row><row role="data"><cell cols="1" rows="1" role="data">In a parabola susp. by its vertex,</cell><cell cols="1" rows="1" role="data">5/7 axis + 1/3 param.</cell></row><row role="data"><cell cols="1" rows="1" role="data">The same susp. by mid. of base,</cell><cell cols="1" rows="1" role="data">4/7 axis + 1/2 param.</cell></row><row role="data"><cell cols="1" rows="1" role="data">In a sector of a circle</cell><cell cols="1" rows="1" role="data">(3 arc X radius)/(4 chord):</cell></row><row role="data"><cell cols="1" rows="1" role="data">In a cone</cell><cell cols="1" rows="1" role="data">4/5 axis +(radius base<hi rend="sup">2</hi>)/(5 axis):</cell></row><row role="data"><cell cols="1" rows="1" role="data">In a sphere</cell><cell cols="1" rows="1" role="data"><hi rend="italics">g</hi> + 2<hi rend="italics">r</hi><hi rend="sup">2</hi>/5<hi rend="italics">g</hi>, where <hi rend="italics">r</hi> is</cell></row></table> the radius, and the rad. added to the length of the thread. <hi rend="center">See also Simpson's Fluxions, art. 183 &c.</hi></p><p><hi rend="italics">To find the Centre of Oscillation Mechanically or Experimentally.</hi> Make the body oscillate about its point of suspension; and hang up also a simple pendulum of such a length that it may vibrate or just keep time with the other body: then the length of the simple pendulum is equal to the distance of the centre of oscillation of the body below the point of suspension.</p><p>Or it will be still better found thus: Suspend the body very freely by the given point, and make it vibrate <pb n="269"/><cb/> in small arcs, counting the vibrations it makes in any portion of time, as a minute, by a good stop watch; and let that number of oscillations made in a minute be called <hi rend="italics">n</hi>: then shall the distance of the centre of oscillation be SO=(140850)/<hi rend="italics">nn</hi> inches. For, the length of the pendulum vibrating seconds, or 60 times in a minute, being 39 1/8 inches, and the lengths of pendulums being reciprocally as the square of the number of vibrations made in the same time, therefore <hi rend="italics">n</hi><hi rend="sup">2</hi> : 60<hi rend="sup">2</hi> :: 39 1/8 : 140850/<hi rend="italics">nn</hi> the length of the pendulum which vibrates <hi rend="italics">n</hi> times in a minute, or the distance of the centre of oscillation below the axis of motion.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Percussion,</hi> in a moving body, is that point where the percussion or stroke is the greatest, in which the whole percutient force of the body is supposed to be collected; or about which the impetus of the parts is balanced on every side, so that it may be stopt by an immoveable obstacle at this point, and rest on it, without acting on the centre of suspension.</p><p>1. When the percutient body revolves about a sixed point, the centre of percussion is the same with the centre of oscillation; and is determined in the same manner, viz, by considering the impetus of the parts as so many weights applied to an inflexible right line void of gravity; namely, by dividing the sum of the products of the forces of the parts multiplied by their distances from the point of suspension, by the sum of the forces. And therefore what has been above shewn of the centre of oscillation, will hold also of the centre of percussion when the body revolves about a fixed point. For instance, that the centre of percussion in a cylinder is at 2/3 of its length from the point of suspension, or that a stick of a cylindrical figure, supposing the centre of motion at the hand, will strike the greatest blow at a point about two-thirds of its length from the hand.</p><p>2. But when the body moves with a parallel motion, or all its parts with the same celerity, then the centre of percussion is the same as the centre of gravity. For the momenta are the products of the weights and celerities; and to multiply equiponderating bodies by the same velocity, is the same thing as to take equimultiples; but the equimultiples of equiponderating bodies do also equiponderate; therefore equivalent momenta are disposed about the centre of gravity, and consequently in this case the two centres coincide, and what is shewn of the one will hold in the other.</p><p>Centre of Percussion in a fluid, is the same as out of it.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Parallelogram,</hi> the point in which its diagonals intersect.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of Pressure,</hi> of a fluid against a plane, is that point against which a force being applied equal and contrary to the whole pressure, it will just sustain it, so as that the body pressed on will not incline to either side.—This is the same as the centre of percussion, supposing the axis of motion to be at the intersection of this plane with the surface of the fluid; and the centre of pressure upon a plane parallel to the horizon, or upon any plane where the pressure is uniform, is the same as the centre of gravity of that plane. Emerson's Mechanics, prop. 91. <cb/></p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Regular Polygon,</hi> or <hi rend="italics">Regular Body,</hi> is the same as that of the inscribed, or circumscribed circle or sphere.</p><p><hi rend="smallcaps">Centre</hi> <hi rend="italics">of a Sphere,</hi> is the same as that of its generating semicircle, or the middle point of the sphere, from whence all right lines drawn to the superficies, are equal.</p><p><hi rend="smallcaps">Centring</hi> <hi rend="italics">of an Optic Glass,</hi> the grinding it so as that the thickest part be exactly in the middle.</p><p>Cassini the younger has a discourse expressly on the necessity of well centring the object glass of a large telescope, that is, of grinding it so as that the centre may fall exactly in the axis of the telescope. Mem. Acad. 1710.</p><p>Indeed one of the greatest difficulties in grinding large optic glasses is, that in figures so little convex, the least difference will throw the centre two or three inches out of the middle. And yet Dr. Hook remarks, that though it were better the thickest part of a long object glass were exactly in the middle, yet it may be a very good one when it is an inch or two out of it. Philos. Trans. N° 4.</p><p>CENTRIFUGAL <hi rend="smallcaps">Force</hi>, is that by which a body revolving about a centre, or about another body, endeavours to recede from it. And</p><p>CENTRIPETAL <hi rend="smallcaps">Force</hi>, is that by which a moving body is perpetually urged towards a centre, and made to revolve in a curve, instead of a right line.</p><p>Hence, when a body revolves in a circle, these two forces, viz, the centrifugal and centripetal, are equal and contrary to each other, since neither of them gains upon the other, the body being in a manner equally balanced by them. But when, in revolving, the body recedes sarther from the centre, then the centrifugal exceeds the centripetal force; as in a body revolving from the lower to the higher apsis, in an ellipse, and respecting the focus as the centre. And when the revolving body approaches nearer to the centre, the centrifugal is less than the centripetal force; as while the body moves from the farther to the nearer extremity of the transverse axis of the ellipse: the two forces being equal to each other only at the very extremities of that axis.</p><p>It is one of the established laws of nature, that all motion is of itself rectilinear, and that the moving body never recedes from its first right line, till some new impulse be superadded in a different direction: after that new impulse the motion becomes compounded, but it is still rectilinear, though not in the same line or direction as before. To move in a curve, it must receive a new impulse in a different direction every moment; a curve not being reducible to any number of finite right lines. If then a body, continually drawn towards a centre, be projected in a line that does not pass through that centre, it will describe a curve; in each point of which, as A, it will endeavour to recede from the curve, and proceed in the tangent AD; and if nothing hindered, it would actually proceed in it; so as in the same time, in which it describes the arch AE, it would recede the length of the line DE, perpendicular to AD, by its centrifugal force: Or being projected in the direction AD, but being continually drawn out of its direction into a curve by a centripetal <pb n="270"/><cb/> force, so as to fall below the line of direction by the perpendicular space DE: Then the centrifugal or centripetal force is as this line of deviation DE; supposing the arch AE indefinitely small. <figure/></p><p>The doctrine of centrifugal forces was first mentioned by Huygens, at the end of his Horologium Oscillatorium, published in 1673, and demonstrated in the volume of his Posthumous Works, as also by Guido Grando; where he has given a few easy cases in bodies revolving in the circumference of circles. But Newton, in his Principia, was the first who fully handled this doctrine; at least as far as regards the conic sections. After him there have been several other writers upon this subject; as Leibnitz, Varignon in the Mem. de l'Acad. Keil in the Philos. Trans. and in his Physics, Bernoulli, Herman, Cotes in his Harmonia Mensurarum, Maclaurin in his Geometrica Organica, and in his Fluxions, and Euler in his book de Motu, where he considers the curves described by a body acted on by centripetal forces tending to several fixed points.</p><p>See also the art. <hi rend="italics">Central Forces,</hi> where this doctrine is more fully explained.</p></div1><div1 part="N" n="CENTROBARICO" org="uniform" sample="complete" type="entry"><head>CENTROBARICO</head><p>, the same as centre of gravity.</p><p><hi rend="smallcaps">Centrobaric</hi>-<hi rend="italics">Method,</hi> is a method of determining the quantity of a sursace or solid, by means of the generating line or plane, and its centre of gravity. The doctrine is chiefly comprized in this theorem:</p><p>Every figure, whether supersicial or solid, generated by the motion of a line or plane, is equal to the product of the generating magnitude and the path of its centre of gravity, or the line which its centre of gravity describes.</p><p>See more of this subject under the article <hi rend="italics">Centre of Gravity,</hi></p></div1><div1 part="N" n="CENTRUM" org="uniform" sample="complete" type="entry"><head>CENTRUM</head><p>, in Geometry, Mechanics, &c. See <hi rend="smallcaps">Centre.</hi></p><p><hi rend="smallcaps">Centrum</hi> <hi rend="italics">Phonicum,</hi> in Acoustics, is the place where the speaker stands in polysyllabical and articulate echoes.</p><p><hi rend="smallcaps">Centrum</hi> <hi rend="italics">Phonocampticum,</hi> is the place or object that returns the voice in an echo.</p></div1><div1 part="N" n="CEPHEUS" org="uniform" sample="complete" type="entry"><head>CEPHEUS</head><p>, a constellation of the northern hemisphere, being one of the 48 old asterisms. The Greeks fable that Cepheus was a king of Ethiopia, and the father of Andromeda, the princess who was delivered up to be devoured by a sea monster, from which she was rescued by Perseus.</p><p>The stars of this constellation, in Ptolomy's catalogue are 13, in Tycho's 11, in Hevelius's 51, and in the Britannic catalogue 35.</p></div1><div1 part="N" n="CERBERUS" org="uniform" sample="complete" type="entry"><head>CERBERUS</head><p>, one of the new constellations formed by Hevelius out of the unformed stars, and added to the 48 old asterisms. It contains only 4 stars, which are enumerated under Hercules in the Britannic catalogue. <cb/></p></div1><div1 part="N" n="CETUS" org="uniform" sample="complete" type="entry"><head>CETUS</head><p>, <hi rend="italics">the Whale,</hi> a southern constellation, and one of the 48 old asterisms. The Greeks pretend that it was the sea-monster sent by Neptune to devour Andromeda, but was killed by Perseus.</p><p>In the neck of the whale is a remarkable star, Collo Ceti, which appears and disappears periodically, or rather grows brighter and fainter by turns, owing it is supposed to the alternate turning of its bright and dark sides towards us, as it revolves upon its axis, or else owing to the star having a flattish form. The period of its changes is about 312 days. Bullialdus in Phil. Trans. vol. 2, Hevelius ibid. vol. 6, Herschel ibid. vol. 70, Marald. in Mem. Acad. 1719.</p><p>The stars in the constellation Cetus, in Ptolomy's catalogue, are 22, in Tycho's 21, in Hevelius's 45, and in the Britannic catalogue 97.</p></div1><div1 part="N" n="CHAIN" org="uniform" sample="complete" type="entry"><head>CHAIN</head><p>, in Surveying, is a lineal measure, consisting of a certain number of iron links, usually 100: serving to take the dimensions of fields &c.</p><p>At every 10th link is usually fastened a small brass plate, with a figure engraven upon it, or else cut into different shapes, to shew how many links it is from one end of the chain.</p><p>Chains are of various kinds and lengths; as</p><p>1. A chain of 100 feet long, each link one foot, for measuring of large distances only, when regard is not proposed to be had to acres &c, in the superficial content.</p><p>2. A chain of one pole or 16 feet and a half in length; especially useful in measuring and laying out gardens and orchards, or the like, by the pole or rod measure.</p><p>3. A chain of 4 poles, or 66 feet, or 22 yards, in length, called Gunter's chain, and is peculiarly adapted to the business of Surveying or Land-measuring, because that 10 square chains just make an English acre of land; so that the dimensions being taken in these chains, and thence the contents computed in square chains, they are readily turned into acres by dividing by 10, or barely cutting off the last figure from the square chains. But it is still better in practice to proceed thus, viz, count the dimensions, not in chains, but all in links; then the contents are in square links; and sive sigures being cut off for decimals, the rest are acres; that is four figures to bring the square links to square chains, and one more to bring the square chains to acres.</p><p>In this chain, the links are each 7 inches and 92/100, or 7.92 inches in length, which is very nearly 2/3 of a foot. And hence any number of chains or links are easily brought to feet or inches, or the contrary: the best way of doing which is this: multiply the number of links by 66, then cut off two figures for decimals, and the rest are feet: or multiply links by 22 for yards, cutting off two figures.</p></div1><div1 part="N" n="CHALDRON" org="uniform" sample="complete" type="entry"><head>CHALDRON</head><p>, of Coals, an English dry measure of capacity consisting of 36 bushels heaped up.</p><p>The chaldron of coals is accounted to weigh about 2000 pounds.—On ship board, 21 chaldrons of coals are allowed to the score.</p><p>CHAMBER <hi rend="italics">of a Mortar,</hi> or <hi rend="italics">some cannon,</hi> is a cell or cavity at the bottom of the bore, to receive the charge of powder.</p><p>It is not found by experience that chambers have <pb n="271"/><cb/> any sensible effect on the velocity of the shot, unless in the largest ordnance, as mortars or very large cannon. Neither is it found that the form of them is very material; a small cylinder is as good as any; though mathematical speculations may shew a preference of one form over another. But in practice, the chief point to be observed, is to have the chamber of a size just to contain the charge of powder, and no more, that the ball may lie colse to the charge; and that its entrance may point exactly to the centre of the ball.</p></div1><div1 part="N" n="CHAMBERS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CHAMBERS</surname> (<foreName full="yes"><hi rend="smallcaps">Ehhraim</hi></foreName>)</persName></head><p>, author of the dictionary of sciences called the <hi rend="italics">Cyclopædia.</hi> He was born at Milton in the county of Westmoreland, where he received the common education for qualifying a youth for trade and commerce. When he became of a proper age, he was put apprentice to Mr. Senex the globemaker, a business which is connected with literature, especially with geography and astronomy. It was during Mr. Chambers's residence with this skilful artist, that he acquired that taste for literature which accompanied him through life, and directed all his pursuits. It was even at this time that he formed the design of his grand work, the Cyclopædia; some of the first articles of which were written behind the counter. To have leisure to pursue this work, he quitted Mr. Senex, and took chambers at Gray's-Inn, where he chiefly resided during the rest of his life. The first edition of the Cyclopædia, which was the result of many years intense application, appeared in 1728, in 2 vols. folio. The reputation that Mr. Chambers acquired by the execution of this work, procured him the honour of being elected F. R. S. Nov. 6, 1729. In less than ten years time, a second edition became necessary; which accordingly was printed, with corrections and additions, in 1738; and this was followed by a third edition the very next year.</p><p>Although the Cyclopædia was the chief business of mr. Chambers's life, and may be regarded as almost the sole foundation of his fame, his attention was not wholly confined to this undertaking. He was concerned in a periodical publication, called, <hi rend="italics">The Literary Magazine,</hi> which was begun in 1735. In this work he wrote a variety of articles; particularly a review of Morgan's <hi rend="italics">Moral Philosopher.</hi> He was also concerned with Mr. John Martyn, professor of botany at Cambridge, in preparing for the press a translation and abridgment of the <hi rend="italics">Philosophical History and Memoirs of the R. Acad. of Sciences at Paris;</hi> which work was not published till 1742, some time after our author's decease, in 5 volumes 8vo. Mr. Chambers was also author of the translation of the <hi rend="italics">Jesuit's Perspective,</hi> from the French, in 4to; which has gone through several editions.</p><p>Mr. Chambers's close and unremitting attention to his studies at length impaired his health, and obliged him occasionally to take a country lodging, but without much benefit; he afterwards visited the south of France, but still with little effect; he therefore returned to England, where he soon after died, at Islington, May 15, 1740, and was buried at Westminster Abbey.</p><p>After the author's death, two more editions of his Cyclopædia were published. The proprietors afterwards procured a supplement to be compiled, by Mr. <cb/> Scott and Dr. Hill, but chiefly by the latter, which extended to two volumes more; and the whole has smce been reduced into one alphabet in 4 volumes, by Dr. Rees, forming a very valuable body of the sciences.</p></div1><div1 part="N" n="CHAMBRANLE" org="uniform" sample="complete" type="entry"><head>CHAMBRANLE</head><p>, the border, frame, or ornament of stone or wood, surrounding the three sides of doors, windows, and chimneys. This is different in the different orders: when it is plain, and without mouldings, it is called simply and properly, <hi rend="italics">band, case,</hi> or <hi rend="italics">srame.</hi> In an ordinary door, it is mostly called <hi rend="italics">door-case</hi>; in a window, the <hi rend="italics">window-frame.</hi></p><p>The Chambranle consists of three parts; the two sides, called ascendants; and the top, called the traverse or supercilium.</p></div1><div1 part="N" n="CHAMFER" org="uniform" sample="complete" type="entry"><head>CHAMFER</head><p>, or <hi rend="smallcaps">Chamferet</hi>, an ornament, in architecture, consisting of half a scotia; being a kind of small furrow or gutter on a column.</p></div1><div1 part="N" n="CHAMFERING" org="uniform" sample="complete" type="entry"><head>CHAMFERING</head><p>, is used for cutting the edge or the end of any thing bevel, or aslope.</p></div1><div1 part="N" n="CHANCE" org="uniform" sample="complete" type="entry"><head>CHANCE</head><p>, <hi rend="italics">the Doctrine and Laws of,</hi> are the same as those of expectation, or probability, &c; which see. Chances, in play, consist of the number of ways by which events may happen. Thus, if a halfpenny, or other piece of money, be tossed up, there are two events, or chances, or sides that may turn up, namely one chance for turning up a head, and one for the contrary; that is, it is an equal chance to throw a head or not. And in throwing a common die, which has 6 faces, there are in all 6 chances, that is one chance for throwing an ace or any other single point, and 5 chances against it; or it is 5 to 1 that such assigned point does not come up.</p><p>Upon this subject, see De Moivre, Simpson, &c.</p></div1><div1 part="N" n="CHANDELIERS" org="uniform" sample="complete" type="entry"><head>CHANDELIERS</head><p>, in Fortification, a kind of wooden parapet, consisting of upright timbers supporting others laid across the tops of them, 6 feet high, and fortified with faseines &c. They are used to cover the workmen in approaches, galleries, and mines. And they differ from blinds only in this, that the former serve to cover the men before, and the latter over head.</p></div1><div1 part="N" n="CHANGES" org="uniform" sample="complete" type="entry"><head>CHANGES</head><p>, the permutations or variations of any number of things, with regard to their position, order, &c; as how many changes may be rung on a number of bells, or how many different ways any number of persons may be placed, or how many several variations may be made of any number of letters, or any other things proposed to be varied.</p><p><hi rend="italics">To sind out such number of changes,</hi> multiply continually together all the terms in a series of arithmetical progression, whose first term and common difference are each unity or 1, and the last term the number of things proposed to be varied, thus 1 X 2 X 3 X 4 X 5 &c. till the last number be the proposed number of things. For,</p><p>If there be only two things, as <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> they admit of a double order or position only; for they may be placed either thus <hi rend="italics">ab</hi> or thus <hi rend="italics">ba,</hi> viz, . If there be three things, <hi rend="italics">a, b,</hi> and <hi rend="italics">c,</hi> they <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell></row></table> will admit of 6 variations = 1 X 2 since as in the margin, and no more; three each of the three may be combined there different ways with each of the other two. <pb n="272"/><cb/></p><p>And if there be 4 things, each of them may be combined 4 ways with each order of the other three, that is 4 times 6 ways, or ways.</p><p>In like manner, the combinations of 5 things are of 6 things are &c.</p><p>So that if it be proposed to assign how many different ways a company of 6 persons may be placed, at table for instance, the answer will be 720 ways. Also the number of changes that can be rung on 7 bells, are changes.</p></div1><div1 part="N" n="CHAPITERS" org="uniform" sample="complete" type="entry"><head>CHAPITERS</head><p>, the crowns or upper parts of a pillar or column.</p></div1><div1 part="N" n="CHAPPE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CHAPPE</surname> (<foreName full="yes"><hi rend="smallcaps">Jean d'Auteroche</hi></foreName>)</persName></head><p>, a French astronomer, was born at Mauriac, in Auvergne, March 2, 1728. A taste for drawing and mathematics appeared in him at a very tender age; and he owed to Dom Germain a knowledge of the first elements of mathematics and astronomy. M. Cassini, after assuring himself of the genius of this young man, undertook to improve it. He employed him upon the map of France, and the translation of Halley's tables, to which he made considerable additions. The king charged him in 1753 with drawing the plan of the county of Bitche, in Lorraine, all the elements of which he determined geographically. He occupied himself greatly with the two comets of 1760; and the fruit of his labour was his Elementary treatise on the theory of those comets, enriched with observations on the zodiacal light, and on the aurora borealis. He soon after went to Tobolsk, in Siberia, to observe the transit of Venus over the sun; a journey which greatly impaired his health. After two years absence he returned to France in 1762, where he occupied himself for some time in putting in order the great quantity of observations he had made. M. Chappe also went to observe the next transit of Venus, viz that of 1769, at California, on the west side of North America, where he died of a dangerous epidemic disease, the 1st of August 1769. He had been named Adjunct Astronomer to the Academy the 17th of January 1759.</p><p>The published works of M. Chappe, are,</p><p>1. The Astronomical Tables of Dr. Halley; with observations and additions: in 8vo, 1754.</p><p>2. Voyage to California to observe the transit of Venus over the sun, the 3d of June 1769: in 4to, 1772.</p><p>3. He had a considerable number of papers inserted in the Memoirs of the Academy, for the years 1760, 1761, 1764, 1765, 1766, 1767, and 1768; chiefly relating to astronomical matters.</p></div1><div1 part="N" n="CHAPTREL" org="uniform" sample="complete" type="entry"><head>CHAPTREL</head><p>, the same with Impost.</p></div1><div1 part="N" n="CHARACTERISTIC" org="uniform" sample="complete" type="entry"><head>CHARACTERISTIC</head><p>, <hi rend="italics">of a Logarithm,</hi> the same as Index, or Exponent. This term was first used by Briggs in the 4th section of his Arithmetica Logarithmica, where he treats particularly of it; meaning by it, the integral or first part of a logarithm towards the left hand, which expresses 1 less than the integer places or figures in the number answering to that logarithm, or how far the first figure of this number is removed from the place of units; namely, that 0 is the characteristic of all numbers from 1 to 10; and 1 the characteristic <cb/> of all those from 10 to 100; and 2 the characteristic of all those from 100 to 1000; and so on.</p></div1><div1 part="N" n="CHARACTERS" org="uniform" sample="complete" type="entry"><head>CHARACTERS</head><p>, are certain marks used by Astronomers, Mathematicians, &c, to denote certain things, whether for the sake of brevity, or perspicuity, in their operations. <hi rend="center">1. <hi rend="smallcaps">Astronomical Characters.</hi></hi> <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Planets &c.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">The twelve Signs or Constellations of the Zodiac.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> The Sun</cell><cell cols="1" rows="1" role="data"><figure/> Aries, the Ram</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> The Moon</cell><cell cols="1" rows="1" role="data"><figure/> Taurus, the Bull</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> The Earth</cell><cell cols="1" rows="1" role="data"><figure/> Gemini, the Twins</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Mercury</cell><cell cols="1" rows="1" role="data"><figure/> Cancer, the Crab</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Venus</cell><cell cols="1" rows="1" role="data"><figure/> Leo, the Lion</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Mars</cell><cell cols="1" rows="1" role="data"><figure/> Virgo, the Maid</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Jupiter</cell><cell cols="1" rows="1" role="data"><figure/> Libra, the Balance</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Saturn</cell><cell cols="1" rows="1" role="data"><figure/> Scorpio, the Scorpion</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Herschel, or the</cell><cell cols="1" rows="1" role="data"><figure/> Sagittary, the Archer</cell></row><row role="data"><cell cols="1" rows="1" role="data">Georgian Planet</cell><cell cols="1" rows="1" role="data"><figure/> Capricorn, the Goat</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Ascending Node</cell><cell cols="1" rows="1" role="data"><figure/> Aquarius, the Water-bearer</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Descending Node</cell><cell cols="1" rows="1" role="data"><figure/> Pisces, the Fishes</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">The Aspects, Time, Motion, &c.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Conjunction</cell><cell cols="1" rows="1" role="data">° Degrees</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Opposition</cell><cell cols="1" rows="1" role="data">′ Minutes or Primes</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Sextile</cell><cell cols="1" rows="1" role="data">″ Seconds, &c.</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Quartile</cell><cell cols="1" rows="1" role="data">A. M. Ante merid. or m morn.</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> Trine</cell><cell cols="1" rows="1" role="data">P. M. Post merid. or a aftern.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">h, m, s, Hours, min. sec.</cell></row></table> <hi rend="center">2. <hi rend="smallcaps">Mathematical</hi> <hi rend="italics">&c.</hi> <hi rend="smallcaps">Characters.</hi></hi> <hi rend="center"><hi rend="italics">Numeral Characters used by different Nations.</hi></hi></p><p>The most common numeral characters, are those called Arabic or Indian, viz. 1, 2, 3, 4, 5, 6, 7, 8, 9, with 0 or cipher for nothing.</p><p>The Roman numeral characters are seven, viz, I one, V sive, X ten, L sifty, C a hundred, D or I[C] five hundred, M or D[D] or CI[C] a thousand. Other combinations are as in the following synopsis of the Roman Notation. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">I</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">II: As often as any character is repeated,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">III so many times its value is repeated.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">IIII or IV: A less character before a</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">V greater diminishes its value.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">VI: A less character after a greater in-</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">VII creases its value.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">VIII</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">IX</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">X</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">L</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">100</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="valign=top align=right" role="data">500</cell><cell cols="1" rows="1" rend="valign=top" role="data">=</cell><cell cols="1" rows="1" role="data">D or I[C]: For every [C] added, this becomes 10 times as many.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">M or CI[C]: For every C and [C], set one at</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">MM [each end, it becomes 10 times as much.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">I[C][C] or ―V: A line over any number in-</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">―VI creases it 1000 fold.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">―X or CCI[C][C]</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">I[C][C][C]</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">60000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">―LX</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">100000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">―C or CCCI[C][C][C]</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1000000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">―M or CCCCI[C][C][C][C]</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2000000</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">―MM, &c.</cell></row></table> <pb n="273"/><cb/> <hi rend="center"><hi rend="italics">Greek Numerals.</hi></hi></p><p>The Greeks had three ways of expressing numbers. First, The most simple was, for every single letter, according to its place in the alphabet, to denote a number from <foreign xml:lang="greek">a</foreign> 1 to <foreign xml:lang="greek">w</foreign> 24; in which manner the books of Homer's Ilias are distinguished. Secondly, Another way was by dividing the alphabet into <hi rend="italics">(first)</hi> 8 units, <foreign xml:lang="greek">a</foreign> 1, <foreign xml:lang="greek">b</foreign> 2, &c; <hi rend="italics">(</hi>2<hi rend="italics">nd)</hi> 8 tens, <foreign xml:lang="greek">i</foreign> 10, <foreign xml:lang="greek">k</foreign> 20, &c; <hi rend="italics">(</hi>3<hi rend="italics">d)</hi> 8 hundreds, <foreign xml:lang="greek">r</foreign> 100, <foreign xml:lang="greek">s</foreign> 200, &c: And thousands they expressed by a point or accent under a letter, as <foreign xml:lang="greek">a</foreign> 1000, <foreign xml:lang="greek">b</foreign> 2000, &c. Thirdly, A third way was by six capital letters, thus, I (<foreign xml:lang="greek">ia</foreign> for <foreign xml:lang="greek">mia</foreign>) 1, <foreign xml:lang="greek">*p</foreign> (<foreign xml:lang="greek">wente</foreign>) 5, <foreign xml:lang="greek">*d</foreign> (<foreign xml:lang="greek">deka</foreign>) 10, <foreign xml:lang="greek">*h</foreign> (<foreign xml:lang="greek">*heka<*>on</foreign>) 100, <foreign xml:lang="greek">*x</foreign> (<foreign xml:lang="greek">xilia</foreign>) 1000, <foreign xml:lang="greek">*m</foreign> (<foreign xml:lang="greek">mueia|</foreign>) 10000: and when the letter <foreign xml:lang="greek">*p</foreign> inclosed any of these, except I, it shewed that the inclosed letter was five times its own value, as <06> 50, <07> 500, <08> 5000, <09> 50000. <hi rend="center"><hi rend="italics">Hebrew Numerals.</hi></hi></p><p>The Hebrew alphabet was divided into, Nine Units, as <figure/> 1, <figure/> 2, &c; Nine Tens, as <figure/> 10, <figure/> 20, &c; Nine Hundreds, as <figure/> 100, <figure/> 200, &c, <figure/> 500, <figure/> 600, <figure/> 700, <figure/> 800, <figure/> 900. Thousands were sometimes expressed by the units prefixed to hundreds, as <figure/> 1534, &c; and even to tens, as <figure/> 1070, &c. But more commonly thousands were expressed by the word <figure/> 1000, <figure/> 2000; and <figure/> with the other numerals prefixed to signify the number of thousands, as <figure/> 3000, &c. <hi rend="center"><hi rend="italics">Characters used in Arithmetic and Algebra.</hi></hi></p><p>The first letters of the alphabet, <hi rend="italics">a, b, c,</hi> &c, denote given quantities; and the last letters <hi rend="italics">z, y, x,</hi> &c, denote such as are unknown or sought. Stifelius first used the capitals A, B, C, &c, for the unknown or required quantities. After that, Vieta used the capital vowels A, E, I, O, U, Y for the unknown or required quantities, and the consonants B, C, D, &c, for known or given numbers. Harriot changed Vieta's capitals into the small letters, viz <hi rend="italics">a, e, i, o, u,</hi> for unknown, and <hi rend="italics">b, c, d,</hi> &c, for known quantities. And Descartes changed Harriot's vowels for the latter letters <hi rend="italics">z, y, x,</hi> &c, and the consonants for the leading letters <hi rend="italics">a, b, c, d,</hi> &c.</p><p>Newton denotes the several orders of the fluxions of variable quantities by as many points over the latter letters; <hi rend="center">as <hi rend="italics">x<hi rend="sup">.</hi>, y<hi rend="sup">.</hi>, z<hi rend="sup">.</hi></hi> are the 1st fluxions,</hi> <hi rend="center"><hi rend="italics">x<hi rend="sup">..</hi>, y<hi rend="sup">..</hi>, z<hi rend="sup">..</hi></hi> are the 2d fluxions,</hi> <hi rend="center"><hi rend="italics">x<hi rend="sup">∴</hi>, y<hi rend="sup">∴</hi>, z<hi rend="sup">∴</hi></hi> are the 3d fluxions,</hi> &c, of <hi rend="italics">x, y, z.</hi> And Leibnitz denotes the differentials of the same quantities by prefixing <hi rend="italics">d</hi> to each of them, thus <hi rend="italics">dx, dy, dz.</hi></p><p>Powers of quantities are denoted by placing the index or exponent after them, towards the upper part; thus <hi rend="italics">a</hi><hi rend="sup">2</hi> is the 2d power, <hi rend="italics">a</hi><hi rend="sup">3</hi> the third power, and <hi rend="italics">a</hi><hi rend="sup">n</hi> the <hi rend="italics">n</hi> power of <hi rend="italics">a.</hi> Diophantus marked the powers by their initials, thus <foreign xml:lang="greek">d<hi rend="sup">n</hi>, k<hi rend="sup">n</hi>, dd<hi rend="sup">n</hi>, dk<hi rend="sup">n</hi>, kk<hi rend="sup">n</hi></foreign>, &c, for dynamis, cubus, dynamodynamis, &c, or the 2d, 3d, 4th, &c powers; and the same method has been used by several of the early writers, since the introduction of Algebra into Europe: but the first of them, as Paciolus, Cardan, &c, used no mark for powers, but the words <cb/> themselves. Stifel, and others about his time, used the initials or abbreviations, <figure/>, [dram], <figure/>, [dram] [dram], &c, of res or coss, zenzus, cubus, zenzizenzus, &c, barbarous corruptions of the Italian cosa, census, cubo, censi-census, &c. But he used also numeral exponents, both positive and negative, to the general characters or roots A, B, C, &c. Bombelli used a half circle thus <*> as a general character for the unknown or quantity required to be found in any question, and the several powers of it he denoted by figures set above it; thus <figure/>, are the 1st, 2d, 3d powers of <figure/>; which powers he called dignities. Stevinus used a whole circle for the same unknown quantity, with the numeral index within it, and that both integral and fractional; thus ○0, ○1, ○2, ○3, are the 0, 1, 2, 3 powers of the general quantity ○; also ○<*>, ○1/3, ○1/4, he uses as the square root, cubic root, 4th root of the same; and ○2/3, the cube root of the square, and ○3/2, the square root of the cube, and so on. And these fractional exponents were adopted and farther used by his commentator Albert Girard. So that Stevinus ought to be esteemed the first person who rendered general the notation of all powers and roots in the same way, the former by integral, and the latter by fractional exponents. Harriot denoted his powers by a repetition of the letters; thus <hi rend="italics">a, aa, aaa,</hi> &c. And Descartes, instead of this, set the numeral index at the upper part of the letters, as at present thus <hi rend="italics">a, a</hi><hi rend="sup">2</hi>, <hi rend="italics">a</hi><hi rend="sup">3</hi>, &c. Though, I am informed, by such as have seen Harriot's posthumous papers, that he also there makes use of exponents.</p><p>The character √ is the sign of radicality, or of a root, being derived from the initial R or <hi rend="italics">r,</hi> which was used at first by Paciolus, Cardan, &c. This character √ I first find used by Stifel, in 1544, and by Robert Recorde in 1557. The character √ alone denotes the square root only; but at first they used the initial of the name after it, to denote the several roots: as √<hi rend="italics">q</hi> the quadrate or square root, and √<hi rend="italics">c</hi> the cubic root. But the numeral indices of the root were prefixed by Albert Girard, exactly the same as they are used at present, viz √<hi rend="sup">2</hi>, √<hi rend="sup">3</hi>, √, the 2d, 3d, or 4th root.</p><p>The character + denotes addition, and a positive quantity. At first the word itself was used, plus, piu, or the initial <hi rend="italics">p.</hi> by Paciolus, Cardan, Tartalea, &c. And the character + for addition occurs in Stifelius.</p><p>The character - denotes subtraction, and a negative quantity; which also first occurs in the same author Stifelius. Before that, the word minus, mene, or the initial <hi rend="italics">m.</hi> was used. Other characters have also been sometimes used by other authors, for addition and subtraction; but they are now no longer in use.</p><p>X denotes multiplication, and was introduced by Oughtred.</p><p>÷ denoting division, was introduced by Dr. Pell. Division is also denoted like a fraction, thus <hi rend="italics">a/b</hi> or 6/3 = 2.</p><p>= denotes equality, and was used by Robert Recorde. Descartes uses <figure/> for the same purpose.</p><p>The character :: for proportionality, or equality of ratios, was introduced by Oughtred; as was also the mark <04> for continued proportion. <pb n="274"/><cb/></p><p>> for greater, and [angle] for less, were used by Harriot.</p><p>And <figure/> and <figure/> were used by Oughtred for the same purposes.</p><p>Dr. Pell used <figure/> for involution, and <figure/> for evolution.</p><p><01> denotes a general difference between any two quantities, and was introduced by Dr. Wallis.</p><p>The Parenthesis ( ), as a vinculum, was invented by Albert Girard, and used in such expressions as these, √<*> (72 + √5120), and B (B<hi rend="italics">q</hi> + C<hi rend="italics">q</hi>), both for universal roots, and multiplication, &c.</p><p>The straight-lined vinculum, ―, was used by Victa for the fame purpose; thus ―(A - B) in ―(B + C). <hi rend="center"><hi rend="italics">Characters in Geometry and Trigonometry.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data"><figure/> A Square</cell><cell cols="1" rows="1" role="data">[angle] An Angle</cell></row><row role="data"><cell cols="1" rows="1" role="data">▵ A Triangle</cell><cell cols="1" rows="1" role="data"><figure/> A Rightangle</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/> A Rectangle</cell><cell cols="1" rows="1" role="data"><figure/> Perpendicular</cell></row><row role="data"><cell cols="1" rows="1" role="data">Θ or Θ A Circle</cell><cell cols="1" rows="1" role="data"><figure/> Parallel.</cell></row></table></p></div1><div1 part="N" n="CHARGE" org="uniform" sample="complete" type="entry"><head>CHARGE</head><p>, in Electricity, in a strict sense, imports the accumulation of the electric matter on one surface of an electric, as the Leyden phial, a pane of glass, &c, whilst an equal quantity passes off from the opposite surface. Or, more generally, electrics are said to be charged, when the equilibrium of the electric matter on the opposite surface is destroyed, by communicating one kind of electricity to one side, and the contrary kind to the opposite side: nor can the equilibrium be restored till a communication be made by means of conducting substances between the two opposite surfaces: and when this is done, the electric is said to be discharged. The charge properly refers to one side, in contradistinction from the other; since the whole quantity in the electric is the same before and after the operation of charging; and the operation cannot succeed, unless what is gained on one side is lost by the other, by means of conductors applied to it, and communicating either with the earth, or with a sufficient number of non-electrics. To facilitate the communication of electricity to an electric plate &c, the opposite surfaces are coated with some conducting substance, usually with tin-foil, within some distance of the edge; in consequence of which the electricity communicated to one part of the coating, is readily diffused through all parts of the surface of the electric in contact with it; and a discharge is easily made by forming a communication with any conductor from one coating to the other. If the opposite coatings approach too near each other, the electric matter forces a passage from one surface to the other before the charge is complete. And some kinds of glass have the property of conducting the electricity over the surface, so that they are altogether unsit for the operation of charging and discharging. If indeed the charge be too high, and the glass plate or phial too thin, the attraction between the two opposite electricities forces a passage through the glass, making a spontaneous discharge, and the glass becomes unfit for farther use. See <hi rend="italics">Conductors, Electrics, Leyden Phial,</hi> &c.</p><div2 part="N" n="Charge" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Charge</hi></head><p>, in Gunnery, the load of a piece of ordnance, or the quantity of powder and ball, or shot, with which it is prepared for execution.</p><p>The charge of powder, for proving guns, is equal to the weight of the ball; but forservice, the charge is 1/2 or 1/3 <cb/> the weight of the ball, or still less; and indeed in most cases of service, the quantity of powder used is too great for the intended execution. In the British navy, the allowance for 32 pounders is but 7/16 of the weight of the ball. But it is probable that, if the powder in all ship guns was reduced to 1/3 the weight of the ball, or even less, it would be a considerable advantage, not only by saving ammunition, but by keeping the guns cooler and quieter, and at the same time more effectually injuring the vessels of the enemy. With the present allowance of powder, the guns are heated, and their tackle and furniture strained, and all this only to render the ball less efficacious: for a ball which can but just pass through a piece of timber, and in the passage loses almost all its motion, is found to rend and fracture it much more, than when it passes through with a much greater velocity. See Robins's Tracts, vol. 1. pa. 290, 291.</p><p>Again, the same author observes, that the charge is not to be determined by the greatest velocity that may be produced; but that it should be such a quantity of powder as will produce the least velocity necessary for the purpose in view; and if the windage be moderate, no field-piece should ever be loaded with more than 1/6, or at the utmost 1/5 of the weight of its ball in powder; nor should the charge of any battering piece exceed 1/3 of the weight of its bullet. Ib. pa. 266.</p><p>Different charges of powder, with the same weight of ball, produce different velocities in the ball, which are in the subduplicate ratio of the weights of powder; and when the weight of powder is the same, and the ball varied, the velocity produced is in the reciprocal subduplicate ratio of the weight of the ball: which is agreeable both to theory and practice. See my paper on Gunpowder in the Philos. Trans. 1778, pa. 50; and my Tracts, vol. 1. pa. 266.</p><p>But this is on a supposition that the gun is of an indefinite length; whereas, on account of the limited length of guns, there is some variation from this law in practice, as well as in theory; in consequence of which it appears that the velocity of the ball increases with the charge only to a certain point, which is peculiar to each gun, where the velocity is the greatest; and that by farther increasing the charge, the velocity gradually diminishes, till the bore is quite full of powder. By an easy sluxionary process it appears that, calling the length of the bore of the gun <hi rend="italics">b,</hi> the length of the charge producing the greatest velocity, ought to be <hi rend="italics">b</hi>/2.718281828, or about 3/8 of the length of the bore; where 2.718281828 is the number whose hyp. log. is 1. But, for several reasons, in practice the length of the charge producing the greatest velocity, falls short of that above mentioned, and the more so as the gun is longer. From many experments I have found the length of the chgre producing the greatest velocity, in guns of various lengths of bore, from 15 to 40 calibres, as follows. <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Length of Bore in Calibres.</cell><cell cols="1" rows="1" rend="align=center" role="data">Length of Charge for greatest Veloc.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">15</cell><cell cols="1" rows="1" role="data">3/10</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">20</cell><cell cols="1" rows="1" role="data">3/12</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" role="data">3/16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">40</cell><cell cols="1" rows="1" role="data">3/20.</cell></row></table> <pb n="275"/><cb/></p><p>CHARLES's WAIN, a name by which some of the astronomical writers, in our own language, have called Ursa Major, or the great bear; though some writers say the lesser bear. Indeed both of the two bears have been called waggons or wains, and by the Latins, who have followed the Arabians, two biers, Feretrum majus & minus.</p></div2></div1><div1 part="N" n="CHART" org="uniform" sample="complete" type="entry"><head>CHART</head><p>, or <hi rend="smallcaps">Sea-Chart</hi>, a hydrographical or seamap, for the use of navigators; being a projection of some part of the sea in plano, shewing the sea coasts, rocks, sands, bearings, &c. Fournier ascribes the invention of sea-charts to Henry son of John king of Portugal. These charts are of various kinds, the Plain chart, Mercator's or Wright's chart, the Globular chart, &c.</p><p>In the construction of charts, great care should be taken that the several parts of them preserve their position to one another, in the same order as on the earth; and it is probable that the finding out of proper methods to do this, gave rise to the various modes of projection.</p><p>There are many ways of constructing maps and charts; but they depend chiefly on two principles. First, by considering the earth as a large extended flat surface; and the charts made on this supposition are usually called Plain Charts. Secondly, by considering the earth as a sphere; and the charts made on this principle are sometimes called Globular Charts, or Mercator's Charts, or Reduced Charts, or Projected Charts.</p><p><hi rend="italics">Plain Charts</hi> have the meridians, as well as the parallels of latitude, drawn parallel to each other, and the degrees of longitude and latitude everywhere equal to those at the equator. And therefore such charts must be deficient in several respects. For, 1st, since in reality all the meridians meet in the poles, it is absurd to represent them, especially in large charts, by parallel right lines. 2dly, As plain charts shew the degrees of the several parallels as equal to those of the equator, therefore the distances of places lying east and west, must be represented much larger than they really are. And 3dly, In a plain chart, while the same rhumb is kept, the vessel appears to sail on a great circle, which is not really the case. Yet plain charts made for a small extent, as a few degrees in length and breadth, may be tolerably exact, especially for any part within the torrid zone; and even a plain chart made for the whole of this zone will differ but little from the truth.</p><p><hi rend="italics">Mercator's Chart,</hi> like the plain charts, has the meridians represented by parallel right lines, and the degrees of the paralleis, or longitude, everywhere equal to those at the equator, so that they are increased more and more, above their natural size, as they approach towards the pole; but then the degrees of the meridians, or of latitude, are increased in the same proportion at the same part; so that the same proportion is preserved between them as on the globe itself. This chart has its name from that of the author, Girard Mercator, who first proposed it for use in the year 1556, and made the first charts of this kind; though they were not altogether on true or exact principles, nor does it appear that he perfectly understood them. Neither indeed was the thought originally his own, viz. of lengthening the degrees of the meridian in some proportions for this was hinted by Ptolemy near two thousand year; ago. It was not perfected however till Mr. Wright <cb/> first demonstrated it about the year 1590, and shewed a ready way of constructing it, by enlarging the meridian line by the continual addition of the secants. See his Correction of Errors in Navigation, published in 1599.</p><p><hi rend="italics">Globular Chart,</hi> is a projection so called from the conformity it bears to the globe itself; and was proposed by Messrs Senex, Wilson, and Harris. This is a meridional projection, in which the parallels are equidistant circles, having the pole for their common centre, and the meridians curvilinear and inclined, so as all to meet in the pole, or common centre of the parallels. By which means the several parts of the earth have their proper proportion of magnitude, distance, and situation, nearly the same as on the globe itself; which renders it a good method for geographical maps.</p><p><hi rend="italics">Hydrographical Charts,</hi> are sheets of large paper, on which several parts of the land and sea are described, with their respective coasts, harbours, sounds, flats, rocks, shelves, sands, &c, also the points of the compass, and the latitudes and longitudes of the places.</p><p><hi rend="italics">Selenographic Charts,</hi> are particular descriptions of the appearances, spots and maculæ of the moon.</p><p><hi rend="italics">Topographic Charts,</hi> are draughts of some small parts only of the earth, or of some particular place, without regard to its relative situation, as London, York, &c.</p><p>For the Construction of Charts, see <hi rend="smallcaps">Geography, Maps, Projection</hi>, &c.</p></div1><div1 part="N" n="CHASE" org="uniform" sample="complete" type="entry"><head>CHASE</head><p>, of a Gun, is its bore or cylinder.</p></div1><div1 part="N" n="CHAULNES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CHAULNES</surname> (<foreName full="yes"><hi rend="smallcaps">The Duke De</hi></foreName>)</persName></head><p>, a peer of France, but more honourable and remarkable as an astronomer and mathematician. He was born at Paris Dec. 30, 1714. He soon discovered a singular taste and genius for the sciences; and in the tumults of armies and camps, he cultivated mathematics, astronomy, mechanics, &c. He was named honorary-academician the 27th of February 1743, and few members were more punctual in attending the meetings of that body; where he often brought different constructions and corrections of instruments of astronomy, of dioptrics, and achromatic telescopes. These researches were followed with a new parallactic machine, more solid and convenient than those that were in use; as also with many reflections on the manner of applying the micrometer to those telescopes, and of measuring exactly the value of the parts of that instrument. The duke of Chaulnes proposed many other works of the same kind, when death surprised him the 23d Sept. 1769.</p><p>He had several papers published in the volumes of Memoirs of the Academy of Sciences, as follow:</p><p>1. Observations on some Experiments in the 4th part of the 2d book of Newton's Optics: an. 1755.</p><p>2. Observations on the Platform for dividing mathematical instruments: 1765.</p><p>3. Determination of the distance of Arcturus from the Sun's limb, at the summer solstice: 1765.</p><p>4. On some means of perfecting astronomical instruments: 1765.</p><p>5. O some experiments relative to dioptrics: 1767.</p><p>6. The art of dividing mathematical instruments: 1768.</p><p>7. Observations of the Transit of Venus, June 3, 1769: 1769.</p><p>8. New method of dividing mathematical and astronomical instruments. <pb n="276"/><cb/></p><p>CHAUSE TRAPPES, or <hi rend="italics">Caltrops,</hi> or <hi rend="italics">Crowsfeet,</hi> are iron instruments of spikes about 4 inches long, made like a star, in such a manner that whichever way they fall, one point stands always upwards, like a nail. They are usually thrown and scattered into moats and breaches, to gall the horses feet, and stop the hasty approach of the enemy.</p></div1><div1 part="N" n="CHAZELLES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CHAZELLES</surname> (<foreName full="yes"><hi rend="smallcaps">John Matthew</hi></foreName>)</persName></head><p>, a French mathematician and engineer, was born at Lyons in 1657, and educated there in the college of Jesuits, from whence he removed to Paris in 1675. He first became acquainted with Du Hamel, secretary to the Academy of Sciences, and through him with Cassini, who employed him with himself at the Observatory, where Chazelles greatly improved himself, and also assisted Cassini in the measurement of the southern part of the meridian of France. Having, in 1684, instructed the duke of Montemar in the mathematical sciences, this nobleman procured him the appointment of hydrography-professor to the galleys of Marseilles. In discharging the duties of this department, he made numerous geometrical and astronomical observations, from which he drew a new map of the coast of Provence.— He also performed many other services in that department, and as an engineer, along with the armies and naval expeditions. To make observations in Geography and Astronomy, he undertook also a voyage to the Levant, and among other things he measured the pyramids of Egypt, and found the four sides of the largest of them exactly to face the four cardinal points of the compass. He made a report of his voyage, on his return, to the Academy of Sciences, upon which he was uamed a member of their body in 1695, and had many papers inserted in the volumes of their Memoirs, from 1693 to 1708. Chazelles died at Marseilles the 16th of January 1710.</p><p>CHEMIN <hi rend="italics">des Ronds,</hi> in Fortification, the way of the rounds, or a space between the rampart and the low parapet under it, for the rounds to go about it.</p></div1><div1 part="N" n="CHEMISE" org="uniform" sample="complete" type="entry"><head>CHEMISE</head><p>, a wall that lines a bastion, or ditch, or the like, for its greater support and strength.</p></div1><div1 part="N" n="CHERSONESUS" org="uniform" sample="complete" type="entry"><head>CHERSONESUS</head><p>, a peninsula, or part of the land almost encompassed round with the sea, only joining to the main land by a narrow neck or isthmus. Varenius enumerates 14 of these.</p><p>CHEVAL <hi rend="italics">de Frise,</hi> pl. <hi rend="italics">Chevaux de Frise,</hi> or Friseland horse, so called because it was first used in that country. It consists of a joist or piece of timber, about a foot in diameter, and 10 or 12 long, pierced and transversed with a great number of wooden spikes of 5 or 6 feet long, and armed or pointed with iron. It is sometimes also called <hi rend="italics">turnpike,</hi> or <hi rend="italics">tourniquet.</hi> It is chiefly used to stop a breach, defend a passage, or make a retrenchment to stop the cavalry.</p></div1><div1 part="N" n="CHEVRETTE" org="uniform" sample="complete" type="entry"><head>CHEVRETTE</head><p>, in Artillery, an engine to raise guns or mortars into their carriage. It is formed of two pieces of wood of about 4 feet long, standing upright upon a third, which is square. The uprights are about a foot asunder, and pierced with holes exactly opposite to one another, to receive a bolt of iron, which is put in either higher or lower at pleasure, to serve as a support to a handspike by which the gun is raised up. <cb/></p></div1><div1 part="N" n="CHEYNE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CHEYNE</surname> (<foreName full="yes"><hi rend="smallcaps">George</hi></foreName>)</persName></head><p>, a British physician, and mathematician, was born in Scotland, 1671, and educated at Edinburgh under Dr. Pitcairn. He passed his youth in close study and great abstemiousness; but coming to London, when about 30 years of age, he fuddenly changed his whole manner of living; which had such an effect upon his constitution, that his body grew to a most enormous bulk, weighing it is said about 448 pounds. From this load of oppression however he was afterward in a great measure relieved, by means of a milk and vegetable diet, which reduced his weight to about one-third of what it had been, and restored him to a good state of health; by which his life was prolonged to the 72d year of his age.</p><p>Dr. Cheyne was author of various medical and other tracts, and of a treatise on the Inverse method of Fluxions, under the title of <hi rend="italics">Fluxionum Methodus Inversa; sive quantitatum fluentium leges generaliores:</hi> in 4to, 1703. Upon this book De Moivre wrote some animadversions in an 8vo vol. 1704; which were replied to by Cheyne in 1705.</p></div1><div1 part="N" n="CHILIAD" org="uniform" sample="complete" type="entry"><head>CHILIAD</head><p>, an assemblage of several things ranged by thousands. It was particularly applied to tables of logarithms, because they were at first divided into thousands. Thus, in the year 1624, Mr. Briggs published a table of logarithms for 20 chiliads of absolute numbers; afterward, he published 10 chiliads more; and lastly, one more; making in all 31 chiliads.</p></div1><div1 part="N" n="CHILIAGON" org="uniform" sample="complete" type="entry"><head>CHILIAGON</head><p>, a regular plane figure of a thousand sides and angles. <figure/></p></div1><div1 part="N" n="CHORD" org="uniform" sample="complete" type="entry"><head>CHORD</head><p>, a right line connecting the two extremes of an arch; so called from its resemblance to the chord or string of a bow; as AB, which is common to the two parts or arches ADB, AEB that make up the whole circle. The chords have several properties:</p><p>1. The Chord is bisected by a perpendicular CF drawn to it from the centre.</p><p>2. Chords of equal arcs, in the same or equal circles, are themselves equal.</p><p>3. Unequal Chords have to one another a less ratio than that of their arcs.</p><p>4. The chord of an arc, is a mean proportional between the diameter and the versed sine of that arc.</p><p><hi rend="italics">Scale or Line of Chords.</hi> See <hi rend="italics">Plane Scale.</hi></p><div2 part="N" n="Chord" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Chord</hi></head><p>, or <hi rend="smallcaps">Cord</hi>, in Music, denotes the string or line by whose vibrations the sensation of sound is excited; and by whose divisions the several degrees o tune are determined.</p><p><hi rend="italics">To divide a Chord</hi> AB <hi rend="italics">in the most simple manner, so as to exhibit all the original concords.</hi> <figure/> <pb n="277"/><cb/></p><p>Divide the given line into two equal parts at C; then subdivide the part CB equally in two at D, and again the part CD into two equal parts at E. Here AC to AB is an octave; AC to AD a fifth; AD to AB a fourth; AC to AE a greater third, and AE to AD a less third; AE to EB a greater sixth, and AE to AB a less sixth. Malcolm's Treatise of Music, ch. 6. sec. 3. See <hi rend="smallcaps">Monochord.</hi></p><p><hi rend="italics">To find the number of Vibrations made by a Musical Chord or String in a given time;</hi> having given its weight, length, and tension. Let <hi rend="italics">l</hi> be the length of the chord in feet, 1 its weight, or rather a small weight fixed to the middle and equal to that of the whole chord, and <hi rend="italics">w</hi> the tension, or a weight by which the chord is stretched. Then shall the time of one vibration be expressed by 11/7√<hi rend="italics">l</hi>/(32 1/6<hi rend="italics">w</hi>), and consequently the number of vibrations per second is equal to 7/11√(32 1/6<hi rend="italics">w./l</hi>)</p><p>For example, suppose <hi rend="italics">w</hi> = 28800, or the tension equal to 28800 times the weight of the chord, and the length of it 3 feet; then the last theorem gives 354 nearly for the number of vibrations made in each second of time.</p><p>But if <hi rend="italics">w</hi> were 14400, there would be made but 250 vibrations per second; and if <hi rend="italics">w</hi> were only 288, there would be no more than 35 16/45 vibrations per second. See my Select Exerc. prob. 21. pag. 200.</p></div2><div2 part="N" n="Chord" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Chord</hi></head><p>, in Music, is used for the union of two or more sounds uttered at the same time, and forming together a complete harmony.</p><p>Chords are divided into perfect and imperfect. The perfect chord is composed of the fundamental sound below, of its third, its fifth, and its octave: they are likewise subdivided into major and minor, according as the thirds which enter into their composition are flat or sharp. Imperfect chords are those in which the sixth, instead of the fifth, prevails; and in general all those whose lowest are not their fundamental sounds.</p><p>Chords are again divided into consonances and dissonances. The consonances are the perfect chord, and its derivatives. Every other chord is a dissonance. A table of both, according to the system of M. Rameau, may be seen in Rousseau's Musical Dictionary, vol. 1, pa. 27.</p></div2></div1><div1 part="N" n="CHOROGRAPHY" org="uniform" sample="complete" type="entry"><head>CHOROGRAPHY</head><p>, the art of delineating or describing some particular country or province.</p><p>This differs from geography as the description of a particular country differs from that of the whole earth: And from topography, as the description of a country differs from that of a town or a district.</p></div1><div1 part="N" n="CHROMATIC" org="uniform" sample="complete" type="entry"><head>CHROMATIC</head><p>, a species of music which proceeds by semitones and minor thirds. The word is derived from the Greek <foreign xml:lang="greek">xrwma</foreign>, which signifies colour, and perliaps the shade or intermediate shades of colour, which mingle and connect colours, like as the small intervals in this scale easily slide or run into each other.</p><p>Boethius and Zarlin ascribe the invention of the chromatic genus to Timotheus, a Milesian, in the time of Alexander the Great. The Spartans banished it their city on account of its softness. The character of this genus, according to Aristides Quintillianus, was sweetness and pathos. <cb/></p></div1><div1 part="N" n="CHROMATICS" org="uniform" sample="complete" type="entry"><head>CHROMATICS</head><p>, is that part of optics which explains the several properties of the colours of light, and of natural bodies.</p><p>Before the time of Sir I. Newton, the notions concerning colour were very vague and wild. The Pythagoreans called colour the superficies of bodies: Plato said that it was a flame issuing from them: According to Zeno, it is the first consiguration of matter: And Aristotle said it was that which made bodies actually transparent. Descartes accounted colour a modification of light, and he imagined that the difference of colour proceeds from the prevalence of the direct or rotatory motion of the particles of light. Grimaldi, Dechales, and many others, imagined that the differences of colour depended upon the quick or slow vibrations of a certain elastic medium with which the universe is silled. Rohault conceived, that the different colours were made by the rays of light entering the eye at different angles with respect to the optic axis. And Dr. Hooke imagined that colour is caused by the sensation of the oblique or uneven pulse of light; which being capable of no more than two varieties, he concluded there could be no more than two primary colours.</p><p>Sir I. Newton, in the year 1666, began to investigate this subject; when finding that the coloured image of the sun, formed by a glass prism, was of an oblong, and not of a circular form, as, according to the laws of equal refraction, it ought to be, he conjectured that light is not homogeneal; but that it consists of rays of different colours, and endued with divers degrees of refrangibility. And, from a farther prosecution of his experiments, he concluded that the different colours of bodies arise from their reflecting this or that kind of rays most copiously. This method of accounting for the different colours of bodies soon became generally adopted, and still continues to be the most prevailing opinion. It is hence agreed that the light of the sun, which to us seems white and perfectly homogeneal, is composed of no fewer than seven different colours, viz red, orange, yellow, green, blue, purple, and violet or indigo: that a body which appears of a red colour, has the property of reflecting the red rays more plentifully than the rest; and so of the other colours, the orange, yellow, green, &c: also that a body which appears black, instead of reflecting, absorbs all or the most part of the rays that fall upon it; while, on the contrary, a body which appears white, reflects the greatest part of all the rays indiscriminately, without separating them one from another.</p><p>The foundation of a rational theory of colours being thus laid, the next inquiry was, by what peculiar mechanism, in the structure of each particular body, it was fitted to reflect one kind of rays more than another; and this is attributed, by Sir I. Newton, to the density of these bodies. Dr. Hooke had remarked, that thin transparent substances, particularly soap-water blown into bubbles, exhibited various colours, according to their thinness; and yet, when they have a considerable degree of thickness, they appear colourless. And Sir Isaac himself had observed, that as he was compressing two prisms hard together, in order to make their sides (which happened to be a little convex) to touch one another, in the place of contact they were <pb n="278"/><cb/> both perfectly transparent, as if they had been but one continued piece of glass: but round the point of contact, where the glasses were a little separated from each other, rings of different colours appeared. And when he afterwards, farther to elucidate this matter, employed two convex glasses of telescopes, pressing their convex sides upon one another, he observed several series of circles or rings of such colours, different, and of various intensities, according to their distance from the common central pellucid point of contact.</p><p>As the colours were thus found to vary according to the different distances between the glass plates, Sir Isaac conceived that they proceeded from the different thickness of the plate of air intercepted between the glasses; this plate of air being, by the mere circumstance of thinness or thickness, disposed to reflect or transmit the rays of this or that particular colour. Hence therefore he concluded, that the colours of all natural bodies depend on their density, or the magnitude of their component particles: and hence also he constructed a table, in which the thickness of a plate necessary to reflect any particular colour, was expressed in millionth parts of an inch.</p><p>From a great variety of such experiments, and observations upon them, our author deduced his theory of colours. And hence it seems that every substance in nature is transparent, provided it be made sufficiently thin; as gold, the densest substance we know of, when reduced into thin leaves, transmits a bluish-green light through it. If we suppose any body therefore, as gold for instance, to be divided into a vast number of plates, so thin as to be almost perfectly transparent; it is evident that all, or the greatest part of the rays, will pass through the upper plates, and when they lose their force will be reflected from the under ones. They will then have the same number of plates to pass through which they had penetrated before; and thus, according to the number of those plates through which they are obliged to pass, the object appears of this or that colour, just as the rings of colours appeared different in the experiment of the two plates, according to their distance from one another, or the thickness of the plate of air between them.</p><p>This theory of the colours has been illustrated and confirmed by various experiments, made by other phylosophers. Mr. E. H. Delaval produced similar effects by the infusions of slowers of different colours, and by the intimate mixture of the metals with the substance of glass, when they are reduced to very fine parts; the more dense metals imparting to the glass the less refrangible colours, and the lighter ones those colours that are more easily refrangible. Dr. Priestley and Mr. Canton also, by laying very thin leaves or slips of the metals upon glass, ivory, wood, or metal, and passing an electrical stroke through them, found that the same effect was produced, viz, that the substrated was tinged with different colours, according to the distance from the point of explosion.</p><p>However, the Abbe Mazeas and M. du Tour contended, that the colours between the glasses are not to be ascribed to the thin stratum of air, since they equally produced them by rubbing and pressing together two flat glasses, which cohered so closely that it required <cb/> the greatest force to move or slide them over one another. See Priestley's History of Vision. <hi rend="center"><hi rend="italics">Of Newton's</hi> 8<hi rend="italics">th Exper. in the</hi> 2<hi rend="italics">d Book of Optics.</hi></hi></p><p>The event of this experiment, which has been contradicted by repetitions of the same by other philosophers, having been the occasion of much controversy; and relating to a material part of the doctrine of chromatics, it will not be improper here to give an account of what has passed concerning it. Newton found, he says, that when light, by contrary refractions through different mediums, is so corrected, that it emerges in lines parallel to the incident rays, it continues ever after to be white. But that if the emergent rays be inclined to the incident ones, the whiteness of the emerging light will, by degrees, in passing on from the place of emergence, become tinged at its edges with colours. And these laws he inferred from experiments made by refracting light with prisms of glass, placed within a prismatic vessel of water.</p><p>By theorems deduced from this experiment he infers, that the refraction of the rays of every sort, made out of any medium into air, are known by having the refraction of the rays of any one sort: and also, that the refraction out of one medium into another is found, whenever we have the refractions out of them both, into any third medium.</p><p>Now the same experiment, when since performed by other persons, turning out contrary to what is stated above, some rather free reflections have been thrown upon Newton concerning it; but which however have been very satisfactorily obviated by Mr. Peter Dollond, in a late pamphlet on this subject; as we shall shew below.</p><p>In the first place then, M. Klingenstierna, a Swedish philosopher, having in the year 1755 considered the controversy between Euler and Mr. John Dollond, relative to the refraction of light, formed a theorem of his own, from geometrical reasoning, by which he was induced to believe that the result of Newton's experiment could not be as he had related it; except when the angles of the refracting mediums are small. See the paper on this matter by Klingenstierna in the pamphlet above cited by Mr. Peter Dollond.</p><p>This paper of Klingenstierna being communicated to Mr. John Dollond by Mr. Mallet, to whom it was sent for that purpose, made Dollond entertain doubts concerning Newton's report of the result of his experiment, and determined him to have recourse to experiments of his own, which he did in the year 1757, as follows.</p><p>He cemented two glass planes together by their edges, so as to form a prismatic vessel when closed at the ends or bases; and the edge being turned downward, he placed it in a glass prism with one of its edges upward, filling up the vacancy with clear water; so that the refraction of the prism was contrived to be contrary to that of the water, in order that a ray of light, transmitted through both these refracting mediums, might be affected by the difference only between the two refractions. As he found the water to refract more or less than the glass prism, he diminished or augmented the angle between the glass plates, till the two contrary refractions became equal, which he discovered by view- <pb n="279"/><cb/> ing an object through this double prism. For when it appeared neither raised nor depressed, he was satisfied that the refractions were equal, and that the emergent rays were parallel to the incident ones.</p><p>Now, according to the prevailing opinion, he observes, that the object ought to have appeared through this double prism in its natural colour; for if the difference of refrangibility had been in all respects equal, in the two equal refractions, they would have rectified each other. But this experiment fully proved the fallacy of the received opinion, by shewing that the divergency of the light by the glass prism, was almost double of that by the water; for the image of the object, though not at all refracted, was yet as much infected with prismatic colours, as if it had been seen through a glass wedge only, having its angle of near 30 degrees.</p><p>This experiment is the very same with that of Sir Isaac Newton above-mentioned, not withstanding the result was so remarkably different. Mr. Dollond plainly saw however, that if the refracting angle of the watervessel could have admitted of a sufficient increase, the divergency of the coloured rays would have been greatly diminished, or entirely rectified; and that there would have been a very great refraction without colour, as he had already produced a great discolouring without refraction: but the inconveniency of so large an angle as that of the prismatic vessel must have been, to bring the light to an equal divergency with that of the glass prism, whose angle was about 60 degrees, made it necessary to try some experiments of the same kind with smaller angles.</p><p>Accordingly he procured a wedge of plate-glass, whose angle was only 9 degrees; and, using it in the same circumstances, he increased the angle of the water-wedge, in which it was placed, till the divergency of the light by the water was equal to that by the glass; that is, till the image of the object, though considerably refracted by the excess of the refraction of the water, appeared nevertheless quite free from any colours proceeding from the different refrangibility of the light.</p><p>Many conjectures were made as to the cause of so striking a difference in the results of the same experiment; but none that gave any great satisfaction, till lately that it has been shewn to be probably owing to the nature of the glass then used by Newton. This conjecture is made by Mr. Peter Dollond, son of John, the inventor of the achromatic telescope, in a pamphlet by him lately published in defence of his father's invention, against the misrepresentations of some persons who have unjustly attempted to give the invention to other philosophers, who themselves never imagined that they had any right to it. After a full and satisfactory vindication of his father, Mr. P. Dollond then adds,</p><p>“I now come to a more agreeable part of this paper, which is, to endeavour to reconcile the different results of the 8th experiment of the 2d part of the 1st book of Newton's Optics, as related by himself, and as it was found by Dollond, when he tried the same experiment, in the year 1757. Newton says, that light, as often as by contrary refractions it is so corrected, that it emergeth in lines parallel to the incident, continues ever after to be white. Now Dollond says, when he tried the same <cb/> experiment, and made the mean refraction of the water equal to that of the glass prism, so that the light emerged in lines parallel to the incident, he found the divergency of the light by the glass prism to be nearly double to what it was by the water prism. The light appeared to be so evidently coloured, that it was directly said by some persons, that if Newton had actually tried the experiment, he must have perceived it to have been so. Yet who could for a moment doubt the veracity of such a character? Therefore different conjectures were made by different persons. Mr. Murdoch in particular gave a paper to the Royal Society in defence of Newton; but it was such as very little tended to clear up the matter. Philos. Trans. vol. 53. pa. 192.—Some have supposed that Newton made use of water strongly impregnated with saccharum saturni, because he mentions sometimes using such water, to increase the refraction, when he used water prisms instead of glass prisms. Newton's Opt. pa. 62.—And others have supposed, that he tried the experiment with so strong a persuasion in his own mind that the divergency of the colours was always in the same proportion to the mean refraction, in all sorts of refracting mediums, that he did not attend so much to that experiment as he ought to have done, or as he usually did. None of these suppositions having appeared at all satissactory, I have therefore endeavoured to find out the true cause of the difference, and thereby shew, how the experiment may be made to agree with Newton's description of it, and to get rid of those doubts, which have hitherto remained to be cleared up.</p><p>“It is well known, that in Newton's time the English were not the most famous for making optical instruments: Telescopes, opera-glasses, &c, were imported from Italy in great numbers, and particularly from Venice; where they manufactured a kind of glass which was much more proper for optical purposes than any made in England at that time. The glass made at Venice was nearly of the same refractive quality as our own crown-glass, but of a much better colour, being sufficiently clear and transparent for the purpose of prisms. It is probable that Newton's prisms were made with this kind of glass; and it appears to be the more so, because he mentions the specific gravity of common glass to be to water as 2.58 to 1, Newton's Opt. pa. 247, which nearly answers to the specisic gravity we sind the Venetian glass generally to have. Having a very thick plate of this kind of glass, which was presented to me about 25 years ago by the late professor Allemand, of Leyden, and which he then informed me had been made many years; I cut a piece from this plate of glass to form a prism, which I conceived would be similar to those made use of by Newton himself. I have tried the Newtonian experiment with this prism, and find it answers so nearly to what Newton relates, that the difference which remains may very easily be supposed to arise from any little difference which may and does often happen in the same kind of glass made at the same place at different times. Now the glass prism made use of by Dollond to try the same experiment, was made of English flint-glass, the specific gravity of which I have never known to be less than 3.22. This difference in the densities of the prisms, <pb n="280"/><cb/> used by Newton and Dollond, was sufficient to cause all the difference which appeared to the two experimenters in trying the same experiment.</p><p>“From this it appears, that Newton was accurate in this experiment as in all others, and that his not having discovered that, which was discovered by Dollond so many years afterwards, was owing entirely to accident; for if his prism had been made of glass of a greater or less density, he would certainly have then made the discovery, and refracting telescopes would not have remained so long in their original imperfect state.” See <hi rend="italics">Achromatic,</hi> and <hi rend="italics">Telescope.</hi></p><p><hi rend="italics">Mr. Delaval's experiments on the colours of opaque bodies.</hi>—Beside the experiments of this gentleman, before-mentioned, on the colours of transparent bodies, he has lately published an account of some made upon the permanent colours of opaque substances, the discovery of which must be of the utmost consequence in the arts of colour-making and dyeing.</p><p>The changes of colour in permanently coloured bodies, our author observes, are produced by the same laws that take place in transparent colourless substances; and the experiments by which they are investigated consist chiefly of various methods of uniting the colouring particles into larger masses, or dividing them into smaller ones. Sir Isaac Newton made his experiments chiefly on transparent substances; and in the few places where he treats of others, he acknowledges his want of experiments. He makes the following remark however on those bodies which reflect one kind of light and transmit another, viz, that if these glasses or liquors were so thick and massy that no light could get through them, he questioned whether they would not, like other opaque bodies, appear of one and the same colour in all positions of the eye; though he could not yet affirm it from experience. Indeed it was the opinion of this great philosopher, that all coloured matter reflects the rays of light, some reflecting the more refrangible rays most copiously, and others those that are less so; and that this is at once the true and only reason of these colours. He was likewise of opinion that opaque bodies reflect the light from their anterior surface, by some power of the body evenly disfused over and external to it. With respect to transparent coloured bodies he thus expresses himself: “A transparent body which looks of any colour by transmitted light, may also look of the same colour by reflected light; the light of that colour being reflected by the farther surface of that body, or by the air beyond it: and then the reflected colour will be diminished, and perhaps cease, by making the body very thick, and pitching it on the back-side to diminish the reflection of its farther surface, so that the light reflected from the tinging particles may predominate. In such cases the colour of the reflected light will be apt to vary from that of the light transmitted.”</p><p>To search out the truth of these opinions, Mr. Delaval entered upon a course of experiments with transparent coloured liquors and glasses, as well as with opaque and semitransparent bodies. And from these experiments he discovered several remarkable properties of the colouring matter; particularly, that in transparent coloured substances it does not reflect any light; <cb/> and when, by intercepting the light which was transmitted, it is hindered from passing, through such substances, they do not vary from their former colour to any other, but become entirely black.</p><p>This incapacity of the colouring particles of transparent bodies to reflect light, being deduced from very numerous experiments, may therefore be taken as a general law. It will appear the more extensive, if it be considered that, for the most part, the tinging particles of liquors, or other transparent substances, are extracted from opaque bodies; that the opaque bodies owe their colours to those particles, in like manner as the transparent substances do; and that by the loss of them they are deprived of their colours.</p><p>Notwithstanding these and many other experiments, the theory of colours seems not yet determined with certainty; and it must be acknowledged that very strong objections might be brought against every hypothesis on this subject that has been invented. The discoveries of Sir Isaac Newton however are sufficient to justify the following Aphorisms.</p><p><hi rend="italics">Aphorism</hi> 1. All the colours in nature arise from the rays of light.</p><p>2. There are seven primary colours, namely red, orange, yellow, green, blue, indigo, and violet.</p><p>3. Every ray of light may be separated into these seven primary colours.</p><p>4. The rays of light, in passing through the same medium, have different degrees of refrangibility.</p><p>5. The difference in the colours of light arises from its different refrangibility: that which is the least refrangible producing red; and that which is the most refrangible, violet.</p><p>6. By compounding any two of the primary, as red and yellow, or yellow and blue, the intermediate colour, orange or green, may be produced.</p><p>7. The colours of bodies arise from their dispositions to reflect one sort of rays, and to absorb the others: those that reflect the least refrangible rays appearing red; and those that reflect the most refrangible, violet.</p><p>8. Such bodies as reflect two or more sorts of rays, appear of various colours.</p><p>9. The whiteness of bodies arises from their disposition to reflect all the rays of light promiscuously.</p><p>10. The blackness of bodies proceeds from their incapacity to reflect any of the rays of light.—And from their thus absorbing all the rays of light that are thrown upon them, it arises, that black bodies, when exposed to the sun, become hot sooner than all others.</p><p><hi rend="italics">Of the Diatonic Scale of Colours.</hi>—Sir Isaac Newton, in the course of his investigations of the properties of light, discovered that the lengths of the spaces occupied in the spectrum by the seven primary colours, exactly correspond to the lengths of chords that sound the seven notes in the diatonic scale of music: which is made evident by the following experiment. <figure/> <pb n="281"/><cb/></p><p>On a paper, in a dark chamber, let a ray of light be largely refracted into the spectrum ABCDEF, marking upon it the precise boundaries of the several colours, as <hi rend="italics">a, b, c,</hi> &c; and across the spectrum draw the perpendicular lines <hi rend="italics">ag, bh,</hi> &c. Then it will be found that the spaces, by which the several colours are bounded, viz, B<hi rend="italics">ag</hi>F containing the red, <hi rend="italics">abhg</hi> containing the orange, <hi rend="italics">bcih</hi> containing the yellow, &c, will be in exact proportion to the divisions of a musical chord for the notes of an octave; that is, as the intervals of these numbers 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/16, 1/2.</p></div1><div1 part="N" n="CHRONOLOGICAL" org="uniform" sample="complete" type="entry"><head>CHRONOLOGICAL</head><p>, belonging to chronology.</p><p><hi rend="italics">Chronological Characters,</hi> are characters by which times are distinguished. Of these, some are natural, or astronomical; others are artificial or historical. The natural characters are such as depend on the motions of the stars or luminaries; as eclipses, solstices, equinoxes, the different aspects of planets, &c. And the artificial characters are those that have been invented and established by men; as the solar cycle, the lunar cycle, &c. Historical Chronological Characters are those supported by the testimonies of historians, when they fix the dates of certain events to certain periods.</p></div1><div1 part="N" n="CHRONOLOGY" org="uniform" sample="complete" type="entry"><head>CHRONOLOGY</head><p>, the art of measuring and distinguishing time; with the doctrine of dates, epochs, eras, &c.</p><p>The measurement of time in the most early periods, was by means of the seasons, or the revolutions of the sun and moon. The succession of Juno's priestesses at Argos served Hellanicus for the regulation of his narrative; while Ephorus reckoned his matters by generations. Even in the histories of Herodotus and Thucydides, there are no regular dates for the events recorded; nor were there any endeavours to establish a fixed era until the time of Ptolomy Philadelphus, who attempted it by comparing and correcting the dates of the olympiads, the kings of Sparta, and the succession of the priestesses of Juno at Argos. Eratosthenes and Apollodorus digested the events related by them, according to the succession of the olympiads and of the Spartan kings.</p><p>The chronology of the Latins is still more uncertain. The records of the Romans were destroyed by the Gauls; and Fabius Pictor, the most ancient of their historians, was obliged to borrow the chief part of his information from the Greeks. In other European nations the chronology is still more imperfect, and of a later date: and even in modern times a considerable degree of confusion and inaccuracy has arisen, from the want of attention in the historians to ascertain the dates and epochs with precision.</p><p>Hence it is evident, how necessary a proper system of chronology must be for the right understanding of history, and also how difficult it must be to establish such a system. For this purpose, however, several learned men have spent much time, particularly Julius Africanus, Eusebius of Cæsarea, George Cyncelle, John of Antioch, Dennis, Petau, Clavius, Calvisius, Scaliger, Vieta, Newton, Usher, Simson, Marsham, Helvicus, Vossius, Strauchius, Blair, and Playfair.</p><p>Such a system is founded, 1st, On astronomical observations, especially of the eclipses of the sun and moon, combined with calculations of the years and eras of different nations. 2d, The testimonies of cre- <cb/> dible authors. 3d, Such epochs in history as are so well attested and determined, that they have never been controverted. 4th, Ancient medals, coins, monuments, and insoriptions.</p><p>The most obvious division of time, as has been observed, is derived from the apparent or real revolutions of the luminaries, the sun and moon. Thus, the apparent revolution of the sun, or the real rotation of the earth on her axis causing the sun to appear to rise and set, constitutes the vicissitudes of day and night, which must be evident to the most barbarous and ignorant nations. The moon, by her revolution about the earth, and her changes, as naturally and obviously forms months; while the great annual course of the sun through the several constellations of the zodiac, points out the larger division of the year.</p><p>The Day is divided into hours, minutes, &c; while the month is divided into weeks, and the year into months, having particular names, and a certain number of days.—See a particular account of each of these under the respective words.</p><p>Beside the natural divisions of time arising immediately from the revolutions of the heavenly bodies, there are others which are formed from some of the less obvious consequences of these revolutions, and are called <hi rend="italics">cycles,</hi> or circles. The most remarkable of these are, 1, The <hi rend="italics">Solar Cycle,</hi> or cycle of the sun, a period or revolution of 28 years, in which time the days of the months return to the same days of the week, the sun's place to the same signs and degrees of the ecliptic on the same months and days, and the leap-years begin the same course over again with respect to the days of the week on which the days of the months fall. 2, The <hi rend="italics">Lunar Cycle,</hi> or cycle of the moon, commonly called the Golden Number, is a revolution of 19 years; in which time the conjunctions, oppositions, and other aspects of the moon, are on the same days of the months as they were 19 years before, and within an hour and a half of the same time of the day.</p><p>The <hi rend="italics">Indiction,</hi> or Roman Indiction, is a period of 15 years, used only by the Romans for indicating the times of certain payments made by the subjects to the republic.</p><p>The <hi rend="italics">Cycle of Easter,</hi> called also the <hi rend="italics">Dionysian Period,</hi> is a revolution of 532 years, and is produced by multiplying the solar cycle 28, by the lunar cycle 19.</p><p>The <hi rend="italics">Julian Period,</hi> is a revolution of 7980 years, and is produced from the continual multiplication of the three numbers 28, 19, 15, of the three former cycles, viz, the solar, lunar, and indiction.</p><p>As there are certain fixed points in the heavens, from which astronomers begin their computations, so there are certain points of time, from which historians begin to reckon; and these points or roots of time are called <hi rend="italics">eras</hi> or <hi rend="italics">epochs.</hi> The most remarkable of these are, those of the Creation, the Greek Olympiads, the building of Rome, the era of Nabonnassar, the death of Alexander, the birth of Christ, the Arabian Hegira, or flight of Mahomet, and the Persian Jesdegird. All which, with some others of less note, have their beginnings fixed by chronologers to the years of the Julian period, to the age of the world, and to the years before and after the birth of Christ.</p><p>The testimony of authors is the second principal <pb n="282"/><cb/> part of historical chronology. Though no man has a right to be considered as infallible, it would however be making a very unfair judgment of mankind, to treat them all as dupes or impostors; and it would be an injury offered to public integrity, to doubt the veracity of authors universally esteemed, and facts that are truly worthy of belief. When the historian is allowed to be completely able to judge of an event, and to have no intent of deceiving by his relation, his testimony cannot be refused.</p><p>The <hi rend="italics">Epochs</hi> form the 3d principal part of chronology; being those fixed points in history that have never been contested, and of which there cannot reasonably be any doubt. Notwithstanding that chronologers fix upon the events which are to serve as epochs, in a manner quite arbitrary; yet this is of little consequence, provided the dates of these epochs agree, and that there is no contradiction in the facts themselves.</p><p>Medals, Monuments, and Inscriptions, form the last of the four principal parts of chronology; and this study is but of very modern date, scarce more than 150 years having elapsed since close application has been made to the study of these. To the celebrated Spanheim we owe the greatest obligations, for the progress that is made in this method; and it is by the aid of medals that M. Vaillant has composed his judicious history of the kings of Syria, from the time of Alexander the Great to that of Pompey. Nor have they been of less service in elucidating all ancient history, especially that of the Romans; and even sometimes that of the middle ages.</p><p>Besides the foregoing general account, there are some few systems of chronology which may deserve some more particular notice, as follows.</p><p><hi rend="italics">Sacred Chronology.</hi> There have been various systems relating to sacred chronology; which is not to be wondered at, as the three chief copies of the Bible give a very different account of the first ages of the world. For while the Hebrew text reckons about 4000 years from the creation to the birth of Christ, and to the flood 1656 years; the Samaritan makes the former much longer, though it counts from the creation to the flood only 1307 years; and the Septuagint removes the creation of the world to 6000 years before Christ, and 2250 years before the flood. Many attempts have been made to reconcile these differences; though none of them are quite satisfactory. Walton and Vossius give the preference to the accountof the Septuagint; while others have defended the Hebrew text. See an abstract of the different opinions of learned men on this subject, in Strauchius's Brev. Chron. translated by Sault, p. 166 and 176.</p><p><hi rend="italics">The Chinese Chronology.</hi> No nation has boasted more of its antiquity than the Chinese: but though they be allowed to trace their origin as far back as the deluge, they have few or no authentic records of their history for so long a period as 500 years before the Christian era. This indeed may be owing to the general destruction of ancient remains by the tyrant Tsin-chi-hoang, in the year 213, or some say 246, before Christ. From a chronology of the Chinese history (for which we are obliged to an illustrious Tartar who was viceroy of Canton in the year 1724, <cb/> and of which a Latin translation was published at Rome in 1730), we learn that the most remote epoch of the Chinese chronology does not surpass the first year of Guei-lie-wang, or 424 years before our vulgar era. And this opinion is confirmed by the practice of two of the most approved historians of China, who admit nothing into their histories previous to this period.</p><p>The Chinese, in their computation, make use of a cycle of 60 years, called kia-tse, from the name given to the first year of it, which serves as the basis of their whole chronology. Every year of this cycle is marked with two letters which distinguish it from the others; and all the years of the emperors, for upwards of 2000 years, have names in history common to them with the corresponding years of the cycle. Philos. Trans. Abridg. vol. 8, part 4, pag. 13.</p><p>According to M. Freret, in his Essays, the Chinese date the epocha of Yao, one of their first emperors, about the year 2145, or 2057, before Christ; and they reckon that their first astronomical observations, and the composition of their calendar, preceded Yao 150 years: from whence it is inferred that the era of their astronomical observations coincides with that of the Chaldeans. But later authors date the rise and progress of the sciences in China from the grand dynasty of Tcheou, about 1200 years before the Christian era, and shew that all historical relations of events prior to the reign of Yao are fabulous. Mem. de l'Histoire des Sciences &c. Chinois. vol. 1, Paris 1776.</p><p><hi rend="italics">Babylonian, Egyptian, and Chaldean Annals.</hi> These, M. Gibert has attempted to reduce to our chronology, in a letter published at Amsterdam in 1743. He begins with shewing, by the authorities of Macrobius, Eudoxus, Varro, Diodorus Siculus, Pliny, Plutarch, St. Augustin, &c, that by a year, the ancients meant the revolution of any planet in the heavens; so that it might consist sometimes of only one day. Thus, according to him, the solar day was the astronomical year of the Chaldeans; and so the boasted period of 473,000 years, assigned to their observations, is reduced to 1297 years 9 months; the number of years which, according to Eusebius, elapsed from the first discoveries of Atlas in astronomy, in the 384th year of Abraham, to the march of Alexander into Asia in the year 1682 of the same era. And the 17,000 years added by Berosus to the observations of the Chaldeans, reduced in the same manner, will give 46 years and 6 or 7 months; being the exact interval between Alexander's march, and the first year of the 123d Olympiad, or the time to which Berosus carried his history.</p><p>Epigenius ascribes 720,000 years to the observations preserved at Babylon; but these, according to M. Gibert's system, amount only to 1971 years 3 months; which differ from Callisthenes's period of 1903 years, allotted to the same observations, only by 68 years, the period elapsed from the taking of Babylon by Alexander, which terminated the latter account, and to the time of Ptolomy Philadelphus, to which Epigenius extended his account.</p><p><hi rend="italics">The Newtonian Principles of Chronology.</hi>—Sir Isaac Newton has shewn, that the chronology of ancient kingdoms is involved in the greatest uncertainty: that <pb n="283"/><cb/> the Europeans in particular had no chronology before the Persian empire, which commenced 536 years before the birth of Christ, when Cyrus conquered Darius the Mede: that the antiquities of the Greeks are full of fables, because their writings were in verse only, till the conquest of Asia by Cyrus the Persian; about which time prose was introduced by Pherecides Syrius and Cadmus Milesius. After this time several of the Greek historians introduced the computation by generations. The chronology of the Latins was still more uncertain: their old records having been burnt by the Gauls 120 years after the expulsion of their kings, or 64 years before the death of Alexander the Great, answering to 388 before the birth of Christ. The chronologers of Gaul, Spain, Germany, Scythia, Sweden, Britain, and Ireland, are of a still later date. For Scythia, beyond the Danube, had no letters till Ulphilas, their bishop, formed them, about the year 276. Germany had none till it received them from the western empire of the Latins, about the year 400. The Huns had none in the days of Procopius, about the year 526. And Sweden and Norway received them still later.</p><p>Sir Isaac Newton, after a general account of the obscurity and defects of the ancient chronology, observes that, though many of the ancients computed by successions and generations, yet the Egyptians, Greeks, and Latins, reckoned the reigns of kings equal to generations of men, and three of them to a hundred, and sometimes to 120 years; and this was the foundation of their technical chronology. He then proceeds, from the ordinary course of nature, and a detail of historical facts, to shew the difference between reigns and generations; and that, though a generation from father to son may at an average be reckoned about 33 years, or three of them equal to 100 years, yet, when they are taken by the eldest sons, three of them cannot be estimated at more than about 75 or 80 years; and the reigns of kings are still shorter; so that 18 or 20 years may be allowed a just medium. Sir Isaac then fixes on four remarkable periods, viz, the return of the Heraclidæ into the Peloponnesus, the taking of Troy, the Argonautic expedition, and the return of Sesostris into Egypt, after his wars in Thrace; and he settles the epoch of each by the true value of a generation. To instance only his estimate of that of the Argonautic expedition: Having fixed the return of the Heraclidæ to about the 159th year after the death of Solomon, and the destruction of Troy to about the 76th year after the same period, he observes, that Hercules the Argonaut was the father of Hyllus, the father of Clerdius, the father of Aristomachus, the father of Aristodemus, who conducted the Heraclidæ into Peloponnesus; so that, reckoning by the chief of the family, their return was four generations later than the Argonautic expedition, which therefore happened about 43 years after the death of Solomon. This is farther confirmed by another argument: Æsculapius and Hercules were Argonauts: Hippocrates was the 18th inclusively from the former by the father's side, and the 19th from the latter by the mother's side: now, allowing 28 or 30 years to each of them, the 17 intervals by the father, and the 18 intervals by the mo- <cb/> ther, will on a medium give 507 years; and these, reckoning back from the commencement of the Peloponnesian war, or the 431st year before Christ, when Hippocrates began to flourish, will place the Argonautic expedition in the 43d year after the death of Solomon, or 937 years before Christ.</p><p>The other kind of reasoning by which Newton endeavours to establish this epoch, is purely astronomical. The sphere was formed by Chiron and Musæus for the use of the Argonautic expedition, as is plainly shewn by several of the asterisins referring to that event: and at the time of the expedition the cardinal points of the equinoxes and solstices were placed in the middle of the constellations Aries, Cancer, Chelæ, and Capricorn. This point is established by Newton from the consideration of the ancient Greek calendar, which consisted of 12 lunar months, and each month of 30 days, which required an intercalary month. Of course this lunisolar year, with the intercalary month, began sometimes a week or two before or after the equinox or solstice; and hence the first astronomers were led to the before-mentioned disposition of the equinoxes and solstices: and that this was really the case, is confirmed by the testimonies of Eudoxus, Aratus, and Hipparchus. Upon these principles Sir Isaac proceeds to argue in the following manner. The equinoctial colure in the end of the year 1689 cut the ecliptic in <figure/> 6° 44′; and by this reckoning the equinox had then gone back 36° 44′ since the time of the Argonautic expedition. But it recedes 50′ in a year, or 1° in 72 years, and consequently 36° 44′ in 2645 years; and this, counted backwards from the beginning of 1690, will place this expedition about 25 years after the death of Solomon. But as there is no necessity for allowing that the middle of the constellations, according to the general account of the ancients, should be precisely the middle between the prima Arietis and ultima Caudæ, our author proceeds to “examine what were those stars through which Eudoxus made the colures to pass in the primitive sphere, and in this way to fix the position of the cardinal points.” Now from the mean of five places he finds, that the great circle, which in the primitive sphere, described by Eudoxus, or which at the time of the Argonautic expedition was the equinoctial colure, did in the end of 1689 cut the ecliptic in <figure/> 6° 29′ 15″. In the same manner our author determines that the mean place of the solstitial colure is <figure/> 6° 28′ 46″, and as it is at right angles with the other, he concludes that it is rightly drawn. And hence he infers that the cardinal points, in the interval between that expedition and the year 1689, have receded from those colures 1<hi rend="sup">s</hi> 6° 29′; which, allowing 72 years to a degree, amounts to 2627 years; and these counted backwards, as above, will place the Argonautic expedition 43 years after the death of Solomon. Our author has, by other methods also of a similar nature, established this epoch, and reduced the age of the world 500 years.</p><p>This elaborate system has not escaped censure; Mess. Freret and Souciet having both attacked it, and on much the same ground: but the sormer has confounded reigns and generations, which are carefully distinguished in this system. The aftronomical objections of both <pb n="284"/><cb/> have been answered by Sir Isaac Newton himself, and by Dr. Halley. Philos. Trans. abr. vol. 8, part 4, pa. 4. Newton's Chronol. ch. 1.</p></div1><div1 part="N" n="CHRONOMETER" org="uniform" sample="complete" type="entry"><head>CHRONOMETER</head><p>, is any instrument or machine used in measuring time; such as dials, clocks, watches, &c.</p><p>The term is however more particularly used for a kind of clock, so contrived as to measure a small portion of time, even to the 16th, or the 40th part of a second; one of this latter kind I have seen, made by an ingenious artist; but it could not be stopped to the 10th part of the proposed degree of accuracy. There is a description of one also in Desaguliers's Experimental Philosophy, invented by the late ingenious Mr. George Graham; which might be of great use for measuring small portions of time in astronomical observations, the time of the fall of bodies, the velocity of running waters, &c. But long intervals of time cannot be measured by it with sufficient exactness, unless its pendulum be made to vibrate in a cycloid; for otherwise it is liable to err considerably, as is the case of all clocks with short pendulums that swing in large arches of the circle.</p><p>Various other contrivances, besides clocks, have been used for measuring time for some particular purposes. See a musical chronometer described in Malcolm's Treatise of Music, pa. 407.</p></div1><div1 part="N" n="CHRONOSCOPE" org="uniform" sample="complete" type="entry"><head>CHRONOSCOPE</head><p>, a word sometimes used for a pendulum, or machine, to measure time.</p><p>CHRYSTALLINE. See <hi rend="smallcaps">Crystalline.</hi></p></div1><div1 part="N" n="CIMA" org="uniform" sample="complete" type="entry"><head>CIMA</head><p>, or <hi rend="smallcaps">Sima</hi>, in Architecture, a member, or moulding, called also ogee, and cimatium.</p></div1><div1 part="N" n="CINCTURE" org="uniform" sample="complete" type="entry"><head>CINCTURE</head><p>, in Architecture, is a ring or list around the shaft of a column, at its top and bottom.</p></div1><div1 part="N" n="CINTRE" org="uniform" sample="complete" type="entry"><head>CINTRE</head><p>, in Building, the mould on which an arch is turned or built; popularly called centre, and sometimes a cradle.</p></div1><div1 part="N" n="CIPHER" org="uniform" sample="complete" type="entry"><head>CIPHER</head><p>, or Cypher, one of the numeral characters, or sigures, thus formed 0. The word comes from the Hebrew <hi rend="italics">saphar,</hi> to number.</p><p>The cipher of itself fignifies nothing, or implies a privation of value; but when combined with other numeral characters, it alters their value in a tenfold proportion, for every cipher so annexed; viz, when set after a figure in common integral arithmetic, it increases its value in that proportion, though it has no effect when set before or to the left hand side of figures; but on the contrary, in decimal arithmetic, it decreases their value in that proportion when set before the figures, but has no effect when set after them. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Thus,</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">is five,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">but</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">is fifty,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and</cell><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">is five hundred;</cell></row></table> whereas 05, or 005, &c, is still but 5 or five. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Also</cell><cell cols="1" rows="1" role="data">.5</cell><cell cols="1" rows="1" role="data">is five tenths,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">but</cell><cell cols="1" rows="1" role="data">.05</cell><cell cols="1" rows="1" role="data">is five hundredths,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and</cell><cell cols="1" rows="1" role="data">.005</cell><cell cols="1" rows="1" role="data">is five thousandths;</cell></row></table> whereas .50, or .500, &c, is still but .5 or five tenths.</p><p>The invention and use of the cipher, as in the common arithmetic and notation of numbers, is one of the happiest devices that can be imagined; and is ascribed to the Indians, by the Arabians, through whom it came into Europe, along with the revival of literature.</p></div1><div1 part="N" n="CIRCLE" org="uniform" sample="complete" type="entry"><head>CIRCLE</head><p>, a plane figure, bounded by a curve line <cb/> which returns into itself, called its circumference, and which is every where equally distant from a point within, called its centre.</p><p>The circumference or periphery itself is called the circle, though improperly, as that name denotes the space contained within the circumference.</p><p>A circle is described with a pair of compasses, fixing one foot in the centre, and turning the other round to trace out the circumference.</p><p>The circumference of every circle is supposed to be divided into 360 equal parts, called degrees, and marked °; each degree into 60 minutes or primes, marked ; each minute into 60 seconds, marked ″; and so on. So 24° 12′ 15″ 20‴, is 24 degrees 12 minutes 15 seconds and 20 thirds.</p><p>Circles have many curious properties, some of the most important of which are these:</p><p>1. The circle is the most capacious of all plain figures, or contains the greatest area within the same perimeter, or has the least perimeter about the same area; being the limit and last of all regular polygons, having the number of its sides infinite.</p><p>2. The area of a circle is always less than the area of any regular polygon circumscribed about it, and its circumference always less than the perimeter of the polygon. But on the other hand, its area is always greater than that of its inscribed polygon, and its circumference greater than the perimeter of the said inscribed polygon. However, the area and perimeter of the circle approach always nearer and nearer to those of the two polygons, as the number of the sides of these is greater; the circle being always limited between the two polygons.</p><p>3. The area of a circle is equal to that of a triangle whose base is equal to the circumference, and perpendicular equal to the radius. And therefore the area of the circle is found by drawing half the circumference into half the diameter, or the whole circumference into the whole diameter, and taking the 4th part of the product. Demonstrated by Euclid.</p><p>4. Circles, like other similar plane figures, are to one another, as the squares of their diameters. And the area of the circle is to the square of the diameter, as 11 to 14 nearly, as proved by Archimedes; or as .7854 to 1 more nearly; or still more nearly as .7853981633,9744830961,5660845819,8757210492, 9234984377,6455243736,1480769541,0157155224, 9657008706,3355292669,9553702162,8318076661, 7734611 + to 1; as it has been found by modern mathematicians.</p><p>In Wallis's Arithmetic of Infinites are contained the first infinite series for expressing the ratio of a circle to the square of its diameter: viz,</p><p>1st, The circle is to the square of its diameter, <pb n="285"/><cb/> or as , by Sir I. Newton; or as , by Gregory and Leibnitz; and a great many other forms of series have been invented by different authors, to express the same ratio between the circle and circumscribed square.</p><p>5. The circumferences of circles are to one another, as their diameters, or radii. And as the areas of circles are proportional to the rectangles of their radii and circumferences; therefore the quadrature of the circle will be effected by the rectification of its circumference; that is, if the true length of the circumference could be found, the true area could be found also. But whilst several mathematicians have endeavoured to determine the true area and circumference, others have even doubted of the possibility of the same. Of this latter opinion is Dr. Isaac Barrow: towards the end of his 15th Mathematical Lecture he says, he is of opinion, that the radius and circumference of a circle, are lines of such a nature, as to be not only incommensurable in length and square, but even in length, square, cube, biquadrate, and all other powers to infinity: for, continues he, the side of the inscribed square is incommensurable to the radius, and the square of the side of the inscribed octagon is incommensurable to the square of the radius; and consequently the square of the octagonal perimeter is incommensurable to the square of the radius: and thus the ambits of all regular polygons, inscribed in a circle, may have their superior powers incommensurate with the co-ordinate powers of the radius; from whence the last polygon, that is, the circle itself, seems to have its periphery incommensurate with the radius. Which, if true, will put a final stop to the quadrature of the circle, since the ratio of the circumference to the radius is altogether inexplicable from the nature of the thing, and consequently the problem requiring the explication of such a ratio is impossible to be solved, or rather it requires that for its solution which is impossible to be apprehended. But, concludes he, this great mystery cannot be explained in a few words: But if time and opportunity had permitted, I would have endeavoured to produce many things for the explication and confirmation of this conjecture. Sir Isaac Newton too, in book 1 of his Principia, has attempted to demonstrate the impossibility of the general quadrature of oval figures, by the description of a spiral, and the impossibility of determining, by a finite equation, the intersections of that oval and spiral, which must be the case, if the oval be quadrable. And several other authors have attempted to demonstrate the impossibility of the general quadrature of the circle by any means whatever. On the other hand, many authors not only believe in the possibility of the quadrature of the circle, but some have even pretended to have discovered the same, and have published to the world their pretended discoveries: of which no one has rendered himself more remarkable than our countryman Mr. Hobbes, though a great scholar, and of excellent understanding in other matters. See <hi rend="smallcaps">Quadrature.</hi></p><p>The approximate quadrature of the circle however, or <cb/> the determination of the ratio between the diameter and the circumference, is what the mathematicians of all ages have successfully attempted, and with different degrees of accuracy, according to the improved state of the science. Archimedes, in his book <hi rend="italics">de Dimensione Circuli,</hi> first gave a near value of that ratio in small numbers, being that of 7 to 22, which are still used as very convenient numbers for this purpose in common measurements. Other and nearer ratios have since been successively assigned, but in larger numbers, <hi rend="center">as 106 to 333,</hi> <hi rend="center">or 113 to 355,</hi> <hi rend="center">or 1702 to 5347,</hi> <hi rend="center">or 1815 to 5702, &c,</hi> which are each more accurate than the foregoing. Vieta, in his <hi rend="italics">Universalium Inspectionum ad Canonum Mathematicum,</hi> published 1579, by means of the inscribed and circumscribed polygons of 393216 sides, has carried the ratio to ten places of figures, shewing that if the diameter of a circle be 1000 &c, the circumference will be greater than 314,159,265,35, but less than 314,159,265,37. And Van Colen, in his book <hi rend="italics">de Circulo & Adscriptis,</hi> has, by the same means, carried that ratio to 36 places of figures; which were also recomputed and confirmed by Willebrord Snell. After these, that indefatigable computer, Mr. Abraham Sharp, extended the ratio to 72 places of figures, in a sheet of paper, published about the year 1706, by means of the series of Dr. Halley, from the tangent of an arc of 30 degrees. And the ingenious Mr. Machin carried the same to a hundred places, by other series, depending on the differences of arcs whose tangents have certain relations to one another. See this method explained in my Mensuration, pa. 120 second edit. And, finally, M. De Lagny, in the Memoirs de l'Acad. 1719, by means of the tangent of the arc of 30 degrees, has extended the same ratio to the amazing length of 128 places of sigures; finding, that, if the diameter be 1000 &c, the circumference will be 31415,92653,58979,32384,62643,38327,95028, 84197,16939,93751,05820,97494,45923,07816, 40628,62089,98628,03482,53421,17067,98214, 80865,13272,30664,70938,446 + or 447 -</p><p>From such methods as the foregoing, a variety of series have been discovered for the length of the circumference of a circle, such as the following, viz, If the diameter be 1, the circumference <hi rend="italics">c</hi> will be variously expressed thus, <pb n="286"/><cb/> And many other series might here be added. See my Mensuration in several places; also my paper on such series in the Philos. Trans. 1776; Euler's <hi rend="italics">Introductio in Analysin Infinitorum</hi>; and many other authors.</p><p>6. Some of the more remarkable properties of the circle, are as follow.</p><p>If two lines AB, CD cut the <figure/> circle, and intersect within it, the angle of intersection E is measured by half the <hi rend="italics">sum</hi> of the intercepted arcs AC, DB.</p><p>But if the lines intersect without the circle, the angle E is measured by half the <hi rend="italics">difference</hi> of the intercepted arcs AC, DB. <figure/></p><p>7. The angle at the centre of a circle is double the angle at the circumference, standing on the same arc; and all angles in the same segment are equal. Also the angle at the centre is measured by the arc it stands upon, and the angle at the circumference by half the same arc. <figure/></p><p>8. If the chords FG, HI cross at right angles, the sums of the opposite arcs are equal; viz . <figure/></p><p>9. If one side NO of a trapezium inscribed in a circle be continued out, the external angle, LOP will be equal to the opposite internal angle M. <figure/></p><p>10. An angle, as RQS, formed by a tangent QR and chord QS, is measured by half the arc of the chord QS, and is equal to any angle T formed in the opposite arc QTS. <figure/></p><p>11. If VW be a diameter, and XYZ a chord perpendicular to it; then is XZ or ZY a mean proportional between the segments YZ, ZW. So that if <hi rend="italics">d</hi> denote the diameter VW, <hi rend="italics">x</hi> the absciss VZ, and <hi rend="italics">y</hi> the ordinate ZX; then is ; which is called the equation to the circle.</p><p>The chord VX is a mean proportional between the diameter VW and the versed sine VZ; and the chord WX is a mean proportional between the diameter and the versed sine WZ; also each versed sine is proportional to the square of the corresponding chord; viz VZ : WZ :: VX<hi rend="sup">2</hi> : WX<hi rend="sup">2</hi>.</p><p>12. When two lines cut the circle, whether they intersect within the circle, or without it, as in the two figures to article 6, the segments between the common intersection and the two points where each line cuts the curve, are reciprocally proportional, and their rectan- <cb/> gles are equal; viz, EA : EC :: ED : EB, or .</p><p>13. In a trapezium inscribed in <figure/> a circle, the rectangle of the two diagonals is equal to the sum of the two rectangles of the two pairs of opposite sides; viz, . <figure/></p><p>14. If any chords EF, EG, drawn from the same point E in the circumference, be cut by any other line <figure/> HI, the rectangles will be all equal which are made of each chord and the part intercepted by this line, viz, .</p><p>15. In a circle whose centre is N <figure/> and radius NO, if two points M, P, in the radius produced, be so placed that the three NM, NO, MP, be in continued proportion; then if from the points M and P lines be drawn to any, or every point in the circumference, as Q; these lines will be always in the given ratio of MO to PO; viz, MQ : PQ :: MO : PO.</p><p>16. If VW, be two points in the diameter, equidistant from the centre T; and if two <figure/> lines be drawn from these to any point X in the circumference; the sum of their squares will be equal to the sum of the squares of the segments of the diameter made by either point; viz, . <figure/></p><p>17. If a line FE perpendicular to the diameter AB, meet any other <figure/> chord CD in the point E; then is .</p><p>18. If upon the diameter GH of <figure/> a circle there be formed a rectangle GHKI, whose breadth GI or HK is equal to GL or HL, the chord of a quadrant, or side of the inscribed square; then if from I and K lines be drawn to any point M in the circle GMH, they will cut the diameter GH in such a manner that .</p><p>19. If the arcs PQ, QR, RS, &c, be equal, and there be drawn the chords PQ, PR, PS, PT, &c, then it <pb n="287"/><cb/> will be PQ : PR :: PR : PQ + PS :: PS : PR + PT :: PT : PS + PV, &c. <figure/></p><p>20. The centre of a circle being O, and P a point in the radius, or in the radius produced; if the circumference be divided into as many equal parts AB, BC, CD, &c, as there are units in 2n, and lines be drawn from P to all the points of division; then shall the continual product of all the alternate lines viz PA X PC X PE &c be = <hi rend="italics">r</hi><hi rend="sup">n</hi> - <hi rend="italics">x</hi><hi rend="sup">n</hi> when P is within the circle, or = <hi rend="italics">x</hi><hi rend="sup">n</hi> - <hi rend="italics">r</hi><hi rend="sup">n</hi> when P is without the circle; and the product of the rest of the lines, viz PB X PD + PF &c = <hi rend="italics">r</hi><hi rend="sup">n</hi> + <hi rend="italics">x</hi><hi rend="sup">n</hi>: where <hi rend="italics">r</hi> = AO the radius, and <hi rend="italics">x</hi> = OP the distance of P from the centre. <figure/></p><p>21. A circle may thus be divided into any number of parts that shall be equal to one another both in area and perimeter. Divide the diameter QR into the same number of equal parts at the points S, T, V, &c; then, on one side of the diameter describe semicircles on the diameters QS, QT, QV, and on the other side of it describe semicircles on RV, RT, RS; so shall the parts 1 7, 3 5, 5 3, 7 1 be all equal, both in area and perimeter. See my Tracts, pa. 93. <figure/></p><p>22. <hi rend="italics">To describe a Circle either about or within a given Regular Polygon.</hi> Bisect two of its angles, or two of its sides, with perpendiculars, and the intersection of the bisecting lines will, in either case, be the centre of the circles.</p><p><hi rend="italics">Parallel,</hi> or <hi rend="italics">Concentric</hi> <hi rend="smallcaps">Circles</hi>, are such as are equally distant from each other in every point of their peripheries; or that have the same centre. As, on the other hand, those are called the <hi rend="italics">eccentric</hi> circles, that have not the same point for their centres.</p><p><hi rend="italics">The Quadrature of the</hi> <hi rend="smallcaps">Circle</hi>, is the manner o sdeseribing, or assigning, a square, whose surface shall be perfectly equal to that of a circle. This problem has exercised the geometricians of all ages, but it is now generally given up as a problem impossible to be effected, <cb/> by most persons that have any just claim to that rank. Des Cartes insists on the impossibility of it, for this reason, that a right line and a circle being of different natures, there can be no strict proportion between them. Dr. Barrow shews the strongest probability of the same thing; and not only that the diameter and circumference themselves, but that all powers of them to infinity, are incommensurate.</p><p>The Emperor Charles V offered a reward of 100,000 crowns to any person who should resolve this celebrated problem: and the States of Holland also proposed a reward for the same thing. See <hi rend="italics">Quadrature.</hi></p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of the Higher Orders,</hi> are curves in which WY<hi rend="sup">m</hi> : YZ<hi rend="sup">m</hi> :: YZ : YX, or WY<hi rend="sup">m</hi> : YZ<hi rend="sup">m</hi> :: YZ<hi rend="sub">n</hi> : YX<hi rend="sup">n</hi>. <figure/></p><p>When <hi rend="italics">m</hi> and <hi rend="italics">n</hi> are each equal to 1, then WY : YZ :: YZ : YX, which is the property of the common circle.</p><p>Put WY = <hi rend="italics">x,</hi> YZ = <hi rend="italics">y,</hi> WX = <hi rend="italics">a</hi>; then is YX - <hi rend="italics">a</hi> - <hi rend="italics">x,</hi> and the above proportions become , and , the equations to curves of this kind.</p><p>Curves defined by this equation will be ovals when <hi rend="italics">m</hi> is an odd number. Thus suppose <hi rend="italics">m</hi> = 1, then the equation becomes <hi rend="italics">y</hi><hi rend="sup">2</hi> = <hi rend="italics">x.</hi> ―(<hi rend="italics">a</hi>-<hi rend="italics">x</hi>) or <hi rend="italics">ax</hi>-<hi rend="italics">x</hi><hi rend="sup">2</hi>, the equation to the common circle. And if <hi rend="italics">m</hi> = 3, it becomes <hi rend="italics">y</hi><hi rend="sup">4</hi> = <hi rend="italics">x</hi><hi rend="sup">3</hi>. ―(<hi rend="italics">a</hi>-<hi rend="italics">x</hi>) or <hi rend="italics">ax</hi><hi rend="sup">3</hi>-<hi rend="italics">x</hi><hi rend="sup">4</hi>, which denotes a curve of this form AB. <figure/></p><p>But when <hi rend="italics">m</hi> denotes an even number, the curve will have two insinite legs. So if <hi rend="italics">m</hi> = 2, the equation becomes <hi rend="italics">y</hi><hi rend="sup">3</hi> = <hi rend="italics">x</hi><hi rend="sup">2</hi>. ―(<hi rend="italics">a</hi>-<hi rend="italics">x</hi>) or <hi rend="italics">ax</hi><hi rend="sup">2</hi>-<hi rend="italics">x</hi><hi rend="sup">3</hi>, for a circle of the 2d order, and which defines one of Newton's defective hyperbolas, being his 37th species of curves, whose asymptote is the right line EF, making an angle of 40 degrees with the absciss AB.</p><p><hi rend="smallcaps">Circle</hi> <hi rend="italics">of Curvature,</hi> or circle of equi-curvature, is that circle which has the same curvature with a given curve at a certain point; or that circle whose radius is equal to the radius of curvature of the given curve at that point.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of the Sphere,</hi> are such as cut the mundane sphere, and have their circumference in its surface.</p><p>These circles are either fixed or moveable.</p><p>The latter are those whose peripheries are in the <pb n="288"/><cb/> moveable or revolving surface; and which therefore move or turn with it; as the meridians, &c. The former, having their periphery in the immoveable surface, do not revolve; as the ecliptic, equator, and its parallels.</p><p>The circles of the sphere are either great or little.</p><p><hi rend="italics">A Great Circle</hi> of the Sphere, is that which divides it into two equal parts or hemispheres, having the same centre and diameter with it. As the horizon, meridian, equator, ecliptic, the colures, and the a<*>muths.</p><p><hi rend="italics">A Little,</hi> or <hi rend="italics">Lesser Circle</hi> of the Sphere, divides the sphere into two unequal parts, having neither the same centre nor diameter with the sphere; its diameter being only some chord of the sphere less than its axis. Such as the parallels of latitude, &c.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of Altitude,</hi> or <hi rend="italics">Almucantars,</hi> are little circles parallel to the horizon, having their common pole in the zenith, and still diminishing as they approach it. They are so called from their use, which is to shew the altitude of a star above the horizon.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of Declination,</hi> are great circles intersecting each other in the poles of the world.</p><p><hi rend="smallcaps">Circle</hi> <hi rend="italics">of Dissipation,</hi> in Optics. See the article <hi rend="smallcaps">Dissipation.</hi></p><p><hi rend="italics">Diurnal</hi> <hi rend="smallcaps">Circles</hi>, are parallels to the equinoctial, supposed to be described by the several stars, and other points of the heavens, in their apparent diurnal rotation about the earth.</p><p><hi rend="smallcaps">Circle</hi> <hi rend="italics">Equant,</hi> in the Ptolomaic Astronomy, is a circle described on the centre of the equant. Its chief use is, to find the variation of the first inequality.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of Excursion,</hi> are little circles parallel to the ecliptic, and at such a distance from it, as that the excursions of the planets towards the poles of the ecliptic, may be included within them; being usually fixed at about 10 degrees.</p><p>It may here be observed, that all the circles of the sphere, described above, are transferred from the heavens to the earth; and so come to have a place in geography as well as in astronomy: all the points of each circle being conceived as let fall perpendicularly on the surface of the terrestrial globe, and thus tracing out circles perfectly similar to them. So, the terrestrial equator is a circle conceived precisely under the equinoctial line, which is in the heavens: and so of the rest.</p><p><hi rend="italics">Horary</hi> <hi rend="smallcaps">Circles</hi>, in Dialling, are the lines which shew the hours on dials. These are straight lines on the dials, but called circles as being the projections of the meridians.</p><p><hi rend="italics">Horary</hi> <hi rend="smallcaps">Circle</hi>, or <hi rend="italics">Hour Circle,</hi> on the globe, is a small brazen circle fixed to the north pole, divided into 24 hours, and furnished with an index to point them out, thereby shewing the difference of meridians in time, and serving for the solution of many problems, on the artificial globes.</p><p><hi rend="smallcaps">Circle</hi> <hi rend="italics">of Illumination,</hi> is that imaginary circle on the surface of the earth, which separates the illuminated side or hemisphere of the earth from the dark side: and all lines passing from the sun to the earth, being physically parallel, are perpendicular to the plane of this circle.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of Latitude,</hi> or <hi rend="italics">Secondaries of the Ecliptic,</hi> are great circles perpendicular to the plane of the ecliptic, <cb/> intersecting one another in its poles, and passing through every star and planet, &c.—These are so-called, because they serve to measure the latitude of the stars, which is an arch of one of these circles, intersected between the star and the ecliptic.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of Longitude,</hi> are lesser circles parallel to the ecliptic, diminishing more and more as they recede from it, or as they approach the pole of that circle.</p><p>They are so called, because the longitudes of the stars are counted upon them.</p><p><hi rend="smallcaps">Circle</hi> <hi rend="italics">of Perpetual Apparition,</hi> one of the lesser circles parallel to the equator, described by the most northern point of the horizon, as the sphere revolves round by its diurnal motion.—All the stars included within this circle, are continually above the horizon, and so never set.</p><p><hi rend="smallcaps">Circle</hi> <hi rend="italics">of Perpetual Occultation,</hi> is another lesser circle at a like distance from the equator, but on the other side of it, being described by the most southern point of the horizon, and contains all those stars which never appear in our hemisphere, or which never rise.</p><p>All other stars, being contained between these two circles, do alternately rise and set, at certain moments of the diurnal rotation.</p><p><hi rend="italics">Polar</hi> <hi rend="smallcaps">Circles</hi>, are immoveable circles, parallel to the equator, and at such a distance from the pole as is equal to the greatest declination of the ecliptic, which now is 23° 28′. That next the northern pole is called the arctic, and that next the southern one the antarctic.</p><p><hi rend="smallcaps">Circles</hi> <hi rend="italics">of Position,</hi> are circles passing through the common intersections of the horizon and meridian, and through any degree of the ecliptic, or the centre of any star, or other point in the heavens; and are used for finding out the situation or position of any star. These are usually six in number, cutting the equinoctial into 12 equal parts, which the astrologers call the Celestial Houses, and hence they are sometimes called Circles of the Celestial Houses.</p></div1><div1 part="N" n="CIRCUIT" org="uniform" sample="complete" type="entry"><head>CIRCUIT</head><p>, <hi rend="italics">Electrical,</hi> denotes the course of the electric fluid from the charged surface of an electric body, to the opposite surface, into which the discharge is made. Some electricians at first apprehended, that the same particles of the electric fluid that were thrown on one side of the charged glass, actually made the whole circuit of the intervening conductors, and arrived at the opposite side: whereas Dr. Franklin's theory only requires, that the redundancy of electric matter on the charged surface should pass into those bodies which form that part of the circuit which is contiguous to it, driving forward that part of the fluid which they naturally possess; and that the desiciency of the exhausted surface should be supplied by the neighbouring conductors, which form the last part of the circuit. On this supposition, a vibrating motion is successively communicated through the whole length of the circuit.</p><p>Many attempts were made, both in France and England, at an early period in the practice of electricity, to ascertain the distance to which the electric shock might be carried, and the velocity of its motion. The French philosophers, at different times, caused it to pals through circuits of 900 and even 2000 toises, or about 2 English miles and a half; and they discharged the Leyden phial through a bason of water, whose surface <pb n="289"/><cb/> was equal to about one acre. M. Monier found that, in passing through an iron wire of 950 toises in length, it did not spend a quarter of a second; and that its motion was instantaneous through a wire of 1319 feet. In 1747, Dr. Watson, and other English philosophers, after many experiments of a similar kind, conveyed the electric matter through a circuit of 4 miles; and, from two several trials, they concluded that its passage is instantaneous. By all which doubtless is meant, that its motion is too rapid to be measured. Priestley's Hist. of Elect. vol. 1, pa. 128, 8vo edit.</p></div1><div1 part="N" n="CIRCULAR" org="uniform" sample="complete" type="entry"><head>CIRCULAR</head><p>, appertaining to a circle; as a circular form, circular motion, &c.</p><p><hi rend="smallcaps">Circular</hi> <hi rend="italics">Lines,</hi> a name given by some authors to such straight lines as are divided by means of the divisions made in the arch of a circle. Such as the Sines, Tangents, Secants, &c.</p><p><hi rend="smallcaps">Circular</hi> <hi rend="italics">Numbers,</hi> are such as have their powers ending in the roots themselves. As the number 5, whose square is 25, and its cube 125, &c.</p><p>Napier's <hi rend="smallcaps">Circular</hi> <hi rend="italics">Parts,</hi> are five parts of a rightangled or a quadrantal spherical triangle; they are the two legs, the complement of the hypothenuse, and the complements of the two oblique angles.</p><p>Concerning these circular parts, Napier gave a general rule in his <hi rend="italics">Logarithmorum Canonis Descriptio,</hi> which is this; “<hi rend="italics">The rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts, and to the rectangle under the cosines of the opposite parts.</hi> The right angle or quadrantal side being neglected, the two sides and the complements of the other three natural parts are called the circular parts; as they follow each other as it were in a circular order. Of these, any one being fixed upon as the middle part, those next it are the adjacent, and those farthest from it the opposite parts.” Lord Buchan's Life of Napier, pa. 98.</p><p>This rule contains within itself all the particular rules for the solution of right-angled spherical triangles, and they were thus brought into one general comprehensive theorem, for the sake of the memory; as thus, by charging the memory with this one rule alone: All the cases of right-angled spherical triangles may be resolved, and those of oblique ones also, by letting fall a perpendicular, excepting the two cases in which there are given either the three sides, or the three angles.— And for these a similar expedient has been devised by Lord Buchan and Dr. Minto. “M. Pingre, in the Memoires de Mathematique et de Physique for the year 1756, reduces the solution of all the cases of spherical triangles to four analogies. These four analogies are in fact, under another form, Napier's rule of the circular parts, and his second or fundamental theorem, with its application to the supplemental triangle. Although it would be no difficult matter to get by heart the four analogies of M. Pingre, yet there are few persons blessed with a memory capable of retaining them for any considerable time. For this reason, the rule for the circular parts ought to be kept under its present form. If the reader attends to the circumstance of the second letters of the words <hi rend="italics">tangents</hi> and <hi rend="italics">cosines</hi> being the same with the first of the words <hi rend="italics">adjacent</hi> and <hi rend="italics">opposite,</hi> he will sind it almost impossible to forget the rule. And <cb/> the rule for the solution of the two cases of spherical triangles, for which the former of itself is insufficient, may be thus expressed: <hi rend="italics">Of the circular parts of an oblique spherical triangle, the rectangle under the tangents of half the sum and half the difference of the segments at the middle part</hi> (formed by a perpendicular drawn from an angle to the opposite side), <hi rend="italics">is equal to the rectangle under the tangents of half the sum and half the difference of the opposite parts.</hi> By the circular parts of an oblique spherical triangle are meant its three sides and the supplements of its three angles. Any of these six being assumed as a middle part, the opposite parts are those two of the same denomination with it, that is, if the middle part is one of the sides, the opposite parts are the other two, and, if the middle part is the supplement of one of the angles, the opposite parts are the supplements of the other two.—Since every plane triangle may be considered as described on the surface of a sphere of an infinite radius, these two rules may be applied to plane triangles, provided the middle part be restricted to a side.</p><p>“Thus it appears that two simple rules suffice for the solution of all the possible cases of plane and spherical triangles. These rules, from their neatness and the manner in which they are expressed, cannot fail of engraving themselves deeply on the memory of every one who is a little versed in trigonometry. It is a circumstance worthy of notice, that a person of a very weak memory may carry the whole art of trigonometry in his head.” Napier's Life, pa. 102.</p><p><hi rend="smallcaps">Circular</hi> <hi rend="italics">Sailing,</hi> is that performed in the arch of a great circle.—It is chiefly on account of the shortest distance that this method of sailing has been proposed; and for the most part it is advantageous for a ship to reach her port by the shortest course.</p><p>As the solutions of the cases in Mercator's sailing are performed by plane triangles; so the cases in greatcircle sailing are resolved by the solution of spherical triangles. But, after all, the several cases in this kind of sailing serve rather for exercises in the solution of spherical triangles, than for any real use towards the navigating of a ship.</p><p><hi rend="smallcaps">Circular</hi> <hi rend="italics">Spots</hi> are made on pieces of metal by large electrical explosions. See Philos. Trans. vol. 58, pa. 68; also Priestley's Hist. of Electricity, vol. 2, sect. 9, edit. 8vo.</p><p>These beautiful spots, produced by the moderate charge of a large battery, discharged between two smooth surfaces of metals, or semi-metals, lying at a small distance from each other, consist of one central spot, and several concentric circles, which are more or less distinct, and more or fewer in number, as the metal upon which they are marked is more easy or difficult of fusion, and as a greater or less force is employed. They are composed of dots or cavities, which indicate a real fusion. If the explosion of a battery, issuing from a pointed body, be repeatedly taken on the plain surface of a piece of metal near the point, or be received from the surface on a point, the metal will be marked with a spot, consisting of all the prismatic colours disposed in circles, and formed by scales of the metal separated by the force of the explosion.</p><p><hi rend="smallcaps">Circular</hi> <hi rend="italics">Velocity,</hi> a term in astronomy signifying <pb n="290"/><cb/> that velocity of a planet, or revolving body, which is measured by the arch of a circle.</p><p><hi rend="smallcaps">Circulating</hi> <hi rend="italics">Decimals,</hi> called also recurring or repeating decimals, are those in which a sigure or several figures are continually repeated. They are distinguished into <hi rend="italics">single</hi> and <hi rend="italics">multiple,</hi> and these again into <hi rend="italics">pure</hi> and <hi rend="italics">mixed.</hi></p><p>A <hi rend="italics">pure single</hi> circulate, is that in which one figure only is repeated; as .222 &c, and is marked thus .2.</p><p>A <hi rend="italics">pure multiple</hi> circulate, is that in which several figures are continually repeated; as .232323 &c, marked .2<hi rend="sup">.</hi>3<hi rend="sup">.</hi>; and .524524 &c, marked .5<hi rend="sup">.</hi>24<hi rend="sup">.</hi>.</p><p>A <hi rend="italics">mixed single</hi> circulate, is that which consists of a terminate part, and a single repeating figure; as 4.222 &c, or 4.2<hi rend="sup">.</hi>. And</p><p>A <hi rend="italics">mixed multiple</hi> circulate is that which contains a terminate part with several repeating figures; as 45.5<hi rend="sup">.</hi>24<hi rend="sup">.</hi>.</p><p>That part of the circulate which repeats, is called the <hi rend="italics">repetend:</hi> and the whole repetend, supposed infinitely continued, is equal to a vulgar fraction, whose numerator is the repeating number, or figures, and its denominator the same number of nines: so .2 is = 2/9; and .2<hi rend="sup">.</hi>3<hi rend="sup">.</hi> is = 23/99; and .5<hi rend="sup">.</hi>24<hi rend="sup">.</hi> is = 524/999.</p><p>It seems it was Dr. Wallis who first distinctly considered, or treated of infinite circulating decimals, as he himself informs us in his Treatise of Infinites. Since his time many other authors have treated on this part of arithmetic; the principal of these however, to whom the art is mostly indebted, are Messrs. Brown, Cunn, Martin, Emerson, Malcolm, Donn, and Henry Clarke, in whose writings the nature and practice of this art may be fully seen, especially in the last mentioned ingenious author.</p></div1><div1 part="N" n="CIRCUMFERENCE" org="uniform" sample="complete" type="entry"><head>CIRCUMFERENCE</head><p>, in a general sense denotes the line or lines bounding any figure. But it is commonly used in a more limited sense, to denote the curve line which bounds a circle, and which is otherwise called the <hi rend="italics">periphery;</hi> the boundary of a right-lined figure being expressed by the term <hi rend="italics">perimeter.</hi></p><p>The circumference of a circle is every where equidistant from the centre. And the circumferences of different circles are to one another as their radii or diameters, or the ratio of the diameter to the circumference is a constant ratio, in every circle, which is nearly as 7 to 22, as it was found by Archimedes, or, more nearly, as 1 to 3.1416. Under the article <hi rend="italics">Circle</hi> may be seen various other approximations to that ratio, one of which is carried to 128 places of figures, viz by M. De Lagny.</p><p>The Circumference of every circle is supposed to be divided into 360 equal parts, called <hi rend="italics">degrees.</hi>—Any part of a circumference is called an <hi rend="italics">arc</hi> or <hi rend="italics">arch;</hi> and a right line drawn from one end of an arc to the other, is called its <hi rend="italics">chord.</hi>—The <hi rend="italics">angle</hi> at the circumference is equal to half the angle at the centre, standing on the same arc; and therefore it is measured by the half of that arc.</p></div1><div1 part="N" n="CIRCUMFERENTOR" org="uniform" sample="complete" type="entry"><head>CIRCUMFERENTOR</head><p>, a particular instrument used by surveyors for taking angles. It consists of a brass circle and index all of a piece; the diameter of the circle is commonly about 7 inches; the index about 14 inches long, and an inch and a half broad. On the circle is a card or compass, divided into 360 degrees; the <cb/> meridian line of which answers to the middle of the breadth of the index. On the limb or circumference of the circle is soldered a brass ring; which, with another fitted with a glass, forms a kind of box for the needle, which is suspended on a pivot in the centre of the circle. There are also two sights to serew on, and slide up and down the index, as also a spangle and socket screwed on the under side of the circle, to receive the head of the three-legged staff. <figure/></p><p><hi rend="italics">To take, or observe the Quantity of an Angle by the Circumferentor.</hi> The angle proposed being EKG; place the instrument at K, with the flower-de-luce of the card towards you; then direct the sights to E, and observe what degrees are cut by the south end of the needle, which let be 295; then, turning the instrument about on its stand, direct the sights to G, noting again what degrees are cut by the south end of the needle, which suppose are 213. This done, subtract the less number from the greater, viz, 213 from 295, and the remainder, or 82 degrees, is the quantity of the angle EKG sought.</p></div1><div1 part="N" n="CIRCUMGYRATION" org="uniform" sample="complete" type="entry"><head>CIRCUMGYRATION</head><p>, is the whirling motion of any body about a centre; as of the planets about the sun, &c.</p><p>CIRCUM POLAR <hi rend="italics">Stars,</hi> are those stars which, by reason of their vicinity to the pole, move round it, without setting.</p><p>CIRCUMSCRIBED <hi rend="italics">Figure,</hi> is a figure which is drawn about another, so that all its sides or planes touch the latter or inscribed figure.</p><p>The area and perimeter of every polygon that can be circumscribed about a circle, are greater than those of the circle; and the area and perimeter of every inscribed polygon, are less than those of the circle; but they approach always nearer to equality as the number of sides is more. And on these principles Archimedes, and some other authors since his time, attempted the quadrature of the circle; which is nothing else, in effect, but the measuring the area or capacity of a circle.</p><p><hi rend="smallcaps">Circumscribed</hi> <hi rend="italics">Hyperbola,</hi> is one of Newton's hyperbolas of the 2d order, that cuts its asymptotes, and contains the parts cut off within its own space.</p></div1><div1 part="N" n="CIRCUMVALLATION" org="uniform" sample="complete" type="entry"><head>CIRCUMVALLATION</head><p>, or <hi rend="italics">Line of Circumvallation,</hi> in the Art of War, is a trench, bordered with a parapet, thrown up around the besieger's camp, as a security against any army that may attempt to relieve the place, as well as to prevent desertion.</p></div1><div1 part="N" n="CIRCUMVOLUTION" org="uniform" sample="complete" type="entry"><head>CIRCUMVOLUTION</head><p>, in Architecture, the torus of the spiral line of the Ionic Order.</p></div1><div1 part="N" n="CISSOID" org="uniform" sample="complete" type="entry"><head>CISSOID</head><p>, is a curve line of the second order, in- <pb n="291"/><cb/> vented by Diocles for the purpose of sinding two continued mean proportionals between two other given lines. The generation or description of this curve is as follows:</p><p>On the extremity B of the di- <figure/> ameter AB of the circle AOB, erect the indefinite perpendicular CBD, to which from the other extremity A draw several lines, cutting the circle in I, O, N, &c; and upon these lines set off the corresponding equal distances, viz, HM = AI, and FO = AO, and CL = AN, &c; then the curve line drawn through all the points M, O, L, &c, is the cissoid of Diocles, who was an ancient Greek geometrician.</p><p>This curve is, by Newton, reckoned among the defective hyperbolas, being the 42d species in his <hi rend="italics">Enumeratio Linearum tertii ordinis.</hi> And in his appendix <hi rend="italics">de Æquationum Constructione Lineari,</hi> at the end of his <hi rend="italics">Arithmetica Universalis,</hi> he gives another elegant method of describing this curve by the continual motion of a square ruler. Other methods have also been devised by different authors for the same thing. <hi rend="center"><hi rend="italics">The Properties of the Cissoid</hi> are the following:</hi></p><p>1. The curve has two insinite legs AMOL, A<hi rend="italics">mol</hi> meeting in a cusp A, and tending continually towards the indefinite line CBD, which is their common asymptote.</p><p>2. The curve passes through O and <hi rend="italics">o,</hi> points in the circle equally distant from A and B; or it bisects each semicircle.</p><p>3. Letting fall perpendiculars MP, IK from any corresponding points I, M; then is AP = BK, and AM = HI, because AI = MH.</p><p>4. AP : PB :: MP<hi rend="sup">2</hi> : AP<hi rend="sup">2</hi>. So that, if the diameter AB be = <hi rend="italics">a,</hi> the absciss AP = <hi rend="italics">x,</hi> and the ordinate PM = <hi rend="italics">y</hi>; then is ; which is the equation of the curve.</p><p>5. Sir Isaac Newton, in his last letter to M. Leibnitz, has shewn how to find a right line equal to one of the legs of this curve, by means of the hyperboia; but he suppressed the investigation, which however may be seen in his Fluxions.</p><p>6. The whole infinitely long cissoidal space, contained between the infinite asymptote BCD, and the curves LOA<hi rend="italics">ol</hi> &c, of the cissoid, is equal to triple the generating circle AOB<hi rend="italics">o</hi>A.</p><p>See more of this curve in Dr. Wallis, vol. 1, pa. 545.</p><p>CIVIL <hi rend="italics">Day.</hi> See <hi rend="italics">Day.</hi></p><p><hi rend="smallcaps">Civil</hi> <hi rend="italics">Month.</hi> See <hi rend="italics">Month.</hi></p><p><hi rend="smallcaps">Civil</hi> <hi rend="italics">Year,</hi> is the legal year, or annual account of time, which every government appoints to be used within its own dominions.</p><p>It is so called in contradistinction to the natural year, <cb/> which is measured exactly by the revolution of the heavenly bodies.</p></div1><div1 part="N" n="CLAIRAULT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CLAIRAULT</surname> (<foreName full="yes"><hi rend="smallcaps">Alexis-Claude</hi></foreName>)</persName></head><p>, a celebrated French mathematician and academician, was born at Paris the 13th of May 1713, and died the 17th of May 1765, at 52 years of age. His father, a teacher of mathematics at Paris, was his sole instructor, teaching him even the letters of the alphabet on the figures of Euclid's Elements, by which he was able to read and write at 4 years of age. By a similar stratagem it was that calculations were rendered samiliar to him. At 9 years of age he put into his hands Guisnée's Application of Algebra to Geometry; at 10 he studied l'Hopital's Conic Sections; and between 12 and 13 he read a memoir to the Academy of Sciences concerning four new Geometrical curves of his own invention. About the same time he laid the first foundation of his work upon curves that have a double curvature, which he finished in 1729, at 16 years of age. He was named Adjoint-Mechanician to the Academy in 1731, at the age of 18, Associate in 1733, and Pensioner in 1738; during his connection with the Academy, he had a great multitude of learned and ingenious communications inserted in their Memoirs, beside several other works which he published separately; the list of which is as follows.</p><p>1. On Curves of a Double Curvature; in 1730, 4to.</p><p>2. Elements of Geometry; 1741, 8vo.</p><p>3. Theory of the Figure of the Earth; 1743, 8vo.</p><p>4. Elements of Algebra; 1746, 8vo.</p><p>5. Tables of the Moon; 1754, 8vo.</p><p>His papers inserted in the Memoirs of the Academy are too numerous to be particularised here; but they may be found from the year 1727, for almost every year till 1762; being upon a variety of subjects, astronomical, mathematical, optical, &c.</p></div1><div1 part="N" n="CLAVIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CLAVIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Christopher</hi></foreName>)</persName></head><p>, a German Jesuit, was born at Bamberg in Germany, in 1537. He became a very studious mathematician, and elaborate writer; his works, when collected, and closely printed, making 5 large folio volumes; being a complete body or course of the mathematics. They are mostly elementary, and commentaries on Euclid and others; having very little of invention of his own. His talents and writings have been variously spoken of, and it must be acknowledged that they are heavy and elaborate. He was sent for to Rome, to assist, with other learned men, in the reformation of the calendar, by pope Gregory; which he afterward undertook a defence of, against Scaliger, Vieta, and others, who attacked it. He died at Rome, the 6th of February, 1612, at 75 years of age, after more than 50 years close application to the mathematical sciences.</p><p>CLEFF. See <hi rend="italics">Cliff.</hi></p></div1><div1 part="N" n="CLERC" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CLERC</surname> (<foreName full="yes"><hi rend="smallcaps">John le</hi></foreName>)</persName></head><p>, a celebrated writer and universal scholar, was born at Geneva in 1657. After passing through the usual course of study at Geneva, he went to France in 1678; but returning the year after, he took holy orders. In 1682 Le Clerc visited England, to learn the language: but the smoky air of London not agreeing with his lungs, he soon returned to Holland, where he settled; and was appointed professor of philosophy, polite literature, and the Hebrew tongue, in the school at Amsterdam. Here he long continued <pb n="292"/><cb/> to read lectures; for which purpose he drew up and published his Logic, Ontology, Pneumatology, and Natural Philosophy. He published also <hi rend="italics">Ars Critica</hi>; a Commentary on the Old Testament; a Compendium of Universal History; an Ecclesiastical History of the two first centuries; a French translation of the New Testament, and other works. In 1686, he began, jointly with M. de la Crose, his <hi rend="italics">Bibliotheque Universelle et Historique,</hi> in imitation of other literary journals; which was continued to the year 1693, making 26 volumes. In 1703 he began his <hi rend="italics">Bibliotheque Choisie,</hi> and continued it to 1714, when he commenced another work on the same plan, called <hi rend="italics">Bibliotheque Ancienne et Moderne,</hi> which he continued to the year 1728; all of them justly esteemed excellent stores of useful knowledge. He published also, in 1713, a neat little treatise on Practical Geometry, in 2 vols. small 8vo, called, <hi rend="italics">Pratique de la Geometric, sur le papier et sur le terrain.</hi> In 1728 he was seized with a palsy and fever; and, after spending the last six years of his life with little or no understanding, he died in 1736, at 79 years of age.</p></div1><div1 part="N" n="CLEPSYDRA" org="uniform" sample="complete" type="entry"><head>CLEPSYDRA</head><p>, a kind of water clock, or an hourglass serving to measure time by the fall of a certain quantity, commonly out of one vessel into another.— There have been also clepsydræ made with quicksilver; and the term is also used for hour-glasses of sand.</p><p>By this instrument the Egyptians measured their time and the course of the sun. Also Tycho Brahe, in modern times, made use of it to measure the motion of the stars, &c; and Dudley used the same contrivance in making all his maritime observations.</p><p>The use of Clepsydræ is very ancient. They were probably invented in Egypt under the Ptolemys; though some authors ascribe the invention of them to the Greeks, and others to the Romans. Pliny informs us, that Scipio Nasica, about 150 years before Christ, gave the first hint for the construction of them: and Pancirollus has particularly described them. According to his account, the clepsydra was a vessel made of glass, with a small hole in the bottom, edged with gold: in the upper part of this vessel a line was drawn, and marked with the 12 hours: the vessel was silled with water, and a cork with a pin fixed in it floated on the surface, pointing to the first hour; and as the water sunk in the vessel by issuing out of the small hole, the pin indicated the other hours as it descended.</p><p>Clepsydræ were chiefly used in the winter; as sundials served for the summer. They had however two great defects; the one, that the water ran out more or less easily, as the air was more or less dense; the other, that the water flowed more rapidly at the beginning, than towards the conclusion when its quantity and pressure were much decreased. Amontons has invented a clepsydra which, it is said, is free from both these inconveniences; and the same effect is produced by one described by Mr. Hamilton, in the Philos. Trans. vol. 44, pa. 171, or Abridg. vol. 10, pa. 248. Varignon too, in the <hi rend="italics">Memoires de l'Acad.</hi> 1699, delivers a general geometrical method of making clepsydræ, or waterclocks, with any kind of vessels, and with any given orisices for the water to run through.</p><p>Vitruvius, in lib. 9 of his Achitecture, treats of these instruments; and Pliny in chap. 60, lib. 7, says that Scipio Nasica was the first who measured time at Rome <cb/> by clepsydræ, or water-clocks. Gesner, in his Pandects pa. 91, gives several contrivances for these instruments. Solomon de Caus also treats on this subject in his Reasons of Moving Forces &c. So also does Ozanam, in his Mathematical Recreations, in which is contained a Treatise on Elementary Clocks, translated from the Italian of Dominique Martinelli. There is likewise a treatise on Hour-Glasses by Arcangelo Maria Radi, called <hi rend="italics">Nova Sceinza de Horologi Polvere.</hi> See also the <hi rend="italics">Technica Curiosa</hi> of Gasper Schottus; and Amonton's Remarques & Experiences Physiques sur la Construction d'une nouvelle Clepsydre, exempte des défauts des autres.</p></div1><div1 part="N" n="CLIFF" org="uniform" sample="complete" type="entry"><head>CLIFF</head><p>, or <hi rend="smallcaps">Cleff</hi>, a term in Music, for a certain mark, from the position of which the proper places of all other notes in a piece of music are known.</p></div1><div1 part="N" n="CLIMACTERIC" org="uniform" sample="complete" type="entry"><head>CLIMACTERIC</head><p>, a critical year in a person's life.</p><p>According to some, this is every 7th year: but others allow it only to those years produced by multiplying 7 by the odd numbers 3, 5, 7, 9. These years, say they, bring with them some remarkable change with respect to health, life, or fortune: the grand climacteric is the 63d year; but some add also the 81st to it: the other remarkable climacterics are the 7th, 21st, 35th, 49th, and 56th.</p></div1><div1 part="N" n="CLIMATE" org="uniform" sample="complete" type="entry"><head>CLIMATE</head><p>, or <hi rend="italics">Clime,</hi> in Geography, a part of the surface of the earth, bounded by two lesser circles parallel to the equator; and of such a breadth, as that the longest day in the parallel nearer the pole exceeds the longest day in that next the equator, by some certain space, as half an hour, or an hour, or a month.</p><p>The beginning of a climate, is a parallel circle in which the day is the shortest; and the end of the climate, is that in which the day is the longest. The climates therefore are reckoned from the equator to the pole; and are so many zones or bands, terminated by lines parallel to the equator: though, in strictness, there are several climates, or different degrees of light or temperature, in the breadth of one zone. Each climate only differs from its contiguous ones, in that the longest day in summer is longer or shorter, by half an hour, for instance, in the one place than in the other.</p><p>As the climates commence at the equator, at the beginning of the first climate, that is at the equator, the day is just 12 hours long; but at the end of it, or at the beginning of the 2d climate, the longest day is 12 hóurs and a half long; and at the end of the 2d, or beginning of the 3d climate, the longest day is 13 hours long; and so of the rest, as far as the polar circles, where the hour climates terminate, and month climates commence. And as an hour climate is a space comprised between two parallels of the equator, in the first of which the longest day exceeds that in the latter by half an hour; so the month climate is a space contained between two circles parallel to the polar circles, and having its longest day longer or shorter than that of its contiguous one, by a month, or 30 days. But some authors, as Ricciolus, make the longest day of the contiguous climates to differ by half hours, to about the latitude of 45 degrees; then to differ by an hour, or sometimes 2 hours, to the polar circle; and after that by a month each. See tables of climates in Varenius, chap. 25, prop. 13.</p><p>The ancients, who confined the climates to what they thought the habitable parts of the earth, reckoned only <pb n="293"/><cb/> seven, the middles of which they made to pass through some remarkable places; as the 1st through Meroe, the 2d through Sienna, the 3d through Alexandria, the 4th through Rhodes, the 5th through Rome, the 6th through Pontus, and the 7th through the mouth of the Borysthenes. But the moderns, who have sailed farther toward the poles, make 30 climates on each side.</p><p>Vulgarly the term Climate is bestowed on any country or region differing from another either in respect of the seasons, the quality of the soil, or even the manners of the inhabitants; without any regard to the length of the longest day. Abulfeda, an Arabic author, distinguishes the first kind of climates by the term <hi rend="italics">real climates,</hi> and the latter by that of <hi rend="italics">apparent climates.</hi></p></div1><div1 part="N" n="CLOCK" org="uniform" sample="complete" type="entry"><head>CLOCK</head><p>, a machine now constructed in such a manner, and so regulated by the uniform motion of a pendulum, as to measure time, and all its subdivisions, with great exactness. Before the invention of the pendulum, a balance, not unlike the fly of a kitchen-jack, was used instead of it.—They were at first called nocturnal dials to distinguish them from sun-dials, which shewed the hour by the shadow of the sun.</p><p>The invention of clocks with wheels is ascribed to Pacificus, archdeacon of Verona, in the 9th century, on the credit of an epitaph quoted by Ughelli, and borrowed by him from Panvinius. Others attribute the invention to Boethius, about the year 510.</p><p>Mr. Derham, however, makes clock-work of a much older date; ranking Archimedes's sphere, mentioned by Claudian, and that of Posidonius, mentioned by Cicero, among machines of this kind: not that either their form or use were the same with those of ours; but that they had their motion from some hidden weights, or springs, with wheels or pulleys, or some such clockwork principle.</p><p>In the <hi rend="italics">Disquisitiones Monasticæ</hi> of Benedictus Haëften, published in the year 1644, he says, that clocks were invented by Silvester the 4th, a monk of his order, about the year 998, as Dithmarus and Bozius have shewn; for before that time, they had nothing but sundials and clepsydræ to shew the hour.—Conrade Gesner, in his Epitome, pa. 604, says, that Richard Wallingford, an English abbot of St. Albans, who flourished in the year 1326, made a wonderful clock by a most excellent art, the like of which could not be produced by all Europe.—Moreri, under the word <hi rend="italics">Horologe</hi> du Palais, says, that Charles the Fifth, called the wise king of France, ordered at Paris the first large clock to be made by Henry de Vie, whom he sent for from Germany, and set it upon the tower of his palace, in the year 1372.— John Froissart, in his <hi rend="italics">Histoire & Chronique,</hi> vol. 2, ch. 28, says, the duke of Bourgogne had a clock, which sounded the hour, taken away from the city of Courtray, in the year 1382: and the same thing is said by Wm. Paradin, in his <hi rend="italics">Annales de Bourgogne.</hi></p><p>Clock-makers were first introduced into England in 1368, when Edward the 3d granted a licence for three artists to come over from Delst in Holland, and practise their occupation in this country.</p><p>The water-clocks, or clepsydræ, and sun-dials, have both á much better claim to antiquity. The French annals mention one of the former kind, sent by Aaron, king of Persia, to Charlemagne, about the year 807, which it would seem bore some resemblance to the <cb/> modern clocks: it was of brass, and shewed the hours by 12 little balls of the same metal, which at the end of each hour fell upon a bell, and made a sound. There were also figures of 12 cavaliers, which at the end of each hour came out through certain apertures, or windows, in the side of the clock, and shut them again, &c.</p><p>The invention of pendulum clocks is owing to the happy industry of the last age; and the honour of that discovery is disputed between Galileo and Huygens. The latter, who wrote an excellent volume on the subject, declares it was first put in practice in the year 1657, and the description of it printed in 1658. Becher, <hi rend="italics">De Nova Temporis dimetiendi Theoria, anno</hi> 1680, contends for Galileo; and relates, though at second-hand, the whole history of the invention; adding that one Trefler, clock-maker to the father of the then grand duke of Tuscany, made the first pendulum clock at Florence, under the direction of Galileo Galilei, a pattern of which was brought to Holland. And the Academy del Cimento say expressly, that the application of the pendulum to the movement of a clock was first proposed by Galileo, and put in practice by his son Vincenzo Galilei, in 1649. But whoever may have been the inventor, it is certain that the invention never flourished till it came into the hands of Huygens, who insists on it, that if ever Galileo thought of such a thing, he never brought it to any degree of perfection. The first pendulum clock made in England was in the year 1662, by one Fromantil, a Dutchman.</p><p>Among the modern clocks, those of Strasburg and Lyons are very eminent for the richness of their furniture, and the variety of their motions and figures. In the former, a cock claps his wings, and proclaims the hour: the angel opens a door, and salutes the Virgin; and the holy spirit descends on her, &c. In the latter, two horsemen encounter, and beat the hour upon each other: a door opens, and there appears on the theatre the Virgin, with Jesus Christ in her arms; the Magi, with their retinue, marching in order, and presenting their gifts; two trumpeters sounding all the while to proclaim the procession.</p><p>These, however, are far excelled by two that have lately been made by English artists, as a present from the East-India company to the emperor of China. These two clocks are in the form of chariots, in each of which a lady is placed, in a fine attitude, leaning her right hand upon a part of the chariot, under which appears a clock of curious workmanship, little larger than a shilling, that strikes and repeats, and goes for eight days. Upon the lady's finger sits a bird, finely modelled, and set with diamonds and rubies, with its wings expanded in a flying posture, and actually flutters for a considerable time, on touching a diamond button below it; the body of the bird, in which are contained part of the wheels that animate it as it were, is less than the 16th part of an inch. The lady holds in her left hand a golden tube little thicker than a large pin, on the top of which is a small round box, to which is fixed a circular ornament not larger than a sixpence, set with diamonds, which goes round in near three hours in a constant regular motion. Over the lady's head is a double umbrella, supported by a small fluted pillar not thicker than a quill, and under the larger of which a bell is fixed at a considerable distance from the <pb n="294"/><cb/> clock, with which it seems not to have any connection; but from which a communication is secretly conveyed to a hammer, that regularly strikes the hour, and repeats the same at pleasure, by touching a diamond button fixed to the clock below. At the feet of the lady is a golden dog; before which, from the point of the chariot, are two birds fixed on spiral springs, the wings and feathers of which are set with stones of various colours, and they appear as if flying away with the chariot, which, from another secret motion, is contrived to run in any direction, either straight or circular, &c; whilst a boy, that lays hold of the chariot behind, seems also to push it forwards. Above the umbrella are flowers and ornaments of precious stones; and it terminates with a flying dragon set in the same manner. The whole is of gold, most curiously executed, and embellished with rubies and pearls.</p><p>The ingenious Dr, Franklin contrived a clock to shew the hours, minutes, and seconds, with only three wheels and two pinions in the whole movement. The dial-plate has the hours engraven upon it in spiral spaces along two diameters of a circle, containing four times 60 minutes. The index goes round in four hours, and counts the minutes from any hour by which it has passed to the next following hour. The small hand, in an arch at top, goes round once in a minute, and shews the seconds. The clock is wound up by a line going over a pulley, on the axis of the great wheel, like a common 30 hour clock. Many of these very simple machines have since been constructed, that measure time exceedingly well. This clock is subject, however, to the inconvenience of requiring frequent winding, by drawing up the weight; as also to some uncertainty as to the particular hour shewn by the index.</p><p>Mr. Ferguson has proposed to remedy these inconveniences by another construction, which is described in his Select Exercises, pa. 4. This clock will go a week without winding, and always shews the precise hour; but, as Mr. Ferguson acknowledges, it has two disadvantages which do not belong to Dr. Franklin's clock: when the minute hand is adjusted, the hour plate must also be set right, by means of a pin; and the smallness of the teeth in the pendulum wheel will cause the pendulum ball to describe but small arcs in its vibrations; and therefore the momentum of the ball will be less, and the times of the vibrations will be more affected by any unequal impulse of the pendulum wheel on the pallets. Besides, the weight of the flat ring, on which the seconds are engraven, will load the pivots of the axis of the pendulum wheel with a great deal of friction, which ought by all possible means to be avoided. To remedy this inconvenience, the seconds plate might be omitted.</p><p>Mr. Ferguson also contrived a clock, shewing the apparent diurnal motions of the sun and moon, the age and phases of the moon, with the time of her coming to the meridian, and the times of high and low water; and all this by having only two wheels and a pinion added to the common movement. See his Select Exercises before mentioned. In this clock the figure of the sun serves as an hour index, by going round the dialplate in 24 hours; and a figure of the moon goes round in 24 h. 50 1/2 min. the time of her going round in the heavens from any meridian to the same meridian again. <cb/> A clock of this kind was adapted by Mr. Ferguson to the movement of an old watch. See also a description and drawing of an astronomical clock, shewing the apparent daily motions of the sun, moon, and stars, with the times of their rising, southing, and setting; the places of the sun and moon in the ecliptic, and the age and phases of the moon for every day of the year, in the same book, pa. 19.</p><p>There have been several treatises upon clocks; the principal of which are the following. Hieronymus Cardan, de Varietate Rerum libri 17.—Conrade Dasypodius, Descriptio Horologii Astronomici Argentinensis in summa Templi erecti.—Guido Pancirollus, Antiqua deperdita & nova reperta.—L'Usage du Cadran, ou de l'Horloge Physique Universelle, par Galilée, Mathematicien du Duc de Florence.——Oughtred's Opuscula Mathematica.—Huygens's Horologium Oscillatorium.—Pendule perpetuelle, par l'Abbé de Hautefeuille.—J. J. Becheri Theoria & Experientia de nova Temporis dimetiendi Ratione & Horologiorum Constructione.——Clark's Oughtredus explicatus, ubi de Constructione Horologiorum.—Horological Disquisitions.—Huygens's Posthumous Works.—Sully's Regle Artisicielle du Temps, &c.—Serviere's Recueil d'Ouvrages Curieux.—Derham's Artificial Clock-maker.— Camus's Traités des Forces Mouvantes.—Alexandre's Traité Général des Horologies.—Also Treatises and Principles of Clock-making, by Hatton, Cuming, &c. &c.</p></div1><div1 part="N" n="CLOUD" org="uniform" sample="complete" type="entry"><head>CLOUD</head><p>, a collection of vapours suspended in the atmosphere, and rendered visible.</p><p>Although it be generally allowed that the clouds are formed from the aqueous vapours, which before were so closely united with the atmosphere as to be invisible: it is, however, not easy to account for the long continuance of some very opaque clouds without dissolving; or to assign the reason why the vapours, when they have once begun to condense, do not continue to do so till they at last fall to the ground in the form of rain or snow, &c. It is now known that a separation of the latent heat from the water of which vapour is composed is attended with a condensation of that vapour in some degree; in such case, it will first appear as a smoke, mist, or fog; which, if interposed between the sun and earth, will form a cloud; and the same causes continuing to operate, the cloud will produce rain or snow. It is however abundantly evident that some other cause besides mere heat or cold is concerned in the formation of clouds, and the condensation of atmospherical vapours. This cause is esteemed in a great measure the electrical fluid; indeed electricity is now so generally admitted as an agent in all the great operations of nature, that it is no wonder to find the formation of clouds attributed to it; and this has accordingly been given by Beccaria as the cause of the formation of all clouds whatsoever, whether of thunder, rain, hail, or snow.</p><p>But whether the clouds are produced, that is, the atmospheric vapours rendered visible, by means of electricity or not, it is certain that they do often contain the electric fluid in prodigious quantities, and many terrible and destructive accidents have been occasioned by clouds very highly electrified. The most extraordinary instance of this kind perhaps on record happened <pb n="295"/><cb/> in the island of Java, in the East-Indies, in August, 1772. On the 11th of that month, at midnight, a bright cloud was observed covering a mountain in the district called Cheribou, and several reports like those of a gun were heard at the same time. The people who dwelt upon the upper parts of the mountain not being able to fly fast enough, a great part of the cloud, eight or nine miles in circumference, detached itself under them, and was seen at a distance, rising and falling like the waves of the sea, and emitting globes of fire so luminous, that the night became as clear as day. The effects of it were astonishing; every thing was destroyed for 20 miles round; the houses were demolished; plantations were buried in the earth; and 2140 people lost their lives, besides 1500 head of cattle, and a vast number of horses, goats, &c. Another remarkable instance of the dreadful effects of electric clouds, which happened at Malta the 29th of October 1757, is related in Brydone's Tour through Malta.</p><p>The height of the clouds is not usually great: the summits of high mountains being commonly quite free from them, as many travellers have experienced in passing these mountains. It is found that the most highly electrified clouds descend lowest, their height being often not more than 7 or 800 yards above the ground; and sometimes thunder-clouds appear actually to touch the ground with one of their edges: but the generality of clouds are suspended at the height of a mile, or little more, above the earth.</p><p>The motions of the clouds, though often directed by the wind, are not always so, especially when thunder is about to ensue. In this case they are seen to move very slowly, or even to appear quite stationary for some time. The reason of this probably is, that they are impelled by two opposite streams of air nearly of equal strength; and in such cases it seems that both the aerial currents ascend to a considerable height; for Mess. Charles and Robert, when endeavouring to avoid a thunder cloud, in one of their aerial voyages with a balloon, could find no alteration in the course of the current, though they ascended to the height of 4000 feet above the earth. In some cases the motions of the clouds evidently depend on their electricity, independent of any current of air whatever. Thus, in a calm and warm day, small clouds are often seen meeting each other in opposite directions, and setting out from such short distances, that it cannot be supposed that any opposite winds are the cause. Such clouds, when they meet, instead of forming a larger one, become much smaller, and sometimes quite vanish; a circumstance most probably owing to the discharge of opposite electricities into each other. And this serves also to throw some light on the true cause of the formation of clouds; for if two clouds, the one electrified positively, and the other negatively, destroy each other on contact, it follows that any quantity of vapour suspended in the atmosphere, while it retains its natural quantity of electricity, remains invisible, but becomes a cloud when electrified either plus or minus.</p><p>The shapes of the clouds are also probably owing to their electricity; for in those seasons in which a great commotion has been excited in the atmospherical electricity, the clouds are seen assuming strange and whimsical shapes, that are continually varying. This, as <cb/> well as the meeting of small clouds in the air, and vanishing upon contact, is a sure sign of thunder.</p><p>The uses of the clouds are evident, as from them proceeds the rain that refreshes the earth, and without which, according to the present state of nature, the whole surface of the earth must become a mere desert. They are likewise useful as a screen interposed between the earth and the scorching rays of the sun, which are often so powerful as to destroy the grass and other tender vegetables. In the more secret operations of nature too, where the electric fluid is concerned, the clouds bear a principal share; and chiefly serve as a medium for conveying that fluid from the atmosphere into the earth, and from the earth into the atmosphere: in doing which, when electrified to a great degree, they sometimes produce very terrible effects; an instance of which is related above.</p></div1><div1 part="N" n="CLOUTS" org="uniform" sample="complete" type="entry"><head>CLOUTS</head><p>, in Artillery, are thin plates of iron nailed on that part of the axle-tree of a gun-carriage which comes through the nave, and through which the linspin goes.</p></div1><div1 part="N" n="CLUVIER" org="uniform" sample="complete" type="entry"><head>CLUVIER</head><p>, <hi rend="italics">or</hi> <hi rend="smallcaps">Cluverius, (Philip</hi>), a celebrated geographer, was born at Dantzic in 1580. After an education at home, he travelled into Poland, Germany, and the Netherlands, to improve himself in the knowledge of the law. But, when at Leyden, Joseph Scaliger persuaded him to give way to his genius for geography. In pursuance of this advice, Cluvier visited the greatest part of the European states. He was well skilled in many languages, speaking half a score with facility, viz, Greek, Latin, German, French, English, Dutch, Italian, Hungarian, Polish, and Bohemian. On his return to Leyden, he taught there with great applause; and died in 1623, being only 43 years of age, justly esteemed the first geographer who had put his researches in order, and reduced them to certain principles. He was author of several ingenious works in geography, viz:</p><p>1. <hi rend="italics">De Tribus Rheni Alveis.</hi></p><p>2. <hi rend="italics">Germania Antiqua.</hi></p><p>3. <hi rend="italics">Italia Antiqua, Sicilia, Sardinia, & Corsica.</hi></p><p>4. <hi rend="italics">Introductio in Universam Geographiam.</hi></p></div1><div1 part="N" n="COASTING" org="uniform" sample="complete" type="entry"><head>COASTING</head><p>, is that part of Navigation in which the places are not far asunder, so that a ship may sail in sight of land, or within soundings between them.</p></div1><div1 part="N" n="COCHLEA" org="uniform" sample="complete" type="entry"><head>COCHLEA</head><p>, one of the five Mechanical powers, otherwise called the Screw; being so named from the resemblance a screw bears to the spiral shell of a snail, which the Latins call Cochlea. See <hi rend="smallcaps">Screw</hi>, and M<hi rend="smallcaps">ECHANICAL</hi> <hi rend="italics">Powers.</hi></p><p>COCK <hi rend="italics">of a Dial,</hi> the pin, style, or gnomon.</p></div1><div1 part="N" n="COEFFICIENTS" org="uniform" sample="complete" type="entry"><head>COEFFICIENTS</head><p>, in Algebra, are numbers, or given quantities, usually prefixed to letters, or unknown quantities, by which it is supposed they are multiplied; and so, with such letters, or quantities, making a product, or <hi rend="italics">coefficient</hi> production; whence the name.</p><p>Thus, in 3<hi rend="italics">a</hi> the coefficient is 3, in <hi rend="italics">bx</hi> it is <hi rend="italics">b,</hi> and in <hi rend="italics">cx</hi><hi rend="sup">2</hi> it is <hi rend="italics">c.</hi> If a quantity have no number prefixed, unity or 1 is understood; as <hi rend="italics">a</hi> is the same as 1<hi rend="italics">a,</hi> and <hi rend="italics">be</hi> the same as 1<hi rend="italics">bc.</hi> The name <hi rend="italics">coefficient</hi> was first given by Vieta.</p><p>In any equation so reduced as that its highest power or term has 1 for its coefficient; then the coefficient of the 2d term is equal to the sum of all the roots, both <pb n="296"/><cb/> positive and negative; so that if the 2d term is wanting in an equation, the sum of the positive roots of that equation is equal to the sum of the negative roots; as they mutually balance and cancel each other. Also the coefficient of the 3d term of an equation is equal to the sum of all the rectangles arising by the multiplication of every two of the roots, how many ways soever they can be combined by twos; as once in the quadratic, 3 times in the cubic, 6 times in the biquadratic equation, &c. And the coefficient of the 4th term of an equation, is the sum of all the solids made by the continual multiplication of every three of the roots, how often soever such a ternary can be had; as once in a cubic, 4 times in a biquadratic, 10 times in an equation of 5 dimensions, &c. And thus it will go on infinitely.</p><p><hi rend="smallcaps">Coefficients</hi> <hi rend="italics">of the same Order,</hi> is a term sometimes used for the coefficients prefixed to the same unknown quantities, in different equations. the coefficients <hi rend="italics">a, d, g,</hi> are of the same order, being the coefficients of the same letter <hi rend="italics">x</hi>; also <hi rend="italics">b, e, h</hi> are of the same order, being the coefficients of <hi rend="italics">y</hi>; and so on.</p><p><hi rend="italics">Opposite</hi> <hi rend="smallcaps">Coefficients</hi>, such as are taken each from a different equation, and from a different order of coefficients. Thus, in the foregoing equations, <hi rend="italics">a, e, k,</hi> or <hi rend="italics">a, h, f,</hi> or <hi rend="italics">d, b, k,</hi> &c, are opposite coefficients.</p><p>COELESTIAL. See <hi rend="smallcaps">Celestial.</hi></p></div1><div1 part="N" n="COFFER" org="uniform" sample="complete" type="entry"><head>COFFER</head><p>, in Architecture, a square depressure or sinking, in each interval between the modillions of the Corinthian cornice; usually filled up with a rose; sometimes with a pomegranate, or other enrichment.</p></div1><div1 part="N" n="COFFER" org="uniform" sample="complete" type="entry"><head>COFFER</head><p>, in Fortification, denotes a hollow lodgment, athwart a dry moat, 6 or 7 feet deep, and 16 or 18 broad. The upper part of it is made of pieces of timber, raised 2 feet above the level of the moat; the elevation having hurdles laden with earth for its covering, and serving as a parapet with embrazures.</p><p>The coffer is nearly the same with the caponiere, excepting that this last is sometimes made beyond the counterscarp on the glacis, and the coffer always in the moat, taking up its whole breadth, which the caponiere does not.</p><p>It differs from the traverse and gallery, in that these are made by the besiegers, and the coffer by the besieged.</p><p>The besieged commonly make use of coffers to repulse the besiegers, when they endeavour to pass the ditch. And, on the other hand, the besiegers, to save themselves from the fire of these coffers, throw up the earth on that side towards the coffer.</p><p>COFFER-<hi rend="italics">Dams,</hi> or <hi rend="italics">Batardeaux,</hi> in Bridge-building, are inclosures formed for laying the foundation of piers, and for other works in water, to exclude the surrounding water, and so prevent it from interrupting the workmen.</p><p>These inclosures are sometimes single, and sometimes double, with clay rammed between them; sometimes they are made with piles driven close by one another, and sometimes the piles are notched or dove-tailed into one another; but the most usual method is to drive piles with grooves in them, at the distance of five or six feet from each other, and then boards are let down <cb/> between them, after which the water is pumped out.</p><p>COGGESHALL's <hi rend="italics">Sliding-Rule,</hi> an instrument used in Gauging, and so called from its inventor. See the description and use under <hi rend="smallcaps">Sliding</hi>-<hi rend="italics">Rule.</hi></p></div1><div1 part="N" n="COHESION" org="uniform" sample="complete" type="entry"><head>COHESION</head><p>, one of the four species of attraction, denoting that force by which the parts of bodies adhere or stick together.</p><p>This power was first considered by Newton as one of the properties essential to all matter, and the cause of all that variety observed in the texture of different terrestrial bodies. He did not, however, absolutely determine that the power of cohesion was an immaterial one; but that it might possibly arise, as well as that of gravitation, from the action of another. His doctrine of cohesion Newton delivers in these words: “The particles of all hard homogeneous bodies, which touch one another, cohere with a great force; to account for which, some philosophers have recourse to a kind of hooked atoms, which in effect is nothing else but to beg the thing in question. Others imagine that the particles of bodies are connected by rest, i. e. in effect by nothing at all; and others by conspiring motions, i. e. by a relative rest among themselves. For myself, it rather appears to me, that the particles of bodies cohere by an attractive force, whereby they tend mutually toward each other: which force, in the very point of contact, is very great; at little distances is less, and at a little farther distance is quite insensible.”</p><p>It is uncertain in what proportion this force decreases as the distance increases: Desaguliers conjectures, from some phenomena, that it decreases as the biquadratic or 4th power of the distance, so that at twice the distance it acts 16 times more weakly, &c.</p><p>“Now if compound bodies be so hard, as by experience we find some of them to be, and yet have a great many hidden pores within them, and consist of parts only laid together; no doubt those simple particles which have no pores within them, and which were never divided into parts, must be vastly harder. For such hard particles, gathered into a mass, cannot possibly touch in more than a few points: and therefore much less force is required to sever them, than to break a solid particle, whose parts touch throughout all their surfaces, without any intermediate pores or interstices. But how such hard particles, only laid together, and touching only in a few points, should come to cohere so firmly, as in fact we find they do, is inconceivable; unless there be some cause, whereby they are attracted and pressed together. Now the smallest particles of matter may cohere by the strongest attractions, and constitute larger, whose attracting force is feebler: and again, many of these larger particles cohering, may constitute others still larger, whose attractive force is still weaker; and so on for several successions, till the progression end in the biggest particle, on which the operations in chemistry, and the colours of natural bodies, do depend; and which by cohering compose bodies of a sensible magnitude.”</p><p>Again, the opinion maintained by many is that which is so strongly defended by J. Bernoulli, <hi rend="italics">De Gravitate Ætheris</hi>; who attributes the cohesion of the parts of matter to the uniform pressure of the atmosphere; confirming this opinion by the known experiment of two polished marble planes, which cohere very strongly <pb n="297"/><cb/> in the open air, but easily drop asunder in an exhausted receiver. However, if two plates of this kind be smeared with oil, to fill up the pores in their surfaces, and prevent the lodgment of air, and one of them be gently rubbed upon the other, they will adhere so strongly, even when suspended in an exhausted receiver, that the weight of the lower plate will not be able to separate it from the upper one. But although this theory might serve tolerably well to explain the cohesion of compositions, or greater collections of matter; yet it falls far short of accounting for that first cohesion of the atoms, or primitive corpuscles, of which the particles of hard bodies are composed.</p><p>Again, some philosophers have positively asserted, that the powers, or means, are immaterial, by which matter coheres; and, in consequence of this supposition, they have so refined upon attractions and repulsions, that their systems seem but little short of scepticism, or denying the existence of matter altogether. A system of this kind is adopted by Dr. Priestley, from Messrs. Boscovich and Michell, to solve some difficulties concerning the Newtonian doctrine of light. See his History of Vision, vol. 1. pa. 392. “The easiest method,” says he, “of solving all difficulties, is to adopt the hypothesis of Mr. Boscovich, who supposes that matter is not impenetrable, as has been perhaps universally taken for granted; but that it consists of physical points only, endued with powers of attraction and repulsion in the same manner as solid matter is generally supposed to be: provided therefore that any body move with a sufficient degree of velocity, or have a sufficient momentum to overcome any powers of repulsion that it may meet with, it will sind no difficulty in making its way through any body whatever; for nothing else will penetrate one another but powers, such as we know do in fact exist in the same place, and counterbalance or over-rule one another. The most obvious difficulty, and indeed almost the only one, that attends this hypothesis, as it supposes the mutual penetrability of matter, arises from the idea of the nature of matter, and the difficulty we meet with in attempting to force two bodies into the same place. But it is demonstrable, that the first obstruction arises from no actual contact of matter, but from mere powers of repulsion. This difficulty we can overcome; and having got within one sphere of repulsion, we fancy that we are now impeded by the solid matter itself. But the very same is the opinion of the generality of mankind with respect to the first obstruction. Why, therefore, may not the next be only another sphere of repulsion, which may only require a greater force than we can apply to overcome it, without disordering the arrangement of the constituent particles; but which may be overcome by a body moving with the amazing velocity of light?”</p><p>Other philosophers have supposed that the powers both of gravitation and cohesion are material; and that they are only different actions of the etherial fluid, or elementary sire. In proof of this doctrine, they allege the experiment with the Magdeburg hemispheres, as they are called. The pressure of the asmosphere we see is, in this case, capable of producing a very strong cohesion; and if there be in nature any fluid more penetrating, as well as more powerful in its effects, than the air we breath, it is possible that what is called <cb/> the attraction of cohesion may in some measure be an effect of the action of that fluid. Such a fluid as this is the element of fire. Its activity is such as to penetrate all bodies whatever; and in the state in which it is commonly called fire, it acts according to the quantity of solid matter contained in the body. In this state, it is capable of dissolving the strongest cohesions observed in nature. Fire, therefore, being able to dissolve cohesions, must also be capable of causing them, provided its power be exerted for that purpose, which possibly it may be, when we consider its various modes or appearances, viz, as fire or heat, in which state it consumes, destroys, and dissolves; or as light, when it seems deprived of that destructive power; and as the electric fluid, when it attracts, repels, and moves bodies, in a great variety of ways. In the Philos. Trans. for 1777 this hypothesis is noticed, and in some measure adopted by Mr. Henly. “Some gentlemen (says he) have supposed that the electric matter is the cause of the cohesion of the particles of bodies. If the electric matter be, as I suspect, a real elementary sire inherent in all bodies, that opinion may probably be well founded; and perhaps the soldering of metals, and the cementation of iron, by fire, may be considered as strong proofs of the truth of their hypothesis.”</p><p>But whatever the cause of cohesion may be, its effects are evident and certain. The different degrees of it constitute bodies of different forms and properties. Thus, Newton observes, the particles of fluids, which do not cohere too strongly, and are small enough to render them susceptible of those agitations which keep liquors in a fluor, are most easily separated and rarefied into vapour, and make what the chemists call <hi rend="italics">volatile bodies</hi>; being rarefied with an easy heat, and again condensed with a moderate cold. Those that have grosser particles, and so are less susceptible of agitation, or cohere by a stronger attraction, are not separable without a greater degree of heat; and some of them not without fermentation: and these make what the chemists call <hi rend="italics">fixt bodies.</hi></p><p>Air, in its fixed state, possesses the interstices of solid substances, and probably serves as a bond of union to their constituent parts; for when these parts are separated, the air is discharged, and recovers its elasticity. And this kind of attraction is evinced by a variety of familiar experiments; as, by the union of two contiguous drops of mercury; by the mutual approach of two pieces of cork, floating near each other in a bason of water; by the adhesion of two leaden balls, whose surfaces are scraped and joined together with a gentle twist, which is so considerable, that, if the surfaces are about a quarter of an inch in diameter, they will not be separated by a weight of 100 lb; by the ascent of oil or water between two glass planes, so as to form the hyperbolic curve, when they are made to touch on one side, and kept separate at a small distance on the other; by the depression of mercury, and by the rise of water in capillary tubes, and on the sides of glass vessels; also in sugar, sponge, and all porous substances. And where this cohesive attraction ends, a power of repulsion begins.</p><p><hi rend="italics">To determine the force of cohesion,</hi> in a variety of different substances, many experiments have been made, and particularly by professor Muschenbroek. The ad- <pb n="298"/><cb/> hesion of polished planes, about two inches in diameter, heated in boiling water, and smeared with grease, required the following weights to separate them: <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Cold grease</cell><cell cols="1" rows="1" rend="align=center" role="data">Hot grease</cell></row><row role="data"><cell cols="1" rows="1" role="data">Planes of Glass</cell><cell cols="1" rows="1" role="data">130 lb</cell><cell cols="1" rows="1" role="data">300 lb</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brass</cell><cell cols="1" rows="1" role="data">150</cell><cell cols="1" rows="1" role="data">800</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper</cell><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">850</cell></row><row role="data"><cell cols="1" rows="1" role="data">Marble</cell><cell cols="1" rows="1" role="data">225</cell><cell cols="1" rows="1" role="data">600</cell></row><row role="data"><cell cols="1" rows="1" role="data">Silver</cell><cell cols="1" rows="1" role="data">150</cell><cell cols="1" rows="1" role="data">250</cell></row><row role="data"><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">950</cell></row></table></p><p>But when the Brass planes were made to adhere by other sorts of matter, the results were as in the following table: <table><row role="data"><cell cols="1" rows="1" role="data">With</cell><cell cols="1" rows="1" role="data">Water</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">oz</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Oil</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Venice Turpentine</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Tallow Candle</cell><cell cols="1" rows="1" rend="align=right" role="data">800</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Rosin</cell><cell cols="1" rows="1" rend="align=right" role="data">850</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Pitch</cell><cell cols="1" rows="1" rend="align=right" role="data">1400</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>In estimating the <hi rend="italics">Absolute Cohesion</hi> of solid pieces of bodies, he applied weights to separate them according to their length: his pieces of wood were long square parallelopipedons, each side of which was .26 of an inch, and they were drawn asunder by the following weights: <table><row role="data"><cell cols="1" rows="1" role="data">Fir</cell><cell cols="1" rows="1" rend="align=right" role="data">600</cell><cell cols="1" rows="1" role="data">lb</cell></row><row role="data"><cell cols="1" rows="1" role="data">Elm</cell><cell cols="1" rows="1" rend="align=right" role="data">950</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Alder</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Linden tree</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Oak</cell><cell cols="1" rows="1" rend="align=right" role="data">1150</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Beech</cell><cell cols="1" rows="1" rend="align=right" role="data">1250</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Ash</cell><cell cols="1" rows="1" rend="align=right" role="data">1250</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>He tried also wires of metal, 1-10th of a Rhinland inch in diameter: the metals and weights were as follow: <table><row role="data"><cell cols="1" rows="1" role="data">Of</cell><cell cols="1" rows="1" role="data">Lead</cell><cell cols="1" rows="1" role="data"> 29 1/4</cell><cell cols="1" rows="1" role="data">lb</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Tin</cell><cell cols="1" rows="1" role="data"> 40 1/4</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Copper</cell><cell cols="1" rows="1" role="data">299 1/4</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Yellow Brass</cell><cell cols="1" rows="1" role="data">360</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Silver</cell><cell cols="1" rows="1" role="data">370</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" role="data">450</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Gold</cell><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>He then tried the <hi rend="italics">Relative Cohesion,</hi> or the force with which bodies resist an action applied to them in a direction perpendicular to their length. For this purpose he fixed pieces of wood by one end into a square hole in a metal plate, and hung weights towards the other end, till they broke at the hole: the weights and distances from the hole are exhibited in the following table. <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Distance</cell><cell cols="1" rows="1" rend="align=center" role="data">Weight</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Pine</cell><cell cols="1" rows="1" role="data">9 1/2 inc</cell><cell cols="1" rows="1" role="data">36 1/2</cell><cell cols="1" rows="1" role="data">oz</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fir</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Beech</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">56 1/2</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Elm</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Oak</cell><cell cols="1" rows="1" role="data">8 1/2</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Alder</cell><cell cols="1" rows="1" role="data">9 1/4</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/></row></table> See his Elem. Nat. Philos. cap. 19.</p></div1><div1 part="N" n="COLD" org="uniform" sample="complete" type="entry"><head>COLD</head><p>, the privation of heat, or the opposite to it.</p><p>As it is supposed that heat consists in a particular motion of the parts of the hot body, hence the nature of cold, which is its opposite, is deduced; for it is found that cold extinguishes, or rather abates heat; <cb/> hence it would seem to follow, that those bodies are cold, which check and restrain the motion of the particles in which heat consists.</p><p>In general, cold contracts most bodies, and heat expands them: though there are some instances to the contrary, especially in the extreme cases or states of these qualities of bodies. Thus, though iron, in common with other bodies, expand with heat, yet, when melted, it is always found to expand in cooling again. So also, though water always is found to expand gradually as it is heated, and to contract as it cools, yet in the act of freezing, it suddenly expands again, and that with a most enormous force, capable of rending rocks, or bursting the very thick shells of metal, &c. A computation of the force of freezing water has been made by the Florentine Academicians, from the bursting of a very strong brass globe or shell, by freezing water in it; when, from the known thickness and tenacity of the metal, it was found that the expansive power of a spherule of water only one inch in diameter, was sufficient to overcome a resistance of more than 27,000 pounds, or 13 tons and a half. See also experiments on bursting thick iron bomb-shells by freezing water in them by Major Edward Williams of the Royal Artillery, in the Edinb. Philos. Trans. vol. 2.</p><p>Such a prodigious power of expansion, almost double that of the most powerful steam engines, and exerted in so small a mass, seemingly by the force of cold, was thought a very powerful argument in favour of those who supposed that cold, like heat, is a positive substance. Dr. Black's discovery of latent heat, however, has now afforded a very easy and natural explication of this phenomenon. He has shewn, that, in the act of congelation, water is not cooled more than it was before, but rather grows warmer: that as much heat is discharged, and passes from a latent to a sensible state, as, had it been applied to water in its fluid state, would have heated it to 135°. In this process, the expansion is occasioned by a great number of minute bubbles suddenly produced. Formerly these were supposed to be cold in the abstract; and to be so subtle, that, insinuating themselves into the substances of the fluid, they augmented its bulk, at the same time that, by impeding the motion of its particles upon each other, they changed it from a fluid to a solid. But Dr. Black shews that these are only air extricated during the congelation; and to the extrication of this air he ascribes the prodigious expansive force exerted by freezing water. The only question therefore now remaining, is, By what means this air comes to be extricated, and to take up more room than it naturally does in the fluid. To this it may be answered, that perhaps part of the heat which is discharged from the freezing water, combines with the air in its unelastic state, and, by restoring its elasticity, gives it that extraordinary force, as is seen also in the case of air suddenly extricated in the explosion of gun-powder.</p><p>Cold also usually tends to make bodies electric, which are not so naturally, and to increase the electric properties of such as are so. And it is farther found that all substances do not transmit cold equally well; but that the best conductors of electricity, viz metals, are likewise the best conductors of cold. It may farther be added, that when the cold has been carried to <pb n="299"/><cb/> such an extremity as to render any body an electric, it then ceases to conduct the cold so well as before. This is exemplified in the practice of the Laplanders and Siberians; where, to exclude the extreme cold of the winters from their habitations the more effectually, and yet to admit a little light, they cut pieces of ice, which in the winter time must always be electric in those countries, and put them into their windows; which they find to be much more effectual in keeping out the cold than any other substance.</p><p>Cold is the destroyer of all vegetable life, when increased to an excessive degree. It is found that many garden plants and flowers, which seem to be very stout and hardy, go off at a little increase of cold beyond the ordinary standard. And in severe winters, nature has provided the best natural defence for the corn fields and gardens, namely, a covering of snow, which preserves such parts green and healthy as are under it, while such as are uncovered by it are either killed or greatly injured.</p><p>Dr. Clarke is of opinion, that cold is owing to certain nitrous, and other saline particles, endued with particular figures proper to produce such effects. Hence, sal-ammoniac, saltpetre, or salt of urine, and many other volatile and alkalizate salts, mixed with water, very much increase its degree of cold. In the Philos. Trans. number 274, M. Geoffroy relates some remarkable experiments with regard to the production of cold. Four ounces of sal-ammoniac dissolved in a pint of water, made his thermometer descend 2 inches and 3/4 in less than 15 minutes. An ounce of the same salt put into 4 or 5 ounces of distilled water, made the thermometer descend 2 inches and 1/4. Half an ounce of sal-ammoniac mixed with 3 ounces of spirit of nitre, made the thermometer descend 2 inches and 5/12; but, on using spirit of vitriol instead of nitre, it sunk 2 inches and 1/2. In this last experiment it was remarked, that the vapours raised from the mixture had a considerable degree of heat, though the liquid itself was so extremely cold. Four ounces of saltpetre mixed with a pint of water, sunk the thermometer an inch and 1/4; but a like quantity of sea salt sunk it only 1/6 of an inch. Acids always produced heat, even common salt with its own spirit. Volatile alkaline salts produced cold in proportion to their purity, but fixed alkalies heat.</p><p>But the greatest degree of cold produced by the mixture of salts and aqueous fluids, was that shewn by Homberg; who gives the following receipt for making the experiment: Take a pound of corrosive sublimate, and as much sal-ammoniac; powder them separately, and mix the powders well; put the mixture into a vial, pouring upon it a pint and a half of distilled vinegar, shaking all well together. This composition grows so cold, that it can scarce be held in the hand in summer; and it happened, as M. Homberg was making the experiment, that the matter froze. The same thing once happened to M. Geoffroy, in making an experiment with sal-ammoniac and water, but it never was in his power to make it succeed a second time.</p><p>If, instead of making these experiments with fluid water, it be taken in its congealed state of ice, or rather snow, degrees of cold will be produced greatly superior to any that have yet been mentioned. A mixture of snow and common salt sinks Fahrenheit's thermometer to 0; pot ashes and pounded ice sunk it 8 degrees far- <cb/> ther; two affusions of spirit of salt on pounded ice sunk it 14 1/2 below 0; and by repeated assusions of spirit of nitre M. Fahrenheit sunk it to 40° below 0. This is the ultimate degree of cold which the mercurial thermometer will measure; for the mercury itself begins then to congeal; and therefore recourse must afterwards be had to spirit of wine, naptha, or some other fluid that will not congeal. The greatest degree of cold hitherto produced by artificial means, has been 80° below 0; which was done at Hudson's Bay by means of snow and vitriolic acid, the thermometer standing naturally at 20° below 0. Indeed greater degrees of cold than this have been supposed: Mr. Martin, in his Treatise on Heat, relates, that at Kirenga in Siberia, the mercurial thermometer sunk to 118° below 0; and professor Brown at Petersburg, when he made the first experiment of congealing quicksilver, fixed the point of congelation at 350° below 0; but from later experiments it has been more accurately determined, that 40° below 0 is the freezing point of quicksilver.</p><p>The most remarkable experiment however was made by Mr. Walker of Oxford, with spirit of nitre poured on Glauber's salt, the effect of which was found to be similar to that of the same spirit poured on ice or snow; and the addition of sal-ammoniac rendered the cold still more intense. The proportions of these ingredients recommended by Mr. Walker, are concentrated nitrous acid two parts by weight, water one part; of this mixture, cooled to the temperature of the atmosphere, 18 ounces; of Glauber's salt, a pound and a half avoirdupois; and of sal-ammoniac, 12 ounces. On adding the Glauber's salt to the nitrous acid, the thermometer fell 52°, viz from 50 to - 2; and on the addition of the salammoniac, it fell to - 9°. Thus Mr. Walker was able to freeze quicksilver without either ice or snow, when the thermometer stood at 45°; viz, by putting the ingredients in 4 different pans, and inclosing these within each other.</p><p>Excessive degrees of cold occur naturally in many parts of the globe in the winter season.</p><p>Although the thermometer in this country hardly ever descends so low as 0, yet in the winter of 1780, Mr, Wilson of Glasgow observed, that a thermometer laid on the snow sunk to 25° below 0; and Mr. Derham, in the year 1708, observed in England, that the mercury stood within one-tenth of an inch of its station when plunged into a mixture of snow and salt. At Petersburg, in 1732, the thermometer stood at 28° below 0; and when the French academicians wintered near the polar circle, the thermometer sunk to 33° below 0; and in the Asiatic and American continents, still greater degrees of cold are often observed.</p><p>The effects of these extreme degrees of cold are very surprising. Trees are burst, rocks rent, and rivers and lakes frozen several feet deep: metallic substances blister the skin like red-hot iron: the air, when drawn in by breathing, hurts the lungs, and excites a cough: even the effects of fire in a great measure seem to cease; and it is observed, that though metals are kept for a considerable time before a strong fire, they will still freeze water when thrown upon them. When the French mathematicians wintered at Tornea in Lapland, the external air, when suddenly admitted into their rooms, converted the moisture of the air into whirls of snow; their breasts seemed to be rent when they breathed <pb n="300"/><cb/> it, and the contact of it was intolerable to their bodies; and the spirit of wine, which had not been highly rectified, burst some of their thermometers by the congelation of the aqueous part.</p><p>Extreme cold too often proves fatal to animals in those countries where the winters are very severe; thus 7000 Swedes perished at once in attempting to pass the mountains which divide Norway from Sweden. But it is not necessary that the cold, in order to prove fatal to human life, should be so very intense as has just been mentioned; it is only requisite to be a little below 32° of Fahrenheit, or the freezing point, accompanied with snow or hail, from which shelter cannot be obtained. The snow which falls upon the clothes, or the uncovered parts of the body, then melts, and by a continual evaporation carries off the animal heat to such a degree, that a sufficient quantity is not left for the support of life. In such cases, the person first feels himself extremely chill and uneasy; he turns listless, unwilling to walk or use exercise to keep himself warm, and at last turns drowsy, sits down to refresh himself with sleep, but wakes no more.</p></div1><div1 part="N" n="COLLIMATION" org="uniform" sample="complete" type="entry"><head>COLLIMATION</head><p>, <hi rend="italics">Line of,</hi> in a telescope, is a line passing through the intersection of those wires that are fixed in the focus of the object-glass, and the centre of the same glass.</p></div1><div1 part="N" n="COLLINS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">COLLINS</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent accountant and mathematician, was born at Wood Eaton near Oxford, March 5, 1624. At 16 years of age he was put apprentice to a bookseller at Oxford; but his genius appeared so remarkable for the study of the mechanical and mathematical sciences, that he was taken under the tuition of Mr. Marr, who drew several curious dials, which were placed in different positions in the king's gardens; under whom Mr. Collins made no small progress in the mathematics. In the course of the civil wars, he travelled abroad, to prosecute his favourite study; and on his return he took upon him the profession of an accountant, and published, in the year 1652, a large work entitled, <hi rend="italics">An Introduction to Merchants Accompts;</hi> which was followed by several other publications on different branches of accounts. In 1658, he published a treatise called <hi rend="italics">The Sector on a Quadrant;</hi> containing the description and use of four several quadrants, each accommodated to the making of sun-dials, &c; to which he afterward added an appendix concerning reflected dialling, from a glass placed reclining.—In 1659, he published his <hi rend="italics">Geometrical Dialling;</hi> and the same year also his <hi rend="italics">Mariner's Plain Scale new plained.</hi>—Collins now became a fellow of the Royal Society in London, to which he made various communications; particularly some ingenious chronological rules for the calendar, printed in the Philos. Trans. number 46, for April 1669: also a curious dissertation concerning the resolution of equations in numbers, in number 69, for March 1671: an elegant construction of the curious problem, having given the mutual distances of three objects in a plane, with the angles made by them at a fourth place in that plane, to find the distance of this place from each of the three former, vol. 6. pa. 2093: and thoughts about some defects in algebra, vol. 14. pa. 375.</p><p>Collins wrote also several commercial tracts, highly acceptable to the public; viz, A Plea for bringing over Irish cattle, and keeping out the fish caught by foreign- <cb/> ers:—For the promotion of the English fishery:—For the working the Tin-mines:—A Discourse of Salt and Fishery. He was frequently consulted in nice and critical cases of accounts, of commerce, and engineering. On one of these occasions, being appointed to inspect the ground for cutting a canal or river between the Isis and the Avon, he contracted a disorder by drinking cyder when he was too warm, which ended in his death, the 10th of November 1683, at 59 years of age.</p><p>Mr. Collins was a very useful man to the sciences, keeping up a constant correspondence with the most learned men, both at home and abroad, and promoting the publication of many valuable works, which, but for him, would never have been seen by the public; particularly Dr. Barrow's optical and geometrical lectures; his abridgment of the works of Archimedes, Apollonius, and Theodosius; Branker's translation of Rhronius's algebra, with Dr. Pell's additions, &c; which were procured by his frequent solicitations.</p><p>It was a considerable time after, that his papers were all delivered into the hands of the learned and ingenious Mr. William Jones, F. R. S. among which were found manuscripts, upon mathematical subjects, of Briggs, Oughtred, Barrow, Newton, Pell, and many others. From a variety of letters from these, and many other celebrated mathematicians, it appears that Collins spared neither pains nor cost to procure what tended to promote real science: and even many of the late discoveries in physical knowledge owe their improvement to him; for while he excited some to make known every new and useful invention, he employed others to improve them. Sometimes he was peculiarly useful, by shewing where the defect was in any useful branch of science, pointing out the difficulties attending the enquiry, and at other times setting forth the advantages, and keeping up a spirit and warm desire for improvement. Mr. Collins was also as it were the register of all the new improvements made in the mathematical sciences; the magazine to which the curious had frequent recourse: in so much that he acquired the appellation of the English Mersennus. If some of his correspondents had not obliged him to conceal their communications, there could have been no dispute about the priority of the invention of a method of analysis, the honour of which evidently belongs to Newton; as appears undeniably from the papers printed in the <hi rend="italics">Commercium Epistolicum D. Joannis Collins & aliorum de Analysi promota; jussu Societatis Regiæ in lucem editum,</hi> 1712; a work that was made out from the letters in the possession of our author.</p><p>COLLINS's <hi rend="italics">Quadrant.</hi> See <hi rend="smallcaps">Quadrant.</hi></p></div1><div1 part="N" n="COLLISION" org="uniform" sample="complete" type="entry"><head>COLLISION</head><p>, is the friction, percussion, or striking of bodies against one another.</p><p>Striking bodies are considered either as elastic, or non-elastic. They may also be either both in motion, or one of them in motion, and the other at rest.</p><p>When non-elastic bodies strike, they unite together as one mass; which, after collision, either remains at rest, or moves forward as one body. But when elastic bodies strike, they always separate after the stroke.</p><p>The principal theorems relating to the collision of bodies, are the following:</p><p>1. If any body impinge or act obliquely on a plane surface; the force or energy of the stroke, or action, is as the sine of the angle of incidence. Or the force <pb n="301"/><cb/> upon the surface, is to the same when acting perpendicularly, as the sine of incidence is to radius.</p><p>2. If one body act on another, in any direction, and by any kind of force; the action of that force on the second body is made only in a direction perpendicular to the surface on which it acts.</p><p>3. If the plane, acted on, be not absolutely fixed, it will move, after the stroke, in the direction perpendicular to its surface.</p><p>4. If a body A strike another body B, which is either at rest, or else in motion, either towards A or from it; then the momenta, or quantities of motion, of the two bodies, estimated in any one direction, will be the very same after the stroke that they were before it. <figure/></p><p>Thus, first, if A with a momentum of 10, strike B at rest, and communicate to it a momentum of 4, in the direction AB. Then there will remain in A only a momentum of 6 in that direction: which together with the momentum of B, viz 4, makes up still the same momentum between them as before.—But if B were in motion before the stroke, with a momentum of 5, in the same direction, and receive from A an additional momentum of 2: then the motion of A after the stroke will be 8, and that of B, 7; which between them make up 15, the same as 10 and 5, the motions before the stroke.—Lastly, if the bodies move in opposite directions, and meet one another, namely A with a motion of 10, and B, of 5; and A communicate to B a motion of 6 in the direction AB of its motion: then, before the stroke, the whole motion from both, in the direction AB, is 10-5, or 5: but after the stroke the motion of A is 4 in the direction AB, and the motion of B is 6 - 5, or 1 in the same direction AB; therefore the sum 4+1, or 5, is still the same motion from both as it was before.</p><p>5. If a hard and fixed plane be struck either by a soft or a hard unelastic body; the body will adhere to it. But if the plane be struck by a perfectly elastic body, it will rebound from it with the same velocity with which it struck the plane.</p><p>6. The effect of the blow of the elastic body, upon the plane, is double to that of the non-elastic one; the velocity and mass being the same in both.</p><p>7. Hence, non-elastic bodies lose, by their collision, only half the motion that is lost by elastic bodies; the masses and velocities being equal. <figure/></p><p>8. If an elastic body A impinge upon a firm plane DE at the point B, it will rebound from it in an angle equal to that in which it struck it; or the angle of incidence will be equal to the angle of reflection: namely, the angle ABD = CBE. <figure/></p><p>9. If the non-elastic body B, moving with the velocity V in the direction B<hi rend="italics">b,</hi> and the body <hi rend="italics">b</hi> with the velocity <hi rend="italics">v,</hi> strike each other, the direction of the mo- <cb/> tion being in the line BC; then they will move after the stroke with a common velocity, which will be more or less according as, before the stroke, <hi rend="italics">b</hi> moved towards B, or from B, or was at rest; and that common velocity, in each of these cases, will be as follows: viz, it will be (BV + <hi rend="italics">bv</hi>)/(B + <hi rend="italics">b</hi>) when <hi rend="italics">b</hi> moved from B, (BV - <hi rend="italics">bv</hi>)/(B + <hi rend="italics">b</hi>) when <hi rend="italics">b</hi> moved towards B, BV/(B + <hi rend="italics">b</hi>) when <hi rend="italics">b</hi> was at rest.</p><p>For example, if the bodies or weights, B and <hi rend="italics">b,</hi> be 5lb and 3lb; and their velocities V and <hi rend="italics">v,</hi> 60 feet and 40 feet per second; then 300 and 120 will be their momenta BV and <hi rend="italics">bv,</hi> and the sum of the weights. Consequently the common velocity after the stroke, in the three cases above mentioned, will be thus, viz, in the first case, in the second case, 300/18 or 16 2/3 in the third case.</p><p>10. If two perfectly elastic bodies impinge on each other; their relative velocity is the same both before and after the impulse; that is, they will recede from each other with the same velocity with which they approached and met.</p><p>It is not meant however by this theorem, that each body will have the very same velocity after the impulse as it had before; but that the velocity of the one, after the stroke, will be as much increased, as that of the other is decreased, in one and the same direction. So, if the elastic body B move with the velocity V, and overtake the elastic body <hi rend="italics">b,</hi> moving the same way, with the velocity <hi rend="italics">v</hi>; then their relative velocity, or that with which they strike, is only V-<hi rend="italics">v</hi>; and it is with this same velocity that they separate from each other after the stroke: but if they meet each other, or the body <hi rend="italics">b</hi> move contrary to the body B; then they meet and strike with the velocity V + <hi rend="italics">v,</hi> and it is with the same velocity that they separate again, and recede from each other after the stroke: in like manner, they would separate with the velocity V of B, if <hi rend="italics">b</hi> were at rest before the stroke. Also the sum of the velocities of the one body, is equal to the sum of the others. But whether they move forwards or backwards after the impulse, and with what particular velocities, are circumstances that depend on the various masses and velocities of the bodies before the stroke, and are as specified in the next theorem.</p><p>11. If the two elastic bodies B and <hi rend="italics">b</hi> move directly towards each other, or directly from each other, the former with the velocity V, and the latter with the velocity <hi rend="italics">v;</hi> then, after their meeting and impulse, the respective velocities of B and <hi rend="italics">b</hi> in the direction BC, in the three cases of motion, will be as follow: viz, the velocity of B, the velocity of <hi rend="italics">b,</hi> when the bodies both moved towards C before the stroke; and <pb n="302"/><cb/> the velocity of B, the velocity of <hi rend="italics">b,</hi> when B moved towards C, and <hi rend="italics">b</hi> towards B before the stroke; the velocity of B, the velocity of <hi rend="italics">b,</hi> when <hi rend="italics">b</hi> was at rest before the stroke.</p><p>12. The motions of bodies after impact, that ftrike each other obliquely, are thus determined. <figure/></p><p>Let the two bodies B, <hi rend="italics">b,</hi> move in the oblique directions BA, <hi rend="italics">b</hi>A, and strike each other at A with velocities which are in proportion to the lines BA, <hi rend="italics">b</hi>A. Let CAH represent the plane in which the bodies touch in the point of concourse; to which draw the perpendiculars BC, <hi rend="italics">b</hi>D, and complete the rectangles CE, DF. Now the motion in BA is resolved into the two BC, CA; and the motion in <hi rend="italics">b</hi>A is resolved into the two <hi rend="italics">b</hi>D, DA; of which the antecedents BC, <hi rend="italics">b</hi>D are the velocities with which they irectly meet, and the consequents CA, DA are parallel, and therefore by these the bodies do not impinge on each other, and consequently the motions according to these directions will not be changed by the impulse; so that the velocities with which the bodies meet, are as BC or EA, and <hi rend="italics">b</hi>D or FA. The motions therefore of the bodies B, <hi rend="italics">b,</hi> directly striking each other with the celerities EA, FA, will be determined by art. 11 or 9, according as the bodies are elastic or non-elastic; which being done, let AG be the velocity, so determined, of one of them, as A; and since there remains also in the same body a force of moving in the direction parallel to BE, with a velocity as BE, make AH equal to BE, and complete the rectangle GH: then the two motions in AH and AG, or HI, are compounded into the diagonal AI, which therefore will be the path and celerity of the body B after the stroke. And after the same manner is the motion of the other body <hi rend="italics">b</hi> determined after the impact.</p><p>13. The state of the common centre of gravity of bodies is not affected by the collision or other actions of those bodies on one another. That is, if it were at rest before their collision, so will it be also at rest after collision: and if it were moving in any direction, and with any velocity, before collision; it will do the very same after it.</p><p>See more upon this subject under the article P<hi rend="smallcaps">ERCUSSION.</hi> <cb/></p></div1><div1 part="N" n="COLONNADE" org="uniform" sample="complete" type="entry"><head>COLONNADE</head><p>, a Peristyle of a circular figure; or a series of columns disposed in a circle, and insulated within-side.</p></div1><div1 part="N" n="COLOUR" org="uniform" sample="complete" type="entry"><head>COLOUR</head><p>, a property inherent in light, by which, according to the various sizes of its parts, or from some other cause, it excites different vibrations in the optic nerve; which, propagated to the sensorium, affect the mind with different sensations. See the doctrine of colours fully explained under <hi rend="italics">Chromatics.</hi> See also <hi rend="italics">Optics, Achromatic,</hi> and <hi rend="italics">Telescope.</hi></p><p>COLUMBA <hi rend="italics">Noachi,</hi> Noah's Dove, a small constellation in the southern hemisphere, consisting of 10 stars.</p></div1><div1 part="N" n="COLUMN" org="uniform" sample="complete" type="entry"><head>COLUMN</head><p>, in Architecture, a round pillar, made to support or adorn a building.</p><p>The column is the principal part of an architectonical order, and is composed of three parts, the <hi rend="italics">base,</hi> the <hi rend="italics">shaft,</hi> and the <hi rend="italics">capital</hi>; each of which is subdivided into a number of lesser parts, called members, or mouldings.</p><p>Columns are different according to the different orders they are used in; and also according to their matter, construction, form, disposition, and use. The proportion of the length of each to its diameter, and the diminution of the diameter upwards, are diversly stated by different authors. The medium of them is nearly as follows:</p><p>The <hi rend="italics">Tuscan</hi> is the simplest and shortest of all; its height 3 1/2 diameters, or 7 modules; and it diminishes 1/4 part of its diameter.</p><p>The <hi rend="italics">Doric</hi> is more delicate, and adorned with flutings; its height 7 1/2 or 8 diameters.</p><p>The <hi rend="italics">Ionic</hi> is more delicate still, being 9 diameters long. It is distinguished from the rest by the volutes, or curled serolls in its capital, and by its base which is peculiar to it.</p><p>The <hi rend="italics">Corinthian</hi> is the richest and most delicate of all the columns, being 10 diameters in length, and adorned with two rows of leaves, and stalks or stems, from whence spring out small volutes.</p><p>The <hi rend="italics">Composite</hi> Column is also 10 diameters long, its capital adorned with rows of leaves like the Corinthian, and with angular volutes like the Ionic.</p></div1><div1 part="N" n="COLURES" org="uniform" sample="complete" type="entry"><head>COLURES</head><p>, are two great circles imagined to intersect at right angles in the poles of the world, and to pass, the one through the equinoctial points Aries and Libra, and the other through the solstitial points Cancer and Capricorn; from whence they are called the Equinoctial and Solstitial Colures. By thus dividing the ecliptic into four equal parts, they mark the four seasons, or quarters of the year.</p><p>It is disputed over what part of the back of Aries the equinoctial colure passed in the time of Hipparchus. Newton, in his Chronology, takes it to have been over the middle of the constellation. Father Souciet insists that it passed over the dodecatemorion of Aries, or midway between the rump and first of the tail. There are some observations in the Philos. Trans. number 466, concerning the position of this colure in the ancient sphere, from a draught of the constellation Aries, in the Aratæa published at Leyden and Amsterdam in 1652, which seem to confirm Newton's opinion; but the antiquity and authority of the original draught may still remain in question.</p><p>COMA <hi rend="smallcaps">Berenices</hi>, <hi rend="italics">Berenice's Hair,</hi> a modern con- <pb n="303"/><cb/> stellation of the northern hemisphere; composed of unformed stars between the Lion's tail and Bootes.</p><p>It is said that this constellation was formed by Conon, an astronomer, to console the queen of Ptolomy Euergetes, for the loss of a lock of her hair, which was stolen out of the temple of Venus, where she had dedicated it on account of a victory obtained by her husband.</p><p>The stars in this constellation are, in Tycho's catalogue 14, in Hevelius's 21, and in the Britannic catalogue 43.</p></div1><div1 part="N" n="COMBINATIONS" org="uniform" sample="complete" type="entry"><head>COMBINATIONS</head><p>, denote the alternations or variations of any number of quantities, letters, sounds, or the like, in all possible ways.</p><p>Father Mersenne gives the combinations of all the notes and sounds in music, as far as 64; the sum of which amounts to a number expressed by 90 places of figures. And the number of possible combinations of the 24 letters of the alphabet, taken first two by two, then three by three, and so on, according to Prestet's calculation, amounts to <hi rend="center">1391724288887252999425128493402200.</hi></p><p>Father Truchet, in Mem. de l'Acad. shews, that two square pieces, each divided diagonally into two colours, may be arranged and combined 64 different ways, so as to form so many different kinds of chequer-work: a thing that may be of use to masons, paviours, &c.</p><p><hi rend="italics">Doctrine of</hi> <hi rend="smallcaps">Combinations.</hi></p><p>I. <hi rend="italics">Having given any number of things, with the number in each combination; to find the number of combinations.</hi> <hi rend="center">1. <hi rend="italics">When only two are combined together.</hi></hi></p><p>One thing admits of no combination.</p><p>Two, <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> admit of one only, viz <hi rend="italics">ab.</hi></p><p>Three, <hi rend="italics">a, b, c,</hi> admit of three, viz <hi rend="italics">ab, ac, bc.</hi></p><p>Four admit of six, viz, <hi rend="italics">ab, ac, ad, bc, bd, cd.</hi></p><p>Five admit of 10, viz, <hi rend="italics">ab, ac, ad, ae, bc, bd, be, cd, ce, de.</hi></p><p>Whence it appears that the numbers of combinations, of two and two only, proceed according to the triangular numbers 1, 3, 6, 10, 15, 21, &c, which are produced by the continual addition of the ordinal series 0, 1, 2, 3, 4, 5, &c. And if <hi rend="italics">n</hi> be the number of things, then the general formula for expressing the sum of all their combinations by twos, will be . <table><row role="data"><cell cols="1" rows="1" role="data">Thus, if <hi rend="italics">n</hi> = 2; this becomes 2.1/2</cell><cell cols="1" rows="1" role="data">= 1.</cell></row><row role="data"><cell cols="1" rows="1" role="data">If <hi rend="italics">n</hi> = 3; it is 3.2/2 "</cell><cell cols="1" rows="1" role="data">=3.</cell></row><row role="data"><cell cols="1" rows="1" role="data">If <hi rend="italics">n</hi> = 4; it is 4.3/2 "</cell><cell cols="1" rows="1" role="data">= 6. &c.</cell></row></table> <hi rend="center">2. <hi rend="italics">When three are combined together; then</hi></hi></p><p>Three things admit of one order, <hi rend="italics">abc.</hi></p><p>Four admit of 4; viz <hi rend="italics">abc, abd, acd, bcd.</hi></p><p>Five admit of 10; viz <hi rend="italics">abc, abd, abe, acd, ace, ade, bcd, bce, bde, cde.</hi> And so on according to the first pyramidal numbers 1, 4, 10, 20, &c, which are formed by the continual addition of the former, or triangular numbers 1, 3, 6, 10, &c. And the general formula for any number <hi rend="italics">n</hi> of combinations, taken by threes, is . <cb/></p><p>So, if . &c. Proceeding thus, it is found that a general formula for any number <hi rend="italics">n</hi> of things, combined by <hi rend="italics">m</hi> at each time, is , continued to <hi rend="italics">m</hi> factors, or terms, or till the last factor in the denominator be <hi rend="italics">m.</hi></p><p>So, in 6 things, combined by 4's, the number of combinations is .</p><p>3. By adding all these series together, their sum will be the whole number of possible combinations of <hi rend="italics">n</hi> things combined both by twos, by threes, by fours, &c. And as the said series are evidently the coefficients of the power <hi rend="italics">n</hi> of a binomial, wanting only the first two 1 and <hi rend="italics">n</hi>; therefore the said sum, or whole number of all such combinations, will be ―(1 + 1) - <hi rend="italics">n</hi> - 1, or 2<hi rend="sup">n</hi> - <hi rend="italics">n</hi> - 1. Thus if the number of things be 5; then .</p><p>II. <hi rend="italics">To find the number of Changes and Alterations which any number of quantities can undergo, when combined in all possible varieties of ways, with themselves and each other, both as to the things themselves, and the Order or Position of them.</hi></p><p>One thing admits but of one order or position.</p><p>Two things may be varied four ways; thus, <hi rend="italics">aa, ab, ba, bb.</hi></p><p>Three quantities, taken by twos, may be varied nine ways; thus <hi rend="italics">aa, ab, ac, ba, ca, bb, bc, cb, cc.</hi></p><p>In like manner four things, taken by twos, may be varied 4<hi rend="sup">2</hi> or 16 ways; and 5 things, by twos, 5<hi rend="sup">2</hi> or 25 ways; and, in general, <hi rend="italics">n</hi> things, taken by twos, may be changed or varied <hi rend="italics">n</hi><hi rend="sup">2</hi> different ways.</p><p>For the same reason, when taken by threes, the changes will be <hi rend="italics">n</hi><hi rend="sup">3</hi>; and when taken by fours, they will be <hi rend="italics">n</hi><hi rend="sup">4</hi>; and so generally, when taken by <hi rend="italics">n</hi>'s, the changes will be <hi rend="italics">n</hi><hi rend="sup">n</hi>.</p><p>Hence, then, adding all these together, the whole number of changes, or combinations in <hi rend="italics">n</hi> things, taken both by 2's, by 3's, by 4's, &c, to <hi rend="italics">n</hi>'s, will be the sum of the geometrical series <hi rend="italics">n</hi> + <hi rend="italics">n</hi><hi rend="sup">2</hi> + <hi rend="italics">n</hi><hi rend="sup">3</hi> + <hi rend="italics">n</hi><hi rend="sup">4</hi> <hi rend="italics">n</hi><hi rend="sup">n</hi>, which sum is .</p><p>For example, if the number of things <hi rend="italics">n</hi> be 4; this gives .</p><p>And if <hi rend="italics">n</hi> be 24, the number of letters in the alphabet; the theorem gives . In so many different ways, therefore, may the 24 letters of the alphabet be varied or combined among themselves, or so many different words may be made out of them.</p></div1><div1 part="N" n="COMBUST" org="uniform" sample="complete" type="entry"><head>COMBUST</head><p>, or <hi rend="smallcaps">Combustion</hi>, is said of a planet <pb n="304"/><cb/> when it is in conjunction with the sun, or not distant from it above half their disc.</p><p>But according to Argol, a planet is Combust, or in Combustion, when it is within eight degrees and a half of the sun.</p></div1><div1 part="N" n="COMET" org="uniform" sample="complete" type="entry"><head>COMET</head><p>, a heavenly body in the planetary region, appearing suddenly, and again disappearing; and during the time of its appearance moving in a proper, though very eccentric orbit, like a planet.</p><p>Comets are vulgarly called <hi rend="italics">Blazing Stars,</hi> and have this to distinguish them from other stars, that they are usually attended with a long train of light, tending always opposite to the sun, and being of a fainter lustre the farther it is from the body of the comet. And hence arises a popular division of comets, into three kinds; viz <hi rend="italics">bearded, tailed,</hi> and <hi rend="italics">hairy</hi> comets; though in reality, this division rather relates to the several circumstances of the same comet, than to the phenomena of several. Thus, when the comet is eastward of the sun, and moves from him, it is said to be <hi rend="italics">bearded,</hi> because the light precedes it in the manner of a beard: When the comet is westward of the sun, and sets after him, it is said to be <hi rend="italics">tailed,</hi> because the train of light follows it in the manner of a tail: And lastly, when the sun and comet are diametrically opposite, the earth being between them, the train is hid behind the body of the comet, excepting the extremities, which, being broader than the body of the comet, appear as it were around it, like a border of hair, or <hi rend="italics">coma,</hi> from which it is called <hi rend="italics">hairy,</hi> and a comet.</p><p>But there have been comets whose disc was as clear, round, and well desined, as that of Jupiter, without either tail, beard or coma.</p><p><hi rend="italics">Of the Nature of Comets.</hi>—Philosophers and Astronomers, of all ages, have been much divided in their opinions as to the nature of comets. Their strange appearance has in all ages been matter of terror to the vulgar, who have uniformly considered them as evil omens, and forerunners of war, pestilence, &c. Diodorus Siculus and Appollinus Myndius, in Seneca, inform us, that many of the Chaldeans held them to be lasting bodies, having stated revolutions as well as the planets, but in orbits vastly more extensive; on which account they are only visible while near the earth, but disappear again when they go into the higher regions. Others of them were of opinion, that the comets were only meteors raised very high in the air, which blaze for a while, and disappear when the matter of which they consist is consumed or dispersed.</p><p>Some of the Greeks, before Aristotle, supposed that a comet was a vast heap or assemblage of very small stars meeting together, by reason of the inequality of their motions, and so uniting into a visible mass, by the union of all their small lights; which must again disappear, as those stars separated, and each proceeded in its course. Pythagoras, however, accounted them a kind of planets or wandering stars, disappearing in the superior parts of their orbits, and becoming visible only in the lower parts of them.</p><p>But Aristotle held, that comets were only a kind of transient fires, or meteors, consisting of exhalations raised to the upper region of the air, and there set on fire; far below the course of the moon. <cb/></p><p>Seneca, who lived in the first century, and who had seen two or three comets himself, plainly intimates that he thought them above the moon; and argues strongly against those who supposed them to be meteors, or who held other absurd opinions concerning them; declaring his belief that they were not fires suddenly kindled, but the eternal productions of nature. He points out also the only way to come at a certainty on this subject, viz, by collecting a number of observations concerning their appearance, in order to discover whether they return periodically or not. “For this purpose, says he, one age is not sufficient; but the time will come when the nature of comets and their magnitudes will be demonstrated, and the routes they take, so different from the planets, explained. Posterity will then wonder, that the preceding ages should be ignorant of matters so plain and easy to be known.”</p><p>For a long time this prediction of Seneca seemed not likely to be fulfilled; and Tycho Brahe was the first among the moderns, who restored the comets to their true rank in the creation; for after diligently observing the comet of 1577, and finding that it had no sensible diurnal parallax, he assigned it its true place in the planetary regions. See his book De Cometa, anni 1577.</p><p>Before this however, there were various opinions concerning them. In the dark and superstitious ages, comets were held to be forerunners of every kind of calamity, and it was supposed they had different degrees of malignity, according to the shape they assumed; from whence also they were differently denominated. Thus, it was said that some were bearded, some hairy; that some represented a beam, sword, or spear; others a target, &c; whereas modern astronomers acknowledge only one species of comets, and account for their different appearances from their different situations with respect to the sun and earth.</p><p>Kepler, in other respects a very great genius, indulged the most extravagant conjectures, not only concerning comets, but the whole system of nature in general. The planets he imagined were huge animals swimming round the sun; and the comets monstrous and uncommon animals generated in the celestial spaces.</p><p>A still more ridiculous opinion, if possible, was that of John Bodin, a learned Frenchman in the 16th century; who maintained that comets “are spirits, which having lived on the earth innumerable ages, and being at last arrived on the confines of death, celebrate their last triumph, or are recalled to the firmament like shining stars! This is followed by samine, plague, &c, because the cities and people destroy the governors and chiefs who appease the wrath of God.”—Others again have denied even the existence of comets, and maintained that they were only false appearances, occasioned by the refraction or reflection of light.</p><p>Hevelius, from a great number of observations, proposed it as his opinion, that the comets, like the solar maculæ or spots, are formed or condensed out of the grosser exhalations of his body; in which he differs but little from the opinion of Kepler.</p><p>James Bernoulli, in his Systema Cometarum, imagined that comets were no other than the satellites of <pb n="305"/><cb/> some very distant planet, which was itself invisible to us on account of its distance, as were also the satellites unless when in a certain part of their orbits.</p><p>Des Cartes advances another opinion: He conjectures that comets are only stars, formerly fixed, like the rest, in the heavens; but which becoming gradually covered with maculæ or spots, and at length wholly deprived of their light, cannot keep their places, but are carried off by the vortices of the circumjacent stars; and in proportion to their magnitude and solidity, moved in such a manner, as to be brought nearer the orb of Saturn; and thus coming within reach of the sun's light, rendered visible.</p><p>But the vanity of all these hypotheses now abundantly appears from the observed phenomena of comets, and from the doctrine of Newton, which is as follows:</p><p>The comets, he says, are compact, solid, fixed, and durable bodies; in fact a kind of planets, which move in very oblique and eccentric orbits, every way with the greatest freedom; persevering in their motions, even against the course and direction of the planets: and their tail is a very thin and slender vapour, emitted by the head or nucleus of the comet, ignited or heated by the sun. This theory of the comets at once solves their principal phenomena, which are as below. <hi rend="center"><hi rend="italics">The Principal Phenomena of the Comets.</hi></hi></p><p>1. First then, those comets which move according to the order of the signs, do all, a little before they disappear, either advance slower than usual, or else go retrograde, if the earth be between them and the sun; but more swiftly, if the earth be placed in a contrary part. On the other hand, those which proceed contrary to the order of the signs, move more swiftly than usual, if the earth be between them and the sun; and more slowly, or else retrograde, when the earth is in a contrary part.—For since this course is not among the fixed stars, but among the planets; as the motion of the earth either conspires with them, or goes against them; their appearance, with respect to the earth, mnst be changed; and, like the planets, they must sometimes appear to move swifter, sometimes slower, and sometimes retrograde.</p><p>2. So long as their velocity is increased, they nearly move in great circles; but towards the end of their course, they deviate from those circles; and when the earth proceeds one way, they go the contrary way. Because, in the end of their course, when they recede almost directly from the sun, that part of the apparent motion which arises from the parallax, must bear a greater proportion to the whole apparent motion.</p><p>3. The comets move in ellipses, having one of their foci in the centre of the sun; and by radii drawn to the sun, describe areas proportional to the times. Because they do not wander precariously from one fictitious vortex to another; but, making a part of the solar system, return perpetually, and run a constant round. Hence, their elliptic orbits being very long and eccentric, they become invisible when in that part which is most remote from the sun. And from the curvity of the paths of comets, Newton concludes, that when they disappear, they are much beyond the orbit of Jupiter; and that in their perihelion they frequently descend within the orbits of Mars and the inferior planets. <cb/></p><p>4. The light of their nuclei, or bodies, increases as they recede from the earth toward the sun; and on the contrary, it decreases as they recede from the sun. Because, as they are in the regions of the planets, their access towards the sun bears a oonsiderable proportion to their whole distance.</p><p>5. Their tails appear the largest and brightest, immediately after their transit through the region of the sun, or after their perihelion. Because then, their heads being the most heated, will emit the most vapours.—From the light of the nucleus we infer their vicinity to the earth, and that they are by no means in the region of the sixed stars, as some have imagined; since, in that case, their heads would be no more illuminated by the sun, than the planets are by the fixed stars.</p><p>6. The tails always <*>cline from a just opposition to the sun towards those parts which the nuclei or bodies pass over, in their progress through their orbits. Because all smoke, or vapour, emitted srom a body in motion, tends upwards obliquely, still receding from that part towards which the smoking body proceeds.</p><p>7. This declination, cæteris paribus, is the smallest when the nuclei approach nearest the sun; and it is also less near the nucleus, or head, than towards the extremity of the tail. Because the vapour ascends more swiftly near the head of the comet, than in the higher extremity of its tail; and also when the comet is nearer the sun, than when it is farther off.</p><p>8. The tails are somewhat brighter, and more distinctly defined in their convex, than in their concave part. Because the vapour in the convex part, which goes first, being somewhat nearer and denser, reflects the light more copiously.</p><p>9. The tails always appear broader at their upper extremity, than near the centre of the comet. Because the vapour in a free space continually rarefies and dilates.</p><p>10. The tails are always transparent, and the smallest stars appear through them. Because they consist of infinitely thin vapour.</p><p><hi rend="italics">The Phases of Comets.</hi>—The nuclei, which are also called the heads, and bodies, of comets, viewed through a telescope, shew a face very different from those of the fixed stars or planets. They are liable to apparent changes, which Newton ascribes to changes in the atmosphere of comets: and this opinion was consirmed by observations of the comet in 1744. Hist. Acad. Scienc. 1744. Sturmius says that, observing the comet of 1680 with a telescope, it appeared like a coal dimly glowing, or a rude mass of matter illuminated with a dusky fumid light, less sensible to the extremes than in the middle; whereas a star appears with a round disc, and a vivid light.</p><p>Of the comet of 1661, Hevelius observes, that its body was of a yellowish colour, very bright and conspicuous, but without any glittering light: in the middle was a dense ruddy nucleus, almost equal to Jupiter, encompassed by a much fainter, thinner matter. February 5th, its head was somewhat larger and brighter, and of a gold colour; but its light more dusky than the stars: and here the nucleus appeared divided into several parts. Feb. 6th, the disc was lessened; the parts of the nucleus still existed, though <pb n="306"/><cb/> less than before: one of them, on the lower part of the disc, on the left, much denser and brighter than the rest; its body round, and representing a very lucid little star: the nuclei still encompassed with another kind of matter. Feb. 10th, the head somewhat more obscure, and the nuclei more confused, but brighter at top than bottom. Feb. 13th, the head diminished much both in size and splendor. March 2d, its roundness a little impaired, and its edges lacerated, &c. March 28th, very pale, and exceeding thin; its matter much dispersed; and no distinct nucleus at all appearing.</p><p>Weigelius, who saw the comet of 1664, as also the moon, and a small cloud in the horizon illuminated by the sun at the same time, observed, that through the telescope the moon appeared of a continued luminous surface: but the comet very different; being exactly like the little cloud. And from these observations it was that Hevelius formed his opinion, that comets are like maculæ or spots formed out of the solar exhalations.</p><p><hi rend="italics">Of the Magnitude of Comets.</hi>—The estimates that have been given of the magnitude of comets by Tycho Brahe, Hevelius, and some others, are not very accurate; as it does not appear that they distinguished between the nucleus and the surrounding atmosphere. Thus Tycho computes that the true diameter of the comet in 1577 was in proportion to the diameter of the earth, as 3 is to 14; and Hevelius made the diameter of the comet of 1652 to that of the earth, as 52 to 100. But the diameter of the atmosphere is often 10 or 15 times as great as that of the nucleus: the former, in the comet of 1682, was measured by Flamsteed, and found to be 2′, when the diameter of the nucleus alone was only 11 or 12″. Though some comets, estimated by a comparison of their distance and apparent magnitude, have been judged much larger than the moon, and even equal to some of the primary planets. The diameter of that of 1744, when at the distance of the sun from us, measured about 1′, which makes its diameter about three times that of the earth: at another time the diameter of its nucleus was nearly equal to that of the planet Jupiter.</p><p><hi rend="italics">Of the Tails of Comets.</hi>—There have been various conjectures about the nature of the tails of comets, the principal of which are those of Newton, and the others that follow. Newton shews that the atmospheres of comets will furnish vapour sufficient to form their tails. This he argues from that wonderful rarefaction in our air at a distance from the earth; which is such, that a cubic inch of common air, expanded to the rarity of that at the distance of half the earth's diameter, or 4000 miles, would fill a space larger than the whole region of the stars. Since then the coma, or atmosphere of a comet, is 10 times higher than the surface of the nucleus, from the centre; the tail, ascending still much higher, must necessarily be immensely rare: so that it is no wonder the stars are visible through it.</p><p>Now the ascent of vapours into the tail of the comet, he supposes occasioned by the rarefaction of the matter of the atmosphere at the time of the perihelion. Smoke, it is observed, ascends the chimney by the impulse of the air in which it floats; and air, rarefied by <cb/> heat, ascends by the diminution of its specific gravity, carrying up the smoke along with it: in the same manner then it may be supposed that the tail of a comet is raised by the sun.</p><p>The tails therefore thus produced in the perihelions of comets, will go off along with their head into remote regions; and either return from thence, together with the comets, after a long series of years; or rather be there lost, and vanish by little and little, and the comet be left bare; till at its return, descending towards the sun, some short tails are again gradually produced from the head; which afterwards, in the perihelion, descending down into the sun's atmosphere, will be immensely increased.</p><p>Newton farther observes, that the vapours, when thus dilated, rarefied, and dissused through all the celestial regions, may probably, by means of their own gravity, be gradually attracted down to the planets, and become intermingled with their atmospheres. He adds that this intermixture may be useful and necessary for the conservation of the water and moisture of the planets, dried up or consumed in various ways. And I suspect, adds our author, that the spirit, which makes the finest, subtilest, and best part of our air, and which is absolutely requisite for the life and being of all things, comes principally from the comets.—On this principle there may seem to be some foundation for the popular opinion of presages from comets; since the tail of a comet thus intermingled with our atmosphere, may produce changes very sensible in animal and vegetable bodies.</p><p>It may here be added that another use which Newton conjectures comets may be designed to serve, is that of recruiting the sun with fresh fuel, and repairing the consumption of his light by the streams continually sent forth in every direction from that luminary. In support of this conjecture he observes, that comets in their perihelion may suffer a diminution of their projectile force, by the resistance of the solar atmosphere; so that by degrees their gravitation towards the sun may be so far increased, as to precipitate their fall into his body.</p><p>Other opinions on the tails of comets, are the following.</p><p>Apian, Tycho Brahe, and some others, think they were produced by the sun's rays transmitted through the nucleus of the comet, which they supposed was transparent, and there refracted as in a glass lens, so as to form a beam of light behind the comet. Des Cartes accounted for the phenomenon of the tail by the refraction of light from the head of the comet to the spectator's eye. Mairan supposes that the tails are formed out of the luminous matter composing the sun's atmosphere: and M. De la Lande combines this hypothesis with that of Newton recited above. But Mr. Rowning, not satisfied with Newton's opinion, accounts for the tails of comets in the following manner: It is well known, says he, that when the sun's light passes through the atmosphere of any body, as the earth, that which passes on one side, is by the refraction made to converge towards that which passes on the opposite side; and this convergency is not wholly effected either at the entrance of the light into the atmosphere, or at its exit on going out; but beginning at <pb n="307"/><cb/> its entrance, it increases in every point of its progress: It is also agreed that the atmospheres of the comets are very large and dense: he therefore supposes that by such time as the light of the sun has passed through a considerable part of the atmosphere of the comet, the rays are so far refracted towards each other, that they then begin sensibly to illuminate it, or rather the vapours floating in it; and so render that part they have yet to pass through, visible to us: and that this portion of the atmosphere of a comet thus illuminated, appears to us in form of a beam of the sun's light, and passes under the denomination of a comet's tail. Rowning's Nat. Philos. part 4. chap. 11.</p><p>M. Euler, Mem. Berlin tom. 2. pa. 117, thinks there is a great affinity between the tails of comets, the zodiacal light, and the aurora borealis, and that the common cause of all of them, is the action of the sun's light on the atmospheres of the comets, of the sun, and of the earth. He supposes that the impulse of the rays of light on the atmosphere of comets, may drive some of the finer particles of that atmosphere far beyond its limits; and that this force of impulse combined with that of gravity towards the comet, would produce a tail, which would always be in opposition to the sun, if the comet did not move. But the motion of the comet in its orbit, and about an axis, must vary the position and figure of the tail, giving it a curvature, and deviation from a line joining the centres of the sun and comet; and that this deviation will be greater, as the orbit of the comet has the greater curvature, and as the motion of the comet is more rapid. It may even happen, that the velocity of the comet, in its perihelion, may be so great, that the force of the sun's rays may produce a new tail, before the old one can follow; in which case the comet might have two or more tails. The possibility of this is confirmed by the comet of 1744, which was observed to have several tails while it was in its perihelion.</p><p>Dr. Hamilton urges several objections against the Newtonian hypothesis; and concludes that the tail of a comet is formed of matter which has not the power of refracting or reflecting the rays of light; but that it is a lucid or self-shining substance: and from its similarity to the Aurora borealis, that it is produced by the same cause, and is properly an electrical phenomenon. Dr. Halley too seemed inclined to this hypothesis, when he said, that the streams of light in an Aurora borealis so much resembled the long tails of comets, that at sirst sight they might well be taken for such: and that this light seems to have a greater affinity to that which the effluvia of electric bodies emit in the dark. Philos. Trans. N° 347. Hamilton's Philos. Essays, pa. 91.</p><p><hi rend="italics">The Motion of Comets.</hi>—If it be supposed that the paths of comets are perfectly parabolical, as some have imagined, it will follow that, being impelled towards the sun by a ceatripetal force, they descend as from spaces infinitely distant; and that by their falls they acquire such a velocity as will carry them off again into the remotest regions, never more to return. But the frequency of their appearance, and their degree of velocity, which does not exceed what they might acquire by their gravity towards the sun, seem to put it past doubt that they move like the planets, in elliptic orbits, <cb/> though exceedingly eccentric; and so return again after very long periods.</p><p>The apparent velocity of the comet of 1472, as observed by Regiomontanus, was such as to carry it through 40° of a great circle in 24 hours: and it was observed that the comet of 1770 moved through more than 45° in the last 25 hours.</p><p>About the return of comets there have been different opinions. Newton, Flamsteed, Halley, and other English astronomers, seem satisfied of the return of comets: Cassini and some of the French think it highly probable; but De la Hire and others oppose it. Those on the affirmative side suppose that the comets describe orbits prodigiously eccentric, insomuch that we can see them only in a very small part of their revolution: out of this, they are lost in the immensity of space; hid not only from our eyes, but our telescopes: that little part of their orbit next us passing sometimes within those of all the inferior planets.</p><p>M. Cassini gives the following reasons in favour of the return of comets. 1. It is found that they move a considerable time in the arch of a great circle, when referred to the fixed stars, that is a circle whose plane passes through the centre of the earth; deviating but a little from it chiefly towards the end of their appearance; a deviation however common to them with the planets.—2. Comets, as well as planets, appear to move so much the faster as they are nearer the earth; and when they are at equal distances from their perigee, their velocities are nearly the same. By subtracting from their motion the apparent inequality of velocity occasioned by their different distance from the earth, their equal motion might be found: but we should not still be certain that this is their true motion; because they might have considerable inequalities, not distinguishable in that small part of their orbit visible to us. It is rather probable that their real motion, as well as that of the planets, is unequal in itself; and hence we have a reason why the observations made during the appearance of a comet, cannot give the just period of their revolution.—3. There are no two different planets whose orbits cut the ecliptic in the same angle, whose nodes are in the same points of the ecliptic, and having the same apparent velocity in their perigee: consequently, two comets seen at different times, yet agreeing in all those three circumstances, can only be one and the same comet. Not that this exact agreement, in these circumstances, is absolutely necessary to determine their identity: for the moon herself is irregular in all of them, so that it seems there may be cases in which the same comet, at different periods of revolution, may disagree in these points.</p><p>As to the objections against the return of comets, the principal is that of the rarity of their appearance, with regard to the number of revolutions assigned to them. In 1702 there was a comet, or rather the tail of one, seen at Rome, which M. Cassini takes to be the same with that observed by Aristotle, and again lately in the year 1668; which would imply a period of 34 years: Now, it may seem strange that a star which has so short a revolution, and of consequence such frequent returns, should be so seldom seen. Again, in April of the same year 1702, a comet was observed by Messrs. Bianchini and Maraldi, which the latter sup- <pb n="308"/><cb/> posed was the same with that of 1664, both on account of its motion, velocity, and direction. M. de la Hire thought it had some relation to another he had observed in 1698, which Cassini refers to that of 1652; which would make it a period of 43 months, and the number of revolutions between 1652 and 1692, 14: now, it is hard to suppose, that in this age, when the heavens are so narrowly watched, a star should make 14 revolutions unperceived; especially such a star as this, which might appear above a month together; and consequently be often disengaged from the crepuscula. For this reason M. Cassini is very reserved in maintaining the hypothesis of the return of comets, and only proposes those for planets where the motions are easy and simple, and are solved without straining, or allowing any irregularities.</p><p>M. de la Hire proposes one general difficulty against the whole system of the return of comets, which would seem to prevent any comet from returning as a planet: which is this; that by the disposition necessarily given to their courses, they ought to appear as small at first as at last; and always increase till they arrive at their nearest proximity to the earth; or if they should chance not to be observed, as soon as they are capable of being seen, it is yet hardly possible but they must often shew themselves before they have arrived at their full magnitude and brightness: but, adds he, none were ever yet observed till they had arrived at it. However, the appearance of a comet in the month of October 1723, while at a great distance, so as to be too small and dim to be viewed without a telescope, as well as the observations of several others since, may serve to remove this obstacle, and set the comets still on the same footing with the planets.</p><p>It is a conjecture of Newton, that as those planets which are nearest to the sun, and revolve in the least orbits, are the smallest; so among the comets, such as in their perihelion come nearest the sun, are the smallest, and revolve in the least orbits. <hi rend="center"><hi rend="italics">Of the Writings and Lists of Comets.</hi></hi></p><p>There have been many writings upon the subject of comets, beside the notices of historians as to the appearance of certain particular ones.</p><p>Regiomontanus first shewed how to find the magnitudes of comets, their distance from the earth, and their true place in the heavens. His 16 problems De Co- <cb/> metæ Magnitudine, Longitudine, ac Loco, are to be found in a book published in the year 1544, with the title of Scripta Joannis Regiomontani.</p><p>Peter Apian observed and wrote upon the comets of 1631, 1632, &c. Other writers are Tycho Brahe, in his Progymnasmata Astronomiæ Instauratæ.—Kepler, of the comet in the year 1607, and de Cometis Libelli tres.—Ricciolus, in his Almagestum Novum, published 1651, enumerates 154 comets cited by historians down to the year 1618.—Hevelius's Prodromus Cometicus, containing the history of the comet of the year 1664. Also his Cometographia.—Lubienietz, in a large folio work expressly on this subject, published 1667, extracts, with immense labour, from the passages of all historians, an account of 415 comets, ending with that of 1665.— Dr. Hook, in his Posthumous works.—M. Cassini's little Tract of Comets.—Sturmius's Dissertatio de Cometarum Natura.—Newton, in his Principia, lib. 3; who first assigned their proper orbits, and by calculations compared the observations of the great comet of 1680 with his theory.—Dr. Halley, his Synopsis Cometica, in the Philos. Trans. number 218, &c; who computed the elements and orbits of 24 comets, and who first ventured to predict the return of one in 1759, which happened accordingly.—De la Lande, Théorie des Comètes, 1759; also, in his Astronomie, vol. 3.—Clairaut, Théorie du mouvement des Comètes, 1760—D'Alembert, Opuscules Mathématiques, vol. 2 pa. 97.—M. Albert Euler, 1762.—Séjour, Essai sur les Comètes, 1775.— Besides Boscovich, De la Grange, De la Place, Frisi, Lexel, Barker, Hancocks, Cole, with many others.— And M. Pingré's Cométographie, in 2 vols. 4to, 1784; in which is contained the most ample list of such comets as have been well observed, and their elements computed, to the number of 67. And accounts of a very few more that have been observed since that time, may be seen in the Mem. de l'Acad. and in the Philos. Trans.—And while this work is printing, there has just come out a very ingenious and ample work upon comets, by Sir Henry Englefield, entitled, “On the Determination of the Orbits of Comets.”</p><p>The whole list of comets that have been noticed, on record, amount to upwards of 500, but the following is a complete list of all that have been properly observed, and their elements computed, the mean distance of the earth from the sun being 100000. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=15" role="data">TABLE OF THE ELEMENTS OF COMETS.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Year</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Ascending Node</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Inclin. of Orbit</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Perihel. Dist.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Time of Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Motion</cell><cell cols="1" rows="1" rend="align=center" role="data">Calculated by</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">d</cell><cell cols="1" rows="1" rend="align=right" role="data">h</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">837</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">abt.</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">58000</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1231</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">94776</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1264</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">44500</cell><cell cols="1" rows="1" role="data">July</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Duntherne</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">41081</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1299</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">31793</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1301</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">about</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">abt.</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">about</cell><cell cols="1" rows="1" role="data">45700</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">about</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1337</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">40666</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">64450</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1456</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">58550</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1472</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">54273</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row></table><pb n="309"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Year</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Ascending Node</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Inclin. of Orbit</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Perihel. Dist.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Time of Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Motion</cell><cell cols="1" rows="1" rend="align=center" role="data">Calculated by</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">d</cell><cell cols="1" rows="1" rend="align=right" role="data">h</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1531</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">56700</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1532</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">50910</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1533</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">20280</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Douwes</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1556</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">66390</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1577</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">74</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">18342</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1580</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">59628</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">59553</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1582</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">22570</cell><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">4006</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3 align=center" role="data"><hi rend="smallcaps">New Style</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1585</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" role="data">109358</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1590</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">57661</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1593</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">87</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">8911</cell><cell cols="1" rows="1" role="data">July</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1596</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">51293</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">54942</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1607</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">58680</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1618</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">51298</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1618</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">37975</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1652</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">84750</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1661</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">44851</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1664</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">102575</cell><cell cols="1" rows="1" role="data">Dec.</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1665</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">76</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">10649</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1672</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">83</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">69739</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1677</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" role="data">28059</cell><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1678</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">123801</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Douwes</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1680</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">612 1/2</cell><cell cols="1" rows="1" role="data">Dec.</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">617</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">656</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Euler</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">592</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Newton</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">603</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1682</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">58328</cell><cell cols="1" rows="1" role="data">Sept.</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">58250</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1683</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">83</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">56020</cell><cell cols="1" rows="1" role="data">July</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1684</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">96015</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1686</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">32500</cell><cell cols="1" rows="1" role="data">Sept.</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1689</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">69</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">1689</cell><cell cols="1" rows="1" role="data">Dec.</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1698</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" role="data">69129</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Halley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1699</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">69</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">74400</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1702</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">64590</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1706</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">42581</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">42686</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Struyck</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1707</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">88</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">86350</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Houtteryn</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">88</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">85974</cell><cell cols="1" rows="1" role="data">Dec.</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">88</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">85904</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Struyck</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1718</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">102655</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">102565</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Douwes</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1723</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">99865</cell><cell cols="1" rows="1" role="data">Sept.</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Bradley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1729</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">406980</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Douwes</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">76</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">426140</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">76</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">416927</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Maraldi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">394927</cell><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Kies</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" role="data">408165</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">De l'Isle</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1737</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">22282</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Bradley</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1739</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">67160</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Zanotti</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">67358</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1742</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">67</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">76555</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Struyck</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">76550</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Le Monnier</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">76568</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">76530</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Zanotti</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">73766</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Euler</cell></row></table><pb n="310"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Year</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Ascending Node</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Inclin. of Orbit</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Perihel. Dist.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Time of Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Motion</cell><cell cols="1" rows="1" rend="align=center" role="data">Calculated by</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">d</cell><cell cols="1" rows="1" rend="align=right" role="data">h</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1742</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" role="data">73668</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Euler</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">76890</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">. .</cell><cell cols="1" rows="1" role="data">Wrigt</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">67</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">76620</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Klinkenberg</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" role="data">76545</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Houtteryn</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1743</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">83811</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Struyck</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">83501</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1743</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">52057</cell><cell cols="1" rows="1" role="data">Sept.</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Klinkenberg</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1744</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">22206</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Bets and Bliss</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">22322</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Maraldi</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1744</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">22250</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">22156</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Zanotti</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">22192</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Chéseaux</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">22222</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Euler</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">22223</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">22200</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Klinkenberg</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">22176</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Hiorter</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1747</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">229388</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Cheseaux</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">219859</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Maraldi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">219851</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1748</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">84066</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Maraldi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">84150</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Le Monnier</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">84040</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1748</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">65525</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Struyck</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1757</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">33754</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Bradley</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">33907</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" role="data">33797</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">33932</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">De Ratte</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1758</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" role="data">21535</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1759</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">58255</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Messier</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">58490</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">De la Lande</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">58360</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Maraldi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">58380</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">13 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">58350</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">13 2/3</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">58298</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Klinkenberg</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">59708</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Klinkenberg</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">58234</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Bailly</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1759</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">80139</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">78</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">79851</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" role="data">80208</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Chappe</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1759</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">96599</cell><cell cols="1" rows="1" role="data">Dec.</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">La Caille</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1759</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">96180</cell><cell cols="1" rows="1" role="data">Dec.</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Chappe</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1762</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">101415</cell><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Maraldi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">84</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">101249</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">De la Lande</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">101065</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Bailly</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">100686</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Klinkenberg</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">100986</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Struyck</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1763</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">72</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">49876</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1764</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">55522</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1766</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">50532</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1766</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">33275</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1769</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">12376</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">De la Lande</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">12287</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Wallot</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">12258</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Cassin, jun.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">12272</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Prosperin</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">12289</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Audiffrédi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">12100</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Slop</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">15880</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Zanotti</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">12308</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Asclépi</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">11640</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Lambert</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">12280</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Widder</cell></row></table><pb n="311"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Year</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Ascending Node</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Inclin. of Orbit</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Perihel. Dist.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Time of Perihelion</cell><cell cols="1" rows="1" rend="align=center" role="data">Motion</cell><cell cols="1" rows="1" rend="align=center" role="data">Calculated by</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">s</cell><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">d</cell><cell cols="1" rows="1" rend="align=right" role="data">h</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1769</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">12264</cell><cell cols="1" rows="1" role="data">Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Euler</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">12269</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Lexell</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">12274</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1770</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">62959</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">65800</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">62955</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Prosperin</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" role="data">64456</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Prosperin</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">71717</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Prosperin</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">64946</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Widder</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1770</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">67438</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Lexell</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">67689</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">62872</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Slop</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" role="data">63100</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Lambert</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">62758</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Rittenhouse</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1770</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">52824</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1771</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">90576</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">90188</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Prosperin</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1772</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">101814</cell><cell cols="1" rows="1" role="data">Feb.</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">La Lande</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1773</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">113390</cell><cell cols="1" rows="1" role="data">Sept.</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">62</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">123800</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Lambert</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">62</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">121550</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Schultz</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">113010</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Lexell</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">112530</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Pingré</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1774</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">82</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">142525</cell><cell cols="1" rows="1" role="data">Aug.</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">De Saron</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">82</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">142525</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">De Saron</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">82</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">142600</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Boscowich</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">83</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">142860</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Mechain</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1779</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">71322</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">De Saron</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">71313</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Mechain</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">71319</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">D'Angos</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1780</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">9781</cell><cell cols="1" rows="1" role="data">Sept.</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Lexell</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">10047</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Lexell</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">9926</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">Mechain</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1781</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">1027558</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Boscowich</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">944040</cell><cell cols="1" rows="1" role="data">Jan.</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">. . .</cell><cell cols="1" rows="1" role="data">La Place</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1781</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">81</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">77586</cell><cell cols="1" rows="1" role="data">July</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">Dir.</cell><cell cols="1" rows="1" role="data">Mechain</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1781</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">96101</cell><cell cols="1" rows="1" role="data">Nov.</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">Ret.</cell><cell cols="1" rows="1" role="data">Mechain</cell></row></table><cb/></p><p>M. Facio has suggested, that some of the comets have their nodes so very near the annual orbit of the earth, that if the earth should happen to be found in that part next the node at the time of a comet's passing by; as the apparent motion of the comet will be immensely swift, so its parallax will become very sensible; and its proportion to that of the sun will be given: whence, such transits of comets will afford the best means of determining the distance between the earth and sun.</p><p>The comet of 1472, for instance, had a parallax above 20 times greater than the sun's: and if that of 1618 had come down in the beginning of March to its descending node, it would have been much nearer the earth, and its parallax much more notable. But hitherto none has threatened the earth with a nearer appulse than that of 1680: for, Dr. Halley finds, by calculation, that Nov. 11th, at 1 h. 6 min. afternoon, that comet was not more than one semidiameter of the earth to the northward of the earth's path; at which time had the earth been in that part of its orbit, the comet would have had a parallax equal to that of the moon.— <cb/> What might have been the consequence of so near an appulse, a contact, or lastly, a shock of these bodies? Mr. Whiston says, a deluge!</p><p><hi rend="italics">To determine the Place and Course of a Comet.</hi>—Observe the distance of the comet from two fixed stars, whose longitudes and latitudes are known: then from the distances thus known, calculate the place of the comet by spherical trigonometry.</p><p>Longomontanus shews an easy method of finding and tracing out the places of a comet mechanically, which is, to find two stars in the same line with the comet, by stretching a thread before the eye over all the three; then do the same by two other stars and the comet: this done, take a celestial globe, or a planisphere, and draw a line upon it first through the former two stars, and then through the latter two; so shall the intersection of the two lines be the place of the comet at that time. If this be repeated from time to time, and all the points of intersection connected, it will shew the path of the comet in the heavens.</p></div1><div1 part="N" n="COMETARIUM" org="uniform" sample="complete" type="entry"><head>COMETARIUM</head><p>, a machine adapted to give a representation of the revolution of a comet about the sun. <pb n="312"/><cb/> It is so contrived as, by elliptical wheels, to shew the unequal motion of a comet in every part of its orbit. The comet is represented by a small brass ball, carried by a radius vector, or wire, in an elliptical groove about the sun in one of its foci; and the years of its period are shewn by an index moving with an equable motion over a graduated silvered circle. See a particular description, with a figure of it, in Ferguson's Astron. 8vo. pa. 400.</p></div1><div1 part="N" n="COMMANDINE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">COMMANDINE</surname> (<foreName full="yes"><hi rend="smallcaps">Frederick</hi></foreName>)</persName></head><p>, a celebrated mathematician and linguist, was born at Urbino in Italy, in 1509; and died in 1575; consequently at 66 years of age. He was famous for his learning and knowledge in the sciences. To a great depth, and just taste in the mathematics, he joined a critical skill in the Greek language; a happy conjunction which made him very well qualified for translating and expounding the writings of the Greek mathematicians. And accordingly, with a most laudable zeal and industry, he translated and published several of their works, to which no former writer had done that good office. On which account, Francis Moria, duke of Urbino, who was very conversant in those sciences, proved a very affectionate patron to him. He is greatly applauded by Bianchanus, and other writers; and he justly deserved their encomiums.</p><p>Of his own works Commandine published the following:</p><p>1. Commentarius in Planisphærium Ptolomæi: 1558, in 4to.</p><p>2. De Centro Gravitatis Solidorum: Bonon. 1565, in 4to.</p><p>3. Horologiorum Descriptio: Rom. 1562, in 4to.</p><p>He translated and illustrated with notes the following works, most of them beautifully printed, in 4to. by the celebrated printer Aldus:</p><p>1. Archimedis Circuli Dimensio; de Lineis Spiralibus; Quadratura Parabolæ; de Conoidibus & Sphæroidibus; de Arenæ Numero: 1558.</p><p>2. Ptolomæi Planisphærium; & Planisphærium Jordani: 1558.</p><p>3. Ptolomæi Analemma: 1562.</p><p>4. Archimedis de iis quæ vehuntur in aqua: 1565.</p><p>5. Apollonii Pergæi Conicorum libri quatuor, una cum Pappi Alexandrini Lemmatibus, & Commentariis Eutocii Ascalonitæ, &c: 1566.</p><p>6. Machometes Bagdadinus de Superficierum Divisionibus: 1570.</p><p>7. Elementa Euclidis: 1572.</p><p>8. Aristarchus de Magnitudinibus & Distantiis Solis & Lunæ: 1572.</p><p>9. Heronis Alexandrini Spiritualium liber: 1583:</p><p>10. Pappi Alexandrini Collectiones Mathematicæ: 1588.</p><p>COMMANDING <hi rend="italics">Ground,</hi> in Fortification, an eminence, or rising ground, overlooking any post or strong place. This is of three sorts. 1st, <hi rend="italics">A Front Commanding Ground,</hi> or a height opposite to the face of the post, which plays upon its front. 2dly, <hi rend="italics">A Reverse Commanding Ground,</hi> or an eminence that can play upon the rear or back of the post. 3dly, <hi rend="italics">An Enfilade Commanding Ground,</hi> or an eminence in flank which can, with its shot, scour all the length of a straight line. <cb/></p><p>COMMENSURABLE <hi rend="italics">Quantities,</hi> or <hi rend="italics">Magnitudes,</hi> are such as have some common aliquot part, or which may be measured or divided, without a remainder, by one and the same measure or divisor, called their common measure. Thus, a foot and a yard are commensurable, because there is a third quantity that can measure each, viz an inch; which is 12 times contained in the foot, and 36 times in the yard.—Commensurables are to each other, as one rational whole number is to another; but incommensurables are not so: And therefore the ratio of commensurables is rational; but that of incommensurables, irrational: hence also the exponent of the ratio of commensurables, is a rational number.</p><p>COMMENSURABLE <hi rend="italics">Numbers,</hi> whether integers, or fractions, or surds, are such as have some other number, which will measure or divide them exactly, or without a remainder. Thus, 6 and 8 are-commensurable, because 2 measures or divides them both. And 2/3 and 3/4, or 8/12 and 9/12 are commensurable fractions, because the fraction 1/6, or 1/12, &c, will measure them both: and in this sense, all fractions may be said to be commensurable. Also, the surds 2√2 and 3√2 are commensurable, being measured by √2, or being to each other as 2 to 3.</p><p><hi rend="smallcaps">Commensurable</hi> <hi rend="italics">in Power.</hi> Euclid says, right lines are commensurable in power, when their squares are measured by one and the same space or superficies.</p></div1><div1 part="N" n="COMMON" org="uniform" sample="complete" type="entry"><head>COMMON</head><p>, is applied to an angle, line, measure, or the like, that belongs to two or more figures, or other things. As, a common angle, a common side, a common base, a common measure, &c.</p><p><hi rend="smallcaps">Common Measure</hi>, or divisor, is that which measures two or more things without a remainder. So of 8 and 12, a common measure is 2, and so is 4.</p><p><hi rend="italics">The greatest</hi> <hi rend="smallcaps">Common</hi> <hi rend="italics">Measure,</hi> is the greatest number that can measure two other numbers. So, of 8 and 12, the greatest common measure is 4.</p><p><hi rend="italics">To find the greatest common measure</hi> of two numbers. Divide the greater term by the less; then divide the divisor by the remainder, if there be any; and so on continually, always dividing the last divisor by the last remainder, till nothing remains; and then is the last divisor the greatest common measure sought.</p><p>Thus, to find the greatest common measure of 816 and 1488. <figure/> <pb/><pb/><pb n="313"/><cb/> Therefore 48 is the greatest common measure of 816 and 1488, thus: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48) 816</cell><cell cols="1" rows="1" role="data">(17</cell><cell cols="1" rows="1" rend="align=right" role="data">48) 1488</cell><cell cols="1" rows="1" role="data">(31</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">144 </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">336</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">336</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>The common measure is useful in fractions, to reduce a fraction to its least terms, by dividing those that are given by their greatest common measure. So 816/1488 reduces to 17/31, by dividing 816 and 1488 both by their greatest common measure 48.</p><p>COMMUNICATION <hi rend="italics">of Motion,</hi> that act of a moving body, by which it gives motion, or transfers its motion to another body.</p><p>Father Mallebranche considers the communication of motion, as something metaphysical; that is, as not necessarily arising from any physical principles, or any properties of bodies, but flowing from the immediate agency of God.</p><p>The communication of motion results from, and is an evidence of the impenetrability and inertia of matter, as such; unless we admit the hypothesis of the penetrability of matter, advanced by Boscovich and Michell, and ascribe to the power of repulsion those effects which have been usually ascribed to its solidity and actual resistance.</p><p>Newton shews that action and reaction are equal and opposite; so that one body striking or acting against another, and thence causing a change in its motion, does itself undergo the very same change in its own motion, the contrary way. And hence, a moving body striking directly another at rest, it loses just as much of its motion as it communicates to the other. For the laws and quantity of motion so communicated, either in elastic or nonelastic bodies, see <hi rend="smallcaps">Collision.</hi></p></div1><div1 part="N" n="COMMUTATION" org="uniform" sample="complete" type="entry"><head>COMMUTATION</head><p>, <hi rend="italics">Angle of,</hi> is the distance between the sun's true place seen from the earth, and the place of a planet reduced to the ecliptic: which therefore is found by taking the difference between the sun's longitude and the heliocentric longitude of the planet.</p></div1><div1 part="N" n="COMPANY" org="uniform" sample="complete" type="entry"><head>COMPANY</head><p>, <hi rend="italics">Rule of,</hi> or <hi rend="italics">Rule of Fellowship,</hi> in Arithmetic, is a rule by which are determined the true shares of profit or loss, due to the several partners, or associates, in any enterprize, or trade, in due proportion to the stock contributed by each, and the time it was employed. To do which properly, see the <hi rend="italics">Rule of Fellowship.</hi></p></div1><div1 part="N" n="COMPARTMENT" org="uniform" sample="complete" type="entry"><head>COMPARTMENT</head><p>, a design composed of several different figures, disposed with symmetry; to adorn a parterre, a cieling, pannel of joinery, or the like.</p></div1><div1 part="N" n="COMPARTITION" org="uniform" sample="complete" type="entry"><head>COMPARTITION</head><p>, the useful and graceful distribution of the whole ground-plot of an edifice, into rooms of office, and of reception, or entertainment.</p></div1><div1 part="N" n="COMPASS" org="uniform" sample="complete" type="entry"><head>COMPASS</head><p>, or <hi rend="italics">Mariner's Compass,</hi> is an instrument used at sea by mariners to direct and ascertain the course of their ships. It consists of a circular brass box, which contains a paper card with the 32 points of the compass, or winds, fixed on a magnetic needle that always turns to the north, excepting a small deviation, which is variable at different places, and at the same <cb/> place at different times. See <hi rend="smallcaps">Variation</hi> <hi rend="italics">of the Compass.</hi></p><p>The needle with the card turns on an upright pin fixed in the centre of the box. To the middle of the needle is fixed a brass conical socket or cap, by which the card hanging on the pin turns freely round the centre.</p><p>The top of the box is covered with a glass, to prevent the wind from disturbing the motion of the card. The whole is inclosed in another box of wood, where it is suspended by brass hoops or gimbals, to keep the card in a horizontal position during the motions of the ship. The whole is to be so placed in the ship, that the middle section of the box, parallel to its sides, may be parallel to the middle section of the ship along its keel.</p><p>The invention of the compass is usually ascribed to Flavio Gioia, or Flavio of Malphi, about the year 1302; and hence it is that the territory of Principato, the part of the kingdom of Naples where he was born, has a compass for its arms. He divided his compass only into 8 points. Others ascribe the invention to the Chinese; and Gilbert, in his book <hi rend="italics">de Magnete,</hi> affirms that Marcus Paulus, a Venetian, making a journey to China, brought back the invention with him in 1260. What strengthens this conjecture is, that at first they used the compass, in the same manner as the Chinese still do, viz, letting it float on a small piece of cork, instead of suspending it on a pivot. It is added, that their emperor Chiningus, a celebrated astrologer, had a knowledge of it 1120 years before Christ. The Chinese divide their compass only into 24 points. But Ludi Vertomanus affirms, that when he was in the East-Indies, about the year 1500, he saw a pilot of a ship direct his course by a compass, fastened and framed as those now commonly used. And Barlow, in his book called The Navigator's Supply, anno 1597, says, that in a personal conference with two East-Indians, they affirmed, that instead of our compass, they use a magnetical needle of 6 inches, and longer, upon a pin in a dish of white China earth, filled with water; in the bottom of which they have two cross lines for the 4 principal winds, the rest of the divisions being left to the skill of their pilots. Also in the same book he says that the Portuguese, in their first discovery of the EastIndies, got a pilot of Mahinde, who brought them from thence in 33 days, within sight of Calicut.</p><p>But Fauchette relates some verses of Guoyot de Provence, who lived in France about the year 1200, which seem to make mention of the compass under the name of <hi rend="italics">marinette,</hi> or <hi rend="italics">mariner's stone</hi>; which shew it was used in France near 100 years before either the Malfite or Venetian one. The French even lay claim to the invention, from the fleur de lys with which most people distinguish the north point of the card. With as much reason Dr. Wallis ascribes it to the English, from its name <hi rend="italics">compass,</hi> by which name most nations call it, and which he observes is used in many parts of England to signify a circle.</p><p>The mariner's compass was long very rude and imperfect, but at length received great improvement from the invention and experiments of Dr. Knight, who discovered the useful practice of making artificial magnets; and the farther emendations of Mr. Smeaton, and <pb n="314"/><cb/> Mr. M'Culloch, by which the needles are larger and stronger than formerly, and instead of swinging in gimbals, the compass is supported in its very centre upon a prop, and the centres of motion, gravity, and magnetism are brought almost all to the same point.</p><p>After the discovery of that most useful property of the magnet, or loadstone, viz, its giving a polarity to hardened iron or steel, the compass was many years in use before it was known in anywise to deviate from the poles of the world. About the middle of the 16th century, so confident were some persons that the needle invariably pointed due north, that they treated with contempt the notion of the variation, which about that time began to be suspected. However, careful observations soon discovered that in England, and its neighbourhood, the needle pointed to the eastward of the true north line; and the quantity of this deviation being known, mariners became as well satisfied as if the compass had none; because the true course could be obtained by making allowance for the true variation.</p><p>From succeeding observations it was afterwards found, that the deviation of the needle from the north was not a constant quantity, but that it gradually diminished, and at last, namely, about the year 1657, it was found that the needle pointed due north at London, and has ever since been going to the westward, till now the variation is upwards of two points of the compass: indeed it was 22° 41′ about the middle of the year 1781, as appears by the Philos. Trans. pa. 225, for that year, and is probably now somewhat more, which it would be of consequence to know; but why such useful observations and experiments, as those of the variation and dip of the magnetic needle, have been so long discontinued, to the prejudice of science, is best known to the learned President of that Society, and his Council. So that in any one place it may be suspected the variation has a kind of libratory motion, traversing through the north, to unknown limits eastward and westward. But the settling of this point must be left to time and suture experiments. See <hi rend="italics">Variation,</hi> also <hi rend="italics">Inclination,</hi> and <hi rend="italics">Dip.</hi> Also for a farther description of different compasses and their uses, see that useful book, Robertson's Navigation, vol. 2. p. 231.</p><p><hi rend="italics">The Azimuth</hi> <hi rend="smallcaps">Compass</hi> differs from the common sea compass in this; that the circumference of the card or box is divided into degrees; and there is fitted to the box an index with two sights, which are upright pieces of brass, placed diametrically opposite to each other, having a slit down the middle of them, through which the sun or star is to be viewed at the time of observation.</p><p><hi rend="italics">The Use of the Azimuth Compass,</hi> is to take the bearing of any celestial object, when it is in, or above the horizon, that from the magnetical azimuth or amplitude, the variation of the needle may be known. See <hi rend="italics">Azimuth,</hi> and <hi rend="italics">Amplitude.</hi></p><p>The figure of the compass card, with the names of the 32 points or winds, are as in fig. 5, plate vi; where other compasses are also exhibited.</p><p>As there are 32 whole points quite around the circle, which contains 360 degrees, therefore each point of the compass contains the 32d part of 360, that is 11 1/4 degrees, or 11° 15′; consequently the half point is 5° 37′ 30″, and the quarter point 2° 48′ 45″. <cb/></p><p>The points of the compass are otherwise called Rhumbs; and the numbers of degrees, minutes and seconds made by every quarter point with the meridian, are exhibited in the following table. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=11" role="data">A TABLE of Rhumbs, shewing the Degrees, Minutes, and Seconds, that every Point and Quarter-point of the Compass makes with the Meridian.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">North</cell><cell cols="1" rows="1" rend="align=center" role="data">Pts.</cell><cell cols="1" rows="1" rend="align=center" role="data">qr.</cell><cell cols="1" rows="1" rend="align=center" role="data">°</cell><cell cols="1" rows="1" rend="align=center" role="data">′</cell><cell cols="1" rows="1" rend="align=center" role="data">″</cell><cell cols="1" rows="1" rend="align=center" role="data">Pts.</cell><cell cols="1" rows="1" rend="align=center" role="data">qr.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">South</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=7 align=center" role="data">N b E</cell><cell cols="1" rows="1" rend="rowspan=7 align=center" role="data">N b W</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="rowspan=7 align=center" role="data">S b E</cell><cell cols="1" rows="1" rend="rowspan=7 align=center" role="data">S b W</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">N N E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">N NW</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">S S E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">S S W</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">NE b N</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">NW b N</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">SE b S</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">SW b S</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">N E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">N W</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">S E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">S W</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">NE b E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">NWbW</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">SE b E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">SW b W</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">N NE</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">WNW</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">67</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">E S E</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">W S W</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">73</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">75</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">E b N</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">W b N</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">78</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">E b S</cell><cell cols="1" rows="1" rend="rowspan=4 align=center valign=top" role="data">W b S</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">81</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">84</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">87</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">East</cell><cell cols="1" rows="1" rend="align=center" role="data">West</cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">East</cell><cell cols="1" rows="1" rend="align=center" role="data">West</cell></row></table></p><p><hi rend="smallcaps">Compass</hi> <hi rend="italics">Dials,</hi> are small dials, fitted in boxes, for the pocket, to shew the hour of the day by the direction of the needle that indicates how to place them right, by turning the dial about till the cock or style stand directly over the needle. But these can never be very exact, because of the variation of the needle itself; unless that variation be allowed for, in making and placing the instrument.</p></div1><div1 part="N" n="COMPASSES" org="uniform" sample="complete" type="entry"><head>COMPASSES</head><p>, <hi rend="italics">or Pair of</hi> <hi rend="smallcaps">Compasses</hi>, a mathematical instrument for describing circles, measuring and dividing lines, or figures, &c.</p><p>The common compasses consist of two sharp-pointed branches or legs of iron, steel, brass, or other metal, joined together at the top by a rivet, about which they move as on a centre. Those compasses are of the best fort in which the pin or axle, on which the joint turns, is made of steel, and also half the joint itself, as the opposite metals wear more equally: the points should also be made of hard steel, well polished; and the joint should open and shut with a smooth, easy, and uniform motion. In some compasses, the points are both fixed; but in others, one is made to take out occasionally, and a drawing-pen, or pencil, put in its place.</p><p>There are in use compasses of various kinds and contrivances, adapted to the various purposes they are intended for; as,</p><p><hi rend="smallcaps">Compasses</hi> <hi rend="italics">of three Legs,</hi> or Triangular Compasses; the construction of which is like that of the common compasses, with the addition of a third leg or point, <pb n="315"/><cb/> which has a motion every way. Their use is to take three points at once, and so to form triangles, and lay down three positions of a map to be copied at once.</p><p><hi rend="italics">Beam</hi> <hi rend="smallcaps">Compasses</hi> consist of a long straight beam or bar, carrying two brass cursors; one of these being fixed at one end, the other sliding along the beam, with a screw to fasten it on occasionally. To the cursors may be screwed points of any kind, whether steel, pencils, or the like. To the sixed cursor is sometimes applied an adjusting or micrometer screw, by which an extent is obtained to very great nicety. The beam compasses are used to draw large circles, to take great extents, or the like.</p><p><hi rend="italics">Bow</hi> <hi rend="smallcaps">Compasses</hi>, or Bows, are a small sort of compasses, that shut up in a hoop, which serves for a handle. Their use is to describe arcs or circumserences with a very small radius.</p><p><hi rend="italics">Caliber</hi> <hi rend="smallcaps">Compasses.</hi> See <hi rend="smallcaps">Caliber.</hi></p><p><hi rend="italics">Clockmakers</hi> <hi rend="smallcaps">Compasses</hi> are jointed like the common compasses, with a quadrant or bow, like the spring compasses; only of different use, serving here to keep the instrument firm at any opening. They are made very strong, with the points of their legs of well-tempered steel, as being used to draw or cut lines in pasteboard, or copper, &c.</p><p><hi rend="italics">Cylindrical and Spherical</hi> <hi rend="smallcaps">Compasses</hi>, consist of four branches, joined in a centre, two of them being circular and two flat, a little bent at the ends. The use of them is to take the diameter, thickness, or caliber of round or cylindrical bodies; as cannons, balls, pipes, &c.</p><p>There are also spherical compasses, differing in nothing from the common ones, but that their legs are arched; serving to take the diameters of round bodies.</p><p>There is also another sort of compasses lately invented, for measuring the diameter of round bodies, as balls, &c, which consist of two slat pieces of metal set at right angles on a straight bar or beam of the same; the one piece being fixed, and the other sliding along it, so far as just to receive the round body between them; and then its diameter, or distance between the two pieces, is shewn by the divisions marked on the beam.</p><p><hi rend="italics">Elliptical</hi> <hi rend="smallcaps">Compasses</hi>, are used to draw ellipses or ovals of any kind. The instrument consists of a beam AB (Plate vi. fig. 6.) about a foot long, bearing three cursors; to one of which may be screwed points of any kind; and to the bottom of the other two are rivetted two sliding dove-tails, adjusted in grooves made in the cross branches of the beam. The dove-tails having a motion every way, by turning about the long branch, they go backward and forward along the cross; so that when the beam has gone half way round, one of these will have moved the whole length of one of the branches; and when the beam has gone quite round, the same dove-tail has gone back the whole length of the branch. Understand the same of the other dovetail.</p><p>Note, the distance between the two sliding dovetails, is the distance between the two foci of the ellipse; so that by changing that distance, the ellipse will be rounder or flatter. Under the ends of the branches of the cross, are placed four steel points to keep it fast.</p><p>The use of this compass is easy: by turning round <cb/> the long branch, the pen, pencil, or other points will draw the ellipse required.</p><p>Its figure shews both its use and construction.</p><p><hi rend="italics">German</hi> <hi rend="smallcaps">Compasses</hi> have their legs a little bent outwards, near the top; so that when shut, the points only meet.</p><p><hi rend="italics">Hair</hi> <hi rend="smallcaps">Compasses</hi> are so contrived within side by a small adjusting screw to one of the legs, as to take an extent to a hair's breadth, or great exactness.</p><p><hi rend="italics">Proportional</hi> <hi rend="smallcaps">Compasses</hi> are those whose joint lies, not at the end of the legs, but between the points terminating each leg. These are either simple, or compound. In the former sort the centre, or place of the joint is fixed; so that one pair of these serves only for one proportion.</p><p><hi rend="italics">Compound Proportional</hi> <hi rend="smallcaps">Compasses</hi> have the joint or centre moveable. They consist of two parts or sides of brass, which lie upon each other so nicely as to seem but one when they are shut. These sides easily open, and move about the centre, which is itself moveable in a hollow canal cut through the greatest part of their length. To this centre on each side is sixed a sliding piece, of a small length, with a sine line drawn on it serving as an index, to be set against other lines or divisions placed upon the compasses on both sides. These lines are, 1, A line of lines; 2, a line of supersicies, areas, or planes, the numbers on which answer to the squares of those on the line of lines; 3, a line of solids, the numbers on which answer to the cubes of those on the line of lines; 4, a line of circles, or rather of polygons to be inscribed in circles. These lines are all unequally divided, the first three from 1 to 20, and the last from 6 to 20. The use of the first is to divide a line into any number of equal parts; by the 2d and 3d are found the sides of like planes or solids in any given proportion; and by the 4th, circles are divided into any number of equal parts, or any polygons inscribed in them. See Plate vi. fig. 7.</p><p><hi rend="italics">Spring</hi> <hi rend="smallcaps">Compasses</hi>, or <hi rend="italics">Dividers,</hi> are made of hardened steel, with an arched head, which by its spring opens the legs; the opening being directed by a circular screw fastened to one of the legs, let through the other, and worked with a nut.</p><p><hi rend="italics">Trisecting</hi> <hi rend="smallcaps">Compasses</hi>, for the trisecting of angles geometrically, for which purpose they were invented by M. Tarragon.</p><p>The instrument consists of two central rules, and an arch of a circle of 120 degrees, immoveable, with its radius: the radius is fastened with one of the central rules, like the two legs of a sector, that the central rule may be carried through all the points of the circumference of the arch. The radius and rule should be as thin as possible; and the rule fastened to the radius should be hammered cold, to be more elastic; and the breadth of the other central rule must be triple the breadth of the radius: in this rule also is a groove, with a dove-tail fastened on it, for its motion; there must also be a hole in the centre of each rule.</p><p><hi rend="italics">Turn-up</hi> <hi rend="smallcaps">Compasses</hi>, a late contrivance to save the trouble of changing the points: the body is like the common compasses; and towards the bottom of the legs without side, are added two other points, besides the usual ones; the one carrying a drawing pen point, and <pb n="316"/><cb/> the other a port-crayon; both being adjusted to turn up, to be used or not, as occasion may require.</p><p>COMPLEMENT in general, is what is wanting, or necessary, to complete some certain quantity or thing. As, the</p><p><hi rend="smallcaps">Complement</hi> <hi rend="italics">of an arch or angle,</hi> as of 90° or a quadrant, is what any given arch or angle wants of it; so the complement of 50° is 40°, and the complement of 100 degrees is —10°, a negative quantity.—The complement to 180° is usually called the supplement, to distinguish it from the complement to 90°, properly so called.—The sine of the complement of an arc, is contracted into the word cosine; the tangent of the complement, into cotangent; &c.</p><p><hi rend="italics">Arithmetical</hi> <hi rend="smallcaps">Complement</hi>, is what a number or logarithm wants of unity or 1 with some number of ciphers. It is best found, by beginning at the left-hand side, and subtracting every figure from 9, except the last, or right-hand figure, which must be subtracted from 10. So, the arithmetical comp. of the log. 9.5329714, by subtracting from 9's, &c, is 0.4670286.</p><p>The arithmetical complements are much used in operations by logarithms, to change subtractions into additions, which are more conveniently performed, especially when there are more than one of them in the operation.</p><div2 part="N" n="Complement" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Complement</hi></head><p>, in Astronomy, is used for the distance of a star from the zenith; or the arc contained between the zenith and the place of a star which is above the horizon. It is the same as the complement of the altitude, or co-altitude, or the zenith distance.</p><p><hi rend="smallcaps">Complement</hi> <hi rend="italics">of the Course,</hi> in Navigation, is the quantity which the course wants of 90°, or 8 points, viz, a quarter of the compass.</p><p><hi rend="smallcaps">Complement</hi> <hi rend="italics">of the Curtain,</hi> in Fortisication, is that part of the anterior side of the curtain, which makes the demigorge.</p><p><hi rend="smallcaps">Complement</hi> <hi rend="italics">of the Line of Defence,</hi> is the remainder of that line, after the angle of the flank is taken away.</p><p><hi rend="smallcaps">Complements</hi> <hi rend="italics">of a Parallelogram,</hi> or <hi rend="italics">in a Parallelogram,</hi> are the two lesser parallelograms, made by drawing two right lines parallel to each side of the given parallelo- <figure/> gram, through the same point in the diagonal. So P and Q are the complements in the parallelogram ABCD.</p><p>In every case, these complements are always equal, viz, the parallelogram P = Q.</p><p><hi rend="smallcaps">Complement</hi> <hi rend="italics">of Life,</hi> a term much used, in the doctrine of Life Annuities, by De Moivre, and, according to him, it denotes the number of years which a given life wants of 86, this being the age which he considered as the utmost probable extent of life. So 56 is the complement of 30, and 30 is the complement of 56.</p><p>That author supposed an equal annual decrement of lise through all its stages, till the age of 86. Thus, if there be 56 persons living at 30 years of age, it is supposed that one will die every year, till they be all dead in 56 years. This hypothesis in many cases is very near the truth; and it agrees so nearly with Halley's <cb/> table, formed from his observations of the mortuary bills of Breslaw, that the value of lives deduced either from the hypothesis, or the table, need not be distinguished; hence it very much eases the labour of calculating them. See <hi rend="italics">Life</hi> <hi rend="smallcaps">Annuities</hi>, also De Moivre's Treatise on Annuities, pa. 83, and Price on Reversionary Payments, pa. 2.</p><p>COMPOSITE <hi rend="italics">Number,</hi> is one that is compounded of, or made up by the multiplication of two other numbers, greater than 1, or which can be measured by some other number greater than 1. As 12, which is composed, or compounded of 2 and 6, or 3 and 4, viz by multiplying together 2 and 6, or 3 and 4, both products making the same number 12; which therefore is a composite number.</p><p>Composites are opposed to prime numbers, or primes, which cannot be exactly measured by any other number, and cannot be produced by multiplying together two other factors.</p><p><hi rend="italics">Composite Numbers between themselves,</hi> are the same with commensurable numbers, or such as have a common measure or factor; as 15 and 12, which have the common term 3.</p><p>The doctrine of Prime and Composite numbers is pretty fully treated in the 7th and 8th books of Euclid's Elements.</p><p><hi rend="smallcaps">Composite</hi> <hi rend="italics">Order,</hi> in Architecture, is the last of the five orders of columns; and is so called because its capital is composed out of those of the other orders. Thus, it borrows a quarter-round from the Tuscan and Doric; a double row of leaves from the Corinthian; and volutes from the Ionic. Its cornice has single modillions, or dentils; and its column is 10 diameters in height.</p><p>This order is also called the Roman order, and Italic order, as having been invented by the Romans, like as the other orders are denominated from the people among whom they had their rise.</p></div2></div1><div1 part="N" n="COMPOSITION" org="uniform" sample="complete" type="entry"><head>COMPOSITION</head><p>, is a species of reasoning by which we proceed from things that are known and given, step by step, till we arrive at such as were before unknown or required; viz, procceding upon principles self-evident, on definitions, postulates, and axioms, with a previously demonstrated series of propositions, step by step, till it gives a clear knowledge of the thing to be known or demonstrated. Composition, otherwise called the synthetical method, is opposed to Resolution, or the analytical method, and is chiefly used by the ancients, Euclid, Apollonius, &c. See Pappus; also the term <hi rend="italics">Analysis.</hi></p><p><hi rend="smallcaps">Composition</hi> <hi rend="italics">of forces,</hi> or <hi rend="italics">of motion,</hi> is the union or assemblage of several forces or motions that are oblique to one another, into an equivalent one in another direction.</p><p>1. When several forces or motions are united, that act in the same line of direction, the combined force or motion will be in the same line of direction still. But when oblique forces are united, the compounded force takes a new direction, different from both, and is either a right line or a curve, according to the nature of the forces compounded.</p><p>2. If two compounding motions be both equable ones, whether equal to each other or not, the line of the <pb n="317"/><cb/> compound motion will be a straight line. Thus, if the one equable in the direction AB be <figure/> sufficient to carry a body over the space AB in any time, and the other motion sufficient to pass over AC in the same time; then by the compound motion, or both acting on the body together, it would in the same time pass over the diagonal AD of the parallelogram ABDC. For because the motions are uniform, any spaces A<hi rend="italics">b,</hi> A<hi rend="italics">c</hi> passed over in the same time, are proportional to the velocities, or to AB and AC; and consequently all the points A, <hi rend="italics">d, d,</hi> D, of the path are in the same right line.</p><p>3. And though the compounding motions be not equable, but variable, either accelerated or retarded, provided they do but vary in a similar manner, the compounded motion will still be in a straight line. Thus, suppose, for instance, that the motions both vary in such a manner, as that the spaces passed over in the same time, whether they be equal to each other or not, are both as the same power <hi rend="italics">n</hi> of the time; then AB<hi rend="sup">n</hi> : AC<hi rend="sup">n</hi> :: Ab<hi rend="sup">n</hi> : Ac<hi rend="sup">n</hi>, and hence AB : AC :: A<hi rend="italics">b</hi> : A<hi rend="italics">c,</hi> and therefore, as before, A<hi rend="italics">dd</hi>D is still a right line.</p><p>4. But if the compounding motions be not similar to each other, as when the one is equable and the other variable, or when they are varied in a dissimilar manner; then the compounded motion is in some curve line. So if the motion in the one direction EF be in a less proportion, with respect to the time, than that in the direction EG is, then the path will be a curve line E<hi rend="italics">i</hi>H concave to <figure/> wards EF; but if the motion in EF be in a grearer proportion than that in EG, then the path of the compound motion will be a curve E<hi rend="italics">h</hi>H convex towards EF: that is, in general, the curvilineal path is convex towards that direction in which the motion is in the less proportion to the time. Hence, for a particular instance, if the motion in the direction EF be a motion of projection, which is an equable motion, and the motion in the direction EG that arising from gravity, which is a uniformly accelerated motion, or in proportion to the squares of the times; then is EG as GH<hi rend="sup">2</hi>, and E<hi rend="italics">g</hi> as <hi rend="italics">gh</hi><hi rend="sup">2</hi>, that is EG : E<hi rend="italics">g</hi> :: GH<hi rend="sup">2</hi> : <hi rend="italics">gh</hi><hi rend="sup">2</hi>, which is the property of the parabola; and therefore the path E<hi rend="italics">h</hi>H of any body projected, is the common parabola.</p><p>5. If there be three forces united, or acting against the same point A at the same time, viz, the force or weight B in the direction AB, and the forces or tensions in the directions AC, AD; and if these three forces mutually balance each other, so as to keep the common point A in equilibrio; then are these forces directly proportional to the respective sides of a triangle formed by drawing lines parallel to the directions of these forces; or indeed perpendicular to those directions, or making any one and the same angle with them. So that, if BE be drawn parallel, for instance, to AD, and meet CA produced in E, forming the triangle <cb/> ABE; then are the three forces in the directions AB, AC, AD, respectively proportional to the sides AB, AE, BE. <figure/></p><p>And this theorem, with its corollaries, Dr. Keil observes, is the foundation of all the new mechanics of M. Varignon: by help of which may the force of the muscles be computed, and most of the mechanic theorems in Borelli, De Motu Animalium, may immediately be deduced.</p><p>See more of the Composition of Forces under the article <hi rend="smallcaps">Collision.</hi></p><p><hi rend="smallcaps">Composition</hi> <hi rend="italics">of Numbers and Quantities.</hi> See C<hi rend="smallcaps">OMBINATION.</hi></p><p><hi rend="smallcaps">Composition</hi> <hi rend="italics">of Proportion,</hi> according to the 15th definition of the 5th book of Euclid's Elements, is when, of four proportionals, the sum of the 1st and 2d is to the 2d, as the sum of the 3d and 4th is to the 4th: as if it be <hi rend="italics">a : b :: c : d,</hi> then by composition <hi rend="italics">a</hi>+<hi rend="italics">b : b :: c</hi>+<hi rend="italics">d : d.</hi> Or, in numbers, if 2 : 4 :: 9 : 18, then by composition 6 : 4 :: 27 : 18.</p><p><hi rend="smallcaps">Composition</hi> <hi rend="italics">of Ratios,</hi> is the adding of ratios together: which is performed by multiplying together their corresponding terms, viz, the antecedents together, and the consequents together, for the antecedent and consequent of the compounded ratio; like as the addition of logarithms is the same thing as the multiplication of their corresponding numbers. Or, if the terms of the ratios be placed fraction-wise, then the addition or composition of the ratios, is performed by multiplying the fractions together. Thus, the ratio of <hi rend="italics">a : b,</hi> or of 2 : 4, added to the ratio of <hi rend="italics">c : d,</hi> or of 6 : 8, makes the ratio of <hi rend="italics">ac : bd,</hi> or of 12 : 32; and so the ratio of <hi rend="italics">ac</hi> to <hi rend="italics">bd</hi> is said to be compounded of the ratios of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> and <hi rend="italics">c</hi> to <hi rend="italics">d.</hi> So likewise, if it were required to compound together the three ratios, viz, of <hi rend="italics">a</hi> to <hi rend="italics">b, c</hi> to <hi rend="italics">d,</hi> and <hi rend="italics">e</hi> to <hi rend="italics">f</hi>; then are the terms of the compound ratio; or the ratio of <hi rend="italics">ace</hi> to <hi rend="italics">bdf</hi> is compounded, or made up of the ratios of <hi rend="italics">a</hi> to <hi rend="italics">b, c</hi> to <hi rend="italics">d,</hi> and <hi rend="italics">e</hi> to <hi rend="italics">f.</hi></p><p>Hence, if the consequent of each ratio be the same as the antecedent of the preceding ratio, then is the ratio of the first term to the last, compounded, or made up of all the other ratios, viz, the ratio of <hi rend="italics">a</hi> to <hi rend="italics">e,</hi> equal to the sum of all the ratios of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> of <hi rend="italics">b</hi> to <hi rend="italics">c,</hi> of <hi rend="italics">c</hi> to <hi rend="italics">d,</hi> and of <hi rend="italics">d</hi> to <hi rend="italics">e;</hi> for the terms or exponents of the compounded ratio. <pb n="318"/><cb/></p><p>Hence also, in a series of continual proportionals, the ratio of the first term to the third is double of the ratio of the first to the second, and the ratio of the 1st to the 4th is triple of it, and the ratio of the 1st to the 5th is quadruple of it, and so on; that is, the exponents are double, triple, quadruple, &c, of the first exponent: as in the series 1, <hi rend="italics">a, a</hi><hi rend="sup">2</hi>, <hi rend="italics">a</hi><hi rend="sup">3</hi>, <hi rend="italics">a</hi><hi rend="sup">4</hi>, &c; where the ratio 1 to <hi rend="italics">a</hi><hi rend="sup">2</hi> is double, of 1 to <hi rend="italics">a</hi><hi rend="sup">3</hi> triple, &c, of the ratio of 1 to <hi rend="italics">a;</hi> or the exponent of <hi rend="italics">a</hi><hi rend="sup">2</hi>, <hi rend="italics">a</hi><hi rend="sup">3</hi>, <hi rend="italics">a</hi><hi rend="sup">4</hi>, &c, double, triple, quadruple, &c, of <hi rend="italics">a.</hi></p><p>COMPOUND <hi rend="smallcaps">Interest</hi>, called also <hi rend="italics">Interest upon Interest,</hi> is that which is reckoned not only upon the principal, but upon the interest itself forborn, which thus becomes a sort of secondary principal.</p><p>If <hi rend="italics">r</hi> be the amount of 1 pound for 1 year, that is the sum of the principal and interest together for one year; then is <hi rend="italics">r</hi><hi rend="sup">2</hi> the amount for 2 years, and <hi rend="italics">r</hi><hi rend="sup">3</hi> the amount for 3 years, and in general <hi rend="italics">r</hi><hi rend="sup">t</hi> the amount for <hi rend="italics">t</hi> years; that is <hi rend="italics">r</hi><hi rend="sup">t</hi> is the sum or total amount of all the principals and interests together of 1l. for the whole time or number of years <hi rend="italics">t</hi>; consequently, if <hi rend="italics">p</hi> be any other principal sum, forborn sor <hi rend="italics">t</hi> years, then its amount in that time at compound interest, is .</p><p><hi rend="italics">The Rule</hi> therefore in words is this, to one pound add its interest for one year, or half year, or for the first time at which the interest is reckoned; raise the sum <hi rend="italics">r</hi> to the power denoted by the time or number of terms; then this power multiplied by the principal, or first sum lent, will produce the whole amount.</p><p>For example, To find how much 50l. will amount to in 5 years at 5 per cent. per annum, compound interest.——Here the interest of 1l. for 1 year is <table><row role="data"><cell cols="1" rows="1" role="data">.05, and therefore <hi rend="italics">r</hi></cell><cell cols="1" rows="1" role="data">=1.05 ; hence the 5th power</cell></row><row role="data"><cell cols="1" rows="1" role="data">of it for 5 years, is <hi rend="italics">r</hi><hi rend="sup">5</hi></cell><cell cols="1" rows="1" role="data">=1.27628 &c;</cell></row><row role="data"><cell cols="1" rows="1" role="data">multiply this by <hi rend="italics">p</hi> or</cell><cell cols="1" rows="1" role="data">-        50</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">gives the amount <hi rend="italics">pr</hi><hi rend="sup">t</hi> or 63.814l.</cell></row></table> or 63l. 16s. (3 1/4)d. for the amount sought.</p><p>But Compound Interest is best computed by means of such a table as the following, being the amounts of 1 pound for any number of years, and at several rates of compound interest.</p><p>As an example of the use of this table, suppose it were required to sind the amount of 250l. for 35 years at 4 per cent. compound interest. <table><row role="data"><cell cols="1" rows="1" role="data">In the column of 4 per cent, and line of 35 years, is</cell><cell cols="1" rows="1" rend="align=right" role="data">3.94609,</cell></row><row role="data"><cell cols="1" rows="1" role="data">which multiplied by the principal</cell><cell cols="1" rows="1" rend="align=right" role="data">250 </cell></row><row role="data"><cell cols="1" rows="1" role="data">gives</cell><cell cols="1" rows="1" rend="align=right" role="data">986.52250 </cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" rend="align=right" role="data">986<hi rend="italics">l</hi> 10<hi rend="italics">s</hi> (5 1/4)<hi rend="italics">d,</hi></cell></row></table> which is the amount sought.</p><p><hi rend="italics">Note,</hi> By a bare inspection of this table, it appears how many years are required for any sum of money to double itself, at any rate of compound interest; viz, by looking in the columns when the amount becomes the number 2. So it is found that at the several rates the respective times requisite for doubling any sum, are nearly thus: viz, <table><row role="data"><cell cols="1" rows="1" role="data">Rate</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">3 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Years</cell><cell cols="1" rows="1" rend="align=center" role="data">23 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">20 1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">17 3/4</cell><cell cols="1" rows="1" rend="align=right" role="data">15 3/4</cell><cell cols="1" rows="1" rend="align=center" role="data">14 1/4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row></table> <cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=7 align=center" role="data">TABLE of the Amount of 1l. at Compound Interest for many Years and several Rates of Interest.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Yrs.</cell><cell cols="1" rows="1" rend="align=center" role="data">at 3 per cent</cell><cell cols="1" rows="1" rend="align=center" role="data">at 3 1/2 per cent</cell><cell cols="1" rows="1" rend="align=center" role="data">at 4 per cent</cell><cell cols="1" rows="1" rend="align=center" role="data">at 4 1/2 per cent</cell><cell cols="1" rows="1" rend="align=center" role="data">at 5 per cent</cell><cell cols="1" rows="1" rend="align=center" role="data">at 6 per cent</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">1.03000</cell><cell cols="1" rows="1" role="data">1.03500</cell><cell cols="1" rows="1" role="data">1.04000</cell><cell cols="1" rows="1" role="data">1.04500</cell><cell cols="1" rows="1" rend="align=right" role="data">1.05000</cell><cell cols="1" rows="1" rend="align=right" role="data">1.06000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">1.06090</cell><cell cols="1" rows="1" role="data">1.07123</cell><cell cols="1" rows="1" role="data">1.08160</cell><cell cols="1" rows="1" role="data">1.09203</cell><cell cols="1" rows="1" rend="align=right" role="data">1.10250</cell><cell cols="1" rows="1" rend="align=right" role="data">1.12360</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">1.09273</cell><cell cols="1" rows="1" role="data">1.10872</cell><cell cols="1" rows="1" role="data">1.12486</cell><cell cols="1" rows="1" role="data">1.14117</cell><cell cols="1" rows="1" rend="align=right" role="data">1.15763</cell><cell cols="1" rows="1" rend="align=right" role="data">1.19102</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">1.12551</cell><cell cols="1" rows="1" role="data">1.14752</cell><cell cols="1" rows="1" role="data">1.16986</cell><cell cols="1" rows="1" role="data">1.19252</cell><cell cols="1" rows="1" rend="align=right" role="data">1.21551</cell><cell cols="1" rows="1" rend="align=right" role="data">1.26248</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">1.15927</cell><cell cols="1" rows="1" role="data">1.18769</cell><cell cols="1" rows="1" role="data">1.21665</cell><cell cols="1" rows="1" role="data">1.24618</cell><cell cols="1" rows="1" rend="align=right" role="data">1.27628</cell><cell cols="1" rows="1" rend="align=right" role="data">1.33823</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">1.19405</cell><cell cols="1" rows="1" role="data">1.22926</cell><cell cols="1" rows="1" role="data">1.26532</cell><cell cols="1" rows="1" role="data">1.30226</cell><cell cols="1" rows="1" rend="align=right" role="data">1.34010</cell><cell cols="1" rows="1" rend="align=right" role="data">1.41852</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">1.22987</cell><cell cols="1" rows="1" role="data">1.27228</cell><cell cols="1" rows="1" role="data">1.31593</cell><cell cols="1" rows="1" role="data">1.36086</cell><cell cols="1" rows="1" rend="align=right" role="data">1.40710</cell><cell cols="1" rows="1" rend="align=right" role="data">1.50363</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">1.26677</cell><cell cols="1" rows="1" role="data">1.31681</cell><cell cols="1" rows="1" role="data">1.36857</cell><cell cols="1" rows="1" role="data">1.42210</cell><cell cols="1" rows="1" rend="align=right" role="data">1.47746</cell><cell cols="1" rows="1" rend="align=right" role="data">1.59385</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">1.30477</cell><cell cols="1" rows="1" role="data">1.36290</cell><cell cols="1" rows="1" role="data">1.42331</cell><cell cols="1" rows="1" role="data">1.48610</cell><cell cols="1" rows="1" rend="align=right" role="data">1.55133</cell><cell cols="1" rows="1" rend="align=right" role="data">1.68948</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">1.34392</cell><cell cols="1" rows="1" role="data">1.41060</cell><cell cols="1" rows="1" role="data">1.48024</cell><cell cols="1" rows="1" role="data">1.55297</cell><cell cols="1" rows="1" rend="align=right" role="data">1.62890</cell><cell cols="1" rows="1" rend="align=right" role="data">1.79085</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">1.38423</cell><cell cols="1" rows="1" role="data">1.45997</cell><cell cols="1" rows="1" role="data">1.53945</cell><cell cols="1" rows="1" role="data">1.62285</cell><cell cols="1" rows="1" rend="align=right" role="data">1.71034</cell><cell cols="1" rows="1" rend="align=right" role="data">1.89830</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">1.42576</cell><cell cols="1" rows="1" role="data">1.51107</cell><cell cols="1" rows="1" role="data">1.60103</cell><cell cols="1" rows="1" role="data">1.69588</cell><cell cols="1" rows="1" rend="align=right" role="data">1.79586</cell><cell cols="1" rows="1" rend="align=right" role="data">2.01220</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">1.46853</cell><cell cols="1" rows="1" role="data">1.56396</cell><cell cols="1" rows="1" role="data">1.66507</cell><cell cols="1" rows="1" role="data">1.77220</cell><cell cols="1" rows="1" rend="align=right" role="data">1.88565</cell><cell cols="1" rows="1" rend="align=right" role="data">2.13293</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">1.51259</cell><cell cols="1" rows="1" role="data">1.61869</cell><cell cols="1" rows="1" role="data">1.73168</cell><cell cols="1" rows="1" role="data">1.85194</cell><cell cols="1" rows="1" rend="align=right" role="data">1.97993</cell><cell cols="1" rows="1" rend="align=right" role="data">2.26090</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">1.55797</cell><cell cols="1" rows="1" role="data">1.67535</cell><cell cols="1" rows="1" role="data">1.80094</cell><cell cols="1" rows="1" role="data">1.93528</cell><cell cols="1" rows="1" rend="align=right" role="data">2.07893</cell><cell cols="1" rows="1" rend="align=right" role="data">2.39656</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">1.60471</cell><cell cols="1" rows="1" role="data">1.73399</cell><cell cols="1" rows="1" role="data">1.87298</cell><cell cols="1" rows="1" role="data">2.02237</cell><cell cols="1" rows="1" rend="align=right" role="data">2.18287</cell><cell cols="1" rows="1" rend="align=right" role="data">2.54035</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">1.65285</cell><cell cols="1" rows="1" role="data">1.79468</cell><cell cols="1" rows="1" role="data">1.94790</cell><cell cols="1" rows="1" role="data">2.11338</cell><cell cols="1" rows="1" rend="align=right" role="data">2.29202</cell><cell cols="1" rows="1" rend="align=right" role="data">2.69277</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">1.70243</cell><cell cols="1" rows="1" role="data">1.85749</cell><cell cols="1" rows="1" role="data">2.02582</cell><cell cols="1" rows="1" role="data">2.20848</cell><cell cols="1" rows="1" rend="align=right" role="data">2.40662</cell><cell cols="1" rows="1" rend="align=right" role="data">2.85434</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">1.75351</cell><cell cols="1" rows="1" role="data">1.92250</cell><cell cols="1" rows="1" role="data">2.10685</cell><cell cols="1" rows="1" role="data">2.30786</cell><cell cols="1" rows="1" rend="align=right" role="data">2.52695</cell><cell cols="1" rows="1" rend="align=right" role="data">3.02560</cell></row><row role="data"><cell 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cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">2.22129</cell><cell cols="1" rows="1" role="data">2.53157</cell><cell cols="1" rows="1" role="data">2.88337</cell><cell cols="1" rows="1" role="data">3.28201</cell><cell cols="1" rows="1" rend="align=right" role="data">3.73346</cell><cell cols="1" rows="1" rend="align=right" role="data">4.82235</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">2.28793</cell><cell cols="1" rows="1" role="data">2.62017</cell><cell cols="1" rows="1" role="data">2.99870</cell><cell cols="1" rows="1" role="data">3.42970</cell><cell cols="1" rows="1" rend="align=right" role="data">3.92013</cell><cell cols="1" rows="1" rend="align=right" role="data">5.11169</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">2.35657</cell><cell cols="1" rows="1" 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rend="align=right" role="data">4.53804</cell><cell cols="1" rows="1" rend="align=right" role="data">6.08810</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">3.57508</cell><cell cols="1" rows="1" role="data">3.00671</cell><cell cols="1" rows="1" role="data">3.50806</cell><cell cols="1" rows="1" role="data">4.08998</cell><cell cols="1" rows="1" rend="align=right" role="data">4.76494</cell><cell cols="1" rows="1" rend="align=right" role="data">6.45339</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">2.65234</cell><cell cols="1" rows="1" role="data">3.11194</cell><cell cols="1" rows="1" role="data">3.64838</cell><cell cols="1" rows="1" role="data">4.27403</cell><cell cols="1" rows="1" rend="align=right" role="data">5.00319</cell><cell cols="1" rows="1" rend="align=right" role="data">6.84059</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">2.73191</cell><cell cols="1" rows="1" role="data">3.22086</cell><cell cols="1" rows="1" role="data">3.79432</cell><cell cols="1" rows="1" role="data">4.46636</cell><cell cols="1" rows="1" rend="align=right" role="data">5.25335</cell><cell cols="1" rows="1" rend="align=right" role="data">7.25103</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">2.81386</cell><cell cols="1" rows="1" role="data">3.33359</cell><cell cols="1" rows="1" role="data">3.94609</cell><cell cols="1" rows="1" role="data">4.66735</cell><cell cols="1" rows="1" rend="align=right" role="data">5.51602</cell><cell cols="1" rows="1" rend="align=right" role="data">7.68609</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">2.89828</cell><cell cols="1" rows="1" 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rend="align=right" role="data">6.38548</cell><cell cols="1" rows="1" rend="align=right" role="data">9.15425</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">3.16703</cell><cell cols="1" rows="1" role="data">3.82537</cell><cell cols="1" rows="1" role="data">4.61637</cell><cell cols="1" rows="1" role="data">5.56590</cell><cell cols="1" rows="1" rend="align=right" role="data">6.70475</cell><cell cols="1" rows="1" rend="align=right" role="data">9.70351</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">3.26204</cell><cell cols="1" rows="1" role="data">3.95926</cell><cell cols="1" rows="1" role="data">4.80102</cell><cell cols="1" rows="1" role="data">5.81636</cell><cell cols="1" rows="1" rend="align=right" role="data">7.03999</cell><cell cols="1" rows="1" rend="align=right" role="data">10.28572</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">3.35990</cell><cell cols="1" rows="1" role="data">4.09783</cell><cell cols="1" rows="1" role="data">4.99306</cell><cell cols="1" rows="1" role="data">6.07810</cell><cell cols="1" rows="1" rend="align=right" role="data">7.39199</cell><cell cols="1" rows="1" rend="align=right" role="data">10.90286</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">3.46070</cell><cell cols="1" rows="1" role="data">4.24126</cell><cell cols="1" rows="1" role="data">5.19278</cell><cell cols="1" rows="1" role="data">6.35162</cell><cell cols="1" rows="1" rend="align=right" role="data">7.76159</cell><cell cols="1" rows="1" rend="align=right" role="data">11.55703</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">3.56452</cell><cell cols="1" rows="1" role="data">4.38970</cell><cell cols="1" rows="1" role="data">5.40050</cell><cell cols="1" rows="1" role="data">6.63744</cell><cell cols="1" rows="1" rend="align=right" role="data">8.14967</cell><cell cols="1" rows="1" rend="align=right" role="data">12.25045</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">3.67145</cell><cell cols="1" rows="1" role="data">4.54334</cell><cell cols="1" rows="1" role="data">5.61652</cell><cell cols="1" rows="1" role="data">6.93612</cell><cell cols="1" rows="1" rend="align=right" role="data">8.55715</cell><cell cols="1" rows="1" rend="align=right" role="data">12.98548</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">3.78160</cell><cell cols="1" rows="1" role="data">4.70236</cell><cell cols="1" rows="1" role="data">5.84118</cell><cell cols="1" rows="1" role="data">7.24825</cell><cell cols="1" rows="1" rend="align=right" role="data">8.98501</cell><cell cols="1" rows="1" rend="align=right" role="data">13.76461</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">3.89504</cell><cell cols="1" rows="1" role="data">4.86694</cell><cell cols="1" rows="1" role="data">6.07482</cell><cell cols="1" rows="1" role="data">7.57442</cell><cell cols="1" rows="1" rend="align=right" role="data">9.43426</cell><cell cols="1" rows="1" rend="align=right" role="data">14.59049</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" role="data">4.01190</cell><cell cols="1" rows="1" role="data">5.03728</cell><cell cols="1" rows="1" role="data">6.31782</cell><cell cols="1" rows="1" role="data">7.91527</cell><cell cols="1" rows="1" rend="align=right" role="data">9.90597</cell><cell cols="1" rows="1" rend="align=right" role="data">15.46592</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">4.13225</cell><cell cols="1" rows="1" role="data">5.21359</cell><cell cols="1" rows="1" role="data">6.57053</cell><cell cols="1" rows="1" role="data">8.27146</cell><cell cols="1" rows="1" rend="align=right" role="data">10.40127</cell><cell cols="1" rows="1" rend="align=right" role="data">16.39387</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" role="data">4.25622</cell><cell cols="1" rows="1" role="data">5.39606</cell><cell cols="1" rows="1" role="data">6.83335</cell><cell cols="1" rows="1" role="data">8.64367</cell><cell cols="1" rows="1" rend="align=right" role="data">10.92133</cell><cell cols="1" rows="1" rend="align=right" role="data">17.37750</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">4.38391</cell><cell cols="1" rows="1" role="data">5.58493</cell><cell cols="1" rows="1" role="data">7.10668</cell><cell cols="1" rows="1" role="data">9.03264</cell><cell cols="1" rows="1" rend="align=right" role="data">11.46740</cell><cell cols="1" rows="1" rend="align=right" role="data">18.42015</cell></row></table></p><p>COMPOUND <hi rend="italics">Motion,</hi> that motion which is the effect of several conspiring powers or forces, viz, such forces as are not directly opposite to each other: as when the radius of a circle is considered as revolving about a centre, and at the same time a point as moving straight along it; which produces a kind of a spiral for the path of the point. And hence it is easily perceived, that all curvilinear motion is compound, or the effect of two or more forces; although every compound motion is not curvilinear.</p><p>It is a popular theorem in Mechanics, that in uniform compound motions, the velocity produced by the conspiring powers or forces, is to that of either of the two compounding powers separately, as the diagonal <pb n="319"/><cb/> of a parallelogram, according to the direction of whose sides they act separately, is to either of the sides. See <hi rend="smallcaps">Composition</hi> <hi rend="italics">of Motion,</hi> and <hi rend="smallcaps">Collision.</hi></p><p><hi rend="smallcaps">Compound</hi> <hi rend="italics">Numbers,</hi> those composed of the multiplication of two or more numbers; as 12, composed of 3 times 4. See <hi rend="smallcaps">Composite.</hi></p><p><hi rend="smallcaps">Compound</hi> <hi rend="italics">Pendulum,</hi> that which consists of several weights constantly keeping the same distance, both from each other, and from the centre about which they oscillate.</p><p><hi rend="smallcaps">Compound</hi> <hi rend="italics">Quantities,</hi> are such as are connected together by the signs + or-. Thus, <hi rend="italics">a</hi> + <hi rend="italics">b,</hi> or <hi rend="italics">a</hi>-<hi rend="italics">c</hi> + <hi rend="italics">d,</hi> or <hi rend="italics">aa</hi> - 2<hi rend="italics">a,</hi> are compound quantities.</p><p>Compound quantities are disting uished into binomials, trinomials, quadrinomials, &c, according to the number of terms in them; viz, the binomial having two terms; the trinomial, three; the quadrinomial, 4; &c. Also, those that have more than two terms, are called by the general name of multinomials, as also polynomials.</p><p><hi rend="smallcaps">Compound</hi> <hi rend="italics">Ratio,</hi> is that which is made by adding two or more ratios together; viz, by multiplying all their antecedents together for the antecedent, and all the consequents together for the consequent of the compound ratio. So 6 to 72 is a ratio compounded of the ratios of 2 to 6, and 3 to 12; because : also <hi rend="italics">ab</hi> to <hi rend="italics">cd</hi> is a ratio compounded of the ratio of <hi rend="italics">a</hi> to <hi rend="italics">c,</hi> and <hi rend="italics">b</hi> to <hi rend="italics">d</hi>; for . See C<hi rend="smallcaps">OMPOSITION</hi> <hi rend="italics">of Ratios.</hi></p></div1><div1 part="N" n="COMPRESSION" org="uniform" sample="complete" type="entry"><head>COMPRESSION</head><p>, the act of pressing, or squeezing something, so as to bring its parts nearer together, and make it occupy less space.</p><p>Compression differs from condensation as the cause from the effect, compression being the action of any force on a body, without regarding its effects; whereas condensation denotes the state of a body that is actually reduced into a less bulk, and is an effect of compression, though it may be effected also by other means. Nevertheless, compression and condensation are often confounded.</p><p>Pumps, which the ancients imagined to act by suction, do in reality act by compression; the piston, in working in the narrow pipe, compresses the inclosed air, so as to enable it, by the force of its increased elasticity, to raise the valve, and make its escape; upon which, the balance being destroyed, the pressure of the atmosphere on the stagnant surface, forces up the water in the pipe, thus evacuated of its air.</p><p>It was long thought that water was not compressible into less bulk: and it was believed, till lately, that after the air had been purged out of it, no art or violence was able to press it into less space. In an experiment made by the Academy del Cimento, water, when violently squeezed, made its way through the fine pores of a globe of gold, rather than yield to the compression.</p><p>But the ingenious Mr. Canton, attentively considering this experiment, found that it was not sufficiently accurate to justify the conclusion which had always been drawn from it; since the Florentine philosophers had no method of determining that the alteration of sigure in their globe of gold, occasioned <cb/> such a diminution of its internal capacity, as was exactly equal to the quantity of water forced into its pores. To bring this matter therefore to a more accurate and decisive trial, he procured a small glafs tube of about two feet long, with a ball at one end, of an inch and a quarter in diameter. Having filled the ball and part of the tube with mercury, and brought it exactly to the heat of 50° of Fahrenheit's thermometer, he marked the place where the mercury stood in the tube, which was about six inches and a half above the ball; he then raised the mercury by heat to the top of the tube, and there sealed the tube hermetically; then upon reducing the mercury to the same degree of heat as before, it stood in the tube 32/100 of an inch higher than the mark. The same experiment was repeated with water exhausted of air, instead of mercury, and the water stood in the tube 43/100 of an inch above the mark. Since the weight of the atmosphere on the outside of the ball, without any counterbalance from within, will compress the ball, and equally raise both the mercury and water, it appears that the water expands 11/100 of an inch more than the mercury by removing the weight of the atmosphere. Having thus determined that water is really compressible, he proceeded to estimate the degree of compression corresponding to any given weight. For this purpose he prepared another ball, with a tube joined to it; and finding that the mercury in 23/100 of an inch of the tube was the hundred thousandth part of that contained in the ball, he divided the tube accordingly. He then filled the ball and part of the tube with water exhausted of air; and leaving the tube open, placed this apparatus under the receiver of an air-pump, and observed the degree of expansion of the water answering to any degree of rarefaction of the air: and again by putting it into the glass receiver of a condensing engine, he noted the degree of compression of the water corresponding to any degree of condensation of the air. He thus sound, by repeated trials, that, in a temperature of 50°, and when the mercury has been at its mean height in the barometer, the water expands one part in 21740; and is as much compressed by the weight of an additional atmosphere; or the compression of water by twice the weight of the atmosphere, is one part in 10870 of its whole bulk. Should it be objected, that the compressibility of the water was owing to any air which it might be supposed to contain, he answers, that more air would make it more compressible; he therefore let into the ball a bubble of air, and found that the water was not more compressed by the same weight than before.</p><p>In some farther experiments of the same kind, Mr. Canton found that water is more compressible in winter than in summer; but he observed the contrary in spirit of wine, and oil of olives.</p><p>The following table was formed, when the barometer was at 29 inches and a half, and the thermometer at 50 degrees. <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Compression of</cell><cell cols="1" rows="1" rend="align=center" role="data">Millionth parts.</cell><cell cols="1" rows="1" rend="align=center" role="data">Spec. grav.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of wine</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">846</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of olives</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">918</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rain water</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sea water</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">1028</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">13595</cell></row></table> <pb n="320"/><cb/> He infers that these fluids are not only compressible, but elastic; and that the compressions of them, by the same weight, are not in the inverse ratio of their densities, or specific gravities, as might be supposed. Phil. Trans. vol. lii. 1762. art. 103. and vol. liv. 1764. art. 47.</p><p>The compression of the air, by its own weight, is surprisingly great: but the air may be still further compressed by art. See <hi rend="italics">Elasticity of</hi> <hi rend="smallcaps">Air.</hi></p><p>This immense compression and dilatation, Newton observes, cannot be accounted for in any other way, but by a repelling force, with which the particles of air are endued; by virtue of which, when at liberty, they mutually fly each other.</p><p>This repelling power, he adds, is stronger and more sensible in air, than in other bodies; because air is generated out of very fixed bodies, but not without great difficulty, and by the help of fermentation: now those particles always recede from each other with the greatest violence, and are compressed with the greatest difficulty, which, when contiguous, cohere the most strongly. See <hi rend="smallcaps">Air, Attraction, Cohesion</hi>, D<hi rend="smallcaps">ILATATION</hi>, and <hi rend="smallcaps">Repulsion.</hi></p></div1><div1 part="N" n="COMPUTATION" org="uniform" sample="complete" type="entry"><head>COMPUTATION</head><p>, the manner of accounting and estimating time, weights, measures, money, &c. See <hi rend="smallcaps">Calculation</hi>, which it is also used for.</p></div1><div1 part="N" n="CONCAVE" org="uniform" sample="complete" type="entry"><head>CONCAVE</head><p>, an appellation used in speaking of the inner surface of hollow bodies, more especially of spherical or circular ones.</p><p><hi rend="smallcaps">Concave</hi> glasses, lenses, and mirrors, have either one side or both sides concave.</p><p>The property of all concave lenses is, that the rays of light, in passing through them, are deflected, or made to recede from one another; as in convex lenses they are inflected towards each other; and that the more as the concavity or convexity has a small<*> radius. Hence parallel rays, as those of the sun, by passing through a concave lens, become diverging; diverging rays are made to diverge more; and converging rays are made either to converge less, or to become parallel, or go out diverging. And hence it is, that objects viewed through concave lenses, appear diminished; and the more so, as they are portions of less spheres. See <hi rend="smallcaps">Lens.</hi></p><p>Concave mirrors have the contrary effect to lenses: they reflect the rays which fall on them, so as to make them approach more to, or recede from each other, than before, according to the situation of the object; and that the more as the concavity is greater, or as the radius of concavity is less. Hence it is that concave mirrors magnifying objects that are presented to them; and that in a greater proportion, as they are portions of greater spheres. And hence also concave mirrors have the effect of burning glasses. See <hi rend="smallcaps">Mirror</hi>, and <hi rend="smallcaps">Burning Glass.</hi></p></div1><div1 part="N" n="CONCAVITY" org="uniform" sample="complete" type="entry"><head>CONCAVITY</head><p>, that side of a figure or body which is hollow.</p><p>An arch of a curve has its concavity turned all one way, when the right lines that join any two of its points are all on the same side of the arch.</p><p>Archimedes, intending to include in his definition such lines as have rectilinear parts, says, a line has its concavity turned one way, when the right lines that join any two of its points, are either all upon one side <cb/> of it, or while some fall upon the line itself, none fall upon the opposite side. Archim. de Sphær. et Cyl. Def. 2, and Maclaurin's Fluxions, art. 180.</p><p>When two lines that have their concavity turned the same way, have the same extremes, and the one includes the other, or has its concavity towards it, the perimeter of that which includes, is greater than that which is included. Archim. ib. ax. 2.</p></div1><div1 part="N" n="CONCENTRIC" org="uniform" sample="complete" type="entry"><head>CONCENTRIC</head><p>, having the same centre. It is opposed to excentric, or having different centres.</p><p>The word is chiefly used in speaking of round bodies and figures, such as circular, and elliptic ones; but it may likewise be used for polygons that are drawn parallel to each other, from the same centre.</p><p>Nonnius's method of graduating instruments consists in describing with the same quadrant 45 concentric arches, dividing the outermost into 90 equal parts, the next into 89, and so on.</p></div1><div1 part="N" n="CONCHOID" org="uniform" sample="complete" type="entry"><head>CONCHOID</head><p>, or <hi rend="smallcaps">Conchiles</hi>, the name of a curve invented by Nicomedes. It was much used by the ancients in the construction of solid problems, as appears by what Pappus says.</p><p>It is thus constructed: AP and BD being two lines intersecting at right angles; from P draw a number of other lines PFDE, &c, on which take always DE = DF = AB or BC; so shall the curve line drawn through all the points E, E, E, be the first conchoid, or that of Nicomedes; and the curve drawn through all the other points F, F, F, is called the second conchoid; though in reality, they are both but parts of the same curve, having the same pole P, and four infinite legs, to which the line DBD is a common asymptote. <figure><head><hi rend="italics">Fig. 1</hi></head></figure> <figure><head><hi rend="italics">Fig. 2</hi></head></figure> <figure><head><hi rend="italics">Fig. 3</hi></head></figure></p><p>The inventor, Nicomedes, contrived an instrument for describing his conchoid by a mechanical motion: thus, in the rule AD is a channel or groove cut, so that a smooth nail, firmly fixed in the moveable rule CB, in the point F, may slide freely within it: into the rule EG is fixed another nail at K, for the moveable rule CB to slide upon. If therefore the rule BC be so moved, as that the nail F passes along the canal AD; the style, or point in C, will describe the first conchoid.</p><p>To determine the equation of the curve: put AB = BC = DE = DF = <hi rend="italics">a,</hi> PB= <hi rend="italics">b,</hi> BG = EH = <hi rend="italics">x,</hi> and GE = BH = <hi rend="italics">y</hi>; then the equation to the first con- <pb n="321"/><cb/> choid will be ; and, changing only the sign of <hi rend="italics">x,</hi> as being negative in the other curve, the equation to the 2d conchoid will be .</p><p>Of the whole conchoid, expressed by these two equations, or rather one equation only, with different signs, there are three cases or species; as first, when BC is less than BP, the conchoid will be as in fig. 1; when BC is equal to BP, the conchoid will be as in fig. 2; and when BC is greater than BP, the conchoid will be as in fig. 3.</p><p>Newton approves of the use of the conchoid for trisecting angles, or finding two mean proportionals, or for constructing other solid problems. Thus, in the Linear Construction of equations, towards the end of his Universal Arithmetic, he says, “The antients at first endeavoured in vain at the trisection of an angle, and the finding of two mean proportionals by a right line and a circle. Afterwards they began to consider several other lines, as the conchoid, the cissoid, and the conic sections, and by some of these to solve these problems.” A gain, “Either therefore the trochoid is not to be admitted at all into geometry, or else, in the construction of problems, it is to be preferred to all lines of a more difficult description: and there is the same reason for other curves; for which reason we approve of the trisections of an angle by a conchoid, which Archimedes in his Lemmas, and Pappus in his Collections, have preferred to the inventions of all others in this case; because we ought either to exclude all lines, besides the circle and right line, out of geometry, or admit them according to the simplicity of their descriptions, in which case the conchoid yields to none, except the circle.” Lastly, “That is <hi rend="italics">arithmetically</hi> more simple which is determined by the more simple equations, but that is <hi rend="italics">geometrically</hi> more simple which is determined by the more simple drawing of lines; and in geometry, that ought to be reckoned best which is geometrically most simple: wherefore I ought not to be blamed, if, with that prince of mathematicians, Archimedes, and other antients, I make use of the conchoid for the construction of solid problems.”</p><p>CONCRETE <hi rend="italics">Numbers</hi> are those that are applied to express or denote any particular subject; as 3 men, 2 pounds, &c. Whereas, if nothing be connected with a number, it is taken abstractedly or universally: thus, 4 signifies only an aggregate of 4 units, without any regard to a particular subject, whether men or pounds, or any thing else.</p></div1><div1 part="N" n="CONCURRING" org="uniform" sample="complete" type="entry"><head>CONCURRING</head><p>, or <hi rend="smallcaps">Congruent</hi> <hi rend="italics">Figures,</hi> in Geometry, are such as, being laid upon one another, do exactly correspond to, and cover one another, and consequently must be equal among themselves. Thus, triangles having two sides and the contained angle equal, each to each, are equal to each other in all respects.</p></div1><div1 part="N" n="CONDENSATION" org="uniform" sample="complete" type="entry"><head>CONDENSATION</head><p>, is the compressing or reducing of a body into a less bulk or space; by which means it is rendered more dense and compact.</p><p>Wolfius, and some other writers, restrain the use of the word <hi rend="italics">condensation</hi> to the action of cold: that which is done by external application, they call compression.</p><p>Condensation however, in general, consists in bring- <cb/> ing the parts closer to each other, and increasing their contact, whatever be the means by which it is effected: in opposition to rarefaction, which renders the body lighter and looser, by setting the parts farther asunder, and diminishing their contact, and of consequence their cohesion.</p><p>Air easily condenses, either by cold, or by pressure, but much more by the latter; but most of all by chemical process. Water condenses also both by cold and by pressure; but it suddenly expands by congelation: indeed almost all matter, both solids and fluids, has the same property of condensation by those means. See <hi rend="smallcaps">Compression.</hi> So also vapour is condensed, or converted into water, by distillation, or naturally in the clouds. The way in which vapour commonly condenses, is by the application of some cold substance. On touching it, the vapour parts with its heat which it had before absorbed: and on doing so, it immediately loses the proper characteristics of vapour, and becomes water. But though this be the most common and usual way in which we observe vapour to be condensed, nature certainly proceeds after another manner; since we often observe the vapours most plentifully condensed when the weather is really warmer than at other times.</p></div1><div1 part="N" n="CONDENSER" org="uniform" sample="complete" type="entry"><head>CONDENSER</head><p>, a pneumatic engine, or syringe, by which an extraordinary quantity of air may be crowded or pushed into a given space; so that frequently ten atmospheres, or ten times as much air as the space naturally contains, without the engine, may be thrown in by means of it, and its egress prevented by valves properly disposed.</p><p>The condenser is made either of metal, or glass, and either in a cylindrical or globular form, into which the air is thrown with an injecting syringe.</p><p>The receiver, or vessel containing the condensed air, should be made very strong, to bear the force of the air's elasticity thus increased; for which reason it is commonly made of brass. When glass is used, it will not bear so great a condensation of air; but then the experiment will be more entertaining, as the effect may be viewed of the condensed air upon any subject put within it.</p></div1><div1 part="N" n="CONDUCTOR" org="uniform" sample="complete" type="entry"><head>CONDUCTOR</head><p>, in Electricity, a term first introduced in this science by Dr. Desaguliers, and used to denote those substances which are capable of receiving and transmitting electricity; in opposition to electrics, in which the matter or virtue of electricity may be excited and accumulated, or retained. The former are also called <hi rend="italics">non-electrics,</hi> and the latter <hi rend="italics">non-conductors.</hi> And all bodies are ranked under one or other of these two classes, though none of them are <hi rend="italics">persect electrics,</hi> nor <hi rend="italics">perfect conductors,</hi> so as wholly to retain, or freely and without resistance to transmit the electric sluid.</p><p>To the class of conductors, belong all metals and semi-metals, ores, and all fluids (except air and oils), together with the substances containing them, the effluvia of flaming bodies, ice (unless very hard frozen), and snow, most saline and stony substances, charcoals, of which the best are those that have been exposed to the greatest heat, smoke, and the vapour of hot water.</p><p>It seems probable that the electric fluid passes through the substance, and not merely over the surfaces of metallic conductors; because, if a wire of any kind of metal be covered with some electric substance, as resin, <pb n="322"/><cb/> sealing-wax, &c, and a jar be discharged through it, the charge will be conducted as well as without the electric coating.</p><p>It has also been alleged, that electricity will pervade a vacuum, and be transmitted through it almost as freely as through the substance of the best conductor: but Mr. Walsh found, that the electrie spark or shock would no more pass through a perfect vacuum, than through a stick of solid glass. In other instances however, when the vacuum has been made with all possible eare, the experiment has not succeeded.</p><p>It may also be observed, that many of the forementioned substances are capable of being electrified, and that their conducting power may be destroyed and recovered by different processes: for example, green wood is a conductor; but baked, it becomes a non-conductor: again its conducting power is restored by charring it; and lastly it is destroyed by reducing this to ashes.</p><p>Again, many electric substances, as glass, resin, air, &c, become conductors by being made very hot: however, air heated by glass must be excepted.</p><p>See, on this subject, Priestley's Hist. of Electricity, vol. 1; Franklin's Letters &c, pa. 96 and 262 edit. 1769; Cavallo's Complete Treat. of Electr. chap. 2; Henley's Exper. and Obser. in Electr. also Philos. Trans. vol. 67 pa. 122; and elsewhere in the different volumes of the Transactions.</p><p><hi rend="italics">Prime</hi> <hi rend="smallcaps">Conductor</hi>, is an insulated conductor, so connected with the electrical machine, as to receive the electricity immediately from the excited electric.</p><p>Mr. Grey first employed metallic conductors in this way, in 1734; and these were several pieces of metal suspended on silken strings, which he charged with electricity. Mr. Du Fay fastened to the end of an iron bar, which he used as his prime conductor, a bundle of linen threads, to which he applied the excited tube: but these were afterwards changed for small wires suspended from a common gun-barrel, or other metallic rod.</p><p>In the present advanced state of the science, this part of the electrical apparatus has been considerably improved. The prime conductor is made of hollow brass, and usually of a cylindrical form. Care should be taken, that it be perfectly smooth and round, without points and sharp edges. The ends of the conductor are spherical; and it is necessary, that the part most remote from the electric should be made round and much larger than the rest, the better to prevent the electric matter from escaping, which it always endeavours most to do at the greatest distance from the electric: and the other end should be furnished with several pointed wires or needles, either suspended from, or fixed to an open metallic ring, and pointing to the globe or cylinder, or plate, to collect the fire. It is best supported by pillars of solid glass, covered with sealing-wax or good varnish. Prime conductors of a large size are usually made of paste-board, covered with tin-foil or gilt paper; these being useful for throwing off a longer and denser spark than those of a smaller size: they should terminate in a smaller knob or obtuse edge, at which the sparks should be solicited. Mr. Nairne prepared a conductor 6 feet in length, and 1 foot in diameter, from which he drew electrical sparks at the distance of 16, 17, or 18 inches; and Dr. Van Marum <cb/> still far exceeded this, with a conductor of 8 inches diameter, and upwards of 20 feet long, formed of different pieces, and applied to the large electrical machine in Teyler's Museum at Harlem, the most powerful machine of the kind ever yet constructed. But the size of the conductor is always limited by that of the electric, there being a maximum which the size of the former should not exceed; for it may be so large, that the dissipation of the electricity from its surface may be greater than that which the electric is capable of supplying.</p><p>Dr. Priestley recommends a prime conductor of polished copper, in the form of a pear, supported by a pillar and a firm basis of baked wood: this receives its fire by a long arched wire of soft brass, which may be easily bent, and raised or lowered to the globe: it is terminated by an open ring, in which some sharp-pointed wires are hung. In the body of this conductor are holes for the insertion of metalline rods. This, he says, collects the fire perfectly well, and retains it equally everywhere. Philos. Trans. vol. 64, art. 7. Hist. Elect. vol. 2, § 2.</p><p>Mr. Henly has contrived a new kind of prime conductor, which, from its use, is called the <hi rend="italics">luminous</hi> conductor. It consists of a glass tube 18 inches long, and 2 inches diameter. The tube is furnished at both ends with brass caps and ferules about 2 inches long, cemented and made air-tight, and terminated by brass balls. In one of these caps is drilled a small hole, which is covered by a strong valve, and serves for exhausting the tube of its air. Within the tube at each end there is a knobbed wire, projecting to the distance of 2 inches and a half from the brass caps. To one of the balls is annexed a fine-pointed wire for receiving and collecting the electricity, and to the other a wire with a knob or ball for discharging it. The conductor, thus prepared, is supported on pillars of sealing-wax or glass. Beside the common purposes of a prime conductor to an electrical machine, this apparatus serves to exhibit and ascertain the direction of the electric matter in its passage through it. See a figure of this conductor in the Philos. Trans. with a description of experiments, &c. with it, vol. 64, pa. 403.</p><p><hi rend="smallcaps">Conductors</hi> <hi rend="italics">of Lightning,</hi> are pointed metallic rods fixed to the upper parts of buildings, to secure them from strokes of lightning. These were invented and proposed by Dr. Franklin for this purpose, soon after the identity of electricity and lightning was ascertained; and they exhibit a very important and useful application of modern discoveries in this science. This ingenious philosopher, having found that pointed bodies are better fitted for receiving and throwing off the electric fire, than such as are terminated by blunt ends or flat surfaces, and that metals are the readiest and best conductors, soon discovered that lightning and electricity resembled each other in this and other distinguishing properties: he therefore recommended a pointed metalline rod, to be raised some feet above the highest part of a building, and to be continued down into the ground, or the nearest water. The lightning, should it ever come within a certain distance of this rod or wire, would be attracted by it, and pass through it preferably to any other part of the building, and be conveyed into the earth or water, and there dissipated, without doing any damage to the building. Many facts have occur- <pb n="323"/><cb/> red to evince the utility of this simple and seemingly trifling apparatus. And yet some electricians, of whom Mr. Wilson was the chief, have objected to the pointed termination of this conductor; preferring rather a blunt end: because, they pretend, a point invites the electricity from the clouds, and attracts it at a greater distance than a blunt conductor. Philos. Trans. vol. 54, pa. 234; vol. 63, pa. 49; and vol. 68, pa. 232.</p><p>This subject has indeed been very accurately examined and discussed; and pointed conductors are almost universally, and for the best reasons, recommended as the most proper and eligible. A sharp-pointed conductor, as it attracts the electric fire of a cloud at a greater distance than the other, draws it off gradually: and by conveying it away gently, and in a continued stream, prevents an accumulation and a stroke; whereas a conductor with a blunt termination receives the whole discharge of a cloud at once, and is much more likely to be exploded, whenever a cloud comes within a striking distance. To this may be added experience; for buildings guarded by either natural or artificial conductors terminating in a point, have very seldom been struck by lightning; but others, having flat or blunt terminations, have often been struck and damaged by it.</p><p>The best conductor for this purpose, is a rod of iron, or rather of copper, as being a better conductor of electricity, and less liable to rust, about 3 quarters of an inch thick, which is either to be fastened to the walls of a building by wooden cramps, or supported by wooden posts, at the distance of a foot or two from the wall; though less may do: the upper end of it should terminate in a pyramidal form, with a sharp point and edges; and, when made of iron, gilt or painted near the top, or else pointed with copper; and be elevated 5 or 6 feet above the highest part of the building, or chimneys, to which it may be fastened. The lower end should be driven 5 or 6 feet into the ground, and directed away from the foundations of the building, or continued till it communicates with the nearest water: and if this part be made of lead, it will be less apt to decay. When the conductor is formed of different pieces of metal, care should be taken that they are well joined: and it is farther recommended, that a communication be made from the conductor by plates of lead, 8 or 10 inches broad, with the lead on the ridges and gutters, and with the pipes that carry down the rain water, which should be continued to the bottom of the building, and be made to communicate either with water or moist earth, or with the main pipe which serves the house with water. If the building be large, two, three, or more conductors should be applied to different parts of it, in proportion to its extent. Philos. Trans. vol. 64, pa. 403.</p><p>Chains have been used as conductors for preserving ships; but as the electric matter does not pass readily through the links of it, copper wires, a little thicker than a goose quill, have been preferred, and are now generally used. They should reach 2 or 3 feet above the highest mast, and be continued down in any convenient direction, so as always to touch the sea water. Philos. Trans. vol. 52, pa. 633. See also Franklin's Letters &c 1769, pa. 65, 124, 479, &c; and Cavallo's Electr. chap. 9. <cb/></p><p>For the <hi rend="italics">Construction</hi> and management of <hi rend="italics">Electrical Kites,</hi> and <hi rend="italics">Cenductors</hi> or <hi rend="italics">Machines</hi> for drawing electricity from the clouds, see Priestley's Hist. of Electr. vol. 2, pa. 103 edit. 1775.</p></div1><div1 part="N" n="CONE" org="uniform" sample="complete" type="entry"><head>CONE</head><p>, a kind of round pyramid, or a solid body having a circle for its base, and its sides formed by right lines drawn from the circumference of the base to a point at top, being the vertex or apex of the cone.</p><p>Euclid desines a cone to be a solid figure, whose base is a circle, and is produced by the entire revolution of a right-angled triangle about its perpendicular leg, called the axis of the cone. If this leg, or axis, be greater than the base of the triangle, or radius of the circular base of the cone, then the cone is <hi rend="italics">acute-angled,</hi> that is, the angle at its vertex is an acute angle; but if the axis be less than the radius of the base, it is an <hi rend="italics">obtuseangled</hi> cone; and if they are equal, it is a <hi rend="italics">rig<hi rend="sup">.</hi>ht-angled</hi> cone.</p><p>But Euclid's definition only extends to a <hi rend="italics">right cone,</hi> that is, to a cone whose axis is perpendicular or at rightangles to its base; and not to oblique ones, in which the axis is oblique to the base, the general definition, or description of which may be this: <figure/> If a line VA continually pass through the point V, turning upon that point as a joint, and the lower part of it be carried round the circumference ABC of a circle; then the space inclosed between that circle and the path of the line, is a cone. The circle ABC is the base of the cone; V is its vertex; and the line VD, from the vertex to the centre of the base, is the axis of the cone. Also the other part of the revolving line, produced above V, will describe another cone V<hi rend="italics">acb,</hi> called the opposite cone, and having the same common axis produced DV<hi rend="italics">d,</hi> and vertex V.</p><p><hi rend="italics">Properties of the</hi> <hi rend="smallcaps">Cone.</hi>—1. The area or surface of every right cone, exclusive of its base, is equal to a triangle whose base is the periphery, and its height the slant side of the cone. Or, the curve superficies of a right cone, is to the area of its circular base, as the slant side is to the radius of the base. And therefore, the same curve surface of the cone is equal to the sector of a circle whose radius is the slant side, and its arch equal to the circumference of the base of the cone.</p><p>2. Every cone, whether right or oblique, is equal to one-third part of a cylinder of equal base and altitude; and therefore the solid content is found by multiplying the base by the altitude, and taking 1/3 of the product; and hence also all cones of the same or equal base and altitude, are equal.</p><p>3. Although the solidity of an oblique cone be obtained in the same manner with that of a right one, it is otherwise with regard to the surface, since this cannot be reduced to the measure of a sector of a circle, because all the lines drawn from the vertex to the base are not equal. See a Memoir on this subject, by M. Euler, in the Nouv. Mem. de Petersburgh vol. 1. Dr. Barrow has demonstrated, in his Lectiones Geometricæ, that the solidity of a cone with an elliptic base, forming part of a right cone, is equal to the product of its surface by a third part of one of the perpendiculars <pb n="324"/><cb/> drawn from the point in which the axis of the right cone intersects the ellipse; and that it is also equal to 1/3 of the height of the cone multiplied by the elliptic base: consequently that the perpendicular is to the height of the cone, as the elliptic base is to the curve surface. For the curve surface of all the oblique parts of a cone, see my Mensur. pa. 234 &c.</p><p>4. <hi rend="italics">To find the Curve Surface of the Frustum of a Cone.</hi> Multiply half the sum of the circumferences of the two ends, by the slant side, or distance between these circumferences.</p><p>5. <hi rend="italics">For the Solidity of the Frustum of a Cone,</hi> add into one sum the areas of the two ends and the mean proportional between them, multiply that sum by the perpendicular height, and 1/3 of the product will be the solidity. See also my Mensuration, pa. 189.</p><p>6. The Centre of Gravity of a cone is 3/4 of the axis distant from the vertex.</p><p><hi rend="smallcaps">Cones</hi> <hi rend="italics">of the Higher Kinds,</hi> are those whose bases are circles of the higher kinds; and are generated, like the common cone, by conceiving a line turning on a point or vertex on high, and revolving round the circle of the higher kind.</p><p><hi rend="smallcaps">Cone</hi> <hi rend="italics">of Rays,</hi> in Optics, includes all the several rays which fall from any point of a radiant object, on the surface of a glass.</p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Cone</hi>, or <hi rend="italics">Spindle,</hi> in Mechanics, is a solid formed of two equal cones joined at their bases. If this be laid on the lower part of two rulers, making an angle with each other, and elevated in a certain degree above the horizontal plane, the cones will roll up towards the raised ends, and seem to ascend, though in reality its centre of gravity descends perpendicularly lower.</p></div1><div1 part="N" n="CONFIGURATION" org="uniform" sample="complete" type="entry"><head>CONFIGURATION</head><p>, the exterior surface or shape that bounds bodies, and gives them their particular figure.</p><p><hi rend="smallcaps">Configuration</hi> <hi rend="italics">of the planets,</hi> in Astrology, is a certain distance or situation of the planets in the zodiac, by which it is supposed that they assist or oppose each other.</p><p>CONFUSED <hi rend="italics">Vision.</hi> See <hi rend="smallcaps">Vision.</hi></p></div1><div1 part="N" n="CONGELATION" org="uniform" sample="complete" type="entry"><head>CONGELATION</head><p>, or <hi rend="smallcaps">Freezing</hi>, the act of fixing the fluidity of any liquid, by cold, or the application of cold bodies: in which it differs from coagulation, which is produced by other causes. See F<hi rend="smallcaps">REEZING, Frost</hi>, and <hi rend="smallcaps">Ice.</hi></p></div1><div1 part="N" n="CONGRUITY" org="uniform" sample="complete" type="entry"><head>CONGRUITY</head><p>, in Geometry, is applied to lines and figures, which exactly correspond when laid over one another; as having the same terms, or bounds. It is assumed, as an axiom, that those things are equal and similar, between which there is a congruity. Euclid, and most geometricians after him, demonstrate great part of their elements from the principle of congruity: though Leibnitz and Wolfius substitute the notion of Similitude instead of that of congruity.</p><p>CONIC <hi rend="smallcaps">Sections</hi>, are the figures made by cutting a cone by a plane.</p><p>2. According to the different positions of the cutting plane, there arise five different figures or sections, viz, a <hi rend="italics">triangle,</hi> a <hi rend="italics">circle,</hi> an <hi rend="italics">ellipse,</hi> a <hi rend="italics">parabola,</hi> and an <hi rend="italics">hyperbola:</hi> the last three of which only are peculiarly called conic sections. <cb/></p><p>3. If the cutting plane pass <figure/> through the vertex of the cone, and any part of the base, the section will evidently be a <hi rend="italics">triangle;</hi> as VAB.</p><p>4. If the plane cut the cone <figure/> parallel to the base, or make no angle with it, the section will be a <hi rend="italics">circle,</hi> as ABD.</p><p>5. The section DAB is an <hi rend="italics">el-</hi> <figure/> <hi rend="italics">lipse,</hi> when the cone is cut obliquely through both sides, or when the plane is inclined to the base in a less angle than the side of the cone is.</p><p>6. The section is a <hi rend="italics">parabola,</hi> <figure/> when the cone is cut by a plane parallel to the side, or when the cutting plane and the side of the cone make equal angles with the base.</p><p>7. The section is an <hi rend="italics">hyperbola,</hi> <figure/> when the cutting plane makes a greater angle with the base than the side of the cone makes. And if the plane be continued to cut the opposite cone, this latter section is called the opposite hyperbola to the former; as <hi rend="italics">d</hi>B<hi rend="italics">e.</hi></p><p>8. The <hi rend="italics">vertices</hi> of any section, are the points where the cutting plane meets the opposite sides of the cone, or the sides of the vertical triangular section; as A and B. —Hence, the ellipse and the opposite hyperbolas have each two vertices; but the parabola only one; unless we consider the other as at an infinite distance.</p><p>9. The <hi rend="italics">axis,</hi> or <hi rend="italics">transverse diameter</hi> of a conic section, is the line or distance AB between the vertices.— Hence the axis of a parabola is infinite in length. <figure><head>Ellipse.</head></figure> <figure><head>Oppos. Hyperb.</head></figure> <figure><head>Parabola.</head></figure></p><p>10. The <hi rend="italics">centre</hi> C is the middle of the axis.—Hence the centre of a parabola is infinitely distant from the vertex. And of an ellipse, the axis and centre lie within the curve; but of an hyperbola, without. <pb n="325"/><cb/></p><p>11. A <hi rend="italics">Diameter</hi> is any right line, as AB or DE, drawn through the centre, and terminated on each side by the curve: and the extremities of the diameter, or its intersections with the curve, are its <hi rend="italics">vertices.</hi>—Hence all the diameters of a parabola are parallel to the axis, and infinite in length; because drawn through the centre, a point at an infinite distance. And hence also every diameter of the ellipse and hyperbola have two vertices; but of the parabola, only one; unless we consider the other as at an infinite distance.</p><p>12. The <hi rend="italics">conjugate</hi> to any diameter, is the line drawn through the centre, and parallel to the tangent of the curve at the vertex of the diameter. So FG, parallel to the tangent at D, is the conjugate to DE; and HI, parallel to the tangent at A, is the conjugate to AB. —Hence the conjugate HI, of the axis AB, is perpendicular to it; but the conjugates of other diameters are oblique to them.</p><p>13. An <hi rend="italics">ordinate</hi> to any diameter, is a line parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter and curve. So DK and EL are ordinates to the axis AB; and MN and NO ordinates to the diameter DE.—Hence the ordinates of the axis are perpendicular to it; but of other diameters, the ordinates are oblique to them.</p><p>14. An <hi rend="italics">absciss</hi> is a part of any diameter, contained between its vertex and an ordinate to it; as AK or BK, and DN or EN.—Hence, in the ellipse and hyperbola, every ordinate has two abscisses; but in the parabola, only one; the other vertex of the diameter being infinitely distant.</p><p>15. The <hi rend="italics">parameter</hi> of any diameter, is a third proportional to that diameter and its conjugate.</p><p>16. The <hi rend="italics">focus</hi> is the point in the axis where the ordinate is equal to half the parameter: as K and L, where DK or EL is equal to the semiparameter.—— Hence, the ellipse and hyperbola have each two foci; but the parabola only one.—The foci, or burning points, were so called, because all rays are united or reflected into one of them, which proceed from the other focus, and are reflected from the curve. <figure/></p><p>17. If DAE, FBG be two opposite hyperbolas, having AB for their first or transverse axis, and <hi rend="italics">ab</hi> for their second or conjugate axis; and if <hi rend="italics">dae, fbg</hi> be two other opposite hyperbolas, having the same axis, but in the contrary order, viz, <hi rend="italics">ab</hi> their first axis, and AB their second; then these two latter curves <hi rend="italics">dae, fbg,</hi> are called the <hi rend="italics">conjugate hyperbolas</hi> to the two former DAE, FBG; and each pair of opposite curves mutually conjugate to the other.</p><p>18. And if tangents be drawn to the four vertices of the curves, or extremities of the axis, forming the inscribed rectangle HIKL; the diagonals HCK and ICL, of this rectangle, are called the <hi rend="italics">asymptotes</hi> of the curves.</p><p>19. <hi rend="italics">Scholium.</hi> The rectangle inscribed between the <cb/> four conjugate hyperbolas, is similar to a rectangle circumscribed about an ellipse, by drawing tangents, in like manner, to the four extremities of the two axes; also the asymptotes or diagonals in the hyperbola, are analogous to those in the ellipse, cutting this curve in similar points, and making the pair of equal conjugate diameters. Moreover, the whole figure, formed by the four hyperbolas, is, as it were, an ellipse turned inside out, cut open at the extremities D, E, F, G, of the said equal conjugate diameters, and those four points drawn out to an infinite distance, the curvature being turned the contrary way, but the axes, and the rectangle passing through their extremities, remaining fixed, or unaltered.</p><p>From the foregoing definitions are easily derived the following general corollaries to the sections. <figure><head>Ellipse.</head></figure> <figure><head>Hyperbola.</head></figure> <figure><head>Parabola.</head></figure></p><p>20. <hi rend="italics">Corol.</hi> 1. In the ellipse, the semiconjugate axis, CD or CE, is a mean proportional between CO and CP, the parts of the diameter OP of a circular section of the cone, drawn through the centre C of the ellipse, and parallel to the base of the cone. For DE is a double ordinate in this circle, being perpendicular to OP as well as to AB.</p><p>21. In like manner, in the hyperbola, the length of the semiconjugate axis, CD or CE, is a mean proportional between CO and CP, drawn parallel to the base, and meeting the sides of the cone in O and P. Or, if AO′ be drawn parallel to the side VB, and meet PC produced in O′, making CO′=CO; and on this diameter O′P a circle be drawn parallel to the base: then the semiconjugate CD or CE will be an ordinate of this circle, being perpendicular to O′P as well as to AB.</p><p>Or, in both figures, the whole conjugate axis DE is a mean proportional between QA and BR, parallel to the base of the cone. See my Conic Sections, pa. 6.</p><p>In the parabola, both the transverse and conjugate are infinite; for AB and BR are both infinite.</p><p>22. <hi rend="italics">Corol.</hi> 2. In all the sections, AG will be equal to the parameter of the axis, if QG be drawn making the angle AQG equal to the angle BAR. In like manner B<hi rend="italics">g</hi> will be equal to the same parameter, if R<hi rend="italics">g</hi> be drawn to make the angle BR<hi rend="italics">g</hi>=the angle ABQ.</p><p>23. <hi rend="italics">Corol.</hi> 3. Hence the upper hyperbolic section, or section of the opposite cone, is equal and similar to the lower one. For the two sections have the same transverse or first axis AB, and the same conjugate or second axis DE, which is the mean proportional between AQ and RB; and they have also equal parame- <pb n="326"/><cb/> ters AG, B<hi rend="italics">g.</hi> So that the two opposite sections make, as it were, but the two opposite ends of one entire section or hyperbola, the two being every where mutually equal and similar. Like the two halves of an ellipse, with their ends turned the contrary way.</p><p>24. <hi rend="italics">Corol.</hi> 4. And hence, although both the transverse and conjugate axis in the parabola be infinite, yet the former is infinitely greater than the latter, or has an insinite ratio to it. For the transverse has the same ratio to the conjugate, as the conjugate has to the parameter, that is, as an infinite to a finite quantity, which is an infinite ratio.</p><p>The peculiar properties of each particular curve, will be best referred to the particular words <hi rend="smallcaps">Ellipse</hi>, H<hi rend="smallcaps">YPERBOLA, Parabola;</hi> and therefore it will only be proper here to lay down a few of the properties that are common to all the conic sections. <hi rend="center"><hi rend="italics">Some other General Properties.</hi></hi></p><p>25. From the foregoing desinitions, &c, it appears, that the conic sections are in themselves a system of regular curves, naturally allied to each other; and that one is changed into another perpetually, when it is either increased, or diminished, in infinitum. Thus, the curvature of a circle being ever so little increased or diminished, passes into an ellipse; and again, the centre of the ellipse going off infinitely, and the curvature being thereby diminished, is changed into a parabola; and lastly, the curvature of a parabola being ever so little changed, there ariseth the first of the hyperbolas; the innumerable species of which will all of them arise orderly by a gradual diminution of the curvature; till this quite vanishing, the last hyperbola ends in a right line. From whence it is manifest, that every regular curvature, like that of a circle, from the circle itself to a right line, is a conical curvature, and is distinguished with its peculiar name, according to the divers degrees of that curvature.</p><p>26. That all diameters in a circle and ellipse intersect one another in the centre of the figure within the section: that in the parabola they are all parallel among themselves, and to the axis: but in the hyperbola, they intersect one another, without the figure, in the common centre of the opposite and conjugate sections.</p><p>27. In the circle, the <hi rend="italics">latus rectum,</hi> or parameter, is double the distance from the vertex to the focus, which is also the centre. But in ellipses, the parameters are in all proportions to that distance, between the double and quadruple, according to their different species. While, in the parabola, the parameter is just quadruple that distance. And, lastly in hyperbolas, the parameters are in all proportions beyond the quadruple, according to their various kinds.</p><p>28. The first general property of the conic sections, with regard to the abscisses and ordinates of any diameter, is, that the rectangles of the abscisses are to each other, as the squares of their corresponding ordinates. Or, which is the same thing, that the square of any diameter is to the square of its conjugate, as the rectangle of two abscisses of that diameter, to the square of the ordinate which divides them. That is, in all the figures, the rect. AC. CB: rect. AE. EB :: CD<hi rend="sup">2</hi> : EF<hi rend="sup">2</hi> : <cb/> <figure/> But as, in the parabola the infinites CB and EB are in a ratio of equality, for this curve the same property becomes AC : AE :: CD<hi rend="sup">2</hi> : EF<hi rend="sup">2</hi>, that is, in the parabola, the abscisses are as the squares of their ordinates.</p><p>Or, when one of the ordinates is the semiconjugate GH, dividing the diameter equally in the centre, the same general property becomes, AG . GB or AG<hi rend="sup">2</hi> : AC . CB :: GH<hi rend="sup">2</hi> : CD<hi rend="sup">2</hi>, or AB<hi rend="sup">2</hi> : HI<hi rend="sup">2</hi> :: AC . CB : CD<hi rend="sup">2</hi>.</p><p>29. From hence is derived the equation of the curves of the conic sections; thus, putting the diameter AB =<hi rend="italics">d,</hi> its conjugate HI=<hi rend="italics">c,</hi> abscrss AC=<hi rend="italics">x,</hi> and its ordinate CD=<hi rend="italics">y;</hi> then is the other absciss CB=<hi rend="italics">d</hi>-<hi rend="italics">x</hi> in the ellipse, or <hi rend="italics">d</hi>+<hi rend="italics">x</hi> in the hyperbola, or <hi rend="italics">d</hi> in the parabola; and hence the last analogy above, becomes <hi rend="italics">y</hi><hi rend="sup">2</hi>=((<hi rend="italics">c</hi><hi rend="sup">2</hi>)/<hi rend="italics">d</hi>) <hi rend="italics">x</hi> or=<hi rend="italics">px</hi> in the parabola, where the parameter <hi rend="italics">p</hi>=((<hi rend="italics">c</hi><hi rend="sup">2</hi>)/<hi rend="italics">d</hi>) the third proportional to the diameter and its conjugate, by the definition of it.</p><p>And from this one general proposition alone, which is easily derived from the section in the solid cone itself, together with the definitions only, as laid down above, all the other properties of all the sections may easily be derived, without any farther reference to the cone, and without mechanical descriptions of the curves in plano; as is done in my Treatise on Conic Sections, for the use of the Royal Mil. Acad.; in which also all the similar propositions in the ellipse and hyperbola are carried on word for word in them both.</p><p>The more ancient mathematicians, before the time of Apollonius Pergæus, admitted only the right cone into their geometry, and they supposed the section of it to be made by a plane perpendicular to one of its sides; and as the vertical angle of a right cone may be either right, acute, or obtuse, the same method of cutting these several cones, viz, by a plane perpendicular to one side, produced all the three conic sections. The parabola was called the section of a right-angled cone; the ellipse, the section of the acute-angled cone; and the hyperbola, the section of the obtuse-angled cone. But Apollonius, who, on account of his writings on this <pb n="327"/><cb/> subject, obtained the appellation of <hi rend="italics">Magnus Geometra,</hi> the Great Geometrician, observed, that these three sections might be obtained in every cone, both oblique and right, and that they depended on the different inclinations of the plane of the section to the cone itself. Apollon. Con. Halley's edit. lib. 1, p. 9.</p><p>Instead of considering these curves as sections cut from the solid cone, which is the true genuine way of all the ancients, and of the most elegant writers among the moderns, Descartes, and some others of the moderns, have given arbitrary constructions of curves on a plane, from which constructions they have demonstrated the properties of these, and have afterwards proved that some principal property of them belongs to such curves or sections as are cut from a cone; and hence it is inferred by them that those curves, so described on a plane, are the same with the conic sections.</p><p>The doctrine of the conic sections is of great use in physical and geometrical astronomy, as well as in the physico-mathematical sciences. The doctrine has been much cultivated by both ancient and modern geometricians, who have left many good treatises on the subject. The most ancient of these is that of Apollonius Pergæus, containing 8 books, the first 4 of which have often been published; but Dr. Halley's edition has all the eight. Pappus, in his Collect. Mathem. lib. 7, says that the first four of these were written by Euclid, though perfected by Appollonius, who added the other 4 to them. Among the moderns, the chief writers are Mydorgius de Sectionibus Conicis; Gregory St. Vincent's Quadratura Circuli & Sectionum Coni; De la Hire de Sectionibus Conicis; Trevigar Elem. Section. Con.; De Witt's Elementa Curvarum; Dr. Wallis's Conic Sections; De l'Hospital's Anal. Treat. of Conic Sections; Dr. Simson's Section. Con.; Milne's Elementa Section. Conicarum; Muller's Conic Sections; Steel's Conic Sections; Dr. Hamilton's elegant treatise; my own treatise, above cited; and at the writing of this, my friend Mr. Abram Robertson of Oxford is preparing a curious work on this subject, containing at the same time a treatise on the science, and a history of the writings relating to it.</p></div1><div1 part="N" n="CONICS" org="uniform" sample="complete" type="entry"><head>CONICS</head><p>, that part of the higher geometry, or geometry of curves, which considers the cone, and the several curve lines arising from the sections of it.</p><p>CONJUGATE <hi rend="italics">Axis,</hi> or <hi rend="italics">Diameter,</hi> in the Conic Sections, is the axis, or a diameter parallel to a tangent to the curve at the vertex of another axis, or diameter, to which that is a conjugate. Indeed the two are mutually conjugates to each other, and each is parallel to the tangent at the vertex of the other.</p><p><hi rend="smallcaps">Conjugate Hyperbolas</hi>, also called <hi rend="italics">Adjacent Hyperbolas,</hi> are such as have the same axes, but in the contrary order, the first or principal axis of the one being the 2d axis of the other, and the 2d axis of the former, the 1st axis of the latter. See art. 17 of <hi rend="smallcaps">Conic</hi> S<hi rend="smallcaps">ECTIONS.</hi></p></div1><div1 part="N" n="CONJUNCTION" org="uniform" sample="complete" type="entry"><head>CONJUNCTION</head><p>, in Astronomy, is the meeting of the stars and planets in the same point or place in the heavens; and is either true or apparent.</p><p><hi rend="italics">True</hi> <hi rend="smallcaps">Conjunction</hi> is when the line drawn through the centres of the two stars passes also through the centre of the earth. And <hi rend="italics">Apparent</hi> <hi rend="smallcaps">Conjunction</hi> is when that line does not pass through the earth's centre. <cb/></p></div1><div1 part="N" n="CONOID" org="uniform" sample="complete" type="entry"><head>CONOID</head><p>, is a figure resembling a cone, except that the slant sides from the base to the vertex are not straight lines as in the cone, but curved. It is generated by the revolution of a conic section about its axis; and it is therefore threefold, answering to the three sections of the cone, viz, the <hi rend="italics">Elliptical Conoid,</hi> or spheroid, the <hi rend="italics">Hyperbolic Conoid,</hi> and the <hi rend="italics">Parabolic Conoid.</hi></p><p>If a conoid be cut by a plane in any position, the section will be of the figure of some one of the conic sections; and all parallel sections, of the same conoid, are like and similar figures. When the section of the solid returns into itself, it is an ellipse; which is always the case in the sections of the spheroid, except when it is perpendicular to the axis; which position is also to be excepted in the other solids, the section being always a circle in that position. In the parabolic conoid, the section is always an ellipse, except when it is parallel to the axis. And in the hyperbolic conoid, the section is an ellipse, when its axis makes with the axis of the solid, an angle greater than that made by the said axe of the solid and the asymptote of the generating hyperbola; the section being an hyperbola in all other cases, but when those angles are equal, and then it is a parabola.</p><p>But when the section is parallel to the fixed axis, it is of the same kind with, and similar to the generating plane itself; that is, the section parallel to the axis, in the spheroid, is an ellipse similar to the generating ellipse; in the parabolic conoid it is a parabola, sunilar to the generating one; and in the hyperbolic conoid, it is an hyperbola similar to the generating one.</p><p>The section through the axis, which is the generating plane, is, in the spheroid the greatest of the parallel sections, but in the hyperboloid it is the least, and in the paraboloid those parallel sections are all equal.</p><p>The analogy of the sections of the hyperboloid to those of the cone, are very remarkable, all the three conic sections being formed by cutting an hyperboloid in the same positions as the cone is cut. Thus, let an hyperbola and its asymptote be revolved together about the transverse axis, the sormer describing an hyperboloid, and the latter a cone circumscribing it: then let it be supposed that they are both cut by one plane in any position; so shall the two sections be like, similar, and concentric figures: that is, if the plane cut both the sides of each, the sections will be concentric and similar ellipses; but if the cutting plane be parallel to the asymptote, or to the side of the cone, the sections will be parabolas; and in all other positions, the sections will be similar and concentric hyperbolas.</p><p>And this analogy of the sections will not seem strange, when it is considered that a cone is a species of the hyperboloid; or a triangle a species of the hyperbola, the axes being infinitely small. See my Mensuration, prop. 1, part 3, sect. 4, pag. 265 edit. 8vo.</p></div1><div1 part="N" n="CONON" org="uniform" sample="complete" type="entry"><head>CONON (<hi rend="italics">of</hi> <hi rend="smallcaps">Samos</hi>)</head><p>, a respectable mathematician and philosopher, who flourished about the 130th olympiad, being a contemporary and friend of Archimedes, to whom Conon communicated his writings, and sent him some problems, which Archimedes received with approbation, saying they ought to be published while Conon was living, for he comprehends them with ease, and can give a proper demonstration of them.</p><p>At another time he laments the loss of Conon, thus <pb n="328"/><cb/> admiring his genius. “How many theorems in geometry, says he, which at first seemed impossible, would in time have been brought to persection! Alas! Conon, though he invented many, with which he enriched geometry, had not time to perfect them, but left many in the dark, being prevented by death.” He had an uncommon skill in mathematics, joined to an extraordinary patience and application. This is farther confirmed by a letter sent to Archimedes by a friend of Conon's. “Having heard of Conon's death, with whose friendship I was honoured, and with whom you kept an intimate correspondence; as he was thoroughly versed in geometry, I greatly lament the loss of a sincere friend, and a person of surprising knowledge in mathematics. I then determined to send to you, as I had before done to him, a theorem in geometry, hitherto observed by no one.”</p><p>Conon had some disputes with Nicoteles, who wrote against him, and treated him with too much contempt. Apollonius confesses it; though he acknowledges that Conon was not fortunate in his demonstrations.</p><p>Conon invented a kind of volute, or spiral, different from that of Dynostratus; but because Archimedes explained the properties of it more clearly, the name of the inventor was forgotten, and it was hence called Archimedes's volute or spiral.</p><p>As to Conon's astrological or astronomical knowledge, it may in some measure be gathered from the poem of Catullus, who describes it in the beginning of his verses on the hair of Berenice, the sister and wife of Ptolomy Euergetes, upon the occasion of Conon having given out that it was changed into a constellation among the stars, to console the queen for the loss, when it was stolen out of the temple, where she had consecrated it to the gods.</p></div1><div1 part="N" n="CONSECTARY" org="uniform" sample="complete" type="entry"><head>CONSECTARY</head><p>, or <hi rend="italics">Corollary,</hi> a consequence deduced from some foregoing principles.</p></div1><div1 part="N" n="CONSEQUENT" org="uniform" sample="complete" type="entry"><head>CONSEQUENT</head><p>, is the latter of the two terms of a ratio; or that to which the antecedent is referred and compared. Thus, in the ratio <hi rend="italics">a : b,</hi> or <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> the latter term <hi rend="italics">b</hi> is the consequent, and <hi rend="italics">a</hi> is the antecedent.</p><p>CONSISTENT <hi rend="italics">Bodies,</hi> is a term much used by Mr. Boyle, for such as are usually called <hi rend="italics">firm,</hi> or <hi rend="italics">fixed bodies;</hi> in opposition to <hi rend="italics">sluid</hi> ones.</p></div1><div1 part="N" n="CONSOLE" org="uniform" sample="complete" type="entry"><head>CONSOLE</head><p>, in Architecture, is an ornament cut upon the key of an arch, having a projecture or jetting, and occasionally serving to support small cornices, busts, and bases.</p></div1><div1 part="N" n="CONSONANCE" org="uniform" sample="complete" type="entry"><head>CONSONANCE</head><p>, in Music, is commonly used in the same sense with <hi rend="italics">concord,</hi> viz, for the union or agreement of two sounds produced at the same time, the one grave, the other acute, which is compounded together by such a proportion of each, as proves agreeable to the ear.</p><p>An unison is the first consonance, an eighth is the 2d, a fifth is the 3d; and then follow the fourth, with the third and sixths, major and minor.</p><p>CONSTANT <hi rend="italics">Quantities</hi> are such as remain invariably the same, while others increase or decrease. Thus, the diameter of a circle is a constant quantity; for it remains the same while the abscisses and ordinates, or the sines, tangents, &c, are variable.</p><p>These are sometimes called <hi rend="italics">given,</hi> or <hi rend="italics">invariable</hi> or <hi rend="italics">per-</hi> <cb/> <hi rend="italics">manent</hi> quantities; and in algebra it is now usual to represent them by the leading letters of the alphabet, <hi rend="italics">a, b, c,</hi> &c; while the variable ones are denoted by the last latters, <hi rend="italics">z, y, x,</hi> &c.</p></div1><div1 part="N" n="CONSTELLATIONS" org="uniform" sample="complete" type="entry"><head>CONSTELLATIONS</head><p>, certain imaginary sigures of birds, beasts, fishes, and other things in the heavens, within which are arranged certain stars. These assemblages are also sometimes called asterisms.</p><p>The ancients portioned out the firmament into several parts, or constellations; reducing a certain number of stars under the representation of certain images, to assist the imagination and memory, to conceive or retain their number, order, and disposition, or even to distinguish the virtues they attributed to them.</p><p>The division of the heavens into constellations is very ancient; being known to the most early authors, whether sacred or profane. In the book of Job the names of some of them are mentioned; witness that sublime expostulation, <hi rend="italics">Canst thou restrain the sweet influence of the</hi> Pleiades, <hi rend="italics">or loosen the bands of</hi> Orion? And the same may be observed of the oldest among the heathen writers, Hesiod and Homer.</p><p>The division of the ancients took in only the visible firmament, or so much as came under their notice, as visible to the naked eye. The first or earliest of these, is contained in the catalogue of Ptolomy, given in the 7th book of his Almagest, prepared, as he assures us, from his own observations, compared with those of Hipparchus, and the other ancient astronomers. In this catalogue Ptolomy has formed 48 constellations. Of these, 12 are about the ecliptic, commonly called the 12 signs; 21 to the north of it; and 15 to the south. The northern constellations are, the Little Bear, the Great Bear, the Dragon, Cepheus, Bootes, the Northern Crown, Hercules, the Harp, the Swan, Cassiopeia, Perseus, Auriga, Ophiucus or Serpentary, the Serpent, the Arrow, the Eagle, the Dolphin, the Horse, Pegasus, Andromeda, and the Triangle.</p><p>The constellations about the ecliptic are Aries, Taurus, Gemini, Cancer, Leo, Virgo, Libra, Scorpio, Sagittarius, Capricorn, Aquarius, and Pisces: or according to the English names, the Ram, the Bull, the Twins, the Crab, the Lion, the Virgin, the Balance, the Scorpion, the Archer, the Goat, the Water bearer, and the Fishes.</p><p>The Southern constellations are, the Whale, Orion, the Eridanus, the Hare, the Great Dog, the Little Dog, the Ship, the Hydra, the Cup, the Raven, the Centaur, the Wolf, the Altar, the Southern Crown, and the Southern Fish.</p><p>The other stars not comprehended under these constellations, yet visible to the naked eye, the ancients called <hi rend="italics">informes,</hi> or <hi rend="italics">sporades,</hi> some of which the modern astronomers have since reduced into new figures, or constellations. Ptolomy has set down the longitude and latitude of all these stars to about the year of Christ 137, amounting to the number of 1022, viz, <table><row role="data"><cell cols="1" rows="1" role="data">in the northern constellations</cell><cell cols="1" rows="1" rend="align=right" role="data">360</cell></row><row role="data"><cell cols="1" rows="1" role="data">in the zodiacal constellations</cell><cell cols="1" rows="1" rend="align=right" role="data">346</cell></row><row role="data"><cell cols="1" rows="1" role="data">in the southern constellations</cell><cell cols="1" rows="1" rend="align=right" role="data">316</cell></row><row role="data"><cell cols="1" rows="1" role="data">in all of Ptolomy's catalogue</cell><cell cols="1" rows="1" rend="align=right" role="data">1022</cell></row></table></p><p>Among the modern astronomers, Tycho Brahe is the <pb n="329"/><cb/> first who determined, with exactness, and in consequence of his own observations, the long. and lat. of the fixed stars, out of which he formed 45 constellations; of these, 43 were of the old ones described by Ptolomy, to which Tycho added the Coma Berenices, and Antinous; but he omits 5 of the old southern constellations, viz, the Centaur, the Wolf, the Altar, the Southern Crown, and Southern Fish; which he could not observe, because of the high northern latitude of Uranibourg.</p><p>After Tycho, Bayer gave the figures of 60 constellations, very exactly represented, and with tables annexed, having added, to the 48 old ones of Ptolomy, the following 12 about the south pole, viz, the Peacock, the Toucan, the Crane, the Phœnix, the Dorado, the Flying Fish, the Hydra, the Chameleon, the Bee, the Bird of Paradise, the Triangle, and the Indian. Besides accurately distinguishing the relative size and the situation of every star, Bayer marks the stars in each constellation with the letters of the Greek and Roman alphabets, setting the first letter <foreign xml:lang="greek">a</foreign> to the first or principal star in each constellation, <foreign xml:lang="greek">b</foreign> to the 2d in order, <foreign xml:lang="greek">g</foreign> to the 3d, and so on; a very useful method of noting and describing the stars, which has been used by all astronomers since, and who have farther enlarged this method, by adding the ordinal numbers 1, 2, 3, &c, to the other stars discovered since his time, when any constellation contains more than can be marked by the two alphabets. The number and order of the stars, as mentioned by Bayer, are, <table><row role="data"><cell cols="1" rows="1" role="data">of the 1st magnitude</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell></row><row role="data"><cell cols="1" rows="1" role="data">of the 2d magnitude</cell><cell cols="1" rows="1" rend="align=right" role="data">63</cell></row><row role="data"><cell cols="1" rows="1" role="data">of the 3d magnitude</cell><cell cols="1" rows="1" rend="align=right" role="data">196</cell></row><row role="data"><cell cols="1" rows="1" role="data">of the 4th magnitude</cell><cell cols="1" rows="1" rend="align=right" role="data">415</cell></row><row role="data"><cell cols="1" rows="1" role="data">of the 5th magnitude</cell><cell cols="1" rows="1" rend="align=right" role="data">348</cell></row><row role="data"><cell cols="1" rows="1" role="data">of the 6th magnitude</cell><cell cols="1" rows="1" rend="align=right" role="data">341</cell></row><row role="data"><cell cols="1" rows="1" role="data">of the unformed stars</cell><cell cols="1" rows="1" rend="align=right" role="data">326</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">in all</cell><cell cols="1" rows="1" rend="align=right" role="data">1706</cell></row></table></p><p>After Bayer, a catalogue, with new constellations, was published by Schiller, in 1627, in a work called Coelum Stellatum Christianum, the Christian Starry Heaven, in which he substitutes, very improperly, other sigures of the constellations, and names, taken from the sacred scriptures, instead of the old ones.</p><p>In the year 1665, Riccioli published his Astronomy Reformed, containing a catalogue of the stars in 62 constellations, viz, the 60 of Bayer, with the Coma Berenices and Antinous of Tycho. He distributes the stars in all the constellations into four classes. In the first of these classes are contained those stars determined by his own observations, and those of Grimaldi. In the second are those stars which had been ascertained by Tycho Brahe and Kepler. In the 3d are the stars determined by Hipparchus and Ptolomy. And the 4th class consists of those of the southern hemisphere discovered by Navigators, who have ascertained their places in a more or less accurate manner; in which he has marked the longitudes and latitudes for the year 1700, the period to which he has reduced all his observations. This catalogue was followed by a number of celestial schemes and maps of the heavens, published in 1673 by Pardies, who has represented very carefully all the constellations, with the stars they contain. After this, Vi- <cb/> talis published a catalogue of the sixed stars in his Tables of the Primum Mobile, in which their longitudes and latitudes, with the right-ascensions and declinations are set down for the year 1675.</p><p>Some time after this, Royer published maps of the heavens, reduced into 4 tables, with a catalogue of the fixed stars for the year 1700. To the stars marked by Bayer, he adds a number of stars not before seen, with others taken from the tables of Riccioli, and not mentioned by Bayer: he also forms, out of the unformed stars, eleven other constellations. Five of these are to the north, and are called the Giraffe, the River Jordan, the River Tigris, the Sceptre, and the Flower-de-luce; with 6 on the south part, which are the Dove, the Unicorn, the Cross, the Great Cloud, the Little Cloud, and the Rhomboide. To this work Royer has joined the catalogue of the southern stars observed by Dr. Halley at the island of St. Helena.</p><p>Hevelius has also improved upon the labours of those who went before him, and collected together several stars of the before unformed class into some new constellations. These are, the Unicorn, the Camelopardalis, described by Bartschius, the Sextant of Urania, the Dogs, the Little Lion, the Lynx, the Fox and Goose, the Sobieski's Crown, the Lizard, the Little Triangle, and the Cerberus; to which Gregory has added the Ring and the Armilla. Some of these new constellations however answer to those of Royer, as the Camelopardal to the Giraffe, the Dogs to the River Jordan, and the Fox to the River Tigris. The latitudes and longitudes are added for the year 1700.</p><p>Finally, Flamsteed has given a catalogue of the fixed stars, not only much more correct, but much larger than those of all that went before him. He has set down the longitude, latitude, right ascension, and polar distance of 2934 stars, as they were at the beginning of 1690, all determined from his own observations. He distinguishes all the stars into seven classes, or orders of magnitude, distinguishing those of Bayer by his letters, and marking their variation in right ascension, for shewing their situation in the succeeding years. See the term <hi rend="smallcaps">Catalogue.</hi></p><p>This catalogue was followed by an Atlas Cœlestis, published at London in the year 1729, describing, in several schemes, the figures of the constellations seen in our hemisphere, with the exact position of the fixed stars, with respect to the circles of the sphere, as resulting from the last catalogue corrected by Flamsteed. And still later observations, made with farther improved telescopes, have greatly enlarged the number and accuracy of the stars; but the number of the constellations remains the same as above described, except that an attempt has lately been made by Dr. Hill to add to the list 14 new ones, formed out of more of the clusters of unformed stars.</p><p>Beside the literal marks of the stars introduced by Bayer, it is usual also to distinguish them by that part of the constellation in which they are placed; and many of them again have their peculiar names; as Arcturus, between the knees of Bootes; Gemina, or Lucida, in the Corona Septentrionalis, or Northern Crown; Palilitium, or Aldebaran, in the Bull's eye, Pleiades in his neck, and Hyades in his forehead; Castor and Pollux in the heads of Gemini; Capella, with the Hœdi in the <pb n="330"/><cb/> shoulder of Auriga; Regulus, or Cor Leonis, the Lion's Heart; Spica Virginis in the hand, and Vindemiatrix in the shoulder of Virgo; Antares, or Cor Scorpionis, the Scorpion's Heart; Fomalhaut, in the mouth of Piscis Australis, or Southern Fish; Regel, in the foot of Orion; Sirius, in the mouth of Canis Major, the Great Dog; Procyon, in the back of Canis Minor, the Little Dog; and the Pole Star, the last in the tail of Ursa Minor, the Little Bear.</p><p>The Greek and Roman poets, from the ancient theology, give wild and romantic fables about the origin of the constellations, probably derived from the hieroglyphics of the Egyptians, and transmitted, with some alterations, from them to the Greeks, who probably obscured them greatly with their own fables. See Hyginus's Poeticon Astron.; Riccioli Almagest. lib. 6. cap. 3, 4, 5; Shelburne's Notes upon Manilius; Bailly's Antient Astronomy; and Gebelin's Monde Primitif, vol. 4: from the whole of which it is made probable, that the invention of the signs of the zodiac, and probably of most of the other constellations of the sphere, is to be ascribed to some very ancient nation, inhabiting the northern temperate zone, probably what is now called Tartary, or the parts to the northward of Persia and China; and from thence transmitted through China, India, Babylon, Arabia, Egypt, Greece, &c.</p><p>It is a very probable conjecture, that the figures of the signs in the zodiac, are descriptive of the seasons of the year, or months, in the sun's path: thus, the first sign Aries, denotes, that about the time when the sun enters that part of the ecliptic, the lambs begin to follow the sheep; that on the sun's approach to the 2d constellation, Taurus, the Bull, is about the time of the cows bringing forth their young. The third sign, now Gemini, was originally two kids, and signified the time of the goats bringing forth their young, which are usually two at a birth, while the former, the sheep and cow, commonly produce only one. The 4th sign, Cancer, the Crab, an animal that goes side-ways and backwards, was placed at the northern solstice, the point where the sun begins to return back again from the north to the southward. The 5th sign, Leo, the Lion, as being a very furious animal, was thought to denote the heat and fury of the burning sun, when he has left Cancer, and entered the next sign Leo. The succeeding constellation, the 6th in order, received the sun at the time of ripening corn and approaching harvest; which was aptly expressed by one of the female reapers, with an ear of corn in her hand; viz, Virgo the maid. The ancients gave to the next sign Scorpio, two of the 12 divisions of the zodiac: Autumn, which affords fruits in great abundance, affords the means and causes of diseases, and the succeeding time is the most unhealthy of the year; expressed by this venemous animal, here spreading out his long claws into the one sign, as threatening mischief, and in the other brandishing his tail to denote the completion of it. The fall of the leaf was the season of the ancient hunting; for which reason the stars which marked the sun's place at this season, into the constellation Sagittary, a huntsman with his arrows and his club, the weapons of destruction for the large creatures he pursued. The reason of the Wild Goat's being chosen to mark the southern sol- <cb/> stice, when the sun has attained his extreme limit that way, and begins to return and mount again to the northward, is obvious enough; the character of that animal being, that it is mostly climbing, and ascending some mountain as it browzes. There yet remain two of the signs of the zodiac to be considered with regard to their origin, viz, Aquarius and Pisces. As to the former, it is to be considered that the winter is a wet and uncomfortable season; this therefore was expressed by Aquarius, the figure of a man pouring out water from an urn. The last of the zodiacal constellations was Pisces, a couple of fishes, tied together, that had been caught: The lesson was, the severe season is over, your flocks do not yet yield their store; but the seas and rivers are open, and there you may take fish in abundance.</p><p>Through a vain and blind zeal, rather than through any love for the science, some persons have been moved to alter either the figures of the constellations, or their names. Thus, venerable Bede, instead of the profane names and figures of the twelve zodiacal constellations, substituted those of the 12 apostles; which example was followed by Schiller, who completed the reformation, and gave scripture names to all the constellations in the heavens. Thus, Aries, or the Ram, was changed into Peter; Taurus, or the Bull, into St. Andrew; Andromeda, into the Sepulchre of Christ; Lyra, into the Manger of Christ; Hercules, into the Magi coming from the East; the Great Dog, into David; and so on. And Weigelius, professor of Mathematics in the uni versity of Jena, made a new order of constellations; changing the firmament into a Cœlum Heraldicum; and introducing the arms of all the princes in Europe, by way of constellations. Thus Ursa major, the Great Bear, he transformed into the elephant of the kingdom of Denmark; the Swan, into the Ruta with swords of the House of Saxony; Ophiuchus, into the Cross of Cologne; the Triangle, into Compasses, which he calls the Symbol of Artificers; and the Pleiades into the Abacus Pythagoricus, which he calls that of merchants; &c.</p><p>But the more judicious among astronomers never approved of such innovations; as they only tend to introduce confusion into astronomy. The old constellations are therefore still retained; both because better could not be substituted, and likewise to keep up the greater correspondence and uniformity between the old astronomy and the new. See <hi rend="smallcaps">Catalogue.</hi></p></div1><div1 part="N" n="CONSTRUCTION" org="uniform" sample="complete" type="entry"><head>CONSTRUCTION</head><p>, in Geometry, the art or manner of drawing or describing a figure, scheme, the lines of a problem, or such like.</p><p><hi rend="smallcaps">Construction</hi> <hi rend="italics">of Equations,</hi> in Algebra, is the finding the roots or unknown quantities of an equation, by geometrical construction of right lines or curves; or the reducing given equations into geometrical figures. And this is effected by lines or curves according to the order or rank of the equation.</p><p>The roots of any equation may be determined, that is, the equation may be constructed, by the intersections of a straight line with another line or curve of the same dimensions as the equation to be constructed: for the roots of the equation are the ordinates of the curve at the points of intersection with the right line; and it is well known that a curve may be cut by a right line in as many points as its dimensions amount to. Thus, <pb n="331"/><cb/> then, a simple equation will be constructed by the intersection of one right line with another: a quadratic equation, or an affected equation of the 2d rank, by the intersections of a right line with a circle, or any of the conic sections, which are all lines of the 2d order; and which may be cut, by the right line, in two points, thereby giving the two roots of the quadratic equation. A cubic equation may be constructed by the intersection of the right line with a line of the 3d order: and so on.</p><p>But if, instead of the right line, some other line of a higher order be used; then the 2d line, whose intersections with the former are to determine the roots of the equation, may be taken as many dimensions lower, as the former is taken higher. And, in general, an equation of any height will be constructed by the intersections of two lines whose dimensions, multiplied together, produce the dimension of the given equation. Thus, the intersections of a circle with the conic sections, or of these with each other, will construct the biquadratic equations, or those of the 4th power, because ; and the intersections of the circle or conic sections with a line of the 3d order, will construct the equations of the 5th and 6th power; and so on.—For example,</p><p><hi rend="italics">To construct a Simple Equation.</hi> This is done by resolving the given simple equation into a proportion, or finding a third or 4th proportional, &c. Thus, 1. If the equation be <hi rend="italics">ax</hi> = <hi rend="italics">bc</hi>; then , the fourth proportional to <hi rend="italics">a, b, c.</hi></p><p>2. If <hi rend="italics">ax</hi> = <hi rend="italics">b</hi><hi rend="sup">2</hi>; then , a third proportional to <hi rend="italics">a</hi> and <hi rend="italics">b.</hi></p><p>3. If ; then, since , it will be , a fourth proportional to <hi rend="italics">a, b</hi> + <hi rend="italics">c</hi> and <hi rend="italics">b</hi> - <hi rend="italics">c.</hi></p><p>4. If ; then construct <figure/> theright-angled triangle ABC, whose base is <hi rend="italics">b,</hi> and perpendicular is <hi rend="italics">c,</hi> so shall the square of the hypothenuse be <hi rend="italics">b</hi><hi rend="sup">2</hi> + <hi rend="italics">c</hi><hi rend="sup">2</hi>, which call <hi rend="italics">h</hi><hi rend="sup">2</hi>; then the equation is <hi rend="italics">ax</hi> = <hi rend="italics">b</hi><hi rend="sup">2</hi>, and <hi rend="italics">x</hi> = <hi rend="italics">h</hi><hi rend="sup">2</hi>/<hi rend="italics">a</hi> a third proportional to <hi rend="italics">a</hi> and <hi rend="italics">h.</hi> <hi rend="center"><hi rend="italics">To construct a Quadratic Equation.</hi></hi></p><p>1. If it be a simple quadratic, it may be reduced to this form <hi rend="italics">x</hi><hi rend="sup">2</hi> = <hi rend="italics">ab</hi>; and hence <hi rend="italics">a : x :: x : b,</hi> or <hi rend="italics">x</hi> = √<hi rend="italics">ab</hi> a mean proportional between <hi rend="italics">a</hi> and <hi rend="italics">b.</hi> Therefore upon a straight line take <figure/> AB = <hi rend="italics">a,</hi> and BC = <hi rend="italics">b</hi>; then upon the diameter AC describe a semicircle, and raise the perpendicular BD to meet it in D; so shall BD be = <hi rend="italics">x</hi> the mean proportional sought between AB and BC, or between <hi rend="italics">a</hi> and <hi rend="italics">b.</hi></p><p>2. If the quadratic be affected, let it first be ; then form the right-angled triangle whose base AB is <hi rend="italics">a,</hi> and perpendicular BC is <hi rend="italics">b</hi>; and with the centre A and radius AC describe the semi- <cb/> circle DCE; so shall DB and BE be the two roots of the given quadratic equation .</p><p>3. If the quadratic be , then the construction will be the very same as of the preceding one . <figure/></p><p>4. But if the form be : form a rightangled triangle whose hypothenuse FG is <hi rend="italics">a,</hi> and perpendicular GH is <hi rend="italics">b</hi>; then with the radius FG and centre F describe a semi-circle IGK; so shall IH and HK be the two roots of the given equation , or . See Maclaurin's Algebra, part 3, cap. 2, and Simpson's Algebra, pa. 267.</p><p><hi rend="italics">To construct Cubic and Biquadratic Equations.</hi>— These are constructed by the intersections of two conic sections; for the equation will rise to 4 dimensions, by which are determined the ordinates from the 4 points in which these conic sections may cut one another; and the conic sections may be assumed in such a manner, as to make this equation coincide with any proposed biquadratic: so that the ordinates from these 4 intersections will be equal to the roots of the proposed biquadratic. When one of the intersections of the conic section falls upon the axis, then one of the ordinates vanishes, and the equation, by which these ordinates are determined, will then be of 3 dimensions only, or a cubic; to which any proposed cubic equation may be accommodated. So that the three remaining ordinates will be the roots of that proposed cubic. The conic sections for this purpose should be such as are most easily described; the circle may be one, and the parabola is usually assumed for the other.</p><p>Vieta, in his Canonica Recensione Effectionum Geometricarum, and Ghetaldus, in his Opus Posthumum de Resolutione & Compositione Mathematica, as also Des Cartes, in his Geometria, have shewn how to construct simple and quadratic Equations. Des Cartes has also shewn how to construct cubic and biquadratic equations, by the intersection of a circle and a parabola: And the same has been done more generally by Baker in his Clavis Geometrica, or Geometrical Key. But the genuine foundation of all these constructions was first laid and explained by Slusius in his Mesolabium, part 2. This doctrine is also pretty well handled by De la Hire, in a small treatise, called La Construction des Equations Analytiques, annexed to his Conic Sections. Newton, at the end of his Algebra, has given the construction of cubic and biquadratic equations mechanically; as also by the conchoid and cissoid, as well as the conic sections. See also Dr. Halley's Construction of Cubic and Biquadratic Equations; Colson's, in the Philos. Trans.; the Marquis de l'Hospital's Traité Analytique des Sections Coniques; Maclaurin's Algebra, part 3, c. 3 &c.</p></div1><div1 part="N" n="CONTACT" org="uniform" sample="complete" type="entry"><head>CONTACT</head><p>, the relative state of two things that touch each other, but without cutting or entering; or whose surfaces join to each other without any interstice. <pb n="332"/><cb/></p><p>The contact of curve lines or surfaces, with either straight or curved ones, is only in points; and yet these points have different proportions to one another, as is shewn by Mr. Robartes, in the Philos. Trans. vol. 27 pa. 470; or Abr. vol. 4. pa. 1.</p><p>Because few or no surfaces are capable of touching in all points, and the cohesion of bodies is in proportion to their contact, those bodies will adhere fastest together, that are capable of the greatest contact.</p><p><hi rend="italics">Angle of</hi> <hi rend="smallcaps">Contact</hi> is the opening <figure/> between a curve line and a tangent to it, particularly the circle and its tangent; as the angle formed at A between BA and AC, at the point of contact A.</p><p>It is demonstrated by Euclid, that the line CA standing perpendicular to the radius DA, touches the circle only in one point: and that no other right line can be drawn between the tangent and the circle.</p><p>Hence, the angle of contact is less than any rectilinear angle; and the angle of the semi-circle between the radius DA and the arch AB, is greater than any rectilinear acute angle.</p><p>This seeming paradox of Euclid has exercised the wits of mathematicians: it was the subject of a long controversy between Peletarius and Clavius; the former of whom maintained that the angle of contact is heterogeneous to a rectilinear one; as a line is to a surface; the latter maintained the contrary.</p><p>Dr. Wallis has a formal treatise on the angle of contact, and of the semi-circle; where, with other great mathematicians, he approves of the opinion of Peletarius.</p></div1><div1 part="N" n="CONTENT" org="uniform" sample="complete" type="entry"><head>CONTENT</head><p>, a term often used for the measurement of bodies and surfaces, whether solid or superficial; or the capacity of a vessel and the area of a space; being the quantity either of matter or space included within certain bounds or limits.</p></div1><div1 part="N" n="CONTIGUITY" org="uniform" sample="complete" type="entry"><head>CONTIGUITY</head><p>, the relation of bodies touching one another.</p></div1><div1 part="N" n="CONTIGUOUS" org="uniform" sample="complete" type="entry"><head>CONTIGUOUS</head><p>, a relative term, understood of things disposed so near each other, that they join their surfaces, or touch.</p><p><hi rend="smallcaps">Contiguous</hi> <hi rend="italics">Angles,</hi> are such as have one leg or fide common to each angle; and are otherwise called <hi rend="italics">adjoining angles</hi>; in contradistinction to those made by continuing their legs through the point of contact, which are called <hi rend="italics">opposite</hi> or <hi rend="italics">vertical angles.</hi></p></div1><div1 part="N" n="CONTINENT" org="uniform" sample="complete" type="entry"><head>CONTINENT</head><p>, a terra firma, main-land, or a large extent of country, not interrupted by seas: so called, in opposition to island, peninsula, &c.</p><p>The world is usually divided into two grand continents, the old and the new: the old continent comprehends Europe, Asia, and Africa; the new, North and South America. Since the discovery of New Holland and New South Wales, it is a doubt with many whether to call that vast country an island or a continent.</p><p>CONTINGENT <hi rend="italics">Line,</hi> the same with tangent line in Dialling, being the intersection of the planes of the dial and equinoctial, and at right angles to the substilar line. <cb/></p><p>CONTINUAL <hi rend="smallcaps">Proportionals</hi>, are a series of three or more quantities compared together, so that the ratio is the same between every two adjacent terms, viz between the 1st and 2d, the 2d and 3d, the 3d and 4th, &c. As 1, 2, 4, 8, 16, &c, where the terms continually increase in a double ratio; or 12, 4, 4/3, 4/9, where the terms decrease in a triple ratio.</p><p>A series of continual or continued proportionals, is otherwise called a <hi rend="italics">progression.</hi></p><p>CONTINUED <hi rend="italics">Quantity,</hi> or <hi rend="italics">Body,</hi> is that whose parts are joined and united together.</p><p>CONTINUED <hi rend="italics">Proportion,</hi> is that in which the consequent of the first ratio is the same with the antecedent of the second; as in these, 3 : 6 :: 6: 12. See <hi rend="smallcaps">Continual</hi> <hi rend="italics">Proportion.</hi></p><p>On the contrary, if the consequent of the first ratio be different from the antecedent of the second, the proportion is called <hi rend="italics">discrete:</hi> as 3 : 6 :: 4 : 8.</p><p>CONTRA-<hi rend="smallcaps">Harmonical</hi> <hi rend="italics">Proportion,</hi> that relation of three terms, in which the difference of the first and second is to the difference of the 2d and 3d, as the 3d is to the first. Thus, for instance, 3, 5, and 6, are numbers contra-harmonically proportional; for 2 : 1 :: 6 : 3.</p><p>CONTRA-<hi rend="smallcaps">Mure</hi>, in Fortification, is a little wall built before another partition wall, to strengthen it, so that it may receive no damage from the adjacent buildings.</p><p>CONTRATE-<hi rend="smallcaps">Wheel</hi>, is that wheel in watches which is next to the crown, whose teeth and hoop lie contrary to those of the other wheels; from whence comes its name.</p></div1><div1 part="N" n="CONTRAVALLATION" org="uniform" sample="complete" type="entry"><head>CONTRAVALLATION</head><p>, <hi rend="italics">Line of,</hi> in Fortification, is a trench, guarded with a parapet; being made by the besiegers, between them and the place besieged, to secure themselves on that side, and stop the sallies of the garrison. It is made without musket-shot of the town; sometimes going quite around it, and sometimes not, as occasion may require. The besiegers lie between the lines of circumvallation and contravallation: but it is now seldom used.</p><p>CONVERGING <hi rend="italics">Curves.</hi> See <hi rend="smallcaps">Curve.</hi></p><div2 part="N" n="Converging" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Converging</hi></head><p>, or <hi rend="smallcaps">Convergent</hi> <hi rend="italics">Lines,</hi> in Geometry, are those that continually approximate, or whose distance becomes continually less and less the farther they are continued, till they meet: in opposition to <hi rend="italics">divergent</hi> lines, whose distance becomes continually greater.</p><p>Lines that converge the one way, diverge the other.</p><p><hi rend="smallcaps">Converging</hi> <hi rend="italics">Rays,</hi> in Optics, are such as incline towards one another in their passage, and in Dioptrics, are those rays which, in their passage out of one medium into another of a different density, are refracted towards one another; so that, if far enough continued, they will meet in a point or focus.</p><p><hi rend="smallcaps">Converging</hi> <hi rend="italics">Series,</hi> a series of terms or quantities, that always decrease the farther they proceed, or which tend to a certain magnitude or limit: in opposition to diverging series, or such as become larger and larger continually. See <hi rend="smallcaps">Series.</hi></p><p>CONVERSE. A proposition is said to be the converse of another, when, after drawing a conclusion from something first supposed, we return again, and, making a supposition of what had before been concluded, draw from thence as a conclusion what before was made the <pb n="333"/><cb/> supposition. Thus, when it is supposed that the two sides of a triangle are equal, and thence demonstrate or conclude that the two angles opposite to those fides are equal also; then the converse is to suppose that the two angles of a triangle are equal, and thence to prove or conclude that the sides opposite to those angles are also equal.</p><p><hi rend="smallcaps">Converse</hi> <hi rend="italics">Direction,</hi> in Astrology, is used in opposition to <hi rend="italics">direct</hi> direction; that is, by the latter the promoter is carried to the significator, according to the order of the signs: whereas by the other it is carried from east to west, contrary to the order of the signs.</p></div2></div1><div1 part="N" n="CONVERSION" org="uniform" sample="complete" type="entry"><head>CONVERSION</head><p>, or <hi rend="smallcaps">Convertendo</hi>, is when there are four proportionals, and it is inferred, that the first is to its excess above the 2d, as the third to its excess above the 4th: according to Euclid, lib. 5, def. 17. <table><row role="data"><cell cols="1" rows="1" role="data">Thus, if it be</cell><cell cols="1" rows="1" role="data">8 : 6 :: 4 : 3,</cell></row><row role="data"><cell cols="1" rows="1" role="data">then convertendo, or by conversion,</cell><cell cols="1" rows="1" role="data">8 : 2 :: 4 : 1,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Or if there be</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a : b :: c : d,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">then convertendo, or by conversion,</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">a : a</hi>-<hi rend="italics">b :: c : c</hi>-<hi rend="italics">d.</hi></cell></row></table></p></div1><div1 part="N" n="CONVEX" org="uniform" sample="complete" type="entry"><head>CONVEX</head><p>, round or curved and protuberant outwards, as the outside of a globular body.</p><p><hi rend="smallcaps">Convex</hi> <hi rend="italics">Lens, Mirror,</hi> &c. See <hi rend="smallcaps">Lens, Mirror</hi>, &c.</p></div1><div1 part="N" n="CONVEXITY" org="uniform" sample="complete" type="entry"><head>CONVEXITY</head><p>, the exterior or outward surface of a convex or round body.</p></div1><div1 part="N" n="COPERNICAN" org="uniform" sample="complete" type="entry"><head>COPERNICAN</head><p>, something relating to Copernicus. As, the</p><p><hi rend="smallcaps">Copernican</hi> <hi rend="italics">Sphere.</hi> See <hi rend="smallcaps">Sphere.</hi></p><p><hi rend="smallcaps">Copernican</hi> <hi rend="italics">System,</hi> is that system of the world, in which it is supposed that the sun is at rest in the centre, and the earth and planets all moving around him in their own orbits.</p><p>Here it is supposed, that the heavens and stars are at rest; and the diurnal motion which they appear to have, from east to west, is imputed to the earth's diurnal motion from west to east.</p><p>This system was maintained by many of the ancieñts; particularly Ecphantus, Seleucus, Aristarchus, Philolaus, Cleanthes Samius, Nicetas, Heraclides Ponticus, Plato, and Pythagoras; from the last of whom it was anciently called the Pythagoric, or Pythagorean System.</p><p>This system was also held by Archimedes, in his book of the number of the Grains of Sand; but after him it became neglected, and even forgotten, for many ages; till about 300 years since, when Copernicus revived it; from whom it took the new name of the Copernican System. See the next article.</p></div1><div1 part="N" n="COPERNICUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">COPERNICUS</surname> (<foreName full="yes"><hi rend="smallcaps">Nicholas</hi></foreName>)</persName></head><p>, an eminent astronomer, was born at Thorn in Prussia, January 19, 1473. He was instructed in the Latin and Greek languages at home; and afterward sent to Cracow, where he studied philosophy, mathematics, and medicine: though his genius was naturally turned to mathematics, which he chiesly studied, and purfued through all its various branches.</p><p>He set out for Italy at 23 years of age; stopping at Bologna, that he might converse with the celebrated astronomer of that place, Dominic Maria, whom he assisted for some time in making his observations. From hence he passed to Rome, where he was presently considered as not inferior to the famous Regiomontanus. Here he soon acquired so great a reputation, that he <cb/> was chosen professor of mathematics, which he taught there for a long time with the greatest applause; and here also he made some astronomical observations about the year 1500.</p><p>Afterward, returning to his own country, he began to apply his fund of observations and mathematical knowledge, to correcting the system of astronomy which then prevailed. He set about collecting all the books that had been written by philosophers and astronomers, and to examine all the various hypotheses they had invented for the solution of the celestial phenomena; to try if a more symmetrical order and constitution of the parts of the world could not be discovered, and a more just and exquisite harmony in its motions established, than what the astronomers of those times so easily admitted. But of all their hypotheses, none pleased him so well as the Pythagorean, which made the sun to be the centre of the system, and supposed the earth to move both round the sun, and also round its own axis. He thought he discerned much beautiful order and proportion in this; and that all the embarrassment and perplexity, from epicycles and excentries, which attended the Ptolemaic hypotheses, would here be entirely removed.</p><p>This system he began to consider, and to write upon, when he was about 35 years of age. He carefully contemplated the phenomena; made mathematical calculations; examined the observations of the antients, and made new ones of his own; till, after more than 20 years chiefly spent in this manner, he brought his scheme to perfection, establishing that system of the world which goes by his name, and is now universally received by all philosophers.</p><p>This system however was at first looked upon as a most dangerous heresy, and his work had long been finished and perfected, before he could be prevailed upon to give it to the world, being strongly urged to it by his friends. At length yielding to their intreaties, it was printed, and he had but just received a perfect copy, when he died the 24th of May 1543, at 70 years of age; by which it is probable he was happily relieved from the violent fanatical persecutions of the church, which were but too likely to follow the publication of his astronomical opinions; and which indeed was afterward the sate of Galileo, for adopting and defending them.</p><p>The above work of Copernicus, first printed at Norimberg in folio, 1543, and of which there have been other editions since, is intitled <hi rend="italics">De Revolutionibus Orbium Cælestium,</hi> being a large body of astronomy, in 6 books.</p><p>When Rheticus, the disciple of our author, returned out of Prussia, he brought with him a tract of Copernicus, on plane and spherical trigonometry, which he had printed at Norimberg, and which contained a table of sines. It was afterward printed at the end of the first book of the Revolutions. An edition of our author's great work was also published in 4to at Amsterdam in 1617, under the title of <hi rend="italics">Astronomia Instaurata,</hi> illustrated with notes by Nicolas Muler of Groningen.</p></div1><div1 part="N" n="COPERNICUS" org="uniform" sample="complete" type="entry"><head>COPERNICUS</head><p>, the name of an astronomical instrument, invented by Whiston, to shew the motion and phenomena of the planets, both primary and secondary. It is founded upon the Copernican system, and therefore ealled by his name. <pb n="334"/><cb/></p><p>COR <hi rend="smallcaps">Caroli</hi>, Charles's Heart, an extra-constellated star of the 2d magnitude in the northern hemisphere, between the Coma Berenices and Ursa Major; so called by Sir Charles Scarborough, in honour of king Charles I.</p><p><hi rend="smallcaps">Cor Hydræ</hi>, the Hydra's Heart, a star of the 2d magnitude, in the Heart of the constellation Hydra.</p><p><hi rend="smallcaps">Cor Leonis</hi>, Lion's Heart, or Regulus, a star of the first magnitude in the constellation Leo.</p><p><hi rend="smallcaps">Cor Scorpii.</hi> See <hi rend="smallcaps">Antares.</hi></p></div1><div1 part="N" n="CORBEILS" org="uniform" sample="complete" type="entry"><head>CORBEILS</head><p>, in Fortification, are little baskets about a foot and a half high, 8 inches broad at the bottom, and 12 at the top; which being filled with earth, are set against one another on the parapet, or elsewhere, leaving certain port-holes, from whence to fire under cover upon the enemy.</p></div1><div1 part="N" n="CORBEL" org="uniform" sample="complete" type="entry"><head>CORBEL</head><p>, in Architecture, the representation of a basket, sometimes seen on the heads of caryatides.</p><div2 part="N" n="Corbel" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Corbel</hi></head><p>, or <hi rend="smallcaps">Corbil</hi>, is also used, in Building, for a short piece of timber placed in a wall, with its end projecting out 6 or 8 inches, as occasion serves, in the manner of a shouldering-piece.</p></div2></div1><div1 part="N" n="CORBET" org="uniform" sample="complete" type="entry"><head>CORBET</head><p>, the same as <hi rend="smallcaps">Corbel.</hi></p></div1><div1 part="N" n="CORDON" org="uniform" sample="complete" type="entry"><head>CORDON</head><p>, in Fortisication, a row of stones jutting out between the rampart and the basis of the parapet, like the tore of a column. The cordon ranges round the whole fortress, and serves to join the rampart, which is aslope, and the parapet, which is perpendicular, more agreeably together.</p><p>In fortifications raised of earth, this space is filled up with pointed stakes, instead of a cordon.</p></div1><div1 part="N" n="CORDS" org="uniform" sample="complete" type="entry"><head>CORDS</head><p>, in Music, are the sounds produced by an instrument or the voice.</p></div1><div1 part="N" n="CORIDOR" org="uniform" sample="complete" type="entry"><head>CORIDOR</head><p>, or <hi rend="smallcaps">Corridor</hi>, in Fortification, is the covert-way lying round about the whole compass of the works of a place, between the outside of the moat and the pallisadoes, being about 20 yards broad.</p><p><hi rend="smallcaps">Coridor</hi> is also used, in Architecture, for a gallery. or long aile, around a building, leading to several chambers at a distance from each other, sometimes wholly inclosed, and sometimes open on one side.</p><p>CORINTHIAN <hi rend="italics">Order,</hi> of Architecture, is the 4th in order, or the 5th and last according to Scamozzi and Le Clerc.</p><p>This order was invented by an Athenian Architect, and is the richest and most delicate of them all; its capital being adorned with rows of leaves, and of 8 volutas, which support the abacus. The height of its column is 10 diameters, and its cornice is supported by modillions.</p><p>CORNEA <hi rend="italics">Tunica,</hi> the second coat of the eye; so called from its substance resembling the horn of a lantern. This is situated in the fore-part; and is surrounded by the sclerotica. It has a greater convexity than the re<*> of the eye, and is a portion of a small sphere, or rather spheroid, and consolidates the whole eye.</p></div1><div1 part="N" n="CORNICE" org="uniform" sample="complete" type="entry"><head>CORNICE</head><p>, <hi rend="smallcaps">Corniche</hi>, or <hi rend="smallcaps">Cornish</hi>, the third and uppermost part of the entablature of a column, or the uppermost ornament of any wainscotting, &c.</p></div1><div1 part="N" n="COROLLARY" org="uniform" sample="complete" type="entry"><head>COROLLARY</head><p>, or <hi rend="smallcaps">Consectary</hi>, a consequence drawn from some proposition or principles already advanced or demonstrated, and without the aid of any other proposition: as if from this theorem, <hi rend="italics">That a triangle which has two equal sides, has also two equal</hi> <cb/> <hi rend="italics">angles,</hi> this consequence should be drawn, <hi rend="italics">that a triangle which hath the three sides equal, has also its three angles equal.</hi></p></div1><div1 part="N" n="CORONA" org="uniform" sample="complete" type="entry"><head>CORONA</head><p>, <hi rend="italics">Crown</hi> or <hi rend="italics">Crowning,</hi> in Architecture, the flat and most advanced part of the cornice; so called, because it crowns the cornice and entablature: by the workmen it is called the <hi rend="italics">drip,</hi> as serving by its projecture to screen the rest of the building from the rain.</p></div1><div1 part="N" n="CORONA" org="uniform" sample="complete" type="entry"><head>CORONA</head><p>, in Optics, a luminous circle, usually coloured, round the sun, moon, or largest planets. See <hi rend="smallcaps">Halo.</hi></p><p><hi rend="smallcaps">Corona</hi> <hi rend="italics">Borealis,</hi> or <hi rend="italics">Septentrionalis,</hi> the <hi rend="italics">Northern Crown</hi> or <hi rend="italics">Garland,</hi> a constellation of the northern hemisphere, being one of the 48 old ones. It contains 8 stars according to the catalogue of Ptolomy, Tycho, and Hevelius; but according to the Britannic Catalogue, 21.</p><p><hi rend="smallcaps">Corona</hi> <hi rend="italics">Australis,</hi> or <hi rend="italics">Meridianalis,</hi> the <hi rend="italics">Southern Crown,</hi> a constellation of the southern hemisphere, whose stars in Ptolomy's catalogue are 13; in the British Catalogue, 12.</p><p>CORPUSCLE the diminutive of corpus, used to express the minute parts, or particles, that constitute natural bodies; meaning much the same as <hi rend="italics">atoms.</hi></p><p>Newton shews a method of determining the sizes of the corpuscles of bodies, from their colours.</p><p>CORPUSCULAR <hi rend="italics">Philosophy,</hi> that scheme or system of physics, in which the phenomena of bodies are accounted for, from the motion, rest, position, &c, of the corpuscles or atoms of which bodies consist.</p><p>The Corpuscular philosophy, which now flourishes under the name of the mechanical philosophy, is very ancient. Leucippus and Democritus taught it in Greece; from them Epicurus received it, and improved it; and from him it was called the <hi rend="italics">Epicurean Philosophy.</hi></p><p>Leucippus, it is said, received it from one Mochus, a Phenician phisiologist, before the time of the Trojan war, and the first who philosophized about atoms: which Mochus is, according to the opinion of some, the Moses of the Scriptures.</p><p>After Epicurus, the corpuscular philosophy gave way to the peripatetic, which became the popular system. Thus, instead of atoms, were introduced specific and substantial forms, qualities, sympathies, &c, which amused the world, till Gassendus, Charleton, Descartes, Boyle, Newton, and others, retrieved the corpuscularian hypotheses; which is now become the basis of the mechanical and experimental philosophy.</p><p>Boyle reduces the principles of the corpuscular philosophy to the 4 following heads.</p><p>1. That there is but one universal kind of matter, which is an extended, impenetrable, and divisible substance, common to all bodies, and capable of all ferms. —On this head, Newton sinely remarks thus: “All things considered, it appears probable to me, that God in the beginning created matter in solid, hard, in penetrable, moveable particles; of such sizes and figures, and with such other properties, as most conduced to the end for which he formed them: and that these primitive particles, being solids, are incomparably harder than any of the sensible porous bodies compounded of them; even so hard as never to wear, or break in pieces: no other power being able to divide what God made one in the first creation. While these corpuscles remain entire, they may compose bodies of one <pb n="335"/><cb/> and the same nature and texture in all ages: but should they wear away, or break in pieces, the nature of things depending on them would be changed: water and earth, composed of old worn particles, of fragments of particles, would not be of the same nature and texture now, with water and earth composed of entire particles at the beginning. And therefore, that nature may be lasting, the changes of corporeal things are to be placed only in the various separations, and new associations, of these permanent corpuscles.”</p><p>2. That this matter, in order to form the vast variety of natural bodies, must have motion in some, or all its assignable parts; and that this motion was given to matter by God, the creator of all things; and has all manner of directions and tendencies.— “These corpuscles, says Newton, have not only a vis inertiæ, accompanied with such passive laws of motion as naturally result from that force; but also are moved by certain active principles; such as that of gravity, and that which causes fermentation, and the cohesion of bodies.”</p><p>3. That matter must also be actually divided into parts; and each of these primitive particles, fragments, or atoms of matter, must have its proper magnitude, figure, and shape.</p><p>4. That these differently sized and shaped particles, have different orders, positions, situations, and postures, from whence all the variety of compound bodies arises.</p><p>CORRIDOR. See <hi rend="smallcaps">Coridor.</hi></p></div1><div1 part="N" n="CORVUS" org="uniform" sample="complete" type="entry"><head>CORVUS</head><p>, the <hi rend="italics">Raven,</hi> a southern constellation, fabled by the Greeks, as taken up to heaven by Apollo, to whom it tattled that the beautiful maid Coronis, the daughter of Phlegeos, and mother of Esculapius by Apollo, played the deity false with Ischys, under a tree upon which the animal happened to be perched.</p><p>The stars in this constellation, in Ptolomy's and Tycho's catalogues are 7; but in the Britannic catalogue, 9.</p></div1><div1 part="N" n="COSECANT" org="uniform" sample="complete" type="entry"><head>COSECANT</head><p>, COSINE, COTANGENT, COVERSED SINE, are the secant, sine, tangent, and versed sine of the complement of an arch or angle; <hi rend="italics">Co</hi> being, in this case, a contraction of the word complement, and was first introduced by Gunter.</p><p>COSMICAL <hi rend="smallcaps">Aspect</hi>, among astrologers, is the aspect of a planet with respect to the earth.</p><p><hi rend="smallcaps">Cosmical</hi> <hi rend="italics">Rising,</hi> or <hi rend="italics">Setting,</hi> is said of a star when it rises or sets at the same time when the sun rises.</p><p>But, according to Kepler, to rise or set cosmically, is only simply to rise or set, that is, to ascend above, or descend below, the horizon; as much as to say, to rise or set to the world.</p></div1><div1 part="N" n="COSMOGONY" org="uniform" sample="complete" type="entry"><head>COSMOGONY</head><p>, the science of the formation of the universe; as distinguished from cosmography, which is the science of the parts of the universe, supposing it formed, and in the state as we behold it; and from cosmology, which reasons on the actual and permanent state of the world as it now is; whereas cosmogony reasons on the variable state of the world at the time of its formation.</p></div1><div1 part="N" n="COSMOGRAPHY" org="uniform" sample="complete" type="entry"><head>COSMOGRAPHY</head><p>, the description of the world; or the art that teaches the construction, figure, disposition, and relation of all the parts of the world, with <cb/> the manner of representing them on a plane. It consists chiefly of two parts; viz, <hi rend="italics">Astronomy,</hi> which shews the structure of the heavens, with the disposition of the stars; and <hi rend="italics">Geography,</hi> which shews those of the earth.</p></div1><div1 part="N" n="COSMOLOGY" org="uniform" sample="complete" type="entry"><head>COSMOLOGY</head><p>, the science of the world in general.</p></div1><div1 part="N" n="COSS" org="uniform" sample="complete" type="entry"><head>COSS</head><p>, <hi rend="italics">Rule of,</hi> meant the same as Algebra, by which name it was for some time called, when first introduced into Europe through the Italians, who named it <hi rend="italics">Regola de Cosa, the Rule of the thing</hi>; the unknown quantity, or that which was required in any question, being called <hi rend="italics">cosa,</hi> the <hi rend="italics">thing</hi>; from whence we have Coss, and Cossic numbers, &c.</p></div1><div1 part="N" n="COTES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">COTES</surname> (<foreName full="yes"><hi rend="smallcaps">Roger</hi></foreName>)</persName></head><p>, a very eminent mathematician, philosopher, and astronomer, was born July 10, 1682, at Burbach in Leicestershire, where his father Robert was rector. He was first placed at Leicester school; where, at 12 years of age, he discovered a strong inclination to the mathematics. This being observed by his uncle, the Rev. Mr. John Smith, he gave him all the encouragement he could; and prevailed on his father to send him for some time to his house in Lincolnshire, that he might assist him in those studies: and here he laid the foundation of that deep and extensive knowledge in that science, for which he was afterward so deservedly famous. He was hence removed to St. Paul's school, London, where he made a great progress in classical learning; and yet he found so much leisure as to support a constant correspondence with his uncle, not only in mathematics, but also in metaphysics, philosophy, and divinity. His next remove was to Trinity College Cambridge, where he took his degrees, and became fellow.</p><p>Jan. 1706, he was appointed professor of astronomy and experimental philosophy, upon the foundation of Dr Thomas Plume, archdeacon of Rochester; being the first that enjoyed that office, to which he was unanimously chosen, on account of his high reputation and merits. He entered into orders in 1713; and the same year, at the desire of Dr. Bentley, he published at Cambridge the second edition of Newton's Mathematica Principia; inserting all the improvements which the author had made to that time. To this edition he prefixed a most admirable preface, in which he pointed out the true method of philosophising, shewing the foundation on which the Newtonian philosophy was raised, and refuting the objections of the Cartesians and all other philosophers against it.</p><p>The publication of this edition of Newton's Principia added greatly to his reputation; nor was the high opinion the public now conceived of him in the least diminished, but rather much increased, by several productions of his own, which afterward appeared. He gave in the Philos. Transactions, two papers, viz, 1, Logometria, in vol. 29; and a Description of the great fiery meteor that was seen March 6, 1716, in vol. 31.</p><p>This extraordinary genius in the mathematics died, to the great regret of the university, and all the lovers of the sciences, June 5, 1716, in the very prime of his life, being not quite 34 years of age.</p><p>Mr. Cotes left behind him some very ingenious, and indeed admirable tracts, part of which, with the Lo- <pb n="336"/><cb/> gometria above mentioned, were published, in 1722, by Dr. Robert Smith, his cousin and successor in his professorship, afterward master of Trinity College, under the title of <hi rend="italics">Harmonia Mensurarum,</hi> which contains a number of very ingenious and learned works: see the Introduction to my Logarithms. He wrote also a <hi rend="italics">Compendium of Arithmetic;</hi> of the <hi rend="italics">Resolution of Equations;</hi> of <hi rend="italics">Dioptrics;</hi> and of the <hi rend="italics">Nature of Curves.</hi> Beside these pieces, he drew up, in the time of his lectures, a course of <hi rend="italics">Hydrostatical and Pneumatical Lectures,</hi> in English, which were published also by Dr. Smith in 8vo, 1737, and are held in great estimation.</p><p>So high an opinion had Sir Isaac Newton of our author's genius, that he used to say, “If Cotes had lived, we had known something.”</p><p>COTESIAN <hi rend="italics">theorem,</hi> in Geometry, an appellation used for an elegant property of the circle discovered by Mr. Cotes. The theorem is this:</p><p>If the factors of the binomial <hi rend="italics">a</hi><hi rend="sup">c</hi><01> <hi rend="italics">x</hi><hi rend="sup">c</hi> be required, the index <hi rend="italics">c</hi> being an integer number. With the centre O, and radius AO = <hi rend="italics">a,</hi> describe a circle, and di- <figure/> vide its circumferance into as many equal parts as there are units in 2<hi rend="italics">c,</hi> at the points A, B, C, D, &c; then in the radius, produced if necessary, take OP = <hi rend="italics">x,</hi> and from the point P, to all the points of division in the circumference, draw the lines PA, PB, PC, &c; so shall these lines taken alternately be the factors sought; viz, , according as the point P is within or without the circle.</p><p>For instance, if <hi rend="italics">c</hi> = 5, divide the circumference into 10 equal parts, and the point P being within the circle, then will .</p><p>In like manner, if <hi rend="italics">c</hi> = 6, having divided the circumference into 12 equal parts, then will .</p><p>The demonstration of this theorem may be seen in Dr. Pemberton's Epist. de Cotesii inventis. See also Dr. Smith's Theoremata Logometrica and Trigonometrica, added to Cotes's Harm. Mens. pa. 114; De Moivre Miscel. Analyt. pa. 17; and Waring's Letter to Dr. Powell, pa. 39.</p><p>By means of this theorem, the acute and elegant author was enabled to make a farther progress in the inverse method of Fluxions, than had been done before. But in the application of his discovery there <cb/> still remained a limitation, which was removed by Mr. De Moivre. Vide ut supra.</p><p>COVERT-<hi rend="smallcaps">Way</hi>, in fortification, a space of ground level with the adjoining country, on the outer edge of the ditch, ranging quite round all the works. This is otherwise called the <hi rend="italics">corridor,</hi> and has a parapet with its banquette and glacis, which form the height of the parapet. It is sometimes also called the counterscarp, because it is on the edge of the scarp.</p><p>One of the greatest difficulties in a siege, is to make a lodgment on the covert-way; because it is usual for the besieged to palisade it along the middle, and undermine it on all sides.</p></div1><div1 part="N" n="COVING" org="uniform" sample="complete" type="entry"><head>COVING</head><p>, in Building, si when houses are built projecting over the ground plot, and the turned projecture formed into an arch.</p><p><hi rend="smallcaps">Coving</hi> <hi rend="italics">Cornice,</hi> is one that has a large casemate or hollow in it.</p><p>COUNT-<hi rend="smallcaps">Wheel</hi>, is a wheel in the striking part of a clock, moving round once in 12 or 24 hours. It is sometimes called the <hi rend="italics">locking-wheel,</hi> because it has usually 11 notches in it at unequal distances from one another, to make the clock strike.</p><p>COUNTER-<hi rend="smallcaps">Approaches</hi>, in Fortification, lines or trenches made by the besieged, where they come out to attack the lines of the besiegers in form.</p><p><hi rend="smallcaps">Counter-Battery</hi>, a battery raised to play on another, to dismount the guns, &c.</p><p><hi rend="smallcaps">Counter-Breast-Work</hi>, the same as <hi rend="italics">Fausse-Braye.</hi></p><p><hi rend="smallcaps">Counter-Forts</hi>, <hi rend="italics">Buttresses,</hi> or <hi rend="italics">Spurs,</hi> are pillars of masonry serving to prop or sustain walls, or terraces, subject to bulge, or be thrown down.</p><p><hi rend="smallcaps">Counter-Fugue</hi>, in Music, is when fugues proceed contrary to one another</p><p><hi rend="smallcaps">Counter-Guard</hi>, in Fortification, a work commonly serving to cover a bastion. It is composed of two faces, forming a salient angle before the flanked angle of a bastion.</p><p><hi rend="smallcaps">Counter-Harmonical.</hi> See <hi rend="smallcaps">Contra</hi>-H<hi rend="smallcaps">ARMONICAL.</hi></p><p><hi rend="smallcaps">Counter-Mine</hi>, a subterraneous passage, made by the besieged, in search of the enemy's mine, to give air to it, to take away the powder; or by any other means to frustrate the effect of it.</p><p><hi rend="smallcaps">Counter-Part</hi>, a term in Music, only denoting that one part is opposite to another, so, the bass and treble are counterparts to each other.</p><p><hi rend="smallcaps">Counter-Point</hi>, in Music, the art of composing harmony; or disposing and concerting several parts so together, as that they may make an agreeable whole.</p><p><hi rend="smallcaps">Counter-Poise</hi>, any thing serving to weigh against another; particularly a piece of metal, ufually of brass or iron, making an appendage to the Roman <hi rend="italics">statera,</hi> or steel-yard. It is contrived to slide along the beam; and from the division at which it keeps the balance in equilibrio, the weight of the body is determined. It is sometimes called the <hi rend="italics">pear,</hi> on account of its sigure; and <hi rend="italics">mass,</hi> by reason of its weight.</p><p>Rope-dancers make use of a pole by way of counterpoise, to keep their bodies in equilibrio.</p></div1><div1 part="N" n="COUNTERSCARP" org="uniform" sample="complete" type="entry"><head>COUNTERSCARP</head><p>, is that side of the ditch that is next the country; or properly the talus that supports the earth of the covert-way: though by this word is often understood the whole covert-way, with <pb n="337"/><cb/> its parapet and glacis. And so it must be understood when it is said, The enemy lodged themselves on the counterscarp.</p><div2 part="N" n="Counter- swallows-tail" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Counter- swallows-tail</hi></head><p>, is an outwork in Fortification, in form of a single tenaille, wider towards the place, or at the gorge, than at the head, or next the country.</p></div2><div2 part="N" n="Counter-tenor" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Counter-tenor</hi></head><p>, one of the mean or middle parts of music: so called, as being opposite to the tenor.</p></div2></div1><div1 part="N" n="COURSE" org="uniform" sample="complete" type="entry"><head>COURSE</head><p>, in Navigation, the point of the compass, or horizon, which a ship steers on; or the angle which the rhumb line on which it sails makes with the meridian; being sometimes reckoned in degrees, and sometimes in points of the compass.</p><p>When a ship sails either due north or south, she sails on a meridian, makes no departure, and her distance and difference of latitude are the same.</p><p>When she sails due east or west, her course makes right-angles with the meridian, and she sails either upon the equator, or a parallel to it; in which case she makes no difference of latitude, but her distance and departure are the same.</p><p>But when the ship sails between the cardinal points, on a course making always the same oblique angle with the meridians, her path is then the loxodromic curve, being a spiral cutting all the meridians in the same angle, and terminating in the pole.</p><p>COURTAIN. See <hi rend="smallcaps">Curtin.</hi></p></div1><div1 part="N" n="CRAB" org="uniform" sample="complete" type="entry"><head>CRAB</head><p>, in Mechanics, an engine used for mounting guns on their carriages. See <hi rend="smallcaps">Gin.</hi></p><div2 part="N" n="Crab" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Crab</hi></head><p>, on ship-board, is a wooden pillar, whose lower end is let down through the ship's decks, and refts upon a socket like the capstan: in its upper end are three or four holes, at different heights, through the middle of it, above one another, to receive long bars, against which men act by pushing or thrusting.— It is employed to wind in the cable, and for other purposes requiring a great mechanical power.</p><p>The Crab with three claws is used to launch ships, and to heave them into the dock, or off the key.</p></div2></div1><div1 part="N" n="CRANE" org="uniform" sample="complete" type="entry"><head>CRANE</head><p>, a machine used in building, and in commerce, for raising large stones and other weights.</p><p>M. Perrault, in his notes on Vitruvius, makes the crane the same with the corvus, or raven, of the ancients.</p><p>The modern crane consists of several members or pieces, the principal of which is a strong upright beam, or arbor, firmly fixed in the ground, and sustained by eight arms, coming from the extremities of four pieces of wood laid across, through the middle of which the foot of the beam passes. About the middle of the arbor the arms meet, and are mortised into it: its top ends in an iron pivot, on which is borne a transverse piece, advancing out to a good distance like a crane's neck; whence the name. The middle and extremities of this are again sustained by arms from the middle of the arbor: and over it comes a rope, or cable, to one end of which the weight is fixed; the other is wound about the spindle of a wheel, which, turned, draws the rope, and that heaves up the weight; to be afterwards applied to any side or quarter, by the mobility of the transverse piece on the pivot.</p><p>Several improvements of this useful machine are mentioned in Desaguliers's Exper. Philos. pa. 178 & <cb/> seq. particularly how to prevent the inconveniences arising from sudden jerks, as well as to increase its force by using a double axis in peritrochio, and two handles.</p><p>The Crane is of two kinds; in the first kind, called the rat-tailed crane, the whole machine, with the load, turns upon a strong axis: in the second kind, the gibbet alone moves on its axis. See Desaguliers, as above, for a particular account of the different cranes, and of the gradual improvements they have received. See also the Supplement to Ferguson's Lectures, pa. 3, &c, or Philos. Trans. vol. 54, pa. 24, for a description of a new and safe crane, with four different powers adapted to different weights.</p><p><hi rend="smallcaps">Crane</hi> is the name of a southern constellation. See <hi rend="smallcaps">Grus.</hi></p><p><hi rend="smallcaps">Crane</hi> is also a popular name for a syphon.</p></div1><div1 part="N" n="CRANK" org="uniform" sample="complete" type="entry"><head>CRANK</head><p>, a contrivance in machines, in manner of an elbow, only of a square form; projecting out from an axis, or spindle; and serving, by its rotation, to raise and fall the pistons of engines for raising water, or the like.</p></div1><div1 part="N" n="CRATER" org="uniform" sample="complete" type="entry"><head>CRATER</head><p>, the Cup, a constellation in the southern hemisphere; whose stars, in Ptolomy's catalogue, are 7; in Tycho's, 8; in Hevelius's, 10; and in the Britannic catalogue, 31.</p></div1><div1 part="N" n="CREEK" org="uniform" sample="complete" type="entry"><head>CREEK</head><p>, a part of a haven, where any thing is landed from the sea. It is also said to be a shore or bank on which the water beats, running in a small channel from any part of the sea.</p></div1><div1 part="N" n="CREPUSCULUM" org="uniform" sample="complete" type="entry"><head>CREPUSCULUM</head><p>, <hi rend="italics">Twilight;</hi> the time from the first dawn or appearance of the morning, to the rising of the sun; and again, between the setting of the sun, and the last remains of day.</p><p>The Crepusculum, or twilight, it is supposed, usually begins and ends when the sun is about 18 degrees below the horizon; for then the stars of the 6th magnitude disappear in the morning, and appear in the evening. It is of longer duration in the solstices than in the equinoxes, and longer in an oblique sphere, than in a right one; because, in those cases the sun, by the obliquity of his path, is longer in ascending through 18 degrees of altitude. <figure/></p><p>Twilight is occasioned by the sun's rays refracted in our atmosphere, and reflected from the particles of it to the eye. For let A be the place of an observer on the earth ADL, AB the sensible horizon, meeting in B the circle CBM bounding that part of the atmosphere which is capable of refracting and reflecting light to the eye. It is plain that when the sun is under this <pb n="338"/><cb/> horizon, no direct rays can come to the eye at A: but the sun being in the refracted line CG, the particle C will be illuminated by the direct rays of the sun; and that particle may reflect those rays to A, where they enter the eye of the spectator. And thus the sun's light illuminating an innumerable multitude of particles, may be all reflected to the spectator at A.— From B draw BD touching the circle ADL in D; and let the sun be in the line BD at S: Then the ray SB will be reflected into BA, and will enter the eye, because the angle of incidence DBE is equal to the angle of reflection ABE: And that will be the first ray that reaches the eye in the morning, when the dawning begins; or the last that falls upon the eye at night, when the twilight ends: for when the sun goes lower down, the particles at B can be no longer illuminated.</p><p>Kepler indeed assigns another cause of the crepusculum, viz, the luminous matter or atmosphere about the sun; which, arising near the horizon, in a circular figure, exhibits the crepusculum; in no wise, he thinks, owing to the refraction of the atmosphere.— The sun's luminous atmosphere indeed, though neither the sole nor principal cause of twilight, may lengthen its duration, by illuminating our air, when the sun is too low to reach it with his own light. Gregor. Astr. lib. 2, prop. 8.</p><p><hi rend="italics">The depth of the sun below the horizon, at the beginning of the morning, or end of the evening twilight,</hi> is determined in the same manner as the arch of vision; viz, by observing the moment when the air first begins to shine in the morning, or ceases to shine in the evening; then finding the sun's place for that moment, and thence the time till his rising in the horizon, or from his setting in it in the evening. It is now generally agreed that this depth is about 18 degrees upon an average.—Alhazen found it to be 19°; Tycho, 17°; Rothmann, 24°; Stevenius, 18°; Cassini, 15°; Riccioli, in the equinox in the morning 16°, in the evening 20° 30′; in the summer solstice in the morning 21° 25′, in the winter solstice in the morning 17° 25′.</p><p>Nor is this difference among the determinations of astronomers to be wondered at; the cause of the crepusculum being inconstant: for, if the exhalations in the atmosphere be either more copious, or higher, than ordinary; the morning twilight will begin sooner, and the evening hold longer than ordinary: for the more copious the exhalations are, the more rays will they reflect, consequently the more will they shine; and the higher they are, the sooner will they be illuminated by the sun. On this account too, the evening twilight is longer than the morning, at the same time of the year in the same place. To this it may be added, that in a denser air, the refraction is greater; and that not only the brightness of the atmosphere is variable, but also its height from the earth: and therefore the twilight is longer in hot weather than in cold, in summer than in winter, and also in hot countries than in cold, other circumstances being the same. But the chief differences are owing to the different situations of places upon the earth, or to the difference of the sun's place in the heavens. Thus, the twilight is longest in a parallel sphere, and shortest in a right sphere, and longer to places in an oblique sphere in proportion as they <cb/> are nearer to one of the poles; a circumstance which affords relief to the inhabitants of the more northern countries, in their long winter nights. And the twilights are longest in all places of north latitude, when the sun is in the tropic of cancer; and to those in south latitude, when he is in the tropic of capricorn. The time of the shortest twilight is also different in different latitudes; in England, it is about the beginning of October and of March, when the sun is in the signs <figure/> and <figure/>. For the method of determining it by trigonometry, see Gregor. Astron. lib. 2, prob. 41. See also Robertson's Navigation, book 5, prob. 12.—Hence, when the difference between the sun's declination and the depth of the equator is less than 18°, so that the sun does not descend more than 18° below the horizon; the crepusculum will continue the whole night, as is the case in England from about the 22d of May to the 22d of July.</p><p><hi rend="italics">Given the latitude of the place, and the sun's declination; to find the beginning of the morning, and end of the evening twilight.</hi>—In the oblique-angled spherical triangle ZPS, are given ZP the colatitude, PS the codeclination, and ZS = 108°, being the sum of 90° the quadrant and 18° the depression at the extremity of the twilight. Then, by spherical trigonometry, calculate the angle ZPS the hour-angle from noon; which changed into time, at the rate of 15° to the hour, gives the time from noon at the beginning or end of twilight. See Robertson, ubi supra.</p><p>Of the Height of the sensible Atmosphere, as determined from the duration of twilight, see Keil's Astron. Lect. lect. 20, pa. 235, ed. 1721; or Long's Astron. vol. 1, pa. 260; where it is determined that the height where the atmosphere is dense enough to reflect the rays of light, is about 42 miles.</p><div2 part="N" n="Crescent" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Crescent</hi></head><p>, the new moon, which, as it begins to recede from the sun, shews a small rim of light, terminating in horns or points, which are still increasing, till it becomes full, and round in the opposition.</p><p>The term is sometimes also used for the same sigure of the moon in her wane, or decrease, but improperly; both because the horns are then turned towards the west, and because the figure is on the decrease; the <hi rend="italics">crescent</hi> properly signifying increase, from <hi rend="italics">cresco, I grow.</hi></p><p>CRONICAL. See <hi rend="smallcaps">Acronical.</hi></p></div2></div1><div1 part="N" n="CRONOS" org="uniform" sample="complete" type="entry"><head>CRONOS</head><p>, a name given to Saturn by some of the old astronomical writers.</p></div1><div1 part="N" n="CROSIER" org="uniform" sample="complete" type="entry"><head>CROSIER</head><p>, four stars, in form of a cross; by the help of which, those that sail in the southern hemisphere sind the antarctic pole.</p></div1><div1 part="N" n="CROSS" org="uniform" sample="complete" type="entry"><head>CROSS</head><p>, in Surveying, is an instrument consisting of a brass circle, divided into 4 equal parts, by two lines crossing each other in the centre. At each extremity of these lines is fixed a perpendicular sight, with small holes below each slit, for the better discovering of distant objects. The cross is mounted on a stasf, or stand, to fix it in the ground, and is very useful for measuring small pieces of land, and taking offsets, &c.</p><p><hi rend="italics">Ex.</hi> Suppose it be required to survey the field ABCDE with the Cross. Measure along the diagonal line AC, and observe, with the Cross, when you are perpendicularly opposite to the corners, as at F, G, H, and from thence measure the perpendiculars FE, GB, <pb n="339"/><cb/> HD. When you think you are nearly opposite a corner, set up the cross, with one of the bars or cross lines in the direction AC; then look through the sights of the other cross bar for the corner, as B; if it be seen through them, the cross is fixed in the right place; if not, take it up and move it backward or forward in the line AC, till the point B be seen through those sights; and then you have the true place of the perpendicular. <figure/></p><p><hi rend="italics">Invention of the</hi> <hi rend="smallcaps">Cross</hi>, <hi rend="italics">Inventio Crucis,</hi> an ancient feast, which is still retained in our calendar, and solemnized on the 3d of May, in memory of the sinding of the true Cross of Christ, deep in the ground, on Mount Calvary, by St. Helena, the mother of Constantine; where she erected a church for the preservation of part of it: the rest being brought to Rome, and deposited in the church of the Holy Cross of Jerusalem.</p><p><hi rend="italics">Exaltation of the</hi> <hi rend="smallcaps">Cross</hi>, an ancient feast, held on the 14th of September, in memory of this, that Heraclitus restored to Mount Calvary the true cross, in 642, which had been carried off, 14 years before, by Cosroes king of Persia, upon his taking Jerusalem from the emperor Phocas. This feast is still retained in our calendar, on Sept. 14, under the denomination of <hi rend="italics">Holy Rood,</hi> or <hi rend="italics">Holy Cross.</hi></p><p><hi rend="smallcaps">Cross-Multiplication</hi>, a method used chiefly by artificers in multiplying feet and inches by feet and inches, or the like; so called, because the factors are multiplied cross-wise, thus: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">f.</cell><cell cols="1" rows="1" rend="align=right" role="data">i.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">8′</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell></row></table></p><p><hi rend="smallcaps">Cross-Staff</hi>, or <hi rend="italics">Fore-Staff,</hi> is a mathematical instrument of box, or pear tree, consisting of a square staff, of about 3 feet long, having each of its faces divided like a line of tangents, and having 4 cross pieces of unequal lengths to fit on to the staff, the halves of these being as the radii to the tangent lines on the faces of the staff.—The instrument was used in taking the altitudes of the celestial bodies at sea. <figure/> <cb/></p></div1><div1 part="N" n="CROUSAZ" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CROUSAZ</surname> (<foreName full="yes"><hi rend="smallcaps">John Peter de</hi></foreName>)</persName></head><p>, a learned philosopher and mathematician, was born at Lausanne in Switzerland, April 13, 1663; where he died in 1748, at 85 years of age. Having made great progress in mathematics and the philosophy of Des Cartes, he travelled into Geneva, Holland, and France. He was successively professor in several universities; and at length was chosen governor to Prince Frederick of HesseCassel, nephew to the king of Sweden.</p><p>Crousaz was author of many works, in various branches; belles-lettres, logic, philosophy, divinity, &c, &c; but the most esteemed of them are, 1. His Logic; the best edition of which is that of 1741, in 6 vols. 8vo.—2. A Treatise on Beauty.—3. A Treatise on Education, 2 vols, 12mo.—4. A Treatise on the Human Understanding.—5. Several Treatises on Philosophy and Mathematics; as a Treatise on Motion, &c. with several papers inserted in the Memoirs of the French Academy of Sciences.</p></div1><div1 part="N" n="CROW" org="uniform" sample="complete" type="entry"><head>CROW</head><p>, in Mechanics, an iron lever, made with a sharp point at one end, and two claws at the other; being used in heaving and purchasing great weights, &c.</p></div1><div1 part="N" n="CROWN" org="uniform" sample="complete" type="entry"><head>CROWN</head><p>, in Astronomy, a name given to two con stellations, the southern and the northern.</p><div2 part="N" n="Crown" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Crown</hi></head><p>, in Geomctry, a plane ring included between two parallel or concentric peripheries, of unequal circles.</p><p>The area of this is had, by multiplying its breadth by the length of a middle periphery, which is an arithmetical mean between the two perpheries that bound it; or by multiplying half the sum of the circumferences by half the difference of the diameters; or lastly by multiplying the sum of the diameters by the difference of the diameters, and this last product by .7854. See my Mensuration, pa. 148, 2d ed.</p><p><hi rend="smallcaps">Crown</hi>-<hi rend="italics">Post,</hi> is a post in some buildings standing upright in the middle, between two principal rafters; and from which proceed struts or braces to the middle of each rafter. It is otherwise called a <hi rend="italics">king-post,</hi> or <hi rend="italics">king's-piece,</hi> or <hi rend="italics">joggle-piece.</hi></p><p><hi rend="smallcaps">Crown</hi>-<hi rend="italics">Wheel,</hi> of a Watch, is the upper wheel next the balance, or that which drives the balance.</p><p><hi rend="smallcaps">Crown</hi>-<hi rend="italics">Work,</hi> in Fortification, is an out-work running into the field; designed to keep off the enemy, gain some hill, or advantageous post, and cover the other works of the place. It consists of two demi-bastions at the extremities, and an entire bastion in the middle, with curtains.</p><p><hi rend="smallcaps">Crowned</hi> <hi rend="italics">Horn-work,</hi> is a Horn-work with a crownwork before it.</p><p>CRYSTALLINE <hi rend="italics">Humour,</hi> is a thick compact humour of the eye, in form of a flattish convex lens, placed in the middle of the eye, and serving to make that refraction of the rays of light which is necessary to have them meet in the retina, and form an image there, by which vision may be performed.</p><p><hi rend="smallcaps">Crystalline</hi> <hi rend="italics">Heavens,</hi> in the Old Astronomy, two orbs imagined between the primum mobile and the firmament, in the Ptolomaic system, which supposed the heavens solid, and only susceptible of a single motion.</p><p>King Alphonsus of Arragon, it is said, introduced the Crystallines, to explain what they called the <hi rend="italics">motion of trepidation,</hi> or <hi rend="italics">titubation.</hi></p><p>The first Crystalline, according to Regiomontanus, <pb n="340"/><cb/> &c, serves to account for the slow motion of the fixed stars; by which they advance a degree in about 70 years, according to the order of the signs, or from west to east; which occasions a precession of the equinox. The 2d serves to account for the motion of libration, or trepidation; by which the celestial sphere librates from one pole towards the other, causing a difference in the sun's greatest declination.</p></div2></div1><div1 part="N" n="CUBATURE" org="uniform" sample="complete" type="entry"><head>CUBATURE</head><p>, or <hi rend="smallcaps">Cubation</hi>, of a solid, is the measuring the space contained in it, or finding the solid content of it, or sinding a cube equal to it.</p><p>The cubature regards the content of a body, as the quadrature does the supersicies or area of a figure.</p></div1><div1 part="N" n="CUBE" org="uniform" sample="complete" type="entry"><head>CUBE</head><p>, a regular or solid body, consisting of six equal sides or faces, which are squares.—A die is a small cube.</p><p>It is also called a <hi rend="italics">hexaedron,</hi> because of its six sides, and is the 2d of the five Platonic or Regular bodies.</p><p>The cube is supposed to be generated by the motion of a square plane, along a line equal and perpendicular to one of its sides.</p><p><hi rend="italics">To describe a Rete, or Net, for forming a cube, or with which it may be covered.</hi>—Describe six squares as in the annexed figure, upon card paper, paste-board, or the like, of the size of the faces of the proposed cube; and cut it half through by the lines AB, CD, EF, AC, BD; then fold up the several squares till their edges meet, and so form the cube, or a covering over one, as in the figure annexed. <figure/></p><p><hi rend="italics">To determine the Surface and Solidity of a Cube.</hi>—Multiply one side by itself, which will give one square or face; then this multiplied by 6, the number of faces, will give the whole surface. Also multiply one side twice by itself, that is, cube it, and that will be the solid content.</p><p><hi rend="italics">Duplication of a</hi> <hi rend="smallcaps">Cube.</hi> See <hi rend="smallcaps">Duplication.</hi></p><div2 part="N" n="Cubes" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cubes</hi></head><p>, or <hi rend="smallcaps">Cubic</hi> <hi rend="italics">Numbers,</hi> are formed by multiplying any numbers twice by themselves. So the cubes of 1, 2, 3, 4, 5, 6, &c, are 1, 8, 27, 64, 125, 216, &c.</p><p>The third differences of the eubes of the natural numbers are all equal to each other, being the constant number 6. For, let <hi rend="italics">m</hi><hi rend="sup">3</hi>, <hi rend="italics">n</hi><hi rend="sup">3</hi>, <hi rend="italics">p</hi><hi rend="sup">3</hi> be any three adjacent cubes in the natural series as above, that is, whose roots <hi rend="italics">m, n, p</hi> have the common difference 1; then because ; so that <cb/> the difference between the 1st and 2d, and between the 2d and 3d cubes, are and the dif. of these differences, is the 2d difference.</p><p>In like manner the next 2d dif. is : hence the dif. of these two 2d diffs. is , which is therefore the constant 3d difference of all the series of cubes. And hence that series of cubes will be formed by addition only, viz, adding always the 3d dif. 6 to find the column or series of 2d diffs, then these added always for the 1st diffs, and lastly these always added for the cubes themselves, as below: <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3d Difs.</cell><cell cols="1" rows="1" rend="align=center" role="data">2d Difs.</cell><cell cols="1" rows="1" rend="align=center" role="data">1st Difs.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">91</cell><cell cols="1" rows="1" rend="align=right" role="data">125</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">127</cell><cell cols="1" rows="1" rend="align=right" role="data">216</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">169</cell><cell cols="1" rows="1" rend="align=right" role="data">343</cell></row></table> Peletarius, among various speculations concerning square and cubic numbers, shews that the continual sums of the cubic numbers, whose roots are 1, 2, 3, &c, form the series of squares whose roots are 1, 3, 6, 10, 15, 21, &c.</p><p>It is also a pretty property, that any number, and the cube of it, being divided by 6, leave the same remainder; the series of remainders being 0, 1, 2, 3, 4, 5, continually repeated. Or that the differences between the numbers and their cubes, divided by 6, leave always o remaining; and the quotients, with their successive differences, form the several orders of figurate numbers. Thus, <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes.</cell><cell cols="1" rows="1" rend="align=center" role="data">Difs.</cell><cell cols="1" rows="1" rend="align=center" role="data">Quot.</cell><cell cols="1" rows="1" rend="align=center" role="data">1 Dif.</cell><cell cols="1" rows="1" rend="align=center" role="data">2 Dif.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">125</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">216</cell><cell cols="1" rows="1" rend="align=right" role="data">210</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">343</cell><cell cols="1" rows="1" rend="align=right" role="data">336</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell></row></table></p><p>The following is a Table of the first 1000 cubic numbers. <pb n="341"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=12" role="data">TABLE OF CUBES.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Num</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">216000</cell><cell cols="1" rows="1" role="data">119</cell><cell cols="1" rows="1" role="data">1685159</cell><cell cols="1" rows="1" role="data">178</cell><cell cols="1" rows="1" rend="align=right" role="data">5639752</cell><cell cols="1" rows="1" role="data">237</cell><cell cols="1" rows="1" role="data">13312053</cell><cell cols="1" rows="1" role="data">296</cell><cell cols="1" rows="1" role="data">25934336</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">226981</cell><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data">1728000</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" rend="align=right" role="data">5735339</cell><cell cols="1" rows="1" role="data">238</cell><cell cols="1" rows="1" role="data">13481272</cell><cell cols="1" rows="1" role="data">297</cell><cell cols="1" rows="1" role="data">26198073</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">62</cell><cell cols="1" rows="1" rend="align=right" role="data">238328</cell><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" role="data">1771561</cell><cell cols="1" rows="1" role="data">180</cell><cell cols="1" rows="1" rend="align=right" role="data">5832000</cell><cell cols="1" rows="1" role="data">239</cell><cell cols="1" rows="1" role="data">13651919</cell><cell cols="1" rows="1" role="data">298</cell><cell cols="1" rows="1" role="data">26463592</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">63</cell><cell cols="1" rows="1" rend="align=right" role="data">250047</cell><cell cols="1" rows="1" role="data">122</cell><cell cols="1" rows="1" role="data">1815848</cell><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" rend="align=right" role="data">5929741</cell><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" role="data">13824000</cell><cell cols="1" rows="1" role="data">299</cell><cell cols="1" rows="1" role="data">26730899</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">125</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">262144</cell><cell cols="1" rows="1" role="data">123</cell><cell cols="1" rows="1" role="data">1860867</cell><cell cols="1" rows="1" role="data">182</cell><cell cols="1" rows="1" rend="align=right" role="data">6028568</cell><cell cols="1" rows="1" role="data">241</cell><cell cols="1" rows="1" role="data">13997521</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">27000000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">216</cell><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" rend="align=right" role="data">274625</cell><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" role="data">1906624</cell><cell cols="1" rows="1" role="data">183</cell><cell cols="1" rows="1" rend="align=right" role="data">6128487</cell><cell cols="1" rows="1" role="data">242</cell><cell cols="1" rows="1" role="data">14172488</cell><cell cols="1" rows="1" role="data">301</cell><cell cols="1" rows="1" role="data">27270901</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">343</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">287496</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">1953125</cell><cell cols="1" rows="1" role="data">184</cell><cell cols="1" rows="1" rend="align=right" role="data">6229504</cell><cell cols="1" rows="1" role="data">243</cell><cell cols="1" rows="1" role="data">14348907</cell><cell cols="1" rows="1" role="data">302</cell><cell cols="1" rows="1" role="data">27543608</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">512</cell><cell cols="1" rows="1" rend="align=right" role="data">67</cell><cell cols="1" rows="1" rend="align=right" role="data">300763</cell><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" role="data">200376</cell><cell cols="1" rows="1" role="data">185</cell><cell cols="1" rows="1" rend="align=right" role="data">6331625</cell><cell cols="1" rows="1" role="data">244</cell><cell cols="1" rows="1" role="data">14526784</cell><cell cols="1" rows="1" role="data">303</cell><cell cols="1" rows="1" role="data">27818127</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">729</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell><cell cols="1" rows="1" rend="align=right" role="data">314432</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">2048383</cell><cell cols="1" rows="1" role="data">186</cell><cell cols="1" rows="1" rend="align=right" role="data">6434856</cell><cell cols="1" rows="1" role="data">245</cell><cell cols="1" rows="1" role="data">14706125</cell><cell cols="1" rows="1" role="data">304</cell><cell cols="1" rows="1" role="data">28094464</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" rend="align=right" role="data">69</cell><cell cols="1" rows="1" rend="align=right" role="data">328509</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">2097152</cell><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" rend="align=right" role="data">6539203</cell><cell cols="1" rows="1" role="data">246</cell><cell cols="1" rows="1" role="data">14886936</cell><cell cols="1" rows="1" role="data">305</cell><cell cols="1" rows="1" role="data">28372625</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">1331</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">343000</cell><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" role="data">2146689</cell><cell cols="1" rows="1" role="data">188</cell><cell cols="1" rows="1" rend="align=right" role="data">6644672</cell><cell cols="1" rows="1" role="data">247</cell><cell cols="1" rows="1" role="data">15069223</cell><cell cols="1" rows="1" role="data">306</cell><cell cols="1" rows="1" role="data">28652616</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">1728</cell><cell cols="1" rows="1" rend="align=right" role="data">71</cell><cell cols="1" rows="1" rend="align=right" role="data">357911</cell><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" role="data">2197000</cell><cell cols="1" rows="1" role="data">189</cell><cell cols="1" rows="1" rend="align=right" role="data">6751269</cell><cell cols="1" rows="1" role="data">248</cell><cell cols="1" rows="1" role="data">15252992</cell><cell cols="1" rows="1" role="data">307</cell><cell cols="1" rows="1" role="data">28934443</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">2197</cell><cell cols="1" rows="1" rend="align=right" role="data">72</cell><cell cols="1" rows="1" rend="align=right" role="data">373248</cell><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data">2248091</cell><cell cols="1" rows="1" role="data">190</cell><cell cols="1" rows="1" rend="align=right" role="data">6859000</cell><cell cols="1" rows="1" role="data">249</cell><cell cols="1" rows="1" role="data">15438249</cell><cell cols="1" rows="1" role="data">308</cell><cell cols="1" rows="1" role="data">29218112</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">2744</cell><cell cols="1" rows="1" rend="align=right" role="data">73</cell><cell cols="1" rows="1" rend="align=right" role="data">389017</cell><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" role="data">2299968</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" rend="align=right" role="data">6967871</cell><cell cols="1" rows="1" role="data">250</cell><cell cols="1" rows="1" role="data">15625000</cell><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" role="data">29503629</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">3375</cell><cell cols="1" rows="1" rend="align=right" role="data">74</cell><cell cols="1" rows="1" rend="align=right" role="data">405224</cell><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" role="data">2352637</cell><cell cols="1" rows="1" role="data">192</cell><cell cols="1" rows="1" rend="align=right" role="data">7077888</cell><cell cols="1" rows="1" role="data">251</cell><cell cols="1" rows="1" role="data">15813251</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">29791000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">4096</cell><cell cols="1" rows="1" rend="align=right" role="data">75</cell><cell cols="1" 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rows="1" rend="align=right" role="data">118</cell><cell cols="1" rows="1" rend="align=right" role="data">1643032</cell><cell cols="1" rows="1" role="data">177</cell><cell cols="1" rows="1" role="data">5545233</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" rend="align=right" role="data">13144256</cell><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" role="data">25672375</cell><cell cols="1" rows="1" role="data">354</cell><cell cols="1" rows="1" role="data">44361864</cell></row></table><pb n="342"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Num</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell cols="1" rows="1" rend="align=center" role="data">Num.</cell><cell cols="1" rows="1" rend="align=center" role="data">Cubes</cell><cell 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role="data">996</cell><cell cols="1" rows="1" rend="align=right" role="data">988047936</cell></row><row role="data"><cell cols="1" rows="1" role="data">777</cell><cell cols="1" rows="1" role="data">469097433</cell><cell cols="1" rows="1" role="data">832</cell><cell cols="1" rows="1" role="data">575930368</cell><cell cols="1" rows="1" role="data">887</cell><cell cols="1" rows="1" role="data">697864103</cell><cell cols="1" rows="1" role="data">942</cell><cell cols="1" rows="1" role="data">835896888</cell><cell cols="1" rows="1" rend="align=right" role="data">997</cell><cell cols="1" rows="1" rend="align=right" role="data">991026973</cell></row><row role="data"><cell cols="1" rows="1" role="data">778</cell><cell cols="1" rows="1" role="data">470910952</cell><cell cols="1" rows="1" role="data">833</cell><cell cols="1" rows="1" role="data">578009537</cell><cell cols="1" rows="1" role="data">888</cell><cell cols="1" rows="1" role="data">700227072</cell><cell cols="1" rows="1" role="data">943</cell><cell cols="1" rows="1" role="data">838561807</cell><cell cols="1" rows="1" rend="align=right" role="data">998</cell><cell cols="1" rows="1" rend="align=right" role="data">994011992</cell></row><row role="data"><cell cols="1" rows="1" role="data">779</cell><cell cols="1" rows="1" role="data">472729139</cell><cell cols="1" rows="1" role="data">834</cell><cell cols="1" rows="1" role="data">580093704</cell><cell cols="1" rows="1" role="data">889</cell><cell cols="1" rows="1" role="data">702595369</cell><cell cols="1" rows="1" role="data">944</cell><cell cols="1" rows="1" role="data">841232384</cell><cell cols="1" rows="1" rend="align=right" role="data">999</cell><cell cols="1" rows="1" rend="align=right" role="data">997002999</cell></row><row role="data"><cell cols="1" rows="1" role="data">780</cell><cell cols="1" rows="1" role="data">474552000</cell><cell cols="1" rows="1" role="data">835</cell><cell cols="1" rows="1" role="data">582182875</cell><cell cols="1" rows="1" role="data">890</cell><cell cols="1" rows="1" role="data">704969000</cell><cell cols="1" rows="1" role="data">945</cell><cell cols="1" rows="1" role="data">843908625</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" rend="align=right" role="data">100000000</cell></row><row role="data"><cell cols="1" rows="1" role="data">781</cell><cell cols="1" rows="1" role="data">476379541</cell><cell cols="1" rows="1" role="data">836</cell><cell cols="1" rows="1" role="data">584277056</cell><cell cols="1" rows="1" role="data">891</cell><cell cols="1" rows="1" role="data">707347971</cell><cell cols="1" rows="1" role="data">946</cell><cell cols="1" rows="1" role="data">846590536</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table><pb n="344"/><cb/></p><p><hi rend="italics">The Cube of a Binomial,</hi> is equal to the cubes of the two parts or members, together with triple of the two parallelopipedons under each part and the square of the other; viz, . And hence the common method of extracting the cube root.</p><p><hi rend="smallcaps">Cubic</hi> <hi rend="italics">Equations,</hi> are those in which the unknown quantities rise to three dimensions; as <hi rend="italics">x</hi><hi rend="sup">3</hi>=<hi rend="italics">a,</hi> or , or , &c.</p><p>All cubic equations may be reduced to this form, ; viz, by taking away the 2d term.</p><p>All cubic equations have three roots; which are either all real, or else one only is real, and the other two imaginary; for all roots become imaginary by pairs.</p><p>But the nature of the roots as to real and imaginary, is known partly from the sign of the co-efficient <hi rend="italics">p,</hi> and partly from the relation between <hi rend="italics">p</hi> and <hi rend="italics">q:</hi> for the equation has always two imaginary roots when <hi rend="italics">p</hi> is positive; it has also two imaginary roots when <hi rend="italics">p</hi> is negative, provided ―(1/3)<hi rend="italics">p</hi>)<hi rend="sup">3</hi> is less ―(1/2)<hi rend="italics">q</hi>)<hi rend="sup">2</hi>, or 4<hi rend="italics">p</hi><hi rend="sup">3</hi> less than 27<hi rend="italics">q</hi><hi rend="sup">2</hi>; otherwise the roots are all real, namely, whenever <hi rend="italics">p</hi> is negative, and 4<hi rend="italics">p</hi><hi rend="sup">3</hi> either equal to, or greater than 27<hi rend="italics">q</hi><hi rend="sup">2</hi>.</p><p>Every cubic equation of the above form, viz, wanting the 2d term, has both positive and negative roots, and the greatest root is always equal to the sum of the two less roots; viz, either one positive root equal to the sum of the two negative ones, or else one negative root equal to the sum of two smaller and positive ones. And the sign of the greatest, or single root, is positive or negative, according as <hi rend="italics">q</hi> is positive or negative when it stands on the right-hand side of the equation, thus ; and the two smaller roots have always the contrary sign to <hi rend="italics">q.</hi></p><p>So that, in general, the sign of <hi rend="italics">p</hi> determines the nature of the roots, as to real and imaginary; and the sign of <hi rend="italics">q</hi> determines the affection of the roots, as to positive and negative. See my Tract on Cubic Equations in the Philos. Trans. for 1780.</p><p><hi rend="italics">To find the Values of the Roots of Cubic Equations.</hi> Having reduced the equation to this form , its root may be found in various ways; the first of these, is that which is called Cardan's Rule, by whom it was first published, but invented by Ferreus and Tartalea. See <hi rend="smallcaps">Algebra.</hi> The rule is this: Put <hi rend="italics">a</hi>=(1/3)<hi rend="italics">p,</hi> and <hi rend="italics">b</hi>=(1/2)<hi rend="italics">q;</hi> then is Cardan's root ; or if there be put , and ; then , the 1st or Cardan's root, also is the 2d root, and is the 3d root.</p><p>Now the first of these, or Cardan's root, is always a real root, though it is not always the greatest root, as it has been commonly mistaken for. And yet this rule always exhibits the root in the form of an imaginary quantity when the equation has no imaginary roots at all; but in the form of a real quantity when the equation has two imaginary roots. See the reason of this explained in my Tract above cited, pa. 407. As to <cb/> the other two roots, viz, though, in their general form, they have an imaginary appearance; yet, by substituting certain particular numbers, they come out in a real form in all such cases as they ought to be so.</p><p>But, after the first root is found, by Cardan's rule, the other two roots may be found, or exhibited, in several other different ways; some of which are as follow:</p><p>Let <hi rend="italics">r</hi> denote the 1st, or Cardan's root, and <hi rend="italics">v</hi> and <hi rend="italics">w</hi> the other two roots: then is , and <hi rend="italics">vwr</hi>=<hi rend="italics">q</hi>; and the resolution of these two equations will give the other two roots <hi rend="italics">v</hi> and <hi rend="italics">w.</hi></p><p>Or resolve the quadratic equation , and its two roots will be those sought. Or the same two roots will be either.</p><p><hi rend="italics">Ex.</hi> 1. If the equation be : here ; hence : therefore , the 1st root; and , the other two roots.</p><p><hi rend="italics">Ex.</hi> 2. If : here <hi rend="italics">a</hi>=-2, and <hi rend="italics">b</hi>=2; therefore : hence then , the first root; and 1±√3 the other two roots.</p><p><hi rend="italics">Ex.</hi> 3. If : here <hi rend="italics">a</hi>=6, and <hi rend="italics">b</hi>=3; then , and : therefore , the 1st root, and are the two other roots.</p><p>2. Another method for the roots of the equation , is by means of infinite series, as shewn at pa. 415 and seq. of my Tract above cited; whence it appears that the roots are exhibited in various forms of series as follow: viz, &c for the 1st root, and &c for the two other roots: where , and . <pb n="345"/><cb/></p><p>And various other series for the same purpose may also be seen in my Tract, so often before cited.</p><p>3. A third method for the roots of cubic equations, is by angular sections, and the table of sines. It was first hinted by Bombelli, in his Algebra, that angles are trisected by the resolution of cubic equations. Afterwards, Vieta gave the resolution of cubics, and the higher equations, by angular sections. Next, Albert Girard, in his Invention Nouvelle en l'Algebre, shews how to resolve the irreducible case in cubics by a table of sines: and he also constructs the same, or finds the roots, by the intersection of the hyperbola and circle. Halley and De Moivre also gave rules and examples of the same sort of resolutions by a table of sines. And, lastly, Mr. Anthony Thacker invented, and Mr. William Brown computed, a large set of similar tables, for resolving affected quadratic and cubic equations, with their application to the resolution of biquadratic ones.</p><p>4. Lastly, the several methods of approach, or approximation, for the roots of all affected equations, which have been used in various ways by Stevin, Vieta, Newton, Halley, Raphson, and others.</p><p>To these may be added the method of Trial-and-error, or of Double Position, one of the easiest and best of any. Of this method, let there be taken the last example, viz, , in which it is evident that <hi rend="italics">x</hi> is very nearly equal to 1/3, but a little less; take it therefore <hi rend="italics">x</hi>=.33; then , but should be 6, and therefore the error is .024063 in defect.</p><p>Again suppose <hi rend="italics">x</hi>=.34; then , which is .159304 in excess. Therefore , the root as before very nearly.</p><p>For the construction of cubic equations, see C<hi rend="smallcaps">ONSTRUCTION.</hi></p><p><hi rend="smallcaps">Cubic</hi> <hi rend="italics">Foot,</hi> of any thing, is so much of it as is contained in a cube whose side is one foot.</p><p><hi rend="smallcaps">Cubic</hi> <hi rend="italics">Hyperbola,</hi> is a figure expressed by the equation <hi rend="italics">xy</hi><hi rend="sup">2</hi>=<hi rend="italics">a,</hi> having two asymptotes, and consisting of two hyperbolas, lying in the adjoining angles of the asymptotes, and not in the opposite angles, like the Apollonian hyperbola; being otherwise called by Newton, in his Enumeratio Linearum Tertii Ordinis, an hyperbolismus of a parabola; and is the 65th species of those lines according to him.</p><p><hi rend="smallcaps">Cubic</hi> <hi rend="italics">Numbers.</hi> See <hi rend="smallcaps">Cubes.</hi> <figure><head><hi rend="italics">Fig 2</hi></head></figure></p><p><hi rend="smallcaps">Cubic</hi> <hi rend="italics">Parabola,</hi> a curve, as BCD, of the 2d order, having two infinite legs CB, CD, tending contrary <cb/> ways. And if the absciss, AP or <hi rend="italics">x,</hi> touch the curve in C, the relation between the absciss and ordinate, viz, AP=<hi rend="italics">x,</hi> and PM=<hi rend="italics">y,</hi> is expressed by the equation ; or when A coincides with C, by the equation <hi rend="italics">y</hi>=<hi rend="italics">ax</hi><hi rend="sup">3</hi>, which is the simplest form of the equation of this curve.</p><p>If the right line AP (fig. 2) cut the cubical parabola in three points A, B, C; and from any point P there be drawn the right line or ordinate PM, cutting the curve in one point M only: then will PM be always as the solid APXBPXCP; which is an essential property of this curve.</p><p>And hence it is easy to construct a cubic equation, as , by the intersection of this curve and a right line. See the Construction of a cubic equation by means of the cubic parabola and a right line, by Dr. Wallis, in his Algebra: As also the Construction of equations of 6 dimensions, by means of the cubic parabola and a circle, by Dr. Halley, in a lecture formerly read at Oxford.</p><p><hi rend="italics">The curve of this parabola</hi> cannot be rectisied, not even by means of the conic sections. But a circle may be found equal to the <hi rend="italics">Curve Surface,</hi> generated by the rotation of the curve AM about the tangent AP to the principal vertex A. Let MN be an ordinate, and MT a tangent at the point M; and let PM be parallel to AN. Divide MN in the point O, so that MO be to ON as TM to MN. Then a mean proportional between TM+ON and 1/3 of AN will be the radius of a circle, whose area is equal to the superficies described by that rotation, viz, of AM about AP. <figure/></p><p><hi rend="italics">The Area of a Cubic Parabola</hi> is 3/4 of its circumscribing parallelogram.</p><p><hi rend="smallcaps">Cubic</hi> <hi rend="italics">Root,</hi> of any number, or quantity, is such a quantity as being cubed, or twice multiplied by itself, shall produce that which was given. So, the cubic root of 8 is 2, because 2<hi rend="sup">3</hi> or 2X2X2 is equal to 8.</p><p>The common method of extracting the cube root, founded on the property given above, viz, , is found in every book of common arithmetic, and is as old at least as Lucas de Burgo, where it is first met with in print. Other methods for the cube root may be seen under the article E<hi rend="smallcaps">XTRACTION</hi> <hi rend="italics">of Roots,</hi> particularly this one, viz, the cube root of <hi rend="italics">n,</hi> or , very nearly, or the cube root of <hi rend="italics">n</hi> nearly; where <hi rend="italics">n</hi> <pb n="346"/><cb/> is any number given whose cube root is sought, and <hi rend="italics">a</hi><hi rend="sup">3</hi> is the nearest complete cube to <hi rend="italics">n,</hi> whether greater or less.</p><p>For example, suppose it were proposed to double the cube, or, which comes to the same thing, to extract the cube root of the number 2. Here the nearest cube is 1, whose cube root is 1 also, that is, <hi rend="italics">a</hi><hi rend="sup">3</hi>=1, and <hi rend="italics">a</hi>=1, also <hi rend="italics">n</hi>=2; therefore nearly.</p><p>But, for a nearer value, assume now ; then is ; hence , or the cube root of 2, which is true in the last place of decimals.</p><p>And this is the simplest and easiest method for the cube root of any number. See its investigation in my Tracts, vol. 1, pa. 49.</p><p>Every number or quantity has three cubic roots, one that is real, and two imaginary: So, the cube root of 1 is either 1, or ; and if <hi rend="italics">r</hi> be the real root of any cube <hi rend="italics">r</hi><hi rend="sup">3</hi>, the two imaginary cubic roots of it will be for any one of these being cubed, gives the same cube <hi rend="italics">r</hi><hi rend="sup">3</hi>.</p><p><hi rend="smallcaps">Cubing</hi> <hi rend="italics">of a Solid.</hi> See <hi rend="smallcaps">Cubature.</hi></p></div2><div2 part="N" n="Cubo-cube" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cubo-cube</hi></head><p>, the 6th power.</p></div2><div2 part="N" n="Cubo-cubo-cube" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cubo-cubo-cube</hi></head><p>, the 9th power.</p></div2></div1><div1 part="N" n="CULMINATION" org="uniform" sample="complete" type="entry"><head>CULMINATION</head><p>, the passage of a star or planet over the meridian, or that point of its orbit which it is in at its greatest altitude. Hence a star is said to culminate, when it passes the meridian.</p><p><hi rend="italics">To find the time of a Star's culminating.</hi> Estimate the time nearly; and find the right ascension both of the sun and star corrected for this estimated time; then the disference between these right ascensions, converted into solar time, at the rate of 15 degrees to the hour, gives the time of southing. See an example of this calculation every year in White's Ephemeris, pa. 45.</p></div1><div1 part="N" n="CULVERIN" org="uniform" sample="complete" type="entry"><head>CULVERIN</head><p>, was the name of a piece of ordnance; but is now disused.</p><p>CUNETTE. See <hi rend="smallcaps">Cuvette.</hi></p><p>CUNEUS. See <hi rend="smallcaps">Wedge.</hi></p></div1><div1 part="N" n="CUNITIA" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CUNITIA</surname> (<foreName full="yes"><hi rend="smallcaps">Maria</hi></foreName>)</persName></head><p>, a lady of Silesia, who was famous for her extensive knowledge in many branches of learning, but more particularly in mathematics and astronomy, upon which she wrote several ingenious treatises; particularly one, entitled <hi rend="italics">Urania Propitia,</hi> printed in 1650, in Latin and German, and dedicated to Ferdinand the third, emperor of Germany. In this work are contained astronomical tables of great ease and accuracy, founded upon Kepler's hypothesis. But notwithstanding her merit shines with such peculiar lustre as to reflect honour on her sex, history does not inform us of the time of her birth or death.</p></div1><div1 part="N" n="CURRENT" org="uniform" sample="complete" type="entry"><head>CURRENT</head><p>, a stream or flux of water in any direction. The <hi rend="italics">setting</hi> of the current, is that point of the <cb/> compafs towards which the waters run; and the <hi rend="italics">drift</hi> of a current is the rate it runs an hour.</p><p>Currents in the sea, are either natural and general, as arising from the diurnal rotation of the earth on its axis, or the tides, &c; or accidental and particular, caused by the waters being driven against promontories, or into gulphs and streights; from whence they are forced back, and thus disturb the natural flux of the sea.</p><p>The currents are so violent under the equator, where the motion of the earth is greatest, that they hurry vessels very speedily from Africa to America; but absolutely prevent their return the same way: so that ships are forced to run as far as the 40th or 45th degree of north latitude, to fall into the return of the current again, to bring them home to Europe. It is shewn by Governor Pownal, that this current performs a continual circulation, setting out from the Guinea coast in Africa, for example, from thence crossing straight over the Atlantic ocean, and so setting into the Gulph of Mexico by the south side of it; then sweeping round by the bottom of the Gulph, it issues out by the north side of it, and thence takes a direction north-easterly along the coast of North America, till it arrives near Newfoundland, where it is turned by a rounding motion backward across the Atlantic again, upon the coasts of Europe, and from thence southward again to the coast of Africa, from whence it set out.</p><p>In the streights of Gibraltar, the currents set in by the south side, sweep along the coast of Africa to Egypt, by Palestine, and so return by the northern side, or European coasts, and issue out again by the northern side of the streights. In St. George's Channel too they usually set eastward. The great violence and danger of the sea in the Streights of Magellan, is attributed to two contrary currents setting in, one from the south, and the other from the north sea.</p><p>Currents are of some consideration in the art of navigation, as a ship is by them either accelerated or retarded in her course, according as the set is with or against the ship's motion. If a ship sail along the direction of a current, it is evident that the velocity of the current must be added to that of the vessel: but if her course be directly against the current, it must be subtracted: and if she sail athwart it, her motion will be compounded with that of the current; and her velocity augmented or retarded, according to the angle of her direction with that of the current; that is, she will proceed in the diagonal of the parallelogram formed according to the two lines of direction, and will describe or pass over that diagonal in the same time in which she would have described either of the sides, by the separate velocities.</p><p>For suppose ABDC be a parallelogram, the diagonal of which is <figure/> AD. Now if the wind alone would drive the ship from A to B, in the same time as the current alone would drive it from A to C: Then, as the wind neither helps nor hinders the ship from coming towards the line CD, the current will bring it there in the same time as if the wind did not act: And as the current neither helps nor hinders the ship from <pb n="347"/><cb/> coming towards the line BD, the wind will bring it there in the same time as if the current did not act: Therefore the ship must, at the end of that time, be found in both those lines, that is, in their meeting D: Consequently the ship must have passed from A to D in the diagonal AD. See <hi rend="smallcaps">Composition</hi> <hi rend="italics">of Forces.</hi></p><p>See the Sailing in Currents largely exemplified in Robertson's Navigation, vol. 2, book 7, sect. 8.</p></div1><div1 part="N" n="CURSOR" org="uniform" sample="complete" type="entry"><head>CURSOR</head><p>, a small piece of brass &c that slides; as, the piece in an equinoctial ring dial that slides to the day of the month; or the little ruler or label of brass sliding in a groove along the middle of another label, representing the horizon in the analemma; or the point that slides along the beam compass; &c.</p><p>CURTATE <hi rend="italics">Distance,</hi> is the distance of a planet's place from the sun, reduced to the ecliptic; or, the interval between the sun and that point where a perpendicular, let fall from the planet, meets the ecliptic.</p></div1><div1 part="N" n="CURTATION" org="uniform" sample="complete" type="entry"><head>CURTATION</head><p>, the interval between a planet's distance from the sun, and the curtate distance.</p><p>From the foregoing article it is easy to find the curtate distance; whence the manner of constructing tables of curtation is obvious; the quantity of inclination, reduction, and curtation of a planet, depending on the argument of latitude. Kepler, in his Rodolphine Tables, reduces the tables of them all into one, under the title of <hi rend="italics">Tabulæ Latitudinariæ.</hi></p></div1><div1 part="N" n="CURTIN" org="uniform" sample="complete" type="entry"><head>CURTIN</head><p>, <hi rend="smallcaps">Curtain</hi>, or <hi rend="smallcaps">Courtine</hi>, in Fortification, that part of a wall or rampart that joins two bastions, or lying between the flank of one and that of another.—The curtain is usually bordered with a parapet 5 feet high; behind which the soldiers stand to fire upon the covert-way, and into the moat.</p><p>CURVATURE <hi rend="italics">of a Line,</hi> is its bending, or flexure; by which it becomes a curve, of any peculiar form and properties. Thus, the nature of the curvature of a circle is such, as that every point in the periphery is equally distant from a point within, called the centre; and so the curvature of the same circle is every where the same; but the curvature in all other curves is continually varying.—The curvature of a circle is so much the more, as its radius is less, being always reciprocally as the radius; and the curvature of other curves is measured by the reciprocal of the radius of a circle having the same degree of curvature as any curve has, at some certain point.</p><p>Every curve is bent from its tangent by its curvature, the measure of which is the same as that of the angle of contact formed by the curve and tangent. Now the same tangent AB is common to an infinite number of circles, or other curves, all touching it and each other in the same point of contact C. So that any curve DCE may be touched by an infinite number of different circles at the same point C; and some of these circles fall wholly within it, being more curved, or having a greater curvature than that curve; while others fall without it near the point of contact, or between the curve and tangent at that point, and so, being less deflected from the tangent than the curve is, they have a less degree of curvature there. Consequently there is one, of this infinite number of circles, which neither falls below it nor above it, but, being equally deflected from the tangent, coincides with it most intimately of all the circles; and the radius of this circle is called the <cb/> <hi rend="italics">radius of curvature</hi> of the curve; also the circle itself is called the <hi rend="italics">circle of curvature,</hi> or the <hi rend="italics">osculatory</hi> circle of that curve, because it touches it so closely that no other circle can be drawn between it and the curve. <figure/></p><p>As a curve is separated from its tangent by its flexure or curvature, so it is separated from the osculatory circle by the increase or decrease of its curvature; and as its curvature is greater or less, according as it is more or less deflected from the tangent, so the variation of curvature is greater or less, according as it is more or less separated from the circle of curvature.</p><p>It appears, however, from the demonstration of geometricians, that circles may touch curve lines in such a manner, that there may be indefinite degrees of more or less intimate contact between the curve and its osculatory circle; and that a conic section may be described that shall have the same curvature with a given line at a given point, and the same variation of curvature, or a contact of the same kind with the circle of curvature.</p><p>If we conceive the tangent of any proposed curve to be a base, and that a new line or curve be described, whose ordinate, upon the same base or absciss, is a 3d proportional to the ordinate and base of the first; this new curve will determine the chord of the circle of curvature, by its intersection with the ordinate at the point of contact; and it will also measure the variation of curvature, by means of the tangent of the angle in which it cuts that circle: the less this angle is, the closer is the contact of the curve and circle of curvature; and of this contact there may be indesinite degrees.</p><p>For example, let EMH be any curve, to which ET is a tangent at the point E; then let there be always taken MT : ET :: ET : TK, and through all the points K draw the curve BKF; then from the point of contact E draw EB parallel to the ordinate TK, meeting the last curve in B; and finally, describe a circle ERQB through the point B and touching ET in E; and it shall be the osculatory circle to the given curve EMH. And the contact of EM and ER is always the closer, the less the angle KBQ is. See Maclaurin's Fluxions, art. 366.</p><p>Hence it follows, that the contact of the curve EMH and its osculatory circle is closest, when the curve BK touches the arch BQ in B, the angle BET being given; and it is farthest from this, or most open, when BK touches the right line BE in B.</p><p>Hence also there may be indefinite degrees of more and more intimate contact between a circle and a curve. The first degree is when the same right line <pb n="348"/><cb/> touches them both in the same point; and a contact of this sort may take place between any circle and any arch of a curve. The 2d is when the curve EMH and circle ERB have the same curvature, and the tangents of the curve BKF and circle BQE intersect each other at B in any assignable angle. The contact of the curve EM and circle of curvature ER at E, is of the 3d degree, or order, and their osculation is of the 2d, when the curve BKF touches the circle BQE at B, but so as not to have the same curvature with it. The contact is of the 4th degree, or order, and their osculation of the 3d, when the curve BKF has the same curvature with the circle BQE at B, but so as that their contact is only of the 2d degree. And this gradation of more and more intimate contact, or of approximation towards coincidence, may be continued indefinitely, the contact of EM and ER at E being always of an order two degrees closer than that of BK and BQ at B. There is also an indefinite variety comprehended under each order: thus, when EM and ER have the same curvature, the angle formed by the tangents of BK and BQ admits of indefinite variety, and the contact of EM and ER is the closer the less that angle is. And when that angle is of the same magnitude, the contact of EM and ER is the closer, the greater the circle of curvature is. When BK and BQ touch at B, they may touch on the same or on different sides of their common tangent; and the angle of contact KBQ may admit of the same variety with the angle of contact MER; but as there is seldom occasion for considering those higher degrees of more intimate contact of the curve EMH and circle of curvature ERB, Mr. Maclaurin calls the contact or osculation of the same kind, when, the chord EB and angle BET being given, the angle contained by the tangents of BK and BQ is of the same magnitude.</p><p>When the curvature of EMH increases from E towards H, and consequently corresponds to that of a circle gradually less and less, the arch EM falls within ER, the arch of the osculatory circle, and BK is within BQ. The contrary happens when the curvature of EM decreases from E towards H, and consequently corresponds to that of a circle which is gradually greater and greater, the arch EM falls without ER, and BK is without BQ. And according as the curvature of EM varies more or less, it is more or less unlike to the uniform curvature of a circle; the arch of the curve EMH separates more or less from the arch of the osculatory circle ERB, and the angle contained by the tangents of BKF and BQE at B, is greater or less. Thus the quality of curvature, as it is called by Newton, depends on the angle contained by the tangents of BK and BQ at B; and the measure of the inequability or variation of curvature, is as the tangent of this angle, the radius being given, and the angle BET being a right one.</p><p>The radii of curvature of similar arcs in similar figures, are in the same ratio as any homologous lines of these sigures; and the variation of curvature is the same. See Maclaurin, art. 370.</p><p>When the proposed curve EMH is a conic section, the new line BKF is also a conic section; and it is a right line when EMH is a parabola, to the axis of which the ordinates TK are parallel. BKF is also a <cb/> right line when EMH is an hyperbola, to one asymptote of which the ordinate TK is parallel.</p><p>When the ordinate EB, at the point of contact E, instead of meeting the new curve BK, is an asymptote to it, the curvature of EM will be less than in any circle; and this is the case in which it is said to be infinitely little, or that the radius of curvature is infinitely great. And of this kind is the curvature at the points of contrary flexure in lines of the 3d order.</p><p>When the curve BK passes through the point of contact E, the curvature is greater than in any circle, or the radius of curvature vanishes; and in this case the curvature is said to be infinitely great. Of this kind is the curvature at the cusps of the lines of the 3d order.</p><p>As to the degree of curvature in lines of the 3d and higher orders, see Maclaurin, art. 379; also art. 380, when the proposed curve is mechanical.</p><p>As curves which pass through the same point have the same tangent when the first fluxions of the ordinates are equal, so they have the same curvature when the 2d fluxions of the ordinate are likewise equal; and half the chord of the osculatory circle that is intercepted between the points where it intersects the ordinate, is a 3d proportional to the right lines that measure the 2d fluxion of the ordinate and first fluxion of the curve, the base being supposed to flow uniformly. When a ray revolving about a given point, and terminated by the curve, becomes perpendicular to it, the first fluxion of the radius vanishes; and if its 2d fluxion vanish at the same time, that point must be the centre of curvature. The same may be said, when the angular motion of the ray about that point is equal to the angular motion of the tangent of the curve; as the angular motion of the radius of a circle about its centre is always equal to the angular motion of the tangent of the circle. Hence the various properties of the circle may suggest several theorems for determining the centre of curvature.</p><p>See art. 396 of the said book, also the following, concerning the curvature of lines that are described by means of right lines revolving about given poles, or of angles that either revolve about such poles, or are carried along fixed lines.</p><p>It is to be observed that, as when a right line intersects an arc of a curve in two points, if by varying the position of that line the two intersections unite in one point, it then becomes the tangent of the arc; so when a circle touches a curve in one point, and intersects it in another, if, by varying the centre, this intersection joins the point of contact, the circle has then the closest contact with the arc, and becomes the circle of curvature; but it still continues to intersect the curve at the same point where it touches it, that is, where the same right line is their common tangent, unless another intersection join that point at the same time. In general, the circle of curvature intersects the curve at the point of osculation, only when the number of the successive orders of fluxions of the radius of curvature, that vanishes when this radius comes to the point of osculation, is an even number.</p><p>It has been supposed by some, that two points of contact, or four intersections of the curve and circle of curvature, must join to form an osculation. But Mr. <pb n="349"/><cb/> James Bernoulli justly insisted, that the coalition of one point of contact and one intersection, or of three interfections, was sufficient. In which case, and in general, when an odd number of intersections only join each other, the point where they coincide continues to be an intersection of the curve and circle of curvature, as well as a point of their mutual contact and osculation. See Maclaurin's Flux. art. 493. <figure/></p><p>From these principles may be determined the circle of curvature at any point of a conic section. Thus, <*>uppose AEMHG be any conic section, to the point E of which the circle or radius of curvature is to be found. Let ET be a tangent at E, and draw EGB and the tangent HI parallel to the chords of the circle of curvature; then take EB to EG as EI<hi rend="sup">2</hi> to HI<hi rend="sup">2</hi>; or, when the section has a centre O, as in the ellipse and hyperbola, as the square of the semi-diameter O<hi rend="italics">a</hi> parallel to ET, is to the square of the semi-diameter OA parallel to EB; and a circle EB described upon the chord EB that touches ET, will be the circle of curvature sought.</p><p>When BET is a right angle, or EB is the diameter of the circle of curvature, EG will be the axis of the conic section, and EB will be the parameter of this axis; also when the point G, where the conic section cuts EB, and the point B, are on the same side of E, then EMG will be an ellipsis, and EG the greater or less axis, according as EG is greater or less than EB.</p><p>The propositions relating to the curvature of the conic sections, commonly given by authors, follow with out much difficulty from this construction.</p><p>1. When the chord of curvature, thus found, passes through the centre of the conic section, it will then be equal to the parameter of the diameter that passes through the point of contact.</p><p>2. The square of the semi-diameter O<hi rend="italics">a,</hi> is to the rectangle of half the transverse and half the conjugate axis, as the radius of curvature CE is to O<hi rend="italics">a.</hi> And therefore the cube of the semi-diameter O<hi rend="italics">a,</hi> parallel to the tangent ET, is equal to the solid contained by the radius of curvature CE, and the rectangle of the two axes.</p><p>3. The perpendicular to either axis bisects the angle made by the chord of curvature, and the common tangent of the conic section and circle of curvature.</p><p>4. The chord of the osculatory circle that passes through the focus, the diameter conjugate to that which passes through the point of contact, and the transverse axis of the figure, are in continued proportion. <cb/></p><p>5. When the section is an ellipse, if the circle of curvature at E meet O<hi rend="italics">a</hi> in <hi rend="italics">d,</hi> the square of E<hi rend="italics">d</hi> will be equal to twice the square of O<hi rend="italics">a.</hi> Hence E<hi rend="italics">d</hi> : O<hi rend="italics">a</hi> :: √2 : 1. Which gives an easy method of determining the circle of curvature to any point E, when the semidiameter O<hi rend="italics">a</hi> is given in magnitude and position.</p><p>Several other properties of the circle of curvature, and methods of determining it when the section is given; or vice versa, of determining the section when the circle of curvature is given, may be seen in Maclaurin's Flux. art. 375. See also the Appendix to Maclaurin's Algebra, sect. 1.</p><p><hi rend="italics">To determine the Radius and Circle of Curvature by the Method of Fluxions.</hi> Let AE<hi rend="italics">e</hi> be any curve, coneave towards its axis AD; draw an ordinate DE to the point E where the curve is required to be found; and suppose EC perpendicular to the curve, and equal to the radius of the circle BE<hi rend="italics">e</hi> of curvature sought; lastly, draw E<hi rend="italics">d</hi> parallel to AD, and <hi rend="italics">de</hi> parallel and indefinitely near to DE; thereby making E<hi rend="italics">d</hi> the fluxion or increment of the absciss AD, also <hi rend="italics">de</hi> the fluxion of the ordinate DE, and E<hi rend="italics">e</hi> that of the curve AE. Now put <hi rend="italics">x</hi>=AD, <hi rend="italics">y</hi>=DE, <hi rend="italics">z</hi>=AE, and <hi rend="italics">r</hi>=CE the radius of curvature; then is ED=<hi rend="italics">x<hi rend="sup">.</hi>, de</hi>=<hi rend="italics">y<hi rend="sup">.</hi>,</hi> and E<hi rend="italics">e</hi>=<hi rend="italics">z<hi rend="sup">.</hi>.</hi> <table><row role="data"><cell cols="1" rows="1" role="data">Now, by sim. tri. the 3 lines</cell><cell cols="1" rows="1" rend="align=right" role="data">E<hi rend="italics">d , de,</hi> E<hi rend="italics">e,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">x<hi rend="sup">.</hi> , y<hi rend="sup">.</hi> , z<hi rend="sup">.</hi>,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">are respectively as the three</cell><cell cols="1" rows="1" rend="align=right" role="data">GE, GC, CE;</cell></row><row role="data"><cell cols="1" rows="1" role="data">therefore</cell><cell cols="1" rows="1" rend="align=right" role="data">GC.<hi rend="italics">x<hi rend="sup">.</hi></hi>=GE <hi rend="italics">y<hi rend="sup">.</hi></hi>;</cell></row></table> and the flux. of this equa. is , or because GC<hi rend="sup">.</hi>=-BG<hi rend="sup">.</hi>, it is But, since the two curves AE and BE have the same curvature at the point E, their abscisses and ordinates have the same fluxions at that point, that is E<hi rend="italics">d</hi> or <hi rend="italics">x<hi rend="sup">.</hi></hi> is the fluxion both of AD and BG, and <hi rend="italics">de</hi> or <hi rend="italics">y<hi rend="sup">.</hi></hi> is the fluxion both of DE and GE. In the above equation therefore substitute <hi rend="italics">x<hi rend="sup">.</hi></hi> for BG<hi rend="sup">.</hi>, and <hi rend="italics">y<hi rend="sup">.</hi></hi> for G<hi rend="sup">.</hi>E, and it <table><row role="data"><cell cols="1" rows="1" role="data">becomes</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" role="data">.</cell></row></table> Now mult. the three terms of this equation respectively by these three quantities, , which are equal, and it becomes ; and hence , which is the general value of the radius of curvature for all curves whatever, in terms of the fluxions of the absciss and ordinate.</p><p>Farther, as in any case either <hi rend="italics">x</hi> or <hi rend="italics">y</hi> may be supposed to flow equably, that is, either <hi rend="italics">x<hi rend="sup">.</hi></hi> or <hi rend="italics">y<hi rend="sup">.</hi></hi> constant quantities, or <hi rend="italics">x<hi rend="sup">..</hi></hi> or <hi rend="italics">y<hi rend="sup">..</hi></hi>=to nothing, by this supposition either of the terms in the denominator of the value of <hi rend="italics">r</hi> may be made to vanish. So that when <hi rend="italics">x<hi rend="sup">.</hi></hi> is constant, the value of <hi rend="italics">r</hi> is , but <hi rend="italics">r</hi> is when <hi rend="italics">y<hi rend="sup">.</hi></hi> is constant.</p><p>For example, suppose it were required to find the radius or circle of curvature to any point of a parabola, its vertex being A, and axis AD.—Now the equa- <pb n="350"/><cb/> tion of the curve is <hi rend="italics">ax</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>; hence <hi rend="italics">ax<hi rend="sup">..</hi></hi> = 2<hi rend="italics">yy<hi rend="sup">.</hi>,</hi> and <hi rend="italics">ax<hi rend="sup">..</hi></hi> = 2<hi rend="italics">y</hi><hi rend="sup">.</hi><hi rend="sup">2</hi>, supposing <hi rend="italics">y<hi rend="sup">.</hi></hi> constant, also <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">2</hi> = 4<hi rend="italics">y</hi><hi rend="sup">2</hi><hi rend="italics">y</hi><hi rend="sup">2</hi>; hence <hi rend="italics">r</hi> or , the general value of the radius of curvature for any point E, the ordinate to which cuts offthe absciss AD = <hi rend="italics">x.</hi></p><p>Hence, when <hi rend="italics">x</hi> or the absciss is nothing, the last expression becomes barely for the radius of curvature at the vertex of the parabola; that is, the diameter of the circle of curvature at the vertex of a parabola, is equal to <hi rend="italics">a</hi> the parameter of axis.</p><p><hi rend="italics">Variation of</hi> <hi rend="smallcaps">Curvature.</hi> See <hi rend="smallcaps">Variation.</hi></p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Curvature</hi>, is used for the curvature of a line, which twists so that all the parts of it do not lie in the same plane.</p></div1><div1 part="N" n="CURVE" org="uniform" sample="complete" type="entry"><head>CURVE</head><p>, a line whose several parts p[rdot]oceed bowing, or tend different ways; in opposition to a straight line, all whose parts have the same course or direction.</p><p>The doctrine of curves, and of the figures and solids generated from them, constitute what is called the higher geometry. <figure/></p><p>In a curve, the line AD, which bisects all the parallel lines MN, is called a <hi rend="italics">diameter;</hi> and the point A, where the diameter meets the curve, is called the <hi rend="italics">vertex:</hi> if AD bisect all the parallels at right angles, it is called the <hi rend="italics">axis.</hi> The parallel lines MN are called <hi rend="italics">ordinates,</hi> or <hi rend="italics">applicates;</hi> and their halves, PM, or PN, <hi rend="italics">semi-ordinates.</hi> The portion of the diameter AP, between the vertex, or any other fixed point, and an ordinate, is called the absciss; also the concourse of all the diameters, if they meet all in one point, is the <hi rend="italics">centre.</hi> This definition of the diameter, as bisecting the parallel ordinates, respects only the conic sections, or such curves as are cut only in two points by the ordinates; but in the lines of the 3d order, which may be cut in three points by the ordinates, then the diameter is that line which cuts the ordinates so, that the sum of the two parts that lie on one side of it, shall be equal to the part on the other side: and so on for curves of higher orders, the sum of the parts of the ordinates on one side of the diameter, being always equal to the sum of the parts on the other side of it.</p><p>Curve lines are distinguished into <hi rend="italics">algebraical</hi> or <hi rend="italics">geometrical,</hi> and <hi rend="italics">transcendental</hi> or <hi rend="italics">mechanical.</hi></p><p><hi rend="italics">Algebraical</hi> or <hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Curves</hi>, are those in which the relation of the abscisses AP, to the ordinates PM, can be expressed by a common algebraic equation <cb/></p><p>And <hi rend="italics">Transcendental</hi> or <hi rend="italics">Mechanical Curves,</hi> are such as cannot be so defined or expressed by an algebraical equation. See <hi rend="smallcaps">Transcendental</hi> <hi rend="italics">Curve.</hi></p><p>Thus, suppose, for instance, the curve be the circle; and that the radius AC = <hi rend="italics">r,</hi> the absciss AP = <hi rend="italics">x,</hi> and the ordinate PM = <hi rend="italics">y</hi>; then, because the nature of the circle is such, that the rectangle AP X PB is always = PM<hi rend="sup">2</hi>, therefore the equation is , defining this curve, which is therefore an algebraical or geometrical line. Or, suppose CP = <hi rend="italics">x</hi>; then is , that is ; which is another form of the equation of the curve.</p><p>The doctrine of curve lines in general, as expressed by algebraical equations, was first introduced by Des Cartes, who called algebraical curves geometrical ones; as admitting none else into the construction of problems, nor consequently into geometry. But Newton, and after him Leibnitz and Wolfius, are of another opinion; and think, that in the construction of a problem, one curve is not to be preferred to another for its being defined by a more simple equation, but for its being more easily described.</p><p>Algebraical or geometrical lines are best distinguished into orders according to the number of dimensions of the equation expressing the relation between its ordinates and abscisses, or, which is the same thing, according to the number of points in which they may be cut by a right line. And curves of the same kind or order, are those whose equations rise to the same dimension. Hence, of the first order, there is the right line only; of the 2d order of lines, or the first order of curves, are the circle and conic sections, being 4 species only, viz, the circle, the ellipse, the hyperbola, and <hi rend="italics">dx</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi> the parabola: the lines of the 3d order, or curves of the 2d order, are expressed by an equation of the 3d degree, having three roots; and so on. Of these lines of the 3d order, Newton wrote an express treatise, under the title of Enumeratio Linearum Tertii Ordinis, shewing their distinctive characters and properties, to the number of 72 different species of curves: but Mr. Stirling afterwards added four more to that number; and Mr. Nic. Bernoulli and Mr. Stone added two more.</p><p><hi rend="italics">Curves of the</hi> 2<hi rend="italics">d and other higher kinds,</hi> Newton observes, have parts and properties similar to those of the 1st kind: Thus, as the conic sections have diameters and axes; the lines bisected by these are ordinates; and the intersection of the curve and diameter, the vertex: so, in curves of the 2d kind, any two parallel right lines being drawn to meet the curve in 3 points; a right line cutting these parallels so, as that the sum of the two parts between the secant and the curve on one side, is equal to the 3d part terminated by the curve on the other side, will cut, in the same manner, all other right lines parallel to these, and that meet the curve in three points, that is, so as that the sum of the two parts on one side, will still be equal to the 3d part on the other side. These three parts therefore thus equal, may be called <hi rend="italics">ordinates,</hi> or <hi rend="italics">applicates;</hi> the cutting line, the <hi rend="italics">dia-</hi> <pb n="351"/><cb/> <hi rend="italics">meter;</hi> and where it cuts the ordinates at right angles, the <hi rend="italics">axis;</hi> the intersection of the diameter and the curve, the <hi rend="italics">vertex;</hi> and the concourse of two diameters, the <hi rend="italics">centre;</hi> also the concourse of all the diameters, the <hi rend="italics">common</hi> or <hi rend="italics">general centre.</hi></p><p>Again, as an hyperbola of the first kind has two asymptotes; that of the 2d has 3; that of the 3d has 4; &c: and as the parts of any right line between the conic hyperbola and its two asymptotes, are equal on either side; so, in hyperbolas of the 2d kind, any right line cutting the curve and its three asymptotes in three points; the sum of the two parts of that right line, extended from any two asymptotes, the same way, to two points of the curve, will be equal to the 3d part extended from the 3d asymptote, the contrary way, to the 3d point of the curve.</p><p>Again, as in the conic sections that are not parabolical, the square of an ordinate, i. e. the rectangle of the ordinates drawn on the contrary sides of the diameter, is to the rectangle of the parts of the diameter terminated at the vertices of an ellipse or hyperbola, in the same proportion as a given line called the latus rectum, is to that part of the diameter which lies between the vertices, and called the latus transversum: so, in curves of the 2d kind, not parabolical, the parallelopiped under three ordinates, is to the parallelopiped under the parts of the diameter cut off at the ordinates and the three vertices of the figure, in a given ratio: in which, if there be taken three right lines situate at the three parts of the diameter between the vertices of the figure, each to each; then these three right lines may be called the <hi rend="italics">latera recta</hi> of the figure; and the parts of the diameter between the vertices, the <hi rend="italics">latera transversa.</hi></p><p>And, as in a conic parabola, which has only one vertex to one and the same diameter, the rectangle under the ordinates is equal to the rectangle under the part of the diameter cut off at the ordinates and vertex, and a given right line called the latus rectum: so, in curves of the 2d kind, which have only two vertices to the same diameter, the parallelopiped under three ordinates, is equal to the parallelopiped under two parts of the diameter cut off at the ordinates and the two vertices, and a given right line, which may therefore be called the <hi rend="italics">latus transversum.</hi></p><p>Further, as in the conic sections, where two parallels, terminated on each side by a curve, are cut by two other parallels terminated on each side by a curve, the 1st by the 3d, and the 2d by the 4th; the rectangle of the parts of the 1st is to the rectangle of the parts of the 3d, as that of the 2d is to that of the 4th: so, when four such right lines occur in a curve of the 2d kind, each in three points; the parallelopiped of the parts of the 1st, will be to that of the parts of the 3d, as that of the 2d to that of the 4th.</p><p>Lastly, the legs of curves, both of the 1st, 2d, and higher kinds, are either of the parabolic or hyperbolic kind: an hyperbolic leg being that which approaches infinitely towards some asymptote; and a parabolic one, that which has no asymptote.</p><p>These legs are best distinguished by their tangents; for, if the point of contact go off to an infinite distance, the tangent of the hyperbolic leg will coincide with the asymptote; and that of the parabolic leg, recede <cb/> infinitely, and vanish. Therefore the asymptote of any leg is found, by seeking the tangent of that leg to a point infinitely distant; and the direction of an infinite leg is found, by seeking the position of a right line parallel to the tangent, where the point of contact is infinitely remote, for this line tends that way towards which the infinite leg is directed. <hi rend="center"><hi rend="italics">Reduction of <hi rend="smallcaps">Curves</hi> of the</hi> 2<hi rend="italics">d kind.</hi></hi></p><p>Newton reduces all curves of the 2d kind to four cases of equations, expressing the relation between the ordinate and absciss, viz, in the 1st case, ; in the 2d, " ; in the 3d, " ; in the 4th, " . See Newton's Enumeratio, sect. 3; and Stirling's Lineæ, &c, pa. 83. <hi rend="center"><hi rend="italics">Enumeration of the <hi rend="smallcaps">Curves</hi> of the</hi> 2<hi rend="italics">d kind.</hi></hi></p><p>Under these four cases, the author brings a great number of different forms of curves, to which he gives different names. An hyperbola lying wholly within the angle of the asymptotes, like a conic hyperbola, he calls an <hi rend="italics">inscribed hyperbola;</hi> that which cuts the asymptotes, and contains the parts cut off within its own periphery, a <hi rend="italics">circumscribed hyperbola;</hi> that which has one of its infinite legs inscribed and the other circumscribed, he calls <hi rend="italics">ambigenal;</hi> that whose legs look towards each other, and are directed the same way, <hi rend="italics">converging;</hi> that where they look contrary ways, <hi rend="italics">diverging;</hi> that where they are convex different ways, <hi rend="italics">crosslegged;</hi> that applied to its asymptote with a concave vertex, and diverging legs, <hi rend="italics">conchoidal</hi>; that which cuts its asymptote with contrary flexures, and is produced each way into contrary legs, <hi rend="italics">anguineous,</hi> or <hi rend="italics">snake-like;</hi> that which cuts its conjugate across, <hi rend="italics">cruciform;</hi> that which returning around cuts itself, <hi rend="italics">nodated</hi>; that whose parts concur in the angle of contact, and there terminate, <hi rend="italics">cuspidated;</hi> that whose conjugate is oval, and infinitely small, i. e. a point, <hi rend="italics">pointed;</hi> that which, from the impossibility of two roots, is without either oval, node, cusp, or point, <hi rend="italics">pure.</hi> And in the same manner he denominates a parabola <hi rend="italics">converging, diverging, cruciform,</hi> &c. Also when the number of hyperbolic legs exceeds that of the conic hyperbola, that is more than two, he calls the hyperbola <hi rend="italics">redundant.</hi></p><p>Under those 4 cases the author enumerates 72 different curves: of these, 9 are redundant hyperbolas, without diameters, having three asymptotes including a triangle; the first consisting of three hyperbolas, one inscribed, another circumscribed, and the third ambigenal, with an oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; the 5th and 6th, pure; the 7th and 8th, cruciform; the 9th or last, anguineal. There are 12 redundant hyperbolas, having only one diameter: the 1st, oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; 5th, 6th, 7th, and 8th, pure; the 9th and 10th, cruciform; the 11th and 12th, conchoidal. And to this class Stirling adds 2 more. There are 2 redundant hyperbolas, with three diameters. There are 9 redundant hyperbolas, with three asymptotes converging to a common point; the 1st being formed of the 5th and 6th redundant parabolas, <pb n="352"/><cb/> whose asymptotes include a triangle; the 2d formed of the 7th and 8th; the 3d and 4th, of the 9th; the 5th is formed of the 5th and 7th of the redundant hyperbolas, with one diameter; the 6th, of the 6th and 7th; the 7th, of the 8th and 9th; the 8th, of the 10th and 11th; the 9th, of the 12th and 13th: all which conversions are effected, by diminishing the triangle comprehended between the asymptotes, till it vanish into a point.</p><p>Six are defective parabolas, having no diameters: the 1st, oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; the 5th, pure; &c.</p><p>Seven are defective hyperbolas, having diameters: the 1st and 2d, conchoidal, with an oval; the 3d, nodated; the 4th, cuspidated, which is the cissoid of the ancients; the 5th and 6th, pointed; the 7th, pure.</p><p>Seven are parabolic hyperbolas, having diameters: the 1st, oval; the 2d, nodated; the 3d, cuspidated; the 4th, pointed; the 5th, pure; the 6th, cruciform; the 7th, anguineous.</p><p>Four are parabolic hyperbolas: four are hyperbolisms of the hyperbola: three, hyperbolas of the ellipsis: two, hyperbolisms of the parabola.</p><p>Six are diverging parabolas; one, a trident; the 2d, oval; the 3d, nodated; the 4th, pointed; the 5th, cuspidated (which is Neil's parabola, usually called the semi-cubical parabola); the 6th, pure.</p><p>Lastly, one, commonly called the cubical parabola.</p><p>Mr. Stirling and Mr. Stone have shewn that this enumeration is imperfect, the former having added four new species of curves to the number, and the latter two, or rather these two were first noticed by Mr. Nic. Bernoulli. Also Mr. Murdoch and Mr. Geo. Sanderson have found some new species; though some persons dispute the realíty of them. See the Genesis Curvarum per umbras, and the Ladies' Diary 1788 and 1789, the prize question.</p><p><hi rend="italics">Organical Description of</hi> <hi rend="smallcaps">Curves.</hi>—Sir Isaac Newton shews that curves may be generated by shadows. He says, if upon an infinite plane, illuminated from a lucid point, the shadows of figures be projected; the shadows of the conic sections will always be conic sections; those of the curves of the 2d kind, will always be curves of the 2d kind; those of the curves of the 3d kind, will always be curves of the 3d kind; and so on ad infinitum</p><p>And, like as the projected shadow of a circle generates all the conic sections, so the 5 diverging parabolas, by their shadows, will generate and exhibit all the rest of the curves of the 2d kind: and thus some of the most simple curves of the other kinds may be found, which will form, by their shadows upon a plane, projected from a lucid point, all the other curves of that same kind. And in the French Memoirs may be seen a demonstration of this projection, with a specimen of a few of the curves of the 2d order, which may be generated by a plane cutting a solid formed from the motion of an infinite right line along a diverging parabola, having an oval, always passing through a given or fixed point above the plane of that parabola. The above method of Newton has also been pursued and illustrated with great elegance by Mr. Murdoch, in his treatise entitled <hi rend="italics">Newtoni Genesis Curvarum per umbras, seu Perspectivæ Universalis Elementa.</hi> <cb/></p><p>Mr. Maclaurin, in his <hi rend="italics">Geometria Organica,</hi> shews how to describe several of the species of curves of the 2d order, especially those having a double point, by the motion of right lines and angles: but a good commodious description by a continued motion of those curves which have no double point, is ranked by Newton among the most difficult problems. Newton gives also other methods of description, by lines or angles revolving above given poles; and Mr. Brackenridge has given a general method of describing curves, by the intersection of right lines moving about points in a given plane. See Philos. Trans. No. 437, or Abr vol. 8, pa. 58; and some particular cases are demonstrated in his <hi rend="italics">Exerc. Geometrica de Curvarum Descriptione.</hi></p><p><hi rend="smallcaps">Curves</hi> <hi rend="italics">above the</hi> 2<hi rend="italics">d Order.</hi> The number of species in the higher orders of curves increase amazingly, those of the 3d order only it is thought amounting to some thousands, all comprehended under the following ten particular equations, viz, .</p><p>Those who wish to see how far this doctrine has been advanced, with regard to curves of the higher orders, as well as those of the 1st and 2d orders, may consult Mr. Maclaurin's Geometria Organica, and Brackenridge's Exerc. Geom.</p><p>All geometrical lines of the odd orders, viz, the 3d, 5th, 7th, &c, have at least one leg running on infinitely; because all equations of the odd dimensions have at least one real root. But vast numbers of the lines of the even orders are only ovals; among which there are several having very pretty figures, some being like single hearts, some double ones, some resembling fiddles, and others again single knots, double knots, &c.</p><p>Two geometrical lines of any order will cut one another in as many points, as are denoted by the product of the two numbers expressing those orders.</p><p>The theory of curves forms a considerable branch of the mathematical sciences. Those who are curious of advancing beyond the knowledge of the circle and the conic sections, and to consider geometrical curves of a higher nature, and in a general view, will do well to study Cramer's Introduction à l'Analyse des Lignes Courbes Algebraiques, which the learned and ingenious author composed for the use of beginners. There is an excellent posthumous piece too of Maclaurin's, printed as an Appendix to his Algebra, and entitled De Linearum Geometricarum Proprietatibus Generalibus. The same author, at a very early age, gave a remarkable specimen of his genius and knowledge in his Geometria Organica; and he carried these speculations farther afterwards, as may be seen in the theorems he <pb n="353"/><cb/> has given in the Philos. Trans. See Abr. vol. 8, pa. 62. Other writings on this subject, beside the Treatises on the Conic Sections, are Archimedes de Spiralibus; Des Cartes Geometria; Dr. Barrow's Lectiones Geometricæ; Newton's Enumeratio Linearum Tertii Ordinis; Stirling's Illustratio Tractatûs Newtoni de Lineis Tertii Ordinis; Maclaurin's Geometria Organica; Brackenridge's Descriptio Linearum Curvarum; M. De Gua's Usages de l'Analyse de Des Cartes; beside many other Tracts on Curves in the Memoirs of several Academies &c.</p><p><hi rend="italics">Use of</hi> <hi rend="smallcaps">Curves</hi> <hi rend="italics">in the Construction of Equations.</hi> One great use of curves in Geometry is, by means of their intersections, to give the solution of problems. See <hi rend="smallcaps">Construction.</hi></p><p>Suppose, <hi rend="italics">ex. gr.</hi> it were required to construct the following equation of 9 dimensions, : assume the equation to a cubic parabola <hi rend="italics">x</hi><hi rend="sup">3</hi>=<hi rend="italics">y</hi>; then, by writing <hi rend="italics">y</hi> for <hi rend="italics">x</hi><hi rend="sup">3</hi>, the given equation will become ; an equation to another curve of the 2d kind, where <hi rend="italics">m</hi> or <hi rend="italics">f</hi> may be assumed = 0 or any thing else: and by the descriptions and intersections of these curves will be given the roots of the equation to be constructed. It is sufficient to describe the cubic parabola once. When the equation to be constructed, by omitting the two last terms <hi rend="italics">hx</hi> and <hi rend="italics">k,</hi> is reduced to 7 dimensions; the other curve, by expunging <hi rend="italics">m,</hi> will have the double point in the beginning of the absciss, and may be easily described as above: If it be reduced to 6 dimensions, by omitting the last three terms, <hi rend="italics">gx</hi><hi rend="sup">2</hi>+<hi rend="italics">hx</hi>+<hi rend="italics">k</hi>; the other curve, by expunging <hi rend="italics">f,</hi> will become a conic section. And if, by omitting the last three terms, the equation be reduced to three dimensions, we shall fall upon Wallis's construction by the cubic parabola and right line.</p><p><hi rend="italics">Rectification, Inflection, Quadrature, &c of</hi> <hi rend="smallcaps">Curves.</hi> See the respective terms.</p><p><hi rend="smallcaps">Curve</hi> <hi rend="italics">of a Double Curvature,</hi> is such a curve as has not all its parts in the same plane.</p><p>M. Clairaut has published an ingenious treatise on curves of a double curvature. See his Recherches sur les Courbes à Double Courbure. Mr. Euler has also treated this subject in the Appendix to his Analysis Infinitorum, vol. 2, pa. 323.</p><p><hi rend="italics">Family of</hi> <hi rend="smallcaps">Curves</hi>, is an assemblage of several curves of different kinds, all defined by the same equation of an indeterminate degree; but differently, according to the diversity of their kind. For example, Suppose an equation of an indeterminate degree, : if <hi rend="italics">m</hi>=2, then will <hi rend="italics">ax</hi>=<hi rend="italics">y</hi><hi rend="sup">2</hi>; if <hi rend="italics">m</hi>=3, then will <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">3</hi>; if <hi rend="italics">m</hi>=4, then is <hi rend="italics">a</hi><hi rend="sup">3</hi><hi rend="italics">x</hi>=<hi rend="italics">y</hi><hi rend="sup">4</hi>; &c: all which curves are said to be of the same family or tribe.</p><p>The equations by which the families of curves are defined, are not to be confounded with the transcendental ones: for though with regard to the whole family, they be of an indeterminate degree; yet with respect to each several curve of the family, they are determinate; whereas transcendental equations are of an indefinite degree with respect to the same curve.</p><p>All Algebraical curves therefore compose a certain family, consisting of innumerable others, each of which comprehends infinite kinds. For the equations by <cb/> which curves are defined involve only products, either of powers of the abscisses and ordinates by constant coefficients; or of powers of the abscisses by powers of the ordinates; or of constant, pure, and simple quantities by one another. Moreover, every equation to a curve may have 0 for one member or side of it; for example, <hi rend="italics">ax</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, by transposition becomes . Therefore the equation for all algebraic curves will be</p><p><hi rend="italics">Catacaustic, and Diacaustic</hi> <hi rend="smallcaps">Curves.</hi> See C<hi rend="smallcaps">ATACAUSTIC</hi>, and <hi rend="smallcaps">Diacaustic.</hi></p><p><hi rend="italics">Exponential</hi> <hi rend="smallcaps">Curve</hi>, is that which is defined by an exponential equation, as <hi rend="italics">ax</hi><hi rend="sup">z</hi>=<hi rend="italics">y,</hi> &c.</p><p><hi rend="smallcaps">Curves</hi> <hi rend="italics">by the Light,</hi> or <hi rend="smallcaps">Courbes</hi> <hi rend="italics">a la Lumiere,</hi> a name given to certain curves by M. Kurdwanowski, a Polish gentleman. He observed that any line, straight or curved, exposed to the action of a luminous point, received the light differently in its different parts, according to their distance from the light. These different effects of the light upon each point of the line, may be represented by the ordinates of some curve, which will vary precisely with these effects. Priestley's Hist. of Vision, pa. 752.</p><p><hi rend="italics">Logarithmic</hi> <hi rend="smallcaps">Curve.</hi> See <hi rend="smallcaps">Logarithmic</hi> <hi rend="italics">Curve.</hi></p><p><hi rend="smallcaps">Curve</hi> <hi rend="italics">Reflectoire,</hi> so called because it is the appearance of the plane bottom of a bason covered with water, to an eye perpendicularly over it. In this position, the bottom of the bason will appear to rise upwards, from the centre outwards; but the curvature will be less and less, and at last the surface of the water will be an asymptote to it. M. Mairan, who first conceived this idea from the phenomena of light, found also several kinds of these curves; and he gives a geometrical deduction of their properties, shewing their analogy to caustics by refraction. Mem. Ac. 1740; Priestley's Hist. of Vision, pa. 752.</p><p><hi rend="italics">Radical</hi> <hi rend="smallcaps">Curves</hi>, a name given by some authors to curves of the spiral kind, whose ordinates, if they may be so called, do all terminate in the centre of the including circle, and appear like so many radii of that circle: whence the name.</p><p><hi rend="italics">Regular</hi> <hi rend="smallcaps">Curves</hi>, are such as have their curvature turning regularly and continually the same way; in opposition to such as bend contrary ways, by having points of contrary flexure, which are called irregular curves.</p><p><hi rend="italics">Characteristic Triangle of a</hi> <hi rend="smallcaps">Curve</hi>, is the differential or elementary right-angled triangle whose three sides are, the fluxions of the absciss, ordinate, and curve; the fluxion of the curve being the hypothenuse. So, if <hi rend="italics">pq</hi> be parallel to, and indesi- <figure/> nitely near to the ordinate PQ, and Q<hi rend="italics">r</hi> parallel to the absciss AP; then Q<hi rend="italics">r</hi> is the fluxion of the absciss AP, and <hi rend="italics">qr</hi> the fluxion of the ordinate PQ, and Q<hi rend="italics">q</hi> the fluxion of the curve AQ; hence the elementary triangle Q<hi rend="italics">qr</hi> is the characteristic triangle of the curve AQ; and the three sides are <hi rend="italics">x<hi rend="sup">.</hi>, y<hi rend="sup">.</hi>, z<hi rend="sup">.</hi></hi>; in which .</p><p>CURVILINEAR <hi rend="italics">Angle, Figure, Superficies, &c,</hi> <pb n="354"/><cb/> are such as are formed or bounded by curves; in opposition to rectilinear ones, which are formed by straight lines or planes.</p></div1><div1 part="N" n="CUSP" org="uniform" sample="complete" type="entry"><head>CUSP</head><p>, in Astronomy, is used to express the points or horns of the moon, or other luminary.</p><div2 part="N" n="Cusp" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cusp</hi></head><p>, in Astrology, is used for the 1st point of each of the twelve houses, in a figure or scheme of the heavens. See <hi rend="smallcaps">House.</hi></p></div2><div2 part="N" n="Cusp" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cusp</hi></head><p>, in the Higher Geometry, is used for the point or corner formed by two parts of a curve meeting and terminating there. See <hi rend="smallcaps">Curve.</hi></p><p><hi rend="smallcaps">Cuspidated</hi> <hi rend="italics">Hyperbola,</hi> &c. See <hi rend="smallcaps">Curve.</hi></p><p>CUT-<hi rend="smallcaps">Bastion.</hi> See <hi rend="smallcaps">Bastion.</hi></p></div2></div1><div1 part="N" n="CUVETTE" org="uniform" sample="complete" type="entry"><head>CUVETTE</head><p>, or <hi rend="smallcaps">Cunette</hi>, in Fortification, is a kind of ditch within a ditch, being a pretty deep trench, about four fathoms broad, sunk and running along the middle of the great dry ditch, to hold water; serving both to keep off the enemy, and prevent him from mining.</p></div1><div1 part="N" n="CYCLE" org="uniform" sample="complete" type="entry"><head>CYCLE</head><p>, a certain period or series of numbers proceeding orderly from first to last, then returning again to the first, and so circulating perpetually.</p><p>Cycles have chiefly arisen from the incommensurability of the revolutions of the earth and celestial bodies to one another. The apparent revolution of the sun about the earth, has been arbitrarily divided into 24 hours, which is the basis or foundation of all our mensuration of time, whether days, years, &c. But neither the annual motion of the sun, nor that of the other heavenly bodies, can be measured exactly, and without any remainder, by hours, or their multiples. That of the sun, for example, is 365 days 5 hours 49 minutes nearly; that of the moon, 29 days 12 hours 44 minutes nearly.</p><p>Hence, to swallow up these fractions in whole numbers, and yet in numbers which only express days and years, cycles have been invented; which, comprehending several revolutions of the same body, replace it, after a certain number of years, in the same points of the heaven whence it first departed; or, which is the same thing, in the same place of the civil calendar.</p><p>There are various cycles; as, the cycle of Indiction, the cycle of the moon, the cycle of the sun, &c.</p><p><hi rend="smallcaps">Cycle</hi> <hi rend="italics">of Indiction,</hi> is a series of 15 years, returning constantly around like the other cycles; and commenced from the third year before Christ; whence it happens that if 3 be added to any given year of Christ, and the sum be divided by 15, what remains is the year of the indiction. See <hi rend="smallcaps">Indiction.</hi></p><p><hi rend="smallcaps">Cycle</hi> <hi rend="italics">of the Moon,</hi> or the <hi rend="italics">Lunar Cycle,</hi> is a period of 19 years; in which time the new and full moons return to the same day of the Julian year. See C<hi rend="smallcaps">ALIPPIC.</hi></p><p>This cycle is also called the <hi rend="italics">Metonic period</hi> or <hi rend="italics">cycle,</hi> from its inventor Meton, the Athenian; and also the <hi rend="italics">Golden Number,</hi> from its excellent use in the calendar: though, properly speaking, the golden number is rather the particular number which shews the year of the lunar cycle, which any given year is in. This cycle of the moon only holds true for 310 7/10 years: for, though the new moons do return to the same day after 19 years; yet not to the same time of the day, but near an hour and a half sooner; an error which in 310 7/10 <cb/> years amounts to an entire day. Yet those employed in reforming the calendar went on a supposition that the lunations return precisely from 19 years to 19 years, for ever.</p><p>The use of this cycle, in the ancient calendar, is to shew the new moon of each year, and the time of Easter. In the new one, it only serves to find the Epacts; which shew, in either calendar, that the new moon falls 11 days too late.</p><p>As the Crientals began the use of this cycle at the time of the Council of Nice in 325, they assumed, that the first year of the cycle the paschal new moon fell on the 13th of March: on which account the lunan cycle 3 fell on the 1st of January in the third year.</p><p>The Occidentals, on the contrary, placed the number 1 to the 1st of January, which occasioned a considerable difference in the time of Easter. Hence, Dionysius Exiguus, on framing a new calendar, persuaded the Christians of the west to salve the difference, and come into the practice of the church of Alexandria.</p><p><hi rend="italics">To find the Year of the Lunar Cycle,</hi> is to find the golden number. See <hi rend="smallcaps">Golden</hi>-<hi rend="italics">Number.</hi></p><p><hi rend="smallcaps">Cycle</hi> <hi rend="italics">of the Sun,</hi> or <hi rend="italics">Solar Cycle,</hi> is a period or revolution of 28 years; beginning with 1, and ending with 28; which elapsed, the Dominical or Sundayletters, and those that express the other feasts, &c, return into their former place, and proceed in the same order as before. The days of the month return again to the same days of the week; the sun's place to the same signs and degrees of the ecliptic on the same months and days, so as not to differ one degree in a hundred years; and the leap years begin the same course with respect to the days of the week on which the days of the month fall.</p><p>This is called the cycle of the <hi rend="italics">sun,</hi> or the <hi rend="italics">solar</hi> cycle, not from any regard to the sun's course, which has no concern in it; but from <hi rend="italics">Sunday,</hi> anciently called <hi rend="italics">dies solis,</hi> the <hi rend="italics">sun's day</hi>; as the dominical or sunday letter is chiefly sought for from this revolution.</p><p>The reformation of the calendar under pope Gregory the 13th, occasioned a considerable alteration of this cycle: In the Gregorian calendar, the solar cycle is not constant and perpetual; because every 4th secular year is common; whereas, in the Julian, it is bissextile. The epoch, or beginning of the solar cycle, both Julian and Gregorian, is the 9th year before Christ. And therefore,</p><p><hi rend="italics">To find the Cycle of the Sun for any given year:</hi> add 9 to the number given, and divide the sum by 28; the remainder will be the number of the cycle, and the quotient the number of revolutions since Christ. If there be no remainder, it will be the 28th or last year of the cycle. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=14 align=center" role="data">CYCLE of the Sun, with the correspondent Sunday letters, in Julian Years.</cell></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">GF</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">BA</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" rend="align=center" role="data">DC</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">FE</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" rend="align=center" role="data">AG</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">CB</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=center" role="data">ED</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">E</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">G</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">B</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">F</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">A</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" rend="align=center" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">F</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">A</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">C</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">E</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">G</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" rend="align=center" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">C</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">E</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">G</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">B</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">F</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=center" role="data">A</cell></row></table> <pb n="355"/><cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=14 align=center" role="data">CYCLE of the Sun, and Sunday Letters, from the Gregorian year 1700 to the year 1800.</cell></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">DC</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">FE</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" rend="align=center" role="data">AG</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">CB</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" rend="align=center" role="data">ED</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">GF</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=center" role="data">BA</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">B</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">F</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">A</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">C</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">E</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" rend="align=center" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">A</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">C</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">E</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">G</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">B</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" rend="align=center" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">G</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">B</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">F</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">A</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">C</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=center" role="data">E</cell></row></table></p><p><hi rend="italics">Great Pascal</hi> <hi rend="smallcaps">Cycle</hi>, is another name for the Victorian or Dionysian Period. Which see.</p></div1><div1 part="N" n="CYCLOID" org="uniform" sample="complete" type="entry"><head>CYCLOID</head><p>, or <hi rend="smallcaps">Trochoid</hi>, a mechanical or transcendental curve, which is thus generated: Suppose a wheel, or a circle, AE, to roll along a straight line AB, beginning at the point A, and ending at B, where it has completed just one revolution, thereby measuring out a right line AB exactly equal to the circumference of the generating circle AE, whilst a nail or point A in the circumference of the wheel, or circle, traces out or describes a curvilineal path ADB; then this curve ADB is the cycloid, or trochoid. <figure/></p><p>Schooten, in his Commentary on Des Cartes, says that Des Cartes first conceived the notion of this elegant curve, and after him it was first published by Father Mersenne, in the year 1615. But Torricelli, in the Appendix de Dimensione Cycloidis, at the end of his treatise De Dimensione Parabolæ, published 1644, says that this curve was considered and named a cycloid, by his predecessors, and particularly by Galileo about 45 years before, i. e. about 1599. And Dr. Wallis shews that it is of a much older standing, having been known to Bovilli about the year 1500, and even considered by cardinal Cusanus much earlier, viz, before the year 1451. Philos. Trans. Abr. vol. 1, pa. 116. It would seem however that Torricelli's was the first regular treatise on the Cycloid; though several particular properties of it might be known prior to his work. He first shewed, that the cycloidal space is equal to triple the generating circle, (though Pascal contends that Roberval shewed this): also that the solid generated by the rotation of that space about its base, is to the circumscribing cylinder, as 5 to 8: about the tangent parallel to the base, as 7 to 8: about the tangent parallel to the axis, as 3 to 4: &c.</p><p>Honoratus Fabri, in his Synopsis Geom. has a short treatise on the cycloid, containing demonstrations of the above, and many other theorems concerning the centres of gravity of the cycloidal space, &c; which he says he found out before the year 1658.</p><p>From the preface to Dr. Wallis's treatise on the eycloid we learn, that, in the year 1658, M. Pascal publicly proposed at Paris, under the name of D'Ettonville, the two following problems as a challenge, to <cb/> be solved by the mathematicians of Europe, with a reward of 20 pistoles for the solution: viz, to find the area of any segment of the cycloid, cut off by a right line parallel to the base; also the content of the solid generated by the rotation of the same about the axis, and about the base of that segment. This challenge set the Doctor upon writing that treatise upon the cycloid, which is a much better and compleater piece than had been given before upon this curve. He here gives the curve surfaces of the solids generated by the rotation of the cycloidal space about its axis, and about its base, with determinations of the centres of gravity, &c. He here asserts too, that Sir Christopher Wren, in 1658, was the first who found out a right line equal to the curve of the cycloid; and Mr. Huygens, in his Herolog. Oscillat. says that he himself was the first inventor of the segment of a cycloidal space, cut off by a right line parallel to the base at the distance of 1/4 the axis of the curve from the centre, being equal to a rectilinear space, viz, to a regular hexagon inscribed in the generating circle; the demonstration of which may be seen in Wallis's treatise.</p><p>Several other authors have spoken or treated of the cycloid: as Pascal, in his treatise, under the name of D'Ettonville: Schooten in his Commentary on Des Cartes's Geometry, near the end of the 2d book; M. Reinau, in his Analyse Démontrée, tom. 2, pa. 595: also Newton, Leibnitz, de la Loubere, Roberval, Des Cartes, Wren, Fabri, the Bernoulli's, De la Hire, Cotes, &c, &c.</p><p><hi rend="italics">Properties of the</hi> <hi rend="smallcaps">Cycloid.</hi>—The circle AE, by whose revolution the cycloid is traced out, is called the <hi rend="italics">generating circle;</hi> the line AB, which is equal to the circumference of the circle, is the <hi rend="italics">base</hi> of the cycloid; and the perpendicular DC on the middle of the base, is its <hi rend="italics">axis.</hi> The properties of the cycloid are among the most beautiful and useful of all curve lines: some of the most remarkable of which are as follow:</p><p>1. The circular arc DG = the line GH parallel to AB.</p><p>2. The semicircumf. DGC = the semibase AC.</p><p>3. The arc DH = double the chord DG.</p><p>4. The arc DA = double the diam. DC.</p><p>5. The tang. HI is parallel to the chord DG.</p><p>6. The space ADBA = triple the circle AE or CGD &c.</p><p>7. The space ADGCA = the same circle AE, &c. <figure/></p><p>8. A body falls through any arc KL of a cycloid reversed, in the same time, whether that arc be great or small; that is, from any point L, to the lowest point K, which is the vertex reversed: and that time <pb n="356"/><cb/> is to the time of falling perpendicularly through the axis MK, as the semicircumference of a circle is to its diameter, or as 3.1416 to 2. And hence it follows that, if a pendulum be made to vibrate in the arc LKN of a cycloid, all the vibrations will be performed in the same time.</p><p>9. The evolute of a cycloid, is another equal cycloid. So that if two equal semicycloids OP, OQ, be joined at O, so that OM be = MK the diameter of the generating circle, and the string of a pendulum hung up at O, having its length = OK or = the curve OP; then, by plying the string round the curve OP, to which it is equal, and then the ball let go, it will describe, and vibrate in the other cycloid PKQ.</p><p>10. The cycloid is the curve of swiftest descent: or a heavy body will fall from one given point to another, by the way of the arc of a cycloid passing through those two points, in a less time, than by any other rout. See the Works of James and John Bernoulli for many other curious properties concerning the descents in cycloids, &c.</p><p><hi rend="smallcaps">Cycloids</hi> are also either <hi rend="italics">curtate</hi> or <hi rend="italics">prolate.</hi> <figure/></p><div2 part="N" n="Cycloid" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cycloid</hi></head><p>, <hi rend="italics">Curtate,</hi> or <hi rend="italics">contracted,</hi> is the path described by some point without the circle, while the circumference rolls along a straight line; and a <figure/></p></div2><div2 part="N" n="Cycloid" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Cycloid</hi></head><p>, <hi rend="italics">Prolate,</hi> or <hi rend="italics">Inflected,</hi> is in like manner the path of some point taken within the generating circle.</p><p>Thus, if, while the circle rolls along the line AB, the point R be taken without the circle, it will describe or trace out the curtate or contracted cycloid RST; but the point being taken within the circle, it will describe the prolate or inflected cycloid RVW.</p><p>These two curves were both noticed by Torricelli and Schooten, and more fully treated of by Wallis, in his Treatise on the Cycloid, printed at Oxford in 1659; where he shews that these have properties similar to the first or primary cycloid; only the last of these is a curve having a point of inflection, and the other crossing itself, and forming a node.</p><p>By continuing the motion of the wheel, or circle, so as to describe a right line equal to the generating circumference several times repeated, there will be produced as many repetitions of the cycloids, which so united together will appear as in these figures following: <cb/> <figure><head>Curtate Cycloid.</head></figure> <figure><head>Common Cycloid.</head></figure> <figure><head>Inflected Cycloid.</head></figure></p></div2></div1><div1 part="N" n="CYGNUS" org="uniform" sample="complete" type="entry"><head>CYGNUS</head><p>, the <hi rend="italics">Swan,</hi> a constellation of the northern hemisphere, being one of the 48 old ones, and fabled by the Greeks to be the swan, under the form of which Jupiter deceived Leda or Nemesis, from which embrace sprung the beauteous Helen.</p><p>The stars in the constellation Cygnus, in Ptolomy's catalogue are 19, in Tycho's 18, in Hevelius's 47, and in the Britannic catalogue 81.</p></div1><div1 part="N" n="CYLINDER" org="uniform" sample="complete" type="entry"><head>CYLINDER</head><p>, a solid having two equal circular ends, and every plane section parallel to the ends a circle equal to them also.</p><p>The cylinder may be conceived to be thus generated: <figure/> Suppose two parallel circles AB and CD, and a right line carried continually round them, always parallel to itself; this line will describe the curve surface of a cylinder, ABDC, of which the two parallel circles AB and CD form the two ends. When the line, or sides is perpendicular to the ends, the cylinder is a right or perpendicular one; otherwise it is oblique.</p><p>Or the right cylinder may be conceived to be generated by the rotation of a rectangle about one of its sides. The <hi rend="italics">axis</hi> of the cylinder is the line connecting the centres of its two parallel circular ends; and is equal to the altitude of the cylinder when this is a right one, but exceeds the altitude in the oblique cylinder, in the proportion of radius to the sine of the angle of its inclination to the base.</p><p>The convex surface of a cylinder is equal to the product of the axis multiplied by the circumference of its base.</p><p>The solidity of a cylinder is equal to the area of its base multiplied by its perpendicular altitude.</p><p>Cylinders of equal bases and altitudes, are equal.</p><p>Cylinders are to each other, as the product of their bases and altitudes. And equal cylinders have their bases reciprocally as their altitudes.</p><p>A cylinder is to its inscribed sphere, or spheroid, as 3 to 2 <*> and to its inscribed cone as 3 to 1. <pb n="357"/><cb/></p><p>The oblique plane sections of a cylinder, are ellipses; but all the sections parallel to the ends, are circles.</p><p>For the surfaces and solidities of the ungulas, or oblique slices, of a cylinder, see my Mensuration, pa. 218, 2d edition.</p></div1><div1 part="N" n="CYLINDRICAL" org="uniform" sample="complete" type="entry"><head>CYLINDRICAL</head><p>, pertaining to a cylinder.</p></div1><div1 part="N" n="CYLINDROID" org="uniform" sample="complete" type="entry"><head>CYLINDROID</head><p>, a solid resembling the figure of a cylinder; but differing from it as having ellipses for its ends or bases, instead of circles, in the cylinder.</p><p>In the cylindroid, the solidity and curve superficies are found the same way as those of the cylinder; viz, <cb/> by multiplying the circumference of the base by the length or axis, for the surface; and the area of the base by the altitude, for the solidity.</p></div1><div1 part="N" n="CYMATIUM" org="uniform" sample="complete" type="entry"><head>CYMATIUM</head><p>, <hi rend="smallcaps">Cimatium</hi>, or <hi rend="smallcaps">Cima</hi>, in Architecture, a member, or moulding of the cornice, whose profile is waved; i. e. concave at top, and convex at bottom.</p></div1><div1 part="N" n="CYNOSURA" org="uniform" sample="complete" type="entry"><head>CYNOSURA</head><p>, a name given by the Greeks, to ursa minor, or little bear, otherwise called Charles's wain; the star in the extremity of the tail being called the pole star.</p></div1><div1 part="N" n="CYPHER" org="uniform" sample="complete" type="entry"><head>CYPHER</head><p>, or nought. See Cipher. </p></div1></div0><div0 part="N" n="D" org="uniform" sample="complete" type="alphabetic letter"><head>D</head><cb/><div1 part="N" n="DACTYLONOMY" org="uniform" sample="complete" type="entry"><head>DACTYLONOMY</head><p>, the art of counting or numbering by the fingers.—The rule is this: the left thumb is reckoned 1, the index or fore-finger 2, and so on to the right thumb, which is the tenth or last, and consequently is denoted by the cipher 0.</p></div1><div1 part="N" n="DADO" org="uniform" sample="complete" type="entry"><head>DADO</head><p>, that part in the middle of the pedestal of a column &c, between its base and cornice.</p></div1><div1 part="N" n="DAILY" org="uniform" sample="complete" type="entry"><head>DAILY</head><p>, in Astronomy. See <hi rend="smallcaps">Diurnal.</hi></p></div1><div1 part="N" n="DARCY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DARCY</surname> (<foreName full="yes"><hi rend="smallcaps">Count</hi></foreName>)</persName></head><p>, an ingenious, philosopher and mathematician, was born in Ireland in 1725; but his friends being attached to the Stuart family, he was sent to France, at 14 years of age, where he spent the rest of his life. Being put under the care of the celebrated Clairaut, he improved so rapidly in the mathematics, that at 17 years of age he gave a new solution of the problem concerning the curve of equal pressure in a resisting medium. This was followed the year after by a determination of the curve described by a heavy body, sliding by its own weight along a moveable plane, at the same time that the pressure of the body causes a horizontal motion in the plane. Darcy served in the war of 1744, and was taken prisoner by the English: and yet, during the course of the war he gave two memoirs to the academy; the first of these contained a general principle in mechanics, that of <hi rend="italics">the preservation of the rotatory motion;</hi> a principle which he again brought forward in 1750, by the name of <hi rend="italics">the principle of the preservation of action.</hi></p><p>In 1760, Darcy published <hi rend="italics">An Essay on Artillery,</hi> containing some curious experiments on the charges of gunpowder, &c, &c, and improvements on those of the ingenious Robins; a kind of experiments which our author carried on occasionally to the end of his life.</p><p>In 1765, he published his <hi rend="italics">Memoir on the Duration of the Sensation of Sight,</hi> the most ingenious of his works: the result of these researches was, that a body may sometimes pass by our eyes without being seen, or marking its presence, otherwise than by weakening the brightness of the object it covers.</p><p>All Darcy's works bear the character which results <cb/> from the union of genius and philosophy; but as he measured every thing upon the largest scale, and required extreme accuracy in experiment, neither his time, fortune, nor avocations, allowed him to execute more than a very small part of what he projected. In his disposition, he was amiable, spirited, lively, and a lover of independence, a passion to which he nobly sacrificed, even in the midst of literary society.—He died of a cholera morbus in 1779, at 54 years of age.</p><p>Darcy was admitted of the French academy in 1749, and was made pensioner-geometrician in 1770.—His essays, printed in the Memoirs of the Academy of Sciences, are various and very ingenious, and are contained in the volumes for the years 1742, 1747, 1749, 1750, 1751, 1752, 1753, 1754, 1758, 1759, 1760, 1765, and in tom. 1, of the Savans Etrangers.</p><p>DARK <hi rend="italics">Chamber.</hi> See <hi rend="smallcaps">Camera</hi> <hi rend="italics">Obscura.</hi></p><p>DARK <hi rend="italics">Tent,</hi> a portable camera obscura, made somewhat like a desk, and fitted with optic glasses, to take prospects of landscapes, buildings, &c.</p></div1><div1 part="N" n="DATA" org="uniform" sample="complete" type="entry"><head>DATA</head><p>, in General Mathematics, denote certain things or quantities, supposed given or known, from which other quantities are discovered that were unknown, or sought. A problem or question usually consists of two parts, <hi rend="italics">data</hi> and <hi rend="italics">quæsita.</hi></p><p>Euclid has an express and excellent treatise of <hi rend="italics">Data;</hi> in which he uses the word for such spaces, lines, angles, &c, as are given; or to which others can be found equal.</p><p>Euclid's <hi rend="italics">Data</hi> is the first in order of the books that have been written by the ancient geometricians, to facilitate and promote the method of resolution or analysis. In general a thing is said to be given which is either actually exhibited, or can be found out, that is, which is either known by hypothesis, or that can be demonstrated to be known: and the propositions in the book of Euclid's Data shew what things can be found out or known, from those that by hypothesis are already known: so that in the analysis or investigation of a problem, from the things that are laid down as given <pb n="358"/><cb/> or known, by the help of these propositions, it is demonstrated that other things are given, and from these last that others again are given, and so on, till it is demonstrated that that which was proposed to be found out in the problem is given; and when this is done, the problem is solved, and its composition is made and derived from the compositions of the Data which were employed in the analysis. And thus the Data of Euclid are of the most general and necessary use in the solution of problems of every kind.</p><p>Marinus, at the end of his Preface to the Data, is mistaken in asserting that Euclid has not used the synthetical, but the analytical method in delivering them: for, though in the analysis of a theorem, the thing to be demonstrated is assumed in the analysis; yet, in the demonstrations of the Data, the thing to be demonstrated, which is, that something is given, is never once assumed in the demonstration; from which it is manifest that every one of them is demonstrated synthetlcally: though indeed if a proposition of the Data be turned into a problem, the demonstration of the proposition becomes the analysis of the problem. See Simson's edition of Euclid's Data, which is esteemed the best.</p><p>DAVIS's <hi rend="italics">Quadrant,</hi> the common sea quadrant, or back-staff. See <hi rend="smallcaps">Back-Staff.</hi> See also Robertson's Navigation, book 9, sect. 7.</p></div1><div1 part="N" n="DAY" org="uniform" sample="complete" type="entry"><head>DAY</head><p>, a division of time arising from the appearance and disappearance of the sun.</p><p><hi rend="smallcaps">Day</hi> is either <hi rend="italics">natural</hi> or <hi rend="italics">artificial.</hi></p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Day</hi> is that which is primarily meant by the word Day, and is the time of its being light, or the time while the sun is above the horizon. Though sometimes the twilight is included in the term daylight; in opposition to night or darkness, being the time from the end of twilight to the beginning of daylight.</p><p><hi rend="italics">Natural</hi> <hi rend="smallcaps">Day</hi> is the portion of time in which the sun performs one revolution round the earth; or rather the time in which the earth makes a rotation on its axis. And this is either astronomical or civil.</p><p><hi rend="italics">Astronomical</hi> <hi rend="smallcaps">Day</hi> begins at noon, or when the sun's centre is on the meridian, and is counted 24 hours to the following noon.</p><p><hi rend="italics">Civil</hi> <hi rend="smallcaps">Day</hi> is the time allotted for day in civil purposes, and begins differently in different nations, but still including one whole rotation of the earth on its axis; beginning either at sun-rise, sun-set, noon, or midnight.</p><p>1st, At sun-rising, among the ancient Babylonians, Persians, Syrians, and most other eastern nations, with the present inhabitants of the Balearic islands, the Greeks, &c. 2dly, At sun-setting, among the ancient Athenians and Jews, with the Austrians, Bohemians, Marcomanni, Silesians, modern Italians, and Chinese. 3dly, At noon, with astronomers, and the ancient Umbri and Arabians. And 4thly, at midnight, among the ancient Egyptians and Romans, with the modern English, French, Dutch, Germans, Spaniards, and Portuguese.</p><p>The day is divided into hours; and a certain number of days makes a week, a month, or a year.</p><p>The different length of the natural day in different climates, has been matter of controversy, viz, whether the natural days be all equally long throughout the year; and if not, what their difference is? A professor <cb/> of mathematics at Seville, in the Philos. Trans. vol. 10, pa. 425, asserts, from a continued series of observations for three years, that they are all equal. But Mr. Flamsteed, in the same Trans. pa. 429, refutes the opinion; and shews that one day, when the sun is in the equinoctial, is shorter than when he is in the tropics, by 40 seconds; and that 14 tropical days are longer than so many equinoctial ones, by 10 minutes. This inequality of the days flows from two several principles: the one, the eccentricity of the earth's orbit; the other, the obliquity of the ecliptic with regard to the equator, which is the measure of time. As these two causes happen to be differently combined, the length of the day is varied. See <hi rend="smallcaps">Equation</hi> <hi rend="italics">of time.</hi></p><p><hi rend="smallcaps">Day's</hi>-<hi rend="italics">Work,</hi> in Navigation, denotes the reckoning or account of the ship's course, during 24 hours, or between noon and noon.</p></div1><div1 part="N" n="DECAGON" org="uniform" sample="complete" type="entry"><head>DECAGON</head><p>, a plane geometrical figure of ten sides and ten angles. When all the sides and angles are equal, it is a <hi rend="italics">regular</hi> decagon, and may be inscribed in a circle; otherwise, not.</p><p>If the radius of a circle, or the side of the inscribed hexagon, be divided in extreme and mean proportion, the greater segment will be the side of a decagon inscribed in the same circle. And therefore, as the side of the decagon is to the radius, so is the radius to the sum of the two. Whence, if the radius of the circle be <hi rend="italics">r,</hi> the side of the inscribed decagon will be (√(5-1))/2 X <hi rend="italics">r.</hi></p><p>If the side of a regular decagon be 1, its area will be 5/2 √(5+2√5)=7.6942088; therefore as 1 is to 7.6942088, so is the square of the side of any regular decagon, to the area of the same: so that, if <hi rend="italics">s</hi> be the side of such a decagon, its area will be equal to 7.6942088<hi rend="italics">s</hi><hi rend="sup">2</hi>. See <hi rend="smallcaps">Regular</hi> <hi rend="italics">Figure.</hi></p><p>To inscribe a decagon in a circle geometrically. See my Mensuration, prob. 35, pa. 25, 2d edit.</p></div1><div1 part="N" n="DECEMBER" org="uniform" sample="complete" type="entry"><head>DECEMBER</head><p>, the last month of the year; in which the sun enters the tropic of Capricorn, making the winter solstice.</p><p>In the time of Romulus, December was the 10th month; whence the name, viz, from decem, ten; for the Romans began their year in March, from which December is the 10th month.</p><p>The month of December was under the protection of Vesta. Romulus assigned it 30 days; Numa reduced it to 29; which Julius Cæsar increased to 31.</p><p>At the latter part of this month they had the <hi rend="italics">Juveniles Ludi,</hi> and the country people kept the feast of the goddess Vacuna in the fields, having then gathered in their fruits, and sown their corn; whence it seems is derived our popular festival called Harvest-home.</p></div1><div1 part="N" n="DECHALES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DECHALES</surname> (<foreName full="yes"><hi rend="smallcaps">Claud-Francis-Milliet</hi></foreName>)</persName></head><p>, an excellent mathematician, mechanist, and astronomer, was born at Chambery, the capital of Savoy, in 1611. He chiefly excelled in a just knowledge of the mathematical and mechanical sciences: not that he was bent upon new discoveries, or happy in making them; as his talent rather lay in explaining those sciences with ease and accuracy; which perhaps rendered him equally useful and deserving of esteem. Indeed it was generally allowed that he made the best use of the productions of other men, and that he drew the several parts of the mathematical sciences together with great judgment <pb n="359"/><cb/> and perspicuity. It is also said of him, that his probity was not inferior to his learning; and that both these qualities made him generally admired and beloved at Paris, where for four years together he read public mathematical lectures in the college of Clermont.—— From hence he removed to Marseilles, where he taught the art of navigation and the practical mathematical sciences.—He afterward became professor of mathematics in the university of Turin, where he died March 28, 1678, at 67 years of age.</p><p>Among other works which do honour to his memory, are,</p><p>1. An edition of Euclid's Elements; in which he has omitted the less important propositions, and explained the uses of those he has retained.</p><p>2. A Discourse on Fortification; and another on Navigation.</p><p>3. These performanoes, with some others, were collected in 3 volumes folio, under the title of <hi rend="italics">Mundus Mathematicus,</hi> being indeed a complete course of mathematics. And the same was afterward much enlarged, and published at Lyons, 1690, in 4 large volumes, folio.</p></div1><div1 part="N" n="DECIL" org="uniform" sample="complete" type="entry"><head>DECIL</head><p>, <hi rend="italics">Decilis,</hi> an aspect or position of two planets, when they are distant from each other a 10th part of the zodiac, or 36 degrees; and is one of the new aspects invented by Kepler.</p></div1><div1 part="N" n="DECIMALS" org="uniform" sample="complete" type="entry"><head>DECIMALS</head><p>, any thing proceeding by tens; as Decimal arithmetic, Decimal fractions, Decimal scales, &c.</p><p><hi rend="smallcaps">Decimal</hi> <hi rend="italics">Arithmetic,</hi> in a general sense, may be considered as the common arithmetical computation in use, in which the decimal scale of numbers is used, or in which the places of the figures change their value in a tenfold proportion, being 10 times as much for every place more towards the left hand, or 10 times less for every place more towards the right hand; the places being supposed indefinitely continued, both to the right and left. In this sense, the word includes both the arithmetic of integers, and decimal fractions. In a more restrained sense however, it means only.</p><p><hi rend="smallcaps">Decimal</hi> <hi rend="italics">Fractions,</hi> which are fractions whose denominator is always a 1 with some number of ciphers annexed, more or fewer according to the value of the fraction, the numerator of which may be any number whatever; as 1/10, 3/100, 7/1000, &c.</p><p>As the denominator of a decimal is always one of the numbers 10, 100, 1000, &c, the inconvenience of writing these denominators down may be saved, by placing a proper distinction before the figures of the numerator only, to distinguish them from integers, for the value of each place of figures will be known in decimals, as well as in integers, by their distance from the 1st or unit's place of integers, having similar names at equal distances, as appears by the following scale of places, both in decimals and integers: <table><row role="data"><cell cols="1" rows="1" role="data">&c</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">&c</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">millions</cell><cell cols="1" rows="1" role="data">hund. of thousands</cell><cell cols="1" rows="1" role="data">tens of thousands</cell><cell cols="1" rows="1" role="data">thousands</cell><cell cols="1" rows="1" role="data">hundreds</cell><cell cols="1" rows="1" role="data">tens</cell><cell cols="1" rows="1" role="data">units</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">tenths</cell><cell cols="1" rows="1" role="data">hundredths</cell><cell cols="1" rows="1" role="data">thousandths</cell><cell cols="1" rows="1" role="data">ten thousandths</cell><cell cols="1" rows="1" role="data">hund. thousandths</cell><cell cols="1" rows="1" role="data">millionths</cell><cell cols="1" rows="1" role="data"/></row></table> <cb/></p><p>The mark of distinction for decimals, called the <hi rend="italics">separatrix,</hi> has been various at different times, according to the fancy of different authors; sometimes a semiparenthesis, or a semicrotchet, or a perpendicular bar, or the same with a line drawn under the figures, or simply this line itself, &c; but it is usual now to write either a comma or a full point near the bottom of the figures; I place the point near the upper part of the figures, as was done also by Newton; a method which prevents the separatrix from being confounded with mere marks of punctuation.</p><p>In setting down a decimal fraction without its denominator, the numerator must consist of as many places as there are ciphers in the denominator; and if it has not so many figures, the defect must be supplied by setting before them as many ciphers as will make them up so many: thus 3/10 is .3; and 14/100 is .14; and 14/1000 is .014; and 3/1000 is .003; &c.</p><p>So that, as ciphers on the right-hand side of integers <hi rend="italics">increase</hi> their value decimally, or in a tenfold proportion, as 2, 20, 200, &c; so, when set on the left-hand of decimal fractions, they <hi rend="italics">decrease</hi> the value decimally, or in a tenfold proportion, as .2, .02, .002, &c. But ciphers set on the other sides of these numbers, make no alteration in their value, neither of increase nor decrease, viz, on the left-hand of integers, or on the righthand of decimals; so 2, or 02, or 002, &c, are all the same; as are also .2, or .20, or .200, &c.</p><p>Decimal fractions may be considered as having been introduced by Regiomontanus, about the year 1464, viz, when he transformed the tables of sines from a sexagesimal to a decimal scale. They were also used by Ramus, in his Arithmetic, written in 1550; and before his time by our countrymen Buckley and Recorde. But it was Stevinus who first wrote an express treatise on decimals, viz, about the year 1582, in <hi rend="italics">La Practique d'Arithmetique;</hi> since which time, this has commonly made a part in most treatises on arithmetic.</p><p><hi rend="italics">To reduce any Vulgar fraction,</hi> or parts of any thing, as suppose 3/8, to a decimal fraction of the same value; add ciphers at pleasure to the numerator, and divide by the denominator: thus, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8)3.000</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">.375</cell><cell cols="1" rows="1" role="data">=3/8;</cell></row></table> and therefore .375 or 375/1000 is a decimal of the same value with the proposed vulgar fraction 3/8.</p><p>Some vulgar fractions can never be reduced into decimals without defect; as 1/3, which by division is .33333 &c infinitely.</p><p>Such numbers are very properly called circulating decimals, and repetends, because of the continual return of the same figures. See <hi rend="smallcaps">Repetends</hi> and C<hi rend="smallcaps">IRCULATES.</hi></p><p>The common arithmetical operations are performed the same way in decimals, as they are in integers; regard being had only to the particular notation, to distinguish the fractional from the integral part of a sum. Thus,</p><p>In Addition and Subtraction, all figures of the same place or denomination are set straight under each other, the separatrix, or decimal points, forming a straight column.</p><p>In Multiplication, set down the numbers, and mul- <pb n="360"/><cb/> tiply them as integers; and point off from the product as many places of decimals as there are in both factors; prefixing ciphers if there be any defect of figures.</p><p>In Division, set down the numbers and divide also as in integers; making as many decimals in the quotient, as those in the dividend are more than those in the divisor.</p><p>Examples are numerous and common in most books of arithmetic.</p><p><hi rend="smallcaps">Decimal</hi> <hi rend="italics">Scales,</hi> are any scales divided decimally, or by tens.</p></div1><div1 part="N" n="DECLINATION" org="uniform" sample="complete" type="entry"><head>DECLINATION</head><p>, in Astronomy, is the distance of the sun, star, planet, &c, from the equinoctial, either northward or southward; being the same with latitude in geography, or distance from the equator.</p><p>Declination is either <hi rend="italics">real</hi> or <hi rend="italics">apparent,</hi> according as the real or apparent place of the point or object is considered.</p><p>The declination of any point S is an arch of the meridian SE, contained between the given point and the equinoctical EQ. The declination of a star &c, is found by knowing or observ- <figure/> ing the latitude of the place, i. e. the height of the pole, and then the meridian altitude of the star, &c; hence the difference between the co-latitude and the altitude of the star &c, is the declination, viz, the difference between EH the co-latitude, and SH the altitude, is ES the declination. For ex. Tycho found at Uranibourg the meridian <table><row role="data"><cell cols="1" rows="1" role="data">altitude of Cauda Leonis, viz,</cell><cell cols="1" rows="1" role="data">HS=</cell><cell cols="1" rows="1" role="data">50°</cell><cell cols="1" rows="1" role="data">59′</cell><cell cols="1" rows="1" role="data">00″</cell></row><row role="data"><cell cols="1" rows="1" role="data">the co-latitude is</cell><cell cols="1" rows="1" role="data">HE=</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">45</cell></row><row role="data"><cell cols="1" rows="1" role="data">rem. declin. north</cell><cell cols="1" rows="1" role="data">ES=</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">15</cell></row></table></p><p><hi rend="italics">To find the Sun's Declination at any time;</hi> having given his place in the ecliptic; the rule is, as radius is to the sine of the sun's longitude, so is the sine of the greatest declination, or obliquity of the ecliptic, to his present declination sought.</p><p>In constructing tables of declination of the sun, planets, and stars, regard should be had to refraction, aberration, nutation, and parallax.</p><p>By comparing ancient observations with the modern, it appears that the declination of the fixed stars is variable; and that differently in different stars; for in some it increases, and in others decreases, and that in different quantities.</p><p><hi rend="italics">Circles of</hi> <hi rend="smallcaps">Declination</hi>, are great circles of the sphere passing through the poles of the world, on which the declination is measured; and consequently are the same as meridians in geography.</p><p><hi rend="italics">Parallax of</hi> <hi rend="smallcaps">Declination</hi>, is an arch of the circle of declination, by which the parallax in altitude increases or diminishes the declination of a star.</p><p><hi rend="italics">Parallels of</hi> <hi rend="smallcaps">Declination</hi>, are lesser circles parallel to the equinoctial. The tropic of Cancer is a parallel of declination at 23° 28′ distance from the equinoctial northward; and the tropic of Capricorn is the parallel of declination as far distant southward.</p><p><hi rend="italics">Refraction of the</hi> <hi rend="smallcaps">Declination</hi>, an arch of the circle of declination, by which the declination of a star is increased or diminished by means of the refraction. <cb/></p><p><hi rend="smallcaps">Declination</hi> <hi rend="italics">of the Compass,</hi> or <hi rend="italics">Needle,</hi> is its deviation from the true meridian. See <hi rend="smallcaps">Variation.</hi></p><p><hi rend="smallcaps">Declination</hi> <hi rend="italics">of a Vertical Plane,</hi> or <hi rend="italics">Wall,</hi> in Dialling, is an arch of the horizon, comprehended either between the plane and the prime vertical circle, when it is counted from the east or west; or between the plane and the meridian, if it be accounted from the north or south.</p></div1><div1 part="N" n="DECLINATOR" org="uniform" sample="complete" type="entry"><head>DECLINATOR</head><p>, or <hi rend="smallcaps">Declinatory</hi>, an instrument in dialling, by which the declination, inclination, and reclination of planes are determined.</p></div1><div1 part="N" n="DECLINERS" org="uniform" sample="complete" type="entry"><head>DECLINERS</head><p>, or <hi rend="smallcaps">Declining</hi> <hi rend="italics">Dials,</hi> are those which cut obliquely, either the plane of the prime vertical circle, or the plane of the horizon.</p><p>The use of declining vertical dials is very frequent; because the erect walls of houses, on which dials are commonly drawn, mostly decline from the cardinal points. But incliners and recliners are very rare.</p></div1><div1 part="N" n="DECLIVITY" org="uniform" sample="complete" type="entry"><head>DECLIVITY</head><p>, a sloping or oblique descent.</p></div1><div1 part="N" n="DECREMENT" org="uniform" sample="complete" type="entry"><head>DECREMENT</head><p>, <hi rend="italics">Equal, of Life.</hi> See C<hi rend="smallcaps">OMPLEMENT</hi> <hi rend="italics">of life.</hi></p><p><hi rend="smallcaps">Decrements</hi> are the small parts by which a variable and decreasing quantity becomes less and less. The indefinitely small decrements are proportional to the fluxions, which in this case are negative. See <hi rend="smallcaps">Fluxions</hi>, also <hi rend="smallcaps">Increments.</hi></p></div1><div1 part="N" n="DECUPLE" org="uniform" sample="complete" type="entry"><head>DECUPLE</head><p>, a term of relation or proportion in arithmetic, implying a tenfold change or scale of variation, or one thing 10 times as much as another.</p></div1><div1 part="N" n="DECUSSATION" org="uniform" sample="complete" type="entry"><head>DECUSSATION</head><p>, a term in geometry and optics, signifying the crossing of any two lines or rays &c: or the action itself of crossing.</p><p>The rays of light decussate in the chrystalline, before they reach the retina.</p><p>Many of the lines of the 3d order decussate themselves. See Newton's Enumeratio, &c.</p></div1><div1 part="N" n="DEE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DEE</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a famous mathematician and astrologer, was born at London 1527. In 1542 he was sent to St. John's College, Cambridge. After five years close application to study, chiesly in the mathematical and astronomical sciences, he went over to Holland, to visit some mathematicians on the Continent; whence, after a year's absence, he returned to Cambridge, and was there elected one of the Fellows of Trinity College, then first erected by King Henry the 8th. In 1548 he left England a second time, his stay at home being rendered uneasy to him, by the suspicions that were entertained of his being a conjurer, arising chiefly from his application to astronomy, and from some mechanical inventions of his.</p><p>He now visited the university of Louvain; where he was much caressed, and visited by several persons of high rank. After two years he went into France, and read lectures, in the college of Rheims, upon Euclid's Elements. In 1551, he returned to England, and was introduced to King Edward, who assigned him a pension of 100 crowns, which he afterward relinquished for the rectory of Upton upon Severn. But soon after the accession of Queen Mary, having some correspondence with her sister Elizabeth, he was accused of practising against the queen's life by enchantment: on which account he suffered a tedious confinement, and was several times examined; till, in the year 1555, he obtained his liberty by an order of council. <pb n="361"/><cb/></p><p>When Queen Elizabeth ascended the throne, Dee was consulted concerning a propitious day for the coronation: on which occasion he was introduced to the queen, who made him great promises, which were but ill performed. In 1564, he made another voyage to the continent, to present a book which he had dedicated to the Emperor Maximilian. He returned to England the same year; but in 1571 we find him in Lorrain; where, being dangerously ill, the queen sent over two physicians to his relief. Having once more returned to his native country, he settled at Mortlake in Surry, where he continued his studies with much ardour, and collected a great library of printed books and manuscripts, with a number of instruments; most of which were afterward destroyed by the mob, as belonging to one who dealt with the devil.</p><p>In 1578, the queen being much indisposed, Mr. Dee was sent abroad to consult with German physicians and philosophers (astrologers no doubt) on the occasion; though some have said she employed him as a spy; probably he acted in a double capacity. We next find him again in England, where he was soon after employed in a more rational service. The queen, desirous to be informed concerning her title to those countries which had been discovered by Englishmen, ordered Dee to consult the ancient records, and to furnish her with proper geographical descriptions. Accordingly, in a short time, he presented to the queen, at Richmond, two large rolls, in which the discovered countries were geographically described and historically illustrated. His next employment was the reformation of the calendar, on which subject he wrote a rational and learned treatise, preserved in the Ashmolean library at Oxford.</p><p>Hitherto the extravangancies of our eccentrical philosopher seem to have been tempered with a tolerable proportion of reason and science; but henceforward he is to be considered as a mere necromancer and credulous alchymist. In the year 1581 he became acquainted with one Edward Kelly, by whose assistance he performed divers incantations, and maintained a frequent imaginary intercourse with spirits and angels; one of whom made him a present of a black speculum (a polished piece of cannel-coal), in which these appeared to him as often as he had occasion for them, answering his questions, &c. Hence Butler says, Kelly did all his feats upon<lb/> The devil's looking-glass, a stone.<lb/> <hi rend="smallcaps">Hudibras.</hi><lb/></p><p>In 1583 they became acquainted with a certain Polish nobleman, then in England, named Albert Laski, a person equally addicted to the same ridiculous pursuits: he was so charmed with Dee and Kelly, that he persuaded them to accompany him to his native country; by whose means they were introduced to Rodolph king of Bohemia; who, though a credulous man, was soon disgusted with their nonsense. They were afterward introduced to the king of Poland, but with no better success. Soon after this they were entertained at the castle of a rich Bohemian nobleman, where they lived for some time in great affluence; owing, as they asserted, to their art of transmutation by means of a certain powder in the possession of Kelly. <cb/></p><p>Dee, now quarrelling with his companion, quitted Bohemia, and returned to England, where he was once more graciously received by the queen; who, in 1595, made him warden of Manchester college, in which town he resided several years. In 1604 he returned to his house at Mortlake, where he died in 1608, at 81 years of age; leaving a large family and many works behind him.</p><p>The books that were printed and published by Dee, are, 1. <hi rend="italics">Propœdumata Aphoristica, &c.</hi> in 1558, in 12mo. —2. <hi rend="italics">Monas Hieroglyphica ad Regem Romanorum Maximilianum;</hi> 1564.—3. <hi rend="italics">Epistola ad eximium ducis Urbini mathematicum, Fredericum Commandinum, prefixa libello Machometi Bagdadini de Superficierum Divisionibus &c;</hi> 1570.—4. <hi rend="italics">The British Monarchy, othcrwise called, The Petty Navy Royal;</hi> 1576.—5. <hi rend="italics">Preface Mathematical to the English Euclid,</hi> published by Henry Billingsley, 1570: certainly a very curious and elaborate composition, and where he says, many more arts are wholly invented by name, definition, property, and use, than either the Grecian or Roman mathematicians have left to our knowledge.—6. <hi rend="italics">Divers and many annotations and inventions dispersed and added after the</hi> 10<hi rend="italics">th book of English Euclid;</hi> 1570.—7. <hi rend="italics">Epistola prefixa Ephemeridibus Joannis Feldi à</hi> 1557, <hi rend="italics">cui rationem declaraverat Ephemerides conscribendi.</hi>—8. <hi rend="italics">Parallaticæ Commentationis Paxeosque Nucleus quidam</hi>; 1573.</p><p>This catalogue of Dee's printed and published works is to be found in his <hi rend="italics">Compendious Rehearsal &c,</hi> as well as in his letter to Abp. Whitgift: and from the same places might be transcribed more than 40 titles of books unpublished, that were written by him.</p></div1><div1 part="N" n="DEFENCE" org="uniform" sample="complete" type="entry"><head>DEFENCE</head><p>, in Sieges, is used for any thing that serves to preserve or screen the soldiers, or the place. So the parapets, flanks, casemates, ravelins, and outworks, that cover the place, are called the defences, or covers of the place: and when the cannon have beaten down or ruined these works, so that the men cannot fight under cover, the defences of the place are said to be demolished.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Defence</hi>, is that which flanks a bastion, being drawn from the flank opposite to it.</p><p>The line of defence should not exceed a musket shot, i. e. 120 fathoms: indeed Melder allows 130, Scheiter 140, Vauban and Pagan 150.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Defence</hi>, <hi rend="italics">greater,</hi> or <hi rend="italics">fichant,</hi> is a line drawn from the point of the bastion to the concourse of the opposite flank and curtin.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Defence</hi>, <hi rend="italics">lesser,</hi> or <hi rend="italics">rasant,</hi> or <hi rend="italics">flanquant,</hi> is the face of the bastion continued to the curtin.</p></div1><div1 part="N" n="DEFERENT" org="uniform" sample="complete" type="entry"><head>DEFERENT</head><p>, or <hi rend="smallcaps">Deferens</hi>, in the ancient astronomy, an imaginary circle, which, as it were, carries about the body of a planet, and is the same with the eccentric; being invented to account for the eccentricity, perigee, and apogee of the planets.</p><p>DEFICIENT <hi rend="italics">Hyperbola,</hi> is a curve having only one asymptote, though two hyperbolic legs running out infinitely by the side of the asymptote, but contrary ways. See <hi rend="smallcaps">Curve.</hi></p><p>This name was given to the curves by Newton, in his Enumeratio Linearum tertii Ordinis. There are 6 different species of them, which have no diameters, expressed by the equation , the term <hi rend="italics">ax</hi><hi rend="sup">3</hi> being negative. When the equation <pb n="362"/><cb/> has all its roots real and unequal, the curve has an oval joined to it. When the two middle roots are equal, the oval joins to the legs, which then cut one another in shape of a noose. When three roots are equal, the nodus is changed into a very acute cusp or point. When, of three roots with the same sign, the two greatest are equal, the oval vanishes into a point. When any two roots are imaginary, there is only a pure serpentine hyperbola, without any oval, decussation, cusp, or conjugate point; and when the terms <hi rend="italics">b</hi> and <hi rend="italics">d</hi> are wanting, it is of the 6th species.</p><p>There are also 7 different species of these curves, having each one diameter, expressed by the above equation when the term <hi rend="italics">ey</hi> is wanting: according to the various conditions of the roots of the equation , as to their reality, equality, their having the same signs, or two of them being imaginary.</p><p><hi rend="smallcaps">Deficient</hi> <hi rend="italics">Numbers,</hi> are those whose aliquot parts added together, make a sum less than the whole number: as 8, whose parts 1, 2, 4, make only 7; or the number 16, whose parts 1, 2, 4, 8 make only 15.</p></div1><div1 part="N" n="DEFILE" org="uniform" sample="complete" type="entry"><head>DEFILE</head><p>, <hi rend="italics">in Fortification,</hi> a narrow line or passage through which troops can pass only in file, making a small front, so that the enemy may easily stop their march, and charge them with the more advantage, as the front and rear cannot come to the relief of one another.</p></div1><div1 part="N" n="DEFINITION" org="uniform" sample="complete" type="entry"><head>DEFINITION</head><p>, an enumeration, or specification of the chief simple ideas of which a compound idea consists, in order to ascertain or explain its nature and character.</p><p>Definitions are of two kinds; the one <hi rend="italics">nominal,</hi> or of the <hi rend="italics">name;</hi> the other <hi rend="italics">real,</hi> or of the <hi rend="italics">thing.</hi></p><p><hi rend="italics">Nominal</hi> <hi rend="smallcaps">Definition</hi>, is an enumeration of such known characters as are sufficient for distinguishing any proposed thing from others; as is that of a square, when it is said that it is a quadrilateral, equilateral, rectangular figure.</p><p><hi rend="italics">Real</hi> <hi rend="smallcaps">Definition</hi>, a distinct notion, explaining the genesis of a thing; that is, how the thing is made or done: as is this definition of a circle, viz, that it is a figure described by the motion of a right line about a fixed point.</p></div1><div1 part="N" n="DEFLECTION" org="uniform" sample="complete" type="entry"><head>DEFLECTION</head><p>, the turning any thing aside from its former course, by some adventitious or external cause.</p><p>The word is often applied to the tendency of a ship from her true course, by reason of currents, &c, which turn her out of her right way.</p><p><hi rend="smallcaps">Deflection</hi> <hi rend="italics">of the Rays of Light,</hi> is a property which Dr. Hook observed in 1675. He found it different both from reflection and refraction; and that it was made perpendicularly towards the surface of the opacous body.</p><p>This is the same property which Newton calls <hi rend="italics">inflection.</hi> And by others it is called <hi rend="italics">diffraction.</hi></p></div1><div1 part="N" n="DEGREE" org="uniform" sample="complete" type="entry"><head>DEGREE</head><p>, <hi rend="italics">in Algebra,</hi> is used in speaking of equations, when they are said to be of such a degree according to the highest power of the unknown quantity. If the index of that power be 2, the equation is of the 2d degree; if 3, it is of the 3d degree, and so on.</p><div2 part="N" n="Degree" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Degree</hi></head><p>, in Geometry or Trigonometry, is the 360th part of the circumference of any circle; for every <cb/> circle is considered as divided into 360 parts, called degrees; which are marked by a small ° near the top of the figure; thus 45° is 45 degrees.</p><p>The degree is subdivided into 60 smaller parts, called minutes, meaning first minutes; the minute into 60 others, called seconds; the second into 60 thirds; &c. Thus 45° 12′ 20″ are 45 degrees, 12 minutes, 20 seconds.</p><p>The magnitude or quantity of angles is accounted in degrees; for because of the uniform curvature of a circle in all its parts, equal angles at the centre are subtended by equal arcs, and by similar arcs in peripheries of different diameters; and an angle is said to be of so many degrees, as are contained in the arc of any circle comprehended between the legs of the angle, and having the angular point for its centre. Thus we say an angle of 90°, or of 45° 24′, or of 12° 20 30″. It is also usual to say, such a star is mounted so many degrees above the horizon, or declines so many degrees from the equator; or such a town is situate in so many degrees of latitude or longitude.—A sign of the ecliptic, or zodiac, contains 30 degrees.</p><p>The division of the circle into 360 degrees is usually ascribed to the Egyptians, probably from the circle of the sun's annual course, or according to their number of days in the year, allotting a degree to each day. It is a convenient number too, as admitting of a great many aliquot parts, as 2, 3, 4, 5, 6, 8, 9, &c. The sexagesimal subdivision, however, has often been condemned as improper, by many eminent mathematicians, as Stevinus, Oughtred, Wallis, Briggs, Gellibrand, Newton, &c; who advise a decimal division instead of it, or else that of centesms; as the degree into 100 parts, and each of these into 100 parts again, and so on. Stevinus even holds, that this division of the circle which he contends for, obtained in the wise age, <hi rend="italics">in sæculo sapienti.</hi> Stev. Cosmog. lib. 1, def. 6. And several large tables of sines &c have been constructed according to that plan, and published, by Briggs, Newton, and others. And I myself have carried the idea still much farther, in a memoir published in the Philos. Trans. of 1783, containing a proposal for a new division of the quadrant, viz, into equal decimal parts of the radius; by which means the degrees or divisions of the arch would be the real lengths of the arcs, in terms of the radius: and I have since computed those lengths of the arcs, with their sines, &c, to a great extent and accuracy.</p><p><hi rend="smallcaps">Degree</hi> <hi rend="italics">of Latitude,</hi> is the space or distance on the meridian through which an observer must move, to vary his latitude by one degree, or to increase or diminish the distance of a star from the zenith by one degree; and which, on the supposition of the perfect sphericity of the earth, is the 360th part of the meridian.</p><p>The quantity of a degree of a meridian, or other great circle, on the surface of the earth, is variously determined by different observers: and the methods made use of are also various.</p><p>Eratosthenes, 250 years before Christ, first determined the magnitude of a degree of the meridian, between Alexandria and Syene on the borders of Ethiopia, by measuring the distance between those places, and comparing it with the difference of a star's zenith distances at those places; and found it 694 4/9 stadia. <pb n="363"/><cb/></p><p>Posidonius, in the time of Pompey the Great, by means of the different altitudes of a star near the horizon, taken at different places under the same meridian, compared in like manner with the distance between those places, determined the length of a degree only 600 stadia.</p><p>Ptolomy fixes the degree at 68 2/3 Arabic miles, counting 7 1/2 stadia to a mile. The Arabs themselves, who made a computation of the diameter of the earth, by measuring the distance of two places under the same meridian, in the plains of Sennar, by order of Almamon, make it only 56 miles. Kepler, determining the diameter of the earth by the distance of two mountains, makes a degree 13 German miles; but his method is far from being accurate. Snell, seeking the diameter of the earth from the distance between two parallels of the equator, finds the quantity of a degree, by one method 57064 Paris toises, or 342384 feet; by another meth. 57057 toises, or 342342 feet. The mean between which two numbers, M. Picard found by mensuration, in 1669, from Amiens to Malvoisin, the most certain, and he makes the quantity of a degree 57060 toises, or 342360 feet. However, M. Cassini, at the king's command, in the year 1700, repeated the same labour, and measuring the space of 6° 18′, from the observatory at Paris, along the meridian, to the city of Collioure in Roussillon, that the greatness of the interval might diminish the error, found the length of the degree equal to 57292 toises, or 343742 Paris feet, amounting to 365184 English feet.</p><p>And with this account nearly agrees that of our countryman Norwood, who, about the year 1635, measured the distance between London and York, and found that distance 905751 English feet; the difference of latitude being 2° 28′, hence, he determined the quantity of one degree at 367196 English feet, or 57300 Paris toises, or 69 miles, 288 yards. See Newt. Princ. Phil. prop. 19; and Hist. Acad. Scienc. anno 1700, pa. 153.</p><p>M. Cassini, the son, completed the work of measuring the whole arc of the meridian through France, in 1718. For this purpose he divided the meridian of France into two arcs, which he measured separately. The one from Paris to Collioure gave him 57097 toises; the other from Paris to Dunkirk 56960; and the whole are from Dunkirk to Collioure 57060, the same as M. Picard's.</p><p>M. Muschenbroek, in 1700, resolving to correct the errors of Snell, found by particular observations, that the degree between Alcmaer and Bergen-op-zoom contained 57033 toises.</p><p>Messieurs Maupertuis, Clairaut, Camus, Monnier, and Outheir of France, were sent on a northern expedition, and began their operations, assisted by M. Celsus, an eminent astronomer of Sweden, in Swedish Lapland, in July 1736, and sinished them by the end of May following. They obtained the measure of that degree, whose middle point was in lat. 66° 20′ north, and found it 57439 toises, when reduced to the level of the sea. About the same time another company of philosophers was sent to South America, viz, Messieurs Godin, Bouguer, and Condamine of France, to whom were joined Don Jorge Juan, and Don An- <cb/> tonio de Ulloa of Spain. They left Europe in 1735, and began their operations in the province of Quito in Peru, about October 1736, and finished them, after many interruptions, about 8 years after. The Spanish gentlemen published a separate account, and assign for the measure of a degree of the meridian at the equator 56768 toises. M. Bouguer makes it 56753 toises, when reduced to the level of the sea; and M. Condamine states it at 56749 toises.</p><p>M. Caille, being at the Cape of Good Hope in 1752, found the length of a degree of the meridian in lat. 33° 18′ 30″ south, to be 57037 toises. In 1755, father Boscovich found the length of a degree in lat. 43° north to be 56972 toises, as measured between Rome and Rimini in Italy. In the year 1740, Messrs Cassini and La Caille again examined the former measures in France, and, after making all the necessary corrections, found the measure of a degree, whose middle point is in lat. 49° 22′ north, to be 57074 toises; and in the lat. of 45°, it was 57050 toises.</p><p>In 1764, F. Beccaria completed the measurement of a portion of the meridian near Turin; from which it is deduced that the length of a degree, whose middle lat. is 44° 44′ north, is 57024 Paris toises.</p><p>At Vienna, 3 degrees of the meridian were measured; and the medium, for the latitude of 47° 40′ north may be taken at 57091 Paris toises. See an account of this measurement, by father Joseph Liesganig, in the Philos. Trans. 1768, pa. 15.</p><p>Finally, in the same vol. too is an account of the measurement of a part of the meridian in Maryland and Pensilvania, North America, 1766, by Messrs Mason and Dixon; from which it follows that the length of a degree whose middle point is 39° 12′ north, was 363763 English feet, or 56904 1/2 Paris toises.</p><p>Hence, from the whole we may collect the following table of the principal measures of a degree in different parts of the earth, as measured by different persons, viz, <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Mean Latitude.</cell><cell cols="1" rows="1" rend="align=center" role="data">Length of a Degree in Paris toises.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Names of the Measurers.</cell><cell cols="1" rows="1" rend="align=center" role="data">Years of Measurement.</cell></row><row role="data"><cell cols="1" rows="1" role="data">66°</cell><cell cols="1" rows="1" rend="align=right" role="data">20′</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">57422</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Maupertuis &c</cell><cell cols="1" rows="1" role="data">1736 & 1737</cell></row><row role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">57074</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Maupertuis &c and Cassini</cell><cell cols="1" rows="1" role="data">1739 & 1740</cell></row><row role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">57091</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Liesganig</cell><cell cols="1" rows="1" role="data">1766</cell></row><row role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">57028</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Cassini</cell><cell cols="1" rows="1" role="data">1739 & 1740</cell></row><row role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">57069</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Beccaria</cell><cell cols="1" rows="1" role="data">1760 to 1764</cell></row><row role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">56979</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Boscovich and Le Maire</cell><cell cols="1" rows="1" role="data">1752</cell></row><row role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">56888</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Mason & Dixon</cell><cell cols="1" rows="1" role="data">1764 to 1768</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">56750</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Bouguer and Condamine</cell><cell cols="1" rows="1" role="data">1736 to 1744</cell></row><row role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data">57037</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">La Caille</cell><cell cols="1" rows="1" role="data">1752</cell></row></table></p><p>The method of obtaining the length of a degree of the terrestrial meridian, is to measure a certain distance upon it by a series of triangles, whose angles may be found by actual observation, connected with a base, whose length may be taken by an actual survey, or otherwise; and then to observe the different altitudes <pb n="364"/><cb/> of some star at the two extremities of that distance, which gives the difference of latitude between them: then, by proportion, as this difference of latitude is to one degree, so is the measured length to the length of one degree of the meridian sought. This method was first practised by Eratosthenes, in Egypt. See G<hi rend="smallcaps">GEOGRAPHY</hi>, and the beginning of this article.</p><p><hi rend="smallcaps">Degree</hi> <hi rend="italics">of longitude,</hi> is the space between two meridians that make an angle of 1° with each other at the poles; the quantity or length of which is variable, according to the latitude, being every where as the cosine of the latitude; viz, as the cosine of one lat. is to the cosine of another, so is the length of a degree in the former lat. to that in the latter; and from this theorem is computed the following Table of the length of a degree of long. indifferent latitudes, supposing the earth to be a globe. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Degr. lat.</cell><cell cols="1" rows="1" rend="align=center" role="data">English miles.</cell><cell cols="1" rows="1" rend="align=center" role="data">Degr. lat.</cell><cell cols="1" rows="1" rend="align=center" role="data">English miles.</cell><cell cols="1" rows="1" rend="align=center" role="data">Degr. lat.</cell><cell cols="1" rows="1" rend="align=center" role="data">English miles.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">69.07</cell><cell cols="1" rows="1" rend="align=center" role="data">31</cell><cell cols="1" rows="1" rend="align=center" role="data">59.13</cell><cell cols="1" rows="1" rend="align=center" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">33.45</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">69.06</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" rend="align=center" role="data">58.51</cell><cell cols="1" rows="1" rend="align=center" role="data">62</cell><cell cols="1" rows="1" rend="align=right" role="data">32.40</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">69.03</cell><cell cols="1" rows="1" rend="align=center" role="data">33</cell><cell cols="1" rows="1" rend="align=center" role="data">57.87</cell><cell cols="1" rows="1" rend="align=center" role="data">63</cell><cell cols="1" rows="1" rend="align=right" role="data">31.33</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">68.97</cell><cell cols="1" rows="1" rend="align=center" role="data">34</cell><cell cols="1" rows="1" rend="align=center" role="data">57.20</cell><cell cols="1" rows="1" rend="align=center" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">30.24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">68.90</cell><cell cols="1" rows="1" rend="align=center" role="data">35</cell><cell cols="1" rows="1" rend="align=center" role="data">56.51</cell><cell cols="1" rows="1" rend="align=center" role="data">65</cell><cell cols="1" rows="1" rend="align=right" role="data">29.15</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">68.81</cell><cell cols="1" rows="1" rend="align=center" role="data">36</cell><cell cols="1" rows="1" rend="align=center" role="data">55.81</cell><cell cols="1" rows="1" rend="align=center" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">28.06</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">68.62</cell><cell cols="1" rows="1" rend="align=center" role="data">37</cell><cell cols="1" rows="1" rend="align=center" role="data">55.10</cell><cell cols="1" rows="1" rend="align=center" role="data">67</cell><cell cols="1" rows="1" rend="align=right" role="data">26.96</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">68.48</cell><cell cols="1" rows="1" rend="align=center" role="data">38</cell><cell cols="1" rows="1" rend="align=center" role="data">54.37</cell><cell cols="1" rows="1" rend="align=center" role="data">68</cell><cell cols="1" rows="1" rend="align=right" role="data">25.85</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">68.31</cell><cell cols="1" rows="1" rend="align=center" role="data">39</cell><cell cols="1" rows="1" rend="align=center" role="data">53.62</cell><cell cols="1" rows="1" rend="align=center" role="data">69</cell><cell cols="1" rows="1" rend="align=right" role="data">24.73</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">68.15</cell><cell cols="1" rows="1" rend="align=center" role="data">40</cell><cell cols="1" rows="1" rend="align=center" role="data">52.85</cell><cell cols="1" rows="1" rend="align=center" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">23.60</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">67.95</cell><cell cols="1" rows="1" rend="align=center" role="data">41</cell><cell cols="1" rows="1" rend="align=center" role="data">52.07</cell><cell cols="1" rows="1" rend="align=center" role="data">71</cell><cell cols="1" rows="1" rend="align=right" role="data">22.47</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">67.73</cell><cell cols="1" rows="1" rend="align=center" role="data">42</cell><cell cols="1" rows="1" rend="align=center" role="data">51.27</cell><cell cols="1" rows="1" rend="align=center" role="data">72</cell><cell cols="1" rows="1" rend="align=right" role="data">21.32</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">67.48</cell><cell cols="1" rows="1" rend="align=center" role="data">43</cell><cell cols="1" rows="1" rend="align=center" role="data">50.46</cell><cell cols="1" rows="1" rend="align=center" role="data">73</cell><cell cols="1" rows="1" rend="align=right" role="data">20.17</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">67.21</cell><cell cols="1" rows="1" rend="align=center" role="data">44</cell><cell cols="1" rows="1" rend="align=center" role="data">49.63</cell><cell cols="1" rows="1" rend="align=center" role="data">74</cell><cell cols="1" rows="1" rend="align=right" role="data">19.02</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">66.95</cell><cell cols="1" rows="1" rend="align=center" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">48.78</cell><cell cols="1" rows="1" rend="align=center" role="data">75</cell><cell cols="1" rows="1" rend="align=right" role="data">17.86</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">66.65</cell><cell cols="1" rows="1" rend="align=center" role="data">46</cell><cell cols="1" rows="1" rend="align=center" role="data">47.93</cell><cell cols="1" rows="1" rend="align=center" role="data">76</cell><cell cols="1" rows="1" rend="align=right" role="data">16.70</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">66.31</cell><cell cols="1" rows="1" rend="align=center" role="data">47</cell><cell cols="1" rows="1" rend="align=center" role="data">47.06</cell><cell cols="1" rows="1" rend="align=center" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">15.52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=center" role="data">65.98</cell><cell cols="1" rows="1" rend="align=center" role="data">48</cell><cell cols="1" rows="1" rend="align=center" role="data">46.16</cell><cell cols="1" rows="1" rend="align=center" role="data">78</cell><cell cols="1" rows="1" rend="align=right" role="data">14.35</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">65.62</cell><cell cols="1" rows="1" rend="align=center" role="data">49</cell><cell cols="1" rows="1" rend="align=center" role="data">45.26</cell><cell cols="1" rows="1" rend="align=center" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">13.17</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">65.24</cell><cell cols="1" rows="1" rend="align=center" role="data">50</cell><cell cols="1" rows="1" rend="align=center" role="data">44.35</cell><cell cols="1" rows="1" rend="align=center" role="data">80</cell><cell cols="1" rows="1" rend="align=right" role="data">11.98</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">64.84</cell><cell cols="1" rows="1" rend="align=center" role="data">51</cell><cell cols="1" rows="1" rend="align=center" role="data">43.42</cell><cell cols="1" rows="1" rend="align=center" role="data">81</cell><cell cols="1" rows="1" rend="align=right" role="data">10.79</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">64.42</cell><cell cols="1" rows="1" rend="align=center" role="data">52</cell><cell cols="1" rows="1" rend="align=center" role="data">42.48</cell><cell cols="1" rows="1" rend="align=center" role="data">82</cell><cell cols="1" rows="1" rend="align=right" role="data">9.59</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">63.97</cell><cell cols="1" rows="1" rend="align=center" role="data">53</cell><cell cols="1" rows="1" rend="align=center" role="data">41.53</cell><cell cols="1" rows="1" rend="align=center" role="data">83</cell><cell cols="1" rows="1" rend="align=right" role="data">8.41</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">63.51</cell><cell cols="1" rows="1" rend="align=center" role="data">54</cell><cell cols="1" rows="1" rend="align=center" role="data">40.56</cell><cell cols="1" rows="1" rend="align=center" role="data">84</cell><cell cols="1" rows="1" rend="align=right" role="data">7.21</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">63.03</cell><cell cols="1" rows="1" rend="align=center" role="data">55</cell><cell cols="1" rows="1" rend="align=center" role="data">39.58</cell><cell cols="1" rows="1" rend="align=center" role="data">85</cell><cell cols="1" rows="1" rend="align=right" role="data">6.00</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=center" role="data">62.53</cell><cell cols="1" rows="1" rend="align=center" role="data">56</cell><cell cols="1" rows="1" rend="align=center" role="data">38.58</cell><cell cols="1" rows="1" rend="align=center" role="data">86</cell><cell cols="1" rows="1" rend="align=right" role="data">4.81</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=center" role="data">62.02</cell><cell cols="1" rows="1" rend="align=center" role="data">57</cell><cell cols="1" rows="1" rend="align=center" role="data">37.58</cell><cell cols="1" rows="1" rend="align=center" role="data">87</cell><cell cols="1" rows="1" rend="align=right" role="data">3.61</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=center" role="data">61.48</cell><cell cols="1" rows="1" rend="align=center" role="data">58</cell><cell cols="1" rows="1" rend="align=center" role="data">36.57</cell><cell cols="1" rows="1" rend="align=center" role="data">88</cell><cell cols="1" rows="1" rend="align=right" role="data">2.41</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=center" role="data">60.93</cell><cell cols="1" rows="1" rend="align=center" role="data">59</cell><cell cols="1" rows="1" rend="align=center" role="data">35.54</cell><cell cols="1" rows="1" rend="align=center" role="data">89</cell><cell cols="1" rows="1" rend="align=right" role="data">1.21</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=center" role="data">60.35</cell><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=center" role="data">34.50</cell><cell cols="1" rows="1" rend="align=center" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">0.00</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">59.75</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>Note, This table is computed on the supposition that the length of the degrees of the equator are equal to those of the meridian at the medium latitude of 45°, which length is 69 1/15 English miles.</p><p>The expressions <hi rend="italics">Latitude</hi> and <hi rend="italics">Longitude,</hi> are borrowed from the ancients, who happened to be acquainted with a much larger extent of the earth in the direction east and west, than in that of north and south; the former of which therefore passed, with them, for the length of the earth, or longitude, and the latter for the breadth or shorter dimension, viz, the latitude. <cb/></p></div2></div1><div1 part="N" n="DEJECTION" org="uniform" sample="complete" type="entry"><head>DEJECTION</head><p>, in Astrology, is applied to the planets when in their detriment, as astrologers speak, i. e. when they have lost their force, or influence, as is pretended, by reason of their being in opposition to some others, which check and counteract them.</p><p>Or, it is used when a planet is in a sign opposite to that in which it has its greatest effect, or influence, which is called its exaltation. Thus, the sign Aries being the exaltation of the Sun, the opposite sign Libra is its dejection.</p></div1><div1 part="N" n="DEINCLINERS" org="uniform" sample="complete" type="entry"><head>DEINCLINERS</head><p>, or <hi rend="smallcaps">Deinclining</hi> <hi rend="italics">Dials,</hi> are such as both decline and incline, or recline, at the same time. Suppose, for instance, a plane cutting the prime vertical circle at an angle of 30 degrees, and the horizontal plane at an angle of 24 degrees, the latitude of the place being 52 degrees; a dial drawn on this plane, is called a deincliner.</p><p>DELIACAL <hi rend="italics">Problem,</hi> a celebrated problem among the ancients, concerning the duplication of the cube.</p></div1><div1 part="N" n="DELPHINUS" org="uniform" sample="complete" type="entry"><head>DELPHINUS</head><p>, the <hi rend="italics">Dolphin,</hi> a constellation of the northern hemisphere; whose stars, in Ptolomy's catalogue, are 10; in Tycho's the same; in Hevelius's 14; and in Flamsteed's 18.</p></div1><div1 part="N" n="DEMETRIUS" org="uniform" sample="complete" type="entry"><head>DEMETRIUS</head><p>, a Cynic philosopher, and disciple of Apollonius Thyaneus, in the age of Caligula. That emperor wishing to gain the philosopher to his interest by a large present, he refused it with indignation, saying, If Caligula wishes to bribe me, let him send me his crown. Vespasian was displeased with his insolence, and banished him to an island. The cynic derided the punishment, and bitterly inveighed against the emperor.</p><p>Demetrius lived to a very great age. And Seneca observes, that “nature had brought him forth to shew mankind that an exalted genius can live securely without being corrupted by the vice of the surrounding world.”</p><p>DEMI-<hi rend="italics">Bastion,</hi> in Fortification, one that has only one face and one flank.</p><p><hi rend="smallcaps">Demi</hi>-<hi rend="italics">Cannon,</hi> and <hi rend="smallcaps">Demi</hi>-<hi rend="italics">Culverin,</hi> names of certain species of cannon, now no longer used.</p><p><hi rend="smallcaps">Demi</hi> <hi rend="italics">Cross,</hi> an instrument used by the Dutch to take the altitude of the sun or a star at sea; instead of which we use the cross staff, or forestaff.</p><p><hi rend="smallcaps">Demi</hi>-<hi rend="italics">Gorge,</hi> is half the gorge or entrance into the bastion; not taken directly from angle to angle, where the bastion joins to the curtin, but from the angle of the flank to the centre of the bastion; or the angle the two curtins would make, were they thus protracted to meet in the bastion.</p><p><hi rend="smallcaps">Demi</hi>-<hi rend="italics">Lune,</hi> or <hi rend="italics">Half-moon,</hi> an outwork consisting of two faces, and two little flanks. It is often built before the angle of a bastion, and sometimes also before the curtin; though now it is very seldom used.</p></div1><div1 part="N" n="DEMOCRITUS" org="uniform" sample="complete" type="entry"><head>DEMOCRITUS</head><p>, one of the greatest philosophers of antiquity, was born at Abdera, a town of Thrace, about the 80th olympiad, or about 400 years before Christ. His father, says Valerius Maximus, was able to entertain the army of Xerxes; and Diogenes Laertius adds, upon the testimony of Herodotus, that the king, in requital, presented him with some Magi and Chaldeans. From these, it seems, Democritus received the first part of his education; and from them, whilst yet a boy, he learned theology and astronomy. He next applied to Leucippus, from whom he learned the <pb n="365"/><cb/> system of atoms and a vacuum. His father dying, he and his two brothers divided his effects. Democritus made choice of that part which consisted in money, as being, though the least share, the most convenient for travelling; and it is said that his portion amounted to more than 100 talents, which is near 20 thousand pounds sterling. His extraordinary inclination for knowledge and the sciences, induced him to travel into all parts of the world where he might find learned men. He went to visit the priests of Egypt, from whom he learned geometry: He consulted the Chaldean and Persian philosophers: and it is said that he penetrated even into India and Ethiopia, to confer with the Gymnosophists. In these travels he wasted his substance; after which, at his return he was obliged for some time to be maintained by his brother. Settling himself at Abdera, he there governed in the most absolute manner, by virtue of his consummate wisdom. The magistrates of that city made him a present of 500 talents, and erected statues to him, even in his lifetime: but being naturally more inclined to contemplation than delighted with public honours and employments, he withdrew into solitude and retirement.</p><p>Democritus always laughed at human life, as a continued farce, which made the people think he was mad; on which they sent for Hippocrates to cure him: but that celebrated physician having discoursed with the philosopher, told the people that he had a great veneration for Democritus; and that, in his opinion, those who esteemed themselves the most healthy, were the most distempered.</p><p>It is said, though with little probability, that Democritus put out his own eyes, that he might meditate more profoundly on philosophical subjects. He died, according to Diogenes Laertius, in the 361st year before the Christian era, at 109 years of age. He was the author of many books, which are lost; from which Epicurus borrowed his philosophy.</p></div1><div1 part="N" n="DEMOIVRE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DEMOIVRE</surname> (<foreName full="yes"><hi rend="smallcaps">Abraham</hi></foreName>)</persName></head><p>, a celebrated mathematician, of French original, but who spent most of his life in England. He was born at Vitri in Champagne 1667. The revocation of the edict of Nantz, in 1685, determined him, with many others, to take shelter in England; where he perfected his mathematical studies, the foundation of which he had laid in his own country. A mediocrity of fortune obliged him to employ his talent in this way in giving lessons, and reading public lectures, for his better support: in the latter part of his life too, he chiefly subsisted by giving answers to questions in chances, play, annuities, &c, and it is said most of these responses were delivered at a Coffee-house in St. Martin's lane, where he spent most of his time. The Principia Mathematica of Newton, which chance is said to have thrown in his way, soon convinced Demoivre how little he had advanced in the science he professed. This induced him to redouble his application; which was attended by a considerable degree of success; and he soon became connected with, and celebrated among, the first rate mathematicians. His eminence and abilities in this line, opened him an entrance into the Royal Society of London, and into the academies of Berlin and Paris. By the former his merit was so well known and esteemed, that they judged <cb/> him a fit person to decide the famous contest between Newton and Leibnitz, concerning the invention of Fluxions.</p><p>The collection of the Academy of Paris contains no memoir of this author, who died at London Nov. 1754, at 87 years of age, soon after his admission into it. But the Philosophical Transactions of London have several, and all of them interesting, viz, in the volumes 19, 20, 22, 23, 25, 27, 29, 30, 32, 40, 41, 43.</p><p>He published also some very respectable works, viz,</p><p>1. <hi rend="italics">Miscellanea Analytica, de Seriebus & Quadraturis &c;</hi> 1730, in 4to. But perhaps he has been more generally known by his</p><p>2. <hi rend="italics">Doctrine of Chances; or, Method of Calculating the Probabilities of Events at Play.</hi> This work was first printed, 1718, in 4to, and dedicated to Sir Isaac Newton: it was reprinted in 1738, with great alterations and improvements; and a third edition was afterwards printed.</p><p>3. <hi rend="italics">Annuities on Lives;</hi> first printed 1724, in 8vo.— In 1742 the ingenious Thomas Simpson (then only 33 years of age) published his <hi rend="italics">Doctrine of Annuities and Reversions;</hi> in which he paid some handsome compliments to our author. Notwithstanding which, Demoivre presently brought out a second edition of his Annuities, in the preface to which he passed some harsh reflections upon Simpson. To these the latter gave a handsome and effectual answer, 1743, in <hi rend="italics">An Appendix, containing some Remarks on a late book on the same subject, with answers to some persoual and malignant misrepresentations, in the preface thereof.</hi> At the end of this answer, Mr. Simpson concludes, “Lastly, I appeal to all mankind, whether, in his treatment of me, he has not discovered an air of self-sufficiency, ill-nature and inveteracy, unbecoming a gentleman.” Here it would seem the controversy dropped: Mr. Demoivre published the 3d edition of his book in 1750, without any farther notice of Simpson, but omitted the offensive reflections that had been in the preface.</p></div1><div1 part="N" n="DEMONSTRATION" org="uniform" sample="complete" type="entry"><head>DEMONSTRATION</head><p>, a certain or convincing proof of some proposition: such as the demonstrations of the propositions in Euclid's Elements.</p><p>The method of demonstrating in mathematics, is the same with that of drawing conclusions from principles in logic. Indeed, the demonstrations of mathematicians are no other than series of enthymemes; every thing is concluded by force of syllogism, only omitting the premises, which either occur of their own accord, or are recollected by means of quotations.</p></div1><div1 part="N" n="DENDROMETER" org="uniform" sample="complete" type="entry"><head>DENDROMETER</head><p>, an instrument lately invented by Messrs Duncombe and Whittel; so called, from its use in measuring trees.</p></div1><div1 part="N" n="DENEB" org="uniform" sample="complete" type="entry"><head>DENEB</head><p>, an Arabic term, signifying tail; used by astronomers as a name to some of the fixed stars, but especially for the bright star in the Lion's tail.</p></div1><div1 part="N" n="DENOMINATOR" org="uniform" sample="complete" type="entry"><head>DENOMINATOR</head><p>, <hi rend="italics">of a fraction,</hi> is the number or quantity placed below the line, which shew<*> the whole integer, or into how many parts the integer is supposed to be divided by the fraction; as that which gives denomination or name to the parts of the fraction. Thus, in the fraction 5/12, five-twelfths, the number 12 is the denominator, and shews that the integer is here divided into 12 parts, or that it consists of 12 of those <pb n="366"/><cb/> parts of which the numerator contains 5. Also <hi rend="italics">b</hi> is the denominator of the fraction <hi rend="italics">a/b.</hi></p><p><hi rend="smallcaps">Denomina tor</hi> <hi rend="italics">of a ratio,</hi> is the quotient arising fron the division of the antecedent by the consequent. Thus, 6 is the denominator of the ratio 30 to 5, because 30 divided by 5 gives 6. It is otherwise called the <hi rend="italics">exponent</hi> of the ratio.</p></div1><div1 part="N" n="DENSITY" org="uniform" sample="complete" type="entry"><head>DENSITY</head><p>, that property of bodies, by which they contain a certain quantity of matter, under a certain bulk or magnitude. Accordingly a body that contains more matter than another, under the same bulk, is said to be denser than the other, and that in proportion to the quantity of matter; or if the quantity of matter be the same, but under a less bulk, it is said to be denser, and so much the more so as the bulk is less. So that, in general, the density is directly proportional to the mass or quantity of matter, and reciprocally or inversely proportional to the bulk or magnitude under which it is contained.</p><p>The quantities of matter in bodies, or at least the proportions of them, are known by their gravity or weight; every equal particle of matter being endowed with an equal gravity, it is inferred that equal masses or quantities of matter have an equal weight or gravity; and unequal masses have proportionally unequal weights. So that, when body, or mass, or quantity of matter is spoken of, we are to understand their weight or gravity.</p><p>From the foregoing general proportion of the density of bodies, viz, that it is as the mass directly, and as the bulk inversely, may be inferred the proportion of the masses, or of the magnitudes; viz, that the mass or quantity of matter, is in the compound ratio of the bulk and density; and that the bulk or magnitude, is as the mass directly, and the density inversely. Hence, if B, <hi rend="italics">b</hi> be two bodies, or masses, or weights; and D, <hi rend="italics">d</hi> their respective densities; also M, <hi rend="italics">m</hi> their magnitudes, or bulks: Then the theorems above are thus expressed, viz, D <figure/> B/M, and B <figure/> DM, and M <figure/> B/D; or D : <hi rend="italics">d.</hi> : B/M : <hi rend="italics">b/m,</hi> and B : <hi rend="italics">b</hi> :: DM : <hi rend="italics">dm,</hi> &c; or .</p><p>No body is absolutely or perfectly dense; or no space is perfectly full of matter, so as to have no vacuity or interstices; on the contrary, it is the opinion of Newton, that even the densest bodies, as gold &c, contain but a small portion of matter, and a very great portion of vacuity; or that it contains a great deal more of pores or empty space, than of real substance.</p><p>It has been observed above, that the relative density of bodies may be known by their weight or gravity; and hence the most general way of knowing those densities, is by actually weighing an equal bulk or magnitude of the bodies, whether solid or fluid; if solid, by shaping them to the same figure and dimensions; if fluid, by filling the same vessel with them, and weighing it.</p><p>For fluids, there are also other methods of finding <cb/> their density: as 1st, by making an equilibrium between them in tubes that communicate; for, the diameters of the tubes being equal, and the weights or quantities of matter also equal, the densities will be inversely as the altitudes of the liquids in them, that is inversely as the bulk.</p><p>2dly, The densities of fluids are also compared together by immerging a solid in them; for if the solid be lighter than the liquids, the part immerged by its own weight, will be inversely as the density of the fluid; or if it be heavier, and sink in the liquids, by weighing it in them; then the weights lost by the body will be directly proportional to the densities of the fluids.</p><p><hi rend="smallcaps">Density</hi> <hi rend="italics">of the Air,</hi> is a property that has much employed the later philosophers, since the discovery of the Torricellian experiment, and the air-pump. By means of the barometer it is demonstrated that the air is of the same density at all places at the same distance from the level of the sea; provided the temperature, or degree of heat, be the same. Also the density of the air always increases in proportion to the compression, or the compressing forces. And hence the lower parts of the atmosphere are always denser than the upper: yet the density of the lower air is not exactly proportional to the weight of the atmosphere, by reason of heat and cold, which make considerable alterations as to rarity and density; so that the barometer measures the elasticity of the air, rather than its density. If the height of the barometer be considered as the measure both of the density and elasticity of the air, when the thermometer is at 31°, and <hi rend="italics">b</hi> be any other height of the barometer, when the thermometer is at <hi rend="italics">t</hi> degrees; then in this case, <hi rend="italics">b</hi> is the measure of the elasticity, and ((466 - <hi rend="italics">t</hi>)/435)<hi rend="italics">b</hi> is the measure of the density of the air.</p><p><hi rend="smallcaps">Density</hi> <hi rend="italics">of the Planets.</hi> In homogeneous, unequal, spherical bodies, the gravities on their surfaces, are as their diameters when the densities are equal, or the gravities are as the densities when the bulks are equal; therefore, in spheres of unequal magnitude and density, the gravity is in the compound ratio of the diameters and densities, or the densities are as the gravities divided by the diameters. Knowing therefore the diameters of the planets by observation and comparison, and the gravities at their surface by means of the revolution of the satellites, the relation of their densities becomes known. And as I have found the mean density of the earth to be about 4 1/2 times that of water, Philos. Trans. 1778; hence the densities of the planets, with respect to water, become known, and are as below: <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Densities.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Water</cell><cell cols="1" rows="1" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Sun</cell><cell cols="1" rows="1" role="data">1 2/15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data">9 1/6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data">5 11/15</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Earth</cell><cell cols="1" rows="1" role="data">4 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data">3 2/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Moon</cell><cell cols="1" rows="1" role="data">3 1/11</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data">1 1/24</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data">0 13/32</cell></row><row role="data"><cell cols="1" rows="1" role="data">Georgian Planet</cell><cell cols="1" rows="1" role="data">0 99/100</cell></row></table></p><p>As it is not likely that any of these bodies are homo- <pb n="367"/><cb/> geneal, the densities here determined are supposed to be the mean densities, or such as the bodies would have if they were homogeneal, and of the same mass of matter and magnitude.</p></div1><div1 part="N" n="DENTICLES" org="uniform" sample="complete" type="entry"><head>DENTICLES</head><p>, or <hi rend="smallcaps">Dentils</hi>, are ornaments in a cornice, cut after the manner of teeth. These are mostly affected in the Ionic and Corinthian orders; and of late also in the Doric. The square member on which they are cut, is called the Denticule.</p></div1><div1 part="N" n="DEPARTURE" org="uniform" sample="complete" type="entry"><head>DEPARTURE</head><p>, in Navigation, is the easting or westing of a ship, with regard to the meridian she <hi rend="italics">departed</hi> or sailed from. Or, it is the difference in longitude, either east or west, between the present meridian the ship is under, and that where the last reckoning or observation was made. This departure, any where but under the equator, must be accounted according to the number of miles in a degree proper to the parallel the ship is in.</p><p>The Departure, in Plane and Mercator's Sailing, is always represented by the base of a right-angled plane triangle, where the course is the angle opposite to it, and the distance sailed is the hypothenuse, the perpendicular or other leg being the difference of latitude. And then the theorem for finding it, is always this: As radius is to the sine of the course, so is the distance sailed, to the departure sought.</p><p>DEPRESSION <hi rend="italics">of the Pole.</hi> So many degrees &c as you sail or travel from the pole towards the equator, so many it is said you depress the pole, because it becomes so much lower, or nearer the horizon.</p><p><hi rend="smallcaps">Depression</hi> <hi rend="italics">of a Star,</hi> or <hi rend="italics">of the Sun,</hi> is its distance below the horizon; and is measured by an arc of a vertical circle, intercepted between the horizon and the place of the star.</p><p><hi rend="smallcaps">Depression</hi> <hi rend="italics">of the Visible Horizon,</hi> or <hi rend="italics">Dip of the Horizon,</hi> denotes its sinking or dipping below the true horizontal plane, by the observer's eye being raised above the surface of the sea; in consequence of which, the observed altitude of an object is by so much too great.</p><p>Thus, the eye being at E, the height AE above the surface <figure/> of the earth, whose centre is C; then EH is the real horizon, and E<hi rend="italics">h</hi> the visible one, below the former by the angle HE<hi rend="italics">h,</hi> by reason of the elevation AE of the eye. <hi rend="center"><hi rend="italics">To compute the Depression or Dip of the Horizon.</hi></hi></p><p>In the right-angled triangle CE<hi rend="italics">h,</hi> are given C<hi rend="italics">h</hi> the earth's radius = 21000000 feet, and the hypothenuse CE = the radius increased by the height AE of the eye; to find the angle C which is = the angle HE<hi rend="italics">h,</hi> or depression sought; viz, as C<hi rend="italics">h</hi> : CE :: radius : sec. < C, or as CE : C<hi rend="italics">h</hi> :: radius : cosin. < C.</p><p>By either of these theorems are computed the numbers in the following table, which shews the depression or dip of the horizon of the sea for different heights of the eye, from 1 foot to 100 feet. <cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Height of the eye</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Dip of the horizon</cell><cell cols="1" rows="1" rend="align=center" role="data">Height of the eye</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Dip of the horizon</cell><cell cols="1" rows="1" rend="align=center" role="data">Height of the eye</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Dip of the horizon</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">feet</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">″</cell><cell cols="1" rows="1" rend="align=center" role="data">feet</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">″</cell><cell cols="1" rows="1" rend="align=center" role="data">feet</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">″</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=center" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" rend="align=center" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=center" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=center" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=center" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=center" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">80</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">100</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell></row></table></p><p>See Robertson's Navigation, book 9 appendix; and Tables requisite to be used with the Nautical Ephemeris, pa. 1. See also <hi rend="smallcaps">Levelling.</hi></p></div1><div1 part="N" n="DEPTH" org="uniform" sample="complete" type="entry"><head>DEPTH</head><p>, the opposite of Height, and one of the dimensions of bodies, or of space. See <hi rend="smallcaps">Height</hi>, A<hi rend="smallcaps">LTITUDE, Elevation</hi>, &c.</p></div1><div1 part="N" n="DERHAM" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DERHAM</surname> (Doctor <foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an eminent English philosopher and divine, was born at Stowton, near Worcester, 1657, and educated in Trinity College, Oxford. In 1682, he was presented to the vicarage of Wargrave in Berkshire; and, in 1689, to the valuable rectory of Upminster in Essex; which, lying at a convenient distance from London, afforded him an opportunity of conversing and corresponding with the priucipal literary geniuses of the nation. Applying himself there with great eagerness to the pursuit of his studies in natural and experimental philosophy, he soon became a distinguished and useful member of the Royal Society, whose Philosophical Transactions contain a great variety of curious and valuable pieces, the fruits of his laudable industry, in all or most of the volumes, from the 20th to the 39th, both inclusive; the principal of which are:</p><p>1. Experiments on the Motion of Pendulums in vacuo.</p><p>2. A Description of an instrument for finding the Meridian.</p><p>3. Experiments and Observations on the Motion of Sound.</p><p>4. On the Migration of Birds.</p><p>5. History of the Spots in the Sun, from 1703 to 1711.</p><p>6. Observations on the Northern Lights, Oct. 8, 1726, and Oct. 13, 1728.</p><p>7. Tables of the Eclipses of Jupiter's Satellites.</p><p>8. The difference of Time in the meridian of different places.</p><p>9. Of the meteor called Ignis Fatuus.</p><p>10. The History of the Death Watch.</p><p>11. Meteorological Diaries for several years.</p><p>In his younger days he published his Artificial Clockmaker, a very useful little work, that has gone through several editions. In 1711, 1712, 1714, he preached those sermons at Boyle's lecture, which he afterward digested under the well-known titles of <hi rend="italics">Physico-Theology</hi> and <hi rend="italics">Astro-Theology,</hi> or Demonstrations of the being and <pb n="368"/><cb/> attributes of God, from his works of creation, and a survey of the heavens.</p><p>In 1716 he was made a canon of Windsor, being at that time chaplain to the Prince of Wales; and in 1730 received, from the university of Oxford, the degree of Doctor of Divinity. He revised the <hi rend="italics">Miscellanea Curiosa,</hi> in 3 vols 8vo, containing many curious papers of Dr. Halley and several other ingenious philosophers. To him also the world is indebted for the publication of the Philosophical Experiments of the late eminent Dr. Hooke, and other ingenious men of his time; as well as notes and illustrations of several other works.</p><p>Dr. Derham was very well skilled in medical as well as in physical knowledge; and was constantly a physician to the bodies as well as the souls of his parishioners. This great and good man, after spending his life in the most agreeable and improving study of nature, and the diligent and pious discharge of his duty, died at Upminster in 1735, at 78 years of age.</p></div1><div1 part="N" n="DESAGULIERS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DESAGULIERS</surname> (<foreName full="yes"><hi rend="smallcaps">John Theophilus</hi></foreName>)</persName></head><p>, an eminent experimental philosopher, was the son of the Rev. John Desaguliers, a French Protestant refugee, and born at Rochelle in 1683. His father brought him to England an infant; and having taught him the classics himself, he sent him at a proper age to Christ-church College, Oxford; where in 1702 he succeeded Dr. Keil in reading lectures on experimental philosophy at Hart Hall. In 1712 he married, and settled in London, when he first of any introduced the reading of lectures in experimental philosophy in the metropolis, which he continued during the rest of his life with the greatest applause, having several times the honour of reading his lectures before the king and royal family. In 1714 he was elected F. R. S. and proved a very useful member, as appears from the great number of his papers that are printed in their Philos. Trans. on the subjects of optics, mechanics, and meteorology. The magnificent duke of Chandos made Dr. Desaguliers his chaplain, and presented him to the living of Edgware, near his seat at Cannons; and he became afterward chaplain to Frederick prince of Wales. In the latter part of his life, he removed to lodgings over the Great Piazza in Covent Garden, where he carried on his lectures with great success till the time of his death in 1749, at 66 years of age.</p><p>He was a member of several foreign academies, and corresponding member of the Royal Academy of Sciences at Paris; from which academy he obtained the prize, proposed by them for the best account of electricity. He communicated a multitude of curious and valuable papers to the Royal Society, for the year 1714 to 1743, or from vol. 29 to vol. 42.</p><p>Beside those numerous communications, he published a valuable <hi rend="italics">Course of Experimental Philosophy,</hi> 1734, in 2 large vols. 4to; and gave an edition of <hi rend="italics">Gregory's Elements of Catoptrics and Dioptrics,</hi> with an Appendix on Reflecting Telescopes, 8vo, 1735. This appendix contains some Original Letters that passed between Sir Isaac Newton and Mr. James Gregory, relating to those telescopes.</p></div1><div1 part="N" n="DESCENDING" org="uniform" sample="complete" type="entry"><head>DESCENDING</head><p>, a going or moving from above, downwards.</p><p>There are ascending and descending stars, and ascending and descending degrees, &c. <cb/></p><p><hi rend="smallcaps">Descending</hi> <hi rend="italics">Latitude,</hi> is the latitude of a planet in its return from the nodes to the equator.</p></div1><div1 part="N" n="DESCENSION" org="uniform" sample="complete" type="entry"><head>DESCENSION</head><p>, in Astronomy, is either <hi rend="italics">right,</hi> or <hi rend="italics">oblique.</hi></p><p><hi rend="italics">Right</hi> <hi rend="smallcaps">Descension</hi> is a point, or arch, of the equator, which descends with a star, or sign, below the horizon, in a <hi rend="italics">right</hi> sphere, and</p><p><hi rend="italics">Oblique</hi> <hi rend="smallcaps">Descension</hi> is a point, or arch, of the equator, which descends at the same time with a star, or sign, below the horizon, in an oblique sphere.</p><p>Descensions, both right and oblique, are counted from the first point of Aries, or the vernal intersection, according to the order of the signs, i. e. from west to east. And, as they are unequal, when it happens that they answer to equal arcs of the ecliptic, as for example to the 12 signs of the zodiac, it follows, that sometimes a greater part of the equator rises or descends with a sign, in which case the sign is said to ascend or descend rightly: and sometimes again, a less part of the equator rises or sets with the same sign, in which case it is said to ascend or descend obliquely. See A<hi rend="smallcaps">SCENSION.</hi></p><p><hi rend="italics">Refraction of the</hi> <hi rend="smallcaps">Descension.</hi> See <hi rend="smallcaps">Refraction.</hi></p><p><hi rend="smallcaps">Descensional</hi> <hi rend="italics">Difference,</hi> is the difference between the right and oblique descension of the same star, or point of the heavens.</p><div2 part="N" n="Descent" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Descent</hi></head><p>, or <hi rend="italics">Fall,</hi> in mechanics, &c, is the motion, or tendency, of a body towards the centre of the earth, either directly or obliquely.</p><p>The descent of bodies may be considered either as freely, like as in a vacuum, or as clogged or resisted by some external force, as an opposing body, or a fluid medium, &c.</p><p>1st, If the body <hi rend="italics">b</hi> descend freely, and perpendicularly, by the force of gravity; then the motive force urging it downwards, is equal to its whole weight <hi rend="italics">b</hi>; and the quantity of matter being <hi rend="italics">b</hi> also, the accelerative force will be <hi rend="italics">b/b</hi> or 1.</p><p>2dly, If the body <hi rend="italics">b</hi> descending, be opposed by some mechanical power, suppose a wedge or inclined plane, that is, instead of pursuing the perpendicular line of gravity, it is made to descend in a sloping direction down the inclined plane: then if the sine of the angle the plane makes with the horizon be <hi rend="italics">s,</hi> to the radius 1, the motive force urging the body down the plane will be <hi rend="italics">bs</hi>; and therefore the accelerative force is <hi rend="italics">bs/b</hi> or <hi rend="italics">s;</hi> which is less than in the former case in the proportion of <hi rend="italics">s</hi> to 1.</p><p>3dly, In a medium, a body suspended loses as much of its weight, as is the weight of a like bulk of the medium; and when descending, it loses the same, beside the obstruction arifing from the cohesion of the parts of the medium, and the opposing force of the particles struck, which last produces a greater or less resistance, according to the velocity of the motion. But, the weight of the body being <hi rend="italics">b,</hi> and that of a like bulk of the fluid medium <hi rend="italics">m,</hi> the motive force urging the body to descend, is only <hi rend="italics">b</hi>-<hi rend="italics">m</hi>; that is, the body only falls by the excess of its weight above that of an equal bulk of the medium.</p><p>Hence, the power that sustains a body in a medium, <pb n="369"/><cb/> is equal to the excess of the absolute weight of the body above an equal bulk of the medium. Thus, a piece of copper weighing (47 1/3)lb, loses (5 1/3)lb of its weight in water: and therefore a power of 42lb will sustain it in the water.</p><p>4thly, If two bodies have the same specisic gravity, the less the bulk of the descending body is, the more of its gravity does it lose, and the slower does it descend, in the same medium. For, though the proportion of the specisic gravity of the body to that of the fluid be still the same, whether the bulk be greater or less, yet the smaller the body, the more the surface is, in proportion to the mass; and the more the surface, the more the resistance of the parts of the fluid, in proportion.</p><p>5thly, If the specific gravities of two bodies be different; that which has the greatest specific gravity will descend with greater velocity in the air, or resisting medium, than the other body. Thus, a ball of lead descends swifter than wood or cork, because it loses less of its weight, though in a vacuum they both fall equally swift.</p><p>The cause of this descent, or tendency downwards, has been greatly controverted. Two opposite hypotheses have been advanced; the one, that it proceeds from an internal principle, and the other from an external one: the first is maintained by the Peripatetics, Epicureans, and the Newtonians; and the latter by the Cartesians and Gassendists. See also <hi rend="smallcaps">Acceleration.</hi> <hi rend="center"><hi rend="italics">Laws of <hi rend="smallcaps">Descent</hi> of Bodies.</hi></hi></p><p>1st, Heavy bodies, in an unresisting medium, fall with an uniformly accelerated motion. For, it is the nature of all constant and uniform forces, such as that of gravity at the same distance from the centre of the earth, to generate or produce equal additions of velocity in equal times. So that, if in one second of time there be produced 1 degree of velocity, in 2 seconds there will be 2 degrees of velocity, in 3 seconds 3 degrees, and so on, the degree or quantity of velocity being always proportional to the length of the time.</p><p>2nd, The space descended by an uniform gravity, in any time, is just the half of the space that might be uniformly described in the same time by the last velocity acquired at the end of that time, if uniformly continued. For, as the velocity increases uniformly in an arithmetic progression, the whole space descended by the variable velocity, will be equal to the space that would be described with the middle velocity uniformly continued for the same time; and this again will be only half the space that would be described with the last velocity, also uniformly continued for the same time, because the last velocity is double of the middle velocity, being produced in a double time.</p><p>3d, The spaces descended by an uniform gravity, in different times, are proportional to the squares of the times, or to the squares of the velocities. For the whole space descended in any number of particles of time, consists of the sums of all the particular spaces, or velocities, which are in arithmetical progression; but the sum of such an arithmetical progression, beginning at 0, and having the last term and the number of terms the same quantity, is equal to half the square of the last term, or of the number of terms; therefore <cb/> the whole sums are as the squares of the times, or of the velocities.</p><p>This theory of the descents by gravity was first discovered and taught by Galileo, who afterwards confirmed the same by experiments; which have often been repeated in various ways by many other persons since his time, as Grimaldi, Riccioli, Huygens, Newton, and many others, all consirming the same laws.</p><p>The experiments of Grimaldi and Riccioli were made by dropping a number of balls, of half a pound weight, from the tops of several towers, and measuring the times of falling by a pendulum. Ricciol. Almag. Nov. tom. 1 lib. 2, cap. 21, prop. 4. An abstract of their experiments is exhibited here below: <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Vibrations of the pendulum</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">The time</cell><cell cols="1" rows="1" rend="align=center" role="data">Space at the end of the time</cell><cell cols="1" rows="1" rend="align=center" role="data">Space descended each time</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">″</cell><cell cols="1" rows="1" rend="align=right" role="data">‴</cell><cell cols="1" rows="1" rend="align=center" role="data">Rom. feet</cell><cell cols="1" rows="1" rend="align=center" role="data">Rom. feet</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">160</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">250</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">135</cell><cell cols="1" rows="1" rend="align=right" role="data">75</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">240</cell><cell cols="1" rows="1" rend="align=right" role="data">105</cell></row></table></p><p>The space descended by a heavy body in any given time, being determined by experiment, is sufficient, in connection with the preceding theorems, for determining every inquiry concerning the times, velocities, and spaces descended, depending on an uniform force of gravity. From many accurate experiments made in England, it has been found that a heavy body descends freely through 16 feet 1 inch, or 16 1/12 feet, in the first second of time; and consequently, by theorem 2, the velocity gained at the end of 1 second, is 32 1/6 feet per second. Hence, by the same, and theorem 3, the velocity gained in any other time <hi rend="italics">t</hi> is (32 1/6)<hi rend="italics">t,</hi> and the space descended is (16 1/12)<hi rend="italics">t</hi><hi rend="sup">2</hi>. So that, if <hi rend="italics">v</hi> denote the velocity, and <hi rend="italics">s</hi> the space due to the time <hi rend="italics">t,</hi> and there be put <hi rend="italics">g</hi> = 16 1/12; then is .</p><p>The experiments with pendulums give also the same space for the descent of a heavy body in a second of time. Thus, in the latitude of London, it is found by experiment, that the length of a pendulum vibrating seconds is just 39 1/8 inches; and it being known that the circumference of a circle is to its diameter, as the time of one vibration of any pendulum, is to the time in which a heavy body will fall through half the length of the pendulum; therefore as 3.1416 : 1 :: 1 : 1/3.1416 which is the time of descending through 19 9/16 inches, or <pb n="370"/><cb/> half the length of the pendulum; then, spaces being as the squares of the times, as 1/(3.1416<hi rend="sup">2</hi>) : 1<hi rend="sup">2</hi> :: 19 9/16 : 193 inches, or 16 feet 1 inch, which therefore is the space a heavy body will descend through in one second; the very same as before.</p><p>4th, For any other constant force, instead of the perpendicular free descent by gravity, find by experiment, or otherwise, the space descended in one second by that force, and substitute that instead of 16 1/12 for the value of <hi rend="italics">g</hi> in these formulæ: or, if the proportion of the force to the force of gravity be known, let the value of <hi rend="italics">g</hi> be altered in the same proportion, and the same formulæ will still hold good. So, if the descent be on an inclined plane, making, for instance, an angle of 30° with the horizon; then, the force of descent upon the plane being always as the sine of the angle it makes with the horizon, in the present case it will be as the sine of 30°, that is, as 1/2 the radius; therefore in this case the value of <hi rend="italics">g</hi> will be but half the former, 8 1/24, in all the foregoing formulæ.</p><p>Or, if one body descending perpendicularly draw another after it, by means of a cord sliding over a pulley; then it will be, as the sum of the two bodies is to the descending body, so is 16 1/12 to the value of <hi rend="italics">g</hi> in this case; which value of it being ufed in the said formulæ, they will still hold good. And in like manner for any other constant forces whatever.</p><p>5th, The time of the oblique descent down any chord of a circle, drawn either from the uppermost point or lowermost point of the circle, is equal to the perpendicular descent through the diameter of the circle.</p><p>6th. The descent, or vibration, through all arcs of the same cycloid are equal, whether great or small.</p><p>7th. But the descent, or vibration, through unequal arcs of a circle, are unequal; the times being greater in the greater arcs, and less in the less.</p><p>8th, For Descents by Forces that are variable, see <hi rend="smallcaps">Forces</hi>, &c. See also <hi rend="smallcaps">Inclined Plane, Cycloid, Pendulum</hi>, &c.</p><p><hi rend="italics">Line of Swiftest</hi> <hi rend="smallcaps">Descent</hi>, is that which a body, falling by the action of gravity, describes in the shortest time possible, from one given point to another. And this line, it is proved by philosophers, is the arc of a cycloid, when the one point is not perpendicularly over the other. See <hi rend="smallcaps">Cycloid.</hi></p></div2></div1><div1 part="N" n="DESCRIBENT" org="uniform" sample="complete" type="entry"><head>DESCRIBENT</head><p>, a term in Geometry, signifying a line or superficies, by the motion of which a superficies or solid is described.</p></div1><div1 part="N" n="DETENTS" org="uniform" sample="complete" type="entry"><head>DETENTS</head><p>, in a clock, are those stops which, by being lifted up or let down, lock and unlock the clock in striking.</p><p><hi rend="smallcaps">Detent</hi>-<hi rend="italics">Wheel,</hi> or <hi rend="smallcaps">Hoop</hi>-<hi rend="italics">Wheel,</hi> that wheel in a clock which has a hoop almost round it, in which there is a vacancy, where the clock locks.</p><p>DETERMINATE <hi rend="italics">Number.</hi> See <hi rend="smallcaps">Number.</hi></p><p><hi rend="smallcaps">Determinate</hi> <hi rend="italics">Problem,</hi> is that which has but one solution, or a certain limited number of solutions; in contradistinction to an indeterminate problem, which admits of infinite solutions.</p><p>Such, for instance, is the problem, To form an isosceles triangle on a given line, so that each of the angles at the base shall be double of that at the ver- <cb/> tex; which has only one solution: or this, To find an isosceles triangle whose area and perimeter are given; which admits of two solutions.</p><p><hi rend="smallcaps">Determinate</hi> <hi rend="italics">Section,</hi> the name of a Tract, or General Problem, written by the ancient geometrician Apollonius. None of this work has come down to us, excepting some extracts and an account of it by Pappus, in the preface to the 7th book of his Mathematical Collections. He there says that the general problem was, “To cut an infinite right line in one point so, that, of the segments contained between the point of section sought, and given points in the said line, either the square on one of them, or the rectangle contained by two of them, may have a given ratio, either to the rectangle contained by one of them and a given line, or to the rectangle contained by two of them.”</p><p>Pappus farther informs us, that this Tract of Apollonius was divided into two books; that the first book contained 6 problems, and the second 3; that the 6 problems of the first book contained 16 epitagmas, or cases, respecting the dispositions of the points; and the second book 9. Farther, that of the epitagmas of the 6 problems of the first book, 4 were maxima, and one a minimum: that the maxima are at the 2d epitagma of the 2d problem, at the 3d of the 4th, the 3d of the 5th, and the 3d of the 6th; but that the minimum was at the 3d epitagma of the 3d problem. Also, that the second book eontained three determinations; of which the 3d epitagma of the 1st problem, and the 3d of the 2d were minima, and the 3d of the 3d a maximum. Moreover, that the first book had 27 lemmas, and the second book 24; and lastly, that both books contained 83 theorems.</p><p>From such account of the contents of this Tract, and the lemmas also given by Pappus, several persons have attempted to restore, or recompose what they thought might be nearly the form of Apollonius's tract, or the subject of each problem, case, determination, &c; among whom are, Snellius, an eminent Dutch mathematician of the last century; a translation of whose work was published in English by Mr. John Lawson, in 1772, together with a new restoration of the whole work by his friend Mr. William Wales.</p></div1><div1 part="N" n="DEW" org="uniform" sample="complete" type="entry"><head>DEW</head><p>, a thin light insensible mist, or rain, ascending with a slow motion, and falling while the sun is below the horizon.</p><p>To us it appears to differ from rain, as less from more. Its origin and matter are doubtless from the vapours and exhalations that rise from the earth and water. See <hi rend="smallcaps">Exhalation.</hi> Some define it a vapour liquesied, and let fall in drops. M. Huet, in one of his letters, shews that dew does not fall, but rises; and others have adopted the same opinion.</p><p>M. du Fay made several experiments, first with glasses, then with pieces of cloth stretched horizontally at different heights; and he found that the lower bodies, with their under surfaces, were wetted before those that were placed higher, or their upper surfaces. And Du Fay and Muschenbroek both found, that different substances, and even different colours, receive the dew differently, and some little or not at all.</p><p>From the principles laid down under the article E<hi rend="smallcaps">VAPORATION</hi>, the several phenomena of dews are easily accounted for. Such as, for instance, that dews are <pb n="371"/><cb/> more copious in the spring, than in the other seasons of the year; there being then a greater stock of vapour in readiness, than at other times, by reason of the small expence of it in the winter's cold and frost. Hence it is too, that Egypt, and some other hot countries, abound with dews throughout all the heats of summer; for the air there being too hot to constipate the vapours in the day-time, they never gather into clouds; and hence they have no rain: but in climates that are excessively hot, the nights are remarkahly cold; so that the vapours raised after sun-set, are readily condensed into dews.</p><p>It is natural to conclude, from the different substances which are combined with dew, that it must be either salutary or injurious, both to plants and animals.</p><p>It is not easy to ascertain the quantity of dew that rises every night, or in the whole year, because of the winds which disperse it, the rains which carry it down, and other inconveniences: but it is known that it rises in greater abundance after rain than after dry weather, and in warm countries than in cold ones. There are some places in which dew is observed only to ascend, and not to fall; and others again in which it is carried upwards in greater plenty than downwards, being dispersed by the winds.</p><p>Dr. Hales made some experiments, to determine the quantity of dew that falls in the night. For this purpose, on the 15th of August, at 7 in the evening, he filled two glazed earthen pans with moist earth; the dimensions of the pans being, 3 inches deep, and 12 inches diameter: and he observes, that the moister the earth, the more dew falls on it in a night; and that more than a double quantity of dew falls on a surface of water, than on an equal surface of moist earth. These pans increased in weight by the night's dew, 180 grains; and decreased in weight by the evaporation of the day, 1 oz 282grs: so that 540 grains more are evaporated from the earth every 24 hours in summer, than the dew that falls in the night; i. e. in 21 days near 26 ounces from a circular area of a foot diameter. Now if 180 grains of dew, falling in one night on such an area, which is equal to 113 square inches, be equally spread on the surface, its depth will be the 159th part of an inch. He likewise found that the depth of dew in a winter's night was the 90th part of an inch. If therefore we allow 159 nights for the extent of the summer's dew, it will in that time amount to one inch in depth; and reckoning the remaining 206 nights for the extent of the winter's dew, it will produce 2.28 inches depth; and the dew of the whole year will amount to 3.28 inches depth. But the quantity which evaporated in a fair summer's day from the same surface, being 1 oz and 282 grs, gives the 40th part of an inch deep for evaporation, which is 4 times as much as fell at night. Dr. Hales observes that the evaporation of a winter's day is nearly the same as in a summer's day; the earth's greater moisture in winter compensating for the sun's greater heat in summer. Hales's Vegetable Statics, vol. 1, pa. 52 of 4th edit. See <hi rend="smallcaps">Evaporation.</hi></p><p>Signor Beccaria made several experiments to demonstrate the existence of the electricity that is produced by dew. He observes in general, that such electricity took place in clear and dry weather, during which no strong wind prevailed; and that it depends on the <cb/> quantity of the dew, as the electricity of the rain depends on the quantity of the rain. He sometimes found that it began before sun-set; at other times not till 11 o'clock at night. <hi rend="italics">Artificial Electricity,</hi> Appendix, letter 3.</p><p>DE WIT (<hi rend="smallcaps">John</hi>), the famous Dutch pensionary, was born at Dort, in 1625; where he prosecuted his studies so diligently, that at 23 years of age, he published <hi rend="italics">Elementa Curvarum Linearum,</hi> one of the deepest books in mathematics at that time. After taking his degrees, and travelling, he, in 1650, became pensionary of Dort, and distinguished himself very early in the management of public affairs, which soon after raised him to the rank of pensionary of Holland. After rendering the greatest benesits to his country in many important instances, and serving it in several high capacities, with the greatest ability, diligence, and integrity, by some intrigues of the court, it is said, he and his brother were thrown into prison, from whence they were dragged by the mob, and butchered with the most cruel and savage barbarity.</p><p>DIACAUSTIC <hi rend="italics">Curve,</hi> or the <hi rend="italics">Caustic by Refraction,</hi> is a species of caustic curves, the genesis of which is in the following manner. Imagine an infinite number of rays BA, BM, BD, &c, issuing from the same luminous point B, refracted to or from the perpendicular MC, by the given curve AMD; and so, that CE the sines of the angles of incidence CME, be always to CG the sines of the refracted angles CMG, in a given ratio: then the curve HFN that touches all the refracted rays AH, MF, DN, &c, is called the Diacaustic, or Caustic by Refraction. <figure/></p><p>DIACOUST ICS, or <hi rend="smallcaps">Diaphonics</hi>, the considera tion of the properties of sound refracted in passing through different mediums; that is, out of a denser into a more subtile, or out of a more subtile into a denser medium. See <hi rend="smallcaps">Sound.</hi></p></div1><div1 part="N" n="DIADROME" org="uniform" sample="complete" type="entry"><head>DIADROME</head><p>, a term sometimes used for the vibration, motion, or swing of a pendulum.</p></div1><div1 part="N" n="DIAGONAL" org="uniform" sample="complete" type="entry"><head>DIAGONAL</head><p>, is a right line drawn across a figure, from one angle to another; and is sometimes called a diameter. It is used chiefly in quadrilateral sigures, viz, in parallelograms and trapeziums.</p><p>1. Every diagonal, as AC, di- <figure/> vides a parallelogram into two equal parts or triangles ABC, ADC.</p><p>2. Two diagonals, AC, BD, drawn in a parallelogram, do mutually bisect each other; as in the point E. <pb n="372"/><cb/></p><p>3. Any line, as FG, drawn through the middle of the diagonal of a parallelogram, is bisected by it at the point E; and it divides the parallelogram into two equal parts, BFGC and AFGD.</p><p>4. The diagonal of a square is incommensurable with its side.</p><p>5. In any parallelogram, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals.</p><p>6. In any trapezium, the sum of the squares of the four sides is equal to the sum of the squares of the two diagonals together with 4 times the square of the distance between the middle points of the diagonals.</p><p>7. In any trapezium, the sum of the squares of the two diagonals is double the sum of the squares of two lines bisecting the two pairs of opposite sides.</p><p>8. In any quadrilateral inscribed in a circle, the rectangle of the two diagonals is equal to the sum of the two rectangles under the two pairs of opposite sides.</p><p><hi rend="smallcaps">Diagonal</hi> <hi rend="italics">Scale.</hi> See <hi rend="smallcaps">Scales.</hi></p></div1><div1 part="N" n="DIAGRAM" org="uniform" sample="complete" type="entry"><head>DIAGRAM</head><p>, is a scheme for the explanation or demonstration of any figure, or of its properties.</p></div1><div1 part="N" n="DIAL" org="uniform" sample="complete" type="entry"><head>DIAL</head><p>, or <hi rend="smallcaps">Sun-Dial</hi>, an instrument for measuring time by means of the sun's shadow. Or, it is a draught or description of certain lines on the surface of a body, so that the shadow of a style, or ray of the sun through a hole, should touch certain marks at certain hours.</p><p>Sun-Dials are doubtless of great antiquity. But the first upon record is, it seems, the dial of Ahaz, who began to reign 400 years before Alexander, and within 12 years of the Building of Rome: it is mentioned in Isaiah, chap. 38, ver. 8.</p><p>Several of the antients are spoken of, as makers of dials; as Anaximenes Milesius, Thales. Vitruvius mentions one made by the ancient Chaldee historian Berosus, on a reclining plane, almost parallel to the equator. Aristarchus Samius invented the hemispherical dial. And there were at the same time some spherical ones, with a needle for a gnomon. The discus of Aristarchus was an horizontal dial, with its rim raised up all around, to prevent the shadow from stretching too far.</p><p>It was late before the Romans became acquainted with dials. The first sun-dial at Rome was set up by Papyrius Cursor, about the 460th year of the city; before which time, Pliny says there is no mention of any account of time but by the sun's rifing and setting: the first dial was set up near the temple of Quirinus; but being inaccurate, about 30 years after, another was brought out of Sicily by the consul M. Valerius Messala, which he placed on a pillar near the Rostrum; but neither did this shew time truly, because not made for that latitude; and, after using it 99 years, Martius Philippus set up another more exact.</p><p>The diversity of sun-dials arises from the different situation of the planes, and from the different figure of the surfaces upon which they are described; whence they become denominated <hi rend="italics">equinoctial, horizontal, vertical, polar, direct, erect, declining, inclining, reclining, cylindrical,</hi> &c. For the general principles of their construction, see <hi rend="smallcaps">Dialling.</hi></p><p>Dials are sometimes distinguished into <hi rend="italics">primary</hi> and <hi rend="italics">secondary.</hi></p><p><hi rend="italics">Primary</hi> <hi rend="smallcaps">Dials</hi> are such as are drawn either on the <cb/> plane of the horizon, and thence called <hi rend="italics">horizontal</hi> dials; or perpendicular to it, and called <hi rend="italics">vertical</hi> dials; or else drawn on the polar and equinoctial planes, though neither horizontal nor vertical. And</p><p><hi rend="italics">Secondary</hi> <hi rend="smallcaps">Dials</hi> are all those that are drawn on the planes of other circle<*>, beside those last mentioned; or those which either decline, incline, recline, or deincline.</p><p>Each of these again is divided into several others, as follow:</p><p><hi rend="italics">Equinoctial</hi> <hi rend="smallcaps">Dial</hi>, is that which is described on an equinoctial plane, or one parallel to it.</p><p><hi rend="italics">Horizontal</hi> <hi rend="smallcaps">Dial</hi>, is described on an horizontal plane, or a plane parallel to the horizon.—This dial shews the hours from sun-rise to sun-set.</p><p><hi rend="italics">South</hi> <hi rend="smallcaps">Dial</hi>, or an <hi rend="italics">Erect, direct South Dial,</hi> is that described on the surface of the prime vertical circle looking towards the south.—This dial shews the time from 6 in the morning till 6 at night.</p><p><hi rend="italics">North</hi> <hi rend="smallcaps">Dial</hi>, or an <hi rend="italics">Erect, direct North Dial,</hi> is that which is described on the surface of the prime vertical looking northward. This dial only shews the hours before 6 in the morning, and after 6 in the evening.</p><p><hi rend="italics">East</hi> <hi rend="smallcaps">Dial</hi>, or <hi rend="italics">Erect, direct East Dial,</hi> is that drawn on the plane of the meridian, looking to the east.—This can only shew the hours till 12 o'clock.</p><p><hi rend="italics">West Dial,</hi> or <hi rend="italics">Erect, direct West Dial,</hi> is that described on the western side of the meridian.—This can only shew the hours after noon. Consequently this, and the last preceding one, will shew all the hours of the day between them.</p><p><hi rend="italics">Polar</hi> <hi rend="smallcaps">Dial</hi>, is that which is described on a plane passing through the poles of the world, and the east and west points of the horizon. It is of two kinds; the first looking up towards the zenith, and called the <hi rend="italics">upper;</hi> the latter, down towards the nadir, called the <hi rend="italics">lower.</hi> The polar dial therefore is inclined to the horizon in an angle equal to the elevation of the pole.—The upper polar dial shews the hours from 6 in the morning till 6 at night, and the lower one shews the hours before 6 in the morning, and after 6 in the evening, viz, from sun-rise and till sun-set.</p><p><hi rend="italics">Declining</hi> <hi rend="smallcaps">Dials</hi>, are erect or vertical dials which decline from any of the cardinal points; or they are such as cut either the plane of the prime vertical, or of the horizon, at oblique angles.</p><p>Declining dials are of very frequent use; as the walls of houses, on which dials are mostly drawn, commonly deviate from the cardinal points.</p><p>Of declining dials there are several kinds, which are denominated from the cardinal points which they are nearest to; as decliners from the south, and from the north, and even from the zenith.</p><p><hi rend="italics">Inclined</hi> <hi rend="smallcaps">Dials</hi>, are such as are drawn on planes not erect, but inclining, or leaning forward towards the south, or southern side of the horizon, in an angle, either greater or less than the equinoctial plane.</p><p><hi rend="italics">Reclining</hi> <hi rend="smallcaps">Dials</hi>, are those drawn on planes not erect, but reclined, or leaning backwards from the zenith towards the north, in an angle greater or less than the polar plane.</p><p><hi rend="italics">Deinclined</hi> <hi rend="smallcaps">Dials</hi>, are such as both decline and incline, or recline.—These last three sorts of dials are very rare. <pb/><pb/><pb n="373"/><cb/></p><p><hi rend="smallcaps">Dials</hi> <hi rend="italics">without Centres,</hi> are those whose hour lines converge so slowly, that the centre, or point of their concourse, cannot be expressed on the given plane.</p><p><hi rend="italics">Quadrantal</hi> <hi rend="smallcaps">Dial.</hi> See <hi rend="italics">Horodictical</hi> <hi rend="smallcaps">Quadrant.</hi></p><p><hi rend="italics">Reflecting</hi> <hi rend="smallcaps">Dial.</hi> See <hi rend="smallcaps">Reflecting</hi> <hi rend="italics">Dial.</hi></p><p><hi rend="italics">Cylindrical</hi> <hi rend="smallcaps">Dial</hi>, is one drawn on the curve surface of a cylinder. This may first be drawn on a paper plane, and then pasted round a cylinder of wood, &c. It will shew the time of the day, the sun's place in the ecliptic, and his altitude at any time of observation.</p><p>There are also <hi rend="italics">Portable</hi> <hi rend="smallcaps">Dials</hi>, or <hi rend="italics">on a Card,</hi> and <hi rend="italics">Universal</hi> <hi rend="smallcaps">Dials</hi> <hi rend="italics">on a Plain Cross,</hi> &c.</p><p><hi rend="italics">Refracted</hi> <hi rend="smallcaps">Dials</hi>, are such as shew the hour by means of some refracting transparent duid.</p><p><hi rend="italics">Ring</hi> <hi rend="smallcaps">Dial</hi>, is a small portable dial, consisting of a brass ring or rim, about 2 inches in diameter, and onethird of an inch in breadth. In a point of this rim there is a hole, through which the sun beams pass, and form a bright speck in the concavity of the opposite semi-circle, which gives the hour of the day in the divisions marked within it.</p><p>When the hole is fixed, the dial only shews true about the time of the equinox. But to have it perform throughout the whole year, the hole is made moveable, the signs of the zodiac, or the days of the month, being marked on the convex side of the ring; hence, in using it, the moveable hole is set to the day of the month, or the degree of the zodiac the sun is in; then suspending the dial by the little ring, turn it towards the sun, and his rays through the hole will shew the hour on the divisions within side.</p><p><hi rend="italics">Universal,</hi> or <hi rend="italics">Astronomical Ring</hi> <hi rend="smallcaps">Dial</hi>, is a ring dial which shews the hour of the day in any part of the earth; whereas the former is confined to a certain latitude. Its figure see represented below. <figure/></p><p>It consists of two rings or flat circles, from 2 to 6 inches in diameter; and of a proportionable breadth &c. The outward ring A represents the meridian of any place you are at, and contains two divisions of 90 degrees each, diametrically opposite to one another, the one serving from the equator to the north pole, the <cb/> other to the south pole. The inner ring represents the equator, and turns exactly within the outer, by means of two pivots in each ring at the hour of 12.</p><p>Across the two circles goes a thin reglet or bridge, with a cursor C, sliding along the middle of the bridge, and having a small hole for the sun to shine through. The middle of this bridge is conceived as the axis of the world, and the extremities as the poles: on the one side are drawn the signs of the zodiac, and on the other the days of the month. On the edge of the meridian slides a piece, to which is fitted a small ring to suspend the instrument by.</p><p>In this dial, the divisions on the axis are the tangents of the angles of the sun's declination, adapted to the semidiameter of the equator as radius, and placed on either side of the centre: but instead of laying them down from a line of tangents, a scale of equal parts may be made, of which 1000 shall answer exactly to the length of the semi-axis, from the centre to the inside of the equinoctial ring; and then 434 of these parts may be laid down from the centre towards each end, which will limit all divisions on the axis, because 434 is the natural tangent of 23° 28′. And thus, by a nonius fixed to the sliding piece, and taking the sun's declination from an ephemeris, and the tangent of that declination from the table of natural tangents, the slider might be always set true within 2 minutes of a degree. This scale of 434 equal parts might be placed right against the 23° 28′ of the sun's declination, on the axis, instead of the sun's place, which is there of little use. For then the slider might be set in the usual way, to the day of the month, for common use; or to the natural tangent of the declination, when great accuracy is required.</p><p><hi rend="italics">To use this Dial:</hi> Place the line <hi rend="italics">a</hi> (on the middle of the sliding piece) over the degree of latitude of the place, as for instance 51 1/2 degrees for London: put the line which crosses the hole of the cursor to the degree of the sign, or day of the month. Open the instrument so as that the two rings be at right angles to each other, and suspend it by the ring H, that the axis of the dial, represented by the middle of the bridge, may be parallel to the axis of the world. Then turn the flat side of the bridge towards the sun, so that his rays, striking through the small hole in the middle of the cursor, may fall exactly on a line drawn round the middle of the concave surface of the inner ring; in which case the bright spot shews the hour of the day in the said concave surface of the ring.</p><p><hi rend="italics">Nocturnal</hi> or <hi rend="italics">Night</hi>-<hi rend="smallcaps">Dial</hi>, is that which shews the hour of the night, by the light, or shadow projected from the moon or stars.</p><p>Lunar or Moon Dials may be either purposely described and adapted to the moon's motion; or the hour may be found on a sun-dial by the moon shining upon it, thus: Observe the hour which the shadow of the index points at by moon light; find the days of the moon's age in the calendar, and take 3-4ths of that number, for the hours to be added to the hour shewn by the shadow, to give the hour of the night. The reason of which is, that the moon comes to the same horary circle later than the sun by about three quarters of an hour every day; and at the time of new moon the solar and lunar hour coincide. <pb n="374"/><cb/></p><p><hi rend="smallcaps">Dial</hi> <hi rend="italics">Planes,</hi> are the plane superficies upon which the hour lines of dials are drawn.</p><p><hi rend="italics">Tide</hi> <hi rend="smallcaps">Dial.</hi> See <hi rend="smallcaps">Tide</hi> <hi rend="italics">Dial.</hi></p></div1><div1 part="N" n="DIALLING" org="uniform" sample="complete" type="entry"><head>DIALLING</head><p>, the art of drawing sun, moon, and star-dials on any sort of surface, whether plane or curved.</p><p>Dialling is wholly founded on the first motion of the heavenly bodies, and chiefly the sun; or rather on the diurnal rotation of the earth: so that the elements of spherics, and spherical trigonometry, should be understood, before a person advances to the doctrine of dialling.</p><p>The principles of dialling may be aptly deduced from, and illustrated by, the phenomena of a hollow or transparent sphere, as of glass. Thus, suppose <hi rend="italics">a</hi>P<hi rend="italics">cp</hi> to re- <figure/> present the earth as transparent; and its equator as divided into 24 equal parts by so many meridian semicircles <hi rend="italics">a, b, c, d, e,</hi> &c, one of which is the geographical meridian of any given place, as London, which it is supposed is at the point <hi rend="italics">a</hi>; and if the hour of 12 were marked at the equator, both upon that meridian and the opposite one, and all the rest of the hours in order on the other meridians, those meridians would be the hour circles of London: because, as the sun appears to move round the earth, which is in the centre of the visible heavens, in 24 hours, he will pass from one meridian to another in an hour. Then, if the sphere had an opake axis, as PE<hi rend="italics">p,</hi> terminating in the poles P and <hi rend="italics">p,</hi> the shadow of the axis, which is in the same plane with the sun and with each meridian, would fall upon every particular meridian and hour, when the sun came to the plane of the opposite meridian, and would consequently shew the time at London, and at all other places on the same meridian. If this sphere were cut through the middle by a solid plane ABCD in the rational horizon of London, one half of the axis EP would be above the plane, and the other half below it; and if straight lines were drawn from the centre of the plane to those points where its circumference is cut by the hour circles of the sphere, those lines would be the hour lines of an horizontal dial for London; for the shadow of the axis would fall upon each particular hour line of the dial, when it fell upon the like hour circle of the sphere. <cb/></p><p>If the plane which cuts the sphere be upright, as AFCG, touching the given place, for ex. London, at F, <figure/> and directly facing the meridian of London, it will then become the plane of an erect direct south dial; and if right lines be drawn from its centre E, to those points of its circumference where the hour circles of the sphere cut it, these will be the hour lines of a vertical or direct south dial for London, to which the hours are to be set in the figure, contrary to those on an horizontal dial; and the lower half E<hi rend="italics">p</hi> of the axis will cast a shadow on the hour of the day in this dial, at the same time that it would fall upon the like hour circle of the sphere, if the dial plane was not in the way.</p><p>If the plane, still facing the meridian, be made to incline, or recline, any number of degrees, the hour circles of the sphere will still cut the edge of the plane in those points to which the hour lines must be drawn straight from the centre; and the axis of the sphere will cast a shadow on these lines at the respective hours. The like will still hold, if the plane be made to decline by any number of degrees from the meridian towards the east or west; provided the declination be less than 90 degrees, or the reclination be less than the co-latitude of the place; and the axis of the sphere will be the gnomon: otherwise, the axis will have no elevation above the plane of the dial, and cannot be a gnomon.</p><p>Thus it appears that the plane of every dial represents the plane of some great circle on the earth, and the gnomon the earth's axis; the vertex of a right gnomon the centre of the earth or visible heavens; and the plane of the dial is just as far from this centre as from the vertex of this stile. The earth itself, compared with its distance from the sun, is considered as a point; and therefore, if a small sphere of glass be placed upon any part of the earth's surface, so that its axis be parallel to the axis of the earth, and the sphere have such lines upon it, and such planes within it, as above described; it will shew the hours of the day as truly as if it were placed at the earth's centre, and the shell of the earth were as transparent as glass. Ferguson, lect. 10.</p><p>The principal writers on Dials, and Dialling, are the following: Vitruvius, in his Architecture, cap. 4 <pb n="375"/><cb/> and 7, lib. 9: Sebastian Munster, his Horolographia: John Dryander de Horologiorum varia Compositione: Conrade Gesner's Pandectæ: Andrew Schoner's Gnomonicæ: Fred. Commandine de Horologiorum Descriptione: Joan. Bapt. Benedictus de Gnomonum Umbrarumque Solarium Usu: Joannes Georgius Schomberg, Exegesis Fundamentorum Gnomonicorum: Solomon de Caus, Traité des Horologes Solaires: Joan. Bapt. Trolta, Praxis Horologiorum: Desargues, Maniere Universelle pour poser l'Essieu & placer les Heures & autres Choses aux Cadrans Solaires: Ath. Kircher, Ars magna Lucis & Umbræ: Hallum, Explicatio Horologii in Horto Regio Londini: Tractatus Horologiorum Joannis Mark: Clavius, Gnomonices de Horologiis; in which he demonstrates both the theory and the operations after the rigid manner of the ancient mathematicians: Dechales, Ozanam, and Schottus, gave much easier treatises on this subject; as did also Wolfius in his Elementa: M. Picard gave a new method of making large dials, by calculating the hour lines; and M. De la Hire, in his Dialling, printed in 1683, gave a geometrical method of drawing hour lines from certain points, determined by observation. Everhard Walper, in 1625, published his Dialling, in which he lays down a method of drawing the primary dials on a very easy foundation; and the same foundation is also described at length by Sebastian Munster, in his Rudimenta Mathematica, published in 1651. In 1672, Sturmius published a new edition of Walper's Dialling, with the addition of a whole second part, concerning inclining and declining dials, &c. In 1708, the same work, with Sturmius's additions, was re-published, with the addition of a 4th part, containing Picard's and De la Hire's methods of drawing large dials, which makes much the best and fullest book on the subject. Peterson, Michael, and Muller, have each written on Dialling, in the German language: Coetfius, in his Horologiographia Plana, printed in 1689: Gauppen, in his Gnomonica Mechanica: Leybourn, in his Dialling: Bion, in his Use of Mathematical Instruments: Wells, in his Art of Shadows. There is also a treatise by M. Deparceux, 1740. Mr. Ferguson has also written on the same subject in his Lectures on Mechanics; besides Emerson, in his Dialling; and Mr. W. Jones, in his Instrumental Dialling.</p><p><hi rend="italics">Universal</hi> <hi rend="smallcaps">Dialling</hi> <hi rend="italics">Cylinder,</hi> is represented in fig. 2, plate vii, where ABCD is a glass cylindrical tube, closed at both ends with brass plates, in the centres of which a wire or axis EFG is fixed. The tube is either fixed to an horizontal board H, so that its axis may make an angle with the board equal to that which the earth's axis makes with the horizon of any given place, and be parallel to the axis of the world; or it may be made to move on a joint, and elevated for any particular latitude. The twenty-four hour lines are drawn with a diamond on the outside of the glass, equidistant from each other, and parallel to the axis. The XII next B stands for midnight, and the XII next the board H for noon. When the axis of this instrument is elevated according to the latitude, and the board set level, with the line HN in the plane of the meridian, and the end towards the north; the axis EFG will serve as a gnomon or stile, and cast a shadow on the hour of the day among the pa- <cb/> rallel hour lines, when the sun shines on the instrument. As the plate AD at the top is parallel to the equator, and the axis EFG perpendicular to it, right lines drawn from the centre to the extremities of the parallels will be the hour lines of an equinoctial dial, and the axis will be the stile. An horizontal plate <hi rend="italics">ef</hi> put down into the tube, with lines drawn from the centre to the several parallels, cutting its edge, will be an horizontal dial for the given latitude; and a vertical plate <hi rend="italics">gc,</hi> fronting the meridian, and touching the tube with its edge, with lines drawn from its centre to the parallels, will be a vertical south dial: the axis of the instrument serving in both cases for the stile of the dial: and if a plate be placed within the tube, so as to decline, incline, or recline, by any given number of degrees, and lines be drawn, as above, a declining, inclining, or reclining dial will be formed for the given latitude. If the axis with the several plates sixed to it be drawn out of the tube, and set up in sunshine in the same position as they were in the tube, AD will be an equinoctial dial, <hi rend="italics">ef</hi> an horizontal dial, and <hi rend="italics">ge</hi> a vertical south dial; and the time of the day will be shewn by the axis EFG. If the cylinder were wood, instead of glass, and the parallel lines drawn upon it in the same manner, it would serve to facilitate the operation of making these several dials. The upper plate with lines drawn to the several intersections of the parallels, which appears obliquely in fig. 2, would be an equinoctial dial as in fig. 3, and the axis perpendicular to it be its stile. An horizontal dial for the latitude of the elevation of the axis might be made, by drawing out the axis and cutting the cylinder, as at <hi rend="italics">efgh,</hi> parallel to the horizontal board H; the section would be elliptic as in fig. 4. A circle might be described on the centre, and lines drawn to the divisions of the ellipse would be the hour lines; and the wire put in its place again, as E, would be the stile. If this cylinder were cut by a plane perpendicular to the horizontal board H, or to the line SHN, beginning at <hi rend="italics">g,</hi> the plane of the section would be elliptical as in fig. 5, and lines drawn to the points of intersection of the parallels on its edge would be the hour lines of a vertical direct south dial, which might be made of any shape, either circular or square, and F the axis of the cylinder would be its stile. Thus also inclining, declining, or reclining dials might be easily constructed, for any given latitude. Ferguson, ubi supra.</p><p><hi rend="smallcaps">Dialling</hi> <hi rend="italics">Globe,</hi> is an instrument made of brass, or wood, with a plane fitted to the horizon, and an index; particularly contrived to draw all sorts of dials, and to give a clear exhibition of the principles of that art.</p><p><hi rend="smallcaps">Dialling</hi> <hi rend="italics">Lines,</hi> or <hi rend="italics">Scales,</hi> are graduated lines, placed on rules or the edges of quadrants, and other instruments, to expedite the construction of dials. The principal of these lines are, 1. A scale of six hours, which is only a double tangent, or two lines of tangents each of 45 degrees, joined together in the middle, and equal to the whole line of fines, with the declination set against the meridian altitudes in the latitude of London, suppose, or any place for which it is made: the radius of which line of sines is equal to the dialling scale of six hours. 2. A line of latitudes, which is sitted to the hour scale, and is made by this <pb n="376"/><cb/> canon: as the radius is to the chord of 90 degrees; so are the tangents of each respective degree of the line of latitudes, to the tangents of other arches: and then the natural sines of those arches are the numbers, which, taken from a diagonal scale of equal parts, will graduate the divisions of the line of latitude to any radius. The line of hours and latitudes is generally for pricking down all dials with centres. For the method of constructing these scales, see <hi rend="smallcaps">Scale.</hi></p><p><hi rend="smallcaps">Dialling</hi> <hi rend="italics">Sphere,</hi> is an instrument made of brass, with several semicircles sliding over one another, on a moving horizon, to demonstrate the nature of the doctrine of spherical triangles, and to give a true idea of the drawing of dials on all manner of planes.</p></div1><div1 part="N" n="DIAMETER" org="uniform" sample="complete" type="entry"><head>DIAMETER</head><p>, <hi rend="italics">of a circle,</hi> is a right line passing through the centre, and terminated at the circumference on both sides.</p><p>The diameter divides the circumference, and the area of the circle, into two equal parts. And hals the diameter, or the semi-diameter, is called the radius.</p><p>For the proportion between the diameter and the circumference of a circle, see <hi rend="smallcaps">Circle</hi> and C<hi rend="smallcaps">IRCUMFERENCE.</hi></p><p><hi rend="smallcaps">Diameter</hi> <hi rend="italics">of a Conic Section,</hi> or <hi rend="italics">Transverse Diameter,</hi> is a right-line passing through the centre of the section, or the middle of the axis.—The diameter bisects all ordinates, or lines drawn parallel to the tangent at its vertex. See <hi rend="smallcaps">Conic</hi> <hi rend="italics">Sections.</hi></p><p><hi rend="italics">Conjugate</hi> <hi rend="smallcaps">Diameter</hi>, is a diameter, in Conic Sections, parallel to the ordinates of another diameter, called the transverse; or parallel to the tangent at the vertex of this other.</p><div2 part="N" n="Diameter" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Diameter</hi></head><p>, <hi rend="italics">of any Curve,</hi> is a right line which divides two other parallel right lines, in such manner that, in each of them, all the segments or ordinates on one side, between the diameter and different points of the curve, are equal to all those on the other side. This is Newton's sense of a Diameter.</p><p>But, according to some, a diameter is that line, whether right or curved, which bisects all the parallels drawn from one point to another of a curve. So that in this way every curve will have a diameter; aud hence the curves of the 2d order, have, all of them, either a right-lined diameter, or else the curves of some one of the conic sections for diameters. And many geometrical curves of the higher orders, may also have for diameters, curves of more inferior orders.</p><p><hi rend="smallcaps">Diameter</hi> <hi rend="italics">of Gravity,</hi> is a right line passing through the centre of gravity.</p><p><hi rend="smallcaps">Diameter</hi> <hi rend="italics">in Astromony.</hi> The diameters of the heavenly bodies are either <hi rend="italics">apparent,</hi> i. e. such as they appear to the eye; or <hi rend="italics">real,</hi> i. e. such as they are in themselves.</p><p>The apparent diameters are best measured with a micrometer, and are estimated by the measure of the angle they subtend at the eye. These are different in different circumstances and parts of the orbits, or according to the various distances of the luminary; being in the inverse ratio of the distance.</p><p>The sun's vertical diameter is found by taking the height of the upper and lower edge of his disk, when he is in the meridian, or near it; correcting the altitude of each edge on account of refraction and parallax; then the difference between the true altitudes of <cb/> the two, is the true apparent diameter sought. Or the apparent diameter may be determined by observing, with a good clock, the time which the sun's disc takes in passing over the meridian: and here, when the sun is in or near the equator, the following proportion may be used; viz, as the time between the sun's leaving the meridian and returning to it again, is to 360 degrees, so is the time of the sun's passing over the meridian, to the number of minutes and seconds of a degree contained in his apparent diameter: but when the sun is in a parallel at some distance from the equator, his diameter measures a greater number of minutes and seconds in that parallel than it would do in a great circle, and takes up proportionally more time in passing over the meridian; in which case say, as radius is to the cosine of the sun's declination, so is the time of the sun's passing the meridian reduced to minutes and seconds of a degree, to the ar<*> of a great circle which measures the sun's apparent horizontal diameter. See <hi rend="smallcaps">Transit.</hi></p><p>The sun's apparent diameter may also be taken by the projection of his image in a dark room.</p><p>There are several ways of finding the apparent diameters of the planets: but the most certain method is that with the micrometer.</p><p>The following is a table of the apparent diameters of the sun and planets, in different circumstances, and as determined by different astronomers. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=12" role="data"><hi rend="smallcaps">Table of Apparent Diameters.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 valign=top" role="data">1. <hi rend="italics">Of the Sun, according to</hi></cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Greatest</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Mean</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Least</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">″</cell><cell cols="1" rows="1" rend="align=right" role="data">‴</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">″</cell><cell cols="1" rows="1" rend="align=right" role="data">‴</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="align=right" role="data">″</cell><cell cols="1" rows="1" rend="align=right" role="data">‴</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Aristarchus and</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Archimedes</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Ptolomy</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Albategnius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Regiomontanus</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Copernicus</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Tycho</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Kepler</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Riccioli</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">J. D. Cassini</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Gascoigne</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Flamsteed</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Mouton</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">De la Hire</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Louville</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">M. Cassini, jun.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Short</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">2. <hi rend="italics">Of the Moon.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Ptolomy</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Tycho,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data">in Conjunc.</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">in Opposit.</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Kepler</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">De La Hire</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Newton,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data">in Syzygy</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">in Quadrat.</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4" role="data">Mouton, Full, in Perigee</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Monnier,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data">in Syzygy</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">in Quadrat</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row></table> <pb n="377"/><cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Greatest</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Mean</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Least</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">3. <hi rend="italics">Of Mercury.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Albategnius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Alfraganus</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Tycho</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hevelius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hortensius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Riccioli</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Bradley</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">4. <hi rend="italics">Of Venus.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Albategnius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Alfraganus</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Tycho</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hevelius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hortensius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Kepler</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Riccioli</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Huygens</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Flamsteed</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Horrox</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Crabtree</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Transit of 1761</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Transit of 1769</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">5. <hi rend="italics">Of Mars.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=4" role="data">Albateg. and Alfrag.</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Tycho</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hevelius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hortensius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Kepler</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Riccioli</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Huygens</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Flamsteed</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Herschel,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data">Polar Diam.</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Equat. Diam.</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">6. <hi rend="italics">Of Jupiter.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=4" role="data">Albateg. and Alfrag.</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Tycho</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hevelius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hortensius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Kepler</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Riccioli</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Huygens</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Flamsteed</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Newton, from</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">37</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">15</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Pound's Obs.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">7. <hi rend="italics">Of Saturn.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=4" role="data">Albateg and Alfrag.</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Tycho</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hevelius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Hortensius</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Kepler</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row></table> <cb/> <table rend="BORDER"><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Greatest</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Mean</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Least</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Riccioli</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Huygens</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Flamsteed</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Newton, from</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">16</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Pound's Observ.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Huygens, <figure/> 's Ring</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Newton, from</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">40</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Pound's Obs.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Monnier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">8. <hi rend="italics">New Planet.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Herschel</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell><cell cols="1" rows="1" rend="align=center" role="data">-</cell></row></table></p><p>The Mean Apparent diameters of the planets, as seen from the sun, are as follow: <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Mercury,</cell><cell cols="1" rows="1" rend="align=center" role="data">Venus,</cell><cell cols="1" rows="1" rend="align=center" role="data">Earth,</cell><cell cols="1" rows="1" rend="align=center" role="data">Moon,</cell><cell cols="1" rows="1" rend="align=center" role="data">Mars,</cell><cell cols="1" rows="1" rend="align=center" role="data">Jup.</cell><cell cols="1" rows="1" rend="align=center" role="data">Sat.</cell><cell cols="1" rows="1" rend="align=center" role="data">Hersch.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">20″</cell><cell cols="1" rows="1" rend="align=center" role="data">30″</cell><cell cols="1" rows="1" rend="align=center" role="data">17″</cell><cell cols="1" rows="1" rend="align=center" role="data">6″</cell><cell cols="1" rows="1" rend="align=center" role="data">11″</cell><cell cols="1" rows="1" rend="align=center" role="data">37″</cell><cell cols="1" rows="1" rend="align=center" role="data">16″</cell><cell cols="1" rows="1" rend="align=center" role="data">4′</cell></row></table></p><p>For the true diameters of the sun and planets, and their proportions to each other, see <hi rend="smallcaps">Planets</hi>, S<hi rend="smallcaps">EMIDIAMETER</hi>, and <hi rend="smallcaps">Solar System.</hi></p><p><hi rend="smallcaps">Diameter</hi> <hi rend="italics">of a Column,</hi> is its thickness just above the base. From this the module is taken, which measures all the other parts of the column.</p><p><hi rend="smallcaps">Diameter</hi> <hi rend="italics">of the Diminution,</hi> is that taken at the top of the shaft.</p><p><hi rend="smallcaps">Diameter</hi> <hi rend="italics">of the Swelling,</hi> is that taken at the height of one third from the base.</p></div2></div1><div1 part="N" n="DIAPASON" org="uniform" sample="complete" type="entry"><head>DIAPASON</head><p>, a musical interval, otherwise called an octave, or eighth: so called because it contains all the possible diversities of sound.</p><p>If the lengths of two strings be to each other as 1 to 2, the tensions being equal, their tones will produce an octave.</p></div1><div1 part="N" n="DIAPENTE" org="uniform" sample="complete" type="entry"><head>DIAPENTE</head><p>, is the perfect fifth or second of the concords, making an octave with the Diatessaron. The lengths of the chords are as 3 to 2.</p><p>DIAPHONOUS Body or Medium, one that is translucent, or through which the rays of light easily pass; as water, air, glass, talc, fine porcelain, &c. See <hi rend="smallcaps">Transparency.</hi></p></div1><div1 part="N" n="DIAPHONICS" org="uniform" sample="complete" type="entry"><head>DIAPHONICS</head><p>, is sometimes used for the science of refracted sound, as it passes through different mediums.</p></div1><div1 part="N" n="DIASTYLE" org="uniform" sample="complete" type="entry"><head>DIASTYLE</head><p>, a sort of edifice, in which the pillars stand at such a distance from one another, that three diameters of their thickness are allowed for the intercolumnation.</p></div1><div1 part="N" n="DIATESSARON" org="uniform" sample="complete" type="entry"><head>DIATESSARON</head><p>, is the perfect fourth; or a musical interval, consisting of one greater tone, one lesser, and one greater semitone. The lengths of strings to sound the diatessaron, are as 3 to 4.</p></div1><div1 part="N" n="DIATONIC" org="uniform" sample="complete" type="entry"><head>DIATONIC</head><p>, a term signifying the ordinary sort of music, which proceeds by tones or degrees, both ascending and descending. It contains or admits only the greater and lesser tone, and the greater semitone.</p></div1><div1 part="N" n="DIESIS" org="uniform" sample="complete" type="entry"><head>DIESIS</head><p>, a division of a tone, less than a semitone; or an interval consisting of a lesser, or imperfect semitone. The Diesis is the smallest and softest change, or <pb n="378"/><cb/> inflexion, of the voice imaginable. It is also called a <hi rend="italics">feint,</hi> and is expressed by a St. Andrew's Cross, or saltier,</p></div1><div1 part="N" n="DIFFERENCE" org="uniform" sample="complete" type="entry"><head>DIFFERENCE</head><p>, is the excess by which one magnitude or quantity exceeds another. When a less quantity is subtracted from a greater, the remainder is otherwise called difference.</p><p><hi rend="italics">Ascensional</hi> <hi rend="smallcaps">Difference.</hi> See <hi rend="smallcaps">Ascensional.</hi></p><p><hi rend="smallcaps">Difference</hi> <hi rend="italics">of Longitude,</hi> of two places, is an arch of the equator contained between the meridians of those two places, or the measure of the angle formed by their meridians.</p></div1><div1 part="N" n="DIFFERENTIAL" org="uniform" sample="complete" type="entry"><head>DIFFERENTIAL</head><p>, an indefinitely small quantity, part, or difference. By some, the Differential is considered as infinitely small, or less than any assignable quantity; and also as of the same import as fluxion.</p><p>It is called a Differential, or Differential Quantity, because often considered as the difference between two quantities; and as such it is the foundation of the Differential Calculus. Newton used the term <hi rend="italics">moment</hi> in a like sense, as being the momentary increase or decrease of a variable quantity. M. Leibnitz and others call it also an infinitesimal.</p><p><hi rend="smallcaps">Differential</hi> <hi rend="italics">of the</hi> 1<hi rend="italics">st,</hi> 2<hi rend="italics">d,</hi> 3<hi rend="italics">d, &c degree.</hi> See <hi rend="italics">Differentio</hi>-<hi rend="smallcaps">Differential.</hi></p><p><hi rend="smallcaps">Differential</hi> <hi rend="italics">Calculus,</hi> or <hi rend="italics">Method,</hi> is a method of differencing quantities. See <hi rend="smallcaps">Differential</hi> <hi rend="italics">Method,</hi> <hi rend="smallcaps">Calculus</hi>, and <hi rend="smallcaps">Fluxions.</hi></p><p><hi rend="smallcaps">Differentio-Differential</hi> <hi rend="italics">Calculus,</hi> is a method of differencing differential quantities.</p><p>As the sign of a differential is the letter <hi rend="italics">d</hi> prefixed to the quantity, as <hi rend="italics">dx</hi> the differential of <hi rend="italics">x</hi>; so that of a differential of <hi rend="italics">dx</hi> is <hi rend="italics">ddx,</hi> and the differential of <hi rend="italics">ddx</hi> is <hi rend="italics">dddx,</hi> &c; similar to the fluxions <hi rend="italics">x<hi rend="sup">.</hi>, x<hi rend="sup">..</hi>, x<hi rend="sup">∴</hi>,</hi> &c.</p><p>Thus we have degrees of differentials. The differential of an ordinary quantity, is a differential of the first order or degree, as <hi rend="italics">dx</hi>; that of the 2d degree is <hi rend="italics">ddx;</hi> that of the 3d degree, <hi rend="italics">dddx,</hi> &c. The rules for differentials, are the very same as those for fluxions. See <hi rend="smallcaps">Fluxions.</hi></p><div2 part="N" n="Differential" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Differential</hi></head><p>, in Logarithms. Kepler calls the logarithms of tangents, <hi rend="italics">differentiales;</hi> which we usually call <hi rend="italics">artificial tangents.</hi></p><p><hi rend="smallcaps">Differential</hi> <hi rend="italics">Equation,</hi> is an equation involving or containing differential quantities; as the equation . Some mathematicians, as Stirling, &c, have also applied the term differential equation in another sense, to certain equations defining the nature of series.</p><p><hi rend="smallcaps">Differential</hi> <hi rend="italics">Method,</hi> a method of finding quantities by means of their successive differences.</p><p>This method is of very general use and application, but especially in the construction of tables, and the summation of series, &c. This method was first used, and the rules of it laid down, by Briggs, in his Construction of Logarithms and other Numbers, much the same as they were afterwards taught by Cotes, in his Constructio Tabularum per Differentias; as I have shewn in the Introduction to my Logs. pa. 69 & seq. See Briggs's Arithmetica Logarithmica, cap. 12 and 13, and his Trigonometria Britannica.</p><p>The method was next treated in another form by Newton in the 5th Lemma of the 3d book of his Principia, and in his Methodus Differentialis, published <cb/> by Jones in 1711, with the other tracts of Newton. This author here treats it as a method of describing a curve of the parabolic kind, through any given number of points. He distinguishes two cases of this problem; the first, when the ordinates drawn from the given points to any line given in position, are at equal distances from one another; and the 2d, when these ordinates are not at equal distances. He has given a solution of both cases, at first without demonstration, which was afterwards supplied by himself and others: see his Methodus Differentialis above mentioned; and Stirling's Explanation of the Newtonian Differential Method, in the Philos. Trans. N° 362; Cotes, De Methodo Differentiali Newtoniana, published with his Harmonia Mensurarum; Herman's Phoronomia; and Le Seur & Jacquier, in their Commentary on Newton's Principia. It may be observed, that the methods there demonstrated by some of these authors extend to the description of any algebraic curve through a given number of points, which Newton, writing to Leibnitz, mentions as a problem of the greatest use.</p><p>By this method, some terms of a series being given, and conceived as placed at given intervals, any intermediate term may be found nearly; which therefore gives a method for interpolations. Briggs's Arith. Log. ubi supra; Newton Meth. Differ. prop. 5; Stirling, Methodus Differentialis.</p><p>Thus also may any curvilinear figure be squared nearly, having some few of its ordinates. Newton, ibid. prop 6; Cotes De Method. Differ.; Simpson's Mathematical Dissert. pa. 115. And thus may mathematical tables be constructed by interpolation: Briggs, ibid. Cotes Canonotechnia.</p><p>The successive differences of the ordinates of parabolic curves, becoming ultimately equal, and the intermediate ordinate required, being determined by these differences of the ordinates, is the reason for the name Differential Method.</p><p>To be a little more particular.—The first case of Newton's problem amounts to this: A series of numbers, placed at equal intervals, being given, to find any intermediate number of that series, when its interval or distance from the first term of the series is given.—— Subtract each term of the series from the next following term, and call the remainders first differences; then subtract in like manner each of these differences from the next following one, calling these remainders 2d differences; again, subtract each 2d difference from the next following, for the 3d differences; and so on: then if A be the 1st term of the series, <hi rend="italics">d</hi>′ the first of the 1st differences, <hi rend="italics">d</hi>″ the first of the 2d differences, <hi rend="italics">d</hi>‴ the first of the 3d differences, &c; and if <hi rend="italics">x</hi> be the interval or distance between the first term of the series and any term sought, T, that is, let the number of terms from A to T, both included, be = <hi rend="italics">x</hi>+1; then will the term sought, T, be = A+(<hi rend="italics">x</hi>/1)<hi rend="italics">d</hi>′+((<hi rend="italics">x</hi>/1).((<hi rend="italics">x</hi>-1)/2))<hi rend="italics">d</hi>″+((<hi rend="italics">x</hi>/1).((<hi rend="italics">x</hi>-1)/2).((<hi rend="italics">x</hi>-2)/3))<hi rend="italics">d</hi>‴ &c.</p><p>Hence, if the differences of any order become equal, that is, if any of the diffs. <hi rend="italics">d</hi>″, <hi rend="italics">d</hi>‴, &c, become = 0, the above series will give a finite expression for T the term sought; it being evident, that the series must terminate when any of the diffs. <hi rend="italics">d</hi>″, <hi rend="italics">d</hi>‴, &c, become = 0. <pb n="379"/><cb/></p><p>It is also evident that the co-efficients <hi rend="italics">x</hi>/1, (<hi rend="italics">x</hi>/1).((<hi rend="italics">x</hi>-1)/2), &c, of the differences, are the same as to the terms of the binomial theorem.</p><p><hi rend="italics">For ex.</hi> Suppose it were required to find the log. tangent of 5′ 1″ 12‴ 24′′′′, or 5′ 1″ 62/300, or 5′ 1″ .2066 &c.</p><p>Take out the log. tangents to several minutes and seconds, and take their first and second differences, as below: <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Tang.</cell><cell cols="1" rows="1" rend="align=center" role="data">d′</cell><cell cols="1" rows="1" rend="align=center" role="data">d″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">5′</cell><cell cols="1" rows="1" role="data">0″</cell><cell cols="1" rows="1" role="data">7.1626964</cell><cell cols="1" rows="1" role="data">14453</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="rowspan=4" role="data">-48</cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7.1641417</cell><cell cols="1" rows="1" role="data">14404</cell><cell cols="1" rows="1" role="data">-49</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(7)">}</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7.1655821</cell><cell cols="1" rows="1" role="data">14357</cell><cell cols="1" rows="1" role="data">-47</cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7.1670178</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>Here A = 7.1641417; <hi rend="italics">x</hi> = 62/300; <hi rend="italics">d</hi>′ = 14404; and the mean 2d difference <hi rend="italics">d</hi>″ = -48. Hence <table><row role="data"><cell cols="1" rows="1" role="data">A</cell><cell cols="1" rows="1" rend="align=right" role="data">7.1641417</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">xd</hi>′</cell><cell cols="1" rows="1" rend="align=right" role="data">2977</cell></row><row role="data"><cell cols="1" rows="1" role="data">(<hi rend="italics">x</hi>/1).((<hi rend="italics">x</hi>-1)/1)<hi rend="italics">d</hi>″</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Theref. the tang. of 5′ 1″ 12‴ 24′′′′ is</cell><cell cols="1" rows="1" rend="align=right" role="data">7.1644398</cell></row></table></p><p>Hence may be deduced a method of finding the sums of the terms of such a series, calling its terms A, B, C, D, &c. For, conceive a new series having its 1st term = 0, its 2d = A, its , its , its , and so on; then it is plain that assigning one term of this series, is finding the sum of all the terms A, B, C, D, &c. Now since these terms are the differences of the sums, 0, A, A+B, A+B+C, &c; and as some of the differences of A, B, C, &c, are = 0 by supposition; it follows that some of the differences of the sums will be = 0; and since in the series A + (<hi rend="italics">x</hi>/1)<hi rend="italics">d</hi>′+(<hi rend="italics">x</hi>/1).((<hi rend="italics">x</hi>-1)/2)<hi rend="italics">d</hi>″ &c, by which a term was assigned, A represented the 1st term; <hi rend="italics">d</hi>′ the 1st of the 1st differences, and <hi rend="italics">x</hi> the interval between the first term and the last; we are to write 0 instead of A, A instead of <hi rend="italics">d</hi>′, <hi rend="italics">d</hi>′ instead of <hi rend="italics">d</hi>″, <hi rend="italics">d</hi>″ instead of <hi rend="italics">d</hi>‴, &c, also <hi rend="italics">x</hi>+1 instead of <hi rend="italics">x</hi>; which being done, the series expressing the sums will be 0 + ((<hi rend="italics">x</hi>+1)/1)A + ((<hi rend="italics">x</hi>+1)/1).(<hi rend="italics">x</hi>/2)<hi rend="italics">d</hi>′ + ((<hi rend="italics">x</hi>+1)/1).(<hi rend="italics">x</hi>/2).((<hi rend="italics">x</hi>-1)/3)<hi rend="italics">d</hi>″, &c. Or, if the real number of terms of the lines be called <hi rend="italics">z,</hi> that is, if , or , the sum of the series will be &c. See De Moivre's Doct. of Chances, pa. 59, 60; or his Miscel. Analyt. pa. 153; or Simpson's Essays, pa. 95.</p><p><hi rend="italics">For ex.</hi> to find the sum of six terms of the series of squares 1 + 4 + 9 + 16 + 25 + 36, of the natural numbers. <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Terms</cell><cell cols="1" rows="1" role="data">d′</cell><cell cols="1" rows="1" role="data">d″</cell><cell cols="1" rows="1" role="data">d‴</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> <cb/></p><p>Here A = 1, <hi rend="italics">d</hi>′ = 3, <hi rend="italics">d</hi>″ = 2, <hi rend="italics">d</hi>‴ &c = 0, and <hi rend="italics">z</hi> = 6; therefore the sum is 6 + (6/1).(5/2).3 + (6/1).(5/3).(4/3).2 = 6 + 45 + 40 = 91 the sum required, viz. of 1 + 4 + 9 + 16 + 25 + 36.</p><p>A variety of examples may be seen in the places above cited, or in Stirling's Methodus Differentialis, &c.</p><p>As to the Differential method, it may be observed, that though Newton and some others have treated it as a method of describing an algebraic curve, at least of the parabolic kind, through any number of given points; yet the consideration of curves is not at all essential to it, though it may help the imagination. The description of a parabolic curve through given points, is the same problem as the finding of quantities from their given differences, which may always be done by Algebra, by the resolution of simple equations. See Stirling's Method. Differ. pa. 97. This ingenious author has treated very fully of the differential method, and shewn its use in the solution of some very difficult problems. See also <hi rend="smallcaps">Series.</hi></p><p><hi rend="smallcaps">Differential</hi> <hi rend="italics">Scale,</hi> in Algebra, is used for the scale of relation subtracted from unity. See <hi rend="italics">Recurring</hi> <hi rend="smallcaps">Series.</hi></p></div2></div1><div1 part="N" n="DIFFRACTION" org="uniform" sample="complete" type="entry"><head>DIFFRACTION</head><p>, a term first used by Grimaldi, to denote that property of the rays of light, which others have called Inflection; the discovery of which is attributed by some to Grimaldi, and by others to Dr. Hook.</p></div1><div1 part="N" n="DIGBY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DIGBY</surname> (<hi rend="italics">Sir</hi> <foreName full="yes"><hi rend="smallcaps">Kenelm</hi></foreName>)</persName></head><p>, a famous English philosopher, was born at Gothurst in Buckinghamshire, 1603. He was descended of an ancient family: his great grandfather, with six of his brothers, fought valiantly at Bosworth-field on the side of Henry the 7th, against Richard the 3d. His father, Everard, engaged in the gunpowder plot against James the 1st, for which he was beheaded. His son, however, was restored to his estate; and had afterwards several appointments under king Charles the 1st. He granted him letters of reprisal against the Venetians, from whom he took several prizes with a small fleet which he commanded. He fought the Venetians near the port of Scanderoon, and bravely made his way through them with his booty.</p><p>In the beginning of the civil wars, he exerted himself greatly in the king's cause. He was afterwards imprisoned, by order of the parliament; but was set at liberty in 1643. He afterward compounded for his estate; but being banished from England, he retired to France, and was sent on two embassies to Pope Innocent the 10th, from the queen, widow to Charles the 1st, whose chancellor he then was. On the restoration of Charles the 2d, he returned to London; where he died in 1665, at 62 years of age.</p><p>Digby was a great lover of learning, and translated several authors into English, as well as published several works of his own; as, 1. <hi rend="italics">Observations upon Dr. Brown's Religio Medici,</hi> 1643.—2. <hi rend="italics">Observations on part of Spenser's Fairy Queen,</hi> 1644.—3. <hi rend="italics">A Treatise of the Nature of Bodies,</hi> 1644.—4. <hi rend="italics">A Treatise declaring the Operations and Nature of Man's Soul, out of which the Immortality of reasonable souls is evinced:</hi> works that discover great penetration and extensive knowledge.</p><p>He applied much to chemistry; and found out seve- <pb n="380"/><cb/> ral useful medicines, which he distributed with a liberal hand. He particularly distinguished himself by his sympathetic powder for the cure of wounds at a distance; his discourse concerning which made great noise for a while. He held several conferences with Des Cartes, about the nature of the soul, and the principles of things. At the beginning of the Royal Society, he became a distinguished member, being one of the first council. And he had at his own house regular levees or meetings of learned men, to improve themselves in knowledge, by conversing with one another.</p><p>This eminent person was, for the early pregnancy of his talents, and his great proficiency in learning, compared to the celebrated Picus de Mirandola, who was one of the wonders of human nature. Yet his knowledge, though various and extensive, probably appeared greater than it really was; as he had all the powers of elocution and address to recommend it. He knew how to shine in a circle, either of ladies or philosophers; and was as much attended to when he spoke on the most trivial subjects, as when he spoke on the most important. It has been said that one of the princes of Italy, who had no child, was desirous that his princess should bring him a son by Sir Kenelm, whom he esteemed a just model of perfection.</p></div1><div1 part="N" n="DIGGES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DIGGES</surname> (<foreName full="yes"><hi rend="smallcaps">Leonard</hi></foreName>)</persName></head><p>, a considerable mathematician in the 16th century, was descended from an ancient family, and born at Digges-court in the parish of Barham in Kent; but in what year is not known; and died about the year 1574. He was educated for some time at Oxford, where he laid a good foundation of learning. Retiring from thence, he prosecuted his studies, and became an excellent mathematician, a skilful architect, and an expert surveyor of land, &c. He composed several books: as, 1. <hi rend="italics">Tectonicum: briefly shewing the exact Measuring, and speedy Reckoning of all manner of Lands, Squares, Timber, Stones, Steeples, &c;</hi> 1556, 4to. Augmented and published again by his son Thomas Digges, in 1592; and also reprinted in 1647. —2. <hi rend="italics">A Geometrical Practical Treatise, named Pantometria, in three books.</hi> This he left in manuscript; but after his death, his son supplied such parts of it as were obscure and imperfect, and published it in 1591, folio; subjoining, “A Discourse Geometrical of the sive regular and Platonic bodies, containing sundry theoretical and practical propositions, arising by mutual conference of these solids, Inseription, Circumscription, and Transformation.”—3. <hi rend="italics">Prognostication Everlasting of right good effect: or Choice Rules to judge the Weather by the Sun, Moon, and Stars, &c;</hi> in 4to. 1555, 1556, and 1564: corrected and augmented by his son, with divers general tables, and many compendious rules, in 4to, 1592.</p></div1><div1 part="N" n="DIGGES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DIGGES</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>)</persName></head><p>, only son of Leonard Digges, after a liberal education from his tenderest years, went and studied for some time at Oxford; and by the improvements he made there, and the subsequent instructions of his learned father, became one of the best mathematicians of his age. When queen Elizabeth sent some forces to assist the oppressed inhabitants of the Netherlands, Mr. Digges was appointed muster-master general of them; by which he became well skilled in military affairs; as his writings afterward shewed. He died in 1595.</p><p>Mr. Digges, beside revising, correcting, and enlarg- <cb/> ing some pieces of his father's, already mentioned, wrote and published the following learned works himself: viz, 1. <hi rend="italics">Alæ sive Scalæ Mathematicæ: or Mathematical Wings or Ladders,</hi> 1573, 4to. A book which contains several demonstrations for finding the parallaxes of any comet, or other celestial body, with a correction of the errors in the use of the radius astronomicus.—2. <hi rend="italics">An Arithmetical Military Treatise, containing so much of Arithmetic as is necessary towards military disciplinc,</hi> 1579, 4to.—3. <hi rend="italics">A Geometrical Treatise, named Stratioticos, requisite for the perfection of Soldiers,</hi> 1579, 4to. This was begun by his father, but finished by himself. They were both reprinted together in 1590, with several additions and amendments, under this title: “An Arithmetical Warlike Treatise, named Stratioticos, compendiously teaching the science of Numbers, as well in Fractions as Integers, and so much of the Rules and Equations Algebraical, and art of Numbers Cossical, as are requisite for the profession of a souldier. Together with the Moderne militaire discipline, offices, lawes, and orders in every well-governed campe and armie, inviolably to be observed.” At the end of this work there are two pieces; the first, “A briefe and true report of the proceedings of the Earle of Leycester, for the reliefe of the towne of Sluce, from his arrival at Vlishing, about the end of June 1587, untill the surrendrie thereof 26 Julii next ensuing. Whereby it shall plainelie appear, his excellencie was not in anie fault for the losse of that towne:” the second, “A briefe discourse what orders were best for repulsing of foraine forces, if at any time they should invade us by sea in Kent, or elsewhere.”—4. <hi rend="italics">A perfect Description of the Celestial Orbs, according to the most ancient doctrine of the Pythagoreans, &c.</hi> This was placed at the end of his father's “Prognostication Everlasting, &c.” printed in 1592, 4to.—5. <hi rend="italics">A humble Motive for Association to maintain the religion established,</hi> 1601, 8vo. To which is added, <hi rend="italics">his Letter to the same purpose to the archbishops and bishops of England.</hi>—6. <hi rend="italics">England's Defence: or, A Treatise concerning Invasion.</hi> This is a tract of the same nature with that printed at the end of his Stratioticos, and called, A briefe Discourse, &c. It was written in 1599, but not published till 1686.—7. <hi rend="italics">A Letter printed before Dr. John Dee's Parallaticæ Commentationis praxeosque nucleus quidam,</hi> 1573, 4to.—Beside these, and his <hi rend="italics">Nova Corpora,</hi> he left several mathematical treatises ready for the press; which, by reason of lawsuits and other avocations, he was hindered from publishing.</p><p>If our author was great in himself, he was not less so in his son, Sir Dudley Digges, so celebrated as a politician and elegant writer.</p></div1><div1 part="N" n="DIGIT" org="uniform" sample="complete" type="entry"><head>DIGIT</head><p>, in Arithmetic, one of the ten characters 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, by means of which all numbers are expressed.</p><div2 part="N" n="Digit" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Digit</hi></head><p>, in Astronomy, is the measure by which the part of the luminaries in eclipses is estimated, being the 12th part of the diameter of the luminary. Thus, an eclipse is said to be of 10 digits, when 10 parts out of 12 of the diameter are in the eclipsed part; when the whole of the luminary is just all covered, the digits eclipsed are just 12; and when the luminary is more than covered, as often happens in lunar eclipses, then more than 12 digits are said to be eclipsed: Thus, if the diameter or breadth of the earth's shadow, where <pb n="381"/><cb/> the moon passes through, be equal to one diameter and a half of the moon, then 18° or digits are said to be eclipsed.</p><p>These digits are by Wolsius, and some others, called <hi rend="italics">digiti ecliptici.</hi></p><p><hi rend="smallcaps">Digit</hi> is also a measure taken from the breadth of the finger; being estimated at 3-4ths of an inch, and equal to 4 grains of barley, laid breadth-ways, so as to touch each other.</p></div2></div1><div1 part="N" n="DILATATION" org="uniform" sample="complete" type="entry"><head>DILATATION</head><p>, a motion of the parts of a body by which it expands, or opens itself, so as to occupy a greater space.</p><p>Many authors confound dilatation with rarefaction; but the more accurate writers distinguish between them; defining dilatation as the expansion of a body into a greater bulk, by its own elastic power; and rarefaction, the like expansion produced by means of heat.</p><p>The moderns have observed, that bodies which, after being compressed, and again left at liberty, restore themselves perfectly, do endeavour to dilate themselves with the same force by which they are compressed; and accordingly they sustain a force, and raise a weight equal to that with which they are compressed.</p><p>Again, bodies, in dilating by their elastic power, exert a greater force at the beginning of their dilatation, than towards the end; as being at first more compressed; and the greater the compression, the greater the elastic power and endeavour to dilate. So that these three, the compressing power, the compression, and the elastic power, are always equal.</p><p>Finally, the motion by which compressed bodies restore themselves, is usually accelerated: thus, when compressed air begins to restore itself, and dilate into a greater space, it is still compressed; and consequently a new impetus is still impressed upon it, from the dilatative cause; and the former remaining, with the increase of the cause, the effect, that is the motion and velocity, must be increased likewise. Indeed it may happen, that where the compression is only partial, the motion of dilatation shall not be accelerated, but retarded <*>s is evident in the compression of a spunge, soft bread, gauze, &c.</p><p>DILUTE. To dilute a body, is to render it liquid; or, if it were liquid before, to render it more so, by the addition of a thinner to it.</p></div1><div1 part="N" n="DIMENSION" org="uniform" sample="complete" type="entry"><head>DIMENSION</head><p>, the extension of a body, considered as measurable. Hence, as we conceive a body extended, and measurable in length, breadth, and depth, dimension is considered as threefold, viz, length, breadth, and thickness. So a line has one dimension only, viz length; a supersicies two, length and breadth; and a body or solid has three, viz, length, breadth, and thickness.</p><p><hi rend="smallcaps">Dimension</hi> is also particularly used with regard to the powers of quantities in equations. Thus, in a simple equation, , the unknown quantity is only of one dimension; in a quadratic equation, , it is of two dimensions; in a cubic, , it is of three dimensions; and so on.</p></div1><div1 part="N" n="DIMETIENT" org="uniform" sample="complete" type="entry"><head>DIMETIENT</head><p>, has sometimes been used for diameter.</p><p>DIMINISHED <hi rend="italics">Angle,</hi> a term in Fortification. See <hi rend="smallcaps">Angle.</hi> <cb/></p></div1><div1 part="N" n="DIMINUTION" org="uniform" sample="complete" type="entry"><head>DIMINUTION</head><p>, in Music, is the abating something of the full value or quantity of any note.</p><div2 part="N" n="Diminution" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Diminution</hi></head><p>, in Architecture, is a contraction of the upper part of a column, by which its diameter is made less than that of the lower part.</p></div2></div1><div1 part="N" n="DINOCRATES" org="uniform" sample="complete" type="entry"><head>DINOCRATES</head><p>, a celebrated ancient architect of Macedonia, of whom several extraordinary things are related. He was taken, by Alexander the Great, into Egypt, where he employed him in marking out and building the city of Alexandria. He formed a design, in which Mount Athos was to be laid out into the form of a man, in whose left hand were designed the walls of a great city, and all the rivers of the mount flowing into his right, and from thence into the sea. Another memorable instance of Dinocrates's architectonic skill, is his restoring and building, in a more august and magnificent manner than before, the celebrated temple of Diana at Ephesus, after Herostratus, for the sake of immortalizing his name, had destroyed it by fire. A third instance, more extraordinary and wonderful than either of the former, is related by Pliny in his Natural History; who says he had formed a scheme, by building the dome of the temple of Arsinoë at Alexandria of loadstone, to make her image all of iron to hang in the middle of it, as if it were in the air; but the king's death, and his own, prevented the execution or attempt of this project.</p></div1><div1 part="N" n="DIONYSIUS" org="uniform" sample="complete" type="entry"><head>DIONYSIUS</head><p>, the <hi rend="italics">Periegetic,</hi> an ancient geographer and poet. Pliny says, he was a native of the Persian Alexandria, afterwards called Antioch, and at last Charrax; that he was sent by Augustus, to survey the eastern part of the world. Dionysius wrote a great number of pieces, enumerated by Suidas and his commentator Eustathius: but his <hi rend="italics">Periegesis,</hi> or <hi rend="italics">Survey of the World,</hi> is the only one now extant; which may be well esteemed one of the most exact systems of ancient geography, since Pliny himself proposed it as his pattern.</p></div1><div1 part="N" n="DIONYSIAN" org="uniform" sample="complete" type="entry"><head>DIONYSIAN</head><p>, or <hi rend="italics">Victorian Period.</hi> See <hi rend="smallcaps">Period.</hi></p></div1><div1 part="N" n="DIOPHANTUS" org="uniform" sample="complete" type="entry"><head>DIOPHANTUS</head><p>, a celebrated mathematician of Alexandria, who has been reputed to be the inventor of Algebra; at least his is the earliest work extant on that science. It is not certain when Diophantus lived. Some have placed him before Christ, and some after, in the reigns of Nero and the Antonines; but all with equal uncertainty. It seems he is the same Diophantus who wrote the <hi rend="italics">Canon Astronomicus,</hi> which Suidas says was commented on by the celebrated Hypatia, daughter of Theon of Alexandria. His reputation must have been very high among the ancients, since they ranked him with Pythagoras and Euclid in mathematical learning. Bachet, in his notes upon the 5th book <hi rend="italics">De Arithmeticis,</hi> has collected, from Diophantus's epitaph in the Anthologia, the following circumstances of his life; namely, that he was married when he was 33 years old, and had a son born 5 years after; that this son died when he was 42 years of age, and that his father did not survive him above 4 years; from which it appears, that Diophantus was 84 years old when he died.</p><p>DIOPHANTUS wrote 13 books of Arithmetic, or Algebra, which Regiomontanus in his preface to Alfraganus, tells us, are still preserved in manuscript in the Vatican library. Indeed Diophantus himself tells us <pb n="382"/><cb/> that his work consisted of 13 books, viz, at the end of his address to Dionysius, placed at the beginning of the work; and from hence Regiomontanus might be led to say the 13 books were in that library. No more than 6 whole books, with part of a seventh, have ever been published; and I am of opinion there are no more in being; indeed Bombelli, in the preface to his Algebra, written 1572, says there were but 6 of the books then in the library, and that he and another were about a translation of them.</p><p>Those 6 books, with the imperfect 7th, were first published at Basil by Xylander in 1575, but in a Latin version only, with the Greek scholia of Maximus Planudes upon the two first books, and observations of his own. The same books were afterwards published in Greek and Latin at Paris in 1621, by Bachet, an ingenious and learned Frenchman, who made a new Latin version of the work, and enriched it with very learned commentaries. Bachet did not entirely neglect the notes of Xylander in his edition, but he treated the scholiast Planudes with the utmost contempt. He seems to intimate, in what he says upon the 28th question of the 2d book, that the 6 books which we have of Diophantus, may be nothing more than a collection made by some novice, of such propositions as he judged proper, out of the whole 13: but Fabricius thinks there is no just ground for such a supposition.</p><p>DIOPHANTINE <hi rend="italics">Problems,</hi> are certain questions relating to square and cubic numbers, and to rightangled triangles, &c; the nature of which were first and chiesly treated of by Diophantes, in his Arithmetic, or rather Algebra.</p><p>In these questions, it is chiefly intended to find commensurable numbers to answer indeterminate problems; which often bring out an infinite number of incommensurable quantities. For example, let it be proposed to find a right-angled triangle, whose three sides <hi rend="italics">x, y, z</hi> are expressed by rational numbers; from the nature of the figure it is known that , where <hi rend="italics">z</hi> denotes the hypothenuse. Now it is plain that <hi rend="italics">x</hi> and <hi rend="italics">y</hi> may also be so taken, that <hi rend="italics">z</hi> shall be irrational; for if <hi rend="italics">x</hi> = 1, and <hi rend="italics">y</hi> = 2, then is <hi rend="italics">z</hi> = √5.</p><p>Now the art of resolving such problems, consists in ordering the unknown quantity or quantities, in such a manner, that the square or higher power may vanish out of the equation, and then by means of the unknown quantity in its first dimension, the equation may be resolved without having recourse to incommensurables. For ex. in the equation above, , suppose , then is , out of which equation <hi rend="italics">x</hi><hi rend="sup">2</hi> vanishes, and then it is , which gives . Hence, assuming <hi rend="italics">y</hi> and <hi rend="italics">u</hi> equal to any numbers at pleasure, the three sides of the triangle will be <hi rend="italics">y,</hi> (<hi rend="italics">y</hi><hi rend="sup">2</hi> - <hi rend="italics">u</hi><hi rend="sup">2</hi>)/2<hi rend="italics">u</hi>, and (<hi rend="italics">y</hi><hi rend="sup">2</hi> + <hi rend="italics">u</hi><hi rend="sup">2</hi>)/2<hi rend="italics">u</hi>, which are all rational whenever <hi rend="italics">y</hi> and <hi rend="italics">u</hi> are rational. For ex. if <hi rend="italics">y</hi> = 3, and <hi rend="italics">u</hi> = 1, then , and <hi rend="italics">x</hi> + <hi rend="italics">u</hi> or . It is evident that this problem admits of infinite numbers of solutions, as <hi rend="italics">y</hi> or <hi rend="italics">u</hi> may be assumed infinitely various. See <hi rend="smallcaps">Algebra</hi>, and <hi rend="smallcaps">Diophantus.</hi></p><p>Abundant information on this sort of problems may <cb/> be found in the writings of a great many authors, particularly Fermat, Bachet, Ozanam, Kersey, Saunderson, Euler, &c.</p></div1><div1 part="N" n="DIOPTER" org="uniform" sample="complete" type="entry"><head>DIOPTER</head><p>, or <hi rend="smallcaps">Dioptra</hi>, the same with the index or alhidade of an astrolabe, or other such instrument.</p><p>DIOPTRA was an instrument invented by Hipparchus, which served for several uses; as, to level water-courses; to take the height of towers, or places at a distance; to determine the places, magnitudes, and distances of the planets, &c.</p></div1><div1 part="N" n="DIOPTRICS" org="uniform" sample="complete" type="entry"><head>DIOPTRICS</head><p>, called also <hi rend="italics">anaclastics,</hi> is the doctrine of refracted vision; or that part of Optics which explains the effects of light as refracted by passing through different mediums, as air, water, glass, &c, and especially lenses.</p><p>Dioptrics is one of the most useful and pleasant of all the human sciences; bringing the remotest objects near hand, enlarging the smallest objects so as to shew their minute parts, and even giving sight to the blind; and all this by the simple means of the attractive power in glass and water, causing the rays of light in their passage through them to alter their course according to the different substances of the medium; whence it happens, that the object seen through them, do, in appearance, alter their magnitude, distance, and situation.</p><p>The ancients have treated of direct and reflected vision; but what we have of refracted vision, is very imperfect. J. Baptista Porta wrote a treatise on refraction, in 9 books, but without any great improvement. Kepler was the first who succeeded in any great degree, on this subject; having demonstrated the properties of spherical lenses very accurately, in a treatise first published anno 1611. After Kepler, Galileo gave somewhat of this doctrine in his Letters; as also an Examination of the Preface of Johannes Pena upon Euclid's Optics, concerning the use of optics in astronomy. Des Cartes also wrote a treatise on Dioptrics, commonly annexed to his Principles of Philosophy, which is one of his best works; in which the true manner of vision is more distinctly explained than by any former writer, and in which is contained the true law of refraction, which was found out by Snell, though the name of the inventer is suppressed: here are also laid down the properties of elliptical and hyperbolical lenses, with the practice of grinding glasses. Dr. Barrow has treated on Dioptrics in a very elegant manner, though rather too briefly, in his Optical Lectures, read at Cambridge. There are also Huygens's Dioptrics, an excellent work of its kind. Molyneux's Dioptrics, a work rather heavy and dull. Hartsoeker's Essai de Dioptrique. Cherubin's Dioptrique Oculaire, et La Vision Parfaite. David Gregory's Elements of Dioptrics. Traber's Nervus Opticus. Zahn's Oculus Artificiah Teledioptricus. Dr. Smith's Optics, a complete work of its kind. Wolsius's Dioptrics, contained in his Elementa Matheseos Universalis. But over all, the Treatise on Optics, and the Optical Lectures of Newton, in whose experiments are contained far more discoveries than in all the former writers. Lastly, this science was perfected by Dollond's discovery of the acromatic glasses, by which the colours are obviated in refracting telescopes.</p><p>The laws of Dioptrics see delivered under the article <hi rend="smallcaps">Refraction, Lens</hi>, &c; and the application of it in the construction of telescopes, miscroscopes, and other <pb n="383"/><cb/> dioptrical instruments, under the articles <hi rend="smallcaps">Telescope</hi>, and <hi rend="smallcaps">Microscope.</hi></p><p>DIP <hi rend="italics">of the Horizon.</hi> See <hi rend="smallcaps">Depression.</hi></p><p><hi rend="smallcaps">Dipping</hi>-<hi rend="italics">Needle,</hi> or <hi rend="italics">Inclinatory Needle,</hi> a magnetical needle, so hung, as that, instead of playing horizontally, and pointing out north and south, one end dips, or inclines to the horizon, and the other points to a certain degree of elevation above it.</p><p>The inventor of the Dipping-needle was one Robert Norman, a compass-maker at Ratcliffe, about the year 1580; this is not only testified by his own account, in his New Attractive, but also by Dr. Gilbert, Mr. William Burrowes, Mr. Henry Bond, and other writers of that time, or soon after it. The occasion of the discovery he himself relates, viz, that it being his custom to finish, and hang the needles of his compasses, before he touched them, he always found that, immediately after the touch, the north point would dip or decline downward, pointing in a direction under the horizon; so that, to balance the needle again, he was always forced to put a piece of wax on the south end, as a counterpoise. The constancy of this effect led him, at length, to observe the precise quantity of the dip, or to measure the greatest angle which the needle would make with the horizon. This, in the year 1576, he found at London was 71° 50′. It is not quite certain whether the dip varies, as well as the horizontal direction, in the same place. Mr. Graham made a great many experiments with the dipping-needle in 1723, and found the dip between 74 and 75 degrees. Mr. Nairne, in 1772, found it somewhat above 72°. And by many observations made since that time at the Royal Society, the medium quantity is 72°1/2. The trifling difference between the first observations of Norman, and the last of Mr. Nairne and the Royal Society, lead to the opinion that the dip is unalterable; and yet it may be difficult to account for the great difference between these and Mr. Graham's numbers, considering the well-known accuracy of that ingenious gentleman. Philos. Trans. vol. 45, pa. 279, vol. 62, pa. 476. vol. 69, 70, 71.</p><p>It is certain however, from many experiments and observations, that the dip is different in different latitudes, and that it increases in going northward. It appears from a table of observations made with a marine dipping needle of Mr. Nairne's, in a voyage towards the north pole, in 1773, that <table><row role="data"><cell cols="1" rows="1" role="data">in latitude</cell><cell cols="1" rows="1" role="data">60°</cell><cell cols="1" rows="1" role="data">18′</cell><cell cols="1" rows="1" role="data">the Dip was</cell><cell cols="1" rows="1" role="data">75°</cell><cell cols="1" rows="1" rend="align=right" role="data">0′,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">in latitude</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">the Dip was</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">52,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">in latitude</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">the Dip was</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" rend="align=right" role="data">52,</cell><cell cols="1" rows="1" role="data">and</cell></row><row role="data"><cell cols="1" rows="1" role="data">in latitude</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">the Dip was</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" rend="align=right" role="data">2 1/2.</cell><cell cols="1" rows="1" role="data"/></row></table> See Phipps's Voyage, pa. 122. See also the Observations of Mr. Hutchins, made in Hudson's Bay and Straits, Philos. Trans. vol. 65, pa. 129.</p><p>Burrowes, Gilbert, Ridley, Bond, &c, endeavoured to apply this discovery of the dip to the finding of the latitude; and Bond, going still farther, first of any proposed finding the longitude by it; but for want of observations and experiments, he could not go any length. Mr. Whiston, being furnished with the farther observations of colonel Windham, Dr. Halley, Mr Pound, Mr. Cunningham, M. Noel, M. Feuille, and his own, made great improvements in the doctrine and use of the dipping-needle, brought it to more certain rules, and endeavoured in good earnest to find the longitude by it<*> For <cb/> this purpose, he observes, 1st, That the true tendency of the north or south end of every magnetic needle, is not to that point of the horizon, to which the horizontal needle points, but towards another, directly under it, in the same vertical, and in different degrees under it, in different ages, and at different places. 2dly, That the power by which the horizontal needle is governed, and all our navigation usually directed, it is proved is only one quarter of the power by which the dippingneedle is moved; which should render the latter far the more effectual and accurate instrument. 3dly, That a dipping-needle of a foot long will plainly shew an alteration of the angle of inclination, in these parts of the world, in half a quarter of a degree, or 7 1/2 geographical miles; and a needle of 4 feet, in 2 or 3 miles; i. e. supposing these distances taken along, or near a meridian. 4thly, A dipping-needle, 4 feet long, in these parts of the world, will shew an equal alteration along a parallel, as another of a foot long will shew along a meridian; i. e. that will, with equal exactness, shew the longitude, as this the latitude.</p><p>This depends on the position of the lines of equal dip, in these parts of the world, which it is found do lie about 14 or 15 degrees from the parallels. Hence he argues, that as we can have needles of 5, 6, 7, 8, or more feet long, which will move with strength sufficient for exact observation; and since microscopes may be applied for viewing the smallest divisions of degrees on the limb of the instrument, it is evident that the longitude at land may thus be found to less than 4 miles.</p><p>And as there have been many observations made at sea with the same instrument by Noel, Feuille, &c, which have determined the dip usually within a degree, sometimes within 1/2 or 1/3 of a degree, and this with small needles, of 5 or 6, or at the most 9 inches long; it is inferred, that the longitude may be found, even at sea, to less than half a quarter of a degree. This premised, the observation itself follows.</p><p><hi rend="italics">To find the Longitude or Latitude by the DippingNeedle.</hi>—If the lines of equal dip, below the horizon, be drawn on maps, or sea-charts, from good observations, it will be easy, from the longitude known, to find the latitude; and from the latitude known, to find the longitude either at sea or land.</p><p>Suppose, for example, a person travelling or sailing along the meridian of London, should find that the angle of dip, with a needle of one foot, was 75°; the chart will shew that this meridian, and the line of dip, meet in the latitude of 53° 11′; which is therefore the latitude sought.</p><p>Or if he be travelling or sailing along the parallel of London, i. e. in 51° 31 north latitude, and find the angle of dip 74°; then this parallel, and the line of this dip, will meet on the map in 1° 46′ of east longitude from London; which therefore is the longitude sought.</p></div1><div1 part="N" n="DIRECT" org="uniform" sample="complete" type="entry"><head>DIRECT</head><p>, in Arithmetic, is when the proportion of any terms, or quantities, is in the natural or direct order in which they stand; being the opposite to <hi rend="italics">inverse,</hi> which considers the proportion in the inverted order of the terms. So, 3 : 4 :: 6 : 8 <hi rend="italics">directly</hi>; or 3 : 4 :: 8 : 6 inversely.</p><p><hi rend="italics">Rule of Three</hi> <hi rend="smallcaps">Direct</hi>, is when both pairs of terms are in <hi rend="italics">direct</hi> proportion.</p><div2 part="N" n="Direct" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Direct</hi></head><p>, in Astronomy. A planet is said to be di- <pb n="384"/><cb/> rect, or its motion direct, when it goes forward by its proper motion in the zodiac, according to the succession or order of the signs; or when it appears so to do, to an observer standing upon the earth. Whereas it is said to be, or to move <hi rend="italics">retrograde,</hi> when it appears to go the contrary way, or backward; and to be <hi rend="italics">stationary,</hi> when it seems not to move either way.</p><p><hi rend="smallcaps">Direct</hi> <hi rend="italics">Dials.</hi> See <hi rend="smallcaps">Dial</hi> and <hi rend="smallcaps">Dialling.</hi></p></div2><div2 part="N" n="Direct" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Direct</hi></head><p>, in Optics. <hi rend="italics">Direct</hi> vision is that performed by direct rays; in contradistinction to vision by refracted, or reflected rays. Direct vision is the subject of Optics, which prescribes the laws and rules of it.</p><p><hi rend="smallcaps">Direct</hi> <hi rend="italics">Rays,</hi> are those which pass on in right lines from the object to the eye, without being turned out of their rectilinear direction by any intervening body, either opaque or pellucid; or without being either reflected or refracted.</p><p><hi rend="smallcaps">Direct</hi> <hi rend="italics">Sphere.</hi> See <hi rend="smallcaps">Right</hi> <hi rend="italics">Sphere.</hi></p></div2></div1><div1 part="N" n="DIRECTION" org="uniform" sample="complete" type="entry"><head>DIRECTION</head><p>, in Astronomy, the motion and other phenomena of a planet, when direct.</p><div2 part="N" n="Direction" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Direction</hi></head><p>, in Astrology, is a kind of calculus, by which they pretend to find the time in which any notable accident shall befall the person whose horoscope is drawn.</p><p>For instance, having established the sun, moon, or ascendant, as masters or significators of life; and Mars or Saturn as promisers or portenders of death; the direction is a calculation of the time in which the significator shall meet the portender.</p><p>The significator they likewise call <hi rend="italics">apheta,</hi> or giver of life; and the promiser, <hi rend="italics">anereta, promissor,</hi> or giver of death.</p><p>They work the directions of all the principal points of the heavens and stars, as the ascendant, mid-heaven, sun, moon, and part of fortune. The like is done for the planets and fixed stars; but all differently, according to the different authors.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Direction</hi>, in Gunnery, is the direct line in which a piece is pointed. Sometimes a line of direction is marked on the upper side of the gun, by a small notch or slit, or knob, in the base and muzzle rings: but, unless the two wheels of the carriage stand equally high, this line will be fallacious; for which reason the gunners commonly find a new line of direction every time, by means of a plummet.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Direction</hi>, in Mechanics, denotes the line in which a body moves, or endeavours to proceed.</p><p><hi rend="italics">Angle of</hi> <hi rend="smallcaps">Direction</hi>, is that comprehended between the lines of direction of two conspiring powers.</p><p><hi rend="italics">Quantity of</hi> <hi rend="smallcaps">Direction</hi>, is used for the product arising from multiplying the velocity of the common centre of gravity, in a system of bodies, by the sum of their masses. In the collision of bodies the quantity of direction is the same both before and after the impulse.</p><p><hi rend="smallcaps">Direction</hi> <hi rend="italics">of the Load Stone,</hi> that property by which the magnet, or a needle touched by it, always presents one of its ends toward one of the poles of the world, and the opposite end to the other pole. This is also called the <hi rend="italics">polarity</hi> of the magnet or needle.</p><p>The attractive property of the magnet was known long before its directive; and the directive long before the inclinatory.</p><p><hi rend="italics">Number of</hi> <hi rend="smallcaps">Direction</hi>, is the number of days that Septuagesima Sunday falls after the 17th of January. See <hi rend="smallcaps">Number.</hi> <cb/></p></div2></div1><div1 part="N" n="DIRECTLY" org="uniform" sample="complete" type="entry"><head>DIRECTLY</head><p>, in Geometry: we say two lines lie directly against each other, when they are parts of the same right line. Also quantities are said to be directly proportional, when the proportion is according to the order of the terms; in contradistinction to <hi rend="italics">inversely,</hi> or <hi rend="italics">reciprocally</hi> proportional, which is taking the proportion contrary to the order of the terms.</p><p>In Mechanics, a body is said to strike or impinge <hi rend="italics">directly</hi> against another body, when the stroke is in a direction perpendicular to the surface at the point of impact.</p><p>And a sphere in particular strikes directly against another, when the line of direction passes through both their centres.</p><p>DIRECTRIX. See <hi rend="smallcaps">Dirigent.</hi></p><div2 part="N" n="Directrix" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Directrix</hi></head><p>, in a Parabola, a line perpendicular to the axis produced, at the distance of the focus without the vertex.</p></div2></div1><div1 part="N" n="DIRIGENT" org="uniform" sample="complete" type="entry"><head>DIRIGENT</head><p>, a term expressing the line of motion, along which a describent line, or surface, is carried in the genesis of any plane or solid figure.</p><p>Thus, if the line AB move <figure/> parallel to itself, and along the line AD, so that the point A always keeps in the line AD, and the point B in the line BC; a parallelogram ABCD will be formed; of which the line AB is the describent, and the line AD the dirigent. So also, if the surface ABEG be supposed carried along the line AD, in a position always parallel to itself at its first situation, the solid AF will be formed; where the surface AE is the describent, and the line AD is the dirigent.</p></div1><div1 part="N" n="DISC" org="uniform" sample="complete" type="entry"><head>DISC</head><p>, or <hi rend="smallcaps">Disk</hi>, the body or face of the sun or moon; such as it appears to us; for though they be really spherical bodies, they are apparently circular planes.</p><p>The diameter of the disc is considered as divided into 12 equal parts, called digits; by means of which it is, that the magnitude of an eclipse is measured, or estimated.—In a total eclipse of either of those luminaries, the whole disc is obscured, or darkened; in a partial eclipse, only part of them.</p><p><hi rend="italics">Illuminated</hi> <hi rend="smallcaps">Disc</hi> of the Earth. See <hi rend="smallcaps">Circle</hi> <hi rend="italics">of Illumination.</hi></p><div2 part="N" n="Disc" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Disc</hi></head><p>, in Optics, the magnitude of a telescope glass, or the width of its aperture, whatever its figure be, whether a plane, convex, meniscus, or the like.</p></div2></div1><div1 part="N" n="DISCHARGER" org="uniform" sample="complete" type="entry"><head>DISCHARGER</head><p>, or <hi rend="smallcaps">Discharging</hi> <hi rend="italics">Rod,</hi> in Electricity, consists of a handle of glass or baked wood, A, <figure/> and two bent metal rods BB, terminating in points, and capable of being screwed into the knobs DD, which <pb n="385"/><cb/> move by a joint C, fixed to the handle A. Thus it may be used either with the points or balls, as occasion requires; and by being made moveable on a joint, it may be applied to larger or smaller jars at pleasure. By bringing one of these knobs or points to one coated side of a charged electric, and the other to the other side, or to any conductor connected with it, the communication is completed between the two sides, and the electric is discharged.</p><p>For the description and use of an Universal Discharger, by Mr. Henly, with which many curious experiments may be performed, see Cavallo's Electricity, pa. 164.</p></div1><div1 part="N" n="DISCORD" org="uniform" sample="complete" type="entry"><head>DISCORD</head><p>, the relation of two sounds which are always, and of themselves, disagreeable, whether applied in succession or consonance.</p></div1><div1 part="N" n="DISCOUNT" org="uniform" sample="complete" type="entry"><head>DISCOUNT</head><p>, or <hi rend="italics">Rebate,</hi> is used for an allowance made on a bill, or any other debt not yet become due, in consideration of making present payment of the bill or debt.</p><p>Among merchants and traders, it is usual to allow a sum for discount that is equal to the interest of the debt, calculated for the time till it becomes due: but this is not just; for as the true value of the discount is equal to the difference between the debt and its present worth, it is equal only to the interest of that present worth, instead of the interest on the whole debt. And therefore the rule for finding the true discount is this:</p><p>As the amount of 100l, for the given rate and time:</p><p>Is to the given sum or debt::</p><p>So is the interest of 100l, for the given rate and time:</p><p>To the discount of the debt.</p><p>So that, if <hi rend="italics">p</hi> be the principal or debt, <hi rend="italics">r</hi> the rate of interest per cent. and <hi rend="italics">t</hi> the time; then as , which is the true discount. Hence also is the present worth, or sum to be received.</p><p><hi rend="italics">For ex.</hi> Suppose it be required to find the discount of 250l, for five months, at the rate of 5 per cent. per annum interest. Here <hi rend="italics">p</hi>=250, <hi rend="italics">r</hi>=5, and <hi rend="italics">t</hi> = 5/12 or 5 months; then the discount sought.</p><p>A Table of Discounts may be seen in Smart's Tables of Interest, the use of which makes calculations of discount very easy.</p><p><hi rend="smallcaps">Discount</hi> is also the name of a rule in books of Arithmetic, by which calculations of Discount are made.</p></div1><div1 part="N" n="DISCRETE" org="uniform" sample="complete" type="entry"><head>DISCRETE</head><p>, or <hi rend="smallcaps">Disjunct</hi>, <hi rend="italics">Proportion,</hi> is that in which the ratio between two or more pairs of numbers is the same, and yet the proportion is not continued, so as that the ratio may be the same between the consequent of one pair and the antecedent of the next pair.</p><p>Thus, if the numbers or proportion 6 : 8 :: 3 : 4 be considered: the ratio of 6 to 8 is the same as that of 3 to 4, and therefore these four numbers are proportional: but it is only <hi rend="italics">discretely</hi> or <hi rend="italics">disjunctly,</hi> and not continued; for 8 to 3 is not the same ratio as the former; that is, the proportion is broken off between 8 and 3, and not continued all along, as it is in these fol- <cb/> lowing four numbers, which are called <hi rend="italics">continual proportionals,</hi> viz, 3 : 6 :: 12 : 24.</p><p><hi rend="smallcaps">Discrete</hi> <hi rend="italics">Quantity,</hi> is such as is not continued and joined together. Such for instance is any number; for its parts, being distinct units, cannot be united into one <hi rend="italics">continuum;</hi> for in a <hi rend="italics">continuum</hi> there are no actual determinate parts before division, but they are potentially infinite: so that it is usually and truly said that continued quantity is divisible <hi rend="italics">in infinitum.</hi></p></div1><div1 part="N" n="DISDIAPASON" org="uniform" sample="complete" type="entry"><head>DISDIAPASON</head><p>, in Music, a compound concord, in the quadruple ratio of 4 to 1, being a sifteenth or double eighth, and is produced when the voice goes from the first tone to the sifteenth.</p><p>DISJUNCT <hi rend="italics">Proportion.</hi> See <hi rend="smallcaps">Discrete</hi> <hi rend="italics">Proportion.</hi></p><p>DISK. See <hi rend="smallcaps">Disc.</hi></p></div1><div1 part="N" n="DISPART" org="uniform" sample="complete" type="entry"><head>DISPART</head><p>, a term in Gunnery, used for a mark set upon the muzzle-ring of a piece of ordnance, of such height, that a sight-line taken from the top of the base ring near the vent or touch-hole to the top of the dispart near the muzzle, may be parallel to the axis of the concave cylinder; for which reason it is evident that the height of the dispart is equal to the difference between the radii of the piece at the base and muzzle-rings, or to half the difference of the diameters there. Hence comes the common method of disparting the gun, which is this: Take, with the calipers, the two diameters, viz, of the base ring and the place where the dispart is to stand, subtract the lesa from the greater, and take half the difference, which will be the length of the dispart, which is commonly cut to that length from a small bit of wood, and so fixed upright in its place with a bit of wax or pitch.</p></div1><div1 part="N" n="DISPERSION" org="uniform" sample="complete" type="entry"><head>DISPERSION</head><p>, in Dioptrics, is the divergency of refracted rays of light.</p><p><hi rend="italics">Point of</hi> <hi rend="smallcaps">Dispersion</hi>, is a point from which refracted rays begin to diverge, when their refraction renders them divergent.—It is called <hi rend="italics">point of dispersion</hi> in opposition to the <hi rend="italics">point of concourse,</hi> or point in which converging rays concur after refraction. But it is more usual to call the latter the <hi rend="italics">focus,</hi> and the former the <hi rend="italics">virtual focus.</hi></p><p><hi rend="smallcaps">Dispersion</hi> <hi rend="italics">of Light,</hi> occasioned by the refrangibility of the rays, or the nature of the refracting medium. See <hi rend="smallcaps">Aberration</hi>, and <hi rend="smallcaps">Inflection.</hi></p></div1><div1 part="N" n="DISSIPATION" org="uniform" sample="complete" type="entry"><head>DISSIPATION</head><p>, in Physics, a gradual, slow, insensible loss or consumption of the minute parts of a body; or, more properly, the flux by which they fly off and are lost. See <hi rend="smallcaps">Effluvia.</hi></p><p><hi rend="italics">Circle of</hi> <hi rend="smallcaps">Dissipation</hi>, or <hi rend="smallcaps">Aberration</hi>, in Optics, denotes that circular space upon the retina of the eye, which is occupied by the rays of each pencil in indistinct vision: thus, if the distance of the object, or the constitution of the eye, be such, that the image falls beyond the retina, as when objects are too near; or before the retina, when the rays have not a sufficient divergency, the rays of a pencil, instead of being collected into a central point, will be dissipated over this circular space: and, all other circumstances being alike, this circle will be greater or less, according to the distances from the retina of the foci of refracted rays. But this circle causes no perceptible difference in the distinctness of vision, unless it exceed a certain magnitude; as soon as that is the case, we begin to perceive an indistinct- <pb n="386"/><cb/> ness, which increases as that circle increases, till at length the object is lost in confusion. This circle is also greater or less, according to the greater or less magnitude of the visible object: and though it be not easy to assign the diameter of the said circle, it seems very probable that vision continues distinct for all such distances, or so long as these circles, or the pencils of light from them, do not touch one another upon the retina; and the indistinctness begins when the said circles begin to interfere. It has been often observed, that a precise union of the respective rays upon the retina, is not necessary to distinct vision; but the sirst author who ascertained the fact beyond all doubt, was Dr. Jurin. See a variety of observations and experiments on this subject, in his Essay on Distinct and Indistinct Vision, in Smith's Optics, Appendix. In the Philos. Trans. for 1789, pa. 256, is an excellent paper on this subject by Dr. Maskelyne; in which he computes the diameter of the circle of dissipation at .002667 of an inch, making it answer to an external angle of 15°, which he shews is very compatible with distinct vision. See also <hi rend="smallcaps">Moon</hi>, and <hi rend="smallcaps">Vision.</hi></p><p><hi rend="italics">Radius of</hi> <hi rend="smallcaps">Dissipation</hi>, is the radius of the circle of dissipation.</p></div1><div1 part="N" n="DISSOLVENT" org="uniform" sample="complete" type="entry"><head>DISSOLVENT</head><p>, something that dissolves; i. e. divides, and reduces a body into its smallest parts.</p></div1><div1 part="N" n="DISSOLUTION" org="uniform" sample="complete" type="entry"><head>DISSOLUTION</head><p>, is a separation of the structure of a body, into small or minute parts.</p><p>According to Newton, and others, this is effected by certain powerful attractions.</p></div1><div1 part="N" n="DISSONANCE" org="uniform" sample="complete" type="entry"><head>DISSONANCE</head><p>, or <hi rend="smallcaps">Discord</hi>, is a false consonance or concord; being produced by the mixture or meeting of two sounds which are disagreeable to the ear.</p></div1><div1 part="N" n="DISTANCE" org="uniform" sample="complete" type="entry"><head>DISTANCE</head><p>, properly speaking, denotes the shortest line between two points, objects, &c.</p><div2 part="N" n="Distance" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Distance</hi></head><p>, in Astronomy, as of the sun, planets, comets, &c.</p><p>The Real Distances are found from the parallaxes of the planets, &c. See <hi rend="smallcaps">Parallax</hi>, and <hi rend="smallcaps">Planet</hi>, and <hi rend="smallcaps">Transit.</hi> The distance of the earth from the sun has been determined at 95 millions of miles, by the late transits of Venus; and from this one real distance, and the several relative distances, by analogy are found all the other real distances, as in the table below.</p><p>The Proportional or Relative Distances of the planets are very well deduced from the theory of gravity: for Kepler has long since discovered, and Newton has demonstrated, that the squares of their periodical times are proportional to the cubes of their distances. Kepler's Epit. Astron. lib. 4; Newton's Principia, lib. 3, phæn. 4; and Gregory's Astron. book 1, prop. 40. If therefore the mean distance of the earth from the sun be assumed, or supposed 10000, we shall then, from the foregoing analogy, and the known periodical times, obtain the relative distances of the other planets: thus,</p><p>The Periodical Revolutions in Days and Parts. Mercury. Venus. Earth. Mars. Jupiter. Saturn. Herschel. <table><row role="data"><cell cols="1" rows="1" role="data">87 23/24</cell><cell cols="1" rows="1" role="data">224 17/24</cell><cell cols="1" rows="1" role="data">365 1/4</cell><cell cols="1" rows="1" role="data">686 23/24</cell><cell cols="1" rows="1" role="data">4332 1/2</cell><cell cols="1" rows="1" role="data">10759 7/24</cell><cell cols="1" rows="1" role="data">30445</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=7 align=center" role="data">Relative Mean Distances from the Sun.</cell></row><row role="data"><cell cols="1" rows="1" role="data">3871</cell><cell cols="1" rows="1" role="data">7233</cell><cell cols="1" rows="1" role="data">10000</cell><cell cols="1" rows="1" role="data">15237</cell><cell cols="1" rows="1" role="data">52010</cell><cell cols="1" rows="1" role="data">95401</cell><cell cols="1" rows="1" role="data">190818</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=7 align=center" role="data">Real Distance in Millions of Miles.</cell></row><row role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">145</cell><cell cols="1" rows="1" role="data">493 1/2</cell><cell cols="1" rows="1" role="data">903 1/2</cell><cell cols="1" rows="1" role="data">1813</cell></row></table> <cb/></p><p>For the distances of the secondary planets from the centres of their respective primaries, see <hi rend="smallcaps">Satellites.</hi></p><p>As to that of the fixed stars, as having no sensible parallax, we can do little more than guess at.</p><p><hi rend="smallcaps">Distance</hi> <hi rend="italics">of the Sun from the Moon's node,</hi> or <hi rend="italics">apogee,</hi> is an arch of the ecliptic, intercepted between the sun's true place and the moon's node, or apogee. See <hi rend="smallcaps">Node.</hi></p><p><hi rend="italics">Curtate</hi> <hi rend="smallcaps">Distance.</hi> See <hi rend="smallcaps">Curtate.</hi></p><p><hi rend="smallcaps">Distance</hi> <hi rend="italics">of the Bastions,</hi> in Fortification, is the side of the exterior polygon.</p><p><hi rend="italics">Accessible</hi> <hi rend="smallcaps">Distances</hi>, in Geometry, are measured with the chain, decempeda or ten-foot rod, or the like.</p><p><hi rend="italics">Inaccessible</hi> <hi rend="smallcaps">Distances</hi>, are found by taking bearings to them, from the two extremities of a line whose length is given. Various ways of performing this may be seen in my Treatise on Mensuration, sect. 3, on Heights and Distances.</p></div2><div2 part="N" n="Distance" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Distance</hi></head><p>, in Geography, is the arch of a great circle intercepted between two places.</p><p>To find the distance of two <figure/> places, A and B, far remote from each other. Assume two stations, C and D, from which both the places A and B may be seen; and there, with a theodolite, observe the quantity of the angles ACD, BCD, ADC, BDC, and measure any distance as AC.</p><p>Then, in the triangle ACD, there are given the angles ACD, ADC, and the side AC; to find the side CD.—Next, in the triangle BCD, there are given the angles BCD, BDC, and the side CD; to find the side BC.—Lastly, in the triangle ABC, there are given the angle ACB, and the sides AC, CB; to find the side AB, which is the distance sought.</p><p>And in these operations, the triangles may be computed either as plane triangles, or as spherical ones, as the case may require, or according to the magnitude of the distances.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Distance</hi> <hi rend="italics">of a remote object may also be found from its height.</hi> This admits of several cases, according as the distances are large or small, &c. 1st, Suppose that from the top of a tower at A, whose height AB is 120 feet, there be taken the angle BAC = 33°, and the angle BAD = 64° 1/2, to two trees, or other objects, C, D; to find the distance between them CD, and the distance of each from the bottom of the tower at B. <figure/> <table><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">First, rad. : tang. [angle] BAD :: AB : BD =</cell><cell cols="1" rows="1" rend="align=right" role="data">251.585,</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=right" role="data">next, rad. : tang. [angle] BAC :: AB : BC =</cell><cell cols="1" rows="1" rend="align=right" role="data">77.929,</cell></row><row role="data"><cell cols="1" rows="1" role="data">their difference is the dist.</cell><cell cols="1" rows="1" rend="align=right" role="data">CD =</cell><cell cols="1" rows="1" rend="align=right" role="data">173.656.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2 align=right" role="data">2d, Suppose</cell></row></table> <pb n="387"/><cb/></p><p>2d, Suppose it be required to find the distance to which an object can be seen, by knowing its altitude; <hi rend="italics">ex. gr.</hi> the Pike of Teneriffe, whose height is accounted 3 miles above the level of the sea, supposing the circumference of the earth 25,000 miles, or the diameter 7958 miles. Let FG be the radius = 3979, EF = 3 the height of the mountain, and EI a tangent to the earth at the point H, which is the farthest distant point to which the top of the mountain E canbe seen. Here in the right angled triangle EGH, are given the hypothenuse EG = 3982, and the leg FG = 3979; to sind the other leg HE = 154 1/2 miles = the distance sought nearly. Or, rather, as EG : GH :: rad. : cosin. [angle] G=2° 13′ 1/2; then as 360° : 2° 13′ 1/2 :: 25,000 : 154 1/2 miles = the arch of dist. HF sought, the same as before.</p><p>3d, If the eye, instead of being in the horizon at H, were elevated above it at I, any known height, as suppose 264 feet, or (1/20)th of a mile, as on the top of a ship's mast, &c; then the mountain can be seen much farther off along the line IE, and the distance will be the two tangents IH and HE, or rather the two arcs KH and HF. Hence, as above, as IG : GH :: rad. cosin. [angle] IGH = 17′ 2/7; then as 360° : 17′ 2/7 :: 25,000 : 20 miles = the are KH : this added to the former are HF = 154 1/2, makes the whole are KF = 174 1/2 miles, for the whole distance to which the top of the mountain can be seen in this case.</p><p><hi rend="italics">Apparent</hi> <hi rend="smallcaps">Distance</hi>, in Optics, that distance which we judge an object is placed at when seen afar off, being usually very different from the true distance; because we are apt to think that all very remote objects, whose parts cannot well be distinguished, and which have no other object in view near them, are at the same distance from us, though perhaps the one is thousands of miles nearer than the other, as is the case with regard to the sun and moon.</p><p>M. De la Hire enumerates sive circumstances, which assist us in judging of the distance of objects; viz, their apparent magnitude, the strength of the colouring, the direction of the two eyes, the parallax of the objects, and the distinctness of their small parts. On the contrary, Dr. Smith maintains, that we judge of distance principally, or solely, by the apparent magnitude of objects; and concludes universally, that the apparent distance of an object seen in a glass, is to its apparent distance seen by the naked eye, as the apparent magnitude to the naked eye is to its apparent magnitude in the glass: But it was long since observed by Alhazen, that we do not judge of distance merely by the angle under which objects are seen; and Mr. Robins clearly shews that Dr. Smith's hypothesis is contrary to fact, in the most common and simple cases. Thus, if a double convex glass be held upright before some luminous object, as a candle, there will be seen two images, one erect, and the other inverted; the first is made simply by reflexion from the nearest surface; the second by reflexion from the farther surface, the rays undergoing a refraction from the first surface both before and after the reflexion. If this glass has not too short a focal distance, when it is held near the object, the inverted image will appear larger than the other, and also nearer; but if the glass be carried off from the object, though the eye remain as near to it as before, <cb/> the inverted image will be diminished so much father than the other, that at length it will appear much less than it, but still nearer. Here, says Mr. Robins, two images of the same object are seen under one view, and their apparent distances immediately compared; and it is evident that those distances have no necessary connexion with the apparent magnitude. This experiment may be made still more convincing, by sticking a piece of paper on the middle of the lens, and viewing it through a short tube. He observes farther, that the apparent magnitude of very distant objects is neither determined by the magnitude of the angle only under which they are seen, nor is the exact proportion of that angle compared with their true distance, but is compounded also with a deception concerning that distance; so that if we had no idea of difference in the distance of objects, each would appear in magnitude proportional to the angle under which it was seen; and if our apprehension of the distance were always just, our idea of their magnitude would be unvaried, in all distances; but in proportion as we err in our conception of their distance, the greater angle suggests a greater magnitude. By not attending to this compound effect, Mr. Robins apprehends that Dr. Smith was led into his mistake.</p><p>Dr. Porterfield has made several remarks on the sive methods of judging concerning the distance of objects above recited from M. De la Hire; and he has also added to them one more, viz, the conformation of each eye. See <hi rend="italics">Circle of</hi> <hi rend="smallcaps">Dissipation.</hi> This, he says, can be of no use to us, with respect to objects that are placed without the limits of distinct vision. But the greater or less confusion with which the object appears, as it is more or less removed from those limits, will assist the mind in judging of its distance: the more confused it appears, the farther will it be thought distant. However, this confusion has its limits; for when an object is placed at a certain distance from the eye, to which the breadth of the pupil bears no sensible proportion, the rays proceeding from a point in the object may be considered as parallel; in which case, the picture on the retina will not be sensibly more confused, though the object be removed to a much greater distance. The most universal, and often the most sure means of judging of the distance of objects, he says, is the angle made by the optic axes: our two eyes are like two different stations, by the assistance of which, distances are taken; and this is the reason why those persons who have lost the sight of one eye, so frequently miss their mark in pouring liquor into a glass, snuffing a candle, and such other actions as require that the distance be exactly distinguished. With respect to the method of judging by the apparent magnitude of objects, he observes that this can only serve when we are otherwise acquainted with their real magnitude. Thus he accounts for the deception to which we are liable in estimating distances, by any extraordinary magnitudes that terminate them; as, in travelling towards a large city, castle, or cathedral, we fancy they are nearer than they really are. Hence also, animals and small objects seen in a valley contiguous to large mountains, or on the top of a mountain or high building, appear exceedingly small. Dr. Jurin accounts for the last recited phenomenon, by observing that we have no distinct idea of <pb n="388"/><cb/> distance in that oblique direction, and therefore judge of them merely by their pictures on the eye.</p><p>Dr. Porterfield observes, with respect to the strength of colouring, that if we are assured they are of a similar colour, and one appears more bright and lively than the other, we judge that the brighter object is the nearer. When the small parts of objects appear confused, or do not appear at all, we judge that they are at a great distance, and <hi rend="italics">vice versa;</hi> because the image of any object, or part of an object, diminishes as the distance of it increases. Finally, we judge of the distance of objects by the number of intervening bodies, by which it is divided into separate and distinct parts; and the more this is the case, the greater will the distance appear. Thus distances upon uneven surfaces appear less than upon a plane, because the inequalities do not appear, and the whole apparent distance is diminished by the parts that do not appear in it: and thus the banks of a river appear contiguous to a distant eye, when the river is low and not seen. Accidens de la Vue, pa. 358. Smith's Optics, vol. 1, pa. 52, and Rem. pa. 51. Robins's Tracts, vol. 2, pa. 230, 247, 251. Porterfield on the Eye, vol. 1, pa. 105, vol. 2, pa. 387. See Priestley's Hist. of Vision, pa. 205, and pa. 689.</p></div2><div2 part="N" n="Distance" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Distance</hi></head><p>, in Navigation, is the number of miles or leagues that a ship has sailed from any point or place. See <hi rend="smallcaps">Sailing.</hi></p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Distance</hi>, in Per- <figure/> spective, is a right line drawn from the eye to the principal point: as the line OF, drawn between the eye at O, and the principal point F. As this is perpendicular to the plane, or table, it is therefore the distance of the eye from the table.</p><p><hi rend="italics">Point of</hi> <hi rend="smallcaps">Distance</hi>, in Perspective, is a point in the horizontal line at the same distance from the principal point, as the eye is from the same. Such are the points P and Q, in the horizontal line PQ, whose distance from the principal point F, is equal to that of the eye from the same F.</p><p>DISTINCT <hi rend="italics">Base,</hi> in Optics, is that distance from the pole of a convex glass, at which objects, beheld through it, appear distinct and well defined: so that the Distinct base is the same with what is otherwise called the focus.</p><p>The Distinct base is caused by the collection of the rays proceeding from a single point in the object, into a single point in the representation: and therefore concave glasses, which do not unite, but scatter and dissipate the rays, can have no real Distinct base.</p><p><hi rend="smallcaps">Distinct</hi> <hi rend="italics">Vision.</hi> See <hi rend="smallcaps">Vision.</hi></p></div2></div1><div1 part="N" n="DITCH" org="uniform" sample="complete" type="entry"><head>DITCH</head><p>, in Fortification, called also <hi rend="italics">Foss,</hi> and <hi rend="italics">Moat,</hi> is a trench dug round the rampart, or wall of a fortified place, between the scarp and counterscarp.</p><p>Ditches are either <hi rend="italics">dry,</hi> or <hi rend="italics">wet,</hi> that is having water in them; both of which have their particular advantages. The earth dug out of the ditch serves to raise the rampart.</p><p>The ditch in front should be of such breadth as that tall trees may not reach over it, being from 12 to 24 fathoms wide, and 7 or 8 feet deep. The ditches on <cb/> the sides are made smaller. But the most general rule is perhaps, that the dimensions of the ditch be such as that the earth dug out may be sufficient to build the rampart of a proper magnitude. The space sometimes left between the rampart and ditch, being about 6 or 8 feet, is called the <hi rend="italics">berm,</hi> or <hi rend="italics">list,</hi> serving to pass and repass, and to prevent the earth from rolling into the ditch.</p></div1><div1 part="N" n="DITONE" org="uniform" sample="complete" type="entry"><head>DITONE</head><p>, in Music, an interval comprehending two tones, a greater and a less. The ratio of the sounds that form the Ditone, is of 4 to 5; and that of the semi-ditone, of 5 to 6.</p></div1><div1 part="N" n="DITTON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DITTON</surname> (<foreName full="yes"><hi rend="smallcaps">Humphrey</hi></foreName>)</persName></head><p>, an eminent mathematician, was born at Salisbury, May 29, 1675. Being an only son, and his father observing in him an extraordinary good capacity, determined to cultivate it with a good education. For this purpose he placed him in a reputable private academy; upon quitting of which, he, at the desire of his father, though against his own inclination, engaged in the profession of divinity, and began to exercise his function at Tunbridge in the county of Kent, where he continued to preach some years; during which time he married a lady of that place.</p><p>But a weak constitution, and the death of his father, induced Mr. Ditton to quit that profession. And at the persuasion of Dr. Harris and Mr. Whiston, both eminent mathematicians, he engaged in the study of mathematics, a science to which he had always a strong inclination. In the prosecution of this science, he was much encouraged by the success and applause he received: being greatly esteemed by the chief professors of it, and particularly by Sir Isaac Newton, by whose interest and recommendation he was elected master of the new Mathematical School in Christ's Hospital; where he continued till his death, which happened in 1715, in the 40th year of his age, much regretted by the philosophical world, who expected many useful and ingenious discoveries from his assiduity, learning, and penetrating genius.</p><p>Mr. Ditton published several mathematical and other tracts, as below.—1. <hi rend="italics">Of the Tangents of Curves,</hi> &c. Philos. Trans. vol. 23.</p><p>2. A Treatise on <hi rend="italics">Spherical Catoptrics,</hi> published in the Philos. Trans. for 1705; from whence it was copied and reprinted in the Acta Eruditorum 1707, and also in the Memoirs of the Academy of Sciences at Paris.</p><p>3. <hi rend="italics">General Laws of Nature and Motion;</hi> 8vo, 1705. Wolfius mentions this work, and says, that it illustrates and renders easy the writings of Galileo, Huygens, and the Principia of Newton. It is also noticed by La Roche, in the Memoires de Literature, vol. 8, pa. 46.</p><p>4. <hi rend="italics">An Institution of Fluxions, containing the first Principles, Operations, and Applications, of that admirable Method, as invented by Sir Isaac Newton;</hi> 8vo, 1706. This work, with additions and alterations, was again published by Mr. John Clarke, in the year 1726.</p><p>5. In 1709 he published the <hi rend="italics">Synopsis Algebraica</hi> of John Alexander, with many additions and corrections.</p><p>6. His <hi rend="italics">Treatise on Perspective</hi> was published in 1712. In this work he explained the principles of that art mathematically; and besides teaching the methods then generally practised, gave the first hints of the new <pb n="389"/><cb/> method afterward enlarged upon and improved by Dr. Brook Taylor; and which was published in the year 1715.</p><p>7. In 1714, Mr. Ditton published several pieces, both theological and mathematical; particularly his <hi rend="italics">Discourse on the Resurrection of Jesus Christ; and The New Law of Fluids, or a Discourse concerning the Ascent of Liquids, in exact Geometrical Figures, between two nearly contiguous Surfaces.</hi> To this was annexed a tract, to demonstrate the impossibility of thinking or perception being the result of any combination of the parts of matter and motion: a subject much agitated about that time. To this work also was added an advertisement from him and Mr. Whiston, concerning a method for discovering the longitude, which it seems they had published about half a year before. This attempt probably cost our author his life; for although it was approved and countenanced by Sir Isaac Newton, before it was presented to the Board of Longitude, and the method has been successfully put in practice, in finding the longitude between Paris and Vienna, yet that Board then determined against it: so that the disappointment, together with some public ridicule (particularly in a poem written by Dean Swift), affected his health, so that he died the ensuing year, 1715.</p><p>In an account of Mr. Ditton, prefixed to the German translation of his Discourse on the Resurrection, it is said that he had published, in his own name only, another method for finding the longitude; but which Mr. Whiston denied. However, Raphael Levi, a learned Jew, who had studied under Leibnitz, informed the German editor, that he well knew that Ditton and Leibnitz had corresponded upon the subject; and that Ditton had sent to Leibnitz a delineation of a machine he had invented for that purpose; which was a piece of mechanism constructed with many wheels like a clock, and which Leibnitz highly approved of for land use; but doubted whether it would answer on shipboard, on account of the motion of the ship.</p><p>DIVERGENT <hi rend="italics">Point.</hi> See <hi rend="italics">Virtual</hi> <hi rend="smallcaps">Focus.</hi></p><div2 part="N" n="Divergent" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Divergent</hi></head><p>, or <hi rend="smallcaps">Diverging</hi> <hi rend="italics">Lines,</hi> in Geometry, are those whose distance is continually increasing.— Lines which diverge one way, converge the other way.</p></div2><div2 part="N" n="Divergent" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Divergent</hi></head><p>, or <hi rend="smallcaps">Diverging</hi>, in Optics, is particularly applied to rays which, issuing from a radiant point, or having, in their passage, undergone a refraction, or reflexion, do continually recede farther from each other.</p><p>In this sense the word is opposed to convergent, which implies that the rays approach each other, or that they tend to a centre, where they intersect, and, being continued, go on diverging. Indeed all intersecting rays, or lines, diverge both ways from the centre, or point of intersection.</p><p>Concave glasses render the rays diverging; and convex ones, converging.—Concave mirrors make the rays converge; and convex ones, diverge.—It is demonstrated in Optics, that as the diameter of a pretty large pupil does not exceed 1/5 of a digit; diverging rays, flowing from a radiant point, will enter the pupil as parallel, to all intents and purposes, if the distance of the radiant from the eye amount to 40,000 feet. See <hi rend="smallcaps">Focus, Light</hi>, and <hi rend="smallcaps">Vision.</hi></p><p><hi rend="smallcaps">Diverging Hyperbola</hi>, is one whose legs turn <cb/> their convexities toward each other, and run out quite contrary ways. See <hi rend="smallcaps">Hyperbola.</hi></p><p><hi rend="smallcaps">Diverging</hi> <hi rend="italics">Parabola.</hi> See <hi rend="italics">Diverging</hi> <hi rend="smallcaps">Parabola.</hi></p></div2></div1><div1 part="N" n="DIVIDEND" org="uniform" sample="complete" type="entry"><head>DIVIDEND</head><p>, in Arithmetic, is the number given to be divided by some other number, called the <hi rend="italics">divisor.</hi> Or it is the number given to be divided, or separated, into a certain number of equal parts, viz. as many as the divisor contains units; and the number of such equal parts is called the <hi rend="italics">quotient.</hi> Or, more gonerally, the dividend contains the divisor, as many times as the quotient contains unity.</p><p>The Dividend is the numerator of a fraction, whose denominator is the divisor, and the quotient is the value of the fraction. Thus, 8/2 = 4, and 3/4 = .75.</p><p>DIVIDUAL. By this name some authors distinguish the several parts of a dividend, from which each separate figure of the quotient is found.</p></div1><div1 part="N" n="DIVING" org="uniform" sample="complete" type="entry"><head>DIVING</head><p>, the art, or act of descending under water, to considerable depths, and remaining there a competent time.</p><p>The uses of Diving are very considerable, particularly in the fishing for pearls, corals, sponges, &c.</p><p>Various methods have been proposed, and engines contrived, to render the bufiness of diving more safs and easy. The great point in all these, is to furnish the diver with fresh air, without which he must either make but a short stay, or perish.</p><p>Those who dive for sponges in the Mediterranean, help themselves by carrying down sponges dipt in oil in their mouths. But considering the small quantity of air that can be contained in the pores of a sponge, and how much that little will be contracted by the pressure of the incumbent water, such a supply cannot long subsist the diver. For it is found by experiment, that a gallon of air included in a bladder, and by a pipe reciprocally inspired and expired by the lungs, becomes unfit for respiration in little more than one minute of time. For though its elasticity be but little altered in passing the lungs, yet it loses its vivifying spirit, and is rendered effete. In effect, a naked diver, Dr. Halley assures us, without a sponge, cannot remain above two minutes inclosed in water; nor much longer with one, without suffocating; nor without long practice, near so long; ordinary persons beginning to be suffocated in about half a minute. Besides, if the depth be confiderable, the pressure of the water on the vessels makes the eyes blood shotten, and frequently occasions a spitting of blood. Hence, where there has been occasion to continue long at the bottom, some have contrived double flexible pipes, to circulate air down into a cavity inclosing the diver, as with armour, both to furnish air, and to bear off the pressure of the water, and give leave to his breast to dilate upon inspiration; the fresh air being forced down one of the pipes with bellows, and returning by the other, not unlike to an artery, and vein.</p><p>But this method is impracticable when the depth ex ceeds three fathoms; the water embracing the bare limbs so closely, as to obstruct the circulation of the blood in them; and withal pressing so strongly on all the junctures where the armour is made tight with leather; that if there be the least defect in any of them, the water rushes in, and instantly sills the whole engine, to the great danger of the diver's life. <pb n="390"/><cb/></p><p><hi rend="smallcaps">Diving</hi>-<hi rend="italics">Bell,</hi> is a machine contrived to remedy all these inconveniencies. In this the diver is safely conveyed to any reasonable depth, and may stay more or less time under the water, as the bell is greater or less. It is most conveniently made in form of a truncated cone, the smallest base being closed, and the larger open. It is to be poised with lead, and so suspended, that it may sink full of air, with its open basis downward, and as near as may be in a situation parallel to the horizon, so as to close with the surface of the water all at once.</p><p>Under this covercle the diver sitting, sinks down with the included air to the depth desired; and if the cavity of the vessel can contain a ton of water, a single man may remain a full hour, without much inconvenience, at five or six fathoms deep. But the lower he goes, still the more the included air contracts itself, according to the weight of the water that compresses it; so that at thirty-three feet deep, the bell becomes half full of water; the pressure of the incumbent water being then equal to that of the atmosphere; and at all other depths, the space occupied by the compressed air in the upper part of its capacity, is to the space filled with water, as thirty-three feet to the depth of the surface of the water in the bell below the common surface of it. And this condensed air, being taken in with the breath, soon insinuates itself into all the cavities of the body, and has no ill effect, provided the bell be permitted to descend so slowly as to allow time for that purpose.</p><p>One inconvenience that attends it, is found in the ears, within which there are cavities which open only outwards, and that by pores so small, as not to give admission even to the air itself, unless they be dilated and distended by a considerable force. Hence, on the first descent of the bell, a pressure begins to be felt on each ear, which, by degrees, grows painful, till the force overcoming the obstacle, what constringes these pores, yields to the pressure, and letting some condensed air slip in, presently ease ensues. The bell descending lower, the pain is renewed, and afterwards it is again eased in the same manner. But the greatest inconvenience of this engine is, that the water entering it, contracts the bulk of air into so small a compass, that it soon heats, and becomes unfit for respiration: so that there is a necessity for its being drawn up to recruit it; besides the uncomfortable abiding of the diver, who is almost covered with water.</p><p>To obviate the difficulties of the diving-bell, Dr. Halley, to whom we owe the preceding account, contrived some further apparatus, by which not only to recruit and refresh the air from time to time, but also to keep the water wholly out of it at any depth; which he effected after the following manner:</p><p>His diving-bell (plate vii, fig. 6.) was of wood, three feet wide at top, five feet at bottom, and eight feet high, containing about fixty-three cubic feet in its concavity, coated externally with lead so heavy, that it would sink empty; a particular weight being distributed about its bottom R, to make it descend perpendicularly, and no otherwise. In the top was fixed a meniscus glass D, concave downwards, like a window, to let in light from above; with a cock, as at B, to let out the hot air; and a circular seat, as at LM, for the divers to sit on: and, below, about a yard under the bell, was a stage suspended from it by three ropes, each charged <cb/> with a hundred weight, to keep it steady, and for the divers to stand upon to do their business. The machine was suspended from the mast of a ship by a sprit, which was secured by stays to the mast-head, and was directed by braces to carry it overboard clear of the side of the ship, and to bring it in again.</p><p>To supply air to this bell when under water, he had a couple of barrels, as C, holding thirty-six gallons each, cased with lead, so as to sink empty, each having a bung-hole at bottom, to let in the water as they descended, and let it out again as they were drawn up. In the top of the barrels was another hole, to which was fixed a leathern pipe, or hose, well prepared with bees wax and oil, long enough to hang below the bunghole; being kept down by a weight appended. So that the air driven to the upper part of the barrel by the encroachment of the water, in the defcent, could not escape up this pipe, unless the lower end were lifted up.</p><p>These air-barrels were fitted with tackle, to make them rise and fall alternately, like two buckets; being directed in their descent by lines fastened to the under edge of the bell: so that they came readily to the hand of a man placed on the stage, to receive them; and who taking up the ends of the pipes, as soon as they came above the surface of the water in the barrels, all the air included in the upper part of it was blown forcibly into the bell; the water taking its place.</p><p>One barrel thus received, and emptied; upon a signal given, it was drawn up, and at the same time the other let down; by which alternate succession, fresh air was furnished so plentifully, that the learned Doctor himself was one of five, who were all together in nine or ten fathoms deep of water for above an hour and a half, without the least inconvenience; the whole cavity of the bell being perfectly dry.</p><p>All the precaution he observed, was, to be let down gradually about twelve feet at a time, and then to stop, and drive out the water that had entered, by taking in three or four barrels of fresh air, before he descended farther. And, being arrived at the depth intended, he let out as much of the hot air that had been breathed, as each barrel would replace with cold, by means of the cock B, at the top of the bell, through whose aperture, though very small, the air would rush with so much violence, as to make the surface of the sea boil.</p><p>Thus, he found, any thing could be done that was required to be done underneath. And by taking off the stage, he could, for a space as wide as the circuit of the bell, lay the bottom of the sea so far dry as not to be over shoes in water. Besides, by the glass window so much light was transmitted, that, when the sea was clear, and especially when the sun shone, he could see perfectly well to write or read, much more to fasten, or lay hold of any thing under him that was to be taken up. And by the return of the air barrel he often sent up orders written with an iron pen on a plate of lead, directing how he would be moved from place to place.</p><p>At other times, when the water was troubled and thick, it would be as dark as night below; but in such cases he was able to keep a candle burning in the bell.</p><p>Dr. Halley observes, that they were subject to one inconvenience in this bell; they felt at first a small pain <pb n="391"/><cb/> in their ears, as if the end of a tobacco pipe were thrust into them; but after a little while there was a small puff of air, with a little noise, and they were easy.</p><p>This he supposes to be occasioned by the condensed air shutting up a valve leading from some cavity in the ear, full of common air; but when the condensed air pressed harder, it forced the valve to yield, and filled every cavity. One of the divers, in order to prevent this pressure, stopped his ear with a pledget of paper; which was pushed in so far, that a surgeon could not extract it without great difficulty.</p><p>The same author intimates, that by an additional contrivance he has found it practicable for a diver to go out of the bell to a good distance from it; the air being conveyed to him in a continued stream by small flexible pipes, which serve him as a clue to direct him back again to the bell. For this purpose, one end of these pipes, kept open against the pressure of the sea, by a small spiral wire, and made tight without by painted leather, and sheep's guts drawn over it, being open, was fastened in the bell, as at P, to receive air, and the other end was fixed to a leaden cap on the man's head, reaching down below his shoulders, open at bottom, to serve him as a little bell, full of air, for him to breathe at his work, which would keep out the water from him, when at the level of the great bell, because of the same density as the air in the great bell. But when he stooped down lower than the level of the great bell, he shut the cock F, to cut off the communication between the two bells. Phil. Trans. abr. vol. iv. part ii. p. 188, &c. vol. vi. p. 550, &c.</p><p>The air in this bell would serve him for a minute or two; and he might instantly change it, by raising himself above the great bell, and opening the cock F. The diver was furnished with a girdle of large leaden weights, and clogs of lead for the seet, which, with the weight of the leaden cap, kept him firm on the ground; he was also well clothed with thick flannels, which being first made wet, and then warmed in the bell by the heat of his body, kept off the chill of the cold water for a considerable time, when he was out of the bell.</p><p>Mr. Martin Triewald, F. R. S. and military architect to the king of Sweden, contrived to construct a diving-bell on a smaller scale, and less expence, than that of Dr. Halley, and yet capable of answering the same intents and purposes. This bell, AB (fig. 7.) sinks with leaden weights DD, suspended from the bottom of it. It is made of copper, and tinned all over on the inside; three strong convex lenses GGG, defended by the copper lids HHH, illuminate this bell. The iron plate E serves the diver to stand upon, when he is at work; this is suspended by chains FFF, at such a distance from the bottom of the bell, that when he stands upright, his head is just above the water in the bell, where he has the advantage of air fitter for respiration, than when he is much higher up; but as there is occasion for the diver to be wholly in the bell, and consequently his head in the upper part of it, Mr. Triewald has contrived, that, even there, after he has breathed the hot air as long as he well can, by means of a spiral copper tube placed close to the inside of the bell, he may draw the cooler and fresher air from the lowermost parts; for which purpose a flexible leather <cb/> pipe, about two feet long, is fixed to the upper end of the tube at <hi rend="italics">b</hi>; and to the other end of the pipe is fastened an ivory mouth-piece, for the diver to hold in his mouth, by which to respire the air from below. We shall only remark, that as air rendered effete by respiration is somewhat heavier than common air, it must naturally subside in the bell; but it may probably be restored by the agitation of the sea-water, and thus become fitter for respiration. See Fixed Air. Phil. Trans. abr. vol. viii. p. 634. Or Desaguliers's Exper. Phil. vol. ii. p. 220, &c.</p><p>The famous Corn. Drebell had an expedient in some respects superior even to the diving bell, if what is related of it be true. He contrived not only a vessel to be rowed under water, but also a liquor to be carried in the vessel, which supplied the place of fresh air.</p><p>The vessel was made for king James I. carrying twelve rowers, besides the passengers. It was tried in the river Thames; and one of the persons in that submarine navigation, then living, told it one, from whom Mr. Boyle had the relation.</p><p>As to the liquor, Mr. Boyle assures us, he discovered by a physician, who married Drebell's daughter, that it was used from time to time, when the air in that submarine boat was clogged by the breath of the company, and rendered unfit for respiration: at which time, by unstopping the vessel full of this liquor, he could speedily restore to the troubled air such a proportion of vital parts, as would make it serve again a good while. The secret of this liquor Drebell would never disclose to above one person, who himself assured Mr. Boyle what it was. Boyle's Exp. Phys. Mech. of the Spring of the Air.</p><p>We have had many projects of diving machines, and diving ships of various kinds, which have proved abortive.</p><p><hi rend="smallcaps">Diving</hi>-<hi rend="italics">Bladder,</hi> a term used by Borelli for a machine which he contrived for diving under the water to great depths, with great facility, which he prefers to the common diving-bell. The vesica, or bladder, as it is usually called, is to be of brass or copper, and about two feet in diameter. This is to contain the diver's head, and is to be fixed to a goat's skin habit, exactly fitted to the shape of the body of the person. Within this vesica there are pipes, by means of which a circulation of air is contrived; and the person carries an air pump by his side, by means of which he may make himself heavier or lighter, as the fishes do, by contracting or dilating their air bladder: by this means, the objections all other diving machines are liable to are obviated, and particularly that of the air; the moisture by which it is clogged in respiration, and by which it is rendered unfit for the same use again, being here taken from it by its circulation through the pipes, to the sides of which it adheres, and leaves the air as sree as before. Borelli Opera Posthuma.</p></div1><div1 part="N" n="DIVISIBILITY" org="uniform" sample="complete" type="entry"><head>DIVISIBILITY</head><p>, a property in quantity, body, or extension, by which it becomes separable into parts; either actually, or at least mentally.</p><p>Such divisibility is infinite, if not actually, at least potentially; for no part can be conceived so small, but another may be conceived still smaller; for every part of matter must have some finite extension, and that extension may be bisected, or otherwise divided; for the <pb n="392"/><cb/> same reason, these parts may be divided again, and so on without end.</p><p>We are not here contending for the possibility of an actual division <hi rend="italics">in infinitum:</hi> it is only asserted that however small a body is, it may be still farther divided; which it is presumed may be called a division <hi rend="italics">in infinitum,</hi> because what has no limits, is called <hi rend="italics">infinite.</hi></p><p>The infinite, or indefinite divisibility of mathematical quantity is thus proved, and illustrated by mathematicians: Suppose a line AD <figure/> perpendicular to BF; and another as GH also perpendicular to the same BF; with the centres C, C, C, &c, and distances CA, CA, &c, describe circles cutting the line GH in the points <hi rend="italics">e, e,</hi> &c. Now, the greater the radius AC is, the lefs is the part <hi rend="italics">e</hi>G; but the radius may be augmented in infinitum, and therefore the part <hi rend="italics">e</hi>G may be diminished in the same manner; and yet it can never be reduced to nothing, because the circle can never coincide with the right line BF. Consequently the parts of any magnitude may be diminished in infinitum.</p><p>All that is supposed, in strict geometry, concerning the divisibility of magnitude, amounts to no more, than that a given magnitude may be conceived as divided into a number of parts, equal to any given or proposed number.</p><p>It is true that there are no such things as parts infinitely small; yet the subtilty of the particles of several bodies is such, that they far surpass our conception; and there are innumerable instances in nature of such parts actually separated from one another.</p><p>Several instances of this are given by Mr. Boyle. He speaks of a silken thread 300 yards long, that weighed but two grains and a half. He measured leaf-gold, and found by weighing it, that 50 square inches weighed but one grain: if the length of an inch be divided into 200 parts, the eye may distinguish them all; therefore in one square inch there are 40,000 visible parts; and in one grain of it there are two millions of such parts; which visible parts no one will deny are still farther divisible.</p><p>Again, an ounce weight of silver may be gilt over with 8 grains of gold, which may be afterwards drawn into a wire 13,000 feet long, and still be all covered with the same gilding.</p><p>In odoriferous bodies a still greater subtilty of parts is perceived, and even such as are actually separated from one another: several bodies scarce lose any sensible part of their weight in a long time, and yet continually fill a very large space with odoriferous particles. Dr. Keil, in his Vera Physica, Lect. 5, has calculated the magnitude of a particle of Assafœtida, which will be the (57/10000000000000000)th part of a cubic inch. And in the same Lecture he shews that the particles of the blood in animalculæ, observed in fluids by means of microscopes, must be less than that part of a cubic inch which is expressed by a fraction whose numerator is 8, and denominator unity with 30 ciphers after it. <cb/></p><p>The particles of light, if light consist of real particles, furnish another surprising instance of the minuteness of some parts of matter. A small lighted candle placed on a plain, will be visible two miles, and consequently its light fills a sphere of 4 miles diameter, before it has lost any sensible part of its weight. Now, as the force of any body is directly in proportion to its quantity of matter multiplied by its velocity; and since it is demonstrated that the velocity of the particles of light is at least a million of times greater than the velocity of a cannon-ball, it is plain, that if a million of these particles were round, and of the size of a small grain of sand, we durst no more open our eyes to the light, than expose them to sand shot point-blank from a cannon.</p><p>By help of microscopes, such objects as would otherwise escape our sight, appear very large: there are some small animals scarce visible with the best microscopes; and yet these have all the parts necessary for life, as blood, and other fluids. How wonderful then must the subtilty of the parts be, which make up such fluids!</p><p>Whence is deducible the following theorem:</p><p>Any particle of matter, how small soever, and any finite space, how large soever, being given; it is possible for that small sand, or particle of matter, to be diffused through all that great space, and to fill it in such manner, as that there shall be no pore in it, whose diameter shall exceed any given line; as is demonstrated by Dr. Keil. Introduct. ad Ver. Phys.</p><div2 part="N" n="Divisible" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Divisible</hi></head><p>, the faculty or quality of being capable of being divided.</p></div2></div1><div1 part="N" n="DIVISION" org="uniform" sample="complete" type="entry"><head>DIVISION</head><p>, is one of the four principal Rules of Arithmetic, being that by which we find out how often one quantity is contained in another, so that Division is in reality only a compendious method of Subtraction; its effect being to take one number from another as often as possible; that is, as often as it is contained in it. There are therefore three numbers concerned in Division: 1st, That which is given to be divided, called the <hi rend="italics">dividend;</hi> 2d, That by which the dividend is to be divided, called the <hi rend="italics">divisor;</hi> 3d, That which expresses how often the divisor is contained in the dividend; or the number resulting from the division of the dividend by the divisor, called the <hi rend="italics">quotient.</hi></p><p>There are various ways of performing Division, one called the English, another the Flemish, another the Italian, another the Spanish, another the German, and another the Indian way, all equally just, as finding the quotient with the same certainty, and only differing in the manner of arranging and disposing the numbers.</p><p>There is also division in integers, division in fractions, and division in species, or algebra.</p><p>Division is performed by seeking how often the divisor is contained in the dividend; and when the latter consists of a greater number of figures than the former, the dividend must be taken into parts, beginning on the left, and proceeding to the right, and seeking how often the divisor is found in each of those parts.</p><p>For ex. If it be required to divide 6758 by 3. First seek how often 3 is contained in 6, which is 2 times; then how often in 7, which is likewise 2 times, with 1 remaining; which joined to the next figure 5 makes 15, then the 3's in 15 are 5 times; and lastly <pb n="393"/><cb/> the 3's in 8 are 2 times, and 2 remaining. All the numbers expressing how often 3 is contained in each of those parts, are to be written down according to the order of the parts of the dividend, or from left to right, and separated from the dividend itself by a crooked line, thus: <hi rend="center">Divisor Dividend Quotient</hi> <hi rend="center">3 ) 6758 ( 2252 2/3</hi></p><p>It appears therefore, that 3 is contained 2252 times in 6758, with 2 remaining over; or that 6758 being divided into 3 parts, each part will be 2252 2/3, viz, the figures of the quotient before found, together with the fraction 2/3 formed of the remainder and the divisor.</p><p>When the divisor is a single digit, or even as large as the number 12, the division is easily performed by setting down only the quotient as above. But when the divisor is a larger number, it is necessary to set down the several remainders and products &c. This process may be seen at large in most books of arithmetic, as well as various contractions adapted to particular cases: such as, 1st, when the divisor has any number of ciphers at the end of it, they are cut off, as well as the same number of figures from the end of the dividend, and then the work is performed without them both, annexing only the figures last cut off, to the last remainder; 2d, when the divisor is equal to the product of several single digits, it is easier to divide successively by those digits, instead of the divisor at once; 3d, when it is required to continue a quotient to a great many places of figures, as in decimals, a very expeditious method of performing it, is as follows: Suppose it were required to divide 1 by 29, to a great many places of decimals. Adding ciphers to the 1, first divide 10000 by 29 in the common way, till the remainder become a single figure, and annex the fractional supplement to complete the quotient, which gives 1/29 = 0.03448 8/29: next multiply each of these by the numerator 8, so shall 8/29 = 0.27584 64/29 or rather 0.27586 6/29; which figures substituted instead of the fraction 8/29 in the first value of 1/29, it becomes 1/29 = 0.0344827586 6/29: again, multiply both of these by the last numerator 6, and it will be 6/29 = 0.2068965517 7/29; which figures substituted for 6/29 in the last-found value of 1/29, it becomes 1/29 = 0.03448275862068965517 7/29: and again, multiplying these by the numerator 7, gives 7/29 = 0.24137931034482758620 10/29; which figures substituted instead of 7/29 in the last-found value of 1/29, it becomes 1/29 = 0.0344827586206896551724137931034482758620 10/29 and so on; where every operation will at least double the number of figures before found by the last one.</p><p><hi rend="italics">Proof of</hi> <hi rend="smallcaps">Division.</hi> In every example of division, unity is always in the same proportion to the divisor, as the quotient is to the dividend; and therefore the product of the divisor and quotient is equal to the product of 1 and the dividend, that is, the dividend itself. Hence, to prove division, multiply the divisor by the quotient, to the product add the remainder, and the sum will be equal to the dividend when the work is right; if not, there is a mistake.</p><div2 part="N" n="Division" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Division</hi></head><p>, <hi rend="italics">in Decimal Fractions,</hi> is performed the same way as in integers, regard being had to the number of decimals, viz, making as many in the quotient as those of the dividend exceed those in the divisor. <cb/></p></div2><div2 part="N" n="Division" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Division</hi></head><p>, <hi rend="italics">in Vulgar Fractions,</hi> is performed by dividing the numerators by each other, and the denominators by each other, if they will exactly divide; but if not, then the dividend is multiplied by the reciprocal of the divisor, that is, having its terms inverted; for, taking the reciprocal of any quantity, converts it from a divisor to a multiplier, and from a multiplier to a divisor. For ex. (15/16)÷by 5/8 gives 3/2, by dividing the numerators and denominators; but 15/16÷by 4/3 is the same as 15/16 X 3/4, which is = 45/64. Where X is the sign of multiplication, and the character ÷ is the mark of division. Or division is also denoted like a vulgar fraction; so 3 divided by 2, is 3/2.</p></div2><div2 part="N" n="Division" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Division</hi>, in <hi rend="italics">Algebra</hi></head><p>, or <hi rend="italics">Species,</hi> is performed like that of common numbers, either making a fraction of the dividend and divisor, and cancelling or dividing by the terms or parts that are common to both; or else dividing after the manner of long division, when the quantities are compound ones. Thus, <hi rend="italics">ab</hi> divided by <hi rend="italics">a,</hi> gives <hi rend="italics">b</hi> for the quotient: and 12<hi rend="italics">ab</hi> divided by 4<hi rend="italics">b,</hi> gives 3<hi rend="italics">a</hi> for the quotient: and 16<hi rend="italics">abc</hi><hi rend="sup">2</hi> divided by 8<hi rend="italics">ac,</hi> gives 2<hi rend="italics">bc</hi>: and <hi rend="italics">a</hi> divided by 3<hi rend="italics">b,</hi> gives <hi rend="italics">a</hi>/3<hi rend="italics">b</hi>: and 15<hi rend="italics">abc</hi><hi rend="sup">3</hi> divided by 12<hi rend="italics">bc</hi><hi rend="sup">2</hi>, gives </p><p>In some cases, the quotient will run out to an infinite series; and then, after continuing it to any certain number of terms, it is usual to annex, by way of a fraction, the remainder with the divisor set under it.</p><p>It is to be noted that, in dividing any terms by one another, if the signs be both alike, either both plus, or both minus, the sign of the quotient will be plus; but when the signs are different, the one plus and the other minus, the sign of the quotient will be minus.</p><p><hi rend="smallcaps">Division</hi> <hi rend="italics">by Logarithms.</hi> See <hi rend="smallcaps">Logarithms.</hi></p><p><hi rend="smallcaps">Division</hi> <hi rend="italics">of Mathematical Instruments.</hi> See G<hi rend="smallcaps">RADUATION</hi>, and <hi rend="smallcaps">Mural</hi> <hi rend="italics">Arc</hi> or <hi rend="italics">Quadrant.</hi></p><p><hi rend="smallcaps">Division</hi> in Music, is the dividing the interval of an octave into a number of lesser intervals.</p><p><hi rend="smallcaps">Division</hi> <hi rend="italics">by Napier's Bones.</hi> See <hi rend="smallcaps">Napier's</hi> <hi rend="italics">Bones.</hi></p><p><hi rend="smallcaps">Division</hi> <hi rend="italics">of Powers,</hi> is performed by subtracting their exponents. Thus, ; and . <pb n="394"/><cb/></p><p><hi rend="smallcaps">Division</hi> <hi rend="italics">of Proportion,</hi> is comparing the difference between the antecedent and consequent, with either of them. Thus,</p></div2><div2 part="N" n="Divisor" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Divisor</hi></head><p>, is the dividing number; or that which shews how many parts the dividend is to be divided into.</p><p><hi rend="italics">Common</hi> <hi rend="smallcaps">Divisor.</hi> See <hi rend="smallcaps">Common</hi> <hi rend="italics">Divisor.</hi></p></div2></div1><div1 part="N" n="DIURNAL" org="uniform" sample="complete" type="entry"><head>DIURNAL</head><p>, something relating to the day; in opposition to nocturnal, relating to the night.</p><p><hi rend="smallcaps">Diurnal</hi> <hi rend="italics">Arch,</hi> is the arch described by the sun, moon, or stars, between their rising and setting.</p><p><hi rend="smallcaps">Diurnal</hi> <hi rend="italics">Circle,</hi> is the apparent circle described by the sun, moon, or stars, in consequence of the rotation of the earth.</p><p><hi rend="smallcaps">Diurnal</hi> <hi rend="italics">Motion of a Planet,</hi> is so many degrees and minutes &c as any planet moves in 24 hours.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Diurnal</hi> <hi rend="italics">Motion of the Earth,</hi> is its rotation round its axis, the duration of which constitutes the natural day.—The reality of the diurnal rotation of the earth is now past all dispute.</p><p><hi rend="smallcaps">Diurnal</hi> <hi rend="italics">Parallax.</hi> See <hi rend="smallcaps">Parallax.</hi></p><p><hi rend="smallcaps">Diurnal</hi> is also used in speaking of what belongs to the nycthemeron, or natural day of 24 hours: in which sense it is opposed to annual, menstrual, &c.</p><p>The diurnal phenomena of the heavenly bodies are solved from the diurnal revolution of the earth; that is, from the rotation of the earth round its own axis in 24 hours. This rotation is equable, and from west to east, about an axis whose inclination to the ecliptic is now 66° 32′. Since the earth is an opaque body, that small part of its surface which comes at the same time under the confined view of the spectator, though really spherical, will seem to be extended like a plane: and the eye, taking a view of the heavens all around, defines a concave spherical superficies, concentric with the earth, or rather with the eye, which the said plane of the earth's supersicies will divide into two equal parts, the one of which is visible, but the other, because of the earth's opacity, hid from the view.</p><p>And as the earth revolves about its axis, the spectator, standing upon it, together with the said plane he stands upon, called his horizon, dividing the visible from the invisible hemisphere of the heavens, is carried round the same way, viz, towards the east. From hence it is, that the sun and stars, placed towards the east, being before hid, now become visible, the horizon as it were sinking below them; and the stars &c towards the west are covered or hid, and become invisible, the horizon being elevated above them. So that the former stars, to the spectator, who reckons the place he stands on as immoveable, will appear to ascend above the horizon, or rise; and the latter to descend below the horizon, or set.</p><p>Since the earth, with the horizon of the spectator fixed to it, continues to move always towards the same parts, and about the same axis equally; all bodies, and <cb/> all phenomena, that do not partake of the said motion, (that is, all such things as are entirely separate from the earth) will seem to move in the same time uniformly, but towards the opposite parts, or from east to west: and every one of these objects, according to sense, will describe the circumference of a circle, whose plane is perpendicular to the axis of the earth. And because all these circles, together with the visible objects describing them, appear to be in the concave spherical superficies of the heavens, every visible object will seem to describe a greater or less circle, according to its greater or less distance from the poles, or extremities of the earth's axis produced; the middle circle between these poles, called the equator, is consequently the greatest.</p><p>It may farther be observed, that whereas, by the diurnal revolution of the earth, all the several luminaries seem to move in the heavens from east to west, hence this seeming diurnal motion of the celestial lights is called their common motion, as being common to all of them. Besides which, all the luminaries, except the sun, have a proper motion; from which arise their proper phenomena: as for the proper phenomena of the sun, they likewise seem to arise from the proper motion of the sun; though they are really produced by another motion, which the earth has, and by which it moves round the sun once every year, and thence called the annual motion of the earth.</p></div1><div1 part="N" n="DODECAGON" org="uniform" sample="complete" type="entry"><head>DODECAGON</head><p>, a regular polygon of 12 equal sides and angles.</p><p>If the side of a Dodecagon be 1, its area will be equal to 3 times the tang. of nearly; and, the areas of plane figures being as the squares of their sides, therefore 11.1961524 multiplied by the square of the side of any Dodecagon, will give its area. See my Mensuration, pa. 114, 2d ed.</p><p><hi rend="italics">To inscribe a Dodecagon in a given Circle.</hi> Carry the radius 6 times round the circumference, which will divide it into 6 equal parts, or will make a hexagon; then bisect each of those parts, which will divide the whole into 12 parts, for the Dodecagon. See also other methods of describing the same figure in my Mensur. pa. 26, &c. See <hi rend="smallcaps">Polygon.</hi></p></div1><div1 part="N" n="DODECAHEDRON" org="uniform" sample="complete" type="entry"><head>DODECAHEDRON</head><p>, one of the Platonic bodies, or five regular solids, being contained under a surface composed of twelve equal and regular pentagons.</p><p><hi rend="italics">To form a Dodecahedron.</hi> See <hi rend="italics">Regular</hi> <hi rend="smallcaps">Body.</hi></p><p>If the side, or linear edge, of a Dodecahedron be <hi rend="italics">s,</hi> its surface will be and its solidity </p><p>If the radius of the sphere that circumscribes a Dodecahedron be <hi rend="italics">r,</hi> then is its side or linear edge , its superficies , and its solidity .</p><p>The side of a Dodecahedron inscribed in a sphere, is <pb n="395"/><cb/> equal to the greater part of the side of a cube inscribed in the same sphere, and cut according to extreme and mean proportion.</p><p>If a line be cut according to extreme and mean proportion, and the lesser segment be taken for the side of a Dodecahedron, the greater segment will be the side of a cube inscribed in the same sphere.</p><p>The side of the cube is equal to the right line which subtends the angle of a pentagon of the Dodecahedron, inscribed in the same sphere. See <hi rend="smallcaps">Polyhedron;</hi> also my Mensur. pa. 253, &c.</p></div1><div1 part="N" n="DODECATEMORY" org="uniform" sample="complete" type="entry"><head>DODECATEMORY</head><p>, the 12 houses or parts of the zodiac of the primum mobile. Also the 12 signs of the zodiac are sometimes so called, because they contain each the 12th part of the zodiac.</p></div1><div1 part="N" n="DOG" org="uniform" sample="complete" type="entry"><head>DOG</head><p>, a name common to two constellations, called the Great and Little Dog; but more usually Canis Major, and Canis Minor.</p></div1><div1 part="N" n="DOME" org="uniform" sample="complete" type="entry"><head>DOME</head><p>, is a round, vaulted, or arched roof, of a church, hall, pavilion, vestibule, stair-case, &c, by way of crowning, or acroter.</p><p>DOMINICAL <hi rend="italics">Letter,</hi> otherwise called the <hi rend="smallcaps">Sunday</hi> <hi rend="italics">Letter,</hi> is one of these first seven letters of the alphabet ABCDEFG, used in almanacs &c, to mark or denote the Sundays throughout the year.</p><p>The reason for using seven letters, is because that is the number of days in a week; and the method of using them is this: the first letter A is set opposite the 1st day of the year, the 2d letter B opposite the 2d day of the year, the 3d letter C opposite the 3d day of the year, and so on through the seven letters; after which they are repeated over and over again, all the way to the end of the year, the letter A denoting the 8th day, the letter B the 9th, &c. Then whichever of the letters so placed, falls opposite the first Sunday in the year, the same letter, it is evident, will fall opposite every future Sunday throughout the year, because the number of the letters is the same as the number of days in the week, being both 7 in number; that is, in common years; for as to leap years, an interruption of the order takes place in them. For, on account of the intercalary day, either the letters must be thrust out of their places for the whole year afterwards, so as that the letter, for ex. which answers to the 1st of March, shall likewise answer to the 2d &c; or else the intercalary day must be denoted by the same letter as the preceding one. This latter expedient was judged the better, and accordingly all the Sundays in the year after the intercalary day have another Dominical letter.</p><p>The Dominical letters were introduced into the calendar by the primitive christians, instead of the nundinal letters in the Roman calendar; and in this manner were those seven letters set opposite the days of the year, to denote the days in the week, in most of our common almanacs, till the year 1771, when the initial letters of the days of the week were generally introduced instead of them, excepting the Sunday letter itself, which is still retained.</p><p>From the foregoing account it follows that,</p><p>1st, As the common year consists of 365 days, or 52 weeks, and one day over; the letters go one day backwards every common year: so that in such a year, if the beginning or first day fall on a Sunday, the next year it will fall on Saturday, the next on Friday, and <cb/> so on. Consequently, if G be the Dominical letter for the present year, F will be that for the next year; and so on, in a retrograde order.</p><p>2d, As the bissextile or leap year consists of 366 days, or 52 weeks, and 2 days over, the beginning of the year next after bissextile goes back 2 days. Whence, if in the beginning of the bissextile year, the Dominical letter were G, that of the following year will be E.</p><p>3d, Since in leap-years the intercalary day falls on the 24th of February, in which case the 24th and 25th days are considered as one day, and denoted by the same letter; after the 24th day of February the Dominical letter goes back one place: thus, if in the beginning of the year the Dominical letter be G, it will afterwards change to the letter F for the remaining Sundays of the year. With us however, this day is now added at the end of February, and from thence it is that the change takes place.</p><p>4th, As every fourth year is bissextile, or leap-year, and as the number of letters is 7; the same order of Dominical letters only returns in 4 times 7, or 28 years; which without the interruption of bissextiles, would return in 7 years.</p><p>5th, Hence the invention of the solar cycle of 28 years; upon the expiration of which the Dominical letters are restored successively to the same days of the month, or the same order of the letters returns.</p><p><hi rend="italics">To find the <hi rend="smallcaps">Dominical</hi> Letter of any given year.</hi> Find the cycle of the sun for that year, as directed under cycle; and the Dominical letter is found corresponding to it. When there are two letters, the proposed year is bissextile; the former of them serving till the end of February, and the latter for the rest of the year.</p><p>The Dominical letter for any year of the present century may be found by this canon:</p><p><hi rend="italics">Divide the odd years, their fourth and</hi> 4, <hi rend="italics">by</hi> 7,</p><p><hi rend="italics">What is left take from</hi> 7, <hi rend="italics">the letter is given.</hi> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Thus, for the year 1794, the odd years</cell><cell cols="1" rows="1" rend="align=right" role="data">94</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">their 4th</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">divided by 7)</cell><cell cols="1" rows="1" rend="align=right" role="data">121</cell><cell cols="1" rows="1" role="data">(17</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">remains</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">from</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">leaves</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data"/></row></table> which answers to E the 5th letter in the alphabet.</p><p>The Dominical letter may be found universally, for any year of any century, thus:</p><p><hi rend="italics">Divide the centuries by</hi> 4; <hi rend="italics">and twice what does remain</hi></p><p><hi rend="italics">Take from</hi> 6; <hi rend="italics">and then add to the number you gain</hi></p><p><hi rend="italics">Their odd years and their</hi> 4<hi rend="italics">th; which dividing by</hi> 7,</p><p><hi rend="italics">What is left take from</hi> 7, <hi rend="italics">the letter is given.</hi></p><p>Thus, for the year 1878 the letter is F.</p><p>For the centuries 18 divided by 4, leave 2; the double of which taken from 6 leaves 2 again; to which add the odd years 78, and their 4th part 19, the sum 99 divided by 7 leaves 1; which taken from 7, leaves 6 answering to F the 6th letter in the alphabet.</p><p>By the reformation of the calendar under pope Gregory the 13th, the order of the Dominical letters was again disturbed in the Gregorian year; for the year <pb n="396"/><cb/> 1582, which had G for its Dominical letter at the beginning; by retrenching 10 days after the 4th of October, came to have C for its Dominical letter: by which means the Dominical letter of the ancient Julian calendar is 4 places before that of the Gregorian, the letter A in the former answering to D in the latter.</p></div1><div1 part="N" n="DONJON" org="uniform" sample="complete" type="entry"><head>DONJON</head><p>, in Fortification, usually denotes a large strong tower, or redoubt, of a fortress, where the garrison may retreat in case of necessity, and capitulate with greater advantage. See <hi rend="smallcaps">Dungeon.</hi></p></div1><div1 part="N" n="DORADO" org="uniform" sample="complete" type="entry"><head>DORADO</head><p>, a southern constellation, not visible in our latitude; called also <hi rend="italics">Xiphias,</hi> or the <hi rend="italics">Sword-fish.</hi> The stars of this constellation, in Sharp's catalogue, are six.</p><p>DORIC <hi rend="italics">Order,</hi> of Architecture, is the second of the five orders, being placed between the Tuscan and the Ionic. The Doric seems the most natural, and best proportioned, of all the orders; all its parts being founded on the natural position of solid bodies; for which reason it is most proper to be used in great and massy buildings, as the outside of churches and public places.</p><p>The Doric order has no ornaments on its base, nor its capital. Its column is 8 diameters high, and its freeze is divided between triglyphs and metopes.</p></div1><div1 part="N" n="DORMER" org="uniform" sample="complete" type="entry"><head>DORMER</head><p>, or <hi rend="smallcaps">Dormant</hi>, in Architecture, denotes a window made in the roof of a building, or above the entablature; being raised upon the rafters.</p><p>DOUBLE <hi rend="italics">Aspect, Bastion, Concave, Convex, Cone, Descant, Eccentricity, Position, Ratio, Tenaille, &c.</hi> See the respective words.</p><p><hi rend="smallcaps">Double</hi> <hi rend="italics">Horizontal Dial,</hi> one with a double gnomon, the one pointing out the hour on the outer circle, the other the hour on the stereographic projection drawn upon it. This dial finds the meridian, the hour, the sun's place, rising, setting, &c, and many other problems of the globe.</p><p><hi rend="smallcaps">Double</hi> <hi rend="italics">Point,</hi> in the Higher Geometry, is a point which is common to two parts or legs or branches of some curve of the 2d or higher order: such as, an infinitely small oval, or a cusp, or the cruciform intersection, &c, of such curves. See Newton's Enumeratio Linearum &c, de Curvarum Punctis Duplicibus.</p><p><hi rend="smallcaps">Doubling</hi> <hi rend="italics">a Cape,</hi> or <hi rend="italics">Point of Land,</hi> in Navigation, signifies the coming up with it, passing by it, and leaving it behind the ship.</p><p>The Portuguese pretend that they first doubled the Cape of Good Hope, under their admiral Vasquez de Gama; but there are accounts in history, particularly in Herodotus, that the Egyptians, Carthaginians, &c, had done the same long before them.</p></div1><div1 part="N" n="DOUCINE" org="uniform" sample="complete" type="entry"><head>DOUCINE</head><p>, in Architecture, is an ornament on the highest part of the cornice, or a moulding cut in the figure of a wave, half convex, and half concave.</p><p>DOVE-<hi rend="italics">tail,</hi> in Carpentry, is a method of fastening boards or timbers together, by letting or indenting one piece into another, with a Dove-tail joint, or a joint in form of a Dove's tail.</p></div1><div1 part="N" n="DRACHM" org="uniform" sample="complete" type="entry"><head>DRACHM</head><p>, or <hi rend="smallcaps">Dram</hi>, is the name of a small weight used with us, and is of two kinds, viz, the 8th part of an ounce in Apothecaries weight, and the 16th part of an ounce in Avoirdupois weight.</p></div1><div1 part="N" n="DRACO" org="uniform" sample="complete" type="entry"><head>DRACO</head><p>, the <hi rend="smallcaps">Dragon</hi>, a constellation of the northern hemisphere, whose origin is variously fabled by the Greeks; some of them representing it as the Dragon <cb/> which guarded the Hesperian fruit, or golden apples, but being killed by Hercules, Juno, as a reward for its faithful services, took it up to heaven, and so formed this constellation; while others say, that in the war of the giants, this Dragon was brought into the combat, and opposed to Minerva, when the goddess taking the Dragon in her hand, threw him, twisted as he was, up to the skies, and fixed him to the axis of the heavens, before he had time to unwind his contortions.</p><p>The stars in this constellation are, according to Ptolomy, 31; according to Tycho, 32; according to Hevelius, 40; according to Bayer, 33; and according to Flamsteed, 80.</p></div1><div1 part="N" n="DRAGON" org="uniform" sample="complete" type="entry"><head>DRAGON</head><p>, in Astronomy. See <hi rend="smallcaps">Draco.</hi></p><p><hi rend="smallcaps">Dragon's</hi> <hi rend="italics">Head,</hi> and <hi rend="italics">Tail,</hi> are the nodes of the planets, but more particularly of the moon, being the points in which the ecliptic is intersected by her orbit, in an angle of about 5° 18′.</p><p>One of these points looks northward, the moon beginning then to have north latitude; and the other southward, where she commences south latitude; the former point being represented by the knot <figure/> for the head, and the other by the same reversed, or <figure/> for the tail. And near these points it is that all eclipses of the sun and moon happen.</p><p>This deviation of the path of the moon from the ecliptic seems, according to the fancy of some, to make a figure like that of a dragon, whose belly is at the part where she has the greatest latitude; the intersections representing the head and tail, from which resemblance the denomination arises.</p><p>These intersections are not always in the same two points of the ecliptic, but shift by a retrograde motion, at the rate of 3′ 11″ per day, and completing their circle in 18 years 225 days.</p><p><hi rend="smallcaps">Dragon</hi>-<hi rend="italics">Beams,</hi> in Architecture, are two strong braces or struts, standing under a breast-summer, and meeting in an angle on the shoulder of the king-piece.</p><p>DRAM. See <hi rend="smallcaps">Drachm.</hi></p><p>DRAUGHT-<hi rend="italics">Compasses,</hi> those provided with several moveable points, to draw fine draughts in architecture &c. See <hi rend="smallcaps">Compass.</hi></p><p>DRAUGHT-<hi rend="italics">Hooks,</hi> are large hooks of iron, fixed on the cheeks of a<*>gun-carriage, two on each side, one near the trunnion hole, and the other at the train.</p><p>DRAW-<hi rend="italics">Bridge,</hi> a bridge made after the manner of a floor, to be drawn up, or let down, as occasion requires, before the gate of a town or castle.</p></div1><div1 part="N" n="DRIFT" org="uniform" sample="complete" type="entry"><head>DRIFT</head><p>, in Navigation, denotes the angle which the line of a ship's motion makes with the nearest meridian, when she drives with her side to the wind and waves, and is not governed by the power of the helm; and also the distance which the ship drives on that line, so called only in a storm.</p></div1><div1 part="N" n="DRIP" org="uniform" sample="complete" type="entry"><head>DRIP</head><p>, in Architecture. See <hi rend="smallcaps">Larmier.</hi></p><p>DRY-<hi rend="italics">Moat.</hi> See <hi rend="smallcaps">Moat.</hi></p></div1><div1 part="N" n="DUCTILITY" org="uniform" sample="complete" type="entry"><head>DUCTILITY</head><p>, a property of certain bodies, by which they are capable of being beaten, pressed, drawn, or stretched forth, without breaking; or by which they are capable of great alterations in their figure and dimensions, and of gaining in one way as they lose in another.</p><p>Such are metals, which, being urged by the hammer, gain in length and breadth what they lose in thickness; or, being drawn into wire through holes in iron, grow <pb n="397"/><cb/> longer as they become more slender. Such also are gums, glues, resins, and some other bodies; which, though not malleable, may yet be denominated ductile, in as much as, when softened by water, fire, or some other menstruum, they may be drawn into threads.</p><p>Some bodies are ductile both when they are hot and cold, and in all circumstances: such are metals, and especially gold and silver; other bodies are ductile only when they have a certain degree of heat; such as glass, and wax, and such like substances: others again are ductile only when cold, and brittle when hot; as some kinds of iron, viz, those called by workmen redshort, as also brass, and some metallic alloys.</p><p>The cause of ductility is very obscure; as depending much on hardness, a quality whose nature we know very little about. It is true, it is usual to account for hardness from the force of attraction between the particles of the hard body; and for ductility, from the particles of the ductile body being, as it were jointed, and entangled with each other. But without dwelling on any fanciful hypotheses about ductility, we may amuse ourselves with some truly amazing circumstances and phenomena of it, in the instances of gold, glass, and spider's-webs. Observing however that the ductility of metals decreases in the following order: gold, silver, copper, iron, tin, lead.</p><p><hi rend="smallcaps">Ductility</hi> <hi rend="italics">of Gold.</hi> One of the properties of gold is, to be the most ductile of all bodies; of which the gold beaters and gold wire-drawers, furnish us with abundant proof.</p><p>Fa. Mersenne, M. Rohault, Dr. Halley, &c, have made computations of it: but they trusted to the reports of the workmen. M. Reaumur, in the Memoires de l'Academie Royale des Sciences, an. 1713, took a surer way; he made the experiment himself. A single grain of gold, he found, even in the common gold leaf, used in most of our gildings, is extended into 36 and a half square inches; and an ounce of gold, which, in form of a cube, is not half an inch either high, broad, or long, is beat under the hammer into a surface of 146 and a half square feet; an extent almost double to what could be done in former times. In Fa. Mersenne's time, it was deemed prodigious, that an ounce of gold should form 1600 leaves; which, together, only made a surface of 105 square feet.</p><p>But the distension of gold under the hammer (how considerable soever) is nothing to that which it undergoes in the drawing-iron. There are gold leaves, in some parts scarce the 1/360000 part of an inch thick; but 1/360000 part of an inch is a considerable thickness, in comparison of that of the gold spun on silk in our gold thread.</p><p>To conceive this prodigious ductility, it is necessary to have some idea of the manner in which the wire drawers proceed. The wire, and thread we commonly call gold thread, &c, (which is only silver wire gilt, or covered over with gold), is drawn from a large ingot of silver, usually about thirty pounds weight. This they round into a cylinder, or roll, about an inch and a half in diameter, and twenty-two inches long, and cover it with the leaves prepared by the gold beater, laying one over another, till the cover is a good deal thicker than that in our ordinary gilding; and yet, even then, it is <cb/> very thin; as will be easily conceived from the quantity of gold that goes to gild the thirty pounds of silver: two ounces ordinarily do the business; and, frequently, little more than one.</p><p>In effect, the full thickness of the gold on the ingot rarely exceeds 1/400 or 1/<*>00 part; and, sometimes not 1/1000 part of an inch.</p><p>But this thin coat of gold must be yet vastly thinner: the ingot is successively drawn through the holes of several irons, each smaller than the other, till it be as sine as, or siner than a hair. Every new hole lessens its diameter; but it gains in length what it loses in thickness; and, of consequence, increases in surface: yet the gold still covers it; it follows the silver in all its extension, and never leaves the minutest part bare, not even to the microscope. Yet, how inconceivably must it be attenuated while the ingot is drawn into a thread, whose diameter is 9000 times less than that of the ingot.</p><p>M. Reaumur, by exact weighing, and rigorous calculation, found, that one ounce of the thread was 3232 feet long; and the whole ingot 1163520 feet, Paris measure, or 96 French leagues; equal to 1264400 English feet, or 240 miles English; an extent which far surpasses what Fa. Mersenne, Furetiere, Dr. Halley, &c, ever dreamt of.</p><p>Mersenne says, that half an ounce of the thread is 100 toises, or fathoms long; on which footing, an ounce would only be 1200 feet: whereas, M. Reaumur finds it 3232. Dr. Halley makes 6 feet of the wire one grain in weight, and one grain of the gold 98 yards; and, consequently, the ten thousandth part of a grain, above one third of an inch. The diameter of the wire he found one-186th part of an inch; and the thickness of the gold one-154500th part of an inch. But this, too, comes short of M. Reaumur; for, on this principle, the ounce of wire would only be 2680 feet.</p><p>But the ingot is not yet extended to its full length. The greatest part of our gold thread is spun, or wound on silk; and, before it is spun, they flat it, by passing it between two rolls, or wheels of exceedingly well polished steel; which wheels, in flatting it, lengthen it by above one seventh. So that our 240 miles are now got to 274.</p><p>The breadth, now, of these laminæ, or plates, M. Reaumur finds, is only one-8th of a line, or one-96th of an inch; and their thickness one-3072d. The ounce of gold, then, is here extended to a surface of 1190 square feet; whereas, the utmost the gold beaters can do, we have observed, is to extend it to 146 square feet. But the gold, thus exceedingly extended, how thin must it be! From M. Reaumur's calculus, it is found to be one-175000th of a line, or one-2100000th of an inch; which is scarce one-13th of the thickness of Dr. Halley's gold.</p><p>But he adds, that this supposes the thickness of the gold every where equal, which is no ways probable; for in beating the gold leaves, whatever care they can bestow, it is impossible to extend them equally. This we easily find, by the greater opacity of some parts than others; for, where the leaf is thickest, it will gild the wire the thickest. <pb n="398"/><cb/></p><p>M. Reaumur, computing what the thickness of the gold must be where thinnest, finds it only one-3150000th part of an inch. But what is the one-3150000th part of an inch? Yet this is not the utmost ductility of gold. for, instead of two ounces of gold to the ingot, which we have here computed upon, a single one might have been used; and, then, the thickness of the gold, in the thinnest places, would only be the 6300000th part of an inch.</p><p>And yet, as thin as the plates are, they might be made twice as thin, yet still be gilt; by only pressing them more between the flatter's wheels, they are extended to double the breadth and proportionably in length. So that their thickness, at last, will be reduced to one thirteen or fourteen millionth part of an inch.</p><p>Yet, with this amazing thinness of the gold, it is still a perfect cover for the silver: the best eye, or even the best microscope, cannot discover the least chasm, or discontinuity. There is not an aperture to admit alcohol of wine, the subtilest fluid in nature, or even light itself, unless it be owing to cracks occasioned by repeated strokes of the hammer. Add, that if a piece of this gold thread, or gold plate be laid to dissolve in aquafortis, the silver will be all excavated, or eat out, and the gold left entire, in little tubules.</p><p>It should be observed, that gold, when it has been struck for some time by a hammer, or violently compressed, as by gold wire drawers, becomes more hard, elastic and stiff, and less ductile, so that it is apt to be cracked or torn: the same thing happens to the other metals by percussion and compression. But ductility and tractability may be restored to metals in that state, by annealing them, or making them red hot. Gold seems to be more affected by percussion and annealing, than any other metals.</p><p>As to the <hi rend="smallcaps">Ductility</hi> <hi rend="italics">of soft bodies,</hi> it is not yet carried to that pitch. The reader, however, must not be surprised that, among the ductile bodies of this class, we give the first place to the most brittle of all other, glass.</p><p><hi rend="smallcaps">Ductility</hi> <hi rend="italics">of Glass.</hi> We all know, that, when well penetrated with the heat of the fire, the workmen can figure and manage glass like sost wax; but what is most remarkable, it may be drawn, or spun out into threads exceedingly fine and long.</p><p>Our ordinary spinners do not form their threads of silk, flax, or the like, with half the ease, and expedition, as the glass spinners do threads of this brittle matter. We have some of them used in plumes for children's heads, and divers other works, much finer than any hair, and which bend and wave like it with every wind.</p><p>Nothing is more simple and easy than the method of making them: there are two workmen employed; the first holds one end of a piece of glass over the flame of a lamp; and, when the heat has softened it, a second operator applies a glass hook to the metal thus in fusion; and, withdrawing the hook again, it brings with it a thread of glass, which still adheres to the mass: then, fitting his hook on the circumference of a wheel about two feet and a half in diameter, he turns the wheel as fast as he pleases; which, drawing out the thread winds it on its rim; till, after a certain number of revolutions, it is covered with a skain of glass thread. <cb/></p><p>The mass in fusion over the lamp diminishes insensibly: being wound out, as it were, like a pelotoon, or clue of silk, upon the wheel; and the parts, as they recede from the flame, cooling, become more coherent to those next to them; and this by degrees: the parts nearest the fire are always the least coherent, and, of consequence, must give way to the effort the rest make to draw them towards the wheel.</p><p>The circumference of these threads is usually a flat oval, being three or four times as broad as thick: some of them seem scarce bigger than the thread of a silk worm, and are surprisingly flexible. If the two ends of such threads be knotted together, they may be drawn and bent, till the aperture, or space in the middle of the knot, doth not exceed one-4th of a line, or one-48th of an inch diameter.</p><p>Hence M. Reaumur advances, that the flexibility of glass increases in proportion to the fineness of the threads; and that, probably, had we but the art of drawing threads as fine as a spider's web, we might weave stuffs and cloths of them for wear. Accordingingly, he made some experiments this way: and found he could make threads fine enough, viz, as fine, in his judgment, as spider's thread, but he could never make them long enough to do any thing with them.</p><p><hi rend="smallcaps">Ductility</hi> <hi rend="italics">of Spider's-webs.</hi> See <hi rend="smallcaps">Web.</hi></p></div1><div1 part="N" n="DUNGEON" org="uniform" sample="complete" type="entry"><head>DUNGEON</head><p>, <hi rend="smallcaps">Donjon</hi>, in Fortification, the highest part of a castle built after the ancient mode; serving as a watch-tower, or place of observation; and also for the retreat of a garrison, in case of necessity, so that they may capitulate with greater advantage.</p></div1><div1 part="N" n="DUPLE" org="uniform" sample="complete" type="entry"><head>DUPLE</head><p>, or <hi rend="smallcaps">Double</hi> <hi rend="italics">Ratio,</hi> is that in which the antecedent term is double the consequent; or, where the exponent of the ratio is 2. Thus, 6 to 3 is in a duple ratio.</p><p><hi rend="italics">Sub</hi>-<hi rend="smallcaps">Duple</hi> <hi rend="italics">Ratio,</hi> is that in which the consequent is double the antecedent; or, in which the exponent of the ratio is 1/2. As in 3 to 6, which is in subduple ratio.</p><p>DUPLICATE <hi rend="italics">Ratio,</hi> is the square of a ratio, or the ratio of the squares of two quantities. Thus, the duplicate ratio of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> is the ratio of <hi rend="italics">a</hi><hi rend="sup">2</hi> to <hi rend="italics">b</hi><hi rend="sup">2</hi>, or of the square of <hi rend="italics">a</hi> to the square of <hi rend="italics">b.</hi>—In a series of geometrical proportionals, the 1st term is to the 3d, in a duplicate ratio of the 1st to the 2d, or as the square of the first to the square of the 2d: Thus, in the geometricals 2, 4, 8, 16, &c, the ratio of 2 to 8, is the duplicate of that of 2 to 4, or as the square of 2 to the square of 4, that is, as 4 to 16. So that duplicate ratio is the ratio of the squares, as triplicate ratio is the ratio of the cubes, &c.</p></div1><div1 part="N" n="DUPLICATION" org="uniform" sample="complete" type="entry"><head>DUPLICATION</head><p>, is the doubling of a quantity, or multiplying it by 2, or adding it to itself.</p><p><hi rend="smallcaps">Duplication</hi> <hi rend="italics">of a Cube,</hi> is finding out the side of a cube that shall be double in solidity to a given cube: which is a celebrated problem, much cultivated by the ancient geometricians, about 2000 years ago.</p><p>It was first proposed by the oracle of Apollo at Delphos; which, being consulted about the manner of stopping a plague then raging at Athens, returned for answer, that the plague should cease when Apollo's altar, which was cubical, should be doubled. Upon this, they applied themselves in good earnest, to seek <pb n="399"/><cb/> the duplicature of the cube, which from thence was called the <hi rend="italics">Delian problem.</hi></p><p>This problem cannot be effected geometrically, as it requires the solution of a cubic equation, or requires the finding of two mean proportionals, viz, between the side of the given cube and the double of the same, the first of which two mean proportionals is the side of the double cube, as was first observed by Hippocrates of Chios. For, let <hi rend="italics">a</hi> be the side of the given cube, and <hi rend="italics">z</hi> the side of the double cube sought; then it is <hi rend="italics">z</hi><hi rend="sup">3</hi> = 2<hi rend="italics">a</hi><hi rend="sup">3</hi>, or <hi rend="italics">a</hi><hi rend="sup">2</hi> : <hi rend="italics">z</hi><hi rend="sup">2</hi> :: <hi rend="italics">z</hi> : 2<hi rend="italics">a</hi>; so that, if <hi rend="italics">a</hi> and <hi rend="italics">z</hi> be the first and 2d terms of a set of continued proportionals, then <hi rend="italics">a</hi><hi rend="sup">2</hi> : <hi rend="italics">z</hi><hi rend="sup">2</hi> is the ratio of the square of the 1st to the square of the 2d, which, it is known, is the same as the ratio of the 1st term to the 3d, or of the 2d to the 4th, that is of <hi rend="italics">z</hi> to 2<hi rend="italics">a</hi>; therefore <hi rend="italics">z</hi> being the 2d term, 2<hi rend="italics">a</hi> will be the 4th. So that <hi rend="italics">z,</hi> the side of the cube sought, is the 2d of four terms in continued proportion, the 1st and 4th being <hi rend="italics">a</hi> and 2<hi rend="italics">a,</hi> that is, the side of the double cube is the first of two mean proportionals between <hi rend="italics">a</hi> and 2<hi rend="italics">a.</hi></p><p>Eutocius, in his Commentaries on Archimedes, gives several ways of performing this by the mesolabe. In Pappus too are found three different ways; the first according to Archimedes, the second according to Hero, and the 3d by an instrument invented by Pappus, which gives all the proportions required. The sieur de Comiers has likewise published a demonstration of the same problem, by means of a compass with three legs. But all these methods are only mechanical. See Valerius Maximus, lib. 8; also Eutocius's Com. on lib. 2. Archimedes de Sphæra & Cylindro; and Pappus, lib. 3, prop. 5, & lib. 4, prop. 22.</p></div1><div1 part="N" n="DURER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">DURER</surname> (<foreName full="yes"><hi rend="smallcaps">Albert</hi></foreName>)</persName></head><p>, descended of an Hungarian family, but born at Nuremberg, in 1471, was one of the best engravers, painters, and practical geometricians of his age. He was at the same time a man of letters and a philosopher; and he was an intimate friend of Erasmus, who revised some of the pieces which he published. He was also a man of business, and for many years the leading magistrate of Nuremberg.</p><p>Though not the inventor, he was one of the first and greatest improvers of the art of engraving. He was however the inventor of cutting in wood, which he devised and practised in great perfection, using this way for expedition, as he had a multitude of designs to execute; and as his work was usually done in the most exquisite manner, his pieces took him up much time. For in many of those prints which he executed on copper, the engraving is elegant to a great degree. His HellScene in particular, engraven in the year 1513, is as highly finished a print as ever was engraved, and as hap- <cb/> pily executed. In his wooden prints too it is surprifing to see so much meaning in so early a master. In fact, Durer was a man of a very extensive genius. His pictures were excellent; as well as his prints, which were very numerous. They were much admired, from the first, and eagerly bought up; which put his wife, who was another Xantippe, upon urging him to spend more time upon engraving than he was inclined to do: for he was rich; and chose rather to practise his art as an amusement, than as a business. He died at Nuremberg, in 1528, at 57 years of age.</p><p>Albert Durer wrote several books, in the German language, which were translated into Latin by other persons, and published after his death. viz,</p><p>1. His book upon the rules of painting, intitled, <hi rend="italics">De Symmetria Partium in rectis formis Humanorum Corporum,</hi> is one of them: printed in folio, at Nuremberg, in 1532, and at Paris in 1557. An Italian version also was published at Venice, in 1591.</p><p>2. <hi rend="italics">Institutiones Geometricæ;</hi> Paris 1532.</p><p>3. <hi rend="italics">De Urbibus, Arcibus, Castellisque condendis & muniendis;</hi> Paris 1531.</p><p>4. <hi rend="italics">De Varietate Figurarum, et Flexuris Partium, et Gestibus Imaginum;</hi> Nuremberg 1534.</p><p>The figures in these books, which are from wooden plates, are very numerous, and most admirably well executed, indeed far beyond any thing of the kind done in our own days. Some of them also are of a very large size, as much as 16 inches in length, and of a proportional breadth, which being exquisitely worked, must have cost great labour. His geometry is chiefly of the practical kind, consisting of the most curious descriptions, inscriptions, and circumscriptions of geometrical lines, planes, and solids. We here meet, for the first time, with the plane figures, which folded up make the five regular or platonic bodies, as well as that curious construction of a pentagon, being the last method in prob. 23 of my Mensuration.</p></div1><div1 part="N" n="DYE" org="uniform" sample="complete" type="entry"><head>DYE</head><p>, in Architecture, the trunk of the pedestal, or that part between the base and the cornice, being so called, because it is often made in the form of a dye or cube.</p></div1><div1 part="N" n="DYMANICS" org="uniform" sample="complete" type="entry"><head>DYMANICS</head><p>, is the science of moving powers; more particularly of the motion of bodies that mutually act on one another. See <hi rend="smallcaps">Mechanics, Motion, Communication</hi> <hi rend="italics">of Motion,</hi> <hi rend="smallcaps">Oscillation</hi>, P<hi rend="smallcaps">ERCUSSION</hi>, &c.</p></div1><div1 part="N" n="DYPTERE" org="uniform" sample="complete" type="entry"><head>DYPTERE</head><p>, or <hi rend="smallcaps">Diptere</hi>, was a kind of temple, encompassed round with a double row of columns; and the pseudo-diptere, or false diptere, was the same, only this was encompassed with a single row of columns, instead of a double row. <pb n="400"/></p></div1></div0><div0 part="N" n="E" org="uniform" sample="complete" type="alphabetic letter"><head>E</head><cb/><div1 part="N" n="EAGLE" org="uniform" sample="complete" type="entry"><head>EAGLE</head><p>, <hi rend="italics">Aquila,</hi> is a constellation of the northern hemisphere, having its right wing contiguous to the equinoctial. For the stars in this constellation, see <hi rend="smallcaps">Aquila.</hi></p></div1><div1 part="N" n="EARTH" org="uniform" sample="complete" type="entry"><head>EARTH</head><p>, <hi rend="italics">Terra,</hi> in Natural Philosophy, one of the four vulgar, or Peripatetical elements; defined a simple, dry, and cold substance; and, as such, an ingredient in the composition of all natural bodies.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Earth</hi>, in Geography, this terraqueous globe or ball, which we inhabit, consisting of land and sea.</p><p><hi rend="italics">Figure of the</hi> <hi rend="smallcaps">Earth.</hi> The ancients had various opinions as to the figure of the earth: some, as Anaximander and Leucippus, held it cylindrical, or in form of a drum: but the principal opinion was, that it was flat; that the visible horizon was the bounds of the earth, and the ocean the bounds of the horizon; that the heavens and earth above this ocean were the whole visible universe; and that all beneath the ocean was Hades: and of this same opinion were also some of the Christian fathers, as Lactantius, St. Augustine, &c. See Lactan. lib. 3, cap. 24; St. Aug. lib. 16, de Civitate Dei; Aristotle de Cœlo, lib. 2, cap. 13.</p><p>Such of the ancients however as understood any thing of astronomy, and especially the doctrine of eclipses, must have been acquainted with the round figure of the earth; as the ancient Babylonian astronomers, who had calculated eclipses long before the time of Alexander, and Thales the Grecian, who predicted an eclipse of the sun. It is now indeed agreed on all hands, unless perhaps by the most vulgar and ignorant, that the form of the terraqueous globe is globular, or very nearly so.</p><p>That the exterior of the earth is round, or rotund, is manifest to the most common perception, in the case of a ship sailing either from the land, or towards it; for when a person stands upon the shore, and sees a ship sail from the land, out to sea; at first he loses sight of the hull and lower parts of the ship, next the rigging and middle parts, and lastly of the tops of the masts themselves, in every case the rotundity of the sea between the ship and the eye being very visible: the contrary happens when a ship sails towards us; we first see the tops of the masts appear just over the rotundity of the sea; next we perceive the rigging, and lastly the hull of the ship itself: all which is well illustrated by the following figure. <cb/> <figure/></p><p>The round figure of the earth is also evident from the eclipses of the sun and moon; for in all eclipses of the moon, which are caused by the moon passing through the earth's shadow, that shadow always appears circular upon the face of the moon, what way soever it be projected, whether east, west, north, or south, and howsoever its diameter vary, according to the greater or less distance from the earth. Hence it follows, that the shadow of the earth, in all situations, is really conical; and consequently the body that projects it, i. e. the earth, is at least nearly spherical.</p><p>The spherical figure of the earth is also evinced from the rising and setting of the sun, moon, and stars; all which happen sooner to those who live to the east, and later to those living westwardly; and that more or less so, according to the distance and roundness of the earth.</p><p>So also, going or sailing to the northward, the north pole and northern stars become more elevated, and the south pole and southern stars more depressed; the elevation northerly increasing equally with the depression southerly; and either of them proportionably to the distance gone. The same thing happens in going to the southward. Besides, the oblique ascensions, de- <pb n="401"/><cb/> scensions, emersions, and amplitudes of the rising and setting of the sun and stars in every latitude, are agreeable to the supposition of the earth's being of a spherical form: all which could not happen if it was of any other figure.</p><p>Moreover, the roundness of the earth is farther confirmed by its having been often sailed round: the first time was in the year 1519, when Ferd. Magellan made the tour of the whole globe in 1124 days. In the year 1557 Francis Drake performed the same in 1056 days: in the year 1586, Sir Tho. Cavendish made the same voyage in 777 days; Simon Cordes, of Rotterdam, in the year 1590; in the year 1598, Oliver Noort, a Hollander, in 1077 days; Van Schouten, in the year 1615, in 749 days; Jac. Heremites and Joh. Huygens, in the year 1623, in 802 days: and many others have since performed the same navigation, particularly Anson, Bougainville, and Cook. Sometimes failing round by the eastward, sometimes to the westward; till at length they arrived again in Europe, from whence they set out; and in the course of their voyage, observed that all the phenomena, both of the heavens and earth, correspond to, and evince this spherical figure.</p><p>The same globular figure is likewise inferred from the operation of Levelling, in which it is found necessary to make an allowance for the difference between the apparent and the true level.</p><p>The natural cause of this sphericity of the globe is, according to Sir Isaac Newton, the great principle of attraction, which the Creator has stamped on all the matter in the universe; and by which all bodies, and all the parts of bodies, mutually attract one another.— And the same is the cause of the sphericity of the drops of rain, quicksilver, &c.</p><p>What the earth loses of its sphericity by mountains and valleys, is nothing considerable; the highest eminence being scarce equivalent to the minutest protuberance on the surface of an orange. Its difference from a perfect sphere however is more considerable in another respect, by which it approaches nearly to the shape of an orange, or to an oblate spheroid, being a little flatted at the poles, and raised about the equatorial parts, so that the axis from pole to pole is less than the equatorial diameter. What gave the first occasion to the discovery of this figure of the earth, was the observations of some French and English philosophers in the East-Indies, and other parts, who found that pendulums, the nearer they came to the equator, performed their vibrations slower: from whence it follows, that the velocity of the descent of bodies by gravity, is less in countries nearer to the equator; and consequently that those parts are farther removed from the centre of the earth, or from the common centre of gravity. See the History of the Royal Academy of Sciences, by Du Hamel, p. 110, 156, 206; and l'Hist. de l'Acad. Roy. 1700 and 1701.—This circumstance put Huygens and Newton upon sinding out the cause, which they attributed to the revolution of the earth about its axis. If the earth were in a fluid state, its rotation round its axis would necessarily make it put on such a figure, because the centrifugal force being greatest towards the equator, the fluid would there rise and swell most; and that its figure really should be so now, seems necessary, to keep the sea in the equinoctial re- <cb/> gions from overflowing the earth about those parts. See this curious subject well handled by Huygens, in his discourse De Causa Gravitatis, pa. 154, where he states the ratio of the polar diameter to that of the equator, as 577 to 578. And Newton, in his Principia, first published in 1686, demonstrates, from the theory of gravity, that the figure of the earth must be that of an oblate spheroid generated by the rotation of an ellipse about its shortest diameter, provided all the parts of the earth were of an uniform density throughout, and that the proportion of the polar to the equatorial diameter of the earth, would be that of 689 to 692, or nearly that of 229 to 230, or as .9956522 to 1.</p><p>This proportion of the two diameters was calculated by Newton in the following manner. Having found that the centrifugal force at the equator is 1/289th of gravity, he assumes, as an hypothesis, that the axis of the earth is to the diameter of the equator as 100 to 101, and thence determines what must be the centrifugal force at the equator to give the earth such a form, and finds it to be 4/505ths of gravity: then, by the rule of proportion, if a centrifugal force equal to 4/805ths of gravity would make the earth higher at the equator than at the poles by 1/100th of the whole height at the poles, a centrifugal force that is the 1/289th of gravity will make it higher by a proportional excess, which by calculation is 1/229th of the height at the poles; and thus he discovered that the diameter at the equator is to the diameter at the poles, or the axis, as 230 to 229. But this computation supposes the earth to be every where of an uniform density; whereas if the earth is more dense near the centre, then bodies at the poles will be more attracted by this additional matter being nearer; and therefore the excess of the semi-diameter of the equator above the semi-axis, will be different. According to this proportion between the two diameters, Newton farther computes, from the different measures of a degree, that the equatorial diameter will exceed the polar, by 34 miles and 1/5.</p><p>Nevertheless, Messrs Cassini, both father and son, the one in 1701, and the other in 1713, attempted to prove, in the Memoirs of the Royal Academy of Sciences, that the earth was an oblong spheroid; and in 1718, M. Cassini again undertook, from observations, to shew that, on the contrary, the longest diameter passes through the poles; which gave occasion for Mr. John Bernoulli, in his Essai d'une Nouvelle Physique Celeste, printed at Paris in 1735, to triumph over the British philosopher, apprehending that these observations would invalidate what Newton had demonstrated. And in 1720, M. De Mairan advanced arguments, supposed to be strengthened by geometrical demonstrations, farther to confirm the assertions of Cassini. But in 1735 two companies of mathematicians were employed, one for a northern, and another for a southern expedition, the result of whose observations and measurement plainly proved that the earth was flatted at the poles. See <hi rend="smallcaps">Degree.</hi></p><p>The proportion of the equatorial diameter to the polar, as stated by the gentlemen employed on the northern expedition for measuring a degree of the meridian, is as 1 to 0.9891; by the Spanish mathematicians as 266 to 265, or as 1 to 0.99624; by M. Bouguer as 179 to 178, or as 1 to 0.99441.</p><p>As to all conclusions however deduced from the <pb n="402"/><cb/> length of pendulums in different places, it is to be observed that they proceed upon the supposition of the uniform density of the earth, which is a very improbable circumstance; as justly observed by Dr. Horsley in his letter to Capt. Phipps. “You sinish your article, he concludes, relating to the pendulum with saying, ‘that these observations give a figure of the earth nearer to Sir Isaac Newton's computation, than any others that have hitherto been made;’ and then you state the several figures given, as you imagine, by former observations, and by your own. Now it is very true, that <hi rend="italics">if</hi> the meridians be ellipses, or <hi rend="italics">if</hi> the figure of the earth be that of a spheroid generated by the revolution of an ellipsis, turning on its shorter axis, the particular figure, or the ellipticity of the generating ellipsis, which your observations give, is nearer to what Sir Isaac Newton saith it should be, if the globe were homogeneous, than any that can be derived from former observations. But yet it is not what you imagine. Taking the gain of the pendulum in latitude 79° 50′ exactly as you state it, the difference between the equatorial and the polar diameter, is about as much less than the Newtonian computation makes it, and the hypothesis of homogeneity would require, as you reckon it to be greater. The proportion of 212 to 211 should indeed, according to your observations, be the proportion of the force that acts upon the pendulum at the poles, to the force acting upon it at the equator. But this is by no means the same with the proportion of the equatorial diameter to the polar. If the globe were homogeneous, the equatorial diameter would exceed the polar by 1/230 of the length of the latter: and the polar force would also exceed the equatorial by the like part. But if the difference between the polar and equatorial force be greater than 1/230, (which may be the case in an heterogeneous globe, and seems to be the case in ours,) then the difference of the diameters should, according to theory, be less than 1/230, and vice versa.</p><p>“I confess this is by no means obvious at first sight; so far otherwise, that the mistake, which you have fallen into, was once very general. Many of the best mathematicians were misled by too implicit a reliance upon the authority of Newton, who had certainly consined his investigations to the homogeneous spheroid, and had thought about the heterogeneous only in a loofe and general way. The late Mr. Clairault was the first who set the matter right, in his elegant and subtle treatise on the figure of the earth. That work hath now been many years in the hands of mathematicians, among whom I imagine there are none, who have considered the subject attentively, that do not acquiesce in the author's conclusions.</p><p>“In the 2d part of that treatise, it is proved, that putting <foreign xml:lang="greek">*r</foreign> for the polar force, <foreign xml:lang="greek">*p</foreign> for the equatorial, <foreign xml:lang="greek">d</foreign> for the true ellipticity of the earth's figure, and <foreign xml:lang="greek">e</foreign> for the ellipticity of the homogeneous spheriod, : therefore ; and therefore, according to your observation, <foreign xml:lang="greek">d</foreign>=1/251. This is the just conclusion from your observations of the pendulum, taking it for granted, that the meridians are ellipses: which is an hypothesis, upon which all the reasonings of theory have hitherto proceeded. But plausible as it may seem, I must say, that there is much <cb/> reason from experiment to call it in question. If it were true, the increment of the force which actuates the pendulum, as we approach the poles, should be as the square of the sine of the latitude: or, which is the same thing, the decrement, as we approach the equator, should be as the square of the cosine of the latitude. But whoever takes the pains to compare together such of the observations of the pendulum in different latitudes, as seem to have been made with the greatest care, will find that the increments and decrements do by no means follow these proportions; and in those which I have examined, I sind a regularity in the deviation which little resembles the mere error of observation. The unavoidable conclusion is, that the true figure of the meridians is not elliptical. If the meridians are not ellipses, the difference of the diameters may indeed, or it may not, be proportional to the difference between the polar and the equatorial force; but it is quite an uncertainty, what relation subsists between the one quantity and the other; our whole theory, except so far as it relates to the homogeneous spheroid, is built upon false assumptions, and there is no saying what figure of the earth any observations of the pendulum give.”</p><p>He then lays down the following table, which shews the different results of observations made in different latitudes; in which the first three columns contain the names of the several observers, the places of observation, and the latitude of each; the 4th column shews the quantity of <foreign xml:lang="greek">*r</foreign>-<foreign xml:lang="greek">*p</foreign> in such parts as <foreign xml:lang="greek">*p</foreign> is 100000, as deduced from comparing the length of the pendulum at each place of observation, with the length of the equatorial pendulum as determined by M. Bouguer, upon the supposition that the increments and decrements of force, as the latitude is increased or lowered, observe the proportion which theory assigns. Only the 2d and the last value of <foreign xml:lang="greek">*r</foreign>-<foreign xml:lang="greek">*p</foreign> are concluded from comparisons with the pendulum at Greenwich and at London, not at the equator. The 5th column shews the value of <foreign xml:lang="greek">d</foreign> corresponding to every value of <foreign xml:lang="greek">*r</foreign>-<foreign xml:lang="greek">*p</foreign>, according to Clairault's theorem: <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">Observers.</hi></cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">Places.</hi></cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">Latitudes.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><foreign xml:lang="greek">*r</foreign>-<foreign xml:lang="greek">*p</foreign></cell><cell cols="1" rows="1" rend="align=center" role="data"><foreign xml:lang="greek">d</foreign></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Bouguer</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Equator</cell><cell cols="1" rows="1" rend="align=right" role="data">0°</cell><cell cols="1" rows="1" rend="align=right" role="data">0′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Bouguer</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Porto Bello</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">741.8</cell><cell cols="1" rows="1" role="data">1/784</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Green</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Otaheitee</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">563.2</cell><cell cols="1" rows="1" role="data">1/326</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Bouguer</cell><cell cols="1" rows="1" rend="colspan=2" role="data">San Domingo</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">591.0</cell><cell cols="1" rows="1" role="data">1/368</cell></row><row role="data"><cell cols="1" rows="1" role="data">Abbé de la Caille</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" role="data">Cape of Good Hope</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">731.5</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">"</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Paris</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">585.1</cell><cell cols="1" rows="1" role="data">1/361</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Academicians</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data">Pello</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">565.9</cell><cell cols="1" rows="1" role="data">1/329</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Capt. Phipps</cell><cell cols="1" rows="1" rend="colspan=2" role="data">"</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">471.2</cell><cell cols="1" rows="1" role="data">1/2<*>1</cell></row></table></p><p>“By this table it appears, that the observations in the middle parts of the globe, setting aside the single one at the Cape, are as consistent as could reasonably be expected; and they represent the ellipticity of the earth as about 1/340. But when we come within 10 degrees of the equator, it should seem that the force of gravity suddenly becomes much less, and within the like <pb n="403"/><cb/> distance of the poles much greater than it could be in such a spheroid.”</p><p>The following problem, communicated by Dr Leatherland to Dr. Pemberton, and published by Mr. Robertson, serves for finding the proportion between the axis and the equatorial diameter, from measures taken of a degree of the meridian in two different latitudes, supposing the earth an oblate spheroid. <figure/></p><p>Let AP<hi rend="italics">ap</hi> be an ellipse representing a section of the earth through the axis P<hi rend="italics">p</hi>; the equatorial diameter, or the greater axis of the ellipse, being A<hi rend="italics">a</hi>; let E and F be two places where the measure of a degree has been taken; these measures are proportional to the radii of curvature in the ellipse at those places; and if CQ, CR be conjugates to the diameters whose vertices are E and F, CQ will be to CR in the subtriplicate ratio of the radius of curvature at E to that at F, by Cor. 1, prop. 4, part 6 of Milnes's Conic Sections, and therefore in a given ratio to one another; also the angles QCP, RCP are the latitudes of E and F; so that, drawing QV parallel to P<hi rend="italics">p,</hi> and QXYW to A<hi rend="italics">a,</hi> these angles being given, as well as the ratio of CQ to CR, the rectilinear figure CVQXRY is given in species; and the ratio of to is given, which is the ratio of CA<hi rend="sup">2</hi> to CP<hi rend="sup">2</hi>; therefore the ratio of CA to CP is given.</p><p>Hence, if the sine and cosine of the greater latitude be each augmented in the subtriplicate ratio of the measure of the degree in the greater latitude to that in the lesser, then the difference of the squares of the augmented sine, and the sine of the lesser latitude, will be to the difference of the squares of the cosine of the lesser latitude and the augmented cosine, in the duplicate ratio of the equatorial to the polar diameter. For, C<hi rend="italics">q</hi> being taken in CQ equal to CR, and <hi rend="italics">qv</hi> drawn parallel to QV, C<hi rend="italics">v</hi> and <hi rend="italics">vq,</hi> CZ and ZR will be the signs and cosines of the respective latitudes to the same radius; and CV, VQ will be the augmentations of C<hi rend="italics">v</hi> and C<hi rend="italics">q</hi> in the ratio named.</p><p>Hence, to find the ratio between the two axes of the earth, let E denote the greater, and F the lesser of the two latitudes, M and N the respective measures taken in each; and let P denote √<hi rend="sup">3</hi>M/N: then . <cb/></p><p>It also appears by the above problem, that when one of the degrees measured, is at the equator, the cosine of the latitude of the other being augmented in the subtriplicate ratio of the degrees, the tangent of the latitude will be to the tangent answering to the augmented cosine, in the ratio of the greater axis to the less. For supposing E the place out of the equator; then if the semi-circle P<hi rend="italics">lmnp</hi> be described, and <hi rend="italics">l</hi>C joined, and <hi rend="italics">mo</hi> drawn parallel to <hi rend="italics">a</hi>C: C<hi rend="italics">o</hi> is the cosine of the latitude to the radius CP, and CY that cosine augmented in the ratio before-named; YQ being to Y<hi rend="italics">l,</hi> that is <hi rend="italics">Ca</hi> to C<hi rend="italics">n</hi> or CP, as the tangent of the angle YCQ, the latitude of the point E, to the tangent of the angle YC<hi rend="italics">l,</hi> belonging to the augmented cosine. Thus, if M represent the measure in a latitude denoted by E, and N the measure at the equator, let A denote an angle whose measure is Then .</p><p>But M, or the length of a degree, obtained by actual mensuration in different latitudes, is known from the following table: <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">Names.</hi></cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">Latit.</hi></cell><cell cols="1" rows="1" rend="colspan=3 align=left" role="data"><hi rend="italics">Value of M.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">°</cell><cell cols="1" rows="1" rend="align=right" role="data">′</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">toises</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Maupertuis and Assoc.</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">M = 57438</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">Cassini and La Caille</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">M = 57074</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">M = 57050</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Boscovich</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">M = 56972</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">De la Caille</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">M = 57037</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Juan and Ulloa</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">M = 56768</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(7)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=3" role="data">at the equator.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Bouguer</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">M = 56753</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Condamine</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">M = 56749</cell></row></table></p><p>Now, by comparing the 1st with each of the following ones; the 2d with each of the following; and in like manner the 3d, 4th, and 5th, with each of the following; there will be obtained 25 results, each shewing the relation of the axes or diameters; the arithmetical means of all of which will give that ratio as 1 to 0.9951989.</p><p>If the measures of the latitudes of 49° 22′, and of 45°, which fall within the meridian line drawn through France, and which have been re-examined and corrected since the northern and southern expedition, be compared with those of Maupertuis and his associates in the north, and that of Bouguer at the equator, there will result 6 different values of the ratio of the two axes, the arithmetical mean of all which is that of 1 to 0.9953467, which may be considered as the ratio of the greater axis to the less; which is as 230 to 228.92974, or 215 to 214, or very near the ratio as assigned by Newton.</p><p>Now, the magnitude as well as the sigure of the earth, that is the polar and equatorial diameters, may be deduced from the foregoing problem. For, as half the latus rectum of the greater axis A<hi rend="italics">a</hi> is the radius of curvature at A, it is given in magnitude from the degree measured there, and thence the axes themselves are given. Thus, the circular are whose length is equal to the radius being 57.29578 degrees, if this number be multiplied by 56750 toises, the measure of a degree at <pb n="404"/><cb/> the equator, as Bouguer has stated it, the product will be the radius of curvature there, or half the latus rectum of the greater axis; and this is to half the lesser axis in the ratio of the less axis to the greater, that is, as 0.9953467 to 1; whence the two axes are 6533820 and 6564366 toises, or 7913 and 7950 English miles; and the difference between the two axes about 37 miles. See Robertson's Navigation, vol. 2, pa. 206 &c. See also Suite des Mem. de l'Acad. 1718, pa. 247, and Maclaurin's Fluxions, vol. 2, book 1, ch. 14.</p><p>And very nearly the same ratio is deduced from the lengths of pendulums vibrating in the same time, in different latitudes; provided it be again allowed that the meridians are real ellipses, or the earth a true spheroid, which however can only take place in the case of an uniform gravity in all parts of the earth.</p><p>Thus, in the new Petersburgh Acts, for the years 1788 and 1789, are accounts and calculations of experiments relative to this subject, by M. Krafft. These experiments were made at different times and in various parts of the Russian empire. This gentleman has collected and compared them, and drawn the proper conclusions from them: thus he infers that the length <hi rend="italics">x</hi> of a pendulum that swings seconds in any given latitude <foreign xml:lang="greek">l</foreign>, and in a temperature of 10 degrees of Reaumur's thermometer, may be determined by this equation:</p><p> sine <hi rend="sup">2</hi><foreign xml:lang="greek">l</foreign>, lines of a French foot, or sine <hi rend="sup">2</hi><foreign xml:lang="greek">l</foreign>, in English inches, in the temperature of 53 of Fahrenheit's thermometer.</p><p>This expression nearly agrees, not only with all the experiments made on the pendulum in Russia, but also with those of Mr. Graham in England, and those of Mr. Lyons in 79° 50′ north latitude, where he found its length to be 431.38 lines. It also shews the augmentation of gravity from the equator to the parallel of a given latitude <foreign xml:lang="greek">l</foreign>: for, putting <hi rend="italics">g</hi> for the gravity under the equator, G for that under the pole, and <hi rend="italics">y</hi> for that under the latitude <foreign xml:lang="greek">l</foreign>, M. Krafft finds ; and theref. G= 1.0052848 <hi rend="italics">g.</hi></p><p>From this proportion of gravity under different latitudes, the same author infers that, in case the earth is a homogeneous ellipsoid, its oblateness must be 1/191; instead of 1/2<*>0, which ought to be the result of this hypothesis: but on the supposition that the earth is a heterogeneous ellipsoid, he finds its oblateness, as deduced from these experiments, to be 1/297; which agrees with that resulting from the measurement of some of the degrees of the meridian. This confirms an observation of M. De la Place, that, if the hypothesis of the earth's homogeneity be given up, then theory, the measurement of degrees of latitude, and experiments with the pendulum, all agree in their result with respect to the oblateness of the earth. See Memoires de l' Acad. 1783, pa. 17.</p><p>In the Philos. Trans. for 1791, pa. 236, Mr. Dalby has given some calculations on measured degrees of the meridian, from whence he infers, that those degrees measured in middle latitudes, will answer nearly to an ellipsoid whose axes are in the ratio assigned by Newton, viz, that of 230 to 229. And as to the deviations of some of the others, viz, towards the poles <cb/> and equator, he thinks they are caused by the errors in the observed celestial arcs.</p><p>Tacquet draws some pretty little inferences, in the form of paradoxes, from the round figure of the earth; as, 1st, That if any part of the surface of the earth were quite plane, a man could no more walk upright upon it, than on the side of a mountain. 2d, That the traveller's head goes a greater space than his feet; and a horseman than a footman; as moving in a greater circle. 3d, That a vessel, full of water, being raised perpendicularly, some of the water will be continually flowing out, yet the vessel still remain full; and on the contrary, if a vessel of water be let perpendicularly down, though nothing flow out, yet it will cease to be full: consequently there is more water contained in the same vessel at the foot of a mountain, than on the top; because the surface of the water is compressed into a segment of a smaller sphere below than above. Tacq. Astron. lib. 1, cap. 2.</p><p><hi rend="italics">Changes of the</hi> <hi rend="smallcaps">Earth.</hi> Mr. Boyle suspects that there are great, though slow internal changes, in the mass of the earth. He argues from the varieties observed in the change of the magnetic needle, and from the observed changes in the temperature of climates. But as to the latter, there is reason to doubt that he could not have diaries of the weather sufficient to direct his judgment. Boyle's Works abr. vol. 1, pa. 292, &c.</p><p><hi rend="italics">Magnetism of the</hi> <hi rend="smallcaps">Earth.</hi> The notion of the magnetism of the earth was started by Gilbert; and Boyle supposes magnetic effluvia moving from one pole to the other. See his Works abr. vol. 1, pa. 285, 290.</p><p>Dr. Knight also thinks that the earth may be considered as a great loadstone, whose magnetical parts are disposed in a very irregular manner; and that the south pole of the earth is analogous to the north pole in magnets, that is, the pole by which the magnetical stream enters. See <hi rend="smallcaps">Magnet.</hi></p><p>He observes, that all the phenomena attending the direction of the needle, in different parts of the earth, in great measure correspond with what happens to a needle, when placed upon a large terrella; if we make allowances for the different dispositions of the magnetical parts, with respect to each other, and consider the south pole of the earth as a north pole with regard to magnetism. The earth might become magnetical by the iron ores it contains, for all iron ores are capable of magnetism. It is true, the globe might notwithstanding have remained unmagnetical, unless some cause had existed capable of making that repellent matter producing magnetism move in a stream through the earth.</p><p>Now the doctor thinks that such a cause does exist. For if the earth revolves round the sun in an ellipsis, and the south pole of the earth is directed towards the sun, at the time of its descent towards it, a stream of repellent matter will thence be made to enter at the south pole, and issue out at the north. And he suggests, that the earth's being in its perihelion in winter may be one reason why magnetism is stronger in this season than in summer.</p><p>This cause here assigned for the earth's magnetism must continue, and perhaps improve it, from year to year. Hence the doctor thinks it probable, that the earth's magnetism has been improving ever since the creation, and that this may be one reason why the use <pb n="405"/><cb/> of the compass was not discovered sooner. See Dr. Knight's Attempt to demonstrate, that all the phenomena in nature may be explained by Attraction and Repulsion, prop. 87.</p><p><hi rend="italics">Magnitude and Constitution of the</hi> <hi rend="smallcaps">Earth.</hi> This has been variously determined by different authors, both ancient and modern. The usual way has been, to measure the length of one degree of the meridian, and multiply it by 360, for the whole circumference. See <hi rend="smallcaps">Degree.</hi> Diogenes Laertius informs us, that Anaximander, a scholar of Thales, who lived about 550 years before the birth of Christ, was the first who gave an account of the circumference of the sea and land; and it seems his measure was used by the succeeding mathematicians, till the time of Eratosthenes. Aristotle, at the end of lib. 2 De Cœlo, says, the mathematicians who have attempted to measure the circuit of the earth, make it 40000 stadiums: which it is thought is the number determined by Anaximander.</p><p>Eratosthenes, who lived about 200 years before Christ, was the next who undertook this business; which, as Cleomedes relates, he performed by taking the sun's zenith distances, and measuring the distance between two places under the same meridian; by which he deduced for the whole circuit about 250000 stadiums, which Pliny states at 31500 Roman miles, reckoning each at 1000 paces. But this measure was accounted false by many of the ancient mathematicians, and particularly by Hipparchus, who lived 100 years afterwards, and who added 25000 stadiums to the circuit of Eratosthenes.</p><p>Possidonius, in the time of Cicero and Pompey the Great, next measured the earth, viz, by means of the altitudes of a star, and measuring a part of a meridian; and he concluded the circumference at 240000 stadiums, according to Cleomedes, but only at 180000 according to Strabo.</p><p>Ptolomy, in his Geography, says that Marinus, a celebrated geographer, attempted something of the same kind; and, in lib. 1, cap. 3, he mentions that he himself had tried to perform the business in a way different from any other before him, which was by means of places under different meridians: but he does not say how much he made the number; for he still made use of the 180000, which had been found out before him.</p><p>Snellius relates, from the Arabian Geographer Abelfedea, who lived about the 1300th year of Christ, that about the 800th year of Christ, Almaimon, an Arabian king, having collected together some skilful mathematicians, commanded them to find out the circumference of the earth. Accordingly these made choice of the fields of Mesopotamia, where they measured under the same meridian from north to south, till the pole was depressed one degree lower: which measure they found equal to 56 miles, or 56 1/2: so that according to them the circuit of the earth is 20160 or 20340 miles.</p><p>It was a long time after this before any more attempts were made in this business. At length however, the same Snell, above mentioned, professor of mathematies at Leyden, about the year 1620, with great skill and labour, by measuring large distances between two parallels, found one degree equal to 28500 perches, each of which is 12 Rhinland feet, amounting to 19 <cb/> Dutch miles, and so the whole periphery 6840 miles; a mile being, according to him, 1500 perches, or 18000 Rhinland feet. See his treatise called <hi rend="italics">Eratosthenes Batavus.</hi></p><p>The next that undertook this measurement, was Richard Norwood, who in the year 1635, by measuring the distance from London to York with a chain, and taking the sun's meridian altitude, June 11th old style, with a sextant of about 5 feet radius, found a degree contained 367200 feet, or 69 miles and a half and 14 poles; and thence the circumference of a great circle of the earth is a little more than 25036 miles, and the diameter a little more than 7966 miles. See the particulars of this measurement in his <hi rend="italics">Seaman's Practice.</hi></p><p>The measurement of the earth by Snell, though very ingenious and troublesome, and much more accurate than any of the ancients, being still thought by some French mathematicians, as liable to certain small errors, the business was renewed, after Snell's manner, by Picard and other mathematicians, by the king's command; using a quadrant of 3 1/6 French feet radius; by which they found a degree contained 342360 French feet. See Picard's treatise, <hi rend="italics">La Mesure de la Terre.</hi></p><p>M. Cassini the younger, in the year 1700, by the king's command also, renewed the business with a quadrant of 10 feet radius, for taking the latitude, and another of 3 1/8 feet for taking the angles of the triangles; and found a degree, from his calculation, contained 57292 toises, or almost 69 1/2 English miles.</p><p>See the results of many other measurements under the article <hi rend="smallcaps">Degree.</hi> From the mean of all which, the following dimensions may be taken as near the truth: <table><row role="data"><cell cols="1" rows="1" role="data">the circumference</cell><cell cols="1" rows="1" rend="align=center" role="data">25000 miles,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the diameter</cell><cell cols="1" rows="1" rend="align=center" role="data">7957 3/4 miles,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the superficies</cell><cell cols="1" rows="1" rend="align=center" role="data">198944206 square miles,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the solidity</cell><cell cols="1" rows="1" rend="align=center" role="data">263930000000 cubic miles.</cell></row></table> Also the seas and unknown parts of the earth, by a measurement of the best maps, contain 160522026 square miles, the inhabited parts 38922180; of which Europe contains 4456065; Asia, 10768823; Africa, 9654807; and America, 14110874.</p><p>It is now generally granted that the terraqueous globe has two motions, besides that on which the precession of the equinoxes depends; the one diurnal around its own axis in the space of 24 hours, which constitutes the natural day or nycthemeron; the other annual, about the sun, in an elliptical orbit or track, in 365 days 6 hours, constituting the year. From the former arise the diversities of night and day; and from the latter, the vicissitudes of seasons, spring, summer, autumn, winter.</p><p>The terraqueous globe is distinguished into three parts or regions, viz, 1st, The external part or crust, being that from which vegetables spring and animals are nursed. 2d, The middle, or intermediate part, which is possessed by fossils, extending farther than human labour ever yet penetrated. 3d, The internal or central part, which is utterly unknown to us, though by many authors supposed of a magnetic nature; by others, a mass or sphere of fire; by others, an abyss or collection of waters, surrounded by the strata of earth; and by others, a hollow, empty space, inhabited by animals, who have their sun, moon, planets, and <pb n="406"/><cb/> other conveniences within the same. But others divide the body of the globe into two parts, viz, the external part, called the cortex, including the internal, which they call the nucleus, being of a different nature from the former, and possessed by fire, water, or more probably by a considerable portion of metals, as it has been found, by calculation, that the mean density of the whole earth is near double the density of common stone. See my determination of it, Philos. Trans. 1778, pa. 781.</p><p>The external part of the globe either exhibits inequalities, as mountains and valleys; or it is plane and level; or dug in channels, fissures, beds, &c, for rivers, lakes, seas, &c. These inequalities in the face of the earth most naturalists suppose have arisen from a rupture or subversion of the earth, by the force either of the subterraneous sires or waters. The earth, in its natural and original state, it has been supposed by Des Cartes, and after him, Burnet, Steno, Woodward, Whiston, and others, was perfectly round, smooth, and equable; and they account for its present rude and irregular form, principally from the great deluge. See <hi rend="smallcaps">Deluge.</hi></p><p>In the external, or cortical part of the earth, there appear various strata, supposed the sediments of several sloods; the waters of which, being replete with matters of divers kinds, as they dried up, or oozed through, deposited these different matters, which in time hardened into strata of stone, sand, coal, clay, &c.</p><p>Dr. Woodward has considered the circumstances of these strata with great attention, viz, their order, number, situation with respect to the horizon, depth, intersections, fissures, colour, consistence, &c. He ascribes the origin and formation of them all, to the great flood or cataclysmus. At that terrible revolution he supposed that all sorts of terrestrial bodies had been dissolved and mixed with the waters, forming all together a chaos or confused mass. This mass of terrestrial particles, intermixed with water, he supposes was at length precipitated to the bottom; and that generally according to the order of gravity, the heaviest sinking first, and the lightest afterwards. By such means were the strata formed of which the earth consists; which, attaining their solidity and hardness by degrees, have continued so ever since. These sediments, he farther concludes, were at first all parallel and concentrical; and the surface of the earth formed of them, perfectly smooth and regular; but that in course of time, divers changes happening, from earthquakes, volcanos, &c, the order and regularity of the strata was disturbed and broken, and the surface of the earth by such means brought to the irregular form in which it now appears.</p><p>M. De Buffon surmises that the earth, as well as the other planets, are parts struck off from the body of the sun by the collision of comets; and that when the earth assumed its form, it was in a state of liquefaction by fire. But that could not be the method of producing the planets; for if they were struck off from the body of the sun, they would move in orbits that would always pass through the sun, instead of having the sun for their socus, or centre, as they are now <cb/> found; so that having been struck off they would fall down into the sun again, terminating their career as it were after one revolution only.</p><div2 part="N" n="Earth" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Earth</hi></head><p>, <hi rend="italics">in Astronomy,</hi> is one of the primary planets, according to the system of Copernicus, or Pythagoras; its astronomical character or mark being <figure/>: but according to the Ptolomaic hypothesis, the earth is the centre of the system. For, whether the earth move or remain at rest, that is, whether it be fixed in the centre, having the sun, the heavens and stars moving round it from east to west; or, these being at rest, whether the earth only moves from west to east, is the great article that distinguishes the Ptolomaic system from the Copernican.</p><p><hi rend="italics">Motion of the</hi> <hi rend="smallcaps">Earth.</hi> It is now universally agreed that, besides the small motion of the earth which causes the precession of the equinoxes, the earth has two great and independent motions; viz, the one by which it turns round its own axis, in the space of 24 hours nearly, and causing the continual succession of day and night; and the other an absolute motion of its whole mass in a large orbit about the sun, having that luminary for its centre, in such manner that its axis keeps always parallel to itself, inclined in the same angle to its path, and by that means causing the vicissitudes of seasons, spring, summer, autumn, winter.</p><p>It is indeed true that, as to sense, the earth appears to be fixed in the centre, with the sun, stars and heavens moving round it every day; and such doubtless would be considered as the true nature of the motions in the rude ages of mankind, as they are still by the rude and unlearned. But to a thinking and learned mind, the contrary will soon appear.</p><p>Indeed there are traces of the knowledge of these motions in the earliest age of the sciences. Cicero, in his Tusc. Quæst. says that Nicetas of Syracuse first discovered that the earth had a diurnal motion, by which it revolved round its axis every 24 hours; and Plutarch, de Placit. Philosoph. informs, that Philolaus discovered its annual motion round the sun; and Aristarchus, about 100 years after Philolaus, proposed the motion of the earth in stronger and clearer terms, as we are assured by Archimedes, in his Arenarius. And the same, we are farther assured, was the opinion and doctrine of Pythagoras.</p><p>But the religious opinions of the heathen world prevented this doctrine from being more cultivated. For, Aristarchus being accused of sacrilege by Cleanthes for moving Vesta and the tutelar deities of the universe out of their places, the philosophers were obliged to dissemble, and seem to relinquish so perilous a position.</p><p>Many ages afterwards, Nic. Cusanus revived the ancient system, in his Doct. de Pignorant. and asserted the motion of the earth: but the doctrine gained very little ground till the time of Copernicus, who shewed its great use and advantages in astronomy; and who had immediately all the philosophers and astronomers on his side, who dared to differ from the crowd, and were not afraid of ecclesiastical censure, which was not less dangerous under the christian dispensation, than it had been under that of the heathen. For, because certain parts of scripture make mention of the stability <pb n="407"/><cb/> of the earth, and of the motion of the sun, as the rising and setting, &c, the fathers of the church thought their religion required that they should defend, with all its power, what they conceived to be its doctrines, and to censure and punish every attempt at innovation on such points. They have now however been pretty generally convinced that in such instances the expressions are only to be considered as accommodated to appearances, and the vulgar notions of things.</p><p>By the diurnal rotation of the earth on its axis, the same phenomena will take place as if it had no such motion, and as if the sun and stars moved round it. For, turning round from west to east, causes the sun and all the visible heavens to seem to move the contrary way, or from east to west, as we daily see them do. So, when in its rotation it has brought the sun or a star to appear just in the horizon in the east, they are then said to be rising; and as the earth continues to revolve more and more towards the east, other stars seem to rise and advance westwards, passing the meridian of the observer, when they are due south from him, and at their greatest altitude above his horizon; after which, by a continuance of the same motions, viz, of the earth's rotation eastwards, and the luminaries apparent counter motion westwards, these decline from the meridian, or south point, towards the west, where being arrived, they are said to set and descend below it; and so on continually from day to day; thus making it day while the sun is above the horizon, and night while he is below it.</p><p>While the earth is thus turning on its axis, it is at the same time carried by its proper motion in its orbit round the sun, as one of the planets, namely, between the orbits of Venus and Mars, having the orbits of Venus and Mercury within its own, or between it and the sun, in the centre, and those of Mars, Jupiter, Saturn, &c, without or above it; which are therefore called superior planets, and the others the inferior ones. This is called the annual motion of the earth, because it is performed in a year, or 365 days 6 hours nearly; or rather 365 days 5<hi rend="sup">h</hi> 49<hi rend="sup">m</hi>, from any equinox or solstice to the same again, making the tropical year; but from any fixed star to the same again, as seen from the sun, in 365 days 6 hours 9 minutes, which is called the sidereal year. The figure of this orbit is elliptical, having the sun in one focus, the mean distance being about 95 millions of miles, which is upon the supposition that the sun's parallax is about 8″ <*>, or the angle under which the earth's semi-diameter would appear to an observer placed in the sun: and the eccentricity of the orbit, or distance of the sun, in the focus, from the centre of this elliptic orbit, is about 1/60th of the mean distance.</p><p>Now this annual motion is performed in such a manner, that the earth's axis is every where parallel, or in the same direction in every part of the orbit; by which means it happens, that at one time of the year the sun enlightens more of the north polar parts, and at the opposite season of the year more of the southern parts, thus shewing all the varieties of seasons, spring, summer, autumn and winter; which may be illustrated in the following manner: Let the candle I (fig. 1, plate viii) represent the sun, about which the <cb/> earth E, or F, &c, is moved in its elliptical orbit ABCD, or ecliptic, and cutting the equator <hi rend="italics">abcd</hi> in the nodes E and G: then, suspending the terrella by its north pole, and moving it, so suspended, round the ecliptic, its axis will always be parallel to its first position, and the various seasons will be represented at the different parts of the path. Thus, when the earth is at <figure/> or F, the enlightened half of it includes the south pole, and leaves the north pole in darkness, making our winter; at G it is spring, and the two poles are equally illuminated, and the days are every where of the same length; at H or <figure/> it is our summer, the north polar parts being in the illuminated hemisphere, and the southern in the dark one; lastly at E it is autumn, the poles being equally illuminated again, and the days of equal length every where.</p></div2><div2 part="N" n="Earth" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Earth</hi></head><p>, <hi rend="italics">its Quantity of Matter, Density, and Attractive Power.</hi> Although the relative densities of the earth and most of the other planets have been known a considerable time, it is but very lately that we have come to the knowledge of the absolute gravity or density of the whole mass of the earth. This I have calculated and deduced from the observations made by Dr. Maskelyne, Astronomer Royal, at the mountain Schehallien, in the years 1774, 5, and 6. The attraction of that mountain on a plummet, being observed on both sides of it, and its mass being computed from a number of sections in all directions, and consisting of stone; these data being then compared with the known attraction and magnitude of the earth, gave by proportion its mean density, which is to that of water as 9 to 2, and to common stone as 9 to 5: from which very considerable mean density, it may be presumed that the internal parts contain some great quantities of metals.</p><p>From the density, now found, its quantity of matter becomes known, being equal to the product of its density by its magnitude. From various experiments too, we know that its attractive force, at the surface, is such, that bodies fall there through a space of 16 1/12 feet in the first second of time: from whence the force at any other place, either within or without it, becomes known; for the force at any part within it, is directly as its distance from the centre; but the force of any part without it, reciprocally as the square of its distance from the centre.</p></div2></div1><div1 part="N" n="EAST" org="uniform" sample="complete" type="entry"><head>EAST</head><p>, one of the cardinal points of the horizon, or of the compass, being the middle point of it between north and south, on that side where the sun rises, or the point in which it is intersected on that side by the prime vertical.</p></div1><div1 part="N" n="EASTER" org="uniform" sample="complete" type="entry"><head>EASTER</head><p>, a feast of the church, held in memory of our Saviour's resurrection. This feast has been annually celebrated ever since the time of the apostles, and is one of the most considerable festivals in the christian calendar; being that which regulates and determines the times of all the other moveable feasts.</p><p>The rule for the celebration of Easter, fixed by the council of Nice, in the year 325, is, that it be held on the Sunday which falls upon, or next after, the full moon which happens next after the 21st of March; that is, the Sunday which falls upon, or next after the first full moon after the vernal equinox. The reason of <pb n="408"/><cb/> which decree was, that the christians might avoid celebrating their Easter at the same time with the Jewish Passover, which, according to the institution of Moses, was held the very day of the full moon. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Gold. Num.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Paschal full moons.</cell><cell cols="1" rows="1" rend="align=center" role="data">Dom. letter.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">C</cell></row></table></p><p><hi rend="italics">To find</hi> <hi rend="smallcaps">Easter</hi> <hi rend="italics">according to the Old, or Julian Style.</hi></p><p>In the annexed table, find the golden number, with the day of the paschal full moon, and the Sunday letter annexed; compare this letter with the dominical letter of the given year, that it may appear how many days are to be added to the day of the paschal full moon, to give Easter-day.</p><p><hi rend="italics">For ex.</hi> In the year 1715, the dominical letter is B, and the golden number is 6, opsite to which stands April 10 for the day of the paschal full moon; opposite to which is the Sunday letter B, which happening to be the same with that of the year given, that day is a Sunday; and therefore Easter will fall 7 days after, viz, on the 17th of April.</p><p>But in this computation, the vernal equinox is supposed fixed to the 21st of March; and the cycle of 19 years, or golden numbers, is supposed to point out the places of the new and full moons exactly; both which suppositions are erroneous: so that the Julian Easter never happens at its due time, unless by accident. For instance, in the above example the vernal equinox falls on the 10th of March, eleven days before the rule supposes it; and the paschal full moon on the 7th of April, or 3 days earlier than was supposed: and therefore Easter-day should be held on the 10th of April, instead of the 17th.</p><p>This error had grown to such a height, that pope Gregory the 13th thought it necessary to correct it; and accordingly, in the year 1582, by the advice of Aloysius Lilius and others, he ordered 10 days to be thrown out of October, to bring the vernal equinox back again to the 21st of March: and hence arise the terms Gregorian calendar, Gregorian year, &c.</p><p>This correction however did not entirely remove the error; for the equinoxes and solstices still anticipate 28′ 20″ in every 100 Gregorian years; but the difference is so inconsiderable as not to amount to a whole day, or 24 hours, in less than 5082 Gregorian years.</p><p>The Gregorian, or New Style, was not introduced into England till the year 1752, when eleven days were thrown out, viz, between the 3d and 14th of September, the error amounting then to that quantity. <cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Gold. Num.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Paschal full moons.</cell><cell cols="1" rows="1" rend="align=center" role="data">Sund. letter.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">C</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">D</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">E</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">F</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">G</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">B</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">C</cell></row></table></p><p><hi rend="italics">To find</hi> <hi rend="smallcaps">Easter</hi> <hi rend="italics">according to the New or Gregorian Style, till the year</hi> 1900 <hi rend="italics">exclusive.</hi> Look for the golden number of the year in the first column of the table, against which stands the day of the paschal full moon; then look in the 3d column for the Sunday letter, next after the day of the full moon, and the day of the month standing against that Sunday letter is Easter-day. When the full moon happens on a Sunday, then the next Sunday after is Easter-day.</p><p><hi rend="italics">For Ex.</hi> For the year 1790, the golden number is 5; against which stands March the 30th, and the next Sunday letter, which is C, below that, stands opposite April 4, which is therefore the Easter day for the year 1790.</p><p>Though the Gregorian calendar be much preferable to the Julian, it is yet not without its defects. It cannot, for instance, keep the equinox sixed on the 21st of March, but it will sometimes fall on the 19th, and sometimes on the 23d. Add, that the full moon happening on the 20th of March, might sometimes be paschal; yet it is not allowed as such in the Gregorian computation; as on the contrary, the full moon of the 22d of March may be allowed for paschal, which it is not. Scaliger and Calvisius have also pointed out other inaccuracies in this calendar. An excellent paper on this subject by the earl of Macclesfield, may be seen in the Philos. Trans. vol. 40, pa. 417.</p><p><hi rend="smallcaps">Easter</hi> <hi rend="italics">Term.</hi> See <hi rend="smallcaps">Term.</hi></p></div1><div1 part="N" n="EAVES" org="uniform" sample="complete" type="entry"><head>EAVES</head><p>, the margin or lower edge of the roof of a house, being the lowest course of tiles, slates, or the like, which hang over the walls to throw off the water to a distance from the wall.</p><p><hi rend="smallcaps">Eaves</hi>-<hi rend="italics">Board,</hi> or <hi rend="smallcaps">Eaves</hi>-<hi rend="italics">Lath,</hi> a thick featheredged board, usually nailed round the eaves of a house for the lowermost tiles, slate, or shingles, to rest upon.</p><p>EBBING <hi rend="italics">and</hi> <hi rend="smallcaps">Flowing</hi> <hi rend="italics">of the Sea.</hi> See <hi rend="smallcaps">Tides.</hi></p><p>ECCENTRIC. See <hi rend="smallcaps">Excentric.</hi></p><p>ECCENTRICITY. See <hi rend="smallcaps">Excentricity.</hi></p></div1><div1 part="N" n="ECHO" org="uniform" sample="complete" type="entry"><head>ECHO</head><p>, or <hi rend="smallcaps">Eccho</hi>, a sound reflected, or reverberated from some body, and thence returned or repeated to the ear.</p><p>For an echo to be heard, the ear must be in the line of reflection; that the person who made the sound, may hear the echo, it is necessary he should be in a perpendicular to the place which reflects it; and for a <pb n="409"/><cb/> multiple or ta<*>tological echo, it is necessary there be a number of walls and vaults, rocks, and cavities, either placed behind each other, or fronting each other. Those murmurs in the air, that are occasioned by the discharge of great guns, &c, are a kind of indefinite echoes, and are produced from the vaporous particles suspended in the atmosphere, which resist the undulations of sound, and reverberate them to the ear.</p><p>There can be no echo, unless the direct and reflex sounds follow one another at a sufficient distance of time; for if the reflex sound arrive at the ear before the impression of the direct sound ceases, the sound will not be doubled, but only rendered more intense. Now it we allow that 9 or 10 syllables can be pronounced in a second, in order to preserve the sounds articulate and distinct, there should be about the 9th part of a second between the times of their appulse to the ear; or, as sound flies about 1142 feet in a second, the said difference should be 1/9 of 1142, or 127 feet; and therefore every syllable will be reflected to the ear at the distance of about 70 feet from the reflecting body; but as, in the ordinary way of speaking, 3 or 4 syllables only are uttered in a second, the speaker, that he may have the echo returned as soon as they are expressed, should stand about 500 feet from the reflecting body; and so in proportion for any other number of syllables. Mersenne allows for a monosyllable the distance of 69 feet; Morton, 90 feet; for a dissyllable 105 feet, a trisyllable 160 feet, a tetrasyllable 182 feet, and a pentasyllable 204 feet. Nat. Hist. Northampton, cap. 5, pa. 358.</p><p>From what has been said, it follows that echoes may be applied for measuring inaccessible distances. Thus, Mr. Derham, standing upon the banks of the Thames, opposite to Woolwich, observed that the echo of a single sound was reflected back from the houses in 3 seconds; consequently the sum of the direct and reflex rays must have been 1142 X 3 = 3426 feet, and the half of it, 1713 feet, the breadth of the river in that place.</p><p>It also follows that the echoing body being removed farther off, it reflects more of the sound than when nearer; which is the reason why some echoes repeat but one syllable, or one word, and some many. Of these, some are tonical, which only return a voice when modulated into some particular musical tone; and others polysyllabical. That fine echo in Woodstock park, Dr. Plot assures us, in the day-time will return very distinctly 17 syllables, and in the night 20. Nat. Hist. Oxf. cap. 1, pa. 7.</p><p>Echoing bodies may be so contrived, and placed, as that reflecting the sound from one to the other, a multiple echo, or many echoes, shall arise.—At Rosneath, near Glasgow, in Scotland, there is an echo that repeats a tune played with a trumpet three times completely and distinctly.—At the sepulchre of Metella, wife of Crassus, there was an echo, which repeated what a man said five times.—Authors mention a tower at Cyzicus, where the echo repeated seven times.— There is an echo at Brussels, that answers 15 times.</p><p>One of the finest echoes we read of, is that mentioned by Barthius, in his notes on Statius's Thebais, lib. 6, ver. 30, which repeated the words a man uttered 17 times. This was on the banks of the Naha, between Coblentz and Bingen. And whereas, in common echoes, the repetition is not heard till some time after hearing the words <cb/> spoken, or the notes sung; in this, the person who speaks, or sings, is scarce heard at all; but the repetition very clearly, and always in surprising varieties; the echo seeming sometimes to approach nearer, and sometimes farther off; sometimes the voice is heard very distinctly, and sometimes scarce at all: one person hears only one voice, and another several; one hears the echo on the right, and the other on the left, &c.</p><p>Addison, and other travellers in Italy, mention an echo at Simonetta palace, near Milan, still more extraordinary, returning the sound of a pistol 56 times. The echo is heard behind the house, which has two wings; the pistol is discharged from a window in one of these wings, the sound is returned from a dead wall in the other wing, and heard from a window in the back-front. See Addis. Travels, pa. 32; Misson, Voyag. d'Ital. tom. 2, pà. 196; Philos. Trans. N° 480, pa. 220.</p><p>Farther, a multiple echo may be made, by so placing the echoing bodies, at unequal distances, as that they may reflect all one way, and not one on the other; by which means, a manifold successive sound will be heard; one clap of the hands like many; one <hi rend="italics">ha</hi> like a laughter; one single word like many of the same tond and accent; and so one musical instrument like many of the same kind, imitating each other.</p><p>Lastly, echoing bodies may be so ordered, that from any one sound given, they shall produce many echoes, different both as to tone and intension. By which means a musical room may be so contrived, that not only one instrument playing in it shall seem many of the same sort and size, but even a concert of different ones; this may be contrived by placing certain echoing bodies so, as that any note played, shall be returned by them in 3ds, 5ths, and 8ths.</p><p><hi rend="smallcaps">Echo</hi> is also used for the place where the repetition of the sound is produced, or heard. This is either natural or artificial.</p><p>In echoes, the place where the speaker stands, is called the <hi rend="italics">centrum phonicum;</hi> and the object or place that returns the voice, the <hi rend="italics">centrum phonocampticum.</hi></p><div2 part="N" n="Echo" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Echo</hi></head><p>, in Architecture, is applied to certain vaults and arches, mostly of elliptical or parabolical figures; used to redouble sounds, and produce artificial echoes. —The method of making them is taught by F. Blancani, in his Echometria, at the end of his book on the Sphere.</p><p>Vitruvius tells us, that in divers parts of Greece and Italy there were brazen vessels, artfully ranged under the seats of the theatres, to render the sound of the actors' voices more clear, and make a kind of echo; by which means, every one of the prodigious multitude of persons, present at those spectacles, might hear with ease and pleasure.</p></div2></div1><div1 part="N" n="ECLIPSAREON" org="uniform" sample="complete" type="entry"><head>ECLIPSAREON</head><p>, an instrument invented by Mr. Ferguson for shewing the phenomena of eclipses; as their time, quantity, duration, progress, &c. Ferguson's Astron., or Philos. Trans. vol. 48, pa. 520.</p></div1><div1 part="N" n="ECLIPSE" org="uniform" sample="complete" type="entry"><head>ECLIPSE</head><p>, a privation of the light of one of the luminaries, by the interposition of some opaque body, either between it and the eye, or between it and the sun.</p><p>The ancients had terrible ideas of eclipses; supposing them presages of some dreadful events. Plutarch assures <pb n="410"/><cb/> us, that at Rome it was not allowed to talk publicly of any natural causes of eclipses; the popular opinion running so strongly in favour of their supernatural production, at least those of the moon; for as to those of the sun, they had some idea that they were caused by the interposition of the moon between us and the sun; but were at a loss for a body to interpose between us and the moon, which they thought must be the way, if the eclipses of the moon were produced by natural causes. They therefore made a great noise with brazen instruments, and set up loud shouts, during eclipses of the moon; thinking by that means to ease her in labour: whence Juvenal, speaking of a talkative woman, says, <hi rend="italics">Una laboranti poterit succurrere lunæ.</hi> Others attributed the eclipse of the moon to the arts of magicians, who, by their inchantments, plucked her out of heaven, and made her skim over the grass.</p><p>The natives of Mexico keep fast during eclipses; and particularly their women, who beat and abuse themselves, drawing blood from their arms, &c; imagining the moon has been wounded by the sun, in some quarrel between them.</p><p>The Chinese fancy that eclipses are occasioned by great dragons, who are ready to devour the sun and moon; and therefore when they perceive an eclipse, they rattle drums and brass kettles, till they think the monster, terrified by the noise, lets go his prey.</p><p>The superstitious notions entertained of eclipses, were once of considerable advantage to Christopher Columbus, the discoverer of America, who being driven on the island of Jamaica in the year 1493, and distressed for want of provisions, was refused relief by the natives; but having threatened them with a plague, and foretelling an eclipse as a token of it, which happened according to his prediction, the barbarians were so terrified, that they strove who should be the first in bringing him supplies, throwing them at his feet, and imploring forgiveness.</p><p><hi rend="italics">Duration of an</hi> <hi rend="smallcaps">Eclipse</hi>, is the time of its continuance, or between the immersion and emersion.</p><p><hi rend="italics">Immersion, or Incidence, of an</hi> <hi rend="smallcaps">Eclipse</hi>, is the moment when the eclipse begins, or when part of the sun, moon, or planet first begins to be obscured.</p><p><hi rend="italics">Emersion, or Expurgation, of an</hi> <hi rend="smallcaps">Eclipse</hi>, is the time when the eclipsed luminary begins to re-appear, or emerge out of the shadow.</p><p><hi rend="italics">Quantity of an</hi> <hi rend="smallcaps">Eclipse</hi>, is the part of the luminary eclipsed. To determine the quantity eclipsed, it is usual to divide the diameter of the luminary into 12 equal parts, called digits; whence the eclipse is said to be of so many digits according to the number of them contained in that part of the diameter which is eclipsed or obseured.</p><p>Eclispses are divided, with respect to the luminary eclipsed, into <hi rend="italics">Eclipses of the sun, of the moon,</hi> and <hi rend="italics">of the satellites;</hi> and with respect to the circumstances, into <hi rend="italics">total, partial, annular, central,</hi> &c.</p><p><hi rend="italics">Annular</hi> <hi rend="smallcaps">Eclipse</hi>, is when the whole is eclipsed, except a ring, or annulus, which appears round the border or edge.</p><p><hi rend="italics">Central</hi> <hi rend="smallcaps">Eclipse</hi>, is one in which the centres of the two luminaries and the earth come into the same straight line. <cb/></p><p><hi rend="italics">Partial</hi> <hi rend="smallcaps">Eclipse</hi>, is when only a part of the luminary is eclipsed. And a</p><p><hi rend="italics">Total</hi> <hi rend="smallcaps">Eclipse</hi>, is that in which the whole luminary is darkened.</p><p><hi rend="smallcaps">Eclipse</hi> <hi rend="italics">of the Moon,</hi> is a privation of the light of the moon, occasioned by an interposition of the body of the earth directly between the sun and moon, and so intercepting the sun's rays that they cannot arrive at the moon, to illuminate her. Or, the obscuration of the moon may be considered as a section of the earth's conical shadow, by the moon passing through some part of it.</p><p>The manner of this eclipse is represented in this figure, where S is the sun, E the earth, and M or M the moon. <figure/></p><p>Lunar Eclipses only happen at the time of full moon; because it is only then the earth is between the sun and moon: nor do they happen every full moon, because of the obliquity of the moon's path with respect to the sun's; but only in such full moons as happen either at the intersection of those two paths, called the moon's nodes, or very near them; viz, when the moon's latitude, or distance between the centres of the earth and moon, is less than the sum of the apparent semidiameters of the moon and the earth's shadow.</p><p><hi rend="italics">The chief Circumstances in Lunar Eclipses,</hi> are the following:—1. All lunar Eclipses are universal, or visible in all parts of the earth which have the moon above their horizon; and are every where of the same magnitude, with the same beginning and end.—2. In all lunar eclipses, the eastern side is what first immerges and emerges again, i. e. the left-hand side of the moon as we look towards her, from the north; for the proper motion of the moon being swifter than that of the earth's shadow, the moon approaches it from the west, overtakes and passes through it with the moon's east side foremost, leaving the shadow behind, or to the westward.—3. Total eclipses, and those of the longest duration, happen in the very nodes of the ecliptic; because the section of the earth's shadow, then falling on the moon, is considerably larger than her disc. There may however be total eclipses within a small distance of the nodes; but their duration is the less as they are farther from it; till they become only partial ones, and at last, none at all.—4. The moon, even in the middle of an eclipse, has usually a faint appearance of light, resembling tarnished copper; which Gassendus, Ricciolus, Kepler, &c, attribute to the light of the sun, refracted by the earth's atmosphere, and so transmitted thither. —Lastly, she grows sensibly paler and dimmer, before entering into the real shadow; owing to a penumbra which surrounds that shadow to some distance.</p><p><hi rend="italics">Astronomy of Lunar</hi> <hi rend="smallcaps">Eclipses</hi>, <hi rend="italics">or the method of calculating their times, places, magnitudes, and other phenomena.</hi> <pb n="411"/><cb/> The 1st preliminary is to find the length of the earth's conical shadow. This may be found either from the distance between the earth and sun, and the proportion of their diameters, or from the angle of the sun's apparent magnitude at the time. Thus, suppose the semiaxis of the earth's orbit 95,000000 miles, and the eccentricity of the orbit 1,377000 miles, making the greatest distance 96,377000 miles, or 24194 semidiameters of the earth; and the sun's semidiameter being to the earth's, as 112 to 1; then as AD : BE :: DB : EC, that is, 111 : 1 :: 24194 : 218 semidiameters of the earth = EC the length of the earth's shadow. Otherwise, suppose the angle AES, or the sun's apparent semidiameter be 15′ 56″, and the angle BAE, or the sun's parallax 8.6″, then is their difference, or the angle ACE = 15′ 47.4″; hence, as tang. 15′ 47.4″: radius :: BE or 1 : 218 nearly = CE, the same distance as before. Hence, as the moon's least distance from the earth is scarce 56 semidiameters, and the greatest not more than 64, the moon, when in opposition to the sun, in or near the nodes, will fall into the earth's shadow, and will be eclipsed, as the length of the shadow is almost 4 times the moon's distance.</p><p>2. <hi rend="italics">To sind the apparent semidiameter of the earth's shadow,</hi> in the place where the moon passes through it, at any given time. Add together the sun and moon's parallaxes, and from the sum subtract the apparent semidiameter of the sun; so shall the remainder be the apparent semidiameter of the shadow at the place of the moon's passage. For ex. the 28th of April 1790, at midnight, the moon's parallax is 61′ 9″, to which add 8.6″, or 9″, for the sun's parallax, from the sum 61′ 18″ take 15′ 56″, the sun's apparent semidiameter, and the remainder 45′ 22″ is the semidiameter of the shadow at the place where the moon passes through at that time. N. B. Some omit the sun's parallax, as of no consequence; but increase the apparent semidiameter of the shadow by one whole minute, for the shadow of the atmosphere; which would give the semidiameter of the shadow, in the case above, 46′ 13″.</p><p>3. There must also be had, the true distance of the moon from the node, at the mean opposition; also the true time of the opposition, with the true place of the sun and moon, reduced to the ecliptic; likewise the moon's true latitude at the time of the true opposition; the angle of the moon's way with the ecliptic, and the true horary motions of the sun and moon: from which all the circumstances of her eclipse may be computed by common arithmetic and trigonometry. <hi rend="center"><hi rend="italics">To Construct an</hi> <hi rend="smallcaps">Eclipse</hi> <hi rend="italics">of the Moon.</hi></hi></p><p>Let EW be a part of the ecliptic, and C the centre of the earth's shadow, through which draw perpendicular to EW, the line CN towards the north, if the moon have north latitude at the time of the eclipse, or CS southward, if she have south latitude. Make the angle NCD equal to the angle of the moon's way with the ecliptic, which may be always taken at 5° 35′, on an average, without any sensible error; and bisect this angle by the right line CF; in which line it is that the true equal time of opposition of the sun and moon falls, as given by the tables. <cb/> <figure/></p><p>From a convenient scale of equal parts, representing minutes of a degree, take the moon's latitude at the true time of full moon, and set it from C to G, on the line CF; and through the point G, at right-angles to CD, draw the right line HKGLI for the path of the moon's centre. Then is L the point in the earth's shadow, where the moon's centre is at the middle of the eclipse; G the point where her centre is at the tabular time of her being full; and K the point where her centre is at the instant of her ecliptic opposition: also I the moon's centre at the moment of immersion, and H her centre at the end of the eclipse.</p><p>With the moon's semidiameter as a radius, and the points I, L, H, as centres, describe circles for the moon at the beginning, middle, and end of the eclipse-</p><p>Finally, the length of the line of path IH, measured on the same scale, will serve to determine the duration of the eclipse, viz, by saying, As the moon's horary motion from the sun is to IH :: 1 hour or 60 min. to the whole duration of the eclipse.</p><p><hi rend="italics">To Compute a Lunar</hi> <hi rend="smallcaps">Eclipse.</hi> This will be very easy from the foregoing construction. For, 1st, in the triangle CGL, right-angled at L, there are given the hypothenuse CG=the moon's latitude at the time of full moon, and the angle GCL=the half of 5° 35′; to find the legs CL and LG.—2d, In the right-angled triangle CHL or CIL, are given the leg CL, and CH or CI, the sum of the semidiameters of the moon and the earth's shadow; to find LH or LI, half the difference of the sun's and moon's motions during the time of the eclipse.—3d, As the difference of the horary motions of the luminaries is to one hour, or 60 min. :: HL to the semiduration of the eclipse, and :: GL to the difference between the opposition and middle of the eclipse; this last therefore taken from the time of full moon, gives the time of the middle of the eclipse; from which subtracting the time in LI, or semiduration before found, gives the beginning of the eclipse; or add the same, and it gives the end of it.—Lastly, from CO the semidiameter of the shadow, take CL, leaves LO; to which add LP, the moon's semidiameter, when necessary, gives OP the quantity eclipsed.</p><p><hi rend="italics">Note,</hi> When the moon's distance from the node exceeds 12°, there can be no eclipse of the moon; or, more accurately, the limit is from 10 1/2 and 12 1/30 degrees, according to the distances of the sun, earth, and moon.</p><p><hi rend="smallcaps">Eclipse</hi> <hi rend="italics">of the Sun,</hi> is an occultation of the sun's <pb n="412"/><cb/> body, occasioned by an interposition of the moon between the earth and sun. On which account it is by some considered as an eclipse of the earth, since the light of the sun is hid from the earth by the moon, whose shadow involves a part of the earth.</p><p>The manner of a solar eclipse is represented in this figure; where S is the sun, <hi rend="italics">m</hi> the moon, and CD the <figure/> earth, <hi rend="italics">rmso</hi> the moon's conical shadow, travelling over a part of the earth C<hi rend="italics">o</hi>D, and making a complete eclipse to all the inhabitants residing in that track, but no where else; excepting that for a large space around it there is a fainter shade, included within all the space CD<hi rend="italics">s,</hi> which is called the <hi rend="italics">Penumbra.</hi></p><p>Hence, Solar Eclipses happen when the moon is in conjunction with the sun, or at the new moon, and also in the nodes or near them, the limit being about 17 degrees on each side of it; and such eclipses only happen when the latitude of the moon, viewed from the earth, is less than the sum of the apparent semidiameters of the sun and moon; because the moon's way is oblique to the ecliptic, or sun's path, making an angle of nearly 5° 35′ with it.</p><p>In the nodes, when the moon has no visible latitude, the occultation is total; and with some continuance, when the disc of the moon in perigee appears greater than that of the sun in apogee, and its shadow is extended beyond the surface of the earth; and without continuance at moderate distances when the cusp, or point of the moon's shadow, barely touches the earth. Lastly, out of the nodes, but near them, the eclipses are partial.</p><p>The other circumstances of solar eclipses are, 1. That none of them are universal; that is, none of them are seen throughout the whole hemisphere which the sun is then above; the moon's disc being much too little, and much too near the earth, to hide the sun from the whole disc of the earth. Commonly the moon's dark shadow covers only a spot on the earth's surface, about 180 miles broad when the sun's distance is greatest, and the moon's least. But her partial shadow, or penumbra, may then cover a circular space of 4900 miles in diameter, within which the sun is more or less eclipsed, as the places are nearer to or farther from the centre of the penumbra. In this case the axis of the shade passes through the centre of the earth, or the new moon happens exactly in the node, and then it is evident that the section of the shadow is circular; but in every other case the conical shadow is cut obliquely by the surface of the earth, and the section will be an oval, and very nearly a true ellipsis.</p><p>2. Nor does the Eclipse appear the same in all parts <cb/> of the earth, where it is seen; but when in one place it is total, in another it is only partial. Farther, when the moon appears much less than the sun, as is chiefly the case when she is in apogee and he in perigee, the vertex of the lunar shadow is then too short to reach the earth, and though she be in a central conjunction with the sun, is yet not large enough to cover his whole disc, but lets his limb appear like a lucid ring or bracelet, and so causes an <hi rend="italics">Annular Eclipse.</hi></p><p>3. A Solar Eclipse does not happen at the same time, in all places where it is seen; but appears more early to the western parts, and later to the eastern; as the motion of the moon, and consequently of her shadow, is from west to east.</p><p>4. In most Solar Eclipses the moon's disc is covered with a faint light; which is attributed to the reflexion of the light from the illuminated part of the earth.</p><p>Lastly, in total Eclipses of the sun, the moon's limb is seen surrounded by a pale circle of light; which some astronomers consider as an indication of a lunar atmosphere; but others as the atmosphere of the sun, because it has been observed to move equally with the sun, and not with the moon; and besides, it is generally believed that the moon is without any atmosphere, unless it be one that is very small, and very rare. <hi rend="center"><hi rend="italics">To determine the Bounds of a Solar <hi rend="smallcaps">Eclipse.</hi></hi></hi></p><p>If the moon's parallax were insensible, the bounds of a solar eclipse would be determined after the same manner as those of a lunar; but because here is a sensible parallax, the method is a little altered, viz.</p><p>1. Add together the apparent semidiameters of the luminaries, both in apogee and perigee; which gives 33′ 6″ for the greatest sum of them, and 30′ 31″ for the least sum.</p><p>2. Since the parallax diminishes the northern latitude and augments the southern, therefore let the greatest parallax in latitude be added to the former sums, and also subtracted from them: Thus in each case there will be had the true latitude, beyond which there can be no eclipse. This latitude being given, the moon's distance from the nodes, beyond which eclipses cannot happen, is found as for a lunar eclipse. This limit is nearly between 16 1/2 and 18 1/3 degrees distance from the nodes.</p><p><hi rend="italics">To find the Digits eclipsed.</hi> Add the apparent semidiameters of the luminaries into one sum; from which subtract the moon's apparent latitude; the remainder is the scruples, or parts of the diameter eclipsed. Then say, As the semidiameter of the sun is to the scruples eclipsed; so are 6 digits reduced into scruples, viz 360 scruples or minutes, to the digits &c eclipsed.</p><p><hi rend="italics">To determine the Duration of a Solar</hi> <hi rend="smallcaps">Eclipse.</hi> Find the horary motion of the moon from the sun, for one hour before the conjunction, and another hour after; then say, As the former horary motion is to the seconds in an hour, so are the scruples of half duration (found as in a lunar eclipse) to the time of immersion; and as the latter horary motion is to the same seconds, so are the same scruples of half duration to the time of emersion. Lastly, adding the times of immersion and emersion together, the aggregate is the total duration.</p><p>The moon's apparent diameter when largest, ex- <pb n="413"/><cb/> ceeds the sun's when least, only 2′ of a degree; and at the greatest solar eclipse that can happen at any time and place, the total darkness cannot continue any longer than whilst the moon is moving through this 2′ from the sun in her orbit, which is about 4 minutes of time: for the motion of the shadow on the earth's disc is equal to the moon's motion from the sun, which on account of the earth's rotation on its axis towards the same way, or eastward, is about 30 1/2 minutes of a degree every hour, at a mean rate; but so much of the moon's orbit is equal to 30 1/2 degrees of a great circle on the earth, because the circumference of the moon's orbit is about 60 times that of the earth; and therefore the moon's shadow goes 30 1/2 degrees, or 1830 geographical miles in an hour, or 30 1/2 miles in a minute.</p><p><hi rend="italics">To determine the Beginning, Middle, and End, of a Solar Eclipse.</hi> From the moon's latitude, for the time of conjunction, find the arch GL (last fig. but one), or the distance of the greatest obscurity. Then say, as the horary motion of the moon from the sun, before the conjunction, is to 1 hour; so is the distance of the greatest darkness, to the interval of time between the greatest darkness and the conjunction. Subtract this interval, in the 1st and 3d quarter of the anomaly, from the time of the conjunction; and in the other quarters, add it to the same; the result is the time of the greatest darkness. Lastly, from the time of the greatest darkness subtract the time of incidence, and add it to the time of emersion; the difference in the first case will be the beginning; and the sum, in the latter case, the end of the eclipse.</p><p><hi rend="italics">To Calculate</hi> <hi rend="smallcaps">Eclipses</hi> <hi rend="italics">of the Sun.</hi> First, find the mean new moon, and thence the true one; with the place of the luminaries for the apparent time of the true one.—2. For the apparent time of the true new moon, compute the apparent time of the new moon observed.—3. For the apparent time of the new moon seen, compute the latitude seen.—4. Thence determine the digits eclipsed.—5. Find the times of the greatest darkness, immersion, and emersion.—6. Thence determine the beginning, and ending of the eclipse.</p><p>From the foregoing problems, it is evident that all the trouble and fatigue of the calculus arises from the parallaxes of longitude and latitude, without which, the calculation of solar eclipses would be the same with that of lunar ones.</p><p>See the Construction and Calculation of Eclipses by Flamsteed in Sir Jonas Moor's System of Mathematics, and in Ferguson's Astronomy, &c.</p><p>In the Philos. Trans. N° 461 is a contrivance to represent solar eclipses, by means of the terrestrial globe, by M. Seguer, professor of Mathematics at Gottingen. And Mr. Ferguson has fitted a terrestrial globe, so as to shew the time, quantity, duration, and progress of solar eclipses, at any place of the earth where they are visible; which he calls the Eclipsareon. He has also given a large catalogue of ancient and modern eclipses, including those recorded in history, from 721 years before Christ, to A. D. 1485; also computed eclipses from 1485 to 1700, and all the eclipses visible in Europe from 1700 to 1800. See his Astron.</p><p><hi rend="italics">The Number of</hi> <hi rend="smallcaps">Eclipses</hi>, of both luminaries, in any year, cannot be less than two, nor more than seven; the most usual number is 4, and it is rare to have more <cb/> than 6. The reason is obvious; because the sun passes by both the nodes but once in a year, unless he pass by one of them in the beginning of the year; in which case he will pass by the same again a little before the end of the year; because the nodes move backwards 19 1/3 degrees every year, and therefore the sun will come to either of them 173 days after the other. And if either node be within 17° of the sun at the time of new moon, the sun will be eclipsed; and at the subsequent opposition, the moon will be eclipsed in the other node, and come round to the next conjunction before the former node be 17° beyond the sun, and eclipse him again. When three eclipses happen about either node, the like number commonly happens about the opposite one; as the sun comes to it in 173 days afterward, and 6 lunations contain only 4 days more. Thus there may be two eclipses of the sun, and one of the moon, about each of the nodes. But when the moon changes in either of the nodes, she cannot be near enough the other node at the next full, to be eclipsed; and in 6 lunar months afterward she will change near the other node; in which case there cannot be more than two eclipses in a year, both of the sun.</p><p><hi rend="italics">Period of</hi> <hi rend="smallcaps">Eclipses</hi>, is the period of time in which the same eclipses return again; and as the nodes move backwards 19 1/3 degrees every year, they would shift through every point of the ecliptic in 18 years and 225 days; and this would be the regular period of their return, if any complete number of lunations were finished without a fraction; but this is not the case. However, in 223 mean lunations, after the sun, moon, and nodes have been once in a line of conjunction, they return so nearly to the same state again, as the the same node which was in conjunction with the sun and moon at the beginning of the first of these lunations, will be within 28′ 12″ of the line of conjunction with the sun and moon again, when the last of these lunations is completed; and in this period there will be a regular return of eclipses for many ages. To the mean time of any solar or lunar eclipse, by adding this period, or 18 Julian years 11 days 7 hours 43 minutes 20 seconds, when the last day of February in leap years is 4 times included, or a day less when it occurs 5 times, we shall have the mean time of the return of the same eclipse. In an interval of 6890 mean lunations, containing 557 years 21 days 18 hours 30 minutes 11 seconds, the sun and node meet so nearly, as to be distant only 11 seconds.</p><p><hi rend="italics">The Use of</hi> <hi rend="smallcaps">Eclipses.</hi> In Astronomy, eclipses of the moon determine the spherical figure of the earth; they also shew that the sun is larger than the earth, and the earth than the moon. Eclipses also, that are similar in all circumstances, and that happen at considerable intervals of time, serve to ascertain the period of the moon's motion. In Geography, eclipses discover the longitude of different places; for which purpose those of the moon are the more useful, because they are more often visible, and the same lunar eclipse is of equal magnitude and duration at all places where it is seen. In Chronology, both solar and lunar eclipses serve to determine exactly the time of any past event.</p><p><hi rend="smallcaps">Eclipses</hi> <hi rend="italics">of the Satellites.</hi> See <hi rend="smallcaps">Satellites</hi> <hi rend="italics">of Jupiter.</hi></p><p>The chief circumstances here observed, are, 1. That the satellites of Jupiter undergo two or three kinds of <pb n="414"/><cb/> eclipses; the first of which are proper, being such as happen when Jupiter's body is directly interposed between them and the sun: and these happen almost every day. Various authors have given tables for computing eclipses of the satellites of Jupiter; as Flamsteed, Cassini, &c, but the latest and best of all, are those of professor Wargentin of Upsal.</p><p>The second sort are occultations, rather than observations; when the satellites, coming too near the body of Jupiter, are lost in his light; which Riccioli calls <hi rend="italics">occidere zeusiace, setting jovially.</hi> In which case, the nearest or first satellite exhibits a third kind of eclipse; being observed like a round macula, or dark spot, transiting the disc of Jupiter, with a motion contrary to that of the satellite; like as the moon's shadow projected on the earth, will appear to do, to the lunar inhabitants.</p><p>The eclipses of Jupiter's satellites furnish very good means of finding the longitude at sea. Those especially of the first satellite are much surer than the eclipses of the moon, and they also happen much oftener: the manner of applying them is also very easy. See L<hi rend="smallcaps">ONGITUDE.</hi></p></div1><div1 part="N" n="ECLIPTIC" org="uniform" sample="complete" type="entry"><head>ECLIPTIC</head><p>, in Astronomy, a great circle of the sphere conceived to pass through the middle of the zodiac. It is sometimes called the <hi rend="italics">via solis,</hi> or <hi rend="italics">sun's path,</hi> being the track which he appears to describe among the fixed stars; though more properly it is the apparent path of the earth, as viewed from the sun, and thence called the heliocentric circle of the earth. It is called the ecliptic, because all the eclipses of the sun or moon happen when the moon crosses it, or is nearly in one of those two parts of her orbit where it crosses the ecliptic, which points are called the moon's nodes.</p><p>Upon the ecliptic are marked and counted the 12 celestial signs, Aries, Taurus, Gemini, &c; and upon it is counted the longitude of the planets and stars. It is placed obliquely with respect to the equator, which it cuts in two opposite points, viz, the beginning of Aries and Libra, which are directly opposite to each other, and called the equinoxes, making the one half of the ecliptic to the north, and the other half on the south side of the equator; the two extreme points of it, to the north and south, which are opposite to each other, and at a quadrant distance from the equinoctial points both ways, are called the solstices, or solstitial points, or also the two tropics, which are at the beginning of Cancer and Capricorn, and which are at the farthest distance of any points of it from the equator, which distance is the measure of the sun's greatest declination, which is the same with the obliquity of the ecliptic, or the angle it makes with the equator.</p><p><hi rend="italics">This obliquity of the ecliptic</hi> is not permanent, but is continually diminishing, by the ecliptic approaching nearer and nearer to a parallelism with the equator, at the rate of half a second in a year nearly, or from 50″ to 55″ in 100 years, as is deduced from ancient and modern observations compared together; and as the mean obliquity of the ecliptic was 23° 28′ about the end of the year 1788, or beginning of 1789, by adding half a second for each preceding year, or subtracting the same for each following year, the mean obliquity will be found nearly for any year either before or since that period. The quantity however of this change is variously stated by different authors, from 50″ to 60″ or <cb/> 70″ for each century or 100 years. Hipparchus, almost two thousand years since, observed the obliquity of the ecliptic, and found it about 23° 51′; and all succeeding astronomers, to the present time, having observed the same, have found it always less and less; being now rather under 23° 28′; a difference of about 23′ in 1950 years; which gives a medium of 70″ in 100 years. There is great reason however to think that the diminution is variable.</p><p>This diminution of the obliquity of the ecliptic to the equator, according to Mr. Long and some others, is chiefly owing to the unequal attraction of the sun and moon on the protuberant matter about the earth's equator. For if it be considered, say they, that the earth is not a perfect sphere, but an oblate spheroid, having its axis shorter than its equatorial diameter; and that the sun and moon are constantly acting obliquely upon the greater quantity of matter about the equator, drawing it, as it were, towards a nearer and nearer coincidence with the ecliptic; it will not appear strange that these actions should gradually diminish the angle between the planes of those two circles. Nor is it less probable that the mutual attractions of all the planets should have a tendency to bring their orbits to a coincidence: though this change is too small to become sensible in many ages.</p><p>It is now however well known that this change in the obliquity of the ecliptic, is wholly owing to the actions of the planets upon the earth, and especially the planets Venus and Jupiter, but chiefly the former. See La Grange's excellent paper upon this subject in the Memoirs of the French Academy for 1774; Cassini's in 1778; and La Lande's Astron. vol. 3, art. 2737. According to La Grange, who proceeds upon theory, the annual change of obliquity is variable, and has its limits: about 2000 years ago, he thinks it was after the rate of about 38″ in 100 years; that it is now, and will be for 400 years to come, 56″ per century; but 2000 years hence, 49″ per century. According to Cassini, who computes from observations of the obliquity between the years 1739 and 1778, the annual change at present is 60″ or 1′ in 100 years. But according to La Lande, the diminution is at the rate of 88″ per century; while Dr. Maskelyne makes it only 50″ in the same time.</p><p>Beside the regular diminution of the obliquity of the ecliptic, at the rate of near 50 seconds in a century, or half a second a year, which arises from a change of the ecliptic itself, it is subject to two periodical inequalities, the one produced by the unequal force of the sun in causing the precession of the equinoxes, and the other depending on the nutation of the earth's axis. See the Explanation and Use of Dr. Maskelyne's Tables and Observations, pa. vi, where we are shewn how to calculate those inequalities, and where he shews that, from his own observations, the mean obliquity of the ecliptic to the beginning of the year 1769, was 23° 28′ 9″.7.</p><p><hi rend="italics">To find the Obliquity of the Ecliptic,</hi> or the greatest declination of the sun: about the time of the summer solstice observe very carefully the sun's zenith distance for several days together; then the difference between this distance and the latitude of the place, will be the obliquity sought, when the sun and equator are both on one side of the place of observation; but their sum <pb n="415"/><cb/> will be the obliquity when they are on different sides of it. Or, it may be found by observing the meridian altitude, or zenith distance, of the sun's centre, on the days of the summer and winter solstice; then the difference of the two will be the distance between the tropics, the half of which will be the obliquity sought.</p><p>By the same method too, the declination of the sun from the equator for any other day may be found; and thus a table of his declination for every day in the year might be constructed. Thus also the declination of the stars might be found. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Authors' Names</cell><cell cols="1" rows="1" rend="align=center" role="data">Years before Christ</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Obliquity</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">°</cell><cell cols="1" rows="1" rend="align=center" role="data">′</cell><cell cols="1" rows="1" rend="align=center" role="data">″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pytheas</cell><cell cols="1" rows="1" rend="align=center" role="data">324</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell></row><row role="data"><cell cols="1" rows="1" rend="valign=top" role="data">Eratosthenes and Hipparchus</cell><cell cols="1" rows="1" rend="align=center" role="data">230 & 140 after Christ</cell><cell cols="1" rows="1" rend="valign=top" role="data">23</cell><cell cols="1" rows="1" rend="valign=top" role="data">51</cell><cell cols="1" rows="1" rend="align=right valign=top" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ptolomy</cell><cell cols="1" rows="1" rend="align=center" role="data">140</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Almahmon</cell><cell cols="1" rows="1" rend="align=center" role="data">832</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Albategnius</cell><cell cols="1" rows="1" rend="align=center" role="data">880</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Thebat</cell><cell cols="1" rows="1" rend="align=center" role="data">911</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Abul Wasi and Hamed</cell><cell cols="1" rows="1" rend="align=center" role="data">999</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Persian Tables in Chrysococea</cell><cell cols="1" rows="1" rend="align=center" role="data">1004</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Albatrunius</cell><cell cols="1" rows="1" rend="align=center" role="data">1007</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Arzachel</cell><cell cols="1" rows="1" rend="align=center" role="data">1104</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Almæon</cell><cell cols="1" rows="1" rend="align=center" role="data">1140</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Choja Nassir Oddin</cell><cell cols="1" rows="1" rend="align=center" role="data">1290</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Prophatius the Jew</cell><cell cols="1" rows="1" rend="align=center" role="data">1300</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Ebn Shattir</cell><cell cols="1" rows="1" rend="align=center" role="data">1363</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Purbach and Regiomontanus</cell><cell cols="1" rows="1" rend="align=center" role="data">1460</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Ulugh Beigh</cell><cell cols="1" rows="1" rend="align=center" role="data">1463</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell></row><row role="data"><cell cols="1" rows="1" role="data">Walther</cell><cell cols="1" rows="1" rend="align=center" role="data">1476</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Do. corrected by refraction &c</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Werner</cell><cell cols="1" rows="1" rend="align=center" role="data">1510</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copernicus</cell><cell cols="1" rows="1" rend="align=center" role="data">1525</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" role="data">Egnatio Danti</cell><cell cols="1" rows="1" rend="align=center" role="data">1570</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Prince of Hesse</cell><cell cols="1" rows="1" rend="align=center" role="data">1570</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Rothmann and Byrge</cell><cell cols="1" rows="1" rend="align=center" role="data">1570</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tycho Brahe</cell><cell cols="1" rows="1" rend="align=center" role="data">1584</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ditto corrected</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Wright</cell><cell cols="1" rows="1" rend="align=center" role="data">1594</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Kepler</cell><cell cols="1" rows="1" rend="align=center" role="data">1627</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gassendus</cell><cell cols="1" rows="1" rend="align=center" role="data">1630</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Ricciolus</cell><cell cols="1" rows="1" rend="align=center" role="data">1646</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ditto corrected</cell><cell cols="1" rows="1" rend="align=center" role="data">1655</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Hevelius</cell><cell cols="1" rows="1" rend="align=center" role="data">1653</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ditto corrected</cell><cell cols="1" rows="1" rend="align=center" role="data">1661</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cassini</cell><cell cols="1" rows="1" rend="align=center" role="data">1655</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Montons, corrected, &c</cell><cell cols="1" rows="1" rend="align=center" role="data">1660</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Richer corrected</cell><cell cols="1" rows="1" rend="align=center" role="data">1672</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" role="data">De la Hire</cell><cell cols="1" rows="1" rend="align=center" role="data">1686</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Ditto corrected</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" role="data">Flamsteed</cell><cell cols="1" rows="1" rend="align=center" role="data">1690</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Bianchini</cell><cell cols="1" rows="1" rend="align=center" role="data">1703</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell></row><row role="data"><cell cols="1" rows="1" role="data">Roemer</cell><cell cols="1" rows="1" rend="align=center" role="data">1706</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell></row><row role="data"><cell cols="1" rows="1" role="data">Louville</cell><cell cols="1" rows="1" rend="align=center" role="data">1715</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" role="data">Godin</cell><cell cols="1" rows="1" rend="align=center" role="data">1730</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bradley</cell><cell cols="1" rows="1" rend="align=center" role="data">1750</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mayer</cell><cell cols="1" rows="1" rend="align=center" role="data">1756</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" role="data">Maskelyne</cell><cell cols="1" rows="1" rend="align=center" role="data">1769</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hornsby</cell><cell cols="1" rows="1" rend="align=center" role="data">1772</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row></table> <cb/></p><p>The observations of astronomers of all ages, on the obliquity of the ecliptic, have been collected together; and although some of them may not be quite accurate, yet they sufficiently shew the gradual and continual decrease of the obliquity from the times of the earliest observations down to the present time. The chief of those observations may be seen in the foregoing table; where the first column contains the name of the observer, or author, the 2d the year before or after Christ, and the 3d the obliquity of the ecliptic for that time.</p><p>See Ptolom. Alm. lib. 1, cap. 10; Ri<*>cioli Ahr. vol. 1, lib. 3, cap. 27; Flamsteed Proleg. Hist. Cœl. vol. 3; Philos. Trans. number 163; ib. vol. 63, pt. 1; Long's Astron. vol. 1, cap. 16; Memoirs of the Acad. an. 1716, 1734, 1762, 1767, 1774, 1778; Acta Erud. Lipsiæ 1719; Naut. Alm. 1779; Maskelyne's Observ. Explan. pa. vi; &c.</p><p>According to an ancient cradition of the Egyptians, mentioned by Herodotus, the ecliptic had formerly been perpendicular to the equator: they were led into this notion by observing, for a long series of years, that the obliquity was continually diminishing; or, which amounts to the same thing, that the ecliptic was continually approaching to the equator. From thence they took occasion to suspect that those two circles, in the beginning, had been as far off each other as possible, that is, perpendicular to each other. Diodorus Siculus relates, that the Chaldeans reckoned 403,000 years from their first observations to the time of Alexander's entering Babylon. This enormous account may have some foundation, on the supposition that the Chaldeans built on the diminution of the obliquity of the ecliptic at the rate of a minute in 100 years. M. de Louville, taking the obliquity such as it must have been at the time of Alexander's entrance into Babylon, and going back to the time when the ecliptic, at that rate, must have been perpendicular to the equator, actually finds 402,942 Egyptian, or Chaldean years; which is only 58 years short of that epocha. Indeed there is no way of accounting for the fabulous antiquity of the Egyptians, Chaldeans, &c, so probable, as from the supposition of long periods of very slow celestial motions, a small part of which they had observed, and from which they calculated the beginning of the period, making the world and their own nation to commence together. Or perhaps they sometimes counted months or days for years.</p><p>Should the diminution always continue at the rate it has lately done, viz at 50″ or 56″ a century, it would take 96,960 years, from the year 1788, to bring the ecliptic exactly to coincide with the equator.</p><div2 part="N" n="Ecliptic" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ecliptic</hi></head><p>, <hi rend="italics">in Geography,</hi> a great circle on the terrestrial globe, in the plane of, or directly under, the celestial ecliptic.</p></div2><div2 part="N" n="Ecliptic" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ecliptic</hi></head><p>, <hi rend="italics">Eclipticus,</hi> something belonging to the ecliptic, or to eclipses; as ecliptic conjunction, opposition, &c.</p><p><hi rend="smallcaps">Ecliptic</hi> <hi rend="italics">Bounds,</hi> or <hi rend="italics">Limits,</hi> are the greatest distances from the nodes at which the sun or moon can be eclipsed; namely, near 18 degrees for the sun, and 12 degrees for the moon.</p><p><hi rend="smallcaps">Ecliptic</hi> <hi rend="italics">Digits, digiti ecliptici.</hi> See <hi rend="smallcaps">Digits.</hi></p><p><hi rend="italics">Poles of the</hi> <hi rend="smallcaps">Ecliptic</hi>, are the two opposite points <pb n="416"/><cb/> of the sphere which are each everywhere equally distant from the ecliptic quite around, or 90° distant from it. The distance of the poles of the ecliptic from the poles of the equator, or of the world, is always equal to the varying distance of the obliquity of the ecliptic, and at the beginning of the year 1789 it was just 23° 28′.</p><p><hi rend="italics">Reduction to the</hi> <hi rend="smallcaps">Ecliptic.</hi> See <hi rend="smallcaps">Reduction.</hi></p></div2></div1><div1 part="N" n="EFFECT" org="uniform" sample="complete" type="entry"><head>EFFECT</head><p>, the result or consequence of the application of a cause, or agent, on some subject. It is one of the great axioms in philosophy, that full effects are always proportional to the powers of their adequate causes.</p></div1><div1 part="N" n="EFFECTION" org="uniform" sample="complete" type="entry"><head>EFFECTION</head><p>, denotes the geometrical construction of a proposition. The term is also used in reference to problems and practices; which, when they are deducible from, or founded upon some general propositions, are called the <hi rend="italics">geometrical effection</hi> of them.</p></div1><div1 part="N" n="EFFERVESCENCE" org="uniform" sample="complete" type="entry"><head>EFFERVESCENCE</head><p>, is popularly used for a light ebullition, or a brisk intestine motion, produced in a liquor by the first action of heat, with any remarkable separation of its parts.</p><p>EFFICIENT <hi rend="italics">Cause,</hi> is that which produces an effect. See <hi rend="smallcaps">Cause</hi> and <hi rend="smallcaps">Effect.</hi></p><div2 part="N" n="Efficients" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Efficients</hi></head><p>, in Arithmetic, are the numbers given for an operation of multiplication, and are otherwise called the factors. Hence the term coefficients in Algebra, which are the numbers prefixed to, or that multiply the letters or algebraic quantities.</p></div2></div1><div1 part="N" n="EFFLUVIUM" org="uniform" sample="complete" type="entry"><head>EFFLUVIUM</head><p>, a flux or exhalation of minute particles from any body; or an emanation of subtile corpuscles from a mixed, sensible body, by a kind of motion of transpiration.</p></div1><div1 part="N" n="ELASTIC" org="uniform" sample="complete" type="entry"><head>ELASTIC</head><p>, an appellation given to all bodies endowed with the property of elasticity or springiness.</p><p><hi rend="smallcaps">Elastic</hi> <hi rend="italics">Body,</hi> is that which changes its figure, and yields to any impulse or pressure, but endeavours by its own nature and force to restore the same again; or, it is a springy body, which, when compressed or condensed, or the like, makes an effort to set itself at liberty, and to repel the body that constrained it. Such, for instance, as a bow, or a sword blade, &c, which are easily bent, but presently return to their former figure and extension. All bodies partake of this property in some degree, though perhaps none are perfectly elastic, as none are found to restore themselves with a force equal to that with which they are compressed.</p><p>The principal phenomena observable in Elastic bodies, are 1, That an elastic body (i. e. a body perfectly elastic, if any such there be) endeavours to restore itself with the same force with which it is pressed or bent. 2, An elastic body exerts its force equally towards all sides; though the effect is chiefly found on that side where the resistance is weakest; as is evident in the case of a gun exploding a ball, a bow shooting out an arrow, &c.—3, Elastic bodies, in what manner soever struck, or impelled, are inflected and rebound after the same manner: thus a bell yields the same musical sound, in what manner, or on what side soever it be struck; the same of a tense or musical chord; and a body rebounds from a plane in the same angle in which it meets or strikes it, making the angle of incidence equal to the angle of reflection, whether the intensity of the stroke be greater or less.—4, A body perfectly fluid, if any such <cb/> there be, cannot be elastic, if it be allowed that its parts cannot be compressed.—5, A body perfectly solid, if any such there be, cannot be elastic; because, having no pores, it is incapable of being compressed.—6, The elastic properties of bodies seem to differ, according to their greater or less density or compactness, though not in an equal degree: thus, metals are rendered more compact and elastic by being hammered: tempered steel is much more elastic than soft steel; and the density of the former is to that of the latter as 7809 to 7738: cold condenses solid bodies, and renders them more elastic; whilst heat, that relaxes them, has the opposite effect: but, on the contrary, air, and other elastic fluids, are expanded by heat, and rendered more elastic.—For the laws of Motion and Percussion in Elastic bodies, see <hi rend="smallcaps">Motion</hi>, and <hi rend="smallcaps">Percussion.</hi></p><p><hi rend="smallcaps">Elastic</hi> <hi rend="italics">Curve.</hi> See <hi rend="smallcaps">Catenaria.</hi></p><p><hi rend="smallcaps">Elastic</hi> <hi rend="italics">Fluids.</hi> See <hi rend="smallcaps">Air, Electricity, Gas, Elastic Vapours</hi>, &c.</p><p><hi rend="smallcaps">Elastic</hi> <hi rend="italics">Gum.</hi> The same as <hi rend="smallcaps">Caoutchouc</hi>, or <hi rend="italics">Indian Rubber.</hi></p><p><hi rend="smallcaps">Elastic</hi> <hi rend="italics">Vapours,</hi> or <hi rend="italics">Fluids,</hi> are such as may be compressed mechanically into a less space, and which resume their former state when the compressing force is withdrawn. Such as atmospherical air, and all the aerial fluids, with all kinds of fumes raised by means of heat, whether from solid or fluid bodies.</p><p>Of these, some remain elastic only while a considerable degree of heat is applied to them, or to the substance which produces them; while others continue elastic in every degree of cold that has yet been observed. Of the former kind, are the vapours of water, spirit of wine, mercury, sal-ammoniac, and all kinds of sublimable salts: of the latter, those of spirit of salt, mixtures of vitriolic acid and iron, nitrous acid, and various other metals, and in short the several species of aerial fluids indiscriminately.</p><p>The elastic force with which any one of these fluids is endowed, has not yet been calculated, as being ultimately greater than any obstacle we can put in its wayThus, on compressing the atmospherical air, we find that for some little time at first it easily yields to any force applied; but at every succeeding moment the resistance becomes always the stronger, and a greater and greater force must be applied, to compress it farther. As the compression goes on, the vessel containing the air becomes hot; but no power whatever has yet been able in any degree to destroy the elasticity of the contained fluid; for, upon removing the pressure, it is always found to occupy the very same space that it did before. The case is the same with the steam of water, to which a sufficient heat is applied to keep it from condensing into water.</p><div2 part="N" n="Elasticity" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elasticity</hi></head><p>, or <hi rend="smallcaps">Elastic</hi> <hi rend="italics">Force,</hi> that property of bodies by which they restore themselves to their former figure, after any external pressure.</p><p>The cause or principle of this important property, elasticity, is variously accounted for. The Cartesians ascribe it to their subtile matter making an effort to pass through pores that are too narrow for it. Thus, say they, in bending or compressing a hard elastic body, as a bow, for instance, its parts recede from each other on the convex side, and approach on the concave one: <pb n="417"/><cb/> consequently the pores are contracted or straitened on the concave side; and, if they were before round, are now perhaps oval: so that the materia subtilis, or matter of the second element, endeavouring to pass out of the pores thus straitened, must make an effort, at the same time, to restore the body to the state it was in when the pores were rounder, i. e. before the bow was bent: and in this consists its Elasticity.</p><p>Other later philosophers account for Elasticity-much after the same manner as the Cartesians; with this only difference, that instead of the subtile matter of the Cartesians, these substitute <hi rend="italics">Ether,</hi> or a fine ethereal medium that pervades all bodies.</p><p>Others, setting aside the precarious notion of a materia subtilis, account for Elasticity from the great law of nature, Attraction, or the cause of the cohesion of the parts of solid and firm bodies. Thus, say they, when a hard body is struck or bent, so that the component parts are moved a little from each other, but not quite disjointed or broken off, or separated so far as to be out of the power of that attracting force by which they cohere; they must, on removing the external violence, spring back to their former natural state.</p><p>Others again resolve Elasticity into the pressure of the atmosphere: for a violent tension, or compression, though not so great as to separate the constituent particles of bodies far enough to let in any foreign matter, must yet occasion many little vacuola between the separated surfaces; so that on the removal of the force they will close again by the pressure of the aerial fluid upon the external parts.</p><p>Lastly, others attribute the Elasticity of all hard bodies to the power of resilition in the air included within them; and so make the elastic force of the air the principle of Elasticity in all other bodies. See Desaguliers's Exper. Philos. vol. 2, pa. 38, &c.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Elasticity</hi> <hi rend="italics">of Fluids</hi> is accounted for from their particles being all endowed with a centrifugal force; whence Sir Isaac Newton demonstrates, prop. 23, lib. 2, that particles, which naturally avoid or fly off from one another by such forces as are reciprocally proportional to the distances of their centres, will compose an elastic fluid, whose density shall be proportional to its compression; and vice versa, if any fluid be composed of particles that fly off or avoid one another, and have its density proportional to its compression, then the centrifugal forces of those particles will be reciprocally proportional to the distances of their centres.</p><p><hi rend="smallcaps">Elasticity</hi> <hi rend="italics">of the Air</hi> is the force with which that element endeavours to expand, and with which it does actually dilate itself, on removing the force that compressed it. See <hi rend="smallcaps">Air</hi>, and <hi rend="smallcaps">Atmosphere.</hi></p><p>The Elasticity or spring of the air was first discovered by lord Bacon, and farther established by Galileo. Its existence is proved by this experiment of that philosopher: An extraordinary quantity of air being intruded, by means of a syringe, into a hollow ball or shell of glass or metal, till such time as the ball, with this accession of air, weigh considerably more in the balance than it did before; then, opening the mouth of the ball, the air rushes out, till the ball sink to its former weight. From hence we infer, that there is just as much air gone out, as compressed air had been crowded <cb/> in. Air therefore returns to its former degree of expansion, upon removing the force that compressed or resisted its expansion; and consequently it is endowed with an elastic force. It may be added, that as the air is found to rush out in every situation or direction of the orifice, the elastic force acts every way, or in every direction alike.</p><p>The cause of Elasticity in air hath been usually ascribed to a repulsion between its particles; but what is the cause of that repulsion? The term repulsion, like that of attraction, requires to be defined; and probably it will be found in most cases to be the effect of the action of some other fluid. Thus, it is found that the Elasticity of the atmosphere is very considerably affected by heat. Supposing a quantity of air heated to such a degree as to raise Fahrenheit's thermometer to 212, it will then occupy a considerable space; but if it be cooled again to such a degree, as to sink the thermometer to o, it will shrink up to less than half the former bulk. The quantity of repulsive power therefore acquired by the air, while passing from one of these states to the other, is evidently owing to the heat added to it, or taken away from it. Nor does there seem to be any reason to suppose, that the quantity of Elasticity or repulsive power it still possesses, is owing to any other cause than the fire contained in it. The supposition that repulsion is a primary cause, independent of all others, has given rise to many erroneous theories, and very much embarrassed philosophers in accounting for the phenomena of Elasticity.</p><p>The Elasticity of the air is not only proportional to its density, but is always equal to the force which compresses it, because these two exactly balance each other. This Elasticity, in the atmospheric air, is meafured by the height of the barometer at any time, allowing for its heat or temperature, after this rate, viz, the 434th part for each degree of Fahrenheit's thermometer, above or below some mean temperature, as 55°; for by that part of the whole it is that air expands or contracts, or else increases or decreases in its Elasticity, for each degree of the thermometer. Sir Geo. Shuckburgh, in the Philos. Trans. for 1777, pa. 561.</p></div2></div1><div1 part="N" n="ELECTIONS" org="uniform" sample="complete" type="entry"><head>ELECTIONS</head><p>, or <hi rend="italics">Choice,</hi> signify the several different ways of taking any number of things proposed, either separately, or as combined in pairs, in threes, in fours, &c; not as to the order, but only as to the number and variety of them. Thus, of the things <hi rend="italics">a, b, c, d, e,</hi> &c, the elections of &c; and of any number, <hi rend="italics">n,</hi> all the elections are 2<hi rend="sup">n</hi>-1; that is, one less than the power of 2 whose exponent is <hi rend="italics">n,</hi> the number of single things to be chosen, either separately or in combination.</p></div1><div1 part="N" n="ELECTRIC" org="uniform" sample="complete" type="entry"><head>ELECTRIC</head><p>, in Physics, is a term applied to those substances, in which the electric fluid can be excited, and accumulated, without transmitting it; and which are therefore called <hi rend="italics">non-conductors.</hi> They are also called <hi rend="italics">original Electrics,</hi> and <hi rend="italics">Electrics per se.</hi></p><p>The word is derived from <foreign xml:lang="greek">plextzon</foreign>, <hi rend="italics">amber,</hi> one of the most observable non-conductors. To this class also belong glass, and all vitrifications, even of metals; all precious stones, of which the most transparent are the <pb n="418"/><cb/> best; all resins, and resinous compositions; also sulphur, baked wood, all bituminous substances, wax, silk, cotton, all dry animal substances, as feathers, wool, hair, &c; also paper, white sugar, and sugarcandy; likewise air, oils, chocolate, calces of metals and semi-metals, the ashes of animal and vegetable substances, the rust of metals, all dry vegetable substances, and stones, of which the hardest are the best.</p><p>Substances of this kind may be excited, so as to exhibit the Electric appearances of attracting and repelling light bodies, emitting a spark of light, attended with a snapping noise, and yielding a current of air, the sensation of which resembles that of a spider's web drawn over the face, &c, and a smell like that of phosphorus; and this exciting may be either by friction, or by heating and cooling, or by melting, and pouring one melted substance into another.</p><p>The term is peculiarly applied to the electric, viz. the globe, or cylinder, &c, used in electrical machines, to collect the electrical matter by rubbing it.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Air Thermometer,</hi> an instrument contrived by Mr. Kinnersley of Philadelphia, and used in determining the effects of the electrical explosion upon air. The description may be seen in Franklin's Letters, &c, pa. 389, 4to, 1769.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Apparatus,</hi> consists of glass tubes, about 3 feet long, and an inch and a half in diameter, one of which should be closed at one end, and furnished at the other end with a brass cap and stop-cock, to rarefy or condense the inclosed air; sticks of sealing wax, or tubes of rough glass, or glass tubes covered with sealing-wax, or cylinders of baked wood for producing the negative electricity; with proper rubbers, as black oiled silk, with amalgam upon it for the former, and soft new flannel, or hare skins, or cat skins, tanned with the hair on, for the latter; coated jars, or plates of glass, either single, or combined in a battery, for accumulating electricity; metal rods, as dischargers; an electrical machine; electrometers, and insulated stools, supported by pillars of glass, covered with sealing-wax, or baked wood, varnished or boiled in linseed oil.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Atmosphere,</hi> is a stream or mass of the Electrical fluid which surrounds an excited or electrified body, to some distance.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Balls.</hi> See <hi rend="smallcaps">Balls</hi> and E<hi rend="smallcaps">LECTROMETER.</hi></p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Battery,</hi> consists of a large quantity of coated jars, placed near each other in a convenient manner. These being charged, or electrified, and connected with each other, are then suddenly exploded or discharged, with a prodigious effect.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Fluid,</hi> is a fine rare fluid which issues from, and surrounds electrified bodies.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Kite,</hi> was contrived by Dr. Franklin, to verify his hypothesis of the identity of electricity and lightning. It consisted of a large thin silk handkerchief, extended and fastened at the four corners to two slender strips of cedar, and accommodated with a tail, loop, and string, so as to rise in the air like a paper kite. To the top of the upright stick of the cross was fixed a very sharp-pointed wire, rising a foot or more above the wood; and to the end of the twine, next the hand, a silk ribband was tied. From a key <cb/> suspended at the junction of the twine and silk, when the kite is raised during a thunder-storm, a phial may be charged, and electric fire collected, as is usually done by means of a rubbed glass tube or globe. Philos. Trans. vol. 47, pa. 565, or Franklin's Letters, pa. 111 and 112.</p><p>Kites made of paper, covered with varnish, or with well boiled linseed oil, to preserve them from the rain, with a stick and cane bow, like the common ones used by boys, will answer the purpose extremely well, and are very useful in determining the electricity of the atmosphere. See <hi rend="smallcaps">Conductor.</hi></p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Machine,</hi> is a part of the Electrical apparatus, contrived for collecting a great quantity of electricity, and exhibiting its effects in a very sensible manner. It consists of the electric, the moving engine, the rubber, and the prime conductor. In the early state of this science, for the electric, was used sealing-wax, sulphur, or rough glass; but, since the method of insulating the rubber, and so producing negative electricity, was introduced, smooth glass has been used. The form is commonly either that of a globe, or of a cylinder. Each figure has its advantages, and its inconveniences. Dr. Van Marum, a late German writer, has constructed a machine, in which gumlac, in the form of a disc, is used as an electric instead of glass; which has the effect of depending very little on the temperature of the air; described in his Verhandeling over het Electrizeeren, &c, or a Treatise concerning the method of electrifying. Groningen, 1776. But he has since procured some others to be made by Mr. Cuthbertson, a very ingenious artist, of large discs, or round plates of glass: one of these is now placed in Teyler's Museum at Harlem, having two of these glass plates, of 65 inches diameter, excited, on both sides of them, by rubbers of waxed taffaty; with which, effects are produced that are truly astonishing and tremendous. See his Description of this machine, and its effects, published in 4to, at Harlem, 1785, &c.</p><p>There have been various contrivances for giving motion to the electric of a machine. The common method is by a wheel turned by a winch or handle; a cord going round a groove in the periphery of the wheel, and over a pulley in the neck of the globe or cylinder. Others have used multiplying wheels, which are easily turned by a winch; and others again make use of a wheel and pinion, or a wheel and endless screw. But Van Marum's machine it seems has the completest movement, its operation being very uniform, and easily worked; it is kept in motion by a weight, which, after being wound up to the height of 12 feet, will continue the motion uniformly for 6 hours; yielding also a negative power, as well as the positive; and the conductors annexed to it serving easily to convey the electrical power wherever it is required, without the addition of any chain, or wires, &c.</p><p>The Rubber is the next material part of a machine. These were formerly made of red basil skins, stuffed with hair, wool, flax, or bran: Dr. Nooth introduced silk cushions stuffed with hair, over which is laid a piece of leather, rubbed with amalgam, which are better than the others. The rubber may be insulated in any way that best suits the construction of the ma- <pb/><pb/><pb n="419"/><cb/> chine: and a chain or wire may easily be suspended from it, to communicate with the floor, whenever the insulation is not necessary; and thus positive and negative electricity may be produced at pleasure. Van Marum uses mercury to his rubbers.</p><p>The Prime Conductor is another necessary appendage to the Electrical Machine: its use is to receive the electricity from the electric, as it is produced, and accumulate it as in a magazine, ready to be drawn off and employed on all occasions. See <hi rend="italics">Prime</hi> <hi rend="smallcaps">Conductor.</hi> <hi rend="center"><hi rend="italics">Description of the most useful Electrical Machines.</hi></hi></p><p>Fig. 1, plate ix, represents Dr. Priestley's Machine, a very extensively useful one, described in his History of Electricity; in which <hi rend="italics">g</hi> is the globe, or electric; <hi rend="italics">f</hi> the rubber; in the two pillars <hi rend="italics">d, d,</hi> of baked wood, are several holes to receive the spindles of different globes or cylinders, several of which may be put on together, to increase the electricity: <hi rend="italics">klm</hi> is the prime conductor, being a copper tube, supported on a stand of glass or baked wood.</p><p>Fig. 2 is Dr. Watson's Machine, for using several globes at once, to accumulate a great quantity of electricity.</p><p>Fig. 3 represents a very portable Electrical Machine invented by Mr. Read, and improved by Mr. Lane. A is the glass cylinder, moved vertically by means of the pulley at the lower end of the axis, the pulley being turned by the large wheel B parallel to the table; there are several pulleys, of different sizes, either of which may be used, according as the motion is required to be quicker or slower. The conductor C is furnished with points to collect the fluid, and is screwed to the wire of a coated jar D. The figure shews also the manner of applying Mr. Lane's electrometer to this machine.</p><p>Electrical Machines have of late years undergone some very essential alterations and improvements; both from the suggestions of private electricians, and the inventions of Messrs. Adams, Nairne, and Jones, instrument makers in London; some of which are as follow:</p><p>Fig. 4 represents a very convenient machine for practice. The frame of this machine consists of the bottom board ABCD; which, when the machine must be used, is fastened to the table by two metal cramps. EF are two round pillars, of baked wood, which support the cylinder G by the axles of the brass or wooden caps H, turned sometimes by a simple winch I, and sometimes by a pulley and wheel, as in the next fig. The rubber is fixed to a glass pillar K, which is fastened to a wooden basis L at the bottom. The conductor N is usually made of brass or tin japanned, and is insulated by a glass pillar, screwed into a wooden basis or foot, which is most conveniently placed parallel to the cylinder.</p><p>Fig. 5 represents an Electrical Machine, with a conductor in the shape of a T; and an improved medical apparatus, where it is necessary to give the shock in the arms.</p><p>Fig. 6 shews Mr. Nairne's patent machine for medical purposes. Its glass cylinder is about 7 inches in diameter, and 12 long, with two conductors parallel to it. The rubber is fastened to the conductor R; and consists of a cushion of leather stuffed, having a piece <cb/> of silk glewed to its under part. The conductors are of tin covered with black lacker, each of them containing a large coated glass jar, and likewise a smaller one, or a coated tube, which are visible when the caps NN are removed. To each conductor is fixed a knob O, for the occasional suspension of a chain to produce positive or negative electricity. That part of the winch C which acts as a lever in turning the cylinder, is of glass. Thus every part of the machine is insulated, the cylinder itself and its brass caps not excepted; by which means very little of the electricity is dissipated, and hence of course the effects are likely to be the more powerful. And to this the inventor has adapted some flexible conducting joints, a discharging electrometer, &c, for the practice of medical electricity.</p><p>The large Electrical Machine placed in Teyler' Museum at Harlem, has been partly described above. It was constructed by Mr. John Cuthbertson, an English instrument maker; and it has, for the electric, two glass plates of 65 inches diameter, made of French glass, as this is found to produce the most electricity next to English flint glass, which could not be made of a sufficient size: these plates are set on the same horizontal axis, at the distance of 7 1/2 inches, and are excited by 8 rubbers, each 15 1/2 inches long; and both sides of the plates are covered with a resinous substance to the distance of 16 1/2 inches from the centre, both to strengthen the plates, and to prevent any electricity from being carried off by the axis. Its battery of jars contains 225 square feet of coated surface, and its effects are astonishingly great.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Phial.</hi> See <hi rend="smallcaps">Leyden</hi> <hi rend="italics">Phial.</hi></p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Rubber.</hi> See <hi rend="smallcaps">Electrical</hi> <hi rend="italics">Apparatus,</hi> and <hi rend="smallcaps">Electrical</hi> <hi rend="italics">Machine.</hi></p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Shock,</hi> is the sudden explosion between the opposite sides of a charged electric; so called because if the discharge be made through the body of an animal, it occasions a sudden motion by the contraction of the muscles through which it passes, accompanied with a disagreeable sensation. The force of this shock is proportioned to the quantity of coated surface, the thinness of the glass, and the power of the machine by which it is charged. Its velocity is almost instantaneous, and it has not been found to take up the least sensible time in passing to the greatest distances.</p><p>It has been observed that the Electrical Shock is weakened by being communicated through several persons in contact with one another. Indeed it is obstructed in its passage, even through the best conductors, as it will prefer a short passage through the air to a long one through the most perfect conductors; and if the circuit be interrupted, either by electrics, or very imperfect conductors of a moderate thickness, the shock will rend them in its passage, disperse them in every direction, and exhibit the appearance of a sudden expansion of the air about the centre of the shock. A strong shock made to pass through or over the belly of a muscle, forces it to contract; and sent through a small animal body, deprives it instantly of life, and hastens putrefaction. It gives polarity to magnetic needles, reverses their poles, and produces effects precisely similar, though inferior in degree, to those of lightning.</p><p><hi rend="smallcaps">Electrical</hi> <hi rend="italics">Star.</hi> See <hi rend="smallcaps">Star.</hi> <pb n="420"/><cb/></p></div1><div1 part="N" n="ELECTRICITY" org="uniform" sample="complete" type="entry"><head>ELECTRICITY</head><p>, or <hi rend="smallcaps">Electrical</hi> <hi rend="italics">Force,</hi> is that power or property, which was first observed in amber, the lyneurium, or tourmalin, and which sealing-wax, glass, and a variety of other substances, called electrics, are now known to possess, of attracting light bodies, when excited by heat or friction; and which is also capable of being communicated in particular circumstances to other bodies.</p><p><hi rend="smallcaps">Electricity</hi> also denotes the science, or that part of natural philosophy, which proposes to investigate the nature and effects of this power. From <foreign xml:lang="greek">plextzon</foreign>, the Greek name for amber, is derived the term Electricity, which is now very extensively applied, not only to the power of attracting light bodies inherent in amber, but to other similar powers, and their various effects, in whatever bodies they reside, or to whatever bodies they may be communicated.</p><p>Muschenbroek and Æpinus have observed a considerable analogy, in a variety of particulars, between the powers of Electricity and Magnetism; and they have also pointed out many instances in which they differ.</p><p><hi rend="italics">History of</hi> <hi rend="smallcaps">Electricity.</hi>——The property which amber possesses of attracting light bodies, was very anciently observed. Thales of Miletus, 600 years before Christ, concluded from hence that it was animated. But the first person who expressly mentioned this substance, was Theophrastus, about 300 years before Christ. The attractive property of amber is also occasionally noticed by Pliny, and other later naturalists, particularly Gassendus, Kenelm Digby, and Sir Thomas Brown. But it was generally apprehended that this quality was peculiar to amber and jet, and perhaps agate, till W. Gilbert, a native of Colchester, and a physician in London, published his treatise <hi rend="italics">De Magnete,</hi> in the year 1600. Dr. Gilbert made many considerable experiments and discoveries, considering the then infant state of the science. He enlarged the lift both of electrics, and of the bodies on which they act: he remarked, that a dry air was most favourable to electrical appearances, whilst a moist air almost annihilates the electric virtue: he also observed the conical figure assumed by electrified drops of water: he considered electrical attraction separately from repulsion, which he thought had no place in Electricity, as a phenomenon similar to the attraction of cohefion, and he imagined, that electrics were brought into contact with the bodies on which they act by their effluvia, excited by friction.</p><p>The ingenious Mr. Boyle added to the catalogue of electric substances; but he thought that glass possessed this power in a very low degree: he found, that the Electricity of all bodies, in which it might be excited, was increased by wiping and warming them before they were rubbed; that an excited electric was acted upon by other bodies as strongly as it acted upon them; that diamonds rubbed against any kind of stuff, emitted light in the dark; and that feathers would cling to the fingers, and to other substances, after they had been attracted by electrics. He accounted for electrical attraction, by supposing a glutinous effluvia emitted from electrics, which laid hold of small bodies, in its way, and carried them back to the body from which it proceeded. <cb/></p><p>Otto Guericke, the celebrated inventor of the airpump, lived about the same time. This ingenious philosopher discovered, by means of a globe of sulphur, that a body once attracted by an electric, was next repelled, and continued in this state of repulsion till it should be touched by some other body: he also observed the sound and light produced by the excitation of his globe; and that bodies immerged in electrical atmospheres are themselves electrified with an electricity opposite to that of the atmosphere.</p><p>The light emitted by electrical bodies was, not long after, observed to much greater advantage by Dr. Wall, who ascribes to light the electrical property which they possess; and he suggests a similarity between the effects of electrieity and lightning.</p><p>Sir Isaac Newton was not inattentive to this subject: he observed that excited glass attracts light bodies on the side opposite to that on which it is rubbed; and he ascribes the action of electric bodies to an elastic fluid, which freely penetrates glass, and the emission of it to the vibratory motions of the parts of excited bodies.</p><p>Mr. Hawksbee wrote on this subject in the year 1709, when a new æra commenced in the history of this science. He first took notice of the great electrical power of glass, and the light proceeding from it; though others had before observed the light proceeding from other electrified substances: he also noted the noise occasioned by it, with a variety of phenomena relating to electrical attraction and repulsion. He first introduced a glass globe into the electrical apparatus, to which circumstance it was that many of his important discoveries were owing.</p><p>After his time there was an interval of near 20 years in the progress of this science, till Mr. Stephen Grey established a new æra in the history of Electricity. To him we owe the capital discovery of communicating the power of native electrics to other bodies, in which it cannot be excited, by supporting them on silken lines, hair lines, cakes of resin or glass; and a more accurate distinction than had hitherto obtained between electrics and non-electrics: he also shewed the effect of electricity on water much more obviously than Gilbert had done in the infancy of this science.</p><p>The experiments of Mr. Grey were repeated by M. du Fay, member of the Academy of Sciences at Paris, to which he added many new experiments and discoveries of his own. He observed, that electrical operations are obstructed by great heat, as well as by a moist air; that all bodies, both solid and fluid, would receive electricity, when placed on warm or dry glass, or sealing-wax; that those bodies which are naturally the least electric, have the greatest degree of electricity communicated to them by the approach of the excited tube. He transmitted the electric virtue through a distance of 1256 feet; and first observed the electric spark from a living body, suspended on silken lines, and noted several circumstances attending it. M. du Fay also established a principle, first suggested by Otto Guericke, that electric bodies attract all those that are not so, and repel them as soon as they are become electric, by the vicinity or contact of the electric body. He likewise inferred from other experiments, <pb n="421"/><cb/> that there were two kinds of electrieity; one of which he called the <hi rend="italics">vitreous,</hi> belonging to glass, rock crystal, &c; and the other <hi rend="italics">resinous,</hi> as that of amber, gumlac, &c, distinguished by their repelling those of the same kind, and attracting each other. He farther observed, that communicated electricity had the same property as the excited; and that electric substances attract the dew more than conductors.</p><p>Mr. Grey, resuming his experiments in 1734, suspended several pieces of metal on silken lines, and found that by electrifying them they gave sparks; which was the origin of metallic conductors: and on this occasion he discovered a cone or pencil of electric light, such as is now known to issue from an electrified point. From other experiments he concludes, that the electric power seems to be of the same nature with that of thunder and lightning.</p><p>Dr. Desaguliers succeeded Mr. Grey in the prosecution of this science. The account of his first experiments is dated in 1739. To him we owe those technical terms of <hi rend="italics">conductors</hi> or <hi rend="italics">non-electrics,</hi> and <hi rend="italics">electrics per se;</hi> and he first ranked pure air among the electrics per se, and supposed its Electricity to be of the vitreous kind.</p><p>After the year 1742, in which Dr. Desaguliers concluded his experiments, the subject was taken up and pursued in Germany: the globe was substituted for the tube, which had been used ever since the time of Hawksbee, and a cushion was soon after used as a rubber, instead of the hand. About this time too, some used cylinders instead of the globes; and some of the German electricians made use of more globes than one at the same time. By thus increasing the electrical power, they were the first who succeeded in setting fire to inflammable substances: this was first done by Dr. Ludolf, in the beginning of the year 1744, who, with sparks excited by the friction of a glass tube, kindled the ethereal spirit of Frobenius. Winkler did the same by a spark from his own finger, by which he kindled French brandy, and other spirits, after previously heating them. Mr. Gralath fired the smoke of a candle just blown out, and so lighted it again; and Mr. Boze fired gun-powder, by means of its inflammable vapour. About this time Ludolf the younger demonstrated, that the luminous barometer was made perfectly electrical by the motion of the quicksilver. The electrical star and electrical bells were also of German invention.</p><p>In England Dr. Watson made a distinguished figure from this period in the history of Electricity: he fired a variety of substances by the electrical spark, and first discovered that they are capable of being fired by the repulsive power of Electricity. In the year 1745, the accumulation of the electrical power in glass, by means of the Leyden phial, was first discovered. See L<hi rend="smallcaps">EYDEN</hi> <hi rend="italics">phial:</hi> and for the method practised about this time, of measuring the distance to which the electrical shock may be conveyed, see <hi rend="italics">Electrical</hi> <hi rend="smallcaps">Circuit.</hi> Dr. Watson discovered that the glass tubes and globes do not contain the electric matter in themselves, but only serve as <hi rend="italics">first-movers</hi> or <hi rend="italics">determiners,</hi> as he expresses it, of that power; which was also confirmed towards the end of 1746, by Mr. Benjamin Wilson, who made the same discovery, that the electric fluid does not come <cb/> from the globe, but from the earth, and other nonelectric bodies about the apparatus. Dr. Watson also discovered what Dr. Franklin observed about the same time in America, and called the <hi rend="italics">plus</hi> and <hi rend="italics">minus</hi> in Electricity. He likewise shewed that the electric matter passed through the substance of the metal of communication, and not merely over the surface. The history of medical Electricity commenced in the year 1747. We must omit other experiments, and conclusions drawn from them, by Mr. Wilson, Mr. Smeaton, and Dr. Miles in England, and by the Abbé Nollet, with regard to the effect of Electricity on the evaporation of fluids, on solids, and on animal and other organized bodies, in France.</p><p>Whilst the philosophers of Europe were busily employed in electrical experiments and pursuits, those of America, and Dr. Franklin in particular, were equally industrious, and no less successful. His discoveries and observations in Electricity were communicated in several letters to a friend; the first of which is dated in 1747, and the last in 1754; and the particulars of his system may be seen under the articles, <hi rend="italics">Theory of</hi> E<hi rend="smallcaps">LECTRICITY, Leyden</hi> <hi rend="italics">Phial,</hi> <hi rend="smallcaps">Points, Charging</hi>, C<hi rend="smallcaps">ONDUCTORS, Electrics</hi>, &c.</p><p>The similarity between Electricity and Lightning had been suggested by several writers: Dr. Franklin first proposed a method of bringing the matter to the test of experiment, by raising an electrical kite; and he succeeded in collecting electrical fire by this means from the clouds, in 1752, one month after the same theory had been verified in France, and without knowing what had been done there: and to him we owe the practical application of this discovery, in securing buildings from the damage of lightning, by erecting metallic conductors. See <hi rend="smallcaps">Conductors</hi>, and <hi rend="smallcaps">Lightning.</hi></p><p>In the subsequent period of the history of this science, Mr. Canton in England, and Signior Beccaria in Italy, acquired distinguished reputation. They both discovered, independently of each other, that air is capable of receiving Electricity by communication, and of retaining it when received. Mr. Canton also, towards the latter end of the year 1753, pursued a series of experiments, which prove that the appearances of positive and negative Electricity, which had hitherto been deemed essential and unchangeable properties of different substances, as of glass and sealing-wax for instance, depend upon the surface of the electrics, and that of the rubber.</p><p>This hypothesis, verified by numerous experiments, occasioned a controversy between Mr. Canton, and Mr. Delaval, who still maintained that these different powers depended entirely on the substances theniselves. About this time too, some curious experiments were performed by four of the principal electricians of that period, viz. Dr. Franklin, and Messrs Canton, Wilcke, and Æpinus, to ascertain the nature of electric atmospheres; the result of which see under that article.</p><p>The theory of two electric fluids, always co-existent and counteracting each other, though not absolutely independent, was maintained by a course of experiments on silk stockings of different colours, communicated to the Royal Society by Mr. Symmer, in the year 1759, which were farther pursued by Mr. Cigna <pb n="422"/><cb/> of Turin, who published an account of them in the Memoirs of the Academy at Turin for the year 1765.</p><p>Many instances occur in the history of the science about this period, of the astonishing force of the electric shock, in melting wires, and producing other similar effects: but the most remarkable is an experiment of S. Beccaria, in which he thus revivified metals. Several experiments were also made by Dr. Watson, Mr. Smeaton, Mr. Canton, and others, on the passage of the electric fluid through a vacuum, and its luminous appearance, and on the power possessed by certain substances of retaining the light communicated to them by an electric explosion. Mr. Canton, S. Beccaria, and others, made many experiments to identify Electricity and lightning, to ascertain the state of the atmosphere at different times, and to explain the various phenomena of the Aurora Borealis, Water-Spouts, Hurricanes, &c, on the principles of this science.</p><p>Those who are desirous of farther information with respect to the history of electrical experiments and discoveries, may consult Dr. Priestley's History and Present State of Electricity. This author however is not merely an historian: his work contains many original experiments and discoveries made by himself. He ascertained the conducting power of charcoal, and of hot glass; the Electricity of fixed and inflammable air, and of oil; the difference between new and old glass, with respect to the diffusion of Electricity over its surface; the lateral explosion in electrical discharges; a new method of fixing circular-coloured spots on the surfaces of metals, and the most probable difference between electrics and conductors, &c. The science is also greatly indebted to many other persons, elther for their experiments and improvements of it, or for treatises and other writings upon it; as Mr. Henley, to whom we owe several curious experiments and observations on the electrical and conducting quality of different substances, as chocolate, vapour, &c, with the reason of the difference between them; the fusion of platina; the nature of the electric fluid, and its course in a discharge; the method of estimating the quantity of it in electrical bodies by an electrometer; the influence of points; &c, &c. Also Messrs Van Marum, Van Swinden, Ferguson, Cavallo, Lord Mahon, Nairne, &c, &c, for their several treatises on the subject of Electricity, any of which may be consulted with advantage for the experiments and principles of the science.</p><p><hi rend="italics">Medical</hi> <hi rend="smallcaps">Electricity.</hi> It is natural to imagine that a power of such efficacy as that of electricity would be applied to medical purposes; especially, since it has been found invariably to increase the sensible perspiration, to quicken the circulation of the blood, and to promote the glandular secretion: accordingly, many instances occur in the latter period of the history of this science, in which it has been applied with considerable advantage and success. And among the variety of cases in which it has been tried, there are none in which it has been found prejudicial except those of pregnancy and the venereal disease. In most disorders, in which it has been used with perseverance, it has given at least a temporary and partial relief, and in many it has effected a total cure. Of which numerous inftances may be seen in the Philos. Trans. and the writ- <cb/> ings on this science by Messrs Lovet, Westley, Ferguson, Cavallo, &c. &c.</p><p><hi rend="italics">Theory of</hi> <hi rend="smallcaps">Electricity.</hi> It is hardly necessary to recite the ancient hypotheses on this subject; such as that of the sympathetic powder of the Peripatetics; that of unctuous effluvia emitted by excited bodies, and returning to them again, adopted by Gilbert, Gassendus, Sir Kenelm Digby, &c; or that of the Cartesians, who ascribed electricity to the globules of the first elements, discharged through the pores of the rubbed substance, and in their return carrying with them those light bodies, in whose pores they were entangled: these hypotheses were framed in the infancy of the science, and of philosophy in general, and have long since been exploded. In the more advanced state of electricity there have been two principal theories, each of which has had its advocates. The one, is that of two distinct electric fluids, repulsive with respect to themselves, and attractive of one another, adopted by M. du Fay, on discovering the two opposite species of electricity, viz, the vitreous and resinous, and since new-modelled by Mr. Symmer. It is supposed that these two fluids are equally attracted by all bodies, and exist in intimate union in their pores; and that in this state they exhibit no mark of their existence. But that the friction of an electric by a rubber separates these fluids, and causes the vitreous electricity of the rubber to pass to the electric, and then to the prime conductor of a machine, while the resinous electricity of the conductor and electric is conveyed to the rubber: and thus the quality of the electric fluid, possessed by the conductor and the rubber, is changed, while the quantity remains the same in each. In this state of separation, the two electric fluids will exert their respective powers; and any number of bodies charged with either of them will repel each other, attract those bodies that have less of each particular fluid than themselves, and be still more attracted by bodies that are wholly destitute of it, or that are loaded with the contrary. According to this theory, the electric spark makes a double current; one fluid passing to an electrified conductor from any substance presented to it, whilst the same quantity of the other fluid passes from it; and when each body receives its natural quantity of both fluids, the balance of the two powers is restored, and both bodies are unelectrified. For a surther account of the explication of some of the principal phenomena of electricity by this theory, see Dr. Priestley's History, vol. 2, § 3.</p><p>The other theory is commonly distinguished under the denomination of <hi rend="italics">positive and negative electricity,</hi> being first suggested by Dr. Watson, but digested, illustrated, and confirmed by Dr. Franklin; and since that it has been known by the appellation of the Franklinian hypothesis. It is here supposed that all the phenomena of electricity depend on one fluid, <hi rend="italics">sui generis,</hi> extremely subtile and elastic, dispersed through the pores of all bodies, by which the particles of it are as strongly attracted as they are repelled by one another. When bodies possess their natural share of this fluid, or such a quantity as they can retain by their non-attraction, it is then said they are in an unelectrified state; but when the equilibrium is disturbed, and they either acquire an additional quantity <hi rend="italics">from</hi> other bodies, or lose part of their own natural share by communication <hi rend="italics">to</hi> other bodies, <pb n="423"/><cb/> they exhibit electrical appearances. In the former case it is said they are electrified positively, or <hi rend="italics">plus;</hi> and in the other negatively, or <hi rend="italics">minus.</hi> This electric fluid, it is supposed, moves with great ease in those bodies that are called conductors, but with extreme difficulty and slowness in the pores of electrics; whence it comes to pass, that all electrics are impermeable to it. It is farther supposed that electrics contain always an equal quantity of this fluid, so that there can be no surcharge or increase on one side without a proportionable decrease or loss on the other, and vice versa; and as the electric does not admit the passage of the fluid through its pores, there will be an accumulation on one side, and a corresponding deficiency on the other. Then when both sides are connected together by proper conductors, the equilibrium will be restored by the rushing of the redundant fluid from the overcharged surface to the exhausted one. Thus also, if an electric be rubbed by a conducting substance, the electricity is only conveyed from one to the other, the one giving what the other receives; and if one be electrified positively, the other will be electrified negatively, unless the loss be supplied by other bodies connected with it, as in the case of the electric and insulated rubber of a machine. This theory serves likewise to illustrate the other phenomena and operations in the science of electricity. Thus, bodies differently electrified will naturally attract each other, till they mutually give and receive an equal quantity of the electric fluid, and the equilibrium is restored between them. Beccaria supposes, that this effect is produced by the electric matter making a vacuum in its passage, and the contiguous air afterwards collapsing, and so pushing the bodies together.</p><p>The influence of points, in drawing or throwing off the electric fluid, depends on the less resistance it finds to enter or pass off through fewer particles than through a greater number, whose resistance is united in flat or round surfaces. The electric light is supposed to be part of the electric fluid, which appears when it is properly agitated; and the sound of an explosion is produced by vibrations, occasioned by the air's being displaced by the electric fluid, and again suddenly collapsing.</p><p>As to the nature of the electric fluid, philosophers have entertained very different sentiments: some, and among them Mr. Wilson, have supposed that it is the same with the ether of Sir Isaac Newton, to which the phenomena of attraction and repulsion are ascribed; whilst the light, smell, and other sensible qualities of the electric fluid, are referred to the grosser particles of bodies, driven from them by the forcible action of this ether; and other appearances are explained by means of a subtile medium diffused over the surfaces of all bodies, and resisting the entrance and exit of the ether; which medium, it is supposed, is the same with the electric fluid, and is more rare on the surfaces of conductors, and more dense and resisting on those of electrics: but Dr. Priestley remarks that, though they may possess some common properties, they have others essentially distinct; the ether is repelled by all other matter, whereas the electric fluid is strongly attracted by it. Others have had recourse to the element of fire; and from the supposed identity of fire and the electric fluid, as well as from the similarity of some of their effects, the <cb/> latter has been usually called the electric fire: but most electricians have supposed that it is a fluid <hi rend="italics">sui generis.</hi> Mr. Cavendish has published an attempt to deduce and explain some of the principal phenomena of electricity in a mathematical and systematic manner, from the nature of this fluid, considered as composed of particles that repel each other, and attract the particles of all other matter, with a force inversely as some less power of the distance than the cube, whilst the particles of all other matter repel each other, and attract those of the electric fluid, according to the same law. Philos. Trans. vol. 61, pa. 584—677. And a similar hypothesis and method of reasoning was also proposed by M. Æpinus, in his Tentamen Theoriæ Electricitatis & Magnetismi.</p><p>Dr. Priestley concludes, from experiments, that the electric matter either is phlogiston, or contains it, since he found that both produced similar effects. Mr. Henley also apprehends, that the electric fluid is a modification of that element, which, in its quiescent state, is called phlogiston; in its first active state, electricity; and when violently agitated, fire. Perhaps we may be allowed to enlarge our views, and consider the sun as the fountain of the electric fluid, and the zodiacal light, the tails of comets, the aurora borealis, lightning, and artificial electricity, as its various and not very dissimilar modifications. On this subject, see Priestley's Hist. of Electr. vol. 2, part 3, § 1, 2, 3; Wilson's Essay towards an Explication of the Phenomena of Electricity, &c; Wilson and Hoadley's Obs. &c, pa. 55, 1759; Freke's Essay on the Cause of Electricity, 1746; Priestley on Air, vol 1, pa. 186, 274, &c; Philos. Trans. vol. 67, pa. 129; and Mr. Eeles's Letters, on the same subject.</p></div1><div1 part="N" n="ELECTROMETER" org="uniform" sample="complete" type="entry"><head>ELECTROMETER</head><p>, is an instrument that measures the quantity, and determines the quality of electricity, in any electrified body. Previous to the invention of instruments of this kind, Mr. Canton estimated the quantity of electricity in a charged phial, by presenting the phial with one hand to an insulated conductor, and giving it a spark, which he took off with the other; proceeding in this manner till the phial was discharged, when he determined the height of the charge by the number of sparks. Electrometers are of 4 kinds: 1, the single thread; 2, the cork or pith balls; 3, the quadrant; and 4, the discharging Electrometer.</p><p>The 1st, or most simple Electrometer, is a linen thread, called by Dr. Desaguliers, the <hi rend="italics">thread of trial;</hi> which, being brought near an electrified body, is attracted by it: but this does little more than determine whether the body is in any degree electrified or not; without determining with any precision its quantity, much less the quality of it. The Abbé Nollet used two threads, shewing the degrec of electricity by the angle of their divergency exhibited in their shadow on a board placed behind them.</p><p>Mr. Canton's Electrometer consisted of two balls of cork, or pith of elder, about the size of a small pea, suspended by fine linen threads, about 6 inches long, which may be wetted in a weak solution of salt. See sig. 7. If the box containing these balls be insulated, by placing it on a drinking glass, &c, and an excited smooth glass tube be brought near them, they will first be attracted by it, and then be repelled both from the <pb n="424"/><cb/> glass, and from each other; but on the approach of excited wax, they will gradually approach and come together; and vice versa. This apparatus will also serve to determine the electricity of the clouds and air, by holding them at a sufficient distance from buildings, trees, &c; for if the electricity of the clouds or air be positive, their mutual repulsion will increase by the approach of excited glass, or decrease by the approach of amber or sealing-wax; on the contrary, if it be negative, their repulsion will be diminished by the former, and increased by the latter. See Philos. Trans. vol. 48, part 1 and 2, for an account of Mr. Canton's curious experiments with this apparatus.</p><p>If two balls of this kind be annexed to a prime conductor, they will serve to determine both the degree and quality of its electrification, by their mutual repulsion and divergency.</p><p>The Discharging Electrometer, sig. 3, plate ix, was invented by Mr. Lane. It consists of brass work G, the lower part of which is inclosed in the pillar F, made of baked wood, and boiled in linseed oil, and bored cylindrically about two-thirds of its length; the brass work is fixed to the pillar by the screw H, moveable in the groove I; and through the same is made to pass a steel screw L, to the end of which, and opposite to K, a polished hemispherical piece of brass, attached to the prime conductor, is fixed a ball of brass M well polished. To this screw is annexed a circular plate O, divided into 12 equal parts. The use of this Electrometer is to discharge a jar D, or any battery connected with the conductor, without a discharging rod, and to give shocks successively of the same degree of strength; on which account it is very fit for medical purposes. Then, if a person holds a wire fastened to the screw H in one hand, and another wire fixed to E, a loop of brass wire passing from the frame of the machine to a tin plate, on which the phial D stands, he will perceive no shock, when K and M are in contact; and the degree of the explosion, as well as the quantity of electricity accumulated in the phial, will be regulated by the distance between K and M. Philos. Trans. vol. 57, pa. 451.— Mr. Henley much improved Mr. Lane's Electrometer, by taking away the screw, the double milled nut, and the sharp-edged graduated plate, and adding other contrivances in their stead. Mr. Henley's discharger of this kind has two tubes, one sliding within the other, to lengthen and accommodate it to larger apparatus.</p><p>The Quadrant Electrometer of Mr. Henley, consists of a stem, terminating at its lower end with a brass ferrule and screw, for fastening it upon any occasion; and its upper part ends in a ball. Near the top is fixed a graduated semicircle of ivory, on the centre of which the index, being a very light rod with a cork ball at its extremity, reaching to the brass ferrule of the stem, is made to turn on a pin in the brass piece, so as to keep near the graduated limb of the semicircle. When the Electrometer is not electrisied, the index hangs parallel to the stem; but as soon as it begins to be electrisied, the index, repelled by the stem, will begin to move along the graduated edge of the semicircle, and so mark the degree to which the conductor is electrified, or the height to which the charge of any jar or battery is advanced.</p><p>Mr. Cavallo has also contrived several ingenious Elec- <cb/> trometers, for disserent uses; as may be seen in his Treatise on Electricity, pa. 370, &c, and in the Philos, Trans. vol. 67, pa. 48 and 399.</p></div1><div1 part="N" n="ELECTROPHOR" org="uniform" sample="complete" type="entry"><head>ELECTROPHOR</head><p>, or <hi rend="smallcaps">Electrophorus</hi>, an instrument for shewing perpetual electricity; which was invented by Mr. Volta, of Como, near Milan, in Italy. The machine consists of two plates, fig. 8, one of which B is a circular plate of glass, covered on one side with some resinous electric, and the other A is a plate of brass, or a circular board, coated with tinfoil, and furnished with a glass handle I, which may be screwed into its centre by means of a socket. If the plate B be excited by rubbing it with new white flannel, and the plate A be applied to its coated side, a finger, or any other conductor, will receive a spark on touching this plate; and if the plate A be then separated, by means of the handle I, it will be found strongly electrified, with an electricity contrary to that of the plate B. By replacing the plate A, touching it with the finger, and separating it again, it will be found electrified as before, and give a spark to any conductor, attended with a snapping noise; and by this means a coated phial may be charged. The same phenomena may repeatedly be exhibited, without any renewed excitation of the electric plate B; the electric power of B having continued for several days, and even weeks, after excitation; though there is no reason to imagine that it is perpetual.</p><p>Mr. Cavallo prepares this machine by coating the glass plate with sealing wax; and Mr. Adams, philosophical instrument maker, prepares them with plates formed from a composition of two parts of shell-lac, and one of Venice turpentine, without any glass plate.</p><p>The action of this plate depends on a principle discovered and illustrated by the experiments of Franklin, Canton, Wilcke, and Æpinus, viz, that an excited electric repels the electricity of another body, brought within its sphere of action, and gives it a contrary electricity. Thus the plate A, touched by a conductor, whilst in contact with the plate B, electrified negatively, will acquire an additional quantity of the electric fluid from the conductor; but if it were in contact with a plate electrified positively, it would part with its electricity to the conductor connected with it. See an account of several curious experiments with this machine, by Mr. Henley, Mr. Cavallo, and Dr. Ingenhousz, in the Philos. Trans. vol. 66, pa. 513; vol. 67, pa. 116 and 389; and vol. 68, pa. 1027 and 1049.</p></div1><div1 part="N" n="ELEMENTARY" org="uniform" sample="complete" type="entry"><head>ELEMENTARY</head><p>, something that relates to the principles or elements of bodies, or sciences; as Elementary Air, Fire, Geometry, Music, &c.</p></div1><div1 part="N" n="ELEMENTS" org="uniform" sample="complete" type="entry"><head>ELEMENTS</head><p>, the first principles, of which all bodies and things are composed. These are supposed few in number, unchangeable, and by their combinations producing that extensive variety of objects to be met with in the works of nature.</p><p>Democritus stands at the head of the Elementary Philosophers, in which he is followed by Epicurus, and many others after them, of the Epicurean and corpuscular philosophers.</p><p>Among those who hold the Elements corruptible, some will have only one, and some several. Of the former, the principal are Heraclitus, who held fire; Anaximenes, air; Thales Milesius, water; and Hesiod, <pb n="425"/><cb/> earth; as the only Element. Hesiod is followed by Bernardin, Telesius; and Thales by many of the chemists.</p><p>Among those who admit several corruptible Elements, the principal are the Peripatetics; who, after their leader Aristotle, contend for four Elements, viz, fire, air, water, and earth. Aristotle took the notion from Hippocrates; Hippocrates from Pythagoras; and Pythagoras from Ocellus Lucanus, who it seems was the first author of it.</p><p>The Cartesians admit only three Elements, fire, air, and earth. See <hi rend="smallcaps">Cartesian</hi> <hi rend="italics">Philosophy.</hi></p><p>Newton observes, that it seems probable that God, in the beginning, formed matter in solid, massive, hard, impenetrable, moveable particles, of such sizes and sigures, &c, as most conduced to the end for which he formed them; and that these primitive particles, being solids, are incomparably harder than any porous body compounded of them; even so hard as never to wear out; no ordinary power being able to divide what God made one in the first creation. While the particles remain entire, they may compose bodies of one and the same nature and texture in all ages; but should they wear away, or break in pieces, the nature of things, depending on them, would be changed; water and earth, composed of old worn particles, and fragments of particles, would not be of the same nature and texture now, with water and earth composed of entire particles in the beginning. And therefore, that things may be lasting, the changes of corporeal things are to be placed only in the various separations, and new associations and motions of those permanent particles; compound bodies being apt to break, not in the midst of solid particles, but where those particles are laid together, and only touch in a few points. It seems to him likewise, that these particles have not only a vis inertiæ, with the passive laws of motion thence resulting, but are also moved by certain active principles; such as gravity, and the cause of fermentation, and the cohesion of bodies.</p><div2 part="N" n="Elements" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elements</hi></head><p>, a term also used for the first grounds and principles of arts and sciences; as the Elements of geometry, Elements of mathematics, &c. So Euclid's Elements, or simply the Elements, as they were anciently and peculiarly named, denotes the treatise on the chief properties of geometrical figures by that author.</p><p>The <hi rend="smallcaps">Elements</hi> of Mathematics have been delivered by several authors in their courses, systems, &c. The first work of this kind is that of Herigon, in Latin and French, and published in 1664, in 10 tomes; which contains Euclid's Elements and Data, Apollonius, Theodosius, &c; with the modern Elements of arithmetic, algebra, trigonometry, architecture, geography, navigation, optics, spherics, astronomy, music, perspective, &c. The work is remarkable for this, that a kind of real and universal characters are used throughout; so that the demonstrations may be understood by such as only remember the characters, without any dependence on language or words at all.</p><p>Since Herigon, the Elements of the several parts of mathematics have been also delivered by others; particularly the Jesuit Schottus, in his Cursus Mathematicus, in 1674; De Chales, in his Cursus, 1674; Sir Jonas Moore, in his New System of Mathematics, in <cb/> 1681; Ozanam, in his Cours de Mathematique, in 1699; Jones, in his Synopsis Palmariorum Matheseos, in 1706; and many others, but above all, Christ. Wolfius, or Wolf, in his Elementa Matheseos Universæ, in 2 vols 4to, the 1st published in 1713, and the 2d in 1715; a very excellent work of the kind. Another edition of the work was published at Geneva, in 5 vols 4to, of the several dates 1732, 1733, 1735, 1738, and 1741.</p><p>The <hi rend="smallcaps">Elements</hi> of Euclid, as they were the first, so they continue still the best system of geometry, are in 15 books. There have been numerous editions and commentaries of this work. Proclus wrote a commentary on it. Orontius Fineus first gave a printed edition of the sirst 6 books, in 1530, with notes, to explain Euclid's sense. Peletarius did the same in 1557. Nic. Tartaglia, about the same time, made a comment on all the 15 books, with the addition of many things of his own. And the same was also done by Billingsley in 1570; and by Flussates Candalla, a noble Frenchman, in the year 1578, with considerable additions as to the comparison and inscriptions of solid bodies; which work was afterwards republished with a prolix commentary, by Clavius. Commandine gave also a good edition of it. In 1703, Dr. Gregory published an edition of the whole works of Euclid, in Greek and Latin, including his Elements. But it would be endless to relate all the other editions of these Elements, either the whole, or in part, that have been given; some of the best of which are those of De Chales, Tacquet, Ozanam, Whiston, Stone, and most especially that of Dr. Rob. Simson, of Glasgow.</p><p>Other writers on the Elements of Geometry are almost out of number, in all nations.</p></div2><div2 part="N" n="Elements" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elements</hi></head><p>, in the Higher or Sublime Geometry, are the insinitely small parts, or differentials, of a right line, curve, surface, or solid.</p></div2><div2 part="N" n="Elements" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elements</hi></head><p>, in Astronomy, are those principles deduced from astronomical observations and calculations, and those fundamental numbers, which are employed in the construction of tables of the planetary motions. Thus, the Elements of the theory of the sun, or rather of the earth, are his mean motion and eccentricity, with the motion of the aphelia. And the Elements of the theory of the moon, are her mean motion, that of the node and apogee, the eccentricity, the inclination of her orbit to the plane of the ecliptic; &c.</p></div2></div1><div1 part="N" n="ELEVATION" org="uniform" sample="complete" type="entry"><head>ELEVATION</head><p>, the height or altitude of any thing.</p><div2 part="N" n="Elevation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elevation</hi></head><p>, in Architecture, denotes a draught or description of the principal face or side of a building; called also its upright or orthography.</p></div2><div2 part="N" n="Elevation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elevation</hi></head><p>, in Astronomy and Geography, is various; as Elevation of the equator, of the pole, of a star, &c.</p><p><hi rend="smallcaps">Elevation</hi> <hi rend="italics">of the Equator,</hi> is the height of the equator above the horizon; or an arc of the meridian intercepted between the equator and the horizon of the place.—The Elevation of the equator and of the pole together always make up a quadrant; the one being the complement of the other. Therefore, the Elevation of the pole being found, and subtracted from 90°, leaves the elevation of the equator.</p><p><hi rend="smallcaps">Elevation</hi> <hi rend="italics">of the Pole,</hi> is its height above the <pb n="426"/><cb/> horizon; or an arc of the meridian comprehended between the equator and the horizon of the place.</p><p>The elevation of the pole is always equal to the latitude of the place; that is, the arc of the meridian intercepted between the pole and the horizon, is every where equal to the are of the same meridian intercepted between the equator and the zenith. Thus the north pole is elevated 51° 31′ above the horizon of London; and the distance, or number of degrees, is the same between London and the equator; so that London is also in 51° 31′ of north latitude.</p><p><hi rend="smallcaps">Elevation</hi> <hi rend="italics">of a Star,</hi> or of any other point in the sphere, is the angular height above the horizon; or an are of the vertical circle intercepted between the star and the horizon. The meridian altitude of any such point, or its altitude when in the meridian, is the greatest of all.</p><p><hi rend="smallcaps">Elevation</hi> <hi rend="italics">of a Cannon,</hi> or <hi rend="italics">Mortar,</hi> is the angle which the bore or the axis of the piece makes with the horizontal plane.</p><p><hi rend="italics">Angle of</hi> <hi rend="smallcaps">Elevation</hi>, is the angle which any line of direction makes above a horizontal line.</p><p><hi rend="smallcaps">Elevation</hi> is also used by some writers on Perspective, for the scenography, or perspective representation of the whole body or building.</p></div2></div1><div1 part="N" n="ELLIPSE" org="uniform" sample="complete" type="entry"><head>ELLIPSE</head><p>, or <hi rend="smallcaps">Ellipsis</hi>, is one of the conic sections, popularly called an oval; being called an Ellipse or Ellipsis by Apollonius, the first and principal author on the conic sections, because in this figure the fquares of the ordinates are <hi rend="italics">less</hi> than, or <hi rend="italics">defective</hi> of, the rectangles under the parameters and abscisses.</p><p>This figure is differently defined by different authors; either from some of its properties, or from mechanical construction, or from the section of a cone, which is the best and most natural way. Thus; <figure><head>Fig. 1.</head></figure> <figure><head>Fig. 2.</head></figure></p><p>1. An Ellipse is a plane figure made by cutting a cone by a plane obliquely through the opposite sides of it; or so as that the plane makes a less angle with the base than the side of the cone makes with it; as ABD fig. 1.</p><p>The line AB connecting the uppermost and lowest points of the section, is the transverse axis; the middle of it C, is the centre; and the perpendicular to it DCE, through the centre, is the conjugate axis. The parameter, or latns rectum, is a 3d proportional to the transverse and conjugate axes; and the foci are two points in the transverse axis, at such equal distances from the centre, that the double ordinates passing through those points, and perpendicular to the transverse, are equal to the parameter.</p><p>2. The Ellipse is also variously described from some of its properties. As first, That it is a figure of such a <cb/> nature that if two lines be drawn from two certain points C and D in the axis, fig. 2, to any point E in the circumference, the sum of those two lines CE and DE will be every where equal to the same constant quantity, viz, the axis AB. Or, secondly, that it is <figure><head>Fig. 3.</head></figure> <figure><head>Fig. 4.</head></figure> a figure of such a nature, that the rectangle AGXGB (fig. 3) of the abscisses, tending contrary ways, is to GH<hi rend="sup">2</hi> the square of the ordinate, as AB<hi rend="sup">2</hi> to IK<hi rend="sup">2</hi>, the square of the transverse axis to the square of the conjugate, or, which is the same thing, as the transverse axis is to the parameter. And so of other properties.</p><p>3. Or the Ellipse is also variously described from its mechanical constructions, which also depend on some of its chief properties. Thus; 1st, If in the axis AB, there be taken any point I (fig. 4); and if with the radii AI, BI, and centres F and <hi rend="italics">f,</hi> the two foci, arcs be described, these arcs will intersect in certain points E, E, <hi rend="italics">e, e,</hi> which will be in the curve or circumference of the figure: and thus several points I being taken in the axis AB, as many more points E, <hi rend="italics">e,</hi> &c, will be found; then the curve line drawn through all these points E, <hi rend="italics">e,</hi> will be an Ellipse. Or, thus; if there be taken a thread of the exact length of the transverse axis AB, and the ends of the thread be fixed by pins in the two foci F and <hi rend="italics">f;</hi> (fig. 5) then moving a pen or pencil within the thread, so as to keep it always stretched out, it will describe the curve called an Ellipse. <figure><head>Fig. 5.</head></figure> <figure><head>Fig. 6.</head></figure></p><p><hi rend="italics">To Construct an</hi> <hi rend="smallcaps">Ellipse.</hi> There are many other ways of describing or constructing an Ellipse, besides those just now given: as</p><p>1st. If upon the given transverse axis there be described a circle AGB (fig. 6), to which draw any ordinate DG, and DE a 4th proportional to the transverse, the conjugate, and the ordinate DG; then E is a point in the curve. Or if the circle <hi rend="italics">agb</hi> be described on the conjugate axis <hi rend="italics">ab,</hi> to which any ordinate <hi rend="italics">dg</hi> is drawn, in which taking <hi rend="italics">d</hi>E in like manner a 4th proportional to the conjugate, the transverse, and ordinate <hi rend="italics">dg,</hi> then <pb n="427"/><cb/> shall E be in the curve. Or, having described the two circles, and drawn the common radius C<hi rend="italics">g</hi>G cutting them in G and <hi rend="italics">g</hi>; then <hi rend="italics">dg</hi>E drawn parallel to the transverse, and DGE parallel to the conjugate, the intersection E of these two lines will be in the curve of the ellipse. And thus several points E being found, the curve may be drawn through them all with a steady hand. <figure><head>Fig. 7.</head></figure> <figure><head>Fig. 8.</head></figure></p><p>2. If there be provided three rulers, of which the two GH and FI (fig. 7) are of the length of the transverse axis LK; and the third FG equal to HI the distance between the foci; then connecting these rulers so as to be moveable about the foci H and I, and about the points F and G, their intersection E will always be in the curve of the ellipse; so that by moving the rulers about the joints, with a pencil passed through the slits made in them, it will trace out the Ellipse. <figure><head>Fig. 9.</head></figure> <figure><head>Fig. 10.</head></figure></p><p>3. If one end A of any two equal rulers AB, DB, (fig. 9 and 10) which are moveable about the point B, like a carpenter's joint-rule, be fastened to the ruler LK, so as to be moveable about the point A; and if the end D of the ruler DB be drawn along the side of the ruler LK; then any point E, taken in the side of the ruler DB, will describe an ellipse, whose centre is A, conjugate axis = 2DE, and transverse = 2AB+2BE.</p><p>Another method of description is by the Elliptical Compass. See that article, below.</p><p><hi rend="italics">Some of the more Remarkable Properties of the Ellipse.</hi> —1. The rectangles under the abscisses are proportional to the squares of their ordinates; or as the square of any axis, or any diameter, is to the square of its conjugate, so is the rectangle under two abscisses of the former, to the square of their ordinate parallel to the latter; or again, as any diameter is to its parameter, so is the said rectangle under two abscisses of that diameter, to the square of their ordinate. So that if <hi rend="italics">d</hi> be <cb/> any diameter, <hi rend="italics">c</hi> its conjugate, <hi rend="italics">p</hi> its parameter = <hi rend="italics">c</hi><hi rend="sup">2</hi>/<hi rend="italics">d</hi><*> <hi rend="italics">x</hi> the one absciss, <hi rend="italics">d</hi>-<hi rend="italics">x</hi> the other, and <hi rend="italics">y</hi> the ordinate; then, as . From either of which equations, called the equation of the curve, any one of the quantities may be found, when the other three are given.</p><p>2. The sum of two lines drawn from the foci to meet in any point of the curve, is always equal to the transverse axis; that is, CE + DE = AB, in the 2d fig. Consequently the line CG drawn from the focus to the end of the conjugate axis, is equal to AI the semitransverse. <figure><head>Fig. 11.</head></figure></p><p>3. If from any point of the curve, there be an ordinate to either axis, and also a tangent meeting the axis produced; then half that axis will be a mean proportional between the distances from the centre to the two points of intersection; viz, CA a mean proportional between CD and CT. And consequently all the tangents TE, TE, meet in the same point of the axis produced, which are drawn from the extremities E, E, of the common ordinates DE, DE, of all Ellipses described on the same axis AB.</p><p>4. Two lines drawn from the foci to any point of the curve, make equal angles with the tangent at that point: that is, the [angle] FET = [angle]<hi rend="italics">f</hi>E<hi rend="italics">t.</hi></p><p>5. All the parallelograms are equal to each other, that are circumscribed about an Ellipsis; and every such parallelogram is equal to the rectangle of the two axes.</p><p>6. The sum of the squares of every pair of conjugate diameters, is equal to the same constant quantity, viz, the sum of the squares of the two axes.</p><p>7. If a circle be described upon either axis, and from any point in that axis an ordinate be drawn both to the circle and ellipsis; then shall the ordinate of the circle be to the ordinate of the Ellipse, as that axis is to the other axis: viz, AB : <hi rend="italics">ab</hi> :: DG : DE, and <hi rend="italics">ab</hi> : AB :: <hi rend="italics">dg : d</hi>E. (in the 6th fig.) And in the same proportion is the area of the circle to the area of the Ellipse, or any corresponding segments ADG, ADE. Also the area of the Ellipse is a mean proportional between the areas of the inscribed and circumscribed circles. Hence therefore,</p><p>8. <hi rend="italics">To find the Area of an Ellipse.</hi> Multiply the two axes together, and that product by .7854, for the area. Or</p><p>9. <hi rend="italics">To find the Area of any Segment</hi> ADE. Find the <pb n="428"/><cb/> area of the corresponding segment ADG of a circle on the same diameter AB; then say, as the axis AB: its conj. <hi rend="italics">ab</hi> :: circ. seg. ADG: elliptic seg. ADE.</p><p>10. <hi rend="italics">To find the length of the whole circumference of the Ellipse.</hi> Multiply the circumference of the circumscribing circle by the sum of the series &c, for the area: where <hi rend="italics">d</hi> is the difference between an unit and the square of the less axis divided by the square of the greater.</p><p>Or, for a near approximation, take the circumference of the circle whose diameter is an arithmetical mean between the two axes, or half their sum: that is, (<hi rend="italics">t</hi> + <hi rend="italics">c</hi>)/2 X 3.1416 = the perimeter nearly; being about the 200th part too little; where <hi rend="italics">t</hi> denotes the transverse, and <hi rend="italics">c</hi> the conjugate axis.</p><p>Or, again, take the circumference of the circle, the square of whose diameter is half the sum of, or an arithmetical mean between the squares of the two axes: that is, the perimeter nearly; being about the 200th part too great.</p><p>Hence combining these two approximate rules together, the periphery of the Ellipse will be very nearly equal to half their sum, or equal to , within about the 30000th part of the truth.</p><p>For the length of any particular arc, and many other parts about the Ellipse, see my Mensuration, pa. 283, &c, 2d edit. See also my Conic Sections, for many other properties of the Ellipse, especially such as are common to the hyperbola also, or to the conic sections in general.</p><p><hi rend="italics">Infinite</hi> <hi rend="smallcaps">Ellipses.</hi> See <hi rend="smallcaps">Elliptoide.</hi></p></div1><div1 part="N" n="ELLIPSOID" org="uniform" sample="complete" type="entry"><head>ELLIPSOID</head><p>, is an elliptical spheroid, being the solid generated by the revolution of an ellipse about either axis. See <hi rend="smallcaps">Spheroid.</hi></p><p>ELLIPTIC or <hi rend="smallcaps">Elliptical</hi>, something relating to an ellipse.</p><p><hi rend="smallcaps">Elliptic</hi> <hi rend="italics">Compasses.</hi> or <hi rend="smallcaps">Elliptical</hi> <hi rend="italics">Compass,</hi> is an instrument for describing ellipses at one revolution of the index. It consists of a cross ABGH (fig. 8.) with grooves in it, and an index CE, sliding in dovetail grooves; by which motion the end E describes the curve of an ellipse.</p><p><hi rend="smallcaps">Elliptical</hi> <hi rend="italics">Conoid,</hi> is sometimes used for the spheroid.</p><p><hi rend="smallcaps">Elliptical</hi> <hi rend="italics">Dial,</hi> an instrument usually made of brass, with a joint to fold together, and the gnomon to fall flat, for the convenience of the pocket. By this instrument are found the meridian, the hour of the day, the rising and setting of the sun, &c.</p><div2 part="N" n="Elliptoide" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Elliptoide</hi></head><p>, an infinite or indefinite Ellipsis, defined by the indefinite equation when <hi rend="italics">m</hi> or <hi rend="italics">n</hi> are greater than 1: for when they are each = 1, it denotes the common ellipse.</p><p>There are several kinds or degrees of Elliptoides, denominated from the exponent <hi rend="italics">m</hi> + <hi rend="italics">n</hi> of the ordinate <hi rend="italics">y.</hi> <cb/></p><p>As the cubical Elliptoide, expressed by ; the biquadratic, or sursolid ; &c.</p></div2></div1><div1 part="N" n="ELONGATION" org="uniform" sample="complete" type="entry"><head>ELONGATION</head><p>, in Astronomy, the distance of a planet from the sun, with respect to the earth; or the angle formed by two lines drawn from the earth, the one to the sun, and the other to the planet; or the arc measuring that angle: Or it is the difference between the sun's place and the geocentric place of the planet.</p><p>The Greatest Elongation, is the greatest distance to which the planets recede from the sun, on either side. This is chiefly considered in the inferior planets, Venus and Mercury; the Greatest Elongation of Venus being about 48 degrees, and of Mercury only about 28 degrees; which is the reason that this planet is so rarely seen, being usually lost in the light of the sun.</p><p>EMBER-<hi rend="italics">Days,</hi> are certain days observed by the church at four different seasons of the year; viz, the Wednesday, Friday, and Saturday next after Quadragesima Sunday, or the 1st Sunday in Lent; after Whitsunday; after Holyrood, or Holycross, the 14th day of September; and after St. Lucy, the 13th day of December. The name, it seems, is derived from Embers, or ashes, which it is supposed were strewed on the head, on these solemn fasts.</p><p><hi rend="smallcaps">Ember</hi>-<hi rend="italics">Weeks,</hi> are those weeks in which the Emberdays fall. These Ember-weeks are now chiefly noticed on account of the ordination of priests and deacons; because the canon appoints the Sundays next after the Ember-weeks for the solemn times of ordination; though the bishops, if they please, may ordain on any Sunday or holiday.</p><p>EMBOLIMÆAN, and <hi rend="smallcaps">Embolismic</hi>, <hi rend="italics">Intercalary,</hi> is chiefly used in speaking of the additional months inserted by chronologists to form the lunar cycle of 19 years.</p><p>The 19 solar years consisting of 6939 days and 18 hours, and the 19 lunar years only making 6726 days, it was found necessary to intercalate or insert 7 lunar months, containing 209 days; which, with the 4 bissextile days happening in the lunar cycle, make 213 days, and the whole 6939 days, the same as the 19 solar years, which make the lunar cycle.</p><p>In the course of 19 years there are 228 common moons, and 7 Embolismic moons, which are distributed in this manner, viz, the 3d, 6th, 9th, 11th, 14th, 17th, and 19th years, are Embolismic, and so contain 384 days each. And this was the method of computing time among the Greeks: though they did not keep regularly to it, as it seems the Jews did. And the method of the Greeks was followed by the Romans till the time of Julius Cæsar.</p><p>The Embolismic months, like other lunar months, are sometimes of 30 days, and sometimes only 29 days.</p><p>The <hi rend="italics">Embolismic Epacts</hi> are those between 19 and 29; which are so called, because, with the addition of the epact 11, they exceed the number 30: or rather, because the years which have these epacts, are Embolismic; having 13 moons each, the 13th being the Embolismic.</p></div1><div1 part="N" n="EMBOLISMUS" org="uniform" sample="complete" type="entry"><head>EMBOLISMUS</head><p>, in Chronology, signifies intercalation. As the Greeks used the lunar year, which contains only 354 days, that they might bring it to the <pb n="429"/><cb/> solar year, of 365 days, they had an Embolism every two or three years, when they added a 13th lunar month.</p></div1><div1 part="N" n="EMBOLUS" org="uniform" sample="complete" type="entry"><head>EMBOLUS</head><p>, the moveable part of a pump or syringe; called also the piston, and popularly the sucker. The pipe or barrel of a syringe, &c, being close shut, the embolus cannot be drawn up without a very considerable force; which force being withdrawn, the embolus returns again with violence; owing to the greater pressure of air above than below it.</p></div1><div1 part="N" n="EMBRASURE" org="uniform" sample="complete" type="entry"><head>EMBRASURE</head><p>, in Architecture, an enlargement of the aperture or opening of a door, or window, within side the wall, sloping back inwards, to give the greater play for the opening of the door, casement, &c, or to take in the more light.</p><div2 part="N" n="Embrasures" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Embrasures</hi></head><p>, in Fortification, are the apertures or holes through which the cannon are pointed, whether in casemates, batteries, or in the parapets of walls. In the navy, these are called port-holes. The Embrasures are placed 12 or 15 feet apart from each other; being made sloping or opening outwards, from 6 to 9 feet wide on the outside of the wall, and from 2 to 3 within, to allow the gun to traverse from side to side. Their base is about 2 1/2 or 3 feet above the platform on the inside of the wall, but sloping down outwards, so as to be only about 1 1/2 above it on the outside; in order that the muzzle on occasion may be depressed, and so the gun shoot low, or downwards.</p><p>EMERGENT <hi rend="italics">Year,</hi> in Chronology, is the epoch, or date, from whence any people begin to compute their time or dates. So, our Emergent year is sometimes the year of the creation, but more usually the year of the birth of Christ. The Jews used that of the Deluge, or the Exodus, &c. The Emergent year of the Greeks, was the beginning of the Olympic games; while that of the Romans was the date of the building of their city.</p></div2></div1><div1 part="N" n="EMERSION" org="uniform" sample="complete" type="entry"><head>EMERSION</head><p>, in Astronomy, is the re-appearance of the sun, moon, or other planet, after having been eclipsed, or hid by the interposition of the moon, earth, or other body.</p><p>The Emersions and immersions of Jupiter's first satellite, are particularly useful for finding the longitudes of places; the immersions being observed from Jupiter's conjunction with the sun, till his opposition; and the Emersions from the opposition till the conjunction. But within 15 days of the conjunction, both before and after it, they cannot be observed, because the planet and his satellites are then lost in the sun's light.</p><p><hi rend="smallcaps">Emersion</hi> is also used when a star, aster being hid by the sun, begins to re-appear, and to get out of his rays.</p><p><hi rend="italics">Minutes or Scruples of</hi> <hi rend="smallcaps">Emersion</hi>, an arc of the moon's orbit, which her centre passes over, from the time she begins to emerge out of the earth's shadow, to the end of the eclipse.</p><div2 part="N" n="Emersion" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Emersion</hi></head><p>, in Physics, the rising of any solid above the surface of a fluid that is specifically heavier than the solid, into which it had been violently immerged, or pushed.</p><p>It is one of the known laws of hydrostatics, that a lighter solid, being forced down into a heavier fluid, immediately endeavours to emerge; and that with a force equal to the excess of the weight of a quantity of <cb/> the fluid above that of an equal bulk of the solid. Thus, if the body be immerged in a fluid of double its specific gravity, it will emerge again till half its bulk be above the surface of the fluid.</p></div2></div1><div1 part="N" n="EMERSON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">EMERSON</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, a late eminent mathematician, was born in June 1701, at Hurworth, a village about three miles south of Darlington, on the borders of the county of Durham; at least it is certain that he resided here from his childhood. His father Dudley Emerson taught a school, and was a tolerable prosicient in the mathematics; and without his books and instructions, perhaps his son's genius, though eminently fitted for mathematical studies, might never have been unfolded. Beside his father's instructions, our author was assisted in the learned languages by a young clergyman, then curate of Hurworth, who was boarded at his father's house. In the early part of his life he attempted to teach a few scholars: but whether from his concise method (for he was not happy in explaining his ideas), or the warmth of his natural temper, he made no progress in his school; he therefore soon left it off; and satisfied with a moderate competence left him by his parents, he devoted himself to a studious retirement, which he thus closely pursued, in the same place, through the course of a long life, being mostly very healthy, till towards the latter part of his days, when he was much asflicted with the stone. Toward the close of the year 1781, being sensible of his approaching dissolution, he disposed of the whole of his mathematical library to a bookseller at York; and on May the 20th, 1782, his lingering and painful disorder put an end to his life at his native village; being near 81 years of age.</p><p>Mr. Emerson, in his person, was rather short, but strong and well made, with an open countenance and ruddy complexion, being of a healthy and hardy disposition. He was very singular in his behaviour, dress, and conversation. His manner and appearance were that of a rude and rather boorish country man; he was of very plain conversation, and indeed seemingly rude, commonly mixing oaths in his sentences, though without any ill intention. He had strong good natural mental parts, and could discourse sensibly on any subject, but was always positive and impatient of any contradiction. He spent his whole life in close study, and writing books, from the profits of which, he redeemed his little patrimony from some original incumbrance. In his dress he was as singular as in every thing else. He possessed commonly but one suit of clothes at a time, and those very old in their appearance. He seldom used a waistcoat; and his coat he wore open before, except the lower button; and his shirt quite the reverse of one in common use, the hind-side turned foremost, to cover his breast, and buttoned close at the collar behind. He wore a kind of rusty coloured wig, without a crooked hair in it, which probably had never been tortured with a comb from the time of its being made. A hat he would make to last him the best part of a life time; gradually lessening the flaps, bit by bit, as it lost its elasticity and hung down, till little or nothing but the crown remained.</p><p>He often walked up to London when he had any book to be published, revising sheet by sheet himself:— trusting no eye but his own was always a favourite <pb n="430"/><cb/> maxim with him. In mechanical subjects, he always tried the propositions practically, making all the different parts himself on a small scale; so that his house was filled with all kinds of mechanical instruments, together or disjointed. He would frequently stand up to his middle in water while sishing; a diversion he was remarkably fond of. He used to study incessantly for some time, and then for relaxation take a ramble to any pot-alchouse where he could get any body to drink with and talk to. The late Mr. Montagu was very kind to Mr. Emerson, and often visited him, being pleased with his conversation, and used often to come to him in the fields where he was working, and accompany him home, but could never persuade him to get into a carriage: on these occasions he would sometimes exclaim, “Damn your whim-wham! I had rather walk.” He was a married man, and his wife used to spin on an old-fashioned wheel, of his own making, a drawing of which is given in his Mechanics.</p><p>Mr. Emerson, from his strong vigorous mind and close application, had acquired a deep knowledge of all the branches of mathematics and physics, upon all parts of which he wrote good treatises, though in a rough and unpolished style and manner. He was not remarkable however for genius or discoveries of his own, as his works shew hardly any traces of original invention. He was well skilled in the science os music, the theory of sounds, and the various scales both ancient and modern; but he was a very poor performer, though he could make and repair some instruments, and sometimes went about the country tuning harpsichords.</p><p>The following is a list of Mr. Emerson's works; all of them printed in 8vo, excepting his Mechanics and his Increments in 4to, and his Navigation in 12mo. 1. The Doctrine of Fluxions.—2. The Projection of the Sphere, orthographic, stereographic, and gnomonical. —3. The Elements of Trigonometry.—4. The Principles of Mechanics.—5. A Treatise of Navigation on the Sea.—6. A Treatise on Arithmetic.—7. A Treatise on Geometry.—8. A Treatise of Algebra, in 2 books.—9. The Method of Increments.—10. Arithmetic of Infinites, and the Conic Sections, with other Curve Lines.—11. Elements of Optics and Perspective.—12. Astronomy.—13. Mechanics, with Centripetal and Centrifugal Forces.—14. Mathematical Principles of Geography, Navigation, and Dialling.—15. Commentary on the Principia, with the Defence of Newton.—16. Tracts.—17. Miscellanies.</p><p><hi rend="smallcaps">Eminential</hi> <hi rend="italics">Equation,</hi> a term used by some algebraists, in the investigation of the areas of curvilineal figures, for a kind of assumed equation that contains another equation Eminently, the latter being a particular case of the former. Hayes's Flux. pa. 97.</p></div1><div1 part="N" n="ENCEINTE" org="uniform" sample="complete" type="entry"><head>ENCEINTE</head><p>, a French term, in Fortification, signifying the whole inclosure, circumference, or compass of a fortified place, whether built with stone or brick, or only made of earth, and whether with or without bastions, &c.</p><p>ENCYCLOPÆDIA, the circle or chain of arts and sciences; sometimes also written Cyclopædia.</p></div1><div1 part="N" n="ENDECAGON" org="uniform" sample="complete" type="entry"><head>ENDECAGON</head><p>, a plane geometrical figure of eleven sides and angles, otherwise called Undecagon. If each side of this figure be 1, its area will be 9.3656399 <cb/> = 11/4 of the tang. of 73 7/11 degrees, to the radius 1. See my Mensuration, pa. 114, &c, 2d edit. See also <hi rend="smallcaps">Regular</hi> <hi rend="italics">Figure.</hi></p></div1><div1 part="N" n="ENFILADE" org="uniform" sample="complete" type="entry"><head>ENFILADE</head><p>, a French term, applied to those trenches, and other lines, that are ranged in a right line, and so may be scoured or swept by the cannon lengthways, or in the direction of the line.</p><p><hi rend="italics">To</hi> <hi rend="smallcaps">Enfilade</hi>, is to sweep lengthways by the firing of cannon, &c.</p><p><hi rend="italics">A Battery d'</hi><hi rend="smallcaps">Enfilade</hi>, is that where the cannon sweep a right line.</p><p>A <hi rend="italics">Post,</hi> or <hi rend="italics">Command d'</hi> <hi rend="smallcaps">Enfilade</hi>, is a height from whence a whole line may be swept at once.</p></div1><div1 part="N" n="ENGINE" org="uniform" sample="complete" type="entry"><head>ENGINE</head><p>, in Mechanics, a compound machine, consisting of several simple ones, as wheels, screws, levers, or the like, combined together, in order to lift, cast, or sustain a weight, or produce some other considerable effect, so as to save either force or time.</p><p>There are numberless kinds of engines; of which some are for war, as the Balista, Catapulta, Scorpio, Aries or Ram, &c; others for the arts of peace, as Mills, Cranes, Presses, Clocks, Watches, &c, &c.</p></div1><div1 part="N" n="ENGINEER" org="uniform" sample="complete" type="entry"><head>ENGINEER</head><p>, or <hi rend="smallcaps">Ingineer</hi>, is applied to a contriver or maker of any kind of useful engines or machines; or who is particularly skilled or employed in them. And he is denominated either a civil or military Engineer, according as the objects of his profession respect civil or military purposes.</p><p>A military Engineer should be an expert mathematician and draughtsman, and particularly versed in fortification and gunnery, being the person officially employed to direct the operations both for attacking and defending works. When at a siege the Engineers have narrowly surveyed the place, they are to make their report to the general, or commander, by acquainting him which part they judge the weakest, and where approaches may be made with most success. It is their business also to draw the lines of circumvallation and contravallation; also to mark out the trenches, places of arms, batteries, and lodgments, and in general to direct the workmen in all such operations.</p></div1><div1 part="N" n="ENGONASIS" org="uniform" sample="complete" type="entry"><head>ENGONASIS</head><p>, in Astronomy, the same as Hercules, one of the northern constellations; which see.</p></div1><div1 part="N" n="ENGYSCOPE" org="uniform" sample="complete" type="entry"><head>ENGYSCOPE</head><p>, the same as <hi rend="smallcaps">Microscope.</hi></p></div1><div1 part="N" n="ENHARMONIC" org="uniform" sample="complete" type="entry"><head>ENHARMONIC</head><p>, the last of the three kinds of music. It abounds in dieses, or the least sensible divisions of a tone. See Philos. Trans. number 481; also Wallis Appendix ad Ptolom. pa. 165, 166.</p></div1><div1 part="N" n="ENNEADECATERIS" org="uniform" sample="complete" type="entry"><head>ENNEADECATERIS</head><p>, in Chronology, a cycle or period of 19 solar years, being the same as the Golden Number and Lunar Cycle, or Cycle of the Moon; which see; as also <hi rend="smallcaps">Embolismic.</hi></p></div1><div1 part="N" n="ENNEAGON" org="uniform" sample="complete" type="entry"><head>ENNEAGON</head><p>, a plane geometrical figure of 9 sides and angles; and is otherwise called a Nonagon. If each side of this figure be 1, its area will be 6.1818242 = 9/4 of the tang. of 70 degrees, to the radius 1. See my Mensuration, pa. 114, 2d edit. See also the article <hi rend="smallcaps">Regular</hi> <hi rend="italics">Figure.</hi></p></div1><div1 part="N" n="ENTABLATURE" org="uniform" sample="complete" type="entry"><head>ENTABLATURE</head><p>, in Architecture, is that part of an order of column which is over the capital, comprehending the Architrave, Frize, and Corniche.</p><div2 part="N" n="Entablature" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Entablature</hi></head><p>, or <hi rend="smallcaps">Entablement</hi>, is sometimes also used for the last row of stones on the top of the wall of a building, on which the timber and cover- <pb n="431"/><cb/> ing rest; sometimes also called the Drip, because it projects a little, to throw the water off.</p></div2></div1><div1 part="N" n="ENVELOPE" org="uniform" sample="complete" type="entry"><head>ENVELOPE</head><p>, in Fortification, is a mound of earth, sometimes raised in the ditch of a place, and sometimes beyond it, being either in form of a single parapet, or of a small parapet bordered with a parapet. These Envelopes are made only to cover weak parts with single lines, without advancing towards the field, which cannot be done without works that require a great deal of room, such as horn-works, half-moons, &c. Envelopes are sometimes called Sillons, Contregards, Conserves, Lunettes, &c.</p></div1><div1 part="N" n="ENUMERATION" org="uniform" sample="complete" type="entry"><head>ENUMERATION</head><p>, a numbering or counting. Sir Isaac Newton wrote an ingenious treatise, being an Enumeration of the lines of the 3d order.</p><p>EOLIPILE. See Æ<hi rend="smallcaps">OLIPILE.</hi></p></div1><div1 part="N" n="EPACT" org="uniform" sample="complete" type="entry"><head>EPACT</head><p>, in Chronology, the excess of the solar month above the lunar synodical month; or of the solar year above the lunar year of 12 synodical months; or of several solar months above as many synodical months; or of several solar years above as many dozen of synodical months.</p><p>The Epacts then are either Annual or Menstrual.</p><p><hi rend="italics">Menstrual</hi> <hi rend="smallcaps">Epacts</hi>, are the excesses of the civil calendar month above the lunar month. Suppose, for example, it were new moon on the first day of January: then since the month of January contains 31 days, <table><row role="data"><cell cols="1" rows="1" role="data">and the lunar month</cell><cell cols="1" rows="1" role="data">29<hi rend="sup">ds</hi></cell><cell cols="1" rows="1" role="data">12<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">44<hi rend="sup">m</hi></cell><cell cols="1" rows="1" role="data"> 3<hi rend="sup">s</hi>;</cell></row><row role="data"><cell cols="1" rows="1" role="data">the menstrual Epact is</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">57</cell></row></table></p><p><hi rend="italics">Annual</hi> <hi rend="smallcaps">Epacts</hi>, are the excesses of the solar year above the lunar. Hence, <table><row role="data"><cell cols="1" rows="1" role="data">as the Julian solar year is</cell><cell cols="1" rows="1" role="data">365<hi rend="sup">ds</hi></cell><cell cols="1" rows="1" role="data"> 6<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data"> 0<hi rend="sup">m</hi></cell><cell cols="1" rows="1" role="data"> 0<hi rend="sup">s</hi>,</cell></row><row role="data"><cell cols="1" rows="1" role="data">and the Julian lunar year</cell><cell cols="1" rows="1" role="data">354</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">38,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the annual Epact will be</cell><cell cols="1" rows="1" role="data"> 10</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">22,</cell></row></table> that is, almost 11 days. Consequently the Epact of 2 years, is 22 days; of 3 years, 33 days; or rather 3, since 30 days make an embolismic, or intercalary month. Then, adding still 11, the Epact of 4 years is 14 days; and so of the rest as in the following table, where they do not become 30, or 0 again, till the 19th year; so that at the 20th year the Epact is 11 again; and hence the cycle of Epacts expires with the Golden Number, or Lunar Cycle of 19 years, and begins with the same again. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=6 align=center" role="data"><hi rend="italics"><hi rend="smallcaps">Table</hi> of Julian Epacts.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Golden Numb.</cell><cell cols="1" rows="1" rend="align=center" role="data">Epacts</cell><cell cols="1" rows="1" rend="align=center" role="data">Golden Numb.</cell><cell cols="1" rows="1" rend="align=center" role="data">Epacts</cell><cell cols="1" rows="1" rend="align=center" role="data">Golden Numb.</cell><cell cols="1" rows="1" rend="align=center" role="data">Epacts</cell></row><row role="data"><cell cols="1" rows="1" role="data">I</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">VIII</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">XV</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell></row><row role="data"><cell cols="1" rows="1" role="data">II</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">IX</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">XVI</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" role="data">III</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">X</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">XVII</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" role="data">IV</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">XI</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">XVIII</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell></row><row role="data"><cell cols="1" rows="1" role="data">V</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">XII</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">XIX</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">VI</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">XIII</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">or 0</cell></row><row role="data"><cell cols="1" rows="1" role="data">VII</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">XIV</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>Again, as the new moons are the same, or fall on the same day, every 19 years, so the difference between the solar and lunar years is the same every 19 years. And because the said difference is always to be added to the lunar year, to adjust or make it equal to the solar <cb/> year; hence the said difference respectively belonging to each year of the moon's cycle, is called the <hi rend="italics">Epact of the said year,</hi> that is, the number to be added to the said year, to make it equal to the solar year. Upon this mutual respect between the cycle of the moon and the cycle of the Epacts, is founded this <hi rend="center"><hi rend="italics">Rule for finding the Julian Epact, belonging to any year of the Moon's Cycle.</hi></hi></p><p>Multiply the Golden Number, or the given year of the Moon's Cycle, by 11, and the product will be the Epact if it be less than 30; but if it exceed 30, then throw out as many 30's as the product contains, and the remainder will be the Epact. <hi rend="center"><hi rend="italics">Rule to find the Gregorian Epact.</hi></hi></p><p>1st, The difference between the Julian and Gregorian years being equal to the difference between the solar and lunar year, or 11 days, therefore the Gregorian Epact for any year is the same with the Julian Epact for the preceding year; and hence the Gregorian Epact will be found, by subtracting 1 from the golden number, multiplying the remainder by 11, and rejecting the 30's. This rule will serve till the year 1900; but after that year, the Gregorian Epact will be found by this rule: Divide the centuries of the given year by 4; multiply the remainder by 17; then to this product add 43 times the quotient, and also the number 86, and divide the whole sum by 25, reserving the quotient: next multiply the golden number by 11, and from the product subtract the reserved quotient, so shall the remainder, after rejecting all the 30's contained in it, be the Epact sought.</p><p>The following table contains the Golden Numbers, with their corresponding Epacts, till the year 1900. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=6 align=center" role="data"><hi rend="italics"><hi rend="smallcaps">Table</hi> of Gregorian Epacts.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Golden Numb.</cell><cell cols="1" rows="1" rend="align=center" role="data">Epacts</cell><cell cols="1" rows="1" rend="align=center" role="data">Golden Numb.</cell><cell cols="1" rows="1" rend="align=center" role="data">Epacts</cell><cell cols="1" rows="1" rend="align=center" role="data">Golden Numb.</cell><cell cols="1" rows="1" rend="align=center" role="data">Epacts</cell></row><row role="data"><cell cols="1" rows="1" role="data">I</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">VIII</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">XV</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">II</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">IX</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">XVI</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell></row><row role="data"><cell cols="1" rows="1" role="data">III</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">X</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">XVII</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" role="data">IV</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">XI</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">XVIII</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" role="data">V</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">XII</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">XIX</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data">VI</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">XIII</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">I</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">VII</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">XIV</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>On the subject of Epacts, see Wolfius's Elementa Chronologiæ, apud Opera, tom. 4, pa. 133; also Philos. Trans. vol. 46. pa. 417, or numb. 495, art. 5.</p></div1><div1 part="N" n="EPAULE" org="uniform" sample="complete" type="entry"><head>EPAULE</head><p>, or <hi rend="smallcaps">Espaule</hi>, in Fortification, the shoulder of the bastion, or the angle made by the face and flank, otherwise called the Angle of the Epaule.</p></div1><div1 part="N" n="EPAULEMENT" org="uniform" sample="complete" type="entry"><head>EPAULEMENT</head><p>, in Fortification, a side-work hastily thrown up, to cover the cannon or the men; and is made either of earth thrown up, or bags filled with earth or sand, or of gabions, or fascines, &c, with earth: of which latter sort are commonly the Epaulements of the places of arms for the cavalry behind the trenches.</p><div2 part="N" n="Epaulement" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Epaulement</hi></head><p>, is also used for a demi-bastion, con- <pb n="432"/><cb/> sisting of a face and flank, placed at the point of a horn-work or crown-work. Also for a little flank added to the sides of horn-works, to defend them when they are too long. Also for the redoubts made on a right line, to fortify it. And lastly, for an orillon, or mass of earth almost square, faced and lined with a wall, and designed to cover the cannon of a casemate.</p></div2></div1><div1 part="N" n="EPHEMERIS" org="uniform" sample="complete" type="entry"><head>EPHEMERIS</head><p>, <hi rend="smallcaps">Ephemerides</hi>, tables calculated by astronomers, shewing the present state of the heavens for every day at noon; that is, the places in which all the planets are found at that time; differing but little from an Astronomical Almanac. It is from such tables as these that the eclipses, conjunctions, and aspects of the planets are made out; as also horoscopes, or celestial schemes, constructed, &c.</p><p>There have been Ephemerides of Origan, Kepler, Argoli, Heckerus, Mezzarachis, Wing, Gadbury, Parker, De la Hire, &c.</p><p>In France the Academy of Sciences have published annually, from the beginning of the present century, a kind of Ephemeris, under the title of Connoissance des Temps, which is still continued, and is in great esteem; as are also the Ephemerides, published there every ten years, by M. Desplaces, and De la Lande.</p><p>There are now published such Ephemerides by the Academies of several other nations; but that which is in most esteem for its accuracy and use in finding the longitude, is the Nautical Almanac, or Astronomical Ephemeris, published in England by the Board of Longitude, under the direction of the Rev. Dr. Maskelyne, Astronomer Royal, which commenced with the year 1767.</p></div1><div1 part="N" n="EPICHARMUS" org="uniform" sample="complete" type="entry"><head>EPICHARMUS</head><p>, an ancient poet and philosopher, born in Sicily, was a scholar of Pythagoras, and flourished in the time of Hiero, in whose reign it is said he introduced comedy at Syracuse. He wrote also treatises concerning philosophy and medicine; but none of his works have been preserved. He died at 90 years of age, according to Laortius, who has preserved four verses inscribed on his statue.</p></div1><div1 part="N" n="EPICURUS" org="uniform" sample="complete" type="entry"><head>EPICURUS</head><p>, a celebrated ancient philosopher, was born at Gargettium in Attica, in the 109th Olympiad, or about 340 years before Christ. He settled at Athens in a fine garden he had bought; where he lived with his friends in much tranquillity, and educated a great number of disciples; who lived all in common with their master. His school was never divided, but his doctrine was followed as an oracle; and the respect which his followers paid to his memory is admirable; his birth day being still kept in Pliny's time, and even the very month he was born in observed as a continued festival, and his picture placed every where. He wrote a great many books, and valued himself upon making no quotations. He raised the atomical system to great reputation, though he was not the inventor of it, but only made some change in that of Democritus. As to his doctrine concerning the supreme good or happiness, it was very liable to be misrepresented, and some ill effects proceeded from thence, which discredited his sect, though undeservedly. He was charged with perverting the worship of the gods, and inciting men to debauchery. But he did not forget himself on this occasion: he published his opinions to the whole world; wrote some books of devotion; recommended the ve- <cb/> neration of the gods, sobriety, and chastity, living in an exemplary manner, and conformably to the rules of philosophical wisdom and frugality. He died of a suppression of urine, at 72 years of age.—Gassendus has given us all he could collect from the ancients concerning the person and doctrine of this philosopher.</p><p><hi rend="smallcaps">Epicurean</hi> <hi rend="italics">Philosophy,</hi> the doctrine, or system of philosophy maintained by Epicurus and his followers. This consisted of three parts; canonical, physical, and ethical. The first respected the canons or rules of judging; in which soundness and simplicity of sense, assisted by some natural reflections, chiefly formed his art. His search after truth proceeded only by the senses; to the evidence of which he gave so great a certainty, that he considered them as an infallible rule of truth, and termed them the First natural light of mankind.</p><p>In the 2d part of his philosophy he laid down atoms, space, and gravity, as the first principles of all things. He asserted the existence of God, whom he accounted a blessed immortal being, but who did not concern himself with human affairs.</p><p>As to his ethics, he made the supreme good of man to consist in pleasure, and consequently supreme evil in pain. Nature itself, says he, teaches us this truth; and prompts us from our birth to procure whatever gives us pleasure, and avoid what gives us pain. To this end he proposes a remedy against the sharpness of pain, which was to divert the mind from it, by turning our whole attention upon the pleasures we have formerly enjoyed. He held that the wise man must be happy, as long as he is wise: the pain, not depriving him of his wisdom, cannot deprive him of his happiness: from which it would seem that his pleasure consisted rather in intellectual than in sensual enjoyments: though this is a point strongly contested.</p></div1><div1 part="N" n="EPICUREANS" org="uniform" sample="complete" type="entry"><head>EPICUREANS</head><p>, the sect of philosophers holding or following the principles and doctrine of Epicurus. As the nature of the pleasure, in which the chief happiness i<*> supposed to be seated, is a great problem in the morals of Epicurus, there hence arise two kinds of Epicureans, the rigid and the remiss: the first were those who understood Epicurus's notion of pleasure in the best sense, and placed all their happiness in the pure pleasures of the mind, arising from the practice of virtue: while the loose or remiss Epicureans, taking the words of that philosopher in a gross sense, placed all their happiness in bodily pleasures or debauchery.</p></div1><div1 part="N" n="EPICYCLE" org="uniform" sample="complete" type="entry"><head>EPICYCLE</head><p>, in the ancient astronomy, a little circle having its centre in the circumference of a greater one: or a small orb or sphere, which being fixed in the deferent of a planet, is carried along with it, and yet, by its own peculiar motion, carries the planet fastened to it round its proper centre.</p><p>It was by means of Epicycles that Ptolomy and his followers solved the various phenomena of the planets, but more especially their stations and retrogradations.</p></div1><div1 part="N" n="EPICYCLOID" org="uniform" sample="complete" type="entry"><head>EPICYCLOID</head><p>, is a curve generated by the revolution of a point of the periphery of a circle, which rolls along or upon the circumference of another circle, either on the convex or concave side of it.</p><p>When a circle rolls along a straight line, a point in its circumference describes the curve called a cycloid. <pb n="433"/><cb/> But if, instead of the right line, the circle roll along the circumference of another circle, either equal to the former or not, then the curve described by any point in its circumference is what is called the Epicycloid.</p><p>If the generating circle roll <figure/> along the convexity of the circumference, the curve is called an <hi rend="italics">Upper,</hi> or <hi rend="italics">Exterior</hi> Epicycloid; but if along the concavity, it is called a <hi rend="italics">Lower,</hi> or <hi rend="italics">Interior</hi> Epicycloid. Also the circle that revolves is called the <hi rend="italics">Generant;</hi> and the arc of the other circle along which it revolves, is called the <hi rend="italics">Base</hi> of the Epicycloid. Thus, ABC or BLV is the Generant; DPVE the Exterior Epicycloid, its axis BV; DPUE the Interior Epicycloid; and DBE their common Base. <hi rend="center"><hi rend="italics">For the Length of the Curve.</hi></hi></p><p>The length of any part of the curve of an Epicycloid, which any given point in the revolving circle has described, from the position where it touched the circle upon which it revolved, is to double the versed side of half the arc which all the time of revolving touched the quiescent circle, as the sum of the diameters of the circles, is to the semidiameter of the quiescent circle in the Exterior Cycloid; or as the difference of the diameters is to that semidiameter, for the Interior one. <hi rend="center"><hi rend="italics">For the Area of the Epicycloid.</hi></hi></p><p>Dr. Halley has given a general proposition for the measuring of all cycloids and Epicycloids: thus, the area of a cycloid, or Epicycloid, either primary, or contracted, or prolate, is to the area of the generating circle; and also the areas of the parts generated in those curves, to the areas of analogous segments of the circle; as the sum of double the velocity of the centre and the velocity of the circular motion, is to this velocity of the circular motion. See the Demonstr. in the Philos. Trans. number 218.</p><p><hi rend="italics">Spherical</hi> <hi rend="smallcaps">Epicycloids</hi> are formed by a point of the revolving circle, when its plane makes a constant angle with the plane of the circle on which it revolves. Messrs. Bernoulli, Maupertuis, Nicole, and Clairaut, have demonstrated several properties of these Epicycloids, in Hist. Acad. Sci. for 1732.</p><p><hi rend="italics">Parabolic, Elliptic,</hi> &c. <hi rend="smallcaps">Epicycloids.</hi></p><p>If a parabola roll upon another equal to it; its focus will describe a right line perpendicular to the axis of the quiescent parabola: also the vertex of the rolling parabola will describe the cissoid of Diocles; and any other point of it will describe some one of Newton's defective hyperbolas, having a double point in the like point of the quiescent parabola.</p><p>In like manner, if an ellipse revolve upon another ellipse, equal and similar to it, its focus will describe a circle, whose centre is in the other focus, and consequently the radius is equal to the axis of the ellipsis; and any other point in the plane of the ellipse will describe a line of the 4th order. <cb/></p><p>The same may be said also of an hyperbola, revolving upon another, equal and similar to it; for one of the foci will describe a circle, having its centre in the other focus, and the radius will be the principal axis of the hyperbola; and any other point of the hyperbola will describe a line of the 4th order.</p><p>Concerning these lines, see Newton's Principia, lib. 1; also De la Hire's Memoires de Mathematique &c, where he shews the nature of this line, and its use in Mechanics; see also Maclaurin's Geometria Organica.</p><p>EP<*>HANY, a christian festival, otherwise called the Manifestation of Christ to the Gentiles, observed on the 6th of January, in honour of the appearance of our Saviour to the three magi or wise men, who came to adore him and bring him presents.</p></div1><div1 part="N" n="EPISTYLE" org="uniform" sample="complete" type="entry"><head>EPISTYLE</head><p>, in the ancient Architecture, a term used by the Greeks for what we call Architrave, viz a massive stone, or a piece of wood, laid immediately over the capital of a column.</p></div1><div1 part="N" n="EPOCHA" org="uniform" sample="complete" type="entry"><head>EPOCHA</head><p>, or <hi rend="smallcaps">Epoch</hi>, a term or fixed point of time, from whence the succeeding years are numbered or reckoned.</p><p>Different nations make use of different Epochs. The christians chiefly use the Epoch of the nativity or incarnation of Jesus Christ; the Mahometans, that of the Hegira; the Jews, that of the creation of the world, or that of the Deluge; the ancient Greeks, that of the Olympiads; the Romans, that of the building of their city; the ancient Persians and Assyrians, that of Nabonassar; &c.</p><p>The doctrine and use of Epochs is of very great extent in chronology. To reduce the years of one Epoch to those of another, i. e. to find what year of one corresponds to a given year of another; a period of years has been invented, which, commencing before all the known Epochs, is, as it were, a common receptacle of them all, called the Julian Period. To this period all the Epochs are reduced; i. e. the year of this period when each Epoch commences, is determined. So that, adding the given year of one Epoch to the year of the period corresponding with its rise, and from the sum subtracting the year of the same period corresponding to the other Epoch, the remainder is the year of that other Epoch.</p><p><hi rend="smallcaps">Epoch</hi> <hi rend="italics">of Christ,</hi> is the common Epoch throughout Europe, commencing at the supposed time of our Saviour's nativity, December 25; or rather, according to the usual account, from his circumcision, or the 1st of January. The author of this Epoch was an Abbot of Rome, one Dionysius Exiguus, a Scythian, about the year 507 or 527. Dionysius began his account from the conception or incarnation, usually called the Annunciation, or Lady Day; which method obtained in the dominions of Great Britain till the year 1752, before which time the Dionysian was the same as the English Epoch: but in that year the Gregorian calendar having been admitted by act of parliament, they now reckon from the first of January, as in the other parts of Europe, except in the court of Rome, where the Epoch of the Incarnation still obtains for the date of their bulls. <pb n="434"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data"><hi rend="italics">ATABLE of the Years of the most remarkable Epochs or Eras and Events.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">N. B. The years before Christ, are those before the reputed year of his birth, and not reckoned back from the first year of his age, as is generally done in such tables.</cell><cell cols="1" rows="1" rend="align=center" role="data">Julian Period.</cell><cell cols="1" rows="1" rend="align=center" role="data">Year of the World.</cell><cell cols="1" rows="1" rend="align=center" role="data">Years before Christ.</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Creation of the World</cell><cell cols="1" rows="1" rend="align=right" role="data">706</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">4007</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Deluge, or Noah's flood</cell><cell cols="1" rows="1" rend="align=right" role="data">2362</cell><cell cols="1" rows="1" rend="align=right" role="data">1656</cell><cell cols="1" rows="1" rend="align=right" role="data">2351</cell></row><row role="data"><cell cols="1" rows="1" role="data">Assyrian monarchy founded by Nimrod</cell><cell cols="1" rows="1" rend="align=right" role="data">2537</cell><cell cols="1" rows="1" rend="align=right" role="data">1831</cell><cell cols="1" rows="1" rend="align=right" role="data">2176</cell></row><row role="data"><cell cols="1" rows="1" role="data">The birth of Abraham</cell><cell cols="1" rows="1" rend="align=right" role="data">2714</cell><cell cols="1" rows="1" rend="align=right" role="data">2008</cell><cell cols="1" rows="1" rend="align=right" role="data">1999</cell></row><row role="data"><cell cols="1" rows="1" role="data">Kingdom of Athens founded by Cecrops</cell><cell cols="1" rows="1" rend="align=right" role="data">3157</cell><cell cols="1" rows="1" rend="align=right" role="data">2451</cell><cell cols="1" rows="1" rend="align=right" role="data">1556</cell></row><row role="data"><cell cols="1" rows="1" role="data">Entrance of the Israelites into Canaan</cell><cell cols="1" rows="1" rend="align=right" role="data">3262</cell><cell cols="1" rows="1" rend="align=right" role="data">2556</cell><cell cols="1" rows="1" rend="align=right" role="data">1451</cell></row><row role="data"><cell cols="1" rows="1" role="data">The destruction of Troy</cell><cell cols="1" rows="1" rend="align=right" role="data">3529</cell><cell cols="1" rows="1" rend="align=right" role="data">2823</cell><cell cols="1" rows="1" rend="align=right" role="data">1184</cell></row><row role="data"><cell cols="1" rows="1" role="data">Solomon's temple founded</cell><cell cols="1" rows="1" rend="align=right" role="data">3701</cell><cell cols="1" rows="1" rend="align=right" role="data">2995</cell><cell cols="1" rows="1" rend="align=right" role="data">1012</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Argonautic expedition</cell><cell cols="1" rows="1" rend="align=right" role="data">3776</cell><cell cols="1" rows="1" rend="align=right" role="data">3070</cell><cell cols="1" rows="1" rend="align=right" role="data">937</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lycurgus formed his laws</cell><cell cols="1" rows="1" rend="align=right" role="data">3829</cell><cell cols="1" rows="1" rend="align=right" role="data">3103</cell><cell cols="1" rows="1" rend="align=right" role="data">884</cell></row><row role="data"><cell cols="1" rows="1" role="data">Arbaces, 1st king of the Medes</cell><cell cols="1" rows="1" rend="align=right" role="data">3838</cell><cell cols="1" rows="1" rend="align=right" role="data">3132</cell><cell cols="1" rows="1" rend="align=right" role="data">875</cell></row><row role="data"><cell cols="1" rows="1" role="data">Olympiads of the Greeks began</cell><cell cols="1" rows="1" rend="align=right" role="data">3938</cell><cell cols="1" rows="1" rend="align=right" role="data">3232</cell><cell cols="1" rows="1" rend="align=right" role="data">775</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rome built, or Roman Era</cell><cell cols="1" rows="1" rend="align=right" role="data">3961</cell><cell cols="1" rows="1" rend="align=right" role="data">3255</cell><cell cols="1" rows="1" rend="align=right" role="data">752</cell></row><row role="data"><cell cols="1" rows="1" role="data">Era of Nabonassar</cell><cell cols="1" rows="1" rend="align=right" role="data">3967</cell><cell cols="1" rows="1" rend="align=right" role="data">3261</cell><cell cols="1" rows="1" rend="align=right" role="data">746</cell></row><row role="data"><cell cols="1" rows="1" role="data">First Babylonish captivity, by Nebuchadnezzar</cell><cell cols="1" rows="1" rend="align=right" role="data">4107</cell><cell cols="1" rows="1" rend="align=right" role="data">3401</cell><cell cols="1" rows="1" rend="align=right" role="data">606</cell></row><row role="data"><cell cols="1" rows="1" role="data">The 2d ditto, and birth of Cyrus</cell><cell cols="1" rows="1" rend="align=right" role="data">4114</cell><cell cols="1" rows="1" rend="align=right" role="data">3408</cell><cell cols="1" rows="1" rend="align=right" role="data">599</cell></row><row role="data"><cell cols="1" rows="1" role="data">Solomon's temple destroyed</cell><cell cols="1" rows="1" rend="align=right" role="data">4125</cell><cell cols="1" rows="1" rend="align=right" role="data">3419</cell><cell cols="1" rows="1" rend="align=right" role="data">588</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cyrus began to reign in Babylon</cell><cell cols="1" rows="1" rend="align=right" role="data">4177</cell><cell cols="1" rows="1" rend="align=right" role="data">3471</cell><cell cols="1" rows="1" rend="align=right" role="data">536</cell></row><row role="data"><cell cols="1" rows="1" role="data">Peloponnesian war began</cell><cell cols="1" rows="1" rend="align=right" role="data">4282</cell><cell cols="1" rows="1" rend="align=right" role="data">3576</cell><cell cols="1" rows="1" rend="align=right" role="data">431</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alexander the great died</cell><cell cols="1" rows="1" rend="align=right" role="data">4390</cell><cell cols="1" rows="1" rend="align=right" role="data">3684</cell><cell cols="1" rows="1" rend="align=right" role="data">323</cell></row><row role="data"><cell cols="1" rows="1" role="data">Captivity of 100,000 Jews by Ptolomy</cell><cell cols="1" rows="1" rend="align=right" role="data">4393</cell><cell cols="1" rows="1" rend="align=right" role="data">3687</cell><cell cols="1" rows="1" rend="align=right" role="data">320</cell></row><row role="data"><cell cols="1" rows="1" role="data">Archimedes killed at Syracuse</cell><cell cols="1" rows="1" rend="align=right" role="data">4506</cell><cell cols="1" rows="1" rend="align=right" role="data">3800</cell><cell cols="1" rows="1" rend="align=right" role="data">207</cell></row><row role="data"><cell cols="1" rows="1" role="data">Julius Cæsar invaded Britain</cell><cell cols="1" rows="1" rend="align=right" role="data">4659</cell><cell cols="1" rows="1" rend="align=right" role="data">3953</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell></row><row role="data"><cell cols="1" rows="1" role="data">He corrected the calendar</cell><cell cols="1" rows="1" rend="align=right" role="data">4667</cell><cell cols="1" rows="1" rend="align=right" role="data">3961</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell></row><row role="data"><cell cols="1" rows="1" role="data">The true year of Christ's birth</cell><cell cols="1" rows="1" rend="align=right" role="data">4709</cell><cell cols="1" rows="1" rend="align=right" role="data">4003</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">The Christian Era begins here.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Years since Christ.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dionysian, or vulgar era of Christ's birth</cell><cell cols="1" rows="1" rend="align=right" role="data">4717</cell><cell cols="1" rows="1" rend="align=right" role="data">4007</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Christ crucified, Friday April 3d</cell><cell cols="1" rows="1" rend="align=right" role="data">4746</cell><cell cols="1" rows="1" rend="align=right" role="data">4040</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jerusalem destroyed</cell><cell cols="1" rows="1" rend="align=right" role="data">4783</cell><cell cols="1" rows="1" rend="align=right" role="data">4077</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell></row><row role="data"><cell cols="1" rows="1" role="data">Adrian's wall built in Britain</cell><cell cols="1" rows="1" rend="align=right" role="data">4833</cell><cell cols="1" rows="1" rend="align=right" role="data">4127</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dioclesian Epoch, or that of Martyrs</cell><cell cols="1" rows="1" rend="align=right" role="data">4997</cell><cell cols="1" rows="1" rend="align=right" role="data">4291</cell><cell cols="1" rows="1" rend="align=right" role="data">284</cell></row><row role="data"><cell cols="1" rows="1" role="data">The council of Nice</cell><cell cols="1" rows="1" rend="align=right" role="data">5038</cell><cell cols="1" rows="1" rend="align=right" role="data">4332</cell><cell cols="1" rows="1" rend="align=right" role="data">325</cell></row><row role="data"><cell cols="1" rows="1" role="data">Constantine the great died</cell><cell cols="1" rows="1" rend="align=right" role="data">5050</cell><cell cols="1" rows="1" rend="align=right" role="data">4344</cell><cell cols="1" rows="1" rend="align=right" role="data">337</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Saxons invited into Britain</cell><cell cols="1" rows="1" rend="align=right" role="data">5158</cell><cell cols="1" rows="1" rend="align=right" role="data">4452</cell><cell cols="1" rows="1" rend="align=right" role="data">445</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hegira, or flight of Mohammed</cell><cell cols="1" rows="1" rend="align=right" role="data">5335</cell><cell cols="1" rows="1" rend="align=right" role="data">4629</cell><cell cols="1" rows="1" rend="align=right" role="data">622</cell></row><row role="data"><cell cols="1" rows="1" role="data">Death of Mohammed</cell><cell cols="1" rows="1" rend="align=right" role="data">5343</cell><cell cols="1" rows="1" rend="align=right" role="data">4637</cell><cell cols="1" rows="1" rend="align=right" role="data">630</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Persian Yesdegird</cell><cell cols="1" rows="1" rend="align=right" role="data">5344</cell><cell cols="1" rows="1" rend="align=right" role="data">4638</cell><cell cols="1" rows="1" rend="align=right" role="data">631</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sun, Moon, and Planets <figure/> in <figure/>, seen from the earth</cell><cell cols="1" rows="1" rend="align=right" role="data">5899</cell><cell cols="1" rows="1" rend="align=right" role="data">5193</cell><cell cols="1" rows="1" rend="align=right" role="data">1186</cell></row><row role="data"><cell cols="1" rows="1" role="data">Art of printing discovered</cell><cell cols="1" rows="1" rend="align=right" role="data">6153</cell><cell cols="1" rows="1" rend="align=right" role="data">5447</cell><cell cols="1" rows="1" rend="align=right" role="data">1440</cell></row><row role="data"><cell cols="1" rows="1" role="data">The reformation begun by Martin Luther</cell><cell cols="1" rows="1" rend="align=right" role="data">6230</cell><cell cols="1" rows="1" rend="align=right" role="data">5524</cell><cell cols="1" rows="1" rend="align=right" role="data">1517</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Calendar corrected by pope Gregory</cell><cell cols="1" rows="1" rend="align=right" role="data">6295</cell><cell cols="1" rows="1" rend="align=right" role="data">5589</cell><cell cols="1" rows="1" rend="align=right" role="data">1582</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oliver Cromwell died</cell><cell cols="1" rows="1" rend="align=right" role="data">6371</cell><cell cols="1" rows="1" rend="align=right" role="data">5665</cell><cell cols="1" rows="1" rend="align=right" role="data">1658</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sir Isaac Newton born, Dec. 25</cell><cell cols="1" rows="1" rend="align=right" role="data">6355</cell><cell cols="1" rows="1" rend="align=right" role="data">5649</cell><cell cols="1" rows="1" rend="align=right" role="data">1642</cell></row><row role="data"><cell cols="1" rows="1" role="data">Made President of the Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">6416</cell><cell cols="1" rows="1" rend="align=right" role="data">5710</cell><cell cols="1" rows="1" rend="align=right" role="data">1703</cell></row><row role="data"><cell cols="1" rows="1" role="data">Died, March 20th</cell><cell cols="1" rows="1" rend="align=right" role="data">6440</cell><cell cols="1" rows="1" rend="align=right" role="data">5734</cell><cell cols="1" rows="1" rend="align=right" role="data">1727</cell></row><row role="data"><cell cols="1" rows="1" role="data">New Planet discovered by Herschel</cell><cell cols="1" rows="1" rend="align=right" role="data">6494</cell><cell cols="1" rows="1" rend="align=right" role="data">5788</cell><cell cols="1" rows="1" rend="align=right" role="data">1781</cell></row></table><pb n="435"/><cb/></p><p>EQUABLE <hi rend="italics">Motion,</hi> Celerity, Velocity, &c, is that which is uniform, or without alteration, or by which equal spaces are passed over in equal times. Hence, the spaces, passed over in Equable motions, are proportional to the times. So that if a body pass over 20 feet in 1 second of time, it will pass over 40 feet in 2 seconds, and so on.</p><p><hi rend="smallcaps">Equably</hi> <hi rend="italics">Accelerated</hi> or <hi rend="italics">Retarded,</hi> &c, is when the motion or change is increased or decreased by equal quantities or degrees in equal times.</p></div1><div1 part="N" n="EQUAL" org="uniform" sample="complete" type="entry"><head>EQUAL</head><p>, a term of relation between different things, but of the same kind, magnitude, quantity, or quality.——Wolfius defines Equals to be those things that may be substituted for each other, without any alteration of their quantity.—It is an axiom in mathematics &c, that two things which are equal to the same third, are also equal to each other. And if Equals be equally altered, by Equal addition, subtraction, multiplication, division, &c, the results will be also Equal.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Circles,</hi> are those whose diameters are equal.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Angles,</hi> are those whose sides are equally inclined, or which are measured by similar arcs of circles.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Lines,</hi> are lines of the same length.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Plane Figures,</hi> are those whose areas are equal; whether the figures be of the same form or not.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Solids,</hi> are such as are of the same space, capacity, or solid content; whether they be of the same kind or not.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Curvatures,</hi> are such as have the same or equal radii of curvature.</p><p><hi rend="smallcaps">Equal</hi> <hi rend="italics">Ratios,</hi> are those whose terms are in the same proportion.</p><div2 part="N" n="Equal" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equal</hi></head><p>, in Optics, is said of things that are seen under equal angles.</p></div2></div1><div1 part="N" n="EQUALITY" org="uniform" sample="complete" type="entry"><head>EQUALITY</head><p>, the exact agreement of two things in respect of quantity. Those figures are equal which may occupy the same space, or may be conceived to possess the same space, by the flexion or transposition of their parts. See a learned discourse upon this, by Dr. Barrow, in the 11th and 12th of his Mathematical Lectures.</p><div2 part="N" n="Equality" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equality</hi></head><p>, in Algebra, the relation or comparison between two quantities that are really or effectually equal. See <hi rend="smallcaps">Equation.</hi></p><p>Equality, in Algebra, is usually denoted by two equal parallel lines, as = : thus , i. e. 2 plus 3, are equal to 5. This character =, was first introduced by Robert Recorde. Des Cartes, and some others after him, use the mark <figure/> instead of it: as 2 + 3 <figure/> 5.</p></div2><div2 part="N" n="Equality" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equality</hi></head><p>, in Astronomy. <hi rend="italics">Circle of</hi> <hi rend="smallcaps">Equality</hi>, or the <hi rend="smallcaps">Equant.</hi> See <hi rend="smallcaps">Circle</hi> and <hi rend="smallcaps">Equant.</hi></p><p><hi rend="italics">Ratio</hi> or <hi rend="italics">Proportion</hi> of <hi rend="smallcaps">Equality</hi>, is that between two equal numbers or quantities.</p><p><hi rend="italics">Proportion</hi> of <hi rend="smallcaps">Equality</hi> <hi rend="italics">evenly ranged,</hi> or <hi rend="italics">ex æquo ordinata,</hi> is that in which two terms, in a rank or series, are proportional to as many terms in another series, compared to each other in the same order, i. e. the first of one rank to the first of another, the 2d to the 2d, &c.</p><p><hi rend="italics">Proportion of</hi> <hi rend="smallcaps">Equality</hi> <hi rend="italics">evenly disturbed,</hi> called also <cb/> <hi rend="italics">ex æquo perturbata,</hi> is that in which more than two terms of one rank, are proportional to as many terms of another, compared to each other in a different and interrupted order; viz, the 1st of one rank to the 2d of another, the 2d to the 3d, &c.</p></div2></div1><div1 part="N" n="EQUANT" org="uniform" sample="complete" type="entry"><head>EQUANT</head><p>, or Æ<hi rend="smallcaps">QUANT</hi>, in Astronomy, a circle formerly conceived by astronomers, in the plane of the deferent, or eccentric; for regulating and adjusting certain motions of the planets, and reducing them more easily to a calculus.</p><p>EQUATED <hi rend="italics">Anomaly.</hi> See <hi rend="smallcaps">Anomaly.</hi></p><p><hi rend="smallcaps">Equated</hi> <hi rend="italics">Bodies.</hi> On Gunter's Sector there are sometimes placed two lines, answering to one another, and called the Lines of Equated Bodies. They are situated between the lines of superficies and solids, and are marked with the letters D, I, C, S, O, T, to signify dodecahedron, icosahedron, cube, sphere, octahedron, and tetrahedron.</p><p>The uses of these lines are, 1st, When the diameter of the sphere is given, to find the sides of the five regular bodies, each equal to that sphere; 2d, From the side of any one of those bodies being given, to find the diameter of the sphere, and the sides of the other bodies, which shall be each equal to the first given body. So, when the sphere is given, take its diameter, and apply it over on the sector in the points S, S; but when one of the other five bodies is given, apply its side over in its proper points; then the parallels taken from between the points of the other bodies, or sphere, shall be the sides or diameter, equal severally to the sphere or body first given.</p></div1><div1 part="N" n="EQUATION" org="uniform" sample="complete" type="entry"><head>EQUATION</head><p>, in Algebra, an expression of equality between two different quantities; or two quantities, whether simple or compound, with the mark of equality between them: as ; &c. When the things, or two sides of the equation, are the same, the expression becomes an Identity, as 5 = 5, or <hi rend="italics">a</hi> = <hi rend="italics">a,</hi> &c. Sometimes the quantities are placed all on one side, and made equal to 0 or nothing on the other side; as : which is no more than setting down the difference of two equal quantities equal to nothing.</p><p>The character or sign usually employed to denote an Equation, is =, which is placed between the two equal quantities, called the two sides of the Equation.</p><p>The <hi rend="italics">Terms</hi> of an <hi rend="smallcaps">Equation</hi>, are the several quantities or parts of which it is composed. Thus, of the Equation , the terms are <hi rend="italics">a, b,</hi> and <hi rend="italics">c:</hi> and the tenor or import of the expression is, that some quantity represented by <hi rend="italics">c,</hi> is equal to two others represented by <hi rend="italics">a</hi> and <hi rend="italics">b.</hi></p><p>Equations are either Simple or Affected.</p><p>A <hi rend="italics">Simple</hi> <hi rend="smallcaps">Equation</hi> is that which has only one power of the unknown quantity: as , &c; where <hi rend="italics">x</hi> denotes the unknown quantity, and the other letters known ones. But</p><p>An <hi rend="italics">Assected,</hi> or <hi rend="italics">Adsected</hi> <hi rend="smallcaps">Equation</hi> contains two or more different powers: as , &c. <pb n="436"/><cb/></p><p>Again, Equations are denominated from the highest power contained in them; as quadratic, cubic, biquadratic, &c. Thus,</p><p>A <hi rend="italics">Quadratic</hi> <hi rend="smallcaps">Equation</hi>, is that in which the unknown quantity rises to two dimensions, or to the square or 2d power: as .</p><p>A <hi rend="italics">Cubic</hi> <hi rend="smallcaps">Equation</hi>, is that in which the unknown quantity is of three dimensions, or rises to the cube or 3d power: as .</p><p>A <hi rend="italics">Biquadratic</hi> <hi rend="smallcaps">Equation</hi>, is that in which the unknown quantity is of 4 dimensions, or rises to the 4th or biquadratic power: as .</p><p>And so for other higher orders of Equations.</p><p>The <hi rend="italics">Root</hi> of an <hi rend="smallcaps">Equation</hi>, is the value of the unknown letter or quantity contained in it. And this value being substituted in the terms of the Equation instead of that letter or quantity, will cause both sides to vanish, or will make the one side exactly equal to the other. So the root of the Equation , is 10; because that using 10 for <hi rend="italics">x,</hi> it becomes .</p><p>Every Equation has as many roots as it has dimensions, or as it contains units in the index of the highest power, when the powers are all reduced to integral exponents. So the simple Equation of the 1st power, has only one root; but the quadratic has 2, the cubic 3, the biquadratic 4, &c. Thus the two roots of this equation are 1 and 3; for either of these substituted for <hi rend="italics">x</hi> makes <hi rend="italics">x</hi><hi rend="sup">2</hi>-4<hi rend="italics">x</hi> come out equal to - 3. Also the three roots of , are 2, 5, and -3; as will appear by substituting each of these instead of <hi rend="italics">x</hi> in the equation, which will make all the terms on one side equal to the other side. And so of others.</p><p><hi rend="italics">The Relation between the Roots of Equations, and the Coefficients of their Terms.</hi>——In every Equation, when the terms are ranged in order according to the order of the powers, the greater before the less; the first term or highest power freed from its coefficient, by dividing all the terms by it, and all brought to one side, and made equal to nothing on the other side, when it will appear in this form, ; then the relations between the roots and coefficients, are as follow:</p><p>1st, The coefficient <hi rend="italics">a</hi> of the 2d term, is equal to the sum of all the roots.</p><p>2d, The coefficient <hi rend="italics">b</hi> of the 3d term, is equal to the sum of all the products of the roots that can be made by multiplying every two of them together. In like manner,</p><p>3d, The coefficients <hi rend="italics">c, d, e,</hi> &c, of the following terms, are respectively equal to the sum of the products of the roots made by multiplying every three together, or every four together, or every five together, &c, the signs of all the roots being changed. All which will appear below, in the Generation of Equations.</p><p>The Roots of Equations are Positive or Negative, <cb/> and Real or Imaginary. Thus, the two roots of the Equation , are 1 and 3, real and both positive; but the roots of the Equation , are 2, 5, & -3, are real, two positive and one negative; and the roots of the equation , are 1 and - 1/2 ± 1/2 √(-39), one real and two imaginary.</p><p>The <hi rend="italics">Generation</hi> of <hi rend="smallcaps">Equations</hi>, is the multiplying of certain assumed simple equations together, to produce compound ones, with intent to shew the nature of these; a method invented by Harriot, which is this: Suppose <hi rend="italics">x</hi> to denote the unknown quantity of any equation, and let the roots of that equation, or the values of <hi rend="italics">x,</hi> be, <hi rend="italics">a, b, c, d,</hi> &c; that is <hi rend="italics">x</hi> = <hi rend="italics">a,</hi> and <hi rend="italics">x</hi> = <hi rend="italics">b,</hi> and <hi rend="italics">x</hi> = <hi rend="italics">c,</hi> &c; or , &c; then multiply these last equations together, thus,</p><p>Now the roots of these equations are <hi rend="italics">a, b, c, d,</hi> &c; and it is obvious that the sum of all the roots is the coefficient of the 2d term, the sum of all the products of every two is the coefficient of the 3d term, the sum of all the products of every three that of the 4th term, and so on, to the last term, which is the continual product of all the roots.</p><p><hi rend="italics">Reduction of</hi> <hi rend="smallcaps">Equations</hi>, is the transforming or changing them to their simplest and most commodious form, to prepare them for finding or extracting their roots. The most convenient form is, that the terms be ranged according to the powers of the unknown letter, the highest power foremost next the left hand, and that term to have only + 1 for its coefficient; also all the terms containing the unknown letter to be on one side of the equation, and the absolute known term only on the other side.</p><p>Now this reduction chiefly respects the first term, or that which contains the highest power of the unknown quantity; and the general rule for reducing it is, to consider in what manner it is involved or connected with other quantities, and then perform the counter or op- <pb n="437"/><cb/> posite relation or operation; for every operation is undone or counteracted by the reverse of it; as addition by subtraction, multiplication by division, involution by evolution, &c: then bring all the unknown terms to one side, and the known term to the other side, changing the signs, from + to -, or from - to +, of those terms which are changed from one side to the other; and lastly divide by the coefficient of the first term, with its sign.</p><p/><p/><p/><p/><p><hi rend="center"><hi rend="italics">Extracting or finding the Roots of <hi rend="smallcaps">Equations.</hi></hi></hi></p><p>This is finding the value or values of the unknown letter in an Equation, the rules for which are various, according to the degree of the Equation. <hi rend="center">1. <hi rend="italics">For the Root of a Simple <hi rend="smallcaps">Equation.</hi></hi></hi></p><p>Having reduced the equation as above, by bringing the unknown terms to one side, and the known ones to the other, freeing the former from radicals and fractions, by their counter operations, and lastly dividing by the coefficients of the unknown quantity, the value of it is then found: as in the first and 2d examples of reduction above given. <hi rend="center">2. <hi rend="italics">For the Roots of Quadratic <hi rend="smallcaps">Equations.</hi></hi></hi></p><p>These are usually found by what is called completing the square; which consists in squaring half the coefficient of the 2d term, and adding it to both sides of the equation; for then the unknown side is a complete square of a binomial, and the other side consists only of known quantitles. Therefore, extract the root on both sides, so shall the root of the first side be a binomial, one part of which is the unknown letter, and the <cb/> other a known or given quantity, and the root of the other side is taken either + or -, since the square of either of these is the same given quantity: lastly, bringing over the known part of the binomial root to the other side, with a contrary sign, gives the two roots or values of the unknown letter sought.</p><p>Thus, if be a general quadratic Equation, 2<hi rend="italics">a</hi> being the coefficient of the 2d term, and <hi rend="italics">b</hi><hi rend="sup">2</hi> the absolute known term, both with their signs. Then, <hi rend="italics">a</hi> is half that coefficient, and <hi rend="italics">a</hi><hi rend="sup">2</hi> its square; which being added, gives ; and the root extracted gives ; then, transposing <hi rend="italics">a,</hi> it is , the two roots, or values of <hi rend="italics">x.</hi></p><p/><p/><p><hi rend="center">3. <hi rend="italics">For the Roots of Cubic <hi rend="smallcaps">Equations.</hi></hi></hi></p><p><hi rend="italics">A Cubic</hi> <hi rend="smallcaps">Equation</hi> is that in which the unknown letter ascends to the 3d power; as .</p><p>The 2d term of every Cubic Equation being taken away, those equations may all be reduced to this form, ; and the general value of one root is . This rule is usually called Cardan's, because first published by him, but it was invented both by Scipio Ferreus, and Nich. Tartalea, by whom it was communicated to Cardan. See the article <hi rend="smallcaps">Algebra.</hi></p><p>When the 2d term is negative, or the equation of this form, , the radical √((1/4)<hi rend="italics">b</hi><hi rend="sup">2</hi> + (1/27)<hi rend="italics">a</hi><hi rend="sup">3</hi>) becomes √((1/4)<hi rend="italics">b</hi><hi rend="sup">2</hi>-(1/27)<hi rend="italics">a</hi><hi rend="sup">3</hi>), which will be imaginary or impossible when (1/27)<hi rend="italics">a</hi><hi rend="sup">3</hi> is greater than (1/4)<hi rend="italics">b</hi><hi rend="sup">2</hi>, for √((1/4)<hi rend="italics">b</hi><hi rend="sup">2</hi>-(1/27)<hi rend="italics">a</hi><hi rend="sup">3</hi>) will then be the square root of a negative quantity, which is impossible: and yet, in this case, the root <hi rend="italics">x</hi> is a real quantity; though algebraists have never been able to sind a real finite general expression for it. And this is called the Irreducible or Impracticable Case.</p><p>This case may indeed be resolved by the trisection of an arc or angle; or by any of the usual methods of converging; or by general expressions in infinite series. See Saunderson's Algebra, pa. 713; Philos. Trans. vol. 18, pa. 136, or Abr. vol. 1, pa. 87; also vol. 70, pa. 415. See also the article <hi rend="smallcaps">Cubic</hi> <hi rend="italics">Equations.</hi></p><p>Mr. Cotes observes, in his Logometria, pa. 29, that the solution of all cubic Equations depends either upon the trisection of a ratio, or of an angle. See this method explained in Saunderson's Alg. p. 718. <pb n="438"/><cb/></p><p><hi rend="italics">Biquadratic</hi> <hi rend="smallcaps">Equations</hi>, or those that are of 4 dimensions, are resolved after various methods. The first rule was given by Lewis Ferrari, the companion of Cardan, which is one of the best. A 2d method was given by Des Cartes, and another by Mr. Simpson and Dr. Waring. For the explanation of which, see B<hi rend="smallcaps">IQUADRATIC</hi> <hi rend="italics">Equations.</hi> <hi rend="center"><hi rend="italics"><hi rend="smallcaps">Equations</hi> of the Higher Degrees or Orders.</hi></hi></p><p>There is no general rule to express algebraically the roots of Equations above those of the 4th degree; and therefore methods of approximation are here made use of, which, though not accurately, are yet practically true. Some of these excel in ease and simplicity, and others in quickness of converging. Among these may be reckoned first, Double Position, or Trial-and-Error, both in respect of ease and universality, as it applies in the simplest manner to all sorts of Equations whatever, not excepting even exponential ones, radical expressions of ever so complex a form, expressions of logarithms, of arches by the sines or tangents, of arcs of curves by the abscisses, or any other fluents, or roots of fluxional Equations. For an explanation of this and other methods of converging to the roots of equations, by Halley, Newton, Raphson, &c, &c, see A<hi rend="smallcaps">PPROXINATION</hi>, and <hi rend="smallcaps">Converging.</hi></p><p>Besides the methods above adverted to, there have been some others, given in the Memoirs of several Academies, and elsewhere. As, by M. Daniel Bernoulli, in the Acta Petropolitana, tom. 3, p. 92; and by M. L. Euler, in the same, vol. 6, New Series, and tom. 5, p. 63 & 82; by Mr. Thos. Simpson, in his Essays, p. 82; in his Dissertations, p. 102; in his Algebra, p. 158; and in his Select Exercises, p. 215.</p><p><hi rend="italics">Absolute</hi> <hi rend="smallcaps">Equation.</hi> See <hi rend="smallcaps">Absolute.</hi></p><p><hi rend="italics">Adfected,</hi> or <hi rend="italics">Affected</hi> <hi rend="smallcaps">Equation.</hi> See <hi rend="smallcaps">Affected.</hi></p><p><hi rend="italics">Differential</hi> <hi rend="smallcaps">Equation</hi>, is the Equation of Differences or Fluxions.</p><p><hi rend="italics">Eminential</hi> <hi rend="smallcaps">Equation.</hi> See <hi rend="smallcaps">Eminential.</hi></p><p><hi rend="italics">Exponential</hi> <hi rend="smallcaps">Equation</hi>, one in which the exponents of the powers are variable or unknown quantities. See <hi rend="smallcaps">Exponential.</hi></p><p><hi rend="italics">Fluential</hi> <hi rend="smallcaps">Equation</hi>, is the Equation of the fluents.</p><p><hi rend="italics">Fluxional</hi> <hi rend="smallcaps">Equation</hi>, is the Equation of the fluxions.</p><p><hi rend="italics">Literal</hi> <hi rend="smallcaps">Equation</hi>, is a general Equation expressed in letters, as contradistinguished from a</p><p><hi rend="italics">Numeral</hi> Equation, one expressed in numbers.</p><p><hi rend="italics">Transcendental</hi> <hi rend="smallcaps">Equation.</hi> See <hi rend="smallcaps">Transcendental.</hi></p><div2 part="N" n="Equation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equation</hi></head><p>, in Astronomy, as <hi rend="italics">Annual</hi> <hi rend="smallcaps">Equation</hi>, is either of the mean motion of the sun and moon, or of the moon's apogee and nodes.</p><p>The Annual Equation of the sun's centre being given, the other three corresponding annual Equations, will be also given, and therefore a table of the first will serve for all of them. Thus, if the annual Equation of the sun's centre, taken from such a table for any time, be called <hi rend="italics">s;</hi> and if (1/10)<hi rend="italics">s</hi> = A, and (1/6)<hi rend="italics">s</hi>= B; then shall the other annual Equations for that time be thus,</p><p>, that of the moon's mean motion; and , that of the moon's apogee; and , that of her nodes.</p><p>And here note, that when <hi rend="italics">s,</hi> or the Equation of the sun's centre, is additive; then <hi rend="italics">m</hi> is negative, <hi rend="italics">a</hi> is positive, and <hi rend="italics">n</hi> is negative. But on the contrary, when <hi rend="italics">s</hi> <cb/> is negative or subductive; then <hi rend="italics">m</hi> is positive, <hi rend="italics">a</hi> negative, and <hi rend="italics">n</hi> positive.</p><p>There is also an <hi rend="italics">Equation of the moon's mean motion,</hi> depending on the situation of her apogee in respect of the sun; which is greatest when the moon's apogee is in an octant with the sun, and is nothing at all when it is in the quadratures of syzygies. This Equation when greatest, and the sun in perigee, is 3′ 56″. But it is never above 3′ 34″ when the sun is in apogee. At other distances of the sun from the earth, this Equation when greatest, is reciprocally as the cube of that distance. But when the moon's apogee is any where out of the octants, this Equation grows less, and is mostly, at the same distance between the earth and sun, as the sine of double the distance of the moon's apogee from the next quadrature or syzygy, is to radius. This is to be added to the moon's motion while her apogee passes from a quadrature with the sun to a syzygy; but is to be subtracted from it, while the apogee moves from the syzygy to the quadrature.</p><p>There is moreover another <hi rend="italics">Equation of the moon's motion,</hi> which depends on the aspect of the nodes of the moon's orbit with respect to the sun: and this is greatest when her nodes are in octants to the sun, and quite vanishes when they come to their quadratures or syzygies. This Equation is proportional to the sine of double the distance of the node from the next syzygy or quadrature; and at the greatest is only 47″. This must be added to the moon's mean motion while the nodes are passing from the syzygies with the sun to their quadratures; but subtracted while they pass from the quadratures to the syzygies.</p><p>From the sun's true place subtract the equated mean motion of the lunar apogee, as was shewn above, the remainder will be the annual argument of the said apogee; from whence the eccentricity of the moon and the 2d Equation of her apogee may be compared. See <hi rend="italics">Theory of the</hi> <hi rend="smallcaps">Moon's</hi> <hi rend="italics">motions,</hi> &c.</p><p><hi rend="smallcaps">Equation</hi> <hi rend="italics">of the Centre,</hi> called also <hi rend="italics">Prosthapheresis,</hi> and <hi rend="italics">Total Prosthapheresis,</hi> is the difference between the true and mean place of a planet, or the angle made by the lines of the true and mean place; or, which amounts to the same, between the mean and equated anomaly.</p><p>The greatest Equation of the Centre may be obtained by finding the sun's longitude at the times when he is near his mean distances, for then the difference will give the true motion for that interval of time: next find the sun's mean motion for the same interval of time; then half the difference between the true and mean motions will shew the greatest Equation of the Centre.</p><p>For Example, by observations made at the Royal Observatory at Greenwich, it appears that at the following mean times the sun's longitudes were as annexed; viz, <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Mean times.</cell><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">Sun's longitudes.</cell></row><row role="data"><cell cols="1" rows="1" role="data">1769 Oct. 1 at 23<hi rend="sup">h</hi> 49<hi rend="sup">m</hi> 12<hi rend="sup">s</hi></cell><cell cols="1" rows="1" role="data">6<hi rend="sup">s</hi></cell><cell cols="1" rows="1" role="data"> 9°</cell><cell cols="1" rows="1" role="data">32′</cell><cell cols="1" rows="1" role="data"> 0.6″</cell></row><row role="data"><cell cols="1" rows="1" role="data">1770 Mar. 29 at 0 4 50</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">27.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">dif. of time 178<hi rend="sup">d</hi> 0 15 38; true dif. lon.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">27</cell></row></table> tropical year = 365<hi rend="sup">d</hi> 5<hi rend="sup">h</hi> 48<hi rend="sup">m</hi> 42<hi rend="sup">s</hi> = 365.2421527; observed interval = 178 0 15 38 = 178.0108565: <pb n="439"/><cb/> Then 365.2421527 : 178.01085648 :: 360° : 175.455948 or 175° 27′ 21″ the mean motion. <table><row role="data"><cell cols="1" rows="1" role="data">Therefore</cell><cell cols="1" rows="1" role="data">175°</cell><cell cols="1" rows="1" role="data">27′</cell><cell cols="1" rows="1" role="data">21″</cell><cell cols="1" rows="1" role="data">of mean motion,</cell></row><row role="data"><cell cols="1" rows="1" role="data">answers to</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">of true motion;</cell></row><row role="data"><cell cols="1" rows="1" role="data">their difference is</cell><cell cols="1" rows="1" role="data">  3</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">and its half</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/></row></table> is the greatest Equation of the Centre according to these observations.</p><p><hi rend="italics">To find the Equation of the Centre,</hi> or to resolve Kepler's problem, is a very troublesome operation, especially in the more eccentric orbits. How this is to be done, has been shewn by Newton, Gregory, Keil, Machin, La Caille, and others, by methods little differing from one another; which consist chiefly in finding a certain intermediate angle, called the eccentric anomaly; having known the mean anamoly, and the dimensions of the sun's orbit. The mean anomaly is easily found, by determining the exact time when the sun is in the aphelion, and using the following proportion, viz, As the time of a tropical revolution, or solar year, Is to the interval between the aphelion and given time, So is 360 degrees, to the degrees of the mean anomaly. Or it may be found by taking the sun's mean motion at the given time out of tables.</p><p><hi rend="italics">To find the Eccentric Anomaly,</hi> say, As the aphelion distance, Is to the perihelion distance; So is the tangent of half the mean anomaly, To the tangent of an arc.</p><p>Which arc added to half the mean anomaly, gives the eccentric anomaly. Then,</p><p><hi rend="italics">To find the True Anomaly,</hi> say, As the square root of the aphelion distance, Is to the square root of the perihelion distance; So is the tangent of half the eccentric anomaly, To the tangent of half the true anomaly.</p><p>Then, the difference between the true and mean anomaly, gives the Equation of the Centre, sought. Which is subtractive, from the aphelion to the perihelion, or in the first 6 signs of anomaly; and additive, from the perihelion to the aphelion, or in the last 6 signs of anomaly; and hence called <hi rend="italics">Prosthapheresis.</hi></p><p>By this problem a table may easily be formed. When the Equations of the Centre for every degree of the first 6 signs of mean anomaly are found, they will serve also for the degrees of the last 6 signs, because equal anomalies are at equal distances on both sides of their apses. Then set these equations orderly to their signs and degrees of anomaly; the first 6 being reckoned srom the top of the table downwards, and signed <hi rend="italics">subtract;</hi> the last 6, for which the same Equations serve, in a contrary order, being reckoned from the bottom upwards, and marked <hi rend="italics">add.</hi> Let also the difference between every adjacent two Equations, called Tabular Differences, be set in another column. Hence, from these Equations of the Centre, augmented or diminished by the proportional parts of their respective tabular differences, for any given minutes and seconds, may easily be deduced Equations of the Centre to any mean anomaly proposed. Robertson's Elem. of Navig. book 5, p. 286, 290, 295, and 308, where such a table of Equations is given. <cb/></p><p>The late excellent Mr. Euler has particularly considered this subject, in the Mem. de l'Acad. de Berlin, tom. 2, p. 225 & seq. where he resolves the following problems:</p><p>1. To find the true and mean anomaly corresponding to the planet's mean distance from the sun; that is, where the planet is in the extremity of the conjugate axis of its orbit.</p><p>2. The eccentricity of a planet being given, to find the eccentric anomaly corresponding to the greatest Equation.</p><p>3. The eccentricity being given, to find the mean anomaly corresponding to the greatest Equation.</p><p>4. From the same data, to find the true anomaly corresponding to this Equation.</p><p>5. From the same data, to find the greatest Equation.</p><p>6. The greatest Equation being given, to find the eccentricity.</p><p>Mr. Euler observes, that this problem is very difficult, and that it can only be resolved by approximation and tentatively, in the manner he mentions: but if the eccentricity be not great, it may then be found directly from the greatest Equation. Thus, if the greatest Equation be = <hi rend="italics">m,</hi> and the eccentricity = <hi rend="italics">n</hi>; then is Whence by reversion Where the greatest equation <hi rend="italics">m</hi> must be expressed in parts of the radius, which may be done by reducing the angle <hi rend="italics">m</hi> into seconds, and adding 4.6855749 to the log. of the resulting number, which will be the log. of the number <hi rend="italics">m.</hi></p><p>The mean anomaly to which this greatest equation corresponds, will be Whence, if 90° be added to 5/8 of the greatest Equation, the sum will be the mean anomaly sufficiently exact.</p><p>Mr. Euler subjoins a table, by which may be found the greatest Equations, with the mean and eccentric anomalies corresponding to these greatest Equations for every 100th part of unity, which he supposes equal to the greatest eccentricity, or when the transverse and distance of the foci become infinite. The last column of the table gives also the logarithm of that distance of the planet from the sun where its Equation is greatest. By means of this table, any eccentricity being given, by interpolation will be found the corresponding greatest Equation. But the chief use of the table is to determine the eccentricity when the greatest Equation is known; and without this help Mr. Euler thinks the problem cannot be resolved.</p><p><hi rend="smallcaps">Equation</hi> <hi rend="italics">of Time,</hi> denotes the difference between mean and apparent time, or the reduction of the apparent unequal time, or motion of the sun or a planet, to equal and mean time, or motion; or the Equation of time is the difference between the sun's mean motion, and his right ascension. Apparent time is that which takes its beginning from the passage of the sun's centre over the meridian of any place; and had the sun no <pb n="440"/><cb/> motion in the ecliptic, or was his motion reduced to the equator or in right ascension uniform, he would always return to the meridian after equal intervals of time. But his apparent motion in the ecliptic being continually varying, and his motion in right ascenfion being rendered farther unequal on account of the obliquity of the ecliptic to the equator, from these causes it arises that the intervals of his return to the meridian become unequal, and the sun will gradually come too slow or too soon to the meridian for an equable motion, such as that of clocks and watches ought to be; and this retardation or acceleration of the sun's coming to the meridian, is called the equation of time.</p><p>Now, computing the celestial motions according to equal time, it is necessary to turn that time back again into apparent time, that they may correspond to observation: on the contrary, any phenomenon being observed, the apparent time of it must be converted into equal time, to have it correspond with the times marked in the astronomical tables.</p><p>The Equation of time is nothing at four different times in the year, when the whole mean and unequal motions exactly agree; viz, about the 15th of April, <cb/> the 15th of June, the 31st of August, and the 24th of December: at all other times the sun is either too fast or too slow for mean, equal, or clock time, by a certain number of minutes and seconds, which at the greatest is 16′ 14″, and happens about the 1st of November; every other day throughout the year having a certain quantity of this difference belonging to it; which however is not exactly the same every year, but only every 4th year; for which reason it is necessary to have 4 tables of this Equation, viz, one for each of the four years in the period of leap years. Instead of these, may be here inserted, as follows, one general equation of time, according to the place of the sun, in every point of the ecliptic: where it is to be observed, that the sign of the ecliptic is placed at the tops of the columns, and the particular degree of the sun's place, in each sign, in the first and last columns; and in the angle of meeting in all the other columns, is the equation of time, in minutes and seconds, when the sun has any particular longitude: supposing the obliquity of the ecliptic 23° 28′, and the sun's apogee in 9° of <figure/>. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=26 align=center" role="data"><hi rend="italics">A Table of the Equation of Time, for every Degree of the Sun's Longitude.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">Deg.</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data"><figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">11</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">Deg.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell><cell cols="1" rows="1" rend="align=center" role="data">m</cell><cell cols="1" rows="1" rend="align=center" role="data">s</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">+36</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">-9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">-51</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">+13</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">+57</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">+20</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">-38</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">-31</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">-33</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">-11</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">+28</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">+19</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">+17</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" 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role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">7</cell><cell 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role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell 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role="data">33</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">+5</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">-11</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">29</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row></table><pb n="441"/><cb/></p><p>The Equations with +, are to be added to the apparent time, to have the mean time, those with -, are to be subtracted from apparent time, to give the mean time.</p><p>The preceding mark, whether + or -, at the top of any column, belongs to all the numbers or equations in that column till the sign changes; after which, the remainder of the column belongs to the contrary sign.</p><p>The Equation answering to any point of longitude between one degree and another, or any number of minutes or parts of a degree, is to be found by proportion in the usual way, viz, as 1° or 60′, to that number of minutes, so is the whole difference in the Equation from the given whole degree of longitude to the next degree, to the proportional part of it answering to the given number of minutes.</p><p>See Tables of the Equation of Time computed for every year, in the Nautical Almanac, by a method proposed and illustrated by Dr. Maskelyne, the astronomer royal, viz, by taking the difference between the sun's true right ascension and his mean longitude, corrected by the Equation of the equinoxes in right ascension, and turning it into time at the rate of 1 minute of time to 15′ of right ascension. Philos. Trans. vol. 54, p. 336.</p><p><hi rend="smallcaps">Equation</hi> <hi rend="italics">of a Curve,</hi> is an Equation shewing the nature of a curve by expressing the relation between any absciss and its corresponding ordinate, or else the relation of their fluxions, &c. Thus, the Equation to the circle, is , where <hi rend="italics">a</hi> is its diameter, <hi rend="italics">x</hi> any absciss, or part of that diameter, and <hi rend="italics">y</hi> the ordinate at that point of the diameter; the meaning being, that whatever abseiss is denoted by <hi rend="italics">x,</hi> then the square of its corresponding ordinate will be <hi rend="italics">ax</hi>-<hi rend="italics">x</hi><hi rend="sup">2</hi>. In like manner the Equation of the ellipse is , of the hyperbola is , of the parabola is <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>. Where <hi rend="italics">a</hi> is an axis, and <hi rend="italics">p</hi> the parameter.</p><p>And in like manner for any other curves.</p><p>This method of expressing the nature of curves by algebraical equations, was first introduced by Des Cartes, who, by thus connecting together the two sciences of algebra and geometry, made them mutually assisting to each other, and so laid the foundation of the greatest improvements that have been made in every branch of them since that time. See Des Cartes's Geometry; also Newton's Lines of the 3d Order, and many other similar works on curve lines, by several authors.</p></div2></div1><div1 part="N" n="EQUATOR" org="uniform" sample="complete" type="entry"><head>EQUATOR</head><p>, in Geography, a great circle of the earth, equally distant from its two poles, and dividing it into two equal parts, or hemispheres, the northern and southern. The Equator is sometimes simply called the Line.</p><p>The circle in the heavens conceived directly over the Equator, is the Equinoctial. See <hi rend="smallcaps">Equinoctial.</hi></p><p>The greatest height of the Equator above the horizon, is equal to the latitude of the place. <cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=9 align=center" role="data"><hi rend="italics"><hi rend="smallcaps">Table</hi> for turning Degrees and Minutes into Time, and the Contrary.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" rend="align=center" role="data">H</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" rend="align=center" role="data">H</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">D</cell><cell cols="1" rows="1" rend="align=center" role="data">H</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">S</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">S</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">M</cell><cell cols="1" rows="1" rend="align=center" role="data">S</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">62</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">122</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">63</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">123</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">67</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">69</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">71</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">72</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">73</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">74</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">134</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">75</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">76</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">136</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">77</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">137</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">78</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">138</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">79</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">139</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">80</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">81</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">141</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">82</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">83</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">143</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">84</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">144</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">145</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">86</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">87</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">88</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">148</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">89</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">149</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">150</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">31</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">91</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">92</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">33</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">93</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">153</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">94</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">154</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">95</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">155</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">96</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">156</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">97</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">157</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">98</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">158</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">99</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">159</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">100</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">160</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">101</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">161</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">102</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">103</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">163</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">104</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">164</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">105</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">106</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">166</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">47</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">107</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">167</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">108</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">49</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">109</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">169</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">110</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">170</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">51</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">111</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">112</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">172</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" rend="align=right" role="data">113</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">173</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">54</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">114</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">115</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" rend="align=right" role="data">116</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">176</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">44</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">117</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">177</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" rend="align=right" role="data">118</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">178</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">119</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">56</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">180</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row></table> <pb n="442"/><cb/></p><p>As one whole revolution of the earth, or of the 360° of the Equator, is performed in 24 hours, which is at the rate of 15° to the hour; hence the number of degrees of the Equator, answering to any other given time, or the time answering to any given degrees of the Equator, will be easily found by proportion, viz, as 1hr. : 15° :: any time : its degrees, or as 15° : 1hr. :: any degrees : their time. And thus is computed the foregoing Table for turning time into degrees of the Equator, and the contrary.</p></div1><div1 part="N" n="EQUATORIAL" org="uniform" sample="complete" type="entry"><head>EQUATORIAL</head><p>, <hi rend="italics">Universal,</hi> or <hi rend="smallcaps">Portable</hi> O<hi rend="smallcaps">BSERVATORY</hi>, is an instrument intended to answer a number of useful purposes in practical astronomy, independent of any particular observatory. It may be employed in any steady room or place, and it performs most of the useful problems in the science of astronomy. The following is the description of one lately invented, and named the <hi rend="italics">Universal Equatorial.</hi></p><p>The principal parts of this instrument (sig. 2, plate viii.) are, 1st, The azimuth or horizontal circle A, which represents the horizon of the place, and moves on a long axis B, called the vertical axis. 2d, The Equatorial or hour-circle C, representing the Equator, placed at right angles to the polar axis D, or the axis of the earth, upon which it moves. 3d, The semicircle of declination E, on which the telescope is placed, and moving on the axis of declination, or the axis of motion of the line of collimation F. Which circles are measured and divided as in the following Table: <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Measures of the several circles and divisions on them.</cell><cell cols="1" rows="1" role="data">Radius Inches</cell><cell cols="1" rows="1" role="data">Limb divided to</cell><cell cols="1" rows="1" role="data">Non. of 3.0 gives seconds</cell><cell cols="1" rows="1" role="data">Divid. on limb into pts.of Inc.</cell><cell cols="1" rows="1" role="data">Divid. by Non. into pts.of Inc.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Azimuth or horizontal circle</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">5.1</cell><cell cols="1" rows="1" rend="align=center" role="data">15′</cell><cell cols="1" rows="1" role="data">30″</cell><cell cols="1" rows="1" role="data">45th</cell><cell cols="1" rows="1" role="data">1350th</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Equatorial or hour circle</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">5.1</cell><cell cols="1" rows="1" rend="rowspan=2 align=center" role="data">15′, or 1 m. in time</cell><cell cols="1" rows="1" role="data">30″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">2″</cell><cell cols="1" rows="1" role="data">45th</cell><cell cols="1" rows="1" role="data">1350th</cell></row><row role="data"><cell cols="1" rows="1" role="data">Vertical semicircle for declination or latitude</cell><cell cols="1" rows="1" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">5.5</cell><cell cols="1" rows="1" rend="align=center" role="data">15′</cell><cell cols="1" rows="1" role="data">30″</cell><cell cols="1" rows="1" role="data">42d</cell><cell cols="1" rows="1" role="data">1260th</cell></row></table></p><p>4th, The telescope, which is an achromatic refractor with a triple object-glass, whose focal distance is 17 inches, and its aperture 2.45 inc., and it is furnished with 6 different eye-tubes; so that its magnifying powers extend from 44 to 168. The telescope in this Equatorial may be brought parallel to the polar axis, as in the figure, so as to point to the pole-star in any part of its diurnal revolution; and thus it has been observed near noon, when the sun has shone very bright. 5th, The apparatus for correcting the error in altitude occasioned by refraction, which is applied to the eye-end of the telescope, and consists of a slide G moving in a groove or dove-tail, and carrying the several eye-tubes of the telescope, on which slide there is an index corresponding <cb/> to five small divisions engraved on the dove-tail; a very small circle, called the refraction circle H, moveable by a finger-serew at the extremity of the eye-end of the telescope; which circle is divided into half minutes, one whole revolution of it being equal to 3′ 18″, and by its motion it raises the centre of the cross-hairs on a circle of altitude; and also a quadrant I of 1 1/2 inc. radius, with divisions on each side, one expressing the degree of altitude of the object viewed, and the other expressing the minutes and seconds of error occasioned by refraction, corresponding to that degree of altitude. To this quadrant is joined a small round level K, which is adjusted partly by the pinion that turns the whole of this apparatus, and partly by the index of the quadrant; for which purpose the refraction circle is set to the same minute &c, which the index points to on the limb of the quadrant; and if the minute &c, given by the quadrant, exceed the 3′ 18″ contained in one entire revolution of the refraction circle, this must be set to the excess above one or more of its entire revolutions; then the centre of the cross-hairs will appear to be raised on a circle of altitude to the additional height which the error of refraction will occasion at that altitude.</p><p>The principal adjustment in this instrument, is that of making the line of collimation to describe a portion of an hour circle in the heavens: in order to which, the azimuth circle must be truly level; the line of collimation, or some corresponding line represented by the small brass rod M parallel to it, must be perpendicular to the axis of its own proper motion; and this last axis must be perpendicular to the polar axis. On the brass rod M there is occasionally placed a hanging level N, the use of which will appear in the following adjustments:</p><p>The azimuth circle may be made level by turning the instrument till one of the levels be parallel to an imaginary line joining two of the feet screws; then adjust that level with these two feet screws; turn the circle 180°, or half round; and if the bubble be not then right, correct half the error by the screw belonging to the level, and the other half error by the two foot screws, repeating this operation till the bubble come right; then turn the circle 90° from the two former positions, and set the bubble right, if it be wrong, by the foot screw at the end of the level; when this is done, adjust the other level by its own screw, and the azimuth circle will be truly level. The hanging level must then be fixed to the brass rod by two hooks of equal length, and made truly parallel to it: for this purpose, make the polar axis perpendicular or nearly perpendicular to the horizon; then adjust the level by the pinion of the declination semicircle: reverse the level, and if it be wrong, correct half the error by a small steel screw that lies under one end of the level, and the other half error by the pinion of the declinationsemicircle, repeating the operation till the bubble be right in both positions. To make the brass rod, on which the level is suspended, at right angles to the axis of motion of the telescope, or line of collimation, make the polar axis horizontal, or nearly so; set the declination semicircle to 0°, and turn the hour-circle till the bubble come right; then turn the declination- <pb/><pb/><pb n="443"/><cb/> circle to 90°; adjust the bubble by raising or depressing the polar axis (first by hand till it be nearly right, afterwards tighten with an ivory key the socket which runs on the arch with the polar axis, and then apply the same ivory key to the adjusting screw at the end of the said arch till the bubble come quite right); then turn the declination-circle to the opposite 90°; if the level be not then right, correct half the error by the aforesaid adjusting screw at the end of the arch, and the other half error by the two screws that raise or depress the end of the brass rod. The polar axis remaining nearly horizontal as before, and the declinationsemicircle at 0°, adjust the bubble by the hour-circle; then turn the declination-semicircle to 90°, and adjust the bubble by raising or depressing the polar axis; then turn the hour-circle 12 hours; and if the bubble be wrong, correct half the error by the polar axis, and the other half error by the two pair of capstan screws at the feet of the two supports on one side of the axis of motion of the telescope; and thus this axis will be at right angles to the polar axis. The next adjustment, is to make the centre of the cross hairs remain on the same object, while the eye-tube is turned quite round by the pinion of the refraction apparatus: for this adjustment, set the index on the slide to the first division on the dove-tail; and set the division marked 18″ on the refraction-circle to its index; then look through the telescope, and with the pinion turn the eye-tube quite round; then if the centre of the hairs does not remain on the same spot during that revolution, it must be corrected by the four small screws, 2 and 2 at a time, which will be found upon unscrewing the nearest end of the eye-tube that contains the first eye-glass; repeating this correction till the centre of the hairs remain on the spot looked at during a whole revolution. To make the line of collimation parallel to the brass rod on which the level hangs, set the polar axis horizontal, and the declination-circle to 90°, adjust the level by the polar axis; look through the telescope on some distant horizontal object, covered by the centre of the cross hairs: then invert the telescope, which is done by turning the hour-circle half round; and if the centre of the cross hairs does not cover the same object as before, correct half the error by the uppermost and lowermost of the 4 small screws at the eyeend of the large tube of the telescope; this correction will give a second object now covered by the centre of the hairs, which must be adopted instead of the first object; then invert the telescope as before; and if the second object be not covered by the centre of the hairs, correct half the error by the same two screws as were used before: this correction will give a third object, now covered by the centre of the hairs, which must be adopted instead of the second object; repeat this operation till no error remain; then set the hour-circle exactly to 12 hours, the declination-circle remaining a 90° as before; and if the centre of the cross hairs do not cover the last object fixed on, set it to that object by the two remaining small screws at the eye-end of the large tube, and then the line of collimation will be parallel to the brass rod. For rectifying the nonius of the declination and Equatorial circles, lower the telescope as many degrees &c below 0° or Æ on the decli- <cb/> nation-semicircle as are equal to the complement of the latitude; then elevate the polar axis till the bubble be horizontal; and thus the Equatorial circle will be elevated to the co-latitude of the place: set this circle to 6 hours; adjust the level by the pinion of the declination-circle; then turn the Equatorial circle exactly 12 hours from the last position; and if the level be not right, correct one half of the error by the Equatorial circle, and the other half by the declination-circle: then turn the Equatorial circle back again exactly 12 hours from the last position; and if the level be still wrong, repeat the correction as before, till it be right, when turned to either position: that being done, set the nonius of the Equatorial circle exactly to 6 hours, and the nonius of the declination-circle exactly to 0°.</p><p>The chief uses of this Equatorial are,</p><p>1st, To find the meridian by one observation only: for this purpose, elevate the Equatorial circle to the co-latitude of the place, and set the declination-semicircle to the sun's declination for the day and hour of the day required; then move the azimuth and hourcircles both at the same time, either in the same or contrary direction, till you bring the centre of the cross hairs in the telescope exactly to cover the centre of the sun; when that is done, the index of the hour-circle will give the apparent or solar time at the instant of observation; and thus the time is gained, though the sun be at a distance from the meridian; then turn the hour-circle till the index points precisely at 12 o'clock, and lower the telescope to the horizon, in order to observe some point there in the centre of the glass; and that point is the meridian mark, found by one observation only. The best time for this operation is 3 hours before, or 3 hours after 12 at noon.</p><p>2d, To point the telescope on a star, though not on the meridian, in full day-light. Having elevated the equatorial circle to the co-latitude of the place, and set the declination-semicircle to the star's declination, move the index of the hour-circle till it shall point to the precise time at which the star is then distant from the meridian, found in the tables of the right ascension of the stars, and the star will then appear in the glass.</p><p>Besides these uses, peculiar to this instrument, it may also be applied to all the purposes to which the principal astronomical instruments are applied; such as a transit instrument, a quadrant, and an equal-altitude instrument.</p><p>See the description and drawing of an Equatorial telescope, or portable observatory, invented by Mr. Short, in the Philos. Trans. number 493, or vol. 46, p. 242; and another by Mr. Nairne, vol. 61, p. 107.</p><p>EQUIANGULAR <hi rend="italics">Figure,</hi> is one that has all its angles equal among themselves; as the square, and all the regular figures.</p><p>An equilateral figure inscribed in a circle, is always Equiangular. But an Equiangular figure inscribed in a circle, is not always equilateral, except when it has an odd number of sides: If the sides be of an even number, then they may either be all equal, or else half of them will always be equal to each other, and the other half to each other, the equals being placed alternately. See the demonstration in my Mathematical Miscellany, pa. 272. <pb n="444"/><cb/></p><div2 part="N" n="Equiangular" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equiangular</hi></head><p>, is also said of any two figures of the same kind, when each angle of the one is equal to a corresponding angle in the other, whether each figure, separately considered in itself, be an equiangular figure or not, that is, having all its angles equal to each other. Thus, two triangles are Equiangular to each other, if, ex. gr. one angle in each be of 30°, a second angle in each of 50°, and the third angle of each equal to 100 degrees.</p><p>Equiangular triangles have not their like sides necessarily equal, but only proportional to each other; and such triangles are always similar to each other.</p><p><hi rend="smallcaps">Equicrural</hi> <hi rend="italics">Triangle,</hi> is one that has two of its sides equal to each other; and is more usually called an Isosceles triangle.</p></div2><div2 part="N" n="Equiculus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equiculus</hi></head><p>, <hi rend="smallcaps">Equuleus</hi>, or <hi rend="smallcaps">Equus</hi> <hi rend="italics">Minor,</hi> a constellation of the northern hemisphere. See E<hi rend="smallcaps">QUULEUS.</hi></p></div2></div1><div1 part="N" n="EQUIDIFFERENT" org="uniform" sample="complete" type="entry"><head>EQUIDIFFERENT</head><p>, are such things as have equal differences, or arithmetically proportional. If the terms have all the same difference, viz, the 1st and 2d, the 2d and 3d, the 3d and 4th, &c, they are said to be Continually Equidifferent; as the numbers 3, 6, 9, 12, &c, where the common difference is 3. But if the several different couplets only have the same difference, as the 1st and 2d, the 3d and 4th, the 5th and 6th, &c, they are said to be Discretely Equidifferent; as the terms 3 and 6, 7 and 10, 9 and 12, &c. See A<hi rend="smallcaps">RITHMETICAL</hi> <hi rend="italics">Progression</hi> and <hi rend="italics">Proportion.</hi></p><p>EQUILATERAL <hi rend="italics">Figure,</hi> is one that has all its sides equal to each other. Such as the square, and all the regular figures or polygons, or a triangle that has all its sides equal. See <hi rend="smallcaps">Equiangular.</hi></p><p><hi rend="smallcaps">Equilateral</hi> <hi rend="italics">Hyperbola,</hi> is that which has the two axes equal to each other, and every pair of conjugate diameters also equal to each other. The asymptotes also are at right angles to each other, and make each half a right angle with either axis. Also, such an hyperbola is equal to its opposite hyperbola, and likewise to its conjugate hyperbola; so that all the four conjugate hyperbolas are mutually equal to each other.</p><p>Moreover, as the 3d proportional to the two axes is the parameter, therefore in such a sigure, the parameter and two axes are all three equal to one another. Hence, as the general equation to hyperbolas is , where <hi rend="italics">t</hi> is the transverse axis, <hi rend="italics">c</hi> the conjugate, <hi rend="italics">p</hi> the parameter, <hi rend="italics">x</hi> the absciss, and <hi rend="italics">y</hi> the ordinate; then making <hi rend="italics">t, c,</hi> and <hi rend="italics">p</hi> all equal, the equation, for the Equilateral Hyperbola, becomes ; differing from the equation of the circle only in the sign of the term <hi rend="italics">x</hi><hi rend="sup">2</hi>, which in the circle is —.</p></div1><div1 part="N" n="EQUILIBRIUM" org="uniform" sample="complete" type="entry"><head>EQUILIBRIUM</head><p>, is an equality between two equal forces acting in opposite directions; so that they mutually balance each other; like the two equal arms, or scales, of a balance, &c.</p><div2 part="N" n="Equilibrium" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Equilibrium</hi></head><p>, in solid bodies, forms a considerable part of the science of Statics. And Equilibrium of fluids, a considerable part of the doctrine of Hydrostatics. <cb/></p></div2></div1><div1 part="N" n="EQUIMULTIPLES" org="uniform" sample="complete" type="entry"><head>EQUIMULTIPLES</head><p>, the products of quantities equally multiplied. Thus, 3<hi rend="italics">a</hi> and 3<hi rend="italics">b</hi> are Equimultiples of <hi rend="italics">a</hi> and <hi rend="italics">b;</hi> and <hi rend="italics">ma</hi> and <hi rend="italics">mb</hi> are Equimultiples of <hi rend="italics">a</hi> and <hi rend="italics">b.</hi></p><p>Equimultiples of any quantities, have the same ratio as the quantities themselves. Thus <hi rend="italics">a : b</hi> :: 3<hi rend="italics">a</hi> : 3<hi rend="italics">b :: ma : mb.</hi></p></div1><div1 part="N" n="EQUINOCTIAL" org="uniform" sample="complete" type="entry"><head>EQUINOCTIAL</head><p>, a great circle in the heavens under which the equator moves in its diurnal motion. The poles of this circle are the poles of the world. It divides the sphere into two equal parts, the northern and southern. It cuts the horizon, of any place, in the east and west points; and at the meridian its elevation above the horizon is equal to the co-latitude of the place. The Equinoctial has also various other properties; as,</p><p>1. Whenever the sun comes to this circle, he makes equal days and nights all round the globe; because he then rises due east, and sets due west. Hence it has the name Equinoctial. All stars which are under this circle, or have no declination, do also rise due east, and set due west.</p><p>2. All people living under this circle, or upon the equator, or line, have their days and nights at all times equal to each other.</p><p>3. From this circle, on the globe, is counted, upon the meridian, the declination in the heavens, and the latitude on the earth.</p><p>4. Upon the Equinoctial, or equator, is counted the longitude, making in all 360° quite round, or else 180° east, and 180° west.</p><p>5. And as the time of one whole revolution is divided into 24 hours; therefore 1 hour anfwers to 15°, or the 24th part of 360°. Hence, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1°</cell><cell cols="1" rows="1" role="data">of longitude answers</cell><cell cols="1" rows="1" role="data">to</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">min.</cell><cell cols="1" rows="1" role="data">of time,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15′</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">to</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">min.</cell><cell cols="1" rows="1" role="data">of time,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1′</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">to</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">sec.</cell><cell cols="1" rows="1" role="data">of time,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table></p><p>6. The shadows of those people who live under this circle are cast to the southward of them for one half of the year, and to the northward of them during the other half; and twice in a year, viz, at the time of the equinoxes, the sun at noon casts no shadow, being exactly in their zenith.</p><p><hi rend="smallcaps">Equinoctial</hi> <hi rend="italics">Colure,</hi> is the great circle passing through the poles of the world and the Equinoctial points, or first points of Aries and Libra.</p><p><hi rend="smallcaps">Equinoctial</hi> <hi rend="italics">Dial,</hi> is one whose plane is parallel to the Equinoctial. The properties or principles of this dial are,</p><p>1. The hour lines are all equally distant from one another, quite round the circumference of a circle; and the style is a straight pin, or wire, set up in the centre of the circle, perpendicular to the plane of the dial.</p><p>2. The sun shines upon the upper part of this dialplane from the 21st of March to the 22d of September, and upon the under part of the plane the other half of the year.</p><p>Some of these dials are made of brass, &c; and set up in a frame, to be elevated to any given latitude.</p><p><hi rend="smallcaps">Equinoctial</hi> <hi rend="italics">Points,</hi> are the two opposite points where the Ecliptic and Equinoctial cross each other; <pb n="445"/><cb/> the one point being in the beginning of Aries, and called the Vernal point, or Vernal Equinox; and the other in the beginning of Libra, and called the Autumnal point, or Autumnal Equinox.</p><p>It is found by observation, that the Equinoctial points, and all the other points of the ecliptic, are continually moving backwards, or in autecedentia, i. e. westwards. This retrograde motion of the Equinoctial points, is that phenomenon called the Precession of the Equinoxes, and is made at the rate of 50″ every year nearly.</p></div1><div1 part="N" n="EQUINOXES" org="uniform" sample="complete" type="entry"><head>EQUINOXES</head><p>, the times when the sun enters the Equinoctial points, or about the 21st of March and 22d of September: the former being the Vernal or Spring Equinox, and the latter time the Autumnal Equinox.</p><p>As the sun's motion is unequal, being sometimes quicker and sometimes slower, it hence happens that there are about 8 days more from the vernal to the autumnal Equinox, or while the sun is on the northern side of the equator, than while he is in moving through the southern figns from the autumnal to the vernal Equinox, or on the southern side of the equator. According to the obfervations of M. Cassini, the <table><row role="data"><cell cols="1" rows="1" role="data">sun is</cell><cell cols="1" rows="1" role="data">186<hi rend="sup">d</hi></cell><cell cols="1" rows="1" role="data">14<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">53<hi rend="sup">m</hi></cell><cell cols="1" rows="1" role="data">in the northern signs,</cell></row><row role="data"><cell cols="1" rows="1" role="data">and only</cell><cell cols="1" rows="1" role="data">178</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">in the southern signs,</cell></row><row role="data"><cell cols="1" rows="1" role="data">so that</cell><cell cols="1" rows="1" role="data">  7</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">is the difference of them, or</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">nearly 8 days.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>EQUINUS <hi rend="italics">Barbatus,</hi> a kind of comet. See H<hi rend="smallcaps">IPPEUS.</hi></p></div1><div1 part="N" n="EQUULEUS" org="uniform" sample="complete" type="entry"><head>EQUULEUS</head><p>, <hi rend="smallcaps">Equiculus</hi>, and <hi rend="smallcaps">Equus</hi> <hi rend="italics">Minor, Equi sectio,</hi> the <hi rend="italics">Horse's Head,</hi> one of the 48 old constellations, in the northern hemisphere. Its stars, in Ptolomy's catalogue, are 4, in Tycho's 4, in Hevelius's 6, and in Flamsteed's 10.</p><p>ERECT <hi rend="italics">Vision.</hi> See <hi rend="smallcaps">Vision.</hi></p><p><hi rend="smallcaps">Erect</hi> <hi rend="italics">Dials,</hi> such as stand perpendicular to the horizon, and are of various kinds; as Erect Direct, when they face exactly one of the four cardinal points, east, west, north, south; and Erect Declining, when declined from the cardinal points. See <hi rend="smallcaps">Dial.</hi></p><p><hi rend="smallcaps">To Erect</hi> <hi rend="italics">a Perpendicular,</hi> is a popular problem in practical geometry, and denotes to raise a perpendicular from a given line &c, as distinguished from Demitting or letting a perpendicular fall upon a line &c, from some point out of it. See <hi rend="smallcaps">Perpendicular.</hi></p></div1><div1 part="N" n="ERIDANUS" org="uniform" sample="complete" type="entry"><head>ERIDANUS</head><p>, the <hi rend="italics">River,</hi> a constellation of the southern hemisphere, and one of the 48 old asterisms. The stars in this constellation, in Ptolomy's catalogue, are 34, in Tycho's 19, and in the British catalogue 84.</p></div1><div1 part="N" n="ERRATIC" org="uniform" sample="complete" type="entry"><head>ERRATIC</head><p>, an epithet applied to the planets, which are called Erratic or wandering stars, in contradistinction to the fixed stars.</p></div1><div1 part="N" n="ESCALADE" org="uniform" sample="complete" type="entry"><head>ESCALADE</head><p>, or <hi rend="smallcaps">Scalade</hi>, a furious attack of a wall or a rampart; carried on with ladders, to pass the ditch, or mount the rampart; without proceeding in form, breaking ground, or carrying on regular works to secure the men.</p></div1><div1 part="N" n="ESPAULE" org="uniform" sample="complete" type="entry"><head>ESPAULE</head><p>, or <hi rend="smallcaps">Epaule.</hi> See <hi rend="smallcaps">Epaule.</hi></p></div1><div1 part="N" n="FSPLANADE" org="uniform" sample="complete" type="entry"><head>FSPLANADE</head><p>, in Fortification, called also Glacis, a part which serves as a parapet to the counterscarp, or covert way; being a declivity or slope of earth, commencing from the top of the counterscarp, and losing itself insensibly in the level of the champaign. <cb/></p><p>Esplanade also means the ground which has been levelled from the glacis of the counterscarp, to the sirst houses; or the vacant space between the works and the houses of the town.</p><p>The term is also applied, in the general, to any piece of ground that is made flat or level, and which before had some eminence that incommoded the place.</p><p>ESTIVAL <hi rend="italics">Occident, Orient,</hi> or <hi rend="italics">Solstice.</hi> See O<hi rend="smallcaps">CCIDENT, Orient, Solstice.</hi></p></div1><div1 part="N" n="EVAPORATION" org="uniform" sample="complete" type="entry"><head>EVAPORATION</head><p>, the act of dissipating the humidity of a body in fumes or vapour; differing from exhalation, which is properly a dispersion of dry particles issuing from a body.</p><p>Evaporation is usually produced by heat, and by the change of air: thus, common salt is formed by evaporating all the humidity in the brine or salt water; which evaporation is either performed by the heat of the sun, as in the salt-works on the sea-coast, &c; or by means of fire, as at the salt-springs, &c: and it is well known how useful a brisk wind is in drying wet clothes, or the surface of the ground; while in a calm, still atmosphere, they dry extreme slowly.</p><p>But, though Evaporation be generally considered as an effect of the heat and motion of the air, yet M. Gauteron, in the Memoires de l'Acad. des Scienc. an. 1705, shews, that a quite opposite cause may have the same effect, and that sluids lose more of their parts in the severest frost than when the air is moderately warm: thus, in the great frost of the year 1708, he found that the greater the cold, the more considerable the evaporation; and that ice itself lost full as much as the warmer liquors that did not freeze.</p><p>There are indeed few subjects of philosophical investigation that have occasioned a greater variety of opinion than the theory of Evaporation, or of the ascent of water in such a fluid as air, between 8 and 9 hundred times lighter than itself, to different heights according to the different densities of the air; in which case it must be specifically lighter than the air through which it ascends. The Cartesians account for it by supposing, that by the action of the sun upon the water, small particles of the water are formed into hollow spheres and filled with the <hi rend="italics">materia subtilis,</hi> which renders them specifically lighter than the ambient air, so that they are buoyed up by it.</p><p>Dr. Nieuwentyt, in his Religious Philosopher, cont. 19, and several others, have alleged, that the sun emits particles of fire which adhere to those of water, and form moleculæ, or small bodies, lighter than an equal bulk of air, which consequently ascend till they come to a height where the air is of the same specific gravity with themselves; and that these particles being separated from the fire with which they are incorporated, coalesce and descend in dew or rain.</p><p>Dr. Halley has advanced another hypothesis, which has been more generally received: he imagined, that by the action of the sun on the surface of the water, the aqueous particles are formed into hollow spherules, that are filled with a finer air highly rarefied, so as to become specifically lighter than the external air. Philos. Trans. number 192, or Abr. vol. 2, p. 126.</p><p>Dr. Desaguliers, dissatisfied with these two hypotheses, proposes another in the Philos. Trans. number 407, or Abr. vol. 7, pa. 61. See also his Course of <pb n="446"/><cb/> Experimental Philosophy, vol. 2, p. 336. He supposes that heat acts more powerfully on water than on common air; that the same degree of heat which rarefies air two-thirds, will rarefy water near 14,000 times; and that a very small degree of heat will raise a steam or vapour from water, even in winter, whilst it condenses the air; and thus the particles of water are converted into vapour by being made to repel each other strongly, and, deriving electricity from the particles of air to which they are contiguous, are repelled by them and by each other, so as to form a fluid which, being lighter than the air, rises in it, according to their relative gravities. The particles of this vapour retain their repellent force for a considerable time, till, by some diminution of the density of the air in which they float, they are precipitated downwards, and brought within the sphere of each other's attraction of cohesion, and so join again into drops of water.</p><p>Many objections have been urged against this opinion, by Mr. Clare in his Treatise of the Motion of Fluids, pa. 294, and by Mr. Rowning in his System of Philosophy, part 2, diss. 6; to which Dr. Hamilton has added the two following, viz, that if heat were the only cause of evaporation, water would evaporate faster in a warm close room, than when exposed in a colder place, where there is a constant current of air; which is contrary to experience; and that the evaporation of water is so far from depending on its being rarefied by heat, that it is carried on even whilst water is condensed by the coldness of the air, till it freezes; and since it evaporates even when frozen into hard ice, it must also evaporate in all the lesser degrees of cold. And therefore heat does not seem to be the principal, much less the only cause of Evaporation.</p><p>Others have more successfully accounted for the phenomena of Evaporation on another principle, viz that of solution; and shewn, from a variety of experiments, that what we call Evaporation, is nothing more than a gradual solution of water in air, produced and supported by the same means, viz, attraction, heat, and motion, by which other solutions are effected.</p><p>It seems the Abbé Nollet first started this opinion, though without much pursuing it, in his Leçons de Physique Experimentale, first published in 1743: he offers it as a conjecture, that the air of the atmosphere serves as a solvent or sponge, with regard to the bodies that encompass it, and receives into its pores the vapours and exhalations that are detached from the masses to which they belong in a fluid state; and he accounts for their ascent on the same principles with the ascent of liquors in capillary tubes. On his hypothesis, the condensation of the air contributes, like the squeezing of a sponge, to their descent.</p><p>Dr. Franklin, in a paper of Philosophical and Meteorological Observations, Conjectures and Suppositions, delivered to the Royal Society about the year 1747, and read in 1756, suggested a similar hypothesis: he observes, that air and water mutually attract each other; and hence he concludes, that water will dissolve in air, as salt in water; every particle of air assuming one or more particles of water; and when too much is added, it precipitates in rain. But as there is not the same contiguity between the particles of air as of water, the solution of water in air is not carried on without <cb/> a motion of the air, so as to cause a fresh accession of dry particles. A small degree of heat so weakens the cohesion of the particles of water, that those on the surface easily quit it, and adhere to the particles of air: a greater degree of heat is necessary to break the cohesion between water and air; for its particles being by heat repelled to a greater distance from each other, thereby more easily keep the particles of water that are annexed to them from running into cohesions that would obstruct, refract, or reflect the heat: and hence it happens that when we breathe in warm air, though the same quantity of moisture may be taken up from the lungs as when we breathe in cold air, yet that moisture is not so visible. On these principles he accounts for the production and different appearances of fogs, mists, and clouds. He adds, that if the particles of water bring electrical fire when they attach themselves to air, the repulsion between the particles of water electrisied, joins with the natural repulsion of the air to force its particles to a greater distance, so that the air being more dilated, rises and carries up with it the water; which mutual repulsion of the particles of air is increased by a mixture of common fire in the particles of water. When air, loaded with surrounding particles of water, is compressed by adverse winds, or by being driven against mountains, &c, or condensed by taking away the fire that assisted it in expanding, the particles will approach one another, and the air with its water will descend as a dew; or if the water surrounding one particle of air come in contact with the water surrounding another, they coalesce and form a drop, producing rain; and since it is a well-known fact, that vapour is a good conductor of electricity, as well as of common fire, it is reasonable to conclude with Mr. Henley, that Evaporation is one great cause of the clouds becoming at times surcharged with this fluid. Philos. Trans. vol. 67, pa. 134. See also vol. 55, p. 182, or Franklin's Letters and Papers on Philosophical Subjects, p. 42 &c, and pa. 182, ed. 1769.</p><p>M. le Roi, of the Acad. of Sciences at Paris, has also advanced the same opinion, and supported it by a variety of facts and observations in the Memoirs for the year 1751. He shews, that water does undergo in the air a real dissolution, forming with it a transparent mixture, and possessing the same properties with the solutions of most salts in water; and that the two principal causes which promote the solution of water in the air, are heat and wind; that the hotter the air is, within a certain limit, the more water it will dissolve; and that at a certain degree of heat the air will be saturated with water; and by determining at different times the degree of the air's saturation, he estimates the influence of those causes on which the quantity depends that is suspended in the air in a state of solution. Accordingly, the air, heated by evaporating substances to which it is contiguous, becomes more rare and light, rises and gives way to a denser air; and, by being thus removed, contributes to accelerate the Evaporation. The fixed air contained in the internal parts of evaporating bodies, put into action by heat, seems also to increase their Evaporation. The wind is another cause of the increase of Evaporation, chiefly by changing and renewing the air which immediately encompasses the evaporating substances; and from the consideration <pb n="447"/><cb/> of these two causes combined, it appears why the quantity of vapour raised in the night is less than that of the day, since the air is then both less heated and less agitated. To the objection urged against this hypothesis, on account of the Evaporation of water in a vacuum, this ingenious writer replies, that the water itself contains a great quantity of air, which gradually disengages itself, and causes the Evaporation; and that it is impossible that a space containing water which evaporates should remain perfectly free from air. To this objection a late writer, Dr. Dobson of Liverpool, replies, that though air appears, by unquestionable experiments, to be a chemical solvent of water, and as such, is to be considered as one cause of its Evaporation, heat is another cause, acting without the intervention of air, and producing a copious Evaporation in an exhausted receiver; agrecably to an experiment of Dr. Irving, who says, that in an exhausted receiver water rises in vapour more copiously at 180° of Fahrenheit's thermometer, than in the open air at 212°, its boiling point. Dr. Dobson farther adds, that water may exist in air in three different states; in a state of perfect solution, when the air will be clear, dry, and heavy, and its powers of solution still active; in a state of beginning precipitation, when it becomes moist and foggy, its powers of solution are diminished, and it becomes lighter in proportion as its water is deposited; and also, when it is completely precipitated, which may happen either by a slower process, when the dissolved water falls in a drizzling rain, or by a more sudden process, when it descends in brisk showers. Philos. Trans. vol. 67, p. 257, and Phipps's Voyage towards the North Pole, p. 211.</p><p>Dr. Hamilton, professor of philosophy in the university of Dublin, transmitted to the Royal Society in 1765, a long Dissertation on the nature of Evaporation, in which he proposes and establishes this theory of solution; and though other writers had been prior in their conjectures, and even in their reasoning on this subject, Dr. Hamilton assures us, that he has not represented any thing as new which he was conscious had ever been proposed by any one before him, even as a conjecture. Dr. hamilton having evinced the agreement between Solution and Evaporation, concludes, that Evaporation is nothing more than a gradual solution of water in air, produced and promoted by attraction, heat, and motion, just as other solutions are effected.</p><p>To account for the ascent of aqueous vapours into the atmosphere, this ingenious writer observes, that the lowest part of the air being pressed by the weight of the upper against the surface of the water, and continually rubbing upon it by its motion, attracts and dissolves those particles with which it is in contact, and separates them from the rest of the water. And since the cause of solution in this case is the stronger attraction of the particles of water towards the air, than towards each other, those that are already dissolved and taken up, will be still farther raised by the attraction of the dry air that lies over them, and thus will diffuse themselves, rising gradually higher and higher, and so leave the lowest air not so much saturated but that it will still be able to dissolve and take up fresh particles of water; which process is greatly promoted by <cb/> the motion of the wind. When the vapours are thus raised and carried by the winds into the higher and colder parts of the atmosphere, some of them will coalesce into small particles, which slightly attracting each other, and being intermixed with air, will form clouds; and these clouds will float at different heights, according to the quantity of vapour borne up, and the degree of heat in the upper parts of the atmosphere: and thus clouds are generally higher in summer than in winter. When the clouds are much increased by a continual addition of vapours, and their particles are driven close together by the force of the winds, they will run into drops heavy enough to fall down in rain. If the clouds be frozen before their particles are gathered into drops, small pieces of them being condensed and made heavier by the cold, fall down in thin flakes of snow. When the particles are formed into drops before they are frozen, they become hailstones. When the air is replete with vapours, and a cold breeze springs up, which checks the solution of them, clouds are formed in the lower parts of the atmosphere, and compose a mist or fog, which usually happens in a cold morning, and is dispersed when the sun has warmed the air, and made it capable of dissolving these watry particles. Southerly winds commonly bring rain, because, being warm and replete with aqueous vapours, they are cooled by coming into a colder climate; and therefore they part with some of them, and suffer them to precipitate in rain: whereas northerly winds, being cold, and acquiring additional heat by coming into a warmer climate, are ready to dissolve and receive more vapour than they before contained; and therefore, by long continuance, they are dry and parching, and commonly attended with fair weather.</p><p>Changes of the air, with respect to its density and rarity, as well as its heat and cold, will produce contrary effects in the solution of water, and the consequent ascent or fall of vapours. Several experiments prove that air, when raresied, cannot keep so much water dissolved as it does in a more condensed state; and therefore when the atmosphere is saturated with water, and changes from a denser to a rarer state, the high and colder parts of it will let go some of the water before dissolved, forming new clouds, and disposing them to fall down in rain: but a change from a rarer to a denser state will stop the precipitation of the water, and enable the air to dissolve, either in whole or in part, some of those clouds that were formed before, and render their particles less apt to run into drops and fall down in rain: on this account, we generally sind that the rarefied and condensed states of the atmosphere are respectively attended with rain or fair weather. See more on this subject in the Philos. Trans. vol. 55, pa. 146, or Hamilton's Philosophical Essays, p. 33.</p><p>Dr. Halley, before mentioned, has furnished some experiments on the Evaporation of water; the result of which is contained in the following articles: 1. That water salted to about the same degree as sea-water, and exposed to a heat equal to that of a summer's day, did, from a circular surface of about 8 inches diameter, evaporate at the rate of 6 ounces in 24 hours: whence by a calculus he finds that, in such circumstances, the<*>water evaporates 1-10th of an inch deep in 12 hours: <pb n="448"/><cb/> which quantity, he observes, will be found abundantly sofficient to furnish all the rains, springs, dews, &c. By this experiment, every 10 square inches of surface of the water yield in vapour <hi rend="italics">per diem</hi> a cubic inch of water: and each square foot half a wine pint; every space of 4 feet square, a gallon; a mile square, 6914 tuns; and a square degree, of 69 English miles, will evaporate 33 millions of tuns a day; and the whole Mediterranean, computed to contain 160 square degrees, at least 5280 millions of tuns each day. Philos. Trans. number 189, or Abridg. vol. 2, pa. 108.—2. A surface of 8 square inches, evaporated purely by the natural warmth of the weather, without either wind or sun, in the course of a whole year, 16292 grains of water, or 64 cubic inches; consequently, the depth of water thus evaporated in one year, amounts to 8 inches. But this being too little to answer the experiments of the French, who found that it rained 19 inches of water in one year at Paris; or those of Mr. Townley, who found the annual quantity of rain in Lancashire above 40 inches; he concludes, that the sun and wind contribute more to Evaporation than any internal heat or agitation of the water. In effect, Dr. Halley fixes the annual Evaporation of London at 48 inches; and Dr. Dobson states the same for Liverpool at 36 3/4 inches. Philos. Trans. vol. 67, p. 252.</p><p>3. The effect of the wind is very considerable, on a double account; for the same observations shew a very odd quality in the vapours of water, viz, that of adhering and hanging to the surface that exhaled them, which they clothe as it were with a fleece of vapourous air; which once investing the vapour, it afterwards rises in much less quantity. Whence, the quantity of water lost in 24 hours, when the air is very still, was very small, in proportion to what went off when there was a strong gale of wind abroad to dissipate the fleece, and make room for the emission of vapour; and this, even though the experiment was made in a place as close from the wind as could be contrived. Add, that this fleece of water, hanging to the surface of waters in still weather, is the occasion of very strange appearances, by the refraction of the vapours differing from and exceeding that of common air: whence every thing appears raised, as houses like steeples, ships as on land above the water, the land raised, and as it were lifted from the sea, &c.</p><p>4. The same experiments shew that the Evaporation in May, June, July, and August, which are nearly equal, are about three times as great as those in the months of November, December, January, and February. Philos. Trans. numb. 212, or Abr. vol. 2, pa. 110.</p><p>Dr. Brownrigg, in his Art of making common salt, pa. 189, fixes the Evaporation of some parts of England at 73.8 inches during the months of May, June, July, and August; and the Evaporation of the whole year at more than 140 inches. But the Evaporation of the four summer months at Liverpool, on a medium of 4 years, was found to be only 18.88 inches. Also Dr. Hales calculates the greatest annual Evaporation from the surface of the earth in England at 6.66 inches; and therefore the annual Evaporation from a surface of water, is to the annual Evaporation from the surface of the earth at Liverpool, nearly as 6 to 1. Philos. Trans. vol. 67, ubi supra. <cb/></p><p>In the Transactions of the American Philosophical Society, vol. 3, pa. 125, there is an ingenious paper on Evaporation, by Dr. Wistar. It is there shewn, that evaporation arises when the moist body is warmer than the medium it is inclosed in. And, on the contrary, it acquires moisture from the air, when the body is the colder. This carrying off, and acquiring of moisture, it is shewn, is by the passage of heat out of the body, or into it.</p></div1><div1 part="N" n="EUCLID" org="uniform" sample="complete" type="entry"><head>EUCLID</head><p>, of Megara, a celebrated philosopher and logician; he was a disciple of Socrates, and flourished about 400 years before Christ. The Athenians having prohibited the Megarians from entering their city on pain of death, this philosopher disguised himself in women's clothes to attend the lectures of Socrates. After the death of Socrates, Plato and other philosophers went to Euclid at Megara, to shelter themselves from the tyrants who governed Athens.—This philosopher admitted but one chief good; which he at different times called <hi rend="italics">God,</hi> or the <hi rend="italics">Spirit,</hi> or <hi rend="italics">Providence.</hi></p><div2 part="N" n="Euclid" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Euclid</hi></head><p>, the celebrated mathematician, according to the account of Pappus and Proclus, was born at Alexandria, in Egypt, where he flourished and taught mathematics, with great applause, under the reign of Ptolomy Lagos, about 280 years before Christ. And here, from his time, till the conquest of Alexandria by the Saracens, all the eminent mathematicians were either born, or studied; and it is to Euclid, and his scholars, we are beholden for Eratosthenes, Archimedes, Apollonius, Ptolomy, Theon, &c, &c. He reduced into regularity and order all the fundamental principles of pure mathematics, which had been delivered down by Thales, Pythagoras, Eudoxus, and other mathematicians before him, and added many others of his own discovering: on which account it is said he was the first who reduced arithmetic and geometry into the form of a science. He likewise applied himself to the study of mixed mathematics, particularly to astronomy and optics.</p><p>His works, as we learn from Pappus and Proclus, are the <hi rend="italics">Elements, Data, Introduction to Harmony, Phenomena, Optics, Catoprics, a Treatise of the Division of Supersicies, Porisms, Loci ad Supersiciem, Fallacies, and Four books of Conics.</hi> The most celebrated of these, is the sirst work, the Elements of Geometry; of which there have been numberless editions, in all languages; and a fine edition of all his works, now extant, was printed in 1703, by David Gregory, Savilian professor of astronomy at Oxford.</p><p>The Elements, as commonly published, consist of 15 books, of which the two last it is suspected are not Euclid's, but a comment of Hypsicles of Alexandria, who lived 200 years after Euclid. They are divided into three parts, viz, the Contemplation of Superficies, Numbers, and Solids: The first 4 books treat of planes only; the 5th of the proportions of magnitudes in general; the 6th of the proportion of plane figures; the 7th, 8th, and 9th give us the fundamental properties of numbers; the 10th contains the theory of commensurable and incommensurable lines and spaces; the 11th, 12th, 13th, 14th, and 15th, treat of the doctrine of solids.</p><p>There is no doubt but, before Euclid, Elements of Geometry were compiled by Hippocrates of Chius, <pb n="449"/><cb/> Eudoxus, Leon, and many others, mentioned by Proclus, in the beginning of his second book; for he affirms that Euclid new ordered many things in the Elements of Eudoxus, completed many things in those of Theatetus, and besides strengthened such propositions as before were too slightly, or but superficially established, with the most firm and convincing demonstrations.</p><p>History is silent as to the time of Euclid's death, or his age. He is represented as a person of a courteous and agreeable behaviour, and in great esteem and familiarity with king Ptolomy; who once asking him, whether there was any shorter way of coming at geometry than by his Elements, Euclid, as Proclus testifies, made answer, that there was no royal way or path to geometry.</p></div2></div1><div1 part="N" n="EUDIOMETER" org="uniform" sample="complete" type="entry"><head>EUDIOMETER</head><p>, an instrument for determining the purity of the air, or the quantity of pure and dephlogisticated or vital air contained in it, chiefly by means of its diminution on a mixture with nitrous air.</p><p>Instruments of this kind have been but lately made, and that in consequence of the experiments and discoveries of Dr. Priestley, for determining the salubrity of different kinds of air. That writer having discovered, that when nitrous air is mixed with any other air, their original bulk is diminished; and that the diminution is nearly, if not exactly, in proportion to its salubrity; he was hence led to adopt nitrous air as a true test of the purity of respirable air; and nothing more seemed to be necessary but an easy, expeditious, and accurate method of estimating the degree of diminution in different cases; and for this purpose, the Eudiometer was contrived; of which several kinds have been invented, the principal of which are the following.</p><p>I. The Eudiometer originally used by Dr. Priestley is a divided glass tube, into which, after having filled it with common water, and inverted it into the same, one measure or more of common air, and an equal quantity of nitrous air, are introduced by means of a small phial, which is called the measure; and thus the diminution of the volume of the mixture, which is seen at once by means of the graduations of the tube, instantly discovers the purity of the air required.</p><p>II. The discovery of Dr. Priestley was announced to the public in the year 1772; and several persons, both at home and abroad, presently availed themselves of it, by framing other more accurate instruments. The sirst of these was contrived by M. Landriani; an account of which is published in the 6th volume of Rosier's Journal, for the year 1775. It consists of a glass tube, fitted by grinding to a cylindrical vessel, to which are joined two glass cocks and a small bason; the whole being fitted to a wooden frame. In this instrument quicksilver is used instead of water; though that is attended with an inconvenience, because the nitrous air acts upon the metal, and renders the experiment ambiguous.</p><p>III. In 1777, Mr. Magellan published an account of three Eudiometers invented by himself, consisting of glass vessels of rather difficult construction, and trouble some use. Mr. Cavallo observes, that the construction of all the three is founded on a supposition, that the mixture of nitrous and atmospherical air, having continued for some time to diminish, afterwards increases <cb/> again; which it seems is a mistake: neither do they give accurate or uniform results in any two experiments made with nitrous and common air of precisely the same quality.</p><p>IV. A preferable method of discovering the purity of the air by means of an Eudiometer, is recommended by M. Fontana, of very great accuracy. The instrument is originally nothing more than a divided glass tube, though the inventor afterwards added to it a complicated apparatus, perhaps of little or no use. The first simple Eudiometer consisted only of a glass tube, uniformly cylindrical, about 18 inches long, and 3-4ths of an inch diameter within side, the outside being marked with a diamond at such distances as are exactly filled by equal measures of elastic fluids: and when any parts of these divisions are required, the edge of a ruler, divided into inches and smaller parts, is held against the tube, so as that the first division of the ruler may coincide with one of the marks on the tube. The nitrous and atmospherical air are introduced into this tube, in order to be diminished, and thence the purity of the atmospheric air ascertained.</p><p>V. M. Saussure of Geneva has also invented an Eudiometer, which he thinks is more exact than any of those before described; the apparatus of which is as follows: 1. A cylindrical glass bottle, with a ground stopple, containing about 5 1/2 ounces, which serves as a receiver for mixing the two airs.—2. A small glass phial, to serve as a measure, and is about one-third the size of the receiver.—3. A small pair of scales that may weigh very exactly.—4. Several glass bottles, for containing the nitrous or other air to be used, and which may supply the place of the recipient when broken. The method of using it is as follows: The receiver is to be filled with water, closed exactly with its glass stopper, wiped dry on the outside, and then weighed very nicely. Being then immerged in a vessel of water, and held with the mouth downwards, the stopple is removed, and, by means of a funnel, two measures of common and one of nitrous air are introduced into it, one after another: these diminish as soon as they come into contact; in consequence of which the water enters the recipient in proportionable quantity. After being stopped and well shaken, to promote the diminution, the receiver is to be opened again under water; then stopped and shaken again, and so on for three times successively, after which the bottle is stopped for the last time under water, then taken out, wiped very clean and dry, and exactly weighed as before. It is plain that now, the bottle being filled partly with elastic fluid and partly with water, it must be lighter than when quite full of water; and the difference between those two weights, shows nearly what quantity of water would fill the space occupied by the diminished elastic fluid. Now, in making experiments with airs of different degrees of purity, the said difference will be greater when the diminution is less, or when the air is less pure, and vice versa; by which means the comparative purity between two different kinds of air is determined.</p><p>VI. But as this method, notwithstanding the encomiums bestowed on it by the inventor, is subject to several errors and inconveniences; to remedy all these, another instrument was invented by Mr. Cavallo; the descrip- <pb n="450"/><cb/> tion of which, being long, may be seen in his Treatise on the Nature and Properties of Air, pa. 344.</p><p>Other constructions of the Eudiometer have also been given by Mr. Cavendish and Mr. Scheele. For farther information, see Magellan's Letter to Dr. Priestley, containing the Description of a Glass Apparatus, &c, and of New Eudiometers &c, 1777, pa. 15 &c; Priestley's Exp. and Obs. on Air, vol. 3, preface and appendix; the methods of Dr. Ingenhousz in Philos. Trans. vol. 66, art. 15; see also the Philos. Trans. vol. 73; and Cavallo's Treatise on Air, pa. 274, 315, 316, 317, 328, 340, 344, and 834.</p></div1><div1 part="N" n="EUDOXUS" org="uniform" sample="complete" type="entry"><head>EUDOXUS</head><p>, of Cnidus, a city of Caria in Asia Minor, slourished about 370 years before Christ. He learned geometry from Archytas, and afterwards travelled into Egypt to learn astronomy and other sciences. There he and Plato studied together, as Laertius informs us, for the space of 13 years; and afterwards came to Athens, fraught with all sorts of knowledge, which they had imbibed from the mouths of the priests. Here Eudoxus opened a school; which he supported with so much glory and renown, that even Plato, though his friend, is said to have envied him. Eudoxus composed Elements of Geometry, from whence Euclid liberally borrowed, as mentioned by Proclus. Cicero calls Eudoxus the greatest astronomer that had ever lived: and Petronius says, he spent the latter part of his life upon the top of a very high mountain, that he might contemplate the stars and the heavens with more convenience and less interruption: and we learn from Strabo, that there were some remains of his observatory at Cnidus, to be seen even in his time. He died in the 53d year of his age.</p></div1><div1 part="N" n="EVECTION" org="uniform" sample="complete" type="entry"><head>EVECTION</head><p>, is used by some astronomers for the Libration of the moon; being an inequality in her motion, by which, at or near the quadratures, she is not in a line drawn through the centre of the earth to the sun, as she is at the syzygies, or conjunction and opposition, but makes an angle with that line of about 2° 51′. The motion of the moon about her axis only is equable, which rotation is performed exactly in the same time as she revolves about the earth; for which reason it is that she turns always the same face towards the earth nearly, and would do so exactly were it not that her menstrual motion about the earth, in an elliptic orbit, is not equable; on which account the moon, seen from the earth, appears to librate a little upon her axis, sometimes from east to west, and sometimes from west to east; or some parts in the eastern limb of the moon go backwards and forwards a small space, and some that were conspicuous, are hid, and then appear again.</p><p>The term <hi rend="smallcaps">Evection</hi> is used by some astronomers to denote that equation of the moon's motion, which is proportional to the sine of double the distance of the moon from the sun, diminished by the moon's anomaly: this equation is not yet accurately determined; some state it at 1° 30′, others at 1° 16′, &c. It is the greatest of all the moon's equations, except the Equation of the Centre.</p><p>EVEN <hi rend="italics">Number,</hi> is that which can be divided into two equal whole numbers; such as the series of alternate numbers 2, 4, 6, 8, 10, &c.</p><p>EVENLY <hi rend="italics">Even Number,</hi> is that which an even num- <cb/> ber measures by an even number; as 16, which the even number 8 measures by the even number 2.</p><p><hi rend="smallcaps">Evenly</hi> <hi rend="italics">Odd Number,</hi> is that which an even number measures by an odd one; as 30, which the even number 6 measures by the odd number 5.</p><p>EVERARD's <hi rend="italics">Sliding Rule,</hi> a particular sort of one invented by Mr. Thomas Everard, for the purpose of gauging. See <hi rend="smallcaps">Sliding Rule.</hi></p></div1><div1 part="N" n="EULER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">EULER</surname> (<foreName full="yes"><hi rend="smallcaps">Leonard</hi></foreName>)</persName></head><p>, one of the most extraordinary, and even prodigious, mathematical geniuses, that the world ever produced. He was a native of Basil, and was born April 15, 1707. The years of his infancy were passed at Richen, where his father was minister. He was afterwards sent to the university of Basil; and as his memory was prodigious, and his application regular, he performed his academical tasks with great rapidity; and all the time that he saved by this, was consecrated to the study of mathematics, which soon became his favourite science. The early progress he made in this study, added fresh ardour to his application; by which too he obtained a distinguished place in the attention and esteem of professor John Bernoulli, who was then one of the chief mathematicians in Europe.</p><p>In 1723, M. Euler took his degree as master of arts; and delivered on that occasion a Latin discourse, in which he drew a comparison between the philosophy of Newton and the Cartesian system, which was received with the greatest applause. At his father's desire, he next applied himself to the study of theology and the oriental languages: and though these studies were foreign to his predominant propensity, his success was considerable even in this line: however, with his father's consent, he afterward returned to mathematics as his principal object. In continuing to avail himself of the counsels and instructions of M. Bernoulli, he contracted an intimate friendship with his two sons Nicholas and Daniel; and it was chiefly in consequence of these connections that he afterwards became the principal ornament of the philosophical world.</p><p>The project of erecting an academy at Petersburg, which had been formed by Peter the Great, was executed by Catharine the 1st; and the two young Bernoullis being invited to Petersburg in 1725, promised Euler, who was desirous of following them, that they would use their endeavours to procure for him an advantageous settlement in that city. In the mean time, by their advice, he made close application to the study of philosophy, to which he made happy applications of his mathematical knowledge, in a dissertation on the nature and propagation of sound, and an answer to a prize question concerning the masting of ships; to which the Academy of Sciences adjudged the <hi rend="italics">accessit,</hi> or second rank, in the year 1727. From this latter discourse, and other circumstances, it appears that Euler had very early embarked in the curious and useful study of naval architecture, which he afterward enriched with so many valuable discoveries. The study of mathematics and philosophy however did not solely engage his attention, as he in the mean time attended the medical and botanical lectures of the professors at Basil.</p><p>Euler's merit would have given him an easy admission to honourable preferment either in the magistracy or university of his native city, if both civil and acade- <pb n="451"/><cb/> mical honours had not been there distributed by lot. The lot being against him in a certain promotion, he left his country, set out for Petersburg, and was made joint professor with his countrymen Hermann and Daniel Bernoulli in the university of that city.</p><p>At his first setting out in his new career, he enriched the academical collection with many memoirs, which excited a noble emulation between him and the Bernoullis; an emulation that always continued, without either degenerating into a selfish jealousy, or producing the least alteration in their friendship. It was at this time that he carried to new degrees of perfection the integral calculus, invented the calculation by sines, reduced analytical operations to a greater simplicity, and thus was enabled to throw new light on all the parts of mathematical science.</p><p>In 1730, M. Euler was promoted to the professorship of natural philosophy; and in 1733 he succeeded his friend D. Bernoulli in the mathematical chair. In 1735, a problem was proposed by the academy, which required expedition, and for the calculation of which some eminent mathematicians had demanded the space of some months. The problem was undertaken by Euler, who completed the calculation in three days, to the great astonishment of the academy: but the violent and laborious efforts it cost him threw him into a fever, which endangered his life, and deprived him of the use of his right eye, which afterward brought on a total blindness.</p><p>The Academy of Sciences at Paris, which in 1738 had adjudged the prize to his memoir Concerning the Nature and Properties of Fire, proposed for the year 1740 the important subject of the Tides of the Sea; a problem whose solution comprehended the theory of the solar system, and required the most arduous calculations. Euler's solution of this question was adjudged a master-piece of analysis and geometry; and it was more honourable for him to share the academical prize with such illustrious competitors as Colin Maclaurin and Daniel Bernoulli, than to have carried it away from rivals of less magnitude. Seldom, if ever, did such a brilliant competition adorn the annals of the academy; and perhaps no subject, proposed by that learned body, was ever treated with such force of genius and accuracy of investigation, as that which here displayed the philosophical powers of this extraordinary triumvirate.</p><p>In the year 1741, M. Euler was invited to Berlin to direct and assist the academy that was there rising into fame. On this occasion he enriched the last volume of the Miscellanies <hi rend="italics">(Melanges)</hi> of Berlin with five memoirs, which form an eminent, perhaps the principal, figure in that collection. These were followed, with amazing rapidity, by a great number of important researches, which are dispersed through the memoirs of the Prussian academy; a volume of which has been regularly published every year since its establishment in 1744. The labours of Euler will appear more especially astonishing, when it is considered, that while he was enriching the academy of Berlin with a prosusion of memoirs, on the deepest parts of mathematical science, containing always some new points of view, often sublime truths, and sometimes discoveries of great importance; he still continued his philosophical contributions to the Petersburg academy, whose memoirs display the <cb/> marvellous fecundity of his genius, and which granted him a pension in 1742.</p><p>It was with great difficulty that this extraordinary man, in 1766, obtained permission from the king of Prussia to return to Petersburg, where he wished to pass the remainder of his days. Soon after his return, which was graciously rewarded by the munificence of Catharine the 2d, he was seized with a violent disorder, which ended in the total loss of his sight. A cataract, formed in his left eye, which had been essentially damaged by the loss of the other eye, and a too close application to study, deprived him entirely of the use of that organ. It was in this distressing situation that he dictated to his servant, a taylor's apprentice, who was absolutely devoid of mathematical knowledge, his Elements of Algebra; which by their intrinsic merit in point of perspicuity and method, and the unhappy circumstances in which they were composed, have equally excited wonder and applause. This work, though purely elementary, plainly discovers the proofs of an inventive genius; and it is perhaps here alone that we meet with a complete theory of the analysis of Diophantus.</p><p>About this time M. Euler was honoured by the Academy of Sciences at Paris with the place of one of the foreign members of that learned body; after which, the academical prize was adjudged to three of his memoirs, <hi rend="italics">Concerning the Inequalities in the Motions of the Planets.</hi> The two prize questions proposed by the same Academy sor 1770 and 1772 were designed to obtain from the abours of astronomers <hi rend="italics">a more perfect Theory of the Moon.</hi> M. Euler, assisted by his eldest son, was a competitor for these prizes, and obtained them both. In this last memoir, he reserved for farther consideration several inequalities of the moon's motion, which he could not determine in his first theory, on account of the complicated calculations in which the method he then employed had engaged him. He afterward revised his whole theory, with the assistance of his son and Messrs Krafft and Lexell, and pursued his researches till he had constructed the new tables, which appeared, together with the great work, in 1772. Instead of confining himself, as before, to the fruitless integration of three differential equations of the second degree, which are furnished by mathematical principles, he reduced them to the three ordinates, which determine the place of the moon: he divided into classes all the inequalities of that planet, as far as they depend either on the elongation of the sun and moon, or upon the eccentricity, or the parallax, or the inclination of the lunar orbit. All these means of investigation, employed with such art and dexterity as could only be expected from a genius of the first order, were attended with the greatest success; and it is impossible to observe without admiration, such immense calculations on the one hand, and on the other the ingenious methods employed by this great man to abridge them, and to facilitate their application to the real motion of the moon. But this admiration will become astonishment, when we consider at what period and in what circumstances all this was effectuated. It was when our author was totally blind, and consequently obliged to arrange all his computations by the sole powers of his memory and his genius: it was when he was embarrassed in his domestic affairs by a dreadful sire, <pb n="452"/><cb/> that had consumed great part of his substance, and forced him to quit a ruined house, every corner of which was known to him by habit, which in some measure supplied the want of sight. It was in these circumstances that Euler composed a work which alone was sufficient to render his name immortal.</p><p>Some time after this, the famous oculist Wentzell, by couching the cataract, restored sight to our author; but the joy produced by this operation was of short duration. Some instances of negligence on the part of his surgeons, and his own impatience to use an organ, whose cure was not completely sinished, deprived him a second time and for ever of his sight: a relapse which was also accompanied with tormenting pain. With the assistance of his sons, however, and of Messrs Krafft and Lexell, he continued his labours: neither the infirmities of old age, nor the loss of his sight, could quell the ardour of his genius. He had engaged to furnish the academy of Petersburg with as many memoirs as would be sufficient to complete its acts for 20 years after his death. In the space of 7 years he transmitted to the Academy above 70 memoirs, and above 200 more, left behind him, were revised and completed by a friend. Such of these memoirs as were of ancient date were separated from the rest, and form a collection that was published in the year 1783, under the title of <hi rend="italics">Analytical Works.</hi></p><p>The general knowledge of our author was more extensive than could well be expected in one who had pursued, with such unremitting ardour, mathematics and astronomy as his favourite studies. He had made a very considerable progress in medical, botanical, and chemical science. What was still more extraordinary, he was an excellent scholar, and possessed in a high degree what is generally called <hi rend="italics">erudition.</hi> He had attentively read the most eminent writers of ancient Rome; the civil and literary history of all ages and all nations was familiar to him; and foreigners, who were only acquainted with his works, were astonished to find in the conversation of a man, whose long life seemed solely occupied in mathematical and physical researches and discoveries, such an extensive acquaintance with the most interesting branches of literature. In this respect, no doubt, he was much indebted to a very uncommon memory, which seemed to retain every idea that was conveyed to it, either from reading or from meditation. He could repeat the Æneid of Virgil, from the beginning to the end, without hesitation, and indicate the first and last line of every page of the edition he used.</p><p>Several attacks of a vertigo, in the beginning of September 1783, which did not prevent his computing the motions of the aerostatic globes, were however the forerunners of his mild passage out of this life. While he was amusing himself at tea with one of his grandchildren, he was struck with an apoplexy, which terminated his illustrious career at 76 years of age.</p><p>M. Euler's constitution was uncommonly strong and vigorous. His health was good; and the evening of his long life was calm and serene, sweetened by the fame that follows genius, the public esteem and respect that are never withheld from exemplary virtue, and several domestic comforts which he was capable of feeling, and therefore deserved to enjoy. <cb/></p><p>The catalogue of his works has been printed in 50 pages, 14 of which contain the manuscript works.— The printed ones consist of works published separately, and works to be found in the memoirs of several Academies, viz, in 38 volumes of the Petersburg Acts, (from 6 to 10 papers in each volume);—in several volumes of the Paris Acts;—in 26 volumes of the Berlin Acts, (about 5 papers to each volume);—in the Acta Eruditorum, in 2 volumes;—in the Miscellanea Taurinensia;—in vol. 9 of the Society of Ulyssingue; —in the Ephemerides of Berlin;—and in the Memoires de la Société Oeconomique for 1766.</p></div1><div1 part="N" n="EVOLVENT" org="uniform" sample="complete" type="entry"><head>EVOLVENT</head><p>, in the Higher Geometry, a term used by some writers for the Involute, or curve resulting from the evolution of a curve, in contradistinction to that evolute, or curve supposed to be opened or evolved. See <hi rend="smallcaps">Evolute</hi>, and <hi rend="smallcaps">Involute.</hi></p></div1><div1 part="N" n="EVOLUTE" org="uniform" sample="complete" type="entry"><head>EVOLUTE</head><p>, in the Higher Geometry, a curve first proposed by M. Huygens, and since much studied by the later mathematicians. It is any curve supposed to be evolved or opened, by having a thread wrapped close upon it, fastened at one end, and beginning to evolve or unwind the thread from the other end, keeping the part evolved, or wound off, tight stretched; then this end of the thread will describe another curve called the Involute. Or the same involute is described the contrary way, by wrapping the thread upon the Evolute, keeping it always stretched.</p><p>Thus, if EFGH be any curve, and AE either a part of the curve, or a right line; then if a thread be wound close upon the curve from A to H, where it is fixed, and then be unwound from A; the curve AEFGH, from which it is evolved, is called the Evolute; and the other curve ABCD described by the end of the thread, as it evolves or unwinds, is the Involute. Or, if the thread HD, fixed at H, be wound or wrapped upon the Evolute HGFEA, keeping it always tight, as at the several positions of it HD, GC, FB, EA, the extremity will describe the Involute curve DCBA. <figure/></p><p>From this description it appears, 1. That the parts of the thread at any positions, as EA, FB, GC, HD, &c, are radii of curvature, or osculatory radii, of the involute curve, at the points A, B, C, D.</p><p>2. The same parts of the thread are also equal to the corresponding lengths AE, AEF, AEFG, &c, of the Evolute; that is, <table><row role="data"><cell cols="1" rows="1" role="data">AE = AE</cell><cell cols="1" rows="1" role="data">is the rad. of curvature to the point</cell><cell cols="1" rows="1" role="data">A,</cell></row><row role="data"><cell cols="1" rows="1" role="data">BF = AF</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">B,</cell></row><row role="data"><cell cols="1" rows="1" role="data">CG = AG</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">C,</cell></row><row role="data"><cell cols="1" rows="1" role="data">DH = AH</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">D.</cell></row></table> <pb n="453"/><cb/></p><p>3. Any radius of curvature BF, is perpendicular to the involute at the point B, and is a tangent to the Evolute curve at the point F.</p><p>4. The Evolute is the locus of the centre of curvature of the involute curve.</p><p>The finding the radii of Evolutes, is a matter of great importance in the higher speculations of geometry; and is even sometimes useful in practice; as is shewn by Huygens, the inventor of this theory, in applying it to the pendulum. Horol. Oscil. part 3. The doctrine of the Oscula of Evolutes is owing to M. Leibnitz, who sirst shewed the use of Evolutes in the measuring of curvatures.</p><p><hi rend="italics">To find the</hi> <hi rend="smallcaps">Evolute</hi> <hi rend="italics">and Involute Curves,</hi> the one from the other.</p><p>For this purpose, put <hi rend="italics">x</hi> = AD the absciss of the involute, <hi rend="italics">y</hi> = DB its ordinate, <hi rend="italics">z</hi> = AB the involute curve, <hi rend="italics">r</hi> = BC its radius of curvature, <hi rend="italics">v</hi> = EF the absciss of the Evolute, <hi rend="italics">u</hi> = FC its ordinate, and <hi rend="italics">a</hi> = AE a given line, (fig. 2 above). Then, by the nature of the radius of curvature, it is ; also by sim. triangles, , . Hence ; and ; which are the values of the absciss and ordinate of the Evolute curve EC; and therefore these may be found when the involute is given.</p><p>On the other hand, if <hi rend="italics">v</hi> and <hi rend="italics">u,</hi> or the Evolute be given: then, putting the given curve EC = <hi rend="italics">s</hi>; since , or , this gives <hi rend="italics">r</hi> the radius of curvature. Also, by similar triangles, there result these proportions, viz, , ; theref. , and ; which are the absciss and ordinate of the involute curve, and which may therefore be found when the Evolute is given. Where it may be noted that , and . Also either of the quantities <hi rend="italics">x, y,</hi> may be supposed to flow equably, in which case the respective second fluxion <hi rend="italics">x<hi rend="sup">..</hi></hi> or <hi rend="italics">y<hi rend="sup">..</hi></hi> will be nothing, and the corresponding term in the denominator <hi rend="italics">y<hi rend="sup">.</hi>x<hi rend="sup">..</hi></hi>-<hi rend="italics">x<hi rend="sup">.</hi>y<hi rend="sup">..</hi></hi> will vanish, leaving only the other term in it; which <cb/> will have the effect of rendering the whole operation simpler.</p><p><hi rend="italics">For Ex.</hi> Suppose it were required to find the Evolution EC when the given involute AB is the common parabola, whose equation is <hi rend="italics">px</hi>=<hi rend="italics">y</hi><hi rend="sup">2</hi>, the parameter being <hi rend="italics">p.</hi></p><p>Here , making <hi rend="italics">x<hi rend="sup">..</hi></hi>=o. Then, to find first AE the radius of curvature of the parabola AB at the vertex, when <hi rend="italics">x<hi rend="sup">..</hi></hi>=o, the general value of the radius of curvature above given becomes =(by substituting the value of <hi rend="italics">y<hi rend="sup">.</hi></hi> and <hi rend="italics">y<hi rend="sup">..</hi></hi> &c) (―(<hi rend="italics">p</hi>+4<hi rend="italics">x</hi>)<hi rend="sup">3/2</hi>)/(2√<hi rend="italics">p</hi>) which is the general value of <hi rend="italics">r</hi> or BC, the radius of curvature, for any value of <hi rend="italics">x</hi> or AD; and when <hi rend="italics">x</hi> or AD is = 0 or nothing, the value of <hi rend="italics">r,</hi> or AE, becomes then only; that is half the parameter of the axis is the radius of curvature at the vertex of the parabola.</p><p>Again, in the general values of <hi rend="italics">v</hi> and <hi rend="italics">u</hi> above given, by substituting the values of <hi rend="italics">y<hi rend="sup">.</hi>, y<hi rend="sup">..</hi>,</hi> and <hi rend="italics">z<hi rend="sup">.</hi>,</hi> also o for <hi rend="italics">x<hi rend="sup">..</hi>,</hi> and (1/2)<hi rend="italics">p</hi> for <hi rend="italics">a</hi>; those quantities become ; and . Hence then, comparing the values of <hi rend="italics">v</hi> and <hi rend="italics">u,</hi> there is found 3<hi rend="italics">p</hi><hi rend="sup">1/2</hi><hi rend="italics">v</hi>=4<hi rend="italics">x</hi><hi rend="sup">1/2</hi><hi rend="italics">u,</hi> and 27<hi rend="italics">pv</hi><hi rend="sup">2</hi>=16<hi rend="italics">u</hi><hi rend="sup">3</hi>; which is the equation between the absciss and ordinate of the Evolute curve EC, shewing it to be the semicubical parabola.</p><p>In like manner the Evolute to any other curve is found.—The Evolute to the common cycloid, is an equal cycloid; a property first demonstrated by Huygens, and which he used as a contrivance to make a pendulum vibrate in the curve of a cycloid. See his Horolog. Oscil. See also, on the subject of Evolute and Involute Curves, the Fluxions of Newton, Maclaurin, Simpson, De l'Hôpital, &c, Wolf. Elem. Math. tom. 1, &c, &c.</p><p>M. Varignon has applied the doctrine of the radius of the Evolute, to that of central forces; so that having the radius of the Evolute of any curve, there may be found the value of the central force of a body; which, moving in that curve, is found in the same point where that radius terminates; or reciprocally, having the central force given, the radius of the Evolute may be determined. Hist. de l'Acad. an. 1706.</p><p>The variation of curvature of the line described by the Evolution of a curve, is measured by the ratio of the radius of curvature of the Evolute, to the radius of curvature of the line described by the Evolution. See Maclaurin's Flux. art. 402, prop. 36.</p><p><hi rend="italics">Imperfect</hi> <hi rend="smallcaps">Evolute</hi>, a name given by M. Reaumur to a new kind of Evolute. The mathematicians had hitherto only considered the perpendiculars let sall from the Involute on the convex side of the Evolute: but if <pb n="454"/><cb/> other lines not perpendicular be drawn upon the same points, provided they be all drawn under the same angle, the effect will still be the same; that is, the oblique lines will all intersect in the curve, and by their intersections form the infinitely small sides of a new curve, to which they would be so many tangents.— Such a curve is a kind of Evolute, and has its radii; but it is an Imperfect one, since the radii are not perpendicular to the sirst curve, or Involute, Hist. de l'Acad. &c, an. 1709.</p></div1><div1 part="N" n="EVOLUTION" org="uniform" sample="complete" type="entry"><head>EVOLUTION</head><p>, in Arithmetic and Algebra, denotes the Extraction of the roots out of powers. In which sense it stands opposed to Involution, which is the raising of powers. The note or character that has been used by some Algebraists, to denote Evolution, is <figure/>; as the sign of involution is <figure/>: characters I think sirst used by Dr. Pell.</p><div2 part="N" n="Evolution" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Evolution</hi></head><p>, in Geometry, the opening, or unfolding of a curve, and making it describe an Evolvent.</p><p>The equable Evolution of the periphery of a circle, or other curve, is such a gradual approach of the circumference to rectitude, as that its parts do all concur, and equally evolve or unbend; so that the same line becomes successively a less arc of a reciprocally greater circle; till at last they change into a straight line.— In the Philos. Trans. N° 260, a new quadratix to the circle is found by this means, being the curve described by the equable Evolution of its periphery.</p></div2></div1><div1 part="N" n="EURYTHMY" org="uniform" sample="complete" type="entry"><head>EURYTHMY</head><p>, in Architecture, Painting, and Sculpture, is a kind of majesty, elegance, and easiness appearing in the composition of certain members or parts of a body, building, or painting, and resulting from the sine and exact proportions of them.</p></div1><div1 part="N" n="EUSTYLE" org="uniform" sample="complete" type="entry"><head>EUSTYLE</head><p>, is the best manner of placing columns, with regard to their distance; which, according to Vitruvius, should be four modules, or two diameters and a quarter.</p><p>EXAGON. See <hi rend="smallcaps">Hexagon.</hi></p></div1><div1 part="N" n="EXALTATION" org="uniform" sample="complete" type="entry"><head>EXALTATION</head><p>, in Astrology, is a dignity which a planet acquires in certain signs of the zodiac; which dignity, it is supposed, gives the planet an extraordinary virtue, efficacy, and influence. The opposite side of the zodiac is called the Dejection of the planet.—— Thus, the 15th degree of Cancer is the Exaltation of Jupiter, according to Albumazar, because it was the ascendant of that planet at the time of the creation; that of the sun is in the 19th degree of Aries; and its dejection in Libra; that of the moon is in Taurus, &c. Ptolomy gives the reason of this in his first book De Quadripartita.</p></div1><div1 part="N" n="EXCENTRIC" org="uniform" sample="complete" type="entry"><head>EXCENTRIC</head><p>, is applied to such figures, circles, spheres, &c, as have not the same centre; as opposed to Concentric, which have the same centre.</p><div2 part="N" n="Excentric" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Excentric</hi></head><p>, or <hi rend="italics">Excentric Circle,</hi> in the ancient Ptolomaic astronomy, was the very orbit of the planet itself, which it was supposed to describe about the earth, and which was conceived Excentric with it; called also the Deferent.</p><p>Instead of these Excentric Circles round the earth, the moderns make the planets describe elliptic orbits about the sun; which accounts for all the irregularities of their motions, and their various distances from the earth, &c, more justly and naturally. <cb/></p></div2><div2 part="N" n="Excentric" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Excentric</hi></head><p>, or <hi rend="italics">Excentric Circle,</hi> in the New Astronomy, is the circle described from the centre of the orbit of a planet, with half the greatest axis as a radius; or it is the circle that circumscribes the elliptic orbit of the planet; as the circle AQB.</p><p><hi rend="smallcaps">Excentric</hi> <hi rend="italics">Anomaly,</hi> or <hi rend="italics">A- <figure/> nomaly of the Centre,</hi> is an are AQ of the Excentric circle, intercepted between the aphelion A, and the right line QH, drawn through the centre P of the planet perpendicular to the line of the apses AB.</p><p><hi rend="smallcaps">Excentric</hi> <hi rend="italics">Equation,</hi> in the Old Astronomy, is an angle made by a line drawn from the centre of the earth, with another line drawn from the centre of the Excentric, to the body or place of any planet. This is the same with the prosthapheresis; and is equal to the difference, accounted in an arch of the ecliptic, between the real and apparent place of the sun or planet. See <hi rend="smallcaps">Equation</hi> <hi rend="italics">of the Centre.</hi></p><p><hi rend="smallcaps">Excentric</hi> <hi rend="italics">Place of a planet,</hi> in its orbit, is the Heliocentric place, or that in which it appears as seen from the sun.</p><p><hi rend="smallcaps">Excentric</hi> <hi rend="italics">Place in the ecliptic,</hi> is the point of the ecliptic to which the planet is referred as viewed from the sun; and which coincides with the heliocentric longitude.</p></div2></div1><div1 part="N" n="EXCENTRICITY" org="uniform" sample="complete" type="entry"><head>EXCENTRICITY</head><p>, is the distance between the centres of two circles, or spheres, which have not the same centre.</p><div2 part="N" n="Excentricity" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Excentricity</hi></head><p>, <hi rend="italics">in the Old Astronomy,</hi> is the distance between the centre of a planet and the centre of the earth.—That the planets have such an Excentricity, is allowed on all sides, and may be evinced from various circumstances; and especially this, that the planets at some times appear larger, and at others less; which can only proceed from hence, that their orbits being Excentric to the earth, in some parts of those orbits the planets are nearer to us, and in others more remote. And as to the Excentricities of the sun and moon, it is thought they are sufficiently proved, both from eclipses, from the moon's greater and less parallax at the same distance from the zenith, and from the sun's continuing longer by 8 days in the northern hemisphere than in the southern one.</p></div2><div2 part="N" n="Excentricity" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Excentricity</hi></head><p>, <hi rend="italics">in the New Astronomy,</hi> is the distance CS between the sun S and the centre C of a planet's orbit; or the distance of the centre from the focus of the elliptic orbit; called also the <hi rend="italics">Simple</hi> or <hi rend="italics">Single Excentricity.</hi></p><p>When the greatest Equation of the centre is given, the Excentricity of the earth's orbit may be found by the following proportion; viz, As the diameter of a circle in degrees, Is to the diameter in equal parts; So the greatest equat. of the centre in degrees, To the Excentricity in equal parts. Thus, Greatest equat. of the cent. 1°55′33″=1°.9258333 &c. The diam. of a circ. being 1, its circumf. is 3.1415926. Then 3.1415926 : 1 :: 360° : 114°.5915609 diam. in deg. And 114.5915609 : 1 :: 1.9258333 : 0.016806, the Ex <pb n="455"/><cb/> Hence, by adding this to 1, and subtracting it from 1, gives 1.016806 = AS the aphelion distance, and 0.983194 = BS the perihelion distance. See Robertson's Elem. of Navig. book 5, pa. 286.</p><p><hi rend="italics">Otherwise,</hi> thus: Since it is found that the sun's greatest apparent semi-diameter is to his least, as 32′ 43″ to 31′ 38″, or as 1963″ to 1898″; the sun's greatest distance from the earth will be to his least, or AS to SB, as 1963 to 1898; of which, the half dif. is 32 1/2 = CS, and half sum 1930 1/2 = CB; wherefore, as 1930 1/2 : 32 1/2 :: 1 : .016835 = CS the Excentricity, to the mean distance or semi-axis 1; which is nearly the same as before.</p><p>The Excentricities of the orbits of the several planets, in parts of their own mean distances 1000, and also in English miles, are as below, viz, the Excentricity of the orbit of <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">Parts.</cell><cell cols="1" rows="1" rend="align=center" role="data">Miles.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" rend="align=right" role="data">210</cell><cell cols="1" rows="1" rend="align=right" role="data">7,730,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">482,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Earth</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">1,618,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" rend="align=right" role="data">93</cell><cell cols="1" rows="1" rend="align=right" role="data">13,486,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" rend="align=right" role="data">23,760,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">49,940,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Georgian</cell><cell cols="1" rows="1" rend="align=right" role="data">47 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">86,000,000</cell></row></table></p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Excentricity</hi>, is the distance between the two foci of the elliptic orbit, and is equal to double the Single Excentricity above given.</p></div2></div1><div1 part="N" n="EXCHANGE" org="uniform" sample="complete" type="entry"><head>EXCHANGE</head><p>, in Arithmetic, is the bartering or exchanging the money of one place for that of another; or the finding what quantity of the money of one place is equal to a given sum of another, according to a given course of exchange.</p><p>The several operations in this case are only different applications of the Rule of Three. See most books of Arithmetic.</p><p><hi rend="italics">Arbitration of</hi> <hi rend="smallcaps">Exchange</hi>, is the method of remitting to, and drawing upon, foreign places, in such a manner as shall turn out the most profitable.</p><p><hi rend="italics">Arbitration</hi> is either Simple or Compound.</p><p><hi rend="italics">Simple Arbitration</hi> respects three places only. Here, by comparing the par of arbitration between a first and second place, and between the 1st and a 3d, the rate between the 2d and 3d is discovered; from whence a person can judge how to remit or draw to the most advantage, and to determine what that advantage is.</p><p><hi rend="italics">Compound Arbitration</hi> respects the cases in which the exchanges among three, four, or more places are concerned. A person who knows at what rate he can draw or remit directly, and also has advice of the course of exchange in foreign parts, may trace out a path for circulating his money, through more or fewer of such places, and also in such order, as to make a benefit of his skill and credit: and in this lies the great art of such negociations. See my Arithmetic, pa. 105, &c.</p></div1><div1 part="N" n="EXCURSION" org="uniform" sample="complete" type="entry"><head>EXCURSION</head><p>, in Astronomy. See E<hi rend="smallcaps">LONGATION.</hi></p><p><hi rend="italics">Circles of</hi> <hi rend="smallcaps">Excursion.</hi> See <hi rend="smallcaps">Circles.</hi></p></div1><div1 part="N" n="EXEGESIS" org="uniform" sample="complete" type="entry"><head>EXEGESIS</head><p>, or <hi rend="smallcaps">Exegetica</hi>, in Algebra, is the finding, either in numbers or lines, the roots of the equation of a problem, according as the problem is either numeral or geometrical. <cb/></p></div1><div1 part="N" n="EXHALATION" org="uniform" sample="complete" type="entry"><head>EXHALATION</head><p>, a fume or steam Exhaling, or issuing, from a body, and diffusing itself in the atmosphere.</p><p>The terms Exhalation and Vapour are often used indifferently; but the more accurate writers distinguish them, appropriating the term Vapour to the moist fumes raised from water and other liquid bodies; and the term Exhalation to the dry ones emitted from solid bodies; as earth, sire, minerals, &c. In this sense, Exhalations are dry and subtle corpuscles, or effluvia, loosened from hard terrestrial bodies, either by the heat of the sun, or the action of the air, or some other cause: being emitted upwards to a certain height in the atmosphere, where, mixing with the vapours, they help to constitute clouds, and return back in dews, mists, rains, &c.</p><p>Sir Isaac Newton thinks, that true and permanent air is formed from the Exhalations raised from the hardest and most compact bodies.</p><p>EXHAUSTED <hi rend="italics">Receiver,</hi> is a glass, or other vessel, applied on the plate of an air-pump, to have the air extracted out of it by the working of the pump.— Things placed in such an Exhausted Receiver, are said to be <hi rend="italics">in vacuo.</hi></p></div1><div1 part="N" n="EXHAUSTIONS" org="uniform" sample="complete" type="entry"><head>EXHAUSTIONS</head><p>, or the <hi rend="italics">Method of</hi> E<hi rend="smallcaps">XHAUSTIONS</hi>, a method of demonstration founded upon a kind of Exhausting a quantity by continually taking away certain parts of it.</p><p>The method of Exhaustions was of frequent use among the ancient mathematicians; as Euclid, Archimedes, &c. It is founded on what Euclid says in the 10th book of his Elements; viz, that those quantities are equal, whose difference is less than any assignable quantity. Or thus, two quantities A and B are equal, when, if to or from one of them as A, any other quantity as <hi rend="italics">d</hi> be subtracted, however small it be, then the sum or difference is respectively greater or less than the other quantity B: viz, <hi rend="italics">d</hi> being an indefinitely small quantity, if A + <hi rend="italics">d</hi> be greater than B, and A - <hi rend="italics">d</hi> less than B, then is A equal to B.</p><p>This principle is used in the 1st prop. of the 10th book, which imports, that if from the greater of two quantities be taken more than its half, and from the remainder more than its half, and so on; there will at length remain a quantity less than either of those proposed. On this foundation it is demonstrated, that if a regular polygon of infinite sides be inscribed in a circle, or circumscribed about it; then the space, which is the difference between the circle and the polygon, will by degrees be quite exhausted, and the circle become ultimately equal to the polygon. And in this way it is that Archimedes demonstrates, that a circle is equal to a right-angled triangle, whose two sides about the right angle, are equal, the one to the semidiameter, and the other to the perimeter of the circle. Prop. 1 De Dimensione Circuli.</p><p>Upon the Method of Exhaustions depends the Method of Indivisibles introduced by Cavalerius, which is but a shorter way of expressing the method of Exhaustions; as also Wallis's Arithmetic of Insinites, which is a farther improvement of the Method of Indivisibles; and hence also the Methods of Increments, Differentials, Fluxions, <pb n="456"/><cb/> and Infinite Series. See some account of the Method of Exhaustions in Wallis's Algebra, chap. 73, and in Ronayne's Algebra, part 3, pa. 395.</p></div1><div1 part="N" n="EXPANSION" org="uniform" sample="complete" type="entry"><head>EXPANSION</head><p>, is the dilating, stretching, or spreading out of a body; whether from any external cause, as the cause of rarefaction, or from an internal cause, as elasticity. Bodies naturally expand by heat beyond their dimensions when cold; and hence it happens that their dimensions and specific gravities are different in different temperatures and seasons of the year. Air compressed or condensed, as soon as the compressing or condensing force is removed, expands itself by its elastic power to its former dimensions.</p><p>In some few cases indeed bodies seem to expand as they grow cold, as water in the act of freezing: but it seems this is owing to the extrication of a number of air bubbles from the fluid at a certain time; and is not at all a regular and gradual expansion like that of metals, &c, by means of heat. Mr. Boyle, in his History of Cold, says that ice takes up one 12th part more space than water; but by Major Williams's experiments on the force of freezing water, I have found it occupies but about the 17th or 18th part more space. Transac. of the R. Soc. of Edinb. vol. 2, pa. 28. In certain metals also, an Expansion takes place when they pass from a fluid to a solid state: but this too is not to be accounted any proper effect of cold, but of the arrangement of the parts of the metal in a certain manner; and is therefore to be accounted a kind of crystallization, rather than any thing else.</p><p>The Expansion of different bodies by heat is very various; and many experiments upon it are to be met with in the volumes of the Philos. Trans. and elsewhere. In the 48th vol. in particular, Mr. Smeaton has given a table of the Expansion of many different substances, as determined by experiment, from which the following particulars are extracted. Where it is to be noted, that the quantities of Expansion which answer to 180 degrees of Fahrenheit's thermometer, are expressed in ten-thousandth parts of an English inch, each substance being 1 foot or 12 inches in length. <table><row role="data"><cell cols="1" rows="1" role="data">White glass barometer tube</cell><cell cols="1" rows="1" role="data">100</cell></row><row role="data"><cell cols="1" rows="1" role="data">Martial regulus of antimony</cell><cell cols="1" rows="1" role="data">130</cell></row><row role="data"><cell cols="1" rows="1" role="data">Blistered steel</cell><cell cols="1" rows="1" role="data">138</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hard steel</cell><cell cols="1" rows="1" role="data">147</cell></row><row role="data"><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" role="data">151</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bismuth</cell><cell cols="1" rows="1" role="data">167</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper hammered</cell><cell cols="1" rows="1" role="data">204</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper 8 parts, mixed with 1 of tin</cell><cell cols="1" rows="1" role="data">218</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cast brass</cell><cell cols="1" rows="1" role="data">225</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brass 16 parts, with tin 1</cell><cell cols="1" rows="1" role="data">229</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brass wire</cell><cell cols="1" rows="1" role="data">232</cell></row><row role="data"><cell cols="1" rows="1" role="data">Speculum metal</cell><cell cols="1" rows="1" role="data">232</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spelter solder, viz. brass 2 parts, zink 1</cell><cell cols="1" rows="1" role="data">247</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fine pewter</cell><cell cols="1" rows="1" role="data">274</cell></row><row role="data"><cell cols="1" rows="1" role="data">Grain tin</cell><cell cols="1" rows="1" role="data">298</cell></row><row role="data"><cell cols="1" rows="1" role="data">Soft solder, viz. lead 2, tin 1</cell><cell cols="1" rows="1" role="data">301</cell></row><row role="data"><cell cols="1" rows="1" role="data">Zink 8 parts, tin 1, a little hammered</cell><cell cols="1" rows="1" role="data">323</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lead</cell><cell cols="1" rows="1" role="data">344</cell></row><row role="data"><cell cols="1" rows="1" role="data">Zink or Spelter</cell><cell cols="1" rows="1" role="data">353</cell></row><row role="data"><cell cols="1" rows="1" role="data">Zink hammer'd 1/2 an inch per foot</cell><cell cols="1" rows="1" role="data">373</cell></row></table></p><p>By other experiments too it has been found that, for each degree of heat of the thermometer, mercury, wa- <cb/> ter, and air, expand by the following parts of their own bulk, viz,</p><p>From the foregoing table it appears, that there is no general rule for the degree of Expansion to which bodies are subject by the same degree of heat, either from their specific gravity or otherwise. Zink, which is much lighter than lead, expands more with heat; while glass, which is lighter than either, expands much less; and copper, which is heavier than a mixture of brass and tin, expands less.</p><p>It seems too that metals observe a proportion of Expansion in a fluid state, quite different from what they do in a solid one: For regulus of antimony seemed to shrink in fixing, after being melted, considerably more than zink.</p><p>But of all known substances, those of the aërial kind expand most by an equal degree of heat; and in general the greater quantity of latent heat that any substance contains, the more easily is it expanded; though even here no general rule can be formed. It is indeed certain that the densest fluids, such as mercury, oil of vitriol, &c, are less expansible than water, spirit of wine, or ether. Which last is so easily expanded, that were it not for the pressure of the atmosphere it would be in a continual state of vapour. And indeed this is the case, in some measure, with perhaps all fluids; as it has been found, by experiments with the best air-pumps, that water, and other fluids, ascend in vapours the more as the exhaustion is the more perfect; from which it would seem that water would wholly rise in vapour, in any temperature, if the pressure of the atmosphere was entirely taken off.</p><p>After bodies are reduced to a vaporous state, their Expansion seems to go on without any limitation, in proportion to the degree of heat applied; though it may be impossible to say what would be the ultimate effects of that principle upon them in this way. The force with which these vapours expand on the application of high degrees, is very great; nor does it appear that any obstacle whatever is insuperable by them.</p><p>On this principle depend the steam engines, so much used in various mechanical operations; likewise some hydraulic machines; and the instruments called manometers, which shew the variation of gravity in the external atmosphere, by the expansion or condensation of a small quantity of air confined in a proper vessel. On this principle also, perpetual movements might be constructed similar to those invented by Mr. Coxe, on the principle of the barometer. And a variety of other curious machines may be constructed on the principle of aërial expansion; an account of some of which is given under <hi rend="smallcaps">Hydrostatics</hi> and <hi rend="smallcaps">Pneumatics.</hi></p><p>On the principle of the Expansion of sluids are constructed Thermometers. And for the effects of the different Expansions of metals in correcting the errors of machines for measuring time, see the article P<hi rend="smallcaps">ENDULUM.</hi></p><p>The Expansion of solid bodies is measured by an instrument called the Pyrometer; and the force with which they expand is still greater than that of aërial vapours; the flame of a farthing candle produces an <pb n="457"/><cb/> Expansion in a bar of iron capable of counteracting a weight of 500 pounds. The quantity of expansion however is so small, that it has never been applied to the movement of any mechanical engine.</p></div1><div1 part="N" n="EXPECTATION" org="uniform" sample="complete" type="entry"><head>EXPECTATION</head><p>, in the Doctrine of Chances, is applied to any contingent event, upon the happening of which some benefit &c is expected. This is capable of being reduced to the rules of computation: for a sum of money in Expectation when a particular event happens, has a determinate value before that event happens. Thus, if a person is to receive any sum, as 10l, when an event takes place which has an equal chance or probability of happening and failing, the value of the Expectation is half that sum or 5l.: but if there are 3 chances for failing, and only 1 for its happening, or one chance only in its favour out of all the 4 chances; then the probability of its happening is only 1 out of 4, or 1/4, and the value of the Expectation is but 1/4 of 10l. which is only 2l. 10s. or half the former sum. And in all cases, the value of the Expectation of any sum is found by multiplying that sum by the fraction expressing the probability of obtaining it. So the value of the Expectation on 100l. when there are 3 chances out of 5 for obtaining it, or when the probability of obtaining it is 3/5, is 3/5 of 100l. which is 60l. And if <hi rend="italics">s</hi> be any sum expected on the happening of an event, <hi rend="italics">h</hi> the chances for that event happening, and <hi rend="italics">f</hi> the chances for its failing; then, there being <hi rend="italics">h</hi> chances out of <hi rend="italics">f</hi> + <hi rend="italics">h</hi> for its happening, the probability will be <hi rend="italics">h</hi>/(<hi rend="italics">f</hi> + <hi rend="italics">h</hi>), and the value of the expectation is <hi rend="italics">h</hi>/(<hi rend="italics">f</hi> + <hi rend="italics">h</hi>) X <hi rend="italics">s.</hi> See Simpson's or De Moivre's Doctrine of Chances.</p><p><hi rend="smallcaps">Expectation</hi> <hi rend="italics">of Life,</hi> in the Doctrine of Life Annuities, is the share, or number of years of life, which a person of a given age may, upon an equality of chance, expect to enjoy.</p><p>By the Expectation or share of life, says Mr. Simpson (Select Exercises pa. 273), is not here to be understood that particular period which a person hath an equal chance of surviving; this last being a different, and more simple consideration. The Expectation of a life, to put it in the most familiar light, may be taken as the number of years at which the purchase of an annuity, granted upon it, without discount of money, ought to be valued. Which number of years will differ more or less from the period above-mentioned, according to the different degrees of mortality to which the several stages of life are incident. Thus it is much more than an equal chance, according to the table of the probability of the duration of life (p. 254 ut supra), that an infant, just come into the world, arrives not to the age of 10 years; yet the Expectation or share of life due to it, upon an average, is near 20 years. The reason of which wide difference, is the great excess of the probability of mortality in the first tender years of life, above that respecting the more mature and stronger ages. Indeed if the numbers that die at every age were to be the same, the two quantities above specified would also be equal; but when the said numbers become continually less and less, the Expectation must of consequence be the greater of the two. <cb/></p><p>Mr. Simpson has given a table and rule for finding this Expectation, pa. 255 and 273 as above. Thus, <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=6 align=center" role="data"><hi rend="italics">A Table of the <hi rend="smallcaps">Expectations</hi> of Life in London.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">Age</cell><cell cols="1" rows="1" rend="align=center" role="data">Expectation</cell><cell cols="1" rows="1" rend="align=center" role="data">Age</cell><cell cols="1" rows="1" rend="align=center" role="data">Expectation</cell><cell cols="1" rows="1" rend="align=center" role="data">Age</cell><cell cols="1" rows="1" rend="align=center" role="data">Expectation</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">27.0</cell><cell cols="1" rows="1" rend="align=center" role="data">28</cell><cell cols="1" rows="1" rend="align=center" role="data">24.6</cell><cell cols="1" rows="1" rend="align=center" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">14.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">32.0</cell><cell cols="1" rows="1" rend="align=center" role="data">29</cell><cell cols="1" rows="1" rend="align=center" role="data">24.1</cell><cell cols="1" rows="1" rend="align=center" role="data">56</cell><cell cols="1" rows="1" rend="align=right" role="data">13.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">34.0</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">23.6</cell><cell cols="1" rows="1" rend="align=center" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">13.4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">35.6</cell><cell cols="1" rows="1" rend="align=center" role="data">31</cell><cell cols="1" rows="1" rend="align=center" role="data">23.1</cell><cell cols="1" rows="1" rend="align=center" role="data">58</cell><cell cols="1" rows="1" rend="align=right" role="data">13.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">36.0</cell><cell cols="1" rows="1" rend="align=center" role="data">32</cell><cell cols="1" rows="1" rend="align=center" role="data">22.7</cell><cell cols="1" rows="1" rend="align=center" role="data">59</cell><cell cols="1" rows="1" rend="align=right" role="data">12.7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=center" role="data">36.0</cell><cell cols="1" rows="1" rend="align=center" role="data">33</cell><cell cols="1" rows="1" rend="align=center" role="data">22.3</cell><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">12.4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">35.8</cell><cell cols="1" rows="1" rend="align=center" role="data">34</cell><cell cols="1" rows="1" rend="align=center" role="data">21.9</cell><cell cols="1" rows="1" rend="align=center" role="data">61</cell><cell cols="1" rows="1" rend="align=right" role="data">12.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=center" role="data">35.6</cell><cell cols="1" rows="1" rend="align=center" role="data">35</cell><cell cols="1" rows="1" rend="align=center" role="data">21.5</cell><cell cols="1" rows="1" rend="align=center" role="data">62</cell><cell cols="1" rows="1" rend="align=right" role="data">11.6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=center" role="data">35.2</cell><cell cols="1" rows="1" rend="align=center" role="data">36</cell><cell cols="1" rows="1" rend="align=center" role="data">21.1</cell><cell cols="1" rows="1" rend="align=center" role="data">63</cell><cell cols="1" rows="1" rend="align=right" role="data">11.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">34.8</cell><cell cols="1" rows="1" rend="align=center" role="data">37</cell><cell cols="1" rows="1" rend="align=center" role="data">20.7</cell><cell cols="1" rows="1" rend="align=center" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">10.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=center" role="data">34.3</cell><cell cols="1" rows="1" rend="align=center" role="data">38</cell><cell cols="1" rows="1" rend="align=center" role="data">20.3</cell><cell cols="1" rows="1" rend="align=center" role="data">65</cell><cell cols="1" rows="1" rend="align=right" role="data">10.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">33.7</cell><cell cols="1" rows="1" rend="align=center" role="data">39</cell><cell cols="1" rows="1" rend="align=center" role="data">19.9</cell><cell cols="1" rows="1" rend="align=center" role="data">66</cell><cell cols="1" rows="1" rend="align=right" role="data">10.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">33.1</cell><cell cols="1" rows="1" rend="align=center" role="data">40</cell><cell cols="1" rows="1" rend="align=center" role="data">19.6</cell><cell cols="1" rows="1" rend="align=center" role="data">67</cell><cell cols="1" rows="1" rend="align=right" role="data">9.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">32.5</cell><cell cols="1" rows="1" rend="align=center" role="data">41</cell><cell cols="1" rows="1" rend="align=center" role="data">19.2</cell><cell cols="1" rows="1" rend="align=center" role="data">68</cell><cell cols="1" rows="1" rend="align=right" role="data">9.4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">31.9</cell><cell cols="1" rows="1" rend="align=center" role="data">42</cell><cell cols="1" rows="1" rend="align=center" role="data">18.8</cell><cell cols="1" rows="1" rend="align=center" role="data">69</cell><cell cols="1" rows="1" rend="align=right" role="data">9.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">31.3</cell><cell cols="1" rows="1" rend="align=center" role="data">43</cell><cell cols="1" rows="1" rend="align=center" role="data">18.5</cell><cell cols="1" rows="1" rend="align=center" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">8.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=center" role="data">30.7</cell><cell cols="1" rows="1" rend="align=center" role="data">44</cell><cell cols="1" rows="1" rend="align=center" role="data">18.1</cell><cell cols="1" rows="1" rend="align=center" role="data">71</cell><cell cols="1" rows="1" rend="align=right" role="data">8.4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">30.1</cell><cell cols="1" rows="1" rend="align=center" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">17.8</cell><cell cols="1" rows="1" rend="align=center" role="data">72</cell><cell cols="1" rows="1" rend="align=right" role="data">8.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">29.5</cell><cell cols="1" rows="1" rend="align=center" role="data">46</cell><cell cols="1" rows="1" rend="align=center" role="data">17.4</cell><cell cols="1" rows="1" rend="align=center" role="data">73</cell><cell cols="1" rows="1" rend="align=right" role="data">7.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">28.9</cell><cell cols="1" rows="1" rend="align=center" role="data">47</cell><cell cols="1" rows="1" rend="align=center" role="data">17.0</cell><cell cols="1" rows="1" rend="align=center" role="data">74</cell><cell cols="1" rows="1" rend="align=right" role="data">7.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=center" role="data">28.3</cell><cell cols="1" rows="1" rend="align=center" role="data">48</cell><cell cols="1" rows="1" rend="align=center" role="data">16.7</cell><cell cols="1" rows="1" rend="align=center" role="data">75</cell><cell cols="1" rows="1" rend="align=right" role="data">7.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">27.7</cell><cell cols="1" rows="1" rend="align=center" role="data">49</cell><cell cols="1" rows="1" rend="align=center" role="data">16.3</cell><cell cols="1" rows="1" rend="align=center" role="data">76</cell><cell cols="1" rows="1" rend="align=right" role="data">6.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=center" role="data">27.2</cell><cell cols="1" rows="1" rend="align=center" role="data">50</cell><cell cols="1" rows="1" rend="align=center" role="data">16.0</cell><cell cols="1" rows="1" rend="align=center" role="data">77</cell><cell cols="1" rows="1" rend="align=right" role="data">6.4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">26.6</cell><cell cols="1" rows="1" rend="align=center" role="data">51</cell><cell cols="1" rows="1" rend="align=center" role="data">15.6</cell><cell cols="1" rows="1" rend="align=center" role="data">78</cell><cell cols="1" rows="1" rend="align=right" role="data">6.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=center" role="data">26.1</cell><cell cols="1" rows="1" rend="align=center" role="data">52</cell><cell cols="1" rows="1" rend="align=center" role="data">15.2</cell><cell cols="1" rows="1" rend="align=center" role="data">79</cell><cell cols="1" rows="1" rend="align=right" role="data">5.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=center" role="data">25.6</cell><cell cols="1" rows="1" rend="align=center" role="data">53</cell><cell cols="1" rows="1" rend="align=center" role="data">14.9</cell><cell cols="1" rows="1" rend="align=center" role="data">80</cell><cell cols="1" rows="1" rend="align=right" role="data">5.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=center" role="data">25.1</cell><cell cols="1" rows="1" rend="align=center" role="data">54</cell><cell cols="1" rows="1" rend="align=center" role="data">14.5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>For Example, if it be required to find the Expectation or share of life, due to a person of 30 years old. Opposite the given age in the first column of the table, stands 23.6 in the second col. for the years in the Expectation sought.</p><p>See De Moivre's Doctrine of Chances applied to the Valuation of Annuities, p. 288; or Dr. Price's Observations on Reversionary Payments, p. 168, 364, 374, &c; or Philos. Trans. vol. 59, p. 89.</p></div1><div1 part="N" n="EXPERIMENT" org="uniform" sample="complete" type="entry"><head>EXPERIMENT</head><p>, in Philosophy, a trial of the effect or result of certain applications and motions of natural bodies, in order to discover something of their laws and relations, &c.</p><p>The making of experiments is grown into a kind of art; and there are now many collections of them, mostly under the denomination of Courses of Experimental Philosophy. Sturmius made a curious collection of the principal Discoveries and Experiments of the last age, under the title of Collegium Experimentale. Other Courses of Experiments have been published by Gravesande, Desaguliers, Helsham, Cotes, and others.</p><p>EXPERIMENTAL <hi rend="italics">Philosophy,</hi> is that which proceeds on Experiments, or which deduces the laws of nature and the properties and powers of bodies, and their actions upon each other, from sensible Experiments and observations.</p><p>Experiments are of the utmost importance in philosophy; and the great advantages the modern physics <pb n="458"/><cb/> have over the ancient, is chiefly owing to this, viz, that we abound much more in Experiments, and that we make more use of the Experiments we have. The method of the ancients, was chiefly to begin with the causes of things, and thence argue to the phenomena and essects; on the contrary, that of the moderns proceeds from Experiments and Observations, from whence the properties and laws of natural things are deduced, and general theories are formed.</p><p>Several of the ancients indeed thought as highly of Experiments as the moderns, and practised them also. Plato omits no occasion of speaking of the advantages of them; and Aristotle's history of animals bears ample testimony for <hi rend="italics">him.</hi> Democritus's great employment was to make experiments; and even Epicurus himself owes part of his glory to the same cause.</p><p>Among the moderns, the making of Experiments was chiefly begun by Friar Bacon, in the 13th century, who it seems spent a great deal of money and labour in this way. After him, the lord chancellor Bacon is looked upon as the founder of the present mode of philosophising by Experiments. And his method has been prosecuted with laudable emulation by the Academy del Cimento, the Royal Society, the Royal Academy at Paris; by Mr. Boyle, and, over all, by Sir Isaac Newton, with many other illustrious names.</p><p>Indeed, Experiments, within the last century, are come so much into vogue, that nothing will pass in philosophy, but what is either founded on Experiments, or confirmed by them; so that the new philosophy is almost wholly Experimental.</p><p>Yet there are some, even among the learned, who speak of Experiments in a different manner, or perhaps rather of the abuse of them, and in derision of the pretenders to this practice. Thus, though Dr. Keil allows that philosophy has received very considerable advantages from the makers of Experiments; yet he complains of their disingenuity, in too often wresting and distorting their Experiments and Observations to favour some darling theories they had espoused. Nay more, M. Hartsoeker, in his Recueil de plusieurs Pieces de Physique, undertakes to shew, that such as employ themselves in the making of Experiments, are not properly philosophers, but as it were the labourers or operators of philosophers, that work under them, and for them, furnishing them with materials to build their systems and hypotheses upon. And the learned M. Dacier, in the beginning of his discourse on Plato, at the head of his translation of the works of that philosopher, deals still more severely with the makers of Experiments. He breaks out with a kind of indignation at a tribe of idly curious people, whose sole employment consists in making Experiments on the gravity of the air, the equilibrium of fluids, the loadstone, &c, and yet arrogate to themselves the noble title of philosophers. But his honest indignation would have exceeded all bounds, had he lived to see the contemptible fall of one of the principal societies above-mentioned; while its members first amuse themselves with magnetical conundrums, spinning electrical wheels, torturing the unseen and unknown phlogistic particles; and finally polluting the source of science, and the streams of wisdom, with the folly of hunting after cockle-shells, caterpillars, and butterflies! <cb/></p></div1><div1 part="N" n="EXPLOSION" org="uniform" sample="complete" type="entry"><head>EXPLOSION</head><p>, a sudden and violent expansion of an elastic fluid, by which it instantly throws off any obstacle that happens to be in the way, sometimes with astonishing force and rapidity, as the Explosion of fired gun-powder, &c.</p><p>Explosion differs from expansion, in that the latter is a gradual and continued power, acting uniformly for some certain time; whereas the former is always sudden, and only of momentary or immensurably short duration. The expansions of solid substances do not terminate in violent explosions, on account of their slowness, and the small space through which the expanding substance moves; though their strength may be equally great with that of the most active aerial fluids. Thus we find that though wedges of wood, when wetted, will cleave solid blocks of stone, they never throw them to any distance, as is the case with gunpowder. On the other hand, it is seldom that the expansion of any elastic fluid bursts a solid substance without throwing the fragments of it to a considerable distance, with effects that are often very terrible.</p><p>The most part of explosive substances are either aerial, or convertible into such, and raised into an elastic fluid. Thus gun-powder, whose essence seems to consist in common air fixed in the nitre, or at least an air of similar elasticity, where it is condensed into a bulk many hundred times less than the natural state of the atmosphere; which air being suddenly disengaged by the firing of the gun-powder, and the decomposition of its parts, it rapidly expands itself again with a force proportioned to the degree of its condensation when fixed in the gun-powder, and so explodes, and produces all those terrible effects that attend the explosion. The elastic fluid generated by the fired gun-powder expands itself with a velocity of about 10,000 feet per second, and with a force more than 1000 times greater than the pressure of the atmosphere on the same base.</p><p>The Electric Explosions seem to be still much more strong and astonishing; as in the cases of lightning, earthquakes, and volcanoes; and even in the artificial electricity produced by the ordinary machines. The astonishing strength of electric explosions, which is beyond all possible means of measuring it, manifests itself by the many tremendous effects we hear of fire-balls and lightning.</p><p>In cases where the electric matter acts like common fire, the force of the explosions, though very great, is capable of measurement, by comparing the distances to which bodies are thrown, with their weight. This is most evident in volcanoes, where the projections of the burning rocks and lava manifest the greatness of the power, at the same time that they afford a method of measuring it: and these explosions are owing to the extrication of aerial vapours, and their rarefaction by intense heat.</p><p>Next in strength to the aerial vapours, are those of aqueous and other liquids. Very remarkable effects of these are observed in steam-engines; and there is one case from which it has been inferred that aqueous steam is even vastly stronger than fired gun-powder. This is when water is thrown upon melted copper: for here the explosion is so strong as almost to exceed imagination; and the most terrible accidents have happened, even from so slight a cause as one of the workmen spit- <pb n="459"/><cb/> ting in the furnace where copper was melting; arising probably from a sudden decomposition of the water. Explosions happen also from the application of water to other melted metals, though in a lower degree, when the fluid is applied in small quantities, and even to common fire itself, as every person's own experience must have informed him; and this seems to be occasioned by the sudden rarefaction of the water into steam. Examples of this kind often occur when workmen are fastening cramps of iron into stones; where, if there happen to be a little water in the hole into which the lead is poured, this will fly out in such a manner as sometimes to burn them severely. Terrible accidents of this kind have sometimes happened in founderies, when large quantities of melted metal have been poured into wet or damp moulds. In these cases, the sudden expansion of the aqueous steam has thrown out the metal with great violence; and if any decomposition has taken place at the same time, so as to convert the aqueous vapour into an aerial one, the explosion must be still greater.</p><p>To this last kind of explosion must be referred that which takes place on pouring cold water into boiling or burning oil or tallow, or in pouring the latter upon the former; the water however being always used in a small quantity.</p><p>Another remarkable kind of Explosion is that produced by inflammable and dephlogisticated air, when mixed together, and set on fire; a kind of explosion that often happens in coal mines, &c. This differs from any of the cases before mentioned; for here is an absolute condensation rather than an expansion throughout the whole of the operation; and could the airs be made to take fire throughout their whole substance absolutely at the same instant, there would be no Explosion, but only a sudden production of heat.</p><p>Though Explosions be sometimes very destructive, they are likewise of confiderable use in life, as in removing obstacles that could scarcely be overcome by any mechanical power whatever. The principal of these are the blowing up of rocks, the separating of stones in quarries, and other purposes of that kind. The destruction occasioned by them in times of war, and the machines formed upon the principle of Explosion for the destruction of the human race, are well known; and if we cannot call these useful, they must be allowed at least to be necessary evils.</p><p>The effects of Explosions, when violent, are felt at a considerable distance, by reason of the concussions they give to the atmosphere. Sir Wm. Hamilton relates, that at the explosions of Vesuvius, in 1767, the doors and windows of the houses at Naples flew open if unbolted, and one door was burst open that had been locked, though at the distance of 6 miles: and the explosion of a powder-magazine, or a powder-mill, it is well known, spreads destruction for many miles round; and even kills people by the mere concussion of the air. A curious effect of them too is, that they electrify the air, and even glass windows, at a considerable distance. This is always observable in firing the guns at the Tower of London: and some years ago, after an Explosion of some powder-mills near that city, many people were alarmed by a rattling and breaking of their china-ware. In this respect however, the effects of electrical Explo- <cb/> sions are the most remarkable, though not in the uncommon way just mentioned; but it is certain that the influence of a flash of lightning is diffused for a great way round the place where the Explosion happens, producing very perceptible changes both on the animul and vegetable creation.</p><p>EXPONENT <hi rend="italics">of a Power,</hi> in Arithmetic and Algebra, denotes the number or quantity expressing the degree or elevation of the power, or which shews how often a given power is to be divided by its root before it be brought down to unity or 1. Thus, the Exponent or index of a square number, or the 2d power, is 2; of a cube 3; and so on; the square being a power of the 2d degree; the cube, of a 3d, &c. It is otherwise called the Index.</p><p>Exponents, as now used, are rather of modern invention. Diophantus, with the Arabian and the first European authors, denoted the powers of quantities by subjoining an abbreviation of the name of the power; though with some variation, and difference from one another. The names of the powers, and the marks for denoting them, according to Diophantus, are as follow: viz. Names, <figure/> Marks, <figure/> which we now denote by</p><p>1, <hi rend="italics">a, a</hi><hi rend="sup">2</hi>, <hi rend="italics">a</hi><hi rend="sup">3</hi>, <hi rend="italics">a</hi><hi rend="sup">4</hi>, <hi rend="italics">a</hi><hi rend="sup">5</hi>, <hi rend="italics">a</hi><hi rend="sup">6</hi>, &c.</p><p>F. Lucas Paciolus, or De Burgo, for the root, square, cube, &c, uses the terms <hi rend="italics">cosa, censo, cubo, relato (primo, secundo, tertio,</hi> &c), or the abbreviations <hi rend="italics">co. ce. cu.</hi>; and R for root or radicality.</p><p>Cardan used the Latin contractions of the names of the powers; and other contemporary, as well as succeeding, authors, especially the Germans, as Stifelius, Scheubelius, Pelitarius, &c, used the like contractions, but somewhat varied, as thus: <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data">[dram],</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data">[dram][dram],</cell><cell cols="1" rows="1" role="data">∫[dram],</cell><cell cols="1" rows="1" role="data">[dram]<figure/>,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data">[dram],</cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data">[dram][dram],</cell><cell cols="1" rows="1" role="data">∫<hi rend="italics">s,</hi></cell><cell cols="1" rows="1" role="data">[dram]<figure/>,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">℞,</cell><cell cols="1" rows="1" role="data"><hi rend="italics">q,</hi></cell><cell cols="1" rows="1" role="data"><figure/>,</cell><cell cols="1" rows="1" role="data"><hi rend="italics">qq,</hi></cell><cell cols="1" rows="1" role="data">∫<hi rend="italics">s,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">q</hi><figure/>,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Exp.</cell><cell cols="1" rows="1" role="data">0,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" role="data">6,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table></p><p>But besides that way, the same authors also made use of the numbers as in the last line here above, and it was Stifelius who first called them by the name <hi rend="italics">Exponent.</hi></p><p>Bombelli, whose Algebra was published in 1579, denotes the <hi rend="italics">res,</hi> or unknown quantity, by this mark <*>, and the powers by numeral Exponents set over it, thus: <figure/>, &c. And</p><p>Stevinus, who published his Arithmetic in 1585, and his Algebra soon afterwards, has such another me thod, but instead of <*> he uses a small circle ○, within which he places the numeral Exponent of the power; thus ○0, ○1, ○2, ○3, &c: and in this way he extends his notation to fractional Exponents, and even to radical ones; thus ○1/2, ○1<*>, ○3/4, ○2<*>, &c.</p><p>Vieta after this used words again to denote the powers. Afterwards Harriot denoted the powers by a repetition of the root; as <hi rend="italics">a, aa, aaa,</hi> for the 1st, 2d, and 3d powers. Instead of which, Des Cartes again restored the numeral Exponents, placing them after the root, when the power is high, to avoid a too frequent repetition of the letter of the root; as <hi rend="italics">a</hi><hi rend="sup">3</hi> <hi rend="italics">a</hi><hi rend="sup">4</hi>, &c, as at <pb n="460"/><cb/> present. Also Albert Girard, in 1629, used the Exponents to roots, thus; √, √<hi rend="sup">2</hi>, √<hi rend="sup">3</hi>, &c.</p><p>The notation of powers and roots by the present way of Exponents, has introduced a new and general arithmetic of Exponents or powers; for hence powers are multiplied by only adding their Exponents, divided by subtracting the Exponents, raised to other powers, or roots of them extracted, by multiplying or dividing the Exponent by the index of the power or root.—</p><p>So ; the 2d power of <hi rend="italics">a</hi><hi rend="sup">3</hi> is <hi rend="italics">a</hi><hi rend="sup">6</hi>, and the 3d root of <hi rend="italics">a</hi><hi rend="sup">6</hi> is <hi rend="italics">a</hi><hi rend="sup">2</hi>.</p><p>This algorithm of powers led the way to the invention of logarithms, which are only the indices or Exponents of powers: and hence the addition and subtraction of logarithms, answer to the multiplication and division of numbers; while the raising of powers, and extracting of roots, is effected by multiplying the logarithm by the index of the power, or dividing the logarithm by the index of the root.</p><p><hi rend="smallcaps">Exponent</hi> <hi rend="italics">of a Ratio,</hi> is, by some, understood as the quotient arising from the division of the antecedent of the ratio by the consequent: in which sense, the Exponent of the ratio of 3 to 2 is 3/2; and that of the ratio of 2 to 3 is 2/3.</p><p>But others, and those among the best mathematicians, understand logarithms as the Exponents of ratios; in which sense they coincide with the idea of measures of ratios, as delivered by Kepler, Mercator, Halley, Cotes, &c.</p><p>EXPONENTIAL <hi rend="italics">Calculus,</hi> the method of differencing, or finding the fluxions of, Exponential quantities, and of summing up those differences, or finding their fluents. See <hi rend="smallcaps">Calculus, Fluxions</hi>, and F<hi rend="smallcaps">LUENTS.</hi></p><p><hi rend="smallcaps">Exponential</hi> <hi rend="italics">Curve,</hi> is that whose nature is defined or expressed by an Exponential equation; as the curve denoted by <hi rend="italics">a</hi><hi rend="sup">x</hi> = <hi rend="italics">y,</hi> or by <hi rend="italics">x</hi><hi rend="sup">x</hi> = <hi rend="italics">y.</hi></p><p><hi rend="smallcaps">Exponential</hi> <hi rend="italics">Equation,</hi> is one in which is contained an exponential quantity: as the equation <hi rend="italics">a</hi><hi rend="sup">x</hi> = <hi rend="italics">b,</hi> or <hi rend="italics">x</hi><hi rend="sup">x</hi> = <hi rend="italics">ab,</hi> &c.</p><p>Exponential Equations are commonly best resolved by means of logarithms, viz, first taking the log. of the given equation: thus, taking the log. of the equation <hi rend="italics">a</hi><hi rend="sup">x</hi> = <hi rend="italics">b,</hi> it is <hi rend="italics">x</hi> X log. of <hi rend="italics">a</hi> = log. of <hi rend="italics">b</hi>; and hence <hi rend="italics">x</hi> = (log. <hi rend="italics">b</hi>)/(log. <hi rend="italics">a</hi>)</p><p>Also, the log. of the equation <hi rend="italics">x</hi><hi rend="sup">x</hi> = <hi rend="italics">ab,</hi> is <hi rend="italics">x</hi> X log. <hi rend="italics">x</hi> = log. <hi rend="italics">ab</hi>; and then <hi rend="italics">x</hi> is easily found by trial-and-error, or the double rule of position.</p><p><hi rend="smallcaps">Exponential</hi> <hi rend="italics">Quantity,</hi> is that whose power is a variable quantity; as the expression <hi rend="italics">a</hi><hi rend="sup">x</hi>, or <hi rend="italics">x</hi><hi rend="sup">x</hi>.</p><p>Exponential quantities are of several degrees, and orders, according to the number of exponents or powers, one over another. Thus, <hi rend="italics">a</hi><hi rend="sup">x</hi> is an Exponential of the 1st order, <hi rend="italics">a</hi><hi rend="sup">x<hi rend="sup">y</hi></hi>, is one of the 2d order, <hi rend="italics">a</hi><hi rend="sup">x<hi rend="sup">y<hi rend="sup">z</hi></hi></hi> is one of the 3d order, and so on. See Bernoulli Oper. tom. 1, pa. 182, &c. <cb/></p></div1><div1 part="N" n="EXPRESSION" org="uniform" sample="complete" type="entry"><head>EXPRESSION</head><p>, in Algebra, is any algebraical quantity, simple or compound: as the expression, 3<hi rend="italics">a,</hi> or 2<hi rend="italics">ab,</hi> or √(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">c</hi><hi rend="sup">2</hi>.)</p></div1><div1 part="N" n="EXTENSION" org="uniform" sample="complete" type="entry"><head>EXTENSION</head><p>, one of the common and essential properties of body; or that by which it possesses or takes up some part of universal space, called the place of that body.</p><p>The extension of a body, is properly in every direction whatever; but it is usual to consider it as extended only in length, breadth, and thickness.</p><p>EXTERIOR <hi rend="italics">Polygon,</hi> or <hi rend="italics">Talus,</hi> is the outer or circumscribing one. See <hi rend="smallcaps">Polygon</hi> and <hi rend="smallcaps">Talus.</hi></p></div1><div1 part="N" n="EXTERMINATION" org="uniform" sample="complete" type="entry"><head>EXTERMINATION</head><p>, or EXTERMINATING, in Algebra, is the taking away, or expelling of something from an expression, or from an equation: as to Exterminate surds, fractions, or any particular letter or quantity out of equations.</p><p>Thus, to take away the fractional form from this equation ; multiply each numerator by the other's denominator, and the equation becomes <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">d</hi> + <hi rend="italics">dx</hi><hi rend="sup">2</hi> = 2<hi rend="italics">abc,</hi> out of fractions.</p><p>Also, to take away the radicality from the equation , raise each to the 2d power, and it becomes 9<hi rend="italics">a</hi><hi rend="sup">2</hi>-9<hi rend="italics">x</hi><hi rend="sup">2</hi> = 4<hi rend="italics">c</hi><hi rend="sup">2</hi>.</p><p>For Exterminating any quantity out of equations, there are various rules and methods, according to the form of the equations; of which many excellent specimens may be seen in Newton's Algebra, pa. 60, ed. 1738; or in Maclaurin's Algebra, part 1, chap. 12. For example, to Exterminate <hi rend="italics">y</hi> out of these two equations, ; subtract the upper equation from the under, so shall there arise 3<hi rend="italics">b</hi>-<hi rend="italics">a</hi>-<hi rend="italics">x</hi>=2<hi rend="italics">x</hi>-<hi rend="italics">b</hi>; then, by the known methods of transposition &c, there is obtained 4<hi rend="italics">b</hi>-<hi rend="italics">a</hi>=3<hi rend="italics">x,</hi> and hence <hi rend="italics">x</hi> = (4<hi rend="italics">b</hi>-<hi rend="italics">a</hi>)/3.</p><p>EXTERNAL <hi rend="italics">Angles,</hi> are the angles formed withoutside of a figure, by producing its sides out.</p><p>In a triangle, any External angle is equal to the sum of both the two internal opposite angles taken together: and, in any right-lined figure, the sum of all the external angles, is always equal to 4 right angles.</p><p>EXTRA-<hi rend="italics">Constellary Stars,</hi> such as are not properly included in any constellation.</p><p>EXTRA-<hi rend="italics">Mundane Space,</hi> is the infinite, empty, void space, which is by some supposed to be extended beyond the bounds of the universe, and consequently in which there is really nothing at all.</p><p>EXTRACTION <hi rend="italics">of Roots,</hi> is the finding the roots of given numbers, or quantities, or equations.</p><p>The roots of quantities are denominated from their powers; as the square or 2d root, the cubic or 3d root, the biquadratic or 4th root, the 5th root, &c; which are the roots of the 2d, 3d, 4th, 5th, &c powers. The Extraction of roots has always made a part of arithmetical calculation, at least as far back as the composition of powers has been known: for the composition of powers always led to their resolution, or Extraction of roots, which is performed by the rules exactly reverse of the former. Thus, if any root be con- <pb n="461"/><cb/> sidered as consisting of two parts <hi rend="italics">a</hi> + <hi rend="italics">x,</hi> of which the former <hi rend="italics">a</hi> is known, and the latter <hi rend="italics">x</hi> unknown, then the square of this root being <hi rend="italics">a</hi><hi rend="sup">2</hi> + 2<hi rend="italics">ax</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi>, which is its composition, this indicated the method of resolution, so as to find out the unknown part <hi rend="italics">x</hi>; for having subtracted the nearest square <hi rend="italics">a</hi><hi rend="sup">2</hi> from the given quantity, there remains 2<hi rend="italics">ax</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi> or ―(2<hi rend="italics">a</hi> + <hi rend="italics">x</hi>) X <hi rend="italics">x</hi>; therefore divide this remainder by 2<hi rend="italics">a,</hi> the double of the first member of the root, the quotient will be nearly <hi rend="italics">x</hi> the other member; then to 2<hi rend="italics">a</hi> add this quotient <hi rend="italics">x,</hi> and multiply the sum 2<hi rend="italics">a</hi> + <hi rend="italics">x</hi> by <hi rend="italics">x,</hi> and the product will make up the remaining part 2<hi rend="italics">ax</hi>+<hi rend="italics">x</hi><hi rend="sup">2</hi> of the given power.</p><p>The composition of the cubic or 3d power next presented itself, which consists of these four terms <hi rend="italics">a</hi><hi rend="sup">3</hi> + 3<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> + 3<hi rend="italics">ax</hi><hi rend="sup">2</hi> + <hi rend="italics">x</hi><hi rend="sup">3</hi>; by means of which the cubic roots of numbers have been extracted; viz, by subtracting the nearest cube <hi rend="italics">a</hi><hi rend="sup">3</hi> from the given power, dividing the remainder by 3<hi rend="italics">a</hi><hi rend="sup">2</hi>, which gives <hi rend="italics">x</hi> nearly for the quotient; then completing the divisor up to 3<hi rend="italics">a</hi><hi rend="sup">2</hi> + 3<hi rend="italics">ax</hi>+<hi rend="italics">x</hi><hi rend="sup">2</hi>, multiply it by <hi rend="italics">x</hi> for the other part of the power to be subtracted. And this was the extent of the <*>xtraction of roots in the time of Lucas de Burgo, who, from 1470 to 1500, wrote several pieces on arithmetic and algebra, which were the first works of this kind printed in Europe.</p><p>It was not long however before the nature and composition of all the higher powers became known, and general tables of coefficients formed for raising them; the first of which is contained in Stifelius's arithmetic, printed at Norimberg in 1543, where he fully explains their use in Extracting the roots of all powers whatever, by methods similar to those for the square and cubic roots, as above described; and thus completed the Extraction of all sorts of roots of numbers, at least so far as respects that method of resolution. Since that time, however, many new methods of Extraction have been devised, as well as improvements made in the old way.</p><p>The Extraction of roots of equations followed closely that of known numbers. In De Burgo's time they extracted the roots of quadratic equations, the same way as at present. Ferreus, Tartalea, and Cardan extracted the roots of cubic equations, by general rules. Soon afterwards the roots of higher equations were extracted, at least in numbers, by approximation. And the late improvements in analytics have furnished general rules for Extracting the roots, in infinite series, of all equations whatever. All which methods may be seen in most books of arithmetic and algebra. Of which it may suffice to give here a short specimen of some of the easiest rules for Extracting the roots of quantities and equations, as they here follow.</p><p>I. <hi rend="italics">To Extract the Square Root of any Number.</hi>— Point off, or divide the number, from the place of units, into portions of two figures each, as here of the number 99856, <figure/> setting a point or mark over the space between each portion of two sigures. Then, beginning at the left hand, take the greatest root 3, of the first part 9, placing it on the xight hand for the first figure of the root, and subtracting its square 9 from the said first part; to the re- <cb/> mainder, which here is o, bring down the 2d part 98, and on the left hand of it place 6 the double of the first figure 3, for a divisor; conceive a cipher added to this, making it 60, and then divide the 98 by the 60, the quotient is 1 for the second figure of the root, which is accordingly placed there, after the 3, also in the divisor after the 6 and below the same; then multiply these as they stand, the 61 by the 1, and the product 61 set below the 98, and subtract it from the same, which leaves 37 for the next remainder; to this bring down the 3d period 56, making 3756 for the next resolvend: then form its divisor as before, viz, doubling the root 31, or adding, as they stand in the divisor, the 1 to the 61, either way making 62, which with a cipher makes 620, by which divide the resolvend 3756; the quotient of this division is 6, to be placed, as before, both as the next figure of the root, and at the end of the divisor 62, and below itself there; then multiply as they stand the whole divisor 626 by the 6, the product 3756 is exactly the same as the resolvend, and therefore the number 316 is accurately the square root of the given number 99856, as required.</p><p>When the root is to be carried into deccimals, couplets of ci- <figure/> phers are to be added, instead of figures, as far as may be wanted. In which case too, a good abbreviation is made, after the work has been carried on to half the number of figures, by continuing it to the other half only by the contracted way of division; as here in the annexed example for the square root of 2 to eight decimals, or nine places of figures in all.</p><p>II. <hi rend="italics">To extract the cubic root, or any other root whatever.</hi> This is easiest done by one general rule, which I have invented, and published in my Tracts, vol. 1, pa. 49, which is to this essect: Let N be any number or power, whose <hi rend="italics">n</hi>th root is to be extracted; and let R be the nearest rational root of N, of the same kind, or R<hi rend="sup">n</hi> the nearest rational power to N, either greater or less than it; then shall the true root be very nearly equal to ; which rule is general for any root whose index is denoted by <hi rend="italics">n.</hi> And by expounding <hi rend="italics">n</hi> successively by all the numbers 2, 3, 4, 5, &c, this theorem will give the following particular rules for the several roots, viz, the <table><row role="data"><cell cols="1" rows="1" role="data">2d or squ. root,</cell><cell cols="1" rows="1" role="data">(3N+R<hi rend="sup">2</hi>)/(N+3R<hi rend="sup">2</hi>)</cell><cell cols="1" rows="1" role="data">X R;</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">3d or cube root,</cell><cell cols="1" rows="1" role="data">(4N+2R<hi rend="sup">3</hi>)/(2N+4R<hi rend="sup">3</hi>)</cell><cell cols="1" rows="1" role="data">X R, or</cell><cell cols="1" rows="1" role="data">(2N+ R<hi rend="sup">3</hi>)/(N+2R<hi rend="sup">3</hi>)</cell><cell cols="1" rows="1" role="data">X R;</cell></row><row role="data"><cell cols="1" rows="1" role="data">4th root</cell><cell cols="1" rows="1" role="data">(5N+3R<hi rend="sup">4</hi>)/(3N+5R<hi rend="sup">4</hi>)</cell><cell cols="1" rows="1" role="data">X R;</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row> <pb n="462"/><cb/> <row role="data"><cell cols="1" rows="1" role="data">5th root</cell><cell cols="1" rows="1" role="data">(6N+4R<hi rend="sup">5</hi>)/(4N+6R<hi rend="sup">5</hi>)</cell><cell cols="1" rows="1" role="data">X R, or</cell><cell cols="1" rows="1" role="data">(3N+2R<hi rend="sup"><*></hi>)/(2N+3R<hi rend="sup"><*></hi>)</cell><cell cols="1" rows="1" role="data">X R;</cell></row><row role="data"><cell cols="1" rows="1" role="data">6th root</cell><cell cols="1" rows="1" role="data">(7N+5R<hi rend="sup">6</hi>)/(5N+7R<hi rend="sup">6</hi>)</cell><cell cols="1" rows="1" role="data">X R;</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">7th root</cell><cell cols="1" rows="1" role="data">(8N+6R<hi rend="sup">7</hi>)/(6N+8R<hi rend="sup">7</hi>)</cell><cell cols="1" rows="1" role="data">X R, or</cell><cell cols="1" rows="1" role="data">(4N+3R<hi rend="sup">7</hi>)/(3N+4R<hi rend="sup">7</hi>)</cell><cell cols="1" rows="1" role="data">X R;</cell></row><row role="data"><cell cols="1" rows="1" role="data">&c.</cell><cell cols="1" rows="1" rend="align=center" role="data">&c.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> Or the theorem may be stated in the form of a proportion, thus: as the root sought very nearly.</p><p>For ex. suppose the problem proposed, of doubling the cube, or to sind the cube root of the number 2. Here N = 2, <hi rend="italics">n</hi> = 3, and the nearest power, and root too, is 1: Hence ; then the first approximation.</p><p>Again, taking R = 5/4, and conseq. R<hi rend="sup">3</hi> = 125/64 : Hence ; then , for the cube root of 2, which is exact in the very last figure.</p><p>And again by taking 635/504 for the value of R, a great many more figures may be found.</p><p>III. <hi rend="italics">To Extract the Roots of Algebraic Quantities.</hi>— This is done by the same rules, and in the same manner as for the roots of numbers in arithmetic, as above taught. Thus, to Extract the square root of 4<hi rend="italics">a</hi><hi rend="sup">2</hi> + 12<hi rend="italics">ax</hi> + 9<hi rend="italics">x</hi><hi rend="sup">2</hi>. <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4<hi rend="italics">a</hi><hi rend="sup">2</hi>+</cell><cell cols="1" rows="1" role="data">12<hi rend="italics">ax</hi>+9<hi rend="italics">x</hi><hi rend="sup">2</hi> (2<hi rend="italics">a</hi>+3<hi rend="italics">x</hi> the root</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4<hi rend="italics">a</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">4<hi rend="italics">a</hi>+3<hi rend="italics">x</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12<hi rend="italics">ax</hi>+9<hi rend="italics">x</hi><hi rend="sup">2</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3<hi rend="italics">x</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12<hi rend="italics">ax</hi>+9<hi rend="italics">x</hi><hi rend="sup">2</hi></cell></row></table></p><p>So also the root is carried out in an infinite series, in imitation of the like Extraction of numbers in infinite decimals: thus, for the square root of <hi rend="italics">a</hi><hi rend="sup">2</hi>+<hi rend="italics">x</hi><hi rend="sup">2</hi>. <figure/></p><p>To extract the cube root of <hi rend="italics">a</hi><hi rend="sup">3</hi>-<hi rend="italics">x</hi><hi rend="sup">3</hi> by the general rule in the 2d article.—Here ; therefore, by the rule, <cb/> &c, which is the cube root of <hi rend="italics">a</hi><hi rend="sup">3</hi>-<hi rend="italics">x</hi><hi rend="sup">3</hi> very nearly.</p><p>But these sorts of roots are best extracted by the Binomial Theorem; which see.</p><p>IV. <hi rend="italics">To Extract the Roots of Equations.</hi>—This is the same thing as to find the value of the unknown quantity in an equation; which is effected by various means, depending on the form of the equation, and the height of the highest power of the unknown quantity in it: for which, see the respective terms, <hi rend="smallcaps">Equation, Root, Quadratic, Cubic</hi>, &c.</p><p>The most general, as well as the most easy, method of Extracting the roots of all equations, is by Double Position, or Trial-and-Error; as it easily applies to all sorts of equations whatever, be they ever so complex, even logarithmic and exponential ones. There are also several other good methods of approximating to the roots of equations, given by Newton, Halley, Raphson, De Moivre, &c; of which the most general is a rule for Extracting the root of the following indefinite equation,</p><p>viz, , given by M. De Moivre in the Philos. Trans. vol. 20. p. 190, or Abr. vol. 1, pa. 101.</p></div1><div1 part="N" n="EXTRADOS" org="uniform" sample="complete" type="entry"><head>EXTRADOS</head><p>, the outside of an arch of a bridge, vault, &c.</p><p>EXTREME-<hi rend="italics">and-Mean Proportion,</hi> is when a line, or any quantity is so divided, as that the whole line is to the greater part, as that greater part is to the less part. Hence, in any line so divided, the rectangle of the whole line and the less segment, is equal to the square of the greater segment.</p><p>Euclid shews how to divide a line in Extreme-andmean ratio, in his Elements, book 2, prop. 11, to this effect: Let AB be the given line; to which draw AE perpendicular and equal to half AB; in EA produced take EF = EB, so shall AF be equal to the greater part; consequently if AG be taken equal to AF, the line AB will be divided in G as required. <figure/></p><p>The same may be done otherwise thus:</p><p>As before, make AE (fig. 2.) perpendicular and = (1/2)AB; join EB, on which take EC = EA, and then take BD = BC, so shall the line be divided in D as required.</p><p>No number can be divided into extreme and mean proportion, so that its two parts shall be rational; as is well demonstrated by Clavius, in his Commentary upon the 9th book of Euclid's Elements; and the same thing will also appear from the following algebraical solution of the same problem: Let <hi rend="italics">a</hi> denote the whole line, and <hi rend="italics">x</hi> the greater part; then shall <hi rend="italics">a</hi>-<hi rend="italics">x</hi> be the less part, and <pb n="463"/><cb/> the rectangle of the whole and less part being put equal to the square of the greater part, gives this equation, ; hence and by completing the square, and extracting the root, &c, there is at last the greater part; consequently is the less part. And as the square root of 5, which cannot be exactly extracted, makes a portion of both these parts, it is manifest that neither of them can be obtained in rational numbers.</p><p>Euclid makes great use of this problem, viz, in several parts of the 13th book of the Elements; and by means of it he constructs that notable proposition, viz the 10th of the 4th book, which is to construct an isosceles triangle having each angle at the base double the angle at the vertex.</p><p>EXTREMES <hi rend="italics">Conjunct,</hi> and <hi rend="smallcaps">Extremes</hi> Disjunct, in Spherical Trigonometry, are the former the two circular parts that lie next the assumed middle part, and the latter are the two that lie remote from the middle part. These were terms applied by lord Napier, in his universal theorem for resolving all right-angled and quadrantal spherical triangles, and published in his Logarithmorum Canonis Descriptio, an. 1614. In this theorem, Napier condenses into one rule, in two parts, the rules for all the cases of right-angled spherical triangles, which had been separately demonstrated by Pitiscus, Lansbergius, Copernicus, Regiomontanus, and others. In this theorem, neglecting the right angle, Napier calls the other five parts, circular parts, which are, the two legs about the right angle, and the complements of the other three, viz of the hypothenufe, and the two oblique angles. Then, taking any three of these five parts, one of them will be in the middle between the other two, and these two are the Extremes Conjunct when they are immediately adjacent to that middle part, or they are the Extremes Disjunct when they are each separated from the middle one by another part. Thus, the five parts being AB, AC, and <figure/> the complements of BC and of the two angles B and C: then if the three parts be AB, and the complements of the angle B and hypothenuse BC be taken, these three are contiguous to each other, the angle B lying in the middle between the other two; therefore the comp. of B is middle part, and AB with the comp. of BC the Extremes Conjunct. But if the three sides be taken; BC is equally separated from the two legs AB and AC, by two angles B and C; and therefore these two legs AB and AC are Extremes Disjunct, and the comp. of BC the middle part.</p><p>Napier's rule for resolving each case is in two parts, as below:</p><p>The rectangle contained by radius and the sine of the middle part, is equal to the rectangle of the tangents of the Extremes conjunct, or equal to the rectangle of the fines of the Extremes disjunct. Which rule comprehends all the cases that can happen in right-angled spherical triangles; in the application of which rule, the equal rectangles are divided into a proportion or analogy, in such manner that the term sought may be the last of <cb/> the four terms that are concerned, and consequently i<*>s corresponding term in the same rectangle must be the first of those terms.</p></div1><div1 part="N" n="EYE" org="uniform" sample="complete" type="entry"><head>EYE</head><p>, the organ of sight, consisting of several parts, and of such forms as best to answer the purpose for which it was formed.</p><p>As vision or sight is effected by a refraction of light through the humours of the eye to the bottom or farther internal part of it, where the images of external objects are formed on a fine expansion of the optic nerve, called the <hi rend="italics">Retina,</hi> and therefore the sorepart of the eye must be of a convex figure, and of such a precise degree of convexity as the particular refractive power of the several humours require for forming the image of an object at a given focal distance, viz, the diameter of the eye. Hence we sind,</p><p>1st; The external part of the eye-ball CD (Plate 2, fig. 8.) is a strong pellucid substance, properly convex, and which, when dried, has some resemblance to a piece of transparent horn, for which reason it is called the <hi rend="italics">Cornea,</hi> or horny coat of the eye.</p><p>2dly; Immediately behind this coat there is a fine clear humour which, from its likeness to water, is called the <hi rend="italics">Aqueous</hi> or watery humour, and is contained in the space between CD and GFE.</p><p>3dly; In this space there is a membrane or diaphragm, called the <hi rend="italics">Uvea,</hi> with a hole in the middle as at F, called the <hi rend="italics">Pupil,</hi> of a muscular contexture for altering the dimensions of that hole, for the adjusting or admitting a due quantity of light.</p><p>4thly; Just behind this diaphragm is placed a lenticular-formed substance GE, of a considerable consistence, called from its transparency the <hi rend="italics">Crystalline</hi> humour. This is contained in a sine tunic called the <hi rend="italics">Choroides,</hi> and is suspended in the middle of the eye by a ring of muscular fibres called the <hi rend="italics">Ligamentum Ciliare,</hi> as at G and E; by which means it is moved a little nearer to, or farther from, the bottom of the eye, to alter the focal distance.</p><p>5thly; All the remaining interior part of the eye, constituting the great body of it, from GHE to IMK, is made up of a large quantity of a jelly-like substance, called the <hi rend="italics">Vitreous</hi> or glassy humour; though it resembles glass in nothing except its transparency; it being most like the white of an egg of any thing.</p><p>6thly; On one side of the hinder part of the Eye, as at K, the optic nerve enters it from the brain, and is expanded over all the interior part of the eye to C and E quite around, the expansion being named the <hi rend="italics">Retina.</hi> On this delicate membrane, the image IM of every external object OB, is formed according to the optic laws of nature, in the following manner.</p><p>Let OB be any very distant object. Then a pencil of rays proceeding from any point L, will fall on the cornea CD, and be refracted by the aqueous humour under it, to a point in the axis of that pencil continued out. Then the radius of convexity of the cornea being nearly 1/3 of an inch; and the sine of incidence in air to that of refraction in the aqueous humour, being nearly as 4 to 3, supposing the rays parallel, or the object very far distant, the focal distance after the first refraction, by the proper theorem <hi rend="italics">mr</hi>/(<hi rend="italics">m</hi>-<hi rend="italics">n</hi>), will be found 1 1/3 inch from the cornea: <hi rend="italics">r</hi> being the radius 1/3, and <hi rend="italics">m</hi> to <hi rend="italics">n</hi> as 4 to 3. <pb n="464"/><cb/></p><p>The rays thus refracted by the cornea, fall converging on the crystalline humour, and tend to a point 1.228 inch behind it; also the radii of convexity in the said humour are 1/3 and 1/4 respectively; and the sine of incidence to that of refraction of the aqueous into the crystalline humour, being as 13 to 12; therefore, by this theorem <hi rend="italics">mdr</hi>/(<hi rend="italics">md</hi>-<hi rend="italics">nd</hi>+<hi rend="italics">nr</hi>), the focal distance after refraction in the crystalline, will be 1.02 inch from the fore part of it: where <hi rend="italics">m</hi> = 13, <hi rend="italics">n</hi> = 12, <hi rend="italics">r</hi> = 1/3, and <hi rend="italics">d</hi> = 1.228.</p><p>The rays now pass from the crystalline to the vitreous humour still in a converging state, and the sines of incidence and refraction being here as 12 to 13, as found by experiment; and since the surface of the vitreous humour is concave which receives the rays, and is the same with the convexity of the hinder surface of the crystalline, the radius will be the same, viz 1/4 of an inch. Therefore the focal distance after this third refraction will be found, by this theorem, <hi rend="italics">mdr</hi>/(<hi rend="italics">nd</hi>-<hi rend="italics">md</hi>+<hi rend="italics">nr</hi>), to be 6 tenths of an inch nearly from the hinder part of the cornea: where <hi rend="italics">m</hi>=12, <hi rend="italics">n</hi>=13, <hi rend="italics">r</hi>=1/4, and <hi rend="italics">d</hi>=.82; the thickness of the lens of the cornea being nearly 1/5 of an inch.</p><p>Now experience shews that the distance of the retina in the back part of the eye, behind the cornea, is nearly equal to that focal distance; and therefore it follows that all objects at a great distance have their images formed on the retina in the bottom or hinder part of the eye, and thus distinct vision is produced by this wonderful organ of optic sensation.</p><p>When the distance of objects is not very great, the focal distance, after the last refraction in the vitreous humour, will be a little increased; and to do this we can move the crystalline a little nearer the cornea by means of the ligamentum ciliare, and thus on all occasions it may be adjusted for a due focal distance for every distance of objects, excepting that which is less than 6 or 7 inches, in good eyes. Many are of opinion, however, that this is effected by a power in the eye to alter the convexity of the crystalline humour as occasion requires; though this is rather doubtful.</p><p>By what has been said it appears, that rays of light flowing from every part of an object OB, placed at a proper distance from the eye, will have an image IM thereby formed on the retina in the bottom of the eye; and since the rays OM, BI, which come from the extreme parts of the object, cross each other in the middle of the pupil, the position of the image IM will be contrary to that of the object, or inverted, as in the case of a lens.</p><p>The apparent place of any part of an object is in the axis, and conjugate focus of that pencil of rays by which that part or point is formed in the image. Thus, OM is the axis, and O the focus proper to the rays by which the point M in the image is made; therefore the sensation of the place of that part will be conceived in the mind to be at O; in like manner the idea of place belonging to the point I, will be referred in the axis IB, to the proper focus B; therefore the apparent place of the whole image IM, will be conceived in the mind to occupy all the space between O and B, and at the distance AL from the eye. <cb/></p><p>Hence likewise appears the reason, why we see an object upright by means of an inverted image; for since the apparent place of every point M will be in the axis MO at O; and this axis crossing the axis of the eye HL in the pupil, it follows, that the sensible place O of that point will lie, without the eye, on the contrary side of the axis of the eye, to that of the point in the eye; and since this is true of all other parts or points in the image, it is evident that the position of every part of the object will be on the contrary side of the axis to every corresponding part in the image, and therefore the whole object OB will have a contrary position to that of the image IM, or will appear upright.</p><p>If the convexity of the cornea CD happens not exactly to correspond to the diameter of the eye, considered as the natural focal distance, then the image will not be formed on the retina, and consequently no distinct vision can be effected in such an eye.</p><p>If the cornea be too convex, the focal distance in the eye will be less than its diameter, and the image will be formed short of the retina. Hence the reason why people having such eyes are obliged to hold things very near them, to lengthen the focal distances; and also why they use concave glasses to counteract or remedy the excess of convexity, in order to view distant objects distinctly.</p><p>When the eye has less than a just degree of convexity, or is too flat, as is generally the case with old eyes, by a natural deficiency of the aqueous humour, then the rays tend to a point or focus beyond the retina or bottom of the eye; and to supply this want of convexity in the cornea, we use convex lenses in those frames called spectacles, or visual glasses.</p><p>Since the rays of light OA, BA, which constitute the visual angle OAB, will, when they are intercepted by a lens, be refracted sooner to the axis; the said angle will thereby be enlarged, and the object of course become magnisied; which is the reason why those lenses are called magnisiers, or reading-glasses.</p><p>The dimensions, or magnitude, of an object OB, are judged of by the quantity of the angle OAB which it subtends at the eye. For if the same object be placed at two different distances L and N, the angles OAB, <hi rend="italics">o</hi>A<hi rend="italics">b,</hi> which in these two places it subtends at the eye, will be of different magnitude; and the lineal dimensions, viz length and breadth, will be at N and at L, as the angle <hi rend="italics">o</hi>A<hi rend="italics">b</hi> is to the angle OAB. But the surfaces of the objects will be as the squares of those angles, and the solidities as the cubes of them.</p><p>It is found by experience, that two points O, L, in any object, will not be distinctly seen by the Eye, till they are near enough to subtend an angle OAL of one minute. And hence when objects, however large they may be, are so remote as not to be seen under an angle of one minute, they cannot properly be said to have any apparent dimensions or magnitude at all; such as is the case of the large bodies of the planets, comets, and fixed stars. But the optic science has supplied means of enlarging this natural small angle under which most distant objects appear, and thereby increasing their apparent magnitudes to a very surprising degree, in the instance of that noble instrument the telescope.</p><p>On the other hand, there are in creation an infinity <pb n="465"/><cb/> of objects, of such small dimensions, that they will not subtend the requisite angle, if brought to the nearest limits of distinct vision, viz 6, 7, or 8 inches from the Eye, as found by experience; and therefore to render them visible at a very near distance, we have a variety of glasses, and instruments of different constructions, usually called microscopes, by which those minute objects appear many thousand times larger than to the naked Eye; and thereby enrich the mind with discoveries of the sublimest nature, in regard to creative power, wisdom, and œconomy.</p><p>EYE-<hi rend="italics">glass,</hi> in Optical Instruments, is that which is next the Eye in using the machine. This is usually a lens convex on both sides; but Eustachia Divini long <cb/> since invented a microscope of this kind, the power of which he places very greatly above that of the common sort; and this chiefly depending on the Eye-glass, which was double, consisting of two plano-convex glasses, so placed as to touch one another in the middle of their convex surface. This instrument is well spoken of by Fabri in his Optics, and as possessing this peculiar excellence, that it shews all the objects flat, and not crooked, and takes in a large area, though it magnifies very much.</p><p><hi rend="italics">Bull's</hi> EYE, a star of the first magnitude, in the Eye of the constellation Taurus, the bull, and by the Arabs called <hi rend="italics">Aldebaran.</hi> </p></div1></div0><div0 part="N" n="F" org="uniform" sample="complete" type="alphabetic letter"><head>F</head><cb/><div1 part="N" n="FACE" org="uniform" sample="complete" type="entry"><head>FACE</head><p>, or <hi rend="smallcaps">Façade</hi>, in Architecture, is sometimes used for the front or outward part of a building, which immediately presents itself to the eye; or the side where the chief entrance is, or next the street, &c.</p><div2 part="N" n="Face" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Face</hi></head><p>, <hi rend="smallcaps">Facia</hi>, or <hi rend="smallcaps">Fascia</hi>, also denotes a flat member, having a considerable breadth, and but a small projecture. Such are the bands of an architrave, larmier, &c.</p></div2><div2 part="N" n="Face" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Face</hi></head><p>, in Astrology, is used for the 3d part of a sign. —Each sign is supposed to be divided into three faces, of 10 degrees each: the first 10 degrees compose the first face; the next 10 degrees, the 2d face; and the last 10, the 3d face.—Venus is in the 3d face of Taurus; that is, in the last 10 degrees of it.</p><p><hi rend="smallcaps">Faces</hi> <hi rend="italics">of a Bastion,</hi> in Fortification, are the two foremost sides, reaching from the flanks to the outermost point of the bastion, where they meet, and form the saliant angle of the bastion. These are usually the first parts that are undermined, or beaten down; because they reach the farthest out, are the least flanked, and are therefore the weakest.</p><p><hi rend="smallcaps">Face</hi> <hi rend="italics">of a Place,</hi> is the extent between the outermost points of two adjacent bastions; containing the curtain, the two flanks, and the two faces of those bastions that look towards each other. This is otherwise called the Tenaille of the place.</p><p><hi rend="smallcaps">Face</hi> <hi rend="italics">Prolonged,</hi> is that part of a line of defence rasant, which is between the angle of the epaule or shoulder of a bastion and the curtain; or the line of a defence rasant diminished by the face of the bastion. <cb/></p></div2></div1><div1 part="N" n="FACIA" org="uniform" sample="complete" type="entry"><head>FACIA</head><p>, in Architecture. See <hi rend="smallcaps">Face</hi>, and F<hi rend="smallcaps">ASCIA.</hi></p></div1><div1 part="N" n="FACTORS" org="uniform" sample="complete" type="entry"><head>FACTORS</head><p>, in Multiplication, in Arithmetic, a name given to the two numbers that are multiplied together, viz, the multiplicand and multiplier; so called because they are to facere productum, make or constitute the factum or product.</p></div1><div1 part="N" n="FACTUM" org="uniform" sample="complete" type="entry"><head>FACTUM</head><p>, the product of two quantities multiplied together. As, the factum of 3 and 4 is 12; and the factum of 2<hi rend="italics">a</hi> and 5<hi rend="italics">b</hi> is 10<hi rend="italics">ab.</hi></p><p>FACULÆ, in Astronomy, a name given by Scheiner, and others after him, to certain bright spots on the sun's disc, that appear more bright and lucid than the rest of his body.</p><p>Hevelius assures us that, on July 20, 1634, he observed a facula whose breadth was equal to a 3d part of the sun's diameter. He says too that the maculæ often change into Faculæ; but these seldom or never into maculæ. And some authors even contend that all the maculæ degenerate into Faculæ before they quite disappear. Many authors, after Kircher and Scheiner, have represented the sun's body full of bright, fiery spots, which they conceive to be a sort of volcanos in the body of the sun: but Huygens, and others of the latest and best observers, sinding that the best telescopes discover nothing of the matter, agree entirely to explode the phenomena of Faculæ. All the foundation he could see for the notion of Faculæ, he says, was, that in the darkish clouds which frequently surround the maculæ, there are sometimes seen little points or sparks <pb n="466"/><cb/> brighter than the rest. Their cause is attributed by these authors to the tremulous agitation of the vapours near our earth; the same as sometimes shews a little unevenness in the circumference of the sun's disc when viewed through a telescope. Strictly then, the Faculæ are not eructations of fire and flame, but refractions of the sun's rays in the rarer exhalations, which, being condensed near that shade, seem to exhibit a light greater than that of the sun.</p></div1><div1 part="N" n="FACULTY" org="uniform" sample="complete" type="entry"><head>FACULTY</head><p>, denotes the several parts of an university, divided according to the arts or sciences taught or professed there. In most universities there are four Faculties; that of arts, which includes philosophy and the humanities or languages, and is the most ancient and extensive; the 2d is that of theology; the 3d, that of medicine; and the 4th, jurisprudence, or laws.</p></div1><div1 part="N" n="FALCATED" org="uniform" sample="complete" type="entry"><head>FALCATED</head><p>, one of the phases of the planets, vulgarly called horned. The astronomers say, the moon, or any planet, is Falcated, when the enlightened part appears in form of a crescent, like a sickle, or reaping-hook, which by the Latins is called falx. The moon is Falcated while she moves from the 3d quarter to the conjunction, and so on from hence to the first quarter; the bright part appearing then like a crescent, viz during the first and last quarters. But during the 2d and 3d quarters, the light part appears gibbous, and the dark part Falcated.</p><p>FALCON or <hi rend="smallcaps">Faucon</hi>, and <hi rend="smallcaps">Falconet</hi> or F<hi rend="smallcaps">AUCONET</hi>, certain old species of cannon, now long disused.</p></div1><div1 part="N" n="FALL" org="uniform" sample="complete" type="entry"><head>FALL</head><p>, the descent or natural motion of bodies towards the centre of the earth, &c. Galileo first discovered the ratio of the acceleration of falling bodies; viz, that the spaces descended from rest are as the squares of the times of descent; or, which comes to the same thing, that if the whole time of Falling be divided into any number of equal parts, whatever space it falls through in the first part of the time, it will Fall 3 times as far in the 2d part of time, and 5 times as far in the 3d portion of time, and so on, according to the uneven numbers 1, 3, 5, 7, &c. See A<hi rend="smallcaps">CCELERATION, Descent, Gravity</hi>, &c.</p><p>FALSE-<hi rend="smallcaps">BRAYE</hi>, in Fortification. See F<hi rend="smallcaps">AUSSEBRAYE.</hi></p><p><hi rend="smallcaps">False</hi> <hi rend="italics">Position,</hi> in Arithmetic. See <hi rend="smallcaps">Position.</hi></p><p><hi rend="smallcaps">False</hi> <hi rend="italics">Root,</hi> a name given by Cardan, to the negative roots of equations, and numbers. So the root of 9 may be either 3 or - 3, the former he calls the true, and the latter the false or fictitious root; also of this equation <hi rend="italics">x</hi><hi rend="sup">2</hi> - <hi rend="italics">x</hi> = 6, the two roots are 3 and - 2, the former true, and the latter False.</p></div1><div1 part="N" n="FASCIA" org="uniform" sample="complete" type="entry"><head>FASCIA</head><p>, in Architecture. See <hi rend="smallcaps">Facia</hi> and <hi rend="smallcaps">Face.</hi></p><p>FASCIÆ, in Astronomy, are certain stripes or rows of bright parts, observed on the bodies of some of the planets, like swathes, bands, or belts; especially on the planet Jupiter.</p><p>The Fasciæ, or belts of Jupiter, are more lucid than the rest of the dise, and are terminated by parallel lines. They are sometimes broader and sometimes narrower; nor do they always possess the same part of the disc.</p><p>M. Huygens also observed a very large kind of Fascia in Mars, in the year 1656; but it was darker than the rest of the disc, and occupied the middle part of it.</p></div1><div1 part="N" n="FASCINES" org="uniform" sample="complete" type="entry"><head>FASCINES</head><p>, in Fortification, are faggots made of <cb/> the twigs and small branches of trees and brush wood, bound up in bundles; these, being mixed with earth, serve to fill up ditches, to make the parapets of trenches, batteries, &c.</p></div1><div1 part="N" n="FATHOM" org="uniform" sample="complete" type="entry"><head>FATHOM</head><p>, an English measure of the length of 6 feet or 2 yards; and is taken from the utmost extent of both arms when stretched into a right line.</p><p>FATUUS <hi rend="italics">Ignis.</hi> See <hi rend="smallcaps">Ignis</hi> <hi rend="italics">Fatuus.</hi></p></div1><div1 part="N" n="FAUCON" org="uniform" sample="complete" type="entry"><head>FAUCON</head><p>, and <hi rend="smallcaps">Fauconet</hi>, the same as Falcon and Falconet; the old names of certain species of ordnance; which, as well as many other names; are now no longer in use, as it has been for some time the practice to denominate the several sizes of cannon from the weight of their ball, instead of calling them by those fanciful and unmeaning names.</p><p>FAUSSE-<hi rend="smallcaps">Braye</hi>, in Fortification, an elevation of earth, about three feet above the level ground; round the foot of the rampart on the outside, defended by a parapet about four or five fathoms distant from the upper parapet, which parts it from the berme, and the edge of the ditch. The Fausse-braye is the same with what is otherwise called Chemin des rondes, and Basse enceinte; and its use is for the defence of the ditch.</p><p>FEATHER-<hi rend="smallcaps">EDGED</hi>, is a term used by workmen, for such boards as are thicker on one edge, or side, than on the other.</p></div1><div1 part="N" n="FEBRUARY" org="uniform" sample="complete" type="entry"><head>FEBRUARY</head><p>, the 2d month of the year, containing 28 days for three years, and every fourth year 29 days.—In the first ages of Rome, February was the last month of the year, and preceded January, till the Decemviri made an order that February should be the 2d month of the year, and come after January.</p></div1><div1 part="N" n="FELLOWSHIP" org="uniform" sample="complete" type="entry"><head>FELLOWSHIP</head><p>, <hi rend="smallcaps">Company</hi>, or <hi rend="smallcaps">Partnership</hi>, is a rule in arithmetic, of great use in balancing accounts among merchants, and partners in trade, teaching how to assign to every one of them his due share of the gain or loss, in proportion to the stock he has contributed, and the time it has been employed, or according to any other conditions. Or, more generally, it is a method of dividing a given number, or quantity, into any number of parts, that shall have any assigned ratios to one another. And hence comes this general rule: Having added into one sum the several numbers that express the proportions of the parts, it will be, As that sum of the proportional numbers: Is to the given quantity that is to be divided:: So is each proportional number: To the corresponding share of the given quantity.</p><p><hi rend="italics">For Ex.</hi> Suppose it be required to divide the number 120 into three parts that shall be in proportion to each other as the numbers 1, 2, 3.—Here 120 is the quantity to be divided, and 6 is the sum of the numbers 1, 2, and 3, which express the proportions of the parts; therefore as</p><p>This rule is usually distinguished into two cases, one in which time is concerned, or in which the stocks of partners are continued for different times; and the other in which time is not considered; this latter being called Single Fellowship, and the former Double Fellowship. <pb n="467"/><cb/></p><p><hi rend="italics">Single</hi> <hi rend="smallcaps">Fellowship</hi>, or <hi rend="smallcaps">Fellowship</hi> <hi rend="italics">without Time,</hi> is the case in which the times of continuance of the shares of partners are not considered, because they are all the same; and in this case, the rule will be as above, viz, As the whole stock of the partners: Is to the whole gain or loss:: So is each one's particular stock: To his share of the gain or loss.</p><p><hi rend="italics">Ex.</hi> Two partners, A and B, form a joint stock, of which A contributed 75l, and B 45l; with which they gain 30l: how much of it must each one have?</p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Fellowship</hi>, or <hi rend="smallcaps">Fellowship</hi> <hi rend="italics">with Time,</hi> is the case in which the times of the stocks continuing are considered, because they are not all the same.</p><p>In this case, the shares of the gain or loss must be proportional, both to the several shares of the stock, and to the times of their continuance, and therefore proportional to the products of the two. Hence this Rule: Multiply each particular share of the stock by the time of its continuance, and add all the products together into one sum; then say, As that sum of the products: Is to the whole gain or loss:: So is each several product: To the corresponding share of the gain or loss.</p><p><hi rend="italics">For Ex.</hi> A had in company 50l. for 4 months, and B 60l. for 5 months; and their gain was 24l: how must it be divided between them? <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">200</cell><cell cols="1" rows="1" rend="align=right" role="data">300</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">300</cell><cell cols="1" rows="1" role="data"/></row></table></p></div1><div1 part="N" n="FERGUSON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FERGUSON</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, an eminent experimental philosopher, mechanist, and astronomer, was born in Bamffshire, in Scotland, 1710, of very poor parents. At the very earliest age his extraordinary genius began to unfold itself. He first learned to read, by overhearing his father teach his elder brother: and he had made this acquisition before any one suspected it. He soon discovered a peculiar taste for mechanics, which first arose on seeing his father use a lever. He pursued this study a considerable length, while he was yet very young; and made a watch in wood-work, from having once seen one. As he had at first no instructor, nor any help from books, every thing he learned had all the merit of an original discovery; and such, with inexpressible joy, he believed it to be.</p><p>As soon as his age would permit, he went to service; in which he met with hardships, which rendered his constitution feeble through life. While he was servant to a farmer (whose goodness he acknowledges in the modest and humble account of himself which he prefixed to one of his publications), he contemplated and learned to know the stars, while he tended the sheep; and began the study of astronomy, by laying down, from his <cb/> own observations only, a celestial globe. His kind master, observing these marks of his ingenuity, procured him the countenance and assistance of some neighbouring gentlemen. By their help and instructions he went on gaining farther knowledge, having by their means been taught arithmetic, with some algebra, and practical geometry. He had got some notion of drawing, and being sent to Edinburgh, he there began to take portraits in miniature, at a small price; an employment by which he supported himself and family for several years, both in Scotland and England, while he was pursuing more serious studies. In London he sirst published some curious astronomical tables and calculations; and afterwards gave public lectures in experimental philosophy, both in London and most of the country towns in England, with the highest marks of general approbation. He was elected a fellow of the Royal Society, and was excused the payment of the admission fee and the usual annual contributions. He enjoyed from the king a pension of 50 pornds a year, besides other occasional presents, which he privately accepted and received from different quarters, till the time of his death; by which, and the fruits of his own labours, he left behind him a sum to the amount of about six thousand pounds, instead of which all his friends had always entertained an idea of his great poverty. He died in 1776, at 66 years of age, though he had the appearance of many more years.</p><p>Mr. Ferguson must be allowed to have been a very uncommon genius, especially in mechanical contrivances and executions, for he executed many machines himself in a very neat manner. He had also a good taste in astronomy, with natural and experimental philosophy, and was possessed of a happy manner of explaining himself in an easy, clear, and familiar way. His general mathematical knowledge, however, was little or nothing. Of algebra he understood but little more than the notation; and he has often told me he could never demonstrate one proposition in Euclid's Elements; his constant method being to satisfy himself, as to the truth of any problem, with a measurement by scale and compasses. He was a man of a very clear judgment in any thing that he professed, and of unwearied application to study: benevolent, meek, and innocent in his manners as a child: humble, courteous, and communicative: instead of pedantry, philosophy seemed to produce in him only diffidence and urbanity.</p><p>The list of Mr. Ferguson's public works, is as follows:</p><p>1. Astronomical Tables and Precepts, for calculating the true times of New and Full Moons, &c; 1763. —2. Tables and Tracts, relative to several arts and sciences; 1767.—3. An Easy Introduction to Astronomy, for Young Gentlemen and Ladies; 2d edit. 1769.—4. Astronomy explained upon Sir Isaac Newton's Principles; 5th edit. 1772.—5. Lectures on Select Subjects in Mechanics, Hydrostatics, Pneumatics, and Optics; 4th edit. 1772.—6. Select Mechanical Exercises; with a short Account of the life of the author, by himself; 1773.—7. The Art of Drawing in Perspective made easy; 1775.—8. An Introduction to Electricity; 1775.—9. Two Letters to the Rev. Mr. John Kennedy; 1775.—10. A Third Letter to the Rev. Mr. John Kennedy; 1775. <pb n="468"/><cb/></p></div1><div1 part="N" n="FERMAT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FERMAT</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, counsellor of the parliament of Toulouse, in France, who flourished in the 17th century, and died in 1663. He was a general scholar, and a universal genius, cultivating jurisprudence, poetry, and mathematics, but especially the latter, for his amusement. He was contemporary and intimately connected with Mersenne, Torricelli, Des Cartes, Pascal, Roberval, Huygens, Frenicle, and Carcavi, and several others the most celebrated philosophers of their time. He was a first-rate mathematician, and possessed the finest taste for pure and genuine geometry, which he contributed greatly to improve, as well as algebra.</p><p>Fermat was author of 1. A Method for the Quadrature of all sorts of Parabolas.—2. Another on Maximums and Minimums: which serves not only for the determination of plane and solid problems; but also for drawing tangents to curve lines, finding the centres of gravity in solids, and the resolution of questions concerning numbers: in short a method very similar to the Fluxions of Newton.—3. An Introduction to Geometric Loci, plane and solid.—4. A Treatise on Spherical Tangencies: where he demonstrates in the solids, the same things as Vieta demonstrated in planes.—5. A Restoration of Apollonius's two books on Plane Loci. —6. A General Method for the dimension of Curve Lines. Besides a number of other smaller pieces, and many letters to learned men; several of which are to be found in his <hi rend="italics">Opera Varia Mathematica,</hi> printed at Toulouse, in folio, 1679.</p></div1><div1 part="N" n="FERMENTATION" org="uniform" sample="complete" type="entry"><head>FERMENTATION</head><p>, an intestine motion, arising spontaneously among the small and insensible particles of a mixed body, thereby producing a new disposition, and a different combination of those parts. Fermentation differs from dissolution, as the cause from its effect, the latter being only a result or effect of the former.</p></div1><div1 part="N" n="FESTOON" org="uniform" sample="complete" type="entry"><head>FESTOON</head><p>, in Architecture &c, a decoration in form of a garland or cluster of flowers.</p><p>FICHANT <hi rend="smallcaps">Flank.</hi> See <hi rend="smallcaps">Flank.</hi></p><p><hi rend="smallcaps">Fichant</hi> <hi rend="italics">Line of Defence.</hi> See <hi rend="smallcaps">Fixed</hi> <hi rend="italics">Line of Defence.</hi></p><p>FIELD-<hi rend="smallcaps">Fort.</hi> See <hi rend="smallcaps">Fortine.</hi></p><p><hi rend="smallcaps">Field</hi>-<hi rend="italics">Pieces,</hi> are small cannon, usually carried along with an army in the field: such as, one pounders, one and a half, two, three, four, six, nine, and 12 pounders; which, being light and small, are easily carried.</p><p><hi rend="smallcaps">Field</hi>-<hi rend="italics">Staff,</hi> is a staff carried by the gunners, in which they screw lighted matches, when they are on service; which is called arming the Field-staffs. See <hi rend="smallcaps">Linstock.</hi></p><p><hi rend="smallcaps">Field</hi> <hi rend="italics">of View,</hi> or <hi rend="italics">of Vision,</hi> is the whole space or extent within which objects can be seen through an optical machine, or at one view of the eye without turning it.</p><p>The precise limits of this space are not easily ascertained, for the natural view of the eye. In looking at a small distance, we have an imperfect glimpse of objects through almost the extent of a hemisphere, or at least for above 60 degrees each way from the optic axis; but towards the extremity of this space, objects are very imperfectly seen; and the diameter of the field of distinct vision does not subtend an angle of more than 5 degrees at most, so that the diameter of a distinct image on the retina is less than 6/100 of an inch; but it is probably much less. <cb/></p><p><hi rend="smallcaps">Field</hi>-<hi rend="italics">Book,</hi> in Surveying, a book used for setting down angles, distances, and other things, remarkable in taking surveys.</p><p>The pages of the Field-book may be conveniently divided into three columns. In the middle column are to be entered the angles taken at the several stations by the theodolite, with the distances measured from station to station. And the offsets, taken with the offset-staff, on either side of the station line, are to be entered in the columns on either side of the middle column, according to their position, on the right or left, with respect to that line: also on the right or left of these are to be set down the names and characters of the objects, with proper remarks, &c. See a specimen in my Treatise on Mensuration, pa. 517, ed. 2d.</p></div1><div1 part="N" n="FIFTH" org="uniform" sample="complete" type="entry"><head>FIFTH</head><p>, in Music, one of the harmonical intervals or concords; called by the ancients Diapente.</p><p>The Fifth is the 3d in order of the concords, and the ratio of the chords that produce it, is that of 3 to 2. It is called Fifth, because it contains five terms, or sounds, between its extremes, and four degrees; so that in the natural scale of music it comes in the 5th place, or order, from the fundamental.—The imperfect, or defective Fifth, by the ancients called Semidiapente, is less than the Fifth by a mean semitone.</p></div1><div1 part="N" n="FIGURAL" org="uniform" sample="complete" type="entry"><head>FIGURAL</head><p>, the same as <hi rend="smallcaps">Figurate</hi> numbers; which see.</p><p>FIGURATE <hi rend="italics">Numbers,</hi> such as do or may represent some geometrical figure, in relation to which they are always considered; as triangular, pentagonal, pyramidal, &c, numbers.</p><p>Figurate numbers are distinguished into orders, according to their place in the scale of their generation, being all produced one from another, viz, by adding continually the terms of any one, the successive sums are the terms of the next order, beginning from the first order which is that of equal units 1, 1, 1, 1, &c; then the 2d order consists of the successive sums of those of the 1st order, forming the arithmetical progression 1, 2, 3, 4, &c; those of the 3d order are the successive sums of those of the 2d, and are the triangular numbers 1, 3, 6, 10, 15, &c; those of the 4th order are the successive sums of those of the 3d, and are the pyramidal numbers 1, 4, 10, 20, 35, &c; and so on, as below: <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Order.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Name.</hi></cell><cell cols="1" rows="1" rend="colspan=6 align=center" role="data"><hi rend="italics">Numbers.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1.</cell><cell cols="1" rows="1" rend="align=center" role="data">Equals.</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" rend="align=right" role="data">1,</cell><cell cols="1" rows="1" rend="align=right" role="data">1,</cell><cell cols="1" rows="1" rend="align=right" role="data">1,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2.</cell><cell cols="1" rows="1" role="data">Arithmeticals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" rend="align=right" role="data">3,</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">5,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3.</cell><cell cols="1" rows="1" role="data">Triangulars,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell><cell cols="1" rows="1" rend="align=right" role="data">10,</cell><cell cols="1" rows="1" rend="align=right" role="data">15,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4.</cell><cell cols="1" rows="1" role="data">Pyramidals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">10,</cell><cell cols="1" rows="1" rend="align=right" role="data">20,</cell><cell cols="1" rows="1" rend="align=right" role="data">35,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5.</cell><cell cols="1" rows="1" role="data">2d Pyramidals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" rend="align=right" role="data">15,</cell><cell cols="1" rows="1" rend="align=right" role="data">35,</cell><cell cols="1" rows="1" rend="align=right" role="data">70,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6.</cell><cell cols="1" rows="1" role="data">3d Pyramidals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">6,</cell><cell cols="1" rows="1" rend="align=right" role="data">21,</cell><cell cols="1" rows="1" rend="align=right" role="data">56,</cell><cell cols="1" rows="1" rend="align=right" role="data">126,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">7.</cell><cell cols="1" rows="1" role="data">4th Pyramidals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" rend="align=right" role="data">28,</cell><cell cols="1" rows="1" rend="align=right" role="data">84,</cell><cell cols="1" rows="1" rend="align=right" role="data">210,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table></p><p>The above are all considered as different sorts of triangular numbers, being formed from an arithmetical progression whose common difference is 1. But if that common difference be 2, the successive sums will be the series of square numbers: if it be 3, the series will be pentagonal numbers, or pentagons; if it be 4, the series will be hexagonal numbers, or hexagons; and so on. Thus: <pb n="469"/><cb/> <table><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data"><hi rend="italics">Arithmeticals.</hi></cell><cell cols="1" rows="1" rend="colspan=5 align=center" role="data">1st <hi rend="italics">Sums,</hi> or <hi rend="italics">Polygons.</hi></cell><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">2d <hi rend="italics">Sums,</hi> or 2d <hi rend="italics">Polygons.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell><cell cols="1" rows="1" role="data">Tri.</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" role="data">10,</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" rend="align=right" role="data">7,</cell><cell cols="1" rows="1" role="data">Sqrs.</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">9,</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" role="data">14,</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" rend="align=right" role="data">10,</cell><cell cols="1" rows="1" role="data">Pent.</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" rend="align=right" role="data">12,</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">6,</cell><cell cols="1" rows="1" role="data">18,</cell><cell cols="1" rows="1" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" rend="align=right" role="data">13,</cell><cell cols="1" rows="1" role="data">Hex.</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">6,</cell><cell cols="1" rows="1" rend="align=right" role="data">15,</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" role="data">22,</cell><cell cols="1" rows="1" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4 align=center" role="data">&c.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>And the reason of the names triangles, squares, pentagons, hexagons, &c, is, that those numbers may be placed in the form of these regular figures or polygons, as here below: <figure><head><hi rend="italics">Triangles.</hi></head></figure> <figure><head><hi rend="italics">Squares.</hi></head></figure> <figure><head><hi rend="italics">Pentagons.</hi></head></figure> <figure><head><hi rend="italics">Hexagons.</hi></head></figure></p><p>But the Figurate numbers of any order may also be found without computing those of the preceding orders; which is done by taking the successive products of as many of the terms of the arithmeticals 1, 2, 3, 4, 5, &c, in their natural order, as there are units in the number which denominates the order of Figurates required, and dividing those products always by the first product: thus, the triangular numbers are found by dividing the products 1 X 2, 2 X 3, 3 X 4, 4 X 5, &c, each by the 1st pr. 1 X 2; the first pyramids by dividing the products 1 X 2 X 3, 2 X 3 X 4, 3 X 4 X 5, &c, by the first 1 X 2 X 3. And, in general, the figurate numbers of any order <hi rend="italics">n,</hi> are found by substituting successively 1, 2, 3, 4, 5, &c, instead of <hi rend="italics">x</hi> in this general expression ; where the factors in the numerator and denominator are supposed to be multiplied together, and to be continued till the number in each be less by 1 than that which expresses the <cb/> order of the Figurates required. See Maclaurin's Fluxions, art. 351, in the notes; also Simpson's Algebra, pa. 213; or Malcolm's Arithmetic, pa. 396, where the subject of Figurates is treated in a very extensive and perspicuous manner.</p></div1><div1 part="N" n="FIGURE" org="uniform" sample="complete" type="entry"><head>FIGURE</head><p>, in general, denotes the surface or terminating extremes of a body.——All finite bodies have some figure, form, or shape; whence, figurability is reckoned among the essential properties of body, or matter: abody without Figure, would be an infinite body.</p><div2 part="N" n="Figures" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figures</hi></head><p>, in Architecture and Sculpture, denote representations of things made in solid matter; such as statures, &c.</p></div2><div2 part="N" n="Figures" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figures</hi></head><p>, in Arithmetic, are the numeral characters, by which numbers are expressed or written, as the ten digits, 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These are usually called the Arabic, and Indian figures, from which people it is supposed they have been derived. They were brought into Europe by the Moors of Spain, and into England about 1130, as Dr. Wallis apprehends: see his Algebra, pa. 9. However, from some ancient dates, supposed to consist wholly or in part of Arabian figures, some have concluded that these Figures, originally Indian, were known and used in this country at least as early as the 10th century. The oldest date discovered by Dr. Wallis, was on a chimney piece, at Helmdon, in Northamptonshire, thus M133, that is 1133. Other dates discovered fince, are 1090, at Colchester, in Essex; M16 or 1016, at Widgel-hall, near Buntingford, in Hertfordshire; 1011 on the north front of the parish church of Rumsey in Hampshire; and 975 over a gate-way at Worcester.</p><p>Dr. Ward, however, has urged several objections against the antiquity of these dates. As no example occurs of the use of these figures in any ancient manuscript, earlier than some copies of Johannes de Sacro Bosco, who died in 1256, he thinks it strange that these Figures should have been used by artificers so long before they appear in the writings of the learned; and he also disputes the fact. The Helmdon date, according to him, should be 1233; the Colchester date 1490; that at Widgel-hall has in it no Arabic Figures, the 1 and 6 being I and G, the initial letters of a name; and the date at Worcester consists, he supposes, of Roman numerals, being really MXV. Martyn's Abridg. Philos. Trans. vol. 9, pa. 420.</p><p>Mr. Gibbon observes (in his History of the Decline and Fall of the Roman Empire, vol. v. pa. 321). that “under the reign of the caliph Waled, the Greek language and characters were excluded from the accounts of the public revenue. If this change was productive of the invention or familiar use of our present numerals, the Arabic characters or cyphers, as they are commonly styled, a regulation of office has promoted the most important discoveries of arithmetic, algebra, and the mathematical sciences.”</p><p>On the other hand it may be observed that, “according to a new, though probable notion, maintained by M. de Villaison) Anecdota Græca, tom. ii. p. 152, 157), our cyphers are not of Indian or Arabic invention. They were used by the Greek and Latin arithmeticians long before the age of Boethius. After the extinction of science in the west, they were adopted in the Arabic versions from the original manuscripts, <pb n="470"/><cb/> and restored to the Latins about the eleventh century.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, in Astrology, a description, draught, or construction of the state and disposition of the heavens, at a certain point of time; containing the places of the planets and stars, marked down in a Figure of 12 triangles, called houses. This is also called a Horoscope, and Theme.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, <hi rend="italics">of an Eclipsc,</hi> in Astronomy, denotes a representation upon paper &c, of the path or orbit of the sun or moon, during the time of the eclipse; with the different phases, the digits eclipsed, and the beginning, middle, and end of darkness, &c.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, or <hi rend="italics">Delineation,</hi> of the full moon, such as, viewed through a telescope with two convex glasses, is of considerable use in observations of eclipses, and conjunctions of the moon with other luminaries. In this Figure are usually represented the maculæ or spots of the moon, marked by numbers; beginning with the spots that usually enter first within the shade at the time of the eclipses, and also emerge the first.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, in Conic Sections, according to Apollonius, is the rectangle contained under the latus-rectum and the transverse axis, in the ellipse and hyperbola.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, in Fortification, is the plan of any fortified place; or the interior polygon, &c. When the sides, and the angles, are all equal, it is called a regular Figure; but when unequal, an irregular one.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, in Geomancy, is applied to the extremes of points, lines, or numbers, thrown or cast at random: on the combinations or variations of which, the sages of this art found their divinations.</p></div2><div2 part="N" n="Figure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Figure</hi></head><p>, in Geometry, denotes a surface or space inclosed on all sides; and is either superficial or solid; superficial when it is inclosed by lines, and solid when it is inclosed or bounded by surfaces.</p><p>Figures are either straight, curved, or mixed, according as their bounds are straight, or curved, or both.— The exterior bounds of a Figure, are called its sides; the lowest side, its base; and the angular point opposite the base, the vertex of the Figure; also its height, is the distance of the vertex from the base, or the perpendicular let fall upon it from the vertex.</p><p>For <hi rend="italics">Figures,</hi> equal, equiangular, equilateral, circumscribed, inscribed, plane, regular, irregular, similar, &c; see the respective adjectives.</p><p><hi rend="italics">Apparent</hi> <hi rend="smallcaps">Figure</hi>, in Optics, that Figure, or shape, under which an object appears, when viewed at a distance. This is often very different from the true figure; for a straight line viewed at a distance may appear but as a point; a surface as a line; a solid as a surface; and a crooked figure as a straight one. Also, each of these may appear of different magnitudes, and some of them of different shapes, according to their situation with regard to the eye. Thus an arch of a circle may appear a straight line; a square or parallelogram, a trapezium, or even a triangle; a circle, an ellipsis; angular magnitudes, round; a sphere, a circle; &c.</p><p>Also any small light, as a candle, seen at a distance in the dark, will appear magnified, and farther off than it really is. Add to this, that when several objects are seen at a distance, under angles that are so small as to be insensible, as well as each of the angles subtended by <cb/> any one of them, and that next to it; then all these objects appear not only as contiguous, but as constituting and seeming but one continued magnitude.</p><p><hi rend="smallcaps">Figure</hi> <hi rend="italics">of the Sines, Cosines, Versed-sines, Tangents,</hi> or <hi rend="italics">Secants, &c,</hi> are Figures made by conceiving the circumference of a circle extended out in a right line, upon every point of which are erected perpendicular ordinates equal to the Sines, Cosines, &c, of the corresponding arcs; and then drawing the curve line through the extremity of all these ordinates; which is then the Figure of the Sines, Cosines, &c.</p><p>It would seem that these Figures took their rise from the circumstance of the extension of the meridian line by Edward Wright, who computed that line by collecting the successive sums of the secants, which is the same thing as the area of the Figure of the secants, this being made up of all the ordinates, or secants, by the construction of the Figure. And in imitation of this, the Figures of the other lines have been invented. By means of the Figure of the secants, James Gregory shewed how the logarithmic tangents may be constructed, in his Exercitationes Geometricæ, 4to, 1668. <hi rend="center"><hi rend="italics">Construction of the Figures of Sines, Cosines, &c.</hi></hi></p><p>Let ADB &c (fig. 1) be the circle, AD an arc, DE its sine, CE its cosine, AE the versed sine, AF the tangent, GH the cotangent, CF the secant, and CH the cosecant. Draw a right line <hi rend="italics">aa</hi> equal to the whole circumference ADGBA of the circle, upon which lay off also the lengths of several arcs, as the arcs at every 10°, from 0 at <hi rend="italics">a,</hi> to 360° at the other end at <hi rend="italics">a;</hi> upon these points raise perpendicular ordinates, upwards or downwards, according as the sine, cosine, &c, is affirmative or negative in that part of the circle; lastly, upon these ordinates set off the length of the sines, cosines, &c, corresponding to the arcs at those points of the line or circumference <hi rend="italics">aa,</hi> drawing a curve line through the extremities of all these ordinates; which will be the Figure of the sines, cosines, versedsines, tangents, cotangents, secants, and cosecants, as in the annexed Figures. Where it may be observed, that the following curves are the same, viz, those of the sines and cosines, those of the tangents and cotangents, and those of the secants and cosecants; only some of their parts a little differently placed. <figure><head>Fig. 1.</head></figure> <figure><head>Sines.</head></figure> <figure><head>Cosines.</head></figure> <figure><head>Versedsines.</head></figure> <pb n="471"/><cb/> <figure><head>Tangents.</head></figure> <figure><head>Cotangents.</head></figure> <figure><head>Secants.</head></figure> <figure><head>Cosecants.</head></figure></p><p>It may be known when any of these lines, viz, the sines, cosines, &c, are affirmative or negative, i. e. to be set upwards or downwards, by observing the following general rules for those lines in the 1st, 2d, 3d, and 4th quadrants of the circle. <table><row role="data"><cell cols="1" rows="1" role="data">The sines</cell><cell cols="1" rows="1" role="data">in the 1st and 2d</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">affirmative,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the 3d and 4th</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">negative:</cell></row><row role="data"><cell cols="1" rows="1" role="data">The cosines</cell><cell cols="1" rows="1" role="data">in the 1st and 4th</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">affirmative,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the 2d and 3d</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">negative:</cell></row><row role="data"><cell cols="1" rows="1" role="data">The tangents</cell><cell cols="1" rows="1" role="data">in the 1st and 3d</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">affirmative,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the 2d and 4th</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">negative:</cell></row><row role="data"><cell cols="1" rows="1" role="data">The cotangents</cell><cell cols="1" rows="1" role="data">in the 1st and 3d</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">affirmative,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the 2d and 4th</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">negative:</cell></row><row role="data"><cell cols="1" rows="1" role="data">The secants</cell><cell cols="1" rows="1" role="data">in the 1st and 4th</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">affirmative,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the 2d and 3d</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">negative:</cell></row><row role="data"><cell cols="1" rows="1" role="data">The cosecants</cell><cell cols="1" rows="1" role="data">in the 1st and 2d</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">affirmative,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the 3d and 4th</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">negative:</cell></row></table> And all the versedsines are affirmative. <hi rend="center"><hi rend="italics">To find the Equation and Area, &c, to each of these Curves.</hi></hi></p><p>Draw any ordinate <hi rend="italics">de;</hi> putting <hi rend="italics">r</hi> = the radius AC of the given circle, <hi rend="italics">x</hi> = <hi rend="italics">ad</hi> or AD any absciss or arc, and <hi rend="italics">y</hi> = <hi rend="italics">de</hi> its ordinate, which will be either the sine DE = <hi rend="italics">s,</hi> cosine CE = <hi rend="italics">c,</hi> versedsine AE = <hi rend="italics">v,</hi> tangent AF = <hi rend="italics">t,</hi> cotangent GH = <foreign xml:lang="greek">t</foreign>, secant CF = <hi rend="italics">s,</hi> or cosecant CH = <foreign xml:lang="greek">s</foreign>, according to the nature of the particular construction. Now, from the nature of the circle, are obtained these following general equations, expressing the relations between the fluxions of a circular arc and its sine, or cosine, &c. . And these also express the relation between the absciss and ordinate of the curves in question, each in the order in which it stands; where <hi rend="italics">x</hi> is the common absciss to all of them, and the respective ordinates are <hi rend="italics">s, c, v, t,</hi> <foreign xml:lang="greek">t</foreign>, <hi rend="italics">s,</hi> and <foreign xml:lang="greek">s</foreign>. And hence the area &c, of any of these curves may be found, as follows: <cb/></p><p>1. <hi rend="italics">In the <hi rend="smallcaps">Figure</hi> of Sines.</hi>—Here <hi rend="italics">x</hi> = <hi rend="italics">ad,</hi> and <hi rend="italics">s</hi> = the ordinate <hi rend="italics">de</hi>; and the equation of the curve, as above, is . Hence the fluxion of the area, or <hi rend="italics">sx<hi rend="sup">.</hi></hi> is ; the correct fluent of which is the rectangle of radius and vers. i. e. - or + as <hi rend="italics">s</hi> is increasing or decreasing; which is a general expression for the area <hi rend="italics">ade</hi> in the Figure of sines. When <hi rend="italics">s</hi> = 0, as at <hi rend="italics">a</hi> or <hi rend="italics">b,</hi> this expression becomes 0 or 2<hi rend="italics">r</hi><hi rend="sup">2</hi>; that is 0 at <hi rend="italics">a,</hi> and 2<hi rend="italics">r</hi><hi rend="sup">2</hi> = the area <hi rend="italics">aeb;</hi> or <hi rend="italics">r</hi><hi rend="sup">2</hi> = the area of <hi rend="italics">afg</hi> when <hi rend="italics">ad</hi> becomes a quadrant <hi rend="italics">af.</hi></p><p>2. <hi rend="italics">In the <hi rend="smallcaps">Figure</hi> of Cosines.</hi>—Here <hi rend="italics">x</hi> = <hi rend="italics">ad</hi> and <hi rend="italics">c</hi> = <hi rend="italics">de;</hi> and the equation of the curve is . Hence the fluxion of the area is ; and the fluent of this is , the rectangle of radius and sine, for the general area <hi rend="italics">adec.</hi> When <hi rend="italics">s</hi> =<hi rend="italics">r,</hi> or <hi rend="italics">c</hi> = 0, this becomes <hi rend="italics">r</hi><hi rend="sup">2</hi> = the area <hi rend="italics">afc,</hi> whose absciss <hi rend="italics">af</hi> is equal to a quadrant of the circumference; the same as in the Figure of the sines, upon an equal absciss.</p><p>3. <hi rend="italics">In the <hi rend="smallcaps">Figure</hi> of Versedsines.</hi>—Here <hi rend="italics">x</hi> = <hi rend="italics">ad,</hi> and <hi rend="italics">v</hi> = <hi rend="italics">de;</hi> and the equation of the curve is . Hence the fluxion of the area is ; and the fluent of this is for the area <hi rend="italics">ade</hi> in the Figure of versed sines. When AD or <hi rend="italics">ad</hi> is a quadrant AG or <hi rend="italics">af,</hi> this becomes the area <hi rend="italics">afg.</hi> And when AD or <hi rend="italics">ad</hi> is a semicircle <hi rend="italics">ab,</hi> it becomes 3.1416<hi rend="italics">r</hi><hi rend="sup">2</hi> = the area <hi rend="italics">abg</hi> in the Figure of versedsines.</p><p>4. <hi rend="italics">In the <hi rend="smallcaps">Figure</hi> of Tangents.</hi>—Here <hi rend="italics">x</hi> = <hi rend="italics">ad,</hi> and <hi rend="italics">t</hi> = <hi rend="italics">de;</hi> and the equation of the curve is . Hence the fluxion of the area is ; and the correct fluent of this is (1/2)<hi rend="italics">r</hi><hi rend="sup">2</hi> X hyp. log. of hyp. log. of hyp. log. of <hi rend="italics">s/r</hi>. And hence the Figure of the tangents may be used for constructing the logarithmic secants; a property that was remarked by Gregory at the end of his Exercit. Geomet.</p><p>When <hi rend="italics">ad</hi> becomes a quadrant <hi rend="italics">af, t</hi> being then insinite, this becomes infinite for the area <hi rend="italics">afg.</hi> And the same for the Figure of cotangents, beginning at <hi rend="italics">f</hi> instead of <hi rend="italics">a.</hi> <pb n="472"/><cb/></p><p>5. <hi rend="italics">For the <hi rend="smallcaps">Figure</hi> of the Secants.</hi>—Here <hi rend="italics">x</hi> = <hi rend="italics">ad,</hi> and <hi rend="italics">s</hi> = <hi rend="italics">de;</hi> and the equation of the curve is . Hence the fluxion of the area is ; the fluent of which is <hi rend="italics">r</hi><hi rend="sup">2</hi> X hyp. log. of for the general area <hi rend="italics">ade.</hi> And when <hi rend="italics">ad</hi> becomes the quadrant <hi rend="italics">af,</hi> this expression becomes infinite for the area <hi rend="italics">afg.</hi></p><p>The same process will serve for the Figure of cosecants, beginning at <hi rend="italics">f</hi> instead of <hi rend="italics">a.</hi></p><p>From hence the meridional parts in Mercator's chart may be calculated for any latitude AD or <hi rend="italics">ad:</hi> For the merid. parts: are to the arc of latitude AD :: as the sum of the secants: to the sum of as many radii or :: as the area <hi rend="italics">ade</hi>: to <hi rend="italics">ad</hi> X radius <hi rend="italics">ac</hi> or AD X AC in the first sigure.</p></div2></div1><div1 part="N" n="FILLET" org="uniform" sample="complete" type="entry"><head>FILLET</head><p>, in Architecture, any little square member or ornament used in crowning a larger moulding.</p><p>FINÆUS (<hi rend="smallcaps">Orontius</hi>), in French <hi rend="italics">Finé,</hi> professor of mathematics in the Royal-college of Paris, was the son of a physician, and was born at Briançon in Dauphiné in 1494. He went young to Paris, where his friends procured him a place in the college of Navarre. He applied himself there to philosophy and polite literature; but more especially to mathematics, in which, having a natural propensity, he made a considerable prosiciency. Particularly he made a good progress in mechanics; in which, having both a genius to invent instruments, and a skilful hand to make them, he gained much reputation by the specimens he gave of his ingenuity.</p><p>Finæus first made himself publicly known by correcting and publishing Siliceus's <hi rend="italics">Arithmetic,</hi> and the <hi rend="italics">Margarita Philosophica.</hi> He afterwards read private lectures in mathematics, and then taught that science publicly in the college of Gervais: from the reputation of which, he was recommended to Francis the 1st, as the properest person to teach mathematics in the new college which that prince had founded at Paris. And here, though he spared no pains to improve his pupils, he yet found time to publish a great many books upon most parts of the mathematics. But neither his genius, his labours, his inventions, and the esteem which numberless persons shewed him, could secure him from that fate which so often befalls men of letters. He was obliged to struggle all his life time with poverty; and when he died, left a numerous family deeply in debt. However, as merit must always be esteemed in secret, though it seldom has the luck to be rewarded openly; so Finæus's children found Mecænases, who for their father's sake assisted his family.—He died in 1555, at 61 years of age.</p><p>Like all other mathematicians and astronomers of those times, he was greatly addicted to astrology; and had the misfortune to be a long time imprisoned for having predicted some things, that were not acceptable to the court of France. He was also one of those, who vainly boasted of having found out the quadrature of the circle. An edition of his works, translated into the <cb/> Italian language, was published in 4to, at Venice, 1587; consisting of Arithmetic, Practical Geometry, Cosmography, Astronomy, and Dialling.</p></div1><div1 part="N" n="FINITE" org="uniform" sample="complete" type="entry"><head>FINITE</head><p>, the property of any thing that is bounded or limited, either in its power, or extent, or duration, &c; as distinguished from the property of infinite, or without bounds.</p></div1><div1 part="N" n="FINITOR" org="uniform" sample="complete" type="entry"><head>FINITOR</head><p>, the horizon; being so called, because it sinishes or bounds the sight or prospect.</p></div1><div1 part="N" n="FIRE" org="uniform" sample="complete" type="entry"><head>FIRE</head><p>, is that subtile invisible cause by which bodies are made hot to the touch, and expauded or enlarged in bulk; by which fluids are rarefied into vapour; or solid bodies become fluid, and at last either dissipated and carried off in vapour, or else melted into glass. It seems also to be the chief agent in nature on which animal and vegetable life have an immediate dependence.</p><p>The disputes concerning fire, which long existed among philosophers, have now in a great measure subsided. Those celebrated philosophers of the last century, Bacon, Boyle, and Newton, were of opinion, that Fire was not a substance distinct from other bodies, but that it entirely consisted in the violent motion of the parts of any body, which was produced by the mechanical force of impulsion, or of attrition. So Boyle says, when a piece of iron becomes hot by hammering, “there is nothing to make it so, except the forcible motion of the hammer impressing a vehement and variously determined agitation on the small parts of the iron.” And Bacon defines heat, which he makes synonymous with Fire, an “expansive undulatory motion in the minute particles of a body, whereby they tend with some rapidity from a centre towards a circumference, and at the same time a little upwards.” And according to Newton, Fire is a body heated so hot as to emit light copiously; for what else, says he, is red-hot iron, but Fire? and what else is a fiery coal than redhod wood? by which he suggests, that bodies which are not Fire, may be changed and converted into it.</p><p>On the other hand, the chemists strenuously contended that Fire was a fluid of a certain kind, distinct from all others, and universally present throughout the whole globe. Boerhaave particularly maintained this doctrine; and in support of it brought this argument, that flint and steel would strike fire, and produce the same degree of heat in Nova Zembla as they would do under the equator. Other arguments were drawn from the increased weight of metallic calces, which they thought proceeded from the fixing of the element of Fire in the substance whose weight was thus increased. For a long time however, the matter was most violently disputed; but the mechanical philosophers at last prevailed through the deference paid to the principles of Newton, though he himself had scarcely taken any active part in the contest.</p><p>The experiments of Dr. Black however seemed to bring the dispute to a decision, and that in favour of the chemists, concerning what he called latent-heat. From these discoveries it appears, that Fire may exist in bodies in such a manner, as not to discover itself in any other way than by its action on the minute parts of the body; but that suddenly this action may be changed in such a manner, as no longer to be directed upon the particles of the body itself, but upon external objects; in which case we then perceive its action by our sense of <pb n="473"/><cb/> feeling, or discover it by the thermometer, and call it heat, or sensible Fire.</p><p>From this discovery, and others in electricity, it is now pretty generally allowed, that Fire is a distinct fluid, capable of being transferred from one body to another. But when this was discovered, another question no less perplexing arose, viz, what kind of a fluid it was; or whether it bears any analogy to those with which we are better acquainted. Now there are found two fluids, viz, the solar light, and the electric matter, both of which occasionally act as fire, and which therefore seem likely to be all the same at the bottom; and popularly the matter has been long since determined; the solar rays and the electric fluid having been indifferently accounted elementary Fire. Some indeed have imagined both these fluids to be mere phlogiston itself, or at least containing a large portion of it; and Mr. Scheele went so far in this way as to form an hypothesis, which he endeavoured to support by experiments, that Fire is composed of phlogiston and dephlogisticated air. But it is now ascertained beyond doubt, that the result of such a combination is not Fire, but sixed air.</p><p>It was long since observed by Newton, that heat was certainly conveyed by a medium more subtile than the common air; for two thermometers, one included in the vacuum of an air-pump, the other placed in the open air, at an equal distance from the fire, would grow equally hot in nearly the same time. This and other experiments shew, that Fire exists and acts where there is no other matter, and of consequence it is a fluid per se, independent of every terrestrial substance, without being generated or compounded of any thing we are yet acquainted with. To determine the nature of the fluid, we have only to consider whether any other can be discovered which will pass through the perfect vacuum just mentioned, and act there as Fire. Such a fluid is found in the solar light, which is well known to act even in vacuo as the most violent Fire. The solar light will likewise act in the very same manner in the most intense cold; for M. de Saussure has found, that on the cold mountain top the sun-beams are equally powerful as on the plain below, if not more so. It appears therefore, that the solar light will produce heat independent of any other substance whatever; that is, where no other body is present, at least as far as we can judge, except the light itself, and the body to be acted upon. We cannot therefore avoid concluding, that a certain modification of the solar light is the cause of heat, expansion, vapour, &c, and answers to the rest of the characters given in the foregoing definition of Fire, and that independent of any other substance whatever.</p><p>It is very probable too, that the electric matter is no other than the solar light absorbed by the earth, and thus becoming subject to new laws, and assuming many properties apparently different from what it has when it acts as light. Even in this case it manifests its identity with Fire or light, viz, by producing a most intense heat where a large quantity of it passes through a small space. So that at any rate, the experiments which have already been made, and the proofs drawn from the phenomena of nature, shew such a strong affinity between the elements of Fire, light and electricity, that we may not only assert their identity upon the <cb/> most probable grounds, but lay it down as a position against which at present no argument of any weight has an existence.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Arrow,</hi> is a small iron dart, furnished with springs and bars, and also a match impregnated with sulphur and powder, which is wound about its shaft. It is chiefly used by privateers and pirates to fire the sails of the enemy's ship, and for this purpose it is discharged from a musketoon, or a swivel gun. The match being kindled by the explosion, it communicates the flame to the sail against which it is directed, where the arrow fastens itself by means of its bars and springs. This weapon is peculiar to hot climates, particularly the West Indies; the sails being very dry, are quickly set on Fire, and the Fire is soon conveyed to the masts, rigging, and finally to the vessel itself.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Balls,</hi> in Artillery, are certain balls composed of combustible matters, such as fine or mealed powder, sulphur, saltpetre, rosin, pitch, &c. These are thrown into the enemy's works in the night time, to discover where they are; or to set Fire to houses, galleries, or blinds of the besiegers, &c. They are sometimes armed with spikes or hooks of iron, that they may not roll off, but stick or hang where they are to take effect.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Balls,</hi> or <hi rend="italics">Fiery-Meteors,</hi> in Meteorology, a kind of luminous bodies usually appearing at a great height above the earth, with a splendor surpassing that of the moon, and sometimes apparently as large. They have not been found to observe any regular course or motion, but, on the contrary, moving in all directions, and with very different degrees of celerity; frequently breaking into several smaller ones; sometimes making a strong hissing sound, sometimes bursting or vanishing with a loud report, and sometimes not.</p><p>These luminous appearances doubtless constitute one part of the ancient prodigies, blazing stars, or comets, which last they sometimes resemble in being attended with a train; but more often they appear round. The first of these of which we have any accurate account, was observed by Dr. Halley, and some other philosophers at different places, in the year 1719, the height of which above the surface of the earth was computed at more than 70 miles. Many others have been pretty accurately observed since that time, and described by philosophers, as in the French Memoirs, and in the Philos. Trans. vols. 30, 41, 42, 43, 46, 47, 48, 51, 53, 54, 63, 74, &c. The velocities, directions, appearances, and heights of all these were found to be very various; though the height of all of them was supposed above the limits assigned to our atmosphere, or where it loses its refractive power. The most remarkable of those on record, appeared on the 18th of August 1783, about 9 o'clock in the evening. It was seen to the northward of Shetland, and took a southcasterly direction for an immense space, being observed as far as the southern provinces of France, and by some it was said to have been seen at Rome, passing over a space of 1000 miles in about half a minute of time, and at a very great height. During its course it appeared several times to change its shape; sometimes appearing in the form of one ball, sometimes of two or more; sometimes with a train, and sometimes without one.</p><p>There are divers opinions concerning the nature and <pb n="474"/><cb/> origin of these Fire-balls. The first thing that occurred to philosophers on this subject was, that the meteors in question were burning bodies rising from the surface of the earth, and flying through the atmosphere with great rapidity. But this hypothesis was soon rejected, on considering that there was no power known by which such bodies could either be raised to a sufficient height, or projected with the velocity of the meteors. The next hypothesis was, that, instead of one single body, they consist of a train of sulphureous vapours, extending a vast way through the atmosphere, and being kindled at one end, display the luminous appearances in question by the fire running from one end of the train to the other. But it is not easy to conceive how such matters can exist and be disposed in such lines in so rare a part of the atmosphere, and even to burn there, in an almost perfect vacuum. For which reason this hypothesis was abandoned, for another, which was, that those meteors are permanent solid bodies, not rising from the earth, but revolving round it in very excentric orbits, and thus in their perigeon moving with vast rapidity. But as the various appearances of one and the same meteor, to observers at different places, are not compatible with the idea of a single body so revolving, this hypothesis has also been given up in its turn. Another hypothesis that has sometimes been advanced is this; viz, that these meteors are a kind of bodies that take fire as soon as they come within the atmosphere of the earth. But this cannot be supposed without implying a previous knowledge of these bodies, which it is impossible we can have. The only opportunity we can have of seeing them, is when they are on fire. Before that time they are in an invisible and unknown state; and it is surely improper to argue concerning them in this state, or pretend to determine any one of their properties, when it is not in our power at all to see or investigate them. As these meteors therefore never manifest themselves to our senses but when they are on fire, the only rational conclusion we can draw from thence is, that they have no existence in any other state; and consequently that their substance must be composed of that fluid which, when acting after a certain manner, becomes luminous and shews itself as fire; remaining invisible and eluding our researches in every other case. On this hypothesis, it is now pretty generally concluded, that the Fire-balls are great bodies of electric matter, moving from one part of the heavens where, to our conception, it is superabundant, to another where it is desicient: a conclusion attended with much probability, from the analogy observed between electricity and the phenomena of these meteors: and hence these Fireballs appear to be of the same family with shootingstars, lightning, the aurora-borealis, &c, being all referred to the same origin, viz, the electricity of the atmosphere.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Engine,</hi> is a machine for extinguishing accidental Fires by means of a stream or jet of water. The common squirting Fire-engine consists of a lifting pump placed in a vessel of water, and wrought by two levers that act always together. During the stroke, the water raised by the piston of the pump spouts forcibly through a pipe joined to the pump-barrel, and made capable of any degree of elevation by means of a yield- <cb/> ing leather pipe, or by a ball and socket turning every way, screwed on the top of the pump. The vessel containing the water is covered with a strainer, to prevent the mud, &c, which is poured into it with the water, from choking the pump-work. Between the strokes of this engine the stream is discontinued, for want of an air vessel. However, in some cases, Engines of this construction have their use, because the stream, though interrupted, is much smarter than when the engine is made to throw water in a continued stream. See these Engines particularly described in Desaguliers's Exper. Philos. vol. 2, pa. 505; or Martin's Philos. Britan. vol. 2, pa. 69. See also the figure of them, plate viii. fig. 3.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Engine,</hi> is also sometimes used for the machine employed in raising water by steam, and more properly called <hi rend="smallcaps">Steam</hi>-<hi rend="italics">Engine</hi>; which see.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Lock,</hi> or <hi rend="italics">Fusil,</hi> a small gun or musket, which fires with a flint and steel; as distinguished from the old musket, or match-lock, which was fired with a match. The Fire-lock is now in common use with the European armies, and carried by the foot-soldiers. It is usually about 3 feet 8 inches in the barrel; and, including the stock, 4 feet 8 inches, carrying a leaden bullet, of which 29 make 2lb. The diameter of the ball is .55, and that of the barrel .56 parts of an inch. The time of the invention of Fire-locks is uncertain; but they were used in 1690.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Places,</hi> are contrivances for communicating heat to rooms, and also for answering various purposes of art and manufacture.</p><p>The general properties of air and fire, on which their construction chiefly depends, are the following, viz, that air is rarefied by heat, and condensed by cold; i. e. the same quantity of air takes up more space when warm than when cold. Air rarefied and expanded by heat, is specifically lighter than it was before, and will rise in other air of greater density: so that a sire being made in any chimney, the air about and over the fire is rarefied by the heat, thence becomes lighter, and so rises in the funnel, and goes out at the top of the chimney: the other air in the room, flowing towards the chimney, supplies its place, is then rarefied in its turn, and rises likewise; and the place of the air thus carried out of the room, is supplied by fresh air coming in through doors and windows, or, if they be shut, through every crevice with violence; or if the avenues to the room be so closed up, that little or no fresh supply of air can be obtained, the current up the funnel must flag; and the smoke, no longer driven up, float about in the room.</p><p>Upon these principles, various contrivances and kinds of Fire-grates and stoves have been devised, from the old very open and wide chimney places, down to the present modish ones, which are much narrowed in the front, opening, by side and back jambs, and a low breast or mantle, besides the convenience of a flap, called a register, that covers the top of the Fire-stove, but opening to any degree with a small winch, which lifts the back part sloping upwards, and so throws the smoke freely up the funnel, and yet admitting as little air to pass as you please; by which simple means the warm air is kept very much in the room, while the very narrow and sloping orifice promotes the brisk ascent of <pb n="475"/><cb/> the smoke, and yet prevents its return down again, for the same reason.</p><p>Another very ingenious, but more complex, apparatus, called the Pensylvania Fire-place, was invented by Dr. Franklin, by which a room is kept very warm by a constant supply of fresh hot air, that passes into it through the stove itself. See the description in his Letters and Papers on Philosophical Subjects.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Pot,</hi> in the Military Art, is a small earthen pot, into which is put a grenade, filled with fine powder till the grenade be covered; the pot is then covered with a piece of parchment, and two pieces of match laid across and lighted. This pot being thrown where it is designed to do execution, breaks and fires the powder, and this again fires the powder in the grenade, which ought to have no fuze, that its operation may be the quicker.</p><p><hi rend="italics">Rasant,</hi> or <hi rend="italics">Razant</hi> <hi rend="smallcaps">Fire</hi>, is a fire from the artillery and small arms, directed parallel to the horizon, or to those parts of the works of a place that are defended.</p><p><hi rend="italics">Running</hi> <hi rend="smallcaps">Fire</hi>, is when ranks of men fire one after another; or when the lines of an army are drawn out to fire on account of a victory; in which case each squadron or battalion takes the fire from that on its right, from the right of the first line to the left, and from the lest to the right of the second line, &c.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Ships,</hi> in the Navy, are vessels charged with combustible materials or artificial Fire-works; which having the wind of an enemy's ship, grapple her, and set her on fire.</p><p>Anderson, in his History of Commerce, vol. 1, pa. 432, ascribes the invention to the English, in this instance, viz, some vessels being filled with combustible matter, and sent among the Spanish ships composing the Invincible Armada in 1588; and hence arose it is said the terrible invention of Fire-ships.</p><p>But Livy informs us, that the Rhodians had invented a kind of Fire-ships, which were used in junction with the Roman fleet in their engagement with the Syrians, in the year 190 before Christ: cauldrons of combustible and burning materials were hung out at their prows, so that none of the enemies' ships durst approach them: for these fell on the enemies' gallies, struck their beaks into them, and at the same time set them on fire. Livy, lib. 37, cap. 30.</p><p><hi rend="italics">Wild</hi>-<hi rend="smallcaps">Fire</hi>, is a kind of artificial or factitious fire, that burns even under water, and that with greater violence than out of it. It is composed of sulphur, naphtha, pitch, gum, and bitumen, and it is only extinguishable by vinegar, mixed with sand and urine, or by covering it with raw hides. It is said its motion is contrary to that of natural fire, always following the direction in which it is thrown, whether it be downwards, sideways, or otherwise.</p><p>The French call it Greek Fire, or Feu Gregeois, because first used by the Greeks about the year 660, as is observed by the Jesuit Petavius, on the authority of Nicetas, Theophanes, Cedrenus, &c. The inventor, according to the same author, was an engineer of Heliopolis, in Syria, named Callinicus, who first applied it in the sea-fight commanded by Constantine Pogonates, against the Saracens, near Cyzicus, in the Hellespont; and with such effect, that he burnt the whole fleet, which contained 30,000 men. But others refer it to <cb/> a much older date, and ascribe the invention to Marcus Gracchus; an opinion which is supported by several passages, both in the Greek and Roman writers; which shew that it was anciently used by both these nations in their wars. See Scaliger against Cardan.</p><p>The successors of Constantine used it on several occasions, with great advantage: and it is remarkable that they were able to keep the secret of the composition to themselves; so that no other nation knew it in the year 960.</p><p>It is recorded by Chorier, in his Hist. de Dauph. that Hugh, king of Burgundy, demanding ships of the emperor Leo for the siege of Fresne, desired also the Greek Fire.</p><p>And F. Daniel gives a good description of the Greek Fire, in his account of the siege of Damietta, under St. Louis. Every body, says he, was astonished with the Greek Fire, which the Turks then prepared; and the secret of which is now lost. They threw it out of a kind of mortar, and sometimes shot it with an odd sort of cross-bow, which was strongly bent by means of a handle, or winch, of much greater force than the bare arm. That which was thrown from the mortar sometimes appeared in the air of the size of a tun, with a long tail, and a noise like that of thunder. The French, by degrees, got the secret of extinguishing it; in which they succeeded several times.</p><p>After all, perhaps the invention of the Wild-Fire is to be ascribed to other nations, and to a still older date, and that it was the same as that used among the Indians in Alexander's invasion, when it was said they fought with thunder and lightning, or shot Fire with a terrible noise.</p><p><hi rend="smallcaps">Fire</hi>-<hi rend="italics">Works,</hi> otherwise called Pyrotechnia, are artificial Fires, or preparations made of gunpowder, sulphur, and other inflammable and combustible ingredients, used on occasion of public rejoicings, and other solemnities. The principal of these are rockets, serpents, stars, hail, mines, bombs, garlands, letters, and other devices.</p><p>The invention of Fire-works is attributed, by M. Mahudel, to the Florentines and people of Sienna; who found out likewise the method of adding to them decorations of statues, with fire issuing from their eyes and mouths.</p></div1><div1 part="N" n="FIRKIN" org="uniform" sample="complete" type="entry"><head>FIRKIN</head><p>, an English measure of capacity; being the 4th part of a barrel; and containing 8 gallons of ale, soap, butter, or herrings; or 9 gallons of beer.</p></div1><div1 part="N" n="FIRLOT" org="uniform" sample="complete" type="entry"><head>FIRLOT</head><p>, a dry measure used in Scotland. The oat-firlot contains 21 1/4 pints of that country, or about 85 English pints; and the barley-sirlot, 31 standard pints. The wheat firlot contains about 2211 cubic inches; and therefore exceeds the English bushel by 60 cubic inches, or almost an English quart.</p></div1><div1 part="N" n="FIRMAMENT" org="uniform" sample="complete" type="entry"><head>FIRMAMENT</head><p>, by some old astronomers, is the orb of the fixed stars, or the highest of all the heavens. But in scripture and common language it is used for the middle regions, or the space or expanse appearing like an arch quite around or above us in the heavens. Many ancients and moderns also accounted the Firmament a fluid matter; but those who gave it the name of Firmament must have taken it for a solid one.</p></div1><div1 part="N" n="FIRMNESS" org="uniform" sample="complete" type="entry"><head>FIRMNESS</head><p>, is the consistence of a body; or that state when its sensible parts cohere, or are united together, so that the motion of one part induces a mo- <pb n="476"/><cb/> tion of the rest. In which sense firmness stands opposed to fluidity.</p><p>The sirmness of bodies then depends on the connexion or cohesion of their particles; and the cause of cohesion the Newtonians hold to be an attractive force, inherent in bodies, which binds their small particles together; exerting itself only at the points of contact, or extremely near them, and vanishing at greater distances.</p><p>FIRST <hi rend="italics">Mover,</hi> in the old Astronomy, is the Primum Mobile, or that which gives motion to the other parts of the universe.</p></div1><div1 part="N" n="FISSURES" org="uniform" sample="complete" type="entry"><head>FISSURES</head><p>, in the History of the Earth, are certain interruptions, mostly parallel to each other, that divide or separate the strata of it from one another, in nearly horizontal directions; and the parts of the same stratum in nearly vertical directions.</p><p>FIXED <hi rend="italics">Line of Desence,</hi> a line drawn along the face of the bastion, and terminating in the curtain.</p><p><hi rend="smallcaps">Fixed</hi> <hi rend="italics">Signs of the Zodiac,</hi> according to some, are the four signs Taurus, Leo, Scorpio, Aquarius. They are so called because the sun passes them respectively in the middle of each quarter, when that season is more settled and Fixed than under the signs which begin and end it.</p><p><hi rend="smallcaps">Fixed</hi> <hi rend="italics">Stars,</hi> are such as constantly retain the same position and distance with respect to each other; by which they are contradistinguished from erratic or wandering stars, which are continually varying their situation and distance.—The Fixed Stars only are properly and absolutely called stars; the rest having their peculiar denomination of planet or comet.</p></div1><div1 part="N" n="FIXITY" org="uniform" sample="complete" type="entry"><head>FIXITY</head><p>, or <hi rend="smallcaps">Fixedness</hi>, the quality of a body which determines it fixed; or a property which enables it to endure the fire and other violent agents.</p><p>A body may be said to be fixed in two respects: 1st, When on being exposed to the fire, or a corrosive menstruum, its particles are indeed separated, and the body rendered fluid, but without being resolved into its first elements. The 2d, when the body sustains the active force of the fire or menstruums whilst its integral parts are not carried off in fumes. Each kind of Fixity is the result of a strong or intimate cohesion between the particles.</p></div1><div1 part="N" n="FLAME" org="uniform" sample="complete" type="entry"><head>FLAME</head><p>, the subtlest and brightest part of the fuel, ascending above it in a pyramidal or conical figure, and heated red-hot. Sir Isaac Newton defines Flame as only red-hot smoke, or the vapour of any substance raised from it by fire, and heated to such a degree as to emit light copiously. Is not Flame, says he, a vapour, fume, or exhalation, heated red-hot; that is, so hot as to shine? For bodies do not Flame without emitting a copious fume; and this fume burns in the Flame. The ignis fatuus is a vapour shining without heat; and is there not the same difference between this vapour and Flame, as between rotten wood shining without heat, and burning coals of fire? In distilling hot spirits, if the head of the still be taken off, the vapour which ascends will take fire at the Flame of a candle, and turn into Flame. Some bodies, heated by motion or fermentation, if the heat grow intense, fume copiously; and if the heat be great enough, the fumes will shine, and become Flame. Metals in fusion do not Flame, for want of a copious fume. All flaming bodies, as oil, tallow, wax, wood, fossil coal, pitch, sul- <cb/> phur, &c, by burning, waste in smoke, which at first is lucid; but at a little distance from the body ceases to be so, and only continues hot. When the Flame is put out, the smoke is thick, and frequently smells strongly: but in the Flame it loses its smell; and, according to the nature of the fuel, the Flame is of divers colours. That of sulphur and spirit of wine is blue; that of copper opened with sublimate, green; that of tallow, yellow; of camphire, white; &c. Newton's Optics, p. 318.</p></div1><div1 part="N" n="FLAMSTEED" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FLAMSTEED</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent English astronomer, being indeed the first astronomer royal, for whose use the royal observatory was built at Greenwich, thence called Flamsteed House. He was born at Denby in Derbyshire the 19th of August 1646. He was educated at the free school of Derby, where his father lived; and at 14 years of age was afflicted with a severe illness, which rendered his constitution tender ever after, and prevented him then from going to the university, for which he was intended. He nevertheless prosecuted his school education with the best effect; and then, in 1662, on quitting the grammar school, he pursued the natural bent of his genius, which led him to the study of astronomy, and closely perused Sacrobosco's book <hi rend="italics">De Sphæra,</hi> which fell in his way, and which laid the groundwork of all that mathematical and astronomical knowledge, for which he became afterward so justly famous. He next procured other more modern books of the same kind, and among them Streete's <hi rend="italics">Astronomia Carolinæ,</hi> then lately published, from which he learned to calculate eclipses and the planets' places. Some of these being shewn to a Mr. Halton, a considerable mathematician, he lent him Riccioli's <hi rend="italics">Almagestum Novum,</hi> and Kepler's <hi rend="italics">Tabulæ Rudolphinæ,</hi> which he profited much by. In 1669, having calculated some remarkable eclipses of the moon, he sent them to lord Brouncker, president of the Royal Society, which were greatly approved by that learned body, and procured him a letter of thanks from Mr. Oldenburg their secretary, and another from Mr. John Collins, with whom, and other learned men, Mr. Flamsteed for a long time afterwards kept up a correspondence by letters on literary subjects. In 1670, his father, observing he held correspondence with these ingenious gentlemen, advised him to take a journey to London, to make himself personally acquainted with them; an offer which he gladly embraced, and visited Mr. Oldenburg and Mr. Collins, who introduced him to Sir Jonas Moore, which proved the means of his greatest honour and preferment. He here got the knowledge and practice of astronomical instruments, as telescopes, micrometers, &c. On his return, he called at Cambridge, and visited Dr. Barrow, Mr. Isaac Newton, and other learned men there, and entered himself a student of Jesus College. In 1672 he extracted several observations from Mr. Gascoigne's and Mr. Crabtree's letters, which improved him greatly in dioptrics. In this year he made many celestial observations, which, with calculations of appulses of the moon and planets to fixed stars for the year following, he sent to Mr. Oldenburg, who published them in the Philos. Trans.</p><p>In 1673, Mr. Flamsteed wrote a small tract concerning the true diameters of all the planets, when at their greatest and least distances from the earth; which he lent to Mr. Newton in 1685, who made some use of <pb n="477"/><cb/> it in the 4th book of his Principia.—In 1674 he wrote an ephemeris, to shew the falsity of Astrology, and the ignorance of those who pretended to it: with calculations of the moon's rising and setting; also occultations and appulses of the moon and planets to the fixed stars. To which, at Sir Jonas Moore's request, he added a table of the moon's southings for that year; from which, and from Philips's theory of the tides, the high-waters being computed, he found the times come very near. In 1674 too, he drew up an account of the tides, for the use of the king. Sir Jonas also shewed the king, and the duke of York, some barometers and thermometers that Mr. Flamsteed had given him, with the necessary rules for judging of the weather; and otherwise took every opportunity of speaking favourably of Flamsteed to them, till at length he brought him a warrant to be the king's astronomer, with a salary of 100l. per annum, to be paid out of the office of Ordnance, because Sir Jonas was then Surveyor General of the Ordnance. This however did not abate our author's propensity for holy orders, and he was accordingly ordained at Ely by bishop Gunning.</p><p>On the 10th of August, 1675, the foundation of the Royal Observatory at Greenwich was laid; and during the building of it, Mr. Flamsteed's temporary observatory was in the queen's house, where he made his observations of the appulses of the moon and planets to the fixed stars, and wrote his Doctrine of the Sphere, which was afterward published by Sir Jonas, in his System of the Mathematics.</p><p>About the year 1684 he was presented to the living of Burslow in Surry, which he held as long as he lived. Mr. Flamsteed was equally respected by the great men his contemporaries, and by those who have succeeded since his death. Dr. Wotton, in his Reflections upon Ancient and Modern Learning, styles our author one of the most accurate Observers of the Planets and Stars, and says he calculated tables of the eclipses of the several satellites, which proved very useful to the astronomers. And Mr. Molyneux, in his <hi rend="italics">Dioptrica Nova,</hi> gives him a high character; and, in the admonition to the reader prefixed to the work, observes, that the geometrical method of calculating a ray's progress is quite new, and never before published; and for the first hint of it, says he, I must acknowledge myself obliged to my worthy friend Mr. Flamsteed. He wrote several small tracts, and had many papers inserted in the Philosophical Transactions, viz, several in almost every volume, from the 4th to the 29th, too numerous to be mentioned in this place particularly.</p><p>But his great work, and that which contained the main operations of his life, was the <hi rend="italics">Historia Cœlestis Britannica,</hi> published in 1725, in 3 large folio volumes. The first of which contains the observations of Mr. William Gascoigne, the first inventor of the method of measuring angles in a telescope by means of screws, and the first who applied telescopical sights to astronomical instruments, taken at Middleton, near Leeds in Yorkshire, between the years 1638 and 1643; extracted from his letters by Mr. Crabtree; with some of Mr. Crabtree's observations about the same time; and also those of Mr. Flamsteed himself, made at Derby between the years 1670 and 1675; besides a multitude of curious observations, and necessary tables to be used with <cb/> them, made at the Royal Observatory, between the years 1675 and 1689.—The 2d volume contains his observations, made with a mural arch of near 7 feet radius, and 140 degrees on the limb, of the meridional zenith distances of the fixed stars, sun, moon, and planets, with their transits over the meridian; also observations of the diameters of the sun and moon, with their eclipses, and those of Jupiter's satellites, and variations of the compass, from 1689 to 1719: with tables shewing how to render the calculation of the places of the stars and planets easy and expeditious. To which are added, the moon's place at her oppositions, quadratures, &c; also the planets' places, derived from the observations.—The 3d volume contains a catalogue of the right-ascensions, polar-distances, longitudes, and magnitudes of near 3000 fixed stars, with the corresponding variations of the same. To this volume is prefixed a large preface, containing an account of all the astronomical observations made before his time, with a description of the instruments employed; as also of his own observations and instruments; with a new Latin version of Ptolomy's catalogue of 1026 fixed stars; and Ulegh-beig's places annexed on the Latin page, with the corrections: a small catalogue of the Arabs: Tycho Brahe's of about 780 fixed stars: the Landgrave of Hesse's of 386: Hevelius's of 1534: and a catalogue of some of the southern sixed stars not visible in our hemisphere, calculated from the observations made by Dr. Halley at St. Helena, adapted to the year 1726.</p><p>This work he prepared in a great measure for the press, with much care and accuracy: but through a natural weakness of constitution, and the declines of age, he died of a strangury before he had finished it, December the 19th, 1719, at 73 years of age; leaving the care of finishing and publishing his work to his friend Mr. Hodgsou.—A less perfect edition of the <hi rend="italics">Historia Cælestis</hi> had before been published, without his consent, viz, in 1712, in one volume folio, containing his observations to the year 1705.</p><p>Thus then, as Dr. Keil observed, our author, with indefatigable pains, for more than 40 years watched the motions of the stars, and has given us innumerable observations of the sun, moon, and planets, which he made with very large instruments, accurately divided, and fitted with telescopic sights; whence we may rely much more on the observations he has made, than on former astronomers, who made their observations with the naked eye, and without the like assistance of telescopes.</p></div1><div1 part="N" n="FLANK" org="uniform" sample="complete" type="entry"><head>FLANK</head><p>, in Fortification, is that part of the bastion which reaches from the curtain to the face; and it defends the curtain, with the opposite face and flank.</p><p><hi rend="italics">Oblique</hi> or <hi rend="italics">Second</hi> <hi rend="smallcaps">Flank</hi>, or <hi rend="smallcaps">Flank</hi> <hi rend="italics">of the Curtan,</hi> is that part of the curtain from whence the face of the opposite bastion can be seen, being contained between the lines rasant and sichant, or the greater and less lines of defence; or the part of the curtain between the Flank and the point where the fichant line of defence terminates.</p><p><hi rend="italics">Covered, Low,</hi> or <hi rend="italics">Retired</hi> <hi rend="smallcaps">Flank</hi>, is the platform of the casemate, which lies hid in the bastion; and is otherwise called the Orillon.</p><p><hi rend="italics">Fichant</hi> <hi rend="smallcaps">Flank</hi>, is that from whence a cannon <pb n="478"/><cb/> playing, fires directly on the face of the opposite bastion.</p><p><hi rend="italics">Rasant</hi> or <hi rend="italics">Razant</hi> <hi rend="smallcaps">Flank</hi>, is the point from whence the line of defence begins, from the conjunction of which with the curtain, the shot only raseth the face of the next bastion, which happens when the face cannot be discovered but from the Flank alone.</p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Flanks</hi>, are lines going from the angle of the shoulder to the curtain; the chief office of which is for the defence of the moat and place.</p><div2 part="N" n="Flank" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Flank</hi></head><p>, is also a term of war, used by way of analogy for the side of a battalion, army, &c, in contradistinction to the front and rear. So, to attack the enemy in Flank, is to discover and fire upon them on one side.</p></div2></div1><div1 part="N" n="FLANKED" org="uniform" sample="complete" type="entry"><head>FLANKED</head><p>, is used of something that has Flanks, or may be approached on the Flank. As</p><p><hi rend="smallcaps">Flanked</hi> <hi rend="italics">Angle,</hi> which is that formed by the two faces of the bastion, and forming its point or angle. Also,</p><p><hi rend="smallcaps">Flanked</hi> <hi rend="italics">Line of Defence,</hi> <hi rend="smallcaps">Flanked</hi> <hi rend="italics">Tenaille,</hi> &c.</p></div1><div1 part="N" n="FLANKING" org="uniform" sample="complete" type="entry"><head>FLANKING</head><p>, in general, is the discovering and firing upon the side of a place, body, battalion, &c. To Flank a place, or other work, is to dispose it in such a manner as that every part of it may be played upon both in front and rear.</p><p><hi rend="smallcaps">Flanking</hi> means also defending. Any fortification that has no defence, but just right forwards, is faulty; and to render it complete, one part ought to be made to Flank the other. Hence the curtain is always the strongest part of any place, because it is flanked at each end.</p><p>Battalions also are said to be Flanked by the wings of the cavalry. And a house is sometimes said to be Flanked with two pavilions, or two galleries; meaning it has a gallery, &c, on each side.——There are also Flanking Angle, Flanking Line of Defence, &c.</p><p>FLAT <hi rend="italics">Bastion,</hi> is that which is built on a right line, as on the middle of the curtain, &c.</p></div1><div1 part="N" n="FLEXIBLE" org="uniform" sample="complete" type="entry"><head>FLEXIBLE</head><p>, is the property or quality of a body that may be bent.</p></div1><div1 part="N" n="FLEXION" org="uniform" sample="complete" type="entry"><head>FLEXION</head><p>, the same as Flexure.</p></div1><div1 part="N" n="FLEXURE" org="uniform" sample="complete" type="entry"><head>FLEXURE</head><p>, or <hi rend="smallcaps">Flexion</hi>, is the bending or curving of a line or figure.</p><p>When a line first bends one way, and then gradually changes to a bend the contrary way, the point where the two parts join, or where the bending changes to the other side, is called the point of inflexion, or of contrary Flexure.</p></div1><div1 part="N" n="FLIE" org="uniform" sample="complete" type="entry"><head>FLIE</head><p>, or <hi rend="smallcaps">Fly</hi>, that part of the mariner's compass, on which the 32 points of the wind are drawn, and over which the needle is placed, and fastened underneath.</p><p>FLOAT-<hi rend="italics">Boards,</hi> the boards fixed to the outer rim of undershot water wheels, serving to receive the impulse of the stream, by which the wheel is carried round.—There may be too many of these boards on a wheel. It is thought to be the best rule, to have their distance asunder such, that each of them may come out of the water as soon as possible, aster it has received and acted with its full impulse; or, which comes to the same thing, when the succeeding one is in a direction perpendicular to the surface of the water.</p><p><hi rend="smallcaps">Floating</hi> <hi rend="italics">Bridge,</hi> is a bridge of boats, casks, &c, covered with planks, firmly bound together for the passage of men, horses, or carriages, &c. <cb/></p></div1><div1 part="N" n="FLOOD" org="uniform" sample="complete" type="entry"><head>FLOOD</head><p>, a Deluge or inundation of water.</p><div2 part="N" n="Flood" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Flood</hi></head><p>, is also used in speaking of the tide, when it is rising or flowing up; in contradistinction to the Ebb, which is when it is decreasing or running out.</p></div2></div1><div1 part="N" n="FLOORING" org="uniform" sample="complete" type="entry"><head>FLOORING</head><p>, in Carpentry, is commonly understood of the boarding of the Floors. The measurement of Flooring is estimated in squares, of 100 square feet each, or of 10 feet on each side every way; for 10 times 10 are 100. Hence the length of the floor being multiplied by its breath, in feet, and two sigures cut off on the right-hand, gives the squares, and feet, or decimals cut off. Thus, a Floor being 22 feet long, and 16 wide; <table><row role="data"><cell cols="1" rows="1" role="data">then</cell><cell cols="1" rows="1" role="data"> 22</cell><cell cols="1" rows="1" role="data">length</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 16</cell><cell cols="1" rows="1" role="data">breadth</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3,52</cell><cell cols="1" rows="1" role="data">the content is therefore 3 squares, and 52</cell></row></table> feet, or decimals.</p></div1><div1 part="N" n="FLUENT" org="uniform" sample="complete" type="entry"><head>FLUENT</head><p>, or <hi rend="italics">Flowing Quantity,</hi> in the Doctrine of Fluxions, is the variable quantity which is considered as increasing or decreasing; or the Fluent of a given fluxion, is that quantity whose fluxion being taken, according to the rules of that doctrine, shall be the same with the given fluxion. See <hi rend="smallcaps">Fluxions.</hi></p><p><hi rend="italics">Contemporary</hi> <hi rend="smallcaps">Fluents</hi>, are such as flow together or for the same time. And the same is to be understood of Contemporary Fluxions.——When Contemporary Fluents are always equal, or in any constant ratio; then also are their fluxions respectively either equal, or in that same constant ratio. That is, if <hi rend="italics">x</hi> = <hi rend="italics">y,</hi> then is <hi rend="italics">x<hi rend="sup">.</hi></hi> = <hi rend="italics">y<hi rend="sup">.</hi></hi>; or if <hi rend="italics">x : y :: n</hi> : 1, then is <hi rend="italics">x<hi rend="sup">.</hi> : y<hi rend="sup">.</hi> :: n</hi> : 1; or if <hi rend="italics">x</hi> = <hi rend="italics">ny,</hi> then is <hi rend="italics">x<hi rend="sup">.</hi></hi> = <hi rend="italics">ny<hi rend="sup">.</hi>.</hi></p><p>It is easy to find the fluxions to all the given forms of Fluents; but, on the contrary, it is difficult to find the Fluents of many given fluxions; and indeed there are numberless cases in which this cannot at all be done, excepting by the quadrature and rectification of curve lines, or by logarithms, or infinite series.</p><p>This doctrine, as it was sirst invented by Sir Isaac Newton, so it was carried by him to a considerable degree of perfection, at least as to the most frequent, and most useful forms of fluents; as may be seen in his Fluxions, and in his Quadrature of Curves. Maclaurin, in his Treatise of Fluxions, has made several inquiries into Fluents, reducible to the rectification of the ellipse and hyperbola: and D'Alembert has pusued the same subject, and carried it farther, in the Memoires de l'Acad. de Berlin, tom. 2, p. 200. To the celebrated Mr. Euler this doctrine is greatly indebted, in many parts of his various writings, as well as in the Institutio Calculi Integralis, in 3 vols 4to, Petr. 1768. The ingenious Mr. Cotes contributed very much to this doctrine, in his Harmonia Mensurarum, concerning the measures of ratios and angles, in a large collection of different forms of fluxions, with their corresponding Fluents. And this subject was farther prosecuted in the same way by Walmesley, in his Analyse des Mesures des Rapportes et des Angles, a large vol. in 4to, 1749. Besides many other Authors who, by their ingenious labours, have greatly contributed to facilitate and extend the doctrine of Fluents; as Emerson, Simpson, Landen, Waring, &c, in this country; with l'Hô- <pb n="479"/><cb/> pital, and many other learned foreigners. Lastly, in 1785 was published at Vienna, by M. Paccassi, a German nobleman, Udhandlung uber eine neue Methode zu Integriren, being a method of integrating, or sinding the Fluents of given fluxions, by the rules of the direct method, or by taking again the fluxion of the given fluxion, or the 2d fluxion of the fluent sought; and then making every flowing quantity its fluxion, and 2d fluxion, in geometrical progression; a method however, which, it seems, only holds true in the easiest cases or forms, whose fluents are easily had by the most common methods. See this method farther explained in the rules following.</p><p>As it is only in certain particular forms and cases that the Fluents of given fluxions can be found; there being no method of performing this universally a priori, by a direct investigation; like finding the fluxion of a given fluent quantity; we can do little more than lay down a few rules for such forms of fluxions as are known, from the direct method, to belong to such and such kinds of Fluents or flowing quantities: and these rules, it is evident, must chiefly consist in performing such operations as are the reverse of those by which the fluxions are found to given flowing quantities. The principal cases of which are as follow:</p><p>I. <hi rend="italics">To find the Fluent of a simple fluxion</hi>; or that in which there is no variable quantity, and only one fluxional quantity. This is done by barely substituting the variable or flowing quantity instead of its fluxion, and is the result or reverse of the notation only. Thus, The Fluent of <hi rend="italics">ax<hi rend="sup">.</hi></hi> is <hi rend="italics">ax.</hi> The Fluent of <hi rend="italics">ay<hi rend="sup">.</hi></hi> + 2<hi rend="italics">y<hi rend="sup">.</hi></hi> is <hi rend="italics">ay</hi> + 2<hi rend="italics">y.</hi> The Fluent of √(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi>) is √(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi>).</p><p>II. <hi rend="italics">When any power of a flowing quantity is multip'ied by the fluxion of the root.</hi> Then, having substituted, as before, the flowing quantity for its fluxion, divide the result by the new index of the power. Or, which is the same thing, take out, or divide by, the fluxion of the root; add 1 to the index of the power; and divide by the index so increased. <table><row role="data"><cell cols="1" rows="1" role="data">So if the fluxion proposed be</cell><cell cols="1" rows="1" role="data">3<hi rend="italics">x</hi><hi rend="sup">5</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>;</cell></row><row role="data"><cell cols="1" rows="1" role="data">Strike out <hi rend="italics">x<hi rend="sup">.</hi>,</hi> then it is</cell><cell cols="1" rows="1" role="data">3<hi rend="italics">x</hi><hi rend="sup">5</hi>;</cell></row><row role="data"><cell cols="1" rows="1" role="data">add 1 to the index, and it is</cell><cell cols="1" rows="1" role="data">3<hi rend="italics">x</hi><hi rend="sup">6</hi>;</cell></row><row role="data"><cell cols="1" rows="1" role="data">divide by the index 6, and it is</cell><cell cols="1" rows="1" role="data">(3/6)<hi rend="italics">x</hi><hi rend="sup">6</hi> or (1/2)<hi rend="italics">x</hi><hi rend="sup">6</hi>;</cell></row></table> which is the Fluent of the proposed fluxion 3<hi rend="italics">x</hi><hi rend="sup">5</hi><hi rend="italics">x<hi rend="sup">.</hi>.</hi></p><p>In like manner, the Fluent of 4<hi rend="italics">axx<hi rend="sup">.</hi></hi> is 2<hi rend="italics">ax</hi><hi rend="sup">2</hi>; of 3<hi rend="italics">x</hi><hi rend="sup">1/2</hi><hi rend="italics">x</hi> is 2<hi rend="italics">x</hi><hi rend="sup">3/2</hi>; of <hi rend="italics">ax</hi><hi rend="sup">n</hi><hi rend="italics">x<hi rend="sup">.</hi></hi> is <hi rend="italics">a</hi>/(<hi rend="italics">n</hi> + 1)<hi rend="italics">x</hi><hi rend="sup">n + 1</hi>; of <hi rend="italics">z<hi rend="sup">.</hi>/z</hi><hi rend="sup">2</hi> or <hi rend="italics">z</hi><hi rend="sup">-2</hi><hi rend="italics">z<hi rend="sup">.</hi></hi> is - <hi rend="italics">z</hi><hi rend="sup">-1</hi> or -1/<hi rend="italics">z</hi>. of .</p><p>III. <hi rend="italics">When the root under a vinculum is a compound quantity; and the index of the part or factor without the vinculum increased by</hi> 1, <hi rend="italics">is some multiple of that under the vinculum:</hi> Put a single variable letter for the compound root; and substitute its powers and fluxion instead of those, of the same value, in the given quantity; so will it be reduced to a simpler form, to which the preceding rule can then be applied. <cb/></p><p>So, if the given fluxion be ; where 3, the index of the quantity without the vinculum, increased by 1, makes 4, which is double of 2, the exponent of <hi rend="italics">x</hi><hi rend="sup">2</hi> within the same; therefore putting , thence , and its fluxion is 2<hi rend="italics">xx<hi rend="sup">.</hi></hi> = <hi rend="italics">z<hi rend="sup">.</hi></hi>; hence then , and the given quantity F<hi rend="sup">.</hi> or ; and the Fluent of each term gives ; or, by substituting the value of <hi rend="italics">z</hi> instead of it, the same Fluent is</p><p>IV. <hi rend="italics">When there are several terms involving two or more variable quantities, having the fluxion of each multiplied by the other quantity or quantities:</hi> Take the Fluent of each term, as if there was only one variable quantity in it, namely that whose fluxion is contained in it, supposing all the others to be constant in that term; then if the Fluents of all the terms so found, be the very same quantity, that quantity will be the Fluent of the whole.</p><p>Thus, if the given fluxion be <hi rend="italics">x<hi rend="sup">.</hi>y</hi> + <hi rend="italics">xy<hi rend="sup">.</hi>.</hi> Then, the Fluent of <hi rend="italics">x<hi rend="sup">.</hi>y</hi> is <hi rend="italics">xy,</hi> supposing <hi rend="italics">y</hi> constant; and the Fluent of <hi rend="italics">xy<hi rend="sup">.</hi></hi> is also <hi rend="italics">xy,</hi> when <hi rend="italics">x</hi> is constant; therefore the common resulting quantity <hi rend="italics">xy</hi> is the required Fluent. of the given fluxion <hi rend="italics">x<hi rend="sup">.</hi>y</hi> + <hi rend="italics">xy<hi rend="sup">.</hi>.</hi></p><p>And, in like manner, the Fluent of <hi rend="italics">x<hi rend="sup">.</hi>yz</hi> + <hi rend="italics">xy<hi rend="sup">.</hi>z</hi> + <hi rend="italics">xyz<hi rend="sup">.</hi></hi> is <hi rend="italics">xyz.</hi></p><p>V. <hi rend="italics">When the given fluxional expression is in this form</hi> (<hi rend="italics">x<hi rend="sup">.</hi>y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi></hi>)/<hi rend="italics">y</hi><hi rend="sup">2</hi>, <hi rend="italics">viz, a sraction including two quantities, being the fluxion of the former drawn into the latter, minus the fluxion of the latter drawn into the former, and divided by the square of the latter:</hi> then the Fluent is the fraction <hi rend="italics">x/y,</hi> or of the former quantity divided by the latter. That is, The Fluent of (<hi rend="italics">x<hi rend="sup">.</hi>y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi></hi>)/<hi rend="italics">y</hi><hi rend="sup">2</hi> is <hi rend="italics">x/y</hi>; and the Fluent of (2<hi rend="italics">xx<hi rend="sup">.</hi>y</hi><hi rend="sup">2</hi> - 2<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">yy<hi rend="sup">.</hi></hi>)/<hi rend="italics">y</hi><hi rend="sup">4</hi> is <hi rend="italics">x</hi><hi rend="sup">2</hi>/<hi rend="italics">y</hi><hi rend="sup">2</hi>.</p><p>Though the examples of this case may be performed by the foregoing one. Thus the given fluxion (<hi rend="italics">x<hi rend="sup">.</hi>y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi></hi>)/<hi rend="italics">y</hi><hi rend="sup">2</hi> reduces to <hi rend="italics">x<hi rend="sup">.</hi>/y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi>/y</hi><hi rend="sup">2</hi> or <hi rend="italics">x<hi rend="sup">.</hi>/y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi>y</hi><hi rend="sup">-2</hi>; of which the Fluent of <hi rend="italics">x<hi rend="sup">.</hi>/y</hi> is <hi rend="italics">x/y</hi> when <hi rend="italics">y</hi> is constant; and the Fluent of <hi rend="italics">xy<hi rend="sup">.</hi>y</hi><hi rend="sup">-2</hi> is + <hi rend="italics">xy</hi><hi rend="sup">-1</hi> or <hi rend="italics">x/y</hi> when <hi rend="italics">x</hi> is constant; and therefore, by that case, <hi rend="italics">x/y</hi> is the Fluent on the whole (<hi rend="italics">x<hi rend="sup">.</hi>y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi></hi>)/<hi rend="italics">y</hi><hi rend="sup">2</hi>.</p><p>VI. <hi rend="italics">When the fluxion of a quantity is divided by the quantity itself:</hi> Then the Fluent is equal to the hyperbolic logarithm of that quantity; or, which is the same thing, the Fluent is equal to 2.30258509 &c, multiplied by the common log. of the same quantity. <pb n="480"/><cb/></p><p>So, the Fluent of <hi rend="italics">x<hi rend="sup">.</hi>/x</hi> or <hi rend="italics">x</hi><hi rend="sup">-1</hi> <hi rend="italics">x<hi rend="sup">.</hi></hi> is the hyp. log. of <hi rend="italics">x</hi>; of 2<hi rend="italics">x<hi rend="sup">.</hi>/x</hi> is 2 X hyp. log. of <hi rend="italics">x,</hi> or = h. l. of <hi rend="italics">x</hi><hi rend="sup">2</hi>; of <hi rend="italics">ax<hi rend="sup">.</hi>/x</hi> is <hi rend="italics">a</hi> X h. l. of <hi rend="italics">x,</hi> or h. l. of <hi rend="italics">x</hi><hi rend="sup">a</hi>; of <hi rend="italics">x<hi rend="sup">.</hi></hi>/(<hi rend="italics">a</hi> + <hi rend="italics">x</hi>) is the h. l. of <hi rend="italics">a</hi> + <hi rend="italics">x</hi>; of 3<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>/(<hi rend="italics">a</hi> + <hi rend="italics">x</hi><hi rend="sup">3</hi>) is the h. l. of <hi rend="italics">a</hi> + <hi rend="italics">x</hi><hi rend="sup">3</hi>.</p><p>VII. Many Fluents may be found by the direct method of fluxions, thus: Take the fluxion again of the given fluxional expression, or the 2d fluxion of the Fluent sought; into which substitute <hi rend="italics">x</hi><hi rend="sup">.2</hi>/<hi rend="italics">x</hi> for <hi rend="italics">x<hi rend="sup">..</hi>,</hi> and <hi rend="italics">y</hi><hi rend="sup">.2</hi>/<hi rend="italics">y</hi> for <hi rend="italics">y<hi rend="sup">..</hi>,</hi> &c, that is, make <hi rend="italics">x, x<hi rend="sup">.</hi>, x<hi rend="sup">..</hi>,</hi> as also <hi rend="italics">y, y<hi rend="sup">.</hi>, y<hi rend="sup">..</hi>,</hi> &c, in continual proportion, or <hi rend="italics">x : x<hi rend="sup">.</hi> :: x<hi rend="sup">.</hi> : x<hi rend="sup">..</hi>,</hi> and <hi rend="italics">y : y<hi rend="sup">.</hi> :: y<hi rend="sup">.</hi> : y<hi rend="sup">..</hi>,</hi> &c; then divide the square of the given fluxional expression by the 2d fluxion, just found, and the quotient will be the Fluent sought in many cases.</p><p><hi rend="italics">Or the same rule may be otherwise delivered thus:</hi> In the given fluxion F<hi rend="sup">.</hi>, write <hi rend="italics">x</hi> for <hi rend="italics">x<hi rend="sup">.</hi>, y</hi> for <hi rend="italics">y<hi rend="sup">.</hi>,</hi> &c, and call the result G, taking also the fluxion of this quantity, G<hi rend="sup">.</hi>; then make G<hi rend="sup">.</hi> : F<hi rend="sup">.</hi> :: G : F, so shall the 4th proportional F be the Fluent as before. And this is the rule of M. Paccassi.</p><p>It may be proved if this be the true Fluent, by taking the fluxion of it again, which, if it agree with the proposed fluxion, will shew that the Fluent is right; otherwise, it is wrong. <cb/></p><p>Thus, if it be proposed to find the Fluent of <hi rend="italics">nx</hi><hi rend="sup">n-1</hi><hi rend="italics">x<hi rend="sup">.</hi>.</hi> Here F<hi rend="sup">.</hi> = <hi rend="italics">nx</hi><hi rend="sup">n-1</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>; write first <hi rend="italics">x</hi> for <hi rend="italics">x<hi rend="sup">.</hi>,</hi> and it is <hi rend="italics">nx</hi><hi rend="sup">n-1</hi><hi rend="italics">x</hi> or <hi rend="italics">nx</hi><hi rend="sup">n</hi>=G; the fluxion of this is G<hi rend="sup">.</hi> = <hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">n-1</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>; therefore G<hi rend="sup">.</hi> : F<hi rend="sup">.</hi> :: G : F becomes <hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">n-1</hi><hi rend="italics">x<hi rend="sup">.</hi> : nx</hi><hi rend="sup">n-1</hi><hi rend="italics">x<hi rend="sup">.</hi> :: nx</hi><hi rend="sup">n</hi> : <hi rend="italics">x</hi><hi rend="sup">n</hi> = F, the Fluent sought.</p><p>For a 2d ex. suppose it be proposed to find the Fluent of <hi rend="italics">x<hi rend="sup">.</hi>y</hi> + <hi rend="italics">xy<hi rend="sup">.</hi>.</hi> Here ; then, writing <hi rend="italics">x</hi> for <hi rend="italics">x<hi rend="sup">.</hi>,</hi> and <hi rend="italics">y</hi> for <hi rend="italics">y<hi rend="sup">.</hi>,</hi> it is <hi rend="italics">xy</hi> + <hi rend="italics">xy</hi> or 2<hi rend="italics">xy</hi> = G; the fluxion of which is ; then G<hi rend="sup">.</hi> : F<hi rend="sup">.</hi> :: G : F becomes , the Fluent sought.</p><p>VIII. <hi rend="italics">To find Fluents by means of a table of forms of Fluxions and Fluents.</hi></p><p>In the following table are contained the most usual forms of fluxions that occur in the practical solution of problems, with their corresponding Fluents set opposite to them; by means of which, viz, comparing any proposed fluxion with the corresponding form here, the Fluent of it will be found.</p><p>Where it is to be noted, that the logarithms in the said forms, are the hyperbolic ones, which are found by multiplying the common logs. by 2.3025850929940 &c. Also the arcs whose sine, or tangent, &c, are mentioned, have the radius 1, and are those in the common tables of sines, tangents, &c.—And the numbers <hi rend="italics">m, n,</hi> &c. are to be some quantities, as the forms fail when <hi rend="italics">n</hi> = o, or <hi rend="italics">m</hi> = o, &c. <pb n="481"/><cb/></p><p><hi rend="italics">The Use of the foregoing Table of Forms of Fluxions and Fluents.</hi>—In the use of this table, it is to be observed, that the first column serves only to shew the number of the form, as a mark of reference; in the 2d column are the several forms of fluxions, which are of different kinds or classes; and in the 3d or last column are the corresponding Fluents.</p><p>The method of using the table is this. Having any fluxion given, whose Fluent it is proposed to find: First, compare the given fluxion with the several forms <cb/> of fluxions in the 2d column of the table, till one of the forms be found that agrees with it; which is done by comparing the terms of the given fluxion with the like parts of the tabular fluxion, viz, the radical quan<*> tity of the one, with that of the other; and the exponents of the variable quantities of each, both within and without the vinculum; all which, being found to agree or correspond, will give the particular values of the general quantities in the tabular form. Then substitute these particular values, for the same quantities <pb n="482"/><cb/> in the general or tabular form of the Fluent, and the result will be the particular Fluent sought; after it is multiplied by any coefficient the proposed fluxion may have.</p><p><hi rend="italics">For Ex.</hi> To find the Fluent of the given fluxional expression 3<hi rend="italics">x</hi>(5/3)<hi rend="italics">x<hi rend="sup">.</hi>.</hi> This agrees with the first form; and by comparing the fluxions, it appears that <hi rend="italics">x</hi> = <hi rend="italics">x,</hi> and , or <hi rend="italics">n</hi> = <*>/3; which being substituted in the tabular Fluent, or (1/<hi rend="italics">n</hi>)<hi rend="italics">x</hi><hi rend="sup">n</hi>, gives, after multiplying by 3 the coefficient, 3 X (3/8)<hi rend="italics">x</hi><hi rend="sup">8/3</hi> or (9/8)<hi rend="italics">x</hi><hi rend="sup">8/3</hi> for the Fluent sought.</p><p>Again, To find the Fluent of 5<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>√(<hi rend="italics">c</hi><hi rend="sup">3</hi> - <hi rend="italics">x</hi><hi rend="sup">3</hi>), or 5<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi>.</hi>―(<hi rend="italics">c</hi><hi rend="sup">3</hi> - <hi rend="italics">x</hi><hi rend="sup">3</hi>)<hi rend="sup">1/2</hi>. This belongs to the 2d form; for under the vinculum, , and the exponent<hi rend="sup">n - 1</hi> of <hi rend="italics">x</hi><hi rend="sup">n - 1</hi> without the vinculum, by using 3 for <hi rend="italics">n,</hi> is <hi rend="italics">n</hi> - 1 =2, which agrees with <hi rend="italics">x</hi><hi rend="sup">2</hi> in the fluxion given; and therefore all the parts of the form are found to answer. Then, substituting these values into the general Fluent, , it becomes .</p><p>Thirdly, To find the Fluent of <hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>/(1 + <hi rend="italics">x</hi><hi rend="sup">3</hi>). This agrees with the 8th form; where in the denominator, or <hi rend="italics">n</hi> = 3; and the numerator <hi rend="italics">x</hi><hi rend="sup">n - 1</hi> then becomes <hi rend="italics">x</hi><hi rend="sup">2</hi>, which agrees with the numerator in the given fluxion; also <hi rend="italics">a</hi> = 1. Hence then, by substituting in the general form of the Fluent 1/<hi rend="italics">n</hi> logarithm of <hi rend="italics">a</hi> + <hi rend="italics">x</hi><hi rend="sup">n</hi>, it becomes 1/3 logarithm of 1 + <hi rend="italics">x</hi><hi rend="sup">3</hi>.</p><p>IX. <hi rend="italics">To find Fluents by means of Infinite Series.</hi>— When a finite form cannot be found to agree with a proposed fluxion, it is then usual to throw it into an infinite series, either by division, or extraction of roots, or by the binomial theorem, &c; after which, the Fluents of all the terms are taken separately.</p><p><hi rend="italics">For Ex.</hi> To find the Fluent of (1 - <hi rend="italics">x</hi>)/(1 + <hi rend="italics">x</hi> - <hi rend="italics">x</hi><hi rend="sup">2</hi>)<hi rend="italics">x<hi rend="sup">.</hi>.</hi> Here, by dividing the numerator by the denominator, this becomes <hi rend="italics">x<hi rend="sup">.</hi></hi> - 2<hi rend="italics">xx<hi rend="sup">.</hi></hi> + 3<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi> - 5<hi rend="italics">x</hi><hi rend="sup">3</hi><hi rend="italics">x<hi rend="sup">.</hi></hi> + 8<hi rend="italics">x</hi><hi rend="sup">4</hi><hi rend="italics">x<hi rend="sup">.</hi></hi> &c; and, the Fluents of all the terms being taken, give <hi rend="italics">x</hi> - <hi rend="italics">x</hi><hi rend="sup">2</hi> + <hi rend="italics">x</hi><hi rend="sup">3</hi> - (5/4)<hi rend="italics">x</hi><hi rend="sup">4</hi> + (8/5)<hi rend="italics">x</hi><hi rend="sup">5</hi> &c, for the Fluent sought.</p><p><hi rend="italics">To Correct a</hi> <hi rend="smallcaps">Fluent.</hi>—The Fluent of a given fluxion, found as above, sometimes wants a correction, to make it contemporary with that required by the problem under consideration, &c: for the Fluent of any given fluxion, as <hi rend="italics">x<hi rend="sup">.</hi>,</hi> may be either <hi rend="italics">x</hi> (which is found by the rule) or it may be <hi rend="italics">x</hi> ± <hi rend="italics">c,</hi> that is <hi rend="italics">x</hi> plus or minus some constant quantity <hi rend="italics">c</hi>; because both <hi rend="italics">x</hi> and <hi rend="italics">x</hi> ± <hi rend="italics">c</hi> have the same fluxion <hi rend="italics">x<hi rend="sup">.</hi></hi>: and the finding of this constant quantity, is called correcting the Fluent. Now this correction is to be determined from the nature of the problem in hand, by which we come to know the relation which the Fluent quantities have to each other at some certain point or time. Reduce therefore the general Fluential equation, found by the rules above, to that point or time; then if the equation be true at that point, it is correct; but if not, it wants a correction, and the quantity of that correction is the dif- <cb/> ference between the two general sides of the equation when reduced to that particular state. Hence the general rule for the correction is this:</p><p>Connect the constant, but indeterminate, quantity <hi rend="italics">c</hi> with one side of the Fluential equation, as determined by the foregoing rules; then, in this equation, substitute for the variable quantities such values as they are known to have at any particular state, place, or time; and then from that particular state of the equation find the value of <hi rend="italics">c,</hi> the constant quantity of the correction.</p><p><hi rend="italics">Ex.</hi> To find the Correct Fluent of <hi rend="italics">z<hi rend="sup">.</hi></hi> = <hi rend="italics">ax</hi><hi rend="sup">3</hi><hi rend="italics">x<hi rend="sup">.</hi>.</hi> First the general Fluent of this is <hi rend="italics">z</hi> = <hi rend="italics">ax</hi><hi rend="sup">4</hi>, or , taking in the correction <hi rend="italics">c.</hi></p><p>Now if it be known that <hi rend="italics">z</hi> and <hi rend="italics">x</hi> begin together, or that <hi rend="italics">z</hi> = 0, when <hi rend="italics">x</hi> = 0; then writing 0 both for <hi rend="italics">x</hi> and <hi rend="italics">z,</hi> the general equation becomes , or <hi rend="italics">c</hi> = 0; so that, the value of <hi rend="italics">c</hi> being 0, the Correct Fluents are <hi rend="italics">z</hi> = <hi rend="italics">ax</hi><hi rend="sup">4</hi>.</p><p>But if <hi rend="italics">z</hi> be = 0, when <hi rend="italics">x</hi> is = <hi rend="italics">b,</hi> any known quantity; then substituting 0 for <hi rend="italics">z,</hi> and <hi rend="italics">b</hi> for <hi rend="italics">x,</hi> in the general equation, it becomes , from which is found <hi rend="italics">c</hi> = - <hi rend="italics">ab</hi><hi rend="sup">4</hi>; and this being written for it in the general equation, this becomes , for the correct, or contemporary Fluents.</p><p>Or lastly, if it be known that <hi rend="italics">z</hi> is = some quantity <hi rend="italics">d,</hi> when <hi rend="italics">x</hi> is equal some other quantity, as <hi rend="italics">b</hi>; then substituting <hi rend="italics">d</hi> for <hi rend="italics">z,</hi> and <hi rend="italics">b</hi> for <hi rend="italics">x,</hi> in the general Fluential equation , it becomes ; and hence is deduced the value of the correction, viz, ; consequently, writing this value for <hi rend="italics">c</hi> in the general equation, it becomes , for the Correct equation of the Fluents in this case.</p><p>And hence arises another easy and general way of correcting the Fluents, which is this: In the general equation of the Fluents, write the particular values of the quantities which they are known to have at any certain time; then subtract the sides of the resulting particular equation, from the corresponding sides of the general one, and the remainders will give the Correct equation of the Fluents sought. So, as above, the general equation being ; write <hi rend="italics">d</hi> for <hi rend="italics">z,</hi> and <hi rend="italics">b</hi> for <hi rend="italics">x,</hi> then ; hence by subtraction , or , the Correct Fluents as before.</p></div1><div1 part="N" n="FLUID" org="uniform" sample="complete" type="entry"><head>FLUID</head><p>, or <hi rend="smallcaps">Fluid</hi> <hi rend="italics">Body,</hi> according to Newton, is that whose parts yield to the smallest force impressed, and by yielding are easily moved among each other; in which sense it stands opposed to a solid. This is the definition of a perfect fluid: if the Fluid require some sensible force to move its parts, it is imperfect in proportion to that force; such as perhaps all the fluids we know of in nature.</p><p>That Fluids have vacuities in their substance is evident, because certain bodies may be dissolved in them without increasing their bulk. Thus, water will dissolve a certain quantity of salt; after which it will receive a little sugar, and after that a little alum; and all this without incerasing its first dimensions. Which shews that the particles of these solids are so far separated as to become smaller than those of the Fluid, and to be received and contained in the interstices between them.</p><p>Fluids are either elastic, such as air; or non-elastic, <pb n="483"/><cb/> as water, mercury, &c. These latter occupy the same space, or are of the same bulk, under all pressures or forces; but the former dilate and expand themselves continually by taking off the external pressure from them; for which reason it is that the density and elasticity of such fluids, are proportional to the force or weight that compresses them. The doctrine and laws of Fluids are of the greatest extent in philosophy: the properties of elastic Fluids constituting the doctrine of Pneumatics; those of the non-elastic ones, that of Hydrostatics; and their motions, Hydraulics. For which see these respective articles. Also,</p><p>For the laws of the pressure and gravitation in Fluids specifically heavier or lighter than the bodies immerged, see <hi rend="smallcaps">Specific</hi> <hi rend="italics">Gravity.</hi></p><p>For the laws of the resistance of Fluids, or the retardation of solid bodies moving in Fluids, see R<hi rend="smallcaps">ESISTANCE.</hi> And</p><p>For the ascent of Fluids in capillary tubes, or between glass planes, &c, see <hi rend="smallcaps">Ascent</hi>, and <hi rend="smallcaps">Capillary</hi> <hi rend="italics">Tubes.</hi></p></div1><div1 part="N" n="FLUTES" org="uniform" sample="complete" type="entry"><head>FLUTES</head><p>, or <hi rend="smallcaps">Flutings</hi>, are certain channels or cavities cut along the shaft of a column, or pilaster.</p></div1><div1 part="N" n="FLUIDITY" org="uniform" sample="complete" type="entry"><head>FLUIDITY</head><p>, that state or affection of bodies, which denominates or renders them Fluid; or that property by which they yield to the smallest force impressed: in contradistinction to Solidity or Firmness.</p><p>Fluidity is to be carefully distinguished from Liquidity or Humidity, which latter implies wetting or adhering. Thus, air, ether, mercury, and other melted metals, and even smoke and flame itself, are Fluid bodies, but not Liquid ones; whilst water, beer, milk, urine, &c, are both Fluids and Liquids at the same time.</p><p>The nature and causes of Fluidity have been variously assigned. The Gassendists, and ancient corpuscularians, require only three conditions as necessary to it; viz, a smallness and smoothness of the particles of the body, vacuities interspersed between them, and a spherical figure. The Cartesians, and after them Dr. Hook, Mr. Boyle, &c, beside these circumstances, require also a certain internal or intestine motion of the particles as chiefly contributing to Fluidity. Thus, Mr. Boyle, in his History of Fluidity, argues from various experiments: for example, a little dry powder of alabaster, or plaister of Paris, finely sifted, being put into a vessel over the fire, soon begins to boil like water; exhibiting all the motions and phenomena of a boiling liquor: it will tumble variously in great waves like that; will bear stirring with a stick or ladle like that, without resisting; and if strongly stirred near the side of the vessel, its waves will apparently dash against it: yet it is all the while a dry parched powder.</p><p>The like is observed in sand; a dish of which being set on a drum-head, briskly beaten by the sticks, or on the upper stone of a mill, it in all respects emulates the properties of a Fluid body. A heavy body, ex. gr. will immediately sink in it to the bottom, and a light one emerge to the top: each grain of sand has a constant vibratory and dancing motion; and if a hole be made in the side of the dish, the sand will spin out like water.</p><p>The Cartesians bring divers considerations to prove that the parts of Fluids are in continual motion: as 1st, <cb/> The change of solids into Fluids, ex. gr. ice into water, and vice versa; the chief difference between the body in those two states consisting in this, that the parts, being fixed and at rest in the one, resist the touch; whereas in the other, being already in motion, they give way to the slightest impulse. 2dly, The effects of Fluids, which commonly proceed from motion: such are the insinuation of Fluids among the pores of bodies; the softening and dissolving hard bodies; the actions of corrosive menstruums; &c: Add, that no solid can be brought to a state of Fluidity, without the intervention of some moving or moveable body, as fire, air, or water. Air, the same gentlemen hold to be the first spring of these causes of Fluidity, it being this that gives motion to fire and water, though itself receives its motion and action from the ether, or subtle medium.</p><p>But Boerhaave pleads strenuously that fire is the first mover, and the cause of all Fluidity in other bodies, as air, water, &c: without this, he shews that the atmosphere itself would fix into one solid mass. And in like manner, Dr. Black, of Edinburgh, mentions Fluidity as an effect of heat. The different degrees of heat which are required to bring different bodies into a state of Fluidity, he supposes may depend on some particulars in the mixture and composition of the bodies themselves: which is rendered farther probable from considering that the natural state of bodies in this respect is changed by certain mixtures; thus, when two metals are compounded, the mixture is commonly more fusible than either of them separately.</p><p>Newton's idea of the cause of Fluidity is different: he makes it to be the great principle of attraction. The various intestine motion and agitation among the particles of Fluid bodies, he thinks is naturally accounted for, by supposing it a primary law of nature, that as all the particles of matter attract each other when within a certain distance; so at all greater distances, they avoid and fly from one another. For then, though their common gravity, together with the pressure of other bodies upon them, may keep them together in a mass, yet their continual endeavour to avoid one another singly, and the adventitious impulses of heat and light, or other external causes, may make the particles of Fluids continually move round about one another, and so produce this quality.</p><p>As therefore the cause of cohesion of the parts of solid bodies appears to be their mutual attraction; so, on this principle, the chief cause of Fluidity seems to be a contrary motion, impressed on the particles of Fluids; by which they avoid and fly from one another, as soon as they come at, and as long as they keep at, such a distance from each other.</p><p>It is observed also in all Fluids, that the direction of their pressure against the vessels which contain them, is in lines perpendicular to the sides of such vessels; which property, being the necessary result of the spherical sigure of the particles of any Fluid, shews that the parts of all Fluids are so, or of a figure very nearly approaching to it.</p></div1><div1 part="N" n="FLUX" org="uniform" sample="complete" type="entry"><head>FLUX</head><p>, in Hydrography, a regular and periodical motion of the sea, happening twice in 24 hours and 48 minutes, nearly; in which time the water is raised, and driven violently against the shores. The Flux, or Flow, <pb n="484"/><cb/> is one of the motions of the tide: the other, by which the water sinks and retires, being called the Reflux, or Ebb. See <hi rend="smallcaps">Tide.</hi></p><p>Between the Flux and Reflux there is a kind of rest or cessation, of about half an hour; during which time the water is at its greatest height, called High-water.</p><p>The Flux of the sea follows chiefly the course of the moon; and is always highest and greatest at new and full moons, particularly near the time of the equinoxes. In some parts, as at Mount St. Michael, it rises 80 or 90 feet, though in the open sea it never rises above a foot or two; and in some places, as about the Morea, there is no flux at all. It runs up some rivers above 120 miles: though up the river Thames it goes only about 80, viz, near to Kingston in Surry. Above London-bridge, the water flows 4 hours, and ebbs 8; and below the bridge, it flows 5 hours, and ebbs 7.</p></div1><div1 part="N" n="FLUXION" org="uniform" sample="complete" type="entry"><head>FLUXION</head><p>, in the Newtonian analysis, denotes the rate or proportion at which a flowing or varying quantity increases its magnitude or quantity; and it is proportional to the magnitude by which the flowing quantity would be uniformly increased, in a given time, by the generating quantity continuing of the invariable magnitude it has at the moment of time for which the Fluxion is taken: by which it stands contradistinguished from fluent, or the flowing quantity, which is gradually, and indefinitely increasing, after the manner of a space which a body in motion describes.</p><p>Mr. Simpson observes, that there is an advantage in thus considering Fluxions, not as mere velocities of increase at a certain point, but as the magnitudes which would be uniformly generated in a given sinite time: the imagination is not here confined to a single point, and the higher orders of Fluxions are rendered much more easy and intelligible. And though Sir Isaac Newton defines Fluxions to be the velocities of motions, yet he has recourse to the moments or increments, generated in equal particles of time, to determine those velocities, which he afterwards directs to expound by finite magnitudes of other kinds.</p><p>As to the illustration of this definition, and the rules for finding the Fluxions of all sorts of fluent quantities, see the following article, or the Method of Fluxions.</p><p><hi rend="italics">Method of</hi> <hi rend="smallcaps">Fluxions</hi>, is the algorithm and analysis of Fluxions, and fluents or flowing quantities.</p><p>Most foreigners define this as the method of differences or differentials, being the analysis of indefinitely small quantities. But Newton, and other English authors, call these infinitely small quantities, moments; considering them as the momentary increments of variable quantities; as of a line considered as generated by the flux or motion of a point, or of a surface generated by the flux of a line. Accordingly, the variable quantities are called Fluents, or flowing quantities; and the method of sinding either the Fluxion, or the fluent, the method of Fluxions.</p><p>M. Leibnitz considers the same infinitely small quantities as the differences, or differentials, of quantities; and the method of finding those differences, he calls the Differential Calculus.</p><p>Besides this difference in the name, there is another in the notation. Newton expresses the Fluxion of a quantity, as of <hi rend="italics">x,</hi> by a dot placed over it, thus <hi rend="italics">x<hi rend="sup">.</hi>;</hi> <cb/> while Leibnitz expresses his differential of the same <*>, by prefixing the initial letter <hi rend="italics">d,</hi> as <hi rend="italics">dx.</hi> But, setting aside these circumstances, the two methods are just alike.</p><p>The Method of Fluxions is one of the greatest, most subtle, and sublime discoveries of perhaps any age: it opens a new world to our view, and extends our knowledge, as it were, to infinity; carrying us beyond the bounds that seemed to have been prescribed to the human mind, at least infinitely beyond those to which the ancient geometry was confined.</p><p>The history of this important discovery, recent as it is, is a little dark, and embroiled. Two of the greatest men of the last age have both of them claimed the invention, Sir I. Newton, and M. Leibnitz; and nothing can be more glorious for the method itself, than the zeal with which the partisans of either side have asserted their title.</p><p>To exhibit a just view of this dispute; and of the pretensions of each party, we may here advert to the origin of the discovery, and mark where each claim commenced, and how it was supported.</p><p>The principles upon which the Method of Fluxions is founded, or which conducted to it, had been laying, and gradually developing, from the beginning of the last century, by Fermat, Napier, Barrow, Wallis, Slusius, &c, who had methods of drawing tangents, of maxima and minima, of quadratures, &c, in certain particular cases, as of rational quantities, upon nearly the same principles. And it was not wonderful that such a genius as Newton should soon after raise those faint beginnings into a regular and general system of science, which he did about the year 1665, or sooner.</p><p>The first time however that the method appeared in print, was in 1684, when M. Leibnitz gave the rules of it in the Leipsic Acts of that year; but without the demonstrations. The two brothers however, John and James Bernoulli, being greatly struck with this new method, applied themselves diligently to it, found out the demonstrations, and applied the calculus with great success.</p><p>But before this, M. Leibnitz had proposed his Differential Method, viz, in a letter, dated Jan. 21, 1677, in which he exactly pursues Dr. Barrow's method of tangents, which had been published in 1670: and Newton communicated his method of drawing tangents to Mr. Collins, in a letter dated Dec. 10, 1672; which letter, together with another dated June 13, 1676, were sent to Mr. Leibnitz by Mr. Oldenburgh, in 1676. So that there is a strong presumption that he might avail himself of the information contained in these letters, and other papers transmitted with them, and also in 1675, before the publication of his own letter, containing the first hint of his differential method. Indeed it sufficiently appears that Newton had invented his method before the year 1669, and that he actually made use of it in his Compendium of Analysis and Quadrature of Curves before that time. His attention seems to have been directed this way, even before the time of the plague which happened in London in 1665 and 1666, when he was about 28 years of age.</p><p>This is all that is heard of the method, till the year 1687, when Newton's admirable Principia came out, <pb n="485"/><cb/> which is almost wholly built on the same calculus. The common opinion then was, that Newton and Leibnitz had each invented it about the same time: and what seemed to confirm it was, that neither of them made any mention of the other; and that, though they agreed in the substance of the thing, yet they differed in their ways of conceiving it, calling it by different names, and using different characters. However, foreigners having first learned the method through the medium of Leibnitz's publication, which spread the method through Europe, those geometricians were insensibly accustomed to look upon him as the sole, or principal inventor, and became ever after strongly prejudiced in favour of his notation and mode of conceiving it.</p><p>The two great authors themselves, without any seeming concern, or dispute, as to the property of the invention, enjoyed the glorious prospect of the progresses continually making under their auspices, till the year 1699, when the peace began to be disturbed.</p><p>M. Facio, in a treatise on the Line of Swiftest Descent, declared, that he was obliged to own Newton as the first inventor of the differential calculus, and the first by many years; and that he left the world to judge, whether Leibnitz, the second inventor, had taken any thing from him. This precise distinction between first and 2d inventor, with the suspicion it insinuated, raised a controversy between M. Leibnitz, supported by the editors of the Leipfic Acts, and the English mathematicians, who declared for Newton. Sir Isaac himself never appeared on the scene; his glory was become that of the nation; and his adherents, warm in the cause of their country, needed not his assistance to animate them.</p><p>Writings succeeded each other but slowly, on either side; probably on account of the distance of places; but the controversy grew still hotter and hotter: till at length M. Leibnitz, in the year 1711, complained to the Royal Society, that Dr. Keil had accused him of publishing the Method of Fluxions invented by Sir I. Newton, under other names and characters. He insisted that nobody knew better than Sir Isaac himself, that he had stolen nothing from him; and required that Dr. Keil should disavow the ill construction which might be put upon his words.</p><p>The Society, thus appealed to as a judge, appointed a committee to examine all the old letters, papers, and documents, that had passed among the several mathematicians, relating to the point; who, after a strict examination of all the evidence that could be procured, gave in their report as follows: “That Mr. Leibnitz “was in London in 1673, and kept a correspondence “with Mr. Collins by means of Mr. Oldenburgh, till “Sept. 1676, when he returned from Paris to Hano“ver, by way of London and Amsterdam: that it did “not appear that Mr. Leibnitz knew any thing of the “differential calculus before his letter of the 21st of “June, 1677, which was a year after a copy of a let“ter, written by Newton in the year 1672, had been “sent to Paris to be communicated to him, and above “4 years after Mr. Collins began to communicate that “letter to his correspondents; in which the Method of “Fluxions was sufficiently explained, to let a man of his “sagacity into the whole matter: and that Sir I. New- <cb/> “ton had even invented his method before the year “1669, and consequently 15 years before M. Leibnitz “had given any thing on the subject in the Leipsic “Acts.” From which they concluded that Dr. Keil had not at all injured M. Leibnitz in what he had said.</p><p>The Society printed this their determination, together with all the pieces and materials relating to it, under the title of Commercium Epistolicum de Analysi Promota, 8vo, Lond. 1712. This book was carefully distributed through Europe, to vindicate the title of the English nation to the discovery; for Newton himself, as already hinted, never appeared in the affair: whether it was that he trusted his honour with his compatriots, who were zealous enough in the cause; or whether he felt himself even superior to the glory of it.</p><p>M. Leibnitz and his friends however could not shew the same indifference: he was accused of a theft; and the whole Commercium Epistolicum either expresses it in terms, or insinuates it. Soon after the publication therefore, a loose sheet was printed at Paris, in behalf of M. Leibnitz, then at Vienna. It is written with great zeal and spirit; and it boldly maintains that the Method of Fluxions had not preceded the Method of Differences; and even insinuates that it might have arisen from it. The detail of the proofs however, on each side, would be too long, and could not be understood without a large comment, which must enter into the deepest geometry.</p><p>M. Leibnitz had begun to work upon a Commercium Epistolicum, in opposition to that of the Royal Society; but he died before it was completed.</p><p>A second edition of the Commercium Epistolicum was printed at London in 1722; when Newton, in the preface, account, and annotations, which were added to that edition, particularly answered all the objections which Leibnitz and Bernoulli were able to make since the Commercium first appeared in 1712; and from the last edition of the Commercium, with the various original papers contained in it, it evidently appears that Newton had discovered his Method of Fluxions many years before the pretensions of Leibnitz. See also Raphson's History of Fluxions.</p><p>There are however, according to the opinion of some, strong presumptions in favour of Leibnitz; i. e. that he was no plagiary: for that Newton was at least the first inventor, is past all dispute; his glory is secure; the reasonable part, even among the foreigners, allow it: and the question is only, whether Leibnitz took it from him, or fell upon the same thing with him; for, in his theory of abstract notions, which he dedicated to the Royal Academy in 1671, before he had seen any thing of Newton's, he already supposed infinitely small quantities, some greater than others; which is one of the great principles of his system.</p><p>Before prosecuting farther the history and improvements of this science, it will be proper to premise somewhat of the principles and practice of it, according to the ideas of the inventor. <hi rend="center"><hi rend="italics">Principles of the Method of <hi rend="smallcaps">Fluxions.</hi></hi></hi></p><p>1. In the doctrine of Fluxions, magnitudes or quantities, of all kinds, are considered, not as made up of a <pb n="486"/><cb/> number of small parts, but as generated by continued motion, by means of which they increase or decrease: as a line by the motion of a point; a surface by the motion of a line; and a solid by the motion of a surface: which is no new principle in geometry; having been used by Euclid and Archimedes. So likewise, time may be considered as represented by a line, increasing uniformly by the motion of a point. And quantities of all kinds whatever, which are capable of increase and decrease, may in like manner be represented by lines, surfaces, or solids, considered as generated by motion.</p><p>2. Any quantity, thus generated, and variable, is called a Fluent, or a flowing quantity. And the rate or proportion according to which any flowing quantity increases, at any position or instant, is the Fluxion of the said quantity, at that position or instant: and it is proportional to the magnitude by which the flowing quantity would be uniformly increased, in a given time, with the generating celerity uniformly continued during that time.</p><p>3. The small quantities that are actually generated or described, in any small given time, and by any continued motion, either uniform or variable, are called Increments.</p><p>4. Hence, if the motion of increase be uniform, by which increments are generated, the increments will in that case be proportional, or equal, to the measures of the Fluxions: but if the motion of increase be accelerated, the increments so generated, in a given finite time, will exceed the Fluxion; and if it be a decreasing motion, the increment so generated, will be less than the Fluxion. But if the time be indefinitely small, so that the motion be considered as uniform for that instant; then these nascent increments will always be proportional or equal to the Fluxions, and may be substituted for them, in any calculation.</p><p>5. To illustrate these definitions: Suppose a point <hi rend="italics">m</hi> be conceived to move from the position A, and to <figure/> generate a line AP, with a motion any-how regulated; and suppose the celerity of the point <hi rend="italics">m,</hi> at any position P, to be such, as would, if from thence it should become, or continue, uniform, be sufficient to describe, or pass uniformly over, the distance P<hi rend="italics">p,</hi> in the given time allowed for the Fluxion: then will the said line P<hi rend="italics">p</hi> represent the Fluxion of the said fluent or flowing line AP, at that position.</p><p>6. Again, suppose the right line <hi rend="italics">mn</hi> to move, from the position AB, continually parallel to itself, <figure/> with any continued motion, so as to generate the fluent, or flowing rectangle ABQP, whilst the point <hi rend="italics">m</hi> describes the line AP; also let the distance P<hi rend="italics">p</hi> be taken, <cb/> as above, to express the Fluxion of the line or base AP; and complete the rectangle PQ <hi rend="italics">qp.</hi> Then, like as P<hi rend="italics">p</hi> is the Fluxion of the line AP, so is the small parallelogram P<hi rend="italics">q</hi> the Fluxion of the flowing parallelogram, AQ; both these Fluxions or increments being uniformly described in the same time.</p><p>7. In like manner, if the solid AERP be conceived as generated by the plane PQR moving, <figure/> from the position ABE, always parallel to itself, along the line AD; and if P<hi rend="italics">p</hi> denote the Fluxion of the line AP. Then, like as the parallelogram P<hi rend="italics">q,</hi> or P<hi rend="italics">p</hi> X PQ, expresses the Fluxion of the flowing rectangle AQ, so likewise shall the Fluxion of the variable solid or prism AR be expressed by the prism P<hi rend="italics">r,</hi> or P<hi rend="italics">p</hi> X the plane PR. And in both these last two cases, it appears that the Fluxion of the generated rectangle, or prism, is equal to the product of the generant, whether line or plane, drawn into the Fluxion of the line along which it moves.</p><p>8. Hitherto the generant, or generating line or plane, has been considered as of a constant or invariable magnitude; in which case the fluent, or quantity generated, is a parallelogram, or a prism, the former being described by the motion of a line, and the latter by the motion of a plane. In like manner are other figures, whether plane or solid, conceived to be described, by the motion of a variable magnitude, whether it be a line or a plane. Thus, let a variable line PQ be carried with a parallel motion along AP, or whilst a point P is carried along, and describes, the line AP, <figure/> suppose another point Q to be carried by a motion perpendicular to the former, and to describe the line PQ: let <hi rend="italics">pq</hi> be another position of PQ, indefinitely near to the former; and draw Q<hi rend="italics">r</hi> parallel to AP. Now in this case there are several fluents or flowing quantities, with their respective Fluxions: viz, the line or fluent AP, the Fluxion of which is P<hi rend="italics">p,</hi> or Q<hi rend="italics">r;</hi> the line or fluent PQ, the Fluxion of which is <hi rend="italics">qr;</hi> the curve, or oblique line AQ, described by the oblique motion of the point, the Fluxion of which is Q<hi rend="italics">q;</hi> and lastly the surface APQ, described by the variable line PQ, and the Fluxion of which is the rectangle PQ<hi rend="italics">rp,</hi> or PQ X P<hi rend="italics">p.</hi> And in the same manner may any solid be conceived to be described, by the motion of a variable plane parallel to itself, substituting the variable plane for the variable line; in which case, the Fluxion of the solid, at any position, is represented by the va- <pb n="487"/><cb/> riable plane, at that position, drawn into the Fluxion of the line along which it is carried.</p><p>9. Hence then it follows generally, that the Fluxion of any figure, whether plane or solid, at any position, is equal to the section of it, at that position, drawn into the Fluxion of the axis, or line along which the variable section is supposed to be perpendicularly carried; i. e. the Fluxion of the figure AQP, is equal the plane PQ X P<hi rend="italics">p</hi> when that figure is a solid, or to the ordinate PQ X P<hi rend="italics">p</hi> when the figure is a surface.</p><p>10. It also follows, from the same premises, that, in any curve, or oblique line, AQ, whose absciss is AP, and ordinate is PQ, the Fluxions of these three form a small right-angled plane triangle Q <hi rend="italics">qr;</hi> for Q<hi rend="italics">r</hi> = P<hi rend="italics">p</hi> is the Fluxion of the absciss AP, <hi rend="italics">qr</hi> the Fluxion of the ordinate PQ, and Q<hi rend="italics">q</hi> the Fluxion of the curve or right line AQ. And consequently that, in any curve, the square of the Fluxion of the curve, is equal to the sum of the squares of the Fluxions of the absciss and ordinate, when these two lines are at right angles to each other.</p><p>11. From the premises it also appears, that contemporaneous fluents, or quantities that flow or increase together, which are always in a constant ratio to each other, have their Fluxions also in the same constant ratio at every position. For, let AP and BQ be two <figure/> contemporaneous fluents, described in the same time by the motion of the points P and Q, the contemporaneous positions being P, Q, and <hi rend="italics">p, q;</hi> and let AP be to BQ, or A<hi rend="italics">p</hi> to B<hi rend="italics">q,</hi> in the constant ratio of <hi rend="italics">n</hi> to 1. <table><row role="data"><cell cols="1" rows="1" role="data">Then is</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" role="data">;</cell></row><row role="data"><cell cols="1" rows="1" role="data">therefore by subtraction,</cell><cell cols="1" rows="1" role="data">;</cell></row><row role="data"><cell cols="1" rows="1" role="data">that is, the Fluxion P<hi rend="italics">p</hi> : Fluxion</cell><cell cols="1" rows="1" role="data">Q<hi rend="italics">q :: n</hi> : 1,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the same as Fluent AP : Fluent</cell><cell cols="1" rows="1" role="data">BQ :: <hi rend="italics">n</hi> : 1;</cell></row></table> or the Fluxions and Fluents are in the same constant ratio.</p><p>But if the ratio of the fluents be variable, so will that of the Fluxions be also, though not in the same variable ratio with the former, at every position. <hi rend="center"><hi rend="italics">The Notation, &c, in <hi rend="smallcaps">Fluxions.</hi></hi></hi></p><p>12. To apply the foregoing principles to the determination of the Fluxions of algebraic quantities, by means of which those of all other kinds are determined, it will be necessary first to premise the notation used in this science, with some observations. As, first, that the final letters of the alphabet <hi rend="italics">z, y, x, w,</hi> &c, are used to denote variable or flowing quantities; and the initial letters <hi rend="italics">a, b, c, d,</hi> &c, constant or invariable ones: Thus, the variable base AP of the flowing rectangular figure ABQP, at art. 6, may be represented by <hi rend="italics">x;</hi> and the invariable altitude PQ, by <hi rend="italics">a</hi> : also the variable base or absciss AP, of the figures in art. 8, may be represented by <hi rend="italics">x</hi>; the variable ordinate PQ, by <hi rend="italics">y;</hi> and the variable curve or line AQ, by <hi rend="italics">z.</hi></p><p>Secondly, that the Fluxion of a quantity denoted by a single letter, is represented by the same letter with a point over it: Thus the Fluxion of <hi rend="italics">x</hi> is expressed <cb/> by <hi rend="italics">x<hi rend="sup">.</hi>,</hi> that of <hi rend="italics">y</hi> by <hi rend="italics">y<hi rend="sup">.</hi>,</hi> and that of <hi rend="italics">z</hi> by <hi rend="italics">z<hi rend="sup">.</hi>.</hi> As to the Fluxions of constant or invariable quantities, as of <hi rend="italics">a, b, c,</hi> &c, they are equal to 0 or nothing, because they do not slow, or change their magnitude.</p><p>Thirdly, that the increments of variable or flowing quantities, are also denoted by the same letters with a small (′) over them: So the increments of <hi rend="italics">x, y, z,</hi> are <hi rend="italics">x′, y′, z′.</hi></p><p>13. From these notations, and the foregoing principles, the quantities and their Fluxions, there considered, will be denoted as below.</p><p>In all the foregoing figures, put <table><row role="data"><cell cols="1" rows="1" role="data">the variable or flowing line</cell><cell cols="1" rows="1" role="data">AP = <hi rend="italics">x,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">in art. 6, the constant line</cell><cell cols="1" rows="1" role="data">PQ= <hi rend="italics">a,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">in art. 8, the variable ordinate</cell><cell cols="1" rows="1" role="data">PQ= <hi rend="italics">y,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">the variable curve or right line</cell><cell cols="1" rows="1" role="data">AQ= <hi rend="italics">z;</hi></cell></row></table> Then shall the several Fluxions be thus represented, viz, <hi rend="italics">x<hi rend="sup">.</hi></hi> = P<hi rend="italics">p</hi> the Fluxion of the line AP, <hi rend="italics">ax<hi rend="sup">.</hi></hi> = PQ <hi rend="italics">qp</hi> the Fluxion of ABQP in art. 6, <hi rend="italics">yx<hi rend="sup">.</hi></hi> = PQ<hi rend="italics">rp</hi> the Fluxion of APQ in art. 8, the Fluxion of AQ, and <hi rend="italics">ax<hi rend="sup">.</hi></hi> = P<hi rend="italics">r</hi> the Flux. of the solid in art. 7, if <hi rend="italics">a</hi> denote the constant generating plane PQR. Also <hi rend="italics">nx</hi> = BQ in the figure to art. 11, and <hi rend="italics">nx<hi rend="sup">.</hi></hi> = Q<hi rend="italics">q</hi> the Fluxion of the same.</p><p>14. The principles and notation being now laid down, we may proceed to the practice and rules of this doctrine, which consists of two principal parts, called the direct and inverse method of Fluxions; viz, the direct method, which consists in finding the Fluxion of any proposed fluent or flowing quantity; and the inverse method, which consists in finding the fluent of any proposed Fluxion. As to the former of these two problems, it can always be determined, and that in finite algebraic terms; but the latter, or finding of fluents, only in some certain cases, except by means of infinite series.—First then, of <hi rend="center"><hi rend="italics">The Direct Method of <hi rend="smallcaps">Fluxions.</hi></hi></hi></p><p>15. To find the Fluxion of the product or rectangle of two variable quantities; <figure/> let ARQP = <hi rend="italics">xy</hi> be the flowing or variable rectangle, generated by two lines RQ and PQ moving always perpendicular to each other, from the positions AP and AR; denoting the one by <hi rend="italics">x,</hi> and the other by <hi rend="italics">y</hi>; and suppose <hi rend="italics">x</hi> and <hi rend="italics">y</hi> to be so related, that the curve AQ always passes through their intersection Q, or the opposite angle of the rectangle.</p><p>Now this rectangle consists of the two trilineal spaces APQ, ARQ, of which the Fluxion of the former is PQ X P<hi rend="italics">p</hi> or <hi rend="italics">x<hi rend="sup">.</hi>y,</hi> and that of the latter is RQ X R<hi rend="italics">r</hi> or <hi rend="italics">xy<hi rend="sup">.</hi>,</hi> by art. 8; therefore the sum of the two, <hi rend="italics">x<hi rend="sup">.</hi>y</hi> + <hi rend="italics">xy<hi rend="sup">.</hi>,</hi> is the Fluxion of the whole rectangle <hi rend="italics">xy</hi> or ARQP.</p><p><hi rend="italics">The same otherwise.</hi>—Let the sides of the rectangle, <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> by flowing, become <hi rend="italics">x</hi> + <hi rend="italics">x′</hi> and <hi rend="italics">y</hi> + <hi rend="italics">y′</hi>: then the product of the two, or <hi rend="italics">xy</hi> + <hi rend="italics">x′y</hi> + <hi rend="italics">xy′</hi> + <hi rend="italics">yy′</hi> will be the new or contemporaneous value of the flowing rectangle PR or <hi rend="italics">xy;</hi> subtract the one value from the other, <pb n="488"/><cb/> and the remainder <hi rend="italics">xy′</hi> + <hi rend="italics">x′y</hi> + <hi rend="italics">x′y′,</hi> will be the increment generated in the same time as <hi rend="italics">x′</hi> or <hi rend="italics">y′</hi>; of which the last term <hi rend="italics">x′y′</hi> is nothing, or indefinitely small in respect of the other two terms, because <hi rend="italics">x′</hi> and <hi rend="italics">y′</hi> are indefinitely small in respect of <hi rend="italics">x</hi> and <hi rend="italics">y</hi>; which term being therefore omitted, there remains <hi rend="italics">xy′</hi> + <hi rend="italics">x′y</hi> for the value of that increment: and hence, by substituting <hi rend="italics">x<hi rend="sup">.</hi></hi> and <hi rend="italics">y<hi rend="sup">.</hi></hi> for <hi rend="italics">x′</hi> and <hi rend="italics">y′,</hi> to which they are proportional, there arises <hi rend="italics">xy<hi rend="sup">.</hi></hi> + <hi rend="italics">x<hi rend="sup">.</hi>y</hi> for the value of the Fluxion of <hi rend="italics">xy;</hi> the same as before.</p><p>17. Hence may be derived the Fluxions of all powers and products, and of all other forms of algebraic quantities whatever. And first for the continual products, of any number of quantities, as <hi rend="italics">xyz,</hi> or <hi rend="italics">wxyz,</hi> or <hi rend="italics">vwxyz,</hi> &c. For <hi rend="italics">xyz</hi> put <hi rend="italics">q</hi> or <hi rend="italics">pz,</hi> so that <hi rend="italics">p</hi> = <hi rend="italics">xy,</hi> and <hi rend="italics">xyz</hi> = <hi rend="italics">pz</hi> = <hi rend="italics">q.</hi> Now, taking the Fluxion of <hi rend="italics">q</hi> = <hi rend="italics">pz,</hi> by the last article, it is ; but <hi rend="italics">p</hi> = <hi rend="italics">xy,</hi> and so by the same article; substituting therefore these values of <hi rend="italics">p</hi> and <hi rend="italics">p<hi rend="sup">.</hi></hi> instead of them, in the value of <hi rend="italics">q<hi rend="sup">.</hi>,</hi> this becomes , the Fluxion of <hi rend="italics">xyz</hi> required; which is therefore equal to the sum of the products arising from the Fluxion of each letter or quantity multiplied by the product of the other two.</p><p>Again, to determine the Fluxion of <hi rend="italics">wxyz,</hi> the continual product of four variable quantities; put this product, viz, <hi rend="italics">wxyz</hi> or <hi rend="italics">qw</hi> = <hi rend="italics">r,</hi> where <hi rend="italics">q</hi> = <hi rend="italics">xyz</hi> as above; then, taking the Fluxion by the last article, ; and this, by substituting for <hi rend="italics">q</hi> and <hi rend="italics">q<hi rend="sup">.</hi></hi> their values as above, becomes , the Fluxion of <hi rend="italics">wxyz</hi> as required; consisting of the Fluxion of each quantity drawn into the products of the other three.</p><p>In the very same manner it is found that the Fluxion of <hi rend="italics">vwxyz</hi> is <hi rend="italics">v<hi rend="sup">.</hi>wxyz</hi> + <hi rend="italics">vw<hi rend="sup">.</hi>xyz</hi> + <hi rend="italics">vwx<hi rend="sup">.</hi>yz</hi> + <hi rend="italics">vwxy<hi rend="sup">.</hi>z</hi> + <hi rend="italics">vwxyz<hi rend="sup">.</hi>;</hi> and so on, for any number of quantities whatever; in which it is always found that there as many terms as there are variable quantities in the proposed fluent, and that these terms consift of the Fluxion of each variable quantity multiplied by the product of all the rest of the quantities.</p><p>18. From hence is easily derived the Fluxion of any power of a variable quantity, as of <hi rend="italics">x</hi><hi rend="sup">2</hi>, or <hi rend="italics">x</hi><hi rend="sup">3</hi>, or <hi rend="italics">x</hi><hi rend="sup">4</hi>, &c. For, in the rectangle or product <hi rend="italics">xy,</hi> if <hi rend="italics">x</hi> = <hi rend="italics">y,</hi> then is the product <hi rend="italics">xy</hi> = <hi rend="italics">xx</hi> or <hi rend="italics">x</hi><hi rend="sup">2</hi>, and also its Fluxion , the Fluxion of <hi rend="italics">x</hi><hi rend="sup">2</hi>.</p><p>Again, if all the three <hi rend="italics">x, y, z</hi> be equal; then is the product of the three <hi rend="italics">xyz</hi> = <hi rend="italics">xxx</hi> or <hi rend="italics">x</hi><hi rend="sup">3</hi>; and its Fluxion , the Fluxion of <hi rend="italics">x</hi><hi rend="sup">3</hi>.</p><p>And in the same manner it will appear that the Fluxion of <hi rend="italics">x</hi><hi rend="sup">4</hi> is = 4<hi rend="italics">x</hi><hi rend="sup">3</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>, that of <hi rend="italics">x</hi><hi rend="sup">5</hi> is = 5<hi rend="italics">x</hi><hi rend="sup">4</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>, that of <hi rend="italics">x</hi><hi rend="sup">n</hi> is = <hi rend="italics">nx</hi><hi rend="sup">n - 1</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>; where <hi rend="italics">n</hi> is any positive whole number. That is, the Fluxion of any positive integral power, is equal to the exponent of the power (<hi rend="italics">n</hi>), multiplied by the next less power of the same quantity (<hi rend="italics">x</hi><hi rend="sup">n - 1</hi>), and by the Fluxion of the root (<hi rend="italics">x</hi>). <cb/></p><p>19. Next, for the Fluxion of any fraction, as <hi rend="italics">x<*>/y</hi> of one variable quantity divided by another; put the proposed fraction <hi rend="italics">x/y</hi> = <hi rend="italics">q;</hi> then multiplying by the denominator, <hi rend="italics">x</hi> = <hi rend="italics">qy;</hi> and, taking the Fluxions, ; and, by division, (by substituting the value of <hi rend="italics">q,</hi> or the Fluxion of <hi rend="italics">x<hi rend="sup">.</hi>/y,</hi> as required. That is, the Fluxion of any fraction, is equal to the Fluxion of the numerator drawn into the denominator, minus the Fluxion of the denominator drawn into the numerator, and the remainder divided by the square of the denominator.</p><p>20. Hence too is easily derived the Fluxion of any negative integer power of a variable quantity, as of <hi rend="italics">x</hi><hi rend="sup">- n</hi> or 1/<hi rend="italics">x</hi><hi rend="sup">n</hi>, which is the same thing. For here the numerator of the fraction is 1, whose Fluxion is nothing; and therefore, by the last article, the Fluxion of such a fraction, or negative power, is barely equal to minus the Fluxion of the denominator, divided by the square of the said denominator. That is, the Fluxion of ; which is the same rule as before for integral powers.</p><p>Or, the same thing is otherwise derived immediately from the Fluxion of a rectangle or product, thus: put the proposed fraction, or quotient, 1/<hi rend="italics">x</hi><hi rend="sup">n</hi> = <hi rend="italics">q;</hi> then is <hi rend="italics">qx</hi><hi rend="sup">n</hi> = 1; and, taking the Fluxions, ; hence , and (dividing by <hi rend="italics">x</hi><hi rend="sup">n</hi>), (by substituting 1/<hi rend="italics">x</hi><hi rend="sup">n</hi> for <hi rend="italics">q</hi>), - <hi rend="italics">nx<hi rend="sup">.</hi></hi>/(<hi rend="italics">x</hi><hi rend="sup">n</hi>+1) or -<hi rend="italics">nx</hi><hi rend="sup">-n-1</hi><hi rend="italics">x<hi rend="sup">.</hi>;</hi> the same as before.</p><p>21. Much in the same manner is obtained the Fluxion of any surd, or fractional power of a fluent quantity, as of <hi rend="italics">x</hi><hi rend="sup">m/n</hi> or √<hi rend="sup">n</hi><hi rend="italics">x</hi><hi rend="sup">m</hi>. For, putting the proposed quantity <hi rend="italics">x</hi><hi rend="sup">m/n</hi> = <hi rend="italics">q,</hi> then, raising each to the <hi rend="italics">n</hi> power, <hi rend="italics">x</hi><hi rend="sup">m</hi> = <hi rend="italics">q</hi><hi rend="sup">n</hi>; take the Fluxions, ; divide by <hi rend="italics">nq</hi><hi rend="sup">n-1</hi>, : which is still the same rule as before, for finding the Fluxion of any power of a sluent quantity, and which is therefore general, whether the exponent be positive or negative, or integral or fractional. <pb n="489"/><cb/></p><p>22. For the Fluxions of <figure/> Logarithms: Let A be the principal vertex of an hyperbola, having its asymptotes CD, CP, with the ordinates DA, BA, PQ, &c, parallel to them. Then, from the nature of the hyperbola, and of logarithms, it is known that any space ABPQ is the log. of the ratio of CB to CP, to the modulus ABCD. Now put 1 = CB or BA the side of the square or rhombus DB; <hi rend="italics">m</hi> = the modulus, or area of DB, or sine of the angle C to the radius 1; also the absciss CP = <hi rend="italics">x,</hi> and the ordinate PQ = <hi rend="italics">y.</hi> Then, by the nature of the hyperbola, CP X PQ is always equal to DB, that is, <hi rend="italics">xy</hi> = <hi rend="italics">m;</hi> hence <hi rend="italics">y</hi> = <hi rend="italics">m/x,</hi> and the fluxion of the space, or <hi rend="italics">x<hi rend="sup">.</hi>y</hi> is <hi rend="italics">mx<hi rend="sup">.</hi>/x</hi> = PQ <hi rend="italics">qp</hi> the fluxion of the log. of <hi rend="italics">x,</hi> to the modulus <hi rend="italics">m.</hi> And in the ordinary hyp. logarithms the modulus <hi rend="italics">m</hi> being 1, therefore <hi rend="italics">x<hi rend="sup">.</hi>/x</hi> is the fluxion of the hyp. log. of <hi rend="italics">x;</hi> which is therefore equal to the Fluxion of the quantity, divided by the quantity itself. And the same might be brought out in several other ways, independent of the figure of the hyperbola.</p><p>23. By means of the Fluxions of logarithms, are determined those of exponential quantities, i. e. quantities which have their exponent also a flowing or variable quantity. These exponentials are of two kinds, viz, when the root is a constant quantity, as <hi rend="italics">e</hi><hi rend="sup">x</hi>; and when the root is variable, as <hi rend="italics">y</hi><hi rend="sup">x</hi>.</p><p>In the former case, put the proposed exponential <hi rend="italics">e</hi><hi rend="sup">x</hi> = <hi rend="italics">z,</hi> a single variable quantity; then take the logarithm of each, so shall log. of ; take the fluxions of these, so shall ; hence , the fluxion of the proposed exponential <hi rend="italics">e</hi><hi rend="sup">x</hi>; and which therefore is equal to the said proposed quantity, drawn into the fluxion of the exponent, and also into the log. of the root.</p><p>24. Also in the 2d case, put the exponential <hi rend="italics">y</hi><hi rend="sup">x</hi>=<hi rend="italics">z</hi>; then the logarithms give log. , and the fluxions give ; hence (by substituting <hi rend="italics">y</hi><hi rend="sup">x</hi> for <hi rend="italics">z</hi>) , is the fluxion of the proposed exponential <hi rend="italics">y</hi><hi rend="sup">x</hi>; which therefore consists of two terms, of which the one is the fluxion of the proposed quantity considering the exponent only as constant, and the other is the fluxion of the same quantity considering the root as constant.</p><p><hi rend="italics">Of Second, Third, &c</hi> <hi rend="smallcaps">Fluxions.</hi>—Having explained the manner of considering and determining the first fluxions of flowing or variable quantities; it remains now to consider those of the higher orders, as 2d, 3d, 4th, &c, fluxions.</p><p>25. If the rate or celerity with which any flowing quantity changes its magnitude, be constant, or the same, at every position; then is the fluxion of it also <cb/> constantly the same. But if the variation of magnitude be continually changing, either increasing or decreasing; then will there be a certain degree of fluxion peculiar to every point or position; and the rate of variation or change in the fluxion, is called the <hi rend="italics">Fluxion of the Fluxion,</hi> or the <hi rend="italics">second Fluxion</hi> of the given fluent quantity. In like manner, the variation or fluxion of this 2d fluxion is called the <hi rend="italics">third Fluxion</hi> of the first proposed fluent quantity; and so on.</p><p>And these orders of fluxions are denoted by the fluent letter or quantity, with the corresponding number of points over it; viz, 2 points for the 2d fluxion, 3 for the 3d fluxion, 4 for the 4th fluxion, and so on. So the different orders of the fluxions of <hi rend="italics">x,</hi> are <hi rend="italics">x<hi rend="sup">.</hi>, x<hi rend="sup">..</hi>, x<hi rend="sup">∴</hi> x<hi rend="sup">....</hi>,</hi> &c; where each is the fluxion of the one next before it.</p><p>26. This description of the higher orders of fluxions may be illustrated by the three figures at the 8th article; where, if <hi rend="italics">x</hi> denote the absciss AP, and <hi rend="italics">y</hi> the ordinate PQ; and if the ordinate PQ or <hi rend="italics">y</hi> flow along the absciss AP or <hi rend="italics">x,</hi> with an uniform motion; then the fluxion of <hi rend="italics">x,</hi> viz <hi rend="italics">x<hi rend="sup">.</hi></hi> = P<hi rend="italics">p</hi> or Q<hi rend="italics">r</hi> is a constant quantity, or <hi rend="italics">x<hi rend="sup">..</hi></hi>=o, in all the figures. Also, in fig. 1, in which AQ is a right line, <hi rend="italics">y<hi rend="sup">.</hi></hi> is = <hi rend="italics">rq,</hi> or the fluxion of PQ, is a constant quantity, or <hi rend="italics">y<hi rend="sup">..</hi></hi>=o; for, the angle Q, = the angle A, being constant, Q<hi rend="italics">r</hi> is to <hi rend="italics">rq,</hi> or <hi rend="italics">x<hi rend="sup">.</hi></hi> to <hi rend="italics">y<hi rend="sup">.</hi>,</hi> in a constant ratio. But in the 2d figure, <hi rend="italics">rq,</hi> or the fluxion of PQ, continually increases more and more; and in fig. 3 it continually decreases more and more; and therefore in both these cases <hi rend="italics">y</hi> has a 2d fluxion, being positive in fig. 2, but negative in fig. 3: and so on for the other orders of fluxions.</p><p>27. Thus, if for instance, the nature of the curve be such, that <hi rend="italics">x</hi><hi rend="sup">3</hi> is everywhere equal to <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">y</hi>; then, taking the fluxions, it is <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">.</hi></hi> = 3<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi></hi>; and, considering <hi rend="italics">x<hi rend="sup">.</hi></hi> always as a constant quantity, and taking always the fluxions, the equations of the several orders of fluxions will be as below; viz, the 1st fluxions <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">.</hi></hi> = 3<hi rend="italics">x</hi><hi rend="sup">2</hi><hi rend="italics">x<hi rend="sup">.</hi>,</hi> the 2d fluxions <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">..</hi></hi> = 6<hi rend="italics">xx<hi rend="sup">.</hi></hi><hi rend="sup">2</hi>, the 3d fluxions <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">∴</hi></hi> = 6<hi rend="italics">x<hi rend="sup">.</hi></hi><hi rend="sup">3</hi>, the 4th fluxions <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">....</hi></hi> = o, and all the higher fluxions = o or nothing.</p><p>Also the higher orders of fluxions are found in the same manner as the lower ones. Thus, The 1st flux. of <hi rend="italics">y</hi><hi rend="sup">3</hi> is 3<hi rend="italics">y</hi><hi rend="sup">2</hi><hi rend="italics">y<hi rend="sup">.</hi></hi>;</p><p>28. In the foregoing articles, it has been supposed that the fluents increase; or that their fluxions are positive; but it often happens that some fluents decrease, and that therefore their fluxions are negative: and whenever this is the case, the sign of the fluxion must be changed, or made contrary to that of the fluent. So, of the rectangle <hi rend="italics">xy,</hi> when both <hi rend="italics">x</hi> and <hi rend="italics">y</hi> increase together, the fluxion is <hi rend="italics">x<hi rend="sup">.</hi>y</hi> + <hi rend="italics">xy<hi rend="sup">.</hi></hi>: but if one of them, as <hi rend="italics">y,</hi> decrease, while the other, <hi rend="italics">x,</hi> increases; then the fluxion of <hi rend="italics">y</hi> being -<hi rend="italics">y<hi rend="sup">.</hi>,</hi> the fluxion of <hi rend="italics">xy</hi> will in that case be <hi rend="italics">x<hi rend="sup">.</hi>y</hi> - <hi rend="italics">xy<hi rend="sup">.</hi>.</hi> This may be illustrated by the annexed rec- <pb n="490"/><cb/> tangle APQR = <hi rend="italics">xy,</hi> supposed to <figure/> be generated by the motion of the line PQ from A towards C, and by the motion of the line RQ from B towards A: For, by the motion of PQ, from A towards C, the rectangle is increased, and its fluxion is + <hi rend="italics">x<hi rend="sup">.</hi>y</hi>; but by the motion of RQ, from B towards A, the rectangle is decreased, and the fluxion of the decrease is <hi rend="italics">xy<hi rend="sup">.</hi></hi>; therefore, taking the fluxion of the decrease from that of the increase, the fluxion of the rectangle <hi rend="italics">xy,</hi> when <hi rend="italics">x</hi> increases and <hi rend="italics">y</hi> decreases, is <hi rend="italics">x<hi rend="sup">.</hi>y</hi>-<hi rend="italics">xy<hi rend="sup">.</hi>.</hi></p><p>For the Inverse Method, or the finding of fluents, see <hi rend="smallcaps">Fluent.</hi> And for the several applications of this science to <hi rend="smallcaps">Maxima</hi> and <hi rend="smallcaps">Minima</hi>, the drawing of T<hi rend="smallcaps">ANCENTS</hi>, &c, see the respective articles.</p><p>An idea of the principles of Fluxions being now delivered, as above, we may next consider somewhat of the chief writings and improvements that have been made by divers authors, since the first discovery of them: indeed some of the chief improvements may be learned by consulting the preface to Dr. Waring's Meditationes Analyticæ.</p><p>The inventor himself brought the doctrine of Fluxions to a considerable degree of perfection; as may be seen by many specimens of this science, given by him; particularly in his Principia, in his Tract on Quadratures, and in his Treatise on Fluxions, published by Mr. Colson; from all which it will appear, that he not only laid down the whole theory of this method, both direct and inverse; but also applied it in practice, to the solution of many of the most useful and important problems in mathematics and philosophy.</p><p>Various improvements however have been made by many illustrious authors on this science; particularly by John Bernoulli, who treated of the fluents belonging to the fluxions of exponential expressions; James Bernoulli, Craig, Cheyne, Cotes, Manfredi, Riccati, Taylor, Fagnanus, Clairaut, D'Alembert, Euler, Condorcet, Walmesley, Le Grange, Emerson, Simpson, Landen, Waring, &c. There are several other treatises also on the principles of Fluxions, by Hayes, Newyentyt, L'Hôpital, Hodson, Rowe, &c, &c, delivering the elements of this science in an easy and familiar way.</p></div1><div1 part="N" n="FLY" org="uniform" sample="complete" type="entry"><head>FLY</head><p>, in Mechanics, a heavy weight applied to certain machines, to regulate their motions, as in a jack, or to increase their effect, as in the coining engine, &c; by means of which the force of the power is not only preserved, but equally distributed in all the parts of the revolution.</p><p>The Fly is either like a cross, with heavy weights at the ends of its arms; or like a heavy wheel at rightangles to the axis of motion. It may be applied to various sorts of engines, whether moved by men, horses, wind, or water; and is of great use in those parts of an engine having a quick circular motion, and where the power or resistance acts unequally in the different parts of a revolution. In this case the Fly becomes a moderator, making the motion of revolution almost everywhere equal.</p></div1><div1 part="N" n="FLYERS" org="uniform" sample="complete" type="entry"><head>FLYERS</head><p>, in Architecture, such stairs as go straight, and do not wind round, nor have the steps made taper- <cb/> ing, but equally broad at both ends. Hence, if one flight do not rise to the top of the story &c, there is a broad half pace, and then commonly another set of flyers.</p></div1><div1 part="N" n="FLYING" org="uniform" sample="complete" type="entry"><head>FLYING</head><p>, the progressive motion of a bird, or other winged animal, through the air.</p><p>The parts of birds chiefly concerned in Flying, are the wings and tail: by the former, the bird sustains and wafts himself along; and by the latter he is assisted in ascending and descending, to keep his body poised and upright, and steady. The wings are extended or stretched quite out, and then struck forcibly downwards against the air, which by its resistance raises the bird upwards; then to make another stroke, the wing, by means of its joints, readily closes in some degree, presenting the sharp edge of the pinion foremost to cut the air, and drawing the collapsed feathers after it like a flag, to diminish the resistance to the ascent as much as possible; the wing and feathers are then stretched out horizontally again, and another downward stroke made, which raises the bird still higher; and so on as far as he pleases, or as the density of the air will sustain him; performing those motions of the wings very rapidly, that the flight may be the quicker.</p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Flying</hi>, is that attempted by men, &c, by the assistance of mechanics.</p><p>The art of flying has been attempted by several persons in all ages. Friar Bacon, about 500 years ago, not only asserts the possibility of flying, but affirms that he himself knew how to make a machine with which a man might be able to convey himself through the air like a bird; and further adds, that it had been tried with success. Though the fact is to be doubted, if, as it was said, it consisted in the following method; viz, in a couple of large thin hollow copper globes, exhausted of air; which being much lighter than air, would sustain a chair on which a person might sit. Father Francisco Lana, in his Prodromo, proposes the same thing, as his own thought. He computes, that a round vessel of plate-brass, 14 feet in diameter, weighing 3 ounces the square foot, will only weigh 1848 ounces; whereas a quantity of air of the same bulk will weigh near 2156 ounces; so that the globe will not only be sustained in the air, but will carry with it a weight of 304 ounces; and by increasing the size of the globe, the thickness of the metal remaining the same, he adds, a vessel might be made to carry a much greater weight. But the fallacy is obvious: a globe of the dimensions he describes, as Dr. Hook observes, would not sustain the pressure of the air, but be crushed inwards. Indeed it is not probable that such a globe can be made of a thinness sufficient to float in the atmosphere after it is exhausted of air, and yet be strong enough to sustain the compressing force of the atmosphere. But for this purpose it seems that the globe should be silled with an air as elastic or strong as the atmofphere, and yet be very much lighter; such as has lately been used in the Mongolsiers and Balloons; the former of which is filled with common air heated, so as to be more elastic, and less heavy; and the latter with inflammable air, which is as elastic as the common air, with only about one tenth of its weight. And thus the idea of flying, or rather floating, in the air, has been lately realized by the moderns, using however a different sort of air. See <hi rend="smallcaps">Aerostation.</hi> <pb n="491"/><cb/></p><p>The same author describes a machine for Flying, invented by the Sieur Besnier, a smith of Sable, in the county of Main. See Philos. Collec. numb. 1.</p><p>By the foregoing method however, at best, only a method of floating can be obtained, like a log floating in a current; but not of Flying, which consists in moving through the air, independent of any current; and which must be effected by something in the nature of wings. Attempts of this latter kind also have indeed been made by several persons of late years; but it does not appear that any of them have been attended with such success as to induce the authors of those attempts to make them public. The philosophers of king Charles the second's reign were much busied about this art; and the celebrated bishop Wilkins was so confident of success in it, that he says, he does not question but in future ages it will be as usual to hear a man call for his wings, when he is going a journey, as it is now to call for his boots.</p><p>The story of the flight of Dædalus is well known.</p><p><hi rend="smallcaps">Flying</hi> <hi rend="italics">Pinion,</hi> is part of a clock having a fly or fan with which to gather the air, and so bridle the rapidity of the clock's motion, when the weight descends in the striking part.</p><p>FOCAL <hi rend="italics">Distance,</hi> the Distance of the Focus, which is sometimes understood as its distance from the vertex, as in the parabola; and sometimes its distance from the centre, as in the ellipse or hyperbola.</p></div1><div1 part="N" n="FOCUS" org="uniform" sample="complete" type="entry"><head>FOCUS</head><p>, in Geometry and the Conic Sections, is applied to certain points in the Ellipse, Hyperbola, and Parabola, where the rays reflected from all parts of these curves do concur or meet; that is, rays issuing from a luminous point in the one focus, and falling on all points of the curves, are reflected into the other Focus, or into the line directed to the other Focus, viz, into the other Focus in the ellipse and parabola, and directly from it in the hyperbola. Which is the reason of the name Focus, or Burning-point. Hence, as the one Focus of the parabola is at an infinite distance; and consequently all rays drawn from it, to any finite part of the curve about the vertex, are parallel to one another; therefore if rays from the sun, or any other object so distant as that those rays may be accounted parallel, fall upon the curve of a parabola or concave surface of a paraboloidal figure, those rays will all be reflected into its Focus.</p><p>Thus, the rays P<hi rend="italics">f,</hi> from the Focus <hi rend="italics">f,</hi> are reflected in the direction PF, into the other Focus F, in the ellipse and parabola, and form the Focus F, into FQ, in the hyperbola.</p><p>In all the three curves, the double ordinate CD drawn through the Focus F, is the parameter of the axis, or a 3d proportional to AB and <hi rend="italics">ab,</hi> the transverse and conjugate axes.</p><p>In the ellipse and parabola, the transverse axis is equal to the sum of the two lines PF + P<hi rend="italics">f,</hi> drawn from the Foci to any point P in the curve; but in the hyperbola, the transverse is equal to the difference of those two lines. That is, in the ellipse and parabola, in the hyperbola.</p><p>In the ellipse and parabola, the square of the distance between the Foci, is equal to the difference of the squares of the two axes; and in the hyperbola, it is equal to the sum of their squares: that is <cb/> in the ellipse and parabola. in the hyperbola.</p><p>Therefore the two semi-axes, with the distance of the Focus from the centre, form always a right-angled triangle F<hi rend="italics">a</hi>E, or A<hi rend="italics">a</hi>E. <figure/></p><p>In all the curves, the conjugate semi-axis is a mean proportional between the distances of either Focus from either end of the transverse axis: that is, AF : E<hi rend="italics">a</hi> :: E<hi rend="italics">a</hi> : FB, or E<hi rend="italics">a</hi><hi rend="sup">2</hi>=AF . FB.</p><p>If there be any tangent to these curves, and two lines drawn from the Foci to the point of contact; these two lines will make equal angles with that tangent. So, if GPG touch the curve at P; then is the angle FPG = [angle]<hi rend="italics">f</hi>PG.</p><p>If a line be drawn from either Focus, perpendicularly upon a tangent; the distance of their intersection from the centre will be equal to the semi-transverse axis. So, if FH or <hi rend="italics">f</hi>H be perpendicular to the tangent PH; then is EH=EA or EB. Consequently, the circle described on the diameter AB, will pass through all the points H.</p><p>The foregoing are the chief properties that are common to the Foci of all the three conic sections. To which may be added the following properties which are peculiar to the parabola: viz,</p><p>In the parabola, the distance from the Focus to the vertex, is equal to 1/4 of the parameter, or half the ordinate at the Focus: viz, AF=1/2 FC.</p><p>Also, a line drawn from the Focus to any point in the curve, is equal to the sum of the Focal distance from the vertex and the absciss of the ordinate to that point: i. e. </p><p>If from any point of a parabola there be drawn a tangent, and a perpendicular to it PK, both to meet the axis produced; then the focus will be equally distant from the two intersections and the point of contact: i. e. FG=FP=FK. <pb n="492"/><cb/></p><p>Hence also the subnormal IK is=2AF or=FC the semi-parameter.</p><p>The line drawn from the Focus to any point of the curve, is equal to 1/4 the parameter of the diameter to that point: i. e. FP=1/4 the parameter of the diameter P<hi rend="italics">f.</hi></p><p>If an ordinate to any diameter pass through the Focus, it will be equal to half its parameter; and its absciss equal to 1/4 of the same parameter; or the absciss equal to half the ordinate: i. e. PL=(1/4)MN=(1/2)LM or (1/2)LN.</p><div2 part="N" n="Focus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Focus</hi></head><p>, in Optics, is a point in which several rays meet, and are collected, after being either reflected or refracted. It is so called, because the rays being here brought together and united, their force and effect are increased, insomuch as to be able to burn; and therefore it is that bodies are placed in this point to be burnt, or to shew the effect of burning glasses, or mirrors.—It is to be observed however, that in practice, the Focus is not an absolute point, but a space of some small breadth, over which the rays are scattered; owing to the different nature and refrangibility of the rays of light, and to the imperfections in the figure of the lens, &c. However, the smaller this space is, the better, or the nearer to perfection the machine approaches. Huygens shews that the Focus of a lens convex on both sides, has its breadth equal to 5/8 of the thickness of the lens.</p><p><hi rend="italics">Virtual</hi> <hi rend="smallcaps">Focus</hi>, or <hi rend="italics">Point of Divergence,</hi> so called by Mr. Molynenx, is the point from whence rays tend, after refraction or reflection; being in this respect opposed to the ordinary Focus, or Point of Concurrence, where rays are made to meet after refraction or reflection. Thus, the Foci of an hyperbola are mutually Virtual Foci to each other; but, in an ellipse, they are common Foci to each other: for the rays are reflected <hi rend="italics">from</hi> the other Focus in the hyperbola, but <hi rend="italics">towards</hi> it in the ellipse; as appears by the figures at the beginning of this article. <figure/></p><p>And, in Dioptrics, let ABC be the concavity of a glass, whose centre is D, and axis DE: Let FG be a ray of light falling on the glass, parallel to the axis DE; this ray FG, after it has passed through the glass, at its emersion at G will not proceed directly to H, but be refracted from the perpendicular DG, and will become the ray GK, which being produced to meet the axis in E, this point E is the Virtual Focus, as the ray is refracted directly <hi rend="italics">from</hi> this point. <hi rend="center"><hi rend="italics">Rules for the <hi rend="smallcaps">Foci</hi> of Lenses and Mirrors.</hi></hi> <hi rend="center">I. <hi rend="italics">In Catoptrics,</hi> or <hi rend="italics">Lenses.</hi></hi></p><p>1. The Focus of a convex glass, i. e. the point where parallel rays transmitted through a convex glass, <cb/> whose surface is the segment of a sphere, do unite, is distant from the pole or vertex of the glass, almost a diameter and half of the convexity.—2. In a PlanoConvex glass, the Focus of parallel rays is distant from the pole of the glass a diameter of the convexity, if the segment do not exceed 30 degrees. Or the rule in Plano-Convex glasses is, As 107 : 193 :: so is the radius of convexity: to the refracted ray taken to ita concourse with the axis; which in glasses of larger spheres is almost equal to the distance of the Focus taken in the axis.—3. In Double Convex glasses of the same sphere, the Focus is distant from the pole of the glass about the radius of the convexity, if the segment be but 30 degrees. But when the two convexities are unequal, or segments of different spheres, then the rule is, As the sum of the radii of both convexities: to the radius of either convexity alone :: so is double the radius of the other convexity: to the distance of the Focus. —Here observe, that the rays which fall nearer the axis of any glass, are not united with it so near the pole of the glass as those farther off: nor will the Focal distance be so great in a plano-convex glass, when the convex side is towards the object, as when the plane side is towards it. And hence it is truly concluded, that, in viewing any object by a plano-convex glass, the convex side should always be turned outward; as also in burning by such a glass. <hi rend="center">II. <hi rend="italics">For the Virtual <hi rend="smallcaps">Focus</hi>,</hi> observe</hi></p><p>1. That in Concave glasses, when a ray falls from air parallel to the axis, the Virtual Focus, by its first refraction, becomes at the distance of a diameter and a half of the concavity.—2. In Plano-Concave glasses, when the rays fall parallel to the axis, the Virtual Focus is distant from the glass, the diameter of the concavity.—3. In Plano-Concave glasses, as 107 : 193 :: so is the radius of the concavity: to the distance of the Virtual Focus.—4. In Double Concaves of the same sphere, the Virtual Focus of parallel rays is at the distance of the radius of the concavity. But, whether the concavities be equal or unequal, the Virtual Focus, or point of divergency of the parallel rays, is determined by this rule; As the sum of the radii of both concavities: is to the radius of either concavity :: so is double the radius of the other concavity : to the distance of the Virtual Focus.—5. In Concave glasses, exposed to converging rays, if the point to which the incident ray converges, be farther distant from the glass than the Virtual Focus of parallel rays, the rule for finding the Virtual Focus of this ray, is this; As the difference between the distanoe of this point from the glass, and the distance of the Virtual Focus from the glass: is to the distance of the Virtual Focus :: so is the distance of this point of convergence from the glass: to the distance of the Virtual Focus of this converging ray.—6. In Concave glasses, if the point to which the incident ray converges, be nearer to the glass than the Virtual Focus of parallel rays, the rule to find where it crosses the axis, is this; As the excess of the Virtual Focus, more than this point of-convergency: is to the Virtual Focus :: so is the distance of this point of convergency from the glass: to the distance of the point where this ray crosses the axis. <pb n="493"/><cb/> <hi rend="center">III. <hi rend="italics">Practical Rules for finding the <hi rend="smallcaps">Foci</hi> of Glasses.</hi></hi></p><p>1. To find, by experiment, the Focus of a convex spherical glass, being of a small sphere; apply it to the end of a scale of inches and decimal parts, and expose it before the sun; upon the scale may be seen the bright intersection of the rays measured out: or, expose it in the hole of a dark chamber; and where a white paper receives the distinct representation of distant objects, there is the Focus of the glass.—2. For a glass of a pretty long Focus, observe some distant object through it, and recede from the glass till the eye perceives all in confusion, or the object begins to appear inverted; then the eye is in the Focus.—3. For a Plano-Convex glass: make it reflect the sun against the wall; on the wall will then be seen two sorts of light, a brighter within another more obscure: withdraw the glass from the wall, till the bright image be in its least dimensions; then is the glass distant from the wall about a fourth part of its Focal length.—4. For a Double Convex: expose each side to the sun in like manner; and observe both the distances of the glass from the wall : then is the first distance about half the radius of the convexity turned from the sun; and the second is about half the radius of the other convexity. The radii of the two convexities being thus known, the Focus is then found by this rule; As the sum of the radii of both convexities : is to the radius of either convexity :: so is double the radius of the other convexity : to the distance of the Focus. <hi rend="center">IV. <hi rend="italics">To find the <hi rend="smallcaps">Foci</hi> of all Glasses Geometrically.</hi></hi></p><p>Dr. Halley has given a general method for finding the Foci of spherical glasses of all kinds, both concave and convex; exposed to any kind of rays, either parallel, converging, or diverging; as follows: To find the Focus of any parcel of rays diverging from, or converging to, a given point in the axis of a spherical lens, and making the same angle with it; the ratio of the sines of refraction being given. <figure/></p><p>Suppose GL a lens; P a point in its surface; V its pole; C the centre of the spherical segment; O the object, or point in the axis, to or from which the rays proceed; and OP a given ray : and suppose the ratio of refraction to be as <hi rend="italics">r</hi> to <hi rend="italics">s.</hi> Then making CR to CO, as <hi rend="italics">s</hi> to <hi rend="italics">r</hi> for the immersion of a ray, or as <hi rend="italics">r</hi> to <hi rend="italics">s</hi> for the emersion (i. e. as the sines of the angles in the medium which the ray enters, to the corresponding sines in the medium out of which it comes); and laying CR from C towards O, the point R will be the same for all the rays of the point O. Lastly, drawing the radius PC, continued if necessary; with the centre R, and distance OP, describe an are intersecting PC in <cb/> Q. The line QR, being drawn, shall be parallel to the reflected ray; and PF, being made parallel to it, shall intersect the axis in the point F, the Focus sought. —Or, make as CQ : CP :: CR : CF, which will be the distance of the Focus from the centre of the sphere. —And from this general construction, he adverts to a number of particular simple cases.</p><p>Dr. Halley gave also an universal algebraical theorem to find the Focus of all sorts of optic glasses, or lenses. See the Philos. Trans. N° 205, or Abr. vol. 1, pa. 191. <hi rend="center">V. <hi rend="italics">In Catoptrics, or <hi rend="smallcaps">Foci</hi> by Reflection.</hi></hi></p><p>These are easily found for any known curve, from this principle, that the angle of reflection is always equal to the angle of incidence.</p><p>The same are also easily found by experiment, being exposed to any object.</p><p>The increase of heat from collecting the sun's rays into a Focus, has been found in many cases of burning glasses, to be astonishingly great; the effect being increased as the square of the diameter of the glass exceeds that of the Focus. If, for instance, there be a burning glass of 12 inches diameter; this will collect or crowd together all the rays of the sun which fall upon the glass into the compass of about 1/8 part of an inch : then, the areas of the two spaces being as the square of 12 to the square of 1/8, or as the square of 96 to the square of 1, that is, as 9216 to 1; it follows, that the heat in the Focus will be 9216 times greater than the sun's common heat. And this will have an effect as great as the direct rays of the sun would have on a body placed at the 96th part of the earth's distance from the sun; or the same as on a planet that should move round the sun at but a very little more than a diameter of the sun's distance from him, or that would never appear farther from him than about 36 minutes.</p><p>Besides Dr. Halley in his method for finding the Foci, several other authors have written upon this subject; as Mr. Ditton, in his Fluxions; Dr. Gregory, in his Elements of Dioptrics; M. Carré, and Guisnée, in the Memoires de l'Acad.: Dr. Barrow and Sir I. Newton have also neat and elegant ways of finding geometrically the Foci of spherical glasses; which may be seen in Barrow's Optical Lectures.</p></div2></div1><div1 part="N" n="FOLIATE" org="uniform" sample="complete" type="entry"><head>FOLIATE</head><p>, a name given by some to a curve of the 2d order, expressed by the equation , being one species of defective hyperbolas, with one asymptote, and consisting of two infinite legs crossing each other, forming a sort of leaf. It is the 42d species of Newton's Lines of the 3d Order.</p></div1><div1 part="N" n="FOLKES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FOLKES</surname> (<foreName full="yes"><hi rend="smallcaps">Martin</hi></foreName>)</persName></head><p>, an English mathematician, philosopher, and antiquary, was born at Westminster about 1690; and was greatly distinguished as a member of the Royal Society in London, and of the Academy of Sciences at Paris. He was admitted into the former at 24 years of age; made one of their council two years after; named vice-president by Sir Isaac Newton himself; and, after Sir Hans Sloane, became president. Coins, ancient and modern, were a great object with him; and his last production was a book upon the English Silver Coin, from the Conquest to his own times. He died at London in 1754. Dr. Birch had drawn up materials for a life of Mr. Folkes, which <pb n="494"/><cb/> are preser<*>ed at large in the Anecdotes of Bowyer, p. 562. There are many memoirs of Mr. Folkes's in the Philos. Trans. from vol. 30 to vol. 46, both inclusive; viz, 1. Account of an Aurora Borealis, vol. 30.—2. Of Lieuwenhoek's curious Microscope, vol. 32.—3. On the Standard Measures in the Capitol at Rome, vol. 39.—4. Observations of three Mock-suns, vol. 40.—5. On the fresh-water Polypus, vol. 42.—6. On human bones petrisied, vol. 43.—7. On a passage in Pliny's Natural History, vol. 44.—8. On an Earthquake at London, vol. 46.—9. Ditto at Ke<*>sington, vol. 46.—10. Ditto at Newton, vol. 46.</p></div1><div1 part="N" n="FOMAHAUT" org="uniform" sample="complete" type="entry"><head>FOMAHAUT</head><p>, or <hi rend="smallcaps">Fomalhaut</hi>, a star of the first magnitude in the water of the constellation Aquarius, or in the mouth of the southern fish. Its latitude is 21° 6′ 28″ south, and mean longitude to the beginning of 1760, 11<hi rend="sup">s</hi> 0° 28′ 55″.</p></div1><div1 part="N" n="FONTENELLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FONTENELLE</surname> (<foreName full="yes"><hi rend="smallcaps">Bernard de</hi></foreName>)</persName></head><p>, a celebrated French author, was born at Rouen in 1657, and died in 1756, when he was near 100 years old. He was a universal genius: at a very early age he wrote several comedies and tragedies of considerable merit; and he did the same at a very advanced age. Voltaire declares him the most universal genius the age of Lewis the 14th produced; and compares him to lands situated in so happy a climate, as to produce all sorts of fruits. His last comedies, though they shewed the elegance of Fontenelle, were however, little fitted for the stage; he then also produced an Apology for the Vortices of Des Cartes; upon which Voltaire says, “We must excuse his comedies, on account of his great age; and his Cartesian opinions, as they were those of his youth, when they were universally received all over Europe.”</p><p>In his poetical performances and Dialogues of the Dead, the spirit of Voiture was discerned, though more extended and more philosophical. His Plurality of Worlds is a work singular in its kind : his design in this was, to present that part of philosophy to view in a gay and pleasing dress.</p><p>Fontenelle applied himself also to mathematics and natural philosophy; in which he proved not less successful than he had been in polite literature. Having been appointed perpetual secretary to the Academy of Sciences, he discharged that trust above 40 years with universal applause: his History of the Academy often throws great light upon their memoirs, which are sometimes obscure; and it has been said, he was the first who introduced elegance into the sciences. The Eloges, which he pronounced on the deceased members of the Academy, have this peculiar merit, that they excite a respect for the sciences as well as for the authors.</p><p>Upon the whole, Fontenelle must be looked upon as the great master of the new art of treating abstract sciences, in a manner that make their study at once easy and agreeable: nor are any of his works of other kinds void of merit. All those talents which he possessed from nature, were assisted by a good knowledge of history and languages: and he perhaps surpasses all men of learning who have not had the gift of invention.</p><p>Beside his poetical and theatrical works, with those of Belles lettres, &c, he published <hi rend="italics">Elemens de Geometrie de l'Infini,</hi> in 4to, 1727; also the <hi rend="italics">Theorie des Tourbillens Cartesiens;</hi> and <hi rend="italics">Discours moraux & philosophiques.</hi> <cb/> All his different works were collected in eleven volumes, 12mo, under the title of <hi rend="italics">Oeuvres Diverses.</hi></p></div1><div1 part="N" n="FOOT" org="uniform" sample="complete" type="entry"><head>FOOT</head><p>, a measure of length, divided into 12 inches, and each inch supposed to contain 3 barley-corns in length. Geometricians divide the Foot into 10 digits, and the digit into 10 lines, &c. The French divide their Foot, as we do, into 12 inches; but their inch they divide into 12 lines.</p><p>It seems this measure has been taken from the length of the human Foot; but it is of different lengths in different countries. The Paris royal Foot is to the English Foot, as 4263 to 4000, and exceeds the English by 9 1/2 lines; the ancient Roman Foot of the Capitol consisted of 4 palms; equal to 11 inches and 7/10 English; the Rhinland, or Leyden Foot, used by the northern nations, is to the Roman Foot, as 19 to 20. For the proportions of the Foot of several nations, compared with the English, see the article <hi rend="smallcaps">Measure.</hi></p><p><hi rend="italics">Square</hi> <hi rend="smallcaps">Foot</hi>, is a square whose side is 1 foot, or 12 inches, and consequently its area is 144 square inches.</p><p><hi rend="italics">Cubic</hi> <hi rend="smallcaps">Foot</hi>, is a cube whose side is one Foot, or 12 inches, and consequently it contains 12<hi rend="sup">3</hi> or 1728 cubic inches.</p><p><hi rend="smallcaps">Foot</hi>-<hi rend="italics">bank,</hi> or <hi rend="smallcaps">Foot</hi>-<hi rend="italics">step,</hi> in Fortification. See B<hi rend="smallcaps">ANQUETTE.</hi></p></div1><div1 part="N" n="FORCE" org="uniform" sample="complete" type="entry"><head>FORCE</head><p>, <hi rend="italics">Vis,</hi> or <hi rend="italics">Power,</hi> in Mechanics, Philosophy, &c, denotes the cause of the change in the state of a body, with respect to motion, rest, pressure, &c; as well as its endeavour to oppose or resist any change made in such its state. Thus, whenever a body, which was at rest, begins to move; or when its motion is either not uniform, or not direct; the cause of this change in the state of the body, is what is called <hi rend="italics">Force,</hi> and is an external Force. Or, while a body remains in the same state, either of rest, or of uniform and rectilinear motion, the cause of its remaining in such state, is in the nature of the body, being an innate internal Force, and is called its <hi rend="italics">Inertia.</hi></p><p>Mechanical Forces may be reduced to two sorts; one of a body at rest, the other of a body in motion.</p><p>The Force of a body at rest, is that which we conceive to be in a body lying still on a table, or hanging by a rope, or supported by a spring, &c; and this is called by the names of Pressure, Tension, Force, or Vis Mortua, Solicitatio, Conatus Movendi, Conamen, &c; which kind of Force may be always measured by a weight, viz, the weight that sustains it. To this class of Forces may also be referred Centripetal and Centrifugal Forces, though they reside in a body in motion; because these Forces are homogeneous to weights, pressures, or tensions of any kind. The pressure, or Force of gravity in any body, is proportional to the quantity of matter in it.</p><p>The Force of a body in motion, is a power residing in that body, so long as it continues its motion; by means of which, it is able to remove obstacles lying in its way; to lessen, destroy, or overcome the Force of any other moving body, which meets it in an opposite direction; or to surmount any the largest dead pressure or resistance, as tension, gravity, friction, &c, for some time; but which will be lessened or destroyed by such resistance as lessens or destroys the motion of the body. This is called Vis Motrix, Moving Force, or Motive Force, and by some late writers Vis Viva, to distinguish <pb n="495"/><cb/> it from the Vis Mortua spoken of before; and by these appellations, however different, the same thing is understood by all mathematicians; namely, that power of displacing, of withstanding opposite moving Forces, or of overcoming any dead resistance, which resides in a moving body, and which, in whole or in part, continues to accompany it, as long as the body moves; and may be otherwise called Percussive Force, or Momentum.</p><p>But concerning the measure of this kind of Force, mathematicians have been divided into two parties. It is allowed by both sides, that the measure of this Force depends partly upon the mass of matter in the body, or its weight, and partly upon the velocity of its motion; so that upon any increase of either weight or velocity, the moving Force will become greater. It is also agreed, that the velocity being given, or being the same in two moving bodies, their Forces will be in proportion to their masses or weights. But, when two bodies are equal, and the velocities with which they move are different, the two parties no longer agree about the measure of the moving Force.</p><p>The Cartesians and Newtonians maintain, that, in this case, the moving Force is in proportion simply as the velocity with which a body moves; so that with a double velocity it has a double Force, &c: But the Leibnitians assert, that the moving Force is proportional to the square of the velocity; so as, with a double velocity to have a quadruple Force, &c. Or, when the bodies are different, the former hold, that the momentum or moving Force of bodies, is in the compound ratio of their weights and velocities: But the Leibnitians hold, that it is in the compound ratio of the weights and squares of the velocities.</p><p>Though Leibnitz was the first who expressly asserted, that the Force of a body in motion is as the square of its velocity, which was in a paper inserted in the Leipsic Acts for the year 1686, yet it is thought that Huygens led him into that notion, by some demonstrations in the 4th part of his book De Horologio Oscillatorio, relating to the centre of oscillation, and by his dissertations, in answer to the objections of the abbot Catalan, one of which was published in 1684. This eminent mathematician had demonstrated, that in the collision of two bodies that are perfectly elastic, the sum of the products of each body multiplied by the square of its velocity, was the same after the shock as before; (though the same thing is true of the sums of the products of the bodies multiplied simply by their velocities). Now that proposition is so far general as to obtain in all collisions of bodies that are perfectly elastic: and it is also true, when bodies of a perfect elasticity strike any immoveable obstacle, as well as when they strike one another; or when they are constrained by any power or resistance to move in directions different from those in which they impel one another. These considerations might have induced Huygens to lay it down as a general rule, that bodies constantly preserve their Ascensional Force, i. e. the product of their mass by the height to which their centre of gravity can ascend, which is as the square of the velocity; and therefore, in a given system of bodies, the sum of the squares of their velocities will remain the same, and not be altered by the action of the bodies among <cb/> themselves, nor against immoveable obstacles. Leibnitz's metaphysical system led him to think that the same quantity of action or Force subsisted in the universe; and finding this impossible, if Force were estimated by the quantity of motion, he adopted Huygens's principle of the preservation of the Ascensional Force, and made it the measure of moving Forces. But it is to be observed that Huygens's principle, above-mentioned, is general only when bodies are perfectly elastic; and in some other cases which Maclaurin has endeavoured to distinguish: shewing at the same time that no useful conclusion in mechanics is affected by the disputes concerning the measure of the Force of bodies in motion, which have been objected to mathematicians. Analyst, Query 9. See Maclaurin's Fluxions, vol. 2, art. 533; Huygens Oper. tom. 1, pa. 248; &c.</p><p>Leibnitz's principle was adopted by several persons; as Wolfius, the Bernoullis, &c. Mr. Dan. Bernoulli, in his Treatise, has assumed the preservation of the Vis Ascendens of Huygens, or, as others express it, the Conservatio Virium Vivarum; and, in Bernoulli's own expression, <hi rend="italics">æqualitas inter descensum actualem ascensumque potentialem,</hi> as an hypothesis of wonderful use in mechanics. But a late author contends, that the conclusions drawn from this principle are oftener false than true. See De Conservat. Virium Vivarum Dissert. Lond. 1744.</p><p>Catalan and Papin answered Leibnitz's paper published in 1686; and from that time the controversy became more general, and was carried on for several years by Leibnitz, John and Daniel Bernoulli, Poleni, Wolfius, s'Gravesande, Camus, Muschenbroek, &c, on one side; and Pemberton, Eames, Desaguliers, Dr. S. Clark, M. de Mairan, Jurin, Maclaurin, Robins, &c, on the other. See Act. Erud. 1686, 1690, 1691, 1695; Nouv. de la Rep. des Let. Sept. 1686, 1687, art. 2; Comm. Epist. inter Leibn. et Bern. Ep. 24, p. 143; Discourses sur les Loix de la Comm. du Mouvement, Oper. tom. 3, & Diss. de vera Notione Virium Vivarum, ib.; Act. Petropol. tom. 1, p. 131, &c; Hydronamica, sect. 1; Herman, in Act. Petrop. tom. 1, p. 2, &c; Polen. de Castellis; Wolf. in Act. Petrop. tom. 1, p. 217, &c, and in Cosmol. Gener.; Graves. in Journ. Lit. and Phys. Elem. Math. 1742, lib. 2, cap. 2 and 3; Memoir. de l'Acad. des Sciences 1728; Muschenbr. Int. ad Phil. Nat. 1762, vol. 1, p. 83 &c; Pemb. &c, in Phil. Trans. number 371, 375, 376, 396, 400, 401, or Abridg. vol. 6, p. 216 &c. Mairan in Memo. de l'Acad. des Sc. 1728, Phil. Trans. numb. 459, or Abridg. vol. 8, p. 236, Philos. Trans. vol. 43, p. 423 &c; Maclaurin's Acc. of Newton's Discoveries, p. 117 &c, Flux. ubi supr. & Recueil des Pieces qui ont emporté le Prix &c. tom. 1 : Desagul. Course Exp. Philos. vol. 1, p. 393 &c, vol. 2, p. 49 &c; and Robins's Tracts, vol. 2, p. 135.</p><p>The nature and limits of this work will not admit of a full account of the arguments and experiments that have been urged on both sides of this question; but they may be found chiefly in the preceding references. A few of them however may be considered, as follows.</p><p>The defenders of Leibnitz's principle, beside the arguments above-mentioned, refer to the spaces that bodies ascend to, when thrown upwards, or the penetrations of bodies let fall into soft wax, tallow, clay, snow, <pb n="496"/><cb/> and other soft substances, which spaces are always as the squares of the velocities of the bodies. On the other hand, their opponents retort, that such spaces are not the measures of the force in question, which is rather percussive and momentary, as those above are passed over in unequal times, and are indeed the joint effect of the Forces and times.</p><p>Desaguliers brings an argument from the familiar experiment of the balance, and the other simple mechanic powers, shewing that the effect is in proportion to the velocity multiplied by the weight; for example, 4 pounds being placed at the distance of 6 inches from the centre of motion of a balance, and 2 pounds at the distance of 12 inches; these will have a Vis Viva if the balance be put into a swinging motion. Now it appears that these Forces are equal, because, with contrary directions, they soon destroy each other; and they are to each other in the simple ratio of the velocity multiplied by the mass, viz , and also.</p><p>Mr. Robins, in his remarks on J. Bernoulli's treatise, entitled, Discours sur les Loix de la Communication du Mouvement, informs us, that Leibnitz adopted this opinion through mistake; for though he maintained that the quantity of Force is always the same in the universe, he endeavours to expose the error of Des Cartes, who also asserted, that the quantity of motion is always the same; and in his discourse on this subject in the Acta Eruditorum for 1686, he says that it is agreed on by the Cartesians, and all other philosophers and mathematicians, that there is the same Force requisite to raise a body of 1 pound to the height of 4 yards, as to raise a body of 4 pounds to the height of 1 yard; but being shewn how much he was mistaken in taking that for the common opinion, which would, if allowed, prove the force of the body to be as the square of the velocity it moved with, he afterwards, rather than own himself capable of such a mistake, endeavoured to defend it as true; since he found it was the necessary consequence of what he had once asserted; and maintained, that the force of a body in motion was proportional to the height from which it must fall, to acquire that velocity; and the heights being as the squares of the velocities, the Forces would be as the masses multiplied by them; whereas, when a body descends by its gravity, or is projected perpendicularly upwards, its motion may be considered as the sum of the uniform and continual impulses of the power of gravity, during its falling in the former case, and till they extinguish it in the latter. Thus when a body is projected upwards with a double velocity, these uniform impulses must be continued for a double time, in order to destroy the motion of the body; and hence it follows, that the body, by setting out with a double velocity, and ascending for a double time, must arise to a quadruple height, before its motion is exhausted. But this proves that a body with a double velocity moves with a double Force, since it is produced or destroyed by the same uniform power continued for a double time, and not with a quadruple force, though it rises to a quadruple height; so that the error of Leibnitz consisted in his not considering the time, since the velocities alone are not the causes of the spaces described, but the times and the velocities together; yet this is the fallacious argument on which he first built his new doctrine; <cb/> and those which have been since much insisted on, and derived from the indentings or hollows produced in soft bodies by others falling into them, are much of the same kind. Robins's Tracts, vol. 2, p. 178.</p><p>But many of the experiments and reasonings, that have been urged on both sides in this controversy, have been founded in the different senses applied to the term Force. The English and French philosophers, by the word Force, mean the same thing as they do by momentum, motion, quantity of motion, percussion, or instantaneous pressure, which is measured by the mass drawn into the velocity, and may be known by its effect; and when they consider bodies as moving through a certain space, they allow for the time in which that space is described: whereas the Dutch, Italian, and German philosophers, who have espoused the new opinion, mean by the word Force, or Force inherent in a body in motion, that which it is able to produce; or, in other words, the Force is always measured by the whole effect produced by the body in motion; until its whole Force be entirely communicated or destroyed, without any regard to the time employed in producing this total effect. Thus, say they, if a point runs through a determinate space, and presses with a certain given Force, or intensity of pressure, it will perform the same action whether it move fast or slow, and therefore the time of the action in this case ought not to be regarded. s'Gravesande, Phys. El. Math. § 723—728.</p><p>Mr. Euler observes, with regard to this dispute concerning the measure of vivid Force, or living Force, as it is sometimes called, that we cannot absolutely ascribe any Force to a body in motion, whether we suppose this Force proportional to the velocity, or to the square of the velocity: for the Force exerted by a body, striking another at rest, is different from that which it exerts in striking the same body in motion; so that this Force cannot be ascribed to any body considered in itself, but only relatively to the other bodies it meets with. There is no force in a body absolutely considered, but its inertia, which is always the same, whether the body be in motion or at rest. But if this body be forced by others to change its state, its inertia then exerts itself as a Force, properly so called, which is not absolutely determinable; because it depends on the changes that happen in the state of the body.</p><p>A second observation which has been made by some eminent writers, is, that the effect of a shock of two or more bodies, is not produced in an instant, but requires a certain interval of time. If this be so, the heterogeneity between the Vis Viva and Mortua, or Living and Dead Force, will vanish; since a pressure may always be assigned, which in the same time, however little, shall produce the same effect. If then the Vis Viva be homogeneous to the Vis Mortua, and having a perfect measure and knowledge of the latter, we need require no other measure of the former than that which is derived from the Vis Mortua equivalent to it.</p><p>Now that the change in the state of two bodies, by their shock, does not happen in an instant, appears evidently from the experiments made on soft bodies: in these, percussion forms a small cavity, visible after the shock, if the bodies have no elasticity. Such a cavity cannot certainly be made in an instant. And if the <pb n="497"/><cb/> shock of soft bodies require a determinate time, we must certainly say as much of the hardest, though this time may be so small as to be beyond all our ideas. Neither can an instantaneous shock agree with that constant law of nature, by virtue of which nothing is performed per saltum. But it is needless to insist farther upon this, since the duration of any shock may be determined from the most certain principles.</p><p>There can be no shock or collision of bodies, without their making mutual impressions on each other: these impressions will be greater or less, according as the bodies are more or less soft, other circumstances being the same. In bodies called hard, the impressions are small; but a perfect hardness, which admits of no impression, seems inconsistent with the laws of nature; so that while the collision lasts, the action of bodies is the result of their mutually pressing each other. This pressure changes their state; and the Forces exerted in percussion are really pressures, and truly Vires Mortuæ, if we will use this expression, which is no longer proper, since the pretended infinite difference between the Vires Vivæ and Mortuæ ceases.</p><p>The Force of percussion, resulting from the pressures that bodies exert on each other, while the collision lasts, may be perfectly known, if these pressures be determined for every instant of the shock. The mutual action of the bodies begins the first moment of their contact; and is then least; after which this action increases, and becomes greatest when the reciprocal impressions are strongest. If the bodies have no elasticity, and the impressions they have received remain, the Forces will then cease. But if the bodies be elastic, and the parts compressed restore themselves to their former state, then will the bodies continue to press each other till they separate. To comprehend therefore perfectly the Force of percussion, it is requisite first to define the time the shock lasts, and then to assign the pressure corresponding to each instant of this time; and as the effect of pressures in changing the state of any body may be known, we may thence come at the true cause of the change of motion arising from collision. The Force of percussion therefore is no more than the operation of a variable pressure during a given time; and to measure this Force, we must have regard to the time, and to the variations according to which the pressure increases and decreases.</p><p>Mr. Euler has given some calculations relative to these particulars; and he illustrates their tendency by this instance: Suppose that the hardness of the two bodies, A and B, is equal; and such, that being pressed together with the force of 100lb, the impression made on each is of the depth of (1/1000)th part of a foot. Suppose also that B is fixed, and that A strikes it with the velocity of 100 feet in a second; according to Mr. Euler, the greatest Force of compression will be equivalent to 400lb, and this Force will produce in each of these bodies an impression equal to 1/25 of a foot; and the duration of the collision, that is, till the bodies arrive at their greatest compression, will be about 1/800 of a second. Mr. Euler, in his calculations, supposes the hardness of a body to be proportional to the Force or pressure requisite to make a given impression on it; so that the Force by which a given impression is made on a body, is in a compound ratio of the hardness of <cb/> the body and of the quantity of the impression. But he observes, that regard must be had to the magnitude of the bodies, as the same impression cannot be made on the least bodies as on the greatest, from the defect of space through which their component particles must be driven: he considers therefore only the least impressions, and supposes the bodies of such magnitudes, that with respect to them the impressions may be looked upon as nothing. What he supposes concerning the hardness of bodies, neither implies elasticity nor the want of it, as elasticity only produces a restitution of figure and impression when the pressing Force ceases; but this restitution need not be here considered. It is also supposed, that the bodies which strike each other, have plane and equal bases, by which they touch each other in the collision; so that the impression hereby made diminishes the length of each body. It is farther to be observed, that in Mr. Euler's calculations, bodies are supposed so constituted, that they may not only receive impressions from the Forces pressing them, but that a greater Force is requisite to make a greater impression. This excludes all bodies, fluid or solid, in which the same Force may penetrate farther and farther, provided it have time, without ever being in equilibrio with the resistance: thus a body may continually penetrate farther into soft wax, although the force impelling it be not increased: in these, and the like cases, nothing is required but to surmount the first obstacles; which being once done, and the connection of parts broken, the penetrating body always advances, meeting with the same obstacles as before, and destroying them by an equal Force. But Mr. Euler only considers the first obstacles which exist before any separation of parts, and which are doubtless such, that a greater impression requires a greater Force. Indeed this chiefly takes place in elastic bodies; but it seems likewise to obtain in all bodies when the impressions made on them are small, and the contexture of their parts is not altered.</p><p>These things being premised, let the mass or weight of the body A be expressed in general by A, and let its velocity before the shock be that which it might acquire by falling from the height <hi rend="italics">a.</hi> Farther, let the harduess of A be expressed by M, and that of B by N, and let the area of the base, on which the impression is made, be <hi rend="italics">cc;</hi> then will the greatest compression be made with the Force Therefore if the hardness of the two bodies, and the plane of their contact during the whole time of their collision be the same, this Force will be as √A<hi rend="italics">a,</hi> that is, as the square root of the Vis Viva of the striking body A. And as √<hi rend="italics">a</hi> is proportional to the velocity of the body A, the Force of percussion will be in a compound ratio of the velocity and of the subduplicate ratio of the mass of the body striking; so that in this case neither the Leibnitian nor the Cartesian propositions take place. But this Force of percussion depends chiefly on the hardness of the bodies; the greater this is, the greater will the Force of percussion be. If M = N, this Force will be as √(M<hi rend="italics">cc</hi> X A<hi rend="italics">a,</hi>) that is, in a compound subduplicate ratio of the Vis Viva of the striking body, of the hardness, and of the plane of contact. But if <pb n="498"/><cb/> M, the hardness of one of the bodies, be infinite, the Force of percussion will be as √(N<hi rend="italics">cc</hi> X A<hi rend="italics">a</hi>;) at the same time, if M =N, this Force will be as √((1/2)N<hi rend="italics">cc</hi> X A<hi rend="italics">a.</hi>) Therefore, all other things being equal, the Force of percussion, if the striking body be infinitely hard, will be to the Force of percussion when both the bodies are equally hard, as √2 to 1.</p><p>Mr. Euler farther deduces from his calculation, that the impression received by the bodies A and B will be as follows; viz, as and respectively. If therefore the hardness of A, that is M, be infinite, it will suffer no impression; whereas that on B will extend to the depth of √A<hi rend="italics">a</hi>/N<hi rend="italics">cc</hi>. But if the hardness of the two bodies be the same, or M = N, they will each receive equal impressions of the depth √A<hi rend="italics">a</hi>/2N<hi rend="italics">cc</hi>. So that the impression received by the body B in this case, will be to the impression it receives in the former, as 1 to √2.</p><p>Mr. Euler has likewise considered and computed the case where the striking body has its anterior surface convex, with which it strikes an immoveable body whose surface is plane. He has also examined the case when both bodies are supposed immoveable; and from his formulæ he deduces the known laws of the collision of elastic and non-elastic bodies. He has also determined the greatest pressures the bodies receive in these cases; and likewise the impressions made on them. In particular he shews, that the impressions received by the body struck, or B, if moveable, is to the impression received by the same body when immoveable, as √B to √(A+B).</p><p>There are several curious as well as useful observations in Desaguliers's Experimental Philosophy, concerning the comparative Forces of men and horses, and the best way of applying them. A horse draws with the greatest advantage when the line of direction is level with his breast; in such a situation, he is able to draw 200lb for 8 hours a day, walking about 2 1/2 miles an hour. But if the same horse be made to draw 240lb, he can work only 6 hours a day, and cannot go quite so fast. On a carriage, indeed, where friction alone is to be overcome, a middling horse will draw 1000lb. But the best way to try the Force of a horse, is to make him draw up out of a well, over a single pulley or roller; and in that case, an ordinary horse will draw about 200lb, as besore observed.</p><p>It is found that 5 men are of equal Force with one horse, and can with equal ease push round the horizontal beam of a mill, in a walk 40 feet wide; whereas 3 men will do it in a walk only 19 feet wide.</p><p>The worst way of applying the Force of a horse is to make him carry or draw up hill: for if the hill be steep, 3 men will do more than a horse, each man climbing up faster with a burden of 100lb weight, than a horse that is loaded with 300lb: a difference which is owing to the position of the parts of the human body being better adapted to climb, than those of a horse. <cb/></p><p>On the other hand, the best way of applying the Force of a horse, is the horizontal direction, in which a man can exert the least Force: thus, a man that weighs 140lb, when drawing a boat along by means of a rope coming over his shoulders, cannot draw above 27lb, or exert above 1-7th part of the Force of a horse employed to the same purpose; so that in this way the Force of a horse is equal to that of 7 men.</p><p>The best and most effectual posture in a man, is that of rowing; when he not only acts with more muscles at once for overcoming the resistance, than in any other position; but also as he pulls backwards, the weight of his body assists by way of lever. See Desaguliers's Exp. Philos. vol. 1, p. 241, where several other observations are made relative to Force acquired by certain positions of the body; from which that author accounts for most feats of strength and activity. See also a Memoir on this subject by M. De la Hire, in the Mem. Roy. Acad. 1729; or in Desaguliers's Exp. &c. p. 267 &c, who has published a translation of part of it with remarks.</p><p><hi rend="smallcaps">Force</hi> is distinguished into Motive and Accelerative or Retardive.</p><p><hi rend="italics">Motive</hi> <hi rend="smallcaps">Force</hi>, otherwise called Momentum, or Force of Percussion, is the absolute Force of a body in motion, &c; and is expressed by the product of the weight or mass of matter in the body multiplied by the velocity with which it moves. But</p><p><hi rend="italics">Accelerative</hi> <hi rend="smallcaps">Force</hi>, or <hi rend="italics">Retardive</hi> <hi rend="smallcaps">Force</hi>, is that which respects the velocity of the motion only, accelerating or retarding it; and it is denoted by the quotient of the motive Force divided by the mass or weight of the body. So, if <hi rend="italics">m</hi> denote the motive Force, and <hi rend="italics">b</hi> the body, or its weight, and <hi rend="italics">f</hi> the accelerating or retarding Force, then is <hi rend="italics">f</hi> as <hi rend="italics">m</hi>/<hi rend="italics">b</hi>.</p><p>Again, Forces are either Constant or Variable.</p><p><hi rend="italics">Constant</hi> <hi rend="smallcaps">Forces</hi> are such as remain and act continually the same for some determinate time. Such, for example, is the Force of gravity, which acts constantly the same upon a body while it continues at the same distance from the centre of the earth, or from the centre of Force, wherever that may be. In the case of a constant Force F acting upon a body <hi rend="italics">b,</hi> for any time <hi rend="italics">t,</hi> we have these following theorems; putting <hi rend="italics">f</hi> = the constant accelerating Force = F ÷ <hi rend="italics">b,</hi> <hi rend="italics">v</hi> = the velocity at the end of the time <hi rend="italics">t,</hi> <hi rend="italics">s</hi> = the space passed over in that time, by the constant action of that Force on the body: and <hi rend="italics">g</hi> = 16 1/12 feet, the space generated by gravity in 1 second, and calling the accelerating Force of gravity 1; then is</p><p><hi rend="italics">Variable</hi> <hi rend="smallcaps">Forces</hi> are such as are continually chang- <pb n="499"/><cb/> ing in their effect and intensity; such as the Force of gravity at different distances from the centre of the earth, which decreases in proportion as the square of the distance increases. In variable Forces, theorems similar to those above may be exhibited by using the fluxions of quantities, and afterwards taking the fluents of the given fluxional equations. And herein consists one of the great excellencies of the Newtonian or modern analysis, by which we are enabled to manage, and compute the effects of all kinds of variable Forces, whether accelerating or retarding. Thus, using the same notation as above for constant forces, viz, <hi rend="italics">f</hi> the accelerating Force at any instant, <hi rend="italics">t</hi> the time a body has been in motion by the action of the variable Force, <hi rend="italics">v</hi> the velocity generated in that time, <hi rend="italics">s</hi> the space run over in that time, and <hi rend="italics">g</hi> = 16 1/12 feet; then is</p><p>In these four theorems, the Force <hi rend="italics">f,</hi> though variable, is supposed to be constant for the indefinitely small time <hi rend="italics">t<hi rend="sup">.</hi>;</hi> and they are to be used in all cases of variable Forces, as the former ones in constant Forces; viz, from the circumstances of the problem under consideration, deduce a general expression for the value of the force <hi rend="italics">f,</hi> at any indefinite time <hi rend="italics">t;</hi> then substitute it in one of these theorems, which shall be proper to the case in hand; and the equation thence resulting will determine the corresponding values of the other quantities in the problem.</p><p>It is also to be observed, that the foregoing theorems equally hold good for the destruction of motion and velocity, by means of retarding or resisting Forces, as for the generation of the same by means of accelerating Forces.</p><p>Many applications of these theorems may be seen in my Select Exercises, p. 172 &c.</p><p>There are many other denominations and kinds of Forces; such as attractive, central, centrifugal, &c, &c; for which see the respective words.</p></div1><div1 part="N" n="FORCER" org="uniform" sample="complete" type="entry"><head>FORCER</head><p>, in Mechanics, is properly a piston without a valve. For, by drawing up such a piston, the air is drawn up, and the water follows; then pushing the piston down again, the water, being prevented from descending by the lower valve, is forced up to any height above, by means of a side branch between the two.</p><p>See the ways of making these in Desaguliers's Exper. Philos. vol. 2, p. 161 &c. See also Clare's Motion of Fluids, p. 60.</p><p><hi rend="smallcaps">Forcing</hi> <hi rend="italics">Pump,</hi> one that acts, or raises water, by a Forcing piston. See above.</p></div1><div1 part="N" n="FORELAND" org="uniform" sample="complete" type="entry"><head>FORELAND</head><p>, or <hi rend="smallcaps">Foreness</hi>, in Navigation, a point of land jutting out into the sea. <cb/></p><div2 part="N" n="Foreland" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Foreland</hi></head><p>, in Fortification, is a small piece of ground between the wall of a place and the moat; called also Berme and Liziere.</p><p>FORE-STAFF, an instrument used at sea, for taking the altitudes of the heavenly bodies; being so called, because the observer, in using it, turns his face forward or towards the object, in contradistinction to the Back-staff, with which he turns his back to the object. It is also called the Cross-staff, because it consists of several pieces set across a staff. <figure/></p><p>The Fore-staff is formed of a straight square staff AB, of about 3 feet long, having each of its four sides graduated like a line of tangents, and four crosses, or vanes, FF, EE, DD, CC, sliding upon it, of unequal lengths, the halves of which represent the radii to the lines of tangents on the different sides of the staff. The first or shortest of these vanes, FF, is called the Ten Cross, or Ten Vane, and belongs to the 10 scale, or that side of the instrument on which divisions begin at 3 degrees, and end at 10. The next longer cross, EE, is called the 30 Cross, belonging to that side of the staff where the divisions begin at 10 degrees, and end at 30, called the 30 scale. The third vane DD, is called the 60 Cross, and belongs to that side where the divisions begin at 20 degrees, and end at 60. The last or longest vane CC, called the 90 Cross, belongs to the side where the divisions begin at 30 degrees, and end at 90.</p><p>The chief use of this instrument, is to take the height of the sun, and stars, or the distance between two stars: and the 10, 30, 60, or 90 Cross is to be used, according as the altitude is more or less; that is, if the altitude be less than 10 degrees, the 10 Cross is to be used; if above 10, but less than 30, the 30 Cross is to be used; and so on.</p><p><hi rend="italics">To observe an Altitude with the Fore-staff.</hi> Apply the flat end of the staff to the eye, and slide one of the crosses backwards and forwards upon it, till over the upper end of the Cross be just seen the centre of the sun or star, and over the under end the extreme horizon; then the degrees and minutes cut by the cross on the side of the staff proper to the vane in use, gives the altitude above the horizon.</p><p><hi rend="italics">In like manner, for the Distance between two luminaries;</hi> the Staff being set to the eye, bring the cross just to subtend or cover that distance, by having the one luminary just at the one end of it, and the other luminary at the other end of it; and the degrees and minutes, in the distance, will be cut on the proper side of the staff, as before.</p></div2></div1><div1 part="N" n="FORMULA" org="uniform" sample="complete" type="entry"><head>FORMULA</head><p>, a theorem or general rule, or expression, for resolving certain particular cases of some problem, &c. So (1/2)<hi rend="italics">s</hi> + (1/2)<hi rend="italics">d</hi> is a general Formula for <pb n="500"/><cb/> the greater of two quantities whose sum is <hi rend="italics">s</hi> and difference <hi rend="italics">d</hi>; and (1/2)<hi rend="italics">s</hi> - (1/2)<hi rend="italics">d</hi> is the Formula, or general value, for the less quantity. Also √(<hi rend="italics">dx</hi> - <hi rend="italics">x</hi><hi rend="sup">2</hi>) is the Formula, or general value, of the ordinate to a circle, whose diameter is <hi rend="italics">d,</hi> and absciss <hi rend="italics">x.</hi></p></div1><div1 part="N" n="FORT" org="uniform" sample="complete" type="entry"><head>FORT</head><p>, a little castle or fortress; or a place of small extent, fortified either by nature or art.</p><p>The Fort is usually encompassed with a moat, rampart, and parapet, to secure some high ground, or passage of a river; to make good or strengthen an advantageous post; or to fortify the lines and quarters of a siege.</p><p><hi rend="italics">Field</hi> <hi rend="smallcaps">Fort</hi>, otherwise called <hi rend="italics">Fortin,</hi> or <hi rend="italics">Fortlet,</hi> and sometimes <hi rend="italics">Sconce,</hi> is a small Fort, built in haste, for the defence of a pass or post; but particularly constructed for the defence of a camp in the time of a siege, where the principal quarters are usually joined, or made to communicate with each other, by lines defended by fortins and redoubts. Their figure and size are various, according to the nature of the situation, and the importance of the service for which they are intended; but they are most commonly made square, each side about 100 toises, the perpendicular 10, and the faces 25; the ditch about this Fort may be 10 or 12 toises wide; the parapet is made of turf, and fraised, and the ditch pallisadoed when dry. There may be made a covert-way about this Fort, or else a row of pallisades might be placed on the outside of the ditch. Some of these are fortified with bastions, and some with demibastions.——A Fort differs from a citadel, as this last is erected to command and guard some town; and from a redoubt, as it is closed on all sides, while the redoubt is open on one side.</p><p><hi rend="italics">Royal</hi> <hi rend="smallcaps">Fort</hi>, is one whose line of defence is at least 26 fathoms long.</p><p><hi rend="italics">Star</hi> <hi rend="smallcaps">Fort</hi>, is a sconce or redoubt, constituted by reentering and saliant angles, having commonly from sive to eight points, and the sides flanking each other.</p><p><hi rend="italics">Forts</hi> are sometimes made triangular, only with half bastions; or of various other figures, regular or irregular, and sometimes in the form of a semicircle, especially when they are situated near a river, or the sea, as at the entrance of a harbour, for the convenience of firing at ships quite around them on that side. In the construction of all Forts, it should be remembered, that the figure of fewest sides and bastions, that can probably answer the proposed defence, is always to be preferred; as works on such a plan are sooner executed, and with less expence; besides, fewer troops will serve, and they are more readily brought together in case of necessity.</p><div2 part="N" n="Fortification" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Fortification</hi></head><p>, called also Military Architecture, is the art of fortifying or strengthening a town or other place, by making certain works around it, to secure it and defend it from the attacks of enemies.</p><p>Fortification has been undoubtedly practised by all nations, and in all ages; being at first doubtless very rude and simple, and varying in its nature and manner, according to the mode of attack, and the weapons made use of. Thus, when villages and towns were first formed, it was found necessary, for the common safety, to encompass them with walls and ditches, to prevent all violence and sudden surprises from their <cb/> neighbours. When offensive and missive weapons came to be used, walls were made as a defence against the assailants, and look holes or loop holes made through the walls to annoy the enemy, by shooting arrows &c through them. But finding that as soon as the enemy got close to the walls, they could no longer be seen nor annoyed by the besieged, these added square towers along the wall, at proper distances from each other, so that all the intervening parts of the wall might be seen and defended from the adjacent sides of the towers. However, this manner of inclosing towns was found to be rather imperfect, because there remained still the outer face of the towers which fronted the field, that could not be seen and defended from any other part. To remedy this imperfection, they next made the towers round instead of square, as seeming better adapted both for strength to resist the battering engines, and for being defended from the other parts of the wall. Nevertheless, a small part of these towers still remained unseen, and incapable of being defended, for which reason they were again changed for square ones, as before, but with this difference, that now they presented an angle of the square outwards to the field, instead of a face or side; and thus such a disposition of the works was obtained, as that no part could be approached by the enemy without being seen and attacked.</p><p>Since the use of gun-powder, it has been found necessary to add thick ramparts of earth to the walls, and the towers have been enlarged into bastions, as well as many other things added, that have given a new appearance to the whole art of defence, and the name of Fortification, on account of the strength afforded by it, which was about the year 1500, when the round towers were changed into bastions.</p><p>But notwithstanding all the improvements made in this art since the invention of gun-powder, that of attacking is still superior to it: the superiority of the besiegers' fire, together with the greater number of men, soonet or later compels the besieged to submit. A special advantage was added to the art of attacking by M. Vauban, at the siege of Ath, in the year 1697, viz, in the use of ricochet firing, or at a low elevation of the gun, by which the shot was made to run and roll a great way along the inside of the works, to the great annoyance of the besieged.</p><p>The chief authors who have treated of Fortification, since it has been considered as a particular art, are the following, and mostly in the order of time: viz, La Treille, Alghisi, Marchi, Pasino, Ramelli, Cataneo, and Speckle, who, as Mr. Robins says, was one of the greatest geniuses that has applied to this art: he was architect to the city of Strasburgh, and died in the year 1589: he published a treatise on Fortification in the German language, which was reprinted at Leipsic in 1736. Afterwards, Errard, who was engineer to Henry the Great of France; Stevinus, engineer to the prince of Orange; with Marolois, the chevalier de Ville, Lorini, Coehorn, the count de Pagan, and the marshal de Vauban; which last two noble authors have contributed greatly to the perfection of the art; besides Scheiter, Mallet, Belidor, Blondel, Muller, Montalambert, &c. Also a list of several works on the art of Fortification may be added, as follows: viz, <pb n="501"/><cb/> Melder's Praxis Fortificatoria: Les Fortifications du Comte de Pagan: L'Ingénieur Parfait du Sieur de Ville: Sturmy's Architectura Militaris Hypothetical.: Blondel's Nouvelle Maniere de Fortifier les Places: The Abbé de Fay's Veritable Maniere de Bien Fortifier: Vauban's Ingénieur François: Coehorn's Nouvelle Fortification tant pour un Terrain bas & humide, que sec & elevé: Alexander de Grotte's Fortification: Donatus Roselli's Fortification: Medrano's Ingénieur François: The chevalier de St. Julien's Architecture Militaire: Lansberg's Nouvelle Maniere de Fortifier les Places: An anonymous treatise in French, called Nouvelle Maniere de Fortifier les Places, tirée des Methodes du chevalier de Ville, &c: Ozanam's Traité de Fortification: Memoires de l'Artillerie de Surirey de St. Remy: Muller's treatises of Elementary and Practical Fortification: and Montalambert's Fortification Perpendiculaire.</p><p><hi rend="italics">Maxims in</hi> <hi rend="smallcaps">Fortification.</hi> From the nature and circumstances of this art, certain general rules, or maxims, have been drawn, and laid down. These may indeed be multiplied to any extent, but the principal of them, are the following: viz,</p><p>1. That the manner of fortifying should be accommodated to that of attacking. So that no one manner can be assured always to hold, unless it be assured that the manner of besieging is incapable of being altered. Also, to judge of the perfection of a Fortification, the method of besieging at the time when it was built must be considered.</p><p>2. All the parts of a Fortification should be equally strong on all sides, where there is equal danger; and they should be able to resist the most powerful machines used in besieging.</p><p>3. A Fortification should be so contrived, as to be defended with the fewest men possible: which confideration, when well attended to, saves a great deal of expence.</p><p>4. That the defendants may be in the better condition, they must not be exposed to the enemies' artillery; but the aggressors must be exposed to theirs. Hence,</p><p>5. All the parts of a Fortification should be so disposed, as that they may defend each other. In order to this, every part ought to be flanked, i. e. seen sideways, capable of being seen and defended from some other part; so that there be no place where an enemy can lodge himself, either unseen, or under shelter.</p><p>6. All the campaign around must lie open to the defendants; so that no hill or eminence must be allowed, behind which the enemy might shelter himself from the guns of the Fortification; or from which he might annoy them with his own. Hence, the fortress is to command all the place round about; and consequently, the outworks must all be lower than the body of the place.</p><p>7. No line of defence must exceed the point-blank musket-shot, which is from 120 to 150 fathoms.</p><p>8. The more acute the angle at the centre, the stronger is the place; as consisting of the more sides, and consequently more defensible.</p><p>9. All the defences should be as nearly direct as possible.</p><p>10. The works that are most remote from the centre<cb/> of the place, ought always to be open to those that are more near.</p><p>These are the general laws and views of Fortification. As to the particular ones, or such as respect the several members or parts of the work, they are given under those articles respectively.</p><p>Fortification is either theoretical or practical.</p><p><hi rend="italics">Thcoretical</hi> <hi rend="smallcaps">Fortification</hi>, consists in tracing the plans and profiles of a work on paper, with scales and compasses; and in examining the systems proposed by different authors, to discover their advantages and defects. And</p><p><hi rend="italics">Practical</hi> <hi rend="smallcaps">Fortification</hi>, consists in forming a project of a work according to the nature of the ground, and other necessary circumstances, tracing it on the ground, and executing the project, together with all the military buildings, such as magazines, storehouses, bridges, &c.</p><p>Again, Fortification is either Defensive or Offensive.</p><p><hi rend="italics">Defensive</hi> <hi rend="smallcaps">Fortification</hi> is the art of defending a town that is besieged, with all the advantages the Fortification of it will admit. And</p><p><hi rend="italics">Offensive</hi> <hi rend="smallcaps">Fortification</hi> is the same with the attack of a place, being the art of making and conducting all the different works in a siege, in order to gain possession of the place.</p><p><hi rend="smallcaps">Fortification</hi> is also used for the place fortified; or the several works raised to defend and flank it, and keep off the enemy.</p><p>All Fortifications consist of lines and angles, which have names according to their various offices. The principal lines are those of circumvallation, of contravallation, of the capital, &c. The principal angles are those of the centre, the flanking angle, flanked angle, angle of the epaule, &c.</p><p>Fortifications are either durable or temporary.</p><p><hi rend="italics">Durable</hi> <hi rend="smallcaps">Fortification</hi>, is that which is built and intended to remain a long time. Such are the usual Fortifications of cities, frontier places, &c. And a</p><p><hi rend="italics">Temporary</hi> <hi rend="smallcaps">Fortification</hi>, is that which is erected on some emergent occasion, and only for a short time. Such are field-works, thrown up for the seizing and maintaining a post, or passage; those about camps, or in sieges; as circumvallations, contravallations, redoubts, trenches, batteries, &c.</p><p>Again, Fortifications are either regular or irregular. A</p><p><hi rend="italics">Regular</hi> <hi rend="smallcaps">Fortification</hi>, is that in which the bastions are all equal; or which is built in a regular polygon, the sides and angles of which are usually about a musket shot from each other. A Regular Fortification, having the parts all equal, has the advantage of being equally defensible; so that there are no weak places. And an</p><p><hi rend="italics">Irregular</hi> <hi rend="smallcaps">Fortification</hi>, is that in which the bastions are unequal, and unlike; or the sides and angles not all equal, and equidistant.</p><p>In an Irregular Fortification, the defence and strength being unequal, it is necessary to reduce the irregular shape of the ground, as near as may be, to a regular figure: i. e. by inscribing it in an oval, instead of a circle; so that one half may be similar and equal to the other half.<pb n="502"/><cb/></p><p><hi rend="italics">Marine</hi> <hi rend="smallcaps">Fortification</hi>, is sometimes used for the art of raising works on the sea coast &c, to defend harbours against the attacks of shipping.——See a neat treatise on Marine Fortisication, at the end of Robertson's Elements of Navigation.</p><p>There are many modes of Fortification that have been much esteemed and used; a small specimen of a comparative view of the principal of these, is represented in plate x, viz, those of Count Pagan, and Mess. Vauban, Coehorn, Belidor, and Blondel; the explanation of which is as follows: <hi rend="center">1. <hi rend="italics">Pagan's System.</hi></hi></p><p>A Half Bastions.</p><p>B Ravelin and Counterguard.</p><p>C Counterguards before the bastions.</p><p>D The Ditch.</p><p>E The Glacis.</p><p>G The place of Arms.</p><p>H Retired Flanks.</p><p><hi rend="italics">a</hi> Line of Defence. <hi rend="center">2. <hi rend="italics">Vauban's System.</hi></hi></p><p><hi rend="italics">b</hi> Angle of the Bastion, or Flanked angle.</p><p><hi rend="italics">c</hi> Angle of the Shoulder.</p><p><hi rend="italics">d</hi> Angle of the Flank.</p><p><hi rend="italics">e</hi> Saliant Angle.</p><p><hi rend="italics">f</hi> Face of the bastion.</p><p><hi rend="italics">g</hi> The Flank.</p><p><hi rend="italics">h</hi> The Curtain.</p><p><hi rend="italics">i</hi> Tenailles.</p><p><hi rend="italics">k</hi> Traverses in the covert way. <hi rend="center">3. <hi rend="italics">Coehorn's System.</hi></hi></p><p>1 Concave Flanks.</p><p>2 The Curtains.</p><p>3 Redoubts in the re-entering angles.</p><p>4 Traverses.</p><p>5 Stone lodgments.</p><p>6 Round Flanks.</p><p>7 Redoubt.</p><p>8 Coffers planked on the sides, and above covered overhead with a foot of earth. <hi rend="center">4 <hi rend="italics">Belidor's System.</hi></hi></p><p>I Cavaliers.</p><p>K Rams-horns, or Tenailles.</p><p>L Retrenchments within the detached bastions.</p><p>M Circular Curtain.</p><p>N The Ravelin.</p><p>P Lunettes with retired batteries.</p><p>Q Redoubt.</p><p>R Detached Redoubt.</p><p>S An Arrow.</p><p>P Small Traverses. <hi rend="center">5. <hi rend="italics">Blondel's System.</hi></hi></p><p>I Retired Battery.</p><p><hi rend="italics">m</hi> Lunettes.</p><p><hi rend="italics">n</hi> Ravelin, with Retired Bastion.</p><p><hi rend="italics">o</hi> Orillons.</p><p>Another, or new method of Fortification has lately been proposed by M. Montalambert, called <hi rend="italics">Fortification Perpendic<*>laire,</hi> because the faces of the works are<cb/> made by a series of lines running zigzag perpendicular to one another.</p><p><hi rend="italics">Profile of a</hi> <hi rend="smallcaps">Fortification</hi>, is a representation of a vertical section of a work; serving to shew those dimensions which cannot be represented in plans, and are necessary in the building of a Fortification. The names and dimensions of the principal parts are as follow (see fig. 8, pl. vii), where the numbers or dimensions are all expressed in feet.</p><p>AB The level of the ground plane,</p><p>AC = 27,</p><p>CD = 18, and CW = 16 1/2; also DN is paral. to AB.</p><p>DE = 30,</p><p>EF = 2, FG = 3, GH = 3, HI = 4 1/2, IL = 1 1/2,</p><p>LK = 18, KM = 2 1/2, NP = 36, NO = 5,</p><p>PR = 7,</p><p>RS = 1, ST = 12 or 18, OV = 9, P<hi rend="italics">n</hi> = 120,</p><p><hi rend="italics">mz</hi> = 3, <hi rend="italics">nu</hi> = 3, <hi rend="italics">mc</hi> = 30, <hi rend="italics">cd</hi> = 2,</p><p><hi rend="italics">de</hi> = 3, <hi rend="italics">ef</hi> = 3, <hi rend="italics">fl</hi> = 4 1/2, <hi rend="italics">rg</hi> = 120, <hi rend="italics">lh</hi> = 1.</p><p>AW the interior talus or slope of the rampart,</p></div2></div1><div1 part="N" n="WE" org="uniform" sample="complete" type="entry"><head>WE</head><p>, or DE the terre-plein of ditto,</p><p>FG the talus, and GH the upper part, of the banquette,</p><p>HL the interior side of the parapet,</p><p>LM the upper part of ditto,</p><p>N the cordon of 1 foot radius,</p><p>NP the depth, and P<hi rend="italics">n</hi> the breadth of the ditch,</p><p>OQ interior side of revetement,</p><p>NR the scarp or exterior side of ditto,</p><p>ST the depth of the foundation,</p><p>YZ revetement of the parapet,</p><p><hi rend="italics">nu</hi> the counterscarp,</p><p><hi rend="italics">mc</hi> the covert-way,</p><p><hi rend="italics">ce</hi> talus of the banquette,</p><p><hi rend="italics">ef</hi> the upper part of ditto,</p><p><hi rend="italics">fh</hi> parapet of the covert-way,</p><p><hi rend="italics">hg</hi> the glacis.</p><p>Other sections are at fig. 2, pl. 12.</p><p><hi rend="smallcaps">Fortified</hi> <hi rend="italics">Place,</hi> a Fortress, or Fortification, i. e. a place well flanked, and sheltered with works.</p><div2 part="N" n="Fortin" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Fortin</hi></head><p>, or <hi rend="italics">Fortlet,</hi> a diminitive of the word Fort, meaning a little fort, or sconce, called also Field Fort.</p><p><hi rend="italics">Star</hi> <hi rend="smallcaps">Fortin</hi>, is that whose sides flan keach other, &c.</p></div2></div1><div1 part="N" n="FORTRESS" org="uniform" sample="complete" type="entry"><head>FORTRESS</head><p>, the same as Fort, or a Fortification.</p></div1><div1 part="N" n="FOSTER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FOSTER</surname> (<foreName full="yes"><hi rend="smallcaps">Samuel</hi></foreName>)</persName></head><p>, an English mathematician, and astronomy professor of Gresham-college, was born in Northamptonshire, and admitted a Sizer at Emanuelcollege Cambridge in 1616. He took the degree of bachelor of arts in 1619, and of master in 1623. He applied early to the mathematics, and attained a great proficiency in it, of which he gave the first specimen in 1624, in a treatise on <hi rend="italics">The Use of the Quadrant.</hi></p><p>On the death of Mr. Gellibrand, he was chosen to succeed him, in 1636, as astronomy professor in Gresham-college, London. He quitted it again however the same year, though for what reason does not appear, and was succeeded by Mr. Mungo Murray, professor of philosophy at St. Andrews in Scotland. But this gentleman marrying, the professorship again became vacant, and Mr. Foster was re-elected in 1641.</p><p>Mr. Foster was one of those gentlemen who held private meetings for cultivating philosophy and useful <pb/><pb/><pb/><pb/><pb n="503"/><cb/> knowledge, which afterwards gave rise to the Royal Society. In 1646, Dr. Wallis, who was one of those associating gentlemen, received from Mr. Foster a theorem <hi rend="italics">de triangulo sphærico,</hi> which he published in his <hi rend="italics">Mechanics.</hi> Neither was it only in this branch of science that he excelled, but he was likewise well versed in the ancient languages; as appears from his revising and correcting the <hi rend="italics">Lemmata</hi> of Archimedes, which had been translated into Latin from an Arabic manuscript by Mr. John Greaves. He made also several curious observations upon eclipses of the sun and moon, in various places; and was particularly noted for inventing, as well as improving, astronomical and other mathematical instruments. After a long declining state of health, he died at Gresham-college in 1652.</p><p>His printed works are as follow; of which the first two articles were published before his death, and the rest of them after it.</p><p>1. <hi rend="italics">The Description and Use of a small Portable Quadrant, for the easy finding the hour of Azimuth;</hi> 4to, 1624. Originally published at the end of Gunter's Description of the Cross-Staffe, as an appendix to it.</p><p>2. <hi rend="italics">The Art of Dialling;</hi> 4to, 1638. Reprinted, with additions, in 1675.</p><p>3. <hi rend="italics">Posthuma Fosteri;</hi> by Wingate; 4to, 1652.</p><p>4. <hi rend="italics">Four Treatises of Dialling;</hi> 4to, 1654.</p><p>5. <hi rend="italics">Miscellanies, or Mathematical Lucubrations.</hi> Published by John Twysden, with additions of his own; and an appendix by Leybourne; folio, 1659.</p><p>6. <hi rend="italics">The Sector altered, and other Scales added, &c.</hi> Published by Leybourne in 1661, in 4to.</p><p>There have been two other persons of the same name, who have published some mathematical pieces. The first was,</p><p><hi rend="smallcaps">William</hi> FOSTER, who was a disciple of Mr. Oughtred, and afterward a teacher of the Mathematics in London. He distinguished himself by a book, which he dedicated to Sir Kenelm Digby, entitled, <hi rend="italics">The Circles of Proportion, and the Horizontal Instrument,</hi> &c; 4to, 1633.—The other was</p><p><hi rend="smallcaps">Mark</hi> FOSTER, who lived later in point of time than either of the other two, and published a treatise entitled, <hi rend="italics">Arithmetical Trigonometry: being the solution of all the usual Cases in Plain Trigonometry by Common Arithmetic, without any tables whatsoever.</hi> 12mo, 1690.</p></div1><div1 part="N" n="FOUNDATION" org="uniform" sample="complete" type="entry"><head>FOUNDATION</head><p>, that part of a building which is underground: or the mass which supports a building, and upon which it stands: or it is the coffer or bed dug below the level of the ground, to raise a building upon.</p></div1><div1 part="N" n="FOUNTAIN" org="uniform" sample="complete" type="entry"><head>FOUNTAIN</head><p>, in Philosophy, a spring or source of water rising out of the Ground. See <hi rend="smallcaps">Spring.</hi></p><div2 part="N" n="Fountain" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Fountain</hi></head><p>, or <hi rend="italics">Artificial</hi> <hi rend="smallcaps">Fountain</hi>, in Hydraulics, a machine or contrivance by which water is violently spouted or darted up; called also a Jet d'eau.</p><p>There are various kinds of Artificial Fountains, but all formed by a pressure of one sort or another upon the water &c, viz, either the pressure or weight of a head of water, or the pressure arising from the spring and elasticity of the air, &c. When these are formed by the pressure of a head of water, or any other fluid of the same kind with the Fountain, or jet, then will this spout up nearly to the same height as that head, abating only a little for the resistance of the air, with that of<cb/> the adjutage &c, in the fluid's rushing through; but, when the Fountain is produced by any other force than the pressure of a column of the same fluid with itself, it will rise to such a height as may be nearly equal to the altitude of a column of the same fluid whose pressure is equal to the given force that produces the fountain.</p><p><hi rend="italics">To Construct an Artificial Fountain, playing by the pressure of the water.</hi> This is to be effected by making a close connection between a head or elevated piece of water, and the lower place, where the Fountain is to play; which may be done in this manner: Having a head of water, naturally, or, for want of such, make an artificial one, raising the water by pumps, or other machinery: from this head convey the water in close pipes, in any direction, down to the place where the Fountain is to play; and there let it issue through an adjutage, or small hole, turned upwards, by which means it will spout up nearly as high as the head of the water comes from, as above mentioned.</p><p><hi rend="italics">To Construct an Artisicial Fountain, playing by the spring or elasticity of the air.</hi> A vessel proper for a reservoir, as AB, fig. 4, plate viii, is provided either of metal, or glass, or the like, ending in a small neck <hi rend="italics">c,</hi> at the top: through this neck is put a tube <hi rend="italics">cd,</hi> till the lower end come near the bottom of the vessel, this being about half full of water. The neck is so contrived, as that a syringe, or condensing pipe, may be screwed upon the tube; by means of which a large quantity of air may be intruded through the tube into the water, out of which it will disengage itself, and emerge into the vacant part of the vessel, and lie over the surface of the water CD.</p><p>Now, the water in the vessel being thus pressed by the air, which is, for ex. double the density of the external air; and the elastic force of air being proportional to its density, or to its gravitating force, the effect will be the same, as if the weight of the column of air, over the surface of the water, were double that of the column pressing in the tube; so that the water must be forced to spout up through, when the syringe is removed, with a force equal to the excess of pressure of the included air, above that of the external, that is, in this instance, with a force equal to the pressure of an entire column of the atmosphere; which being equal to the pressure of a column of 33 or 34 feet of water, it follows that the Fountain will play to nearly 33 or 34 feet high.</p><p>These aereal or aquatic Fountains may be applied in different ways, so as to exhibit various appearances; and from these alone arises the greatest part of our artificial water-works: even the engine for extinguishing fire, is a Fountain playing by the force of confined air.</p><p><hi rend="italics">A Fountain spouting the water in various directions.</hi> Suppose AB the vertical tube, or spout, in which the water rises, (fig. 5, pl. viii): into this let several other tubes be sitted; some horizontal, others oblique, or inclining, or reclining, &c, as at E, H, L, N, P, &c. Then, as all water retains the direction of the aperture through which it comes, that issuing through A will rise perpendicularly; and the rest will tend different ways, describing arches of different magnitudes.</p><p>Or thus: Suppose the vertical tube AB (fig. 6) through which the water rises, to be stopped at the<pb n="504"/><cb/> top, as in A; and, instead of pipes or cocks, let it be only perforated with little holes all around or only half round its surface: then will the water spin out, in all directions, through the little holes, to different distances.</p><p>And hence, if the tube AB be about the height of a man, and having a turn-cock at C; upon opening this cock, the spectators will be sprinkled unexpectedly with a shower.</p><p><hi rend="smallcaps">Fountain</hi> <hi rend="italics">playing by drawing the breath.</hi> Suppose AB (fig. 8, plate vi), a globe of glass, or metal, in which is fitted a tube CD, having a small orifice in C, and reaching almost to D, the bottom of the globe. Then if the air be sucked, or drawn with the mouth, out of the tube CD, and the orifice C be immediately immerged under cold water, the water will ascend through the tube into the sphere. Thus proceeding, by repeated exsuctions, till the vessel be above half full of water; then applying the mouth to C, and blowing air into the tube, upon removing the mouth, the water will spout forth.</p><p>Or, if the globe be put into hot water, the air being thus rarefied, will make the water spout as before.</p><p>And this kind of Fountain is called Pila Heronis, or Hero's Ball, from the name of its inventor.</p></div2><div2 part="N" n="Fountain" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Fountain</hi></head><p>, <hi rend="italics">whose stream raises and plays a brass ball.</hi> Provide a hollow brass ball A (fig. 9, pl. vi), made very thin, that its weight may not be too great for the force of the water; and let the tube BC, through which the water rises, be exactly perpendicular to the horizon. Then the ball, being laid in the bottom of the cup or bason B, will be taken up in the stream, and sustained at a considerable height, playing a little up and down.</p><p><hi rend="smallcaps">Fountain</hi> <hi rend="italics">which spouts water in form of a shower.</hi> To the tube in which the water is to rise, fit a spherical, or lenticular head AB (fig. 1, pl. xi) made of a plate of metal, and perforated at the top with a great number of little holes. The water rising violently towards AB, will be there divided into innumerable little threads, and afterwards broken, and dispersed into the finest drops.</p><p><hi rend="smallcaps">Fountain</hi> <hi rend="italics">which spreads the water in form of a tablecloth.</hi> To the tube AB (fig. 2, pl. xi) solder two spherical segments, C and D, almost touching each other, with a screw E, to contract or amplify the interstice or chink, at pleasure. Some choose to make a smooth and even cleft, in a spherical or lenticular head, fitted upon the tube. The water spouting through this chink or cleft will expand itself like a cloth.—And thus, the fountain may be made to spout out in the figure of men, or other animals.</p></div2><div2 part="N" n="Fountain" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Fountain</hi></head><p>, <hi rend="italics">which, when it has done spouting, may be turned like an hour-glass.</hi> Provide two glasses, A and B, (fig. 3, pl. xi) to be so much the larger as the fountain is to play the longer, and placed at so much the greater distance from each other as the water is desired to spout the higher. Let CDE be a crooked tube, furnished in E with a jet; and GHI another bent tube, furnished with a jet in I: GF and KL are to be other lesser tubes, open at both ends, and reaching near the<cb/> bottom of the vessels A and B, to which the tubes CD and GH are likewise to reach.</p><p>Now if the vessel A be filled with water, it will descend through the tube CD; and it will spout up through the jet E, by the pressure of the column of water CD. But unless the pipe GF were open at G, to let the air run up to F, and press at the top of the surface of the water in the cavity A, the water could not run down and spout at E. After its fall again, it will sink through the little tube KL, into the vessel B, and expel the air through the tube GI. At length, when all the water is emptied out of the vessel A, by turning the machine upside down, the vessel B will be the reservoir, and make the water spout up through the jet I, the pipe KL supplying B with air to let the water descend in the direction GHI.</p><p>Hence, if the vessels A and B contain just as much water as will be spouted up in an hour's time, we shall have a spouting clipsydra, or water-clock; which may be graduated or divided into quarters, minutes, &c.</p><p><hi rend="smallcaps">Fountain</hi> <hi rend="italics">of Command.</hi> This depends on the same principles with those of the former: CAE (fig. 4, pl. xi) is a vessel of water secured against the entrance of the air, except through the pipe GF, when the cock C, by which it is filled, is shut. There is another pipe EDHB, going from the bottom of the water to the jet B in the bason DB; but this is stopped by the cock H. At the lowest part of the bason DB, there is a small hole at I, to let the water of the bason DB run into the bason GH under it; there is also a small triangular hole or notch, in the bottom of the pipe FG, at G. Turn the cock H, and the fountain will play for some time, then stop, then play again alternately for several times together. When those times of playing and stopping are known before hand, you may command the Fountain to play or stop; whence its name. The cause of this phenomenon is as follows: the water coming down the pipe EDHB, would not come out at B, if the air S<hi rend="italics">s,</hi> above the water, were not supplied as it dilated: now it is supplied by the pipe GF, which takes it in at the notch G, and delivers it out at F; but after some time the water, which was spouted out at B, falling down into the bason DB, rises high enough to come above the notch G, which stops the passage of the air; so that the air S<hi rend="italics">s,</hi> above the water in the vessel CAE, wanting a supply, cannot sufficiently press, and the Fountain ceases playing: But when the water of DB has run down into the lower bason GH, through the hole I, till it falls below the top of the notch G, the air runs up into the upper receptacle, and supplies that at S<hi rend="italics">s,</hi> and the Fountain plays again. This is seen a little before hand, by a skin of water on the notch G, before the air finds a passage, and then you may command the Fountain to play. It is evident that the hole I must be less than the hole of the jet, or else all the water would run out into the lower bason, without rising high enough to stop the notch G.</p><p><hi rend="smallcaps">Fountain</hi> <hi rend="italics">that begins to play upon the lighting of candles, and ceases as they go out.</hi> Provide two cylindrical vessels, AB and CD, (fig. 7, pl. xi.) connect them by tubes, open at both ends, KL, BE, &c, so that the air may descend out of the higher into the lower:<pb n="505"/><cb/> to the tubes solder candlesticks, H, &c, and to the hollow cover of the lower vessel CE, fit a little tube or jet, FG, furnished with a cock G, and reaching almost to the bottom of the vessel. In G let there be an aperture, furnished with a screw, by which water may be poured into CD. Then, upon lighting the candles H, &c, the air in the contiguous tubes becoming rarefied by the heat, the water will begin to spout through GF.</p><p>By the same contrivance may a statue be made to shed tears upon the presence of the sun, or on the lighting of a candle, &c: all that is here required, being only to lay tubes from the cavity where the air is raresied, to some other cavities placed near the eyes, full of water.</p><p><hi rend="italics">A</hi> <hi rend="smallcaps">Fountain</hi> <hi rend="italics">by the Rarefaction of the air,</hi> may be made in the following manner: Let AB and CD, fig. 5, pl. xi, be two pipes fixed to a brass head C, made to screw into a glass vessel E, which having a little water in it, is inverted till the pipes are screwed on; then reverting it suddenly, so as to put A, the lower end of the spouting pipe AB, into a jar of water A, and the lower end of the descending pipe CD, into a receiving vessel D, the water will spout up from the jar A into the tall glass vessel E, from which it will go down at the mouth C, through CD, into the vessel D, till the water is wholly emptied out of A, making a Fountain in E, into D. The reason of the play of the Fountain is this: the pipe CD, being 2 feet 9 inches long, lets down a column of water, which rarefies the air 1-12th part in the vessel E, where it presses against the water spouting at B with 1-12th of the force by which the water is pushed up at the hole A, by the pressure of the common air on the water in the vessel A; so that the water spouts up into E, when the air is rarefied 1-12th, with the difference of the pressure of the atmosphere, and the forementioned rarefied air; i. e. of 33 to 2 3/4, or of 12 to 1. This would raise the water 2 feet 9 inches; but the length of the pipe A, of 9 inches, being deducted, the jet will only rise 2 feet. This, says Desaguliers, may be called a syphon Fountain, where AB is the driving leg, and CD the issuing leg.</p><p><hi rend="smallcaps">Fountain</hi> <hi rend="italics">of Hero of Alexandria,</hi> so called, because it was contrived by him. In the second Fountain above described, the air is compressed by a syringe; in this, (see fig. 6, pl. xi) the air, being only compressed by the concealed fall of water, makes a jet, which, after some continuance, is considered by the ignorant as a perpetual motion; because they imagine that the same water which fell from the jet rises again. The boxes CE and DYX, being close, we see only the bason ABW, with a hole at W, into which the water spouting at B falls; but that water does not come up again; for it runs down through the pipe WX into the box DYX, from whence it drives out the air, through the ascending pipe YZ, into the cavity of the box CE, where, pressing upon the water that is in it, it forces it out through the spouting pipe OB, as long as there is any water in CE; so that this whole play is only whilst the water contained in CE, having spouted out, falls down through the pipe WX, or of the boxes CE and DY above one another: the height of the water, measured from the bason ABW to the surface of the water in the lower box DYX, is always equal to the<cb/> height measured from the top of the jet to the surfac<*> of the water in the middle cavity at CE. Now, since the surface CE is always falling, and the water in DY always rising, the height of the jet must continually decrease, till it is shorter by the depth of the cavity CE, which is emptying, added to the depth of the cavity DY, which is always filling; and when the jet is fallen so low, it immediately ceases. The air is represented by the points in this figure.</p><p>To prepare this Fountain for playing, which should be done unobserved, pour in water at W, till the cavity DXY is filled; then invert the Fountain, and the water will run from the cavity DXY into the cavity CE, which may be known to be full, when the water runs out at B held down. Set the Fountain up again, and, to make it play, pour in about a pint of water into the bason ABW; and as soon as it has filled the pipe WX it will begin to play, and continue as long as there is any water in CE. You may then pour back the water left in the bason ABW, into any vessel, and invert the Fountain, which, being set upright again, will be made to play, by putting back the water poured out into ABW; and so on as often as you please.</p><p><hi rend="italics">Spouting</hi> <hi rend="smallcaps">Fountain</hi>, or Jet d'Eau, is any Fountain whose water is darted forth impetuously through jets, or ajutages, and returns in form of rains, nets, folds, or the like.</p><p><hi rend="smallcaps">Fountain</hi>-<hi rend="italics">Pen,</hi> is a pen contrived to contain a quantity of ink, and let it flow very gently, so as to supply the writer a long time without the necessity of taking fresh ink.</p><p>The Fountain-pen, represented fig. 8, pl. xi, consists of divers pieces of metal, F, G, H, the middle piece F carrying the pen, which is screwed into the inside of a little pipe; and this again is soldered into another pipe of the same size as the lid G; in which lid is soldered a male screw, for screwing on the cover; as also for stopping a little hole at the place, and hindering the ink from passing through it: at the other end of the piece F is a little pipe, on the outside of which may be screwed the top cover H. A porte-craion goes in the cover, to be screwed into the last mentioned pipe, to stop the end of the pipe into which the ink is to be poured by a funnel.</p><p>To use the Pen, the cover G must be taken off, and the pen a little shaken, to make the ink run more freely.</p></div2></div1><div1 part="N" n="FOURTH" org="uniform" sample="complete" type="entry"><head>FOURTH</head><p>, in Music, one of the harmonic intervals, or concords. It consists in the mixture of two sounds, which are in the ratio of 4 to 3; i. e. of two sounds produced by chords, whose lengths are to each other as 4 to 3.</p></div1><div1 part="N" n="FRACTION" org="uniform" sample="complete" type="entry"><head>FRACTION</head><p>, or Broken Number, in Arithmetic and Algebra, is a part, or some parts, of another number or quantity considered as a whole, but divided into a certain number of parts; as 3-4ths, which denotes 3 parts out of 4, of any quantity.</p><p>Fractions are usually divided into Vulgar, Decimal, Duodecimal, and Sexagesimal. For the last three sorts, see the respective words.</p><p><hi rend="italics">Vulgar</hi> <hi rend="smallcaps">Fractions</hi>, called also simple <hi rend="italics">Fractions,</hi> are usually denoted by two numbers, the one set under the other, with a small line between them: thus 3/4 denotes the Fraction three-fourths, or 3 parts out of 4, of some<pb n="506"/><cb/> whole quantity considered as divided into 4 equal parts.</p><p>The lower number 4, is called the Denominator of the Fraction, shewing into how many parts the whole or integer is divided; and the upper number 3, is called the Numerator, and shews how many of those equal parts are contained in the Fraction. Hence it follows, that as the numerator is to the denominator, so is the Fraction itself, to the whole of which it is a Fraction; or as the denominator is to the numerator, so is the whole or integer, to the Fraction: thus, the integer being denoted by 1, as the Fraction.— And hence there may be innumerable Fractions all of the same value, as there may be innumerable quantities all in the same ratio, viz, of 4 to 3; such as 8 to 6, or 12 to 9, &c. So that if the two terms of any Fraction i. e. the numerator and denominator, be either both multiplied or both divided by any number, the resulting Fraction will still be of the same value: thus, 3/4 or 6/8 or 9/12 or 12/16 &c, are all of the same value with each other.</p><p>Fractional expressions are usually distinguished into Proper and Improper, Simple and Compound, and Mixt Numbers.</p><p><hi rend="italics">A Proper</hi> <hi rend="smallcaps">Fraction</hi>, is that whose numerator is less than the denominator; and consequently the Fraction is less than the whole or integer; as 3/4.</p><p><hi rend="italics">Improper</hi> <hi rend="smallcaps">Fraction</hi>, is when the numerator is either equal to, or greater than, the denominator; and consequently the Fraction either equal to, or greater than, the whole integer, as 4/4, which is equal to the whole; or 5/4, which is greater than the whole.</p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Fractions</hi>, or <hi rend="italics">Single</hi> <hi rend="smallcaps">Fractions</hi>, are such as consist of only one numerator, and one denominator; as 3/4, or 5/4, or 12/25.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Fractions</hi> are Fractions of Fractions, and consist of several Fractions, connected together by the word <hi rend="italics">of:</hi> as 2/3 of 3/4, or 1/2 of 2/3 of 3/4.</p><p><hi rend="italics">A Mixt Number</hi> consists of an integer and a Fraction joined together: as 1 3/4, or 12 2/3.</p><p>The arithmetic of Fractions consists in the Reduction, Addition, Subtraction, Multiplication, and Division of them.</p><p><hi rend="italics">Reduction of</hi> <hi rend="smallcaps">Fractions</hi> is of several sorts; as 1. <hi rend="italics">To reduce a given whole number into a Fraction of any given denominator.</hi> Multiply the given integer by the proposed denominator, and the product will be the numerator. Thus, it is found that 3 = 6/2, and 5 = 20/4, or 7 = 35/5.</p><p>If no denominator be given, or it be only proposed to express the integer Fraction-wise, or like a Fraction; set 1 beneath it, for its denominator. So 3 = 3/1, and 5 = 5/1, and 7 = 7/1.</p><p>2. <hi rend="italics">To reduce a given Fraction to another Fraction equal to it, that shall have a given denominator.</hi> Multiply the numerator by the proposed denominator, and divide the product by the former denominator, then the quotient set over the proposed denominator will form the Fraction required. Thus, if it be proposed to reduce 3/4 to an equal Fraction whose denominator shall be 8; then , and the numerator, so that 6/8 is the Fraction sought, being = 3/4, and having 8 for its denominator.</p><p>3. <hi rend="italics">To Abbreviate, or reduce Fractions to lower terms.</hi><cb/> Divide their terms, i. e. numerator and denominator, by any number that will dividē them both without a remainder, so shall the quotients be the corresponding terms of a new Fraction, equal to the former, but in smaller numbers. In like manner abbreviate these new terms again, and so on till there be no number greater than 1 that will divide them without a remainder, and then the Fraction is said to be in its least terms. Thus, to abbreviate 15/60; first divide both terms by 5, and the Fraction becomes 3/12; next divide these by 3, and it becomes 1/4: so that 15/60 = <*>/12 = 1/4, which is in its least terms.</p><p>4. <hi rend="italics">To reduce Fractions to other equivalent ones of the same denominator.</hi> Multiply each numerator, separately taken, by all the denominators except its own, and the products will be the new numerators; then multiply all the denominators continually together, for the common denominator, to these numerators. Thus, 2/3 and 4/5 reduce to 10/15 and 12/15; and 2/3, 3/4, and 4/5 reduce to 40/60, 45/60, and 48/60. <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">16)</cell><cell cols="1" rows="1" rend="align=right" role="data">180</cell><cell cols="1" rows="1" role="data">(11 <hi rend="italics">s</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16 </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">16)</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data">(3 <hi rend="italics">d</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">48</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>5. <hi rend="italics">To find the value of a Fraction, in the known parts of its integer.</hi> Multiply always the numerator by the number of parts of the next inferior denomination, and divide the products by the denominator. So, to find the value of 9/16 of a pound sterling; multiply 9 by 20 for shillings, and dividing by 16, gives 11 for the shillings; then multiply the remainder 4 by 12 pence, and dividing by 16 gives 3 for pence: so that 11s. 3d. is the value of 9/16l. as required.</p><p>6. <hi rend="italics">To reduce a mixt number to an equivalent improper Fraction.</hi> Multiply the integer by the denominator, and to the product add the numerator, for the new numerator, to be set over the same denominator as before. Thus 3 5/8 becomes 29/8.</p><p>7. <hi rend="italics">To reduce an improper Fraction to its equivalent whole or mixt number.</hi> Divide the numerator by the denominator; so shall the quotient be the integral part, and the remainder set over the denominator will form the fractional part of the equivalent mixt number. Thus 29/8 reduces to 3 5/8, and 32/4 = 8.</p><p>8. <hi rend="italics">To reduce a compound Fraction to a simple one.</hi> Multiply all the numerators together for the numerator, and all the denominators together for the denominator, of the simple Fraction sought. Thus, 1/2 of 3/4 = 3/8, and 2/3 of 4/5 of 7/9 = 56/135.</p><p>To reduce a Vulgar Fraction to a decimal. See <hi rend="smallcaps">Decimals.</hi> And for several other particulars concerning Reduction, as well as the other operations in Fractions; see my Arithmetic.</p><p><hi rend="italics">Addition of</hi> <hi rend="smallcaps">Fractions.</hi> First reduce the Fractions to their simplest form, and reduce them also to a common denominator, if their denominators are different; then add all the numerators together, and set the sum over the common denominator, for the sum of all the Fractions as required. Thus, ; And .</p><p><hi rend="italics">Subtraction of</hi> <hi rend="smallcaps">Fractions.</hi> Reduce the Fractions the same as for addition; then subtract the one numerator<pb n="507"/><cb/> from the other, and set the difference over the common denominator. So ; And .</p><p><hi rend="italics">To Multiply</hi> <hi rend="smallcaps">Fractions</hi> <hi rend="italics">together.</hi> Reduce them all to the form of simple Fractions, if they are not so; then multiply all the numerators together for the numerator, and all the denominators together for the denominator of the product sought. Thus ; And .</p><p><hi rend="italics">To Divide</hi> <hi rend="smallcaps">Fractions.</hi> Divide the numerator by the numerator, and the denominator by the denominator, if they will exactly divide. Thus, .</p><p>But if they will not divide without a remainder, then multiply the dividend by the reciprocal of the divisor, that is, by the Fraction obtained by inverting or changing its terms. Thus, .</p><p><hi rend="italics">Algebraic</hi> <hi rend="smallcaps">Fractions</hi>, or <hi rend="smallcaps">Fractions</hi> <hi rend="italics">in Species,</hi> are exactly similar to vulgar Fractions, in numbers, and all the operations are performed exactly in the same way; therefore the rules need not be repeated, and it may be sufficient here to set down a few examples to the foregoing rules. Thus,</p><p>1. The Fraction <hi rend="italics">aab</hi>/<hi rend="italics">bc</hi> abbreviates to <hi rend="italics">aa</hi>/<hi rend="italics">c</hi>.</p><p>2. , by dividing by 3<hi rend="italics">a</hi>.</p><p>3. , by dividing by <hi rend="italics">a</hi> - <hi rend="italics">x.</hi> See <hi rend="smallcaps">Common</hi> <hi rend="italics">Measure.</hi></p><p>4. <hi rend="italics">a</hi>/<hi rend="italics">b</hi> and <hi rend="italics">c</hi>/<hi rend="italics">d</hi> become <hi rend="italics">ad</hi>/<hi rend="italics">bd</hi> and <hi rend="italics">bc</hi>/<hi rend="italics">bd,</hi> when reduced to a common denominator.</p><p>5. .</p><p>6. .</p><p>7. .</p><p>8. .</p><p><hi rend="italics">Continued</hi> <hi rend="smallcaps">Fraction</hi>, is used for a Fraction whose denominator is an integer with a Fraction, which latter Fraction has for its denominator an integer and a Fraction, and the same for this last Fraction again, and so on, to any extent, whether supposed to be infinitely continued, or broken off after any number of terms. Euler, Analys. Inf. vol. 1, p. 295. As , or , or . Or, using letters instead of numbers, .<cb/> or .</p><p>When these series are not far extended, it is not difficult to collect them by common arithmetic.</p><p>Lord Brounker, it seems, was the first who considered Continued Fractions, or at least, who applied them to the quadrature of curves, in Wallis's Arith. Infin. prop. 191, vol. 1, p. 469 &c, where this author explains the manner of forming them, giving several numeral examples, in approximating ratios, as well as the geneneral series &c, as he denotes it. Huygens also used it for the like purpose, viz, to approximate the ratios of large numbers, in his Descrip. Autom. Planet. in Oper. Relig. p. 173 &c, edit. Amst. 1728. And a special treatise on Continued Fractions was given by Euler, in his Analys. Infin. vol. 1, pa. 295 &c.</p><p>This subject is perhaps capable of much improvement, though it has been rather neglected, as very little use has been made of it, except, by those authors, in approximating to the value of Fractions, and ratios, that are expressed in large numbers; besides a method of Goniometry by De Lagny, explained in the Introduction to my Logarithms, pa. 78; as also some use I have made of it in summing very slowly converging series, in my Tracts, p. 38 & seq.</p><p>As to the reducing of common Fractions, and ratios, that are expressed in large numbers, to Continued Fractions, it is no more than the common method of finding the greatest common measure of those two numbers, by dividing the greater by the less, and the last divisor always by the last remainder; for then the several quotients are the denominators of the Fractions, the numerators being always 1 or unity. Thus, to find approximating values of the Fraction 31415926535/10000000000, or to the ratio of 31415926535 to 10000000000, being the ratio of the circumference of a circle to its diameter, by means of a Continued Fraction; or, to change the said Common Fraction to a Continued Fraction: Dividing the greater term always by the less, the same as to find the greatest common measure of the said numbers or terms, the several quotients will be 3, 7, 15, 1, 292, 1, 1, &c, which, after the first, will be the denominators, to the common numerator 1; and therefore the said Fraction will be changed into this Continued Fraction, . Hence, stopping at any part of these single Fractions, one after another, will give several values of the proposed ratio, all successively nearer and nearer the truth, but alternately too great and too little. So, stopping<pb n="508"/><cb/> at 1/7, it is 3 1/7 = 22/7 = 3.142857 too great, or 22 to 7, the ratio of the circumference to the diameter as given by Archimedes. Again, stopping at 1/15, it is 3 1/(7 1/15) = 3 15/106 = 333/106 = 3.141509 &c, too little. But stopping at 1/1, it is (the ratio of Metius) = 3.1415929 &c, which is rather too great. And so on, always nearer and nearer, but alternately too great and too little.</p><p>And, in like manner is any algebraic Fraction thrown into a Continued Fraction. As the Fraction , which being in like manner divided, the quotients are <foreign xml:lang="greek">a</foreign>/<hi rend="italics">a,</hi> <foreign xml:lang="greek">b</foreign>/<hi rend="italics">b,</hi> <foreign xml:lang="greek">g</foreign>/<hi rend="italics">c,</hi> <foreign xml:lang="greek">d</foreign>/<hi rend="italics">d</hi>; which single Fractions being considered as denominators to other Fractions whose common numerator is 1, these will be the reciprocals of the former, and so will become <hi rend="italics">a</hi>/<foreign xml:lang="greek">a</foreign>, <hi rend="italics">b</hi>/<foreign xml:lang="greek">b</foreign>, <hi rend="italics">c</hi>/<foreign xml:lang="greek">g</foreign>, <hi rend="italics">d</hi>/<foreign xml:lang="greek">d</foreign>; and hence the proposed common Fraction is equal to this terminate Continued Fraction, .</p><p>On the other hand, any Continued Fraction being given, its equivalent common Fraction will be found, by beginning at the last denominator, or lowest end of the given Continued Fraction, and gradually collecting the Fractions backwards, till we arrive at the first, when the whole will thus be collected together into one common Fraction; as was done above in collecting the Fractions And in like manner the Continued Fraction collects into the Fraction .</p><p>When the given Continued Fraction is an infinite one, collect it successively, first one term, then two together, three together, &c, till the sum is sufficiently exact. Or, if these collected sums converge too slowly to the true value, having collected a few of the terms into successive sums, these being alternately too great and too little, the true value will be found as near as you please by the method of arithmetical means, explained in my Tracts, vol. 1, Tract 2, pa. 11.</p><p><hi rend="italics">Vanishing</hi> <hi rend="smallcaps">Fractions.</hi> Such Fractions as have both their numerator and denominator vanish, or equal to 0, at the same time, may be called <hi rend="italics">Vanishing Fractions.</hi> We are not to conclude that such Fractions are equal to nothing, or have no value; for that they have a certain determinate value, has been shewn by the best ma-<cb/> thematicians. The idea of such Fractions as these, first originated in a very severe contest among some French mathematicians, in which Varignon and Rolle were the two chief opposite combatants, concerning the then new or differential calculus, of which the latter gentleman was a strenuous opponent. Among other arguments against it, he proposed an example of drawing a tangent to certain curves at the point where the two parts cross each other; and as the fractional expression for the subtangent, by that method, had both its numerator and denominator equal to 0 at the point proposed, Rolle looked upon it as an absurd expression, and as an argument against the method of solution itself. The seeming mystery however was soon explained, and first of all by John Bernoulli. See an account of this affair in Montucla, Hist. Math. vol. 2, pa. 366.</p><p>Since that time, such kind of fractions have often been contemplated by mathematicians. As, by Maclaurin, in his Fluxions, vol. 2, pa. 698: Saunderson, in his Algebra, vol. 2, art. 469: De Moivre, in Miscel. Anal. pa. 165: Emerson, in his Algebra, pa. 212: and by many others. The same fractions have also proved a stumbling-block to more mathematicians than one, and the cause of more violent controversies: witness that between Powell and Waring, when they were competitors for the professorship at Cambridge. In the specimen of a work published on occasion of that competition, by Waring, was the fraction , which he said became 4 when <hi rend="italics">p</hi> was = 1. This was struck at by Powell, as absurd, because when <hi rend="italics">p</hi> = 1, then the fraction , which was one chief cause of his not succeeding to the professorship. Waring replied that (by common division) , when <hi rend="italics">p</hi> is = 1. See the controversial pamphlets that passed between those two gentlemen at that time.</p><p>There are two modes of finding the value of such fractions, that have been given by the gentlemen above quoted. The one is by considering the terms of the fraction as two variable quantities, continually decreasing, till they both vanish together; or finding the ultimate value of the ratio denoted by the fraction. In this way of considering the matter, it appears that, as the terms of the fraction are supposed to decrease till they vanish, or become only equal to their fluxions or their increments, the value of the fraction at that state, will be equal to the fluxion or increment of the numerator divided by that of the denominator. Hence then, taking the example when <hi rend="italics">x</hi> = 1; the fluxion of the numerator is , and of the denominator - <hi rend="italics">x</hi><hi rend="sup">.</hi>; therefore , the value of the fraction when <hi rend="italics">x</hi> = 1.——Or, thus, because <hi rend="italics">x</hi> = 1, therefore ; then the fluxion of the numerator, - 4<hi rend="italics">x</hi><hi rend="sup">3</hi><hi rend="italics">x</hi><hi rend="sup">.</hi>, divided by<pb n="509"/><cb/> the fluxion of the denominator, or - <hi rend="italics">x</hi><hi rend="sup">.</hi>, gives 4<hi rend="italics">x</hi><hi rend="sup">3</hi> or 4, the same as before.</p><p>The other method is by reducing the given expression to another, or simple form, and then substituting the values of the letters. So in the above example , or , when <hi rend="italics">x</hi> = 1; divide the numerator by the denominator, and it becomes , which when <hi rend="italics">x</hi> = 1, becomes 4, for the given fraction, the same as before.—Again, to find the value of when <hi rend="italics">x</hi> is = <hi rend="italics">a,</hi> in which case both the numerator and denominator become = 0. Divide the numerator by the denominator, and the quotient is ; which when <hi rend="italics">x</hi> = <hi rend="italics">a,</hi> becomes , for the value of the fraction in that state of it.</p></div1><div1 part="N" n="FRAISE" org="uniform" sample="complete" type="entry"><head>FRAISE</head><p>, in Fortification, a kind of defence, consisting of pointed stakes, driven almost parallel to the horizon into the retrenchments of a camp, &c, to ward off and prevent any approach or scalade.</p></div1><div1 part="N" n="FRANKLIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FRANKLIN</surname> (<foreName full="yes"><hi rend="smallcaps">Dr. Benjamin</hi></foreName>)</persName></head><p>, one of the most celebrated philosophers and politicians of the 18th century, was born in Boston in North America in the year 1706, being the youngest of 13 children. His father was a tallow-chandler in Boston, and young Franklin was taken from school at 10 years of age, to assist him in his business. In this situation he continued two years; but disliking that occupation, he was bound apprentice to an elder brother, who was then a printer in Boston, but had learned that business in London, and who in the year 1721 began to print a newspaper, being the second ever published in America; the copies of which our author was sent to distribute, after having assisted in composing and printing it. Upon this occasion, our young philosopher enjoyed the secret and singular pleasure of being the much admired author of many essays in this paper; a circumstance which he had the address to keep a secret even from his brother himfelf; and this when he was only 15 years of age. The frequent ill usage from his brother, induced young Franklin to quit his service, which he did, at the age of 17, and went to New York. But not meeting employment here, he went forward to Philadelphia, where he worked with a printer a short time; after which, at the instance of Sir William Keith, governor of the province, he returned to Boston to solicit pecuniary assistance from his father to set up a printing-house for himself at Philadelphia, upon the promise of great encouragement from Sir William, &c. His father however thought fit to refuse such aid, alleging that he was yet too young (18 years old) to be entrusted with such a concern; so our author again returned to Philadelphia without it. Upon this, Sir William said he would advance the sum that might be necessary, and our young philosopher should go to England, and purchase all the types and materials, for which purpose he would give him letters of credit. He could never however get these letters, yet by dint of fair promises of their being sent on board the ship after him, he failed for England, expecting these letters of credit were in the governor's packet, which he was to receive upon its<cb/> being opened. In this however he was cruelly deceived, and thus he was sent to London without either money, friends, or credit, at 18 years of age.</p><p>He soon found employment, however, as a journeyman printer, first at a Mr. Palmer's, and afterward with Mr. Watts, with whom he worked a considerable time, and by whom he was greatly esteemed, being also treated with such kindness, that it was always most gratefully remembered by our philosopher.</p><p>After a stay of 18 months in London, he returned to Philadelphia, viz in 1726, along with a merchant of that town, as his clerk, on a salary of 50 pounds a year. But his master dying the year after, he again engaged to direct the printing business of the same person with whom he had worked before. After continuing with him the best part of a year, our philosopher, in partnership with another young man, at length set up a printing-house himself.</p><p>A little before this time, young Franklin had gradually associated a number of persons, like himself, of a rational and philosophical turn of mind, and formed them into a club or society, to hold meetings, to converse and communicate their sentiments together, for their mutual improvement in all kinds of useful knowledge, which was in high repute for many years after. Among many other useful regulations, they agreed to bring such books as they had into one place, to form a common library. This resource being found defective, at Franklin's persuasion they resolved to contribute a small sum monthly towards the purchase of books for their use from London. Thus their stock began to increase rapidly; and the inhabitants of Philadelphia, being desirous of having a share in their literary knowledge, proposed that the books should be lent out on paying a small sum for the indulgence. Thus in a few years the society became rich, and possessed more books than were perhaps to be found in all the other colonies: the collection was advanced into a public library; and the other colonies, sensible of its advantages, began to form similar plans; from whence originated the libraries at Boston, New York, Charlestown, &c; that of Philadelphia being now not inferior to any in Europe.</p><p>About 1728 or 1729, young Franklin set up a newspaper, the second in Philadelphia, which proved very profitable, and otherwise useful, as affording an opportunity of making himself known as a political writer, by inserting several of his writings of that kind into it. In addition to his printing-house, he set up a shop to sell books and stationary; and in 1730 he married his wife, who proved very usesul in assisting to manage the shop, &c. He afterward began to have some leisure, both for reading books, and writing them, of which he gave many specimens from time to time. In 1732 he began to publish Poor Richard's Almanac, which was continued for many years. It was always remarkable for the numerous and valuable concise maxims which it contained, for the œconomy of human life, all tending to exhort to industry and frugality: and in the almanac for the last year, all the maxims were collected in an address to the reader, entitled, The Way to Wealth. This has been translated into various languages, and inserted in different publications. It has also been printed on a large sheet, proper to be framed, and hung up in conspicuous places in all houses, as it<pb n="510"/><cb/> very well deserves to be. Mr. Franklin became gradually more known for his political talents, and in the year 1736, he was appointed clerk to the General Assembly of Pennsylvania; and was re-elected by succeeding assemblies for several years, till he was chosen a representative for the city of Philadelphia; and in 1737 he was appointed post-master of that city. In 1738, he formed the first fire-company there, to extinguish and prevent fires and the burning of houses: an example which was soon followed by other persons, and other places. And soon after, he suggested the plan of an association for insuring houses and ships from losses by sire, which was adopted; and the association continues to this day. In the year 1744, during a war between France and Great Britain, some French and Indians made inroads upon the frontier inhabitants of the province, who were unprovided for such an attack: the situation of the province was at this time truly alarming, being destitute of every means of defence. At this crisis Franklin stepped forth, and proposed to a meeting of the citizens of Philadelphia, a plan of a voluntary association for the defence of the province. This was approved of, and signed by 1200 persons immediately. Copies of it were circulated through the province; and in a short time the number of signers amounted to 10,000. Franklin was chosen colonel of the Philadelphia regiment; but he did not think proper to accept of the honour.</p><p>Pursuits of a different nature now occupied the greatest part of his attention for some years. Being always much addicted to the study of natural philosophy; and the discovery of the Leyden experiment in electricity having rendered that science an object of general curiosity; Mr. Franklin applied himself to it, and soon began to distinguish himself eminently in that way. He engaged in a course of electrical experiments with all the ardour and thirst for discovery which characterized the philosophers of that day. By these, he was enabled to make a number of important discoveries, and to propose theories to account for various phenomena; which have been generally adopted, and which will probably endure for ages. His observations he communicated, in a series of letters, to his sriend Mr. Collinson; the first of which is dated March 28, 1747. In these he makes known the power of points in drawing and throwing off the electric matter, which had hitherto escaped the notice of electricians. He also made the discovery of a <hi rend="italics">plus</hi> and <hi rend="italics">minus,</hi> or of a <hi rend="italics">positive</hi> and <hi rend="italics">negative</hi> state of electricity; from whence in a satisfactory manner he explained the phenomena of the Leyden phial, first observed by Cuneus or Muschenbroeck, which had much perplexed philosophers. He shewed that the bottle, when charged, contained no more electricity than before, but that as much was taken from one side as was thrown on the other; and that, to discharge it, it was only necessary to make a communication between the two sides, by which the equilibrium might be restored, and that then no signs of electricity would remain. He afterwards demonstrated by experiments, that the electricity did not reside in the coating, as had been supposed, but in the pores of the glass itself. After a phial was charged, he removed the coating, and found that upon applying a new coating the shock might still be received. In the year 1749, he first suggested his idea of explaining the phenomena<cb/> of thunder-gusts, and of the aurora borealis, upon electrical principles. He points out many particulars in which lightning and electricity agree; and he adduces many facts, and reasoning from facts, in support of his positions. In the same year he conceived the bold and grand idea of ascertaining the truth of his doctrine, by actually drawing down the forked lightning, by means of sharp-pointed iron rods raised into the region of the clouds; from whence he derived his method of securing buildings and ships from being damaged by lightning. It was not until the summer of 1752 that he was enabled to complete his grand discovery the experiment of the electrical kite, which being raised up into the clouds, brought thence the electricity or lightning down to the earth; and M. D'Alibard made the experiment about the same time in France, by following the track which Franklin had before pointed out. The letters which he sent to Mr. Collinson, it is said, were refused a place among the papers of the Royal Society of London; and Mr. Collinson published them in a separate volume, under the title of <hi rend="italics">New Experiments and Observations on Electricity, made at Philadelphia, in America;</hi> which were read with avidity, and soon translated into different languages. His theories were at first opposed by several philosophers, and by the members of the Royal Society of London; but in 1755, when he returned to that city, they voted him the gold medal which is annually given to the person who presents the best paper on some interesting subject. He was also admitted a member of the Society, and had the degree of doctor of laws conferred upon him by different universities: but at this time, by reason of the war which broke out between Britain and France, he returned to America, and interested himself in the public affairs of that country. Indeed he had done this long before; for although philosophy was a principal object of Franklin's pursuit for several years, he did not confine himself to it alone. In the year 1747 he became a member of the General Assembly of Pennsylvania, as a burgess for the city of Philadelphia. Being a friend to the rights of man from his infancy, he soon distinguished himself as a steady opponent of the unjust schemes of the proprietaries. He was soon looked up to as the head of the opposition; and to him have been attributed many of the spirited replies of the assembly, to the messages of the governors. His influence in the body was very great. This arose not from any superior powers of eloquence; he spoke but seldon and he never was known to make any thing like an elaborate harangue. His speeches often consisted of a single sentence, or of a well-told story, the moral of which was always obviously to the point. He never attempted the flowery fields of oratory. His manner was plain and mild. His style in speaking was, like that of his writings, simple, unadorned, and remarkably concise. With this plain manner, and his penetrating and solid judgment, he was able to confound the most eloquent and subtle of his adversaries, to confirm the opinions of his friends, and to make converts of the unprejudiced who had opposed him. With a single observation, he has rendered of no avail a long and elegant discourse, and determined the fate of a question of importance.</p><p>In the year 1749, he proposed a plan of an academy,<pb n="511"/><cb/> to be erected in the city of Philadelphia, as a foundation for posterity to erect a seminary of learning, more extensive and suitable to future circumstances; and in the beginning of 1750, three of the schools were opened, namely, the Latin and Greek school, the Mathematical, and the English schools. This foundation soon after gave rise to another more extensive college, incorporated by charter May 27, 1755, which still subsists, and in a very flourishing condition. In 1752, he was instrumental in the establishment of the Pennsylvania Hospital, for the cure and relief of indigent invalids, which has proved of the greatest use to that class of persons. Having conducted himself so well as Postmaster of Philadelphia, he was, in 1753, appointed Deputy Postmaster-general for the whole British colonies.</p><p>The colonies being much exposed to depredations in their frontier by the Indians and the French, at a meeting of commissioners from several of the provinces, Mr. Franklin proposed a plan for the general defence, to establish in the colonies a general government, to be administered by a president-general, appointed by the crown, and by a grand council, consisting of members chosen by the representatives of the different colonies; a plan which was unanimously agreed to by the commissioners present. The plan however had a singular fate: It was disapproved of by the ministry of Great Britain, because it gave too much power to the representatives of the people; and it was rejected by every assembly, as giving to the president general, who was to be the representative of the crown, an influence greater than appeared to them proper, in a plan of government intended for freemen. Perhaps this rejection, on both sides, is the strongest proof that could be adduced of the excellence of it, as suited to the situation of Great Britain and America at that time. It appears to have steered exactly in the middle, between the opposite interests of both. Whether the adoption of this plan would have prevented the separation of America from Great Britain, is a question which might afford much room for speculation.</p><p>In the year 1755, General Braddock, with some regiments of regular troops, and provincial levies, was sent to dispossess the French of the posts upon which they had seized in the back settlements. After the men were all ready, a difficulty occurred, which had nearly prevented the expedition. This was the want of waggons. Franklin now stepped forward, and, with the assistance of his son, in a little time procured 150. After the defeat of Braddock, Franklin introduced into the assembly a bill for organizing a militia, and had the dexterity to get it passed. In consequence of this act a very respectable militia was formed; and Franklin was appointed colonel of a regiment in Philadelphia, which confisted of 1200 men; in which capacity he acquitted himself with much propriety, and was of singular service; though this militia was soon after disbanded by order of the English ministry.</p><p>In 1757, he was sent to England, with a petition to the king and council, against the proprietaries, who refused to bear any share in the public expences and assessments; which he got settled to the satisfaction of the state. After the completion of this business, Franklin remained at the court of Great Britain for some time, as agent for the province of Pennsylvania;<cb/> and also for those of Massachusetts, Maryland, and Georgia. Soon after this, he published his Canada pamphlet, in which he pointed out, in a very forcible manner, the advantages that would result from the conquest of this province from the French. An expedition was accordingly planned, and the command given to General Wolfe; the success of which is well known. He now divided his time indeed between philosophy and politios, rendering many services to both. Whilst here, he invented the elegant musical instrument called the <hi rend="italics">Armonica,</hi> formed of glasses played on by the fingers. In the summer of 1762 he returned to America; on the passage to which he observed the singular effect produced by the agitation of a vessel, containing oil floating on water: the upper surface of the oil remained smooth and undisturbed, whilst the water was agitated with the utmost commotion. On his return he received the thanks of the Assembly of Pennsylvania, which having annually elected him a member in his absence, he again took his seat in this body, and continued a steady defender of the liberties of the people.</p><p>In 1764, by the intrigues of the proprietaries, Franklin lost his seat in the assembly, which he had possessed for 14 years; but was immediately appointed provincial agent to England, for which country he presently set out. In 1766 he was examined before the parliament relative to the stamp-act; which was soon after repealed. The same year he made a journey into Holland and Germany; and another into France; being everywhere received with the greatest respect by the literati of all nations. In 1773 he attracted the public attention by a letter on the duel between Mr. Whately and Mr. Temple, concerning the publication of Governor Hutchinson's letters, declaring that he was the person who had discovered those' letters. On the 29th of January next year, he was examined before the privy-council on a petition he had presented long before as agent for Massachusetts Bay against Mr. Hutchinson: but this petition being disagreeable to ministry, it was precipitately rejected, and Dr. Franklin was soon after removed from his office of Postmaster-general for America. Finding now all efforts to restore harmony between Great Britain and her colonies useless, he returned to America in 1775; just after the commencement of hostilities. Being named one of the delegates to the Continental Congress, he had a principal share in bringing about the revolution and declaration of independency on the part of the colonies. In 1776 he was deputed by Congress to Canada, to negociate with the people of that country, and to persuade them to throw off the British yoke; but the Canadians had been so much disgusted with the hot-headed zeal of the New Englanders, who had burnt some of their chapels, that they refused to listen to the proposals, though enforced by all the arguments Dr. Franklin could make use of. On his return to Philadelphia, Congress, sensible how much he was esteemed in France, sent him thither to put a sinishing hand to the private negociations of Mr. Silas Deane; and this important commission was readily accepted by the Doctor, though then in the 71st year of his age. The event is well known; a treaty of alliance and commerce was signed between France and America; and M. Le Roi asserts, that the Doctor had a great share in the transaction, by strongly advising M. Maurepas not<pb n="512"/><cb/> to lose a single moment, if he wished to secure the friendship of America, and to detach it from the mother-country. In 1777 he was regularly appointed plenipotentiary from Congress to the French court; but obtained leave of dismission in 1780. Having at length seen the full accomplishment of his wishes by the conclusion of the peace in 1783, which gave independency to America, he became desirous of revisiting his native country. He therefore requested to be recalled; and, after repeated solicitations, Mr. Jefferson was appointed in his stead. On the arrival of his successor, he repaired to Havre de Grace, and crossing the channel, landed at Newport in the Isle of Wight; from whence, after a favourable passage, he arrived safe at Philadelphia in September 1785. He was received amidst the acclamations of a vast multitude who flocked from all parts to see him, and who conducted him in triumph to his own house; where in a few days he was visited by the members of the Congress and the principal inhabitants of Philadelphia. He was afterward twice chosen president of the Assembly of Philadelphia; but his increasing infirmities obliged him to ask permission to retire, and to spend the remainder of his life in tranquillity; which was granted, in 1788. After this, the infirmities of age increased fast upon him; he became more and more afflicted with the gout and the stone, till the time of his death, which happened the 17th of April 1790, about 11 at night, at 84 years of age; leaving one son, governor William Franklin, a zealous loyalist, who now resides in London; and a daughter, married to Mr. William Bache, merchant in Philadelphia.</p><p>Doctor Franklin was author of many tracts on electricity, and other branches of natural philosophy, as well as on politics, and miscellaneous subjects. He had also many papers inserted in the Philosophical Transactions, from the year 1757 to 1774.</p></div1><div1 part="N" n="FREEZE" org="uniform" sample="complete" type="entry"><head>FREEZE</head><p>, or <hi rend="smallcaps">Frize</hi>, in Architecture, a large flat member, being that part of the entablature of columns that separates the architrave from the cornice.</p></div1><div1 part="N" n="FREEZING" org="uniform" sample="complete" type="entry"><head>FREEZING</head><p>, or Congelation, the fixing of a fluid body into a firm or solid mass by the action of cold: in which sense the term is applied to water when it freezes into ice; to metals when they resume their solid form after being melted by heat; or to glass, wax, pitch, tallow, &c, when they harden again after having been rendered fluid by heat. But it differs from crystallization, which is rather a separation of the particles of a solid from a fluid in which it had been dissolved more by the moisture than the action of heat.</p><p>The process of congelation is always attended with the emission of heat, as is found by experiments on the freezing of water, wax, spermaceti, &c; for in such cases it is always found that a thermometer dipt into the fluid mass, keeps continually descending as this cools, till it arrive at a certain point, being the point of freezing, which is peculiar to each fluid, where it is rather stationary, and then rises for a little, while the congelation goes on. But by what means it is that fluid bodies should thus be rendered solid by cold, or fluid by heat, or what is introduced into the bodies by either of those principles, are matters the learned have never yet been able to discover, or to satisfy them-<cb/> selves upon. The following phenomena however are usually observed.</p><p>Water, and some other fluids, suddenly dilate and expand in the act of Freezing, so as to occupy a greater space in the form of ice than before, in consequence of which it is that ice is specifically lighter than the same fluid, and floats in it. And the degree of expansion of water, in the state of ice, is by some authors computed at about 1/10 of its volume. Oil however is an exception to this property, and quicksilver too, which shrinks and contracts still more after Freezing. Mr. Boyle relates several experiments of vessels made of metal, very thick and strong; in which, when filled with water, close stopped, and exposed to the cold, the water being expanded in Freezing, and not sinding either room or vent, burst the vessels. A strong barrel of a gun, with water in it close stopped and frozen, was rent the whole length. Huygens, to try the force with which it expands, filled a cannon with it, whose sides were an inch thick, and then closed up the mouth and vent, so that none could escape; the whole being exposed to a strong Freezing air, the water froze in about 12 hours, and burst the piece in two places. Mathematicians have computed the force of the ice upon this occasion; and they say, that such a force would raise a weight of 27720 pounds. Lastly, Major Edward Williams, of the Royal Artillery, made many experiments on the force of it, at Quebec, in the years 1784 and 1785. He filled all sizes of iron bomb-shells with water, then plugged the fuze hole close up, and exposed them to the strong Freezing air of the winter in that climate; sometimes driving in the iron plugs as hard as possible with a sledge hammer; and yet they were always thrown out by the sudden expansion of the water in the act of Freezing, like a ball shot by gunpowder, sometimes to the distance of between 400 and 500 feet, though they weighed near 3 pounds; and when the plugs were screwed in, or furnished with hooks or barbs, to lay hold of the inside of the shell by, so that they could not possibly be forced out, in this case the shell was always split in two, though the thickness of the metal of the shell was about an inch and three quarters. It is farther remarkable, that through the circular crack, round about the shells, where they burst, there stood out a thin film or sheet of ice, like a fin; and in the cases when the plugs were projected by Freezing water, there suddenly issued out from the fuze-hole, a bolt of ice, of the same diameter, and stood over it to the height sometimes of 8 inches and a half. And hence we need not be surprised at the effects of ice in destroying the substance of vegetables and trees, and even splitting rocks, when the frost is carried to excess.</p><p>It is also observed that water loses of its weight by Freezing, being found lighter after thawing again, than before it was frozen. And indeed it evaporates almost as fast when frozen, as when it is fluid.</p><p>It is said too that water does not freeze in vacuo; requiring for that purpose the presence and contiguity of the air. But this circumstance is liable to some doubt, and it may be suspected that the degree of cold has not been carried far enough in these instances; as it is found that mercury in thermometers has even been<pb n="513"/><cb/> frozen, though it requires a vastly greater degree of cold to freeze mercury, than water.</p><p>That water which has been boiled freezes more readily than that which has not been boiled; and that a slight disturbance of the fluid disposes it to freeze more speedily; having sometimes been cooled several degrees below the Freezing point, without congealing when kept quite still, but suddenly freezing into ice on the least motion or disturbance.</p><p>That the water, being covered over with a surface of oil of olives, does not freeze so readily as without it; and that nut oil absolutely preserves it under a strong frost, when olive oil would not.</p><p>That rectified spirit of wine, nut oil, and oil of turpentine, seldom freeze.</p><p>That the surface of the water, in Freezing, appears all wrinkled; the wrinkles being sometimes in parallel lines, and sometimes like rays, proceeding from a centre to the circumference.</p><p><hi rend="smallcaps">Freezing</hi> <hi rend="italics">Mixture,</hi> a preparation for the artificial congelation of water, and other fluids.</p><p>According to Mr. Boyle, all kinds of salts, whether alkaline or acid; and even all spirits, as spirit of wine, &c; as also sugar, and saccharum saturni, mixed with snow, are capable of Freezing most fluids; and the same effect is produced, in a very high degree, by a mixture of oil of vitriol, or spirit of nitre, with snow.</p><p>M. Homberg remarks the same of equal quantities of corrosive sublimate, and sal ammoniac, with four times the quantity of distilled vinegar.</p><p>Boerhaave gives a method of producing artisicial frost without either snow or ice: we must have for this purpose, at any season of the year, the coldest water that can be procured; this is to be mixed with a proper quantity of any salt (sal ammoniac will answer the intention best), at the rate of about 3 ounces to a quart of water. Another quart of water must be prepared in the same manner with the first; the salt, by being dissolved in each, will make the water much colder than it was before. The two quarts are then to be mixed together, and this will make them colder still. Two quarts more of water prepared and mixed in the same manner are to be mixed with these, which will increase the cold to a much higher degree in all. The whole of this operation is to be carried on in a cold cellar; and a glass of common water is then to be placed in the vessel of the fluid thus artificially cooled, and it will be turned into ice in the space of 12 hours.</p><p>There is also a method of making artisicial ice by means of snow, without any kind of salt. For this purpose fill a small pewter dish with water, and upon that set a common pewter plate filled, but not heaped, with snow. Bring this simple apparatus near the fire, and stir the snow in the plate: the snow will dissolve, and the ice will be formed on the back of the plate, which was set in the dish of water.</p><p>Mr. Reaumur tried the effect of several salts, and examined the various degrees of cold by an ice thermometer, which being placed in the fluid to be frozen, shewed very exactly the degree of cold by the descent of the spirit.</p><p>Nitre, or saltpetre, usually passes for a salt that may be very serviceable in these artisicial congelations; but the experiments of this gentleman prove that this opi-<cb/> nion is erroneous. The most perfectly resined saltpetre employed in the operation sunk the spirit in the thermometer only 3 degrees and a half below the fixed point. Less refined nitre sunk the thermometer lower, and gave a greater degree of cold; owing to the common or sea-salt that it contains when less pure, which has a greater effect than the pure saltpetre itself.</p><p>Two parts of common salt being mixed with three parts of powdered ice in very hot weather, the spirit in the thermometer immediately descended 15 degrees, which is half a degree lower than it would have descended in the severest cold of our winters. Mr. Reaumur then tried the salts all round, determining with great regularity and exactness, what was the degree of cold occasioned by each in a given dose. Among the neutral salts, none produced a greater degree of cold than the common sea salt. Among the alkalies, sal ammoniac sunk the thermometer only to 13 degrees. Pot-ashes sunk it just as low as well refined saltpetre.</p><p>For the common uses of the table, the ice is not required to be very hard, or such as is produced by long continuance of violent cold: it is rather desired to be like snow. Saltpetre, which is no very powerful freezer, is therefore more fit for the purpose than a more potent salt. It is not necessary that the congelation should be very suddenly made; but that it may retain its sorm as long as may be, when made, is of great importance.</p><p>If it be desired to have ices very hard and sirm, and very suddenly prepared, then sea salt is of all others most to be chosen for the operation. The ices thus made will be very hard, but they will soon run. Pot-ashes afford an ice of about the hardness that is usually required. This forms indeed very slowly, but then it will preserve a long time. And common wood ashes will perform the business very nearly in the same manner as the pot-ashes; but for this purpose, the wood which is burnt, ought to be fresh.</p><p>The strong acid spirits of the neutral salts act much more powerfully in these congelations than the salts themselves, or indeed than any simple salt can do. Thus, spirit of nitre mixed with twice its quantity of powdered ice, immediately sinks the spirit in the thermometer to 19 degrees, or 4 degrees more than that obtained by means of sea salt, the most powerful of all the salts in making artisicial cold. A much greater degree of cold may be given to this mixture, by piling it round with more ice mixed with sea salt. This gives a redoubled cold, and sinks the thermometer to 24 degrees. If this whole matter be covered with a fresh mixture of spirit of nitre and ice, a still greater degree of cold is produced, and so on; the cold being by this method of fresh additions to be increased almost without bounds: but it is to be observed, that every addition gives a smaller increase than the former.</p><p>It is very remarkable in the acid spirits, that though sea salt is so much more powerful than nitre in substance in producing cold, yet the spirit of nitre is much stronger than that of sea salt; and another not less wonderful phenomenon is, that spirit of wine, which is little else than liquid fire, has as powerful an effect in congelations, or very nearly so, as the spirit of nitre itself.<pb n="514"/><cb/> The several liquid substances which produce cold, in the same manner as the dry salts on being mixed with ice, are much more speedy in their action than the salts: because they immediately and much more intimately come into contact with the particles of the ice, than the salts can. Of this nature are spirit of nitre, spirit of wine, &c. To produce the expected degree of cold, it is always necessary that the ice and the added matter, whatever it be, should both run together, and, intimately uniting, form one clear fluid. It is hence that no new cold is produced with oil, which, though it melts the ice, yet cannot mix itself into a homogeneous liquid with it, but must always remain floating on the surface of the water that is produced by the melting of the ice. Mem. Acad. Scienc. Par. 1734.</p><p>It has been discovered, that sluids standing in a current of air, grow by this means much colder than before. Fahrenheit had long since observed, that a pond, which stands quite calm, often acquires a degree of cold much beyond what is sufficient for Freezing, and yet no congelation ensued: but if a slight breath of air happens in such a case to brush over the surface of the water, it stiffens the whole in an instant. It has also been discovered, that all substances grow colder by the evaporation of the fluids which they contain, or with which they are mixed. If both these methods therefore be practised upon the same body at the same time, they will increase the cold to almost any degree of intenseness we please.</p><p>But the most extraordinary instances of artificial Freezing, have since been made in Russia, at Hudson's bay, and other parts, by which quicksilver was frozen into a solid mass of metal. And the same thing had before happened from the natural cold of the atmosphere alone, in Siberia. In the winter of 1733, Professor Gmelin, with two other gentleman of the Russian Academy, were sent by Anne Ivanouna, the new empress, to explore and describe the different parts of her Asiatic dominions, with the communication of Asia and America. In the winter of 1734-5, Mr. Gmelin being at Yeneseisk in 58° 30′ north lat. and 92° long. east from Greenwich, first observed such a descent of the mercury, as must have been attended with congelation, being far below its Freezing point, now fixed at - 40 of Fahrenheit's thermometer. “ Here, says he, we first experienced the truth of what various travellers have related with respect to the extreme cold of Siberia; for, about the middle of December, such severe weather set in, as we were sure had never been known in our time at Petersburg. The air seemed as if it were frozen, with the appearance of a fog, which did not suffer the smoke to ascend as it issued from the chimneys. Birds fell down out of the air as dead, and froze immediately, unless they were brought into a warm room. Whenever the door was opened, a fog suddenly formed round it. During the day, short as it was, parhelia and haloes round the sun were frequently seen; and in the night mock moons, and haloes about the moon. Finally, our thermometer, not subject to the same deception as the senses, left us no doubt of the excessive cold; for the quicksilver in it was reduced, on the 5th of January, old style, to - 120° of Fahrenheit's scale, lower than it had ever been observed in na- ture.”<cb/></p><p>The next instance of congelation happened at Yakutsk, in 62° north lat. and 150° east longitude. The weather here was unusually mild for the climate, yet the thermometer fell to - 72°; and one person informed the professor by a note, that the mercury in his barometer was frozen. He hastened immediately to his house to behold such a surprising phenomenon; but though he was witness to the fact, observing that the mercury did not continue in one column, but was divided in different places as into little cylinders, which appeared frozen, yet the prejudice he had entertained against the possibility of the congelation, would not allow him to believe it.</p><p>Another set of observations, in the course of which the mercury must frequently have been cougealed, were made by professor Gmelin at Kirenga fort, in 57 1/2 north lat., and 108 east long.; his thermometer, at different times, standing at - 108, - 86, - 100, - 113, and many other intermediate degrees; in the course of the winter of 1737-8. On the 27th of November, after the thermometer had been standing for two days at - 46°, he found it sunk at noon to - 108. Suspecting some mistake, after he had noted down the observation, he instantly ran back, and found it at - 102; but ascending with such rapidity, that in the space of half an hour it had risen to - 19°. This phenomenon, which appeared so surprising, doubtless depended on the expansion of the mercury frozen in the bulb of the thermometer, and which now melting, forced upwards the small thread in the stem. And similar appearances were observed on other days afterwards, when the thread of quicksilver in the thermometer was separated about 6 degrees.</p><p>A second instance where a natural congelation of mercury has certainly been observed, is recorded in the transactions of the Royal Academy of Sciences at Stockholm, as made by Mr. Andrew Hellant. The weather, in January 1760, was remarkably cold in Lapland; so that on the 5th of that month, the thermometers fell to - 76, - 128, or lower; on the 23d and following days they fell to - 58, - 79, - 92, and below - 238 entirely into the ball. This was observed at four different places in Lapland, situated between the 65th and 78th degrees of north lat. and the 21st and 28th degree of east longitude.</p><p>But the congelation of quicksilver, by an artificial Freezing mixture, was sirst observed, and put beyond doubt, by Mr. Joseph Adam Braun, professor of philosophy at Petersburg. This gentleman wishing to try how many degrees of cold he could produce, availed himself of a good opportunity which offered for that purpose on the 14th of December 1759, when the mercury in the thermometer stood in the natural cold at - 34, which it is now known is only 5 or 6 degrees above its point of congelation. Assisting this natural cold therefore with a mixture prepared of aquafortis and pounded ice, his thermometer was sunk to - 69. Part of the quicksilver must now have been really congealed, but unexpected by him, and he only thought of pursuing his object of producing still greater degrees of cold; and having expended all his pounded ice, he was obliged to use snow instead of it. With this fresh mixture the mercury sunk to - 100, - 240, and - 350°. Taking the thermometer out, he found it<pb n="515"/><cb/> whole, but the quicksilver fixed, and it continued so for 12 minutes. On repeating the experiment, with another thermometer which had been graduated no lower than - 220, all the mercury sunk into the ball, and became solid as before, and did not re-ascend till after a still longer interval of time. Mr. Braun now suspected that the quicksilver was really frozen, and prepared for making a decisive experiment. This was accomplished on the 25th of the same month, and the bulb of the thermometer broken as soon as the metal was congealed; when it appeared that the mcrcuty was changed into a solid and shining metallic mass, which flatted and extended under the strokes of a pestle, being rather less hard than lead, and yielding a dull sound like that metal. Mr. Æpinus made similar experiments at the same time, employing as well thermometers as tubes of a larger bore; in which last he remarked, that the quicksilver fell sensibly on being frozen, assuming a concave surface, and likewise that the congealed pieces sunk in fluid mercury: also, in their farther experiments, they invariably found that the mercury sunk lower when the whole of it was congealed, than if any part of it remained fluid: all shewing that, contrary to water, mercury contracted in Freezing. It was farther observed, that the mercury when congealed looked like the most polished silver, and when beaten flat, it was easily cut with a penknife, like soft thin sheet lead.</p><p>The fact being thus established, and fluidity no longer to be considered as an essential property of quicksilver, Mr. Braun communicated an account of his experiments to the Petersburg academy, on the 6th of September 1760; of which a large extract was inserted in the Philos. Trans. vol. 52, pa. 156. He afterwards declared that he never suffered a winter to pass without repeating the experiment of Freezing quicksilver, and never failed of success when the natural cold was of a sufficient strength for the purpose; and this degree of natural cold he supposes at - 10 of Fahrenheit; though some commencement of the congelation might be perceived when the temperature of the air was as high as + 2.</p><p>The results of all his experiments were, that with the abovementioned frigorific mixtures, and once with rectified spirits and snow, when the natural cold was at - 28°, he congealed the quicksilver, and discovered that it is a real metal that melts with a very sinall degree of heat. However, not perceiving the necessary consequence of its great contraction in Freezing, he always confounded its point of congelation with that of its greatest contraction in Freezing, and thus marked the former a great deal too low.</p><p>In the process of his observations, Mr. Braun found that double aquafortis was more effectual than spirit of nitre; but with this simple spirit, which seldom brings the mercury lower than - 148, this metal may be frozen in the following manner: Six glasses being filled with snow as usual, and the thermometer put in one of them, the spirit of nitre was poured upon it; when the mercury would fall no lower in this, the thermometer was removed to the second, and so on to the third and fourth, in which fourth immersion the mercury was congealed.</p><p>Mr. Æpinus gives the following direction for using<cb/> the fuming spirit of nitre: Take some of this spirit, cooled as much as possible, and put it into a wine glass till it be about half full, filling it up with snow, and stirring them till the mixture become of the consistence of pap; by which means you obtain, almost in an instant, the necessary degree of cold for the Freezing of quicksilver.</p><p>It is remarked by Mr. Braun, that by the mixture of snow and spirit of nitre, which froze the mercury, he never was able to bring thermometers, filled with the most highly rectified spirit of wine, lower than - 148: so that the cold which will freeze mercury, will not freeze spirit of wine; and therefore spirit thermometers are the most sit to determine the degree of coldness in frigorific mixtures, till we can construct solid metallic thermometers with sufficient accuracy. Mr. Braun tried the effects of different Fluids in his frigorific mixtures: he always found that Glauber's spirit of nitre and double aquafortis were the most powerful; and from a number of experiments made when the temperature of the air was between 21 and 28 of Fahrenheit, he concludes, that spirit of salt pounded upon snow increased the natural cold 36°; spirit of sal ammoniac, 12; oil of vitriol, 42; Glauber's spirit of nitre, 70; aquafortis, 48; simple spirit of nitre, 36; dulcified spirit of vitriol, 24; Hoffman's anodyne liquor, 38; spirit of hartshorn, 12; spirit of sulphur, 12; spirit of wine rectified, 24; camphorated spirit, 18; French brandy, 14; and several kinds of wine increased the natural cold to 7, 8, or 9 degrees.</p><p>The most remarkable congelation of mercury, by natural cold, that has ever been observed, was that related by Dr. Peter Simon Pallas, who had been sent by the empress of Russia, with some other gentlemen, on an expedition similar to that of Mr. Gmelin. Being at Krasnoyarsk in the year 1772, in north lat. 56° 30′, and east long. 93°, he had an opportunity of observing the phenomenon we speak of. On the 6th and 7th of December that year, says he, there happened the greated cold I have ever experienced in Siberia: the air was calm at the time, and seemingly thickened; so that, though the sky was in other respects clear, the sun appeared as through a fog. I had only one small thermometer left, in which the scale went no lower than - 7°; and on the 6th in the morning, I remarked that the quicksilver in it sunk into the ball, except some short columns which stuck fast in the tube. When the ball of the thermometer, as it hung in the open air, was touched with the finger, the quicksilver rose; and it could plainly be seen that the solid columns stuck and resisted a good while, and were at length pushed upward with a sort of violence. He also placed upon the gallery, on the north side of his house, some quicksilver in an open bowl. Within an hour he found the edges and surface of it frozen solid; and some minutes afterward the whole was condensed by the natural cold into a soft mass very much like tin. While the inner part was still fluid, the frozen surface exhibited a great variety of branched wrinkles; but in general it remained pretty smooth in Freezing. The congealed mercury was more flexible than lead; but on being bent short, it was found more brittle than tin; and when hammered out thin, it seemed somewhat granulated. When the hammer was not perfectly cooled, the<pb n="516"/><cb/> quicksilver melted away under it in drops; and the same thing happened when the metal was touched with the finger, by which also the finger was immediately benumbed. When the frozen mass was broken to pieces in the cold, the fragments adhered to each other and to the bowl in which they lay. In the warm room it thawed on its surface gradually, by drops, like wax on the fire, and did not melt all at once. Although the frost seemed to abate a little towards night, yet the congealed quicksilver remained unaltered, and the experiment with the thermometer could still be repeated. On the 7th of December he had an opportunity of making the same observations all day; but some hours after sunset, a northwest wind sprung up, which raised the thermometer to - 46°, when the mass of quicksilver began to melt.</p><p>The experiments of Mr. Braun were successfully repeated at Gottingen, in 1774, by Mr. John Frederick Blumenbach; being encouraged to this attempt by the exceffive cold of the winter that year, especially the night of January the 11th, when he made the experiment, the thermometer standing at - 10 in the open air. Mr. Blumenbach at 5 in the evening, put 3 drachms of quicksilver into a small sugar glass, and covered it with a mixture of snow and Egyptian sal ammoniac, setting the glass out in the air upon a mixture also of sal ammoniac. At one the next morning, the mercury was found frozen quite solid, and hard to the glass; and did not melt again till 7 or 8 the next morning. The colour of the frozen mercury was a dull pale white with a blueish cast, like zinc, very different from the natural appearance of quicksilver.</p><p>In the year 1775, by similar means, quicksilver was twice frozen by Mr. Hutchins, governor of Albany fort, in Hudson's bay, viz, in the months of January and February of that year. And the same was done on the 28th of January 1776, by Dr. Lambert Bicker, secretary of Rotterdam. The temperature of the atmosphere was then at + 2°; and the lowest it could reduce the thermometer by artificial cold, was - 94; when, on breaking the glass, the mercury was found frozen.</p><p>In the beginning of the year 1780 M. Von Elterlein of Vytegra, a town of Russia, in lat. 61° north, and long. 36° east, froze quicksilver by natural cold. On the 4th of January 1780, the cold being increased to - 34 that evening at Vytegra, he exposed to the open air 3 ounces of very pure quicksilver in a china teacup, covered with paper pierced full of holes. Next day, at 8 in the morning, he found it solid, and looking like a piece of cast lead, with a considerable depression in the middle. On attempting to loosen it in the cup, his knife raised shavings from it as if it had been lead, which remained sticking up; and at length the metal separated from the bottom of the cup in one mass. He then took it in his hand to try if it would bend: it was stiff like glue, and broke into two pieces; but his fingers immediately lost all feeling, and could scarcely be restored in an hour and a half by rubbing with snow. At 8 o'clock the thermometer stood at - 57; but half after 9 it was risen to - 40; and then the two pieces of mercury which lay in the cup had lost so much of their hardness, that they could no longer be broken, or cut into shavings, but resembled<cb/> a thick amalgam, which, though it became fluid when pressed by the fingers, immediately afterwards resumed the consistence of pap. With the thermometer at - 39, the quicksilver became fluid. The cold was never less on the 5th than - 28, and by 9 in the evening it had increased again to - 33. This experiment seems to fix the Freezing point of mercury at - 40 of Fahrenheit's thermometer, or 40 below 0; which is 72° below the Freezing point of water.</p><p>In the winter of 1781 and 82, Mr. Hutchins resumed the subject of Freezing quicksilver by artificial cold, with such success, that from his experiments and those of M. Von Elterlein, last mentioned, the Freezing point of mercury is now almost as well fettled, viz at - 40, as that of water is at + 32. Other philosophers indeed had not been altogether inattentive to this subject. Professor Braun himself had taken great pains to investigate it; but for want of a proper attention to the difference between the contraction of the fluid mercury by cold, and that of the congealing metal by Freezing, he could not determine any thing certain concerning it.</p><p>An instance of the natural congelation of quicksilver also occurred in Jemptland, one of the provinces of Sweden, on the 1st of January 1782; and lastly, on the 26th of the same month, Mr. Hutchins observed the same effect of the cold at Hudson's bay; when he found that at the point of its Freezing a mercurial thermometer stood at - 40, and a spirit thermometer at - 30.</p><p>On this subject, see the Philos. Trans. vol. 51, pa. 672; vol. 52, pa. 156; vol. 66, pa. 174; vol. 73, pa. 303 and 325; vol. 76, pa. 241; vol. 77, pa. 285; vol. 78, pa. 43; and several others, particularly vol. 79, pa. 199, &c, being experiments on the congelation of quicksilver in England, by Mr. Richard Walker, where he proves that mercury may be frozen not only in England in summer, but even in the hottest climate, at any season of the year, and without the use of ice or snow.</p><p><hi rend="smallcaps">Freezing</hi> <hi rend="italics">Point,</hi> denotes the point or degree of cold, shewn by a mercurial thermometer, at which certain sluids begin to freeze, or, when frozen, at which they begin to thaw again. On Fahrenheit's thermometer, this point is at + 32 for water, and at - 40 for quicksilver, these fluids freezing at those two points respectively. It would also be well if the Freezing points for other fluids were ascertained, and the whole arranged in a table.</p><p><hi rend="smallcaps">Freezing</hi> <hi rend="italics">Rain,</hi> or <hi rend="italics">Raining Ice,</hi> a very uncommon kind of shower which sometimes falls, particularly one in December 1672, in the west of England: of which some accounts are given in the Philos. Trans. number 90.</p><p>This rain, as soon as it touched any thing above ground, as a bough, or the like, immediately settled into ice; and by enlarging and multiplying the icicles, it broke all down with its weight. The rain that fell on the snow, immediately froze into ice, without sinking in the snow at all.</p><p>It made an amazing destruction of trees, beyond any thing in all history. “ Had it concluded with some gust of wind, says a gentleman on the spot, it might have been of terrible consequence. Having weighed<pb n="517"/><cb/> the sprig of an ash tree, the wood of which was just three quarters of a pound, the ice upon it amounted to 16 pounds. Some were frighted with the noise in the air; till they discerned it was the clatter of icy boughs dashed against each other.” Dr. Beale observes, that there was no considerable frost perceived on the ground during the whole; from which he concludes, that a frost may be very fierce and dangerous on the tops of some hills, while in other places it keeps at some feet above the ground; and may wander about very furious in some places, and be remiss in others not far off. This rain was followed by glowing heats, and a wonderful forwardness of vegetation.</p></div1><div1 part="N" n="FRENICLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">FRENICLE</surname> (<foreName full="yes"><hi rend="smallcaps">Bernard</hi></foreName>)</persName></head><p>, a celebrated French mathematician of the 17th century. He was the contemporary and companion of Des Cartes, Fermat, and the other learned mathematicians of their time. He was admitted Geometrician of the French Academy in 1666; and died in 1675.</p><p>He had many papers inserted in the Ancient Memoirs of the Academy, of 1666, particularly in vol. 5, of that collection, viz, 1. A method of resolving problems by Exclusions.—2. Treatise of right-angled Triangles in Numbers.—3. Short tract on Combinations. —4. Tables of Magic Squares.—5. General method of making Tables of Magic Squares.</p></div1><div1 part="N" n="FRESCO" org="uniform" sample="complete" type="entry"><head>FRESCO</head><p>, is a sort of painting which is made upon the plastering of walls before it is dry.</p></div1><div1 part="N" n="FRIABILITY" org="uniform" sample="complete" type="entry"><head>FRIABILITY</head><p>, the property of a body that is Friable.</p></div1><div1 part="N" n="FRIABLE" org="uniform" sample="complete" type="entry"><head>FRIABLE</head><p>, a quality of bodies by which they are rendered tender and brittle, easily crumbled or reduced to powder between the fingers; their force of cohesion being such as easily exposes them to such solution. Such are pumice, and all calcined stones, burnt allum, &c.</p><p>It is supposed that Friability arises from hence, that the body consists wholly of dry parts irregularly combined, and which are readily separated, as having nothing unctuous or glutinous to bind them together.</p></div1><div1 part="N" n="FRICTION" org="uniform" sample="complete" type="entry"><head>FRICTION</head><p>, the act of rubbing or grating the furfaces of bodies against or over each other, called also Attrition.</p><p>The phenomena arising from the Friction of divers bodies, under different circumstances, are very numerous and considerable. Mr. Hawksbee gives a number of experiments of this kind; particularly of the attrition or Friction of glass, under various circumstances; the result of which was, that it yielded light, and became electrical. Indeed all bodies by Friction are brought to conceive heat; many of them to emit light; particularly a cat's back, sugar, beaten sulphur, mercury, sea water, gold, copper, &c, but above all diamonds, which when briskly rubbed against glass, gold, or the like, yield a light equal to that of a live coal when blowed by the bellows.</p><div2 part="N" n="Friction" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Friction</hi></head><p>, in Mechanics, denotes the resistance a moving body meets with from the surface on which it moves.</p><p>Friction arises from the roughness or asperity of the surface of the body moved on, and that of the body moving: for such surfaces consisting alternately of eminences and cavities, either the eminences of the one must be raised over those of the other, or they must be<cb/> both broken and worn off: but neither can happen without motion, nor can motion be produced without a force impressed. Hence the force applied to move the body is either wholly or partly spent on this effect; and consequently there arises a resistance, or Friction, which will be greater as the eminences are greater, and the substance the harder; and as the body, by continual Friction, becomes more and more polished, the Friction diminishes.</p><p>As the Friction is less in a body that rolls, than when it slides, hence in machines, lest the Friction should employ a great part of the power, care is to be taken that no part of the machine slide along another, if it can be avoided; but rather that they roll, or turn upon each other. With this view it may be proper to lay the axes of cylinders, not in a groove or concave matrix, as usual, but between little wheels, called Friction wheels, moveable on their respective axes: for by this contrivance, the Friction is transferred from the circumference of those wheels to their pivots. And in like manner the Friction may be still farther diminished, by making the axis of those wheels rest upon other Friction wheels that turn round with them. This was long since recommended by P. Casabus; and experience confirms the truth of it. Hence also it is, that a pulley moveable on its axis resists less than if it were fixed, and the cord sliding over the circumference. And the same may be observed of the wheels of coaches, and other carriages. Indeed about 20 years ago Friction balls or rollers were placed within the naves of carriage wheels by some persons, particularly a Mr. Varlo; and lately Mr. Garnett had a patent for an improved manner of applying Friction wheels to any axis, as of carriages, blocks or pulleys, scale beams, &c, in which the inclosed wheels or rollers are kept always at the same distance by connecting rods or bars.</p><p>From these principles, with the assistance of the higher geometry, Olaus Roemer determined the figure of the teeth of wheels, that should make the least resistance possible, which he found should be epicycloids. And the same was afterwards demonstrated by De la Hire, and Camus.</p><p>M. Amontons, by experiment, attempted to settle a foundation for the precise calculation of the quantity of Friction; which M. Parent endeavoured to confirm from reasoning and geometry. M. Amontons' principle is, that the Friction of two bodies depends only on the weight or force with which they press each other, being always more or less in proportion to that pressure; esteeming it a vulgar error, that the quantity of Friction has any dependence on the extent of the surface that is rubbed, or that the Friction increases with the surface: arguing that it will require the same weight to draw along a plane, a piece of wood on its narrow edge, as on its broad and flat side; because, though on the broad side there be 4 times the number of touching particles, yet each particle is pressed with but 1/4 of the weight bearing on those of the narrow side; and since 4 times the number multiplied by 1/4 of the weight is equal to 1/4 of the number multiplied by 4 times the weight, it is plain that the effect, that is, the resistance, is equal in both cases, and therefore requires the same force to overcome it.</p><p>On the first proposal of this paradox, M. De la Hire<pb n="518"/><cb/> very properly had recourse to experiments, as the best test, had they been judiciously performed: such as they were however, they succeeded in favour of this system. He laid several pieces of rough wood on a rough table; their sizes were unequal; but he laid weights on them, so as to render them all equally heavy: and he found that the same precise sorce, or weight, applied to them by a little pulley, was required to put each in motion, notwithstanding all the inequality of the surfaces. The experiment succeeded in the same manner with pieces of marble, laid on a marble table. After this, by reasoning, M. de la Hire gave a physical solution of the effect. And M. Amontons settled a calculus of the value of Friction, with the loss sustained by it in machines, on the foundation of this new principle. In wood, iron, lead, and brass, which are the chief materials used in machines, he makes the resistance caused by Friction to be nearly the same in all, when those materials are anointed with oil or fat: and the quantity of this resistance, independent of the magnitude of the surface, he makes nearly equal to a third part of the weight of the body moved, or of the force with which the two bodies are pressed together. Others have observed, that if the surfaces be hard and well polished, the Friction will be less than a third part of the weight; but if the parts be soft or rugged, it will be much greater. It was farther observed, that in a cylinder moved on two small gudgeons, or on a small axis, the Friction would be diminished in the same proportion as the diameter of these gudgeons is less than the diameter of the cylinder; because in this case, the parts on which the cylinder moves and rubs, will have less velocity than the power which moves it in the same proportion, which is in effect making the Friction to be proportional to the velocity. So that, from the whole of their observations, this general proposition is deduced, viz, That the resistances arising from Friction, are to one another in a ratio compounded of the pressures of the rubbing parts, and the velocities of their motions. Principles which, it is now known from better experiments, are both erroneous; notwithstanding the hypothesis of M. Amontons has been adopted, and attempted to be consirmed by Camus, Desaguliers, and others.</p><p>M. Muschenbroek and the abbé Nollet, however, on the other hand, have concluded from experiments, that the Friction of bodies depends on the magnitude of their surface, as well as on their weight. Though the former says, that in small velocities the Friction varies very nearly as the velocity, but that in great velocities the proportion increases faster: he has also attempted to prove, that by increasing the weight of a body, the Friction does not always increase exactly in the same ratio. Introd. ad Phil. Nat. vol. 1, c. 9, and Lect. Phys. Exp. tom. 1, p. 241. Helsham and Ferguson, from the same kind of experiments, have endeavoured to prove, that the Friction does not vary by changing the quantity of surface on which the body moves; and the latter of these asserts, that the Friction increases very nearly as the velocity; and that by increasing the weight, the Friction is increased in the same ratio. Indeed there is scarce any subject of experiment, with regard to which, different persons have formed such various conclusions. Of those who have written on the theory, no one has established it altogether on true principles, till the experiments<cb/> lately made by Mr. Vince of Cambridge: Euler, whose theory is extremely elegant, and would have been quite satisfactory had his principles been founded on good experiments, supposes the Friction to vary in proportion to the velocity of the body, and its pressure upon the plane; neither of which is true: and others, though they have justly imagined that Friction is a uniformly retarding force, have yet retained the other supposition, and so rendered their solutions not at all applicable to the cases for which they were intended.</p><p>For these reasons a new and ingenious set of experiments was successfully instituted by the rev. Samuel Vince, A. M. of Cambridge, which are published in the 75th vol. of the Philos. Trans. p. 165. The object of these experiments was to determine,</p><p>1st, Whether Friction be a uniformly retarding sorce.</p><p>2d, The quantity of Friction.</p><p>3d, Whether Friction varies in proportion to the pressure or weight.</p><p>4th, Whether the Friction be the same on whichever of its surfaces a body moves.</p><p>Mr. Vince says, “the experiments were made with the utmost care and attention, and the several results agreed so very exactly with each other, that I do not scruple to pronounce them to be conclusive.”—“ A plane was adjusted parallel to the horizon, at the extremity of which was placed a pulley, which could be elevated or depressed in order to render the string which connected the body and the moving force parallel to the plane or horizon. A scale accurately divided was placed by the side of the pulley perpendicular to the horizon, by the side of which the moving force descended; upon the scale was placed a moveable stage, which could be adjusted to the space through which the moving force descended in any given time, which time was measured by a well regulated pendulum clock vibrating seconds. Every thing being thus prepared, the following experiments were made to ascertain the law of Friction. But let me first observe, that if Friction be a uniform force, the difference between it and the given force of the moving power must be also uniform, and therefore the moving body must descend with a uniformly accelerated velocity, and consequently the spaces described from the beginning of the motion must be as the squares of the times, just as when there was no Friction, only they will be diminished on account of the Friction.” Accordingly the experiments are then related, which are performed agreeably to these ingenious and philosophical ideas, and from them are deduced these general conclusions, which may be considered as established and certain facts or maxims. viz,</p><p>1st, That Friction is a uniformly retarding force in hard bodies, not subject to alteration by the velocity; except when the body is covered with cloth, woollen, &c, and in this case the Friction increases a little with the velocity.</p><p>2dly, Friction increases in a less ratio than the quantity of matter, or weight of the body. This increase however is different for the different bodies, more or less; nor is it yet sufficiently known, for any one<pb n="519"/><cb/> body, what proportion the increase of Friction bears to the increase of weight.</p><p>3dly, The smallest surface has the least Friction; the weight being the same. But the ratio of the Friction to the surface is not yet accurately known.</p><p>Mr. Vince's experiments consisted in determining how far the sliding bodies would be drawn, in given times, by a weight hanging freely over a pulley. This method would both shew him if the Friction were a constant retarding force, and the other conclusions above stated. For as the spaces described by any constant force, in given times, are as the squares of the times; and as the weight drawing the body is a constant force, if the Friction, which acts in opposition to the weight, should also be a constant force, then their difference, or the force by which the body is urged, will also be constant, in which case the spaces described ought to be as the squares of the times; which happened accordingly in the experiments.</p><p>Mr. Vince adds some remarks on the nature of the experiments which have been made by others. These, he observes, the authors “have instituted, To find what moving force would <hi rend="italics">just</hi> put a body at rest in motion: and they concluded from thence, that the accelerative force was then equal to the Friction; but it is manifest, that any force which will put a body in motion must be greater than the force which opposes its motion, otherwise it could not overcome it; and hence, if there were no other objection than this, it is evident, that the Friction could not be very accurately obtained; but there is another objection, which totally destroys the experiment, so far as it tends to shew the quantity of Friction, which is the strong cohesion of the body to the plane when it lies at rest.” This he confirms by several experiments, and then adds, “From these experiments therefore it appears, how very considerable the cohesion was in proportion to the Friction when the body was in motion; it being, in one case almost <*>/3, and in another it was found to be very nearly equal to the whole Friction. All the conclusions therefore deduced from the experiments, which have been instituted to determine the Friction from the force necefsary to <hi rend="italics">put</hi> a body in motion (and I have never seen any described but upon such a principle) have manifestly been totally false; as such experiments only shew the resistance which arises from the cohesion and Friction conjointly.” Philos. Trans. vol. 75, pa. 165.</p><p>Mr. Emerson, in his Principles of Mechanics, deduces from experiments the following remarks relating to the quantity of Friction: When a cubic piece of soft wood of 8 pounds weight, moves upon a smooth plane of soft wood, at the rate of 3 feet per second, its Friction is about 1/3 of the weight; but if it be rough, the Friction is little less than half the weight: on the same supposition, when both the pieces of wood are very smooth, the Friction is about 1/4 of the weight: the Friction of soft wood on hard, or of hard wood upon soft, is 1/5 or 1/6 of the weight; of hard wood upon hard wood, 1/7 or 1/<*>; of polished steel moving on steel or pewter, 1/4; moving on copper or lead, 1/<*> of the weight. He observes in general, that metals of the same sort have more Friction than those of different sorts; that lead makes much resistance; that iron or steel running in brass makes the least Friction of any; and that metals oiled make the Friction less than when<cb/> polished, and twice as little as when unpolished. Desaguliers observes that, in M. Camus's experiments on small models of sledges in actual motion, there are more cases in which the Friction is less than where it is more than 1/3 of the weight. See a table, exhibiting the Friction between various substances, formed from his experiments in Desag. Exp. Philos. vol. 1, p. 193 &c. also p. 133 to 138, and p. 182 to 254, and p. 458 to 460. On the subject of Friction, see several vols. of the Philos. Trans. as vol. 1, p. 206; vol. 34, p. 77; vol. 37, p. 394; vol. 53, p. 139, &c.</p></div2></div1><div1 part="N" n="FRIDAY" org="uniform" sample="complete" type="entry"><head>FRIDAY</head><p>, the 6th day of the week, so called from Friga, or Friya, a goddess worshipped by the Saxons on this day. It is a fast-day in the church of England, in memory of our Saviour's crucisixion, unless Christmas-day happen to fall on Friday, which is always a festival.</p><p><hi rend="italics">Good</hi> <hi rend="smallcaps">Friday</hi>, the Friday next before Easter, representing the day of our Saviour's crucifixion.</p><p>FRIGID <hi rend="italics">Zone,</hi> the space about either pole of the earth to which the sun never rises for one whole day at least in their winter. These two zones extend to about 23 1/2 degrees every way from the pole, as their centre.</p></div1><div1 part="N" n="FRIGORIFIC" org="uniform" sample="complete" type="entry"><head>FRIGORIFIC</head><p>, in Physics, something belonging to, or that occasions cold.—Some philosophers, as Gassendus, and other corpuscularians, denying cold to be a mere privation, or absence of heat, contend that there are actual Frigorific corpuscles or particles, as well as fiery ones: whence proceed cold and heat. But later philosophers allow of no other Frigorific particles beside those nitrous salts &c, which float in the air in cold weather, and occasion freezing.</p></div1><div1 part="N" n="FRIZE" org="uniform" sample="complete" type="entry"><head>FRIZE</head><p>, <hi rend="smallcaps">Frieze</hi>, or <hi rend="smallcaps">Freeze</hi>, in Architecture, a part of the entablature of columns, between the architrave and cornice.</p></div1><div1 part="N" n="FRONT" org="uniform" sample="complete" type="entry"><head>FRONT</head><p>, in Architecture, denotes, the principal face or side of a building; or that presented to their chief aspect and view.</p><div2 part="N" n="Front" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Front</hi></head><p>, in Perspective, a projection or representation of the face, or forepart of an object, or of that part directly opposite to the eye, called also more usually <hi rend="italics">orthography.</hi></p></div2></div1><div1 part="N" n="FRONTISPIECE" org="uniform" sample="complete" type="entry"><head>FRONTISPIECE</head><p>, in Architecture, the portale, or principal face of a fine building.</p></div1><div1 part="N" n="FRONTON" org="uniform" sample="complete" type="entry"><head>FRONTON</head><p>, in Architecture, an ornament among us more usually called Pediment.</p></div1><div1 part="N" n="FROST" org="uniform" sample="complete" type="entry"><head>FROST</head><p>, such a state of the atmosphere as causes the congelation or freezing of water or other fluids into ice.</p><p>The nature and effects of Frost in different countries, are mentioned under the articles Congelation, and Freezing. In the more northern parts of the world, even solid bodies are affected by Frost, though this is only or chiefly in consequence of the moisture they contain, which being frozen into ice, and so expanding as water is known to do when frozen, it bursts and rends any thing in which it is contained, as plants, trees, stones, and large rocks. Some fluids expand by Frost, as water, which expands about <*>th part, for which reason ice floats in water; but others again contract, as quicksilver, and hence frozen quicksilver sinks in the fluid metal.</p><p>Frost, being derived from the atmosphere, naturally proceeds from the upper parts of bodies downwards, as the water and the earth: so, the longer a Frost is<pb n="520"/><cb/> continued, the thicker the ice becomes upon the water in ponds, and the deeper into the earth the ground is frozen. In about 16 or 17 days Frost, Mr. Boyle found it had penetrated 14 inches into the ground. At Moscow, in a hard season, the Frost will penetrate 2 seet deep in the ground; and Capt. James found it penetrated 10 feet deep in Charlton island, and the water in the same island was frozen to the depth of 6 feet. Scheffer assures us, that in Sweden the Frost pierces 2 cubits, or Swedish ells into the earth, and turns what moisture is found there into a whitish substance, like ice; and standing waters to 3 ells, or more. The same author also mentions sudden cracks or rifts in the ice of the lakes of Sweden, 9 or 10 feet deep, and many leagues long; the rupture being made with a noise not less loud than if many guns were discharged together. By such means however the fishes are surnished with air; so that they are rarely found dead.</p><p>The natural histories of Frosts furnish very extraordinary esfects of them. The trees are often scorched, and burnt up, as with the most excessive heat; and split or shattered. In the great Frost in 1683, the trunks of oak, ash, walnut, &c, were miserably split and cleft, so that they might be seen through, and the cracks often attended with dreadful noises like the explosion of fire-arms. Philos. Trans. number 165.</p><p>The close of the year 1708, and the beginning of 1709, were remarkable, throughout the greatest part of Europe, for a severe Frost. Dr. Derham says, it was the greatest in degree, if not the most universal, in the memory of man; extending through most parts of Europe, though scarcely felt in Scotland or Ireland.</p><p>In very cold countries, meat may be preserved by the Frost 6 or 7 months, and prove tolerable good eating. See Capt. Middleton's observations made in Hudson's bay, in the Philos. Trans. no. 465, sect. 2. In that climate the Frost seems never out of the ground, it having been found hard frozen in the two summer months. Brandy and spirit of wine, set out in the open air, freeze to solid ice in 3 or 4 hours. Lakes and standing waters, not above 10 or 12 feet deep, are frozen to the ground in winter, and all their fish perish. But in rivers, where the current of the tide is strong, the ice does not reach so deep, and the fish are preserved. Id. ib.</p><p>Some remarkable instances of Frost in Europe, and chiesly in England, are recorded as below: In the year</p><p>220, Frost in Britain that lasted 5 months.</p><p>250, The Thames frozen 9 weeks.</p><p>291, Most rivers in Britain srozen 6 weeks.</p><p>359, Severe Frost in Scotland for 14 weeks.</p><p>508, The rivers in Britain frozen for 2 months.</p><p>558, The Danube quite frozen over.</p><p>695, Thames frozen 6 weeks; booths built on it.</p><p>759, Frost from Oct. 1 till Feb. 26, 760.</p><p>827, Frost in England for 9 weeks.</p><p>859, Carriages used on the Adriatic sea.</p><p>908, Most rivers in England frozen 2 months.</p><p>923, The Thames frozen 13 weeks.</p><p>987, Frost lasted 120 days: began Dec. 22.</p><p>998, The Thames frozen 5 weeks.<cb/></p><p>1035, Severe Frost on June 24: the corn and fruits destroyed.</p><p>1063, The Thames frozen 14 weeks.</p><p>1076, Frost in England from Nov. till April.</p><p>1114, Several wooden bridges carried away by ice.</p><p>1205, Frost from Jan. 14 till March 22.</p><p>1407, Frost that lasted 15 weeks.</p><p>1434, From Nov. 24 till Feb. 10. Thames frozen down to Gravesend.</p><p>1683, Frost for 13 weeks.</p><p>170 8/9, Severe Frost for many weeks.</p><p>1715, The same for many weeks.</p><p>1739, One for 9 weeks. Began Dec. 24.</p><p>1742, Severe Frost for many weeks.</p><p>1747, Severe Frost in Russia.</p><p>1754, Severe one in England.</p><p>1760, The same in Germany.</p><p>1776, The same in England.</p><p>1788, Thames frozen below bridge; booths on it.</p><p><hi rend="italics">Hoar</hi> <hi rend="smallcaps">Frost</hi>, is the dew frozen or congealed, early in cold mornings; chiefly in autumn. Though many Cartesians will have it formed of a cloud; and either congealed in the cloud, and so let fall; or ready to be congealed as soon as it arrives at the earth.</p><p>Hoar Frost, M. Regis observes, consists of an assemblage of little parcels of ice crystals; which are of various figures, according to the different disposition of the vapours, when met and condensed by the cold.</p></div1><div1 part="N" n="FRUSTUM" org="uniform" sample="complete" type="entry"><head>FRUSTUM</head><p>, in Geometry, is the part of a solid next the base, left by cutting off the top, or segment, by a plane parallel to the base: as the Frustum of a pyramid, of a cone, of a conoid, of a spheroid, or of a sphere, which is any part comprised between two parallel circular sections; and the Middle Frustum of a sphere, is that whose ends are equal circles, having the centre of the sphere in the middle of it, and equally distant from both ends.</p><p><hi rend="italics">For the Solid Content of the Frustum of a cone, or of any pyramid, whatever figure the base may have.</hi> Add into one sum, the areas of the two ends and the mean proportional between them; then 1/3 of that sum will be a mean area, or the area of an equal prism, of the same altitude with the Frustum; and consequently that mean area being multiplied by the height of the Frustum, the product will be the solid content of it. That is, if A denote the area of the greater end, <hi rend="italics">a</hi> that of the less, and <hi rend="italics">h</hi> the height; then is the solidity.</p><p>Other rules for pyramidal or conic Frustums may be seen in my Mensuration, p. 189, 2d edit. 1788.</p><p><hi rend="italics">The curve Surface of the Zone or Frustum of a sphere,</hi> is had by multiplying the circumference of the sphere by the height of the Frustum. Mensur. p. 197.</p><p><hi rend="italics">And the Solidity of the same Frustum is found,</hi> by adding together the squares of the radii of the two ends. and 1/3 of the square of the height of the Frustum, then multiplying the sum by the said height and by the number 1.5708. That is, is the solid content of the spheric Frustum, whose height is <hi rend="italics">h,</hi> and the radii of its ends R and <hi rend="italics">r,</hi> <hi rend="italics">p</hi> being = 3.1416. Mensur. p. 209.</p><p>For the Frustums of spheroids, and conoids, either parabolic or hyperbolic, see Mensur. p. 326, 328, 332, 382, 435. And in p. 486 &c, are general theorem<*> concerning the Frustum of a sphere, cone, spheroid, or<pb n="521"/><cb/> conoid, terminated by parallel planes, when compared with a cylinder of the same altitude, on a base equal to the middle section of the Frustum made by a parallel plane. The difference between the Frustum and the cylinder is always the same quantity, in different parts of the same, or of similar solids, or whatever the magnitude of the two parallel ends may be; the inclination of those ends to the axis, and the altitude of the Frustum being given; and the said constant difference is 1/4 part of a cone of the same altitude with the Frustum, and the radius of its base is to that altitude, as the fixed axis is to the revolving axis of the Frustum. Thus, if BEC be any conic section, or a right line, or a circle, whose <figure/> axis, or a part of it, is AD; AB and CD the extreme ordinates, FE the middle ordinate, AF being = FD; then taking, as AD to DK, so is the whole fixed axis, of which AD is a part, to its conjugate axis; and completing the parallelogram AGHD: then if the whole figure revolve about the axis AD, the line BEC will generate the Frustum of the cone or conoid, according as it is a right line or a conic section, or it will generate the whole solid when AB vanishes, or A and B meet in the same point; likewise AGHD will generate a cylinder, and ADK a cone: then is the 4th part of this cone always equal to the difference between the said cylinder generated by AGHD and the solid or Frustum generated by ABECD; having all the same altitude or axis AD.</p><p>In the parabolic conoid, this difference and the cone vanish, and the Frustum, or whole conoid ABECD, is always equal to the cylinder AGHD, of the same altitude.</p><p>In the sphere, or spheroid, the Frustum ABECD is <hi rend="italics">less</hi> than the cylinder AGHD, by 1/4 of the cone AKD. And</p><p>In the cone or hyperboloid, that Frustum is <hi rend="italics">greater</hi> than the cylinder, by 1/4 of the said cone AKD, which is similar to the other cone IBCD.</p><p>It may be observed, that the same relations are true, whether the ends of the Frustum are perpendicular or oblique to the axis. And the same will hold for the Frustum of any pyramid, whether right or oblique; and such a Frustum of a pyramid will exceed the prism, of the same altitude, and upon the middle section of the Frustum, by 1/4 of the same cone.</p><p>It has been observed, that the difference, or 1/4 of the cone AKD, is the same, or constant, when the altitude and inclination of the ends of the Frustum remain the same. But when the inclination of the ends varies, the altitude being constant; then the said difference varies so as to be always reciprocally as the cube of the conjugate to the diameter AD. And when both the altitude and inclination of the ends vary, the differential cone is as the cube of the altitude directly, and the cube of the said conjugate diameter reciprocally: but if they vary so, as that the altitude is always reciprocally as that diameter, then the difference is a constant quantity.</p><p><hi rend="italics">Another general theorem for Frustums, is this.</hi> In the Frustum of any solid, generated by the revolution of any conic section about its axis, if to the sum of the<cb/> two ends be added 4 times the middle fection, 1/<*> of the last sum will be a mean area, and being drawn into the altitude of the solid, will produce the content. That is, is the content of ABCD.</p><p>And this theorem is general for all Frustums, as well as the complete solids, whether right or oblique to the axis, and not only of the solids generated from the circle or conic sections, but also of all pyramids, cones, and in short of any solid whose parallel sections are similar figures.</p><p>The same theorem also holds good for any parabolic <hi rend="italics">area</hi> ABECD, and is very <hi rend="italics">nearly true</hi> for the area of any other curve whatever, or for the content of any other solid than those above mentioned.</p></div1><div1 part="N" n="FUGUE" org="uniform" sample="complete" type="entry"><head>FUGUE</head><p>, in Music, is when the different parts of a musical composition follow each other, each repeating in order what the first had performed.</p></div1><div1 part="N" n="FULCRUM" org="uniform" sample="complete" type="entry"><head>FULCRUM</head><p>, or <hi rend="italics">Prop,</hi> in Mechanics, is the fixed point about which a lever &c turns and moves.</p><p>FULGURATING <hi rend="italics">Phosphorus,</hi> a term used by some English writers, to express a substance of the phosphorus kind. It was prepared both in a dry and liquid state, but the preparation it seems was not well known to any but the inventor of it. This matter not only shone in the dark in both states, but communicated its light to any thing it was rubbed on. When inclosed in a glass vessel well stopped, it sometimes would Fulgurate, or throw out little flashes of light, and sometimes fill the whole phial with waves of flame. It does not need recruiting its light at the fire, or in the sunshine, like the phosphorus of the Bolognian stone, but of itself continues in a state of shining for several years together, and is seen as soon as exposed in the dark; the solid or dry matter always resembling a burning coal of fire, though not consuming itself. Philos. Trans. N° 134.</p></div1><div1 part="N" n="FULIGINOUS" org="uniform" sample="complete" type="entry"><head>FULIGINOUS</head><p>, an epithet applied to thick smoke or vapour replete with soot or other crass matter.</p><p>In the first fusion of lead, there exhales a great deal of Fuliginous vapour, which being retained and collected, makes what is called Litharge. And Lampblack is what is gathered from the Fuliginous vapours of pines, and other resinous wood, when burnt.</p></div1><div1 part="N" n="FULMINANT" org="uniform" sample="complete" type="entry"><head>FULMINANT</head><p>, <hi rend="smallcaps">Fulminans</hi>, or <hi rend="smallcaps">Fulminating</hi>, an epithet applied to something that thunders, or makes a noise like thunder.</p><p><hi rend="italics">Aurum</hi> <hi rend="smallcaps">Fulminans.</hi> See <hi rend="smallcaps">Aurum.</hi></p><p><hi rend="italics">Pulvis</hi> <hi rend="smallcaps">Fulminans</hi>, is a composition of 3 parts of nitre, 2 parts of salt of tartar, and 1 of sulphur.—Both the Aurum and Pulvis Fulminans produce their effect chiefly downwards; in which they disfer from gunpowder, which acts in orbem, or all around, but principally upwards. When the composition is laid in brass ladles, and so set on sire, aster fulmination, the ladles are often found perforated. It differs also from gunpowder in this, that it does not require to be confined, in order to fulminate, and it must be slowly and gradually heated. Some instants before explosion, a light blue flame appears on its surface, proceeding from the vapours beginning to kindle. No more fire or flame is perceived during the fulmination, being suffocated and extinguished by the quickness and violence of the commotion. Nor does the Fulminating powder<pb n="522"/><cb/> generally kindle the combustible bodies in contact with it, because the time of its inflammation is too short.</p><p><hi rend="smallcaps">Fulminating</hi> <hi rend="italics">Damp.</hi> See <hi rend="smallcaps">Damp.</hi></p></div1><div1 part="N" n="FULMINATION" org="uniform" sample="complete" type="entry"><head>FULMINATION</head><p>, or <hi rend="smallcaps">Fulguration</hi>, a vehement noise or shock resembling thunder, caused by the sudden explosion and inflammation of divers preparations; as aurum fulminans, &c, when set on fire.</p></div1><div1 part="N" n="FUNCTION" org="uniform" sample="complete" type="entry"><head>FUNCTION</head><p>, a term used in analytics, for an algebraical expression any how compounded of a certain letter or quantity with other quantities or numbers: and the expression is said to be a Function of that letter or quantity. Thus , or , or , or <hi rend="italics">x</hi><hi rend="sup">c</hi>, or <hi rend="italics">c</hi><hi rend="sup">x</hi>, is each of them a Function of the quantity <hi rend="italics">x.</hi></p><p>On the subject of Functions, their divisions, transformations, explication by insinite series, &c, see Euler's Analys. Infinitorum, c. 1, where the subject is fully treated.</p></div1><div1 part="N" n="FURLONG" org="uniform" sample="complete" type="entry"><head>FURLONG</head><p>, an English long measure, containing 660 feet, or 220 yards, or 40 poles or perches, or the 8th part of a mile.</p></div1><div1 part="N" n="FURNITURE" org="uniform" sample="complete" type="entry"><head>FURNITURE</head><p>, in Dialling, certain additional points and lines drawn on a dial, by way of ornament. Such as the signs of the zodiac, length of days, parallels of declination, azimuths, points of the compass, meridians of chief cities, Babylonic, Jewish, or Italian hours, &c.</p></div1><div1 part="N" n="FUSAROLE" org="uniform" sample="complete" type="entry"><head>FUSAROLE</head><p>, in Architecture, a small round member cut in form of a collar, with oval beads, under the <*>chinus, or quarter-round, in the Doric, Ionic, and Composite capitals.</p></div1><div1 part="N" n="FUSEE" org="uniform" sample="complete" type="entry"><head>FUSEE</head><p>, or <hi rend="smallcaps">Fusy</hi>, in Watch-work, is that part resembling a low cone with its sides a little sunk or concave, which is drawn by the spring, and about which the chain or string is wound.</p><p>The spring of a watch is the first mover. It is rolled up in a cylindrical box, against which it acts, and which it turns round in unbending itself. The chain, which at one end is wound about the Fusee, and at the other fastened to the spring-box, disengages itself from the Fusee in proportion as the box is turned. And hence the motion of all the other parts of the spring-watch. Now the effort or action of the spring is continually diminishing from first to last; and unless that inequality was rectified, it would draw the chain with more force, and wind a greater quantity of it upon the box, at one time than another; so that the movement would never keep equal time.</p><p>To correct this irregularity of the spring, it was very happily contrived to have the spring applied to the arms of levers, which are continually longer as the force of the spring is weaker: this foreign assistance, always increasing as it is most needed, maintains the action and essect of the spring in an equality.</p><p>It is for this reason then that the Fusee is made tapering somewhat conical, its radius at every point of the axis answering to the corresponding strength of the spring.</p><p>Now if the action of the spring diminished equally, as the parallels to the base of a triangle do; the cone,<cb/> which is generated of a triangle, would be the precise figure required for the Fusee; but it is certain that the weakening of the spring is not in that proportion; and therefore the Fusee should not be exactly conical; and in fact experience shews that it should be a little hollowed about the middle, because the action of the spring is not there sufficiently diminished of itself. <figure/></p><p>Mr. Varignon has investigated the figure of the Fusee, or the nature of the curve by whose revolution about its axis, shall be produced the solid whose figure the Fusee is to have. This curve it may easily be shewn is an hyperbola whose asymptote is the axis of the Fusee. Thus, let DFE be the curve of the Fusee, its axis being ABC: let AD express the greatest strength of the spring when the watch is quite wound up, o<*> when the spring acts at D, and BG the least strength when the watch is down, or when the spring acts at E; so as that , or ; join DG, produeing it to meet the axis produced in C; then shall HI denote the strength of the spring acting at the corresponding point F of the Fusee; and the nature of it must be such that the rectangle be equal to a constant quantity, or HF must be reciprocally as HI, or ; but because - - AD, HI, BG, are directly proportional to - CA, CH, CB, theref. these are reciprocally propor. to AD, HF, BE, and consequently the curve DFE is an hyperbola, whose centre is C, and asymptotes AC and KL: so that the figure of the Fusee is the solid generated by an equilateral hyperbola revolved about its asymptote. See also Martin's Mathem. Instit. vol. 2, p. 364.</p><div2 part="N" n="Fusee" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Fusee</hi></head><p>, <hi rend="smallcaps">Fuse</hi>, or <hi rend="smallcaps">Fuze</hi>, in Artillery, is a woode<*> tap or tube used to set fire to the powder in a bombshell. The bore of this tube is filled with a composition, of sulphur one part, saltpetre 3 parts, and mealed powder 3, 4, or 5 parts. The tube is driven hard into the hole in the shell, having first cut it to the exact length answering to the time of the intended flight of the shell, so that, the composition in the Fuse catching fire by the discharge of the shell from the mortar, it just burns down to its lower end, and so sets fire to the powder in the shell, and thereby bursts it, at the moment when it arrives at the end of its range or flight.</p></div2></div1><div1 part="N" n="FUST" org="uniform" sample="complete" type="entry"><head>FUST</head><p>, in Architecture, the shaft of a column, or the part comprehended between the base and the capital, called also the Naked.</p></div1><div1 part="N" n="FUZE" org="uniform" sample="complete" type="entry"><head>FUZE</head><p>, or <hi rend="smallcaps">Fuzee.</hi> See <hi rend="smallcaps">Fusee.</hi><pb n="523"/></p></div1></div0><div0 part="N" n="G" org="uniform" sample="complete" type="alphabetic letter"><head>G</head><cb/><div1 part="N" n="GABIONS" org="uniform" sample="complete" type="entry"><head>GABIONS</head><p>, in Fortisication, are large cylindrical baskets, open at both ends, made of ozier twigs, of 3 or 4 feet in diameter, and from 3 to 6 feet high. Thefe, being filled with earth, are sometimes used as merlons for the batteries, and sometimes as a parapet for the lines of approach, when the attacks are carried on through a stony or rocky ground, and to advance them with extraordinary vigour. They serve also to make lodgments in some posts, and to secure other places from the shot of the enemy; who, on their part, endeavour to burn and destroy the Gabions, by throwing pitched faggots among them.</p></div1><div1 part="N" n="GABLE" org="uniform" sample="complete" type="entry"><head>GABLE</head><p>, or <hi rend="smallcaps">Gable</hi>-<hi rend="italics">end,</hi> of a house, is the upright triangular end, from the cornice or eaves to the top of its roof.</p></div1><div1 part="N" n="GAGE" org="uniform" sample="complete" type="entry"><head>GAGE</head><p>, in Hydrostatics, Pneumatics, &c, is an instrument for ascertaining measures of various kinds. As</p><p><hi rend="smallcaps">Gage</hi> <hi rend="italics">of the Air-pump,</hi> is adapted for shewing the degree to which the air is rarefied, or the receiver is exhausted, at any time by the air-pump. This is either the common barometer-gage, both long and short, or the pear gage, which at first was thought a great improvement, but afterwards it was discovered that its seeming accuracy was founded on a fallacy, which gave an erroneous indication of exhaustion. See <hi rend="smallcaps">Air</hi>-<hi rend="italics">pump.</hi></p><p><hi rend="smallcaps">Gage</hi> <hi rend="italics">of the Barometer,</hi> is a contrivance for estimating the exact degree of the rise and fall of the mercury in the tube of that instrument. It is well known that whilst the mercury rises in the tube, it sinks in the cistern, and vice versa; and consequently the divisions on the scale fixed near the top of the tube had their distance from the surface of the mercury in the cistern always various; from which there must often happen errors in determining the height of the mercury in the tube. To remedy this inconvenience, a line is cut upon a round piece of ivory, which is fixed near the cistern: this line is accurately placed at a given distance from the scale; for example at 27 inches; and a small float of cork, with a cylindrical piece of ivory fixed to its upper surface, on which a line is cut at the exact distance of 2 inches from the under side of the cork, is left to play freely on the quicksilver, and the cylinder works in a groove made in the other piece. From this construction it appears, that if these marks are made to coincide, by raising or lowering the screw which acts on the quicksilver, then the divisions on the scale will express the true measure of the distance from the surface.</p><p><hi rend="smallcaps">Gage</hi> <hi rend="italics">of the Condenser,</hi> is a glass tube of a particular construction, adapted to the condensing engine, and de- <*>gned to shew the exact density and quantity of the<cb/> air contained at any time in the condenser. See Desaguliers's Exper. Philos. vol. 2, p. 394.</p><p><hi rend="italics">Sea</hi> <hi rend="smallcaps">Gage</hi>, an instrument for finding the depth of the sea. Several sorts of these have been invented by Dr. Hales, Dr. Desaguliers, and others. Formerly, the machines for this purpose consisted of two bodies, the one specisically lighter, and the other specifically heavier than the water, so joined together, that as soon as the heavy one came to the bottom, the lighter should get loose from it, and emerge; and the depth was to be estimated by the time the compound was in falling from the top to the bottom of the water, together with the time the lighter body was in rising, reckoned from the disappearing of the machine, till the emergent body was seen again; but no certain conclusion could be drawn from so precarious and incomplete an experiment.</p><p>But that invented by Drs. Hales and Desaguliers was of a more exact nature, depending on the pressure of the fluid only. For as the pressure of fluids in all directions is the same at the same depth, a Gage which discovers what the pressure is at the bottom of the sea, will shew what the true depth of the sea is in that place, whether the time of the machine's descent be longer or shorter.</p><p>Dr. Hales, in his Vegetable Statics, describes his Gage for estimating the pressures made in opaque vessels; where honey being poured over the surface of mercury in an open vessel, rises upon the surface of the mercury as it is pressed up into a tube whose lower orifice is immersed into the honey and mercury, and whose top is hermetically sealed. Now as by the pressure, the air in the tube is condensed, and the mercury rises, so the mercury comes down again when the pressure is taken off, and would leave no mark of the height to which it had risen; but the honey (or treacle, which does better) which is upon the mercury, sticking to the inside of the tube, leaves a mark, which shews the height to which it had risen, and consequently gives the quantity of pressure, and the height of the surface of the fluid.</p><p>Desaguliers's addition to this machine, consisted in a contrivance to carry it down to the bottom of the sea by means of a heavy weight, which was immediately disengaged by striking the bottom, and the Gage, made very light for the purpose, re-ascended to the top.</p><p>Dr. Hales afterwards made more experiments of this sort, and proposed another Sea Gage for vast depths, which is described in the Philos. Trans. N° 405, and is to this effect. Suppose a pretty long tube of copper or iron, close at the upper end, to be let down into the<pb n="524"/><cb/> sea, to any depth, the water will rise in the tube to a height bearing a certain proportion to the depth of the sea to which the machine is sunk. And this proportion is as follows: 33 feet of sea water being nearly equal to the mean pressure of the atmosphere, therefore at 33 feet deep, the air in the tube will be compressed into half the length of the tube, or the water will rise and fill half way up the tube; in like manner at 66 feet deep, the water will occupy 2/3 of the tube; at 99 feet deep it will fill 3/4 of the tube; at 132 feet de<*>p it will fill 4/5 of the tube; and so on. Hence therefore, by knowing the height to which the water rises in the tube, there will be known the consequent depth of the sea.</p><p>But, in very great depths, the scale near the top of the tube would be so small, and the divisions so close, that there would be no accuracy in the experiment, unless the tube were of a very great length, and this again would render it both liable to be broken, and quite impracticable.</p><p>To remedy this inconvenience, he made the following contrivance: To the bottom of the tube he screwed a large hollow globe of copper, with a small orisice, or a short pipe at bottom of the globe, to let in the water; by which means he had a very great quantity of air, and the scale enlarged. See also Desagul. Exp. Phil. vol. 2, p. 224 and 241.</p><p><hi rend="italics">Bucket Sea</hi> <hi rend="smallcaps">Gage</hi>, is an instrument contrived by Dr. Hales to find the different degrees of coolness and saltness of the sea at different depths. This Gage consists of a common pale or bucket, with two heads: these heads have each a round hole in the middle, about 4 inches in diameter, covered with square valves opening upward; and that they may both open and shut together, there is a small iron rod, having one end fixed to the upper side of the lower valve, and the other end to the lower side of the upper valve. So that as the bucket descends with its sinking weight into the sea, both the valves may open by the force of the water, which by that means has a free passage through the bucket. But when the bucket is drawn up, then both the valves shut by the force of the water at the upper end of the bucket; so that the bucket is drawn up full of the lowest sea water to which it has descended, and immediately the mercurial thermometer, fixed within it, is examined, to see the degree of temperature; and the degree of saltness is afterwards examined at leisure. Philos. Trans. numb. 9, p. 149, and numb. 24, p. 447, or Abridg. vol. 2, p. 260.</p><p>Lord Charles Cavendish adapted a thermometer for the temperature of the sea water, at different depths. See Philos. Trans. vol. 50, p. 300, and Phipps's Voyage towards the North Pole, p. 142 &c.</p><p><hi rend="italics">Aqueo-mercurial</hi> <hi rend="smallcaps">Gage</hi> is the name of an apparatus contrived by Dr. Hales, and applied, in various forms, to the branches of trees, to determine the force with which they imbibe moifture. Vegetable Statics, vol. 1, ch. 2, p. 84.</p><p><hi rend="italics">Sliding</hi> <hi rend="smallcaps">Gage</hi>, a tool used by mathematical instrument makers, for measuring and fetting off distances; consisting of a beam, tooth, sliding socket, and the shoulder of the socket.</p><p><hi rend="italics">Tide</hi> <hi rend="smallcaps">Gage</hi>, an instrument used for determining the height of the tides by Mr. Bayly, in the course of a<cb/> voyage towards the south pole &c, in the Resolution and Adventure, in the years 1772, 1773, 1774, and 1775. This instrument consists of a glass tube, whose internal diameter was 7-10ths of an inch, lashed fast to a 10 foot fir rod, divided into feet, inches, and parts; the rod being fastened to a strong post fixed firm and upright in the water. At the lower end of the tube was an exceeding small aperture, through which the water was admitted. In consequence of this construction, the surface of the water in the tube was so little affected by the agitation of the sea, that its height was not altered the 10th part of an inch when the swell of the sea was 2 feet; and Mr. Bayly was certain, that with this instrument he could discern a difference of the 10th of an inch in the height of the tide.</p><p><hi rend="italics">Water</hi> <hi rend="smallcaps">Gage.</hi> See <hi rend="smallcaps">Altitude</hi>, and <hi rend="smallcaps">Hydrometer.</hi></p><p><hi rend="italics">Wind</hi> <hi rend="smallcaps">Gage</hi>, an instrument for measuring the force of the wind upon any given surface. Several have been invented formerly, and one was lately invented by Dr. Lind, which is described in the Philos. Trans. vol. 65. See several also under the article <hi rend="smallcaps">Anemometer.</hi></p></div1><div1 part="N" n="GAGER" org="uniform" sample="complete" type="entry"><head>GAGER</head><p>, see <hi rend="smallcaps">Gauger.</hi></p></div1><div1 part="N" n="GAGING" org="uniform" sample="complete" type="entry"><head>GAGING</head><p>, see <hi rend="smallcaps">Gauging.</hi></p></div1><div1 part="N" n="GALAXY" org="uniform" sample="complete" type="entry"><head>GALAXY</head><p>, or <hi rend="italics">Milky-Way,</hi> or <hi rend="italics">Via Lactea,</hi> in Astronomy, that long, whitish, luminous track, which seems to encompass the heavens like a swath, scarf, or girdle; and which is easily seen in a clear night, especially when the moon is not up. It is of a considerable, though unequal breadth; being also in some parts double, but in others single.</p><p>The Galaxy passes through many of the constellations in its circuit round the heavens, and keeps its exact place or position with respect to them.</p><p>There have been various strange and fabulous stories and opinions concerning the Galaxy.</p><p>The ancient poets, and even some of the philosophers, speak of it as the road or way by which the heroes went to heaven. But the Egyptians called it the Way of Straw, from the story of its rising from burning straw, thrown behind the goddess Isis in her flight from the giant Typhon. While the Greeks, who affect to derive every thing in the heavens from some of their own fables, have two origins for it; the one, that Juno, without perceiving it, accidentally gave suck to Mercury when an infant, but that as soon as she turned her eyes upon him, she threw him from her, and as the nipple was drawn from his mouth, the milk ran about for a moment; and the other, that the infant Hercules being laid by the side of Juno when asleep, on waking she gave him the breast; but soon perceiving who it was, she threw him from her, and the heavens were marked by the wasted milk.</p><p>Some other philosophers however gave it a different turn, and different origin: these esteemed it to be a tract of liquid fire, spread in this manner along the skies: and others again, supposing a celestial region beyond all that was visible, and imagining that fire, at some time let loose from thence, was to consume the world, made this a part of that celestial fire, and appealed to it as a presage of what would surely happen. This diffused brightness they considered as a crack in the vault or wall of heaven, and fancied this a glimmering of the celestial fire through it, and that there required nothing more than the undoing of this<pb n="525"/><cb/> crack by some accident in nature, or by the will of the Gods, to make the whole frame start, and let out the fire of destruction.</p><p>Aristotle makes the Galaxy a kind of meteor, formed of a crowd of vapours, drawn into that part by certain large stars disposed in the region of the heavens answering to it. Others, finding that the Galaxy was seen all over the globe, that it always corresponded to the same fixed stars, and that it was far above the highest planets, set Aristotle's opinion aside, and placed the Galaxy in the sirmament or region of the fixed stars; and concluded that it was nothing else but an assemblage of an infinite number of minute stars. And since the invention of telescopes, this opinion has been abundantly confirmed. For, by directing a good telescope to any part of the milky way, we perceive an innumerable multitude of very small stars, where before we only observed a confused whiteness, arising from the assemblage and union of their joint light; like as any thing powdered with fine white powder, at a distance we only observe the confused whiteness, but on examining it very near we perceive all the small particles of the powder separately; as Milton finely expresses it, A broad and ample road, whose dust is gold,<lb/> And pavement stars, as stars to thee appear,<lb/> Seen in the Galaxy, that milky way,<lb/> Which nightly, as a circling Zone thou seest<lb/> Powder'd with stars.<lb/></p><p>There are other such marks in the heavens; as the nebulæ, or, nebulous stars, and certain whitish parts about the south pole, called Magellanic clouds, which are all of the same nature, appearing to be vast clusters of small stars when viewed through a telescope, which are too faint to affect the eye singly.</p><p>M. le Monnier however, not being able to discover more stars in this space than in other parts of the heavens, disputes the opinion above recited as to the reason of the whiteness, and supposes that this and the nebulous stars are occasioned by some other kind of matter. Inst. Ast. p. 60.</p><p>GALILEI (<hi rend="smallcaps">Galileo</hi>,) a most excellent philosopher, mathematician and astronomer, was born at Pisa in Italy, in 1564. From his infancy he had a strong propensity to philosophy and mathematics, and soon made a great progress in these sciences. So that in 1592 he was chosen professor of mathematics at Padua. While he was professor there, visiting Venice, then famous for the art of glass-making, he heard that in Holland a glass had been invented, through which very distant objects were seen distinctly as if near at hand. This was sufficient for Galileo; his curiosity was raised, and put him upon considering what must be the form of such a glass, and the manner of making it. The result of his enquiry was the invention of the telescope, produced from this hint, without having seen the Dutch glass. All the discoveries he made in astronomy were easy and natural consequences of this invention, which opening a way, till then unknown, into the heavens, thence brought the finest discoveries. One of the first of these, was that of 4 of Jupiter's satellites, which he called the Medicean stars or planets, in honour of Cosmo the 2d, grand-duke of Tuscany, who was of<cb/> that family. Cosmo sent for our astronomer from Padua, and made him professor of mathematics at Pisa in 1611; and soon after inviting him to Florence, gave him the office and title of <hi rend="italics">principal philosopher and mathematician to his highness.</hi></p><p>He had been but a few years at Florence, before the Inquisition began to be very busy with him. Having observed some solar spots in 1612, he printed that discovery the following year at Rome; in which, and in some other pieces, he ventured to assert the truth of the Copernican system, and brought several new arguments to confirm it. For these he was cited before the Inquisition at Rome, in 1615: after some months imprisonment, he was released, and sentence pronounced against him, that he should renounce his heretical opinions, and not defend them by word or writing, or insinuate them into the minds of any persons. But having afterwards, in 1632, published at Florence his Dialogues of the two Great Systems of the World, the Ptolomaic and Copernican, he was again cited before the holy-office, and committed to the prison of that ecclesiastical court at Rome. The inquisitors convened in June that year; and in his presence pronounced sentence against him and his books, obliging him to abjure his errors in the most solemn manner; committed him to the prison of their office during pleasure; and enjoined him, as a saving penance, for three years to come, to repeat once a week the seven penitential psalms: reserving to themselves, however, the power of moderating, changing, or taking away altogether or in part, the said punishment and penance. On this sentence, he was detained in prison till 1634; and his Dialogues of the System of the World were burnt at Rome.</p><p>Galileo lived ten years after this; seven of which were employed in making still further discoveries with his telescope. But by the continual application to that instrument, added to the damage his sight received from the nocturnal air, his eyes grew gradually weaker, till he became totally blind in 1639. He bore this calamity with patience and resignation, worthy of a great philosopher. The loss neither broke his spirit, nor stopped the course of his studies. He supplied the defect by constant meditation; by which means he prepared a large quantity of materials, and began to arrange them by dictating his ideas; when, by a distemper of three months continuance, wasting away by degrees, he expired at Arcetri near Florence, in January 1642, being the 78th year of his age.</p><p>Galileo was in his person of small stature, though of a venerable aspect, and vigorous constitution. His conversation was affable and free, and full of pleasantry. He took great delight in architecture and painting, and designed extremely well. He played exquisitely on the lute; and whenever he spent any time in the country, he took great pleasure in husbandry. His learning was very extensive; and he possessed in a high degree a clearness and acuteness of wit. From the time of Archimedes, nothing had been done in mechanical geometry, till Galileo, who being possessed of an excellent judgment, and great skill in the most abstruse points of geometry, first extended the boundaries of that science, and began to reduce the resistance of solid<pb n="526"/><cb/> bodies to its laws. Besides applying geometry to the doctrine of motion, by which philosophy became established on a sure foundation, he made surprising discoveries in the heavens by means of his telescope. He made the evidence of the Copernican system more sensible, when he shewed from the phases of Venus, like to those of the moon, that Venus actually revolves about the sun. He proved the rotation of the sun on his axis, from his spots; and thence the diurnal rotation of the earth became more credible. The satellites that attend Jupiter in his revolution about the sun, represented, in Jupiter's smaller system, a just image of the great solar system: and rendered it more easy to conceive how the moon might attend the earth, as a satellite, in her annual revolution. By discovering hills and cavities in the moon, and spots in the sun constantly varying, he shewed that there was not so great a difference between the celestial bodies and the earth as had been vainly imagined.</p><p>He rendered no less service to science by treating, in a clear and geometrical manner, the doctrine of motion, which has justly been called the key of nature. The rational part of mechanics had been so much neglected, that hardly any improvement was made in it for almost 2000 years. But Galileo has given us fully the theory of equable motion<*>, and of such as are uniformly accelerated or retarded, and of these two compounded together. He, first of any, demonstrated that the spaces described by heavy bodies, from the beginning of their descent, are as the squares of the times; and that a body, projected in any direction not perpendicular to the horizon, describes a parabola. These were the beginnings of the doctrine of the motion of heavy bodies, which has been since carried to so great a height by Newton. In geometry, he invented the cycloid, or trochoid; though the properties of it were afterwards chiefly demonstrated by his pupil Torricelli. He invented the simple pendulum, and made use of it in his astronomical experiments: he had also thoughts of applying it to clocks; but did not execute that design: the glory of that invention was reserved for his son Vicenzio, who made the experiment at Venice in 1649; and Huygens afterward carried this invention to perfection. Of Galileo's invention also, was the machine, with which the Venetians render their Laguna fluid and navigable. He also discovered the gravíty of the air, and endeavoured to compare it with that of water; besides, opening up several other enquiries in natural philosophy. In short, he was not esteemed and followed by philosophers only, but was honoured by persons of the greatest distinction of all nations.</p><p>Galileo had scholars too that were worthy of so great a master, by whom the gravitation of the atmosphere was fully established, and its varying pressure accurately and conveniently measured, by the column of quicksilver of equal weight sustained by it in the barometrical tube. The elasticity of the air, by which it perpetually endeavours to expand itself, and, while it admits of condensation, resists in proportion to its density, was a phenomenon of a new kind (the common fluids having no such property), and was of the utmost importance to philosophy. These principles opened a vast <*>eld of new and useful knowledge, and explained a<cb/> great variety of phenomena, which had been accounted for before that time in a very absurd manner. It seemed as if the air, the fluid in which men lived from the beginning, had been then but first discovered. Philosophers were every where busy enquiring into its various properties and their effects: and valuable discoveries rewarded their industry. Of the great number who distinguished themselves on this occasion, may be mentioned Torricelli and Viviani in Italy, Pascal in France, Otto Guerick in Germany, and Boyle in England.</p><p>Galileo wrote a number of treatises, many of which were published in his life-time. Most of them were also collected after his death, and published by Mendessi in 2 vols 4to, under the title-of <hi rend="italics">L'Opere di Galileo Galilei Lynceo,</hi> in 1656. Some of these, with others of his pieces, were translated into English and published by Thomas Salisbury, in his Mathematical Collections, in 2 vols folio. A volume also of his letters to several learned men, and solutions of several problems, were printed at Bologna in 4to. His last disciple, Vincenzo Viviani, who proved a very eminent mathematician, methodized a piece of his master's, and published it under this title, <hi rend="italics">Quinto libro de gli Elententi d'Euclidi, &c;</hi> at Florence in 1674, 4to. Viviani published some more of Galileo's things, being Extracts from his letters to a learned Frenchman, where he gives an account of the works which he intended to have published, and a passage from a letter of Galileo dated at Arcetri, Oct. 30, 1635, to John Camillo, a mathematician of Naples, concerning the angle of contact. Besides all these, he wrote many other pieces, which were unfortunately lost through his wife's devotion; who, solicited by her confessor, gave him leave to peruse her husband's manuscripts; of which he tore and took away as many as he said were not sit to be published.</p></div1><div1 part="N" n="GALLERY" org="uniform" sample="complete" type="entry"><head>GALLERY</head><p>, in Architecture, a covered place in a building, much longer than broad; which is usually placed in the wings of the building, and serving to walk in, and to place pictures in. It denotes a little aisle, or walk, serving as a common passage to several rooms placed in a line, or row.</p><div2 part="N" n="Gallery" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gallery</hi></head><p>, in Fortification, a covered walk, or passage, made across the ditch of a besieged town, with timbers fastened in the ground and covered over,</p></div2></div1><div1 part="N" n="GALLON" org="uniform" sample="complete" type="entry"><head>GALLON</head><p>, an English measure of capacity, for things both liquid and dry, containing 2 pottles, or 4 quarts, or 8 pints. But those pints and quarts, and consequently the Gallon itself, are different, according to the quality of the things measured: the wine Gallon, for instance, contains 231 cubic inches, and holds 8lb 5 2/3 oz, avoirdupois, of pure water; the beer and ale Gallon contains 282 cubic inches, and holds 10lb 3 1/3 oz of water; and the Gallon dry measure, for grain, meal, &c, contains 268 4/5 cubic inches, and holds 9lb 11 1/2 oz of water.</p></div1><div1 part="N" n="GALLOPER" org="uniform" sample="complete" type="entry"><head>GALLOPER</head><p>, in Artillery, the name of a carriage serving for the very small guns, and having shafts so as to be drawn without a limber.</p><p>GAMING. See <hi rend="smallcaps">Chances</hi>, and <hi rend="smallcaps">Laws</hi> <hi rend="italics">of</hi> <hi rend="smallcaps">Chance.</hi></p></div1><div1 part="N" n="GARDECAUT" org="uniform" sample="complete" type="entry"><head>GARDECAUT</head><p>, or <hi rend="smallcaps">Guard du Cord</hi>, in a watch, is that which stops the fusee, when wound up, and for that end is driven up by the spring. Some call it Guard-cock; others Guard du Gut.<pb n="527"/><cb/></p><p>GARRISON <hi rend="smallcaps">Guns</hi>, such as are mounted and used in a Garrison, consisting of the following weights, viz the 42, 32, 24, 18, 12, 9, and 6 pounders; being made either of brass or iron. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=12 align=center" role="data"><hi rend="italics">Table of the Weight and Dimensions of Garrison Guns.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=6" role="data">Brass Garrison Guns.</cell><cell cols="1" rows="1" rend="colspan=6" role="data">Iron Garrison Guns.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Shot</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Length</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Weight.</cell><cell cols="1" rows="1" role="data">Shot</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Length</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Weight</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">lb</cell><cell cols="1" rows="1" role="data">f.</cell><cell cols="1" rows="1" role="data">in.</cell><cell cols="1" rows="1" role="data">Cw.</cell><cell cols="1" rows="1" role="data">qr.</cell><cell cols="1" rows="1" role="data">lb.</cell><cell cols="1" rows="1" role="data">lb.</cell><cell cols="1" rows="1" role="data">f.</cell><cell cols="1" rows="1" role="data">in.</cell><cell cols="1" rows="1" role="data">Cw.</cell><cell cols="1" rows="1" role="data">qr.</cell><cell cols="1" rows="1" role="data">lb.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row></table></p></div1><div1 part="N" n="GASSENDI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GASSENDI</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, one of the most celebrated philosophers France has produced, was born at Chantersier, about 3 miles from Digne in Provence, in the year 1592. When a child, he took great delight in gazing at the moon and stars whenever they appeared. This pleasure often drew him into bye-places, that he might feast his eyes freely and undisturbed; by which means his parents had him often to seek, not without many anxious fears and apprehensions. Inconsequence of this promising disposition, he was sent to the best schools, to cultivate it with the instructions of the first masters. He profited so well of these aids, that he was invited to be professor of rhetoric at Digne, before he was quite 16 years of age. After filling this office three years, upon the death of his master at Aix, he was appointed to succeed him as professor of philosophy. After a few years residence here, he composed his Paradoxical Exercitations; which coming to the hands of Nicholas Peiresc, that great patron of learning joined with Joseph Walter, prior of Valette, in promoting him; and, having entered into holy orders, he was first made canon of the church of Digne and doctor of divinity, and then warden or rector of the same church.</p><p>Gassendi's fondness for astronomy grew up with his years; and his reputation daily increasing, he was appointed the king's professor of mathematics at Paris in 1645. This institution being chiefly intended for astronomy, our author read lectures on that science to a crowded audience. However, he did not long enjoy this situation; for a dangerous cough and inflammation of the lungs obliged him, in 1647, to return to Digne for the benefit of his native air. Having thus, and by the intermission of his studies, recovered his health, he again returned to Paris in 1653; where, after first writing and publishing the lives of Tycho Brahe, Copernicus, Purbach, and Regiomontanus, in 1654 he again renewed his astronomical labours, with the design of completing the system of the heavens.<cb/> But while he was thus employed, too intensely for the feeble state of his health, he relapsed into his former disorder, under which, with the aid of too copious and numerous bleedings, by order of three physicians, he sunk in the year 1655, at 63 years of age.</p><p>Gassendi wrote against the metaphysical meditations of Des Cartes; and divided with that great man the philosophers of his time, almost all of whom were either Cartesians or Gassendists. To his knowledge in philosophy and mathematics, he joined profound erudition and deep skill in the languages. He wrote, 1. Three volumes on Epicurus's philosophy; and six others, which contain his own philosophy.—2. Astronomical Works.—3. The lives of Nicholas de Peirese, Epicurus, Copernicus, Tycho Brahe, Purbach, and Regiomontanus.—4. Epistles, and other treatises. All his works were collected together, and printed at Lyons in 1658, in 6 volumes folio.</p><p>Gassendi was the first person that saw the transit of Mercury over the sun, viz, Nov. 7, 1631; as Horrox first predicted and shewed the transit of Venus.—His library was large and valuable: to which he added an astronomical and philosophical apparatus, which, for their accuracy and magnitude, were purchased by the emperor Ferdinand the 3d.—It appears by his letters, printed in the 6th volume of his works, that he was often consulted by the most celebrated astronomers of his time, as<*>Kepler, Longomontanus, Snell, Hevelius, Galileo, Kircher, Bulliald, and others: and he has generally been esteemed one of the founders of the reformed philosophy, in opposition to the groundless hypotheses and empty subtleties of Aristotle and the schoolmen.</p><p>GAUGE-<hi rend="italics">Line,</hi> a line on the common Gauging rod, used for the purpose of gauging liquids. See <hi rend="smallcaps">Gauging</hi>-<hi rend="italics">Rod.</hi></p><p><hi rend="smallcaps">Gauge</hi>-<hi rend="italics">Point,</hi> of a solid measure, is the diameter of a circle, whose area is expressed by the same number as the solid content of that measure. Or it is the diameter of a cylinder, whose altitude is 1, and its content the same as of that measure.</p><p>Thus, the solid content of a wine gallon being 231 cubic inches; if a circle be conceived to contain so many square inches, its diameter will be 17.15; which is therefore the Gauge-point for wine measure. And an ale gallon containing 282 cubic inches; by the same rule, the Gauge-point for ale measure will be found to be 18.95. And after the same manner may the Gauge-point for any other measure be determined.</p><p>Hence it follows, that when the diameter of a cylinder in inches is equal to the Gauge-point in any measure, given likewise in inches, every inch in its length will contain an integer of the same measure. So in a cylinder whose diameter is 17.15 inches, every inch in height contains one entire gallon in wine measure; and in another, whose diameter is 18 95, every inch in length contains one ale gallon.</p></div1><div1 part="N" n="GAUGER" org="uniform" sample="complete" type="entry"><head>GAUGER</head><p>, an officer appointed by the commission. ers of excise, to Gauge, measure, or examine, all casks, tuns, pipes, barrels, hogsheads, of beer, wine, oil, &c.</p></div1><div1 part="N" n="GAUGING" org="uniform" sample="complete" type="entry"><head>GAUGING</head><p>, the art or act of measuring the capacities or contents of all kinds of vessels, and determining the quantity of fluids, or other matters contained in them. These are principally pipes, tuns, barrels, rundlets, and other casks; also backs, coolers, vats, &c.<pb n="528"/><cb/></p><p>As to the solid contents of all prismatical vessels, as cubes, parallelopipedons, cylinders, &c, they are found by multiplying the area of the base by their altitude. And the contents of all pyramidal bodies, and cones, are equal to 1 3d of the same.</p><p>In Gauging, it has been usual to divide casks into four varieties or forms, denominated as follows, from the supposed resemblance they bear to the frustums of solids of the same names: viz,</p><p>1. The middle frustum of a spheroid,</p><p>2. The middle frustum of a parabolic spindle,</p><p>3. The two equal frustums of a paraboloid,</p><p>4. The two equal frustums of a cone.</p><p>And particular rules, adapted to each of these forms, may be found in most books of Gauging, and in my Mensuration, p. 575 &c. But as the form is imaginary, and only guessed at, it hardly ever happens that a true solution is brought out in this way; beside which, it is very troublesome and inconvenient to have so many rules to put in practice. I shall therefore give here one rule only, from p. 592 of that book, which is not only general for all casks that are commonly met with, but quite easy, and very accurate, as having been often verified and proved by filling the casks with a true gallon measure.</p><p><hi rend="italics">General Rule.</hi> Add into one sum, 39 times the square of the bung diameter, 25 times the square of the head diameter, and 26 times the product of those diameters; multiply the sum by the length of the cask, and the product by the number .00034; then this last product divided by 9 will give the wine gallons, and divided by 11 will give the ale gallons. Or, is the content in inches; which being divided by 231 for wine gallons, or by 282 for ale gallons, will be the content.</p><p><hi rend="italics">For Ex.</hi> If the length of a cask be 40 inches, the bung diameter 32, and the head diameter 24. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Here</cell><cell cols="1" rows="1" role="data">   32<hi rend="sup">2</hi> × 39</cell><cell cols="1" rows="1" rend="align=right" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">39936</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and</cell><cell cols="1" rows="1" role="data">   24<hi rend="sup">2</hi> × 25 </cell><cell cols="1" rows="1" rend="align=right" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">14400</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and</cell><cell cols="1" rows="1" role="data">  32 × 24 × 26</cell><cell cols="1" rows="1" rend="align=right" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">19968</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">the sum</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">74304</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">multiplied by</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">and divid. by</cell><cell cols="1" rows="1" rend="align=right" role="data">114)</cell><cell cols="1" rows="1" rend="align=right" role="data">2972160</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">gives</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">26071</cell><cell cols="1" rows="1" role="data">cubic inches;</cell></row></table> this divided by 231 gives 112 wine gallons, or divided by 282 gives 92 ale gallons.</p><p>But the common practice of Gauging is performed mechanically, by means of the Gauging or Diagonal Rod, or the Gauging Sliding Rule, the description and use of which here follow.</p><div2 part="N" n="Gauging" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gauging</hi></head><p>, or <hi rend="italics">Diagonal, Rod,</hi> is a rod or rule adapted for determining the contents of casks, by measuring the diagonal only, viz the diagonal from the bung to the extremity of the opposite stave next the head. It is a square rule, having 4 sides or faces, being usually 4 feet long, and folding together by means of joints.</p><p>Upon one face of the rule is a scale of inches, for taking the measure of the diagonal; to these are adapted the areas, in ale gallons, of circles to the corresponding diameters, like the lines on the under sides<cb/> of the three slides in the sliding rule, described below. And upon the opposite face are two scales, of ale and wine gallons, expressing the contents of casks having the corresponding diagonals; and these are the lines which chiefly constitute the difference between this instrument and the sliding rule; for all the other lines upon it are the same with those in that instrument, and are to be used in the same manner.</p><p><hi rend="italics">To use the Diagonal Rod.</hi> Unfold the rod straight out, and put it in at the bung hole of the cask to be gauged, till its end arrive at the intersection of the head and opposite stave, or to the farthest possible distance from the bung-hole, and note the inches and parts cut by the middle of the bung; then draw out the rod, and look for the same inches and parts on the opposite face of it, and annexed to them are found the contents of the cask, both in ale and wine gallons.</p><p><hi rend="italics">For Ex.</hi> Let it be required to find, by this rod, the content of a cask whose diagonal measures 34.4 inches; which answers to the cask in the foregoing example, whose head and bung diameters are 32 and 24, and length 40 inches; for if to the square of 20, half the length, be added the square of 28, half the sum of the diameters, the square root of the sum will be 34.4 nearly.</p><p>Now, to this diagonal 34.4, corresponds, upon the rule, the content 91 ale gallons, or 111 wine gallons; which are but 1 less than the content brought out by the former general rule above given.</p><p><hi rend="smallcaps">Gauging</hi> <hi rend="italics">Rule,</hi> or <hi rend="italics">Sliding Rule,</hi> is a sliding rule particularly adapted to the purposes of Gauging. It is a square rule, of four faces or sides, three of which are furnished with sliding pieces running in grooves. The lines upon them are mostly logarithmic ones, or distances which are proportional to the logarithms of the numbers placed at the ends of them; which kind of lines was placed upon rulers, by Mr. Edmund Gunter, for expeditiously performing arithmetical operations, using a pair of compasses for taking off and applying the several logarithmic distances: but instead of the compasses, sliding pieces were added, by Mr. Thomas Everard, as more certain and convenient in practice, from whom this sliding rule is often called Everard's Rule. For the more particular description and uses of this rule, see my Mensuration, p. 564, 2d edition.</p><p>The writers on Gauging are, Beyer, Kepler, Dechales, Hunt, Everard, Dougherty, Shettleworth, Shirtcliffe, Leadbetter, &c.</p></div2></div1><div1 part="N" n="GAZONS" org="uniform" sample="complete" type="entry"><head>GAZONS</head><p>, in Fortification, turfs, or pieces of fresh earth covered with grass, cut in form of a wedge, about a foot long, and half a foot thick, to line or face the outside of works made of earth, to keep them up, and prevent their mouldering.</p></div1><div1 part="N" n="GELLIBRAND" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GELLIBRAND</surname> (<foreName full="yes"><hi rend="smallcaps">Henry</hi></foreName>)</persName></head><p>, professor of astronomy at Gresham-college, was born in London the 27th of Nov. 1597. He was sent to Trinity-college, Oxford, in 1615, and took his degree in arts 1619. He then entered into orders, and became curate of Chiddingstone in Kent. Afterwards, taking a great fancy to mathematics, by happening to hear one of Sir Henry Saville's lectures in that science, he immediately set himself to the close study of that noble science, and relinquished his fair prospects in the church. Contenting himself there-<pb n="529"/><cb/> fore with his private patrimony, which was now come into his hands by the death of his father, the same year he entered again a student at Oxford, making mathematics his sole employment. He made such proficiency in this science before he proceeded A. M., which was in 1623, that he drew the attention and intimate friendship of Mr. Henry Briggs, then lately removed from the geometry professorship in Gresham-college to that of Savilian professor of geometry at Oxford, by the founder Sir Henry Savile, and who, upon the death of Mr. Gunter, procured for our author the professorship of astronomy in Gresham-college, to which he was elected in the beginning of the year 1627. His friend Mr. Briggs dying in 1630, before he had finished the introduction to his Trigonometria Britannica, he recommended the completing and publishing of that work to our author. Gellibrand accordingly added a preface, and the application of the logarithms to plane and spherical trigonometry, &c, and the whole was printed at Gouda, under the care of Adrian Vlacq, in 1633.</p><p>While Mr. Gellibrand was preparing that work, he was brought into trouble in the high-commission court, by Dr. Laud, then bishop of London, on account of an almanac, published by William Beale, servant to Mr. Gellibrand, for the year 1631, with the approbation of his master. In this almanac, the popish saints, then usually put into calendars, were omitted, and the names of other saints and martyrs, mentioned in the Book of Martyrs, were placed in their stead, as they stand in Fox's calendars. This it seems gave offence to the bishop, and occasioned the prosecution. But when the cause came to be heard, it appeared that other almanacs of the same kind had formerly been printed; upon which, both master and man were acquitted by Abp. Abbot and the whole court, Laud only excepted; which was afterward made one of the articles against him on his own trial.</p><p>It seems Gellibrand was strongly attached to the old Ptolomaic system. For when he went over to Holland, about the printing of Briggs's book abovementioned, he had some discourse with Lansberg, an eminent brother astronomer in Zealand, who affirming that he was fully persuaded of the truth of the Copernican system; our author observes, “that this so styled a truth he should “receive as an hypothesis; and so be easily led on to “the consideration of the imbecility of man's appre“hension, as not able rightly to conceive of this ad“mirable opifice of God, or frame of the world, with“out falling foul of so great an absurdity:” so firmly was he fixed in his adherence to the Ptolomaic system. Gellibrand wrote several things after this, chiefly tending to the improvement of navigation, which would probably have been further advanced by him, had his life been continued longer; but he was untimely carried off by a fever, in 1636, at 39 years of age.</p><p>The character of Mr. Gellibrand is that of a plain, plodding, industrious, well-intentioned man, with little invention or genius. His writings are chiefly as below:</p><p>1. <hi rend="italics">Trigonometria Britannica;</hi> or the Doctrine of Triangles, being the 2d part of Briggs's work abovementioned.</p><p>2. A small Tract concerning the longitude.<cb/></p><p>3. A Discourse on the <hi rend="italics">Variation of the Magnetic Needle;</hi> annexed to Wright's Errors in Navigation detected.</p><p>4. <hi rend="italics">Institution Trigonometrical,</hi> with its application to astronomy and navigation; 8vo, 1635.</p><p>5. <hi rend="italics">Epitome of Navigation,</hi> with the necessary tables; 8vo.</p><p>6. Several manuscripts never published; as, The Doctrine of Eclipses.—A Treatise of Lunar Astronomy.—A Treatise of Ship-building, &c.</p></div1><div1 part="N" n="GEMINI" org="uniform" sample="complete" type="entry"><head>GEMINI</head><p>, a constellation of the northern hemisphere, one of the 48 old constellations, and the 3d in order of the zodiacal signs, Aries, Taurus, Gemini, &c. This constellation consists of two children, twins, called Castor and Pollux, and denoted by the mark, 11, being a rude drawing of the same.</p><p>This constellation was, more anciently, depicted by a couple of young kids, by the Egyptians and eastern nations, as denoting that part of the spring when these animals appear; but the Greeks altered them to two children, which some of them make to be Castor and Pollux, some of them again Hercules and Apollo, and others Triptolemus and Jasion; but the Arabians afterwards changed the figures into two peacocks, their religion not allowing them to paint or draw any human figure. Sir Isaac Newton thinks the figures had some reference to the Argonautic expedition.</p><p>The ancients attributed to every sign of the zodiac one of the principal deities for its tutelary power. Phœbus had the care of Gemini, and hence all the jargon of astrologers about the agreement of the sun and this constellation.</p><p>The stars in the sign Gemini are, in Ptolomy's catalogue 25, in Tycho's 25, in Hevelius's 38, and in the Britannic catalogue 85.</p></div1><div1 part="N" n="GENERATED" org="uniform" sample="complete" type="entry"><head>GENERATED</head><p>, is used by some mathematical writers for whatever is produced by arithmetical operation, or in geometry by the motion of other magnitudes. Thus 20 is the product Generated of 4 and 5; <hi rend="italics">ab</hi> that of <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> 4, 8, 16, &c, the powers generated of or from the root 2, and <hi rend="italics">a</hi><hi rend="sup">2</hi>, <hi rend="italics">a</hi><hi rend="sup">3</hi>, <hi rend="italics">a</hi><hi rend="sup">4</hi>, &c, those from the root <hi rend="italics">a.</hi> So also, a circle is Generated by the revolution of a line about one of its extremities, a cone by the rotation of a right-augled triangle about its perpendicular, a cylinder by the rotation of a rectangle about one of its sides, or, otherwise, by the motion of a circle in the direction of a right line, and keeping always parallel to itself.</p><p>GENERATING <hi rend="italics">Line</hi> or <hi rend="italics">Figure,</hi> in Geometry, is that which, by any kind of supposed motion, may generate, or produce, any other figure, plane, or solid.</p><p>Thus a line, according to Euclid, generates a circle; or a right-angled triangle, a cone &c; 2nd thus also Archimedes supposes his spirals to be generated by the motions of Generating points and lines; the figure thus generated, is called the <hi rend="italics">Generant.</hi></p><p>It is a general theorem in geometry, that the measure of any generant, or sigure produced by any kind of motion of any other figure, or Generating quantity, is equal to the product of this Generating quantity drawn into the length of the path described by its centre of gravity, whatever the kind of motion may be, whether rotatory, or direct, &c.<pb n="530"/><cb/></p><p>In the modern analysis, or ssuxions, all sorts of quantities are considered as Generated by some such motion, and the quantity hereby generated is called a Fluent.</p></div1><div1 part="N" n="GENERATION" org="uniform" sample="complete" type="entry"><head>GENERATION</head><p>, in Mathematics, is used for the formation or production of any geometrical figure, or other quantities. Such as of the figures mentioned in the foregoing articles, or the Generation of equations, curves, solids, &c.</p></div1><div1 part="N" n="GENESIS" org="uniform" sample="complete" type="entry"><head>GENESIS</head><p>, in Geometry, means much the same as Generation mentioned above, being the formation of a line, surface, or solid, by the motion or flux of a point, line, or surface; as of a globe by the rotation of a semi-circle about its diameter, &c.</p><p>In the Genesis of figures, the line or surface which moves, is called the <hi rend="italics">Describent;</hi> and the line round which, or according to which, the revolution or other motion is made, the <hi rend="italics">Dirigent.</hi></p></div1><div1 part="N" n="GENETHLIACI" org="uniform" sample="complete" type="entry"><head>GENETHLIACI</head><p>, in Astrology, are persons who erect horoscopes, or pretend to foretell what shall befall a person, by means of the stars which presided at his nativity.</p><p>The ancients called them Chaldæi, and by the general name Mathematici: accordingly, the several civil and canon laws, which we find made against the mathematicians, only respect the Genethliaci, or astrologers.</p><p>They were expelled Rome, by a formal decree of the senate; and yet found so much protection from the credulity of the people, that they remained in the city unmolested.</p><p>Antipater and Archinapolus have shewn that Genethliology should rather be founded on the time of the conception than on that of the birth.</p></div1><div1 part="N" n="GEOCENTRIC" org="uniform" sample="complete" type="entry"><head>GEOCENTRIC</head><p>, is said of a planet or its orbit, to denote its having the earth for its centre. The moon alone is properly geocentric. And yet the motions of all the planets may be considered in respect of the earth, or as they appear from the earth, and thence called their Geocentric motions.—Hence also the terms Geocentric place, or latitude, or longitude, &c, as explained below.</p><p><hi rend="smallcaps">Geocentric</hi> <hi rend="italics">Place,</hi> of a planet, is the place where it appears to us, from the earth; or it is a point in the ecliptic, to which a planet, seen from the earth, is referred.</p><p><hi rend="smallcaps">Geocentric</hi> <hi rend="italics">Latitude,</hi> of a planet, is its latitude as seen from the earth; or the inclination of a line, connecting the planet and the earth, to the plane of the earth's (or true) ecliptic. Or it is the angle which the said line (connecting the planet and the earth) makes with a line drawn to meet a perpendicular let fall from the planet to the plane of the ecliptic.</p><p><hi rend="smallcaps">Geocentric</hi> <hi rend="italics">Longitude,</hi> of a planet, is the distance measured on the ecliptic, in the order of the signs, between the Geocentric place and the first point of Aries.</p></div1><div1 part="N" n="GEODESIA" org="uniform" sample="complete" type="entry"><head>GEODESIA</head><p>, is properly that part of practical geometry that teaches how to divide or lay out lands and fields, among several owners.</p><p><hi rend="smallcaps">Geodesia</hi> is also applied, by some writers, to all measurements in the field, and as synonymous with surveying.</p><p><hi rend="smallcaps">Geodesia</hi> is defined by Vitalis, as the art of mea-<cb/> suring surfaces and solids, not by imaginary lines, as is done in geometry, but by sensible and visible things; on by the sun's rays, &c.</p></div1><div1 part="N" n="GEOGRAPHER" org="uniform" sample="complete" type="entry"><head>GEOGRAPHER</head><p>, a person skilled in Geography.</p></div1><div1 part="N" n="GEOGRAPHICAL" org="uniform" sample="complete" type="entry"><head>GEOGRAPHICAL</head><p>, something relating to Geography, as</p><p><hi rend="smallcaps">Geographical</hi> <hi rend="italics">Mile,</hi> which is the sea-mile or minute, being the 60th part of a degree of a great circle.</p><p><hi rend="smallcaps">Geographical</hi> <hi rend="italics">Table.</hi> See <hi rend="smallcaps">Map.</hi></p></div1><div1 part="N" n="GEOGRAPHY" org="uniform" sample="complete" type="entry"><head>GEOGRAPHY</head><p>, the science that teaches and explains the nature and properties of the earth, as to its figure, place, magnitude, motions, celestial appearances, &c, with the various lines, real or imaginary, on its surface.</p><p>Geography is distinguished from Cosmography, as a part from the whole; this latter considering the whole visible world, both heaven and earth. And from Topography and Chorography, it is distinguished, as the whole from a part.</p><p>Golnitz considers Geography as either exterior or interior: but Varenius more justly divides it into General and Special; or Universal and Particular.</p><p><hi rend="italics">General</hi> or <hi rend="italics">Universal</hi> <hi rend="smallcaps">Geography</hi>, is that which considers the earth in general, without any regard to particular countries, or the affections common to the whole globe: as its sigure, magnitude, motion, land, sea, &c.</p><p><hi rend="italics">Special</hi> or <hi rend="italics">Particular</hi> <hi rend="smallcaps">Geography</hi>, is that which contemplates the constitution of the several particular regions, or countries; their bounds, figure, climate, seasons, weather, inhabitants, arts, customs, language, &c.</p><p><hi rend="italics">History of</hi> <hi rend="smallcaps">Geography.</hi> The study and practice of Geography must have commenced at very early ages of the world. By the accounts we have remaining, it seems this science was in use among the Babylonians and Egyptians, from whom it passed to the Greeks first of any Europeans, and from these successively to the Romans, the Arabians, and the western nations of Europe. Herodotus says the Greeks first learned the pole, the gnomon, and the 12 divisions of the day, from the Babylonians. But Pliny and Diogenes Laertius assert, that Thales of Miletus, in the 6th century before Christ, first found out the passage of the sun from tropic to tropic, and it is said was the author of two books, the one on the tropic, and the other on the equinox; both probably determined by means of the gnomon; whence he was led to the discovery of the four seasons of the year, which are determined by the equinoxes and solstices; all which however it is likely he learned of the Egyptians, as well as his division of the year into 365 days. This it is said was invented by the second Mercury, surnamed Trismegistus, who, according to Eusebius, lived about 50 years after the Exodus. Pliny expressly says that this discovery was made by observing when the shadow returned to its marks; a clear proof that it was done by the gnomon. It is farther said that Thales constructed a globe, and represented the land and sea upon a table of brass. Farther that Anaximander, a disciple of Thales, first drew the figure of the earth upon a globe; and that Hecate, Democritus, Eudoxus, and others, formed Geogra-<pb n="531"/><cb/> phical maps, and brought them into common use in Greece.</p><p>Meton and Euctemon observed the summer solstice at Athens, on the 27th of June 432 years before Christ, by watching narrowly the shadow of the gnomon, with the design of fixing the beginning of their cycle of 19 years.</p><p>Timocharis and Aristillus, who began their observations about 295 B. C., it seems first attempted to fix the latitudes and longitudes of the fixed stars, by considering their distances from the equator, &c. One of their observations gave rise to the discovery of the precession of the equinoxes, which was first remarked by Hipparchus about 150 years after; who also made use of their method, for delineating the parallels of latitude and the meridians, on the surface of the earth; thus laying the foundation of this science as it now appears.</p><p>The latitudes and longitudes, thus introduced by Hipparchus, were not however much attended to till Ptolomy's time. Strabo, Vitruvius, and Pliny, have all of them entered into a minute geographical description of the situation of places, according to the length of the shadows of the gnomon, without noticing the longitudes and latitudes.</p><p>Maps at first were little more than rude outlines, and topographical sketches of different countries. The earliest on record were those of Sesostris, mentioned by Eustathius; who says, that “this Egyptian king, having traversed great part of the earth, recorded his march in maps, and gave copies of them not only to the Egyptians, but to the Scythians, to their great astonishment.” Some have imagined with much probability, that the Jews made a map of the Holy Land, when they gave the different portions to the nine tribes at Shiloh: for Joshua tells us that they were sent to walk through the land, and that they <hi rend="italics">described it in seven parts in a book;</hi> and Josephus relates that when Joshua sent out people from the different tribes to measure the land, he gave them, as companions, persons well skilled in geometry, who could not be mistaken in the truth.</p><p>The first Grecian map on record, was that of Anaximander, mentioned by Strabo, lib. 1, p. 7, supposed to be the one referred to by Hipparchus under the designation of the <hi rend="italics">ancient map.</hi> Herodotus minutely describes a map made by Aristagoras tyrant of Miletus, which will serve to give some idea of the maps of those times. He relates, that Aristagoras shewed it to Cleomenes king of Sparta, to induce him to attack the king of Persia at Susa, in order to restore the Ionians to their ancient liberty. It was traced upon brass or copper, and seems to have been a mere itinerary, containing the route through the intermediate countries which were to be traversed in that march, with the rivers Halys, the Euphrates, and Tigris, which Herodotus mentions as necessary to be crossed in that expedition. It contained one straight line called the <hi rend="italics">Royal Road</hi> or <hi rend="italics">Highway,</hi> which took in all the stations or places of encampment from Sardis to Susa; being 111 in the whole journey, and containing 13,500 stadia, or 1687 1/2 Roman miles of 5000 feet each.</p><p>These itinerary maps of the places of encampment were indispensably necessary in all armies and marches;<cb/> and indeed war and navigation seem to be the two grand causes of the improvements both in Geography and astronomy. Athenæus quotes Bæton as author of a work intitled, <hi rend="italics">The encampments of Alexander's march;</hi> and likewife Amyntas to the same purpose. Pliny observes that Diognetus and Bæton were the surveyors of Alexander's marches, and then quotes the exact number of miles according to their mensuration; which he afterwards confirms by the letters of Alexander himself. The same author also remarks that a copy of this great monarch's surveys was given by Xenocles his treasurer to Patrocles the geographer, who was admiral of the fleets of Selencus and Antiochus. His book on geography is often quoted both by Strabo and Pliny; and it seems that this author furnished Eratosthenes with the principal materials for constructing his map of the oriental part of the world.</p><p>Eratosthenes first attempted to reduce Geography to a regular system, and introduced a regular parallel of latitude, which began at the straits of Gibraltar, passed eastwards through the isle of Rhodes, and so on to the mountains of India, noting all the intermediate places through which it passed. In drawing this line, he was not regulated by the same latitude, but by observing where the longest day was 14 hours and a half, which Hipparchus afterwards determined was the latitude of 36 degrees.</p><p>This first parallel through Rhodes was ever after considered with a degree of preference, in constructing all the ancient maps; and the longitude of the then known world was often attempted to be measured in stadia and miles, according to the extent of that line, by many succeeding geographers.</p><p>Eratosthenes soon after attempted not only to draw other parallels of latitude, but also to trace a meridian at right angles to these, passing through Rhodes and Alexandria, down to Syene and Meroë; and at length he undertook the arduous task of determining the circumference of the globe, by an actual measurement of a segment of one of its great circles. To find the magnitude of the earth, is indeed a problem which has engaged the attention of astronomers and geographers ever since the spherical sigure of it was known. It seems Anaximander was the first among the Greeks who wrote upon this subject. Archytas of Tarentum, a Pythagorean, famous for his skill in mathematics and mechanics, also made some attempts in this way; and Dr. Long conjectures that these are the authors of the most ancient opinion that the circumference of the earth is 400,000 stadia: and Archimedes makes mention of the ancients who estimated the circumference of the earth at only 30,000 stadia.</p><p>As to the methods of measuring the circumference of the earth, it would seem, from what Aristotle says in his treatise De Cœlo, that they were much the same as those used by the moderns, deficient only in the accuracy of the instruments. That philosopher there says, that different stars pass through our zenith, according as our situation is more or less northerly; and that in the southern parts of the earth stars come above our horizon, which are no longer visible if we go northward. Hence it appears that there are two ways of measuring the circumference of the earth; one by observing sta<*> which pass through the zenith of one place, and do<pb n="532"/><cb/> not pass through that of another; the other, by observing some stars which come above the horizon of one place, and are observed at the same time to be in the horizon of another. The former of these methods, which is the best, was followed by Eratofthenes at Alexandria in Egypt, 250 years before Christ. He knew that at the summer solstice, the sun was vertical to the inhabitants of Syene, a town on the confines of Ethiopia, under the tropic of cancer, where they had a well made to observe it, at the bottom of which the rays of the sun fell perpendicularly the day of the summer solstice: he observed by the shadow of a wi<*>e set perpendicularly in an hemispherical bason, how far the sun was on that day at noon distant from the zenith of Alexandria; when he found that distance was equal to the 50th part of a great circle in the heavens. Then supposing Syene and Alexandria under the same meridian, he inferred that the distance between them was the 50th part of a great circle upon the earth; and this distance being by measure 5000 stadia, he concluded that the whole circumference of the earth was 250,000 stadia. But as this number divided by 360 would give 694 4/<*> stadia to a degree, either Eratosthenes himself or some of his followers assigned the round number 700 stadia to a degree; which multiplied by 360, makes the circumference of the earth 252,000 stadia; whence both these measures are given by different authors, as that of Eratosthenes.</p><p>In the time of Pompey the Great, Posidonius determined the measure of the circumference of the earth by the 2d method above hinted by Aristotle, viz, the horizontal observations. Knowing that the star called Canopus was but just visible in the horizon of Rhodes, and at Alexandria finding its meridian height was the 48th part of a great circle in the heavens, or 7 1/2 deg., answering to the like quantity of a circle on the earth: Then, supposing these two places under the same meridian, and the distance between them 5000 stadia, the circumference of the earth will be 240,000 stadia; which is the first measure of Posidonius. But according to Strabo, Posidonius made the measure of the earth to be 180,000 stadia, at the rate of 500 stadia to a degree. The reason of this difference is thought to be, that Eratosthenes measured the distance between Rhodes and Alexandria, and found it only 3750 stadia: taking this for a 48th part of the earth's circumference, which is the measure of Posidonius, the whole circumference will be 180,000 stadia. This measure was received by Marinus of Tyre, and is usually ascribed to Ptolomy. But this measurement is subject to great uncertainty, both on account of the great refraction of the stars near the horizon, the difficulty of measuring the distance at sea between Rhodes and Alexandria, and by supposing those places under the same meridian, when they are really very different.</p><p>Several geographers afterwards made use of the different heights of the pole in distant places under the same meridian, to find the dimensions of the earth. About the year 800, the khalif Almamun had the distance measured between two places that were 2 degrees asunder, and under the same meridian in the plains of Sinjar near the Red Sea. And the result was, that the degree at one time was found equal to 56 miles, and at another 56 1/3 or 56 2/3 miles.<cb/></p><p>The next attempt to find out the circumference of the earth, was in 1525, by Fernelius, a learned philosopher of France. For this purpose, he took the height of the pole at Paris, going from thence directly northwards, till he came to the place where the height of the pole was one degree more than at that city. The length of the way was measured by the number of revolutions made by one of the wheels of his carriage; and after proper allowances for the declivities and turnings of the road, he concluded that 68 Italian miles were equal to a degree on the earth.</p><p>According to these methods many other measurements of the earth's circumference have since that time been made, with much greater accuracy: a particular account of which is given under the article <hi rend="smallcaps">Degree.</hi></p><p>Though the maps of Eratosthenes were the best of his time, they were yet very imperfect and inaccurate. They contained little more than the states of Greece, and the dominions of the successors of Alexander, digested according to the surveys abovementioned. He had indeed seen, and has quoted, the voyages of Pythias into the great Atlantic ocean, which gave him some faint idea of the western parts of Europe; but so imperfect, that they could not be realized into the outlines of a chart. Strabo says he was very ignorant of Gaul, Spain, Germany, and Britain; and he was equally ignorant of Italy, the coasts of the Adriatic, Pontus, and all the countries towards the north.</p><p>Such was the state of Geography, and the nature of the maps, before the time of Hipparchus. He made a closer connection between Geography and astronomy, by determining the latitudes and longitudes from celestial observations.</p><p>War has usually been the occasion of making or improving the maps of countries; and accordingly Geography made great advances from the progress of the Roman arms. In all the provinces occupied by that people, camps were every where constructed at proper intervals, and good roads made for communication between them; and thus civilization and surveying were carried on according to system, through the whole extent of that large empire. Every new war produced a new survey and itinerary of the countries where the scenes of action passed; so that the materials of Geography were accumulated by every additional conquest. Polybius says, that at the beginning of the second Punic war, when Hannibal was preparing his expedition against Rome, the countries through which he was to pass were carefully measured by the Romans. And Julius Cæsar caused a general survey of the Roman Empire to be made, by a decree of the senate. Three surveyors had this task assigned them, which they completed in 25 years. The Roman itineraries that are still extant, also shew what care and pains they had been at in making surveys in all the different provinces of their empire; and Pliny has filled the 3d, 4th, and 5th books of his Natural History with the Geographical distances that were thus measured. Other maps are also still preserved, known by the name of the Pentingerian Tables, published by Welser and Bertius, which give a good specimen of what Vegetius calls the <hi rend="italics">Itinera Picta,</hi> for the better direction of their armies in their march.</p><p>The Roman empire had been enlarged to its greatest extent, and all its provinces well known and surveyed,<pb n="533"/><cb/> when Ptolomy, about 150 years after Christ, composed his system of Geography. The chief materials he employed in composing this work, were the proportions of the gnomon to its shadow, taken by different astronomers at the times of the equinoxes and solstices; calculations founded on the length of the longest days; the measured or computed distances of the principal roads contained in their surveys and itineraries; and the various reports of travellers and navigators. All these were compared together, and digested into one uniform body or system; and afterwards were translated by him into a new mathematical language, expressing the different degrees of latitude and longitude, after the invention of Hipparchus, which had been neglected for 250 years.</p><p>Ptolomy's system of Geography, notwithstanding it was still very imperfect, continued in vogue till the last three or four centuries, within which time the great improvements in astronomy, the many discoveries of new countries by voyagers, and the progress of war and arms, have contributed to bring it to a very considerable degree of perfection; the particulars of which will be found treated under their respective articles in this work.</p><p>Among the moderns, the chief authors on the subject of Geography are Johannes de Sacrobosco, or John Hallifax, who wrote a treatise on the sphere; Sebastian Munster, in his Cosmographia Universalis, in 1559; Clavius, on the sphere of Sacrobosco; Piccioli's Geographia et Hydrographia Reformata; Weigelius's Speculum Terræ; De Chales's Geography, in his Mundus Mathematicus; Cellarius's Geography; Cluverius's Introductio in Universam Geographiam; Leibnecht's Elementa Geographiæ generalis; Stevenius's Compendium Geographicum; Wolfius's Geographia, in his Elementa Matheseos; Busching's New System of Geography; Gordon's, Salmon's, and Guthrie's Grammars; and above all, Varenius's Geographia generalis, with Jurin's additions, the most scientific and systematical of any.</p></div1><div1 part="N" n="GEOMETER" org="uniform" sample="complete" type="entry"><head>GEOMETER</head><p>, more properly <hi rend="smallcaps">Geometrician;</hi> which see.</p></div1><div1 part="N" n="GEOMETRICAL" org="uniform" sample="complete" type="entry"><head>GEOMETRICAL</head><p>, something that has a relation to Geometry, or done after the manner, or by the means of Geometry. As, a Geometrical construction, a Geometrical curve, a Geometrical demonstration, genius, line, method, Geometrical strictness, &c.</p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Consiruction,</hi> of an equation, is the drawing of lines and figures, so as to express by them the same general property and relation, as are denoted by the algebraical equation. See <hi rend="smallcaps">Construction</hi> <hi rend="italics">of Equations.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Curve</hi> or <hi rend="italics">Line,</hi> called also an <hi rend="italics">Algebraical one,</hi> is that in which the relations between the abscisses and ordinates may be expressed by a finite algebraical equation. See <hi rend="smallcaps">Algebraical</hi> <hi rend="italics">Curves.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Lines,</hi> as observed by Newton, are distinguished into classes, orders, or genera, according to the number of the dimensions of the equation that expresses the relation between the ordinates and abscisses; or, which comes to the same thing, according to the number of points in which they may be cut by a right line.</p><p>Thus, a line of the first order, is a right line, since it<cb/> can be only once cut by another right line, and is expressed by the simple equation : those of the 2d, or quadratic order, will be the circle, and the conic sections, since all of these may be cut in two points by a right line, and expressed by the equation : those of the 3d or cubic order, will be such as may be cut in 3 points by a right line, whose most general equation is ; as the cubical and Neilian parabola, the cissoid, &c. And a line of an infinite order, is that which a right line may cut in infinite points; as the spiral, the cycloid, the quadratrix, and every line that is generated by the infinite revolutions of a radius, or circle, or wheel, &c.</p><p>In each of those equations, <hi rend="italics">x</hi> is the absciss, <hi rend="italics">y</hi> its corresponding ordinate, making any given angle with it; and <hi rend="italics">a, b, c,</hi> &c, are given or constant quantities, affected with their signs + and -, of which one or more may vanish, be wanting or equal to nothing, provided that by such defect the line or equation does not become one of an inferior order.</p><p>It is to be noted that a curve of any kind is denominated by a number next less than the line of the same kind: thus, a curve of the 1st order, (because the right line cannot be reckoned among curves) is the same with a line of the 2d order; and a curve of the 2d kind, the same with a line of the 3d order; &c.</p><p>It is to be observed also, that it is not so much the equation, as the construction or description, that makes any curve, geometrical, or not. Thus, the circle is a geometrical line, not because it may be expressed by an equation, but because its description is a postulate: and it is not the simplicity of the equation, but the easiness of the description, that is to determine the choice of the lines for the construction of a problem. The equation that expresses a parabola, is more simple than that which expresses a circle; and yet the circle, by reason of its more simple construction, is admitted before it. Again, the circle and the conic sections, with respect to the dimensions of the equations, are of the same order; and yet the circle is not numbered with them in the construction of problems, but by reason of its simple description is depressed to a lower order, viz, that of a right line; so that it is not improper to express that by a circle, which may be expressed by a right line; but it is a fault to construct that by the conic sections, which may be constructed by a circle.</p><p>Either, therefore, the law must be taken from the dimensions of equations, as observed in a circle, and so the distinction be taken away between plane and solid problems: or the law must be allowed not to be strictly observed in lines of superior kinds, but that some, by reason of their more simple description, may be preferred to others of the same order, and be numbered with lines of inferior orders.</p><p>In constructions that are equally Geometrical, the most simple are always to be preferred: and this law is so universal as to be without exception. But algebraical expressions add nothing to the simplicity of the construction; the bare descriptions of the lines here are only to be considered; and these alone were considered by those geometricians who joined a circle with a right<pb n="534"/><cb/> line. And as these are easy or hard, the construction becomes easy or hard: and therefore it is foreign to the nature of the thing, from any other circumstance to establish laws relating to constructions.</p><p>Either, therefore, with the ancients, we must exclude all lines beside the circle, and perhaps the conic sections, out of geometry; or admit all, according to the simplicity of the description. If the trochoid were admitted into geometry, by means of it we might divide an angle in any given ratio; would it be right therefore to blame those who would make use of this line to divide an angle in the ratio of one number to another; and contend, that you must make use only of such lines as are defined by equations, and therefore not of this line, which is not so defined? If, when an angle is proposed to be divided, for instance, into 10001 parts, we should be obliged to bring a curve defined by an equation of more than 100 dimensions to do the business; which nobody could describe, much less understand; and should prefer this to the trochoid, which is a line well known, and easily described by the motion of a wheel, or circle: who would not see the absurdity?</p><p>Either therefore the trochoid is not to be admitted at all in geometry; or else, in the construction of problems, it is to be preferred to all lines of a more difficult description; and the reason is the same for other curves. Hence the trisections of an angle by a conchoid, which Archimedes in his Lemmas, and Pappus in his Collections, have preferred to the inventions of all others in this case, must be allowed as good; because we must either exclude all lines, beside the circle and right line, out of geometry, or admit them according to the simplicity of their descriptions; in which case the conchoid yields to none, except the circle. Equations are expressions of arithmetical computation, and properly have no place in geometry, excepting so far as quantities truly Geometrical (that is, lines, surfaces, solids, and proportions) may be said to be some equal to others. Multiplications, divisions, and such like computations, are newly received into Geometry, and that unwarily, and contrary to the first design of this science. For whoever considers the construction of problems by a right line and a circle, found out by the first geometricians, will easily perceive that geometry was introduced, that by drawing lines, we might easily avoid the tediousness of computation. For which reason the two sciences ought not to be confounded together: the ancients so carefully distinguished between them, that they never introduced arithmetical terms into geometry; and the moderns, by confounding them, have lost the simplicity in which all the elegance of geometry consists. In short, that is arithmetically more simple, which is determined by the more simple equations; but that is Geometrically more simple, which is determined by the more simple drawing of lines; and in geometry that ought to be reckoned best, which is geometrically most simple. Newton's Arith. Univers. appendix. See <hi rend="smallcaps">Curves.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Locus,</hi> or <hi rend="italics">Place,</hi> called also simply <hi rend="italics">Locus,</hi> is the path or track of some certain Geometrical determination, in which it always falls. See <hi rend="smallcaps">Locus.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Medium.</hi> See <hi rend="smallcaps">Medium.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Method of the Ancients.</hi> The ancients established the higher parts of their geometry on<cb/> the same principles as the elements of that science, by demonstrations of the same kind: and they were careful not to suppose any thing done, till by a previous problem they had shewn that it could be done by actually performing it. Much less did they suppose any thing to be done that cannot be conceived; such as a line or series to be actually continued to infinity, or a magnitude diminished till it become infinitely less than what it is. The elements into which they resolved magnitudes were sinite, and such as might be conceived to be real. Unbounded liberties have of late been introduced; by which geometry, which ought to be perfectly clear, is filled with mysteries. Maclaurin's Fluxions, Introd. p. 39.</p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Pace,</hi> is a measure of 5 feet long.</p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Plan,</hi> in Architecture. See <hi rend="smallcaps">Plan.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Plane.</hi> See <hi rend="smallcaps">Plane.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Progression,</hi> a progression in which the terms have all successively the same ratio: as 1, 2, 4, 8, 16, &c, where the common ratio is 2.</p><p>The general and common property of a Geometrical progression is, that the product of any two terms, or the square of any one single term, is equal to the product of every other two terms that are taken at an equal distance on both sides from the former. So of these terms, 1, 2, 4, 8, 16, 32, 64, &c, .</p><p>In any Geometrical Progression, if <hi rend="italics">a</hi> denote the least term, <hi rend="italics">z</hi> the greatest term, <hi rend="italics">r</hi> the common ratio, <hi rend="italics">n</hi> the number of the terms, <hi rend="italics">s</hi> the sum of the series, or all the terms; then any of these quantities may be found from the others, by means of these general values, or equations, viz, . When the series is infinite, then the least term a is nothing, and the sum . See also P<hi rend="smallcaps">ROGRESSION.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Proportion,</hi> called also simply <hi rend="italics">Proportion,</hi> is the similitude or equality of ratios.</p><p>Thus, if , or , the terms <hi rend="italics">a, b, c, d</hi> are in Geometrical Proportion; also 6, 3, 14, 7, are in Geometrical Proportion, because , or .<pb n="535"/><cb/></p><p>In a Geometrical Proportion, the product of the extremes, or 1st and 4th terms, is equal to the product of the means, or the 2d and 3d terms: so <hi rend="italics">ad</hi> = <hi rend="italics">bc,</hi> and . See <hi rend="smallcaps">Proportion.</hi></p><p><hi rend="smallcaps">Geometrical</hi> <hi rend="italics">Solution,</hi> of a problem, is when the problem is directly resolved according to the strict rules and principles of geometry, and by lines that are truly Geometrical. This expression is used in contradistinction to an arithmetical, or a mechanical, or instrumental solution; the problem being resolved only by a ruler and compasses.</p><p>The same term is likewise used in opposition to all indirect and inadequate kinds of solutions, as by approximation, infinite series, &c. So, we have no Geometrical way of finding the quadrature of the circle, the duplicature of the cube, or two mean proportionals; though there are mechanical ways, and others, by infinite series, &c.</p><p>Pappus informs us, that the ancients endeavoured in vain to trisect an angle, and to find out two mean proportionals, by means of the right line and circle. Afterwards they began to consider the properties of several other lines; as the conchoid, the cissoid, and the conic sections; and by some of these they endeavoured to resolve some of those problems. At length, having more thoroughly examined the matter, and the conic sections being received into geometry, they distinguished Geometrical problems and solutions into three kinds; viz,</p><p>1. <hi rend="italics">Plane</hi> ones, which, deriving their origin from lines on a plane, may be properly resolved by a right line and a circle.—2. <hi rend="italics">Solid</hi> ones, which are resolved by lines deriving their original from the consideration of a solid; that is, of a cone.—3. <hi rend="italics">Linear</hi> ones, to the solution of which are required lines more compounded.</p><p>According to this distinction, we are not to resolve solid problems by other lines than the conic sections; especially if no other lines beside the right line, circle, and the conic sections, must be received into geometry.</p><p>But the moderns, advancing much farther, have received into geometry all lines that can be expressed by equations; and have distinguished, according to the dimensions of the equations, those lines into classes or orders; and have laid it down as a law, not to construct a problem by a line of a higher order, that may be constructed by one of a lower.</p></div1><div1 part="N" n="GEOMETRICIAN" org="uniform" sample="complete" type="entry"><head>GEOMETRICIAN</head><p>, a person skilled in Geometry.</p></div1><div1 part="N" n="GEOMETRY" org="uniform" sample="complete" type="entry"><head>GEOMETRY</head><p>, the science or doctrine of local extension, as of lines, surfaces, and solids, with that of ratios, &c.</p><p>The name Geometry literally signifies measuring of the earth, as it was the necessity of measuring the land that first gave occasion to contemplate the principles and rules of this art; which has since been extended to numberless other speculations; insomuch that together with arithmetic, Geometry forms now the chief foundation of all the mathematics.</p><p>Herodotus (lib. 2), Diodorus (lib 1), Strabo (lib. 17), and Proclus, ascribe the invention of Geometry to the Egyptians, and assert that the annual inundations of the Nile gave occasion to it; for those waters bearing away the bounds and land-marks of estates and farms, covering the face of the ground uniformly with<cb/> mud, the people, say they, were obliged every year to distinguish and lay out their lands by the consideration of their figure and quantity; and thus by experience and habit they formed a method or art, which was the origin of Geometry. A farther contemplation of the draughts of figures of fields thus laid down, and plotted in proportion, might naturally lead them to the discovery of some of their excellent and wonderful properties; which speculation continually improving, the art continually gained ground, and made advances more and more towards perfection.</p><p>Josephus however seems to ascribe the invention to the Hebrews: and others of the ancients make Mercury the inventor. Polyd. Virgil, de Invent. Rer. lib. 1, cap. 18.</p><p>From Egypt, this science passed into Greece, being carried thither by Thales; where it was much cultivated and improved by himself, as also by Pythagoras, Anaxagoras of Clazomene, Hippocrates of Chios, and Plato, who testified his conviction of the necessity and importance of Geometry to the successful study of philosophy, by this inscription over the door of his academy, <hi rend="italics">Let no one ignorant of Geometry enter here.</hi> Plato thought the word Geometry too mean a name for this science, and substituted instead of it the more extensive name of <hi rend="italics">Mensuration;</hi> and after him others gave it the title of <hi rend="italics">Pantometry.</hi> But even these are now become too scanty in their import, fully to comprehend its extent; for it not only inquires into, and demonstrates the quantities of magnitudes, but also their qualities, as the species, figures, ratios, positions, transformations, descriptions, divisions, the finding of their centres, diameters, tangents, asymptotes, curvature, &c. Some again define it as the science of inquiring, inventing, and demonstrating all the affections of magnitude. And Proclus calls it the knowledge of magnitudes and figures, with their limitations; as also of their ratios, affections, positions, and motions of every kind.</p><p>About 50 years after Plato, lived Euclid, who collected together all those theorems which had been invented by his predecessors in Egypt and Greece, and digested them into 15 books, called the Elements of Geometry; demonstrating and arranging the whole in a very accurate and perfect manner. The next to Euclid, of those ancient writers whose works are extant, is Apollonius Pergæus, who flourished in the time of Ptolomy Euergetes, about 230 years before Christ, and about 100 years after Euclid. He was author of the first and principal work on Conic Sections; on account of which, and his other accurate and ingenious geometrical writings, he acquired from his patron the emphatical appellation of <hi rend="italics">The Great Geometrician.</hi> Contemporary with Apollonius, or perhaps a few years before him, flourished Archimedes, celebrated for his mechanical inventions at the siege of Syracuse, and not less so for his very many ingenious Geometrical compositions.</p><p>We can only mention Eudoxus of Cnidus, Archytas of Tarentum, Philolaus, Eratosthenes, Aristarchus of Samos, Dinostratus, the inventor of the quadratrix, Menechmus, his brother and the disciple of Plato, the two Aristeus's, Conon, Thracidius, Nicoteles, Leon, Theudius, Hermotimus, Hero, and Nicomedes the in-<pb n="536"/><cb/> ventor of the conchoid: besides whom, there are many other ancient geometricians, to whom this science has been indebted.</p><p>The Greeks continued their attention to it, even after they were subdued by the Romans. Whereas the Romans themselves were so little acquainted with it, even in the most flourishing time of their republic, that Tacitus informs us they gave the name of mathematicians to those who pursued the chimeras of divination and judicial astrology. Nor does it appear they were more disposed to cultivate Geometry during the decline, and after the fall of the Roman empire. But the case was different with the Greeks; among whom are found many excellent Geometricians since the commencement of the Christian era, and after the translation of the Roman empire. Ptolomy lived under Marcus Aurelius; and we have still extant the works of Pappus of Alexandria, who lived in the time of Theodosius; the commentary of Eutocius, the Ascalonite, who lived about the year of Christ 540, on Archimedes's mensuration of the circle; and the commentary on Euclid, by Proclus, who lived under the empire of Anastasius.</p><p>The consequent inundation of ignorance and barbarism was unfavourable to Geometry, as well as to the other sciences; and the few who applied themselves to this science, were calumniated as magicians. However, in those times of European darkness, the Arabians were distinguished as the guardians and promoters of science; and from the 9th to the 14th century, they produced many astronomers, geometricians, geographers, &c; from whom the mathematical sciences were again received into Spain, Italy, and the rest of Europe, somewhat before the year 1400. Some of the carliest writers after this period, are Leonardus Pisanus, Lucas Paciolus or De Burgo, and others between 1400 and 1500. And after this appeared many editions of Euclid, or commentaries upon him: thus, Orontius Finæus, in 1530, published a commentary on the first 6 books; as did James Peletarius, in 1557; and about the same time Nicholas Tartaglia published a commentary on the whole 15 books. There have been also the editions, or commentaries, of Commandine, Clavius, Billingsly, Scheubelius, Herlinus, Dasypodius, Ramus, Herigon, Stevinus, Saville, Barrow, Taquet, Dechales, Furnier, Scarborough, Keill, Stone, and many others; but the completest edition of all the works of Euclid, is that of Dr. Gregory, printed at Oxford 1703, in Greek and Latin: the edition of Euclid, by Dr. Robert Simson of Glasgow, containing the first 6 books, with the 11th and 12th, is much esteemed for its correctness. The principal other elementary writers, besides the editors of Euclid, are Borelli, Pardies, Marchetti, Wolfius, Simpson, &c. And among those who have gone beyond Euclid in the nature of the Elementary parts of Geometry, may be chiefly reckoned, Apollonius, in his Conics, his Loci Plani, De Sectione Determinata, his Tangencies, Inclinations, Section of a Ratio, Section of a Space, &c; Archimedes, in his treatises of the Sphere and Cylinder, the Dimension of the Circle, of Conoids and Spheroids, of Spirals, and the Quadrature of the Parabola; Theodosius, in his Spherics; Serenius, in his Sections of the Cone and Cylinder; Kepler's Nova Stereometria; Cavalerius's<cb/> Geometria Indivisibilium; Torricelli's Opera Geometrica; Viviani, in his Divinationes Geometricæ, Exercitatio Mathematica, De Locis Solidis, De Maximis & Minimis, &c; Vieta, in his Effectio Geometrica, Supplement. Geometriæ, Sectiones Angulares, Responsum ad Problema, Apollonius Gallus, &c; Gregory St. Vincent's Quadratura Circuli; Fermat's Varia Opera Mathematica; Dr. Barrow's Lectiones Geometricæ; Bulliald de Lineis Spiralibus; Cavalerius; Schooten and Gregory's Exercitationes Geometricæ, and Gregory's Pars Universalis, &c; De Billy's treatise De Proportione Harmonica; La Lovera's Geometria veterum promota; Slusius's Mesolabium, Problemata Solida, &c; Wallis, in his treatises De Cycloide, Cissoide, &c; De Proportionibus, De Sectionibus Conicis, Arithmetica Infinitorum, De Centro Gravitatis, De Sectionibus Angularibus, De Angulo Contactus, Cuno-Cuneus, &c, &c; Hugo De Omerique, in his Analysis Geometrica; Pascal on the Cycloid; Step. Angeli's Problemata Geometrica; Alex. Anderson's Suppl. Apollonii Redivivi, Variorum Problematum Practice, &c; Baronius's Geomet. Prob. &c; Guido Grandi Geometr. Demonstr. &c; Ghetaldi Apollonius Redivivus, &c; Ludolph van Colen or a Ceulen, de Circulo et Adscriptis, &c; Snell's Apollonius Batavus, Cyclometricus, &c; Herb<*>rstein's Diotome Circulorum; Palma's Exercit. in Geometriam; Guldini Centro-Baryca<*> with several others equally eminent, of more modern date, as Dr. Rob. Simson, Dr. Mat. Stewart, Mr. Tho. Simpson, &c. Since the introduction of the new Geometry, or the Geometry of Curve Lines, as expressed by algebraical equations, in this part of Geometry, the following names, among many others, are more especially to be respected, viz, Des Cartes, Schooten, Newton, Maclaurin, Brackenridge, Cramer, Cotes, Waring, &c, &c.</p><p>As to the subject of Practical Geometry, the chief writers are Beyer, Kepler, Ramus, Clavius, Mallet, Tacquet, Ozanam, Wolfius, Gregory, with innumerable others.</p><p>Geometry is distinguished into Theoretical or Speculative, and Practical.</p><p><hi rend="italics">Theoretical</hi> or <hi rend="italics">Speculative</hi> <hi rend="smallcaps">Geometry</hi>, treats of the various properties and relations in magnitudes, demonstrating the theorems, &c. And</p><p><hi rend="italics">Practical</hi> <hi rend="smallcaps">Geometry</hi>, is that which applies those speculations and theorems to particular uses in the solution of problems, and in the measurements in the ordinary concerns of life.</p><p>Speculative Geometry again may be divided into Elementary and Sublime.</p><p><hi rend="italics">Elementary</hi> or <hi rend="italics">Common</hi> <hi rend="smallcaps">Geometry</hi>, is that which is employed in the consideration of right lines and plane surfaces, with the solids generated from them. And the</p><p><hi rend="italics">Higher</hi> or <hi rend="italics">Sublime</hi> <hi rend="smallcaps">Geometry</hi>, is that which is employed in the consideration of curve lines, conic sections, and the bodies formed of them. This part has been chiefly cultivated by the moderns, by help of the improved state of Algebra, and the modern analysis or Fluxions.</p></div1><div1 part="N" n="GIBBOUS" org="uniform" sample="complete" type="entry"><head>GIBBOUS</head><p>, is used for the shape of one state of the enlightened part of the moon, being that in which she<pb n="537"/><cb/> appears more than half full or enlightened, which is the time between the first quarter and the full moon, and from the full moon to the last quarter; appearing then Gibbous, that is, bunched out, or convex on both sides of the enlightened part; as contradistinguished from the state when she is less than half full, when she is said to be horned, or a crescent.</p></div1><div1 part="N" n="GIMBOLS" org="uniform" sample="complete" type="entry"><head>GIMBOLS</head><p>, are the brass rings by which a seacompass is suspended in its box that usually stands in the binacle.</p></div1><div1 part="N" n="GIN" org="uniform" sample="complete" type="entry"><head>GIN</head><p>, in Artillery and Mechanics, is a machine for raising great weights, usually composed of three long legs, &c.</p></div1><div1 part="N" n="GIRDERS" org="uniform" sample="complete" type="entry"><head>GIRDERS</head><p>, in Architecture, are the largest beams or pieces of timber supporting the floors. Their ends are usually fastened into the summers, or breast-summers; and the joists are framed in at one end to the Girders. By the statute for rebuilding London, no Girder is to lie less than 10 inches into the wall, and their ends to be always laid in loam, &c. The shorter bearings a Girder has, and the oftener it is supported by the internal or partition walls, so much the better. The established breadth and depth of a Girder, according to its length of bearing, are as in the following tablet: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=7" role="data">Girders and Summers in length</cell><cell cols="1" rows="1" role="data">From</cell><cell cols="1" rows="1" role="data">to</cell><cell cols="1" rows="1" rend="colspan=2" role="data">must be in</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Breadth</cell><cell cols="1" rows="1" role="data"> Depth</cell></row><row role="data"><cell cols="1" rows="1" role="data">10 seet</cell><cell cols="1" rows="1" role="data">15 ft.</cell><cell cols="1" rows="1" role="data">11 inc.</cell><cell cols="1" rows="1" role="data"> 8 inc.</cell></row><row role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> 9</cell></row><row role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">14</cell></row></table></p></div1><div1 part="N" n="GIRT" org="uniform" sample="complete" type="entry"><head>GIRT</head><p>, in Timber-measuring, is the circumference of a tree; though some use this word for the quarter or 4th part of the circumference only, on account of the great use that is made of it; for the square of this 4th part is esteemed and used as equal to the area of the section of the tree; which square therefore multiplied by the length of the tree, is accounted the solid content. This content however is always about one-fourth part less than the true quantity; being nearly equal to what this will be after the tree is hewed square in the usual way: so that it seems intended to make an allowance for the squaring of the tree.</p><p><hi rend="smallcaps">Girt</hi>-<hi rend="italics">Line,</hi> is a line on the common or carpenter's sliding rule, employed in casting up the contents of trees by means of their Girt.</p></div1><div1 part="N" n="GIVEN" org="uniform" sample="complete" type="entry"><head>GIVEN</head><p>, <hi rend="italics">Datum,</hi> a term very often used in mathematics, and signifies something that is supposed to be known.</p><p>Thus, if a magnitude be known, or if we can find another equal to it, it is said to be Given in magnitude. Or when the position of any thing is known, it is said to be Given in position. And when the diameter of <*> circle is known, the circle is Given in magnitude. Or the circle is Given in position when its centre is Given in position. When the kind or species of a figure is known, or remains the same, it is Given in specie. And so on.<cb/></p><p>Euclid wrote a book of <hi rend="italics">Data,</hi> or concerning things Given, in 95 propositions, usually accompanying his Elements, in the best editions, and which Pappus reckons as one of the best specimens of the analytical works of the ancients.</p></div1><div1 part="N" n="GLACIS" org="uniform" sample="complete" type="entry"><head>GLACIS</head><p>, in Fortification, a sloping bank reaching from the parapet of the counterscarp, or covered-way, to the level side of the field, commonly at the distance of about 40 yards.</p></div1><div1 part="N" n="GLOBE" org="uniform" sample="complete" type="entry"><head>GLOBE</head><p>, a round or spherical body, more usually called a sphere, bounded by one uniform convex surface, every point of which is equally distant from a point within called its centre. Euclid defines the Globe, or sphere, to be a solid figure described by the revolution of a semi-circle about its diameter, which remains unmoved. Also, its axis is the fixed line or diameter about which the semi-circle revolves; and its centre is the same with that of the revolving semicircle, a diameter of it being any right line that passes through the centre, and terminated both ways by the superficies of the sphere. Elem. 11. def. 14, 15, 16, 17.</p><p>Euclid, at the end of the 12th book, shews that spheres are to one another in the triplicate ratio of their diameters, that is, their solidities are to one another as the cubes of their diameters. And Archimedes determines the real magnitudes and measures of the surfaces and solidities of spheres and their segments, in his treatise de Sphæra et Cylindro: viz, 1, That the superficies of any Globe is equal to 4 times a great circle of it.—2, That any sphere is equal to 2/3 of its circumscribing cylinder, or of the cylinder of the same diameter and altitude.—3, That the curve surface of the segment of a globe, is equal to the circle whose radius is the line drawn from the vertex of the segment to the circumference of the base.—4, That the content of a solid sector of the Globe, is equal to a cone whose altitude is the radius of the Globe, and its base equal to the curve superficies or base of the sector. With many other properties. And from hence are easily deduced these practical rules for the surfaces and solidities of Globes and their segments; viz,</p><p>1. <hi rend="italics">For the Surface of a Globe,</hi> multiply the square of the diameter by 3.1416; or multiply the diameter by the circumference.</p><p>2. <hi rend="italics">For the Solidity of a Globe,</hi> multiply the cube of the diameter by .5236 (viz 1/<*> of 3.1416); or multiply the surface by 1/6 of the diameter.</p><p>3. <hi rend="italics">For the Sursace of a Segment,</hi> multiply the diameter of the Globe by the altitude of the segment and the product again by 3.1416.</p><p>4. <hi rend="italics">For the Sol<*>dity of a Segment,</hi> multiply the square of the diameter of the Globe by the difference between 3 times that diameter and 2 times the altitude of the segment, and the product again by .5236, or 1/6 of 3.1416.</p><p>Hence, if <hi rend="italics">d</hi> denote the diameter of the Globe, <hi rend="italics">c</hi> the circumference, <hi rend="italics">a</hi> the altitude of any segment, and <hi rend="italics">p</hi> = 3.1416; then <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">The surface.</cell><cell cols="1" rows="1" role="data"> The solidity</cell></row><row role="data"><cell cols="1" rows="1" role="data">In the Globe</cell><cell cols="1" rows="1" role="data">  <hi rend="italics">pd</hi><hi rend="sup">2</hi> = <hi rend="italics">cd</hi></cell><cell cols="1" rows="1" role="data">1/<*> <hi rend="italics">pd</hi><hi rend="sup">3</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">In the Segt.</cell><cell cols="1" rows="1" role="data">  <hi rend="italics">pad</hi></cell><cell cols="1" rows="1" role="data">1/6<hi rend="italics"> pa</hi><hi rend="sup">2</hi> × ―(3<hi rend="italics">d</hi> - 2<hi rend="italics">a</hi>)</cell></row></table></p><p>See the art. <hi rend="smallcaps">Sphere</hi>, and my Mensuration, p. 197 &c, 2d edit.<pb n="538"/><cb/></p><p>The <hi rend="smallcaps">Globe</hi>, or <hi rend="italics">Terraque<*>s</hi> <hi rend="smallcaps">Globe</hi>, is the body or mass of the earth and water together, which is nearly globular.</p><div2 part="N" n="Globe" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Globe</hi></head><p>, or <hi rend="italics">Artificial</hi> <hi rend="smallcaps">Globe</hi>, is more particularly used for a Globe of metal, plaister, paper, pasteboard, &c, on the surface of which is drawn a map, or representation of either the heavens or the earth, with the several circles conceived upon them. And hence</p><p><hi rend="smallcaps">Globes</hi> are of two kinds, Terrestrial, and Celestial; which are of considerable use in geography and astronomy, by serving to give a lively representation of their principal objects, and for performing and illustrating many of their operations in a manner easy to be perceived by the senses, and so as to be conceived even without any knowledge of the mathematical grounds of those sciences. <hi rend="center"><hi rend="italics">Description of the Globes.</hi></hi></p><p>The fundamental parts that are common to both Globes, are an axis, representing the axis of the world, passing through the two poles of a spherical shell, representing those of the world, which shell makes the body of the Globe, upon the external surface of which is drawn the representation of the whole surface of the earth, sea, rivers, islands, &c, for the Terrestrial Globe, and the stars and constellations of the heavens, for the Celestial one; besides the equinoctial and ecliptic lines, the zodiac, the two tropics and polar circles, and a number of meridian lines. There is next a brazen meridian, being a strong circle of brass, circumscribing the Globe, at a small distance from it quite round, in which the globe is hung by its two poles, upon which it turns round within this circle, which is divided into 4 times 90 degrees, beginning at the equator on both sides, and ending with 90 at the two poles. There are also two small hour circles, of brass, divided into twice 12 hours, and fitted on the meridian round the poles, which carry an index pointing to the hour. The whole is set in a wooden ring, placed parallel to, and representing the horizon, in which the Globe slides by the brass meridian, elevating or depressing the pole according to any proposed latitude. There is also a thin slip of brass, called a Quadrant of Altitude, made to fit on occasionally upon the brass meridian, at the highest or vertical point, to measure the altitude of any thing above the horizon. A magnetic compass is sometimes set underneath. See the figure of the Globes so mounted, at fig. 1, plate xii.</p><p>Such is the plain and simple construction of the artificial Globe, whether celestial or terrestrial, as adapted to the time only for which it is made. But as the angle formed by the equator and ecliptic, as well as their point of intersection, is always changing; to remedy these inconveniences, several contrivances have been made, so as to adapt the same Globes to any other time, either past or to come; as well as other contrivances to answer particular purposes.</p><p>Thus, Mr. Senex, a celebrated maker of Globes, had a contrivance which, by means of a nut and serew, caused the pole of the equator to revolve about the pole of the ecliptic, by any quantity answering to the precession of the equinoxes, since the time for which the Globe was made. Philos. Trans. number 447, or Abr. vol. 8, p. 217, also Philos. Trans. vol. 46, p. 290.</p><p>Mr. Joseph Harris, late assay-master of the Mint,<cb/> made some contrivances to shew the effects of the <*>arth's motions. He fixed two horary circles under the brass meridian, to the axis, one at each pole, so as to turn round with the Globe, and that meridian served as an index to cut the horary divisions. The Globe in this state serves equally for resolving problems in both north and south latitudes, as also in places near the equator; whereas, in the common construction, the axis and horary circle prevent the brass meridian from being moveable quite round in the horizon. This Globe is also adapted for shewing how the vicissitudes of day and night, and the alteration of their lengths, are really occasioned by the motion of the earth: for this purpose, he divides the brass meridian, at one of the poles, into months and days, according to the sun's declination, reckoning from the pole. Therefore, by bringing the day of the month to the horizon, and rectifying the Globe according to the time of the day, the horizon will represent the circle separating light and darkness, and the upper half of the Globe the illuminated hemisphere, the sun being in the zenith. Mr. Harris also gives an account of a cheap machine for shewing how the annual motion of the earth in its orbit causes the change of the sun's declination, without the great expence of an orrery. Philos. Trans. number 456, or Abr. vol. 8, p. 352.</p><p>The late Mr. George Adams made also some useful improvements in the construction of the Globes. Besides what is usual, his Globes have a thin brass semicircle moveable about the poles, with a small thin sliding circle upon it. On the terrestrial Globe, the former of these is a moveable meridian, and the latter is the visible horizon of any particular place to which it is set. But on the celestial Globe, the semi-circle is a moveable circle of declination, and its small annexed circle an artisicial sun or planet. Each Globe has a brass wire circle, placed at the limits of the twilight. The terrestrial Globe has many additional circles, as well as the rhumb-lines, for resolving all the necessary geographical and nautical problems: and on the celestial Globe are drawn, on each side of the ecliptic, 8 parallel circles, at the distance of one degree from each other, including the zodiac; which are crossed at rightangles by segments of great circles at every 5th degree of the ecliptic, for the more readily noting the place of the moon or of any planet upon the Globe. On the strong brass circle of the terrestrial Globe, and about 23 1/2 degrees on each side of the north pole, the days of each month are laid down according to the sun's declination: and this brass circle is so contrived, that the Globe may be placed with the north and south poles in the plane of the horizon, and with the south pole elevated above it. The equator, on the surface of either Globe, serves the purpose of the horary circle, by means of a semi-circular wire placed in the plane of the equator, carrying two indices, one of which is occasionally to be used to point out the time. For a farther account of these Globes, with the method of using them, see Mr. Adams's treatise on their construction and use.</p><p>There are also what are called Patent Globes, made by Mr. Neale; by means of which he resolves several astronomical problems, which do not admit of solution by the common Globes.<pb n="539"/><cb/></p><p>Mr. Ferguson likewise made several improvements of the Globes, particularly one for constructing dials, and another called a planetary Globe. See Philos. Trans. vol. 44, p. 535, and Ferguson's Astron. p. 291, and 292.</p><p>Lastly, in the Philos. Trans. for 1789, vol. 79, p. 1, Mr. Smeaton has proposed some improvements of the celestial Globe, especially with respect to the quadrant of altitude, for the resolution of problems relating to the azimuth and altitude. The difficulty, he observes, that has occurred in fixing a semicircle, so as to have a centre in the zenith and nadir points of the Globe, at the same time that the meridian is left at liberty to raise the pole to its desired elevation, I suppose, has induced the Globe-makers to be contented with the strip of thin flexible brass, called the quadrant of altitude; and it is well known how imperfectly it performs its office. The improvement I have attempted, is in the application of a quadrant of altitude of a more solid construction; which being affixed to a brass socket of some length, and this ground, and made to turn upon an upright steel spindle, fixed in the zenith, steadily directs the quadrant, or rather arc, of altitude to its true azimuth, without being at liberty to deviate from a vertical circle to the right hand or left: by which means the azimuth and altitude are given with the same exactness as the measure of any other of the great circles. For a more particular description of this improvement, illustrated with figures, see the place above quoted. <hi rend="center"><hi rend="italics">The use of the Terrestrial <hi rend="smallcaps">Globe.</hi></hi></hi></p><p><hi rend="smallcaps">Prob.</hi> I. <hi rend="italics">To find the latitude and longitude of any place.</hi> —Bring the place to the graduated side of the first meridian: then the degree of the meridian it cuts is the latitude sought; and the degree of the equator then under the meridian is the longitude.</p><p>II. <hi rend="italics">To find a place, having a given latitude and longitude.</hi>—Find the degree of longitude on the equator, and bring it to the brass meridian; then find the degree of latitude on the meridian, either north or south of the equator, as the given latitude is north or south; then the point of the Globe just under that degree of latitude is the place required.</p><p>III. <hi rend="italics">To find all the places on the Globe that have the same latitude, and the same longitude, or hour, with a given place, as suppose London.</hi>—Bring the given place London to the meridian, and observe what places are just under the edge of it, from north to south; and all those places have the same longitude and hour with it. Then turn the Globe quite round; and all those places which pass just under the given degree of latitude on the meridian, have the same latitude with the given place.</p><p>IV. <hi rend="italics">To find the Antœci, Periœci and Antipodes, of any given place, suppose London.</hi>—Bring the given place London to the meridian, then count 51 1/2 the same degree of latitude southward, or towards the other pole, and the point thus arrived at will be the Antœci, or where the hour of the day or night is always the same at both places at the same time, and where the seasons and lengths of days and nights are also equal, but at half a year distance from each other, because their sea-<cb/> sons are opposite or contrary. London being still under the meridian, set the hour index to 12 at noon, or pointing towards London; then turn the Globe just half round, or till the index point to the opposite hour, or 12 at night; and the place that comes under the same degree of the meridian where London was, shews where the Periœci dwell, or those people that have the same seasons and at the same time as London, as also the same length of days and nights &c at that time, but only their time or hour is just opposite, or 12 hours distant, being day with one when night with the other, &c. Lastly, as the Globe stands, count down by the meridian the same degree of latitude south, and that will give the place of the Antipodes of London, being diametrically under or opposite to it; and so having all its times, both hours and seasons opposite, being day with the one when night with the other, and summer with the one when winter with the other.</p><p>V. <hi rend="italics">To find the Distance of two places on the Globe.</hi>— If the two places be either both on the equator, or both on the same meridian, the number of degrees in the distance between them, reduced into miles, at the rate of 70 English miles to the degree, (or more exact 69 1/3), will give the distance nearly. But in any other situations of the two places, lay the quadrant of altitude over them, and the degrees counted upon it, from the one place to the other, and turned into miles as above, will give the distance in this case.</p><p>VI. <hi rend="italics">To find the Difference in the Time of the day at any two given places, and thence the Difference of Longitude.</hi>—Bring one of the places to the meridian, and set the hour index to 12 at noon; then turn the Globe till the other place comes to the meridian, and the index will point out the difference of time; then by allowing 15° to every hour, or 1° to 4 minutes of time, the difference of longitude will be known.—Or the difference of longitude may be found without the time, thus:</p><p>First bring the one place to the meridian, and note the degree of longitude on the equator cut by it; then do the same by the other place; which gives the longitudes of the two places; then subtracting the one number of degrees from the other, gives the difference of longitude sought.</p><p>VII. <hi rend="italics">The time being known at any given place, as suppose London, to find what hour it is in any other part of the world.</hi>—Bring the given place, London, to the meridian, and set the index to the given hour; then turn the Globe till the other place come to the meridian, and look at what hour the index points, which will be the time sought.</p><p>VIII. <hi rend="italics">To find the Sun's place in the ecliptic, and also on the Globe, at any given time.</hi>—Look into the calendar on the wooden horizon for the month and day of the month proposed, and immediately opposite stands the sign and degree which the sun is in on that day. Then in the ecliptic drawn upon the Globe, look for the same sign and degree, and that will be the place of the sun required.</p><p>IX. <hi rend="italics">To find at what place on the earth the sun is vertical, at a given moment of time at another place, as suppose London.</hi>—Find the sun's place on the Globe by the last problem, and turn the Globe about till<pb n="540"/><cb/> that place come to the meridian, and note the degree of the meridian just over it. Then turn the Globe till the given place, London, come to the meridian, and set the index to the given moment of time. Lastly, turn the Globe till the index points to 12 at noon; then the place of the earth, or Globe, which stands under the before noted degree, has the sun at that moment in the zenith.</p><p>X. <hi rend="italics">To find how long the sun shines without setting, in any given place in the frigid zones.</hi>——Subtract the degrees of latitude of the given place from 90, which gives the complement of the latitude, and count the number of this complement upon the meridian from the equator towards the pole, marking that point of the meridian; then turn the Globe round, and carefully observe what two degrees of the ecliptic pass exactly under the point marked on the meridian. Then look for the same degrees of the ecliptic on the wooden horizon, and just opposite to them stand the months and days of the months corresponding, and between which two days the sun never sets in that latitude.</p><p>If the beginning and end of the longest night be required, or the period of time in which the sun never rises at that place; count the same complement of latitude towards the south or farthest pole, and then the rest of the work will be the same in all respects as above.</p><p>Note, that this solution is independent of the horizontal refraction of the sun, which raises him rather more than half a degree higher, by that means making the day so much longer, and the night the shorter; therefore in this case, set the mark on the meridian half a degree higher up towards the north pole, than what the complement of latitude gives; then proceed with it as before, and the more exact time and length of the longest day and night will be found.</p><p>XI. <hi rend="italics">A place being given in the torrid zone, to find on what two days of the year the sun is vertical at that place.</hi>——Turn the Globe about till the given place come to the meridian, and note the degree of the meridian it comes under. Next turn the Globe round again, and note the two points of the ecliptic passing under that degree of the meridian. Lastly, by the wooden horizon, find on what days the sun is in those two points of the ecliptic; and on these days he will be vertical to the given place.</p><p>XII. <hi rend="italics">To find those places in the torrid zone to which the sun is vertical on a given day.</hi>——Having found the sun's place in the ecliptic, as in the 8th problem, turn the Globe to bring the same point of the ecliptic on the Globe to the meridian; then again turn the Globe round, and note all the places which pass under that point of the meridian; which will be the places sought.</p><p>After the same manner may be found what people are Ascii for any given day. And also to what place of the earth, the moon, or any other planet, is vertical on a given day; finding the place of the planet on the globe by means of its right ascension and declination, like finding a place from its longitude and latitude given.</p><p>XIII. <hi rend="italics">To rectify the Globe for the latitude of any place.</hi> ——By sliding the brass meridian in its groove, elevate<cb/> the pole as far above the horizon as is equal to the latitude of the place; so for London, raise the north pole 51 1/2 degrees above the wooden horizon: then turn the Globe on its axis till the place, as London, come to the meridian, and there set the index to 12 at noon. Then is the place exactly on the vertex, or top point of the Globe, at 90° every way round from the wooden horizon, which represents the horizon of the place. And if the frame of the Globe be turned about till the compass needle point to 22 1/2 degrees, or two points west of the north point (because the variation of the magnetic needle is nearly 22 1/2 degrees west), so shall the Globe then stand in the exact position of the earth, with its axis pointing to the north pole.</p><p>XIV. <hi rend="italics">To find the length of the day or night, or the sun's rising or setting, in any latitude; having the day of the month given.</hi>——Rectify the Globe for the latitude of the place; then bring the sun's place on the globe to the meridian, and set the index to 12 at noon, or the upper 12, and then the Globe is in the proper position for noon day. Next turn the Globe about towards the east till the sun's place come just to the wooden horizon, and the index will then point to the hour of sunrise; also turn the Globe as far to the west side, or till the sun's place come just to the horizon on the west side, and then the index will point to the hour of sunset. These being now known, double the hour of setting will be the length of the day, and double the rising will be the length of the night.—And thus also may the length of the longest day, or the shortest day, be found for any latitude.</p><p>XV. <hi rend="italics">To find the beginning and end of Twilight on any day of the year, for any latitude.</hi>——It is twilight all the time from sunset till the sun is 18° below the horizon, and the same in the morning from the time the sun is 18° below the horizon till the moment of his rise<*> Therefore, rectify the Globe for the latitude of the place, and for noon by setting the index to 12, and screw on the quadrant of altitude. Then take the point of the ecliptic opposite the sun's place, and turn the Globe on its axis westward, as also the quadrant of altitude, till that point cut this quadrant in the 18th degree below the horizon, then the index will shew the time of dawning in the morning; next turn the Globe and quadrant of altitude towards the east, till the said point opposite the sun's place meet this quadrant in the same 18th degree, and then the index will shew the time when twilight ends in the evening.</p><p>XVI. <hi rend="italics">At any given day, and hour of the day, to find all those places on the Globe where the sun then rises, on sets, as also where it is noon day, where it is day light, and where it is in darkness.</hi>——Find what place the sun is vertical to, at that time; and elevate the Globe according to the latitude of that place, and bring the place also to the meridian; in which state it will also be in the zenith of the Globe. Then is all the upper hemisphere, above the wooden horizon, enlightened, or in day light; while all the lower one, below the horizon, is in darkness, or night: those places by the edge of the meridian, in the upper hemisphere, have noon day, or 12 o'clock; and those by the meridian below, have it midnight: lastly, all those places by the eastern side of the horizon, have the sun just set-<pb n="541"/><cb/> ting, and those by the western horizon have him just rising.</p><p>Hence, as in the middle of a lunar eclipse the moon is in that degree of the ecliptic opposite to the sun's place; by the present problem it may be shewn what places of the earth then see the middle of the eclipse, and what the beginning or ending; by using the moon's place instead of the sun's place in the problem.</p><p>XVII. <hi rend="italics">To find the bearing of one place from another, and their ang'e of position.</hi>——Bring the one place to the zenith, by rectifying the Globe for its latitude, and turning the Globe till that place come to the meridian; then screw the quadrant of altitude upon the meridian at the zenith, and make it revolve till it come to the other place on the Globe; then look on the wooden horizon for the point of the compass, or number of degrees from the fouth, where the quadrant of altitude cuts it, and that will be the bearing of the latter place from the former, or the angle of position sought. <hi rend="center"><hi rend="italics">The Use of the Celcstial <hi rend="smallcaps">Globe.</hi></hi></hi></p><p>The Celestial Globe differs from the terrestrial only in this; instead of the several parts of the earth, the images of the stars and constellations are designed. The meridian circle drawn through the two poles and through the point Cancer, represents the solstitial colure; but that through the point Arics, represents the equinoctial colure.</p><p><hi rend="smallcaps">Pros.</hi> XVIII. <hi rend="italics">To exhibit the true representation of the face of the heavens at any given time and place.</hi>—— Rectify for the lat. of the place, by prob. 13, setting the Globe with its pole pointing to the pole of the world, by means of a compass. Find the sun's place in the ecliptic, and turn the Globe to bring it to the meridian, and there set the index to 12 at noon. Again revolve the Globe on its axis, till the index point to the given hour of the day or night: so shall the Globe in this position exactly represent the face of the heavens as it appears at that time, every constellation and star, in the heavens, answering in position to those on the Globe; so that, by examining the Globe, it will immediately appear which stars are above or below the horizon, which on the east or western parts of the heavens, which lately risen, and which going to set, &c. And thus the positions of the sevcral planets, or comets, may also be exhibited; having marked the places of the Globe where they are, by means of their declination and right ascension.</p><p>XIX. <hi rend="italics">To find the Declination and Right-ascension of any star upon the Globe.</hi>——Turn the Globe till the star come to the meridian: then the number of degrees on the meridian, between the equator and the star, is its declination; and the degree of the equator cut by the meridian, is the right-ascension of the star.——In like manner are found the declination and right-ascension of the sun, or any other point.</p><p>XX. <hi rend="italics">To find the Latitude and Longitude of any star drawn upon the Globe.</hi>——Bring the solstitial colure to the meridian, and there fix the quadrant of altitude over the pole of the ecliptic in the same hemisphere with the star, and bring its graduated edge to the star: then the degree on the quadrant cut by the star is its latitude, counted from the ecliptic; and the degree of the ecliptic cut by the quadrant its longitude.<cb/></p><p>XXI. <hi rend="italics">To find the place of a star, planet, comet, &c. on the Globe; its declination and right-ascension being known.</hi>——Find the given point of right-ascension on the equinoctial, and bring it to the meridian; then count the degrees of declination upon the meridian from the equinoctial, and there make a mark on the Globe, which will be the place of the planet, &c, sought.</p><p>XXII. <hi rend="italics">To find the place of a star, planet, comet, or other object on the Globe; its latitude and longitude being given.</hi>——Bring the pole of the ecliptic to the meridian, and there fix the quadrant of altitude, which turn round till its edge cut the given longitude on the ecliptic; then count the given latitude, from the ecliptic, upon the quadrant of altitude, and there make a mark on the Globe, which will be the place of the planet, &c, sought.——The place on the Globe, of any such planet, &c, being found by this or the foregoing problem, its rising, or setting, or any other circumstance concerning it, may then be found, the same as the sun, by the proper problems.</p><p>XXIII. <hi rend="italics">To find the rising, setting, and culminating of a star, planet, sun, &c; with its continuance above the horizon, for any place and day; as also its oblique ascension and descension, with its eastern and western amplitude and azimuth.</hi>——Adjust the Globe to the state of the heavens at 12 o'clock that day. Bring the star, &c, to the eastern side of the horizon: which will give its eastern amplitude and azimuth, and the time of rising, as for the sun. Again, turn the Globe to bring the same star to the western side of the horizon: so will the western amplitude and azimuth, with the time of setting, be found. Then, the time of rising, subtracted from that of setting, leaves the continuance of the star above the horizon: this continuance above the horizon taken from 24 hours, leaves the time it is below the horizon. Lastly, bring the star to the meridian, and the hour to which the index then points is the time of its culmination, or southing.</p><p>XXIV. <hi rend="italics">To find the altitude of the sun, or star, &c, for any given hour of the day or night.</hi>——Adjust the Globe to the position of the heavens, and turn it till the index point at the given hour. Then six on the quadrant of altitude, at 90 degrees from the horizon, and turn it to the place of the sun or star: so shall the degrees of the quadrant, intercepted between the horizon and the sun or star, be the altitude sought.</p><p>XXV. <hi rend="italics">Given the altitude of the sun by day, or of a star by night, to find the hour of the day or night.</hi>——Rectify the Globe as in the foregoing problem; and turn the Globe and quadrant, till such time as the star or degree of the ecliptic the sun is in, cut the quadrant in the given degree of altitude; then will the index point at the hour required.</p><p>XXVI. <hi rend="italics">Given the azimuth of the sun or a star, to find the time of the day or night.</hi>——Rectify the Globe, and bring the quadrant to the given azimuth in the horizon; then turn the Globe till the sun or star come to the quadrant, and the index will then shew the time of the day or night.</p></div2></div1><div1 part="N" n="GLOBULAR" org="uniform" sample="complete" type="entry"><head>GLOBULAR</head><p>, relating to, or partaking the property or shape of, the Globe. As Globular chart, Globular projection, or Globular sailing, &c.</p><p><hi rend="smallcaps">Globular</hi> <hi rend="italics">Chart,</hi> is a representation of the surface<*><pb n="542"/><cb/> or part of the surface, of the terraqueous Globe upon a plane; in which the parallels of latitude are circles nearly concentric; and the meridians are curves bending towards the poles; the rhumb-lines being curves also.</p><p>The merits of this chart consist in these particulars, viz, that the distances between places on the same rhumb are all measured by the same scale of equal parts; and the distance of any two places in the arch of a great circle, is nearly represented in this chart by a straight line.</p><p>Land maps also made according to this projection would have great advantages over those made in any other way. But for sea charts for the use of navigation, Mercator's are preferable, as both the meridians and parallels, as also the rhumbs, are all straight lines.</p><p>This projection is not new, though not much noticed till of late. It is mentioned by Ptolomy, in his Geography; and also by Blundeville, in his Exercises.</p><p>For Globular projection of maps or charts, see <hi rend="smallcaps">Map.</hi></p><p><hi rend="smallcaps">Globular</hi> <hi rend="italics">Sailing,</hi> is the method of resolving the cases of sailing upon principles deduced from the spherical figure of the earth. Such as Mercator's sailing, or Great-circle sailing; which see.</p></div1><div1 part="N" n="GLOSSOCOMON" org="uniform" sample="complete" type="entry"><head>GLOSSOCOMON</head><p>, in Mechanics, is a name given by Heron to a machine composed of divers dented wheels with pinions, serving to raise huge weights.</p></div1><div1 part="N" n="GNOMON" org="uniform" sample="complete" type="entry"><head>GNOMON</head><p>, in Astronomy, is an instrument or apparatus for measuring the altitudes, declinations, &c, of the sun and stars. The Gnomon is usually a pillar, or column, or pyramid, erected upon level ground, or a pavement. For making the more considerable observations, both the ancients and moderns have made great use of it, especially the former; and many have preferred it to the smaller quadrants, both as more accurate, easier made, and more easily applied.</p><p>The mostancient observation of this kind extant, is that made by Pytheas, in the time of Alexander the Great, at Marseilles, where he found the height of the Gnomon was in proportion to the meridian shadow at the summer solstice, as 213 1/8 to 600; just the same as Gassendi found it to be, by an observation made at the same place, almost 2000 years after, viz, in the year 1636. Ricciol. Almag. vol. 1, lib. 3, cap. 14.</p><p>Ulugh Beigh, king of Parthia, &c, used a Gnomon in the year 1437, which was 180 Roman feet high. That erected by Ignatius Dante, in the church of St. Petronius, at Bologna, in the year 1576, was 67 feet high. M. Cassini erected another of 20 feet high, in the same church, in the year 1655.</p><p>The Egyptian obelisks were also used as Gnomons; and it is thought by some modern travellers that this was the very use they were designed and built for; it has also been found that their four sides stand exactly facing the four cardinal points of the compass. It may be added, that the Spaniards in their conquest of Peru, found pillars of curious and costly workmanship, set up in several places, by the meridian shadows of which their amatas or philosophers had, by long experience and repeated observations, learned to determine the times of the equinoxes; which seasons of the year were celebrated with great festivity and rich offerings, in honour of the sun. Garcillasso de la Vega, Hist. Peru. lib. 2, cap. 22.<cb/></p><p><hi rend="smallcaps">Use</hi> <hi rend="italics">of the</hi> <hi rend="smallcaps">Gnomon</hi>, <hi rend="italics">in taking the meridian altitude of the Sun, and thence finding the Latitude of the place.</hi>—— A meridian line being drawn through the centre of the Gnomon, note the point where the shadow of the Gnomon terminates when projected along the meridian line, and measure the distance of that extreme point from the centre of the Gnomon, which will be the length of its shadow. Then having the height of the Gnomon, and the length of the shadow, the sun's altitude is thence easily found. <figure/></p><p>Suppose, ex. gr. AB the Gnomon, and AC the length of the shadow. Here in the right-angled triangle ABC, are given the base AC, and the perpendicular AB, to find the angle C, or the sun's altitude, which will be found by this analogy, as CA : AB : : radius: the tang. of [angle] C, that is, as the length of the shadow is to the height of the Gnomon, so is radius to the tangent of the sun's altitude above the horizon.</p><p>The following example will serve to illustrate this proposition: Pliny says, Nat. Hist. lib. 2, cap. 72, that at Rome, at the time of the equinoxes, the shadow is to the Gnomon as 8 to 9; therefore as or radius : a tangent, to which answers the angle 48° 22′, which is the height of the equator at Rome, and its complement 41° 38′ is therefore the height of the pole, or the latitude of the place.</p><p>Riccioli remarks the following defects in the observations of the sun's height, made with the Gnomon by the ancients, and some of the moderns: viz, that they neglected the sun's parallax, which makes his apparent altitude less, by the quantity of the parallax, than it would be, if the Gnomon were placed at the centre of the earth: 2d, they neglected also the refraction, by which the apparent height of the sun is a little increased: and 3dly, they made the calculations from the length of the shadow, as if it were terminated by a ray coming from the centre of the sun's disc, whereas the shadow is really terminated by a ray coming from the upper edge of the sun's disc; so that, instead of the height of the sun's centre, their calculations gave the height of the upper edge of his disc. And therefore, to the altitude of the sun found by the Gnomon, the sun's parallax must be added, and from the sum must be subtracted the sun's semidiameter, and refraction, which is different at different altitudes; which being done, the correct height of the equator at Rome will be 48° 4′ 13″, the complement of which, or 41° 55′ 46″, is the latitude. Ricciol. Geogr. Refor. lib. 7, cap. 4.<pb n="543"/><cb/></p><p>The preceding problem may be resolved more accurately by means of a ray of light let in through a small hole, than by a shadow, thus: Make a circular perforation in a brass plate, to transmit enough of the sun's rays to exhibit his image on the floor, or a stage; fix the plate parallel to the horizon in a high place, proper for observation, the height of which above the floor let be accurately measured with a plummet. Let the floor, or stage, be perfectly plane and horizontal, and coloured over with some white substance, to shew the sun more distinctly. Upon this horizontal plane draw a meridian line passing through the foot or centre of the Gnomon, i. e. the point upon which the plummet falls from the centre of the hole; and upon this line note the extreme points I and K of the sun's image or diameter, and from each end subtract the image of half the diameter of the aperture, viz KH and LI: then will HL be the image of the sun's diameter, which, when bisected in B, gives the point on which the rays fall from the centre of the sun. <figure/></p><p>Now having given the line AB, and the altitude of the Gnomon AG, beside the right angle A, the angle B, or the apparent altitude of the sun's centre, is easily found, thus: as AB : AG : : radius: tang. angle B.</p><div2 part="N" n="Gnomon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gnomon</hi></head><p>, in Dialling, is the style, pin, or cock of a dial, the shadow of which points out the hours. This is always supposed to represent the axis of the world, to which it is therefore parallel, or coincident, the two ends of it pointing straight to the north and south poles of the world.</p></div2><div2 part="N" n="Gnomon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gnomon</hi></head><p>, in Geometry, is a figure formed of the two complements, in a parallelogram, together with either of the parallelograms about the diameter. Thus the parallelo- <figure/> gram AC being divided into four parallelograms by the two lines DG, EF parallel to the sides, forming the two complements AB and BC, with the two DE, FG about the diameter HI: then the two Gnomons are , and .</p><p><hi rend="smallcaps">Gnomonic</hi> <hi rend="italics">Projection of the Sphere,</hi> is the representation of the circles of an hemisphere upon a plane touching it in the vertex, by the eye in the centre, or by lines or rays issuing from the centre of the hemisphere, to all the points in the surface.</p><p>In this projection of the sphere, all the great circles are projected into right lines, on the plane, of an indefinite length; and all lesser circles that are parallel to<cb/> the plane, into circles; but if oblique to the plane, then are they projected either into ellipses or hyperbolas, according to their different obliquity. It has its name from Gnomonics, or Dialling, because the lines on the face of every dial are from a projection of this sort: for if the sphere be projected on any plane, and upon that side of it on which the sun is to shine; also the projected pole be made the centre of the dial, and the axis of the globe the style or Gnomon, and the radius of projection its height; you will have a dial drawn with all its furniture. See Emerson's Projection of the Sphere.</p></div2></div1><div1 part="N" n="GNOMONICS" org="uniform" sample="complete" type="entry"><head>GNOMONICS</head><p>, the same as <hi rend="smallcaps">Dialling;</hi> or the art of drawing sun and moon dials, on any given plane; being so called, because it shews how to find the hour of the day or night by the shadow of a Gnomon or style.</p><p>GOLDEN <hi rend="italics">Number,</hi> is the particular year of the Metonic or Lunar Cycle. See <hi rend="italics">Lunar</hi> <hi rend="smallcaps">Cycle.</hi> <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1791</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19)</cell><cell cols="1" rows="1" rend="align=right" role="data">1792</cell><cell cols="1" rows="1" role="data">(94</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">171 </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">82</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">76</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Golden</cell><cell cols="1" rows="1" rend="align=right" role="data">No. 6</cell><cell cols="1" rows="1" role="data"/></row></table></p><p><hi rend="italics">To find the Golden Number:</hi> Add 1 to the given year, and divide the sum by 19, and what remains will be the Golden Number; unless 0 remain, for then 19 is the Golden Number.</p><p>Thus, the Golden Number for the year 1791, is 6; as by the operation in the margin.</p><p><hi rend="smallcaps">Golden</hi> <hi rend="italics">Rule,</hi> a rule so called on account of its excellent use, in arithmetic, and especially in ordinary calculations, by which numbers are found in certain proportions, viz, having three numbers given, to find a 4th number, that shall have the same proportion to the 3d as the 2d hath to the 1st. On this account, it is otherwise called <hi rend="italics">The Rule of Three,</hi> and <hi rend="italics">The Rule of Proportion.</hi> See <hi rend="smallcaps">Rule</hi> <hi rend="italics">of Three.</hi></p><p>Having stated, or set down in a line, the three terms, in the order in which they are proportional, multiply the 2d and 3d together, and divide the product by the 1st, so shall the quotient be the answer, or the 4th term sought.</p><p>Thus, if 3 yards of cloth cost a guinea or 21 shillings, what will 20 yards cost. Here the two prices or values must bear the same proportion to each other as the two quantities, or number of yards of cloth, i. e. 3 must bear the same proportion to 20, as 21<hi rend="italics">s,</hi> the value of the former, must bear to the value of the latter: and therefore the stating and operation of the numbers will be thus, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3 : 20 : :</cell><cell cols="1" rows="1" role="data"> 21 :</cell><cell cols="1" rows="1" role="data">140<hi rend="italics">s.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 20</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3)</cell><cell cols="1" rows="1" role="data">420</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Quot.</cell><cell cols="1" rows="1" role="data">140<hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">7£.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>Then multiplying the 2d and 3d together, and dividing the product by the 1st, it gives 140<hi rend="italics">s.</hi> or 7<hi rend="italics">l.</hi> for the answer, being the cost of 20 yards.</p><p>GONIOMETRICAL <hi rend="italics">Lines,</hi> are lines used for measuring or determining the quantity of angles: such as sines, tangents, secants, versed sines, &c.</p><p>Mr. Jones, in the Philos. Trans. number 483, sect. 26, gave a paper, containing a commodious disposition of equations for exhibiting the relations of Goniometrical Lines; from which a multitude of curious theo-<pb n="544"/><cb/> rems may be derived. See also Robertson's Elem. of Navigation, vol. 1, p. 181, Edit. 4.</p></div1><div1 part="N" n="GONIOMETRY" org="uniform" sample="complete" type="entry"><head>GONIOMETRY</head><p>, a method of measuring angles, so called by M. de Lagny, who gave several papers, on this method, in the Memoirs of the Royal Acad. an. 1724, 1725, 1729. M. de Lagny's method of Goniometry consists in measuring the angles with a pair of compasses, and that without any scale whatever, except an undivided semicircle. Thus, having any angle drawn upon paper, to be measured; produce one of the sides of the angle backwards behind the angular point; then with a pair of fine compasses describe a pretty large semicircle from the angular point as a centre, cutting the sides of the proposed angle, which will intercept a part of the semicircle. Take then this intercepted part very exactly between the points of the compasses, and turn them successively over upon the arc of the semicircle, to find how often it is contained in it, after which there is commonly some remainder: then take this remainder in the compasses, and in like manner find how often it is contained in the last of the integral parts of the 1st arc, with again some remainder: find in like manner how often this last remainder is contained in the former; and so on continually, till the remainder become too small to be taken and applied as a measure. By this means he obtains a series of quotients, or fractional parts, one of another, which being properly reduced into one fraction, give the ratio of the first arc to the semicircle, or of the proposed angle to two right angles, or 180 degrees, and consequently that angle itself in degrees and minutes. <figure/></p><p>Thus, suppose the angle BAC be proposed to be measured. Produce BA out towards <hi rend="italics">f;</hi> and from the centre A describe the semicircle <hi rend="italics">abcf,</hi> in which <hi rend="italics">ab</hi> is the measure of the proposed angle. Take <hi rend="italics">ab</hi> in the compasses, and apply it 4 times on the semicircle, as at <hi rend="italics">b, c, d,</hi> and <hi rend="italics">e;</hi> then take the remainder <hi rend="italics">fe,</hi> and apply it back upon <hi rend="italics">ed,</hi> which is but once, viz at <hi rend="italics">g;</hi> again take the remainder <hi rend="italics">gd,</hi> and apply it 5 times on <hi rend="italics">ge,</hi> as at <hi rend="italics">h, i, k, l,</hi> and <hi rend="italics">m;</hi> lastly, take the remainder <hi rend="italics">me,</hi> and it is contained just 2 times in <hi rend="italics">ml.</hi> Hence the series of quotients is 4, 1, 5, 2; consequently the 4th or last arc <hi rend="italics">em</hi> is 1/2 the third <hi rend="italics">ml</hi> or <hi rend="italics">gd,</hi> and therefore the 3d arc <hi rend="italics">gd</hi> is 1/(5 1/2) or 2/11 of the 2d arc <hi rend="italics">ef;</hi> and therefore again this 2d arc <hi rend="italics">ef</hi> is 1/(1 2/11) or 11/1<*> of the 1st arc <hi rend="italics">ab;</hi> and consequently this 1st arc <hi rend="italics">ab</hi> is 1/(4 11/13) or 13/63 of the whole semicircle <hi rend="italics">af.</hi> But 13/63 of 180° are 37 1/7 degrees, or 37° 8′ 34″ 2/7, which therefore is the measure of the angle sought. When the operation is nicely performed, this angle may be within 2 or 3 minutes of the truth; though M. de Lagny pretends to measure muchnearer than that.<cb/></p><p>It may be added, that the series of fractions forms what is called a continued fraction. Thus, in the example above, the continued fraction, and its reduction, will be as follow: ; the quotients being the successive denominators, and 1 always for each numerator.</p></div1><div1 part="N" n="GORGE" org="uniform" sample="complete" type="entry"><head>GORGE</head><p>, or <hi rend="italics">Neck,</hi> in Architecture, is the narrowest part of the Tuscan or Doric capitals, lying above the shaft of the pillar, between the astragal and the annulets.</p><p>It is also a kind of concave moulding, serving for compartments &c, larger than a scotia, but not so deep.</p><div2 part="N" n="Gorge" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gorge</hi></head><p>, in Fortification, is the entrance into a bastion, or a ravelin, or other out-work.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Gorge</hi> <hi rend="italics">of a Bastion,</hi> is what remains of the sides of the polygon of a place, after cutting off the curtains; in which case it makes an angle in the centre of the bastion, viz, the angle made by two adjacent curtains produced to meet within the bastion.</p><p>In flat bastions, the Gorge is a right line on the curtain, reaching between the two flanks.</p><p><hi rend="smallcaps">Gorge</hi> <hi rend="italics">of a Half-moon,</hi> or <hi rend="italics">of a Ravelin,</hi> is the space between the two ends of their faces next the place.</p><p><hi rend="smallcaps">Gorge</hi> of the other out-works, is the interval between their sides next the great ditch.</p><p>All the Gorges are to be made without parapets: otherwise the besiegers, having taken possession of a work, might make use of them to defend themselves from the shot of the place. So that they are only fortified with pallisadoes, to prevent a surprize.</p><p>The <hi rend="italics">Demi</hi>-<hi rend="smallcaps">Gorge</hi>, or <hi rend="italics">Half</hi> the <hi rend="smallcaps">Gorge</hi>, is that part of the polygon between the flank and the centre of the bastion.</p><p>GOTHIC <hi rend="italics">Architecture,</hi> is that which deviates from the manner, character, proportions, &c, of the antique; having its ornaments wild and chimerical, and its profiles incorrect. This manner of building came originally from the North, whence it was brought, in the 5th century, by the Goths into Germany, and has since been introduced into other countries. The first or most ancient style of Gothic building was very solid, heavy, massive and simple, with semicircular arches, &c: but the more modern style of the Gothic is exceedingly rich, light, and delicate; having an abundance of little whimsical ornaments, with sharp-pointed arches formed by the intersections of different circular segments; also lofty and light spires and steeples, large ramisied windows, clustered pillars, &c. Of this kind are our English cathedrals, and many other old buildings.</p></div2></div1><div1 part="N" n="GRADUATION" org="uniform" sample="complete" type="entry"><head>GRADUATION</head><p>, is used for the act of Graduating, or dividing any thing into degrees.</p><p>For an account of the various methods of Graduating mathematical and astronomical instruments, by straight and circular diagonals, and by concentric arcs, &c; see <hi rend="italics">Plain</hi> <hi rend="smallcaps">Scale</hi>, <hi rend="italics">Nonius,</hi> and <hi rend="italics">Vernier.</hi> And for an account of Mr. Bird's improved method of dividing astronomical instruments, see <hi rend="smallcaps">Mural</hi> <hi rend="italics">Arch.</hi></p><p>Mr. Ramsden, an ingenious mathematical instrumentmaker of London, has lately published, by encouragement of the commissioners of longitude, an explanation<pb n="545"/><cb/> and description of an engine contrived by him for dividing mathematical instruments, accompanied with proper drawings; in consideration of which, the said commissioners have granted to him the sum of 615<hi rend="italics">l.</hi> See his book, 4to, 1777.</p><p>On the subject of dividing a foot into many thousand parts, for mathematical purposes, see Philos. Trans. vol. 2, p. 457, 459, 541, or Abr. vol. 1, pa. 218, 220, &c. And for an account of various other methods and Graduations, see a paper of Mr. Smeaton's in the Philos. Trans. vol. 76, for the year 1786, p. 1; being “Observations on the Graduation of astronomical instruments; with an explanation of the method invented by the late Mr. Henry Hindley, of York, clockmaker, to divide circles into any given number of parts.”</p></div1><div1 part="N" n="GRAHAM" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GRAHAM</surname> (<foreName full="yes"><hi rend="smallcaps">George</hi></foreName>)</persName></head><p>, clock and watch maker, the most ingenious and accurate artist in his time, was born at Gratwick, a village in the north of Cumberland, in 1675. In 1688 he came up to London, and was put apprentice to a person in that profession; but after being some time with his master, he was received, purely on account of his merit, into the family of the celebrated Mr. Tompion, who treated him with a kind of parental affection as long as he lived. That Mr. Graham was, without competition, the most eminent of his profession, is but a small part of his character: he was the best general mechanic of his time, and had a complete knowledge of practical astronomy; so that he not only gave to various movements for measuring time a degree of perfection which had never before been attained, but invented several astronomical instruments, by which considerable advances have been made in that science: he made great improvements in those which had before been in use; and, by a wonderful manual dexterity, constructed them with greater precision and accuracy than any other person in the world.</p><p>A great mural arch in the observatory at Greenwich was made for Dr. Halley, under Mr. Graham's immediate inspection, and divided by his own hand: and from this incomparable original, the best foreign instruments of the kind are copies made by English artists. The sector by which Dr. Bradley first discovered two new motions in the fixed stars, was of his invention and fabric. He comprised the whole planetary system within the compass of a small cabinet; from which, as a model, all the modern orreries have been constructed. And when the French Academicians were sent to the north, to make observations for ascertaining the figure of the earth, Mr. Graham was thought the fittest person in Europe to supply them with instruments; by which means they finished their operations in one year; while those who went to the south, not being so well furnished, were very much embarrassed and retarded in their operations.</p><p>Mr. Graham was many years a member of the Royal Society, to which he communicated several ingenious and important discoveries, viz, from the 31st to the 42d volume of the Philos. Transactions, chiefly on astronomical and philosophical subjects; particularly a kind of horary alteration of the magnetic needle; a quicksilver pendulum, and many curious particulars relating to the true length of the simple pendulum, upon<cb/> which he continued to make experiments till almost the year of his death, which happened in 1751, at 76 years of age.</p><p>His temper was not less communicative than his genius was penetrating; and his principal view was the advancement of science, and the benesit of mankind. As he was perfectly sincere, he was above suspicion; and as he was above envy, he was candid.</p></div1><div1 part="N" n="GRANADO" org="uniform" sample="complete" type="entry"><head>GRANADO</head><p>, in Artillery, is a little shell or hollow globe of iron, or other matter, which, being filled with powder, is fired by means of a small fusee, and thrown either by the hand, or a piece of ordnance. As soon as it is kindled, the case slies in pieces, to the great danger of all that stand near it. Granadoes serve to set fire to close and narrow passages, and are often thrown with the hand among the soldiers, to disorder their ranks; more especially in those posts where they stand thickest, as in trenches, redoubts, lodgments, &c.</p></div1><div1 part="N" n="GRAVESANDE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GRAVESANDE</surname> (<foreName full="yes"><hi rend="smallcaps">William James</hi></foreName>)</persName></head><p>, a very celebrated Dutch mathematician and philosopher, was born at Bois-le-duc, Sept. 27, 1688. He studied the civil law at Leyden, but mathematical learning was his favourite amusement. When he had taken his doctor's degree, in 1707, he went and settled at the Hague, where he practised at the bar, and cultivated an acquaintance with learned men; with a Society of whom he published a periodical Review, entitled <hi rend="italics">Le Journal Litteraire,</hi> which was continued without interruption from the year 1713 to the year 1722. The parts of it written or extracted by Gravesande were chiefly those relating to geometry and physics. But he en<*> riched it also with several original pieces entirely of his own composition; viz, Remarks on the Construction of Pneumatical Engines: A Moral Essay on Lying: And a celebrated Essay on the Collision of Bodies; which, as it opposed the Newtonian philosophy, was attacked by Dr. Clarke, and many other learned men.</p><p>In 1715, when the States sent to congratulate George the 1st, on his accession to the throne, Dr. Gravesande was appointed secretary to the embassy. During his stay in England he was admitted a member of the Royal Society, and became intimately acquainted with Sir Isaac Newton. On his return to Holland, he was chosen professor of mathematics and astronomy at Leyden; where he had the honour of first teaching the Newtonian philosophy, which was then in its infancy. He died in 1742, at 54 years of age.</p><p>Gravesande was a man amiable in his private character, and respectable in his public one; for few men of letters have rendered more eminent services to their country. The ministers of the republic consulted him on all occasions when his talents were requisite to assist them, which his skill in calculation often enabled him to do in money matters. He was of great service as a decypherer, in detecting the secret correspondence of their enemies. And in his own profession none ever applied the powers of nature with more success, or to more useful purposes.</p><p>Of his publications, the principal are.</p><p>1. An Introduction to the Newtonian Philosophy; or, a Treatise on the Elements of Physics, coasirmed by Experiments. This performance, being only a more<pb n="546"/><cb/> perfect copy of his public lectures, was first printed in 1720; and hath since gone through many editions, with considerable improvements: the 6th edit. is in English, in 2 large vols. 4to, by Dr. Desaguliers, in 1747, under the title of Mathematical Elements of Natural Philosophy, confirmed by Experiments.</p><p>2. A treatise on the Elements of Algebra, for the use of Young Students; to which is added a Specimen of a Commentary on Newton's Universal Arithmetic; as also, A New Rule for determining the Form of an Assumed Infinite Series.</p><p>3. An Essay on Perspective. This was written at 19 years of age.</p><p>4. A New Theory of the Collision of Bodies.</p><p>5. A Course of Logic and Metaphysics.</p><p>With several smaller pieces.</p><p>His whole mathematical and philosophical works, except the first article above, were collected and published at Amsterdam, in 2 vols. 4to, to which is presixed a critical account of his life and writings, by Professor Allamand.</p></div1><div1 part="N" n="GRAVITATION" org="uniform" sample="complete" type="entry"><head>GRAVITATION</head><p>, the exercise of gravity, or the pressure a body exerts on another body beneath it by the power of gravity.</p><p>This is sometimes distinguished from gravity. Thus M. Maupertuis, in his <hi rend="italics">Figure de la Terre,</hi> takes gravity for that force by which a body would fall to the earth supposed at rest; and Gravitation for the same, but diminished by the centrifugal force. It is only Gravitation, or gravity thus blended with the centrifugal force, that we can usually measure by our experiments. Methods however have been found to distinguish what remains of the primitive gravity, and what has been destroyed by the centrifugal force.</p><p>It is one of the laws of nature, discovered by Newton, and now received by all philosophers, that every particle of matter in nature gravitates towards every other particle; which law is the main principle in the Newtonian philosophy. But what is called Gravitation with respect to the gravitating body, is usually called attraction with respect to the body gravitated to. The planets, both primary and secondary, as also the comets, do all gravitate towards the sun, and towards each other; as well as the sun towards them; and that in proportion to the quantity of matter in each of them.</p><p>The Peripatetics &c hold, that bodies only gravitate or weigh when out of their natural places, and that Gravitation ceases when they are restored to the same, the purpose of nature being then fulfilled: and they maintain that the final cause of this faculty is only to bring elementary bodies to their proper place, where they may rest. But the moderns shew that bodies exercise gravity even when at rest, and in their proper places. This is particularly shewn of sluids; and it is one of the laws of hydrostatics, demonstrated by Boyle and others, that fluids gravitate in proprio loco, the upper parts pressing on the lower, &c.</p><p>For the laws of Gravitation of bodies in fluids specifically lighter or heavier than themselves, see S<hi rend="smallcaps">PECIFIC Gravity.</hi> Also for the centre or line or plane of Gravitation, see <hi rend="smallcaps">Centre, Line</hi>, or <hi rend="smallcaps">Plane.</hi></p></div1><div1 part="N" n="GRAVITY" org="uniform" sample="complete" type="entry"><head>GRAVITY</head><p>, in Physics, the natural tendency or inclination of bodies towards the centre. And in this sense Gravity agrees with Centripetal force.<cb/></p><p>Gravity however is, by some, defined more generally as the natural tendency of one body towards another; and again by others still more generally as the mutual tendency of each body, and each particle of a body, towards all others: in which sense the word answers to what is more usually called Attraction. Indeed the terms Gravity, weight, centripetal force, and attraction, denote in effect all the same thing, only in different views and relations; all which however it is very common to confound, and use promiscuously. But, in propriety, when a body is considered as tending towards the earth, the force with which it so tends is called Gravity, Force of Gravity, or Gravitating Force; when the body is considered as immediately tending to the centre of the earth, it is called Centripetal Force; but when we consider the earth, or mass to which the body tends, it is called Attraction, or Attractive Force; and when it is considered in respect of an obstacle or another body in the way of its tendency, upon which it acts, it is called Weight.</p><p>Philosophers think differently on the subject of Gravity. Some consider it as an inactive property or innate power in bodies, by which they endeavour to join their centre. Others hold Gravity in this sense to be an occult quality, and to be exploded as such out of all sound philosophy. Newton, though he often calls it a vis, power, or property in bodies, yet explains himself, that he means nothing more by the word but the effect or phenomenon: he does not consider the principle, the cause by which bodies tend downwards, but the tendency itself, which is no occult quality, but a sensible phenomenon, be its causes what they may; whether a property essential to body, as some make it, or superadded to it, as others; or even an impulse of some body from without, as others.</p><p>It is a law of nature long observed, that all bodies near the earth have a Gravity or weight, or a tendency towards its centre, or at least perpendicular to its surface; which law the moderns, and especially Sir I. Newton, from certain observations have found to be much more extensive, and holding universally with respect to all known bodies and matter in nature. It is therefore at present acknowledged as a principle or law of nature, that all bodies, and all the particles of all bodies, mutually gravitate towards each other: from which single principle it is that Newton has happily deduced all the great phenomena of nature.</p><p>Hence Gravity may be distinguished into Particular and General.</p><p><hi rend="italics">Particular</hi> <hi rend="smallcaps">Gravity</hi>, is that which respects the earth, or by which bodies descend, or tend towards the centre of the earth; the phenomena or properties of which are as follow:</p><p>1. All circumterrestrial bodies do hereby tend towards a point, which is either accurately or very nearly the centre of magnitude of the terraqueous globe. No<*> that it is meant that there is really any virtue or charm in the point called the centre, by which it attracts bodies; but because this is the result of the gravitation of bodies towards all the parts of which the earth consists.</p><p>2. This point or centre is fixed within the earth, or at least has been so far as any authentic history reaches. For a consequence of its shifting, though ever so little,<pb n="547"/><cb/> would be the overflowing of the low lands on that side of the globe towards which it should approach. Dr. Halley suggests, that it would well account for the universal deluge, to have the centre of gravitation removed for a time towards the middle of the then inhabited world; for the change of its place but the 2000th part of the radius of the earth, or about 2 miles, would be sufficient to lay the tops of the highest hills under water.</p><p>3. In all places equidistant from the centre of the earth, the force of Gravity is nearly equal. Indeed all parts of the earth's surface are not at equal distances from the centre, because the equatorial parts are higher than the polar parts by about 17 miles; as has been proved by the necessity of making the pendulum shorter in those places, before it will swing seconds. In the new Petersburg Transactions, vol. 6 and 7, M. Krafft gives a formula for the proportion of Gravity in different latitudes on the earth's surface, which is this: ; where <hi rend="italics">g</hi> denotes the Gravity at the equator, and <hi rend="italics">y</hi> the Gravity under any other latitude <foreign xml:lang="greek">l</foreign>. On this subject, see also the articles <hi rend="smallcaps">Degree</hi>, and <hi rend="smallcaps">Earth.</hi></p><p>4. Gravity equally affects all bodies, without regard either to their bulk, figure, or matter: so that, abstracting from the resistance of the medium, the most compact and loose, the greatest and smallest bodies would all descend through an equal space in the same time; as appears from the quick descent of very light bodies in an exhausted receiver. The space which bodies do actually fall, in vacuo, is 16 1/12 feet in the first second of time, in the latitude of London; and for other times, either greater or less than that, the spaces descended from rest are directly proportional to the squares of the times, while the falling body is not far from the earth's surface.</p><p>5. This power is the greatest at the earth's surface, from whence it decreases both upwards and downwards, but not both ways in the same proportion; for upwards the force of Gravity is less, or decreases, as the square of the distance from the centre increases, so that at a double distance from the centre, above the surface, the force would be only 1-4th of what it is at the surface; but below the surface, the power decreases in such sort that its intensity is in the direct ratio of the distance from the centre; so that at the distance of half a semidiameter from the centre, the force would be but half what it is at the surface; at 1/3 of a semidiameter the force would be 1/3, and so on.</p><p>6. As all bodies gravitate towards the earth, so does the earth equally gravitate towards all bodies; as well as all bodies towards particular parts of the earth, as hills, &c, which has been proved by the attraction a hill has upon a plumb line, insensibly drawing it aside.— Hence the gravitating force of entire bodies consists of those of all their parts: for by adding or taking away any part of the matter of a body, its Gravity is increased or decreased in the proportion cf the quantity of such portion to the whole mass. Hence also the gravitating powers of bodies, at the same distance from the centre, are proportional to the quantities of matter in the bodies.</p><p><hi rend="italics">General or Universal</hi> <hi rend="smallcaps">Gravity</hi>, is that by which all the planets tend to one another, and indeed by which<cb/> all the bodies and particles of matter in the universe tend towards one another.</p><p>The existence of the same principle of Gravitation in the superior regions of the heavens, as on the earth, is one of the great discoveries of Newton, who made the proof of it as easy as that on the earth. At first it would seem this was only conjecture with him: he observed that all bodies near the earth, and in its atmosphere, had the property of tending directly towards it; he soon conjectured that it probably extended much higher than any distance to which we could reach, or make experiments; and so on, from one distance to another, till he at length saw no reason why it might not extend as far as to the moon, by means of which she might be retained in her orbit as a stone in a sling is retained by the hand; and if so, he next inferred why might not a similar principle exist in the other great bodies in the universe, the sun and all the other planets, both primary and secondary, which might all be retained in their orbits, and perform their revolutions, by means of the same universal principle of gravitation.</p><p>These conjectures he soon realized and verified by mathematical proofs. Kepler had found out, by contemplating the motions of the planets about the sun, that the area described by a line connecting the sun and planet, as this revolved in its orbit, was always proportional to the time of its description, or that it described equal areas in equal times, in whatever part of its orbit the planet might be, moving always so much the quicker as its distance from the sun was less. And it is also found that the satellites, or secondary planets, respect the same law in revolving about their primaries. But it was soon proved by Newton, that all bodies moving in any curve line described on a plane, and which, by radii drawn to any certain point, describe areas about the point proportional to the times, are impelled or acted on by some power tending towards that point. Confequently the power by which all these planets revolve, and are retained in their orbits, is directed to the centre about which they move, viz, the primary planets to the sun, and the satellites to their several primaries.</p><p>Again, Newton demonstrated, that if several bodies revolve with an equable motion in several circles about the same centre, and that if the squares<*> of their periodical times be in the same proportion as the cubes of their distances from the common centre, then the centripetal forces of the revolving bodies, by which they tend to their central body, will be in the reciprocal or inverse ratio of the squares of the distances. Or if bodies revolve in orbits approaching to circles, and the apses of those orbits be at rest, then also the centripetal forces of the revolving bodies will be reciprocally proportional to the squares of the distances. But it had been agreed on by the astronomers, and particularly Kepler, that both these cases obtain in all the planets. And therefore he inferred that the centripetal forces of all the planets are reciprocally proportional to the squares of the distances from the centres of their orbits.</p><p>Upon the whole it appears, that the planets are retained in their orbits by some power which is continually acting upon them: that this power is directed towards the centre of their orbits: that the intensity or efficacy of this power increases upon an approach<pb n="548"/><cb/> towards the centre, and diminishes on receding from the same, and that in the reciprocal duplicate ratio of the distances: and that, by comparing this centripetal force of the planets with the force of gravity on the earth, they are found to be perfectly alike, as may easily be shewn in various instances. For example, in the case of the moon, the nearest of all the planets. The rectilinear spaces described in any given time by a falling body, urged by any powers, reckoning from the beginning of its descent, are proportional to those powers. Consequently the centripetal force of the moon revolving in her orbit, will be to the force of Gravity on the surface of the earth, as the space which the moon would describe in falling during any small time, by her centripetal force towards the earth, if she had no circular motion at all, to the space a body near the earth would describe in falling by its Gravity towards the same.</p><p>Now by an easy calculation of those two spaces, it appears that the former force is to the latter, as the square of the semi-diameter of the earth is to the square of that of the moon's orbit. The moon's centripetal force therefore is equal to the force of Gravity; and consequently these forces are not different, but they are one and the same: for if they were different, bodies acted on by the two powers conjointly would fall towards the earth with a velocity double to that arising from the sole power of Gravity.</p><p>It is evident therefore that the moon's centripetal force, by which she is retained in her orbit, and prevented from running off in tangents, is the very power of Gravity of the earth extended thither. See Newton's Princip. lib. 1, prop. 45, cor. 2, and lib. 3, prop. 3; where the numeral calculation may be seen at full length.</p><p>The moon therefore gravitates towards the earth, and reciprocally the earth towards the moon. And this is also farther consirmed by the phenomena of the tides.</p><p>The like reasoning may also be applied to the other planets. For as the revolutions of the primary planets round the sun, and those of the satellites of Jupiter and Saturn round their primaries, are phenomena of the same kind with the revolution of the moon about the earth; and as the centripetal powers of the primary are directed towards the centre of the sun, and those of the satellites towards the centres of their primaries; and lastly as all these powers are reciprocally as the squares of the distances from the centres, it may safely be concluded that the power and cause are the same in all.</p><p>Therefore, as the moon gravitates towards the earth, and the earth towards the moon; so do all the secondaries to their primaries, and these to their secondaries; and so also do the primaries to the sun, and the sun to the primaries. Newton's Princip. lib. 3, prop. 4, 5, 6; Greg. Astron. lib. 1, sect. 7, prop. 46 and 47.</p><p>The laws of Universal Gravity are the same as thofe of bodies gravitating towards the earth, before laid down.</p><p><hi rend="italics">Cause of</hi> <hi rend="smallcaps">Gravity.</hi> Various theories have been advanced by the philosophers of different ages to account for this grand principle of Gravitation. The ancients, who were only acquainted with particular Gravity, or the tendency of sublu<*>ar bodies towards the earth, aimed no farther than a system that might answer the<cb/> more obvious phenomena of it. However, some hints are sound concerning the Gravitation of celestial bodies in the account given of the doctrine of Thales and his successors; and it would seem that Pythagoras was still better acquainted with it, to which it is supposed he had a view in what he taught concerning the Harmony of the Spheres.</p><p>Aristotle and the Peripatetics content themselves with referring Gravity or weight to a native inclination in heavy bodies to be in their proper place or sphere, the centre of the earth. And Copernicus ascribes it to an innate principle in all parts of matter, by which, when separated from their wholes, they endeavour to return to them again the nearest way. In answer to Aristotle and his followers, who considered the centre of the earth as the centre of the universe, he observed that it was reasonable to think there was nothing peculiar to the earth in this principle of Gravity; that the parts of the sun, moon, and stars tended likewise to each other, and that their spherical figure was preserved in their various motions by this power. Copern. Revol. lib. 1, cap. 9. But neither of these systems assigns any physical cause of this great effect: they only amount to this, that bodies descend because they are inclined to descend.</p><p>Kepler, in his preface to the commentaries concerning the planet Mars, speaks of Gravity as of a power that was mutual between bodies, and says that the earth and moon tend towards each other, and would meet in a point so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds, that the tides arise from the Gravity of the waters towards the moon. To him we also owe the important discovery of the analogy between the distances of the several planets from the sun, and the periods in which they complete their revolutions, viz, that the squares of their periodic times are always in the same proportion as the cubes of their mean distances from the sun. However, Kepler, Gassendi, Gilbert, and others, ascribe Gravity to a certain magnetic attraction of the earth; conceiving the earth to be one great magnet continually emitting effluvia, which take hold of all bodies, and draw them towards the earth. But this is inconfistent with the several phenomena.</p><p>Des Cartes and his followers, Rohault &c, attribute Gravity to an external impulse or trusion of some subtle matter. By the rotation of the earth, say they, all the parts and appendages of it necessarily endeavour to recede from the centre of rotation; but whence they cannot all actually recede, as there is no vacuum or space to receive them. But this hypothesis, founded on the supposition of a plenum, is overthrown by what has been since proved of the existence of a vacuum.</p><p>Dr. Hook inclines to an opinion much like that of Des Cartes. Gravity he thinks deducible from the action of a most subtle medium, which easily pervades and penetrates the most solid bodies; and which, by some motion it has, detrudes all earthly bodies from it, towards the centre of the earth. Vossius too, and many others, give partly into the Cartesian notion, and suppose Gravity to arise from the diurnal rotation of the earth round its axis.</p><p>Dr. Halley, despairing of any satisfactory theory,<pb n="549"/><cb/> chooses to have immediate recourse to the agency of the Deity. So Dr. Clarke, from a view of several properties of Gravity, concludes that it is no adventitious effect of any motion, or subtle matter, but an original and general law impressed by God on all matter, and preserved in it by some efficient power penetrating the very solid and intimate substance of it; being found always proportional, not to the surfaces of bodies or corpuscles, but to their solid quantity and contents. It should therefore be no more inquired why bodies gravitate, than how they came to be first put in motion. Annot. in Rohault. Phys. part 1, cap. 11.</p><p>Gravesande, in his Introduct. ad Philos. Newton. contends that the cause of Gravity is utterly unknown; and that we are to consider it no otherwise than as a law of nature originally and immediately impressed by the Creator, without any dependence on any second law or cause at all. Of this he thinks the three following considerations sufficient proof. 1. That Gravity requires the presence of the gravitating or attracting body: so the satellites of Jupiter, for ex. gravitate towards Jupiter, wherever he may be. 2. That the distance being supposed the same, the velocity with which bodies are moved by the force of Gravity, depends on the quantity of matter in the attracting body: and the velocity is not changed, whatever the mass of the gravitating body may be. 3. That if Gravity do depend on any known law of motion, it must be some impulse from an extraneous body; so that as Gravity is continual, a continual stroke must also be required. Now if there be any such matter continually striking on bodies, it must be fluid, and subtle enough to penetrate the substance of all bodies: but how shall a body subtle enough to penetrate the substance of the hardest bodies, and so rare as not sensibly to hinder the motion of bodies, be able to impel vast masses towards each other with such force? how does this force increase the ratio of the mass of the body, towards which the other body is moved? whence is it that all bodies move with the same velocity, the distance and body gravitated to being the same? can a fluid which only acts on the surface either of the bodies themselves, or their internal particles, communicate such a quantity of motion to bodies, which in all bodies shall exactly follow the proportion of the quantity of matter in them?</p><p>Mr. Cotes goes yet farther. Giving a view of Newton's philosophy, he asserts that Gravity is to be ranked among the primary qualities of all bodies; and deemed equally essential to matter as extension, mobility, or impenetrability. Præfat. ad Newt. Princip. But Newton himself disclaims this notion; and to shew that he does not take Gravity to be essential to bodies, he declares his opinion of the cause; choosing to propose it by way of query, not being yet sufficiently satissied about its experiments. Thus, after having shewn that there is a medium in nature vastly more subtle than air, by whose vibrations sound is propagated, by which light communicates heat to bodies, and by the different densities of which the refraction and reslection of light are performed; he proceeds to inquire: “Is not this medium much rarer within the dense bodies of the sun, stars, planets, and comets, than<cb/> in the empty celestial spaces between them? and in passing from them to greater distances, doth it not grow denser and denser perpetually, and thereby cause the Gravity of those great bodies towards one another, and of their parts towards the bodies; every body endeavouring to recede from the denser parts of the medium towards the rarer?</p><p>For if this medium be supposed rarer within the sun's body than at its surface, and rarer there than at the hundredth part of an inch from his body, and rarer there than at the fiftieth part of an inch from his body, and rarer there than at the orb of Saturn; I see no reason why the increase of density should stop any where, and not rather be continued through all distances from the Sun to Saturn, and beyond.</p><p>And though this increase of density may at great distances be exceeding slow; yet if the elastic force of this medium be exceeding great, it may suffice to impel bodies from the denser parts of the medium towards the rarer with all that power which we call Gravity.</p><p>And that the elastic force of this medium is exceeding great, may be gathered from the swiftness of its vibrations. Sounds move about 1140 English feet in a second of time, and in seven or eight minutes of time, they move about one hundred English miles: light moves from the Sun to us in about seven or eight minutes of time, which distance is about 70000000 English miles, supposing the horizontal parallax of the Sun to be about twelve seconds; and the vibrations, or pulses of this medium, that they may cause the alternate fits of easy transmission, and easy reslection, must be swifter than light, and by consequence above 700000 times swifter than sounds; and therefore the elastic force of this medium, in proportion to its density, must be above 700000 × 700000 (that is, above 490000000000) times greater than the elastic force of the air is in proportion to its density: for the velocities of the pulses of elastic mediums are in a subduplicate ratio of the elasticities and the rarities of the mediums taken together.</p><p>As Magnetism is stronger in small loadstones than in great ones, in proportion to their bulk; and Gravity is stronger on the surface of small planets, than those of great ones, in proportion to their bulk; and small bodies are agitated much more by electric attraction than great ones: so the smallness of the rays of light may contribute very much to the power of the agent by which they are refracted; and if any one should suppose, that æther (like our air) may contain particles which endeavour to recede from one another (for I do not know what this æther is), and that its particles are exceedingly smaller than those of air, or even than those of light; the exceeding smallness of such particles may contribute to the greatness of the force, by which they recede from one another, and thereby make that medium exceedingly more rare and elastic than air, and of consequence, exceedingly less able to resist the motions of projectiles, and exceedingly more able to press upon gross bodies by endeavouring to expand itself.” Optics, p. 325 &c.</p><div2 part="N" n="Gravity" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gravity</hi></head><p>, in Mechanics, denotes the conatus or tendency of bodies towards the centre of the earth.— That part of mechanics which considers the equilibrium<pb n="550"/><cb/> or motion of bodies arising from Gravity or weight, is particularly called statics.</p><p>Gravity in this view is distinguished into Absolute and Relative.</p><p><hi rend="italics">Absolute</hi> <hi rend="smallcaps">Gravity</hi> is that with which a body descends freely and perpendicularly through an unresisting medium. The laws of which see under <hi rend="smallcaps">Descent of Bodies, Acceleration, Motion</hi>, &c.</p><p><hi rend="italics">Relative</hi> <hi rend="smallcaps">Gravity</hi> is that with which a body descends on an inclined plane, or through a resisting medium, or as opposed by some other resistance. The laws of which see under the articles <hi rend="smallcaps">Inclined Plane, Descent, Fluid, Resistance</hi>, &c.</p></div2><div2 part="N" n="Gravity" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Gravity</hi></head><p>, in Hydrostatics. The laws of bodies gravitating in Fluids make the business of Hydrostatics.</p><p>Gravity is here divided into Absolute and Specific.</p><p><hi rend="italics">Absolute</hi> or <hi rend="italics">True</hi> <hi rend="smallcaps">Gravity</hi>, is the whole force with which the body tends downwards.</p><p><hi rend="italics">Specific</hi> <hi rend="smallcaps">Gravity</hi>, is the relative, comparative, or apparent Gravity in any body, in respect of that of an equal bulk or magnitude of another body; denoting that Gravity or weight which is peculiar to each species or kind of body, and by which it is distinguished from all other kinds.</p><p>In this sense a body is said to be Specifically Heavier than another, when under the same bulk it contains a greater weight than that other; and reciprocally the latter is said to be Specifically Lighter than the former. Thus, if there be two equal spheres, each one foot in diameter; the one of lead, and the other of wood: since the leaden one is found heavier than the wooden one, it is said to be Specifically, or in Specie, Heavier; and the wooden one Specifically Lighter.</p><p>This kind of Gravity is by some called Relative; in opposition to Absolute Gravity, which increases in proportion to the quantity or mass of the body. <hi rend="center"><hi rend="italics">Laws of the <hi rend="smallcaps">Specific Gravity</hi> of bodies.</hi></hi></p><p>I. If two bodies be equal in bulk, their specific gravities are to each other as their weights, or as their densities.</p><p>II. If two bodies be of the same specific gravity or density, their absolute weights will be as their magnitudes or bulks.</p><p>III. In bodies of the same weight, the specific gravities are reciprocally as their bulks.</p><p>IV. The specific gravities of all bodies are in a ratio compounded of the direct ratio of their weights, and the reciprocal ratio of their magnitudes. And hence again the specific gravities are as the densities.</p><p>V. The absolute gravities or weights of bodies are in the compound ratio of their specific gravities and magnitudes or bulks.</p><p>VI. The magnitudes of bodies are directly as their weights, and reciprocally as their specific gravities.</p><p>VII. A body specifically heavier than a fluid, loses as much of its weight when immersed in it, as is equal to the weight of a quantity of the fluid of the same bulk or magnitude.</p><p>Hence, since the Specific Gravities are as the abso-<cb/> lute gravities under the same bulk; the Specific Gravity of the fluid, will be to that of the body immerged, as the part of the weight lost by the solid, is to the whole weight.</p><p>And hence the Specific Gravities of fluids are as the weights lost by the same solid immerged in them.</p><p>VIII. <hi rend="italics">To find the Specific Gravity of a Fluid, or of a Solid.</hi>—On one arm of a balance suspend a globe of lead by a fine thread, and to the other fasten an equal weight, which may just balance it in the open air. Immerge the globe into the fluid, and observe what weight balances it then, and consequently what weight is lost, which is proportional to the Specific Gravity as above. And thus the proportion of the Specific Gravity of one fluid to another is determined by immersing the globe successively in all the fluids, and observing the weights lost in each, which will be the proportions of the Specific Gravities of the fluids sought.</p><p>This same operation determines also the Specific Gravity of the solid immerged, whether it be a globe or of any other shape or bulk, supposing that of the fluid known. For the Specific Gravity of the fluid is to that of the solid, as the weight lost is to the whole weight.</p><p>Hence also may be found the Specific Gravity of a body that is lighter than the fluid, as follows:</p><p>IX. <hi rend="italics">To find the Specific Gravity of a Solid that is lighter than the fluid, as water, in which it is put.</hi>—Annex to the lighter body another that is much heavier than the fluid, so as the compound mass may sink in the fluid. Weigh the heavier body and the compound mass separately, both in water and out of it; then find how much each loses in water, by subtracting its weight in water from its weight in air; and subtract the less of these remainders from the greater.</p><p>Then, As this last remainder, Is to the weight of the light body in air,<lb/> So is the Specific Gravity of the fluid,<lb/> To the Specific Gravity of that body.<lb/></p><p>X. The Specific Gravities of bodies of equal weight, are reciprocally proportional to the quantities of weight lost in the same fluid. And hence is found the ratio of the Specific Gravities of solids, by weighing in the same fluids, masses of them that weigh equally in air, and noting the weights lost by each.</p><p>The Specific Gravities of many kinds of bodies, both solid and fluid, have been determined by varioüs authors. Marinus Ghetaldus particularly tried the Specific Gravities of various bodies, especially metals; which were taken from thence by Oughtred. In the Philos. Trans. are several ample tables of them, by various authors, particularly those of Mr. Davis, vol. 45, p. 416, or Abr. vol. 10, p. 206. Some tables of them were also published by P. Mersenne, Muschenbroeck, Ward, Cotes, Emerson, Martin, &c.</p><p>It will be sufficient here to give those of some of the most usual bodies, that have been determined with the greater certainty. The numbers in this table express the number of Avoirdupois ounces in a cubic foot of each body, that of common water being just 1000 ounces, or 62 1/2 lb.<pb n="551"/><cb/> <hi rend="center"><hi rend="italics"><hi rend="smallcaps">Table</hi> of <hi rend="smallcaps">Specific Gravities.</hi></hi></hi> <hi rend="center">I. <hi rend="italics">Solids.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data">Platina, pure</cell><cell cols="1" rows="1" rend="align=right" role="data">23000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fine gold</cell><cell cols="1" rows="1" rend="align=right" role="data">19640</cell></row><row role="data"><cell cols="1" rows="1" role="data">Standard gold</cell><cell cols="1" rows="1" rend="align=right" role="data">18888</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lead</cell><cell cols="1" rows="1" rend="align=right" role="data">11325</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fine Silver</cell><cell cols="1" rows="1" rend="align=right" role="data">11091</cell></row><row role="data"><cell cols="1" rows="1" role="data">Standard Silver</cell><cell cols="1" rows="1" rend="align=right" role="data">10535</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper</cell><cell cols="1" rows="1" rend="align=right" role="data">9000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copper halfpence</cell><cell cols="1" rows="1" rend="align=right" role="data">8915</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gun metal</cell><cell cols="1" rows="1" rend="align=right" role="data">8784</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fine brass</cell><cell cols="1" rows="1" rend="align=right" role="data">8350</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cast brass</cell><cell cols="1" rows="1" rend="align=right" role="data">8000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Steel</cell><cell cols="1" rows="1" rend="align=right" role="data">7850</cell></row><row role="data"><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" rend="align=right" role="data">7645</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pewter</cell><cell cols="1" rows="1" rend="align=right" role="data">7471</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cast Iron</cell><cell cols="1" rows="1" rend="align=right" role="data">7425</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tin</cell><cell cols="1" rows="1" rend="align=right" role="data">7320</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lapis calaminaris</cell><cell cols="1" rows="1" rend="align=right" role="data">5000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Loadstone</cell><cell cols="1" rows="1" rend="align=right" role="data">4930</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mean of the whole Earth</cell><cell cols="1" rows="1" rend="align=right" role="data">4500</cell></row><row role="data"><cell cols="1" rows="1" role="data">Crude Antimony</cell><cell cols="1" rows="1" rend="align=right" role="data">4000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diamond</cell><cell cols="1" rows="1" rend="align=right" role="data">3517</cell></row><row role="data"><cell cols="1" rows="1" role="data">Granite</cell><cell cols="1" rows="1" rend="align=right" role="data">3500</cell></row><row role="data"><cell cols="1" rows="1" role="data">White lead</cell><cell cols="1" rows="1" rend="align=right" role="data">3160</cell></row><row role="data"><cell cols="1" rows="1" role="data">Island crystal</cell><cell cols="1" rows="1" rend="align=right" role="data">2720</cell></row><row role="data"><cell cols="1" rows="1" role="data">Marble</cell><cell cols="1" rows="1" rend="align=right" role="data">2705</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pebble stone</cell><cell cols="1" rows="1" rend="align=right" role="data">2700</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jasper</cell><cell cols="1" rows="1" rend="align=right" role="data">2666</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rock crystal</cell><cell cols="1" rows="1" rend="align=right" role="data">2650</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pearl</cell><cell cols="1" rows="1" rend="align=right" role="data">2630</cell></row><row role="data"><cell cols="1" rows="1" role="data">Green glass</cell><cell cols="1" rows="1" rend="align=right" role="data">2600</cell></row><row role="data"><cell cols="1" rows="1" role="data">Flint</cell><cell cols="1" rows="1" rend="align=right" role="data">2570</cell></row><row role="data"><cell cols="1" rows="1" role="data">Onyx stone</cell><cell cols="1" rows="1" rend="align=right" role="data">2510</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common stone</cell><cell cols="1" rows="1" rend="align=right" role="data">2500</cell></row><row role="data"><cell cols="1" rows="1" role="data">Crystal</cell><cell cols="1" rows="1" rend="align=right" role="data">2210</cell></row><row role="data"><cell cols="1" rows="1" role="data">Clay</cell><cell cols="1" rows="1" rend="align=right" role="data">2160</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oyster shells</cell><cell cols="1" rows="1" rend="align=right" role="data">2092</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brick</cell><cell cols="1" rows="1" rend="align=right" role="data">2000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common earth</cell><cell cols="1" rows="1" rend="align=right" role="data">1984</cell></row><row role="data"><cell cols="1" rows="1" role="data">Nitre</cell><cell cols="1" rows="1" rend="align=right" role="data">1900</cell></row><row role="data"><cell cols="1" rows="1" role="data">Vitriol</cell><cell cols="1" rows="1" rend="align=right" role="data">1880</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alabaster</cell><cell cols="1" rows="1" rend="align=right" role="data">1874</cell></row><row role="data"><cell cols="1" rows="1" role="data">Horn</cell><cell cols="1" rows="1" rend="align=right" role="data">1840</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ivory</cell><cell cols="1" rows="1" rend="align=right" role="data">1825</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sulphur</cell><cell cols="1" rows="1" rend="align=right" role="data">1810</cell></row><row role="data"><cell cols="1" rows="1" role="data">Chalk</cell><cell cols="1" rows="1" rend="align=right" role="data">1793</cell></row><row role="data"><cell cols="1" rows="1" role="data">Solid gunpowder</cell><cell cols="1" rows="1" rend="align=right" role="data">1745</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alum</cell><cell cols="1" rows="1" rend="align=right" role="data">1714</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dry bone</cell><cell cols="1" rows="1" rend="align=right" role="data">1660</cell></row><row role="data"><cell cols="1" rows="1" role="data">Human calculus</cell><cell cols="1" rows="1" rend="align=right" role="data">1542</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sand</cell><cell cols="1" rows="1" rend="align=right" role="data">1520</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lignum vitæ</cell><cell cols="1" rows="1" rend="align=right" role="data">1327</cell></row><row role="data"><cell cols="1" rows="1" role="data">Coal</cell><cell cols="1" rows="1" rend="align=right" role="data">1250</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jet</cell><cell cols="1" rows="1" rend="align=right" role="data">1238</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ebony</cell><cell cols="1" rows="1" rend="align=right" role="data">1177</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pitch</cell><cell cols="1" rows="1" rend="align=right" role="data">1150</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rosin</cell><cell cols="1" rows="1" rend="align=right" role="data">1100</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mahogany</cell><cell cols="1" rows="1" rend="align=right" role="data">1063</cell></row><row role="data"><cell cols="1" rows="1" role="data">Amber</cell><cell cols="1" rows="1" rend="align=right" role="data">1040</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brazil wood</cell><cell cols="1" rows="1" rend="align=right" role="data">1031</cell></row><row role="data"><cell cols="1" rows="1" role="data">Boxwood</cell><cell cols="1" rows="1" rend="align=right" role="data">1030</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common water</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell></row></table><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">Bees wax</cell><cell cols="1" rows="1" rend="align=right" role="data">955</cell></row><row role="data"><cell cols="1" rows="1" role="data">Butter</cell><cell cols="1" rows="1" rend="align=right" role="data">940</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oak</cell><cell cols="1" rows="1" rend="align=right" role="data">925</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gunpowder, shaken</cell><cell cols="1" rows="1" rend="align=right" role="data">922</cell></row><row role="data"><cell cols="1" rows="1" role="data">Logwood</cell><cell cols="1" rows="1" rend="align=right" role="data">913</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ice</cell><cell cols="1" rows="1" rend="align=right" role="data">908</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ash</cell><cell cols="1" rows="1" rend="align=right" role="data">800</cell></row><row role="data"><cell cols="1" rows="1" role="data">Maple</cell><cell cols="1" rows="1" rend="align=right" role="data">755</cell></row><row role="data"><cell cols="1" rows="1" role="data">Beech</cell><cell cols="1" rows="1" rend="align=right" role="data">700</cell></row><row role="data"><cell cols="1" rows="1" role="data">Elm</cell><cell cols="1" rows="1" rend="align=right" role="data">600</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fir</cell><cell cols="1" rows="1" rend="align=right" role="data">550</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sassafras wood</cell><cell cols="1" rows="1" rend="align=right" role="data">482</cell></row><row role="data"><cell cols="1" rows="1" role="data">Charcoal</cell><cell cols="1" rows="1" rend="align=right" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Cork</cell><cell cols="1" rows="1" rend="align=right" role="data">240</cell></row><row role="data"><cell cols="1" rows="1" role="data">New fallen snow</cell><cell cols="1" rows="1" rend="align=right" role="data">86</cell></row></table> <hi rend="center">II. <hi rend="italics">Fluids.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data">Quicksilver</cell><cell cols="1" rows="1" rend="align=right" role="data">13600</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of Vitriol</cell><cell cols="1" rows="1" rend="align=right" role="data">1700</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of Tartar</cell><cell cols="1" rows="1" rend="align=right" role="data">1550</cell></row><row role="data"><cell cols="1" rows="1" role="data">Honey</cell><cell cols="1" rows="1" rend="align=right" role="data">1450</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of Nitre</cell><cell cols="1" rows="1" rend="align=right" role="data">1315</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aqua Fortis</cell><cell cols="1" rows="1" rend="align=right" role="data">1300</cell></row><row role="data"><cell cols="1" rows="1" role="data">Treacle</cell><cell cols="1" rows="1" rend="align=right" role="data">1290</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aqua Regia</cell><cell cols="1" rows="1" rend="align=right" role="data">1234</cell></row><row role="data"><cell cols="1" rows="1" role="data">Human blood</cell><cell cols="1" rows="1" rend="align=right" role="data">1054</cell></row><row role="data"><cell cols="1" rows="1" role="data">Urine</cell><cell cols="1" rows="1" rend="align=right" role="data">1032</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cow's milk</cell><cell cols="1" rows="1" rend="align=right" role="data">1031</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sea Water</cell><cell cols="1" rows="1" rend="align=right" role="data">1030</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ale</cell><cell cols="1" rows="1" rend="align=right" role="data">1028</cell></row><row role="data"><cell cols="1" rows="1" role="data">Vinegar</cell><cell cols="1" rows="1" rend="align=right" role="data">1026</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tar</cell><cell cols="1" rows="1" rend="align=right" role="data">1015</cell></row><row role="data"><cell cols="1" rows="1" role="data">Water</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Distilled Water</cell><cell cols="1" rows="1" rend="align=right" role="data">993</cell></row><row role="data"><cell cols="1" rows="1" role="data">Red Wine</cell><cell cols="1" rows="1" rend="align=right" role="data">990</cell></row><row role="data"><cell cols="1" rows="1" role="data">Proof Spirits</cell><cell cols="1" rows="1" rend="align=right" role="data">931</cell></row><row role="data"><cell cols="1" rows="1" role="data">Olive Oil</cell><cell cols="1" rows="1" rend="align=right" role="data">913</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pure Spirits of Wine</cell><cell cols="1" rows="1" rend="align=right" role="data">866</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of Turpentine</cell><cell cols="1" rows="1" rend="align=right" role="data">800</cell></row><row role="data"><cell cols="1" rows="1" role="data">Æther</cell><cell cols="1" rows="1" rend="align=right" role="data">726</cell></row><row role="data"><cell cols="1" rows="1" role="data">Common Air</cell><cell cols="1" rows="1" rend="align=right" role="data">1.232</cell></row><row role="data"><cell cols="1" rows="1" role="data">   or very nearly</cell><cell cols="1" rows="1" rend="align=right" role="data">1 7/50. </cell></row></table></p><p>These numbers being the weight of a cubic foot, or 1728 cubic inches, of each of the bodies, in Avoirdupois ounces, by proportion the quantity in any other weight, or the weight of any other quantity, may be readily known.</p><p>For ex. Required the content of an irregular block of common stone which weighs 1 cwt, or 112lb, or 1792 ounces. Here, as cubic inches the content.</p><p>Ex. 2. To find the weight of a block of granite, whose length is 63 feet, and breadth and thickness each 12 feet; being the dimensions of one of the stones, of granite, in the walls of Balbeck. Here, feet is the content of the stone; therefore as or 885 tons 18 cwt. 3 qrs. the weight of the stone.</p><p>XI. A body descends in a fluid specifically lighter. or ascends in a fluid specifically heavier, with a force equal to the difference between its weight and that of an equal bulk of the fluid.</p><p>XII. A body sinks in a fluid specifically heavier, so<pb n="552"/><cb/> far as that the weight of the body is equal to the weight of a quantity of the fluid of the same bulk as the part immersed. Hence, as the Specific Gravity of the fluid is to that of the body, so is the whole magnitude of the body, to the magnitude of the part immersed.</p><p>XIII. The Specific Gravities of equal solids are as their parts immerged in the same fluid.</p><p>The several theorems here delivered, are both demonstrable from the principles of mechanics, and are also equally conformable to experiment, which answers exactly to the calculation; as is abundantly evident from the courses of philosophical experiments, so frequently exhibited; where the laws of specific gravitation are well illustrated.</p><p>GREAT BEAR, one of the constellations in the northern hemisphere. See <hi rend="smallcaps">Ursa Major.</hi></p><p><hi rend="smallcaps">Great Circles</hi>, of the Globe or Sphere, are those whose planes pass through the centre, dividing it into two equal parts or hemispheres, and therefore having the same centre and diameter with the sphere itself. The principal of these are, the equator, the ecliptic, the horizon, the meridians, and the two colures.</p><p><hi rend="smallcaps">Great-Circle Sailing</hi>, is the art of conducting a ship along the arc of a great circle. And it is also that part of the theory of navigation which treats of sailing in the arc of a great circle. See <hi rend="smallcaps">Navigation.</hi></p></div2></div1><div1 part="N" n="GREAVES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GREAVES</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent astronomer, antiquary and linguist, was born in 1602, being the eldest son of John Greaves rector of Colemore, near Alresford in Hampshire, and master of a grammar school, where his son of course was well grounded in the primary rules of literature. He then went to Baliol college in Oxford, in 1617; but afterward, on account of his skill in philosophy and polite literature, he was the first of five that were elected into Merton college. Having read over all the ancient Greek and Latin writers, he applied to the study of natural philosophy and mathematics; and having contracted an intimacy with Mr. Briggs, Savilian professor of geometry at Oxford, and Dr. Bainbridge, Savilian professor of astronomy there, he was animated by their examples to prosecute that study with the greatest industry; and not content with reading the writings of Purbach, Regiomontanus, Copernicus, Tycho Brahe, Kepler, and other celebrated astronomers of that and the preceding age, he made the ancient Greek, Arabian, and Persian authors familiar to him, having before gained an accurate skill in the oriental languages. These accomplishments procured him the professorship of geometry in Greshamcollege London, in 1630; and at the same time he held his fellowship of Merton-college.</p><p>In a journey to the Continent, in 1635, he visited the celebrated Golius, professor of Arabic at Leyden, and Claud Hardy at Paris, to converse about the Persian language. Hence he passed through Italy, and accurately surveyed the venerable remains of antiquity at Rome, visiting and corresponding everywhere with the most learned men of every nation. After visiting Padua, Florence, and Leghorn, he hence embarked for Constantinople, where he arrived in 1638. From thence he passed over to Rhodes, and Alexandria in Egypt, where he staid four or five months, and made a great number of curious observations. He next went to<cb/> Grand Cairo, measured the pyramids; and while there he adjusted the measure of the foot, observed by all nations. From hence he returned again through Italy, and arrived in England in the year 1640, after storing his mind with a variety of curious knowledge, and collecting many valuable oriental manuscripts and ancient curiosities; and while at Rome he made a particular inquiry into the true state of the ancient weights and measures.</p><p>On the death of Dr. John Bainbridge, in 1643, he was chosen Savilian professor of astronomy at Oxford, and principal reader of Linacre's lecture in Merton college; an appointment for which he was eminently qualisied, from his critical acquaintance with the works of the ancient and modern astronomers. In 1645 he proposed a method of reforming the calendar, by omitting the intercalary day for 40 years to come: the paper relating to which, was published by Dr. Thomas Smith, in the Philos. Trans. for 1699. In 1646, he published his Pyramidographia, or a Description of the Pyramids of Egypt; and, in 1647, his Discourse on the Roman Foot and Denarius; from which, as from two principles, the measures and weights used by the Ancients may be deduced. He also published several other curious works concerning antiquities, &c.</p><p>Soon after publishing the last mentioned book, he was ejected, by the parliament visitors, from the professorship of astronomy and fellowship of Merton-college; and the soldiers committed many outrages, breaking open his chests, and destroying many of his manuscripts; which greatly affected him. On this occasion he retired to London, where he afterwards married, and prosecuted his studies with great vigour, as appears from several of his philosophical and theological writings. This however proved but a transient happiness to him; for he died at London, the 8th of October 1652, before he was quite 50 years of age; and left his astronomical instruments to the Savilian library in Oxford, where they are deposited.</p><p>GREEK <hi rend="smallcaps">Orders</hi>, in Architecture, are the Doric, Ionic and Corinthian; in contradistinction to the two Latin orders, viz the Tuscan and Composite.</p></div1><div1 part="N" n="GREEN" org="uniform" sample="complete" type="entry"><head>GREEN</head><p>, One of the original colours of the rays of light, or of the prismatic colours exhibited by the refraction of the rays of light.</p><p><hi rend="italics">Green</hi> is the pleasantest of all the colours to the sight. And hence it has been inferred as a proof of the wisdom and goodness of the Deity, that almost all vegetables, cloathing the surface of the earth, are green; which they are when growing in the open air; though those in subterraneous places, or places inaccessible to fresh air, are white or yellow. See <hi rend="smallcaps">Chromatics</hi>, and <hi rend="smallcaps">Colours.</hi></p><p>GREGORIAN <hi rend="smallcaps">Calendar</hi>, so called from Pope Gregory the 13th, is the new or reformed Calendar, shewing the new and full moons, with the time of Easter, and the other moveable feasts depending upon it, by means of epacts disposed through the several months of the Gregorian year.</p><p><hi rend="smallcaps">Gregorian</hi> <hi rend="italics">Epoch,</hi> is the epoch or time, from which the Gregorian calendar, or computation, took place. This began in the year 1582; so that the year 1800 is the 218th of this epoch.<pb n="553"/><cb/></p><p><hi rend="smallcaps">Gregorian</hi> <hi rend="italics">Telescope,</hi> a particular sort of telescope, invented by Mr. James Gregory. See <hi rend="smallcaps">Telescope.</hi></p><p><hi rend="smallcaps">Gregorian</hi> <hi rend="italics">Year,</hi> the new account, or new style, introduced upon the reformation of the calendar, by Pope Gregory the 13th, in the year 1582, and from whom it took its name. This was introduced to reform the old, or Julian year, established by Julius Cæsar, which consisted of 365 days 6 hours, or 365 days and a quarter, that is three years of 365 days each, and the fourth year of 366 days. But as the mean tropical year consists only of 365ds 5hrs 48m 57sec. the former lost 11min. 3sec. every year, which in the time of Pope Gregory had amounted to 10 days, and who, by adding these 10 days, brought the account of time to its proper day again, and at the same time appointed that every century after, a day more should be added, thereby making the years of the complete centuries, viz 1600, 1700, 1800, &c, to be common years of 365 days each, instead of leap years of 366 days, which makes the mean Gregorian year equal to 365ds 5hrs 45m. 36sec.</p><p>This computation was not introduced into the account of time in England, till the year 1752, when the Julian account had lost 11 days, and therefore the 3d of September was in that year, by act of parliament, accounted the 14th, thereby restoring the 11 days which had thus been omitted. See <hi rend="smallcaps">Year.</hi></p></div1><div1 part="N" n="GREGORY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GREGORY</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, professor of mathematics, first in the university of St. Andrews, and afterwards in that of Edinburgh, was one of the most eminent mathematicians of the 17th century. He was a son of the Rev. Mr. John Gregory minister of Drumoak in the county of Aberdeen, and was born at Aberdeen in November 1638. His mother was a daughter of Mr. David Anderson of Finzaugh, or Finshaugh, a gentleman who possessed a singular turn for mathematical and mechanical knowledge. This mathematical genius was hereditary in the family of the Andersons, and from them it seems to have been transmitted to their descendants of the names of Gregory, Reid, &c. Alexander Anderson, cousin-german of the said David, was professor of mathematics at Paris in the beginning of the 17th century, and published there several valuable and ingenious works; as may be seen in the memoirs of his life and writings, under the article <hi rend="smallcaps">Anderson.</hi> The mother of James Gregory inherited the genius of her family; and observing in her son, while yet a child, a strong propensity to mathematics, she instructed him herself in the elements of that science. His education in the languages he received at the grammar school of Aberdeen, and went through the usual course of academical studies in the Marischal college; but he was chiefly delighted with philosophical researches, into which a new door had lately been opened by the key of the mathematics. Galileo, Kepler, Des Cartes, &c, were the great masters of this new method: their works therefore became the principal study of young Gregory, who soon began to make improvements upon their discoveries in Optics. The first of these improvements was the invention of the reflecting telescope; the construction of which instrument he published in his <hi rend="italics">Optica Promota,</hi> in 1663, at 24 years of age. This discovery soon attracted the attention of the mathematicians, both of our own and of foreign countries, who<cb/> immediately perceived its great importance to the sciences of optics and astronomy. But the manner of placing the two specula upon the same axis appearing to Newton to be attended with the disadvantage of losing the central rays of the larger speculum, he proposed an improvement on the instrument, by giving an oblique position to the smaller speculum, and placing the eye-glass in the side of the tube. It is observable however, that the Newtonian construction of that instrument was long abandoned for the original or Gregorian, which is now always used when the instrument is of a moderate size; though Herschel has preferred the Newtonian form for the construction of those immense telescopes, which he has of late so successfully employed in observing the heavens.</p><p>About the year 1664 or 1665, coming to London, he became acquainted with Mr. John Collins, who recommended him to the best optic glass-grinders there, to have his telescope executed. But as this could not be done for want of skill in the artists to grind a plate of metal for the object speculum into a true parabolic concave, which the design required, he was much discouraged with the disappointment; and after a few imperfect trials made with an ill-polished spherical one, which did not succeed to his wish, he dropped the pursuit, and resolved to make the tour of Italy, then the mart of mathematical learning, that he might prosecute his favourite study with greater advantage. And the university of Padua being at that time in high reputation for mathematical studies, Mr. Gregory fixed his residence there for some years. Here it was that he published, in 1667, <hi rend="italics">Vera Circuli et Hyperbola Quadra<*>ura;</hi> in which he propounded another discovery of his own, the invention of an infinitely converging series for the areas of the circle and hyperbola. He sent home a copy of this work to his friend Mr. Collins, who communicated it to the Royal Society, where it met with the commendations of lord Brounker and Dr. Wallis. He reprinted it at Venice the year following, to which he added a new work, entitled <hi rend="italics">Geometriæ Pars Universalis, inserviens Quantitatum Curvarum Transmutationi et Mensuræ;</hi> in which he is allowed to have shewn, for the first time, a method for the transmutation of curves. These works engaged the notice, and procured the author the correspondence of the greatest mathematicians of the age, Newton, Huygens, Wallis, and others. An account of this piece was also read by Mr. Collins before the Royal Society, of which Mr. Gregory, being returned from his travels, was chosen a member the same year, and communicated to them an account of a controversy in Italy about the motion of the earth, which was denied by Riccioli and his followers.— Through this channel, in particular, he carried on a dispute with Mr. Huygens on the occasion of his treatise on the quadrature of the circle and hyperbola, to which that great man had started some objections; in the course of which our author produced some improvements of his series. But in this dispute it happened, as it generally does on such occafions, that the antagonists, though setting out with temper enough, yet grew too warm in the combat. This was the case here, especially on the side of Gregory, whose defence was, at his own request, inserted in the Philosophical Transactions. It is unnecossary to enter into particulars:<pb n="554"/><cb/> suffice it therefore to say that, in the opinion of Leibnitz, who allows Mr. Gregory the highest merit for his genius and discoveries, M. Huygens has pointed out, though not errors, some considerable deficiencies in the treatise above mentioned, and shewn a much simpler method of attaining the same end.</p><p>In 1668, our author published at London another work, entitled, <hi rend="italics">Exercitationes Geometricæ,</hi> which contributed still much farther to extend his reputation. About this time he was elected professor of mathematics in the university of St. Andrew's, an office which he held for six years. During his residence there, he married, in 1669, Mary, the daughter of George Jameson, the celebrated painter, whom Mr. Walpole has termed the Vandyke of Scotland, and who was fellow disciple with that great artist in the school of Rubens at Antwerp.</p><p>In 1672, he published “The Great and New Art of Weighing Vanity: or a Discovery of the Ignorance and Arrogance of the Great and New Artist, in his Pseudo-philosophical Writings. By M. Patrick Mathers, Arch-bedal to the University of St. Andrews. To which are annexed some Tentamina de Motu Penduli & Projectorum.” Under this asumed name, our author wrote this little piece to expose the ignorance of Mr. Sinclare, professor at Glasgow, in his hydrostatical writings, and in return for some ill usage of that author to a colleague of Mr. Gregory's. The same year, Newton, on his wonderful discoveries in the nature of light, having contrived a new reflecting telescope, and made several objections to Mr. Gregory's, this gave birth to a dispute between those two philosophers, which was carried on during this and the following year, in the most amicable manner on both sides; Mr. Gregory defending his own construction, so far, as to give his antagonist the whole honour of having made the catoptric telescopes preferable to the dioptric; and shewing, that the imperfections in these instruments were not so much owing to a defect in the object speculum, as to the different refrangibility of the rays of light. In the course of this dispute, our author described a burning concave mirror, which was approved by Newton, and is still in good esteem. Several letters that passed in this dispute, are printed by Dr. Desaguliers, in an Appendix to the English edition of Dr. David Gregory's Elements of Catoptrics and Dioptrics.</p><p>In 1674, Mr. Gregory was called to Edinburgh, to <*>ill the chair of mathematics in that university. This place he had held but little more than a year, when, in October 1675, being employed in shewing the satellites of Jupiter through a telescope to some of his pupils, he was suddenly struck with total blindness, and died a few days after, to the great loss of the mathematical world, at only 37 years of age.</p><p>As to his character, Mr. James Gregory was a man of a very acute and penetrating genius. His temper seems to have been warm, as appears from his conduct in the dispute with Huygens; and, conscious perhaps of his own merits as a discoverer, he seems to have been jealous of losing any portion of his reputation by the improvements of others upon his inventions. He possessed one of the most amiable characters of a true philosopher, that of being content with his fortune in his situation. But the most brilliant part of his character<cb/> is that of his mathematical genius as an inventor, which was of the first order; as will appear by the following list of his inventions and discoveries. Among many others may be reckoned, his Reflecting Telescope;— Burning Concave Mirror;—Quadrature of the Circle and Hyperbola, by an insinite converging series;— his method for the Transformation of Curves;—a Geometrical Demonstration of lord Brounker's series for Squaring the Hyperbola—his Demonstration that the Meridian Line is analogous to a scale of Logarithmic Tangents of the Half Complements of the Latitude;— he also invented and demonstrated geometrically, by help of the hyperbola, a very simple converging feries for making the logarithms;—he sent to Mr. Collins the solution of the famous Keplerian problem by an infinite series;—he discovered a method of drawing Tangents to Curves geometrically, without any previous calculations;—a rule for the Direct and Inverse method of Tangents, which stands upon the same principle (of exhaustions) with that of fluxions, and differs not much from it in the manner of application; a Series for the length of the Arc of a Circle from the Tangent, and vice versa; as also for the Secant and Logarithmic Tangent and Secant, and vice versa:— These, with others, for measuring the length of the elliptic and hyperbolic curves, were sent to Mr. Collins, in return for some received from him of Newton's, in which he followed the elegant example of this author, in delivering his series in simple terms, independent of each other. These and other writings of our author are mostly contained in the following works, viz,</p><p>1. <hi rend="italics">Optica Promota;</hi> 4to, London 1663.</p><p>2. <hi rend="italics">Vera Circuli et Hyperbolæ Quadratura;</hi> 4to, Padua 1667 and 1668.</p><p>3. <hi rend="italics">Geometriæ Pars Universalis;</hi> 4to, Padua 1668.</p><p>4. <hi rend="italics">Exercitationes Geometricæ;</hi> 4to, London 1668.</p><p>5. <hi rend="italics">The Great and New Art of Weighing Vanity,</hi> &c. 8vo, Glasgow 1672.</p><p>The rest of his inventions make the subject of several letters and papers, printed either in the <hi rend="italics">Philos. Trans.</hi> vol. 3; the <hi rend="italics">Commerc. Epistol. Job. Collins et Aliorum,</hi> 8vo, 1715; in the Appendix to the English edition of Dr. David Gregory's <hi rend="italics">Elements of Optics,</hi> 8vo, 1735, by Dr. Desaguliers; and some series in the <hi rend="italics">Exercitatio Geometrica</hi> of the same author, 4to, 1684, Edinburgh; as well as in his little piece on Practical Geometry.</p><p><hi rend="smallcaps">Gregory</hi> <hi rend="italics">(Dr. David),</hi> Savilian professor of astronomy at Oxford, was nephew of the above-mentioned Mr. James Gregory, being the eldest son of his brother Mr. David Gregory of Kinardie, a gentleman who had the singular fortune to see three of his sons all professors of mathematics, at the same time, in three of the British Universities, viz, our author David at Oxford, the second son James at Edinburgh, and the third son Charles at St. Andrews. Our author David, the eldest son, was born at Aberdeen in 1661, where he received the early parts of his education, but completed his studies at Edinburgh; and, being possessed of the mathematical papers of his uncle, soon distinguished himself likewise as the heir of his genius. In the 23d year of his age, he was elected professor of mathematics in the university of Edinburgh; and, in the same year, he published <hi rend="italics">Exercitatio Geometrica de Dimensione</hi><pb n="555"/><cb/> <hi rend="italics">Figurarum, sive Specimen Methodi generalis Dimetiendi quasvis Figuras,</hi> Edinb. 1684, 4to. He very soon perceived the excellence of the Newtonian philosophy; and had the merit of being the first that introduced it into the schools, by his public lectures at Edinburgh. “He had (says Mr. Wiston, in the Memoirs of his own Life, i. 32) already caused several of his scholars to keep acts, as we call them, upon several branches of the Newtonian philosophy; while we at Cambridge, poor wretches, were ignominiously studying the fictitious hypothesis of the Cartesian.”</p><p>In 1691, on the report of Dr. Bernard's intention of resigning the Savilian professorship of astronomy at Oxford, our author went to London; and being patronised by Newton; and warmly befriended by Mr. Flamsteed the astronomer royal, he obtained the vacant professorship, though Dr. Halley was a competitor. This rivalship, however, instead of animosity, laid the foundation of friendship between these eminent men; and Halley soon after became the colleague of Gregory, by obtaining the professorship of geometry in the same university. Soon after his arrival in London, Mr. Gregory had been elected a fellow of the Royal Society; and, previously to his election into the Savilian professorship, had the degree of doctor of physic conferred on him by the university of Oxford.</p><p>In 1693, he published in the Philos. Trans. a resolution of the Florentine problem <hi rend="italics">de Testudine veliformi quadrabili;</hi> and he continued to communicate to the public, from time to time, many ingenious mathematical papers by the same channel.</p><p>In 1695, he printed at Oxford, <hi rend="italics">Catoptricæ et Dioptricæ Sphæricæ Elementa;</hi> a work which, we are informed in the preface, contains the substance of some of his public lectures read at Edinburgh, eleven years before. This valuable treatise was republished in English, first with additions by Dr. William Brown, with the recommendation of Mr. Jones and Dr. Desaguliers; and afterwards by the latter of these gentlemen, with an appendix containing an account of the Gregorian and Newtonian telescopes, together with Mr. Hadley's tables for the construction of both those instruments. It is not unworthy of remark, that, in the conclusion of this treatise, there is an observation which shews, that the construction of achromatic telescopes, which Mr. Dollond has carried to such great perfection, had occurred to the mind of David Gregory, from reflecting on the admirable contrivance of nature in combining the different humours of the eye. The passage is as follows: “Perhaps it would be of service to make the object lens of a different medium, as we see done in the fabric of the eye; where the crystalline humour (whose power of refracting the rays of light differs very little from that of glass) is by nature, who never does any thing in vain, joined with the aqueous and vitreous humours (not differing from water as to their power of refraction) in order that the image may be painted as distinct as possible upon the bottom of the eye.”</p><p>In 1702 our author published at Oxford, in folio, <hi rend="italics">Astronomiæ Physicæ et Geometricæ Elementa;</hi> a work which is accounted his master-piece. It is founded on the Newtonian doctrines, and was esteemed by New-<cb/> ton himself as a most excellent explanation and defence of his philosophy. In the following year he gave to the world an edition, in folio, of the works of Euclid, in Greek and Latin; being done in prosecution of a design of his predecessor Dr. Bernard, of printing the works of all the ancient mathematicians. In this work, which contains all the treatises that have been attributed to Euclid, Dr. Gregory has been careful to point out such as he found reason, from internal evidence, to believe to be the productions of some inferior geometrician. In prosecution of the same plan, Dr. Gregory engaged soon after, with his colleague Dr. Halley, in the publication of the Conics of Apollonius; but he had proceeded only a little way in this undertaking, when he died at Maidenhead in Berkshire, in 1710, being the 49th year of his age only.</p><p>Besides those works published in our author's life time, as mentioned above, he had several papers inserted in the Philos. Trans. vol. 18, 19, 21, 24, and 25, particularly a paper on the catenarian curve, first considered by our author. He left also in manuscript, <hi rend="italics">A Short Treatise of the Nature and Arithmetic of Logarithms,</hi> which is printed at the end of Keill's translation of Commandine's Euclid; and a <hi rend="italics">Treatise of Practical Geometry,</hi> which was afterwards translated, and published in 1745, by Mr. Maclaurin.</p><p>Dr. David Gregory married, in 1695, Elizabeth, the daughter of Mr. Oliphant of Langtown in Scotland. By this lady he had four sons, of whom, the eldest, David, was appointed regius professor of modern history at Oxford by king George the 1st, and died at an advanced age in 1767, after enjoying for many years the dignity of dean of Christchurch in that university.</p><p>When David Gregory quitted Edinburgh, he was succeeded in the professorship at that university by his brother <hi rend="italics">James,</hi> likewise an eminent mathematician; who held that office for 33 years, and, retiring in 1725, was succeeded by the celebrated Maclaurin. A daughter of this professor James Gregory, a young lady of great beauty and accomplishments, was the victim of an unfortunate attachment, that furnished the subject of Mallet's well known ballad of <hi rend="italics">William and Margaret.</hi></p><p>Another brother, Charles, was created professor of mathematics at St. Andrews by Queen Anne, in 1707. This office he held with reputation and ability for 32 years; and, resigning in 1739, was succeeded by his son, who eminently inherited the talents of his family, and died in 1763.</p><p><hi rend="italics">Some farther Particulars of the Family of the Gregorys and Andersons, communicated by Dr. Thomas Reid, Professor of Moral Philosophy in the University of Glasgow, a Nephew of the late Dr. David Gregory Savilian Professor at Oxford.</hi></p><p>Some account of the family of the Gregorys at Aberdeen, is given in the Life of the late Dr. John Gregory prefixed to his works, printed at Edinburgh for A. Strahan and T. Cadell, London, and W. Creech, Edinburgh, 1788, in four small 8vo volumes.</p><p>Who was the author of that Life, or whence he had his information, I do not know. I have heard it ascribed to Mr. Tytler the younger, whose father was appointed<pb n="556"/><cb/> one of the guardians of Dr. John Gregory's children. Some additions to what is contained in it, and remarks upon it, is all I can furnish upon this subject.</p><p>Page 3. I know nothing of the education of David Anderson of Finzaugh. He seems to have been a selftaught Engineer. Every public work which surpassed the skill of common artists, was committed to the management of David. Such a reputation he acquired by his success in works of this kind, that with the vulgar he got the by-name of <hi rend="italics">Davie do a' thing,</hi> that is in the Scottish dialect, <hi rend="italics">David who could do every thing.</hi> By this appellation he is better known than by his proper name. He raised the great bells into the steeple of the principal church: he cut a passage for ships of burden through a ridge of rock under water, which crossed the entrance into the harbour of Aberdeen. In a long picture gallery at Cullen House, the seat of the earl of Findlater, the wooden ceiling is painted with several of the fables of Ovid's Metamorphosis. The colours are still bright, and the representation lively. The present earl's grandfather told me that this painting was the work of David Anderson my ancestor, whom he acknowledged as a friend and relation of his family.</p><p>Such works, while they gave reputation to David, suited ill with his proper business, which was that of a merchant in Aberdeen. In that he succeeded ill; and having given up mercantile business, from a small remainder of his fortune began a trade of making malt; and having instructed his wife in the management of it, left it to her care, and went into England to try his fortune as an engineer; an employment which in his own country he had practised gratuitously. Having in that way made a fortune which satisfied him, he returned to Aberdeen, where his wife had also made money by her malting business.</p><p>After making such provision for their children as they thought reasonable, they agreed that the longest liver of the two should enjoy the remainder, and at death should bequeath it to certain purposes in the management of the magistrates of Aberdeen.</p><p>The wife happened to live longeft, and fulfilled what had been concerted with her husband. Her legacies, well known in Aberdeen, are called after her name <hi rend="italics">Jane Gu<*>ld's Mortifications,</hi> a mortification in Scots law signifying a bequeathment for some charitable purpofe. They consist of sums for different purposes. For orphans, for the education of boys and girls, for unmarried gentlewomen, and for widows; and they still continue to be useful to many in indigent circumstances. She was the daughter of Dr. Guild a minister of Aberdeen. Besides her money, she bequeathed a piece of tapestry, wrought by her own hand, and representing the history of queen Esther, from a drawing made by her husband. The tapestry continues to ornament the wall of the principal church.</p><p>In the same page it is said that Alexander Anderson, professor of mathematics at Paris, was the cousingerman of David above-mentioned. I know not the writer's authority for this: I have always heard that they were brothers; but for this I have only family tradition.</p><p>P. 4. It is here said that James Gregory was in-<cb/> structed in the Elements of Euclid by his mother, the daughter of David Anderson.</p><p>The account I have heard differs from this. It is, that his brother David, being ten or eleven years older, had the direction of his education after their father's death, and, when James had sinished his course of philosophy, was at a loss to what literary profession he should direct him. After some unsuccessful trials, he put Euclid's Elements into his hand, and finding that he applied to Euclid with great avidity and success, he encouraged and assisted him in his mathematical studies.</p><p>This tradition agrees with what James Gregory says in the preface to his <hi rend="italics">Optica Promota;</hi> where after mentioning his advance to the 26th proposition, he adds, <hi rend="italics">Ubi diu hæsi omne spe progrediendi orbatus, sed continuis hortatibus et auxiliis sratris mei Davidis Gregorii, in Mathematicis non parum versati (cui si quid in hisce Scientiis præstitero, me illud debere non inficias ibo) animatus, tandem incidi &c.</hi> Whether David had been instructed in mathematics by his mother, or had any living instructor, I know not.</p><p>P. 5, 6. In these two pages I think the merit of Gregory compared with that of Newton in the invention of the catoptric telescope, is put in a light more unfavourable to Newton than is just. Gregory believing that the imperfection of the dioptric telescope arose solely from the spherical figure of the glasses, invented his telescope to remedy that imperfection. Being less conversant in the practice of mechanics, he did not attempt to make any model. The specula of his telescope required a degree of polish and a figure which the best opticians of that age were unable to execute. Newton demonstrated that the imperfection of the dioptric telescope arose chiefly from the different refrangibility of the rays of light; he demonstrated also that the catopric telescope required a degree of polish far beyond what was necessary for the dioptric. He made a model of his telescope; and finding that the best polish which the opticians could give, was insufficient, he improved the polish with his own hand, so as to make it answer the purpose, and has described most accurately the manner in which he did this. And, had he not given this example of the practicability of making a reflecting telescope, it is probable that it would have passed as an impracticable idea to this day.</p><p>P. 11. To what is said of this James Gregory might have been added, that he was led by analogy to the true law of Refraction, not knowing that it was discovered by Des Cartes before (see Preface to <hi rend="italics">Optica Promota);</hi> and that in 1670 having received in a letter from Collins, a Series for the Area of the Zone of a Circle, and as Newton had invented an universal method by which he could square all Curves Geometrical and Mechanical by Infinite Series of that kind; Gregory after much thought discovered this universal method, or an equivalent one. Of this he perfectly satisfied Newton and the other mathematicians of that time, by a letter to Collins in Feb. 1671. He was strongly solicited by his brother David to publish his Universal Method of Series without delay, but excused himself upon a point of honour; that as Newton was the first inventor, and as he had<pb n="557"/><cb/> been led to it by an account of Newton's having such a method, he thought himself bound to wait till Newton should publish his method. I have seen the letters that passed between the brothers on this subject.</p><p>With regard to the controversy between James Gregory and Huygens, I take the subject of that controversy to have been, not whether J. Gregory's Quadrature of the Circle by a converging series was just, but whether he had demonstrated, as in one of his propositions he pretended to do, That it is impossible to express perfectly the Area of a Circle in any known Algebraical form, besides that of an infinite converging series. Huygens excepted to the demonstration of this proposition, and Gregory defended it; neither of them convinced his antagonist, nor do I know that Leibnitz improved upon what Gregory had done.</p><p>P. 12. David Gregory of Kinardie deserved a more particular account than is here given.</p><p>It is true that he served an apprenticeship to a mercantile house in Holland, but he followed that profession no longer than he was under authority, having a stronger passion for knowledge than for money. He returned to his own country in 1655, being about 28 years of age, and from that time led the life of a philosopher. Having succeeded to the estate of Kinardie by the death of an elder brother, he lived there to the end of that century. There all his children were born, of whom he had thirty-two by two wives.</p><p>Kinardie is above 40 English miles north from Aberdeen, and a few miles from Bamf, upon the river Diveron. He was a jest among the neighbouring gentlemen for his ignorance of what was doing about his own farm, but an oracle in matters of learning and philosophy, and particularly in medicine, which he had studied for his amusement, and begun to practise among his poor neighbours. He acquired such a reputation in that science, that he was employed by the nobility and gentlemen of that county, but took no fees. His hours of study were singular. Being much occupied through the day with those who applied to him as a physician, he went early to bed, rose about two or three in the morning, and, after applying to his studies for some hours, went to bed again and slept an hour or two before breakfast.</p><p>He was the first man in that country who had a barometer; and by some old letters which I have seen, it appeared, that he had corresponded with some philosophers on the continent about the changes in the barometer and in the weather, particularly with Mariotte the French philosopher. He was once in danger of being prosecuted as a conjurer by the Presbytery on account of his barometer. A deputation of that body having waited upon him to enquire into the ground of certain reports that had come to their ears, he satisfied them so far as to prevent the profecution of a man known to be so extensively useful by his knowledge of medicine.—About the beginning of this century he removed with his family to Aberdeen, and in the time of queen Anne's war employed his thoughts upon an improvement in artillery, in order to make the shot of great guns more destructive to the enemy, and executed a model of the engine he had conceived. I have conversed with a clock-maker in Aberdeen who was em-<cb/> ployed in making this model; but having made many different pieces by direction without knowing their intention, or how they were to be put together, he could give no account of the whole. After making some experiments with this model, which satisfied him, the old gentleman was so sanguine in the hope of being useful to the allies in the war against France, that he set about preparing a field equipage with a view to make a campaign in Flanders, and in the mean time sent his model to his son the Savilian professor, that he might have his and Sir Isaac Newton's opinion of it. His son shewed it to Newton, without letting him know that his own father was the inventor. Sir Isaac was much displeased with it, saying, that if it tended as much to the prefervation of mankind as to their destruction, the inventor would have deserved a great reward; but as it was contrived solely for destruction, and would soon be known by the enemy, he rather deserved to be punished, and urged the professor very strongly to destroy it, and if possible to suppress the invention. It is probable the professor followed this advice. He died soon after, and the model was never found.</p><p>When the rebellion broke out in 1715, the old gentleman went a second time to Holland, and returned when it was over to Aberdeen, where he died about 1720, aged 93.</p><p>He left an historical manuscript of the Transactions of his own Time and Country, which my father told me he had read.</p><p>I was well acquainted with two of this gentleman's sons, and with several of his daughters, besides my own mother. The facts abovementioned are taken from what I have occasionally heard from them, and from other persons of his acquaintance.</p><p>P. 14. In confirmation of what is said in this page, that the two brothers David and James were the first who taught the Newtonian philosophy in the Scotch Universities; I have by me a <hi rend="italics">Thesis,</hi> printed at Edinburgh in 1690, by James Gregory, who was at that time a professor of philosophy at St. Andrews, and succeeded his brother David in the profession of mathematics at Edinburgh. In this <hi rend="italics">Thesis,</hi> after a dedication to Viscount Tarbet, follow the names of twenty-one of his scholars who were candidates for the degree of A. M. then twenty-five positions or <hi rend="italics">Theses.</hi> The first three relate to logic, and the abuse of it in the Aristotelian and Cartesian philosophy. He desines logic to be the art of making a proper ufe of things granted, in order to find what is sought, and therefore admits only two <hi rend="italics">Categories</hi> in logic, viz, <hi rend="italics">Data</hi> and <hi rend="italics">Quasita.</hi> The remaining twenty-two positions are a compend of Newton's Principia. This <hi rend="italics">Thesis,</hi> as was the custom at that time in the Scotch universities, was to be defended in a public disputation, by the candidates, previous to their taking their degree.</p><p>The famous Dr. Pitcairn was a fellow student and intimate companion of these two Gregories, and during the vacation of the college was wont to go north with them to Kinardie, their father's house.</p><p>David Gregory was appointed a preceptor to the duke of Gloucester, queen Anne's son; but his entering upon that office was prevented by the death of that prince in the eleventh year of his age.</p><p>P. 19. D. Gregory's Euclid is said to have been wrote<pb n="558"/><cb/> in prosecution of a design of his predecessor Dr. Bernard, of printing the works of all the antient mathematicians. This design ought to have been ascribed to Savile, who left in charge to the two professors of his foundation, to print the mathematical works of the antients, and I think left a fund for defraying the expence. Wallis did something in consequence of this charge; Gregory and Halley did a great deal; but I think nothing has been done in this design by the Savilian professors since their time.</p><p>P. 20. Besides what is mentioned, Dr. Gregory left in manuscript a Commentary on Newton's Principia, which Newton valued, and kept by him for many years after the author's death. It is probable that in what relates to astronomy, this commentary may coincide in a great measure with the author's astronomy, which indeed is an excellent Commentary upon that part of the Principia.</p><p>P. 24. This David Gregory published in Latin, a very good compend of arithmetic and algebra, with the title <hi rend="italics">Arithmeticæ et Algebræ Compendium, in Ufum Juventutis Academicæ.</hi> Edinb. 1736. He had a design of publishing his uncle's Commentary on the Principia, with extracts from the papers left by James Gregory his grand uncle; but the expence being too great for his fortune, and he too gentle a solicitor of the assistance of others, the design was dropped. His son David, yet alive, was master of an East India ship.</p><p>P. 40. To the projectors of the society at Aberdeen, ought to have been added John Stewart professor of mathematics in the Marischal college at Aberdeen. He published an explanation of two treatises of Sir Isaac Newton, viz, his Quadrature of Curves, and his Analysis by Equations of an infinite number of terms. He was an intimate friend of Dr. Reid's.</p><p>Another of the first members of that society was Dr. David Skene, who, besides his eminence in the practice of medicine, had applied much to all parts of natural history, particularly to botany, and was a correspondent of the celebrated Linnæus.</p><p>Dr. John Gregory and Dr. David Skene were the first who attempted a college of medicine at Aberdeen. The first gave lectures to his pupils in the theory and practice of medicine, and in chemistry; the last, in anatomy, materia medica, and midwifery, in order to prepare them for attending the medical college at Edinburgh. T. R. <hi rend="center"><hi rend="italics">The following additional lines by Mr. James Millar, Professor of Mathematics, Glasgow.</hi></hi></p><p>Another instance of the prevalence of mathematical genius in the family of Gregory or Anderson, whether produced by an original and inexplicable determinatiòn of the mind, or communicated by the force of example, and the consciousness of an intimate connection with a reputation already acquired in a particular line, is the celebrated Dr. Reid, professor of moral philosophy in the university of Glasgow; a nephew, by his mother, of the late Dr. David Gregory, Savilian professor at Oxford.</p><p>This gentleman, well known to the public by his moral and metaphysical writings, and remarkable for that liberality, and that ardent spirit of enquiry, which neither overlooks nor undervalues any branch of science, is<cb/> peculiarly distinguished by his abilities and proficiency in mathematical learning. The objects of literary pursuit are often directed by accidental occurrences. And apprehension of the bad consequences which might result from the philosophy of the late Mr. Hume, induced Dr. Reid to combat the doctrines of that eminent author; and produced a work, which has excited universal attention, and seems to have given a new turn to speculations upon that subject. But it is well known to Dr. Reid's literary acquaintance, that these exertions have not diminished the original bent of his genius, nor blunted the edge of his inclination for mathematical researches; which, at a very advanced age, he still continues to prosecute with a youthful attachment, and with unremitting assiduity.</p><p>It may farther be observed, of the extraordinary family above mentioned, that Dr. James Gregory, the present learned professor of physic and medicine in the university of Edinburgh, is the son of the late Dr. John Gregory, upon the memoirs of whose life the above remarks have been written by Dr. Reid; the said James has lately published a most ingenious work, intitled, <hi rend="italics">Philosophical and Literary Essays,</hi> in 2 volumes 8vo, Edinb. 1792; and he seems to be another worthy inheritant of the singular genius of his family.</p><p><hi rend="smallcaps">Gregory</hi> <hi rend="italics">(St. Vincent),</hi> a very respectable Flemish geometrician, was born at Bruges in 1584, and became a Jesuit at Rome at 20 years of age. He studied mathematics under the learned Jesuit Clavius. He afterward became a reputable professor of those sciences himself, and his instructions were solicited by several princes: he was called to Prague by the emperor Ferdinand the 2d; and Philip the 4th, king of Spain, was desirous of having him to teach mathematics to his son the young prince John of Austria. He was not less estimable for his virtues than his skill in the sciences. His well-meant endeavours were very commendable, when his holy zeal, though for a false religion, led him to follow the army in Flanders one campaign, to confess the wounded and dying soldiers, in which he received several wounds himself. He died of an apoplexy at Ghent, in 1667, at 83 years of age.</p><p>As a writer, Gregory St. Vincent was very diffuse and voluminous, but he was an excellent geometrician. He published, in Latin, three mathematical works, the principal of which was his <hi rend="italics">Opus Geometricum Quadraturæ Circuli, et Sectionum Coni,</hi> Antwerp, 1647, 2 vol. folio. Although he has not demonstrated, in this work, the Quadrature of the circle, as he pretends to have done, the book nevertheless contains a great number of truths and important discoveries; one of which is this, viz, that if one asymptote of an hyperbola be divided into parts in geometrical progression, and from the points of division ordinates be drawn parallel to the other asymptote, they will divide the space between the asymptote and curve into equal portions; from whence it was shewn by Mersenue, that, by taking the continual sums of those parts, there would be obtained areas in arithmetical progression, adapted to abscisses in geometrical progression, and which therefore were analogous to a system of logarithms.</p></div1><div1 part="N" n="GRENADE" org="uniform" sample="complete" type="entry"><head>GRENADE</head><p>, or <hi rend="smallcaps">Grenado.</hi> See <hi rend="smallcaps">Granade.</hi></p></div1><div1 part="N" n="GRUS" org="uniform" sample="complete" type="entry"><head>GRUS</head><p>, the <hi rend="italics">Crane,</hi> one of the new constellations, in<pb n="559"/><cb/> the southern hemisphere; containing, according to Mr. Sharp's catalogue, 13 stars.</p><p><hi rend="smallcaps">Grus</hi> is also one of the Arabian constellations, and answers to our Ophiucus, to which they changed this constellation, their religion prohibiting them from drawing any human figures.</p></div1><div1 part="N" n="GRY" org="uniform" sample="complete" type="entry"><head>GRY</head><p>, a measure containing one-tenth of a line. A line is one-tenth of a digit, and a digit is one-tenth of a foot, and a philosophical foot, one-third of a pendulum, whose diadromes, or vibrations, in the latitude of 45 degrees, are each equal to one second of time, or one-sixtieth of a minute.</p></div1><div1 part="N" n="GUARDS" org="uniform" sample="complete" type="entry"><head>GUARDS</head><p>, a name that has been sometimes applied to the two stars nearest the north pole; being in the hind part of the chariot, at the tail of Ursa Minor or little bear; one of them being also called the pole star.</p></div1><div1 part="N" n="GUERICKE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GUERICKE</surname> (<foreName full="yes"><hi rend="smallcaps">Otto</hi></foreName> or <hi rend="smallcaps">Otho</hi>)</persName></head><p>, counsellor to the elector of Brandenbourg and burgomaster of Magdebourg, was born in 1602, and died in 1686 at Hambourg. He was one of the greatest philosophers of his time. It was Guericke that invented the air-pump; the two brass hemispheres, which being applied to each other, and the air exhausted, 16 horses were not able to draw them asunder; the marmouset of glass which descended in a tube in rainy weather, and rose again on the return of serene weather. This last machine fell into disuse on the invention of the barometer, especially after Huygens and Amontons gave theirs to the worldGuericke made use of his marmouset to foretell storms; from whence he was looked upon as a sorcerer by the people; so that the thunder having one day fallen upon his house, and shivered to pieces several machines which he had employed in his experiments, they failed not to say it was a punishment from heaven that was angry with him.——Guericke was author of several works in natural philosophy, the principal of which was his <hi rend="italics">Ex<*> perimenta Magdeburgica,</hi> in folio, which contains his experiments on a vacuum.</p></div1><div1 part="N" n="GUERITE" org="uniform" sample="complete" type="entry"><head>GUERITE</head><p>, in Fortification, a centry-box; being a small tower of wood, or stone, usually placed on the point of a bastion, or on the angles of the shoulder, to hold a centinel, who is to take care of the ditch, and watch against a surprise.</p></div1><div1 part="N" n="GUEULE" org="uniform" sample="complete" type="entry"><head>GUEULE</head><p>, in Architecture. See <hi rend="smallcaps">Gula.</hi></p></div1><div1 part="N" n="GUINEA" org="uniform" sample="complete" type="entry"><head>GUINEA</head><p>, a gold coin struck in England.——The value or rate of the guinea has varied. It was at first equal to 20 shillings; but by the scarcity of gold it was afterwards advanced to 21 <hi rend="italics">s.</hi> 6<hi rend="italics">d.;</hi> though it is now sunk to 21<hi rend="italics">s.</hi></p><p>The pound weight troy of gold is cut into 44 parts and a half, and each part makes a guinea, which is therefore equal to 2/89lb, or 24/89oz, or 5dwts 9 3<*>/89 gr.</p><p>This coin took its name, Guinea, from the circumstance of the gold of which it was sirst struck being brought from that part of Africa called Guinea, for which reason also it bore the impression of an elephant.</p></div1><div1 part="N" n="GULA" org="uniform" sample="complete" type="entry"><head>GULA</head><p>, <hi rend="smallcaps">Gueule</hi>, or <hi rend="smallcaps">Gola</hi>, in Architecture, a wavy member whose contour resembles the letter S, commonly called an Ogee.</p></div1><div1 part="N" n="GULBE" org="uniform" sample="complete" type="entry"><head>GULBE</head><p>, in Architecture, the same as Gorge.</p></div1><div1 part="N" n="GULF" org="uniform" sample="complete" type="entry"><head>GULF</head><p>, or <hi rend="smallcaps">Gueph</hi>, in Geography, a part of the<cb/> ocean running up into the land through a narrow passage, or strait, and forming a bay within. As, the Gulf of Venice, or Adriatic sea; the Gulf of Arabia, or of Persia, which is the Red Sea; the Gulf of Constantinople, or the Black Sea; the Gulf of Mexico; &c.</p></div1><div1 part="N" n="GUN" org="uniform" sample="complete" type="entry"><head>GUN</head><p>, a fire-arm, or weapon of offence, which forcibly discharges a ball or other matter through a cylindrical tube, by means of inflamed gun-powder.</p><p>The word Gun now includes most of the species of fire-arms; mortars and pistols being almost the only ones excepted from this denomination. They are divided into great and small guns: the former including all that are usually called cannon, ordnance, or artillery; and the latter includes musquets, firelocks, carabines, musquetoons, blunderbusses, fowling-pieces, &c.</p><p>It is not certainly known at what time these weapons were first invented. And though the introduction of guns into the western part of the world is but of modern date, comparatively speaking; yet it is certain that in some parts of Asia they have been used for many ages, though in a very rude and imperfect manner. Philostratus speaks of a city near the river Hyphafis in the Indies, which was said to be impregnable, and that its inhabitants were relations of the gods, because they threw thunder and lightning upon their enemies; and other Greek authors, as also Quintus Curtius, speak of the same thing having happened to Alexander the Great. Hence some have imagined that guns were used by the eastern nations in his time, while others suppose the thunder and lightning alluded to by those authors, were only certain artificial fire-works, or rockets, such as we know are used in the wars by the Indians even in the present day against the Europeans. Be this however as it may, it is asserted by many modern travellers, that Guns were used in China as far back as the year of Christ 85, and have continued in use ever fince.</p><p>The first hint of the invention of Guns in Europe, is in the works of Roger Bacon, who flourished in the 13th century. In a treatise written by him about the year 1280, he proposes to apply the violent explosive force of gun-powder for the destruction of armies. And though it is certainly known that the composition of gun-powder is described by Bacon in the said work, yet the invention has usually, though improperly, been ascribed to Bartholdus Schwartz, a German monk, who it is said discovered it only in the year 1320; and the invention is related in the following manner. Schwartz having, for some purpose, pounded nitre, sulphur, and charcoal together, in a mortar, which he afterwards covered imperfectly with a stone, a spark of fire accidentally fell into the mortar, which setting the mixture on fire, the explosion blew the stone to a considerable distance. Hence it is probable that Schwartz might be taught the simplest method of applying it in war; for it rather seems that Bacon conceived the manner of using it to be by the violent effort of the flame unconfined, and which is indeed capable of producing as<*>onishing effects. (See <hi rend="smallcaps">Gunpowder.</hi>) And the figure and name of <hi rend="italics">mortars</hi> given to a species of old artillery, and their employment, in throwing large stone bullets at an elevation, very much favour this conjecture.<pb n="560"/><cb/></p><p>Soon after the time of Schwartz, we sind Guns commonly used as instruments of war. Great guns were first used. These were originally made of iron-bars soldered together, and fortified with strong iron hoops or rings; several of which are still to be seen in the Tower of London, and in the Warren at Woolwich. Others were made of thin sheets of iron rolled up together and hooped: and on particular emergencies some have been made of leather, and of lead, with plates of iron or copper. These sirst pieces were executed in a rude and imperfect manner, like the sirst essays of most new inventions. Stone balls were thrown out of them, and a small quantity of powder used on account of their weakness. They were of a cylindrical form, without ornaments, and were placed on their carriages by rings.</p><p>When, or by whom they were first made, is uncertain. It is known however that the Venetians used eannon at the <*>iege of Claudia Jessa, now called Chioggia, in 1366, which were brought thither by two Germans, with some powder and leaden balls; as likewise ín their wars with the Genoese in 1379. But before that, king Edward the 3d made use of cannon at the battle of Cressy in 1346, and at the siege of Calais in 1347. Cannon were employed by the Turks at the siege of Constantinople, then in possession of the Christians, in 1394, and in that of 1452, which threw a weight of 100lb; but they commonly burst at the 1st, 2d, or 3d firing. Louis the 12th had one cast at Tours, of the same size, which threw a ball from the Bastile to Charenton: one of these extraordinary cannon was taken at the siege of Dieu in 1546, by Don John de Castro, and is now in the castle of St. Julian da Barra, 10 miles from Lisbon: the length of is 20 feet 7 inches, its diameter at the middle 6 feet 3 inches, and it threw a ball of 100lb weight. It has neither dolphins, rings, nor button; is of an unusual kind of metal; and it has a large Indostan inscription upon it, which says it was cast in 1400.</p><p>Formerly, cannon were dignisied with uncommon names. Thus, Lewis the 12th, in 1503, had 12 brass cannon cast, of an extraordinary size, called after the names of the 12 peers of France. The Spanish and Portuguese called them after their saints. The emperor Charles the 5th, when he marched against Tunis, founded the 12 apostles. At Milan there is a 70 pounder, called the Pimontelle; and one at Bois-le-duc, called the Devil. A 60 pounder at Dover-castle, called Queen Elizabeth's pocket-pistol. An 80 pounder in the Tower of London, brought there from Edinburghcastle, called Mounts-meg. An 80 pounder in the royal arsenal at Berlin, called the Thunderer. An 80 pounder at Malaga, called the Terrible. Two curious 60 pounders in the arsenal at Bremen, called the Messenger of bad news. And lastly an uncommon 70 pounder in the castle of St. Angelo at Rome, made of the nails that fastened the copper-plates which covered the ancient Pantheon, with this inscription upon it, Ex clavis trabalibus porticus Agrippæ.</p><p>In the beginning of the 15th century these uncommon names were generally abolished, and the following more universal ones took place, viz,<cb/> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Names.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Wt. of ball, Pounders.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Wt. of piece in cwts, about Cwt.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Cannon royal, or carthoun,</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">90</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bastard cannon, or 3/4 carthoun,</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">79</cell></row><row role="data"><cell cols="1" rows="1" role="data">Demi-carthoun,</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">60</cell></row><row role="data"><cell cols="1" rows="1" role="data">Whole culverins,</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" role="data">Demi-culverins,</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Falcon,</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">25</cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">largest size</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sacker</cell><cell cols="1" rows="1" role="data">ordinary</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">15</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">lowest sort</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data">Basilisk</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">85</cell></row><row role="data"><cell cols="1" rows="1" role="data">Serpentine</cell><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data"> 8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aspic</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data"> 7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dragon</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data">Syren</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">81</cell></row><row role="data"><cell cols="1" rows="1" role="data">Falconet</cell><cell cols="1" rows="1" role="data"> 3, 2, and 1</cell><cell cols="1" rows="1" role="data">15, 10, 5</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rabinet</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Moyens</cell><cell cols="1" rows="1" role="data">10 or 12 oz.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>These curious names of beasts and birds of prey were adopted on account of their swiftness in motion, or of their cruelty; as the falconet, falcon, sacker, and culverin, &c, for their swiftness in flying; the basilisk, serpentine, aspic, dragon, syren, &c, for their cruelty.</p><p>But, at present, cannon take their names from the weight of their proper ball. Thus, a piece that discharges a cast-iron ball of 24 pounds, is called a 24 pounder; one that carries a ball of 12 pounds, is called a 12 pounder; and so of the rest, divided into the following sorts, viz,</p><p>Ship-guns, consisting in 42, 36, 32, 24, 18, 12, 9, 6, and 3 pounders.</p><p>Garrison-guns, in 42, 32, 24, 18, 12, 9, and 6 pounders.</p><p>Battering-guns, in 24, 18, and 12 pounders.</p><p>Field-pieces, in 12, 9, 6, 3, 2, 1 1/2, 1, and 1/2 pounders.</p><p><hi rend="italics">Mortars,</hi> it is thought, have been at least as ancient as cannon. They were employed in the wars of Italy, to throw balls of red-hot iron, stones, &c, long before the invention of shells. These last, it is supposed, were of German invention, and the use of them in war shewn by the following accident; viz, a citizen of Venlo, at a festival celebrated in honour of the duke of Cleves, throwing a number of shells, one of them fell on a house and set it on fire, by which misfortune the greatest part of the town was reduced to ashes. The first account of shells used for military purposes, is in 1435, when Naples was besieged by Charles the 8th. History informs us, with more certainty, that shells were thrown out of mortars at the siege of Wachtendonk, in Guelderland, in 1588, by the earl of Mansfield; and Cyprian Lucar wrote upon the method of filling and throwing such shells in his Appendix to the Colloquies of Tartaglia, printed at London in 1588; where also the compounding and throwing of carcasses and various sorts of fire-works are shewn.</p><p>Mr. Malter, an English engineer, first taught the French the art of throwing shells, which they practised at the siege of Motte in 1634. The method of throwing red-hot balls out of mortars was first certainly put<pb n="561"/><cb/> in practice at the siege of Stralsund in 1675 by the elector of Brandenburgh: though some say in 1653 at the siege of Bremen.</p><p>Another species of ordnance has been long in use, by the name of Howitzer, which is a kind of medium as to its length, between the cannon and the mortar, and is a very useful piece, for discharging either shells or large balls, which is done either at point-blanc, or at a small elevation.</p><p>A new species of ordnance has lately been introduced by the Carron company, and thence called a Carronade, which is only a very short howitzer, and which possesses the advantage of being very light and easy to work.</p><p>The species of Guns before mentioned, are now made chiefly of cast iron; except the howitzer, which is of brass, as well as some cannon and mortars.</p><p>Muskets were first used at the siege of Rhege in the year 1521. The Spaniards were the first who armed part of their foot with these weapons. At first they were very heavy, and could not be used without a rest. They had match-locks, and did execution at a great distance. On their march the soldiers carried only the rests and ammunition, having boys to bear their muskets after them. They were very slow in loading, not only by reason of the unwieldiness of their pieces, and because they carried the powder and ball separate, but from the time it took to prepare and adjust the match; so that their fire was not near so brisk as ours is now. Afterwards a lighter match-lock musket came in use: and they carried their ammunition in bandeliers, to which were hung several little cases of wood covered with leather, each containing a charge of powder. The muskets with rests were used as late as the beginning of the civil wars in the time of Charles the 1st. The lighter kind succeeded them, and continued till the beginning of the present century, when they also were disused, and the troops throughout Europe armed with firelocks. These are usually made of hammered iron. For the dimensions, construction, and practice of every species of Gun, &c, see the several articles <hi rend="smallcaps">Cannon, Mortar</hi>, &c. See also <hi rend="smallcaps">Gunnery.</hi></p></div1><div1 part="N" n="GUNNERY" org="uniform" sample="complete" type="entry"><head>GUNNERY</head><p>, the art of charging, directing, and exploding fire-arms, as cannon, mortars, muskets, &c, to the best advantage.</p><p>Gunnery is sometimes considered as a part of the military art, and sometimes as a part of pyrotechny. To the art of Gunnery too belongs the knowledge of the force and effect of gunpowder, the dimensions of the pieces, and the proportions of the powder and ball they carry, with the methods of managing, charging, pointing, spunging, &c. Also some parts of Gunnery are brought under mathematical consideration, which among mathematicians are called absolutely by the name Gunnery, viz, the rules and method of computing the range, elevation, quantity of powder, &c, so as to hit a mark or object proposed, and is more particularly called <hi rend="smallcaps">Projectiles;</hi> which see.</p><p><hi rend="smallcaps">History</hi> <hi rend="italics">of</hi> <hi rend="smallcaps">Gunnery.</hi></p><p>Long before the invention of gunpowder, and of Gunnery, properly so called, the art of artillery, or projectiles, was actually in practice. For, not to mention the use of spears, javelins, or stones thrown with the hand, or of bows and arrows, all which are found among the most barbarous and ignorant people, ac-<cb/> counts of the larger machines for throwing stones, darts, &c, are recorded by the most ancient writers. Thus, one of the kings of Judah, 800 years before the christian æra, erected engines of war on the towers and bulwarks of Jerusalem, for shooting arrows and great stones for the defence of that city. 2 Chron. xxvi. 15. Such machines were afterwards known among the Greeks and Romans by the names of Ballista, Catapulta, &c, which produced effects by the action of a spring of a strongly twisted cordage, formed of tough and elastic animal substances, no less terrible than the artillery of the moderns. Such warlike instruments continued in use down to the 12th and 13th centuries, and the use of bows still longer; nor is it probable that they were totally laid aside till they were superseded by gunpowder and the modern ordnance.</p><p>The first application of gunpowder to military affairs, it seems, was made soon after the year 1300, for which the proposal of friar Bacon, about the year 1280, for applying its enormous explosion to the destruction of armies, might give the first hint; and Schwartz, to whom the invention of gunpowder has been erroneously ascribed, on account of the accident abovementioned under the article <hi rend="smallcaps">Gun</hi>, might have been the first who actually applied it in this way, that is in Europe; for as to Asia, it is probable that the Chinese and Indians had something of the kind many ages before. Thus, only to mention the prohibition of fire-arms in the co<*>e of Gentoo laws, printed by the East India Company in 1776, which seems to confirm the suspicion suggested by a passage in Quintus Curtius, that Alexander the Great found some weapons of that kind in India: Cannon in the Shansorit idiom is called shet-aghnee, or the weapon that kills a hundred men at once.</p><p>However, the first pieces of artillery, which were charged with gunpowder and stone bullets of a prodigious size, were of very clumsy and inconvenient structure and weight. Thus, when Mahomet the 2d besieged Constantinople in 1453, he battered the walls with stones of this kind, and with pieces of the calibre of 1200 pounds; which could not be fired more than four times a day. It was however soon discovered that iron bullets, of much less weight than stone ones, would be more efficacious if impelled by greater quantities of stronger powder. This occasioned an alteration in the matter and form of the cannon, which were now cast of brass. These were lighter and more manageable than the former, at the same time that they were stronger in proportion to their bore. This change took place about the close of the 15th century.</p><p>By this means came first into use such powder as is now employed over all Europe, by varying the proportion of the materials. But this change of the proportion was not the only improvement it received. The practice of graining it is doubtless of considerable advantage. At first the powder had been always used in the form of fine meal, such as it was reduced to by grinding the materials together. And it is doubtful whether the first graining of powder was intended to increase its strength, or only to render it more convenient for filling into small charges and the charging of small arms, to which alone it was applied for many years, whilst meal-powder was still used for cannon.<pb n="562"/><cb/> But at last the additional strength which the grained powder was found to possess, doubtless from the free passage of the air between the grains, occasioned the meal-powder to be entirely laid aside.</p><p>For the last 200 years, the formation of cannon has been very little improved; the best pieces of modern artillery differing little in their proportions from those used in the time of Charles the 5th. Indeed lighter and shorter pieces have been often proposed and tried; but though they have their advantages in particular cases, it is agreed they are not sufficient for general service. Yet the size of the pieces has been much decreased; the same purposes being now accomplished, by smaller pieces than what were formerly thought necessary. Thus the battering cannon now approved, are those that formerly were called demi cannon, carrying a ball of 24 pounds weight; this weight having been sound fully sufficient. The method also of making a breach, by first cutting off the whole wall as low as possible before its upper part is attempted to be beaten down, seems to be a considerable modern improvement in the practical part of gunnery. But the most considerable improvement in the practice, is the method of firing with small quantities of powder, and elevating the piece but a little, so that the bullet may just go clear of the parapet of the enemy, and drop into their works, called ricochet firing: for by this means the ball, coming to the ground at a small angle, and with a small velocity, does not bury itself, but bounds or rolls along a great way, destroying all before it. This method was first practised by M. Vauban at the siege of Aeth, in the year 1692. A practice of this kind was successfully used by the king of Prussia at the battle of Rosbach in 1757. He had several six-inch mortars, made with trunnions, and mounted on travelling carriages, which were fired obliquely on the enemy's lines, and among their horse. These being charged with only 8 ounces of powder, and elevated at one degree and a quarter, did great execution: for the shells rolling along the lines with burning fuses made the stoutest of the enemy not wait for their bursting.</p><p>The use of fire-arms was however long known before any theory of projectiles was formed. The Italians were the first people that made any attempts at the theory, which they did about the beginning of the 16th century, and amongst them it seems the first who wrote professedly on the flight of cannon shot, was Nicholas Tartalia, of Brescia, the same author who had so great a share in the invention of the rules for cubic equations. In 1537 he published, at Venice, his <hi rend="italics">Nova Scientia,</hi> and in 1546 his <hi rend="italics">Quesiti & Inventioni diversi,</hi> in both which he treats professedly on these motions, as well as in another work, translated into English with additions by Cyprian Lucar, under the title of Colloquies concerning the Art of Shooting in great and small Pieces of Artillery, and published at London in 1588. He determined, that the greatest range of a shot was when discharged at an elevation of 45°: and he asserted, contrary to the opinion of his contemporaries, that no part of the path described by a ball is a right line; although the curvature in the first part of it is so small, that it need not be attended to. He compared it to the surface of the sea; which, though it appears to be a plane, is yet doubtless incurvated round the centre of<cb/> the earth. He says he invented the gunner's quadran<*> for laying a piece of ordnance at any point or degree of elevation; and though he had but little opportunity of acquiring any practical knowledge by experiments, he yet gave shrewd guesses at the event of some untried methods.</p><p>The philosophers of those times also took part in the questions arising upon this subject; and many disputes on motion were held, especially in Italy, which continued till the time of Galileo, and probably gave rise to his celebrated Dialogues on Motion. These were not published till the year 1638; and in the interval there were published many theories of the motion of military projectiles, as well as many tables of their comparative ranges; though for the most part very fallacious, and inconsistent with the motions of these bodies.</p><p>It is remarkable however that, during these contests, so few of those who were intrusted with the care of artillery, thought it worth while to bring their theories to the test of experiment. Mr. Robins informs us, in the preface to his New Principles of Gunnery, that he had met with no more than four authors who had treated experimentally on this subject. The first of these is Collado, in 1642, who has given the ranges of a falconet, carrying a three-pound shot, to every point of the gunner's quadrant, each point being the 12th part, or 7° and a half. But from his numbers it is manifest that the piece was not charged with its usual allotment of powder. The result of his trials shews the ranges at the point-blanc, and the several points of elevation, as below. <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Collado's Experiments.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Elevation at</cell><cell cols="1" rows="1" role="data">Range in</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Points.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Deg.</cell><cell cols="1" rows="1" role="data">paces.</cell></row><row role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">or</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" rend="align=center" role="data">268</cell></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data"> 7 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">594</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">794</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">22 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">954</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">1010</cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">37 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">1040</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">1053</cell></row><row role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">52 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">between the 3d and 4th</cell></row><row role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" rend="align=center" role="data">between the 2d and 3d</cell></row><row role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">67 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">between the 1st and 2d</cell></row><row role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" rend="align=center" role="data">between the 0 and 1st</cell></row><row role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">82 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">fell very near the piece.</cell></row></table></p><p>The next was by Wm. Bourne, in 1643, in his <hi rend="italics">Art of Shooting in Great Ordnance.</hi> His elevations were not regulated by the points of the Gunner's quadrant, but by degrees; and he gives the proportions between the ranges at different elevations and the extent of the pointblanc shot, thus: if the extent of the point-blanc shot be represented by 1, then the proportions of the ranges at several elevations will be as below, viz. <table><row role="data"><cell cols="1" rows="1" rend="colspan=3 align=center" role="data"><hi rend="italics">Bourne's Proportìon of Ranges.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Elevation.</cell><cell cols="1" rows="1" role="data">Range.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">0°</cell><cell cols="1" rows="1" rend="align=light" role="data">1</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=light" role="data">2 2/<*></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=light" role="data">3 1/3</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=light" role="data">4 1/3</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=light" role="data">4 5/6</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">and the greatest random 5 1/2;</cell></row></table><pb n="563"/><cb/> which greatest random, he says, in a calm day is at 42° elevation; but according to the strength of the wind, and as it favours or opposes the slight of the shot, the elevation may be from 45° to 36°.—He does not say with what piece he made his trials; though from his proportions it seems to have been a smail one. This however ought to have been mentioned, as the relation between the extent of different ranges varies extremely according to the velocity and density of the bullet.</p><p>After him, Eldred and Anderson, both Englishmen, also published treatises on this subject. The former of these was many years gunner of Dover Castle, where most of his experiments were made, the earliest of which are dated in 1611, though his book was not published till 1646, and was intitled <hi rend="italics">The Gunner's Glass.</hi> His principles were sufficiently simple, and within certain limits very near the truth, though they were not rigorously so. He has given the actual ranges of different pieces of artillery at small elevations, all under 10 degrees. His experiments are numerous, and appear to be made with great care and caution; and he has honestly set down some, which were not reconcilable to his method: upon the whole he seems to have taken more pains, and to have had a juster knowledge of his business, than is to be found in most of his practical brethren.</p><p>Galileo printed his Dialogues on Motion in the year 1646. In these he pointed out the general laws observed by nature in the production and composition of motion, and was the first who described the action and effects of gravity on falling bodies: on these principles he determined, that the flight of a cannon-shot, or of any other projectile, would be in the curve of a parabola, unless so far as it should be diverted from that track by the resistance of the air. He also proposed the means of examining the inequalities which arise from thence, and of discovering what sensible effects that resistance would produce in the motion of a bullet at some given distance from the piece.</p><p>Notwithstanding these determinations and hints of Galileo, it seems that those who came after him never imagined that it was necessary to consider how far the operations of Gunnery were affected by this resistance. Instead of this, they boldly asserted, without making the experiment, that no great variation could arise from the resistance of the air in the flight of shells or cannon shot. In this persuasion they supported themselves chiefly by considering the extreme rarity of the air, compared with those dense and ponderous bodies; and at last it became an almost generally established maxim, that the flight of these bodies was nearly in the curve of a parabola.</p><p>Thus, Robert Anderson, in his <hi rend="italics">Genuine Use and Effects of the Gunne,</hi> published in 1674, and again in his book, <hi rend="italics">To hit a Mark,</hi> in 1690, relates a great many experiments; but proceeding on the principles of Galileo, he ftrenuously asserts that the flight of all bullets is in the curve of a parabola; undertaking to answer all objections that could be brought to the contrary. The same thing was also undertaken by Blondel, in his <hi rend="italics">Art de jetter les Bombes,</hi> published in 1683; where, after long discussion, he concludes, that the variations from the air's resistance are so slight as not to deserve any notice. The same subject is treated of in the Philos.<cb/> Trans. N° 216, p. 68, by Dr. Halley; who also, swayed by the very great disproportion between the density of the air and that of iron or lead, thought it reasonable to believe that the opposition of the air to large metal-shot is scarcely discernible; although in small and light shot he owns that it must be accounted for.</p><p>But though this hypothesis went on smoothly in speculation; yet Anderson, who made a great number of trials, found it impossible to support it without some new modification. For though it does not appear that he ever examined the comparative ranges of either cannon or musket shot when fired with their usual velocities, yet his experiments on the ranges of shells thrown with velocities that were but sinall, in comparison of those above mentioned, convinced him that their whole track was not parabolical. But instead of making the proper inferences from hence, and concluding that the resistance of the air was of considerable efficacy, he framed a new hypothesis; which was, that the shell or bullet at its first discharge flew to a certain distance in a right line, from the end of which line only it began to describe a parabola: and this right line, which he calls the line of the impulse of the fire, he supposes is the same for all elevations. So that, by assigning a proper length to this line of impulse, it was always in his power to reconcile any two shots made at any two different angles; though the same method could not succeed with three shots; nor indeed does he ever inform us of the event of his experiments when three ranges were tried at one time.</p><p>But after the publication of Newton's Principia, it might have been expected, that the defects of the theory would be ascribed to their true cause, which is the great resistance of the air to such swift motions; as in that work he particularly considered the subject of such motions, and related the result of experiments, made on slow motions at least; by which it appeared, that in such motions the resistance increases as the square of the velocities, and he even hints a suspicion that it will increase above that law in swifter motions, as is now known to be the case. So far however were those who treated this subject scientifically, from making a proper allowance for the resistance of the atmosphere, that they still neglected it, or rather opposed it, and their theories still differed most egregiously from the truth. Huygens alone seems to have attended to this principle: for in the year 1690 he published a treatise on gravity, in which he gave an account of some experiments tending to prove that the track of all projectiles, moving with very swift motions, was widely different from that of a parabola. The rest of the learned generally acquiesced in the justness and sufficiency of Galileo's doctrine, and accordingly very erroneous calculations concerning the ranges of cannon were given. Nor was any farther notice taken of these errors till the year 1716, at which time Mr. Ressons, a French officer of artillery, of great merit and experience, gave in a memoir to the Royal Academy, importing that, “although it was agreed that theory joined with practice did constitute the perfection of every art; yet experience had taught him that theory was of very little service in the use of mortars: That the works of M. Blondel had justly enough described the several parabolic lines, according<pb n="564"/><cb/> to the different degrees of the elevation of the piece; but that practice had convinced him there was no theory in the effect of gunpowder; for having endeavoured, with the greatest precision, to point a mortar according to these calculations, he had never been able to establish any solid foundation upon them.”—— One instance only occurs in which D. Bernoulli applies the doctrine of Newton to the motions of projectiles, in the Com. Acad. Petrop. tom. 2, pa. 338 &c. Besides which nothing farther was done in this business till the time of Mr. Benjamin Robins, who published a treatise in 1742, intitled <hi rend="italics">New Principles of Gunnery,</hi> in which he treated particularly, not only of the resistance of the atmosphere, but also of the force of gunpowder, the nature and effects of different guns, and almost every thing else relating to the flight of military projectiles; and indeed he carried the theory of gunnery nearly to its utmost perfection.</p><p>The sirst thing considered by Mr. Robins, and which is indeed the foundation of all other particulars relating to Gunnery, is the explosive force of gunpowder. M. De la Hire, in the Hist. of the Acad. of Sciences for the year 1702, supposed that this force may be owing to the increased elasticity of the air contained in, and between the grains, in consequence of the heat and fire produced at the time of the explosion: a cause not adequate to the 200th part of the effect. On the other hand, Mr. Robins determined, by irrefragable experiments, that this force was owing to an elastic fluid, similar to our atmosphere, existing in the powder in an extremely condensed state, which being suddenly freed from the powder by the combustion, expanded with an amazing force, and violently impelled the bullet, or whatever may oppose its expansion.</p><p>The intensity of this force of exploded gunpowder Mr. Robins ascertained in different ways, after the example of Mr. Hawksbee, related in the Philos. Trans. N° 295, and his Physico-Mechan. Exper. pa. 81. One of these is by firing the powder in the air thus: A small quantity of the powder is placed in the upper part of a glass tube, and the lower part of the tube is immerged in water, the water being made to rise so near the top, that only a small portion of air is left in that part where the powder is placed: then in this situation the communication between the upper part of the tube and the external air being closed, the powder is fired by means of a burning glass, or otherwise; the water descends upon the explosion, and stands lower in the tube than before, by a space proportioned to the quantity of powder fired.</p><p>Another way was by firing the powder in vacuo, viz, in an exhausted receiver, by dropping the grains of powder upon a hot iron included in the receiver. By this means a permanent elastic fluid was generated from the fired gunpowder, and the quantity of it was always in proportion to the quantity of powder that was used, as was found by the proportional sinking of the mercurial gage annexed to the air pump. The result of these experiments was, that the weight of the elastic air thus generated, was equal to 3/10 of the compound mass of the gunpowder which yielded it; and that its bulk, when cold and expanded to the rarity of common atmospheric air, was about 240 times the bulk of the powder; and consequently in the same proportion<cb/> would such fluid at first, if it were cold, exceed the force or clasticity of the atmosphere. But as Mr. Robins found, by another ingenious-experiment, that air heated to the extreme degree of the white heat of iron, has its elasticity quadrupled, or is 4 times as strong; he thence inferred that the force of the elastic air generated as above, at the moment of the explosion, is at least 4 times 240, or 960, or in round numbers about 1000 times as strong as the elasticity or pressure of the atmosphere, on the same space.</p><p>Having thus determined the force of the gunpowder, or intensity of the agent by which the projectile is to be urged, Mr. Robins next proceeds to determine the effects it will produce, or the velocity with which it will impel a shot of a given weight from a piece of ordnance of given dimensions; which is a problem strictly limited, and perfectly soluble by mathematical rules, and is in general this: Given the first force, and the law of its variation, to determine the velocity with which it will impel a given body in passing through a given space, which is the length of the bore of the gun.</p><p>In the solution of this problem, Mr. Robins assumes these two postulates, viz, 1, That the action of the powder on the bullet ceases as soon as the bullet is out of the piece; and 2d, That all the powder of the charge is fired and converted into elastic fluid before the bullet is sensibly moved from its place: assumptions which for good reasons are found to be in many cases very near the truth. It is to be noted also, that the law by which the force of the elastic fluid varies, is this, viz, that its intensity is directly as its density, or reciprocally proportional to the space it occupies, being so much the stronger as the space is less: a principle well known, and common to all elastic fluids. Upon these principles then Mr. Robins resolves this problem, by means of the 39th prop. of Newton's Principia in a direct way, and the result is equivalent to this theorem, when the quantities are expressed by algebraic symbols; viz, the velocity of the ball where <hi rend="italics">v</hi> is the velocity of the ball, <hi rend="italics">a</hi> the length of the charge of powder, <hi rend="italics">b</hi> the whole length of the bore, <hi rend="italics">c</hi> the spec. grav. of the ball, or wt. of a cubic foot of the same matter in ounces, <hi rend="italics">d</hi> the diam. of the bore, <hi rend="italics">w</hi> the wt. of the ball in ounces.</p><p>For example, Suppose <hi rend="italics">a</hi> = 2 5/8 inc., <hi rend="italics">b</hi> = 45 inches, <hi rend="italics">c</hi> = 11345 oz, for a ball of lead, and <hi rend="italics">d</hi> = 3/4 inches; then feet per second, the velocity of the ball.</p><p>Or, if the wt. of the bullet be <hi rend="italics">w</hi> = 1 9/20 oz = 20/20 oz. Then feet, as before.</p><p>“Having in this proposition, says Mr. Robins, shewn how the velocity, which any bullet acquires from<pb n="565"/><cb/> the force of powder, may be computed upon the principles of the theory laid down in the preceding propositions; we shall next shew, that the actual velocities, with which bullets of different magnitudes are impelled from different pieces, with different quantities of powder, are really the same with the velocities assigned by these computations; and consequently that this theory of the force of powder, here delivered, does unquestionably ascertain the true action and modification of this enormous power.</p><p>“But in order to compare the velocities communicated to bullets by the explosion with the velocities resulting from the theory by computation; it is necessary that the actual velocities with which bullets move, should be capable of being discovered, which yet is impossible to be done by any methods hitherto made public. The only means hitherto practised by others for that purpose, have been either by observing the time of the flight of the shot through a given space, or by measuring the range of the shot at a given elevation; and thence computing, on the parabolic hypothesis, what velocity would produce this range. The first method labours under this insurmountable difficulty, that the velocities of these bodies are often so swift, and consequently the time observed is so short, that an imperceptible error in that time may occasion an error in the velocity thus found, of 2, 3, 4, 5, or 600 feet in a second. The other method is so fallacious, by reason of the resistance of the air (to which inequality the first is also liable), that the velocities thus assigned may not be perhaps the 10th part of the actual velocities sought.</p><p>“To remedy then these inconveniences, I have invented a new method of finding the real velocities of bullets of all kinds; and this to such a degree of exactness (which may be augmented too at pleasure), that in a bullet moving with a velocity of 1700 feet in 1″, the error in the estimation of it need never amount to its 500th part; and this without any extraordinary nicety in the construction of the machine.”</p><p>Mr. Robins then gives an account of the machine by which he measures the velocities of the balls, which machine is simply this, viz, a pendulous block of wood suspended freely by a horizontal axis, against which block are to be fired the balls whose velocities are to be determined.</p><p>“This instrument thus fitted, if the weight of the pendulum be known, and likewise the respective distances of its centre of gravity, and of its centre of oscillation, from its axis of suspension, it will thence be known what motion will be communicated to this pendulum by the percussion of a body of a known weight moving with a known degree of celerity, and striking it in a given point; that is, if the pendulum be supposed at rest before the percussion, it will be known what vibration it ought to make in consequence of such a determined blow; and, on the contrary, if the pendulum, being at rest, is struck by a body of a known weight, and the vibration, which the pendulum makes after the blow, is known, the velocity of the striking body may from thence be determined.</p><p>“Hence then, if a bullet of a known weight strikes the pendulum, and the vibration, which the pendulum makes in consequence of the stroke, be ascertained;<cb/> the velocity with which the ball moved, is thence to be known.”</p><p>Mr. Robins then explains his method of computing velocities from experiments with this machine; which method is rather troublesome and perplexed, as well as the rules of Euler and Antoni, who followed him in this business, but a much simpler rule is given in my Tracts, vol. 1, p. 119, where such experiments are explained at full length, and this rule is expressed by either of the two following formulas, , the velocity; where <hi rend="italics">v</hi> denotes the velocity of the ball when it strikes the pendulum, <hi rend="italics">p</hi> the weight of the pendulum, <hi rend="italics">b</hi> the weight of the ball, <hi rend="italics">c</hi> the chord of the arc described by the vibration to the radius <hi rend="italics">r, g</hi> the distance below the axis of motion to the centre of gravity, <hi rend="italics">o</hi> the distance to the centre of oscillation, <hi rend="italics">i</hi> the distance to the point of impact, and <hi rend="italics">n</hi> the number of oscillations the pendulum will perform in one minute, when made to oscillate in small arcs. The latter of these two theorems is much the easiest, both because it is free of radicals, and because the value of the radical √<hi rend="italics">o,</hi> in the former, is to be first computed from the number <hi rend="italics">n,</hi> or number of oscillations the pendulum is observed to make.</p><p>With such machines Mr. Robins made a great number of experiments, with musket barrels of different lengths, with balls of various weights, and with different charges or quantities of powder. He has set down the results of 61 of these experiments, which nearly agree with the corresponding velocities as computed by his theory of the force of powder, and which therefore establish that theory on a sure foundation.</p><p>From these experiments, as well as from the preceding theory, many important conclusions were deduced by Mr. Robins; and indeed by means of these it is obvious that every thing may be determined relative both to the true theory of projectiles, and to the practice of artillery. For, by firing a piece of ordnance, charged in a similar manner, against such a ballistic pendulum from different distances, the velocity lost by passing through such spaces of air will be found, and consequently the resistance of the air, the only circumstance that was wanting to complete the theory of Gunnery, or military projectiles; and of this kind I have since made a great number of experiments with cannon balls, and have thereby obtained the whole series of resistances to such a ball when moving with every degree of velocity, from 0 up to 2000 feet per second of time. In the structure of artillery, they may likewise be of the greatest use: For hence may be determined the best lengths of guns; the proportions of the shot and powder to the several lengths; the thickness of a piece, so as it may be able to confine, without bursting, any given charge of powder; as also the effect of wads, chambers, placing of the vent, ramming the powder, &c. For the many other curious circumstances relating to this subject, and the various other improvements in the theory and practice of Gunnery, made by Mr. Robins, consult the first vol. of his Tracts, collected and published by Dr. Wilson, in the year 1761, where ample information may be found.</p><p>Soon after the first publication of Robins's New Principles of Gunnery, in 1742, the learned in several<pb n="566"/><cb/> other nations, treading in his steps, repeated and farther extended the same subject, sometimes varying and enlarging the machinery; particularly Euler in Germany, D'Antoni in Italy, and Messrs. D'Arcy and Le Roy in France. But most of these, like Mr. Robins, with small fire-arms, such as muskets, and fusils.</p><p>But in the year 1775, in conjunction with several able officers of the Royal Artillery, and other ingenious gentlemen, I undertook a course of experiments with the ballistic pendulum, in which we ventured to extend the machinery to cannon shot of 1, 2, and 3 pounds weight. An account of these experiments was published in the Philos. Trans. for 1778, and for which the Royal Society honoured me with the prize of the gold medal. “These were the only experiments that I know of which had been made with cannon balls for this purpose, although the conclusions to be deduced from such, are of the greatest importance to those parts of natural philosophy which are dependent on the effects of fired gunpowder; nor do I know of any other practical method of ascertaining the initial velocities within any tolerable degree of the truth. The knowledge of this velocity is of the utmost consequence in Gunnery: by means of it, together with the law of the resistance of the medium, every thing is determinable relative to that business; for, besides its being an excellent method of trying the strength of different sorts of powder, it gives us the law relative to the different quantities of powder, to the different weights of shot, and to the different lengths and sizes of guns. Besides these, there does not seem to be any thing wanting to answer any inquiry that can be made concerning the flight and ranges of shot, except the effects arising from the resistance of the medium. In these experiments the weights of the pendulums employed were from 300 to near 600 pounds. In that paper is described the method of constructing the machinery, of finding the centres of gravity and oscillation of the pendulum, and of making the experiments, which are all set down in the form of a journal, with all the minute and concomitant circumstances; as also the investigation of the new and easy rule, set down just above, for computing the velocity of the ball from the experiments. The charges of powder were varied from 2 to 8 ounces, and the shot from 1 to near 3 pounds. And from the whole were clearly deduced these principal inferences, viz,</p><p>“1. First, That gunpowder fires almost instantaneously.—2. That the velocities communicated to balls or shot, of the same weight, by different quantities of powder, are nearly in the subduplicate ratio of those quantities: a small variation, in defect, taking place when the quantities of powder became great.—3. And when shot of different weights are employed, with the same quantity of powder, the velocities communicated to them, are nearly in the reciprocal subduplicate ratio of their weights.—4. So that, universally, shot which are of different weights, and impelled by the firing of different quantities of powder, acquire velocities which are directly as the square roots of the quantities of powder, and inversely as the square roots of the weights of the shot, nearly.—5. It would therefore be <*> great improvement in artillery, to make use of shot of <*> long form, or of heavier matter; for thus the mo-<cb/> mentum of a shot, when fired with the same weight of powder, would be increased in the ratio of the square root of the weight of the shot.—6. It would also be an improvement to diminish the windage; for by so doing, one-third or more of the quantity of powder might be saved.—7. When the improvements mentioned in the last two articles are considered as both taking place, it is evident that about half the quantity of powder might be saved, which is a very considerable object. But important as this saving may be, it seems to be still exceeded by that of the article of the guns; for thus a small gun may be made to have the effect and execution of another of two or three times its size in the present mode, by discharging a shot of two or three times the weight of its natural ball or round shot. And thus a small ship might discharge shot as heavy as those of the greatest now made use of.</p><p>“Finally, as the above experiments exhibit the regulations with regard to the weights of powder and balls, when fired from the same piece of ordnance, &c; so by making similar experiments with a gun, varied in its length, by cutting off from it a certain part before each course of experiments, the effects and general rules for the different lengths of guns may be certainly determined by them. In short, the principles on which these experiments were made, are so fruitful in consequences, that, in conjunction with the effects resulting from the resistance of the medium, they seem to be sufsicient for answering all the enquiries of the speculative philosopher, as well as those of the practical artillerist.</p><p>In the year 1786 was published the first volume of my Tracts, in which is detailed, at great length, another very extensive course of experiments which were carried on at Woolwich in the years 1783, 1784, and 1785, by order of the Duke of Richmond, Master Genearl of the Ordnance. The objects of this course were very numerous, but the principal of them were the following:</p><p>“1. The velocities with which balls are projected by equal charges of powder, from pieces of the same weight and calibre, but of different lengths.</p><p>“2. The velocities with different charges of powder, the weight and length of the gun being the same.</p><p>“3. The greatest velocity due to the different lengths of guns, to be obtained by increasing the charge as far as the resistance of the piece is capable of sustaining.</p><p>“4. The effect of varying the weight of the piece; every thing else being the same.</p><p>“5. The penetration of balls into blocks of wood.</p><p>“6. The ranges and times of flight of balls; to compare them with their initial velocities for determining the resistance of the medium.</p><p>“7. The effect of wads; of different degrees of ramming; of different degrees of windage; of different positions of the vent; of chambers, and trunnions, and every other circumstance necessary to be known for the improvement of artillery.”</p><p>All these objects were obtained in a very perfect and accurate manner; excepting only the article of ranges, which were not quite so regular and uniform as might<pb n="567"/><cb/> be wished. The balls too were most of them of one pound weight; but the powder was increased from 1 ounce, up till the bore was quite full; and the pendulum was from 600 to 800lb. weight. The conclusions from the whole were as follow:</p><p>“1. That the former law, between the charge and velocity of ball, is again confirmed, viz, that the velocity is directly as the square root of the weight of powder, as far as to about the charge of 8 ounces: and so it would continue for all charges, were the guns of an indefinite length. But as the length of the charge is increased, and bears a more considerable proportion to the length of the bore, the velocity falls the more short of that proportion.</p><p>“2. That the velocity of the ball increases with the charge to a certain point, which is peculiar to each gun, where it is greatest; and that by farther increasing the charge, the velocity gradually diminishes, till the bore is quite full of powder. That this charge for the greatest velocity is greater as the gun is longer, but not greater however in so high a proportion as the length of the gun is, so that the part of the bore filled with powder bears a less proportion to the whole in the long guns, than it does in the short ones; the part of the whole which is filled being indeed nearly in the reciprocal subduplicate ratio of the length of the empty part. And the other circumstances are as in this table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=5 align=center" role="data"><hi rend="italics"><hi rend="smallcaps">Table</hi> of Charges producing the Greatest Velocity.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Gun Num.</cell><cell cols="1" rows="1" role="data">Length of the bore.</cell><cell cols="1" rows="1" role="data">Length filled.</cell><cell cols="1" rows="1" role="data">Part of the whole.</cell><cell cols="1" rows="1" role="data">Wt. of the powder.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">inches.</cell><cell cols="1" rows="1" role="data">inches.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">oz.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">28.2</cell><cell cols="1" rows="1" rend="align=right" role="data">8.2</cell><cell cols="1" rows="1" role="data">3/10</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">38.1</cell><cell cols="1" rows="1" rend="align=right" role="data">9.5</cell><cell cols="1" rows="1" role="data">3/12</cell><cell cols="1" rows="1" role="data">14</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">57.4</cell><cell cols="1" rows="1" rend="align=right" role="data">10.7</cell><cell cols="1" rows="1" role="data">3/1<*></cell><cell cols="1" rows="1" role="data">16</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">79.9</cell><cell cols="1" rows="1" rend="align=right" role="data">12.1</cell><cell cols="1" rows="1" role="data">3/20</cell><cell cols="1" rows="1" role="data">18</cell></row></table></p><p>“3. It appears that the velocity continually increases as the gun is longer, though the increase in velocity is but very small in respect of the increase in length, the velocities being in a ratio somewhat less than that of the square roots of the length of the bore, but somewhat greater than that of the cube roots of the length, and is indeed nearly in the middle ratio between the two.</p><p>“4. The range increases in a much less ratio than the velocity, and indeed is nearly as the square root of the velocity, the gun and elevation being the same. And when this is compared with the property of the velocity and length of gun in the foregoing paragraph, we perceive that very little is gained in the range by a great increase in the length of the gun, the charge being the same. And indeed the range is nearly as the 5th root of the length of the bore; which is so small an increase, as to amount only to about 1/7th part more range for a double length of gun.</p><p>“5. It also appears that the time of the ball's flight<cb/> is nearly as the range; the gun and elevation being the same.</p><p>“6. It appears that there is no sensible difference caused in the velocity or range, by varying the weight of the gun, nor by the use of wads, nor by different degrees of ramming, nor by firing the charge of powder in different parts of it.</p><p>“7. But a great difference in the velocity arises from a small degree of windage. Indeed with the usual established windage only, namely, about 1/20th of the caliber, no less than between 1/3 and 1/4 of the powder escapes and is lost. And as the balls are often smaller than that size, it frequently happens that half the powder is lost by unnecessary windage.</p><p>“8. It appears that the resisting force of wood, to balls fired into it, is not constant. And that the depths penetrated by different velocities or charges, are nearly as the logarithms of the charges, instead of being as the charges themselves, or, which is the same thing, as the square of the velocity.</p><p>“9. These, and most other experiments, shew that balls are greatly deflected from the direction they are projected in; and that so much as 300 or 400 yards in a range of a mile, or almost 1/4th of the range, which is nearly a deflection of an angle of 15 degrees.</p><p>“10. Finally, these experiments furnish us with the following concomitant data, to a tolerable degree of accuracy, namely, the dimensions and elevation of the gun, the weight and dimensions of the powder and shot, with the range and time of flight, and the first velocity of the ball. From which it is to be hoped that the measure of the resistance of the air to projectiles, may be determined, and thereby lay the foundation for a true and practical system of Gunnery, which may be as well useful in service as in theory.”</p><p>Since the publication of those Tracts, we have prosecuted the experiments still farther, from year to year, gradually extending our aim to more objects, and enlarging the guns and machinery, till we have arrived at experiments with the 6 pounder guns, and pendulums of 1800 pounds weight. One of the new objects of enquiry, was the resistance the atmosphere makes to military projectiles; to obtain which, the guns have been placed at many different distances from the pendulum, against which they are fired, to get the velocity lost in passing through those spaces of air; by which, and the use of the whirling machine, described near the end of the 1st vol. of Robins's Tracts, for the slower motions, I have investigated the resistance of the air to given balls moving with all degrees of velocity, from 0 up to 2000 feet per second<*> as well as the resistance for many degrees of velocity, to planes and figures of other shapes, and inclined to their path in all varieties of angles; from which I have deduced general laws and formulas for all such motions.</p><p>Mr. Robins made also similar experiments on the resistance of the air; but being only with musket bullets, on account of their smallness, and of their change of figure by the explosion of the powder, I find they are very inaccurate, and considerably different from those above mentioned, which were accurately made with pretty considerable cannon balls, of iron. For which reason we may omit here the rules and theory deduced from them by Mr. Robins, till others more correct shall<pb n="568"/><cb/> have been established. All these experiments indeed agree in evincing the very enormous resistance the air makes to the swift motions of military projectiles, amounting in some cases to 20 or 30 times the weight of the ball itself; on which account the common rules for projectiles, deduced from the parabolic theory, are of little or no use in real practice; for, from these experiments it is clearly proved, that the track described by the flight even of the heaviest shot, is neither a parabola, nor yet approaching any thing near it, except when they are projected with very small velocities; in so much that some balls, which in the air range only to the distance of one mile, would in vacuo, when projected with the same velocity, range above 10 or 20 times as far. For the common rules of the parabolic theory, see <hi rend="smallcaps">Projectiles.</hi> And for a small specimen of my experiments on resistances, see the 2d vol. of the Edinburgh Philos. Trans.; as also my Conic Sections and Select Exercises, at the end, also the articles <hi rend="smallcaps">Force</hi>, and <hi rend="smallcaps">Resistance</hi>, in this Dictionary.</p><p>Mr. Benjamin Thompson instituted a <*>ry considerable course of experiments of the same kind as those of Mr. Robins, with musket barrels, which was published in the Philos. Trans. vol. 71, for the year 1781. In these experiments, the conclusions of Mr. Robins are generally confirmed, and several other curious circumstances in this business are remarked by Mr. Thompson. This gentleman also pursues a hint thrown out by Mr. Robins relative to the determining the velocity of a ball from the recoil of the pendulous gun itself. Mr. Robins, in prop. 11, remarks that the effect of the exploded powder upon the recoil of the gun, is the same, whether the gun is charged with a ball, or without one; and that the chord, or velocity, of recoil with the powder alone, being subtracted from that of the recoil when charged with both powder and ball, leaves the velocity which is due to the ball alone. From whence Mr. Thompson observes, that the inference is obvious, viz, that the momentum thus communicated to the gun by the ball alone, being equal to the momentum of the ball, this becomes known; and therefore being divided by the known weight of the ball, the quotient will be its velocity. Mr. Thompson sets a great value on this new rule, the velocities by means of which, he found to agree nearly with several of those deduced from the motion of the pendulum; and in the other cases in which they differed greatly from these, he very inconsistently supposes that these latter ones are erroneous. In the experiments however contained in my Tracts, a great multitude of those cases are compared together, and the inaccuracy of that new rule is fully proved.</p><p>Having in the 9th prop. compared together a number of computed and experimented velocities of balls, to verify his theory: in the 10th prop. Mr. Robins assigns the changes in the force of powder, which arise from the different state of the atmosphere, as to heat and moisture, both which he finds have some effect on it, but especially the latter. In prop. 11 he investigates the velocity which the flame of gunpowder acquires by expanding itself, supposing it fired in a given piece of artillery, without either a bullet or any other body before it. This velocity he finds is upwards of 7000 feet per second. But the celebrated Euler, in his commentary on this part of Mr. Robins's book, thinks it may be<cb/> still much greater. And in this prop. too it is that Mr. Robins declares his opinion, above alluded to, viz, that the effect of the powder upon the recoil of the gun is the same, in all cases, whether fired with a ball, or without one.—In prop. 12 he ascertains the manner in which the flame of powder impels a ball which is laid at a considerable distance from the charge; shewing here that the sudden accumulation and density of the fluid against the ball, is the reason that the barrel is so often burst in those cases.—In prop. 13 he enumerates the various kinds of powder, and describes the properest methods of examining its goodness. He here shews that the best proportion of the ingredients, is when the saltpetre is 3/4 of the whole compound mass of the powder, and the sulphur and charcoal the other 1/4 between them, in equal quantities. In this prop. Mr. Robins takes occasion to remark upon the use of eprouvettes, or methods of trying powder; condemning the practice of the English in using what is called the vertical eprouvette; as well as that of the French, in using a small mortar, with a very large ball, and a small charge of powder: and instead of these, he strongly recommends the use of his ballistic pendulum, for its great accuracy: But for still more dispatch, he says he should use another method, which however he reserves to himself, without giving any particular description of it. From what has been done by Mr. Robins upon this head, several persons have introduced his method of suspending the gun as a pendulum, and noting the quantity of its oscillating recoil when fired with a certain quantity of powder; and of this kind I have contrived a machine, which possesses several advantages over all others, being extremely simple, accurate, and expeditious; so much so indeed, that the weighing out of the powder is the chief part of the trouble. See <hi rend="smallcaps">Gunpowder</hi>, and <hi rend="smallcaps">Eprouvette.</hi></p><p>The other or 2d chapter of Mr. Robins's work, in 8 propositions, treats “of the resistance of the air, and of the track described by the flight of shot and shells.” And of these, prop. 1 describes the general principles of the resistance of fluids to solid bodies moving in them. Here Mr. Robins discriminates between continued and compressed fluids, which immediately rush into the space quitted by a body moving in them, and whose parts yield to the impulse of the body without condensing and accumulating before it; and such fluids as are imperfectly compressed, rushing into a void space with a limited velocity, as in the case of our atmosphere, which condenses more and more before the ball as this moves quicker, and also presses the less behind it, by following it always with only a given velocity: hence it happens that the former fluid will resist moving bodies in proportion to the square of the velocity, while the latter resists in a higher proportion.—Proposition 2 is “to determine the resistance of the air to projectiles by experiments.” One of the methods for this purpose, is by the ballistic pendulum, placing the gun at different distances from it, by which he finds the velocity lost in passing through certain spaces of air, and consequently the force of resistance to such velocities as the body moves with in the several parts of its path. And another way was by firing balls, with a known given velocity, over a large piece of water, in which the fall and plunge of the ball<pb n="569"/><cb/> could be seen, and consequently the space it passed over in a given time. By these means Mr. Robins determined the resistances of the air to several different velocities, all which shewed that there was a gradual increase of the resistance, over the law of the square of the velocity, as the body moved quicker.—In the remaining propositions of this chapter, he proceeds a little farther in this subject of the resistance of the air; in which he lays down a rule for the proportion of the resistance between two assigned velocities; and he shews that when a 24 pound ball, fired with its full charge of powder, first issues from the piece, the resistance it meets with from the air is more than 20 times its weight. He farther shews that “the track described by the flight of shot or shells is neither a parabola, nor nearly a parabola, unless they are projected with small velocities;” and that “bullets in their flight are not only depressed beneath their original direction by the action of gravity, but are also frequently driven to the right or left of that direction by the action of some other force: and in the 8th or last proposition, he pretends to shew that the depths of penetration of balls into firm substances, are as the squares of the velocities. But this is a mistake; for neither does it appear that his trials were sufficiently numerous or various, nor were his small leaden balls fit for this purpose; and I have found, from a number of trials with iron cannon balls, that the penetrations are in a much lower proportion, and that the resisting force of wood is not uniform. See my <hi rend="smallcaps">Tracts.</hi></p><p>In the following small tracts, added to the principles, in this volume, Mr. Robins prosecutes the subject of the resistance of the air much farther, and lays down rules for computing ranges made in the air. But these must be far from accurate, as they are founded on the two following principles, which I know, from numerous experiments, are erroneous: viz, 1st, “That till the velocity of the projectile surpasses that of 1100 feet in a second, the resistance may be esteemed to be in the duplicate proportion of the velocity. 2d, That if the velocity be greater than that of 11 or 1200 feet in a second, then the absolute quantity of that resistance in these greater velocities will be near 3 times as great, as it should be by a comparison with the smaller velocities.” For, instead of leaping at once from the law of the square of the velocities, and ever after being about 3 times as much, my experiments prove that the increase of the resistance above the law of the square of the velocity, takes place at first in the smallest motions, and increases gradually more and more, to a certain point, but never rises so high as to be 3 times that quantity, after which it decreases again. To render this evident, I have inserted the following table of the actual quantities of resistances, which are deduced from accurate experiments, and which shew also the nature of the law of the variations, by means of the columns of differences annexed; reserving the detail of the experiments themselves to another occasion. These resistances are, upon a ball of 1.965 inc. diameter, in avoirdupois ounces, and are for all velocities, from 0, up to that of 2000 feet per second of time.<cb/> <hi rend="center"><hi rend="italics">The quantity of the resistance of the air to a ball of 1.965 inc. diameter.</hi></hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Veloc. in feet</cell><cell cols="1" rows="1" role="data">Resist. in ounces</cell><cell cols="1" rows="1" role="data">1st Differenes</cell><cell cols="1" rows="1" role="data">2d Differences</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.000</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0.006</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.025</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">0.054</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0.100</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">0.155</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">0.23 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">0.42 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">0.67 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">2 3/4 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8 1/4</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">11 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5 3/4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">25 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">45 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">72 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">107 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">700</cell><cell cols="1" rows="1" role="data">151 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">800</cell><cell cols="1" rows="1" role="data">205 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">900</cell><cell cols="1" rows="1" role="data">271 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">350 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1100</cell><cell cols="1" rows="1" role="data">442 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">104</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1200</cell><cell cols="1" rows="1" role="data">546 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">115</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1300</cell><cell cols="1" rows="1" role="data">661 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1400</cell><cell cols="1" rows="1" role="data">785 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1500</cell><cell cols="1" rows="1" role="data">916 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ 4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1600</cell><cell cols="1" rows="1" role="data">1051 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1700</cell><cell cols="1" rows="1" role="data">1186 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- 2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1800</cell><cell cols="1" rows="1" role="data">1319 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1900</cell><cell cols="1" rows="1" role="data">1447 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">122</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2000</cell><cell cols="1" rows="1" role="data">1569 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>The additional tracts of Mr. Robins, in the latter part of this volume, which contain many useful and important matters, are numbered and titled as follows, viz, Number 1, “Of the resistance of the air. Number 2, Of the resistance of the air; together with the method of computing the motions of bodies projected in that medium. Number 3, An account of the experiments, relating to the resistance of the air, exhibited at different times before the Royal Society, in the year 1746. Number 4, Of the force of fired gunpowder, together with the computation of the velocities thereby communicated to military projectiles. Number 5, A comparison of the experimental ranges of cannon and mortars with the theory contained in the preceding papers.—Practical Maxims relating to the effects and management of artillery, and the flight of shells and shot.—A proposal for increasing the strength of the British navy, by changing all the guns, from the 18 pounders downwards, into others of equal weight, but of a greater bore.” With several letters, and other papers, “On pointing, or the directing of cannon to strike distant objects; Of the nature and advantage of reifled barrel picces,” &c.<pb n="570"/><cb/></p><p>I have dwelt thus long on Mr. Robins's New Principles of Gunnery, because it is the first work that can be considered as attempting to establish a practical system of gunnery, and projectiles, on good experiments, on the force of gunpowder, on the resistance of the air, and on the effects of different pieces of artillery. Those experiments are however not sufficiently perfect, both on account of the smallness of the bullets, and for want of good ranges, to form a proper theory upon. I have supplied some of the necessary desiderata for this purpose, viz, the resistance of the air to cannon balls moving with all degrees of velocity, and the velocities communicated by given charges of powder to different balls, and from different pieces of artillery. But there are still wanting good experiments with different pieces of ordnance, giving the ranges and times of flight, with all varieties of charges, and at all different angles of elevation. A few however of those I have obtained, as in the following small table, which are derived from experiments made with a medium onepounder gun, the iron ball being nearly 2 inches in diameter. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Powder</cell><cell cols="1" rows="1" role="data">Elev. of gun</cell><cell cols="1" rows="1" role="data">Veloc. of ball</cell><cell cols="1" rows="1" role="data">Range</cell><cell cols="1" rows="1" role="data">Time of flight</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">feet</cell><cell cols="1" rows="1" role="data">feet</cell><cell cols="1" rows="1" role="data">″</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">860</cell><cell cols="1" rows="1" rend="align=right" role="data">4100</cell><cell cols="1" rows="1" role="data"> 9</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">1230</cell><cell cols="1" rows="1" rend="align=right" role="data">5100</cell><cell cols="1" rows="1" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">1640</cell><cell cols="1" rows="1" rend="align=right" role="data">6000</cell><cell cols="1" rows="1" role="data">14 1/2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">1680</cell><cell cols="1" rows="1" rend="align=right" role="data">6700</cell><cell cols="1" rows="1" role="data">15 1/2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">860</cell><cell cols="1" rows="1" rend="align=right" role="data">5100</cell><cell cols="1" rows="1" role="data">21</cell></row></table></p><p>The celebrated Mr. Euler added many excellent dissertations on the subject of Gunnery, in his translation of Robins's Gunnery into the German language; which were again farther improved in Brown's translation of the same into English in the year 1777. See also Antoni's <hi rend="italics">Examen de la Poudre;</hi> the experiments of MM. D'Arcy and Le Roy, in the Memoirs of the Academy in 1751; and D'Arcy's <hi rend="italics">Essai d'une theorie d'artillerie</hi> in 1760: my Tracts; and paper on the force of fired gunpowder in the Philof. Trans. for 1778: and Thompson's paper on the same subject in 1781. Of the common or parabolic theory of Gunnery, Mr. Simpson gave a very neat and concise treatise in his Select Exercises. And other authors on this part, are Starrat, Gray, Williams, Glenie, &c.</p></div1><div1 part="N" n="GUNPOWDER" org="uniform" sample="complete" type="entry"><head>GUNPOWDER</head><p>, a composition of nitre, sulphur, and charcoal, mixed together, and usually granulated. This easily takes fire; and when fired, it rarefies or expands with great vehemence, by means of its elastic force.—It is to this powder that we owe all the effect and action of guns, and ordnance of all sorts. So that fortification, with the modern military art, &c, in a great measure depends upon it. The above definition however is not general, for instead of the nitre, it has lately been discovered that the marine acid answers much better.</p><p>The invention of Gunpowder is ascribed, by Polydore Virgil, to a chemist; who having accidentally put<cb/> some of this composition in a mortar, and covered it with a stone, it happened to take sire, and blew up the stone. Thevet says that the person here spoken of was a monk of Fribourg, named Constantine Anelzen; but Belleforet and other authors, with more probability, hold it to be Bartholdus Schwartz, or the black, who discovered it, as some say, about the year 1320; and the first use of it is ascribed to the Venetians, in the year 1380, during the war with the Genoese. But there are earlier accounts of its use, after the accident of Schwartz, as well as before it. For Peter Mexia, in his Various Readings, mentions that the Moors being besieged in 1343, by Alphonsus the 11th, king of Castile, discharged a kind of iron mortars upon them, which made a noise like thunder; and this is seconded by what is related by Don Pedro, bishop of Leon, in his chronicle of king Alphonsus, who reduced Toledo, viz, that in a sea-combat between the king of Tunis and the Moorish king of Seville, about that time, those of Tunis had certain iron tubs or barrels, with which they threw thunderbolts of fire.</p><p>Du-Cange adds, that there is mention made of gunpowder in the registers of the chambers of accounts in France as early as the year 1338.</p><p>But it appears that Roger Bacon knew of Gunpowder near 100 years before Schwartz was born. He tells us, in his Treatise De Secretis Operibus Artis. & Naturæ, & de Nullitate Magiæ, cap. 6, (which is supposed by some to have been published at Oxford in 1216, and which was undoubtedly written before his Opus Majus, in 1267), “that from saltpetre, and other ingredients, we are able to make a fire that shall burn at what distance we please.” And Dr. Plott, in his History of Oxfordshire, pa. 236, assures us that these “other ingredients were explained in a MS. copy of the same treatise, in the hands of Dr. G. Langbain, and seen by Dr. Wallis, to be sulphur and wood coal.” Farther, in the life of Friar Bacon in Biographia Britannica, vol. 1, we are told that Bacon himself has divulged the secret of this composition in a cypher, by transposing the letters of the two words in chap. xi. of the said treatise; where it is thus expressed: <hi rend="italics">sed tamen salis petræ <hi rend="smallcaps">Lura mope can ubre</hi></hi> (i. e. <hi rend="italics">carbonum pulvere) et sulphuris; et sic facies tonitrum & corruscationem, si scias artificium:</hi> and from hence the biographer apprehends the words <hi rend="italics">carbonum pulvere</hi> were transferred to the 6th chapter of Langbain's MS. In this same chapter Bacon expressly says that sounds like thunder, and corruscations, may be formed in the air, much more horrible than those that happen naturally. And farther adds, that there are many ways of doing this, by which a city or an army might be destroyed: and he supposes that by an artifice of this kind Gideon defeated the Midianites with only 300 men: Judges, chap. 7. There is also another passage to the same purpose, in the treatise De Scientia Experimentali. See Dr. Jebb's edition of the Opus Majus, p. 474.</p><p>Mr. Robins, in the preface to his Gunnery, apprehends that Bacon describes Gunpowder not as a new composition first proposed by himself, but as the application of an old one to military purposes, and that it was known long before his time.</p><p>But Mr. Dutens carries the antiquity of Gunpow-<pb n="571"/><cb/></p><p>der still much higher, and refers to the writings of the ancients themselves for the proof of it. “Virgil, says he, and his Commentator Servius (Æneid, lib. 6, v. 585), Hyginus (Fabul. 61 and 650), Eustathius (ad Odyss, <foreign xml:lang="greek">l</foreign> 234, pa. 1682, lib. 1), La Cerda (in Virgil. loc. cit.), Valerius Flaccus (lib. i. 662), and many other authors (as Raphael Volatarran. in Commentar. Cornelius Agrippa poster. Oper. de Verbo Dei, c. 100, p. 237.—Gruteri Fax Artium Liberal tom. 2, p. 1236), speak in such a manner of Salmoneus's attempts to imitate thunder, as suggest to us that this prince used for that purpose a composition of the nature of gunpowder. Eustathius in particular speaks of him on this occasion, as being so very expert in mechanics, that he formed machines, which imitated the noise of thunder; and the writers of fable, whose surprise in this respect may be compared to that of the Mexicans when they first beheld the fire-arms of the Spaniards, give out that Jupiter, incensed at the audacity of this prince, slew him with lightning, as he was employing himself in launching his thunder. But it is much more natural to suppose that this unfortunate prince, the inventor of Gunpowder, gave rise to these fables, by having accidentally fallen a victim to his own experiments. Dion (Hist. Rom. in Caligula, p. 662) and Joannes Antiochenus (in Chronico, &c. a Valesio edita, Paris 1634, p. 804), report the very same thing of Caligula, assuring us that this emperor imitated thunder and lightning by means of certain machines, which at the same time emitted stones. Themistius informs us that the Brachmans encountered one another with thunder and lightning, which they had the art of launching from on high at a considerable distance; (Themist. Oratio 27, p. 337). And in another place he relates, that Hercules and Bacchus, attempting to assail them in a fort where they were entrenched, were so roughly received by reiterated strokes of thunder and lightning, launched upon them from on high by the besieged, that they were obliged to retire, leaving behind them an everlasting monument of the rashness of their enterprise. Agathias the historian reports of Anthemius Traliensis, that having fallen out with his neighbour Zeno the rhetorician, he set sire to his house with thunder and lightning. It appears from all these passages, that the effects ascribed to these engines of war, especially those of Caligula, Anthemius, and the Indians, could be only brought about by Gunpowder. And what is still more, we find in Julius Africanus a receipt for an ingenious composition to be thrown upon an enemy, which very nearly resembles that powder. But what places this beyond all doubt, is a clear and positive passage of an author called Marcus Græcus, whose work in manuscript is in the royal library at Paris, intitled Liber Ignium. Dr. Mead had the same also in manuscript, and a copy of that is now in my hands. (See above). The author describes several ways of encountering an enemy, by launching fire upon him; and among others gives the following. Mix together one pound of live sulphur, two of charcoal of willow, and 6 of saltpetre; reducing them to a very fine powder in a marble mortar. He adds, that a certain quantity of this is to be put into a long, narrow, and well compacted cover, and so discharged into the air. Here we have the description<cb/> of a rocket. The cover with which thunder is imitated, he represents as short, thick, but half-silled, and strongly bound with packthread; which is exactly the form of a cracker. He then treats of different methods of preparing the match, and how one squib may set fire to another in the air, by having it inclosed within it. In short, he speaks as clearly of the composition and effects of Gunpowder, as any person in our times could do. I own I have not yet been able precisely to determine when this author lived, but probably it was before the time of the Arabian physician Mesue, who speaks of him, and who flourished in the beginning of the 9th century. Nay, there is reason to believe that he is the same of whom Galen speaks; in which case he will be of antiquity sufficient to support what I advance.” It appears too, from many authors, and many circumstances, that this composition has been known to the Chinese and Indians for thousands of years. See what is said on this head under the article <hi rend="smallcaps">Gun.</hi></p><p>To this history of Gunpowder it may be added, that it has lately been discovered that saltpetre or nitre is not essential to this composition, but that its place may be supplied by other substances; for new Gunpowder, of double the strength of the old, has lately been made in France, by the chemists in that country, without any nitre at all; and in the year 1790 I tried some of this new powder, that was made at Woolwich, with my eprouvette, when I found it about double the strength of the ordinary sort. This is effected by substituting, instead of the nitre, the like quantity of the marine acid.</p><p>But perhaps this new composition may not come into common and general use; both because of the great expence in procuring or making the acid, and of the trouble and danger of preventing it from taking sire by the heat of making it; for it is found to catch fire and explode from a very small degree of heat, and without the aid of a spark.</p><p><hi rend="italics">As to the Prcparation of</hi> <hi rend="smallcaps">Gunpowder;</hi> there are divers compositions of it, with respect to the proportions of the three ingredients, to be met with in pyrotechnical writings; but the process of making it up is much the same in all.</p><p>For some time after the invention of artillery, Gunpowder was of a much weaker composition than that now in use, or that deseribed by Marcus Græcus; which was chiefly owing to the weakness of their first pieces. See <hi rend="smallcaps">Gun</hi> and <hi rend="smallcaps">Cannon.</hi> Of 23 different compositions, used at different times, and mentioned by Tartaglia in his Ques. and Inv. lib. 3, ques. 5, the first, which was the oldest, contained equal parts of the three ingredients. But when guns of modern structure were introduced, Gunpowder of the same composition as the present came also into use. In the time of Tartaglia the cannon powder was made of 4 parts of nitre, one of sulphur, and one of charcoal; and the <*>usket powder of 48 parts of nitre, 7 parts of sulphur, and 8 parts of charcoal; or of 18 parts of nitre, 2 parts of sulphur, and 3 parts of charcoal. But the modern composition is 6 parts of nitre, to one of each of the other two ingredients. Though Mr. Napier says, he finds the strength commonly to be greatest when the proportions are, nitre 3lb, charcoal about 90z, and sulphur about 30z. See his paper on Gunpowder in the Transactions of the Royal Irish Academy, vol. 2. The can-<pb n="572"/><cb/> non powder was in meal, and the musket powder grained. And it is certain that the graining of powder, which is a very considerable advantage, is a modern improvement. See the presace to Robins's Math. Tracts, pa. 32.</p><p>To make Gunpowder duly, regard is to be had to the purity or goodness of the ingredients, as well as the proportions of them; for the strength of the powder depends much on that circumstance, and also on the due working or mixing of them together.</p><p>To purify the nitre, by taking away the fixt or common salt, and earthy part. Dissolve it in a quantity of hot water over the fire; then filtrate it through a flannel bag, into an open vessel, and set it aside to cool, and to crystallize. These crystals may in like manner be dissolved and crystallized again; and so on, till they become quite pure and white. Then put the crystals into a dry kettle over a moderate fire, which gradually increase till it begins to smoke, evaporate, lose its humidity, and grow very white: it must be kept continually stirring with a ladle, lest it should return to its former figure, by which its greasiness would be taken away: after that, so much water is to be poured into the kettle as will cover the nitre; and when it is dissolved, and reduced to the consistency of a thick liquor, it must be continually stirred with a ladle till all the moisture is again evaporated, and it be reduced to a dry and white meal.</p><p>The like regard is to be had to the sulphur; choosing that which is in large lumps, clear and perfectly yellow; not very hard, nor compact, but porous; nor yet too much shining; and if, when set on fire, it freely burns all away, it is a sign of its goodness: so likewise, if it be pressed between two iron plates that are hot enough to make it run, and in the running appear yellow, and that which remains of a reddish colour, it is then fit for the purpose. But in case it be foul, it may be purisied in this manner: melt the sulphur in a large iron ladle, or pot, over a very gentle coal fire, well kindled, but not flaming; then scum off all that rises on the top, and swims upon the sulphur; take it presently after from <*>he fire, and strain it through a double linen cloth, letting it pass leisurely; so will it be pure, the gross matter remaining behind in the cloth.</p><p>For the charcoal, the third ingredient, such should be chosen as is large, clear, and free from knots, well burnt, and cleaving. The charcoal of light woods is mostly preferred, as of willow, and that of the branches or twigs of a moderate thickness, as of an inch or two in diameter. Dogwood is now much esteemed for this purpose. And a method of charring the wood in a large iron cylinder has lately been recommended, and indeed proved, as yielding better charcoal than formerly.</p><p>The charcoal not only concurs with the sulphur in supplying the inslammable matter, which causes the detonation of the nitre, but also greatly adds to the explosive power of it by the quantity of elastic vapour expelled during its combustion.</p><p>These three ingredients, in their purest state, being procured, long experience has shewn that they are then to be mixed together in the proportion before mentioned, to have the best effect, viz, three-quarters of the composition to be nitre, and the other quarter made up<cb/> of equal parts of the other two ingredients; or, which is the same thing, 6 parts nitre, 1 part sulphur, and 1 part charcoal.</p><p>But it is not the due proportion of the materials only, which is necessary to the making of good powders another circumstance, not less essential, is the mixing them well together: if this be not effectually done, some parts of the composition will have too much nitre in them, and others too little; and in either case there will be a defect of strength in the powder. Robins, pa. 119.</p><p>After the materials have been reduced to sine dust, they are mixed together, and moistened with water, or vinegar, or urine, or spirit of wine, &c, and then beaten together with wooden pestles for 24 hours, either by hand, or by mills, and afterwards pressed into a hard, firm, and solid cake. When dry, it is grained or corned; which is done by breaking the cake of powder into small pieces, and so running it through a sieve; by which means the grains may have any size given them, according to the nature of the sieve employed, either finer or coarser; and thus also the dust is separated from the grains, and again mixed with other manufacturing powder, or worked up into cakes again.</p><p>Powder is smoothed, or glazed, as it is called, for small arms, by the following operation: a hollow cylinder or cask is mounted on an axis, turned by a wheel; this cask is half filled with powder, and turned for 6 hours; and thus by the mutual friction of the grains of powder it is smoothed, or glazed. The fine mealy part, thus separated or worn off from the rest, is again granulated.</p><p><hi rend="italics">The Nature, Effects, &c, of Powder.</hi>—When the powder is prepared as above, if the least spark be struck upon it from a steel and flint, the whole will immediately inflame, and burst out with extreme violence.—The effect is not hard to account for: the charcoal part of the grain upon which the spark falls, catching fire like tinder, the sulphur and nitre are readily melted, and the former also breaks into flame; the contiguous grains at the same time undergoing the same fate.</p><p>Sir Isaac Newton reasons thus upon the point: The charcoal and sulphur in Gunpowder easily take fire, and kindle the nitre; and the spirit of the nitre, being thereby rarefied into vapour, rushes out with an explosion much after the manner that the vapour of water rushes out of an eolipile; the sulphur also, being volatile, is converted into vapour, and augments the explosion: add, that the acid vapour of the sulphur, namely that which distils under a bell into oil of sulphur, entering violently into the fixt body of the nitre, lets loose the spirit of the nitre, and excites a greater fermentation, by which the heat is farther augmented, and the fixt body of the nitre is also rarefied into fume; and the explosion is thereby made more vehement and quick.</p><p>For if salt of tartar be mixed with Gunpowder, and that mixture be warmed till it takes fire, the explosion will be far more violent and quick than that of Gunpowder alone; which cannot proceed from any other cause, than the action of the vapour of the Gunpowder upon the salt of tartar, by which that salt is rarefied.</p><p>The explosion of Gunpowder therefore arises from the violent action, by which all the mixture <*>eing quickly<pb n="573"/><cb/> and vehemently heated, is rarefied and converted into fume and vapour; which vapour, by the violence of that action becoming so hot as to shine, appears in the form of a flame.</p><p>M. De la Hire, in the History of the French Academy for 1702, ascribes all the force and effect of Gunpowder to the spring or elasticity of the air inclosed in the several grains of it, and in the intervals or spaces between the grains: the powder being kindled, sets the springs of so many little parcels of air a-playing, and dilates them all at once, whence the effect; the powder itself only serving to light a fire which may put the air in action; after which the whole is done by the air alone.</p><p>But it appears from the experiments and observations of Mr. Robins, that if this air be in its natural state at the time when the powder is fired, the greatest addition its elasticity could acquire from the flame of the explosion, would not amount to five times its usual quantity, and therefore could not suffice for the 200th part of the effort which is exerted by fired powder.</p><p>To understand the force of Gunpowder, it must be considered that, whether it be fired in a vacuum or in air, it produces by its explosion a permanent elastic fluid. See Philos. Trans. number 295; also Hauksbee's Phys. Mechan. Exp. p, 81. It also appears from experiment, that the elasticity or pressure of the fluid produced by the firing of Gunpowder, is, cæteris paribus, directly as its density.</p><p>To determine the elasticity and quantity of this elastic fluid, produced from the explosion of a given quantity of Gunpowder, Mr. Robins premises, that the elasticity of this fluid increases by heat, and diminishes by cold, in the same manner as that of the air; and that the density of this fluid, and consequently its weight, is the same with the weight of an equal bulk of air, having the same elasticity and the same temperature. From these principles, and from the experiments by which they are established (for a detail of which we must refer to the book itself, so often cited in these articles), he concludes that the fluid produced by the firing of Gunpowder is nearly 3/10 of the weight of the generating powder itself; and that the volume or bulk of this air or fluid, when expanded to the rarity of common atmospheric air, is about 244 times the bulk of the said generating powder.—Count Saluce, in his Miscel. Phil. Mathem. Soc. Priv. Taurin. p. 125, makes the proportion as 222 to 1; which he says agrees with the computation of Mess. Hauksbee, Amontons, and Belidor.</p><p>Hence it appears, that any quantity of powder fired in any confined space, which it adequately fills, exerts at the instant of its explosion against the sides of the vessel containing it, and the bodies it impels before it, a force at least 244 times greater than the elasticity of common air, or, which is the same thing, than the pressure of the atmosphere; and this without considering the great addition arising from the violent degree of heat with which it is endued at that time; the quantity of which augmentation is the next head of Mr. Robins's enquiry. He determines that the elasticity of the air is augmented in a proportion somewhat greater than that of 4 to 1, when heated to the extremest heat of red hot iron; and supposing that the<cb/> flame of sired Gunpowder is not of a less degree of heat, increasing the former number a little more than 4 times, makes nearly 1000; which shews that the elasticity of the flame, at the moment of explosion, is about 1000 times stronger than the elasticity of common air, or than the pressure of the atmosphere. But, from the height of the barometer, it is known that the pressure of the atmosphere upon every square inch, is on a medium 14 3/4 lb; and therefore 1000 times this, or 14750lb, is the force or pressure of the flame of Gunpowder, at the moment of explosion, upon a square inch, which is very nearly equivalent to 6 tons and a half.</p><p>This great force however diminishes as the fluid dilates itself, and in that proportion, viz, in proportion to the space it occupies, it being only half the strength when it occupies a double space, one third the strength when triple the space, and so on.</p><p>Mr. Robins farther supposes the degree of heat above mentioned to be a kind of medium heat; but that in the case of large quantities of powder the heat will be higher, and in very small quantities lower; and that therefore in the former case the force will be somewhat more, and in the latter somewhat less, than 1000 times the force of the atmosphere.</p><p>He farther found that the strength of powder is the same in all variations in the density of the atmosphere. But that the moisture of the air has a great effect upon it; for the same quantity which in a dry season would discharge a bullet with a velocity of 1700 feet in one second, will not in damp weather give it a velocity of more than 12 or 1300 feet in a second, or even less, if the powder be bad, and negligently kept. Robins's Tracts, vol. 1, p. 101, &c. Farther, as there is a certain quantity of water which, when mixed with powder, will prevent its firing at all, it cannot be doubted but every degree of moisture must abate the violence of the explosion; and hence the effects of damp powder are not difficult to account for.</p><p>It is to be observed, that the moisture imbibed by powder does not render it less active when dried again. Indeed, if powder be exposed to very great damps without any caution, or when common salt abounds in it, as often happens through negligence in refining the nitre, in such cases the moisture it imbibes may perhaps be sufficient to dissolve some part of the nitre; which is a permanent damage that no drying can retrieve. But when tolerable care is taken in preserving powder, and the nitre it is composed of has been well purged from common salt, it will retain its force for a long time; and it is said that powder has been known to have been preserved for 50 years without any apparent damage from its age.</p><p>The velocity of expansion of the flame of Gunpowder, when fired in a piece of artillery, without either bullet or other body before it, is prodigiously great, viz, 7000 feet per second, or upwards, as appears from the experiments of Mr. Robins. But Mr. Bernoulli and Mr. Euler suspect it is still much greater. And I suspect it may not be less, at the moment of explosion, than 4 times as much.</p><p>It is this prodigious celerity of expansion of the flame of fired Gunpowder, which is its peculiar excellence, and the circumstance in which it so eminently<pb n="574"/><cb/> surpasses all other inventions, either ancient or modern: for as to the momentum of these projectiles only, many of the warlike machines of the ancients produced this in a degree far surpassing that of our heaviest cannon shot or shells; but the great celerity given to these bodies cannot be in the least approached by any other means but the flame of powder.</p><p><hi rend="italics">To prove Gunpowder.</hi> There are several ways of doing this. 1, By sight: thus, if it be too black, it is a sign that it is moist, or else that it has too much charcoal in it; so also, if rubbed upon white paper, it blackens it more than good powder does: but if it be of a kind of azure colour, somewhat inclining to red, it is a sign of good powder. 2, By touching: for if in crushing it with the fingers ends, the grains break easily, and turn into dust, without feeling hard, it has too much coal in it; or if, in pressing it under the fingers upon a smooth hard board, some grains feel harder than the rest, it is a sign the sulphur is not well mixed with the nitre. Also by thrusting the hand into the parcel of powder, and grasping it, as if to take out a handful, you will feel if it is dry and equal grained, by its evading the grasp, and running mostly out of the hand. 3, By burning; and here the method most commonly followed for this purpose with us, says Mr. Robins, is to fire a small heap of it on a clean board, and to attend nicely to the flame and smoke it produces, and to the marks it leaves behind on the board: but besides this uncertain method, there are other contrivances made use of, such as powder-triers acting by a spring, commonly sold at the shops, and others again that move a great weight, throwing it upwards, which is a very bad sort of eprouvette. But these machines, says Mr. Robins, though more perfect than the common powder-triers, are yet liable to great irregularities; for as they are all moved by the instantaneous stroke of the flame, and not by its continued pressure, they do not determine the force of the fired powder with sufficient certainty and uniformity. Another method is to judge from the range given to a large solid ball, thrown from a very short mortar, charged with a small quantity of powder; which is also an uncertain way, both on account of the great disproportion between the weight of the ball and powder, and the unequal resistance of the air; not to mention that it is too tedious to prove large quantities of powder in this way; for, “if each barrel of powder was to be proved in this m<*>ner, the trouble of charging the mortar, and bringing back the ball each time, would be intolerable, and the delay so great, that no bu iness of this kind could ever be finished; and if a number of barrels are received on the merit of a few, it is great odds, but some bad ones would be amongst them, which may prove a great disappointment in time of service.” These exceptions do “noways hold, continues Mr. Robins, against the method by which I have tried the comparative strength of different kinds of powder, which has been by the actual velocity given to a bullet, by such a quantity of powder an is usually esteemed a proper charge for the piece: and as this velocity, however great, is easily discovered by the motion which the pendulum acquires from the stroke of the bullet, it might seem a good amendment to the method used by the French (viz, that of the small mortar above<cb/> mentioned) to introduce this trial by the pendulum instead of it. But though I am satisfied, that this would be much more accurate, less laborious, and readier than the other, yet, as there is some little attention and caution required in this practice, which might render it of less dispatch than might be convenient, when a great number of barrels were to be separately tried, I should myself choose to practise another method not less certain, but prodigiously more expeditious; so that I could engage, that the weighing out of a small parcel of powder from each barrel should be the greatest part of the labour; and, doubtless, three or four hands could, by this means, examine 500 barrels in a morning: besides, the machines for this purpose, as they might be made of cast iron, would be so very cheap, that they might be multiplied at pleasure.” Robins, page 123. It is not certainly known what might be the particular construction of the eprouvette here hinted at, but it was probably a piece of ordnance suspended like a pendulum, as he had made several experiments with a barrel in that manner. Be this however as it may, several persons, from those ideas and experiments of Mr. Robins, have made eprouvettes on this principle, which seems to be the best of any; and on this idea also I have lately made a machine for this purpose, which has several peculiar contrivances, and advantages over all others, both in the nature of its motion, and the divisions on its arc, &c. It is a small cannon, the bore of which is about one inch in diameter, and is usually charged with 2 ounces of powder, and with powder only, as a ball is not necessary, and the strength of the powder is accurately shewn by the arc of the gun's recoil. The whole machine is so simple, easy, and expeditious, that, as Mr. Robins observed above, the weighing of the powder is the chief part of the trouble; and so accurate and uniform, that the successive repetitions or firings with the same quantity of the same sort of powder, hardly ever yield a difference in the recoil of the 100th part of itself.</p><p><hi rend="italics">To recover damaged Powder.</hi> The method of the powder merchants is this; they put part of the powder on a sail-cloth, to which they add an equal weight of what is really good; then with a shovel they mingle it well together, dry it in the sun, and barrel it up, keeping it in a dry and proper place.</p><p>Others again, if it be very bad, restore it by moistening it with vinegar, water, urine, or brandy; then they beat it fine, sift it, and to every pound of powder add an ounce, or an ounce and a half, or two ounces (according as it is decayed), of melted nitre; and afterwards these ingredients are to be moistened and well mixed, so that nothing may be discerned in the composition; which may be known by cutting the mass, and then they granulate it as useful.</p><p>In case the powder be quite spoiled, the only way is to extract the saltpetre with water, in the usual way, by boiling, filtrating, evaporating, and crystallizing; and then, with fresh sulphur and charcoal, to make it up afresh.</p><p>On the subject of Gunpowder, see also Euler on Robins's Gunnery, Antoni Examen de la Poudre, Baumé's Chemistry, and Thompson's Experiments in the Philos. Trans for 1781.</p></div1><div1 part="N" n="GUNTER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">GUNTER</surname> (<foreName full="yes"><hi rend="smallcaps">Edmund</hi></foreName>)</persName></head><p>, an excellent English ma-<pb n="575"/><cb/> thematician, was born in Hertfordshire in 1581. He was educated at Westminster school under Dr. Busby, and from thence was elected to Christ-church College, Oxford, in 1599, where he took the degree of master of arts in 1606, and afterwards entered into holy orders; and in 1615 he took the degree of bachelor of divinity. But being particularly distinguished for his mathematical talents, when Mr. Williams resigned the professorship of astronomy in Gresham College, London, Mr. Gunter was chosen to succeed him, the 6th of March, 1619; where he greatly distinguished himself by his lectures and writings, and where he died in 1626, at only 45 years of age, to the great loss of the mathematical world.</p><p>Mr. Gunter was the author of many useful inventions and works. About the year 1606, he merited the title of an inventor, by the new projection of his Sector, which he then described in a Latin treatise, not printed however till some time afterwards.—In 1618 he had invented a small portable quadrant, for the more easy finding the hour and azimuth, and other useful purposes in astronomy.—And in 1620 or 1623, he published his <hi rend="italics">Canon Triangulorum,</hi> or Table of Artificial Sines and Tangents, to the radius, 10,000,000 parts, to every minute of the quadrant, being the first tables of this kind published; together with the first 1000 of Briggs's logarithms of common numbers, which were in later editions extended to 10,000 numbers.—In 1622, he discovered, by experiment made at Deptford, the variation or changeable declination of the magnetic needle; his experiment shewing that the declination had changed by 5 degrees in the space of 42 years; and the same was confirmed and established by his successor Mr. Gellibrand.—He applied the logarithms of numbers, and of fines and tangents, to straight lines drawn on a scale or rule; with which proportions in common numbers and trigonometry were resolved by the mere application of a pair of compasses; a method founded on this property, that the logarithms of the terms of equal ratios are equidisserent. This was called Gunter's Proportion, and Gunter's Line; and the instrument, in the form of atwo-foot scale, is now in common use for navigation and other purposes, and is commonly called the Gunter. He also greatly improved the Sector and other instruments for the same uses; the description of all which he published in 1624.—He introduced the common measuring chain, now constantly used in land-surveying, which is thence called Gunter's Chain.—Mr. Gunter drew the lines on the dials in Whitehall-garden, and wrote the description and use of them, by the direction of prince Charles, in a small tract; which he afterwards printed at the desire of king James, in 1624.—He was the first who used the word <hi rend="italics">co-sine,</hi> for the sine of the complement of an arc. He also introduced the use of Arithmetical Complements into the logarithmical arithmetic, as is witnessed by Briggs, cap. 15, Arith. Log. And it has been said that he first started the idea of the Logarithmic Curve, which was so called, because the segments of its axis are the logarithms of the corresponding ordinates.</p><p>His works have been collected, and various editions of them have been published; the 5th is by Mr. William Leybourn, in 1673, containing the Defcription and Use of the Sector, Cross-staff, Bow, Quadrant, and<cb/> other instruments; with several pieces added by Samuel Foster, Henry Bond, and William Leybourn.</p><p><hi rend="smallcaps">Gunter's Chain</hi>, the chain in common use for measuring land, according to true or statute measure; so called from Mr. Gunter its reputed inventor.</p><p>The length of the chain is 66 feet, or 22 yards, or 4 poles of 5 1/2 yards each; and it is divided into 100 links, of 7.92 inches each.</p><p>This Chain is the most convenient of any thing for measuring land, because the contents thence computed are so easily turned into acres. The reason of which is, that an acre of land is just equal to 10 square chains, or 10 chains in length and 1 in breadth, or equal to 100000 square links. Hence, the dimensions being taken in chains, and multiplied together, it gives the content in square chains; which therefore being divided by 10, or a figure cut off for decimals, brings the content to acres; after which the decimals are reduced to roods and perches, by multiplying by 4 and 40. But the better way is to set the dimensions down in links as integers, considering each chain as 100 links; then, having multiplied the dimensions together, producing square links, divide these by 100000, that is, cut off five places for decimals, the rest are acres, and the decimals are reduced to roods and perches, as before.</p><p><hi rend="italics">Ex.</hi> Suppose in measuring a rectangular piece of ground, its length be 795 links, and its breadth 480 links. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">795</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">480</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">63600</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3180  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Ac.</cell><cell cols="1" rows="1" role="data">3.81600</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Ro.</cell><cell cols="1" rows="1" role="data">3.264  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Per.</cell><cell cols="1" rows="1" role="data">10.560  </cell></row></table> So the content is 3 ac. 3 roods 10 perches.</p><p><hi rend="smallcaps">Gunter's Line</hi>, a Logarithmic line, usually graduated upon scales, sectors, &c; and so called from its inventor Mr. Gunter.</p><p>This is otherwise called the <hi rend="italics">line of lines,</hi> or <hi rend="italics">line of numbers,</hi> and consists of the logarithms transferred upon a ruler, &c, from the tables, by means of a scale of equal parts, which therefore serves to resolve problems instrumentally, in the same manner as logarithms do arithmetically. For, whereas logarithms resolve proportions, or perform multiplication and division, by only addition and subtraction, the same are performed on this line by turning a pair of compasses over this way or that, or by sliding one slip of wood by the side of another, &c.</p><p>This line has been contrived various ways, for the advantage of having it as long as possible. As, first, on the two feet ruler or scale, by Gunter. Then, in 1627 the logarithms were drawn by Wingate, on two separate rulers, sliding against each other, to save the use of compasses in resolving proportions. They were also in 1627 applied to concentric circles by Oughtred. Then in a spiral form by Mr. Milburne of Yorkshire, about the year 1650. Also, in 1657, on the present common sliding rule, by Seth Partridge.</p><p>Lastly, Mr. William Nicholson has proposed another disposition of them, on concentric circles, in the Philos. Trans. an. 1787, pa. 251. His instrument is equivalent<pb n="576"/><cb/> to a straight rule of 28 1/2 inches long. It consists of three concentric circles, engraved and graduated on a plate of about 1 1/2 inch in diameter. From the centre proceed two legs, having right-lined edges in the direction of radii; which are moveable either singly, or together. To use this instrument; place the edge of one leg at the antecedent of any proportion, and the other at the consequent, and fix them to that angle: the two legs being then moved together, and the antecedent leg placed at any other number, the other leg gives its consequent in the like position or situation on the lines.</p><p>The whole length of the line is divided into two equal intervals, or radii, of 9 larger divisions in each radius, which are numbered from 1 to 10, the 1 standing at the beginning of the line, because the logarithm of 1 is 0, and the 10 at the end of each radius; also each of these 9 spaces is subdivided into 10 other parts, unequal according to the logarithms of numbers; the smaller divisions being always 10ths of the larger; thus, if the large divisions be units or ones, the smaller are tenth-parts; if the larger be tens, the smaller are ones; and if the larger be 100's, the smaller are 10's; &c.</p><p><hi rend="italics">Use of Gunter's Line.</hi> 1. <hi rend="italics">To find thē product of two numbers.</hi> Extend the compasses from 1 to either of the numbers, and that extent will reach the same way from the other number to the product. Thus, to multiply 7 and 5 together; extend the compasses from 1 to 5, and that extent will reach from 7 to 35, which is the product.</p><p>2. <hi rend="italics">To divide one number by another.</hi> Extend the compasses from the divisor to 1, and that extent will reach the same way from the dividend to the quotient. Thus, to divide 35 by 5; extend the compasses from 5 to 1, and that extent will reach from 35 to 7, which is the quotient.</p><p>3. <hi rend="italics">To find a 4th Proportional to three given Numbers;</hi> as suppose to 6, 9, and 10. Extend from 6 to 9, and that extent will reach from 10 to 15, which is the 4th proportional sought. And the same way a 3d proportional is found to two given terms, extending from the 1st to the 2d, and then from the 2d to the 3d.</p><p>4. <hi rend="italics">To find a Mean Proportional between two given numbers,</hi> as suppose between 7 and 28. Extend from 7 to 28, and bisect that extent; then its half will reach from 7 forward, or from 28 backward, to 14, the mean proportional between them.—Also, to extract the square root, as of 25, which is only finding a mean proportional between 1 and the given square 25, bisect the distance between 1 and 25, and the half will reach from 1 to 5, the root sought.—In like manner the cubic or 3d root, or the 4th, 5th, or any higher root, is found, by taking the extent between 1 and the given power; then take such part of it as is denoted by the index of the root, viz, the 3d part for the cube root, the 4th part for the 4th root, and so on, and that part will reach from 1 to the root sought.</p><p>If the Line on the Scale or Ruler have a slider, this is to be used instead of the compasses.</p><p><hi rend="smallcaps">Gunter's Quadrant</hi>, is a quadrant made of wood, brass, or some other substance; being a kind of stereographic prejection on the plane of the equinoctial, the eye being supposed in one of the poles: so that the tropic, ecliptic, and horizon, form the arches of circles,<cb/> but the hour circles other curves, drawn by means of several altitudes of the sun, for some particular latitude every day in the year.</p><p>The use of this instrument, is to find the hour of the day, the sun's azimuth, &c, and other common problems of the sphere or globe; as also to take the altitude of an object in degrees. See <hi rend="smallcaps">Quadrant.</hi></p><p><hi rend="smallcaps">Gunter's Scale</hi>, usually called by seamen <hi rend="italics">the Gunter,</hi> is a large plain scale, having various lines upon it, of great use in working the cases or questions in Navigation.</p><p>This Scale is usually 2 feet long, and about an inch and a half broad, with various lines upon it, both natural and logarithmic, relating to trigonometry, navigation, &c.</p><p>On the one side are the natural lines, and on the other the artificial or logarithmic ones. The former side is first divided into inches and tenths, and numbered from 1 to 24 inches, running the whole length near one edge. One half the length of this side consists of two plane diagonal scales, for taking off dimensions to three places of figures. On the other half or foot of this side, are contained various lines relating to trigonometry, in the natural numbers, and marked thus, viz, <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Rumb,</hi></cell><cell cols="1" rows="1" role="data">the rumbs or points of the compass,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Chord,</hi></cell><cell cols="1" rows="1" role="data">the line of chords,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Sine,</hi></cell><cell cols="1" rows="1" role="data">the line of sines,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Tang.</hi></cell><cell cols="1" rows="1" role="data">the tangents,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">S. T.</hi></cell><cell cols="1" rows="1" role="data">the semitangents,</cell></row></table> and at the other end of this half are <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Leag.</hi></cell><cell cols="1" rows="1" role="data">leagues, or equal parts,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Rumb.</hi></cell><cell cols="1" rows="1" role="data">another line of rumbs,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">M. L.</hi></cell><cell cols="1" rows="1" role="data">miles of longitude,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Chor.</hi></cell><cell cols="1" rows="1" role="data">another line of chords.</cell></row></table></p><p>Also in the middle of this foot are <hi rend="italics">L.</hi> and <hi rend="italics">P.</hi> two other lines of equal parts. And all these lines on this side of the scale serve for drawing or laying down the figures to the cases in trigonometry and navigation.</p><p>On the other side of the scale are the following artisicial or logarithmic lines, which serve for working or resolving those cases; viz, <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">S. R.</hi></cell><cell cols="1" rows="1" role="data">the fine rumbs,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">T. R.</hi></cell><cell cols="1" rows="1" role="data">the tangent rumbs,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Numb.</hi></cell><cell cols="1" rows="1" role="data">line of numbers,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Sine,</hi></cell><cell cols="1" rows="1" role="data">Sines,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">V. S.</hi></cell><cell cols="1" rows="1" role="data">the versed sines,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Tang.</hi></cell><cell cols="1" rows="1" role="data">the tangents,</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Meri.</hi></cell><cell cols="1" rows="1" role="data">Meridional parts.</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">E. P.</hi></cell><cell cols="1" rows="1" role="data">Equal parts.</cell></row></table></p><p>The late Mr. John Robertson, librarian to the Royal Society, greatly improved this scale, both as to size and accuracy, for the use of mariners. He extended it to 30 inches long, 2 inches broad, and half an inch thick; upon which the several lines are very accurately laid down by Messrs. Nairne and Blunt, ingenious instrument makers. Mr. Robertson died before his improved scales were published; but the account and description of them were supplied and drawn up by his friend Mr. William Mountaine, and published in 1778.</p><p>GUTTÆ, or <hi rend="italics">Drops,</hi> in Architecture, are ornaments in form of little bells or cones, used in the Doric order, on the architrave, below the tryglyphs. There are usually fix of them.<pb n="577"/></p></div1></div0><div0 part="N" n="H" org="uniform" sample="complete" type="alphabetic letter"><head>H</head><cb/><p>HADLEY's <hi rend="italics">Quadrant, Sextant,</hi> &c, an excellent instrument so called from its inventor John Hadley, Esq. See its description and use under the article <hi rend="smallcaps">Quadrant.</hi></p><div1 part="N" n="HAIL" org="uniform" sample="complete" type="entry"><head>HAIL</head><p>, or <hi rend="smallcaps">Hailstones</hi>, an aqueous concretion, usually in form of white or pellucid spherules, descending out of the atmosphere.</p><p>Hailstones assume various shapes, being sometimes round, at other times pyramidal, crenated, angular, thin, and flat, and sometimes stellated, with six radii like the small crystals of snow.</p><p>It is very difficult to account for the phenomena of hail in a satisfactory manner; and there are various opinions upon this head. It is usually conceived that hail is formed of drops of rain, frozen in their passage through the middle region. Others, as the Cartesians, take it for the fragments of a frozen cloud, half melted, and thus precipitated and congealed again. Signior Beccaria supposes, that it is formed in the higher regions of the air, where the cold is intense, and where the electric matter is very copious. In these circumstances, a great number of particles of water are brought near together, where they are frozen, and in their descent they collect other particles; so that the density of the substance of the Hailstone grows less and less from the centre; this being formed first in the higher regions, and the surface being collected in the lower. Accordingly, in mountains, Hailstones as well as drops of rain, are very small; and both agree in this circumstance, that the more intense is the electricity that forms them, the larger they are.</p><p>It is frequently observed that Hail attends thunder and lightning; and hence Beccaria observes, that as motion promotes freezing, so the rapid motion of the electrisied clouds may promote that effect in the air.</p><p>Natural histories furnish us with a great variety of curious instances of extraordinary showers of Hail. See the Philos. Trans. number 203, 229; and Hist. de France, tom. 2, pa. 339.</p><p>HALF-<hi rend="smallcaps">Moon</hi>, in Fortification, is an outwork having only two faces, forming together a saliant angle, which is flanked by some part of the place, and of the other bastions. See <hi rend="smallcaps">Demilune</hi> and <hi rend="smallcaps">Ravelin.</hi></p><p><hi rend="smallcaps">Half-Tangents</hi>, are the tangents of the half arcs. See <hi rend="smallcaps">Scale</hi> and <hi rend="smallcaps">Semitangents.</hi></p></div1><div1 part="N" n="HALLEY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HALLEY</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">Edmund</hi></foreName>)</persName></head><p>, a most eminent English mathematician, philosopher and astronomer, was born in the parish of St. Leonard, Shoreditch, near<cb/> London, Oct. 29, 1656. His father, a wealthy citizen and soap-boiler, resolving to improve the promising disposition observed in his son, put him first to St. Paul's school, where he soon excelled in all parts of classical learning, and made besides a considerable advance in the mathematics; so that, as Wood observes, he had perfectly learnt the use of the celestial globe, and could make a complete dial; and we are informed by Halley himself, that he observed the change of the variation of the magnetic needle at London in 1672, one year before he left school. In 1673 he was sent to Oxford, where he chiefly applied himself to mathematics and astronomy, in which he was greatly assisted by a curious apparatus of instruments, which his father, willing to encourage his son's genius, had purchased for him. At 19 years of age he began to oblige the world with new observations and discoveries (which he continued to do to the end of a very long life), by publishing “A Direct and Geometrical method of finding the Aphelia and Excentricity of the planets.” Besides various particular observations, made from time to time upon the celestial phenomena; he had, from his first admission into college, pursued a general scheme for ascertaining the true places of the fixed stars, and so to correct the errors of Tycho Brahe. His original view in this was, to carry on the design of that first restorer of astronomy, by completing the catalogue of those stars from his own observations; but upon farther enquiry, finding this province taken up by Hevelius and Flamsteed, he dropped that pursuit, and formed another; which was, to perfect the whole scheme of the heavens, by the addition of the stars which lie so near the south pole, that they could not be observed by those astronomers, as never rising above the horizon either at Dantzick or at Greenwich. With this view he left the University, before he had taken any degree, and embarked for the island of St. Helena in Nov. 1676, when he was only 20 years of age, and arrived there after a voyage of three months. He immediately set about his task with such diligence, that he completed his catalogue, and, returning home, landed in England in Nov. 1678, after an absence of two years only. The university of Oxford immediately conferred upon him the degree of A. M. and the Royal Society of London elected him one of their members.</p><p>In 1679 he was pitched upon by the Royal Society to go to Dantzick, to endeavour to adjust a dispute between Hevelius and Mr. Hooke, concerning the pre-<pb n="578"/><cb/> ference as to plain and glass sights in astroscopical instruments. He arrived at Dantzick the 26th of May, when he immediately, in conjunction with Hevelius, set about their astronomical observations, which they closely continued till the 18th of July, when Halley left Dantzick, and returned to England.</p><p>In the year 1680 he undertook what is called the grand tour, accompanied by his friend the celebrated Mr. Nelson. In the way from Calais to Paris, Mr. Halley had a sight of a remarkable comet, as it then appeared a second time that year, in its return from the sun. He had the November before seen it in its descent; and he now hastened to complete his observations upon it, by viewing it from the royal observatory of France. His design in this part of his tour was, to settle a friendly correspondence between the two royal astronomers of Greenwich and Paris; and in the mean time to improve himself under so great a master as Cassini. From thence he went to Italy, where he spent great part of the year 1681; but his affairs calling him home, he then returned to England.</p><p>Soon after his return, he married the daughter of Mr. Tooke, auditor of the exchequer, and took up his residence at Islington, where he set up his tube and sextant, and eagerly pursued his favourite study: in the society of this amiable lady he lived happily for five-andfifty years. In 1683 he published his <hi rend="italics">Theory of the Variation of the Magnetical Compass;</hi> in which he supposes the whole globe of the earth to be one great magnet, having four magnetical poles or points of attraction, &c. The same year also he entered upon a new method of finding out the longitude, by an accurate observation of the moon's motion. His pursuits it seems were now a little interrupted by the death of his father, who having suffered greatly by the fire of London, as well as by a second marriage, into which he had imprudently entered, was found to have wasted his fortunes. Our author soon resumed his pursuits however; for in the beginning of 1684 he turned his thoughts to the subject of Kepler's sesqui-alterate proportion; when, after some meditation, he concluded from it, that the centripetal force must decrease in proportion to the square of the distance reciprocally. He found himself unable to make it out in any geometrical way; and therefore, after applying in vain for assistance to Mr. Hooke and Sir Christopher Wren, he went to Cambridge to Mr. Newton, who fully supplied him with what he so ardently sought. But Halley having now found an immense treasure in Newton, could not rest, till he had prevailed with the owner to enrich the public with it; and to this interview the world is in some measure indebted for the <hi rend="italics">Principia Mathematica Philosophiæ Naturalis.</hi> That great work was published in 1686; and Halley, who had the whole care of the impression, prefixed to it a discourse of his own, giving a general account of the astronomical part of the book; and also a very elegant copy of verses in Latin.</p><p>In 1687 he undertook to explain the cause of a natural phenomenon, which had till then baffled the researches of the ablest geographers. It is observed that the Mediterranean sea never swells in the least, although there is no visible discharge of the prodigious quantity of water that runs into it from nine large rivers, besides several small ones, and the constant setting<cb/> in of the current at the mouth of the Streights. His solution of this difficulty gave so much satisfaction to the Society, that he was requested to prosecute these enquiries. He did so; and having shewn, by accurate experiments, how that vast accession of water was actually carried off in vapours raised by the action of the sun and wind upon its surface, he proceeded with the like success to point out the method used by nature to return the said vapours into the sea. This circulation he supposes to be carried on by the winds driving these vapours to the mountains; where being collected, they form springs, which uniting become rivulets or brooks, and many of these again meeting in the valleys, grow into large rivers, emptying themselves at last into the sea: thus demonstrating, in the most beautiful manner, the way in which the equilibrium of receipt and expence is continually preserved in the universal ocean.</p><p>He next ranged in the field of speculative geometry, where, observing some imperfections in the methods before laid down for constructing solid problems, or equations of the 3d and 4th powers, he furnished new rules, which were both more easy and more elegant than any of the former; together with a new method of finding the number of roots of such equations, and the limits of the same.</p><p>Mr. Halley next undertook to publish a more correct Ephemeris for the year 1688, there being then great want of proper ephemerides of any tolerable exactness, the common ones being justly complained of by Mr. Flamsteed.—In 1691 he published exact tables of the conjunctions of Venus and Mercury; and he afterwards shewed one extraordinary use to be made of those tables, viz, for discovering the sun's parallax, and thence the true distance of the earth from the sun.—In 1692, our author produced his tables for shewing the value of annuities on lives, calculated from bills of mortality; and his universal theorem for finding the foci of optic glasses.</p><p>But it would be endless to enumerate all his valuable discoveries now communicated to the Royal Society, and published in the Philos. Trans. of which, for many years, his pieces were the chief ornament and support. Their various merit is thrown into one view by the writer of his eloge in the Paris Memoirs; who, having mentioned his History of the Trade-winds and Monsoons, proceeds in these terms: “This was immediately followed by his estimation of the quantity of vapours which the sun raises from the sea; the circulation of vapours; the origin of fountains; questions on the nature of light and transparent bodies; a determination of the degrees of mortality, in order to adjust the valuation of annuities on lives; and many other works, all the sciences relating to astronomy, geometry, and algebra, optics and dioptrics, balistics and artillery, speculative and experimental philosophy, natural history, antiquities, philology, and criticism; being about 25 or 30 dissertations, which he produced during the 9 or 10 years of his residence at London; and all abounding with ideas new, singular, and useful.”</p><p>In 1691, the Savilian professorship of astronomy at Oxford being vacant, he applied for that office, but without success. Whiston, in the Memoirs of his own Life, tells us from Dr. Bentley, that Halley “being thought of for successor to the mathematical chair at<pb n="579"/><cb/> Oxford, bishop Stillingfleet was desired to recommend him at court; but hearing that he was a sceptic and a banterer of religion, the bishop scrupled to be concerned, till his chaplain Bentley should talk with him about it, which he did. But Halley was so sincere in his insidelity, that he would not so much as pretend to believe the christian religion, though he thereby was likely to lose a professorship; which he did accordingly, and it was then given to Dr. Gregory.”</p><p>Halley had published his Theory of the Variation of the Magnetical Compass, as has been already observed, in 1683; which, though it was well received both at home and abroad, he found, upon a review, liable to great and insuperable objections. Yet the phenomena of the variation of the needle, upon which it is raised, being so many certain and indisputed facts, he spared no pains to possess himself of all the observations relating to it he could possibly come at. To this end he procured an application to be made to king William, who appointed him commander of the Paramour pink, with orders to search out by observations the discovery of the rule of variations, and to lay down the longitudes and latitudes of the English settlements in America.— He set out on this attempt on the 24th of November, 1698: but having crossed the line, his men grew sickly; and his first lieutenant mutinying, he returned home in June 1699. Having got the lieutenant tried and cashiered, he set sail a second time in September following, with the same ship, and another of less bulk, of which he had also the command. He now traversed the vast Atlantic ocean from one hemisphere to the other, as far as the ice would permit him to go; and having made his observations at St. Helena, Brazil, Cape Verde, Barbadoes, the Madeiras, the Canaries, the coast of Barbary, and many other latitudes, he arrived in England in September 1700; and the next year published a general chart, shewing at one view the variation of the compass in all those places.</p><p>Captain Halley, as he was now called, had been at home little more than half a year, when he was sent by the king, to observe the course of the tides, with the longitude and latitude of the principal head-lands in the British channel; which having executed with his usual expedition and accuracy, he published a large map of the channel.</p><p>Soon after, the emperor of Germany resolving to make a convenient harbour for shipping in the Adriatic, captain Halley was sent by queen Anne to view the two ports on the coast of Dalmatia. He embarked on the 22d of November 1702; passed over to Holland; and going through Germany to Vienna, he proceeded to Istria: but the Dutch opposing the design, it was laid aside; yet the emperor made him a present of a rich diamond ring from his finger, and honoured him with a letter of recommendation, written with his own hand, to queen Anne. Presently after his return, he was sent again on the same business; when passing through Hanover, he supped with the electoral prince, who was afterward king George the 1st, and his sister the queen of Prussia. On his arrival at Vienna, he was the same evening presented to the emperor, who sent his chief engineer to attend him to Istria, where they repaired the fortifications of Trieste, and added new ones.<cb/></p><p>Mr. Halley returned to England in Nov. 1703: and the same year he was made professor of geometry in the university of Oxford, instead of Dr. Wallis then just deceased, and he was at the same time honoured by the university with the degree of doctor of laws. He was scarcely settled in Oxford, when he began to translate into Latin, from the Arabic, <hi rend="italics">Apollonius de Sectione Rationis;</hi> and to restore the two books <hi rend="italics">De Sectione Spatii</hi> of the same author, which are lost, from the account given of them by Pappus; and he published the whole work in 1706. He afterwards had a share in preparing for the press Apollonius's Conics; and ventured to supply the whole 8th book, the original of which is also lost. To this work he added Serenus on the Section of the Cylinder and Cone, printed from the original Greek, with a Latin translation, and published the whole in folio 1710. Beside these, the <hi rend="italics">Miscellanea Curiosa,</hi> in 3 volumes 8vo, had come out under his direction in 1708.</p><p>In 1713, he succeeded Doctor, afterwards Sir, Hans Sloane, in the office of Secretary to the Royal Society. And, upon the death of Mr. Flamsteed in 1719, he was appointed to succeed him at Greenwich as Astronomer Royal; upon which occasion, that he might be more at leisure to attend the duties of this office, he resigned that of secretary to the Royal Society in 1721. Although he was 63 or 64 years of age when he entered upon his office at Greenwich, for the space of 18 years he watched the heavens with the closest attention, hardly ever missing an observation during all that time, and, without any assistant, performed the whole business of the observatory himself.</p><p>Upon the accession of the late king, his consort queen Caroline made a visit at the Royal Observatory; and being pleased with every thing she saw, took notice that Dr. Halley had formerly served the crown as a captain in the navy: and she soon after obtained a grant of his half-pay for that commission, which he accordingly enjoyed from that time during his life. An offer was also made him of being appointed mathematical preceptor to the duke of Cumberland; but he declined that honour, on account of his advanced age, and the duties of his office. In 1729 he was chosen a foreign member of the Academy of Sciences at Paris.</p><p>About 1737 he was seized with a paralytic disorder in his right hand, which, it is said, was the first attack he ever felt upon his constitution: however, he came as usual once a week, till within a very short time of his death, to meet his friends in town on Thursdays, before the meeting of the Royal Society, at what is yet called Dr. Halley's club. His paralytic disorder increasing, his strength gradually wore away, till he expired Jan. 14, 1742, in the 86th year of his age; and his corps was interred in the church-yard of Lee near Blackheath.——Beside the works before mentioned, his principal publications are, 1. <hi rend="italics">Catalogus Stellarum Australium.</hi> 2. <hi rend="italics">Tabulæ Astronomicæ.</hi> 3. The Astronomy of Comets. With a great multitude of Papers in the Philos. Trans. from vol. 11 to vol. 60.</p><p>HALIFAX (<hi rend="smallcaps">John</hi>). See <hi rend="smallcaps">Sacrobosco.</hi></p></div1><div1 part="N" n="HALO" org="uniform" sample="complete" type="entry"><head>HALO</head><p>, or <hi rend="smallcaps">Corona</hi>, a coloured circle appearing round the body of the sun, moon, or any of the larger stars.</p><p>Naturalists conceive the Halo to arise from a refrac-<pb n="580"/><cb/> tion of the rays of light in passing through the fine rare vesiculæ of a thin vapour towards the top of the atmosphere.</p><p>Des Cartes observes, that the Halo never appears when it rains; whence he concludes that this phenomenon is occasioned by the refraction of light in the round particles of ice, which are then floating in the atmosphere; and to the different protuberance of these particles he ascribes the variation in the diameter of the Halo. Gassendi supposes, that a Halo is occasioned in the same manner as the rainbow; the rays of light being, in both eases, twice refracted and once reflected within each drop of rain or vapour, and that the difference between them is wholly owing to their different situation with respect to the observer, Dechales also endeavours to shew that the generation of the Halo is similar to that of the rainbow; and that the reason why the colours of the Halo are more dilute than those of the rainbow, is owing chiefly to their being formed, not in large drops of rain, but in very small vapour. But the most considerable and generally received theory, relating to the generation of Halos, is that of Mr. Huygens. This celebrated author supposes Halos, or circles round the sun, to be formed by small round grains of hail, composed of two different parts, the one of which is transparent, inclosing the other, which is opaque; which is the general structure actually observed in hail. He farther supposes that the grains or globules, that form these Halos, consisted at first of sost snow, and that they have been rounded by a continual agitation in the air, and thawed on their outside by the heat of the sun, &c. And he illustrates his ideas of their formation by geometrical figures.</p><p>Mr. Weidler endeavours to refute Huygens's manner of accounting for Halos, by a vast number of small vapours, each with a snowy nucleus, coated round with a transparent covering. He says, that when the sun paints its image in the atmosphere, and by the force of its rays puts the vapours in motion, and drives them toward the surface, till they are collected in such a quantity, and at such a distance from the sun on each side, that its rays are twice refracted, and twice reflected, when they reach the eye they exhibit the appearance of a Halo, adorned with the colours of the rainbow; which may happen in globular pellucid vapours without snowy nuclei, as appears by the experiment of hollow glass spheres filled with water: therefore, whenever those spherical vapours are situated as before mentioned, the refractions and reflections will happen every where alike, and the figure of a circular crown, with the usual order of colours, will be the consequence. Philos. Trans. number 458.</p><p>Newton's theory of Halos may be seen in his Optics, p. 155. And this curious theory was confirmed by actual observation in June 1692, when the author saw by reflection, in a vessel of stagnated water, three Halos, crowns, or rings of colours, about the sun, like three little rainbows concentric to his body. These crowns inclosed one another immediately, so that their colours proceeded in this continual order from the sun outward: blue, white, red; purple, blue, green, pale yellow, and red; pale blue, pale red. The like crowns sometimes appear about the moon. The more equal the globules of water or ice are to one another, the<cb/> more crowns of colours will appear, and the colours will be the more lively. Optics, p. 288.</p><p>There are several ways of exhibiting phenomena similar to these. The slame of a candle, placed in the midst of a steam in cold weather, or placed at the distance of some feet from a glass window that has been breathed upon, while the spectator is also at the distance of some feet from another part of the window, or placed behind a glass receiver, when air is admitted into the vacuum within it to a certain density, in each of these circumstances will appear to be encompassed by a coloured Halo. Also, a quantity of water being thrown up against the sun, as it breaks and disperses into drops, forms a kind of Halo or iris, exhibiting the colours of the natural rainbow. Musschenbroek observed, that when the glass windows of his room were covered with a thin plate of ice on the inside, the moon seen through it was surrounded with a large and variously coloured Halo; which, upon opening the window, he found arose entirely from that thin plate of ice, because none was seen except through this plate. Musschenbroek concludes his account of coronas with observing, that some density of vapour, or some thickness of the plates of ice, divides the light in its transmission either through the small globules or their interstices, into its separate colours; but what that density is, or what the size of the particles which compose the vapour, he does not pretend to determine. Introd. ad Phil. Nat. p. 1037.</p><p>It has often been observed that a Halo about the sun or moon, does not appear circular and concentric to the luminary, but oval and excentric, with its longest diameter perpendicular to the horizon, and extended from the moon farther downward than upward. Dr. Smith ascribes this phenomenon to the apparent concave of the fky being less than a hemisphere. When the angle which the diameter of a Halo subtends at the eye is 45° or 46°, and the bottom of the Halo is near the horizon, and consequently its apparent figure is most oval, the apparent vertical diameter is divided by the moon in the proportion of about 2 to 3 or 4, and is to the horizontal diameter drawn through the moon, as 4 to 3, nearly.—See farther on the subject of this article, Priestley's Hist. of Discoveries relating to Vision, p. 596—613; and Smith's Optics, art. 167, 513, 526, 527, &c.</p></div1><div1 part="N" n="HAMEL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HAMEL</surname> (<foreName full="yes"><hi rend="smallcaps">John Baptiste du</hi></foreName>)</persName></head><p>, a very learned French philosopher and writer, in the 17th century, was born in lower Normandy in 1614. At 18 years of age he published a treatise, in which he explained, in a very simple manner, and by one or two figures, Theodosius's 3 books upon Spherics; to which he added a tract upon trigonometry extremely perspicuous, and designed as an introduction to astronomy. He published afterwards various other works on astronomy and philosophy. Natural philosophy, as it was then taught, was only a collection of vague, knotty, and barren questions; when our author undertook to establish it upon right principles, and published his <hi rend="italics">Astronomia Physica.</hi></p><p>In 1666 M. Colbert proposed to Louis the 14th a scheme, which was approved of, for establishing a royal academy of sciences; and appointed our author secretary of it. In 1678, his <hi rend="italics">Philosophia Vetus & Nova</hi><pb n="581"/><cb/> was printed at Paris in 4 vols, 12mo; and in 1681 it was enlarged and printed there in 6 vols. He wrote several other pieces; and his works in this way were collected and published at Nuremberg 1681, in 4 volumes 4to, under the title of <hi rend="italics">Opera Philosophica & Astronomica.</hi> These were highly valued then, though the improvements in philosophy since that time have rendered them of little or no use now.</p><p>In 1697 he resigned his place of secretary of the Royal Academy of Sciences; in which he was succeeded by M. Fontenelle. However, he published, in 1698, <hi rend="italics">Regiæ Scientiarum Academiæ Historia,</hi> 4to, in four books; which being much liked, he afterwards augmented with two books more. This work contains an account of the foundation of the Royal Academy of Sciences, and its transactions, from 1666 to 1700, and is now the most useful of all h<*>s works. He was Regius Professor of Philosophy, in which office he was succeeded by M. Varignon, at his death, which happened Aug. 6, 1706, in the 93d year of his age.</p></div1><div1 part="N" n="HANCES" org="uniform" sample="complete" type="entry"><head>HANCES</head><p>, <hi rend="smallcaps">Hanches</hi>, or <hi rend="smallcaps">Hanses</hi>, in architecture, are certain small intermediate parts of arches between the key or crown and the spring at the bottom, being perhaps about one-third of the arch, and situated nearer the bottom than the top or crown; and are otherwise called the <hi rend="italics">spandrels.</hi></p></div1><div1 part="N" n="HANDSPIKE" org="uniform" sample="complete" type="entry"><head>HANDSPIKE</head><p>, or <hi rend="smallcaps">Handspec</hi>, a lever or piece of ash, elm, or other strong wood, for raising by the hand great weights, &c. It is 5 or 6 feet long, cut thin and crooked at the lower end, that it may get the easier between things that are to be separated, or under any thing that is to be raised. It is better than a crow of iron, because its length allows a better poise.</p><p>HARD <hi rend="italics">Bodies,</hi> are such as are absolutely inflexible to any pressure or percussion whatever: differing from soft bodies, whose parts yield and are easily moved amongst one another, without restoring themselves again; and from elastic bodies, the parts of which also yield and give way, but presently restore themselves again to their former state and situation. Hence, hard bodies do not bend, or indent, but break. It is probable however there are no bodies in nature that are absolutely or perfectly either hard, soft, or elastic; but all possessing these qualities, more or less, in some degree. M. Bernoulli goes so far as to say that Hardness, in the common sense, is absolutely impossible, being contrary to the law of continuity.</p><p>The laws of motion for hard bodies are the same as for soft ones, both being supposed to adhere together on their impact. And these two sorts of bodies might be comprized under the common name of Unelastic.</p></div1><div1 part="N" n="HARDENING" org="uniform" sample="complete" type="entry"><head>HARDENING</head><p>, the giving a greater degree of hardness to bodies than they had before.</p><p>There are several ways of Hardening iron and steel; as by hammering them, quenching them in cold water, &c.</p><p><hi rend="italics">Case</hi>-<hi rend="smallcaps">Hardening</hi>, is a superficial conversion of iron into steel, as if it were casing it, or covering it with a thin coat of harder matter. It is thus performed: Take cow horn or hoof, dry it well in an oven, and beat it to powder; put equal quantities of this powder and of bay salt into stale urine, or white wine vinegar, and mix them well together; cover the iron or steel all<cb/> over with this mixture, and wrap it up in loam, or plate iron, so as the mixture touch every part of the work; then put it in the fire, and blow the coals to it, till the whole lump have a blood red heat, but no higher; lastly, take it out, and quench it.—See <hi rend="smallcaps">Steel</hi>, under which article are described other processes for this purpose.</p></div1><div1 part="N" n="HARDNESS" org="uniform" sample="complete" type="entry"><head>HARDNESS</head><p>, or Rigidity, that quality in bodies by which their parts so cohere as not to yield inward, or give way to an external impulse, without instantly going beyond the distance of their mutual attraction; and therefore are not subject to any motion in respect of each other, without breaking the body.</p><p>There were many fanciful opinions among the ancients concerning the cause of hardness; such as, heat, cold, dryness, the hooked figure of the particles of matter. The Cartesians make the Hardness of bodies to consist in rest, as that of soft and fluid ones in the motion of their particles.</p><p>Newton shews that the primary particles of all bodies, whether solid or fluid, are perfectly hard; and are not capable of being broken or divided by any power in nature. These particles, he maintains, are connected together by an attractive power; and according to the circumstances of this attraction, the body is either hard, or soft, or even fluid. If the particles be so disposed or fitted for each other, as to touch in large surfaces, the body will be hard; and the more so as those surfaces are the larger. If, on the contrary, they only touch in small surfaces, the body, by the weakness of the attraction, will remain soft.</p><p>At present, many philosophers think that Hardness consists in the absence or want of the action of the universal fluid, or elementary fire, among the particles of the body, or a deficiency of what is called latent heat; while on the contrary, fluidity, according to them, consists in the motion of the particles, in consequence of the action of that elementary fire.</p><p>Hardness appears to diminish the cohesion of bodies, in some degree, though their frangibility or brittleness does not by any means keep pace with their hardness. Thus, though glass be very hard and very brittle; yet flint is still harder, though less brittle. Among the metals, these two properties seem to be more connected, though even here the connection is by no means complete: for though steel be both the hardest and most brittle of all the metals; yet lead, which is the softest, is not the most ductile. Neither is Hardness connected with the specific gravity of bodies; for a diamond, the hardest substance in nature, is little more than half the weight of the lightest metal. And as little is it connected with the coldness, or electrical properties, or any other quality with which we are acquainted. Some bodies are rendered hard by cold, and others by different degrees of heat.</p><p>Mr. Quist and others have constructed tables of the Hardness of different substances. And the manner of constructing these tables, was by observing the order in which they were able to cut or make any impression upon one another. The following table, extracted from Magellan's edition of Cronstedt's Mineralogy, was taken from Quist, Bergman, and Kirwan. The first column shews the Hardness, and the second the specific gravity.<pb n="582"/><cb/> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">Hardness.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Spec. Grav.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Diamond from Ormus</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">3.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pink diamond</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Blueish diamond</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3.3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Yellowish diamond</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3.3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cubic diamond</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3.2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ruby</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">4.2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pale blue sapphire</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pale ruby from Brazil</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">Deep blue sapphire</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Topaz</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">4.2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Whitish ditto</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">3.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ruby spinell</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Emerald</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Garnet</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">4.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Agate</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Onyx</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sardonyx</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bohemian topaz</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Occid. amethyst</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Crystal</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Carnelian</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Green jasper</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Schoerl</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tourmaline</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Quartz</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Opal</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Chrysolite</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Reddish yellow jasper</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">2.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Zeolyte</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">2.1</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fluor</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">3.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">Calcareous spar</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">2.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gypsunr</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">2.3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Chalk</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">2.7</cell></row></table></p></div1><div1 part="N" n="HARMONICA" org="uniform" sample="complete" type="entry"><head>HARMONICA</head><p>, <hi rend="smallcaps">Harmonics</hi>, a branch or division of the ancient music; being that part which considers the differences and proportions of sounds, with respect to acute and grave; as distinguished from Rhythmica, and Metrica.</p><p>Mr. Malcolm has made a very industrious and learned enquiry into the Harmonica, or harmonic principles, of the ancients.</p><div2 part="N" n="Harmonica" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Harmonica</hi></head><p>, the name of a musical instrument invented by Dr. Franklin, consisting of the glasses, called musical glasses.</p><p>It is said that the first hint of musical glasses is to be found in an old English book, in which a number of various amusements were described. That author directs his pupil to choose half a dozen drinking-glasses; to fill each of them with water in proportion to the gravity or acuteness of the sound which he intended it to give; and having thus adjusted them one to another, he might entertain the company with a church tune. These were perhaps the rude hints which Mr. Puckeridge, an Irish gentleman, afterwards improved, and after him, Mr. E. Delaval, an ingenious member of the Royal Society; and finally brought to perfection by the celebrated Franklin. See the history and description in his Letters, particularly in that to Beccaria.</p></div2></div1><div1 part="N" n="HARMONICAL" org="uniform" sample="complete" type="entry"><head>HARMONICAL</head><p>, or <hi rend="smallcaps">Harmonic</hi>, something relating to Harmony. Thus,</p><p><hi rend="smallcaps">Harmonical</hi> <hi rend="italics">Arithmetic,</hi> is so much of the theory<cb/> and doctrine of numbers, as relates to making the comparisons and reductions of musical intervals, which are expressed by numbers, for finding their mutual relations, compositions, and resolutions.</p><p><hi rend="smallcaps">Harmonical</hi> <hi rend="italics">Composition,</hi> in its general sense, includes the composition both of harmony and melody; i. e. of music, or song, both in a single part, and in several parts.</p><p><hi rend="smallcaps">Harmonical</hi> <hi rend="italics">Interval,</hi> the difference between two sounds, in respect of acute and grave: or that imaginary space terminated with two sounds differing in acuteness or gravity.</p><p><hi rend="smallcaps">Harmonical</hi> <hi rend="italics">Proportion,</hi> or <hi rend="italics">Musical Proportion,</hi> is that in which the first term is to the third, as the difference of the first and second is to the difference of the 2d and 3d; or when the first, the third, and the said two differences, are in geometrical proportion. Or, four terms are in Harmonical proportion, when the 1st is to the 4th, as the difference of the 1st and 2d is to the difference of the 3d and 4th. Thus, 2, 3, 6, are in harmonical proportion, because . And the four terms 9, 12, 16, 24 are in harmonical proportion, because .—If the proportional terms be continued in the former case, they will form an harmonical progression, or series.</p><p>1. The reciprocals of an arithmetical progression are in Harmonical progression; and, conversely, the reciprocals of Harmonicals are arithmeticals. Thus, the reciprocals of the Harmonicals 2, 3, 6, are 1/2, 1/3, 1/6, which are arithmeticals; for , and also: and the reciprocals of the arithmeticals 1, 2, 3, 4, &c, are 1/1, 1/2, 1/3, 1/4, &c, which are Harmonicals; for ; and so on. And, in general, the reciprocals of the arithmeticals <hi rend="italics">a, a</hi> + <hi rend="italics">d, a</hi> + 2<hi rend="italics">d, a</hi> + 3<hi rend="italics">d,</hi> &c, viz, , &c, are Harmonicals; et e contra.</p><p>2. If three or four numbers in Harmonical proportion be either multiplied or divided by some number, the products, or the quotients, will still be in Harmonical proportion. Thus, <table><row role="data"><cell cols="1" rows="1" role="data">the Harmonicals</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell><cell cols="1" rows="1" rend="align=right" role="data">8,</cell><cell cols="1" rows="1" rend="align=right" role="data">12,</cell></row><row role="data"><cell cols="1" rows="1" role="data">multiplied by 2 give</cell><cell cols="1" rows="1" rend="align=right" role="data">12,</cell><cell cols="1" rows="1" rend="align=right" role="data">16,</cell><cell cols="1" rows="1" rend="align=right" role="data">24,</cell></row><row role="data"><cell cols="1" rows="1" role="data">or divided by 2 give</cell><cell cols="1" rows="1" rend="align=right" role="data">3,</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell></row></table> which are also Harmonicals.</p><p>3. To find a Harmonical mean proportional between two terms: Divide double their product by their sum.</p><p>4. To find a 3d term in Harmonical proportion to two given terms: Divide their product by the difference between double the 1st term and the 2d term.</p><p>5. To find a 4th term in Harmonical proportion to three terms given: Divide the product of the 1st and 3d by the difference between double the 1st and the 2d term.<pb n="583"/><cb/> <hi rend="center">Hence, of the two terms <hi rend="italics">a</hi> and <hi rend="italics">b;</hi></hi> <hi rend="center">the Harmonical mean is ,</hi> <hi rend="center">the 3d Harmonical propor. is ,</hi> <hi rend="center">also to <hi rend="italics">a, b, c,</hi> the 4th Harm. is .</hi></p><p>6. If there be taken an arithmetical mean, and a Harmonical mean, between any two terms, the four terms will be in geometrical proportion. Thus, between 2 and 6, <hi rend="center">the arithmetical mean is 4, and</hi> <hi rend="center">the Harmonical mean is 3;</hi> <hi rend="center">and hence .</hi></p><p>Also, between <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> <hi rend="center">the arithmetical mean is , and</hi> <hi rend="center">the Harmonical mean is ;</hi> <hi rend="center">but .</hi></p></div1><div1 part="N" n="HARMONY" org="uniform" sample="complete" type="entry"><head>HARMONY</head><p>, in Music, the agreeable result of an union of several musical sounds, heard at one and the same time; or the mixture of divers sounds, which together have an agreeable effect on the ear.</p><p>As a continued succession of musical sounds produces melody, so a continued combination of them produces Harmony.</p><p>Among the ancients however, as sometimes also among the moderns, Harmony is used in the strict sense of consonance; and so is equivalent to the symphony.</p><p>The words <hi rend="italics">concord</hi> and Harmony do really signify the same thing; though custom has made a little difference between them. Concord is the agreeable effect of two sounds in consonance; and Harmony the effect of any greater number of agreeable sounds in consonance.</p><p>Again, Harmony always implies consonance; but concord is also applied to sounds in succession; though never but where the terms can stand agreeably in consonance. The effect of an agreeable succession of several sounds, is called <hi rend="italics">melody;</hi> as that of an agreeable consonance is called Harmony.</p><p>Harmony is well defined, the sum or result of the combination of two or more concords; that is, of three or more simple sounds striking the ear all together; and diffevent compositions of concords make different Harmony.</p><p>The ancients seem to have been entirely unacquainted with Harmony, the soul of the modern music. In all their explications of the melopœia, they say not one word of the concert or Harmony of parts. We have instances, indeed, of their joining several voices, or instruments, in consonance; but then these were not so joined, as that each had a distinct and proper melody, so making a succession of various concords; but they were either unisons, or octaves, in every note; and so all performed the same individual melody, and constituted one song.</p><p>When the parts differ, not in the tension of the whole, but in the different relations of the successive notes, it is this that constitutes the modern art of Har- mony.<cb/></p><p><hi rend="smallcaps">Harmony</hi> <hi rend="italics">of the Spheres,</hi> or <hi rend="italics">Celestial Harmony,</hi> a kind of music much spoken of by many of the ancient philosophers and fathers, supposed to be produced by the sweetly tuned motions of the stars and planets. This Harmony they attributed to the various proportionate impressions of the heavenly bodies upon one another, acting at proper intervals. They think it impossible that such prodigious bodies, moving with such rapidity, should be silent: on the contrary, the atmosphere, continually impelled by them, must yield a set of sounds proportionate to the impression it receives; and that consequently, as they run all in different circuits, and with various degrees of velocity, the different tones arising from the diversity of motions, directed by the hand of the Almighty, must form an agreeable symphony or concert.</p><p>They therefore supposed, that the moon, as being the lowest of the planets, corresponded to <hi rend="italics">mi;</hi> Mercury, to <hi rend="italics">fa;</hi> Venus, to <hi rend="italics">sol;</hi> the Sun, to <hi rend="italics">la;</hi> Mars, to <hi rend="italics">si;</hi> Jupiter, to <hi rend="italics">ut;</hi> Saturn, to <hi rend="italics">re;</hi> and the orb of the fixed stars, as being the highest of all, to <hi rend="italics">mi,</hi> or the octave.</p><p>It is though that Pythagoras had a view to the gravitation of celestial bodies, in what he taught concerning the Harmony of the spheres.</p><p>A musical chord gives the same note as one double in length, when the tension or force with which the latter is stretched is quadruple; and the gravity of a planet is quadruple of the gravity of a planet at a double distance. In general, that any musical chord may become unison to a lesser chord of the same kind, its tension must be increased in the same proportion as the square of its length is greater; and that the gravity of a planet may become equal to the gravity of another planet nearer the sun, it must be increased in proportion as the square of its distance from the sun is greater. If therefore we should suppose musical chords extended from the sun to each planet, that all these chords might become unison, it would be requisite to increase or diminish their tensions in the same proportions as would be sufficient to render the gravities of the planets equal; and from the similitude of those proportions, the celebrated doctrine of the Harmony of the spheres is supposed to have been derived.</p><p>Kepler wrote a large work, in folio, on the Harmonies of the world, and particularly of that of the celestial bodies. He first endeavoured to find out some relation between the dimensions of the five regular solids and the intervals of the planetary spheres; and imagining that a cube, inscribed in the sphere of Saturn, would touch by its six planes the sphere of Jupiter, and that the other four regular solids in like manner fitted the intervals that are between the spheres of the other planets, he became persuaded that this was the true reason why the primary planets were precisely six in number, and that the author of the world had determined their distances from the sun, the centre of the system, from a regard to this analogy. But afterwards finding that the disposition of the five regular solids amongst the planetary spheres, was not agreeable to the intervals between their orbits, he endeavoured to discover other schemes of Harmony. For this purpose he compared the motions of the same planet at its greatest and least distances, and of the different planets in their several orbits, as they would appear viewed<pb n="584"/><cb/></p><p>from the sun; and here he fancied that he found a similitude to the divisions of the octave in music. Lastly, he imagined that if lines were drawn from the earth, to each of the planets, and the planets appended to them, or stretched by weights proportional to the planets, these lines would then sound all the notes in the octave of a musical chord.</p><p>See his Harmonics; also Plin. lib. 2, cap. 22; Macrob. in Somn. Scip. lib. 2, cap. 1; Plutarch de Animal. Procreatione, è Timæo; and Maclaurin's View of Newton's Discov. book 1, chap. 2.</p></div1><div1 part="N" n="HARQUEBUSS" org="uniform" sample="complete" type="entry"><head>HARQUEBUSS</head><p>, a hand-gun, or a fire-arm of a proper length and weight to be borne in the arm. Hanzelet prescribes its proper length to be 40 calibres, or diameters of its bore; and the weight of its ball <*> oz. and 7/8; its charge of powder as much.</p></div1><div1 part="N" n="HARRIOT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HARRIOT</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>)</persName></head><p>, a very eminent English mathematician and astronomer, was born at Oxford in 1560, and died at London July 2, 1621, in the 61st year of his age. Harriot has hitherto been known to the world only as an algebraist, though a very eminent one; but from his manuscript papers, that have been but lately discovered by Dr. Zach, astronomer to the duke of Saxe-Gotha, it appears that he was not less eminent as an astronomer and geometrician. Dr. Zach has printed an account of those papers, in the Astronomical Ephemeris of the Royal Academy of Sciences at Berlin, for the year 1788; of which, as it is very curious, and contains a great deal of information, I shall here give a translation, to serve as memoirs concerning the life and writings of this eminent man; afterwards adding only some necessary remarks of my own.</p><p>“I here present to the world (says Dr. Zach), a short account of some valuable and curious manuscripts, which I found in the year 1784, at the seat of the earl of Egremont, at Petworth in Sussex, in hopes that this learned and inquisitive age will either think my endeavours about them worthy of its assistance, or else will be thereby induced to attempt some other means of publishing them. The only undeniable proof I can now produce of the usefulness of such an undertaking, is by giving a succinct report of the contents of these materials, and briefly shewing what may be effected by them. And although I come to the performance of such an enterprize with much less abilities than the different parts of it require, yet I trust that my love for truth, my design and zeal to vindicate the honour due to an Englishman, the author of these manuscripts, which are the chief reasons that have influenced me in this undertaking, will serve as my excuse.</p><p>“A predecessor of the family of lord Egremont, viz, that noble and generous earl of Northumberland, named Henry Percy, was not only a generous favourer of all good learning, but also a patron and Mæcenas of the learned men of his age. Thomas Harriot, the author of the said manuscripts, Robert Hues (well known by his Treatise upon the Globes), and Walter Warner, all three eminent mathematicians, who were known to the earl, received from him yearly pensions; so that when the earl was committed prisoner to the Tower of London in the year 1606, our author, with Hues and Warner, were his constant companions; and were usually called the earl of Northumberland's three Magi.<cb/></p><p>“Thomas Harriot is a known and celebrated mathematician among the learned of all nations, by his excellent work, <hi rend="italics">Artis Analyticæ Praxis, ad æquationes algebraicas nova expedita & generali methodo, resolvendas, Tractatus posthumus;</hi> Lond. 1631: dedicated to Henry earl of Northumberland; published after his death by Walter Warner. It is remarkable, that the fame and the honour of this truly great man were constantly attacked by the French mathematicians; who could not endure that Harriot should in any way diminish the fame of their Vieta and Des Cartes, especially the latter, who was openly accused of plagiarism from our author. [<hi rend="italics">See Montucla's Histoire des Mathematiques, part</hi> 3, <hi rend="italics">p.</hi> 485 <hi rend="italics">& seq.—Lettres de M. Des Cartes, tom.</hi> 3, <hi rend="italics">pa.</hi> 457, <hi rend="italics">edit. Paris</hi> 1667, <hi rend="italics">in</hi> 4<hi rend="italics">to.—Dictionnaire de Moreri, word</hi> Harriot.—<hi rend="italics">Encyclopedie, word</hi> Algebra.—<hi rend="italics">Lettres de M. de Voltaire, sur la nation Angloise, lettre</hi> 14. —<hi rend="italics">Memoire de l'Abbé de Gua dans les Mem. de l'Acad. des Sciences de Paris pour</hi> 1741.—<hi rend="italics">Jer. Collier's great Historical Dictionary, word</hi> Harriot.—<hi rend="italics">Dr. Wallis's preface to his Algebra.—To which may be added the article</hi> Algebra, <hi rend="italics">in this dictionary.</hi>]</p><p>“Des Cartes published his Geometry 6 years after Harriot's work appeared, viz, in the year 1637. Sir Charles Cavendish, then ambassador at the French court at Paris, when Des Cartes's Geometry made its first appearance in public, observed to the famous geometrician Roberval, that these improvements in Analysis had been already made these 6 years in England, and shewed him afterwards Harriot's <hi rend="italics">Artis Analyticæ Praxis,</hi> which as Roberval was looking over, at every page he cried out, <hi rend="italics">Oui! oui! il l'a vu! Yes! yes! he has seen it!</hi> Des Cartes had also been in England before Harriot's death, and had heard of his new improvements and inventions in Analysis. A critical life of this man, which his papers would enable me to publish, will shew more clearly what to think upon this matter, which I hope may be discussed to the due honour of our author.</p><p>“Now all this relates to Harriot the celebrated analyst; but it has not hitherto been known that Harriot was an eminent astronomer, both theoretical and practical, which first appears by these manuscripts; among which, the most remarkable are 199 observations of the Sun's Spots, with their drawings, calculations and determinations of the sun's rotation about his axis. There is the greatest probability that Harriot was the first discoverer of these spots, even before either Galileo or Scheiner. The earliest intelligence we have of the first discovered solar spots, is of one Joh. Fabricius Phrysius, who in the year 1611 published at Wittemberg a small treatise, intitled, <hi rend="italics">De Maculis in Sole observatis & apparente eorum cum Sole conversione narratio.</hi> Galileo, who is commonly accounted the first discoverer of the Solar Spots, published his book, <hi rend="italics">Istoria e Dimonstrazioni intorne alle Machie Solare e loro accidenti,</hi> at Rome, in the year 1613. His first observation in this work, is dated June 2, 1612. Angelo de Filiis, the editor of Galileo's work, who wrote the dedication and preface to it, mentions, pa. 3, that Galileo had not only discovered these spots in the month of April in the year 1611, at Rome, in the Quirinal Garden, but had shewn them several months before <hi rend="italics">(molti mesi innanzi)</hi> to his friends in Florence. And that the observations of the disguised Apelles (the Jesuit Scheiner, a pre-<pb n="585"/><cb/> tender to this first discovery) were not later than the month of October in the same year; by which the epoch of this discovery was fixed to the beginning of the year 1611. But a passage in the first letter of Galileo's works, pa. 11, gives a more precise term to this discovery. Galileo there says in plain terms, that he had observed the Spots in the Sun 18 months before. The date of this letter is May 24, 1612; which brings the true epoch of this discovery to the month of November, 1610. However, Galileo's first produced observations are only from June 2, 1612, and those of father Scheiner of the month of October, in the same year. But <*>now it appears from Harriot's manuscripts, that his first observations of these Spots are of Dec. 8, 1610. It is not likely that Harriot could have this notice from Galileo, for I do not find this mathematician's name ever quoted in Harriot's papers. But I find him mentioning Josephus a Costa's book 1, chap. 2, of his <hi rend="italics">Natural and Moral History of the West Indies,</hi> in which he relates that in Peru there are Spots to be seen in the Sun which are not to be seen in Europe. It rather seems that Harriot had taken the hint from thence. Besides, it is very likely that Harriot, who lived with so generous a patron to all good learning and improvements, had got the new invention of telescopes in Holland much sooner in England, than they could reach Galileo, who at that time lived at Venice. Harriot's very careful and exact observations of these Spots, shew also that he was in possession of the best and most improved telescopes of that time; for it appears he had some with magnifying powers of 10, 20, and 30 times. At least there are no earlier observations of the Solar Spots extant than his: they run from December 8, 1610, till January 18, 1613. I compared the corresponding ones with these observed by Galileo, between which I found an exact agreement. Had Harriot had any notion about Galileo's discoveries, he certainly would have also known something about the phases of Venus and Mercury, and especially about the singular shape of Saturn, first discovered by Galileo; but I find not a word in all his papers concerning the particular figure of that planet.</p><p>“<hi rend="italics">Of Jupiter's Satellites.</hi> I found among his papers a large set of observations, with their drawing, position, and calculations of their revolutions and periods. His first observation of those discovered Satellites, I find to be of January 16, 1610; and they go till February 26, 1612. Galileo pretends to have discovered them January 7, 1610; so that it is not improbable that Harriot was likewise the first discoverer of these attendants of Jupiter.</p><p>“Among his other observations of the Moon, of some Eclipses, of the planet Mars, of Solstices, of Refraction, of the Declination of the Needle; there are most remarkable ones of the noted Comets of 1607, and of 1618, the latter, for there were two this year <hi rend="italics">(see Kepler de Cometis, pa.</hi> 49). They were all observed with a cross-staff, by measuring their distances from fixed stars; whence these observations are the more valuable, as comets had before been but grossly observed: Kepler himself observed the comet of 1607 only with the naked eye, pointing out its place by a coarse estimation, without the aid of an instrument; and the elements of their orbits could, in defect of better observa-<cb/> tions, be only calculated by them. The observations of the comet of the year 1607, are of the more importance, even now for modern astronomy, as this is the same comet that fulsilled Dr. Halley's prediction of its return in the year 1759. That prediction was only grounded upon the elements afforded him by these coarse observations; for which reason he only assigned the term of its return to the space of a year. The very intricate calculations of the perturbations of this comet, afterwards made by M. Clairaut, reduced the limits to a month's space. But a greater light may now be thrown upon this matter by the more accurate observations on this comet by Mr. Harriot. In the month of October 1785, when I conversed upon the subject of Harriot's papers, and especially on this comet, with the very celebrated mathematician M. de la Grange, director of the Royal Academy of Sciences at Berlin; he then suggested to me an idea, which, if brought into execution, will clear up an important point in astronomy. It is well known to astronomers how difficult a matter it is, to determine the mass, or quantity of matter, in the planet Saturn; and how little satisfactory the notions of it are, that have hitherto been formed. The whole theory of the perturbations of comets depending upon this uncertain datum, several attempts and trials have been made towards a more exact determination of it by the most eminent geometricians of this age, and particularly by la Grange himself; but never having been satisfied with the few and uncertain data heretofore obtained for the resolution of this problem, he thought that Harriot's observations on the comet of 1607, and the modern ones of the same comet in 1759, would suggest a way of resolving the problem <hi rend="italics">a posteriori;</hi> that of determining by them the elements of its ellipsis. The retardation of the comet compared to its period, may clearly be laid to the account of the attraction and perturbation it has suffered in the region of Jupiter and Saturn; and as the part of it belonging to Jupiter is very well known, the remainder must be the share which is due to Saturn; from whence the mass of the latter may be inferred. In consequence of this consideration I have already begun to reduce most of Harriot's observations of this comet, in order to calculate by them the true elements of its orbit on an elliptical hypothesis, to complete M. la Grange's idea upon this matter.</p><p>“I forbear to mention here any more of Harriot's analytical papers, which I found in a very great number. They contain several elegant solutions of quadratic, cubic, and biquadratic equations; with some other solutions and <hi rend="italics">loca geometrica,</hi> that shew his eminent qualifications, and will serve to vindicate them against the attacks of several French writers, who refuse him the justice due to his skill and accomplishments, merely to save Des Cartes's honour, who yet, by some impartial men of his own nation, was accused of public plagiarism.</p><p>“Thomas Harriot was born at Oxford, in the year 1560. After being instructed in the rudiments of languages, he became a commoner of St. Mary's-Hall, where he took the degree of bachelor of arts in 1579. He had then so distinguished himself by his uncommon skill in mathematics, as to be recommended soon after to Sir Walter Raleigh, as a proper preceptor to him in that science. Accordingly that noble knight became his first patron, took him into his family, and allowed<pb n="586"/><cb/> him a handsome pension. In 1584 he went over with Sir Walter's first colony to Virginia; where he was employed in discovering and surveying the country, &c; maps of which I have found (says Dr. Zach) very n<*>atly done among his papers. After his return he published <hi rend="italics">A Brief and True Report of the Newfoundland of Virginia, of the Commodities there sound to be raised,</hi> &c; Lond. 1588. This was reprinted in the 3d volume of Hakluyt's Voyages: it was also translated into Latin, and printed at Frankfort in the year 1590. Sir Walter introduced him to the acquaintance of the earl of North<*>berland, who allowed him a yearly pension; Wood says, of 120l. only; but by some of his receipts, which I have found among his papers, it appears that he had 300l, which indeed was a very large sum at that time. Wood, in his <hi rend="italics">Athen. Oxon.</hi> mentions nothing of Harriot's papers, except a manuscript in the library at Sion College, London, entitled <hi rend="italics">Ephemeris Chyrometrica.</hi> I got access to this library and manuscripts, and was indeed in hopes of finding something more of Harriot's; for most of his observations are dated from Sion College; but I could not find any thing from Harriot himself. I found indeed some other papers of his friends: he mentions, in his observations, one Mr. Standish, at Oxford, and Nicol. Torperly, who also was of the acquaintance of the earl of Northumberland, and had a yearly pension: from the former I found two observations of the same comet of 1618, made in Oxford, which he communicated to Mr. Harriot.</p><p>“Thomas Harriot died July 2, 1621. His disease was a cancerous ulcer in the lip, which some pretend he got by a custom he had of holding the mathematical brass instruments, when working, in his mouth. I found several of his letters, and answers to them, from his physician Dr. Alexander Rhead, who in his treatise mentions Harriot's disease. His body was conveyed to St. Christopher's church, in London. Over his grave was soon after erected a monument, with a large inscription, which was destroyed with the church itself by the dreadful fire of September 1666. He was but 60 years of age.”</p><p>The peculiar nature and merits of Harriot's Algebra, we have spoken largely and particularly of, under the art. <hi rend="italics">Algebra,</hi> page 89. As to his manuscripts lately discovered by Dr. Zach, as above mentioned, it is with pleasure I can announce, that they are in a fair train to be published: they have been presented to the university of Oxford, on condition of their printing them; with a view to which, they have been lately put into the hands of an ingenious member of that learned body, to arrange and prepare them for the press.</p></div1><div1 part="N" n="HARRISON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HARRISON</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a most accurate mechanic, the celebrated inventor of the famous <hi rend="italics">time-keeper</hi> for ascertaining the longitude at sea, and also of the compound or <hi rend="italics">gridiron-pendulum;</hi> was born at Foulby, near Pontesract in Yorkshire, in 1693. His father was a carpenter, in which profession the son assisted; occasionally also, according to the miscellaneous practice of country artists, surveying land, and repairing clocks and watches; and young Harrison always was, from his early childhood, greatly attached to any machinery moving by wheels. In 1700 he removed with his father to Barrow in Lincolnshire; where, though his opportunities of acquiring knowledge were very few, he eagerly improved every incident from which he might<cb/> collect information; frequently employing all or great part of his nights in writing or drawing: and he always acknowledged his obligations to a clergyman who came every Sunday to officiate in the neighbourhood, who lent him a MS. copy of professor Sanderson's lectures; which he carefully and neatly transcribed, with all the diagrams. His native genius exerted itself superior to these solitary disadvantages; for in the year 1726, he had constructed two clocks, mostly of wood, in which he applied the escapement and compound pendulum of his own invention: these surpassed every thing then made, scarcely erring a second in a month. In 1728, he came up to London with the drawings of a machine for determining the longitude at sea, in expectation of being enabled to execute one by the Board of Longitude. Upon application to Dr. Halley, the astronomer royal, he referred him to Mr. George Graham; who advised him to make his machine, before applying to that Board. He accordingly returned home to perform his task; and in 1735 came to London again with his first machine; with which he was sent to Lisbon the next year, to make trial of it. In this short voyage, he corrected the dead reckoning about a degree and a half; a success which procured him both public and private encouragement. About the year 1739 he completed his second machine, of a construction much more simple than the former, and which answered much better: this, though not sent to sea, recommended Mr. Harrison yet stronger to the patronage of his friends and the public. His third machine, which he produced in 1749, was still less complicated than the second, and more accurate, as erring only 3 or 4 seconds in a week. This he conceived to be the <hi rend="italics">ne plus ultra</hi> of his attempts; but by endeavouring to improve pocket-watches, he found the principles he applied to surpass his expectations, so much, as to encourage him to make his fourth time-keeper, which is in the form of a pocket-watch, about 6 inches diameter. With this time-keeper his son made two voyages, the one to Jamaica, and the other to Barbadoes; in which experiments it corrected the longitude within the nearest limits required by the act of the 12th of queen Anne; and the inventor had therefore, at different times, more than the proposed reward, receiving from the Board of Longitude at different times almost 24,000l. besides a few hundreds from the East India Company, &c. These four machines were given up to the Board of Longitude. The three former were not of any use, as all the advantages gained by making them, were comprehended in the last: being worthy however of preservation, as mechanical curiosities, they are deposited in the Royal Observatory at Greenwich. The fourth machine, emphatically distinguished by the name of <hi rend="italics">The time-keeper,</hi> was copied by the ingenious Mr. Kendal; and that duplicate, during a three years circumnavigation of the globe in the southern hemisphere by captain Cook, answered as well as the original.</p><p>The latter part of Mr. Harrison's life was employed in making a fifth improved time-keeper, on the same principles with the preceding one; which, after a tenweeks trial, in 1772, at the king's private observatory at Richmond, erred only 4 1/2 seconds. Within a few years of his death, his constitution visibly declined; and he had frequent fits of the gout, a disorder that<pb n="587"/><cb/> never attacked him before his 77th year. His constitution at last yielding to the infirmities of old age, he died at his house in Red-Lion Square, in 1776, at 83 years of age.</p><p>Like many other mere mechanics, Mr. Harrison found a difficulty in delivering his sentiments in writing (at least in the latter periods of his life, when his faculties were much impaired) in which he adhered to a peculiar and uncouth phraseology. This was but too evident in his <hi rend="italics">Description concerning such Mechanism as will afford a nice or true Mensuration of Time,</hi> &c. 8vo, 1775. This small work includes also an account of his new musical scale; being a mechanical division of the octave, according to the proportion which the radius and diameter of the circle have respectively to the circumference. He had in his youth been the leader of a band of church-singers; had a very delicate ear for music; and his experiments on sound, with a curious monochord of his own improvement, it has been said were not less accurate than those he was engaged in for the mensuration of time.</p></div1><div1 part="N" n="HAUTEFEUILLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HAUTEFEUILLE</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an ingenious mechanic, born at Orleans in 1674. He made a great progress in mechanics in general, but had a particular taste for clock-work, and made several discoveries in it that were of singular use. It was he it seems who found out the secret of moderating the vibration of the balance by means of a small steel-spring, which has since been made use of. This discovery he laid before the members of the Academy of Sciences in 1694; and these watches are, by way of eminence, called <hi rend="italics">pendulum-watches;</hi> not that they have real pendulums, but because they nearly approach to the justness of pendulums. M. Huygens perfected this happy invention; but having declared himself the inventor and obtained a patent for making watches with spiral springs, the abbé Feuille opposed the registering of it, and published a piece on the subject against Huygens. He died in 1724, at 50 years of age. Befides the above,</p><p>He wrote a great many other pieces, most of which are small pamphlets, but very curious: as, 1. His Perpetual Pendulum. 2. New Inventions. 3. The Art of Breathing under Water, and the means of preserving a Flame shut up in a Small Place. 4. Reflections on machines for Raising Water. 5. His Opinion on the different Sentiments of Mallebranche and Regis, relating to the Appearance of the Moon when seen in the Horizon. 6. The Magnetic Balance. 7. A Placet to the king on the Longitude. 8. Letter on the Secret of the Longitude. 9. A New System on the Flux and Reflux of the Sea. 10. The means of making Sensible Experiments that prove the Motion of the Earth: and many other pieces.</p></div1><div1 part="N" n="HAYES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HAYES</surname> (<foreName full="yes"><hi rend="smallcaps">Charles</hi></foreName>, Esq.)</persName></head><p>, a very singular person, whose great erudition was so concealed by his modesty, that his name is known to very few, though his publications are many. He was born in 1678, and died in 1760, at 82 years of age. He became distinguished in 1704 by a Treatise of Fluxions, in folio, being we believe the fir<*> treatise on that science ever published in the English language; and the only work to which he ever set his name. In 1710 came out a small 4to pamphlet, in 19 pages, intitled, A New and easy Method to find out the Longitude from observing the Altitudes of the Celestial<cb/> bodies. Also, in 1723, he published, The Moon, a Philosophical Dialogue; tending to shew, that the moon is not an opaque body, but has native light of her own.</p><p>To a skill in the Greek and Latin, as well as the modern languages, he added the knowledge of the Hebrew: and he published several pieces relating to the translation and chronology of the scriptures. During a long course of years he had the chief management of the late African company, being annually elected subgovernor. But on the dissolution of that company in 1752, he retired to Down in Kent, where he gave himself up to study; from whence however he returned in 1758, to chambers in Gray's inn, London, where he died in 1760, as mentioned above.</p><p>He left a posthumous work, that was published in 8vo, under the title of <hi rend="italics">Chronographia Asiatica et Ægyptiaca &c.</hi></p></div1><div1 part="N" n="HEAT" org="uniform" sample="complete" type="entry"><head>HEAT</head><p>, the opposite to cold, being a relative term denoting the property of fire, or of those bodies we denominate hot; being in us a sensation excited by the action of fire.</p><p>Heat, as it exists in the hot body, or that which constitutes and denominates a body hot, and enables it to produce such effects on our organs, is variously considered by the philosophers: some making it a quality, others a substance, and others only a mechanical affection. The former principle is laid down by Aristotle and the Peripatetics. While the Epicureans, and other corpuscularians, define Heat not as an accident of fire, but as an essential power or property of it, the same in reality with it, and only distinguished from it in the manner of our conception. So that Heat, on their principles, is no other than the volatile substance of fire itself, reduced into atoms, and emitted in a continual stream from ignited bodies; so as not only to warm the objects within its reach, but also, if they be inflammable, to kindle them, turn them into fire, and conspire with them to make flame. In effect, these corpuscles, say they, flying off from the ignited body, constitute fire while yet contained within the sphere of its flame; but when fled, or got beyond the same, and dispersed every way, so as to escape the apprehension of the eye, and only to be perceived by the feeling, they take the denomination of Heat, inasmuch as they excite in us that sensation.</p><p>The Cartesians, improving on this doctrine, assert that Heat consists in a certain motion of the insensible particles of a body, resembling the motion by which the several parts of our body are agitated by the motion of the heart and blood.</p><p>Our latest and best writers of mechanical, experimental, and chemical philosophy, differ very considerably about Heat. The chief difference is, whether it be a peculiar property of one certain immutable body, called fire, or phlogiston, or electricity; or whether it may be produced mechanically in other bodies, by inducing an alteration in their particles. The former tenet, which is as ancient as Democritus, and the system of atoms, had given way to that of the Cartesians, and other mechanists; but is now with great addres<*> retrieved, and improved on, by some of the latest writers, particularly Homberg, the younger Lemery, Gravesande, Boerhaave in his lectures on fire, Black, Crawford, and other chemical philosophers.<pb n="588"/><cb/></p><p>The thing called fire, according to Boerhaave, is a body sui generis, created such ab origine, unalterable in its nature and properties, and not either producible de novo from any other body, nor capable of being reduced into any other body, or of ceasing to be fire. This fire, he contends, is diffused equably every where, and exists alike, or in equal quantity, in all the parts of space, whether void, or possessed by bodies; but that naturally, and in itself, it is perfectly latent and imperceptible; being only discovered by certain effects which it produces, and which are cognizable by our senses. These effects are Heat, light, colour, rarefaction, and burning, which are all indications of fire, as being none of them producible by any other cause: so that whereever we observe any of these, we may safely infer the action and presence of fire. But though the effect cannot be without the cause, yet the fire may remain without any of these effects; any, we mean, gross enough to affect our senses, or become objects of them: and this, he adds, is the ordinary case; there being a concurrence of other circumstances, which are often wanting, necessary to the production of such sensible effects.</p><p>The mechanical philosophers, particularly Bacon, Boyle, and Newton, conceive otherwise of Heat; considering it not as an original inherent property of any particular sort of body; but as mechanically producible in any body. The former, in an express treatise De Forma Calidi, from a particular enumeration of the several phenomena and effects of Heat, deduces several general properties of it; and hence he defines Heat, an expansive undulatory motion in the minute particles of the body; by which they tend, with some rapidity, towards the circumference, and at the same time incline a little upwards.</p><p>Mr. Boyle, in a Treatise on the Mechanical Origin of Heat and Cold, strongly supports the doctrine of the producibility of Heat, with new observations and experiments; as in the instance of a smith briskly hammering a small piece of iron, which, though cold before, soon becomes exceedingly hot.</p><p>This system is also farther supported by Newton, who does not conceive fire as any particular species of body, originally endued with such and such properties. Fire, according to him, is only a body much ignited, that is heated hot, so as to emit light copiously: what else, says he, is red-hot iron but fire? and what else is a burning charcoal but red-hot wood? or flame itself, but red-hot smoke? It is certain that flame is only the volatile part of the fuel heated red-hot, i. e. so hot as to shine; and hence only such bodies as are volatile, that is, such as emit a copious fume, will flame; nor will they flame longer than they have fume to burn. In distilling hot spirits, if the head of the still be taken off, the ascending vapours will catch fire from a candle, and turn into a flame. And in the same manner several bodies, much heated by motion, attrition, fermentation, or the like, will emit lucid fumes, which, if they be copious enough, and the heat sufficiently great, will be flame; and the reason why fused metals do not flame, is the smallness of their fume; this is evident, because spilter, which fumes most copiously, does likewise flame. Add, that all flaming bodies, as oil, tallow, wax, wood, pitch, sulphur, &c, by flaming, waste and vanish into burning smoke. And do not all fixed bodies, when heated beyond a certain degree, emit light,<cb/> and shine? and is not this emission performed by the vibrating motion of their parts? and do not all bodies, which abound with terrestrial and sulphureous parts, emit light as often as those parts are sufficiently agitated, whether that agitation be made by external fire, or by friction, or percussion, or putrefaction, or by any other cause? Thus, sea water, in a storm; quicksilver agitated in vacuo; the back of a cat, or the neck of a horse, obliquely rubbed in a dark place; wood, fles<*>, and fish, while they putrefy; vapours from putrefying waters, usually called ignes fatui; stacks of moist hay or corn; glow-worms; amber and diamonds by rubbing; fragments of steel struck off with a flint, &c, all emit light. Are not gross bodies and light convertible into one another? and may not bodies receive much of their activity from the particles of light which enter their composition? I know no body less apt to shine than water; and yet water, by frequent distillations, changes into fixed earth, which, by a sufficient Heat, may be brought to shine like other bodies.</p><p>Add, that the sun and stars, according to Newton's conjecture, are no other than great earths vehemently heated: for large bodies, he observes, preserve their Heat the longest, their parts heating one another; and why may not great, dense, and fixed bodies, when heated beyond a certain degree, emit light so copiously, as by the emission and reaction of it, and the reflections and refractions of the rays within the pores, to grow still hotter, till they arrive at such a period of Heat as is that of the sun? Their parts also may be farther preserved from fuming away, not only by their fixity, but by the vast weight and density of their atmospheres incumbent on them, thus strongly compressing them, and condensing the vapours and exhalations arising from them. Hence we see warm water, in an exhausted receiver, shall boil as vehemently as the hottest water open to the air; the weight of the incumbent atmosphere, in this latter case, keeping down the vapours, and hindering the ebullition, till it has conceived its utmost degree of Heat. So also a mixture of tin and lead, put on a red-hot iron in vacuo, emits a fume and flame; but the same mixture in the open air, by reason of the incumbent atmosphere, does not emit the least sensible flame.</p><p>Thus much for the system of the producibility of Heat.</p><p>On the other hand, M. Homberg, in his Essai du Soufre Principe, holds, that the chemical principle or element, sulphur, which is supposed one of the simple, primary, pre-existent ingredients of all natural bodies, is real fire; and consequently that fire is co-eval with body. Mem. de l'Acad. an. 1705.</p><p>Dr. Gravesande goes upon much the same principle. According to him, fire enters the composition of all bodies, is contained in all bodies, and may be separated or procured from all bodies, by rubbing them against each other, and thus putting their fire in motion. But fire, he adds, is by no means generated by such motion. Elem. Phys. tom. 2, cap. 1. Heat, in the hot body, he says, is an agitation of the parts of the body, made by means of the fire contained in it; by such agitation a motion is produced in our bodies, which excites the idea of Heat in our minds: so that Heat, in respect of us, is nothing but that idea, and in the hot body nothing but motion. If such motion expel<pb n="589"/><cb/> the fire in right lines, it may give us the idea of light; if in a various and irregular motion, only of Heat.</p><p>Lemery, the younger, agrees with these two authors, in asserting this absolute and ingenerable nature of fire; but he extends it farther. Not contented with confining it as an element to bodies, he endeavours to shew, that it is equably diffused through all space; that it is present in all places, even in the void spaces between the bodies, as well as in the insensible interstices between their parts. And this last sentiment falls in with that of Boerhaave above delivered. Mem. de l'Acad. an. 1713.</p><p>Philosophers have lately distinguished Heat into Absolute, and Sensible. By Absolute Heat, or fire, they mean that power or element which, when it is in a certain degree, excites in animals the sensation of Heat; and by Sensible Heat, the same power considered in its relation to the effects which it produces: thus, two bodies are said to have equal quantities of sensible Heat, when they produce equal effects upon the mercury in the thermometer; but as bodies of different kinds have different capacities for containing Heat, the absolute Heat in such bodies will be different, though the sensible Heat be the same. Thus, a pound of water and a pound of antimony, of the same temperature, have equal sensible Heat; but the former contains a much greater quantity of absolute Heat than the latter.</p><p>M. De Luc has evinced, by a variety of experiments, that the expansions of mercury between the freezing and boiling points of water, correspond precisely to the quantities of absolute Heat applied, and that its contractions are proportionable to the diminution of this element within these limits. And from hence it may be inferred, that if the mercury were to retain its fluid form, its contractions would be proportionable to the decrements of the absolute Heat, though the diminution were continued to the point of total privation. But the comparative quantities of absolute Heat, which are communicated to different bodies, or separated from them, cannot be determined in a direct manner by the thermometer.</p><p>Some philosophers have apprehended that the quantities of absolute Heat in bodies, are in proportion to their densities. While others, as Boerhaave, imagined that Heat is equally diffused through all bodies, the densest as well as the rarest, and therefore that the quantities of Heat in bodies are in proportion to their bulk or magnitude: and, at his desire, Fahrenheit attempted to determine the fact by experiment. For this purpose, he took equal quantities of the same fluid, and gave them different degrees of Heat, then upon mixing them intimately together, he found that the temperature of the mixture was a just medium, or arithmetical mean, between the two. But if this experiment be made with water and mercury, in the same circumstances, viz in equal bulks, the result will be different, as the temperature of the mixture will not be a mean between the two, but always nearer to that of the water than to the quicksilver; so that, when the water is the hotter, the temperature of the mixture is above the mean, and below it when the water is the colder. And from experiments of this kind it has been inferred, that the comparative quantities of the absolute Heat<*> of these fluids, are reciprocally proportional to the changes which<cb/> are produced in their sensible Heats, when they are mixed together at different temperatures: and this fact has been publicly taught, for several years, by Dr. Black, and Dr. Irvine, in the universities of Edinburgh and Glasgow. This rule however does not apply to those substances which, in mixture, excite sensible Heat by chemical action.</p><p>From the experiments and reasoning employed by Dr. Crawford, it more fully appears, that the quantities of absolute Heat in different bodies, are not as their densities; or that equal weights of heterogeneous substances, as air and water, having the same temperature<*> may contain unequal quantities of absolute Heat: he also shews, that if phlogiston be added to a body, a quantity of the absolute Heat of that body will be extricated; and if the phlogiston be separated again, an equal quantity of Heat will be absorbed. So that Heat and phlogiston appear to be two opposite principles in nature. But this ingenious writer has not presumed absolutely to decide the question that has been long agitated, whether Heat be a substance or a quality.— He inclines to the former opinion however, and observes, that if we adopt the opinion, that Heat is a distinct substance, or an element sui generis, the phenomena will be found to admit of a simple and obvious interpretation, and to be perfectly agreeable to the analogy of nature. See Crawford's Experiments and Observations<*> on Animal Heat and the Inflammation of Combustible Bodies.</p><p><hi rend="italics">Animal</hi> <hi rend="smallcaps">Heat</hi>. The Heat of animals is very various, both according to the variety of their kinds, and the difference of the seasons: accordingly, zoologists have divided them into hot and cold blooded, reckoning those to be hot that are near or above our own temperature, and all others cold whose Heat is below ours, and consequently affect us with the sense of cold; thus making the human species a medium between the hot and cold blooded animals, or at least the lowest order of the hot blooded.</p><p>The Heat of the human body, in its natural state, according to Dr. Boerhaave, is such as to raise the mercury in the thermometer to 92° or at most to 94°; and Dr. Pitcairn makes the heat of the human skin the same. Indeed it is evident that different parts of the human body, and its different states, as well as the different seasons, will make it shew of different temperatures. Thus, by various experiments at different times, the Heat of the human body is made various by the following authors: <table><row role="data"><cell cols="1" rows="1" role="data">Boerhaave and Pitcairn</cell><cell cols="1" rows="1" rend="align=right" role="data">92°</cell></row><row role="data"><cell cols="1" rows="1" role="data">Amontons</cell><cell cols="1" rows="1" rend="align=right" role="data">91, 92, or 93</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sir Isaac Newton</cell><cell cols="1" rows="1" rend="align=right" role="data">95 1/2</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Fahrenheit and Musschenbro<*>k, the blood,</cell><cell cols="1" rows="1" rend="align=right" role="data">96</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dr. Martine, the skin</cell><cell cols="1" rows="1" rend="align=right" role="data">97 or 98</cell></row><row role="data"><cell cols="1" rows="1" role="data">-----------, the urine</cell><cell cols="1" rows="1" rend="align=right" role="data">99</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dr. Hales, the skin</cell><cell cols="1" rows="1" rend="align=right" role="data">97</cell></row><row role="data"><cell cols="1" rows="1" role="data">---------, the urine</cell><cell cols="1" rows="1" rend="align=right" role="data">103</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mr. John Hunter, under his tongue,</cell><cell cols="1" rows="1" rend="align=right" role="data">97</cell></row><row role="data"><cell cols="1" rows="1" role="data">---------------, in his rectum</cell><cell cols="1" rows="1" rend="align=right" role="data">98 1/2</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">---------------, his urethra at 1 inch,</cell><cell cols="1" rows="1" rend="align=right" role="data">92</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">                            at 2 inches,</cell><cell cols="1" rows="1" rend="align=right" role="data">93</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">                            at 4 inches,</cell><cell cols="1" rows="1" rend="align=right" role="data">9<*></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">        the ball of the thermom. at the bulb</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data"><hi rend="size(6)">}97</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">             of the urethra</cell></row></table><pb n="590"/><cb/></p><p>For the powers of animals to bear various degrees of Heat, see the Philof. Trans. vol. 65, 68, &c.</p><p>There is hardly any subject of philosophical investigation that has afforded a greater variety of hypotheses, conjectures, and-experiments, than the cause of animal Heat. The first opinion which has very generally obtained, is, that the Heat of animal bodies is owing to the attrition between the arteries and the blood. All the observations and reasoning brought in favour of this opinion however, only shew that the Heat and the motion of the arteries are generally proportional to each other; without shewing which is the cause, and which the effect; or indeed that either is the cause or effect of the other, since both may be the effects of some other cause.</p><p>Dr. Douglas, in his Essay on the Generation of Heat in animals, ascribes it solely to the friction of the globules of blood in their circulation through the capillary vessels.</p><p>Another opinion is, that the lungs are the fountain of Heat in the human body: and this opinion is supported by much the same sort of arguments as the former, and seemingly to little better purpose.</p><p>A third opinion is, that the cause of animal Heat is owing to the action of the solid parts upon one another. And as the heart and arteries move most, it has been thought natural to expect that the Heat should be owing to this motion. But even this does not seem very plausible, from the following considerations: 1st, The moving parts, however we term them solid, are neither hard nor dry; which two conditions are absolutely requisite to make them sit to generate Heat by attrition. 2d, None of their motions are swift enough to promise Heat in this way. 3d, They have but little change of surface in their attritions. And 4thly, The moveable fibres have fat, mucilage, or liquors everyway surrounding them, to prevent their being destroyed, or heated by attrition.</p><p>A fourth cause assigned for the Heat of our bodies, is that process by which our aliment and fluids are perpetually undergoing some alteration. And this opinion is chiefly supported by Dr. Stevenson, in the Edinburgh Medical Essays, vol. 5, art. 77.</p><p>The late ingenious Dr. Franklin inclines to this opinion, when he says, that the fluid fire, as well as the fluid air, is attracted by plants in their growth, and becomes consolidated with the other materials of which they are formed, and makes a great part of their substance; that when they come to be digested, and to undergo a kind of fermentation in the vessels, part of the fire, as well as part of the air, recovers its fluid active state again, and diffuses itself on the body digesting and separating it; &c. Exper. and Obs. on Electricity, p. 346.</p><p>Dr. Mortimer thinks the Heat of animals explicable from the phosphorus and air they contain. Phosphorus exists, at least in a dormant state, in animal fluids; and it is also known that they all contain air; it is therefore only necessary to bring the phosphoreal and aereal particles into contact, and Heat must of consequence be generated; and were it not for the quantity of aqueous humours in animals, fatal accensions would frequently happen. Philos. Trans. number 476.</p><p>Dr. Black supposes, that animal Heat is generated<cb/> altogether in the lungs, by the action of the air on the principle of inflammability, and is thence diffused over the rest of the body by means of the circulation. But Dr. Leslie urges several arguments against this hypothesis, tending to shew that it is repugnant to the known laws of the animal machine; and he advances another hypothesis instead of it, viz, that the subtle principle by chemists termed phlogiston, which enters into the composition of natural bodies, is in consequence of the actio<*> of the vascular system gradually evol<*>ed through every part of the animal machine, and that during this evolution Heat is generated. This opinion, he candidly acknowledges, was first delivered by Dr. Duncan of Edinburgh; and that something similar to it is to be found in Dr. Franklin's works, and in a paper of Dr. Mortimer's in the Philos. Trans.</p><p>The last hypothesis we shall mention, is the very plausible one of Dr. Crawford, lately published in his Experiments and Observations on Animal Heat. This isgenious gentleman has inferred, from a variety of experiments, that Heat and phlogiston, so far from being connected, as most philosophers have imagined, act in some measure in opposition to each other. By the action of Heat on bodies, the force of their attraction of phlogiston is diminished, and by the action of phlogiston, a part of their absolute Heat is expelled. He has also demonstrated, that atmospherical air contains a greater quantity of absolute Heat than the air which is expired from the lungs of animals: he makes the proportion of the absolute Heat of atmospherical air to that of fixed air, as 67 to 1; and the Heat of dephlogisticated air to that of atmospherical air as 4.6 to 1; and observing that Dr. Priestley has proved, that the power of this dephlogisticated air in supporting animal life, is 5 times as great as that of atmospherical air, he concludes that the quantity of absolute Heat contained in any kind of air fit for respiration, is very nearly in proportion to its purity, or to its power of supporting animal life; and since the air exhaled by respiration, is found to contain only the 67th part of the Heat which was contained in the atmospherical air, previous to inspiration, it is very reasonably inferred, that the latter must necessarily deposit a very great proportion of its absolute Heat in the lungs. Dr. Crawford has also shewn, that the blood which passes from the lungs to the heart, by the pulmonary vein, contains more absolute Heat than that which passes from the heart to the lungs, by the pulmonary artery; the absolute Heat of florid arterial blood being to that of venous blood, as 11 1/2 to 10: therefore, since the blood which is returned by the pulmonary vein to the heart has the quantity of its absolute Heat increased, it must have acquired this Heat in its passage through the lungs; so that in the process of respiration a quantity of absolute Heat is separated from the air, and absorbed by the blood. Dr. Priestley has also proved, that in respiration, phlogiston is separated from the blood, and combined with air.</p><p>This theory however has been contested and disputed, and, it has been said, Dr. Crawford's experiments repeated, with contrary results. Though no regular and systematical theory has yet been formed in its stead.</p><p><hi rend="smallcaps">Heat</hi> <hi rend="italics">of Combustible and Inflammable Bodies.</hi> Dr. Crawford's theory with respect to the inflammation of<pb n="591"/><cb/> combustible bodies, is founded on the same principles as his doctrine concerning the Heat of animals. According to him, the Heat which is produced by combustion, is derived from the air, and not from the inflammable body. Inflammable bodies, he says, abound with phlogiston, and contain little absolute Heat: the atmosphere, on the contrary, abounds with absolute Heat, and contains little phlogiston. In the process of inflammation, the phlogiston is separated from the inflammable body, and combined with the air; the air is phlogisticated, and gives off a great proportion of its absolute Heat, which, when extricated suddenly, bursts forth into flame, and produces an intense degree of sensible Heat. And since it appears by calculation, that the Heat produced by converting atmospherical into fixed air, is such, if it were not dissipated, as would be sufficient to raise the air so changed, to more than 12 times the Heat of red-hot iron, it follows, that in the process of inflammation a very great quantity of Heat is derived from the air. But, on the contrary, no part of the Heat can be derived from the combustible body; because this body, during the inflammation, being deprived of its phlogiston, undergoes a change similar to that of the blood by the process of respiration, in consequence of which its capacity of containing Heat is increased; and therefore it will not give off any part of its absolute Heat, but, like the blood in its passage through the lungs, it will absorb Heat.</p><p>A similar theory of Heat has lately been published by Mr. Elliot. See his Philosophical Observations on the senses of Vision and Hearing; to which is added an Essay on Combustion and Animal Heat. 8vo, 1780.</p><p><hi rend="smallcaps">Heat</hi>, <hi rend="italics">in Geography,</hi> is that which relates to the earth. There is a great variety in the Heat of different places and seasons. Naturalists have commonly laid it down, that the nearer any place is to the centre of the earth, the hotter it is found; but this does not hold strictly true. And if it were, the effect might be otherwise accounted for, and more satisfactorily, than from their imagined central fire.</p><p>Mr. Boyle, who had been at the bottom of some mines himself, with more probability fuspects that this degree of Heat, at least in some of them, may arise from the peculiar nature of the minerals there produced. And he instances a mineral of the vitriolic kind, dug up in large quantities, in several parts of England, which, by the bare effusion of common water, will grow so hot as almost to take fire. To which may be added, that such places, in the bowels of the earth, usually feel hot, from the confined and stagnant state of the air in them, in which the heat is retained, through the want of a current or change of air to carry the Heat off.</p><p>On the other hand, on ascending high mountains, the air grows more and more cold and piercing. Thus, the tops of the Pike of Teneriffe, the Alps, and several other mountains, even in the most sultry countries, are found always invested with snow and ice, which the Heat is never sufficient to thaw. In some of the mountains of Peru there is no such thing as running water, but all ice: plants vegetate a little about the bottom of the mountains, but near the top no vegetable can live, for the intenseness of the cold. This effect is attributed to<cb/> the thinness of the air, and the little surface of the earth there is to reflect the rays, as well as the great distance of the general surface of the earth which reflects the rays back into the atmosphere.</p><p>As to the diversity in the Heat of different climes and seasons, it arises <*>rom the different angles under which the sun's rays strike upon the surface of the earth. In the Philos. Trans. Abr. vol. 2, p. 165, Dr. Halley has given a computation of this Heat, on the principle, that the simple action of the sun's rays, like other impulses or strokes, is more or less forcible, according to the <*>ines of the angles of incidence, or to the sines of the sun's altitudes, at different times or places.</p><p>Hence it follows, that, the time of continuance, or the sun's shining on any place, being taken for a basis, and the sines of the sun's altitudes perpendicularly erected upon it, and a curve line drawn through the extremities of those perpendiculars, the area thus comprehended will be proportional to the collection of all the Heat of the sun's beams in that space of time.</p><p>Hence it will likewise follow, that at the pole, the collection of all the Heat of a tropical day, is proportional to the rectangle or product of the fine of 23 1/2 degrees into 24 hours, or the circumference of a circle, or as <*>/10 into 12 hours, the sine of 23 1/2 degrees being nearly 4/10 of radius. Or the polar Heat will be equal to that of the sun continuing 12 hours above the horizon at 53 degrees height; and the sun is not 5 hours more elevated than this under the equinoctial.</p><p>But as it is the nature of Heat to remain in the subject, after the luminary is removed, and particularly in the air, under the equinoctial the 12 hours absence of the sun abates but little from the effect of his Heat in the day; but under the pole, the long absence of the sun for 6 months has so chilled the air, that it is in a manner frozen, and after the sun has risen upon the pole again, it is long before his beams can make any impression, being obstructed by thick clouds and fogs.</p><p>From the foregoing principle Dr. Halley computes the following table, exhibiting the Heat to every 10th degree of latitude, for the equinoctial and tropical sun, and from which an estimate may easily be made for the intermediate degrees. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Lat.</cell><cell cols="1" rows="1" rend="colspan=3 align=center" role="data">Sign that the Sun is in.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><*> or <*></cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data"><*></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">20000</cell><cell cols="1" rows="1" role="data">18341</cell><cell cols="1" rows="1" role="data">18341</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">19696</cell><cell cols="1" rows="1" role="data">20290</cell><cell cols="1" rows="1" role="data">15834</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">18794</cell><cell cols="1" rows="1" role="data">21737</cell><cell cols="1" rows="1" role="data">13166</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">17321</cell><cell cols="1" rows="1" role="data">22651</cell><cell cols="1" rows="1" role="data">10124</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">15321</cell><cell cols="1" rows="1" role="data">23048</cell><cell cols="1" rows="1" role="data">6944</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">12855</cell><cell cols="1" rows="1" role="data">22991</cell><cell cols="1" rows="1" role="data">3798</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">10000</cell><cell cols="1" rows="1" role="data">22773</cell><cell cols="1" rows="1" role="data">1075</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">6840</cell><cell cols="1" rows="1" role="data">23543</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">3473</cell><cell cols="1" rows="1" role="data">24673</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">25055</cell><cell cols="1" rows="1" role="data">0</cell></row></table></p><p>From the same principles, and table, also are deduced the following corollaries, viz,<pb n="592"/><cb/></p><p>1, That the equatorial Heat, when the sun becomes vertical, is as twice the square of the radius.—2, That at the equator the Heat is as the sine of the sun's declination.—3, That in the frigid zones, when the sun sets not, the Heat is as the circumference of a circle into the sine of the altitude at 6: And consequently that in the same latitude, these aggregates of Heat are as the fines of the sun's declination; and at the same declination of the sun, they are as the sines of the latitudes; and generally they are as the sines of the latitudes into the sines of declination.—4, That the equatorial day's Heat is everywhere as the cosine of the latitude.—5, In all places where the sun sets, the difference between the summer and winter Heats, when the declinations are contrary, is equal to a circle into the sine of the altitude at 6, in the summer parallel; and consequently those differences are as the rectangles of the sines of the latitude and declination. —6, The tropical sun has the least force of any at the equator; and at the pole it is greatest of all.</p><p>Many objections have been urged against this theory of Dr. Halley. Some have objected, that the effect of the sun's Heat is not in the simple, but in the duplicate ratio of the sines of the angles of incidence; like the law of the impulse of fluids. And indeed, the quantity of the sun's direct rays received at any place, being evidently as the sine of the angle of incidence, or of the sun's altitude, <hi rend="italics">if</hi> the Heat be also proportional to the force with which a ray strikes, like the mechanical action or impulse of any body, then it will follow that the Heat must be in the compound ratio of both, that is, as the square of the sine of the sun's altitude. But this last principle is here only assumed gratis, as we do not know a priori that the Heat is proportional to the force of a striking body; and it is only experiment that can determine this point.</p><p>It is certain that Heat communicated by the sun to bodies on the earth, depends also much upon other circumstances beside the direct force of his rays. These must be modified by our atmosphere, and variously reflected and combined by the action of the earth's surface itself, to produce any remarkable effects of Heat. So that if it were not for these additional circumstances, it is probable the naked Heat of the sun would not be very sensible.</p><p>Dr. Halley himself was well apprised, that many other circumstances, besides the direct force of the sun's rays, contributed to augment or diminish the effect of this, and the Heat resulting from it, in different climates; and therefore no calculation, formed on the preceding theory, can be supposed to correspond exactly with observation and experiment. It has also been objected that, according to the foregoing theory, the greatest Heat in the same place should be at the summer solstice, and the most extreme cold at the winter solstice; which is contrary to experience. To this objection it may be replied, that Heat is not produced in bodies by the sun instantaneously, nor do the effects of his Heat cease immediately when his rays are withdrawn; and therefore those parts which are once heated, retain the Heat for some time; which, with the additional Heat daily imparted, makes it continue to increase, though the sun declines from us: and this is the reason why July is hotter than June, although the sun has<cb/> withdrawn from the summer tropic: as we also sind it is generally hotter at one, two, or three in the afternoon, when the sun has declined towards the west, than at noon, when he is on the meridian. As long as the heating particles, which are constantly received, are more numerous than those which fly away or lose their force, the Heat of bodies must continually increase. So, after the sun has left the tropic, the number of particles, which Heat our atmosphere and earth, constantly increases, because we receive more in the day than we lose at night, and therefore our Heat must also increase. But as the days decrease again, and the action of the sun becomes weaker, more particles will fly off in the night time than are received in the day, by which means the earth and air will gradually cool. Farther, those places which are well cooled, require time to be heated again; and therefore January is mostly colder than December, although the sun has withdrawn from the winter tropic, and begun to emit his rays more perpendicularly upon us.</p><p>But the chief cause of the difference between the Heat of summer and winter is, that in summer the rays fall more perpendicularly, and pass through a less dense part of the atmosphere; and therefore with greater force, or at least in greater number in the same place: and besides, by their long continuance, a much greater degree of Heat is imparted by day than can fly off by night.</p><p>For the calculations and opinions of several other philosophers on this head, see Keill's Astron. Lect. 8; Ferguson's Astron. chap. 10; Long's Astron. § 777; Memo. Acad. Scienc. 1719.</p><p>As to the temperature or Heat of our atmosphere, it may be observed that the mercury seldom falls under 16° in Fahrenheit's thermometer; but we are apt to reckon it very cold at 24°, and it continues coldish to 40° and a little above. However, such colds have been often known as bring it down to 0°, the beginning of the scale, or nearly the cold produced by a mixture of snow and salt, often near it, and in some places below it. Thus, the degree of the thermometer has been observed at various times and places as follows: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Places</cell><cell cols="1" rows="1" role="data">Latit.</cell><cell cols="1" rows="1" role="data">Year</cell><cell cols="1" rows="1" role="data">Thermom.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pensylvania</cell><cell cols="1" rows="1" role="data">40°  0′</cell><cell cols="1" rows="1" role="data">1732</cell><cell cols="1" rows="1" rend="align=right" role="data">5°</cell></row><row role="data"><cell cols="1" rows="1" role="data">Paris</cell><cell cols="1" rows="1" role="data">48 50</cell><cell cols="1" rows="1" role="data">1709 & 1710</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Leyden</cell><cell cols="1" rows="1" role="data">52 10</cell><cell cols="1" rows="1" role="data">1729</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" role="data">Utrecht</cell><cell cols="1" rows="1" role="data">52  8</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">London</cell><cell cols="1" rows="1" role="data">51 31</cell><cell cols="1" rows="1" role="data">1709 & 1710</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copenhagen</cell><cell cols="1" rows="1" role="data">55 43</cell><cell cols="1" rows="1" role="data">1709</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Upsal</cell><cell cols="1" rows="1" role="data">59 56</cell><cell cols="1" rows="1" role="data">1732</cell><cell cols="1" rows="1" rend="align=right" role="data">-1</cell></row><row role="data"><cell cols="1" rows="1" role="data">Petersburg</cell><cell cols="1" rows="1" role="data">59 56</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="align=right" role="data">-28</cell></row><row role="data"><cell cols="1" rows="1" role="data">Torneo</cell><cell cols="1" rows="1" role="data">65 51</cell><cell cols="1" rows="1" role="data">1736-7</cell><cell cols="1" rows="1" rend="align=right" role="data">-33</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hudson's Bay</cell><cell cols="1" rows="1" role="data">52 24</cell><cell cols="1" rows="1" role="data">1775</cell><cell cols="1" rows="1" rend="align=right" role="data">-37</cell></row></table></p><p>The middle temperature of our atmosphere is about 48°, being nearly a medium of all the seasons. The French make it somewhat higher, reckoning it equal to the cave of their royal observatory, or 53°. In cold countries, the air is found agreeable enough to the inhabitants while it is between 40 and 50°. In our climate we are best pleased with the heat of the air from 50 to 60° while in the hot countries the air is gene rally at a medium about 70°. With us, the air is not<pb n="593"/><cb/> reckoned warm till it arrives at about 64°, and it is very warm and sultry at 80°. It is to be noted that the foregoing observations are to be understood of the state of the air in the shade; for as to the Heat of bodies acted upon by the direct rays of the sun, it is much greater: thus, Dr. Martine found dry earth heated to above 120°; but Dr. Hales found a very hot sun-shine Heat in 1727 to be about 140°; and Musschenbroek once observed it so high as 150°; but at Montpelier the sun was so very hot, on one day in the year 1705, as to raise M. Amontons's thermometer to the mark of boiling water itself, which is our 212°.</p><p>It appears from the register of the thermometer kept at London by Dr. Heberden for 9 years, viz, from the end of 1763 to the end of 1772, that the mean Heat at 8 in the morning was 47°.4; and by another register kept at Hawkhill, near Edinburgh, that the mean Heat in that place, during the same period of time, was 46°. Also by registers kept in London and at Hawkhill, for the three years 1772, 1773, 1774, it appears, that the mean Heat of these three years in London, at 8 in the morning, was 48°.5, and at 2 in the afternoon 56°, but the mean of both morning and afternoon 52°.2; while the mean Heat at Hawkhill for the same time, <table><row role="data"><cell cols="1" rows="1" role="data">at 8 in the morning was</cell><cell cols="1" rows="1" rend="align=right" role="data">45°.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">and at 2 in the afternoon</cell><cell cols="1" rows="1" rend="align=right" role="data">50.1</cell></row><row role="data"><cell cols="1" rows="1" role="data">and the mean of both</cell><cell cols="1" rows="1" rend="align=right" role="data">47.7.</cell></row></table> The mean Heat of springs near Edinburgh seems to be 47°, and at London 51°. Philos. Trans. vol. 65, art. 44.</p><p>Lastly, from the meteorological journals of the Royal Society, published in the Philos. Trans. it appears that the mean heights of the thermometer, for the whole years, kept without and within the house, are as below: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Therm. Without</cell><cell cols="1" rows="1" role="data">Therm. Within</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">For 1775  </cell><cell cols="1" rows="1" role="data">51.5</cell><cell cols="1" rows="1" role="data">52.7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1776  </cell><cell cols="1" rows="1" role="data">51.1</cell><cell cols="1" rows="1" role="data">52.9</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1777  </cell><cell cols="1" rows="1" role="data">51.0</cell><cell cols="1" rows="1" role="data">53.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1778  </cell><cell cols="1" rows="1" role="data">52.0</cell><cell cols="1" rows="1" role="data">53.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">mean of all</cell><cell cols="1" rows="1" role="data">51.4</cell><cell cols="1" rows="1" role="data">52.9</cell></row></table></p></div1><div1 part="N" n="HEAVEN" org="uniform" sample="complete" type="entry"><head>HEAVEN</head><p>, an azure transparent orb investing our earth, where the celestial bodies perform their motions. It is of divers denominations, as the highest or empyrean Heaven, the etherial or starry Heaven, the planetary Heaven, &c.</p><p>Formerly the Heavens were considered as solid substances, or else as spaces full of solid matter; but Newton has abundantly shewn that the Heavens are void of almost all resistance, and consequently of almost all matter: this he proves from the phenomena of the celestial bodies; from the planets persisting in their motions, without any sensible diminution of their velocity; and the comets freely passing in all directions towards all parts of the Heavens.</p><p>Heaven, taken in this general sense, or the whole expanse between our earth and the remotest regions of the fixed stars, may be divided into two very unequal parts, according to the matter occupying them; viz, the atmosphere or aereal Heaven, possessed by air; and the ethereal Heaven, possessed by a thin and unresisting medium, called ether.<cb/></p><p><hi rend="smallcaps">Heaven</hi> is more particularly used, in Astronomy, for an orb, or circular region, of the ethereal Heaven.</p><p>The ancient astronomers assumed as many different Heavens as they observed different celestial motions. All these they made solid, thinking they could not otherwise sustain the bodies fixed in them; and of a spherical form, as being the most proper for motion. Thus they had seven Heavens for the seven planets; viz, the Heavens of the Moon, Mercury, Venus, the Sun, Mars, Jupiter, and Saturn. The 8th was for the fixed stars, which they particularly called the firmament. Ptolomy added a 9th Heaven, which he called the primum mobile. After him two crystalline Heavens were added by king Alphonsus, &c, to account for some irregularities in the motions of the other Heavens. And lastly an empyrean Heaven was drawn over the whole for the residence of the Deity; which made the number 12.</p><p>But others admitted many other Heavens, according as their different views and hypotheses required. Eudoxus supposed 23, Calippus 30, Regiomontanus 33, Aristotle 47, and Fracastor no less than 70.</p><p>The astronomers however did not much concern themselves whether the Heavens they thus allowed, were real or not; provided they served a purpose in accounting for any of the celestial motions, and agreed with the phenomena.</p></div1><div1 part="N" n="HEAVINESS" org="uniform" sample="complete" type="entry"><head>HEAVINESS</head><p>, the same as Gravity, which see.</p><p>Heavy bodies do not tend precisely to the very centre of the earth, except at the poles, and the equator, on account of the spheroidal figure of the earth; their direction being every where perpendicular to the surface of the spheroid.</p></div1><div1 part="N" n="HEGIRA" org="uniform" sample="complete" type="entry"><head>HEGIRA</head><p>, a term in Chronology, signifying the epoch, or account of time, used by the Mahomedans, who begin their computation from the day that Mahomet was forced to make his escape from the city of Mecca, which happened on Friday the 16th of July 622.</p><p>The years of the Hegira are lunar ones, consisting only of 354 days. Hence, to reduce these years to the Julian calendar, that is, to find what Julian year a given year of the Hegira answers to: reduce the year of the Hegira into days, by multiplping by 354, divide the product by 365 1/4, and to the quotient add 622, the year the Hegira commenced.</p></div1><div1 part="N" n="HEIGHT" org="uniform" sample="complete" type="entry"><head>HEIGHT</head><p>, the third dimension of a body, considered with regard to its elevation above the ground.</p><p><hi rend="smallcaps">Height</hi>, of a figure, the same as its altitude, being the perpendicular from its vertex to the base.</p><p><hi rend="smallcaps">Height</hi> of the Pole, &c. See Altitude of the Pole, &c.</p></div1><div1 part="N" n="HELIACAL" org="uniform" sample="complete" type="entry"><head>HELIACAL</head><p>, something relating to the sun. Thus,</p><p><hi rend="smallcaps">Heliacal</hi> <hi rend="italics">Rising,</hi> of a star or planet, is when it rises with, or at the same time, as the sun. And Heliacal setting, the same as the setting with the sun.</p><p>Or, a star rises Hcliacally, when, after it has been in conjunction with the sun, and so invisible, it gets at such a distance from him as to be seen in the morning before the sun's rising. And it is said to set Heliacally, when it approaches so near the sun as to be hid by his beams. So that, in strictness, the Heliacal rising and setting are only an apparition and occultation.</p><p>HELICE <hi rend="italics">Major</hi> and <hi rend="italics">Minor;</hi> the same as Ursa Major and Minor.<pb n="594"/><cb/></p><p>HELICOID <hi rend="italics">Parabola,</hi> or the <hi rend="italics">Parabolic Spiral,</hi> is a curve arising from the supposition that the common or Apollonian parabola is bent or twisted, till the axis come into the periphery of a circle, the ordinates still retaining their places and perpendicular positions with respect to the circle, all these lines still remaining in the same plane. Thus, the axis of a parabola being bent into the circumference BCDM, and the ordinates CF, DG, &c, still perpendicular to it, then the parabola itself, passing through the extremities of the ordinates, is twisted into the curve BFG, &c, called the Helicoid, or Parabolic Spiral. <figure/></p><p>Hence all the ordinates CF, DG, &c, tend to the <*>entre of the circle, being perpendicular to the circumference</p><p>Also, the equation of the curve remains the same as when it was a parabola; viz, putting <hi rend="italics">x</hi> = any circular absciss BC, and <hi rend="italics">y</hi> = CF the corresponding ordinate, then is , where <hi rend="italics">p</hi> is the parameter of the parabola.</p><p>HELIOCENTRIC <hi rend="italics">Place of a Planet,</hi> is the place in which a planet would appear to be when viewed from the sun; or the point of the ecliptic, in which a planet viewed from the sun would appear to be. And therefore the Heliocentric place coincides with the longitude of a planet viewed from the sun.</p><p><hi rend="smallcaps">Heliocentric</hi> <hi rend="italics">Latitude of a Planet,</hi> is the inclination of the line drawn between the centre of the sun and the centre of a planet, to the plane of the ecliptic. The greatest Heliocentric Latitude is equal to the inclination of the planet's orbit to the plane of the ecliptic.</p></div1><div1 part="N" n="HELIOCOMETES" org="uniform" sample="complete" type="entry"><head>HELIOCOMETES</head><p>, Comet of the Sun, a phenomenon sometimes observed at the setting of the sun; thus denominated by Sturmius and Pylen, who had seen it, because it seems to make a comet of the sun, being a large tail, or column of light, fixed or hung to that luminary, and dragging after it, at its setting, like the tail of a comet.</p></div1><div1 part="N" n="HELIOMETER" org="uniform" sample="complete" type="entry"><head>HELIOMETER</head><p>, or <hi rend="smallcaps">Astrometer</hi>; an instrument for measuring, with particular exactness, the diameters of the sun, moon, and stars.</p><p>This instrument was invented by M. Bouguer in 1747, and is a kind of telescope, consisting of two object glasses of equal focal distance, placed by the side of each other, so that the same eye-glass serves for both. The tube of this instrument is of a conical form, larger at the upper end, which receives the two object-glasses, than at the lower, which is furnished with an eye-glass and micrometer. By the construction of this instrument,<cb/> two distinct images of an object are formed in the focus of the eye-glass, whose distance, depending on that of the two object-glasses from one another, may be measured with great accuracy. Mem. Acad. Sci. 1748.</p><p>Mr. Servington Savery discovered a similar method of improving the micrometer, which was communicated to the Royal Society in 1743.</p></div1><div1 part="N" n="HELIOSCOPE" org="uniform" sample="complete" type="entry"><head>HELIOSCOPE</head><p>, a kind of telescope peculiarly adapted for viewing and observing the sun without hurting the eye.</p><p>There are various kinds of this instrument, usually made by employing coloured glass for the object or eyeglass, or both; and sometimes only using an eye-glass blacked by holding it over the smoke or flame of a lamp or candle, which is Huygens's way.—See Dr<*> Hooke's treatise on Helioscopes.</p></div1><div1 part="N" n="HELIOSTATA" org="uniform" sample="complete" type="entry"><head>HELIOSTATA</head><p>, an instrument invented by Dr. Gravesande, and so called from its property of fixing the sun-beam in one position, viz, in a horizontal direction across the dark chamber while it is used. See Gravesande's Physices Element. Mathematica, tom. 2, p. 715 ed. 3tia 1742, for an account of the principles, construction and use of this instrument.</p><p>HELISPHERICAL <hi rend="italics">Line,</hi> is the Rhumb-line in Navigation; being so called, because on the globe it winds round the pole helically or spirally, coming still nearer and nearer to it.</p></div1><div1 part="N" n="HELIX" org="uniform" sample="complete" type="entry"><head>HELIX</head><p>, a Spiral line. See <hi rend="smallcaps">Spiral</hi>.</p></div1><div1 part="N" n="HEMISPHERE" org="uniform" sample="complete" type="entry"><head>HEMISPHERE</head><p>, the half of a sphere or globe, when divided in two by a plane passing through its centre.</p><p>Hemisphere is also used for a map or projection of half the terrestrial globe, or of half the celestial spher<*>, on a plane; being more frequently called a planisphere.</p><p>The centre of gravity of a Hemisphere, is five-eighths of the radius distant from the vertex.</p><p>A glass Hemisphere unites the parallel rays at the distance of four-thirds of a diameter from the pole of the glass.</p></div1><div1 part="N" n="HEMITONE" org="uniform" sample="complete" type="entry"><head>HEMITONE</head><p>, in Music, a half note.</p></div1><div1 part="N" n="HENDECAGON" org="uniform" sample="complete" type="entry"><head>HENDECAGON</head><p>, a figure of eleven sides, or the Endecagon; which see.</p></div1><div1 part="N" n="HENIOCHAS" org="uniform" sample="complete" type="entry"><head>HENIOCHAS</head><p>, or <hi rend="smallcaps">Heniochus</hi>, a northern constellation, the same as Auriga, which see.</p></div1><div1 part="N" n="HEPTAGON" org="uniform" sample="complete" type="entry"><head>HEPTAGON</head><p>, in Geometry, a figure of seven sides and seven angles.—When those sides and angles are all equal, the Heptagon is said to be regular, otherwise it is irregular. <figure/></p><p>In a regular Heptagon, the angle Cat the centre is = 51° 3/7, the angle DAB of the polygon is = 128° 4/7, and its half CAB = 64° 2/7. Also the area is = the square of the side or , where <hi rend="italics">t</hi> is the tangent of the angle CAB of 64° 2/7 to the radius 1; or <hi rend="italics">t</hi> is the root of the equation; ; or ,<pb n="595"/><cb/> where the value of <hi rend="italics">x</hi> and <hi rend="italics">y</hi> are the roots of the equations .</p><p>See my Mensuration, p. 21, 114, and 116, 2d edition.</p><div2 part="N" n="Heptagon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Heptagon</hi></head><p>, in Fortification, a place fortified or strengthened with seven bastions for its defence.</p><p><hi rend="smallcaps">Heptagonal</hi> <hi rend="italics">Numbers,</hi> are a kind of polygonal numbers in which the difference of the terms of the corresponding arithmetical progression is 5. Thus, <table><row role="data"><cell cols="1" rows="1" role="data">Arithmeticals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">6,</cell><cell cols="1" rows="1" role="data">11,</cell><cell cols="1" rows="1" role="data">16,</cell><cell cols="1" rows="1" role="data">21,</cell><cell cols="1" rows="1" role="data">26,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Heptagonals,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" role="data">18,</cell><cell cols="1" rows="1" role="data">34,</cell><cell cols="1" rows="1" role="data">55,</cell><cell cols="1" rows="1" role="data">81,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> where the Heptagonals are formed by adding continually the terms of the arithmeticals, above them, whose common difference is 5.</p><p>One property, among many others, of these Heptagonal numbers is, that if any one of them be multiplied by 40, and to the product add 9, the sum will be a square number. Thus ; and ; and ; and ; &c. Where it is remarkable that the series of squares so formed is 7<hi rend="sup">2</hi>, 17<hi rend="sup">2</hi>, 27<hi rend="sup">2</hi>, 37<hi rend="sup">2</hi>, &c, the common difference of whose roots is 10, the double of the common difference of the arithmetical series from which the Heptagonals are formed.—See <hi rend="smallcaps">Polygonals.</hi></p><p>HEPTANGULAR <hi rend="italics">Figure,</hi> in Geometry, is one that has seven angles; and therefore also seven sides.</p></div2></div1><div1 part="N" n="HERCULES" org="uniform" sample="complete" type="entry"><head>HERCULES</head><p>, in Astronomy, a constellation of the northern hemisphere, and one of the 48 old constellations mentioned by ancient writers.</p><p>It is not known by what name it was distinguished by the Egyptians and others before the Greeks. These latter, perhaps not knowing its real name, first called it simply the kneeling man, because he is drawn in that posture; but they afterwards successively ascribed it to, and called it by the names of, Cetheus, Theseus, and lastly Hercules, which it still retains.</p><p>The stars in this constellation, in Ptolomy's catalogue, are 29; in Tycho's 28; and in the Britannic catalogue, 113.</p></div1><div1 part="N" n="HERISSON" org="uniform" sample="complete" type="entry"><head>HERISSON</head><p>, in Fortification, a beam armed with iron spikes, having their points turned outward. It is supported in the middle by a stake, having a pivot on which it turns; and serves as a barrier to block up a passage.—Herissons are frequently placed before gates, especially the posterns of a town or sortress, to secure those passages which must of necessity be often opened.</p></div1><div1 part="N" n="HERMANN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HERMANN</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, a learned mathematician of the Academy of Berlin, and member of the Academy of Sciences at Paris, was born at Basil in 1678. He was a great traveller; and for 6 years was professor of mathematics at Padua. He afterwards went to Russia, being invited thither by the Czar in 1724, as well as his compatriot Daniel Bernoulli. On his return to his native country, he was appointed professor of morality and natural law at Basil; where he died in 1733, at 55 years of age.</p><p>He wrote several mathematical and philosophical pieces, in the Memoirs of different Academies, and elsewhere; but his principal work, is the <hi rend="italics">Phoronomia,</hi> or two Books on the Forces and Motions of both Solid and Fluid bodies; 4to, 1716: a very learned work on the new mathematical physics.<cb/></p></div1><div1 part="N" n="HERMETIC" org="uniform" sample="complete" type="entry"><head>HERMETIC</head><p>, or <hi rend="smallcaps">Hermetical</hi> <hi rend="italics">Art,</hi> a name given to chemistry, on a supposition that Hermes Trismegistus was its inventor.</p><p>HERMETICAL <hi rend="italics">Philosophy,</hi> is that which undertakes to solve and explain all the phenomena of nature, from the three chemical principles, salt, sulphur, and mercury.— —A considerable addition was made to the ancient Hermetical Philosophy, by the modern doctrine of alcali and acid.</p><p><hi rend="smallcaps">Hermetical</hi> <hi rend="italics">Seal,</hi> or <hi rend="italics">Hermetical Sealing,</hi> a manner of stopping or closing glass vessels, for chemical and other operations, so very closely, that no substance can possibly exhale or escape. This is usually done by heating the neck of the vessel in the slame of a lamp, with a blow-pipe, till it be ready to melt, and then with a pair of hot pincers twisting it close together.</p></div1><div1 part="N" n="HERSCHEL" org="uniform" sample="complete" type="entry"><head>HERSCHEL</head><p>, the name by which the French, and most other European nations, call the new planet, discovered by Dr. Herschel in the year 1781. Its mark or character is <*>. The Italians call it Ouranos, or Urania, but the English, the <hi rend="smallcaps">Georgian</hi> <hi rend="italics">Planet,</hi> which see.</p></div1><div1 part="N" n="HERSE" org="uniform" sample="complete" type="entry"><head>HERSE</head><p>, in Fortification, a lattice or portcullice, in the form of a harrow, beset with iron spikes, to block up a gate way, &c.</p></div1><div1 part="N" n="HERSILLON" org="uniform" sample="complete" type="entry"><head>HERSILLON</head><p>, or little Herse, in Fortification, is a plank armed with iron spikes, for the same use as the Herse, and also to impede the march of the infantry or cavalry.</p></div1><div1 part="N" n="HESSE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HESSE</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName> Prince of)</persName></head><p>, rendered his name immortal by his encouragement of learning, by his studies, and by his observations, for many years, of the celestial bodies. For this purpose, he erected an observatory at Cassel, and furnished it with good instruments, well adapted to that design; calling also to his assistance two eminent artists, Christopher Rothmann and Juste Byrge. His observations, which are of a very curious nature, were published at Leyden, in the year 1618, by Willebrord Snell; and are in part mentioned by Tycho Brahe, as well in his epistles as in the 2d volume of his <hi rend="italics">Progymnasmata;</hi> a signal example to all princely and heroic minds, to undertake the promoting the arts of peace, and advancing this truly noble and celestial science. This prince died in the year 1597.</p><p>HETERODROMUS <hi rend="italics">Vectis,</hi> or <hi rend="italics">Lever,</hi> in Mechanics, a lever in which the fulcrum, or point of suspension, is between the weight and the power; being the same as what is otherwise called a lever of the first kind.</p></div1><div1 part="N" n="HETEROGENEAL" org="uniform" sample="complete" type="entry"><head>HETEROGENEAL</head><p>, the same as H<hi rend="smallcaps">ETEROGENEOUS;</hi> which see.</p></div1><div1 part="N" n="HETEROGENEOUS" org="uniform" sample="complete" type="entry"><head>HETEROGENEOUS</head><p>, literally imports things of different natures, or something that consists of parts of different or dissimilar kinds; in opposition to Homogeneous. Thus,</p><p><hi rend="smallcaps">Heterogeneous</hi> <hi rend="italics">Bodies,</hi> are such as have their parts of unequal density.</p><p><hi rend="smallcaps">Heterogeneous</hi> <hi rend="italics">Line,</hi> is that which consists of parts or rays of different refrangibility, reflexibility, and colour.</p><p><hi rend="smallcaps">Heterogeneous</hi> <hi rend="italics">Numbers,</hi> are mixed numbers, consisting of integers and fractions.</p><p><hi rend="smallcaps">Heterogeneous</hi> <hi rend="italics">Particles,</hi> are such as are of different kinds, natures, and qualities; of which generally all bodies consist.</p><p><hi rend="smallcaps">Heterogeneous</hi> <hi rend="italics">Quantities,</hi> in Mathemati<*>s, are<pb n="596"/><cb/> those which cannot have proportion, or be compared together as to greater and less; being of such different kind and consideration, as that one of them taken any number of times, never equals or exceeds the other. As lines, surfaces, and solids in geometry.</p><p><hi rend="smallcaps">Heterogeneous</hi> <hi rend="italics">Surd<*>,</hi> are such as have different radical signs; as √<hi rend="italics">a</hi> and √<hi rend="sup">3</hi>(<hi rend="italics">b</hi><hi rend="sup">2</hi>); or √<hi rend="sup">5</hi>10 and √<hi rend="sup">7</hi>20.</p></div1><div1 part="N" n="HETEROSCII" org="uniform" sample="complete" type="entry"><head>HETEROSCII</head><p>, in Geography, are such inhabitants of the earth as have their shadows at noon projected al. ways the same way with regard to themselves, or always contrary ways with respect to each other. Thus, all the inhabitants without the torrid zone are Heteroscii, with regard to themselves, since any one such inhabitant has his shadow at noon always the same way, viz, always north of him in north latitude, and always south of him in south latitude; or these two situations are Heteroseii to each other, having such shadows projected contrary ways at all times of the year.</p></div1><div1 part="N" n="HEVELIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HEVELIUS</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a very celebrated astronomer, and a burgomaster of Dantzick, was born in that city in 1611. He studied mathematics under Peter Cruger, in which he made a wonderful progress. He afterwards spent several years on his travels through Holland, England, Germany, and France, for his improvement in the sciences. On his return, he constructed excellent telescopes himself, and began diligently to observe the heavens, an employment he closely followed during the course of a long life, which was terminated only in 1687, at 76 years of age. Hevelius was author of several notable discoveries in the heavens. He was the first that observed the phenomenon called the libration of the moon, and made several other important observations on the other planets. He also discovered several fixed stars, which he named the <hi rend="italics">firmament of Sobieski,</hi> in honour of John the 3d, king of Poland. He framed a large catalogue of the stars, and collected multitudes of the unformed ones into new constellations of his own framing. His wife was also well skilled in astronomy, and made a part of the observations that were published by her husband. His principal publications are, his <hi rend="italics">Selenographia,</hi> or an exact description of the moon; in which he has engraved all her phases, and remarkable parts, distinguished by names, and ascertained their respective bounds by the help of telescopes; containing also a delineation of the several visible spots, with the various motions, changes, and appearances, discovered by the telescope, as also in the sun and other planets, 1647.—In 1654, two epistles; one to the celebrated astronomer Ricciolt, concerning the Libration of the Moon; and the other to Bulliald, on the Eclipses of both luminaries.—In 1656, a Differtation. <hi rend="italics">De Natura Saturni faciei, &c.</hi>—In 1668, his <hi rend="italics">Cometogrophia,</hi> representing the whole nature of comets, their situation, parallaxes, distances, diverse appearances, and surprising motions, with a history of all the comets from the beginning of the world down to the present time; being enriched with curious sculpture of his own execution: to which he added a treatise on the planet Mercury, seen in the sun at Dantzick, May 3, 1661; with the history of a new star appearing in the neck of Cetus, and another in the beak of Cygnus; besides an Illustration of some astronomical discoveries of the late Mr. Horrox, in his treatise on Venus seen in the sun, Nov. 24, 1639; with a discourse of some curious Paras<*>lena and Parhelia observed at Dantzick. He sent copies of this work to<cb/> several members of the Royal Society at London, and among them to Mr. Hooke, in return for which, this gentleman sent to Hevelius a description of the Dioptric Telescope, with an account of the manner of using it; and recommending it to him, as much preferable to telescopes with plain sights. This gave rise to a dispute between them, viz, “whether distances and altitudes could be taken with plain sights any nearer than to a minute.” Hooke asserted that they could not; but that, with an instrument of a span radius, by the help of a telescope, they might be determined to the exactness of a second. Hevelius, on the other hand, insisted that, by the advantage of a good eye and long practice, he was able with his instruments to come up even to that exactness; and, appealing to experience and facts, sent by way of challenge 8 distances, each between two different stars, to be examined by Hooke. Thus the affair rested for some time with outward decency, but not without some inward grudge between the parties.</p><p>In 1673, Hevelius published the first part of his <hi rend="italics">Machina Cœlestis,</hi> as a specimen of the exactness both of his instruments and observations; and sent several copies as presents to his friends in England, but omitting Mr. Hooke. This, it is supposed, occasioned that gentleman to print, in 1674, <hi rend="italics">Animadversions on the First Part of the Machina Cœlestis;</hi> in which he treated Hevelius with a very magisterial air, and threw out several unhandsome reflections, which were greatly resented; and the dispute grew aftewards to such a height, and became so notorious, that in 1679 Dr. Halley went, at the request of the Royal Society, to examine both the instruments and the observations made with them. Of both these, Halley gave a favourable account, in a letter to Hevelius; and Hooke managed the controversy so ill, that he was universally condemned, though the preference has since been given to telescopic sights. However, Hevelius could not be prevailed on to make use of them: whether he thought himself too experienced to be informed by a young astronomer, as he considered Hooke; or whether, having made so many observations with plain sights, he was unwilling to alter his method, left he might bring their exactness into question; or whether, being by long practice accustomed to the use of them, and not thoroughly apprehending the use of the other, nor well understanding the difference, is uncertain. Besides Halley's letter, Hevelius received many others in his favour, which he took the opportunity of inserting among the astronomical observations in his <hi rend="italics">Annus Climactericus,</hi> printed in 1685. In a long preface to this work, he speaks with more confidence and greater indignation than he had done before, and particularly exclaimed against Hooke's dogmatical and magisterial manner of assuming a kind of dictatorship over him. This revived the contest, and occasioned several learned men to engage in it. The book itself being sent to the Royal Society, at their request an account of it was given by Dr. Wallis; who, among other things, took notice, that “Hevelius's observations had been misrepresented, since it appeared from this book, that he could distinguish by plain sights to a small part of a minute.” About the same time Mr. Molyneux also wrote a letter to the society, in vindication of Hevelius, against Hooke's animadversions. To all which, Hooke drew up a letter in answer, which was read before the society, containing many qualifying<pb n="597"/><cb/> and <*>accommodating expressions, but still at least expressing the superiority of telescopic sights over plain ones, excellent as the observations were that had been made with these.</p><p>In 1679, Hevelius had published the second part of his <hi rend="italics">Machina Cœlestis;</hi> but the same year, while he was at a seat in the country, he had the misfortune to have his house at Dantzic burnt down. By this calamity it is said he sustained several thousand pounds damage; having not only his observatory and all his valuable instruments and astronomical apparatus destroyed, but also a great many copies of his <hi rend="italics">Machina Cœlestis,</hi> an accident which has made this second part very scarce, and consequently very dear.</p><p>In 1690, were published a description of the heavens, called, <hi rend="italics">Firmamentum Sobiescianum,</hi> in honour of John the 3d, king of Poland, as above mentioned; and also <hi rend="italics">Prodromus Astronomiæ, & Novæ Tabulæ Solares, una cum Catalogo Fixarum;</hi> in which he lays down the necessary preliminaries for taking an exact catalogue of the stars.</p><p>But both these works were posthumous; for Hevelius died the 28th of January 1687, exactly 76 years of age, as above said, and universally admired and respected; abundant evidence of which appears in a collection of letters between him and many other persons, that was printed at Dantzic in 1683.</p></div1><div1 part="N" n="HEXACHORD" org="uniform" sample="complete" type="entry"><head>HEXACHORD</head><p>, a certain interval or musical concord, usually called a sixth.</p></div1><div1 part="N" n="HEXAEDRON" org="uniform" sample="complete" type="entry"><head>HEXAEDRON</head><p>, or <hi rend="smallcaps">Hexahedron</hi>, one of the five regular or Platonic bodies; being indeed the same as the cube; and is so called from its having 6 faces.— The square of the side or edge of a Hexahedron, is onethird of the square of the diameter of the circumscribing sphere: and hence the diameter of a sphere is to the side of its inscribed Hexahedron, as √3 to 1.</p><p>In general, if A, B, and C be put to denote respectively the linear side, the surface, and the solidity of a Hexahedron or cube, also <hi rend="italics">r</hi> the radius of the inscribed sphere, and R the radius of the circumscribed one; then we have these general equations or relations: 1. . 2. . 3. . 4. . 5. .</p></div1><div1 part="N" n="HEXAGON" org="uniform" sample="complete" type="entry"><head>HEXAGON</head><p>, in Geometry, a figure of six sides, and consequently of as many angles. When these are equal, it is a regular Hexagon.—The angles of a Hexagon are each equal to 120°, and its sides are each equal to the radius of its circumscribing circle. Hence a regular Hexagon is inscribed in a circle, by setting the radius off 6 times upon the periphery. And hence also, to describe a regular Hexagon upon a given line, describe an equilateral triangle upon it, the vertex of which will be the centre of the circumscribing circle.</p><p>The side of a Hexagon being <hi rend="italics">s,</hi> its area will be .</p></div1><div1 part="N" n="HEXASTYLE" org="uniform" sample="complete" type="entry"><head>HEXASTYLE</head><p>, in the Ancient Architecture, a building with 6 columns in front.</p><p>HIERO's <hi rend="italics">Crown,</hi> in Hydrostatics. The history of this crown, and of the important hydrostatical proposition which it gave occasion to, is as follows: Hiero,<cb/> king of Syracuse, having furnished a workman with a quantity of gold for making a crown, suspected that he had been cheated, by the workman using a greater alloy of silver than was necessary in making it; and he applied to Archimedes to discover the fraud, without defacing the crown.</p><p>This celebrated mathematician was led by chance to a method of detecting the imposture, and of determining precisely the quantities of gold and silver composing the crown: for he observed, when bathing in a tub of water, that the water ran over as his body entered it, and <*>e presently concluded that the quantity so running over was equal to the bulk of his body that was immersed. He was so pleased with the discovery, that it is said he ran about naked crying out, <foreign xml:lang="greek">eu(/rhxa, eu(/rhxa</foreign>, <hi rend="italics">I have found it;</hi> and some affirm that he offered a hetacomb to Jupiter for having inspired him with the thought.</p><p>On this principle he procured a ball or mass of gold, and another of silver, exactly of the same weight with the crown; considering, that, if the crown were of pure gold, it would be of equal bulk and expel an equal quantity of water as the golden ball; and if it were of silver, then it would be of equal bulk and expel an equal quantity of water with the ball of silver; but of intermediate quantity, if it consisted of a mixture of the two, gold and silver; which, upon trial, he found to be the case; and hence, by a comparison of the quantities of water displaced by the three masses, he discovered the exact portions of gold and silver in the crown.</p><p>Now, suppose, for example, that each of the three masses weighed 100 ounces; and that on immersing them severally in water, there were displaced 5 ounces of water by the golden ball, 9 ounces by the silver, and 6 ounces by the compound, or crown; that is, their respective or comparative bulks are as 5, 9, and 6, the sum of which is 20.</p><p>Then the method of operation is this: <table><row role="data"><cell cols="1" rows="1" role="data">From</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Táke</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" role="data">rem.</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1, whose sum is 4.</cell></row></table> Therefore oz. of gold, and oz. of silver. That is, the crown consisted of 75 ounces of gold, and 25 ounces of silver.</p><p>See Cotes Hydros. Lect. p. 81; or Martin's Philos. Britan, vol. 1, p. 305, &c. See also <hi rend="smallcaps">Specific</hi> <hi rend="italics">Gravity.</hi></p><p>HIGH-<hi rend="italics">Water,</hi> that state of the tides when they have flowed to their greatest height, or have ceased to flow or rise. At High-water the motion commonly ceases for a quarter or half an hour, before it begin to ebb again. The times of High-water of every day of the moon's age, is usually computed from that which is observed on the day of the full or change; viz, by taking 4-5ths of the moon's age on any day of the month, and adding it to the time of High-water on the day of the full or change; then is the sum nearly equal to the time of High-water on the day of the month proposed. And as to the times of High-water, on the day of the full and change of the moon, at many different places; they have been observed as they are set down in the followiug table.<pb n="598"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=8 align=center" role="data"><hi rend="italics"><hi rend="smallcaps">Table</hi> of the Times of High-water on the Days of the New and Full Moons, at many different Places.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Names of Places.</cell><cell cols="1" rows="1" role="data">Countries.</cell><cell cols="1" rows="1" role="data">High-w.</cell><cell cols="1" rows="1" role="data">Names of Places.</cell><cell cols="1" rows="1" role="data">Countries.</cell><cell cols="1" rows="1" role="data">High-w.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aberdeen</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">0h 45 m</cell><cell cols="1" rows="1" role="data">Dort</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">3h 0m</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aldborough</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell><cell cols="1" rows="1" role="data">Dover</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alderney I.</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">12  0</cell><cell cols="1" rows="1" role="data">Downs</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">1 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Amazons River</cell><cell cols="1" rows="1" role="data">South America</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Dublin</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">9 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Amsterdam</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Dunbar</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">2 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Amsterdam I. of</cell><cell cols="1" rows="1" role="data">South Seas</cell><cell cols="1" rows="1" rend="align=right" role="data">8 30</cell><cell cols="1" rows="1" role="data">Dundee</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">2 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Andrew's St.</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">2 15</cell><cell cols="1" rows="1" role="data">Dungarvan</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Anholt I.</cell><cell cols="1" rows="1" role="data">Denmark</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell><cell cols="1" rows="1" role="data">Dungeness</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Antwerp</cell><cell cols="1" rows="1" role="data">Flanders</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Dunkirk</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Archangel</cell><cell cols="1" rows="1" role="data">Russia</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Dunnose</cell><cell cols="1" rows="1" role="data">I. of Wight</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Arran I.</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">11  0</cell><cell cols="1" rows="1" role="data">Dusky Bay</cell><cell cols="1" rows="1" role="data">N. Zealand</cell><cell cols="1" rows="1" rend="align=right" role="data">10 57</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ashley Riv.</cell><cell cols="1" rows="1" role="data">Carolina</cell><cell cols="1" rows="1" rend="align=right" role="data">0 45</cell><cell cols="1" rows="1" role="data">Easter Isle</cell><cell cols="1" rows="1" role="data">Chili</cell><cell cols="1" rows="1" rend="align=right" role="data">2  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Augustine St.</cell><cell cols="1" rows="1" role="data">Florida</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell><cell cols="1" rows="1" role="data">Edystone</cell><cell cols="1" rows="1" role="data">English Channel</cell><cell cols="1" rows="1" rend="align=right" role="data">5 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bajador Ca.</cell><cell cols="1" rows="1" role="data">Negroland</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell><cell cols="1" rows="1" role="data">Elbe R.</cell><cell cols="1" rows="1" role="data">Germany</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Baltimore</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell><cell cols="1" rows="1" role="data">Embden</cell><cell cols="1" rows="1" role="data">Germany</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Barfleur Ca.</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell><cell cols="1" rows="1" role="data">Estaples</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">11  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bayonne</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3 30</cell><cell cols="1" rows="1" role="data">Falmouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Beachy-head</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell><cell cols="1" rows="1" role="data">Flamborough H.</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">4  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">N. and S. Bear</cell><cell cols="1" rows="1" role="data">Labradore</cell><cell cols="1" rows="1" rend="align=right" role="data">12  0</cell><cell cols="1" rows="1" role="data">C. Florida</cell><cell cols="1" rows="1" role="data">Florida</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Belfast</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">10  0</cell><cell cols="1" rows="1" role="data">Flushing</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">0 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bellisle</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3 30</cell><cell cols="1" rows="1" role="data">N. Foreland</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bermudas I.</cell><cell cols="1" rows="1" role="data">Bahama I.</cell><cell cols="1" rows="1" rend="align=right" role="data">7  0</cell><cell cols="1" rows="1" role="data">Foulness</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Berwick</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">2 30</cell><cell cols="1" rows="1" role="data">Fowey</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Blackney</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Fayal Isl.</cell><cell cols="1" rows="1" role="data">Azores</cell><cell cols="1" rows="1" rend="align=right" role="data">2 20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Blanco Cap.</cell><cell cols="1" rows="1" role="data">Negroland</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell><cell cols="1" rows="1" role="data">Garonne R.</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Blavet</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Gibraltar</cell><cell cols="1" rows="1" role="data">Spain</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bourdeaux</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">C. Good Hope</cell><cell cols="1" rows="1" role="data">Caffers</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Boulogne</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">10 30</cell><cell cols="1" rows="1" role="data">Goree (Isle)</cell><cell cols="1" rows="1" role="data">Atlantic Ocean</cell><cell cols="1" rows="1" rend="align=right" role="data">1 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bremen</cell><cell cols="1" rows="1" role="data">Germany</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Granville</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">7  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brest</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell><cell cols="1" rows="1" role="data">Gravelines</cell><cell cols="1" rows="1" role="data">Flanders</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bridlington B.</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell><cell cols="1" rows="1" role="data">Gravesend</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">1 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brill</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">1 30</cell><cell cols="1" rows="1" role="data">Groin</cell><cell cols="1" rows="1" role="data">Spain</cell><cell cols="1" rows="1" rend="align=right" role="data">3  3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bristol</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6 45</cell><cell cols="1" rows="1" role="data">Guernsey I.</cell><cell cols="1" rows="1" role="data">English Channel</cell><cell cols="1" rows="1" rend="align=right" role="data">1 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Buchaness</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Hague</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">8 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Button's Isles</cell><cell cols="1" rows="1" role="data">New Brit.</cell><cell cols="1" rows="1" rend="align=right" role="data">6 50</cell><cell cols="1" rows="1" role="data">Halifax</cell><cell cols="1" rows="1" role="data">Nova Scotia</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cadiz</cell><cell cols="1" rows="1" role="data">Spain</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell><cell cols="1" rows="1" role="data">Hamburgh</cell><cell cols="1" rows="1" role="data">Germany</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Caen</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell><cell cols="1" rows="1" role="data">Hare Isle</cell><cell cols="1" rows="1" role="data">Canada</cell><cell cols="1" rows="1" rend="align=right" role="data">3 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Calais</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">11 30</cell><cell cols="1" rows="1" role="data">Harlem</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Canaria I.</cell><cell cols="1" rows="1" role="data">Canaries</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Hartlepool</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">C. Cantin</cell><cell cols="1" rows="1" role="data">Barbary</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell><cell cols="1" rows="1" role="data">Harwich</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cape Town</cell><cell cols="1" rows="1" role="data">Caffers</cell><cell cols="1" rows="1" rend="align=right" role="data">2 30</cell><cell cols="1" rows="1" role="data">Havre de Grace</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Caskets</cell><cell cols="1" rows="1" role="data">Guernsey</cell><cell cols="1" rows="1" rend="align=right" role="data">8 15</cell><cell cols="1" rows="1" role="data">Holy Head</cell><cell cols="1" rows="1" role="data">Wales</cell><cell cols="1" rows="1" rend="align=right" role="data">1 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cathness Po.</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell><cell cols="1" rows="1" role="data">Honfleur</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Charles Town</cell><cell cols="1" rows="1" role="data">Carolina</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Hull</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Q. Charlotte's S.</cell><cell cols="1" rows="1" role="data">New Zealand</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell><cell cols="1" rows="1" role="data">Humber R.</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 13</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cherbourg</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell><cell cols="1" rows="1" role="data">St. John's</cell><cell cols="1" rows="1" role="data">Newfoundland</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Churchill R.</cell><cell cols="1" rows="1" role="data">Hudson's Bay</cell><cell cols="1" rows="1" rend="align=right" role="data">7 20</cell><cell cols="1" rows="1" role="data">St. Julian (Port)</cell><cell cols="1" rows="1" role="data">Patagonia</cell><cell cols="1" rows="1" rend="align=right" role="data">4 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ca. Cleare</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell><cell cols="1" rows="1" role="data">Kentishnock</cell><cell cols="1" rows="1" role="data">English coast</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Concarneau</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Kinsale</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">5 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Conquet</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">2 15</cell><cell cols="1" rows="1" role="data">Land's End</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Coquet Isle</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Leith</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Corke</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">6 30</cell><cell cols="1" rows="1" role="data">Leostoff</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">C. Corse</cell><cell cols="1" rows="1" role="data">Guinea</cell><cell cols="1" rows="1" rend="align=right" role="data">3 30</cell><cell cols="1" rows="1" role="data">Lisbon</cell><cell cols="1" rows="1" role="data">Portugal</cell><cell cols="1" rows="1" rend="align=right" role="data">2 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cromer</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">7  0</cell><cell cols="1" rows="1" role="data">Liverpool</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dartmouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6 30</cell><cell cols="1" rows="1" role="data">Lizard</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">St. David's H.</cell><cell cols="1" rows="1" role="data">Wales</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Loire (Riv.)</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dieppe</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">10 30</cell><cell cols="1" rows="1" role="data">London</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row></table><pb n="599"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Names of Places.</cell><cell cols="1" rows="1" role="data">Countries.</cell><cell cols="1" rows="1" role="data">High-w.</cell><cell cols="1" rows="1" role="data">Names of Places.</cell><cell cols="1" rows="1" role="data">Countries.</cell><cell cols="1" rows="1" role="data">High-w.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lundy (Isle)</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 h 15 m</cell><cell cols="1" rows="1" role="data">Sandwich</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 h 30m</cell></row><row role="data"><cell cols="1" rows="1" role="data">Madeira</cell><cell cols="1" rows="1" role="data">Atl. Ocean</cell><cell cols="1" rows="1" rend="align=right" role="data">12  4</cell><cell cols="1" rows="1" role="data">Scarborough H.</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">St. Maloes</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">Scilly Isles</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Isle of Man</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell><cell cols="1" rows="1" role="data">Senegal</cell><cell cols="1" rows="1" role="data">Negroland</cell><cell cols="1" rows="1" rend="align=right" role="data">10 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Margate</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 15</cell><cell cols="1" rows="1" role="data">Severn, (Mouth.)</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">St. Mary's (Isle)</cell><cell cols="1" rows="1" role="data">Scilly Isles</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell><cell cols="1" rows="1" role="data">Sheerness</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Milford</cell><cell cols="1" rows="1" role="data">Wales</cell><cell cols="1" rows="1" rend="align=right" role="data">5 15</cell><cell cols="1" rows="1" role="data">Sierra Leona</cell><cell cols="1" rows="1" role="data">Guinea</cell><cell cols="1" rows="1" rend="align=right" role="data">8 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mount's Bay</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell><cell cols="1" rows="1" role="data">Shetland I.</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Nantes</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Isle of Sky</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">5 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Naze</cell><cell cols="1" rows="1" role="data">Norway</cell><cell cols="1" rows="1" rend="align=right" role="data">11 15</cell><cell cols="1" rows="1" role="data">Spurn</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Needles</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">10 15</cell><cell cols="1" rows="1" role="data">Start Point</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Newcastle</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3 15</cell><cell cols="1" rows="1" role="data">Stockton</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Nieuport</cell><cell cols="1" rows="1" role="data">Flanders</cell><cell cols="1" rows="1" rend="align=right" role="data">12  0</cell><cell cols="1" rows="1" role="data">Sunderland</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3 20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Nore</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell><cell cols="1" rows="1" role="data">Tanna</cell><cell cols="1" rows="1" role="data">Pacific Ocean</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">North Cape</cell><cell cols="1" rows="1" role="data">Lapland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Teneriff</cell><cell cols="1" rows="1" role="data">Canaries</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Orfordness</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell><cell cols="1" rows="1" role="data">Texel (Isle)</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Orkneys</cell><cell cols="1" rows="1" role="data">Scotland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Thames Mouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">1 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ostend</cell><cell cols="1" rows="1" role="data">Flanders</cell><cell cols="1" rows="1" rend="align=right" role="data">12  0</cell><cell cols="1" rows="1" role="data">Tinmouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Placentia</cell><cell cols="1" rows="1" role="data">Newfoundland</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell><cell cols="1" rows="1" role="data">Torbay</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">5 15</cell></row><row role="data"><cell cols="1" rows="1" role="data">Plymouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell><cell cols="1" rows="1" role="data">St. Valery</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">10 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Portland</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">8 15</cell><cell cols="1" rows="1" role="data">Vannes</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Porto Praya</cell><cell cols="1" rows="1" role="data">Cape Verdes</cell><cell cols="1" rows="1" rend="align=right" role="data">11  0</cell><cell cols="1" rows="1" role="data">Ushant</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Portsmouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 15</cell><cell cols="1" rows="1" role="data">Waterford</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">6 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Quebec</cell><cell cols="1" rows="1" role="data">Canada</cell><cell cols="1" rows="1" rend="align=right" role="data">7 30</cell><cell cols="1" rows="1" role="data">Wells</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">6  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rhée (Isle)</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Weymouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">7 20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Resolution (Bay)</cell><cell cols="1" rows="1" role="data">Ohitahoo</cell><cell cols="1" rows="1" rend="align=right" role="data">2 30</cell><cell cols="1" rows="1" role="data">Whitby</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Robin Hood's B.</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">Isle of Wight</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">0  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rochefort</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">4 15</cell><cell cols="1" rows="1" role="data">Winchelsea</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">0 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rochelle</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">3 45</cell><cell cols="1" rows="1" role="data">Wintertoness</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rochester</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">0 45</cell><cell cols="1" rows="1" role="data">Yarmouth</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">9 45</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rotterdam</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell><cell cols="1" rows="1" role="data">New York</cell><cell cols="1" rows="1" role="data">America</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rouen</cell><cell cols="1" rows="1" role="data">France</cell><cell cols="1" rows="1" rend="align=right" role="data">1 15</cell><cell cols="1" rows="1" role="data">Youghall</cell><cell cols="1" rows="1" role="data">Ireland</cell><cell cols="1" rows="1" rend="align=right" role="data">4 30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rye</cell><cell cols="1" rows="1" role="data">England</cell><cell cols="1" rows="1" rend="align=right" role="data">11 15</cell><cell cols="1" rows="1" role="data">Zuric Sea</cell><cell cols="1" rows="1" role="data">Holland</cell><cell cols="1" rows="1" rend="align=right" role="data">3  0</cell></row></table><cb/></p></div1><div1 part="N" n="HIPS" org="uniform" sample="complete" type="entry"><head>HIPS</head><p>, in Architecture, are those pieces of timber placed at the corners of a roof. These are much longer than the rafters, because of their oblique position.</p><p><hi rend="smallcaps">Hip</hi> means also the angle formed by two parts of the roof, when it rises outwards.</p><p><hi rend="smallcaps">Hip</hi>-<hi rend="italics">Roof,</hi> called also Italian Roof, is one in which two parts of the roof meet in an angle, rising outwards: the same angle being called a valley when it sinks inwards.</p></div1><div1 part="N" n="HIPPARCHUS" org="uniform" sample="complete" type="entry"><head>HIPPARCHUS</head><p>, a celebrated astronomer among the ancients, was born at Nice in Bithynia, and flourished between the 154th and the 163d olympiads; that is, between 160 and 135 years before Christ; for in this space of time it is that his observations are dated. He is accounted the first, who from vague and scattered observations, reduced astronomy into a science, and prosecuted the study of it systematically. Pliny often mentions him, and always with great commendation. He was the first, he tells us, who attempted to count the number of the fixed stars; and his catalogue is preserved in Ptolomy's Almagest, where they are all noted according to their longitudes and apparent magnitudes. Pliny places him among those men of a sublime gen<*>us, who, by foretelling the eclipses, taught mankind, that they ought not to be frightened at these phenomena. Thales was the first among the Greeks, who could discover when there was to be an eclipse. Sulpitius Gallus among the Romans<cb/> began to succeed in this kind of prediction; and he gave an essay of his skill very seasonably, the day before a battle was fought. After these two, Hipparchus improved that science very much; making ephemerides, or catalogues of eclipses, for 600 years. He admires him for making a review of all the stars, acquainting us with their situations and magnitudes; for by these means, says he, posterity will be able to discover, not only whether they are born and die, but also whether they change their places, and whether they increase or decrease. He mentioned a new star which was produced in his days; and by its motion, at its first appearance, he began to doubt whether this did not frequently happen, and whether those stars, which we call fixed, do not likewise move. Hipparchus is also memorable for being the first who discovered the precession of the equinoxes, or a very slow apparent motion of the fixed stars from east to west, by which in a great number of years they will seem to have performed a complete revolution. He endeavoured also to reduce to rule the many discoveries he made, and invented new instruments, by which he marked their magnitudes and places in the heavens; so that by means of them it might be easily observed, not only whether they appear and disappear, but likewise whether they pass by one another, or move, and whether they increase or decrease.</p><p>The first observations he made, were in the isle of<pb n="600"/><cb/> Rhodes; whence he got the name Rhodius; but afterwards he cultivated this science in Bithynia and Alexandria only. One of his works is still extant, viz, his Commentary upon Aratus's Phenomena. He composed several other works; and upon the whole it is agreed, that astronomy is greatly indebted to him, for laying that rational and solid foundation, upon which all succeeding astronomers have since built their superstructure.</p></div1><div1 part="N" n="HIRCUS" org="uniform" sample="complete" type="entry"><head>HIRCUS</head><p>, in Astronomy, a fixed star of the first magnitude, the same with Capella.</p><p><hi rend="smallcaps">Hircus</hi> is also used by some writers for a comet, encompassed as it were with a mane, seemingly rough and hairy.</p></div1><div1 part="N" n="HIRE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HIRE</surname> (<foreName full="yes"><hi rend="smallcaps">Philip de la</hi></foreName>)</persName></head><p>, an eminent French mathematician and astronomer, was born at Paris in 1640. His father, who was painter to the king, intending him for the same occupation, taught him drawing and such branches of mathematics as relate to it: but died when the son was only 17 years of age. Three years after this, he travelled into Italy for improvement in that art, where he spent 4 years. He applied himself also to mathematics, which gradually engrossed all his attention. On his return to Paris, he continued his mathematical studies with great eagerness, and he afterwards published some works, which gained him so much reputation, that he was named a member of the Academy of Sciences in 1678.</p><p>The minister Colbert having formed a design for a better chart or map of France than any former ones, De la Hire was appointed, with Picard, to make the necessary observations for that purpose. This occupied him some years in several of the provinces; and, beside the main object of his peregrinations, he was not unmindful of other branches of knowledge, but philosophized upon every thing that occurred, and particularly upon the variations of the magnetic needle, upon refractions, and upon the height of mountains, as determined by the barometer.</p><p>In 1683, de la Hire was employed in continuing the meridian line, which Picard had begun in 1669. He continued it from Paris northward, while Cassini carried it on to the south: but Colbert dying the same year, the work was dropped before it was finished. De la Hire was next employed, with other members of the academy, in taking the necessary levels for the grand aqueducts, which Louis the 14th was about to make.</p><p>The great number of works published by our author, together with his continual employments, as professor of the Royal College and of the Academy of Architecture, give us some idea of the great labours he underwent. His days were always spent in study; his nights very often in astronomical observations; seldom seeking any other relief from his labours, than a change of one for another. In his manner, he had the exterior politeness, circumspection, and prudence of Italy; on which account he appeared too reserved in the eyes of his countrymen; though he was always esteemed as a very honest disinterested man. He died in 1718, at 78 years of age.</p><p>Of the numerous works which he published, the principal are, 1. Traité de Mechanique; 1665.— 2. Nouvelle Methode en Geometrie pour les Sections des Supersicies Coniques & Cylindriques; 1673, 4to.<cb/> —3. De Cycloide; 1677, 12mo.—4. Nouveaux Elemens des Sectiones Coniques: les Lieux Geometriques: la Construction, ou Effection des Equations; 1678, 12mo.—5. La Gnomonique, &c; 1682. 12mo.—6. Traité du Nivellement de M. Picard, avec des additions; 1684. —7. Sectiones Conicæ in novem libros distributæ; 1685, folio. This was considered as an original work, and gained the author great reputation all over Europe —8. Traité du Mouvement des Eaux, &c; 1686.— 9. Tabulæ Astronomicæ; 1687 and 1702, 4to.—10. Ecole des Arpenteurs; 1689.—11. Veterum Mathematicorum Opera, Græcè & Latinè, pleraque nunc primùm edita; 1693, folio. This edition had been begun by Thevenot; who dying, the care of finishing it was committed to de la Hire. It shews that our author's strong application to mathematical and astronomical studies had not hindered him from acquiring a very competent knowledge of the Greek tongue. Beside these, and other smaller works, there are a vast number of his pieces scattered up and down in Journals, and particularly in the Memoirs of the Academy of Sciences, viz, from 1666 till the year 1718.</p></div1><div1 part="N" n="HOBBES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HOBBES</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>)</persName></head><p>, a famous writer and philopher, was born at Malmsbury in Wiltshire, in 1588, being the son of a clergyman of that place. He completed his studies at Oxford, and was afterwards governor to the eldest son of William Cavendish earl of Devonshire, with whom he travelled through France and Italy, applying himself closely to the study of polite literature. In 1626 his patron the earl of Devonshire died; and 1628 his son also; the same year Mr. Hobbes published his translation of Thucydides in English. He soon after went abroad a second time as governor to the son of Sir Gervase Clifton; but shortly after returned, to resume his concern for the hopes of the Devonshire family, to whom he had so early attached himself; the countess dowager having desired to put the young earl under his care, then about 13 years of age. This charge was very agreeable to Mr. Hobbes's inclinations, and he discharged the trust with great diligence and fidelity. In 1634 he accompanied his young pupil to Paris, where he employed his own vacant hours in the study of natural philosophy, frequently conversing with Father Mersenne, Gassendi, and other eminent philosophers there. From Paris he attended his pupil into Italy, where he became acquainted with the celebrated Galileo, who freely communicated his notions to him; and from hence he returned with his ward into England. But afterwards, f<*>reseeing the civil wars, he went to seek a retreat at Paris; where he was soon made acquainted with Des Cartes and the other learned philosophers there, with whom he afterwards held a correspondence upon several mathematical subjects, as appears from the letters of Mr. Hobbes published in the works of Des Cartes.</p><p>In 1642, Mr. Hobbs printed his famous book <hi rend="italics">De Cive,</hi> which raised him many adversaries, who charged him with instilling principles of a dangerous tendencyAmong many illustrious persons who, from the troubles in England, retired to France for safety, was Sir Charles Cavendish, brother to the Duke of Newcastle <*> and this gentleman, being well skilled in the mathematics, proved a constant friend and patron to Mr. Hobbes; who, by embarking in 1645 in a controversy<pb n="601"/><cb/> about squaring the circle, was grown so famous by it, that in 1647 he was recommended to instruct Charles prince of Wales, afterwards king Charles the 2d, in mathematical learning. During this he employed his vacant time in composing his Leviathan, which was published in England in 1651. After the publication of this work, he returned to England, and passed the remainder of his long life in a very retired and studious manner, in the house of the Earl of Devonshire, mostly at his seat in Derbyshire, but accompanying the earl always to London, fearing to be left out of his immediate protection, lest he should be seized by officers from the parliament or government, on account of the freedom of his opinions in politics and religion. He received great marks of respect from king Charles the 2d at the restoration in 1660, with a pension of 100l. a year. From that time, till his death, he applied himself to his studies, and in opposing the attacks of his adversaries, who were very numerous: in mathematical subjects disputes rose to a great height between him and Dr. Wallis, on account of his pretended Quadrature of the Circle, Cubature of the Sphere, and Duplication of the Cube, which he obstinately defended without ever acknowledging his error.</p><p>His long life was that of a perfectly honest man; a lover of his country, a good friend, charitable and obliging. He accustomed himself much more to thinking, than reading; and was fond of a well-selected, rather than a large library. He had a hatred to the clergy, having been persecuted by them, on account of the freedom of his doctrine, and having a very indifferent opinion of their knowledge and their principles. In his last sickness he was very anxious to know whether his disease was curable; and when intimations were given, that he might have ease, but no remedy, he said, ‘ I shall be glad to find a hole to creep out of the world at.’ He died the 4th of Dec. 1679, at 91 years of age.</p><p>His chief publications were,</p><p>1. An English translation of Thucydides's History of the Grecian war.</p><p>2. De Mirabilibus Pecci, and Memoirs of his own Life, both in Latin verse.</p><p>3. Elements of Philosophy.</p><p>4. Answer to Sir William Davenant's Epistle, or Preface to Gondibert.</p><p>5. Human Nature, or the Fundamental Elements of Policy.</p><p>6. Elements of Law.</p><p>7. Leviathan; or the Matter, Form, and Power of a Commonwealth.</p><p>8. A Compendium of Aristotle's Rhetoric.</p><p>9. A Letter on Liberty and Necessity.</p><p>10. The Questions, concerning Necessity and Chance, stated.</p><p>11. Six Lessons to the Professors of Mathematics, of the Institution of Sir Henry Saville.</p><p>12. The marks of Absurd Geometry, &c.</p><p>13. Dialogues of Natural Philosophy.</p><p>Besides many other pieces on Polity, Theology, Mathematics, and other miscellaneous subjects, to the number of 41.</p></div1><div1 part="N" n="HOBITS" org="uniform" sample="complete" type="entry"><head>HOBITS</head><p>, in Gunnery. See <hi rend="smallcaps">Howitz.</hi></p></div1><div1 part="N" n="HOGSHEAD" org="uniform" sample="complete" type="entry"><head>HOGSHEAD</head><p>, a measure, or vessel, of wine or<cb/> oil; containing the 4th part of a tun, the half of a pipe, or 63 gallons.</p></div1><div1 part="N" n="HOLDER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HOLDER</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, a learned and philosophical Englishman, was born in Nottinghamshire, educated at Cambridge, and in 1642 became rector of Blechingdon in Oxfordshire. In 1660 he proceeded D. D. he became afterwards canon of Ely, Fellow of the Royal Society, canon of St. Paul's, sub-dean of the royal chapel, and sub-almoner to the king. He was a general scholar, a very accomplished person, and a great virtuoso.</p><p>Dr. Holder greatly distinguished himself, by giving speech to a young gentleman of the name of Popham, who was born deaf. This was effected at his own house at Blechingdon in 1659; but the young man losing what he had been taught by Holder after he was called home to his friends, he was sent to Dr. Wallis, who brought him to his speech again. Holder published a book, intitled “the Elements of Speech; an essay or inquiry into the natural Production of Letters: with an appendix, concerning persons that are deaf and dumb, 1669,” 8vo. In the appendix he relates how soon, and by what methods, he brought young Popham to speak. In the Philos. Trans. for July 1670, was inserted a letter from Dr. Wallis, in which he claims to himself the honour of bringing that gentleman to speak. By way of answer to which, in 1678, Dr. Holder published in 4to, “A Supplement to the Philos. Trans. of July 1670, with some reffections on Dr. Wallis's letter there inserted.” Upon which the latter soon after published “A Defence of the Royal Society, and the Philosophical Transactions, particularly those of July 1670, in answer to the cavils of Dr. William Holder, 1678,” 4to.</p><p>Dr. Holder's accomplishments were very general. He was skilled in the theory and practice of music, and wrote “A Treatise of the Natural Grounds and Principles of Harmony, 1694,” 8vo. He wrote also “A Treatise concerning Time, with applications of the Natural Day, Lunar Month, and Solar Year, &c, 1694,” 8vo. He died at Amen Corner in London, Jan. 24, 1697, and was buried in St. Paul's.</p></div1><div1 part="N" n="HOLLOW" org="uniform" sample="complete" type="entry"><head>HOLLOW</head><p>, in Architecture, a concave moulding, about a quarter of a circle, by some called a Casement, by others an Abacus.</p><p><hi rend="smallcaps">Hollow</hi>-<hi rend="italics">Tower,</hi> in Fortification, is a rounding made of the remainder of two brisures, to join the curtin to the crillon, where the small shot are played, that they may not be so much exposed to the view of the enemy.</p><p>HOLY <hi rend="italics">Thursday,</hi> otherwise called Ascension day, being the 39th day after Easter Sunday, and kept in commemoration of Christ's ascension up into heaven.</p><p><hi rend="smallcaps">Holy</hi> <hi rend="italics">Rood,</hi> or <hi rend="italics">Holy Cross,</hi> a feftival kept on the 14th of September, in memory of the exaltation of our Saviour's cross.</p><p><hi rend="smallcaps">Holy</hi> <hi rend="italics">Week,</hi> is the last week of Lent, called also Passion Week,</p></div1><div1 part="N" n="HOLYWOOD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HOLYWOOD</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, or <hi rend="smallcaps">Halifax</hi>, or <hi rend="italics">Sacrobosco,</hi> was, according to Leland, Bale, and Pitts, born at Halifax in Yorkshire: according to Stainhurst, at Holywood near Dublin; and according to Dempster and Mackenzie, in Nithsdale in Scotland. Though<pb n="602"/><cb/> there may perhaps have been more than one of the name. Mackenzie informs us, that having finished his studies, he entered into orders, and became a canon regular of the order of St. Augustin in the famous monastery of Holywood in Nithsdale. The English biographers, on the contrary, tell us that he was educated at Oxford. They all agree however in asserting, that he spent most of his life at Paris; where, says Mackenzie, he was admitted a member of the university, June 5, 1221, under the syndics of the Scotch nation; and soon after was elected professor of mathematics, which he taught with applause for many years. According to the same author, he died in 1256, as appears from the inscription on his monument in the cloisters of the convent of St. Maturine at Paris.</p><p>Holywood was contemporary with Roger Bacon, but probably older by about 20 years. He was certainly the first mathematician of his time; and he wrote, 1. <hi rend="italics">De Sphæra Mundi;</hi> a work often reprinted, and illustrated by various commentators.—2. <hi rend="italics">De Anni Ratione, seu de Computo Ecclesiastico.</hi>—3. <hi rend="italics">De Algorismo,</hi> printed with <hi rend="italics">Comm. Petri Cirvilli Hisp:</hi> Paris, 1498.</p></div1><div1 part="N" n="HOMOCENTRIC" org="uniform" sample="complete" type="entry"><head>HOMOCENTRIC</head><p>, the same as Concentric.</p><p>HOMODROMUS <hi rend="italics">Vectis,</hi> or <hi rend="italics">Lever,</hi> in Mechanics, is a lever in which the weight and power are both on the same side of the fulcrum, as in the lever of the 2d and 3d kind; being so called because here the weight and power move both in the same direction, whereas in the Heterodromus they move in opposite directions.</p></div1><div1 part="N" n="HOMOGENEAL" org="uniform" sample="complete" type="entry"><head>HOMOGENEAL</head><p>, or <hi rend="smallcaps">Homogeneous</hi>, consisting of similar parts, or of the same kind and nature, in contradistinction from heterogeneous, where the parts are of different kinds.——Natural bodies are usually composed of Homogeneous parts, as a diamond, a metal, &c. But artificial bodies, on the contrary, are assemblages of heterogeneous parts, or parts of different kinds; as a building, of stone, wood, &c.</p><p><hi rend="smallcaps">Homogeneal</hi> <hi rend="italics">Light,</hi> is that whose rays are all of one and the same colour, refrangibility, &c.</p><p><hi rend="smallcaps">Homogeneal</hi> <hi rend="italics">Numbers,</hi> are those of the same kind and nature.</p><p><hi rend="smallcaps">Homogeneal</hi> <hi rend="italics">Surds,</hi> are such as have one common radical sign; as √27 and √30, or √<hi rend="italics">a</hi> and √<hi rend="italics">b,</hi> or 2√<hi rend="sup">3</hi><hi rend="italics">c</hi> and √<hi rend="sup">3</hi><hi rend="italics">d.</hi></p><p>HOMOGENEUM <hi rend="italics">Adfectionis,</hi> a name given by Vieta to the second term of a compound or affected equation, being that which makes it adfected.</p><p><hi rend="smallcaps">Homogeneum</hi> <hi rend="italics">Comparationis,</hi> in Algebra, a name given by Vieta to the absolute known number or term in a compound or affected equation. This he places on the right-hand side of the equation, and all the other terms on the left.</p></div1><div1 part="N" n="HOMOLOGOUS" org="uniform" sample="complete" type="entry"><head>HOMOLOGOUS</head><p>, in Geometry, is applied to the corresponding sides of similar figures, or those that are opposite to equal or corresponding angles, and are so called because they are proportional to each other. For all similar figures have their like sides Homologus, or proportional to one another, also their areas or surfaces are Homologous or proportional to the squares of the like sides, and their solid contents Homologous or proportional to the cubes of the same.</p></div1><div1 part="N" n="HOOKE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HOOKE</surname> (<foreName full="yes"><hi rend="smallcaps">Robert</hi></foreName>)</persName></head><p>, a very eminent mathematician<cb/> and philosopher, was born, 1635, at Freshwater in the Isle of Wight, where his father was minister. He was intended for the church; but being of a weakly constitution, and very subject to the head-ach, all thoughts of that nature were laid aside. Thus left to himself, the boy followed the bent of his genius, which was turned to mechanics; and employed his time in making little toys, which he did with wonderful art and dexterity. He had also a good turn for drawing; for which reason, after his father's death, which happened in 1648, he was placed with Sir Peter Lely; but the smell of the oil-colours increasing his head-ach, he quitted painting in a very short time. He was afterwards kindly taken by Dr. Busby into his house, and supported there, while he attended Westminster-school; where he not only acquired a competent share of Greek and Latin, together with an insight into Hebrew and some other Oriental languages, but also made himself master of a good part of Euclid's Elements; and, as Wood asserts, invented 30 different ways of flying.</p><p>About the year 1653 he went to Christ-church in Oxford; and in 1655 was introduced to the Philosophical Society there; where, discovering his mechanic genius, he was first employed to assist Dr. Willis in his chemical operations, and was afterwards recommended to Mr. Robert Boyle, whom he served several years in the same capacity. He was also instructed in astronomy about this time by Dr. Seth Ward, Savilian Professor of that science; and from henceforward distinguished himself by many noble inventions and improvements of the mechanic kind. He also invented several astronomical instruments, for making observations both at sea and land, and was particularly serviceable to Mr. Boyle in completing the invention of the air-pump. In 1662 he was appointed Curator of Experiments to the Royal Society; and when that body was established by royal charter, he was in the list of those, who were first named by the council in May 20, 1663; and he was admitted accordingly June 3, with a peculiar exemption from all payments. Sept. 28 of the same year, he was named by lord Clarendon, chancellor of Oxford, for the degree of M. A.; and Oct. 19 it was ordered that the repository of the Royal Society should be committed to his care: the white gallery in Gresham-college being appropriated to that use. In May 1664, he began to read the astronomy lecture at Gresham college for the professor Dr. Pope, then in Italy; and the same year he was made Professor of Mechanics to the Royal Society by Sir John Cutler, with a salary of 50l. per annum, which that gentleman, the founder, settled upon him for life. Jan. 11, 1665, that society granted a salary also of 30l. a year, for his office of Curator of Experiments for life; and the month of March the same year he was elected professor of geometry in Gresham-college.</p><p>In 1665 too, he published in folio, his “Micrographia, or some Philosophical Descriptions of Minute Bodies, made by Magnifying Glasses, with Observations and Enquiries thereupon.” And the same year, during the recess of the Royal Society on account of the plague, he attended Dr. Wilkins and other ingenious gentlemen into Surry, where they made several experiments. In 1666 he produced to the Royal So-<pb n="603"/><cb/> ciety a model for rebuilding the city of London, then destroyed by the great fire, with which the Society was well pleased; and the Lord Mayor and Aldermen preferred it to that of the city surveyor, though it happened not to be carried into execution. The rebuilding of the city according to the act of parliament requiring able persons to s<*>t out the ground for the proprietors, Mr. Hooke was appointed one of the surveyors; an employment in which he got most part of his estate, as appeared from a large iron chest of money found after his death, locked down with a key in it, and a date of the time, which shewed it to have been so shut up above 30 years. From 1668 he was engaged for many years in a warm contest with Hevelius, concerning the difference in accuracy between observing with astronomical instruments with plain and telescopic sights; in which dispute many learned men afterwards engaged, and in which Hooke managed so ill, as to be universally condemned, though it has since been agreed that he had the better side of the question.—In 1<*>71 he attacked Newton's “New Theory of Light and Colours;” where, though he was obliged to submit in respect to the argument, it is said he came off with more credit. The Royal Society having commenced their meetings at Gresham-college, November 1674, the Committee in December allowed him 40l. to erect a turret over part of his lodgings, for trying his instruments, and making astronomical observations: and the year following he published “A Description of Telescopes, and some other instruments made by R. H. with a Postscript,” complaining of some injustice done him by their secretary Mr. Oldenburg, who published the Philosophical Transactions, in regard to his invention of pendulum watches. This charge drew him into a dispute with that gentleman, which ended in a declaration of the Royal Society in their secretary's favour.—Mr. Oldenburg dying in 1677, Mr. Hooke was appointed to supply his place, and began to take minutes at the meeting in October, but did not publish the Transactions.—Soon after this, he grew more reserved than formerly; and though he read his Cutlerian Lectures, often made experiments, and shewed new inventions before the Royal Society, yet he seldom left any account of them to be entered in their registers; designing, as he said, to publish them himself, which however he never performed.—In 1686, when Newton's work the Principia was published, Hooke laid claim to his discovery concerning the force and action of gravity, which was warmly resented by that great philosopher. Hooke, though a great inventor and discoverer himself, was yet so envious and ambitious, that he would fain have been thought the only man who could invent and discover. This made him often lay claim to the inventions and discoveries of other persons; on which occasions however, as well as in the present case, the thing was generally carried against him.</p><p>In the beginning of the year 1687, his brother's daughter, Mrs. Grace Hooke, who had lived with him several years, died: and he was so affected with grief at her death, that he hardly ever recovered it, but was observed from that time to become less active, more melancholy, and more cynical than ever. At the same time, a chancery suit in which he was concerned with Sir John Cutler, on account of his salary for reading<cb/> the Cutlerian Lectures, made him uneasy, and increased his disorder.—In 1691, he was employed in forming the plan of the hospital near Hoxton, founded by Robert Ask, alderman of London, who appointed archbishop Tillotson one of his executors; and in December the same year, Hooke was created M. D. by a warrant from that prelate. In July 1696, the chancery suit with Sir John Cutler was determined in his favour, to his inexpressible satisfaction. His joy on that occawas found in his diary thus expressed; <hi rend="smallcaps">DOMSHLGISSA;</hi> that is, <hi rend="italics">Deo, Optimo, Maximo, sit honor, laus, gloria, in sæcula sæculorum, Amen.</hi> “I was born on this day of July 1635, and God hath given me a new birth: may I never forget his mercies to me! while he gives me breath may I praise him!”—In the same year 1696, an order was granted to him for repeating most of his experiments at the expence of the Royal Society, upon a promise of his finishing the accounts, observations, and deductions from them, and of perfecting the description of all the instruments contrived by him: but his increasing illness and general decay rendered him unable to perform it. He continued some years in this wasting condition; and thus languishing till he was quite emaciated, he died March 3, 1702, in his 67th year, at his lodgings in Gresham college, and was buried in St. Helen's church, Bishopsgate street; his corps being attended by all the members of the Royal Society then in London.</p><p>As to Mr. Hooke's character, it is not in all respects one of the most amiable. In his person he made rather a despicable figure, being but of a short stature, very crooked, pale, lean, and of a meagre aspect, with dark-brown hair, very long, and hanging over his face lank and uncut. Suitable to his person, his temper was penurious, melancholy, and mistrustful: and, though possessed of great philosophical knowledge, he had so much ambition, that he would be thought the only man who could invent or discover; and hence he often laid claim to the inventions and discoveries of others, while he boasted of many of his own which he never communicated. In the religious part of his character, he was so exemplary, that he always expressed a great veneration for the Deity; and seldom received any remarkable benefit in life, or made any considerable discovery in nature, or invented any useful contrivance, or found out any difficult problem, without setting down his acknowledgment to God, as many places in his diary plainly shew.—His chief publications are,</p><p>1. <hi rend="italics">Lectiones Cutlerianæ,</hi> or the Cutlerian Lectures.</p><p>2. <hi rend="italics">Micrographia,</hi> or Descriptions of Minute Bodies made by Magnifying Glasses.</p><p>3. A Description of Helioscopes.</p><p>4. A Description of some Mechanical Improvements of Lamps and Water-poises.</p><p>5. Philosophical Collections.</p><p>6. Posthumous Works, collected from his papers by Richard Waller secretary to the Royal Society. Besides a number of papers in the Philos. Trans. volumes 1, 2, 3, 5, 6, 9, 16, 17, 22.</p></div1><div1 part="N" n="HORARY" org="uniform" sample="complete" type="entry"><head>HORARY</head><p>, something relating to Hours. As,</p><p><hi rend="smallcaps">Horary</hi> <hi rend="italics">Circles,</hi> hour lines or circles, marking the hours, or drawn at the distance of hours from one another.</p><p><hi rend="smallcaps">Horary</hi> <hi rend="italics">Motion,</hi> is the motion or space moved in an<pb n="604"/><cb/> hour. Thus, the Horary motion of the earth on her axis, is 15°; for, completing her revolution of 360°, in 24 hours, therefore the motion in one hour will be the 24th part of 360°, which is 15 degrees.</p></div1><div1 part="N" n="HORIZON" org="uniform" sample="complete" type="entry"><head>HORIZON</head><p>, in Astronomy, a great circle of the sphere, dividing the world into two parts, or hemispheres; the one upper, and visible; the other lower, and hid.</p><p>The Horizon is either Rational or Sensible.</p><div2 part="N" n="Horizon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Horizon</hi></head><p>, <hi rend="italics">Rational, True,</hi> or <hi rend="italics">Astronomical,</hi> called also simply and absolutely the Horizon, is a great circle having its plane passing through the centre of the earth, and its poles are the zenith and nadir. Hence all the points of the Horizon, quite around, are at a quadrant distance from the zenith and nadir. Also the meridian and vertical circles cut the Horizon at right angles, and into two equal parts.</p></div2><div2 part="N" n="Horizon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Horizon</hi></head><p>, <hi rend="italics">Apparent, Sensible,</hi> or <hi rend="italics">Visible,</hi> is a lesser circle of the sphere, parallel to the rational Horizon, dividing the visible part of the sphere from the invisible, and whose plane touches the spherical surface of the earth.</p><p>The sensible Horizon is divided into Eastern and Western; the Eastern or Ortive being that in which the heavenly bodies rise; and the Western, or Occidual, being that in which they set.</p></div2><div2 part="N" n="Horizon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Horizon</hi></head><p>, in Geography, is a circle dividing the visible part of the earth and heavens from that which is invisible. This is peculiarly called Sensible or Apparent Horizon, to distinguish it from the Rational or True, which passes through the centre of the earth; as already observed. These two Horizons, though distant from one another by the semidiameter of the earth, will appear to coincide when continued to the sphere of the fixed stars; because the earth compared with this sphere is but a point.</p><p>By Sensible Horizon is also often meant a circle which determines the segment of the surface of the earth, over which the eye can reach; called also the Physical Horizon. And in this sense we say, a spacious Horizon, a narrow or scanty Horizon, &c; depending chiefly on the height the eye is elevated above the earth. <figure/></p><p>For, it is evident that the higher the eye is placed, the farther is the visible Horizon extended. Thus, if the eye be at A, at the height AD above the earth; draw the two tangents A<hi rend="italics">h,</hi> A<hi rend="italics">r;</hi> and let one of these lines A<hi rend="italics">h,</hi> be moved round the point A, and in its revolution always touch the surface of the earth; then<cb/> the other point <hi rend="italics">h</hi> will describe the visible Horizon <hi rend="italics">hor,</hi> &c. But if the eye be placed higher as at B, the tangents BH and BR will reach farther, and the visible Horizon HOR will be larger.</p><p>The visible Horizon is most accurately observed at sea, and is therefore sometimes called the Horizon of the sea. In observing this Horizon, the visual rays A<hi rend="italics">h</hi> and A<hi rend="italics">r</hi> will, on account of the curve surface of the sea, always point a little below the true sensible Horizon SS or EF, and consequently below the rational Horizon TT, which is parallel to it.</p><p>To find the Depression of the Horizon of the sea below the true Horizon, which varies with the height of the eye, and in a small degree with the variation of the refractive power of the atmosphere, see D<hi rend="smallcaps">EPRESSION.</hi></p><p>As to the right-lined distance, or tangent E<hi rend="italics">h,</hi> it may be found thus; as radius : sin. [angle] C : : CA : A<hi rend="italics">h,</hi> or thus; as radius : tan. [angle] C : : C<hi rend="italics">h</hi> : A<hi rend="italics">h,</hi> either of which will be nearly the same as the arc or curved distance D<hi rend="italics">h.</hi> Or, without finding the angle C, thus; the square of A<hi rend="italics">h</hi> is equal to the difference of the squares of CA and C<hi rend="italics">h,</hi> i. e. , and hence , which is also equal to A<hi rend="italics">h</hi> nearly.</p><p>The distance on a perfect globe, if the visual rays came to the eye in a straight line, would be as above stated: but by means of the refraction of the atmosphere, distant objects on the Horizon appear higher than they really are, or appear less depressed below the true Horizon SS, and may be seen at a greater distance, especially on the sea. M. Legendre, in his Memoir on Measurements of the Earth, in the Mem. Acad. Sci. for the year 1787, says that, from several experiments, he is induced to allow for refraction a 14th part of the distance of the place observed, expressed in degrees and minutes of a great circle. Thus, if the distance be 14000 toises, the refraction will be 1000 toises, equal to the 57th part of a degree, or 1′ 3″.</p><p><hi rend="smallcaps">Horizon</hi> <hi rend="italics">of the Globe,</hi> a broad wooden circle. See <hi rend="smallcaps">Globe.</hi></p></div2></div1><div1 part="N" n="HORIZONTAL" org="uniform" sample="complete" type="entry"><head>HORIZONTAL</head><p>, something that relates to the Horizon, or that is taken in the Horizon, or on a level with or parallel to it. Thus, we say, a Horizontal plane, Horizontal line, Horizontal distance, &c.</p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Dial,</hi> is one drawn on a plane parallel to the horizon; having its gnomon or style elevated according to the altitude of the pole of the place it is designed for.</p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Distance,</hi> is that estimated in the direction of the horizon.</p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Line,</hi> in Perspective, is a right line drawn through the principal point, parallel to the horizon; or it is the intersection of the Horizontal and perspective planes.</p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Line,</hi> or base of a hill, in Surveying, a line drawn on the Horizontal plane of the hill, or that on which it stands.</p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Moon.</hi> See <hi rend="italics">Apparent</hi> <hi rend="smallcaps">Magnitude.</hi></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Parallax.</hi> See <hi rend="smallcaps">Parallax.</hi></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Plane,</hi> is that which is parallel to the horizon of the place, or not inclined to it.<pb n="605"/><cb/></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Plane,</hi> in Perspective. See <hi rend="smallcaps">Plane.</hi></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Projection.</hi> See <hi rend="smallcaps">Projection</hi>, and <hi rend="smallcaps">Map.</hi></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Range,</hi> of a piece of ordnance, is the distance at which it falls on, or strikes the horizon, or on a Horizontal plane, whatever be the angle of elevation or direction of the piece. When the piece is pointed parallel to the horizon, the range is then called the point-blank or point-blanc range.</p><p>The greatest Horizontal range, in the parabolic theory, or in a vacuum, is that made with the piece elevated to 45 degrees, and is equal to double the height from which a heavy body must freely fall to acquire the velocity with which the shot is discharged. Thus, a shot being discharged with the velocity of <hi rend="italics">v</hi> feet per second; because gravity generates the velocity 2<hi rend="italics">g</hi> or 32 1/6 feet in the first second of time, by falling 16 1/12 or <hi rend="italics">g</hi> feet, and because the spaces descended are as the squares of the velocities, therefore as the space a body must descend to acquire the velocity <hi rend="italics">v</hi> of the shot or the space due to the velocity <hi rend="italics">v;</hi> consequently the double of this, or is the greatest Horizontal range with the velocity <hi rend="italics">v,</hi> or at an elevation of 45 degrees; which is nearly half the square of a quarter of the velocity.</p><p>In other elevations, the Horizontal range is as the sine of double the angle of elevation; so that, any other elevation being <hi rend="italics">e,</hi> it will be, as radius , the range at the elevation <hi rend="italics">e,</hi> with the velocity <hi rend="italics">v.</hi></p><p>But in a resisting medium, like the atmosphere, the actual ranges fall far short of the above theorems, in so much that with the great velocities, the actual or real ranges may be less than the 10th part of the potential ranges; so that some balls, which actually range but a mile or two, would in vacuo range 20 or 30 miles. And hence also it happens that the elevation of the piece, to shoot farthest in the resisting medium, is always below 45°, and gradually the more below it as the velocity is greater, so that the greater velocities with which balls are discharged from cannon with gunpowder, require an elevation of the gun equal to but about 30°, or even less. And the less the size of the balls is too, the less must this angle of elevation be, to shoot the farthest with a given velocity. See P<hi rend="smallcaps">ROJECTILE</hi>, and <hi rend="smallcaps">Gunnery.</hi></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Refraction.</hi> See <hi rend="smallcaps">Refraction.</hi></p><p><hi rend="smallcaps">Horizontal</hi> <hi rend="italics">Speculum,</hi> one to find a horizon at Sea, &c, when the atmosphere is hazy near the horizon, by which the fight of it is prevented.</p><p>A speculum of this kind was invented by a Mr. Serson, on the principle of a top spinning, which always keeps its upright position, notwithstanding the motion of the substance it spins upon. This curious instrument, as it has since been improved by Mr. Smeaton, consists of a well polished metal speculum, of about 3 inches and a half in diameter, inclosed within a circular rim of brass; so fitted that the centre of gravity of the whole shall fall near the point on which it spins. This is the end of a steel axis running through the<cb/> centre of the speculum, above which it finishes in <*> square, for the conveniency of fitting a roller on it, which sets it in motion by means of a piece of tape wound about the roller.</p><p>Various other contrivances to form artificial horizon<*> have been invented by different persons, as glass planes floating on mercury, &c. See <hi rend="smallcaps">Hadley's Quadrant</hi>, and several inventions of this sort in the Philos. Trans. by Elton, Halley, Leigh, &c. vol. xxxvii, p. 273, vol. xxxviii, p. 167, vol. xl, p. 413, 417, &c.</p><p>HORN-<hi rend="smallcaps">Work</hi>, in Fortification, a sort of out-work, advancing toward the field, to cover and defend a curtin, bastion, or other place, suspected to be weaker than the rest; as also to possess a height; carrying in the fore-part, or head, two demi-bastions, resembling horns: these horns, epaulments, or shoulderings, being joined by a curtin, shut up on the side by two wings, parallel to one another, are terminated at the gorge of the work, and so present themselves to the enemy.</p></div1><div1 part="N" n="HOROGRAPHY" org="uniform" sample="complete" type="entry"><head>HOROGRAPHY</head><p>, the art of making or constructing dials; called also Dialling, Horologiography, Gnomonica, Sciatherica, Photosciatherica, &c.</p></div1><div1 part="N" n="HOROLOGIUM" org="uniform" sample="complete" type="entry"><head>HOROLOGIUM</head><p>, a common name, among ancient writers, for any instrument or machine for measuring the hours. See <hi rend="smallcaps">Clock, Watch, Sun-Dial, Chronometer, Clepsydra</hi>, &c.</p></div1><div1 part="N" n="HOROMETRY" org="uniform" sample="complete" type="entry"><head>HOROMETRY</head><p>, the art of measuring or dividing time by hours, and keeping the account of time. <figure/></p></div1><div1 part="N" n="HOROPTER" org="uniform" sample="complete" type="entry"><head>HOROPTER</head><p>, in Optics, is a right line drawn through the point where the two optic axes meet, parallel to that which joins the centres of the two eyes, or the two pupils. As the line AB drawn through C the point of concourse of the optic axes of the eyes, and parallel to HI joining the centres of the eyes.— This line is called the Horopter, because it is found to be the limit of distinct vision. It has several properties in Optics, which are described at large in Aguillonius, Opt. lib. 2, diss. 10.</p></div1><div1 part="N" n="HOROSCOPE" org="uniform" sample="complete" type="entry"><head>HOROSCOPE</head><p>, in Astrology, is the ascendant or first house, being that part of the zodiac which is just rising in the eastern side of the horizon at any proposed time, when a scheme is to be set or calculated, or a prediction made of any event. See <hi rend="smallcaps">Ascendant.</hi></p><p><hi rend="smallcaps">Horoscope</hi> is also used for a scheme or figure of the 12 celestial houses; i. e. the 12 signs of the zodiac, in which is marked the disposition of the heavens for any given time. Thus it is said, To draw or construct a Horoscope or scheme, &c. And it is more peculiarly<pb n="606"/><cb/></p><p>called, Calculating a nativity, when the life and fortune of a person are the subject of prediction.</p><p><hi rend="italics">Lunar</hi> <hi rend="smallcaps">Horoscope</hi>, is the point the moon issues out of, when the sun is in the ascending point of the east; and is also called the Part of Fortune.</p><p><hi rend="smallcaps">Horoscope</hi> was also a mathematical instrument, in manner of a planisphere; but now disused. It was invented by J. Paduanus, who wrote a special treatise upon it.</p><p>HORROR <hi rend="italics">of a Vacuum,</hi> an imaginary principle among the more ancient philosophers, to which they ascribed the ascent of water in pumps, and other similar phenomena, which are now known to be occasioned by the weight of the air.</p></div1><div1 part="N" n="HORROX" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HORROX</surname> (<foreName full="yes"><hi rend="smallcaps">Jeremiah</hi></foreName>)</persName></head><p>, an eminent English astronomer, was born at Toxteth in Lancashire, about the year 1619. From a grammar school in the country, he was sent to Cambridge, where he spent some time in academical studies. About 1633 he began to apply himself to the study of astronomy: but living at that time with his father at Toxteth, in very moderate circumstances, and being destitute of books and other assistances for such studies, he could not make any considerable progress in it. About the year 1636, he formed an acquaintance with Mr. William Crabtree, of Broughton near Manchester, who was engaged in the same studies, with whom a mutual correspondence was carried on till his death; sometimes communicating their improvements to Mr. Samuel Foster, professor of geometry at Gresham College in London. Having now obtained a companion in his studies, Mr. Horrox assumed new vigour, procured other instruments and books, and was pursuing his studies and observations with great assiduity, when he was suddenly cut off by death, the 3d of January 1640, in the 22d year of his age.</p><p>What we have of his writings is sufficient to shew how great a loss the world had by his death. He had just finished his <hi rend="italics">Venus in Sole visa,</hi> 1639, a little before, as appears by some of the letters to his friend Mr. Crabtree, by which also it appears that he made his observations on that phenomenon at Hool near Liverpool. This tract, of Venus seen in the Sun, was published at Dantzick in 1668, by Hevelius, together with his own <hi rend="italics">Mercurius in Sole visus</hi> May 3, 1661. His other posthumous works, or rather his imperfect papers, were published by Dr. Wallis, in 1673, 4to, with some account of his life; in which we find he first asserts and promotes the Keplerian astronomy against the hypothesis of Lansberg; which he proves to be inconsistent with itself, and neither agreeing with observations nor theory. He likewise reasons very justly concerning the celestial bodies and their motions, vindicates Tycho Brah from some objections made to his hypothesis, and gives a new theory of the moon: to which are added the Lunar Numbers of Mr. Flamsteed. There are also extracts from several letters between him and Mr. Crabtree, upon various astronomical subjects; with a catalogue of astronomical observations.</p><p>There are two things particularly which will perpetuate the memory of this very extraordinary young man. The one is, that he was the first that ever predicted or saw the planet Venus in the sun; for we do not find<cb/> that any persons, besides himself and Mr. Crabtree, ever beheld such a phenomenon. Though he was not apprised of the great use that was to be made of it, in discovering the parallax and distance of the sun and planets, yet he made from it many useful observations, corrections, and improvements in the theory of the motions of Venus.—Secondly, his New Theory of Lunar Motions, which Newton himself made the ground work of all his astronomy, relative to the moon, who always spoke of our author as a genius of the first rank.</p><p>HORSE-<hi rend="smallcaps">Shoe</hi>, in Fortisication, is a work sometimes of a round, sometimes of an oval sigure, inclosed with a parapet, raised in the ditch of a marshy place, or in low grounds; sometimes also to cover a gate; or to serve as a lodgment for soldiers, to prevent surprises, or relieve an over-tedious defence.</p></div1><div1 part="N" n="HOSPITAL" org="uniform" sample="complete" type="entry"><head>HOSPITAL</head><p>, <hi rend="italics">or</hi> <hi rend="smallcaps">Hopital (William-FrancisAnthony</hi>, <hi rend="italics">marquis of),</hi> a celebrated French mathematician, was born of an ancient family in 1661. He was a mathematician almost from his infancy; for being one day at the duke of Rohan's, where some able mathematicians were speaking of a problem of Pascal's, which appeared to them very difficult, he ventured to say, that he believed he could resolve it. They were surprised at such presumption in a boy of 15, for he was then no more; however, in a few days he sent them the solution.</p><p>M. l'Hospital entered early into the army, and was a captain of horse; but being very short-sighted, and on that account exposed to perpetual inconveniences and errors, he at length quitted the army, and applied himself entirely to his favourite amusement.—He contracted a friendship with Malbranche, and took his opinion upon all occasions.—In 1699 he was received an honorary member of the Academy of Sciences at Paris.</p><p>He was the first person in France who wrote upon Newton's analysis, and on this account was regarded almost as a prodigy. His work was entitled <hi rend="italics">l'Analyse des Infinimens Petits,</hi> 1696. He engaged afterwards in another mathematical work, in which he included <hi rend="italics">Les Sections Coniques, les Lieux Géometriques, la Construction des Equations, et une Théorie des Courbes Mechaniques:</hi> but, a little before he had finished it he was seized with a fever, which carried him off, the 2d of February 1704, at 43 years of age. The work was published after his death, viz, in 1707. There are also six of his pieces inserted in different volumes of the Memoirs of the Academy of Sciences.</p></div1><div1 part="N" n="HOUR" org="uniform" sample="complete" type="entry"><head>HOUR</head><p>, in Chronology, an aliquot part of a natural day, usually the 24th, but sometimes a 12th part. With us, it is the 24th part of the earth's diurnal rotation, or the time from noon to noon, and therefore it answers to 15 degrees of the whole circle of longitude, or of 360°. The hour is divided by 60ths, viz, first into 60 minutes, then each minute into 60 seconds, &c.</p><p>The division of the day into Hours is very ancient; as is shewn by Kircher, Oedip. Ægypt. tom. 2, par. 2, class 7, cap. 8. The most ancient Hour is that of the 12th part of the day. Herodotus, lib. 2, observes, that the Greeks learnt from the Egyptians, among other things, the method of dividing the day into 12 parts. And the astronomers of Cathaya, &c, still retain this division.<pb n="607"/><cb/></p><p>The division of the day into 24 hours, was not known to the Romans before the Punic war. Till that time they only regulated their days by the rising and setting of the sun. They divided the 12 hours of their day into four; viz, Prime, which commenced at 6 o'clock, Third at 9, Sixth at 12, and None at 3. They also divided the night into four watches, each containing 3 Hours.</p><p>There are various kinds of Hours, used by chronologers, astronomers, dialists, &c. Sometimes too,</p><p>Hours are divided into Equal and Unequal.</p><p><hi rend="italics">Equal</hi> <hi rend="smallcaps">Hours</hi>, are the 24th parts of a day and night precisely; that is, the time in which the 15 degrees of the equator pass the meridian. These are also called Equinoctial Hours, because measured on the equinoctial; and Astronomical, because used by astronomers.</p><p><hi rend="italics">Astronomical</hi> <hi rend="smallcaps">Hours</hi>, are equal Hours, reckoned from noon to noon, in a continued series of 24.</p><p><hi rend="italics">Babylonish</hi> <hi rend="smallcaps">Hours</hi>, are equal Hours, reckoned from sun-rise in a continued series of 24.</p><p><hi rend="italics">European</hi> <hi rend="smallcaps">Hours</hi>, used in civil computation, are equal Hours, reckoned from midnight; 12 from thence till noon, and 12 more from noon till midnight.</p><p><hi rend="italics">Jewish,</hi> or <hi rend="italics">Planetary,</hi> or <hi rend="italics">Ancient</hi> <hi rend="smallcaps">Hours</hi>, are 12th parts of the artificial day and night. They are called Ancient or Jewish Hours, because used by the ancients, and still among the Jews. They are called Planetary Hours, because the astrologers pretend, that a new planet comes to predominate every Hour; and that the day takes its denomination from that which predominates the first Hour of it; as Monday from the moon, &c.</p><p><hi rend="italics">Italian</hi> Hours, are equal Hours, reckoned from sunset, in a continued series of 24.</p><p><hi rend="italics">Unequal</hi> or <hi rend="italics">Temporary</hi> <hi rend="smallcaps">Hours</hi>, are 12th parts of the artificial day and night. The obliquity of the sphere renders these more or less unequal at different times; so that they only agree with the equal Hours at the times of the equinoxes.</p><p><hi rend="smallcaps">Hour</hi>-<hi rend="italics">Circles,</hi> or <hi rend="smallcaps">Horary</hi>-<hi rend="italics">Circles,</hi> are great Circles, meeting in the poles of the globe or world, and crossing the equinoctial or equator at right angles; the same as meridians. They are supposed to be drawn through every 15th degree of the equinoctial and equator, each answering to an hour, and dividing them into 24 equal parts; and on both globes they are supplied by the meridian Hour-circle and index.</p><p><hi rend="smallcaps">Hour</hi>-<hi rend="italics">Glass,</hi> a popular kind of chronometer or clepsydra, serving to measure time by the descent or running of sand, water, &c, out of one glass vessel into another. —The best, it is said, are such as, instead of sand, have egg-shells, well dried in the oven, then beaten fine and sifted.</p><p><hi rend="smallcaps">Hour</hi>-<hi rend="italics">Lines,</hi> on a Dial, are lines which arise from the intersections of the plane of the Dial, with the several planes of the Hour-circles of the sphere; and therefore must be all right lines on a plane Dial.</p><p><hi rend="smallcaps">Hour</hi>-<hi rend="italics">Scale,</hi> a divided line on the edge of Collins's quadrant, being only two lines of tangents of 45 degrees each, set together in the middle. Its use, together with the lines of latitude, is to draw the Hour-lines of Dials that have centres, by means of an equilateral triangle, drawn on the dial-planes.</p></div1><div1 part="N" n="HOWITZ" org="uniform" sample="complete" type="entry"><head>HOWITZ</head><p>, or <hi rend="smallcaps">Howitzer</hi>, in Artillery, a kind of<cb/> mortar, or something between a cannon and mortar, partaking of the nature of both, being either a very short gun or a long mortar. It is of German invention, and is mounted upon a carriage like a travelling guncarriage, with its trunnions placed nearly in the middle. The Howitz is one of the most useful kinds of ordnance, as it can be employed occasionally either as a cannon or mortar, discharging either shells or grape shot, as well as balls, and so doing great execution. They are also very easily travelled about from place to place.</p></div1><div1 part="N" n="HUMIDITY" org="uniform" sample="complete" type="entry"><head>HUMIDITY</head><p>, or moisture, the power or quality of wetting or moistening other bodies, and adhering to them.</p><p>Fluids are moist to some bodies, and not to others. Thus, quicksilver is not moist in respect to our hands or clothes, and other things, which it will not stick to; but it may be called Humid in reference to gold, tin, or lead, to the surfaces of which it will presently adhere, and render them soft and moist. Even water itself, which wets almost every thing, and is the great standard of moisture and Humidity, is not capable of wetting all things; for it stands or runs off in globular drops from any thing greased or oiled, or the leaves of cabbages, and many other planets; and it will not wet the feathers of ducks, geese, swans, and other water-fowl.</p></div1><div1 part="N" n="HUNDRED" org="uniform" sample="complete" type="entry"><head>HUNDRED</head><p>, the number of ten times ten, or the square of 10. The place of Hundreds makes the third in order in the Arabic or modern numeration, being denoted thus 100. In the Roman notation it is denoted by the letter C, being the initial of its name, Centum.</p><p><hi rend="smallcaps">Hundred</hi> <hi rend="italics">Weight,</hi> or the great Hundred, contains 112 pounds weight. It is subdivided into 4 quarters, and each quarter into 28 lbs.</p></div1><div1 part="N" n="HURTERS" org="uniform" sample="complete" type="entry"><head>HURTERS</head><p>, in Fortification, denote pieces of timber, about 6 inches square, placed at the lower end of the platform, next to the parapet, to prevent the wheel<*> of the gun-carriages from damaging the parapet.</p></div1><div1 part="N" n="HUYGENS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">HUYGENS</surname> (<foreName full="yes"><hi rend="smallcaps">Christian</hi></foreName>)</persName></head><p>, a very eminent astronomer and mathematician, was born at the Hague in Holland, in 1629, being the son of Constantine Huygens, lord of Zuylichem, who had served three successive princes of Orange in the quality of secretary. He spent his whole life in cultivating the mathematics; and not in the speculative way only, but also in making them subservient to the uses of life. From his infancy he discovered an extraordinary fondness for the mathematics; in a short time made a great progress in them; and perfected himself in those studies under professor Schooten, at Leyden. In 1649 he went to Holstein and Denmark, in the retinue of Henry count of Nassau; and was extremely desirous of going to Sweden, to visit Des Cartes who was then in that country with the queen Christina, but the count's short stay in Denmark would not permit him.—In 1651 he gave the world a specimen of his genius for mathematics, in a treatise intitled, <hi rend="italics">Theoremata de Quadratura Hyperboles, Ellipsis, & Circuli, ex dato Portionum Gravitatis Centro;</hi> in which he clearly shewed what might be expected from him afterwards.——In 1655 he travelled into France, and took the degree of LL.D. at Angers.— In 1658 he published his <hi rend="italics">Horologium Oscillatorium, sive de Motu Pendulorum,</hi> &c, at the Hague. He had ex-<pb n="608"/><cb/> hibited in a former work, intitled, <hi rend="italics">Brevis Institutio de Usu Horologiorum ad inveniendas Longitudines,</hi> a model of a new invented pendulum; but as some persons, envious of his reputation, were labouring to deprive him of the honour of the invention, he wrote this book to explain the construction of it; and to shew that it was very different from the pendulum of astronomers invented by Galileo.—In 1659 he published his <hi rend="italics">Systema Saturninum,</hi> &c; in which he first of any one explained the ring of Saturn, and discovered also one of the satellites belonging to that planet, which had hitherto escaped the eyes of astronomers: new discoveries, made with glasses of his own forming, which gained him a high rank among the astronomers of his time.</p><p>In 1660, he took a second journey into France, and the year after passed over into England, where he communicated his art of polishing glasses for telescopes, and was made Fellow of the Royal Society. About this time the air-pump was invented, which received considerable improvements from him. This year also he discovered the laws of the collision of elastic bodies; as did also about this time Wallis and Wren, with whom he had a dispute about the honour of this discovery. Upon his return to France, in 1663, the minister Colbert, being informed of his great merit, settled a considerable pension upon him, to engage him to fix at Paris; to which Mr. Huygens consented, and staid there from the year 1666 to 1681, where he was admitted a member of the Academy of Sciences. All this time he spent in mathematical pursuits, wrote several books, which were published from time to time, and invented and perfected several useful instruments and machines: particularly he had a d<*>spute, about the year 1668, with Mr. James Gregory, concerning the Quadrature of the Circle and Hyperbola of the latter, then just published, in which Huygens it seems had the better side of the question. But continual application gradually impaired his health; and though he had visited his native country twice, viz, in 1670 and 1675, for the recovery of it, he was now obliged to betake himself to it altogether. Accordingly he left Paris in 1681, and retired to his own country, where he spent the remainder of his life in the same pursuits and employments. He died at the Hague, June 8, 1695, in the 67th year of his age, while his <hi rend="italics">Cosmotheoros,</hi> or treatise concerning a plurality of worlds, was printing; so that this work did not appear till 1698.</p><p>Mr. Huygens loved a quiet and studious manner of life, and frequently retired into the country to avoid interruption, but did not contract that moroseness which is so commonly the effect of solitude and retirement. He was one of the purest and most ingenious mathematicians of his age, and indeed of any other; and made many valuable discoveries. He was the first who discovered Saturn's ring, and a third satellite of that planet, as mentioned above. He invented the means of rendering clocks exact, by applying the pendulum, and of rendering all its vibrations equal, by the cycloid. He brought telescopes to perfection, and made many other useful discoveries.</p><p>He was the author of many excellent works. The principal of these are now contained in two collections, of 2 volumes each, printed in 4to, under the care of professor Gravesande. The first was at Leyden<cb/> in 1682, under the title of <hi rend="italics">Opera Varia;</hi> and the second at Amsterdam, in 1728, entitled <hi rend="italics">Opera Reliqua.</hi></p></div1><div1 part="N" n="HYADES" org="uniform" sample="complete" type="entry"><head>HYADES</head><p>, a cluster of 5 stars in the face of the constellation Taurus, or the Bull.</p></div1><div1 part="N" n="HYALOIDES" org="uniform" sample="complete" type="entry"><head>HYALOIDES</head><p>, the vitreous humour of the eye contained between the tunica-retina and the uvea.</p><p>HYBERNAL <hi rend="italics">Occident.</hi> See <hi rend="italics">Occident.</hi></p><p><hi rend="smallcaps">Hybernal</hi> <hi rend="italics">Orient.</hi> See <hi rend="italics">Orient.</hi></p></div1><div1 part="N" n="HYDATOIDES" org="uniform" sample="complete" type="entry"><head>HYDATOIDES</head><p>, the watery humour of the eye contained between the cornea and the uvea.</p></div1><div1 part="N" n="HYDRA" org="uniform" sample="complete" type="entry"><head>HYDRA</head><p>, a southern constellation, consisting of a number of stars, imagined to represent a water serpent. The stars in the constellation Hydra, in Ptolomy's catalogue, are 27; in Tycho's, 19; in Hevelius's, 31; and in the Britannic catalogue, 60.</p></div1><div1 part="N" n="HYDRAULICS" org="uniform" sample="complete" type="entry"><head>HYDRAULICS</head><p>, the science of the motion of water and other fluids, with its application in artificial water-works of all sorts.—As to what respects merely the equilibrium of fluids, or their gravitation or action at rest, belongs to Hydrostatics. Upon removing or destroying that equilibrium, motion ensues; and here Hydraulics commence. Hydraulics therefore suppose Hydrostatics; and many writers, from the near relation between them, like mechanics and statics, join the two together, and treat of them conjointly as one science.</p><p>The laws of Hydraulics are given under the word Fluid. And the art of raising water, with the several machines employed for that purpose, are described under their several names, Fountain, Hydrocanisterium Pump, Siphon, Syringe, &c.</p><p>The principal writers who have cultivated and improved Hydraulics and Hydrostatics, are Archimedes, in his Libris de Insidentibus Humido; Hero of Alexandria, in his Liber Spiritualium; Marinus Ghetaldus, in his Archimedes promotus; Mr. Oughtred; Jo. Ceva, in his Geometria Motus; Jo. Bap. Balianus, De Motu Naturali Gravium, Solidorum et Liquidorum; Mariotte, in his treatise of the Motion of Water and other fluids; Boyle, in his Hydrostatical Paradoxes; Fran. Tertius de Lanis, in his Magisterium Naturæ et Artis; Lamy, in his Traité de l'Equilibre des Liqueurs; Rohault; Dr. Wallis, in his Mechanics; Dechales; Newton, in his Principia; Gulielmeni, in his Mensura Aquarum Fluentium; Herman; Wolsius; Gravesande; Musschenbroek; Leopold; Schottus, in his Mechanica Hydraulico Pneumatica; Geo. Andr. Bockler, in his Architectura Curiosa Germanica; August. Rammilleis; Lucas Antonius Portius; Sturmy, in his treatise on the Construction of Mills; Switzer's Hydrostatics; Varignon, in the Mem. Acad. Sci.; Jurin; Belidor; Bernoulli; Desaguliers; Clare; Emerson; Ferguson; Ximenes; Bossu; D'Alembert; Buat; &c, &c.</p><p>HYDRAULICO-<hi rend="smallcaps">Pneumatical</hi>, a term applied by some authors to such engines as raise water by means of the weight or spring of the air.</p><p>HYDROGRAPHICAL <hi rend="italics">Charts</hi> or <hi rend="italics">Maps,</hi> more usually called sea-charts, are projections of some part of the sea, or coast, for the use of navigation-</p><p>In these are laid down all the rhumbs or points of the compass, the meridians, parallels, &c, with the coasts, capes, islands, rocks, shoals, shallows, &c, in their proper places, and proportions. <pb/><pb/><pb n="609"/><cb/></p><p>The making and selling these charts was for some time the employment of Columbus, the first discoverer of America. The story goes, that happening to be heir to the memoirs and journals of one Alonzo Sanchez de Huelva, a noted pilot and captain of a ship, who by chance had been driven by a storm to the island of St. Domingo, and dying at Columbus's house soon after his return, this gave Columbus the first hint to attempt a discovery of the West Indies.</p><p>For the construction and use of the several kinds of Hydrographical Maps, see <hi rend="smallcaps">Chart</hi>, and <hi rend="smallcaps">Sailing.</hi></p></div1><div1 part="N" n="HYDROLOGY" org="uniform" sample="complete" type="entry"><head>HYDROLOGY</head><p>, is that part of natural history which examines and explains the nature and properties of water in general.</p></div1><div1 part="N" n="HYDROMANCY" org="uniform" sample="complete" type="entry"><head>HYDROMANCY</head><p>, the act or art of divining or foretelling future events by means of water.</p><p>This is one of the four general kinds of divination: the other three respecting the other elements, fire, air, and earth, are denominated respectively pyromancy, aeromancy, and geomancy.</p><p>Varro mentions the Persians as the first inventors of Hydromancy, adding, that Numa Pompilius and Pythagoras made use of it.</p><p>The writers on optics furnish us with divers Hydromantic machines, vessels, &c. For example,</p><p>To construct an Hydromantic machine, by means of which an image or object shall be removed out of the sight of the spectator, and restored again at pleasure, without altering the position, either of the one or the other. Provide two vessels, ABF, CGLK (Plate 12, fig. 3), the uppermost filled with water, and supported by three little pillars, one of which BC is hollow, and furnished with a cock B. Let the lower vessel CL be divided by a partition HI into two parts, the lower of which may be opened or closed by means of a cock at P. Upon the partition place an object, or image, which the spectator at O cannot see by a direct ray GL.</p><p>If now the cock B be opened, the water descending into the cavity CI, the ray GL will be refracted from the perpendicular GR to O; so that the spectator will now see the object by the refracted ray OG. And again, shutting the cock B, and opening the other P, the water will descend into the lower cavity HL; where, the refraction ceasing, no rays will now come from the object to the eye: but upon shutting the cock P, and opening the other B, the water will fill the cavity again, and bring the object in sight of O afresh.</p><p>To make an Hydromantic vessel, which shall exhibit the images of external objects as if swimming in water. Provide a cylindrical vessel ABCD (fig. 4, pl. 12.) divided into two cavities by a glass EF, not perfectly polished: in G apply a lens convex on both sides; and in H incline a plane mirror, of an elliptic figure, to an angle of 45 degrees; and let IH and HG be something less than the distance of the fccus of the lens G; so that the place of the images of an object radiating through the same may fall within the cavity of the upper vessel: let the inner cavity be blacked, and the upper filled with clear water.—If now the vessel be disposed in a dark place, so as the lens be turned towards an object illuminated by the sun, its image will be seen as swimming in the water.</p></div1><div1 part="N" n="HYDROMETER" org="uniform" sample="complete" type="entry"><head>HYDROMETER</head><p>, an instrument for measuring<cb/> the properties and effects of water, as its density, gravity, force, velocity, &c.</p><p>That with which the specifie gravity of water is determined, is often called an aerometer, or water-poise.</p><p>The general principle on which the construction and use of the Hydrometer depends, has been illustrated under the article <hi rend="italics">Spccific</hi> <hi rend="smallcaps">Gravity;</hi> where it is shewn that a body specifically lighter than several fluids, will serve to sind out their specific gravities; because it will sink deepest in the fluids whose specific gravity is the least. So if AB (fig. 5, pl. 12) be a small even glass tube, hermetically sealed, having a scale of equal divisions marked upon it, with a hollow ball of about an inch in diameter at bottom, and a smaller ball C under it, communicating with the first; into the little ball is put mercury or small shot, before the tube is sealed, so that it may sink in water below the ball, and float or stand upright, the divisions on the stem skewing how far it. sinks.—If this instrument be dipped in common water, and sink to D, it will sink only to some lower point E in salt water; but in port wine it will sink to some higher point F, and in brandy perhaps to B.</p><p>It is evident that an Hydrometer of this kind will only shew that one liquid is specifically heavier than another; but the true specific weight of any liquid cannot be determined without a calculation for this particular instrument, the tube of which should be truly cylindrical. Besides, these instruments will not serve for fluids whose densities are much different.</p><p>Mr. Clarke constructed a new Hydrometer, shewing whether any spirits be proof, or above or below proof, and in what degree. This instrument was made of a ball of copper (because ivory imbibes spirituous liquors, and glass is apt to break), to which is soldered a brass wire about a quarter of an inch thick; upon this wire is marked the point to which it exactly sinks in proo<*> spirits; as also two other marks, one above and one below the former, exactly answering to one-tenth above proof and one-tenth below proof. There are also a number of small weights made to add to it, so as to answer to the other degrees of strength besides those above, and for determining the specific gravities of different fluids. Philos. Trans. Abr. vol. vi, p. 326.</p><p>Dr. Desaguliers contrived an Hydrometer for determining the specific gravities of different waters, to such a degree of nicety, that it would shew when one kind of water was but the 40,000th part heavier than another. It consists of a hollow glass ball of about 3 inches in diameter, charged with shot to a proper degree, and having fixed in it a long and very slender wire, of only the 40th part of an inch in diameter, and divided into tenths of inches, each tenth answering to the 40,000th part, as above. See his Exper. Philos. vol. 2, p. 234.</p><p>Mr. Quin and other persons have also constructed Hydrometers, with other and various contrivances, and with different degrees of accuracy; but all nearly on the same general principles.</p><p>But there is one circumstance which deserves particular attention in the construction and graduation of Hydrometers, for determining the precise strength of different brandies, and other spirituous liquors. Mr. Reaumur discovered, in making his spirit thermometers, that when rectified spirit and water, or phlegm,<pb n="610"/><cb/> the other constituent part of brandy, are mixed together, there appears to be a mutual penetration of the two liquors, and not merely juxtaposition of parts; so that a part of the one fluid seems to be received into the pores of the other; by which it happens, that if a pint of rectified spirit be added to a pint of water, the mixture will be sensibly less than a quart. The variations hence produced in the bulk of the mixed fluid render the Hydrometer, when graduated in the usual way by equal divisions, an erroneous measure of its strength; because the specific gravity of the compound is found not to correspond to the mean gravity of the two ingredients. M. Montigny constructed a scale for this instrument in the manner before suggested by Dr. Lewis, on actual observation of the sinking or rising of the Hydrometer in various mixtures of alcohol and water, made in certain known proportions. Hist. de l'Acad. Roy. des Sci. 1768; also Neumann's Chem. by Lewis, p. 450, note <hi rend="italics">r.</hi></p><p>M. De Luc has lately published a scheme for the construction of a comparable Hydrometer, so that a workman, after having constructed one upon his principles, may make all others similar to each other, and capable of indicating the same degree on the scale, when immersed in the same liquor of the same temperature. This instrument is proposed to be constructed of a ball of flint glass, communicating with a small hollow cylinder, containing such a quantity of quicksilver for a ballast, that the instrument may sink nearly to the top, in the most spirituous liquor, made as hot as possible; to which is also attached a thin silvered tube, for a scale, &c. The whole description may be seen at large in the Philos. Trans. vol. 68, p. 500.</p><p>M. Le Roi also published a proposal for constructing comparable Hydrometers. See Hist. de l'Acad. des Scien. for 1770, Mem. 7.</p></div1><div1 part="N" n="HYDROMETRIA" org="uniform" sample="complete" type="entry"><head>HYDROMETRIA</head><p>, <hi rend="smallcaps">Hydrometry</hi>, the mensuration of water and other fluid bodies, their gravity, force, velocity, quantity, &c; including both hydrostatics and hydraulics.</p></div1><div1 part="N" n="HYDROSCOPE" org="uniform" sample="complete" type="entry"><head>HYDROSCOPE</head><p>, an instrument anciently used for the measure of time. It was a kind of water-clock, consisting of a cylindrical tube, conical at bottom: the cylinder was graduated with divisions, to which the top of the water becoming successively contiguous, as it trickled out of the vertex of the cone, pointed out the hour.</p><p>HYDROSTATICAL <hi rend="italics">Balance,</hi> a kind of balance contrived for the exact and easy finding the specific gravities of bodies, both solid and fluid, and thereby of estimating the degree of purity of bodies of all kinds, with the quality and richness of metals, ores, minerals, &c, and the proportions in any mixture, adulteration, or the like.</p><p>This is effected by weighing the body both in water, or other fluid, and out of it; and for this purpose one of the scales has usually a hook at the bottom, for suspending the body by some very fine thread. And the use of the instrument is founded on this theorem of Archimedes, that any body weighed in water, loses as much of its weight as is equal to the weight of the same bulk of the water. Thus then is known the proportion of the specisic gravities of the solid and fluid, or the proportion of their weights under the same bulk,<cb/> viz, the proportion of the weight of the body weighed out of water, to the difference between the same and its weight in water. Hence also, by doing the same thing for several different solids, with the same fluid, or different fluids with the same solid, all their specific gravities become known.</p><p>The instrument needs but little description. AB is <figure/> a nice balance beam, with its scales C and D, turning with the small part of a grain, the one of them, D, having a hook in the bottom, to receive the loop of a horse hair &c, E, by which the body F is suspended. GH is a jar of water, in which the body is immersed when weighing.</p><p>The pieces in the scale C denote the weight of the body out of water; then, upon immerging it, put weights in the scale D to restore the balance again, and they will shew the specific gravity of the body.</p><p>There have been various kinds of the Hydrostatical balance, and improvements made on it, by different persons. Thus, Dr. Desaguliers set three screws in the foot of the stand, to move any side higher or lower, till the stem be quite upright, which is known by a plummet hanging over a fixed point in the pedestal. Desag. Exp. Philos. vol. 2, p. 196. And for sundry other constructions of this instrument, designed for greater accuracy than the common sort, see Martin's Phil. Britan. or Gravesande's Physices Elem. Math. tom. 1, lib. 3, cap. 3, &c.</p><p>The specific gravities of small weights may be determined by suspending them in loops of horse hair, or fine silken threads, to the hook at the bottom of the scale. Thus, if a guinea suspended in air weigh 129 grains, and upon being immersed in water require 7 1/5 grains to be put in the scale over it, to restore the equilibrium; we thus find that a quantity of water of equal bulk with the guinea, weighs 7 1/5 grains, or 7.2; therefore dividing the 129 by the 7.2, the quotient 17.88 shews that the guinea is so many times heavier than its bulk of water. Whence, if any piece of gold be tried, by weighing it first in air, then in water, and if, upon dividing the weight in air by the loss in water, the quotient be 17.88, the gold is good; if the quotient be 18 or more, the gold is more fine; but if it be less than 17.88, the gold is too much alloyed with other metal. If silver be tried in the same manner, and found to be 11 times heavier than water, it is very fine; if it be 10 1/2 times heavier, it is standard; but if less, it is mixed with some lighter metal, such as tin.</p><p>When the body, whose specific gravity is sought, is lighter than water, so that it will not quite sink; annex<pb n="611"/><cb/> to it a piece of another body heavier than water, so that the mass compounded of the two may sink together. Weigh the denser body, and the compound mass, separately, both in water and out of it, thereby finding how much each loses in water; and subtract the less of these two losses from the greater; then say, As the remainder<lb/> is to the weight of the light body in air,<lb/> so is the specific gravity of water<lb/> to the specific gravity of the light body.<lb/></p><p>HYDROSTATICAL <hi rend="italics">Bellows,</hi> a machine for shewing the upward pressure of fluids <figure/> and the Hydrostatical paradox. It consists of two thick boards, A, D, each about 16 or 18 inches diameter, more or less, covered or connected firmly with leather round the edges, to open and shut like a common bellows, but without valves; only a pipe, B, about 3 feet high is fixed into the bellows at <hi rend="italics">e,</hi> Now let water be poured into the pipe at C, and it will run into the bellows, gradually separating the boards, by raising the upper one. Then if several weights, as three hundred weights, be laid upon the upper board, by pouring the water in at the pipe till it be full, it will sustain all the weights, though the water in the pipe should not weigh a quarter of a pound; for the pipe or tube may be as small as we please, provided it be but long enough, the whole effect depending upon the height, and not at all on the width of the pipe: for the proportion is always this, As the area of the orifice of the pipe<lb/> is to the area of the bellows board,<lb/> so is the weight of water in the pipe<lb/> to the weight it will sustain on the board.<lb/></p><p>Hence if a man stand upon the upper board, and blow into the pipe B, he will raise himself upon the board; and the smaller the pipe, the easier he will be able to raise himself; and then by putting his singer upon the top of the pipe, he can support himself as long as he pleases, provided the bellows be air-tight.</p><p>Mr. Ferguson has described another machine, which may be substituted instead of this common Hydrostatical bellows. It is however on the same principle of the Hydrostatical paradox; and may be seen in the Supplement to his Lectures, p. 19.</p><p>HYDROSTATICAL <hi rend="italics">Paradox,</hi> is a principle in Hydrostatics, so called because it has a paradoxical appearance at first view, and it is this; that any quantity of water, or other fluid, how small soever, may be made to balance and support any quantity, or any weight, how great soever. This is partly illustrated in the last article, on the Hydrostatical bellows, where it appears that any weight whatever may be blown up and supported by the breath from a person's mouth. And the principle may be explained as follows: It is well-known that water in a pipe or canal, open at both ends, always rises to the same height at both ends, whether those ends be wide or narrow, equal or unequal. Thus, the small pipe GH being close joined to another open vessel AI, of any size whatever; then<cb/> pouring water into the one of these, it will rise up in the other, and stand at the same height, or horizontal line DF in both of them, and that whether they are upright, or inclined in any position. So that all the water that is in the large vessel from A to I, is supported by that which is in the small vessel from D to I only. And as there is no limit to this latter one, but that it may be made as fine even as a hair, it hence evidently appears that any quantity of water may be thus supported by any other the smallest quantity. <figure/></p><p>Since then the pressure of fluids is directly as their perpendicular heights, without any regard to their quantities, it appears that whatever the figure or size of the vessels may be, if they are but of equal heights, and the areas of their bottoms equal, the pressures of equal heights of water are equal upon the bottoms of these vessels; even though the one should contain a thousand or ten thousand times as much as the other.</p><p>Mr. Ferguson confirms and illustrates this paradox by the following experiment. <figure/></p><p>Let two vessels be prepared of equal heights, but very unequal contents, such as AB and CD; each vessel being open at both ends, and their bottoms E and F of equal widths. Let a brass bottom G and H be exactly fitted to each vessel, not to go into it, but for it to stand upon; and let a piece of wet leather be put<pb n="612"/><cb/> between each vessel and its brass bottom, for the sake of closeness. Join each bottom to its vessel by a hinge D, so that it may open like the lid of a box; and let each bottom be kept up to its vessel by equal weights W, hung to lines which go over the pulleys P, whose blocks are fixed to the sides of the vessel at <hi rend="italics">f,</hi> and the lines tied to hooks at <hi rend="italics">d,</hi> fixed in the brass bottoms opposite to the hinges D. Things being thus prepared and fitted, hold one vessel upright in the hands over a bason on a table, and cause water to be poured slowly into it, till the pressure of the water bears down its bottom at the side <hi rend="italics">d,</hi> and raises the weight E; and then part of the water will run out at <hi rend="italics">d.</hi> Mark the height at which the surface H of the water stood in the vessel, when the bottom began to give way at <hi rend="italics">d;</hi> and then, holding up the other vessel in the same manner, cause water to be poured into it; and it will be seen that when the water rises in this vessel just as high as it did in the former, its bottom will also give way at <hi rend="italics">d,</hi> and it will lose part of the water.</p><p>The natural reason of this surprising phenomenon is, that since all parts of a fluid at equal depths below the surface, are equally pressed in all manner of directions, the water immediately below the fixed part B<hi rend="italics">f</hi> will be pressed as much upward against its lower surface within the vessel, by the action of the column A<hi rend="italics">g,</hi> as it would be by a column of the same height, and of any diameter whatever; and therefore, since action and reaction are equal and contrary to each other, the water immediately below the surface B<hi rend="italics">f</hi> will be pressed as much downward by it, as if it were immediately touched and pressed by a column of the height A<hi rend="italics">g,</hi> and of the diameter B<hi rend="italics">f;</hi> and therefore the water in the cavity BD<hi rend="italics">df</hi> will be pressed as much downward upon its bottom G, as the bottom of the other vessel is pressed by all the water above it. Lectures, p. 105.</p></div1><div1 part="N" n="HYDROSTATICS" org="uniform" sample="complete" type="entry"><head>HYDROSTATICS</head><p>, is the science which treats of the nature, gravity, pressure, and equilibrium of fluids; and of the weighing of solids in them.</p><p>That part of the science of fluids which treats of their motions, being included under the head of Hydraulics.</p><p>Hydrostatics and hydraulics together constitute a branch of philosophy that is justly considered as one of the most curious, ingenious, and useful of any; affording theorems and phenomena not only of the first use and importance, but also surprisingly amusing and pleasant; as appears in the numberless writings upon the subject; the principal points of which may be found under the several particular articles of this work; and the chief writings on this science may be seen under the article Hydraulics.</p></div1><div1 part="N" n="HYDRUS" org="uniform" sample="complete" type="entry"><head>HYDRUS</head><p>, or <hi rend="italics">Water Serpent,</hi> one of the new south- <*>rn constellations, including only ten stars.</p><p>HYEMAL <hi rend="italics">Solstice,</hi> the same with Winter Solstice. See <hi rend="smallcaps">Solstice.</hi></p></div1><div1 part="N" n="HYGROMETER" org="uniform" sample="complete" type="entry"><head>HYGROMETER</head><p>, or <hi rend="smallcaps">Hygroscope</hi>, or N<hi rend="smallcaps">OTIOMETER</hi>, an instrument for measuring the degrees of <*>oisture in the air.</p><p>There are various kinds of Hygrometers; for whatever body either swells by moisture, or shrinks by dryness, is capable of being formed into an Hygrometer. Such are woods of most kinds, particularly deal, ash, poplar, &c. Such also is catgut, the beard of a wild<cb/> oat, and twisted cord, &c. The best and most usual contrivances for this purpose are as follow. <figure/></p><p>1. Stretch a common cord, or a fiddle-string, ABD along a wall, passing it over a pulley B; fixing it at one end A, and to the other end hanging a weight E, carrying a style or index F. Against the same wall fit a plate of metal HI, graduated, or divided into any number of equal parts; and the Hygrometer is complete.</p><p>For it is matter of constant observation, that moisture sensibly shortens cords and strings; and that, as the moisture evaporates, they return to their former length again. The like may be said of a fiddle-string: and from hence it happens that such strings are apt to break in damp weather, if they are not slackened by the screws of the violin. Hence it follows, that the weight E will ascend when the air is more moist, and descend again when it becomes drier. By which means the index F will be carried up and down, and, by pointing to the several divisions on the scale, will shew the degrees of moisture or dryness.</p><p>2. Or thus, for a more sensible and accurate Hygro- <figure/> meter: strain a whipcord, or catgut, over several pulleys B, C, D, E, F; and proceed as before for the rest of the construction. Nor does it matter whether the several parts of the cord be parallel to the horizon, as expressed in the annexed figure, or perpendicular to the same, or in any other position; the advantage of this, over the former method, being merely the having a greater length of cord in the same compass; for the longer the cord, the greater is the contraction and dilatation, and consequently the degrees of variation of the index over the scale, for any given change of moisture in the air.<pb n="613"/><cb/></p><p>3. Or thus: Fasten a twisted <figure/> cord, or fiddle-string, AB, by one end at A, sustaining a weight at B, carrying an index C round a circular scale DE described on a horizontal board or table.—For a cord or catgut twists itself as it moistens, and untwists again as it dries. Hence, upon an increase or decrease of the humidity of the air, the index will shew the quantity of twisting or untwisting, and consequently the increase or decrease of moisture or dryness.</p><p>4. Those Dutch toys, called weather houses, where a small image of a man, and one of a woman, are fixed upon the ends of an index, are constructed upon this principle. For the index, being sustained by a cord or twisted catgut, turns backwards and forwards, bringing out the man in wet weather, and the woman in dry. <figure/></p><p>5. Or thus: Fasten one end of a cord, or catgut, AB, to a hook at A; and to the other end a ball D of about one pound weight; upon which draw two concentric circles, and divide them into any number of equal parts, for a scale; then fit a style or index EC into a proper support at E, so as the extremity C may almost touch the divisions of the ball.——Here the cord twisting or untwisting, as in the former case, will indicate the change of moisture, by the successive application of the divisions of the circular scale, as the ball turns round, to the index C.</p><p>6. Or an Hygrometer may be made of the thin boards of ash or <*>ir, by their swelling or contracting. But this, and all the other kinds of this instrument, above described, become in time sensibly less and less accurate; till at last they lose their effect entirely, and suffer no alteration from the weather. But the following sort is much more durable, serving for many years with tolerable accuracy. <figure/></p><p>7. Take the Manoscope, described under that article, and instead of the exhausted ball E, substitute a sponge, or other body, that easily imbibes moisture. To prepare the sponge, it may be proper first to wash it in water<cb/> very clean; and, when dry again, in water or vinegar in which there has been dissolved sal ammoniac, or salt of tartar; after which let it dry again.——Now, if the air become moist, the sponge will imbibe it and grow heavier, and consequently will preponderate, and turn the index towards C; on the contrary, when the air becomes drier, the sponge becomes lighter, and the index turns towards A; and thus shewing the state of the air.</p><p>8. In the last mentioned Hygrometer, Mr. Gould, in the Philos. Trans. instead of a sponge, recommends oil of vitriol, which grows sensibly lighter or heavier from the degrees of moisture in the air; so that being saturated in the moistest weather, it afterwards retains or loses its acquired weight, as the air proves more or less moist. The alteration in this liquor is so great, that in the space of 57 days it has been known to change its weight from 3 drachms to 9; and has shifted a tongue or index of a balance 30 degrees. So that in this way a pair of scales may afford a very nice Hygrometer. The same author suggests, that oil of sulphur or campanam, or oil of tartar per deliquium, or the liquor of fixed nitre, might be used instead of the oil of vitriol.</p><p>9. This balance may be contrived in two ways; by either having the pin in the middle of the beam, with a slender tongue a foot and a half long, pointing to the divisions on an arched plate, as represented in the last figure above. Or the scale with the liquor may be hung to the point of the beam near the pin, and the other <figure/> extremity made so long, as to describe a large arch on a board placed for the purpose; as in the figure here annexed.</p><p>10. Mr. Arderon has proposed some improvement in the sponge Hygrometer. He directs the sponge A to <figure/> be so cut, as to contain as large a superficies as possible, and to hang by a fine thread of silk upon the beam of a<pb n="614"/><cb/> balance B, and exactly balanced on the other side by another thread of silk at D, strung with the smallest lead shot, at equal distances, so adjusted as to cause an index E to point at G, the middle of a graduated arch FGH, when the air is in a middle state between the greatest moisture and the greatest dryness. Under this silk so strung with shot, is placed a little table or shelf I, for that part of the silk or shot to rest upon which is not suspended. When the moisture imbibed by the sponge increases its weight, it will raise the index, with part of the shot, from the table, and vice versa when the air is dry. Philos. Trans. vol. 44, p. 96.</p><p>11. From a series of Hygroscopical observations, made with an apparatus of deal wood, described in the Philos. Trans. number 480, Mr. Coniers concludes, 1st, That the wood shrinks most in summer, and swells most in winter, but is most liable to change in the spring and fall. 2d, That this motion happens chiefly in the day time, there being scarce any variation in the night. 3d, That there is a motion even in dry weather, the wood swelling in the morning, and shrinking in the afternoon. 4th, That the wood, by night as well as by day, usually shrinks when the wind is in the north, north-east, and east, both in summer and winter. 5th, That by constant observation of the motion and rest of the wood, with the help of a thermometer, the direction of the wind may be told nearly without a weather-cock. He adds, that even the time of the year may be known by it; for in spring it moves more and quicker than in winter; in summer it is more shrunk than in spring; and has less motion in autumn than in summer.</p><p>See an account of a method of constructing these and other Hygrometers, in Phil. Trans. Abr. vol. 2, p. 30, &c, and plate 1 annexed. See also Philos. Trans. vol. 11. p. 647 and 715, vol. 15, p. 1032, vol. 43, p. 6, vol. 44, p. 95, 169 and 184, vol. 54, p. 259, vol. 61, p. 198, vol. 63, p. 404, &c.</p><p>12. Dr. Hook's Hygrometer was made of the beard of a wild oat, set in a small box, with a dial plate and an index. See his Micrographia, p. 150.</p><p>13. The Doctors Hales and Desaguliers both contrived another form of sponge Hygrometer, on this principle. They made an horizontal axis, having a small <figure/> part of its length cylindrical, and the remainder tapering conically with a spiral thread cut in it, after the manner of the fuzee of a watch. The sponge is suspended by a fine silk thread to the cylindrical part of the axis, upon which it winds. This is balanced by a small weight W, suspended also by a thread, which<cb/> winds upon the spiral fuzee. Then when the sponge grows heavier, in moist weather; it descends and turns the axis, and so draws up the weight, which coming to a thicker part of the axis it becomes a balance to the sponge, and its motion is shewn by an attached scale. And vice versa when the air becomes drier.——Salt of tartar, or any other salt, or pot ashes, may be put into the scale of a balance, and used instead of the sponge. Desag. Exper. Philos. vol. 2, p. 300.</p><p>14. Mr. Ferguson made an Hygrometer of a thin deal pannel; and to enlarge the scale, and so render its variations more sensible, he employed a wheel and axle, making one cord pass over the axle, which turned a wheel ten times as large, over which passed a line with a weight at the end of it, whose motion was therefore ten times as much as that of the deal pannel. The board should be changed in 3 or 4 years. See Philos. Trans. vol. 54, art. 47.</p><p>15. Mr. Smeaton gave also an ingenious and elaborate construction of an Hygrometer; which may be seen in the Philos. Trans. vol. 61, art. 24.</p><p>16. Mr. De Luc's contrivance for an Hygrometer is very ingenious, and on this principle. Finding that even ivory swells with moisture, and contracts with dryness, he made a small and very thin hollow cylinder of ivory, open only at the upper end, into which is fitted the under or open end of a very fine long glass tube, like that of a thermometer. Into these is introduced some quicksilver, silling the ivory cylinder, and a small part of the length up the glass tube. The consequence is this: when moisture swells the ivory cylinder, its bore or capacity grows larger, and consequently the mercury sinks in the fine glass tube; and vice versa, when the air is drier, the ivory contracts, and forces the mercury higher up the tube of glass. It is evident that an instrument thus constructed is in fact also a thermometer, and must necessarily be affected by the vicissitudes of heat and cold, as well as by those of dryness and moisture; or that it must act as a thermometer as well as an Hygrometer. The ingenious contrivances in the structure and mounting of this instrument may be seen in the Philos. Trans. vol. 63, art. 38; where it may be seen how the above imperfection is corrected by some simple and ingenious expedients, employed in the original construction and subsequent use of the instrument; in consequence of which, the variations in the temperature of the air, though they produce their full effects on the instrument, as a thermometer, do not interfere with or embarrass its indications as an Hygrometer.</p><p>17. In the Philos. Trans. for 1791, Mr. De Luc has given a second paper on Hygrometry. This has been chiefly occasioned by a Memoir of M. de Saussure on the same subject, entitled Essais sur l'Hygrometrie, in 4to, 1783. In this work M. de S. describes a new Hygrometer of his construction, on the following principle. It is a known fact that a hair will stretch when it is moistened, and contract when dried: and M. de Saussure found, by repeated experiments, that the difference between the greatest extension and contraction, when the hair is properly prepared, and has a weight of about 3 grains suspended by it, is nearly one 40th of its whole length, or one inch in 40. This circumstance suggested the idea of a new Hygrometer. To<pb n="615"/><cb/> render these small variations of the length of the hair perceptible, an apparatus was contrived, in which one of the extremities of the hair is fixed, and the other, bearing the counterpoise abovementioned, surrounds the circumference of a cylinder, which turns upon an axis to which a hand is adapted, marking upon a dial in large divisions the almost insensible motion of this axis. About 12 inches high is recommended as the most convenient and useful: and to render them portable, a contrivance is added, by which the hand and the counterpoise can be occasionally fixed.</p><p>But M. de Luc, in his Idées sur la Meteorologie, vol. 1, anno 1786, shews that hairs, and all the other animal or vegetable hygroscopic substances, taken lengthwise, or in the direction of their fibres, undergo contrary changes from different variations of humidity; that when immersed in water, they lengthen at first, and afterwards shorten; that when they are near the greatest degree of humidity, if the moisture be increased, they shorten themselves; if it be diminished, they lengthen themselves first before they contract again. These irregularities, which render them incapable of being true measures of humidity, he shews to be the necessary consequence of their organic reticular structure De Saussure takes his point of extreme moisture from the vapours of water under a glass bell, keeping the sides of the bell continually moistened; and affirms, that the humidity is, there, constantly the same in all temperatures; the vapours even of boiling water having no other effect than those of cold. De Luc, on the contrary, shews that the differences in humidity under the bell are very great, though De Saussure's Hygrometer was not capable of discovering them; and that the real undecomposed vapour of boiling water has the directly opposite effect to that of cold, the effect of extreme dryness; and on this point he mentions an interesting fact, communicated to him by Mr. Watt, viz, that wood cannot be employed in the steam engine, for any of those parts where the vapour of the boiling water is confined, because it dries so as to crack as if exposed to the fire.</p><p>To these charges of M. De Luc, a reply is made by M. De Saussure, in his Defence of the Hair Hygrometer, in 1788; where he attributes the general disagreement between the two instruments, to irregularities of M. De Luc's; and assigns some aberrations of his own Hygrometer, which could not have proceeded from the above cause, but to its having been out of order; &c.</p><p>This has drawn from M. De Luc a second paper on Hygrometry, published in the Philos. Trans. for 1791, p. 1, and 389. This author here resumes the four fundamental principles which he had sketched out in the former paper, viz, 1st, That fire is a sure, and the only sure means of obtaining extreme dryness. 2d, That water, in its liquid state, is a sure, and the only sure means of determining the point of extreme moisture. 3d, There is no reason, a priori, to expect, from any hygroscopic substance, that the measurable effects, produced in it by moisture, are proportional to the intensities of that cause.—But, 4th, perhaps the comparative changes of the di<*>ensions of a substance, and of the weight of the same or other substances, by the same variations of moisture, may lead to some discovery in that respect. On these heads M. De Luc expatiates<cb/> at large in this paper, shewing the imperfections of M. De Saussure's principles of Hygrometry, and particularly as to a hair, or any such substance when extended lengthwise, being properly used as an Hygrometer. On the other hand, he shews that the expansion of substances across the fibres, or grain, renders them, in that respect, by far the most proper for this purpose. He chooses such as can be made very thin, as ivory, or deal shavings, but over all he finds whalebone to be far the best of any. But, for all the reasonings of these ingenious philosophers on this interesting subject, and complete information, see the publications above quoted, as also the Monthly Review, vol. 51, p. 224, vol. 71, p. 213, vol. 76, p. 316, vol. 78, p. 236, and vol. 6, of the new series for the year 1791, p. 133.</p></div1><div1 part="N" n="HYGROMETRY" org="uniform" sample="complete" type="entry"><head>HYGROMETRY</head><p>, the science of the measurement of the moisture of the atmosphere. The chief writings on this science are those of M. De Luc and M. De Saussure, for which see the last article on Hygrometers.</p></div1><div1 part="N" n="HYGROSCOPE" org="uniform" sample="complete" type="entry"><head>HYGROSCOPE</head><p>, is commonly used in the same sense with Hygrometer. Wolfius, however, regarding the etymology of the word, makes some difference. According to him, the Hygroscope only shews the alterations of the air in respect of humidity and dryness, but the hygrometer measures them. A Hygroscope therefore is only an indefinite or less accurate hygrometer.</p></div1><div1 part="N" n="HYPATIA" org="uniform" sample="complete" type="entry"><head>HYPATIA</head><p>, a very learned and beautiful lady, was born at Alexandria, about the end of the 4th century, as she flourished about the year of Christ 430. She was the daughter of Theon, a celebrated philosopher and mathematician, and president of the famous Alexandrian school. Her father, encouraged by her extraordinary genius, had her not only educated in all the ordinary qualifications of her sex, but instructed in the most abstruse sciences. She made such great progress in philosophy, geometry, astronomy, and the mathematics in general, that she passed for the most learned person of her time. She published commentaries on Apollonius's Conics, on Diophantus's Arithmetic, and other works. At length she was thought worthy to succeed her father in that distinguished and important employment, the government of the school of Alexandria; and to deliver instructions out of that chair where Ammonius, Hierocles, and many other great men, had taught before; and this at a time too when men of great learning abounded both at Alexandria and in many other parts of the Roman empire. Her same being so extensive, and her worth so universally acknowledged, it was no wonder that she had a crowded auditory. “She explained to her hearers (says Socrates, an ecclesiastical historian of the 5th century, born at Constantinople) the several sciences that go under the general name of philosophy; for which reason there was a confluence to her, from all parts, of those who made philosophy their delight and study.”</p><p>Her scholars were not less eminent than they were numerous. One of them was the celebrated Synesius, who was afterwards bishop of Ptolomais. This ancient Christian Platonist always expresses the strongest, as well as the most grateful, testimony of the virtue of his tutoress; never mentioning her without the most profound respect, and sometimes in terms of affection but<pb n="616"/><cb/> little short of adoration. But it was not Synesius only, and the disciples of the Alexandrian school, who admired Hypatia for her virtue and learning: never was woman more caressed by the public, and yet never had woman a more unspotted character. She was held as an oracle for her wisdom, for which she was consulted by the magistrates in all important cases; a circumstance which often drew her among the greatest concourse of men, without the least censure of her manners. In short, when Nicephorus intended to pass the highest compliment on the princess Eudocia, he thought he could not do it better than by calling her another Hypatia.</p><p>While Hypatia thus reigned the brightest ornament of Alexandria, Orestes was governor of that place for the emperor Theodosius, and Cyril was bishop or patriarch. Orestes, having had a liberal education, could not but admire Hypatia; and as a wife governor often consulted her. This, together with an aversion which Cyril had against Orestes, proved fatal to the lady. About 500 monks assembling, attacked the governor one day, and would have killed him, had he not been rescued by the townsmen; and the respect which Orestes had for Hypatia causing her to be traduced among the Christian multitude, they dragged her from her chair, tore her in pieces, and burnt her limbs.</p><p>Cyril is strongly suspected of having fomented this tragedy. Cave indeed endeavours to remove the imputation of so horrid an action from the patriarch; and lays it upon the Alexandrian mob in general, whom he calls “a very trifling inconstant people.” But though Cyril should be allowed neither to have been the perpetrator, nor even the contriver, of it, yet it is much to be suspected that he did not discountenance it in the manner he ought to have done: a suspicion which must needs be greatly confirmed by reflecting, that he was so far from blaming the outrage committed by the monks upon Orestes, that he afterwards received the dead body of Ammonius, one of the most forward in that outrage, who had grievously wounded the governor, and who was justly punished with death. Upon this riotous ruffian Cyril made a panegyric in the church where he was laid, in which he extolled his courage and constancy, as one that had contended for the truth; and changing his name to Thaumasius, or the Admirable, ordered him to be considered as a martyr. “However (continues Socrates), the wisest part of Christians did not approve the zeal which Cyril shewed on this man's behalf, being convinced that Ammonius had justly suffered for his desperate attempt.”</p></div1><div1 part="N" n="HYPERBOLA" org="uniform" sample="complete" type="entry"><head>HYPERBOLA</head><p>, one of the conic sections, being that which is made by a plane cutting a cone so, that, entering one side of the cone, and not being parallel to the opposite side, it may cut the circular base when the opposite side is ever so far produced below the vertex, or shall cut the opposite side of the cone produced above the vertex, or shall make a greater angle with the base than the opposite side of the cone makes; all these three circumstances amounting to the same thing, but in other words.</p><p>1. Thus, the figure DAE is an Hyperbola, made by a plane entering the side VQ of a cone PVQ at A, and either cutting the bafe PEQ when the plane is not parallel to VP, and this is ever so far produced; or when<cb/> the angle ARQ is greater than <figure/> the angle VPQ; or when the plane cuts the opposite side in B above the vertex.</p><p>2. By the Hyperbola is sometimes meant the whole plane of the section, and sometimes only the curve line of the section.</p><p>3. Hence, the cutting plane meets the opposite cone in B, and there forms another Hyperbola <hi rend="italics">d</hi> B <hi rend="italics">e,</hi> equal to the former one, and having the same transverse axis AB; and the same vertices A and B. Also the two are called Opposite Hyperbolas.</p><p>4. The centre C is the middle point of the tranverse axis.</p><p>5. The semi-conjugate axis is CL, a mean proportional between CI and CK, the distances to the sides of the opposite cone, when CI is drawn parallel to the diameter PQ of the base of the cone. Or the whole conjugate axis is a mean proportional between AF and BH, which are drawn parallel to the base of the cone. <figure/></p><p>6. If DAE and FBG be two opposite Hyperbolas, having the same transverse and conjugate axes AB and <hi rend="italics">a b,</hi> perpendicularly bisecting each other; and if <hi rend="italics">d a e</hi> and <hi rend="italics">f b g</hi> be two other opposite Hyperbolas, having the same axes with the two former, but in the contrary order, viz, having <hi rend="italics">a b</hi> for their first or transverse axis, and AB for their second or conjugate axis: then any two adjacent curves are called Conjugate Hyperbolas, and the whole figure formed by all the four curves, the Figure of the Conjugate Hyperbolas. And if the rectangle HIKL be inscribed within the four conjugate Hyperbolas, touching the vertices A, B, <hi rend="italics">a, b,</hi> and having their sides parallel and equal to the two axes; and if then the two diagonals HCK, ICL, of the parallelogram be drawn, these diagonals are the asymptotes of the curves, being lines that continually approach nearer and nearer to the curves, without meeting them, except at an infinite distance, where each asymptote and the two adjacent sides of the two conjugate Hyperbolas may be supposed all to meet; the asymptote being there a common tangent to them both, viz, at that infinite distance.</p><p>7. Hence the four Hyperbolas, meeting and running into each other at the infinite distance, may be cons<*>dered as the four parts of one entire curve, having the same axes, tangents, and other properties.</p><p>8. A Diameter in general, is any line, as MN, drawn through the centre C, and meeting, or termi-<pb n="617"/><cb/> nated by the opposite legs of the opposite Hyperbolas. And if parallel to this diameter there be drawn two tangents, at <hi rend="italics">m</hi> and <hi rend="italics">n,</hi> to the opposite legs of the other two opposite Hyperbolas, the line <hi rend="italics">m</hi>C<hi rend="italics">n</hi> joining the points of contact, is the conjugate diameter to MN, and the two mutually conjugates to each other. Or, if to the points M or N there be drawn a tangent, and through the centre C the line <hi rend="italics">mn</hi> parallel to it, that line will be the conjugate to MN. The points where each of these meet the curves, as M, N, <hi rend="italics">m, n,</hi> are the vertices of the diameters; and the tangents to the curves at the two vertices of any diameter, are parallel to each other, and also to the other or conjugate diameter. <figure/></p><p>9. Moreover, if those tangents to the four Hyperbolas, at the vertices of two conjugate diameters, be produced till they meet, they will form a parallelogram OPQR; and the diagonals OQ and PR of the parallelogram will be the asymptotes of the curves; which therefore pass through the opposite angles of all the parallelograms so inscribed between the curves. Also it is a property of these parallelograms, that they are all equal to each other, and therefore equal to the rectangle of the two axes; as will be farther noticed below. Farther, if these diagonals or asymptotes make a right angle between them, or if the inscribed parallelogram be a square, or if the two axes be equal to each other, then the Hyperbola is called a right-angled or an equilateral one.</p><p>10. An Ordinate to any diameter, is a line drawn parallel to its conjugate, or to the tangent at its vertex, and terminated by the diameter produced and the curve. So MS and TN are ordinates to the axis AB; also AD and BG are ordinates to the diameter MN (last fig. but one). Hence the ordinates to the axis are perpendicular to it; but ordinates to the other diameters are oblique to them.</p><p>11. An absciss is a part of any diameter, contained between its vertex and an ordinate to it; and every ordinate has two abscisses: as AT and BT, or MV and NV.</p><p>12. The Parameter of any diameter, is a third proportional to the diameter and its conjugate.—The Parameter of the axis is also equal to the line AG or B<hi rend="italics">g</hi> (fig. 1), if FG be drawn to make the angle AFG = the angle BAV, or the line H<hi rend="italics">g</hi> to make the angle BH<hi rend="italics">g</hi> = the angle ABV.</p><p>13. The Focus is the point in the axis where the ordinate is equal to half the parameter of the axis; as<cb/> S and T (fig. 2) if MS and TN be half the parámeter, or the 3d proportional to CA and C<hi rend="italics">a.</hi> Hence there are two Foci, one on each side the vertex, or one for each of the opposite Hyperbolas. These two points in the axis are called Foci, or burning points, because it is found by opticians that rays of light issuing from one of them, and falling upon the curve of the Hyperbola, are reflected into lines that verge towards the other point or Focus. <hi rend="center"><hi rend="italics">To describé an Hyperbola, in various ways.</hi></hi></p><p>14. (1st <hi rend="italics">Way by points.)</hi>—In the transverse axis AF produced, take the foci F and <hi rend="italics">f,</hi> by making CF and C<hi rend="italics">f</hi> = A<hi rend="italics">a</hi> or B<hi rend="italics">a,</hi> assume any point I: Then with the <figure/> radii AI, BI, and centres F, <hi rend="italics">s,</hi> describe arcs intersecting in E, which will give four points in the curves. In like manner, assuming other points I, as many other points will be found in the curve. Then, with a steady hand, draw the curve line through all the points of intersection E.—In the same manner are to be constructed the other pair of opposite Hyperbolas, using the axis <hi rend="italics">ab</hi> instead of AB.</p><p>15. (2d <hi rend="italics">Way by points, for a Right-angled Hyperbola only.)</hi>—On, the axis produced if necessary, take any <figure/> point I, through which draw a perpendicular line, upon which set off IM and IN equal to the distance I<hi rend="italics">a</hi> or I<hi rend="italics">b</hi> from I to the extremities of the other axis; and M and N will be points in the curve.</p><p>16. (3d <hi rend="italics">Way by points, to describe the curve through a given point.)</hi>—CG and CH being the asymptotes, and P the given point of the curve; through the point P draw any line GPH between the asymptotes, upon which take GI = PH, so shall I be another point of the curve. And in this manner may any number of points be found, drawing as many lines through the given point P.<pb n="618"/><cb/></p><p>17. (4th <hi rend="italics">Way by a continued Motion.)</hi>—If one end <figure/> of a long ruler <hi rend="italics">f</hi>MO be fastened at the point <hi rend="italics">f,</hi> by a pin on a plane, so as to turn freely about that point as a centre. Then take a thread FMO, shorter than the ruler, and fix one end of it in F, and the other to the end O of the ruler. Then if the ruler <hi rend="italics">f</hi>MO be turned about the fixed point <hi rend="italics">f,</hi> at the same time keeping the thread OMF always tight, and its part MO close to the side of the ruler, by means of the pin M; the curve line AX described by the motion of the pin M is one part of an Hyperbola. And if the ruler be turned, and move on the other side of the fixed point F, the other part AZ of the same Hyperbola may be described after the same manner.—But if the end of the ruler be sixed in F, and that of the thread in <hi rend="italics">f,</hi> the opposite Hyperbola <hi rend="italics">xaz</hi> may be described.</p><p>18. (5th <hi rend="italics">Way, by a continued Motion.)</hi>—Let C and F be the two foci, and E and K the two vertices of the Hyperbola. (See the last fig. above.) Take three rulers CD, DG, GF, so that CD = GF = EK, and DG = CF; the rulers CD and GF being of an indefinite length beyond C and G, and having slits in them for a pin to move in; and the rulers having holes in them at C and F, to fasten them to the foci C and F by means of pins, and at the points D and G they are to be joined by the ruler DG. Then, if a pin be put in the slits, viz, the common intersection of the rulers CD and GF, and moved along, causing the two rulers GF, CD, to turn about the foci C and F, that pin will describe the portion E<hi rend="italics">e</hi> of an Hyperbola.—The foregoing are a few among various ways given by several authors. <hi rend="center"><hi rend="italics">Some of the chief Properties os the Hyperbola.</hi></hi> <figure/></p><p>19. (1st) The squares of the ordinates, of any diameter, are to each other, as the rectangles of their<cb/> abscisses; i. e. .</p><p>20. As the square of any diameter, is to the square of its conjugate; so is the rectangle of two abscisses, to the square of their ordinate. That is, .</p><p>Or, because the rectangle AD . BD is = the difference of the squares CD<hi rend="sup">2</hi> - CB<hi rend="sup">2</hi>, the same property is, , Or ,</p><p>That is , where <hi rend="italics">p</hi> is the parameter of the diameter AB, or the 3d proportional <hi rend="italics">ab</hi><hi rend="sup">2</hi>/(AB).</p><p>And hence is deduced the common equation of the Hyperbola, by which its general nature is expressed. Thus, putting <hi rend="italics">d</hi> = the semidiameter CA or CB, <hi rend="italics">c</hi> = its semiconjugate C<hi rend="italics">a</hi> or C<hi rend="italics">b,</hi> <hi rend="italics">p</hi> = its parameter or 2<hi rend="italics">d</hi><hi rend="sup">2</hi>/<hi rend="italics">c,</hi> <hi rend="italics">x</hi> = the absciss BD from the vertex, <hi rend="italics">y</hi> = the ordinate DE, and <hi rend="italics">v</hi> = the absciss CD from the centre: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Then is </cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">; so that</cell></row></table> <*> any of which equations or proportions express the nature of the curve. And hence arises the name Hyperbola, signifying to exceed, because the ratio of <hi rend="italics">d</hi><hi rend="sup">2</hi> to <hi rend="italics">c</hi><hi rend="sup">2</hi>, or of <hi rend="italics">d</hi> to <hi rend="italics">p,</hi> exceeds that of <hi rend="italics">2dx</hi> to <hi rend="italics">y</hi><hi rend="sup">2</hi>; that ratio being equal in the parabola, and defective in the ellipse, from which circumstances also these take their names.</p><p>21. The distance between the centre and the focus, is equal to the distance between the extremities of the transverse and conjugate axes. That is, CF = A<hi rend="italics">a</hi> or A<hi rend="italics">b,</hi> where F is the focus.</p><p>22. The conjugate semi-axis is a mean proportional between the distances of the focus from both vertices of the transverse. That is, C<hi rend="italics">a</hi> is a mean between AF and BF, or , or .</p><p>23. The difference of two lines drawn from the foci, to meet in any point of the curve, is equal to the transverse axis. That is, <hi rend="italics">f</hi>E - FE = AB, where F and <hi rend="italics">f</hi> are the two foci.</p><p>24. All the parallelograms inscribed between the four conjugate Hyperbolas are equal to one another, and each equal to the rectangle of the two axes. That is, the parallelogram OPQR = AB . <hi rend="italics">ab</hi> (fig. to art. 9).</p><p>25. The difference of the squares of every pair of conjugate diameters, is equal to the same constant quantity, viz, the difference of the squares of the two axes. That is, , (fig. to art. 6)<*> where MN and <hi rend="italics">mn</hi> are any two conjugate diameters.<pb n="619"/><cb/></p><p>26. The rectangles of the parts of two parallel lines, terminated by the curve, are to one another, as the rectangles of the parts of any other two parallel lines, any where cutting the former. Or the rectangles of the parts of two intersecting lines, are as the squares of their parallel diameters, or squares of their parallel tangents.</p><p>27. All the rectangles are equal which are made of the segments of any parallel lines, cut by the curve, and limited by the asymptotes, and each equal to the square of their parallel diameter. That is, HE . EK or or CP<hi rend="sup">2</hi>. <figure/></p><p>28. All the parallelograms are equal, which are formed between the asymptotes and curve, by lines parallel to the asymptotes. That is, the paral. CGEK = CPBQ.—Hence is obtained another method of expressing the nature of the curve by an equation, involving the absciss taken on one asymptote, and ordinate parallel to the other asymptote. Thus, if <hi rend="italics">x</hi> = CK, <hi rend="italics">y</hi> = KE, <hi rend="italics">a</hi> = CQ, and <hi rend="italics">b</hi> = BQ the ordinate at the vertex B of the curve; then, by the property in this article, <hi rend="italics">ab</hi> = <hi rend="italics">xy,</hi> or ; that is, the rectangle of the absciss and ordinate is every where of the same magnitude, or any ordinate is reciprocally as its absciss.</p><p>29. If the abscisses CQ, CK, CL, &c, taken on the one asymptote, be in geometrical progression increasing; then shall the ordinates QB, KE, LM, &c, parallel to the other asymptote, be a like geometrical progression in the same ratio, but decreasing; and all the rectangles are equal, under every absciss and its ordinate, viz, , &c.</p><p>30. The abscisses CQ, CK, CL, &c, being taken in geometrical progression; the spaces or asymptotie areas BQKE, EKLM, &c, will be all equal; or, the spaces BQKE, BQLM, &c, will be in arithmetical progression; and therefore these spaces are the hyperbolic logarithms of those abscisses.</p><p>These, and many other curious properties of the Hyperbola, may be seen demonstrated in my Treatise on Conic Sections, and several others. See also <hi rend="smallcaps">Conic Sections.</hi></p><p><hi rend="italics">Acute</hi> <hi rend="smallcaps">Hyperbola</hi>, one whose asymptotes make an acute angle.</p><p><hi rend="italics">Ambigenal</hi> <hi rend="smallcaps">Hyperbola</hi>, is that which has one of its infinite legs falling within an angle formed by the asymptotes, and the other falling without that angle. This is one of Newton's triple Hyperbolas of the 2d<cb/> order. See his Enumeratio Lin. tert. Ord. See also <hi rend="smallcaps">Ambigenal.</hi></p><p><hi rend="italics">Common,</hi> or <hi rend="italics">Conic</hi> <hi rend="smallcaps">Hyperbola</hi>, is that which arises from the section of a cone by a plane; called also the Apollonian Hyperbola, being that kind treated on by the first and chief author Apollonius.</p><p><hi rend="italics">Conjugate</hi> <hi rend="smallcaps">Hyperbolas</hi>, are those formed or lying together, and having the same axes, but in a contrary order, viz, the transverse of each equal the conjugate of the other; as the two Conjugate Hyperbolas <hi rend="italics">Pee</hi> and EEE in the last figure but one.</p><p><hi rend="italics">Equilateral,</hi> or <hi rend="italics">Rectanglar</hi> <hi rend="smallcaps">Hyperbola</hi>, is that whose two axes are equal to each other, or whose asymptotes make a right angle.—Hence, the property or equation of the equilateral Hyperbola, is , where <hi rend="italics">a</hi> is the axis, <hi rend="italics">x</hi> the absciss, and <hi rend="italics">y</hi> its ordinate; which is similar to the equation of the circle, viz, , differing only in the sign of the second term, and where <hi rend="italics">a</hi> is the diameter of the circle.</p><p><hi rend="italics">Infinite</hi> <hi rend="smallcaps">Hyperbolas</hi>, or <hi rend="smallcaps">Hyperbolas</hi> <hi rend="italics">of the higher kinds,</hi> are expressed or defined by general equations similar to that of the conic or common Hyperbola, but having general exponents, instead of the particular numeral ones, but so as that the sum of those on one side of the equation, is equal to the sum of those on the other side. Such as, , where <hi rend="italics">x</hi> and <hi rend="italics">y</hi> are the absciss and ordinate to the axis or diameter of the curve; or , where the absciss <hi rend="italics">x</hi> is taken on one asymptote, and the ordinate <hi rend="italics">y</hi> parallel to the other.</p><p>As the Hyperbola of the first kind, or order, viz the conic Hyperbola, has two asymptotes; that of the 2d kind or order has three; that of the 3d kind, four; and so on.</p><p><hi rend="italics">Obtuse</hi> <hi rend="smallcaps">Hyperbola</hi>, is that whose asymptotes form an obtuse angle.</p><p><hi rend="italics">Rectangular</hi> <hi rend="smallcaps">Hyperbola</hi>, the same as Equilateral Hyperbola.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Arc,</hi> is the arc of an Hyperbola.</p><p>Put <hi rend="italics">a</hi> = CA the semitransverse axe, <hi rend="italics">c</hi> = C<hi rend="italics">a</hi> the semiconjugate, <hi rend="italics">y</hi> = an ordinate PQ to the axe drawn from the end Q of the arc AQ, beginning at the vertex <figure/> A: then putting , &c; then is the length of the arc AQ expressed by &c;<pb n="620"/><cb/> or by , nearly; where <hi rend="italics">t</hi> is the whole transverse axe 2CA, <hi rend="italics">c</hi> = 2C<hi rend="italics">a</hi> the conjugate, <hi rend="italics">x</hi> = AP the absciss, and <hi rend="italics">y</hi> = PQ the ordinate.</p><p>These and other rules may be seen demonstrated in my Mensuration, p. 408, &c, 2d edit.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Area,</hi> or <hi rend="italics">Space,</hi> the area or space included by the Hyperbolic curve and other lines.</p><p>Putting <hi rend="italics">a</hi> = CA the semitransverse, <hi rend="italics">c</hi> = C<hi rend="italics">a</hi> the semiconjugate, <hi rend="italics">y</hi> = PQ the ordinate, and <hi rend="italics">v</hi> = CP its distance from the centre; then is the area ; sector ; area ; or nearly.</p><p>Let CT and CE be the two asymptotes, and the ordinates DA, EF parallel to the other asymptote CT; then the asymptotic space ADEF or sector CAF is or or &c; and this last series was first given by Mercator in his Logarithmotechnia.</p><p>See my Mensuration, p. 413, &c, 2d edit.</p><p>Generally, if be an equation expressing an Hyperbola of any order; then its asymptotic area will be ; which space therefore is always quadrable, in all the orders of Hyperbolas, except the first or common Hyperbola only, in which <hi rend="italics">m</hi> and <hi rend="italics">n</hi> being each 1, the denominator <hi rend="italics">n</hi> - <hi rend="italics">m</hi> becomes 0 or nothing.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Conoid,</hi> a solid formed by the revolution of an Hyperbola about its axis, otherwise called an Hyperboloid. <hi rend="center"><hi rend="italics">To find the Solid Content of an Hyperboloid.</hi></hi></p><p>Let AC be the semitransverse of the generating Hyperbola, and AH the height of the solid; then as 2AC + AH is to 3AC + AH, so is the cone of the same base and altitude, to the content of the Conoid. <figure/><cb/> <hi rend="center"><hi rend="italics">To find the Curve Surface of an Hyperboloid.</hi></hi></p><p>Let AC be the semitransverse, and AB perpendicular to it, and equal to the semiconjugate of ADE the generating Hyperbola, or section through the axis of the solid. Join CB; make CF = CA, and on CA let fall the perpendicular FG; then with the semitransverse CG, and semiconjugate GH = AB, describe the Hyperbola GIK; then as the diameter of a circle is to its circumference, so is the Hyperbolic frustum ILAMK to the curve surface of the Conoid generated by DAE. See my Mensur. p. 429, &c, 2d edit.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Cylindroid,</hi> a solid formed by the revolution of an Hyperbola about its conjugate axis, or line through the centre perpendicular to the transverse axis. This solid is treated of in the Philos. Trans. by Sir Christopher Wren, where he shews some of its properties, and applies it to the grinding of Hyperbolical Glasses; affirming that they must be formed this way, or not at all. See Philos. Trans. vol. 4, pa. 961.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Leg,</hi> of a curve, is that having an asymptote, or tangent at an infinite distance.—Newton reduces all curves, both of the first and higher kinds, into Hyperbolic and parabolic legs, i. e. such as have asymptotes, and such as have not, or such as have tangents at an infinite distance, and such as have not.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Line,</hi> is used by some authors for what is more commonly called the Hyperbola itself, being the curve line of that figure; in which sense the surface terminated by it is called the Hyperbola.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Logarithm,</hi> a logarithm so called as being similar to the asymptotic spaces of the Hyperbola. The Hyperbolic logarithm of a number, is to the common logarithm, as 2.3025850929940457 to 1, or as 1 to .4342944819032518. The first invented logarithms, by Napier, are of the Hyperbolic kind; and so are Kepler's. See <hi rend="smallcaps">Logarithm.</hi></p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Mirror,</hi> is one ground into that shape.</p><p><hi rend="smallcaps">Hyperbolic</hi> <hi rend="italics">Space,</hi> that contained by the curve of the Hyperbola, and certain other lines. See H<hi rend="smallcaps">YPERBOLIC Area.</hi></p><p>HYPERBOLICUM <hi rend="italics">Acutum,</hi> a solid made by the revolution of the infinite area or space contained between the curve of the Hyperbola, and its asymptote. This produces a solid, which though infinitely long and generated by an infinite area, is nevertheless equal to a finite solid body; as is demonstrated by Torricelli, who gave it this name.</p><p>HYPERBOLIFORM <hi rend="italics">Figures,</hi> are such curves as approach, in their properties, to the nature of the Hyperbola; called also Hyperboloides.</p></div1><div1 part="N" n="HYPERBOLOIDS" org="uniform" sample="complete" type="entry"><head>HYPERBOLOIDS</head><p>, are Hyperbolas of the higher kind, whose nature is expressed by this equation, . See <hi rend="smallcaps">Hyperbola.</hi> It also means the Hyperbolic Conoid. See that article.</p></div1><div1 part="N" n="HYPERBOREANS" org="uniform" sample="complete" type="entry"><head>HYPERBOREANS</head><p>, the most northern nations, or regions, as dwelling beyond or about the wind Boreas: as the Siberians, Samoieds, &c.</p></div1><div1 part="N" n="HYPERTHYRON" org="uniform" sample="complete" type="entry"><head>HYPERTHYRON</head><p>, in Architecture, a sort of table, usually placed over gates or doors of the Doric order, above the chambranle, in form of a frize.</p></div1><div1 part="N" n="HYPETHRE" org="uniform" sample="complete" type="entry"><head>HYPETHRE</head><p>, in Ancient Architecture, two rows<pb n="621"/><cb/> of pillars surrounding, and ten at each face of a temple, &c, with a peristyle within of six columns.</p></div1><div1 part="N" n="HYPOGEUM" org="uniform" sample="complete" type="entry"><head>HYPOGEUM</head><p>, in the ancient Architecture, a name common to all the parts of a building that are under ground; as the cellars, butteries, &c.</p><div2 part="N" n="Hypogeum" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Hypogeum</hi></head><p>, in Astrology, a name given to the celestial houses that are below the horizon; and especially the imum cœli, or bottom point of the heavens.</p></div2></div1><div1 part="N" n="HYPOMOCHLION" org="uniform" sample="complete" type="entry"><head>HYPOMOCHLION</head><p>, the fulcrum or prop of a lever; or the point which sustains its pressure, when employed either in raising or lowering bodies. The Hypomochlion is frequently a roller set under the lever; or under stones or pieces of timber, &c, that they may be the more easily lifted up, or removed.</p></div1><div1 part="N" n="HYPOTENUSE" org="uniform" sample="complete" type="entry"><head>HYPOTENUSE</head><p>, or <hi rend="smallcaps">Hypothenuse</hi>, in a rightangled triangle, is the side which subtends, or is opposite to the right angle, and is always the longest of the three sides; as the side AC, opposite to the right angle B. <figure/></p><p>It is a celebrated theorem in Plane Geometry, being the 47th prop. of the 1st book of Euclid, that in every right-angled triangle ABC, the square formed upon the Hypothenuse AC, is equal to both the two squares formed upon the other two sides AB and BC; or that . This is particularly called the Pythagorean theorem, from its reputed inventor Pythagoras, who it is said sacrificed a whole hecatomb to the muses, in gratitude for the discovery. But the same thing is true of circles or any other similar figures, viz, that any figure described on the Hypotenuse, is equal to the sum of the two similar figures described on both the other two sides.</p><p>HYPOTHENUSE. See <hi rend="smallcaps">Hypotenuse.</hi></p></div1><div1 part="N" n="HYPOTHESIS" org="uniform" sample="complete" type="entry"><head>HYPOTHESIS</head><p>, in Geometry, or Mathematics, means much the same thing with supposition, being a<cb/> supposition or an assumption of something as a condition, upon which to raise a demonstration, or from which to draw an inference.</p><p>Dr. Barrow says, Hypotheses, or postulatums, are propositions assuming or affirming some evidently pos<*> sible mode, action, or motion of a thing, and that there is the same affinity between hypotheses and problems, as between axioms and theorems: a problem shewing the manner, and demonstrating the possibility of some structure, and an Hypothesis assuming some construction which is manifestly possible.</p><div2 part="N" n="Hypothesis" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Hypothesis</hi></head><p>, in Philosophy, denotes a kind of system laid down from our own imagination, by which to account for some phenomenon or appearance of nature. Thus there are Hypotheses to account for the tides, for gravity, for magnetism, for the deluge, &c.</p><p>The real and scientific causes of natural things generally lie very deep: observation and experiment, the proper means of arriving at them, are in most cases extremely slow; and the human mind is very impatient: hence we are often induced to feign or invent something that may seem like the cause, and which is calculated to answer the several phenomena, so that it may possibly be the true cause.</p><p>Philosophers are divided as to the use of such fictions or Hypotheses, which are much less current now than they were formerly. The latest and best writers are for excluding Hypotheses, and standing intirely on observation and experiment. Whatever is not deduced from phenomena, says Newton, is an Hypothesis; and Hypotheses, whether metaphysical, or physical, or mechanical, or of occult qualities, have no place in experimental philosophy. Phil. Nat. Prin. Math. in Calce.</p><p><hi rend="smallcaps">Hypothfsis</hi> is more particularly applied, in Astronomy, to the several systems of the heavens; or the divers manners in which different astronomers have supposed the heavenly bodies to be ranged, or moved. The principal Hypotheses are the Ptolomaic, the Tychonic, and the Copernican. This last is now so generally received, and so well established and warranted by observation, that it is thought derogatory to it to call it an Hypothesis.</p></div2></div1><div1 part="N" n="HYPOTRACHELION" org="uniform" sample="complete" type="entry"><head>HYPOTRACHELION</head><p>, in Architecture, is used for a little frize in the Tuscan and Doric capital, between the astragal and annulets; called also the colerin and gorgerin.</p><p>The word is applied by some authors in a more general sense, to the neck of any column, or that part of its capital below the astragal.<pb n="622"/> <hi rend="center">I. J.</hi><cb/></p></div1><div1 part="N" n="JACK" org="uniform" sample="complete" type="entry"><head>JACK</head><p>, in Mechanics, is an instrument in common use for raising heavy timber, or very great weights of any kind; being a certain very powerful combination of teeth and pinions, and the whole inclosed in a strong wooden stock or frame BC, and moved by a winch or handle HP; the outside appearing as in fig. 1, here annexed. <figure/></p><p>In fig. 6, pl. 12, the wheel or rack-work is shewn, being the view of the inside when the stock is removed. Though it is not drawn in the just proportions and dimensions, for the rack AB must be supposed at least four times as long in proportion to the wheel Q, as the figure represents it; and the teeth, which will be then four times more in number to have about 3 in the inch. Now if the handle HP be 7 inches long, the circumference of this radius will be 44 inches, which is the distance or space the power moves through in one revolution of the handle: but as the pinion of the handle has but 4 leaves, and the wheel Q suppose 20 teeth, or 5 times the number, therefore to make one revolution of the wheel Q, it requires 5 turns of the handle, in which case it passes through 5 times 44 or 220 inches: but the wheel having a pinion R of 3 leaves, these will raise the rack 3 teeth, or one inch, in the same space. Hence then, the handle or power moving 220 times as fast as the weight, will raise or balance a weight of 220 times its own power. And if this be the hand of a man, who can sustain 100 pounds weight, he will, by help of this Jack, be able to raise, or sustain a weight or force of 22000lb, or about 10 tons weight.</p><p>This machine is sometimes open behind from the bottom almost up to the wheel Q, to let the lower claw, which in that case is turned up as at B, draw up any weight. When the weight is drawn or pushed sufficiently high, it is kept from going back by hanging the end of the hook S, fixed to a staple, over the curved part of the handle at <hi rend="italics">h.</hi></p><p><hi rend="smallcaps">Jack</hi> is also the name of a well-known engine in the kitchen, used for turning a spit. Here the weight is the power applied, acting by a set of pulleys; the friction of the parts, and the weight with which the<cb/> spit is charged, make the force to be overcome; and a steady uniform motion is maintained by means of the fly.</p><p>See the fig. of this machine, pl. 12, fig. 7.</p><p><hi rend="italics">Smoke</hi> <hi rend="smallcaps">Jack</hi>, is an engine used for the same purpose with the common Jack, and is so called from its being moved by means of the smoke, or rarefied air, ascending the chimney, and striking against the fails of the horizontal wheel AB (plate 12, fig. 8), which being inclined to the horizon, is moved about the axis of the wheel, together with the pinion C, which carries the wheels D and E; and E carries the chain F, which turns the spit. The wheel AB should be placed in the narrow part of the chimney, where the motion of the smoke is swiftest, and where also the greatest part of it must strike upon the sails.—The force of this machine depends upon the draught of the chimney, and the strength of the sire.</p><p><hi rend="smallcaps">Jack</hi>-<hi rend="italics">arch,</hi> in Architecture, is an arch of one brick thickness.</p><p><hi rend="smallcaps">Jack</hi>-<hi rend="italics">head,</hi> in Hydraulics, a part sometimes annexed to the forcing pump.</p><p>JACOB's-<hi rend="italics">Staff,</hi> a mathematical instrument for taking heights and distances; the same with the Crossstaff; which see.</p></div1><div1 part="N" n="JACOBUS" org="uniform" sample="complete" type="entry"><head>JACOBUS</head><p>, a gold coin, worth 25 shillings; so called from king James the first of England, in whose reign it was struck. They distinguished two kinds of the Jacobus, the old and the new; the former valued at 25 shillings, weighing 6 dwts 10 grs; the latter, called also Carolus, valued at 23 shillings, and weighing 5 dwts 20 grains.</p></div1><div1 part="N" n="JAMBS" org="uniform" sample="complete" type="entry"><head>JAMBS</head><p>, or <hi rend="smallcaps">Jaums</hi>, in Architecture, are the upright sides of chimneys, from the hearth to the mantletree. Also door posts, or the upright posts at the ends of the window frames.</p><p><hi rend="italics">St</hi> JAMES's <hi rend="italics">Day,</hi> a festival in the calendar, observed on the 25th of July, in honour of St. James the apostle.</p></div1><div1 part="N" n="JANUARY" org="uniform" sample="complete" type="entry"><head>JANUARY</head><p>, the first month of the year, according to the computation now used in the West, and containing 31 days; so called by the Romans from Janus, one of their divinities, to whom they gave two faces; because on the one side, the first day of this month looked towards the new year, and on the other towards the old one. The name may also be derived from Janua, a gate; this month, being the first of the year, may be considered as the gate or entrance of it.<pb n="623"/><cb/></p><p>January and February were introduced into the year by Numa Pompilius; Romulus's year beginning with the month of March.</p><p>JAUMS. See <hi rend="smallcaps">Jambs.</hi></p></div1><div1 part="N" n="ICE" org="uniform" sample="complete" type="entry"><head>ICE</head><p>, a brittle transparent body, formed of some fluid, frozen or fixed by cold. The specific gravity of Ice to water, is various, according to the nature and circumstances of the water, degree of cold, &c. Dr. Irving (Phipps's Voyage towards the North Pole) found the densest Ice he could meet with about a 14th part lighter than water. M. de Mairan found it, at different trials, 1-14th, 18th, or 19th lighter than water; and when the water was previously purged of air, only a 22d part.</p><p>The rarefaction of Ice has been supposed owing to the <*>ir-bubbles produced in Ice while freezing; these, being considerably large in proportion to the water frozen, render the Ice so much specifically lighter. It is well known that a considerable quantity of air is lodged in the interstices of water, though it has there little or no elastic property, on account of the disunion of its particles; but upon these particles coming closer together, and uniting as the water freezes, light, expansive, and elastic air-bubbles are thus generated, and increase in bulk as the cold grows stronger, and by their elastic force bursts to pieces any vessel in which the water is closely contained. But snow-water, or any water long boiled over the fire, affords an Ice more solid, and with fewer bubbles. Pure water long kept in vacuo and frozen afterwards there, freezes much sooner, on being exposed to the same degree of cold, than water unpurged of its air and set in the open atmosphere. And the Ice made of water thus divested of its air, is much harder, more solid and transparent, and heavier than common Ice.</p><p>But M. de Mairan, in a dissertation on Ice, attributes the increase of the bulk of the water under this form, chiefly to a different arrangement of its parts: the icy skin on water being composed of filaments which are found to be joined constantly and regularly at an angle of 60°, and which, by this disposition, occupy a greater volume than if they were parallel. Besides, after Ice is formed, he found it continue to expand by cold; a piece of Ice, which was at first only a 14th part specifically lighter than water, on being exposed some days to the frost, became a 12th part lighter; and thus he accounts for the bursting of Ice in ponds.</p><p>It appears from an experiment of Dr. Hooke, in 1663, that Ice refracts the light less than water; whence he infers, that the lightness of Ice, which causes it to swim in water, is not produced merely by the small bubbles which are visible in it, but that it arises from the uniform constitution or general texture of the whole mass: a fact which was afterward confirmed by M. de la Hire. See Hooke's Exper. by Derham, p. 26, Acad. Per. 1693, Mem. p. 25.</p><p>Sir Robert Barker thus describes the process of making Ice in the East Indies, in a country where he never saw any natural Ice. On a large plain they dig three or four pits, each about 30 feet square, and 2 feet deep; the bottoms of which are covered, about 8 or <*>2 inches thick, with sugar-cane, or the stems of the<cb/> large Indian corn, dried. On this bed are placed in rows a number of small shallow unglazed earthen pans, formed of a very porous earth, a quarter of an inch thick, and about an inch and a quarter deep; which, at the dusk of the evening, they fill with soft water that has been boiled. In the morning before sunrise the Ice-makers attend at the pits, and collect what has been frozen in baskets, which they convey to the place of preservation. This is usually prepared in some high and dry situation, by sinking a pit 14 or 15 feet deep, which they line first with straw, and then with a coarse kind of blanketing. The Ice is deposited in this pit, and beaten down with rammers, till at length its own accumulated cold again freezes it, and it forms one solid mass. The mouth of the pit is well secured from the exterior air with straw and blankets, and a thatched roof is thrown over the whole. Philos. Trans. vol. 65, p. 252.</p></div1><div1 part="N" n="ICHNOGRAPHY" org="uniform" sample="complete" type="entry"><head>ICHNOGRAPHY</head><p>, in Architecture, is a transverse or horizontal section of a building, exhibiting the plot of the whole edifice, and of the several rooms and apartments in any story; together with the thickness of the walls and partitions; the dimensions of the doors, windows, and chimneys; the projectures of the columns and piers, with every thing visible in such a section.</p><div2 part="N" n="Ichnography" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ichnography</hi></head><p>, in Fortification, is the plan or representation of the length and breadth of a fortress; the distinct parts of which are marked out, either on the ground itself, or upon paper.</p></div2><div2 part="N" n="Ichnography" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ichnography</hi></head><p>, in Perspective, the view of any thing cut off by a plane parallel to the horizon, just by the base or bottom of it; being the same with what is otherwise called the plan, geometrical plan, or groundplot, of any thing, and is opposed to Orthography or Elevation.</p></div2></div1><div1 part="N" n="ICOSAEDRON" org="uniform" sample="complete" type="entry"><head>ICOSAEDRON</head><p>, or <hi rend="smallcaps">Icosahedron</hi>, one of the five regular bodies or solids, terminated by twenty equilateral and equal triangles. It may be considered as consisting of 20 equal and similar triangular pyramids, whose vertices meet in the centre of a sphere conceived to circumscribe it, and therefore having all their heights and bases equal; therefore the solidity of one of those pyramids multiplied by 20, the number of them, gives the solid content of the Icosaedron.</p><p><hi rend="italics">To form or make the Icosaedron.</hi>—Describe upon <*> card paper, or some other such like substance, 20 equilateral triangles, as in the figure at the article <hi rend="italics">Regular</hi> <hi rend="smallcaps">Body.</hi> Cut it out by the extreme edges, and cut all the other lines half through, then sold the sides up by these edges half cut through, and the solid will be formed.</p><p>The linear edge or side of the Icosaedron being A, then will the surface be , and the solidity = .</p><p>More generally, put A = the linear edge or side, B the surface, and C the solid content of the Icosaedron, also <hi rend="italics">r</hi> the radius of the inscribed, and R the radius of the circumscribing sphere, then we have these general equations, viz,<pb n="624"/><cb/> 1st, . 4th, . See my Mensuration, p. 258, 2d edit.</p></div1><div1 part="N" n="IDES" org="uniform" sample="complete" type="entry"><head>IDES</head><p>, in the Roman Calendar, a name given to a <*>eries of 8 days in each month; which, in the full months, March, May, July, and October, commenced on the 15th day; and in the other months, on the 13th day; from thence reckoned backward, so as in those four months to terminate on the 8th day, and in the rest on the 6th. These came between the calends and the nones. And this way of counting is still used in the Roman Chancery, and in the Calendar of the Breviary.</p><p>The Ides of May were consecrated to Mercury; the Ides of March were always esteemed unhappy, after the death of Cæsar; the time after the Ides of June was reckoned fortunate for those who entered into matrimony; the Ides of August were consecrated to Diana, and were observed as a feast by the slaves; on the Ides of September, auguries were taken for appointing the magistrates, who formerly entered into their offices on the Ides of May, and afterwards on those of March.</p><p>JET <hi rend="smallcaps">D'EAU</hi>, a French word, signifying a fountain that throws up water to some height in the air.</p><p>A Jet of water is thrown up by the weight of the column of water above its ajutage, or orifice, up to its source or reservoir; and therefore it would rise to the same height as the head or reservoir, if certain causes did not prevent it from rising quite so high. For first, the velocity of the lower particles of the Jet being greater than that of the upper, the lower water strikes that which is next above it; and as fluids press every way, by its impulfe it widens, and consequently shortens the column. Secondly, the water at the top of the Jet does not immediately fall off, but forms a kind of ball or head, the weight of which depresses the Jet; but if the Jet be a little inclined, or not quite upright, it will play higher, though it will not be quite so beautiful. Thirdly, the friction against the sides of the pipe and hole of the ajutage, will prevent the Jet from rising quite so high, and a small one will be more impeded than a large one. And th<*> fourth cause is the resistance of the air, which<cb/> is proportional to the square of the velocity of the water nearly; and therefore the defect in the height will be nearly in the same proportion, which is also the same as the proportion of the heights of the reservoirs above the ajutage. Hence, and from experience, it is found that a Jet, properly constructed, will rise to different heights according to the height of the reservoir, as in the following table of the heights of reservoirs and the heights of their corresponding Jets; the former in feet, and the latter in feet and tenths of a foot. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=8" role="data"><hi rend="italics">Heights of Reservoirs and their Jets.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Res.</cell><cell cols="1" rows="1" role="data">Jet.</cell><cell cols="1" rows="1" role="data">Res.</cell><cell cols="1" rows="1" role="data">Jet.</cell><cell cols="1" rows="1" role="data">Res.</cell><cell cols="1" rows="1" role="data">Jet.</cell><cell cols="1" rows="1" role="data">Res.</cell><cell cols="1" rows="1" role="data">Jet.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">4.9</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">28.3</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">49.0</cell><cell cols="1" rows="1" rend="align=right" role="data">82</cell><cell cols="1" rows="1" role="data">67.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">5.9</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">29.2</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">49.7</cell><cell cols="1" rows="1" rend="align=right" role="data">83</cell><cell cols="1" rows="1" role="data">67.7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">6.8</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">30.0</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">50.5</cell><cell cols="1" rows="1" rend="align=right" role="data">84</cell><cell cols="1" rows="1" role="data">68.4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">7.8</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">30.8</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">51.2</cell><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" role="data">69.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" rend="align=right" role="data">8.7</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">31.6</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">52.0</cell><cell cols="1" rows="1" rend="align=right" role="data">86</cell><cell cols="1" rows="1" role="data">69.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">9.7</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">32.5</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">52.7</cell><cell cols="1" rows="1" rend="align=right" role="data">87</cell><cell cols="1" rows="1" role="data">70.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">10.6</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">33.3</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">53.5</cell><cell cols="1" rows="1" rend="align=right" role="data">88</cell><cell cols="1" rows="1" role="data">71.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">11.6</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">34.1</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">54.2</cell><cell cols="1" rows="1" rend="align=right" role="data">89</cell><cell cols="1" rows="1" role="data">71.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">12.5</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">34.9</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">54.9</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" role="data">72.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" rend="align=right" role="data">13.4</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">35.7</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">55.7</cell><cell cols="1" rows="1" rend="align=right" role="data">91</cell><cell cols="1" rows="1" role="data">73.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">14.3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">36.6</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">56.4</cell><cell cols="1" rows="1" rend="align=right" role="data">92</cell><cell cols="1" rows="1" role="data">73.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" rend="align=right" role="data">15.2</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">37.4</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">57.1</cell><cell cols="1" rows="1" rend="align=right" role="data">93</cell><cell cols="1" rows="1" role="data">74.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" rend="align=right" role="data">16.1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">38.1</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">57.8</cell><cell cols="1" rows="1" rend="align=right" role="data">94</cell><cell cols="1" rows="1" role="data">75.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">17.0</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">38.9</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">58.6</cell><cell cols="1" rows="1" rend="align=right" role="data">95</cell><cell cols="1" rows="1" role="data">75.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">17.9</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">39.8</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">59.3</cell><cell cols="1" rows="1" rend="align=right" role="data">96</cell><cell cols="1" rows="1" role="data">76.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=right" role="data">18.8</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">40.5</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">60.0</cell><cell cols="1" rows="1" rend="align=right" role="data">97</cell><cell cols="1" rows="1" role="data">77.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">19.7</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">41.3</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">60.7</cell><cell cols="1" rows="1" rend="align=right" role="data">98</cell><cell cols="1" rows="1" role="data">77.8</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">20.6</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">42.1</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">61.4</cell><cell cols="1" rows="1" rend="align=right" role="data">99</cell><cell cols="1" rows="1" role="data">78.5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">21.5</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">42.9</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">62.1</cell><cell cols="1" rows="1" rend="align=right" role="data">100</cell><cell cols="1" rows="1" role="data">79.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">22.3</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">43.7</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">62.8</cell><cell cols="1" rows="1" rend="align=right" role="data">110</cell><cell cols="1" rows="1" role="data">85.6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">23.2</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">44.4</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">63.5</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell><cell cols="1" rows="1" role="data">91.9</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">24.1</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">45.2</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">64.2</cell><cell cols="1" rows="1" rend="align=right" role="data">130</cell><cell cols="1" rows="1" role="data">98.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">24.9</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">46.0</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">64.9</cell><cell cols="1" rows="1" rend="align=right" role="data">140</cell><cell cols="1" rows="1" role="data">104</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">25.8</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">46.7</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">65.6</cell><cell cols="1" rows="1" rend="align=right" role="data">150</cell><cell cols="1" rows="1" role="data">110</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">26.6</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">47.5</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">66.3</cell><cell cols="1" rows="1" rend="align=right" role="data">160</cell><cell cols="1" rows="1" role="data">116</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">27.5</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">48.2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>By various experiments that have been made by Mariotte, Desaguliers, and others, it has been found, that if the res<*>voir be 5 feet high, a conduct pipe 1 3/4 inch diameter will admit a hole in the ajutage from 1/4 to 3/<*> of an inch; and so on as in the following table: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Height Reservoi<*></cell><cell cols="1" rows="1" role="data">Diam. of the Ajutage.</cell><cell cols="1" rows="1" role="data">Diam. of the Conduct Pipe.</cell></row><row role="data"><cell cols="1" rows="1" role="data">  5 feet</cell><cell cols="1" rows="1" role="data"><*>/4 to 3/8 inch</cell><cell cols="1" rows="1" role="data">1 3/4 inch</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 10</cell><cell cols="1" rows="1" role="data"><*>/4 to 1/2</cell><cell cols="1" rows="1" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 15</cell><cell cols="1" rows="1" role="data">1/2</cell><cell cols="1" rows="1" role="data">2 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 20</cell><cell cols="1" rows="1" role="data">1/2</cell><cell cols="1" rows="1" role="data">2 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 25</cell><cell cols="1" rows="1" role="data">1/2</cell><cell cols="1" rows="1" role="data">2 3/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 30</cell><cell cols="1" rows="1" role="data">1/2 to 3/4</cell><cell cols="1" rows="1" role="data">3 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 40</cell><cell cols="1" rows="1" role="data">3/4</cell><cell cols="1" rows="1" role="data">4 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 50</cell><cell cols="1" rows="1" role="data">3/4</cell><cell cols="1" rows="1" role="data">5 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 60</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">5 3/4 or 6</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 80</cell><cell cols="1" rows="1" role="data">1 1/4</cell><cell cols="1" rows="1" role="data">6 1/2 or 7</cell></row><row role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">1 1/4 to 1 1/2</cell><cell cols="1" rows="1" role="data">7 or 8</cell></row></table><pb n="625"/><cb/> But the size of the pipe will be more or lesa with the distance.</p><p>If it be required to keep any number of Jets of given dimensions playing, by one common conduct-pipe; the diameter of an ajutage must be found that shall be equal to all the small ones that are given, and from this its proper conduct-pipe. Thus, if there be 4. ajutages, each 3/4 of an inch diameter; then the square of 3/4 is 9/16, which multiplied by 4, the number of them, makes 36/16, the square root of which is 6/4 or 1 1/2, the diameter of an ajutage equal to all the other four; to which in the table answe<*>s a pipe of 8 inches diameter. In general, the diameter of the conduct-pipe should be about 6 times that of the ajutage.</p><p>See Mariotte's Mouvement des Eaux; Desaguliers's Exper. Philos. vol. 2, p. 127, &c; Clare's Motion of Fluids, p. 109; &c.</p></div1><div1 part="N" n="JETTE" org="uniform" sample="complete" type="entry"><head>JETTE</head><p>, the border made round the stilts under a pier, in certain old bridges, being the same with starling, consisting of a strong framing of timber filled with stones, chalk, &c; to preserve the foundations of the piers from injury.</p><p>IGNIS <hi rend="smallcaps">Fatuus</hi>, a common met<*>or, chiefly seen in dark nights about meadows, marshes, and other moist places, as also in burying grounds, and near dung-hills. It is known among the people by the appellations, Will with a Wisp, and Jack with a Lantern.</p><p>Dr. Shaw describes a remarkable Ignis Fatuus, which he saw in the Holy Land, that was sometimes globular, or in the form of the flame of a candle; and presently afterward it spread itself so much as to involve the whole company in a pale harmless light, and then contract itself again, and suddenly disappear. But in less than a minute it would become visible as before; or, running along from one place to another, with a swift progressive motion, would expand itself at certain intervals over more than 2 or 3 acres of the adjacent mountains. The atmosphere had been thick and hazy, and the dew on the horses' bridles was uncommonly clammy and unctuous. In the same weather he observed those luminous appearances, which skip about the masts and yards of ships at sea, and which the sailors call corpusanse, by a corruption of the Spanish cuerposanto. Shaw's Travels, p. 363.</p><p>Newton calls it a vapour shining without heat; and supposed it to be of the same nature with the light issuing from putrescent substances. Willughby and Ray were of opinion that it is occasioned by shining insects: but all the appearances of it observed by Derham, Beccaria, and others, sufficiently evince that it must be an ignited vapour. Inflammable air has been found to be the most common of all the factitious airs in nature; and that it is the usual product of the putrefaction and decomposition of vege<*> able substances in water. Signor Volta writes to Dr. Priestley, that he fires inflammable air by the electric spark, even when the electricity is very moderate: and he supposes that this experiment explains the inflammation of the Ignes Fatui, provided they consist of inflammable air, issuing from marshy ground by help of the electricity of fogs, and by falling stars, which have probably an electrical origin. See Priestley's Obs. on Air, vol. 3, p. 382; the Philos. Trans. Abr. vol. 7, p. 147 &c.<cb/></p></div1><div1 part="N" n="ILLUMINATION" org="uniform" sample="complete" type="entry"><head>ILLUMINATION</head><p>, the act or effect of a luminous body, or a body that emits light; sometimes also the state of another body that receives it.</p><p><hi rend="italics">Circle of</hi> <hi rend="smallcaps">Illumination.</hi> See <hi rend="smallcaps">Circle.</hi></p><p>ILLUMINATIVE <hi rend="italics">Lunar Month,</hi> the space of time that the moon is visible, between one conjunction and another.</p></div1><div1 part="N" n="IMAGE" org="uniform" sample="complete" type="entry"><head>IMAGE</head><p>, in Optics, is the spectre or appearance of an object, made either by reflection or refraction.</p><p>In all plane mirrors, the Image is of the same magnitude as the object; and it appears as far behind the mirror as the object is before it. In convex mirrors, the Image appears less than the object; and farther distant from the centre of the convexity, than from the point of reflection. Mr. Molyneux gives the following rule for finding the diameter of an Image, projected in the distinct base of a convex mirror, viz, As the distance of the object from the mirror, is to the distance from the Image to the glass; so is the diameter of the object, to the diameter of the Image. See <hi rend="smallcaps">Lens, mirror, reflection</hi>, and <hi rend="smallcaps">Refraction.</hi></p><p>IMAGINARY <hi rend="italics">Quantities,</hi> or Impossible Quantities, in Algebra, are the even roots of negative quantities; which expressions are Imaginary, or impossible, or opposed to real quantities; as √- <hi rend="italics">aa,</hi> or √<hi rend="sup">4</hi>- <hi rend="italics">a</hi><hi rend="sup">4</hi>, &c. For, as every even power of any quantity whatever, whether positive or negative, is necessarily positive, or having the sign +, because + by +, or - by - give equally +; from hence it follows that every even power, as the square for instance, which is negative, or having the sign -, has no possible root; and therefore the even roots of such powers or quantities are said to be impossible or Imaginary. The mixt expressions arising from Imaginary quantities joined to real ones, are also Imaginary; as <hi rend="italics">a</hi> - √- <hi rend="italics">aa,</hi> or <hi rend="italics">b</hi> + √- <hi rend="italics">aa.</hi></p><p>The roots of negative quantities were, perhaps, first treated of in Cardan's Algebra. As to the uneven roots of such quantities, he shews that they are negative, and he assigns them: but the even roots of them he rejects, observing that they are nothing as to common use, being neither one thing nor another; that is, they are merely Imaginary or impossible. And since his time, it has gradually become a part of Algebra to treat of the roots of negative quantities. Albert Girard, in his <hi rend="italics">Invention Nouvelle en l'Algebre,</hi> p. 42, gives names to the three sorts of roots of equations, calling them, greater than nothing, less than nothing, and <hi rend="italics"><*>nvelopée,</hi> as √- 3: but this was soon after called Imaginary or impossible, as appears by Wallis's Algebra, p. 264, &c; where he observes that the square root of a negative quantity, is a mean proportional between a positive and a negative quantity; as √- <hi rend="italics">bc</hi> is the mean proportional between + <hi rend="italics">b</hi> and - <hi rend="italics">c,</hi> or between - <hi rend="italics">b</hi> and + <hi rend="italics">c;</hi> and this he exemplifies by geometrical constructions. See also p. 313.</p><p>The arithmetic of these Imaginary quantities has not yet been generally agreed upon; viz, as to the operations of multiplication, division, and involution; some authors giving the results with +, and others on the contrary with the negative sign -. Thus, Euler, in his Algebra, p. 106 &c, makes the square of √- 3 to be - 3, of √- 1 to be - 1, &c; and yet he makes the product of two impossibles, when they are unequal, to be possible and real: as <pb n="626"/><cb/> ; and or 2. But how can the equality or inequality of the factors cause any difference in the signs of the products? If be , how can , which is the square of √- 3, be - 3? Again, he makes . Also in division, he makes to be = √+ 4 or 2; and ; also that 1 or ; consequently, multiplying the quotient root √- 1 by the divisor √- 1, must give the dividend √+ 1; and yet, by squaring, he makes the square of √- 1, or the product , equal to - 1.</p><p>But Emerson makes the product of Imaginaries to be Imaginary; and for this reason, that “otherwise a real product would be raised from impossible factors, which is absurd. Thus, and &c. Also and &c.” And thus most of the writers on this part of Algebra, are pretty equally divided, some making the product of impossibles real, and others Imaginary.</p><p>In the Philos. Trans. for 1778, p. 318 &c, Mr. Playfair has given an ingenious dissertation “On the Arithmetic of Impossible Quantities.” But this relates chiefly to the applications and uses of them, and not to the algorithm of them, or rules for their products, quotients, squares, &c. From some operations however here performed, we learn that he makes the product of √- 1 by √- 1, or the square of √- 1, to be - 1; and yet in another place he makes the product of √- 1 and to be Mr. Playfair concludes, “that Imaginary expressions are never of use in investigations but when the subject is a property common to the measures both of ratios and of angles; but they never lead to any consequence which might not be drawn from the affinity between those measures; and that they are indeed no more than a particular method of tracing that affinity. The deductions into which they enter are thus reduced to an argument from analogy, but the fo<*>ce of them is not diminished on that account. The laws to which this analogy is subject; the cases in which it is perfect, in which it suffers certain alterations, and in which it is wholly interrupted, are capable of being precisely ascertained. Supported on so sure a foundation, the arithmetic of impossible quantities will always remain an useful instrument in the discovery of truth, and may be of service when a more rigid analysis can hardly be applied. For this reason, many researches concerning it, which in themselves might be deemed absurd, are nevertheless not destitute of utility. M. Bernoulli has found, for example, that if <hi rend="italics">r</hi> be the radius of a circle, the circumference is . Considered as a quadrature of the circle, this Imaginary theorem is wholly insignificant, and would deservedly pass for an abuse of calculation; at the same time we learn from it, that if in any equation the quantity <cb/> should occur, it may be made to disappear, by the substitution of a circular arch, and a property, common to both the circle and hyperbola, may be obtained. The same is to be observed of the rules which have been invented for the transformation and reduction of impossible quantities*; they facilitate the operations of this imaginary arithmetic, and thereby lead to the knowledge of the most beautiful and extensive analogy which the doctrine of quantity has yet exhibited.</p><p>* The rules chiefly referred to, are those for reducing the impossible roots of an equation to the form .”</p><p><hi rend="smallcaps">Imaginary</hi> <hi rend="italics">Roots,</hi> of an equation, are those roots or values of the unknown quantity in an equation, which contain some Imaginary quantity. So the roots of the equation , are the two Imaginary quantities + √- <hi rend="italics">a a</hi> and - √- <hi rend="italics">a a,</hi> or + <hi rend="italics">a</hi> √- 1 and - <hi rend="italics">a</hi> √- 1; also the two roots of the equation , are the Imaginary quantities ; and the three roots of the equation , or , are 1 and and , the first real, and the two latter Imaginary. Sometimes too the real root of an equation may be expressed by Imaginary quantities; as in the irreducible case of cubic equations, when the root is expressed by Cardan's rule; and that happens whenever the equation has no Imaginary roots at all; but when it has two Imaginary roots, then the only real root is expressed by that rule in an Imaginary form. See my paper on Cubic Equations, in the Philos. Trans. for 1780, p. 406 &c.</p><p>Albert Girard first treated expressly on the impossible or Imaginary roots of equations, and shewed that every equation has as many roots, either real or Imaginary, as the index of the highest power denotes. Thus, the roots of the biquadratic equation , he shews are two real and two Imaginary, viz, 1, 1, , and ; and he renders the relation general, between all the roots and the coeffi<*>ients of the terms of the equation. See his <hi rend="italics">Invention Nouvelle en l'Algebre, anno</hi> 1629, theor. 2, pa. 40 &c.</p><p>M. D'Alembert demonstrated, that every Imaginary root of any equation can always be reduced to the form , where <hi rend="italics">e</hi> and <hi rend="italics">f</hi> are real quantities. And hence it was also shewn, that if one root of an equation be , another root of it will always be <*> and hence it appears that the number of the Imaginary roots in any equation is always even, if any; i. e. either none, or else two, or four, or six, &c. Memoirs of the Academy of Berlin, 1746.</p><p>To discover how many impossible roots are contained in any proposed equation, Newton gave this rule, in his Algebra, viz, Constitute a series of fractions, whose denominators are the series of natural numbers 1, 2, 3, 4, 5, &c, continued to the number shewing the index or exponent of the highest term of the equations, and their numerators the same series of numbers in the contrary order: and divide each of these fractions by that next before it, and place the resulting quotients over the intermediate terms of the equation; then under each of the intermediate terms, if its square multiplied<pb n="627"/><cb/> by the fraction over it, be greater than the product of the terms on each side of it, place the sign +; but if not, the sign -; and under the first and last term place the sign +. Then will the equation have as many Imaginary roots as there are changes of the underwritten signs from + to -, and from - to +. So for the equation , the series of fractions is 3/1, 2/2, 1/3; then the second divided by the first gives 1/6 or 1/3, and the third divided by the second gives 1/3 also; hence these quotients placed over the intermediate terms, the whole will stand thus, . + + - + Now because the square of the 2d term multiplied by its superscribed fraction, is 16/3<hi rend="italics">x</hi><hi rend="sup">4</hi>, which is greater than 4<hi rend="italics">x</hi><hi rend="sup">4</hi> the product of the two adjacent terms, therefore the sign + is set below the 2d term; and because the square of the 3d term multiplied by its overwritten fraction, is 1<*>/3<hi rend="italics">x</hi><hi rend="sup">2</hi>, which is less than 24<hi rend="italics">x</hi><hi rend="sup">2</hi> the product of the terms on each side of it, therefore the sign - is placed under that term; also the sign + is fet under the first and last terms. Hence the two changes of the underwritten signs + + - +, the one from + to -, and the other from - to +, shew that the given equation has two impossible roots.</p><p>When two or more terms are wanting together, under the place of the 1st of the deficient terms write the sign -, under the 2d the sign +, under the 3d -, and so on, always varying the signs, except that under the last of the deficient terms must always be set the sign +, when the adjacent terms on both sides of the deficient terms have contrary <*>igns. As in the equation , + + - + - + which has four Imaginary roots.</p><p>The author remarks, that this rule will sometimes fail of discovering all the impossible roots of an equation, sor some equations may have more of such roots than can be found by this rule, tho' this seldom happens.</p><p>Mr. Maclaurin has given a demonstration of this rule of Newton's, together with one of his own, that will never fail. And the same has also been done by Mr. Campbell. See Philos. Trans. vol. 34, p. 104, and vol. 35, p. 515.</p><p>The real and imaginary roots of equations may be found from the method of fluxions, applied to the doctrine of maxima and minima, that is, to find such a value of <hi rend="italics">x</hi> in an equation, expressing the nature of a curve, made equal to <hi rend="italics">y,</hi> an abscissa which corresponds to the greatest and least ordinate. But when the equation is above 3 dimensions, the computation is very laborious. See Stirling's treatise on the lines of the 3d order, Schol. pr. 8, pa. 59, &c.</p></div1><div1 part="N" n="IMBIBE" org="uniform" sample="complete" type="entry"><head>IMBIBE</head><p>, is commonly used in the same sense as absorb, viz, where a dry porous body takes up another that is moift.</p></div1><div1 part="N" n="IMMENSE" org="uniform" sample="complete" type="entry"><head>IMMENSE</head><p>, that whose amplitude or extension cannot be equalled by any measure whatsoever, or how often soever repeated.</p></div1><div1 part="N" n="IMMERSION" org="uniform" sample="complete" type="entry"><head>IMMERSION</head><p>, the act of plunging into water, or some other fluid.</p><div2 part="N" n="Immersion" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Immersion</hi></head><p>, in Astronomy, is when a star, or<cb/> planet comes so near the sun, that it cannot be seen<*> being as it were enveloped, and hid in the rays of that luminary.</p><p><hi rend="smallcaps">Immersion</hi> also denotes the beginning of an eclipse, or of an occultation, when the body, or any part of it just begins to disappear, either behind the edge of another body, or in its shadow. As, in an eclipse of the moon, when she begins to be darkened by entering into the shadow of the earth: or the beginning of an eclipse of the sun, when the moon's disc just begins to cover him: or the beginning of the eclipses of any of the satellites, as those of Jupiter, by entering into his shadow: or, lastly, the beginning of an occultation of any star or planet, by passing behind the body of the moon or another planet. In all these cases, the darkened body is said to immerge, or to be immerged, or begin to be hid, by dipping as it were into the shade. In like manner, when the darkened body begins to appear again, it is said to emerge, or come out of darkness again.</p></div2></div1><div1 part="N" n="IMPACT" org="uniform" sample="complete" type="entry"><head>IMPACT</head><p>, the simple or single action of one body upon another to put it in motion. Point of Impact, is the place or point where a body acts.</p></div1><div1 part="N" n="IMPENETRABILITY" org="uniform" sample="complete" type="entry"><head>IMPENETRABILITY</head><p>, a quality by which a thing cannot be pierced or penetrated; or a property of body by which it fills up certain spaces, so that there is no room in them for any other body.</p></div1><div1 part="N" n="IMPENETRABLE" org="uniform" sample="complete" type="entry"><head>IMPENETRABLE</head><p>, that cannot be penetrated.</p><p>IMPERFECT <hi rend="italics">Number,</hi> is that whose aliquot parts, taken all together, do not make a sum that is equal to the number itself, but either exceed it, or fall short of it; being an abundant number in the former case, and a defective number in the latter. Thus, 12 is an abundant Imperfect number, because the sum of all its aliquot parts, 1, 2, 3, 4, 6, makes 16, which exceeds the number 12. And 10 is a defective Imperfect number, because its aliquot parts, 1, 2, 5, taken all together, make only 8, which is less than the number 10 itself.</p><p>IMPERIAL <hi rend="italics">Table,</hi> is an instrument made of brass, with a box and needle, and staff, &c, used for measuring of land.</p></div1><div1 part="N" n="IMPERVIOUS" org="uniform" sample="complete" type="entry"><head>IMPERVIOUS</head><p>, not to be pervaded or entered either because of the closeness of the pores, or the particular configuration of its parts.</p></div1><div1 part="N" n="IMPETUS" org="uniform" sample="complete" type="entry"><head>IMPETUS</head><p>, in Mechanics, force, momentum, motion, &c.</p><p>IMPOSSIBLE <hi rend="italics">Quantity,</hi> or <hi rend="italics">Root,</hi> the same as</p><p><hi rend="smallcaps">Imaginary</hi> ones; which see.</p></div1><div1 part="N" n="IMPOST" org="uniform" sample="complete" type="entry"><head>IMPOST</head><p>, in Architecture, a capital or plinth, to a pillar, or pilaster, or pier, that supports an arch, &c.</p><p>IMPROPER <hi rend="italics">Fraction,</hi> is a fraction whose numerator is either equal to, or greater than, its denominator. As 5/5 or 5/3 or 19/6. An Improper fraction is reduced to a whole or mixt number, by dividing the numerator by the denominator; the quotient is the integer, and the remainder set over the divisor makes the fractional part of the value of the original Improper fraction. Thus 5/5 = 1, and 5/3 = 1 2/3, and 19/6 = 3 1/6. So that when the numerator is just equal to the denominator, the Improper fraction is exactly equal to unity or 1; but when the numerator is the greater, the fraction is greater than 1.<pb n="628"/><cb/></p></div1><div1 part="N" n="IMPULSE" org="uniform" sample="complete" type="entry"><head>IMPULSE</head><p>, the single or momentary action or force by which a body is impelled; in contradistinction to continued forces; like the blow of a hammer, &c.</p></div1><div1 part="N" n="IMPULSIVE" org="uniform" sample="complete" type="entry"><head>IMPULSIVE</head><p>, a term applied to actions by impulse.</p><p>INACCESSIBLE <hi rend="italics">Height</hi> or <hi rend="italics">Distance,</hi> is that which cannot be approached, or measured by actual measurement, by reason of some impediment in the way; as water, &c.</p><p>See <hi rend="smallcaps">Heights</hi> and <hi rend="smallcaps">Distances.</hi></p></div1><div1 part="N" n="INCEPTIVE" org="uniform" sample="complete" type="entry"><head>INCEPTIVE</head><p>, of <hi rend="italics">Magnitude,</hi> a term used by Dr. Wallis, to express such moments, or first principles, as, though of no magnitude themselves, are yet capable of producing such as are. See <hi rend="smallcaps">Infinite</hi>, and I<hi rend="smallcaps">NDIVISIBLE.</hi> Thus, a point has no magnitude itself, but is inceptive of a line, which it produces by its motion. Also a line, though it has no breadth, is yet Inceptive of breadth; that is, it is capable, by its motion, of producing a surface, which has breadth.</p></div1><div1 part="N" n="INCH" org="uniform" sample="complete" type="entry"><head>INCH</head><p>, a common English measure, being the 12th part of a foot, or 3 barley corns in length.</p></div1><div1 part="N" n="INCIDENCE" org="uniform" sample="complete" type="entry"><head>INCIDENCE</head><p>, or <hi rend="italics">line of</hi> <hi rend="smallcaps">Incidence</hi>, in Mechanics, implies the direction or inclination in which one body strikes or acts on another.——In the incursions of two moving bodies, their Incidence is said to be Direct or Oblique, as the directions of their motion make a straight line, or an angle at the point of Impact.</p><p><hi rend="italics">Angle of</hi> <hi rend="smallcaps">Incidence</hi>, by some writers, denotes the angle comprehended between the line of Incidence, and a perpendicular to the body acted on at the point of Incidence. Thus, suppose AB an Incident line, and BF a perpendicular to the plane CB at the incident point B; then ABF is the angle of Incidence, or of inclination. <figure/></p><p>But, according to Dr. Barrow, and some other writers, the Angle of Incidence is the complement of the former, or the angle made between the incident line, and the plane acted on, or a tangent at the point of Incidence; as the angle ABC.</p><p>It is demonstrated by optical writers, 1st, That the Angle of Incidence, of the rays of light, is always equal to the angle of reflection; and that they lie in the same plane. And the same is proved by the writers on Mechanics, concer<hi rend="sup">.</hi>ning the reflection of elastic bodies. That is, the [angle]ABF = the [angle]FBD, or the , That the sines of the Angles of Incidence and refraction are to each other, either accurately, or very nearly, in a given or constant ratio.—3dly, That from air into glass, the sine of the Angle of Incidence, is to the sine of the angle of refraction, as 300 to 193, or nearly as 14 to 9: and, on the other hand, that out of glass into air, the sign of the Angle of Incidence, is to the sine of the angle of refraction, as 193 to 300, or as 9 to 14 nearly.</p><p><hi rend="smallcaps">Incidence</hi> <hi rend="italics">of Eclipse.</hi> See <hi rend="smallcaps">Eclipse</hi> and I<hi rend="smallcaps">MMER- SION.</hi><cb/></p><p><hi rend="italics">Axis of</hi> <hi rend="smallcaps">Incidence</hi>, is the line FB perpendicular t<*> the reflecting plane at the point of Incidence B.</p><p><hi rend="italics">Cathetus of</hi> <hi rend="smallcaps">Incidence.</hi> See <hi rend="smallcaps">Cathetus</hi>, and R<hi rend="smallcaps">EFLECTION.</hi></p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Incidence</hi>, in Catoptrics, denotes a right line, as AB, in which light is propagated from a radiant point A, to a point B, in the surface of a speculum. The same line is also called an Incident ray.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Incidence</hi>, in Dioptrics, is a right line, as AB, in which light is propagated unrefracted, in the same medium, from the radiant point to the surface of the refracting body, CBE.</p><p><hi rend="italics">Point of</hi> <hi rend="smallcaps">Incidence</hi>, is the point B on the surface of the reflecting or refracting medium, on which the Incident ray falls.</p><p><hi rend="italics">Scruples of</hi> <hi rend="smallcaps">Incidence.</hi> See <hi rend="smallcaps">Scruples.</hi></p><p>INCIDENT <hi rend="italics">Ray,</hi> is the line or ray AB, falling on the surface of any body, at B.</p></div1><div1 part="N" n="INCLINATION" org="uniform" sample="complete" type="entry"><head>INCLINATION</head><p>, in Geometry, Mechanics, or Physics, denotes the mutual tendency of two lines, planes, or bodies, towards one another; so that their directions make at the point of concourse some certain angle.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of the Axis of the Earth,</hi> is the angle it makes with the plane of the ecliptic; or the angle between the planes of the equator and ecliptic. <figure/></p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of a Line to a plane,</hi> is the acute angle, as CDE, which the line CD makes with another line DE drawn in the plane through the incident point D and the foot of a perpendicular E from any point of the line upon the plane.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of an Incident ray,</hi> is the angle of inclination, or angle of incidence.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of the Magnetical needle.</hi> See <hi rend="smallcaps">Dipping</hi> <hi rend="italics">Needle.</hi></p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of Meridians,</hi> in Dialling, is the angle that the hour-line on the globe, which is perpendicular to the dial-plane, makes with the meridian.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of the Orbit of a planet,</hi> is the angle formed by the planes of the ecliptic and of the orbit of the planet. The quantity of this Inclination for the several planets, is as follows, viz. <table><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" role="data">6°</cell><cell cols="1" rows="1" role="data">54</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Earth</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Moon</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Herschel</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">48</cell></row></table></p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of a Plane,</hi> in Dialling, is the arch of<pb n="629"/><cb/> a vertical circle, perpendicular both to the plane and the horizon, and intercepted between them.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of a Planet,</hi> is the arch or angle comprehended between the ecliptic and the place of the planet in its orbit. The greatest Inclination, or declination, is the same as the Inclination of the orbit; which see above.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of a Reflected ray,</hi> is the angle which a ray after reflection makes with the axis of Inclination; as the angle FBD, in the last fig. but one.</p><p><hi rend="smallcaps">Inclination</hi> <hi rend="italics">of Two Planes,</hi> is the angle made by two lines drawn in those planes perpendicular to their common intersection, and meeting in any point of that intersection.</p><p><hi rend="italics">Angle of</hi> <hi rend="smallcaps">Inclination</hi>, is the same as what is otherwise called the angle of incidence.</p><p><hi rend="italics">Argument of</hi> <hi rend="smallcaps">Inclination.</hi> See <hi rend="smallcaps">Argument.</hi></p><p><hi rend="smallcaps">Inclined</hi> <hi rend="italics">Plan<*>,</hi> in Mechanics, is a plane inclined to the horizon, or making an angle with it. It is one of the simple mechanic powers, and the double inclined plane makes the wedge.</p><p>1. The power gained by the Inclined plane, is in proportion as the length of the plane is to its height, or as radius to the sine of its inclination; that is, a given weight hanging freely, will balance upon the plane another weight, that shall be greater in that proportion. <figure/> So, when the greater weight W on the plane, is balanced by the less weight <hi rend="italics">w</hi> hanging perpendicularly, then is <hi rend="italics">w</hi> : W : : BC : AC : : sin. [angle]A : radius. Or, in other words, the relative gravity of a body upon the plane, or its force in descending down the plane, is to its absolute gravity or weight, in the same proportion of the height of the plane to its length, or of the sine of inclination to radius.</p><p>2. Hence therefore the relative gravities of the same body on different inclined planes, or their forces to descend down the planes, are to each other, as the sines of the angles of inclination, to radius 1, or directly as the heights of the planes, and inversely as their lengths.</p><p>3. Hence, if the planes have the same height, and absolute weights of the bodies be directly proportional to the lengths of the planes, then the forces to descend will be equal. Consequently, if the bodies be then <figure/> connected by a string acting parallel to the planes, they will exactly balance each other; as in the annexed figure.</p><p>4. The relative force of gravity upon the plane being in a constant ratio to the absolute weight of the body, viz, as sine of inclination to radius; therefore all the laws relating to the perpendicular free descents of bodies by gravity, hold equally true for the descents on inclined planes; such as, that the motion is a uniformly accelerated one; that the velocities are directly as the times, and the spaces as the square of either of them; using only the relative force upon the plane for the absolute weight of the body, or instead of 32 1/6 feet, the velocity generated by gravity in the first second of<cb/> time, using 32 1/6<hi rend="italics">s,</hi> where <hi rend="italics">s</hi> is the sine of the inclination to the radius 1.</p><p>5. The velocity acquired by a body in descending down an Inclined plane AC, when the body arrives at A, is the same as the velocity acqu<*>red by descending freely down the perpendicular altitude BC, when it arrives at B. But the times are very different; for the time of descending down the Inclined plane, is greater than down the perpendicular, in the <figure/> same proportion as the length of the plane AC, is to the height CB : and so the time of descending from any point C to a horizontal line or plane ABG &c, down any oblique line, or Inclined plane, is directly proportional to the length of that plane, CA, or CD, or CE, or CB, or CF, &c.</p><p>6. Hence, if there be drawn AH perpendicular to AC, meeting CB produced in H; then the time of descending down any plane CA, is equal to the time of descending down the perpendicular CH. So that, if upon CH as a diameter a circle be described, the times of descent will be exactly equal, down every chord in the circle, beginning at C, and terminating any where in the circumference, as CI, CA, CK, CH, &c, or beginning any where in the circumference, and terminating at the lowest point of the circle, as CH, IH, AH, KH, &c.</p><p>7. When bodies ascend up Inclined planes, their motion is uniformly retarded; and all the former laws for descents, or the generation of motion, hold equally true for ascents, or the destruction of as much motion.</p><p><hi rend="smallcaps">Inclined</hi> <hi rend="italics">Towers,</hi> are towers inclined, or leaning out of the perpendicular. See <hi rend="smallcaps">Towers.</hi></p></div1><div1 part="N" n="INCLINERS" org="uniform" sample="complete" type="entry"><head>INCLINERS</head><p>, in Dialling, are inclined dials. See <hi rend="smallcaps">Dial.</hi></p></div1><div1 part="N" n="INCOMMENSURABLE" org="uniform" sample="complete" type="entry"><head>INCOMMENSURABLE</head><p>, Lines, or Numbers, or Quantities in general, are such as have no common measure, or no line, number, or quantity of the same kind, that will measure or divide them both without a remainder. Thus, the numbers 15 and 16 are Incommensurable, because, though 15 can be measured by 3 and 5, and 16 by 2, 4, and 8, there is yet no single number that will divide or measure them both.</p><p>Euclid demonstrates (prop. 117, lib. 10) that the side of a square and its diagonal are Incommensurable to each other. And Pappus, prop. 17, lib. 4, speaks of Incommensurable angles.</p><p><hi rend="smallcaps">Incommensurable</hi> <hi rend="italics">in Power,</hi> is said of quantities whose 2d powers, or squares, are Incommensurable. As √2 and √3, whose squares are 2 and 3, which are Incommensurable. It is commonly supposed that the diameter and circumference of a circle are Incommensurable to each other; at least their commensurability has never been proved. And Dr. Barrow surmises even that they are insinitely Incommensurable, or that all possible powers of them are Incommensurable.</p><p>INCOMPOSITE <hi rend="italics">Numbers,</hi> are the same with those called by Euclid prime numbers, being such as are not composed by the multiplication together of other numbers. As 3, 5, 7, 11, &c.<pb n="630"/><cb/></p></div1><div1 part="N" n="INCREMENT" org="uniform" sample="complete" type="entry"><head>INCREMENT</head><p>, is the small increase of a variable quantity. Newton, in his Treatise on Fluxions, calls these by the name Moments, and observes that they are proportional to the velocity or rate of increase of the flowing or variable quantities, in an indesinitely small time; he denotes them by subjoining a cipher 0, to the flowing quantity whose moment or Increment it is; thus <*>0 the moment of <hi rend="italics">x.</hi> In the doctrine of Increments, by Dr. Brooke Taylor and Mr. Emerson, they are denoted by points below the variable quantities; as <hi rend="italics">x˙.</hi> Some have also denoted them by accents underneath the letter, as <hi rend="italics">x</hi><hi rend="sub">'</hi> but it is now more usual to express them by accents over the same letter; as <hi rend="italics">x</hi><hi rend="sup">'</hi>.</p></div1><div1 part="N" n="INCREMENTS" org="uniform" sample="complete" type="entry"><head>INCREMENTS</head><p>, <hi rend="italics">Method of,</hi> a branch of Analytics, in which a calculus is founded on the properties of the successive values of variable quantities, and their differences, or Increments.</p><p>The inventor of the Method of Increments was the learned Dr. Taylor, who, in the year 1715, published a treatise upon it; and afterwards gave some farther account and explication of it in the Philos. Trans. as applied to the finding the sums of series. And another ingenious and easy treatise on the same, was published by Mr. Emerson, in the year 1763. The method is nearly allied to Newton's Doctrine of Fluxions, and arises out of it. Also the Differential method of Mr. Stirling, which he applies to the summation and interpolation of series, is of the same nature as the Method of Increments, but not so general and extensive.</p><p>From the Method of Increments, Mr. Emerson observes, “The principal foundation of the Method of Fluxions may be easily derived. For as in the Method of Increments, the Increment may be of any magnitude, so in the Method of Fluxions, it must be supposed infinitely small; whence all preceding and successive values of the variable quantity will be equal, from which equality the rules for performing the principal operations of fluxions are immediately deduced. That I may give the reader, continues he, a more perfect idea of the nature of this method: suppose the abscissa of a curve be divided into any number of equal parts, each part of which is called the Increment of the abscissa; and imagine so many parallelograms to be erected thereon; either circumscribing the curvilineal figure, or inscribed in it; then the finding the sum of all these parallelograms is the business of the Method of Increments. But if the parts of the abscissa be taken infinitely small, then these parallelograms degenerate into the curve; and then it is the business of the Method of Fluxions, to find the sum of all, or the area of the curve. So that the Method of Increments finds the sum of any number of finite quantities; and the Method of Fluxions the sum of any infinite number of infinitely small ones: and this is the essential difference between these two methods.” Again, “There is such a near relation between the Method of Fluxions, and that of Increments, that many of the rules for the one, with little variation, serve also for the other. And here, as in the Method of Fluxions, some questions may be solved, and the integrals found, in finite terms; whilst in others we are forced to have recourse to infinite series for a solution. And the like difficulties will occur in the Method of Increments, as usually happen<cb/> in Fluxio<*>s. For whilst some fluxionary quantities have no fluents, but what are expressed by series; so some Increments have no integrals, but what infinite series afford; which will often, as in fluxions, diverge and become useless.”</p><p>By means of the Method of Increments, many cutious and useful problems are easily resolved, which scarcely admit of a solution in any other way. As, suppose several series of quantities be given, whose terms are all formed according to some certain law, which is given; the Method of Increments will find out a general series, which comprehends all particular cases, and from which all of that kind may be found.</p><p>The Method of Increments is also of great use in sinding any term of a series proposed: for the law being given by which the terms are formed; by means of this general law, the Method of Increments will help us to this term, either expressed in finite quantities, or by an insinite series.</p><p>Another use of the Method of Increments, is to find the sums of series; which it will often do in finite terms. And when the sum of a series cannot be had in finite terms, we must have recourse to infinite series; for the integral being expressed by such a series, the sum of a competent number of its terms will give the sum of the series required. This is equivalent to transforming one series into another, converging quicker: and sometimes a very few terms of this series will give the sum of the series sought. <hi rend="center"><hi rend="italics">Desinitions in the Method of Increments.</hi></hi></p><p>1. When a quantity is considered as increasing, or decreasing, by certain steps or degrees, it is called an Integral.</p><p>2. The increase of any quantity from its present value, to the next succeeding value, is called an Increment: or, if it decreases, a Decrement.</p><p>3. The increase of any Increment, is the Second Increment; and the increase of the 2d Increment, is the 3d Increment; and so on.</p><p>4. Succeeding Values, are the several values of the integral, succeeding one another in regular order, from the present value; and Preceding Values, are such as arise before the present value. All these are called by the general term Factors.</p><p>5. A Perfect quantity is such as contains any number of successive values without intermission; and a Defective quantity, is that which wants some of the successive values. Thus <hi rend="italics">x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">3</hi> <hi rend="italics">x</hi><hi rend="sub">4</hi> <hi rend="italics">x</hi><hi rend="sub">5</hi> is a Perfect quantity; and <hi rend="italics">x</hi><hi rend="sub">2</hi> <hi rend="italics">x</hi><hi rend="sub">4</hi> <hi rend="italics">x</hi><hi rend="sub">5</hi>, an Imperfect or defective one.</p><p><hi rend="italics">Notation.</hi> This, according to Mr. Emerson's method, is as follows:</p><p>1. Simple Integral quantities are denoted by any letters whatever, as <hi rend="italics">z, y, x, u,</hi> &c.</p><p>2. The several values of a simple integral, are denoted by the same letter with small figures under them: so if <hi rend="italics">z</hi> be an integral, then <hi rend="italics">z</hi>, <hi rend="italics">z</hi><hi rend="sub">1</hi>, <hi rend="italics">z</hi><hi rend="sub">2</hi>, <hi rend="italics">z</hi><hi rend="sub">3</hi>, &c are the present value, and the 1st, 2d, 3d, &c, successive values of it; and the preceding values are denoted by figures with negative signs, thus <hi rend="italics">z</hi><hi rend="sub">-1</hi>, <hi rend="italics">z</hi><hi rend="sub">-2</hi>, <hi rend="italics">z</hi><hi rend="sub">-3</hi>, <hi rend="italics">z</hi><hi rend="sub">-4</hi>, are the 1st, 2d,<pb n="631"/><cb/> 3d, 4th preceding values; and the sigure denoting any value, is the characteristic.</p><p>3. The Increments are denoted with the same letters, and points under them: thus, <hi rend="italics">x˙</hi> is the Increment of <hi rend="italics">x,</hi> and <hi rend="italics">z</hi> is the increment of <hi rend="italics">z.</hi> Also <hi rend="italics">x</hi><hi rend="sub">1<*></hi> is the Increment of <hi rend="italics">x</hi><hi rend="sub">1</hi>; and <hi rend="italics">x</hi><hi rend="sub">n.</hi> of <hi rend="italics">x</hi><hi rend="sub">n</hi>, &c.</p><p>4. The 2d, 3d, and other Increments, are denoted with two, three, or more points: so <hi rend="italics">z</hi><hi rend="sub">..</hi> is the 2d Increment of <hi rend="italics">z,</hi> and <hi rend="italics">z</hi><hi rend="sub">...</hi> is the 3d Increment of <hi rend="italics">z,</hi> and so on. And these are denominated Increments of such an order, according to the number of points.</p><p>5. If <hi rend="italics">x</hi> be any Increment, then [<hi rend="italics">x</hi>] is the integral of it; also <hi rend="sup">2</hi>[<hi rend="italics">x</hi>] denotes the integral of [<hi rend="italics">x</hi>], or the 2d integral of <hi rend="italics">x;</hi> and <hi rend="sup">3</hi>[<hi rend="italics">x</hi>] is the 3d integral of <hi rend="italics">x,</hi> or an integral of the 3d order, &c.</p><p>6. Quantities written thus, <hi rend="italics">x</hi><hi rend="sub">1</hi> . . . <hi rend="italics">x</hi><hi rend="sub">5</hi> mean the same as <hi rend="italics">x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">2</hi> <hi rend="italics">x</hi><hi rend="sub">3</hi> <hi rend="italics">x</hi><hi rend="sub">4</hi> <hi rend="italics">x</hi><hi rend="sub">5</hi>, or signify that the quantities are continued from the first to the last, without break or interruption. <hi rend="center"><hi rend="italics">To find the Increment of any Integral, or variable quantity.</hi></hi></p><p><hi rend="italics">Rule</hi> 1. If the proposed quantity be not fractional, and be a perfect integral, cons<*>sting of the successive values of the variable quantity which increases uniformly: Multiply the proposed integral by the number of factors, and change the lowest factor for an Increment. So the Increment of is - 3<hi rend="italics">x˙</hi> + 6<hi rend="italics">z˙</hi>; for the Increment of the constant quantity <hi rend="italics">a</hi> is 0 or nothing. So likewise,</p><p>The Increment of <hi rend="italics">c</hi> <hi rend="italics">x</hi> <hi rend="italics">x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">2</hi> <hi rend="italics">x</hi><hi rend="sub">3</hi>, is 4<hi rend="italics">c x˙ x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">2</hi> <hi rend="italics">x</hi><hi rend="sub">3</hi>.</p><p>The Increment of <hi rend="italics">ax</hi><hi rend="sub">-3</hi> <hi rend="italics">x</hi><hi rend="sub">-2</hi> <hi rend="italics">x</hi><hi rend="sub">-1</hi>, is 3<hi rend="italics">ax˙ x</hi><hi rend="sub">-2</hi> <hi rend="italics">x</hi><hi rend="sub">-1</hi>.</p><p>The Increment of <hi rend="italics">x</hi><hi rend="sub">-m</hi> . . . <hi rend="italics">x</hi><hi rend="sub">n</hi> is .</p><p><hi rend="italics">Rule</hi> 2. In fractional quantities, where the denominator is perfect, and the variable quantity increases uniformly: Multiply the proposed integral by the number of factors, and by the constant Increment with a negative sign, and take the next succeeding value into the denominator. Thus,</p><p>The Increment of , is .</p><p>The Increment of , is .</p><p><hi rend="italics">Rule</hi> 3. The Increment of any power, as <hi rend="italics">x</hi><hi rend="sup">n</hi> is ; that is, the difference between the present value <hi rend="italics">x</hi><hi rend="sup">n</hi> and the next succeeding value ―(<hi rend="italics">x</hi> + <hi rend="italics">x˙</hi>))<hi rend="sup">n</hi>. And generally, the Increment of any quantity whatever, is found by subtracting the present value, or the given quantity, from its next succeeding value. Also by expanding the compound quantity in a series, and subtracting <hi rend="italics">x</hi><hi rend="sup">n</hi> from it, the Increment will be either</p><p>So the Increment of <hi rend="italics">x</hi><hi rend="sup">4</hi>, is .<cb/></p><p>The Increment of 1/<hi rend="italics">x</hi><hi rend="sup">3</hi> or <hi rend="italics">x</hi><hi rend="sup">- 3</hi> is </p><p>The Increment of <hi rend="italics">a</hi><hi rend="sup">x</hi>, <hi rend="italics">a</hi> being constant, is .</p><p>The Increment of 1/<hi rend="italics">a</hi><hi rend="sup">x</hi> is .</p><p>The Increment of .</p><p>And so on for any form of Integral whatever, subtracting the given quantity from its next succeeding value. So,</p><p>The Increment of the log. of <hi rend="italics">x</hi> is , which, by the nature os logarithms, is &c.</p><p><hi rend="italics">Schol.</hi> From hence may be deduced the principles and rules of fluxions; for the method of fluxions is only a particular case of the method of Increments, fluxions being infinitely small Increments; therefore if in any form of Increments the Increment be taken infinitely small, the form or expression will be changed into a fluxional one.</p><p>Thus, in , which is the Increment of the rectangle <hi rend="italics">xz,</hi> if <hi rend="italics">x˙</hi> and <hi rend="italics">z˙</hi> be changed for <hi rend="italics">x</hi><hi rend="sup">.</hi> and <hi rend="italics">z</hi><hi rend="sup">.</hi>, the expression will become for the fluxion of <hi rend="italics">xz,</hi> or only <hi rend="italics">zx</hi><hi rend="sup">.</hi> + <hi rend="italics">xz</hi><hi rend="sup">.</hi>, because <hi rend="italics">x</hi><hi rend="sup">.</hi><hi rend="italics">z</hi><hi rend="sup">.</hi> is insinitely less than the rest.</p><p>So likewise, if <hi rend="italics">x˙</hi> be changed for <hi rend="italics">x</hi><hi rend="sup">.</hi> in this &c, which is the Increment of <hi rend="italics">x</hi><hi rend="sup">n</hi>, it becomes &c, or only <hi rend="italics">nx</hi><hi rend="sup"><hi rend="italics">n</hi> - 1</hi><hi rend="italics">x,</hi> for the fluxion of the power <hi rend="italics">x</hi><hi rend="sup">n</hi>, as all the terms after the first will be nothing, because <hi rend="italics">x</hi><hi rend="sup">.2</hi> and <hi rend="italics">x</hi><hi rend="sup">.3</hi> &c are infinitely less than <hi rend="italics">x</hi><hi rend="sup">.</hi>.</p><p>And thus may all the other forms of fluxions be derived from the corresponding Increments. And in like manner, the finding of the integrals, is only a more general way of finding fluents, as appears in wha<*> follows. <hi rend="center"><hi rend="italics">To find out the Integral of any given Increment.</hi></hi></p><p><hi rend="italics">Rule</hi> 1. When the variable quantity in<*>reases uniformly, and the proposed integral consists of the successive values of it multiplied together, or is a perfect Increment not fractional: Multiply the given Incre-<pb n="632"/><cb/> ment by the next preceding value of the variable quantity, then divide by the new number of factors, and by the constant Increment.</p><p><hi rend="italics">Ex.</hi> Thus, the integral of 4<hi rend="italics">cx˙x</hi><hi rend="sub">1</hi><hi rend="italics">x</hi><hi rend="sub">2</hi><hi rend="italics">x</hi><hi rend="sub">3</hi> is <hi rend="italics">cxx</hi><hi rend="sub">1</hi><hi rend="italics">x</hi><hi rend="sub">2</hi><hi rend="italics">x</hi><hi rend="sub"><*></hi>.</p><p>The integral of 3<hi rend="italics">ax</hi> <hi rend="italics">x</hi><hi rend="sub">-2</hi> <hi rend="italics">x</hi><hi rend="sub">-1</hi> is <hi rend="italics">ax</hi><hi rend="sub">-3</hi> <hi rend="italics">x</hi><hi rend="sub">-2</hi> <hi rend="italics">x</hi><hi rend="sub">-1</hi>.</p><p><hi rend="italics">Rule</hi> 2. In a fractional expression, where the variable quantity increases uniformly, and the denominator is perfect, containing the successive values of the variable quantity: Throw out the greatest value of the variable letter, then divide by the new number of factors, and by the constant Increment with a negative sign. So,</p><p>The integral of is .</p><p>The integral of is .</p><p><hi rend="italics">Rule</hi> 3. Various other particular rules are given, but these and the two foregoing are all best included in the following general table of the most useful forms of Increments and integrals, to be used in the same way as the similar table of fluxions and fluents, to which these correspond. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data"><hi rend="italics">A Table of Increments and their Integrals.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Forms</cell><cell cols="1" rows="1" role="data">Increments</cell><cell cols="1" rows="1" role="data">Integrals</cell></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"><hi rend="italics">x˙</hi> when constant not constant</cell><cell cols="1" rows="1" role="data"><hi rend="italics">x,</hi> or <hi rend="italics">x</hi><hi rend="sub">1</hi>, or <hi rend="italics">x</hi><hi rend="sub">2</hi>, or <hi rend="italics">x</hi><hi rend="sub">3</hi> &c. <hi rend="italics">x</hi> only.</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"><hi rend="italics">x</hi><hi rend="sub">-m</hi> . . . . . . . <hi rend="italics">x</hi><hi rend="sub">n</hi> <hi rend="italics">x˙ x˙</hi> constant</cell><cell cols="1" rows="1" role="data"><hi rend="italics">x</hi><hi rend="sub">-m-1</hi> . . . . . . . . <hi rend="italics">x</hi><hi rend="sub">n</hi>/(<hi rend="italics">m</hi> + <hi rend="italics">n</hi>)</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"><hi rend="italics">ax˙</hi>/<hi rend="italics">x</hi><hi rend="sub">-m</hi> . . . . . . . <hi rend="italics">x</hi><hi rend="sub">n</hi> <hi rend="italics">x˙</hi> constant</cell><cell cols="1" rows="1" role="data">- <hi rend="italics">a</hi>/(―<hi rend="italics">m</hi> + <hi rend="italics">n</hi> - 2. <hi rend="italics">x</hi><hi rend="sub">-m</hi> . . . . <hi rend="italics">x</hi><hi rend="sub">n-1</hi>)</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"><hi rend="italics">zx˙</hi> + <hi rend="italics">x</hi><hi rend="sub">1</hi><hi rend="italics">z˙</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">xz</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">(<hi rend="italics">zx˙</hi> - <hi rend="italics">xz˙</hi>)/<hi rend="italics">zz</hi><hi rend="sub">1</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">x</hi>/<hi rend="italics">z</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">x</hi> <hi rend="italics">x˙</hi> given</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">x</hi>/(<hi rend="italics">a</hi><hi rend="sup"><hi rend="italics">x˙</hi></hi> - 1)</cell></row></table></p><p>Integrals, when found from given Increments, are corrected in the very same way as fluents when found from given fluxions, viz, instead of every several variable quantity in the integral, substituting such a determinate value of them as they are known to have in some particular case; and then subtracting each side of the resulting equation from the corresponding side of the integral, the remaining equation will be the correct form of the integrals.</p><p>For an example of the use of the Method of Increments, fuppose it were required to find the sum of<cb/> any number of terms of the series 1.2 + 2.3 + 3.4 + 4.5 &c. Let <hi rend="italics">x</hi> be the number of the terms, and <hi rend="italics">z</hi> the sum of them. Then, by the progression of the series, the last or the <hi rend="italics">x</hi> term is <hi rend="italics">x x</hi><hi rend="sub">1</hi>, and the next term after that will be <hi rend="italics">x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">2</hi>, that is <hi rend="italics">z˙</hi> = <hi rend="italics">x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">2</hi>, where <hi rend="italics">x˙</hi> = 1. Hence the integral is , which is the sum of <hi rend="italics">x</hi> terms of the given series. So if the number of terms <hi rend="italics">x</hi> be 10, this becomes 1/3 . 10 . 11 . 12 = 440, which is the sum of 10 terms of the given series 1.2 + 2.3 + 3.4 &c. Or, when <hi rend="italics">x</hi> = 100, the sum of 100 terms of the same series is .</p><p>Again, to find the sum <hi rend="italics">z</hi> of <hi rend="italics">n</hi> terms of the series &c.</p><p>Here the <hi rend="italics">n</hi>th term is Put ; then is , and the <hi rend="italics">n</hi>th term is ; and the <hi rend="italics">n</hi> + 1th term or <hi rend="italics">z</hi> is 1/<hi rend="italics">x x</hi><hi rend="sub">1</hi> <hi rend="italics">x</hi><hi rend="sub">2</hi>; the general integral of which is . But this wants a correction; for when <hi rend="italics">n</hi> = 0 or no terms, then <hi rend="italics">x</hi> = - 1, and the sum <hi rend="italics">z</hi> = 0, and the integral becomes <hi rend="italics">z</hi> or ; that is - 1/12 is the correction, and being subtracted, the correct state of the integrals becomes , which is the sum of <hi rend="italics">n</hi> terms of the proposed series. And when <hi rend="italics">n</hi> is insinite, the lat<*>er fraction is nothing, and the sum of the infinite series, or the infinite number of the terms, is accurately 1/12.</p><p>When <hi rend="italics">n</hi> = 100, the sum of 100 terms of the series becomes </p><p>For more ample information and application on this science, see Emerson's Increments, Taylor's M<*>thodus Incrementorum, and Stirling's Summatio & Interpolatio Serierum.</p><p>INCURVATION <hi rend="italics">of the rays of Light.</hi> See <hi rend="smallcaps">Light</hi>, and <hi rend="smallcaps">Refraction.</hi></p></div1><div1 part="N" n="INDEFINITE" org="uniform" sample="complete" type="entry"><head>INDEFINITE</head><p>, <hi rend="italics">Indeterminate,</hi> that which has no certain bounds, or to which the human mind cannot affix any. Des Cartes uses the word, in his Philosophy, instead of infinite, both in numbers and quantities, to signify an inconceivable number, or number so great, that an unit cannot be added to it; and a quantity so great, as not to be capable of any addition. Thus, he says, the stars, visible and invisible, are in number Indefinite, and not, as the Ancients held, infinite; and<pb n="633"/><cb/> that quantity may be divided into an Indefinite number of parts, not an infinite number.</p><p>Indefinite is now commonly used for indeterminate, number or quantity, that is, a number or quantity in general, in contradistinction from some particular known and given one.</p></div1><div1 part="N" n="INDETERMINED" org="uniform" sample="complete" type="entry"><head>INDETERMINED</head><p>, or <hi rend="smallcaps">Indeterminate</hi>, in Geometry, is understood of a quantity, which has no certain or definite bounds.</p><p><hi rend="smallcaps">Indeterminate</hi> <hi rend="italics">Problem,</hi> is that which admits of innumerable different solutions, and sometimes perhaps only of a great many different answers; otherwise called an unlimited problem.</p><p>In problems of this kind the number of unknown quantities concerned, is greater than the number of the conditions or equations by which they are to be found; from which it happens that generally some other conditions or quantities are assumed, to supply the defect, which being taken at pleasure, give the same number of answers as varieties in those assumptions.</p><p>As, if it were required to find two square numbers whose difference shall be a given quantity <hi rend="italics">d.</hi> Here, if <hi rend="italics">x</hi><hi rend="sup">2</hi> and <hi rend="italics">y</hi><hi rend="sup">2</hi> denote the two squares, then will , by the question, which is only one equation, for finding two quantities. Now by assuming a third quantity <hi rend="italics">z</hi> so that the sum of the two roots; then is , and , which are the two roots having the difference of their squares equal to the given quantity <hi rend="italics">d,</hi> and are expressed by means of an assumed quantity <hi rend="italics">z;</hi> so that there will be as many answers to the question, as there can be taken values of the Indeterminate quantity <hi rend="italics">z,</hi> that is, innumerable.</p><p>Diophantus was the first writer on Indeterminate problems, viz, in his Arithmetic or Algebra, which was first published in 1575 by Xilander, and afterwards in 1621 by Bachet, with a large commentary, and many additions to it. His book is wholly upon this subject; whence it has happened, that such kind of questions have been called by the name of Diophantine problems. Fermat, Des Cartes, Frenicle, in France, and Wallis and others in England, particularly cultivated this branch of Algebra, on which they held a correspondence, proposing difficult questions to each other; an instance of which are those two curious ones, proposed by M. Fermat, as a challenge to all the mathematicians of Europe, viz 1st, To find a cube number which added to all its aliquot parts shall make a square number; and 2d, To find a square number which added to all its aliquot parts shall make a cubic number; which problems were answered after several ways by Dr. Wallis, as well as some others of a different nature. See the Letters that passed between Dr. Wallis, the lord Brounker, Sir Kenelm Digby, &c, in the Doctor's Works; and the Works of Fermat, which were collected and published by his son. Most authors on Algebra have also treated more or less on this part of it, but more especially Kersey, Prestet, Ozanam, Kirkby, &c. But afterwards, mathematicians seemed to have forgot such questions, if they did not even despise them as useless, when Euler drew their attention by some excellent compositions, demonstrating some general theorems, which had only been<cb/> known by induction. M. la Grange has also taken up the subject, having resolved very difficult problems in a general way, and discovered more direct methods than heretofore. The 2d volume of the French translation of Euler's Algebra contains an elementary treatise on this branch, and, with la Grange's additions, an excellent theory of it; treating very generally of Indeterminate problems, of the sirst and second degree, of solutions in whole numbers, of the method of Indeterminate coefficients, &c.</p><p>Finally, Mr. John Leslie has given, in the 2d volume of the Edinburgh Philos. Transactions, an ingenious paper on the resolution of Indeterminate problems, resolving them by a new and general principle. “The doctrine of Indeterminate equations,” says Mr. Leslie, “has been seldom treated in a form equally systematic with the other parts of Algebra. The solutions commonly given are devoid of uniformity, and often require a variety of assumptions. The object of this paper is to resolve the complicated expressions which we obtain in the solution of Indeterminate problems, into simple equations, and to do so, without framing a number of assumptions, by help of a single principle, which though extremely simple, admits of a very extensive application.”</p><p>“Let A × B be any compound quantity equal to another, C × D, and let <hi rend="italics">m</hi> be any rational number assumed at pleasure; it is manifest that, taking equimultiples, . If, therefore, we suppose that A = <hi rend="italics">m</hi>D, it must follow that <hi rend="italics">m</hi>B = C, or B = C/<hi rend="italics">m</hi>. Thus two equations of a lower dimension are obtained. If these be capable of farther decomposition, we may assume the multiples <hi rend="italics">n</hi> and <hi rend="italics">p,</hi> and form four equations still more simple. By the repeated application of this principle, an higher equation admitting of divisors, will be resolved into those of the first order, the number of which will be one greater than that of the multiples assumed.”</p><p>For example, resuming the problem at first given, viz, to find two rational numbers, the difference of the squares of which shall be a given number. Let the given number be the product of <hi rend="italics">a</hi> and <hi rend="italics">b;</hi> then by hypothesis, ; but these compound quantities admit of an easy resolution, for . If therefore we suppose , we shall obtain ; where <hi rend="italics">m</hi> is arbitrary, and if rational, <hi rend="italics">x</hi> and <hi rend="italics">y</hi> must also be rational. Hence the resolution of these two equations gives the values of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> the numbers sought, in terms of <hi rend="italics">m;</hi> viz, , and .</p></div1><div1 part="N" n="INDEX" org="uniform" sample="complete" type="entry"><head>INDEX</head><p>, in Arithmetic, is the same with what is otherwise called the characteristic or the exponent of a logarithm; being that which shews of how many places the absolute or natural number belonging to the logarithm confifts, and of what nature it is, whether an integer or a fraction; the Index being less by 1 than the number of integer figures in the natural number, and is positive for integer or whole numbers, but negative in fractions, or in the denominator of a frac-<pb n="634"/><cb/> tion; and in decimals, the negative index is 1 more than the number of ciphers in the decimal, after the point, and before the first significant figure; or, still more generally, the Index shews how far the first figure of the natural number is distant from the place of units, either towards the left hand, as in whole numbers, or towards the right, as in decimals; these opposite cases being marked by the correspondent signs + and —, of opposite affections, the sign — being set over the Index, and not before it, because it is this Index only which is understood as negative, and not the decimal part of the logarithm. Thus, in this logarithm 2.4234097, the figures of whose natural number are 2651, the 2 is the Index, and being positive, it shews that the first figure of the number must be two places removed from the units place, or that there will be three places of integers, the number of these places being always 1 more than the Index; so that the natural number will be 265.1. But if the same Index be negative, thus ―2.4234097, it shews that the natural number is a decimal, and that the first signisicant figure of it is in the 2d place from units, or that there is one cipher at the beginning of the decimal, <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Number.</cell><cell cols="1" rows="1" role="data">Logarithm.</cell></row><row role="data"><cell cols="1" rows="1" role="data">2651</cell><cell cols="1" rows="1" role="data">3.4234097</cell></row><row role="data"><cell cols="1" rows="1" role="data">265.1</cell><cell cols="1" rows="1" role="data">2.4234097</cell></row><row role="data"><cell cols="1" rows="1" role="data">26.51</cell><cell cols="1" rows="1" role="data">1.4234097</cell></row><row role="data"><cell cols="1" rows="1" role="data">2.651</cell><cell cols="1" rows="1" role="data">0.4234097</cell></row><row role="data"><cell cols="1" rows="1" role="data">.2651</cell><cell cols="1" rows="1" role="data">―1.4234097</cell></row><row role="data"><cell cols="1" rows="1" role="data">.02651</cell><cell cols="1" rows="1" role="data">―2.4234097</cell></row><row role="data"><cell cols="1" rows="1" role="data">.002651</cell><cell cols="1" rows="1" role="data">―3.4234097</cell></row></table> being 1 less than the negative Index; and consequently that the natural number of the logarithm in this case is .02651. Hence, by varying the natural number, with respect to the decimal places in it as in the former of the two columns here annexed, the Index of their logarithm will vary as in the 2d column.</p><p>Mr. Townly introduced a peculiar way of noting these Indices, when they become negative, or express decimal figures, which is now much in use, especially in the log. sines and tangents, &c, viz, by taking, instead of the true Index, its arithmetical complement to 10; so that, in this way, the logarithm ―2.4234097 is written 8.4234097.</p><p>For the addition and subtraction of Indices, see <hi rend="smallcaps">Logarithm.</hi></p><p><hi rend="smallcaps">Index</hi> of a Globe, is a little style fitted on to the north-pole, and turning round with it, pointing out the divisions of the hour-circle.</p><p><hi rend="smallcaps">Index</hi> of a Quantity, in Arithmetic and Algebra, otherwise called the exponent, is the number that shews to what power it is understood to be raised: as in 10<hi rend="sup">3</hi>, or <hi rend="italics">a</hi><hi rend="sup">3</hi>, the figure 3 is the Index or exponent of the power, signifying that the root or quantity, 10 or <hi rend="italics">a,</hi> is raised to the 3d power. See this fully treated under <hi rend="smallcaps">Exponent.</hi></p></div1><div1 part="N" n="INDICTION" org="uniform" sample="complete" type="entry"><head>INDICTION</head><p>, or <hi rend="italics">Roman</hi> <hi rend="smallcaps">Indiction</hi>, a kind of <*>poch, or manner of counting time, among the Romans; containing a cycle or revolution of 15 years.</p><p>The popes have dated their acts by the year of the Indiction, which was fixed to the 1st of January anno Domini 313, ever since Charlemagne made them sovereign; before that time, they dated them by the years of the Emperors.<cb/></p><p>At the time of reforming the calendar, the year 1582 was reckoned the 10th year of the Indiction; so that beginning to reckon from hence, and dividing the number of years elapsed between that time and this, by 15, the remainder, with the addition of 10, rejecting 15 if the sum be more, will be the year of the Indiction.</p><p>But the Indiction will be easier found thus: Add 3 to the given year of Christ; divide the sum by 15, and the remainder after the division, will be the year of the Indiction: if there be no remainder, the Indiction is 15. In either of these ways, the Indiction for the year 1795 is 13.</p></div1><div1 part="N" n="INDIVISIBLES" org="uniform" sample="complete" type="entry"><head>INDIVISIBLES</head><p>, are those indefinitely small elements, or principles, into which any body or figure may ultimately be divided.</p><p>A line is said to consist of points, a surface of parallel lines, and a solid of parallel surfaces: and because each of these elements is supposed Indivisible, if in any figure a line be drawn perpendicularly through all the elements, the number of points in that line, will be the same as the number of the elements.</p><p>Whence it appears, that a parallelogram, or a prism, or a cylinder, is resolvable into elements, as Indivisibles, all equal to each other, parallel, and like or similar to the base; for which reason, one of these elements multiplied by the number of them, that is the base of the figure multiplied by its height, gives the area or content. And a triangle is resolvable into lines parallel to the base, but decreasing in arithmetical progression; so also do the circles, which constitute the parabolic conoid, as well as those which constitute the plane of a circle, or the surface of a cone. In all which cases, as the last or least term of the arithmetic progression is 0, and the length of the figure the same thing as the number of the terms, therefore the greatest term, or base, being multiplied by the length of the figure, half the product is the sum of the whole, or the content of the figure.</p><p>And in any other figure or solid, if the law of the decrease of the elements be known, and thence the relation of the sum to the greatest term, which is the base, the whole number of them being the altitude of the sigure, then the said sum of the elements is always the content of the figure.</p><p>A cylinder may also be resolved into cylindrical curve surfaces, having all the same height, and continually decreasing inwards, as the circles of the base do, on which they insist.</p><p>This way of considering magnitudes, is called the Method of Indivisibles, which is only the ancient method of exhaustions, a little disguised and contracted. And it is found of good use, both in computing the contents of figures in a very short and easy way, as above instanced, and in shortening other demonstrations in mathematics; an instance of which may here be given in that celebrated proposition of Archimedes, that a sphere is two-thirds of its circumscribed cylinder. Thus,</p><p>Suppose a cylinder, a hemisphere, and an inverted cone, having all the same base and altitude, and cut by an infinite number of planes all parallel to the base, of which EFGH is one; it is evident that the square of EI, the radius of the cylinder, is every where equal<pb n="635"/><cb/> to the square of SF, the radius of the sphere; and also that the square of EI, or of SF, is <figure/> equal to the sum of the squares of IF and IS, or of IF and IK, because IK = IS; that is, , in every position; but IE is the radius of the cylinder, IF the corresponding radius of the sphere, and IK that of the cone; and the circular sections of these bodies, are as the squares of their radii; therefore the section of the cylinder is every where equal to the sum of the sections of the hemisphere and cone; and, as the number of all those sections, which is the common height of the figures, is the same, therefore all the sections, or elements, of the cylinder, will be equal to the sum of all those of the hemisphere and cone taken together; that is, the cylinder is equal to both the hemisphere and cone: but as the cone itself is equal to one-third part of the cylinder; therefore the hemisphere is equal to the other two-thirds of it.</p><p>The Method of Indivisibles was introduced by Cavalerius, in 1635, in his Geometria Indivisibilium. The same was also pursued by Torricelli in his works, printed 1644: and again by Cavalerius himself in another treatise, published in 1647.</p><p>INERTIÆ <hi rend="italics">Vis.</hi> See Vis <hi rend="italics">Inertiæ.</hi></p></div1><div1 part="N" n="INFINITE" org="uniform" sample="complete" type="entry"><head>INFINITE</head><p>, is applied to quantities which are either greater or less than any assignable ones. In which sense it differs but little from the terms Indefinite and Indeterminate. Thus, an</p><div2 part="N" n="Infinite" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Infinite</hi></head><p>, or <hi rend="italics">Infinitely great</hi> line, denotes only an indefinite or indeterminate line; or a line to which no certain bounds or limits are prescribed.</p><p><hi rend="smallcaps">Infinite</hi> <hi rend="italics">Quantities.</hi> Though the idea of magnitude infinitely great, or such as exceeds any assignable quantity, does include a negation of limits, yet all such magnitudes are not equal among themselves; but besides Infinite length, and Infinite area, there are no less than three several sorts of Infinite solidity; all of which are quantities <hi rend="italics">sui generis;</hi> and those of each species are in given proportions.</p><p>Infinite length, or a line infinitely long, may be considered, either as beginning at a point, and so infinitely extended one way; or else both ways from the same point.</p><p>As to Infinite surface or area, any right line infinitely extended both ways on a plane infinitely extended every way, divides that plane into two equal parts, one on each side of the line. But if from any point in such a plane, two right lines be infinitely extended, making an angle between them; the Infinite area, intercepted between these Infinite right lines, is to the whole Infinite plane, as that angle is to 4 right angles. And if two Infinite and parallel lines be drawn at a given distance on such an Infinite plane, the area intercepted between them will be likewise Infinite; but yet it will be infinitely less than the whole plane; and even infinitely less than the angular or sectoral space, intercepted between two Insinite lines, that are inclined, though at never so small an angle; because in the one case, the given finite distance of the parallel lines diminishes the Infinity in one of the dimensions; whereas<cb/> in a sector, there is Infinity in both dimensions. And thus there are two species of Infinity in surfaces, the one infinitely greater than the other.</p><p>In like manner there are species of Infinites in solids, according as only one, or two, or as all their three dimensions, are Infinite; which, though they be all infinitely greater than a finite solid, yet are they in succession infinitely greater than each other.</p><p>Some farther properties of Infinite quantities are as follow:</p><p>The ratio between a finite and an Infinite quantity, is an Infinite ratio.</p><p>If a finite quantity be multiplied by an infinitely small one, the product will be an infinitely small one; but if the former be divided by the latter, the quotient will be infinitely great.</p><p>On the contrary, a finite quantity being multiplied by an infinitely great one, the product is infinitely great; but the former divided by the latter, the quotient will be infinitely little.</p><p>The product or quotient of an infinitely great or an infinitely little quantity, by a finite one, is respectively infinitely great, or infinitely little.</p><p>An infinitely great multiplied by an infinitely little, is a finite quantity; but the former divided by the latter, the quotient is infinitely Infinite.</p><p>The mean proportional between infinitely great, and infinitely little, is finite.</p><p><hi rend="italics">Arithmetic of</hi> <hi rend="smallcaps">Infinites.</hi> See <hi rend="smallcaps">Arithmetic.</hi> Also Wallis's treatise of this subject; and another by Emerson, at the beginning of his Conic Sections; also Bulliald's treatise <hi rend="italics">Arithmetica Infinitorum.</hi></p><p><hi rend="smallcaps">Infinite</hi> <hi rend="italics">Decimals,</hi> such as do not terminate, but go on without end; as .333 &c = 1/3, or .1<hi rend="sup">.</hi>42857<hi rend="sup">.</hi> &c = 1/7. See <hi rend="smallcaps">Repetend.</hi></p><p><hi rend="smallcaps">Infinitely</hi> <hi rend="italics">Infinite Fractions,</hi> or all the powers of the fractions whose numerator is 1; which are all together equal to unity, as is demonstrated by Dr. Wood, in Hook's Philos. Coll. N<*> 3, p. 45; where some curious properties are deduced from the same.</p><p><hi rend="smallcaps">Infinite</hi> <hi rend="italics">Series,</hi> a series considered as infinitely continued as to the number of its terms. See <hi rend="smallcaps">Series.</hi></p></div2></div1><div1 part="N" n="INFINITESIMALS" org="uniform" sample="complete" type="entry"><head>INFINITESIMALS</head><p>, are certain infinitely or indefinitely small parts; as also the method of computing by them.</p><p>In the method of Infinitesimals, the element by which any quantity increases or decreases, is supposed to be infinitely small, and is generally expressed by two or more terms, some of which are infinitely less than the rest, which being neglected as of no importance, the remaining terms form what is called the <hi rend="italics">difference</hi> of the proposed quantity. The terms that are neglected in this manner, as infinitely less than the other terms of the element, are the very same which arise in consequence of the acceleration, or retardation, of the generating motion, during the infinitely small time in which the element is generated; so that the remaining terms express the element that would have been produced in that time, if the generating motion had continued uniform. Therefore, those <hi rend="italics">differences</hi> are accurately in the same ratio to each other, as the generating motions or fluxions. And hence, though in this method, Infinitesimal parts of the elements are neglected, the conclusions are accurately true, without<pb n="636"/><cb/> even an infinitely small error, and agree precisely with those that are deduced by the method of fluxions.</p><p>But however safe and convenient this method may be, some will always scruple to admit infinitely little quantities, and insinite orders of Infinitesimals, into a science that boasts of the most evident and accurate principles, as well as of the most rigid demonstrations. In order to avoid such suppositions, Newton considers the simultaneous increments of the flowing quantities as finite, and then investigates the ratio which is the limit of the various proportions which those increments bear to each other, while he supposes them to decrease together till they vanish; which ratio is the same with the ratio of the fluxions. See Maclaurin's Treatise of Fluxions, in the Introduc. p. 39 &c, also art. 495 to 502.</p></div1><div1 part="N" n="INFLAMMABILITY" org="uniform" sample="complete" type="entry"><head>INFLAMMABILITY</head><p>, that property of bodies by which they kindle, or catch fire.</p></div1><div1 part="N" n="INFLECTION" org="uniform" sample="complete" type="entry"><head>INFLECTION</head><p>, in Optics, called also Diffraction, and Deflection of the rays of light, is a property of them, by reason of which, when they come within a certain distance of any body, they will either be bent from it, or towards it; being a kind of imperfect reflection or refraction.</p><p>Some writers ascribe the first discovery of this property to Grimaldi, who first published an account of it, in his Treatise De Lumine, Coloribus, & Iride, printed in 1666. But Dr. Hook also claims the discovery of it, and communicated his observations on this subject to the Royal Society, in 1672. He shews that this property differs both from reflection and refraction; and that it seems to depend on the unequal density of the constituent parts of the ray, by which the light is dispersed from the place of condensation, and rarefied or gradually diverged into a quadrant; and this deflection, he says, is made towards the superficies of the opaque body perpendicularly.</p><p>Newton discovered, by experiments, this Inflection of the rays of light; which may be seen in his Optics.</p><p>M. De la Hire observed, that when we look at a candle, or any luminous body, with our eyes nearly shut, rays of light are extended from it, in several directions, to a considerable distance, like the tails of comets. The true cause of this phenomenon, which has exercised the sagacity of Des Cartes, Rohault, and others, seems to be, that the light passing among the eyelashes, in this situation of the eye, is inflected by its near approach to them, and therefore enters the eye in a great variety of directions. He also observes, that he found that the beams of the stars being observed, in a deep valley, to pass near the brow of a hill, are always more refracted than if there were no such hill, or the observation was made on the top of it; as if the rays of light were bent down into a curve, by passing near the surface of the mountain.</p><p><hi rend="italics">Point of</hi> <hi rend="smallcaps">Inflection</hi>, or of <hi rend="italics">contrary flexure,</hi> in a curve, is the point or place in the curve where it begins to bend or turn a contrary way; or which separates the concave part from the convex part, and lying between the two; or where the curve changes from concave to convex, or from convex to concave, on the same side of the curve: such as the point E in the annexed figures; where the former of the two is con-<cb/> cave towards the axis AD from A to E, and convex from E to F; but, on the contrary, the latter figure is convex from A to E, and concave from E to F. <figure/></p><p>There are various ways of finding the point of Inflexion; but the following, which is new, seems to be the simplest and easieft of all. From the nature of curvature it is evident that, while a curve is concave towards an axis, the fluxion of the ordinate decreases, or is in a decreasing ratio, with regard to the fluxion of the absciss; but, on the contrary, that the said fluxion increases, or is in an increafing ratio to the fluxion of the absciss, where the curve is convex towards the axis; and hence it follows that those two fluxions are in a constant ratio at the point of Inflection, where the curve is neither concave nor convex. That is, if <hi rend="italics">x</hi> = AD the absciss, and <hi rend="italics">y</hi> = DE the ordinate, then <hi rend="italics">x</hi><hi rend="sup">.</hi> is to <hi rend="italics">y</hi><hi rend="sup">.</hi> in a constant ratio, or <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi> or <hi rend="italics">y</hi><hi rend="sup">.</hi>/<hi rend="italics">x</hi><hi rend="sup">.</hi> is a constant quantity. But constant quantities have no fluxion, or their fluxion is equal to nothing; so that in this case the fluxion of <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi> or of <hi rend="italics">y</hi><hi rend="sup">.</hi>/<hi rend="italics">x</hi><hi rend="sup">.</hi> is equal to nothing. And hence we have this general rule: viz,</p><p>Put the given equation of the curve into fluxions; from which equation of the fluxions find either <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi> or <hi rend="italics">y</hi><hi rend="sup">.</hi>/<hi rend="italics">x</hi><hi rend="sup">.</hi>; then take the fluxion of this ratio or fraction, and put it equal to 0 or nothing; and from this last equation find also the value of the same <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi> or <hi rend="italics">y</hi><hi rend="sup">.</hi>/<hi rend="italics">x</hi><hi rend="sup">.</hi>: then put this latter value equal to the former, which will be an equation from whence, and the first given equation of the curve, <hi rend="italics">x</hi> and <hi rend="italics">y</hi> will be determined, being the absciss or ordinate answering to the point of Inflection in the curve.</p><p>Or, putting the fluxion of <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi> equal to 0, that is, , or , or , or , that is, the 2d fluxions have the same ratio as the 1st fluxions, which is a constant ratio; and therefore if <hi rend="italics">x</hi><hi rend="sup">.</hi> be constant, or <hi rend="italics">x</hi><hi rend="sup">..</hi> = 0, then shall <hi rend="italics">y</hi><hi rend="sup">..</hi> be = 0 also; which gives another rule, viz; Take both the 1st and 2d fluxions of the given equation of the curve, in which make both <hi rend="italics">x</hi><hi rend="sup">..</hi> and <hi rend="italics">y</hi><hi rend="sup">..</hi> = 0, and the resulting equations will determine the values of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> or absciss and ordinate answering to the point of Inflection.</p><p>For example, if it be required to find the point of Inflection in the curve whose equation is<pb n="637"/><cb/> . Now the fluxion of this is , which gives . Then the fluxion of this again made = 0, gives ; and this gives again . Lastly, this value of <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi> put = the former, gives ; and hence , or , and , the absciss.</p><p>Hence also, from the original equation, , the ordinate to the point of Inflection sought.</p><p>When the curve has but one point of Inflection, it will be determined by a simple equation, as above; but when there are several points of Inflection, by the curve bending several times from the one side to the other, the resulting equation will be of a degree corresponding to them, and its roots will determine the abscisses or ordinates to the same.</p><p>Other methods of determining the points of Inflection in curves, may be seen in most books on the doctrine of fluxions.</p><p>To know whether a curve be concave or convex towards any point assigned in the axis; find the value of <hi rend="italics">y</hi><hi rend="sup">..</hi> at that point; then if this value be positive, the curve will be convex towards the axis, but if it be negative, it will be concave.</p><p>INFORMED <hi rend="italics">Stars,</hi> or <hi rend="smallcaps">Informes</hi> <hi rend="italics">Stellæ,</hi> are such stars as have not been reduced into any constellation; otherwise called Sporades.—There was a great number of this kind left by the ancient astronomers; but Hevelius and some others of the moderns have provided for the greater part of them, by making new constellations.</p><p>INGINEER. See <hi rend="smallcaps">Engineer.</hi></p></div1><div1 part="N" n="INGRESS" org="uniform" sample="complete" type="entry"><head>INGRESS</head><p>, in Astronomy, the sun's entrance into one of the signs, especially Aries.</p><p>INNOCENTS <hi rend="italics">Day,</hi> a feast celebrated on the 28th day of December, in commemoration of the infants murdered by Herod.</p><p>INORDINATE <hi rend="italics">Proportion,</hi> is where the order of the terms compared, is disturbed or irregular. As, for example, in two ranks of numbers, three in each rank, viz, in one rank, - - 2, 3, 9, and in the other rank, - - 8, 24, 36, which are proportional, the former to the latter, but in a different order, viz, - 2 : 3 : : 24 : 36, and - 3 : 9 : : 8 : 24. then, casting out the mean terms in each rank, it is concluded that - - 2 : 9 : : 8 : 36, that is, the first is to the 3d in the first rank, as the first is to the 3d in the 2d rank.</p><p>INSCRIBED <hi rend="italics">Figure,</hi> is one that has all its angular points touching the sides of another figure in which the former is said to be inscribed.</p><p><hi rend="smallcaps">Inscribed</hi> <hi rend="italics">Hyperbola,</hi> is one that lies wholly within the angle of its asymptotes; as the common or conical hyperbola doth.<cb/></p></div1><div1 part="N" n="INSTANT" org="uniform" sample="complete" type="entry"><head>INSTANT</head><p>, otherwise called a Moment, an infinitely small part of duration, or in which we perceive no succession, or which takes up the time of only one idea in our mind.</p><p>It is a maxim in mechanics, that no natural effect can be produced in an Instant, or without some definite time; also that the greater the time, the greater the effect. And hence may appear the reason, why a burt<*>en seems lighter to a person, the faster he carries it; and why, the faster a person slides or scates on the ice, the less liable it is to break, or bend.</p></div1><div1 part="N" n="INSULATE" org="uniform" sample="complete" type="entry"><head>INSULATE</head><p>, or <hi rend="smallcaps">Insulated</hi>, a term applied to a column or other edifice, which stands alone, or free and detached from any adjacent wall, &c, like an island in the sea.</p><div2 part="N" n="Insulated" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Insulated</hi></head><p>, in Electricity, is a term applied to bodies that are supported by electrics, or non-conductors; so that their communication with the earth, by conducting substances, is interrupted.</p><p>INTACTÆ, are right lines to which curves do continually approach, and yet can never meet them more usually called Asymptotes.</p></div2></div1><div1 part="N" n="INTEGERS" org="uniform" sample="complete" type="entry"><head>INTEGERS</head><p>, denote whole numbers: as contradistinguished from fractions.—Integers may be considered as numbers which refer to unity, as a whole to a part.</p><p>INTEGRAL <hi rend="italics">Number,</hi> an integer; not a fraction.</p><p><hi rend="smallcaps">Integral</hi> <hi rend="italics">Calculus,</hi> in the New Analysis, is the reverse of the differential calculus, and is the finding the Integral from a given differential; being similar to the inverse method of fluxions, or the finding the fluent to a given fluxion.</p><p>INTEGRANT <hi rend="italics">Parts,</hi> in <hi rend="italics">Philosophy,</hi> are the similar parts of a body, or parts of the same nature with the whole; as filings of iron are the Integrant parts of iron, having the same nature and properties with the bar or mass they were filed off from.</p></div1><div1 part="N" n="INTENSITY" org="uniform" sample="complete" type="entry"><head>INTENSITY</head><p>, or <hi rend="smallcaps">Intension</hi>, in Physics, is the degree or rate of the power or energy of any quality; as heat, cold, &c. The Intenfity of qualities, as gravity, light, heat, &c, vary in the reciprocal ratio of the squares of the distances from the centre of the radiating quality.</p><p>INTERCALARY <hi rend="italics">Day,</hi> denotes the odd day inserted in the leap-year. See <hi rend="smallcaps">Bissextile.</hi></p><p>INTERCEPTED <hi rend="italics">Axis,</hi> in Conic Sections, the same with what is otherwise called the absciss or abscissa.</p></div1><div1 part="N" n="INTERCOLUMNATION" org="uniform" sample="complete" type="entry"><head>INTERCOLUMNATION</head><p>, or I<hi rend="smallcaps">NTERCOLUMNIATION</hi>, is the space between column and column.</p></div1><div1 part="N" n="INTEREST" org="uniform" sample="complete" type="entry"><head>INTEREST</head><p>, is a sum reckoned for the loan or forbearance of another sum, or principal, lent for, or due at, a certain time, according to some certain rate or proportion; being estimated usually at so much per cent. or by the 100. This forms a particular rule in Arithmetic. The highest legal Interest now allowed in England, is after the rate of 5 per cent. per annum, or the 20th part of the principal for the space of a year, and so in proportion for other times, either greater or less. Except in the case of pawn-brokers, to whom it has lately been made legal to take a higher interest, for one of the worst and most destructive purposes that can be suffered in any state.<pb n="638"/><cb/></p><p>Interest is either Simple or Compound.</p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Interest</hi>, is that which is counted and allowed upon the principal only, for the whole time of forbearance.</p><p>The sum of the Principal and Interest is called the Amount.</p><p>As the Interest of any sum, for any time, is directly proportional to the principal sum and time; therefore the Interest of 1 pound for one year being multiplied by any proposed principal sum, and by the time of its forbearance, in years and parts, will be its Interest for that time. That is, if <hi rend="italics">r</hi> = the rate of Interest of 1<hi rend="italics">l.</hi> per annum, <hi rend="italics">p</hi> = any principal sum lent, <hi rend="italics">t</hi> = the time it is lent for, and <hi rend="italics"><*></hi> = the amount, or sum of principal and Interest; then is <hi rend="italics">prt</hi> = the Interest of the sum <hi rend="italics">p,</hi> for the time <hi rend="italics">t,</hi> at the rate <hi rend="italics">r;</hi> and consequently , the amount of the same for that time. And from this general theorem, other theorems can easily be deduced for sinding any of the quantities above mentioned; which collected all together, will be as follow: 1st, the amount, 2d, the principal, 3d, the rate, 4th, the time.</p><p>For example, let it be required to find in what time any principal sum will double itself, at any rate of Simple Interest. In this case we must use the 1st theorem , in which the amount <hi rend="italics">a</hi> must be = 2<hi rend="italics">p</hi> or double the principal, i. e. ; and hence ; where <hi rend="italics">r</hi> being the interest of 1<hi rend="italics">l.</hi> for one year, it follows that the time of doubling at Simple Interest, is equal to the quotient of any sum divided by its Interest for one year. So that, if the rate of Interest be 5 per cent. then is the time of doubling.</p><p>Or the 4th theorem immediately gives .</p><p>For more readily computing the Interest on money, various Tables of numbers are calculated and formed; such as a Table of Interest of 1<hi rend="italics">l.</hi> for any number of years, and for any number of months, or weeks, or days, &c, and at various rates of Interest.</p><p>Another Table is the following, by which may be readily found the Interest of any sum of money, from 1 to a million of pounds, for any number of days, at any rate of Interest.<cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Numb.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">l.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">q.</hi></cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">l.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">q.</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1000000</cell><cell cols="1" rows="1" role="data">2739</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0.99</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3.01</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">900000</cell><cell cols="1" rows="1" role="data">2465</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3.29</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0.71</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">800000</cell><cell cols="1" rows="1" role="data">2191</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">1.59</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2.41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">700000</cell><cell cols="1" rows="1" role="data">1917</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3.89</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">600000</cell><cell cols="1" rows="1" role="data">1643</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2.19</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1.81</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">500000</cell><cell cols="1" rows="1" role="data">1369</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0.49</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3.51</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">400000</cell><cell cols="1" rows="1" role="data">1095</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2.79</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1.21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300000</cell><cell cols="1" rows="1" role="data">821</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1.10</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2.90</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200000</cell><cell cols="1" rows="1" role="data">547</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3.40</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0.60</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100000</cell><cell cols="1" rows="1" role="data">273</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1.70</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2.30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90000</cell><cell cols="1" rows="1" role="data">246</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0.33</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3.67</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">80000</cell><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2.96</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1.04</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70000</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">1.59</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2.41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60000</cell><cell cols="1" rows="1" role="data">164</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0.22</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3.78</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50000</cell><cell cols="1" rows="1" role="data">136</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2.85</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1.15</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40000</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1.48</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2.55</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30000</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3.89</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20000</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2.74</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1.26</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10000</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">1.37</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2.63</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9000</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3.23</cell><cell cols="1" rows="1" role="data">0.9 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2.37</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8000</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1.10</cell><cell cols="1" rows="1" role="data">0.8 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2.10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7000</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2.96</cell><cell cols="1" rows="1" role="data">0.7 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1.84</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6000</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0.82</cell><cell cols="1" rows="1" role="data">0.6 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1.58</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5000</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.68</cell><cell cols="1" rows="1" role="data">0.5 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1.32</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4000</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0.55</cell><cell cols="1" rows="1" role="data">0.4 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1.05</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3000</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2.41</cell><cell cols="1" rows="1" role="data">0.3 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.79</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2000</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0.27</cell><cell cols="1" rows="1" role="data">0.2 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.53</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2.14</cell><cell cols="1" rows="1" role="data">0.1 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.26</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">900</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3.12</cell><cell cols="1" rows="1" role="data">0.09</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0 24</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">800</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.11</cell><cell cols="1" rows="1" role="data">0.08</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">700</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1.10</cell><cell cols="1" rows="1" role="data">0.07</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2.08</cell><cell cols="1" rows="1" role="data">0.06</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.16</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3.07</cell><cell cols="1" rows="1" role="data">0.05</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0.05</cell><cell cols="1" rows="1" role="data">0.04</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1.04</cell><cell cols="1" rows="1" role="data">0.03</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.08</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.03</cell><cell cols="1" rows="1" role="data">0.02</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0.05</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3.01</cell><cell cols="1" rows="1" role="data">0.01</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0 03</cell></row></table> <hi rend="center"><hi rend="italics">The Rule for using the Table is this:</hi></hi></p><p>Multiply the principal by the rate, both in pounds; multiply the product by the number of days, and divide this last product by 100; then take from the Table the several sums which stand opposite the several parts of the quotient, and adding them together will give the interest required.</p><p><hi rend="italics">Ex.</hi> What is the interest of 225<hi rend="italics">l.</hi> 10<hi rend="italics">s.</hi> for 23 days, at 4 1/2 per cent. per annum? <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">princ. 225.5</cell><cell cols="1" rows="1" rend="rowspan=9" role="data">Then in theTable</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">l.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">q.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">rate     4.5</cell><cell cols="1" rows="1" rend="align=right" role="data">against 200 is</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">2.03</cell></row><row role="data"><cell cols="1" rows="1" role="data">----------</cell><cell cols="1" rows="1" rend="align=right" role="data">30 "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">2.90</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1014.74</cell><cell cols="1" rows="1" rend="align=right" role="data">3 "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">3.89</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">75 days  23</cell><cell cols="1" rows="1" rend="align=right" role="data">0.3 "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0.79</cell></row><row role="data"><cell cols="1" rows="1" role="data">----------</cell><cell cols="1" rows="1" rend="align=right" role="data">0.09 "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">0.24</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">100)23339.25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">---</cell><cell cols="1" rows="1" rend="align=right" role="data">---</cell><cell cols="1" rows="1" rend="align=right" role="data">---</cell><cell cols="1" rows="1" role="data">---</cell></row><row role="data"><cell cols="1" rows="1" role="data">----------</cell><cell cols="1" rows="1" rend="align=right" role="data">Ans.</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">1.85 true</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">233.3925</cell><cell cols="1" rows="1" rend="colspan=5 align=center" role="data">in the last place of decimals.</cell></row></table></p><p>Another ingenious and general method of com-<pb n="639"/><cb/> puting Interest, is by the following small but comprehensive Table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=21 align=center" role="data"><hi rend="italics">A General Interest Table,</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=21 align=center" role="data">By which the Interest of any Sum, at any Rate, and for any Time, may be readily found.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Days.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">3 per Cent.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">3 1/2perCent.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">4 per Cent.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">4 1/2perCe<*>t.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">5 per Cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">l.</cell><cell cols="1" rows="1" role="data">s.</cell><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">q.</cell><cell cols="1" rows="1" role="data">l.</cell><cell cols="1" rows="1" role="data">s.</cell><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">q.</cell><cell cols="1" rows="1" role="data">l.</cell><cell cols="1" rows="1" role="data">s.</cell><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">q.</cell><cell cols="1" rows="1" role="data">l.</cell><cell cols="1" rows="1" role="data">s.</cell><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">q.</cell><cell cols="1" rows="1" role="data">l.</cell><cell cols="1" rows="1" role="data">s.</cell><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">q.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell></row></table></p><p>N. B. This Table contains the interest of 100l. for all the several days in the 1st column, and at the several rates of 3, 3 1/2, 4, 4 1/2, and 5 per cent. in the other 5 columns.</p><p><hi rend="italics">To find the Interest of</hi> 100<hi rend="italics">l. for any other time,</hi> as 1 year and 278 days, at 4 1/2 per cent. Take the sums for the several days as here below. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">The Int. for 1 year</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Against 200 ds. is</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">------- 70 ds. "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">------- 8 ds. "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">---</cell><cell cols="1" rows="1" role="data">---</cell><cell cols="1" rows="1" role="data">---</cell><cell cols="1" rows="1" role="data">---</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Interest required "</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell></row></table></p><p><hi rend="italics">For any other Sum than</hi> 100<hi rend="italics">l.</hi> First find for 100l. as above, and take it so many times or parts as the sum is of 100l. Thus, to find for 355l. at 4 1/2, for 1 year and 278 days.</p><p>First, 3 times the above sum, <table><row role="data"><cell cols="1" rows="1" role="data">(for 300l.) is</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" role="data">1/2 (for 50l.) is</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" role="data">1/10 of this (for 5l.)</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">--</cell><cell cols="1" rows="1" role="data">--</cell><cell cols="1" rows="1" role="data">--</cell><cell cols="1" rows="1" role="data">--</cell></row><row role="data"><cell cols="1" rows="1" role="data">So for 355 it is</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell></row></table></p><p>When the interest is required for any other rate than thosein the Table, it may be easily made out from them. So 1/2 of 5 is 2 1/2, 1/2 of 4 is 2, 1/2 of 3 is 1 1/2, 1/3 of 3 is 1, 1-6th of 3 is 1/2, and 1-12th of 3 is 1/4. And so, by parts, or by adding or subtracting, any rate may be made out.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Interest</hi>, called also <hi rend="italics">Interest-upon-Interest,</hi> is that which is counted not only upon the principal sum lent, but also for its Interest, as it becomes due, at the end of each stated time of payment.</p><p>Although it be not lawful to lend money at Compound Interest, yet in purchasing annuities, pensions,<cb/> &c, and taking leases in reversion, it is usual to allow Compound Interest to the purchaser for his ready money; and therefore it is very necessary to understand this subject.</p><p>Besides the quantities concerned in Simple Interest, viz, the principal <hi rend="italics">p,</hi> the rate or Interest of 1l. for 1 year <hi rend="italics">r,</hi> the amount <hi rend="italics">a,</hi> and the time <hi rend="italics">t,</hi> there is another quantity employed in Compound Interest, viz, the ratio of the rate of Interest, which is the amount of 1l. for 1 time of payment, and which here let be denoted by R, viz, . Then, the particular amounts for the several times may be thus computed, viz, As 1 pound is to its amount for any time, so is any proposed principal sum to its amount for the same time; i. e. the 1st year's amount, the 2d year's amount, the 3d year's amount, and so on. Therefore in general, is the amount for the <hi rend="italics">t</hi> year, or <hi rend="italics">t</hi> time of payment. From whence the following general theorems are deduced: 1st, the amount, 2d, the principal, 3d, the ratio, 4th, the time. From which any one of the quantities may be found, when the rest are given.</p><p>For example, suppose it were required to find in how many years any principal sum will double itself, at any rate of Interest. In this case we must employ the 4th theorem, where <hi rend="italics">a</hi> will be = 2<hi rend="italics">p,</hi> and then it is . So, if the rate of Interest be 5 per cent. per annum; then , and hence nearly; that is, any sum doubles in 14 1/5 years nearly, at the rate of 5 per cent. per annum Compound Interest.</p><p>Hence, and from the like question in Simple Interest, above given, are deduced the times in which any sum doubles itself, at several rates of Interest, both simple and compound: viz, <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">At</cell><cell cols="1" rows="1" role="data"><hi rend="size(20)">}</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">At Simp. Int. Years.</cell><cell cols="1" rows="1" role="data">At Comp. Int. Years.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">35.0028</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">28.0701</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">33 1/3</cell><cell cols="1" rows="1" role="data">23.4498</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">28 4/7</cell><cell cols="1" rows="1" role="data">20.1488</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">per cent. per an.</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">17.6730</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4 1/2</cell><cell cols="1" rows="1" role="data">Interest, 1<hi rend="italics">l.</hi> or</cell><cell cols="1" rows="1" role="data">22 1/9</cell><cell cols="1" rows="1" role="data">15.7473</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">any other sum</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">14.2067</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">will double in</cell><cell cols="1" rows="1" role="data">16 2/3</cell><cell cols="1" rows="1" role="data">11.8957</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">14 2/7</cell><cell cols="1" rows="1" role="data">10.2448</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12 1/2</cell><cell cols="1" rows="1" role="data">9.0065</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">11 1/9</cell><cell cols="1" rows="1" role="data">8.0432</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">7.2725</cell></row></table><pb n="640"/><cb/></p><p>The following Table will facilitate the calculation of Compound Interest for any sum, and any number of years, at various rates of Interest. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=7 align=center" role="data"><hi rend="italics">The Amount of</hi> 1<hi rend="italics">l. in any Number of Years.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Yrs.</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3 1/2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4 1/2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1.0300</cell><cell cols="1" rows="1" role="data">1.0350</cell><cell cols="1" rows="1" role="data">1.0400</cell><cell cols="1" rows="1" role="data">1.0450</cell><cell cols="1" rows="1" role="data">1.0500</cell><cell cols="1" rows="1" role="data">1.0600</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1.0609</cell><cell cols="1" rows="1" role="data">1.0712</cell><cell cols="1" rows="1" role="data">1.0816</cell><cell cols="1" rows="1" role="data">1.0920</cell><cell cols="1" rows="1" role="data">1.1025</cell><cell cols="1" rows="1" role="data">1.1236</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1.0927</cell><cell cols="1" rows="1" role="data">1.1087</cell><cell cols="1" rows="1" role="data">1.1249</cell><cell cols="1" rows="1" role="data">1.1412</cell><cell cols="1" rows="1" role="data">1.1576</cell><cell cols="1" rows="1" role="data">1.1910</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1.1255</cell><cell cols="1" rows="1" role="data">1.1475</cell><cell cols="1" rows="1" role="data">1.1699</cell><cell cols="1" rows="1" role="data">1.1925</cell><cell cols="1" rows="1" role="data">1.2155</cell><cell cols="1" rows="1" role="data">1.2625</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1.1593</cell><cell cols="1" rows="1" role="data">1.1877</cell><cell cols="1" rows="1" role="data">1.2167</cell><cell cols="1" rows="1" role="data">1.2462</cell><cell cols="1" rows="1" role="data">1.2763</cell><cell cols="1" rows="1" role="data">1.3382</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1.1941</cell><cell cols="1" rows="1" role="data">1.2293</cell><cell cols="1" rows="1" role="data">1.2653</cell><cell cols="1" rows="1" role="data">1.3023</cell><cell cols="1" rows="1" role="data">1.3401</cell><cell cols="1" rows="1" role="data">1.4185</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">1.2299</cell><cell cols="1" rows="1" role="data">1.2723</cell><cell cols="1" rows="1" role="data">1.3159</cell><cell cols="1" rows="1" role="data">1.3609</cell><cell cols="1" rows="1" role="data">1.4071</cell><cell cols="1" rows="1" role="data">1.5036</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">1.2668</cell><cell cols="1" rows="1" role="data">1.3168</cell><cell cols="1" rows="1" role="data">1.3686</cell><cell cols="1" rows="1" role="data">1.4221</cell><cell cols="1" rows="1" role="data">1.4775</cell><cell cols="1" rows="1" role="data">1.5939</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1.3048</cell><cell cols="1" rows="1" role="data">1.3629</cell><cell cols="1" rows="1" role="data">1.4233</cell><cell cols="1" rows="1" role="data">1.4861</cell><cell cols="1" rows="1" role="data">1.5513</cell><cell cols="1" rows="1" role="data">1.6895</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1.3439</cell><cell cols="1" rows="1" role="data">1.4106</cell><cell cols="1" rows="1" role="data">1.4802</cell><cell cols="1" rows="1" role="data">1.5530</cell><cell cols="1" rows="1" role="data">1.6289</cell><cell cols="1" rows="1" role="data">1.7909</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">1.3842</cell><cell cols="1" rows="1" role="data">1.4600</cell><cell cols="1" rows="1" role="data">1.5395</cell><cell cols="1" rows="1" role="data">1.6229</cell><cell cols="1" rows="1" role="data">1.7103</cell><cell cols="1" rows="1" role="data">1.8983</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">1.4258</cell><cell cols="1" rows="1" role="data">1.5111</cell><cell cols="1" rows="1" role="data">1.6010</cell><cell cols="1" rows="1" role="data">1.6959</cell><cell cols="1" rows="1" role="data">1.7959</cell><cell cols="1" rows="1" role="data">2.0122</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1.4685</cell><cell cols="1" rows="1" role="data">1.5640</cell><cell cols="1" rows="1" role="data">1.6651</cell><cell cols="1" rows="1" role="data">1.7722</cell><cell cols="1" rows="1" role="data">1.8856</cell><cell cols="1" rows="1" role="data">2.1329</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1.5126</cell><cell cols="1" rows="1" role="data">1.6187</cell><cell cols="1" rows="1" role="data">1.7317</cell><cell cols="1" rows="1" role="data">1.8519</cell><cell cols="1" rows="1" role="data">1.9799</cell><cell cols="1" rows="1" role="data">2.2609</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">1.5580</cell><cell cols="1" rows="1" role="data">1.6753</cell><cell cols="1" rows="1" role="data">1.8009</cell><cell cols="1" rows="1" role="data">1.9353</cell><cell cols="1" rows="1" role="data">2.0789</cell><cell cols="1" rows="1" role="data">2.3966</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">1.6047</cell><cell cols="1" rows="1" role="data">1.7340</cell><cell cols="1" rows="1" role="data">1.8730</cell><cell cols="1" rows="1" role="data">2.0224</cell><cell cols="1" rows="1" role="data">2.1829</cell><cell cols="1" rows="1" role="data">2.5404</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">1.6528</cell><cell cols="1" rows="1" role="data">1.7947</cell><cell cols="1" rows="1" role="data">1.9479</cell><cell cols="1" rows="1" role="data">2.1134</cell><cell cols="1" rows="1" role="data">2.2920</cell><cell cols="1" rows="1" role="data">2.6928</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1.7024</cell><cell cols="1" rows="1" role="data">1.8575</cell><cell cols="1" rows="1" role="data">2.0258</cell><cell cols="1" rows="1" role="data">2.2085</cell><cell cols="1" rows="1" role="data">2.4066</cell><cell cols="1" rows="1" role="data">2.8543</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1.7535</cell><cell cols="1" rows="1" role="data">1.9225</cell><cell cols="1" rows="1" role="data">2.1068</cell><cell cols="1" rows="1" role="data">2.3079</cell><cell cols="1" rows="1" role="data">2.5270</cell><cell cols="1" rows="1" role="data">2.0256</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">1.8061</cell><cell cols="1" rows="1" role="data">1.9898</cell><cell cols="1" rows="1" role="data">2.1911</cell><cell cols="1" rows="1" role="data">2.4117</cell><cell cols="1" rows="1" role="data">2.6533</cell><cell cols="1" rows="1" role="data">2.2071</cell></row></table></p><p>The use of this Table, which contains all the powers R<hi rend="sup">t</hi>, to the 20th power, or the amounts of 1<hi rend="italics">l.</hi> is chiefly to calculate the Interest, or the amount, of any principal sum, for any time, not more than 20 years. For example, required to find to how much 523<hi rend="italics">l.</hi> will amount <*> 15 years, at the rate of 5<hi rend="italics">l.</hi> per cent. per annum Compound Interest.</p><p>In the Table, on the line 15 and column 5 per cent, <table><row role="data"><cell cols="1" rows="1" role="data">is the amount of 1<hi rend="italics">l.</hi> viz.</cell><cell cols="1" rows="1" rend="align=right" role="data">2.0789,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">this multiplied by the principal</cell><cell cols="1" rows="1" rend="align=right" role="data">523,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">gives the amount</cell><cell cols="1" rows="1" rend="align=right" role="data">1087.2647</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">   or</cell><cell cols="1" rows="1" rend="align=right" role="data">1087<hi rend="italics">l.</hi>  5<hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data">3 1/4<hi rend="italics">d.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">and therefore the Interest is</cell><cell cols="1" rows="1" rend="align=right" role="data">564<hi rend="italics">l.</hi>  5<hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data">3 1/4<hi rend="italics">d.</hi></cell></row></table></p><p>See Annuities; Discount; Reversion; Smart's Tables of Interest; the Philos. Trans. vol. 6, p. 508; and most books on Arithmetic.</p><p>INTERIOR <hi rend="italics">Figure, Angle of.</hi> See <hi rend="smallcaps">Angle.</hi></p><p><hi rend="smallcaps">Interior</hi> <hi rend="italics">Polygon.</hi> See <hi rend="smallcaps">Polygon.</hi></p><p><hi rend="smallcaps">Interior</hi> <hi rend="italics">Talus.</hi> See <hi rend="smallcaps">Talus.</hi></p><p><hi rend="smallcaps">Internal</hi> <hi rend="italics">Angles,</hi> are all angles made within any figure, by the sides of it. In a triangle ABC, the two <figure/> angles A and C are peculiarly called Internal and oppo-<cb/> site, in respect of the external angle CBD, which i<*> equal to them both together.</p><p><hi rend="smallcaps">Internal</hi> <hi rend="italics">Angle</hi> is also applied to the two angles formed between two parallels, by a line intersecting those parallels, on each side of the intersecting line. Such are the angles <hi rend="italics">a, b, c, d,</hi> formed between the parallels EF and GH, on each side of the intersecting line.—The two adjacent Internal angles <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> or <hi rend="italics">c</hi> and <hi rend="italics">d,</hi> are together equal to two right angles.</p><p><hi rend="smallcaps">Internal</hi> and Opposite Angles, is also applied to the two angles <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> which are respectively equal to the two <hi rend="italics">n</hi> and <hi rend="italics">m,</hi> called the external and opposite angles.</p><p>Also the alternate Internal angles are equal to one another; viz, <hi rend="italics">a</hi> = <hi rend="italics">d,</hi> and <hi rend="italics">b</hi> = <hi rend="italics">c.</hi></p></div1><div1 part="N" n="INTERPOLATION" org="uniform" sample="complete" type="entry"><head>INTERPOLATION</head><p>, in the modern Algebra, is used for finding an intermediate term of a series, its place in the series being given.</p><p>The Method of Interpolation was first invented by Mr. Briggs, and applied by him to the calculation of logarithms, &c, in his Arithmetica Logarithmica, and his Trigonometria Britannica; where he explains, and fully applies the method of Interpolation by differences. His principles were followed by Reginal and Mouton in France, and by Cotes and others in England. Wallis made use of the method of Interpolation in various parts of his works; as his Arithmetic of Infinites, and his Algebra, for quadratures, &c. The same was also happily applied by Newton in various ways: by it he investigated his binomial theorem, and quadratures of the circle, ellipse, and hyperbola: see Wallis's Algebra, chap. 85, &c. Newton also, in lemma 5, lib. 3 Princip. gave a most elegant solution of the problem for drawing a curve line through the extremities of any number of given ordinates; and in the subsequent proposition, applied the solution of this problem to that of finding from certain observed places of a comet, its place at any given intermediate time. And Dr. Waring, who adds, that a solution still more elegant, on some accounts, has been since discovered by Mess. Nichol and Stirling, has also resolved the same problem, and rendered it more general, without having recourse to finding the successive differences. Philos. Trans. vol. 69, part 1, art. 7.</p><p>Mr. Stirling indeed pursued this branch as a distinct science, in a separate treatise, viz, Tractatus de Summatione et Interpolatione Serierum Infinitarum, in the year 1730.</p><p>When the 1st, 2d, or other successive differences of the terms of a series become at last equal, the Interpolation of any term of such a series may be found by Newton's Differential Method.</p><p>When the Algebraic equation of a series is given, the term required, whether it be a primary or intermediate one, may be found by the resolution of affected equations; but when that equation is not given, as it often happens, the value of the term sought must be exhibited by a converging series, or by the quadrature of curves. See Stirling, ut supra, p. 86. Meyer, in Act. Petr. tom. 2, p. 180.</p><p>A general theorem for Interpolating any term is as follows: Let A denote any term of an equidistant series of terms, and <hi rend="italics">a, b, c,</hi> &c, the first of the 1st, 2d, 3d, &c orders of differences; then the term <hi rend="italics">z,</hi> whose<pb n="641"/><cb/> distance from A is expressed by <hi rend="italics">x,</hi> will be this, viz, <hi rend="italics">Theorem</hi> 1,</p><p>Hence, if any of the orders of differences become equal to one another, or = 0, this series for the interpolated term will break off, and terminate, otherwise it will run out in an insinite series.</p><p><hi rend="italics">Ex.</hi> To find the 20th term of the series of cubes 1, 8, 27, 64, 125, &c, or 1<hi rend="sup">3</hi>, 2<hi rend="sup">3</hi>, 3<hi rend="sup">3</hi>, 4<hi rend="sup">3</hi>, 5<hi rend="sup">3</hi>, &c. <figure/></p><p>Set down the series in a column, and take their continual differences as here annexed, where the 4th differences, and all after it become = 0, also A = 1, <hi rend="italics">a</hi> = 7, <hi rend="italics">b</hi> = 12, <hi rend="italics">c</hi> = 6, and <hi rend="italics">x</hi> = 19; therefore the 20th term sought is barely</p><p><hi rend="italics">Theor.</hi> 2. In any series of equidistant terms, <hi rend="italics">a, b, c, d,</hi> &c, whose first differences are small; to find any term wanting in that series, having any number of terms given. Take the equation which stands against the number of given terms, in the following Table; and by reducing the equation, that term will be found. where it is evident that the coefficients in any equation, are the unciæ of a binomial 1 + 1 raised to the power denoted by the number of the equation.</p><p><hi rend="italics">Ex.</hi> Given the logarithms of 101, 102, 104, and 105; to find the log. of 103.</p><p>Here are 4 quantities given; therefore we must take the 4th equation , in which it is the middle quantity or term <hi rend="italics">c</hi> that is to be found, because 103 is in the middle among the numbers 101, 102, 104, 105; then that equation gives the value of <hi rend="italics">c</hi> as follows, viz .<cb/></p><p>Now the logs. of the given numbers will be thus:</p><p><hi rend="italics">Theor.</hi> 3. When the terms <hi rend="italics">a, b, c, d,</hi> &c, are at unequal distances from each other; to find any intermediate one of these terms, the rest being given.</p><p>Let <hi rend="italics">p, q, r, s,</hi> &c, be the several distances of those terms from each other; then let &c &c &c Then the term <hi rend="italics">z,</hi> whose distance from the beginning is <hi rend="italics">x,</hi> will be to be continued to as many terms as there are terms in the given series.</p><p>By this series may be found the place of a comet, or the sun, or any other object at a given time; by knowing the places of the same for several other given times.</p><p>Other methods of Interpolation may be found in the Philos. Trans. number 362; or Stirling's Summation and Interpolation of Series.</p></div1><div1 part="N" n="INTERSCENDENT" org="uniform" sample="complete" type="entry"><head>INTERSCENDENT</head><p>, in Algebra, is applied to quantities, when the exponents of their powers are radical quantities. Thus <hi rend="italics">x</hi><hi rend="sup">√2</hi>, <hi rend="italics">x</hi><hi rend="sup">√<hi rend="italics">a</hi></hi>, &c, are interscendent quantities. See <hi rend="smallcaps">Function.</hi></p></div1><div1 part="N" n="INTERSECTION" org="uniform" sample="complete" type="entry"><head>INTERSECTION</head><p>, the cutting of one line, or plane, by another; or the point or line in which two lines or two planes cut each other.——The mutual intersection of two planes is a right line. The centre of a circle, or conic section, &c, is in the intersection of two diameters; and the central point of a quadrangle is the Intersection of two diagonals.</p></div1><div1 part="N" n="INTERSTELLAR" org="uniform" sample="complete" type="entry"><head>INTERSTELLAR</head><p>, a word used by some authors, to express those parts of the universe, that are without and beyond the limits of our solar system.</p><p>In the Interstellar regions, it is supposed there are several other systems of planets moving round the fixed stars, as the centres of their respective motions. And if it be true, as it is not improbable, that each fixed star is<pb n="642"/><cb/> thus a sun to some habitable orbs, or earths, that move round it, the Interstellar world will be insinitely the greatest part of the universe.</p></div1><div1 part="N" n="INTERTIES" org="uniform" sample="complete" type="entry"><head>INTERTIES</head><p>, or <hi rend="smallcaps">Interduces</hi>, in Architecture, those small pieces of timber which lie horizontally between the summers, or between them and the cell or raising plate.</p></div1><div1 part="N" n="INTERVAL" org="uniform" sample="complete" type="entry"><head>INTERVAL</head><p>, in Musie, the difference between two sounds, in respect of acute and grave. Authors distinguish several divisions of an Interval, as first into Simple and Compound. The</p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Interval</hi> is that without parts, or division; such are the octave, and all that are within it; as the 2d, 3d, 4th, 5th, 6th, and 7th, with their varieties.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Interval</hi> consists of several lesser Intervals; as the 9th, 10th, 11th, 12th, &c, with their varieties.</p><p>This Simple Interval was by the ancients called a Diastem, and the Compound they called a System.</p><p>An Interval is also divided into Just or True, and into False. The</p><p><hi rend="italics">Just</hi> or <hi rend="italics">True</hi> <hi rend="smallcaps">Intervals</hi>, are such as all those above mentioned, with their varieties, whether major or minor. And the</p><p><hi rend="italics">False</hi> <hi rend="smallcaps">Intervals</hi>, are the diminutive or superfluous ones.</p><p>An Interval is also divided into the Consonance and Dissonance; which see.</p><p>INTESTINE <hi rend="italics">Motion of the parts of fluids,</hi> that which is among its corpuscles or component parts.</p><p>When the attracting corpuscles of any fluid are elastic, they must necessarily produce an intestine motion; and that, greater or less, according to the degrees of their elasticity and attractive force. For, two <*>lastic particles, after meeting, will fly from each other, with the same degree of velocity with which they met; abstracting from the resistance of the medium. But when, in leaping back from each other, they approach other particles, their velocity will be increased.</p></div1><div1 part="N" n="INTRADOS" org="uniform" sample="complete" type="entry"><head>INTRADOS</head><p>, the interior and lower side, or curve, of the arch of a bridge, &c. In contradistinction from the extrados, or exterior curve, or line on the upper side of the arch. See my Treatise on Bridges &c. prop. 5.</p></div1><div1 part="N" n="INVERSE" org="uniform" sample="complete" type="entry"><head>INVERSE</head><p>, is applied to a manner of working the rule of three, or proportion, which seems to go backward, i. e. reverse or contrary to the order of the common and direct rule: So that, whereas, in the direct rule, more requires more, or less requires less; in the Inverse rule, on the contrary, more requires less, or less requires more.</p><p>For instance, in the direct rule it is said, If 3 yards of cloth cost 20 shillings, how much will 6 yards cost? the answer is 40 shillings: where more yards require more money, and less yards require less money. But in the Inverse rule it is said, If 20 men perform a piece of work in 4 days, in how many days will 40 men perform as much? where the answer is 2 days; and here the more men require the less time, and the fewer men the more time.</p><p><hi rend="smallcaps">Inverse</hi> <hi rend="italics">Method of Fluxions,</hi> is the method of finding fluents, from the fluxions being given; and is fimilar to what the foreign mathematicians call the Calculus Integralis. See <hi rend="smallcaps">Fluents.</hi><cb/></p><p><hi rend="smallcaps">Inverse</hi> <hi rend="italics">Method of Tangents,</hi> is the method of finding the curve belonging to a given tangent; as opposed to the direct method, or the finding the tangent to a given curve.</p><p>As, to find a curve whose subtangent is a third proportional to <hi rend="italics">r</hi> - <hi rend="italics">y</hi> and <hi rend="italics">y,</hi> or whose subtangent is equal to the semiordinate, or whose subnormal is a constant quantity.——The solution of this problem depends chiefly on the Inverse method of fluxions. See T<hi rend="smallcaps">ANGENT.</hi></p><p><hi rend="smallcaps">Inverse</hi> <hi rend="italics">Proportion,</hi> or <hi rend="smallcaps">Inverse</hi> <hi rend="italics">Ratio,</hi> is that in which more requires less, or less requires more. As for instance, in the case of light, or heat from a luminous object, the light received is less at a greater distance, and greater at a less distance; so that here more, as to distance, gives less, as to light, and less distance gives more light. This is usually expressed by the term Inversely, or Reciprocally; as in the case above, where the light is Inversely, or Reciprocally as the square of the distance; or in the Inverse or Reciprocal duplicate ratio of the distance.</p></div1><div1 part="N" n="INVERSION" org="uniform" sample="complete" type="entry"><head>INVERSION</head><p>, <hi rend="italics">Invertendo,</hi> or <hi rend="italics">by Inversion,</hi> according to the 14th desinition of Euclid, lib. 5, is Inverting the terms of a proportion, by changing the antecedents into consequents, and the consequents into antecedents. As in these, , then by Inversion .</p></div1><div1 part="N" n="INVESTIGATION" org="uniform" sample="complete" type="entry"><head>INVESTIGATION</head><p>, the searching or finding any thing out, by means of certain steps, traces, or ways.</p><p>INVOLUTE <hi rend="italics">Figure</hi> or <hi rend="italics">Curve,</hi> is that which is traced out by the outer extremity of a string as it is folded or wrapped upon another figure, or as it is unwound from off it.——The Involute of a cycloid, is also a cycloid equal to the former, which was first discovered by Huygens, and by means of which he contrived to make a pendulum vibrate in the curve of a cycloid, and so theoretically at least vibrate always in equal times whether the arch of vibration were great or small, which is a property of that curve.——For the doctrine and nature of Involutes and Evolutes, see <hi rend="smallcaps">Evolute.</hi></p></div1><div1 part="N" n="INVOLUTION" org="uniform" sample="complete" type="entry"><head>INVOLUTION</head><p>, in Arithmetic and Algebra, is the raising of powers from a given root; as opposed to Evolution, which is the extracting, or developing of roots from given powers. So the Involution of the number 3, or its powers, are thus raised: 3 - or 3<hi rend="sup">1</hi> or 3 is the 1st power, or root, 3 × 3 or 3<hi rend="sup">2</hi> or 9 is the 2d power, or square, 3 × 3 × 3 or 3<hi rend="sup">3</hi> or 27 is the 3d power, or cube, and so on.</p><p>And hence, to find any power of a given root, or quantity, let the root be multiplied by itself a number of times which is one less than the number of the index; i. e. onc<*> multiplied for the 2d root, twice for the 3d root, thrice for the 4th root, &c. <table><row role="data"><cell cols="1" rows="1" role="data">Thus, to Involve</cell><cell cols="1" rows="1" rend="align=right" role="data">.12</cell><cell cols="1" rows="1" role="data">to the 3d power.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.12</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.0144</cell><cell cols="1" rows="1" role="data">  square, or 2d power.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.12</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.001728</cell><cell cols="1" rows="1" role="data">  cube, or 3d power.</cell></row></table><pb n="643"/><cb/></p><p>So also, in Algebra, to Involve the binomial <hi rend="italics">a</hi> + <hi rend="italics">b,</hi> or raise its powers.</p><p>And in like manner for any other quantities, whatever the number of their terms may be. But compound algebraic quantities are best involved by the <hi rend="smallcaps">Binomial</hi> <hi rend="italics">Theorem;</hi> which see.</p><p>Simple quantities are Involved, by raising the numeral coefficients to the given power, and the literal quantities are raised by multiplying their indices by that of the root; that is, the raising of powers is performed by the multiplication of indices, the same as the multiplication of logarithms. Thus,</p><p>The 2d power of <hi rend="italics">a</hi> is <hi rend="italics">a</hi><hi rend="sup">2</hi>.</p><p>The 2d power of 2<hi rend="italics">a</hi><hi rend="sup">2</hi> is 2<hi rend="sup">2</hi><hi rend="italics">a</hi><hi rend="sup">2 × 2</hi> or 4<hi rend="italics">a</hi><hi rend="sup">4</hi>.</p><p>The 3d power of 3<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi><hi rend="sup">3</hi> is 27<hi rend="italics">a</hi><hi rend="sup">6</hi><hi rend="italics">l</hi><hi rend="sup">9</hi>.</p><p>The 3d power of <hi rend="italics">a</hi><hi rend="sup">1/2</hi><hi rend="italics">b</hi><hi rend="sup">2/3</hi> is <hi rend="italics">a</hi><hi rend="sup">3/2</hi><hi rend="italics">b</hi><hi rend="sup">2</hi>.</p><p>The <hi rend="italics">n</hi>th power of <hi rend="italics">a</hi><hi rend="sup">m</hi> <hi rend="italics">c</hi><hi rend="sup">p</hi> is <hi rend="italics">a</hi><hi rend="sup">mn</hi><hi rend="italics">c</hi><hi rend="sup">pn</hi> or (―(<hi rend="italics">a</hi><hi rend="sup">m</hi> <hi rend="italics">c</hi><hi rend="sup">p</hi>))<hi rend="sup">n</hi>.</p><p>INWARD <hi rend="italics">Flanking Angle,</hi> in Fortification, is that made by the curtin and the razant flanking line of defence.</p></div1><div1 part="N" n="JOINTS" org="uniform" sample="complete" type="entry"><head>JOINTS</head><p>, in Architecture, are the separations between the stones or bricks; which may be filled with mortar, plaster, or cement.</p><div2 part="N" n="Joint" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Joint</hi></head><p>, in Carpentry, &c, is applied to several manners of assembling, setting, or fixing pieces of wood together. As by a mortise Joint, a dove-tail Joint, &c.</p><p><hi rend="italics">Universal</hi> <hi rend="smallcaps">Joint</hi>, in Mechanics, an excellent invention of Dr. Hook, adapted to all kinds of motions and flexures; of which he has given a large account in his Cutlerian Lectures, printed in 1678. This seems to have g<*>ven occasion to the gimbols used in suspending the sea compasses; the mechanism of which is the same with that of Desaguliers's rolling lamp.</p><p><hi rend="smallcaps">Joint</hi>-<hi rend="italics">Lives,</hi> are such as continue during the same time, or that exist together. See <hi rend="smallcaps">Life</hi>-<hi rend="italics">Annuities.</hi></p></div2></div1><div1 part="N" n="JOISTS" org="uniform" sample="complete" type="entry"><head>JOISTS</head><p>, or <hi rend="smallcaps">Joysts</hi>, those pieces of timber framed into the girders and summers, and on which the boarding of floors is laid.</p><p>“JONES (<hi rend="smallcaps">William</hi>), F. R. S. a very eminent mathematician, was born at the foot of Bodavon mountain [Mynydd Bodafon] in the parish of Llanfihangel tre'r Bard, in the Isle of Anglesy, North Wales, in the year 1675. His father's name was John George,<cb/> his surname being the proper name of his father. For it is a custom in several parts of Wales for the proper name of the father to become the surname of his children. John George the father was commonly called Sion Siors of Llanbabo, to which place he moved, and where his children were brought up. Accordingly our author, whose proper name was William, took the surname of Jones from the proper name of his father, who was a farmer, and of a good family, being descended from Hwfa ap Cynddelw, one of the 15 tribes of North Wales. He gave his two sons the common school education of the country, reading, writing, and accounts, in English, and the Latin Grammar. Harry his second son took to the farming business; but William the eldest, having an extraordinary turn for mathematical studies, determined to try his fortune abroad from a place where the same was but of little service to him. He accordingly came to London, accompanied by a young man, Rowland Williams, afterwards an eminent perfumer in Wych-street. The report in the country is, that Mr. Jones soon got into a merchant's counting house, and so gained the esteem of his master, that he gave him the command of a ship for a West India voyage; and that upon his return he set up a mathematical school, and published his book of Navigation; and that upon the death of the merchant he married his widow: that, lord Macclesfield's son being his pupil, he was made secretary to the chancellor, and one of the deputy tellers of the exchequer:—and they have a story of an Italian wedding which caused great disturbance in lord Macclesfield's family, but was compromised by Mr. Jones; which gave rise to a saying, “that Macclesfield was the making of Jones, and Jones the making of Macclesfield.” The foregoing account of Mr. Jones, I found among the papers of the late Mr. John Robertson, librarian and clerk to the Royal Society, who had been well known to Mr. Jones, and possessed many of his papers.</p><p>Mr. Jones having by his industry acquired a competent fortune, lived upon it as a private gentleman for many years, in the latter part of his life, in habits of intimacy with Sir Isaac Newton and others the most eminent mathematicians and philosophers of his time; and died July 3, 1749, at 74 years of age, being one of the vice-presidents of the Royal Society; leaving at his death one daughter, and his widow with child, which proved a son, who is the present Sir William Jones, now one of the judges in India, and highly esteemed for his great abilities, extensive learning, and eminent patriotism.——Mr. Jones's publications are,</p><p>1. <hi rend="italics">A new Compendium of the Who'e Art of Navigation,</hi> &c; in small 8vo, London, 1702. This is a neat little piece, and dedicated to the Rev. Mr. John Harris, the same I believe who was author of the <hi rend="italics">Lexicon Technicum,</hi> or Universal Dictionary of Arts and Sciences, in whose house Mr. Jones says he composed his book.</p><p>2. <hi rend="italics">Synopsis Palmariorum Matheseos:</hi> Or a New Introduction to the Mathematics, &c; 8vo, London, 1706. Being a very neat and useful compendium of all the mathematical sciences, in about 300 pages.</p><p>His papers in the Philos. Trans. are the following:</p><p>3. A Compendious Disposition of Equations for exhibiting the relations of Goniometrical Lines; vol. 44, p. 560.<pb n="644"/><cb/></p><p>4. A Tract on Logarithms; vol. 61, pa. 455.</p><p>5. Account of the person killed by lightning in Tottenham<*>Court-Chapel, and its effects on the building; vol. 62, pa. 131.</p><p>6. Properties of the Conic Sections, deduced by a compendious method; vol. 63, pa. 340.</p><p>In all these works of Mr. Jones, a remarkable neatness, brevity, and accuracy, every where prevails. He seemed to delight in a very short and comprehensive mode of expression and arrangement; in so much that sometimes what he has contrived to express in two or three pages, would occupy a little volume in the ordinary style of writing.</p><p>Mr. Jones it is said possessed the best mathematical library in England; scarcely any book of that kind but what was there to be found. He had collected also a great quantity of manuscript papers and letters of former mathematicians, which have often proved useful to writers of their lives, &c. After his death, these were dispersed, and fell into different persons hands; many of them, as well as of Mr. Jones's own papers, were possessed by the late Mr. John Robertson, before mentioned, at whose death I purchased a considerable quantity of them. From such collections of these it was that Mr. Jones was enabled to give that first and elegant edition, in 4to, 1711, of several of Newton's papers, that might otherwise have been lost, intitled, <hi rend="italics">Analysis per quantitatum Series, Fluxiones, ac Differentias: cum Enumeratione Linearum Tertii Ordinis.</hi></p><p>IONIC <hi rend="italics">Column,</hi> or <hi rend="italics">Order,</hi> the 3d of the five orders, or columns, of Architecture. The first idea of this order was given by the people of Ionia; who, according to Vitruvius, formed it on the model of a young woman, dressed in her hair, and of an easy elegant shape, as the Doric had been formed on the model of a strong robust man.</p><p>This column is a medium between the massive and the more delicate orders, the simple and the rich. It is distinguished from the Composite, by having none of the leaves of acanthus in its capital; and from the Tuscan, Doric, and Corinthian, by the volutes, or rams horns, which adorn its capital; and from the Tuscan and Doric too, by the channels, or fluting, in its shaft.</p><p>The height of this column is 18 modules, or 9 diameters of the column taken at the bottom: indeed at first its height was but 16 modules; but, to render it more beautiful than the Doric, its height was augmented by adding a base to it, which was unknown in the Doric. M. le Clerc makes its entablature to be 4 modules and 10 minutes, and its pedestal 6 modules; so that the whole order makes 28 modules 10 minutes.</p></div1><div1 part="N" n="JOURNAL" org="uniform" sample="complete" type="entry"><head>JOURNAL</head><p>, in Merchants Accounts, is a book into which every particular article is posted out of the Wafte-book, according to the order of time, specifying the debtor and creditor in each account and transaction.</p><div2 part="N" n="Journal" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Journal</hi></head><p>, in Maritime Affairs, is a reg<*>ster kept by the pilot, and others, noticing every thing that happens to the ship, from day to day, and from hour to hour, with regard to the winds, the rhumbs or courses, the knots or rate of running, the rake, soundings, astronomical observations, for the latitudes and longitudes, &c; to enable them to adjust the reckoning, and determine the place where the ship is.</p><p>In all sea Journals, the day, or what is called the 24 hours, is divided into twice 12 hours, those before<cb/> noon marked A. M. for ante meridiem, and those from noon to midnight marked P. M. post meridiem, or afternoon.</p><p>There are various ways of keeping a sea Journal, according to the different notions of mariners concerning the articles that are to be entered. Some writers direct the keeping such a kind of Journal as is only an abstract of each day's transactions, specifying the weather, what ships or lands were seen, accidents on board, the latitude, longitude, meridional distance, course, and run: these particulars are to be drawn from the ship's log-book, or from that kept by the person himself. Other authors recommend the keeping only of one account, including the log-book, and all the work of each day, with the deductions drawn from it.</p><p><hi rend="smallcaps">Journal</hi> is also used for the title of several books which come out at stated times; and give accounts and abstracts of the new books that are published, with the new improvements daily made in arts and sciences.</p><p>The first Journal of this kind was, the Journal des Sçavans, printed at Paris: the design was set on foot for the ease of such as are too busy, or too lazy, to read the entire books themselves. It seems an excellent way of satisfying a man's curiosity, and becoming learned upon easy terms: and so useful has it been found, that it has been executed in most other countries, though under a great variety of titles.</p><p>Of this kind are the Acta Eruditorum of Leipsic; the Nouvelles de la Republique des lettres of Mr. Bayle, &c; the Bibliotheque Universelle, Choisie, et Ancienne et Moderne, of M. le Clerc; the Memoirs de Trevoux, &c. In 1692, Juncker printed in Latin, An Historical Treatise of the Journals of the Learned, published in the several parts of Europe; and Wolfius, Struvius, Morhoff, Fabricius, &c, have done something of the same kind.</p><p>The Philosophical Transactions of London; the Memoirs of the Royal Academy of Sciences; those of the Academy of Belles Lettres; the Miscellanea Naturæ Curiosorum; the Experiments of the Academy del Cimento, the Acta Philo-exoticorum Naturæ et Artis, which appeared from March 1686 to April 1687, and which are a history of the Academy of Bresse; the Miscellanea Berolinensia, or Memoirs of the Academy of Berlia; the Commentaries of the Academy of Petersburgh; the Memoirs of the Institute at Bologna; the Acta Literaria Sueciæ; the Memoirs of the Royal Academy of Stockholm, begun in 1740; the Commentarii Societatis Regiæ Gottingensis, begun in 1750, &c, &c, are not so properly Journals, though they are frequently ranked in the number.</p><p>Juncker and Wolfius give the honour of the first invention of Journals to Photius. His Bibliotheca, however, is not altogether of the same nature with the modern Journals; nor was his design the same. It consists of abridgments, and extracts of books which he had read during his embassy in Persia. M. Salo first began the Journal des Sçavans at Paris, in 1665, under the name of the Sieur de Hedonville; but his death soon after interrupted the work. The abbé Gallois then took it up, and he, in the year 1674, gave way to the abbé de la Roque, who continued it nine years, and was succeeded by M. Cousin, who carried it on till the year 1702, when the abbé Bignon<pb n="645"/><cb/> instituted a new Society, and committed the care of continuing the Journal to them, who improved and published it under a new form. This Society is still continued, and M. de Loyer has had the inspection of the Journal; which is no longer the work of any single author, but of a great number.</p><p>The other French Journals are the Memoirs and Conferences of Arts and Sciences, by M. Dennis, during the years 1672, 1673, and 1674; New Discoveries in all the parts of Physic, by M. de Blegny; the Journal of Physic, begun in 1684, and some others, discontinued almost as soon as begun.</p><p>Rozier's Journal de Physique, begun in July 1771, and continued, till in the year 1780, there were 19 vols. quarto.</p><p>The Nouvelles de la Republique des Lettres, News from the Republic of Letters, were begun by M. Bayle in 1684, and carried on by him till the year 1687, when M. Bayle being disabled by sickness, his friends, M. Bernard and M. de la Roque, took them up, and continued them till 1699. After an interruption of nine years, M. Bernard resumed the work, and continued it till the year 1710. The History of the Works of the Learned, by M. Basnage, was begun in the year 1686, and ended in 1710. The Universal Historical Library, by M. le Clerc, was continued to the year 1693, and contained twenty-five volumes. The Bibliotheque Choisie of the same author, began in 1703. The Mercury of France is one of the most ancient Journals of that country, and is continued by different hands: the Memoirs of a History of Sciences and Arts, usually called Memoires des Trevoux, from the place where they are printed, began in 1701. The Essays of Literature reached but to a twelfth volume in 1702, 1703, and 1704; these only take notice of ancient authors. The Journal Literaire, by Father Hugo, began and ended in 1705. At Hamburgh they have made two attempts for a French Journal, but the design failed: an Ephemerides Sçavantes has also been undertaken, but that soon disappeared. A Journal des Sçavans, by M. Dartis, appeared in 1694, and was dropt the year following. That of M. Chauvin, begun at Berlin in 1696, held out three years; and an essay of the same kind was made at Geneva. To these may be added, the Journal Literaire begun at the Hague 1715, and that of Verdun, and the Memoires Literaires de la Grande Bretagne by M. de la Roche; the Bibliotheque Angloise, and Journal Britannnique, which are confined to English books alone. The Italian Journals are, that of abbot Nazari, which lasted from 1668 to 1681, and was printed at Rome. That of Venice began in 1671, and ended at the same time with the other: the authors were Peter Moretti, and Francis Miletti. The Journal of Parma, by Roberti and Father Bacchini, was dropped in 1690, and resumed again in 1692. The Journal of Ferrara, by the abbé de la Torre, began and ended in 1691. La Galerio di Minerva, begun in 1696, is the work of a Society of men of letters. Seignior Apostolo Zeno, Secretary to that Society, began another Journal in 1710, under the protection of the Grand Du<*>e: it is printed at Venice, and several persons of distinction have a hand in it.</p><p>The Fasti Euriditi della Bibliotheca Volante, were<cb/> published at Parma. There has appeared since, in Italy, the Giornale dei Letterati.</p><p>The principal among the Latin Journals, is that of Leipsic, under the title of Acta Eruditorum, begun in 1682: P. P. Manzani began another at Parma. The Nova Literaria Maris Balthici lasted from 1698 to 1708. The Nova Literaria Germaniæ, collected at Hamburgh, began in 1703. The Acta Literaria ex Manuscriptis, and the Bibliotheca curiosa, begun in 1705, and ended in 1707, are the work of Struvius. Mess. Kuster and Sike, in 1697, began a Bibliotheca Novorum Librorum, and continued it for two years. Since that time, there have been many Latin Journals; such, besides others, is the Commentarii de Rebus in Scientia Naturali et Medicina gestis, by M. Ludwig. The Swiss Journal, called Nova Literaria Helvetiæ, was begun in 1702, by M. Scheuchzer; and the Acta Medica Hafnensia, published by T. Bartholin, make five volumes from the year 1671 to 1679. There are two Low-Dutch Journals; the one under the title of Boockzal van Europe; it was begun at Rotterdam in 1692, by Peter Rabbus; and continued from 1702 to to 1708, by Sewel and Gavern: the other was done by a physician, called Ruiter, who began it in 1710. The German Journals of best note are, the Monathlichen Unterredungen, which continued from 1689 to 1698. The Bibliotheca Curiosa, began in 1704, and ended in 1707, both by M. Tenzel. The Magazin d'Hambourg, begun in 1748: the Physicalische Belustigunzen, or Philosophical Amusements, begun at Berlin in 1751. The Journal of Hanover began in 1700, and continued for two years by M. Eccard, under the direction of M. Leibnitz, and afterwards carried on by others. The Theological Journal, published by M. Loescher, under the title of Altes und Neues, that is, Old and New. A third at Leipsic and Francfort, the authors Mess. Walterck, Krause, and Groschuffius; and a fourth at Hall, by M. Turk.</p><p>The English Journals are, The History of the Works of the Learned, begun at London in 1699. Censura Temporum, in 1708. About the same time there appeared two new ones, the one under the title of Memoirs of Literature, containing little more than an English translation of some articles in the foreign Journals, by M. de la Roche; the other a collection of loose tracts, entitled, Bibliotheca curiosa, or a Miscellany. These, however, with some others, are now no more, but are succeeded by the Annual Register, which began in 1758; the New Annual Register, begun in 1780; the Monthly Review, which began in the year 1749, and gives a character of all English literary publications, with the most considerable of the foreign ones: the Critical Review, which began in 1756, and is nearly on the same plan: as also the London Review, by Dr. Kenrick, from 1775 to 1780; Maty's Review, from Feb. 1782 to Aug. 1786; the English Review begun in Jan. 1783; and the Analytical Review, begun in May 1788, and still continues with much reputation. Besides these, we have several monthly pamphlets, called Magazines, which, together with a chronological Series of occurrences, contain letters from correspondents, communicating extraordinary discoveries in nature and art, with controversial pieces on all subjects. Of these, the principal are those called, the Gentleman's Magazine,<pb n="646"/><cb/></p><p>which began with the year 1731; the London Magazine, which began a few months after, and has lately been discontinued; the Universal Magazine, which is nearly of as old a date.</p></div2></div1><div1 part="N" n="IRIS" org="uniform" sample="complete" type="entry"><head>IRIS</head><p>, another name for the <hi rend="smallcaps">Rainbow;</hi> which see.</p><p><hi rend="smallcaps">Iris</hi> also denotes the striped variegated circle round the pupil of the eye, formed of a duplicature of the uvea.</p><p>In different subjects, the Iris is of several very different colours; whence the eye is called grey, or black &c. In its middle is a perforation, through which appears a small black speck, called the sight, pupil, or apple of the eye, round which the Iris forms a ring.</p><p><hi rend="smallcaps">Iris</hi> is also applied to those changeable colours, which sometimes appear in the glasses of telescopes, microscopes, &c; so called from their similitude to a rainbow.</p><p>The same appellation is also given to that coloured spectrum, which a triangular prismatic glass will project on a wall, when placed at a proper angle in the sunbeams.</p><p><hi rend="smallcaps">Iris</hi> <hi rend="italics">Marina,</hi> the <hi rend="italics">Sea-Rainbow.</hi> This elegant appearance is generally seen after a violent storm, in which the sea water has been in vast emotions. The celestial rainbow however has great advantage over the marine one, in the brightness and variety of the colours, and in their distinctness one from the other; for in the sea-rainbow, there are scarce any other colours than a dusky yellow on the part next the sun, and a pale green on the opposite side. The other colours are not so bright or so distinct as to be well determined; but the sea-rainbows are more frequent and more numerous than the others: it is not uncommon to see 20 or 30 of them at a time at noon-day.</p><p>IRRATIONAL <hi rend="italics">Numbers,</hi> or <hi rend="italics">Quantities,</hi> are the same as surds, or such roots as cannot be accurately extracted, being incommensurable to unity. See <hi rend="smallcaps">Surds.</hi></p><p>IRREDUCIBLE <hi rend="italics">Case,</hi> in Algebra, is used for that case of cubic equations where the root, according to Cardan's rule, appears under an impossible or imaginary form, and yet is real. Thus, in the equation , the root, according to Cardan's rule, will be , which is in the form of an impossible expression, and yet it is equal to the quantity 4: for , and , therefore there sum is <hi rend="italics">x</hi> = 4. The other two roots of the equation are also real.</p><p>Algebraists, for almost three centuries, have in vain endeavoured to resolve this case, and to bring it under a real form; and the problem is not less celebrated among them, than the squaring of the circle is among geometricians.</p><p>It is to be observed, that, as in some other cases of cubic equations, the value of the root, though rational, is found under an irrational or surd form; because the root in this case is compounded of two equal surds with contrary signs, which destroy each other; as if , then <hi rend="italics">x</hi> = 4. In like manner, in the Irreducible case, where the root is rational, there are two equal imaginary quantities, with contrary signs, joined to real quantities; so that the ima-<cb/> ginary quantities destroy each other; as in the case above of the root of the equation , which was found to be .</p><p>It is remarkable that this case always happens, viz one root, by Cardan's rule, in an impossible form, whenever the equation has three real roots, and no impossible ones, but at no time else.</p><p>If we were possessed of a general rule for accurately extracting the cube root of a binomial radical quantity, it is evident we might resolve the Irreducible case generally, which consists of two of such cubic binomial roots. But the labours of the algebraists, from Cardan's down to the present time, have not been able to remove this difficulty. Dr. Wallis thought that he had discovered such a rule; but, like most others, it is merely tentative, and can only succeed in certain particular circumstances.</p><p>Mr. Maseres, cursitor baron of the exchequer, has lately deduced, by a long train of algebraical reasoning, from Newton's celebrated binomial theorem, an infinite series, which will resolve this case, without any mention of either impossible or negative quantities. And I have also discovered several other series which will do the same thing, in all cases whatever; both inserted in the Phil. Trans. See Cardan's Algebra; the articles Algebra, Cubic Equations; Wallis's Algebra, chap. 48; De Moivre in the Appendix to Sanderson's Algebra, p. 744; Philos. Trans. vol. 68, part 1, art. 42, and vol. 70, p. 387.</p></div1><div1 part="N" n="IRREGULAR" org="uniform" sample="complete" type="entry"><head>IRREGULAR</head><p>, something that deviates from the common forms or rules. Thus, we say an Irregular fortification, an Irregular building, &c.</p><p><hi rend="smallcaps">Irregular</hi> <hi rend="italics">Figure,</hi> in Geometry, whether plane or solid, is that whose sides, as well as angles, are not all equal and similar among themselves.</p><p>IRREGULARITIES <hi rend="italics">in the Moon's motion.</hi> See <hi rend="smallcaps">Moon.</hi></p></div1><div1 part="N" n="ISAGONE" org="uniform" sample="complete" type="entry"><head>ISAGONE</head><p>, in Geometry, is sometimes used for a figure consisting of equal angles.</p></div1><div1 part="N" n="ISLAND" org="uniform" sample="complete" type="entry"><head>ISLAND</head><p>, or <hi rend="smallcaps">Isle</hi>, a tract of dry land encompassed by water; whether by the sea, a river, or lake, &c. In which sense Island stands contradistinguished from Continent, or terra firma; like Great Britain, Ireland, Jersey, Sicily, Minorca, &c.</p><p>Some naturalists imagine that Islands were formed at the deluge: others think they have been rent and separated from the continent by violent storms, inundations, and earthquakes; while others are thrown up by volcanoes, or otherwise grow or emerge from the sea.</p><p>Varenius thinks most of these opinions true in some instances, and believes that there have been Islands produced each of these ways. St. Helena, Ascension, and other steep rocky Islands, he supposes have become so, by the sea's overflowing their neighbouring champaigns. By the heaping up huge quantities of sand &c, he thinks the Islands of Zealand, Japan, &c, were formed: Sumatra and Ceylon, and most of the East India Islands, he rather thinks were rent off from the main land, as England probably was from France. It is also certain that some have emerged from the bottom of the sea; as Santorini formerly, and three other Isles near it lately; the last in 1707, which rose from the bottom of the sea, after an earthquake<*> the ancients had a tradition that Delos rose from the<pb n="647"/><cb/> bottom of the sea; and Seneca observes that the Island Therasia rose out of the Ægean sea in his time, of which the mariners were eye witnesses; as they have been within these 10 years, in the sea between Norway and Iceland, where an Island has just emerged. The late circumnavigators too have made it probable, that many of the South-sea Islands have had their foundations, of coral rock, gradually increasing, and growing out of the sea.</p></div1><div1 part="N" n="ISLES" org="uniform" sample="complete" type="entry"><head>ISLES</head><p>, or rather Ailes, in Architecture, the sides or wings of a building.</p></div1><div1 part="N" n="ISOCHRONAL" org="uniform" sample="complete" type="entry"><head>ISOCHRONAL</head><p>, or <hi rend="smallcaps">Isochronous</hi>, is applied to such vibrations of a pendulum as are performed in equal times. Of which kind are all the vibrations of the same pendulum in a cycloidal curve, and in a circle nearly, whether the arcs it describes be longer or shorter; for when it describes a shorter are, it moves so much the slower; and when a long one, proportionably faster.</p><p><hi rend="smallcaps">Isochronal</hi> <hi rend="italics">Line,</hi> is that in which a heavy body is supposed to descend with a uniform velocity, or without any acceleration.</p><p>Leibnitz, in the Act. Erud. Lips. for April 1689, has a discourse on the Linea Isochrona, in which he shews, that a heavy body, with the velocity acquired by its descent from any height, may descend from the same point by an infinite number of Isochronal curves, which are all of the same species, differing from one another only in the magnitude of their parameters (such as are all the quadratocubical paraboloids), and consequently similar to one another. He shews also, how to find a line, in which a heavy body descending, shall recede uniformly from a given point, or approach uniformly to it.</p></div1><div1 part="N" n="ISOMERIA" org="uniform" sample="complete" type="entry"><head>ISOMERIA</head><p>, in Algebra, a term of Vieta, denoting the-freeing an equation from fractions; which is done by reducing all the fractions to one common denominator, and then multiplying each member of the equation by that common denominator, that is rejecting it out of them all.</p><p>ISOPERIMETRICAL <hi rend="italics">Figures,</hi> are such as have equal perimeters, or circumferences.</p><p>It is demonstrated in geometry, that among Isoperimetrical figures, that is always the greatest which contains the most sides or angles. From whence it follows, that the circle is the most capacious of all figures which have the same perimeter with it.</p><p>That of two Isoperimetrical triangles, which have the same base, and one of them two sides equal, and the other unequal; that is the greater whose sides are equal.</p><p>That of Isoperimetrical figures, whose sides are equal in number, that is the greatest which is equilateral, and equiangular. And hence arises the solution of that popular problem, To make the hedging or walling, which will fence in a certain given quantity of land, also to fence in any other greater quantity of the same. For, let <hi rend="italics">x</hi> be one side of a rectangle that will contain the quantity <hi rend="italics">aa</hi> of acres; then will (<hi rend="italics">aa</hi>)/<hi rend="italics">x</hi> be its other side, and double their sum, viz, , will be the<cb/> perimeter of the rectangle: let also <hi rend="italics">bb</hi> be any greater number of acres, in the form of a square, then is <hi rend="italics">b</hi> one side of it, and 4<hi rend="italics">b</hi> its perimeter, which must be equal to that of the rectangle; and hence the equation , or , in which quadratic equation the two roots are , which are the lengths of the two dimensions of the rectangle, viz, whose area <hi rend="italics">b</hi><hi rend="sup">2</hi> is in any proportion less than the square <hi rend="italics">a</hi><hi rend="sup">2</hi>, of the same perimeter. As, for example, if one side of a square be 10, and one side of a rectangle be 19, but the other only 1; such square and parallelogram will be Isoperimetrical, viz, each perimeter 40; yet the area of the square is 100, and of the parallelogram only 19.</p><p>Isoperimetrical lines and figures have greatly engaged the attention of mathematicians at all times. The 5th book of Pappus's Collections is chiefly upon this subject; where a great variety of curious and important properties are demonstrated, both of planes and solids, some of which were then old in his time, and many new ones of his own. Indeed it seems he has here brought together into this book all the properties relating to Isoperimetrical figures then known, and their different degrees of capacity.</p><p>The analysis of the general problem concerning figures that, among all those of the same perimeter, produce maxima and minima, was given by Mr. James Bernoulli, from computations that involve 2d and 3d fluxions. And several enquiries of this nature have been since prosecuted in like manner, but not always with equal success. Mr. Maclaurin, to vindicate the doctrine of fluxions from the imputation of uncertainty, or obscurity, has illustrated this subject, which is considered as one of the most abstruse parts of this doctrine, by giving the resolution and composition of these problems by first fluxions only; and in a manner that suggests a synthetic demonstration, serving to verify the solution. See Maclaurin's Fluxions, p. 486; Analysis Magni problematis Isoperimetrici Act. Erud. Lips. 1701, p. 213; Mem. Acad. Scienc. 1705, 1706, 1718; and the works of John Bernoulli, tom. 1, p. 202, 208, 424, and tom. 2, p. 235; where is contained what he and his brother James published on this problem. Mr. John Bernoulli, in his first paper, considered only two small successive sides of the curve; whereas the true method of resolving this problem in general, requires the considering three such small sides, as may be perceived by examining the two solutions.</p><p>M. Euler has also published, on this subject, many profound researches, in the Petersburg commentaries; and there was printed at Lausanne, in 1744, a pretty large work upon it, intitled, Methodus inveniendi lineas curvas, maximi minimive proprietate gaudentes: sive Solutio problematis Isoperimetrici in latiss<*>o sensu accepti.</p><p>M. Cramer too, in the Berlin Memoirs for 1752, has given a paper, in which he proposes to demonstrate in general, what can be demonstrated only of regular figures in the elements of geometry, viz, that the circle is the greatest of all Isoperimetrical figures, regular or irregular.<pb n="648"/><cb/></p><p>On this head, see also Simpson's Tracts, p. 98; and the Philos. Trans. vol. 49 and 50.</p><p>ISOSCELES <hi rend="italics">Triangle,</hi> is a triangle that has two sides equal. In the 5th prop. of Euclid's 1st book, which prop. is usually called the Pons Asinorum, or Asses bridge, it is demonstrated, <figure/> that the angles, <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> at the base of the Isosceles triangle, are equal to each other; and that if the equal sides be produced, the two angles, <hi rend="italics">c</hi> and <hi rend="italics">d,</hi> below the base, will also be equal. It is also inferred, that every equilateral triangle is also equiangular.</p><p>Other properties of this figure are, that the perpendicular AP, from the vertex to the base, bisects the base, the vertical angle, and also the whole triangle. And that if the vertical angles of two Isosceles triangles be equal, the two triangles will be equiangular.</p></div1><div1 part="N" n="ISTHMUS" org="uniform" sample="complete" type="entry"><head>ISTHMUS</head><p>, in Geography, a narrow neck or slip of land, that joins two other large tracts together, and separating two seas, or two parts of the same sea.</p><p>The most remarkable Isthmuses are, that of Panama or Straits of Darien, joining north and south America; that of Suez, which connects Asia and Africa; that of Corinth, or Peloponnesus, in the Morea; that of Crim Tartary, otherwise called Taurica Chersonesus; that of the peninsula Romania and Erisso, or the Isthmus of the Thracian Chersonesus, 12 furlongs broad, and which Xerxes undertook to cut through. The Ancients had several designs of cutting the Isthmus of Corinth, which is a rocky hillock, about 10 miles over; but without essect, the invention of sluices being not then known. There have also been attempts for cutting the Isthmus of Suez, to make a communication between the Mediterranean and the Red-sea.</p></div1><div1 part="N" n="JUDICIAL" org="uniform" sample="complete" type="entry"><head>JUDICIAL</head><p>, or <hi rend="smallcaps">Judiciary</hi> <hi rend="italics">Astrology;</hi> that relating to the forming of judgments, and making prognostications. See <hi rend="smallcaps">Astrology.</hi></p><p>JULIAN <hi rend="italics">Calendar,</hi> is that depending on, and connected with the Julian year and account of time; so called from Julius Cæsar, by whom it was established. See <hi rend="smallcaps">Calendar.</hi></p><p><hi rend="smallcaps">Julian</hi> <hi rend="italics">Epoch,</hi> is that of the institution of the Julian reformation of the calendar, which began the 46th year before Christ.</p><p><hi rend="smallcaps">Julian</hi> <hi rend="italics">Period,</hi> is a cycle of 7980 consecutive years, invented by Julius Scaliger, from whom it was named; though some say his name was Joseph Scaliger, and that it was called the Julian Period, because he made use of Julian years. This period is formed by multiplying continually together the three following cycles, viz, that of the sun of 28 years, that of the moon of 19 years, and that of the indiction of 15 years; so that this epoch, although but artificial or feigned, is yet of good use; in that every year within the period is distinguishable by a certain peculiar character; for the year of the sun, moon, and indiction, will not be the same again till the whole 7980 years have revolved. Scaliger fixed the beginning of this period 764 years before the creation, or rather the period naturally reduces to that year, taking the numbers of the three given cycles as he then found them; and accounting<cb/> 3950 years from the creation to the birth of Christ, this makes the 1st year of the Christian era answer to the 4714th year of the Julian period; therefore, to find the year of this period, answering to any proposed year of Christ, to the constant number 4713, add the given year of Christ, and the sum will be the year of the Julian period: thus, to 4713 adding 1791, the sum 6504 is the year of this period for the year of Christ 1791. Hence the first revolution of the Julian period will not be completed till the year of Christ 3267, after which a new revolution of this period will commence.</p><p>But the year of this period may be found for any time, from the numbers of the three cycles that compose it, without making use of the given year of Christ, thus: multiply the <table><row role="data"><cell cols="1" rows="1" role="data">numbers</cell><cell cols="1" rows="1" rend="rowspan=5" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">4845</cell><cell cols="1" rows="1" rend="rowspan=5" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="rowspan=5" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" role="data">sun,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">respectively by</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4200</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">moon,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">the year of the</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6916</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">indiction;</cell></row></table> then add the three products together, and divide the sum by 7980, so shall the remainder after division be the year of the Julian period corresponding to the given years of the other three cycles. Thus, for the year 1791, the years of the solar, lunar, and indiction cycles, are 8, 6, and 9; therefore multiplying by these, &c, according to the rule, thus <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4845</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4200</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6916</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">   9</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">38760</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25200</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">62244</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">38760</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">62244</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7980)</cell><cell cols="1" rows="1" rend="align=right" role="data">126204</cell><cell cols="1" rows="1" role="data">(15</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7980 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">46404</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">39900</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">remains</cell><cell cols="1" rows="1" rend="align=right" role="data">6504</cell><cell cols="1" rows="1" rend="colspan=2" role="data">the year of the Jul. period.</cell></row></table></p><p><hi rend="smallcaps">Julian</hi> <hi rend="italics">Year,</hi> is the old account of the year, established by Julius Cæsar, and consisted of 365 1/4 days. This year continued in use in all Europe till it was superseded in most parts by the new or Gregorian account, in the year 1582. In England however it continued to be used till the year 1752, when it was abolished by act of parliament, and eleven days added to the account, to bring it up to the new style. In Russia, the old or Julian year and style are still in use.</p></div1><div1 part="N" n="JULY" org="uniform" sample="complete" type="entry"><head>JULY</head><p>, the 7th month of the year, consisting of 31 days; about the 21st of which the sun usually enters the sign <*> leo. This month was so named by Mark Antony, from Julius Cæsar, who was born in this month.</p></div1><div1 part="N" n="JUPITER" org="uniform" sample="complete" type="entry"><head>JUPITER</head><p>, <*>, one of the superior planets, remarkable for its brightness, being the brightest of all, except sometimes the planet Venus, and is much the largest of all the planets.</p><p>Jupiter is situated between Mars and Saturn, being the 5th in order of the primary planets from the sun. His diameter is more than 10 times the diameter of the earth, and therefore his magnitude more than 1000<pb n="649"/><cb/> times. His annual revolution about the sun, is performed in 11 years 314 days 12 hours 20 minutes 9 seconds, going at the rate of more than 25 thousand miles per hour; and he revolves about his own axis in the short space of 9 hours 56 minutes, by which his equatorial parts are carried round at the amazing rate of 26 thousand miles per hour, which is about 25 times faster than the like parts of our earth revolve.</p><p>Jupiter is surrounded by faint substances, called zones or belts, in which so many changes appear, that they are generally ascribed to clouds: for some of them have been first interrupted and broken, and then have vanished entirely. They have sometimes been observed of different breadths, and afterwards have all become nearly of the same breadth. Large spots have been seen in these belts; and when a belt vanishes, the contiguous spots disappear with it. The broken ends of some belts have often been observed to revolve in the same time with the spots: only those nearer the equator in somewhat less time than those nearer the poles; perhaps on account of the sun's greater heat near the equator, which is parallel to the belts and course of the spots. Several large spots, which appear round at one time, grow oblong by degrees, and then divide into two or three round spots. The periodical time of the spots near the equator is 9 hours 50 minutes, but of those near the poles 9 hours 56 minutes. See Dr. Smith's Optics, § 1105 and 1109.</p><p>The axis of Jupiter is so nearly perpendicular to his orbit, that he has no sensible change of seasons; which is a great advantage, and wisely ordered by the Author of Nature. For, if the axis of this planet were inclined any considerable number of degrees, just so many degrees round each pole would in their turn be almost 6 years together in darkness. And, as each degree of a great circle on Jupiter contains about 706 miles, it is easy to judge what vast tracts of land would be rendered uninhabitable by any considerable inclination of his axis.</p><p>The difference between the equatorial and polar diameters of Jupiter, is upwards of 6000 miles; the former being to the latter as 13 to 12: so that his poles are more than 3000 miles nearer his centre than the equator is. This happens from his quick motion round his axis; for the fluids, together with the light particles, which they can carry or wash away with them, recede from the poles which are at rest, towards the equator where the motion is quickest, until there be a sufficient number accumulated to make up the deficiency of gravity lost by the centrifugal force, which always arises from a quick motion round an axis: and when the deficiency of weight or gravity of the particles is made up by a sufficient accumulation, there is then an equilibrium, and the equatorial parts rise no higher.</p><p>Jupiter's orbit is 1° 20′ inclined to the ecliptic. The place of his aphelion 9° 10′ of <*>, the place of his ascending node 7° 29′ of <*>, and that of his south or descending node 7° 29′ of <*>. The excentricity of his orbit is 1/20 of his mean distance from the sun.</p><p>The sun appears to Jupiter but the 48th part so large as to us; and his light and heat are in the same<cb/> small proportion, but compensated by the quick returns of them, and by 4 moons, some of them larger than our earth, which revolve about him; so that there is scarce-any part of this huge planet but what is, during the whole night, enlightened by one or more of these moons, except his poles, whence only the farthest moons can be seen, and where their light is not wanted, because the sun conslantly circulates in or near the horizon, and is very probably kept in view of both poles by the refraction of Jupiter's atmofphere, which, if it be like ours, has certainly refractive power enough for that purpose. This planet seen from its nearest moon, appears 1000 times as large as our moon does to us; increasing and waneing in all her monthly shapes, every 42 1/2 hours. The period<*>, distances, in semidiameters of Jupiter, and angles of the orbits of these moons, seen from the earth are as follow: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Periods round Jupiter.</cell><cell cols="1" rows="1" role="data">Distances.</cell><cell cols="1" rows="1" role="data">Angles of orbits.</cell></row><row role="data"><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">----------</cell><cell cols="1" rows="1" role="data">----------</cell><cell cols="1" rows="1" role="data">----------</cell></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"> 1<hi rend="sup">d</hi> 18<hi rend="sup">h</hi> 36<hi rend="sup">m</hi></cell><cell cols="1" rows="1" role="data"> 5 2/3</cell><cell cols="1" rows="1" role="data"> 3′ 55″</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"> 3 13 15</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data"> 6 14</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> 7  3 59</cell><cell cols="1" rows="1" role="data">14 1/3</cell><cell cols="1" rows="1" role="data"> 9 58</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16 18 30</cell><cell cols="1" rows="1" role="data">25 1/3</cell><cell cols="1" rows="1" role="data">17 30</cell></row></table></p><p>The three nearest moons of Jupiter fall into his shadow, and are eclipsed in every revolution: but the orbit of the 4th satellite is so much inclined, that it passeth by its opposition to Jupiter, without falling into his shadow, two years in every six. By these eclipses, astronomers have not only discovered that the sun's light takes up 8 minutes of time in coming to us; but have also by them determined the longitudes of places on this earth, with greater certainty and facility, than by any other method yet known. The outermost of these satellites will appear nearly as large as the moon does to us. See M. De la Place's Theory of Jupiter's Satellites, in the Memoires de l'Acad. and in the Connoissance des Temps for 1792, pa. 273.</p><p>Though there be 4 primary planets below Jupiter, yet an eye placed on his surface would never perceive any of them; unless perhaps as spots passing over the sun's disc, when they happen to come between the eye and the sun.—The parallax of the sun, viewed from Jupiter, will scarce be sensible, being not much above 20 seconds; and the sun's apparent diameter in Jupiter, but about 6 minutes.—Dr. Gregory adds, that an astronomer in Jupiter would easily distinguish two kinds of planets, four nearer him, viz his satellites, and two more remote, viz the sun and Saturn: the former however will fall vastly short of the sun in brightness, notwithstanding the great disproportion in the distances and apparent magnitude.</p></div1><div1 part="N" n="JURIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">JURIN</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, a very distinguished person in several walks of literature, particularly medicine, mathematics, and philosophy, which he cultivated with equal success. He was secretary of the Royal Society in London, as well as president of the College of Physicians there, at the time of his death, which happened March 22, 1750.</p><p>Doctor Jurin was author of several ingenious compositions; particularly “an Essay upon Distinct and In-<pb n="650"/><cb/></p><p>distinct Vision;” printed at the end of the 2d volume of Dr. Smith's System of Optics; also several controversial papers; against Michellotti, upon the momentum of running waters; against Robins, upon distinct vision; and against the partisans of Leibnitz, upon the forces of moving bodies; &c. His papers inserted in the Philos. Trans. are the following:</p><p>1. On the Suspension of Water in Capillary Tubes: vol. 30, p. 739.</p><p>2. Observations on the Motion of Running Water: p. 748.</p><p>3. On an old Roman Inscription: p. 813.</p><p>4. A Discourse on the Power of the Heart: p. 863 and 929.</p><p>5. On the Specisic Gravity of Human Blood: p. 1000.</p><p>6. Defence of his Doctrine of the Power of the Heart against the Objections of Dr. Keill: p. 1039.<cb/></p><p>7. On the Action of Glass-Tubes upon Water and Quicksilver: p. 1083.</p><p>8. On the Specific Gravity of Solids when weighed in Water: vol. 31, p. 223.</p><p>9. On the Motion of Running Water, against Michellotti: vol. 32, p. 179.</p><p>10. Remarkable Instance of the Small-pox: vol. 32, p. 191.</p><p>11. Inoculated and Natural Small-pox compared: vol. 32, pa. 213.</p><p>12. On Meteorological Diaries: vol. 32, p. 422.</p><p>13. On the Measure and Motion of Running Water: vol. 41, p. 5 and 65.</p><p>14. Meteorological Observations in Charles Town: vol. 42, p. 491.</p><p>15. On the Action of Springs: vol. 43, p. 46.</p><p>16. On the Force of Bodies in Motion: p. 423.</p><p>17. Dynamic Principles, or Meteorological Principles of Mechanics: vol. 66, p. 103. <hi rend="center">END OF VOLUME I.</hi></p></div1></div0><pb/><pb/><pb/><div0 part="N" n="K" org="uniform" sample="complete" type="alphabetic letter"><head>K</head><cb/><p>KALENDAR. See <hi rend="smallcaps">Calendar.</hi></p><p>KALENDS. See <hi rend="smallcaps">Calends.</hi></p><div1 part="N" n="KEILL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">KEILL</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent mathematician and philosopher, was born at Edinburgh in 1671, and studied in the university of that city. His genius leading him to the mathematics, he made a great progress under David Gregory the professor there, who was one of the first that had embraced and publicly taught the Newtonian philosophy. In 1694 he followed his tutor to Oxford, where, being admitted of Baliol College, he obtained one of the Scotch exhibitions in that college. It is said he was the first who taught Newton's principles by the experiments on which they are founded: and this it seems he did by an apparatus of instruments of his own providing; by which means he acquired a great reputation in the university. The first public specimen he gave of his skill in mathematical and philosophical knowledge, was his <hi rend="italics">Examination of Dr. Burnet's Theory of the Earth; with Remarks on Mr. Whiston's New Theory;</hi> which appeared in 1698. These theories were defended by their respective authors; which drew from him, in 1699, <hi rend="italics">An Examination of the Reflections on the Theory of the Earth,</hi> together with <hi rend="italics">A Defence of the Remarks on Mr. Whiston's New Theory.</hi> Dr. Burnet was a man of great humanity, moderation, and candour; and it was therefore supposed that Keill had treated him too roughly, considering the great disparity of years between them. Keill however left the doctor in possession of that which has since been thought the great characteristic and excellence of his work; and though he disclaimed him as a philosopher, yet allowed him to be a man of a fine<cb/> imagination. “Perhaps, says he, many of his readers will be sorry to be undeceived about his theory; for, as I believe never any book was fuller of mistakes and errors in philosophy, so none ever abounded with more beautiful scenes and surprising images of nature. But I write only to those who might expect to find a true philosophy in it: they who read it as an ingenious romance, will still be pleased with their entertainment.”</p><p>The year following, Dr. Millington, Sedleian professor of natural philosophy in Oxford, who had been appointed physician to king William, substituted Keill as his deputy, to read the lectures in the public school. This office he discharged with great reputation; and, the term of enjoying the Scotch exhibition at Baliolcollege now expiring, he accepted an invitation from Dr. Aldrich, dean of Christ-church, to reside there.</p><p>In 1701, he published his celebrated treatise, intitled, <hi rend="italics">Introductio ad Veram Physicam,</hi> which is supposed to be the best and most ufeful of all his performances. The first edition of this book contained only fourteen lectures; but to the second, in 1705, he added two more. This work was deservedly esteemed, both at home and abroad, as the best introduction to the Principia, or the new mechanical philosophy, and was reprinted in different places; also a new edition in English was printed at London in 1736, at the instance of M. Maupertuis, who was then in England.</p><p>Being made Fellow of the Royal Society, he published, in the Philos. Trans. 1708, a paper on the Laws of Attraction, and its physical principles: and being offended at a passage in the <hi rend="italics">Acta Eruditorum</hi> of Leipsic, where Newton's claim to the first invention of the me-<pb n="2"/><cb/> thod of Fluxions was called in question, he warmly vindicated that claim against Leibnitz. In 1709 he went to New-England as treasurer of the Palatines; and soon after his return in 1710, he was chosen Savilian professor of astronomy at Oxford. In 1711, being attacked by Leibnitz, he entered the lists with that mathematician, in the dispute concerning the invention of Fluxions. Leibnitz wrote a letter to Dr. Hans Sloane, then secretary to the Royal Society, requiring Keill, in effect, to make him satisfaction for the injury he had done him in his paper relating to the passage in the <hi rend="italics">Acta Eruditorum:</hi> he protested, that he was far from assuming to himself Newton's method of Fluxions; and therefore desired that Keill might be obliged to retract his false assertion. On the other hand, Keill desired that he might be permitted to justify what he had asserted. He made his defence to the approbation of Newton, and other members of the Society. A copy of this was sent to Leibnitz; who, in a second letter, remonstrated still more loudly against Keill's want of candour and sincerity; adding, that it was not sit for one of his age and experience to engage in a dispute with an upstart, who acted without any authority from Newton, and desiring that the Royal Society would enjoin him silence. Upon this, a special committee was appointed; who, after examining the facts, concluded their report with “reckoning Mr. Newton the inventor of Fluxions; and that Mr. Keill, in asserting the same, had been no ways injurious to Mr. Leibnitz.” The whole proceedings upon this matter may be seen in Collins's <hi rend="italics">Commercium Epistolicum,</hi> with many valuable papers of Newton, Leibnitz, Gregory, and other mathematicians. In the mean time Keill behaved himself with great firmness and spirit; which he also shewed afterwards in a Latin epiftle, written in 1720, to Bernoulli, mathematical professor at Basil, on account of the same usage shewn to Newton: in the title-page of which he put the arms of Scotland, viz, a Thistle, with this motto, <hi rend="italics">Nemo me impune lacessit.</hi></p><p>About the year 1711, several objections being urged against Newton's philosophy, in support of Des Cartes's notions of a plenum, Keill published a paper in the Philos. Trans. on the Rarity of Matter, and the Tenuity of its Composition. But while he was engaged in this dispute, queen Anne was pleased to appoint him her Decipherer; and he continued in that place under king George the First till the year 1716. The university of Oxford conferred on him the degree of M. D. in 1713; and, two years after, he published an edition of Commandine's Euclid, with additions of his own. In 1718 he published his <hi rend="italics">Introductio ad Veram Astronomiam:</hi> which was afterwards, at the request of the duchess of Chandos, translated by himself into English; and, with several emendations, published in 1721, under the title of <hi rend="italics">An Introduction to True Astronomy,</hi> &c. This was his last gift to the public; being this summer seized with a violent fever, which terminated his life Sept. 1, in the 50th year of his age.</p><p>His papers in the Philos. Trans. above alluded to, are contained in volumes 26 and 29.</p><p><hi rend="smallcaps">Keill</hi> (Dr. <hi rend="italics">James</hi>), an eminent physician and philosopher, and younger brother of Dr. John Keill above mentioned, was also born in Scotland, in 1673. Having travelled abroad, on his return he read lectures on Anatomy with great applause in the universities of Oxford<cb/> and Cambridge, by the latter of which he had the degree of M. D. conferred upon him. In 1703 he settled at Northampton as a physician, where he died of a cancer in the mouth in 1719. His publications are</p><p>1. An English translation of Lemery's Chemislry.</p><p>2. On Animal Secretion, the quantity of Blood in the Human Body, and on Muscular Motion.</p><p>3. A treatise on Anatomy.</p><p>4. Several pieces in the Philos. Trans. volumes 25 and 30.</p></div1><div1 part="N" n="KEPLER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">KEPLER</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a very eminent astronomer and mathematician, was born at Wiel, in the county of Wirtemberg, in 1571. He was the disciple of Mæstlinus, a learned mathematician and astronomer, of whom he learned those sciences, and became afterwards professor of them to three successive emperors, viz. Matthias, Rudolphus, and Ferdinand the 2d.</p><p>To this sagacious philosopher we owe the first discovery of the great laws of the planetary motions, viz. that the planets describe areas that are always proportional to the times; that they move in elliptical orbits, having the sun in one focus; and that the squares of their periodic times, are proportional to the cubes of their mean dislances; which are now generally known by the name of Kepler's Laws. But as this great man stands as it were at the head of the modern reformed astronomy, he is highly deserving of a pretty large account, which we shall extract chiefly from the words of that great mathematician Mr. Maclaurin.</p><p>Kepler had a particular passion for finding analogies and harmonies in nature, after the manner of the Pythagoreans and Platonists; and to this disposition we owe such valuable discoveries, as are more than sufficient to excuse his conceits. Three things, he tells us, he anxiously sought to find out the reason of, from his early youth; viz, Why the planets were 6 in number? Why the dimensions of their orbits were such as Copernicus had deseribed from observations? And what was the analogy or law of their revolutions? He sought for the reasons of the two first of these, in the properties of numbers and plane figures, without success. But at length reflecting, that while the plane regular sigures may be infinite in number, the regular solids are only five, as Euclid had long ago demonstrated: he imagined, that certain mysteries in nature might correspond with this remarkable limitation inherent in the essences of things; and the rather, as he found that the Pythagoreans had made great use of those sive regular solids in their philosophy. He therefore endeavoured to find some relation between the dimensions of these solids and the intervals of the planetary spheres; thus, imagining that a cube, inscribed in the sphere of Saturn, would touch by its six planes the sphere of Jupiter; and that the other four regular solids in like manner fitted the intervals that are between the spheres of the other planets: he became persuaded that this was the true reason why the primary planets were precisely six in number, and that the author of the world had determined their distances from the sun, the centre of the system, from a regard to this analogy. Being thus possessed, as he thought, of the grand secret of the Pythagoreans, and greatly pleased with his discovery, he published it in 1596, under the title of <hi rend="italics">Mysterium Cosmographicum;</hi> and was for some time so charmed with it, that he said<pb n="3"/><cb/> he would not give up the honour of having invented what was contained in that book, for the electorate of Saxony.</p><p>Kepler sent a copy of this book to Tycho Brahe, who did not approve of those abstract speculations concerning the system of the world, but wrote to Kepler, first to lay a solid foundation in observations, and then, by ascending from them, to endeavour to come at the causes of things. Tycho however, pleased with his genius, was very desirous of having Kepler with him to assist him in his labours: and having settled, under the protection of the emperor, in Bohemia, where he passed the last years of his life, after having left his native country on some ill usage, he prevailed upon Kepler to leave the university os Gratz, and remove into Bohemia, with his family and library, in the year 1600. But Tycho dying the next year, the arranging the observations devolved upon Kepler, and from that time he had the title of Mathematician to the Emperor all his life, and gained continually more and more reputation by his works. The emperor Rudolph ordered him to finish the tables of Tycho Brahe, which were to be called the <hi rend="italics">Rudolphine Tables.</hi> Kepler applied diligently to the work: but unhappy are those learned men who depend upon the good-humour of the intendants of the finances; the treasurers were so ill-affected towards our author, that he could not publish these tables till 1627. He died at Ratisbon, in 1630, where he was soliciting the payment of the arrears of his pension.</p><p>Kepler made many important discoveries from Tycho's observations, as well as his own. He found, that astronomers had erred, from the first rise of the science, in ascribing always circular orbits and uniform motions to the planets; that, on the contrary, each of them moves in an ellipsis which has one of its foci in the sun: that the motion of each is really unequable, and varies so, that a ray supposed to be always drawn from the planet to the sun describes equal areas in equal times.</p><p>It was some years later before he discovered the analogy there is between the distances of the several planets from the sun, and the periods in which they complete their revolutions. He easily saw, that the higher planets not only moved in greater circles, but also more slowly than the nearer ones; so that, on a double account, their periodic times were greater. Saturn, for example, revolves at the distance from the sun 9 1/2 times greater than the earth's distance from it; and the circle described by Saturn is in the same proportion: but as the earth revolves in one year, so, if their velocities were equal, Saturn ought to revolve in 9 years and a half; whereas the periodic time of Saturn is about 29 years. The periodic times of the planets increase, therefore, in a greater proportion than their distances from the sun: but yet not in so great a proportion as the squares of those distances; for if that were the law of the motions, (the square of 9 1/2 being 90 1/4), the periodic time of Saturn ought to be above 90 years. A mean proportion between that of the distances of the planets, and that of the squares of those distances, is the true proportion of the periodic times; as the mean between 9 1/2 and its square 90 1/4, gives the periodic time of Saturn in years. Kepler, after having committed several mistakes in determining this analogy, hit upon it at last, May the 15, 1618; for he is so particular as to mention the precise<cb/> day when he found that “The squares of the periodie times were always in the same proportion as the cubes of their mean distances from the sun.”</p><p>When Kepler saw, according to better observations, that his disposition of the five regular solids among the planetary spheres, was not agreeable to the intervals between their orbits, he endeavoured to discover other schemes of harmony. For this purpose, he compared the motions of the same planet at its greatest and least distances, and of the different planets in their several orbits, as they would appear viewed from the sun; and here he fancied that he found a similitude to the divisions of the octave in music. These were the dreams of this ingenious man, which he was so fond of, that, hearing of the discovery of four new planets (the satellites of Jupiter) by Galileo, he owns that his first reflections were from a concern how he could save his favourite scheme, which was threatened by this addition to the number of the planets. The same attachment led him into a wrong judgment concerning the sphere of the fixed stars: for being obliged, by his doctrine, to allow a vast superiority to the sun in the universe, he restrains the fixed stars within very narrow limits. Nor did he consider them as suns, placed in the centres of their several systems, having planets revolving round them; as the other followers of Copernicus have concluded them to be, from their having light in themselves, from their immense distances, and from the analogy of nature. Not contented with these harmonies, which he had learned from the observations of Tycho, he gave himself the liberty to imagine several other analogies, that have no foundation in nature, and are overthrown by the best observations. Thus from the opinions of Kepler, though most justly admired, we are taught the danger of espousing principles, or hypotheses, borrowed from abstract sciences, and of applying them, with such freedom, to natural enquiries.</p><p>A more recent instance of this fondness, for discovering analogies between matters of abstract speculation, and the constitution of nature, we find in Huygens, one of the greatest geometricians and astronomers any age has produced: when he had discovered that satellite of Saturn, which from him is still called the Huygenian satellite, this, with our moon, and the four satellites of Jupiter, completed the number of six secondary planets then discovered in the system; and because the number of primary planets was also six, and this number is called by mathematicians a perfect number (being equal to the sum of its aliquot parts, 1, 2, 3,) Huygens was hence induced to believe that the number of the planets was complete, and that it was in vain to look for any more. This is not mentioned to lessen the credit of this great man, who never perhaps reasoned in such a manner on any other occasion; but only to shew, by another instance, how ill-grounded reasonings of this kind have always proved. For, not long after, the celebrated Cassini discovered four more satellites about Saturn, not to mention the two more that have lately been discovered to that planet by Dr. Herschel, with another new primary planet and its two satellites, besides many others, of both sorts, as yet unknown, which possibly may belong to our system. The same Cassini having found that the analogy, discovered by Kepler, between the periodic times and the distances from the centre, takes place in<pb n="4"/><cb/> the lesser systems of Jupiter and Saturn, as well as in the great solar system; his observations overturned that groundless analogy which had been imagined between the number of the planets, both primary and secondary, and the number six: but established, at the same time, that harmony in their motions, which will afterwards appear to flow from one real principle extended over the universe.</p><p>But to return to Kepler; his great sagacity, and continual meditations on the planetary motions, suggested to him some views of the true principles from which these motions flow. In his preface to the Commentaries concerning the planet Mars, he speaks of gravity as of a power that was mutual between bodies, and tells us, that the earth and moon tend towards each other, and would meet in a point, so many times nearer to the earth than to the moon, as the earth is greater than the moon, if their motions did not hinder it. He adds, that the tides arise from the gravity of the waters towards the moon. But not having notions sufficiently just of the laws of motion, it seems he was not able to make the best use of these thoughts; nor does it appear that he adhered to them steadily, since in his Epitome of Astronomy, published many years after, he proposes a physical account of the planetary motions, derived from dif ferent principles.</p><p>He supposes, in that treatise, that the motion of the sun on his axis, is preserved by some inherent vital principle; that a certain virtue, or immaterial image of the sun, is diffused with his rays into the ambient spaces, and, revolving with the body of the sun on his axis, takes hold of the planets, and carries them along with it in the same direction; like as a loadstone turned round near a magnetic needle, makes it turn round at the same time. The planet, according to him, by its inertia, endeavours to continue in its place, and the action of the sun's image and this inertia are in a perpetual struggle. He adds, that this action of the sun, like his light, decreases as the distance increases; and therefore moves the same planet with greater celerity when nearer the sun, than at a greater distance. To account for the planet's approaching towards the sun as it descends from the aphelion to the perihelion, and receding from the sun while it ascends to the aphelion again, he supposes that the sun attracts one part of each planet, and repels the opposite part; and that the part attracted is turned towards the sun in the descent, and the other towards the sun in the ascent. By suppositions of this kind, he endeavoured to account for all the other varieties of the celestial motions.</p><p>But, now that the laws of motion are better known than in Kepler's time, it is easy to shew the fallacy of every part of this account of the planetary motions. The planet does not endeavour to stop in consequence of its inertia, but to persevere in its motion in a right line. An attractive force makes it descend from the aphelion to the perihelion in a curve concave towards the sun: but the repelling force, which he supposed to begin at the perihelion, would cause it to ascend in a figure convex towards the sun. There will be occasion to shew afterwards, from Sir Isaac Newton, how an attraction or gravitation towards the sun, alone produces the effects, which, according to Kepler, required both an attractive and repelling force; and that the virtue<cb/> which he ascribed to the sun's image, propagated into the planetary regions, is unnecessary, as it could be of no use for this effect, though it were admitted. For now his own prophecy, with which he concludes his book, is verisied; where he tells us, that “the discovery of such things was reserved for the succeeding ages, when the author of nature would be pleased to reveal these mysteries.”</p><p>The works of this celebrated author are many and valuable; as,</p><p>1. His <hi rend="italics">Cosmographical Mystery,</hi> in 1596.</p><p>2. <hi rend="italics">Optical Astronomy,</hi> in 1604.</p><p>3. <hi rend="italics">Account of a New Star in Sagittarius,</hi> 1605.</p><p>4. <hi rend="italics">New Astronomy;</hi> or, <hi rend="italics">Celestial Physics,</hi> in Commentaries on the planet Mars.</p><p>5. <hi rend="italics">Dissertations;</hi> with the <hi rend="italics">Nuncius Siderius</hi> of Galileo, 1610.</p><p>6. <hi rend="italics">New Gauging of Wine Casks,</hi> 1615. Said to be written on occasion of an erroneous measurement of the wine at his marriage by the revenue officer.</p><p>7. <hi rend="italics">New Ephemerides,</hi> from 1617 to 1620.</p><p>8. <hi rend="italics">Copernican System,</hi> three first books of the, 1618.</p><p>9. <hi rend="italics">Harmony of the World;</hi> and three books of <hi rend="italics">Comets,</hi> 1619.</p><p>10. <hi rend="italics">Cosmographical Mystery,</hi> 2d edit. with Notes, 1621.</p><p>11. <hi rend="italics">Copernican Astronomy;</hi> the three last books, 1622.</p><p>12. <hi rend="italics">Logarithms,</hi> 1624; and the <hi rend="italics">Supplement,</hi> in 1625.</p><p>13. His <hi rend="italics">Astronomical Tables,</hi> called the <hi rend="italics">Rudolphine Tables,</hi> in honour of the emperor Rudolphus, his great and learned patron, in 1627.</p><p>14. <hi rend="italics">Epitome of the Copernican Astronomy,</hi> 1635.</p><p>Beside these, he wrote several pieces on various other branches, as <hi rend="italics">Chronology, Geometry of Solids, Trigonometry,</hi> and an excellent treatise of <hi rend="italics">Dioptrics,</hi> for that time.</p><p><hi rend="smallcaps">Kepler's Laws</hi>, are those laws of the planetary motions discovered by Kepler. These discoveries in the mundane system, are commonly accounted two, viz. 1st, That the planets describe about the sun, areas that are proportional to the times in which they are described, namely, by a line connecting the sun and planet; and 2d, That the squares of the times of revolution, are as the cubes of the mean distances of the planets from the sun. Kepler discovered also that the orbits of the planets are elliptical.</p><p>These discoveries of Kepler, however, were only found out by many trials, in searching among a great number of astronomical observations and revolutions, what rules and laws were found to obtain. On the other hand, Newton has demonstrated, <hi rend="italics">a priori,</hi> all these laws, shewing that they must obtain in the mundane system, from the laws of gravitation and centripetal force; viz, the first of these laws resulting f<*>om a centripetal force urging the planets towards the sun, and the 2d, from the centripetal force being in an inverse ratio of the square of the distance. And the elliptic form of the orbits, from a projectile force regulated by a centripetal one.</p><p><hi rend="smallcaps">Kepler's</hi> <hi rend="italics">Problem,</hi> is the determining the true from the mean anomaly of a planet, or the determining its place, in its elliptic orbit, answering to any given time; and so named from the celebrated astronomer Kepler, who first proposed it. See <hi rend="smallcaps">Anomaly.</hi> <pb/><pb/><pb n="5"/><cb/></p><p>The general state of the problem is this: To find the position of a right line, which, passing through one of the foci of an ellipsis, shall cut off an area which shall be in any given proportion to the whole area of the ellipsis; which results from this property, that such a line sweeps areas that are proportional to the times.</p><p>Many solutions have been given of this problem, some direct and geometrical, others not: viz, by Kepler, Bulliald, Ward, Newton, Keill, Machin, &c. See Newton's Princip. lib. 1. prop. 31, Keill's Astron. Lect. 23, Philos. Trans. abr. vol. 8. pa. 73, &c.</p><p>In the last of these places, Mr. Machin observes, that many attempts have been made at different times, but with no great success, towards the solution of the problem proposed by Kepler: To divide the area of a semici<*>cle into given parts, by a line drawn from a given point in the diameter, in order to find an universal rule for the motion of a body in an elliptic orbit. For among the several methods offered, some are only true in speculation, but are really of no service; others are not different from his own, which he judged improper. And as to the rest, they are all so limited and consined to particular conditions and circumstances, as still to leave the problem in general untouched. To be more particular; it is evident, that all constructions by mechanical curves are seeming solutions only, but in reality unapplicable; that the roots of infinite series are, on account of their known limitations in all respects, so far from being sufficient rules, that they serve for little more than exercises in a method of calculation. And then, as to the universal method, which proceeds by a continued correction of the errors of a false position, it is no method of solution at all in itself; because, unless there be some antecedent rule or hypothesis to begin the operation (as suppose that of an uniform motion about the upper focus, for the orbit of a planet; or that of a motion in a parabola for the perihelion part of the orbit of a comet, or some other such), it would be impossible to proceed one step in it. But as no general rule has ever yet been laid down, to assist this method, so as to make it always operate, it is the same in effect as if there were no method at all. And accordingly in experience it is found, that there is no rule now subsisting but what is absolutely useless in the elliptic orbits of comets; for in such cases there is no other way to proceed but that which was used by Kepler: to compute a table for some part of the orbit, and in it examine if the time to which the place is required, will fall out any where in that part. So that, upon the whole, it appears evident, that this problem, contrary to the received opinion, has never yet been advanced one step towards its true solution.</p><p>Mr. Machin then proceeds to give his own solution of this problem, which is particularly necessary in orbits of a great excentricity; and he illustrates his method by examples for the orbits of Venus, of Mercury, of the comet of the year 1682, and of the great comet of the year 1680, sufficiently shewing the universality of the method.</p></div1><div1 part="N" n="KEY" org="uniform" sample="complete" type="entry"><head>KEY</head><p>, in Music, is a certain sundamental note, or tone, to which the whole piece, be it concerto, sonata, cantata, &c, is accommodated; and with which it usually begins, but always ends.</p><p><hi rend="smallcaps">Keys</hi> denote also, in an organ, barpsichord, &c, the<cb/> pieces of wood or ivory which are struck by the fingers, in playing upon the instrument.</p><div2 part="N" n="Keystone" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Keystone</hi></head><p>, the middle voussoir, or the arch stone in the top, or immediately over the centre of an arch.— The length of the keystone, or thickness of the archivolt at top, is allowed by the best architects, to be about the 15th or 16th part of the span.</p></div2></div1><div1 part="N" n="KILDERKIN" org="uniform" sample="complete" type="entry"><head>KILDERKIN</head><p>, a kind of liquid measure, containing two firkins, or 18 gallons, beer-measure, or 16 alemeasure.</p><p>KING-<hi rend="italics">piece,</hi> or <hi rend="smallcaps">King</hi>-<hi rend="italics">post,</hi> is a piece of timber set upright in the middle, between two principal rafters, and having struts or braces going from it to the middle of each rafter.</p></div1><div1 part="N" n="KIRCH" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">KIRCH</surname> (<foreName full="yes"><hi rend="smallcaps">Christian Frederic</hi></foreName>)</persName></head><p>, of Berlin, a celebrated astronomer, was born at Guben in 1694. He acquired great reputation in the observatories of Dantzic and Berlin. Godfrey Kirch his father, and Mary his mother, also acquired considerable reputation by their astronomical observations. This family corresponded with all the learned societies of Europe, and their astronomical works are in great repute.</p></div1><div1 part="N" n="KIRCHER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">KIRCHER</surname> (<foreName full="yes"><hi rend="smallcaps">Athanasius</hi></foreName>)</persName></head><p>, a famous philosopher and mathematician, was born at Fulde in 1601. He entered into the society of the Jesuits in 1618, and taught philosophy, mathematics, the Hebrew and Syriac Languages, in the university of Wirtsburg, with great applause, till the year 1631. He retired to France on account of the ravages committed by the Swedes in Franconia, and lived some time at Avignon. He was afterwards called to Rome, where he taught mathematics in the Roman college, collected a rich cabinet of machines and antiquities, and died in 1680, in the 80th year of his age.</p><p>The quantity of his works is immense, amounting to 22 volumes in folio, 11 in quarto, and three in octavo; enough to employ a man for a great part of his life even to transcribe them. Most of them are rather curious than useful; many of them visionary and fanciful; and it is not to be wondered at, if they are not always accompanied with the greatest exactness and precision. The principal of them are,</p><p>1. <hi rend="italics">Prælusiones Magneticæ.</hi></p><p>2. <hi rend="italics">Primitiæ Gnomonicæ Catoptricæ.</hi></p><p>3. <hi rend="italics">Ars magna Lucis et Umbræ.</hi></p><p>4. <hi rend="italics">Musurgia Universalis.</hi></p><p>5. <hi rend="italics">Obeliscus Pamphilius.</hi></p><p>6. <hi rend="italics">Oedipus Ægyptiacus;</hi> 4 volumes folio.</p><p>7. <hi rend="italics">Itinerarium Extaticum.</hi></p><p>8. <hi rend="italics">Obeliscus Ægyptiacus;</hi> 4 volumes folio.</p><p>9. <hi rend="italics">Mundus Subterraneu<*>.</hi></p><p>10. <hi rend="italics">China Illustrata.</hi></p></div1><div1 part="N" n="KNOT" org="uniform" sample="complete" type="entry"><head>KNOT</head><p>, a tye, or complication of a rope, cord, or string, or of the ends of two together. There are divers sorts of knots used for different purposes, which may be explained by shewing the figures of them open, or undrawn, thus. 1. Fig. 1, plate xiii. is a <hi rend="italics">Thumb knot.</hi> This is the simplest of all. It is used to tye at the end of a rope, to prevent its opening out: it is also used by taylors &c. at the end of their thread.</p><p>Fig. 2, a <hi rend="italics">Loop knot.</hi> Used to join pieces of rope &c. together.</p><p>Fig. 3, a <hi rend="italics">Draw knot,</hi> which is the same as the last; only one end or both return the same way back, as<pb n="6"/><cb/> <hi rend="italics">a b c d.</hi> By drawing at <hi rend="italics">a,</hi> the part <hi rend="italics">b c d</hi> comes through, and the knot is loosed.</p><p>Fig. 4, a <hi rend="italics">Ring knot.</hi> This serves also to join pieces of cord &c together.</p><p>Fig. 5 is another knot for tying cords together. This is used when any cord is often to be loosed.</p><p>Fig. 6, a <hi rend="italics">Running knot,</hi> to draw any thing close. By pulling at the end <hi rend="italics">a;</hi> the cord is drawn through the loop <hi rend="italics">b,</hi> and the part <hi rend="italics">c d</hi> is drawn close about a beam, &c.</p><p>Fig. 7 is another knot, to tye any thing to a post. And here the end may be put through as often as you please.</p><p>Fig. 8, a <hi rend="italics">Very small knot.</hi> A thumb knot is sirst made at the end of each piece, and then the end of the other is passed through it. Thus, the cord <hi rend="italics">a c</hi> runs through the loop <hi rend="italics">d,</hi> and <hi rend="italics">b d</hi> through <hi rend="italics">c;</hi> and then drawn close by pulling at <hi rend="italics">a</hi> and <hi rend="italics">b.</hi> If the ends <hi rend="italics">e</hi> and <hi rend="italics">f</hi> be drawn, the knot will be loosed again.</p><p>Fig. 9, a <hi rend="italics">Fisher's knot,</hi> or <hi rend="italics">Water knot.</hi> This is the same as the 4th, only the ends are to be put twice through the ring, which in the former was but once; and then drawn close.</p><p>Fig. 10, a <hi rend="italics">Meshing knot,</hi> for nets; and is to be drawn close.</p><p>Fig. 11, a <hi rend="italics">Barber's knot,</hi> or a knot for cawls of wigs; and is to be drawn close.</p><p>Fig. 12, a <hi rend="italics">Bowline knot.</hi> When this is drawn close, it makes a loop that will not slip, as fig. 7; and serves to hitch over any thing.</p><p>Fig. 13, a <hi rend="italics">Wale knot,</hi> which is made with the three strands of a rope, so that it cannot slip. When the rope is put through a hole, this knot keeps it from slipping through. When the three strands are wrought<cb/> round once or twice more, after the same manner, it is called <hi rend="italics">crowning.</hi> By this means the knot is made larger and stronger. A thumb knot, N<*>. 1, may be applied to the same use as this.</p><p><hi rend="smallcaps">Knots</hi> mean also the divisions of the log line, used at sea. These are usually 7 fathom, or 42 feet asunder; but should be 8 1/3 fathom, or 50 feet. And then, as many knots as the log line runs out in half a minute, so many miles does the ship sail in an hour; supposing her to keep going at an equal rate, and allowing for yaws, leeway, &c.</p></div1><div1 part="N" n="KOENIG" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">KOENIG</surname> (<foreName full="yes"><hi rend="smallcaps">Samuel</hi></foreName>)</persName></head><p>, a learned philosopher and mathematician, was a Swiss by birth, and came early into eminence by his mathematical abilities. He was professor of philosophy and natural law at Franeker, and afterwards at the Hague, where he became also librarian to the Stadtholder, and to the Princess of Orange; and where he died in 1757.</p><p>The Academy of Berlin enrolled him among her members; but afterwards expelled him on the following occasion. Maupertuis, the president, had inserted in the volume of the Memoirs for 1746, a discourse upon the Laws of Motion; which Koenig not only attacked, but also attributed the memoir to Leibnitz. Maupertuis, stung with the imputation of plagiarism, engaged the Academy of Berlin to call upon him for his proof; which Koenig failing to produce, he was struck out of the academy. All Europe was interested in the quarrel which this occasioned between Koenig and Maupertuis. The former appealed to the public; and his appeal, written with the animation of resentment, procured him many friends. He was author of some other works, and had the character of being one of the best mathematicians of the age.</p></div1></div0><div0 part="N" n="L" org="uniform" sample="complete" type="alphabetic letter"><head>L</head><cb/><div1 part="N" n="LABEL" org="uniform" sample="complete" type="entry"><head>LABEL</head><p>, a long thin brass ruler, with a small sight at one end, and a central hole at the other; commonly used with a tangent-line on the edge of a circumferentor, to take altitudes, and other angles.</p></div1><div1 part="N" n="LACERTA" org="uniform" sample="complete" type="entry"><head>LACERTA</head><p>, <hi rend="italics">Lizard,</hi> one of the new constellations of the northern hemisphere, added by Hevelius to the 48 old ones, near Cepheus and Cassiopeia.</p><p>This constellation contains, in Hevelius's catalogue 10 stars, and in Flamsteed's 16.</p></div1><div1 part="N" n="LACUNAR" org="uniform" sample="complete" type="entry"><head>LACUNAR</head><p>, an arched roof or cieling; more especially the planking or flooring above the porticos.</p><p>LADY-<hi rend="italics">Day,</hi> the 25th of March, being the Annanciation of the Holy Virgin.</p></div1><div1 part="N" n="LAGNY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LAGNY</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas Fantet</hi></foreName> <hi rend="italics">de</hi>)</persName></head><p>, an eminent French mathematician, was born at Lyons. Fournier's Euclid, and Pelletier's Algebra, by chance falling in<cb/> his way, developed his genius for the mathematics. It was in vain that his father designed him for the law; he went to Paris to deliver himsels wholly up to the study of his favourite science. In 1697, the Abbé Bignon, protector-general of letters, got him appointed professor-royal of Hydrography at Rochfort. Soon after, the duke of Orleans, then regent of France, fixed him at Paris, and made him sub-director of the General Bank, in which he lost the greatest part of his fortune in the failure of the Bank. He had been received into the ancient academy in 1696; upon the renewal of which he was named Associate-geometrician in 1699, and pensioner in 1723. After a life spent in close application, he died, April 12, 1734.</p><p>In the last moments of his life, and when he had lost all knowledge of the persons who surrounded his<pb n="7"/><cb/> bed, one of them, through curiosity, asked him, what is the square of 12? To which he immediately replied, and without seeming to know that he gave any answer, 144.</p><p>De Lagny particularly excelled in arithmetic, algebra, and geometry, in which he made many improvements and discoveries. He, as well as Leibnitz, invented a binary arithmetic, in which only two figures are concerned. He rendered much easier the resolution of algebraic equations, especially the irreducible case in cubic equations; and the numeral resolution of the higher powers, by means of short approximating theorems.—He delivered the measures of angles in a new science, called <hi rend="italics">Goniometry;</hi> in which he measured angles by a pair of compasses, without scales, or tables, to great exactness; and thus gave a new appearance to trigonometry.—<hi rend="italics">Cyclometry,</hi> or the measure of the circle, was also an object of his attention; and he calculated, by means of infinite series, the ratio of the circumference of a circle to its diameter, to 120 places of figures.—He gave a general theorem for the tangents of multiple arcs. With many other curious or useful improvements, which are found in the great multitude of his papers, that are printed in the different volumes of the Memoirs of the Academy of Sciences, viz, in almost every volume, from the year 1699, to 1729.</p></div1><div1 part="N" n="LAKE" org="uniform" sample="complete" type="entry"><head>LAKE</head><p>, a collection of water, inclosed in the cavity of some inland place, of a considerable extent and depth. As the Lake of Geneva, &c.</p><p>LAMMAS-<hi rend="smallcaps">Day</hi>, the 1st of August; so called, according to some, because lambs then grow out of season, as being too large. Others derive it from a Saxon word, signifying <hi rend="italics">loaf-mass,</hi> because on that day our forefathers made an offering of bread prepared with new wheat.</p><p>It is celebrated by the Romish church in memory of St. Peter's imprisonment.</p><p>LAMPÆDIAS, a kind of bearded comet, resembling a burning lamp, being of several shapes; for sometimes its flame or blaze runs tapering upwards like a sword, and sometimes it is double or treble pointed.</p></div1><div1 part="N" n="LANDEN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LANDEN</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent mathematician, was born at Peakirk, near Peterborough in Northamptonshire, in January 1719. He became very early a proficient in the mathematics, for we find him a very respectable contributor to the Ladies Diary in 1744; and he was soon among the foremost of those who then contributed to the support of that small but valuable publication, in which almost every English mathematician who has arrived at any degree of eminence for the best part of this century, has contended for fame at one time or other of his life. Mr. Landen continued his contributions to it at times, under various signatures, till within a few years of his death.</p><p>It has been frequently obferved, that the histories of literary men consist chiefly of the history of their writings; and the observation was never more fully verified, than in the present article concerning Mr. Landen.</p><p>In the 48th volume of the Philosophical Transactions, for the year 1754, Mr. Landen gave “An Investigation of some theorems which suggest several very remarkable properties of the Circle, and are at the same time of considerable use in resolving Fractions,<cb/> the denominators of which are certain Multinomials, into more simple ones, and by that means facilitate the computation of Fluents.” This ingenious paper was delivered to the Society by that eminent mathematician Thomas Simpson of Woolwich, a circumstance which will convey to those who are not themselves judges of it, some idea of its merit.</p><p>In the year 1755, he published a volume of about 160 pages, intitled <hi rend="italics">Mathematical Lucubrations.</hi> The title to this publication was made choice of, as a means of informing the world, that the study of the mathematics was at that time rather the pursuit of his leisure hours, than his principal employment: and indeed it continued to be so, during the greatest part of his life; for about the year 1762 he was appointed agent to Earl Fitzwilliam, an employment which he resigned only two years before his death. These Lucubrations contain a variety of tracts relative to the rectification of curve lines, the summation of series, the finding of fluents, and many other points in the higher parts of the mathematics.</p><p>About the latter end of the year 1757, or the beginning of 1758, he published proposals for printing by subscription, <hi rend="italics">The Residual Analysis,</hi> a new Branch of the Algebraic art: and in 1758 he published a small tract, entitled <hi rend="italics">A Discourse on the Residual Analysis;</hi> in which he resolved a variety of problems, to which the method of fluxions had usually been applied, by a mode of reasoning entirely new: he also compared these solutions with others derived from the fluxionary method; and shewed that the solutions by his new method were commonly more natural and elegant than the fluxionary ones.</p><p>In the 51st volume of the Philosophical Transactions, for the year 1760, he gave <hi rend="italics">A New Method of computing the Sums of a great number of Infinite Series.</hi> This paper was also presented to the Society by his ingenious friend the late Mr. Thomas Simpson.</p><p>In 1764, he published the first book of <hi rend="italics">The Residual Analysis.</hi> In this treatise, besides explaining the principles which his new analysis was founded on, he applied it, in a variety of problems, to drawing tangents, and finding the properties of curve lines; to describing their involutes and evolutes, finding the radius of curvature, their greatest and least ordinates, and points of contrary flexure; to the determination of their cusps, and the drawing of asymptotes: and he proposed, in a second book, to extend the application of this new analysis to a great variety of mechanical and physical subjects. The papers which were to have formed this book lay long by him; but he never found leisure to put them in order for the press.</p><p>In the year 1766, Mr. Landen was elected a Fellow of the Royal Society. And in the 58th volume of the Philosophical Transactions, for the year 1768, he gave <hi rend="italics">A specimen of a New Method of comparing Curvilinear Areas;</hi> by means of which many areas are compared, that did not appear to be comparable by any other method: a circumstance of no small importance in that part of natural philosophy which relates to the doctrine of motion.</p><p>In the 60th volume of the same work, for the year 1770, he gave <hi rend="italics">Some New Theorems</hi> for computing the Whole Areas of Curve Lines, where the Ordinates are<pb n="8"/><cb/> expressed by Fractions of a certain form, in a more concise and elegant manner than had been done by Cotes, De Moivre, and others who had considered the subject before him.</p><p>In the 61st. volume, for 1771, he has investigated several new and useful theorems for computing certain fluents, which are assignable by arcs of the conic sections. This subject had been considered before, both by Maclaurin and d'Alembert; but some of the theorems that were given by these celebrated mathematicians, being in part expressed by the difference between an hyperbolic are and its tangent, and that difference being not directly attainable when the arc and its tangent both become insinite, as they will do when the whole fluent is wanted, although such fluent be finite; these theorems therefore fail in these cases, and the computation becomes impracticable without farther help. This defect Mr. Landen has removed, by assigning the <hi rend="italics">limit</hi> of the difference between the hyperbolic arc and its tangent, while the point of contact is supposed to be removed to an infinite distance from the vertex of the curve. And he concludes the paper with a curious and remarkable property relating to pendulous bodies, which is deducible from those theorems. In the same year he published <hi rend="italics">Animadversions on Dr. Stewart's Computation of the Sun's Distance from the Earth.</hi></p><p>In the 65th volume of the Philosophical Transactions, for 1775, he gave the investigation of a General Theorem, which he had promised in 1771, for finding the Length of any Curve of a Conic Hyperbola by means of two Elliptic Arcs: and he observes, that by the theorems there investigated, both the elastic curve and the curve of equable recess from a given point, may be constructed in those cases where Maclaurin's elegant method fails.</p><p>In the 67th volume, for 1777, he gave “A New Theory of the Motion of bodies revolving about an axis in free space, when that motion is disturbed by some extraneous force, either percussive or accelerative.” At that time he did not know that the subject had been treated by any person before him, and he considered only the motion of a sphere, spheroid, and cylinder. After the publication of this paper however he was informed, that the doctrine of rotatory motion had been considered by d'Alembert; and upon procuring that author's <hi rend="italics">Opuscules Mathematiques,</hi> he there learned that d'Alembert was not the only one who had considered the matter before him; for d'Alembert there speaks of some mathematician, though he does not mention his name, who, after reading what had been written on the subject, doubted whether there be any solid whatever, beside the sphere, in which any line, passing through the centre of gravity, will be a permaneut axis of rotation. In consequence of this, Mr. Landen took up the subject again; and though he did not then give a solution to the general problem, viz, “to determine the motions of a body of any form whatever, revolving without restraint about any axis passing through its centre of gravity,” he fully removed every doubt of the kind which had been started by the person alluded to by d'Alembert, and pointed out several bodies which, under certain dimensions, have that remarkable property. This paper is given, among many others equally curious, in a volume of <hi rend="italics">Memoirs,</hi> which<cb/> he published in the year 1780. That volume is also enriched with a very extensive appendix, containing <hi rend="italics">Theorems for the Calculation of Fluents;</hi> which are more complete and extensive than those that are found in any author before him.</p><p>In 1781, 1782, and 1783, he published three small Tracts on the Summation of Converging Series; in which he explained and shewed the extent of some theorems which had been given for that purpose by De Moivre, Stirling, and his old friend Thomas Simpson, in answer to some things which he thought had been written to the disparagement of those excellent mathematicians. It was the opinion of some, that Mr. Landen did not shew less mathematical skill in explaining and illustrating these theorems, than he has done in his writings on original subjects; and that the authors of them were as little aware of the extent of their own theorems, as the rest of the world were before Mr. Landen's ingenuity made it obvious to all.</p><p>About the beginning of the year 1782, Mr. Landen had made such improvements in his theory of Rotatory Motion, as enabled him, he thought, to give a solution of the general problem mentioned above; but finding the result of it to differ very materially from the result of the solution which had been given of it by d'Alembert, and not being able to see clearly where that gentleman in his opinion had erred, he did not venture to make his own solution public. In the course of that year, having procured the Memoirs of the Berlin Academy for 1757, which contain M. Euler's solution of the problem, he found that this gentleman's solution gave the same result as had been deduced by d'Alembert; but the perspicuity of Euler's manner of writing enabled him to discover where he had differed from his own, which the obscurity of the other did not do. The agreement, however, of two writers of such established reputation as Euler and d'Alembert made him long dubious of the truth of his own solution, and induced him to revise the process again and again with the utmost circumspection; and being every time more convinced that his own solution was right, and theirs wrong, he at length gave it to the public, in the 75th volume of the Philosophical Transactions, for 1735.</p><p>The extreme difficulty of the subject, joined to the concise manner in which Mr. Landen had been obliged to give his solution, to confine it within proper limits for the Transactions, rendered it too difficult, or at least too laborious a task for most mathematicians to read it; and this circumstance, joined to the established reputation of Euler and d'Alembert, induced many to think that their solution was right, and Mr. Landen's wrong; and there did not want attempts to prove it; particularly a long and ingenious paper by the learned Mr. Wildbore, a gentleman of very distinguished talents and experience in such calculations; this paper is given in the 80th volume of the Philosophical Transactions, for the year 1790, in which he agrees with the solutions of Euler and d'Alembert, and against that of Mr. Landen. This determined the latter to revise and extend his solution, and give it at greater length, to render it more generally understood. About this time also he met by chance with the late Fris<*>'s <hi rend="italics">Cosmographi<*> Physicæ et Mathematicæ;</hi> in the second part of<pb n="9"/><cb/> which there is a solution of this problem, agreeing in the result with those of Euler and d'Alembert. Here Mr. Landen learned that Euler had revised the solution which he had given formerly in the Berlin Memoirs, and given it another form, and at greater length, in a volume published at Rostoch and Gryphiswald in 1765, intitled, <hi rend="italics">Theoria Motûs Corporum Solidorum seu Rigidorum.</hi> Having therefore procured this book, Mr. Landen found the same principles employed in it, and of course the same conclusion resulting from them, as in M. Euler's former solution of the problem. But notwithstanding that there were thus a coincidence of at least four most respectable mathematicians against him, Mr. Landen was still persuaded of the truth of his own solution, and prepared to defend it. And as he was convinced of the necessity of explaining his ideas on the subject more fully, so he now found it necessary to lose no time in setting about it. He had for several years been severely afflicted with the stone in the bladder, and towards the latter part of his life to such a degree as to be confined to his bed for more than a month at a time: yet even this dreadful disorder did not extinguish his ardour for mathematical studies; for the second volume of his <hi rend="italics">Memoirs,</hi> lately published, was written and revised during the intervals of his disorder. This volume, besides a solution of the general problem concerning rotatory motion, contains the resolution of the problem relating to the motion of a Top; with an investigation of the motion of the Equinoxes, in which Mr. Landen has first of any one pointed out the cause of Sir Isaac Newton's mistake in his solution of this celebrated problem; and some other papers of considerable importance. He just lived to see this work finished, and received a copy of it the day before his death, which happened on the 15th of January 1790, at Milton, near Peterborough, in the 71st year of his age.</p></div1><div1 part="N" n="LARBOARD" org="uniform" sample="complete" type="entry"><head>LARBOARD</head><p>, the left hand side of a ship, when a person stands with his face towards the head.</p></div1><div1 part="N" n="LARMIER" org="uniform" sample="complete" type="entry"><head>LARMIER</head><p>, in Architecture, a flat square member of the cornice below the cimasium, and jets out farthest; being so called from its use, which is to disperse the water, and cause it to fall at a distance from the wall, drop by drop, or, as it were, by tears; <hi rend="italics">larme</hi> in French signifying a tear.</p><p>LATERAL <hi rend="smallcaps">Equation</hi>, in Algebra, is the same with simple equation. It has but one root, and may be constructed by right lines only.</p></div1><div1 part="N" n="LATION" org="uniform" sample="complete" type="entry"><head>LATION</head><p>, is used by some, for the translation or motion of a body from one place to another.</p></div1><div1 part="N" n="LATITUDE" org="uniform" sample="complete" type="entry"><head>LATITUDE</head><p>, in Geography, or Navigation, the distance of a place from the equator; or an arch of the meridian, intercepted between its zenith and the equator. Hence the Latitude is either north or south, according as the place is on the north or south side of the equator: thus London is said to be in 51° 31′ of north latitude.</p><p>Circles parallel to the equator are called <hi rend="italics">parallels of latitude,</hi> because they shew the latitudes of places by their intersections with the meridian.</p><p>The Latitude of a place is equal to the elevation of the pole above the horizon of the place: and hence these two terms are used indifferently for each other.<cb/></p><p>This will be evident from the <figure/> figure, where the circle ZHQP is the meridian, Z the zenith of the place, HO the horizon, EQ the equator, and P the pole; then is ZE the latitude, and PO the elevation of the pole above the horizon. And because PE is = ZO, being each a quadrant, if the common part PZ be taken from both, there will remain the latitude ZE = PO the elevation of the pole.—Hence we have a method of measuring the circumference of the earth, or of determining the quantity of a degree on its surface; for by measuring directly northward or southward, till the pole be one degree higher or lower, we shall have the number of miles in a degree of a great circle on the surface of the earth; and consequently multiplying that by 360, will give the number of miles round the whole circumference of the earth.</p><p>The knowledge of the Latitude of the place, is of the utmost consequence, in geography, navigation, and astronomy; it may be proper therefore to lay down fome of the best ways of determining it, both by sea and land.</p><p>1st. One method is, to find the Latitude of the pole, to which it is equal, by means of the pole star, or any other circumpolar star, thus: Either draw a true meridian line, or find the times when the star is on the meridian, both above and below the pole; then at these times, with a quadrant, or other fit instrument, take the altitudes of the star; or take the same when the star comes upon your meridian line; which will be the greatest and least altitude of the star: then shall half the sum of the two be the elevation of the pole, or the latitude sought.—For, if <hi rend="italics">abc</hi> be the path of the star about the pole P, Z the zenith, and HO the horizon: then is <hi rend="italics">a</hi>O the altitude of the star upon the meridian when above the pole, and <hi rend="italics">c</hi>O the same when below the pole; hence, because <hi rend="italics">a</hi>P = <hi rend="italics">c</hi>P, therefore , hence the height of the pole OP, or latitude of Z, is equal to half the sum of <hi rend="italics">a</hi>O and <hi rend="italics">c</hi>O.</p><p>2d. A second method is by means of the declination of the sun, or a star, and one meridian altitude of the same, thus: Having, with a quadrant, or other instrument, observed the zenith distance Z<hi rend="italics">d</hi> of the luminary; or else its altitude H<hi rend="italics">d,</hi> and taken its complement Z<hi rend="italics">d;</hi> then to this zenith distance, add the declination <hi rend="italics">d</hi>E when the luminary and place are on the same side of the equator, or subtract it when on different sides, and the sum or difference will be the latitude EZ sought. But note, that all altitudes observed, must be corrected for refraction and the dip of the horizon, and for the semidiameter of the sun, when that is the luminary observed.</p><p>Many other methods of observing and computing the Latitude may be seen in Robertson's Navigation; see book 5 and book 9. See also the Nautical Almanac for 1771.</p><p>Mr. Richard Graham contrived an ingenious instrument for taking the latitude of a place at any time of the day. See Philos. Trans, N°. 435, or Abr. vol. 8. pa. 371.<pb n="10"/><cb/></p><div2 part="N" n="Latitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Latitude</hi></head><p>, in Astronomy, as of a star or planet, is its distance from the ecliptic, being an arch of a circle of latitude, reckoned from the ecliptic towards its poles, either north or south. Hence, the astronomical latitude is quite different from the geographical, the former measuring from the ecliptic, and the latter from the equator, so that this latter answers to the declination in astronomy, which measures from the equinoctial.</p><p>The sun has no latitude, being always in the ecliptic; but all the stars have their several latitudes, and the planets are continually changing their latitudes, sometimes north, and sometimes south, crossing the ecliptic from the one side to the other; the points in which they cross the ecliptic being called the <hi rend="italics">nodes</hi> of the planet, and in these points it is that they can pass over the face of the sun, or behind his body, viz, when they come both to this point of the ecliptic at the same time.</p><p><hi rend="italics">Circle of</hi> <hi rend="smallcaps">Latitude</hi>, is a great circle passing through the poles of the ecliptic, and consequently perpendicular to it, like as the meridians are perpendicular to the equator, and pass through its poles.</p></div2><div2 part="N" n="Latitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Latitude</hi></head><p>, <hi rend="italics">of the Moon, North ascending,</hi> is when she proceeds from the ascending node towards her northern limit, or greatest elongation.</p></div2><div2 part="N" n="Latitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Latitude</hi></head><p>, <hi rend="italics">North descending,</hi> is when the moon returns from her northern limit towards the descending node.</p></div2><div2 part="N" n="Latitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Latitude</hi></head><p>, <hi rend="italics">South descending,</hi> is when she proceeds from the descending node towards her southern limit.</p></div2><div2 part="N" n="Latitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Latitude</hi></head><p>, <hi rend="italics">South ascending,</hi> is when she returns from her southern limit towards her ascending node.</p><p>And the same is to be understood of the other planets.</p><p><hi rend="italics">Heliocentric</hi> <hi rend="smallcaps">Latitude</hi>, of a planet, is its latitude, or distance from the ecliptic, such as it would appear from the sun.—This, when the planet comes to the same point of its orbit, is always the same, or unchangeable.</p><p><hi rend="italics">Geocentric</hi> <hi rend="smallcaps">Latitude</hi>, of a planet, is its latitude as seen from the earth.—This, though the planet be in the same point of its orbit, is not always the same, but alters according to the position of the earth, in respect to the planet.</p><p>The latitude of a star is altered only by the aberration of light, and the secular variation of latitude.</p><p><hi rend="italics">Difference of</hi> <hi rend="smallcaps">Latitude</hi>, is an are of the meridian, or the nearest distance between the parallels of latitude of two places. When the two latitudes are of the same name, either both north or both south, subtract the less latitude from the greater, to give the difference of latitude; but when they are of different names, add them together for the difference of latitude.</p><p><hi rend="italics">Middle</hi> <hi rend="smallcaps">Latitude</hi>, is the middle point between two latitudes or places; and is found by taking half the sum of the two.</p><p><hi rend="italics">Parallax of</hi> <hi rend="smallcaps">Latitude.</hi> See <hi rend="smallcaps">Parallax.</hi></p><p><hi rend="italics">Refraction of</hi> <hi rend="smallcaps">Latitude.</hi> See <hi rend="smallcaps">Refraction.</hi></p><p>LATUS <hi rend="smallcaps">Rectum</hi>, in Conic Sections, the same with parameter; which see.</p><p><hi rend="smallcaps">Latus</hi> <hi rend="italics">Transversum,</hi> of the hyperbola, is the right line between the vertices of the two opposite sections; or that part of their common axis lying between the<cb/> two opposite cones; as the line DE. It is the same as the transverse axis of the hyperbola, or opposite hyperbolas. <figure/></p><p><hi rend="smallcaps">Latus</hi> <hi rend="italics">Primarium,</hi> a right line, DD, or EE, drawn through the vertex of the section of a cone, within the same, and parallel to the base.</p></div2></div1><div1 part="N" n="LEAGUE" org="uniform" sample="complete" type="entry"><head>LEAGUE</head><p>, an extent of three miles in length. A nautical league, or three nautical miles, is the 20th part of a degree of a great circle.</p><p>LEAP-<hi rend="smallcaps">Year</hi>, the same as B<hi rend="smallcaps">ISSEXTILE;</hi> which see. It is so called from its leaping a day more that year than in a common year; consisting of 366 days, and a common year only of 365. This happens every 4th year, except only such complete centuries as are not exactly divisible by 4; such as the 17th, 18th, 19th, 21st &c. centuries, because 17, 18, 19, 21, &c, cannot be divided by 4 without a remainder.</p><p><hi rend="italics">To find Leap Year, &c.</hi> Divide the number of the year by 4; then if o remain, it is leap-year; but if 1, 2, or 3 remain, it is so many after leap-year.</p><p>Or the rule is sometimes thus expressed, in thesetwo memorial verses: Divide by 4; what's left shall be, For leap-year o; for past, 1, 2, or 3.</p><p>Thus if it be required to know what year 1790 is: then 4) 1790 (447 2 remains:</p><p>so that 2 remaining, shews that 1790 is the 2d year after leap-year. And to find what year 1796 is: then 4) 1796 (449 here o remaining, shews that 1796 is a leap-year.</p></div1><div1 part="N" n="LEAVER" org="uniform" sample="complete" type="entry"><head>LEAVER</head><p>, See <hi rend="smallcaps">Lever.</hi></p></div1><div1 part="N" n="LEE" org="uniform" sample="complete" type="entry"><head>LEE</head><p>, a term in Navigation, signifying that side, or quarter, towards which the wind blows.</p><p><hi rend="smallcaps">Lee-Way</hi>, of a Ship, is the angle made by the point of the compass steered upon, and the real line of the ship's way, occasioned by contrary winds and a rough sea.</p><p>All ships are apt to make some lee-way; so that something must be allowed for it, in casting up the log-board. But the lee-way made by different ships, under similar circumstances of wind and sails, is different; and even the same ship, with different lading, and having more or less sail set, will have more or less lee-way. The usual allowances for it are these, as they were given by Mr. John Buckler to the late ingenious Mr. William Jones, who first published them in 1702 in his <hi rend="italics">Compendium of Practical Navigation.</hi> 1st, When a ship is close-hauled, has all her sails set, the sea smooth, and a moderate gale of wind, it is then supposed she makes little or no lee-way. 2d, Allow one point, when it blows so fresh that the small sails are taken in. 3d, Allow two points, when the topsail must be close reefed. 4th, Allow two points and a half, when one topsail must be handed. 5th, Allow three points and a half, when both topsails must be taken in. 6th, Allow four points, when the fore-course is handed. 7th, Allow five points, when trying under the mainsail only. 8th, Allow six points, when both main and fore-courses are taken in.<pb n="11"/><cb/> 9th, Allow seven points, when the ship tries a-hull, or with all sails handed.</p><p>When the wind has blown hard in either quarter, and shifts across the meridian into the next quarter, the lee-way will be leffened. But in all these cases, respect must be had to the roughness of the sea, and the trim of the ship. And hence the mariner will be able to correct his course.</p></div1><div1 part="N" n="LEGS" org="uniform" sample="complete" type="entry"><head>LEGS</head><p>, <hi rend="italics">of a Triangle.</hi> When one side of a triangle is taken as the base, the other two are sometimes called the legs. The term is often used too for the base and perpendicular of a right-angled triangle, or the two sides about the right angle.</p><p><hi rend="italics">Hyperbolic</hi> <hi rend="smallcaps">Legs</hi>, are the ends of a curve line that partake of the nature of the hyperbola, or having asymptotes.</p></div1><div1 part="N" n="LEIBNITZ" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LEIBNITZ</surname> (<foreName full="yes"><hi rend="smallcaps">Godfrey-William</hi></foreName>)</persName></head><p>, an eminent mathematician and philosopher, was born at Leipsic in Saxony in 1646. At the age of 15, he applied himself to mathematics at Leipsic and Jena; and in 1663, maintained a thesis <hi rend="italics">de Principiis Individuationis.</hi> The year following he was admitted Master of Arts. He read with great attention the Greek philosophers; and endeavoured to reconcile Plato with Aristotle, as he afterwards did Aristotle with Des Cartes. But the study of the law was his principal view; in which faculty he was admitted Bachelor in 1665. The year following he would have taken the degree of Doctor; but was refused it on pretence that he was too young, though in reality because he had raised himself several enemies by rejecting the principles of Aristotle and the Schoolmen.</p><p>Upon this he repaired to Altorf, where he maintained a thesis <hi rend="italics">de Casibus Perplexis,</hi> with such applause, that he had the degree of Doctor conferred on him.</p><p>In 1672 he went to Paris, to manage some affairs at the French Court for the baron Boinebourg. Here he became acquainted with all the Literati, and made farther and considerable progress in the study of mathematics and philosophy, chiefly, as he says, by the works of Pascal, Gregory St. Vincent, and Huygens. In this course, having observed the imperfection of Pascal's arithmetical machine, he invented a new one, as he called it, which was approved of by the minister Colbert, and the Academy of Sciences, in which he was offered a seat as a member, but refused the offers made to him, as it would have been necessary to embrace the Catholic religion.</p><p>In 1673, he came over to England; where he became acquainted with Mr. Oldenburg, secretary of the Royal Society, and Mr. John Collins, a distinguished member of the Society; from whom it seems he received some hints of the method of fluxions, which had been invented, in 1664 or 1665, by the then Mr. Isaac Newton.</p><p>The same year he returned to France, where he resided till 1676, when he again passed through England, and Holland, in his journey to Hanover, where he proposed to settle. Upon his arrival there, he applied himself to enrich the duke's library with the best books of all kinds. The duke dying in 1679, his successor Ernest Augustus, then bishop of Osnaburgh, shewed Mr. Leibnitz the same favour as his predecessor<cb/> had done, and engaged him to write the History of the House of Brunswick. To execute this task, he travelled over Germany and Italy, to collect materials, While he was in Italy, he met with a pleasant adventure, which might have proved a more serious affair, Passing in a small bark from Venice to Mesola, a storm arose; during which the pilot, imagining he was not understood by a German, whom, being a heretic, he looked on as the cause of the tempest, proposed to strip him of his cloaths and money, and throw him overboard. Leibuitz hearing this, without discovering the least emotion, drew a set of beads from his pocket, and began turning them over with great seeming devotion. The artisice succeeded; one of the sailors observing to the pilot, that, since the man was no heretic, he ought not to be drowned.</p><p>In 1700 he was admitted a member of the Royal Academy of Sciences at Paris. The same year the elector of Brandenburg, afterwards king of Prussia, founded an academy at Berlin by his advice; and he was appointed perpetual President, though his affairs would not permit him to reside constantly at that place. He projected an academy of the same kind at Dresden; and this design would have been executed, if it had not been prevented by the confusions in Poland. He was engaged likewise in a scheme for an universal language, and other literary projects. Indeed his writings had made him long before famous over all Europe, and he had many honours and rewards conferred on him. Beside the office of Privy Counsellor of Justice, which the elector of Hanover had given him, the emperor appointed him, in 1711, Aulic Counsellor; and the czar made him Privy Counsellor of Justice, with a pension of 1000 ducats. Leibnitz undertook at the same time to establish an academy of sciences at Vienna; but the plague prevented the execution of it. However, the emperor, as a mark of his favour, settled a pension on him of 2000 florins, and promised him one of 4000 if he would come and reside at Vienna; an offer he was inclined to comply with, but was prevented by his death.</p><p>Meanwhile, the History of Brunswick being interrupted by other works which he wrote occasionally, he found, at his return to Hanover in 1714, that the elector had appointed Mr. Eccard for his colleague in writing that history. The elector was then raised to the throne of Great Britain, which place Leibnitz visited the latter end of that year, when he received particular marks of friendship from the king, and was frequently at court. He now was engaged in a dispute with Dr. Samuel Clarke, upon the subjects of free-will, the reality of space, and other philosophical subjects. This was conducted with great candour and learning; and the papers, which were published by Clarke, will ever be esteemed by men of genius and learning. The controversy ended only with the death of Leibnitz, Nov. 14, 1716, which was occasioned by the gout and stone, in the 70th year of his age.</p><p>As to his character and person: He was of a middle stature, and a thin habit of body. He had a studious air, and a sweet aspect, though near-sighted. He was indefatigably industrious to the end of his life. He eat and drank little. Hunger alone marked the time of his meals, and his diet was plain and strong.<pb n="12"/><cb/> He had a very good memory, and it was said could re. peat the Æneid from beginning to end. What he wanted to remember, he wrote down, and never read it afterwards. He always professed the Lutheran religion, but never went to sermons; and when in his last sickness his favourite servant desired to send for a minister, he would not permit it, saying he had no occasion for one. He was never married, nor ever attempted it but once, when he was about 50 years old; and the lady desiring time to consider of it, gave him an opportunity of doing the same: he used to say, “that marriage was a good thing, but a wise man ought to consider of it all his life.”</p><p>Leibnitz was author of a great multitude of writings; several of which were published separately, and many othersin the memoirs of different academies. He invented a binary arithmetic, and many other ingenious matters. His claim to the invention of Fluxions, has been spoken of under that article. Hanschius collected, with great care, every thing that Leibnitz had said, in different passages of his works, upon the principles of philosophy; and formed of them a complete system, under the title of <hi rend="italics">G. G. Leibnitzii Principia Philosaphiæ more geometrico demonstrata</hi> &c, 1728, in 4to. There came out a collection of our author's letters in 1734 and 1735, intitled, <hi rend="italics">Epistolæ ad diversos theologici, juridici, medici, philosophici, mathematici, historici, & philologici argumenti e MSS. auctores : cum annotationibus suis primum divulgavit Christian Cortholtus.</hi> But all his works were collected, distributed into classes by M. Dutens, and published at Geneva in six large volumes 4to, in 1768, intitled, <hi rend="italics">Gothofredi Guillelmi Leibnitii Opera Omnia &c.</hi></p><p><hi rend="smallcaps">Lei<*>nitzian Philosophy</hi>, or the Philosophy of Leibnitz, is a system formed and published by its author in the last century, partly in emendation of the Cartesian, and partly in opposition to the Newtonian philosophy. In this philosophy, the author retained the Cartesian subtile matter, with the vortices and universal plenum; and he represented the universe as a machine that should proceed for ever, by the laws of mechanism, in the most perfect state, by an absolute inviolable necessity. After Newton's philosophy was published, in 1687, Leibnitz printed an Essay on the celestial motions in the Act. Erud. 1689, where he admits the circulation of the ether with Des Cartes, and of gravity with Newton; though he has not reconciled these principles, nor shewn how gravity arose from the impulse of this ether, nor how to account for the planetary revolutions in their respective orbits. His system is also defective, as it does not reconcile the circulation of the ether with the free motions of the comets in all directions, or with the obliquity of the planes of the planetary orbits; nor resolve other objections to which the hypothesis of the vortices and plenum is liable.</p><p>Soon after the period just mentioned, the dispute commenced concerning the invention of the method of Fluxions, which led Mr. Leibnitz to take a very decided part in opposition to the philosophy of Newton. From the goodness and wisdom of the Deity, and his principle of a <hi rend="italics">sufficient reason,</hi> he concluded, that the universe was a perfect work, or the best that could possibly have been made; and that other things, which are evil or incommodious, were permitted as necessary consequences of what was best: that the material sys-<cb/> tem, considered as a perfect machine, can never fall into disorder, or require to be set right; and to suppose that God interposes in it, is to lessen the skill of the author, and the perfection of his work. He expressly charges an impious tendency on the philosophy of Newton, because he asserts, that the fabric of the universe and course of nature could not continue for ever in its present state, but in process of time would<*>require to be re-established or renewed by the hand of its first framer. The perfection of the universe, in consequence of which it is capable of continuing for ever by mechanical laws in its present state, led Mr. Leibnitz to distinguish between the quantity of motion and the force of bodies; and, whilst he owns in opposition to Des Cartes that the former varies, to maintain that the quantity of force is for ever the same in the universe; and to measure the forces of bodies by the squares of their velocities.</p><p>Mr. Leibnitz proposes two principles as the foundation of all our knowledge; the first, that it is impossible for a thing to be, and not to be at the same time, which he says is the foundation of speculative truth; and secondly, that nothing is without a <hi rend="italics">sufficient reason</hi> why it should be so, rather than otherwise; and by this principle he says we make a transition from abstracted truths to natural philosophy. Hence he concludes that the mind is naturally determined, in its volitions and elections, by the greatest apparent good, and that it is impossible to make a choice between things perfectly like, which he calls <hi rend="italics">indiscernibles;</hi> from whence he infers, that two things perfectly like could not have been produced even by the Deity himself: and one reason why he rejects a vacuum, is because the parts of it must be supposed perfectly like to each other. For the same reason too, he rejects atoms, and all similar parts of matter, to each of which, though divisible <hi rend="italics">ad infinitum,</hi> he ascribes a <hi rend="italics">monad</hi> (Act. Lipsiæ 1698, pa. 435) or active kind of principle, endued with perception and appetite. The essence of substance he places in action or activity, or, as he expresses it, in something that is between acting and the faculty of acting. He affirms that absolute rest is impossible, and holds that motion, or a sort of <hi rend="italics">nisus,</hi> is essential to all material substances. Each monad he deseribes as representative of the whole universe from its point of sight; and yet he tells us, in one of his letters, that matter is not a substance, but a <hi rend="italics">substantiatum,</hi> or <hi rend="italics">phenomené bien fondé.</hi> See also Maclaurin's View of Newton's Philosophical Discoveries, book 1, chap. 4.</p></div1><div1 part="N" n="LEMMA" org="uniform" sample="complete" type="entry"><head>LEMMA</head><p>, is a term chiefly used by mathematicians, and signifies a proposition, previously laid down to prepare the way for the more easy apprehension of the demonstration of some theorem, or the construction of some problem.</p></div1><div1 part="N" n="LEMNISCATE" org="uniform" sample="complete" type="entry"><head>LEMNISCATE</head><p>, the name of a curve in the form of the figure of 8. If we call A P, <hi rend="italics">x;</hi> P Q, <hi rend="italics">y,</hi> and the constant line A B or A C, <hi rend="italics">a;</hi> the equation , or , expressing a line of the 4th degree, will denote a lemniscate, having a double point in the point A. There may be other lemniscates, as the ellipse of Cassini, &c; but that above defined is the simplest of them. <figure/><pb n="13"/><cb/></p><p>It easily appears that this curve is quadrable. For since , therefore the fluxion of the curve or ; the fluent of which is for the general area of the curve; which, when <hi rend="italics">x</hi> is = <hi rend="italics">a,</hi> becomes barely .</p></div1><div1 part="N" n="LENS" org="uniform" sample="complete" type="entry"><head>LENS</head><p>, a piece of glass or other transparent substance, having its two surfaces so formed that the rays of light, in passing through it, have their direction changed, and made to converge and tend to a point beyond the lens, or to become parallel after converging or diverging, or lastly to diverge as if they had proceeded from a point before the lens. Some lenses are convex, or thicker in the middle; others concave, or thinner in the middle; while others are plano-convex, or plano-concave; and some again are convex on one side and concave on the other, which are called meniscuses, the properties of which see under that word. When the particular sigure is not considered, a lens that is thickest in the middle is called a convex lens; and that which is thinnest in the middle is called a concave le<*>s, without farther distinction.</p><p>These several forms of lenses are represented in the annexed figure: <figure/> where A, B are convex lenses, and C, D, E are concave ones; also A is a plano-convex, B is convexo-convex, C is plano-concave, D is concavo-concave, and E is a meniscus.</p><p>In every lens, the right line perpendicular to the two surfaces, is called the Axis of the lens, as F G; the points where the axis cuts the surface, are called the Vertices of the l<*>ns; also the middle point between them is called the Centre; and the distance between them, the Diameter.</p><p>Some confine lenses within the diameter of half an inch; and such as exceed that thickness, they call Lenticular Glasses.</p><p>Lenses are either blown or ground.</p><p><hi rend="italics">Blown</hi> <hi rend="smallcaps">Lenses</hi>, are small globules of glass, melted in the flame of a lamp or taper. See Microscope.</p><p><hi rend="italics">Ground</hi> <hi rend="smallcaps">Lenses</hi>, are such as are ground or rubbed into the desired shape, and then polished. For a method of grinding them, and description of a machine for that purpose, see Philos. Trans. vol. xli. pa. 555, or<*>Abr. viii. 281.</p><p>Maurolycus first delivered something relative to the nature of lenses; but we are chiefly indebted to Kepler for explaining the doctrine of refraction through mediums of different forms, the chief substance of which may be comprehended in the cases following.<cb/> <figure/></p><p>Let DA be a ray of light falling upon a conver dense medium, having its centre at E. When the ray arrives at A, it will not proceed in the same direction A<hi rend="italics">t;</hi> but it will be there bent, and thrown into a direction AT, nearer the perpendicular AE. In the same manner, another ray falling on B, at an equal distance on the other side of the vertex C, and parallel to the former ray DA, will be refracted into the same point T. And it will also be found, that all the intermediate parallel rays will converge to the same point, very nearly.</p><p>On the other hand, if the rays fall parallel on the inside of this denser medium, as in the fig. below, they will tend from the perpendicular EA<hi rend="italics">f;</hi> and converge to a point T in the air, or any rarer medium. Also the ray incident on B, at the same distance from the vertex C, will converge to the same place T, together with all the intermediate parallel rays. <figure/></p><p>Since therefore rays are made to converge when they pass either from a rarer or a denser medium terminated by a convex surface, and converge again when they pass from the same medium convex towards the rarer, a lens which is convex on both sides must, on both accounts, make parallel rays converge to a point beyond it. Thus, the parallel rays between A and B, falling upon <figure/> the convex surface of the glass AB, would in that dense medium have converged to T; but that medium being terminated by another convex surface, they will be made more converging, and be collected at some place F, nearer to the lens. <figure/></p><p>Again, to explain the effects of a concave glass, let AB be the concave side of a dense medium, the centre of concavity being at E, In this case, DA will be re-<pb n="14"/><cb/> fracted towards the perpendicular EA; and so likewise will the ray incident at B; in consequence of which they will diverge from one another within the dense medium. The intermediate rays will also diverge more or less, as they recede from the axis TC; which, being in the perpendicular, will go straight on. <figure/></p><p>If the rays be parallel within the dense medium, they will diverge when they pass from thence into a rarer medium, through a concave surface. For the ray DA will be refracted from the perpendicular AE, as will also the ray that is incident at B, together with all the intermediate rays, in proportion to their distance from the axis or central ray TC. <figure/></p><p>Therefore, if a dense medium, as the glass AB, be terminated by two concave surfaces, parallel rays passing through it will be made to diverge by both the sides of it. Thus the first surface AB will make them diverge as if they had come from the point T; and with the effect of the second surface added to this, they will diverge as from a nearer point, F.</p><p>It was Kepler, who by these investigations first gave a clear explanation of the effects of lenses, in making the rays of a pencil of light converge or diverge. He shewed that a plano-convex lens makes rays, that were parallel to its axis, meet at the distance of the diameter of the sphere of convexity; but that if both sides of the lens be equally convex, the rays will have their focus at the distance of the radius of the circle corresponding to that degree of convexity. But he did not investigate any rule for the foci of lenses unequally convex. He only says, in general, that they will fall somewhere in the medium, between the foci belonging to the two different degrees of convexity. It is to Cavalerius that we owe this investigation: he laid down this rule, As the sum of both the diameters is to one of them, so is the other to the distance of the focus. And it is to be noted that all these rules, concerning convex lenses, are applicable to those that are concave, with this difference, that the focus is on the contrary side of the glass. See Montucla, vol. 2, pa. 176; or Priestley's Hist. of Vision, pa. 65, 4to.</p><p>Upon this principle it was not difficult to find the foci of pencils of rays issuing from any point in the axis of the lens; since those that are parallel will meet in the focus; and if they issue from the focus, they will be parallel on the other side. If they issue from a point<cb/> between the focus and the glass, they will continue to diverge after passing the lens, but less than before; while those that come from beyond the focus, will converge after passing the glass, and will meet in a place beyond the opposite focus. This philosopher particularly observed, that rays which issue from twice the distance of the focus, will meet at the same distance on the other side. The most important of these observations have been already illustrated by proper figures, and from them the rest may be easily conceived. Later optical writers have assigned the distances at which rays will meet, that issue from any other place in the axis of a lens; but Kepler was too much intent upon his astronomical and other pursuits, to give much attention to geometry. But, from the whole, Montucla gives the following rule concerning this subject: As the excess of the distance of the object from the glass, above the distance of the focus, is to the distance of the focus; so is this distance, to the place of convergency beyond the glass. And the same rule will find the point of divergency, when the rays issue from any place between the lens and the focus: for then the excess of the distance of the object from the glass, above that of the focus, is negative, which is the same distance taken the contrary way. Montucla, vol. 2, pa. 177.</p><p>And from the principle above-mentioned, it will not be difficult to understand the application of lenses, in the rationale of telescopes and microscopes. On these principles too is founded the structure of refracting burning glasses, by which the sun's light and heat are exceedingly augmented in the focus of the lens, whether convex or plano-convex; since the rays, falling parallel to the axis of the lens, are reduced into a much narrower compass; so that it is no wonder they burn some bodies, melt others, and produce other extraordinary phenomena.</p><p>In the Philos. Trans. vol. xvii. 960, or the Abr. i. 191, Dr. Halley gives an ingenious investigation of the foci of rays refracted through any lenses, nearly as follows: <figure/></p><p>Let BEL be a double convex lens, C the centre of the segment EB, and K the centre of the segment EL; BL the thickness or diameter of the lens, and D a point in the axis; it is required to find the point F, or focus, where the rays proceeding from D shall be collected, after being refracted through the lens at A and <hi rend="italics">a,</hi> points very near to the axis BL. Put the distance DA or DB=<hi rend="italics">d,</hi> the radius CA or CB=<hi rend="italics">r,</hi> and the radius K<hi rend="italics">a</hi> or KL=R; also the thickness of the lens BL=<hi rend="italics">t,</hi> and <hi rend="italics">m</hi> to <hi rend="italics">n</hi> the ratio of the fine of the angle of incidence DAG to the sine of the refracted angle HAG or CAM; or <hi rend="italics">m</hi> to <hi rend="italics">n</hi> will be the ratio of those angles themselves nearly, since very small angles are to each other in the same ratio as their sines. Hence <hi rend="italics">m</hi> is as the angle DAG or DAC, <hi rend="italics">n</hi> is as the angle HAG or MAC, and because in this case the sides are as their oppo-<pb n="15"/><cb/> site angles, therefore , or which is as the [angle]C; from this take <hi rend="italics">n</hi> or the [angle] MAC, and there remains as the [angle]M; hence again , that is ; which shews in what point the rays would be collected after one refraction, viz, when <hi rend="italics">nr</hi> is less than But when , the point would be at an infinite distance, or the rays will be parallel to the axis; and when <hi rend="italics">nr</hi> is greater than , then MB is negative, or M falls on the other side of the lens beyond D, and the rays still continue to diverge after the first refraction.</p><p>The point M being now found, to or from which the rays proceed after the first refraction, and BM - BL being thus given, which call D, by a process like the former it follows that FL, or the focal distance sought, is equal to . And here, instead of D substituting MB - LB or , and putting <hi rend="italics">p</hi> for , the same theorem will become , the focal distance sought, in its most general form, including the thickness of the lens; being the universal rule for the foci of double convex glasses exposed to diverging rays.</p><p>But if <hi rend="italics">t</hi> the thickness of the lens be rejected, as not sensible, the rule will be much shorter, viz, .</p><p>If therefore the lens consist of glass, whose refraction is as 3 to 2, it will be . And if it be of water, whose refraction is as 4 to 3, it will be . But, if the lens could be made of diamond, whose refraction is as 5 to 2, it would be .</p><p>If the incident rays, instead of diverging, be converging, the distance DB or <hi rend="italics">d</hi> will be negative, and then the theorem for a double convex glass lens will be or , in which case therefore the focus is always on the other fide of the glass.</p><p>And if the rays be parallel, as coming from an infinite distance, or nearly so, then will <hi rend="italics">d</hi> be negative, as well as the terms in the theorem in which it is found; and therefore, the other term <hi rend="italics">pr</hi>R will be nothing in respect of those infinite terms; and by omitting it, the<cb/> theorem will be , or for glass .</p><p>And here if <hi rend="italics">r</hi> = R, or the two sides of the glass be of equal convexity, this last will become barely 2<hi rend="italics">r</hi><hi rend="sup">2</hi>/2<hi rend="italics">r</hi> or barely <hi rend="italics">r</hi> = <hi rend="italics">f</hi> the focus, which therefore is in the centre of the convexity of the lens.</p><p>If the lens be a meniscus of glass; then, making <hi rend="italics">r</hi> negative, the theorem is or for diverging rays, or for converging rays, and or for parallel rays.</p><p>If the lens be a double concave glass, <hi rend="italics">r</hi> and R will be both negative, and then the theorem becomes for diverging rays, always negative; for converging rays; and for parallel rays.</p><p>And here, if the radii of curvature <hi rend="italics">r</hi> and R be equal, this last will be barely - <hi rend="italics">r</hi> = <hi rend="italics">f</hi> for parallel rays falling on a double concave glass of equal curvature.</p><p>Lastly, when the lens is a plano-convex glass; then, <hi rend="italics">r</hi> being infinite, the theorem becomes for diverging rays, for converging rays, and for parallel rays.</p><p>The theorems for parallel rays, as coming from an infinite distance, take place in the common resracting telescopes. And those for converging rays are chiefly of use to determine the focus resulting from any sort of lens placed in a telescope, between the focus of the object-glass and the glass itself; the distance between the said focus of the object-glass and the interposed lens being made = - <hi rend="italics">d;</hi> while those for diverging rays are chiefly of use in microscopes, reading glasses, and other cases in which near objects are viewed.</p><p>It is evident that the foregoing general theorem will serve to find any of the other circumstances, as well as the focus, by considering this as given. Thus, for instance, suppose it be required to find the distance at which an object being placed, it shall by a given lens be represented as large as the object itself; which is of singular use in viewing and drawing them, by transmitting the image through a glass in a dark room, as in the camera obscura, which gives not only the true figure and shades, but the colours themselves as vivid as the life. Now in this case <hi rend="italics">d</hi> is = <hi rend="italics">f,</hi> which makes the theorem become , and<pb n="16"/><cb/> this gives . But if the two convexities belong to equal spheres, so as that <hi rend="italics">r</hi> = R, then it is <hi rend="italics">d</hi> = <hi rend="italics">pr,</hi> or = 2<hi rend="italics">r</hi> when the lens is glass. So that if the object be placed at the diameter of the sphere distant from the lens, then the focus will be as far distant on the other side, and the image as large as the object. But if the glass were a plano-convex, the same distance would be just twice as much.</p><p>Again, recurring to the first general theorem, including <hi rend="italics">t,</hi> the thickness of the lens; let the lens be a whole sphere; then <hi rend="italics">t</hi> = 2<hi rend="italics">r,</hi> and <hi rend="italics">r</hi> = R; and hence the theorem reduces to .</p><p>And here if <hi rend="italics">d</hi> be infinite, the theorem contracts to or ; or for glass : shewing that a sphere of glass collects the sun's rays at half the radius of the sphere without it. And for a sphere of water, the focus is at the distance of a whole radius.</p><p>For another example; when a hemisphere is exposed to parallel rays; then <hi rend="italics">d</hi> and R being infinite, and <hi rend="italics">t</hi>=<hi rend="italics">r,</hi> the theorem becomes . That is, in glass it is 4/3<hi rend="italics">r,</hi> and in water <*>/4<hi rend="italics">r.</hi></p><p>Several other corollaries may be deduced from the foregoing principles. As,</p><p>1st. That the thickness of the lens, being very small, the focus will remain the same, whether the one side or the other be exposed to the rays.</p><p>2d. If a luminous body be placed in a focus behind a lens, whether plano-convex, or convex on both sides; or whether equally or unequally so; the rays become parallel after refraction, as the refracted rays become what were before the incident rays. And hence, by means of a convex lens, or a little glass bubble full of water, a very intense light may be projected to a great distance. Which furnishes us with the structure of a lamp or lantern, to throw an intense light to an immense distance: for a lens, convex on both sides, being placed opposite to a concave mirror, if there be placed a lighted candle or wick in the common focus of both, the rays reflected back from the mirror to the lens will be parallel to each other; and after refraction will converge, till they concur at the distance of the radius, after which they will again diverge. But the candle being likewise in the focus of the lens, the rays it throws on the lens will be parallel; and therefore a very intense light meeting with another equally intense, at the distance of the diameter from the lens, the light will be surprising: and though it afterwards decrease, yet the parallel and diverging rays going a long way together, it will be very great at a great distance. Lanterns of this kind are of considerable service in the night time, to discover remote objects; and are used with success by fowlers and fishermen, to collect their prey together, that so it may be taken.</p><p>If it be required to have the light, at the same time, transmitted to several places, as through several streets, &c, the number of lenses and mirrors must be increased.<cb/></p><p>3d. The images of objects are shewn inverted in the focus of a convex lens: nor is the focus of the sun's rays any thing else, in effect, but the image of the sun inverted. Hence, in solar eclipses, the sun's image, eclipsed as it is, may be burnt by a large lens on a board, &c, and exhibit a very entertaining phenomenon.</p><p>4th. If a concave mirror be so placed, as that an inverted image, sormed by refraction through a lens, be found between the centre and the focus, or even beyond the centre, it will again be inverted by reslection, and so appear erect; in the first case beyond the centre, and in the latter between the centre and the focus. And on these principles the camera obscura is constructed.</p><p>5th. The image of an object, delineated beyond a convex lens, is of such a magnitude, as it would be of, were the object to shine into a dark room through a small hole, upon a wall, at the same distance from the hole, as the focus is from the lens.—When an object is less distant from a lens than the focus of parallel rays, the distance of the image is greater than that of the object; otherwise, the distance of the image is less than that of the object: in the former case, therefore, the image is larger than the object; in the latter, it is less.</p><p>When the images are less than the objects, they will appear more distinct and vivid; because then more rays are accumulated into a given space. But if the images be made greater than the objects, they will not appear distinctly; because in that case there are fewer rays which meet after refraction in the same point; whence it happens, that rays proceeding from different points of an object, terminate in the same point of an image, which is the cause of confusion. Hence it appears, that the same aperture of a lens may be admitted in every case, if we would keep off the rays which produce confusion. However, though the image be then more distinct, when no rays are admitted but those near the axis, yet for want of rays the image is apt to be dim.</p><p>6th. If the eye be placed in the focus of a convex lens, an object viewed through it, appears erect, and enlarged in the ratio of the distance of the object from the eye, to that of the eye from the lens, if it be near; but infinitely if remote.</p><p>7th. An object viewed through a concave lens, appears erect, and diminished in a ratio compounded of the ratios of the space in the axis between the point of incidence, and the point to which an oblique ray would pass without refraction, to the space in the axis between the eye and the middle of the object; and the space in the same axis between the eye and the point of incidence, to the space between the middle of the object and the point to which the oblique ray would pass without refraction.</p><p>Finally, it may be observed, that the very small magnifying glasses used in microscopes, most properly come under the denomination of lens, as they most approach to the figure of the lentil, a seed of the vetch or pea kind, from whence the name is derived; but the reading glasses, and burning glasses, and all that magnify, come under the same denomination; for their surfaces are convex, although less so. A drop of water is a lens, and it will serve as one; and many have used it by way of <pb/><pb/><pb n="17"/><cb/> lens in their microscopes. A drop of any transparent fluid, inclosed between two concave glasses, acquires the shape of a lens, and has all its properties. The crystalline humour of the eye is a lens exactly of this kind; it is a small quantity of a translucent fluid, contained between two concave and transparent membranes, called the coats of the eye; and it acts as the lens made of water would do, in an equal degree of convexity.</p></div1><div1 part="N" n="LEO" org="uniform" sample="complete" type="entry"><head>LEO</head><p>, <hi rend="italics">the Lion,</hi> a considerable constellation of the northern hemisphere, being one of the 48 old constellations, and the 5th sign of the zodiac. It is marked thus <*>, as a rude sketch of the animal.</p><p>The Greeks fabled that this was the Nemæan lion, which had dropped from the moon, but being slain by Hercules, was raised to the heavens by Jupiter, in commemoration of the dreadful conflict, and in honour of that hero. But the hieroglyphical meaning of this sign, so depicted by the Egyptians long before the invention of the fables of Hercules, was probably no more than to signify, by the fury of the lion, the violent heats occasioned by the sun when he entered that part of the ecliptic.</p><p>The stars in the constellation Leo, in Ptolomy's catalogue are 27, besides 8 unformed ones, now counted in later times in the constellation Coma Berenices, in Tycho's 30, in that of Hevelius 49, and in Flamsteed's 95; one of them, of the first magnitude, in the breast of the Lion, is called Regulus, and Cor Leonis, or Lion's Heart.</p><p><hi rend="smallcaps">Leo</hi> <hi rend="italics">Minor, the Little Lion,</hi> a constellation of the northern hemisphere, and one of the new ones that were formed out of what were left by the ancients, under the name of Stellæ Informes, or unformed stars, and added to the 48 old ones. It contains 53 stars in Flamsteed's catalogue.</p><p><hi rend="italics">Cor</hi> <hi rend="smallcaps">Leonis</hi>, <hi rend="italics">Lion's heart,</hi> a fixed star, of the first magnitude, in the sign Leo; called also Regulus, Basilicus, &c.</p></div1><div1 part="N" n="LEPUS" org="uniform" sample="complete" type="entry"><head>LEPUS</head><p>, <hi rend="italics">the Hare,</hi> a constellation of the southern hemisphere, and one of the 48 old constellations.</p><p>The Greeks fabled, that this animal was placed in the heavens, near Orion, as being one of the animals which he hunted. But it is probable their masters, the Egyptians, had some other meaning in this hieroglyphic.</p><p>The stars in the constellation Lepus, in Ptolomy's catalogue are 12, in Tycho's 13, and in Flamsteed's 19.</p></div1><div1 part="N" n="LEUCIPPUS" org="uniform" sample="complete" type="entry"><head>LEUCIPPUS</head><p>, a celebrated Greek philosopher and mathematician, who flourished about the 428th year before Christ. He was the first author of the famous system of atoms and vacuums, and of the hypothesis of storms; since attributed to the moderns.</p></div1><div1 part="N" n="LEVEL" org="uniform" sample="complete" type="entry"><head>LEVEL</head><p>, an instrument used to make a line parallel to the horizon, and to continue it out at pleasure; and by this means to find the true level, or the difference of ascent or descent between two or more places, for conveying water, draining sens, &c.</p><p>There are several instruments, of different contrivance and matter, invented for the perfection of levelling, as may be seen in De la Hire's and Picard's treatises of Levelling, in Biron's treatise on Mathematical Instruments, also in the Philos. Trans. and the Memoirs de <*> Acad. &c. But they may be reduced to the following kinds.</p><p><hi rend="italics">Water</hi>-<hi rend="smallcaps">Level</hi>, that which shews the horizontal line by means of a surface of water or other fluid; founded<cb/> on this principle, that water always places itself level or horizontal.</p><p>The most simple kind is made of a long wooden trough or canal; which being equally filled with water, its surface shews the line of level. And this is the chorobates of the ancients, described by Vitruvius, lib. viii. cap. 6.</p><p>The water-level is also made with two cups sitted to the two ends of a straight pipe, about an inch diameter, and 3 or 4 feet long, by means of which the water communicates from the one cup to the other; and this pipe being moveable on its stand by means of a ball and socket, when the two cups shew equally full of water, their two surfaces mark the line of level.</p><p>This instrument, instead of cups, may also be made with two short cylinders of glass three or four inches long, fastened to each extremity of the pipe with wax or mastic. The pipe is filled with common or coloured water, which shews itself through the cylinders, by means of which the line of Level is determined; the height of the water, with respect to the centre of the earth, being always the same in both cylinders. This level, though very simple, is yet very commodious for levelling small distances. See the method of preparing and using a water-level, and a mercurial Level, annexed to Davis's quadrant, for the same purpose, by Mr. Leigh, in Philos. Trans. vol. <hi rend="smallcaps">XL.</hi> 417, or Abr. viii. 362.</p><p><hi rend="italics">Air</hi>-<hi rend="smallcaps">Level</hi>, that which shews the line of Level by means of a bubble of air inclosed with some fluid in a glass tube of an indeterminate length and thickness, and having its two ends hermetically sealed: an invention, it is said, of M. Thevenot. When the bubble fixes itself at a certain mark, made exactly in the middle of the tube, the case or ruler in which it is fixed, is then level. When it is not level, the bubble will rise to one end.— This glass-tube may be set in another of brass, having an aperture in the middle, where the bubble of air may be observed.—The liquor with which the tube is silled, is oil of tartar, or aqua secunda; those not being liable to freeze as common water, nor to rarefaction and condensation as spirit of wine is.</p><p>There is one of these instruments with sights, being an improvement upon that last described, which, by the addition of other apparatus, becomes more exact and commodious. It consists of an air-Level, n° 1, <hi rend="italics">(fig.</hi> 1, <hi rend="italics">Plate XIV)</hi> about 8 inches long, and about two thirds of an inch in diameter, set in a brass tube, 2, having an aperture in the middle, C. The tubes are carried in a strong straight ruler, of a foot long; at the ends of which are fixed two sights, 3, 3, exactly perpendicular to the tubes, and of an equal height, having a square hole, formed by two fillets of brass crossing each other at right angles; in the middle of which is drilled a very small hole, through which a point on a level with the instrument is seen. The brass tube is fastened to the ruler by means of two screws; the one of which, marked 4, serves to raise or depress the tube at pleasure, for bringing it towards a level. The top of the ball and socket is rivetted to a small ruler that springs, one end of which is fastened with springs to the great ruler, and at the other end is a screw, 5, serving to raise and depress the instrument when nearly level.</p><p>But this instrument is still less commodious than the following one: for though the holes be ever so small, yet they will still take in too great a space to determine the point of Level precisely.<pb n="18"/><cb/></p><p><hi rend="italics">Fig.</hi> 2, is a <hi rend="italics">Level with Telescopic Sights,</hi> first invented by Mr. Huygens. It is like the last; with this difference, that instead of plain sights, it carries a telescope, to determine exactly a point of Level at a considerable distance. The screw 3, is for raising or lowering a little fork, for carrying the hair, and making it agree with the bubble of air when the instrument is Level; and the screw 4, is for making the bubble of air, D or E, agree with the telescope. The whole is fitted to a ball and socket, or otherwise moved by joints and screws.—It may be observed that a telescope may be added to any kind of Level, by applying it upon, or parallel to, the base or ruler, when there is occasion to take the level of remote objects: and it possesses this advantage, that it may be inverted by turning the ruler and telescope half round; and if then the hair cut the same point that it did before, the operation is just. Many varieties and improvements of this instrument have been made by the more modern opticians.</p><p>Dr. Desaguliers proposed a machine for taking the difference of Level, which contained the principles both of a barometer and thermometer; but it is not accurate in practice: Philos. Trans. vol. xxxiii. pa. 165, or Abr. vol. vi. 271. <hi rend="italics">Fig.</hi> 3, 4, 5, 6.</p><p>Mr. Hadley too has contrived a Spirit Level to be fixed to a quadrant, for taking a meridian altitude at sea, when the horizon is not visible. See the description and figure of it in the Philos. Trans. vol. xxxviii. 167, or Abr. viii. 357. Various other Spirit Levels, and Mercurial Levels, are also invented and used upon different occasions.</p><p><hi rend="italics">Reflecting</hi> <hi rend="smallcaps">Level</hi>, that made by means of a pretty long surface of water, representing the same object inverted, which we see erect by the eye; so that the point where these two objects appear to meet, is on a Level with the place where the surface of the water is found. This is the invention of M. Mariotte.</p><p>There is another reflecting Level, consisting of a polished metal mirror, placed a little before the object glass of a telescope, suspended perpendicularly. This mirror must be set at an angle of 45 degrees; in which case the perpendicular line of the telescope becomes a horizontal line, or a line of Level. Which is the invention of M. Cassini.</p><p><hi rend="italics">Artillery Foot</hi>-<hi rend="smallcaps">Level</hi>, is in form of a square (fig. 7), having its two legs or branches of an equal length; at the junction of which is a small hole, by which hangs a plummet playing on a perpendicular line in the middle of a quadrant, which is divided both ways from that point into 45 degrees.</p><p>This instrument may be used on other occasions, by placing the ends of its two branches on a plane; for when the plummet plays perpendicularly over the middle division of the quadrant, the plane is then Level.</p><p>To use it in Gunnery, place the two ends on the piece of artillery, which may be raised to any proposed height, by means of the plummet, which will cut the degree above the Level. But this supposes the outside of the cannon is parallel to its axis, which is not always the case; and theresore they use another instrument now, either to set the piece Level, or clevate it at any angle; namely a small quadrant, with one of its radii continued out pretty long, which being put into the inside of the cylindrical<cb/> bore, the plummet shews the angle of elevation, or the line of Level. See <hi rend="italics">Gunner's</hi> <hi rend="smallcaps">Quadrant.</hi></p><p><hi rend="italics">Carpenter's, Bricklayer's,</hi> or <hi rend="italics">Pavior's</hi> <hi rend="smallcaps">Level</hi>, consists of a long ruler, in the middle of which is sitted at right angles another broader piece, at the top of which is fastened a plummet, which when it hangs over the middle line of the 2d or upright piece, shews that the base or long ruler is horizontal or Level. Fig. 8.</p><p><hi rend="italics">Mason's</hi> <hi rend="smallcaps">Level</hi>, is composed of 3 rules, so jointed as to form an isosceles triangle, somewhat like a Roman A; from the vertex of which is suspended a plummet, which hangs directly over a mark in the middle of the base, when this is horizontal or Level. Fig. 8.</p><p><hi rend="italics">Plumb or Pendulum</hi> <hi rend="smallcaps">Level</hi>, said to be invented by M. Picard; fig. 10. This shews the horizontal line by means of another line perpendicular to that described by a plummet or pendulum. This Level consists of two legs or branches, joined at right angles, the one of which, of about 18 inches long, carries a thread and plummet; the thread being hung near the top of the branch, at the point 2. The middle of the branch where the thread passes is hollow, so that it may hang free every where: but towards the bottom, where there is a small blade of silver, on which a line is drawn perpendicular to the telescope, the said cavity is covered by two pieces of brass, with a piece of glass G, to see the plummet through, forming a kind of case, to prevent the wind from agitating the thread. The telescope, of a proper length, is sixed to the other leg of the instrument, at right angles to the perpendicular, and having a hair stretched horizontally across the focus of the object-glass, which determines the point of Level, when the string of the plummet hangs against the line on the silver blade. The whole is fixed by a ball and socket to its stand.</p><p>Fig. 12, is a <hi rend="italics">Balance</hi> <hi rend="smallcaps">Level;</hi> which being suspended by the ring, the two sights, when in equilibrio, will be horizontal, or in a Level.</p><p>Some other Levels are also represented in plate xiv.</p></div1><div1 part="N" n="LEVELLING" org="uniform" sample="complete" type="entry"><head>LEVELLING</head><p>, the art or act of finding a line parallel to the horizon at one or more stations, to determine the height or depth of one place with respect to another; for laying out grounds even, regulating descents, draining morasses, conducting water, &c.</p><p>Two or more places are on a true level when they are equally distant from the centre of the earth. Also one place is higher than another, or out of level with it, when it is farther from the centre of the earth: and a line equally distant from that centre in all its points, is called the <hi rend="italics">line of true level.</hi> Hence, because the earth is round, that line must be a curve, and make a part of the earth's circumference, or at least parallel to it, or concentrical with <figure/> it; as the line BCFG, which has all its points equally distant from A the centre of the earth; considering it as a persect globe.</p><p>But the line of sight BDE &c given by the operations of levels, is a tangent, or a right line perpendicular to the semidiameter of the earth at the point of contact B, rising always higher above the true line of level,<pb n="19"/><cb/> the farther the distance is, is called the <hi rend="italics">apparent line of level.</hi> Thus, CD is the height of the apparent level above the true level, at the distance BC or BD; also EF is the excess of height at F; and GH at G; &c. The difference, it is evident, is always equal to the excess of the secant of the arch of distance above the radius of the earth.</p><p>The common methods of levelling are sufficient for laying pavements of walks, or for conveying water to small distances, &c: but in more extensive operations, as in levelling the bottoms of canals, which are to convey water to the distance of many miles, and such like, the difference between the true and the apparent level must be taken into the account.</p><p>Now the difference CD between the true and apparent level, at any distance BC or BD, may be found thus: By a well known property of the circle ; or because the diameter of the earth is so great with respect to the line CD at all distances to which an operation of levelling commonly extends, that 2AC may be safely taken for in that proportion without any sensible error, it will be which therefore is or nearly; that is, the difference between the true and apparent level, is equal to the square of the distance between the places, divided by the diameter of the earth; and consequently it is always proportional to the square of the distance.</p><p>Now the diameter of the earth being nearly 7958 miles; if we first take BC = 1 mile, then the excess becomes of a mile, which is 7.962 inches, or almost 8 inches, for the height of the apparent above the true level at the distance of one mile. Hence, proportioning the excesses in altítude according to the squares of the distances, the following Table is obtained, shewing the height of the apparent above the true level for every 100 yards of distance on the one hand, and for every mile on the other. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Dist. or BC</cell><cell cols="1" rows="1" role="data">Dif. of Level, or CD</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Yards</cell><cell cols="1" rows="1" role="data">Inches</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">0.026</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">0.103</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">0.231</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">0.411</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">0.643</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">0.925</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">700</cell><cell cols="1" rows="1" role="data">1.260</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">800</cell><cell cols="1" rows="1" role="data">1.645</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">900</cell><cell cols="1" rows="1" role="data">2.081</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">2.570</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1100</cell><cell cols="1" rows="1" role="data">3.110</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1200</cell><cell cols="1" rows="1" role="data">3.701</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1300</cell><cell cols="1" rows="1" role="data">4.344</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1400</cell><cell cols="1" rows="1" role="data">5.038</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1500</cell><cell cols="1" rows="1" role="data">5.784</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1600</cell><cell cols="1" rows="1" role="data">6.580</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1700</cell><cell cols="1" rows="1" role="data">7.425</cell></row></table> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Dist. or BC</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dif. of Level, or CD</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Miles</cell><cell cols="1" rows="1" role="data">Feet</cell><cell cols="1" rows="1" role="data">Inc.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1/4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0 1/2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1/2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2 </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3/4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4 1/2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">112</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" role="data">1</cell></row></table></p><p>By means of these Tables of reductions, we can now<cb/> level to almost any dìstance at one operation, which the ancients could not do but by a great multitude; for, being unacquainted with the correction answering to any distance, they only levelled from one 20 yards to another, when they had occasion to continue the work to some considerable extent.</p><p>This table will answer several useful purposes. Thus, first, to find the height of the apparent level above the true, at any distance. If the given distance be contained in the table, the correction of level is found on the same line with it: thus at the distance of 1000 yards, the correction is 2.57, or two inches and a half nearly; and at the distance of 10 miles, it is 66 feet 4 inches. But if the exact distance be not found in the table, then multiply the square of the distance in yards by 2.57, and divide by 1000000, or cut off 6 places on the right for decimals; the rest are inches: or multiply the square of the distance in miles by 66 feet 4 inches, and divide by 100. 2ndly, To find the extent of the visible horizon, or how far can be seen from any given height, on a horizontal plane, as at sea, &c. Suppose the eye of an observer, on the top of a ship's mast at sea, be at the height of 130 feet above the water, he will then see about 14 miles all around. Or from the top of a cliff by the sea-side, the height of which is 66 feet, a person may see to the distance of near 10 miles on the surface of the sea. Also, when the top of a hill, or the light in a lighthouse, or such like, whose height is 130 feet, first comes into the view of an eye on board a ship; the table shews that the distance of the ship from it is 14 miles, if the eye be at the surface of the water; but if the height of the eye in th<*> ship be 80 feet, then the distance will be increased by near 11 miles, making in all about 25 miles, distance.</p><p>3dly, Suppose a spring to be on one side of a hill, and a house on an opposite hill, with a valley between them; and that the spring seen from the house appears by a levelling instrument to be on a level with the foundation of the house, which suppose is at a mile distance from it; then is the spring 8 inches above the true level of the house; and this difference would be barely sufficient for the water to be brought in pipes from the spring to the house, the pipes being laid all the way in the ground.</p><p>4th, If the height or distance exceed the limits of the table: Then, first, if the distance be given, divide it by 2, or by 3, or by 4, &c, till the quotient come within the distances in the table; then take out the height answering to the quotient, and multiply it by the square of the divisor, that is by 4, or 9, or 16, &c, for the height required: So if the top of a hill be just seen at the distance of 40 miles; then 40 divided by 4 gives 10, to which in the table answers 66 1/3 feet, which being multiplied by 16, the square of 4, gives 1061 1/3 feet for the height of the hill. But when the height is given, divide it by one of these square numbers 4, 9, 16, 25, &c, till the quotient come within the limits of the table, and multiply the quotient by the square root of the divisor, that is by 2, or 3, or 4, or 5, &c, for the distance sought: So when the top of the pike of Teneriff, said to be almost 3 miles or 15840 feet high, just comes into view at sea; divide 15840 by 225, or the square of 15, and the quotient<pb n="20"/><cb/> is 70 nearly; to which in the table answers, by proportion, nearly 10 2/7 miles; then multiplying 10 2/7 by 15, gives 154 miles and 2/7, for the distance of the hill. <hi rend="center"><hi rend="italics">Of the Practice of Levelling.</hi></hi> <figure/></p><p>The operation of Levelling is as follows. Supposc the height of the point A on the top of a mountain, above that of B, at the foot of it, be required. Place the level about the middle distance at D, and set up pickets, poles, or staffs, at A and B, where persons must attend with signals for raising and lowering, on the said poles, little marks of pasteboard or other matter. The level having been placed horizontally by the bubble, &c, look towards the staff AE, and cause the person there to raise or lower the mark, till it appear through the telescope, or sights, &c, at E: then measure exactly the perpendicular height of the point E above the point A, which suppose 5 feet 8 inches, set it down in your book. Then turn your view the other way, towards the pole B, and cause the person there to raise or lower his mark, till it appear in the visual line as before at C; and measuring the height of C above B, which suppose 15 feet 6 inches, set this down in your book also, immediately above the number of the first observation. Then subtract the one from the other, and the remainder 9 feet 10 inches, will be the difference of level between A and B, or the height of the point A above the point B.</p><p>If the point D, where the instrument is sixed, be exactly in the middle between the points A and B, there will be no necessity for reducing the apparent level to the true one, the visual ray on both sides being raised equally above the true level. But if not, each height must be corrected or reduced according to its distance, before the one corrected height is subtracted from the other; as in the case following. <figure/></p><p>When the distance is very considerable, or irregular, so that the operation cannot be effected at once placing of the level; or when it is required to know if there be a sufficient descent for conveying water from the spring A to the point B; it will be necessary to perform this at several operations. Having chosen a proper place for the first station, as at I, fix a pole at the point A near the spring, with a proper mark to slide <*>p and down it, as L; and measure the distance from<cb/> A to I. Then the level being adjusted in the point let the mark L be raised or lowered till it is seen through the telescope or sights of the level, and measure the height AL. Then having fixed another pole at H, direct the level to it, and cause the mark G to be moved up or down till it appear through the instrument: then measure the height GH, and the distance from I to H; noting them down in the book. This done, remove the level forwards to some other eminence as E, from whence the pole H may be viewed, as also another pole at D; then having adjusted the level in the point E, look back to the pole H; and managing the mark as before, the visual ray will give the point F; then measuring the distance HE and the height HF, note them down in the book. Then, turning the level to look at the next pole D, the visual ray will give the point D; there measure the height of D, and the distance EB, entering them in the book as before. And thus proceed from one station to another, till the whole is completed.</p><p>But all these heights must be corrected or reduced by the foregoing table, according to their respective distances; and the whole, both distances and heights, with their corrections, entered in the book in the following manner. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data"><hi rend="italics">Back-sights.</hi></cell><cell cols="1" rows="1" rend="colspan=3" role="data"><hi rend="italics">Fore-sights.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Dists.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Hts.</cell><cell cols="1" rows="1" role="data">Cors.</cell><cell cols="1" rows="1" role="data">Dists.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Hts.</cell><cell cols="1" rows="1" role="data">Cors.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">yds</cell><cell cols="1" rows="1" role="data">ft</cell><cell cols="1" rows="1" role="data">in.</cell><cell cols="1" rows="1" role="data">inc.</cell><cell cols="1" rows="1" role="data">yds</cell><cell cols="1" rows="1" role="data">ft</cell><cell cols="1" rows="1" role="data">in.</cell><cell cols="1" rows="1" role="data">inc.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">IA 1650</cell><cell cols="1" rows="1" rend="align=right" role="data">AL 11</cell><cell cols="1" rows="1" rend="align=right" role="data">3  </cell><cell cols="1" rows="1" rend="align=center" role="data">7.0</cell><cell cols="1" rows="1" rend="align=right" role="data">IH 1265</cell><cell cols="1" rows="1" rend="align=right" role="data">HG 19</cell><cell cols="1" rows="1" rend="align=right" role="data">5  </cell><cell cols="1" rows="1" role="data">4.0</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">EH   940</cell><cell cols="1" rows="1" rend="align=right" role="data">HF 10</cell><cell cols="1" rows="1" rend="align=right" role="data">7  </cell><cell cols="1" rows="1" rend="align=center" role="data">2.2</cell><cell cols="1" rows="1" rend="align=right" role="data">EB   900</cell><cell cols="1" rows="1" rend="align=right" role="data">BD   8</cell><cell cols="1" rows="1" rend="align=right" role="data">1  </cell><cell cols="1" rows="1" role="data">2.1</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2590</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">10  </cell><cell cols="1" rows="1" rend="align=center" role="data">9.2</cell><cell cols="1" rows="1" rend="align=right" role="data">2165</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">6  </cell><cell cols="1" rows="1" role="data">6.1</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9.2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2590</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6.1</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">0.8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">Dist. 4755</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">11.9</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">0.8</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">Whole Dif. of</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">level</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">11.1</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>Having summed up all the columns, add those of the distances together, and the whole distance from A to B is 4755 yards, or 2 miles and 3 quarters nearly. Then, the sums of the corrections taken from the sums of the apparent heights, leave the two corrected heights; the one of which being taken from the other, leaves 5 feet 11.1 inc. for the true difference of level sought between the two places A and B, which is at the rate of an inch and half nearly to every 100 yards, a quantity more than sufficient to cause the water to run from the spring to the house.</p><p>Or, the operation may be otherwise performed, thus: Instead of placing the level between every two poles, and taking both back-sights and fore-sights; plant it first at the spring A, and from thence observe the level to the first pole; then remove it to this pole, and observe the 2d pole; next move it to the 2d pole, and observe the 3d pole; and so on, from one pole to another, always taking foreward sights or observations only. And then at the last, add all the corrected heights to-<pb n="21"/><cb/> gether, and the sum will be the whole difference of level ought.</p><p>Dr. Halley suggested a new method of levelling performed wholly by means of the barometer, in which the mercury is found to be suspended at so much the less height, as the place is farther remote from the centre of the earth; and hence the different heights of the mercury in two places give the difference of level. This method is, in fact, no other than the method of measuring altitudes by the barometer, which has lately been so successfully practised and perfected by M. De Luc and others; but though it serves very well for the heights of hills, and other considerable altitudes, it is not accurate enough for determining small altitudes, to inches and parts. See the Barometrical Measurement of Altitudes.</p><p><hi rend="smallcaps">Levelling</hi> <hi rend="italics">Poles,</hi> or <hi rend="italics">Staves,</hi> are instruments used in levelling, serving to carry the marks to be observed, and at the same time to measure the heights of those marks from the ground. They usually consist each of two long wooden rulers, made to slide over each other, and divided into feet and inches, &c.</p></div1><div1 part="N" n="LEVER" org="uniform" sample="complete" type="entry"><head>LEVER</head><p>, a straight bar of iron or wood, &c, sup posed to be inflexible, supported on a fulcrum or prop by a single point, about which all the parts are moveable.</p><p>The Lever is the first of those simple machines called <hi rend="italics">mechanical powers,</hi> as being the simplest of them all; and is chiefly used for raising great weights to small heights. <figure/></p><p>The Lever is of three kinds. First the common sort, where the weight intended to be raised is at one end of it, our strength or another weight called the power is at the other end, and the prop or fulcrum is between them both. In stirring up the fire with a poker, we make use of this Lever; the poker is the Lever, it rests upon one of the bars of the grate as a prop, the incumbent fire is the weight to be overcome, and the pressure of the hand on the other end is the force or power. In this, as in all the other machines, we have only to increase the distance between the force and the prop, or to decrease the distance between the weight and the prop, to give the operator the greater power or effect. To this kind of Lever may also be referred all scissais, pincers, snuffers, &c. The steel-yard and the common balance are also Levers of this kind. <figure/></p><p>In the Lever of the 2d kind the prop is at one end, the force or power at the other, and the weight to be<cb/> raised is between them. Thus, in raising a waterplug in the streets, the workman puts his iron bar or Lever through the ring or hole of the plug, till the end of it reaches the ground on the other side; then making that the prop, he lifts the plug with his force or strength at the other end of the Lever. In this Lever too, the nearer the weight is to the prop, or the farther the power from the prop, the greater is the effect. To this 2d kind of Lever may also be referred the oars and rudder of a boat, the masts of a ship, cutting knives fixed at one end, and doors, whose hinges serve as a fulcrum.</p><p>In the Lever of the third kind, the power acts between the weight and the prop; such as a ladder raised by a man somewhere between the two ends, to rear it against a wall, or a pair of tongs, &c. <figure/></p><p>It is by this kind of Lever too that the muscular motions of animals are performed, the muscles being inserted much nearer to the centre of motion, than the point where is placed the centre of gravity of the weight to be raised; so that the power of the muscle is many times greater than the weight it is able to sustain. And in this third kind of Lever, to produce a balance between the power and weight, the power or force must <hi rend="italics">exceed</hi> the weight, in the same proportion as it is nearer the prop than the weight is; whereas in the other two kinds, the power is less than the weight, in the same proportion as its distance is greater; that is, universally, the power and weight are each of them reciprocally as their distance from the prop; as is demonstrated below.</p><p>Some authors make a 4th sort of what is called a bended Lever; such as a hammer in drawing a nail, &c.</p><p>In all Levers, the universal property is, that the effect of either the weight or the power, to turn the Lever about the fulcrum, is directly as its intensity and its distance from the prop, that is as <hi rend="italics">di,</hi> where <hi rend="italics">d</hi> denotes the distance, and <hi rend="italics">i</hi> the intensity, strength, or weight, &c, of the agent. For it is evident that at a double distance it will have a double effect, at a triple distance a triple effect, and so on; also that a double intensity produces a double effect, a triple a triple, and so on: therefore universally the effect is as <hi rend="italics">di</hi> the product of the two. In like manner, if D be the distance of another power or agent, whose intensity is I, then is DI the effect of this also to move the Lever. And if these two agents act against each other on the Lever, and their effects be supposed equal, or the Lever kept in equilibrío by the equal and contrary effects of these two agents; th en is , which equation resolves into this analogy, viz, ; that is, the distances of the agents from the prop, are reciprocally<pb n="22"/><cb/> er inversely as their intensities, or the power is to the weight, as the distance of the latter is to the distance of the former.</p><p>Writers on mechanics commonly demonstr<*>te this proportion in a very absurd manner, viz, by supposing the Lever put into motion about the prop, and then inferring that, because the momenta of two bodies are equal, when placed upon the Lever at such distances, that these distances are reciprocally proportional to the weights of the bodies, that therefore this is also the proportion in case of an equilibrium; which is an attempt absurdly to demonstrate a thing supposing the contrary, that a body is at rest, by supposing it to be in motion. I shall therefore give here a new and universal demonstration of the property, on the pure prineiples of rest and pressure, or force <figure/> only. Thus, let PW be a lever, C the prop, and P and W any two forces acting on the lever at the points P and W, in the directions PO, WO; then if CE and CD be the perpendicular distances of the directions of these forces from the prop C, it is to be demonstrated that . In order to which join CO, and draw CB parallel to WO, and CF parallel to PO. Then will CO be the direction of the pressure on the prop, otherwise there could not be an equilibrium, for the directions of three forces that keep each other in equilibrium, must necessarily meet in the same point. And because any three forces that keep each other in equilibrium, are proportional to the three sides of a triangle formed by drawing lines parallel to the directions of these forces; therefore the forces on P, C, and W, are as the three lines BO, CO, CB, which are in the same direction, or parallel to them; that is the force P is to the force W, as BO or its equal CF is to CB. But the two triangles CDF, CEB are similar, and have their like sides proportional, viz, ; and because it was ; theresore by equality ; that is, each force is reciprocally proportional to the distance of its direction from the fulcrum. And it will be found that this demonstration will serve also for the other kinds of Levers, by drawing the lines as directed. Hence if any given force P be applied to a Lever at A; its effect upon the Lever, to turn it about the centre of motion C, is as the length of the arm CA, and the sine of the angle of direction CAE. For the perp. CE is as CA × sin. [angle]A.</p><p>In any analogy, because the product of the extremes is equal to that of the means; therefore the product of the power by the distance of its direction is equal to the product of the weight by the distance of its direction. That is, .</p><p>If the Lever, with the two weights fixed to it, be made to move about the centre C; the momentum of the power will be equal to that of the weight; and the weights will be reciprocally proportional to their velocities. <figure/><cb/></p><p>When the two forces act perpendicularly on the Lever, as two weights &c; then, in case of an equilibrium, E coincides with P, and D with W; and the distances CP, CW, taken on the Lever, or the distances of the power and weight, from the fulcrum, are reciprocally proportional to the power and weight.</p><p>In a straight Lever, kept in equilibrio by a weight and power acting perpendicularly upon it; then, of these three, the power, weight, and pressure on the prop, any one is as the distance of the other two.</p><p>And hence too , and ; that is, the sum of the weights is to either of them, as the sum of their distances is to the distance of the other. <figure/></p><p>Also, if several weights P, Q, R, S, &c, act on a straight Lever, and keep it in equilibrio; then the sum of the products on one side of the prop, will be equal to the sum on the other side, made by multiplying each weight by its distance from the prop; viz,</p><p>Hitherto the Lever has been considered as a mathematical line void of weight or gravity. But when its weight is considered, it is to be done thus: Find the weight and the centre of gravity of the Lever alone, and then consider it as a mathematical line, but having an equal weight suspended by that centre of gravity; and so combine its effect with those of the other weights, as above.</p><p>Upon the foregoing principles depends the nature of scales and beams for weighing all bodies. For, if the distances be equal, then will the weights be equal also; which gives the construction of the common scales. And the Roman statera, or steel-yard, is also a Lever, but of unequal arms or distances, so contrived that one weight only may serve to weigh a great many, by sliding it backwards and forwards to different distances upon the longer arm of the Lever. See <hi rend="smallcaps">Balance</hi>, &c.</p><p>Also upon the principle of the Lever depends almost all other mechanical powers and effects. See <hi rend="smallcaps">Wheel-and-axle, Pulley, Wedge, Screw</hi>, &c.</p></div1><div1 part="N" n="LEVITY" org="uniform" sample="complete" type="entry"><head>LEVITY</head><p>, the privation or want of weight in any body, when compared with another that is heavier; and in this sense it is opposed to gravity. Thus cork, and most sorts of wood that float in water, have Levity with respect to water, that is, are less heavy. The schools maintained that there is such a thing as positive and absolute Levity; and to this they imputed the rise and buoyancy of bodies lighter in specie than the bodies in which they rise and float. But it is now well known that this happens only in consequence of the heavier and denser fluid, which, by its superior gravity, gains the lowest place, and raises up the lighter body by a force which is equal to the difference of their gravities. It was demonstrated by Archimedes, that a solid body will float any where in a fluid of the same specisic<pb n="23"/><cb/> gravity; and that a lighter body will always be raised up in it.</p></div1><div1 part="N" n="LEUWENHOEK" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LEUWENHOEK</surname> (<foreName full="yes"><hi rend="smallcaps">Antony</hi></foreName>)</persName></head><p>, a celebrated Dutch philosopher, was born at Delft in 1632; and acquired a great reputation throughout all Europe, by his experiments and discoveries in Natural History, by means of the microscope. He particularly excelled in making glasses for microscopes and spectacles; and he was a member of most of the literary societies of Europe; to whom he sent many memoirs. Those in the Philosophical Transactions, and in the Paris Memoirs, extend through many volumes; the former were extracted, and published at Leyden, in 1722. He died in 1723, at 91 years of age.</p><p>LEYDEN <hi rend="smallcaps">Phial</hi>, in Electricity, is a glass phial or jar, coated both within and without with tin foil, or some other conducting substance, that it may be charged, and employed in a variety of useful and entertaining experiments. Or even flat glass, or any other shape, so coated and used, has also received the same denomination. Also a vacuum produced in such a jar, &cs has been named the <hi rend="italics">Leyden Vacuum.</hi></p><p>The Leyden Phial has been so called, because it is said that M. Cunæus, a native of Leyden, first contrived, about the close of the year 1745, to accumulate the electrical power in glass, and use it in this way. Bu<*> Dr. Priestley asserts that this discovery was first made by Von Kleist, dean of the cathedral in Camin; who, on the 4th of November 1745, sent an account of it to Dr. Lìeberkuhn at Berlin: however, those to whom Kleist's account was communicated, could not succeed in performing his experiments. The chief cireumstances of this discovery are stated by Dr. Priestley in the following manner.</p><p>Professor Musschenbroek and his friends, observing that electrified bodies, when exposed to the common atmosphere, which is always replete with conducting particles of various kinds, soon lost the most part of their electricity, imagined that if the electrified bodies should be terminated on all sides by original electrics, they might be capable of receiving a stronger power, and retaining it a longer time, Glass being the most convenient electric for this purpose, and water the most convenient non-electric, they at first made these experiments with water in glass bottles; but no considerable discovery was made, till M. Cunæus, happening to hold his glass vessel in one hand, containing water, which had a communication with the prime conductor by means of a wire; and with the other hand disengaging it from the conductor, when he supposed the water had received as much electricity as the machine could give it, was surprised by a sudden and unexpected shock in his arms and breast. This experiment was repeated, and the first accounts of it published in Holland by Messrs. Allamand and Musschenbroek; by the Abbé Nollet and M. Monnier, in France; and by Messrs. Gralath and Rugger, in Germany. M. Gralath contrived to increase the strength of the shock, by altering the shape and size of the phial, and also by charging several phials at the same time, so as to form what is now called the <hi rend="italics">electrical battery.</hi> He likewise made the shock to pass through a number of persons connected in a circuit from the outside to the inside of the phial. He also observed that a cracked phial would not re-<cb/> ceive a charge: and he discovered what is now called the <hi rend="italics">Residuum of a charge.</hi></p><p>Dr. Watson, about this time, observed a circumstance attending the operation of charging the phial, which, if pursued, might have led him to the discovery which was afterwards made by Dr. Franklin. He says, that when the phial is well electrified, and you apply your hand to it, you see the fire flash from the outside of the glass, wherever you touch it, and it crackles in your hand. He also observed, that when a single wire only was fastened about a phial, properly silled with warm water, and charged; upon the instant of its explosion, the electrical corruscations were seen to dart from the wire, and to illuminate the water contained in the phial. He likewise found that the stroke, in the discharge of the phial, was, <hi rend="italics">cateris paribus,</hi> as the points of contact of the non-electrics of the outside of the glass; which led to the method of co<*>ting glass: in consequence of which he made experiments, from whence he concluded, that the effect of the Leyden phial was greatly increased by, if not chiefly owing to, the number of points of non-electric in contact within the glass, and the density of the matter of, which these points consisted; provided the matter was, in its own nature, a ready conductor of electricity. He farther observed, that the explosion was greater from hot water inclosed in glasses, than from cold, and from his coated jars warmed, than when cold.</p><p>Mr. Wilson, in 1746, discovered a method of giving the shock to any particular part of the body, without affecting the rest. He also increased the strength of the shock by plunging the phial in water, which gave it a coat of water on the outside as high as it was silled within. He likewise found, that the law of accumulation of the electric matter in the Leyden phial, was always in proportion to the thinness of the glass, the surface of the glass, and that of the non-electrics in contact with its outside and inside. He made also a variety of other experiments with the Leyden phial, too long here to be related.</p><p>Mr. Canton found, that when a charged phial was placed upon electrics, the wire and coating would give a spark or two alternately, and that by a continuance of the operation the phial would be discharged; though he did not observe that these alternate sparks proceeded from the two contrary electricities discovered by Dr. Franklin.</p><p>The Abbé Nollet made several experiments with this phial. He received a shock from one, out of which the air had been exhausted, and into which the end of his conductor had been inserted. He ascribed the force of the glass, in giving a shock, to that property of it, by which it retains it more strongly than conductors do, and is not so easily divested of it as they are. It was he also who first tried the effect of the electric shock on brute animals: and he enlarged the circuit of its conveyance.</p><p>M. Monnier, it has been said, was the first who disvered that the Leyden phial would retain its electricity for a considerable time after, it was charged; and that in time of frost he found it continued for 36 hours. It is remarkable too that both the French and English philosophers made several experiments, which, with a small degree of attention, would have led them to the discovery of the different qualities of the electricity on<pb n="24"/><cb/> the contrary sides of the glass. But this discovery was reserved for the ingenious Dr. Franklin; who, in explaining the method of charging the Leyden phial, observes, that when one side of the glass is electrified plus, or positively, the other side is electrified minus, or negatively: so that whatever quantity of sire is thrown upon one side of the glass, the same quantity is drawn out of the other; and in an uncharged phial, none can be thrown into the inside, when none can be taken from the outside; and that there is really no more electric fire in the phial after it is charged than before; all that can be done by charging, being only to take from one side, and convey to the other. Dr. Franklin also observed that glass was not impervious to electricity, and that as the equilibrium could not be restored to the charged phial by any internal communication, it must necessarily be done by conductors externally joining the inside and the outside. These capital discoveries he made by observing, that when a phial was charged, a cork ball suspended by silk, was attracted by the outside coating, when it was repelled by a wire communicating with the inside, and <hi rend="italics">vice versa.</hi> But the truth of this principle appeared more evident, when he brought the knob of the wire, communicating with the outside coating, within a few inches of the wire communicating with the inside coating, and suspended a cork ball between them; for then the ball was attracted by them alternately, till the phial was discharged.</p><p>Dr. Franklin also shewed, that when the phial was charged, one side lost exactly as much as the other gained, in restoring the equilibrium. Hanging a fine linen thread near the coating of an electrical phial, he observed that whenever he brought his finger near the wire, the thread was attracted by the coating; for as the fire was drawn from the inside by touching the wire, the outside drew in an equal quantity by the thread. He likewise proved, that the coating on one side of a phial received just as much electricity, as was emitted from the discharge of the other, and that in the following manner:—He insulated his rubber, and then hanging a phial to his conductor, he found it could not be charged, even when his hand was held constantly to it; because, though the electric fire might leave the outside of the phial, there was none collected by the rubber to be conveyed to the inside. He then took away his hand from the phial, and forming a communication by a wire from the outside coating to the insulated rubber, he found that it was charged with ease. In this case it was plain, that the very same fire which left the outside coating, was conveyed to the inside by the way of the rubber, the globe, the conductor, and the wire of the phial. This new theory of charging the Leyden phial, led Dr. Franklin to observe a greater variety of facts, relating both to the charging and discharging it, than other philosophers had attended to. And this maxim, that it takes in at one surface, what it loses at the other, led Dr. Franklin to think of charging several phials together with the same trouble, by connecting the outsrde of one with the inside of another; by which the fire that was driven out of the first would be received by the second, &c. By this means he found, that a great number of jars might be charged with the same labour as one only; and tha<*> they might be charged equally<cb/> high, were it not that every one of them receives the new fire, and loses its old, with some reluctance, or rather that it gives some small resistance to the charging. And on this principle he first constructed an electrical battery.</p><p>When Dr. Franklin first began his experiments on the Leyden phial, he imagined that the electric fire was all crowded into the substance of the non-electric, in contact with the glass. But he afterwards found, that its power of giving a shock lay in the glass itself, and not in the coating, by the following ingenious analysis of the phial. To find where the strength of the charged bottle lay, having placed it upon a glass, he first took out the cork and the wire; but not finding the virtue in them, he touched the outside coating with one hand, and put a finger of the other into the mouth of the bottle; when the shock was felt quite as strong as if the cork and wire had been in it. He then charged the phial again, and pouring out the water into an empty bottle which was insulated, he expected that if the force resided in the water, it would give the shock; but he found it gave none. He therefore concluded that the electric sire must either have been lost in decanting, or must remain in the bottle; and the latter he found to be true; for, upon filling the charged bottle with fresh water, he found the shock, and was satissied that the power of giving it resided in the glass itsels. The same experiment was made with panes of glass, laying the coating on lightly, and charging it, as the water had been before charged in the bottle, when the result was precisely the same. He also proved in other ways that the electric sire resided in the glass. See Franklin's Letters and Observations, &c. Also Priestley's Hist. of Electricity, vol. i, pa. 191, &c.</p><p>From this account of Dr. Franklin's method of analyzing the Leyden phial, the manner of charging and discharging it, with the reason of the process, are easily understood. Thus, placing a coated phial near the prime conductor, so that the knob of its wire may be in contact with it; then upon turning the winch of the machine, the index of the electrometer, E, fixed to the conductor, will gradually rise as far as 90° nearly, and there rest; which shews that the phial has received its full charge: then holding the discharger by its glass handle, and applying one of its knobs to the outside coating of the phial, the other being brought near the knob of the wire, or near the prime conductor which communicates with it, a report will be heard, and luminous sparks will be seen between the discharger and the conducting substances communicating with the sides of the phial; and by this operation the phial will be discharged. But, instead of using the discharger, if a person touch the outside of the phial with one hand, and bring the other hand near the wire of the phial, the same spark and report will take place, and a shock will be felt, affecting the wrists and elbows, and the breast too when the shock is strong: a shock may also be given to any single part of the body, if that part alone be brought into the circuit. If a number of persons join hands, and the first of them touch the outside of the phial, while the last touches the wire communicating with the inside, they will all feel the shock at the same time. If the coated phial be held by the wire, and the outside coating be presented to the prime<pb n="25"/><cb/> conductor, it will be charged as readily; but only with this difference, that in this case the outside will be positive, and the inside negative; also if the prime conductor, by being connected with the rubber of the machine, be electrified negatively, the phial will be charged in the same manner; but the side that touches the conductor will be electrified negatively, and the opposite side will be electrified positively. But, by insulating the phial, and repeating the same process, the index of the electrometer will soon rise to 90°, yet the phial will remain uncharged; because the outside, having no communication with the earth, &c, cannot part with its own electricity, and therefore the inside cannot acquire an additional quantity: but when a chain, or any other conductor, connects the outside of the phial with the table, the phial may be charged as before. Moreover, if a phial be insulated, and one side of it, instead of being connected with the earth, be connected with the insulated rubber, whilst the other side communicates with the prime conductor, the phial will be expeditiously charged; because that whilst the rubber exhausts one side, the other side is supplied by the prime conductor; and thus the phial is charged with its own electricity; or the natural electric matter of one of its sides is thus thrown upon the other side. This last experiment may be diversified by insulating the phial, and placing it with its wire at the distance of about half an inch from the prime conductor, and holding the knob of another wire at the same distance from its outside <*>oating; then, upon turning the machine, a spark will be observed to proceed from the prime conductor to the wire of the phial, and another spark will pass at the same time from the outside coating to the knob of the wire presented towards it: and thus it appears that as a quantity of the electric matter is entering the inside of the phial, an equal quantity of it is leaving the outside. If the wire presented to the outside of the phial be pointed, it will be seen illuminated with a star; but if the pointed wire be connected with the coating of the phial, it will appear illuminated with a brush of rays. See <hi rend="italics">Charge, Electrical Shock, Experiments, &c.</hi></p><p>Mr. Cavallo has described the construction of a phial which, being charged by an electrical kite, in examining the state of the clouds, or in any other way, may be put into the pocket, and which will retain its charge for a considerable time. A phial of this kind has been kept in a charged state for six weeks. See his Electricity, pa. 340. Many other curious experiments with the Leyden phial may be seen in the books above cited, as also in the volumes of the Philos. Trans. and elsewhere. In this last-mentioned work, Mr. Cavallo describes a method of repairing coated phials that have cracked by any means. He first removes the outside coating from the fractured part, and then makes it moderately hot, by holding it to the flame of a candle; and whilst it remains hot, he applies burning sealing-wax to the part, so as to cover the fracture entirely; observing that the thickness of this wax coating may be greater than that of the glass. Lastly, he covers all the sealingwax, and also part of the surface of the glass beyond it, with a composition made with four parts of bees-wax, one of resin, one of turpentine, and a very little oil of olives; this being spread upon a piece of oiled silk, he applies it in the manner of a plaster. In this way seve-<cb/> ral phials have been so effectually repaired, that after being frequently charged, they were at last broken by a spontaneous discharge, but in a different part of the glass. Philos. Trans. vol. 68, pa. 1011.</p></div1><div1 part="N" n="LIBRA" org="uniform" sample="complete" type="entry"><head>LIBRA</head><p>, <hi rend="italics">Balance,</hi> one of the mechanical powers. See <hi rend="smallcaps">Balance.</hi></p><p><hi rend="smallcaps">Libra</hi> is also one of the 48 old constellations, and the 7th sign of the zodiac, being opposite to Aries, and marked like a part of a pair of scales, thus <*>. The figure of the balance was probably given to this part of the ecliptic, because when the sun arrives at this part, which is at the time of the autumnal equinox, the days and nights are equal, as if weighed in a balance.</p><p>The stars in this constellation are, according to Ptolomy 17, Tycho 10, Hevelius 20, and Flamsteed 51.</p><p><hi rend="smallcaps">Libra</hi> also denotes the ancient Roman pound, which was divided into 12 unciæ, or ounces, and the ounce into 24 scruples. It seems the mean weight of the scruple was nearly equal to 17 1/2 grains Troy, and consequently the libra, or pound, 5040 grains. It was also the name of a gold coin, equal in value to 20 denarii. See Philos. Trans. vol. 61, pa. 462.</p><p>The French livre is derived from the Roman libra, this being used in France for the proportions of their coin till about the year 1100, their sols being so proportioned as that 20 of them were equal to the libra. By degrees it became a term of account, and every thing of the value of 20 sols was called a livre.</p></div1><div1 part="N" n="LIBRATION" org="uniform" sample="complete" type="entry"><head>LIBRATION</head><p>, <hi rend="italics">of the Moon,</hi> is an apparent irregularity in her motion, by which she seems to librate, or waver, about her own axis, one while towards the east, and again another while towards the west. See <hi rend="smallcaps">Moon</hi>, and <hi rend="smallcaps">Evection.</hi> Hence it is that some parts near the moon's western edge at one time recede from the centre of the dise, while those on the other or eastern side approach nearer to it; and, on the contrary, at another time the western parts are seen to be nearer the centre, and the eastern parts farther from it: by which means it happens that some of those parts, which were before visible, set and hide themselves in the hinder or invisible side of the moon, and afterwards return and appear again on the nearer or visible side.</p><p>This Libration of the moon was first discovered by Hevelius, in the year 1654; and it is owing to her equable rotation round her own axis, once in a month, in conjunction with her unequal motion in the perimeter of her orbit round the earth. For if the moon moved in a circle, having its centre coinciding with the centre of the earth, whilst it turned on its axis in the precise time of its period round the earth, then the plane of the same lunar meridian would always pass through the earth, and the same face of the moon would be constantly and exactly turned towards us. But since the real motion of the moon is about a point considerably distant from the centre of the earth, that motion is very unequal, as seen from the earth, the plane of no one meridian constantly passing through the earth.</p><p>The Libration of the moon is of three kinds.</p><p>1st, Her libration in longitude, or a seeming to-andagain motion according to the order of the signs of the zodiac. This libration is nothing twice in each periodical month, viz, when the moon is in her apogeum, and when in her perigeum; for in both these cases the<pb n="26"/><cb/> plane of her meridian, which is turned towards us, is directed alike towards the earth.</p><p>2d, Her libration in latitude; which arises from hence, that her axis not being perpendicular to the plane of her orbit, but inclined to it, sometimes one of her poles and sometimes the other will nod, as it were, or dip a little towards the earth, and consequently she will appear to librate a little, and to shew sometimes more of her spots, and sometimes less of them, towards each pole. Which libration, depending on the position of the moon, in respect to the nodes of her orbit, and her axis being nearly perpendicular to the plane of the ecliptic, is very properly said to be in latitude. And this also is completed in the space of the moon's periodical month, or rather while the moon is returning again to the same position, in respect of her nodes.</p><p>3d, There is also a third kind of libration; by which it happens that although another part of the moon be not really turned to the earth, as in the former libration, yet another is illuminated by the sun. For since the moon's axis is nearly perpendicular to the plane of the ecliptic, when she is most southerly, in respect of the north pole of the ecliptic, some parts near to it will be illuminated by the sun; while, on the contrary, the south pole will be in darkness. In this case, therefore, if the sun be in the same line with the moon's southern limit, then, as she proceeds from conjunction with the sun towards her ascending node, she will appear to dip her northern polar parts a little into the dark hemisphere, and to raife her southern polar parts as much into the light one. And the contrary to this will happen two weeks after, while the new moon is descending from her northern limit; for then her northern polar parts will appear to emerge out of darkness, and the southern polar parts to dip into it. And this seeming libration, or rather these effects of the former libration in latitude, depending on the light of the sun, will be completed in the moon's synodical month. Greg. Astron. lib. 4, sect. 10.</p><p><hi rend="smallcaps">Libration</hi> <hi rend="italics">of the Earth,</hi> is a term applied by some astronomers to that motion, by which the earth is so retained in its orbit, as that its axis continues constantly parallel to the axis of the world.</p><p>This Copernicus calls the <hi rend="italics">motion of libration,</hi> which may be thus illustrated: Suppose a globe, with its axis parallel to that of the earth, painted on the flag of a mast, moveable on its axis, and constantly driven by an east wind, while it sails round an island, it is evident that the painted globe will be so librated, as that its axis will be parallel to that of the world, in every situation of the ship.</p><p>LIFE-ANNUITIES, are such periodical payments as depend on the continuance of some particular life or lives. They may be distinguished into Annuities that commence-immediately, and such as commence at some future period, called <hi rend="italics">reversionary life-annuities.</hi></p><p>The value, or present worth, of an annuity for any proposed life or lives, it is evident, depends on two cir-<cb/> cumstances, the interest of money, and the chance or expectation of the continuance of life. Upon the former only, it has been shewn, under the article A<hi rend="smallcaps">NNUITIES</hi>, depends the value or present worth of an annuity certain, or that is not subject to the continuance of a life, or other contingency; but the expectation of life being a thing not certain, but only possessing a certain chance, it is evident that the value of the certain annuity, as stated above, must be diminished in proportion as the expectancy is below certainty: thus, if the present value of an annuity certain be any sum, as suppose 100l. and the value or expectancy of the life be 1/2, then the value of the life-annuity will be only half of the former, or 50l; and if the value of the life be only 1/3, the value of the life-annuity will be but 1/3 of 100l, that is 33l. 6s. 8d; and so on.</p><p>The measure of the value or expectancy of life, depends on the proportion of the number of persons that die, out of a given number, in the time proposed; thus, if 50 persons die, out of 100, in any proposed time, then, half the number only remaining alive, any one person has an equal chance to live or die in that time, or the value of his life for that time is 1/2; but if 2/3 of the number die in the time proposed, or only 1/3 remain alive, then the value of any one's life is 1/3; and if 3/4 of the number die, or only 1/4 remain alive, then the value of any life is but 1/4; and so on. In these proportions then must the value of the annuity certain be diminished, to give the value of the like life annuity.</p><p>It is plain therefore that, in this business, it is necessary to know the value of life at all the different ages, from some table of observations on the mortality of mankind, which may shew the proportion of the persons living, out of a given number, at the end of any proposed time; or from some certain hypothesis, or assumed principle. Now various tables and hypotheses of this sort were given by the writers on this subject, as Dr. Halley, Mr. Demoivre, Mr. Thomas Simpson, Mr. Dodson, Mr. Kersseboom, Mr. Parcieux, Dr. Price, Mr. Morgan, Mr. Baron Maseres, and many others. But the same table of probabilities of life will not suit all places; for long experience has shewn that all places are not equally healthy, or that the proportion of the number of persons that die annually, is different for different places. Dr. Halley computed a table of the annual deaths as drawn from the bills of mortality of the city of Breslaw in Germany, Mr. Smart and Mr. Simpson from those of London, Dr. Price from those of Northampton, Mr. Kersseboom from those of the provinces of Holland and West-Friesland, and M. Parcieux from the lists of the French tontines, or long annuities, and all these are found to differ from one another. It may not therefore be improper to insert here a comparative view of the principal tables that have been given of this kind, as below, where the first column shews the age, and the other columns the number of persons living at that age, out of 1000 born, or of the age 0, in the first line of each column.<pb n="27"/><cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=12" role="data"><hi rend="italics">TABLE I.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=12" role="data"><hi rend="italics">Shewing the Number of Persons living at all Ages, out of</hi> 1000 <hi rend="italics">that had been born at several Places, viz.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Ages.</cell><cell cols="1" rows="1" role="data">Vienna.</cell><cell cols="1" rows="1" role="data">Berlin.</cell><cell cols="1" rows="1" role="data">London.</cell><cell cols="1" rows="1" role="data">Norwich.</cell><cell cols="1" rows="1" role="data">Northampton.</cell><cell cols="1" rows="1" role="data">Breslaw.</cell><cell cols="1" rows="1" role="data">Branden- burg.</cell><cell cols="1" rows="1" role="data">HolyCross.</cell><cell cols="1" rows="1" role="data">Holland.</cell><cell cols="1" rows="1" role="data">France.</cell><cell cols="1" rows="1" role="data">Vaud, Switzerland.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">542</cell><cell cols="1" rows="1" role="data">633</cell><cell cols="1" rows="1" role="data">680</cell><cell cols="1" rows="1" role="data">798</cell><cell cols="1" rows="1" role="data">738</cell><cell cols="1" rows="1" role="data">769</cell><cell cols="1" rows="1" role="data">775</cell><cell cols="1" rows="1" role="data">882</cell><cell cols="1" rows="1" role="data">804</cell><cell cols="1" rows="1" role="data">805</cell><cell cols="1" rows="1" role="data">811</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">471</cell><cell cols="1" rows="1" role="data">528</cell><cell cols="1" rows="1" role="data">548</cell><cell cols="1" rows="1" role="data">651</cell><cell cols="1" rows="1" role="data">628</cell><cell cols="1" rows="1" role="data">658</cell><cell cols="1" rows="1" role="data">718</cell><cell cols="1" rows="1" role="data">762</cell><cell cols="1" rows="1" role="data">768</cell><cell cols="1" rows="1" role="data">777</cell><cell cols="1" rows="1" role="data">765</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">430</cell><cell cols="1" rows="1" role="data">485</cell><cell cols="1" rows="1" role="data">492</cell><cell cols="1" rows="1" role="data">595</cell><cell cols="1" rows="1" role="data">585</cell><cell cols="1" rows="1" role="data">614</cell><cell cols="1" rows="1" role="data">687</cell><cell cols="1" rows="1" role="data">717</cell><cell cols="1" rows="1" role="data">736</cell><cell cols="1" rows="1" role="data">750</cell><cell cols="1" rows="1" role="data">735</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">434</cell><cell cols="1" rows="1" role="data">452</cell><cell cols="1" rows="1" role="data">566</cell><cell cols="1" rows="1" role="data">562</cell><cell cols="1" rows="1" role="data">585</cell><cell cols="1" rows="1" role="data">664</cell><cell cols="1" rows="1" role="data">682</cell><cell cols="1" rows="1" role="data">709</cell><cell cols="1" rows="1" role="data">727</cell><cell cols="1" rows="1" role="data">715</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">377</cell><cell cols="1" rows="1" role="data">403</cell><cell cols="1" rows="1" role="data">426</cell><cell cols="1" rows="1" role="data">544</cell><cell cols="1" rows="1" role="data">544</cell><cell cols="1" rows="1" role="data">563</cell><cell cols="1" rows="1" role="data">642</cell><cell cols="1" rows="1" role="data">659</cell><cell cols="1" rows="1" role="data">689</cell><cell cols="1" rows="1" role="data">711</cell><cell cols="1" rows="1" role="data">701</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">357</cell><cell cols="1" rows="1" role="data">387</cell><cell cols="1" rows="1" role="data">410</cell><cell cols="1" rows="1" role="data">526</cell><cell cols="1" rows="1" role="data">530</cell><cell cols="1" rows="1" role="data">546</cell><cell cols="1" rows="1" role="data">622</cell><cell cols="1" rows="1" role="data">636</cell><cell cols="1" rows="1" role="data">676</cell><cell cols="1" rows="1" role="data">697</cell><cell cols="1" rows="1" role="data">688</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">344</cell><cell cols="1" rows="1" role="data">376</cell><cell cols="1" rows="1" role="data">397</cell><cell cols="1" rows="1" role="data">511</cell><cell cols="1" rows="1" role="data">518</cell><cell cols="1" rows="1" role="data">532</cell><cell cols="1" rows="1" role="data">607</cell><cell cols="1" rows="1" role="data">618</cell><cell 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rows="1" role="data">627</cell><cell cols="1" rows="1" role="data">649</cell><cell cols="1" rows="1" role="data">643</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" role="data">347</cell><cell cols="1" rows="1" role="data">356</cell><cell cols="1" rows="1" role="data">464</cell><cell cols="1" rows="1" role="data">484</cell><cell cols="1" rows="1" role="data">492</cell><cell cols="1" rows="1" role="data">559</cell><cell cols="1" rows="1" role="data">577</cell><cell cols="1" rows="1" role="data">621</cell><cell cols="1" rows="1" role="data">644</cell><cell cols="1" rows="1" role="data">639</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">344</cell><cell cols="1" rows="1" role="data">351</cell><cell cols="1" rows="1" role="data">460</cell><cell 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rows="1" role="data">601</cell><cell cols="1" rows="1" role="data">626</cell><cell cols="1" rows="1" role="data">622</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" role="data">332</cell><cell cols="1" rows="1" role="data">334</cell><cell cols="1" rows="1" role="data">442</cell><cell cols="1" rows="1" role="data">459</cell><cell cols="1" rows="1" role="data">470</cell><cell cols="1" rows="1" role="data">535</cell><cell cols="1" rows="1" role="data">555</cell><cell cols="1" rows="1" role="data">596</cell><cell cols="1" rows="1" role="data">621</cell><cell cols="1" rows="1" role="data">618</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" role="data">329</cell><cell cols="1" rows="1" role="data">437</cell><cell 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rows="1" role="data">571</cell><cell cols="1" rows="1" role="data">598</cell><cell cols="1" rows="1" role="data">602</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">276</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">415</cell><cell cols="1" rows="1" role="data">426</cell><cell cols="1" rows="1" role="data">446</cell><cell cols="1" rows="1" role="data">512</cell><cell cols="1" rows="1" role="data">525</cell><cell cols="1" rows="1" role="data">566</cell><cell cols="1" rows="1" role="data">592</cell><cell cols="1" rows="1" role="data">597</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">273</cell><cell cols="1" rows="1" role="data">305</cell><cell cols="1" rows="1" role="data">305</cell><cell cols="1" rows="1" role="data">409</cell><cell 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role="data">36</cell><cell cols="1" rows="1" role="data">221</cell><cell cols="1" rows="1" role="data">237</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" role="data">338</cell><cell cols="1" rows="1" role="data">342</cell><cell cols="1" rows="1" role="data">370</cell><cell cols="1" rows="1" role="data">456</cell><cell cols="1" rows="1" role="data">454</cell><cell cols="1" rows="1" role="data">460</cell><cell cols="1" rows="1" role="data">514</cell><cell cols="1" rows="1" role="data">533</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">216</cell><cell cols="1" rows="1" role="data">230</cell><cell cols="1" rows="1" role="data">230</cell><cell cols="1" rows="1" role="data">333</cell><cell cols="1" rows="1" role="data">336</cell><cell cols="1" rows="1" role="data">363</cell><cell cols="1" rows="1" role="data">450</cell><cell cols="1" rows="1" role="data">447</cell><cell cols="1" rows="1" role="data">453</cell><cell cols="1" rows="1" role="data">508</cell><cell cols="1" rows="1" role="data">527</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">211</cell><cell cols="1" rows="1" role="data">223</cell><cell cols="1" rows="1" role="data">224</cell><cell cols="1" rows="1" role="data">327</cell><cell cols="1" rows="1" role="data">330</cell><cell cols="1" rows="1" role="data">356</cell><cell cols="1" rows="1" role="data">444</cell><cell cols="1" rows="1" role="data">440</cell><cell cols="1" rows="1" role="data">446</cell><cell cols="1" rows="1" role="data">503</cell><cell cols="1" rows="1" role="data">520</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">205</cell><cell cols="1" rows="1" role="data">216</cell><cell cols="1" rows="1" role="data">218</cell><cell cols="1" rows="1" role="data">322</cell><cell 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role="data">41</cell><cell cols="1" rows="1" role="data">194</cell><cell cols="1" rows="1" role="data">203</cell><cell cols="1" rows="1" role="data">207</cell><cell cols="1" rows="1" role="data">311</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">335</cell><cell cols="1" rows="1" role="data">427</cell><cell cols="1" rows="1" role="data">418</cell><cell cols="1" rows="1" role="data">425</cell><cell cols="1" rows="1" role="data">487</cell><cell cols="1" rows="1" role="data">500</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">189</cell><cell cols="1" rows="1" role="data">197</cell><cell cols="1" rows="1" role="data">201</cell><cell cols="1" rows="1" role="data">306</cell><cell cols="1" rows="1" role="data">303</cell><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" role="data">422</cell><cell cols="1" rows="1" role="data">410</cell><cell cols="1" rows="1" role="data">419</cell><cell cols="1" rows="1" role="data">482</cell><cell cols="1" rows="1" role="data">494</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">185</cell><cell cols="1" rows="1" role="data">192</cell><cell cols="1" rows="1" role="data">194</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">296</cell><cell cols="1" rows="1" role="data">321</cell><cell cols="1" rows="1" role="data">417</cell><cell cols="1" rows="1" role="data">401</cell><cell cols="1" rows="1" role="data">413</cell><cell cols="1" rows="1" role="data">476</cell><cell cols="1" rows="1" role="data">488</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" role="data">294</cell><cell 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role="data">46</cell><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" role="data">177</cell><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" role="data">281</cell><cell cols="1" rows="1" role="data">275</cell><cell cols="1" rows="1" role="data">299</cell><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">379</cell><cell cols="1" rows="1" role="data">393</cell><cell cols="1" rows="1" role="data">460</cell><cell cols="1" rows="1" role="data">469</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" role="data">172</cell><cell cols="1" rows="1" role="data">167</cell><cell cols="1" rows="1" role="data">274</cell><cell cols="1" rows="1" role="data">268</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">394</cell><cell cols="1" rows="1" role="data">372</cell><cell cols="1" rows="1" role="data">386</cell><cell cols="1" rows="1" role="data">455</cell><cell cols="1" rows="1" role="data">461</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">159</cell><cell cols="1" rows="1" role="data">167</cell><cell cols="1" rows="1" role="data">159</cell><cell cols="1" rows="1" role="data">268</cell><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" role="data">388</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data">378</cell><cell cols="1" rows="1" role="data">449</cell><cell cols="1" rows="1" role="data">451</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">153</cell><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" role="data">153</cell><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" role="data">254</cell><cell cols="1" rows="1" role="data">275</cell><cell cols="1" rows="1" role="data">381</cell><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" role="data">370</cell><cell cols="1" rows="1" role="data">443</cell><cell cols="1" rows="1" role="data">441</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" role="data">157</cell><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" role="data">255</cell><cell cols="1" rows="1" role="data">247</cell><cell cols="1" rows="1" role="data">267</cell><cell cols="1" rows="1" role="data">374</cell><cell cols="1" rows="1" role="data">353</cell><cell cols="1" rows="1" role="data">362</cell><cell cols="1" rows="1" role="data">436</cell><cell cols="1" rows="1" role="data">431</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" role="data">141</cell><cell cols="1" rows="1" role="data">248</cell><cell cols="1" rows="1" role="data">239</cell><cell cols="1" rows="1" role="data">259</cell><cell cols="1" rows="1" role="data">367</cell><cell cols="1" rows="1" role="data">347</cell><cell cols="1" rows="1" role="data">354</cell><cell cols="1" rows="1" role="data">429</cell><cell cols="1" rows="1" role="data">422</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">137</cell><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" role="data">242</cell><cell cols="1" rows="1" role="data">232</cell><cell cols="1" rows="1" role="data">250</cell><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" role="data">340</cell><cell cols="1" rows="1" role="data">345</cell><cell cols="1" rows="1" role="data">422</cell><cell cols="1" rows="1" role="data">414</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" role="data">235</cell><cell cols="1" rows="1" role="data">225</cell><cell cols="1" rows="1" role="data">241</cell><cell cols="1" rows="1" role="data">351</cell><cell cols="1" rows="1" role="data">333</cell><cell cols="1" rows="1" role="data">336</cell><cell cols="1" rows="1" role="data">414</cell><cell cols="1" rows="1" role="data">406</cell></row></table><pb n="28"/><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Ages.</cell><cell cols="1" rows="1" role="data">Vienna.</cell><cell cols="1" rows="1" role="data">Berlin.</cell><cell cols="1" rows="1" role="data">London.</cell><cell cols="1" rows="1" role="data">Norwich.</cell><cell cols="1" rows="1" role="data">Northampton.</cell><cell cols="1" rows="1" role="data">Breslaw.</cell><cell cols="1" rows="1" role="data">Branden- burg.</cell><cell cols="1" rows="1" role="data">HolyCross.</cell><cell cols="1" rows="1" role="data">Holland.</cell><cell cols="1" rows="1" role="data">France.</cell><cell cols="1" rows="1" role="data">Vaud, Switzerland.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">137</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">228</cell><cell cols="1" rows="1" role="data">218</cell><cell cols="1" rows="1" role="data">232</cell><cell cols="1" rows="1" role="data">343</cell><cell cols="1" rows="1" role="data">326</cell><cell cols="1" rows="1" role="data">327</cell><cell cols="1" rows="1" role="data">406</cell><cell cols="1" rows="1" role="data">397</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">123</cell><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data">221</cell><cell cols="1" rows="1" role="data">211</cell><cell cols="1" rows="1" role="data">224</cell><cell cols="1" rows="1" role="data">334</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">397</cell><cell cols="1" rows="1" role="data">388</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">117</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">116</cell><cell cols="1" rows="1" role="data">213</cell><cell cols="1" rows="1" role="data">204</cell><cell cols="1" rows="1" role="data">216</cell><cell cols="1" rows="1" role="data">324</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" role="data">388</cell><cell cols="1" rows="1" role="data">377</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data">206</cell><cell cols="1" rows="1" role="data">197</cell><cell cols="1" rows="1" role="data">209</cell><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" role="data">301</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">379</cell><cell cols="1" rows="1" role="data">364</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" role="data">115</cell><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" role="data">199</cell><cell cols="1" rows="1" role="data">190</cell><cell cols="1" rows="1" role="data">201</cell><cell cols="1" rows="1" role="data">304</cell><cell cols="1" rows="1" role="data">292</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">369</cell><cell cols="1" rows="1" role="data">348</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">101</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">101</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data">183</cell><cell cols="1" rows="1" role="data">193</cell><cell cols="1" rows="1" role="data">293</cell><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" role="data">282</cell><cell cols="1" rows="1" 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role="data">14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">5</cell></row></table><cb/></p><p>These tables shew that the mortality and chance of life are very various in different places; and that therefore, to obtain a sufficient accuracy in this business, it is necessary to adapt a table of probabilities or chances of life, to every place for which annuities are to be calculated; or at least one set of tables for large towns, and another for country places, as well as for the supposition of different rates of interest.</p><p>Several of the foregoing tables, as they commenced with numbers different from one another, are here reduced to the same number at the beginning, viz, 1000 persons, by which means we are enabled by inspection, at any age, to compare the numbers together, and immediately perceive the relative degrees of vitality at the several places. The tables are also arranged according to the degree of vitality amongst them; the least, or that at Vienna, first; and the rest in their order, to the highest, which is the province of Vaud in Switzerland. The authorities upon which these tables de-<cb/> pend, are as they here follow. The first, taken from Dr. Price's Observations on Reversionary payments, is formed from the bills at Vienna, for 8 years, as given by Mr. Susmilch, in his <hi rend="italics">Gottliche</hi> Ordnung; the 2d, for Berlin, from the same, as formed from the bills there for 4 years, viz, from 1752 to 1755; the 3d, from Dr. Price, shewing the true probabilities of life in London, formed from the bills for ten years, viz, from 1759 to 1768; the 4th, for Norwich, formed by Dr. Price from the bills for 30 years, viz, from 1740 to 1769; the 5th, by the same, from the bills for Northampton; the 6th, as deduced by Dr. Halley, from the bills of mortality at Breslaw; the 7th shews the probabilities of life in a country parish in Brandenburg, formed from the bills for 50 years, from 1710 to 1759, as given by Mr. Susmilch; the 8th shews the probabilities of life in the parish of Holy-Cross, near Shrewsbury, formed from a register kept by the Rev. Mr. Garsuch, for 20 years, from 1750 to 1770; the 9th, for<pb n="29"/><cb/> Holland, was formed by M. Kersseboom, from the registers of certain annuities for lives granted by the government of Holland, which had been kept there for 125 years, in which the ages of the several annuitants dying during that period had been truly entered; the 10th, for France, were formed by M. Parcieux, from the lists of the French tontines, or long annuities, and verified by a comparison with the mortuary registers of several religious houses for both sexes; and the 11th, or last, for the district of Vaud in Switzerland, was<cb/> formed by Dr. Price from the registers of 43 parishes, given by M. Muret, in the Bern Memoirs for the year 1766.</p><p>Now from such lists as the foregoing, various tables have been formed for the valuation of annuities on single and joint lives, at several rates of interest, in which the value is shewn by inspection. The following are those that are given by Mr. Simpson, in his Select Exercises, as deduced from the London bills of mortality. <hi rend="center"><hi rend="italics">TABLE II.</hi></hi> <hi rend="center"><hi rend="italics">Shewing the Value of an Annuity on One Life, or Number of Years Annuity in the Value, supposing Money to</hi></hi> <hi rend="center"><hi rend="italics">bear Interest at the several Rates of</hi> 3, 4, <hi rend="italics">and</hi> 5 <hi rend="italics">per cent.</hi></hi><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Age.</cell><cell cols="1" rows="1" role="data">Years value at 3 per cent.</cell><cell cols="1" rows="1" role="data">Years value at 4 per cent.</cell><cell cols="1" rows="1" role="data">Years value at 5 per cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">18.8</cell><cell cols="1" rows="1" role="data">16.2</cell><cell cols="1" rows="1" role="data">14.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">18.9</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">19.0</cell><cell cols="1" rows="1" role="data">16.4</cell><cell cols="1" rows="1" role="data">14.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">19.0</cell><cell cols="1" rows="1" role="data">16.4</cell><cell cols="1" rows="1" role="data">14.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">19.0</cell><cell cols="1" rows="1" role="data">16.4</cell><cell cols="1" rows="1" role="data">14.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">19.0</cell><cell cols="1" rows="1" role="data">16.4</cell><cell cols="1" rows="1" role="data">14.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">18.9</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">18.7</cell><cell cols="1" rows="1" role="data">16.2</cell><cell cols="1" rows="1" role="data">14.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">18.5</cell><cell cols="1" rows="1" role="data">16.0</cell><cell cols="1" rows="1" role="data">14.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">18.3</cell><cell cols="1" rows="1" role="data">15.8</cell><cell cols="1" rows="1" role="data">13.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">18.1</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" role="data">13.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">17.9</cell><cell cols="1" rows="1" role="data">15.4</cell><cell cols="1" rows="1" role="data">13.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">17.6</cell><cell cols="1" rows="1" role="data">15.2</cell><cell cols="1" rows="1" role="data">13.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.0</cell><cell cols="1" rows="1" role="data">13.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">17.2</cell><cell cols="1" rows="1" role="data">14.8</cell><cell cols="1" rows="1" role="data">13.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">17.0</cell><cell cols="1" rows="1" role="data">14.7</cell><cell cols="1" rows="1" role="data">12.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">16.8</cell><cell cols="1" rows="1" role="data">14.5</cell><cell cols="1" rows="1" role="data">12.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">16.5</cell><cell cols="1" rows="1" role="data">14.3</cell><cell cols="1" rows="1" role="data">12.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.1</cell><cell cols="1" rows="1" role="data">12.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">16.1</cell><cell cols="1" rows="1" role="data">14.0</cell><cell cols="1" rows="1" role="data">12.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">15.9</cell><cell cols="1" rows="1" role="data">13.8</cell><cell cols="1" rows="1" role="data">12.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" role="data">13.6</cell><cell cols="1" rows="1" role="data">12.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">15.4</cell><cell cols="1" rows="1" role="data">13.4</cell><cell cols="1" rows="1" role="data">11.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">15.2</cell><cell cols="1" rows="1" role="data">13.2</cell><cell cols="1" rows="1" role="data">11.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">15.0</cell><cell cols="1" rows="1" role="data">13.1</cell><cell cols="1" rows="1" role="data">11.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">14.8</cell><cell cols="1" rows="1" role="data">12.9</cell><cell cols="1" rows="1" role="data">11.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">14.6</cell><cell cols="1" rows="1" role="data">12.7</cell><cell cols="1" rows="1" role="data">11.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">14.4</cell><cell cols="1" rows="1" role="data">12.6</cell><cell cols="1" rows="1" role="data">11.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">14.2</cell><cell cols="1" rows="1" role="data">12.4</cell><cell cols="1" rows="1" role="data">11.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">14.1</cell><cell cols="1" rows="1" role="data">12.3</cell><cell cols="1" rows="1" role="data">10.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">13.9</cell><cell cols="1" rows="1" role="data">12.1</cell><cell cols="1" rows="1" role="data">10.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">13.7</cell><cell cols="1" rows="1" role="data">11.9</cell><cell cols="1" rows="1" role="data">10.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">13.5</cell><cell cols="1" rows="1" role="data">11.8</cell><cell cols="1" rows="1" role="data">10.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">13.3</cell><cell cols="1" rows="1" role="data">11.6</cell><cell cols="1" rows="1" role="data">10.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">13.2</cell><cell cols="1" rows="1" role="data">11.5</cell><cell cols="1" rows="1" role="data">10.3</cell></row></table><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Age.</cell><cell cols="1" rows="1" role="data">Years value at 3 per cent.</cell><cell cols="1" rows="1" role="data">Years value at 4 per cent.</cell><cell cols="1" rows="1" role="data">Years value at 5 per cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">13.0</cell><cell cols="1" rows="1" role="data">11.4</cell><cell cols="1" rows="1" role="data">10.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">12.8</cell><cell cols="1" rows="1" role="data">11.2</cell><cell cols="1" rows="1" role="data">10.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">12.6</cell><cell cols="1" rows="1" role="data">11.1</cell><cell cols="1" rows="1" role="data">10.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">12.5</cell><cell cols="1" rows="1" role="data">11.0</cell><cell cols="1" rows="1" role="data">9.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">12.3</cell><cell cols="1" rows="1" role="data">10.8</cell><cell cols="1" rows="1" role="data">9.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">12.1</cell><cell cols="1" rows="1" role="data">10.7</cell><cell cols="1" rows="1" role="data">9.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">11.9</cell><cell cols="1" rows="1" role="data">10.5</cell><cell cols="1" rows="1" role="data">9.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">11.8</cell><cell cols="1" rows="1" role="data">10.4</cell><cell cols="1" rows="1" role="data">9.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">11.6</cell><cell cols="1" rows="1" role="data">10.2</cell><cell cols="1" rows="1" role="data">9.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">11.4</cell><cell cols="1" rows="1" role="data">10.1</cell><cell cols="1" rows="1" role="data">9.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">11.2</cell><cell cols="1" rows="1" role="data">9.9</cell><cell cols="1" rows="1" role="data">9.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">11.0</cell><cell cols="1" rows="1" role="data">9.8</cell><cell cols="1" rows="1" role="data">8.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">10.7</cell><cell cols="1" rows="1" role="data">9.6</cell><cell cols="1" rows="1" role="data">8.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">10.5</cell><cell cols="1" rows="1" role="data">9.4</cell><cell cols="1" rows="1" role="data">8.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">10.3</cell><cell cols="1" rows="1" role="data">9.3</cell><cell cols="1" rows="1" role="data">8.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">10.1</cell><cell cols="1" rows="1" role="data">9.1</cell><cell cols="1" rows="1" role="data">8.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">9.9</cell><cell cols="1" rows="1" role="data">8.9</cell><cell cols="1" rows="1" role="data">8.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">9.6</cell><cell cols="1" rows="1" role="data">8.7</cell><cell cols="1" rows="1" role="data">8.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">9.4</cell><cell cols="1" rows="1" role="data">8.6</cell><cell cols="1" rows="1" role="data">8.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9.2</cell><cell cols="1" rows="1" role="data">8.4</cell><cell cols="1" rows="1" role="data">7.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">8.9</cell><cell cols="1" rows="1" role="data">8.2</cell><cell cols="1" rows="1" role="data">7.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">8.7</cell><cell cols="1" rows="1" role="data">8.1</cell><cell cols="1" rows="1" role="data">7.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">8.5</cell><cell cols="1" rows="1" role="data">7.9</cell><cell cols="1" rows="1" role="data">7.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">8.3</cell><cell cols="1" rows="1" role="data">7.7</cell><cell cols="1" rows="1" role="data">7.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">8.0</cell><cell cols="1" rows="1" role="data">7.5</cell><cell cols="1" rows="1" role="data">7.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">7.8</cell><cell cols="1" rows="1" role="data">7.3</cell><cell cols="1" rows="1" role="data">6.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">7.6</cell><cell cols="1" rows="1" role="data">7.1</cell><cell cols="1" rows="1" role="data">6.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">7.4</cell><cell cols="1" rows="1" role="data">6.9</cell><cell cols="1" rows="1" role="data">6.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">7.1</cell><cell cols="1" rows="1" role="data">6.7</cell><cell cols="1" rows="1" role="data">6.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">6.9</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">6.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">6.7</cell><cell cols="1" rows="1" role="data">6.3</cell><cell cols="1" rows="1" role="data">6.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">6.1</cell><cell cols="1" rows="1" role="data">5.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">6.2</cell><cell cols="1" rows="1" role="data">5.9</cell><cell cols="1" rows="1" role="data">5.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">5.9</cell><cell cols="1" rows="1" role="data">5.6</cell><cell cols="1" rows="1" role="data">5.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.6</cell><cell cols="1" rows="1" role="data">5.4</cell><cell cols="1" rows="1" role="data">5.2</cell></row></table><pb n="30"/> <hi rend="center"><hi rend="italics">TABLE III.</hi></hi> <hi rend="center"><hi rend="italics">Shewing the Value of an Annuity for Two Joint Lives, that is, for as long as they exist together.</hi></hi><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Age of Younger</cell><cell cols="1" rows="1" role="data">Age of Elder</cell><cell cols="1" rows="1" role="data">Value at 3 per cent.</cell><cell cols="1" rows="1" role="data">Value at 4 per cent.</cell><cell cols="1" rows="1" role="data">Value at 5 per cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=14" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">14.7</cell><cell cols="1" rows="1" role="data">13.0</cell><cell cols="1" rows="1" role="data">11.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">14.3</cell><cell cols="1" rows="1" role="data">12.7</cell><cell cols="1" rows="1" role="data">11.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13.8</cell><cell cols="1" rows="1" role="data">12.2</cell><cell cols="1" rows="1" role="data">10.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">13.1</cell><cell cols="1" rows="1" role="data">11.6</cell><cell cols="1" rows="1" role="data">10.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">12.3</cell><cell cols="1" rows="1" role="data">10.9</cell><cell cols="1" rows="1" role="data">9.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">11.5</cell><cell cols="1" rows="1" role="data">10.2</cell><cell cols="1" rows="1" role="data">9.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">10.7</cell><cell cols="1" rows="1" role="data">9.6</cell><cell cols="1" rows="1" role="data">8.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">10.0</cell><cell cols="1" rows="1" role="data">9.0</cell><cell cols="1" rows="1" role="data">8.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">9.3</cell><cell cols="1" rows="1" role="data">8.4</cell><cell cols="1" rows="1" role="data">7.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8.6</cell><cell cols="1" rows="1" role="data">7.8</cell><cell cols="1" rows="1" role="data">7.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.8</cell><cell cols="1" rows="1" role="data">7.2</cell><cell cols="1" rows="1" role="data">6.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.9</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">6.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">6.1</cell><cell cols="1" rows="1" role="data">5.8</cell><cell cols="1" rows="1" role="data">5.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.3</cell><cell cols="1" rows="1" role="data">5.1</cell><cell cols="1" rows="1" role="data">4.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=13" role="data">15</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">13.9</cell><cell cols="1" rows="1" role="data">12.3</cell><cell cols="1" rows="1" role="data">11.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13.3</cell><cell cols="1" rows="1" role="data">11.8</cell><cell cols="1" rows="1" role="data">10.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">12.6</cell><cell cols="1" rows="1" role="data">11.2</cell><cell cols="1" rows="1" role="data">10.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">11.9</cell><cell cols="1" rows="1" role="data">10.6</cell><cell cols="1" rows="1" role="data">9.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">11.2</cell><cell cols="1" rows="1" role="data">10.0</cell><cell cols="1" rows="1" role="data">9.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">10.4</cell><cell cols="1" rows="1" role="data">9.4</cell><cell cols="1" rows="1" role="data">8.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">9.6</cell><cell cols="1" rows="1" role="data">8.8</cell><cell cols="1" rows="1" role="data">8.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8.9</cell><cell cols="1" rows="1" role="data">8.2</cell><cell cols="1" rows="1" role="data">7.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8.2</cell><cell cols="1" rows="1" role="data">7.6</cell><cell cols="1" rows="1" role="data">7.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.5</cell><cell cols="1" rows="1" role="data">7.0</cell><cell cols="1" rows="1" role="data">6.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.8</cell><cell cols="1" rows="1" role="data">6.4</cell><cell cols="1" rows="1" role="data">6.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">6.0</cell><cell cols="1" rows="1" role="data">5.7</cell><cell cols="1" rows="1" role="data">5.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.2</cell><cell cols="1" rows="1" role="data">5.0</cell><cell cols="1" rows="1" role="data">4.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=12" role="data">20</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">12.8</cell><cell cols="1" rows="1" role="data">11.3</cell><cell cols="1" rows="1" role="data">10.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">12.2</cell><cell cols="1" rows="1" role="data">10.8</cell><cell cols="1" rows="1" role="data">9.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">11.6</cell><cell cols="1" rows="1" role="data">10.3</cell><cell cols="1" rows="1" role="data">9.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">10.9</cell><cell cols="1" rows="1" role="data">9.8</cell><cell cols="1" rows="1" role="data">8.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">10.2</cell><cell cols="1" rows="1" role="data">9.2</cell><cell cols="1" rows="1" role="data">8.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">9.5</cell><cell cols="1" rows="1" role="data">8.6</cell><cell cols="1" rows="1" role="data">7.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8.8</cell><cell cols="1" rows="1" role="data">8.0</cell><cell cols="1" rows="1" role="data">7.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8.1</cell><cell cols="1" rows="1" role="data">7.5</cell><cell cols="1" rows="1" role="data">6.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.4</cell><cell cols="1" rows="1" role="data">6.9</cell><cell cols="1" rows="1" role="data">6.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.7</cell><cell cols="1" rows="1" role="data">6.3</cell><cell cols="1" rows="1" role="data">5.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">6.0</cell><cell cols="1" rows="1" role="data">5.7</cell><cell cols="1" rows="1" role="data">5.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.2</cell><cell cols="1" rows="1" role="data">5.0</cell><cell cols="1" rows="1" role="data">4.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=11" role="data">25</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">11.8</cell><cell cols="1" rows="1" role="data">10.5</cell><cell cols="1" rows="1" role="data">9.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">11.3</cell><cell cols="1" rows="1" role="data">10.1</cell><cell cols="1" rows="1" role="data">9.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">10.7</cell><cell cols="1" rows="1" role="data">9.6</cell><cell cols="1" rows="1" role="data">8.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">10.0</cell><cell cols="1" rows="1" role="data">9.1</cell><cell cols="1" rows="1" role="data">8.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">9.4</cell><cell cols="1" rows="1" role="data">8.5</cell><cell cols="1" rows="1" role="data">7.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8.7</cell><cell cols="1" rows="1" role="data">7.9</cell><cell cols="1" rows="1" role="data">7.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8.0</cell><cell cols="1" rows="1" role="data">7.4</cell><cell cols="1" rows="1" role="data">6.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.3</cell><cell cols="1" rows="1" role="data">6.8</cell><cell cols="1" rows="1" role="data">6.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.6</cell><cell cols="1" rows="1" role="data">6.2</cell><cell cols="1" rows="1" role="data">5.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.9</cell><cell cols="1" rows="1" role="data">5.6</cell><cell cols="1" rows="1" role="data">5.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.1</cell><cell cols="1" rows="1" role="data">4.9</cell><cell cols="1" rows="1" role="data">4.7</cell></row></table><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Age of Younger</cell><cell cols="1" rows="1" role="data">Age of Elder</cell><cell cols="1" rows="1" role="data">Value at 3 per cent.</cell><cell cols="1" rows="1" role="data">Value at 4 per cent.</cell><cell cols="1" rows="1" role="data">Value at 5 per cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=10" role="data">30</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">10.8</cell><cell cols="1" rows="1" role="data">9.6</cell><cell cols="1" rows="1" role="data">8.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">10.3</cell><cell cols="1" rows="1" role="data">9.2</cell><cell cols="1" rows="1" role="data">8.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">9.7</cell><cell cols="1" rows="1" role="data">8.8</cell><cell cols="1" rows="1" role="data">8.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">9.1</cell><cell cols="1" rows="1" role="data">8.3</cell><cell cols="1" rows="1" role="data">7.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8.5</cell><cell cols="1" rows="1" role="data">7.8</cell><cell cols="1" rows="1" role="data">7.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7.9</cell><cell cols="1" rows="1" role="data">7.3</cell><cell cols="1" rows="1" role="data">6.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.2</cell><cell cols="1" rows="1" role="data">6.7</cell><cell cols="1" rows="1" role="data">6.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">6.1</cell><cell cols="1" rows="1" role="data">5.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.8</cell><cell cols="1" rows="1" role="data">5.5</cell><cell cols="1" rows="1" role="data">5.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.1</cell><cell cols="1" rows="1" role="data">4.9</cell><cell cols="1" rows="1" role="data">4 7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=9" role="data">35</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">9.9</cell><cell cols="1" rows="1" role="data">8.8</cell><cell cols="1" rows="1" role="data">8.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">9.4</cell><cell cols="1" rows="1" role="data">8.5</cell><cell cols="1" rows="1" role="data">7.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">8.9</cell><cell cols="1" rows="1" role="data">8.1</cell><cell cols="1" rows="1" role="data">7.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8.3</cell><cell cols="1" rows="1" role="data">7.6</cell><cell cols="1" rows="1" role="data">7.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7.7</cell><cell cols="1" rows="1" role="data">7.1</cell><cell cols="1" rows="1" role="data">6.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.1</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">6.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.4</cell><cell cols="1" rows="1" role="data">6.0</cell><cell cols="1" rows="1" role="data">5.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.7</cell><cell cols="1" rows="1" role="data">5.4</cell><cell cols="1" rows="1" role="data">5.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.0</cell><cell cols="1" rows="1" role="data">4.8</cell><cell cols="1" rows="1" role="data">4.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=8" role="data">40</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">9.1</cell><cell cols="1" rows="1" role="data">8.1</cell><cell cols="1" rows="1" role="data">7.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">8.7</cell><cell cols="1" rows="1" role="data">7.8</cell><cell cols="1" rows="1" role="data">7.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8.2</cell><cell cols="1" rows="1" role="data">7.4</cell><cell cols="1" rows="1" role="data">6.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7.6</cell><cell cols="1" rows="1" role="data">6.9</cell><cell cols="1" rows="1" role="data">6.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">7.0</cell><cell cols="1" rows="1" role="data">6.4</cell><cell cols="1" rows="1" role="data">6.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.4</cell><cell cols="1" rows="1" role="data">5.9</cell><cell cols="1" rows="1" role="data">5.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.7</cell><cell cols="1" rows="1" role="data">5.4</cell><cell cols="1" rows="1" role="data">5.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">5.0</cell><cell cols="1" rows="1" role="data">4.8</cell><cell cols="1" rows="1" role="data">4.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=7" role="data">45</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">8.3</cell><cell cols="1" rows="1" role="data">7.4</cell><cell cols="1" rows="1" role="data">6.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">7.9</cell><cell cols="1" rows="1" role="data">7.1</cell><cell cols="1" rows="1" role="data">6.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7.4</cell><cell cols="1" rows="1" role="data">6.7</cell><cell cols="1" rows="1" role="data">6.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">6.8</cell><cell cols="1" rows="1" role="data">6.3</cell><cell cols="1" rows="1" role="data">5.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.3</cell><cell cols="1" rows="1" role="data">5.8</cell><cell cols="1" rows="1" role="data">5.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.6</cell><cell cols="1" rows="1" role="data">5.3</cell><cell cols="1" rows="1" role="data">5.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">4.9</cell><cell cols="1" rows="1" role="data">4.7</cell><cell cols="1" rows="1" role="data">4.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=6" role="data">50</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">7.6</cell><cell cols="1" rows="1" role="data">6.8</cell><cell cols="1" rows="1" role="data">6.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7.2</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">6.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">6.7</cell><cell cols="1" rows="1" role="data">6.1</cell><cell cols="1" rows="1" role="data">5.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.2</cell><cell cols="1" rows="1" role="data">5.7</cell><cell cols="1" rows="1" role="data">5.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.5</cell><cell cols="1" rows="1" role="data">5.2</cell><cell cols="1" rows="1" role="data">4.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">4.8</cell><cell cols="1" rows="1" role="data">4.6</cell><cell cols="1" rows="1" role="data">4.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=5" role="data">55</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">6.9</cell><cell cols="1" rows="1" role="data">6.2</cell><cell cols="1" rows="1" role="data">5.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">6.5</cell><cell cols="1" rows="1" role="data">5.9</cell><cell cols="1" rows="1" role="data">5.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">6.0</cell><cell cols="1" rows="1" role="data">5.6</cell><cell cols="1" rows="1" role="data">5.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.4</cell><cell cols="1" rows="1" role="data">5.1</cell><cell cols="1" rows="1" role="data">4.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">4.7</cell><cell cols="1" rows="1" role="data">4.5</cell><cell cols="1" rows="1" role="data">4.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=4" role="data">60</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">6.1</cell><cell cols="1" rows="1" role="data">5.6</cell><cell cols="1" rows="1" role="data">5.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">5.7</cell><cell cols="1" rows="1" role="data">5.3</cell><cell cols="1" rows="1" role="data">4.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5.2</cell><cell cols="1" rows="1" role="data">4.9</cell><cell cols="1" rows="1" role="data">4.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">4.6</cell><cell cols="1" rows="1" role="data">4.4</cell><cell cols="1" rows="1" role="data">4.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=3" role="data">65</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">5.4</cell><cell cols="1" rows="1" role="data">5.0</cell><cell cols="1" rows="1" role="data">4.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">4.9</cell><cell cols="1" rows="1" role="data">4.6</cell><cell cols="1" rows="1" role="data">4.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">4.4</cell><cell cols="1" rows="1" role="data">4.2</cell><cell cols="1" rows="1" role="data">4.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">70</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">4.6</cell><cell cols="1" rows="1" role="data">4.4</cell><cell cols="1" rows="1" role="data">4.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">4.2</cell><cell cols="1" rows="1" role="data">4.0</cell><cell cols="1" rows="1" role="data">3.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">75</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">3.8</cell><cell cols="1" rows="1" role="data">3.7</cell><cell cols="1" rows="1" role="data">3.6</cell></row></table><pb n="31"/> <hi rend="center"><hi rend="italics">TABLE IV.</hi></hi> <hi rend="center"><hi rend="italics">For the Value of an Annuity upon the Longer of Two Given Lives.</hi></hi><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Age of Younger</cell><cell cols="1" rows="1" role="data">Age of Elder</cell><cell cols="1" rows="1" role="data">Value at 3 per cent.</cell><cell cols="1" rows="1" role="data">Value at 4 per cent.</cell><cell cols="1" rows="1" role="data">Value at 5 per cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=14" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">23.4</cell><cell cols="1" rows="1" role="data">19.9</cell><cell cols="1" rows="1" role="data">17.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">22.9</cell><cell cols="1" rows="1" role="data">19.5</cell><cell cols="1" rows="1" role="data">16.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">22.5</cell><cell cols="1" rows="1" role="data">19.1</cell><cell cols="1" rows="1" role="data">16.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">22.2</cell><cell cols="1" rows="1" role="data">18.8</cell><cell cols="1" rows="1" role="data">16.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">21.9</cell><cell cols="1" rows="1" role="data">18.6</cell><cell cols="1" rows="1" role="data">16.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">21.6</cell><cell cols="1" rows="1" role="data">18.4</cell><cell cols="1" rows="1" role="data">16.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">21.4</cell><cell cols="1" rows="1" role="data">18.3</cell><cell cols="1" rows="1" role="data">16.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">21.2</cell><cell cols="1" rows="1" role="data">18.2</cell><cell cols="1" rows="1" role="data">15.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">20.9</cell><cell cols="1" rows="1" role="data">18.0</cell><cell cols="1" rows="1" role="data">15.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">20.7</cell><cell cols="1" rows="1" role="data">17.8</cell><cell cols="1" rows="1" role="data">15.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">20.4</cell><cell cols="1" rows="1" role="data">17.6</cell><cell cols="1" rows="1" role="data">15.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">20.1</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">19.8</cell><cell cols="1" rows="1" role="data">17.2</cell><cell cols="1" rows="1" role="data">15.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">19.5</cell><cell cols="1" rows="1" role="data">16.9</cell><cell cols="1" rows="1" role="data">14.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=13" role="data">15</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">22.8</cell><cell cols="1" rows="1" role="data">19.3</cell><cell cols="1" rows="1" role="data">16.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">22.3</cell><cell cols="1" rows="1" role="data">18.9</cell><cell cols="1" rows="1" role="data">16.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">21.9</cell><cell cols="1" rows="1" role="data">18.6</cell><cell cols="1" rows="1" role="data">16.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">21.6</cell><cell cols="1" rows="1" role="data">18.3</cell><cell cols="1" rows="1" role="data">16.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">21.3</cell><cell cols="1" rows="1" role="data">18.1</cell><cell cols="1" rows="1" role="data">15.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">21.1</cell><cell cols="1" rows="1" role="data">17.9</cell><cell cols="1" rows="1" role="data">15.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">20.9</cell><cell cols="1" rows="1" role="data">17.8</cell><cell cols="1" rows="1" role="data">15.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">20.7</cell><cell cols="1" rows="1" role="data">17.6</cell><cell cols="1" rows="1" role="data">15.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">20.4</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">20.1</cell><cell cols="1" rows="1" role="data">17.2</cell><cell cols="1" rows="1" role="data">15.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">19.8</cell><cell cols="1" rows="1" role="data">16.9</cell><cell cols="1" rows="1" role="data">15.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">19.4</cell><cell cols="1" rows="1" role="data">16.6</cell><cell cols="1" rows="1" role="data">14.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">18.9</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=12" role="data">20</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">21.6</cell><cell cols="1" rows="1" role="data">18.3</cell><cell cols="1" rows="1" role="data">15.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">21.1</cell><cell cols="1" rows="1" role="data">17.9</cell><cell cols="1" rows="1" role="data">15.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">20.7</cell><cell cols="1" rows="1" role="data">17.6</cell><cell cols="1" rows="1" role="data">15.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">20.4</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">20.1</cell><cell cols="1" rows="1" role="data">17.2</cell><cell cols="1" rows="1" role="data">15.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">19.9</cell><cell cols="1" rows="1" role="data">17.0</cell><cell cols="1" rows="1" role="data">14.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">19.6</cell><cell cols="1" rows="1" role="data">16.8</cell><cell cols="1" rows="1" role="data">14.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">19.4</cell><cell cols="1" rows="1" role="data">16.6</cell><cell cols="1" rows="1" role="data">14.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">19.1</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">18.7</cell><cell cols="1" rows="1" role="data">16.0</cell><cell cols="1" rows="1" role="data">14.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">18.2</cell><cell cols="1" rows="1" role="data">15.7</cell><cell cols="1" rows="1" role="data">13.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">17.7</cell><cell cols="1" rows="1" role="data">15.3</cell><cell cols="1" rows="1" role="data">13.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=11" role="data">25</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">20.3</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">19.8</cell><cell cols="1" rows="1" role="data">17.0</cell><cell cols="1" rows="1" role="data">14.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">19.4</cell><cell cols="1" rows="1" role="data">16.7</cell><cell cols="1" rows="1" role="data">14.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">19.2</cell><cell cols="1" rows="1" role="data">16.5</cell><cell cols="1" rows="1" role="data">14.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">18.9</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">18.7</cell><cell cols="1" rows="1" role="data">16.1</cell><cell cols="1" rows="1" role="data">14.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">18.4</cell><cell cols="1" rows="1" role="data">15.9</cell><cell cols="1" rows="1" role="data">14.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">18.0</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" role="data">13.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">17.6</cell><cell cols="1" rows="1" role="data">15.3</cell><cell cols="1" rows="1" role="data">13.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">17.2</cell><cell cols="1" rows="1" role="data">15.0</cell><cell cols="1" rows="1" role="data">13.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">16.7</cell><cell cols="1" rows="1" role="data">14.6</cell><cell cols="1" rows="1" role="data">12.9</cell></row></table><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Age of Younger</cell><cell cols="1" rows="1" role="data">Age of Elder</cell><cell cols="1" rows="1" role="data">Value at 3 per cent.</cell><cell cols="1" rows="1" role="data">Value at 4 per cent.</cell><cell cols="1" rows="1" role="data">Value at 5 per cent.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=10" role="data">30</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">19.3</cell><cell cols="1" rows="1" role="data">16.6</cell><cell cols="1" rows="1" role="data">14.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">18.8</cell><cell cols="1" rows="1" role="data">16.2</cell><cell cols="1" rows="1" role="data">14.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">18.4</cell><cell cols="1" rows="1" role="data">15.9</cell><cell cols="1" rows="1" role="data">14.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">18.1</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" role="data">13.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">17.8</cell><cell cols="1" rows="1" role="data">15.4</cell><cell cols="1" rows="1" role="data">13.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.1</cell><cell cols="1" rows="1" role="data">13.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">17.0</cell><cell cols="1" rows="1" role="data">14.8</cell><cell cols="1" rows="1" role="data">13.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">16.6</cell><cell cols="1" rows="1" role="data">14.5</cell><cell cols="1" rows="1" role="data">12.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">16.1</cell><cell cols="1" rows="1" role="data">14.1</cell><cell cols="1" rows="1" role="data">12.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" role="data">13.7</cell><cell cols="1" rows="1" role="data">12.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=9" role="data">35</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">18.3</cell><cell cols="1" rows="1" role="data">15.8</cell><cell cols="1" rows="1" role="data">13.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">17.8</cell><cell cols="1" rows="1" role="data">15.4</cell><cell cols="1" rows="1" role="data">13.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">17.4</cell><cell cols="1" rows="1" role="data">15.1</cell><cell cols="1" rows="1" role="data">13.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">17.1</cell><cell cols="1" rows="1" role="data">14.8</cell><cell cols="1" rows="1" role="data">13.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">16.7</cell><cell cols="1" rows="1" role="data">14.5</cell><cell cols="1" rows="1" role="data">12.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.2</cell><cell cols="1" rows="1" role="data">12.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">15.8</cell><cell cols="1" rows="1" role="data">13.8</cell><cell cols="1" rows="1" role="data">12.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">15.3</cell><cell cols="1" rows="1" role="data">13.4</cell><cell cols="1" rows="1" role="data">12.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">14.8</cell><cell cols="1" rows="1" role="data">13.0</cell><cell cols="1" rows="1" role="data">11.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=8" role="data">40</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">17.3</cell><cell cols="1" rows="1" role="data">15.0</cell><cell cols="1" rows="1" role="data">13.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">16.8</cell><cell cols="1" rows="1" role="data">14.6</cell><cell cols="1" rows="1" role="data">13.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">16.3</cell><cell cols="1" rows="1" role="data">14.2</cell><cell cols="1" rows="1" role="data">12.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">15.9</cell><cell cols="1" rows="1" role="data">13.9</cell><cell cols="1" rows="1" role="data">12.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">15.4</cell><cell cols="1" rows="1" role="data">13.5</cell><cell cols="1" rows="1" role="data">12.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">14.9</cell><cell cols="1" rows="1" role="data">13.1</cell><cell cols="1" rows="1" role="data">11.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">14.5</cell><cell cols="1" rows="1" role="data">12.7</cell><cell cols="1" rows="1" role="data">11.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">14.0</cell><cell cols="1" rows="1" role="data">12.3</cell><cell cols="1" rows="1" role="data">11.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=7" role="data">45</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">16.2</cell><cell cols="1" rows="1" role="data">14.2</cell><cell cols="1" rows="1" role="data">12.8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">15.7</cell><cell cols="1" rows="1" role="data">13.8</cell><cell cols="1" rows="1" role="data">12.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">15.2</cell><cell cols="1" rows="1" role="data">13.4</cell><cell cols="1" rows="1" role="data">12.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">14.7</cell><cell cols="1" rows="1" role="data">12.9</cell><cell cols="1" rows="1" role="data">11.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">14.1</cell><cell cols="1" rows="1" role="data">12.5</cell><cell cols="1" rows="1" role="data">11.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">13.6</cell><cell cols="1" rows="1" role="data">12.0</cell><cell cols="1" rows="1" role="data">11.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">13.1</cell><cell cols="1" rows="1" role="data">11.6</cell><cell cols="1" rows="1" role="data">10.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=6" role="data">50</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">15.0</cell><cell cols="1" rows="1" role="data">13.3</cell><cell cols="1" rows="1" role="data">12.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">14.5</cell><cell cols="1" rows="1" role="data">12.9</cell><cell cols="1" rows="1" role="data">11.7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">13.9</cell><cell cols="1" rows="1" role="data">12.4</cell><cell cols="1" rows="1" role="data">11.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">13.3</cell><cell cols="1" rows="1" role="data">12.0</cell><cell cols="1" rows="1" role="data">10.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">12.8</cell><cell cols="1" rows="1" role="data">11.5</cell><cell cols="1" rows="1" role="data">10.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">12.3</cell><cell cols="1" rows="1" role="data">11.0</cell><cell cols="1" rows="1" role="data">10.1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=5" role="data">55</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">13.6</cell><cell cols="1" rows="1" role="data">12.4</cell><cell cols="1" rows="1" role="data">11.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">13.0</cell><cell cols="1" rows="1" role="data">11.9</cell><cell cols="1" rows="1" role="data">10.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">12.4</cell><cell cols="1" rows="1" role="data">11.3</cell><cell cols="1" rows="1" role="data">10.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">11.8</cell><cell cols="1" rows="1" role="data">10.8</cell><cell cols="1" rows="1" role="data">10.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">11.3</cell><cell cols="1" rows="1" role="data">10.3</cell><cell cols="1" rows="1" role="data">9.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=4" role="data">60</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">12.2</cell><cell cols="1" rows="1" role="data">11.2</cell><cell cols="1" rows="1" role="data">10.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">11.5</cell><cell cols="1" rows="1" role="data">10.6</cell><cell cols="1" rows="1" role="data">10.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">10.9</cell><cell cols="1" rows="1" role="data">10.1</cell><cell cols="1" rows="1" role="data">9.5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">10.3</cell><cell cols="1" rows="1" role="data">9.5</cell><cell cols="1" rows="1" role="data">9.0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=3" role="data">65</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">10.7</cell><cell cols="1" rows="1" role="data">10.0</cell><cell cols="1" rows="1" role="data">9.4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">10.0</cell><cell cols="1" rows="1" role="data">9.4</cell><cell cols="1" rows="1" role="data">8.9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">9.3</cell><cell cols="1" rows="1" role="data">8.7</cell><cell cols="1" rows="1" role="data">8.3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">70</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">9.2</cell><cell cols="1" rows="1" role="data">8.6</cell><cell cols="1" rows="1" role="data">8.2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">8.4</cell><cell cols="1" rows="1" role="data">7.9</cell><cell cols="1" rows="1" role="data">7.6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">75</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">7.6</cell><cell cols="1" rows="1" role="data">7.2</cell><cell cols="1" rows="1" role="data">6.9</cell></row></table><pb n="32"/><cb/></p><p>The uses of these tables may be exemplified in the following problems.</p><p><hi rend="smallcaps">Prob.</hi> 1. <hi rend="italics">To find the Probability or Proportion of Chance, that a person of a Given Age continues in being a proposed number of years.</hi>—Thus, suppose the age be 40, and the number of years proposed 15; then, to calculate by the table of the probabilities for London, in tab. 1. against 40 years stands 214, and against 55 years, the age to which the person must arrive, stands 120, which shews that, of 214 persons who attain to the age of 40, only 120 of them reach the age of 55, and consequently 94 die between the ages of 40 and 55: It is evident therefore that the odds for attaining the proposed age of 55, are as 120 to 94, or as 9 to 7 nearly.</p><p><hi rend="smallcaps">Prob.</hi> 2. <hi rend="italics">To find the Value of an Annuity for a proposed Life.</hi>—This problem is resolved from tab. 2, by looking against the given age, and under the proposed rate of interest; then the corresponding quantity shews the number of years-purchase required. For example, if the given age be 36, the rate of interest 4 per cent, and the proposed annuity L250. Then in the table it appears that the value is 12.1 years purchase, or 12.1 times L250, that is L3025.</p><p>After the same manner the answer will be found in any other case falling within the limits of the table. But as there may sometimes be occasion to know the values of lives computed at higher rates of interest than those in the table, the two following practical rules are subjoined; by which the problem is resolved independent of tables.</p><p><hi rend="italics">Rule</hi> 1. When the given age is not less than 45 years, nor greater than 85, subtract it from 92; then multiply the remainder by the perpetuity, and divide the product by the said remainder added to 2 1/2 times the perpetuity; so shall the quotient be the number of years purchase required. Where note, that by the <hi rend="italics">perpetuity</hi> is meant the number of years purchase of the fee-simple; found by dividing 100 by the rate per cent at which interest is reckoned.</p><p><hi rend="italics">Ex.</hi> Let the given age be 50 years, and the rate of interest 10 per cent. Then subtracting 50 from 92, there remains 42; which multiplied by 10 the perpetuity, gives 420; and this divided by 67, the remainder increased by 2 1/2 times 10 the perpetuity, quotes 6.3 nearly, for the number of years purchase. Therefore, supposing the annuity to be L100, its value in present money will be L630.</p><p><hi rend="italics">Rule</hi> 2. When the age is between 10 and 45 years; take 8 tenths of what it wants of 45, which divide by the rate per cent increased by 1.2; then if the quotient be added to the value of a life of 45 years, found by the preceding rule, there will be obtained the number of years purchase in this case. For example, let the proposed age be 20 years, and the rate of interest 5 per cent. Here taking 20 from 45, there remains 25; 8/10 of which is 20; which divided by 6.2, quotes 3.2; and this added to 9.8, the value of a life of 45, found by the former rule, gives 13 for the number of years purchase that a life of 20 ought to be valued at.</p><p>And the conclusions derived by these rules, Mr. Simpson adds, are so near the true values, computed<cb/> from real observations, as seldom to differ from them by more than 1/10 or 2/10 of one year's purchase.</p><p>The observations here alluded to, are those which are founded on the London bills of mortality. And a similar method of solution, accommodated to the Breslaw observations, will be as follows, viz. “Multiply the difference between the given age and 85 years by the perpetuity, and divide the product by 8 tenths of the said difference increased by double the perpetuity, for the answer.” Which, from 8 to 80 years of age, will commonly come within less than 1/8 of a year's purchase of the truth.</p><p><hi rend="smallcaps">Prob.</hi> 3. <hi rend="italics">To find the Value of an Annuity for Two Joint Lives, that is, for as long as they both continue in being together.</hi>—In table 3, find the younger age, or that nearest to it, in column 1, and the higher age in column 2; then against this last is the number of years purchase in the proper column for the interest. <hi rend="italics">Ex.</hi> Suppose the two ages be 20 and 35 years; then the value. <hi rend="center">is 10.9 years purchase at 3 per cent.</hi> <hi rend="center">or 9.8 " at 4 per cent.</hi> <hi rend="center">or 8.8 " at 5 per cent.</hi></p><p><hi rend="smallcaps">Prob.</hi> 4. <hi rend="italics">To find the Value of the Annuity for the</hi> Longest <hi rend="italics">of Two Lives, that is, for as long as</hi> either of them <hi rend="italics">continues in being.</hi>—In table 4, find the age of the youngest life, or the nearest to it, in col. 1, and the age of the elder in col. 2; then against this last is the answer in the proper column of interest.—<hi rend="italics">Ex.</hi> So, if the two ages be 15 and 40; then the value of the annuity upon the longest of two such lives, <hi rend="center">is 21.1 years purchase at 3 per cent.</hi> <hi rend="center">or 17.9 " 4 per cent.</hi> <hi rend="center">or 15.7 " 5 per cent.</hi></p><p>N B. In the last two problems, if the younger age, or the rate of interest, be not exactly found in the tables, the nearest to them may be taken, and then by proportion the value for the true numbers will be nearly found.</p><p>Rules and tables for the values of three lives, &c, may also be seen in Simpson, and in Baron Maseres's Annuities, &c. All these calculations have been made from tables of the real mortuary registers, differing unequally at the several ages. But rules have also been given upon other principles, as by De Moivre, upon the supposition that the decrements of life are equal at all ages; an assumption not much differing from the truth, from 7 to 70 years of age.</p><p><hi rend="italics"><hi rend="smallcaps">Life-Annuities</hi>, payable half-yearly, &c.</hi>—These are worth more than such as are payable yearly, as computed by the foregoing rules and tables, on the two following accounts: First, that parts of the payments are received sooner; and 2dly, there is a chance of receiving some part or parts of a whole year's payment more than when the payments are only made annually. Mr. Simpson, in his Select Exercises, pa. 283, observes, that the value of these two advantages put together, will always amount to 1/4 of a year's purchase for half-yearly payments, and to 3/8 of a year's purchase for quarterly payments; and Mr. Maseres, at page 233 &c of his Annuities, by a very elaborate calculation, finds the former difference to be nearly 1/4 also. But Dr. Price, in an Essay in the Philos. Trans. vol. 66,<pb n="33"/><cb/> pa. 109, states the same differences only at 2/10 for half-yearly payments, and 3/10 for quarterly payments: And the Doctor then adds some algebraical theorems for such calculations.</p><p><hi rend="smallcaps">Life-Annuities</hi>, <hi rend="italics">secured by Land.</hi>—These differ from other life-annuities only in this, that the annuity is to be paid up to the very day of the death of the age in question, or of the person upon whose life the annuity is granted. To obtain the more exact value therefore of such an annuity, a small quantity must be added to the same as computed by the foregoing rules and observations, which is different according as the payments are yearly, half-yearly, or quarterly, &c; and are thus stated by Dr. Price in his Essay quoted above; viz, the addition is <hi rend="italics">y</hi>/(2<hi rend="italics">n</hi>) for annual payments, or <hi rend="italics">b</hi>/(4<hi rend="italics">n</hi>) for half-yearly payments, or <hi rend="italics">q</hi>/(8<hi rend="italics">n</hi>) for quarterly payments <*> where <hi rend="italics">n</hi> is the complement of the given age, or what it wants of 86 years; and <hi rend="italics">y, h, q</hi> are the respective values of an annuity <hi rend="italics">certain</hi> for <hi rend="italics">n</hi> years, payable yearly, half-yearly, or quarterly. And, by numeral examples, it is found that the first of these additional quantities is about 2/10, the second 1/10, and the 3d half a tenth of one year's purchase.</p><p><hi rend="italics">Complement os</hi> <hi rend="smallcaps">Life.</hi> See <hi rend="smallcaps">Complement.</hi></p><p><hi rend="italics">Expectation of</hi> <hi rend="smallcaps">Life.</hi> See <hi rend="smallcaps">Expectation.</hi></p><p><hi rend="italics">Insurance or Assurance on</hi> <hi rend="smallcaps">Lives.</hi> See <hi rend="smallcaps">Assurances</hi> <hi rend="italics">on Lives.</hi></p></div1><div1 part="N" n="LIGHT" org="uniform" sample="complete" type="entry"><head>LIGHT</head><p>, that principle by which objects are made perceptible to our sense of seeing; or the sensation occasioned in the mind by the view of luminous objects.</p><p>The nature of Light has been a subject of speculation from the first dawnings of philosophy. Some of the earliest philosophers doubted whether objects became visible by means of any thing proceeding from them, or from the eye of the spectator. But this opinion was qualified by Empedocles and Plato, who maintained, that vision was occasioned by particles continually flying off from the surfaces of bodies, which meet with others proceeding from the eye; while the effect was ascribed by Pythagoras solely to the particles proceeding from the external objects, and entering the pupil of the eye. But Aristotle defines Light to be the act of a transparent body, considered as such: and he observes that Light is not fire, nor yet any matter radiating from the luminous body, and transmitted through the transparent one.</p><p>The Cartesians have refined considerably on this notion; and hold that Light, as it exists in the luminous body, is only a power or faculty of exciting in us a very clear and vivid sensation; or that it is an invisible fluid present at all times and in all places, but requiring to be set in motion, by a body ignited or otherwise properly qualified to make objects visible to us.</p><p>Father Malbranche explains the nature of Light from a supposed analogy between it and sound.—<cb/> Thus he supposes all the parts of a luminous body are in a rapid motion, which, by very quick pulses, is constantly compressing the subtle matter between the luminous body and the eye, and excites vibrations of pression. As these vibrations are greater, the body appears more luminous; and as they are quicker or slower, the body is of this or that colour.</p><p>But the Newtonians maintain, that Light is not a fluid <hi rend="italics">per se,</hi> but consists of a great number of very small particles, thrown off from the luminous body by a repulsive power with an immense velocity, and in all directions. And these particles, it is also held, are emitted in right lines: which rectilinear motion they preserve till they are turned out of their path by some of the following causes, viz, by the attraction of some other body near which they pass, which is called <hi rend="italics">inflection;</hi> or by paffing obliquely through a medium of different density, which is called <hi rend="italics">refraction;</hi> or by being turned aside by the opposition of some intervening body, which is called <hi rend="italics">reflection;</hi> or, lastly, by being totally stopped by some substance into which they penetrate, and which is called their <hi rend="italics">extinction.</hi> A succession of these particles following one another, in an exact straight line, is called a <hi rend="italics">ray of Light;</hi> and this ray, in whatever manner its direction may be changed, whether by refraction, reflection, or inflection, always preserves a rectilinear course till it be again changed; neither is it possible to make it move in the arch of a circle, ellipsis, or other curve. For the above properties of the rays of Light, see the several words, <hi rend="smallcaps">Refraction, Reflection</hi>, &c.</p><p>The <hi rend="italics">velocity</hi> of the particles and rays of Light is truly astonishing, amounting to near 2 hundred thousand miles in a second of time, which is near a million times greater than the velocity of a cannon-ball. And this amazing motion of Light has been manifested in various ways, and sirst, from the eclipses of Jupiter's satellites. It was first observed by Roemer, that the eclipses of those satellites happen sometimes sooner, and sometimes later, than the times given by the tables of them; and that the observation was before or after the computed time, according as the earth was nearer to, or farther from Jupiter, than the mean distance. Hence Roemer and Cassini both concluded that this circumstance depended on the distance of Jupiter from the earth; and that, to account for it, they must suppose that the Light was aboút 14 minutes in crossing the earth's orbit. This conclusion however was afterward abandoned and attacked by Cassini himself. But Roemer's opinion sound an able advocate in Dr. Halley; who removed Cassini's difficulty, and left Roemer's conclusion in its full force. Yet, in a memoir presented to the Academy in 1707, M. Maraldi endeavoured to strengthen Cassini's arguments; when Roemer's doctrine found a new defender in Mr. Pound. See Philos. Trans. number 136, also Abridg. vol. 1, pa. 409 and 422, and Groves, Phys. Elem. number 2636. It has since been found, by repeated experiments, that when the earth is exactly between Jupiter and the sun, his satellites are seen e<*>lipsed about 8 1/4 minutes <hi rend="italics">sooner</hi> than they could be according to the tables; but when the earth is nearly in the opposite point of its orbit, these eclipses happen about 8 1/4 minutes <hi rend="italics">later</hi> than the tables prcdict them. Hence<pb n="34"/><cb/> then it is certain that the motion of Light is not instantaneous, but that it takes up about 16 1/2 minutes of time to pass over a space cqual to the diameter of the earth's orbit, which is at least 190 millions of miles in length, or at the rate of near 200,000 miles per second, as above-mentioned. Hence therefore Light takes up about 8 1/4 minutes in passing from the sun to the earth; so that, if he should be annihilated, we would see him for 8<*> minutes after that event should happen; and if he were again created, we should not see him till 8 1/4 minutes afterwards. Hence also it is casy to know the time in which Light travels to the earth, from the moon, or any of the other planets, or even from the sixed stars when their distances shall be known; these distances however are so im mensely great, that from the nearest of them, supposed to be Sirius, the dog-star, Light takes up many years to travel to the earth: and it is even suspected that there are many stars whose Light have not yet arrived at us since their creation. And this, by-the-bye, may perhaps sometimes account for the appearance of new stars in the heavens.</p><p>It may be just observed that Galileo first conceived the notion of measuring the velocity of Light; and a description of his contrivance for this purpose, is in his Treatise on Mechanics, pa. 39. He had two men with Lights covered; the one was to observe when the other uncovered his Light, and to exhibit his own the moment he perceived it. This rude experiment was tried at the drstance of a mile, but without success, as may naturally be imagined: and the members of the Academy Del Cimento repeated the experiment, and placed their observers, to as little purpofe, at the distance of 2 miles.</p><p>But our excellent astronomer, Dr. Bradley, afterwards found nearly the same velocity of Light as Roemer, from his accurate observations, and most ingenious theory, to account for some apparent motions in the fixed stars; for an account of which, see <hi rend="smallcaps">Aberration</hi> of Light. By a long series of these observations, he found the difference between the true and apparent place of several fixed stars, for different times of the year; which difference could no otherwise be accounted for, than from the progressive motion of the rays of Light. From the mean quantity of this difference he ingeniously found, that the ratio of the velosity of Light to the velocity of the earth in its orbit, was as 10313 to 1, or that Light moves 10313 times faster than the earth moves in its orbit about the sun; and as this l<*>tter motion is at the rate of 18 11/12 miles per second nearly, it follows that the former, or the velocity of Light, is at the rate of about 195000 miles in a second; a motion according to which it will require just 8′ 7″ to move from the sun to the earth, or about 95 millions of miles.</p><p>It was also inferred, from the foregoing principles, that Light proceeds with the same velocity from all the stars. And hence it follows, if we suppose that all the stars are not equally distant from us, as many arguments prove, that the motion of Light, all the way it passes through the immense space above our atmosphere, is equable or uniform. And since the different methods of determining the velocity of Light thus agree in the result, it is reasonable to conclude<cb/> that, in the same medium, Light is propagated with the same velocity after it has been reflected, as before.</p><p>For an account of Mr. Melville's hypothesis of the different velocities of differently coloured rays, see <hi rend="smallcaps">Colour.</hi></p><p>To the doctrine concerning the materiality of Light, and its amazing velocity, several objections have been made; of which the moit considerable is, That as rays of Light are continually passing in different directions from every visible point, they must necessarily interfere with each other in such a manner, as entirely to confound all distinct perception of objects, if not quite to destroy the whole sense of seeing: not to mention the continual waste of substance which a constant emission of particles must occasion in the luminous body, and thereby since the creation must have greatly diminished the matter in the sun and stars, as well as increased the bulk of the earth and planets by the vast quantity of particles of Light absorbed by them in so long a period of time.</p><p>But it has been replied, that if Light were not a body, but consisted in mere pression or pulsion, it could never be propagated in right lines, but would be continually inflected ad umbram. Thus Sir I. Newton: “A pressure on a fluid medium, i. e. a motion propagated by such a medium, beyond any obstacle, which impedes any part of its motion, cannot be propagated in right lines, but will be always inflecting and diffusing itself every way, to the quiescent medium beyond that obstacle. The power of gravity tends downwards; but the pressure of water arising from it tends every way with an equable force, and is propagated with equal ease and equal strength, in curves, as in strait lines. Waves, on the surface os the water, gliding by the extremes of any very large obstacle, inflect and dilate themselves, still diffusing gradually into the quiescent water beyond that obstacle. The waves, pulses, or vibrations of the air, wherein sound consists, are manifestly inflected, though not so considerably as the waves of water; and sounds are propagated with equal ease, through crooked tubes, and through strait lines; but Light was never known to move in any curve, nor to inslect itself ad umbram.”</p><p>It must be acknowledged, however, that many philosophers, both English and Foreignérs, have recurred to the opinion, that Light consists of vibrations propagated from the luminous body, through a subtle etherial medium.</p><p>The ingenious Dr. Franklin, in a letter dated April 23, 1752, expresses his dissatisfaction with the doctrine, that Light consists of particles of matter continually driven off from the sun's surface, with so enormous a swiftness. “Must not, says he, the smallest portion conceivable, have, with such a motion, a force exceeding that of a 24 pounder discharged from a cannon? Must not the sun diminish exceedingly by such a waste of matter; and the planets, instead of drawing nearer to him, as some have feared, recede to greater distances through the lessened attraction? Yet these particles, with this amazing motion, will not drive before them, or remove, the least and slightest dust they meet with; and the sun appears to continue of his ancient dimensions, and his attendants move in their ancient orbits.” He therefore conjectures that all the phenomena of<pb n="35"/><cb/> Light may be more properly solved, by supposing all space filled with a subtle elastic fluid, which is not visible when at rest, but which, by its vibrations, affects that fine sense in the eye, as those of the air affect the grosser organs of the ear; and even that different degrees of the vibration of this medium may cause the appearances of different colours. Franklin's Exper. and Observ. 1769, pa. 264.</p><p>The celebrated Euler has also maintained the same hypothesis, in his Theoria Lucis & Colorum. In the summary of his arguments against the common opinion, recited in Acad. Berl. 1752, pa. 271, besides the objections above-mentioned, he doubts the possibility, that particles of matter, moving with the amazing velocity of Light, should penetrate transparent substances with so much ease. In whatever manner they are transmitted, those bodies must have pores, disposed in right lines, and in all possible directions, to serve as canals for the passage of the rays: but such a structure must take away all solid matter from those bodies, and all coherence among their parts, if they do contain any solid matter.</p><p>Doctor Horsley, now Bp. of Rochester, has taken considerable pains to obviate the difficulties started by Dr. Franklin. Supposing that the diameter of each particle of Light does not exceed one millionth of one millionth of an inch, and that the density of each particle is even three times that of iron, that the Light of the sun reaches the earth in 7′, at the distance of 22919 of the earth's semidiameters, he calculates that the momentum or force of motion in each particle of Light coming from the sun, is less than that in an iron ball of a quarter of an inch diameter, moving at the rate of less than an inch in 12 thousand millions of millions of years. And hence he concludes, that a particle of matter, which probably is larger than any particle of Light, moving with the velocity of Light, has a force of motion, which, instead of exceeding the force of a 24 pounder discharged from a cannon, is almost infinitely less than that of the smallest shot discharged from a pocket pistol, or less than any that art can create. He also thinks it possible, that Light may be produced by a continual emission of matter from the sun, without any such waste of his substance as should sensibly contract his dimensions, or alter the motions of the planets, within any moderate length of time. In proof of this, he observes that, for the production of any of the phenomena of Light, it is not necessary that the emanation from the sun should be continual, in a strict mathematical sense, or without any interval; and likewise that part of the Light which issues from the sun, is continually returned to him by reflection from the planets, as well as other Light from the suns of other systems. He proceeds, by calculation, to shew that in 385,130,000 years, the sun would lose but the 13232d part of his matter, and consequently of the gravitation towards him, at any given distance; which is an alteration much too small to discover itself in the motion of the earth, or of any of the planets. He farther computes that the greatest stroke which the retina of a common eye sustains, when turned directly to the sun in a bright day, does not exceed that which would be given by an iron shot, a quarter of an inch diameter, and moving only<cb/> at the rate of 16 1/6 inches in a year; whereas the o<*> dinary stroke is less than the 2084th part of this. See Philos. Trans. vol. 60 and 61.</p><p>In answer to the difficulty respecting the non-interference of the particles of Light with each other, Mr. Melville observes (Edinb. Ess. vol. 2), there is probably no physical point in the visible horizon, that does not send rays to every other point, unless where opaque bodies interpose. Light, in its passage from one system to another, often passes through torrents of Light issuing from other suns and systems, without ever interfering, or being diverted from its course, either by it, or by the particles of that elastic medium, which it has been supposed by some is diffused through all the mundane space. To account for this fact, he supposes that the particles of Light are incomparably rare, even when they are the most dense, or that their diameters are incomparably less than their distance from one another: which obviates the objection urged by Euler and others against the materiality of Light, from its influence in disturbing the freedom and perpetuity of the celestial motions. Boscovich and some others solve the difficulty concerning the non-interference of the particles of Light, by supposing that each particle is endued with an insuperable impulsive force; but in this case, their spheres of impulsion would be more likely to interfere, and on that account they be more liable to disturb one another.</p><p>M. Canton shews (Philos. Trans. vol. 58, p. 344), that the difficulty of the interference will vanish, if a very small portion of time be allowed between the emission of every particle and the next that follows in the same direction. Suppose, for instance, that a lucid point in the sun's surface emits 150 particles in a second of time, which, he observes, will be more than sufficient to give continual Light to the eye, without the least appearance of intermission; yet still the particles of such a ray, on account of their great velocity, will be more than 1000 miles behind each other, a space sufficient to allow others to pass in all directions without any perceptible interruption. And if we adopt the conclusions drawn from the experiments on the duration of the sensations excited by Light, by the chevalier D'Arcy, in the Acad. Scienc. 1765, who states it at the 7th part of a second, art interval of more than 20,000 miles may be admitted between every two successive particles.</p><p>The doctrine of the materiality of Light is farther confirmed by those experiments, which shew, that the colour and inward texture of some bodies are changed by being exposed to the Light.</p><p><hi rend="italics">Of the Momentum, or Force, of the Particles of Light.</hi> Some writers have attempted to prove the materiality of Light, by determining the momentum of their component particles, or by shewing that they have a force so as, by their impulse, to give motion to light bodies. M. Homberg, Ac. Par. 1708, Hist. pa. 25, imagined, that he could not only disperse pieces of amianthus, and other light substances, by the impulse of the solar rays, but also that by throwing them upon the end of a kind of lever, connected with the spring of a watch, he could make it move sensibly quicker; from which, and other experiments, he inferred the weight of the particles of Light. And Hartsoecker made preten-<pb n="36"/><cb/> sions of the same nature. But M. Du Fay and M. Mairan made other experiments of a more accurate kind, without the effects which the former had imagined, and which even proved that the effects mentioned by them were owing to currents of heated air produced by the burning glasses used in their experiments, or some other causes which they had overlooked.</p><p>However, Dr. Priestley informs us, that Mr. Michell endeavoured to ascertain the momentum of Light with still greater accuracy, and that his endeavours were not altogether without success. Having found that the instrument he used, acquired, from the impulse of the rays of light, a velocity of an inch in a second of time, be inferred that the quantity of matter contained in the rays falling upon the instrument in that time, amounted to no more than the 12 hundred millionth part of a grain. In the experiment, the Light was collected from a surface of about 3 square feet; and as this surface reflected only about the half of what fell upon it, the quantity of matter contained in the solar rays, incident upon a square foot and a half of surface, in a second of time, ought to be no more than the <*>2 hundred millionth part of a grain, or upon one square foot only, the 18 hundred millionth part of a grain. But as the density of the rays of Light at the surface of the sun, is 45000 times greater than at the earth, there ought to issue from a square foot of the sun's surface, in one second of time, the 40 thousandth part of a grain of matter; that is, a little more than 2 grains a day, or about 4,752,000 grains, which is about 670 pounds avoirdupois, in 6000 years, the time since the creation; a quantity which would have shortened the sun's semidiameter by no more than about 10 feet, if it be supposed of no greater denfity than water only.</p><p>The <hi rend="italics">Expansion</hi> or <hi rend="italics">Extension</hi> of any portion of Light, is inconceivable. Dr. Hook shews that it is as unlimited as the universe; which he proves from the immense distance of many of the fixed stars, which only become visible to the eye by the best telescopes. Nor, adds he, a<*> they only the great bodies of the sun or stars that are thus liable to disperse their Light through the vast expanse of the universe, but the smallest spark of a lucid body must do the same, even the smallest globule struck from a steel by a flint.</p><p>The <hi rend="italics">Intensity</hi> of different Lights, or of the fame Light in different circumstances, affords a curious subject of speculation. M. Bouguer, Traité de Optique, found that when one Light is from 60 to 80 times less than another, its presence or absence will not be perceived by an ordinary eye; that the moon's Light, when she is 19° 16 high above the horizon, is but about 1/3 of her Light at 66° 11′ high; and when one limb just touched the horizon, her Light was but the 2000th part of her Light at 66° 11 high; and that hence Light is diminished in the proportion of 3 to 1 by traversing 7469 toises of dense air. He found also, that the centre of the sun's difc is considerably more luminous than the edges of it; whereas both the primary and secondary planets are more luminous at their edges than near their centres: That, farther, the Light of the sun is about 300,000 times greater than that of the moon; and therefore it is no wonder that philosophers have had so little success in their attempts to collect the Light of the moon with burning-glasses;<cb/> for, should one of the largest of them even increase the Light 1000 times, it will still leave the Light of the moon in the focus of the glass, 300 times less than the intensity of the common Light of the sun.</p><p>Dr. Smith, in his Optics, vol. 1, pa. 29, thought he had proved that the Light of the full moon would be only the 90,900th part of the full day Light, if no rays were lost at the moon. But Mr. Robins, in his Tracts, vol. 2, pa. 225, shews that this is too great by one half. And Mr. Michell, by a more easy and accurate mode of computation, found that the density of the sun's Light on the surface of the moon is but the 45,000th part of the density at the sun; and that therefore, as the moon is nearly of the same apparent magnitude as the sun, if she reflected to us all the Light received on her surface, it would be only the 45,000th part of our day Light, or that which we receive from the sun. Admitting therefore, with M. Bouguer, that the moon Light is only the 300,000th part of the day or sun's Light, Mr. Michell concludes that the moon reflects no more than between the 6th and 7th part of what she receives.</p><p>Dr. Gravesande says, a lucid body is that which emits or gives fire a motion in right lines, and makes the difference between Light and heat to consist in this, that to produce the former, the fiery particles must enter the eye in a rectilinear motion, which is not required in the latter: on the contrary, an irregular motion seems more proper for it, as appears from the rays coming directly from the sun to the tops of mountains, which have not near that effect with those in the valley, agitated with an irregular motion, by several reflections.</p><p>Sir I. Newton observes, that bodies and Light act mutually on one another; bodies on Light, in emitting, reflecting, refracting, and inflecting it; and Light on bodies, by heating them, and putting their parts into a vibrating motion, in which heat principally consists. For all fixed bodies, he observes, when heated beyond a certain degree, do emit Light, and shine; which shining &c appears to be owing to the vibrating motion of their parts; and all bodies, abounding in earthy and sulphureous particles, if sufficiently agitated, emit Light, which way soever that agitation be effected. Thus, sea water shines in a storm; quicksilver, when shaken in vacuo; cats or horses, when rubbed in the dark; and wood, fish, and flesh, when putrefied.</p><p>Light proceeding from putrescent animal and vegetable substances, as well as from glow-worms, is mentioned by Aristotle. And Bartholin mentions four kinds of luminous insects, two of which have wings: but in hot climates it is said they are found in much greater numbers, and of different species. Columna observes, that their Light is not extinguished immediately on the death of the animal. The first distinct account that occurs of Light proceeding from putrescent animal flesh, is that which is given by Fabricius ab Aquapendente in 1592, de Visione &c, pa. 45. And Bartholin gives an account of a similar appearance, which happened at Montpelier in 1641, in his treatise De Luce Animalium.</p><p>Mr. Boyle speaks of a piece of shining rotten wood, which was extinguished in vacuo; but upon re-admitting the air, it revived again, and shone as before;<pb n="37"/><cb/> though he could not perceive that it was increased in condensed air. But in Birch's History of the Royal Soc. vol. 2, pa. 254, there is an account of the Light of a shining fish, which was rendered more vivid by putting the fish into a condensing engine. The fish called Whitings were those commonly used by Mr. Boyle in his experiments: though in a discourse read before the R. Soc. in 1681, it was asserted that, of all fishy subslances, the eggs of lobsters, after they had been boiled, shone the brightest. Birch's Hist. vol. 2, pa. 70. In 1672 Mr. Boyle accidentally observed Light issuing from flesh meat; and, among other remarks on this subject, he observes that extreme cold extinguishes the Light of shining wood; probably because extreme cold checks the putrefaction, which is the cause of the Light. The shell sish called Pholas, is remarkable for its luminous quality. The <hi rend="italics">luminousness of the S<*>a</hi> has been also a subject of frequent observation. See <hi rend="italics">Ignis fatuus, Phosphorus,</hi> and <hi rend="italics">Putrefaction,</hi> &c.</p><p>Mr. Hawksbee, and many writers on the subject of electricity since his time, have produced a great variety of instances of the artificial production of Light, by the attrition of bodies naturally not luminous; as of amber rubbed on woollen cloth in vacuo; of glafs on woollen, of glass on glass, of oyster shells on woollen, and of woollen on woollen, all in vacuo. On the several experiments of this kind, he makes these following reflections: that different sorts of bodies afford Light of various kinds, different both in colour and in force; that the effects of an attrition are various, according to the different preparations and treatment of the bodies that are to endure it; and that bodies which have yielded a particular Light, may be brought by friction to yield no more of that Light.</p><p>M. Bernoulli found by experiment, that mercury amalgamated with tin, and rubbed on glass, produced a considerable Light in the air; that gold rubbed on glass, exhibited the same in a greater degree; but that the most exquisite Light of all was produced by the attrition of a diamond, this being equally vivid with that of a burning coal briskly agitated with the bellows. See <hi rend="smallcaps">Electricity</hi>, &c.</p><p><hi rend="italics">Of the Attraction of Light.</hi> That the particles of Light are attracted by those of other bodies, is evident from numerous experiments. This phenomenon was observed by Sir I. Newton, who found, by repeated trials, that the rays of Light, in their passage near the edges of bodies, are diverted out of the right lines, and always inslected or bent towards those bodies, whether they be opaque or transparent, as pieces of metals, the edges of knives, broken glasses, &c. See <hi rend="smallcaps">Inflection</hi> and <hi rend="smallcaps">Rays.</hi> The curious observations that had been made on this subject by Dr. Hook and Grimaldi, led Sir I. Newton to repeat and diversify their experiments, and to pursue them much farther than they had done. For a particular account of his experiment and observations, see his treatise on Optics, pa. 293 &c.</p><p>This action of bodies on Light is found to exert itself at a sensible distance, though it always increases as the distance is diminished; as appears very sensibly in the passage of a ray between the edges of two thin planes at different apertures; which is attended with this peculiar circumstance, that the attraction of one <*>dge is increased as the other is brought nearer it.<cb/> The rays of Light, in their passage out of glass into a vacuum, are not only inflected towards the glass, but if they fall too obliquely, they will revert back again to the glass, and be totally reflected. Now the cause of this reflection cannot be attributed to any resistance of the vacuum, but must be entirely owing to some force or power in the glass, which attracts or draws back the rays as they were passing into the vacuum. And this appears farther from hence, that if you wet the back surface of the glass with water, oil, honey, or a solution of quicksilver, then the rays which would otherwise have been reflected, will pervade and pass through that liquor; which shews that the rays are not reflected till they come to that back surface of the glass, nor even till they begin to go out of it; for if, at their going out, they fall into any of the aforesaid mediums, they will not then be reflected, but will persist in their former course, the attraction of the glass being in this case counterbalanced by that of the liquor.</p><p>M. Maraldi prosecuted experiments similar to those of Sir I. Newton on inslected Light. And his observations chiesly respect the inflection of Light towards other bodies, by which their shadows are partially illuminated. Acad. Paris 1723, Mem. p. 159. See also Priestley's Hist. pa. 521 &c.</p><p>M. Mairan, without attempting the discovery of new facts, endeavoured to explain the old ones, by the hypothesis of an atmosphere surrounding all bodies; and consequently two reflections and refractions of Light that impinges upon them, one at the surface of the atmosphere, and the other at the surface of the body itself. This atmosphere he supposed to be of a variable density and refractive power, like the air.</p><p>M. Du Tour succeeded Mairan, and imagined that he could account for all the phenomena by the help of an atmosphere of an uniform density, but of a less refractive power than the air surrounding all bodies. Du Tour also varied the Newtonian experiments, and discovered more than three fringes in the colours produced by the inflection of light. He farther concludes that the refracting atmospheres, surrounding all kinds of bodies, are of the same size; for when he used a great variety of substances, and of different sizes too, he always found coloured streaks of the same dimensions. He also observes, that his hypothesis contradicts an observation of Sir I. Newton, viz, that those rays are the most inflected which pass the nearest to any body. Mem. de Math. & de Phys. vol. 5, pa. 650, or Priestley's Hist. pa. 531.</p><p>M. Le Cat found that objects sometimes appear magnisied by means of the inflection of Light. Looking at a distant steeple, when a wire, of a less diameter than the pupil of his eye, was held pretty near to it, and drawing it several times between that object and his eye, he was surprised to find that every time the wire passed before his eye, the steeple seemed to change its place, and some hills beyond the steeple seemed to have the same motion, just as if a lens had been drawn between them and his eye. This discovery led him to several others depending on the inflećtíon of the rays of Light. Thus, he magnified s<*> objects, as the head of a pin, by viewing them thro<*>gh a small hole in a card; so that the rays which fo<*>med the image must<pb n="38"/><cb/> necessarily pass so near the circumference of the hole, as to be attracted by it. He exhibited also other appearances of a similar nature. Traité des Sens, pa. 299. Priestley, ubi supra, pa. 537.</p><p><hi rend="italics">Reflection and Refraction of Light.</hi> From the mutual attraction between the particles of Light and other bodies, arise two other grand phenomena, besides the inflection of Light, which are called the reflection and refraction of Light. It is well known that the determination of bodies in motion, especially elastic ones, is changed by the interposition of other bodies in their way: thus also Light, impinging on the surfaces of bodies, should be turned out of its course, and beaten back or reflected, so as, like other striking bodies, to make the angle of its reflection equal to the angle of incidence. This, it is found by experience, Light does; and yet the cause of this effect is different from that just now assigned: for the rays of Light are not reflected by striking on the very parts of the reflecting bodies, but by some power equally diffused over the whole surface of the body, by which it acts on the Light, either attracting or repelling it, without contact: by which same power, in other circumstances, the rays are refracted; and by which also the rays are first emitted from the luminous body; as Newton abundantly proves by a great variety of arguments, See <hi rend="smallcaps">Reflection</hi> and <hi rend="smallcaps">Refraction.</hi></p><p>That great author puts it past doubt, that all those rays which are reflected, do not really touch the body, though they approash it infinitely near; and that those which strike on the parts of solid bodies, adhere to them, and are as it were extinguished and lost. Since the reflection of the rays is ascribed to the action of the whole surface of the body without contact, if it be asked, how it happens that all the rays are not reflected from every surface; but that, while some are reflected, others pass through, and are refracted? the answer given by Newton is as follows:—Every ray of Light, in its passage through any refracting surface, is put into a certain transient constitution or state, which in the progress of the ray returns at equal intervals, and disposes the ray at every return to be easily transmitted through the next refracting surface, and between the returns to be easily reflected by it: which alteration of reflection and transmiffion it appears is propagated from every surface, and to all distances. What kind of action or disposition this is, and whether it consists in a circulating or vibrating motion of the ray, or the medium, or something else, he does not enquire; but allows those who are fond of hypotheses to suppose, that the rays of Light, by impinging on any reflecting or refracting surface, excite vibrations in the reflecting or refracting medium, and by that means agitate the solid parts of the body. These vibrations, thus produced in the medium, move faster than the rays, so as to overtake them; and when any ray is in that part of the vibration which conspires with its motion, its velocity is increased, and so it easily breaks through a refracting surface; but when it is in a contrary part of the vibration, which impedes its motion, it is easily reflected; and thus every ray is successively disposed to be easily reflected or transmitted by every vibration which meets it. These returns in the disposition of any ray to be reflected, he calls <hi rend="italics">f<*>s of easy reflection;</hi> and the returns<cb/> in the disposition to be transmitted, he calls <hi rend="italics">fits of easy transinission;</hi> also the space between the returns, <hi rend="italics">the interval of the fits.</hi> Hence then the reason why the surfaces of all thick transparent bodies reflect part of the Light incident upon them, and refract the rest, is that some rays at their incidence are in fits of easy reflection, and others of easy transmission. For <hi rend="italics">the properties of reflected Light,</hi> fee <hi rend="smallcaps">Reflection, Mirror</hi>, &c.</p><p>Again, a ray of Light, passing out of one medium into another of different density, and in its passage making an oblique angle with the surface that separates the mediums, will be refracted, or turned out of its direction; because the rays are more strongly attracted by a denser than by a rarer medium. That these rays are not refracted by striking on the solid parts of bodies, but that this is effected without a real contact, and by the same force by which they are emitted and reflected, only exerting itself differently in different circumstances, is proved in a great measure by the same arguments by which it is demonstrated that reflection is performed without contact. See <hi rend="smallcaps">Refraction, Lens, Colour, Vision</hi>, &c.</p></div1><div1 part="N" n="LIGHTNING" org="uniform" sample="complete" type="entry"><head>LIGHTNING</head><p>, a large bright flame, shooting swiftly through the atmosphere, of momentary or very short duration, and commonly attended with thunder.</p><p>Some philosophers accounted for this awful natural phenomenon in this manner, viz, that an inflammable substance is formed of the particles of sulphur, nitre, and other combustible matter, which are exhaled from the earth, and carried into the higher regions of the atmosphere, and that by the collision of two clouds, or otherwise, this substance takes fire, and darts out into a train of Light, larger or smaller according to the strength and quantity of the materials. And others have explained the phenomenon of Lightning by the fermentation of sulphureous substances with nitrous acids. See <hi rend="smallcaps">Thunder.</hi></p><p>But it is now universally allowed, that Lightning is really an electrical explosion or phenomenon. Philosophers had not proceeded far in their experiments and enquiries on this subject, before they perceived the obvious analogy between Lightning and electricity, and they produced many arguments to evince their similarity. But the method of proving this hypothesis beyond a doubt, was first proposed by Dr. Franklin, who, about the close of the year 1749, conceived the practicability of drawing Lightning down from the clouds. Various circumstances of resemblance between Lightning and electricity were remarked by this ingenious philosopher, and have been abundantly confirmed by later discoveries, such as the following: Flashes of Lightning are usually seen crooked and waving in the air; so the electric spark drawn from an irregular body at some distance, and when it is drawn by an irregular body, or through a space in which the best conductors are disposed in an irregular manner, always exhibits the same appearance: Lightning strikes the highest and most pointed objects in its course, in preference to others, as hills, trees, spires, masts of ships, &c; so all pointed conductors receive and throw off the electric fluid more readily than those that are terminated by slat surfaces: Lightning is observed to take and follow the readiest and best conductor; and the same is the case with electricity in the discharge of the Leyden<pb n="39"/><cb/> phial; from whence the doctor infers, that in a thunder-storm, it would be safer to have one's cloaths wet than dry: Lightning burns, dissolves metals, rends some bodies, sometimes strikes persons blind, destroys animal life, deprives magnets of their virtue, or reverses their poles; and all these are well-known properties of electricity.</p><p>But Lightning also gives polarity to the magnetic needle, as well as to all bodies that have any thing of iron in them, as bricks &c; and by observing afterwards which way the magnetic poles of these bodies lie, it may thence be known in what direction the stroke passed. Persons are sometimes killed by Lightning, without exhibiting any visible marks of injury; and in this case Sig. Beccaria supposes that the Lightning does not really touch them, but only produces a sudden vacuum near them, and the air rushing violently out of their lungs to supply it, they cannot recover their breath again: and in proof of this opinion he alleges, that the lungs of such persons are found flaccid; whereas these are found inflated when the persons are really killed by the electric shock. Though this hypothesis is controverted by Dr. Priestley.</p><p>To demonstrate however, by actual experiment, the identity of the electric fluid with the matter of Lightning, Dr. Franklin contrived to bring Lightning from the heavens, by means of a paper kite, properly fitted up for the purpose, with a long fine wire string, and called an electrical kite, which he raised when a thunder-storm was perceived to be coming on: and with the electricity thus obtained, he charged phials, kindled spirits, and performed all other such electrical experiments as are usually exhibited by an excited glass globe or cylinder. This happened in June 1752, a month after the electricians in France, in pursuance of the method which he had before proposed, had verisied the same theory, but without any knowledge of what they had done. The most active of these were Messrs. Dalibard and Delor, followed by M. Mazeas and M. Monnier.</p><p>In April and June 1753, Dr. Franklin discovered that the air is sometimes electrified negatively, as well as sometimes positively; and he even found that the clouds would change from positive to negative electricity several times in the course of one thunder-gust. This curious and important discovery he soon perceived was capable of being applied to practical use in life, and in consequence proposed a method, which he soon accomplished, of securing buildings from being damaged by Lightning, by means of <hi rend="smallcaps">Conductors.</hi> See the word.</p><p>Nor had the English philosophers been inattentive to this subject: but, for want of proper opportunities of trying the necessary experiments, and from some other unfavourable circumstances, they had failed of success. Mr. Canton, however, succeeded in July 1752; and in the following month Dr. Bevis and Mr. Wilson observed near the same appearances as Mr. Canton had done before. By a number of experiments Mr. Canton also soon after observed that some clouds were in a positive, while some were in a negative state of electricity; and that the electricity of his conductor would sometimes change, from one state to the other, five or six times in less than half an hour.<cb/></p><p>But Sig. Beccaria discovered this variable state of thunder clouds, before he knew that it had been observed by Dr. Franklin or any other person; and he has given a very exact and particular account of the external appearances of these clouds. From the observations of his apparatus within doors, and of the Lightning abroad, he inferred, that the quantity of electric matter in a common thunder storm, is inconceivably great, considering how many pointed bodies, as spires, trees, &c, are continually drawing it off, and what a prodigious quantity is repeatedly discharged to or from the earth. This matter is in such abundance, that he thinks it impossible for any cloud or number of clouds to contain it all, so as either to receive or discharge it. He observes also, that during the progress and increase of the storm, though the lightning frequently struck to the earth, the same clouds were the next moment ready to make a still greater discharge, and his apparatus continued to be as much affected as ever; so that the clouds must have received at one part, in the same moment when a discharge was made from them in another. And from the whole he concludes, that the clouds serve as conductors to convey the electric fluid from those parts of the earth that are overloaded with it, to those that are exhausted of it. The same cause by which a cloud is sirst raised, from vapours dispersed in the atmosphere, draws to it those that are already formed, and still continues to form new ones, till the whole collected mass extends so far as to reach a part of the earth where there is a deficiency of the electric sluid, and where the electric matter will discharge itself on the earth. A channel of communication being thus formed, a fresh supply of electric matter is raised from the overloaded part, which continues to be conveyed by the medium of the clouds, till the equilibrium of the fluid is restored between the two places of the earth. Sig. Beccaria observes, that a wind always blows from the place from which the thunder-cloud proceeds; and it is plain that the sudden accumulation of such a prodigious quantity of vapours must displace the air, and repel it on all sides. Indeed many observations of the descent of Lightning, consirm his theory of the manner of its ascent; for it often throws before it the parts of conducting bodies, and distributes them along the resisting medium, through which it must force its passage; and upon this principle the longest flashes of Lightning seem to be made, by forcing into its way part of the vapours in the air. One of the chief reasons why these flashes make so long a rumbling, is that they are occasioned by the vast length of a vacuum made by the passage of the electric matter: for although the air collapses the moment after it has passed, and that the vibration, on which the sound depends, commences at the same moment; yet when the slash is directed towards the person who hears the report, the vibrations excited at the nearer end of the track, will reach his ear much sooner than those from the more remote end; and the sound will, without any echo or repercussion, continue till all the vibrations have successively reached him.</p><p>How it happens that particular parts of the earth, or the clouds, come into the opposite states of positive and negative electricity, is a question not absolutely determined: though it is easy to conceive that when particular clouds, or different parts of the earth, possess op-<pb n="40"/><cb/> posite electricities, a discharge will take place within a certain distance; or the one will strike into the other, and in the discharge a flash of Lightning will be feen. Mr. Canton queries whether the clouds do not become possessed of electricity by the gradual heating and cooling of the air; and whether air suddenly rarefied, may not give electric fire to clouds and vapours passing through it, and air suddenly condensed receive electric fire from them.——Mr. Wilcke supposes, that the air contracts its electricity in the same manner that sulphur and other substances do, when they are heated and cooled in contact with various bodies. Thus, the air being heated or cooled near the earth, gives electricity to the earth, or receives it from it; and the electrified air, being conveyed upwards by various means, communicates its electricity to the clouds.—Others have queried, whether, since thunder commonly happens in a sultry state of the air, when it seems charged with sulphureous vapours, the electric matter then in the clouds may not be generated by the fermentation of sulphureous vapours with mineral or acid vapours in the air.</p><p>With regard to places of safety in times of thunder and Lightning, Dr. Franklin's advice is, to sit in the middle of a room, provided it be not under a metal lustre suspended by a chain, sitting on one chair, and laying the feet on another. It is still better, he says, to bring two or three mattresses or beds into the middle of the room, and folding them double, to place the chairs upon them; for as they are not so good conductors as the walls, the Lightning will not be so likely to pass through them: but the safest place of all, is in a hammock hung by silken cords, at an equal distance from all the sides of a room. Dr. Priestley observes, that the place of most perfect safety must be the cellar, and especially the middle of it; for when a person is lower than the surface of the earth, the Lightning must strike it before it can possibly reach him. In the fields, the place of safety is within a few yards of a tree, but not quite near it. Beccaria cautions persons not always to trust too much to the neighbourhood of a higher or better conductor than their own body; since he has repeatedly found that the Lightning by no means descends in one undivided track, but that bodies of various kinds conduct their share of it at the same time, in proportion to their quantity and conducting power. See Franklin's Letters, Beccaria's Lettre dell' Ellettricessimo, Priestley's Hist. of Electric., and Lord Mahon's Principles of Electricity.</p><p>Lord Mahon observes that damage may be done by Lightning, not only by the main stroke and lateral explosion, but also by what he calls the returning stroke; by which is meant the sudden violent return of that part of the natural share of electricity which had been gradually expelled from some body or bodies, by the superinduced elastic electrical pressure of the electrical atmosphere of a thunder cloud.</p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Lightning</hi>, an imitation of real or natural Lightning by gunpowder, aurum fulminans, phofphorus, &c, but especially the last, between which and Lightning there is much more resemblance than the others.</p><p>Phosphorus, when newly made, gives a sort of arti- <*>cial Lightning visible in the dark, which would sur-<cb/> prise those not used to such a phenomenon. It is usual to keep this preparation under water; and if it is desired to see the corruscations to the greatest advantage, it should be kept in a deep cylindrical glass, not more than three quarters silled with water. At times the phosphorus will send up corruscations, which will pierce through the incumbent water, and expand themselves with great brightness in the upper or empty part of the glass, and much resembling Lightning. The season of the year, as well as the newness of the phosphorus, must concur to produce these flashes; for they are as common in winter as Lightning is, though both are very frequent in warm weather. The phoiphorus, while burning, acts the part of a corrosive, and when it goes out resolves into a menstruum, which dissolves gold, iron, and other metals; and Lightning, in like manner, melts the same substances.</p><p>LIKE <hi rend="smallcaps">Quantities</hi>, or <hi rend="italics">Similar Quantities,</hi> in Algebra, are <*>uch as are expressed by the same letters, to the same power, or equally repeated in each quantity; though the numeral coefficients may be different.</p><p>Thus 4<hi rend="italics">a</hi> and 5<hi rend="italics">a</hi> are Like quantities, as are also 3<hi rend="italics">a</hi><hi rend="sup">2</hi> and 12<hi rend="italics">a</hi><hi rend="sup">2</hi>, and also 6<hi rend="italics">bxy</hi><hi rend="sup">2</hi> and 10<hi rend="italics">bxy</hi><hi rend="sup">2</hi>. But 4<hi rend="italics">a</hi> and 5<hi rend="italics">b,</hi> or 3<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi> and 10<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi><hi rend="sup">2</hi>, &c, are unlike quantities; because they have not every where the same dimensions, nor are the letters equally repeated. —Like quantities can be united into one quantity, by addition or subtraction; but unlike quantities can only be added or subtracted by placing the signs of these operations between them.</p><p><hi rend="smallcaps">Like</hi> <hi rend="italics">Signs,</hi> in Algebra, are the same signs, either both positive or both negative. But when one is positive and the other negative, they are unlike signs.</p><p>So, + 3<hi rend="italics">ab</hi> and + 5<hi rend="italics">cd</hi> have Like signs, as have also - 2<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">c</hi> and - 2<hi rend="italics">ax</hi><hi rend="sup">2</hi>; but + 3<hi rend="italics">ab</hi> and - 5<hi rend="italics">cd</hi> have unlike signs, as also - 2<hi rend="italics">ax</hi> and 3<hi rend="italics">ax.</hi></p><p><hi rend="smallcaps">Like</hi> <hi rend="italics">Figures,</hi> or <hi rend="italics">Arches,</hi> &c, are the same as <hi rend="italics">Similar</hi> sigures, arches, &c. See <hi rend="smallcaps">Similar.</hi></p><p>All Like figures have their homologous lines in the same ratio. Also Like plane figures are in the duplicate ratio, or as the squares of their homologous lines or sides; and Like solid figures are in the triplicate ratio, or as the cubes of their homologous lines or sides.</p></div1><div1 part="N" n="LILLY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LILLY</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, a noted English astrologer, born in Leicestershire in 1602. His father was not able to give him farther education than common reading and writing; but young Lilly being of a forward temper, and endued with shrewd wit, he resolved to push his fortune in London; where he arrived in 1620, and, for a present support, articled himself as a servant to a mantua-maker in the parish of St. Clement Danes. But in 1624 he moved a step higher, by entering into the service of Mr. Wright in the Strand, master of the Salters company, who not being able to write, Lilly among other offices kept his books. On the death of his master, in 1627, Lilly paid his addresses to the widow, whom he married with a fortune of 1000l. Being now his own master, he followed the bent of his inclinations, which led him to follow the puritanical preachers. Afterwards, turning his mind to judicial astrology, in 1632 he became pupil, in that art, to one Evans, a<pb n="41"/><cb/> profligate Welsh parson; and the next year gave the public a specimen of his skill, by an intimation that the king had chosen an unlucky horoscope for the coronation in Scotland. In 1634, getting a manuscript copy of the <hi rend="italics">Ars Noticia</hi> of Cornelius Agrippa, with alterations, he drank in the doctrine of the magic circle, and the invocation of spirits, with great eagerness, and practised it for some time; after which he treated the mystery of recovering stolen goods, &c, with great contempt, claiming a supernatural sight, and the g<*>ft of prophetical predictions; all which he well knew how to turn to good advantage.</p><p>Mean while, he had buried his first wife, purchased a moiety of 13 houses in the Strand, and married a second wife, who, joining to an extravagant temper a termagant spirit, which he could not lay, made him unhappy, and greatly reduced his circumstances. With this uncomfortable yokemate he removed, in 1636, to Hersham in Surrey, where he staid till 1641; when, seeing a prospect of fishing in troubled waters, he returned to London. Here having purchased several curious books in this art, which were found on pulling down the house of another astrologer, he studied them incessantly, finding out secrets contained in them, which were written in an imperfect Greek character; and, in 1644, published his <hi rend="italics">Merlinus Anglicus,</hi> an almanac, which he continued annually till his death, and several other astrological works; devoting his pen, and other labours, sometimes to the king's party, and sometimes to that of the parliament, but mostly to the latter, raising his fortune by favourable predictions to both parties, sometimes by presents, and sometimes by pensions: thus, in 1648, the council of state gave him in money 50l. and a pension of 100l. per annum, which he received for two years, and then resigned it on some disgust. By his advice and contrivance, the king attempted several times to make his escape from his consinement: he procured and sent the aqua-fortis and files to cut the iron bars of his prison windows at Carisbrook castle; but still advising and writing for the other party at the same time. Mean while he read public lectures on astrology, in 1648 and 1649, for the improvement of young students in that art; and in short, plied his business so well, that in 1651 and 1652 he laid out near 2000l. for lands and a house at Hersham.</p><p>During the siege of Colchester, he and Booker were sent for thither, to encourage the soldiers; which they did by assuring them that the town would soon be taken; which proved true in the event.—Having, in 1650, written publicly that the parliament should not continue, but a new government arise; agreeably to which, in his almanac for 1653, he asserted that the parliament stood upon a ticklish foundation, and that the commonalty and soldiery would join together against them. Upon which he was summoned before the committee of plundered ministers; but, receiving notice of it before the arrival of the messenger, he applied to his friend Lenthal the speaker, who pointed out the offensive passages. He immediately altered them; attended the committee next morning, with 6 copies printed, which six alone he acknowledged to be his; and by that means came off with only 13 days custody by the serjeant at arms. This year he was engaged in a dispute with Mr. Thomas Gataker.—In 1665 he was<cb/> indicted at Hicks's-hall, for giving judgment upon stolen goods; but was acquitted. And in 1659, he received, from the king of Sweden, a present of a gold chain and medal, worth about 50l. on account of his having mentioned that monarch with great respect in his almanacs of 1657 and 1658.—After the Restoration, in 1660, being taken into custody, and examined by a committee of the house of commons, touching the execution of Charles the 1st, he declared, that Robert Spavin, then Secretary to Cromwell, dining with him soon after the fact, assured him it was done by cornet Joyce. The same year he sued out his pardon under the broad seal of England; and afterwards continued in London till 1665; when, upon the raging of the plague there, he retired to his estate at Hersham. Here he applied himself to the study of physic, having, by means of his friend Elias Ashmole, procured from archbishop Sheldon a licence to practise it, which he did, a<*> well as astrology, from thence till the time of his death. —In October 1666 he was examined before a committee of the house of commons concerning the sire of London, which happened in September that year. A little before his death, he adopted for his son, by the name of <hi rend="italics">Merlin junior,</hi> one Henry Coley, a taylor by trade; and at the same time gave him the impression of his almanac, which had been printed for 36 years successively. This Coley became afterwards a celebrated astrologer, publishing in his own name, almanacs, and books of astrology, particularly one intitled <hi rend="italics">A Key to Astrology.</hi></p><p>Lilly died of a palsy 1681, at 79 years of age; and his friend Mr. Ashmole placed a monument over his grave in the church of Walton upon Thames.</p><p>Lilly was author of many works. His <hi rend="italics">Observations on the Life and Death of Charles late King of England,</hi> if we overlook the astrological nonsense, may be read with as much satisfaction as more celebrated histories; Lilly being not only very well informed, but strictly impartial. This work, with the Lives of Lilly and Ashmole, written by themselves, were published in one volume, 8vo, in 1774, by Mr. Burman. His other works were principally as follow:</p><p>1. Merlinus Anglicus junior.—2. Supernatural Sight. —3. The White King's Prophecy.—4. England's Prophetical Merlin: all printed in 1644.—5. The Starry Messenger, 1645.—6. Collection of Prophecies, 1646.—7. A Comment on the White King's Prophecy, 1646.—8. The Nativities of Archbishop Laud and Thomas earl of Strafford, 1646.—9. Christian Astrology, 1647: upon this piece he read his lectures in 1648, mentioned above.—10. The third book of Nativities, 1647.—11. The World's Catastrophe, 1647.— 12. The Prophecies of Ambrose Merlin, with a Key, 1647.—13. Trithemius, or the Government of the World by Presiding Angels, 1647.—14. A treatise of the Three Suns seen in the winter of 1647, printed in 1648.—15. Monarchy or no Monarchy, 1651.— 16. Observations on the Life and Death of Charles, late king of England, 1651; and again in 1651, with the title of Mr. William Lilly's True History of king James and king Charles the 1st, &c.—17. Annus Tenebrosus; or, the Black Year. This drew him into the dispute with Gataker, which Lilly carried on in his Almanac in 1654.<pb n="42"/><cb/></p></div1><div1 part="N" n="LIMB" org="uniform" sample="complete" type="entry"><head>LIMB</head><p>, the outermost border, or graduated edge, of a quadrant, astrolabe, or such like mathematical instrument.</p><p>The word is also used for the arch of the primitive circle, in any projection of the sphere in plano.</p><p><hi rend="smallcaps">Limb</hi> also signifies the outermost border or edge of the sun or moon; as the upper Limb, or edge; the lower Limb; the preceding Limb, or side; the following Limb.—Astronomers observe the upper or lower Limb of the sun or moon, to find their true height, or that of the centre, which differs from the others by the semidiameter of the disc.</p></div1><div1 part="N" n="LIMBERS" org="uniform" sample="complete" type="entry"><head>LIMBERS</head><p>, in Artillery, a sort of advanced train, joined to the carriage of a cannon on a march. It is composed of two shafts, wide enough to receive a horse between them, called the <hi rend="italics">fillet horse:</hi> these shafts are joined by two bars of wood, and a bolt of iron at one end, and mounted on a pair of rather small wheels. Upon the axle-tree rises a strong iron spike, which is put into a hole in the hinder part of the train of the gun carriage, to draw it by. But when a gun is in action, the Limbers are taken off, and run out behind it.—See the dimensions and figure of it in Müller's Treatise of Artillery, pa. 187.</p></div1><div1 part="N" n="LIMIT" org="uniform" sample="complete" type="entry"><head>LIMIT</head><p>, is a term used by mathematicians, for some determinate quantity, to which a variable one continually approaches, and may come nearer to it than by any given difference, but can never go beyond it; in which sense a circle may be said to be the Limit of all its inscribed and circumscribed polygons: because these, by increasing the number of their sides, can be made to be nearer equal to the circle than by any space that can be proposed, how small soever it may be.</p><p>In Algebra, the term <hi rend="italics">Limit</hi> is applied to two quantities, of which the one is greater and the other less than some middle quantity, as the root of an equation, &c. And in this sense it is used when speaking of the Limits of equations, a method by which their solution is greatly facilitated.</p><p><hi rend="smallcaps">Limit</hi> <hi rend="italics">of Distinct Vision,</hi> in Optics. See <hi rend="italics">Distinct</hi> <hi rend="smallcaps">Vision.</hi></p><p><hi rend="smallcaps">Limit</hi> <hi rend="italics">of a Planet,</hi> has been sometimes used for its greatest heliccentric latitude.</p><p><hi rend="smallcaps">Limited</hi> <hi rend="italics">Problem,</hi> denotes a problem that has but one solution, or some determinate number of solutions: as to describe a circle through three given points that do not lie in a right line, which is limited to one solution only; to divide a parallelogram into two equal parts by a line parallel to one side, which admits of two solutions, according as the line is parallel to the length or breadth of the parallelogram; or to divide a triangle in any ratio by a line parallel to one side, which is limited to three solutions, as the line may be parallel to any of the three sides.</p></div1><div1 part="N" n="LINE" org="uniform" sample="complete" type="entry"><head>LINE</head><p>, in Geometry, a quantity extended in length only, without either breadth or thickness.</p><p>A Line is sometimes considered as generated by the flux or motion of a point; and sometimes as the limit or termination of a superficies, but not as any part of that surface, however small.</p><p><hi rend="italics">Lines</hi> are either <hi rend="italics">right</hi> or <hi rend="italics">curved.</hi> A <hi rend="italics">right,</hi> or <*>aight Line, is the nearest distance between two points, which are its extremes or ends; or it is a Line which has in every part of it the same direc-<cb/> tion or position. But a <hi rend="italics">curve Line</hi> has in every part of it a different direction, and is not the shortest distance between its extremes or ends.</p><p><hi rend="italics">Right</hi> <hi rend="smallcaps">Lines</hi> are all of the same species; but curves are of an infinite number of different sorts. As many may be conceived as there are different compound motions, or as many as there may be different relations between their ordinates and abscisses. See <hi rend="smallcaps">Curves.</hi></p><p>Again, <hi rend="italics">Curve</hi> <hi rend="smallcaps">Lines</hi> are usually divided into <hi rend="italics">geometrical</hi> and <hi rend="italics">mechanical.</hi></p><p><hi rend="italics">Geometrical Lines,</hi> are those which may be found exactly in all their parts. See <hi rend="smallcaps">Geometrical Line.</hi></p><p><hi rend="italics">Mechanical Lines</hi> are such as are not determined exactly in all their parts, but only nearly, or tentatively. But</p><p>Des Cartes, and his followers, define geometrical Lines to be those which may be expressed by an algebraical equation of a determinate or finite degree; called its <hi rend="italics">locus.</hi> And mechanical Lines, such as cannot be expressed by such an equation.</p><p>But others distinguish the same Lines by the name <hi rend="italics">algebraical</hi> and <hi rend="italics">transcendental.</hi></p><p>Lines are also divided into orders, by Newton, according to the number of intersections which may be made by them and a right Line, viz, the 1st, 2d, 3d, 4th, &c, order, according as they may be cut by a right Line, in 1, or 2, or 3, or 4, &c, points. In this way of considering them, the right Line only is of the 1st order, being but one in number; the 2d order contains 4 curves only, being such as may be cut from a cone by a plane, viz, the circle, the ellipse, the hyperbola, and the parabola; the lines of the 3d order have been enumerated by Newton, in a particular treatise, who makes their number amount to 72; but Mr. Stirling found 4 others, and Mr. Stone 2 more; though it is disputed by some whether these 2 last ought to be accounted different from some of Newton's, or not. See Newton's Enumer. Lin. Tert<*> Ordin. also Stirling's Lineæ Tert. Ordin. Newtonianæ Oxon. 1717, 8vo. and Philos. Trans. number 456, &c. Again,</p><p><hi rend="italics">Algebraical Lines</hi> are divided into different orders according to the power or degree of their equations. So, the simple equation or equation of the 1st degree, denotes the 1st order or right line; the equation , of the 2d degree, denotes the Lines of the 2d order; and the equation of the 3d degree, expresses the Lines of the 3d order; and so on. See Cramer's Introd. à l'Analyse des Lignes Courbes.</p><p>Lines, considered as to their positions, are either <hi rend="italics">parallel, perpendicular,</hi> or <hi rend="italics">oblique.</hi> And the construction and properties of each of these, see under the respective terms.</p><p><hi rend="smallcaps">Line</hi> also denotes a French measure of length, being the 12th part of an inch, or the 144th part of a foot</p><p>In <hi rend="italics">Astronomy,</hi></p><p><hi rend="smallcaps">Line</hi> <hi rend="italics">of the Apses,</hi> or <hi rend="italics">Apsides,</hi> the Line joining the two apses, or the longer axis of the orbit of a planet.</p><p><hi rend="italics">Fiducial Line,</hi> the index line or edge of the ruler, which passes through the middle of an astrolabe, or other instrument, on which the sights are fitted, and marking the divisions.<pb n="43"/><cb/></p><p><hi rend="italics">Horizontal Line,</hi> a Line parallel to the horizon.</p><p><hi rend="smallcaps">Line</hi> <hi rend="italics">of the Nodes,</hi> that which joins the nodes of the orbit of a planet, being the common section of the plane of the orbit with the plane of the ecliptic.</p><p>In <hi rend="italics">Dialling,</hi></p><p><hi rend="italics">Horizontal Line,</hi> is the common section of the horizon and the dial-plate.</p><p><hi rend="italics">Horary,</hi> or <hi rend="italics">Hour Lines,</hi> are the common intersections of the hour-circles of the sphere with the plane of the dial.</p><p><hi rend="italics">Equinoctial Line</hi> is the common intersection of the equinoctial and the plane of the dial.</p><p>In <hi rend="italics">Fortification, Line</hi> is sometimes used for a ditch, bordered with its parapet: and sometimes for a row of gabions, or sacks of earth, extended lengthwise on the ground, to serve as a shelter against the enemy's fire.</p><p>When the trenches were carried on within 30 paces of the glacis, they drew two Lines, one on the right, and the other on the left, for a place of arms.</p><p>Lines are commonly made to shut up an avenue or entrance to some place; the sides of the entrance being covered by rivers, woods, mountains, morasses, or other obstructions, not easy to be passed over by an army. When they are constructed in an open country, they are carried round the place to be defended, and resemble the Lines surrounding a camp, called Lines of circumvallation. Lines are also thrown up to stop the progress of an army; but the term is most used for the Line which covers a pass that can only be attacked in front.</p><p>When lines are made to cover a camp, or a large tract of land, where a considerable body of troops is posted, the work is not made in one straight, or uniformly bending Line; but, at certain distances, the Lines project in saliant angles, called redents, redans, or flankers, towards the enemy. The distance between these angles is commonly between the limits of 200 and 260 yards; the ordinary flight of a musket ball, point blank, being commonly within those limits; though muskets a little elevated will do effectual service at the distance of 360 yards.</p><p><hi rend="italics">Fundamental Line,</hi> is the first Line drawn for the plan of a place, and which shews its area.</p><p><hi rend="italics">Central Line,</hi> is the Line drawn from the angle of the centre to the angle of the bastion.</p><p><hi rend="italics">Line of Defence,</hi> &c. See <hi rend="smallcaps">Defence</hi> &c.</p><p><hi rend="italics">Line of Approach,</hi> or <hi rend="italics">Attack,</hi> signifies the work which the besiegers carry on under cover, to gain the moat, and the body of the place.</p><p><hi rend="italics">Line of Circumvallation,</hi> is a Line or trench cut by the besiegers, within cannon-shot of the place, which ranges round the camp, and secures its quarters against any relief to be brought to the besieged.</p><p><hi rend="italics">Line of Contravallation,</hi> is a ditch bordered with a parapet, serving to cover the besiegers on the side next the place, and to stop the sallies of the garrison.</p><p><hi rend="italics">Lines of Communication</hi> are those which run from one work to another.</p><p><hi rend="italics">Line of the Base,</hi> is that which joins the points of the two nearest bastions.</p><p>To <hi rend="italics">Line</hi> a work, signisies to face it, as with brick or stone; for example, to strengthen a rampart with a firm wall, <*> to encompass a parapet or moat with good <*>urf, &c.<cb/></p><div2 part="N" n="Line" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Line</hi></head><p>, in Geography and Navigation, is emphatically used for the Equator or Equinoctial Line.</p><p>The seamen use to baptize their fresh men, and passengers, the first time they cross the Line: that is, to dip them in the sea, suspended by a rope from the yardarm, unless they compound for it, by giving something to drink.</p><p>In <hi rend="italics">Perspective,</hi></p><p>The <hi rend="italics">Geometrical Line,</hi> is a right Line drawn in any manner on the geometrical plane.</p><p><hi rend="italics">Terrestrial</hi> or <hi rend="italics">Fundamental Line,</hi> is the common intersection of the geometrical plane and plane of the picture.</p><p><hi rend="italics">Line of the Front,</hi> is any Line parallel to the terrestrial Line.</p><p><hi rend="italics">Vertical Line,</hi> is the section of the vertical and draft planes.</p><p><hi rend="italics">Visual Line,</hi> is the Line or ray conceived to pass from the object to the eye.</p><p><hi rend="italics">Objective Line,</hi> is any Line drawn on the geometrical plane, whose representation is sought for in the draught or picture.</p><p><hi rend="italics">Line of Measures,</hi> is used by Oughtred, and others, to denote the diameter of the primitive circle, in the projection of the sphere in plano, or that Line in which falls the diameter of any circle to be projected.</p><p><hi rend="smallcaps">Linear Numbers</hi>, are such as have relation to length only; such, for example, as express one side of a plane figure; and when the plane figure is a square, the linear number is called a root.</p><p><hi rend="smallcaps">Linear Problem</hi>, is one that can be solved geometrically by the intersection of two right lines. This is called a simple problem, and is capable of only one solution.</p></div2></div1><div1 part="N" n="LIQUID" org="uniform" sample="complete" type="entry"><head>LIQUID</head><p>, a fluid which wets or smears such bodies as are immersed in it, arising from some configuration of its particles, which disposes them to adhere to the surfaces of bodies contiguous to them. Thus water, oil, milk, &c, are Liquids, as well as fluids; but quicksilver is not a Liquid, but simply a fluid.</p></div1><div1 part="N" n="LISLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LISLE</surname> (<foreName full="yes"><hi rend="smallcaps">William de</hi></foreName>)</persName></head><p>, a very learned French geographer, was born at Paris in 1675. His father being much occupied in the same way, young Lisle began at 9 years of age to draw maps, and soon made a great progress in this art. In 1699 he first distinguished himself to the public, by giving a map of the world, and other pieces, which procured him a place in the Academy of Sciences, 1702. He was afterwards appointed geographer to the king, with a pension, and had the honour of instructing the king himself in geography, for whose particular use he drew up several works. De Lisle's reputation was so great, that scarcely any history or travels came out without the embellishment of his maps. Nor was his name less celebrated abroad than in his own country. Many sovercigns in vain attempted to draw him out of France. The Czar Peter, when at Paris on his travels, paid him a visit, to communicate to him some remarks upon Muscovy; but more especially, says Fontenell<*>, to learn from him, better than he could any where else, the extent and situation of his own dominions. De Lisle died of an apoplexy in 1726, at 51 years of age. Beside the excellent maps he published, he wrote<pb n="44"/><cb/> many pieces in the Memoirs of the Academy of Sciences.</p></div1><div1 part="N" n="LIST" org="uniform" sample="complete" type="entry"><head>LIST</head><p>, or <hi rend="smallcaps">Listel</hi>, a small square moulding, serving to crown or accompany larger mouldings; or on occasion to separate the flutings of columns.</p><p>LITERAL <hi rend="smallcaps">Algebra.</hi> See <hi rend="smallcaps">Algebra.</hi></p></div1><div1 part="N" n="LIZARD" org="uniform" sample="complete" type="entry"><head>LIZARD</head><p>, in Astronomy. See <hi rend="smallcaps">Lacerta.</hi></p></div1><div1 part="N" n="LOADSTONE" org="uniform" sample="complete" type="entry"><head>LOADSTONE</head><p>, or <hi rend="smallcaps">Magnet;</hi> which see.</p><p>LOCAL <hi rend="italics">Problem,</hi> is one that is capable of an insinite number of different solutions; because the point, which is to solve the problem, may be indi<*>ferently taken within a certain extent; as suppose any where in such a line, within such a plane figure, &c, which is called a <hi rend="italics">geometrical Locus.</hi></p><p>A Local problem is <hi rend="italics">simple,</hi> when the point sought is in a right line; <hi rend="italics">plane,</hi> when the point sought is in the circumference of a circle; <hi rend="italics">solid,</hi> when it is in the circumference of a conic section; or <hi rend="italics">sursolid,</hi> when the point is in the perimeter of a line of a higher kind.</p><p><hi rend="smallcaps">Local Motion</hi>, or <hi rend="italics">Loco-Motion,</hi> the change of place: See <hi rend="smallcaps">Motion.</hi></p></div1><div1 part="N" n="LOCI" org="uniform" sample="complete" type="entry"><head>LOCI</head><p>, the plural of <hi rend="smallcaps">Locus</hi>, which see.</p></div1><div1 part="N" n="LOCUS" org="uniform" sample="complete" type="entry"><head>LOCUS</head><p>, is some line by which a local or indeterminate problem is solved; or a line of which any point may equally solve an indeterminate problem.</p><p>Loci are expressed by algebraic equations of different orders according to the nature of the Locus. If the equation is constructed by a right line, it is called <hi rend="italics">Locus ad rectum;</hi> if by a circle, <hi rend="italics">Locus ad circulum;</hi> if by a parabola, <hi rend="italics">Locus ad parabolam;</hi> if by an ellipsis, <hi rend="italics">Locus ad ellipsim;</hi> and so on.</p><p>The Loci of such equations as are right lines or circles, the ancients called <hi rend="italics">plane loci;</hi> and of those that are conic sections, <hi rend="italics">solid loci;</hi> but such as are curves of a higher order, <hi rend="italics">sursolid loci.</hi> But the moderns distinguish the Loci into orders according to the dimensions of the equations by which they are expressed, or the number of the powers of indeterminate or unknown quantities in any one term: thus, the equation</p><p> denotes a Locus of the 1st order, but , or , &c, a Locus of the 2d order, and , or , &c, a Locus of the 3d order, and so on; where <hi rend="italics">x</hi> and <hi rend="italics">y</hi> are unknown or indeterminate quantities, and the others known or determinate ones; also <hi rend="italics">x</hi> denotes the absciss, and <hi rend="italics">y</hi> the ordinate of the curve or line which is the Locus of the equation.</p><p>Forinstance, suppose two variable or indeterminate right lines AP, AQ, making any given angle PAQ between <figure/> them, where they are supposed to commence, and to extend indefinitely both ways from the point A: then calling any AP, <hi rend="italics">x,</hi> and its corresponding ordinate<cb/> PQ, <hi rend="italics">y,</hi> continually changing its position by moving parallel to itself along the indesinite line AP; also in the line AP assume AB = <hi rend="italics">a,</hi> and from B draw BC parallel to PQ and = <hi rend="italics">b:</hi> then the indefinite line AQ is called in general a geometrical Locus, and in particular the Locus of the equation ; for whatever point Q is, the triangles ABC, APQ are always similar, and therefore , that is , and therefore is the equation to the right line AQ, or AQ is the Locus of the equation .</p><p>Again, if AQ be a parabola, the nature of which is such, that , or , and therefore is the equation which has the parabola for its Locus, or the parabola is the Locus to every equation of this form . <figure/></p><p>Or if AQ be a circle, having its radius AB = <hi rend="italics">a,</hi> the nature of which is this, that , or or ; therefore the Locus of the equation of this form , is always a circle. <figure/></p><p>In like manner it will appear, that the ellipse is the Locus to the equation , and the hyperbola the Locus to the equation ; where <hi rend="italics">t</hi> is the transver<*>e, and <hi rend="italics">c</hi> the conjugate axis of the ellipse or hyperbola.</p><p>All equations, whose Loci are of the first order, may be reduced to one of the 4 following forms: ; ; ; ; where the letter <hi rend="italics">c</hi> denotes the distance that the ordinates commence from the line AP, either on the one side or the other of it, according as the sign of that quantity is + or -.</p><p>All Loci of the 2d degree are conic sections, viz, either the parabola, the circle, ellipsis, or hyperbola. Therefore when an equation is given, whose Locus is of the 2d degree, and it is required to draw that Locus, or, which is the same thing, to construct the equation generally; bring over all the terms of the equation to one side, so that the other side be 0; then to know which of the conic sections it denotes, there will be two general cases, viz, either when the rectangle <hi rend="italics">xy</hi> is in the equation, or when it is not in it.</p><p><hi rend="italics">Case</hi> 1. When the term <hi rend="italics">xy</hi> is not in the proposed equation. Then, 1st, if only one of the squares<pb n="45"/><cb/> <hi rend="italics">x</hi><hi rend="sup">2</hi>, <hi rend="italics">y</hi><hi rend="sup">2</hi> be found in it, the Locus will be a parabola. 2d, If both the squares be in it, and if they have the same sign, the Locus will be a circle or an ellipse. 3d, But if the signs of the squares <hi rend="italics">x</hi><hi rend="sup">2</hi>, <hi rend="italics">y</hi><hi rend="sup">2</hi> be different, the Locus will be an hyperbola, or the opposite hyperbolas.</p><p><hi rend="italics">Case</hi> 2. When the rectangle <hi rend="italics">xy</hi> is in the proposed equation; then 1st, If neither of the squares <hi rend="italics">x</hi><hi rend="sup">2</hi>, <hi rend="italics">y</hi><hi rend="sup">2</hi>, or only one of them be in the equation, the Locus will be an hyperbola between the asymptotes. 2d, If both <hi rend="italics">x</hi><hi rend="sup">2</hi> and <hi rend="italics">y</hi><hi rend="sup">2</hi> be in it, having different signs, the Locus will be an hyperbola, having the abscisses on its diameter. 3d, If both the squares be in it, and with the same sign, then if the coefficient of <hi rend="italics">x</hi><hi rend="sup">2</hi> be greater than the square of half the coefficient of <hi rend="italics">xy,</hi> the Locus will be an ellipse; if equal, a parabola; and if less, an hyperbola.</p><p>This method of determining geometric Loci, by reducing them to the most compound or general equations, was first published by Mr. Craig, in his Treatise on the Quadrature of Curves, in 1693. It is explained at large in the 7th and 8th books of I'Hospital's Conic Sections. See this subject particularly illustrated in Maclaurin's Algebra. The method of Des Cartes, of finding the Loci of equations of the 2d order, is a good one, viz, by extracting the root of the equation. See his Geometry; as also Stirling's Illustratio Linearum Tert<*> Ordinis. The doctrine of these Loci is likewise well treated by De Witt in his Elementa Curvarum. And Bartholomæus Intieri, in his Aditus ad Nova Arcana Geometrica delegenda, has shewn how to find the Loci of equations of the higher orders. Mr. Stirling too, in his treatise above-mentioned, has given an example or two of finding the Loci of equations of 3 dimensions. Euclid, Apollonius, Aristæus, Fermat, Viviani, have also written on the subject of Loci.</p></div1><div1 part="N" n="LOG" org="uniform" sample="complete" type="entry"><head>LOG</head><p>, in Navigation, is a piece of thin board, of a sectoral or quadrantal form, loaded in the circular side with lead sufficient to make it swim upright in the water; to which is fastened a line of about 150 fathoms, or 300 yards long, called the Log-line, which is divided into certain spaces, called Knots, and wound on a reel which turns very freely, for the line to wind easily off.</p><p>The use of the Log, or Log-line, is to measure the velocity of the ship, or rate at which she runs, which is done from time to time, as the foundation upon which the ship's reckoning, or finding her place, is kept; and the practice is to heave the Log into the sea, with the line tied to it, and observe how much of the line is run off the reel, while the ship sails, during the space of half a minute, which time is measured by a sand-glass made to run that time very exactly. About 10 fathoms of stray or waste line is left next the Log before the knotting or counting commence, that space being usually allowed to carry the Log out of the eddy of the ship's wake.</p><p>The using of the Log for finding the velocity of the ship, is called <hi rend="italics">Heaving the Log,</hi> and is thus performed: One man holds the reel, and another the halfminute glass; an officer of the watch throws the Log over the ship's stern, on the lee-side, and when he observes the stray line, and the first mark is going off,<cb/> he cries <hi rend="italics">turn!</hi> when the glass-holder instantly turns the glass crying out <hi rend="italics">done!</hi> then watching the glass, the moment it is run out he says <hi rend="italics">stop!</hi> upon which the reel being quickly stopt, the last mark run off shews the number of knots, and the distance of that mark from the reel is estimated in fathoms: then the knots and fathoms together shew the distance run in half a minute, or the distance per hour nearly, by considering the knots as miles, and the fathoms as decimals of a mile: thus if 7 knots and 4 fathoms be observed, then the ship runs at the rate of 7.4 miles an hour.</p><p>It follows, therefore, that the length of each knot, or division of the line, ought to be the same part of a sea mile, as half a minute is of an hour, that is 1/120 th part. Now it is found that a degree of the meridian contains nearly 366,000 feet, therefore 1/<*>0 of this, or a nautical mile, will be 6100 feet; the 1/120 th of which, or 51 feet nearly, should be the length of each knot, or division of the Log-line. But because it is safer to have the reckoning rather before the ship than after it, therefore it is usual now to make each knot cqual to 8 fathoms or 48 feet. But the knots are made sometimes to contain only 42 feet; and this method of dividing the Log-line was founded on the supposition, that 60 miles, of 5000 feet each, made a degree; for 1/120th of 5000 is 41 2/3, or in round numbers 42 feet. And although many mariners find by experience that this length of the knot is too short, yet rather than quit the old way, they use sand-glasses for half-minute ones that run only 24 or 25 seconds. The sand, or halfminute glass, may be tried by a pendulum vibrating seconds, in the following manner: On a round nail or peg, hang a thread or fine string that has a musket ball fixed to one end, carefully measuring between the centre of the ball and the string's loop over the nail 39 1/8 inches, being the length of a second pendulum; then make it swing or vibrate very small arches, and count one for every time it passes under the nail, beginning at the second time it passes; and the number of swings made during the time the glass is running out, shews the seconds in the glass.</p><p>It is not known who was the inventor of this method of measuring the ship's way, or her rate of sailing; but no mention of it occurs till the year 1607, in an East-India voyage, published by Purchas; and from that time its name occurs in other voyages in his collections; after which it became famous, being noticed both by our own authors, and by foreigners; as by Gunter in 1623; Snellius, in 1624; Metius, in 1631; Oughtred, in 1633; Herigone, in 1634; Saltonstall, in 1636; Norwood, in 1637; Fournier, in 1643; and almost all the succeeding writers on navigation of every country. Various improvements have lately been made of this instrument by different persons.</p></div1><div1 part="N" n="LOGARITHM" org="uniform" sample="complete" type="entry"><head>LOGARITHM</head><p>, from the Greek <foreign xml:lang="greek">logos</foreign> <hi rend="italics">ratio,</hi> and <foreign xml:lang="greek">ari<*>mos</foreign> <hi rend="italics">number;</hi> q. d. <hi rend="italics">ratio of numbers,</hi> or perhaps rather <hi rend="italics">number of ratios;</hi> the indices of the ratios of numbers to one another; or a series of numbers in arithmetical proportion, corresponding to as many others in geometrical proportion, in such sort that 0 corresponds to, or is the index of 1, in the geometricals. They have been devised for the ease of large arithmetical calculations.<pb n="46"/><cb/></p><p>Thus, 0, 1, 2, 3, 4, &c, indices or Logarithms, <hi rend="brace"><note anchored="true" place="unspecified">the geometrical progressions, or common numbers.</note> <hi rend="brace">1, 2, 4, 8, 16, &c, or 2<hi rend="sup">0</hi>, 2<hi rend="sup">1</hi>, 2<hi rend="sup">2</hi>, 2<hi rend="sup">3</hi>, 2<hi rend="sup">4</hi>, &c,</hi> <hi rend="brace">1, 3, 9, 27, 81, &c, or 3<hi rend="sup">0</hi>, 3<hi rend="sup">1</hi>, 3<hi rend="sup">2</hi>, 3<hi rend="sup">3</hi>, 3<hi rend="sup">4</hi>, &c,</hi> <hi rend="brace">1, 10, 100, 1000, 10000, &c, or 10<hi rend="sup">0</hi>, 10<hi rend="sup">1</hi>, 10<hi rend="sup">2</hi>, 10<hi rend="sup">3</hi>, 10<hi rend="sup">4</hi>, &c,</hi></hi> Where the same indices, or Logarithms, serve equally for any geometric series; and from which it is evident, that there may be an endless variety of sets of Logarithms to the same common numbers, by varying the 2d term 2, or 3, or 10, &c of the geometric series; as this will change the original series of terms whose indices are the numbers 1, 2, 3, &c; and by interpolation the whole system of numbers may be made to enter the geometrical series, and receive their proportional Logarithms, whether integers or decimals.</p><p>Or the Logarithm of any given number, is the index of such a power of some other number, as is equal to the given one. So if N be = <hi rend="italics">r</hi><hi rend="sup">n</hi>, then the Logarithm of N is <hi rend="italics">n,</hi> which may be either positive or negative, and <hi rend="italics">r</hi> any number whatever, according to the different systems of Logarithms. When N is 1, then <hi rend="italics">n</hi> is = 0, whatever the value of <hi rend="italics">r</hi> is; and consequently the Logarithm of <*> is always 0 in every system of Logarithms. When <hi rend="italics">n</hi> is = 1, then N is = <hi rend="italics">r;</hi> consequently the root <hi rend="italics">r</hi> is always the number whose Logarithm is 1, in every system. When <hi rend="italics">r</hi> is = 2.718281828459 &c, the indices are the hyperbolic Logarithms; so that <hi rend="italics">n</hi> is always the hyperbolic Logarithm of ―(2.718 &c))<hi rend="sup">n</hi>. But in the common Logarithms, <hi rend="italics">r</hi> is = 10; so that the common Logarithm of any number, is the index of that power of 10 which is equal to the said number; so the common Logarithm of N = 10<hi rend="sup">n</hi>, is <hi rend="italics">n</hi> the index of the power of 10; for example, 1000, being the 3d power of 10, has 3 for its Logarithm; and if 50 be = 10<hi rend="sup">1.69<*>97</hi>, then is 1.69897 the common Logarithm of 50. And hence it follows that this decimal series of terms</p><p>1000, 100, 10, 1, .1, .01, .001, or 10<hi rend="sup">3</hi>, 10<hi rend="sup">2</hi>, 10<hi rend="sup">1</hi>, 10<hi rend="sup">0</hi>, 10<hi rend="sup">-1</hi>, 10<hi rend="sup">-2</hi>, 10<hi rend="sup">-3</hi>, have 3, 2, 1, 0, -1, -2, -3, respectively for the Logarithms of those terms.</p><p>The Logarithm of a number contained between any two terms of the first series, is included between the two corresponding terms of the latter; and therefore that Logarithm will consist of the same index, whether positive or negative, as the smaller of those two terms, together with a decimal fraction, which will always be positive. So the number 50 falling between 10 and 100, its Logarithm will fall between 1 and 2, being indeed equal to -1.69897 nearly: also the number .05 falling between the terms .1 and .01, its Logarithm will fall between - 1 and - 2, and is indeed = - 2 + .69897, the index of the less term together with the decimal .69897. The index is also called the Characteristic of the Logarithms, and is always an integer, either positive or negative, or else = 0; and it shews what place is occupied by the first significant figure of the given number, either above or below the place of units, being in the former case + or positive; in the latter - or negative.<cb/></p><p>When the characteristic of a Logarithm is negative, the sign — is commonly set over it, to distinguish it from the decimal part, which, being the Logarithm found in the tables, is always positive: so - 2 + .69897, or the Logarithm of .05, is written thus ―2.69897. But on some occasions it is convenient to reduce the whole expression to a negative form; which is done by making the characteristic less by 1, and taking the <hi rend="italics">arithmetical complement</hi> of the decimal, that is, beginning at the left hand, subtract each figure from 9, except the last significant figure, which is subtracted from 10; so shall the remainders form the Logarithm wholly negative: thus the Logarithm of .05, which is ―2.69897 or - 2 + .69897, is also expressed by - 1.30103, which is all negative. It is also sometimes thought more convenient to express such Logarithms entirely as positive, namely by only joining to the tabular decimal the complement of the index to 10; and in this way the above Logarithm is expressed by 8.69897; which is only increasing the indices in the scale by 10.</p><p><hi rend="italics">The Properties of Logarithms.</hi>—From the definition of Logarithms, either as being the indices of a series of geometricals, or as the indices of the powers of the same root, it follows that the multiplication of the numbers will answer to the addition of their Logarithms; the division of numbers, to the subtraction of their Logarithms; the raising of powers, to the multiplying the Logarithm of the root by the index of the power; and the extracting of roots, to the dividing the Logarithm of the given number by the index of the root required to be extracted.</p><p>So, 1st, Log. <hi rend="italics">ab</hi> or of , Log. 18 or of , .</p><p>Secondly, , , , Log. 1/2 or , Log. 1/<hi rend="italics">n</hi> or .</p><p>Thirdly, ; Log. <hi rend="italics">r</hi><hi rend="sup">1/n</hi> or of ; ; log. 2<hi rend="sup">1/3</hi> or of ; and . So that any number and its reciprocal have the same Logarithm, but with contrary signs; and the sum of the Logarithms of any number and its reciprocal, or complement, is equal to 0.</p><p><hi rend="italics">History and Construction of Logarithms.</hi>—The properties of Logarithms hitherto mentioned, or of arithmetical indices to powers or geometricals, with their various uses and properties, as above-mentioned, are taken notice of by Stifelius, in his Arithmetic; and indeed they were not unknown to the ancients; but they come all far short of the use of Logarithms in<pb n="47"/><cb/> Trigonometry, as first discovered by John Napier, baron of Merchiston in Scotland, and published at Edinburgh in 1614, in his Mirifici Logarithmorum Canonis De<*>riptio; which contained a large canon of Logarithms, with the description and uses of them; but their construction was reserved till the sense of the Learned concerning his invention should be known. This work was translated into English by the celebrated Mr. Edward Wright, and published by his son in 1616. In the year 1619, Robert Napier, son of the inventor of Logarithms, published a new edition of his late father's work, together with the promised Construction of the Logarithms, with other miscellaneous pieces written by his father and Mr. Briggs. And in the same year, 1619, Mr John Speidell published his New Logarithms, being an improved form of Napier's.</p><p>All these tables were of the kind that have since been called hyperbolical, because the numbers express the areas between the asymptote and curve of the hyperbola. And Logarithms of this kind were also soon after published by several other persons; as by Ursinus in 1619, Kepler in 1624, and some others.</p><p>On the first publication of Napier's Logarithms, Henry Briggs, then professor of Geometry in Gresham College in London, immediately applied himself to the study and improvement of them, and soon published the Logarithms of the first 1000 numbers, but on a new scale, which he had invented, viz, in which the Logarithm of the ratio of 10 to 1 is 1, the Logarithm of the same ratio in Napier's system being 2.30258 &c; and in 1624, Briggs published his Arithmetica Logarithmica, containing the Logarithms of 30,000 natural numbers, to 14 places of figures besides the index, in a form which Napier and he had agreed upon together, which is the present form of Logarithms; also in 1633 was published, to the same extent of figures, his Trigonometria Britannica, containing the natural and logarithmic sines, tangents, &c.</p><p>With various and gradual improvements, Logarithms were also published successively, by Gunter in 1620, Wingate in 1624, Henrion in 1626, Miller and Norwood in 1631, Cavalerius in 1632 and 1643, Vlacq and Rowe in 1633, Frobenius in 1634, Newton in 1658, Caramuel in 1670, Sherwin in 1706, Gardiner in 1742, and Dodson's Antilogarithmic Canon in the same year; besides many others of lesser note; not to mention the accurate and comprehensive tables in the Tables Portative, and in my own Logarithms lately published, where a complete history of this science may be seen, with the various ways of constructing them that have been invented by different authors.</p><p>In Napier's construction of Logarithms, the natural numbers, and their Logarithms, as he sometimes called them, or at other times the artisicial numbers, are supposed to arise, or to be generated, by the motions of points, describing two lines, of which the one is the natural number, and the other its Logarithm, or artificial. Thus, he conceived the line or length of the radius to be described, or run over, by a point moving along it in such a manner, that in equal portions of time it generated, or cut off, parts in a decreasing geometrical progression, leaving the several remainders, or sines, in geometrical progression also; whilst another<cb/> point described equal parts of an indefinite line, in the same equal portions of time; so that the respective sums of these, or the whole line generated, were always the arithmeticals or Logarithms of the aforesaid natural sines. In this idea of the generation of the Logarithms and numbers, Napier assumed 0 as the Logarithm of the greatest sine or radius; and next he limited his system, not by assuming a particular value to some assigned number, or part of the radius, but by supposing that the two generating points, which, by their motions along the two lines, described the natural numbers and Logarithms, should have their velocities equal at the beginning of those lines. And this is the reason that, in his table, the natural sines and their Logarithms, at the complete quadrant, have equal differences or increments; and this is also the reason why his scale of Logarithms happens accidentally to agree with what have since been called the hyperbolical Logarithms, which have likewise numeral differences equal to those of their natural numbers at the beginning; except only that these latter increase with the natural numbers, while his on the contrary decrease; the Logarithm of the ratio of 10 to 1 being thé same in both, namely 2.30258509 &c.</p><p>Having thus limited his system, Napier proceeds, in the posthumous work of 1619, to explain his construction of the Logarithmic canon. This he effects in various ways, but chiefly by generating, in a very easy manner, a series of proportional numbers, and their arithmeticals or Logarithms; and then finding, by proportion, the Logarithms to the natural sines from those of the natural numbers, among the original proportionals; a particular account of which may be seen in my book of Logarithms above mentioned.</p><p>The methods above alluded to, relate to Napier's or the hyperbolical system of Logarithms, and indeed are in a manner peculiar to that sort of them. But in an appendix to the posthumous work, mention is made of other methods, by which the common Logarithms, agreed upon by him and Briggs, may be constructed, and which it appears were written after that agreement. One of these methods is as follows: Having assumed 0 for the Logarithm of 1, and 1000 &c for the Logarithm of 10; this Logarithm of 10, and the successive quotients, are to be divided ten times by 5, by which divisions there will be obtained these other ten Logarithms, namely 2000000000, 400000000, 80000000, 16000000, 3200000, 640000, 128000, 25600, 5120, 1024; then this last Logarithm, and its quotients, being divided ten times by 2, will give these other ten Logarithms,</p><p>viz, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1. And the numbers answering to these twenty Logarithms are to be found in this manner, viz, Extract the 5th root of 10 (with ciphers), then the 5th root of that root, and so on for ten continual extractions of the 5th root: so shall these ten roots be the natural numbers belonging to the first ten Logarithms above found, in dividing continually by 5. Next, out of the last 5th root is to be extracted the square root, then the square root of this last root, and so on for ten successive extractions of the square root: so shall these last ten roots be the natural numbers corresponding to the Logarithms or quotients arising from<pb n="48"/><cb/> the last ten divisions by the number 2. And from these twenty Logarithms, 1, 2, 4, 8, &c, and their natural numbers, the author observes that other Logarithms and their numbers may be formed, namely by adding the Logarithms, and multiplying their corresponding numbers. But, besides the immense labour of this method, it is evident that this process would generate rather an antilogarithmic canon, such as Dodson's, than the table of Briggs.</p><p>Napier next mentions another method of deriving a few of the primitive numbers and their Logarithms, namely, by taking continually geometrical means, first between 10 and 1, then between 10 and this mean, and again between 10 and the last mean, and so on; and then taking the arithmetical means between their corresponèing Logarithms.</p><p>He then lays down various relations between numbers and their Logarithms, such as, that the products and quotients of numbers, answer to the sums and differences of their Logarithms; and that the powers and roots of numbers, answer to the products and quotients of the Logæithms when multiplied or divided by the index of the power or root, &c; as also that, of any two numbers, whose Logarithms are given, if each number be raised to the power denoted by the Logarithm of the other, the two r<*>sults will be equal; thus, if <hi rend="italics">x</hi> be the Logarithm of any number X, and <hi rend="italics">y</hi> the Logarithm of Y, then is . Napier then adverts to another method of making the Logarithms to a few of the prime integer numbers, which is well adapted to the construction of the common table of Logarithms: this method easily follows from what has been said above, and it depends on this property, that the Logarithm of any number in this scale, is one less than the number of places or figures contained in that power of the given number whose exponent is 10000000000, or the Logarithm of 10, at least as to integer numbers, for they really differ by a fraction, as is shewn by Mr. Briggs in his illustrations of these properties; printed at the end of this Appendix to the Construction of Logarithms.</p><p>Kepler gave a construction of Logarithms somewhat varied from Napier's. His work is divided into two parts: In the first, he raises a regular and purely mathematical system of proportions, and the measures of them, demonstrating both the nature and principles of the construction of Logarithms, which he calls the <hi rend="italics">measures of ralios:</hi> and in the second part, he applies those principles in the actual construction of his table, which contains only 1000 numbers and their Logarithms. The fundamental principles are briefly these: That at the beginning of the Logarithms, their increments or differences are equal to those of the natural numbers: that the natural numbers may be considered as the decreasing cosines of increasing arcs: and that the secants of those arcs at the beginning have the same differences as the cosines, and therefore the same differences as the Logarithms. Then, since the secants are the reciprocals of the cosines of the same arcs, from the foregoing principles, he establishes the following method of raising the first 100 Logarithms, to the numbers 1000, 999, 998, &c, to 900; viz, in this manner: Divide the radius 1000, increased with seven ciphers, by each of these numbers separate-<cb/> ly, and the quotients will be the secants of those arcs which have the divisors for their cosines; continuing the division to the 8th figure, as it is in that place only that the arithmetical and geometrical means differ. Then by adding continually the arithmetical means between every two successive secants, the sums will be the the series of Logarithms. Or by adding continually every two secants, the successive sums will be the series of the double Logarithms. He then derives all the other Logarithms from these first 100, by common principles.</p><p>Briggs first adverts to the methods mentioned above, in the Appendix to Napier's Construction, which methods were common to both these authors, and had doubtless been jointly agreed upon by them. He first gives an example of computing a Logarithm by the property, that the Logarithm is one less than the number of places or figures contained in that power of the given number whose exponent is the Logarithm of 10 with ciphers. Briggs next treats of the other general method of finding the Logarithms of prime numbers, which he thinks is an easier way than the former, at least when many figures are required. This method consists in taking a great number of continued geometrical means between 1 and the given number whose Logarithm is required; that is, first extracting the square root of the given number, then the root of the first root, the root of the 2d root, the root of the 3d root, and so on, till the last root shall exceed 1 by a very small decimal, greater or less according to the intended number of places to be in the Logarithm sought: then finding the Logarithm of this small number, by easy methods described afterwards, he doubles it as often as he made extractions of the square root, or, which is the same thing, he multiplies it by such power of 2 as is denoted by the said number of extractions, and the result is the required Logarithm of the given number; as is evident from the nature of Logarithms.</p><p>But as the extraction of so many roots is a very troublesome operation, our author devises some ingenious contrivances to abridge that labour, chiefly by a proper application of the several orders of the differences of numbers, forming the first instance of what may called <hi rend="italics">the differential method;</hi> but for a particular description of these methods, see my Treatise of Logarithms, above quoted, pag. 65 &c.</p><p>Mr. James Gregory, in his Vera Circuli Hyperbolæ Quadratura, printed at Padua in 1667, having approximated to the hyperbolic asymptotic spaces by means of a series of inscribed and circumscribed polygons, from thence shews how to compute the Logarithms, which are analogous to the areas of those spaces: and thus the quadrature of the hyperbolic spaces became the same thing as the computation of the Logarithms. He here also lays down various methods to abridge the computation, with the assistance of some properties of numbers themselves, by which the Logarithms of all prime numbers under 1000 may be computed, each by one multiplication, two divisions, and the extraction of the square root. And the same subject is farther pursued in his Exercitationes Geometricæ. In this latter place, he first finds an algebraic expression, in an insinite series, for the Logarithm of (1 + <hi rend="italics">a</hi>)/1, and then the like for the<pb n="49"/><cb/> Logarithm of ; and as the one series has all its terms positive, while those of the other are alternately positive and negative, by adding the two together, every 2d term is cancelled, and the double of the other terms gives the Logarithm of the product of and , or the Logarithm of the , that is of the ratio of 1 - <hi rend="italics">a</hi> to 1 + <hi rend="italics">a:</hi> thus, he finds, first , and , theref. , Which may be accounted Mr. James Gregory's method of making Logarithms.</p><p>In 1668, Nicholas Mercator published his Logarithmotechnia, five Methodus Construendi Logarithmos, nova, accurata, & facilis; in which he delivers a new and ingenious method for computing the Logarithms upon principles purely arithmetical; and here, in his modes of thinking and expression, he closely follows the <*>elebrated Kepler, in his writings on the same subject; accounting Logarithms as the measures of ratios, or as the number of ratiunculæ contained in the ratio which any number bears to unity. Purely from these principles, then, the number of the equal ratiunculæ contained in some one ratio, as of 10 to 1, being supposed given, our author shews how the Logarithm, or measure, of any other ratio may be found. But this, however, only by-thebye, as not being the principal method he intends to teach, as his last and best. Having shewn, then, that these Logarithms, or numbers of small ratios, or measures of ratios, may be all properly represented by numbers, and that of 1, or the ratio of equality, the Logarithm or measure being always 0, the Logarithm of 10, or the measure of the ratio of 10 to 1, is most conveniently represented by 1 with any number of ciphers; he then proceeds to shew how the measures of all other ratios may be found from this last supposition: and he explains these principles by some examples in numbers.</p><p>In the latter part of the work, Mercator treats of his other method, given by an infinite series of algebraic terms, which are collected in numbers by common addition only. He here squares the hyperbola, and finally finds that the hyperbolic Logarithm of 1 + <hi rend="italics">a,</hi> is equal to the insinite series &c; which may be considered as Mercator's quadrature of the hyperbola, or his general expression of an hyperbolic Logarithm, in an insinite series.</p><p>And this method was farther improved by Dr. Wallis, in the Philos. Trans. for the year 1668. The celebrated Newton invented also the same series for the quadrature of the hyperbola, and the construction of Logarithms, and that before the same were given by Gregory and Mercator, though unknown to one another, as appears by his letter to Mr. Oldenburg, dated October 24, 1676. The explanation and construction of the Logarithms are also farther pursued in his Fluxions, published in 1736 by Mr. Colson.</p><p>Dr. Halley, in the Philos. Trans. for the year 1695,<cb/> gave a very ingenious essay on the construction of Logarithms, intitled, “A most compendious and facile method for constructing the Logarithms, and exemplified and demonstrated from the nature of numbers, without any regard to the hyperbola, with a speedy method for sinding the number from the given Logarithm.”</p><p>Instead of the more ordinary definition of Logarithms, viz, ‘numerorum proportionalium æquidifferentes comites,’ the learned author adopts this other, ‘numeri rationum exponentes,’ as better adapted to the principle on which Logarithms are here constructed, considering them as the number of ratiunculæ contained in the given ratios whose Logarithms are in question. In this way he first arrives at the Logarithmic series before given by Newton and others, and afterwards, by various combinations and sections of the ratios, he derives others, converging still faster than the former. Thus he found the Logarithms of several ratios, as below, viz, when multiplied by the modulus peculiar to the scale of Logarithms, &c, the Log. of 1 to 1 + <hi rend="italics">q,</hi> &c, the Log. of 1 to 1 - <hi rend="italics">q,</hi> &c, the Log. of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> or &c, the same Log. of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> or &c, the same Log. of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> &c, the Log. of √<hi rend="italics">ab</hi> to (1/2)<hi rend="italics">z,</hi> &c, the same Log. of √<hi rend="italics">ab</hi> to (1/2)<hi rend="italics">z;</hi> where <hi rend="italics">a, b, q,</hi> are any quantitics, and the values of <hi rend="italics">x, y, z,</hi> are thus, viz, .</p><p>Dr. Halley also, sirst of any, performed the reverse of the problem, by assigning the number to a given Logarithm; viz, &c, or &c. where <hi rend="italics">l</hi> is the Logarithm of the ratio of <hi rend="italics">a</hi> the less, to <hi rend="italics">b</hi> the greater of any two terms.</p><p>Mr. Abraham Sharp of Yorkshire made many calculations and improvements in Logarithms, &c. The most remarkable of these were, his quadrature of the circle to 72 places of figures, and his computation of Logarithms to 61 figures, viz, for all numbers to 100, and for all prime numbers to 1100.</p><p>The celebrated Mr. Roger Cotes gave to the world a learned tract on the nature and construction of Logarithms: this was first printed in the Philos. Trans. N° 338, and afterwards with his Harmonia Mensurarum in 1722, under the title Logometria. This tract has justly been complained of, as very obscure and intrieate, and the principle is something between that of Kepler and the method of Fluxions. He invented the terms Modulus and Modular ratio, this being the ratio of &c to 1 or of 1 to <pb n="50"/><cb/> &c; that is the ratio of 2.718281828459 &c to 1, or the ratio of 1 to 0.367879441171 &c; the modulus of any system being the measure or Logarithm of that ratio, which in the hyp. Logarithms is 1, and in Briggs's or the common Logarithms is 0.434294481903 &c.</p><p>The learned Dr. Brook Taylor gave another method of computing Logarithms in the Philos. Trans. No. 352, which is founded on these three principles, viz, 1st, That the sum of the Logarithms of any two numbers is the Logarithm of the product of those numbers; 2d, That the Logarithm of 1 is 0, and consequently that the nearer any number is to 1, the nearer will its Logarithm be to 0; 3d, That the product of two numbers or factors, of which the one is greater and the other less than 1, is nearer to 1, than that factor is which is on the same side of 1 with itself; so of the two numbers 2/3 and 4/2, the product <*>/9 is less than 1, but yet nearer to it than 2/3 is, which is also less than 1.— And on these principles he founds an ingenious, though not very obvious, approximation to the Logarithms of given numbers.</p><p>In the Philos, Trans. a Mr. John Long gave a method of constructing Logarithms, by means of a small table, something in the manner of one of Briggs's methods for the same purpose.</p><p>Also in the Philos. Trans. vol. 61, a tract on the construction of Logarithms is given by the ingenious Mr. William Jones. In this method, all numbers are considered as some certain powers of a constant determined root: thus, any number <hi rend="italics">x</hi> is considered as the <hi rend="italics">z</hi> power of any root <hi rend="italics">r,</hi> or <hi rend="italics">x</hi> = <hi rend="italics">r</hi><hi rend="sup">z</hi> is taken as a general expression for all numbers in terms of the constant root <hi rend="italics">r</hi> and a variable exponent <hi rend="italics">z.</hi> Now the index <hi rend="italics">z</hi> being the Logarithm of the number <hi rend="italics">x,</hi> therefore to find this Logarithm, is the same thing as to find what power of the radix <hi rend="italics">r</hi> is equal to the number <hi rend="italics">x.</hi></p><p>An elegant tract on Logarithms, as a comment on Dr. Halley's method, was also given by Mr. Jones in his Synopsis Palmariorum Matheseos, published in the year 1706.</p><p>In the year 1742, Mr. James Dodson published his Anti-logarithmic Canon, containing all Logarithms under 100,000, and their corresponding natural numbers to eleven places of figures, with all their differences and the proportional parts; the whole arranged in the order contrary to that used in the common tables of numbers and Logarithms, the exact Logarithms being here placed first, and their corresponding nearest numbers in the columns opposite to them.</p><p>And in 1767, Mr. Andrew Reid published an “Essay on Logarithms,” in which he shews the computation of Logarithms from principles depending on the binomial theorem, and on the nature of the exponents of powers, the Logarithms of numbers being here considered as the exponents of the powers of 10. In this way he brings out the usual series for Logarithms, and exemplifies Dr. Halley's construction of them. But for the particulars of this, and the methods given by the other authors, we must refer to the historical preface to my treatise on Logarithms.</p><p>Besides the authors above-mentioned, many others have treated on the subject of Logarithms; among the principal of whom are Leibnitz, Euler, Maclaurin,<cb/> Wolfius, Keill, and professor Simson in an ingenious geo metrical tract on Logarithms, contained in his posthumous works, elegantly printed at Glasgow in the year 1776, at the expence of the learned Earl Stanhope, and by his lordship disposed of in presents among gentlemen most eminent for mathematical learning.</p><p>For the description and uses of Logarithms in numeral calculations, with the shortest method of constructing them, see the Historical Introduction to my Logarithms, pa. 124 & seq.</p><p><hi rend="italics">Briggs</hi>'s or <hi rend="italics">Common</hi> <hi rend="smallcaps">Logarithms</hi>, are those that have 1 for the Logarithm of 10, or which have 0.4342944819 &c for the modulus; as has been explained above.</p><p><hi rend="italics">Hyperbolic</hi> <hi rend="smallcaps">Logarithms</hi>, are those that were computed by the inventor Napier, and called also sometimes <hi rend="italics">Natural Logarithms,</hi> having 1 for their modulus, or 2.302585092994 &c for the Logarithm of 10. These have since been called Hyperbolical Logarithms, because they are analogous to the areas of a rightangled hyperbola between the asymptotes and the curve. See <hi rend="smallcaps">Logarithms</hi>, also <hi rend="smallcaps">Hyperbola</hi> and <hi rend="smallcaps">Asymptotic Space.</hi></p><p><hi rend="italics">Logislic</hi> <hi rend="smallcaps">Logarithms</hi>, are certain Logarithms of sexagesimal numbers or fractions, useful in astronomical calculations. The Logistic Logarithm of any number of seconds, is the difference between the common Logarithm of that number and the Logarithm of 3600, the seconds in 1 degree.</p><p>The chief use of the table of Logistic Logarithms, is for the ready computing a proportional part in minutes and seconds, when two terms of the proportion are minutes and seconds, or hours and minutes, or other such sexagesimal numbers. See the Introd. to my Logarithms, pa. 144.</p><p><hi rend="italics">Imaginary</hi> <hi rend="smallcaps">Logarithm</hi>, a term used in the Log. of imaginary and negative quantities; such as - <hi rend="italics">a,</hi> or √- <hi rend="italics">a</hi><hi rend="sup">2</hi> or <hi rend="italics">a</hi> √- 1. The fluents of certain imaginary expressions are also Imaginary Logarithms; as of , or of , &c. See Euler Analys. Insin. vol. i. pa. 72, 74.</p><p>It is well known that the expression <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">x</hi> represents the fluxion of the Logarithm of <hi rend="italics">x,</hi> and therefore the fluent of <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">x</hi> is the Logarithm of <hi rend="italics">x;</hi> and hence the fluent of is the Imaginary Logarithm of <hi rend="italics">x.</hi></p><p>However, when these Imaginary Logarithms occur in the solutions of problems, they may be transformed into circular arcs or sectors; that is, the Imaginary Logarithm, or imaginary hyperbolic sector, becomes a real circular sector. See Bernoulli Oper. tom. i, pa. 400, and pa. 512. Maclaurin's Fluxions, art. 762. Cotes's Harmon. Mens. pa. 45. Walmesley, Anal. des Mes. pa. 63.</p></div1><div1 part="N" n="LOGARITHMIC" org="uniform" sample="complete" type="entry"><head>LOGARITHMIC</head><p>, or <hi rend="smallcaps">Logistic Curve</hi>, a curve so called from its properties and uses, in explaining and constructing the Logarithms, because its ordinates are in geometrical progression, while the abscisses are in arithmetical progression; so that the abscisses are as the Logarithms of the corresponding ordinates. And hence<pb n="51"/><cb/> the curve will be constructed in this manner: Upon any right line, as an axis, take the equal parts AB, BC, CD, &c, or the arithmetical progression AB, AC, AD, &c; and at the points A, B, C, D, &c, erect the perpendicular ordinates AP, BQ, CR, DS, &c, in a geometrical progression; so is the curve line drawn through all the points P, Q, R, S, &c, the Logarithmic, or Logistic Curve; so called, because any absciss AB, is <*> the Logarithm of its ordinate BQ. So that the axis ABC &c is an asymptote to the curve. <figure/></p><p>Hence, if any absciss AN = <hi rend="italics">x,</hi> its ordinate NO = <hi rend="italics">y,</hi> AP = 1, and <hi rend="italics">a</hi> = a certain constant quantity, or the modulus of the Logarithms; then the equation of the curve is <hi rend="italics">x</hi> = <hi rend="italics">a</hi> × log. of <hi rend="italics">y</hi> = log. <hi rend="italics">y</hi><hi rend="sup">2</hi>.</p><p>And if the fluxion of this equation be taken, it will be ; which gives this proportion, but in any curve the subtangent AT; and therefore the subtangent of this curve is everywhere equal to the constant quantity <hi rend="italics">a,</hi> or the modulus of the Logarithms.</p><p><hi rend="italics">To find the Area contained between two ordinates.</hi> Here the fluxion of the area A<hi rend="sup">.</hi> or <hi rend="italics">yx</hi><hi rend="sup">.</hi> is ; and the correct fluent is . That is, the area APON between any two ordinates, is equal to the rectangle of the constant subtangent and the difference of the ordinates. And hence, when the absciss is infinitely long, or the farther ordinate equal to nothing, then the infinitely long area APZ is equal AT × AP, or double the triangle APT.</p><p><hi rend="italics">For the Solid formed by the curve revolved about its axis</hi> AZ. The fluxion of the solid is , where <hi rend="italics">p</hi> is = 3.1416; and the correct fluent is , which is half the difference between two cylinders of the common altitude <hi rend="italics">a</hi> or AT, and the radii of their bases AP, NO. And hence supposing the solid insinitely long towards Z, where <hi rend="italics">y</hi> or the ordinate is nothing, the infinitely long solid will be equal to , or half the cylinder on the same base and its altitude AT.</p><p>It has been said that Gunter gave the first idea of a curve whose abscisses are in arithmetical progression, while the corresponding ordinates are in geometrical progression, or whose absciss are the Logarithms of their ordinates; but I do not find it noticed in any part of his writings. This curve was afterwards considered by others, and named the Logarithmic or Logistic<cb/> Curve by Huygens in his Dissertatio de Causa Gravitatis, where he enumerates all the principal propertics of it, shewing its analogy to Logarithms. Many other learned men have also treated of its properties; particularly Le Seur and Jacquier, in their Comment on Newton's Principia; Dr. John Keill, in the elegant little Tract on Logarithms subjoined to his edition of Euclid's Elements; and Francis Maseres Esq. Cursitor Baron of the Exchequer, in his ingenious Treatise on Trigonometry: see also Bernoulli's Discourse in the Acta Eruditorum for the year 1696, pa. 216; Guido Grando's Demonstratio Theorematum Huygeneanorum circa Logisticam seu Logarithmicam Lineam; and Emerson on Curve Lines, pa. 19.—It is indeed rather extraordinary that this curve was not sooner announced to the public, since it results immediately from Napier's manner of conceiving the generation of Logarithms, by only supposing the lines which represent the natural numbers as placed at right angles to that upon which the Logarithms are taken.</p><p>This curve greatly facilitates the conception of Logarithms to the imagination, and affords an almost intuitive proof of the very important property of their fluxions, or very small increments, namely, that the fluxion of the number is to the fluxion of the Logarithm, as the number is to the subtangent; as also of this property, that if three numbers be taken very nearly equal, so that their ratios may differ but a little from a ratio of equality, as the three numbers 10000000, 10000001, 10000002, their differences will be very nearly proportional to the Logarithms of the ratios os those numbers to each other: all which follows from the Logarithmic arcs being very little different from their chords, when they are taken very small. And the constant subtangent of this curve is what was afterwards by Cotes called the Modulus of the System of Logarithms.</p><div2 part="N" n="Logarithmic" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Logarithmic</hi></head><p>, or <hi rend="italics">Logistic, Spiral,</hi> a curve constructed as follows. Divide the arch of a circle into any <figure/> equal parts AB, BD, DE, &c; and upon the radii drawn to the points of division take C<hi rend="italics">b,</hi> C<hi rend="italics">d,</hi> C<hi rend="italics">e,</hi> &c, in a geometrical progression; so is the curve A<hi rend="italics">bde</hi> &c the Logarithmic Spiral; so called, because it is evident that AB, AD, AE, &c, being arithmeticals, are as the the Logarithms of CA, C<hi rend="italics">b,</hi> C<hi rend="italics">d,</hi> C<hi rend="italics">e,</hi> &c, which are geometricals; and a Spiral, because it winds continually about the centre C, coming continually nearer, but without ever really falling into it.</p><p>In the Philos. Trans. Dr. Halley has happily applied this curve to the division of the meridian line in Merc<*>tor's chart. See also Cotes's Harmonia Mens., Guido Grando's Demonst. Theor. Huygen., the Acta Erudit. 1691, and Emerson's Curves, &c.</p></div2></div1><div1 part="N" n="LOGISTICS" org="uniform" sample="complete" type="entry"><head>LOGISTICS</head><p>, or LOGISTICAL <hi rend="smallcaps">Arithmetic</hi>, a name sometimes employed for the arithmetic of sexagesimal fractions, used.in astronomical computations.</p><p>This name was perhaps taken from a Greek treatise of Barlæmus, a Monk, who wrote a book of Sexagesimal Multiplication, which he called Logistic. Vossius places this author about the year 1350, but he mistakes the work for a Treatise on Algebra.</p><p>The same term however has been used for the rules<pb n="52"/><cb/> of computations in Algebra, and in other species of Arithmetic: witness the Logistics of Vieta and other writers.</p><p>Shakerly, in his Tabulæ Britannicæ, has a Table of Logarithms adapted to sexagesimal fractions, and which he calls Logistical Logarithms; and the expeditious arithmetic, obtained by means of them, he calls Logistical Arithmetic.</p><p><hi rend="smallcaps">Logistical</hi> <hi rend="italics">Curve, Line,</hi> or <hi rend="italics">Spiral,</hi> the same as the Logarithmic, which see.</p></div1><div1 part="N" n="LONG" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LONG</surname> (<foreName full="yes"><hi rend="smallcaps">Roger</hi></foreName>)</persName></head><p>, D. D. master of Pembroke hall in Cambridge, Lowndes's professor of astronomy in that university, &c, was author of a well-known and much approved treatise of astronomy, and the inventor of a remarkably curious astronomical machine. This was a hollow sphere, of 18 feet diameter, in which more than 30 persons might sit conveniently. Within side the surface, which represented the heavens, was painted the stars and constellations, with the zodiac, meridians, and axis parallel to the axis of the world, upon which it was easily turned round by a winch. He died, December 16, 1770, at 91 years of age.</p><p>A few years before his death, Mr. Jones gave some anecdotes of Dr. Long, as follows: “He is now in the 88th year of his age, and for his years vegete and active. He was lately put in nomination for the office of vice-chancellor: he executed that trust once before, I think in the year 1737. He is a very ingenious person, and sometimes very facetious. At the public Commencement, in the year 1713, Dr. Greene (master of Bennet college, and asterwards bishop of Ely) being then vice-chancellor, Mr. Long was pitched upon for the tripos performance; it was witty and humorous, and has passed through divers editions. Some that remembered the delivery of it, told me, that in addressing the vice-chancellor (whom the university wags usually styled <hi rend="italics">Miss Greene),</hi> the tripos-orator, being a native of Norfolk, and assuming the Norfolk dialect, instead of saying, <hi rend="italics">Domine Vice-Cancellarie,</hi> archly pronounced the words thus, <hi rend="italics">Domina Vice-Cancellaria;</hi> which occasioned a general smile in that great auditory. His friend the late Mr. Bonfoy of Ripton told me this little incident: ‘That he and Dr. Long walking together in Cambridge in a dusky evening, and coming to a short <hi rend="italics">post</hi> fixed in the pavement, which Mr. Bonfoy in the midst of chat and inattention, took to be a <hi rend="italics">boy</hi> standing in his way, he said in a hurry, ‘Get out of my way, boy!’ ‘<hi rend="italics">That boy, Sir,</hi> said the Doctor very calmly and slily, <hi rend="italics">is a</hi> post-boy, <hi rend="italics">who turns out of his way for nobody.</hi>’ I could recollect several other ingenious repartees if there were occafion. One thing is remarkable, he never was a hale and hearty man, always of a tender and delicate constitution, yet took great care of it: his common drink water; he always dines with the Fellows in the Hall Of late years he has left off eating slesh-meats; in the room thereof, puddings; vegetables, &c; sometimes a glass or two of wine.”</p></div1><div1 part="N" n="LONGIMETRY" org="uniform" sample="complete" type="entry"><head>LONGIMETRY</head><p>, the art of measuring lengths or distances, both accessible and inaccessible, forming a part of what is called Heights and Distances, being an application of geometry and trigonometry to such measurements.</p><p>As to accessible lengths, they are easily measured by<cb/> the actual application of a rod, a chain, or wheel, or some other measure of length.</p><p>But inaccessible lengths require the practice and properties of geometry and trigonometry, either in the measurement and construction, or in the computation. For example, Suppose it were required to know the length or distance between the two places A and B, to which places there is free access, but not to the intermediate parts, on account of water or some other impediment; measure therefore, from A and B, the distances to any convenient place C, which suppose to be thus, viz, AC = 735, and BC = 840 links; and let the angle at C, taken with a theodolite or other instrument, be 55° 40′. From these measures the length or distance AB may be determined, either by geometrical measurement, or by trigonometrical computation. Thus, first, lay down an angle C = 55° 40′, and upon its legs set off, from any convenient scale of equal parts, CA = 735, and CB = 840; then measure the distance between the points A and B by the same scale of equal parts, which will be found to be 740 nearly. <figure/> Or this by calculation, <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 840</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">, its half 62° 10′,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 735</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Sum</cell><cell cols="1" rows="1" role="data">1575</cell><cell cols="1" rows="1" rend="align=right" role="data">1.1972806</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dif.</cell><cell cols="1" rows="1" role="data"> 105</cell><cell cols="1" rows="1" rend="align=right" role="data">0.0211893</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tang.</cell><cell cols="1" rows="1" role="data">  62° 10′</cell><cell cols="1" rows="1" rend="align=right" role="data">10.2773793</cell></row><row role="data"><cell cols="1" rows="1" role="data">Tang.</cell><cell cols="1" rows="1" role="data">   7 11 11/14</cell><cell cols="1" rows="1" rend="align=right" role="data">9.1012880</cell></row></table> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">s. Sum or </cell><cell cols="1" rows="1" role="data">[angle] A = 99° 21′11/14</cell><cell cols="1" rows="1" role="data">9.9711092</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">to s. </cell><cell cols="1" rows="1" role="data">[angle] C = 55 40</cell><cell cols="1" rows="1" role="data">9.9168593</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">So</cell><cell cols="1" rows="1" role="data"> BC = 840</cell><cell cols="1" rows="1" role="data">0 9242793</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">To</cell><cell cols="1" rows="1" role="data">AB = 741.2</cell><cell cols="1" rows="1" role="data">0.8699404</cell></row></table></p><p>For a 2d Example—Suppose it were required to find the distance between two inaccessible objects, as between the house and mill, H and M; first measure any convenient line on the ground, as AB, 300 yards; then at the station A take the angles BAM = 58° 20′; and MAH = 37°; also at the station B take the angles ABH = 53° 30′, and HBM = 45° 15′; from hence the distance or length MH may be found, either by geometrical construction, or by trigonometrical calculation, thus:</p><p>First draw a line AB of the given length of 300, by a convenient scale of equal parts; then at the point A lay down the angles BAM and MAH of the mag-<pb n="53"/><cb/> <*>itudes above given; and also at the point B the given angles ABH and HBM: then by applying the length HM to the same scale of equal parts, it is found to be nearly 480 yards.</p><p>Otherwise, by calculation. First, by adding and fubtracting the angles, there is found as below: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">37° 00′</cell><cell cols="1" rows="1" rend="align=right" role="data">58° 20′</cell><cell cols="1" rows="1" role="data">       53° 30′</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">58  20</cell><cell cols="1" rows="1" rend="align=right" role="data">53  30</cell><cell cols="1" rows="1" role="data">       45  15</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">53  30</cell><cell cols="1" rows="1" rend="align=right" role="data">45  15</cell><cell cols="1" rows="1" role="data">sum 98  45 [angle] ABM</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">sums    148  50</cell><cell cols="1" rows="1" rend="align=right" role="data">157  05</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">from    180  00</cell><cell cols="1" rows="1" rend="align=right" role="data">180  00</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">[angle] AHB 31  10</cell><cell cols="1" rows="1" rend="align=right" role="data">22  55</cell><cell cols="1" rows="1" role="data">[angle] AMB</cell></row></table> Then, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">as sin. AHB : sin. ABH :: AB : AH =</cell><cell cols="1" rows="1" role="data">465.9776,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and, as sin. AMB : sin. ABM :: AB : AM =</cell><cell cols="1" rows="1" role="data">761.4655;</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">their sum is  </cell><cell cols="1" rows="1" role="data">1227.4431</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and their diff. </cell><cell cols="1" rows="1" role="data"> 295.4879</cell></row></table> Then as sum AM + AH : to dif. AM - AH :: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">tang. 1/2 AHM + 1/2 AMH =</cell><cell cols="1" rows="1" role="data">71° 30′,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">to tang. 1/2 AHM - 1/2 AMH =</cell><cell cols="1" rows="1" role="data">35  44</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">the dif. of which is AMH =</cell><cell cols="1" rows="1" role="data">35  46.</cell></row></table> Lastly,</p><p>as s. , the distance sought.</p><p>LONGITUDE <hi rend="italics">of the Earth,</hi> is sometimes used to denote its extent from west to east, according to the direction of the equator. By which it stands contradistinguished from the Latitude of the earth, which denotes its extent from one pole to the other.</p><p><hi rend="smallcaps">Longitude</hi> <hi rend="italics">of a Place,</hi> in Geography, is its longitudinal distance from some first meridian, or an arch of the equator intercepted between the meridian of that place and the first meridian.</p><p><hi rend="smallcaps">Longitude</hi> <hi rend="italics">in the Heavens,</hi> as of a star, &c, is an arch of the ecliptic, counted from the beginning of Aries, to the place where it is cut by a circle perpendicular to it, and passing through the place of the star.</p><p><hi rend="smallcaps">Longitude</hi> <hi rend="italics">of the Sun or Star from the next equinoctial point,</hi> is the degrees they are distant from the beginning of Aries or Libra, either before or aster them; which can never exceed 180 degrees.</p><div2 part="N" n="Longitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Longitude</hi></head><p>, <hi rend="italics">Geocentric, Heliocentric,</hi> &c, the Longitude of a planet as seen from the earth, or from the sun. See the respective terms.</p></div2><div2 part="N" n="Longitude" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Longitude</hi></head><p>, <hi rend="italics">in Navigation,</hi> is the distance of a ship, or place, east or west, from some other place or meridian, counted in degrees of the equator. When this distance is counted in leagues, or miles, or in degrees of the meridian, and not in those proper to the parallel of Latitude, it is usually called Departure.</p><p>An easy practicable method of finding the Longitude at sea, is the only thing wanted to render the Art of Navigation perfect, and is a problem that has greatly perplexed mathematicians for the last two centuries: accordingly most of the commercial nations of Europe have offered great rewards for the discovery of it; and in consequence very considerable advances have been made towards a perfect solution of the problem, especially by the English.<cb/></p><p>In the year 1598, the government of Spain offered a reward of 1000 crowns for the solution of this problem; and soon after the States of Holland offered 10 thousand florins for the same. Encouraged by such offers, in 1635, M. John Morin, professor of mathematics at Paris, proposed to cardinal Richlieu, a method of resolving it; and though the commissioners, who were appointed to examine this method, on account of the imperfect state of the lunar tables, judged it insufficient, cardinal Mazarin, in 1645, procured for the author a pension of 2000 livres.</p><p>In 1714 an act was passed in the British parliament, allowing 2000l. towards making experiments; and also osfering a reward to the person who should discover the Longitude at sea, proportioned to the degree of accuracy that might be attained by such discovery; viz, a reward of 10,000l. if it determines the Longitude to one degree of a great circle, or 60 geographical miles; 15,000l. if it determines the same to two-thirds of that distance; and 20,000l. if it determines it to half that distance; with other regulations and encouragements. 12 Ann. cap. 15. See also stat. 14 Geo. II, cap. 39, and 26 Geo. II, cap. 25. But, by stat. Geo. III, all former acts concerning the Longitude at sea are repealed, except so much of them as relates to the appointment and authority of the commissioners, and such clauses as relate to the publishing of nautical almanacs, and other useful tables; and it enacts, that any person who shall discover a method for finding the Longitude by means os a time-keeper, the principles of which have not hitherto been made public, shall be entitled to the reward of 5000l. if it shall enable a ship to keep her Longitude, during a voyage of 6 months, within 60 geographical miles, or one degree of a great circle; to 7500l. if within 40 geographical miles, or two-thirds of a degree of a great circle; or to a reward of 10,000l. if within 30 geographical miles, or half a degree of a great circle. But if the method shall be by means of improved solar and lunar tables, the author of them shall be entitled to a reward of 5000l. if they shew the distance of the moon from the sun and stars within 15″ of a degree, answering to about 7′ of Longitude, after making an allowance of half a degree for the errors of observation, and after comparison with astronomical observations for a period of 18 1/2 years, or during the period of the irregularities of the lunar motions. Or that in case any other method shall be proposed for finding the Longitude at sea, besides those bef<*>re-mentioned, the author shall be entitled to 5000l. if it shall determine the Longitude within one degree of a great circle, or 60 geographical miles; to 7500l. if within two-thirds of that distance; and to 10,000l. if within half the said distance.</p><p>Accordingly, many attempts have been made for such discovery, and several ways proposed, with various degrees of success. These however have been chiefly directed to methods of determining the difference of time between any two points on the earth; for the Longitude of any place being an arch of the equator intercepted between two meridians, and this arc being proportional to the time required by the sun to move from the one meridian to the other, at the rate of 4 minutes of time to one degree of the arch, it follows that the difference of time being known, and turned<pb n="54"/><cb/> into degrees according to that proportion, it will give the Longitude.</p><p>This measurement of time has been attempted by some persons by means of clocks, watches, and other automata: for if a clock or watch were contrived to go uniformly at all seasons, and in all places and situations; such a machine being regulated, for instance, to London or Greenwich time, would always shew the time of the day at London or Greenwich, wherever it should be carried to; then the time of the day at this place being found by observations, the difference between these two times would give the difference of Longitude, according to the proportion of one degree to 4 minutes of time.</p><p>Gemma Frisius, in his tract De Principiis Astronomiæ et Geographiæ, printed at Antwerp in 1530, it seems first suggested the method of finding the Longitude at sea by means of watches, or time-keepers; which machines, he says, were then but lately invented. And soon after, the same was attempted by Metius, and some others; but the state of watch-making was then too imperfect for that purpose. Dr. Hooke and Mr. Huygens also, about the year 1664, applied the invention of the pendulum-spring to watches; and employed it for the purpose of discovering the Longitude at sea. Some disputes however between Dr. Hooke and the English Ministry prevented any experiments from being made with watches constructed by him; but many experiments were made with some constructed by Huygens; particularly Major Holmes, in a voyage from the coast of Guinea in 1665, by one of these watches predicted the Longitude of the island of Fuego to a great degree of accuracy. This success encouraged Huygens to improve the structure of his watches, (see Philos. Trans. for May 1669); but experience soon convinced him, that unless methods could be discovered for preserving the regular motion of such machines, and preventing the effects of heat and cold, and other disturbing causes, they could never answer the intention of discovering the Longitude, and on this account his attempts failed.</p><p>The first person who turned his thoughts this way, after the public encouragement held out by the act of 1714, was Henry Sully, an Englishman; who, in the same year, printed at Vienna, a small tract on the subject of watch-making; and afterwards removing to Paris, he employed himself there in improving timekeepers for the discovery of the Longitude. It is said he greatly diminished the friction in the machine, and rendered uniform that which remained: and to him is principally to be attributed what is yet known of watchmaking in France: for the celebrated Julien le Roy was his pupil, and to him owed most of his inventions, which he afterwards perfected and executed: and this gentleman, with his son, and M. Berthoud, are the principal persons in France who have turned their thoughts this way since the time of Sully. Several watches made by these last two artists, have been tried at sea, it is said with good success, and large accounts have been published of these trials.</p><p>In the year 1726 our countryman, Mr. John Harrison, produced a time-keeper of his own construction, which did not err above one second in a month, for 10 years together: and in the year 1736 he had a machine tried in a<cb/> voyage to and from Lisbon; which was the means of correcting an error of almost a degree and a half in the computation of the ship's reckoning. In consequence of this success, Mr. Harrison received public encouragement to proceed, and he made three other time-keepers, each more accurate than the former, which were finished successively in the years 1739, 1758, and 1761; the last of which proved so much to his own satisfaction, that he applied to the commissioners of the Longitude to have this instrument tried in a voyage to some port in the West Indies, according to the directions of the statute of the 12th of Anne above cited. Accordingly, Mr. William Harrison, son of the inventor, embarked in November 1761, on a voyage for Jamaica, with this 4th timekeeper or watch; and on his arrival there, the Longitude, as shewn by the time-keeper, differed but one geographical mile and a quarter from the true Longitude, deduced from astronomical observations. The same gentleman returned to England, with the time-keeper, in March 1762; when he found that it had erred, in the 4 months, no more than 1′ 54″1/2 in time, or 28 5/8 minutes of Longitude; whereas the act requires no greater exactness than 30 geographical miles, or minutes of a great circle, in such a voyage. Mr. Harrison now claimed the whole reward of 20,0001, offered by the said act: but some doubts arising in the minds of the commissioners, concerning the true situation of the island of Jamaica, and the manner in which the time at that place had been found, as well as at Portsmouth; and it being farther suggested by some, that although the time-keeper happened to be right at Jamaica, and after its return to England, it was by no means a proof that it had been always so in the intermediate times; another trial was therefore proposed, in a voyage to the island of Barbadoes, in which precautions were taken to obviate as many of these objections as possible. Accordingly, the commissioners previously sent out proper persons to make astronomical observations at that island, which, when compared with other corresponding ones made in England, would determine, beyond a doubt, its true situation: and Mr. William Harrison again set out with his father's time keeper, in March 1764, the watch having been compared with equal altitudes at Portsmouth, before he set out, and he arrived at Barbadoes about the middle of May; where, on comparing it again by equal altitudes of the sun, it was found to shew the difference of Longitude, between Portsmouth and Barbadoes, to be 3<hi rend="sup">h</hi> 55<hi rend="sup">m</hi> 3<hi rend="sup">s</hi>; the true difference of Longitude between these places, by astronomical observations, being 3<hi rend="sup">h</hi> 54<hi rend="sup">m</hi> 20<hi rend="sup">s</hi>; so that the error of the watch was 43<hi rend="sup">s</hi>, or 10′ 45″ of Longitude. In consequence of this, and the former trials, Mr. Harrison received one moiety of the reward offered by the 12th of Queen Anne, after explaining the principles on which his watch was constructed, and delivering this as well as the three former to the Commissioners of the Longitude, for the use of the public: and he was promised the other moiety of the reward, when other time-keepers should be made, on the same principles, either by himself or others, performing equally well with that which he had last made. In the mean time, this last timekeeper was sent down to the Royal Observatory at Greenwich, to be tried there under the direction of the Rev. Dr. Maskelyne, the Astronomer Royal. But it<pb n="55"/><cb/> did not appear, during this trial, that the watch went with the regularity that was expected; from which it was apprehended, that the performance even of the same watch, was not at all times equal; and consesequently that little certainty could be expected in the performance of different ones. Moreover, the watch was now found to go faster than during the voyage to and from Barbadoes, by 18 or 19 seconds in 24 hours: but this circumstance was accounted for by Mr. Harrison; who informs us that he had altered the rate of its going by trying some experiments, which he had not time to finish before he was ordered to deliver up the watch to the Board. Soon after this trial, the Commissioners of Longitude agreed with Mr. Kendal, one of the watch-makers appointed by them to receive Mr. Harrison's discoveries, to make another watch on the same construction with this, to determine whether such watches could be made from the account which Mr. Harrison had given, by other persons, as well as himself. The event proved the affirmative; for the watch produced by Mr. Kendal, in consequence of this agreement, went consrderably better than Mr. Harrison's did. Mr. Kendal's watch was sent out with Capt. Cook, in his 2d voyage towards the south pole and round the globe, in the year 1772, 1773, 1774, and 1775; when the only fault found in the watch was, that its rate of going was continually accelerated; though in this trial, of 3 years and a half, it never amounted to 14″1/2 a day. The consequence was, that the House of Commons in 1774, to whom an appeal had been made, were pleased to order the 2d moiety of the reward to be given to Mr. Harrison, and to pass the act above mentioned. Mr. Harrison had also at different times received some other sums of money, as encouragements to him to continue his endeavours, from the Board of Longitude, and from the India Company, as well as from many individuals. Mr. Arnold and some other persons have since also made several very good watches for the same purpose.</p><p>Others have proposed various astronomical methods for finding the Longitude These methods/chiefly depend on having an ephemeris or almanac suited to the meridian of some place, as Greenwich for instance, to which the Nautical Almanac is adapted, which shall contain for every day computations of the times of all remarkable celestial motions and appearances, as adapted to that meridian. So that, if the hour and minute be known when any of the same phenomena are observed in any other place, whose Longitude is desired, the difference between this time and that to which the time of the said phenomenon was calculated and set down in the almanac, will be known, and consequently the difference of Longitude also becomes known, between that place and Greenwich, allowing at the rate of 15 degrees to an hour.</p><p>Now it is easy to find the time at any place, by means of the altitude or azimuth of the sun or stars; which time it is necessary to find by such means, both in these astronomical modes of determining the Longitude, and in the former by a time-keeper; and it is the difference between that time, so determined, and the time at Greenwich, known either by the time-keeper or by the astronomical observations of celestial phenomena, which gives the difference of Longitude, at the rate above-<cb/> mentioned. Now the difficulty in these methods lies in the fewness of proper phenomena, capable of being thus observed; for all slow motions, such as belong to the planet Saturn for instance, are quite excluded, as affording too small a difference, in a considerable space of time, to be properly observed; and it appears that there are no phenomena in the heavens proper for this purpose, except the eclipses or motions of Jupiter's satellites, and the eclipses or motions of the moon, viz, such as her distance from the sun or certain fixed stars lying near her path, or her Longitude or place in the zodiac, &c. Now of these methods,</p><p>1st, That by the eclipses of the moon is very easy, and sufficiently accurate, if they did but happen often, as every night. For at the moment when the beginning, or middle, or end of an eclipse is observed by a telescope, there is no more to be done but to determine the time by observing the altitude or azimuth of some known star; which time being compared with that in the tables, set down for the happening of the same phenomenon at Greenwich, gives the difference in time, and consequently of Longitude sought. But as the beginning or end of an eclipse of the moon cannot generally be observed nearer than one minute, and sometimes 2 or 3 minutes of time, the Longitude cannot certainly be determined by this method, from a single observation, nearer than one degree of Longitude. However, by two or more observations, as of the beginning and end &c, a much greater degree of exactness may be attained.</p><p>2d, The moon's place in the zodiac is a phenomenon more frequent than that of her eclipses; but then the observation of it is difficult, and the calculus perplexed and intricate, by reason of two parallaxes; so that it is hardly practicable, to any tolerable degree of accuracy.</p><p>3d, But the moon's distances from the sun, or certain fixed stars, are phenomena to be observed many times in almost every night, and afford a good practical method of determining the Longitude of a ship at almost any time; either by computing, from thence, the moon's true place, to compare with the same in the almanac, or by comparing her observed distance itself with the same as there set down.</p><p>It is said that the first person who recommended the finding the Longitude from this observed distance between the moon and some star, was John Werner, of Nuremberg, who printed his annotations on the first book of Ptolomy's Geography in 1514. And the same thing was recommended in 1524, by Peter Apian, professor of mathematics at Ingolstadt; also about 1530, by Oronce Finé, of Briançon; and the same year by the celebrated Kepler, and by Gemma Frisius, at Antwerp; and in 1560, by Nonius or Pedro Nunez.</p><p>Nor were the English mathematicians behind hand on this head. In 1665 Sir Jonas Moore prevailed on king Charles the 2d to erect the Royal Observatory at Greenwich, and to appoint Mr. Flamsteed his astronomical observer, with this express command, that he should apply himself with the utmost care and diligence to the rectifying the table of the motions of the heavens, and the places of the fixed stars, in order to find out the so much desired Longitude at sea, for perfecting the Art of Navigation. And to the fidelity and industry<pb n="56"/><cb/> with which Mr. Flamsteed executed his commission, it is that we are chiefly indebted for that curious theory of the moon, which was afterwards formed by the immortal Newton. This incomparable philosopher made the best possible use of the observations with which he was furnished; but as these were interrupted and imperfect, his theory would sometimes differ from the heavens by 5 minutes or more.</p><p>Dr. Halley bestowed much time on the same object; and a Starry Zodiac was published under his direction, containing all the stars to which the moon's appulse can be observed; but sor want of correct tables, and proper instruments, he could not proceed in making the necessary observations. In a paper on this subject, in the Philos. Trans. number 421, he expresses his hope, that the instrument just invented by Mr. Hadley might be applied to taking angles at sea with the desired accuracy. This great astronomer, and after him the Abbé de la Caille, and others, have reckoned the best astronomical method for finding the Longitude at sea, to be that in which the distance of the moon from the sun or from a star is used; for the moon's daily motion being about 13 degrees, her hourly mean motion is above half a degree, or one minute of a degree in two minutes of time; so that an error of one minute of a degree in position will produce an error of 2 minutes in time, or half a degree in Longitude. Now from the great improvements made by Newton in the theory of the moon, and more lately by Euler and others on his principles, professor Mayer, of Gottengen, was enabled to calculate lunar tables more correct than any former ones; having so far succeeded as to give the moon's place within one minute of the truth, as has been proved by a comparison of the tables with the observations made at the Greenwich observatory by the late Dr. Bradley, and by Dr. Maskeline, the present Astronomer Royal; and the same have been still farther improved under his direction, by the late Mr. Charles Mason, by several new equations, and the whole computed to tenths of a second. These new tables, when compared with the above-mentioned series of observations, a proper allowance being made for the unavoidable error of observation, seem to give always the moon's Longitude in the heavens correctly within 30 seconds of a degree; which greatest error, added to a possible error of one minute in taking the moon's distance from the sun or a star at sea, will at a medium only produce an error of 42 minutes of Longitude. To facilitate the use of the tables, Dr. Maskelyne proposed a nautical ephemeris, the scheme of which was adopted by the Commissioners of Longitude, and first executed in the year 1767, since which time it has been regularly continued, and published as far as for the year 1800. But as the rules that were given in the appendix to one of those publications, for correcting the effects of refraction and parallax, were thought too difficult for general use, they have been reduced to tables. So that, by the help of the ephemeris, these tables, and others that are also provided by the Board of Longitude, the calculations relating to the Longitude, which could not be performed by the most expert mathematician in less than four hours, may now be completed with great ease and accuracy in half an hour.</p><p>As this method of determining the Longitude depends on the use of the tables annually published for<cb/> this purpose, those who wish for farther information are referred to the instructions that accompany them, and particularly to those that are annexed to the <hi rend="italics">Tables requisite to be used with the Astronomical and Nautical Ephemeris,</hi> 2d edit. 1781.</p><p>4th. The phenomena of Jupiter's satellites have commonly been preferred to those of the moon, for finding the Longitude; because they are less liable to parallaxes than these are, and besides they afford a very commodious observation whenever the planet is above the horizon. Their motion is very swift, and must be calculated for every hour. These satellites of Jupiter were no sooner announced by Galileo, in his Syderius Nuncius, first printed at Venice in 1610, than the frequency of their eclipses recommended them for this purpose; and among those who treated on this subject, none was more successful than Cassini. This great astronomer published, at Bologna, in 1688, tables for calculating the appearances of their eclipses, with directions for finding the Longitudes of places by them; and being invited to France by Louis the 14th, he there, in the year 1693, published more correct tables of the same. But the mutual attractions of the satellites rendering their motions very irregular, those tables soon became useless for this purpose; insomuch that they require to be renewed from time to time; a service which has been performed by several ingenious astronomers, as Dr. Pound, Dr. Bradley, M. Cassini the son, and more especially by Mr. Wargentin, whose tables are much esteemed, which have been published in several places, as also in the Nautical Almanacs for 1771 and 1779.</p><p>Now, to find the Longitude by these satellites; with a good telescope observe some of their phenomena, as the conjunction of two of them, or of one of them with Jupiter, &c; and at the same time find the hour and minute, from the altitudes of the stars, or by means of a clock or watch, previously regulated for the place of observation; then, consulting tables of the satellites, observe the time when the same appearance happens in the meridian of the place for which the tables are calculated; and the difference of time, as before, will give the Longitude.</p><p>The eclipses of the first and second of Jupiter's satellites are the most proper for this purpose; and as they happen almost daily, they afford a ready means of determining the Longitude of places at land, having indeed contributed much to the modern improvements in geography; and if it were possible to observe them with proper telescopes, in a ship under sail, they would be of great service in ascertaining its Longitude from time to time. To obviate the inconvenience to which these observations are liable from the motions of the ship, a Mr. Irwin invented what he called a marine chair; this was tried by Dr. Maskelyne, in his voyage to Barbadoes, when it was not found that any benefit could be derived from the use of it. And indeed, considering the great power requisite in a telescope proper for these observations, and the violence, as well as irregularities in the motion of a ship, it is to be feared that the complete management of a telescope on ship-board, will always remain among the desiderata in this part of nautical science. And farther, since all methods that depend on the phenomena of the heavens have also this other defect, that they cannot be observed at all times, this<pb n="57"/><cb/> renders the improvement of time keepers an object of the greater importance.</p><p>Many other schemes and proposals have been made by different persons, but most of them of very little or no use; such as by the space between the flash and report of a great gun, proposed by Messrs Whiston and Ditton; and another proposed by Mr. Whiston, by means of the inclinatory or dipping needle; besides a method by the variation of the magnetic needle, &c, &c.</p><p><hi rend="smallcaps">Longitude</hi> <hi rend="italics">of Motion,</hi> is a term used by Dr. Wallis for the measure of motion, estimated according to its line of direction; or it is the distance or length gone through by the centre of any moving body, as it moves on in a right line.</p><p>The same author calls the measure of any motion, estimated according to the line of direction of the vis motrix, the <hi rend="italics">Altitude</hi> of it.</p></div2></div1><div1 part="N" n="LONGOMONTANUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LONGOMONTANUS</surname> (<foreName full="yes"><hi rend="smallcaps">Christian</hi></foreName>)</persName></head><p>, a learned astronomer, born in Denmark in 1<*>62, in the village of Longomontum, whence he took his name. Vossius, by mistake, calls him Christopher. Being the son of a poor man, a plowman, he was obliged to suffer, during his studies, all the hardships to which he could be exposed, dividing his time, like the philosopher Cleanthes, between the cultivation of the earth and the lessons he received from the minister of the place. At length, at 15 years old, he stole away from his family, and went to Wiburg, where there was a college, in which he spent 11 years; and though he was obliged to earn his livelihood as he could, his close application to study enabled him to make a great progress in learning, particularly in the mathematical sciences.</p><p>From hence he went to Copenhagen; where the professors of that university soon conceived a very high opinion of him, and recommended him to the celebrated Tycho Brahe; with whom Longomontanus lived 8 years, and was of great service to him in his observations and calculations. At length, being very desirous of obtaining a professor's chair in Denmark, Tycho Brahe consented, with some difficulty, to his leaving him; giving him a discharge filled with the highest testimonies of his esteem, and furnishing him with money for the expence of his long journey from Germany, whither Tycho had retired.</p><p>He accordingly obtained a professorship of mathematics in the university of Copenhagen in 1605; the duty of which he discharged very worthily till his death, which happened in 1647, at 85 years of age.</p><p>Longomontanus was author of several works, which shew great talents in mathematics and astronomy. The most distinguished of them, is his <hi rend="italics">Astronomica Danica,</hi> first printed in 4to, 1621, and afterwards in folio in 1640, with augmentations. He amused himself with endeavouring to square the circle, and pretended that he had made the discovery of it; but our countryman Dr. John Pell attacked him warmly on that subject, and proved that he was mistaken.——It is remarkable that, obscure as his village and father were, he contrived to dignify and eternize them both; for he took his name from his village, and in the title page to some of his works he wrote himself <hi rend="italics">Christianus Longomontanus Severini filius,</hi> his father's name being Severin or Severinus.</p><p>LOXODROMIC <hi rend="smallcaps">Curve</hi>, or <hi rend="smallcaps">Spiral</hi>, is the same<cb/> as the Rhumb line, or path of a ship sailing always on the same course in an oblique direction, or making always the same angle with every meridian. It is a speciea of logarithmic spiral, described on the surface of the sphere, having the meridians for its radii.</p></div1><div1 part="N" n="LOXODROMICS" org="uniform" sample="complete" type="entry"><head>LOXODROMICS</head><p>, the art or method of oblique sailing, by the loxodromic or rhumb line.</p></div1><div1 part="N" n="LOZENGE" org="uniform" sample="complete" type="entry"><head>LOZENGE</head><p>, an oblique-angled parallelogram; being otherwise called a rhombus, or a rhomboides.</p></div1><div1 part="N" n="LUBIENIETSKI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LUBIENIETSKI</surname> (<foreName full="yes"><hi rend="smallcaps">Stanislaus</hi></foreName>)</persName></head><p>, a Polish gentleman, born at Cracow, in 1623, and educated with great care by his father. He was learned in astronomy, and became a celebrated Socinian minister. He took great pains to obtain a toleration from the German princes for his Socinian brethren. His endeavours however were all in vain; being himself persecuted by the Lutheran ministers, and banished from place to place; till at length he was banished out of the world, with his two daughters, by poison, in 1675, his wife narrowly escaping.</p><p>We have, of his writing, <hi rend="italics">A History of the Reformation in Poland;</hi> and a Treatise on Comets, intitled <hi rend="italics">Theatrum Cometicum,</hi> printed at Amsterdam in 2 volumes folio; which is a most elaborate work, containing a minute historical account of every single comet that had been seen or recorded.</p><p>LUCIDA <hi rend="smallcaps">Coronæ</hi>, a fixed star of the 2d magnitude, in the northern crown. See <hi rend="smallcaps">Corona</hi> <hi rend="italics">Borealis.</hi></p><p><hi rend="smallcaps">Lucida Hydræ.</hi> See <hi rend="smallcaps">Cor</hi> <hi rend="italics">Hydræ.</hi></p><p><hi rend="smallcaps">Lucida Lyræ</hi>, a bright star of the first magnitude in the constellation Lyra.</p></div1><div1 part="N" n="LUCIFER" org="uniform" sample="complete" type="entry"><head>LUCIFER</head><p>, a name given to the planet Venus, when she appears in the morning before sunrise.</p></div1><div1 part="N" n="LUMINARIES" org="uniform" sample="complete" type="entry"><head>LUMINARIES</head><p>, a term used for the sun and moon, by way of eminence, for their extraordinary lustre, and the great quantity of light they give us.</p></div1><div1 part="N" n="LUNA" org="uniform" sample="complete" type="entry"><head>LUNA</head><p>, the Moon; which see.</p></div1><div1 part="N" n="LUNAR" org="uniform" sample="complete" type="entry"><head>LUNAR</head><p>, something relating to the moon.</p><p><hi rend="smallcaps">Lunar</hi> <hi rend="italics">Cycle,</hi> or <hi rend="italics">Cycle of the Moon.</hi> See <hi rend="smallcaps">Cycle.</hi></p><p><hi rend="smallcaps">Lunar</hi> <hi rend="italics">Method for the Longitude,</hi> a method of keeping or finding the Longitude by means of the moon's motions, particularly by her observed distances from the sun and stars; for which, see the article L<hi rend="smallcaps">ONGITUDE.</hi></p><p><hi rend="smallcaps">Lunar</hi> <hi rend="italics">Month,</hi> is either Periodical, Synodical, or Illuminative. Which see; also <hi rend="smallcaps">Month.</hi></p><p><hi rend="smallcaps">Lunar</hi> <hi rend="italics">Year,</hi> consists of 354 days, or 12 synodical months, of 29 1/2 days each. See <hi rend="smallcaps">Year.</hi></p><p>In the early ages, the lunar year was used by all nations; the variety of course being more frequent and conspicuous in this planet, and consequently better known to men, than those of any other. The Romans regulated their year, in part, by the moon, even till the time of Julius Cæsar. The Jews too had their lunar month and year.</p><p><hi rend="smallcaps">Lunar</hi> <hi rend="italics">Dial, Eclipse, Horoscope,</hi> and <hi rend="italics">Rainbow.</hi> See the several substantives.</p></div1><div1 part="N" n="LUNATION" org="uniform" sample="complete" type="entry"><head>LUNATION</head><p>, the period or time between one new moon and another; it is also called the synodical month, consisting of 29 days 12 hrs. 44m. 3 sec. 11 thirds; exceeding the periodical month by 2 ds. 5 hrs. 0 m. 55 sec.</p></div1><div1 part="N" n="LUNE" org="uniform" sample="complete" type="entry"><head>LUNE</head><p>, or <hi rend="smallcaps">Lunula</hi>, or little moon, is a geometri-<pb n="58"/><cb/> cal figure, in form of a crescent, terminated by the arcs of two circles that intersect each other within.</p><p>Though the quadrature of the whole circle has never been effected, yet many of its parts have been squared. The first of these partial quadratures was that of the Lunula, given by Hippocrates of Scio, or Chios; who, from being a shipwrecked merchant, commenced geometrician. But although the quadrature of the Lune be generally ascribed to Hippocrates, yet Proclus expressly says it was found out by Oenopidas of the same place. See Heinius in Mem. de l'Acad. de Berlin, tom. ii. pa. 410, where he gives a dissertation concerning this Oenopidas. See also <hi rend="smallcaps">Circle</hi>, and <hi rend="smallcaps">Quadrature.</hi></p><p>The Lune of Hippocrates is this: Let ABC be a semicircle, having its centre E, and ADC a quadrant, having its centre F; then the Figure ABCDA, contained between the arcs of the semicircle and quadrant, is his Lune; and it is equal to the right-angled triangle ACF, as is thus easily proved. Since , that is, the square of the radius of the quadrant equal to double the square of the radius of the semicircle; therefore the quadrantal area ADCFA is = the semicircle ABCEA; from each of these take away the common space ADCEA, and there remains the triangle ACF = the Lune ABCDA. <figure/></p><p>Another property of this Lune, which is the more general one of the former, is, that if FG be any line drawn from the point F, and AH perpendicular to it; then is the intercepted part of the Lune AGIA = the triangle AGH cut off by the chord line AG; or in general, that the small segment AKGA is equal to the trilineal AIHA. For, the angle AFG being at the centre of the one circle, and at the circumference of the other, the arcs cut off AG, AI are similar to the wholes ABC, ADC, therefore the small seg. AKGA is to the semisegment AIH, as the whole semicircle ABCA to the semisegment or quadrant ADCF, that is in a ratio of equality.</p><p>Again, if ABC (fig. 2) be a triangle, right angled at C, and if semicircles be described on the three sides as diameters; then the triangle T (ABC) is equal to the sum of the two Lunes L1, L2. For, the greatest semicircle is equal to the sum of both the other two; from the greatest semicircle take away the segments S1 and S2, and there remains the triangle T; also from the two less semicircles take away the same two segments S1 and S2, and there remains the two Lunes L1 and L2; therefore the triangle the two Lunes.</p></div1><div1 part="N" n="LUNETTE" org="uniform" sample="complete" type="entry"><head>LUNETTE</head><p>, in Fortisication, an inveloped counterguard, or mound of earth, made beyond the second ditch, opposite to the place of arms; differing from the ravelins only in their situation. Lunettes are usually made in wet ditches, and serve the same purpose as fausse-brays, to defend the passage of the ditch.<cb/></p></div1><div1 part="N" n="LUPUS" org="uniform" sample="complete" type="entry"><head>LUPUS</head><p>, the <hi rend="italics">Wolf,</hi> a southern constellation, joined to the Centaur, containing together 19 stars in Ptolomy's catalogue, but 24 in the Britannic catalogue.</p></div1><div1 part="N" n="LYNX" org="uniform" sample="complete" type="entry"><head>LYNX</head><p>, a constellation of the northern hemisphere, composed by Hevelius out of the unformed stars. In his catalogue it consists of 19 stars, but in the Britannic 44.</p></div1><div1 part="N" n="LYONS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">LYONS</surname> (<foreName full="yes"><hi rend="smallcaps">Israel</hi></foreName>)</persName></head><p>, a good mathematician and botanist, was the son of a Polish Jew silversmith, and teacher of Hebrew at Cambridge in England, where he was come to settle, and where young Lyons was born, 1739. He was a very extraordinary young man for parts and ingenuity; and shewed very early in life a great inclination to learning, particularly mathematics, on which account he was much patronized by Dr. Smith, master of Trinity college. About 1755 he began to study botany, which he continued occasionally till his death; in which he made a considerable progress, and could remember not only the Linnæan names of almost all the English plants, but even the synonyma of the old botanists; and he had prepared large materials for a <hi rend="italics">Flora Cantabrigiensis,</hi> describing fully every part of each plant from the specimen, without being obliged to consult, or being liable to be misled by, former authors.</p><p>In 1758, he obtained much celebrity by publishing <hi rend="italics">A Treatise on Fluxions,</hi> dedicated to his patron, Dr. Smith; and in 1763, <hi rend="italics">Fasciculus Plantarum circa Cantabrigiam, &c.</hi> In the same year, or the year before, he read Lectures on Botany at Oxford with great applause, to at least 60 pupils; but he could not be prevailed on to make a long absence from Cambridge.</p><p>Mr. Lyons was some time employed as one of the computers of the Nautical Almanac; and besides he received srequent other presents from the Board of Longitude for his own inventions.——He had studied the English history; and could quote whole passages from the Monkish writers verbatim. He could read Latin and French with ease, but wrote the former ill. He was appointed by the Board of Longitude to sail with Capt. Phipps, in his voyage towards the North Pole, in 1773, as astronomical observator; and he discharged that office to the satisfaction of his employers. After his return from this voyage, he married, and settled in London, where he died of the meazles in about two years.</p><p>At the time of his death he was engaged in preparing for the press, a complete edition of all the works of the late learned Dr. Halley; a work very much wanted. —His <hi rend="italics">Calculations in Spherical Trigonometry abridged,</hi> were printed in the Philos. Trans. vol. 65, for the year 1775, pa. 470.—After his death, his name appeared in the title-page of <hi rend="italics">A Geographical Dictionary,</hi> the astronomical parts of which were said to be “taken from the papers of the late Mr. Israel Lyons of Cambridge, author of several valuable mathematical productions, and astronomer in lord Mulgrave's voyage to the northern hemisphere.”—The astronomical and other mathematical calculations, printed in the account of captain Phipps's voyage towards the north pole, mentioned above, were made by Mr. Lyons. This appeared afterwards, by the acknowledgment of captain Phipps, when Dr. Horsley detected a material error in some part<pb n="59"/><cb/> of them, in his <hi rend="italics">Remarks on the Observations made in the late Voyage, &c,</hi> 1774.</p><p>“The Scholar's Instructor, or Hebrew Grammar, by Israel Lyons, Teacher of the Hebrew Tongue in the University of Cambridge,” the 2d edit. &c, 1757, 8vo, was the production of his father; as was also another Treatise printed at the Cambridge press, under the title<cb/> of “Observations and Enquiries relating to various parts of Scripture History, 1761.”</p></div1><div1 part="N" n="LYRA" org="uniform" sample="complete" type="entry"><head>LYRA</head><p>, the <hi rend="italics">Harp,</hi> a constellation in the northern hemisphere, containing 10 stars in Ptolomy's catalogue, 11 in Tycho's, 17 in Hevelius's, and 21 in the Britannic catalogue.</p></div1></div0><div0 part="N" n="M" org="uniform" sample="complete" type="alphabetic letter"><head>M</head><cb/><p>M, In <hi rend="italics">Astronomical Ta<*>les,</hi> &c, is used for <hi rend="italics">Meridional</hi> or southern; and sometimes for <hi rend="italics">Meridian,</hi> or mid-day.—In the Roman numeration, it denotes 1000, one thousand.</p><div1 part="N" n="MACHINE" org="uniform" sample="complete" type="entry"><head>MACHINE</head><p>, denotes any thing that serves to augment, or to regulate moving powers: or it is any body destined to produce motion, so as to save either time or force. The word, in Greek, signifies an <hi rend="italics">Invention,</hi> or <hi rend="italics">Art:</hi> and hence, in strictness, a machine is something that consists more in art and invention, than in the strength and solidity of the materials; for which reason it is that the inventors of machines are called <hi rend="italics">Ingenieurs,</hi> or <hi rend="italics">engineers.</hi></p><p>Machines are either simple or compound. The simple machines are the seven mechanical powers, viz, the lever, balance, pulley, wheel-and-axle, wedge, screw, and inclined plane; which are otherwise called the simple mechanic powers.</p><p>These simple machines serve for different purposes, according to the different structures of them; and it is the business of the skilful mechanist to choose them, and combine them, in the manner that may be best adapted to produce the desired effect. The lever is a very handy machine for many purposes, and its power immediately varied as the occasion may require; when weights are to be raised only a little way, such as stones out of quarries, &c. On the other hand, the wheel-and-axle serves to raise weights from the greatest depth, or to the greatest height. Pulleys, being easily carried, are therefore much employed in ships. The balance is useful for ascertaining an equality of weight. The wedge is excellent for separating the parts of bodies; and being impelled by the force of percussion, it is incomparably greater than the other powers. The screw is useful for compressing or squeezing bodies together, and also for raifing very heavy weights to a small height; its great friction is even of considerable use, to preserve the effect already produced by the machine.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Machine</hi>, is formed from these simple machines, combined together for different purposes. The number of compound machines is almost infinite; and yet it would seem that the Ancients went far beyond the Moderns in the powers and effects of them; especially their machines of war and architecture.</p><p>Accurate descriptions and drawings of machines<cb/> would be a very curious and useful work. But to make a collection of this kind as beneficial as possible, it should contain also an analysis of them; pointing out their advantages and disadvantages, with the reasons of the constructions; also the general problems implied in these constructions, with their solutions, should be noticed. Though a complete work of this kind be still wanting, yet many curious and useful particulars may be gathered from Strada, Besson, Beroaldus, Augustinus de Ramellis, Bockler, Leupold, Beyer, Limpergh, Van Zyl, Perault, and others; a short account of whose works may be found in Wolfii Commentatio de Præcipuis Scriptis Mathematicis; Elem. Mathes, Univ. tom. 5, pa. 84. To these may be added, Belidor's Architecture Hydraulique, Desaguliers's Course of Experimental Philosophy, and Emerson's Mechanics. The Royal Academy of Sciences at Paris have also given a collection of machines and inventions approved of by them. This work, published by M. Gallon, consists of 6 volumes in quarto, containing engraved draughts of the machines, with their descriptions annexed.</p><div2 part="N" n="Machine" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Machine</hi></head><p>, <hi rend="italics">Architectonical,</hi> is an assemblage of pieces of wood so disposed as that, by means of ropes and pulleys, a small number of men may raise great loads, and lay them in their places: such as cranes, &c.—— It is hard to conceive what sort of machines the Ancients must have used to raise those immense stones found in some of the antique buildings; as some of those still found in the walls of Balbeck in Turkey, the ancient Heliopolis, which are 63 feet long, 12 feet broad, and 12 feet thick, and which must weigh 6 or 7 hundred tons a piece.</p><p><hi rend="italics">Blowing</hi> <hi rend="smallcaps">Machine.</hi> See <hi rend="smallcaps">Bellows.</hi></p><p><hi rend="italics">Boylcian</hi> <hi rend="smallcaps">Machine.</hi> Mr. Boyle's Air-Pump.</p><p><hi rend="italics">Electrical</hi> <hi rend="smallcaps">Machine.</hi> See <hi rend="smallcaps">Electrical</hi> <hi rend="italics">Machine.</hi></p><p><hi rend="italics">Wind</hi> <hi rend="smallcaps">Machine.</hi> See <hi rend="smallcaps">Anemometer</hi>, and <hi rend="smallcaps">Wind</hi> <hi rend="italics">Machine.</hi></p><p><hi rend="italics">Hydraulic,</hi> or <hi rend="italics">Water</hi> <hi rend="smallcaps">Machine</hi>, is used either to signify a simple Machine, serving to conduct or raise water; as a sluice, pump, and the like, or several of these acting together, to produce some extraordinary effect; as the</p><p><hi rend="smallcaps">Machine</hi> <hi rend="italics">of Marli.</hi> See <hi rend="smallcaps">Marli.</hi> See also <hi rend="smallcaps">Fire</hi>- <hi rend="italics">engine,</hi> <hi rend="smallcaps">Steam</hi>-<hi rend="italics">engine,</hi> and <hi rend="smallcaps">Water</hi>-<hi rend="italics">works.</hi></p><p><hi rend="italics">Military</hi> <hi rend="smallcaps">Machines</hi>, among the Ancients, were of<pb n="60"/><cb/> three kinds: the first serving to launch arrows, as the scorpion; or javelins, as the catapult; or stones, as the balista; or fiery darts, as the pyrabolus: the 2d sort serving to beat down walls, as the battering ram and terebra: and the 3d sort to shelter those who approach the enemy's wall, as the tortoise or testudo, the vinea, and the towers of wood. See the respective articles.</p><p>The Machines of war now in use, consist in artillery, including cannon, mortars, petards, &c.</p></div2></div1><div1 part="N" n="MACLAURIN" org="uniform" sample="complete" type="entry"><head>MACLAURIN</head><p>, (<hi rend="smallcaps">Colin</hi>), a most eminent mathematician and philosopher, was the son of a clergyman, and born at Kilmoddan in Scotland, in the year 1698. He was sent to the university of Glasgow in 1709; where he continued sive years, and applied to his studies in a very intense manner, and particularly to the mathematics. His great genius for mathematical learning discovered itself so early as at 12 years of age; when, having accidentally met with a copy of Euclid's Elements in a friend's chamber, he became in a few days master of the first 6 books without any assistance: and it is certain, that in his 16th year he had invented many of the propositions which were afterwards published as part of his work intitled <hi rend="italics">Geometria Organica</hi> In his 15th year he took the degree of Master of Arts; on which occasion he composed and publicly defended a thesis on the power of gravity, with great applause. After this he quitted the university, and retired to a country seat of his uncle, who had the care of his education; his parents being dead some time. Here he spent two or three years in pursuing his favourite studies; but, in 1717, at 19 years of age only, he offered himself a candidate for the professorship of mathematics in the Marischal College of Aberdeen, and obtained it after a ten days trial, against a very able competitor.</p><p>In 1719, Mr. Maclaurin visited London, where he left his <hi rend="italics">Geometria Organica</hi> to print, and where he became acquainted with Dr. Hoadley then bishop of Bangor, Dr. Clarke, Sir Isaac Newton, and other eminent men; at which time also he was admitted a member of the Royal Society: and in another journey, in 1721, he contracted an intimacy with Martin Folkes, Esq. the president of it, which continued during his whole life.</p><p>In 1722, lord Polwarth, plenipotentiary of the Ling of Great Britain at the congress of Cambray, engaged Maclaurin to go as a tutor and companion to his eldest son, who was then to set out on his travels. After a short stay at Paris, and visiting other towns in France, they sixed in Lorrain; where he wrote his piece, On the Percussion of Bodies, which gained him the prize of the Royal Academy of Sciences for the year 1724. But his pupil dying soon after at Montpelier, he returned immediately to his profession at Aberdeen. He was hardly settled here, when he received an invitation to Edinburgh; the curators of that university being desirous that he should supply the place of Mr. James Gregory, whose great age and infirmities had rendered him incapable of teaching. He had here some difficulties to encounter, arising from competitors, who had good interest with the patrons of the university, and also from the want of an additional fund for the new professor; which however at length were all surmounted, principally by the means of Sir Isaac Newton. Accordingly, in Nov. 1725, he was introduced into the university; as was at the same time his learned colleague and inti-<cb/> mate friend, Dr. Alexander Monro, professor of anatomy. After this, the Mathematical classes soon became very numerous, there being generally upwards of 100 students attending his Lectures every year; who being of different standings and proficiency, he was obliged to divide them into four or five classes, in each of which he employed a full hour every day from the first of November to the first of June. In the first class he taught the first 6 books of Euclid's Elements, Plane Trigonometry, Practical Geometry, the Elements of Fortification, and an Introduction to Algebra. The second class studied Algebra, with the 11th and 12th books of Euclid, Spherical Trigonometry, Conic Sections, and the general Principles of Astronomy. The third went on in Astronomy and Perspective, read a part of Newton's Principia, and had performed a course of experiments for illustrating them: he afterwards read and demonstrated the Elements of Fluxions. Those in the fourth class read a System of Fluxions, the Doctrine of Chances, and the remainder of Newton's Principia.</p><p>In 1734, Dr. Berkley, bishop of Cloyne, published a piece called The Analist; in which he took occasion, from some disputes that had arisen concerning the grounds of the fluxionary method, to explode the method itself; and also to charge mathematicians in general with insidelity in religion. Maclaurin thought himself included in this charge, and began an answer to Berkley's book: but other answers coming out, and as he proceeded, so many discoveries, so many new theories and problems occurred to him, that instead of a vindicatory pamphlet, he produced a Complete System of Fluxions, with their application to the most considerable problems in Geometry and Natural Philosophy. This work was published at Edinburgh in 1742, 2 vols 4to; and as it cost him infinite pains, so it is the most considerable of all his works, and will do him immortal honour, being indeed the most complete treatise on that science that has yet appeared.</p><p>In the mean time, he was continually obliging the public with some observation or performance of his own, several of which were published in the 5th and 6th volumes of the Medical Essays at Edinburgh. Many of them were likewise published in the Philosophical Transactions; as the following: 1. On the Construction and Measure of Curves, vol. 30.—2. A New Method of describing all kinds of Curves, vol. 30.—3. On Equations with Impossible Roots, vol. 34.—4. On the Roots of Equations, &c. vol. 34.—5. On the Description of Curve Lines, vol. 39.—6. Continuation of the same, vol. 39.—7. Observations on a Solar Eclipse, vol. 40.—8. A Rule for finding the Meridional Parts of a Spheroid with the same Exactness as in a Sphere, vol. 41.—9. An Account of the Treatise of Fluxions, vol. 42.—10. On the Bases of the Cells where the Bees deposit their Honey, vol. 42.</p><p>In the midst of these studies, he was always ready to lend his assistance in contriving and promoting any scheme which might contribute to the public service. When the earl of Morton went, in 1739, to visit his estates in Orkney and Shetland, he requested Mr. Maclaurin to assist him in settling the geography of those countries, which is very erroneous in all our maps; to examine their natural history, to survey the coasts, and to take the measure of a degree of the meridian. Mac-<pb n="61"/><cb/> laurin's family affairs would not permit him to comply with this request: he drew up however a memorial of what he thought necessary to be observed, and furnished proper instruments for the work, recommending Mr. Short, the noted optician, as a sit operator for the management of them.</p><p>Mr. Maclaurin had still another scheme for the improvement of geography and navigation, of a more extensive nature; which was the opening a passage from Greenland to the South Sea by the North Pole. That such a passage might be sound, he was so fully persuaded, that he used to say, if his situation could admit of such adventures, he would undertake the voyage, even at his own charge. But when schemes for finding it were laid before the parliament in 1741, and he was consulted by several persons of high rank concerning them, and before he could finish the memorials he proposed to send, the premium was limited to the discovery of a north-west passage: and he used to regret that the word West was inserted, because he thought that passage, if at all to be found, must lie not far from the pole.</p><p>In 1745, having been very active in fortifying the city of Edinburgh against the rebel army, he was obliged to fly from thence into England, where he was invited by Dr. Herring, archbishop of York, to reside with him during his stay in this country. In this expedition however, being exposed to cold and hardships, and naturally of a weak and tender constitution, which had been much more enfeebled by close application to study, he laid the foundation of an illness which put an end to his life, in June 1746, at 48 years of age, leaving his widow with two sons and three daughters.</p><p>Mr. Maclaurin was a very good, as well as a very great man, and worthy of love as well as admiration. His peculiar merit as a philosopher was, that all his studies were accommodated to general utility; and we sind, in many places of his works, an application even of the most abstruse theories, to the perfecting of mechanical arts. For the same purpose, he had resolved to compose a course of Practical Mathematics, and to rescue several useful branches of the science from the ill treatment they often met with in less skilful hands. These intentions however were prevented by his death; unless we may reckon, as a part of his intended work, the translation of Dr. David Gregory's Practical Geometry, which he revised, and published with additions, in 1745.</p><p>In his lifetime, however, he had frequent opportunities of serving his friends and his country by his great skill. Whatever difficulty occurred concerning the constructing or perfecting of machines, the working of mines, the improving of manufactures, the conveying of water, or the execution of any public work, he was always ready to resolve it. He was employed to terminate some disputes of consequence that had arisen at Glasgow concerning the gauging of vessels; and for that purpose presented to the commissioners of the excise two elaborate memorials, with their demonstrations, containing rules by which the officers now act. He made also calculations relating to the provision, now established by law, for the children and widows of the Scotch clergy, and of the professors in the universities, entitling them to certain annuities and sums, upon the voluntary<cb/> annual payment of a certain sum by the incumbent. In contriving and adjusting this wise and useful scheme, he bestowed a great deal of labour, and contributed not a little towards bringing it to perfection.</p><p>Of his works, we have mentioned his <hi rend="italics">Geometria Organica,</hi> in which he treats of the description of curve lines by continued motion; as also of his piece which gained the prize of the Royal Academy of Sciences in 1724. In 1740, he likewise shared the prize of the same Academy, with the celebrated D. Bernoulli and Euler, for resolving the problem relating to the motion of the tides srom the theory of gravity: a question which had been given out the former year, without receiving any solution. He had only ten days to draw this paper up in, and could not find leisure to transcribe a fair copy; so that the Paris edition of it is incorrect. He afterwards revised the whole, and inserted it in his Treatise of Fluxions; as he did also the substance of the former piece. These, with the Treatise of Fluxions, and the pieces printed in the Medical Essays and the Philosophical Transactions, a list of which is given above, are all the writings which our author lived to publish. Since his death, however, two more volumes have appeared; his <hi rend="italics">Algebra,</hi> and his <hi rend="italics">Account of Sir Isaac Newton's Philosophical Discoveries.</hi> The Algebra, though not finished by himself, is yet allowed to be excellent in its kind; containing, in no large volume, a complete elementary treatise of that science, as far as it has hitherto been carried; besides some neat analytical papers on curve lines. His Account of Newton's Philosophy was occasioned in the following manner:—Sir Isaac dying in the beginning of 1728, his nephew, Mr. Conduitt, proposed to publish an account of his life, and desired Mr. Maclaurin's assistance. The latter, out of gratitude to his great benefactor, cheerfully undertook, and soon finished, the History of the Progress which Philosophy had made before Newton's time; and this was the first draught of the work in hand; which not going forward, on account of Mr. Conduitt's death, was returned to Mr. Maclaurin. To this he afterwards made great additions, and left it in the state in which it now appears. His main design seems to have been, to explain only those parts of Newton's philosophy, which have been controverted: and this is supposed to be the reason why his grand discoveries concerning light and colours are but transiently and generally touched upon; for it is known, that whenever the experiments, on which his doctrine of light and colours is founded, had been repcated with due care, this doctrine had not been contested; while his accounting for the celestial motions, and the other great appearances of nature, from gravity, had been misunderstood, and even attempted to be ridiculed.</p><p>MACULÆ, in Astronomy, are dark spots appearing on the luminous surfaces of the sun and moon, and even some of the planets.</p><p>The Solar Maculæ are dark spots of an irregular and changeable figure, observed in the face of the sun. These were first observed in November and December of the year 1610, by Galileo in Italy, and Harriot in England, unknown to, and independent of each other, soon after they had made or procured telescopes. They were afterwards also observed by Scheiner, Hevelius, Flamsteed, Cassini, Kirch, and others. See Philos. Trans. vol. 1, pa. 274, and vol. 64, pa. 194.<pb n="62"/><cb/></p><p>There have been various observations made of the phenomena of the solar maculæ, and hypotheses invented for explaining them. Many of these maculæ appear to consist of heterogeneous parts; the darker and denser being called, by Hevelius, nuclei, which are encompassed as it were with atmospheres, somewhat rarer and less obscure; but the figure, both of the nuclei and entire maculæ, is variable. These maculæ are often subject to sudden mutations: In 1644 Hevelius observed a small thin macula, which in two days time grew to ten times its bulk, appearing also much darker, and having a larger nucleus: the nucleus began to fail sensibly beforc the spot disappeared; and before it quite vanished, it broke into four, which re-united again two days after. Some maculæ have lasted 2, 3, 10, 15, 20, 30, but seldom 40 days; though Kirchius observed one in 1681, that was visible from April 26th to the 17th of July. It is found that the spots move over the sun's disc with a motion somewhat slacker near the edge than in the middle parts; that they contract themselves near the limb, and in the middle appear larger; that they often run into one in the disc, though separated near the centre; that many of them first appear in the middle, and many disappear there; but that none of them deviate from their path near the horizon; whereas Hevelius, observing Mercury in the sun near the horizon, found him too low, being depressed 27″ beneath his former path.</p><p>From these phenomena are collected the following consequences. 1. That since Mercury's depression below his path arises from his parallax, the maculæ, having no parallax from the sun, are much nearer him than that planet.</p><p>2. That, since they rise and disappear again in the middle of the sun's disc, and undergo various alterations with regard both to bulk, figure, and density, they must be formed <hi rend="italics">de novo,</hi> and again dissolved about the sun; and hence some have inferred, that they are a kind of solar clouds, formed out of his exhalations; and if so, the sun must have an atmosphere.</p><p>3. Since the spots appear to move very regularly about the sun, it is hence inferred, that it is not that they really move, but that the fun revolves round his axis, and the spots accompany him, in the space of 27 days 12 hours 20 minutes.</p><p>4. Since the sun appears with a circular disc in every situation, his figure, as to sense, must be spherical.</p><p>The magnitude of the surface of a spot may be estimated by the time of its transit over a hair in a sixed telescope. Galilco estimates some spots as larger than both Asia and Africa put together: but if he had known more exactly the sun's parallax and distance, as they are known now, he would have found some of those spots much larger than the whole surface of the earth. For, in 1612, he observed a spot so large as to be plainly visible to the naked eye; and therefore it subtended an angle of about a minute. But the earth, seen at the distance of the sun, would subtend an angle of only about 17″: therefore the diameter of the spot was to the diameter of the earth, as 60 to 17, or 3 1/2 to 1 nearly; and consequently the surface of the spot, if circular, to a great circle of the earth, as 12 1/4 to 1, and to the whole surface of the earth, as 12 1/4 to 4, or nearly 3 <*>o 1. Gassendus observed a spot whose breadth was<cb/> <*> of the sun's diameter, and which therefore subtended an angle at the eye of above a minute and a half; and consequently its surface was above seven times larger than the surface of the whole earth. He says he observed above 40 spots at once, though without sensibly diminishing the light of the sun.</p><p>Various opinions have been formed concerning the nature, origin, and situation of the solar spots; but the most probable seems to be that of Dr. Wilson, professor of practical astronomy in the university of Glasgow. By attending particularly to the different phases presented by the umbra, or shady zone, of a spot of an extraordinary size that appeared on the sun, in the month of November 1769, during its progress over the solar disc, Dr. Wilson was led to form a new and singular conjecture on the nature of these appearances; which he afterwards greatly strengthened by repeated observations. The results of these observations are, that the solar maculæ are cavities in the body of the sun; that the nucleus, as the middle or dark part has usually been called, is the bottom of the excavations; and that the umbra, or shady zone surrounding it, is the shelving sides of the cavity. Dr. Wilson, besides having satisfactorily ascertained the reality of these immense excavations in the body of the sun, has also pointed out a method of measuring the depth of them. He estimates, in particular, that the nucleus, or bottom of the large spot above-mentioned, was not less than a semidiameter of the earth, or about 4000 miles below the level of the sun's surface; while its other dimensions were of a much larger extent. He observed that a spot near the middle of the sun's disc, is surrounded equally on all sides with its umbra; but that when, by its apparent motion over the sun's disc, it comes near the western limb, that part of the umbra which is next the sun's centre gradually diminishes in breadth, till near the edge of the limb it totally disappears; whilst the umbra on the other side of it is little or nothing altered. After a semirevolution of the sun on his axis, if the the spot appear again, it will be on the opposite side of the disc, or on the left hand, and the part of the umbra which had before disappeared, is now plainly to be seen; while the umbra on the other side of the spot, seems to have vanished in its turn; being hid from the view by the upper edge of the excavation, from the oblique position of its sloping sides with respect to the eye. But as the spot advances on the sun's disc, this umbra, or side of the cavity, comes in sight; at first appearing narrow, but afterwards gradually increasing in breadth, as the spot moves towards the middle of the disc. Which appearances perfectly agree with the phases that are exhibited by an excavation in a spherical body, revolving on its axis; the bottom of the cavity being painted black, and the sides lightly shaded.</p><p>From these, and other observations, it is inferred, that the body of the sun, at the depth of the nucleus, emits little or no light, when seen at the same time, and compared with that resplendent, and probably, in some degree, fluid substance, that covers his surface.</p><p>This manner of considering these phenomena, naturally gives rise to many curious speculations and inquiries. It is natural, for instance, to inquire, by what great commotion this refulgent matter is thrown up on all sides, so as to expose to our view the darker part of<pb n="63"/><cb/> the sun's body, which was before covered by it? what is the nature of this shining matter? and why, when an excavation is made in it, is the lustre of this shining substance, which forms the shelving sides of the cavity, so far diminished, as to give the whole the appearanee of a shady zone, or darkish atmosphere, surrounding the denuded part of the sun's body? On these, and many other subjects, Dr. Wilson has advanced some ingenious conjectures; for which see the Philos. Trans. vol. 64, art. 1. See also some remarks on this theory, by Mr. Woolaston, in the same vol. pa. 337, &c.</p></div1><div1 part="N" n="MADRIER" org="uniform" sample="complete" type="entry"><head>MADRIER</head><p>, in Artillery, is a thick plank, armed with plates of iron, and having a cavity sufficient to receive the mouth of a petard, with which it is applied against a gate, or any thing else intended to be broken down.</p><p>This term is also applied to certain flat beams, fixed to the bottom of a moat, to support a wall.</p><p>There are also Madriers lined with tin, and covered with earth; serving as defences against artificial fires, in lodgments, &c, where there is need of being covered overhead.</p><p>MÆSTLIN (<hi rend="smallcaps">Michael</hi>), in Latin Mæstlinus, a noted astronomer of Germany, was born in the duchy of Wittemberg; but spent his youth in Italy, where he made a speech in favour of Copernicus's system, which brought Galileo over from Aristotle and Ptolomy, to whom he was before wholly devoted. He afterwards returned to Germany, and became professor of mathematicsat Tubingen; where, among his other scholars, he taught the celebrated Kepler, who has commended several of his ingenious inventions, in his Astronomia Optica.</p><p>Mæstlin published many mathematical and astronomical works; and died in 1590.—Though Tycho Brahe did not assent to Mæstlin's opinion, yet he allowed him to be an extraordinary person, and deeply skilled in the science of astronomy.</p></div1><div1 part="N" n="MAGAZINE" org="uniform" sample="complete" type="entry"><head>MAGAZINE</head><p>, a place in which stores are kept, of arms, ammunition, provisions, &c.</p><p><hi rend="italics">Artillery</hi> <hi rend="smallcaps">Magazine</hi>, or the Magazine to a field battery, is made about 25 or 30 yards behind the battery, towards the parallels, and at least 3 feet under ground, to receive the powder, loaded shells, port-fires, &c.— Its roof and sides should be well secured with boards, to prevent the earth from falling in: it has a door, and a double trench or passage sunk from the magazine to the battery, the one to enter, and the other to go out at, to prevent confusion. Sometimes traverses are made in the passages, to prevent ricochet shot from entering the magazine.</p><p><hi rend="italics">Powder</hi>-<hi rend="smallcaps">Magazine</hi>, is the place where powder is kept in large quantities. Authors differ very much with regard to the situation and construction of these magazines; but all agree, that they ought to be arched and bomb proof. In fortifications, they were formerly placed in the rampart; but of late they have been built in different parts of the town. The first powder-magazines were made with Gothic arches: but M. Vauban finding these too weak, constructed them of a semicircular form, the dimensions being 60 feet long within, and 25 feet broad; the foundations are 8 or 9 feet thick, and 8 feet high from the foundation to the spring of the arch; also the floor 2 feet from the ground, to keep it from dampness.<cb/></p><p>It is a constant observation, that aster the centering of semicircular arches is struck, they settle at the crown, and rise up at the hances, even with a straight horizontal extrados; and still much more so in powdermagazines, where the outside at top is formed, like the roof of a house, by inclined planes joining in an angle over the top of the arch, to give a proper descent to the rain; which effects are exactly what might be expected from the true theory of arches. Now, this shrinking of the arches, as it must be attended with very bad consequences, by breaking the texture of the cement after it has in some degree been dried, and also by opening the joints of the vousoirs at one end, so a remedy is provided for this inconvenience, with regard to bridges, by the arch of equilibration, in my book on the Principles of Bridges: but as the ill consequences of it are much greater in powder-magazines, in question 96 of my Mathematical Miscellany, I proposed to find an arch of equilibration for them also; which question was there resolved both by Mr. Wildbore and myself, both upon general principles, and which I illustrated by an application to a particular case, which is there constructed, and accompanied with a table of numbers for that purpose. Thus, if ALKMB represent a vertical transverse section of the arch, the roof forming an angle LKM of 112° 37′, also PC an ordmate parallel to the horizon taken in any part, and IC perpendicular to the same; then for properly constructing the curve so as to be the strongest, or an arch of equilibration in all its parts, the corresponding values of PC and CI will be as in the following table, where those numbers may denote any lengths whatever, either inches, or feet, or half-yards. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Value of PC</cell><cell cols="1" rows="1" role="data">Value of IC</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">7.031</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7.125</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7.264</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7.501</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7.789</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8.164</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8.574</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9.078</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9.663</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">10.333</cell></row></table> <figure/></p><div2 part="N" n="Magazine" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Magazine</hi></head><p>, or <hi rend="italics">Powder-Room,</hi> on ship-board, is a close room or store-house, built in the fore or after part of the hold, in which to preserve the gunpowder for the use of the ship. This apartment is strongly secured against fire, and no person is allowed to enter it with a lamp or candle. it is therefore lighted, as occasion requires, by means of the candles or lamps in the lightroom contiguous to it.</p><p>MAGELLANIC-<hi rend="smallcaps">Clouds</hi>, whitish appearances like clouds, seen in the heavens towards the south pole, and having the same apparent motion as the stars. They are three in number, two of them near each other.— The largest lies far from the south pole; but the other two are not many degrees more remote from it than the nearest conspicuous star, that is, about 11 degrees.<pb n="64"/><cb/> Mr. Boyle conjectures that if these clouds were seen through a good telescope, they would appear to be multitudes of small stars, like the milky way.</p><p>MAGIC <hi rend="smallcaps">Lantern</hi>, an optical machine, by means of which small painted images are represented on the wall of a dark room, magnified to any size at pleasure. This machine was contrived by Kircher, (see his Ars Magna Lucis and Umbræ, pa. 768); and it was so called, because the images were made to represent strange phantasms, and terrible apparitions, which have been taken for the effect of magic, by such as were ignorant of the secret.</p><p>This machine is composed of a concave speculum, from 4 to 12 inches diameter, reflecting the light of a candle through the small hole of a tube, at the end of which is fixed a double convex lens of about 3 inches focus. Between the two are successively placed, many small plain glasses, painted with various figures, usually such as are the most formidable and terrifying to the spectators, when represented at large on the opposite wall.</p><p>Thus, (Pl. 13, fig. 14) ABCD is a common tin lantern, to which is added a tube FG to draw out. In H is fixed the metallic concave speculum, from 4 to 12 inches diameter; or else, instead of it, near the extremity of the tube, there must be placed a convex lens, consisting of a segment of a small sphere, of but a few inches in diameter. The use of this lens is to throw a strong light upon the image; and sometimes a concave speculum is used with the lens, to render the image still more vivid. In the focus of the concave speculum or lens, is placed the lamp L; and within the tube, where it is soldered to the side of the lantern, is placed a small lens, convex on both sides, being a portion of a small sphere, having its focus about the distance of 3 inches. The extreme part of the tube FM is square, and has an aperture quite through, so as to receive an oblong frame NO passing into it; in which frame there are round holes, of an inch or two in diameter. Answering to the magnitude of these holes there are drawn circles on a plain thin glass; and in these circles are painted any figures, or images, at pleasure, with transparent water colours. These images fitted into the frame, in an inverted position, at a small distance from the focus of the lens I, will be projected on an opposite white wall of a dark room, in all their colours, greatly magnisied, and in an erect position. By having the instrument so contrived, as that the lens I may move on a slide, the focus may be made, and consequently the image appear distinct, at almost any distance.</p><p>Or thus: Every thing being managed as in the former case, into the sliding tube FG, insert another convex lens K, the segment of a sphere rather larger than I. Now, if the picture be brought nearer to I than the distance of the focus, diverging rays will be propagated as if they proceeded from the object; wherefore, if the lens K be so placed, as that the object be very near its focus, the image will be exhibited on the wall, greatly magnisied.</p><p><hi rend="smallcaps">Magic Square</hi>, is a square figure, formed of a series of numbers in arithmetical progression, so disposed in parallel and equal ranks, as that the sums of each row, taken either perpendicularly, horizontally, or diagonally, are equal to one another. As the annexed square, form-<cb/> ed of these nine numbers, 1, 2, 3, 4, 5, 6, 7, 8, 9, where the sum of the three figures in every row, in all directions, is always the same number, viz 15. But if the same numbers be placed in this natural order, the first being 1, and the last of them a square number, they will form what is called a natural square. As in the first 25 numbers, viz, 1, 2, 3, 4, 5, &c to 25. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell></row></table> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=5 align=center" role="data">Natural Square.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">15</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">25</cell></row></table> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=5 align=center" role="data">Magic Square.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">25</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">17</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">13</cell></row></table></p><p>where every row and diagonal in the magic square makes just the sum 65, being the same as the two diagonals of the natural square.</p><p>It is probable that these magic squares were so called, both because of this property in them, viz, that the ranks in every direction make the same sum, appeared extremely surprising, especially in the more ignorant ages, when mathematics passed for magic, and because also of the superstitious operations they were employed in, as the construction of talismans, &c; for, according to the childish philosophy of those days, which ascribed virtues to numbers, what might not be expected from numbers so seemingly wonderful!</p><p>The Magic Square was held in great veneration among the Egyptians, and the Pythagoreans their disciples, who, to add more efficacy and virtue to this square, dedicated it to the then known seven planets divers ways, and engraved it upon a plate of the metal that was esteemed in sympathy with the planet. The square thus dedicated, was inclosed by a regular polygon, inscribed in a circle, which was divided into as many equal parts as there were units in the side of the square; with the names of the angels of the planet, and the signs of the zodiac written upon the void spaces between the polygon and the circumference of the circumscribed circle. Such a talisman or metal they vainly imagined would, upon occasion, befriend the person who carried it about him.</p><p>To Saturn they attributed the square of 9 places or cells, the side being 3, and the sum of the numbers in every row 15: to Jupiter the square of 16 places, the side being 4, and the amount of each row 34: to Mars the square of 25 places, the side being 5, and the amount of each row 65: to the Sun the square with 36 places, the side being 6, and the sum of each row 111: to Venus the square of 49 places, the side being 7, and the amount of each row 175: to Mercury the square with 64 places, the side being 8, and the sum of <pb/><pb/><pb n="65"/><cb/> each row 260: and to the Moon the square of 81 places, the side being 9, and the amount of each row 369. Finally, they attributed to imperfect matter, the square with 4 divisions, having 2 for its side; and to God the square of only one cell, the side of which is also an unit, which multiplied by itself, undergoes no change.</p><p>However, what was at first the vain practice of conjurers and makers of talismans, has since become the subject of a serious research among mathematicians. Not that they imagine it will lead them to any thing of solid use or advantage; but rather as it is a kind of play, in which the difficulty makes the merit, and it may chance to produce some new views of numbers, which mathematicians will not lose the occasion of.</p><p>It would seem that Eman. Moschopulus, a Greek author of no high antiquity, is the first now known of, who has spoken of magic squares: he has left some rules for their construction; though, by the age in which he lived, there is reason to imagine he did not look upon them merely as a mathematician.</p><p>In the treatise of Cornelius Agrippa, so much accused of magic, are found the squares of seven numbers, viz, from 3 to 9 inclusive, disposed magically; and it is not to be supposed that those seven numbers were preferred to all others without some good reason: indeed it is because their squares, according to the system of Agrippa and his followers, are planetary. The square of 3, for instance, belongs to Saturn; that of 4 to Jupiter; that of 5 to Mars; that of 6 to the Sun; that of 7 to Venus: that of 8 to Mercury; and that of 9 to the Moon.</p><p>M. Bachet applied himself to the study of magic squares, on the hint he had taken from the planetary squares of Agrippa, as being unacquainted with Moschopulus's work, which is only in manuscript in the French king's library; and, without the assistance of any author, he found out a new method for the squares of uneven numbers; for instance, 25, or 49, &c; but he could not succeed with those that have even roots.</p><p>M. Frenicle next engaged in this subject. It was the opinion of some, that although the first 16 numbers might be disposed 20922789888000 different ways in a natural square, yet they could not be disposed more than 16 ways in a magic square; but M. Frenicle shewed, that they might be thus disposed in 878 different ways.</p><p>To this business he thought fit to add a difficulty that had not yet been considered; which was, to take away the marginal numbers quite around, or any other circumference at pleasure, or even several of such circumferences, and yet that the remainder should still be magical.</p><p>Again he inverted that condition, and required that any circumference taken at pleasure, or even several circumferences, should be inseparable from the square; that is, that it should cease to be magical when they were removed, and yet continue magical after the removal of any of the rest. M. Frenicle however gives no general demonstration of his methods, and it often seems that he has no other guide but chance. It is true, his book was not published by himself, nor did it appear till after his death, viz, in 1693.</p><p>In 1703 M. Poignard, canon of Brussels, published a treatise on sublime magic squares. Before his time<cb/> there had been no magic squares made, but for serieses of natural numbers that formed a square; but M. Poignard made two very considerable improvements. 1st, Instead of taking all the numbers that fill a square, for instance, the 36 successive numbers, which would sill all the cells of a natural square whose side is 6, he only takes as many successive numbers as there are units in the side of the square, which in this case are 6; and these six numbers alone he disposes in such manner, in the 36 cells, that none of them occur twice in the same rank, whether it be horizontal, vertical, or diagonal; whence it follows, that all the ranks, taken all the ways possible, must always make the same sum; and this method M. Poignard calls repeated progressions. 2d, Instead of being confined to take these numbers according to the series and succession of the natural numbers, that is in arithmetical progression, he takes them likewise in a geometrical progression; and even in an harmonical progression, the numbers of all the ranks always following the same kind of progression: he makes. squares of each of these three progressions repeated.</p><p>M. Poignard's book gave occasion to M. de la Hire to turn his thoughts to the same subject, which he did with such success, that he greatly extended the theory of magic squares, as well for even numbers as those that are uneven; as may be seen at large in the Memoirs of the Royal Academy of Sciences, for the years 1705 and 1710. See also Saunderson's Algebra, vol. 1, pa. 354, &c; as also Ozanam's Mathematical Recreations, who lays down the following easy method of filling up a magic square.</p><p><hi rend="italics">To form a magic square of an odd number of terms in the arithmetic progression</hi> 1, 2, 3, 4, &c. Place the least term 1 in the cell immediately under the middle, or central one, and the rest of the terms, in their natural order, in a descending diagonal direction, till they run off either at the bottom, or on the side: when the number runs off at the bottom, carry it to the uppermost cell, that is not occupied, of the same column that it would have fallen in below, and then proceed descending diagonalwise again as far as you can, or till the numbers either run off at bottom or side, or are interrupted by coming at a cell already filled: now when any number runs off at the right-hand side, then bring it to the farthest cell on the left-hand of the same row or line it would have fallen in towards the righthand: and when the progress diagonalwise is interrupted by meeting with a cell already occupied by some other number, then descend diagonally to the left from this cell till an empty one is met with, where enter it; and thence proceed as before. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">29</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">37</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">45</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">28</cell></row></table></p><p>Thus, to make a magic square of the 49 numbers 1, 2, 3, 4, &c. First place the 1 next below the centre cell, and thence descend to the right till the 4 runs off at the bottom, which therefore carry to the top corner on the same column as it would have fallen in; but as runs off at the side, bring it to the beginning of the second line,<pb n="66"/><cb/> and thence descend to the right till they arrive at the cell occupied by 1; carry the 8 therefore to the next diagonal cell to the left, and so proceed till 10 run off at the bottom, which carry therefore to the top of its column, and so proceed till 13 runs off at the side, which therefore bring to the beginning of the same line, and thence proceed till 15 arrives at the cell occupied by 8; from this therefore descend diagonally to the left; but as 16 runs off at the bottom, carry it to the top of its proper column, and thence descend till 21 run off at the side, which is therefore brought to the beginning of its proper line; but as 22 arrives at the cell occupied by 15, descend diagonally to the left, which brings it into the 1st column, but off at the bottom, and therefore it is carried to the top of that column; thence descending till 29 runs off both at bottom and side, which therefore carry to the highest unoccupied cell in the last column; and here, as 30 runs off at the side, bring it to the beginning of its proper column, and thence descend till 35 runs off at the bottom, which therefore carry to the beginning or top of its own column; and here, as 36 meets with the cell occupied by 29, it is brought from thence diagonally to the left; thence descending, 38 runs off at the side, and therefore it is brought to the beginning of its proper line; thence descending, 41 runs off at the bottom, which therefore is carried to the beginning or top of its column; from whence descending, 43 arrives at the cell occupied by 36, and therefore it is brought down from thence to the left; thence descending, 46 runs off at the side, which therefore is brought to the beginning of its line; but here, as 47 runs off at the bottom, it is carried to the beginning or top of its column, from whence descending with 48 and 49, the square is completed, the sum of every row and column and diagonal making just 175.</p><p>There are many other ways of filling up such squares, but none that are easier than the above one.</p><p>It was observed before, that the sum of the numbers in the rows, columns and diagonals, was 15 in the square of 9 numbers, 34 in a square of 16, 65 in a square of 25, &c; hence then is derived a method of finding the sums of the numbers in any other square, viz, by taking the successive differences till they become equal, and then adding them successively to produce or find out the amount of the following sums. Thus, <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Side</cell><cell cols="1" rows="1" role="data">Cells</cell><cell cols="1" rows="1" role="data">Sums</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Diffs.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">260</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">369</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">136</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">505</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/></row></table> having ranged the sides and cells in two columns, and a few of the first sums in a third column, take the first differences of these, which will be 1, 4, 10, 19, &c, as in the 4th column; and of these take the differences<cb/> 0, 3, 6, 9, 12, &c, as in the 5th column; and again, of these the differences 3, 3, 3 &c, as in the 6th or last column. Then, returning back again, add always 3, the constant last or 3d difference, to the last found of the 2d differences, which will complete the remainder of the column of these, viz, 15, 18, 21, 24, &c: then add these 2d differences to the last found of the 1st differences, which will complete the column of these, viz, giving 31, 46, 64, &c: lastly, add always these corresponding 1st differences to the last found number or amount of the sums, and the column of sums will thus be completed.</p><p>Again, like as the terms of an arithmetical prog<*>ession arranged magically, give <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">64</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">32</cell></row></table> the same sum in every row &c, so the terms of a geometrical series arranged magically give the same product in every row &c, by multiplying the numbers continually together; so this progression 1, 2, 4, 8, 16, &c, arranged as in the margin, gives, for each continual product, 4096 <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1260</cell><cell cols="1" rows="1" role="data">840</cell><cell cols="1" rows="1" role="data">630</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">504</cell><cell cols="1" rows="1" role="data">420</cell><cell cols="1" rows="1" role="data">360</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">315</cell><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data">252</cell></row></table> in every row &c, which is just the cube of the middle term, 16.</p><p>Also, the terms of an harmonical progression being ranged magically, as in the margin, have the terms in each row &c in harmonical progression.</p><p>The ingenious Dr. Franklin, it seems, carried this curious speculation farther than any of his predecessors in the same way. He constructed both a <hi rend="italics">magic square of squares,</hi> and a <hi rend="italics">magic circle of circles,</hi> the description of which is as follows. The magic square of squares is formed by dividing the great square as in fig. 1, Pl. 15. The great square is divided into 256 little squares, in which all the numbers from 1 to 256, or the square of 16, are placed, in 16 columns, which may be taken either horizontally or vertically. Their chief properties are as follow:</p><p>1. The sum of the 16 numbers in each column or row, vertical or horizontal, is 2056.</p><p>2. Every half column, vertical and horizontal, makes 1028, or just one half of the same sum 2056.</p><p>3. Half a diagonal ascending, added to half a diagonal descending, makes also the same sum 2056; taking these half diagonals from the ends of any side of the square to the middle of it; and so reckoning them either upward or downward; or sideways from right to left, or from left to right.</p><p>4. The same with all the parallels to the half diagonals, as many as can be drawn in the great square: for any two of them being directed upward and downward, from the place where they begin, to that where they end, their sums still make the same 2056. Also the same holds true downward and upward; as well as if taken sideways to the middle, and back to the same side again. Only one set of these half diagonals and their parallels, is drawn in the same square upward and downward; but another set may be drawn from any of the other three sides.</p><p>5. The four corner numbers in the great square added to the four central numbers in it, make 1028, the<pb n="67"/><cb/> half sum of any vertical or horizontal column, which contains 16 numbers; and also equal to half a diagonal or its parallel.</p><p>6. If a square hole, equal in breadth to four of the little squares or cells, be cut in a paper, through which any of the <*>6 little cells in the great square may be seen, and the paper be laid upon the great square; the sum of all the 16 numbers, seen through the hole, is always equal to 2056, the sum of the 16 numbers in any horizontal or vertical column.</p><p>The <hi rend="italics">Magic Circle of Circles,</hi> fig. 2, pl. 15, by the same author, is composed of a series of numbers, from 12 to 75 inclusive, divided into 8 concentric circular spaces, and ranged in 8 radii of numbers, with the number 12 in the centre; which number, like the centre, is common to all these circular spaces, and to all the radii.</p><p>The numbers are so placed, that 1st, the sum of all those in either of the concentric circular spaces above mentioned, together with the central number 12, amount to 360, the same as the number of degrees in a circle.</p><p>2. The numbers in each radius also, together with the central number 12, make just 360.</p><p>3. The numbers in half of any of the above circular spaces, taken either above or below the double horizontal line, with half the central number 12, make just 180, or half the degrees in a circle.</p><p>4. If any four adjoining numbers be taken, as if in a square, in the radial divisions of these circular spaces; the sum of these, with half the central number, make also the same 180.</p><p>5. There are also included four sets of other circular spaces, bounded by circles that are excentric with regard to the common centre; each of these sets containing five spaces; and the centres of them being at A, B, C, D. For distinction, these circles are drawn with different marks, some dotted, others by short unconnected lines, &c; or still better with inks of divers colours, as blue, red, green, yellow.</p><p>These sets of excentric circular spaces intersect those of the concentric, and each other; and yet, the numbers contained in each of the excentric spaces, taken all around through any of the 20, which are excentric, make the same sum as those in the concentric, namely 360, when the central number 12 is added. Their halves also, taken above or below the double horizontal line, with half the central number, make up 180.</p><p>It is observable, that there is not one of the numbers but what belongs at least to two of the circular spaces; some to three, some to four, some to five: and yet they are all so placed, as never to break the required number 360, in any of the 28 circular spaces within the primitive circle. They have also other properties. See Franklin's Exp. and Obs. pa. 350, edit. 4to, 1769; or Ferguson's Tables and Tracts, 1771, pa. 318.</p><p>MAGICAL <hi rend="italics">Picture,</hi> in Electricity, was first contrived by Mr. Kinnersley, and is thus made: Having a large mezzotinto with a frame and glass, as of the king for instance, take out the print, and cut a pannel out of it, near two inches distant from the frame all around; then with thin paste or gum-water, six the border that is cut off on the inside of the glass, pressing it smooth and close; then sill up the vacancy by gilding the glass well with leaf gold or brass. Gild likewise the inner<cb/> edge of the back of the frame all round, except the top part, and form a communication between that gilding and the gilding behind the glass; then put in the boardy and that side is finished. Next turn up the glass, and gild the foreside exactly over the back gilding, and when it is dry, cover it by pasting on the pannel of the picture that has been cut out, observing to bring the corresponding parts of the border and picture together, by which means the picture will appear entire, as at first, only part behind the glass, and part before.</p><p>Hold the picture horizoutally by the top, and place a small moveable gilt crown on the king's head. If now the picture be moderately electrified, and another person take hold of the frame with one hand, so that his fingers touch its inside gilding, and with the other hand endeavour to take off the crown, he will receive a violent blow, and fail in the attempt. If the picture were highly charged, the consequence might be as fatal as that of high treason. The operator, who holds the picture by the upper end, where the inside of the frame is not gilt, to prevent its falling, feels nothing of the shock, and may touch the face of the picture without danger. And if a ring of persons take the shock among them, the experiment is called the conspirators. See Franklin's Exper. and Observ. pa. 30.</p></div2></div1><div1 part="N" n="MAGINI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MAGINI</surname> (<foreName full="yes"><hi rend="smallcaps">John-Anthony</hi></foreName>)</persName></head><p>, or <hi rend="smallcaps">Maginus</hi>, professor of mathematics in the university of Bologna, was born at Padua in the year 1536. Magini was remarkable for his great assiduity in acquiring and improving the knowledge of the mathematical sciences, with several new inventions for these purposes, and for the extraordinary favour he obtained from most princes of his time. This doubtless arose partly from the celebrity he had in matters of astrology, to which he was greatly addicted, making horoscopes, and foretelling events, both relating to persons and things. He was invited by the emperor Rodolphus to come to Vienna, where he promised him a professor's chair, about the year 1597; but not being able to prevail on him to settle there, he nevertheless gave him a handsome pension.</p><p>It is said, he was so much addicted to astrological predictions, that he not only foretold many good and evil events relative to others with success; but even foretold his own death, which came to pass the same year: all which he represented as under the influence of the stars. Tomasini says, that Magini, being advanced to his 61st year, was struck with an apoplexy, which ended his days; and that a long while before, he had told him and others, that he was afraid of that year. And Roffeni, his pupil, says, that Magini died under an aspect of the planets, which, according to his own prediction, would prove fatal to him; and he mentions Riccioli as affirming that he said, the figure of his nativity, and his climacteric year, doomed him to die about that time; which happened in 1618, in the 62d year of his age.</p><p>His writings do honour to his memory, as they were very considerable, and upon learned subjects. The principal were the following: 1. His Ephemeris, in 3 volumes, from the year 1580 to 1630.—2. Tables of Secondary Motions.—3. Astronomical, Gnomonical, and Geographical Problems.—4. Theory of the Planets, according to Copernicus.—5. A Confutation of Scaliger's Dissertation concerning the Precession of the<pb n="68"/><cb/> Equinox.—6. A Primum Mobile, in 12 books.—7. A Treatise of Plane and Spherical Trigonometry.—8. A Commentary on Ptolomy's Geography.—9. A Chorographical Description of the Regions and Cities of Italy, illustrated with 60 maps; with some other papers on Astrological subjects.</p></div1><div1 part="N" n="MAGNET" org="uniform" sample="complete" type="entry"><head>MAGNET</head><p>, <hi rend="smallcaps">Magnes</hi>, the <hi rend="italics">Loadstone;</hi> a kind of ferruginous stone, resembling iron ore in weight and colour, though rather harder and heavier; and is endued with divers extraordinary properties, attractive, directive, inclinatory, &c. See <hi rend="smallcaps">Magnetism.</hi></p><p>The Magnet is also called <hi rend="italics">Lapis Heraclœus,</hi> from Heraclea, a city of Magnesia, a port of the ancient Lydia, where it was said it was first found, and from which it is usually supposed that it took its name. Though some derive the word from a shepherd named <hi rend="italics">Magues,</hi> who first discovered it on Mount lda with the iron of his crook. It is also called <hi rend="italics">Lapis Nauticus,</hi> from its use in navigation; also <hi rend="italics">Siderites,</hi> from its virtue in attracting iron, which the Greeks call <foreign xml:lang="greek">si<*>hros</foreign>.</p><p>The Magnet is usually found in iron mines, and sometimes in very large pieces, half magnet, half iron. Its colour is different, as found in different countries. Norman observes, that the best are those brought from China and Bengal, which are of an irony or sanguine colour; those of Arabia are reddish; those of Macedonia, blackish; and those of Hungary, Germany, England, &c, the colour of unwrought iron. Neither its figure nor bulk are constant or determined; being found of all shapes and sizes.</p><p>The Ancients reckoned five kinds of Magnets, different in colour and virtue: the Ethiopic, Magnesian, Bœotic, Alexandrian, and Natolian. They also took it to be male and female: but the chief use they made of it was in medicine; especially for the cure of burns and defluxions of the eyes.—The Moderns, more happy, take it to conduct them in their voyages.</p><p>The most distinguishing properties of the Magnet are, That it attracts iron, and that it points towards the poles of the world; and in other circumstances also dips or inclines to a point beneath the horizon, directly under the pole; it also communicates these properties, by touch, to iron. By means of which, are obtained the mariner's needles, both horizontal, and inclinatory or dipping needles.</p><p><hi rend="italics">The Attractive Power of the</hi> <hi rend="smallcaps">Magnet</hi>, was known to the Ancients, and is mentioned even by Plato and Euripides, who call it the <hi rend="italics">Herculean stone,</hi> because it commands iron, which subdues every thing else: but the knowledge of its directive power, by which it disposes its poles along the meridian of every place, or nearly so, and causes needles, pieces of iron, &c, touched with it, to point nearly north and south, is of a much later date; though the discoverer himself, and the exact time of the discovery, be not now known. The first mention of it is about 1260, when it has been said that Marco Polo, a Venetian, introduced the mariner's compass; though not as an invention of his own, but as derived from the Chinese, who it seems had the use of it long before; though some imagine that the Chinese rather borrowed it from the Europeans.</p><p>But Flavio de Gira, a Neapolitan, who lived in the 13th century, is the person usually supposed to have the best title to the discovery; and yet Sir G. Wheeler<cb/> mentions, that he had seen a book of astronomy much older, which supposed the use of the needle; though not as applied to the purposes of navigation, but of astronomy. And in Guiot de Provins, an old French poet, who wrote about the year 1180, there is an express mention made of the loadstone and the compass; and their use in navigation obliquely hinted at.</p><p><hi rend="italics">The Variation of the</hi> <hi rend="smallcaps">Magnet</hi>, or needle, or its deviation from the pole, was first discovered by Sebastian Cabot, a Venetian, in 1500; and the variation of that variation, or change in its direction, by Mr. Henry Gellibrand, professor of astronomy in Gresham college, about the year 1625.</p><p>Lastly, the Dip or inclination of the needle, when at liberty to play vertically, to a point beneath the horizon, was first discovered by another of our countrymen, Mr. Robert Norman, about the year 1576.</p><p><hi rend="italics">The Phenomena of the</hi> <hi rend="smallcaps">Magnet</hi>, are as follow: 1, In every Magnet there are two poles, of which the one points northwards, the other southwards; and if the Magnet be divided into ever so many pieces, the two poles will be found in each piece. The poles of a Magnet may be found by holding a very fine short needle over it; for where the poles are, the needle will stand upright, but no where else.—2, These poles, in different parts of the globe, are differently inclined towards a point under the horizon.—3, These poles, though contrary to each other, do help mutually towards the Magnet's attraction, and suspension of iron. —4, If two Magnets be spherical, one will turn or conform itself to the other, so as either of them would do to the earth; and after they have so conformed or turned themselves, they endeavour to approach or join each other; but if placed in a contrary position, they avoid each other.—5, If a Magnet be cut through the axis, the segments or parts of the stone, which before were joined, will now avoid and fly each other.—6, If the Magnet be cut perpéndicular to its axis, the two points, which before were conjoined, will become contrary poles; one in the one, and one in the other segment.—7, Iron receives virtue from the Magnet by application to it, or barely from an approach near it, though it do not touch it; and the iron receives this virtue variously, according to the parts of the stone it is made to touch, or even approach to.— 8, If an oblong piece of iron be anyhow applied to the stone, it receives virtue from it only lengthways.—9, The Magnet loses none of its own virtue by communicating any to the iron; and this virtue it can communicate to the iron very speedily: though the longer the iron joins or touches the stone, the longer will its communicated virtue hold; and a better Magnet will communicate more of it, and sooner, than one not so good. —10, Steel receives virtue from the Magnet better than iron.—11, A needle touched by a Magnet will turn its ends the same way towards the poles of the world, as the Magnet itself does.—12, Neither loadstone nor needles touched by it do conform their poles exactly to those of the world, but have usually some variation from them: and this variation is different in divers places, and at divers times in the same places.— 13, A loadstone will take up much more iron when armed, or capped, than it can alone. (A loadstone is said to be armed, when its poles are surrounded with <pb/><pb/><pb n="69"/><cb/> plates of steel: and to determine the quantity of steel to be applied, try the Magnet with several steel bars; and the greatest weight it takes up, with a bar on, is to be the weight of its armour.) And though an iron ring or key be suspended by the loadstone, yet this does not hinder the ring or key from turning round any way, either to the right or left.—14, The force of a loadstone may be variously increased or lessened by variously applying to it, either iron, or another loadstone.—15, A strong Magnet at the least distance from a smaller or a weaker, cannot draw to it a piece of iron adhering actually to such smaller or weaker stone; but if it come to touch it, it can draw it from the other: but a weaker Magnet, or even a small piece of iron, can draw away or separate a piece of iron contiguous to a larger or stronger Magnet.—16, In these northern parts of the world, the south pole of a Magnet will raise up more iron than its north pole.—17, A plate of iron only, but no other body interposed, can impede the operation of the loadstone, either as to its attractive or directive quality.—18, The power or virtue of a loadstone may be impaired by lying long in a wrong position, as also by rust, wet, &c; and may be quite destroyed by fire, lightning, &c.—19, A piece of iron wire well touched, upon being bent round in a ring, or coiled round on a stick, &c, will always have its directive virtue diminished, and often quite destroyed. And yet if the whole length of the wire were not entirely bent, so that the ends of it, though but for the length of one-tenth of an inch, were left straight, the virtue will not be destroyed in those parts; though it will in all the rest.—20, The sphere of activity of Magnets is greater and less at different times. Also, the variation of the needle from the meridian, is various at different times of the day.—21, By twisting a piece of wire touched with a Magnet, its virtue is greatly diminished; and sometimes so disordered and confused, that in some parts it will attract, and in others repel; and even, in some places, one side of the wire seems to be attracted, and the other side repelled, by one and the same pole of the stone.—22, A piece of wire that has been touched, on being split, or cleft in two, the poles are sometimes changed, as in a cleft Magnet; the north pole becoming the south, and the south the north: and yet sometimes one half of the wire will retain its former poles, and the other half will have them changed. —23, A wire being touched from end to end with one pole of a Magnet, the end at which you begin will always turn contrary to the pole that touched it: and if it be again touched the <*> me way with the other pole of the Magnet, it will then be turned the contrary way.—24, If a piece of wire be touched in the middle with only one pole of the Magnet, without moving it backwards or forwards; in that place will be the pole of the wire, and the two ends will be the other pole.—25, If a Magnet be heated red hot, and again cooled either with its south pole towards the north in a horizontal position, or with its south pole downwards in a perpendicular position, its poles will be changed. —26, Mr. Boyle (to whom we are indebted for the following magnetical phenomena) found he could presently change the poles of a small fragment of a loadstone, by applying them to the opposite vigorous poles of a large one.—27, Hard iron tools well tempered,<cb/> when heated by a brisk attrition, as siling, turning, &c, will attract thin filings or chips of iron, steel, &c; and hence we observe that siles, punches, augres, &c, have a small degree of magnetic virtue.—28, The iron bars of windows, &c, which have stood a long time in an erect position, grow permanently magnetical; the lower ends of such bars being the north pole, and the upper end the south pole.—29, A bar of iron that has not stood long in an erect posture, if it be only held perpendicularly, will become magnetical, and its lower end the north pole, as appears from its attracting the south pole of a needle: but then this virtue is transient, and by inverting the bar, the poles change their places. In order therefore to render the quality permanent in an iron bar, it must continue a long time in a proper position. But fire will produce the effect in a short time: for as it will immediately deprive a loadstone of its attractive virtue; so it soon gives a verticity to a bar of iron; if, being heated red hot, it be cooled in an erect posture, or directly north and south. Even tongs and fireforks, by being often heated, and set to cool again in a posture nearly erect, have gained this magnetic property. Sometimes iron bars, by long standing in a perpendicular position, have acquired the magnetic virtue in a surprising degree. A bar about 10 feet long, and three inches thick, supporting the summer beam of a room, was able to turn the needle at 8 or 10 feet distance, and exceeded a loadstone of 3 1/2 pounds weight: from the middle point upwards it was a north pole, and downwards a south pole. And Mr. Martin mentions a bar, which had been the beam of a large steel-yard that had several poles in it.—30, Mr. Boyle found, that by heating a piece of English oker red-hot, and placing it to cool in a proper posture, it manifestly acquired a magnetic virtue. And an excellent Magnet, belonging to the same ingenious gentleman, having lain near a year in an inconvenient posture, had its virtue greatly impaired, as if it had been by fire.—31, A needle well touched, it is known, will point north and south: if it have one contrary touch of the same stone, it will be deprived of its faculty; and by another such touch, it will have its poles interchanged.—32, If an iron bar have gained a verticity by being heated red-hot and cooled again, north and south, and then hammered at the two ends; its virtue will be destroyed by two or three smart blows on the middle.—33, By drawing the back of a knife, or a long piece of steel-wire, &c, leisurely over the pole of a loadstone, carrying the motion from the middle of the stone to the pole; the knife or wire will attract one end of a needle; but if the knife or wire be passed from the said pole to the middle of the stone, it will repel the same end of the needle.—34, Either a Magnet or a piece of iron being laid on a piece of cork, so as to float freely on water; it will be found, that, whichsoever of the two is held in the hand, the other will be drawn to it: so that iron attracts the Magnet as much as it is attracted by it; action and re-action being always equal. In this experiment, if the Magnet be set afloat, it will direct its two poles to the poles of the world nearly.—35, A knife &c touched with a Magnet, acquires a greater or less degree of virtue, according to the part it is touched on. It receives the strongest virtue, when it is drawn leisurely from the<pb n="70"/><cb/> handle towards the point over one of the poles. And if the same knife thus touched, and thus possessed of a strong attractive power, be retouched in a contrary direction, viz, by drawing it from the point towards the handle over the same pole, it immediately loses all its virtue.—36, A Magnet acts with equal force in vacuo as in the open air.—37, The smallest Magnets have usually the greatest power in proportion to their bulk. A large Magnet will seldom take up above 3 or 4 times its own weight, while a small one will often take up more than ten times its weight. A Magnet worn by Sir Isaac Newton in a ring, and which weighed only 3 grains, would take up 746 grains, or almost 250 times its own weight. A magnetic bar made by Mr. Canton, weighing 10 oz. 12 dwts, took up more than 79 ounces; and a flat semicircular steel Magnet, weighing 1 oz. 13 dwts, took up an iron wedge of 90 ounces.</p><p><hi rend="italics">Armed</hi> <hi rend="smallcaps">Magnet</hi>, denotes one that is capped, cased, or set in iron or steel, to make it take up a greater weight, and also more readily to distinguish its poles. For the methods of doing this, see Mr. Michell's book on this subject.</p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Magnet</hi>, is a bar of iron or steel, impregnated with the magnetic virtue, so as to possess all the properties of the natural loadstone, and be used instead of it. How to make Magnets of this kind, by means of a natural Magnet, and even without the assistance of any Magnet, was suggested many years since by Mr. Savary, and particularly described in the Philos. Trans. number 414. See also Abridgment, vol. 6, pa. 260. But as his method was tedious and operose, though capable of communicating a very considerable virtue, it was little practised. Dr. Gowin Knight first brought this kind of Magnets to their present state of perfection, so as to be even of much greater efficacy than the natural ones. But as he refused to discover his methods upon any terms whatever (even, as he said, though he should receive in return as many guineas as he could carry), these curious and valuable secrets in a great measure died with him. The result of his method however was first published in the Philos. Trans. for 1744, art. 8, and for 1745, art. 3. See also the vol. for 1747, art. 2. And in the 69th vol. Mr. Benjamin Wilson has given a process, which at least discovers one of the leading principles of Dr. Knight's art. The method, according to Mr. Wilson, was as follows. Having provided a great quantity of clean iron filings, he put them into a large tub that was more than one-third filled with clean water; he then, with great labour, shook the tub to and fro for many hours together, that the friction between the grains of iron, by this treatment, might break or rub off such small parts as would remain suspended in the water for some time. The water being thus rendered very muddy, he poured it into a clean iron vessel, leaving the filings behind; and when the water had stood long enough to become clear, he poured it out carefully, without disturbing such of the sediment as still remained, which now appeared reduced almost to impalpable powder. This powder was afterwards removed into another vessel, to dry it. Having, by several repetitions of this process, procured a sufficient quantity<cb/> of this very fine powder, the next thing was to make a paste of it, and that with some vehicle containing a good quantity of the phlogistic principle; for this purpose, he had recourse to linseed oil, in preference to all other fluids. With these two ingredients only, he made a stiff paste, and took great care to knead it well before he moulded it into convenient shapes. Sometimes, while the paste continued in its soft state, he would put the impression of a seal; one of which is in the British Museum. This paste so moulded was then set upon wood, or a tile, to dry or bake it before a moderate fire, being placed at about one foot distance. He found that a moderate fire was most proper, because a greater degree of heat would make the composition crack in many places. The time requisite for the baking or drying of this paste, was usually about 5 or 6 hours, before it attained a sufficient degree of hardness. When that was done, and the several baked pieces were become cold, he gave them their magnetic virtue in any direction he pleased, by placing them between the extreme ends of his large magazine of artisicial magnets, for a few seconds. The virtue they acquired by this method was such, that, when any of those pieces were held between two of his best ten guinea bars, with its poles purposely inverted, it immediately of itself turned about to recover its natural direction, which the force of those very powerful bars was not sufficient to counteract. Philos. Trans. vol. 65, for 1779.</p><p>Methods for artificial Magnets were also discovered and published by the Rev. Mr. John Michell, in a Treatise on Artificial Magnets, printed in 1750, and by Mr. John Canton, in the Philos. Trans. for 1751. The process for the same purpose was also found out by other persons, particularly by Du Hamel, Hist. Acad. Roy. 1745 and 1750, and by Marul Uitgeleeze Natuurkund. Verhand. tom. 2, p. 261.</p><p>Mr. Canton's method is as follows: Procure a dozen of bars; 6 of soft steel, and 6 of hard; the former to be each 3 inches long, a quarter of an inch broad, and 1-20th of an inch thick; with two pieces of iron, each half the length of one of the bars, but of the same breadth and thickness, and the 6 hard bars to be each 5 1/2 inches long, half an inch broad, and 3-20ths of an inch thick, with two pieces of iron of half the length, but the whole breadth and thickness of one of the hard bars; and let all the bars be marked with a line quite around them at one end. Then take an iron poker and tongs (fig. 1, plate 16), or two bars of iron, the larger they are, and the longer they have been used, the better; and fixing the poker upright between the knees, hold to it, near the top, one of the soft bars, having its marked end downwards by a piece of sewing silk, which must be pulled tight by the left hand, that the bar may not slide: then grasping the tongs with the right hand, a little below the middle, and holding them nearly in a vertical position, let the bar be stroked by the lower end, from the bottom to the top, about ten times on each side, which will give it a magnetic power sufficient to lift a small key at the marked end: which end, if the bar were suspended on a point, would turn towards the north, and is therefore called the north pole; and the unmarked end is, for the same reason,<pb n="71"/><cb/> called the south pole. Four of the soft bars being impregnated after this manner, lay the two (fig. 2) parallel to each other, at a quarter of an inch distance, between the two pieces of iron belonging to them, a north and a south pole against each piece of iron: then take two of the four bars already made magnetical, and place them together so as to make a double bar in thickness, the north pole of one even with the south pole of the other; and the remaining two being put to these, one on each side, so as to have two north and two south poles together, separate the north from the south poles at one end by a large pin, and place them perpendicularly with that end downward on the middle of one of the parallel bars, the two north poles towards its south end, and the two south poles towards its north end: slide them three or four times backward and forward the whole length of the bar; then removing them from the middle of this bar, place them on the middle of the other bar as before directed, and go over that in the same manner; then turn both the bars the other side upwards, and repeat the former operation: this being done, take the two from between the pieces of iron; and, placing the two outermost of the touching bars in their stead, let the other two be the outermost of the four to touch these with; and this process being repeated till each pair of bars have been touched three or four times over, which will give them a considerable magnetic power. Put the half-dozen together after the manner of the four (fig. 3), and touch them with two pair of the hard bars placed between their irons, at the distance of about half an inch from each other; then lay the soft bars aside, and with the four hard ones let the other two be impregnated (fig. 4), holding the touching bars apart at the lower end near two-tenths of an inch; to which distance let them be separated after they are set on the parallel bar, and brought together again before they are taken off: this being observed, proceed according to the method described above, till each pair have been touched two or three times over. But as this vertical way of touching a bar, will not give it quite so much of the magnetic virtue as it will receive, let each pair be now touched once or twice over in their parallel position between the irons (fig. 5), with two of the bars held horizontally, or nearly so, by drawing at the same time the north end of one from the middle over the south end, and the south of the other from the middle over the north end of a parallel bar; then bringing them to the middle again, without touching the parallel bar, give three or four of these horizontal strokes to each side. The horizontal touch, after the vertical, will make the bars as strong as they possibly can be made, as appears by their not receiving any additional strength, when the vertical touch is given by a great number of bars, and the horizontal by those of a superior magnetic power.</p><p>This whole process may be gone through in about half an hour; and each of the large bars, if well hardened, may be made to lift 28 Troy ounces, and sometimes more. And when these bars are thus impregnated, they will give to a hard bar of the same size its full virtue in less than two minutes; and therefore will answer all the purposes of Magnetism in navigation and experimental philosophy, much better than the loadstone, which has not a power sufficient to impregnate<cb/> hard bars. The half dozen being put into a case (fig. 6), in such a manner as that no two poles of the same name may be together, and their irons with them as one bar, they will retain the virtues they have received; but if their power should, by making experiments, be ever so far impaired, it may be restored without any foreign assistance in a few minutes. And if, perchance, a much larger set of bars should be required, these will communicate to them a sufficient power to proceed with; and they may, in a short time, by the same method, be brought to their full strength.</p></div1><div1 part="N" n="MAGNETISM" org="uniform" sample="complete" type="entry"><head>MAGNETISM</head><p>, the quality or constitution of a body, by which it is rendered magnetical, or a magnet, sensibly attracting iron, and giving it a meridional direction.</p><p>This is a transient power, capable of being produced, destroyed, or restored. <hi rend="center"><hi rend="italics">The Laws of</hi> <hi rend="smallcaps">Magnetism.</hi></hi></p><p>These laws are laid down by Mr. Whiston in the following propositions.——1, The Loadstone has both an attractive and a directive power united together, while iron touched by it has only the former; i. e. the magnet not only attracts needles, or steel filings, but also directs them to certain different angles, with respect to its own surface and axis; whereas iron, touched with it, does little or nothing more than attract them; still suffering them to lie along or stand perpendicular to its surface and edges in all places, without any such special direction.</p><p>2. Neither the strongest nor the largest magnets give a better directive touch to needles, than those of a less size or virtue: to which may be added, that whereas there are two qualities in all magnets, an attractive and a directive one; neither of them depend on, or are any argument of the strength of the other.</p><p>3. The attractive power of magnets, and of iron, will greatly increase or diminish the weight of needles on the balance; nay, it will overcome that weight, and even sustain some other additional also: while the directive power has a much smaller effect. Gassendus indeed, as well as Mersennus and Gilbert, assert that it has none at all: but by mistake; for Whiston found, from repeated trials on large needles, that after the touch they weighed less than before. One of 4584 1/8 grains, lost 2 5/8 grains by the touch; and another of 65726 grains weight, no less than 14 grains.</p><p>4. It is probable that iron consists almost wholly of the attractive particles; and the magnet, of the attractive and directive together; mixed, probably, with other heterogeneous matter; as having never been purged by the fire, which iron has; and hence may arise the reason why iron, after it has been touched, will lift up a much greater weight than the loadstone that touched it.</p><p>5. The quantity and direction of magnetic powers, communicated to needles, are not properly, after such communication, owing to the magnet which gave the touch; but to the goodness of the steel that receives it, and to the strength and position of the terrestrial load stone, whose influence alone those needles are afterwards subject to, and directed by: so that all such needles, if good, move with the same strength, and point to the same angle, whatever loadstone they may have been excited by, provided it be but a good one. Nor does it<pb n="72"/><cb/> seem that the touch does much more in magnetical cases, than attrition does in electrical ones; i. e. serving to rub off some obstructing particles, that adhere to the surface of the steel, and opening the pores of the body touched, and so make way for the entrance and exit of such effluvia as occasion or assist the powers we are speaking of. Hence Mr. Whiston takes occasion to observe, that the directive power of the loadstone seems to be mechanical, and to be devived from magnetic effluvia, circulating continually round it.</p><p>6. The absolute attractive power of disserent armed loadstones, is, <hi rend="italics">cœteris paribus,</hi> not according to either the diameters or soliditics of the loadstones, but according to the quantity of their surfaces, or in the duplicate proportion of their diameters.</p><p>7. The power of good magnets unarmed, sensibly equal in strength, similar in figure and position, but unequal in magnitude, is sometimes a little greater, sometimes a little less, than in the proportion of their similar diameters.</p><p>8. The loadstone attracts needles that have been touched, and others that have not been touched, with equal force at distances unequal, viz, when the distance of the former is to the distance of the latter, as 5 to 2.</p><p>9. Both poles of a magnet equally attract needles, till they are touched; then it is, and then only, that one pole begins to attract one end, and repel the other: though the repelling pole will still attract upon contact, and even at very small distances.</p><p>10. The attractive power of loadstones, in their similar position to, but different distances from, magnetic needles, is in the sesquiduplicate proportion of the distances of their surfaces from then needles reciprocally; or as the mean proportionals between the squares and the cubes of those distances reciprocally; or as the square roots of the 5th powers of those distances reciprocally. Thus, the magnetic force of attraction, at twice the distance from the surface of the loadstone, is between a 5th and 6th part of the force at the first distance; at thrice the distance, the force is between the 15th and 16th part; at four times the distance, the power is the 32d part of the first; and at six times the distance, it is the 88th part. Where it is to be noted, that the distances are not counted from the centre, as in the laws of gravity, but from the surface: all experience assuring us, that the magnetic power resides chiefly, if not wholly, in the surfaces of the loadstone and iron; without any particular relation to any centre at all. The proportion here laid down was determined by Mr. Whiston, from a great number of experiments by Mr. Hawksbee, Dr. Brook Taylor, and himself; measuring the force by the chords of those arcs by which the magnet at several distances draws the needle out of its natural direction, to which chords, as he demonstrates, it is always proportional. The numbers in some of their most accurate trials, he gives in the following Table, setting down the half chords, or the sines of half those arcs of declination, as the true measures of the force of magnetic at- traction.<cb/> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Distances in inches.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Degrees of inclination.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Sines of 1/2 arcs.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Sesquiduplicate ratio.</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" role="data">466</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14 8/9</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">349</cell><cell cols="1" rows="1" role="data">216</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13 5/8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">523</cell><cell cols="1" rows="1" role="data">170</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12 3/8</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">697</cell><cell cols="1" rows="1" role="data">138</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11 1/8</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">871</cell><cell cols="1" rows="1" role="data">105</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10 1/4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">1045</cell><cell cols="1" rows="1" role="data">87</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9 1/4</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1219</cell><cell cols="1" rows="1" role="data">70</cell></row></table></p><p>Other persons however have found some variations i<*> the proportions of magnetic force with respect to distance: Thus, Newton supposes it to decrease nearly in the triplicate ratio of the distance: Mr. Martin observes, that the power of this loadstone decreases in the sefquiduplicate ratio of the distances inversely: but Dr. Helsham and Mr. Michell found it to be as the square of the distance inversely: while others, as Dr. Brook Taylor and M. Muschenbroek, are of opinion, that this power follows no certain ratio at all, and that the variation is different in different stones.</p><p>11. An inclinatory, or dipping-needle, of 6 inches radius, and of a prismatic or cylindric figure, when it oscillates along the magnetic meridian, performs there every mean vibration in about 6″ or 360‴, and every small oscillation in about 5″1/2, or 330‴: and the same kind of needle, 4 feet long, makes every mean oscillation in about 24″, and every small one in about 22″.</p><p>12. The whole power of Magnetism in this country, as it affects needles a foot long, is to that of gravity nearly as 1 to 300; and as it affects needles 4 feet long, as 1 to 600.</p><p>13. The quantity of magnetic power accelerating the same dipping-needle, as it oscillates in different vertical planes, is always as the cosines of the angles made by those planes with the magnetic meridian, taken on the horizon.</p><p>Thus, in estimating the quantity of force in the horizontal and in the vertical situations of needles at London, it is found that the latter, in needles of a foot long, is to the whole force along the magnetic meridian, as 96 to 100; and in needles 4 feet long, as 9667 to 10000: whereas, in the former, the whole force in needles of a foot long, is as 28 to 100; and in those of 4 feet long, as 256 to 1000. Whence it follows, that the power by which horizontal needles are governed in these parts of the world, is but the quarter of the power by which the dipping-needle is moved.</p><p>Hence also, as the horizontal needle is moved only by a part of the power that moves the dipping-needle; and as it only points to a certain place in the horizon, because that place is the nearest to its original tendency of any that its situation will allow it to tend to; whenever the dipping-needle stands exactly perpendicular to the horizon, the horizontal needle will not respect one point of the compass more than another, but will wheel about any way uncertainly.</p><p>14. The time of oscillation and vibration, both in dipping and horizontal needles, that are equally good, is as their length directly; and the actual velocities of their points along their arcs, are always unequal. And hence, magnetical needles are, <hi rend="italics">cœteris paribus,</hi> still better, the longer they are; and that in the same proportion with their lengths.<pb n="73"/><cb/></p><p><hi rend="italics">Of the Causes of</hi> <hi rend="smallcaps">Magnetism.</hi> Though many authors have proposed hypotheses, or written concerning the cause of Magnetism, as Plutarch, Descartes, Boyle, Newton, Gilbert, H<*>rtsoeker, Halley, Whiston, Knight, Beccaria, &c; nothing however has yet appeared that can be called a fatisfactory solution of its phenomena. It is certain indeed, that both natural and artificial electricity will give polarity to needles, and even reverse their poles; but though from this it may appear probable that the electric fluid is also the cause of Magnetism, yet in what manner the fluid acts while producing the magnetical phenomena, seems to be quite unknown.</p><p>Dr. Knight indeed deduces from several experiments the following propositions, which he offers, not so much to explain the nature of the cause of Magnetism, as the manner in which it acts: the magnetic matter of a loadstone, he says, moves in a stream from one pole to the other internally, and is then carried back in a curve line externally, till it arrive again at the pole where it first entered, to be again admitted: the immediate cause why two or more magnetical bodies attract each other, is the flux of one and the same stream of magnetical matter through them; and the immediate cause of magnetic repulsion, is the conflux and accumulation of the magnetic matter. Philos. Trans. vol. 44, pa. 665.</p><p>Mr. Michell rejects the motion of a subtle fluid; but though he proposed to publish a theory of Magnetism established by experiments, no such theory has appeared.</p><p>Signor Beccaria, from observing that a sudden stroke of lightning gives polarity to Magnets, conjectures, that a regular and constant circulation of the whole mass of the electric fluid from north to south may be the original cause of Magnetism in general. This current he would not suppose to arise from one source, but from several, in the northern hemisphere of the earth: the aberration of the common centre of all the currents from the north point, may be the cause of the variation of the needle; the period of this declination of the centre of the currents, may be the period of the variation; and the obliquity with which the currents strike into the earth, may be the cause of the dipping of the needle, and also why bars of iron more easily receive the magnetic virtue in one particular direction. Lettre dell' Elettricismo, pa. 269; or Priestley's Hist. Elec. vol. 1, pa. 409. See also Cavallo's Treatise on Magnetism.</p></div1><div1 part="N" n="MAGNIFYING" org="uniform" sample="complete" type="entry"><head>MAGNIFYING</head><p>, is the making of objects appear larger than they usually and naturally appear to the eye; whence convex lenses, which have the power of doing this, are called Magnifying Glasses.</p><p>The Magnifying power of dense mediums of certain figures, was known to the Ancients; though they were far from understanding the cause of this effect. Seneca says, that small and obscure letters appear larger and brighter through a glass globe filled with water; and he absurdly accounts for it by saying, that the eye slides in the water, and cannot lay hold of its object. And Alexander Aphrodisensis, about two centuries after Seneca, says, that the reason why apples appear large when immersed in water, is, that the water which is contiguous to any body is affected with<cb/> the same quality and colour; so that the eye is deceived in imagining the body itself larger. But the first distinct account we have of the Magnifying power of glasses, is in the 12th century, in the writings of Roger Bacon, and Alhazen; and it is not improbable that from their observations the construction of spectacles was derived. In the Opus Majus of Bacon, it is demonstrated, that if a transparent body, interspersed between the eye and an object, be convex towards the eye, the object will appear magnified.</p><p><hi rend="smallcaps">Magnifying</hi> <hi rend="italics">Glass,</hi> in Optics, is a small spherical convex lens; which, in transmitting the rays of light, inflects them more towards the axis, and so exhibits objects viewed through them larger than when viewed by the naked eye. See <hi rend="smallcaps">Microscope.</hi></p></div1><div1 part="N" n="MAGNITUDE" org="uniform" sample="complete" type="entry"><head>MAGNITUDE</head><p>, any thing made up of parts locally extended, or continued; or that has several dimensions; as a line, surface, solid, &c. Quantity is often used as synonymous with Magnitude. See <hi rend="smallcaps">Quantity.</hi></p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Magnitudes</hi>, are usually, and most properly, considered as generated or produced by motion; as lines by the motion of points, surfaces by the motion of lines, and solíds by the motion of surfaces.</p><p><hi rend="italics">Apparent</hi> <hi rend="smallcaps">Magnitude</hi>, is that which is measured by the optic or visual angle, intercepted between rays drawn from its extremes to the centre of the pupil of the eye. It is a fundamental maxim in optics, that whatever things are seen under the same or equal angles, appear equal; and vice versa.—The apparent Magnitudes of an object at different distances, are in a ratio less than that of their distances reciprocally.</p><p>The apparent Magnitudes of the two great luminaries, the sun and moon, at rising and setting, are a phenomenon that has greatly embarrassed the modern philosophers. According to the ordinary laws of vision, they should appear the least when nearest the horizon, being then farthest from the eye; and yet it is found that the contrary is true in fact. Thus, it is well known that the mean apparent diameter of the moon, at her greatest height in the meridian, is nearly 31′ in round numbers, subtending then an angle of that quantity as measured by any instrument. But, being viewed when she rises or sets, she seems to the eye as two or three times as large as before; and yet when measured by the instrument, her diameter is not found increased at all.</p><p>Ptolomy, in his Almagest, lib. 1, cap. 3, taking for granted, that the angle subtended by the moon was really increased, ascribed the increase to a refraction of the rays by vapours, which actually enlarge the angle under which the moon appears; just as the angle is enlarged by which an object is seen from under water: and his commentator Theon explains distinctly how the dilatation of the angle in the object immersed in water is caused. But it being afterwards discovered, that there is no alteration in the angle, another solution was started by the Arab Alhazen, which was followed and improved by Bacon, Vitello, Kepler, Peckham, and others. According to Alhazen, the sight apprehends the surface of the heavens as flat, and judges of the stars as it would of ordinary visible objects extended upon a wide plain; the eye sees then under equal angles indeed, but withal perceives a difference in their<pb n="74"/><cb/> distances, and (on account of the semidiameter of the earth, which is interposed in one case, and not in the other) it is hence induced to judge those that appear more remote to be greater. Some farther improvement was made in this explanation by Mr. Hobbes, though he fell into some mistakes in his application of geometry to this subject: for he observes, that this deception operates gradually from the zenith to the horizon; and that if the apparent arch of the sky be divided into any number of equal parts, those parts, in descending towards the horizon, will subtend an angle that is gradually less and less. And he was the first who expressly considered the vaulted appearance of the sky as a real portion of a circle.</p><p>Des Cartes, and from him Dr. Wallis, and most other authors, account for the appearance of a different distance under the same angle, from the long series of objects interposed between the eye and the extremity of the sensible horizon; which makes us imagine it more remote than when in the meridian, where the eye sees nothing in the way between the object and itself. This idea of a great distance makes us imagine the luminary the larger; for an object being seen under any certain angle, and believed at the same time very remote, we naturally judge it must be very large, to appear under such an angle at such a distance. And thus a pure judgment of the mind makes us see the sun, or the moon, larger in the horizon than in the meridian; notwithstanding their diameters measured by any instrument are really less in the former situation than the latter.</p><p>James Gregory, in his Geom. Pars Universalis, pa. 141, subscribes to this opinion: Father Mallebranche also, in the first book of his Recherche de la Verité, has explained this phenomenon almost in the expression of Des Cartes: and Huygens, in his Treatise on the Parhelia, translated by Dr. Smith, Optics, art. 536, has approved, and very clearly illustrated, the received opinion. The cause of this fallacy, says he, in short, is this; that we think the sun, or any thing else in the heavens, farther from us when it is near the horizon, than when it approaches towards the vertex, because we imagine every thing in the air that appears near the vertex to be farther from us than the clouds that fly over our heads; whereas, on the other hand, we are used to observe a large extent of land lying between us and the objects near the horizon, at the farther end of which the convexity of the sky begins to appear; which therefore, with the objects that appear in it, are usually imagined to be much farther from us. Now when two objects of equal magnitude appear under the same angle, we always judge that object to be larger which we think is remoter. And this, according to them, is the true cause of the deception in question. It is really astonishing that an hypothesis so palpably false should ever be held and maintained by such eminent men; for it is daily feen that the moon or sun, when near the horizon, very suddenly change their magnitude, as they ascend or descend, though all the intervening objects are seen just as before; and that the luminary appears largest of all when fewest objects appear on the earth, as in a thick fog or mist. It is no wonder therefore that other reasons have been assigned for this remarkable phenomenon.</p><p>Accordingly Gassendus was of opinion, that this<cb/> effect arises from hence; that the pupil of the eye, being always more open as the place is more dark, as in the morning and evening, when the light is less, and besides the earth being then covered with gross vapours, through a longer column of which the rays must pass to reach the horizon; the image of the luminary enters the eye at a greater angle, and is really painted there larger than when the luminary is higher. See A<hi rend="smallcaps">PPARENT</hi> <hi rend="italics">Diameter</hi> and <hi rend="italics">Magnitude.</hi></p><p>F. Gouge advances another hypothesis, which is, that when the luminaries are in the horizon, the proximity of the earth, and the gross vapours with which they then appear enveloped, have the same effect with regard to us, as a wall, or other dense body, placed behind a column; which in that case appears larger than when insulated, and encompassed on all sides with an illuminated air.</p><p>The commonly received opinion has been disputed, not only by F. Gouge, who observes, Acad. Sci. 1700, pa. 11, that the horizontal moon appears equally large across the sea, where there are no objects to produce the effect ascribed to them; but also by Mr. Molyneux, who says, Philos. Trans. abr. vol. 1, pa. 221, that if this hypothesis be true, we may at any time increase the apparent magnitude of the moon, even in the meridian; for, in order to divide the space between it and the eye, we need only to look at it behind a cluster of chimneys, the ridge of a hill, or the top of a house, &c. He makes also the same observation with F. Gouge, above mentioned, and farther observes, that when the height of all the intermediate objects is cut off; by looking through a tube, the imagination is not helped, and yet the moon seems still as large as before. However, Mr. Molyneux advances no hypothesis of his own.</p><p>Bishop Berkley supposed, that the moon appears larger near the horizon, because she then appears fainter, and her beams affect the eye less. And Mr. Robins has recited some other opinions on this subject, Math. Tracts, vol. 2, pa. 242.</p><p>Dr. Desaguliers has illustrated the doctrine of the horizontal moon, Philos. Trans. abr. vol. 8, pa. 130, upon the supposition of our imagining the visible heavens to be only a small portion of a spherical surface, and consequently supposing the moon to be farther from us in the horizon than near the zenith; and by several ingenious contrivances he demonstrated how liable we are to such deceptions. The same idea is pursued still farther by Dr. Smith, in his Optics, where he determines that, the centre of the apparent spherical segment of the sky lying much below the eye, or the horizon, the apparent distance of its parts near the horizon was about 3 or 4 times greater than the apparent distance of its parts over head; from which reason it is, he infers, that the moon always appears the larger as she is lower, and also that we always think the height of a celestial object to be more than it really is. Thus, he determined, by measuring the actual height of some of the heavenly bodies, when to his eye they seemed to be half way between the horizon and the zenith; that their real altitude was then only 23°: when the sun was about 30° high, the upper always appeared less than the under; and he thought that it was constantly greater when the sun was 18° or 20° high. Mr. Robins, in<pb n="75"/><cb/> his Tracts, vol. 2, pa. 245, shews how to determine the apparent concavity of the sky in a more accurate and geometrical manner; by which it appears, that if the altitude of any of the heavenly bodies be 20°, at the time when it seems to be half way between the horizon and the zenith, <*>he horizontal distance will be hardly less than 4 times the perpendicular distance; but if that altitude be 28°, it will be little more than 2 and a half.</p><p>Dr. Smith, having determined the apparent figure of the sky, thus applies it to explain the phenomenon of the horizontal moon, and other similar appearances in the heavens. Suppose the are ABC to re- <figure/> present that apparent concavity; then the diameter of the sun and moon would seem to be greater in the horizon than at any altitude, measured by the angle AOB, in the ratio of its apparent distances, AO, BO. The numbers that express these proportions he reduced into the annexed table, answering to the corresponding altitudes of the sun or <table rend="border"><row role="data"><cell cols="1" rows="1" role="data">The alt. of the sun or moon in degrees.</cell><cell cols="1" rows="1" role="data">Apparent diameters or distances.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">00</cell><cell cols="1" rows="1" rend="align=right" role="data">100</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">34</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">75</cell><cell cols="1" rows="1" rend="align=right" role="data">31</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row></table> moon, which are also exactly represented to the eye in the figure, in which the moon, placed in the quadrantal arc FG described about the centre O, are all equal to each other, and represent the body of the moon in the heights there noted, and the unequal moons in the concavity ABC are terminated by the visual rays coming from the circumference of the real moon, at those heights to the eye, at O. Dr. Smith also observes, that the apparent concave of the sky, being less than a hemisphere, is the cause that the breadths of the colours in the inward and outward rainbows, and the interval between the bows, appear least at the top, and greater at the bottom. This<cb/> theory of the horizontal moon is also confirmed by the appearances of the tails of comets, which, whatever be their real figure, magnitude, and situation in absolute space, do always appear to be an are of the concave sky. Dr. Smith however justly acknowledges that, at different times, the moon appears of very different magnitudes, even in the same horizon, and occasionally of an extraordinary large size; which he is not able to give a satisfactory explanation of. Smith's Optics, vol. 1, pa. 63, &c, Remarks, pa. 53.</p></div1><div1 part="N" n="MAIGNAN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MAIGNAN</surname> (<foreName full="yes"><hi rend="smallcaps">Emanuel</hi></foreName>)</persName></head><p>, a religious minim, and one of the greatest philosophers of his age, was born at Thoulouse in 1601. Like the famous Pascal, he became a complete mathematician without the assistance of a teacher; and filled the professor's chair at Rome in 1636, where, at the expence of Cardinal Spada, he published his book <hi rend="italics">De Perspectiva Horaria,</hi> in 1648. Upon this book, Baillet, in his Life of Des Cartes, has the following passage: “M. Carcavi acquainted Des Cartes, that there was at Rome one father Maignan, a minim, of greater learning and more depth than father Mersenne, who made him expect some objections against his principles. This father's proper name was Emanuel, and his native place Thoulouse: but he lived at that time at Rome, where he taught divinity in the convent of the Trinity upon Mount Pincio, which they otherwise call the convent of the French minims.” Maignan returned to Thoulouse in 1650, and was created Provincial. His knowledge in mathematics, and physical experiments, were very early known; especially from <*> dispute which arose between him and father Kircher, about the invention of a catoptrical work.</p><p>The king, who in 1660 amused himself with the machines and curiosities in the father's cell, made him offers by Cardinal Mazarin, to draw him to Paris; but he humbly desired to spend the remainder of his days in a cloyster.—He published a Course of Philosophy, in 4 volumes 8vo, at Thoulouse, in 1652; to the second edition of which, in folio, 1673, he added two Treatises; the one against the vortices of Des Cartes, the other upon the speaking trumpet invented by Sir Samuel Morland.—He formed a machine, which shewed, by its movements, that Des Cartes's supposition concerning the manner in which the universe was formed, or might have been formed, and concerning the centrifugal force, was entirely without foundation.</p><p>Thus this great philosopher and divine passed a life of tranquillity, in writing books, making experiments, and reading lectures. He was frequently consulted by the most eminent philosophers; and has had a thousand answers to make, either by writing or otherwise. Never was mortal less inclined to idleness. It is said that he even studied in his sleep; for his very dreams employed him in problems, which he pursued sometimes till he came to a solution or demonstration; and he has frequently been awaked out of his sleep of a sudden, by the exquisite pleasure which he felt upon discovery of it. The excellence of his manners, and his unspotted virtues, rendered him no less worthy of esteem, than his genius and learning.—It is said that he composed with great ease, and without any alterations at all.—He died at Thoulouse in 1676, at 75 years of age.</p></div1><div1 part="N" n="MALLEABLE" org="uniform" sample="complete" type="entry"><head>MALLEABLE</head><p>, the property of a solíd ductile body, from which it may be beaten, forged, and ex-<pb n="76"/><cb/> tended under the hammer, without breaking, which is a property of all metals.</p></div1><div1 part="N" n="MANFREDI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MANFREDI</surname> (<foreName full="yes"><hi rend="smallcaps">Eustachio</hi></foreName>)</persName></head><p>, a celebrated astronomer and mathematician, born at Bologna in 1674. His genius was always above his age. He was a tolerable poct, and wrote ingenious verses while he was but a child. And while very young he formed in his father's house an academy of youth of his own age, who became the Academy of Sciences, or the Institute, there. He became Professor of Mathematics at Bologna in 1698, and Superintendant of the waters there in 1704. The same year he was placed at the head of the College of Montalte, founded at Bologna for young men intended for the church. In 1711 he obtained the office of Astronomer to the Institute of Bologna. He became member of the Academy of Sciences of Paris in 1726, and of the Royal Society of London in 1729; and died the 15th of February 1739.—His works are:</p><p>1. <hi rend="italics">Ephemerides Motuum Cœleslium ab anno</hi> 1715 <hi rend="italics">ad annum</hi> 1750; 4 volumes in 4to.—The first volume is an excellent introduction to astronomy; and the other three contain numerous calculations. His two sisters were greatly assisting to him in composing this work.</p><p>2. <hi rend="italics">De Transitu Mercurii per Solem, anno</hi> 1723. Bologna 1724, in 4to.</p><p>3. <hi rend="italics">De Annuis Inerrantium Stellarum Aberrationibus,</hi> Bologna 1729, in 4to.—Besides a number of papers in the Memoirs of the Academy of Sciences, and in other places.</p></div1><div1 part="N" n="MANILIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MANILIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Marcus</hi></foreName>)</persName></head><p>, a Latin astronomical poet, who lived in the reign of Augustus Cæsar. He wrote an ingenious poem concerning the stars and the sphere, called <hi rend="italics">Astronomicon;</hi> which, not being mentioned by any of the ancient poets, was unknown, till about two centuries since, when it was found buried in some German library, and published by Poggius. There is no account to be found of this author, but what can be drawn from his poem; which contains a system of the ancient astronomy and astrology, together with the philosophy of the Stoics. It consists of five books; though there was a sixth, which has not been recovered. In this work, Manilius hints at some opinions, which later ages have been ready to glory in as their own discoveries. Thus, he defends the fluidity of the heavens, against the hypothesis of Aristotle: he asserts that the fixed stars are not at all in the same concave superficies of the heavens, and equally distant from the centre of the world: he maintains that they are all of the same nature and substance with the sun, and that each of them has a particular vortex of its own: and lastly, he says, that the milky way is only the united lustre of a great many small imperceptible stars; which indeed the Moderns now see to be such through their telescopes.</p><p>The best editions of Manilius are, that of Joseph Scaliger, in 4to, 1600; that of Bentley, in 4to, 1738, and that of Edmund Burton, Esq. in 8vo, 1783.</p></div1><div1 part="N" n="MANOMETER" org="uniform" sample="complete" type="entry"><head>MANOMETER</head><p>, or <hi rend="smallcaps">Manoscope</hi>, an instrument to shew or measure the alterations in the rarity or density of the air.</p><p>The Manometer differs from the barometer in this, That the latter only serves to measure the <hi rend="italics">weight</hi> of the atmosphere, or of the column of air over it; but the former, the density of the air in which it is found;<cb/> which density depends not only on the weight of the atmosphere, but also on the action of heat and cold, &c. Authors however often confound the two together; and Mr. Boyle himself has given a very good Manometer of his contrivance, under the name of a Statical Barometer, consisting of a bubble of thin glass, about the size of an orange, which being counterpoised when the air was in a mean state of density, by means of a nice pair of scales, sunk when the atmosphere became lighter, and rose as it grew heavier.</p><p>The Manometer used by captain Phipps, in his voyage towards the North Pole, consisted of a tube of a small bore, with a ball at the end. The barometer being at 29.7, a small quantity of quicksilver was put into the tube, to take off the communication between the external air, and that confined in the ball and the part of the tube below this quicksilver. A scale is placed on the side of the tube, which marks the degrees of dilatation arising from the increase of heat in this state of the weight of the air, and has the same graduation as that of Fahrenheit's thermometer, the point of freezing being marked 32. In this state therefore it will shew the degrees of heat in the same manner as a thermometer. But when the air becomes lighter, the bubble inclosed in the ball, being less compressed, will dilate itself, and occupy a space as much larger as the compressing force is less; therefore the changes arising from the increase of heat, will be proportionably larger; and the instrument will shew the differences in the density of the air, arising from the changes in its weight and heat. Mr. Ramsden found, that a heat equal to that of boiling water, increased the magnitude of the air, from what it was at the freezing point, by 414/1000 of the whole. Hence it follows, that the ball and the part of the tube below the beginning of the scale, is of a magnitude equal to almost 414 degrees of the scale. If the height of both the Manometer and thermometer be given, the height of the barometer may be thence deduced, by this rule; as the height of the Manometer increased by 414, to the height of the thermometer increased by 414, so is 29.7, to the height of the barometer; or if <hi rend="italics">m</hi> denote the height of the Manometer, and <hi rend="italics">t</hi> the height of the thermometer; then , which is the height of the barometer.</p><p>Another kind of Manometer was made use of by colonel Roy, in his attempts to correct the errors of the barometer; which is described in the Philos. Trans. vol. 67, pa. 689.</p></div1><div1 part="N" n="MANTELETS" org="uniform" sample="complete" type="entry"><head>MANTELETS</head><p>, a kind of moveable parapet, or screen, of about 6 feet high, set upon trucks or little wheels, and guided by a long pole; so that in a siege it may be driven before the pioneers, and serve as blinds, or screens, to shelter them from the enemy's small shot. Mantelets are made of different materials, so as to render them musket proof; as of strong boards nailed together, and covered with tin; or of thick leather, or of layers of rope, &c, firmly bound together.</p><p>There are also other sorts of Mantelets, covered on the top, used by the miners in approaching the walls or works of an enemy. The double Mantelets form an <pb/><pb/><pb n="77"/><cb/> angle, and stand square, making two fronts. It appears from Vegetius, that Mantelets were in use among the Ancients, under the name of Vineæ.</p></div1><div1 part="N" n="MANTLE" org="uniform" sample="complete" type="entry"><head>MANTLE</head><p>, or <hi rend="smallcaps">Mantle</hi>-<hi rend="italics">tree,</hi> is the lower part of the breast or front of a chimney. It was formerly a piece of timber that lay across the jambs, supporting the breastwork; but by a late act of parliament, chimney-breasts are not to be supported by a wooden mantletree, or turning piece, but by an iron bar, or by an arch of brick or stone.</p></div1><div1 part="N" n="MAP" org="uniform" sample="complete" type="entry"><head>MAP</head><p>, a plane figure representing the surface of the earth, or some part of it; being a projection of the globular surface of the earth, exhibiting countries, seas, rivers, mountains, cities, &c, in their due positions, or nearly so.</p><p>Maps are either Universal or Particular, that is Partial.</p><p><hi rend="italics">Universal</hi> <hi rend="smallcaps">Maps</hi> are such as exhibit the whole surface of the earth, or the two hemispheres.</p><p><hi rend="italics">Particular,</hi> or <hi rend="italics">Partial</hi> <hi rend="smallcaps">Maps</hi>, are those that exhibit some particular region, or part of the earth.</p><p>Both kinds are usually called Geographical, or LandMaps, as distinguished from Hydrographical, or SeaMaps, which represent only the seas and sea coasts, and are properly called <hi rend="italics">Charts.</hi></p><p>Anaximander, the scholar of Thales, it is said, about 400 years before Christ, first invented geographical tables, or Maps. The Pentingerian Tables, published by Cornelius Pentinger of Ausburgh, contain an itinerary of the whole Roman Empire; all places, except seas, woods, and desarts, being laid down according to their measured distances, but without any mention of latitude, longitude, or bearing.</p><p>The Maps published by Ptolomy of Alexandria, about the 144th year of Christ, have meridians and parallels, the better to desine and determine the situation of places, and are great improvements on the construction of Maps. Though Ptolomy himself owns that his Maps were copied from some that were made by Marinus, Tirus, &c, with the addition of some improvements of his own. But from his time till about the 14th century, during which, geography and most sciences were neglected, no new Maps were published. Mercator was the first of note among the Moderns, and next to him Ortelius, who undertook to make a new set of Maps, with the modern divisions of countries and names of places; for want of which, those of Ptolomy were become almost useless. After Mercator, many others published Maps, but for the most part they were mere copies of his. Towards the middle of the 17th century, Bleau in Holland, and Sanson in France, published new sets of Maps, with many improvements from the travellers of those times, which were afterwards copied, with little variation, by the English, French, and Dutch; the best of these being those of Vischer and De Witt. And later observations have furnished us with still more accurate and copious sets of Maps, by De Lisle, Robert, Wells, &c, &c. Concerning Maps, see Varenius's Geog. lib. 3, cap. 3, prop. 4; Fournier's Hydrog. lib. 4, c. 24; Wolfius's Elem. Hydrog. c. 9; John Newton's Idea of Navigation; Mead's Construction of Globes and Maps; Wright's Constructions of Maps, &c, &c.</p><p><hi rend="italics">Construction of</hi> <hi rend="smallcaps">Maps.</hi> Maps are constructed by<cb/> making a projection of the globe, either on the plane of some particular circle, or by the eye placed in some particular point, according to the rules of Perspective, &c; of which there are several methods. <hi rend="center"><hi rend="italics">First, to construct a Map of the World, or a general Map.</hi></hi></p><p>1st <hi rend="italics">Method.</hi>—A map of the world must represent two hemispheres; and they must both be drawn upon the plane of that circle which divides the two hemispheres. The first way is to project each hemisphere upon the plane of some particular circle, by the rules of Orthographic projection, forming two hemispheres, upon one common base or circle. When the plane of projection is that of a meridian, the maps will be the east and west hemispheres, the other meridians will be ellipses, and the parallel circles will be right lines. Upon the plane of the equinoctial, the meridians will be right lines crossing in the centre, which will represent the pole, and the parallels of latitude will be circles having that common centre, and the Maps will be the northern and southern hemispheres. The fault of this way of drawing Maps, is, that near the outside the circles are too near one another; and therefore equal spaces on the earth are represented by very unequal spaces upon the Map.</p><p>2d <hi rend="italics">Method.</hi>—Another way is to project the same hemispheres by the rules of Stereographic projection; in which way, all the parallels will be represented by circles, and the meridians by circles or right lines. And here the contrary fault happens, viz, the circles towards the outsides are too far asunder, and about the middle they are too near together.</p><p>3d <hi rend="italics">Method.</hi>—To remedy the faults of the two former methods, proceed as follows. First, for the east and west hemispheres, describe the circle PENQ for the meridian (pl. xvii, fig. 1), or plane of projection; through the centre of which draw the equinoctial EQ, and axis PN perpendicular to it, making P and N the north and south pole. Divide the quadrants PE, EN, NQ, and QP into 9 equal parts, each representing 10 degrees, beginning at the equinoctial EQ: divide also CP and CN into 9 equal parts; beginning at EQ; and through the corresponding points draw the parallels of latitude. Again, divide CE and CQ into 9 equal parts; and through the points of division, and the two poles P and N, draw circles, or rather ellipses, for the meridians. So shall the Map be prepared to receive the several places and countries of the earth.</p><p>Secondly, for the north or south hemisphere, draw AQBE, for the equinoctial (fig. 2), dividing it into the four quadrants EA, AQ, QB, and BE; and each quadrant into 9 equal parts, representing each 10 degrees of longitude; and then, from the points of division, draw lines to the centre C, for the circles of longitude. Divide any circle of longitude, as the first meridian EC, into 9 equal parts, and through these points describe circles from the centre C, for the parallels of latitude; numbering them as in the figure.</p><p>In this 3d method, equal spaces on the earth are represented by equal spaces on the Map, as near as any projection will bear; for a spherical surface can no way be represented exactly upon a plane. Then the several countries of the world, seas, islands, sea-coasts, towns,<pb n="78"/><cb/> &c, are to be entered in the Map, according to their latitudes and longitudes.</p><p>In filling up the Map, all places representing land are silled with such things as the countries contain; but the seas are left white; the shores adjoining to the sea being shaded. Rivers are marked by strong lines, or by double lines, drawn winding in form of the rivers they represent; and small rivers are expressed by small lines. Different countries are best distinguished by different colours, or at least the borders of them. Forests are represented by trees; and mountains shaded to make them appear. Sands are denoted by small points or specks; and rocks under water by a small cross. In any void space, draw the mariner's compass, with the 32 points or winds. <hi rend="center">II. <hi rend="italics">To draw a Map of any particular Country.</hi></hi></p><p>1st <hi rend="italics">Method.</hi>—For this purpose its extent must be known, as to latitude and longitude; as suppose Spain, lying between the north latitudes 36 and 44, and extending from 10 to 23 degrees of longitude; so that its extent from north to south is 8 degrees, and from east to west 13 degrees.</p><p>Draw the line AB for a meridian passing through the middle of the country (fig. 3), on which set off 8 degrees from B to A, taken from any convenient scale; A being the north, and B the south point. Through A and B draw the perpendiculars CD, EF, for the extreme parallels of latitude. Divide AB into 8 parts, or degrees, through which draw the other parallels of latitude, parallel to the former.</p><p>For the meridians; divide any degree in AB into 60 equal parts, or geographical miles. Then, because the length of a degree in each parallel decreases towards the pole, from the table shewing this decrease, under the article <hi rend="smallcaps">Degree</hi>, take the number of miles answering to the latitude of B, which is 48 1/2 nearly, and set it from B, 7 times to E, and 6 times to F; so is EF divided into degrees. Again, from the same table take the number of miles of a degree in the latitude A, viz 43 1/6 nearly; which set off, from A, 7 times to C, and 6 times to D. Then from the points of division in the line CD, to the corresponding points in the line EF, draw so many right lines, for the meridians. Number the degrees of latitude up both sides of the Map, and the degrees of longitude on the top and bottom. Also, in some vacant place make a scale of miles; or of degrees, if the Map represent a large part of the earth; to serve for finding the distances of places upon the Map.</p><p>Then make the proper divisions and subdivisions of the country: and having the latitudes and longitudes of the principal places, it will be easy to set them down in the Map: for any town, &c, must be placed where the circles of its latitude and longitude intersect. For instance, Gibraltar, whose latitude is 36° 11′, and longitude 12° 27′, will be at G: and Madrid, whose lat. is 40° 10′, and long. 14° 44′, will be at M. In like manner the mouth of a river must be set down; but to describe the whole river, the latitude and longitude of every turning must be marked down, and the towns and bridges by which it passes. And so for woods, forests, mountains, lakes, castles, &c. The boundaries will be described, by setting down the re-<cb/> markable places on the sea-coast, and drawing a continued line through them all. And this way is very proper for small countries.</p><p>2d <hi rend="italics">Method.</hi>—Maps of particular places are but portions of the globe, and therefore may be drawn after the same manner as the whole is drawn. That is, such a Map may be drawn either by the orthographic or stereographic projection of the sphere, as in the last prob. But in partial Maps, an easier way is as follows. Having drawn the meridian AB (fig. 3), and divided it into equal parts as in the last method, through all the points of division draw lines perpendicular to AB, for the parallels of latitude; CD, EF being the extreme parallel. Then to divide these, set off the degrees in each parallel, diminished after the manner directed for the two extreme parallels CD, EF, in the last method: and through all the corresponding points draw the meridians, which will be curve lines; which were right lines in the last method; because only the extreme parallels were divided by the table. This method is proper for a large tract, as Europe, &c: in which case the parallels and meridians need only be drawn to every 5 or 10 degrees. This method is much used in drawing Maps; as all the parts are nearly of their due magnitude, but a little distorted towards the outside, from the oblique intersections of the meridians and parallels.</p><p>3d <hi rend="italics">Method.</hi>—Draw PB of a convenient length, for a meridian; divide it into 9 equal parts, and through the points of division, describe as many circles for the parallels of latitude, from the centre P, which reprefents the pole. Suppose AB (fig. 4) the height of the Map; then CD will be the parallel passing through the greatest latitude, and EF will represent the equator. Divide the equator EF into equal parts, of the same size as those in AB, both ways, beginning at B. Divide also all the parallels into the same number of equal parts, but lesser, in proportion to the numbers for the several latitudes, as directed in the last method for the rectilineal parallels. Then through all the corresponding divisions, draw curve lines, which will represent the meridians, the extreme ones being EC and FD. Lastly, number the degrees of latitude and longitude, and place a scale of equal parts, either of miles or degrees, for measuring distances.—This is a very good way of drawing large Maps, and is called the globular projection; all the parts of the earth being represented nearly of their due magnitude, excepting that they are a little distorted on the outsides.</p><p>When the place is but small that a Map is to be made of, as if a county was to be exhibited; the meridians, as to sense, will be parallel to one another, and the whole will differ very little from a plane. Such a Map will be made more easily than by the preceding rules. It will here be sufficient to measure the distances of places in miles, and so lay them down in a plane rectangular map. But this belongs more properly to Surveying.</p><p><hi rend="italics">The Use of</hi> <hi rend="smallcaps">Maps</hi> is obvious from their construction. The degrees of the meridians and parallels shew the latitudes and longitudes of places, and the scale of miles annexed, their distances; the situation of places, with regard to each other, as well as to the cardinal points, appears by inspection; the top of the map being always the north, the bottom the south, the right hand the<pb n="79"/><cb/> east, and the left hand the west; unless the compass, usually annexed, shew the contrary.</p></div1><div1 part="N" n="MARALDI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MARALDI</surname> (<foreName full="yes"><hi rend="smallcaps">James Philip</hi></foreName>)</persName></head><p>, a learned astronomer and mathematician, was born in 1665 at Perinaldo in the county of Nice, a place already honoured by the birth of his maternal uncle the celebrated Cassini. Having made a considerable progress in mathematics, at the age of 22 his uncle, who had been a long time settled in France, invited him there, that he might himself cultivate the promising genius of his nephew. Maraldi no sooner applied himself to the contemplation of the heavens, than he conceived the design of forming a catalogue of the fixed stars, the foundation of all the astronomical edifice. In consequence of this design, he applied himself to observe them with the most constant attention; and he became by this means so intimate with them, that on being shewn any one of them, however small, he could immediately tell what constellation it belonged to, and its place in that constellation. He has been known to discover those small comets, which astronomers often take for the stars of the constellation in which they are seen, for want of knowing precisely what stars the constellation consists of, when others, on the spot, and with eyes directed equally to the same part of the heavens, could not for a long time see any thing of them.</p><p>In 1700 he was employed under Cassini in prolonging the French meridian to the northern extremity of France, and had no small share in completing it. He then set out for Italy, where Clement the 11th invited him to assist at the assemblies of the Congregation then sitting in Rome to reform the calendar. Bianchini also availed himself of his assistance to construct the great meridian of the Carthusian church in that city. In 1718 Maraldi, with three other academicians, prolonged the French meridian to the southern extremity of that country. He was admitted a member of the Academy of Sciences of Paris in 1699, in the department of Astronomy, and communicated a great multitude of papers, which are printed in their memoirs, in almost every year from 1699 to 1729, and usually several papers in each of the years; for he was indefatigable in his observations of every thing that was curious and useful in the motions and phenomena of the heavenly bodies. As to the catalogue of the fixed stars, it was not quite completed: just as he had placed a mural quadrant on the terras of the observatory, to observe some stars towards the north and the zenith, he fell sick, and died the 1st of December 1729.</p></div1><div1 part="N" n="MARCH" org="uniform" sample="complete" type="entry"><head>MARCH</head><p>, the 3d month of the year, according to the common way of computing, and consists of 31 days. The sun enters the sign Aries about the 20th or 21st day of this month.</p><p>Among the Romans, March was the first month; and in some ecclesiastical computations, that order is still preserved. In England, before the alteration of the ftile, March was the 1st month in order, the year always commencing with the 25th day of the month.</p><p>It has been said it was Romulus who first divided the year into months; to the first of which he gave the name of his supposed father Mars. It is observed by Ovid, however, that the people of Italy had the month of March before the time of Romulus; but that they placed it differently; some making it the third, some<cb/> the 4th, some the 5th, and others the 10th month of the year.</p><p>MARINE <hi rend="smallcaps">Barometer.</hi> See <hi rend="smallcaps">Barometer.</hi></p><p>MARINERS-<hi rend="smallcaps">Compass.</hi> See <hi rend="smallcaps">Compass.</hi></p></div1><div1 part="N" n="MARIOTTE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MARIOTTE</surname> (<foreName full="yes"><hi rend="smallcaps">Edme</hi></foreName>)</persName></head><p>, an eminent French philosopher and mathematician, was born at Dijon, and admitted a member of the Academy of Sciences of Paris in 1666. His works however are better known than his life. He was a good mathematician, and the first French philosopher who applied much to experimental physics. The law of the shock or collision of bodies, the theory of the pressure and motion of fluids, the nature of vision, and of the air, particularly engaged his attention. He carried into his philosophical researches, that spirit of scrutiny and investigation so necessary to those who would make any considerable progress in it. He died in 1684.</p><p>He communicated a number of curious and valuable papers to the Academy of Sciences, which were printed in the collection of their Memoirs dated 1666, viz, from volume 1 to volume 10. And all his works were collected into 2 volumes in 4to, and printed at Leyden in 1717.</p></div1><div1 part="N" n="MARS" org="uniform" sample="complete" type="entry"><head>MARS</head><p>, one of the seven primary planets now known, and the first of the four superior ones, being placed immediately next above the earth. It is usually denoted by this character <*>, being a mark rudely formed from a man holding a spear protruded, representing the god of war of the same name.</p><p>The mean distance of Mars from the sun, is 1524 of those parts, of which the distance of the earth from the sun is 1000; his excentricity 141; and his real distance 145 millions of miles. The inclination of his orbit to the plane of the ecliptic, is 1° 52′; the length of his year, or the period of one revolution about the sun, is 686 23/24 of our days, or 667 3/4 of his own days, which are 40 minutes longer than ours, the revolution on his axis being performed in 24 hours 40 minutes. His mean diameter is 4444 miles; and the same seen from the sun is 11″: the inclination of the axis to his orbit 0° 0′; the inclination of his orbit to the ecliptic 1° 52′; place of the aphelion <*> 0° 32′; place of his ascending node <*> 17° 17′; and his parallax, according to Dr. Hook and Mr. Flamsteed, is scarce 30 seconds.</p><p>Dr. Hook, in 1665, observed several spots in Mars; which having a motion, he concluded the planet turned round its centre. In 1666, M. Cassini observed several spots in the two faces or hemispheres of Mars, which he found made one revolution in 24hours 40minutes. These observations were repeated in 1670, and confirmed by Miraldi in 1704, and 1719: whence both the motion and period, or natural day, of that planet, were determined.</p><p>In the Philos. Trans. for 1781, Mr. Herschel gave a series of observations on the rotation of this planet about its axis, from which he concluded that one mean sidereal rotation was between 24 h. 39 m. 5 sec. and 24 h. 39 m. 22 sec.; and in the Philos. Trans. for 1784, is given a paper by the same gentleman, on the remarkable appearances at the polar regions of the planet Mars, the inclination of its axis, the position of its poles, and its spheroidical figure; with a few hints relating to its real diameter and atmosphere, deduced from<pb n="80"/><cb/> his observations taken from the year 1777 to 1783 inclusively. He observed several remarkable bright spots near both poles, which had some small motion; and the results of his observations are as follow; viz,</p><p>“Inclination of axis to the ecliptic, 59° 22′.</p><p>The node of the axis is in <*> 17° 47′.</p><p>Obliquity of the planet's ecliptic 28° 42′.</p><p>The point Aries on Mars's ecliptic answers to our <*> 19° 28′.</p><p>The figure of Mars is that of an oblate spheroid, whose equatorial diameter is to the polar one, as 1355 to 1272, or as 16 to 15 nearly.</p><p>The equatorial diameter of Mars, reduced to the mean distance of the earth from the sun, is 9′ 8‴.</p><p>And the planet has a considerable, but moderate atmosphere, so that its inhabitants probably enjoy a situation in many respects similar to ours.”</p><p>Mars always appears with a ruddy troubled light; owing, it is supposed, to the nature of his atmosphere, through which the light passes.</p><p>In the acronical rising of this planet, or when in opposition to the sun, it is five times nearer to us than when in conjunction with him; and so appears much larger and brighter than at other times.</p><p>Mars, having his light from the sun, and revolving round it, has an increase and decrease like the moon: it may also be observed almost bisected, when in the quadratures, or in perigæon; but is never seen cornicular, as the inferior planets. All which shews both that his orbit includes that of the earth within it, and that he shines not by his own light.</p></div1><div1 part="N" n="MARTIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MARTIN</surname> (<foreName full="yes"><hi rend="smallcaps">Benjamin</hi></foreName>)</persName></head><p>, was born in 1704, and became one of the most celebrated mathematicians and opticians of his time. He first taught a school in the country; but afterwards came up to London, where he read lectures on experimental philosophy for many years, and carried on a very extensive trade as an optician and globe-maker in Fleet-street, till the growing infirmities of old age compelled him to withdraw from the active part of business. Trusting too fatally to what he thought the integrity of others, he unfortunately, though with a capital more than sufficient to pay all his debts, became a bankrupt. The unhappy old man, in a moment of desperation from this unexpected stroke, attempted to destroy himself; and the wound, though not immediately mortal, hastened his death, which happened the 9th of February 1782, at 78 years of age.</p><p>He had a valuable collection of fossils and curiosities of almost every species; which after his death were almost given away by public auction. He was indefatigable as an artist, and as a writer he had a very happy method of explaining his subject, and wrote with clearness, and even considerable elegance. He was chiefly eminent in the science of optics; but he was well skilled in the whole circle of the mathematical and philosophical sciences, and wrote useful books on every one of them; though he was not distinguished by any remark able inventions or discoveries of his own. His publications were very numerous, and generally useful: some of the principal of them were as follow:</p><p>The Philosophical Grammar; being a View of the<cb/> present State of Experimental Physiology, or Natural Philosophy, 1735, 8vo.—A new, complete, and universal System or Body of Decimal Arithmetic, 1735, 8vo.—The Young Student's Memorial Book, or Pocket Library, 1735, 8vo.—Description and Use of both the Globes, the Armillary Sphere and Orrery, Trigonometry, 1736, 2 vols. 8vo.—System of the Newtonian Philosophy, 1759, 3 vols.—New Elements of Optics, 1759.—Mathematical Institutions, 1764, 2 vols. —Philologic and Philosophical Geography, 1759. —Lives of Philosophers, their inventions, &c. 1764. —Young Gentleman and Lady's Philosophy, 1764, 3 vols.—Miscellaneous Correspondence, 1764, 4 vols.— Institutions of Astronomical Calculations, 3 parts, 1765. —Introduction to the Newtonian Philosophy, 1765.— Treatise of Logarithms.—Treatise on Navigation.— Description and Use of the Air-pump.—Description of the Torricellian Barometer.—Appendix to the Use of the Globes.—Philosophia Britannica, 3 vols.—Principles of Pump-work.—Theory of the Hydrometer.— Description and Use of a Case of Mathematical Instruments.—Ditto of a Universal Sliding Rule.—Micrographia, on the Microscope.—Principles of Perspective. —Course of Lectures.—Optical Essays.—Essay on Electricity.—Essay on Visual Glasses or Spectacles.— Horologia Nova, or New Art of Dialling.—Theory of Comets.—Nature and Construction of Solar Eclipses. —Venus in the Sun.—The Mariner's Mirror.—Thermometrum Magnum.—Survey of the Solar System.— Essay on Island Chrystal.—Logarithmologia Nova, &c. &c.</p><p>MASCULINE <hi rend="italics">Signs.</hi> Astrologers divide the Signs, &c, into Masculine and Feminine; by reason of their qualities, which are either active, and hot, or cold, accounted Masculine; or passive, dry, and moist, which are feminine. On this principle they call the Sun, Jupiter, Saturn, and Mars, Masculine; and the Moon and Venus, feminine. Mercury, they suppose, partakes of the two. Among the Signs, they account Aries, Libra, Gemini, Leo, Sagittarius, and Aquarius, Masculine; but Cancer, Capricornus, Taurus, Virgo, Scorpio, and Pisces are feminine.</p></div1><div1 part="N" n="MASS" org="uniform" sample="complete" type="entry"><head>MASS</head><p>, the quantity of matter in any body. This is rightly estimated by its weight; whatever be its figure, or whether its bulk or magnitude be large or small.</p></div1><div1 part="N" n="MATERIAL" org="uniform" sample="complete" type="entry"><head>MATERIAL</head><p>, relating to Matter.</p></div1><div1 part="N" n="MATHEMATICAL" org="uniform" sample="complete" type="entry"><head>MATHEMATICAL</head><p>, relating to Mathematics.</p><p><hi rend="smallcaps">Mathematical</hi> <hi rend="italics">Sect,</hi> is one of the two leading philosophical sects, which arose about the beginning of the 17th century; the other being the Metaphysical sect. The former directed its researches by the principles of Gassendi, and sought after truth by observation and experience. The disciples of this sect denied the possibility of erecting on the basis of metaphysical and abstract truths, a regular and solid system of philosophy, without the aid of assiduous observation and repeated experiments, which are the most natural and effectual means of philosophical progress and improvement. The advancement and reputation of this sect, and of natural knowledge in general, were much owing to the plan of philosophizing proposed by lord Bacon, to the establishment of the Royal Society in<pb n="81"/><cb/> London, to the genius and industry of Mr. Boyle, and to the unparalleled researches and discoveries of Sir Isaac Newton. Barrow, Wallis, Locke, and many other great luminaries in learning, adorned this sect.</p></div1><div1 part="N" n="MATHEMATICS" org="uniform" sample="complete" type="entry"><head>MATHEMATICS</head><p>, the science of quantity; or a science that considers magnitudes either as computable or measurable.</p><p>The word in its original, <foreign xml:lang="greek">maqhs<*>s</foreign>, <hi rend="italics">mathesis,</hi> signifies <hi rend="italics">discipline</hi> or <hi rend="italics">science</hi> in general; and, it seems, has been applied to the doctrine of quantity, either by way of eminence, or because, this having the start of all other sciences, the rest took their common name from it.</p><p>As to the origin of the Mathematics, Josephus dates it before the flood, and makes the sons of Seth observers of the course and order of the heavenly bodies: he adds, that to perpetuate their discoveries, and secure them from the injuries either of a deluge or a conflagration, they had them engraven on two pillars, the one of stone, the other of brick; the former of which, he says, was yet standing in Syria in his time.</p><p>Indeed it is pretty generally agreed that the first cultivators of Mathematics, after the flood, were the Assyrians and Chaldeans; from whom, Josephus adds, the science was carried by Abraham to the Egyptians; who proved such notable proficients, that Aristotle even fixes the first rise of Mathematics among them. From Egypt, 584 years before Christ, Mathematics passed into Greece, being carried thither by Thales; who having learned geometry of the Egyptian priests, taught it in his own country. After Thales, came Pythagoras; who, among other Mathematical arts, paid a particular regard to arithmetic; drawing the greatest part of his philosophy from numbers. He was the first, according to Laertius, who abstracted geometry from matter; and to him we owe the doctrine of incommensurable magnitude, and the five regular bodies, besides the first principles of music and astronomy. To Pythagoras succeeded Anaxagoras, Oenopides, Briso, Antipho, and Hippocrates of Scio; all of whom particularly applied themselves to the quadrature of the circle, the duplicature of the cube, &c; but the last with most success of any: he is also mentioned by Proclus, as the first who compiled elements of Mathematics.</p><p>Democritus excelled in Mathematics as well as physics; though none of his works in either kind are extant; the destruction of which is by some authors ascribed to Aristotle. The next in order is Plato, who not only improved geometry, but introduced it into physics, and so laid the foundation of a solid philosophy. From his school arose a crowd of mathematicians. Proclus mentions 13 of note; among whom was Leodamus, who improved the analysis first invented by Plato; Theætetus, who wrote Elements; and Archytas, who has the credit of being the first that applied Mathematics to use in life. These were succeeded by Neocles and Theon, the last of whom contributed to the elements. Eudoxus excelled in arithmetic and geometry, and was the first founder of a system of astronomy. Menechmus invented the conic sections, and Theudius and Hermotimus improved the elements.</p><p>For Aristotle, his works are so stored witb Mathematics, that Blancanus compiled a whole book of them: o<*>t of his school came Eudemus and Theophrastus;<cb/> the first of whom wrote upon numbers, geometry, and invisible lines; and the latter composed a mathematical history. To Aristeus, Isidorus, and Hypsicles, we owe the books of Solids; which, with the other books of Elements, were improved, collected, and methodised by Euclid, who died 284 years before the birth of Christ.</p><p>A hundred years after Euclid, came Eratosthenes and Archimedes: and contemporary with the latter was Conon, a geometrician and astronomer. Soon after came Apollonius Pergæus; whose excellent conics are still extant. To him are also ascribed the 14th and 15th books of Euclid, and which, it is said, were contracted by Hypsicles. Hipparchus and Menelaus wrote on the subtenses of the arcs in a circle; and the latter also on spherical triangles. Theodosius's 3 books of Spherics are still extant. And all these, Menelaus excepted, lived before Christ.</p><p>Seventy years after Christ, was born Ptolomy of Alexandria; a good geometrician, and the prince of astronomers: to him succeeded the philosopher Plutarch, some of whose Mathematical problems are still extant. After him came Eutocius, who commented on Archimedes, and occasionally mentions the inventions of Philo, Diocles, Nicomedes, Sporus, and Heron, on the duplicature of the cube. To Ctesebes of Alexandria we are indebted for pumps; and Geminus, who lived soon after, is preferred by Proclus to Euclid himself.</p><p>Diophantus of Alexandria was a great master of numbers, and the first Greek writer on Algebra. Among others of the Ancients, Nicomachus is celebrated for his arithmetical, geometrical, and musical works: Serenus, for his books on the section of the cylinder; Proclus, for his commentaries on Euclid; and Theon has the credit among some, of being author of the books of elements ascribed to Euclid. The last to be named among the Ancients, is Pappus of Alexandria, who flourished about the year of Christ 400, and is justly celebrated for his books of Mathematical collections, still extant.</p><p>Mathematics are commonly distinguished into <hi rend="italics">Speculative</hi> and <hi rend="italics">Practical, Pure</hi> and <hi rend="italics">Mixed.</hi></p><p><hi rend="italics">Speculative</hi> <hi rend="smallcaps">Mathematics</hi>, is that which barely contemplates the properties of things: and</p><p><hi rend="italics">Practical</hi> <hi rend="smallcaps">Mathematics</hi>, that which applies the knowledge of those properties to some uses in life.</p><p><hi rend="italics">Pure</hi> <hi rend="smallcaps">Mathematics</hi> is that branch which considers quantity abstractedly, and without any relation to matter or bodies.</p><p><hi rend="italics">Mixed</hi> <hi rend="smallcaps">Mathematics</hi> considers quantity as subsisting in material being; for instance, length in a pole, depth in a river, height in a tower, &c.</p><p><hi rend="italics">Pure Mathematics,</hi> again, either considers quantity as discrete, and so computable, as arithmetic; or as concrete, and so measureable, as geometry.</p><p><hi rend="italics">Mixed Mathematics</hi> are very extensive, and are distinguished by various names, according to the different subjects it considers, and the different views in which it is taken; such as Astronomy, Geography, Optics, Hydrostatics, Navigation, &c, &c.</p><p>Pure Mathematics has one peculiar advantage, that it occasions no contests among wrangling disputants, as happens in other branches of knowledge: and the<pb n="82"/><cb/> reason is, because the definitions of the terms are premised, and every person that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies, by shewing, either that our adversary has not stuck to his definitions, or has not laid down true premises, or else that he has drawn false conclusions from true principles; and in case we are not able to do either of these, we must acknowledge the truth of what he has proved.</p><p>It is true, that in mixed Mathematics, where we reason mathematically upon physical subjects, such just definitions cannot be given as in geometry: we must therefore be content with descriptions; which will be of the same use as definitions, provided we be consistent with ourselves, and always mean the same thing by those terms we have once explained.</p><p>Dr. Barrow gives a very elegant description of the excellence and usefulness of mathematical knowledge, in his inaugural oration, upon being appointed Professor of Mathematics at Cambridge. The Mathematics, he observes, effectually exercise, not vainly delude, nor vexatiously torment studious minds with obscure subtilties, but plainly demonstrate every thing within their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These disciplines likewise enure and corroborate the mind to a constant diligence in study; they wholly deliver us from a credulous simplicity, most strongly fortify us against the vanity of scepticism, effectually restrain us from a rash presumption, most easily incline us to a due assent, and perfectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas, and investigates the harmony of proportions; the manners themselves are sensibly corrected and improved, the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more divine contemplations.</p></div1><div1 part="N" n="MATTER" org="uniform" sample="complete" type="entry"><head>MATTER</head><p>, an extended substance. Other properties of Matter are, that it resists, is solid, divisible, moveable, passive, &c; and it forms the principles of which all bodies are composed.</p><p>Matter and form, the two simple and original principles of all things, according to the Ancients, composing some simple natures, which they called Elements; from the various combinations of which all natural things were afterwards composed.</p><p>Dr. Woodward was of opinion, that Matter is originally and really various, being at first creation divided into several ranks, sets, or kinds of corpuscles, differing in substance, gravity, hardness, flexibility, figure, size, &c; from the various compositions and combinations of which, he thinks, arise all the varieties in bodies as to colour, hardness, gravity, tastes, &c. But it is Sir Isaac Newton's opinion, that all those differences result from the various arrangements of the same Matter; which he accounts homogeneous and uniform in all bodies.</p><p>The quantity of Matter in any body, is its measure arising from the joint consideration of the magnitude and density of the body: as if one body be twice as dense as another, and also occupy twice the space, then will it contain 4 times the Matter of the other. This<cb/> quantity of Matter is best discovered by the weight or gravity of the body, to which it is always proportional.</p><p>Newton observes, that “it seems probable, God, in the beginning, formed Matter in solid, massy, hard, impenetrable, moveable particles, of such sizes, figures, and with such other properties, and in such proportion to space, as most conduced to the end for which he formed them; and that these primitive particles, being solid, are incomparably harder than any porous bodies compounded of them; even so very hard, as never to wear, and break in pieces: no ordinary power being able to divide what God himself made one in the first creation. While the particles continue entive, they may compose bodies of one and the same nature and texture in all ages; but should they wear away, or break in pieces, the nature of things depending on them would be changed. Water and earth, composed of old worn particles, would not be of the same nature and texture now with water and earth composed of entire particles in the beginning. And therefore, that nature may be lasting, the changes of corporeal things are to be placed only in the various separations and new associations and motions of these permanent particles; compound bodies being apt to break, not in the midst of solid particles, but where those particles are laid together, and touch in a few points. It seems farther, he continues, that these particles have not only a vis inertiæ, accompanied with such passive laws of motion as naturally result from that force, but also that they are moved by certain active principles, such as is that of gravity, and that which causeth fermentation, and the cohesion of bodies. These principles are to be considered not as occult qualities, supposed to result from the specific forms of things, but as general laws of nature, by which the things themselves are formed; their truth appearing to us by phenomena, though their causes are not yet discovered.”</p><p>Hobbes, Spinoza, &c, maintain that all the beings in the universe are material, and that their differences arise from their different modifications, motions, &c. Thus they conceive that Matter extremely subtile, and in a brisk motion, may think; and so they exclude spirit out of the world.</p><p>Dr. Berkley, on the contrary, argues against the existence of Matter itself; and endeavours to prove that it is a mere <hi rend="italics">ens rationis,</hi> and has no existence out of the mind.</p><p>Some late philosophers have advanced a new hypothesis concerning the nature and essential properties of Matter. The first of these who suggested, or at least published an account of this hypothesis, was M. Boscovich, in his Theoria Philosophiæ Naturalis. He supposes that Matter is not impenetrable, but that it consists of physical points only, endued with powers of attraction and repulsion, taking place at different distances, that is, surrounded with various spheres of attraction and repulsion; in the same manner as solid Matter is generally supposed to be. Provided therefore that any body move with a sufficient degree of velocity, or have sufficient momentum to overcome any power of repulsion that it may meet with, it will find no difficulty in making its way through any body whatever. If the velocity of such a body in motion be sufficiently great, Bo<*>covich contends, that the particles of any body through which it passes, will not even be<pb n="83"/><cb/> moved out of their place by it. With a degree of velocity something less than this, they will be considerably agitated, and ignition might perhaps be the consequence, though the progress of the body in motion would not be sensibly interrupted; and with a still less momentum it might not pass at all.</p><p>Mr. Michell, Dr. Priestley, and some others of our own country, are of the same opinion. See Priestley's History of Discoveries relating to Light, pa. 390.— In conformity to this hypothesis, this author maintains, that Matter is not that inert substance that it has been supposed to be; that powers of attraction or repulsion are necessary to its very being, and that no part of it appears to be impenetrable to other parts. Accordingly, he defines Matter to be a substance, possessed of the property of extension, and of powers of attraction or repulsion, which are not distinct from Matter, and foreign to it, as it has been generally imagined, but absolutely essential to its very nature and being: so that when bodies are divested of these powers, they become nothing at all. In another place, Dr. Priestley has given a somewhat different account of Matter; according to which it is only a number of centres of attraction and repulsion; or more properly of centres, not divisible, to which divine agency is directed; and as sensation and thought are not incompatible with these powers, solidity, or impenetrability, and consequently a vis inertiæ only having been thought repugnant to them, he maintains, that we have no reason to suppose that there are in man two substances absolutely distinct from each other. See Disquisitions on Matter and Spirit.</p><p>But Dr. Price, in a correspondence with Dr. Priestley, published under the title of A Free Discussion of the Doctrines of Materialism and Philosophical Necessity, 1778, has suggested a variety of unanswerable objections against this hypothesis of the penetrability of Matter, and against the conclusions that are drawn from it. The vis inertiæ of Matter, he says, is the foundation of all that is demonstrated by natural philosophers concerning the laws of the collision of bodies. This, in particular, is the foundation of Newton's philosophy, and especially of his three laws of motion. Solid Matter has the power of acting on other Matter by impulse; but unsolid Matter cannot act at all by impulse; and this is the only way in which it is capable of acting, by any action that is properly its own. If it be said, that one particle of Matter can act upon another without contact and impulse, or that Matter can, by its own proper agency, attract or repel other Matter which is at a distance from it, then a maxim hitherto universally received must be false, that “nothing can act where it is not.” Newton, in his letters to Bentley, calls the notion, that Matter possesses an innate power of attraction, or that it can act upon Matter at a distance, and attract and repel by its own agency, an absurdity into which he thought no one could possibly fall. And in another place he expressly disclaims the notion of innate gravity, and has taken pains to shew that he did not take it to be an essential property of bodies. By the same kind of reasoning pursued, it must appear, that Matter has not the power of attracting and repelling; that this power is the power of some foreign cause, acting upon Mat-<cb/> ter according to stated laws; and consequently that attraction and repulsion, not being actions, much less inherent qualities of Matter, as such, it ought not to be defined by them. And if Matter has no other property, as Dr. Priestley asserts, than the power of attracting and repelling, it must be a non-entity; because this is a property that cannot belong to it. Besides, all power is the power of something; and yet if Matter is nothing but this power, it must be the power of nothing; and the very idea of it is a contradiction. If Matter be not solid extension, what can it be more than mere extension?</p><p>Farther, Matter that is not solid, is the same with pore; and therefore it cannot possess what philosophers mean by the momentum or force of bodies, which is always in proportion to the quantity of Matter in bodies, void of pore.</p><p>MAUNDY <hi rend="smallcaps">Thursday</hi>, is the Thursday in Passion week; which was called <hi rend="italics">Maundy</hi> or <hi rend="italics">Mandate Thursday,</hi> from the command which Christ gave his apostles to commemorate him in the Lord's Supper, which he this day instituted; or from the new commandment which he gave them to love one another, after he had washed their feet as a token of his love to them.</p></div1><div1 part="N" n="MAUPERTUIS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MAUPERTUIS</surname> (<foreName full="yes"><hi rend="smallcaps">Peter Louis Morceau de</hi></foreName>)</persName></head><p>, a celebrated French mathematician and philosopher, was born at St Malo in 1698, and was there privately educated till he attained his 16th year, when he was placed under the celebrated professor of philosophy, M. le Blond, in the college of la Marche, at Paris; while M. Guisnée, of the Academy of Sciences, was his instructor in mathematics. For this science he soon discovered a strong inclination, and particularly for geometry. He likewise practised instrumental music in his early years with great success; but fixed on no profession till he was 20, when he entered into the army; in which he remained about 5 years, during which time he pursued his mathematical studies with great vigour; and it was soon remarked by M. Freret and other academicians, that nothing but mathematics could satisfy his active soul and unbounded thirst for knowledge.</p><p>In the year 1723, he was received into the Royal Academy of Sciences, and read his first performance, which was a memoir upon the construction and form of musical instruments. During the first years of his admission, he did not wholly confine his attention to mathematics; he dipt into natural philosophy, and discovered great knowledge and dexterity in observations and experiments upon animals.</p><p>If the custom of travelling into remote countries, like the sages of antiquity, in order to be initiated into the learned mysteries of those times, had still subsisted, no one would have conformed to it with more eagerness than Maupertuis. His first gratification of this passion was to visit the country which had given birth to Newton; and during his residence at London he became as zealous an admirer and follower of that philosopher as any one of his own countrymen. His next excursion was to Basil in Switzerland, where he formed a friendship with the celebrated John Bernoulli and his family, which continued till his death. At his return to Paris, he applied himself to his favourite studies with greater zeal than ever. And how well he ful-<pb n="84"/><cb/> filled the duties of an academician, may be seen by running over the Memoirs of the Academy from the year 1724 to 1744; where it appears that he was neither idle, nor occupied by objects of small importance. The most sublime questions in the mathematical sciences, received from his hand that elegance, clearness, and precision, so remarkable in all his writings.</p><p>In the year 1736, he was sent to the polar circle, to measure a degree of the meridian, in order to ascertain the sigure of the earth; in which expedition he was accompanied by Mess. Clairault, Camus, Monnier, Outhier, and Celsus the celebrated professor of astronomy at Upsal. This business rendered him so famous, that on his return he was admitted a member of almost every academy in Europe.</p><p>In the year 1740, Maupertuis had an invitation from the king of Prussia to go to Berlin; which was too flattering to be refused. His rank among men of letters had not wholly effaced his love for his first profe<*>ion, that of arms. He followed the king to the field, but at the battle of Molwitz was deprived of the pleasure of being present, when victory declared in favour of his royal patron, by a fingular kind of adventure. His horse, during the heat of the action, running away with him, he fell into the hands of the enemy; and was at first but roughly treated by the Austrian Hussars, to whom he could not make himself known for want of language; but being carried prisoner to Vienna, he received such honours from the emperor as never were effaced from his memory. Maupertuis lamented very much the loss of a watch of Mr. Graham's, the celebrated English artist, which they had taken from him; the emperor, who happened to have another by the same artist, but enriched with diamonds, presented it to him, saying, “the Hussars meant only to jest with you, they have sent me your watch, and I return it to you.”</p><p>He went soon after to Berlin; but as the reform of the academy which the king of Prussia then meditated was not yet mature, he repaired to Paris, where his affairs called him, and was chosen in 1742 director of the Academy of Sciences. In 1743 he was received into the French Academy; which was the first instance of the same person being a member of both the academies at Paris at the same time. Maupertuis again assumed the soldier at the siege of Fribourg, and was pitched upon by marshal Coigny and the count d'Argenson to carry the news to the French king of the surrender of that citadel.</p><p>Maupertuis returned to Berlin in the year 1744, when a marriage was negotiated and brought about, by the good offices of the queen mother, between our author and madamoiselle de Borck, a lady of great beauty and merit, and nearly related to M. de Borck at that time minister of state. This determined him to settle at Berlin, as he was extremely attached to his new spouse, and regarded this alliance as the most fortunate circumstance of his life.</p><p>In the year 1746, Maupertuis was declared, by the king of Prussia, President of the Royal Academy of Sciences at Berlin, and soon after by the same prince was honoured with the Order of Merit. However, all these accumulated honours and advantages, so far<cb/> from lessening his ardour for the sciences, seemed to furnish new allurements to labour and application. Not a day passed but he produced some new project or essay for the advancement of knowledge. Nor did he confine himself to mathematical studies only: metaphysics, chemistry, botany, polite literature, all shared his attention, and contributed to his fame. At the fame time he had, it seems, a strange inquietude of spirit, with a dark atrabilaire humour, which rendered him miserable amidst honours and pleasures. Such a temperament did not promise a pacific life; and he was in fact engaged in several quarrels. One of these was with Koenig the professor of philosophy at Franeker, and another more terrible with Voltaire. Maupertuis had inserted in the volume of Memoirs of the Academy of Berlin for 1746, a discourse upon the laws of motion; which Koenig was not content with attacking, but attributed to Leibnitz. Maupertuis, stung with the imputation of plagiarism, engaged the academy of Berlin to call upon him for his proof; which Koenig failing to produce, his name was struck out of the academy, of which he was a member. Several pamphlets were the consequence of this measure; and Voltaire, for some reason or other, engaged in the quarrel against Maupertuis. We say, for some reason or other; because Maupertuis and Voltaire were apparently upon the most amicable terms; and the latter respected the former as his master in the mathematics. Voltaire upon this occasion exerted all his wit and satire against him; and upon the whole was so much transported beyond what was thought right, that he found it expedient in 1753 to quit the court of Prussia.</p><p>Our philosopher's constitution had long been considerably impaired by the great fatigues of various kinds in which his active mind had involved him; though from the amazing hardshipa he had undergone, in his northern expedition, most of his bodily sufferings may be traced. The intehfe sharpness of the air could only be supported by means of strong liquors; which helped but to lacerate his lungs, and bring on a spitting of blood, which began at least 12 years before he died. Yet still his mind seemed to enjoy the greatest vigour; for the best of his writings were produced, and most sublime ideas developed, during the time of his confinement by sickness, when he was unable to occupy his presidial chair at the academy. He took several journeys to St. Malo, during the last years of his life, for the recovery of his health<*> and though he always received benefit by breathing his native air, yet still, upon his return to Berlin, his disorder likewise returned with greater violence. His last journey into France was undertaken in the year 1757; when he was obliged, soon after his arrival there, to quit his favourite retreat at St. Malo, on account of the danger and confusion which that town was thrown into by the arrival of the English in its neighbourhood. From thence he went to Bourdeaux, hoping there to meet with a neutral ship to carry him to Hamburgh, in his way back to Berlin; but being disappointed in that hope, he went to Toulouse, where he remained seven months. He had then thoughts of going to Italy, in hopes a milder climate would restore him to health; but finding himself grow worse, he rather inclined towards Germany, and went no Neufchatel, where for three<pb n="85"/><cb/> months he enjoyed the conversation of lord Marischal, with whom he had formerly been much connected. At length he arrived at Basil, October 16, 1758, where he was received by his friend Bernoulli and his family with the utmost tenderness and affection. He at first found himself much better here than he had been at Neufchatel: but this amendment was of short duration; for as the winter approached, his disorder returned, accompanied by new and more alarming symptoms. He languished here many months, during which he was attended by M. de la Condamine; and died in 1759, at 61 years of age.</p><p>The works which he published were collected into 4 volumes 8vo, published at Lyons in 1756, where also a new and elegant edition was printed in 1768. These contain the following works:</p><p>1. Essay on Cosmology.—2. Discourse on the different Figures of the Stars.—3. Essay on Moral Philosophy.—4. Philosophical Reflections upon the Origin of Languages, and the Signification of Words— 5. Animal Physics, concerning Generation &c.— 6. System of Nature, or the Formation of bodies— 7. Letters on various subjects.—8. On the Progress of the Sciences.—9. Elements of Geography—10. Account of the Expedition to the Polar Circle, for determining the Figure of the Earth; or the Measure of the Earth at the Polar Circle.—11. Account of a Journey into the Heart of Lapland, to search for an Ancient Monument.—12. On the Comet of 1742.— 13. Various Academical Discourses, pronounced in the French and Prussian Academies.—14. Dissertation upon Languages.—15. Agreement of the Different Laws of Nature, which have hitherto appeared incompatible.—16. Upon the Laws of Motion.—17. Upon the Laws of R<*>st.—18. Nautical Astronomy.—19. On the Parallax of the Moon.—20. Operations for determining the Figure of the Earth, and the Variations of Gravity.—21. Measure of a Degree of the Meridian at the Polar Circle.</p><p>Beside these works, Maupertuis was author of a great multitude of interesting papers, particularly those printed in the Memoirs of the Paris and Berlin Academies, far too numerous here to mention; viz, in the Memoirs of the Academy at Paris, from the year 1724, to 1749; and in those of the Academy of Berlin, from the year 1746, to 1756.</p></div1><div1 part="N" n="MAXIMUM" org="uniform" sample="complete" type="entry"><head>MAXIMUM</head><p>, denotes the greatest state or quantity attainable in any given case, or the greatest value of a variable quantity. By which it stands opposed to Minimum, which is the least possible quantity in any case.</p><p>As in the algebraical expression , where <hi rend="italics">a</hi> and <hi rend="italics">b</hi> are constant or invariable quantities, and <hi rend="italics">x</hi> a variable one. Now it is evident that the value of this remainder or difference, , will increase as the term <hi rend="italics">bx,</hi> or <hi rend="italics">x,</hi> decreases; and therefore that will be the greatest when this is the smallest; that is, is a maximum, when <hi rend="italics">x</hi> is the least, or nothing at all.</p><p>Again, the expression or difference , evidently increases as the fraction <hi rend="italics">b</hi>/<hi rend="italics">x</hi> diminishes; and this diminisnes as <hi rend="italics">x</hi> increases; therefore the given expression will be the greatest, or a maximum, when <hi rend="italics">x</hi> is the greatest, or infinite.<cb/></p><p>Also, if along the diameter KZ <hi rend="italics">(the 3d fig. below)</hi> of a circle, a perpendicular ordinate LM b[ecedil] conceived to move, from K towards Z; it is evident that, from K it increases continually till it arrive at the centre, in the position NO, where it is at the greatest state; and from thence it continually decreases again, as it moves along from N to Z, and quite vanishes at the point Z. So that the maximum state of the ordinate is NO, equal to the radius of the circle.</p><p><hi rend="italics">Methodus de</hi> <hi rend="smallcaps">Maximis</hi> <hi rend="italics">et</hi> <hi rend="smallcaps">Minimis</hi>, a method of finding the greatest or least state or value of a variable quantity. <figure/></p><p>Some quantities continually increase, and so have no maximum but what is insinite; as the ordinates BC, DE of the parabola ACE: Some continually decrease, and so their least or minimum state is nothing; as the ordinates FG, HI, to the asymptotes of the hyperbola. Others increase to a certain magnitude, which is their maximum, and then decrease again; as the ordinates LM &c of the circle. And others again decrease to a certain magnitude TV, which is their minimum, and then increase again; as the ordinates of the curve SVY. While others admit of several maxima and minima; as the ordinates of the curve <hi rend="italics">abcde,</hi> where at <hi rend="italics">b</hi> and <hi rend="italics">d</hi> they are maxima, and <hi rend="italics">a, c, e,</hi> minima. And thus the maxima and minima of all other variable quantities may be conceived; expressing those quantities by the ordinates of some curves.</p><p>The first maxima and minima are found in the Elements of Euclid, or flow immediately from them: thus, it appears, by the 5th prop. of book 2, that the greatest rectangle that can be made of the two parts of a given line, any how divided, is when the line is divided equally in the middle; prob. 7, book 3, shews that the greatest line that can be drawn from a given point within a circle, is that which passes through the centre; and that the least line that can be so drawn, is the continuation of the same to the other side of the circle: prop. 8 ib. shews the same for lines drawn from a point without the circle: and thus other instances might be pointed out in the Elements.— Other writers on the Maxima and Minima, are, Apollonius, in the whole 5th book of his Conic Sections;<pb n="86"/><cb/> and in the Preface or Dedication to that book, he says others had then also treated the subject, though in a slighter manner.—Archimedes; as in prop. 9 of his Treatise on the Sphere and Cylinder, where he demonstrates that, of all spherical segments under equal superficies, the hemisphere is the greatest.—Serenus, in his 2d book, or that on the Conic Sections.— Pappus, in many parts of his Mathematical Collections; as in lib. 3, prop. 28 &c, lib. 6, prop. 31 &c, where he treats of some curious cases of variable geometrical quantities, shewing how some increase and decrease both ways to infinity; while others proceed only one way, by increase or decrease, to infinity, and the other way to a certain magnitude; and others again both ways to a certain magnitude, giving a maximum and minimum; also lib. 7, prop. 13, 14, 165, 166, &c. And all these are the geometrical Maxima and Minima of the Ancients; to which may be added some others of the same kind, viz. Viviani De Maximis & Minimis Geometrica Divinatio in quintum Conicorum Apollonii Pergæi, in fol. at Flor. 1659; also an ingenious little tract in Thomas Simpson's Geometry, on the Maxima and Minima of Geometrical Quantities.</p><p>Other writings on the Maxima and Minima are chiefly treated in a more general way by the modern analysis; and first among these perhaps may be placed that of Fermat. This, and other methods, are best referred to, and explained by the ordinates of curves. For when the ordinate of a curve increases to a certain magnitude, where it is greatest, and afterwards decreases again, it is evident that two ordinates on the contrary sides of the greatest ordinate may be equal to each other; and the ordinates decrease to a certain point, where they are at the least, and afterwards increase again; there may also be two equal ordinates, one on each side of the least ordinate. Hence then an equal ordinate corresponds to two different abscisses, or for every value of an ordinate there are two values of abscisses. Now as the difference between the two abscisses is conceived to become less and less, it is evident that the two equal ordinates, corresponding to them, approach nearer and nearer together; and when the differences of the abscisses are infinitely little, or nothing, then the equal ordinates unite in one, which is either the maximum or minimum. The method hence derived then, is this: Find two values of an ordinate, expressed in terms of the abscisses: put those two values equal to each other, cancelling the parts that are common to both, and dividing all the remaining terms by the difference between the abscisses, which will be a common factor in them: next, supposing the abscisses to become equal, that the equal ordinates may concur in the maximum or minimum, that difference will vanish, as well as all the terms of the equation that include it; and therefore, striking those terms out of the equation, the remaining terms will give the value of the absciss corresponding to the maximum or minimum.</p><p>For example, suppose it were required to find the greatest ordinate in a circle KMQ. Put the diameter KZ = <hi rend="italics">a,</hi> the absciss KL = <hi rend="italics">x,</hi> the ordinate LM = <hi rend="italics">y;</hi> hence the other part of the diameter is , and consequently, by the nature of the circle being equal LM<hi rend="sup">2</hi>, or .<cb/> Again, put another absciss , where <hi rend="italics">d</hi> is the difference LP, the ordinate PQ, being equal to LM or <hi rend="italics">y;</hi> here then again , or : put now these two values of <hi rend="italics">y</hi><hi rend="sup">2</hi> equal to each other, so shall ; cancel the common terms <hi rend="italics">ax</hi> and <hi rend="italics">x</hi><hi rend="sup">2</hi>, then , or ; divide all by <hi rend="italics">d,</hi> so shall , a general equation derived from the equality of the two ordinates. Now, bringing the two equal ordinates together, or making the two abscisses equal, their difference <hi rend="italics">d</hi> vanishes, and the last equation becomes barely 2<hi rend="italics">x</hi> = <hi rend="italics">a,</hi> or <hi rend="italics">x</hi> = (1/2)<hi rend="italics">a,</hi> = KN, the value of the absciss KN when the ordinate NO is a maximum, viz, the greatest ordinate bisects the diameter. And the operation and conclusion it is evident will be the same, to divide a given line into two parts, so that their rectangle shall be the greatest possible. <figure/></p><p>For a second example, let it be required to divide the given line AB into two such parts, that the one line drawn into the square of the other may be the greatest possible. Putting the given line AB = <hi rend="italics">a,</hi> and one part AC = <hi rend="italics">x;</hi> then the other part CB will be , and therefore is the product of one part by the square of the other. Again, let one part be , then the other part is , and . Then, putting these two products equal to each other, cancelling the common terms , and dividing the remainder by <hi rend="italics">d,</hi> there results ; hence, cancelling all the terms that contain <hi rend="italics">d,</hi> there remains , or 3<hi rend="italics">x</hi> = 2<hi rend="italics">a,</hi> and, <hi rend="italics">x</hi> = (2/3)<hi rend="italics">a;</hi> that is, the given line must be divided into two parts in the ratio of 3 to 2. See Fermat's Opera Varia, pa. 63, and his Letters to F. Mersenne.</p><p>The next method was that of John Hudde, given by Schooten among the additions to Des Cartes's Geometry, near the end of the 1st vol. of his edition. This method is also drawn from the property of an equation that has two equal roots. He there demonstrates that, having ranged the terms of an equation, that has two roots equal, according to the order of the exponents of the unknown quantity, taking all the terms over to one side, and so making them equal to nothing on the other side; if then the terms in that order be multiplied by the terms of any arithmetical progression, the resulting equation will still have one of its roots equal to one of the two equal roots of the former equation. Now since, by what has been said of the foregoing method, when the ordinate of a curve, admitting of a maximum or minimum, is expressed in terms of the abscissa, that abscissa, or the value of <hi rend="italics">x,</hi> will be two-fold, because there are two ordinates of the same value; that is, the equation has at least two unequal roots or values of <hi rend="italics">x:</hi> but when the ordinate becomes a maximum or minimum, the two abscisses unite in one, and the two roots, or values of <hi rend="italics">x,</hi> are equal; therefore, from the above said property, the terms of this equation for the maximum or minimum being multiplied by the terms of any arithmetical progression, the root of the resulting equa-<pb n="87"/><cb/> tion will be one of the said equal roots, or the value of the absciss <hi rend="italics">x</hi> when the ordinate is a maximum.</p><p>Although the terms of any arithmetic progression may be used for this purpose, some are more convenient than others; and Mr. Hudde directs to make use of that progression which is formed by the exponents of <hi rend="italics">x,</hi> viz, to multiply each term by the exponent of its power, and putting all the resulting products equal to nothing; which, it is evident, is exactly the same process as taking the fluxions of all the terms, and putting them equal to nothing; being the common process now used for the same purpose.</p><p>Thus, in the former of the two foregoing examples, where , or <hi rend="italics">y</hi><hi rend="sup">2</hi>, is to be a maximum; mult. by 1 2 gives ; hence 2<hi rend="italics">x</hi> = <hi rend="italics">a,</hi> and <hi rend="italics">x</hi> = (1/2)<hi rend="italics">a,</hi> as before.</p><p>And in the 2d example, where , is to be a maximum; mult. by - - 2 3 gives - - - - - ; hence , or 3<hi rend="italics">x</hi> = 2<hi rend="italics">a,</hi> and <hi rend="italics">x</hi> = (2/3)<hi rend="italics">a,</hi> as before.</p><p>The next general method, and which is now usually practised, is that of Newton, or the method of Fluxions, which proceeds upon a principle different from that of the two former methods of Fermat and Hudde. These proceed upon the idea of the two equal ordinates of a curve uniting into one, at the place of the maximum and minimum; but Newton's upon the principle, that the fluxion or increment of an ordinate is nothing, at the point of the maximum or minimum; a circumstance which immediately follows from the nature of that doctrine: for, since a quantity ceafes to increase at the maximum, and to decrease at the minimum, at those points it neither increases nor decreases; and since the fluxion of a quantity is proportional to its increase or decrease, therefore the fluxion is nothing at the maximum or minimum. Hence this rule. Take the fluxion of the algebraical expression denoting the maximum or minimum, and put it equal to nothing; and that equation will determine the value of the unknown letter or quantity in question.</p><p>So in the first of the two foregoing examples, where it is required to determine <hi rend="italics">x</hi> when is a maximum: the fluxion of this is ; divide by <hi rend="italics">x</hi><hi rend="sup">.</hi>, so shall , or <hi rend="italics">a</hi> = 2<hi rend="italics">x,</hi> and <hi rend="italics">x</hi> = (1/2)<hi rend="italics">a.</hi></p><p>Also, in the 2d example, where must be a maximum: here the fluxion is ; hence , or 2<hi rend="italics">a</hi> = 3<hi rend="italics">x,</hi> and <hi rend="italics">x</hi> = (2/3)<hi rend="italics">a.</hi></p><p>When a quantity becomes a maximum or minimum, and is expressed by two or more affirmative and negative terms, in which only one variable letter is contained; it is evident that the fluxion of the affirmative terms will be equal to the fluxion of the negative ones; since their difference is equal to nothing.</p><p>And when, in the expression for the fluxion of a maximum or minimum, there are two or more fluxionary letters, each contained in both affirmative and negative terms; the sum of the terms containing the fluxion of each letter, will be equal to nothing: For, in order that any expression be a maximum or minimum, which contains two or more variable quantities, it must produce a maximum or minimum, if but one of those quantities be supposed variable. So if <cb/> denote a minimum; its fluxion is ; hence , and ; from the former of these <hi rend="italics">y</hi> = (1/2)<hi rend="italics">a,</hi> and from the latter <hi rend="italics">x</hi> = (1/2)<hi rend="italics">b.</hi> Or, in such a case, take the fluxion of the whole expression, supposing only one quantity variable; then take the sluxion again, supposing another quantity only variable: and so on, for all the several variable quantities; which will give the same number of equations for determining those quantities. So, in the above example, , the fluxion is , supposing only <hi rend="italics">x</hi> variable; which gives <hi rend="italics">y</hi> = (1/2)<hi rend="italics">a:</hi> and the fluxion is , when <hi rend="italics">y</hi> only is variable; which gives <hi rend="italics">x</hi> = (1/2)<hi rend="italics">b;</hi> the same as before.</p><p>Farther, when any quantity is a maximum or minimum, all the powers or roots of it will be so too; as will also the result be, when it is increased or decreased, or multiplied, or divided by a given or constant quantity; and the logarithm of the same will be also a maximum or minimum.</p><p><hi rend="italics">To find whether a proposed algebraic quantity admits of a maximum or minimum.</hi>—Every algebraic expression does not admit of a maximum or minimum, properly so called; for it may either increase continually to infinity, or decrease continually to nothing; in both which cases there is neither a proper maximum nor minimum; for the true maximum is that value to which an expression increases, and after which it decreases again; and the minimum is that value to which the expression decreases, and after that it increases again. Therefore when the expression admits of a maximum, its fluxion is positive before that point, and negative after it; but when it admits of a minimum, its fluxion is negative before, and positive after it. Hence, take the fluxion of the expression a little before the fluxion is equal to nothing, and a little after it; if the first fluxion be positive, and the last negative, the middle state is a maximum; but if the first fluxion be negative, and the last positive, the middle state is a minimum. See Maclaurin's Fluxions, book 1, chap. 9, and book 2, chap. 5, art. 859.</p></div1><div1 part="N" n="MAY" org="uniform" sample="complete" type="entry"><head>MAY</head><p>, <hi rend="italics">Maius,</hi> the fifth month in the year, reckoning from our first or January; but the third, counting the year to begin with March, as the Romans did anciently. It was called Maius by Romulus, in respect to the senators and nobles of his city, who were named <hi rend="italics">majores;</hi> as the following month was called <hi rend="italics">Junius,</hi> in honour of the youth of Rome, <hi rend="italics">in honorem juniorum,</hi> who served him in the war. Though some say it has been thus called from <hi rend="italics">Maia,</hi> the mother of Mercury, to whom they offered sacrifice on the first day of this month: and Papias derives the name from <hi rend="italics">Madius, eo quod tunc terra madeat.</hi></p><p>In this month the sun enters the sign Gemini, and the plants of our hemisphere begin mostly to flower.</p></div1><div1 part="N" n="MAYER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MAYER</surname> (<foreName full="yes"><hi rend="smallcaps">Tobias</hi></foreName>)</persName></head><p>, one of the greatest astronomers and mechanists of the 18th century, was born at Maspach, in the duchy of Wirtemberg, 1723. He taught himself mathematics, and at 14 years of age designed machines and instruments with the greatest dexterity and justness. These pursuits did not hinder him from cultivating the Belles Lettres. He acquired the Latin tongue, and wrote it with elegance. In 1750, the university of Gottingen chose him for their mathematical professor; and every year of his short life<pb n="88"/><cb/> was thenceforward marked with some confiderable discoveries in geometry and astronomy. He published several works in this way, which are all accounted excellent of their kind; and some papers are inserted in the second volume of the Memoirs of the University of Gottingen. He was very accurate and indefatigable in his astronomical observations; indeed his labours seem to have very early exhausted him; for he died worn out in 1762, at no more than 39 years of age.</p><p>His Table of Refractions, deduced from his astronomical observations, very nicely agrees with that of Doctor Bradley; and his Theory of the Moon, and Astronomical Tables and Precepts, were so well esteemed, that they were rewarded by the English Board of Longitude, with the premium of three thousand pounds, which sum was paid to his widow after his death. These tables and precepts were published by the Board of Longitude in 1770.</p></div1><div1 part="N" n="MEAN" org="uniform" sample="complete" type="entry"><head>MEAN</head><p>, a middle state between two extremes: as a mean motion, mean distance, arithmetical mean, geometrical mean, &c.</p><p><hi rend="italics">Arithmetical</hi> <hi rend="smallcaps">Mean</hi>, is half the sum of the extremes. So, 4 is an arithmetical mean between 2 and 6, or between 3 and 5, or between 1 and 7; also an arithmetical mean between <hi rend="italics">a</hi> and <hi rend="italics">b</hi> is or .</p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Mean</hi>, commonly called a mean proportional, is the square root of the product of the two extremes; so that, to find a mean proportional between two given extremes, multiply these together, and extract the square root of the product. Thus, a mean proportional between 1 and 9, is ; a mean between 2 and 4 1/2 is also; the mean between 4 and 6 is ; and the mean between <hi rend="italics">a</hi> and <hi rend="italics">b</hi> is √<hi rend="italics">ab.</hi></p><p>The geometrical mean is always less than the arithmetical mean, between the same two extremes. So the arithmetical mean between 2 and 4 1/2 is 3 1/4, but the geometrical mean is only 3. To prove this generally; let <hi rend="italics">a</hi> and <hi rend="italics">b</hi> be any two terms, <hi rend="italics">a</hi> the greater, and <hi rend="italics">b</hi> the less; then, universally, the arithmetical mean shall be greater than the geometrical mean √<hi rend="italics">ab,</hi> or greater than 2√<hi rend="italics">ab.</hi> For, by squaring both, they are ; subtr. 4<hi rend="italics">ab</hi> from each, then , that is - - - . <figure/></p><p><hi rend="italics">To find a Mean Proportional Geometrically,</hi> between two given lines M and N. Join the two given lines together at C in one continued line AB; upon the diameter AB describe a semicircle, and erect the perpendicular CD; which will be the mean proportional between AC and CB, or M and N.</p><p><hi rend="italics">To find two Mean Proportionals</hi> between two given extremes. Multiply each extreme by the square of the other, viz, the greater extreme by the square of the less, and the less extreme by the square of the<cb/> greate<*>; then extract the cube root out of each product, and the two roots will be the two mean proportionals sought. That is, √<hi rend="sup">3</hi><hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi> and √<hi rend="sup">3</hi><hi rend="italics">ab</hi><hi rend="sup">2</hi> are the two means between <hi rend="italics">a</hi> and <hi rend="italics">b.</hi> So, between 2 and 16, the two mean proportionals are 4 and 8; for , and .</p><p>In a similar manner we proceed for three means, or four means, or five means, &c. From all which it appears that the series of the several numbers of mean proportionals between <hi rend="italics">a</hi> and <hi rend="italics">b</hi> will be as follows: viz, one mean, √<hi rend="italics">ab;</hi> two means, √<hi rend="sup">3</hi><hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b,</hi> √<hi rend="sup">3</hi><hi rend="italics">ab</hi><hi rend="sup">2</hi>; three means, √<hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">3</hi><hi rend="italics">b,</hi> √<hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi><hi rend="sup">2</hi>, √<hi rend="sup">4</hi><hi rend="italics">ab</hi><hi rend="sup">3</hi>; four means, √<hi rend="sup">5</hi><hi rend="italics">a</hi><hi rend="sup">4</hi><hi rend="italics">b,</hi> √<hi rend="sup">5</hi><hi rend="italics">a</hi><hi rend="sup">3</hi><hi rend="italics">b</hi><hi rend="sup">2</hi>, √<hi rend="sup">5</hi><hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi><hi rend="sup">3</hi>; √<hi rend="sup">5</hi><hi rend="italics">ab</hi><hi rend="sup">4</hi>; five means, √<hi rend="sup">6</hi><hi rend="italics">a</hi><hi rend="sup">5</hi><hi rend="italics">b,</hi> √<hi rend="sup">6</hi><hi rend="italics">a</hi><hi rend="sup">4</hi><hi rend="italics">b</hi><hi rend="sup">2</hi>, √<hi rend="sup">6</hi><hi rend="italics">a</hi><hi rend="sup">3</hi><hi rend="italics">b</hi><hi rend="sup">3</hi>, √<hi rend="sup">6</hi><hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi><hi rend="sup">4</hi>, √<hi rend="sup">6</hi><hi rend="italics">ab</hi><hi rend="sup">5</hi>; &c, &c.</p><p><hi rend="italics">Harmonical</hi> <hi rend="smallcaps">Mean</hi>, is double a fourth proportional to the sum of the extremes, and the two extremes themselves <hi rend="italics">a</hi> and <hi rend="italics">b:</hi> thus, as the harmonical mean between <hi rend="italics">a</hi> and <hi rend="italics">b.</hi> Or it is the reciprocal of the arithmetical mean between the reciprocals of the given extremes; that is, take the reciprocals of the extremes <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> which will be 1/<hi rend="italics">a</hi> and 1/<hi rend="italics">b;</hi> then take the arithmetical mean between these reciprocals, or half their sum, which will be or ; lastly, the reciprocal of this is the harmonical mean: for, arithmeticals and harmonicals are mutually reciprocals of each other; so that if <hi rend="italics">a, m, b,</hi> &c be arithmeticals, then shall 1/<hi rend="italics">a,</hi> 1/<hi rend="italics">m,</hi> 1/<hi rend="italics">b,</hi> &c be harmonicals; or if the former be harmonicals, the latter will be arithmeticals.</p><p>For example, to find a harmonical mean between 2 and 6; here <hi rend="italics">a</hi> = 2, and <hi rend="italics">b</hi> = 6; therefore the harmonical mean sought between 2 and 6.</p><p>In the 3d book of Pappus's Mathematical Collections we have a very good tract on all the three sorts of mean proportionals, beginning at the 5th proposition. He observes, that the Ancients could not resolve, in a geometrical way, the problem of finding two mean proportionals; and because it is not easy to describe the conic sections in plano, for that purpose, they contrived easy and convenient instruments, by which they obtained good mechanical constructions of that problem; as appears by their writings; as in the Mesolabe of Eratosthenes, of Philo, with the Mechanics and Catapultics of Hero. For these, rightly deeming the problem a solid one, effected the construction only by instruments, and Apollonius Pergæus by means of the conic sections; which others again performed by the <hi rend="italics">loci solidi</hi> of Aristæus; also Nicomedes solved it by the conchoid, by means of<pb n="89"/><cb/> which likewise he trisected an angle: and Pappus himself gave another solution of the same problem.</p><p>Pappus adds definitions of the three foregoing different sorts of means, with many problems and properties concerning them, and, among others, this curious similarity of them, viz, <hi rend="italics">a, m, b,</hi> being three continued terms, either arithmeticals, geometricals, or harmonicals; then in the Arithmeticals, <hi rend="italics">a</hi> : <hi rend="italics">a</hi> :: <hi rend="italics">a</hi> - <hi rend="italics">m</hi> : <hi rend="italics">m</hi> - <hi rend="italics">b;</hi> Geometricals, <hi rend="italics">a</hi> : <hi rend="italics">m</hi> :: <hi rend="italics">a</hi> - <hi rend="italics">m</hi> : <hi rend="italics">m</hi> - <hi rend="italics">b;</hi> Harmonicals, <hi rend="italics">a</hi> : <hi rend="italics">b</hi> :: <hi rend="italics">a</hi> - <hi rend="italics">m</hi> : <hi rend="italics">m</hi> - <hi rend="italics">b.</hi></p><p><hi rend="smallcaps">Mean</hi>-<hi rend="italics">and-Extreme Proportion,</hi> or <hi rend="italics">Extreme-and-Mean Proportion,</hi> is when a line, or any quantity is so divided, that the less part is to the greater, as the greater is to the whole.</p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Anomaly,</hi> of a planet, is an angle which is always proportional to the time of the planet's motion from thé aphelion, or perihelion, or proportional to the area described by the radius vector; that is, as the whole periodic time in one revolution of the planet, is to the time past the aphelion or perihelion, so is 360° to the Mean anomaly. See Anomaly.</p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Axis,</hi> in Optics. See <hi rend="smallcaps">Axis.</hi></p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Conjunction</hi> or <hi rend="italics">Opposition,</hi> is when the mean place of the <*>un is in conjunction, or opposition, with the mean place of the moon in the ecliptic.</p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Diameter,</hi> in Gauging, is a Mean between the diameters at the head and bung of a cask.</p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Distance,</hi> of a Planet from the Sun, is an arithmetical mean between the planet's greatest and least distances; and this is equal to the semitransverse axis of the elliptic orbit in which it moves, or to the right line drawn from the sun or focus to the extremity of the conjugate axis of the same.</p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Motion,</hi> is that by which a planet is supposed to move equably in its orbit; and it is always proportional to the time.</p><p><hi rend="smallcaps">Mean</hi> <hi rend="italics">Time,</hi> or Equal time, is that which is measured by an equable motion, as a clock; as distinguished from apparent time, arising from the unequal motion of the earth or sun.</p></div1><div1 part="N" n="MEASURE" org="uniform" sample="complete" type="entry"><head>MEASURE</head><p>, denotes any quantity, assumed as unity, or one, to which the ratio of other homogeneous or like quantities may be expressed.</p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of an Angle,</hi> is an arc of a circle described from the angular point as a centre, and intercepted between the legs or sides of the angle: and it is usual to estimate and express the Measure of the angle by the number of degrees and parts contained in that arc, of which 360 make up the whole circumference. So, the measure of the angle BAC, is the arc BC to the radius AB, or the arc <hi rend="italics">bc</hi> to the radius A<hi rend="italics">b.</hi> <figure/></p><p>Hence, a right angle is measured by a quadrant, or 90 degrees; and any angle, as BAC, is in proportion to a right angle, as the arc BC is to a quadrant, or as the degrees in BC are to 90 degrees.</p><p><hi rend="italics">Common</hi> <hi rend="smallcaps">Measure.</hi> See <hi rend="smallcaps">Common</hi> <hi rend="italics">Measure.</hi></p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Figure,</hi> or Plane Surface, is a square inch, or square foot, or square yard, &c, that is, a<cb/> square whose side is an inch, or a foot, or a yard, or some other determinate length; and this square is called the <hi rend="italics">measuring unit.</hi></p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Line,</hi> is any right line taken at pleasure, and considered as unity; as an inch, or a foot, or a yard, &c.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Measures.</hi> See <hi rend="smallcaps">Line</hi> <hi rend="italics">of Measures.</hi></p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Mass,</hi> or <hi rend="italics">Quantity of Matter,</hi> is its weight.</p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Number,</hi> is any number that divides it, without leaving a remainder. So, 2 is a Measure of 4, of 8, or of any even number; and 3 is a Measure of 6, or of 9, or of 12, &c.</p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Ratio,</hi> is its logarithm, in any system of logarithms; or it is the exponent of the power to which the ratio is equal, the exponent of some given ratio being assumed as unity. So, if the logarithm or Measure of the ratio of 10 to 1, be assumed equal to 1; then the Measure of the ratio of 100 to 1, will be 2, because 100 is = 10<hi rend="sup">2</hi>, or because 100 to 1 is in the duplicate ratio of 10 to 1; and the Measure of the ratio of 1000 to 1, will be 3, because 1000 is = 10<hi rend="sup">3</hi>, or because 1000 to 1 is triplicate of the ratio of 10 to 1.</p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Solid,</hi> is a cubic inch, or cubic foot, or cubic yard, &c; that is, a cube whose side is an inch, or a foot, or a yard, &c.</p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of a Superficies,</hi> the same as the Measure of a figure.</p><p><hi rend="smallcaps">Measure</hi> <hi rend="italics">of Velocity,</hi> is the space uniformly passed over by a moving body in a given time.</p><p><hi rend="italics">Universal</hi> or <hi rend="italics">Perpetual</hi> <hi rend="smallcaps">Measure</hi>, is a kind of Measure unalterable by time or place, to which the Measures of different ages and nations might be reduced, and by which they may be compared and estimated. Such a Measure would be very useful, if it could be attained; since, being used at all times, and in all places, a great deal of confusion and error would be avoided.</p><p>Huygens, in his Horol. Oscil. proposes, for this purpose, the length of a pendulum that should vibrate seconds, measured from the point of suspension to the point of oscillation: the 3d part of such a pendulum to be called horary foot, and to serve as a standard to which the Measure of all other feet might be referred. Thus, for instance, the proportion of the Paris foot to the horary foot, would be that of 864 to 881; because the length of 3 Paris feet is 864 half lines, and the length of a pendulum, vibrating seconds, contains 881 half lines. But this Measure, in order to its being universal, supposes that the action of gravity is the same on every part of the earth's surface, which is contrary to fact; for which reason it would really serve only for places under the same parallel of latitude: so that, if every different latitude were to have its foot equal to the 3d part of the pendulum vibrating seconds there, any latitude would still have a different length of foot. And besides, the difficulty of measuring exactly the distance between the centres of motion and oscillation are such, that hardly any two measurers would make it the same quantity.</p><p>M. Mouton, canon of Lyons, has also a treatife <hi rend="italics">De Mensura post<*>ris transmittenda.</hi></p><p>Since that time various other expedients have been proposed for establishing an universal Measure, but<pb n="90"/><cb/> hitherto without the perfect effect. In 1779, a method was proposed to the Society of Arts, &c, by a Mr. Hatton, in consequence of a premium, which had been 4 years advertised by that institution, of a gold medal, or 100 guineas, ‘for obtaining invariable standards for weights and Measures, communicable at all times and to all nations.’ Mr. Hatton's plan consisted in the application of a moveable point of suspension to one and the same pendulum, in order to produce the full and absolute effect of two pendulums, the difference of whose lengths was the intended Measure. Mr. Whitehurst much improved upon this idea, by very curious and accurate machinery, in his tract published 1787, intitled ‘An Attempt towards obtaining invariable Measures of Length, Capacity, and Weight, from the Mensuration of time, &c. Mr. Whitehurst's plan is, to obtain a Measure of the greatest length that conveniency will permit, from two pendulums whose vibrations are in the ratio of 2 to 1, and whose lengths coincide with the English standard in whole numbers. The numbers he has chosen shew great ingenuity. On a supposition that the length of a seconds pendulum, in the latitude of London, is 39.2 inches, the length of one vibrating 42 times in a minute, must be 80 inches; and of another vibrating 84 times in a minute, must be 20 inches; their difference, 60 inches or 5 feet, is his standard Measure. By his experiments, however, the difference in the lengths of the two pendulums was found to be 59.892 inches instead of 60, owing to the error in the assumed length of the seconds pendulum, 39.2 inches being greater than the truth. Mr. Whitehurst has fully accomplished his design, and shewn how an invariable standard may, at all times, be found for the same latitude. He has also ascertained a fact, as accurately as human powers feem capable of ascertaining it, of great consequence in natural philosophy. The difference between the lengths of the rods of two pendulums whose vibrations are known, is a datum from which may be derived the true length of pendulums, the spaces through which heavy bodies fall in a given time, with many other particulars relative to the doctrine of gravitation, the figure of the earth, &c, &c. The result deduced from this experiment is, that the length of a seconds pendulum, vibrating in a circular arc of 3° 20, is 39.119 inches very nearly; but vibrating in the arc of a cycloid it would be 39.136 inches; and hence, heavy bodies will fall, in the first second of their descent, 16.094 feet, or 16 feet 1 1/8 inch, very nearly.</p><p>It is said, the French philosophers have a plan in contemplation, to take for a universal Measure, the length of a whole meridian circle of the earth, and take all other Measures from sub-divisions of that; which will be a very good way.—Other projects have also been devised, but of little or no consideration.</p><div2 part="N" n="Measure" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Measure</hi></head><p>, in a legal, commercial, and popular sense, denotes a certain quantity or proportion of any thing, bought, sold, valued, or the like.</p><p>The regulation of weights and Measures ought to be universally the same throughout the nation, and indeed all nations; and they should therefore be reduced to some sixed rule or standard.</p><p>Measures are various, according to the various kinds or dimensions of the things measured. Hence arise<cb/></p><p><hi rend="italics">Lineal</hi> or <hi rend="italics">Longitudinal</hi> <hi rend="smallcaps">Measures</hi>, for lines or lengths:</p><p><hi rend="italics">Square</hi> <hi rend="smallcaps">Measures</hi>, for areas or superficies: and</p><p><hi rend="italics">Solid</hi> or <hi rend="italics">Cubic</hi> <hi rend="smallcaps">Measures</hi>, for the solid contents and capacities of bodies.</p><p>The several Meafures used in England, are as in the following Tables: <hi rend="center">1. <hi rend="italics">Englisb Long Measure.</hi></hi> Barley Corns 3 = 1 Inch 36 = 12 = 1 Foot 108 = 36 = 3 = 1 Yard 594 = 198 = 16 1/2 = 5 1/2 = 1 Pole 23760 = 7920 = 660 = 220 = 40 = 1 Furlong 190082 = 63360 = 5280 = 1760 = 320 = 8 = 1 Mile Also, 4 Inches = 1 Hand 6 Feet, or 2 yds = 1 Fathom 3 Miles = 1 League 60 Nautical or Geograph. Miles = 1 Degree or 69 1/3 Statute Miles = 1 Degree nearly 360 Degrees, or 25000 Miles nearly = the Circumference of the Earth. <hi rend="center">2. <hi rend="italics">Cloth Measure.</hi></hi> Inches 2 1/4 = 1 Nail 9 = 4 = 1 Quarter 36 = 16 = 4 = 1 Yard 27 = 12 = 3 = 1 Ell Flemish 45 = 20 = 5 = 1 Ell English 54 = 24 = 6 = 1 Ell French. <hi rend="center">3. <hi rend="italics">Square Measure.</hi></hi> Inches 144 = 1 Foot 1296 = 9 = 1 Yard 39204 = 272 1/4 = 30 1/4 = 1 Pole 1568160 = 10890 = 1210 = 40 = 1 Rood 6272640 = 43560 = 4840 = 160 = 4 = 1 Acre. <hi rend="center">4. <hi rend="italics">Solid,</hi> or <hi rend="italics">Cubical Measure.</hi></hi> Inches 1728 = 1 Foot 46656 = 27 = 1 Yard. <hi rend="center">5. <hi rend="italics">Wine Measure.</hi></hi> Pints 2 = 1 Quart 8 = 4 = 1 Gallon = 231 Cubic Inches. 336 = 168 = 42 = 1 Tierce 504 = 252 = 63 = 1 1/2 = 1 Hogshead 672 = 336 = 84 = 2 = 1 1/3 = 1 Puncheon 1008 = 504 = 126 = 3 = 2 = 1 1/3 = 1 Pipe 2016 = 1008 = 252 = 6 = 4 = 3 = 2 = 1 Tun. Also, 231 Cubic Inches = 1 Gallon 10 Gallons = 1 Anker 18 Gallons = 1 Runlet 31 1/2 Gallons = 1 Barrel.<pb n="91"/><cb/> <hi rend="center">6. <hi rend="italics">Ale and Beer Measure.</hi></hi> Pints. 2 = 1 Quart. 8 = 4 = 1 Gallon = 282 Cubic Inches. 72 = 36 = 9 = 1 Firkin. 144 = 72 = 18 = 2 = 1 Kilderkin. 288 = 144 = 36 = 4 = 2 = 1 Barrel. 432 = 216 = 54 = 6 = 3 = 1 1/2 = 1 Hogshead. 576 = 288 = 72 = 8 = 4 = 2 = 1 1/3 = 1 Puncheon. 864 = 432 = 108 = 12 = 6 = 3 = 2 = 1 1/2 = 1 Butt.</p><p>Note, The Ale gallon contains 282 cubic inches.<cb/> <hi rend="center">7. <hi rend="italics">Dry Measure.</hi></hi> Pints. 8 = 1 Gallon = 268 4/5 Cubic Inches. 16 = 2 = 1 Peck. 64 = 8 = 4 = 1 Bushel. 256 = 32 = 16 = 4 = 1 Coom. 512 = 64 = 32 = 8 = 2 = 1 Quarter. 2560 = 320 = 160 = 40 = 10 = 5 = 1 Wey. 5120 = 640 = 320 = 80 = 20 = 10 = 2 = 1 Last. Also, 268 4/5 Cubic Inches = 1 Gallon. and 36 Bushels of Coals = 1 Chaldron. <table rend="border width=100%"><row role="data"><cell cols="1" rows="1" rend="colspan=8 align=center" role="data">8. <hi rend="italics">Proportions of the Long Measures of several Nations to the English Foot.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Thousandth Parts.</cell><cell cols="1" rows="1" role="data">Inches.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Thousandth Parts.</cell><cell cols="1" rows="1" role="data">Inches.</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">English</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" rend="align=right" role="data">12.000</cell><cell cols="1" rows="1" role="data">Amsterdam</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2269</cell><cell cols="1" rows="1" rend="align=right" role="data">27.228</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Paris</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1065 3/4</cell><cell cols="1" rows="1" rend="align=right" role="data">12.792</cell><cell cols="1" rows="1" role="data">Antwerp</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2273</cell><cell cols="1" rows="1" rend="align=right" role="data">27.276</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Rynland, or Leyden</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1033</cell><cell cols="1" rows="1" rend="align=right" role="data">12.396</cell><cell cols="1" rows="1" role="data">Rynland, or Leyden</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2260</cell><cell cols="1" rows="1" rend="align=right" role="data">27.120</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Amsterdam</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">942</cell><cell cols="1" rows="1" rend="align=right" role="data">11.304</cell><cell cols="1" rows="1" role="data">Frankfort</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">1826</cell><cell cols="1" rows="1" rend="align=right" role="data">21.912</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Brill</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1103</cell><cell cols="1" rows="1" rend="align=right" role="data">13.236</cell><cell cols="1" rows="1" role="data">Hamburgh</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">1905</cell><cell cols="1" rows="1" rend="align=right" role="data">22.860</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Antwerp</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">946</cell><cell cols="1" rows="1" rend="align=right" role="data">11.352</cell><cell cols="1" rows="1" role="data">Leipsic</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2260</cell><cell cols="1" rows="1" rend="align=right" role="data">27.120</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Dort</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1184</cell><cell cols="1" rows="1" rend="align=right" role="data">14.208</cell><cell cols="1" rows="1" role="data">Lubeck</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">1908</cell><cell cols="1" rows="1" rend="align=right" role="data">22.896</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Lorrain</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">958</cell><cell cols="1" rows="1" rend="align=right" role="data">11.496</cell><cell cols="1" rows="1" role="data">Noremburgh</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2227</cell><cell cols="1" rows="1" rend="align=right" role="data">26.724</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Mechlin</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">919</cell><cell cols="1" rows="1" rend="align=right" role="data">11.028</cell><cell cols="1" rows="1" role="data">Bavaria</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">954</cell><cell cols="1" rows="1" rend="align=right" role="data">11.448</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Middleburgh</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">991</cell><cell cols="1" rows="1" rend="align=right" role="data">11.892</cell><cell cols="1" rows="1" role="data">Vienna</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">1053</cell><cell cols="1" rows="1" rend="align=right" role="data">12.636</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Strasburgh</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">920</cell><cell cols="1" rows="1" rend="align=right" role="data">11.040</cell><cell cols="1" rows="1" role="data">Bononia</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2147</cell><cell cols="1" rows="1" rend="align=right" role="data">25.764</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Bremen</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">964</cell><cell cols="1" rows="1" rend="align=right" role="data">11.568</cell><cell cols="1" rows="1" role="data">Dantzic</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">1903</cell><cell cols="1" rows="1" rend="align=right" role="data">22.836</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Cologn</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">954</cell><cell cols="1" rows="1" rend="align=right" role="data">11.448</cell><cell cols="1" rows="1" role="data">Florence</cell><cell cols="1" rows="1" rend="align=right" role="data">Brace or ell</cell><cell cols="1" rows="1" rend="align=right" role="data">1913</cell><cell cols="1" rows="1" rend="align=right" role="data">22.956</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Frankfort ad Mœnum</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">948</cell><cell cols="1" rows="1" rend="align=right" role="data">11.376</cell><cell cols="1" rows="1" role="data">Spanish, or Castile</cell><cell cols="1" rows="1" rend="align=right" role="data">palm</cell><cell cols="1" rows="1" rend="align=right" role="data">751</cell><cell cols="1" rows="1" rend="align=right" role="data">9.012</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Spanish</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1001</cell><cell cols="1" rows="1" rend="align=right" role="data">12.012</cell><cell cols="1" rows="1" role="data">Spanish</cell><cell cols="1" rows="1" rend="align=right" role="data">vare</cell><cell cols="1" rows="1" rend="align=right" role="data">3004</cell><cell cols="1" rows="1" rend="align=right" role="data">36.040</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Toledo</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">899</cell><cell cols="1" rows="1" rend="align=right" role="data">10.788</cell><cell cols="1" rows="1" role="data">Lisbon</cell><cell cols="1" rows="1" rend="align=right" role="data">vare</cell><cell cols="1" rows="1" rend="align=right" role="data">2750</cell><cell cols="1" rows="1" rend="align=right" role="data">33.000</cell></row><row role="data"><cell cols="1" rows="1" rend="width=20%" role="data">Roman</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">967</cell><cell cols="1" rows="1" rend="align=right" role="data">11.604</cell><cell cols="1" rows="1" role="data">Gibraltar</cell><cell cols="1" rows="1" rend="align=right" role="data">vare</cell><cell cols="1" rows="1" rend="align=right" role="data">2760</cell><cell cols="1" rows="1" rend="align=right" role="data">33.120</cell></row><row role="data"><cell cols="1" rows="1" rend="width=40% rowspan=2 align=left" role="data">On the monument of Cestius Statilius<hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">972</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">11.664</cell><cell cols="1" rows="1" role="data">Toledo</cell><cell cols="1" rows="1" rend="align=right" role="data">vare</cell><cell cols="1" rows="1" rend="align=right" role="data">2685</cell><cell cols="1" rows="1" rend="align=right" role="data">32.220</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">palm</cell><cell cols="1" rows="1" rend="align=right" role="data">861</cell><cell cols="1" rows="1" rend="align=right" role="data">10.322</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bononia</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1204</cell><cell cols="1" rows="1" rend="align=right" role="data">14.448</cell><cell cols="1" rows="1" role="data">Naples</cell><cell cols="1" rows="1" rend="align=right" role="data">brace</cell><cell cols="1" rows="1" rend="align=right" role="data">2100</cell><cell cols="1" rows="1" rend="align=right" role="data">25.200</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mantua</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1569</cell><cell cols="1" rows="1" rend="align=right" role="data">18.838</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6880</cell><cell cols="1" rows="1" rend="align=right" role="data">82.560</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venice</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1162</cell><cell cols="1" rows="1" rend="align=right" role="data">13.944</cell><cell cols="1" rows="1" role="data">Genoa</cell><cell cols="1" rows="1" rend="align=right" role="data">palm</cell><cell cols="1" rows="1" rend="align=right" role="data">830</cell><cell cols="1" rows="1" rend="align=right" role="data">9.960</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dantzic</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">944</cell><cell cols="1" rows="1" rend="align=right" role="data">11.328</cell><cell cols="1" rows="1" role="data">Milan</cell><cell cols="1" rows="1" rend="align=right" role="data">calamus</cell><cell cols="1" rows="1" rend="align=right" role="data">6544</cell><cell cols="1" rows="1" rend="align=right" role="data">78.528</cell></row><row role="data"><cell cols="1" rows="1" role="data">Copenhagen</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">965</cell><cell cols="1" rows="1" rend="align=right" role="data">11.580</cell><cell cols="1" rows="1" role="data">Parma</cell><cell cols="1" rows="1" rend="align=right" role="data">cubit</cell><cell cols="1" rows="1" rend="align=right" role="data">1866</cell><cell cols="1" rows="1" rend="align=right" role="data">22.392</cell></row><row role="data"><cell cols="1" rows="1" role="data">Prague</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1026</cell><cell cols="1" rows="1" rend="align=right" role="data">12.312</cell><cell cols="1" rows="1" role="data">China</cell><cell cols="1" rows="1" rend="align=right" role="data">cubit</cell><cell cols="1" rows="1" rend="align=right" role="data">1016</cell><cell cols="1" rows="1" rend="align=right" role="data">12.192</cell></row><row role="data"><cell cols="1" rows="1" role="data">Riga</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1831</cell><cell cols="1" rows="1" rend="align=right" role="data">21.972</cell><cell cols="1" rows="1" role="data">Cairo</cell><cell cols="1" rows="1" rend="align=right" role="data">cubit</cell><cell cols="1" rows="1" rend="align=right" role="data">1824</cell><cell cols="1" rows="1" rend="align=right" role="data">21.888</cell></row><row role="data"><cell cols="1" rows="1" role="data">Turin</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1062</cell><cell cols="1" rows="1" rend="align=right" role="data">12.744</cell><cell cols="1" rows="1" role="data">Old Babylonian</cell><cell cols="1" rows="1" rend="align=right" role="data">cubit</cell><cell cols="1" rows="1" rend="align=right" role="data">1520</cell><cell cols="1" rows="1" rend="align=right" role="data">18.240</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Greek</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">1007</cell><cell cols="1" rows="1" rend="align=right" role="data">12.084</cell><cell cols="1" rows="1" role="data">Old Greek</cell><cell cols="1" rows="1" rend="align=right" role="data">cubit</cell><cell cols="1" rows="1" rend="align=right" role="data">1511</cell><cell cols="1" rows="1" rend="align=right" role="data">18.132</cell></row><row role="data"><cell cols="1" rows="1" role="data">Old Roman</cell><cell cols="1" rows="1" rend="align=right" role="data">foot</cell><cell cols="1" rows="1" rend="align=right" role="data">970</cell><cell cols="1" rows="1" rend="align=right" role="data">11.640</cell><cell cols="1" rows="1" role="data">Old Roman</cell><cell cols="1" rows="1" rend="align=right" role="data">cubit</cell><cell cols="1" rows="1" rend="align=right" role="data">1458</cell><cell cols="1" rows="1" rend="align=right" role="data">17.496</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lyons</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">3967</cell><cell cols="1" rows="1" rend="align=right" role="data">47.604</cell><cell cols="1" rows="1" role="data">Turkish</cell><cell cols="1" rows="1" rend="align=right" role="data">pike</cell><cell cols="1" rows="1" rend="align=right" role="data">2200</cell><cell cols="1" rows="1" rend="align=right" role="data">26.400</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bologna</cell><cell cols="1" rows="1" rend="align=right" role="data">ell</cell><cell cols="1" rows="1" rend="align=right" role="data">2076</cell><cell cols="1" rows="1" rend="align=right" role="data">24.912</cell><cell cols="1" rows="1" role="data">Persian</cell><cell cols="1" rows="1" rend="align=right" role="data">arash</cell><cell cols="1" rows="1" rend="align=right" role="data">3197</cell><cell cols="1" rows="1" rend="align=right" role="data">38.364</cell></row></table><cb/></p></div2></div1><div1 part="N" n="MEASURING" org="uniform" sample="complete" type="entry"><head>MEASURING</head><p>, the same as <hi rend="smallcaps">Mensuration</hi>, which see.</p></div1><div1 part="N" n="MECHANICS" org="uniform" sample="complete" type="entry"><head>MECHANICS</head><p>, a mixed mathematical science, that treats of forces, motion, and moving powers, with their effects in machines, &c. The science of Mechanics is distinguished, by Sir Isaac Newton, into Prac-<cb/> tical and Rational: the former treats of the Mechanical Powers, and of their various combinations; the latter, or Rational Mechanics, comprehends the whole theory and doctrine of forces, with the motions and effects produced by them.</p><p>That part of Mechanics, which treats of the weight,<pb n="92"/><cb/> gravity, and equilibrium of bodies and powers, is called Statics; as distinguished from that part which considers the Mechanical powers, and their application, which is properly called Mechanics.</p><p>Some of the principles of Statics were established by Archimedes, in his Treatise on the Centre of Gravity of Plane Figures: besides which, little more upon Mechanics is to be found in the writings of the Ancients, except what is contained in the 8th book of Pappus's Mathematical Collections, concerning the five Mechanical Powers. Galileo laid the best foundation of Mechanics, when he investigated the descent of heavy bodies; and since his time, by the assistance of the new methods of computation, a great progress has been made, especially by Newton, in his Principia, which is a general treatise on Rational and Physical Mechanics, in its largest extent. Other writers on this science, or some branch of it, are, Guido Ubaldus, in his Liber Mechanicorum; Torricelli, Libri de Motu Gravium naturaliter Descendentium & Projectorum; Balianus, Tractatus de Motu naturali Gravium; Huygens, Horologium Oscillatorium, and Tractatus de Motu Corporum ex Percussione; Leibnitz, Resistentia Solidorum in Acta Eruditor. an. 1684; Guldinus, De Centro Gravitatis; Wallis, Tractatus de Mechanica; Varignon, Projet d'une Nouvelle Mechanique, and his papers in the Memoir. Acad. an. 1702; Borelli, Tractatus De Vi Percussionis, De Motionibus Naturalibus a Gravitate pendentibus, and De Motu Animalium; De Chales, Treatise on Motion; Pardies, Discourse of Local Motion; Parent, Elements of Mechanics and Physics; Casatus, Mechanica; Oughtred, Mechanical Institutions; Rohault, Tractatus de Mechanica; Lamy, Mechanique; Keill, Introduction to true Philosophy; De la Hire, Mechanique; Mariotte, Traité du Choc des Corps; Ditton, Laws of Motion; Herman, Phoronomia; Gravesande, Physics: Euler, Tractatus de Motu; Musschenbroek, Physics; Bossu, Mechanique; Desaguliers, Mechanics; Rowning, Natural Philosophy; Emerson, Mechanics; Parkinson, Mechanics; La Grange, Mechanique Analytique; Nicholson, Introduction to Natural Philosophy; Enfield, Institutes of Natural Philosophy, &c, &c. As to the Deseription of Machines, see Strada, Zeisingius, Besson, Augustine de Ramellis, Boetler, Leopold, Sturmy, Perrault, Limberg, Emerson, Royal Academy of Scicnces, &c.</p><p>In treating of machines, we should consider the weight that is to be raised, the power by which it is to be raised, and the instrument or engine by which this effect is to be produced. And, in treating of these, there are two principal problems that present themselves: the first is, to determine the proportion which the power and weight ought to have to each other, that they may just be in equilibrio; the second is, to determine what ought to be the proportion between the power and weight, that a machine may produce the greatest effect in a given time. All writers on Mechanics treat on the first of these problems, but few have considered the second, though not less useful than the other.</p><p>As to the first problem, this general rule holds in all<cb/> powers; namely, that when the power and weight are reciprocally proportional to the distances of the directions in which they act, from the centre of motion; or when the product of the power by the distance of its direction, is equal to the product of the weight by the distance of its direction; this is the case in which the power and weight sustain each other, and are in equilibrio; so that the one would not prevail over the other, if the engine were at rest; and if it were in motion, it would continue to proceed uniformly, if it were not for the friction of its parts, and other resistances. And, in general, the effect of any power, or force, is as the product of that force multiplied by the distance of its direction from the centre of motion, o<*> the product of the power and its velocity when in motion, since this velocity is proportional to the distance from that centre.</p><p>The second general problem in Mechanics, is, to determine the proportion between the power and weight, so that when the power prevails, and the machine is in motion, the greatest effect possible may be produced by it in a given time. It is manifest, that this is an enquiry of the greatest importance, though few have treated of it. When the power is only a little greater than what is sufficient to sustain the weight, the motion usually is too slow; and though a greater weight be raised in this case, it is not sufficient to compensate for the loss of time. On the other hand, when the power is much greater than what is sufficient to sustain the weight, this is raised in less time; but it may happen that this is not sufficient to compensate for the loss arising from the smallness of the load. It ought therefore to be determined when the product of the weight multiplied by its velocity, is the greatest possible; for this product measures the effect of the engine in a given time, which is always the greater in proportion both as the weight is greater, and as its velocity is greater. For some calculations on this problem, see Maclaurin's Account of Newton's Discoveries, p. 171, &c; also his Fluxions, art. 908 &c. And, for the various properties in Mechanics, see the several terms <hi rend="smallcaps">Motion, Force, Mechanical Powers, Lever</hi>, &c.</p></div1><div1 part="N" n="MECHANIC" org="uniform" sample="complete" type="entry"><head>MECHANIC</head><p>, or <hi rend="smallcaps">Mechanical</hi>, something relating to Mechanics, or regulated by the nature and laws of motion.</p><p><hi rend="smallcaps">Mechanical</hi> is also used in Mathematics, to signify a construction or proof of some problem, not done in an accurate and geometrical manner, but coarsely and unartfully, or by the assistance of instruments; as are most problems relating to the duplicature of the cube, and the quadrature of the circle.</p><p><hi rend="smallcaps">Mechanical</hi> <hi rend="italics">Assections,</hi> such properties in matter, as result from their figure, bulk, and motion.</p><p><hi rend="smallcaps">Mechanical</hi> <hi rend="italics">Causes,</hi> are such as are founded on Mechanical Affections.</p><p><hi rend="smallcaps">Mechanical</hi> <hi rend="italics">Curve,</hi> called also <hi rend="italics">Transcendental,</hi> is one whose nature cannot be expressed by a finite Algebraical equation.</p><p><hi rend="smallcaps">Mechanical</hi> <hi rend="italics">Philosophy,</hi> also called the <hi rend="italics">Corpuscular Philosophy,</hi> is that which explains the phenomena of nature, and the operations of corporeal things, on the principles of Mechanics; viz, the motion, gravity, figure, arrangement, disposition, greatness,<pb n="93"/><cb/> or smallness of the parts which compose natural bodies.</p><p><hi rend="smallcaps">Mechanical</hi> <hi rend="italics">Solution,</hi> of a Problem, is either when the thing is done by repeated trials, or when the lines used in the solution are not truly geometrical, or by organical construction.</p><p><hi rend="smallcaps">Mechanical</hi> <hi rend="italics">Powers,</hi> are certain simple machines which are used for raising greater weights, or overcoming greater resistances than could be effected by the natural strength without them.</p><p>These simple machines are usually accounted fix in number, viz, the Lever, the Wheel and Axle, or Axis in Peritrochio, the Pulley, the Inclined Plane, the Wedge, and the Screw. Of the various combinations of these simple powers do all engines, or compound machines, consist: and in treating of them, so as to settle their theory and properties, they are considered as mathematically exact, or void of weight and thickness, and moving without friction. See the properties and demonstrations of each of these under the several words <hi rend="smallcaps">Lever</hi>, &c. To which may be added the following general observations on them all, in a connective way.</p><p>1. A <hi rend="italics">Lever,</hi> the most fimple of all the mechanic powers, is an engine chiefly used to raise large weights to small heights; such as a handspike, when of wood; and a crow, when of iron. In theory, a lever is considered as an inslexible line, like the beam of a balance, and subject to the fame proportions; only that the power applied to it, is commonly an animal power; and from the different ways of ufing it, or applying it, it is called a lever of the first, second, or third kind: viz, of the 1st kind, when the weight is on one side of the prop, and the power on the other; of the 2d kind, when the weight is between the prop and the power; and of the 3d kind, when the power is between the prop and the weight.</p><p>Many of the instruments in common use, are levers of one of the three kinds; thus, pincers, sheers, forceps, snuffers, and such like, are compounded of two levers of the first kind; for the joint about which they move, is the fulcrum, or centre of motion; the power is applied to the handles, to press them together; and the weight is the body which they pinch or cut. The outting knives used by druggists, patten-makers, blockmakers, and some other trades, are levers of the 2d kind: for the knife is fixed by a ring at one end, which makes the fulcrum, or fixed point; the other <*>nd is moved by the hand, or power; and the body to be cut, or the resistance to be overcome, is the weight. Doors are levers of the 2d kind; the hinges being the centre of motion; the hand applied to the lock is the power; while the door or weight lies between them. A pair of bellows consists of two levers of the 2d kind; the centre of motion is where the ends of the boards are fixed near the pipe; the power is applied at the handles; and the air pressed out from between the boards, by its resistance, acts against the middle of the boards like a weight. The oars of a boat are levers of the 2d kind: the fixed point is the blade of the oar in the water; the power is the hand acting at the other end; and the weight to be moved is the boat. And the same of the rudder of a vessel. Spring sheers and tongs<cb/> are levers of the 3d kind; where the centre of motion is at the bow-spring at one end; the weight or resistance is acted on by the other end; and the hand or power is applied between the ends. A ladder reared by a man against a wall, is a lever of the 3d kind: and so are also almost all the bones and muscles of animals.</p><p>In all levers, the effect of any power or weight, is both proportional to that power or weight, and also to its distance from the centre of motion. And hence it is that, in raising great weights by a lever, we chuse the longest levers; and also rest it upon a point as far from the hand or power, and as near to the weight, as possible. Hence also there will be an equilibrium between the power and weight, when those two products are equal, viz, the power multiplied by its distance, equal to the weight multiplied by its distance; when, also, the weight and power are to each other reciprocally as their distances from the prop or fixed point.</p><p>2. The Axis in Peritrochio, or Wheel and Axle, is a simple engine consisting of a wheel fixed upon the end of an axle, so that they both turn round together in the same time. This engine may be referred to the lever: for the centre of the axis, or wheel, is the fixed point; the radius of the wheel is the distance of the power, acting at the circumference of the wheel, from that point; and the radius of the axle is the distance of the weight from the same point. Hence the effect of the power, independent of its own natural intensity, is as the radius of the wheel; and the effect of the weight is as the radius of the axle: so that the two will be in equilibrio, when the two products are equal, which are made by multiplying each of these, the weight and power, by the radius, or distance at which it acts; and then also, the weight and power are reciprocally proportional to those radii.</p><p>In practice, the thickness of the rope, that winds upon the axle, and to which the weight is fastened, is to be considered: which is done, by adding half its thickness to the radius of the axis, for its distance from the fixed point, when there is only one fold of rope upon the axle; or as many times the thickness as there are folds, wanting only one half when there are several folds of the rope, one over another: which is the reafon that more power must be applied when the axis is thus thickened; as often happens in drawing water from a deep and narrow well, over which a long axle cannot be placed.</p><p>If the rope to which the power is fastened, be successively applied to different wheels, whose diameters are larger and larger; the axis will be turned with still more and more ease, unless the intensity of the power be diminished in the same proportion; and if so, the axis will always be drawn with the same strength by a power continually diminishing. This is practised in spring clocks and watches; where the spiral spring, which is strongest in its action when first wound up, draws the fuzee, or continued axis in peritrochio, first by the smaller wheels, and as it unbends and becomes weak, draws at the larger wheels, in such manner that the watch work is always carried round with the same force.</p><p>As a very small axis would be too weak for very great weights, or a large wheel would be expensive as<pb n="94"/><cb/> well as cumbersome, and take more room than perhaps can be spared for it; therefore, that the action of the power may be increased, without incurring either of those inconveniences, a compound Axis in Peritrochio is used, which is effected by combining wheels and axles by means of pinions, or small wheels, upon the axles, the teeth of which take hold of teeth made in the large wheels; as is seen in clocks, jacks, and other compound machines. And in such a combination of wheels and axles, the effect of the power is increased in the ratio of the continual product of all the axles, or small wheels, to that of all the large ones. Thus, if there be two small wheels and an axle, turning three large wheels; the axle being 2 inches diameter, and each of the small wheels 4 inches, while the large ones are 2 feet or 24 inches diameter; then is the continual product of the small diameters, and is that of the large ones; therefore 13824 to 32, or 432 to 1, is the ratio in which the power is increased: and if the power be a man, whose natural strength is equal, suppose, to 150 pounds weight, then , or 28 ton 18 cwt 64lb, is the weight he would be able to balance, suspended about the axle.</p><p>3. <hi rend="italics">A Single Pulley,</hi> is a small wheel, moveable round an axis, called its centre pin; which of itself is not properly one of the mechanical powers, because it produces no gain of power; for, as the weight hangs by one end of the cord that passes over the pulley, and the power acts at the other end of the same, these act at equal distances from the centre or axis of motion, and consequently the power is equal to the weight when in equilibrio. So that the chief use of the single pulley is to change the direction of the power from upwards to downwards, &c, and to convey bodies to a great height or distance, without a person moving from his place.</p><p>But by combining several single pulleys together, a considerable gain of power is made, and that in proportion to the additional number of ropes made to pass over them; and yet it enjoys at the same time the properties of a single pulley, by changing the direction of the action in any manner.</p><p>4. <hi rend="italics">The Inclined Plane,</hi> is made by planks, bars, or beams, laid aslope; by which, large and heavy bodies may be more easily raised or lowered, by sliding them up or down the plane; and the gain in power is in proportion as the length of the plane to its height, or as radius to the sine of the angle of inclination of the plane with the horizon.</p><p>In drawing a weight up an inclined plane, the power acts to the greatest advantage, when its direction is parallel to the plane.</p><p>5. <hi rend="italics">The Wedge,</hi> which resembles a double inclined plane, is very useful to drive in below very heavy weights to raise them but a small height, also in cleaving and splitting blocks of wood, and stone &c; and the power gained, is in proportion of the slant side to half the thickness of the back. So that, if the back of a wedge be 2 inches thick, and the side 20 inches long, any weight pressing on the back will balance 20 times as much acting on the side. But the great advantage of a wedge lies in its being urged, not<cb/> by pressure, but usually by percussion, as the blow of a hammer or mallet; by which means a wedge may be driven in below, and so be made to lift, almost any the greatest weight, as the largest ship, by a man striking the back of a wedge with a mallet.</p><p>To the wedge may be referred the axe or hatchet, the teeth of saws, the chisel, the augur, the spade and shovel, knives and swords of all kinds, as also the bodkin and needle, and in a word all sorts of instruments which, beginning from edges or points, become gradually thicker as they lengthen; the manner in which the power is applied to such instruments, being different according to their different shapes, and the various uses for which they have been contrived.</p><p>6. <hi rend="italics">The Screw,</hi> is a kind of perpetual or endless Inclined Plane; the power of which is still farther assisted by the addition of a handle or lever, where the power acts; so that the gain in power, is in the proportion of the circumference described or passed through by the power, to the distance between thread and thread in the screw.</p><p>The uses to which the screw is applied, are various; as, the pressing of bodies close together; such as the press for napkins, for bookbinders, for packers, hotpressers, &c.</p><p>In the screw, and the wedge, the power has to overcome both the weight, and also a very great friction in those machines; such indeed as amounts sometimes to as much as the weight to be raised, or more. But then this friction is of use in retaining the weight and machine in its place, even after the power is taken off.</p><p>If machines or engines could be made without friction, the least degree of power added to that which balances the weight, would be sufficient to raise it. In the lever, the friction is little or nothing; in the wheel and axle, it is but small; in pulleys, it is very considerable; and in the inclined plane, wedge, and screw, it is very great.</p><p>It is a general property in all the Mechanic powers, that when the weight and power are regulated so as to balance each other, in every one of these machines, if they be then put in motion, the power and weight will be to each other reciprocally as the velocities of their motion, or the power is to the weight as the velocity of the weight is to the velocity of the power; so that their two momenta are equal, viz, the product of the power multiplied by its velocity, equal to the product of the weight multiplied by its velocity. And hence too, universally, what is gained in power, is lost in time; for the weight moves as much slower as the power is smaller.</p><p>Hence also it is plain, that the force of the power is not at all increased by engines; only the velocity of the weight, either in lifting or drawing, is so diminished by the application of the instrument, as that the momentum of the weight is not greater than the force of the power. Thus, for instance, if any force can raise a pound weight with a given velocity, it is impossible by any engine to raise 2 pound weight with the same velocity: but by an engine it may be made to raise 2 pound weight with half the velocity, or even 1000 times the weight with the 1000th part of the velocity.<pb n="95"/><cb/></p><p>See Maclaurin's Account of Newton's Philos. Difcov. book 2, chap. 3; Hamilton's Philos. Ess. 1; Philos. Trans. 53, pa. 116; or Landen's Memoirs, vol. 1, pa. 1.</p></div1><div1 part="N" n="MECHANISM" org="uniform" sample="complete" type="entry"><head>MECHANISM</head><p>, either the construction or the machinery employed in any thing; as the Mechanism of the barometer, of the microscope, &c.</p></div1><div1 part="N" n="MEDIUM" org="uniform" sample="complete" type="entry"><head>MEDIUM</head><p>, the same as mean, either arithmetical, geometrical, or harmonical.</p><p><hi rend="smallcaps">Medium</hi> denotes also that space, or region, or fluid, &c, through which a body passes in its motion towards any point. Thus, the air, or atmosphere, is the medium in which birds and beasts live and move, and in which a projectile moves; water is the medium in which fishes move; and æther is a supposed subtile Medium in which the planets move. Glass is also called a Medium, being that through which the rays of light move and pass.</p><p>Mediums resist the motion of bodies moving through them, in proportion to their density or specific gravity.</p><p><hi rend="italics">Subtile</hi> or <hi rend="italics">Ætherial</hi> <hi rend="smallcaps">Medium</hi>, is an universal one whose existence is by Newton rendered probable. He makes it universal; and vastly more rare, subtile, elastic, and active than air; and by that means freely permeating the pores and interstices of all other Mediums, and diffufing itself through the whole creation. By the intervention of this subtile Medium he thinks it is that most of the great phenomena of nature are effected. See Æ<hi rend="smallcaps">THER.</hi></p><p>This Medium it would seem he has recourse to, as the first and most remote physical spring, and the ultimate of all natural causes. By the vibrations of this Medium, he supposes that heat is propagated from lucid bodies; as also the intenseness of heat increased and preserved in hot bodies, and from them communicated to cold ones.</p><p>By this Medium, he supposes that light is reflected, inflected, refracted, and put alternately into fits of easy reflection and transmission; which effects he also elsewhere ascribes to the power of attraction; so that it would seem, this Medium is the source and cause even of attraction itself.</p><p>Again, this Medium being much rarer within the heavenly bodies, than in the heavenly spaces, and growing denser as it recedes farther from them, he supposes this is the cause of the gravitation of these bodies towards each other, and of the parts towards the bodies.</p><p>Again, from the vibrations of this same Medium, excited in the bottom of the eye by the rays of light, and thence propagated through the capillaments of the optic nerves into the sensorium, he supposes that vision is performed: and so likewise hearing, from the vibrations of this or some other Medium, excited in the auditory nerves by the tremors of the air, and propagated through the capillaments of those nerves into the sensorium: and so of the other senses.</p><p>And again, he conceives that muscular motion is performed by the vibrations of the same Medium, excited in the brain at the command of the will, and thence propagated through the capillaments of the nerves into the muscles; and thus contracting and dilating them.<cb/></p><p>The elaftic force of this Medium, he shews, must be prodigiously great. Light moves at the rate of considerably more than 10 millions of miles in a minute; yet the vibrations and pulsations of this Medium, to cause the fits of easy reflection and transmission, must be swifter than light, which is yet 7 hundred thousand times swifter than sound. The elastic force of this Medium, therefore, in proportion to its density, must be above 490000 million of times greater than the elastic force of the air, in proportion to its density; the velocities and pulses of the elastic Mediums being in a subduplicate ratio of the elasticities, and the rarities of the Mediums, taken together. And thus may it be conceived that the vibration of this Medium is the cause also of the elasticity of bodies.</p><p>Farther, the particles of this Medium being supposed indefinitely small, even smaller than those of light; if they be likewise supposed, like our air, endued with a rcpelling power, by which they recede from each other, the smallness of the particles may exceedingly contribute to the increase of the repelling power, and consequently to that of the elasticity and rarity of the Medium; by that means fitting it for the free transmission of light, and the frce motions of the heavenly bodies. In this Medium may the planets and comets roll without any considerable refistance. If it be 700,000 times more elastic, and as many times rarer, than air, its resistance will be above 600 million times less than that of water; a resistance that would cause no sensible alteration in the motion of the planets in ten thousand years.</p><p>MEGAMETER. See <hi rend="smallcaps">Micrometer.</hi></p></div1><div1 part="N" n="MEIBOMIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MEIBOMIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Marcus</hi></foreName>)</persName></head><p>, a very learned person of the 17th century, of a family in Germany which had long been famous for learned men. He devoted himself to literature and criticism, but particularly to the learning of the Ancients; as their music, the structure of their galleys, &c. In 1652 he published a collection of seven Greek authors, who had written upon Ancient Music, to which he added a Latin version by himself. This work he dedicated to queen Christina of Sweden; in consequence of which he received an invitation to that Princess's court, like several other learned men, which he accepted. The queen engaged him one day to sing an air of ancient music, while a person danced the Greek dances to the sound of his voice; and the immoderate mirth which this occasioned in the spectators, so covered him with ridicule, and disgusted him so vehemently, that he abruptly left the court of Sweden immediately, after heartily battering with his fists the face of Bourdelot, the favourite physician and buffon to the queen, who had persuaded her to exhibit that spectacle.</p><p>Meibomius pretended that the Hebrew copy of the Bible was full of errors, and undertook to correct them by means of a metre, which he fancied he had discovered in those ancient writings; but this it seems drew upon him no small raillery from the Learned. Nevertheless, besides the work above mentioned, he produced several others, which shewed him to be a good scholar; witness his Notes upon Diogenes Laertius in Menage's edition; his <hi rend="italics">Liber de Fabrica Triremium,</hi> 1671, in which he thinks he discovered the<pb n="96"/><cb/> method in which the Ancients disposed their ban<*>s of oars; his edition of the Ancient Greek Mythologists; and his Dialogues on Proportions, a curious work, in which the interlocutors, or persons represented as speaking, are Euclid, Archimedes, Apollonius, Pappus, Eutocius, Theo, and Hermotimus. This last work was opposed by Langius, and by Dr. Wallis, in a considerable Tract, printed in the first volume of his works.</p></div1><div1 part="N" n="MELODY" org="uniform" sample="complete" type="entry"><head>MELODY</head><p>, is the agreeable effect of different musical sounds, ranged or disposed in a proper succession, being the effect only of one single part, voice, or instrument; by which it is distinguished from harmony, which properly results from the union of two or more musical sounds heard together.</p></div1><div1 part="N" n="MENISCUS" org="uniform" sample="complete" type="entry"><head>MENISCUS</head><p>, a lens or glass, convex on one side, and concave on the other. Sometimes also called a Lune or Lunula. See its figure under the article <hi rend="smallcaps">Lens.</hi></p><p><hi rend="italics">To find the Focus of a Meniscus,</hi> the rule is, as the difference between the diameters of the convexity and concavity, is to either of them, so is the other diameter, to the focal length, or distance of the focus from the Meniscus. So that, having given the diameter of the convexity, it is easy to find that of the concavity, so as to remove the focus to any proposed distance from the Meniscus. For, if D and <hi rend="italics">d</hi> be the diameters of the two sides, and <hi rend="italics">f</hi> the focal distance; then since, by the rule , therefore , or .</p><p>Hence, if D the diameter of the concavity be double to <hi rend="italics">d</hi> that of the convexity, <hi rend="italics">f</hi> will be equal to D, or the focal distance equal to the diameter; and therefore the Meniscus will be equivalent to a planoconvex lens.</p><p>Again, if D = 3<hi rend="italics">d,</hi> or the diameter of the concavity triple to that of the convexity, then will <hi rend="italics">f</hi> = (1/2)D, or the focal distance equal to the radius of concavity; and therefore the Meniscus will be equivalent to a lens equally convex on either side.</p><p>But if D = 5<hi rend="italics">d,</hi> then will <hi rend="italics">f</hi> = (1/4)D; and therefore the Menlscus will be equivalent to a sphere.</p><p>Lastly, if D = <hi rend="italics">d,</hi> then will <hi rend="italics">f</hi> be infinite; and therefore a ray falling parallel to the axis, will still continue parallel to it after refraction.</p></div1><div1 part="N" n="MENSTRUUM" org="uniform" sample="complete" type="entry"><head>MENSTRUUM</head><p>, <hi rend="smallcaps">Solvent</hi>, or <hi rend="smallcaps">Dissolvent</hi>, any fluid that will dissolve hard bodies, or separate their parts. Sir Isaac Newton accounts for the action of Menstruums from the acids with which they are impregnated; the particles of acids being endued with a strong attractive force, in which their activity consists, and by virtue of which they dissolve bodies. By this attraction they gather together about the particles of bodies, whether metallic, stony, or the like, and adhere very closely to them, so as scarce to be separated from them by distillation, or sublimation. Thus strongly attracting, and gathering together on all sides, they raise, disjoin, and shake asunder the particles of bodies, i. e. they dissolve them; and by the attractive power with which they rush against the particles of the bodies, they move the fluid, and so excite heat, shaking some of the particles to that degree, as to convert them into air, and so generating bubbles.<cb/></p><p>Dr. Keill has given the theory or foundation of the action of Menstruums, in several propositions. See <hi rend="smallcaps">Attraction.</hi> From those propositions are perceived the reasons of the different effects of different Menstruums; why some bodies, as metals, dissolve in a saline Menstruum; others again, as resins, in a sulphureous one; &c<*> particularly why silver dissolves in aqua fortis, and gold only in aqua regis; all the varieties of which are accountable for, from the different degrees of cohesion, or attraction in the parts of the body to be dissolved, the different diameters and figures of its pores, the different degrees of attraction in the Menstruum, and the different diameters and sigures of its parts.</p></div1><div1 part="N" n="MENSURABILITY" org="uniform" sample="complete" type="entry"><head>MENSURABILITY</head><p>, the fitness of a body for being applied, or conformable to a certain measure.</p></div1><div1 part="N" n="MENSURATION" org="uniform" sample="complete" type="entry"><head>MENSURATION</head><p>, the act, or art, of measuring sigured extension and bodies; or of finding the dimensions, and contents of bodies, both superficial and solid.</p><p>Every different species of Mensuration is estimated and measured by others of the same kind; so, the solid contents of bodies are measured by cubes, as cubic inches, or cubic feet, &c; surfaces by squares, as square inches, feet, &c; and lengths or distances by other lines, as inches, feet, &c.</p><p>The contents of rectilinear figures, whether plane or solid, can be accurately determined, or expressed; but of many curved ones, not. So the quadrature of the circle, and cubature of the sphere, are problems that have never yet been accurately solved. See the various kinds of Mensuration, as well as that of the different figures, under their respective terms.</p><p>The first writers on Geometry were chiefly writers on Mensuration; as Euclid, Archimedes, &c. See <hi rend="smallcaps">Quadrature;</hi> also the Preface to my Mensuration, for the most ample information.</p></div1><div1 part="N" n="MERCATOR" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MERCATOR</surname> (<foreName full="yes"><hi rend="smallcaps">Gerard</hi></foreName>)</persName></head><p>, an eminent geographer and mathematician, was born in 1512, at Ruremonde in the Low Countries. He applied himself with such industry to the sciences of geography and mathematics, that it has been said he often forgot to eat and sleep. The emperor Charles the 5th encouraged him much in his labours, and the duke of Juliers made him his cosmographer. He composed and published a Chronology; a larger and smaller Atlas; and some Geographical Tables; beside other books in Philosophy and Divinity. He was also so curious, as well as ingenious, that he engraved and coloured his maps himself. He made various maps, globes, and other mathematical instruments for the use of the emperor; and gave the most ample proofs of his uncommon skill in what he professed. His method of laying down charts is still used, which bear the name of <hi rend="italics">Mercator's Charts;</hi> also a part of navigation is from him called <hi rend="italics">Mercator's Sailing.</hi>—He died at Duisbourg in 1594, at 82 years of age.—See <hi rend="smallcaps">Mercator</hi>'s <hi rend="italics">Chart,</hi> below.</p><p><hi rend="smallcaps">Mercator</hi> <hi rend="italics">(Nicholas),</hi> an eminent mathematician and astronomer, whose name in High-Dutch was <hi rend="italics">Haussman,</hi> was born, about the year 1640, at Holstein in Denmark. From his works we learn, that he had an early and liberal education, suitable to his distinguished genius, by which he was enabled to extend his<pb n="97"/><cb/> researches into the mathematical sciences, and to make very considerable improvements: for it appears from his writings, as well as from the character given of him by other mathematicians, that his talent rather lay in improving, and adapting any discoveries and improvements to use, than invention. However, his genius for the mathematical sciences was very conspicuous, and introduced him to public regard and esteem in his own country, and facilitated a correspondence with such as were eminent in those sciences, in Denmark, Italy, and England. In consequence, some of his correspondents gave him an invitation to this country, which he some time after accepted, and he afterwards continued in England till his death. He had not been long here before he was admitted F. R. S. and gave frequent proofs of his close application to study, as well as of his eminent abilities in improving some branch or other of the sciences. But he is charged sometimes with borrowing the inventions of others, and adopting them as his own. And it appeared upon some occasions that he was not of an over liberal mind in scientific communications. Thus, it had some time before him been observed, that there was an analogy between a scale of logarithmic tangents and Wright's protraction of the nautical meridian line, which consisted of the sums of the secants; though it does not appear by whom this analogy was first discovered. It appears however to have been first published, and introduced into the practice of navigation, by Henry Bond, who mentions this property in an edition of Norwood's Epitome of Navigation, printed about 1645; and he again treats of it more fully in an edition of Gunter's Works, printed in 1653, where he teaches, from this property, to resolve all the cases of Mercator's Sailing by the logarithmic tangents, independent of the table of meridional parts. This analogy had only been found to be nearly true by trials, but not demonstrated to be a mathematical property. Such demonstration seems to have been first discovered by Mercator, who, desirous of making the most advantage of this and another concealed invention of his in navigation, by a paper in the Philosophical Transactions for June 4, 1666, invites the public to enter into a wager with him on his ability to prove the truth or falsehood of the supposed analogy. This mercenary proposal it seems was not taken up by any one, and Mercator reserved his demonstration. Our author however distinguished himself by many valuable pieces on philosophical and mathematical subjects. His first attempt was, to reduce Astrology to rational principles, which proved a vain attempt. But his writings of more particular note, are as follow:</p><p>1. <hi rend="italics">Cosmographia, sive Descriptio Cœli & Terræ in Circulos, qua fundamentum sterniter sequentibus ordine Trigonometriæ Sphericorum Logarithmicæ, &c, a Nicolao Hauffman Holsato;</hi> printed at Dantzick, 1651, 12mo.</p><p>2. <hi rend="italics">Rationes Mathematicæ subductæ anno</hi> 1653; Copenhagen, in 4to.</p><p>3. <hi rend="italics">De Emendatione annua Diatribæ duæ, quibus exponuntur & demonstrantur Cycli So<*>is & Lunæ, &c;</hi> in 4to.</p><p>4. <hi rend="italics">Hypothesis Astronomica nova, et Consensus ejus cum Observationibus;</hi> Lond. 1664, in folio.<cb/></p><p>5. <hi rend="italics">Logarithmotechnia, sive Methodus Construendi Logarithmos nova, accurata, et facilis; scripto antehae communicata anno sc.</hi> 1667 <hi rend="italics">nonis Augusti; cui nunc accedit, Vera Quadratura Hyperbolæ, & Inventio summæ Logarithmorum. Auctore Nicolao Mercatore Holsato è Societate Regia. Huic etiam jungitur Michaelis Angeli Riccii Exercitatio Geometrica de Maximis et Minimis, hic ob argumenti præstantiam & exemplarium raritate<*> recusa:</hi> Lond. 1668, in 4to.</p><p>6. <hi rend="italics">Institutionum Astronomicarum libri duo, de Motu Astrorum communi & proprio, secundum bypotheses veterum & re<*>ntiorum præcipuas; deque Hypotheseon ex observatis constructione, cum tabulis Tychonianis, Solaribus, Lunaribus, Lunæ-solaribus, & Rudolphinis Solis, Fixar<*>m & quinque Errantium, earumque usu præceptis et exemplis commonstrato. Quibus accedit Appendix de iis, quæ novissimis temporibus cœlitus innotuerunt:</hi> Lond. 1676, 8vo.</p><p>7. <hi rend="italics">Euclidis Elementa Geometrica, novo ordine ac methodo fere, demonstrata. Una cum Nic. Mercatoris in Geometriam Introductione brevi, qua Magnitudinum Ortus ex genuinis Principiis, & Ortarum Affectiones ex ipsa Genesi derivantur.</hi> Lond. 1678, 12mo.</p><p>His papers in the Philosophical Transactions, are,</p><p>1. A Problem on some Points in Navigation: vol. 1, pa. 215.</p><p>2. Illustrations of the Logarithmo-technia: vol. 3, pa. 759.</p><p>3. Considerations concerning his Geometrical and Direct Method for finding the Apogees, Excentricities, and Anomalies of the Planets: vol. 5, pa. 1168.</p><p>Mercator died in 1994, about 54 years of age.</p><p>MERCATOR's <hi rend="italics">Chart,</hi> or <hi rend="italics">Projection,</hi> is a projection of the surface of the earth in plano, so called from Gerrard Mercator, a Flemish Geographer, who first published maps of this sort in the year 1556; though it was Edward Wright who first gave the true principles of such charts, with their application to Navigation, in 1599.</p><p>In this chart or projection, the meridians, parallels, and rhumbs, are all straight lines, the degrees of longitude being every where increased so as to be equal to one another, and having the degrees of latitude also increased in the same proportion; namely, at every latitude or point on the globe, the degrees of latitude, and of longitude, or the parallels, are increased in the proportion of radius to the sine of the polar distance, or cosine of the latitude; or, which is the same thing, in the proportion of the secant of the latitude to radius; a proportion which has the effect of making all the parallel circles be represented by parallel and equal right lines, and all the meridians by parallel lines also, but increasing infinitely towards the poles.</p><p>From this proportion of the increase of the degrees of the meridian, viz, that they increase as the secant of the latitude, it is very evident that the length of an arch of the meridian, beginning at the equator, is proportional to the sum of all the secants of the latitude, i. e. that the increased meridian, is to the true arch of it, as the sum of all those secants, to as many times the radius. But it is not so evident that the same increased meridian is also analogous to a scale of the logarithmic tangents, which however it is. “It does not appear by whom, nor by what accident, was discovered the<pb n="98"/><cb/> analogy between a scale of logarithmic tangents and Wright's protraction of the nautical meridian line, which consisted of the sums of the secants. It appears however to have been first published, and introduced into the practice of navigation, by Mr. Henry Bond, who mentions this property in an edition of Norwood's Epitome of Navigation, printed about 1645; and he again treats of it more fully in an edition of Gunter's Works, printed in 1653, where he teaches, from this property, to resolve all the cases of Mercator's Sailing by the logarithmic tangents, independent of the table of meridional parts. This analogy had only been found however to be nearly true by trials, but not demonstrated to be a mathematical property. Such demonstration, it seems, was first discovered by Mr. Nicholas Mercator, which he offered a wager to disclose, but this not being accepted; Mercator reserved his demonstration; as mentioned in the account of his life in the foregoing page. The proposal however excited the attention of mathematicians to the subject, and demonstrations were not long wanting. The first was published about two years after, by James Gregory, in his Exercitationes Geometricæ; from hence, and other similar properties there demonstrated, he shews how the tables of logarithmic tangents and secants may easily be computed from the natural tangents and secants.</p><p>“The same analogy between the logarithmic tangents and the meridian line, as also other similar properties, were afterwards more elegantly demonstrated by Dr. Halley, in the Philos. Trans. for Feb. 1696, and various methods given for computing the same, by examining the nature of the spirals into which the rhumbs are transformed in the stereographic projection of the sphere on the plane of the equator: the doctrine of which was rendered still more easy and elegant by the ingenious Mr. Cotes, in his Logometria, first printed in the Philos. Trans. for 1714, and afterwards in the collection of his works published 1732, by his cousin Dr. Robert Smith, who succeeded him as Plumian professor of philosophy in the University of Cambridge.”</p><p>The learned Dr. Isaac Barrow also, in his Lectiones Geometricæ, Lect. xi, Āppend. first published in 1672, delivers a similar property, namely, “that the sum of all the secants of any arc, is analogous to the logarithm of the ratio of <hi rend="italics">r</hi> + <hi rend="italics">s</hi> to <hi rend="italics">r</hi> - <hi rend="italics">s,</hi> viz, radius plus sine to radius minus sine; or, which is the same thing, that the meridional parts answering to any degree of latitude, are as the logarithms of the ratios of the versed sines of the distances from the two poles.” Preface to my Logarithms, pa. 100.</p><p>The meridian line in Mercator's Chart, is a scale of logarithmic tangents of the half colatitudes. The differences of longitude on any rhumb, are the logarithms of the same tangents, but of a different species; those species being to each other, as the tangents of the angles made with the meridian. Hence any scale of logarithmic tangents is a table of the differences of longitude, to several latitudes, upon some one determinate rhumb; and therefore, as the tangent of the angle of such a rhumb, is to the tangent of any other rhumb, so is the difference of the logarithms of any two tangents, to the difference of longitude<cb/> on the proposed rhumb, intercepted between the two latitudes, of whose half complements the logarithmic tangents were taken.</p><p>It was the great study of our predecessors to contrive such a chart in plano, with straight lines, on which all, or any parts of the world, might be truly set down, according to their longitudes and latitudes, bearings and distances. A method for this purpose was hinted by Ptolomy, near 2000 years since; and a general map, on such an idea, was made by Mercator; but the principles were not demonstrated, and a ready way shewn of describing the chart, till Wright explained how to enlarge the meridian line by the continual addition of secants; so that all degrees of longitude might be proportional to those of latitude, as on the globe: which renders this chart, in several respects, far more convenient for the navigator's use, than the globe itself; and which will truly shew the course and distance from place to place, in all cases of sailing.</p><p><hi rend="smallcaps">Mercator</hi>'s <hi rend="italics">Sailing,</hi> or more properly <hi rend="italics">Wright</hi>'s Sailing, is the method of computing the cases of sailing on the principles of Mercator's chart, which principles were laid down by Edward Wright in the beginning of the last century; or the art of finding on a plane the motion of a ship upon any assigned course, that shall be true as well in longitude and latitude, as distance; the meridians being all parallel, and the parallels of latitude straight lines. <figure/></p><p>In the right-angled triangle A<hi rend="italics">bc,</hi> let A<hi rend="italics">b</hi> be the true difference of latitude between two places, the angle <hi rend="italics">b</hi>A<hi rend="italics">c</hi> the angle of the course sailed, and A<hi rend="italics">c</hi> the true distance sailed; then will <hi rend="italics">bc</hi> be what is called the departure, as in plane sailing: produce A<hi rend="italics">b,</hi> till AB be equal to the meridional difference of latitude, and draw BC parallel to <hi rend="italics">bc;</hi> so shall BC be the difference of longitude.</p><p>Now from the similarity of the two triangles A<hi rend="italics">bc,</hi> ABC, when three of the parts are given, the rest may be found; as in the following analogies: As Radius : sin. course : : distance : departure; Radius : cos. course : : distance : dif. lat.; Radius : tan. course : : merid. dif. lat : dif. longitude.</p><p>And by means of these analogies may all the cases of Mercator's Sailing be resolved.</p></div1><div1 part="N" n="MERCURY" org="uniform" sample="complete" type="entry"><head>MERCURY</head><p>, the smallest of the inferior planets, and the nearest to the sun, about which it is carried with a very rapid motion. Hence it was, that the Greeks called this planet after the name of the nimble messenger of the Gods, and represented it by the figure of a youth with wings at his head and feet; from whence is derived <*>, the character in present use for this planet.</p><p>The mean distance of Mercury from the sun, is to that of the earth from the sun, as 387 to 1000, and therefore his distance is about 36 millions of miles, or little more than one-third of the earth's distance from<pb n="99"/><cb/> the sun. Hence the sun's diameter will appear at Mercury, near 3 times as large as at the earth; and hence also the sun's light and heat received there is about 7 times those at the earth; a degree of heat sufficient to make water boil. Such a degree of heat therefore must render Mercury not habitable to creatures of our constitution: and if bodies on its surface be not inflamed, and set on fire, it must be because their degree of density is proportionably greater than that of such bodies is with us.</p><p>The diameter of Mercury is also nearly one-third of the diameter of the earth, or about 2600 miles. Hence the surface of Mercury is nearly 1-9th, and his magnitude or bulk 1-27th of that of the earth.</p><p>The inclination of his orbit to the plane of the ecliptic, is 6° 54′; his period of revolution round the sun, 87days 23hours; his greatest elongation from the sun 28°; the excentricity of his orbit 1/5 of his mean distance, which is far greater than that of any of the other planets; and he moves in his orbit about the sun at the amazing rate of 95000 miles an hour.</p><p>The place of his aphelion is <*> 23° 8′; place of ascending node <*> 14° 43′, and consequently that of the descending node <*> 14° 43′.</p><p>His Length of day, or rotation on his axis, Inclination of axis to his orbit, Gravity on his surface, Density, and Quantity of matter, are all unknown.</p><p>Mercury changes his phases, like the moon, according to his various positions with regard to the earth and sun; except only, that he never appears quite full, because his enlightened side is never turned directly towards us, unless when he is so near the sun as to be lost to our sight in his beams. And as his enlightened side is always towards the sun, it is plain that he shines not by any light of his own; for if he did, he would constantly appear round.</p><p>The best observations of this planet are those made when it is seen on the sun's disc, called its transit; for in its lower conjunction, it sometimes passes before the sun like a little spot, eclipsing a small part of the sun's body, only observable with a telescope. That node from which Mercury ascends northward above the ecliptic, is in the 15th degree of Taurus, and the opposite in the 15th degree of Scorpio. The earth is in those parts on the 6th of November, and 4th of May, new style; and when Mercury comes to either of his nodes at his inferior conjunction about these times, he will appear in this manner to pass over the disc of the sun. But in all other parts of his orbit, his conjunctions are invisible, because he goes either above or below the sun. The first observation of this kind was made by Gassendi, in November 1631. Several following observations of the like transits are collected in Du Hamel's Hist. of the Royal Acad. of Sciences, pa. 470, ed. 2. And Mr. Whiston has given a list of several periods at which Mercury may be seen on the sun's disc, viz, in 1782, Nov. 12, at 3h 44m afternoon; in 1786, May 4th, at 6h 57m in the forenoon; in 1789, Dec. 6th, at 3h 55m afternoon; and in 1799, May 7th, at 2h 34m afternoon. There are also several intermediate transits, but none of them visible at London. See Dr. Halley's account of the Transits of Mercury and Venus, in the Philos. Trans. n°. 193.<cb/></p></div1><div1 part="N" n="MERIDIAN" org="uniform" sample="complete" type="entry"><head>MERIDIAN</head><p>, in Astronomy, is a great circle of the celestial sphere, passing through the poles of the world, and both the zenith and nadir, crossing the equinoctial at right angles, and dividing the sphere into two equal parts, or hemispheres, the one eastern, and the other western. Or, the Meridian is a vertical circle passing through the poles of the world.</p><p>It is called Meridian, from the Latin <hi rend="italics">meridies,</hi> midday or noon, because when the sun comes to the south part of this circle, it is noon to all those places situated under it.</p><div2 part="N" n="Meridian" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Meridian</hi></head><p>, in Geography, is a great circle passing through the poles of the earth, and any given place whose Meridian it is; and it lies exactly under, or in the plane of, the celestial Meridian.</p><p>These Meridians are various, and change according to the longitude of places; so that their number may be said to be infinite, for that all places from east to west have their several Meridians. Farther, as the Meridian invests the whole earth, there are many places situated under the same Meridian. Also, as it is noon whenever the centre of the sun is in the celestial Meridian; and as the Meridian of the earth is in the plane of the former; it follows, that it is noon at the same time, in all places situated under the same Meridian.</p><p><hi rend="italics">First</hi> <hi rend="smallcaps">Meridian</hi>, is that from which the rest are counted, reckoning both east and west; and is the beginning of longitude.</p><p>The fixing of the First Meridian is a matter merely arbitrary; and hence different persons, nations, and ages, have fixed it differently: from which circumstance some confusion has arisen in geography. The rule among the Ancients was, to make it pass through the place farthest to the west that was known. But the Moderns knowing that there is no such place on the earth as can be esteemed the most westerly, the way of computing the longitudes of places from one fixed point is much laid aside.</p><p>Ptolomy assumed the Meridian that passes through the farthest of the Canary Islands, as his first Meridian; that being the most western place of the world then known. After him, as more countries were discovered in that quarter, the First Meridian was remov farther off. The Arabian geographers chose to the First Meridian upon the utmost shore of the western ocean. Some fixed it to the island of St. Nicholas near the Cape Verd; Hondius to the isle of St. James; others to the island of Del Co<*>vo, one of the Azores; because on that island the magnetic needle at that time pointed directly north, without any variation: and it was not then known that the variation of the needle is itself subject to variation. The latest geographers, particularly the Dutch, have pitched on the Pike of Teneriffe; others on the Isle of Palm, another of the Canaries; and lastly, the French, by order of the king, on the island of Fero, another of the Canaries.</p><p>But, without much regard to any of these rules, geographers and map-makers often assume the Meridian of the place where they live, or the capital of their country, or its chief observatory, for a First Meridian; and from thence reckon the longitudes of places, east and west.</p><p>Astronomers, in their calculations, usually choose<pb n="100"/><cb/> the Meridian of the place where their observations are made, for their First Meridian; as Ptolomy at Alexandria; Tycho Brahe at Uranibourg; Riccioli at Bologna; Flamsteed at the Royal Observatory at Greenwich; and the French at the Observatory at Paris.</p><p>There is a suggestion in the Philos. Trans. that the Meridians vary in time. And it has been said that this is rendered probable, from the old Meridian line in the church of St. Petronio at Bologna, which is said to vary no less than 8 degrees from the true Meridian of the place at this time; and from the Meridian of Tycho at Uranibourg, which M. Picart observes, varies 18 minutes from the modern Meridian. If there be any thing of truth in this hint, Dr. Wallis says, the alteration must arise from a change of the terrestrial poles (here on earth, of the earth's diurnal motion), not of their pointing to this or that of the fixed stars: for if the poles of the diurnal motion remain fixed to the same place on the earth, the Meridians, which pass through these poles, must remain the same.</p><p>But the notion of the changes of the Meridian seems overthrown by an observation of M. Chazelles, of the French Academy of Sciences, who, when in Egypt, found that the four sides of a pyramid, built 3000 years ago, still looked very exactly to the four cardinal points. A position which cannot be considered as merely fortuitous.</p><p><hi rend="smallcaps">Meridian</hi> <hi rend="italics">of a Globe,</hi> or <hi rend="italics">Sphere,</hi> is the brazen circle, in which the globe hangs and turns.</p><p>It is divided into four 90's, or 360 degrees, beginning at the equinoctial: on it, each way, from the equinoctial, on the celestial globes, is counted the north and south declination of the sun, moon, or stars; and on the terrestrial globe, the latitude of places, north and south. There are two points on this circle called the poles; and a diameter, continued from thence through the centre of either globe, is called the axis of the earth, or heavens, on which it is supposed they turn round.</p><p>On the terrestrial globes there are usually drawn 36 Meridians, one through every 10th degree of the equator, or through every 10th degree of longitude.</p><p>The uses of this circle are, to set the globes in any particular latitude, to shew the sun's or a star's declination, right ascension, greatest altitude, &c.</p><p><hi rend="smallcaps">Meridian</hi> <hi rend="italics">Line,</hi> an arch, or part, of the Meridian of the place, terminated each way by the horizon. Or, a Meridian line is the intersection of the plane of the Meridian of the place with the plane of the horizon, often called a north-and-south line, because its direction is from north to south.</p><p>The Meridian line is of most essential use in astronomy, geography, dialling, &c; and the greatest pains are taken by astronomers to fix it at their observatories to the utmost precision. M. Cassini has distinguished himself by a Meridian line drawn on the pavement of the church of St. Petronio, at Bologna; being extended to 120 feet in length. In the roof of this church, 1000 inches above the pavement, is a small hole, through which the sun's image, when in the meridian, falling upon the line, marks his progress all the year. When finished, M. Cassini, by a public writing, quaintly informed the mathematicians of Eu-<cb/> rope, of a new oracle of Apollo, or the sun, established in a temple, which might be consulted, with entire confidence, as to all dissiculties in astronomy. See <hi rend="smallcaps">Gnomon.</hi></p><p><hi rend="italics">To draw a Meridian Line.</hi>—There are many ways of doing this; but some of the easiest and simplest are as follow:</p><p>1. On an horizontal plane describe several concentric <figure/> circles AB, <hi rend="italics">ab,</hi> &c, and on the common centre C erect a stile, or gnomon, perpendicular to the horizontal plane, of about a foot in length. About the 2 1st of June, between the hours of 9 and 11 in the morning, and between 1 and 3 in the afternoon, observe the points A, <hi rend="italics">a,</hi> B, <hi rend="italics">b,</hi> &c, in the circles, where the shadow of the stile terminates. Bisect the arches AB, <hi rend="italics">ab,</hi> &c, in D, <hi rend="italics">d,</hi> &c. If then the same right line DE bisect all these arches, it will be the Meridian line sought.</p><p>As it is not easy to determine precisely the extremity of the shadow, it will be best to make the stile flat at top, and to drill a small hole through it, noting the lucid point projected by it on the arches AB and <hi rend="italics">ab,</hi> instead of marking the extremity of the shadow itself.</p><p>2. Another method is thus: Knowing the south <figure/> quarter pretty nearly, observe the altitude FE of some star on the east side ofit, and not far from the Meridian HZRN: then, keeping the quadrant firm on its axis, so as the plummet may still cut the same degree, direct it to the western side of the Meridian, and wait till you find the star has the same altitude as before, as <hi rend="italics">fe.</hi> Lastly, bisect the angle EC<hi rend="italics">e,</hi> formed by the intersection of the two planes in which the quadrant has been placed at the time of the two observations, by the right line HR, which will be the Meridian sought.</p><p>Many other methods are given by authors, of describing a Meridian line; as by the pole star, or by equal altitudes of the sun, &c; by Schooten in his Exercitationes Geometriæ; Grey, Derham, &c, in the Philos. Trans. and by Ferguson in his Lectures on Select Subjects.</p><p>From what has been said it is evident that whenever the shadow of the stile covers the Meridian line, the centre of the sun is in the Meridian, and therefore it is then noon. And hence the use of a Meridian line in adjusting the motion of clocks to the sun.</p><p>If another stile be erected perpendicularly on any other horizontal plane, and a signal be given when the shadow of the former stile covers the Meridian line drawn on another plane, noting the apex or extremity of the shadow projected by the second stile, a line drawn through that point and the foot of the stile will be a Meridian line at the 2d place.</p><p>Or, instead of the 2d stile, a plumb line may be hung up, and its shadow noted on a plane, upon a signal given that the shadow of another plummet, or<pb n="101"/><cb/> of a stile, falls exactly in another Meridian line, at a little distance; which shadow will give the other Meridian line parallel to the former.</p><p><hi rend="smallcaps">Meridian</hi> <hi rend="italics">Line,</hi> on a Dial, is a right line arising from the intersection of the Meridian of the place with the plane of the dial. This is the line of noon, or 12 o'clock, and from hence the division of the hourline begins.</p><p><hi rend="smallcaps">Meridian</hi> <hi rend="italics">Line,</hi> on Gunter's scale, is divided unequally towards 87 degrees, in such manner as the Meridian in Mercator's chart is divided and numbered.</p><p>This line is very useful in navigation. For, 1st, It serves to graduate a sea-chart according to the true projection. 2d, Being joined with a line of chords, it serves for the protraction and resolution of such rectilineal triangles as are concerned in latitude, longitude, course, and distance, in the practice of sailing; as also in pricking the chart truly at sea.</p><p><hi rend="italics">Magnetical</hi> <hi rend="smallcaps">Meridian</hi>, is a great circle passing through or by the magnetical poles; to which Meridians the magnetical needle conforms itself.</p><p>Meridian Altitude, of the sun or stars, is their altitude when in the meridian of the place where they are observed.</p><p><hi rend="smallcaps">Meridional</hi> <hi rend="italics">Distance,</hi> in Navigation, is the same with the Departure, or easting and westing, or distance between two meridians.</p><p><hi rend="smallcaps">Meridional</hi> <hi rend="italics">Parts, Miles,</hi> or <hi rend="italics">Minutes,</hi> in Navigation, are the parts of the increased or enlarged meridian, in the Mercator's chart. Tables of these parts are in most books of navigation; and they serve both for constructing that sort of charts, and for working that kind of navigation.</p><p>Under the article <hi rend="smallcaps">Mercator</hi>'s <hi rend="italics">Chart,</hi> it is shewn that the parts of the enlarged Meridian increase in proportion as the cosine of the latitude to radius, or, which is the same thing, as radius to the secant of the latitude; and therefore it follows, that the whole length of the enlarged nautical Meridian, from the equator to any point, or latitude, will be proportional to the sum of all the secants of the several latitudes up to that point of the Meridian. And on this principle was the sirst Table of Meridional Parts constructed, by the inventor of it, Mr. Edward Wright, and published in 1599; viz, he took the Meridional parts of 1′ = the sec. of 1′; of 2′ = sec. of 1′ + sec. of 2′; of 3′ = secants of 1, 2, and 3 min. of 4′ = secants of 1, 2, 3, and 4 min. and so on by a constant addition of the secants.</p><p>The Tables of Meridional Parts, so constructed, are perhaps exact enough for ordinary practice in navigation; but they would be more accurate if the Meridian were divided into more or smaller parts than single minutes; and the smaller the parts, so much the greater the accuracy. But, as a continual subdivision would greatly augment the labour of calculation, other ways of computing such a table have been devised, and treated of, by Bond, Gregory, Oughtred, Sir Jonas Moor, Dr. Wallis, Dr. Halley, and others. See M<hi rend="smallcaps">ERCATOR</hi>'s <hi rend="italics">Chart,</hi> and Robertson's Navigation, vol. 2, book 8. The best of these methods was derived from this property, viz, that the Meridian line, in a Mercator's chart, is analogous to a scale of logarithmic tan-<cb/> gents of half the complements of the latitudes; from which property also a method of computing the cases of Mercator's Sailing has been deduced, by Dr. Halley. Vide ut supra, also the Philos. Trans. vol. 46, pa. 559. <hi rend="center"><hi rend="italics">To find the</hi> <hi rend="smallcaps">Meridional Parts</hi> <hi rend="italics">to any Spheroid, with the same exactness as in a Sphere.</hi></hi></p><p>Let the semidiameter of the equator be to the distance of the centre from the focus of the generating ellipse, as <hi rend="italics">m</hi> to 1. Let A represent the latitude for which the meridional parts are required, <hi rend="italics">s</hi> the sine of the latitude, to the radius 1: Find the arc B, whose sine is <hi rend="italics">s</hi>/<hi rend="italics">m;</hi> take the logarithmic tangent of half the complement of B, from the common tables; subtract the log. tangent from 10.0000000, or the log. tangent of 45°; multiply the remainder by the number 7915.7044679, and divide the product by <hi rend="italics">m;</hi> then the quotient subtracted from the Meridienal parts in the sphere, computed in the usual manner for the latitude A, will give the Meridional parts, expressed in minutes, for the same latitude in the spheroid, when it is the oblate one.</p><p><hi rend="italics">Example.</hi> If , then the greatest difference of the Meridional parts in the sphere and spheroid is 76.0929 minutes. In other cases it is found by multiplying the remainder above mentioned by the number 1174.078.</p><p>When the spheroid is oblong, the difference in the Meridional parts between the sphere and spheroid, for the same latitude, is then determined by a circular arc. See Philos. Trans. no. 461, sect. 14. Also Maclaurin's Fluxions, art. 895, 899. And Murdoch's Mercator's Sailing &c.</p></div2></div1><div1 part="N" n="MERLON" org="uniform" sample="complete" type="entry"><head>MERLON</head><p>, in Fortification, that part of the Parapet, which lies between two embrasures.</p></div1><div1 part="N" n="MERSENNE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MERSENNE</surname> (<foreName full="yes"><hi rend="smallcaps">Martin</hi></foreName>)</persName></head><p>, a learned French author, was born at Bourg of Oyse, in the province of Maine, 1588. He studied at La Fleche at the same time with Des Cartes; with whom he contracted a strict friendship, which continued till death. He afterwards went to Paris, and studied at the Sorbonne; and in 1611 entered himself among the Minims. He became well skilled in Hebrew, philosophy, and mathematics. From 1615 to 1619, he taught philosophy and theology in the convent of Nevers; and became the Superior of that convent. But being desirous of applying himself more freely and closely to study, he resigned all the posts he enjoyed in his order, and retired to Paris, where he spent the remainder of his life; excepting some short excursions which he occasionally made into Italy, Germany, and the Netherlands.</p><p>Study and literary conversation were afterwards his whole employment. He held a correspondence with most of the learned men of his time; being as it were the very centre of communication between literary men of all countries, by the mutual correspondence which he managed between them; being in France what Mr. Collins was in England. He omitted no opportunity to engage them to publish their works; and the world is obliged to him for several excellent discoveries, which would probably have been lost, but for his encouragement; and on all accounts he had the reputation of being one of the best men, as well as philosophers,<pb n="102"/><cb/> of his time. No person was more curious in penetrating into the secrets of nature, and carrying all the arts and sciences to perfection. He was the chief friend and literary agent of Des Cartes at Paris; giving him advice and assistance upon all occasions, and informing him of all that passed at Paris and elsewhere. For, being a person of universal learning, but particularly excelling in physical and mathematical knowledge, Des Cartes scarcely ever did any thing, or at least was not perfectly satissied with any thing he had done, without sirst knowing what Mersenne thought of it. It is even said, that when Mersenne gave out in Paris, that Des Cartes was erecting a new system of physics upon the foundation of a vacuum, and found the public very indifferent to it on that very account, he immediately sent notice to Des Cartes, that a vacuum was not then the fashion at Paris; upon which, that philosopher changed his system, and adopted the old doctrine of a plenum.</p><p>Mersenne was a man of good invention also himself; and he had a peculiar talent in forming curious questions, though he did not always succeed in resolving them; however, he at least gave occasion to others to do it. It is said he invented the Cycloid, otherwise called the Roulette. Presently the chief geometricians of the age engaged in the contemplation of this new curve, among whom Mersenne himself held a distinguished rank. After a very studious and useful life, he died at Paris in 1648, at 60 years of age.</p><p>Mersenne was author of many useful works, particularly the following:</p><p>1. <hi rend="italics">Questiones celeberrimæ in Genesim.</hi></p><p>2. <hi rend="italics">Harmonicorum Libri.</hi></p><p>3. <hi rend="italics">De Sonorum Natura, Causis, et Effectibus.</hi></p><p>4. <hi rend="italics">Cogitata Physico-Mathematica;</hi> 2 vols. 4to.</p><p>5. <hi rend="italics">La Verité des Sciences.</hi></p><p>6. <hi rend="italics">Les Questions inouies.</hi></p><p>Besides many letters in the works of Des Cartes, and other authors.</p></div1><div1 part="N" n="MESOLABE" org="uniform" sample="complete" type="entry"><head>MESOLABE</head><p>, or <hi rend="smallcaps">Mesolabium</hi>, a mathematical instrument invented by the Ancients, for finding two mean proportionals mechanically, which they could not perform geometrically. It consists of three parallelograms, moving in a groove to certain intersections. Its figure is described by Eutocius, in his Commentary on Archimedes. See also Pappus, lib. 3.</p><p>MESO-<hi rend="smallcaps">Logarithm</hi>, a term used by Kepler to signify the logarithms of the cosines and cotangents.</p></div1><div1 part="N" n="METO" org="uniform" sample="complete" type="entry"><head>METO</head><p>, or <hi rend="smallcaps">Meton</hi>, the son of Pausanias, a famous mathematician of Athens, who flourished 432 years before Christ. In the first year of the 87th Olympiad, he observed the solstice at Athens: and published his <hi rend="italics">Anneadecatoride,</hi> that is, his <hi rend="italics">Cycle of</hi> 19 <hi rend="italics">Years;</hi> by which he endeavoured to adjust the course of the sun to that of the moon, and to make the solar and lunar years begin at the same point of time. See <hi rend="smallcaps">Cycle.</hi></p><p><hi rend="smallcaps">Metonic Cycle</hi>, called also the <hi rend="italics">Golden Number,</hi> and <hi rend="italics">Lunar Cycle,</hi> or <hi rend="italics">Cycle of the Moon,</hi> that which was invented by Meton the Athenian; being a period of 19 years. See <hi rend="smallcaps">Cycle.</hi></p></div1><div1 part="N" n="METOPE" org="uniform" sample="complete" type="entry"><head>METOPE</head><p>, or <hi rend="smallcaps">Metopa</hi>, in Architecture, the square space between the triglyphs of the Doric Freeze; which among the Ancients used to be adorned with the heads of beasts, basons, vases, and other instruments used in sacrificing.<cb/></p><p>A <hi rend="italics">Demi-Metope</hi> is a space somewhat less than half a Metope, at the corner of the Doric Freeze.</p></div1><div1 part="N" n="MICHAELMAS" org="uniform" sample="complete" type="entry"><head>MICHAELMAS</head><p>, the feast of St. Michael the archangel; held on the 29th of September.</p></div1><div1 part="N" n="MICROCOUSTICS" org="uniform" sample="complete" type="entry"><head>MICROCOUSTICS</head><p>, the same with M<hi rend="smallcaps">ICROPHONES.</hi></p></div1><div1 part="N" n="MICROMETER" org="uniform" sample="complete" type="entry"><head>MICROMETER</head><p>, is an instrument usually fitted to a telescope, in the focus of the object-glass, for measuring small angles or distances; as the apparent diameters of the planets, &c.</p><p>There are several sorts of these instruments, upon different principles; the origin of which has been disputed. The general principle is, that the instrument moves a sine wire parallel to itself, in the plane of the picture of an object, formed in the focus of a telescope, and so with great exactness to measure its perpendicular distance from a fixed wire in the same plane: and thus are measured small angles, subtended by remote objects at the naked eye.</p><p><hi rend="italics">For example,</hi> Let a planet be viewed through the telescope; and when the parallel wires are opened to such a distance as to appear exactly to touch two opposite points in the circumference of the planet, it is evident that the perpendicular distance between the wires is then equal to the diameter of the picture of the planet, formed in the focus of the object-glass. Let this distance, whose measure is given by the mechanism of the micrometer, be represented by the line <figure/> <hi rend="italics">pq;</hi> then, since the measure of the focal distance <hi rend="italics">q</hi>L may be also known, the ratio of <hi rend="italics">q</hi>L to <hi rend="italics">qp,</hi> that is, of radius to the tangent of the angle <hi rend="italics">q</hi>L<hi rend="italics">p,</hi> will give the angle itself, by a table of sines and tangents; and this angle is equal to the opposite angle PLQ, which the real diameter of the planet subtends at L, or at the naked eye.</p><p>With respect to the invention of the Micrometer; Mess. Azout and Picard have the credit of it in common fame, as being the first who published it, in the year 1666; but Mr. Townley, in the Philos. Trans. reclaims it for one of our own countrymen, Mr. Gascoigne. He relates that, from some scattered papers and letters of this gentleman, he had learnt that before our civil wars he had invented a Micrometer, of as much effect as that since made by M. Azout, and had made use of it for some years, not only in taking the diameters of the planets, and distances upon land, but in determining other matters of nice importance in the heavens; as the moon's distance, &c. Mr. Gascoigne's instrument also fell into the hands of Mr. Townley, who says farther, that by the help of it he could make above 40,000 divisions in a foot. This instrument being shewn to Dr. Hook, he gave a drawing and description of it, and proposed several improvements in it; which may be seen in the Philos. Trans. vol. 1, pa. 63, and Abr. vol. 1, pa. 217. Mr. Gascoigne divided the image of an object, in the focus of the object-glass, by the approach of<pb n="103"/><cb/> two pieces of metal, ground to a very fine edge; instead of which, Dr. Hook would substitute two fine hairs, stretched parallel to each other: and two other methods of Dr. Hook, different from this, are described in his posthumous works, pa. 497 &c. An account of several curious observations which Mr. Gascoigne made by the help of his Micrometer, particularly in measuring the diameter of the moon and other planets, may be seen in the Philos. Trans. vol. 48, pa. 190; where Dr. Bevis refers to an original letter of Mr. Gascoigne, to Mr. Oughtred, written in 1641, for an account given by the author of his own invention, &c.</p><p>Mons. De la Hire, in a discourse on the æra of the inventions of the Micrometer, pendulum clock, and telescope, read before the Royal Academy of Sciences in 1717, makes M. Huygens the inventor of the Micrometer. That author, he observes, in his Observations on Saturn's Ring, &c, published in 1659, gives a method of finding the diameters of the planets by means of a telescope, viz, by putting an object, which he calls a virgula, of a size proper to take in the distance to be measured, in the focus of the convex object-glass: in this case, says he, the smallest object will be seen very distinctly in that place of the glass. By such means, he adds, he measured the diameter of the planets, as he there delivers them. See Huygens's System of Saturn.</p><p>This Micrometer, M. De la Hire observes, is so very little different from that published by the marquis De Malvasia, in his Ephemerides, three years after, that they ought to be esteemed the same: and the Micrometer of the marquis differed yet less from that published four years after his, by Azout and Picard. Hence, De la Hire concludes, that it is to Huygens the world is indebted for the invention of the Micrometer; without taking any notice of the claim of our countryman Gascoigne, which however is many years prior to any of them.</p><p>De la Hire says, that there is no method more simple or commodious for observing the digits of an eclipse, than a net in the focus of the telescope. These, he says, were usually made of silken threads; and for this particular purpose six concentric circles had also been used, drawn upon oiled paper; but he advises to draw the circles on very thin pieces of glass, with the point of a diamond. He also gives some particular directions to assist persons in using them. In another memoir, he shews a method of making use of the same net for all eclipses, by using a telescope with two object-glasses, and placing them at different distances from each other. Mem. 1701 and 1717.</p><p>M. Cassini invented a very ingenious method of ascertaining the right ascensions and declinations of stars, by fixing four cross hairs in the focus of the telescope, and turning it about its axis, so as to make them move in a line parallel to one of them. But the later improved Micrometers will answer this purpose with greater exactness. Dr. Maskelyne has published directions for the use of it, extracted from Dr. Bradley's papers, in the Philos. Trans. vol. 62. See also Smith's Optics, vol. 2, pa. 343.</p><p>Wolfius describes a Micrometer of a very easy and simple structure, first contrived by Kirchius.</p><p>Dr. Derham tells us, that his Micrometer is not put<cb/> into a tube, as is usual, but is contrived to measure the spectres of the sun on paper, of any radius, or to measure any part of them. By this means he can easily, and very exactly, with the help os a fine thread, take the declination of a solar spot at any time of the day; and, by his half-seconds watch, measure the distance of the spot from either limb of the sun.</p><p>J. And. Segner proposed to enlarge the field of view in these Micrometers, by making them of a considerable extent, and having a moveable eye-glass, or several eye-glasses, placed opposite to different parts of it. He thought however, that two would be quite sufficient, and he gives particular directions how to make use of such Micrometers in astronomical observations. See Comm. Gotting. vol. 1, pa. 27.</p><p>A considerable improvement in the Micrometer was communicated to the Royal Society, in 1743, by Mr. S. Savary; an account of which, extracted from the minutes by Mr. Short, was published in the Philos. Trans. for 1753. The first hint of such a Micrometer was suggested by M. Roemer, in 1675: and M. Bouguer proposed a construction similar to that of M. Savary, in 1748; for which see <hi rend="smallcaps">Heliometer.</hi> The late Mr. Dollond made a farther improvement in this kind of Micrometer, an account of which was given to the Royal Society by Mr. Short, and published in the Philos. Trans. vol. 48. Instead of two object-glasses, he used only one, which he neatly cut into two semicircles, and fitted each semicircle in a metal frame, so that their diameters sliding in one another, by means of a screw, may have their centres so brought together as to appear like one glass, and so form one image; or by their centres receding, may form two images of the same object: it being a property of such glasses, for any segment to exhibit a perfect image of an object, although not so bright as the whole glass would give it. If proper scales are fitted to this instrument, shewing how far the centres recede, relative to the focal length of the glass, they will also shew how far the two parts of the same object are asunder, relative to its distance from the object-glass; and consequently give the angle under which the distance of the parts of that object are seen. This divided object-glass Micrometer, which was applied by the late Mr. Dollond to the object end of a reflecting telescope, and has been with equal advantage adapted by his son to the end of an achromatic telescope, is of so easy use, and affords so large a scale, that it is generally looked upon by astronomers as the most convenient and exact instrument for measuring small distances in the heavens. However, the common Micrometer is peculiarly adapted for measuring differences of right ascension, and declination, of celestial objects, but less convenient and exact for measuring their absolute distances; whereas the object-glass Micrometer is peculiarly fitted for measuring distances, though generally supposed improper for the former purpose. But Dr. Maskelyne has found that this may be applied with very little trouble to that purpose also; and he has furnished the directions necessary to be followed when it is used in this manner. The addition requisite for this purpose, is a cell, containing two wires, intersecting each other at right angles, placed in the focus of the eye-glass of the telescope, and moveable round about, by the turning of a button. For the description of this apparatus, with the method of applying and using<pb n="104"/><cb/> it, see Dr. Maskelyne's paper on the subject, in the Philos. Trans. vol. 61, pa. 536 &c.</p><p>After all, the use of the object-glass Micrometer is attended with difficulties, arising from the alterations in the focus of the eye, which are apt to cause it to give different measures of the same angle at different times. To obviate these difficulties, Dr. Maskelyne, in 1776, contrived a prismatic Micrometer, or a Micrometer consisting of two achromatic prisms, or wedges, applied between the object-glass and eye-glass of an achromatic telescope, by moving of which wedges nearer to or farther from the object-glass, the two images of an object produced by them appeared to approach to, or recede from, each other, so that the focal length of the object-glass becomes a scale for measuring the angular distance of the two images. The rationale and use of this Micrometer are explained in the Philos. Trans. vol. 67, pa. 799, &c. And a similar invention by the abbé Rochon, and improved by the abbé Boscovich, was also communicated to the Royal Society, and published in the same volume of the Transactions, pa. 789 &c.</p><p>Mr. Ramsden has lately described two new Micrometers, which he has contrived for remedying the defects of the object-glass Micrometer. One of these is a catoptric Micrometer, which, besides the advantage it derives from the principle of reflection, of not being disturbed by the heterogeneity of light, avoids every defect of other Micrometers, and can have no aberration, nor any defect arising from the imperfection of materials, or of execution; as the great simplicity of its construction requires no additional mirrors or glasses, to those required for the telescope; and the separation of the image being effected by the inclination of the two specula, and not depending on the focus of lens or mirror, any alteration in the eye of an observer cannot affect the angle measured. It has peculiar to itself the advantages of an adjustment, to make the images coincide in a direction perpendicular to that of their motion; and also of measuring the diameter of a planet on both sides of the zero; which will appear no inconsiderable advantage to observers who know how much easier it is to ascertain the contact of the external edges of two images than their perfect coincidence.</p><p>The other Micrometer invented and described by Mr. Ramsden, is suited to the principle of refraction. This Micrometer is applied to the erect eye-tube of a refracting telescope, and is placed in the conjugate focus of the first eye-glass, as the image is considerably magnified before it comes to the Micrometer, any imperfection in its glass will be magnified only by the remaining eye-glasses, which in any telescope seldom exceeds 5 or 6 times; and besides, the size of the Micrometer glass will not be the 100th part of the area which would be required, if it were placed at the objectglass; and yet the same extent of scale is preserved, and the images are uniformly bright in every part of the field of the telescope. See the description and construction of these two Micrometers in the Philos. Trans. vol. 69, part 2, art. 27.</p><p>In vol. 72 of the Philos. Trans. for the year 1782, Dr. Herschel, after explaining the defects and imperfections of the parallel-wire Micrometer, especially for measuring the apparent diameter of stars, and the distances between double and multiple stars, describes one,<cb/> for these purposes, which he calls a lamp Micrometer; one that is free from such defects, and has the advantage of a very enlarged scale. In speaking of the application of this instrument, he says, “It is well known to opticians and others, who have been in the habit of using optical instruments, that we can with one eye look into a microscope or telescope, and see an object much magnified, while the naked eye may see a scale upon which the magnified picture is thrown. In this manner I have generally determined the power of my telescopes; and any one who has acquired a facility of taking such observations, will very seldom mistake so much as one in 50 in determining the power of an instrument, and that degree of exactness is fully sussicient for the purpose.</p><p>“The Newtonian form is admirably adapted to the use of this Micrometer; for the observer stands always erect, and looks in a horizontal direction, notwithstanding the telescope should be elevated to the zenith. —The seale of the Mierometer at the convenient distance of 10 feet from the eye, with the power of 460, is above a quarter of an inch to a second; and by putting on my power of 932, I obtain a scale of more than half an inch to a second, without increasing the distance of the Micrometer; whereas the most perfect of my former Micrometers, with the same instrument, had a scale of less than the 2000th part of an inch to a second.</p><p>“The measures of this Micrometer are not confined to double stars only, but may be applied to any other objects that require the utmost accuracy, such as the diameters of the planets or their satellites, the mountains of the moon, the diameters of the fixed stars, &c.”</p><p>The Micrometer has not only been applied to telescopes, and employed for astronomical purposes; but there have been various contrivances for adapting it to microscopical observations. Mr. Leeuwenhoek's method of estimating the size of small objects, was by comparing them with grains of sand, of which 100 in a line took up an inch. These grains he laid upon the same plate with his objects, and viewed them at the same time. Dr. Jurin's method was similar to this; for he found the diameter of a piece of fine silver wire, by wrapping it very close upon a pin, and observing how many rings made an inch: and he used this wire in the same manner as Leeuwenhoek used his sand. Dr. Hook used to look upon the magnified object with one eye, while at the same time he viewed other objects, placed at the same distance, with the other eye. In this manner he was able, by the help of a ruler, divided into inches and small parts, and laid on the pedestal of the microscope, as it were to cast the magnified appearance of the object upon the ruler, and thus exactly to measure the diameter which it appeared to have through the glass; which being compared with the diameter as it appeared to the naked eye, easily shewed the degree in which it was magnified. A little practice, says Mr. Baker, will render this method exceedingly easy and pleasant.</p><p>Mr. Martin, in his Optics, recommends such a Micrometer for a microscope as had been applied to telescopes; for he advises to draw a number of parallel lines on a piece of glass, with the fine point of a diamond, at the distance of one 40th of an inch from one another, and to place it in the focus of the eye-glass. <pb/><pb/><pb n="105"/><cb/> By this method, Dr. Smith contrived to take the exact draught of objects viewed by a double microscope; for he advises to get a lattice, made with small silver wires or squares, drawn upon a plain glass by the strokes of a diamond, and to put it into the place of the image formed by the object-glass. Then, by transferring the parts of the object, seen in the squares of the glass or lattice, upon similar corresponding squares drawn on paper, the picture may be exactly taken. Mr. Martin also introduced into compound microscopes another Micrometer, consisting of a screw. See both these methods described in his Optics, pa. 277.</p><p>A very accurate division of a scale is performed by Mr. Coventry, of Southwark. The Micrometers of his construction are parallel lines drawn on glass, ivory, or metal, from the 10th to the 10,000th part of an inch. These may be applied to microscopes, for measuring the size of minute objects, and the magnifying power of the glasses; and to telescopes, for measuring the size and distance of objects, and the magnifying power of the instrument. To measure the size of an object in a single microscope; lay it on a Micrometer, whose lines are seen magnified in the same proportion with it, and they give at one view the real size of the object. For measuring the magnifying power of the compound microscope, the best and readiest method is the following: On the stage in the focus of the objectglass, lay a Micrometer, consisting of an inch divided into 100 equal parts; count how many divisions of the Micrometer are taken into the field of view; then lay a two-foot rule parallel to the Micrometer: fix one eye on the edge of the field of light, and the other eye on the end of the rule, which move, till the edge of the field of light and the end of the rule correspond; then the distance from the end of the rule to the middle of the stage, will be half the diameter of the field: ex. gr. If the distance be 10 inches, the whole diameter will be 20, and the number of the divisions of the Micrometer contained in the diameter of the field, is the magnifying power of the microscope. For measuring the height and distance of objects by a Micrometer in the telescope, see <hi rend="smallcaps">Telescope.</hi></p><p>Mr. Adams has applied a Micrometer, that instantly shews the magnifying power of any telescope.</p><p>In the Philos. Trans. for 1791, a very simple scale Micrometer for measuring small angles with the telescope is described by Mr. Cavallo. This Micrometer consists of a thin and narrow slip of mother-of-pearl finely divided, and placed in the focus of the eye-glass of a telescope, just where the image of the object is formed; whether the telescope is a reflector or a refractor, provided the eye-glass be a convex lens. This substance Mr. Cavallo, after many trials, found much more convenient than either glass, ivory, horn, or wood, as it is a very steady substance, the divisions very easy marked upon it, and when made as thin as common writing paper it has a very useful degree of transparency.</p><p>Upon this subject, see M. Azout's Tract on it, contained in <hi rend="italics">Divers Ouvrages de Mathematique & de Phisique; par Messieurs de l'Academie Royal des Sciences; M. de la Hire's Astronomicæ Tabulæ;</hi> Mr. <hi rend="italics">Townley,</hi> in the <hi rend="italics">Philos. Trans.</hi> n°. 21; <hi rend="italics">Wolfius,</hi> in his <hi rend="italics">Elem.</hi><cb/> <hi rend="italics">Astron.</hi> § 508; Dr. <hi rend="italics">Hook,</hi> and many others, in the <hi rend="italics">Philos. Trans.</hi> n°. 29 &c; <hi rend="italics">Hevelius,</hi> in the <hi rend="italics">Acta Eruditorum, ann.</hi> 1708; Mr. <hi rend="italics">Balsbaser,</hi> in his <hi rend="italics">Micrometria;</hi> also several volumes of the <hi rend="italics">Paris Memoirs,</hi> &c.</p></div1><div1 part="N" n="MICROPHONES" org="uniform" sample="complete" type="entry"><head>MICROPHONES</head><p>, instruments contrived to magnify small sounds, as microscopes do small objects.</p></div1><div1 part="N" n="MICROSCOPE" org="uniform" sample="complete" type="entry"><head>MICROSCOPE</head><p>, an optical instrument, composed of lenses or mirrors, by means of which small objects are made to appear larger than they do to the naked eye.</p><p><hi rend="smallcaps">Microscopes</hi> are distinguished into simple and compound, or single and double.</p><p><hi rend="italics">Simple,</hi> or <hi rend="italics">Single</hi> <hi rend="smallcaps">Microscopes</hi>, are such as consist of a single lens, or a single spherule. And a</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Microscope</hi> consists of several lenses duly combined.—As optics have been improved, other varieties have been contrived in this instrument: Hence reflecting Microscopes, water Microscopes, &c.</p><p>It is not certainly known when, or by whom, Microscopes were first invented; although it is probable they would soon follow upon the use of telescopes, since a Microscope is like a telescope inverted. We are iuformed by Huygens, that one Drebell, a Dutchman, had the first Microscope, in the year 1621, and that he was reputed the inventor of it: though F. Fontana, a Neapolitan, in 1646, claims the invention to himself, and dates it from the year 1618. Be this as' it may, it seems they were first used in Germany about 1621. According to Borelli, they were invented by Zacharias Jansen and his son, who presented the sirst Microscope<*> they had constructed to prince Maurice, and Albert arch-duke of Austria. William Borelli, who gives this account in a letter to his brother Peter, says, that when he was ambassador in England, in 1619, Cornelius Drebell shewed him a Microscope, which he said was the same that the arch-duke had given him, and had been made by Jansen himself. Borelli De vero Telescopii inventore, pa. 35. See <hi rend="smallcaps">Lens.</hi> <hi rend="center"><hi rend="italics">Theory</hi> and <hi rend="italics">Foundation</hi> of <hi rend="smallcaps">Microscopes.</hi></hi></p><p>If an object be placed in the focus of the convex lens of a single Microscope, and the eye be very near on the other side, the object will appear distinct in an erect situation, and magnified in the ratio of the focal distance of the lens, to the ordinary distance of distinct vision, viz, about 8 inches. <figure/></p><p>So, if the object AB be placed in the focus F, of a small glass sphere, and the eye behind it, as in the focus G, the object will appear distinct, and in an erect posture, increased as to diameter in the ratio of 3/4 of the diameter EI to 8 inches. If, ex. gr. the diameter EI of the small sphere be 1/10 of an inch; then , and , so that ; then as 3/40 : 8, or as 3 : 320, or as 1 : 106 2/3 : : the natural size to the magnified appearance; that is, the object is magnified about 107 times.</p><p>Hence the smaller the spherule or the lens is, so much the more is the object magnified. But then, so<pb n="106"/><cb/> much the less part is comprehended at one view, and so much the less distinct is the appearance of the object.</p><p>Equal appearances of the same object, formed by different combinations, become obscure in proportion as the number of rays constituting each pencil decreases, that is, in proportion to the smallness of the object-glass.</p><p>Wherefore, if the diameter of the object-glass exceeds the diameter of the pupil, as many times as the diameter of the appearance exceeds the diameter of the object; the appearance shall be as clear and bright as the object itself.</p><p>The diameter of the object-glass cannot be so much increased, without increasing at the same time the focal distances of all the glasses, and consequently the length of the instrument: Otherwise the rays would fall too obliquely upon the eye-glass, and the appearance become confused and irregular.</p><p>There are several kinds of single Microscopes; of which the following is the most simple.</p><p>AB (Plate xviii, fig. 1) is a little tube, to one end of which BC, is fitted a plain glass; to which any object, as a gnat, the wing of an insect, or the like, is applied; to the other end AD, at a proper distance from the object, is applied a lens, convex on both sides, of about an inch in diameter: the plane glass is turned to the sun, or the light of a candle, and the object is seen magnified. And if the tube be made to draw out, lenses or segments of different spheres may be used.</p><p>Again, a lens, convex on both sides, is inclosed in a cell AC (fig. 2), and held there by the screw H. Through the stem or pedestal CD passes a long screw EF, carrying a stile or needle EG. In E is a small tube; on which, and on the point G, the various objects are to be disposed. Thus, lenses of various spheres may be applied.</p><p>A good simple instrument of this kind is Mr. Wilson's pocket Microscope, which has 9 different magnifying glasses, 8 of which may be used with two different instruments, for the better applying them to various objects. One of these instruments is represented at AABB (fig. 3), which is made either of brass or ivory. There are three thin brass plates at E, and a spiral spring H of steel wire within it: to one of the thin plates of brass is fixed a piece of leather F, with a small furrow G, both in the leather, and brass to which it is fixed: in one end of this instrument there is a long screw D, with a convex glass C, placed in the end of it: in the other end of the instrument there is a hollow screw <hi rend="italics">oo,</hi> in which any of the magnifying glasses, M, are screwed, when they are to be made use of. The 9 different magnifying glasses are all set in ivory, 8 of which are set in the manner expressed at M. The greatest magnifier is marked upon the ivory, in which it is set, number 1, the next number 2, and so on to number 8; the 9th glass is not marked, but is set in the manner of a little barrel box of ivory, as at <hi rend="italics">b.</hi> At <hi rend="italics">ee</hi> is a flat piece of ivory, of which there are 8 belonging to this sort of Microscopes (though any one who has a mind to keep a register of objects may have as many of them as he pleases); in each of them there are 3 holes <hi rend="italics">fff,</hi> in<cb/> which 3 or more objects are placed between two thin glasses, or talcs, when they are to be used with the greater magnifiers.</p><p>The use of this instrument AABB is this. A handle W, from fig. 4, being screwed upon the button S, take one of the flat pieces of ivory or sliders <hi rend="italics">ee,</hi> and slide it between the two thin plates of brass at E, through the body of the Microscope, so that the object to be viewed be just in the middle; remarking to put that side of the plate <hi rend="italics">ee,</hi> where the brass rings are, farthest from the end AA: then screw into the hollow screw, <hi rend="italics">oo,</hi> the 3d, 4th, 5th, 6th, or 7th magnifying glass M; which being done, put the end AA close to your eye, and while looking at the object through the magnifying glass, screw in or out the long screw D, which moving round upon the leather F, held tight to it by the spiral wire H, will bring your object to the true distance; which may be known by seeing it clearly and distinctly.</p><p>Thus may be viewed all transparent objects, dusts, liquids, crystals of salts, small insects, such as fleas, mites, &c. If they be insects that will creep away, or such objects as are to be kept, they may be placed between the two register glasses <hi rend="italics">ff.</hi> For, by taking out the ring that keeps in the glasses <hi rend="italics">ff,</hi> where the object lies, they will fall out of themselves; so the object may be laid between the two hollow sides of them, and the ring put in again as before; but if the objects be dusts or liquids, a small drop of the liquid, or a little of the dust laid on the outside of the glass <hi rend="italics">ff,</hi> and applied as before, will be seen very easily.</p><p>As to the 1st, 2d, and 3d magnifying glasses, being marked with a + upon the ivory in which they are set, they are only to be used with those plates or sliders that are also marked with a +, in which the objects are placed between two thin talcs; because the thickness of the glasses in the other plates or sliders, hinders the object from approaching to the true distance from these greater magnifiers. But the manner of using them is the same with the former.</p><p>For viewing the circulation of the blood at the extremities of the arteries and vcins, in the transparent parts of fishes tails, &c, there are two glass tubes, a larger and a smaller, as expressed at <hi rend="italics">gg,</hi> into which the animal is put. When these tubes are to be used, unscrew the end screw D in the body of the Microscope, until the tube <hi rend="italics">gg</hi> can be easily received into that little cavity G of the brass plate fastened to the leather F under the other two thin plates of brass at E. When the tail of the fish lies flat on the glass tube, set it opposite to the magnifying glass, and bringing it to the proper distance by screwing in or out the end screw D, when the blood will be seen clearly circulating.</p><p>To view the blood circulating in the foot of a frog; choose such a frog as will just go into the tube; then with a little stick expand its hinder foot, which apply close to the side of the tube, observing that no part of the frog hinders the light from coming on its foot; and when it is brought to the proper distance, by means of the screw D, the rapid motion of the blood will be seen in its vessels, which are very numerous, in the transparent thin membrane or web between the toes. For this object, the 4th and 5th magnifiers will do very <pb/><pb/><pb n="107"/><cb/> well; but the circulation may be seen in the tails of water-newts in the 6th and 7th glasses, because the globules of the blood of those newts are as large again as the globules of the blood of frogs or small fish, as has been remarked in number 280 of the Philos. Trans. pa. 1184.</p><p>The circulation cannot so well be seen by the 1st, 2d, and 3d magnifiers, because the thickness of the glass tube, containing the fish, hinders the approach of the object to the focus of the magnifying glass. Fig. 4 is another instrument for this purpose.</p><p>In viewing objects, one ought to be careful not to hinder the light from falling upon them by the hat, hair, or any other thing, especially in looking at opaque objects; for nothing can be seen with the best of glasses, unless the object be at a due distance, with a sufficient light. The best lights for the plates or sliders, when the object lies between the two glasses, is a clear skylight, or where the sun shines on something white, or the reflection of the light from a looking-glass. The light of a candle is also good for viewing very small objects, though it be a little uneasy to those who are not practised in the use of Microscopes.</p><p><hi rend="italics">To cast small Glass Spherules for</hi> <hi rend="smallcaps">Microscopes.</hi>— There are several methods for this purpose. Hartsoeker first improved single Microscopes by using small globules of glass, melted in the flame of a candle; by which he discovered the animalculæ in semine masculino, and thereby laid the foundation of a new system of generation. Wolfius describes the following method of making such globules: A small piece of very sine glass, sticking to the wet point of a steel needle, is to be applied to the extreme bluish part of the flame of a lamp, or rather of spirits of wine, which will not black it; being there melted, and run into a small round drop, it is to be removed from the flame, on which it instantly ceases to be fluid. Then folding a thin plate of brass, and making very small smooth perforations, so as not to leave any roughness on the surfaces, and also smoothing them over to prevent any glaring, fit the spherule between the plates against the apertures, and put the whole in a frame, with objects convenient for observation.</p><p>Mr. Adams gives another method, thus: Take a piece of fine window-glass, and rase it, with a diamond, into as many lengths as you think needful, not more than 1-8th of an inch in breadth; then holding one of those lengths between the fore finger and thumb of each hand, over a very sine flame, till the glass begins to soften, draw it out till it be as fine as a hair, and break; then applying each of the ends into the purest part of the flame, you presently have two spheres, which may be made greater or less at pleasure: if they remain long in the flame, they will have spots; so they must be drawn out immediately after they are turned round. Break the stem off as near the globule as possible; and, lodging the remainder of the stem between the plates, by drilling the hole exactly round, all the protuberances are buried between the plates; and the Microscope performs to admiration.</p><p>Mr. Butterfield gave another manner of making these globules, in number 141 Philos. Trans.</p><p>In any of these ways may the spherules be made much smaller than any lens; so that the best single Mi-<cb/> croscopes, or such as magnify the most, are made of them. Leeuwenhoeck and Musschenbroek have suc ceeded very well in spherical Microscopes, and their greatest magnifiers enlarged the diameter of an object about 160 times; Philos. Trans. vol. 7, pa. 129, and vol. 8, pa. 121. But the smallest globules, and consequently the highest magnifiers for Microscopes, were made by F. de Torre of Naples, who, in 1765, sent four of them to the Royal Society. The largest of them was only two Paris points in diameter, and magnified a line 640 times; the second was the size of one Paris point, and magnified 1280 times; and the 3d no more than half a Paris point, or the 144th part of an inch in diameter, and magnified 2560 times. But since the focus of a glass globule is at the distance of one-4th of its diameter, and therefore that of the 3d globule of de Torre, above mentioned, only the 576th part of an inch distant from the object, it must be with the utmost difficulty that globules so minute as those can be employed to any purpose; and Mr. Baker, to whose examination they were referred, considers them as matters of curiosity rather than of real use. Philos. Trans. vol. 55, pa. 246, vol. 56, pa. 67.</p><p><hi rend="italics">Water</hi> <hi rend="smallcaps">Microscope.</hi> Mr. S. Gray, and, after him, Wolfius and others, have contrived water Microscopes, consisting of spherules or lenses of water, instead of glass. But since the distance of the focus of a lens or sphere of water is greater than that in one of glass, the spheres of which they are segments being the same, consequently water Microscopes magnify less than those of glass, and therefore are less esteemed. Mr. Gray first observed, that a small drop or spherule of water, held to the eye by candle light or moon light, without any other apparatus, magnified the animalcules contained in it, vastly more than any other Microscope. The reason is, that the rays coming from the interior surface of the first hemisphere, are reflected so as to fall under the same angle on the surface of the hinder hemisphere, to which the eye is applied, as if they came from the focus of the spherule; whence they are propagated to the eye in the same manner as if the objects were placed without the spherule in its focus.</p><p>Hollow glass spheres of about half an inch diameter, filled with spirit of wine, are often used for Microscopes; but they do not magnify near so much.</p><p><hi rend="italics">Theory of Compound or Double</hi> <hi rend="smallcaps">Microscopes.</hi>— Suppose an object-glass ED, the segment of a very small <figure/> sphere, and the object AB placed without the focus F. Suppose an eye-glass GH, convex on both sides, and the segment of a sphere greater than that of DE, though not too great; and, the focus being at K, let it be so disposed behind the object, that . Lastly suppose .<pb n="108"/><cb/> If then O be the place where an object is seen distinct with the naked eye; the eye in this case, being placed in I, will see the object AB distinctly, in an inverted position, and magnisied in the compound ratio of MK × LC to LK × CO; as is proved by the laws of dioptrics; that is, the image is larger than the object, and we are able to view it distinctly at a less distance. For Examp.—If the image be 20 times larger than the object, and by the help of the eye-glass we are able to view it 5 times nearer than we could have done with the naked eye, it will, on both these accounts, be magnified 5 times 20, or 100 times. <hi rend="center"><hi rend="italics">Laws of Double</hi> <hi rend="smallcaps">Microscopes.</hi></hi></p><p>1. The more an object is magnisied by the Micro scope, the less is its field, i. e. the less of it is taken in at one view.</p><p>2. To the same eye-glass may be successively applied object-glasses of various spheres, so as that both the entire objects, but less magnified, and their several parts, much more magnified, may be viewed through the same Microscope. In which case, on account of the different distance of the image, the tube in which the lenses are sitted, should be made to draw out.</p><p>3. Since it is proved, that the distance of the image LK, from the object-glass DE, will be greater, if another lens, concave on both sides, be placed before its focus; it follows, that the object will be magnified the more, if such a lens be here placed between the objectglass DE, and the eye-glass GH. Such a Microscope is much commended by Conradi, who used an objectlens, convex on both sides, whose radius was 2 digits, its aperture equal to a mustard seed; a lens, concave on both sides, from 12 to 16 digits; and an eye-glass, convex on both sides, of 6 digits.</p><p>4. Since the image is projected to the greater distance, the nearer another lens, of a segment of a larger sphere, is brought to the object-glass; a Microscope may be composed of three lenses, which will magnisy prodigiously.</p><p>5. From these considerations it follows, that the object will be magnified the more, as the eye-glass is the segment of a smaller sphere; but the field of vision will be the greater, as the same is a segment of a larger sphere. Therefore if two eye-glasses, the one a segment of a larger sphere, the other of a smaller one, be so combined, as that the object appearing very near through them, i. e. not farther distant than the focus of the first, be yet distinct; the object, at the same time, will be vastly magnified, and the field of vision much greater than if only one lens was used; and the object will be still more magnified, and the field enlarged, if both the object-glass and eye-glass be double. But because an object appears dim when viewed through so many glasses, part of the rays being reflected in passing through each, it is not adviseable greatly to multiply glasses; so that, among compound Microscopes, the best are those which consist of one objectglass, and two eye glasses.</p><p>Dr. Hook, in the preface to his Micrography, tells us, that in most of his observations he used a Microscope of this kind, with a middle eye-glass of a considerable diameter, when he wanted to see much of the object at one view, and took it out when he would ex-<cb/> amine the small parts of an object more accurately: for the fewer refractions there are, the more light and clear the object appears.</p><p>For a Microscope of three lenses De Chales recommends an object glass of 1/3 or 1/4 of a digit; and the first eye-glass he makes 2 or 2 1/2 digits; and the distance between the object-glass and eye-glass about 20 lines. Conradi had an excellent Microscope, whose object-glass was half a digit, and the two eye-glasses (which were placed very near) 4 digits; but it answered best when, instead of the object-glass, he used two glasses, convex on both sides, their sphere about a digit and a half, and at most 2, and their convexities touching each other within the space of half a line. Eustachius de Divinis, instead of an object-glass convex on both sides, used two plano convex lenses, whose convexities touched. Grindelius did the same; only that the convexities did not quite touch. Zahnius made a binocular Microscope, with which both eyes were used. But the most commodious double Microscope, it is said, is that of our countryman Mr. Marshal; though some improvement was made in it by Mr. Culpepper and Mr. Scarlet. These are exhibited in sigures 5 and 6.</p><p>It is observed, that compound Microscopes sometimes exhibit a fallacious appearance, by representing convex objects concave, and vice versa. Philos. Trans. numb. 476, pa. 387.</p><p>To fit Microscopes, as well as Telescopes, to shortsighted eyes, the object-glass and the eye-glass must be placed a little nearer together, so that the rays of each pencil may not emerge parallel, but may fall diverging upon the eye.</p><p><hi rend="italics">Reflecting</hi> <hi rend="smallcaps">Microscope</hi>, is that which magnifies by reflection, as the foregoing ones do by refraction. The inventor of this Microscope was Sir Isaac Newton.</p><p>The structure of such a Microscope may be con ceived thus: near the focus of a concave speculum AB, place a minute <figure/> object C, that its image may be formed larger than itself in D; to the speculum join a lens, convex on both sides, EF, so as the image D may be in its focus.</p><p>The eye will here see the image inverted, but distinct, and enlarged; consequently the object will be larger than if viewed through the lens alone.</p><p>Any telescope is changed into a Microscope, by removing the objectglass to a greater distance from the eye-glass. And since the distance of the image is various, according to the distance of the object from the focus; and it is magnified the more, as its distance from the object-glass is greater; the same telescope may be successively changed into Microscopes which magnify the object in different degrees. See some instruments of this sort described in Smith's Optics, Remarks, pa. 94.</p><p><hi rend="italics">Solar</hi> <hi rend="smallcaps">Microscope</hi>, called also the Camera Obscura Microscope, was invented by Mr. Lieberkuhn in 1738 or 1739, and consists of a tube, a looking-glass, a convex lens, and a Wilson's Microscope. The tube (fig. 7) is brass, near 2 inches in diameter, fixed in a circular collar of mahogany, with a groove on the out-<pb n="109"/><cb/> side of its periphery, denoted by 2, 3, and connected by a cat-gut to the pulley 4 on the upper part; which turning round at pleasure, by the pin 5 within, in a square frame, may be easily adjusted to a hole in the shutter of a window, by the screws 1, 1, so closely that no light can enter the room but through the tube of the instrument. The mirror G is fastened to the frame by hinges, on the side that goes without the window: this glass, by means of a jointed brass wire, 6, 7, and the screw H 8, coming through the frame, may be moved either vertically or horizontally, to throw the sun's rays through the brass tube into the darkened room. The end of the brass tube without the shutter has a convex lens, 5, to collect the rays thrown on it by the glass G, and bring them to a focus in the other part, where D is a tube sliding in and out, to adjust the object to a due distance from the focus. And to the end G of another tube F, is screwed one of Wilson's simple pocket Microlcopes, containing the object to be magnified in a slider; and by tube F, sliding on the small end E, of the other tube D, it is brought to a true focal distance.</p><p>The Solar Microscope has been introduced into the small and portable Camera Obscura, as well as the large one: and if the image be received upon a piece of half-ground glass, shaded from the light of the sun, it will be sufficiently visible. Mr. Lieberkuhn made considerable improvements in his Solar Microscope, particularly in adapting it to the viewing of opaque objects; and M. Aepinus, Nov. Com. Petrop. vol. 9, pa. 326, has contrived, by throwing the light upon the foreside of any object, before it is transmitted through the object lens, to represent all kinds of objects by it with equal advantage. In this improvement, the body of the common Solar Microscope is retained, and only an addition made of two brass plates, AB, AC, (fig 8), joined by a hinge, and held at a proper distance by a screw. A section of these plates, and of all the necessary parts of the instrument, may be seen in fig. 9, where <hi rend="italics">a c</hi> represent rays of the sun converging from the illuminating lens, and falling upon the mirror <hi rend="italics">bd,</hi> which is fixed to the nearer of the brass plates. From this they are thrown upon the object at <hi rend="italics">ef,</hi> and are thence transmitted through the object lens at K, and a perforation in the farther plate, upon a sereen, as usual. The use of the screen <hi rend="italics">n</hi> is to vary the distance of the two plates, and thereby to adjust the mirror to the object with the greatest exactness. M. Euler also contrived a method of introducing vision by reslected light into this Microscope.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Microscope</hi> <hi rend="italics">for Opaque Objects</hi> was also invented by M. Lieberkuhn, about the same time with the former, and remedies the inconvenience of having the dark side of an object next the eye; for by means of a concave speculum of silver, highly polished, having a magnifying lens placed in its centre, the object is so strongly illuminated, that it may be examined with ease. A convenient apparatus of this kind, with 4 different sp culums and magnifiers of different powers, was broug<*> to perfection by Mr. Cuff. Philos. Trans. number 45<*> § 9.</p><p><hi rend="smallcaps">Micros<*>pic</hi> <hi rend="italics">Objects.</hi> All things too minute to be viewed <*>stinctly by the naked eye, are proper objects for the Microscope. Dr. Hook has distinguished them into these three general kinds; viz, exceeding<cb/> small bodies, exceeeding small pores, or exceeding small motions. The small bodies may be seeds, insects, animalcules, sands, salts, &c: the pores may be the interstices between the solid parts of bodies, as in stones, minerals, shells, &c. or the mouths of minute vessels in vegetables, or the pores of the skin, bones, and other parts of animals: the small motions, may be the movements of the several parts or members of minute animals, or the motion of the fluids, contained either in animal or vegetable bodies. Under one or other of these three general heads, almost every thing about us affords matter of observation, and may conduce both to our amusement and instruction.</p><p>Great caution is to be used in forming a judgment on what is seen by the Microscope, if the objects are extended or contracted by force or dryness.</p><p>Nothing can be determined about them, without making the proper allowances; and different lights and positions will often shew the same object as very different from itself. There is no advantage in any greater magnifier than such as is capable of shewing the object in view distinctly; and the less the glass magnisies, the more pleasantly the object is always seen.</p><p>The colours of objects are very little to be depended on, as seen by the Microscope; for their several component particles, being thus removed to great distances from one another, may give reflections very different from what they would, if seen by the naked eye.</p><p>The motions of living creatures too, or of the fluids contained in their bodies, are by no means to be hastily judged of, from what we see by the Microscope, without duc consideration; for as the moving body, and the space in which it moves, are magnified, the motion must also be magnified; and therefore that rapidity with which the blood seems to pass through the vessels of small animals, must be judged of accordingly. Baker on the Microscope, pa. 52, 62, &c. See also an elegant work on this subject, lately published by that ingenious optician Mr. George Adams.</p><p>MIDDLE <hi rend="italics">Latitude,</hi> is half the sum of two given latitudes; or the arithmetical mean, or the middle between two parallels of latitudc. Therefore,</p><p>If the latitudes be of the same name, either both north or both south, add the one number to the other, and divide the sum by 2; the quotient is the middle latitudes, which is of the same name with the two given latitudes. But</p><p>If the latitudes be of different names, the one north and the other south; subtract the less from the greater, and divide the remainder by 2, so shall the quotient be the middle latitude, of the same name with the greater of the two. <table><row role="data"><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Ex.</hi> 1.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2 align=center" role="data"><hi rend="italics">Ex.</hi> 2.</cell></row><row role="data"><cell cols="1" rows="1" role="data">One lat.</cell><cell cols="1" rows="1" role="data">35°</cell><cell cols="1" rows="1" role="data">27′ N.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35°</cell><cell cols="1" rows="1" role="data">27′ S.</cell></row><row role="data"><cell cols="1" rows="1" role="data">the other</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">13   N.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">13   N.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       2 )</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">2 )</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mid. lat.</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">20 N.</cell><cell cols="1" rows="1" rend="align=right" role="data">Mid. lat.</cell><cell cols="1" rows="1" role="data">  7</cell><cell cols="1" rows="1" role="data">  7   S.</cell></row></table></p><p><hi rend="smallcaps">Middle</hi> <hi rend="italics">Latitude Sailing,</hi> is a method of resolving the cases of globular sailing, by means of the Middle Latitude, on the principles of plane and parallel sailing jointly.<pb n="110"/><cb/></p><p>This method is not quite accurate, yet often agrees pretty nearly with Mercator's Sailing, and is founded on the following principle, viz, That the departure is accounted a meridional distance in the middle latitude between the latitude sailed from and the latitude arrived at.</p><p>This artifice seems to have been invented, on account of the easy manner in which the several cases may be resolved by the Traverse Table, and to serve where a table of meridional parts is wanting. It is sufficiently near the truth either when the two parallels are near the equator, or not far distant from one another, in any latitude. It is performed by these two rules: <table><row role="data"><cell cols="1" rows="1" role="data">1.</cell><cell cols="1" rows="1" role="data">As the cosine of the middle latitude</cell><cell cols="1" rows="1" role="data"> :</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Is to radius</cell><cell cols="1" rows="1" role="data"> : :</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">So is the departure</cell><cell cols="1" rows="1" role="data"> :</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">To the difference of longitude</cell><cell cols="1" rows="1" role="data"> .</cell></row><row role="data"><cell cols="1" rows="1" role="data">2.</cell><cell cols="1" rows="1" role="data">As the cosine of the middle latitude</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Is to the tangent of the course</cell><cell cols="1" rows="1" role="data">: :</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">So is the difference of latitude</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">To the difference of longitude</cell><cell cols="1" rows="1" role="data">.</cell></row></table></p><p><hi rend="italics">Ex.</hi> A ship sails from latitude 37° north, steering constantly N. 33° 19′ east, for 8 days, when she was found in latitude 51° 18′ north; required her difference of longitude. <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">51°</cell><cell cols="1" rows="1" role="data">18′</cell><cell cols="1" rows="1" rend="align=center" role="data">51° 18′</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" rend="align=center" role="data">37 00</cell></row><row role="data"><cell cols="1" rows="1" role="data">        2 )</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" rend="align=center" role="data">Diff. lat. 14 18 = 858 m.</cell></row><row role="data"><cell cols="1" rows="1" role="data">As cos. mid. l.</cell><cell cols="1" rows="1" role="data"> 44</cell><cell cols="1" rows="1" role="data">09</cell><cell cols="1" rows="1" rend="align=center" role="data">0.14417</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. cour.</cell><cell cols="1" rows="1" role="data"> 33</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" rend="align=center" role="data">9.81776</cell></row><row role="data"><cell cols="1" rows="1" role="data">So diff. lat.</cell><cell cols="1" rows="1" role="data">858</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">2.93349</cell></row><row role="data"><cell cols="1" rows="1" role="data">To diff. long.</cell><cell cols="1" rows="1" role="data">786</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">2.89542</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=4" role="data">or 13° 6′ diff. of long. sought.</cell></row></table></p><p><hi rend="smallcaps">Middle</hi> <hi rend="italics">Region.</hi> See <hi rend="smallcaps">Region.</hi></p><p>MID-<hi rend="smallcaps">Heaven</hi>, <hi rend="italics">Medium Cœli,</hi> is that point of the ecliptic which culminates, or is highest, or is in the meridian at any time.</p><p>MIDSUMMER-<hi rend="italics">Day,</hi> is held on the 24th of June, the same day as the Nativity of St. John the Baptist is held.</p></div1><div1 part="N" n="MILE" org="uniform" sample="complete" type="entry"><head>MILE</head><p>, a long measure, by which the English, Italians, and some other nations, use to express the distance between places: the same as the French use the word <hi rend="italics">League.</hi></p><p>The Mile is of different lengths in different countries, The geographical, or Italian Mile, contains 1000 geometrical paces, <hi rend="italics">mille passus,</hi> whence the term Mile is derived. The English Mile consists of 8 furlongs, each furlong of 40 poles, and each pole of 16 1/2 feet: so that the Mile is = 8 furlongs = 320 poles = 1760 yards = 5280 feet.</p><p>The following table shews the length of the Mile, or league, in the principal nations of Europe, expressed in geometrical paces; <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Geomet. Paces.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Mile of Russa</cell><cell cols="1" rows="1" role="data">750</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Italy</cell><cell cols="1" rows="1" role="data">1000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of England,</cell><cell cols="1" rows="1" role="data">1200</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Scotland and Ireland</cell><cell cols="1" rows="1" role="data">1500</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Old League of France</cell><cell cols="1" rows="1" role="data">1500</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Small League, ibid.</cell><cell cols="1" rows="1" role="data">2000</cell></row></table><cb/> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Geomet. Paces.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Mean League of France</cell><cell cols="1" rows="1" role="data">2500</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Great League, ibid.</cell><cell cols="1" rows="1" role="data">3000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Mile of Poland</cell><cell cols="1" rows="1" role="data">3000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Spain</cell><cell cols="1" rows="1" role="data">3428</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Germany</cell><cell cols="1" rows="1" role="data">4000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Sweden</cell><cell cols="1" rows="1" role="data">5000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Denmark</cell><cell cols="1" rows="1" role="data">5000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of Hungary</cell><cell cols="1" rows="1" role="data">6000</cell></row></table></p><p>MILITARY <hi rend="italics">Architecture.</hi> The same with Fortification.</p><p>MILKY <hi rend="smallcaps">Way</hi>, <hi rend="italics">Via Lactea,</hi> or <hi rend="italics">Galaxy,</hi> a broad track or path, encompassing the whole heavens, distinguishable by its white appearance, whence it obtains the name. It extends itself in some parts by a double path, but for the most part it is single. Its course lies through the constellations Cassiopeia, Cygnus, Aquila, Perseus, Andromeda, part of Ophiucus and Gemini, in the northern hemisphere; and in the southern, it takes in part of Scorpio, Sagittarius, Centaurus, the Argonavis, and the Ara. There are some traces of the same kind of light about the south pole, but they are small in comparison of this: these are called by some, luminous spaces, and Magellanic clouds; but they seem to be of the same kind with the Milky way.</p><p>The Milky way has been ascribed to various causes. The Ancients fabled, that it proceeded from a stream of milk, spilt from the breast of Juno, when she pushed away the infant Hercules, whom Jupiter laid to her breast to render him immortal. Some again, as Aristotle, &c, imagined that this path consisted only of a certain exhalation hanging in the air; while Metrodorus, and some Pythagoreans, thought the sun had once gone in this track, instead of the ecliptic; and consequently that its whiteness proceeds from the remains of his light. But it is now well known, by the help of telescopes, that this track in the heavens consists of an immense multitude of stars, seemingly very close together, whose mingled light gives this appearance of whiteness; by Milton beautifully described as a path “powdered with stars.”</p></div1><div1 part="N" n="MILL" org="uniform" sample="complete" type="entry"><head>MILL</head><p>, properly denotes a machine for grinding corn, &c; but in a more general signisication, is applied to all machines whose action depends on a circular motion. Of these there are several kinds, according to the various methods of applying the moving power; as water-mills, wind-mills, horse-mills, handmills, &c, and even steam-mills, or such as are worked by the force of steam; as that noble structure that was erected near Blackfriars Bridge, called the Albion Mills, but lately destroyed by sire.</p><p>The water acts both by its impulse and weight in an overshot water-mill, but only by its impulse in an undershot one; but here the velocity is greater, because the water is suffered to descend to a greater depth before it strikes the wheel. Mr. Ferguson observes, that where there is but a small quantity of water, and a fall great enough for the wheel to lie under it, the bucket or overshot wheel is always used: but where there is a large body of water, with a little fall, the breast or float-board wheel must take place: and where there is a large supply of water, as a river, or large stream or brook, with very little fall, then the undershot wheel is the easiest, cheapest, and most simple struc- ture.<pb n="111"/><cb/></p><p>Dr. Desaguliers, having had occasion to examine many undershot and overshot Mills, generally found that a well made overshot Mill ground as much corn, in the same time, as an undershot Mill does with ten times as much water; supposing the fall of water at the overshot to be 20 feet, and at the undershot about 6 or 7 feet: and he generally observed that the wheel of the overshot Mill was of 15 or 16 feet diameter, with a head of water of 4 or 5 feet, to drive the water into the buckets with some momentum.</p><p>In Water-mills, some few have given the preference to the undershot wheel, but most writers prefer the overshot one. M. Belidor greatly preferred the undershot to any other construction. He had even concluded, that water applied in this way will do more than six times the work of an overshot wheel; while Dr. Desaguliers, in overthrowing Belidor's position, determined that an overshot wheel would do ten times the work of an undershot wheel with an equal quantity of water. So that between these two celebrated authors, there is a difference of no less than 60 to <*>. In consequence of such monstrous disagreement, Mr. Smeaton began the course of experiments mentioned below.</p><p>In the Philos. Trans. vol. 51, for the year 1759, we have a large paper with experiments on Mills turned both by water and wind, by that ingenious and experienced engineer Mr. Smeaton. From those experiments it appears, pa. 129, that the effects obtained by the overshot wheel are generally 4 or 5 times as great as those with the undershot wheel, in the same time, with the same expence of water, descending from the same height above the bottom of the wheels; or that the former performs the same effect as the latter, in the same time, with an expence of only one-4th or one- 5th of the water, from the same head or height. And this advantage seems to arise from the water lodging in the buckets, and so carrying the wheel about by their weight. But, in pa. 130, Mr. Smeaton reckons the effect of overshot only double to that of the undershot wheel. And hence he infers, in general, “that the higher the wheel is in proportion to the whole descent, the greater will be the effect; because it depends less upon the impulse of the head, and more upon the gravity of the water in the buckets. However, as every thing has its limits, so has this; for thus much is desirable, that the water should have somewhat greater velocity, than the circumference of the wheel, in coming thereon; otherwise the wheel will not only be retarded, by the buckets striking the water, but thereby dashing a part of it over, so much of the power is lost.” He is farther of opinion, that the best velocity for an overshot wheel is when its circumference moves at the rate of about 3 feet in a second of time. See <hi rend="smallcaps">Wind Mill.</hi></p><p>Considerable differences have also arisen as to the mathematical theory of the force of water striking the floats of a wheel in motion. M. Parent, Maclaurin, Desaguliers, &c, have determined, by calculation, that a wheel works to the greatest effect, when its velocity is equal to one-third of the velocity of the water which strikes it; or that the greatest velocity that the wheel acquires, is one-third of that of the water. And this determination, which has been followed by all mathematicians till very lately, necessarily results from a<cb/> position which they assume, viz, that the force of the water against the wheel, is proportional to the square of its relative velocity, or of the difference between the absolute velocity of the water and that of the wheel. And this position is itself an inference which they make from the force of water striking a body at rest, being as the square of the velocity, because the force of each particle is as the velocity it strikes with, and the number of particles or the whole quantity that strikes is also as the same velocity. But when the water strikes a body in motion, the quantity of it that strikes is still as the absolute velocity of the water, though the force of each particle be only as the relative velocity, or that with which it strikes. Hence it follows, that the whole force or effect is in the compound ratio of the absolute and relative velocities of the water; and therefore is greater than the before mentioned effect or force, in the ratio of the absolute to the relative velocity. The effect of this correction is, that the maximum velocity of the wheel becomes one-half the velocity of the water, instead of one-third of it only: a determination which nearly agrees with the best experiments, as those of Mr. Smeaton.</p><p>This correction has been lately made by Mr. W. Waring, in the 3d volume of the Transactions of the American Philosophical Society, pa. 144. This ingenious writer says, ‘Being lately requested to make some calculations relative to Mills, particularly Dr. Barker's construction as improved by James Rumsey, I found more difficulty in the attempt than I at first expected. It appeared necessary to investigate new theorems for the purpose, as there are circumstances peculiar to this construction, which are not noticed, I believe, by any author; and the theory of Mills, as hitherto published, is very imperfect, which <*> take to be the reason it has been of so little use to practical mechanics.</p><p>‘The first step, then, toward calculating the power of any water-mill (or wind-mill) or proportioning their parts and velocities to the greatest advantage, seems to be, <hi rend="center">‘<hi rend="italics">The Correction of an Essential Mislake adopted by Writers on the Theory of Mills.</hi></hi></p><p>‘This is attempted with all the deference due to eminent authors, whose ingenious labours have justly raised their reputation and advanced the sciences; but when any wrong principles are successively published by a feries of such pens, they are the more implicitly received, and more particularly claim a public rectification; which must be pleasing, even to these candid writers themselves.’</p><p>A very ingenious writer in England, ‘in his masterly treatife on the rectilinear motion and rotation of bodies, published so lately as 1784, continues this oversight, with its pernicious consequences, through his propositions and corollaries (pa. 275 to 284), although he knew the theory was suspected: for he observes (pa. 382) “Mr. Smeaton in his paper on mechanic “power (published in the Philosophical Transactions “for the year 1776) allows, that the theory usually “given will not correspond with matter of fact, when “compared with the motion of machines; and seems “to attribute this d sagreement, rather to desiciency “in the theory, thanito the obstacles which have pre-<pb n="112"/><cb/> “vented the application of it to the complicated mo“tion of engines, &c. In order to satisfy himself con“cerning the reason of this disagreement, he construct“ed a set of experiments, which, from the known “abilities and ingenuity of the author, certainly de“serve great consideration and attention from every “one who is interested in these inquiries.” ‘And notwithstanding the same learned author says, “The evidence upon which the theory rests is scarcely less than mathematical;” I am sorry to find, in the present state of the sciences, one of his abilities concluding (pa. 380) “It is not probable that the theory of motion, however incontestible its principles may be, can afford much assistance to the practical mechanic,” although indeed his theory, compared with the above cited experiments, might suggest such an inference. But to come to the point, I would just premise these <hi rend="center"><hi rend="italics">Desinilions.</hi></hi></p><p>‘If a stream of water impinge against a wheel in motion, there are three different velocities to be considered, appertaining thereto, viz,</p><p>First, the absolute velocity of the water;</p><p>Second, the absolute velocity of the wheel;</p><p>Third, the relative velocity of the water to that of the wheel, i. e. the difference of the absolute velocities, or the velocity with which the water overtakes or strikes the wheel.’</p><p>‘Now the mistake consists in supposing the momentum or force of the water against the wheel, to be in the <hi rend="italics">duplicate ratio of the relative velocity:</hi> Whereas, <hi rend="center"><hi rend="smallcaps">Prop.</hi> I.</hi></p><p>‘The force of an Invariable Stream, impinging against a Mill-wheel in Motion, is in the <hi rend="italics">Simple Direct Proportion of the Relative Velocity.</hi>’</p><p>‘For, if the relative velocity of a fluid against a single plane be varied, either by the motion of the plane, or of the fluid from a given aperture, or both, then, the number of particles acting on the plane in a given time, and likewise the momentum of each particle, being respectively as the relative velocity, the force on both these accounts, must be in the <hi rend="italics">duplicate</hi> ratio of the relative velocity, agreeably to the common theory, with respect to this <hi rend="italics">single plane:</hi> but, the number of these planes, or parts of the wheel acted on in a given time, will be as the velocity of the wheel, or <hi rend="italics">inversely as the relative velocity;</hi> therefore, the moving force of the wheel must be in the simple direct ratio of the relative velocity. Q. E. D.</p><p>‘Or the proposition is manifest from this consideration; that, while the stream is invariable, whatever be the velocity of the wheel, the same number of particles or quantity of the fluid, must strike it somewhere or other in a given time; consequently the variation of force is <hi rend="italics">only</hi> on account of the varied impingent velocity of the same body, occasioned by a change of motion in the wheel; that is, the momentum is as the relative velocity.’</p><p>‘Now, this true principle substituted for the erroneous one in use, will bring the theory to agree remarkably with the notable experiments of the ingenious<cb/> Smeaton, before mentioned, published in the Philosophical Transactions of the Royal Society of London for the year 1751, vol. 51, for which the honorary annual medal was adjudged by the society, and presented to the author by their president. An instance or two of the importance of this correction may be adduced as below.’ <hi rend="center"><hi rend="smallcaps">Prop.</hi> II.</hi></p><p>‘The velocity of a wheel, moved by the impact of a stream, must be half the velocity of the fluid, to produce the greatest possible effect.—For let V = the velocity, <hi rend="italics">m</hi> = the momentum of the fluid; <hi rend="italics">v</hi> = the velocity, <hi rend="italics">p</hi> = the power of the wheel. Then V - <hi rend="italics">v</hi> = the relative velocity, by def. 3d; and as (prop. 1); this multiplied by <hi rend="italics">v,</hi> gives maximum; hence a maximum, and its fluxion (<hi rend="italics">v</hi> being the variable quantity) is ; therefore , that is, the velocity of the wheel = half that of the fluid, at the place of impact, when the effect is a maximum. Q. E. D.’</p><p>‘The usual theory gives ; where the error is not less than one third of the true velocity of the wheel.’</p><p>‘This proposition is applicable to undershot wheels, and corresponds with the accurate experiments before cited, as appears from the author's conclusion (Philos. Trans. for 1776, pa. 457), viz, “The velocity of the “wheel, which according to M. Parent's determina“tion, adopted by Desaguliers and Maclaurin, ought to “be no more than one third of that of the water, varies “at the maximum in the experiments of table 1, be“tween one third and one half; but in all the cases “there related, in which the most work is performed “in proportion to the water expended, and which ap“proach the nearest to the circumstances of great “works when properly executed, the maximum lies “much nearer one half than one third, <hi rend="italics">one half seeming “to be the true maximum,</hi> if nothing were lost by the “resistance of the air, the scattering of the water car“ried up by the wheel, &c.” Thus he fully shews the common theory to have been very defective; but, I believe, none have since pointed out wherein the deficiency lay, nor how to correct it; and now we see the agreement of the true theory with the result of his experiments.’ For another problem, <hi rend="center"><hi rend="smallcaps">Prob.</hi> III.</hi></p><p>‘Given, the momentum (<hi rend="italics">m</hi>) and velocity (V) of the fluid at I, the place of impact; the radius (R = IS) of the wheel ABC; the radius (<hi rend="italics">r</hi> = DS) of the small wheel DEF on the same axle or shaft; the weight (<hi rend="italics">w</hi>) or resistance to be overcome at D, and the friction (<hi rend="italics">f</hi>) or force necessary to move the wheel without the weight; required the velocity (<hi rend="italics">v</hi>) of the wheel &c.”</p><p>‘Here we have the acting force at I in the direction KI, as before (prop. 2). Nov the power<pb n="113"/><cb/> at I necessary to counterpoise the weight <hi rend="italics">w;</hi> hence the whole resistance opposed to the action <figure/> of the fluid at I; which deducted from the moving force, leaves the accelerating for<*>e of the machine; which, when the motion becomes uniform, will be evanescent or = 0; therefore , which gives the true velocity required; or, if we reject the friction, then is the theorem for the velocity of the wheel. This, by the common theory, would be , which is too little by . No wonder why we have hitherto derived so little advantage from the theory.’</p><p>‘<hi rend="smallcaps">Corol.</hi> 1. If the weight (<hi rend="italics">w</hi>) or resistance be required, such as just to admit of that velocity which would produce the greatest effect; then, by substituting (1/2)V for its equivalent <hi rend="italics">v</hi> (by prop. 2), we have ; hence ; or, if <hi rend="italics">f</hi> = 0, ; but theorists make this , where the error is .’</p><p>‘<hi rend="smallcaps">Corol.</hi> 2. We have also ; or, rejecting friction, , when the greatest effect is produced, instead of , as has been supposed: this is an important theorem in the construction of mills.’</p><p>In the same volume of the American Transactions, pa. 185, is another ingenious paper, by the same au-<cb/> thor, on the power and machinery of Dr. Barker's Mill, as improved by Mr. James Rumsey, with a description of it. This is a Mill turned by the resisting force of a stream of water that issues from an orifice, the rotatory part, in which that orifice is, being impelled the contrary way by its reaction against the stream that issues from it.</p><p>Mr. Ferguson has given the following directions for constructing water mills in the best manner; with a table of the several corresponding dimensions proper to a great variety of perpendicular falls of the water.</p><p>When the float-boards of the water-wheel move with a 3d part of the velocity of the water that acts upon them, the water has the greatest power to turn the Mill: and when the millstone makes about 60 turns in a minute, it is found to perform its work the best: for, when it makes but about 40 or 50, it grinds too slowly; and when it makes more than 70, it heats the meal too much, and cuts the bran so small that a great part of it mixes with the meal, and cannot be separated from it by sifting or boulting. Consequently the utmost perfection of mill-work lies in making the train so as that the millstone shall make about 60 turns in a minute when the water wheel moves with a 3d part of the velocity of the water. To have it so, observe the following rules:</p><p>1. Measure the perpendicular height of the fall of water, in feet, above the middle of the aperture, where it is let out to act by impulse against the floatboards on the lowest side of the undershot wheel.</p><p>2. Multiply that height of the fall in feet by the constant number 64 1/3, and extract the square root of the product, which will be the velocity of the water at the bottom of the fall, or the number of feet the water moves per second.</p><p>3. Divide the velocity of the water by 3; and the quotient will be the velocity of the floats of the wheel in feet per second.</p><p>4. Divide the circumference of the wheel in feet, by the velocity of its floats; and the quotient will be the number of seconds in one turn or revolution of the great water-wheel, on the axis of which is fixed the cogwheel that turns the trundle.</p><p>5. Divide 60 by the number of seconds in one turn of the water-wheel or cog-wheel; and the quotient will be the number of turns of either of these wheels in a minute.</p><p>6. Divide 60 (the number of turns the millstone ought to have in a minute) by the abovesaid number of turns; and the quotient will be the number of turns the millstone ought to have for one turn of the water or cog-wheel. Then,</p><p>7. As the required number of turns of the millstone in a minute is to the number of turns of the cogwheel in a minute, so must the number of cogs in the wheel be to the number of staves or rounds in the trundle on the axis of the millstone, in the nearest whole number that can be found.</p><p>By these rules the following table is calculated; in which, the diameter of the water-wheel is supposed 18 feet, and consequently its circumference 56 4/7 feet, and the diameter of the millstone is 5 feet.<pb n="114"/> <hi rend="center"><hi rend="italics">The <hi rend="smallcaps">Mill-Wright</hi>'s Table.</hi></hi> <table rend="border"><row role="data"><cell cols="1" rows="1" role="data">Perpendicular height of the fall of water.</cell><cell cols="1" rows="1" role="data">Velocity of the water in feet per second.</cell><cell cols="1" rows="1" role="data">Velocity of the wheel in feet per second.</cell><cell cols="1" rows="1" role="data">Number of turns of the wheel in a minute.</cell><cell cols="1" rows="1" role="data">Required n°. of turns of the millstone for each turn of the wheel.</cell><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">Nearest number of cogs and staves for that purpose.</cell><cell cols="1" rows="1" role="data">Number of turns of the millstone for one turn of the wheel by these cogs and staves.</cell><cell cols="1" rows="1" role="data">Number of turns of the millstone in a minute by these cogs and staves.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Cogs.</cell><cell cols="1" rows="1" role="data">Staves.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1 </cell><cell cols="1" rows="1" role="data">8.02</cell><cell cols="1" rows="1" role="data">2.67</cell><cell cols="1" rows="1" role="data">2.83</cell><cell cols="1" rows="1" role="data">21.20</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">6 </cell><cell cols="1" rows="1" role="data">21.17</cell><cell cols="1" rows="1" role="data">59.91</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2 </cell><cell cols="1" rows="1" role="data">11.40</cell><cell cols="1" rows="1" role="data">3.78</cell><cell cols="1" rows="1" role="data">4.00</cell><cell cols="1" rows="1" role="data">15.00</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">7 </cell><cell cols="1" rows="1" role="data">15.00</cell><cell cols="1" rows="1" role="data">60.00</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3 </cell><cell cols="1" rows="1" role="data">13.89</cell><cell cols="1" rows="1" role="data">4.63</cell><cell cols="1" rows="1" role="data">4.91</cell><cell cols="1" rows="1" role="data">12.22</cell><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" role="data">8 </cell><cell cols="1" rows="1" role="data">12.25</cell><cell cols="1" rows="1" role="data">60.14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4 </cell><cell cols="1" rows="1" role="data">16.04</cell><cell cols="1" rows="1" role="data">5.35</cell><cell cols="1" rows="1" role="data">5.67</cell><cell cols="1" rows="1" role="data">10.58</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">9 </cell><cell cols="1" rows="1" role="data">10.56</cell><cell cols="1" rows="1" role="data">59.87</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5 </cell><cell cols="1" rows="1" role="data">17.93</cell><cell cols="1" rows="1" role="data">5.98</cell><cell cols="1" rows="1" role="data">6.34</cell><cell cols="1" rows="1" role="data">9.46</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">9 </cell><cell cols="1" rows="1" role="data">9.44</cell><cell cols="1" rows="1" role="data">59.84</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6 </cell><cell cols="1" rows="1" role="data">19.64</cell><cell cols="1" rows="1" role="data">6.55</cell><cell cols="1" rows="1" role="data">6.94</cell><cell cols="1" rows="1" role="data">8.64</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">9 </cell><cell cols="1" rows="1" role="data">8.66</cell><cell cols="1" rows="1" role="data">60.10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7 </cell><cell cols="1" rows="1" role="data">21.21</cell><cell cols="1" rows="1" role="data">7.07</cell><cell cols="1" rows="1" role="data">7.50</cell><cell cols="1" rows="1" role="data">8.00</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">9 </cell><cell cols="1" rows="1" role="data">8.00</cell><cell cols="1" rows="1" role="data">60.00</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8 </cell><cell cols="1" rows="1" role="data">22.68</cell><cell cols="1" rows="1" role="data">7.56</cell><cell cols="1" rows="1" role="data">8.02</cell><cell cols="1" rows="1" role="data">7.48</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">9 </cell><cell cols="1" rows="1" role="data">7.44</cell><cell cols="1" rows="1" role="data">59.67</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9 </cell><cell cols="1" rows="1" role="data">24.05</cell><cell cols="1" rows="1" role="data">8.02</cell><cell cols="1" rows="1" role="data">8.51</cell><cell cols="1" rows="1" role="data">7.05</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">7.00</cell><cell cols="1" rows="1" role="data">59.57</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">25.35</cell><cell cols="1" rows="1" role="data">8.45</cell><cell cols="1" rows="1" role="data">8.97</cell><cell cols="1" rows="1" role="data">6.69</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6.70</cell><cell cols="1" rows="1" role="data">60.09</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">26.59</cell><cell cols="1" rows="1" role="data">8.86</cell><cell cols="1" rows="1" role="data">9.40</cell><cell cols="1" rows="1" role="data">6.38</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6.40</cell><cell cols="1" rows="1" role="data">60.16</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">27.77</cell><cell cols="1" rows="1" role="data">9.26</cell><cell cols="1" rows="1" role="data">9.82</cell><cell cols="1" rows="1" role="data">6.11</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6.10</cell><cell cols="1" rows="1" role="data">59.90</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">28.91</cell><cell cols="1" rows="1" role="data">9.64</cell><cell cols="1" rows="1" role="data">10.22</cell><cell cols="1" rows="1" role="data">5.87</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5.80</cell><cell cols="1" rows="1" role="data">60.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">30.00</cell><cell cols="1" rows="1" role="data">10.00</cell><cell cols="1" rows="1" role="data">10.60</cell><cell cols="1" rows="1" role="data">5.66</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5.60</cell><cell cols="1" rows="1" role="data">59.36</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">31.05</cell><cell cols="1" rows="1" role="data">10.35</cell><cell cols="1" rows="1" role="data">10.99</cell><cell cols="1" rows="1" role="data">5.46</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5.40</cell><cell cols="1" rows="1" role="data">60.48</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">32.07</cell><cell cols="1" rows="1" role="data">10.69</cell><cell cols="1" rows="1" role="data">11.34</cell><cell cols="1" rows="1" role="data">5.29</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5.30</cell><cell cols="1" rows="1" role="data">60.10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">33.06</cell><cell cols="1" rows="1" role="data">11.02</cell><cell cols="1" rows="1" role="data">11.70</cell><cell cols="1" rows="1" role="data">5.13</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5.10</cell><cell cols="1" rows="1" role="data">59.67</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">34.02</cell><cell cols="1" rows="1" role="data">11.34</cell><cell cols="1" rows="1" role="data">12.02</cell><cell cols="1" rows="1" role="data">4.99</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5.00</cell><cell cols="1" rows="1" role="data">60.10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">34.95</cell><cell cols="1" rows="1" role="data">11.65</cell><cell cols="1" rows="1" role="data">12.37</cell><cell cols="1" rows="1" role="data">4.85</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">4.80</cell><cell cols="1" rows="1" role="data">60.61</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">35.86</cell><cell cols="1" rows="1" role="data">11.92</cell><cell cols="1" rows="1" role="data">12.68</cell><cell cols="1" rows="1" role="data">4.73</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">4.70</cell><cell cols="1" rows="1" role="data">59.59</cell></row></table><cb/></p><p>For the theory and construction of Wind-mills, see <hi rend="smallcaps">Wind</hi>-<hi rend="italics">mill.</hi></p></div1><div1 part="N" n="MILLION" org="uniform" sample="complete" type="entry"><head>MILLION</head><p>, the number of ten hundred thousand, or a thousand times a thousand.</p></div1><div1 part="N" n="MINE" org="uniform" sample="complete" type="entry"><head>MINE</head><p>, in Fortification &c, is a subterraneous canal or passage, dug under any place or work intended to be blown up by gunpowder. The passage of a mine leading to the powder is called the <hi rend="italics">Gallery;</hi> and the extremity, or place where the powder is placed, is called the <hi rend="italics">Chamber.</hi> The line drawn from the centre of the chamber perpendicular to the nearest surface, is called the <hi rend="italics">Line of least Resistance;</hi> and the pit or hole, made by the mine when sprung, or blown up, is called the <hi rend="italics">Excavation.</hi></p><p>The Mines made by the besiegers in the attack of a place, are called simply <hi rend="italics">Mines;</hi> and those made by the besieged, <hi rend="italics">Counter-mines.</hi></p><p>The fire is conveyed to the Mine by a pipe or hose, made of coarse cloth, of about an inch and half in diameter, called <hi rend="italics">Saucisson,</hi> extending from the powder in the chamber to the beginning or entrance of the gallery, to the end of which is fixed a match, that the miner who sets fire to it may have time to retire before it reaches the chamber.</p><p>It is found by experiments, that the figure of the excavation made by the explosion of the powder, is nearly a paraboloid, having its focus in the centre of the powder, and its axis the line of least resistance; its diameter being more or less according to the quantity of the powder, to the same axis, or line of least resistance. Thus, M. Belidor lodged seven different<cb/> quantities of powder in as many different mines, of the same depth, or line of least resistance 10 feet; the charges and greatest diameters of the excavation, meaured after the explosion, were as follow: <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Powder.</cell><cell cols="1" rows="1" rend="align=center" role="data">Diam.</cell></row><row role="data"><cell cols="1" rows="1" role="data">1st</cell><cell cols="1" rows="1" role="data">120lb</cell><cell cols="1" rows="1" role="data">22 2/3 feet</cell></row><row role="data"><cell cols="1" rows="1" role="data">2d</cell><cell cols="1" rows="1" role="data">160</cell><cell cols="1" rows="1" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" role="data">3d</cell><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">29</cell></row><row role="data"><cell cols="1" rows="1" role="data">4th</cell><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" role="data">31 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data">5th</cell><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data">33 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">6th</cell><cell cols="1" rows="1" role="data">320</cell><cell cols="1" rows="1" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" role="data">7th</cell><cell cols="1" rows="1" role="data">360</cell><cell cols="1" rows="1" role="data">38</cell></row></table> From which experiments it appears that the excavation, or quantity of earth blown up, is in the same proportion with the quantity of powder; whence the charge of powder necessary to produce any other proposed effect, will be had by the rule of Proportion.</p><p><hi rend="smallcaps">Mine</hi>-<hi rend="italics">Dial,</hi> is a box and needle, with a brass ring divided into 360 degrees, with several dials graduated upon it, commonly made for the use of miners.</p></div1><div1 part="N" n="MINUTE" org="uniform" sample="complete" type="entry"><head>MINUTE</head><p>, is the 60th part of a degree, or of an hour. The minutes of a degree are marked with the acute accent, thus ′; the seconds by two, ″; the thirds by three, ‴. The minutes, seconds, thirds, &c, in time, are sometimes marked the same way; but, to avoid confusion, the better way is, by the initials of the words; as minutes <hi rend="sup">m</hi>, seconds <hi rend="sup">s</hi>, thirds <hi rend="sup">t</hi>, &c.</p><div2 part="N" n="Minute" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Minute</hi></head><p>, in Arehitecture, usually denotes the 60th part of a module, but sometimes only the 30th part.<pb n="115"/><cb/></p></div2></div1><div1 part="N" n="MIRROR" org="uniform" sample="complete" type="entry"><head>MIRROR</head><p>, a speculum, looking-glass, or any polished body, whose use is to sorm the images of distinct objects by reflexion of the rays of light.</p><p>Mirrors are either plane, convex, or concave. The first sort reflects the rays of light in a direction exactly similar to that in which they fall upon it, and therefore represents bodies of their natural magnitude. But the convex ones make the rays diverge much more than before reflexion, and therefore greatly diminish the images of those objects which they exhibit: while the concave ones, by collecting the rays into a focus, not only magnify the objects they shew, but will also burn very fiercely when exposed to the rays of the sun; and hence they are commonly known by the name of <hi rend="italics">burning Mirrors.</hi></p><p>In ancient times the Mirrors were made of some kind of metal; and from a passage in the Mosaic writings we learn, that the Mirrors used by the Jewish women, were made of brass; a practice doubtless learned from the Egyptians.</p><p>Any kind of metal, when well polished, will reflect very powerfully; but of all others, silver reflects the most, though it has always been too expensive a material for common use. Gold is also very powerful; and all metals, or even wood, gilt and polished, will act very powerfully as burning Mirrors. Even polished ivory, or straw nicely plaited together, will form Mirrors capable of burning, if on a large scale.</p><p>Since the invention of glass, and the application of quicksilver to it, have become generally known, it has been universally employed for those plane Mirrors used as ornaments to houses; but in making reflecting telescopes they have been found much inferior to metallic ones. It does not appear however that the same superiority belongs to the metallic burning Mirrors, considered merely as burning speculums; since the Mirror with which Mr. Macquer melted platina, though only 22 inches diameter, and made of quicksilvered glass, produced much greater effects than M. Villette's metal speculum, which was of a much larger size. It is very probable, however, that M. Villette's Mirror was not so well polished as it ought to have been; as the art of preparing the metal for taking the finest polish, has but lately been discovered, and published in the Philos. Transactions, by Dr. Mudge of Plymouth, and, after him, by Mr. Edwards, Dr. Herschel, &c.</p><p>Some of the more remarkable laws and phenomena of plane Mirrors, are as follow:</p><p>1. A spectator will see his image of the same size, and erect, but reversed as to right and left, and as far beyond the speculum as he is before it. As he moves to or from the speculum, his image will, at the same time, move towards or from the speculum also on the other side. In like manner if, while the spectator is at rest, an object be in motion, its image behind the speculum will be seen to move at the same rate. Also when the spectator moves, the images of objects that are at rest will appear to approach or recede from him, after the same manner as when he moves towards real objects.</p><p>2. If several Mirrors, or several fragments or pieces of Mirrors, be all disposed in the same plane, they will only exhibit an object once.</p><p>3. If two plane Mirrors, or speculums, meet in any<cb/> angle, the eye, placed within that angle, will see the image of an object placed within the same, as often repeated as there may be perpendiculars drawn determining the places of the images, and terminated without the angle. Hence, as the more perpendiculars, terminated without the angle, may be drawn as the angle is more acute; the acuter the angle, the more numerous the images. Thus, Z. Traber found, at an angle of one-3d of a circle, the image was represented twice, at 1/4th thrice, at 1/6th five times, and at (1/12)th eleven times.</p><p>Farther, if the Mirrors be placed upright, and so contracted; or if you retire from them, or approach to them, till the images reflected by them coalesce, or run into one, they will appear monstrously distorted. Thus, if they be at an angle somewhat greater than a right one, the image of one's face will appear with only one eye; if the angle be less than a right one, you will see 3 eyes, 2 noses, 2 mouths, &c. At an angle still less, the body will have two heads. At an angle somewhat greater than a right one, at the distance of 4 feet, the body will be headless, &c. Again, if the Mirrors be placed, the one parallel to the horizon, the other inclined to it, or declined from it, it is easy to perceive that the images will be still more romantic. Thus, one being declined from the horizon to an angle of 144 degrees, and the other inclined to it, a man sees himself standing with his head to another's feet.</p><p>Hence it appears how Mirrors may be managed in gardens, &c, so as to convert the images of those near them into monsters of various kinds; and since glass Mirrors will reslect the image of a lucid object twice or thrice, if a candle, &c, be placed in the angle between two Mirrors, it will be multiplied a great number of times. <hi rend="center"><hi rend="italics">Laws of Convex</hi> <hi rend="smallcaps">Mirrors.</hi></hi></p><p>1. In a spherical convex Mirror, the image is less than the object. And hence the use of such Mirrors in the art of painting, where objects are to be represented less than the life.</p><p>2. In a convex Mirror, the more remote the object, the less its image; also the smaller the Mirror, the less the image.</p><p>3. In a convex Mirror, the right hand is turned to the left, and the left to the right; and magnitudes perpendicular to the Mirror appear inverted.</p><p>4. The image of a right line, perpendicular to the Mirror, is a right line; but that of a right line oblique or parallel to the Mirror, is convex.</p><p>5. Rays reflected from a convex Mirror, diverge more than if reflected from a plane Mirror; and the smaller the sphere, the more the rays diverge. <hi rend="center"><hi rend="italics">Laws of Concave</hi> <hi rend="smallcaps">Mirrors.</hi></hi></p><p>The effects of concave Mirrors are, in general, the reverse of those of convex ones; rays being made to converge more, or diverge less than in plane Mirrors; the image is magnified, and the more so as the sphere is smaller; &c, &c.</p></div1><div1 part="N" n="MITRE" org="uniform" sample="complete" type="entry"><head>MITRE</head><p>, in Architecture, is the workmen's term for an angle that is just 45 degrees, or half a right angle. And if the angle be the half of this, or a quarter of a right angle, they call it a <hi rend="italics">half-mitre.</hi><pb n="116"/><cb/></p><p><hi rend="smallcaps">Mixt</hi> <hi rend="italics">Angle,</hi> or <hi rend="italics">Figure,</hi> is one contained by both right and curved lines.</p><p><hi rend="smallcaps">Mixt</hi> <hi rend="italics">Number,</hi> is one that is partly an integer, and partly a fraction; as 3 1/2.</p><p><hi rend="smallcaps">Mixt</hi> <hi rend="italics">Ratio,</hi> or <hi rend="italics">Proportion,</hi> is when the sum of the antecedent and consequent is compared with the difference of the antecedent and consequent; as if then</p></div1><div1 part="N" n="MOAT" org="uniform" sample="complete" type="entry"><head>MOAT</head><p>, in Fortification, a deep trench dug round a town or fortress, to be defended, on the outside of the wall, or rampart.</p><p>The breadth and depth of a Moat often depend on the nature of the soil; according as it is marshy, rocky, or the like. The brink of the Moat next the rampart, is called the Scarp; and the opposite side, the Counterscarp.</p><p><hi rend="italics">Dry</hi> <hi rend="smallcaps">Moat</hi>, is one that is without water; which ought to be deeper than one that has water, called a Wet Moat. A Dry Moat, or one that has a little water, has often a small notch or ditch run all along the middle of its bottom, called a Cuvette.</p><p><hi rend="italics">Flat-bottomed</hi> <hi rend="smallcaps">Moat</hi>, is that which has no sloping, its corners being somewhat rounded.</p><p><hi rend="italics">Lined</hi> <hi rend="smallcaps">Moat</hi>, is that whose scarp and counterscarp are cased with a wall of mason's work lying aslope.</p></div1><div1 part="N" n="MOBILE" org="uniform" sample="complete" type="entry"><head>MOBILE</head><p>, <hi rend="italics">Primum,</hi> in the Ancient Astronomy, was a 9th heaven, or sphere, conceived above those of the planets and sixed stars. It was supposed that this was the first mover, and carried all the lower spheres about with it; by its rapidity communicating to them a motion carrying them round in 24 hours. But the diurnal apparent revolution of the heavens is now better accounted for, by the rotation of the earth on its axis, without the assistance of any such Primum Mobile.</p></div1><div1 part="N" n="MOBILITY" org="uniform" sample="complete" type="entry"><head>MOBILITY</head><p>, an aptitude or facility to be moved.</p><p>The Mobility of Mercury is owing to the smallness and sphericity of its particles; and these also render its sixation so difficult.</p><p>The hypothesis of the Mobility of the earth is the most plausible, and is universally admitted by the later astronomers.</p><p>Pope Paul V. appointed commissioners to examine the opinion of Copernicus touching the Mobility of the earth. The result of their enquiry was, a prohibition to assert, not that the Mobility was possible, but that it was really true: that is, they allowed the Mobility of the earth to be held as an hypothesis, which gives an easy and sensible solution of the phenomena of the heavenly motions; but forbade the Mobility of the earth to be maintained as a thesis, or real effective thing; because they conceived it contrary to Scripture.</p></div1><div1 part="N" n="MODILLIONS" org="uniform" sample="complete" type="entry"><head>MODILLIONS</head><p>, small inverted consoles under the soffit or bottom of the drip, or of the corniche, seeming to support the projecture of the larmier, in the Ionic, Composite, and Corinthian orders.</p></div1><div1 part="N" n="MODULE" org="uniform" sample="complete" type="entry"><head>MODULE</head><p>, or Little Measure, in Architecture, a certain measure, taken at pleasure, for regulating the proportions of columns, and the symmetry or distribu-<cb/> tion of the whole building. Architects <*>ually choose the diameter, or the semidiameter, of the bottom of the column, for their Module; which they subdivide into minutes; for estimating all the other parts of the building by.</p></div1><div1 part="N" n="MOINEAU" org="uniform" sample="complete" type="entry"><head>MOINEAU</head><p>, a flat bastion raised before a curtin when it is too long, and the bastions of the angles too remote to be able to defend one another. Sometimes the Moineau is joined to the curtin, and sometimes it is divided from it by a moat. Here musquetry are placed to fire each way.</p></div1><div1 part="N" n="MOLYNEUX" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MOLYNEUX</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an excellent mathematician and astronomer, was born at Dublin in 1656. After the usual grammar education, which he had at home, he was entered of the university of that city. Here he distinguished himself by the probity of his manners, as well as by the strength of his parts; and having made a remarkable progress in academical learning, and particularly in the new philosophy, as it was then called, after four years spent in this university, he was sent over to London, where he was admitted into the Middle Temple in 1675. Here he spent three years, in the study of the laws of his country. But the bent of his genius lay strongly toward mathematical and philosophical studies; and even at the university he conceived a dislike to scholastic learning, and sell into the methods of lord Bacon.</p><p>Returning to Ireland in 1678, he shortly after married Lucy the danghter of Sir William Domville, the king's attorney-general. Being master of an easy fortune, he continued to indulge himself in prosecuting such branches of natural and experimental philosophy as were most agreeable to his fancy; in which astronomy having the greatest share, he began, about 1681, a literary correspondence with Mr. Flamsteed, the king's astronomer, which he kept up for several years. In 1683 he formed a design of erecting a Philosophical Society at Dublin, in imitation of the Royal Society at London; and, by the countenance and encouragement of Sir William Petty, who accepted the office of president, began a weekly meeting that year, when our author was appointed their first secretary.</p><p>Mr. Molyneux's reputation for learning recommended him, in 1684, to the notice and favour of the first great duke of Ormond, then lord-lieutenant of Ireland; by whose influence chiefly he was appointed that year, jointly with sir William Robinson, surveyor-general of the king's buildings and works, and chief engineer.</p><p>In 1685, he was chosen fellow of the Royal Society at London; and that year he was sent by the government to view the most considerable fortresses in Flanders. Accordingly he travelled through that country and Holland, with part of Germany and France; and carrying with him letters of recommendation from Flamsteed to Cassini, he was introduced to him, and others, the most eminent astronomers in the several places through which he passed.</p><p>Soon after his return from abroad, he printed at Dublin, in 1686, his <hi rend="italics">Sciothericum Telescopium,</hi> containing a Description of the Structure and Use of a Telescopic Dial, invented by him: another edition of which was published at London in 1700.</p><p>In 1688 the Philosophical Society of Dublin was broken up and dispersed by <*>e confusion of the times.<pb n="117"/><cb/> Mr. Molyneux had distinguished himself as a Member of it from the beginning, and presented several discourses upon curious subjects; some of which were transmitted to the Royal Society at London, and afterwards printed in the Philosophical Transactions. In 1689, among great numbers of other Protestants, he withdrew from the disturbances in Ireland, occasioned by the severities of Tyrconnel's government; and after a short stay at London, he fixed himself with his family at Chester. In this retirement, he employed himself in putting together the materials he had some time before prepared for his <hi rend="italics">Dioptrics,</hi> in which he was much assisted by Mr. Flamsteed; and in August 1690, he went to London to put it to the press, where the sheets were revised by Dr. Halley, who, at our author's request, gave leave for printing, in the appendix, his celebrated Theorem for finding the Foci of Optic Glasses. Accordingly the book came out, 1692, in 4to, under the title of “<hi rend="italics">Dioptrica Nova:</hi> a Treatise of Dioptrics, in two parts; wherein the various effects and appearances of spherical glasses, both convex and concave, single and combined, in telescopes and microscopes, together with their usefulness in many concerns of human life, are explained.” He gave it the title of <hi rend="italics">Dioptrica Nova,</hi> both because it was almost wholly new, very little being borrowed from other writers, and because it was the first book that appeared in English upon the subject. The work contains several of the most generally useful propositions for practice, demonstrated in a clear and easy manner, for which reason it was for many years used by the artificers: and the second part is very entertaining, especially in the history which he gives of the several optical instruments, and of the discoveries made by them.</p><p>Before he left Chester he lost his lady, who died soon after she had brought him a son. Illness had deprived her of her eye-sight 12 years before, that is, soon after her marriage; from which time she had been very sickly, and afflicted with great pains in her head.</p><p>As soon as the public tranquillity was settled in his native country, he returned home; and, upon the convening of a new parliament in 1692, was chosen one of the representatives for the city of Dublin. In the next parliament, in 1695, he was chosen to represent the university there, and continued to do so to the end of his life; that learned body having lately conferred on him the degree of doctor of laws. He was likewise nominated by the lord-lieutenant one of the commissioners for the forfeited estates, to which employment was annexed a salary of 5001. a year; but looking upon it as an invidious office, he declined it.</p><p>In 1698, he published “The Case of Ireland stated, in regard to its being bound by Acts of Parliament made in England:” in which it is supposed he has delivered all, or most, that can be said upon this subject, with great clearness and strength of reasoning.</p><p>Among many learned persons with whom he maintained correspondence and friendship, Mr. Looke was in a particular manner dear to him, as appears from their letters. In the above mentioned year, which was the last of our author's life, he made a journey to England, on purpose to pay a visit to that great man; and not long after his return to Ireland, he was seized with a fit of the stone, which terminated his exist- ence.<cb/></p><p>Besides the three works already mentioned, viz, the <hi rend="italics">Sciothericum Telescopium,</hi> the <hi rend="italics">Dioptrica Nova,</hi> and the <hi rend="italics">Case of Ireland stated;</hi> he published a great number of pieces in the Philosophical Transactions, which are contained in the volumes 14, 15, 16, 18, 19, 20, 21, 22, 23, 26, 29, several papers commonly in each volume.</p><p><hi rend="smallcaps">Molyneux</hi> <hi rend="italics">(Samuel</hi>), son of the former, was born at Chester in July 1689; and educated with great care by his father, according to the plan laid down by Locke on that subject. When his father died, he fell under the management of his uncle, Dr. Thomas Molyneux, an excellent scholar and physician at Dublin, and also an intimate friend of Mr. Locke, who executed his trust so well, that Mr. Molyneux became afterwards a most polite and accomplished gentleman, and was made secretary to George the 3d when prince of Wales. Astronomy and Optics being his favourite studies, as they had been his father's, he projected many schemes for the advancement of them, and was particularly employed in the years 1723, 1724, and 1725, in perfecting the method of making telescopes; one of which instruments, of his own making, he had presented to John the 5th, king of Portugal.</p><p>Being soon after appointed a commissioner of the admiralty, he became so engaged in public affairs, that he had not leisure to pursue those enquiries any farther, as he intended. He therefore gave his papers to Dr. Robert Smith, professor of astronomy at Cambridge, whom he invited to make use of his house and apparatus of instruments, in order to finish what he had left imperfect. But Mr. Molyneux dying soon after, Dr. Smith lost the opportunity; he however supplied what was wanting from M. Huygens and others, and published the whole in his “Complete Treatise of Optics.”</p></div1><div1 part="N" n="MOMENT" org="uniform" sample="complete" type="entry"><head>MOMENT</head><p>, in Time, is sometimes taken for an extremely small part of duration; but, more properly, it is only an instant or termination or limit in time, like a point in geometry. Maclaurin's Fluxions, vol. 1, pa. 245.</p><div2 part="N" n="Moments" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Moments</hi></head><p>, in the new Doctrine of Infinites, denote the indefinitely small parts of quantity; or they are the same with what are otherwise called infinitesimals, and differences, or increments and decrements; being the momentary increments-or decrements of quantity considered as in a continual flux.</p><p>Moments are the generative principles of magnitude: they have no determined magnitude of their own; but are only inceptive of magnitude.</p><p>Hence, as it is the same thing, if, instead of these Moments, the velocities of their increases and decreases be made use of, or the finite quantities that are proportional to such velocities; the method of proceeding which considers the motions, changes, or fluxions of quantities, is denominated, by Sir Isaac Newton, the Method of Fluxions.</p><p>Leibnitz, and most foreigners, considering these infinitely small parts, or insinitesimals, as the differences of two quantities; and thence endeavouring to find the differences of quantities, i. e. some Moments, or quantities indefinitely small, which taken an infinite number of times shall equal given quantities; call these Mo-<pb n="118"/><cb/> ments, Differences; and the method of procedure, the Differential Calculus.</p></div2><div2 part="N" n="Moment" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Moment</hi></head><p>, or <hi rend="italics">Momentum,</hi> in Mechanics, is the same thing with Impetus, or the quantity of motion in a moving body.</p><p>In comparing the motions of bodies, the ratio of their Momenta is always compounded of the quantity of matter and the celerity of the moving body: so that the momentum of any such body, may be considered as the rectangle or product of the quantity of matter and the velocity of the motion. As, if <hi rend="italics">b</hi> denote any body, or the quantity or mass of matter, and <hi rend="italics">v</hi> the velocity of its motion; then <hi rend="italics">bv</hi> will express, or be proportional to, its Momentum <hi rend="italics">m.</hi> Also if B be another body, and V its velocity; then its Momentum M, is as BV. So that, in general, , i. e. the Momenta are as the products of the mass and velocity. Hence, if the Momenta M and <hi rend="italics">m</hi> be equal, then shall the two products BV and <hi rend="italics">bv</hi> be equal also; and consequently , or the bodies will be to each other in the inverse or reciprocal ratio of their velocities; that is, either body is so much the greater as its velocity is less. And this force of Momentum is of a different kind from, and incomparably greater than, any mere dead weight, or pressure, whatever.</p><p>The Momentum also of any moving body, may be considered as the aggregate or sum of all the Momenta of the parts of that body; and therefore when the magnitudes and number of particles are the same, and also moved with the same celerity, then will the Momenta of the wholes be the same also.</p><p>MONADES. <hi rend="smallcaps">Digits.</hi></p></div2></div1><div1 part="N" n="MONOCEROS" org="uniform" sample="complete" type="entry"><head>MONOCEROS</head><p>, the <hi rend="italics">Unicorn,</hi> one of the new constellations of the northern hemisphere, or one of those which Hevelius has added to the 48 old asterisms, and formed out of the stellæ informes, or those which were not comprized within the outlines of any of the others. In Hevelius's catalogue, the Unicorn contains 19 stars, but in the Britannic catalogue 31.</p></div1><div1 part="N" n="MONOCHORD" org="uniform" sample="complete" type="entry"><head>MONOCHORD</head><p>, a musical instrument with only one string, used by the Ancients to try the variety and proportion of sounds. It was formed of a rule, divided and subdivided into several parts, on which there is a moveable string stretched over two bridges at the extremes of it. In the interval between these is a sliding or moveable bridge, by means of which, in applying it to the different divisions of the line, the sounds are found to bear the same proportion to each other, as the division of the line cut by the bridge. This instrument is also called the <hi rend="italics">harmonical canon,</hi> or the <hi rend="italics">canouical rule,</hi> because it serves to measure the degrees of gravity or acuteness. Ptolomy examines his harmonical intervals by the Monochord. When the chord was divided into two equal parts, so that the parts were as 1 to 1, they called them <hi rend="italics">unisons;</hi> but if they were as 2 to 1, they called them <hi rend="italics">octaves</hi> or <hi rend="italics">diapasons;</hi> when they were as 3 to 2, they called them <hi rend="italics">diapentes,</hi> or <hi rend="italics">fifths;</hi> if they were as 4 to 3, they called them <hi rend="italics">diatessarons,</hi> or <hi rend="italics">fourths;</hi> if the parts were as 5 to 4, they called them <hi rend="italics">diton,</hi> or <hi rend="italics">majorthird;</hi> but if they were as 6 to 5, they were called a <hi rend="italics">dcmi-diton,</hi> or <hi rend="italics">minor-third;</hi> and lastly, if the parts were as 24 to 25, a <hi rend="italics">demitone,</hi> or <hi rend="italics">dicze.</hi></p><p>The Monochord, being thus divided, was properly what they called a system, of which there were many<cb/> kinds, according to the different divisions of the Monochord.</p><p><hi rend="smallcaps">Monochord</hi> is also used for any musical instrument consisting of only one chord or string. Such is the Trump-marine.</p></div1><div1 part="N" n="MONOMIAL" org="uniform" sample="complete" type="entry"><head>MONOMIAL</head><p>, in Algebra, is a simple or single nomial, consisting of only one term; as <hi rend="italics">a</hi> or <hi rend="italics">ax,</hi> or <hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">bx</hi><hi rend="sup">3</hi>, &c.</p></div1><div1 part="N" n="MONOTRIGLYPH" org="uniform" sample="complete" type="entry"><head>MONOTRIGLYPH</head><p>, a term in Architecture, denoting the space of one triglyph between two pilasters, or two columns.</p></div1><div1 part="N" n="MONSOON" org="uniform" sample="complete" type="entry"><head>MONSOON</head><p>, a regular or periodical wind, that blows one way for 6 months together, and the contrary way the other 6 months of the year. These prevail in several parts of the eastern and southern oceans.</p></div1><div1 part="N" n="MONTH" org="uniform" sample="complete" type="entry"><head>MONTH</head><p>, the 12th part of the year, and is so called from the Moon, by whose motions it was regulated; being properly the time in which the moon runs through the zodiac. The lunar Month is either <hi rend="italics">illuminative, periodical,</hi> or <hi rend="italics">synodical.</hi></p><p><hi rend="italics">Illuminative</hi> <hi rend="smallcaps">Month</hi>, is the interval between the first appearance of one new moon and that of the next following. As the moon appears sometimes sooner after one change than after another, the quantity of the Illuminative Month is not always the same. The Turks and Arabs reckon by this Month.</p><p><hi rend="italics">Lunar Periodical</hi> <hi rend="smallcaps">Month</hi>, is the time in which the moon runs through the zodiac, or returns to the same point again; the quantity of which is 27days 7hrs 43m. 8 sec.</p><p><hi rend="italics">Lunar Synodical</hi> <hi rend="smallcaps">Month</hi>, called also a Lunation, is the time between two conjunctions of the moon with the sun, or between two new moons; the quantity of which is 29 days, 12 hours, 44m. 3 sec. 11 thirds.</p><p>The ancient Romans used Lunar Months, and made them alternately of 29 and 30 days: They marked the days of each Month by three terms, viz, Calends, Nones, and Ides.</p><p><hi rend="italics">Solar</hi> <hi rend="smallcaps">Month</hi>, is the time in which the sun runs through one entire sign of the ecliptic, the mean quantity of which is 30 days 10 hours 29 min. 5 sec. being the 12th part of 365 ds. 5 hrs. 49 min. the mean solar year.</p><p><hi rend="italics">Astronomical</hi> or <hi rend="italics">Natural</hi> <hi rend="smallcaps">Month</hi>, is that measured by some exact interval corresponding to the motion of the sun or moon. Such are the lunar and solar months above-mentioned.</p><p><hi rend="italics">Civil</hi> or <hi rend="italics">Common</hi> <hi rend="smallcaps">Month</hi>, is an interval of a certain number of whole days, approaching nearly to the quantity of some astronomical month. These may be either lunar or solar. The</p><p><hi rend="italics">Civil Lunar</hi> <hi rend="smallcaps">Month</hi>, consists alternately of 29 and 30 days. Thus will two Civil Months be equal to astronomical ones, abating for the odd minutes; and so the new moon will be kept to the first day of such Civil Months for a long time together. This was the Month in Civil or common use among the Jews, Greeks, and Romans, till the time of Julius Cæsar. The</p><p><hi rend="italics">Civil Solar</hi> <hi rend="smallcaps">Month</hi>, consisted alternately of 30 and 31 days, excepting one Month of the twelve, which consisted only of 29 days, but every 4th year of 30 days. And this form of Civil Months was introduced by Julius Cæsar. Under Augustus, the 6th Month,<pb n="119"/><cb/> till then from its place called Sextilis, received the name Augustus, now August, in honour of that prince; and, to make the compliment still the greater, a day was added to it; which made it consist of 31 days, though till then it had only contained 30 days; to compensate for which, a day was taken from February, making it consist of 28 days, and 29 every 4th year. And such are the Civil or Calendar Months now used through Europe.</p></div1><div1 part="N" n="MOON" org="uniform" sample="complete" type="entry"><head>MOON</head><p>, <hi rend="italics">Luna,</hi> <*>, one of the heavenly bodies, being a fatellite, or secondary planet to the earth, considered as a primary planet, about which she revolves in an elliptic orbit, or rather the earth and Moon revolve about a common centre of gravity, which is as much nearer to the earth's centre than to the Moon's, as the mass of the former exceeds that of the latter.</p><p>The mean time of a revolution of the Moon about the earth, from one new moon to another, when she overtakes the sun again, is 29d. 12h. 44m. 3s. 11th.; but she moves oncc round her own orbit in 27d. 7h. 43m. 8s. moving about 2290 miles every hour; and turns once round her axis exactly in the time that she goes round the earth, which is the reason that she shews always the same side towards us; and that her day and night taken together are just as long as our lunar month.</p><p>The mean distance of the Moon from the earth is 60 1/2 radii, or 30 1/4 diameters, of the earth; which is about 240,000 miles. The mean excentricity of her orbit is 55/1000, or 1/18th nearly of her mean distance, amounting to about 13,000 miles.</p><p>The Moon's diameter is to that of the earth, as 20 to 73, or nearly as 3 to 11, or 1 to 3 2/3; and therefore it is equal to 2180 miles: her mean apparent diameter is 31′ 16″ 1/2, that of the sun being 32′ 12″. The surface of the Moon is to the surface of the earth, as 1 to 13 1/4, or as 3 to 40; so that the earth reflects 13 times as much light upon the Moon, as she does upon the earth; and the solid content to that of the earth. as 3 to 146, or as 1 to 48 2/3. The density of the Moon's body is to that of the earth, as 5 to 4; and therefore her quantity of matter to that of the earth, as 1 to 39 very nearly: the force of gravity on her furface, is to that on the earth, as 100 to 293. The Moon has little or no difference of seasons; because her axis is almost perpendicular to the ecliptic.</p><p><hi rend="italics">Phenomena and Phases of the</hi> <hi rend="smallcaps">Moon.</hi> The Moon being a dark, opaque, spherical body, only shining with the light she receives from the sun, hence only that half turned towards him, at any instant, can be illuminated, the opposite half remaining in its native darkness: then as the face of the Moon visible on our earth, is that part of her body turned towards us; whence, according to the various positions of the Moon, with respect to the earth and sun, we perceive different degrees of illumination; sometimes a large and sometimes a less portion of the enlightened surface being visible: And hence the Moon appears sometimes increasing, then waning; sometimes horned, then half round; sometimes gibbous, then full and round. This may be easily illustrated by means of an ivory ball, which being before a candle in various positions, will present a greater or less portion of its illuminated hemisphere to the view of the observer, according to its situation in moving it round the candle.<cb/></p><p>The same phases may be otherwise exhibited thus: Let S represent the sun, T the earth, and ABCD &c the Moo<*>'s orbit. (Plate xv, fig. 3.) Now, when the Moon is at A, in conjunction with the sun S, her dark side being entirely turned towards the earth, she will be invisible, as at <hi rend="italics">a,</hi> and is then called the new Moon. When she comes to her first octant at B, or has run through the 8th part of her orbit, a quarter of her enlightened hemisphere will be turned towards the earth, and she will then appear horned, as at <hi rend="italics">b.</hi> When she has run through the quarter of her orbit, and arrived at C, she shews us the half of her enlightened hemisphere, as at <hi rend="italics">c,</hi> when it is said she is one half full. At D she is in her 2d octant, and by shewing us more of her enlightened hemisphere than at C, she appears gibbous, as at <hi rend="italics">d.</hi> At her opposition at E her whole enlightened side is turned towards the earth, when she appears round, as at <hi rend="italics">e,</hi> and she is said to be full; having increased all the way round from A to E. On the other side she decreases again all the way from E to A: thus, in her 3d octant at F, part of her dark side being turned towards the earth, she again appears gibbous, as at <hi rend="italics">f.</hi> At G she appears still farther decreased, shewing again just one half of her illuminated side, as at <hi rend="italics">g.</hi> But when she comes to her 4th octant at H, she presents only a quarter of her enlightened hemisphere, and she again appears horned, as at <hi rend="italics">h.</hi> And at A, having now completed her course, she again disappears, or becomes a new moon again, as at first. And the earth presents all the very same phases to a spectator in the Moon, as she does to us, but only in a contrary order, the one being full when the other changes, &c.</p><p><hi rend="italics">The Motions of the</hi> <hi rend="smallcaps">Moon</hi> are most of them very irregular, and very considerably so. The only equable motion she has, is her revolution on her own axis, in the space of a month, or time in which she moves round the earth; which is the reason that she always turns the same face towards us.</p><p>This exposure of the same face is not so uniformly so however, but that she turns sometimes a little more of the one side, and sometimes of the other, called the Moon's Libration; and also shews sometimes a little more towards one pole, and sometimes towards the other, by a motion like a kind of Wavering, or Vacillation. The former of these motions happens from this: the Moon's rotation on her axis is equable or uniform; while her motion in her orbit is unequal, being quickest when the Moon is in her perigee, and slowest when in the apogee, like all other planetary motions; which causes that sometimes more of one side is turned to the earth, and sometimes of the other. And the other irregularity arises from this: that the axis of the Moon is not perpendicular, but a little inclined to the plane of her orbit: and as this axis maintains its parallelism, in the Moon's motion round the earth; it must necessarily change its situation, in respect of an observer on the earth; whence it happens that sometimes the one, and sometimes the other pole of the Moon becomes visible.</p><p>The very orbit of the Moon is changeable, and does not always persevere in the same figure: for though her orbit be elliptical, or nearly so, having the earth in one focus, the excentricity of the ellipse is varied, being sometimes increased, and sometimes diminished; viz,<pb n="120"/><cb/> being greatest when the line of the apses coincides with that of the syzygies, and least when these lines are at right angles to each other.</p><p>Nor is the apogee of the Moon without an irregularity; being found to move forward, when it coincides with the line of the syzygies; and backward, when it cuts that line at right angles. Neither is this progress or regress uniform; for in the conjunction or opposition, it goes briskly forward; and in the quadratures, it either moves slowly forward, stands still, or goes backward.</p><p>The motion of the nodes is also variable; being quicker and slower in different positions.</p><p><hi rend="italics">The Physical Cause of the</hi> <hi rend="smallcaps">Moon's</hi> <hi rend="italics">Motion,</hi> about the earth, is the same as that of all the primary planets about the sun, and of the satellites about their primaries, viz, the mutual attraction between the earth and Moon.</p><p>As for the particular irregularities in the Moon's motion, to which the earth and other planets are not subject, they arise from the sun which acts on, and disturbs her in her ordinary course through her orbit; and are all mechanically deducible from the same great law by which her general motion is directed, viz, the law of gravitation and attraction. The other secondary planets, as those of Jupiter, Saturn, &c, are also subject to the like irregularities with the Moon; as they are exposed to the same perturbating or disturbing force of the sun; but their distance secures them from being so greatly affected as the Moon is, and also from being so well observed by us.</p><p>For a familar idea of this matter, it must first be considered, that if the sun acted equally on the earth and Moon, and always in parallel lines, this action would serve only to restrain them in their annual motions round the sun, and no way affect their actions on each other, or their motions about their common centre of gravity. But because the Moon is nearer the sun, in one half of her orbit, than the earth is, but farther off in the other half of her orbit; and because the power of gravity is always less at a greater distance; it follows, that in one half of her orbit the Moon is more attracted than the earth towards the sun, and less attracted than the earth in the other half: and hence irregularities necessarily arise in the motions of the Moon; the excess of attraction in the first case, and the defect in the second, becoming a force that disturbs her motion: and besides, the action of the sun, on the earth and Moon, is not directed in parallel lines, but in lines that meet in the centre of the sun; which makes the effect of the disturbing force still the more complex and embarrassing. And hence, as well as from the various situations of the Moon, arise the numerous irregularities in her motions, and the equations, or corrections, employed in calculating her places, &c.</p><p>Newton, as well as others, has computed the quantities of these irregularities, from their causes. He finds that the force added to the gravity of the Moon in her quadratures, is to the gravity with which she would revolve in a circle about the earth, at her present mean distance, if the sun had no effect on her, as 1 to 178 29/40: he finds that the force subducted from her gravity in the conjunctions and oppositions, is<cb/> double of this quantity; and that the area described in a given time in the quarters, is to the area described in the same time in the conjunctions and oppositions, as 10973 to 11073: and he finds that, in such an orbit, her distance from the earth in her quarters, would be to her distance in the conjunctions and oppositions, as 70 to 69. Upon these irregularities, see Maclaurin's Account of Newton's Discoveries, book 4, chap. 4; as also most books of astronomy. Other particulars relating to the Moon's motions, &c, have been stated as follow: The power of the Moon's influence, as to the tides, is to that of the sun, as 6 1/3 to 1, according to Sir I. Newton; but different according to others.</p><p>As to the figure of the Moon, supposing her at first to have been a fluid, like the sea, Newton calculates, that the earth's attraction would raise the water there near 90 feet high, as the attraction of the Moon raises our sea 12 feet: whence the figure of the Moon must be a spheroid, whose greatest diameter extended, will pass through the centre of the earth; and will be longer than the other diameter, perpendicular to it, by 180 feet; and hence it comes to pass, that we always see the same face of the Moon; for she cannot rest in any other position, but always endeavours to conform herself to this situation: Princip. lib. 3, prop. 38.</p><p>Newton estimates the mean apparent diameter of the Moon at 32′ 12″; as the sun is 31′ 27″.</p><p>The density of the Moon he concludes is to that of the earth, as 9 to 5 nearly; and that the mass, or quantity of matter, in the Moon, is to that of the earth, as 1 to 26 nearly.</p><p>The plane of the Moon's orbit is inclined to that of the ecliptic, and makes with it an angle of about 5 degrees: but this inclination varies, being greatest when she is in the quarters, and least when in her syzygies.</p><p>As to the inequality of the Moon's motion, she moves swifter, and by the radius drawn from her to the earth describes a greater area in proportion to the time, also has an orbit less curved, and by that means comes nearer to the earth, in her syzygies or conjunctions, than in the quadratures, unless the motion of her eccentricity hinders it: which eccentricity is the greatest when the Moon's apogee falls in the conjunction, but least when this falls in the quadratures: her motion is also swifter in the earth's aphelion, than in its perihelion. The apogee also goes forward swifter in the conjunction, and goes slower at the quadratures: but her nodes are at rest in the conjunctions, and recede swiftest of all in the quadratures.</p><p>The Moon also perpetually changes the figure of her orbit, or the species of the ellipse she moves in.</p><p>There are also some other inequalities in the motion of this planet, which it is very difficult to reduce to any certain rule: as the velocities or horary motions of the apogee and nodes, and their equations, with the difference between the greatest eccentricity in the conjunctions, and the least in the quadratures; and that inequality which is called the Variation of the Moon. All these do increase and decrease annually, in a triplicate ratio of the apparent diameter of the sun: and this variation is increased and diminished in a duplicate ratio of the time between the quadratures; as is proved by Newton in many parts of his Principia.<pb n="121"/><cb/></p><p>He also found that the apogees in the Moon's syzygies, go forward in respect of the fixed stars, at the rate of 23′ each day; and backwards in the quadratures 16′ 1/3 per day: and therefore the mean annual motions he estimates at 40 degrees.</p><p>The gravity of the Moon towards the earth, is increased by the action of the sun, when the Moon is in the quadratures, and diminished in the syzygies: and, from the syzygies to the quadrature, the gravity of the Moon towards the earth is continually increased, and she is continually retarded in her motion: but from the quadrature to the syzygy, the Moon's motion is perpetually diminished, and the motion in her orbit is accelerated.</p><p>The Moon is less distant from the earth at the syzygies, and more at the quadratures.</p><p>As radius is to <*> of the sine of double the Moon's distance from the syzygy, so is the addition of gravity in the quadratures, to the force which accelerates or retards the Moon in her orbit.</p><p>And as radius is to the sum or difference of 1/2 the radius and 3/2 the cosine of double the distance of the Moon from the syzygy, so is the addition of gravity in the quadratures, to the decrease or increase of the gravity of the Moon at that distance.</p><p>The apses of the Moon go forward when she is in the syzygies, and backward in the quadratures. But, in a whole revolution of the Moon, the progress exceeds the regress.</p><p>In a whole revolution, the apses go forward the fastest of all when the line of the apses is in the nodes; and in the same case they go back the slowest of all in the same revolution.</p><p>When the line of the apses is in the quadratures, the apses are carried in consequentia, the least of all in the syzygies; but they return the swiftest in the quadratures; and in this case the regress exceeds the progress, in one entire revolution of the Moon.</p><p>The eccentricity of the orbit undergoes various changes every revolution. It is the greatest of all when the line of the apses is in the syzygies, and the least when that line is in the quadratures.</p><p>Considering one entire revolution of the Moon, cæteris paribus, the nodes move in antecedentia swiftest of all when she is in the syzygies; then slower and flower, till they are at rest, when she is in the quadratures.</p><p>The line of nodes acquires successively all possible situations in respect of the sun; and every year it goes twice through the syzygies, and twice through the quadratures.</p><p>In one whole revolution of the Moon, the nodes go back very fast when they are in the quadratures; then slower till they come to rest, when the line of nodes is in the syzygies.</p><p>The inclination of the plane of the orbit is changed by the same force with which the nodes are moved; being increased as the Moon recedes from the node, and diminished as she approaches it.</p><p>The inclination of the orbit is the least of all when the nodes are come to the syzygies. For in the motion of the nodes from the syzygies to the quadratures, and in one entire revolution of the Moon, the force which increases the inclination exceeds that which di-<cb/> minishes it; therefore the inclination is increased; and it is the greatest of all when the nodes are in the quadratures.</p><p>The Moon's motion being considered in general: her gravity towards the earth is diminished coming near the sun, and the periodical time is the greatest; as also the distance of the Moon, cæteris paribus, the greatest when the earth is in the perihelion.</p><p>All the errors in the Moon's motion are something greater in the conjunction than in the opposition.</p><p>All the disturbing forces are inversely as the cube of the distance of the sun from the earth; which when it remains the same, they are as the distance of the Moon from the earth. Considering all the disturbing forces together, the diminution of gravity prevails.</p><p><hi rend="italics">The figure of the</hi> <hi rend="smallcaps">Moon's</hi> <hi rend="italics">path,</hi> about the earth, is, as has been said, nearly an ellipse; but her path, in moving, together with the earth about the sun, is made up of a series or repetition of epicycloids, and is in every point concave towards the earth. See Maclaurin's Account of Newton's Discov. pa. 336, 4to. Ferguson's Astron. pa. 129, &c; and Rowe's Flux. pa. 225, edit. 2. <hi rend="center"><hi rend="italics">Astronomy of the</hi> <hi rend="smallcaps">Moon.</hi></hi></p><p><hi rend="italics">To determine the Periodical and Synodical Months;</hi> or the period of the Moon's revolution about the earth, and the period between one opposition or conjunction and another.</p><p>In the middle of a lunar eclipse, the Moon is in opposition to the sun: compute therefore the time between two such eclipses, at some considerable distance of time from each other; and divide this by the number of lunations that have passed in the mean time; so shall the quotient be the quantity of the synodical month. Compute also the sun's mean motion during the time of this synodical month, which add to 360°. Then, as the sum is to 360°, so is the synodical to the periodical month.</p><p>For example, Copernicus observed two eclipses of the Moon, the one at Rome on November 6, 1500, at 12 at night, and the other at Cracow on August 1, 1523, at 4h. 25 min. the dif. of meridians being oh. 29 min.: hence the quantity of the synodical month is thus determined: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2d Observ.</cell><cell cols="1" rows="1" rend="align=right" role="data">1523<hi rend="sup">y</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">237<hi rend="sup">d</hi></cell><cell cols="1" rows="1" role="data">4<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">25<hi rend="sup"><*></hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1st Observ.</cell><cell cols="1" rows="1" rend="align=right" role="data">1500 </cell><cell cols="1" rows="1" rend="align=right" role="data">310 </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">29</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Difference</cell><cell cols="1" rows="1" rend="align=right" role="data">22 </cell><cell cols="1" rows="1" rend="align=right" role="data">292 </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">56</cell></row><row role="data"><cell cols="1" rows="1" role="data">Add intercalary days</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Exact interval</cell><cell cols="1" rows="1" rend="align=right" role="data">22 </cell><cell cols="1" rows="1" rend="align=right" role="data">297 </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">56</cell></row></table> which divided by 282, the number of lunations in that time, gives the synodical month 29<hi rend="sup">d</hi> 12<hi rend="sup">h</hi> 41<hi rend="sup">m</hi>.</p><p>From two other observations of eclipses, the one at Cracow, the other at Babylon, the same author determines more accurately the quantity of the synodical month to be 29<hi rend="sup">d</hi> 12<hi rend="sup">h</hi> 43<hi rend="sup">m</hi> &c; and from other observations, probably more accurate still, the same is fixed at 29<hi rend="sup">d</hi> 12<hi rend="sup">h</hi> 44<hi rend="sup">m</hi>.</p><p>The sun's mean motion in that time 29° 6′ 24″ 18‴, added to 360°, gives the Moon's motion 389 6 24 18; Therefore the periodical month is 27<hi rend="sup">d</hi> 7<hi rend="sup">h</hi> 43<hi rend="sup">m</hi> 5<hi rend="sup"><*></hi>.<pb n="122"/><cb/></p><p>According to the observations of Kepler, the mean synodical month is 29<hi rend="sup">d</hi> 12<hi rend="sup">h</hi> 44<hi rend="sup">m</hi> 3<hi rend="sup">s</hi> 2<hi rend="sup">th</hi>, and the mean periodical month 27 7 43 8</p><p>Hence, 1, the quantity of the periodical month being given, by the rule of three are found the Moon's diurnal or horary motion, &c: and thus may tables of the mean motion of the Moon be constructed.</p><p>2. If the mean diurnal motion of the sun be subtracted from that of the Moon, the remainder will give the Moon's diurnal motion from the sun: and thus may a table of this motion be constructed.</p><p>3. Since the Moon is in the node at the time of a total eclipse, if the sun's place be found for that time, and 6 signs be added to the same, the sum will give the place of that node.</p><p>4. By comparing the ancient observations with the modern, it appears, that the nodes have a motion, and that they proceed in antecedentia, or backwards from Taurus to Aries, from Aries to Pisces, &c. Therefore if the diurnal motion of the nodes be added to the Moon's diurnal motion, the sum will be the motion of the Moon from the node; and thence by the rule of three, may be found in what time the Moon goes 360° from the dragon's head, or ascending node, or in what time she goes from, and returns to it; that is, the quantity of the Dracontic Month.</p><p>5. If the motion of the apogee be subtracted from the mean motion of the Moon, the remainder will be the Moon's mean motion from the apogee; and hence, by the rule of three, the quantity of the Anomalistic Month is determined.</p><p>Thus, according to Kepler's observations, <table><row role="data"><cell cols="1" rows="1" role="data">The mean synodical month is</cell><cell cols="1" rows="1" rend="align=right" role="data">29<hi rend="sup">d</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">12<hi rend="sup">h</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">44<hi rend="sup">m</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">3<hi rend="sup">s</hi></cell><cell cols="1" rows="1" rend="align=right" role="data">2<hi rend="sup">th</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">The periodical month "</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">The place of the apogee for the</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data"><hi rend="size(6)">}</hi>11<hi rend="sup">s</hi></cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">8°</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">57′</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"> 1″</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"> year 1700 Jan. 1 old style, was</cell></row><row role="data"><cell cols="1" rows="1" role="data">The place of the ascending node</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Mean diurnal motion of the Moon</cell><cell cols="1" rows="1" rend="align=right" role="data">.</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Diurnal motion of the apogee</cell><cell cols="1" rows="1" rend="align=right" role="data">.</cell><cell cols="1" rows="1" rend="align=right" role="data">.</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Diurnal motion of the nodes</cell><cell cols="1" rows="1" rend="align=right" role="data">.</cell><cell cols="1" rows="1" rend="align=right" role="data">.</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Theref. diurnal mot. from the latter</cell><cell cols="1" rows="1" rend="align=right" role="data">.</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" rend="align=right" role="data">46</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">And the diurnal motion from</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data"><hi rend="size(6)">}</hi>.</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">13</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">3</cell><cell cols="1" rows="1" rend="rowspan=2 align=right" role="data">54</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"> the apogee</cell></row></table> Lastly, the eccentricity is 4362, of such parts as the semidiameter of the eccentric is 100,000. <hi rend="center"><hi rend="italics">To find nearly the</hi> <hi rend="smallcaps">Moon's</hi> <hi rend="italics">Age or Change.</hi></hi></p><p>To the epact add the number and day of the month; their sum, abating 30 if it be above, is the Moon's age; and her age taken from 30, shews the day of the change.</p><p>The numbers of the months, or monthly epacts, are the Moon's age at the beginning of each month, when the solar and lunar years begin together; and are thus: 0 2 1 2 3 4 5 6 8 8 10 10 Jan. Feb. Mar. Ap. M<*>. Jun. Jul. Aug. Sep. Oct. Nov. Dec.</p><p>For Ex. To find the Moon's age the 14th of Oct. 1783. <table><row role="data"><cell cols="1" rows="1" role="data">Here, the epact is</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" role="data">Number of the month</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Day of the month</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" role="data">The sum is</cell><cell cols="1" rows="1" rend="align=right" role="data">48</cell></row></table><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">Subtract or abate</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Leaves Moon's age</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data">Taken from</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">Days till the change</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data">Answering to Oct.</cell><cell cols="1" rows="1" rend="align=right" role="data">26</cell></row></table></p><p><hi rend="italics">To find nearly the</hi> <hi rend="smallcaps">Moon's</hi> <hi rend="italics">Southing,</hi> or coming to the Meridian.</p><p>Take 4/5 or 8/10 of her age, for her southing nearly; after noon, if it be less than 12 hours; but if greater, the excess is the time after last midnight. For Ex. Oct. 14, 1783; The Moon's age is 18 days <table><row role="data"><cell cols="1" rows="1" role="data">8/10 of which is 14.4 or</cell><cell cols="1" rows="1" role="data">14<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">24<hi rend="sup">m</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Subtract</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Rem. Moon's fouthing</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">in the morning.</cell></row></table></p><p>Mr. Ferguson, in his Select Exercises, pa. 135 &c, has given very easy tables and rules for finding the new and full Moons near enough the truth for any common almanac. But the Nautical Almanac, which is now always published for several years before hand, in a great measure supersedes the necessity of these and other such contrivances. <hi rend="center"><hi rend="italics">Of the Spots and Mountains &c in the</hi> <hi rend="smallcaps">Moon.</hi></hi></p><p>The face of the Moon is greatly diversified with inequalities, and parts of different colours, some brighter and some darker than the other parts of her disc. When viewed through a telescope, her face is evidently diversified with hills and valleys: and the same is also shewn by the edge or border of the Moon appearing jagged, when so viewed, especially about the consines of the illuminated part when the Moon is either horned or gibbous.</p><p>The astronomers Florenti, Langreni, Hevelius, Grimaldi, Riccioli, Cassini, and De la Hire, &c, have drawn the face of the Moon as viewed through telescopes; noting all the more shining parts, and, for the better distinction, marking them with some proper name; some of these authors calling them after the names of philosophers, astronomers, and other eminent men; while others denominate them from the known names of the different countries, islands, and seas on the earth. The names adopted by Riccioli however are mostly followed, as the names of Hipparchus, Tycho, Copernicus, &c. Fig. 4, plate xv, is a pretty exact representation of the full Moon in her mean libration, with the numbers to the principal spots according to Riccioli, Cassini, Mayer, &c, which denote the names as in the following List of them: also the asterisk refers to one of the volcanoes observed by Herschel. <table><row role="data"><cell cols="1" rows="1" role="data"> * Herschel's Volcano</cell><cell cols="1" rows="1" role="data">12 Helicon</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 1 Grimaldi</cell><cell cols="1" rows="1" role="data">13 Capuanus</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 2 Galileo</cell><cell cols="1" rows="1" role="data">14 Bulliald</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 3 Aristarchus</cell><cell cols="1" rows="1" role="data">15 Eratosthenes</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 4 Kepler</cell><cell cols="1" rows="1" role="data">16 Timocharis</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 5 Gassendi</cell><cell cols="1" rows="1" role="data">17 Plato</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 6 Schikard</cell><cell cols="1" rows="1" role="data">18 Archimedes</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 7 Harpalus</cell><cell cols="1" rows="1" role="data">19 Insula Sinus Medii</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 8 Heraclides</cell><cell cols="1" rows="1" role="data">20 Pitatus</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 9 Lansberg</cell><cell cols="1" rows="1" role="data">21 Tycho</cell></row><row role="data"><cell cols="1" rows="1" role="data">10 Reinhold</cell><cell cols="1" rows="1" role="data">22 Eudoxus</cell></row><row role="data"><cell cols="1" rows="1" role="data">11 Copernicus</cell><cell cols="1" rows="1" role="data">23 Aristotle</cell></row></table><pb n="123"/><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">24 Manilius</cell><cell cols="1" rows="1" role="data">36 Cleomedes</cell></row><row role="data"><cell cols="1" rows="1" role="data">25 Menelaus</cell><cell cols="1" rows="1" role="data">37 Snell and Furner</cell></row><row role="data"><cell cols="1" rows="1" role="data">26 Hermes</cell><cell cols="1" rows="1" role="data">38 Petavius</cell></row><row role="data"><cell cols="1" rows="1" role="data">27 Possidonius</cell><cell cols="1" rows="1" role="data">39 Langrenus</cell></row><row role="data"><cell cols="1" rows="1" role="data">28 Dionysius</cell><cell cols="1" rows="1" role="data">40 Taruntius</cell></row><row role="data"><cell cols="1" rows="1" role="data">29 Pliny</cell><cell cols="1" rows="1" role="data">A Mare Humorum</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">30 <hi rend="size(6)">{</hi> Catharina Cyrillus, Theophilus</cell><cell cols="1" rows="1" role="data">B Mare Nubium</cell></row><row role="data"><cell cols="1" rows="1" role="data">C Mare Imbrium</cell></row><row role="data"><cell cols="1" rows="1" role="data">31 Fracastor</cell><cell cols="1" rows="1" role="data">D Mare Nectaris</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">32 <hi rend="size(6)">{</hi> Promontorium acutum, Censorinus</cell><cell cols="1" rows="1" role="data">E Mare Tranquilitatis</cell></row><row role="data"><cell cols="1" rows="1" role="data">F Mare Serenitatis</cell></row><row role="data"><cell cols="1" rows="1" role="data">33 Messala</cell><cell cols="1" rows="1" role="data">G Mare Fœcunditatis</cell></row><row role="data"><cell cols="1" rows="1" role="data">34 Promontorium Somnii</cell><cell cols="1" rows="1" role="data">H Mare Crisium</cell></row><row role="data"><cell cols="1" rows="1" role="data">35 Proclus</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>That the spots in the Moon, which are taken for mountains and valleys, are really such, is evident from their shadows. For in all situations of the Moon, the elevated parts are constantly found to cast a triangular shadow in a direction from the sun; and, on the contrary, the cavities are always dark on the side next the sun, and illuminated on the opposite one; which is exactly conformable to what we observe of hills and valleys on the earth. And as the tops of these mountains ave considerably elevated above the other parts of the surface; they are often illuminated when they are at a considerable distance from the confines of the enlightened hemisphere, and by this means afford us a method of determining their heights. <figure/></p><p>Thus, let ED be the Moon's diameter, ECD the boundary of light and darkness; and A the top of a hill in the dark part beginning to be illuminated; with a telescope take the proportion of AE to the diameter ED: then there are given the two sides AE, EC of a right angled triangle ACE, the squares of which being added together give the square of the third side AC, and the root extracted is that side itself; from which subtracting the radius BC, leaves AB the height of the mountain. In this way, Riccioli observed the top of the hill called St. Catherine, on the 4th day after the new moon, to be illuminated when it was distant from the confines of the enlightened hemisphere about one 16th part of the Moon's diameter; and thence found its height must be near 9 miles.</p><p>It is probable however that this determination is too much. Indeed, Galileo makes AE to be only one 20th of ED, and Hevelius makes it only one 26th of ED; the former of these would give 5 1/2 miles, and the latter only 3 1/4 miles, for AB, the height of the mountain: and probably it should be still less than either of these.</p><p>Accordingly, they are greatly reduced by the observations of Herschel, whose method of measuring them may be seen in the Philos. Trans. an. 1780, pa. 507. This gentleman measured the height of many of the lunar prominences, and draws at last the following conclusions:—“From these observations I believe it is evident, that the height of the lunar mountains in general is greatly over-rated; and that, when we have excepted a few, the generality do not exceed half a<cb/> mile in their perpendicular elevation.” And this is confirmed by the measurement of several mountains, as may be seen in the place above quoted.</p><p>As the Moon has on her surface mountains and valleys in common with the earth, some modern astronomers have discovered a still greater similarity, viz, that some of these are really volcanoes, emitting fire as those on the earth do. An appearance of this kind was discovered some few years ago by Don Ulloa in an eclipse of the sun. It was a small bright spot like a star near the margin of the Moon, and which he at that time supposed to be a hole or valley with the sun's light shining through it. Succeeding observations, however, have induced astronomers to attribute appearances of this kind to the eruption of volcanic sire; and Mr. Herschel has particularly observed several eruptions of the lunar volcanos, the last of which he gives an account of in the Philos. Trans. for 1787. April 19, 10h. 36m. sidereal time, I perceived, says he, three volcanos in different places of the dark part of the new Moon. Two of them are either already nearly extinct, or otherwise in a state of going to break out; which perhaps may be decided next lunation. The third shews an actual eruption of fire or luminous matter: its light is much brighter than the nucleus of the comet which M. Mechain discovered at Paris the 10th of this month.” The following night he found it burnt with greater violence; and by measurement he found that the shining or burning matter must be more than 3 miles in diameter; being of an irregular round figure, and very sharply defined on the edges. The other two volcanos resembled large faint nebulæ, that are gradually much brighter in the middle; but no well-desined luminous spot was discovered in them. He adds, “the appearance of what I have called the <hi rend="italics">actual fire,</hi> or eruption of a volcano, exactly resembled a small piece of burning charcoal when it is covered by a very thin coat of white ashes, which frequently adhere to it when it has been some time ignited; and it had a degree of brightness about as strong as that with which a coal would be seen to glow in faint day-light.</p><p>It has been disputed whether the Moon has any atmosphere or not. The following arguments have been urged by those who deny it.</p><p>1. The Moon, say they, constantly appears with the same brightness when our atmosphere is clear; which could not be the case if she were surrounded with an atmosphere like ours, so variable in its density, and so often obscured by clouds and vapours. 2. In an appulse of the Moon to a star, when she comes so near it that a part of her atmosphere comes between our eye and the star, refraction would cause the latter to seem to change its place, so that the Moon would appear to touch it later than by her own motion she would do. 3. Some philosophers are of opinion, that because there are no seas or lakes in the Moon, there is therefore no atmosphere, as there is no water to be raised up in vapours.</p><p>But all these arguments have been answered by other astronomers in the following manner. It is denied that the Moon appears always with the same brightness, even when our atmosphere appears equally clear. Hevelius relates, that he has several times found in<pb n="124"/><cb/> skies perfectly clear, when even stars of the 6th and 7th magnitude were visible, that at the same altitude of the Moon with the same elongation from the sun, and with the same telescope, the Moon and her maculæ do not appear equally lucid, clear, and conspicuous at all times; but are much brighter and more distinct at some times than at others. And hence it is inferred that the cause of this phenomenon is neither in our air, in the tube, in the Moon, nor in the spectator's eye; but must be looked for in something existing about the Moon. An additional argument is drawn from the different appearances of the Moon in total eclipses, which it is supposed are owing to the different constitutions of the lunar at mosphere.</p><p>To the 2d argument Dr. Long replies, that Newton has shewn (Princip. prop. 37, cor. 5), that the weight of any body upon the Moon is but a third part of what the weight of the same would be upon the earth: now the expansion of the air is reciprocally as the weight that compresses it; therefore the air surrounding the Moon, being pressed together by a weight of one-third, or being attracted towards the centre of the Moon by a force equal only to one-third of that which attracts our air towards the centre of the earth, it thence follows, that the lunar atmosphere is only onethird as dense as that of the earth, which is too little to produce any sensible refraction of the star's light. Other astronomers have contended, that such refraction was sometimes very apparent. Mr. Cassini says, that he often observed that Saturn, Jupiter, and the fixed stars, had their eircular figures changed into an elliptical one, when they approached either to the Moon's dark or illuminated limb, though they own that, in other occultations, no such change could be observed. And, with regard to the fixed stars, it has been urged that, granting the Moon to have an atmosphere of the same nature and quantity as ours, no such effect as a gradual diminution of light ought to take place; at least none that we could be capable of perceiving. At the height of 44 miles, our atmosphere is so rare as to be incapable of refracting the rays of light: this height is the 180th part of the earth's diameter; but since clouds are never observed higher than 4 miles, it appears that the vapourous or obscure part is only the 1980th part. The mean apparent diameter of the Moon is 31′ 29″, or 1889″: therefore the obscure parts of her atmosphere, when viewed from the earth, must subtend an angle of less than one second; which space is passed over by the Moon in less than two seconds of time. It can therefore hardly be expected that observation should generally determine whether the supposed obscuration takes place or not.</p><p>As to the 3d argument, it concludes nothing, because it is not known that there is no water in the Moon; nor, though this could be proved, would it follow that the lunar atmosphere answers no other purpose than the raising of water into vapour. There is however a strong argument in favour of the existence of a lunar atmosphere, taken from the appearance of a luminous circle round the Moon in the time of total solar eclipses; a circumstance that has been observed by many astronomers; especially in the total eclipse of the sun which happened May 1, 1706.<cb/></p><p><hi rend="italics">Of the Harvest</hi> <hi rend="smallcaps">Moon.</hi> It is remarkable that the <hi rend="smallcaps">Moon</hi>, during the week in which she is full about the time of harvest, rises sooner after sun-setting, than she does in any other full-moon week in the year. By this means she affords an immediate supply of light after sun-set, which is very benesicial for the harvest and gathering in the fruits of the earth: and hence this full Moon is distinguished from all the others in the year, by calling it the Harvest-Moon.</p><p>To conceive the reason of this phenomenon; it may first be considered, that the Moon is always opposite to the sun when she is full; that she is full in the signs Pisces and Aries in our harvest months, those being the signs opposite to Virgo and Libra, the signs occupied by the sun about the same season; and because those parts of the ecliptic rise in a shorter space of time than others, as may easily be shewn and illustrated by the celestial globe: consequently, when the Moon is about her full in harvest, she rises with less difference of time, or more immediately after sun-set, than when she is full at other seasons of the year.</p><p>In our winter, the Moon is in Pisces and Aries about the time of her first quarter, when she rises about noon; but her rising is not then noticed, because the sun is above the horizon.</p><p>In spring, the Moon is in Pisces and Aries about the time of her change; at which time, as she gives no light, and rises with the sun, her rising cannot be perceived.</p><p>In summer, the Moon is in Pisces and Aries about the time of her last quarter; and then, as she is on the decrease, and rises not till midnight, her rising usually passes unobserved.</p><p>But in autumn, the Moon is in Pisces and Aries at the time of her full, and rises soon after sun-set for several evenings successively; which makes her regular rising very conspicuous at that time of the year.</p><p>And this would always be the case, if the Moon's orbit lay in the plane of the ecliptic. But as her orbit makes an angle of 5° 18′ with the ecliptic, and crosses it only in the two opposite points called the nodes, her rising when in Pisces and Aries will sometimes not differ above 1 h. and 40 min. through the whole of 7 days; and at other times, in the same two signs she will differ 3 hours and a half in the time of her rising in a week, according to the different positions of the nodes with respect to these signs; which positions are constantly changing, because the nodes go backward through the whole ecliptic in 18 years 225 days.</p><p>This revolution of the nodes will cause the Harvest Moons to go through a whole course of the most and least beneficial states, with respect to the harvest, every 19 years. The following Table shews in what years the Harvest Moons are least beneficial as to the times of their rising, and in what years they are most beneficial, from the year 1790 to 1861; the column of years under the letter L, are those in which the Harvest-Moons are least of all beneficial, because they fall about the descending node; and those under the letter M are the most of all beneficial, because they fall about the ascending node.<pb n="125"/><cb/> <table><row role="data"><cell cols="1" rows="1" rend="colspan=8 align=center" role="data"><hi rend="italics">Harvest Moons.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">M</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">M</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">M</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">M</cell></row><row role="data"><cell cols="1" rows="1" role="data">1790</cell><cell cols="1" rows="1" role="data">1798</cell><cell cols="1" rows="1" role="data">1807</cell><cell cols="1" rows="1" role="data">1816</cell><cell cols="1" rows="1" role="data">1826</cell><cell cols="1" rows="1" role="data">1835</cell><cell cols="1" rows="1" role="data">1844</cell><cell cols="1" rows="1" role="data">1843</cell></row><row role="data"><cell cols="1" rows="1" role="data">1791</cell><cell cols="1" rows="1" role="data">1799</cell><cell cols="1" rows="1" role="data">1808</cell><cell cols="1" rows="1" role="data">1817</cell><cell cols="1" rows="1" role="data">1827</cell><cell cols="1" rows="1" role="data">1836</cell><cell cols="1" rows="1" role="data">1845</cell><cell cols="1" rows="1" role="data">1854</cell></row><row role="data"><cell cols="1" rows="1" role="data">1792</cell><cell cols="1" rows="1" role="data">1800</cell><cell cols="1" rows="1" role="data">1809</cell><cell cols="1" rows="1" role="data">1818</cell><cell cols="1" rows="1" role="data">1828</cell><cell cols="1" rows="1" role="data">1837</cell><cell cols="1" rows="1" role="data">1846</cell><cell cols="1" rows="1" role="data">1855</cell></row><row role="data"><cell cols="1" rows="1" role="data">1793</cell><cell cols="1" rows="1" role="data">1801</cell><cell cols="1" rows="1" role="data">1810</cell><cell cols="1" rows="1" role="data">1819</cell><cell cols="1" rows="1" role="data">1829</cell><cell cols="1" rows="1" role="data">1838</cell><cell cols="1" rows="1" role="data">1847</cell><cell cols="1" rows="1" role="data">1856</cell></row><row role="data"><cell cols="1" rows="1" role="data">1794</cell><cell cols="1" rows="1" role="data">1802</cell><cell cols="1" rows="1" role="data">1811</cell><cell cols="1" rows="1" role="data">1820</cell><cell cols="1" rows="1" role="data">1830</cell><cell cols="1" rows="1" role="data">1839</cell><cell cols="1" rows="1" role="data">1848</cell><cell cols="1" rows="1" role="data">1857</cell></row><row role="data"><cell cols="1" rows="1" role="data">1795</cell><cell cols="1" rows="1" role="data">1803</cell><cell cols="1" rows="1" role="data">1812</cell><cell cols="1" rows="1" role="data">1821</cell><cell cols="1" rows="1" role="data">1831</cell><cell cols="1" rows="1" role="data">1840</cell><cell cols="1" rows="1" role="data">1849</cell><cell cols="1" rows="1" role="data">1858</cell></row><row role="data"><cell cols="1" rows="1" role="data">1796</cell><cell cols="1" rows="1" role="data">1804</cell><cell cols="1" rows="1" role="data">1813</cell><cell cols="1" rows="1" role="data">1822</cell><cell cols="1" rows="1" role="data">1832</cell><cell cols="1" rows="1" role="data">1841</cell><cell cols="1" rows="1" role="data">1850</cell><cell cols="1" rows="1" role="data">1859</cell></row><row role="data"><cell cols="1" rows="1" role="data">1797</cell><cell cols="1" rows="1" role="data">1805</cell><cell cols="1" rows="1" role="data">1814</cell><cell cols="1" rows="1" role="data">1823</cell><cell cols="1" rows="1" role="data">1833</cell><cell cols="1" rows="1" role="data">1842</cell><cell cols="1" rows="1" role="data">1851</cell><cell cols="1" rows="1" role="data">1860</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1806</cell><cell cols="1" rows="1" role="data">1815</cell><cell cols="1" rows="1" role="data">1824</cell><cell cols="1" rows="1" role="data">1834</cell><cell cols="1" rows="1" role="data">1843</cell><cell cols="1" rows="1" role="data">1852</cell><cell cols="1" rows="1" role="data">1861</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1825</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p><hi rend="italics">As to the Influence of the</hi> <hi rend="smallcaps">Moon</hi>, on the changes of the weather, and the constitution of the human body, it may be observed, that the vulgar doctrine concerning it is very ancient, and has also gained much credit among the Learned, though perhaps without sufficient examination. The common opinion is, that the Lunar Influence is chiefly exerted about the time of the full and change, but more especially the latter; and it would seem that long experience has in some degree established the fact: hence, persons observed at those times to be a little deranged in their intellects, are called Lunatics; and hence many persons anxiously look for the new Moon to bring a change in the weather. The Moon's Influence on the sea, in producing tides, being agreed upon on all hands, it is argued that she must also produce similar changes in the atmosphere, but in a much higher degree; which changes and commotions there, must, it is inferred, have a considerable influence on the weather, and on the human body.</p><p>Beside the observations of the Ancients, which tend to establish this doctrine, several among the Modern Philosophers have defended the same opinion, and that upon the strength of experience and observation; while others as strenuously deny the fact. The celebrated Dr. Mead was a believer in the Influence of the Sun and Moon on the human body, and published a book to this purpose, intitled, De Imperio Solis ac Lunæ in Corpore Humano. The existence of such influence is however opposed by Dr. Horsley, the present bishop of Rochester, in a learned paper upon this subject in the Philos. Trans. for the year 1775; where he gives a specimen of arranging tables of meteorological observations, so as to deduce from them facts, that may either confirm or refute this popular opinion; recommending it to the Learned, to collect a large series of such observations, as no conclusions can be drawn from one or two only. On the other hand professor Toaldo, and some French philosophers, take the opposite side of the question; and, from the authority of a long series of observations, pronounce decidedly in favour of the Lunar Influence.</p><p><hi rend="italics">Acceleration of the</hi> <hi rend="smallcaps">Moon.</hi> See <hi rend="smallcaps">Acceleration.</hi></p><p><hi rend="smallcaps">Moon</hi>-<hi rend="italics">Dial.</hi> See <hi rend="smallcaps">Dial.</hi></p><p><hi rend="italics">Horizontal</hi> <hi rend="smallcaps">Moon.</hi> See <hi rend="italics">Apparent</hi> <hi rend="smallcaps">Magnitude.</hi></p></div1><div1 part="N" n="MOORE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MOORE</surname> (Sir <foreName full="yes"><hi rend="smallcaps">Jonas</hi></foreName>)</persName></head><p>, a very respectable mathematician, Fellow of the Royal Society, and Surveyorgeneral of the Ordnance, was born at Whitby in Yorkshire about the year 1620. After enjoying the advantages of a liberal education, he bent his studies principally to the mathematics, to which he had al-<cb/> ways a strong inclination. In the expeditions of King Charles the 1st into the northern parts of England, our author was introduced to him, as a person studious and learned in those sciences; when the king expressed much approbation of him, and promised him encouragement; which indeed laid the foundation of his fortune. He was afterwards appointed mathematical master to the king's second son James, to instruct him in arithmetic, geography, the use of the globes, &c. During Cromwell's government it seems he followed the profession of a public teacher of mathematics; for I find him styled, in the title-page of some of his publications, “professor of the mathematics.” After the return of Charles the 2d, he found great favour and promotion, becoming at length surveyor-general of the king's ordnance. He was it seems a great favourite both with the king and the duke of York, who often consulted him, and were advised by him upon many occasions. And it must be owned that he often employed his interest with the court to the advancement of learning and the encouragement of merit. Thus, he got Flamsteed house built in 1675, as a public observatory, recommending Mr. Flamsteed to be the king's astronomer, to make the observations there: and being surveyor-general of the ordnance himself, this was the reason why the salary of the astronomer royal was made payable out of the office of ordnance. Being a governor of Christ's hospital, it seems that by his interest the king founded the mathematical school there, allowing a handsome salary for a master to instruct a certain number of the boys in mathematics and navigation, to qualify them for the sea service. Here he soon found an opportunity of exerting his abilities in a manner somewhat answerable to his wishes, namely, that of serving the rising generation. And considering with himself the benefit the nation might receive from a mathematical school, if rightly conducted, he made it his utmost care to promote the improvement of it. The school was settled; but there still wanted a methodical institution from which the youths might receive such necessary helps as their studies required: a laborious work, from which his other great and assiduous employments might very well have exempted him, had not a predominant regard to a more general usefulness engaged him to devote all the leisure hours of his declining years to the improvement of so useful and important a seminary of learning.</p><p>Having thus engaged himself in the prosecution of this general design, he next sketched out the plan of a course or system of mathematics for the use of the school, and then drew up and printed several parts of it himself, when death put an end to his labours, before the work was completed. I have not found in what year this happened; but it must have been but little before 1681, the year in which the work was published by his sons-in-law, Mr. Hanway and Mr. Potinger. Of this work, the Arithmetic, Practical Geometry, Trigonometry, and Cosmography, were written by Sir Jonas himself, and printed before his death. The Algebra, Navigation, and the books of Euclid were supplied by Mr. Perkins, the then master of the mathematical school. And the Astronomy, or Doctrine of the Sphere, was written by Mr. Flamsteed, the astronomer royal.<pb n="126"/><cb/></p><p>The list of Sir Jonas's works, as far as I have seen them, are the following:</p><p>1. The New System of Mathematics; above mentioned, in 2 vols 4to, 1681.</p><p>2. Arithmetic in two books, viz, Vulgar Arithmetic and Algebra. To which are added two Treatises, the one A new Contemplation Geometrical, upon the Oval Figure called the Ellipsis; the other, The two first books of Mydorgius, his Conical Sections analized &c. 8vo, 1660.</p><p>3. A Mathematical Compendium; or Useful Practices in Arithmetic, Geometry, and Astronomy, Geography and Navigation, &c, &c. 12mo, 4th edition in 1705.</p><p>4. A General Treatise of Artillery: or, Great Ordnance, Written in Italian by Tomaso Moretii of Brescia. Translated into English, with notes thereupon, and some additions out of French for Sea-Gunners. By Sir Jonas Moore, Kt. 8vo, 1683.</p><p>MORTALITY. <hi rend="italics">Bills of Mortality,</hi> are accounts or registers specifying the numbers born, and buried, and sometimes married, in any town, parish, or district. These are of great use, not only in the doctrine of Life Annuities, but in shewing the degrees of healthiness and prolificness, with the progress of population in the places where they are kept. It is therefore much to be wished that such accounts had always been correctly kept in every kingdom, and regularly published at the end of every year. We should then have had under inspection the comparative strength of every kingdom, as far as it depends on the number of inhabitants, and its increase or decrease at different periods.</p><p>Such accounts are rendered still more useful, when they include the ages of the dead, and the distempers of which they have died. In this case they convey some of the most important instructions, by furnishing the means of ascertaining the law which governs the waste of human life, the values of annuities dependent on the continuance of any lives, or any survivorships between them, and the favourableness or unfavourableness of different situations to the duration of human life.</p><p>There are but few registers of this kind; nor has this subject, though so interesting to mankind, ever engaged much attention till lately. Indeed, bills of Mortality for the several parishes of the city of London have been kept from the year 1592, with little interruption; and a very ample account of them has been published down to the year 1759, by Dr. Birch, in a large 4to vol. which is perhaps the fullest work of the kind extant; containing besides the bills of Mortality, with the diseases and casualties, several other valuable tracts on the subject of them, and on political arithmetic, by several other authors, as Capt. John Graunt, F. R. S.; Sir William Petty, F. R. S.; Corbyn Morris, Esq. F. R. S.; and J. P. Esq. F. R. S.; the whole forming a valuable repository of materials; and it would be well if a continuation were published down to the present time, and so continued from time to time.</p><p>Bills containing the ages of the dead, were long since published for the town of Breslaw in Silesia. It is well known what use has been made of these by Dr. Halley, and after him by Mr. De Moivre. A table of<cb/> the probabilities of the duration of human life at every age, deduced from them by Dr. Halley, was published in the Philos. Trans. vol. 17, and has been inserted in this work under the article <hi rend="smallcaps">Life</hi>-<hi rend="italics">Annuities;</hi> which is the first table of this kind that has been published. Since the publication of this table, similar bills have been established in many other places, in England, Germany, Switzerland, France, Holland, &c, but most especially in Sweden; the results of some of which may be seen in the large comparative table of the duration of life, under the article <hi rend="smallcaps">Life</hi>-<hi rend="italics">Annuities,</hi> in this work.</p></div1><div1 part="N" n="MORTAR" org="uniform" sample="complete" type="entry"><head>MORTAR</head><p>, or <hi rend="smallcaps">Mortar-Piece</hi>, a short piece of ordnance, thick and wide, proper for throwing bombshells, carcases, stones, grape-shot, &c.</p><p>It is thought that the use of Mortars is older than that of cannon: for they were employed in the wars of Italy, to throw balls of red-hot iron, and stones, long before the invention of shells: and it is generally believed that the Germans were the first inventors. The practice of throwing red-hot balls out of Mortars, was first practised at the siege of Stralfund in 1675, by the elector of Brandenburg; though some say, in 1653, at the siege of Bremen.</p><p>Mortars are made either of brass or iron, and it is usual to distinguish them by the diameter of the bore; as, the 13 inch, the 10 inch, or the 8 inch Mortar: there are some of a smaller sort, as Coehorns of 4.6 inches, and Royals of 5.8 inches in diameter. As to the larger sizes, as 18 inches, &c, they are now disused by the English, as well as most other European nations. For the circumstances reiating to Mortars, see Muller's Artillery.</p><p><hi rend="italics">Coeborn</hi> <hi rend="smallcaps">Mortar</hi>, a small kind of one, invented by the celebrated engineer baron Coehorn, to throw small shells or grenades. These Mortars are often fixed, to the number of a dozen, on a block of oak, at the elevation of 45°.</p></div1><div1 part="N" n="MOTION" org="uniform" sample="complete" type="entry"><head>MOTION</head><p>, or <hi rend="italics">Local</hi> <hi rend="smallcaps">Motion</hi>, is a continued and successive change of place. Borelli defines it, the successive passage of a body from one place to another, in a determinate time, by becoming successively contiguous to all the parts of the intermediate space.</p><p>Motion is considered as of various kinds; as Natural, Violent, Absolute and Relative, &c, &c.</p><p><hi rend="italics">Natural</hi> <hi rend="smallcaps">Motion</hi>, is that which has its principle, or actuating force, within the moving body. Such is that of a stone falling towards the earth. And</p><p><hi rend="italics">Violent</hi> <hi rend="smallcaps">Motion</hi>, is that whose principle is without, and against which the moving body makes a resistance. Such is that of a stone thrown upwards, or of a ball shot off from a gun, &c.</p><p>Motion is again divided into Absolute and Relative.</p><p><hi rend="italics">Absolute</hi> <hi rend="smallcaps">Motion</hi>, is the change of absolute place, in any moving body, considered independently of any other motion; whose celerity therefore will be measured by the quantity of absolute space which the moveable body runs through. And</p><p><hi rend="italics">Relative</hi> <hi rend="smallcaps">Motion</hi>, is the change of the relative place of a moving body, or considered with respect to the motion of some other body; and has its celerity estimated by the quantity of relative space run through.</p><p><hi rend="italics">As to the Continuation of</hi> <hi rend="smallcaps">Motion</hi>, or the cause why a body once in Motion comes to persevere in it: this has<pb n="127"/><cb/> been much controverted among physical writers; and yet it follows very evidently from one of the grand Laws of Nature; viz, that all bodies persevere in their present state, whether of rest or motion, unless disturbed by some foreign powers. Motion therefore, once begun, would be continued in infinitum, were it to meet with no interruption from external causes; as the power of gravity, the resistance of the medium, &c.</p><p>Nor has the communication of motion, or how a moving body comes to affect another at rest, or how much of its motion is communicated by the first to the last, been less disputed. See the Laws of it under the word <hi rend="smallcaps">Percussion.</hi></p><p>Motion is the proper subject of mechanics; and mechanics is the basis of all natural philosophy; which hence becomes denominated Mechanical.</p><p>In effect, all the phenomena of nature, all the changes that happen in the system of bodies, are owing to Motion; and are directed according to the laws of it. Hence the modern philosophers have applied themselves with peculiar ardour to consider the doctrine of Motion; to investigate the properties and laws of it; by observation and experiment, joined to the use of geometry. And to this is owing the great advantage of the modern philosophy above that of the Ancients; who were extremely disregardful of the effects of Motion.</p><p>Among all the Ancients, there is nothing extant on Motion, excepting some things in Archimedes's books, De Æquiponderantibus. To Galileo is owing a great part of the doctrine of Motion: he first discovered the general laws of it, and particularly of the descent of heavy bodies, both perpendicularly and on inclined planes; the laws of the Motio<*> of projectiles; the vibration of pendulums, and of stretched cords, with the theory of resistances, &c: things which the Ancients had little notion of.</p><p>Torricelli polished and improved the discoveries of his master, Galileo; and added many experiments concerning the force of percussion, and the equilibrium of fluids. Huygens improved very considerably on the doctrine of the pendulum; and both he and Borelli on the force of percussion. Lastly, Newton, Leibnitz, Varignon, Mariotte, &c, have brought the doctrine of Motion still much nearer to perfection.</p><p>The general laws of Motion were first brought into a system, and analytically- demonstrated together, by Dr. Wallis, Sir Christopher Wren, and M. Huygens, all much about the same time; the first in bodies not elastic, and the two latter in elastic bodies. Lastly, the whole doctrine of Motion, including all the discoveries both of the Ancients and Moderns on that head, was given by Dr. Wallis in his Mechanica, sive De Motu, published in 1670.</p><p><hi rend="italics">Quantity of</hi> <hi rend="smallcaps">Motion</hi>, is the same as <hi rend="smallcaps">Momentum</hi>, which see. It is a principle maintained by the Cartesians, and some others, that the Creator at the beginning impressed a certain Quantity of Motion on bodies; and that under such laws, as that no part of it should be lost, but the same portion of Motion should be constantly preserved in matter: and hence they conclude, that if any moving body strike another body, the former loses no more of its Motion than it communicates to the latter. This position however has been opposed by other philosophers, and perhaps justly, unless the preservation<cb/> of Motion be understood only of the quantity of it as estimated always in the same direction; for then it seems the principle will hold good. However, the reasoning ought to have proceeded in the contrary order; by first observing from experiment, or otherwise, that when two bodies act upon each other, the one gains exactly the Motion which is lost by the other, in the same direction; and from hence made the inference, that there is therefore the same Quantity of Motion preserved in the universe, as was created by God in the beginning; since no body can act upon another, without being itself equally acted upon in the opposite or contrary direction.</p><p><hi rend="italics">The Continuation of</hi> <hi rend="smallcaps">Motion</hi>, or the cause why a body once in Motion comes to persevere in it, has been much controverted among physical writers; and yet it follows very evidently from one of the grand Laws of Nature; viz, that all bodies persevere in their present state, whether of Motion or rest, unless they are disturbed by some foreign powers. Motion therefore, once begun, would be continued for ever, were it to meet with no interruption from external causes; as the power of gravity, the resistance of the medium, &c.</p><p><hi rend="italics">The Communication of</hi> <hi rend="smallcaps">Motion</hi>, or the manner in which a moving body comes to affect another at rest, or how much of its Motion is communicated by the sirst to the last, has also been the subject of much discussion and controversy. See the Laws of it under the word <hi rend="smallcaps">Percussion.</hi></p><p><hi rend="smallcaps">Motion</hi> may be considered either as Equable, and Uniform; or as Accelerated, and Retarded. Equable Motion, again, may be considered either as Simple, or as Compound; and Compound Motion either as Rectilinear, or as Curvilinear.</p><p>And all these again may be considered either with regard to themselves, or with regard to the manner of their production, and communication, by percussion, &c.</p><p><hi rend="italics">Equable</hi> <hi rend="smallcaps">Motion</hi>, is that by which the moving body proceeds with exactly the same velocity or celerity; passing always over equal spaces in equal times.</p><p><hi rend="italics">The Laws of Uniform Motion,</hi> are these: 1. The spaces described, or passed over, are in the compound ratio of the velocities, and the times of describing those spaces. So that, if V and <hi rend="italics">v</hi> be any two uniform velocities, S and <hi rend="italics">s</hi> the spaces described or passed over by them, in the respective times T and <hi rend="italics">t:</hi> then is , or ; taking T = 4, <hi rend="italics">t</hi> = 3, V = 5, and <hi rend="italics">v</hi> = 4.</p><p>2. In Uniform Motions, the time is as the space directly, and as the velocity reciprocally; or as the space divided by the velocity. So that .</p><p>3. The velocity is as the space directly, and the time reciprocally; or as the space divided by the time. That is, .</p><p><hi rend="italics">Accelerated</hi> <hi rend="smallcaps">Motion</hi>, is that which continually receives fresh accessions of velocity. And it is said to be<pb n="128"/><cb/> uniformly accelerated, when its accessions of velocity are equal in <*>equal times; such as that which is produced by the continual action of one and the same force, like the force of gravity, &c.</p><p><hi rend="italics">Retarded</hi> <hi rend="smallcaps">Motion</hi>, is that whose velocity continually decreases. And it is said to be uniformly Retarded, when its decrease is continually proportional to the time, or by equal quantities in equal times; like that which is produced by the continual opposition of one and the same force; such as the force of gravity, in uniformly retarding the Motion of a body that is thrown upwards.</p><p>The Laws of Motion, uniformly accelerated or retarded, are these:</p><p>1. In uniformly varied motions, the space, S or <hi rend="italics">s,</hi> is as the square of the time, or as the square of the greatest velocity, or as the rectangle or product of the time and velocity.</p><p>That is, .</p><p>2. The velocity is the time, or as the space divided by the the time, or as the square root of the space.</p><p>That is, .</p><p>3. The time is as the velocity, or as the space divided by the velocity, or as the square root of the space.</p><p>That is, .</p><p>4. When a space is described, or passed over, by an uniformly varied Motion, the velocity either beginning at nothing, and continually accelerated; or else beginning at some determinate velocity, and continually retarded till the velocity be reduced to nothing; then the space, so run over by the variable Motion, will be exactly equal to half the space that would be run over in the same time by the greatest velocity if uniformly continued for that time. So, for instance, if <hi rend="italics">g</hi> denote the space run over in one second, or any other time, by such a variable Motion; then 2<hi rend="italics">g</hi> would be the space that would be run over in one second, or the same time, by the greatest velocity uniformly continued for the same time; or 2<hi rend="italics">g</hi> would be the greatest velocity per second which the moving body had. Consequently, if <hi rend="italics">t</hi> be any other time, <hi rend="italics">s</hi> the space run over in that time, and <hi rend="italics">v</hi> the greatest velocity attained in it; then, from the foregoing articles, it will be the velocity, and the space. And hence, for any such uniformly varied Motions, the relations among the several quantities concerned, will be expressed by the following equations: viz, , , , .<cb/> And these equations will hold good in the Motion either generated or destroyed by the force of gravity, or by any other uniform force whatever. See also the articles <hi rend="smallcaps">Gravity, Acceleration, Retardation</hi>, &c. Again,</p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Motion</hi>, is that which is produced by some one power or force only, and is always rectilinear, or in one direction, whether the force be only momentary or continued. And</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Motion</hi>, is that which is produced by two or more powers acting in different directions. See <hi rend="smallcaps">Compound</hi>, and <hi rend="smallcaps">Composition</hi> <hi rend="italics">of Motion.</hi></p><p>If a moving body be acted on by a double power; the one according to the direction AB, the other according to AC; with the Compound Motion, or that which is compounded of these two together, it will describe the diagonal AD of the parallelogram, whose sides AB and AC it would have described in the same time with each of the respective powers apart. <figure/></p><p>And if the radius of a circle be carried round upon the centre C, while a point in the radius sets off from A, and keeps moving along the radius towards the centre; then, by this Compound Motion, the path of the point will be a kind of a spiral ABC.</p><p><hi rend="italics">For the Particular Laws of</hi> <hi rend="smallcaps">Motion</hi>, <hi rend="italics">arising from the Collision of bodies, both Elastic and Non-elastic, and that where the directions are both Perpendicular and Oblique,</hi> see <hi rend="smallcaps">Percussion.</hi></p><p><hi rend="italics">For</hi> <hi rend="smallcaps">Circular</hi> <hi rend="italics">Motion,</hi> and <hi rend="italics">the Laws of</hi> P<hi rend="smallcaps">ROJECTILES</hi>, see the respective words.</p><p><hi rend="italics">For the Motion of Pendulums, and the Laws of Oscillation,</hi> see <hi rend="smallcaps">Pendulum.</hi></p><p><hi rend="italics">Perpetual</hi> <hi rend="smallcaps">Motion</hi>, is a Motion which is supplied and renewed from itself, without the intervention of any external cause.</p><p>The celebrated problem of a Perpetual Motion, consists in the inventing a machine, which has the principle of its Motion within itself; and is a problem that has employed the mathematicians for 2000 years; though none perhaps have prosecuted it with attention and earnestness equal to those of the present age. Infinite are the schemes, designs, plans, engines, wheels, &c, to which this long-desired Perpetual Motion has given birth.</p><p>But M. De la Hire has proved the impossibility of any such machine, and sinds that it amounts to this; viz, to find a body which is both heavier and lighter at the same time; or to find a body which is heavier than itself. Indeed there seems but little in nature to countenance all this assiduity and expectation: among all the laws of matter and Motion, we know of none yet that seem likely to furnish any principle or foundation for such an effect.<pb n="129"/><cb/></p><p>Action and reaction it is allowed are always equal; and a body that gives any quantity of Motion to another, always loses just so much of its own; but under the present state of things, the resistance of the air, the friction of the parts of machines, &c, do necessarily retard every Motion.</p><p>To continue the Motion therefore either, first, there must be a supply from some foreign cause; which in a Perpetual Motion is excluded.</p><p>Or, 2dly, all resistance from the friction of the parts of matter must be removed; which necessarily implies a change in the nature of things.</p><p>Or, 3dly and lastly, there must be some method of gaining a force equivalent to what is lost, by the artful disposition and combination of mechanic powers; to which last point then all endeavours are to be directed: but how, or by what means, such force should be gained, is still a mystery.</p><p>The multiplication of powers or forces, it is certain, avails nothing; for what is gained in power is lost in time, so that the quantity of Motion still remains the same. This is an inviolable law of nature; by which nothing is left to art, but the choice of the several combinations that may produce the same effect.</p><p>There are various ways by which absolute force may be gained; but since there is always an equal gain in opposite directions, and no increase obtained in the same direction; in the circle of actions necessary to make a perpetual movement, this gain must be presently lost, and will not serve for the necessary expence of force employed in overcoming friction, and the resistance of the medium. And therefore, though it could be shewn, that in an infinite number of bodies, or in an infinite machine, there could be a gain of force for ever, and a Motion continued to infinity, it does not follow that a perpetual movement can be made. That which was proposed by M. Leibnitz in the Leipsic Acts of 1690, as a consequence of the common effimation of the forces of bodies in Motion, is of this kind, and for this and other reasons ought to be rejected. See <hi rend="smallcaps">Perpetual</hi> <hi rend="italics">Motion;</hi> also <hi rend="smallcaps">Orffyreus's</hi> <hi rend="italics">Wheel,</hi> &c.</p><p><hi rend="italics">Animal</hi> <hi rend="smallcaps">Motion</hi>, is that by which the situation, figure, magnitude, &c, of the parts and members of animals are changed. Under these Motions, come all the animal functions; as respiration, circulation of the blood, excretion, walking, running, &c.</p><p>Animal Motions are usually divided into two species; viz, Natural and Spontaneous.</p><p><hi rend="italics">Natural</hi> <hi rend="smallcaps">Motion</hi>, is that involuntary one which is effected without the command of the will, by the mere mechanism of the parts. Such as the Motion of the heart and pulse; the Peristaltic Motion of the intestines, &c. But</p><p><hi rend="italics">Spontaneous,</hi> or <hi rend="italics">Muscular</hi> <hi rend="smallcaps">Motion</hi>, is that which is performed by means of the muscles, at the command of the will; which is hence called Voluntary Motion. Borelli has a celebrated treatise on this subject, entitled De Motu Animalium.</p><p><hi rend="italics">Intestine</hi> <hi rend="smallcaps">Motion</hi>, denotes an agitation of the particles of which a body consists.—Some philosophers will have every body, and every particle of a body, in continual Motion. As for fluids, it is the definition they give of them, that their parts are in continual Motion. And as to solids, they infer the like Motion<cb/> from the effluvia continually emitted through their pores. Hence Intestine Motion is represented to be a Motion of the internal and smaller parts of matter, continually excited by some external, latent agent, which of itself is insensible, and only discovers itself by its effects; appointed by Nature to be the great instrument of the changes in bodies.</p><div2 part="N" n="Motion" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Motion</hi></head><p>, in Astronomy, is peculiarly applied to the orderly courses of the heavenly bodies.</p><p><hi rend="italics">Mean</hi> <hi rend="smallcaps">Motion.</hi> See <hi rend="smallcaps">Mean.</hi></p><p>The Motions of the celestial luminaries are of two kinds: Diurnal, or Common; and Secondary, or Proper.</p><p><hi rend="italics">Diurnal,</hi> or <hi rend="italics">Primary</hi> <hi rend="smallcaps">Motion</hi>, is that with which all the heavenly bodies, and the whole mundane sphere, appear to revolve every day round the earth, from east to west. This is also called the Motion of the Primum Mobile, and the Common Motion, to distinguish it from that rotation which is peculiar to each planet, &c.</p><p><hi rend="italics">Secondary,</hi> or <hi rend="italics">Proper</hi> <hi rend="smallcaps">Motion</hi>, is that with which a star, planet, or the like, advances a certain space every day from the west towards the east. See the several Motions of each luminary, with the irregularities, &c. of them, under the proper articles, <hi rend="smallcaps">Earth, Moon, Star</hi>, &c.</p><p><hi rend="italics">Angular</hi> <hi rend="smallcaps">Motion</hi>, is that by which the angular position of any thing varics. See <hi rend="smallcaps">Angular.</hi></p><p><hi rend="italics">Horary</hi> <hi rend="smallcaps">Motion</hi>, is the Motion during each hour. See <hi rend="smallcaps">Horary.</hi></p><p><hi rend="italics">Paracentric</hi> <hi rend="smallcaps">Motion</hi> <hi rend="italics">of Impetus.</hi> See P<hi rend="smallcaps">ARACENTRIC.</hi></p><p><hi rend="smallcaps">Motion</hi> <hi rend="italics">of Trepidation,</hi> &c. See <hi rend="smallcaps">Trepidation</hi> and <hi rend="smallcaps">Libration.</hi></p><p>MOTIVE <hi rend="italics">Power</hi> or <hi rend="italics">Force,</hi> is the whole power or force acting upon any body, or quantity of matter, to move it; and is proportional to the momentum or quantity of motion it can produce in a given time. To distinguish it from the Accelerative force, which is considered as affecting the celerity only.</p></div2></div1><div1 part="N" n="MOTRIX" org="uniform" sample="complete" type="entry"><head>MOTRIX</head><p>, something that has the power or faculty of moving. See <hi rend="italics">Vis Motrix,</hi> and <hi rend="smallcaps">Motion.</hi></p></div1><div1 part="N" n="MOVEABLE" org="uniform" sample="complete" type="entry"><head>MOVEABLE</head><p>, something susceptible of motion, or that is disposed to be moved. A sphere is the most Moveable of all bodies, or is the easiest to be moved on a plane. A door is Moveable on its hinges; the magnetic needle on a pin or pivot, &c. Moveable is often used in contradistinction to Fixed or Fixt.</p><p><hi rend="smallcaps">Moveable</hi> <hi rend="italics">Feasts,</hi> are such as are not always held on the same day of the year or month; though they may be on the same day of the week. Thus, Easter is a Moveable Feast; being always held on the Sunday which falls upon or next after the first full moon following the 21st of March. See Philos. Trans. numb. 240, pa. 185. All the other Moveable Feasts follow Easter, keeping their constant distance from it; so that they are fixed with respect to it, though Moveable through the course of the year. Such are Septuagesima, Sexagesima, Ash-Wednesday, Ascension-Day, Pentecost, Trinity-Sunday, &c.</p></div1><div1 part="N" n="MOVEMENT" org="uniform" sample="complete" type="entry"><head>MOVEMENT</head><p>, a term often used in the same sense with Automaton. The most usual Movements for keeping time, are Clocks and Watches: the latter are such as shew the parts of time by inspection, and are portable in the pocket; the former such as publish it by sounds, and are fixed as furniture.<pb n="130"/><cb/></p><div2 part="N" n="Movement" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Movement</hi></head><p>, in its popular use, signifies all the inner works-of a clock, watch, or other machine, that move, and by that motion carry on the design of the instrument. The Movement of a clock, or watch, is the inside; or that part which measures the time, and strikes, &c; exclusive of the frame, case, dial-plate, &c.</p><p>The parts common to both of these Movements are, the Main-spring with its appurtenances, lying in the spring box, and in the middle of it lapping about the spring-arbor, to which one end of it is fastened. A-top of the spring-arbor is the Endless screw, and its wheel; but in spring clocks this is a ratchet-wheel with its click, that stops it. That which the main-spring draws, and round which the chain or string is wrapped, is called the fusee: this is mostly taper; in large works, going with weights, it is cylindrical, and is called the barrel. The small teeth at the bottom of the fusee or barrel, which stop it in winding up, is called the Ratchet; and that which stops it when wound up, and is for that end driven up by the spring, the Gardegut. The Wheels are various: the parts of a wheel are, the Hoop or Rim; the Teeth, the Cross, and the Collet, or piece of brass soldered on the arbor or spindle on which the wheel is riveted. The little wheels, playing in the teeth of the larger, are called Pinions; and their teeth, which are 4, 5, 6, 8, &c, are called Leves; the ends of the spindle are called Pivots; and the guttured wheel, with iron spikes at bottom, in which the line of common clocks runs, the Pulley.</p><p><hi rend="italics">Theory of Calculating the Numbers for</hi> <hi rend="smallcaps">Movements.</hi></p><p>1. It is first to be observed, that a wheel, divided by its pinion, shews how many turns the pinion has to one turn of the wheel.</p><p>2. That from the fusee to the balance the wheels drive the pinions, consequently the pinions run faster, or make more revolutions, chan the wheel; but it is the contrary from the great wheel to the dial-wheel.</p><p>3. That the wheels and pinions are written down either as vulgar fractions, or in the way of division in common arithmetic: sor example, a wheel of 60, moving a pinion of 5, is set down either thus 60/5, or thus 5)60, which is better. And the number of turns the pinion has in one turn of the wheel, as a quotient, thus 5) 60 (12. A whole Movement may be written as follows: <table><row role="data"><cell cols="1" rows="1" role="data">4 )</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">(9</cell></row><row role="data"><cell cols="1" rows="1" role="data">5 )</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">( 11</cell></row><row role="data"><cell cols="1" rows="1" role="data">5 )</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">(   9</cell></row><row role="data"><cell cols="1" rows="1" role="data">5 )</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">(   8</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">   17</cell></row></table> where the uppermost number expresses the pinion of report 4, the dial-wheel 36, and the turns of the pinion 9; the second, the pinion and great wheel; the third, the second wheel &c; the sourth, the contrate wheel; and the last, 17, the crown-wheel.</p><p>4. Hence, from the number of turns any pinion makes, in one turn of the wheel it works in, may be determined the number of turns a wheel or pinion has at any greater distance, viz, by multiplying the quotients together; the product being the number of turns. Thus, suppose the wheels and pinions as in the case above; the quotient 11 multiplied by 9, gives 99, the<cb/> number of turns in the second pinion 5 to one turn of the wheel 55, which runs concentrical, or on the same spindle, with the pinion 5. Again, 99 multiplied by 8, gives 792, the number of turns the last pinion has to one turn of the first wheel 5. Hence we proceed to find, not only the turns, but the number of beats of the balance, in the time of those turns. For, having found the number of turns the crown-wheel has in one turn of the wheel proposed, those turns multiplied by its notches, give half the number of beats in that one turn of the wheel. Suppose, for example, the crownwheel to have 720 turns, to one of the first wheel; this number multiplied by 15, the notches in the crownwheel, produces 10800, half the number of strokes of the balance in one turn of the first wheel of 80 teeth.</p><p>The general division of a Movement is, into the clock, and watch parts.</p></div2></div1><div1 part="N" n="MOULDINGS" org="uniform" sample="complete" type="entry"><head>MOULDINGS</head><p>, in Architecture, are certain projections beyond the naked of a wall, column, wainscot &c, the assemblage of which forms cornices, door-cases, and other decorations of architecture.</p><div2 part="N" n="Mouldings" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Mouldings</hi></head><p>, are annexed to great guns by way of ornament, and perhaps in some parts for strength; and probably are derived from the hoops or rings which bound the long iron bars together, anciently used in making cannon.</p><p>MOYNEAU. See <hi rend="smallcaps">Moineau.</hi></p></div2></div1><div1 part="N" n="MULLER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MULLER</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, commonly called R<hi rend="smallcaps">EGIOMONTANUS</hi>, from Mons Regius, or Koningsberg, a town in Franconia, where he was born in 1436, and became the greatest astronomer and mathematician of his time. He was indeed a very prodigy for genius and learning. Having first acquired grammatical learning in his own country, he was admitted, while yet a boy, into the academy at Leipsic, where he formed a strong attachment to the mathematical sciences, arithmetic, geometry, astronomy, &c. But not finding proper assistance in these studies at this place, he removed, at only 15 years of age, to Vienna, to study under the famous Purbach, the professor there, who read lectures in those sciences with the highest reputation. A strong and affectionate friendship soon took place between these two, and our author made such rapid improvement in the sciences, that he was able to be assisting to his master, and to become his companion in all his labours. In this manner they spent about ten years together; elucidating obscurities, observing the motions of the heavenly bodies, and comparing and correcting the tables of them; particularly those of Mars, which they found to disagree with the motions, sometimes as much as two degrees.</p><p>About this time there arrived at Vienna the cardinal Bessarion, who came to negociate some affairs for the pope; who, being a lover of astronomy, soon formed an acquaintance with Purbach and Regiomontanus. He had begun to form a Latin Version of Ptolomy's Almagest, or an Epitome of it; but not having time to go on with it himself, he requested Purbach to complete the work, and for that purpose to return with him into Italy, to make himself master of the Greek tongue, which he was as yet unacquainted with. To these proposals Purbach only assented, on condition that Regiomontanus would accompany him, and share in all the labours. They first however, by<pb n="131"/><cb/> means of an Arabic Version of Ptolomy, made some progress in the work; but this was soon interrupted by the death of Purbach, which happened in 1461, in the 39th year of his age. The whole task then devolved upon Regiomontanus, who finished the work, at the request of Purbach, made to him when on his death<*> bed. This work our author afterwards revised and perfected at Rome, when he had learned the Greek language, and consulted the commentator Theon, &c.</p><p>Regiomontanus accompanied the cardinal Bessarion in his return to Rome, being then near 30 years of age. Here he applied himself diligently to the study of the Greek language; not neglecting however to make astronomical observations and compose various works in that science; as his Dialogue against the Theories of Cremonensis. The cardinal going to Greece soon after, Regiomontanus went to Ferrara, where he continued the study of the Greek language under Theodore Gaza; who explained to him the text of Ptolomy, with the commentaries of Theon; till at length he became so perfect in it, that he could compose verses, and read it like a critic.—In 1463 he went to Padua, where he became a member of the university; and, at the request of the students, explained Alfraganus, an Arabian philosopher.—In 1464 he removed to Venice, to meet and attend his patron Bessarion. Here he wrote, with great accuracy, his Treatise of Triangles, and a Refutation of the Quadrature of the Circle, which Cardinal Cusan pretended he had demonstrated. The same year he returned with Bessarion to Rome; where he made some stay, to procure the most curious books: those he could not purchase, he took the pains to transcribe, for he wrote with great facility and elegance; and others he got copied at a great expence. For as he was certain that none of these books could be had in Germany, he thought on his return thither, he would at his leisure translat, and publish some of the best of them. During this time too he had a fierce contest with George Trabezonde, whom he had greatly offended by animadverting on some passages in his translation of Theon's Commentary.</p><p>Being now weary of rambling about, and having procured a great number of manuscripts, which was one great object of his travels, he returned to Vienna, and performed for some time the offices of his professorship, by reading of lectures &c. After being a while thus employed, he went to Buda, on the invitation of Matthias king of Hungary, who was a great lover of letters and the sciences, and had founded a rich and noble library there: for he had bought up all the Greek books that could be found on the sacking of Constantinople; also those that were brought from Athens, or wherever else they could be met with through the whole Turkish dominions, collecting them all together into a library at Buda. But a war breaking out in this country, he looked out for some other place to settle in, where he might pursue his studies, and for this purpose he retired to Noremberg. He tells us, that the reasons which induced him to desire to reside in this city the remainder of his life were, that the artists there were dextrous in fabricating his astronomical machines; and besides, he could from thence easily transmit his letters by the merchants into foreign countries. Being now well versed in all parts<cb/> of learning, and made the utmost proficiency in mathematics, he determined to occupy himself in publishing the best of the ancient authors, as well as his own lucubrations. For this purpose he set up a printinghouse, and formed a nomenclature of the books he intended to publish, which still remains.</p><p>Here that excellent man, Bernard Walther, one of the principal citizens, who was well skilled in the sciences, especially astronomy, cultivated an intimacy with Regiomontanus; and as soon as he understood those laudable designs of his, he took upon himself the expence of constructing the astronomical instruments, and of erecting a printing-house. And first he ordered astronomical rules to be made of tin, for observing the altitudes of the sun, moon and planets. He next constructed a rectangular, or astronomical radius, for taking the distances of those luminaries. Then an armillary astrolabe, such as was used by Ptolomy and Hipparchus, for observing the places and motions of the stars. Lastly, he made other smaller instruments, as the torquet, and Ptolomy's meteoroscope, with some others which had more of curiosity than utility in them. From this apparatus it evidently appears, that Regiomontanus was a most diligent observer of the laws and motions of the celestial bodies, if there were not still stronger evidences of it in the accounts of the observations themselves which he made with them.</p><p>With regard to the printing-house, which was the other part of his design in settling at Noremberg, as soon as he had completed it, he put to press two works of his own, and two others. The latter were, The <hi rend="italics">New Theories</hi> of his master Purbach, and the <hi rend="italics">Astronomicon</hi> of Manilius. And his own were, the <hi rend="italics">New Calendar,</hi> in which were given (as he says in the Index of the books which he intended to publish) the true conjunctions and oppositions of the luminaries, their eclipses, their true places every day, &c. His other work was his <hi rend="italics">Ephemerides,</hi> of which he thus speaks in the said index: “The Ephemerides, which they vulgarly call an Almanac, for 30 years: where you may every day see the true motion of all the planets, of the moon's nodes, with the aspects of the moon to the sun and planets, the eclipses of the luminaries; and in the fronts of the pages are marked the latitudes.” He published also most acute commentaries on Ptolomy's Almagest: a work which cardinal Bessarion so highly valued, that he scrupled not to esteem it worth a whole province. He prepared also new versions of Ptolomy's Cosmography; and at his leisure hours examined and explained works of another nature. He enquired how high the vapours are carried above the earth, which he fixed to be not more than 12 German miles. He set down observations of two comets that appeared in the years 1471 and 1472.</p><p>In 1474, pope Sixtus the 4th conceived a design of reforming the calendar; and sent for Regiomontanus to Rome, as the properest and ablest person to accomplish his purpose. Regiomontanus was very unwilling to interrupt the studies, and printing of books, he was engaged in at Noremberg; but receiving great promises from the pope, who also for the present named him bishop of Ratisbon, he at length consented to go. He arrived at Rome in 1475, but died there the year after, at only 40 years of age; not without <*><pb n="132"/><cb/> suspicion of being poisoned by the sons of George Trabezonde, in revenge for the death of their father, which was said to have been caused by the grief he felt on account of the criticisms made by Regiomontanus on his translation of Ptolomy's Almagest.</p><p>Purbach first of any reduced the trigonometrical tables of fines, from the old sexagesimal division of the radius, to the decimal scale. He supposed the radius to be divided into 600000 equal parts, and computed the sines of the arcs to every ten minutes, in such equal parts of the radius, by the decimal notation. This project of Purbach was perfected by Regiomontanus; who not only extended the sines to every minute, the radius being 600000, as designed by Purbach, but afterwards, disliking that scheme, as evidently imperfect, he computed them likewise to the radius 1000000, for every minute of the quadrant. Regiomontanus also introduced the tangents into trigonometry, the canon of which he called <hi rend="italics">fœcundus,</hi> because of the many great advantages arising from them. Beside these things, he enriched trigonometry with many theorems and precepts. Indeed, excepting for the use of logarithms, the trigonometry of Regiomontanus is but little inferior to that of our own time. His Treatise, on both Plane and Spherical Trigonometry, is in 5 books; it was written about the year 1464, and printed in folio at Noremberg in 1533. In the 5th book are various problems concerning rectilinear triangles, some of which are resolved by means of algebra: a proof that this science was not wholly unknown in Europe before the treatise of Lucas de Burgo.</p><p>Regiomontanus was author of some other works beside those before mentioned. Peter Ramus, in the account he gives of the admirable works attempted and performed by Regiomontanus, tells us, that in his workshop at Noremberg there was an automaton in perpetual motion: that he made an artisicial fly, which taking its flight from his hand, would fly round the room, and at last, as if weary, would return to his master's hand: that he fabricated an eagle, which, on the emperor's approach to the city, he sent out, high in the air, a great way to meet him, and that it kept him company to the gates of the city. Let us no more wonder, adds Ramus, at the dove of Archytas, since Noremberg can shew a fly, and an eagle, armed with geometrical wings. Nor are those famous artificers, who were formerly in Greece, and Egypt, any longer of such account, since Noremberg can boast of her Regiomontanuses. For Wernerus first, and then the Schoneri, father and son, afterwards, revived the spirit of Regiomontanus.</p><p>MULTANGULAR <hi rend="smallcaps">Figure</hi>, is one that has many angles, and consequently many sides also. These are otherwise called polygons.</p><p>MULTILATERAL <hi rend="smallcaps">Figures</hi>, are such as have many sides, or more than four sides.</p></div1><div1 part="N" n="MULTINOMIAL" org="uniform" sample="complete" type="entry"><head>MULTINOMIAL</head><p>, or <hi rend="smallcaps">Multinomial</hi> <hi rend="italics">Roots,</hi> are such as are composed of many names, parts, or members; as, <hi rend="italics">a</hi> + <hi rend="italics">b</hi> + <hi rend="italics">c</hi> + <hi rend="italics">d</hi> &c.</p><p>For the raising an infinite Multinomial to any proposed power, or extracting any root out of such power, see a method by Mr. De Moivre, in the Philos. Trans. numb. 230. See also <hi rend="smallcaps">Polynomial.</hi></p></div1><div1 part="N" n="MULTIPLE" org="uniform" sample="complete" type="entry"><head>MULTIPLE</head><p>, <hi rend="smallcaps">Multiplex</hi>, a number which com-<cb/> prehends some other number several times. Thus, 6 is a Multiple of 2, this being contained in 6 just 3 times. Also 12 is a common Multiple of 6, 4, and 3; comprehending the first twice, the second thrice, and the third four times.</p><p><hi rend="smallcaps">Multiple</hi> <hi rend="italics">Ratio</hi> or <hi rend="italics">Proportion,</hi> is that which is between Multiple numbers &c. If the less term of a ratio be an aliquot part of the greater, the ratio of the greater to the less is called Multiple; and that of the less to the greater Submultiple.</p><p>A Submultiple number, is that which is contained in the Multiple. Thus, the numbers 2, 3, and 4 are Submultiples of 12 and 24.</p><p>Duple, triple, &c ratios; as also subduples, subtriples, &c, are so many species of Multiple and Submultiple ratios.</p><p><hi rend="smallcaps">Multiple</hi> <hi rend="italics">Superparticular Proportion,</hi> is when one number or quantity contains another more than once, and a certain aliquot part; as 10 to 3, or 3 1/3 to 1.</p><p><hi rend="smallcaps">Multiple</hi> <hi rend="italics">Superpartient Proportion,</hi> is when one number or quantity contains another several times, and some parts besides; as 29 to 6, or 4 5/6 to 1.</p></div1><div1 part="N" n="MULTIPLICAND" org="uniform" sample="complete" type="entry"><head>MULTIPLICAND</head><p>, is one of the two factors in the rule of multiplication, being that number given to be multiplied by the other, called the multiplicator, or multiplier.</p></div1><div1 part="N" n="MULTIPLICATION" org="uniform" sample="complete" type="entry"><head>MULTIPLICATION</head><p>, is, in general, the taking or repeating of one number or quantity, called the Multiplicand, as often as there are units in another number, called the Multiplier, or Multiplicator; and the number or quantity resulting from the Multiplication, is called the Product of the two foregoing numbers or factors.</p><p>Multiplication is a compendious addition; performing at once, what in the usual way of addition would require many operations: for the multiplicand is only added to itself, or repeated, as often as is expressed by the units in the multiplier. Thus, if 6 were to be multiplied by 5, the product is 30, which is the sum arising from the addition of the number 6 five times to itself.</p><p>In every Multiplication, 1 is in proportion to the mulplier, as the multiplicand is to the product.</p><p>Multiplication is of various kinds, in whole numbers, in fractions, decimals, algebra, &c.</p><p>1. <hi rend="smallcaps">Multiplication</hi> <hi rend="italics">of Whole Numbers,</hi> is performed by the following rules: When the multiplier consists of only one figure, set it under the first, or righthand figure, of the multiplicand; then, drawing a line underneath, and beginning at the said first figure, multiply every figure of the multiplicand by the multiplier; setting down the several products below the line, proceeding orderly from right to left. But if any of these products amount to 10, or several 10's, either with or without some overplus, then set down only the overplus, or set down 0 if there be no overplus; and carry, to the next product, as many units as the former contained of tens. Thus, to multiply 35092 by 4. <table><row role="data"><cell cols="1" rows="1" role="data">Multiplicand</cell><cell cols="1" rows="1" rend="align=right" role="data">35092</cell></row><row role="data"><cell cols="1" rows="1" role="data">Multiplier</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">  Product</cell><cell cols="1" rows="1" rend="align=right" role="data">140368</cell></row></table><pb n="133"/><cb/></p><p>When the multiplier consists of several figures; multiply the multiplicand by each figure of it, as before, and place the several lines of products underneath each other in such order, that the first figure or cipher of each line may fall straight under its respective multiplier, or multiplying figure; then add these several lines of products together, as they stand, and the sum of them all will be the product of the whole multiplication. Thus, to multiply 63017 by 236: <table><row role="data"><cell cols="1" rows="1" role="data">Multiplicand</cell><cell cols="1" rows="1" rend="align=right" role="data">63017</cell></row><row role="data"><cell cols="1" rows="1" role="data">Multiplier</cell><cell cols="1" rows="1" rend="align=right" role="data">236</cell></row><row role="data"><cell cols="1" rows="1" role="data">Product of 63017 by 6</cell><cell cols="1" rows="1" rend="align=right" role="data">378102</cell></row><row role="data"><cell cols="1" rows="1" role="data">Product of 63017 by 30</cell><cell cols="1" rows="1" rend="align=right" role="data">189051 </cell></row><row role="data"><cell cols="1" rows="1" role="data">Product of 63017 by 200</cell><cell cols="1" rows="1" rend="align=right" role="data">126034  </cell></row><row role="data"><cell cols="1" rows="1" role="data">Whole product</cell><cell cols="1" rows="1" rend="align=right" role="data">14872012</cell></row></table></p><p>The several lines of products may be set down in any order, or any of them first, and any other of them second, &c; for the order of placing them can make no difference in the sum total. There are many abbreviations, and peculiar cases, according to circumstances, which may be seen in most books of arithmetic.</p><p>The mark or character now used for Multiplication, is either the × cross or a single point .; the former being introduced by Oughtred, and the latter I think by Leibnitz.</p><p><hi rend="italics">To Prove</hi> <hi rend="smallcaps">Multiplication.</hi> This may be done various ways; either by dividing the product by the multiplier, then the quotient will be equal to the multiplicand; or divide the same product by the multiplicand, and the quotient will come out equal to the multiplier; or in general divide the product by either of the two factors, and the quotient will come out equal to the other factor, when the operations are all right. But the more usual, and compendious way of proving Multiplication, is by what is called casting out the nines; which is thus performed: Add the sigures of the multiplicand all together, and as often as the sum amounts to 9, reject it always, and set down the last overplus as in the margin; this in the foregoing example is 8. Then do the same by <figure/> the multiplier, setting down the last overplus, which is 2, on the right of the former remainder 8. Next multiply these two remainders, 2 and 8, together, and from their product 16, cast out the 9, and there remains 7, which set down over the two former. Lastly, add up, in the same manner, all the figures of the whole product of the multiplication, viz 14872012, casting out the 9's, and then there remains 7, to be set down under the two first remains. Then when the figure at top, is the same as that at bottom, as they are here both 7's, the work it may be presumed is right; but if these two figures should not be the same, it is certainly wrong.</p><p>2. <hi rend="italics">To Multiply Money, or any other thing, consisting of different Denominations together, by any number, usually called Compound Multiplication.</hi> Beginning at the lowest, multiply the number of each denomination separately by the multiplier, setting down the products below them. But if any of these products amount to as much<cb/> as 1 or more of the next higher denominations, carry so many to the next product, and set down only the overplus. <hi rend="italics">For Ex.</hi> To find the amount of 9 things at 1l 12s 4 1/2d. each; or to multiply 1l 12s 4 1/2d by 9. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">l</cell><cell cols="1" rows="1" role="data">s</cell><cell cols="1" rows="1" role="data">d</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">4 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 9</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">4 1/2</cell></row></table> set the multiplier 9 under the given sum as in the margin, and multiply thus: 9 halfpence make 4d halfpenny, set down 1/2 penny, and carry 4; then 9 times 4 are 36, and 4 to carry make 40 pence, which are 3s and 4d, set down 4 and carry 3; next 9 times 12 are 108, and 3 to carry, make 111 shillings, or 5l 11s, set down 11, and carry 5; lastly 9 times 1 are 9, and 5 to carry, make 14, which set down; and then the whole amount, or product, comes to 14l 11s 4 1/2d.</p><p>3. <hi rend="italics">To Multiply Vulgar Fractions.</hi>—Multiply all the given numerators together for the numerator of the product, and all the denominators together for the denominator of the product sought.</p><p>Thus, 2/3 multiplied by 4/5, or .</p><p>And .</p><p>And here it may be noted that, when there are any common numbers in the numerators and denominators, these may be omitted from both, which will make the operation shorter, and bring out the whole product in a fraction much simpler and in lower terms. Thus, , by leaving out the two 3's, become</p><p>Also, when any numerators and denominators will both abbreviate or divide by one and the same number, let them be divided, and the quotients used instead of them. So, in the above example, after omitting the two 3's, let the 2 and 6 be both divided by 2, and use the quotients 1 and 3 instead of them, so shall the expression become , as before.</p><p>4. <hi rend="italics">To Multiply Decimals.</hi>—Multiply the given numbers together the same as if they were whole numbers, and point off as many decimals in the whole product as there are in both factors together; <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2.305</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">21.86</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13830</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18440 </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2305  </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4610   </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50.38730</cell></row></table> as in the annexed example, where the number of decimals is five, because there are three in the multiplicand, and two in the multiplier.—When it happens that there are not so many figures in the product as there must be decimals, then prefix as many ciphers as will supply the defect.</p><p>5. <hi rend="italics">Gross</hi> <hi rend="smallcaps">Multiplication</hi>, otherwise called <hi rend="italics">Duodecimal Arithmetic,</hi> is the multiplying of numbers together whose subdivisions proceed by 12's; as feet, inches, and parts, that is 12th parts, &c; a thing of very srequent use in squaring, or multiplying toge-<pb n="134"/><cb/> ther the dimensions of the works of bricklayers, carpenters, and other artificers. <hi rend="italics">For Example.</hi> To multiply 5 feet 3 inches by 2 feet 4 inches. Set them down as in the <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">F</cell><cell cols="1" rows="1" role="data">I</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell></row></table> margin, and multiply all the parts of the multiplicand by each part of the multiplier; thus, 2 times 3 make 6 inches, and 2 times 5 make 10 feet; then 4 times 3 make 12 parts, or 1 inch to carry; and 4 times 5 make 20, and 1 to carry makes 21 inches, or 1f. 9inc. to set down below the former line: Lastly adding the two lines together, the whole sum or product amounts to 12f. 3inc.</p><p>6. <hi rend="smallcaps">Multiplication</hi> <hi rend="italics">in Aigebra.</hi> This is performed, 1. When the quantities are simple, by only joining the letters together like a word; and if the simple quantities have any coefsicients or numbers joined with them, multiply the numbers together, and prefix the product of them to the letters so joined together. But, in algebra, we have not only to attend to the quantities themselves, but also to the signs of them; and the general rule for the signs is this: When the signs are alike, or the same, either both + or both -, then the sign of the product will always be + ; but when the signs are different, or unlike, the one +, and the other -, then the sign of the product will be -. Hence these <hi rend="center"><hi rend="smallcaps">Examples.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data">Mult.</cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">- 2<hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">+ 6<hi rend="italics">x</hi></cell><cell cols="1" rows="1" role="data">- 8<hi rend="italics">x</hi></cell><cell cols="1" rows="1" role="data">- 3<hi rend="italics">ab</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">By</cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">- 4<hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">- 3<hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">+ 5<hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">- 5<hi rend="italics">ac</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Products</cell><cell cols="1" rows="1" role="data">+ <hi rend="italics">ab</hi></cell><cell cols="1" rows="1" role="data">+ 8<hi rend="italics">ab</hi></cell><cell cols="1" rows="1" role="data">- 18<hi rend="italics">ax</hi></cell><cell cols="1" rows="1" role="data">- 40<hi rend="italics">ax</hi></cell><cell cols="1" rows="1" role="data">+ 15<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">bc</hi></cell></row></table></p><p>2. In Compound quantities, multiply every term or part of the multiplicand by each term separately of the multiplier, and set down all the products with their signs, collecting always into one sum as many terms as are similar or like to one another. <hi rend="center"><hi rend="smallcaps">Examples.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> - <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> - <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> - <hi rend="italics">b</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">ab</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> - <hi rend="italics">ab</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">ab</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">+ <hi rend="italics">ab</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">- <hi rend="italics">ab</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">- <hi rend="italics">ab</hi> - <hi rend="italics">b</hi><hi rend="sup">2</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> + 2<hi rend="italics">ab</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> - 2<hi rend="italics">ab</hi> + <hi rend="italics">b</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> - <hi rend="italics">b</hi><hi rend="sup">2</hi></cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> - 3<hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 4<hi rend="italics">x</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi> - <hi rend="italics">ax</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">4<hi rend="italics">a</hi> + 5<hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> - 4<hi rend="italics">x</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 2<hi rend="italics">x</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">8<hi rend="italics">a</hi><hi rend="sup">2</hi> - 12<hi rend="italics">ab</hi></cell><cell cols="1" rows="1" role="data">4<hi rend="italics">a</hi><hi rend="sup">2</hi> + 8<hi rend="italics">ax</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi><hi rend="sup">3</hi> - 2<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">+ 10<hi rend="italics">ab</hi> - 15<hi rend="italics">b</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">- 8<hi rend="italics">ax</hi> - 16<hi rend="italics">x</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">+ 2<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi> - 2<hi rend="italics">a</hi><hi rend="italics">x</hi><hi rend="sup">2</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">8<hi rend="italics">a</hi><hi rend="sup">2</hi> - 2<hi rend="italics">ab</hi> - 15<hi rend="italics">b</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">4<hi rend="italics">a</hi><hi rend="sup">2</hi> - 16<hi rend="italics">x</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi><hi rend="sup">3</hi> - 2<hi rend="italics">ax</hi><hi rend="sup">3</hi></cell></row></table></p><p>3. In Surd quantities, if the terms can be reduced to a common surd, the quantities under each may be<cb/> multiplied together, and the mark of the same surd prefixed to the product; but if not, then the different surds may be set down with some mark of multiplication between then, to denote their product. <hi rend="center"><hi rend="smallcaps">Examples.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data">7√(<hi rend="italics">ax</hi>)</cell><cell cols="1" rows="1" role="data">√7</cell><cell cols="1" rows="1" role="data">√<hi rend="sup">3</hi>(7<hi rend="italics">ab</hi>)</cell><cell cols="1" rows="1" role="data">√(12<hi rend="italics">a</hi>)</cell><cell cols="1" rows="1" role="data">6<hi rend="italics">a</hi>√(2<hi rend="italics">cx</hi>)</cell></row><row role="data"><cell cols="1" rows="1" role="data">5√(<hi rend="italics">cx</hi>)</cell><cell cols="1" rows="1" role="data">√5</cell><cell cols="1" rows="1" role="data">√<hi rend="sup">3</hi>(4<hi rend="italics">ac</hi>)</cell><cell cols="1" rows="1" role="data">√(3<hi rend="italics">a</hi>)</cell><cell cols="1" rows="1" role="data">2<hi rend="italics">b</hi>√(3<hi rend="italics">ax</hi>)</cell></row><row role="data"><cell cols="1" rows="1" role="data">35√(<hi rend="italics">acx</hi><hi rend="sup">2</hi>)</cell><cell cols="1" rows="1" role="data">√35</cell><cell cols="1" rows="1" role="data">√<hi rend="sup">3</hi>(28<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">bc</hi>)</cell><cell cols="1" rows="1" role="data">√(36<hi rend="italics">a</hi><hi rend="sup">2</hi>) = 6<hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">12<hi rend="italics">ab</hi>√(6<hi rend="italics">acx</hi><hi rend="sup">2</hi>)</cell></row></table></p><p>4. Powers or Roots of the same quantity are multiplied together, by adding their exponents. Thus, ; and : also ; and </p><p><hi rend="italics">To Multiply Numbers together by Logarithms.</hi>—This is performed by adding together the logarithms of the given numbers, and taking the number answering to that sum, which will be the product sought.</p><p>Des Cartes, at the beginning of his Geometry, performs Multiplication (and indeed all the other common arithmetical rules) in geometry, or by lines; but this is no more than taking a 4th proportional to three given lines, of which the first represents unity, and the 2d and 3d the two factors or terms to be multiplied, the product being expressed by the 4th proportional; because, in every multiplication, unity or 1 is to either of the two factors, as the other factor is to the product.</p></div1><div1 part="N" n="MULTIPLICATOR" org="uniform" sample="complete" type="entry"><head>MULTIPLICATOR</head><p>, is the number or quantity by which another is multiplied; and is otherwise called the multiplier.</p></div1><div1 part="N" n="MULTIPLIER" org="uniform" sample="complete" type="entry"><head>MULTIPLIER</head><p>, or <hi rend="smallcaps">Multiplicator</hi>, is the number or quantity which multiplies another, called the multiplicand, in any operation of multiplication.</p></div1><div1 part="N" n="MUNSTER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MUNSTER</surname> (<foreName full="yes"><hi rend="smallcaps">Sebastian</hi></foreName>)</persName></head><p>, an eminent German divine and mathematician, was born at Ingelheim in 1489. At the age of 14 he was sent to Heidelberg to study. Two years after, he entered the convent of the Cordeliers; where he assiduonsly studied divinity, mathematics, and geography. He was the first who published a Chaldee Grammar and Lexicon; and he shortly after gave the world a Talmudic Dictionary. He afterwards became professor of the Hebrew language at Basil. He was one of the first who attached himself to Luther, and embraced Protestantism: yet behaved himself with great moderation; never concerning himself with their disputes; but shut himself up at home and pursued his favourite studies, which were mathematics, natural philosophy, with the Hebrew and other Oriental languages. He published a great number of books on these subjects; particularly, a Latin version, from the Hebrew, of all the books of the Old Testament, with learned notes, printed at Basil in 1534 and 1546; Josephus's History of the Jews in Latin; a Treatise of Dialling, in folio, 1536; Universal Cosmography, in 6 books folio, Basil 1550. For these works he was styled the German Strabo; as he was the German Esdras, for his Oriental writings.</p><p>Munster was a meek-tempered, pacisic, studious, retired man, who wrote a great number of books, but<pb n="135"/><cb/> never meddled in controversy.—He died of the plague at Basil, in 1552, at 63 years of age.</p></div1><div1 part="N" n="MURDERERS" org="uniform" sample="complete" type="entry"><head>MURDERERS</head><p>, a small species of ordnance once used on shipboard; but now out of use.</p></div1><div1 part="N" n="MUSIC" org="uniform" sample="complete" type="entry"><head>MUSIC</head><p>, the science of sound, considered as capable of producing melody, or harmony.</p><p>Among the Ancients, Music was taken in a much more extensive sense than among the Moderns: what we call the science of Music, was by the Ancients rather called Harmonica.</p><p>Music is one of the seven sciences called liberal, and comprehended also among the mathematical sciences, as having for its object discrete quantity or number; not however considering it in the abstract, like arithmetic; but in relation to time and sound, with intent to constitute a delightful harmony.</p><p>This science is also Theoretical and Practical. Theoretical, which examines the nature and properties of concords and discords, explaining the proportions between them by numbers. And Practical, which teaches not only compofition, or the manner of composing tunes, or airs; but also the art of singing with the voice, and playing on musical instruments.</p><p>It appears that Music was one of the most ancient of the arts; and, of all others, Vocal Music must doubtless have been the first kind. For man had not only the various tones of his own voice to make his observations on, before any other art or instrument was found out, but had the various natural strains of birds to give him occasion to improve his own voice, and the modulations of sounds it was capable of. The first invention of wind instruments Lucretius ascribes to the observation of the winds whistling in the hollow reeds. As for other kinds of instruments, there were so many occasions for cords or strings, that men could not be long in observing their various sounds; which might give rise to stringed instruments. And for the pulsative instruments, as drums and cymbals, they might arise from the observation of the naturally hollow noise of concave bodies.</p><p>As to the inventors and improvers of Music, Plutarch, in one place, ascribes the first invention of it to Apollo; and in another place to Amphion, the son of Jupiter and Antiope. The latter indeed, it is pretty generally allowed, first brought Music into Greece, and invented the lyre.</p><p>To him succeeded Chiron, the demigod; then Demodocus; Hermes Trismegistus: Olympus; and Orpheus, whom some make the first introducer of Music into Greece, and the inventor of the lyre: to whom add Phemius, and Terpander, who was contemporary with Lycurgus, and set his laws to Music; to whom also some attribute the first institution of musical modes, and the invention of the lyre: lastly, Thales; and Thamyris, who, it has been said, was the first inventor of instrumental Music without singing.</p><p>These were the eminent musicians before Homer's time: others of a later date were, Lasus Hermionensis, Melanippides, Philoxenus, Timotheus, Phrynnis, Epigonius, Lysander, Simmicus, and Diodorus; who were all of them considerable improvers of Mufic. Lasus, it is said, was the first author who wrote upon Music, in the time of Darius Hystaspis; Epigonius invented an instrument of 40 strings, called the Epigonium.<cb/> Simmicus also invented an instrument of 35 strings, called a Simmicium; Diodorus improved the Tibia, by adding new holes; and Timotheus the Lyre, by adding a new string; for which he was fined by the Lacedemonians.</p><p>As the accounts we have of the inventors of musical instruments among the Ancients are very obscure, so also are the accounts of those instruments themselves; of most of them indeed we know little more than the bare names.</p><p>The general division of instruments is, into stringed instruments, wind instruments, and those of the pulsatile kind. Of stringed instruments, mention is made of the ly raor cithara, the psalterium, trigonum, sambuca, pectis, magas, barbiton, testudo, epigonium, simmicium, and panderon; which were all struck with the hand, or a plectrum. Of wind instruments, were the tibia, fistula, hydraulic organs, tubæ, cornua, and lituus. And the pulsatile instruments were the tympanum, cymbalum, creptaculum, tintinnabulum, crotalum, and sistrum.</p><p>Music has ever been in the highest esteem in all ages, and among all people; nor could authors express their opinion of it strongly enough, but by inculcating that it was used in heaven, and as one of the principal entertainments of the gods, and the souls of the blessed. The effects ascribed to it by the Ancients are almost miraculous: by its means, it has been said, diseases have been cured, unchastity corrected, seditions quelled, passions raised and calmed, and even madness occasioned. Athenæus assures us, that anciently all laws, divine and civil, exhortations to virtue, the knowledge of divine and human things, with the lives and actions of illustrious men, were written in verse, and publicly sung by a chorus to the sound of instruments; which was found the most effectual means to impress morality on the minds of men, and a right sense of their duty.</p><p>Dr. Wallis has endeavoured to account for the surprising effects attributed to the ancient Music; and ascribes them chiefly to the novelty of the art, and the hyperboles of the ancient writings: nor does he doubt, but the modern Music, in like cases, would produce effects at least as considerable as the ancient. The truth is, we can match most of the ancient stories of this kind in the modern histories. If Timotheus could excite Alexander's fury with the Phrygian mode, and sooth him into indolence with the Lydian; a more modern musician has driven Eric, king of Denmark, into such a rage, as to kill his best servants. Dr. Niewentyt speaks of an Italian who, by varying his Music from brisk to solemn, and the contrary, could so move the soul, as to cause distraction and madness; and Dr. South has founded his poem, called Musica Incantans, on an instance he knew of the same kind.</p><p>Music however is found not only to exert its force on the affections, but on the parts of the body also: witness the Gascon knight, mentioned by Mr. Boyle, who could not contain his water at the playing of a bagpipe; and the woman, mentioned by the same author, who would burst into tears at the hearing of a certain tune, with which other people were but a little affected. To say nothing of the trite story of the Tarantula, we have an instance, in the History of the Academy of Sciences, of a musician being cured of a violent<pb n="136"/><cb/> fever, by a little concert occasionally played in his room.</p><p>Nor are our minds and bodies alone affected with sounds, but even inanimate bodies are so. Kircher speaks of a large stone, that would tremble at the sound of one particular organ pipe; and Morhoff mentions one Petter, a Dutchman, who could break rummer-glasses with the tone of his voice. Mersenne also mentions a particular part of a pavement, that would shake and tremble, as if the earth would open, when the organs played. Mr. Boyle adds, that seats will tremble at the sound of organs; that he has felt his hat do so under his hand, at certain notes both of organs and discourse; and that he was well informed every well-built vault would thus answer to some determinate note.</p><p>It has been disputed among the Learned, whether the Ancients or Moderns best understood and practised Music. Some maintain that the ancient art of Music, by which such wonderful effects were performed, is quite lost; and others, that the true science of harmony is now arrived at much greater perfection than was known or practised among the Ancients. This point seems no other way to be determinable but by comparing the principles and practice of the one with those of the other. As to the theory or principles of harmonics, it is certain we understand it better than the Ancients; because we know all that they knew, and have improved considerably on their foundations. The great dispute then lies on the practice; with regard to which it may be observed, that among the Ancients, Music, in the most limited sense of the word, included Harmony, Rythmus, and Verse; and consisted of verses sung by one or more voices alternately, or in choirs, sometimes with the sound of instruments, and sometimes by voices only. Their musical faculties, we have just observed, were Melopœia, Rythmopœia, and Poesis; the first of which may be considered under two heads, Melody and Symphony. As to the latter, it seems to contain nothing but what relates to the conduct of a single voice, or making what we call Mclody. It does not appear that the Ancients ever thought of the concert, or harmony of parts; which is a modern invention, for which we are beholden to Guido Aretine, a Benedictine friar.</p><p>Not that the Ancients never joined more voices or instruments than one together in the same symphony; but that they never joined several voices so as that each had a distinct and proper melody, which made among them a succession of various concords, and were not in every note unisons, or at the same distance from each other as octaves. This last indeed agrees to the general definition of the word Symphonia; yet it is plain that in such cases there is but one song, and all the voices perform the same individual melody. But when the parts differ, not by the tension of the whole, but by the different relations of the successive notes, this is the modern art, which requires so peculiar a genius, and on which account the modern Music seems to have much the advantage of the ancient. For farther satisfaction on this head, see Kircher, Perrault, Wallis, Malcolm, Cerceau, and others; who unanimously agree, that after all the pains they have taken to know the true state of the Music of the Ancients, they could not find<cb/> the least reason to think there was any such thing in their days as Music in parts.</p><p>The ancient musical notes are very mysterious and perplexed: Boethius and Gregory the Great first put them into a more easy and obvious method. In the year 1204, Guido Aretine, a Benedictine of Arezzo in Tuscany, first introduced the use of a stass with five lines, on which, with the spaces, he marked his notes by setting a point up and down upon them, to denote the rise and fall of the voice: though Kircher says this artifice was in use before Guido's time.</p><p>Another contrivance of Guido's was to apply the fix musical syllables, <hi rend="italics">ut, re, mi, fa, sol, la,</hi> which he took out of the Latin hymn, <table><row role="data"><cell cols="1" rows="1" role="data">UT queant laxis</cell><cell cols="1" rows="1" role="data">REsonare fibris</cell></row><row role="data"><cell cols="1" rows="1" role="data">MIra gestorum</cell><cell cols="1" rows="1" role="data">FAmuli tuorum,</cell></row><row role="data"><cell cols="1" rows="1" role="data">SOLve polluti</cell><cell cols="1" rows="1" role="data">LAbii reatum,</cell></row></table> <hi rend="center">O Pater Alme.</hi></p><p>We find another application of them in the following lines.</p><p>UT RElevit MIserum FAtum, SOLitosque LAbores Aevi, sit dulcis musica noster amor.</p><p>Besides his notes of Music, by which, according to Kircher, he distinguished the tones, or modes, and the seats of the semitones, he also invented the scale, and several musical instruments, called polyplectra, as spinets and harpsichords.</p><p>The next considerable improvement was in 1330, when Joannes Muria, or de Muris, doctor at Paris (or as Bayle and Gesner make him, an Englishman), invented the different figures of notes, which express the times or length of every note, at least their true relative proportions to one another, now called longs, breves, semi-breves, crotchets, quavers, &c.</p><p>The most ancient writer on Music was Lasus Hermionensis; but his works, as well as those of many others, both Greek and Roman, are lost. Aristoxenus, disciple of Aristotle, is the earliest author extant on the subject: after whom came Euclid, author of the Elements of Geometry; and Aristides Quintilianus wrote after Cicero's time. Alypius stands next; after him Gaudentius the philosopher, and Nicomachus the Pythagorean, and Bacchius. Of which seven Greek authors we have a fair copy, with a translation and notes, by Meibomius. Ptolomy, the celebrated astronomer, wrote in Greek on the principles of harmonics, about the time of the emperor Antoninus Pius. This author keeps a medium between the Pythagoreans and Aristoxenians. He was succeeded at a considerable distance by Manuel Bryennius.</p><p>Of the Latins, we have Boetius, who wrote in the time of Theodoric the Goth; and one Cassiodorus, about the same time; Martianus, and St. Augustine, not far remote.</p><p>And of the moderns are Zarlin, Salinas, Vincenzo Galileo, Doni, Kircher, Mersenne, Paran, De Caux, Perrault, Des Cartes, Wallis, Holder, Malcolm, Rousseau, &c.</p><p><hi rend="smallcaps">Musical</hi> <hi rend="italics">Numbers,</hi> are the numbers 2, 3, and 5, together with their composites. They are so called, because all the intervals of music may be expressed by such numbers. This is now generally admitted by<pb n="137"/><cb/> musical theorists. Mr. Euler seems to suppose, that 7 or other primes might be introduced; but he speaks of this as a doubtful and difficult matter. Here 2 corresponds to the octave, 3 to the fifth, or rather to the 12th, and 5 to the third major, or rather the seventeenth. From these three may all other intervals be found.</p><p><hi rend="smallcaps">Musical</hi> <hi rend="italics">Proportion,</hi> or Harmonical Proportion, is when, of four terms, the first is to the 4th, as the difference of the 1st and 2d is to the difference of the 3d and 4th: as 2, 3, 4, and 8 are in Musical proportion, because . And hence, if there be only three terms, the middle term supplying the place of both the 2d and 3d, the 1st is to the 3d, as the difference of the 1st and 2d, is to the difference of the 2d and 3d : as in these 2, 3, 6; where . See <hi rend="smallcaps">Harmonical</hi> <hi rend="italics">Proportion.</hi></p></div1><div1 part="N" n="MUSSCHENBROEK" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">MUSSCHENBROEK</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, a very distinguished natural philosopher and mathematician, was born at Utrecht a little before 1700. He was first professor of these sciences in his own university, and afterwards invited to the chair at Leyden, where he died full of reputation and honours in 1761. He was a member of several academies, particularly the Acade-<cb/> my of Sciences at Paris. He published several works in Latin, all of them shewing his great penetration and accuracy. As,</p><p>1. His Elements of Physico-Mathematics, in 1726.</p><p>2. Elements of Physics, in 1736.</p><p>3. Institutions of Physics; containing an abridgment of the new discoveries made by the Moderns; in 1748.</p><p>4. Introduction to Natural Philosophy; which he began to print in 1760; and which was completed and published at Leyden, in 1762, by M. Lulofs, after the death of the author. It was translated into French by M. Sigaud de la Fond, and published at Paris in 1769, in 3 vols 4to; under the title of A Course of Experimental and Mathematical Physics.</p><p>He had also several papers, chiefly on meteorology, printed in the volumes of Memoirs of the Academy of Sciences, viz, in those of the years 1734, 1735, 1736, 1753, 1756, and 1760.</p></div1><div1 part="N" n="MUTULE" org="uniform" sample="complete" type="entry"><head>MUTULE</head><p>, a kind of square modillion in the Doric frize.</p></div1><div1 part="N" n="MYRIAD" org="uniform" sample="complete" type="entry"><head>MYRIAD</head><p>, the number of 10,000, or ten thou- sand.</p></div1></div0><div0 part="N" n="N" org="uniform" sample="complete" type="alphabetic letter"><head>N</head><cb/><div1 part="N" n="NABONASSAR" org="uniform" sample="complete" type="entry"><head>NABONASSAR</head><p>, first king of the Chaldeans; memorable for the Jewish era which bears his name, which began on Wednesday February 26th in the 3967th year of the Julian period, or 747 years before Christ; the years of this epoch being Egyptian ones, of 365 days each. This is a remarkable era in chronology, because Ptolomy assures us there were astronomical observations made by the Chaldeans from Nabonassar to his time; also Ptolomy, and the other astronomers, account their years from that epoch.</p><p>Nabonassar was the first king of the Chaldeans or Babylonians. These having revolted from the Medes, who had overthrown the Assyrian monarchy, did, under Nabonassar, found a dominion, which was much increased under Nebuchadnezzar. It is probable this Nabonassar is that Baladan in the 2d Book of Kings, xx, 12, father of Merodach, who sent ambassadors to Hezekiah. See 2 Chron. xxii.</p></div1><div1 part="N" n="NADIR" org="uniform" sample="complete" type="entry"><head>NADIR</head><p>, that point of the heavens diametrically under our feet, or opposite to the zenith, which is directly over our heads. The zenith and Nadir are the two poles of the horizon, each being 90° distant from it.</p><p><hi rend="italics">The Sun's</hi> <hi rend="smallcaps">Nadir</hi>, is the axis of the cone projected by the shadow of the earth: so called, because that axis<cb/> being prolonged, gives a point in the ecliptic diametrically opposite to the sun.</p></div1><div1 part="N" n="NAKED" org="uniform" sample="complete" type="entry"><head>NAKED</head><p>, in Architecture, as the Naked of a wall, &c, is the surface, or plane, from whence the projectures arise; or which serves as a ground to the projectures.</p></div1><div1 part="N" n="NAPIER" org="uniform" sample="complete" type="entry"><head>NAPIER</head><p>, or <hi rend="smallcaps">Neper (John</hi>), baron of Merchiston in Scotland, inventor of the logarithms, was the eldest son of Sir Archibald Napier of Merchiston, and born in the year 1550. Having given early indications of great natural parts, his father was careful to have them cultivated by a liberal education. After going through the ordinary course of education at the university of St. Andrew's, he made the tour of France, Italy, and Germany. On his return to his native country, his literature and other fine accomplishments soon rendered him conspicuous; he however retired from the world to pursue literary researches, in which he made an uncommon progress, as appears by the several useful discoveries with which he afterwards favoured mankind. He chiefly applied himself to the study of mathematics; without however neglecting that of the Scriptures; in both of which he discovered the most extensive knowledge and profound penetration. His Essay upon the book of the Apocalypse indicates the most acute<pb n="138"/><cb/> investigation; though time hath discovered that his calculations concerning particular events had proceeded upon fallacious data. But what has chiefly rendered his name famous, was his great and fortunate discovery of logarithms in trigonometry, by which the ease and expedition in calculation have so wonderfully assisted the science of astronomy and the arts of practical geometry and navigation. Napier, having a great attachment to astronomy, and spherical trigonometry, had occasion to make many numeral calculations of such triangles, with sines, tangents, &c; and these being expressed in large numbers, they hence occasioned a great deal of labour and trouble: To spare themselves part of this labour, Napier, and other authors about his time, set themselves to find out certain short modes of calculation, as is evident from many of their writings. To this necessity, and these endeavours it is, that we owe several ingenious contrivances; particularly the computation by Napier's Rods, and several other curious and short methods that are given in his <hi rend="italics">Rabdologia;</hi> and at length, after trials of many other means, the most complete one of logarithms, in the actual construction of a large table of numbers in arithmetical progression, adapted to a set of as many others in geometrical progression. The property of such numbers had been long known, viz, that the addition of the former answered to the multiplication of the latter, &c; but it wanted the necessity of such very troublesome calculations as those above mentioned, joined to an ardent disposition, to make such a use of that property. Perhaps also this disposition was urged into action by certain attempts of this kind which it seems were made elsewhere; such as the following, related by Wood in his Athenæ Oxonienses, under the article Briggs, on the authority of Oughtred and Wingate, viz, “That one Dr. Craig a Scotchman, coming out of Denmark into his own country, called upon John Neper baron of Marcheston near Edinburgh, and told him among other discourses of a new invention in Denmark (by Longomontanus as 'tis said) to save the tedious multiplication and division in astronomical calculations. Neper being solicitous to know farther of him concerning this matter, he could give no other account of it, than that it was by proportionable numbers. Which hint Neper taking, he desired him at his return to call upon him again. Craig, after some weeks had passed, did so, and Neper then shewed him a rude draught of that he called <hi rend="italics">Canon Mirabilis Logarithmorum.</hi> Which draught, with some alterations, he printing in 1614, it came forthwith into the hands of our author Briggs, and into those of William Oughtred, from whom the relation of this matter came.”</p><p>Whatever might be the inducement however, Napier published his invention in 1614, under the title of <hi rend="italics">Logarithmorum Canonis Descriptic, &c,</hi> containing the construction and canon of his logarithms, which are those of the kind that is called hyperbolic. This work coming presently to the hands of Mr. Briggs, then Professor of Geometry at Gresham College in London, he immediately gave it the greatest encouragement, teaching the nature of the logarithms in his public lectures, and at the same time recommending a change in the scale of them, by which they might be advantageously altered to the kind which he afterwards<cb/> computed himself, which are thence called Briggs's Logarithms, and are those now in common use. Mr. Briggs also presently wrote to lord Napier upon this proposed change, and made journeys to Scotland the two following years, to visit Napier, and consult him about that alteration, before he set about making it. Briggs, in a letter to archbishop Usher, March 10, 1615, writes thus: “Napier lord of Markinston hath set my head and hands at work with his new and admirable logarithms. I hope to see him this summer, if it please God; for I never saw a book which pleased me better, and made me more wonder.” Briggs accordingly made him the visit, and staid a month with him.</p><p>The following passage, from the life of Lilly the astrologer, contains a curious account of the meeting of those two illustrious men. “I will acquaint you (says Lilly) with one memorable story related unto me by John Marr, an excellent mathematician and geometrician, whom I conceive you remember. He was servant to King James and Charles the First. At first when the lord Napier, or Marchiston, made public his logarithms, Mr. Briggs, then reader of the astronomy lectures at Gresham College in London, was so surprised with admiration of them, that he could have no quietness in himself until he had seen that noble person the lord Marchiston, whose only invention they were: he acquaints John Marr herewith, who went into Scotland before Mr. Briggs, purposely to be there when these two so learned persons should meet. Mr. Briggs appoints a certain day when to meet at Edinburgh; but failing thereof, the lord Napier was doubtful he would not come. It happened one day as John Marr and the lord Napier were speaking of Mr. Briggs; ‘Ah, John (said Marchiston), Mr. Briggs will not now come.’ At the very instant one knocks at the gate; John Marr hasted down, and it proved Mr. Briggs to his great contentment. He brings Mr. Briggs up into my lord's chamber, where almost one quarter of an hour was spent, each beholding other almost with admiration before one word was spoke. At last Mr. Briggs began: ‘My lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help into astronomy, viz, the logarithms; but, my lord, being by you found out, I wonder no body else found it out before, when now known it is so easy.’ He was nobly entertained by the lord Napier; and every summer after that, during the lord's being alive, this venerable man Mr. Briggs went purposely into Scotland to visit him.”</p><p>Napier made also considerable improvements in spherical trigonometry &c, particularly by his Catholic or Universal Rule, being a general theorem by which he resolves all the cases of right-angled spherical triangles in a manner very simple, and easy to be remembered, namely, by what he calls the Five Circular Parts. His Construction of Logarithms too, beside the labour of them, manifests the greatest ingenuity. Kepler dedicated his Ephemerides to Napier, which were published in the year 1617; and it appears from many passages in his letter about this time, that he accounted Napier to be the greatest man of his age in the particular department to which he applied his abilities.<pb n="139"/><cb/></p><p>The last literary exertion of this eminent person was the publication of his <hi rend="italics">Rabdology and Promptuary,</hi> in the year 1617; soon after which he died at Marchiston, the 3d of April in the same year, is the 68th year of his age.—The list of his works is as follows:</p><p>1. A Plain Discovery of the Revelation of St. John; <*>593.</p><p>2. <hi rend="italics">Logarithmorum Canonis Descriptio;</hi> 1614.</p><p>3. <hi rend="italics">Mirifici Logarithmorum Canonis Constructio; et eorum ad Naturales ipsorum numeros habitudines; una cum appendice, de alia eaque præstantiore Logarithmorum specie condenda. Luibus accessere propositiones ad triangula sphærica faciliore calculo resolvenda. Una cum Annotationibus aliquot doctissimi D. Henrici Briggii in eas, & memoratam appendicem.</hi> Published by the author's son in 1619.</p><p>4. <hi rend="italics">Rabdologia, seu Numerationis per Virgulas, libri duo;</hi> 1617. This contains the description and use of the Bones or Rods; with several other short and ingenious modes of calculation.</p><p>5. His Letter to Anthony Bacon (the original of which is in the archbishop's library at Lambeth), intitled, Secret Inventions, Profitable and Necessary in these days for the Defence of this Island, and withstanding Strangers Enemies to God's Truth and Religion; dated June 2, 1596.</p><p><hi rend="smallcaps">Napier's</hi> <hi rend="italics">Bones,</hi> or <hi rend="italics">Rods,</hi> an instrument contrived by lord Napier, for the more easy performing of the arithmetical operations of multiplication, division, &c. These rods are five in number, made of Bone, ivory, horn, wood, or pasteboard, &c. Their faces are divided into nine little squares (fig. 7, pl. 16); each of which is parted into two triangles by diagonals. In these little squares are written the numbers of the multiplication-table; in such manner as that the units, or right-hand figures, are found in the right-hand triangle: and the tens, or the left-hand figures, in the left-hand triangle; as in the figure.</p><p><hi rend="italics">To Multiply Numbers by</hi> <hi rend="smallcaps">Napier's</hi> <hi rend="italics">Bones.</hi> Dispose the rods in such manner, as that the top figures may exhibit the multiplicand; and to these, on the left-hand, join the rod of units: in which seek the right-hand figure of the multiplier: and the numbers corresponding to it, in the squares of the other rods, write out, by adding the several numbers occurring in the same rhomb together, and their sums. After the same manner write out the numbers corresponding to the other figures of the multiplier; disposing them under one another as in the common multiplication; and lastly add the several numbers into one sum. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5978</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">937</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">41846</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17934 </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">53802  </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5601386</cell></row></table> For example, suppose the multiplicand 5978, and the multiplier 937. From the outermost triangle on the right-hand (fig. 8, pl. 16) which corresponds to the right-hand figure of the multiplier 7, write out the figure 6, placing it under the line. In the next rhomb towards the left, add 9 and 5; their sum being 14, write the right-hand figure 4, against 6; carrying the left-hand figure 1 to 4 and 3, which are found in the next rhomb: oin the sum 8 to 46, already set down. After the same manner, in the last rhomb, add 6 and 5,<cb/> and the latter figure of the sum 11, set down as before, and carry 1 to the 3 found in the left-hand triangle; the sum 4 join as before on the left-hand of 1846. Thus you will have 41846 for the product of 5978 by 7. And in the same manner are to be found the products for the other figures of the multiplier; after which the whole is to be added together as usual.</p><p><hi rend="italics">To perform Division by</hi> <hi rend="smallcaps">Napier's</hi> <hi rend="italics">Bones.</hi> Dispose the rods so, as that the uppermost figures may exhibit the divisor; to these on the left-hand, join the rod of units. Descend under the divisor, till you meet those figures of the dividend in which it is first required how oft the divisor is found, or at least the next less num ber, which is to be subtracted from the dividend; then the number corresponding to this, in the place of units, set down for a quotient. And by determining the other parts of the quotient after the same manner, the division will be completed. <table><row role="data"><cell cols="1" rows="1" role="data">5978)</cell><cell cols="1" rows="1" role="data">5601386</cell><cell cols="1" rows="1" role="data">(937</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">53802</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">  22118</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">  17934</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">    41846</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">    41846</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>For example; suppose the dividend 5601386, and the divisor 5978; since it is first enquired how often 5978 is found in 56013, descend under the divisor (fig. 8) till in the lowest series you find the number 53802, approaching nearest to 56013; the former of which is to be subtracted from the latter, and the figure 9 corresponding to it in the 10d of units set down for the quotient. To the remainder 2211 join the following figure 8 of the dividend; and the number 17934 being found as before for the next less number to it, the corresponding number 3 in the rod of units is to be set down for the next figure of the quotient. After the same manner the third and last figure of the quotient will be found to be 7; and the whole quotient 937.</p></div1><div1 part="N" n="NATIVITY" org="uniform" sample="complete" type="entry"><head>NATIVITY</head><p>, in Astrology, the scheme or figure of the heavens, and particularly of the twelve houses, at the moment when a person was born; called also the Horoscope.</p><p><hi rend="italics">To Cast the</hi> <hi rend="smallcaps">Nativity</hi>, is to calculate the position of the heavens, and erect the figure of them for the time of birth.</p><p>NATURAL <hi rend="italics">Day, Year, &c.</hi> See <hi rend="smallcaps">Day, Year</hi>, &c.</p><p><hi rend="smallcaps">Natural</hi> <hi rend="italics">Horizon,</hi> is the sensible or physical horizon.</p><p><hi rend="smallcaps">Natural</hi> <hi rend="italics">Magic,</hi> is that which only makes use of natural causes; such as the Treatise of J. Bapt. Porta, Magia Naturalia.</p><p><hi rend="smallcaps">Natural</hi> <hi rend="italics">Philosophy,</hi> otherwise called <hi rend="italics">Physics,</hi> is that science which considers the powers of nature, the properties of natural bodies, and their actions upon one another.</p><p><hi rend="italics">Laws of</hi> <hi rend="smallcaps">Nature</hi>, are certain axioms, or general rules, of motion and rest, observed by natural bod es in their actions upon one another. Of these Laws, Sir I. Newton has established three:</p><p>1st <hi rend="smallcaps">Law.</hi>—That every body perseveres in the same state, either of rest, or uniform rectilinear motion; unless it is compelled to change that state by the action of some foreign force or agent. Thus, projectiles persevere in their motions, except so far as they are<pb n="140"/><cb/> retarded by the resistance of the air, and the action of gravity: and thus a top, once set up in motion, only ceases to turn round, because it is resisted by the air, and by the friction of the plane upon which it moves. Thus also the larger bodies of the planets and comets preserve their progressive and circular motions a long time undiminished, in regions void of all sensible resistance.—As body is passive in receiving its motion, and the direction of its motion, so it retains them, or perseveres in them, without any change, till it be acted upon by something external.</p><p>2d <hi rend="smallcaps">Law.</hi>—The Motion, or Change of Motion, is always proportional to the moving force by which it is produced, and in the direction of the right line in which that force is impressed. If a certain force produce a certain motion, a double force will produce double the motion, a triple force triple the motion, and so on. And this motion, since it is always directed to the same point with the generating force, if the body were in motion before, is either to be added to it, as where the motions conspire; or subtracted from it, as when they are opposite; or combined obliquely, when oblique: being always compounded with it according to the determination of each.</p><p>3d <hi rend="smallcaps">Law.</hi>—Re-action is always contrary, and equal to action; or the actions of two bodies upon one another, are always mutually equal, and directed contrary ways; and are to be estimated always in the same right line. Thus, whatever body presses or draws another, is equally pressed or drawn by it. So, if I press a stone with my finger, the finger is equally pressed by the stone: if a horse draw a weight forward by a rope, the horse is equally opposed or drawn back towards the weight; the equal tension or stretch of the rope hindering the progress of the one, as it promotes that of the other. Again, if any body, by striking on another, do in any manner change its motion, it will itself, by means of the other, undergo also an equal change in its own motion, by reason of the equality of the pressure. When two bodies meet, each endeavours to persevere in its state, and resists any change: and because the change which is produced in either may be equally measured by the action which it excites upon the other, or by the resistance which it meets with from it, it follows that the changes produced in the motions of each are equal, but are made in contrary directions: the one acquires no new force but what the other loses in the same direction; nor does this last lose any force but what the other acquires; and hence, though by their collisions, motion passes from the one to the other, yet the sum of their motions, estimated in a given direction, is preserved the same, and is unalterable by their mutual actions upon each other. In these actions the changes are equal; not those, we mean, of the velocities, but those of the motions, or momentums; the bodies being supposed free from any other impediments. For the changes of velocities, which are likewise made contrary ways, inasmuch as the motions are equally changed, are reciprocally proportional to the bodies or masses.</p><p>This law obtains also in attractions.</p></div1><div1 part="N" n="NAVIGATION" org="uniform" sample="complete" type="entry"><head>NAVIGATION</head><p>, is the art of conducting a ship at sea from one port or place to another.</p><p>This is perhaps the most useful of all arts, and is of the highest antiquity. It may be impossible to fay who<cb/> were the inventors of it; but it is probable that many people cultivated it, independent of each other, who inhabited the coasts of the sea, and had occasion, or found it convenient, to convey themselves upon the water from place to place; beginning from rafts and logs of wood, and gradually improving in the structure and management of their vessels, according to the length of time, and extent of their voyages. Writers however ascribe the invention of this art to different persons, or nations, according to their different sources of information. Thus,</p><p>The poets refer the invention of Navigation to Neptune, some to Bacchus, others to Hercules, to Jason, or to Janus, who it is said made the first ship. Historians ascribe it to the Æginetes, the Phœnicians, Tyrians, and the ancient inhabitants of Britain. Some are of opinion that the first hint was taken from the flight of the kite; and some, as Oppian (De Piscibus, lib. 1) from the fish called Nautilus; while others ascribe it to accident; and others again deriving the hint and invention from Noah's ark.</p><p>However, history represents the Phœnicians, especially those of the capital Tyre, as the first navigators that made any extensive progress in the art, so far as has come to our knowledge; and indeed it must have been this very art that made their city what it was. For this purpose, Lebanon, and the other neighbouring mountains, furnishing them with excellent wood for ship-building, they were speedily masters of a numerous fleet, with which constantly hazarding new navigations, and settling new trades, they soon arrived at an incredible pitch of opulence and populousness; so as to be in a condition to send out colonies, the principal of which was that of Carthage; which, keeping up their Phœnician spirit of commerce, in time far surpassed Tyre itself; sending their merchant ships through Hercules's pillars, now the straits of Gibraltar, and thence along the western coasts of Africa and Europe; and even, according to some authors, to America itself. The city of Tyre being destroyed by Alexander the Great, its Navigation and commerce were transferred by the conqueror to Alexandria, a new city, well situated for these purposes, and proposed for the capital of the empire of Asia, the conquest of which Alexander then meditated. And thus arose the Navigation of the Egyptians; which was afterwards so cultivated by the Ptolomies, that Tyre and Carthage were quite forgotten.</p><p>Egypt being reduced to a Roman province after the battle of Actium, its trade and Navigation fell into the hands of Augustus; in whose time Alexandria was only inferior to Rome; and the magazines of the capital of the world were wholly supplied with merchandizes from the capital of Egypt.</p><p>At length, Alexandria itself underwent the fate of Tyre and Carthage; being surprised by the Saracens, who, in spite of the emperor Heraclius, overspread the northern coasts of Africa, &c; whence the merchants being driven, Alexandria has ever since been in a languishing state, though still it has a considerable part of the commerce of the christian merchants trading to the Levant.</p><p>The fall of Rome and its empire drew along with it not only that of learning and the polite arts, but that of<pb n="141"/><cb/> Navigation also; the barbarians, into whose hands it fell, contenting themselves with the spoils of the industry of their predecessors.</p><p>But no sooner were the brave among those nations well settled in their new provinces; some in Gaul, as the Franks; others in Spain, as the Goths; and others in Italy, as the Lombards; but they began to learn the advantages of Navigation and commerce, with the methods of managing them, from the people they subdued; and this with so much success, that in a little time some of them became able to give new lessons, and set on foot new institutions for its advantage. Thus it is to the Lombards we usually ascribe the invention and use of banks, book-keeping, exchanges, rechanges, &c.</p><p>It does not appear which of the European people, after the settlement of their new masters, first betook themselves to Navigation and commerce.—Some think it began with the French; though the Italians seem to have the juster title to it, and are usually considered as the restorers of them, as well as of the polite arts, which had been banished together from the time the empire was torn asunder. It is the people of Italy then, and particularly those of Venice and Genoa, who have the glory of this restoration; and it is to their advantageous situation for Navigation that they in a great measure owe their glory. From about the time of the 6th century, when the inhabitants of the islands in the bottom of the Adriatic began to unite together, and by their union to form the Venetian state, their fleets of merchantmen were sent to all the parts of the Mediterranean; and at last to those of Egypt, particularly Cairo, a new city, built by the Saracen princes on the eastern banks of the Nile, where they traded for their spices and other products of the Indies. Thus they flourished, increased their commerce, their Navigation, and their conquests on the terra firma, till the league of Cambray in 1508, when a number of jealous princes conspired to their ruin; which was the more easily effected by the diminution of their EastIndia commerce, of which the Portuguese had got one part, and the French another. Genoa too, which had cultivated Navigation at the same time with Venice, and that with equal success, was a long time its dangerous rival, disputed with it the empire of the sea, and shared with it, the trade of Egypt; and other parts both of the east and west.</p><p>Jealousy soon began to break out; and the two republics coming to blows, there was almost continual war for three centuries, before the superiority was ascertained; when, towards the end of the 14th century, the battle of Chioza ended the strife: the Genoese, who till then had usually the advantage, having now lost all; and the Venetians almost become desperate, at one happy blow, beyond all expectation, secured to themselves the empire of the sea, and the superiority in commerce.</p><p>About the same time that Navigation was retrieved in the southern parts of Europe, a new society of merchants was formed in the north, which not only carried commerce to the greatest perfection it was capable of, till the discovery of the East and West Indies, but also formed a new scheme of laws for the regulation of it, which still obtain under the name of, <hi rend="italics">Uses and Customs of the Sea.</hi> This society is that ce-<cb/> lebrated league of the Hanse-towns, begun about the year 1164.</p><p>The art of Navigation has been greatly improved in modern times, both in respect of the form of the vessels themselves, and the methods of working or conducting them. The use of rowers is now entirely superceded by the improvements made in the sails, rigging, &c. It is also very probable, that the Ancients were neither so well skilled as the Moderns, in finding the latitudes, nor in steering their vessels in places of difficult Navigation, as the Moderns. But the greatest advantage which these have over the Ancients, is from the mariner's compass, by which they are enabled to find their way with as much facility in the midst of an immeasurable ocean, as the Ancients could have done by creeping along the coast, and never going out of sight of land. Some people indeed contend, that this is no new invention, but that the Ancients were acquainted with it. They say, it was impossible for Solomon's ships to go to Ophir, Tarshish, and Parvaim, which last they will have to be Peru, without this useful instrument. They insist, that it was impossible for the Ancients to be acquainted with the attractive virtue of the magnet, without knowing its polarity. They even affirm, that this property of the magnet is plainly mentioned in the book of Job, where the loadstone is called topaz, or the stone that turns itself. But, not to mention that Mr. Bruce has lately made it appear highly probable that Solomon's ships made no more than coasting voyages, it is certain that the Romans, who conquered Judea, were ignorant of this instrument; and it is very probable, that so useful an invention, if once it had been commonly known to a nation, would never have been forgotten, or perfectly concealed from so prudent a people as the Romans, who were so much interested in the discovery of it.</p><p>Among those who do agree that the mariner's compass is a modern invention, it has been much disputed who was the inventor. Some give the honour of it to Flavio Gioia of Amalfi in Campania, about the beginning of the 14th century; while others say that it came from the east, and was earlier known in Europe. But, at whatever time it was invented, it is certain, that the mariner's compass was not commonly used in Navigation before the year 1420. In that year the science was considerably improved under the auspices of Henry duke of Visco, brother to the king of Portugal. In the year 1485, Roderic and Joseph, physicians to king John the 2d of Portugal, together with one Martin de Bohemia, a Portuguese native of the island of Fayal, and pupil to Regiomontanus, calculated tables of the sun's declination for the use of sailors, and recommended the astrolabe for taking observations at sea. The celebrated Columbus, it is said, availed himself of Martin's instructions, and improved the Spaniards in the knowledge of this art; for the farther progress of which, a lecture was afterwards founded at Seville by the emperor Charles the 5th.</p><p>The discovery of the variation of the compass, is claimed by Columbus, and by Sebastian Cabot. The former certainly did observe this variation without having heard of it from any other person, on the 14th of September 1492, and it is very probable that Cabot might do the same. At that time it was found that there was no variation at the Azores, for which rea-<pb n="142"/><cb/> son some geographers made that the first meridian, though it has since been discovered that the variation alters in time. The use of the cross-staff now began to be introduced among sailors. This ancient instrument is described by John Werner of Nuremberg, in his annotations on the first book of Ptolomy's Geography, printed in 1514: he recommends it for observing the distance between the moon and some star, from which to determine the longitude.</p><p>At this time the art of Navigation was very imperfect, from the use of the plane chart, which was the only one then known, and which, by its gross errors, must have greatly misled the mariner, especially in places far distant from the equator; and also from the want of books of instruction for seamen.</p><p>At length two Spanish treatises came out, the one by Pedro de Medina, in 1545; and the other by Martin Cortes, or Curtis as it is printed in English, in 1556, though the author says he composed it at Cadiz in 1545, containing a complete system of the art as far as it was then known. Medina, in his dedication to Philip prince of Spain, laments that multitudes of ships daily perished at sea, because there were neither teachers of the art, nor books by which it might be learned; and Cortes, in his dedication, boasts to the emperor, that he was the first who had reduced Navigation into a compendium, valuing himself much on what he had performed. Medina defended the plane chart; but he was opposed by Cortes, who shewed its errors, and endeavoured to account for the variation of the compass, by supposing the needle was influenced by a magnetic pole, different from that of the world, and which he called the <hi rend="italics">point attractive:</hi> which notion has been farther prosecuted by others. Medina's book was soon translated into Italian, French, and Flemish, and served for a long time as a guide to foreign navigators. However, Cortes was the favourite author of the English nation, and was translated in 1561, by Richard Eden, while Medina's work was much neglected, though translated also within a short time of the other. At that time a system of Navigation consisted of materials such as the following: An account of the Ptolomaic hypothesis, and the circles of the sphere; of the roundness of the earth, the longitudes, latitudes, climates, &c, and eclipses of the luminaries; a calendar; the method of finding the prime, epact, moon's age, and tides; a description of the compass, an account of its variation, for the discovering of which Cortes said an instrument might easily be contrived; tables of the sun's declination for 4 years, in order to find the latitude from his meridian altitude; directions to find the same by certain stars: of the course of the sun and moon; the length of the days; of time and its divisions; the method of finding the hour of the day and night; and lastly, a description of the sea-chart, on which to discover where the ship is; they made use also of a small table, that shewed, upon an alteration of one degree of the latitude, how many leagues were run on each rhumb, together with the departure from the meridian; which might be called a table of distance and departure, as we have now a table of difference of latitude and departure. Besides, some instruments were deseribed, especially by Cortes; such as, one to find the place and declination of the sun, with the age and place of the moon; certain dials, the astrolabe,<cb/> and cross-staff; with a complex machine to discover the hour and latitude at once.</p><p>About the same time proposals were made for finding the longitude by observations of the moon. In 1530, Gemma Frisius advised the keeping of the time by means of small clocks or watches, then newly invented, as he says. He also contrived a new sort of cross-staff, and an instrument called the Nautical Quadrant; which last was much praised by William Cuningham, in his Cosmographical Glass, printed in the year 1559.</p><p>In the year 1537 Pedro Nunez, or Nonius, published a book in the Portuguese language, to explain a difficulty in Navigation, proposed to him by the commander Don Martin Alphonso de Susa. In this work he exposes the errors of the plane chart, and gives the solution of several curious astronomical problems; among which is that of determining the latitude from two observations of the sun's altitude and the intermediate azimuth being given. He observed, that though the rhumbs are spiral lines, yet the direct course of a ship will always be in the arch of a great circle, by which the angle with the meridians will continually change: all that the steersman can here do for preserving the original rhumb, is to correct these deviations as soon as they appear sensible. But thus the ship will in reality describe a course without the rhumb-line intended; and therefore his calculations for assigning the latitude, where any rhumb-line crosses the several meridians, will be in some measure erroneous. He invented a method of dividing a quadrant by means of concentric circles, which, after being much improved by Dr. Halley, is used at present, and is called a Nonius.</p><p>In 1577, Mr William Bourne published a treatise, in which, by considering the irregularities in the moon's motion, he shews the errors of the sailors in finding her age by the epact, and also in determining the hour from observing on what point of the compass the sun and moon appeared. In sailing towards high latitudes, he advises to keep the reckoning by the globe, as the plane chart is most erroneous in such situations. He despairs of our ever being able to find the longitude, unless the variation of the compass should be occasioned by some such attractive point as Cortes had imagined; of which however he doubts: but as he had shewn how to find the variation at all times, he advises to keep an account of the observations, as useful for finding the place of the ship; which advice was prosecuted at large by Simon Stevin in a treatise published at Leyden in 1599; the substance of which was the same year printed at London in English by Mr. Edward Wright, intitled the <hi rend="italics">Haven-finding Art.</hi> In the same old tract also is described the way by which our sailors estimate the rate of a ship in her course, by the instrument called the Log. The author of this contrivance is not known; neither was it farther noticed till 1607, when it is mentioned in an East-India voyage published by Purchas: but from this time it became common, and mentioned by all authors on Navigation; and it still continues to be used as at first, though many attempts have been made to improve it, and contrivances proposed to supply its place; some of which have succeeded in still water, but proved useless in a stormy sea.<pb n="143"/><cb/></p><p>In 1581 Michael Coignet, a native of Antwerp, published a Treatise, in which he animadverted on Medina. In this he shewed, that as the rhumbs are spirals, making endless revolutions about the poles, numerous errors must arise from their being represented by straight lines on the sea-charts; but though he hoped to find a remedy for these errors, he was of opinion that the proposals of Nonius were scarcely practicable, and therefore in a great measure useless. In treating of the sun's declination, he took notice of the gradual decrease in the obliquity of the ecliptic; he also described the Cross-Staff with three transverse pieces, as it was then in common use among the sailors. He likewise gave some instruments of his own invention; but all of them are now laid aside, excepting perhaps his Nocturnal. He constructed a sea-table, to be used by such as sailed beyond the 60th degree of latitude; and at the end of the book is delivered a Method of Sailing on a Parallel of Latitude, by means of a ring dial and a 24 hour glass.</p><p>In the same year Mr. Robert Norman published his Discovery of the Dipping-needle, in a pamphlet called the New Attractive; to which is always subjoined Mr. William Burroughs's Discourse of the Variation of the Compass.—In 1594, Capt. John Davis published a small treatise, entitled the Seaman's Secrets, which was much esteemed in its time.</p><p>The writers of this period complained much of the errors of the plane chart, which continued still in use, though they were unable to discover a proper remedy: till Gerrard Mercator contrived his Universal Map, which he published in 1569, without clearly understanding the principles of its construction: these were first discovered by Mr. Edward Wright, who sent an account of the true method of dividing the meridian from Cambridge, where he was a Fellow, to Mr. Blundeville, with a short table for that purpose, and a specimen of a chart so divided. These were published by Blundeville in 1594, among his Exercises; to the later editions of which was added his Discourse of Universal Maps, first printed in 1589. However, in 1599 Mr. Wright printed his Correction of certain Errors in Navigation, in which work he shews the reason of this division, the manner of constructing his table, and its uses in Navigation. A second edition of this treatise, with farther improvements, was printed in 1610, and a third edition by Mr. Moxon, in 1657.—The Method of Approximation, by what is called the middle latitude, now used by our sailors, occurs in Gunter's works, first printed in 1623.—About this time Logarithms began to be introduced, which were applied to Navigation in a variety of ways by Mr. Edmund Gunter; though the first application of the Logarithmic Tables to the Cases of Sailing, was by Mr. Thomas Addison, in his Arithmetical Navigation, printed in 1625.—In 1635 Mr. Henry Gellibrand printed a Discourse Mathematical on the Variation of the Magnetical Needle, containing his discovery of the changes to which the variation is subject.—In 1631, Mr. Richard Norwood published an excellent Treatise of Trigonometry, adapted to the invention of logarithms, particularly in applying Napier's general canons; and for the farther improvement of Navigation, he undertook the laborious work of measuring a degree of the<cb/> meridian, for examining the divisions of the log-line. He has given a full and clear account of this operation in his Seaman's Practice, first published in 1637; where he also describes his own excellent method of setting down and perfecting a sea-reckoning, &c. This treatise, and that of Trigonometry, were often reprinted, as the principal books for learning scientifically the art of Navigation. What he had delivered, especially in the latter of them, concerning this subject, was contracted as a manual for sailors in a very small piece, called his Epitome, which has gone through a great number of editions.—About the year 1645, Mr. Bond published, in Norwood's Epitome, a very great improvement in Wright's method, by a property in his meridian line, by which its divisions are more scientifically assigned than the author was able to effect; which he deduced from this theorem, that these divisions are analogous to the excesses of the logarithmic tangents of half the respective latitudes increased by 45 degrees, above the logarithm of the radius: this he afterwards explained more fully in the 3d edition of Gunter's works, printed in 1653; and the demonstration of the general theorem was supplied by Mr. James Gregory of Aberdeen, in his Exercitationes Geometricæ, printed at London in 1668, and afterwards by Dr. Halley, in the Philos. Trans. numb. 219, as also by Mr. Cotes, numb. 388.—In 1700, Mr. Bond, who imagined that he had discovered the longitude, by having discovered the true theory of the magnetic variation, published a general map, on which curve lines were drawn, expressing the paths or places where the magnetic needle had the same variation. The positions of these curves will indeed continually suffer alterations; and therefore they should be corrected from time to time, as they have already been for the years 1744, and 1756, by Mr. William Mountaine, and Mr. James Dodson.—The allowances proper to be made for lee-way, are very particularly set down by Mr. John Buckler, and published in a small tract first printed in 1702, intitled a New Compendium of the whole Art of Navigation, written by Mr. William Jones.</p><p>As it is now generally agreed that the earth is a spheroid, whose axis or polar diameter is shorter than the equatorial diameter, Dr. Murdoch published a tract in 1741, in which he adapted Wright's, or Mercator's sailing to such a figure; and in the same year Mr. Maclaurin also, in the Philos. Trans. numb. 461, for determining the meridional parts of a spheroid; and he has farther prosecuted the same speculation in his Fluxions, printed in 1742.</p><p>The method of finding the longitude at sea, by the observed distances of the moon from the sun and stars, commonly called the Lunar method, was proposed at an early stage in the Art of Navigation, and has now been happily carried into effectual execution by the encouragement of the Board of Longitude, which was established in England in the year 1714, for rewarding any successful endeavours to keep the longitude at sea. In the year 1767, this Board published a Nautical Almanac, which has been continued annually ever since, by the advice, and under the direction of the astronomer royal at Greenwich: this work is purposely adapted to the use of navigators in long voyages, and, among a great many useful articles, contains tables of the<pb n="144"/><cb/> lunar distances accurately computed for every 3 hours in the year, for the purpose of comparing the diftance thus known for any time, with the distance observed in an unknown place, from whence to compute the longitude of that place. Under the auspices of this Board too, besides giving encouragement to the authors of many useful tables and other works, which would otherwise have been lost, time-keepers have been brought to a wonderful degree of perfection, by Mr. Harrison, Mr. Arnold, and many other persons, which have proved highly advantageous in keeping the time during long voyages at sea, and thence giving the longitude.</p><p>Some of the other principal writers on Navigation are Bartholomew Crescenti, of Rome, in 1607; Willebrord Snell, at Leyden, in 1624, his Typhis Batavus; Geo. Fournier, at Paris, 1633; John Baptist Riccioli, at Bologna, in 1661; Dechales, in 1674 and 1677; the Sieur Blondel St. Aubin, in 1671 and 1673; M. Dassier, in 1683; M. Sauveur, in 1692; M. John Bouguer, in 1698; F. Pezenas, in 1733 and 1741; and M. Peter Bouguer, who, in 1753, published a very elaborate treatise on this subject, intitled, Nouveau Traité de Navigation; in which he gives a variation compass of his own invention, and attempts to reform the Log, as he had before done in the Memoirs of the Academy of Sciences for 1747. He is also very particular in determining the lunations more accurately than by the common methods, and in describing the corrections of the dead reckoning. This book was abridged and improved by M. de la Caille, in 1760. To these may be added the Navigation of Don George Juan of Spain, in 1757. And, in our own nation, the several treatises of Messieurs Newhouse, Seller, Hodgson, Atkinson, Harris, Patoun, Hauxley, Wilson, Moore, Nicholson, &c; but, over all, The Elements of Navigation, in 2 vols, by Mr. John Robertson, first printed about the year 1750, and since often re-printed; which is the most complete work of the kind extant; and to which work is prefixed a Dissertation on the Rise and Progress of the modern Art of Navigation, by Dr. James Wilson, containing a very learned and elaborate history of the writings and improvements in this art.</p><p>For an account of the several instruments used in this art, with the methods for the longitude, and the various kinds and methods of Navigation, &c, see the respective articles themselves.</p><p><hi rend="smallcaps">Navigation</hi> is either Proper or Common.</p><div2 part="N" n="Navigation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Navigation</hi></head><p>, <hi rend="italics">Common,</hi> usually called Coasting, in which the places are at no great distance from one another, and the ship sails usually in sight of land, and mostly within soundings. In this, little else is required besides an acquaintance with the lands, the compass, and sounding-line; each of which, see in its place.</p></div2><div2 part="N" n="Navigation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Navigation</hi></head><p>, <hi rend="italics">Proper,</hi> is where the voyage is long, and pursued through the main ocean. And here, besides the requisites in the former case, are likewise required the use of Mercator's Chart, the azimuth and amplitude compasses, the log-line, and other instruments for celestial observations; as forestaffs, quadrants, and other sectors, &c.</p><p><hi rend="italics">Navigation</hi> turns chiefly upon four things; two of which being given or known, the rest are thence easily<cb/> found out. These four things are, the difference of latitude, difference of longitude, the reckoning or distance run, and the course or rhumb sailed on. The latitudes are easily found, and that with sufficient accuracy: the course and distance are had by the log-line, or dead reckoning, together with the compass. Nor is there any thing wanting to the perfection of Navigation, but to determine the longitude. The mathematicians and astronomers of many ages have applied themselves, with great assiduity, to supply this grand desideratum, but not altogether with the success that was desired, considering the importance of the object, and the magnificent rewards offered by several states to the discoverer. See <hi rend="smallcaps">Longitude.</hi></p><p><hi rend="italics">Sub-Marine</hi> <hi rend="smallcaps">Navigation</hi>, or the art of sailing under water, is mentioned by Mr. Boyle, as the desideratum of the art of Navigation. This, he says, was successfully attempted, by Cornelius Drebbel; several persons who were in the boat breathing freely all the time. See <hi rend="smallcaps">Diving</hi>-<hi rend="italics">bell.</hi></p><p><hi rend="italics">Inland</hi> <hi rend="smallcaps">Navigation</hi>, is that performed by small craft, upon canals &c, cut through a country.</p></div2></div1><div1 part="N" n="NAVIGATOR" org="uniform" sample="complete" type="entry"><head>NAVIGATOR</head><p>, a person capable of conducting a ship at sea to any place proposed.</p><p>NAUTICAL <hi rend="italics">Chart,</hi> the same as Sea-Chart.</p><p><hi rend="smallcaps">Nautical</hi> <hi rend="italics">Compass,</hi> the same as Sea-Compass.</p><p><hi rend="smallcaps">Nautical</hi> <hi rend="italics">Planisphere,</hi> a projection or construction of the terrestrial globe upon a plane, for the use of mariners; such as the Plane Chart, and Mercator's Chart.</p></div1><div1 part="N" n="NEAP" org="uniform" sample="complete" type="entry"><head>NEAP</head><p>, or <hi rend="smallcaps">Neep</hi>-<hi rend="italics">Tides,</hi> are those that happen at equal distances between the spring tides. The Neap tides are the lowest, as the spring tides are the highest ones, being the opposites to them. And as the highest of the spring tides happens about three days after the full or change of the moon, so the lowest of the Neap tides fall about three days after the quarters, or four days before the full and change; when the seamen say it is Deep Neap.</p><p>NEAPED. When a ship wants water, so that she cannot get out of the harbour, out of the dock, or off the ground, the seamen say, she is Neaped, or Beneaped.</p></div1><div1 part="N" n="NEBULOUS" org="uniform" sample="complete" type="entry"><head>NEBULOUS</head><p>, or Cloudy, a term applied to certain fixed stars, which shew a dim, hazy light; being less than those of the 6th magnitude, and therefore scarcely visible to the naked eye, to which at best they only appear like little dusky specks or clouds.</p><p>Through a moderate telescope, these Nebulous stars plainly appear to be congeries or clusters of several little stars. In the Nebulous star called Præsepe, in the breast of Cancer, there are reckoned 36 little stars, 3 of which Mr. Flamsteed sets down in his catalogue. In the Nebulous star of Orion, are reckoned 21. F. le Compte adds, that there are 40 in the Pleiades; 12 in the star in the middle of Orion's sword; 500 in the extent of two degrees of the same constellation; and 2500 in the whole constellation. It may farther be observed, that the galaxy, or milky-way, is a continued assemblage of Nebulæ, or vast clusters of small stars.</p></div1><div1 part="N" n="NEEDHAM" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NEEDHAM</surname> (<foreName full="yes"><hi rend="smallcaps">John Tuberville</hi></foreName>)</persName></head><p>, a respectable philosopher and catholic divine, was born at London December 10, 1713. His father possessed a consider-<pb n="145"/><cb/> able patrimony at Hilston, in the county of Monmouth, being of the younger or catholic branch of the Needham family, and who died young, leaving but a small fortune to his four children. Our author, who was the eldest son, studied in the English college of Douai, where he took orders, taught rhetoric for several years, and surpassed all the other professors of that seminary in the knowledge of experimental philosophy.</p><p>In 1740, he was engaged by his superiors in the service of the English mission, and was entrusted with the direction of the school erected at Twyford, near Winchester, for the education of the Roman Catholic youth. —In 1744 he was appointed professor of philosophy in the English college at Lisbon, where, on account of his bad health, he remained only 15 months. After his return, he passed several years at London and Paris, which were chiefly employed in microscopical observations, and in other branches of experimental philosophy. The results of these observations and experiments were published in the Philosophical Transactions of the Royal Society of London in the year 1749, and in a volume in 12mo at Paris in 1750; and an account of them was also given by M. Busson, in the first volumes of his natural history. There was an intimate connection subsisted between Mr. Needham and this illustrious French naturalist: they made their experiments and observations together; though the results and systems which they deduced from the same objects and operations were totally different.</p><p>Mr. Needham was elected a member of the Royal Society of London in the year 1747, and of the Antiquarian Society some time after.—From the year 1751 to 1767 he was chiefly employed in finishing the education of several English and Irish noblemen, by attending them as tutor in their travels through France, Italy, and other countries. He then retired from this wandering life to the English seminary at Paris, and in 1768 was chosen by the Royal Academy of Sciences in that city a corresponding member.</p><p>When the regency of the Austrian Netherlands, for the revival of philosophy and literature in that country, formed the project of an Imperial Academy, which was preceded by the erection of a small literary society to prepare the way for its execution, Mr. Needham was invited to Brussels, and was appointed successively chief director of both these foundations; an appointment which he held, together with some ecclesiastical preferments in the Low Countries, till his death, which happened December the 30th 1781.</p><p>Mr. Needham's papers inserted in the Philosophical Transactions, were the following, viz:</p><p>1. Account of Chalky Tubulous Concretions, called Malm: vol. 42.</p><p>2. Microscopical Observations on Worms in Smutty Corn: vol. 42.</p><p>3. Electrical Experiments lately made at Paris: vol. 44.</p><p>4. Account of M. Buffon's Mirror, which burns at 66 feet: ib.</p><p>5. Observations upon the Generation, Composition, and Decomposition of Animal and Vegetable Substances: vol. 45.<cb/></p><p>6. On the Discovery of Asbestos in France: vol. 51.</p><p>Other works printed at Paris, in French, are,</p><p>1. New Microscopical Discoveries: 1745.</p><p>2. The same enlarged: 1750.</p><p>3. On Microscopical, and the Generation of Organized Bodies: 2 vols, 1769.</p></div1><div1 part="N" n="NEEDLE" org="uniform" sample="complete" type="entry"><head>NEEDLE</head><p>, <hi rend="italics">Magnetical,</hi> denotes a Needle, or a slender piece of iron or steel, touched with a loadstone; which, when sustained on a pivot or centre, upon which it plays round at liberty, it settles at length in a certain direction, either duly, or nearly north-andsouth, and called the magnetic meridian.</p><p>Magnetical Needles are of two kinds; Horizontal and Inclinatory.</p><p><hi rend="italics">Horizontal</hi> <hi rend="smallcaps">Needles</hi>, are those equally balanced on each side of the pivot which sustains them; and which, playing horizontally, with their two extremes point out the north and south parts of the horizon.</p><p><hi rend="italics">Construction of a Horizontal</hi> <hi rend="smallcaps">Needle.</hi> Having procured a thin light piece of pure steel, about 6 inches long, a perforation is made in the middle, over which a brass cap is soldered on, having its inner cavity conical, so as to play freely on the style or pivot, which has a fine steel point. To give the Needle its verticity, or directive faculty, it is rubbed or stroked leisurely on each pole of a magnet, from the south pole towards the north; first beginning with the northern end, and going back at each repeated stroke towards the south; being careful not to give a stroke in a contrary direction, which would take away the power again. Also the hand should not return directly back again the same way it came, but should return in a kind of oval figure, carrying the hand about 6 or 8 inches beyond the point where the touch ended, but not beyond on the side where the touch begins.</p><p>Before touching, the north end of the Needle, in our hemisphere, is made a little lighter than the other end; because the touch always destroys an exact balance, rendering the north end heavier than the south, and thus causing the Needle to dip. And if, after touching, the Needle be out of its equilibrium, something must be filed off from the heavier side, till it be found to balance evenly.</p><p><hi rend="italics">Needles</hi> may also acquire the magnetic virtue by means of artificial magnetic bars in the following manner: Lay two equal Needles parallel and about an inch asunder, with the north end of one and the south end of the other pointing the same way, and apply two conductors in contact with their ends: then, with two magnetic hard bars, one in each hand, and held as nearly horizontal as can be, with the upper ends, of contrary names, turned outwards to the right and left, let a Needle be stroked or rubbed from the middle to both ends at the same time, for ten or twelve times, the north end of a bar going over the south end of a Needle, and the south end of a bar going over the north end of a Needle: then, without moving from the place, change hands with the bars, or in the same hands turn the other ends downwards, and stroke the other Needle in like manner; so will they both be magnetical. But to make them still stronger, repeat the operation three or four times from Needle to Needle, and<pb n="146"/><cb/> at last turn the lower side of each Needle upwards, and repeat the operations of stroking them, as on the former sides.</p><p>The Needles that were formerly applied to the compass, on board merchant ships, were formed of two pieces of steel wire, each being bent in the middle, so as to form an obtuse angle, while their ends, being applied together, made an acute one, so that the whole represented the form of a lozenge. Dr. Knight, who has so much improved the compass, found, by repeated experiments, that partly from the foregoing structure, and partly from the unequal hardening of the ends, these Needles not only varied from the true direction, but from one another, and from themselves.</p><p>Also the Needles formerly used on board the men of war, and some of the larger trading ships, were made of one piece of steel, of a spring temper, and broad towards the ends, but tapering towards the middle. Every Needle of this form is found to have fix poles instead of two, one at each end, two where it becomes tapering, and two at the hole in the middle.</p><p>To remedy these errors and inconveniences, the Needle which Dr. Knight contrived for his compass, is a slender parallelopipedon, being quite straight and square at the ends, and so has only two poles, although the curves are a little confused about the hole in the middle; though it is, upon the whole, the simplest and best.</p><p>Mr. Michell suggests, that it would be useful to increase the weight and length of magnetic Needles, which would render them both more accurate and permanent; also to cover them with a coat of linseed oil, or varnish, to preserve them from any rust.</p><p>A Needle on occasion may be prepared without touching it on a loadstone: for a fine steel sewing Needle, gently laid on the water, or delicately suspended in the air, will take the north-and-south direction.— Thus also a Needle heated in the fire, and cooled again in the direction of the meridian, or only in an erect position, acquires the same faculty.</p><p><hi rend="italics">Declination or Variation of the</hi> <hi rend="smallcaps">Needle</hi>, is the deviation of the horizontal Needle from the meridian; or the angle it makes with the meridian, when freely suspended in an horizontal plane.</p><p>A Needle is always changing the line of its direction, traversing slowly to certain limits towards the east and west sides of the meridian. It was at first thought that the magnetic Needle pointed due north; but it was observed by Cabot and Columbus that it had a deviation from the north, though they did not suspect that this deviation had itself a variation, and was continually changing. This change in the Variation was first found out, according to Bond, by Mr. John Mair, secondly by Mr. Gunter, and thirdly by Mr. Gellibrand, by comparing together the observations made at different times near the same place by Mr. Burrowes, Mr. Gunter, and himself, and he published a Discourfe upon it in 1635. Soon after this, Mr. Bond ventured to deliver the rate at which the Variation changes for several years; by which he foretold that at London in 1657 there would be no Variation of the compass, and from that time it would gradually increase the other way, or towards the west, making certain revolutions; which happened ac-<cb/> cordingly: and upon this Variation he proposed a method of finding the longitude, which has been farther improved by many others since his time, though with very little success. See <hi rend="smallcaps">Variation.</hi></p><p>The period or revolution of the Variation, Henry Philips made only 370 years, but according to Henry Bond it is 600 years, and their yearly motion 36 minutes. The first good observations of the Variation were by Burrowes, about the year 1580, when the Variation at London was 11° 15′ east; and since that time the Needle has been moving to the westward at that place; also by the observations of different persons, it has been found to point, at different times, as below: <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Years.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Observers.</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data"><hi rend="italics">Variat. E. or W.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">1580</cell><cell cols="1" rows="1" role="data">Burrowes</cell><cell cols="1" rows="1" rend="align=right" role="data">11°</cell><cell cols="1" rows="1" role="data">15′ East.</cell></row><row role="data"><cell cols="1" rows="1" role="data">1622</cell><cell cols="1" rows="1" role="data">Gunter</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">56</cell></row><row role="data"><cell cols="1" rows="1" role="data">1634</cell><cell cols="1" rows="1" role="data">Gellibrand</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"> 3</cell></row><row role="data"><cell cols="1" rows="1" role="data">1640</cell><cell cols="1" rows="1" role="data">Bond</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"> <*></cell></row><row role="data"><cell cols="1" rows="1" role="data">1657</cell><cell cols="1" rows="1" role="data">Bond</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data"> 0</cell></row><row role="data"><cell cols="1" rows="1" role="data">1665</cell><cell cols="1" rows="1" role="data">Bond</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">23 West.</cell></row><row role="data"><cell cols="1" rows="1" role="data">1666</cell><cell cols="1" rows="1" role="data">Bond</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" role="data">1672</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">1683</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">1692</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">00</cell></row><row role="data"><cell cols="1" rows="1" role="data">1723</cell><cell cols="1" rows="1" role="data">Graham</cell><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">17</cell></row><row role="data"><cell cols="1" rows="1" role="data">1747</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" role="data">1774</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" role="data">1775</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">43</cell></row><row role="data"><cell cols="1" rows="1" role="data">1776</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">47</cell></row><row role="data"><cell cols="1" rows="1" role="data">1777</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data">1778</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data">1779</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" role="data">1780</cell><cell cols="1" rows="1" role="data">Royal Society</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">41</cell></row></table></p><p>By this Table it appears that, from the first observations in 1580 till 1657, the change in the Variation was 11° 15′ in 77 years, which is at the rate nearly of 9′ a year; and from 1657 till 1780, or the space of 123 years, it changed 22° 41′, which is at the rate of 11′ a year nearly; which it may be presumed is very near the truth.</p><p>The Variation and Dip of the Needle was for many years carefully observed by the Royal Society while they met at Crane Court; and it is a pity that such observations have not been continued since that time.</p><p><hi rend="italics">Dipping,</hi> or <hi rend="italics">Inclinatory</hi> <hi rend="smallcaps">Needle</hi>, is a Needle to shew the Dip of the Magnetic Needle, or how far it points below the horizon.</p><p>The Inclination or Dip of the Needle was first ob. served by Robert Norman, a compass-maker at Ra<*>cliffe; and according to him, the dip at that place, in the year 1576, was 71° 50′; and at the Royal Society it was observed for some years lately as follows: <table><row role="data"><cell cols="1" rows="1" role="data">viz in</cell><cell cols="1" rows="1" role="data">1776</cell><cell cols="1" rows="1" role="data">72° 30′</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1778</cell><cell cols="1" rows="1" role="data">72 25</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1780</cell><cell cols="1" rows="1" role="data">72 17.</cell></row></table></p><p>Mr. Henry Bond makes the Variation and Dip of the Needle depend on the same motion of the magnetic poles in their revolution, and upon it he founded a method of discovering the longitude at sea.<pb n="147"/><cb/></p><p>NEEP <hi rend="italics">Ti<*>des.</hi> See <hi rend="smallcaps">Neap</hi> <hi rend="italics">Tides.</hi></p></div1><div1 part="N" n="NEGATIVE" org="uniform" sample="complete" type="entry"><head>NEGATIVE</head><p>, in Algebra, something marked with the sign -, or minus, as being contrary to such as are positive, or marked with the sign plus +. As Negative powers and roots, Negative quantities, &c. See <hi rend="smallcaps">Power, Root, Quantity</hi>, &c.</p><p><hi rend="smallcaps">Negative</hi> <hi rend="italics">Sign,</hi> the sign of subtraction -, or that which denotes something in defect. Stifel is the first author I find who used this mark - for subtraction, or negation, before his time, the word minus itself was used, or else its initial <hi rend="italics">m.</hi></p><p>The use of the Negative sign in algebra, is attended with several consequences that at first sight are admitted with some difficulty, and has sometimes given occasion to notions that seem to have no real foundation. This sign implies, that the real value of the quantity represented by the letter to which it is prefixed, is to be subtracted; and it serves, with the positive sign, to keep in view what elements or parts enter into the composition of quantities, and in what manner, whether as increments or decrements, that is whether by addition or subtraction, which is of the greatest use in this art.</p><p>Hence it serves to express a quantity of an opposite quality to a positive; such as a line in a contrary position, a motion with opposite direction, or a centrifugal force in opposition to gravity; and thus it often saves the trouble of distinguishing, and demonstrating separately, the various cases of proportions, and preserves their analogy in view. But as the proportions of lines depend on their magnitude only, without regard to their position; and motions and forces are said to be equal or unequal, in any given ratio, without regard to their directions; and in general the proportion of quantities relates to their magnitude only, without determining whether they are to be considered as increments or decrements; fo there is no ground to imagine any other proportion of + <hi rend="italics">a</hi> and - <hi rend="italics">b,</hi> than that of the real magnitudes of the quantities represented by <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> whether these quantities are, in any particular case, to be added or subtracted.</p><p>As to the usual arithmetical operations of addition, subtraction, &c, the case is different, as the effect of the Negative sign is here to be carefully attended to, and is to be considered always as producing, in those operations, an effect just opposite to the positive sign. Thus, it is the same thing to subtract a decrement as to add an equal increment, or to subtract - <hi rend="italics">b</hi> from <hi rend="italics">a</hi> - <hi rend="italics">b,</hi> is to add + <hi rend="italics">b</hi> to it: and because multiplying a quantity by a Negative number, implies only a repeated subtraction of it, the multiplying - <hi rend="italics">b</hi> by - <hi rend="italics">n,</hi> is subtracting - <hi rend="italics">b</hi> as often as there are units in <hi rend="italics">n,</hi> and is therefore equivalent to adding + <hi rend="italics">b</hi> so many times, or the same as adding + <hi rend="italics">nb.</hi> But if we infer from this, that 1 is to - <hi rend="italics">n</hi> as - <hi rend="italics">b</hi> to <hi rend="italics">nb,</hi> according to the rule, that unit is to one of the factors as the other factor is to the product, there is not ground to imagine that there is any mystery in this, or any other meaning than that the real quantities represented by 1, <hi rend="italics">n, b,</hi> and <hi rend="italics">nb</hi> are proportional. For that rule relates only to the magnitude of the factors and product, without determining whether any factor, or the product, is additive or subtractive. But this likewise must be determined in algebraic computations; and this is the proper use concerning the signs, without which the operation could not pro-<cb/> ceed. Because a quantity to be subtracted is never produced, in composition, by any repeated addition of a positive, or repeated subtraction of a Negative, a Negative square number is never produced by composition from a root. Hence the √- 1, or the square root of a Negative, implies an imaginary quantity, and in resolution is a mark or character of the impossible cases of a problem, unless it is compensated by another imaginary symbol or supposition, for then the whole expression may have a real signification. Thus 1 + √- 1, and 1 - √- 1, taken separately, are both imaginary, but yet their sum is the number 2: as the conditions that separately would render the solution of a problem impossible, in some cases destroy each others effect when conjoined. In the pursuit of general conclusions, and of simple forms for representing them, expressions of this kind must sometimes arise, where the imaginary symbol is compensated in a manner that is not always so obvious.</p><p>By proper substitutions, however, the expression may be transformed into another, wherein each particular term may have a real signification, as well as the whole expression.</p><p>The theorems that are sometimes briefly discovered by the use of this symbol, may be demonstrated without it by the inverse operation, or some other way; and though such symbols are of some use in the computations in the method of fluxions, &c, its evidence cannot be said to depend upon any arts of this kind. See Maclaurin's Fluxions, book 2, chap. 1.</p><p>Mr. Baron Maseres published a pretty large book in quarto, on the use of the Negative Sign in algebra.</p><p>For the rules or ways of using the Negative sign in the several rules of Algebra, see those rules severally, viz, <hi rend="smallcaps">Addition, Subtraction, Multiplication</hi>, &c. And for the method of managing the roots of Negative quantities, see <hi rend="smallcaps">Impossibles.</hi></p><p>NEPER. See <hi rend="smallcaps">Napier.</hi></p></div1><div1 part="N" n="NEWEL" org="uniform" sample="complete" type="entry"><head>NEWEL</head><p>, the upright post that stairs turn about; being that part of the staircase which sustains the steps.</p></div1><div1 part="N" n="NEWTON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NEWTON</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent English mathematician and divine, was the grandson of John Newton of Axmouth in Devonshire, and son of Hum phrey Newton of Oundle in Northamptonshire, where he was born in 1622. After receiving the proper foun dation of a grammar education, he was sent to Oxford, where he was entered a commoner of St. Edmund's Hall in 1637. He took the degree of bachelor of arts in 1641; and the year following he was created master, in precedence to many students of quality, on account of his distinguished talents in the great branches of literature. His genius leading him strongly to astronomy and mathematics, he applied himself diligently to those sciences, as well as to divinity, and made a great proficiency in them, which he found of some service to him during Cromwell's government.</p><p>After the restoration of Charles the 2d, he reaped the fruits of his loyalty: being created doctor of divinity at Oxford, Sept. 1661, he was made one of the king's chaplains, and rector of Ross in Herefordshire, instead of Mr. John Toombes, ejected for nonconformity. He held this living till his death, which happened at Ross on Christmas day 1678, at 56 years of age.<pb n="148"/><cb/></p><p>Mr. Wood gave him the character of a capricious and humoursome person. However that be, his writings are a proof of his great application to study, and a sufficient monument of his genius and skill in the mathematical sciences. These are,</p><p>1. Astronomia Britannica, &c: in 4to, 1656.</p><p>2. Help to Calculation; with Tables of Declination, &c: 4to, 1657.</p><p>3. Trigonometria Britannica, in two books; the one composed by our author, and the other translated from the Latin of Henry Gellibrand: folio, 1658.</p><p>4. Chiliades Centum Logarithmorum, printed with,</p><p>5. Geometrical Trigonometry: 1659.</p><p>6. Mathematical Elements, three parts: 4to, 1660.</p><p>7. A Perpetual Diary, or Almanac: 1662.</p><p>8. Description of the Use of the Carpenter's Rule: 1667.</p><p>9. Ephemerides, shewing the interest and rate of money at 6 per cent. &c: 1667.</p><p>10. Chiliades Centum Logarithmorum et Tabula Partium Proportionalium: 1667.</p><p>11. The Rule of Interest, or the Case of Decimal Fractions, &c, part 2: 8vo, 1668.</p><p>12. School-pastimes for young children, &c: 8vo, 1669.</p><p>13. Art of Practical Gauging, &c: 1669.</p><p>14. Introduction to the art of Rhetoric: 1671.</p><p>15. The Art of Natural Arithmetic in Whole Numbers, and Fractions Vulgar and Decimal: 8vo, 1671.</p><p>16. The English Academy: 8vo, 1677.</p><p>17. Cosmography.</p><p>18. Introduction to Astronomy.</p><p>19. Introduction to Geography: 8vo, 1678.</p></div1><div1 part="N" n="NEWTON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NEWTON</surname> (Sir <foreName full="yes"><hi rend="smallcaps">Isaac</hi></foreName>)</persName></head><p>, one of the greatest philosophers and mathematicians the world has produced, was born at Woolstrop in Lincolnshire on Christmas day 1642. He was descended from the eldest branch of the family of Sir John Newton, bart. who were lords of the manor of Woolstrop, and had been possessed of the estate for about two centuries before, to which they had removed from Westley in the same county, but originally they came from the town of Newton in Lancashire. Other accounts say, I think more truly, that he was the only child of Mr. John Newton of Colesworth, near Grantham in Lincolnshire, who had there an estate of about 120l. a year, which he kept in his own hands. His mother was of the ancient and opulent family of the Ayscoughs, or Askews, of the same county. Our author losing his father while he was very young, the care of his education devolved on his mother, who, though she married again after his father's death, did not neglect to improve by a liberal education the promising genius that was observed in her son. At 12 years of age, by the advice of his maternal uncle, he was sent to the grammar school at Grantham, where he made a good proficiency in the languages, and laid the foundation of his future studies. Even here was observed in him a strong inclination to figures and philosophical subjects. One trait of this early disposition is told of him: he had then a rude method of measuring the force of the wind blowing against him, by observing how much farther he could leap in the direction of the wind, or blowing<cb/> on his back, than he could leap the contrary way, or opposed to the wind: an early mark of his original infantine genius.</p><p>After a few years spent here, his mother took him home; intending, as she had no other child, to have the pleasure of his company; and that, after the manner of his father before him, he should occupy his own estate.</p><p>But instead of minding the markets, or the business of the farm, he was always studying and poring over his books, even by stealth, from his mother's knowledge. On one of these occasions his uncle discovered him one day in a hay-loft at Grantham, whither he had been sent to the market, working a mathematical problem; and having otherwise observed the boy's mind to be uncommonly bent upon learning, he prevailed upon his sister to part with him; and he was accordingly sent, in 1660, to Trinity College in Cambridge, where his uncle, having himself been a member of it, had still many friends. Isaac was soon taken notice of by Dr. Barrow, who was soon after appointed the first Lucasian professor of mathematics; and observing his bright genius, contracted a great friendship for him. At his outsetting here, Euclid was first put into his hands, as usual, but that author was soon dismissed; seeming to him too plain and easy, and unworthy of taking up his time. He understood him almost before he read him; and a cast of his eye upon the contents of his theorems, was sufficient to make him master of them: and as the analytical method of Des Cartes was then much in vogue, he particularly applied to it, and Kepler's Optics, &c, making several improvements on them, which he entered upon the margins of the books as he went on, as his custom was in studying any author.</p><p>Thus he was employed till the year 1664, when he opened a way into his new method of Fluxions and Infinite Series; and the same year took the degree of bachelor of arts. In the mean time, observing that the mathematicians were much engaged in the business of improving telescopes, by grinding glasses into one of the figures made by the three sections of a cone, upon the principle then generally entertained, that light was homogeneous, he set himself to grinding of optic glasses, of other figures than spherical, having as yet no distrust of the homogeneous nature of light: but not hitting presently upon any thing in this attempt to satisfy his mind, he procured a glass prism, that he might try the celebrated phenomena of colours, discovered by Grimaldi not long before. He was much pleased at first with the vivid brightness of the colours produced by this experiment; but after a while, considering them in a philosophical way, with that circumspection which was natural to him, he was surprised to see them in an oblong form, which, according to the received rule of refractions, ought to be circular. At first he thought the irregularity might possibly be no more than accidental; but this was what he could not leave without further enquiry: accordingly, he soon invented an infallible method of deciding the question, and the result was, his <hi rend="italics">New Theory of Light and Colours.</hi></p><p>However, the theory alone, unexpected and surprising as it was, did not satisfy him; he rather considered<pb n="149"/><cb/> the proper use that might be made of it for improving telescopes, which was his first design. To this end, having now discovered that light was not homogeneous, but an heterogeneous mixture of differently refrangible rays, he computed the errors arising from this different refrangibility; and, finding them to exceed some hundreds of times those occasioned by the circular figure of the glasses, he threw aside his glass works, and took reflections into consideration. He was now sensible that optical instruments might be brought to any degree of perfection desired, in case there could be found a reflecting substance which would polish as finely as glass, and reflect as much light as glass transmits, and the art of giving it a parabolical figure he also attained: but these seemed to him very great difficulties; nay, he almost thought them insuperable, <*>hen he further considered, that every irregularity in a reflecting superficies makes the rays stray five or six times more from their due course, than the like irregularities in a refracting one.</p><p>Amidst these speculations, he was forced from Cambridge, in 1665, by the plague; and it was more than two years before he made any further progress in the subject. However, he was far from passing his time idly in the country; on the contrary, it was here, at this time, that he first started the hint that gave rise to the system of the world, which is the main subject of the <hi rend="italics">Principia.</hi> In his retirement, he was sitting alone in a garden, when some apples falling from a tree, led his thoughts upon the subject of gravity; and, reflecting on the power of that principle, he began to consider, that, as this power is not found to be sensibly diminished at the remotest distance from the centre of the earth to which we can rise, neither at the tops of the loftiest buildings, nor on the summits of the highest mountains, it appeared to him reasonable to conclude, that this power must extend much farther than is usually thought. “Why not as high as the moon? said he to himself; and if so, her motion must be influenced by it; perhaps she is retained in her orbit by it: however, though the power of gravity is not sensibly weakened in the little change of distance at which we can place ourselves from the centre of the earth, yet it is very possible that, at the height of the moon, this power may differ in strength much from what it is here.” To make an estimate what might be the degree of this diminution, he considered with himself, that if the moon be retained in her orbit by the force of gravity, no doubt the primary planets are carried about the sun by the like power; and, by comparing the periods of the several planets with their distances from the sun, he found, that if any power like gravity held them in their courses, its strength must decrease in the duplicate proportion of the increase of distance. This he concluded, by supposing them to move in perfect circles, concentric to the sun, from which the orbits of the greatest part of them do not much differ. Supposing therefore the force of gravity, when extended to the moon, to decrease in the fame manner, he computed whether that force would be sufficient to keep the moon in her orbit.</p><p>In this computation, being absent from books, he took the common estimate in use among the geographers and our seamen, before Norwood had measured<cb/> the earth, namely that 60 miles make one degree of latitude; but as that is a very erroneous supposition, each degree containing about 69 1/3 of our English miles, his computation upon it did not make the power of gravity, decreasing in a duplicate proportion to the distance, answerable to the power which retained the moon in her orbit: whence he concluded, that some other cause must at least join with the action of the power of gravity on the moon. For this reason he laid aside, for that time, any further thoughts upon the matter. Mr. Whiston (in his Memoirs, pa. 33) says, he told him that he thought Des Cartes's vortices might concur with the action of gravity.</p><p>Nor did he resume this enquiry on his return to Cambridge, which was shortly after. The truth is, his thoughts were now engaged upon his newly projected reflecting telescope, of which he made a small specimen, with a metallic reflector spherically concave. It was but a rude essay, chiefly defective by the want of a good polish for the metal. This instrument is now in the possession of the Royal Society. In 1667 he was chosen Fellow of his college, and took the degree of master of arts. And in 1669 Dr. Barrow resigned to him the mathematical chair at Cambridge, the business of which appointment interrupted for a while his attention to the telescope: however, as his thoughts had been for some time chiefly employed upon optics, he made his discoveries in that science the subject of his lectures, for the first three years after he was appointed Mathematical Professor: and having now brought his <hi rend="italics">Theory of Light and Colours</hi> to a considerable degree of perfection, and having been elected a Fellow of the Royal Society in Jan. 1672, he communicated it to that body, to have their judgment upon it; and it was afterwards published in their Transactions, viz, of Feb. 19, 1672. This publication occasioned a dispute upon the truth of it, which gave him so much uneasiness, that he resolved not to publish any thing further for a while upon the subject; and in that resolution he laid up his <hi rend="italics">Optical Lectures,</hi> although he had prepared them for the press. And the <hi rend="italics">Analysis by Infinite Series,</hi> which he had intended to subjoin to them, unhappily for the world, underwent the same fate, and for the same reason.</p><p>In this temper he resumed his telescope; and observing that there was no absolute necessity for the parabolic figure of the glasses, since, if metals could be ground truly spherical, they would be able to bear as great apertures as men could give a polish to, he completed another instrument of the same kind. This answering the purpose so well, as, though only half a foot in length, to shew the planet Jupiter distinctly round, with his four satellites, and also Venus horned, he sent it to the Royal Society, at their request, together with a description of it, with further particulars; which were published in the Philosophical Transactions for March 1672. Several attempts were also made by that society to bring it to perfection; but, for want of a proper composition of metal, and a good polish<*> nothing succeeded, and the invention lay dormant, till Hadley made his Newtonian telescope in 1723. At the request of Leibnitz, in 1676, he explained his invention of Infinite Series, and took notice how far he had improved it by his Method of Fluxions, which however<pb n="150"/><cb/> he still concealed, and particularly on this occasion, by a transposition of the letters that make up the two fundamental propositions of it, into an alphabetical order; the letters concerning which are inserted in Collins's Commercium Epistolicum, printed 1712. In the winter between the years 1676 and 1677, he found out the grand proposition, that, by a centripetal force acting reciprocally as the square of the distance, a planet must revolve in an ellipsis, about the centre of force placed in its lower focus, and, by a radius drawn to that centre, describe areas proportional to the times. In 1680 he made several astronomical observations upon the comet that then appeared; which, for some considerable time, he took not to be one and the same, but two different comets; and upon this occasion several letters passed between him and Mr. Flamsteed.</p><p>He was still under this mistake, when he received a letter from Dr. Hook, explaining the nature of the line described by a falling body, supposed to be moved circularly by the diurnal motion of the earth, and perpendicularly by the power of gravity. This letter put him upon enquiring anew what was the real figure in which such a body moved; and that enquiry, convincing him of another mistake which he had before fallen into concerning that figure, put him upon resuming his former thoughts with regard to the moon; and Picart having not long before, viz, in 1679, measured a degree of the earth with sufficient accuracy, by using his measures, that planet appeared to be retained in her orbit by the sole power of gravity; and consequently that this power decreases in the duplicate ratio of the distance; as he had formerly conjectured. Upon this principle, he found the line described by a falling body to be an ellipsis, having one focus in the centre of the earth. And finding by this means, that the primary planets really moved in such orbits as Kepler had supposed, he had the satisfaction to see that this enquiry, which he had undertaken at first out of mere curiosity, could be applied to the greatest purposes. Hereupon he drew up about a dozen propositions, relating to the motion of the primary planets round the sun, which were communicated to the Royal Society in the latter end of 1683. This coming to be known to Dr. Halley, that gentleman, who had attempted the demonstration in vain, applied, in August 1684, to Newton, who assured him that he had absolutely completed the proof. This was also registered in the books of the Royal Society; at whose earnest solicitation Newton finished the work, which was printed under the care of Dr. Halley, and came out about midsummer 1687, under the title of, <hi rend="italics">Philosophiæ naturalis Principia mathematica,</hi> containing in the third book, the Cometic Astronomy, which had been lately discovered by him, and now made its sirst appearance in the world: a work which may be looked upon as the production of a celestial intelligence rather than of a man.</p><p>This work however, in which the great author has built a new system of natural philosophy upon the most sublime geometry, did not meet at first with all the applause it deserved, and was one day to receive. Two reasons concurred in producing this effect: Des Cartes had then got full possession of the world. His philosophy was indeed the creature of a fine imagination, gaily<cb/> dressed out: he had given her likewise some of nature's fine features, and painted the rest to a seeming likeness of her. On the other hand, Newton had with an unparalleled penetration, and force of genius, pursued nature up to her most secret abode, and was intent to demonstrate her residence to others, rather than anxious to describe particularly the way by which he arrived at it himself: he finished his piece in that elegant conciseness, which had justly gained the Ancients an universal esteem. In fact, the consequences flow with such rapidity from the principles, that the reader is often left to supply a long chain of reasoning to connect them: so that it required some time before the world could understand it. The best mathematicians were obliged to study it with care, before they could make themselves master of it; and those of a lower rank durst not venture upon it, till encouraged by the testimonies of the more learned. But at last, when its value came to be sufficiently known, the approbation which had been so slowly gained, became universal, and nothing was to be heard from all quarters, but one general burst of admiration. “Does Mr. Newton eat, drink, or sleep like other men?” says the marquis de l'Hospital, one of the greatest mathematicians of the age, to the English who visited him. “I represent him to myself as a celestial genius intirely disengaged from matter.”</p><p>In the midst of these profound mathematical researches, just before his Principia went to the press in 1686, the privileges of the university being attacked by James the 2d, Newton appeared among its most strenuous defenders, and was on that occasion appointed one of their delegates to the high-commission court; and they made such a defence, that James thought proper to drop the affair. Our author was also chosen one of their members for the Convention-Parliament in 1688, in which he sat till it was dissolved.</p><p>Newton's merit was well known to Mr. Montague, then chancellor of the exchequer, and afterwards earl of Halifax, who had been bred at the same college with him; and when he undertook the great work of recoining the money, he fixed his eye upon Newton for an assistant in it; and accordingly, in 1696, he was appointed warden of the mint, in which employment, he rendered very signal service to the nation. And three years after he was promoted to be master of the mint, a place worth 12 or 15 hundred pounds per annum, which he held till his death. Upon this promotion, he appointed Mr. Whiston his deputy in the mathematical professorship at Cambridge, giving him the full prosits of the place, which appointment itself he also procured for him in 1703. The same year our author was chosen president of the Royal Society, in which chair he sat for 25 years, namely till the time of his death; and he had been chosen a member of the Royal Academy of Sciences at Paris in 1699, as soon as the new regulation was made for admitting foreigners into that society.</p><p>Ever since the sirst discovery of the heterogeneous mixture of light, and the production of colours thence arising, he had employed a good part of his time in bringing the experiment, upon which the theory is founded, to a degree of exactness that might satisfy himself. The truth is, this seems to have been his favourite invention; 30 years he had spent in this ardu-<pb n="151"/><cb/> ous task, before he published it in 1704. In infinite series and fluxions, and in the power and rule of gravity in preserving the solar system, there had been some, though distant hints, given by others before him: whereas in diffecting a ray of light into its primary constituent particles, which then admitted of no further separation; in the discovery of the different refrangibility of these particles thus separated; and that these constituent rays had each its own peculiar colour inherent in it; that rays falling in the same angle of incidence have alternate sits of reflection and refraction; that bodies are rendered transparent by the minuteness of their pores, and become opaque by having them large; and that the most transparent body, by having a great thinness, will become less pervious to the light: in all these, which make up his new theory of light and colours, he was absolutely and entirely the first starter; and as the subject is of the most subtle and delicate nature, he thought it necessary to be himself the last finisher of it.</p><p>In fact, the affair that chiefly employed his researches for so many years, was far from being confined to the subject of light alone. On the contrary, all that we know of natural bodies, seemed to be comprehended in it; he had found out, that there was a natural action at a distance between light and other bodies, by which both the reflections and refractions, as well as inflections, of the former, were constantly produced. To ascertain the force and extent of this principle of action, was what had all along engaged his thoughts, and what after all, by its extreme subtlety, escaped his most penetrating spirit. However, though he has not made so full a discovery of this principle, which directs the course of light, as he has in regard to the power by which the planets are kept in their courses; yet he gave the best directions possible for such as should be disposed to carry on the work, and furnished matter abundantly sufficient to animate them to the pursuit. He has indeed hereby opened a way of passing from optics to an entire system of physics; and, if we look upon his queries as containing the history of a great man's first thoughts, even in that view they must be always at least entertaining and curious.</p><p>This same year, and in the same book with his Optics, he published, for the first time, his Method of Fluxions. It has been already observed, that these two inventions were intended for the public so long before as 1672; but were laid by then, in order to prevent his being engaged on that account in a dispute about them. And it is not a little remarkable, that even now this last piece proved the occasion of another dispute, which continued for many years. Ever since 1684, Leibnitz had been artfully working the would into an opinion, that he first invented this method.— Newton saw his design from the beginning, and had sufficiently obviated it in the first edition of the Principia, in 1687 (viz, in the Scholium to the 2d lemma of the 2d book): and with the same view, when he now published that method, he took occasion to acquaint the world, that he invent<*>d it in the years 1665 and 1666. In the Acta Eruditorum of Leipsic, where an account is given of this book, the author of that account ascribed the invention to Leibnitz, intimating that Newton borrowed it from him. Dr. Keill, the<cb/> astronomical professor at Oxford, undertook Newton's defence; and after several answers on both sides, Leibnitz complaining to the Royal Society, this body appointed a committee of their members to examine the merits of the case. These, after considering all the papers and letters relating to the point in controversy, decided in favour of Newton and Keill; as is related at large in the life of this last mentioned gentleman; and these papers themselves were published in 1712, under the title of Commercium Epistolicum Johannis Collins, 8vo.</p><p>In 1705, the honour of knighthood was conferred upon our author by queen Anne, in consideration of his great merit. And in 1714 he was applied to by the House of Commons, for his opinion upon a new method of discovering the longitude at sea by signals, which had been laid before them by Ditton and Whiston, in order to procure their encouragement; but the petition was thrown aside upon reading Newton's paper delivered to the committee.</p><p>The following year, 1715, Leibnitz, with the view of bringing the world more easily into the belief that Newton had taken the method of fluxions from his Differential method, attempted to foil his mathematical skill by the famous problem of the trajectories, which he therefore proposed to the English by way of challenge; but the solution of this, though the most difficult proposition he was able to devise, and what might pass for an arduous affair to any other, yet was hardly any more than an amusement to Newton's penetrating genius: he received the problem at 4 o'clock in the afternoon, as he was returning from the Mint; and, though extremely fatigued with business, yet he finished the solution before he went to bed.</p><p>As Leibnitz was privy-counsellor of justice to the elector of Hanover, so when that prince was raised to the British throne, Newton came more under the notice of the court; and it was for the immediate satisfaction of George the First, that he was prevailed on to put the last hand to the dispute about the invention of Fluxions. In this court, Caroline princess of Wales, afterwards queen consort to George the Second, happened to have a curiosity for philosophical enquiries; no sooner therefore was she informed of our author's attachment to the house of Hanover, than she engaged his conversation, which soon endeared him to her. Here she found in every difficulty that full fatisfaction, which she had in vain sought for elsewhere; and she was often heard to declare publicly, that she thought herself happy in coming into the world at a juncture of time, which put it in her power to converse with him. It was at this princess's solicitation, that he drew up an abstract of his Chronology; a copy of which was at her request communicated, about 1718, to signior Conti, a Venetian nobleman, then in England, upon a promise to keep it secret. But notwithstanding this promise, the abbé, who while here had also affected to shew a particular friend<*>hip for Newton, though privately betraying him as much as lay in his power to Leibnitz, was no sooner got across the water into France, than he dispersed copies of it, and procured an antiquary to translate it into French, as well as to write a confutation of it. This, being printed at Paris in 1725, was delivered as a present from the bookseller that printed it to our author, that he might obtain, as was said, his consent<pb n="152"/><cb/> to the publication; but though he expressly refused such consent, yet the whole was published the same year. Hereupon. Newton found it necessary to publish a Defence of himself, which was inserted in the Philosophical Transactions. Thus he, who had so much all his life long been studious to avoid disputes, was unavoidably all his life time, in a manner, involved in them; nor did this last-dispute even finish at his death, which happened the year following. Newton's paper was republished in 1726 at Paris, in French, with a letter of the abbé Conti in answer to it; and the same year some dissertations were printed there by father Souciet against Newton's Chronological Index, an answer to which was inserted by Halley in the Philosophical Transactions, numb. 397.</p><p>Some time before this business, in his 80th year, our author was seized with an incontinence of urine, thought to proceed from the stone in the bladder, and deemed to be incurable. However, by the help of a strict regimen and other precautions, which till then he never had occasion for, he procured considerable intervals of ease during the five remaining years of his life. Yet he was not free from some severe paroxysms, which even forced out large drops of sweat that ran down his face. In these circumstances he was never observed to utter the least complaint, nor express the least impatience; and as soon as he had a moment's ease, he would smile and talk with his usual chearfulness. He was now obliged to rely upon Mr. Conduit, who had married his niece, for the discharge of his office in the Mint. Saturday morning March 18, 1727, he read the newspapers, and discoursed a long time with Dr. Mead his physician, having then the perfect use of all his senses and his understanding; but that night he entirely lost them all, and, not recovering them afterwards, died the Monday following, March 20, in the 85th year of his age. His corpse lay in state in the Jerusalemchamber, and on the 28th was conveyed into Westminster-abbey, the pall being supported by the lord chancellor, the dukes of Montrose and Roxburgh, and the earls of Pembroke, Sussex, and Macclesfield. He was interred near the entrance into the choir on the left hand, where a stately monument is erected to his memory, with a most elegant inscription upon it.</p><p>Newton's character has been attempted by M. Fontenelle and Dr. Pemberton, the substance of which is as follows. He was of a middle stature, and somewhat inclined to be fat in the latter part of his life. His countenance was pleasing and venerable at the same time; especially when he took off his peruke, and shewed his white hair, which was pretty thick. He never made use of spectacles, and lost but one tooth during his whole life. Bishop Atterbury says, that, in the whole air of Sir Isaac's face and make, there was nothing of that penetrating sagacity which appears in his compositions; that he had something rather languid in his look and manner, which did not raise any great expectation in those who did not know him.</p><p>His temper it is said was so equal and mild, that no accident could disturb it. A remarkable instance of which is related as follows. Sir Isaac had a favourite little dog, which he called Diamond. Being one day called out of his study into the next room, Diamond was left behind. When Sir Isaac returned, having been ab-<cb/> sent but a few minutes, he had the mortification to find, that Diamond having overset a lighted candle among some papers, the nearly finished labour of many years was in flames, and almost consumed to ashes. This loss, as Sir Isaac was then very far advanced in years, was irretrievable; yet, without once striking the dog, he only rebuked him with this exclamation, “Oh Diamond! Diamond! thou little knowest the mischief thou hast done!”</p><p>He was indeed of so meek and gentle a disposition, and so great a lover of peace, that he would rather have chosen to remain in obscurity, than to have the calm of life ruffled by those storms and disputes, which genius and learning always draw upon those that are the most eminent for them.</p><p>From his love of peace, no doubt, arose that unusual kind of horror which he felt for all disputes: a steady unbroken attention, free from those frequent recoilings inseparably incident to others, was his peculiar felicity; he knew it, and he knew the value of it. No wonder then that controversy was looked on as his bane. When some objections, hastily made to his discoveries concerning light and colours, induced him to lay aside the design he had taken of publishing his Optical Lectures, we find him reflecting on that dispute, into which he had been unavoidably drawn, in these terms: “I blamed my own imprudence for parting with so real a blessing as my quiet, to run after a shadow.” It is true this shadow, as Fontenelle observes, did not escape him afterwards, nor did it cost him that quiet which he so much valued, but proved as much a real happiness to him as his quiet itself; yet this was a happiness of his own making: he took a resolution from these disputes, not to publish any more concerning that theory, till he had put it above the reach of controversy, by the exactest experiments, and the strictest demonstrations; and accordingly it has never been called in question since. In the same temper, after he had sent the manuscript to the Royal Society, with his consent to the printing of it by them; yet upon Hook's injuriously insisting that he himself had demonstrated Kepler's problem before our author, he determined, rather than be involved again in a controversy, to suppress the third book; and he was very hardly prevailed upon to alter that resolution. It is true, the public was thereby a gainer; that book, which is indeed no more than a corollary of some propositions in the first, being briginally drawn up in the popular way, with a design to publish it in that form; whereas he was now convinced that it would be best not to let it go abroad without a strict demonstration.</p><p>In contemplating his genius, it presently becomes a doubt, which of these endowments had the greatest share, sagacity, penetration, strength, or diligence; and, after all, the mark that seems most to distinguish it is, that he himself made the justest estimation of it, declaring, that if he had done the world any service, it was due to nothing but industry and patient thought; that he kept the subject of consideration constantly before him, and waited till the first dawning opened gradually, by little and little, into a full and clear light. It is said, that when he had any mathematical problems or solutions in his mind, he would never quit the subject on any account. And his servant has<pb n="153"/><cb/> said, when he has been getting up in a morning, he has sometimes begun to dress, and with one leg in his breechea, sat down again on the bed, where he has remained for hours before he has got his clothes on: and that dinner has been osten three hours ready for him before he could be brought to table. Upon this head several little anecdotes are related; among which is the following: Doctor Stukely coming in accidentally one day, when Newton's dinner was left for him upon the table, covered up, as usual, to keep it warm till he could find it convenient to come to table; the doctor lifting the cover, found under it a chicken, which he presently ate, putting the bones in the dish, and replacing the cover. Some time after Newton came into the room, and after the usual compliments sat down to his dinner; but on taking up the cover, and seeing only the bones of the fowl left, he observed with some little surprise, “I thought I had not dined, but I now find that I have.”</p><p>After all, notwithstanding his anxious care to avoid every occasion of breaking his intense application to study, he was at a great distance from being steeped in philosophy. On the contrary, he could lay aside his thoughts, though engaged in the most intricate researches, when his other affairs required his attention; and, as soon as he had leisure, resume the subject at the point where he had lest off. This he seems to have done not so much by any extraordinary strength of memory, as by the force of his inventive faculty, to which every thing opened itself again with ease, if nothing intervened to ruffle him. The readiness of his invention made him not think of putting his memory much to the trial; but this was the offspring of a vigorous intenseness of thought, out of which he was but a common man. He spent therefore the prime of his age in those abstruse researches, when his situation in a college gave him leisure, and while study was his proper business. But as soon as he was removed to the mint, he applied himself chiefly to the duties of that office; and so far quitted mathematics and philosophy, as not to engage in any pursuits of either kind afterwards.</p><p>Dr. Pemberton observes, that though his memory was much decayed in the last years of his life, yet he perfectly understood his own writings, contrary to what I had formerly heard, says the doctor, in discourse from many persons. This opinion of theirs might arise perhaps from his not being always ready at speaking on these subjects, when it might be expected he should. But on this head it may be observed, that great geniuses are often liable to be absent, not only in relation to common life, but with regard to some of the parts of science that they are best informed of: inventors seem to treasure up in their minds what they have found out, after another manner, than those do the same things, who have not this inventive faculty. The former, when they have occasion to produce their knowledge, are in some measure obliged immediately to investigate part of what they want; and for this they are not equally fit at all times: from whence it has often happened, that such as retain things chiefly by means of a very strong memory, have appeared off-hand more expert than the discoverers themselves.</p><p>It was evidently owing to the same inventive faculty that Newton, as this writer found, had read fewer of<cb/> the modern mathematicians than one could have expected; his own prodigious invention readily supplying him with what he might have occasion for in the pursuit of any subject he undertook. However, he often censured the handling of geometrical subjects by algebraic calculations; and his book of algebra he called by the name of <hi rend="italics">Universal Arithmetic,</hi> in opposition to the injudicious title of <hi rend="italics">Geometry</hi> which Des Cartes had given to the treatise in which he shews how the geometrician may assi<*> his invention by such kind of computations. He frequently praised Slusius, Barrow, and Huygens, for not being influenced by the false taste which then began to prevail. He used to commend the laudable attempt of Hugo d'Omerique to restore the ancient analysis; and very much esteemed Apollonius's book <hi rend="italics">De Se<*>ione Rationis,</hi> for giving us a clearer notion of that analysis than we had before. Dr. Barrow may be esteemed as having shewn a compass of invention equal, if not superior, to any of the Moderns, our author only excepted; but Newton particularly recommended Huygens's style and manner: he thought him the most elegant of any mathematical writer of modern times, and the truest imitator of the Ancients. Of their taste and mode of demonstration our author always professed himself a great admirer; and even censured himself for not following them yet more closely than he did; and spoke with regret of his mistake at the beginning of his mathematical studies, in applying himself to the works of Des Cartes, and other algebraic writers, before he had considered the Elements of Euclid with that attention which so excellent a writer deserves.</p><p>But if this was a fault, it is certain it was a fault to which we owe both his great inventions in speculative mathematics, and the doctrine of Fluxions and Infinite Series. And perhaps this might be one reason why his particular reverence for the Ancients is omitted by Fontenelle, who however certainly makes some amends by that just elogium which he makes of our author's modesty, which amiable quality he represents as standing foremost in the character of this great man's mind and manners. It was in reality greater than can be easily imagined, or will be readily believed: yet it always continued so without any alteration; though the whole world, says Fontenelle, conspired against it; let us add, though he was thereby robbed of his invention of Fluxions. Nicholas Mercator publishing his <hi rend="italics">Logarithmotechnia</hi> in 1668, where he gave the quadrature of the hyperbola by an infinite series, which was the first appearance in the learned world of a series of this sort drawn from the particular nature of the curve, and that in a manner very new and abstracted; Dr. Barrow, then at Cambridge, where Mr. Newton, then about 26 years of age, resided, recollected, that he had met with the same thing in the writings of that young gentleman; and there not consined to the hyperbola only, but extended, by general forms, to all sorts of curves, even such as are mechanical; to their quadratures, their rectifications, and their centres of gravity; to the solids sormed by their rotations, and to the superficies of those solids; so that, when their determinations were possible, the series stopped at a certain point, or at least their sums were given by stated rules: and if the absolute determinations were impossible, they could yet be insinitely approximated; which is the happiest and most<pb n="154"/><cb/> refined method, says Fontenelle, of supplying the defects of human knowledge that man's imagination could possibly invent. To be master of so fruitful and general a theory was a mine of gold to a geometrician; but it was a greater glory to have been the discoverer of so surprising and ingenious a system. So that Newton, finding by Mercator's book, that he was in the way to it, and that others might follow in his track, should naturally have been forward to open his treasures, and secure the property, which consisted in making the discovery; but he contented himself with his treasure which he had found, without regarding the glory. What an idea does it give us of his un<*> paralleled modesty, when we find him declaring, that he thought Mercator had entirely discovered his secret, or that others would, before he should become of a proper age for writing! His manuscript upon Infinite Series was communicated to none but Mr. John Collins and the lord Brounker, then President of the Royal Society, who had also done something in this way himself; and even that had not been complied with, but for Dr. Barrow, who would not suffer him to indulge his modesty so much as he desired.</p><p>It is further observed, concerning this part of his character, that he never talked either of himself or others, nor ever behaved in such a manner, as to give the most malicious censurers the least occasion even to suspect him of vanity. He was candid and affable, and always put himself upon a level with his company. He never thought either his merit or his reputation sufficient to excuse him from any of the common offices of social life. No singularities, either natural or affected, distinguished him from other men. Though he was firmly attached to the church of England, he was averse to the persecution of the non-conformists. He judged of men by their manners; and the true schismatics, in his opinion, were the vicious and the wicked. Not that he confined his principles to natural religion, for it is said he was thoroughly persuaded of the truth of Revelation; and amidst the great variety of books which he had constantly before him, that which he studied with the greatest application was the Bible, at least in the latter years of his life: and he understood the nature and force of moral certainty as well as he did that of a strict demonstration.</p><p>Sir Isaac did not neglect the opportunities of doing good, when the revenues of his patrimony and a profitable employment, improved by a prudent œconomy, put it in his power. We have two remarkable instances of his bounty and generosity; one to Mr. Maclaurin, extra professor of mathematics at Edinburgh, to encourage whose appointment he offered 20 pounds a year to that office; and the other to his niece Barton, upon whom he had settled an annuity of 100 pounds per annum. When decency upon any occasion required expence and shew, he was magnificent without grudging it, and with a very good grace: at all other times, that pomp which seems great to low minds only, was utterly retrenched, and the expence reserved for better uses.</p><p>Newton never married; and it has been said, that “perhaps he never had leisure to think of it; that, being immersed in profound studies during the prime of his age, and afterwards engaged in an employment of great<cb/> importance, and even quite taken up with the company which his merit drew to him, he was not sensible of any vacancy in life, nor of the want of a companion at home.” These however do not appear to be any sufficient reasons for his never marrying, if he had had an inclination so to do. It is much more likely that he had a constitutional indifference to the state, and even to the sex in general; and it has even been said of him, that he never once knew woman.—He left at his death, it seems, 32 thousand pounds; but he made no will; which, Fontenelle tells us, was because he thought a legacy was no gift.—As to his works, besides what were published in his life-time, there were found after his death, among his papers, several discourses upon the subjects of Antiquity, History, Divinity, Chemistry, and Mathematics; several of which were published at different times, as appears from the following catalogue of all his works; where they are ranked in the order of time in which those upon the same subject were published.</p><p>1. Several Papers relating to his <hi rend="italics">Telescope,</hi> and his <hi rend="italics">Theory of Light and Colours,</hi> printed in the Philosophical Transactions, numbs. 80, 81, 82, 83, 84, 85, 88, 96, 97, 110, 121, 123, 128; or vols 6, 7, 8, 9, 10, 11.</p><p>2. <hi rend="italics">Optics,</hi> or a <hi rend="italics">Treatise of the Reflections, Refractions, and Inflections, and the Colours of Light;</hi> 1704, 4to.— A Latin translation by Dr. Clarke; 1706, 4to.—And a French translation by Pet. Coste, Amst. 1729, 2 vols 12mo.—Beside several English editions in 8vo.</p><p>3. <hi rend="italics">Optical Lectures;</hi> 1728, 8vo. Also in several Letters to Mr. Oldenburg, secretary of the Royal Society, inserted in the General Dictionary, under our author's article.</p><p>4. <hi rend="italics">Lectiones Opticæ</hi> 1729, 4to.</p><p>5. <hi rend="italics">Naturalis Philosophiæ Principia Mathematica;</hi> 1687, 4to.—A second edition in 17<*>3, with a Preface, by Roger Cotes.—The 3d edition in 1726, under the direction of Dr. Pemberton.—An English translation, by Motte, 1729, 2 volumes 8vo, printed in several editions of his works, in different nations, particularly an edition, with a large Commentary, by the two learned Jesuits, Le Seur and Jacquier, in 4 volumes 4to, in 1739, 1740, and 1742.</p><p>6. <hi rend="italics">A System of the World,</hi> translated from the Latin original; 1727, 8vo.—This, as has been already observed, was at first intended to make the third book of his Principia.—An English translation by Motte, 1729, 8vo.</p><p>7. <hi rend="italics">Several Letters</hi> to Mr. Flamsteed, Dr. Halley, and Mr. Oldenburg.—See our author's article in the General Dictionary.</p><p>8. <hi rend="italics">A Paper concerning the Longitude;</hi> drawn up by order of the House of Commons; ibid.</p><p>9. <hi rend="italics">Abregé de Chronologie,</hi> &c; 1726, under the direction of the abbé Conti, together with some Observations upon it.</p><p>10. <hi rend="italics">Remarks upon the Observations made upon a Chronological Index of Sir I. Newton, &c.</hi> Philos. Trans. vol. 33. See also the same, vol. 34 and 35, by Dr. Halley.</p><p>11. The Chronology of Ancient Kingdoms amended, &c; 1728, 4to.</p><p>12. <hi rend="italics">Arithmetica Universalis, &c;</hi> under the inspec-<pb n="155"/><cb/> tion of Mr. Whiston, Cantab. 1707, 8vo. Printed I think without the author's consent, and even against his will: an offence which it seems was never forgiven. There are also English editions of the same, particularly one by Wilder, with a Commentary, in 1769, 2 vols 8vo. And a Latin edition, with a Commentary, by Castilion, 2 vols 4to, Amst. &c.</p><p>13. <hi rend="italics">Analysis per Quantitatum Series, Fluxiones, et Differentias, cum Enumeratione Linearum Tertii Ordinis;</hi> 1711, 4to; under the inspection of W. Jones, Esq. F. R. S.—The last tract had been published before, together with another on the <hi rend="italics">Quadrature of Curves,</hi> by the Method of Fluxions, under the title of <hi rend="italics">Tractatus duo de Speciebus & Magnitudine Figurarum Curvilinearum;</hi> subjoined to the sirst edition of his Optics in 1704; and other letters in the Appendix to Dr. Gregory's Catoptrics, &c, 1735, 8vo.—Under this head may be ranked <hi rend="italics">Newtoni Genesis Curvarum per Umbras;</hi> Leyden, 1740.</p><p>14. <hi rend="italics">Several Letters relating to his Dispute with Leibnitz,</hi> upon his Right to the Invention of Fluxions; printed in the <hi rend="italics">Commercium Epistolicum D. Johannis Collins & aliorum de Analysi Promota, jussu Societatis Regiæ editum;</hi> 1712, 8vo.</p><p>15. Postscript and Letter of M. Leibnitz to the Abbé Conti, with Remarks, and a Letter of his own to that Abbé; 1717, 8vo. To which was added, Raphson's History of Fluxions, as a Supplement.</p><p>16. <hi rend="italics">The Method of Fluxions, and Analysis by Infinite Series,</hi> translated into English from the original Latin; to which is added, a Perpetual. Commentary, by the translator Mr. John Colson; 1736, 4to.</p><p>17. <hi rend="italics">Several Miscellaneous Pieces, and Letters,</hi> as follow:—(1). A Letter to Mr. Boyle upon the subject of the Philosopher's Stone. Inserted in the General Dictionary, under the article <hi rend="smallcaps">Boyle.</hi>—(2). A Letter to Mr. Aston, containing directions for his travels; ibid. under our author's article.—(3). An English Translation of a Latin Dissertation upon the Sacred Cubit of the Jews. Inserted among the miscellaneous works of Mr. John Greaves, vol. 2, published by Dr. Thomas Birch, in 1737, 2 vols 8vo. This Dissertation was found subjoined to a work of Sir Isaac's, not finished, intitled <hi rend="italics">Lexicon Propheticum.</hi>—(4). Four Letters from Sir Isaac Newton to Dr. Bentley, containing some arguments in proof of a Deity; 1756, 8vo.—(5). Two Letters to Mr. Clarke, &c.</p><p>18. <hi rend="italics">Observations on the Prophecies of Daniel and the Apocalypse of St. John;</hi> 1733, 4to.</p><p>19. <hi rend="italics">Is. Newtoni Elementa Perspectivæ Universalis;</hi> 1746, 8vo.</p><p>20. <hi rend="italics">Tables for purchasing College Leases;</hi> 1742, 12mo.</p><p>21. Corollaries, by Whiston.</p><p>22. A Collection of several pieces of our author's, under the following title, <hi rend="italics">Newtoni Is. Opuscula Mathematica Philos. & Philol.</hi> collegit J. Castilioneus; Laus. 1744, 4to, 8 tomes.</p><p>23. <hi rend="italics">Two Treatises</hi> of the Quadrature of Curves, and Analysis by Equations of an Infinite Number of Terms, explained: translated by John Stewart, with a large Commentary; 1745, 4to.</p><p>24. <hi rend="italics">Description of an Instrument</hi> for observing the Moon's Distance from the Fixed Stars at Sea. Philos. Trans. vol. 42.<cb/></p><p>25. Newton also published <hi rend="italics">Barrow's Optical Lectures,</hi> in 1699, 4to: and <hi rend="italics">Bern. Varenii Geographia, &c;</hi> 1681, 8vo.</p><p>26. The whole works of Newton, published by Dr. Horsley; 1779, 4to, in 5 volumes.</p><p>The following is a list of the papers left by Newto<hi rend="sup">n</hi> at his death, as mentioned above.</p><p><hi rend="italics">A Catalogue of Sir Isaac Newton's Manuscripts and Papers, as annexed to a Bond, given by Mr. Conduit, to the Administrators of Sir Isaac; by which he obliges himself to account for any profit he shall make by publishing any of the papers.</hi></p><p>Dr. Pellet, by agreement of the executors, entered into Acts of the Prerogative Court, being appointed to peruse all the papers, and judge which were proper for the press.</p><p>No.</p><p>1. Viaticum Nautarum; by Robert Wright.</p><p>2. Miscellanea; not in Sir Isaac's hand writing.</p><p>3. Miscellanea; part in Sir Isaac's hand.</p><p>4. Trigonometria; about 5 sheets.</p><p>5. Desinitions.</p><p>6. Miscellanea; part in Sir Isaac's hand.</p><p>7. 40 sheets in 4to, relating to Church History.</p><p>8. 126 sheets written on one side, being foul draughts of the Prophetic Stile.</p><p>9. 88 sheets relating to Church History.</p><p>10. About 70 loose sheets in small 4to, of Chemical papers; some of which are not in Sir Isaac's hand.</p><p>11. About 62 ditto, in folio.</p><p>12. About 15 large sheets, doubled into 4to; Chemical.</p><p>13. About 8 sheets ditto, written on one side.</p><p>14. About 5 sheets of foul papers, relating to Chemistry.</p><p>15. 12 half-sheets of ditto.</p><p>16. 104 half-sheets, in 4to, ditto.</p><p>17. About 22 sheets in 4to, ditto.</p><p>18. 24 sheets, in 4to, upon the Prophecies.</p><p>19. 29 half-sheets; being an answer to Mr. Hook, on Sir Isaac's Theory of Colours.</p><p>20. 87 half-sheets relating to the Optics, some of which are not in Sir Isaac's hand.</p><p>From No. 1 to No. 20 examined on the 20th of May 1727, and judged not fit to be printed. <hi rend="center"><hi rend="italics">T. Pellet.</hi></hi> <hi rend="center">Witness, <hi rend="italics">Tho. Pilkington.</hi></hi></p><p>21. 328 half-sheets in folio, and 63 in sinall 4to; being loose and foul papers relating to the Revelations and Prophecies.</p><p>22. 8 half-sheets in small 4to, relating to Church Matters.</p><p>23. 24 half-sheets in small 4to; being a discourse relating to the 2d of Kings.</p><p>24. 353 half-sheets in folio, and 57 in small 4to; being foul and loose papers relating to Figures and Mathematics.</p><p>25. 201 half-sheets in folio, and 21 in small 4to; loose and foul papers relating to the Commercium Epistolicum.<pb n="156"/><cb/></p><p>26. 91 half-sheets in small 4to, in Latin, upon the Temple of Solomon.</p><p>27. 37 half-sheets in folio, upon the Host of Heaven, the Sanctuary, and other Church Matters.</p><p>28. 44 half-sheets in folio, upon Ditto.</p><p>29. 25 half-sheets in folio; being a farther account of the Host of Heaven.</p><p>30. 51 half-sheets in folio; being an Historical Account of two notable Corruptions of Scripture.</p><p>31. 88 half-sheets in small 4to; being Extracts of Church History.</p><p>32. 116 half-sheets in folio; being Paradoxical Questions concerning Athanasius, of which several leaves in the beginning are very much damaged.</p><p>33. 56 half-sheets in folio, De Motu Corporum; the greatest part not in Sir Isaac's hand.</p><p>34. 61 half-sheets in small 4to; being various sections on the Apocalypse.</p><p>35. 25 half-sheets in folio, of the Working of the Mystery of Iniquity.</p><p>36. 20 half-sheets in folio, of the Theology of the Heathens.</p><p>37. 24 half-sheets in folio; being an Account of the Contest between the Host of Heaven, and the Transgressors of the Covenant.</p><p>38. 31 half-sheets in folio; being Paradoxical Questions concerning Athanasius.</p><p>39. 107 quarter-sheets in small 4to, upon the Revelations.</p><p>40. 174 half-sheets in folio; being loose papers relating to Church History.</p><p>May 22, 1727, examined from No. 21 to No. 40 inclusive, and judged them not fit to be printed; only No. 33 and No. 38 should be reconsidered. <hi rend="center"><hi rend="italics">T. Pellet.</hi></hi> <hi rend="center">Witness, <hi rend="italics">Tho. Pilkington.</hi></hi></p><p>41. 167 half-sheets in folio; being loose and foul papers relating to the Commercium Epistolicum.</p><p>42. 21 half-sheets in folio; being the 3d letter upon Texts of Scripture, very much damaged.</p><p>43. 31 half-sheets in folio; being foul papers relating to Church Matters.</p><p>44. 495 half-sheets in folio; being loose and foul papers relating to Calculations and Mathematics.</p><p>45. 335 half-sheets in folio; being loose and foul papers relating to the Chronology.</p><p>46. 112 sheets in small 4to, relating to the Revelations and other Church Matters.</p><p>47. 126 half-sheets in folio; being loose papers relating to the Chronology, part in English and part in Latin.</p><p>48. 400 half-sheets in folio; being loose Mathematical papers.</p><p>49. 109 sheets in 4to, relating to the Prophecies, and Church Matters.</p><p>50. 127 half-sheets in folio, relating to the University; great part not in Sir Isaac's hand.</p><p>51. 18 sheets in 4to; being Chemical papers.</p><p>52. 255 quarter-sheets; being Chemical papers.<cb/></p><p>53. An Account of Corruptions of Scripture; no<*> in Sir Isaac's hand.</p><p>54. 31 quarter-sheets; being Flammell's Explication of Hieroglyphical Figures.</p><p>55. About 350 half-sheets; being Miscellaneous papers.</p><p>56. 6 half-sheets; being An Account of the Empires &c represented by St. John.</p><p>57. 9 half-sheets folio, and 71 quarter-sheets 4to; being Mathematical papers.</p><p>58. 140 half-sheets, in 9 chapters, and 2 pieces in folio, titled, Concerning the Language of the Prophets.</p><p>59. 606 half-sheets folio, relating to the Chronology; 9 more in Latin.</p><p>60. 182 half-sheets folio; being loose papers relating to the Chronology and Prophecies.</p><p>61. 144 quarter sheets, and 95 half-sheets folio; being loose Mathematical papers.</p><p>62. 137 half-sheets folio; being loose papers relating to the Dispute with Leibnitz.</p><p>63. A folio Common-place book; part in Sir Isaac's hand.</p><p>64. A bundle of English Letters to Sir Isaac, relating to Mathematics.</p><p>65. 54 half-sheets; being loose papers found in the Principia.</p><p>66. A bundle of loose Mathematical Papers; not Sir Isaac's.</p><p>67. A bundle of French and Latin Letters to Sir Isaac.</p><p>68. 136 sheets folio, relating to Optics.</p><p>69. 22 half-sheets folio, De Rationibus Motuum &c; not in Sir Isaac's hand.</p><p>70. 70 half-sheets folio; being loose Mathematical Papers.</p><p>71. 38 half-sheets folio; being loose papers relating to Optics.</p><p>72. 47 half-sheets folio; being loose papers relating to Chronology and Prophecies.</p><p>73. 40 half-sheets folio; Procestus Mysterii Magni Philosophicus, by Wm. Yworth; not in Sir Isaac's hand.</p><p>74. 5 half-sheets; being a letter from Rizzetto to Martine, in Sir Isaac's hand.</p><p>75. 41 half-sheets; being loose papers of several kinds, part in Sir Isaac's hand.</p><p>76. 40 half-sheets; being loose papers, foul and dirty, relating to Calculations.</p><p>77. 90 half-sheets folio; being loose Mathematical papers.</p><p>78. 176 half-sheets folio; being loose papers relating to Chronology.</p><p>79. 176 half-sheets folio; being loose papers relating to the Prophecies.</p><p>80. <hi rend="brace">12 half-sheets folio; An Abstract of the Chronology. 92 half-sheets, folio; The Chronology.</hi></p><p>81. 40 half-sheets folio; The History of the Prophecies, in 10 chapters, and part of the 11th unfinished.</p><p>82. 5 small bound books in 12mo, the greatest part not in Sir Isaac's hand, being rough Calcu- lations.<pb n="157"/><cb/></p><p>May 26th 1727, Examined from No. 41 to No. 82 inclufive, and judged not fit to be printed, except No. 80, which is agreed to be printed, and part of No. 61 and 81, which are to be reconfidered. <hi rend="center"><hi rend="italics">Th. Pellet.</hi></hi> <hi rend="center">Witness, <hi rend="italics">Tho. Pilkington.</hi></hi></p><p>It is astonishing what care and industry Sir Isaac had employed about the papers relating to Chronology, Church History, &c; as, on examining the papers themselves, which are in the possession of the family of the earl of Portsmouth, it appears that many of them are copies over and over again, often with little or no variation; the whole number being upwards of 4000 sheets in folio, or 8 reams of folio paper; beside the bound books &c in this catalogue, of which the number of sheets is not mentioned. Of these there have been published only the Chronology, and Observations on the Prophecies of Daniel and the Apocalypse of St. John.</p><p>NEWTONIAN <hi rend="italics">Philosophy,</hi> the doctrine of the universe, or the properties, laws, affections, actions, forces, motions, &c of bodies, both celestial and terrestrial, as delivered by Newton.</p><p>This term however is differently applied; which has given occasion to some confused notions relating to it. For, some authors, under this term, include all the corpuscular philosophy, considered as it now stands reformed and corrected by the discoveries and improvements made in several parts of it by Newton. In which sense it is, that Gravesande calls his Elements of Physics, Introductio ad Philosophiam Newtonianam. And in this sense the Newtonian is the same as the new philosophy; and stands contradistinguished from the Cartesian, the Peripatetic, and the ancient Corpuscular.</p><p>Others, by Newtonian Philosophy, mean the method or order used by Newton in philosophising; viz, the reasoning and inferences drawn directly from phenomena, exclusive of all previous hypotheses; the beginning from simple principles, and deducing the first powers and laws of nature from a few select phenomena, and then applying those laws &c to account for other things. In this sense, the Newtonian Philosophy is the same with the Experimental Philosophy, or stands opposed to the ancient Corpuscular, and to all hypothetical and fanciful systems of Philosophy.</p><p>Others again, by this term, mean that Philosophy in which physical bodies are considered mathematically, and where geometry and mechanics are applied to the solution of phenomena. In which sense, the Newtonian is the same with the Mechanical and Mathematical Philosophy.</p><p>Others, by Newtonian Philosophy, understand that part of physical knowledge which Newton has handled, improved, and demonstrated.</p><p>And lastly, others, by this Philosophy, mean the new principles which Newton has brought into Philosophy; with the new system founded upon them, and the new solutions of phenomena thence deduced; or that which characterizes and distinguishes his Philosophy from all others. And this is the sense <*>n which we shall here chiefly consider it.<cb/></p><p>As to the history of this Philosophy, consult the foregoing article. It was first published in the year 1687, the author being then professor of mathematics in the university of Cambridge; a 2d edition, with considerable additions and improvements, came out in 1713; and a 3d in 1726. An edition, with a very large Commentary, came out in 1739, by Le Seur and Jacquier; besides the complete edition of all Newton's works, with notes, by Dr. Horsley, in 1779 &c. Several authors have endeavoured to make it plainer; by setting aside many of the more sublime mathematical researches, and substituting either more obvious reasonings or experiments instead of them; particularly Whiston, in his Prælect. Phys. Mathem.; Gravesande, in Elem. & Inst.; Pemberton, in his View &c; and Maclaurin, in his Account of Newton's Philosophy.</p><p>The chief parts of the Newtonian Philosophy, as delivered by the author, except his Optical Discoveries &c, are contained in his Principia, or Mathematical Principles of Natural Philosophy. He founds his system on the following definitions.</p><p>1. Quantity of Matter, is the measure of the same, arising from its density and bulk conjointly.—Thus, air of a double density, in the same space, is double in quantity; in a double space, is quadruple in quantity; in a triple space, is sextuple in quantity, &c.</p><p>2. Quantity of Motion, is the measure of the same, arising from the velocity and quantity of matter conjunctly.—This is evident, because the motion of the whole is the motion of all its parts; and therefore in a body double in quantity, with equal velocity, the Motion is double, &c.</p><p>3. The Vis Insita, Vis Inertiæ, or innate force of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether it be of rest, or moving uniformly forward in a right line.—This definition is proved to be just by experience, from observing the difsiculty with which any body is moved out of its place, upwards, or obliquely, or even downwards when acted on by a body endeavouring to urge it quicker than the velocity given it by gravity; and any how to change its state of motion or rest. And therefore this force is the same, whether the body have gravity or not; and a cannon ball, void of gravity, if it could be, being discharged horizontally, will go the same distance in that direction, in the same time, as if it were endued with gravity.</p><p>4. An Impressed Force, is an action exerted upon a body, in order to change its state, whether of rest or motion.—This force consists in the action only; and remains no longer in the body when the action is over. For a body maintains every new state it acquires, by its vis inertiæ only.</p><p>5. A Centripetal Force, is that by which bodies are drawn, impelled, or any way tend towards a point, as to a centre.—This may be considered of three kinds, absolute, accelerative, and motive.</p><p>6. The Absolute quantity of a centripetal force, is a measure of the same, proportional to the efficacy of the cause that urges it to the centre.</p><p>7. The Accelerative quantity of a centripetal force, is the measure of the same, proportional to the velocity which it generates in a given time.<pb n="158"/><cb/></p><p>8. The Motive quantity of a centripetal force, is a measure of the same, proportional to the motion which it generates in a given time.—This is always known by the quantity of a force equai and contrary to it, that is just sufficient to hinder the descent of the body.</p><p>After these definitions, follow certain Scholia, treating of the nature and distinctions of Time, Space, Place, Motion, Absolute, Relative, Apparent, True, Real, &c. After which, the author proposes to shew how we are to collect the true motions from their causes, effects, and apparent differences; and vice versa, how, from the motions, either true or apparent, we may come to the knowledge of their causes and effects. In order to this, he lays down the following axioms or laws of motion.</p><p>1st <hi rend="smallcaps">Law.</hi> Every body perseveres in its state of rest, or of uniform motion in a right line, unless it be compelled to change that state by forces impressed upon it. —Thus, “Projectiles persevere in their motions, so far as they are not retarded by the resistance of the air, or impelled downwards by the force of gravity. A top, whose parts, by their cohesion, are perpetually drawn aside from rectilinear motions, does not cease its rotation otherwise than as it is retarded by the air. The greater bodies of the planets and comets, meeting with less resistance in more free spaces, preserve their motions, both progressive and circular, for a much longer time.”</p><p>2d <hi rend="smallcaps">Law.</hi> The Alteration of motion is always proportional to the motive force impressed; and is made in the direction of the right line in which that force is impressed. Thus, if any force generate a certain quantity of motion, a double force will generate a double quantity, whether that force be impressed all at once, or in successive moments.</p><p>3d <hi rend="smallcaps">Law.</hi> To every action there is always opposed an equal re-action: or the mutual actions of two bodies upon each other, are always equal, and directed to contrary parts. Thus, whatever draws or presses another, is as much drawn or pressed by that other. If you press a stone with your finger, the finger is also pressed by the stone: &c.</p><p>From this axiom, or law, Newton deduces the following corollaries.</p><p>1. A body by two forces conjoined will describe the diagonal of a parallelogram, in the same time that it would describe the sides by those forces apart.</p><p>2. Hence is explained the composition of any one direct force out of any two oblique ones, viz, by making the two oblique forces the sides of a parallelogram, and the diagonal the direct one.</p><p>3. The quantity of motion, which is collected by taking the sum of the motions directed towards the same parts, and the difference of those that are directed to contrary parts, suffers no change from the action of bodies among themselves; because the motion which one body loses, is communicated to another.</p><p>4. The common centre of gravity of two or more bodies does not alter its state of motion or rest by the actions of the bodies among themselves; and therefore the common centre of gravity of all bodies, acting upon each other, (excluding external actions and impe-<cb/> diments) is either at rest, or moves uniformly in a right line.</p><p>5. The motions of bodies included in a given space are the same among themselves, whether that space be at rest, or move uniformly forward in a right line without any circular motion. The truth of this is evident from the experiment of a ship; where all motions are just the same, whether the ship be at rest, or proceed uniformly forward in a straight line.</p><p>6. If bodies, any how moved among themselves, be urged in the direction of parallel lines by equal accelerative forces, they will all continue to move among themselves, after the same manner as if they had not been urged by such forces.</p><p>The mathematical part of the Newtonian Philosophy depends chiefly on the following lemmas; especially the first; containing the doctrine of prime and ultimate ratios.</p><p><hi rend="smallcaps">Lem.</hi> 1. Quantities, and the ratios of quantities, which in any finite time converge continually to equality, and before the end of that time approach nearer the one to the other than by any given difference, become ultimately equal.</p><p><hi rend="smallcaps">Lem.</hi> 2 shews, that in a space bounded by two right lines and a curve, if an infinite number of parallelograms be inscribed, all of equal breadth; then the ultimate ratio of the curve space and the sum of the parallelograms, will be a ratio of equality.</p><p><hi rend="smallcaps">Lem.</hi> 3 shews, that the same thing is true when the breadths of the parallelograms are unequal.</p><p>In the succeeding lemmas it is shewn, in like manner, that the ultimate ratios of the sine, chord, and tangent of arcs infinitely diminished, are ratios of equality, and therefore that in all our reasonings about these, we may safely use the one for the other:—that the ultimate form of evanescent triangles, made by the arc, chord, or tangent, is that of similitude, and their ultimate ratio is that of equality; and hence, in reasonings about ultimate ratios, these triangles may safely be used one for another, whether they are made with the sine, the arc, or the tangent.—He then demonstrates some properties of the ordinates of curvilinear figures; and shews that the spaces which a body describes by any finite force urging it, whether that force is determined and immutable, or continually varied, are to each other, in the very beginning of the motion, in the duplicate ratio of the forces:—and lastly, having added some demonstrations concerning the evanescence of angles of contact, he proceeds to lay down the mathematical part of his system, which depends on the following theorems.</p><p><hi rend="smallcaps">Theor.</hi> 1. The areas which revolving bodies describe by radii drawn to an immoveable centre of force, lie in the same immoveable planes, and are proportional to the times in which they are described.—To this prop. are annexed several corollaries, respecting the velocities of bodies revolving by centripetal forces, the directions and proportions of those forces, &c; such as, that the velocity of such a revolving body, is reciprocally as the perpendicular let fall from the centre of force upon the line touching the orbit in the place of the body, &c.</p><p><hi rend="smallcaps">Theor.</hi> 2. Every body that moves in any curve<pb n="159"/><cb/> line described in a plane, and, by a radius drawn to a point either immoveable or moving forward with an uniform rectilinear motion, describes about that point areas proportional to the times, is urged by a centripetal force directed to that point.—With corollaries relatinig to such motions in resisting mediums, and to the direction of the forces when the areas are not proportional to the times.</p><p><hi rend="smallcaps">Theor.</hi> 3. Every body that, by a radius drawn to the centre of another body, any how moved, deseribes areas about that centre proportional to the times, is urged by a force compounded of the centripetal forces tending to that other body, and of the whole accelerative force by which that other body is impelled.—With several corollaries.</p><p><hi rend="smallcaps">Theor.</hi> 4. The centripetal forces of bodies, which by equal motions describe different circles, tend to the centres of the same circles; and are one to the other as the squares of the arcs described in equal times, applied to the radii of the circles.—With many corollaries, relating to the velocities, times, periodic forces, &c. And, in scholium, the author farther adds, Moreover, by means of the foregoing proposition and its corollaries, we may discover the proportion of a centripetal force to any other known force, such as that of gravity. For if a body by means of its gravity revolve in a circle, concentric to the earth, this gravity is the centripetal force of that body. But from the descent of heavy bodies, the time of one entire revolution, as well as the arc described in any given time, is given by a corol. to this prop. And by such propositions, Mr. Huygens, in his excellent book De Horologio Oscillacorio, has compared the force of gravity with the centrifugal forces of revolving bodies.</p><p>On these, and such-like principles, depends the Newtonian Mathematical Philosophy. The author farther shews how to find the centre to which the forces impelling any body are directed, having the velocity of the body given: and finds that the centrifugal force is always as the versed sine of the nascent arc directly, and as the square of the time inversely; or directly as the square of the velocity, and inversely as the chord of the nascent arc. From these premises, he deduces the method of finding the centripetal force directed to any given point when the body revolves in a circle; and this whether the central point be near hand, or at immense distance; so that all the lines drawn from it may be taken for parallels. And he shews the same thing with regard to bodies revolving in spirals, ellipses, hyperbolas, or parabolas. He shews also, having the figures of the orbits given, how to find the velocities and moving powers; and indeed resolves all the most difficult problems relating to the celestial bodies with a surprising degree of mathematical skill. These problems and demonstrations are all contained in the first book of the Principia: but an account of them here would neither be generally understood, nor easily comprized in the limits of this work.</p><p>In the second book, Newton treats of the properties and motion of fluids, and their powers of resistance, with the motion of bodies through such resisting mediums, those resistances being in the ratio of any powers of the velocities; and the motions being either made in right lines or curves, or vibrating like pendulums.<cb/> And here he demonstrates such principles as entirely overthrow the doctrine of Des Cartes's vortices, which was the fashionable system in his time; concluding the book with these words: “So that the hypothesis of vortices is utterly irreconcileable with astronomical phenomena, and rather serves to perplex than explain the heavenly motions. How these motions are performed in free spaces without vortices, may be understood by the first book; and I shall now more fully treat of it in the following book Of the System of the World.”— In this second book he makes great use of the doctrine of Fluxions, then lately invented; for which purpose he lays down the principles of that doctrine in the 2d Lemma, in these words: “The moment of any Genitum is equal to the moments of each of the generating sides drawn into the indices of the powers of those sides, and into their coefficients continually:” which rule he demonstrates, and then adds the following scholium concerning the invention of that doctrine: “In a letter of mine, says he, to Mr. J. Collins, dated December 10, 1672, having described a method of tangents, which I suspected to be the same with Slusius's method, which at that time was not made public; I subjoined these words: ‘This is one particular, or rather a corollary, of a general method which extends itself, without any troublesome calculation, not only to the drawing of tangents to any curve lines, whether geometrical or mechanical, or any how respecting right lines or other curves, but also to the resolving other abstruser kinds of problems about the curvature, areas, lengths, centres of gravity of curves, &c; nor is it (as Hudden's method de Maximis' & Minimis) limited to equations which are free from surd quantities. This method I have interwoven with that other of working in equations, by reducing them to infinite series.’ So far that letter. And these last words relate to a Treatise I composed on that subject in the year 1671.” Which, at least, is therefore the date of the invention of the doctrine of Fluxions.</p><p>On entering upon the 3d book of the Principia, Newton briefly recapitulates the contents of the two former books in these words: “In the preceding books I have laid down the principles of philosophy; principles not philosophical, but mathematical; such, to wit, as we may build our reasonings upon in philosophical enquiries. These principles are, the laws and conditions of certain motions, and powers or forces, which chiefly have respect to philosophy. But lest they should have appeared of themselves dry and barren, I have illustrated them-here and there with some philosophical scholiums, giving an account of such things, as are of a more general nature, and which philosophy seems chiefly to be founded on; such as the density and the resistance of bodies, spaces void of all matter, and the motion of light and sounds. It remains, he adds, that from the same principles I now demonstrate the frame of the system of the world. Upon this subject, I had indeed composed the 3d book in a popular method, that it might be read by many. But afterwards considering that such as had not sufficiently entered into the principles could not easily discern the strength of the consequences, nor lay aside the prejudices to which they had been many years accustomed; therefore to prevent the disputes which<pb n="160"/><cb/> might be raised upon such accounts, I chose to reduce the substance of that book into the form of propositions, in the mathematical way, which should be read by those only, who had first made themselves masters of the principles established in the preceding books.”</p><p>As a necessary preliminary to this 3d part, Newton lays down the following rules for reasoning in natural philosophy:</p><p>1. We are to admit no more causes of natural things, than such as are both true and sufficient to explain their natural appearances.</p><p>2. Therefore to the same natural effects we must always assign, as far as possible, the same causes.</p><p>3. The qualities of bodies which admit neither intension nor remission of degrees, and which are found to belong to all bodies within the reach of our experiments, are to be esteemed the universal qualities of all bodies whatsoever.</p><p>4. In experimental philosophy, we are to look upon propositions collected by general induction from phenomena, as accurately or very nearly true, notwithstanding any contrary hypotheses that may be imagined, till such time as other phenomena occur, by which they may either be made more accurate, or liable to exceptions.</p><p>The phenomena first considered are, 1. That the satellites of Jupiter, by radii drawn to his centre, describe areas proportional to the times of description; and that their periodic times, the fixed stars being at rest, are in the sesquiplicate ratio of their distances from that centre. 2. The same thing is likewise observed of the phenomena of Saturn. 3. The five primary planets, Mercury, Venus, Mars, Jupiter, and Saturn, with their several orbits, encompass the sun. 4. The fixed stars being supposed at rest, the periodic times of the said five primary planets, and of the earth, about the sun, are in the sesquiplicate proportion of their mean distances from the sun. 5. The primary planets, by radii drawn to the earth, describe areas no ways proportional to the times: but the areas which they describe by radii drawn to the sun are proportional to the times of description. 6. The moon, by a radius drawn to the centre of the earth, describes an area proportional to the time of description. All which phenomena are clearly evinced by astronomical observations. The mathematical demonstrations are next applied by Newton in the following propositions.</p><p><hi rend="smallcaps">Prop.</hi> 1. The forces by which the satellites of Jupiter are continually drawn off from rectilinear motions, and retained in their proper orbits, tend to the centre of that planet; and are reciprocally as the squares of the distances of those satellites from that centre.</p><p><hi rend="smallcaps">Prop.</hi> 2. The same thing is true of the primary planets, with respect to the sun's centre.</p><p><hi rend="smallcaps">Prop.</hi> 3. The same thing is also true of the moon, in respect of the earth's centre.</p><p><hi rend="smallcaps">Prop.</hi> 4. The moon gravitates towards the earth; and by the force of gravity is continually drawn off from a rectilinear motion, and retained in her orbit.</p><p><hi rend="smallcaps">Prop.</hi> 5. The same thing is true of all the other planets, both primary and secondary, each with respect to the centre of its motion.<cb/></p><p><hi rend="smallcaps">Prop.</hi> 6. All bodies gravitate towards every planet; and the weights of bodies towards any one and the same planet, at equal distances from its centre, are proportional to the quantities of matter they contain.</p><p><hi rend="smallcaps">Prop.</hi> 7. There is a power of gravity tending to all bodies, proportional to the several quantities of matter which they contain.</p><p><hi rend="smallcaps">Prop.</hi> 8. In two spheres mutually gravitating each towards the other, if the matter in places on all sides, round about and equidistant from the centres, be similar; the weight of either sphere towards the other, will be reciprocally as the square of the distance between their centres.—Hence are compared together the weights of bodies towards different planets: hence also are discovered the quantities of matter in the several planets: and hence likewise are found the desities of the planets.</p><p><hi rend="smallcaps">Prop.</hi> 9. The force of gravity, in parts downwards from the surface of the planets towards their centres, decreases nearly in the proportion of the distances from those centres.</p><p>These, and many other propositions and corollaries, are proved or illustrated by a great variety of experiments, in all the great points of physical astronomy; such as, That the motions of the planets in the heavens may subsist an exceeding long time:—That the centre of the system of the world is immoveable:—That the common centre of gravity of the earth, the sun, and all the planets, is immoveable:—That the sun is agitated by a perpetual motion, but never recedes far from the common centre of gravity of all the planets:—That the planets move in ellipses which have their common focus in the centre of the sun; and, by radii drawn to that centre, they describe areas proportional to the times of description:—The aphelions and nodes of the orbits of the planets are fixt:—To find the aphelions, eccentricities, and principal diameters of the orbits of the planets:—That the diurnal motions of the planets are uniform, and that the libration of the moon arises from her diurnal motion:—Of the proportion between the axes of the planets and the diameters perpendicular to those axes:—Of the weights of bodies in the different regions of our earth:—That the equinoctial points go backwards, and that the earth's axis, by a nutation in every annual revolution, twice vibrates towards the ecliptic, and as often returns to its former position:—That all the motions of the moon, and all the inequalities of those motions, follow from the principles above laid down:—Of the unequal motions of the satellites of Jupiter and Saturn:—Of the flux and reflux of the sea, as arising from the actions of the sun and moon:—Of the forces with which the sun disturbs the motions of the moon; of the varicus motions of the moon, of her orbit, variation, inclinations of her orbit, and the several motions of her nodes:—Of the tides, with the forces of the sun and moon to produce them:—Of the sigure of the moon's body:—Of the precession of the equinoxes:—And of the motions and trajectory of comets. The great author then concludes with a General Scholium, containing reflections on the principal parts of the great and beautiful system of the universe, and of the infinite, eternal Creator and Governor of it.</p><p>“The hypothesis of vortices, says he, is pressed<pb n="161"/><cb/> with many difficulties. That every planet by a radius drawn to the sun may describe areas proportional to the times of description, the periodic times of the several parts of the vortices should observe the duplicate proportion of their distances from the sun. But that the pcriodic times of the planets may obtain the sesquiplicate proportion of their distances from the sun, the periodic times of the parts of the vortex ought to be in the sesquiplicate proportion of their distances. That the smaller vortices may maintain their lesser revolutions about Saturn, Jupiter, and other planets, and swim quietly and undisturbed in the greater vortex of the sun, the periodic times of the parts of the sun's vortex should be equal. But the rotation of the sun and planets about their axes, which ought to correspond with the motions of their vortices, recede far from all these proportions. The motions of the comets are exceeding regular, are governed by the same laws with the motions of the planets, and can by no means be accounted for by the hypothesis of vortices. For comets are carried with very eccentric motions through all parts of the heavens indifferently, with a freedom that is incompatible with the notion of a vortex.</p><p>“Bodies, projected in our air, suffer no resistance but from the air. Withdraw the air, as is done in Mr. Boyle's vacuum, and the resistance ceases. For in this void a bit of fine down and a piece of solid gold descend with equal velocity. And the parity of reason must take place in the celestial spaces above the earth's atmosphere; in which spaces, where there is no air to resist their motions, all bodies will move with the greatest freedom; and the planets and comets will constantly pursue their revolutions in orbits given in kind and position, according to the laws above explained But though these bodies may indeed persevere in their orbits by the mere laws of gravity, yet they could by no means have at first derived the regular position of the orbits themselves from those laws.</p><p>“The six primary planets are revolved about the sun, in circles concentric with the sun, and with motions directed towards the same parts, and almost in the same plane. Ten moons are revolved about the earth, Jupiter and Saturn, in circles concentric with them, with the same direction of motion, and nearly in the planes of the orbits of those planets. But it is not to be conceived that mere mechanical causes could give birth to so many regular motions: since the comets range over all parts of the heavens, in very eccentric orbits, For by that kind of motion they pass easily through the orbs of the planets, and with great rapidity; and in their aphelions, where they move the slowest, and are detained the longest, they recede to the greatest distances from each other, and thence suffer the least disturbance from their mutual attractions. This most beautiful system of the sun, planets, and comets, could only proceed from the counsel and dominion of an intelligent and powerful Being. And if the fixed stars are the centres of other like systems, these being formed by the like wise counsel, must be all subject to the dominion of one; especially, since the light of the fixed stars is of the same nature with the light of the sun, and from every system light passes into all the other systems. And left the system of the fixed stars should, by their<cb/> gravity, fall on each other mutually, he hath placed those systcms at immense distances one from another.”</p><p>Then, after a truly pious and philosophical descant on the attributes of the Being who could give existence and continuance to such prodigious mechanism, and with so much beautiful order and regularity, the great author proceeds,</p><p>“Hitherto we have explained the phenomena of the heavens and of our sea, by the power of gravity, but have not yet assigned the cause of this power. This is certain, that it must proceed from a cause that penetrates to the very centres of the sun and planets, without suffering the least diminution of its force; that operates, not according to the quantity of the surfaces of the particles upon which it acts, (as mechanical causes use to do,) but according to the quantity of the solid matter which they contain, and propagates its virtue on all sides, to immense distances, decreasing always in the duplicate proportion of the distances. Gravitation towards the sun, is made up out of the gravitations towards the several particles of which the body of the sun is composed; and in receding from the sun, decreases accurately in the duplicate proportion of the distances, as far as the orb of Saturn, as evidently appears from the quiescence of the aphelions of the planets; nay, and even to the remotest aphelions of the comets, if those aphelions are also quiescent. But hitherto I have not been able to discover the cause of those properties of gravity from phenomena, and I frame no hypotheses. For whatever is not deduced from the phenomena, is to be called an hypothesis; and hypotheses, whether metaphysical or physical, whether of occult qualities or mechanical, have no place in experimental philosophy. In this philosophy particular propositions are inferred from the phenomena, and afterwards rendered general by induction. Thus it was that the impenetrability, the mobility, and the impulsive force of bodies, and the laws of motion and of gravitation, were discovered. And to us it is enough, that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of the celestial bodies, and of our sea.</p><p>“And now we might add something concerning a certain most subtle spirit, which pervades and lies hid in all gross bodies, by the force and action of which spirit, the particles of bodies mutually attract one another at near distances, and cohere, if contiguous, and electric bodies operate to greater distances, as well repelling as attracting the neighbouring corpuscles; and light is emitted, reflected, refracted, inflected, and heats bodies; and all sensation is excited, and the members of animal bodies move at the command of the will, namely, by the vibrations of this spirit, mutually propagated along the solid filaments of the nerves, from the outward organs of sense to the brain, and from the brain into the muscles. But these are things that cannot be explained in few words, nor are we furnished with that sufficiency of experiments which is required to an accurate determination and demonstration of the laws by which this electric and elastic spirit ope- rates.”<pb n="162"/><cb/></p></div1><div1 part="N" n="NICHE" org="uniform" sample="complete" type="entry"><head>NICHE</head><p>, a cavity, or hollow part, in the thickness of a wall, to place a figure or statue in.</p></div1><div1 part="N" n="NICOLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NICOLE</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, a very celebrated French mathematician, was born at Paris December the 23d, 1683. His early attachment to the mathematics induced M. Montmort to take the charge of his education: and he opened out to him the way to the higher geometry. He first became publicly remarkable by detecting the fallacy of a pretended quadrature of the circle. This quadrature a M. Mathulon so assuredly thought he had discovered, that he deposited, in the hands of a public notary at Lyons, the sum of 3000 livres, to be paid to any person who, in the judgment of the Academy of Sciences, should demonstrate the falsity of his solution. M. Nicole, piqued at this challenge, undertook the task, and exposing the paralogism, the Academy's judgment was, that Nicole had plainly proved that the rectilineal figure which Mathulon had given as equal to the circle, was not only unequal to it, but that it was even greater than the polygon of 32 sides circumscribed about the circle.—The prize of 3000 livres, Nicole presented to the public hospital of Lyons.</p><p>The Academy named Nicole, Eleve-Mechanician, March 12, 1707; Adjunct in 1716, Associate in 1718, and Pensioner in 1724; which he continued till his death, which happened the 18th of January 1758, at 75 years of age.</p><p>His works were all inserted in the different volumes of the Memoirs of the Academy of Sciences; and are as follow:</p><p>1. A General Method for determining the Nature of Curves formed by the Rolling of other Curves upon any Given Curve; in the volume for the year 1707.</p><p>2. A General Method for Rectifying all Roulets upon Right and Circular Bases; 1708.</p><p>3. General Method of determining the Nature of those Curves which cut an Insinity of other Curves given in Position, cutting them always in a Constant Angle; 1715.</p><p>4. Solution of a Problem proposed by M. de Lagny; 1716.</p><p>5. Treatise of the Calculus of Finite Differences; 1717.</p><p>6. Second Part of the Calculus of Finite Differences; 1723.</p><p>7. Second Section of ditto; 1723.</p><p>8. Addition to the two foregoing papers; 1724.</p><p>9. New Proposition in Elementary Geometry; 1725.</p><p>10. New Solution of a Problem proposed to the English Mathematicians, by the late M. Leibnitz; 1725.</p><p>11. Method of Summing an Infinity of New Series, which are not summable by any other known method; 1727.</p><p>12. Treatise of the Lines of the Third Order, or the Curves of the Second Kind; 1729.</p><p>13. Examination and Resolution of some Questions relating to Play; 1730.</p><p>14. Method of determining the Chances at Play.</p><p>15. Observations upon the Conic Sections; 1731.</p><p>16. Manner of generating in a Solid Body, all the Lines of the Third Order; 1731.<cb/></p><p>17. Manner of determining the Nature of Roulets formed upon the Convex Surface of a Sphere; and of determining which are Geometric, and which are Rectifiable; 1732.</p><p>18. Solution of a Problem in Geometry; 1732.</p><p>19. The Use of Series in resolving many Problems in the Inverse Method of Tangents; 1737.</p><p>20. Observations on the Irreducible Case in Cubic Equations; 1738.</p><p>21. Observations upon Cubic Equations; 1738.</p><p>22. On the Trisection of an Angle; 1740.</p><p>23. On the Irreducible Case in Cubic Equations; 1741.</p><p>24. Addition to ditto; 1743.</p><p>25. His Last Paper upon the same; 1744.</p><p>26. Determination, by Incommensurables and Decimals, the Values of the Sides and Areas of the Series in a Double Progression of Regular Polygons, inscribed in and circumscribed about a Circle; 1747.</p></div1><div1 part="N" n="NIEUWENTYT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NIEUWENTYT</surname> (<foreName full="yes"><hi rend="smallcaps">Bernard</hi></foreName>)</persName></head><p>, an eminent Dutch philosopher and mathematician, was born on the 10th of August 1654, at Westgraafdyk in North Holland, where his father was minister. He discovered very early a good genius and a strong inclination for learning; which was carefully improved by a suitable education. He hadalso that prudence and sagacity, which led him to pursue literature by sure and proper steps, acquiring a kind of mastery in one science before he proceeded to another. His father had designed him for the ministry; but seeing his inclination did not lie that way, he prudently left him to pursue the bent of his genius. Accordingly young Nieuwentyt apprehending that nothing was more useful than fixing his imagination and forming his judgment well, applied himself early to logic, and the art of reasoning justly, in which he grounded himself upon the principles of Des Cartes, with whose philosophy he was greatly delighted. From thence he proceeded to the mathematics, in which he made a considerable proficiency; though the application he gave to that branch of learning did not hinder him from studying both law and physic. In fact he succeeded in all these sciences so well, as deservedly to acquire the character of a good philosopher, a great mathematician, an expert physician, and an able and just magistrate.</p><p>Although he was naturally of a grave and serious disposition, yet he was very affable and agreeable in conversation. His engaging manner procured the affection of every one; and by this means he often drew over to his opinion those who before differed very widely from him. Thus accomplished, he acquired a great esteem and credit in the council of the town of Puremerende, where he resided; as he did also in the states of that province, who respected him the more, inasmuch as he never engaged in any cabals or factions, in order to secure it; regarding in his conduct, an open, honest, upright behaviour, as the best source of satisfaction, and relying solely on his merit. In fact, he was more attentive to cultivate the sciences, than eager to obtain the honours of the government; contenting himself with being counsellor and burgomaster, without courting or accepting any other posts, which might interfere with his studies, and draw him too much out of his library.—Nieuwentyt died the 7th of May 1730, at 76<pb n="163"/><cb/> years of age—having been twice married.—He was author of several works, in the Latin, French, and Dutch languages, the principal of which are the following:</p><p>1. A Treatise in Dutch, <hi rend="italics">proving the Existence of God by the Wonders of Nature;</hi> a much esteemed work, and went through many editions. It was translated also into several languages, as the French, and the English, under the title of, <hi rend="italics">The Religious Philosopher, &c.</hi></p><p>2. A Refutation of Spinoza, in the Dutch language.</p><p>3. <hi rend="italics">Analysis Infinitorum;</hi> 1695, 4to.</p><p>4. <hi rend="italics">Considerationes secundæ circa Calculi Differentialis Principia;</hi> 1696, 8vo.—In this work he attacked Leibnitz, and was answered by John Bernoulli and James Herman.</p><p>5. A Treatise on the New Use of the Tables of Sines and Tangents.</p><p>6. A Letter to Bothnia or Burmania, upon the Subject of Meteors.</p></div1><div1 part="N" n="NIGHT" org="uniform" sample="complete" type="entry"><head>NIGHT</head><p>, that part of the natural day, during which the sun is below the horizon: though sometimes it is understood that the twilight is referred to the day, or time the sun is above the horizon; the remainder only being the Night.</p><p>Under the equator, the Nights, in the former sense, are always equal to the days; each being 12 hours long. But under the poles, the Night continues half a year. —The ancient Gauls and Germans divided their time not by days, but Nights; as appears from Cæsar and Tacitus; also the Arabs and the Icelanders do the same. The same may also be observed of our Saxon ancestors: whence our custom of saying, Sevennight, Fortnight, &c.</p></div1><div1 part="N" n="NOCTILUCA" org="uniform" sample="complete" type="entry"><head>NOCTILUCA</head><p>, a species of phosphorus, so called because it shines in the night, without any light being thrown on it: such is the phosphorus made of urine. By which it stands distinguished from some other species of phosphorus, which require to be exposed to the sunbeams before they will shine; as the Bononian-stone, &c. —Mr. Boyle has a particular Treatise on this subject.</p><p>NOCTURNAL <hi rend="italics">Arch,</hi> is the arch of a circle described by the sun, or a star, in the night.</p><div2 part="N" n="Nocturnal" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Nocturnal</hi></head><p>, or <hi rend="smallcaps">Nocturlabium</hi>, denotes an instrument, chiefly used at sea, to take the altitude or depression of the pole star, and some other stars about the pole, for finding the latitude, and the hour of the night.</p><p>There are several kinds of this instrument; some of which are projections of the sphere; such as the hemispheres, or planispheres, on the plane of the equinoctial. The seamen commonly use two kinds; the one adapted to the pole star and the first of the guards of the Little Bear; the other to the pole star and the pointers of the Great Bear.</p><p>The Nocturnal consists of two circular plates (fig. 15, pl. xiii) applied over each other. The greater, which has a handle to hold the instrument, is about 2 1/2 inches diameter, and is divided into 12 parts, answering to the 12 months; also each month subdivided into every 5th day; and in such manner, that the middle of the handle corresponds to that day of the year in which the star here respected has the same right ascension with the sun.</p><p>When the instrument is fitted for two stars, the han-<cb/> dle is made moveable. The upper circle is divided into 24 equal parts, for the 24 hours of the day, and each hour subdivided into quarters, as in the figure. These 24 hours are noted by 24 teeth; to be told in the night. In the centre of the two circular plates is adjusted a long index A, moveable upon the upper plate. And the three pieces, viz, the two circles and index, are joined by a rivet which is pierced through the centre, with a hole 2 inches in diameter, for the star to be observed through.</p><p><hi rend="italics">To Use the</hi> <hi rend="smallcaps">Nocturnal.</hi> Turn the upper plate till the longest tooth, marked 12, be against the day of the month on the under plate; and bringing the instrument near the eye, suspend it by the handle, with the plane nearly parallel to the equinoctial; then viewing the pole-star through the hole in the centre, turn the index about till, by the edge coming from the centre, you see the bright star or guard of the Little Bear, if the instrument be fitted to that star: then that tooth of the upper circle, under the edge of the index, is at the hour of the night on the edge of the hour-circle: which may be known without a light, by counting the teeth from the longest, which is for the hour of 12.</p><p>NODATED <hi rend="italics">Hyperbola,</hi> one, so called by Newton, which by turning round decussates or crosses itself: as in the 2d, and several other species, of his Enumeratio Linearum Tertii Ordinis.</p></div2></div1><div1 part="N" n="NODES" org="uniform" sample="complete" type="entry"><head>NODES</head><p>, the two opposite points where the orbit of a planet intersects the ecliptic. That, where the planet ascends from the south to the north side of the ecliptic, is called the Ascending Node, or the Dragon's Head, and marked thus <*>: and the opposite point, where the planet descends from the north to the south side of the ecliptic, is called the Descending Node, or Dragon's Tail, and is thus marked <*>. Also the right line drawn from the one Node to the other, is called the Line of the Nodes.</p><p>By observation it appears that, in all the planets, the Line of the Nodes continually changes its place, its motion being <hi rend="italics">in antecedentia;</hi> i. e. contrary to the order of the signs, or from east to west; with a peculiar degree of motion for each planet. Thus, by a retrograde motion, the line of the moon's nodes completes its circuit in 18 years and 225 days, in which time the Node returns again to the same point of the ecliptic. Newton has not only shewn, that this motion arises from the action of the sun, but, from its cause, he has with great skill calculated all the elements and varieties in this motion. See his Princip. lib. 3, prop. 30, 31, &c.</p><p>The moon must be in or near one of the Nodes to make an eclipse either of the sun or moon.</p></div1><div1 part="N" n="NODUS" org="uniform" sample="complete" type="entry"><head>NODUS</head><p>, or <hi rend="italics">Node,</hi> in Dialling, denotes a point or hole in the gnomon of a dial, by the shadow or light of which is shewn, either the hour of the day in dials without furniture, or the parallels of the sun's declination, and his place in the ecliptic, &c, in dials with furniture.</p></div1><div1 part="N" n="NOLLET" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NOLLET</surname> (the Abbé <foreName full="yes"><hi rend="smallcaps">John Anthony</hi></foreName>)</persName></head><p>, a considerable French philosopher, and a member of most of the philosophical societies and academies of Europe, was born at Pimpré, in the district of Noyon, the 19th of November 1700. From the profound retreat, in which the mediocrity of his fortune obliged him to live, his reputation continually increased from day to day.<pb n="164"/><cb/> M. Dufay associated him in his Electrical Researches; and M. de Reaumur resigned to him his laboratory. It was under these masters that he developed his talents. M. Dufay took him along with him in a journey he made into England; and Nollet profited so well of this opportunity, as to institute a friendly and literary correspondence with some of the most celebrated men in this country.</p><p>The king of Sardinia gave him an invitation to Turin, to perform a course of experimental philosophy to the duke of Savoy. From thence he travelled into Italy, where he collected some good observations concerning the natural history of the country.</p><p>In France he was master of philosophy and natural history to the royal family; and professor royal of experimental philosophy to the college of Navarre, and to the sehools of artillery and engineers. The Academy of Sciences appointed him adjunct-mechanician in 1739, associate in 1742, and pensioner in 1757. Nollet died the 24th of April 1770, regretted by all his friends, but especially by his relations, whom he always succoured with an affectionate attention. The works published by Nollet, are the following:</p><p>1. Recueils de Lettres sur l'Electricité; 1753, 3 vols in 12mo.</p><p>2. Essai sur l'Electricité des Corps; 1 vol. in 12mo.</p><p>3. Recherches sur les Causes particulieres des Phenomenes Electriques; 1 vol. in 12mo.</p><p>4. L'Art des Experiences; 1770, 3 vols in 12mo.</p><p>His papers printed in the different volumes of the Memoirs of the Academy of Sciences, are much too numerous to be particularized here; they are inserted in all or most of the volumes from the year 1740 to the year 1767 inclusive, mostly several papers in each volume.</p></div1><div1 part="N" n="NONAGESIMAL" org="uniform" sample="complete" type="entry"><head>NONAGESIMAL</head><p>, or <hi rend="smallcaps">Nonagesimal</hi> <hi rend="italics">Degree,</hi> called also the Mid-heaven, is the highest point, or 90th degree of the ecliptic, reckoned from its intersection with the horizon at any time; and its altitude is equal to the angle that the ecliptic makes with the horizon at their intersection, or equal to the distance of the zenith from the pole of the ecliptic. It is much used in the calculation of solar eclipses.</p></div1><div1 part="N" n="NONAGON" org="uniform" sample="complete" type="entry"><head>NONAGON</head><p>, a figure having nine sides and angles. —In a regular Nonagon, or that whose angles, and sides, are all equal, if each side be 1, its area will be 6.1818242 = 9/4 of the tangent of 70°, to the radius 1. See my Mensuration, p. 114, 2d edit.</p></div1><div1 part="N" n="NONES" org="uniform" sample="complete" type="entry"><head>NONES</head><p>, in the Roman Calendar, the 5th day of the months January, February, April, June, August, September, November, and December; and the 7th of the other months March, May, July, and October: these last four months having 6 days before the Nones, and the others only four.—They had this name probably, because they were always 9 days inclusively, from the first of the Nones to the Ides, i. e. reckoning inclusively both those days.</p></div1><div1 part="N" n="NONIUS" org="uniform" sample="complete" type="entry"><head>NONIUS</head><p>, or <hi rend="smallcaps">Nunez (Peter</hi>), a very eminent Portuguese mathematician and physician, was born in 1497, at Alcazar in Portugal, anciently a remarkable city, known by the name of Salacia, from whence he was surnamed Salaciensis. He was professor of mathematics in the university of Coimbra, where he published<cb/> some pieces which procured him great reputation. He was mathematical preceptor to Don Henry, son to king Emanuel of Portugal, and principal cosmographer to the king. Nonius was very serviceable to the designs, which this court entertained of carrying on their maritime expeditions into the East, by the publication of his book <hi rend="italics">Of the Art of Navigation,</hi> and various other works. He died in 1577, at 80 years of age.</p><p>Nonius was the author of several ingenious works and inventions, and justly esteemed one of the most eminent mathematicians of his age. Concerning his <hi rend="italics">Art of Navigation,</hi> father Dechales says, “In the year 1530, Peter Nonius, a celebrated Portuguese mathematician, upon occasion of some doubts proposed to him by Martinus Alphonsus Sofa, wrote a Treatise on Navigation, divided into two books; in the first, he answers some of those doubts, and explains the nature of Loxodromic lines. In the second book, he treats of rules and instruments proper for navigation, particularly sea-charts, and instruments serving to sind the elevation of the pole; but says he is rather obscure in his manner of writing.”—Furetiere, in his Dictionary, takes notice that Peter Nonius was the first who, in 1530, invented the angles which the Loxodromic curves make with each meridian, calling them in his language Rhumbs, and which he calculated by spherical triangles.—Stevinus acknowledges, that Peter Nonius was scarce inferior to the very best mathematicians of the age. And Schottus says, he explained a great many problems, and particularly the mechanical problem of Aristotle on the motion of vessels by oars. His Notes upon Purbach's Theory of the Planets, are very much to be esteemed: he there explains several things, which had either not been noticed before, or not rightly understood.</p><p>In 1542 he published a Treatise on the Twilight, which he dedicated to John the 3d, king of Portugal; to which he added what Alhazen, an Arabian author, has composed on the same subject. In this work he describes the method or instrument called, from him, a Nonius, a particular account of which see in the following article.—He corrected several mathematical mistakes of Orontius Finæus.—But the most celebrated of all his works, or that at least he appeared most to value, was his <hi rend="italics">Treatise of Algebra,</hi> which he had composed in Portuguese, but translated it into the Castilian tongue, when he resolved upon making it public, which he thought would render his book more useful, as this language was more generally known than the Portuguese. The dedication, to his former pupil, prince Henry, was dated from Lisbon, Dec. 1, 1564. This work contains 341 pages in the Antwerp edition of 1567, in 8vo.</p><p>The catalogue of his works, chiefly in Latin, is as follows:</p><p>1. <hi rend="italics">De Arte Navigandi,</hi> libri duo; 1530.</p><p>2. <hi rend="italics">De Crepusculis;</hi> 1542.</p><p>3. <hi rend="italics">Annotationes in Aristotelem.</hi></p><p>4. Problema Mechanicum de Motu Navigii ex Remis</p><p>5. Annotationes in Planetarum Theorias Georgii Purbachii, &c.</p><p>6. Libro de Algebra en Arithmeticay Geometra; 1564.<pb n="165"/><cb/></p><div2 part="N" n="Nonius" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Nonius</hi></head><p>, is a name also erroneously given to the method of graduation now generally used in the division of the scales of various instruments, and which should be called Vernier, from its real inventor. The method of Nonius, so called from its inventor Pedro Nunez, or Nonius, and described in his treatise De Crepusculis, printed at Lisbon in 1542, consists in describing within the same quadrant, 45 concentric circles, dividing the outermost into 90 equal parts, the next within into 89, the next into 88, and so on, till the innermost was divided into 46 only. By this means, in most observations, the plumb-line or index must cross one or other of those circles in or very near a point of division: whence by calculation the degrees and minutes of the arch might easily be obtained. This method is also described by Nunez, in his treatise De Arte et Ratione Navigandi, lib. 2, cap. 6, where he imagines it was not unknown to Ptolomy. But as the degrees are thus divided unequally, and it is very difficult to attain exactness in the division, especially when the numbers, into which the arches are to be divided, are incomposite, of which there are no less than nine, the method of diagonals, first published by Thomas Digges, Esq. in his treatise Alæ seu Scalæ Mathematicæ, printed at Lond. in 1573, and said to be invented by one Richard Chanseler, a very skilful artist, was substituted in its stead. However, Nonius's method was improved at different times; but the admirable division now so much in use, is the most considerable improvement of it. See V<hi rend="smallcaps">ERNIER.</hi></p></div2></div1><div1 part="N" n="NORMAL" org="uniform" sample="complete" type="entry"><head>NORMAL</head><p>, is used sometimes for a perpendicular.</p><p>NORTH <hi rend="italics">Star,</hi> called also the Pole-star, is the last in the tail of the Little Bear.</p><p><hi rend="smallcaps">Northern</hi> <hi rend="italics">Signs,</hi> are those six that are in the north side of the equator; viz, Aries, Taurus, Gemini, Cancer, Leo, Virgo.</p></div1><div1 part="N" n="NORTHING" org="uniform" sample="complete" type="entry"><head>NORTHING</head><p>, in Navigation, is the difference of latitude, which a ship makes in sailing northwards.</p></div1><div1 part="N" n="NOSTRADAMUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">NOSTRADAMUS</surname> (<foreName full="yes"><hi rend="smallcaps">Michel</hi></foreName>)</persName></head><p>, an able physician and celebrated astrologer, was born at St. Remy in Provence in the diocese of Avignon, December 14, 1503. His father was a notary public, and his grandfather a physician, from whom he received some tincture of the mathematies. He afterwards completed his courses of languages and philosophy at Avignon. From hence, going to Montpelier, he there applied himself to physic; but being forced away by the plague, he travelled through different places till he came to Bourdeaux, undertaking all such patients as were willing to put themselves under his care. This course occupied him five years; after which he returned to Montpelier, and was created doctor of his faculty in 1529; after which he revisited the same places where he had practised physic before. At Agen he formed an acquaintance with Julius Cæsar Scaliger, and married his first wife; but having buried her, and two children which she brought him, he quitted Agen after a residence of about four years. He fixed next at Marseilles; but, his friends having provided an advantageous match for him at Salon, he repaired thither about the year 1544,<cb/> and married accordingly his second wife, by whom he had several children.</p><p>In 1546, Aix being afflicted with the plague, he went thither at the solicitation of the inhabitants, to whom he rendered great service, particularly by a powder of his own invention: so that the town, in gratitude, gave him a considerable pension for several years after the contagion ceased. In 1547 the city of Lyons, being visited with the same distemper, had recourse to our physician, who attended them also. Afterwards returning to Salon, he began a more retired course of life, and in this time of leisure applied himself closely to his studies. He had for a long time followed the trade of a conjurer occasionally; and now he began to fancy himself inspired, and miraculously illuminated with a prospect into futurity. As fast as these illuminations had discovered to him any suture event, he entered it in writing, in simple prose, though in enigmatical sentences; but revising them afterwards, he thought the sentences would appear more respectable, and savour more of a prophetic spirit, if they were expressed in verse. This opinion determined him to throw them all into quatrains, and he afterward ranged them into centuries. For some time he could not venture to publish a work of this nature; but afterwards perceiving that the time of many events foretold in his quatrains was very near at hand, he resolved to print them, as he did, with a dedication addressed to his son Cæsar, an infant only some months old, and dated March 1, 1555. To this first edition, which comprises but seven centuries, he presixed his name in Latin, but gave to his son Cæsar the name as it is pronounced in French, Notradame.</p><p>The public were divided in their sentiments of this work: many looked upon the author as a simple visionary; by others he was accused of magic or the black art, and treated as an impious person who held a commerce with the devil; while great numbers believed him to be really endued with the supernatural gif<*> of prophecy. However, Henry the 2d, and queen Catharine of Medicis, his mother, were resolved to see our prophet, who receiving orders to that effect, he presently repaired to Paris. He was very graciously received at court, and received a present of 200 crowns. He was sent afterwards to Blois, to visit the king's children there, and report what he should be able to discover concerning their destinies. It is not known what his sentence was; however he returned to Salon loaded with honour, and good presents.</p><p>Animated with this success, he augmented his work to the number of 1000 quatrains, and published it with a dedication to the king in 1558. That prince dying the next year of a wound which he received at a tournament, our prophet's book was immediately consulted; and this unfortunate event was found in the 35th quatrain of the first century, which runs thus in the London edition of 1672: Le Lion jeune le vieux surmontera,<lb/> En champ bellique, par singulier duelle,<lb/> Dans cage d'or l'œil il lui crevera,<lb/> Deux playes une, puis mourir mort cruelle.<lb/><pb n="166"/><cb/></p><p>In English thus, from the same edition: The young Lion shall overcome the old one,<lb/> In martial field by a single duel,<lb/> In a golden cage he shall put out his eye,<lb/> Two wounds from one, then he shall die a cruel death.<lb/></p><p>So remarkable a prediction added new wings to his fame; and he was honoured soon after with a visit from Emanuel duke of Savoy, and the princess Margaret of France, his consort. From this time Nostradamus found himself even overburdened with visitors, and his fame made every day new acquisitions. Charles the 9th, coming to Salon, was eager above all things to have a sight of him: Nostradamus, who then was in waiting as one of the retinue of the magistrates, being instantly presented to the king, complained of the little esteem his countrymen had for him; upon which the monarch publicly declared that he <*>ould hold the enemies of Nostradamus to be his enemies, and desired to see his children. Nor did that prince's favour stop here; in passing, not long after, through the city of Arles, he sent for Nostradamus, and presented him with a purse of 200 crowns, together with a brevet, constituting him his physician in ordinary, with the same appointment as the rest. But our prophet enjoyed these honours only a short time, as he died 16 months aftèr, viz, July 2, 1566, at Salon, being then in his grand climacteric, or 63d year.—He had published several other pieces, chiefly relating to medicine.</p><p>He left three sons and three daughters. Cæsar the eldest son was born at Salon in 1555, and died in 1629: he left a manuscript, giving an account of the most remarkable events in the history of Provence, from 1080 to 1494, in which he inserted the lives of the poets of that country. These memoirs falling into the hands of his nephew Cæsar Nostradamus, gentleman to the duke of Guise, he undertook to complete the work; and being encouraged by the estates of the country, he carried the account up to the Celtic Gauls: the impression was finished at Lyons in 1614, and published under the title of Chronique de l'Histoire de Provence. —The second son, John, exercised with reputation the business of a proctor in the parliament of Provence.— He wrote the Lives of the Ancient Provençal Poets, called Troubadours, and the work was printed at Lyons in 1575, 8vo.—The youngest son it is said undertook the trade of peeping into futurity after his father.</p></div1><div1 part="N" n="NOTATION" org="uniform" sample="complete" type="entry"><head>NOTATION</head><p>, is the representing of numbers, or any other quantities, by Notes, characters, or marks.</p><p>The choice of arithmetical, and other, characters, is arbitrary; and hence they are various in various nations: the figures 0, 1, 2, 3, &c, in common use, are derived from the Arabs and Indians, from whom they have their name, and the Notation by them, which forms the decimal or decuple scale, is perhaps the most convenient of any for arithmetical computations.</p><p>The Greeks, Hebrews, and other eastern nations, as also the Romans, expressed numbers by the letters of their common alphabet. See <hi rend="smallcaps">Character.</hi></p><p>In Algebra, the quantities are represented mostly by th<*> letters of the alphabet, &c; and that as early as the time of Diophantus. See <hi rend="smallcaps">Algebra.</hi></p></div1><div1 part="N" n="NOTES" org="uniform" sample="complete" type="entry"><head>NOTES</head><p>, in Music, are characters which mark the tones, i. e. the elevations and fallings of the voice, or<cb/> sound, and the swiftness or slowness of its motions, &c; and these have undergone various alterations and improvements, before they arrived at their present state of perfection.</p></div1><div1 part="N" n="NOVEMBER" org="uniform" sample="complete" type="entry"><head>NOVEMBER</head><p>, the eleventh month in the Julian year, but the ninth in the year of Romulus, beginning with March; whence its name. In this month, which contains 30 days, the sun enters the sign <*>, viz, usually about the 21st day of the month.</p></div1><div1 part="N" n="NUCLEUS" org="uniform" sample="complete" type="entry"><head>NUCLEUS</head><p>, the kernel, is used by Hevelius, and some other astronomers, for the body of a comet, which others call its head, as distinguished from its tail, or beard.</p><p><hi rend="smallcaps">Nucleus</hi> is also used by some writers for the central parts of the earth, and other planets, which they suppose firmer, and as it were separated from them, considered as a cortex or shell.</p></div1><div1 part="N" n="NUEL" org="uniform" sample="complete" type="entry"><head>NUEL</head><p>, the same as <hi rend="smallcaps">Newel</hi> of a Staircase.</p></div1><div1 part="N" n="NUMBER" org="uniform" sample="complete" type="entry"><head>NUMBER</head><p>, a collection or assemblage of several units, or several things of the same kind; as 2, 3, 4, &c, exclusive of the number 1: which is Euclid's definition of Number.—Stevinus defines Number as that by which the quantity of anything is expressed: agreeably to which Newton conceives a Number to consist, not in a multitude of units, as Euclid defines it, but in the abstract ratio of a quantity of any kind to another quantity of the same kind, which is accounted as unity: and in this sense, including all these three species of Number, viz, Integers, Fractions, and Surds.</p><p>Wolfius defines Number to be something which refers to unity, as one right line refers to another. Thus, assuming a right line for unity, a Number may likewise be expressed by a right line. And in this way also Des Cartes considers numbers as expressed by lines, where he treats of the arithmetical operations as performed by lines, in the beginning of his Geometry.</p><p><hi rend="italics">For the manner of characterizing</hi> <hi rend="smallcaps">Numbers</hi>, see N<hi rend="smallcaps">OTATION.</hi> And</p><p><hi rend="italics">For reading and expressing</hi> <hi rend="smallcaps">Numbers</hi> in combination, see <hi rend="smallcaps">Numeration.</hi></p><p>Mathematicians consider Number under a great many circumstances, and different relations, accidents, &c.</p><div2 part="N" n="Numbers" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Numbers</hi></head><p>, <hi rend="italics">Absolute, Abstract, Abundant, Amicable, Applicate, Binary, Cardinal, Circular, Composite, Concrete, Defective, Fractional, Homogeneal, Irrational</hi> or <hi rend="italics">Surd, Linear</hi> or <hi rend="italics">Mixt, Ordinal, Polygonal, Prime, Pyramidal, Rational, Similar, &c,</hi> see the respective adjectives.</p><p><hi rend="italics">Broken</hi> <hi rend="smallcaps">Numbers</hi>, or Fractions, are certain parts of unity, or of some other Number.</p><p><hi rend="italics">Cubic</hi> <hi rend="smallcaps">Number</hi>, is the product of a square Number multiplied by its root, or the continual product of a Number twice multiplied by itself; as the Numbers - - 1, 8, 27, 64, 125, &c, which are the cubes of - 1, 2, 3, 4, 5, &c.</p><p>This series of the cubes of the ordinal Numbers, may be raised by addition only, viz, adding always the differences; as was first shewn by Peletarius, at the end of his Algebra, first printed in 1558, where he gives a table of the squares and Cubes of the first 140 numbers. See <hi rend="smallcaps">Cube.</hi></p><p>Every Cubic Number whose root is less than 6, viz, the Cubic Numbers 1, 8, 27, 64, 125, being divided by 6, the remainder is the root itself:<pb n="167"/><cb/> Thus, ; where the remainders, or the numerators of the small fractions, are 0, 1, 2, 3, 4, 5, the same as the roots of the Cubes 0, 1, 8, 27, 64, 125. After these, the next six Cubic Numbers being divided by 6, the remainders will be respectively the same arithmetical series, viz - - - 0, 1, 2, 3, 4, 5; to each of which adding 6, gives 6, 7, 8, 9, 10, 11, for the roots of the next six cubes 216, 343, &c.</p><p>Then, again dividing the next set of six Cubic Numbers, viz, - - 1728, 2197, &c, <hi rend="brace">by 6, the remainders are again the same series, viz,</hi> 0, 1, 2, 3, 4, 5, to each of which adding 12, gives 12, 13, 14 15, 16, 17, for the roots of the said next six cubes. And so on in infinitum, the series of remainders 0, 1, 2, 3, 4, 5, continually recurring, and to each set of these remainders the respective Numbers 0, 6, 12, 18, 24, &c, being added, the sums will be the whole series of roots, 0, 1, 2, 3, 4, 5, 6, &c.</p><p>M. de la Hire, from considering this property of the Number 6, with regard to Cubic Numbers, found that all other Numbers, raised to any power whatever, had each their divisor, which had the same effect with regard to them, that 6 has with regard to Cubes. And the general rule he has discovered is this: if the exponent of the power of a number be even, i. e. if that number be raised to the 2d, 4th, 6th, &c power, it must be divided by 2, then the remainder added to 2, or to a multiple of 2, gives the root of the Number corresponding to its power, i. e. the 2d, or 4th, &c, root. But if the exponent of the power of the Number be uneven, viz the 3d, 5th, 7th, &c power, the double of that exponent shall be the divisor, which shall have the property here required.</p><p><hi rend="italics">A Determinate</hi> <hi rend="smallcaps">Number</hi>, is that which is referred to some given unit; as a ternary or three.</p><p>An <hi rend="italics">Even</hi> <hi rend="smallcaps">Number</hi>, is that which may be divided into two equal parts, without remainder or fraction, as the Numbers 2, 4, 6, 8, 10, &c.—The sums, differences, products, and powers of Even Numbers, are also Even Numbers.</p><p>An <hi rend="italics">Evenly-Even</hi> <hi rend="smallcaps">Number</hi>, is such as being divided by an even Number, the quotient is also an Even Number without a remainder: as 16, which divided by 8 gives 2 for the quotient.</p><p>An <hi rend="italics">Unevenly-Even</hi> <hi rend="smallcaps">Number</hi>, is such as being divided by an Even Number, the quotient is an Uneven one: as 20, which divided by 4, gives 5 for the quotient.</p><p><hi rend="italics">Figurate</hi> or <hi rend="italics">Figural</hi> <hi rend="smallcaps">Numbers</hi>, are certain ranks of Numbers found by adding together first a rank of units, which is the first order, which gives the 2d order; then these added give the 3d order; and so on. Hence, the several orders of Eigurate Numbers, are as follow: <table><row role="data"><cell cols="1" rows="1" role="data">First order</cell><cell cols="1" rows="1" role="data">1 . 1 . 1  . 1  . 1 . &c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">2d order</cell><cell cols="1" rows="1" role="data">1 . 2 . 3  . 4  . 5 . &c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">3d order</cell><cell cols="1" rows="1" role="data">1 . 3 . 6  . 10. 15. &c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">4th order</cell><cell cols="1" rows="1" role="data">1 . 4 . 10. 20. 35. &c.</cell></row><row role="data"><cell cols="1" rows="1" role="data">5th order</cell><cell cols="1" rows="1" role="data">1 . 5 . 15. 35. 70. &c.</cell></row></table></p><p>The first order consists all of equals, and the 2d order of the natural arithmetical progression; the 3d order<cb/> is also called triangular Numbers, the 4th order pyramidals, &c.</p><p>See <hi rend="smallcaps">Figurate</hi> <hi rend="italics">Numbers.</hi></p><p><hi rend="italics">Heterogeneal</hi> <hi rend="smallcaps">Numbers</hi>, are such as are referred to different units. As three men and 4 trees.</p><p><hi rend="italics">Homogeneal</hi> <hi rend="smallcaps">Numbers</hi>, are such as are referred to the same unit. As 3 men and 4 men.</p><p><hi rend="italics">Impersect</hi> <hi rend="smallcaps">Numbers</hi>, are those whose aliquot parts added together, make either more or less than the whole of the number itself; and are distinguished into Abundant and Defective.</p><p><hi rend="italics">Indeterminate</hi> <hi rend="smallcaps">Number</hi>, is that which is referred to unity in the general; which is what we call Quantity.</p><p><hi rend="italics">Irrational</hi> or <hi rend="italics">Surd</hi> <hi rend="smallcaps">Number</hi>, is one that is not commensurable with unity; as √2, or √<hi rend="sup">3</hi>4, &c.</p><p><hi rend="italics">Perfect</hi> <hi rend="smallcaps">Number</hi>, that which is just equal to the sum of its aliquot parts, added together. As, 6, 28, &c: for the aliquot parts of 6 are 1, 2, 3, whose sum is the same 6; and the aliquot parts of 28, are 1, 2, 4, 7, 14, whose sum is 28. See <hi rend="smallcaps">Perfect</hi> <hi rend="italics">Number.</hi></p><p><hi rend="italics">Plane</hi> <hi rend="smallcaps">Number</hi>, that which arises from the multiplication of two other Numbers: so 6 is a plane or rectangle, whose two sides are 2 and 3, for .</p><p><hi rend="italics">Square</hi> <hi rend="smallcaps">Number</hi>, is a Number produced by multiplying any given Number by itself; as the Square Numbers - - 1, 4, 9, 16, 25, &c, produced from the roots - 1, 2, 3, 4, 5, &c.</p><p>Every Square Number added to its root makes an even Number. See <hi rend="smallcaps">Square.</hi></p><p><hi rend="italics">Uneven</hi> <hi rend="smallcaps">Number</hi>, or <hi rend="italics">Odd</hi> <hi rend="smallcaps">Number</hi>, that which differs from an even Number by one, or which cannot be divided into two equal integer parts; such as 1, 3, 5, 7, &c. The sums and differences of Uneven Numbers are even; but all the products and powers of them are Uneven Numbers. On the other hand, the sum or difference of an even and Uneven Number are both Uneven, but their product is even.</p><p><hi rend="italics">Whole</hi> <hi rend="smallcaps">Number</hi>, or <hi rend="italics">Integer,</hi> is unit, or a collection of units.</p><p><hi rend="italics">Golden</hi> <hi rend="smallcaps">Number.</hi> See <hi rend="smallcaps">Golden</hi> <hi rend="italics">Number</hi> and <hi rend="smallcaps">Cycle.</hi></p><p><hi rend="smallcaps">Number</hi> <hi rend="italics">of Direction,</hi> in Chronology, some one of the 35 Numbers between the Easter limits, or between the earliest and latest day on which it can fall, i. e. between March 22 and April 25, which are 35 days; being so called, because it serves as a Direction for finding Easter for any year; being indeed the Number that expresses how many days after March 21, Easterday falls. Thus, Easter-day falling as in the first line below, the Number of Direction will be as on the lower line: <table><row role="data"><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" role="data">April</cell></row></table> Easter-day, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 1, 2, &c. N<*> of Dir. 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, &c and so on, till the Number of Direction on the lower line be 35, which will answer to April 25, being the latest that Easter can happen. Therefore add 21 to the Number of Direction, and the sum will be so many days in March for the Easter-day: if the sum exceed 31, the excess will be the day of April.</p><p><hi rend="italics">To find the</hi> <hi rend="smallcaps">Number</hi> <hi rend="italics">of Direction.</hi> Enter the following table (which is adapted to the New Style), with the Dominical Letter on the left hand, and the Golden Number at the top, then where the columns meet is<pb n="168"/><cb/> the Number of Direction for that year. See Ferguson's Astron. pa. 381, ed. 8vo. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">G. N.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">19</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Do<*> Le<*>.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">A</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">C</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">D</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">E</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2<*></cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">F</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2<*></cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">G</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2<*></cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"><*>1</cell></row></table></p><p>Thus, for the year 1790, the Dominical Letter being C, and the Golden Number 5; on the line of C, and below 5, is 14 for the Number of Direction. To this add 21, the sum is 35 days from the 1st of March, which, deducting the 31 days of March, leaves 4 for the day of April, for Easter-day that year.</p><p><hi rend="smallcaps">Numeral</hi> <hi rend="italics">Characters.</hi> See <hi rend="smallcaps">Characters.</hi></p><p><hi rend="smallcaps">Numeral</hi> <hi rend="italics">Figures.</hi> The antiquity of these in England has, for several reasons, been supposed as high as the eleventh century; in France about the middle of the tenth century; having been introduced into both countries from Spain, where they had been brought by the Moors or Saracens. See Wallis's Algebra, pa. 9 &c, and pa. 153 of additions at the end of the same. See also Philos. Trans. numb. 439 and 475.</p><p><hi rend="smallcaps">Numeral</hi> <hi rend="italics">Letters,</hi> those letters of the alphabet that are commonly used for figures or numbers, as I, V, X, L, C, D, M.</p></div2></div1><div1 part="N" n="NUMERATION" org="uniform" sample="complete" type="entry"><head>NUMERATION</head><p>, in Arithmetic, the art of estimating or pronouncing any number, or series of numbers.</p><p>Numbers are usually expressed by the ten following characters, 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0; the first nine denoting respectively the first nine ordinal numbers; and the last, or cipher 0, joined to any of the others, denotes so many tens. In like manner, two ciphers joined to any one of the first nine significant figures, make it become so many hundreds, three ciphers make it thousands, and so on.</p><p>Weigelius indeed shews how to number, without going beyond a quaternary; i. e. by beginning to repeat at each fourth. And Leibnitz and De Lagny, in what they call their binary arithmetic, begin to repeat<cb/> at every 2d place; using only the two figures 1 and 0. But these are rather matters of curiosity than any real use.</p><p>That the nine significant sigures may express not only units, but also tens, hundreds, thousands, &c, they have a local value given them, as hinted above; so that, though when alone, or in the right-hand place, they denote only units or ones, yet in the 2d place they denote tens, in the 3d place hundreds, in the 4th place thousands, &c; as the number 5555 is five thousand five hundred fifty and five.</p><p>Hence then, to express any written number, or assign the proper value to each character; beginning at the right hand, divide the proposed number into classes, of three characters to each class; and consider two classes as making up a period of six sigures or places. Then every period, of six figures, has a name common to all the figures in it; the sirst being primes or units; the 2d is millions; the 3d is millions of-millions, or billions; the 4th is millions-of-millions-of-millions, or trillions; and so on; also every class, or half-period, of three sigures, is read separately by itself, so many hundreds, tens, and units; only, after the lest hand half of each period, the word thousands is added; and at the end of the 2d, 3d, 4th &c period, its common name millions, billions, &c, is expressed.</p><p>Thus the number 4,591, is 4 thousand 5 hundred and 91.</p><p>The number 210,463, is 2 hundred and 10 thousands, and 463.</p><p>The number 281,427,307, is 281 millions, 427 thousands, and 307.</p></div1><div1 part="N" n="NUMERATOR" org="uniform" sample="complete" type="entry"><head>NUMERATOR</head><p>, of a Fraction, is the number which shews how many of those parts, which the integer is supposed to be divided into, are denoted by the fraction. And, in the notation the Numerator is set over the denominator, or number that shews into how many parts the integer is divided, in the fraction. So, ex. gr. 3/4 denotes three-fourths, or 3 parts out of 4; where 3 is the numerator, and 4 the denominator.</p></div1><div1 part="N" n="NUMERICAL" org="uniform" sample="complete" type="entry"><head>NUMERICAL</head><p>, <hi rend="smallcaps">Numerous</hi>, or <hi rend="italics">Numeral,</hi> something that relates to number.</p><p><hi rend="smallcaps">Numeral</hi> <hi rend="italics">Algebra,</hi> is that which makes use of numbers, in contradistinction from literal algebra, or that in which the letters of the alphabet are used.</p></div1></div0><div0 part="N" n="O" org="uniform" sample="complete" type="alphabetic letter"><head>O</head><cb/><div1 part="N" n="OBELISK" org="uniform" sample="complete" type="entry"><head>OBELISK</head><p>, a kind of quadrangular pyramid, very tall and slender, raised as an ornament in some public place, or to serve as a memorial of some remarkable transaction.<cb/></p></div1><div1 part="N" n="OBJECT" org="uniform" sample="complete" type="entry"><head>OBJECT</head><p>, something presented to the mind, by sensation, or by imagination. Or something that affects us by its presence, that affects the eye, ear, or some other of the organs of sense.<pb n="169"/><cb/></p><p>The objects of the eye, or vision, are painted on the retina; though not there erect, but inverted, according to the laws of optics. This is easily shewn from Des Cartes's experiment, of laying bare the vitreous humour on the back part of the eye, and putting over it a bit of white paper, or the skin of an egg, and then placing the fore part of the eye to the hole of a darkened room. By this means there is obtained a pretty landscape of the external objects, painted invertedly on the back of the eye. In this case, how the Objects thus painted invertedly should be seen erect, is matter of controversy.</p><p><hi rend="smallcaps">Object</hi> is also used for the subject, or matter of an art or science; being that about which it is employed or concerned.</p><p><hi rend="smallcaps">Object</hi>-<hi rend="italics">Glass,</hi> of a telescope or microscope, is the glass placed at the end of the tube which is next or towards the Object to be viewed.</p><p>To prove the goodness and regularity of an Objectglass; on a paper describe two concentric circles, the one having its diameter the same with the breadth of the Object-glass, and the other half that diameter; divide the smaller circumference into 6 equal parts, pricking the points of division through with a sine needle; cover one side of the glass with this paper, and, exposing it to the sun, receive the rays through these 6 holes upon a plane; then by moving the plane nearer to or farther from the glass, it will be found whether the six rays unite exactly together at any distance from the glass; if they do, it is a proof of the regularity and just form of the glass; and the said distance is also the focal distance of the glass.</p><p>A good way of proving the excellency of an Objectglass, is by placing it in a tube, and trying it with small eye-glasses, at several distant objects; for that Object-glass is always the best, which represents objects the brightest and most distinct, and which bears the greatest aperture, and the most convex and concave eyeglasses, without colouring or haziness.</p><p>A circular Object-glass is said to be truly centred, when the centre of its circumference falls exactly in the axis of the glass; and to be ill centred, when it falls out of the axis.</p><p>To prove whether Object-gla<*>es be well centred, hold the glass at a due distance from the eye, and observe the two reflected images of a candle, varying the distance till the two images unite, which is the true centre point: then if this fall in the middle, or central point of the glass, it is known to be truly centred.</p><p>As Object-glasses are commonly included in cells that screw upon the end of the tube of a telescope, it may be proved whether they be well centred, by fixing the tube, and observing while the cell is unscrewed, whether the cross-hairs keep fixed upon the same lines of an object seen through the telescope.</p><p>For various methods of finding the true centre of an Object-glass, see Smith's Optics, book 3, chap. 3; also the Philos. Trans. vol. 48, pa. 177.</p><p>OBJECTIVE <hi rend="italics">Line,</hi> in Perspective, is any line drawr on the geometrical plane, whose representation is sought for in a draught or picture.</p><p><hi rend="smallcaps">Objective</hi> <hi rend="italics">Plane,</hi> in Perspective, is any plane situated in the horizontal plane, whose perspective representation is required.</p></div1><div1 part="N" n="OBLATE" org="uniform" sample="complete" type="entry"><head>OBLATE</head><p>, flatted, or shortened; as an Oblate sphe-<cb/> roid, having its axis shorter than its middle diameter; being formed by the rotation of an ellipse about the shorter axis.</p></div1><div1 part="N" n="OBLATENESS" org="uniform" sample="complete" type="entry"><head>OBLATENESS</head><p>, of the earth, the flatness about the poles, or the diminution of the polar axis in respect of the equatorial. The ratio of these two axes has been determined in various ways; sometimes by the measures of different degrees of latitude, and sometimes by the length of pendulums vibrating seconds in different latitudes, &c; the results of all which, as well as accounts of the means of determining them, see under the articles <hi rend="smallcaps">Earth</hi> and <hi rend="smallcaps">Degree.</hi> To what is there said, may be added the following, from An Account of the Experiments made in Russia concerning the Length of a Pendulum which swings Seconds, by Mr. Krafst, contained in the 6th and 7th volumes of the New Petersburgh Transactions, for the years 1790 and 1793. These experiments were made at different times, and in various parts of the Russian empire: Mr. Krafft has collected and compared them, with a view to investigate the consequences that may be deduced from them. From the whole he concludes, that the length <hi rend="italics">p</hi> of a pendulum, which swings seconds in any given latitude <hi rend="italics">l<*></hi> and in a temperature of 10 degrees of Reaumur's thermometer, may be determined by the following equation, in lines of a French foot: viz, .</p><p>This expression agrees, very nearly, not only with all the experiments made on the pendulum in Russia, but also with those of Mr. Graham, and those of Mr. Lyons in 79° 50′ north latitude, where he found its length to be 441.38 lines.</p><p>It also shews the augmentation of gravity from the equator to the parallel of a given latitude <hi rend="italics">l:</hi> for, putting <hi rend="italics">g</hi> for the gravity under the equator, G for that under the pole, and <hi rend="italics">z</hi> for that under the latitude <hi rend="italics">l;</hi> Mr. Krafft finds ; and consequently :</p><p>From this proportion of Gravity under different latitudes, Mr. Krafft deduces, that on the hypothesis of the earth's being a homogeneous ellipsoid, its oblateness must be 1/190; instead of 1/230, which ought to be the result of this hypothesis: but on adopting the supposition that the earth is a heterogeneous ellipsoid, he finds its Oblateness, as deduced from these experiments, to be 1/297; which agrees with that resulting from the measurement of degrees of the meridian.</p><p>This confirms an observation of M. De la Place, that, if the hypothesis of the earth's homogeneity be given up, then do theory, the measurement of degrees of latitude, and experiments with the pendulum, all agree in their result with respect to the Oblateness of the earth.</p></div1><div1 part="N" n="OBLIQUE" org="uniform" sample="complete" type="entry"><head>OBLIQUE</head><p>, aslant, indirect, or deviating from the perpendicular. As,</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Angle,</hi> one that is not a right angle, but is either greater or less than this, being either obtuse or acute.</p><p><hi rend="smallcaps">Oblique</hi>-<hi rend="italics">angled Triangle,</hi> that whose angles are all oblique.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Ascension,</hi> is that point of the equinoctial which rises with the centre of the sun, or star, or any other point of the heavens, in an Oblique sphere.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Circle,</hi> in the stereographic projection,<pb n="170"/><cb/> is any circle that is Oblique to the plane of projection.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Descension,</hi> that point of the equinoctial which sets with the eentre of the sun, or star, or other point of the heavens in an Obliqne sphere.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Direction,</hi> that which is not perpendicular to a line or plane.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Force,</hi> or Percussion, or <hi rend="italics">Power,</hi> or Stroke, is that made in a direction Oblique to a body or plane. It is demonstrated that the effect of such Oblique force <*>c, upon the body, is to an equal perpendicular one, as the sine of the angle of incidence is to radius.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Line,</hi> that which makes an Oblique angle with some other line.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Planes,</hi> in Dialling, are such as recline from the zenith, or incline towards the horizon.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Projection,</hi> is that where a body is projected or impelled in a line of direction that makes an oblique angle with the horizontal line.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Sailing,</hi> in Navigation, is that part which includes the application and calculation of Obliqueangled triangles.</p><p><hi rend="smallcaps">Oblique</hi> <hi rend="italics">Sphere,</hi> in Geography, is that in which the axis is Oblique to the horizon of a place.—In this sphere, the equator and parallels of declination cut the horizon obliquely. And it is this obliquity that occasions the inequality of days and nights, and the variation of the seasons. See <hi rend="smallcaps">Sphere.</hi></p></div1><div1 part="N" n="OBLIQUITY" org="uniform" sample="complete" type="entry"><head>OBLIQUITY</head><p>, that which denotes a thing Oblique.</p><p><hi rend="smallcaps">Obliquity</hi> <hi rend="italics">of the Ecliptic,</hi> is the angle which the ecliptic makes with the equator. See <hi rend="smallcaps">Ecliptic.</hi></p></div1><div1 part="N" n="OBLONG" org="uniform" sample="complete" type="entry"><head>OBLONG</head><p>, sometimes means any figure that is longer than it is broad; but more properly it denotes a rectangle, or a right-angled parallelogram, whose length exceeds its breadth.</p><div2 part="N" n="Oblong" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Oblong</hi></head><p>, is also used for the quality or species of a figure that is longer than it is broad: as an Oblong spheroid; formed by an ellipse revolved about its longer or tranfverse axis; in contradistinction from the oblate spheroid, or that which is flatted at its poles, being generated by the revolution of the ellipfe about its conjugate or shorter axis.</p><p>OBSCURA <hi rend="italics">Camera.</hi> See <hi rend="smallcaps">Camera</hi> <hi rend="italics">Obscura.</hi></p><p><hi rend="smallcaps">Obscura</hi> <hi rend="italics">Clara.</hi> See <hi rend="smallcaps">Clara</hi> <hi rend="italics">Obscure.</hi></p></div2></div1><div1 part="N" n="OBSERVATION" org="uniform" sample="complete" type="entry"><head>OBSERVATION</head><p>, in Astronomy and Navigation, is the observing with an instrument some celestial phenomenon; as, the altitude of the sun, moon, or stars, or their distances asunder, &c. But by this term the seamen commonly mean only the taking the meridian altitudes, in order to find the latitude. And the finding the latitude from such observed altitude, they call <hi rend="italics">working an observation.</hi></p></div1><div1 part="N" n="OBSERVATORY" org="uniform" sample="complete" type="entry"><head>OBSERVATORY</head><p>, a place destined for observing the heavenly bodies; or a building, usually in form of a tower, erected on some eminence, and covered with a terrace, for making astronomical observations.</p><p>Most nations, at almost all times, have had their observatories, either public or private ones, and in various degrees of perfection. A description of a great many of them may be seen in a dissertation of Weidler's, De præsenti Specularum Astronomicarum Statu, printed in 1727, and in different articles of his History of Astronomy, printed in 1741, viz, pa. 86 &c; as also in La Lande's Astronomy, the preface pa. 34. The chief among these are the following:<cb/></p><p>I. The Greenwich Observatory, or Royal Observatory of England. This was built and endowed in the year 1676, by order of King Charles the 2d, at the instance of Sir Jonas Moore, and Sir Christopher Wren: the former of these gentlemen being Surveyor General of the Ordnance, the office of Astronomer Royal was placed under that department, in which it has continued ever since.</p><p>This observatory was at sirst furnished with several very accurate instruments; particularly a noble sextant of 7 feet radius, with telescopic sights. And the sirst Astronomer Royal, or the person to whom the province of observing was sirst committed, was Mr. John Flamsteed; a man who, as Dr. Halley expresses it, seemed born for the employment. During 14 years he watched the motions of the planets with unwearied diligence, especially those of the moon, as was given him in charge; that a new theory of that planet being found, shewing all her irregularities, the longitude might thence be determined.</p><p>In the year 1690, having provided himself with a mural arch of near 7 feet radius, made by his Assistant Mr. Abraham Sharp, and fixed in the plane of the meridian, he began to verify his catalogue of the fixed stars, which had hitherto depended altogether on the distances measured with the sextant, after a new and very different manner, viz, by taking the meridian altitudes, and the moments of culmination, or in other words the right ascension and declination. And he was so well pleased with this instrument, that he discontinued almost entirely the use of the sextant.</p><p>Thus, in the space of upwards of 40 years, the Astronomer Royal collected an immense number of good observations; which may be found in his Historia Cœlestis Britannica, published in 1725; the principal part of which is the Britannic catalogue of the fixed stars.</p><p>Mr. Flamsteed, on his death in 1719, was succeeded by Dr. Halley, and he by Dr. Bradley in 1742, and this last by Mr. Bliss in 1762; but none of the observations of these gentlemen have yet been given to the public.</p><p>On the demise of Mr. Bliss, in 1765, he was succeeded by Dr. Nevil Maskelyne, the present worthy astronomer royal, whose valuable observations have been published, from time to time, under the direction of the Royal Society, in several folio volumes.</p><p>The Greenwich Observatory is found, by very accurate observations, to lie in 51° 28′ 40″ north latitude, as settled by Dr. Maskelyne, from many of his own observations, as well as those of Dr. Bradley.</p><p>II. The Paris Observatory was built by Louis the 14th, in the fauxbourg St. Jaques, being begun in 1664, and finished in 1672. It is a singular but magnificent building, of 80 feet in height, with a terrace at top; and here M. De la Hire, M. Cassini, &c, the king's astronomers, have made their observations. Its latitude is 48° 50′ 14″ north, and its longitude 9′ 20″ east of Greenwich Observatory.</p><p>In the Observatory of Paris is a cave, or pit, 170 feet deep, with subterraneous passages, for experiments that are to be made out of the reach of the sun, especially such as relate to congelations, refrigerations, &c. In this cave there is an old thermometer of M. De la Hire, which stands always at the same height; thereby<pb n="171"/><cb/> shewing that the temperature of the place remains always the same. From the top of the platform to the bottom of the cave is a perpendicular well or pit, used formerly for experiments on the fall of bodies; being also a kind of long telescopical tube, through which the stars are seen at mid-day.</p><p>III. Tycho Brahe's Observatory was in the little island Ween, or the Scarlet Island, between the coasts of Schonen and Zealand, in the Baltic sea. This Observatory was not well situated for some kinds of observations, particularly the risings and settings; as it lay too low, and was landlocked on all the points of the compass except three; and the land horizon being very rugged and uneven.</p><p>IV. Pekin Observatory. Father Le Compte describes a very magnificent Observatory, erected and furnished by the late emperor of China, in his capital, at the intercession of some Jesuit missionaries, chiefly father Verbiest, whom he appointed his chief observer. The instruments here are exceeding large; but the divisions are less accurate, and in some respects the contrivance is less commodious than in those of the Europeans. The chief are, an armillary zodiacal sphere, of 6 Paris feet diameter, an azimuthal horizon 6 feet diameter, a large quadrant 6 feet radius, a sextant 8 feet radius, and a celestial globe 6 feet diameter.</p><p>V. Bramins' Observatory at Benares, in the East Indies, which is still one of the principal seminaries of the Bramins or priests of the original Gentoos of Hindostan. This Observatory at Benares it is said was built about 200 years since, by order of the emperor Ackbar: for as this wise prince endeavoured to improve the arts, so he wished also to recover the sciences of Hindostan, and therefore ordered that three such places should be erected; one at Delhi, another at Agra, and the third at Benares.</p><p>Wanting the use of optical glasses, to magnify very distant or very small objects, these people directed their attention to the increasing the size of their instruments, for obtaining the greater accuracy and number of the divisions and subdivisions in their instruments. Accordingly, the Observatory contains several huge instruments, of stone, very nicely erected and divided, consisting of circles, columns, gnomons, dials, quadrants, &c, some of them of 20 feet radius, the circle divided first into 360 equal parts, and sometimes each of these into 20 other equal parts, each answering to 3′, and of about two-tenths of an inch in extent. And although these wonderful instruments have been built upwards of 200 years, the graduations and divisions on the several arcs appear as well cut, and as accurately divided, as if they had been the performance of a modern artist. The execution, in the construction of these instruments, exhibits an extraordinary mathematical exactness in the fixing, bearing, sitting of the several parts, in the necessary and sufficient supports to the very large stones that compose them, and in the joining and fastening them into each other by means of lead and iron.</p><p>See a farther description, and drawing, of this Observatory, by Sir Robert Barker, in the Philos. Trans. vol. 67, pa. 598.</p><p><hi rend="smallcaps">Observatory</hi> <hi rend="italics">Portable.</hi> See <hi rend="smallcaps">Equatorial.</hi></p><p>OBTUSE <hi rend="italics">Angle,</hi> one that is greater than a right- angle.<cb/></p><p><hi rend="smallcaps">Obtuse</hi>-<hi rend="italics">angled Triangle,</hi> is a triangle that has one of its angles Obtuse: and it can have only one such.</p><p><hi rend="smallcaps">Obtuse</hi> <hi rend="italics">Cone,</hi> or <hi rend="smallcaps">Obtuse</hi>-<hi rend="italics">Angled Cone,</hi> one whose angle at the vertex, by a section through the axis, is Obtuse.</p><p><hi rend="smallcaps">Obtuse</hi> <hi rend="italics">Hyperbola,</hi> one whose asymptotes form an Obtuse angle.</p><p><hi rend="smallcaps">Obtuse</hi>-<hi rend="italics">angular Section of a Cone,</hi> a name given to the hyperbola by the ancient geometricians, because they considered this section only in the Obtuse cone.</p></div1><div1 part="N" n="OCCIDENT" org="uniform" sample="complete" type="entry"><head>OCCIDENT</head><p>, or <hi rend="smallcaps">Occidental</hi>, west, or westward, in Astronomy; a planet is said to be Occident, when it sets after the sun.</p><div2 part="N" n="Occident" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Occident</hi></head><p>, in Geography, the westward quarter of the horizon, or that part of the horizon where the ecliptic, or the sun's place in it, descends into the lower hemisphere.</p><p><hi rend="smallcaps">Occident</hi> <hi rend="italics">Equinoctial,</hi> that point of the horizon where the sun sets, when he crosses the equinoctial, or enters the sign Aries or Libra.</p><p><hi rend="smallcaps">Occident</hi> <hi rend="italics">Estival,</hi> that point of the horizon where the sun sets at his entrance into the sign Cancer, or in our summer when the days are longest.</p><p><hi rend="smallcaps">Occident</hi> <hi rend="italics">Hybernal,</hi> that point of the horizon where the sun sets at midwinter, when entering the sign Capricorn.</p><p><hi rend="smallcaps">Occidental</hi> <hi rend="italics">Horizon.</hi> See <hi rend="smallcaps">Horizon.</hi></p></div2></div1><div1 part="N" n="OCCULT" org="uniform" sample="complete" type="entry"><head>OCCULT</head><p>, in Geometry, is used for a line that is scarce perceivable, drawn with the point of the compasses, or a black-lead pencil. Occult or dry lines, are used in several operations; as the raising of plans, designs of building, pieces of perspective, &c. They are to be effaced or rubbed out when the work is finished.</p></div1><div1 part="N" n="OCCULTATION" org="uniform" sample="complete" type="entry"><head>OCCULTATION</head><p>, the obscuration, or hiding from our sight, any star or planet, by the interposition of the body of the moon, or of some other planet.—The Occultation of a star by the moon, if observed in a place whose latitude and longitude are well determined, may be applied to the correction of the lunar tables; but if observed in a place whose latitude only is well known, may be applied to the determining the longitude of the place.</p><p><hi rend="italics">Circle of Perpetual</hi> <hi rend="smallcaps">Occultation.</hi> See <hi rend="smallcaps">Circle.</hi></p></div1><div1 part="N" n="OCEAN" org="uniform" sample="complete" type="entry"><head>OCEAN</head><p>, the vast collection of salt and navigable water, which encompasses most parts of the earth.</p><p>By computation it appears that the Ocean takes up considerably more of what we know of the terrestrial globe, than the dry land does. This is perhaps easiest known, by taking a good map of the world, and with a pair of scissars clipping out all the water from the land, and weighing the two parts separately: by which means it has been found, that the water occupies about two-thirds of the whole surface of the globe.</p><p>The great and universal Ocean is sometimes, by geographers, divided into three parts. As, 1st, the Atlantic and European Ocean, lying between part of Europe, Africa, and America; 2d, the Indian Ocean, lying between Africa, the East-Indian islands, and New Holland; 3d, the Pacisic Ocean, or great south sea, which lies between the Philippine islands, China, Japan, and New Holland on the west, and the coast of America on the east. The Ocean also takes divers other names, ac-<pb n="172"/><cb/> cording to the different countries it borders upon: as the British Ocean, German Ocean, &c. Also according to the position on the globe; as the northern, southern, eastern, and western Oceans.</p><p>The Ocean, penetrating the land at several streights, quits its name of Ocean, and assumes that of sea or gulph; as the Mediterranean sea, the Persian gulph, &c. In very narrow places, it is called a streight, &c.</p></div1><div1 part="N" n="OCTAEDRON" org="uniform" sample="complete" type="entry"><head>OCTAEDRON</head><p>, or <hi rend="smallcaps">Octahedron</hi>, one of the five regular bodies; contained under 8 equal and equilateral triangles.—It may be conceived as consisting of two quadrilateral pyramids joined together at their bases. <figure/></p><p><hi rend="italics">To form an Octaedron.</hi> Join together 8 equal and equilateral triangles, as in fig. 1; then cut the lines half through, and fold the figure up by these cut lines, till the extreme edges meet, and form the Octaedron, as in figure 2.</p><p>In an Octaedron, if A be the linear edge or side, B its whole surface, C its solidity, or solid content, R the radius of the circumscribed sphere, and <hi rend="italics">r</hi> the radius of the inscribed sphere: Then .</p><p>See my Mensuration, pa. 251 &c, 2d edition.</p></div1><div1 part="N" n="OCTAGON" org="uniform" sample="complete" type="entry"><head>OCTAGON</head><p>, is a figure of 8 sides and angles; which, when these are all equal, is also called a regular one, or may be inscribed in a circle.</p><p>If the side of a regular Octagon be <hi rend="italics">s;</hi> then</p><p>Its area ; and the Radius of its circumsc. circle .</p><div2 part="N" n="Octagon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Octagon</hi></head><p>, in Fortification, denotes a place that has 8 sides, or 8 bastions.</p></div2></div1><div1 part="N" n="OCTANT" org="uniform" sample="complete" type="entry"><head>OCTANT</head><p>, the 8th part of a circle.</p><div2 part="N" n="Octant" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Octant</hi></head><p>, or <hi rend="smallcaps">Octile</hi>, means also an aspect, or po-<cb/> sition of two planets, when their places are distant by the 8th part of a circle, or 45 degrees.</p></div2></div1><div1 part="N" n="OCTAVE" org="uniform" sample="complete" type="entry"><head>OCTAVE</head><p>, or 8th, in Music, is an interval of 8 sounds; every 8th note in the scale of the gamut being the same, as far as the compass of music requires.</p><p>Tones, or sounds, that are Octaves to each other, or at an Octave's distance, are alike, or the same nearly as the unison. In this case, the more acute of the two makes exactly two vibrations while the deeper or graver makes but one; whence, they coincide at every two vibrations of the acuter, which, being more frequent, makes this concord more perfect than any other, and as it were an unison. Hence also, it happens, that two chords or strings, of the same matter, thickness, and tension, but the one double the length of the other, produce th<*> Octave.</p><p>The Octave containing in it all the other simple concords, and the degrees being the differences of these concords; it is evident, that the division of the Octave comprehends the division of all the rest.</p><p>By joining therefore all the simple concords to a common fundamental, we have the following series: 1 : 5/6 : 4/5 : 3/4 : 2/3 : 5/8 : 3/5 : 1/2 Fund. 3d<hi rend="italics">l,</hi> 3d<hi rend="italics">g,</hi> 4th, 5th, 6th<hi rend="italics">l,</hi> 6th<hi rend="italics">g,</hi> 8ve.</p><p>Mr. Malcolm observes, that any wind instrument being over-blown, the sound will rise to an Octave, and no other concord; which he ascribes to the perfection of the Octave, and its being next to unison.</p><p>Des Cartes, from an observation of the like kind, viz, that the sound of a whistle, or organ pipe, will rise to an Octave, if forcibly blown, concludes, that no sound is heard, but its acute Octave seems some way to echo or resound in the ear.</p><p>OCTILE. See <hi rend="smallcaps">Octant.</hi></p></div1><div1 part="N" n="OCTOBER" org="uniform" sample="complete" type="entry"><head>OCTOBER</head><p>, the 8th month of the year, in Romulus's calendar; but the tenth in that of Numa, Julius Cæsar, &c, after the addition of January and February. This month contains 31 days; about the 22d of which, the sun enters the sign Scorpio <*></p><p>OCTOGON. See <hi rend="smallcaps">Octagon.</hi></p></div1><div1 part="N" n="OCTOSTYLE" org="uniform" sample="complete" type="entry"><head>OCTOSTYLE</head><p>, in Architecture, the face of a building adorned with 8 columns.</p></div1><div1 part="N" n="ODD" org="uniform" sample="complete" type="entry"><head>ODD</head><p>, in Arithmetic, is said of a number that is not even. The series of Odd numbers is 1, 3, 5, 7, &c.</p><p>ODDLY-<hi rend="smallcaps">Odd.</hi> A number is said to be Oddly-Odd, when an Odd number measures it by an Odd number. So 15 is a number Oddly-odd, because the Odd number 3 measures it by the Odd number 5.</p></div1><div1 part="N" n="OFFING" org="uniform" sample="complete" type="entry"><head>OFFING</head><p>, or <hi rend="smallcaps">Offin</hi>, in Navigation, that part of the sea which is at a good distance from shore; where there is deep water, and no need of a pilot to conduct the ship into port.</p></div1><div1 part="N" n="OFFSETS" org="uniform" sample="complete" type="entry"><head>OFFSETS</head><p>, in Surveying are the perpendiculars let fall, and measured from the station lines, to the corners or bends in the hedge, fence, or boundary of any ground.</p><p><hi rend="smallcaps">Offset</hi>-<hi rend="italics">Staff,</hi> a slender rod or ftaff, of 10 links, or other convenient length. Its use is for measuring the Offsets, and other short lines and distances.</p></div1><div1 part="N" n="OFFWARD" org="uniform" sample="complete" type="entry"><head>OFFWARD</head><p>, in Navigation, the same with from the shore, &c.<pb n="173"/><cb/></p></div1><div1 part="N" n="OGEE" org="uniform" sample="complete" type="entry"><head>OGEE</head><p>, or OG, an ornamental moulding in the shape of an S; consisting of two members, the one concave and the cther convex.</p></div1><div1 part="N" n="OLDENBURG" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">OLDENBURG</surname> (<foreName full="yes"><hi rend="smallcaps">Henry</hi></foreName>)</persName></head><p>, who wrote his name sometimes <hi rend="smallcaps">Grubendol</hi>, reversing the letters, was a learned German gentleman, and born in the Duchy of Bremen in the Lower Saxony, about the year 1626, being descended from the counts of Aldenburg in Westphalia; whence his name. During the long English parliament in the time of Charles the 1st, he came to England as consul for his countrymen; in which capacity he remained at London in Cromwell's administration. But being discharged of that employment, he was engaged as tutor to thelord Henry Obryan, an Irish nobleman, whom he attended to the university of Oxford; and in 1656 he entered himself a student in that university, chiefly to have the benefit of consulting the Bodleian library. He was afterwards appointed tutor to lord William Cavendish, and became intimately acquainted with Milton the poet. During his residence at Oxford, he became also acquainted with the members of that society there, which gave birth to the Royal Society; and upon the foundation of this latter, he was elected a member of it: and when the Society found it necessary to have two secretaries, he was chosen assistant to Dr. Wilkins. He applied himself with extraordinary diligence to the duties of this office, and began the publication of the Philosophical Transactions with No. 1, in 1664. In order to discharge this task with more credit to himself and the Society, he held a correspondence with more than seventy learned persons, and others, upon a great variety of subjects, in different parts of the world. This fatigue would have been insupportable, had he not, as he told Dr. Lister, managed it so as to make one letter answer another; and that, to be always fresh, he never read a letter before he was ready immediately to answer it: so that the multitude of his letters did not clog him, nor ever lie upon his hands. Among others, he was a constant correspondent of Mr. Robert Boyle, and he translated many of that ingenious gentleman's works into Latin.</p><p>About the year 1674 he was drawn into a dispute with Mr. Hook, who complained, that the secretary had not done him justice, in the History of the Transactions, with respect to the invention of the spiral spring for pocket watches; the contest was carried on with some warmth on both sides, but was at length terminated to the honour of Mr. Oldenburg; for, pursuant to an open representation of the affair to the Royal Society, the council thought fit to declare, in behalf of their secretary, that they knew nothing of Mr. Hook having printed a book intitled <hi rend="italics">Lampas, &c;</hi> but that the publisher of the Transactions had conducted himself faithfully and honestly in managing the intelligence of the Royal Society, and given no just cause for such reflections.</p><p>Mr. Oldenburg continued to publish the Transactions as before, to No. 136, June 25, 1677; after which the publication was discontinued till the January following; when they were again resumed by his successor in the secretary's office, Mr. Nehemiah Grew, who carried them on till the end of February 1678. Mr. Oldenburg died at his house at Charlton, between Greenwich and<cb/> Woolwich, in Kent, August 1678, and was interred there, being 52 years of age.</p><p>He published, besides what hasbeen already mentioned, 20 tracts, chiefly on theological and political subjects; in which he principally aimed at reconciling difserences, and promoting peace.</p></div1><div1 part="N" n="OLYMPIAD" org="uniform" sample="complete" type="entry"><head>OLYMPIAD</head><p>, in Chronology, a revolution or period of four years, by which the Greeks reckoned their time: so called from the Olympic games, which were celebrated every fourth year, during 5 days, near the summer solstice, upon the banks of the river Alpheus, near Olympia, a town of Elis. As each Olympiad consisted of 4 years, these were called the 1st, 2d, 3d, and 4th year of each Olympiad; the first year commencing with the nearest new moon to the summer solstice.</p><p>The first Olympiad began the 3938 year of the Julian period, the 3208 of the creation, 776 years before the birth of Christ, and 24 years before the foundation of Rome. And the computation by these, ended with the 404th Olympiad, being the 440th year of the present vulgar Christian era.</p></div1><div1 part="N" n="OMBROMETER" org="uniform" sample="complete" type="entry"><head>OMBROMETER</head><p>, a name given by Mr. Roger Pickering (Philos. Trans. No. 473, or Abridg. V, 456) to what is more commonly, though less properly, called a Pluviameter or Rain gage. See P<hi rend="smallcaps">LUVIAMETER.</hi></p></div1><div1 part="N" n="OMPHALOPTER" org="uniform" sample="complete" type="entry"><head>OMPHALOPTER</head><p>, or <hi rend="smallcaps">Omphaloptic</hi>, in Optics, a glass that is convex on both sides, popularly called a Convex Lens.</p></div1><div1 part="N" n="OPACITY" org="uniform" sample="complete" type="entry"><head>OPACITY</head><p>, a quality of bodies which renders them opake, or the contrary of transparency.</p><p>The Cartesians make opacity to consist in this; that the pores of the body are not all straight, or directly before each other; or rather not pervious every way.</p><p>This doctrine however is deficient: for though, to have a body transparent, its pores must be straight, or rather open every way; yet it is inconceivable how it should happen, that not only glass and diamonds, but even water, whose parts are so very moveable, should have all their pores open and pervious every way; while the finest paper, or the thinnest gold leaf, should exclude the light, for want of such pores. So that another cause of Opacity must be fought for.</p><p>Now all bodies have vastly more pores or vacuities than are necessary for an infinite number of rays to pass freely through them in right lines, without striking on any of the parts themselves. For since water is 19 times lighter or rarer than gold; and yet gold itself is so very rare, that magnetic effluvia pass freely through it, without any opposition; and quicksilver is readily received within its pores, and even water itself by compression; it must have much more pores than solid parts: consequently water must have at least 40 times as much vacuity as solidity.</p><p>The cause therefore, why some bodies are opake, does not consist in the want of rectilinear pores, pervious every way; but either in the unequal density of the parts, or in the magnitude of the pores; and to their being either empty, or filled with a different matter; by means of which, the rays of light, in their pas-<pb n="174"/><cb/> sage, are arrested by innumerable refractions and reflections, till at length falling on some solid part, they become quite extinct, and are utterly absorbed.</p><p>Hence cork, paper, wood, &c, are opake; while glass, diamonds, &c, are pellucid. For in the consines or joining of parts alike in density, such as those of glass, water, diamonds, &c, among themselves, no refraction or reflection takes place, because of the equal attraction every way; so that such of the rays of light as enter the first surface, pass straight through the body, excepting such as are lost and absorbed, by striking on solid parts: but in the bordering of parts of unequal density, such as those of wood and paper, both with regard to themselves, and with regard to the air or empty space in their larger pores, the attraction being unequal, the reflections and refractions will be very great; and thus the rays will not be able to pass through such bodies, being continually driven about, till they become extinct.</p><p>That this interruption or discontinuity of parts is the chief cause of Opacity, Sir Isaac Newton argues, appears from hence; that all opake bodies immediately begin to be transparent, when their pores become filled with a substance of nearly equal density with their parts. Thus, paper dipped in water or oil, some stones steeped in water, linen cloth dipped in oil or vinegar, &c, become more transparent than before.</p></div1><div1 part="N" n="OPAKE" org="uniform" sample="complete" type="entry"><head>OPAKE</head><p>, not translucent, nor transoarent, or not admitting a free passage to the rays of light.</p><p>OPEN <hi rend="italics">Flank,</hi> in Fortification, is that part of the flank which is covered by the orillon or shoulder.</p><p>OPENING <hi rend="italics">of the Trenches,</hi> is the first breaking of ground by the besiegers, in order to carry on their approaches towards a place.</p><p><hi rend="smallcaps">Opening</hi> <hi rend="italics">of Gates,</hi> in Astrology, is when one planet separates from another, and presently applies to a third, bearing rule in a sign opposite to that ruled by the planet with which it was before joined.</p><p>OPERA-<hi rend="italics">Glass,</hi> in Optics, is so called from its use in play-houses, and sometimes a <hi rend="italics">Diagonal Perspective,</hi> from its construction, which is as follows. ABCD (fig. 5, pl. xvii) represents a tube about 4 inches long; in each side of which there is a hole EF and GH, exactly against the middle of a plane mirror IK, which reflects the rays falling upon it to the convex glass LM; through which they are refracted to the concave eyeglass NO, whence they emerge parallel to the eye at the hole <hi rend="italics">rs,</hi> in the end of the tube. Let P<hi rend="italics">a</hi>Q be an object to be viewed, from which proceed the rays P<hi rend="italics">c, ab,</hi> and Q<hi rend="italics">d:</hi> these rays, being reflected by the plane mirror IK, will shewthe object in the direction <hi rend="italics">cp, ba, dq,</hi> in the image <hi rend="italics">pq,</hi> equal to the object PQ, and as far behind the mirror as the object is before it: the mirror being placed so as to make an angle of 45 degrees with the sides of the tube. And as, in viewing near objects, it is not necessary to magnify them, the focal distances of both the glasses may be nearly equal; or if that of LM be 3 inches, and that of NO on e inch, the distance between them will be but 2 inches, and the object will be magnified 3 times, being sufficient for the purposes to which this glass is applied.<cb/></p><p>When the object is very near, as XY, it is viewed through a hole <hi rend="italics">xy,</hi> at the other end of the tube AB, without an eye glass; the upper part of the mirror being polished for that purpose, as well as the under. The tube unscrews near the object-glass LM, for taking out and cleansing the glasses and mirror. The position of the object will be erect through the concave eye-glass.</p><p>The peculiar artifice of this glass is to view a person at a small distance, so that no one shall know who is observed; for the instrument points to a different object from that which is viewed; and as there is a hole on each side, it is impossible to know on which hand the object is situated, which you are viewing.</p></div1><div1 part="N" n="OPHIUCUS" org="uniform" sample="complete" type="entry"><head>OPHIUCUS</head><p>, a constellation of the northern hemisphere; called also Serpentarius. <figure/></p><p>OPPOSITE <hi rend="italics">Angles,</hi> or Vertical Angles, are those opposite to each other, made by two intersecting lines; as <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> or <hi rend="italics">c</hi> and <hi rend="italics">d.</hi>—The opposite angles are equal to each other.</p><p><hi rend="smallcaps">Opposite</hi> <hi rend="italics">Cones,</hi> denote two similar cones vertically opposite, having the same common vertex and axis, and the same sides produced; as the cones A and B. <figure/></p><p><hi rend="smallcaps">Opposite</hi> <hi rend="italics">Sections,</hi> or <hi rend="italics">Hyperbolas,</hi> are those made by cutting the Opposite cones by the same plane; as the hyperbolas C and D.—These are always equal and similar, and have the same transverse axis EF, as also the same conjugate axis.</p></div1><div1 part="N" n="OPPOSITION" org="uniform" sample="complete" type="entry"><head>OPPOSITION</head><p>, is that aspect or situation of two planets or stars, when they are diametrically opposite to each other; being 180°, or a semi-circle apart; and marked thus <*>.</p><p>The moon is in Opposition to the sun when she is at the full.</p></div1><div1 part="N" n="OPTIC" org="uniform" sample="complete" type="entry"><head>OPTIC</head><p>, or <hi rend="smallcaps">Optical</hi>, something that relates to vision, or the sense of seeing, or the science of optics.</p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Angle.</hi> See <hi rend="smallcaps">Angle.</hi></p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Axis.</hi> See <hi rend="smallcaps">Axis.</hi></p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Chamber.</hi> See <hi rend="smallcaps">Camera</hi> <hi rend="italics">Obscura.</hi></p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Glasses,</hi> are glasses ground either concave or convex; so as either to collect or disperse the rays of light; by which means vision is improved, and the eye strengthened, preserved, &c.</p><p>Among these, the principal are spectacles, reading glasses, telescopes, microscopes, magic lanterns, &c.</p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Inequality,</hi> in Astronomy, is an apparent irregularity in the motions of far distant bodies; so called, because it is not really in the moving bodies, but arising from the situation of the observer's eye. For if the eye were in the centre, it would always see the motions as they really are.<pb n="175"/><cb/> <figure/></p><p>The Optic Inequality may be thus illustrated. Suppose a body revolving with a real uniform motion, in the periphery of a circle ABD &c; and suppose the eye in the plane of the same circle, but at a distance from it, viewing the motion of the body from O. Now when the body goes from A to B; its apparent motion is measured by the angle AOB or the arch or line HL, which it will seem to describe. But while it moves through the arch BD in an equal time, its apparent motion will be determined by the angle BOD, or the arch or line LM, which is less than the former LH. But it spends the same time in describing DE, as it does in AB or BD; during all which time of describing DE it appears stationary in the point M. When it really describes EFGIQ, it will appear to pass over MLHKN; so that it will seem to have gone retrograde. And lastly, from Q to P it will again appear stationary in the point N.</p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Nerves,</hi> the second pair of nerves, springing from the crura of the medulla oblongata, and passing thence to the eye.</p><p>These are covered with two coats, which they take from the dura and pia mater; and which, by their expansions, form the two membranes of the eye, called the uvea and cornea. And the retina, which is a third membrane, and the immediate organ of sight, is only an expansion of the fibrous, or inner, and medullary part of these nerves.</p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Pencil.</hi> See <hi rend="smallcaps">Pencil</hi> <hi rend="italics">of Rays.</hi></p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Place,</hi> of a star &c, is that point or part of its orbit, which is determined by our sight, when the star is seen there. This is either true or apparent; true, when the observer's eye is supposed to be at the centre of the motion; or apparent, when his eye is at the circumference of the earth. See also <hi rend="smallcaps">Place.</hi></p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Pyramid,</hi> in Perspective, is the pyramid ABCO, whose base is the visible object ABC, and the vertex is in the eye at O; being formed by rays <figure/> drawn from the several points of the perimeter to the eye.</p><p>Hence also may appear what is meant by Optic triangle.</p><p><hi rend="smallcaps">Optic</hi> <hi rend="italics">Rays,</hi> particularly means those by which an Optic pyramid, or Optic triangle, is terminated. As OA, OB, OC, &c.<cb/></p></div1><div1 part="N" n="OPTICS" org="uniform" sample="complete" type="entry"><head>OPTICS</head><p>, the science of vision; including Catoptries, and Dioptrics; and even Perspective; as also the whole doctrine of light and colours, and all the phenomena of visible objects.</p><p>Optics, in its more extensive acceptation, is a mixed mathematical science; which explains the manner in which vision is performed in the eye; treats of sight in general; gives the reasons of the several modifications or alterations, which the rays of light undergo in the eye; and shews why objects appear sometimes greater, sometimes smaller, sometimes more distinct, sometimes more confused, sometimes nearer and sometimes more remote. In this extensive signification it is considered by Newton, in his excellent work called Optics.</p><p>Indeed Optics makes a considerable branch of natural philosophy; both as it explains the laws of nature, according to which vision is performed; and as it accounts for abundance of physical phenomena, otherwise inexplicable.</p><p><hi rend="italics">The Principal Authors and Discoveries in Optics,</hi> are the following:</p><p>Euclid seems to be the earliest author on Optics that we have. He composed a treatise on the ancient Optics and catoptrics; dioptrics being less known to the Ancients; though it was not quite unnoticed by them, for among the phenomena, at the beginning of that work, Euclid remarks the effect of bringing an object into view, by refraction, in the bottom of a vessel, by pouring water into it, which could not be seen over the edge of the vessel, before the water was poured in; and other authors speak of the then known effects of glass globes &c, both as burning glasses, and as to bodies seen through them. Euclid's work however is chiefly on catoptrics, or reflected rays; in which he shews, in 31 propositions, the chief properties of them, both in plane, convex, and concave surfaces, in his usual geometrical manner; beginning with that concerning the equality of the angles of incidence and reflection, which he demonstrates; and in the last proposition, shewing the effect of a concave speculum, as a burning glass, when exposed to the rays of the sun.</p><p>The effects of burning glasses, both by refraction and reflection, are noticed by several others of the Ancients, and it is probable that the Romans had a method of lighting their sacred fire by some such means. Aristophanes, in one of his comedies, introduces a person as making use of a globe filled with water to cancel a bond that was against him, by thus melting the wax of the feal. And if we give but a small degree of credit to what some ancient historians are said to have written concerning the exploits of Archimedes, we shall be induced to think that he constructed some very powerful burning mirrors. It is even allowed that this eminent geometrician wrote a treatise on the subject of them, though it be not now extant; as also concerning the appearance of a ring or circle under water, and therefore could not have been ignorant of the common phenomena of refraction. We find many questions concerning such optical appearances in Aristotle. This author was also sensible that it is the reflection of light from the atmosphere which prevents total darkness after the sun sets, and in places where he does not shine in the day time. He was also of opinion, that rainbows,<pb n="176"/><cb/> halos, and mock suns, were all occasioned by the reflection of the sunbeams in different circumstances, by which an imperfect image of his body was produced, the colour only being exhibited, and not his proper figure.</p><p>The Ancients were not only acquainted with the more ordinary appearances of refraction, but knew also the production of colours by refracted light. Seneca fays, that when the light of the sun shines through an angular piece of glass, it shews all the colours of the rainbow. These colours however, he says, are false, such as are seen in a pigeon's neck when it changes its position; and of the same nature he says is a speculum, which, without having any colour of its own, assumes that of any other body.</p><p>It appears also, that the Ancients were not unacquainted with the magnifying power of glass globes filled with water, though it does not appear that they knew any thing of the reason of this power: and it is supposed that the ancient engravers made use of a glass globe filled with water to magnify their figures, that they might work to more advantage.</p><p>Ptolomy, about the middle of the second century, wrote a considerable treatise on Optics. The work is lost; but from the accounts of others, it appears that he there treated of astronomical refractions. The first astronomers were not aware that the intervals between stars appear less when near the horizon than in the meridian; and on this account they must have been much embarrassed in their observations: but it is evident that Ptolomy was aware of this circumstance by the caution which he gives to allow something for it, whenever recourse is had to ancient observations. This philosopher also advances a very sensible hypothesis to account for the remarkably great apparent size of the sun and moon when seen near the horizon. The mind, he says, judges of the size of objects by means of a preconceived idea of their distance from us: and this distance is fancied to be greater when a number of objects are interposed between the eye and the body we are viewing; which is the case when we see the heavenly bodies near the horizon. In his Almagest, however, he ascribes this appearance to a refraction of the rays by vapours, which actually enlarge the angle under which the luminaries appear; just as the angle is enlarged by which an object is seen from under water.</p><p>Alhazen, an Arabian writer, was the next author of consequence, who wrote about the year 1100. Alhazen made many experiments on refraction, at the surface between air and water, air and glass, and water and glass; and hence he deduced several properties of atmospherical refraction; such as, that it increases the altitudes of all objects in the heavens; and he first advanced that the stars are sometimes seen above the horizon by means of refraction, when they are really below it: which observation was confirmed by Vitello, Walther, and especially by the observations of Tycho Brahe. Alhazen observed, that refraction contracts the diameters and distances of the heavenly bodies, and that it is the cause of the twinkling of the stars. This refractive power he ascribed, not to the vapours contained in the air, but to its different degrees of transparency. And it was his opinion, that so far from being the cause of the heavenly bodies appearing larger near the hori-<cb/> zon, that it would make them appear less; observing that two stars appear nearer together in the horizon, than near the meridian. This phenomenon he ranks among optical deceptions. We judge of distance, he says, by comparing the angle under which objects appear, with their supposed distance; so that if these angles be nearly equal, and the distance of one object be conceived greater than that of the other, this will be imagined to be the larger. And he farther observes, that the sky near the horizon is always imagined to be farther from us than any other part of the concave surface.</p><p>In the writings of Alhazen too, we find the first distinct account of the magnifying power of glasses; and it is not improbable that his writings on this head gave rise to the useful invention of spectacles: for he says, that if an object be applied close to the base of the larger segment of a sphere of glass, it will appear magnified. He also treats of the appearance of an object through a globe, and says that he was the first who observed the refraction of rays into it.</p><p>In 1270, Vitello, a native of Poland, published a treatise on Optics, containing all that was valuable in Alhazen, and digested in a better manner. He observes, that light is always lost by refraction, which makes objects appear less luminous. He gave a table of the results of his experiments on the refractive powers of air, water, and glass, corresponding to different angles of incidence. He ascribes the twinkling of the stars to the motion of the air in which the light is refracted; and he illustrates this hypothesis, by observing that they twinkle still more when viewed in water put in motion. He also shews, that refraction is necessary as well as reflection, to form the rainbow; because the body which the rays fall upon is a transparent substance, at the surface of which one part of the light is always reflected, and another refracted. And he makes some ingenious attempts to explain refraction, or to ascertain the law of it. He also considers the foci of glass spheres, and the apparent size of objects seen through them; though with but little accuracy.</p><p>To Vitello may be traced the idea of seeing images in the air. He endeavours to shew, that it is possible, by means of a cylindrical convex speculum, to see the images of objects in the air, out of the speculum, when the objects themselves cannot be seen.</p><p>The Optics of Alhazen and Vitello were published at Basil in 1572, by Fred. Risner.</p><p>Contemporary with Vitello, was Roger Bacon, a man of very extensive genius, who wrote upon almost every branch of science; though it is thought his improvements in Optics were not carried far beyond those of Alhazen and Vitello. He even assents to the absurd notion, held by all philosophers down to his time, that visible rays proceed <hi rend="italics">srom</hi> the eye, instead of <hi rend="italics">towards</hi> it. From many stories related of him however, it would seem, that he made greater improvements than appear in his writings. It is said he had the use of spectacles: that he had contrivances, by reflection from glasses, to see what was doing at a great distance, as in an enemy's camp. And lord chancellor Bacon relates a story, of his having apparently walked in the air between two steeples, and which he supposed was effected<pb n="177"/><cb/> by reflection from glasses while he walked upon the ground.</p><p>About 1279 was written a treatise on Optics by Peccam, archbishop of Canterbury.</p><p>One of the next who distinguished himself in this way, was Maurolycus, teacher of mathematics at Messina. In a treatise, De Lumine et Umbra, published in 1575, he demonstrates, that the crystalline humour of the eye is a lens that collects the rays of light issuing from the objects, and throws them upon the retina, where the focus of each pencil is. From this principle he discovered the reason why some people are shortsighted, and others long-sighted; also why the former are relieved by concave glasses, and the others by convex ones.</p><p>Contemporary with Maurolycus, was John Baptista Porta, of Naples. He discovered the Camera Obscura, which throws considerable light on the nature of vision. His house was the constant resort of all the ingenious persons at Naples, whom he formed into what he called An Academy of Secrets; each member being obliged to contribute something that was not generally known, and might be useful. By this means he was furnished with materials for his Magia Naturalis, which contains his account of the Camera Obscura, and the first edition of which was published, as he informs us, when he was not quite 15 years old. He also gave the first hint of the Magic Lantern; which Kircher afterwards followed and improved. His experiments with the camera obscura convinced him, that vision is performed by the intromission of something into the eye, and not by visual rays proceeding from it, as had been formerly imagined; and he was the first who fully satisfied himself and others upon this subject. He justly considered the eye as a camera obscura, and the pupil the hole in the window-shutter; but he was mistaken in supposing that the crystalline humour corresponds to the wall which receives the images; nor was it discovered till the year 1604, that this office is performed by the retina. He made a variety of just remarks concerning vision; and particularly explained several cases in which we imagine things to be without the eye, when the appearances are occasioned by some affection of the eye itself, or by some motion within the eye. —He remarked also that, in certain circumstances, vision will be assisted by convex or concave glasses; and he seems even to have made some small advances towards the discovery of telescopes.</p><p>Other treatises on Optics, with various and gradual improvements, were afterwards successively published by several authors: as Aguilon, Opticorum libr. 6, Antv. 1613; L'Optique, Catoptrique, & Dioptrique of Herigone, in his Cursus Math. Paris 1637; the Dioptrics of Des Cartes, 1637; L'Optique & Catoptrique of Mersenne, Paris 1651: Scheiner, Optica, Lond. 1652: Manchini, Dioptrica Practica, Bologna, 1660: Barrow, Lectiones Opticæ, London 1663: James Gregory, Optica Promota, Lond. 1663: Grimaldi, Physico-mathesis de Lumine, Coloribus, & Iride, Bononia, 1665: Scaphusa, Cogitationes Physico-mechanicæ de Natura Visionis, Heidel. 1670: Kircher, Ars Magna Lucis & Umbræ, Rome 1671: Cherubin, Dioptrique Oculaire, Paris 1671: Leibnitz, Principe Generale de l'Optique, Leipsic Acts 1682:<cb/> Newton's Optics and Lectiones Opticæ, 4to and 8vo, 1704 &c: Molyneux, Dioptrics, Lond. 1692: Dr. Jurin's Theory of Distinct and Indistinct Vision.— There is also a large and excellent work on Optics, by Dr. Smith, 2 vols 4to; and an elaborate History of the Present State of Discoveries relating to Vision, Light, and Colours, by Dr. Priestley, 4to, 1772; with a multitude of other authors of inferior note; besides lesser and occasional tracts and papers in the Memoirs of the several learned Academies and Societies of Europe; with improvements by many other persons, among whom are the respectable names of Snell, Fermat, Kepler, Huygens, Hortensius, Boyle, Hook, De la Hire, Lowthorp, Cassini, Halley, Delisle, Euler, Dollond, Clairaut, D'Alembert, Zeiher, Bouguer, Buffon, Nollet, Baume; but the particular improvements by each author must be referred to the history of his life, under the article of their names; while the history and improvements of the several branches are to be found under the various particular articles, as, Light, Colours, Reflection, Refraction, Inflection, Transmission, &c, Spectacles, Telescope, Microscope, &c, &c.</p></div1><div1 part="N" n="ORB" org="uniform" sample="complete" type="entry"><head>ORB</head><p>, a spherical shell, hollow sphere, or space contained between two concentric spherical surfaces.— The ancient astronomers conceived the heavens as consisting of several vast azure transparent Orbs or spheres, inclosing one another, and including the bodies of the planets.</p><p>The <hi rend="smallcaps">Orbis</hi> <hi rend="italics">Magnus,</hi> or <hi rend="italics">Great</hi> <hi rend="smallcaps">Orb</hi>, is that in which the sun is supposed to revolve; or rather it is that in which the earth makes its annual circuit.</p><div2 part="N" n="Orb" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Orb</hi></head><p>, in Astrology, or <hi rend="smallcaps">Orb</hi> <hi rend="italics">of Light,</hi> is a certain sphere or extent of light, which the astrologers allow a planet beyond its centre. They pretend that, provided the aspects do but fall within this Orb, they have almost the same effect as if they pointed directly against the centre of the planet.—The Orb of Saturn's light they make to be 10 degrees; that of Jupiter 12 degrees; that of Mars 7 1/2; that of the Sun 17 degrees; that of Venus 8 degrees; that of Mercury 7 degrees; and that of the Moon 12 1/2 degrees.</p></div2></div1><div1 part="N" n="ORBIT" org="uniform" sample="complete" type="entry"><head>ORBIT</head><p>, is the path of a planet or comet; being the curve line described by its centre, in its proper motion in the heavens. So the earth's Orbit, is the ecliptic, or the curve it describes in its annual revolution about the sun.</p><p>The ancient astronomers made the planets describe circular Orbits, with an uniform velocity. Copernicus himself could not believe they should do otherwise; being unable to disentangle himself entirely from the excentrics and epicycles to which they had recourse, to account for the inequalities in their motions.</p><p>But Kepler found, from observations, that the Orbit of the earth, and that of every primary planet, is an ellipsis, having the sun in one of its foci; and that they all move in these ellipses by this law, that a radius drawn from the centre of the sun to the centre of the planet, always describes equal areas in equal times; or, which is the same thing, in unequal times, it describes areas that are proportional to those times. And Newton has since demonstrated, from the nature of universal gravitation, and projectile motion, that the Orbits must of necessity be ellipses, and the motions observe that<pb n="178"/><cb/> law, both of the primary and secondary planets; excepting in so far as their motions and paths are disturbed by their mutual actions upon one another; as the Orbit of the earth by that of the moon; or that of Saturn by the action of Jupiter; &c.</p><p>Of these elliptic Orbits, there have been two kinds assigned: the first that of Kepler and Newton, which is the common or conical ellipse; for which Seth Ward, though he himself keeps to it, thinks we might venture to substitute circular Orbits, by using two points, taken at equal distances from the centre, on one of the diameters, as is done in the foci of the ellipsis, and which is called his Circular Hypothesis. The second is that of Cassini, of this nature, viz, that the products of the two lines drawn from the two foci, to any point in the circumference, are everywhere equal to the same constant quantity; whereas, in the common ellipse, it is the sum of those two lines that is always a constant quantity.</p><p>The Orbits of the planets are not all in the same plane with the ecliptic, which is the earth's Orbit round the sun, but are variously inclined to it, and to each other: but still the plane of the ecliptic, or carth's Orbit, intersects the plane of the Orbit of every other planet, in a right line which passes through the sun, called the line of the nodes, and the points of intersection of the Orbits themselves are called the nodes.</p><p>The mean semidiameters of the several Orbits, or the mean distances of the planets from the sun, with the excentricities of the Orbits, their inclination to the ecliptic, and the places of their nodes, are as in the following table; where the 2d column contains the proportions of semidiameters of the Orbits, the true semidiameter of that of the earth being 95 millions of miles; and the 3d column shews what part of the semidiameters the excentricities are equal to. <table rend="border"><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Propor. semid.</cell><cell cols="1" rows="1" role="data">Excentr. pts. of semidiam.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Inclina. of Orbit.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Ascending Node, 1790.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" rend="align=right" role="data">387</cell><cell cols="1" rows="1" rend="align=center" role="data">4/19</cell><cell cols="1" rows="1" role="data">6°</cell><cell cols="1" rows="1" role="data">54′</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data">14°</cell><cell cols="1" rows="1" role="data">43</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" rend="align=right" role="data">723</cell><cell cols="1" rows="1" rend="align=center" role="data">1/138</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">59</cell></row><row role="data"><cell cols="1" rows="1" role="data">Earth</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" rend="align=center" role="data">1/59</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" rend="align=right" role="data">1524</cell><cell cols="1" rows="1" rend="align=center" role="data">1/1<*></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">17</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" rend="align=right" role="data">5201</cell><cell cols="1" rows="1" rend="align=center" role="data">1/21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">29</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" rend="align=right" role="data">9539</cell><cell cols="1" rows="1" rend="align=center" role="data">1/18</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">13</cell></row><row role="data"><cell cols="1" rows="1" role="data">Georgian</cell><cell cols="1" rows="1" rend="align=right" role="data">19034</cell><cell cols="1" rows="1" rend="align=center" role="data">1/21</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">54</cell></row></table></p><p>The Orbits of the comets are also very excentric ellipses.</p></div1><div1 part="N" n="ORDER" org="uniform" sample="complete" type="entry"><head>ORDER</head><p>, in Architecture, a system of the several members, ornaments, and proportions of a column and pilaster.</p><p>There are five Orders of columns, of which three are Greek, viz, the Doric, Ionic, and Corinthian; and two Italic, viz, the Tuscan and Composite. The three Greek Orders represent the three different man-<cb/> ners of building, viz, the solid, the delicate, and the middling: the two Italic ones are imperfect productions of these.</p><div2 part="N" n="Order" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Order</hi></head><p>, in Astronomy. A planet is said to go according to the order of the signs, when it is direct; proceeding from Aries to Taurus, thence to Gemini, &c. As, on the contrary, it goes contrary to the Order of the signs, when it is retrograde, or goes backward, from Pisees to Aquarius, &c.</p></div2><div2 part="N" n="Order" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Order</hi></head><p>, in the Geometry of Curve Lines, is denominated from the rank or Order of the equation by which the geometrical line is expressed; so the simple equation, or 1st power, denotes the 1st Order of lines, which is the right line; the quadratic equation, or 2d power, defines the 2d Order of lines, which are the conic sections and circle; the cubic equation, or 3d power, defines the 3d Order of lines; and so on.</p><p>Or, the Orders of lines are denominated from the number of points in which they may be cut by a right line. Thus, the right line is of the 1st Order, because it can be cut only in one point by a right line; the circle and conic sections are of the 2d Order, because they can be cut in two points by a right line; while those of the 3d Order, are such as can be cut in 3 points by a right line; and so on.</p><p>It is to be observed, that the Order of curves is always one degree lower than the corresponding line; because the 1st Order, or right line, is no curve; and the circle and conic sections, which are the 2d Order of lines, are only the 1st Order of curves; &c.</p><p>See Newton's Enumeratio Linearum Tertii Ordinis.</p></div2></div1><div1 part="N" n="ORDINATES" org="uniform" sample="complete" type="entry"><head>ORDINATES</head><p>, in the Geometry of Curve Lines, are right lines drawn parallel to each other, and cutting the curve in a certain number of points.</p><p>The parallel Ordinates are usually all cut by some other line, which is called the absciss, and commonly the Ordinates are perpendicular to the abscissal line. When this line is a diameter of the curve, the property of the Ordinates is then the most remarkable; for, in the curves of the first kind, or the conic sections and circle, the Ordinates are all bisected by the diameter, making the part on one side of it equal to the part on the other side of it; and in the curves of the 2d order, which may be cut in three points by an Ordinate, then of the three parts of the Ordinate, lying between these three intersections of the curve and the intersection with the diameter, the part on one fide the diameter is equal to both the two parts on the other side of it. And so for curves of any order, whatever the number of intersections may be, the sum of the parts of any Ordinate, on one side of the diameter, is equal to the sum of the parts on the other side of it.</p><p>The use of Ordinates in a curve, and their abscisses, is to define or express the nature of a curve, by means of the general relation or equation between them; and the greatest number of factors, or the dimensions of the highest term, in such equation, is always the same as the order of the line; that equation being a quadratic, or its highest term of two dimensions, in the lines of the 2d order, being the circle and conic sections; and a cubic equation, or its highest term containing 3 dimensions, in the lines of the 3d order; and so on.<pb n="179"/><cb/> <figure/></p><p>Thus, <hi rend="italics">y</hi> denoting an Ordinate BC, and <hi rend="italics">x</hi> its absciss AB; also <hi rend="italics">a, b, c,</hi> &c, given quantities: then is the general equation for the lines of the 2d order; and is the equation for the lines of the 3d order; and so on.</p></div1><div1 part="N" n="ORDNANCE" org="uniform" sample="complete" type="entry"><head>ORDNANCE</head><p>, are all sorts of great guns, used in war; such as cannon, mortars, howitzers, &c.</p><p>ORFFYREUS's <hi rend="italics">Wheel,</hi> in Mechanics, is a machine so called from its inventor, which he asserted to be a perpetual motion. This machine, according to the account given of it by Gravesande, in his Oeuvres Philosophiques, published by Allemand, Amst. 1774, consisted externally of a large circular wheel, or rather drum, 12 feet in diameter, and 14 inches deep; being very light, as it was formed of an ass<*>mblage of deals, having the intervals between them covered with waxed cloth, to conceal the interior parts of it. The two extremities of an iron axis, on which it turned, rested on two supports. On giving a slight impulse to the wheel, in either direction, its motion was gradually accelerated; so that after two or three revolutions it acquired so great a velocity as to make 25 or 26 turns in a minute. This rapid motion it actually preserved during the space of 2 months, in a chamber of the landgrave of Hesse, the door of which was kept locked, and sealed with the landgrave's own seal. At the end of that time it was stopped, to prevent the wear of the materials. The professor, who had been an eye-witness to these circumstances, examined all the external parts of it, and was convinced that there could not be any communication between it and any neighbouring room. Orffyreus however was so incensed, or pretended to be so, that he broke the machine in pieces, and wrote on the wall, that it was the impertinent curiosity of professor Gravesande which made him take this step. The prince of Hesse, who had seen the interior parts of this wheel, but sworn to secresy, being asked by Gravesande, whether, after it had been in motion for some time, there was any change observable in it, and whether it contained any pieces that indicated fraud or deception, answered both questions in the negative, and declared that the machine was of a very simple construction.</p><p>ORGANICAL <hi rend="italics">Description of Curves,</hi> is the description of them upon a plane, by means of instruments, and commonly by a continued motion. The most simple construction of this kind, is that of a circle by means of a pair of compasses. The next is that of an ellipse by means of a thread and two pins in the foci, or the ellipse and hyperbola, by means of the elliptical and hyperbolic compasses.</p><p>A great variety of descriptions of this sort are to be found in Schooten De Organica Conic. Sect. in Plano Descriptione; in Newton's Arithmetica Universalis, De Curvarum Descriptione Organica; Maclaurin's Geometria Organica; Brackenridge's Descriptio Linearum Curvarum: &c.</p></div1><div1 part="N" n="ORGUES" org="uniform" sample="complete" type="entry"><head>ORGUES</head><p>, or <hi rend="smallcaps">Organs</hi>, in Fortification, long and thick pieces of wood, shod with pointed iron, and<cb/> h<*>ng each by a separate rope over the gate way of a town, ready on any surprise or attempt of the enemy to be let down to stop up the gate. The ends of the several ropes are wound about a windlass, so as to be let down all together.</p><p><hi rend="smallcaps">Orgues</hi> is also used for a machine composed of several harquebusses or musket-barrels, bound together; so as to make several explosions at the same time. They are used to defend breaches and other places attacked.</p></div1><div1 part="N" n="ORIENT" org="uniform" sample="complete" type="entry"><head>ORIENT</head><p>, the east, or the eastern point of the horizon.</p><p><hi rend="smallcaps">Orient</hi> <hi rend="italics">Equinoctial,</hi> is used for that point of the horizon where the sun rises when he is in the equinoctial, or when he enters the signs Aries and Libra.</p><p><hi rend="smallcaps">Orient</hi> <hi rend="italics">A<*>stival,</hi> is the point where the sun rises in the middle of summer, when the days are longest.</p><p><hi rend="smallcaps">Orient</hi> <hi rend="italics">Hybernal,</hi> is the point where the sun rises in the middle of winter, when the days are shortest.</p></div1><div1 part="N" n="ORIENTAL" org="uniform" sample="complete" type="entry"><head>ORIENTAL</head><p>, situated towards the east with regard to us: in opposition to occidental or the west.</p><p><hi rend="smallcaps">Oriental</hi> <hi rend="italics">Astronomy, Philosophy,</hi> &c, used for those of the east, or of the Arabians, Chaldeans, Persians, Indians, &c.</p></div1><div1 part="N" n="ORILLON" org="uniform" sample="complete" type="entry"><head>ORILLON</head><p>, in Fortification, a small rounding of earth, lined with a wall, raised on the shoulder of chose bastions that have casemates, to cover the cannon in the retired slank, and prevent their being dismounted by the enemy.</p><p>There are other sorts of Orillons, properly called Epaulements, or Shoulderings, which are almost of a square figure.</p></div1><div1 part="N" n="ORION" org="uniform" sample="complete" type="entry"><head>ORION</head><p>, a constellation of the southern hemisphere, with respect to the ecliptic, but half in the northern, and half on the southern side of the equinoctial, which runs across the middle of his body.</p><p>The stars in this constellation are, 38 in Ptolomy's catalogue, 42 in Tycho's, 62 in Hevelius's, and 78 in Flamsteed's. But some telescopes have discovered several thousands of stars in this constellation.</p><p>Of these stars, there are no less than two of the first magnitude, and four of the second, beside a great many of the third and fourth. One of those two stars of the first magnitude is upon the middle of the left foot, and is called <hi rend="italics">Regel;</hi> the other is on the right shoulder, and called <hi rend="italics">Betelguese;</hi> of the four of the second magnitude, one is on the left shoulder, and called <hi rend="italics">Bellatrix,</hi> and the other three are in the belt, lying nearly in a right line and at equal distances from each other, forming what is popularly called the <hi rend="italics">Yardwand.</hi></p><p>This constellation is one of the 48 old asterisms, and one of the most remarkable in the heavens. It is in the figure of a man, having a sword by his side, and seems attacking the bull with a club in his right hand, his left bearing a shield.</p><p>This constellation is particularly mentioned by many of the ancient authors, and even in the Scriptures themselves. The Greeks, according to their custom, give several fabulous accounts of him. One is, that th<*>s Orion was a son of their sea-god Neptune by Euryale, the famous huntress. The son possessed the disposition of his mother, and became the greatest hunter in the world: and Neptune gave him the singular privilege, that he should walk upon the surface of the sea as well<pb n="180"/><cb/> as if it were on dry land. Another account of his origin is, that one Hyreius in Thebes, having entertained Jupiter and Mercury with great hospitality, requested of them the favour that he might have a son. The skin of the ox which he had sacrisiced to them, was buried in the ground, with certain ce<*>emonies, and the son so much desired was produced from it, a youth of promising spirit, and named Orion.</p><p>They farther tell us. that he visited Chios when grown up, and ravished Penelope the daughter of Œnopron, for which the father put out his eyes, and banished him the island: he thence went to Lemnos, where Vulcan received him, and gave him Cedalion for a companion. Afterwards, being restored to sight by the sun, he returned to Chios, and would have revenged himself on the king, but the people hid him. After this it seems he hunted with Diana, and was so exalted with his success, that he used to say he would destroy every creature on the earth: the Earth, irritated at this, produced a Scorpion, which stung him to death, and both he and the reptile were taken up to the skies, the Scorpion making one of the twelve signs of the zodiac.</p><p>Others give a different account of his destruction: they tell us that he would have ravished the goddess of chastity Diana herself, and that she killed him with her arrow. All the writers, however, are not agreed about this: they who make him the sacrifice to the vengeance of the offended goddess, say, that herself afterwards placed his figure in the skies as a memorial of the attempt, and a terror to all ages. But there are some who say she loved him so well that she had thoughts of marrying him: these add, that Apollo could not bear so dishonourable an alliance for his sister, for which reason he killed him; and that Diana, after shedding showers of tears over his corps, obtained of Jupiter a place for him in the heavens.</p><p>No constellation was so terrible to the mariners of the early periods, as this of Orion. He is mentioned in this way by all the Greek and Latin poets, and even by their hiftorians; his rising and setting being attended by storms and tempests: and as the northern constellations are made the followers of the Pleiades; so are the fouthern ones made the attendants of Orion.</p><p>The name of this constellation is also met with in Scripture several times, viz, in the books of Job, Amos, and Isaiah. In Job it is asked, “Canst thou bind the sweet influence of the Pleiades, or loose the bands of Orion?” And Amos says, “ Seek him that maketh the Seven Stars and Orion, and turneth the shadow of death into morning.”</p><p><hi rend="smallcaps">Orion's</hi> <hi rend="italics">River,</hi> the same as the constellation Eridanus.</p></div1><div1 part="N" n="ORLE" org="uniform" sample="complete" type="entry"><head>ORLE</head><p>, <hi rend="smallcaps">Orlet</hi>, or <hi rend="smallcaps">Orlo</hi>, in Architecture, a fillet under the ovolo, or quarter-round of a capital — When it is at the top or bottom of the shaft, it is called the cincture.—Palladio also uses Orlo for the plinth of the bases of columus and pedestals.</p></div1><div1 part="N" n="ORRERY" org="uniform" sample="complete" type="entry"><head>ORRERY</head><p>, an astronomical machine, for exhibiting the various motions and appearances of the sun and planets; and hence often called a Planetarium.</p><p>The reason of the name Orrery was this: Mr. Rowley, a mathematical instrument-maker, having got one from Mr. George Graham, the original inventor, to be<cb/> sent abroad with some of his own instruments, he copied it, and made the first for the earl of Orrery, Sir Richard Steel, who knew nothing of Mr. Graham's machine, thinking to do justice to the first encourager, as well as to the inventor of such a curious instrument, called it au Orrery, and gave Rowley the praise due to Mr. Graham. Desaguliers' Experim. Philos. vol. 1, pa. 430. The figure of this grand Orrery is exhibited at fig. 1, pl. 19. It is since made in various other figures.</p></div1><div1 part="N" n="ORTEIL" org="uniform" sample="complete" type="entry"><head>ORTEIL</head><p>, in Fortification. See <hi rend="smallcaps">Berme.</hi></p></div1><div1 part="N" n="ORTELIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ORTELIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Abraham</hi></foreName>)</persName></head><p>, a celebrated geographer, was born at Antwerp, in 1527. He was well skilled in the languages and mathematics, and acquired such reputation by his skill in geography, that he was surnamed the <hi rend="italics">Ptolomy of his time.</hi> Justus Lipsius, and most of the great men of the 16th century, were our author's intimate friends. He passed some time at Oxford in the reign of Edward the 6th; and he visited England a second time in 1577.</p><p>His <hi rend="italics">Theatrum Orbis Terræ</hi> was the completest work of the kind that had ever been published, and gained our author a reputation adequate to his immense labour in compiling it. He wrote also several other excellent geographical works; the principal of which are, his <hi rend="italics">Thesaurus,</hi> and his <hi rend="italics">Synonyma Geographica.</hi>—The world is also obliged to him for the <hi rend="italics">Britannia,</hi> which was undertaken by Cambden at his request.—He died at Antwerp, 1598, at 71 years of age.</p></div1><div1 part="N" n="ORTHODROMICS" org="uniform" sample="complete" type="entry"><head>ORTHODROMICS</head><p>, in Navigation, is Great-circle sailing, or the art of sailing in the arch of a great circle, which is the shortest course: For the arch of a great circle is Orthodromia, or the shortest distance between two points or places.</p></div1><div1 part="N" n="ORTHOGONIAL" org="uniform" sample="complete" type="entry"><head>ORTHOGONIAL</head><p>, in Geometry, is the same as rectangular, or right-angled.—When the term refers to a plane figure, it supposes one leg or side to stand perpendicular to the other: when spoken of solids, it supposes their axis to be perpendicular to the plane of the horizon.</p><p>ORTHOGRAPHIC or <hi rend="smallcaps">Orthographical</hi> <hi rend="italics">Projection of the Sphere,</hi> is the projection of its surface or of the sphere on a plane, passing through the middle of it, by an eye vertically at an infinite distance. See P<hi rend="smallcaps">ROJECTION.</hi></p></div1><div1 part="N" n="ORTHOGRAPHY" org="uniform" sample="complete" type="entry"><head>ORTHOGRAPHY</head><p>, in Geometry, is the drawing or delineating the fore-right plan or side of any object, and of expressing the heights or elevations of every part. Being so called from its determining things by perpendicular right lines falling on the geometrical plan; or rather, because all the horizontal lines are here straight and parallel, and not oblique as in representations of perspective.</p><div2 part="N" n="Orthography" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Orthography</hi></head><p>, in Architecture, is the profile or elevation of a building, shewing all the parts in their true proportion. This is either external or internal.</p><p><hi rend="italics">External</hi> <hi rend="smallcaps">Orthography</hi>, is a delineation of the outer face or front of a building; shewing the principal wall with its apertures, roof, ornaments, and every thing visible to an eye placed before the building. And</p><p><hi rend="italics">Internal</hi> <hi rend="smallcaps">Orthography</hi>, called also a Section, is a delineation or draught of a building, such as it would appear if the external wall were removed.</p></div2><div2 part="N" n="Orthography" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Orthography</hi></head><p>, in Fortification, is the profile, or <pb/><pb/><pb n="181"/><cb/> representation of a work; or a draught so conducted, as that the length, breadth, height, and thickness of the several parts are expressed, such as they would appear, if it were perpendicularly cut from top to bottom.</p></div2></div1><div1 part="N" n="ORTIVE" org="uniform" sample="complete" type="entry"><head>ORTIVE</head><p>, or <hi rend="italics">Eastern Amplitude,</hi> in Astronomy, is an arch of the horizon intercepted between the point where a star rises, and the east point of the horizon.</p></div1><div1 part="N" n="OSCILLATION" org="uniform" sample="complete" type="entry"><head>OSCILLATION</head><p>, in Mechanics, vibration, or the reciprocal ascent and descent of a pendulum. <figure/></p><p>If a simple pendulum be suspended between two semicycloids BC, CD, that have the diameter CF of the generating circle equal to half the length of the string, so that the string, as the body E Oscillates, folds about them, then will the body Oscillate in another cycloid BEAD, similar and equal to the former. And the time of the Oscillation in any arc AE, measured from the lowest point A, is always the same constant quantity, whether that arc be larger or smaller. But the Oscillations in a circle are unequal, those in the smaller arcs being less than those in the larger; and so always less and less as the arcs are smaller, but still greater than the time of Oscillation in a cycloidal arc; till the circular arc becomes very small, and then the time of Oscillation in it is very nearly equal to the time in the cycloid, because the circle and cycloid have the same curvature at the vertex, the length of the string being the common radius of curvature to them there.</p><p>The time of one whole Oscillation in the cycloid, or of an ascent and descent in any arch of it, is to the time in which a heavy body would fall freely through CF or FA, the diameter of the generating circle, or through half the length of the pendulum string, as the circumference of a circle is to its diameter, that is as 3.1416 to 1. So that if <hi rend="italics">l</hi> denote the length of the pendulum CA, and <hi rend="italics">g</hi> = 16 1/12 feet = 193 inches, the space a heavy body falls in the 1st second of time, and <hi rend="italics">p</hi> = 3.1416 the circumference of a circle whose diameter is 1: then by the laws of falling bodies, it is , the time of falling through CF or (1/2)<hi rend="italics">l;</hi> therefore , which is the time of one vibration in any arch of the cycloid which has the diameter of its generating circle equal to (1/2)<hi rend="italics">l.</hi> Or, by extracting the known numbers, the same time of an Oscillation becomes barely (4/25)√<hi rend="italics">l</hi> or (16/100)√<hi rend="italics">l</hi> very nearly, <hi rend="italics">l</hi> being the length of the pendulum in inches. And therefore this is also very nearly the time of an Oscillation in a small circular arc, whose radius is <hi rend="italics">l</hi> inches.</p><p>Hence the times of the Oscillation of pendulums of<cb/> different lengths, are directly in the subduplicate ratio of their lengths, or as the square roots of their lengths.</p><p>The more exact time of Oscillating in a circular arc, when this is of some finite small length, is ; where <hi rend="italics">h</hi> is the height of the vibration, or the versed sine of the single arc of ascent, or descent, to the radius <hi rend="italics">l.</hi></p><p>The celebrated Huygens first resolved the problem concerning the Oscillations of pendulums, in his book De Horologio Oscillatorio, reducing compound pendulums to simple ones. And his doctrine is founded on this hypothesis, that the common centre of gravity of several bodies, connected together, must ascend exactly to the same height from which it fell, whether those bodies be united, or separated from one another in ascending again, provided that each begin to ascend with the velocity acquired by its descent.</p><p>This supposition was opposed by several, and very much suspected by others. And those even who believed the truth of it, yet thought it too daring to be admitted without proof into a science which demonstrates every thing.</p><p>At length Mr. James Bernoulli demonstrated it, from the nature of the lever; and published his solution in the Mem. Acad. of Scienc. of Paris, for the year 1703. After his death, which happened in 1705, his brother John Bernoulli gave a more easy and simple solution of the same problem, in the same Memoirs for 1714; and about the same time, Dr. Brook Taylor published a similar solution in his Methodus Incrementorum: which gave occasion to a dispute between these two mathematicians, who accused each other of having stolen their solutions. The particulars of which dispute may be seen in the Leipsic Acts for 1716, and in Bernoulli's works, printed in 1743.</p><p><hi rend="italics">Axis of</hi> <hi rend="smallcaps">Oscillation</hi>, is a line parallel to the horizon, supposed to pass through the centre or fixed point about which the pendulum oscillates, and perpendicular to the plane in which the Oscillation is made.</p><p><hi rend="italics">Centre of</hi> <hi rend="smallcaps">Oscillation</hi>, in a suspended body, is a certain point in it, such that the Oscillations of the body will be made in the same time as if that point alone were suspended at that distance from the point of suspension. Or it is the point into which if the whole weight of the body be collected, the several Oscillations will be performed in the same time as before: the Oscillations being made only by the force of gravity of the oscillating body. See <hi rend="smallcaps">Centke</hi> <hi rend="italics">of Oscillation.</hi></p></div1><div1 part="N" n="OSCULATION" org="uniform" sample="complete" type="entry"><head>OSCULATION</head><p>, in Geometry, denotes the contact between any curve and its osculatory circle, that is, the circle of the same curvature with the given curve, at the point of contact or of Osculation. If AC be the evolute of the involute curve AEF, and the tangent CE the radius of curvature at the point E, with which, and the centre C, if the circle BEG be described; this circle is said to osculate or kiss the curve AEF in the point E, which point E Mr. Huygens calls the point of Osculation, or kissing point.</p><p>The line CE is called the osculatory radius, or the radius of curvature; and the circle BEG the osculatory or kissing circle.<pb n="182"/><cb/></p><p>The evolute AC is the locus of the centres of all the circles that osculate the involute curve AEF. <figure/></p><p><hi rend="smallcaps">Osculation</hi> also means the point of concourse of two branches of a curve which touch each other. For example, if the equation of a curve be , it is easy to see that the curve has two branches touching one another at the point where <hi rend="italics">x</hi> = 0, because the roots have each the signs + and -.</p><p>The point of Osculation differs from the cusp or point of retrocession (which is also a kind of point of contact of two branches) in this, that in this latter case the two branches terminate, and pass no farther, but in the former the two branches exist on both sides of the point of Osculation. Thus, in the second figure above, the point B is the Osculation of the two branches ABD, EBF; but C, though it is also a tangent point, is a cusp or point of retrocession, of AC and AB, the branches not passing beyond the point A.</p><p>OSCULATORY <hi rend="italics">Circle,</hi> or <hi rend="italics">Kissing Circle,</hi> is the same as the circle of curvature; that is, the circle having the same curvature with any curve at a given point. See the foregoing article, Osculation, where BEG, in the last figure but one, is the Osculatory circle of the curve AEF at the point E; and CE the Osculatory radius, or the radius of curvature.</p><p>This circle is called Osculatory, or kissing, because that, of all the circles that can touch the curve in the same point, that one touches it the closest, in such manner that no other such tangent circle can be drawn between it and the curve; so that, in touching the curve, it embraces it as it were, both touching and cutting it at the same time, being on one side at the convex part of the curve, and on the other at the concave part of it.</p><p>In a circle, all the Osculatory radii are equal, being the common radius of the circle; the evolute of a circle being only a point, which is its centre. See some properties of the Osculatory circle in Maclaurin's Algebra, Appendix De Linearum Geometricarum Proprietatibus generalibus Tractatus, Theor. 2, § 15 &c, treated in a pure geometrical manner.</p><p><hi rend="smallcaps">Osculatory</hi> <hi rend="italics">Parabola.</hi> See <hi rend="smallcaps">Parabola.</hi></p><p><hi rend="smallcaps">Osculatory</hi> <hi rend="italics">Point,</hi> the Osculation, or point of contact between a curve and its Osculatory circle.</p><p>OSTENSIVE <hi rend="italics">Demonstrations,</hi> such as plainly and directly demonstrate the truth of any propofition. In which they stand distinguished from Apagogical ones, or reductions ad absurdum, or ad impossibile, which prove the truth proposed by demonstrating the absurdity or impossibility of the contrary</p></div1><div1 part="N" n="OTACOUSTIC" org="uniform" sample="complete" type="entry"><head>OTACOUSTIC</head><p>, an instrument that aids or improves the sense of hearing. See <hi rend="smallcaps">Acoustics.</hi><cb/></p></div1><div1 part="N" n="OVAL" org="uniform" sample="complete" type="entry"><head>OVAL</head><p>, an oblong curvilinear figure, having two unequal diameters, and bounded by a curve line returning into itself. Or a figure contained by a single curve line, imperfectly round, its length being greater than its breadth, like an egg: whence its name.</p><p>The proper Oval, or egg-shape, is an irregular figure, being narrower at one end than the other; in which it differs from the ellipse, which is the mathematical Oval, and is equally broad at both ends.—The common people confound the two together: but geometricians call the Oval a False Ellipse.</p><p>The method of describing an Oval chiefly used among artificers, is by a cord or string, as FH<hi rend="italics">f,</hi> whose length is equal to the greater diameter of the intended Oval, and which is fastened by its extremes to two points or pins, F and <hi rend="italics">f,</hi> planted in its longer diameter; then, holding it always stretched out as at H, with a pin or pencil carried round the inside, the Oval is described: which will be so much the longer and narrower as the two fixed points are farther apart. This Oval so described is the true mathematical ellipse, the points F and <hi rend="italics">f</hi> being the two foci. <figure/></p><p>Another popular way to describe an Oval of a given length and breadth, is thus: Set the given length and breadth, AB and CD, to bisect each other perpendicularly at E; with the centre C, and radius AE, describe an arc to cross AB in F and G; then with these centres, F and G, and radii AF and BG, describe two little arcs HI and KL for the smaller ends of the Oval; and lastly, with the centres C and D, and radius CD, describe the arcs HK and IL, for the flatter or longer sides of the Oval.— Sometimes other points, instead of C and D, are to be taken by trial, as centres in the line CD, produced if necessary, so as to make the two last arcs join best with the two former ones.</p><p><hi rend="smallcaps">Oval</hi> denotes also certain roundish figures, of various and pleasant shapes, among curve lines of the higher kinds. These figures are expressed by equations of all dimensions above the 2d, and more especially the even dimensions, as the 4th, 6th, &c. Of this kind is the equation , which denotes the <figure/> Oval B, in shape of the section of a pear through the middle, and is easily described by means of poínts. For, if<pb n="183"/><cb/> <*> circle be described whose diameter AC is = <hi rend="italics">a,</hi> and AD be perpendicular and equal to AC; then taking any point P in AC, joining DP, and drawing PN parallel to AD, and NO parallel to AC; and lastly taking PM = NO, the point M will be one point of the Oval sought.</p><p>In like manner the equation expresses several very pretty Ovals, among which the following 12 are some of the most remarkable. For when the equation has four real unequal roots, the given equation will denote the three following species, in fig. 1, 2, 3: <figure/></p><p>When the two less roots are equal, the three species will be expressed as in fig. 4, 5, 6, thus: <figure/></p><p>When the two less roots become imaginary, it will denote the three species as exhibited in fig. 7, 8, 9: <figure/></p><p>When the two middle roots are equal, the species will be as appears in fig. 10: when two roots are equal, and two more so, the species will be as in fig. 11: and when the two middle roots become imaginary, the species will be as appears in fig. 12: <figure/><cb/></p></div1><div1 part="N" n="OUGHTRED" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">OUGHTRED</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an eminent English mathematician and divine, was born at Eton in Buckinghamshire, 1573, and educated in the school there; whence he was elected to King's-college in Cambridge in 1592, where he continued about 12 years, and became a fellow; employing his time in close application to useful studies, particularly the mathematical sciences, which he contributed greatly, by his example and ex hortation, to bring into vogue among his acquaintances there.</p><p>About 1603 he quitted the university, and was presented to the rectory of Aldbury, near Guildford in Surry, where he lived a long retired and studious life, seldom travelling so far as London once a year; his recreation being a diversity of studies: “as often, says he, as I was tired with the labours of my own profession, I have allayed that tediousness by walking in the pleasant, and more than Elysian Fields of the diverse and various parts of human learning, and not of the mathematics only.” About the year 1628 he was appointed by the earl of Arundel tutor to his son lord William Howard, in the mathematics, and his Clavis was drawn up for the use of that young nobleman. He always kept up a correspondence by letters with some of the most eminent scholars of his time, upon mathematical subjects: the originals of which were preserved, and communicated to the Royal Society, by William Jones, Esq. The chief mathematicians of that age owed much of their skill to him; and his house was always full of young gentlemen who came from all parts to receive his instruction: nor was he without invitations to settle in France, Italy, and Holland. “He was as facetious, says Mr. David Lloyd, in Greek and Latin, as solid in arithmetic, geometry, and the sphere, of all measures, music, &c; exact in his style as in his judgment; handling his tube and other instruments at 80 as steadily as others did at 30; owing this, as he said, to temperance and exercise; principling his people with plain and solid truths, as he did the world with great and useful arts; advancing new inventions in all things but religion, which he endeavoured to promote in its primitive purity, maintaining that prudence, meekness, and simplicity were the great ornaments of his life.</p><p>Notwithstanding Oughtred's great merit, being a strong royalist, he was in danger, in 1646, of a sequestration by the committee for plundering ministers; several articles being deposed and sworn against him: but upon his day of hearing, William Lilly, the famous astrologer, applied to Sir Bulstrode Whitlocke and all his old friends; who appeared so numerous in his behalf, that though the chairman and many other Presbyterian members were active against him, yet he was cleared by the majority. This is told us by Lilly himself, in the History of his own Life, where he styles Oughtred the most famous mathematician then of Europe.—He died in 1660, at 86 years of age, and was buried at Aldbury. It is said he died of a sudden ecstasy of joy, about the beginning of May, on hearing the news of the vote at Westminster, which passed for the restoration of Charles the 2d.—He left one son, whom he put apprentice to a watch-maker, and wrote a book of instructions in that art for his use.</p><p>He published several works in his life time; the principal of which are the following:<pb n="184"/><cb/></p><p>1. <hi rend="italics">Arithmeticæ in Numero & Speciebus Institutio,</hi> in 8vo, 1631. This treatise he intended should serve as a general Key to the Mathematics. It was afterwards reprinted, with considerable alterations and additions, in 1648, under the title of <hi rend="italics">A Key to the Mathematics.</hi> It was also published in English, with several additional tracts; viz, one on the Resolution of all sorts of Affected Equations in Numbers; a second on Compound Interest; a third on the easy Art of Delineating all manner of Plain Sun-dials; also a Demonstration of the Rule of False-Position. A 3d edition of the same work was printed in 1652, in Latin, with the same additional tracts, together with some others, viz, On the Use of Logarithms; A Declaration of the 10th book of Euclid's Elements; a treatise of Regular Solids; and the Theorems contained in the books of Archimedes.</p><p>2. <hi rend="italics">The Circles of Proportion,</hi> and a <hi rend="italics">Horizontal Instrument;</hi> in 1633, 4to; published by his scholar Mr. William Foster.</p><p>3. <hi rend="italics">Description and Use of the Double Horizontal Dial;</hi> 1636, 8vo.</p><p>4. <hi rend="italics">Trigonometria:</hi> his treatise on Trigonometry, in Latin, in 4to, 1657: And another edition in English, together with Tables of Sines, Tangents, and Secants.</p><p>He left behind him a great number of papers upon mathematical subjects; and in most of his Greek and Latin mathematical books, there were found notes in his own hand writing, with an abridgment of almost every proposition and demonstration in the margin, which came into the museum of the late William Jones Esq. F. R. S. These books and manuscripts then passed into the hands of his friend Sir Charles Scarborough the physician; the latter of which were carefully looked over, and all that were found fit for the press, printed at Oxford in 1676, in 8vo, under the title of</p><p>5. <hi rend="italics">Opuscula Mathematica hactenus inedita</hi> This collection contains the following pieces: (1), Institutiones Mechanicæ: (2), De Variis Corporum Generibus Gravitate & Magnitudine comparatis: (3), Automata: (4), Quæstiones Diophanti Alexandrini, libri tres: (5), De Triangulis Planis Rectangulis: (6), De Divisione Superficierum: (7), Musicæ Elementa: (8) De Propugnaculorum Munitionibus: (9), Sectiones Angulares.</p><p>6. In 1660, Sir Jonas Moore annexed to his Arithmetic a treatise entitled, “<hi rend="italics">Conical Sections;</hi> or, The several Sections of a Cone; being an Analysis or Methodical Contraction of the two first books of Mydorgius, and whereby the nature of the Parabola, Hyperbola, and Ellipsis, is very clearly laid down. Translated from the papers of the learned William Oughtred.”</p><p>Oughtred, though undoubtedly a very great mathematician, was yet far from having the happiest method of treating the subjects he wrote upon. His style and manner were very concise, obscure, and dry; and his rules and precepts so involved in symbols and abbreviations, as rendered his mathematical writings very troublesome to read, and difficult to be understood. Beside the characters and abbreviations before made use of in Algebra, he introduced several others; as × to denote multiplication; : : for proportion or similitude of ratios;<cb/> <*> for continued proportion; <hi rend="brace"><*> <*> </hi> for greater and less; &c.</p></div1><div1 part="N" n="OUNCE" org="uniform" sample="complete" type="entry"><head>OUNCE</head><p>, a small weight, being the 16th part of a pound avoirdupois; and the 12th part of a pound troy. —The avoirdupois Ounce is divided into 16 drachms or drams; also the Ounce troy into 24 pennyweights, and the pennyweight into 24 grains.</p></div1><div1 part="N" n="OVOLO" org="uniform" sample="complete" type="entry"><head>OVOLO</head><p>, in Architecture, a round moulding; whose profile or sweep, in the Ionic and Composite capital, is usually a quadrant of a circle; whence it is also popularly called the Quarter round.</p><p>OUTWARD <hi rend="italics">Flanking Angle,</hi> or the <hi rend="italics">Angle of the Tenaille,</hi> is that comprehended by the two flanking lines of defence.</p></div1><div1 part="N" n="OUTWORKS" org="uniform" sample="complete" type="entry"><head>OUTWORKS</head><p>, in Fortification, all those works made on the outside of the ditch of a fortified place, to cover and defend it.</p><p>Outworks, called also Advanced and Detached Works, are those which not only serve to cover the body of the place, but also to keep the enemy at a distance, and prevent them from taking advantage of the cavities and elevations usually found in the places about the counterscarp; which might serve them either as lodgments, or as rideaux, to facilitate the carrying on their trenches, and planting their batteries against the place. Such are ravelins, tenailles, hornworks, queue d'arondes, envelopes, and crownworks. Of these, the most usual are ravelins, or halfmoons, formed between the two bastions, on the flanking angle of the counterscarp, and before the curtain, to cover the gates and bridges.</p><p>It is a general rule in all Outworks, that if there be several of them, one before another, to cover one and the same tenaille of a place, the nearer ones must gradually, and one after another, command those which are farthest advanced out into the campagne; that is, must have higher ramparts, that so they may overlook and fire upon the besiegers, when they are masters of the more outward works.</p><p>The gorges also of all Outworks should be plain, and without parapets; lest, when taken, they should serve to secure the besiegers against the fire of the retiring besieged; whence the gorges of Outworks are only pallisadoed, to prevent a surprize.</p><p>OX-EYE, in Optics. See <hi rend="smallcaps">Scioptic</hi>, and C<hi rend="smallcaps">AMERA</hi> <hi rend="italics">Obscura.</hi></p></div1><div1 part="N" n="OXGANG" org="uniform" sample="complete" type="entry"><head>OXGANG</head><p>, or <hi rend="smallcaps">Oxgate</hi>, of land, is usually taken for 15 acres; being as much land as it is supposed one ox can plow in a year. In Lincolnshire they still corruptly call it Oskin of land.—In Scotland, the term is used for a portion of arable land, containing 13 acres.</p></div1><div1 part="N" n="OXYGONE" org="uniform" sample="complete" type="entry"><head>OXYGONE</head><p>, in Geometry, is acute-angled, meaning a figure consisting wholly of acute angles, or such as are less than 90 degrees each.—The term is chiefly applied to triangles, where the three angles are all acute.</p></div1><div1 part="N" n="OXYGONIAL" org="uniform" sample="complete" type="entry"><head>OXYGONIAL</head><p>, is acute-angular.</p></div1><div1 part="N" n="OZANAM" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">OZANAM</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, an eminent French mathematician, was descended from a family of Jewish extraction, but which had long been converts to the Romi<*>h faith; and some of whom had held considerable places in the parliaments of Provence. He was born at Boligneux in Bressia, in the year 1640; and being a younger<pb n="185"/><cb/> son, though his father had a good estate, it was thought proper to breed him to the church, that he might enjoy some small benefices which belonged to the family, to serve as a provision for him. Accordingly he studied divinity four years; but then, on the death of his father, he devoted himself entirely to the mathematics, to which he had always been strongly attached. Some mathematical books, which fell into his hands, first excited his curiosity; and by his extraordinary genius, without the aid of a master, he made so great a progress, that at the age of 15 he wrote a treatise of that kind.</p><p>For a maintenance, he first went to Lyons to teach the mathematics; which answered very well there; and after some time his generous disposition procured him still better success elsewhere. Among his scholars were two foreigners, who expressing their uneasiness to him, at being disappointed of some bills of exchange for a journey to Paris; he asked them how much would do, and being told 50 pistoles, he lent them the money immediately, even without their note for it. Upon their arrival at Paris, mentioning this generous action to M. Daguesseau, father of the chancellor, this magistrate was touched with it; and engaged them to invite Ozanam to Paris, with a promise of his favour. The opportunity was eagerly embraced; and the business of teaching the mathematics here soon brought him in a considerable income: but he wanted prudence for some time to make the best use of it. He was young, handsome, and sprightly; and much addicted both to gaming and gallantry, which continually drained his purse. Among others, he had a love intrigue with a woman, who lodged in the same house with him, and gave herself out for a person of condition. However, this expence in time led him to think of matrimony, and he soon after married a young woman without a fortune. She made amends for this defect however by her modesty, virtue, and sweet temper; so that though the state of his purse was not amended, yet he had more home-felt enjoyment than before, being indeed completely happy in her, as long as she lived. He had twelve children by her, who mostly all died young; and he was lastly rendered quite unhappy by the death of his wife also, which happened in 1701. Neither did this misfortune come single: for the war breaking out about the same time, on account of the Spanish succession, it swept away all his scholars, who, being foreigners, were obliged to leave Paris. Thus he sunk into a very melancholy state; under which however he received some relief, and amusement, from the honour of being admitted this same year an eleve of the Royal Academy of Sciences.</p><p>He seems to have had a pre-sentiment of his death, from some lurking disorder within, of which no outward symptoms appeared. In that persuasion he refused to engage with some foreign noblemen, who offered to become his scholars; alleging that he should not live long enough to carry them through their intended course. Accordingly he was seized soon after with an apoplexy, which terminated his existence in less than two hours, on the 3d of April 1717, at 77 years of age.<cb/></p><p>Ozanam was of a mild and calm disposition, a chearful and pleasant temper, endeared by a generosity almost unparalleled. His manners were irreproachable after marriage; and he was sincerely pious, and zealously devout, though studiously avoiding to meddle in theological questions. He used to say, that it was the business of the Sorbonne to discuss, of the pope to decide, and of a mathematician to go straight to heaven in a perpendicular line. He wrote a great number of useful books; a list of which is as follows:</p><p>1. A treatise of Practical Geometry; 12mo, 1684.</p><p>2. Tables of Sines, Tangents and Secants; with a treatise of Trigonometry; 8vo, 1685.</p><p>3. A treatise of Lines of the First Order; of the Construction of Equations; and of Geometric Lines, &c; 4to, 1687.</p><p>4. The Use of the Compasses of Proportion, &c; with a treatise on the Division of Lands; 8vo, 1688.</p><p>5. An Universal Instrument for readily resolving Geometrical Problems without calculation; 12mo, 1688.</p><p>6. A Mathematical Dictionary; 4to, 1690.</p><p>7. A General Method for drawing Dials, &c; 12mo, 1693.</p><p>8. A Course of Mathematics, in 5 volumes, 8vo, 1693.</p><p>9. A treatise on Fortisication, Ancient and Modern; 4to, 1693.</p><p>10. Mathematical and Philosophical Recreations; 2 vols 8vo, 1694; and again with additions in 4 vols, 1724.</p><p>11. New Treatise on Trigonometry; 12mo, 1699.</p><p>12. Surveying, and measuring all sorts of Artificers Works; 12mo, 1699.</p><p>13. New Elements of Algebra; 2 vols 8vo, 1702.</p><p>14. Theory and Practice of Perspective; 8vo, 1711.</p><p>15. Treatise of Cosmography and Geography; 8vo, 1711.</p><p>16. Euclid's Elements, by De Chales, corrected and enlarged; 12mo, 1709.</p><p>17. Boulanger's Practical Geometry enlarged, &c; 12mo, 1691.</p><p>18. Boulanger's treatise on the Sphere corrected and enlarged; 12mo.</p><p>Ozanam has also the following pieces in the <hi rend="italics">Journal des Sçavans:</hi> viz, (1), Demonstration of this theorem, that neither the Sum nor the Difference of two Fourth Powers, can be a Fourth Power; Journal of May 1680. —(2), Answer to a Problem proposed by M. Comiers; Journal of Nov. 17, 1681.—(3), Demonstration of a Problem concerning False and Imaginary Roots; Journal of April 2 and 9, 1685.—(4), Method of finding in Numbers the Cubic and Sursolid Roots of a Binomial, when it has one; Journal of April 9, 1691.</p><p>Also in the <hi rend="italics">Memoires de Trevoux,</hi> of December 1703, he has this piece, viz, Answer to certain articles of Objection to the first part of his Algebra.</p><p>And lastly, in the Memoirs of the Academy of Sciences, of 1707, he has Observations on a Problem of Spherical Trigonometry.<pb n="186"/><cb/></p></div1></div0><div0 part="N" n="P" org="uniform" sample="complete" type="alphabetic letter"><head>P</head><div1 part="N" n="PAGAN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PAGAN</surname> (<foreName full="yes"><hi rend="smallcaps">Blaise François</hi></foreName> Comte de)</persName></head><p>, an eminent French mathematician and engineer, was born at Avignon in Provence, 1604; and took to the profession of a soldier at 14 years of age. In 1620 he was employed at the siege of Caen, in the battle of Pont de Cé, and the reduction of the Navareins, and the rest of Béarn; where he signalized himself, and acquired a reputation far above his years. He was present, in 1621, at the siege of St. John d'Angeli, as also that of Clarac and Montauban, where he lost an eye by a musket-shot. After this time, there happened neither siege, battle, nor any other occasion, in which he did not signalize himself by some effort of courage and conduct. At the passage of the Alps, and the barricade of Suza, he put himself at the head of the Forlorn Hope, composed of the bravest youths among the guards; and undertook to arrive the first at the attack, by a private way which was extremely dangerous; when, having gained the top of a very steep mountain, he cried out to his followers, “There lies the way to glory!” Upon which, sliding along this mountain, they came first to the attack; when immediately commencing a furious onset, and the army coming to their assistance, they forced the barricades. When the king laid siege to Nancy in 1633, Pagan attended him, in drawing the lines and forts of circumvallation.—In 1642 he was sent to the service in Portugal, as fieldmarshal; and the same year he unfortunately lost the sight of his other eye by a distemper, and thus became totally blind.</p><p>But though he was thus prevented from serving his country with his conduct and courage in the field, he resumed the vigorous study of fortification and the mathematics; and in 1645 he gave the public a treatise on the former subject, which was esteemed the best extant.—In 1651 he published his <hi rend="italics">Geometrical Theorems,</hi> which shewed an extensive and critical knowledge of his subject.—In 1655 he printed a <hi rend="italics">Paraphrase of the Account of the River of Amazons,</hi> by father de Rennes; and, though blind, it is said he drew the chart of the river and the adjacent parts of the country, as in that work.—In 1657 he published <hi rend="italics">The Theory of the Planets,</hi> cleared from that multiplicity of eccentric cycles and epicycles, which the astronomers had invented to explain their motions. This work distinguished him among astronomers as much as that of Fortification had among engineers. And in 1658 he printed his <hi rend="italics">Astronomical Tables,</hi> which are plain and succinct.</p><p>Few great men are without some foible: Pagan's was that of a prejudice in favour of judicial astrology; and though he is more reserved than most others on that head, yet we cannot place what he did on that subject<cb/> among those productions which do honour to his understanding. He was beloved and respected by all persons illustrious for rank as well as science; and his house was the rendezvous of all the polite and learned both in city and court.—He died at Paris, universally regretted, Nov. 18, 1665.</p><p>Pagan had an universal genius; and, having turned his attention chiefly to the art of war, and particularly to the branch of Fortisication, he made extraordinary progress and improvements in it. He understood mathematics not only better than is usual for a gentleman whose view is to push his fortune in the army, but even to a degree of perfection superior to that of the ordinary masters who teach that science. He had so particular a genius for this kind of learning, that he acquired it more readily by meditation than by reading authors upon it; and accordingly he spent less time in such books than he did in those of history and geography. He had also made morality and politics his particular study; so that he may be said to have drawn his own character in his <hi rend="italics">Homme Heroïque,</hi> and to have been one of the completest gentlemen of his time.— Having never married, t<*>at branch of his family, which removed from Naples to France in 1552, became extinct in his person.</p></div1><div1 part="N" n="PALILICUM" org="uniform" sample="complete" type="entry"><head>PALILICUM</head><p>, the same as Aldebaran, a fixed star of the first magnitude, in the eye of the Bull, or sign Taurus.</p></div1><div1 part="N" n="PALISADES" org="uniform" sample="complete" type="entry"><head>PALISADES</head><p>, or <hi rend="smallcaps">Palisadoes</hi>, in Fortification, stakes or small piles driven into the ground, in various situations, as some defence against the surprize of an enemy. They are usually about 6 or 7 inches square, and 9 or 10 feet long, driven about 3 feet into the ground, and 6 inches apart from each other, being braced together by pieces nailed across them near the tops; and secured by thick posts at the distance of every 4 or 5 yards.</p><p><hi rend="smallcaps">Palisades</hi> are placed in the covert-way, parallel to and at 3 feet distance from the parapet or ridge of the glacis, to secure it against a surprize. They are also used to fortify the avenues of open forts, gorges, halfmoons, the bottoms of ditches, the parapets of covertways; and in general all places liable to surprize, and easy of access.</p><p><hi rend="smallcaps">Palisadoes</hi> are usually planted perpendicularly; though some make an angle inclining out towards the enemy, that the ropes cast over them, to tear them up, may slip.</p></div1><div1 part="N" n="PALLADIO" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PALLADIO</surname> (<foreName full="yes"><hi rend="smallcaps">Andrew</hi></foreName>)</persName></head><p>, a celebrated Italian architect in the 16th century, was a native of Vicenza in Lombardy, and the disciple of Triffin, a learned man, who was a Patrician, or Roman nobleman, of the same<pb n="187"/><cb/> town of Vicenza. Palladio was one of those, who laboured particularly to restore the ancient beauties of architecture, and contributed greatly to revive a true taste in that art. Having learned the principles of it, he went to Rome; where, applying himself with great diligence to study the ancient monuments, he entered into the spirit of their architects, and possessed himself with all their beautiful ideas. This enabled him to restore their rules, which had been corrupted by the barbarous Goths. He made exact drawings of the principal works of antiquity which were to be met with at Rome; to which he added <hi rend="italics">Commentaries,</hi> which went through several impressions, with the figures. This, though a very useful work, yet is greatly exceeded by the four books of architecture, which he published in 1570. The last book treats of the Roman temples, and is executed in such a manner, as gives him the preference to all his predecessors upon that subject. It was translated into French by Roland Friatt, and into English by several authors. Inigo Jones wrote some excellent remarks upon it, which were published in an edition of Palladio by Leoni, 1742, in 2 volumes folio.</p></div1><div1 part="N" n="PALLETS" org="uniform" sample="complete" type="entry"><head>PALLETS</head><p>, in Clock and Watch Work, are those pieces or levers which are connected with the pendulum or balance, and receive the immediate impulse of the swing-wheel, or balance-wheel, so as to maintain the vibrations of the pendulum in clocks, and of the balance in watches.—The Pallets in all the ordinary constructions of clocks and watches, are formed on the verge or axis of the pendulum or balance, and are of various lengths and shapes, according to the construction of the piece, or the fancy of the artist.</p></div1><div1 part="N" n="PALLIFICATION" org="uniform" sample="complete" type="entry"><head>PALLIFICATION</head><p>, or <hi rend="smallcaps">Piling</hi>, in Architecture, denotes the piling of the ground-work, or the strengthening it with piles, or timber driven into the ground; which is practised when buildings are erected upon a moist or marshy soil.</p><p>PALLISADES. See <hi rend="smallcaps">Palisades.</hi></p></div1><div1 part="N" n="PALM" org="uniform" sample="complete" type="entry"><head>PALM</head><p>, an ancient long measure, taken from the extent of the hand.</p><p>The Roman Palm was of two kinds: the great Palm, taken from the length of the hand, answered to our span, and contained 12 fingers, digits, or fingers breadths, or 9 Roman inches, equal to about 8 1/<*> English inches. The small Palm, taken from the breadth of the hand, contained 4 digits or fingers, equal to about 3 English inches.</p><p>The Greek Palm, or Doron, was also of two kinds. The small contained 4 fingers, equal to little more than 3 inches. The great Palm contained 5 fingers. The Greek double Palm, called Dichas, contained also in proportion.</p><p>The Modern Palm is different in different places where it is used. It contains, <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Inc.</cell><cell cols="1" rows="1" role="data">Lines</cell></row><row role="data"><cell cols="1" rows="1" role="data">At Rome</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">At Naples, according to Riccioli,</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ditto, according to others,</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" role="data">At Genoa</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell></row><row role="data"><cell cols="1" rows="1" role="data">At Morocco and Fez</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Languedoc, and some other parts of France,</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell></row><row role="data"><cell cols="1" rows="1" role="data">The English Palm is</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell></row></table></p><p>PALM-SUNDAY, the last Sunday in Lent, or<cb/> the Sunday next before Easter Day. So called, from the primitive days, on account of a pious ceremony then in use, of bearing Palms, in memory of the triumphant entry of Jesus Christ into Jerusalem, eight days before the feast of the passover.</p></div1><div1 part="N" n="PAPPUS" org="uniform" sample="complete" type="entry"><head>PAPPUS</head><p>, a very eminent Greek mathematician of Alexandria towards the latter part of the 4th century, particularly mentioned by Suidas, who says he flourished under the emperor Theodcsius the Great, who reigned from the year 379 to 395 of Christ. His writings shew him to have been a consummate mathematician. Many of his works are lost, or at least have not yet been discovered. Suidas mentions several of his works, as also Vossius <hi rend="italics">de Scientiis Mathematicis.</hi> The principal of these are, his <hi rend="italics">Mathematical Collections,</hi> in 8 books, the first and part of the second being lost. He wrote also a <hi rend="italics">Commentary upon Ptolomy's Almagest;</hi> an <hi rend="italics">Universal Chorography; A Description of the Rivers of Libya;</hi> A Treatise of <hi rend="italics">Military Engines; Commentaries upon Aristarchus of Samos, concerning the Magnitude and Distance of the Sun and Moon;</hi> &c. Of these, there have been published, The Mathematical Collections, in a Latin translation, with a large Commentary, by Commandine, in folio, 1588; and a second edition of the same in 1660. In 1644, Mersenne exhibited a kind of abridgment of them in his Synopsis Mathematica, in 4to: but this contains only such propositions as could be understood without figures. In 1655, Meibomius gave some of the Lemmata of the 7th book, in his Dialogue upon Proportions. In 1688, Dr. Wallis printed the last 12 propositions of the 2d book, at the end of his Aristarchus Samius. In 1703, Dr. David Gregory gave part of the preface of the 7th book, in the Prolegomena to his Euclid. And in 1706, Dr. Halley gave that Preface entire, in the beginning of his Apollonius.</p><p>As the contents of the principal work, the Mathematical Collections, are exceedingly curious, and no account of them having ever appeared in English, I shall here give a very brief analysis of those books, extracted from my notes upon this author.</p><p><hi rend="italics">Of the Third Book</hi>—The subjects of the third book consist chiesly of three principal problems; for the solution of which, a great many other problems are resolved, and theorems demonstrated. The first of these three problems is, To find Two Mean Proportionals between two given lines—The 2d problem is, To sind, what are called, three Medi<*>tates in a semicircle; where, by a Medietas is meant a set of three lines in continued proportion, whether arithmetical, or geometrical, or harmonical; so that to find three medietates, is to find an arithmetical, a geometrical, and an harmonical set of three terms each. And the third problem is, From some points in the base of a triangle, to draw two lines to meet in a point within the triangle, so that their sum shall be greater than the sum of the other two sides which are without them. A great many curious properties are premised to each of these problems; then their solutions are given according to the methods of several ancient mathematicians, with an historical account of them, and his own demonstrations; and lastly, their applications to various matters of great importance. In his historical anecdotes, many curious things are preserved concerning mathematicians that were ancient<pb n="188"/><cb/> even in his time, which we should otherwise have known nothing at all about.</p><p>In order to the solution of the first of the three problems above mentioned, he begins by premising four general theorems concerning proportions. Then follows a dissertation on the nature and division of problems by the Ancients, into Plane, Solid, and Linear, with examples of them, taken out of the writings of Eratofthenes, Philo, and Hero. A solution is then given to the problem concerning two mean proportionals, by four different ways, namely according to Eratosthenes, Nicomedes, Hero, and after a way of his own, in which he not only doubles the cube, but also finds another cube in any proportion whatever to a given cube.</p><p>For the solution of the second problem, he lays down very curious definitions and properties of <hi rend="italics">medietates</hi> of all sorts, and shews how to find them all in a great variety of cases, both as to what the Ancients had done in them, and what was done by others whom he calls the Moderns. <hi rend="italics">Medietas</hi> seems to have been a general term invented to express three lines, having either an arithmetical, or a geometrical, or an harmonical relation; for the words proportion (or ratio), and analogy (or similar proportions), are restricted to a geometrical relation only. But he shews how all the medietates may be expressed by analogies.</p><p>The solution of the 3d problem leads Pappus out into the consideration of a number of admirable and seemingly paradoxical problems, concerning the inflecting of lines to a point within triangles, quadrangles, and other figures, the sum of which shall exceed the sum of the surrounding exterior lines.</p><p>Finally, a number of other problems are added, concerning the inscription of all the regular bodies within a sphere. The whole being effected in a very general and pure mathematical way; making all together 58 propositions, viz, 44 problems and 14 theorems.</p><p><hi rend="italics">Of the 4th Book of Pappus.</hi>— In the 4th book are first premised a number of theorems relating to triangles, parallelograms, circles, with lines in and about circles, and the tangencies of various circles: all preparatory to this curious and general problem, viz, relative to an infinite series of circles inscribed in the space, called <foreign xml:lang="greek">arbelon</foreign>, <hi rend="italics">arbelon,</hi> contained between the circumferences of two circles touching inwardly. Where it is shewn, that if the infinite series of circles be inscribed in the manner of this first figure, where three semicircles are described on the lines PR, PQ, QR, and <figure/> the perpendiculars A<hi rend="italics">a</hi> B<hi rend="italics">b,</hi> C<hi rend="italics">c,</hi> &c, let fall from the centres of the series of inscribed circles; then the<cb/> property of these perpendiculars is this, viz, that the first perpendicular A<hi rend="italics">a</hi> is equal to the diameter or double the radius of the circle A; the second perpendicular B<hi rend="italics">b</hi> equal to double the diameter or 4 times the radius of the second circle B; the third perpendicular C<hi rend="italics">c</hi> equal to 3 times the diameter or 6 times the radius of the third circle C; and so on, the series of perpendiculars being to the series of the diameters, as 1, 2, 3, 4, &c, to 1, or to the series of radii, as 2, 4, 6, 8, &c, to 1.</p><p>But if the several small circles be inscribed in the <figure/> manner of this second circle, the first circle of the series touching the part of the line QR; then the series of perpendiculars A<hi rend="italics">a,</hi> B<hi rend="italics">b,</hi> C<hi rend="italics">c,</hi> &c, will be 1, 3, 5, 7, &c, times the radii of the circles A, B, C, D, &c; viz, according to the series of odd numbers; the former proceeding by the series of even numbers.</p><p>He next treats of the Helix, or Spiral, proposed by Conon, and resolved by Archimedes, demonstrating its principal properties: in the demonstration of some of which, he makes use of the same principles as Cavallerius did lately, adding together an infinite number of infinitely short parallelograms and cylinders, which he imagines a triangle and cone to be composed of.—He next treats of the properties of the Conchoid which Nicomedes invented for doubling the cube: applying it to the solution of certain problems concerning Inclinations, with the finding of two mean proportionals, and cubes in any proportion whatever.—Then of the <foreign xml:lang="greek">tetragwnizousa</foreign>, or Quadratrix, so called from its use in squaring the circle, for which purpose it was invented and employed by Dinostratus, Nicomedes, and others: the use of which however he blames, as it requires postulates equally hard to be granted, as the problem itself to be demonstrated by it.—Next he treats of Spirals, described on planes, and on the convex surfaces of various bodies.—From another problem, concerning Inclinations, he there shews, how to trisect a given angle; to describe an hyperbola, to two given asymptotes, and passing through a given point; to divide a given arc or angle in any given ratio; to cut off arcs of equal lengths from unequal circles; to take arcs and angles in any proportion, and arcs equal to right lines; with parabolic and hyperbolic loci, which last is one of the inclinations of Archimedes.</p><p><hi rend="italics">Of the 5th Book of Pappus.</hi>—This book opens with reflections on the different natures of men and brutes, the former acting by reason and demonstration, the latter by instinct, yet some of them with a certain portion of reason or foresight, as bees, in the curious structure of their cells, which he observes are of such<pb n="189"/><cb/> a form as to complete the space quite around a point, and yet require the least materials to build them, to contain the same quantity of honey. He shews that the triangle, square, and hexagon, are the only regular polygons capable of filling the whole space round a point; and remarks that the bees have chosen the fittest of these; proving afterwards, in the propositions, that of all regular figures of the same perimeter, that is of the largest capacity which has the greatest number of sides or angles, and consequently that the circle is the most capacious of all figures whatever.</p><p>And thus he finishes this curious book on Isoperimetrical figures, both plane and solid; in which many curious and important properties are strictly demonstrated, both of planes and solids, some of them being old in his time, and many new ones of his own. In fact, it seems he has here brought together into this book, all the properties relating to isoperimetrical figures then known, and their different degrees of capacity. In the last theorem of the book, he has a dissertation to shew, that there can be no more regular bodies beside the five Platonic ones, or, that only the regular triangles, squares, and pentagons, will form regular solid angles.</p><p><hi rend="italics">Of the 6th Book of Pappus.</hi>—In this book he treats of certain spherical properties, which had been either neglected, or improperly and imperfectly treated by some celebrated authors before his time.—— Such are some things in the 3d book of Theodosius's Spherics, and in his book on Days and Nights, as also some in Euclid's Phenomena. For the sake of these, he premises and intermixes many curious geometrical properties, especially of circles of the sphere, and spherical triangles. He adverts to some curious cases of variable quantities; shewing how some increase and decrease both ways to infinity; while others proceed only one way by increase or decrease, to insinity, and the other way to a certain magnitude; and others again both ways to a certain magnitude, giving a maximum and minimum.—Here are also some curious properties concerning the perspective of the circles of the sphere, and of other lines. Also the locus is determined of all the points from whence a circle may be viewed, so as to appear an ellipse, whose centre is a given point within the circle; which locus is shewn to be a semicircle passing through that point.</p><p><hi rend="italics">Of the 7th Book of Pappus.</hi>—In the introduction to this book, he describes very particularly the nature of the mathematical composition and resolution of the Ancients, distinguishing the particular process and uses of them, in the demonstration of theorems and solution of problems. He then enumerates all the analytical books of the Ancients, or those proceeding by resolution, which he does in the following order, viz, 1st, Euclid's Data, in one book: 2d, Apollonius's Section of a Ratio, 2 books: 3d, his Section of a Space, 2 books: 4th, his Tangencies, 2 books; 5th, Euclid's Porisms, 3 books: 6th, Apollonius's Inclinations, 2 books: 7th, his Plane Loci, 2 books: 8th, his Conics, 8 books: 9th, Aristæus's Solid Loci, 5 books: 10th, Euclid's Loci in Superficies, 2 books; and 11th, Eratosthenes's Medietates, 2 books. So that all the books are 31, the arguments or contents of which he exhibits, with the number of the Loci, determina-<cb/> tions, and cases, &c; with a multitude of lemmas and propositions laid down and demonstrated; the whole making 238 propositions, of the most curious geometrical principles and properties, relating to those books.</p><p><hi rend="italics">Of the 8th Book of Pappus.</hi>—The 8th book is altogether on Mechanics. It opens with a general oration on the subject of mechanics; defining the science, enumerating the different kinds and branches of it, and giving an account of the chief authors and writings on it. After an account of the centre of gravity, upon which the science of mechanics so greatly depends, he shews in the first proposition, that such a point really exists in all bodies. Some of the following propositions are also concerning the properties of the centre of gravity. He next comes to the Inclined Plane, and in prop. 9, shews what power will draw a given weight up a given inclined plane, when the power is given which can draw the weight along a horizontal plane. In the 10th prop. concerning the moving a given weight with a given power, he treats of what the Ancients called a Glossocomum, which is nothing more than a series of Wheels-and-axles, in any proportions, turning each other, till we arrive at the given power. In this proposition, as well as in several other places, he refers to some books that are now lost; as Archimedes on the Balance, and the Mechanics of Hero and of Philo. Then, from prop. 11 to prop. 19, treats on various miseellaneous things, as, the organical construction of solid problems; the diminution of an architectural column; to describe an ellipse through five given points; to find the axes of an ellipse organically; to find also organically, the inclination of one plane to another, the nearest point of a sphere to a plane, the points in a spherical surface cut by lines joining certain points, and to inscribe seven hexagons in a given circle. Prop. 20, 21, 22, 23, teach how to construct and adapt the <hi rend="italics">Tympani,</hi> or wheels of the Glossocomum to one another, shewing the proportions of their diameters, the number of their teeth, &c. And prop. 24 shews how to construct the spiral threads of a screw.</p><p>He comes then to the <hi rend="italics">Five Mechanical Powers, by which a given weight is moved by a given power.</hi> He here proposes briefly to shew what has been said of these powers by Hero and Philo, adding also some things of his own. Their names are, the Axis-in-peritrochio, the Lever, Pulley, Wedge and Screw; and he observes, those authors shewed how they are all reduced to one principle, though their figures be very different. He then treats of each of these powers separately, giving their figures and properties, their construction and uses.</p><p>He next describes the manner of drawing very heavy weights along the ground, by the machine Chelone, which is a kind of sledge placed upon two loose rollers, and drawn forward by any power whatever, a third roller being always laid under the fore part of the Chelone, as one of the other two is quitted and left behind by the motion of the Chelone. In fact this is the same machine as has always been employed upon many occasions in moving very great weights to moderate distances.</p><p>Finally, Pappus describes the manner of raising great weights to a height by the combination of mechanic<pb n="190"/><cb/> powers, as by cranes, and other machines; illustrating this, and the rormer parts, by drawings of the machines that are described.</p></div1><div1 part="N" n="PARABOLA" org="uniform" sample="complete" type="entry"><head>PARABOLA</head><p>, in Geometry, a figure arising from the section of a cone, when cut by a plane parallel to one of its sides, as the section ADE parallel to the side VB of the cone. See <hi rend="smallcaps">Conic</hi> <hi rend="italics">Sections,</hi> where some general properties are given. <figure/> <hi rend="center"><hi rend="italics">Some other Properties of the Parabola.</hi></hi></p><p>1. From the same point of a cone only one Parabola can be drawn; all the other sections between the Parabola and the parallel side of the cone being ellipses, and all without them hyperbolas. Also the Parabola has but one focus, through which the axis AC passes; all the other diameters being parallel to this, and infinite in length also.</p><p>2. The parameter of the axis is a third proportional to any aosciss and its ordinate; viz, AC : CD : : CD : <hi rend="italics">p</hi> the parameter. And therefore if <hi rend="italics">x</hi> denote any absciss AC, and <hi rend="italics">y</hi> the ordinate CD, it will be the parameter; or, by multiplying extremes and means, <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, which is the equation of the Parabola.</p><p>3. The focus F is the point in the axis where the double ordinate GH is equal to the parameter. Therefore, in the equation of the curve <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, taking <hi rend="italics">p</hi> = 2<hi rend="italics">y,</hi> it becomes 2<hi rend="italics">yx</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, or 2<hi rend="italics">x</hi> = <hi rend="italics">y,</hi> that is 2AF = FH, or AF = (1/2)FH, or the focal distance from a vertex AF is equal to half the ordinate there, or = (1/4)<hi rend="italics">p,</hi> one-fourth of the parameter.</p><p>4. The abscisses of a Parabola are to one another, as the squares of their corresponding ordinates. This is evident from the general equation of the curve <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, where, <hi rend="italics">p</hi> being constant, <hi rend="italics">x</hi> is as <hi rend="italics">y</hi><hi rend="sup">2</hi>.</p><p>5. The line FE (<hi rend="italics">fig.</hi> 2 <hi rend="italics">above</hi>) drawn from the focus to any point of the curve, is equal to the sum of the focal distance and the absciss of the ordinate to that point; that is , taking . Or EF is always = EO, drawn parallel to DG, to meet the perpendicular GO, called the Directrix.</p><p>6. If a line TBC cut the curve of a Parabola in two points, and the axis produced in T, and BH and CI be ordinates at those two points; then is AT a mean proportional between the abscisses AH and AI, or AT<hi rend="sup">2</hi> = AH . AI .—And if TE touch the curve, then is AT = AD = the mean between AH and AI.<cb/></p><p>7. If FE be drawn from the focus to the point of contact of the tangent TE, and EK perpendicular to the same tangent; then is FT = FE = FK; and the subnormal DK equal to the constant quantity 2AF or (1/2)<hi rend="italics">p.</hi></p><p>8. The diameter EL being parallel to the axis AK, the perpendicular EK, to the curve or tangent at E, bisects the angle LEF. And therefore all rays of light LE, MN, &c, coming parallel to the axis, will be reflected into the point F, which is therefore called the focus, or burning point; for the angle of incidence LEK is = the angle of reflection KEF.</p><p>9. If IEK (<hi rend="italics">next fig. below</hi>) be any line parallel to the axis, limited by the tangent TC and ordinate CKL to the point of contact; then shall IE : EK : : CK : KL. And the same thing holds true when CL is also in any oblique position. <figure/></p><p>10. The external parts of the parallels IE, TA, ON, PL, &c, are always proportional to the squares of their intercepted parts of the tangent; that is, the external parts IE, TA, ON, PL, are proportional to CI<hi rend="sup">2</hi>, CT<hi rend="sup">2</hi>, CO<hi rend="sup">2</hi>, CP<hi rend="sup">2</hi>, or to the squares CK<hi rend="sup">2</hi>, CD<hi rend="sup">2</hi>, CM<hi rend="sup">2</hi>, CL<hi rend="sup">2</hi>.</p><p>And as this property is common to every position of the tangent, if the lines IE, TA, ON, &c, be appended to the points I, T, O, &c, of the tangent, and moveable about them, and of such lengths as that their extremities E, A, N, &c, be in the curve of a Parabola in any one position of the tangent; then making the tangent revolve about the point C, the extremities E, A, N, &c, will always form the curve of some Parabola, in every position of the tangent.</p><p>The same properties too that have been shewn of the axis, and its abscisses and ordinates, &c, are true of those of any other diameter. All which, besides many other curious properties of the Parabola, may be seen demonstrated in my Treatise on Conic Sections. <hi rend="center">11. <hi rend="italics">To Construct a Parabola by Points.</hi></hi></p><p>In the axis produced take AG = AF <hi rend="italics">(last fig. above)</hi> the focal distance, and draw a number of lines EE, EE, &c, perpendicular to the axis AD; then with the distances GD, GD, &c, as radii, and the centre F, describe arcs crossing the parallel ordinates in E, E, &c. Then with a steady hand, or by the side of a slip of bent whale-bone, draw the curve through all the points E, E, E, &c. <hi rend="center">12. <hi rend="italics">To describe a Parabola by a continued Motion.</hi></hi></p><p>If the rule or the directrix BC be laid upon a plane, <hi rend="italics">(first fig. below)</hi> with the square GDO, in such manner that one of its sides DG lies along the edge of that rule; and if the thread FMO equal in length to DO, the other side of the square, have one end fixed in the extremity of the rule at O, and the other end in some<pb n="191"/><cb/> point F: Then slide the side of the square DG along the rule BC, and at the same time keep the thread continually tight by means of the pin M, with its part MO close to the side of the square DO; so shall the curve AMX, which the pin describes by this motion, be one part of a Parabola.</p><p>And if the square be turned over, and moved on the other side of the fixed point F, the other part of the same Parabola AMZ will be described. <figure/> <hi rend="center"><hi rend="italics">To draw Tangents to the Parabola.</hi></hi></p><p>13. If the point of contact C be given: <hi rend="italics">(last fig. above)</hi> draw the ordinate CB, and produce the axis till AT be = AB; then join TC, which will be the tangent.</p><p>14. Or if the point be given in the axis produced: Take AB = AT, and draw the ordinate BC, which will give C the point of contact; to which draw the line TC as before.</p><p>15. If D be any other point, neither in the curve nor in the axis produced, through which the tangent is to pass: Draw DEG perpendicular to the axis, and take DH a mean proportional between DE and DG, and draw HC parallel to the axis, so shall C be the point of contact, through which and the given point D the tangent DCT is to be drawn.</p><p>16. When the tangent is to make a given angle with the ordinate at the point of contact: Take the absciss AI equal to half the parameter, or to double the focal distance, and draw the ordinate IE: also draw AH to make with AI the angle HAI equal to the given angle; then draw HC parallel to the axis, and it will cut the curve in C the point of contact, where a line drawn to make the given angle with CB will be the tangent required.</p><p>17. <hi rend="italics">To find the Area of a Parabola.</hi> Multiply the base EG by the perpendicular height AI, and 2/3 of the product will be the area of the space AEGA; because the Parabolic space is 2/3 of its circumscribing parallelogram.</p><p>18. <hi rend="italics">To find the Length of the Curve</hi> AC, commencing at the vertex.—Let <hi rend="italics">y</hi> = the ordinate BC, <hi rend="italics">p</hi> = the parameter, , and ; then shall be the length of the curve AC.</p><p>See various other rules for the areas, and lengths of the curve, &c, in my Treatise on Mensuration, sec. 6, pa. 355, &c, 2d edition.</p><p><hi rend="smallcaps">Parabqlas</hi> <hi rend="italics">of the Higher Kinds,</hi> are algebraic curves, desined by the general equation ;<cb/> that is, either , or , or , &c.</p><p>Some call these by the name of Paraboloids: and in particular, if , they call it a Cubical Paraboloid; if , they call it a Biquadratical Paraboloid, or a Sursolid Paraboloid. In respect of these, the Parabola of the First Kind, above explained, they call the Apollonian, or Quadratic Parabola.</p><p>Those curves are also to be referred to Parabolas, that are expressed by the general equation , where the indices of the quantities on each side are equal, as before; and these are called Semi Parabolas: as the Semi Cubical Parabola; or the Semi Biquadratical Parabola; &c.</p><p>They are all comprehended under the moregeneral equation , where the two indices on one side are still equal to the index on the other side of the equation; which include both the former kinds of equations, as well as such as these following ones, , or , or , &c.</p><p><hi rend="italics">Cartesian</hi> <hi rend="smallcaps">Parabola</hi>, is a curve of the 2d order expressed by the equation , containing four infinite legs, viz two hyperbolic ones <figure/> MM and B<hi rend="italics">m,</hi> to the common asymptote AE, tending contrary ways, and two Parabolic legs MN and DN joining them, being Newton's 66th species of lines of the 3d order, and called by him a Trident. It is made use of by Des Cartes in the 3d book of his Geometry, for finding the roots of equations of 6 dimensions, by means of its intersections with a circle. Its most simple equation is . And points through which it is to pass may be easily found by means of a common Parabola whose absciss is , and an hyperbola whose absciss is <hi rend="italics">d</hi>/<hi rend="italics">x;</hi> for <hi rend="italics">y</hi> will be equal to the sum or difference of the corresponding ordinates of this Parabola and hyperbola.</p><p>Des Cartes, in the place abovementioned, shews how to describe this curve by a continued motion. And Mr. Maclaurin does the same thing in a different way, in his Organica Geometria.</p><p><hi rend="italics">Diverging</hi> <hi rend="smallcaps">Parabola</hi>, is a name given by Newton to a species of five different lines of the 3d order, expressed by the equation .<pb n="192"/><cb/></p><p>The first is a bell-form Parabola, with an oval at its head (<hi rend="italics">fig.</hi> 1.); which is the case when the equation , has three real and unequal roots; so that one of the most simple equations of a eurve of this kind is . <figure/></p><p>The 2d is also a bell-form Parabola, with a conjugate point, or infinitely small oval, at the head (<hi rend="italics">fig.</hi> 1.); being the case when the equation has its two less roots equal; the most simple equation of which is .</p><p>The third is a Parabola, with two diverging legs, crossing one another like a knot (<hi rend="italics">fig.</hi> 2.); which happens when the equation has its two greater roots equal; the more simple equation being .</p><p>The fourth a pure bell-form Parabola (<hi rend="italics">fig.</hi> 3.); being the case when has two imaginary roots; and its most simple equation is , or .</p><p>The fifth a Parabola with two diverging legs, forming at their meeting a cusp or double point (<hi rend="italics">fig.</hi> 4); being the case when the equation has three equal roots; so that is the most simple equation of this curve, which indeed is the Semicubical, or Neilian Parabola.</p><p>If a solid generated by the rotation of a semi-cubical Parabola, about its axis, be cut by a plane, each of these five Parabolas will be exhibited by its sections. For, when the cutting plane is oblique to the axis, but falls below it, the section is a diverging Parabola, with an oval at its head. When oblique to the axis, but passes through the vertex, the section is a diverging Parabola, having an infinitely small oval at its head. When the cutting is oblique to the axis, falls below it, and at the same time touches the curve surface of the solid, as well as cuts it, the section is a diverging Parabola, with a nodus or knot. When the cutting plane falls above the vertex, either parallel or oblique to the axis, the section is a pure diverging Parabola. And lastly when the cutting plane passes through the axis, the section is the semi-cubical Parabola from which the solid was generated.</p><p>PARABOLIC <hi rend="italics">Asymptote,</hi> is used for a Parabolic line approaching to a curve, so that they never meet;<cb/> yet by producing both indefinitely, their distance from each other becomes less than any given line.</p><p>There may be as many different kinds of these Asymptotes as there are parabolas of different orders. When a curve has a common parabola for its Asymptote, the ratio of the subtangent to the absciss approaches continually to the ratio of 2 to 1, when the axis of the parabola coincides with the base; but this ratio of the subtangent to the absciss approaches to that of 1 to 2, when the axis is perpendicular to the base. And by observing the limit to which the ratio of the subtangent and absciss approaches, Parabolic Asymptotes of various kinds may be discovered. See Maclaurin's Fluxions, art. 337.</p><p><hi rend="smallcaps">Parabolic</hi> <hi rend="italics">Conoid,</hi> is a solid generated by the rotation of a parabola about its axis.</p><p>This solid is equal to half its circumscribed cylinder; and therefore if the base be multiplied by the height, half the product will be the solid content. <hi rend="center"><hi rend="italics">To find the Curve Surface of a Paraboloid.</hi></hi></p><p>Let BAD be the generating parabola, AC = AT, and BT a tangent at B. <figure/> Put <hi rend="italics">p</hi> = 3.1416, <hi rend="italics">y</hi> = BC, <hi rend="italics">x</hi> = AC = AT, and then is the curve surface = .</p><p>See various other rules and geometrical constructions for the surfaces and solidities of Parabolic Conoids, in my Mensuration, part 3, sect. 6, 2d edition.</p><p><hi rend="smallcaps">Parabolic</hi> <hi rend="italics">Pyramidoid,</hi> is a solid figure thus named by Dr. Wallis, from its genesis, or formation, which is thus: Let all the squares of the ordinates of a parabola be conceived to be so placed, that the axis shall pass perpendicularly through all their centres; then the aggregate of all these planes will form the Parabolic Pyramidoid.</p><p>This figure is equal to half its circumscribed parallclopipedon. And therefore the solid content is found by multiplying the base by the altitude, and taking half the product; or the one of these by half the other.</p><p><hi rend="smallcaps">Parabolic</hi> <hi rend="italics">Space,</hi> is the space or area included by the curve line and base or double ordinate of the parabola. The area of this space, it has been shewn under the article Parabola, is 2/3 of its circumscribed parallelogram; which is its quadrature, and which was first found out by Archimedes, though some say by Pythagoras.</p><p><hi rend="smallcaps">Parabolic</hi> <hi rend="italics">Spindle,</hi> is a solid figure conceived to be formed by the rotation of a parabola about its base or double ordinate.</p><p>This solid is equal to 8/<*> of its circumscribed cylinder. See my Mensuration, prob. 15, pa. 390, &c, 2d edition.</p><p><hi rend="smallcaps">Parabolic</hi> <hi rend="italics">Spiral.</hi> See <hi rend="smallcaps">Helicoid</hi> <hi rend="italics">Parabola.</hi></p><p><hi rend="smallcaps">Paraboliform</hi> <hi rend="italics">Curves,</hi> a name sometimes given to the parabolas of the higher orders.</p></div1><div1 part="N" n="PARABOLOIDES" org="uniform" sample="complete" type="entry"><head>PARABOLOIDES</head><p>, Parabolas of the higher orders.——The equation for all curves of this kind being , the proportion of the area of any one to the complement of it to the circumscribing parallelogram, will be as <hi rend="italics">m</hi> to <hi rend="italics">n.</hi><pb n="193"/><cb/></p><p>PARACENTRIC <hi rend="italics">Motion,</hi> denotes the space by which a revolving planet approaches nearer to, or recedes farther from, the sun, or centre of attraction.</p><p>Thus, if a planet in A move towards B; then is the Paracentric motion of that planet: where S is the place of the sun. <figure/></p><p><hi rend="smallcaps">Paracentric</hi> <hi rend="italics">Solicitation of Gravity,</hi> is the same as the Vis Centripeta; and is expressed by the line AL drawn from the point A, parallel to the ray SB (infinitely near SA), till it intersect the tangent BL.</p><p>PARALLACTIC <hi rend="italics">Angle,</hi> called also simply P<hi rend="smallcaps">ARALLAX</hi>, is the angle EST (<hi rend="italics">last fig. above</hi>) made at the centre of a star, &c, by two lines, drawn, the one from the centre of the earth at T, and the other from its surface at E.—Or, which amounts to the same thing, the Parallactic angle, is the difference of the two angles CEA and BTA, under which the real and apparent distances from the zenith are seen.</p><p>The sines of the Parallactic angles ELT, EST, at the same or equal distances DS from the zenith, are in the reciprocal ratio of the distances, TL, and TS, from the centre of the earth.</p></div1><div1 part="N" n="PARALLAX" org="uniform" sample="complete" type="entry"><head>PARALLAX</head><p>, is an arch of the heavens intercepted between the true place of a star, and its apparent place.</p><p>The true place of a star S, is that point of the heavens B, in which it would be seen by an eye placed in the centre of the earth at T. And the apparent place, is that point of the heavens C, where a star appears to an eye upon the surface of the earth at E.</p><p>This difference of places, is what is called absolutely the Parallax, or the Parallax of Altitude; which Copernicus calls the Commutation; and which therefore is an angle formed by two visual rays, drawn, the one from the centre, the other from the circumference of the earth, and traversing the body of the star; being measured by an arch of a great circle intercepted between the two points of true and apparent place, B and C.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">of Altitude</hi> CB is properly the difference between the true distance from the zenith AB, and the apparent distance AC. Hence the Parallax diminishes the altitude of a star, or increases its distance from the zenith; and it has therefore a contrary effect to the refraction.</p><p>The Parallax is greatest in the horizon, called the Horizontal Parallax EFT. From hence it decreases all the way to the zenith D or A, where it is nothing; the real and apparent places there coinciding.<cb/></p><p>The Horizontal Parallax is the same, whether the star be in the true or apparent horizon.</p><p>The fixed stars have no sensible Parallax, by reason of their immense distance, to which the semidiameter of the earth is but a mere point.</p><p>Hence also, the nearer a star is to the earth, the greater is its Parallax; and on the contrary, the farther it is off, the less is the Parallax, at an equal elevation above the horizon. So the star at S has a less Parallax than the star at I. Saturn is so high, that it is difficult to observe in him any Parallax at all.</p><p>Parallax increases the right and oblique ascension, and diminishes the descension; it diminishes the northern declination and latitude in the eastern part, and increases them in the western; but it increases the southern declination in the eastern and western part; it diminishes the longitude in the western part, and increases it in the eastern. Parallax therefore has just opposite effects to refraction.</p><p>The doctrine of Parallaxes is of the greatest importance, in astronomy, for determining the distances of the planets, comets, and other phenomena of the heavens; for the calculation of eclipses, and for finding the longitude. <figure/></p><p><hi rend="smallcaps">Parallax</hi> <hi rend="italics">of Right Ascension and Descension,</hi> is an arch of the Equinoctial D<hi rend="italics">d,</hi> by which the Parallax of altitude increases the ascension, and diminishes the descension.</p><p><hi rend="smallcaps">Parallax</hi> <hi rend="italics">of Declination,</hi> is an arch of a circle of declination <hi rend="italics">s</hi>I, by which the Parallax of altitude increases or diminishes the declination of a star.</p><p><hi rend="smallcaps">Parallax</hi> <hi rend="italics">of Latitude,</hi> is an arch of a circle of latitude SI, by which the Parallax of altitude increases or diminishes the latitude.</p><p><hi rend="italics">Menstrual</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">of the Sun,</hi> is an angle formed by two right lines; one drawn from the earth to the sun, and another from the sun to the moon, at either of their quadratures.</p><p><hi rend="smallcaps">Parallax</hi> <hi rend="italics">of the Annual Orbit of the Earth,</hi> is the difference between the heliocentric and geocentric place of a planet, or the angle at any planet, subtended by the distance between the earth and sun.</p><p>There are various methods for finding the Parallaxes of the celestial bodies: some of the principal and easier of which are as follow:</p><p><hi rend="italics">To Observe the</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">of a Celestial Body.</hi>—Observe when the body is in the same vertical with a fixed star which is near it, and in that position measure its<pb n="194"/><cb/> apparent distance from the star. Observe again when the body and star are at equal altitudes from the horizon; and there measure their distance again. Then the difference of these distances will be the Parallax very nearly.</p><p><hi rend="italics">To Observe the Moon's</hi> <hi rend="smallcaps">Parallax.</hi>—Observe very accurately the moon's meridian altitude, and note the mo nent of time. To this time, equated, compute her true latitude and longitude, and from these find her declination; also from her declination, and the elevation of the equator, find her true meridian altitude. Subtract the refraction from the observed altitude: then the difference between the remainder and the true altitude, will be the Parallax sought. If the observed altitude be not meridional, reduce it to the true altitude for the time of observation.</p><p>By this means, in 1583, Oct. 12 day 5 h. 19 m. from the moon's meridian altitude observed at 13° 38′, Tycho found her Parallax to be 54 minutes.</p><p><hi rend="italics">To Observe the Moon's</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">in an Eclipse.</hi>—In an eclipse of the moon observe when both horns are in the same vertical circle, and at that moment take the altitudes of both horns; then half their sum will be nearly the apparent altitude of the moon's centre; from which subtract the refraction, which gives the apparent altitude freed from refraction. But the true altitude is nearly equal to the altitude of the centre of the shadow at that time: now the altitude of the centre of the shadow is known, because we know the sun's place in the ecliptic, and his depression below the horizon, which is equal to the altitude of the opposite point of the ecliptic, in which the centre of the shadow is. Having thus the true and apparent altitudes, their difference is the Parallax sought.</p><p>De la Hire makes the greatest horizontal Parallax 1° 1′ 25″, and the least 54′ 5″. M. le Monnier determined the mean Parallax of the Moon to be 57′ 12″. Others have made it 57′ 18″.</p><p><hi rend="italics">From the Moon's</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">EST, and alitude SF</hi> (last fig. but one); <hi rend="italics">to find her distance from the Earth.</hi> —From her apparent altitude given, there is given her apparent zenith distance, i. e. the angle AES; or by her true altitude, the complement angle ATS. Wherefore, since at the same time, the Parallactic angle S is known, the 3d or supplemental angle TES is also known. Then, considering the earth's semidiameter TE as 1, in the triangle TES are given all the angles and the side TE, to find ES the moon's distance from the surface of the earth, or TS her distance from the centre.</p><p>Thus Tycho, by the observation above mentioned, found the moon's distance at that time from the earth, was 62 of the earth's semidiameters. According to De la Hire's determination, her distance when in the perigee is near 56 semidiameters, but in her apogee near 63 1/2; and therefore the mean nearly 59<hi rend="sup">3</hi>, or in round numbers 60 semidiameters.</p><p>Hence also, since, from the moon's theory, there is given the ratio of her distances from the earth in the several degrees of her anomaly; those distances being found, by the rule of three, in semidiameters of the earth, the Parallax is thence determined to the several degrees of the true anomaly.</p><p><hi rend="italics">To Observe the</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">of Mars.</hi>—1. Suppose<cb/> Mars in the meridian and equator at H; and that the observer, under the equator in A, observes him culminating with some fixed star. 2. If now the observer were in the centre of the earth, he would see Mars constantly in the same point of the heavens with the star; and therefore, together with it, in the plane of the horizon, or of the 6th horary: but since Mars here has some sensible Parallax, and the fixed star has none, Mars will be seen in the horizon, when in P, the plane of the sensible horizon; and the star, when in R, the plane of the true horizon: therefore observe the time between the transit of Mars and of the star through the plane of the 6th hour.—3. Convert this time into minutes of the equator, at the rate of 15 degrees to the hour; by which means there will be obtained the arch PM, to which the angle PAM, and consequently the angle AMD, is nearly equal; which is the horizontal Parallax of Mars. <figure/></p><p>If the observer be not under the equator, but in a parallel IQ, that difference will be a less arch QM: wherefore, since the small arches QM and PM are nearly as their sines AD and ID; and since ADG is equal to the distance of the place from the equator, i. e. to the elevation of the pole, or the latitude; therefore AD to ID, as radius to the cosine of the latitude; say, as the cosine of the latitude ID is to radius, so is the Parallax observed in I, to the Parallax under the equator.</p><p>Since Mars and the fixed star cannot be commodiously observed in the horizon; let them be observed in the circle of the 3d hour: and since the Parallax observed there TO, is to the horizontal one PM, as IS to ID: say, as the sine of the angle IDS, or 45° (since the plane DO is in the middle between the meridian DH and the true horizon DM), is to radius, so is the Parallax TO to the horizontal Parallax PM.</p><p>If Mars be likewise out of the plane of the equator, the Parallax found will be an arch of a parallel; which must therefore be reduced, as above, to an arch of the equator.</p><p>Lastly, if Mars be not stationary, but either direct or retrograde, by observations for several days find out what his motion is every hour, that his true place from the centre may be assigned for any given time.</p><p>By this method Cassini, who was the author of it, observed the greatest horizontal Parallax of Mars to be 25″; but Mr. Flamsteed found it near 30″. Cassini observed also the Parallax of Venus by the same method.</p><p><hi rend="italics">To Find the Sun's</hi> <hi rend="smallcaps">Parallax.</hi>—The great distance of the sun renders his Parallax too small to fall under even the nicest immediate observation. Many attempts have indeed been made, both by the ancients and moderns, and many methods invented for that purpose. The first was that of Hipparchus, which was followed by Ptolomy, &c, and was founded on the observation of lunar<pb n="195"/><cb/> eclipses. The second was that of Aristarchus, in which the angle subtended by the semidiameter of the moon's orbit, seen from the sun, was sought from the lunar phases. But these both proving desicient, astronomers are now forced to have recourse to the Parallaxes of the nearer planets, Mars and Venus. Now from the theory of the motions of the earth and planets, there is known at any time the proportion of the distances of the sun and planets from us; and the horizontal Parallaxes being reciprocally proportional to those distances; by knowing the Parallax of a planet, that of the sun may be thence found.</p><p>Thus Mars, when opposite to the sun, is twice as near as the sun is, and therefore his Parallax will be twice as great as that of the sun. And Venus, when in her inferior conjunction with the sun, is sometimes nearer us than he is; and therefore her Parallax is greater in the same proportion. Thus, from the Parallaxes of Mars and Venus, Cassini found the sun's Parallax to be 10″; from whence his distance comes out 22000 semidiameters of the earth.</p><p>But the most accurate method of determining the Parallaxes of these planets, and thence the Parallax of the sun, is that of observing their transit. However, Mercury, though frequently to be seen on the sun, is not fit for this purpose; because he is so near the sun, that the difference of their Parallaxes is always less than the solar Parallax required. But the Parallax of Venus, being almost 4 times as great as the solar Parallax, will cause very sensible differences between the times in which she will seem to be passing over the sun at different parts of the earth. With the view of engaging the attention of astronomers to this method of determining the sun's Parallax, Dr. Halley communicated to the Royal Society, in 1691, a paper, containing an account of the several years in which such a transit may happen, computed from the tables which were then in use: those at the ascending node occur in the month of November O. S. in the years 918, 1161, 1396, 1631, 1639, 1874, 2109, 2117; and at the descending node in May O. S. in the years 1048, 1283, 1291, 1518, 1526, 1761, 1769, 1996, 2004. Philos. Trans. Abr. vol. 1, p. 435 &c.</p><p>Dr. Halley even then concluded, that if the interval of time between the two interior contacts of Venus with the sun, could be measured to the exactness of a second, in two places properly situated, the sun's Parallax might be determined within its 500dth part. And this conclusion was more fully explained in a subsequent paper, concerning the transit of Venus in the year 1761, in the Philos. Trans. numb. 348, or Abr. vol. 4, p. 213.</p><p>It does not appear that any of the preceding transits had been observed; except that of 1639, by our ingenious countryman Mr. Horrox, and his friend Mr. Crabtree, of Manchester. But Mr. Horrox died on the 3d of January, 1641, at the age of 25, just after he had finished his treatise, <hi rend="italics">Venus in Sole visa,</hi> in which he discovers a more accurate knowledge of the dimensions of the solar system, than his learned commentator Hevelius.</p><p>To give a general idea of this method of determining the horizontal Parallax of Venus, and from thence,<cb/> by analogy, the Parallax and distance of the sun, and of all the planets from him; let DBA be the earth, V Venus, and TSR the eastern limb of the sun. To an observer at B, the <figure/> point <hi rend="italics">t</hi> of that limb will be on the meridian, its place referred to the heavens will be at E, and Venus will appear just within it at S. But to an observer at A, at the same instant, Venus is east of the sun, in the right line AVF; the point <hi rend="italics">t</hi> of the sun's limb appears at <hi rend="italics">e</hi> in the heavens, and if Venus were then visible she would appear at F. The angle CVA is the horizontal Parallax of Venus; which is equal to the opposite angle FVE, measured by the arc FE. ASC is the sun's horizontal Parallax, equal to the opposite angle <hi rend="italics">e</hi>SE, measured by the arc <hi rend="italics">e</hi>E; and FA<hi rend="italics">e</hi> or VA<hi rend="italics">e</hi> is Venus's horizontal Parallax from the sun, which may be found by observing how much later in absolute time her total ingress on the sun is, as seen from A, than as seen from B, which is the time she takes to move from V to <hi rend="italics">v,</hi> in her orbit OV<hi rend="italics">v.</hi></p><p>If Venus were nearer the earth, as at U, her horizontal Parallax from the sun would be the arch <hi rend="italics">fe,</hi> which measures the angle <hi rend="italics">f</hi>A<hi rend="italics">e;</hi> and this angle is greater than the angle FA<hi rend="italics">e,</hi> by the difference of their measures F<hi rend="italics">f.</hi> So that as the distance of the celestial object from the earth is less, its Parallax is the greater.</p><p>Now it has been already observed, that the horizontal Parallaxes of the planets are inversely as their distances from the earth's centre, therefore as the sun's distance at the time of the transit is to Venus's distance, so is the Parallax of Venus to that of the sun: and as the sun's mean distance from the earth's centre, is to his distance on the day of the transit, so is his horizontal Parallax on that day, to his horizontal Parallax at the time of his mean distance from the earth's centre. Hence his true distance in semidiameters of the earth may be obtained by the following analogy, viz, as the sine of the sun's Parallax is to radius, so is unity or the earth's semidiameter, to the number of semidiameters of the earth in the sun's distance from the centre; which number multiplied by the number of<pb n="196"/><cb/> miles in the earth's semidiameter, will give the number of miles in the sun's distance. Then from the proportional distances of the planets, determined by the theory of gravity, their true distances may be found. And from their apparent diameters at these known distances, their real diameters and bulks may be found.</p><p>Mr. Short, with great labour, deduced the quantity of the sun's Parallax from the best observations that were made of the transit of Venus, on the 6th of June, 1761, (for which see Philos. Trans. vol. 51 and 52) both in Britain and in foreign parts, and found it to have been 8″.52 on the day of the transit, when the sun was very nearly at his greatest distance from the earth; and consequently 8″.65 when the sun is at his mean distance from the earth. See Philos. Trans. vol. 52, p. 611 &c. Whence, <table><row role="data"><cell cols="1" rows="1" role="data">As sin. 8″.65</cell><cell cols="1" rows="1" rend="align=right" role="data">log.  5.6219140</cell></row><row role="data"><cell cols="1" rows="1" role="data">to radius</cell><cell cols="1" rows="1" rend="align=right" role="data">10.0000000</cell></row><row role="data"><cell cols="1" rows="1" role="data">So is 1 semidiameter</cell><cell cols="1" rows="1" rend="align=right" role="data">0.0000000</cell></row><row role="data"><cell cols="1" rows="1" role="data">to 23882.84 semidiameters</cell><cell cols="1" rows="1" rend="align=right" role="data">4.3780860</cell></row></table> that is, 23882 84/100 is the number of the earth's semidiameters contained in its distance from the sun; and this number of semidiameters being multiplied by 3985, the number of English miles contained in the earth's semidiameter, (though later observations make this semidiameter only 3956 1/2 miles), there is obtained 95,173,127 miles for the earth's mean distance from the sun. And hence, from the analogies under the article <hi rend="smallcaps">Distance</hi>, the mean distances of all the rest of the planets from the sun, in miles, are found as follow, viz, <table><row role="data"><cell cols="1" rows="1" role="data">Mercury's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">36,841,468</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">68,891,486</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">145,014,148</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">494,990,976</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">907,956,130.</cell></row></table></p><p>In another paper (Philos. Trans. vol. 53, p. 169) Mr. Short states the mean horizontal Parallax of the sun at 8″.69. And Mr. Hornsby, from several observations of the transit of June 3d, 1769 (for which see the Philos. Trans. vol. 59) deduces the sun's Parallax for that day equal to 8.65, and the mean Parallax 8″.78; whence he makes the mean distance of the earth from the sun to be 93,726,900 English miles, and the distances of the other planets thus: <table><row role="data"><cell cols="1" rows="1" role="data">Mercury's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">36,281,700</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">67,795,500</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">142,818,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">487,472,000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn's distance</cell><cell cols="1" rows="1" rend="align=right" role="data">894,162,000</cell></row></table> See the Philos. Trans. vol. 61, p. 572.</p><p>But others, by taking the results of those observations that are most to be depended on, have made the sun's Parallax at his mean distance from the earth to be 8.6045; and some make it only 8.54. According to the former of these, the sun's mean distance from the earth is 95,109,736 miles; and according to the latter it is 95,834,742 miles. Upon the whole there seems<cb/> reason to conclude that the sun's horizontal Parallax may-be stated at 8″.6, and his distance near 95 millions of miles. Hence, the following horizontal Parallaxes: <table><row role="data"><cell cols="1" rows="1" role="data">Mean Parallax of the sun</cell><cell cols="1" rows="1" role="data"> 0′</cell><cell cols="1" rows="1" role="data"> 8″.6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Moon's greatest</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" role="data">Moon's least</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"> 4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Moon's mean</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">48</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars's</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">25</cell></row></table></p><p><hi rend="italics">Of the</hi> <hi rend="smallcaps">Parallax</hi> <hi rend="italics">of the Fixed Stars.</hi> As to the fixed stars, their distance is so great, that it has never been found that they have any sensible Parallax, neither with respect to the earth's diameter, nor even with regard to the diameter of the earth's annual orbit round the sun, although this diameter be about 190 millions of miles. For, any of those stars being observed from opposite ends of this diameter, or at the interval of half a year between the observations, when the earth is in opposite points of her orbit, yet still the star appears in the same place and situation in the heavens, without any change that is sensible, or measurable with the very best instruments, not amounting to a single se<*> cond of a degree. That is, the diameter of the earth's annual orbit, at the nearest of the fixed stars, does not subtend an angle of a single second; or, in comparison of the distance of the fixed stars, the extent of 190 millions of miles is but as a point!</p><p><hi rend="smallcaps">Parallax</hi> is also used, in Levelling, for the angle contained between the line of true level, and that of apparent level. And, in other branches of science, for the difference between the true and apparent places.</p></div1><div1 part="N" n="PARALLEL" org="uniform" sample="complete" type="entry"><head>PARALLEL</head><p>, in Geometry, is applied to lines, figures, and bodies, which are every where equidistant from each other; or which, though infinitely produced, would never either approach nearer, or recede farther from, each other; their distance being every where measured by a perpendicular line between them. Hence,</p><p><hi rend="smallcaps">Parallel</hi> <hi rend="italics">right lines</hi> are those which, though insinitely produced ever so far, would never meet: which is Euclid's definition of them.</p><p>Newton, in Lemma 22, book 1 of his Principia, defines Parallels to be such lines as tend to a point infinitely distant.</p><p>Parallel Lines stand opposed to lines converging, and diverging.</p><p>Some define an inclining or converging line, to be that which will meet another at a finite distance, and a Parallel line, that which will only meet at an insinite distance.</p><p>As a perpendicular is by some said to be the shortest of all lines that can be drawn to another; so a Parallel is said to be the longest.</p><p>It is demonstrated by geometricians, that two lines, AB and CD, that are both Parallel to one and the same right line EF, are also Parallel to each other. And that if two Parallel lines AB and EF be cut by any other line GH; then 1st, the alternate angles are equal; viz the angle <hi rend="italics">a</hi> = [angle] <hi rend="italics">b,</hi> and [angle] <hi rend="italics">c</hi> = [angle] <hi rend="italics">d.</hi> 2d, The external angle is equal to the internal one on the same side of the cutting line; viz the [angle] <hi rend="italics">e</hi> = [angle] <hi rend="italics">d,</hi> and the [angle] <hi rend="italics">f</hi> = [angle] <hi rend="italics">b.</hi> 3d, That the two internal ones on the same side are, taken together, equal to two<pb n="197"/><cb/> right angles; viz, , or . <figure/></p><p><hi rend="italics">To draw a</hi> <hi rend="smallcaps">Parallel</hi> <hi rend="italics">Line.</hi>—If the line to be Parallel to AB must pass through a given point P: Take the nearest distance between the point P and the given line AB, by setting one foot of the compasses in P, and with the other describe an arc just to touch the line in A; then with that distance as a radius, and a centre B taken any where in the line, describe another arc C; lastly, through P draw a line PC just to touch the arc C, and that will be the Parallel sought. <figure/></p><p><hi rend="italics">Otherwise.</hi>—With the centre P, and any radius, describe an arc BC, cutting the given line in B. Next, with the same radius, and centre B, describe another arc PA, cutting also the given line in A. Lastly, take AP between the compasses, and apply it from B to C; and through P and C draw the Parallel PC required.</p><p>Or, draw the line with the Parallel Ruler, described below, by laying one edge of the ruler along AB, and extending the other to the given point or distance.</p><p>When the one line is to be at a given distance from the other; take that distance between the compasses as a radius, and with two centres taken any where in the given line, describe two arcs; then lay a ruler just to touch the arcs, and by it draw the Parallel.</p><p><hi rend="smallcaps">Parallel</hi> <hi rend="italics">Planes,</hi> are every where equidistant, or have all the perpendiculars that are drawn between them, everywhere equal.</p><p><hi rend="smallcaps">Parallel</hi> <hi rend="italics">Rays,</hi> in Optics, are those which keep always at an equal distance in respect to each other, from the visual object to the eye, from which the object is supposed to be infinitely distant.</p><p><hi rend="smallcaps">Parallel</hi> <hi rend="italics">Ruler,</hi> is a mathematical instrument, consisting of two equal rulers, AB and CD, either of wood or metal, connected together by two slender cross bars or blades AC and BD, moveable about the points or joints A, B, C, D.</p><p>There are other forms of this instrument, a little varied from the above; some having the two blades crossing in the middle, and fixed only at one end of them, the other two ends sliding in groovca along the two rulers; &c.</p><p>The use of this instrument is obvious. For the edge of one of the rulers being applied to any line, the other opened to any extent will be always parallel to the<cb/> former; and consequently any Parallels to this may be drawn by the edge of the ruler, opened to any extent.</p><p><hi rend="smallcaps">Parallel</hi> <hi rend="italics">Sailing,</hi> in Navigation, is the sailing on or under a Parallel of latitude, or Parallel to the equator. —Of this there are three cases.</p><p>1. Given the Distance and Difference of Longitude; to find the Latitude.—Rule. As the difference of longitude is to the distance, so is radius to the cosine of the latitude.</p><p>2. Given the Latitude and Difference of Longitude; to find the Distance.—Rule. As radius is to the cosine of the latitude, so is the difference of longitude to the distance.</p><p>3. Given the Latitude and Distance; to find the difference of longitude.—Rule. As the cosine of latitude is to radius, so is the distance to the difference of longitude.</p><p><hi rend="smallcaps">Parallel</hi> <hi rend="italics">Sphere,</hi> is that situation of the sphere where the equator coincides with the horizon, and the poles with the zenith and nadir.</p><p>In this sphere all the Parallels of the equator become Parallels of the horizon; consequently no stars ever rise or set, but all turn round in circles Parallel to the horizon, as well as the sun himself, which when in the equinoctial wheels round the horizon the whole day. Also, After the sun rises to the elevated pole, he never sets for six months; and after his entering again on the other side of the line, he never rises for six months longer.</p><p>This position of the sphere is theirs only who live at the poles of the earth, if any such there be. The greatest height the sun can rise to them, is 23 1/2 degrees. They have but one day and one night, each being half a year long. See <hi rend="smallcaps">Sphere.</hi></p><div2 part="N" n="Parallels" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Parallels</hi></head><p>, or <hi rend="italics">Places of Arms,</hi> in a Siege, are deep trenches, 15 or 18 feet wide, joining the several attacks together; and serving to place the guard of the trenches in, to be at hand to support the workmen when attacked.</p><p>There are usually three in an attack: the first is about 600 yards from the covert-way, the second between 3 and 400, and the third near or on the glacis. —It is said they were first invented or used by Vauban.</p><p><hi rend="smallcaps">Parallels</hi> <hi rend="italics">of Altitude,</hi> or Almacantars, are circles Parallel to the horizon, conceived to pass through every degree and minute of the meridian between the horizon and zenith; having their poles in the zenith.</p></div2><div2 part="N" n="Parallels" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Parallels</hi></head><p>, or <hi rend="smallcaps">Parallel</hi> <hi rend="italics">Circles,</hi> called also Parallels of Latitude, and Circles of Latitude, are lesser circles of the sphere, Parallel to the equinoctial or equator.</p><p><hi rend="smallcaps">Parallels</hi> <hi rend="italics">of Declination,</hi> are lesser circles Parallel to the equinoctial.</p><p><hi rend="smallcaps">Parallels</hi> <hi rend="italics">of Latitude,</hi> in Geography, are lesser circles Parallel to the equator. But in Astronomy they are Parallel to the ecliptic.</p></div2></div1><div1 part="N" n="PARALLELISM" org="uniform" sample="complete" type="entry"><head>PARALLELISM</head><p>, the quality of a parallel, or that which denominates it such. Or it is that by which two things, as lines, rays, or the like, become equidistant from one another.</p><p><hi rend="smallcaps">Parallelism</hi> <hi rend="italics">of the Earth's Axis,</hi> is that invariable situation of the axis, in the progress of the earth through the annual orbit, by which it always keeps parallel to itself; so that if a line be drawn parallel to its axis,<pb n="198"/><cb/> while in any one position; the axis, in all other positions or parts of the orbit, will always be parallel to the same line.</p><p>In consequence of this Parallelism, the axis of the earth points always, as to sense, to the same place or point in the heavens, viz to the poles. Because, though really the axis, in the annual motion, describes the surface of a cylinder, whose base is the circle of the earth's annual orbit, yet this whole circle is but as a point in comparison with the distance of the fixed stars; and therefore all the sides of the cylinder seem to tend to the same point, which is the celestial pole.—To this Parallelism is owing the change and variety of seasons, with the inequality of days and nights.</p><p>This Parallelism is the necessary consequence of the earth's double motion; the one round the sun, the other round its own axis. Nor is there any necessity to imagine a third motion, as some have done, to account for this Parallelism.</p><p><hi rend="smallcaps">Parallelism</hi> <hi rend="italics">of Rows of Trees.</hi> The eye placed at the end of an alley bounded by two rows of trees, planted in parallel lines, never sees them parallel, but always inclining to each other, towards the farther end.</p><p>Hence mathematicians have taken occasion to enquire, in what lines the trees must be disposed, to correct this effect of the perspective, and make the rows still appear parallel. And, to produce this effect, it is evident that the unequal intervals of any two opposite or corresponding trees may be seen under equal visual angles.</p><p>For this purpose, M. Fabry, Tacquet, and Varignon observe, that the rows must be opposite semi-hyperbolas. See the Mem. Acad. Sciences, an. 1717.</p><p>But notwithstanding the ingenuity of their speculations, it has been proved by D'Alembert, and Bouguer, that to produce the effect proposed, the trees are to be ranged merely in two diverging right lines.</p></div1><div1 part="N" n="PARALLELOGRAM" org="uniform" sample="complete" type="entry"><head>PARALLELOGRAM</head><p>, in Geometry, is a quadrilateral right-lined sigure, whose opposite sides are parallel to each other.</p><p>A Parallelogram may be conceived as generated by the motion of a right line, along a plane, always parallel to itself.</p><p>Parallelograms have several particular denominations, and are of several species, according to certain particular circumstances, as follow:</p><p>When the angles of the Parallelogram are right ones, it is called a Rectangle.—When the angles are right, and all its sides equal, it is a square.—When the sides are equal, but the angles oblique ones, the figure is a Rhombus or Lozenge. And when both the sides and angles are unequal, it is a Rhomboides.</p><p>Every other quadrilateral whose opposite sides are neither parallel nor equal, is called a Trapezium.</p><p><hi rend="italics">Properties of the</hi> <hi rend="smallcaps">Parallelogram.</hi>—1. In every Parallelogram ABDC, the <figure/> diagonal divides the figure into two equal triangles, ABD, ACD. Also the opposite angles and sides are equal, viz, the side AB = CD, and AC = BD, also the angle A = [angle] D, and the [angle] B = [angle] C. And the sum of any two succeeding<cb/> angles, or next the same side, is equal to two right angles, or 180 degrees, as [angle] A + [angle] C = [angle] C + [angle] D = [angle] D + [angle] B = [angle] B + [angle] A = two right-angles.</p><p>2. All Parallelograms, as ABDC and <hi rend="italics">ab</hi>DC, are equal, that are on the same base CD, and between th<*> same parallels A<hi rend="italics">b,</hi> CD; or that have either the same or equal bases and altitudes; and each is double a triangle of the same or equal base and altitude.</p><p>3. The areas of Parallelograms are to one another in the compound ratio of their bases and altitudes. If their bases be equal, the areas are as their altitudes; and if the altitudes be equal, the arcas are as the bases. And when the angles of the one Parallelogram are equal to those of another, the areas are as the rectangles of the sides about the equal angles.</p><p>4. In every Parallelogram, the sum of the squares of the two diagonals, is equal to the sum of the squares of all the four sides of the figure, viz, . Also the two diagonals bisect each other; so that AE = ED, and BE = EC.</p><p>5. <hi rend="italics">To find the Area of a</hi> <hi rend="smallcaps">Parallelogram.</hi>—Multiply any one side, as a base, by the height, or perpendicular let fall upon it from the opposite side. Or, multiply any two adjacent sides together, and the product by the sine of their contained angle, the radius being 1 : viz, The area is [angle] C.</p><p><hi rend="italics">Complement of a</hi> <hi rend="smallcaps">Parallelogram.</hi> See C<hi rend="smallcaps">OMPLEMENT.</hi></p><p><hi rend="italics">Centre of Gravity of a</hi> <hi rend="smallcaps">Parallelogram.</hi> See C<hi rend="smallcaps">ENTRE</hi> <hi rend="italics">of Gravity,</hi> and <hi rend="smallcaps">Centrobaric</hi> <hi rend="italics">Method.</hi></p><div2 part="N" n="Parallelogram" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Parallelogram</hi></head><p>, or <hi rend="smallcaps">Parallelism</hi>, or P<hi rend="smallcaps">ENTAGRAPH</hi>, also denotes a machine used for the ready and exact reduction or copying of designs, schemes, plans, prints, &c, in any proportion. See <hi rend="smallcaps">Pentagraph.</hi></p><p><hi rend="smallcaps">Parallelogram</hi> <hi rend="italics">of the Hyperbola,</hi> is the Parallelogram formed by the two asymptotes of an hyperbola, and the parallels to them, drawn from any point of the curve. This term was first used by Huygens, at the end of his Dissertatio de Causa Gravitatis. This Parallelogram, so formed, is of an invariable magnitude in the same hyperbola; and the rectangle of its sides is equal to the power of the hyperbola.</p><p>This Parallelogram is also the modulus of the logarithmic system; and if it be taken as unity or 1, the hyperbolic sectors and segments will correspond to Napier's or the natural logarithms; for which reason these have been called the hyperbolic logarithms. If the Parallelogram be taken = .43429448190 &c, these sectors and segments will represent Briggs's logarithms; in which case the two asymptotes of the hyperbola make between them an angle of 25° 44′ 25″1/2.</p><p><hi rend="italics">Newtonian or Analytic</hi> <hi rend="smallcaps">Parallelogram</hi>, a term us<*>d for an invention of Sir Isaac Newton, to <*>ind the first term of an infinite converging series. It is sometimes called the Method of the Parallelogram and Ruler; because a ruler or right line is also used in-it.</p><p>This Analytical Parallelogram is formed by dividing any geometrical Parallelogram into equal small squares or Parallelograms, by lines drawn horizontally and per<pb n="199"/><cb/> pendicularly through the equal divisions of the sides of the Parallelogram. The small cells, thus formed, are filled with the dimensions o<*> powers of the species <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> and their products.</p><p>For instance, the powers of <hi rend="italics">y,</hi> as <hi rend="italics">y</hi>° or 1, <hi rend="italics">y,</hi> <hi rend="italics">y</hi><hi rend="sup">2</hi>, <hi rend="italics">y</hi><hi rend="sup">3</hi>, <hi rend="italics">y</hi><hi rend="sup">4</hi>, &c, being placed in the lowest horizontal range of cells; and the powers of <hi rend="italics">x,</hi> as <hi rend="italics">x</hi>° = 1, <hi rend="italics">x,</hi> <hi rend="italics">x</hi><hi rend="sup">2</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>, &c, in the vertical column to the left; or vice versa; these powers and their products will stand as in this figure: <figure/></p><p>Now when any literal equation is proposed, involving various powers of the two unknown quantities <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> to find the value of one of these in an infinite series of the powers of the other; mark such of the cells as correspond to all its terms, or that contain the same powers and products of <hi rend="italics">x</hi> and <hi rend="italics">y;</hi> then let a ruler be applied to two, or perhaps more, of the Parallelograms so marked, of which let one be the lowest in the left hand column at AB, the other touching the ruler towards the right hand; and let all the rest, not touching the ruler, lie above it. Then select those terms of the equation which are represented by the cells that touch the ruler, and from them find the first term or quantity to be put in the quotient.</p><p>Of the application of this rule, Newton has given several examples in his Method of Fluxions and Infinite Series, p. 9 and 10, but without demonstration; which has been supplied by others. See Colson's Comment on that treatise, p. 192 & seq. Also Newton's Letter to Oldenburg, Oct. 24, 1676. Maclaurin's Algebra, p. 251. And especially Cramer's Analyses des Lignes Courbes, p. 148.—This author observes, that this invention, which is the true foundation of the method of series, was but imperfectly understood, and not valued as it deserved, for a long time. He thinks it however more convenient in practice to use the Analytical Triangle of the abbé de Gua, which takes in no more than the diagonal cells lying between A and C, and those which lie between them and B.</p><p><hi rend="smallcaps">Parallelogram</hi> <hi rend="italics">Protractor,</hi> a mathematical instrument, consisting of a semicircle of brass, with four ru-<cb/> lers in form of a Parallelogram, made to move to any angle. One of these rulers is an index, which shews on the semicircle the quantity of any inward and outward angle.</p></div2></div1><div1 part="N" n="PARALLELOPIPED" org="uniform" sample="complete" type="entry"><head>PARALLELOPIPED</head><p>, or <hi rend="smallcaps">Parallelopipedon</hi>, is a solid figure contained under six parallelograms, the opposites of which are equal and parallel. Or, it is a prism whose base is a parallelogram.</p><p><hi rend="italics">Properties of the</hi> <hi rend="smallcaps">Parallelopipedon.</hi>—All Parallelopipedons, whether right or oblique, that have their bases and altitudes equal, are equal; and each equal to triple a pyramid of an equal base and altitude.—A diagonal plane divides the Parallelopipedon into two equal triangular prisms.—See other properties under the general term <hi rend="smallcaps">Prism</hi>, of which this is only a particular species.</p><p><hi rend="italics">To Measure the Surface and Solidity of a</hi> P<hi rend="smallcaps">ARALLELOPIPEDON.</hi>—Find the areas of the three parallelograms AD, BE, and BG, which add into one sum; and double that sum will be the whole surface of the Parallelopipedon. <figure/></p><p>For the Solidity; multiply the base by the altitude; that is, any one face or side by its distance from the opposite side; as AD × DE, or AB × BE, or BG × BD.</p></div1><div1 part="N" n="PARAMETER" org="uniform" sample="complete" type="entry"><head>PARAMETER</head><p>, a certain constant right line in each of the three Conic Sections; otherwise called also Latus Rectum.</p><p>This line is called Parameter, or equal measurer, because it measures the conjugate axis by the same ratio which is between the two axes themselves; being indeed a third proportional to them; viz, a third proportional to the transverse and conjugate axes, in the ellipse and hyperbola; and, which is the same thing, a third proportional to any absciss and its ordinate in the parabola. So if <hi rend="italics">t</hi> and <hi rend="italics">c</hi> be the two axes in the ellipse and hyperbola, and <hi rend="italics">x</hi> and <hi rend="italics">y</hi> an absciss and its ordinate in the parabola; then the Param. in the former, and the Param. in the last.</p><p>The Parameter is equal to the double ordinate drawn through the focus of any of the three conic sections.</p></div1><div1 part="N" n="PARAPET" org="uniform" sample="complete" type="entry"><head>PARAPET</head><p>, or <hi rend="italics">Breastwork,</hi> in Fortification, is a defence or screen, on the extreme edge of a rampart, or other work, serving to cover the soldiers and the cannon from the enemy's fire.</p><p>The thickness of the Parapet is 18 or 20 feet, commonly lined with masonry; and 7 or 8 feet high, when the enemy has no command above the battery; otherwise, it should be raised higher, to cover the men while<pb n="200"/><cb/> they load the guns. There are certain openings, called Embrasures, cut in the Parapet, from the top downwards, to within about 2 1/2 or 3 feet of the bottom of it, for the cannon to fire through; the solid pieces of it between one embrasure and another, being called Merlons.</p><p><hi rend="smallcaps">Parapet</hi> is also a little breast-wall, raised on the brinks of bridges, quays, or high buildings; to serve as a stay, and prevent people from falling over.</p></div1><div1 part="N" n="PARDIES" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PARDIES</surname> (<foreName full="yes"><hi rend="smallcaps">Ignatius Gaston</hi></foreName>)</persName></head><p>, an ingenious French mathematician and philosopher, was born at Pau, in the province of Gaseony, in 1636; his father being a counsellor of the parliament of that city.— At the age of 16 he entered into the order of Jesuits, and made so great a proficiency in his studies, that he taught polite literature, and composed many pieces in prose and verse with a distinguished delicacy of thought and style, before he was well arrived at the age of manhood. Propriety and elegance of language appear to have been his first pursuits; for which purpose he studied the Belles Lettres, and other learned productions. But afterwards he devoted himself to mathematical and philosophical studies, and read, with due attention, the most valuable authors, ancient and modern, in those sciences: so that, in a short time he made himself master of the Peripatetic and Cartesian philosophy, and taught them both with great reputation. Notwithstanding he embraced Cartesianism, yet he affected to be rather an inventor in philosophy himself. In this spirit he sometimes advanced very bold opinions in natural philosophy, which met with opposers, who charged him with starting absurdities: but he was ingenious enough to give his notions a plausible turn, so as to clear them seemingly from contradictions. His reputation procured him a call to Paris, as Professor of Rhetoric in the College of Lewis the Great. He also taught the mathematics in that city, as he had before done in other places. He had from his youth a happy genius for that science, and made a great progress in it; and the glory which his writings acquired him, raised the highest expectations from his future labours; but these were all blasted by his early death, in 1673, at 37 years of age; falling a victim to his zeal, he having caught a contagious disorder by preaching to the prisoners in the Bicetre.</p><p>Pardies wrote with great neatness and elegance. His principal works are as follow:</p><p>1. Horologium Thaumaticum duplex; 1662, in 4to.</p><p>2. Dissertatio de Motu et Natura Cometarum; 1665, 8vo.</p><p>3. Discours du Mouvement Local; 1670, 12mo.</p><p>4. Elemens de Geometrie; 1670, 12mo.—This has been translated into several languages; in English by Dr. Harris, in 1711.</p><p>5. Discours de la Connoissance des Betes; 1672, 12mo.</p><p>6. Lettre d'un Philosophe à un Cartesien de ses amis; 1672, 12mo.</p><p>7. La Statique ou la Science des Forces Mouvantes; 1673, 12mo.</p><p>8. Description et Explication de deux Machines propres à faire des Cadrans avec une grande facilité; 1673, 12mo.</p><p>9. Remarques du Mouvement d<*> la Lumiere.<cb/></p><p>10. Globi Cœlestis in tabula plana redacti Descrip. tio; 1675, folio.</p><p>Part of his works were printed together, at the Hague, 1691, in 12mo; and again at Lyons, 1725.— Pardies had a dispute also with Sir Isaac Newton, about his New Theory of Light and Colours, in 1672. His letters are inserted in the Philosophical Transactions for that year.</p></div1><div1 part="N" n="PARENT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PARENT</surname> (<foreName full="yes"><hi rend="smallcaps">Anthony</hi></foreName>)</persName></head><p>, a respectable French mathematician, was born at Paris in 1666. He shewed an early propensity to the mathematics, eagerly perusing such books in that science as fell in his way. His custom was to write remarks in the margins of the books he read; and in this way he had silled a number of books with a kind of commentary by the time he was 13 years of age.</p><p>Soon after this he was put under a master, who taught rhetoric at Chartres. Here he happened to see a dodecaedron, upon every face of which was delineated a sun-dial, except the lowest on which it stood. Struck as it were instantaneously with the curiosity of these dials, he attempted drawing one himself: but having only a book which taught the practical part, without the theory, it was not till after his master came to explain the doctrine of the sphere to him, that he began to understand how the projection of the circles of the sphere formed sun-dials. He then undertook to write a treatise upon gnomonics. To be sure the piece was rude and unpolished enough; however, it was entirely his own, and not borrowed. About the same time he wrote a book of geometry, in the same taste, at Beauvais.</p><p>His friends then sent for him to Paris to study the law; and in obedience to them he went through a course in that faculty: which was no sooner finished than, urged by his passion for mathematics, he shut himself up in the college of Dormans, that no avocation might take him from his beloved study: and, with an allowance of less than 200 livres a-year, he lived content in this retreat, from which he never stirred but to the Royal College, to hear the lectures of M. de la Hire or M. de Sauveur. When he thought himself capable of teaching others, he took pupils: and fortification being a branch of study which the war had brought into particular notice, he had often occasion to teach it: but after some time he began to entertain scruples about teaching a subject he had never seen, knowing it only by imagination. He imparted this scruple to M. Sauveur, who recommended him to the Marquis d'Aligre, who luckily at that time wanted to have a mathematician with him. M. Parent made two campaigns with the marquis, by which he instructed himself sufficiently in viewing fortified places; of which he drew a number of plans, though he had never learned the art of drawing.</p><p>From this period he spent his time in a continual application to the study of natural philosophy, and mathematics in all its branches, both speculative and practical; to which he joined anatomy, botany, and chemistry:—his genius joined with his indefatigable application overcoming every thing.</p><p>M. de Billettes being admitted into the Academy of Sciences at Paris in 1699, with the title of their mechanician, he named M. Parent for his eleve or dif-<pb n="201"/><cb/> ciple, a branch of mathematics in which he chiefly excelled. It was soon discovered in this society, that he engaged in all the different subjects which were brought before them; and indeed that he had a hand in every thing. But this extent of knowledge, joined to a natural warmth and impetuosity of temper, raised a spirit of contradiction in him, which he indulged on all occasions; sometimes to a degree of precipitancy that was highly culpable, and often with but little regard to decency. Indeed the fame behaviour was returned to him, and the papers which he brought to the academy were often treated with much severity. In his productions, he was charged with obscurity; a fault for which he was indeed so notorious, that he perceived it himself, and could not avoid correcting it.</p><p>By a regulation of the academy in 1716, the class of eleves was suppressed, as that distinction seemed to put too great an inequality between the members. M. Parent was made an adjunct or assistant member for the class of geometry: though he enjoyed this promotion but a very short time; being cut off by the smallpox the same year, at 50 years of age.</p><p>M. Parent, besides leaving many pieces in manuscript, published the following works:</p><p>1. Elemens de Mecanique & de Physique; in 12mo, 1700.</p><p>2. Recherches de Mathematiques & de Physique; 3 vols. 4to, 1714.</p><p>3. Arithmetique theorico-pratique; in 8vo, 1714.</p><p>4. A great multitude of papers in the volumes of the Memoirs of the Academy of Sciences, from the year 1700 to 1714, several papers in almost every volume, upon a variety of branches in the mathematics.</p></div1><div1 part="N" n="PARGETING" org="uniform" sample="complete" type="entry"><head>PARGETING</head><p>, in Building, is used for the plaistering of walls; sometimes for plaister itself.</p></div1><div1 part="N" n="PARHELION" org="uniform" sample="complete" type="entry"><head>PARHELION</head><p>, or <hi rend="smallcaps">Parhelium</hi>, denotes a mock sun, or meteor, appearing as a very bright light by the side of the sun; being formed by the reflection of his beams in a cloud properly situated.</p><p>Parhelia usually accompany the coronæ, or luminous circles, and are placed in the same circumference, and at the same height. Their colours resemble those of the rainbow; the red and yellow are on that side towards the sun, and the blue and violet on the other. Though coronæ are sometimes seen entire, without any Parhelia; and sometimes Parhelia without coronæ.</p><p>The apparent size of Parhelia is the same as that of the true sun; but they are not always round, nor always so bright as the sun; and when several appear, some are brighter than others. They are tinged externally with colours like the rainbow, and many of them have a long siery tail opposite to the sun, but paler towards the extremity. Some Parhelia have been observed with two tails and others with three. These tails mostly appear in a white horizontal circle, commonly passing through all the Parhelia, and would go through the centre of the sun if it were entire. Sometimes there are arcs of lesser circles, concentric to this, touching those coloured circles which surround the sun: these are also tinged with colours, and contain other Parhelia.</p><p>Parhelia are generally situated in the intersections of circles; but Cassini says, those which he saw in 1683, were on the outside of the coloured circle, though the<cb/> tails were in the circle that was parallel to the horizon. M. Aepinus apprehends, that Parhelia with elliptical coronæ are more frequent in the northern regions, and those with circular ones in the southern. They have been visible for one, two, three, or four hours together; and it is said that in North America they continue several days, and are visible from sun-rise to sun-set. When the Parhelia disappear, it sometimes rains, or there falls snow in the form of oblong spiculæ. And Mariotte accounts for the appearance of Parhelia from an infinity of small particles of ice floating in the air, which multiply the image of the sun, either by refracting or breaking his rays, and thus making him appear where he is not; or by reflecting them, and serving as mirrors.</p><p>Most philosophers have written upon Parhelia; as Aristotle, Pliny, Scheiner, Gassendi, Des Cartes, Huygens, Hevelius, De la Hire, Cassini, Grey, Halley, Maraldi, Musschenbroek, &c. See Smith's Optics, book 1, chap. 11. Also Priestley's Hist. of Light &c, p. 613. And Musschenbroek's Introduction &c, vol. 2, p. 1038 quarto.</p><p>PARODICAL <hi rend="italics">Degrees,</hi> in an equation, a term that has been sometimes used to denote the several regular terms in a quadratic, cubic, biquadratic, &c, equation, when the indices of the powers ascend or descend orderly in an arithmetical progression. Thus, is a cubic equation where no term is wanting, but having all its Parodic Degrees; the indices of the terms regularly descending thus, 3, 2, 1, 0.</p></div1><div1 part="N" n="PART" org="uniform" sample="complete" type="entry"><head>PART</head><p>, <hi rend="italics">Aliquant, Aliquot, Circular Proportional, Similar,</hi> &c. See the respective adjectives.</p><p><hi rend="smallcaps">Part</hi> <hi rend="italics">of Fortune,</hi> in Judicial Astrology, is the lunar horoscope; or the point in which the moon is, at the time when the sun is in the ascending point of the east.</p><p>The sun in the ascendant is supposed, according to this science, to give life; and the moon dispenses the radical moisture, and is one of the causes of fortune. In horoscopes the Part of Fortune is represented by a circle divided by a cross.</p></div1><div1 part="N" n="PARTICLE" org="uniform" sample="complete" type="entry"><head>PARTICLE</head><p>, the minute part of a body, or an assemblage of several of the atoms of which natural bodies are composed. Particle is sometimes considered as synonymous with atom, and corpuscle; and sometimes they are distinguished.</p><p>Particles are, as it were, the elements of bodies; by the various arrangement and texture of which, with the difference of the cohesion, &c, are constituted the several kinds of bodies, hard, soft, liquid, dry, heavy, light, &c. The smallest Particles or corpuseles cohere with the strongest attractions, and always compose larger Particles of weaker cohesion: and many of these, cohering, compose still larger Particles, whose vigour is still weaker; and so on for divers successions, till the progression end in the largest Particles, upon which the operations in chemistry, and the colours of natural bodies, depend; and which, by cohering, compose bodies of sensible magnitude.</p><p>PARTILE <hi rend="italics">Aspect,</hi> in Astrology, is when the planets are in the exact degree of any particular aspect. In contradistinction to Platic Aspect, or when they do not regard each other with those very degrees. See <hi rend="smallcaps">Aspect.</hi><pb n="202"/><cb/></p><p>PARTY <hi rend="italics">Arches,</hi> in Architecture, are arches built between separate tenures, where the property is intermixed, and apartments over each other do not belong to the same estate.</p><p><hi rend="smallcaps">Party</hi> <hi rend="italics">Walls,</hi> are partitions of brick made between buildings in separate occupations, for preventing the spread of fire. These are made thicker than the external walls; and their thickness in London is regulated by act of parliament of the 14th of George the Third.</p></div1><div1 part="N" n="PASCAL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PASCAL</surname> (<foreName full="yes"><hi rend="smallcaps">Blaise</hi></foreName>)</persName></head><p>, a respectable French mathematician and philosopher, and one of the greatest geniuses and best writers that country has produced. He was born at Clermont in Auvergne, in the year 1623. His father, Stephen Pascal, was president of the Court of Aids in his province: he was also a very learned man, an able mathematician, and a friend of Des Cartes. Having an extraordinary tenderness for this child, his only son, he quitted his office in his province, and settled at Paris in 1631, that he might be quite at leisure to attend to his son's education, which he conducted himself, and young Pascal never had any other master.</p><p>From his infancy Blaise gave proofs of a very extraordinary capacity. He was extremely inquisitive; desiring to know the reason of every thing; and when good reasons were not given him, he would seek for better; nor would he ever yield his assent but upon such as appeared to him well grounded. What is told of his manner of learning the mathematics, as well as the progress he quickly made in that science, seems almost miraculous. His father, perceiving in him an extraordinary inclination to reasoning, was afraid lest the knowledge of the mathematics might hinder his learning the languages, so necessary as a foundation to all sound learning. He therefore kept him as much as he could from all notions of geometry, locked up all his books of that kind, and refrained even from speaking of it in his presence. He could not however prevent his son from musing on that science; and one day in particular he surprised him at work with charcoal upon his chamber floor, and in the midst of figures. The father asked him what he was doing: I am searching, says Pascal, for such a thing; which was just the same as the 32d proposition of the 1st book of Euclid. He asked him then how he came to think of this: It was, says Blaise, because I found out such another thing; and so, going backward, and using the names of <hi rend="italics">bar</hi> and <hi rend="italics">round,</hi> he came at length to the definitions and axioms he had formed to himself. Does it not seem miraculous, that a boy should work his way into the heart of a mathematical book, without ever having seen that or any other book upon the subject, or knowing any thing of the terms? Yet we are assured of the truth of this by his sister, Madam Perier, and several other persons, the credit of whose testimony cannot reasonably be questioned.</p><p>From this time he had full liberty to indulge his genius in mathematical pursuits. He understood Euclid's Elements as soon as he cast his eyes upon them. At 16 years of age he wrote a treatise on Conic Sections, which was accounted a great effort of genius; and therefore it is no wonder that Des Cartes, who had been in Holland a long time, upon reading it, should choose to believe that M. Pascal the father was the<cb/> real author of it. At 19 he contrived an admirable arithmetical machine, which was esteemed a very wonderful thing, and would have done credit as an invention to any man versed in science, and much more to such a youth.</p><p>About this time his health became impaired, so that he was obliged to suspend his labours for the space of four years. After this, having seen Torricelli's experiment respecting a vacuum and the weight of the air, he turned his thoughts towards these objects, and undertook several new experiments, one of which was as follows: Having provided a glass tube, 46 feet in length, open at one end, and hermetically sealed at the other, he filled it with red wine, that he might distinguish the liquor from the tube, and stopped up the orifice; then having inverted it, and placed it in a vertical position, with the lower end immersed into a vessel of water one foot deep, he opened the lower end, and the wine descended to the distance of about 32 feet from the surface of the vessel, leaving a considerable vacuum at the upper part of the tube. He next inclined the tube gradually, till the upper end became only of 32 feet perpendicular height above the bottom, and he observed the liquor proportionally ascend up to the top of the tube. He made also a great many experiments with siphons, syringes, bellows, and all kinds of tubes, making use of different liquors, such as quicksilver, water, wine, oil, &c; and having published them in 1647, he dispersed his work through all countries.</p><p>All these experiments however only ascertained effects, without demonstrating the causes. Pascal knew that Torricelli conjectured that those phenomena which he had observed were occasioned by the weight of the air, though they had formerly been attributed to Nature's abhorrence of a vacuum; but if Torricelli's theory were true, he reasoned that the liquor in the barometer tube ought to stand higher at the bottom of a hill, than at the top of it. In order therefore to discover the truth of this theory, he made an experiment at the top and bottom of a mountain in Auvergne, called<hi rend="italics">le Puy de Dome,</hi> the result of which gave him reason to conclude that the air was indeed heavy. Of this experiment he published an account, and sent copies of it to most of the learned men in Europe. He also renewed it at the top and bottom of several high towers, as those of Notre Dame at Paris, St. Jaques de la Boucherie, &c; and always remarked the same difference in the weight of the air, at different elevations. This fully convinced him of the general pressure of the atmosphere; and from this discovery he drew many useful and important inferences. He composed also a large treatise, in which he fully explained this subject, and replied to all the objections that had been started against it. As he afterwards thought this work rather too prolix, and being fond of brevity and precision, he divided it into two small treatises, one of which he intitled, A Dissertation on the Equilibrium of Fluids; and the other, An Essay on the Weight of the Atmosphere. These labours procured Pascal so much reputation, that the greatest mathematicians and philosophers of the age proposed various questions to him, and consulted him respecting such difficulties as they could not resolve. Upon one<pb n="203"/><cb/> of these occasions he discovered the solution of a problem proposed by Mersenne, which had baffled the penetration of all that had attempted it. This problem was to determine the curve described in the air by the nail of a coach-wheel, while the machine is in motion; which curve was thence called a roullette, but now commonly known by the name of cycloid. Pascal offered a reward of 40 pistoles to any one who should give a satisfactory answer to it. No person having succeeded, he published his own at Paris; but as he began now to be disgusted with the sciences, he would not set his real name to it, but sent it abroad under that of A. d'Ettonville.—This was the last work which he published in the mathematics; his infirmities, from a delicate constitution, though still young, now increasing so much, that he was under the necessity of renouncing severe study, and of living so recluse, that he scarcely admitted any person to see him.—Another subject on which Pascal wrote very ingeniously, and in which he has been spoken of as an inventor, was what has been called his Arithmetical Triangle, b<*>ing a set of figurate numbers disposed in that form. But such a table of numbers, and many properties of them, had been treated of more than a century before, by Cardan, Stifelius, and other arithmetical writers.</p><p>After having thus laboured abundantly in mathematical and philosophical disquisitions, he forsook those studies and all human learning at once, to devote himself to acts of devotion and penance. He was not 24 years of age, when the reading some pious books had put him upon taking this resolution; and he became as great a devotee as any age has produced. He now gave himself up entirely to a state of prayer and mortisication; and he had always in his thoughts these great maxims of renouncing all pleasure and all superfluity; and this he practised with rigour even in his illnesses, to which he was frequently subject, being of a very invalid habit of body.</p><p>Though Pascal had thus abstracted himself from the world, yet he could not forbear paying some attention to what was doing in it; and he even interested himself in the contest between the Jesuits and the Jansenists. Taking the side of the latter, he wrote his <hi rend="italics">Lettres Provinciales,</hi> published in 1656, under the name of <hi rend="italics">Louis de Montalte,</hi> making the former the fubject of ridicule. “These letters, says Voltaire, may be considered as a model of eloquence and humour. The best comedies of Moliere have not more wit than the first part of these letters; and the sublimity of the latter part of them, is equal to any thing in Bossuet. It is true indeed that the whole book was built upon a false foundation; for the extravagant notions of a few Spanish and Flemish Jesuits were artfully ascribed to the whole society. Many absurdities might likewise have been discovered among the Dominican and Franciscan casuists; but this would not have answered the purpose; for the whole raillery was to be levelled only at the Jesuits. These letters were intended to prove, that the Jesuits had formed a design to corrupt mankind; a design which no sect or society ever had, or can have.” Voltaire calls Pascal the sirst of their satirists; for Despréaux, says he, must be considered as only the second. In another place, speaking of this work of Pascal, he says, that “Examples of all<cb/> the various species of eloquence are to be found in it. Though it has now been written almost 100 years, yet not a single word occurs in it, savouring of that vicissitude to which living languages are so subject. Here then we are to six the epoch when our language may be said to have assumed a settled form. The bishop of Lucon, son of the celebrated Bussy, told me, that asking one day the bishop of Meaux what work he would covet most to be the author of, supposing his own performances set aside, Bossu replied. The Provincial Letters,” These letters have been translated into all languages, and printed over and over again. Some have said that there were decrees of formal condemnation against them; and also that Pascal himself, in his last illness, detested them, and repented of having been a Jansenist: but both these particulars are false and without foundation. It was supposed that Father Daniel was the anonymous author of a piece against them, intitled <hi rend="italics">The Dialogues of Cleander and Eudoxus.</hi></p><p>Pascal was but about 30 years of age when these letters were published; yet he was extremely infirm, and his diforders increasing soon after so much, that he conceived his end fast approaching, he gave up all farther thoughts of literary composition. He resolved to spend the remainder of his days in retirement and pious meditation; and with this view he broke off all his former connections, changed his habitation, and spoke to no one, not even to his own servants, and hardly ever even admitted them into his room. He made his own bed, fetched his dinner from the kitchen, and carried back the plates and dishes in the evening; so that he employed his servants only to cook for him, to go to town, and to do such other things as he could not absolutely do himself. In his chamber nothing was to be seen but two or three chairs, a table, a bed, and a few books. It had no kind of ornament whatever; he had neither a carpet on the floor, nor curtains to his bed. But this did not prevent him from sometimes receiving visits; and when his friends appeared surprised to see him thus without furniture, he replied, that he had what was necessary, and that any thing else would be a superfluity, unworthy of a wise man. He employed his time in prayer, and in reading the Scriptures; writing down such thoughts as this exercise inspired. Though his continual infirmities obliged him to use very delicate food, and though his servants employed the utmost care to provide only what was excellent, he never relished what he ate, and seemed quite indifferent whether they brought him good or bad. His indifference in this respect was so great, that though his taste was not vitiated, he forbad any sauce or ragout to be made for him which might excite his appetite.</p><p>Though Pascal had now given up intense study, and though he lived in the most temperate manner, his health continued to decline rapidly; and his disorders had so enfeebled his organs, that his reason became in some measure affected. He always imagined that he saw a deep abyss on one side of him, and he never would sit down till a chair was placed there, to secure him from the danger which he apprehended. At another time he pretended that he had a kind of vision or ecstasy; a memorandum of which he preserved<pb n="204"/><cb/> during the remainder of his life on a bit of paper, put between the cloth and the lining of his coat, and which he always carried about him. After languishing for several years in this imbecile state of body and mind, M. Pascal died at Paris the 19th of August 1662, at 39 years of age.</p><p>In company, Pascal was distinguished by the amiableness of his behaviour; by great modesty; and by his casy, agreeable, and instructive conversation. He possessed a natural kind of eloquence, which was in a manner irresistible. The arguments he employed for the most part produced the effect which he proposed; and though his abilities intitled him to assume an air of superiority, he never displayed that haughty and imperious tone which may often be observed in men of shining talents. The philosophy of this extraordinary man consisted in renouncing all pleasure, and every superfluity. He not only denied himself the most common gratifications; but he took also without reluctance, and even with pleasure, either as nourishment or as medicine, whatever was disagreeable to the senses; and he every day retrenched some part of his dress, food, or other things, which he considered as not absolutely necessary. Towards the close of his life, he employed himself wholly in devout and moral reflections, writing down those which he deemed worthy of being preserved. The first bit of paper he could find was employed for this purpose; and he commonly set down only a few words of each sentence, as he wrote them merely for his own use. The scraps of paper upon which he had written these thoughts, were found after his death filed upon different pieces of string, without any order or connection; and being copied exactly as they were written, they were afterward arranged and published, under the title of <hi rend="italics">Pensées, &c,</hi> or <hi rend="italics">Thoughts upon Religion and other Subjects;</hi> being parts of a work he had intended against atheists and infidels, which has been much admired. After his death appeared also two other little tracts; the one intitled, <hi rend="italics">The Equilibrium of Fluids;</hi> and the other, <hi rend="italics">The Weight of the Mass of Air.</hi></p><p>The works of Pascal were collected in 5 volumes 8vo, and published at the Hague, and at Paris, in 1779. This edition of Paseal's works may be considered as the first published; at least the greater part of them were not before collected into one body, and some of them had remained only in manuscript. For this collection, the public were indebted to the Abbé Bossu, and Pascal was deserving of such an editor. “This extraordinary man, says he, inherited from nature all the powers of genius. He was a mathematician of the first rank, a profound reasoner, and a sublime and clegant writer. If we reflect, that in a very short life, oppressed by continual infirmities, he invented a curious arithmetical machine, the elements of the calculation of chances, and a method of resolving various problems, respecting the cycloid; that he sixed in an irrevocable manner the wavering opinions of the learned concerning the weight of the air; that he wrote one of the completest works existing in the French language; and that in his <hi rend="italics">Thoughts</hi> there are passages the depth and beauty of which are incomparable—we can hardly believe that a greater genius ever existed in any age or nation. All those who had oc-<cb/> casion to frequent his company in the ordinary commerce of the world, acknowledged his superiority; but it excited no envy against him, as he was never fond of shewing it. His conversation instructed, without making those who heard him sensible of their own inferiority; and he was remarkably indulgent towards the faults of others. It may be easily seen by his Provincial Letters, and by some of his other works, that he was born with a great fund of humour, which his infirmities could never entirely destroy. In company, he readily indulged in that harmless and delicate raillery which never gives offence, and which greatly tends to enliven conversation; but its principal object was generally of a moral nature. For example, ridiculing those authors who say, <hi rend="italics">My Book, my Commentary, my History,</hi> they would do better (added he) to say, <hi rend="italics">Our book, our Commentary, our History;</hi> since there is in them much more of other people's than their own.”</p><p>The celebrated Baley too, speaking of this great man, says, a hundred volumes of sermons are not of so much avail as a simple account of the life of Pascal. His humanity. and his devotion mortisied the libertines more than if they had been attacked by a dozen of missionaries. In short, Bayle had so high an idea of this philosopher, that he calls him <hi rend="italics">a paradox in the human species.</hi> “When we consider his character, says he, we are almost inclined to doubt whether he was born of a woman, like the man mentioned by Lucretius; <hi rend="center">“<hi rend="italics">Ut vix humana videatur stirpe creaius.</hi>”</hi></p></div1><div1 part="N" n="PATE" org="uniform" sample="complete" type="entry"><head>PATE</head><p>, in Fortification, a kind of platform, like what is called a Horse-shoe; not always regular, but commonly oval, encompassed only with a parapet, and having nothing to flank it. It is usually erected in marshy grounds, to cover a gate of a town, or the like.</p><p>PATH <hi rend="italics">of the Vertex,</hi> a term frequently used by Mr. Flamsteed, in his Doctrine of the Sphere, denoting a circle, described by any point of the earth's surface, as the earth turns round its axis.</p><p>This point is considered as vertical to the earth's centre; and is the same with what is called the vertex or zenith in the Ptolomaic projection.</p><p>The semidiameter of this Path of the vertex, is always equal to the complement of the latitude of the point or place that describes it; that is, to the place's distance from the pole of the world.</p></div1><div1 part="N" n="PAVILION" org="uniform" sample="complete" type="entry"><head>PAVILION</head><p>, in Architecture, is a kind of turret, or building usually insulated, and contained under a single roof; sometimes square and sometimes in form of a dome: thus called from the resemblance of its roof to a tent.</p></div1><div1 part="N" n="PAVO" org="uniform" sample="complete" type="entry"><head>PAVO</head><p>, <hi rend="italics">Peacock,</hi> a new constellation, in the southern hemisphere, added by the modern astronomers. It contains 14 stars.</p></div1><div1 part="N" n="PAUSE" org="uniform" sample="complete" type="entry"><head>PAUSE</head><p>, or <hi rend="smallcaps">Rest</hi>, in Music, a character of silence and rest; called also by some a Mute Figure; because it shews that some part or person is to be silent, while the others continue the song.</p></div1><div1 part="N" n="PECK" org="uniform" sample="complete" type="entry"><head>PECK</head><p>, a measure or vessel used in measuring grain, pulse, and the like dry substances.</p><p>The standard, or Winchester Peck, contains two gallons, or the 4th part of a bushel.<pb n="205"/><cb/></p></div1><div1 part="N" n="PEDESTAL" org="uniform" sample="complete" type="entry"><head>PEDESTAL</head><p>, in Architecture, the lowest part of an order of columns; being that which sustains the column, and serves it as a foot to stand upon. It is a square body or dye, with a cornice and base.</p><p>The proportions and ornaments of the Pedestal are different in the different orders. Vignola indeed, and most of the moderns, make the Pedestal, and its ornaments, in all the orders, one third of the height of the column, including the base and capital. But some deviate from this rule.</p><p>Perrault makes the proportions of the three constituent parts of Pedestals, the same in all the orders; viz, the base one fourth of the Pedestal; the cornice an eighth part; and the foele or plinth of the base, two thirds of the base itself. The height of the dye is what remains of the whole height of the Pedestal.</p><p>The <hi rend="italics">Tuscan</hi> <hi rend="smallcaps">Pedestal</hi> is the simplest and lowest of all; frõm 3 to 5 modules high. It has only a plinth for its base, and an astragal crowned for its cornice.</p><p>The <hi rend="italics">Doric</hi> <hi rend="smallcaps">Pedestal</hi> is made 4 or 5 modules in height, by the moderns; for no ancient columns, of this order, are found with any Pedestal, or even with any base.</p><p>The <hi rend="italics">Ionic</hi> <hi rend="smallcaps">Pedestal</hi> is from 5 to 7 modules high.</p><p>The <hi rend="italics">Corinthian</hi> <hi rend="smallcaps">Pedestal</hi> is the richest and most delicate of all, and is from 4 to 7 modules high.</p><p>The <hi rend="italics">Composite</hi> <hi rend="smallcaps">Pedestal</hi> is of 6 or 7 modules in height.</p><p><hi rend="italics">Square</hi> <hi rend="smallcaps">Pedestal</hi>, is one whose breadth and height are equal.</p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Pedestal</hi>, is that which supports two columns, being broader than it is high.</p><p><hi rend="italics">Continued</hi> <hi rend="smallcaps">Pedestal</hi>, is that which supports a row of columns without any break or interruption.</p><p><hi rend="smallcaps">Pedestals</hi> <hi rend="italics">of Statues,</hi> are those serving to support figures or statues.</p></div1><div1 part="N" n="PEDIMENT" org="uniform" sample="complete" type="entry"><head>PEDIMENT</head><p>, in Architecture, a kind of low pinnacle; serving to crown porticos, or finish a frontispiece; and placed as an ornament over gates, doors, windows, niches, altars, &c; being usually of a triangular form, but sometimes an arch of a circle. Its height is various, but it is thought most beautiful when the height is one fifth of the length of its base.</p></div1><div1 part="N" n="PEDOMETER" org="uniform" sample="complete" type="entry"><head>PEDOMETER</head><p>, or <hi rend="smallcaps">Podometer</hi>, foot-measurer, or way-wiser; a mechanical instrument, in form of a watch, and consisting of various wheels and teeth; which, by means of a chain, or string, fastened to a man's foot, or to the wheel of a chariot, advance a notch each step, or each revolution of the wheel: by which it numbers the paces or revolutions, and so the distance from one place to another.</p><p><hi rend="smallcaps">Pedometer</hi> is also sometimes used for the common surveying wheel, an instrument chiefly used in measuring roads; popularly called the way-wiser. See <hi rend="smallcaps">Perambulator.</hi></p></div1><div1 part="N" n="PEER" org="uniform" sample="complete" type="entry"><head>PEER</head><p>, in Building. See <hi rend="smallcaps">Pier.</hi></p></div1><div1 part="N" n="PEGASUS" org="uniform" sample="complete" type="entry"><head>PEGASUS</head><p>, the Horse, a constellation of the northern hemisphere, figured in the form of a flying horse; being one of the 48 ancient constellations.</p><p>It is fabled, by the Greeks, to have been the offspring of an amour between Neptune and the Gorgon Medusa; and to have been that on which Bellerophon rode when he overcame the Chimera; and that flying<cb/> from mount Helicon to heaven, he there became a constellation; having thrown his rider in the flight; and that the stroke of his hoof on the mount opened the sacred fountain Hippocrene.</p><p>The stars in this constellation, in Ptolomy's catalogue, are 20, in Tycho's 19, in Hevelius's 38, and in the Britannic catalogue 89.</p></div1><div1 part="N" n="PELECOIDES" org="uniform" sample="complete" type="entry"><head>PELECOIDES</head><p>, or <hi rend="italics">Hatchet-form,</hi> in Geometry, a figure in form of a hatchet. As the figure ABCDA, contained under the sem<*>circle BCD and the two quadrantal arcs AB and AD. <figure/></p><p>The area of the Pelecoides is equal to the square AC, and this again is equal to the rectangle BE. It is equal to the square, because the two segments AB and AD, which it wants of the square on the lower part, are compensated by the two equal segments BC and CD, by which it exceeds on the upper part. And the square is equal to the rectangle BE, because the triangle ABD, which is half the square, is also half the rectangle BE of the same base and height with it.</p></div1><div1 part="N" n="PELL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PELL</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent English mathematician, descended from an ancient family in Lincolnshire, was born at Southwick in Sussex, March 1, 1610, where his father was minister. He received his grammar education at the free-school at Stenning in that county. At the age of 13 he was sent to Trinity College in Cambridge, being then as good a scholar as most masters of arts in that university; but though he was eminently skilled in the Greek and Hebrew languages, he never offered himself a candidate at the election of scholars or fellows of his college. His person was handsome; and being of a strong constitution, using little or no recreations, he prosecuted his studies with the more application and intenseness.</p><p>In 1629 he drew up the “<hi rend="italics">Description and Use of the Quadrant, written for the Use of a Friend,” in two books;</hi> the original manuscript of which is still extant among his papers in the Royal Society. And the same year he held a correspondence with Mr. Briggs on the subject of logarithms.</p><p>In 1630 he wrote, <hi rend="italics">Modus supputandi Ephemerides Astronomicas, &c, ad an.</hi> 1630 <hi rend="italics">accommodatus;</hi> and, <hi rend="italics">A Key to unlock the meaning of Johannes Trithemius, in his Discourse of Steganography:</hi> which Key he imparted to Mr. Samuel Hartlib and Mr. Jacob Homedæ. The same year he took the degree of Master of Arts at Cambridge. And the year following he was incorporated in the University of Oxford. June the 7th, he wrote <hi rend="italics">A Letter to Mr. Edmund Wingate on Logarithms;</hi> and Oct. 5, 1631, <hi rend="italics">Commentationes in Cosmographiam Alstedii.</hi></p><p>In 1632 he married Ithamaria, second daughter of Mr. Henry Reginolles of London, by whom he had four sons and four daughters.—March 6, 1634, he finished his <hi rend="italics">Astronomical History of Observations of Heavenly Motions and Appearances;</hi> and April the 10th, his <hi rend="italics">Ecliptica Prognostica, or Foreknower of the Eclipses, &c.</hi>—In 1634 he translated <hi rend="italics">The Everlasting Tables of</hi><pb n="206"/><cb/> <hi rend="italics">Heavenly Motions,</hi> grounded upon the Observations of all Times, and agreeing with them all, by Philip Lansberg, of Ghent in Flanders. And June the 12th, the same year, he committed to writing, <hi rend="italics">The Manner of Deducing his Astronomical Tables out of the Tables and Axioms of Philip Lansberg.</hi>—March the 9th, 1635, he wrote <hi rend="italics">A Letter of Remarks on Gellibrand's Mathematical Discourse on the Variation of the Magnetic Needle.</hi> And the 3d of June following, another on the same subject.</p><p>His eminence in mathematical knowledge was now so great, that be was thought worthy of a professor's chair in that science; and, upon the vacancy of one at Amsterdam in 1639, Sir William Boswell, the English Resident with the States General, used his interest, that he might succeed in that professorship: it was not filled up however till 1643, when Pell was chosen to it; and he read with great applause public lectures upon Diophantus.—In 1644 he printed at Amsterdam, in two pages 4to, <hi rend="italics">A Refutation of Longomontanus's Discourse, De Vera Circuli Mensura.</hi></p><p>In 1646, on the invitation of the Prince of Orange, he removed to the new college at Breda, as Professor of Mathematics, with a salary of 1000 guilders a year.— His <hi rend="italics">Idea Matheseos,</hi> which he had addressed to Mr. Hartlib, who in 1639 had sent it to Des Cartes and Mersenne, was printed 1650 at London, in 12mo, in English, with the title of <hi rend="italics">An Idea of Mathematies,</hi> at the end of Mr. John Durie's Reformed Librarykeeper. It is also printed by Mr. Hook, in his Philosophical Collections, No. 5, p. 127; and is esteemed our author's principal work.</p><p>In 1652 Pell returned to England: and in 1654 he was sent by the protector Cromwell agent to the Protestant Cantons in Switzerland; where he continued till June 23, 1658, when he se<*> out for England, where he arrived about the time of Cromwell's death. His negociations abroad gave afterwards a general satisfaction, as it appeared he had done no small service to the interest of king Charles the Second, and of the church of England; so that he was encouraged to enter into holy orders; and in the year 1661 he was instituted to the rectory of Fobbing in Essex, given him by the king. In December that year, he brought into the upper house of convocation the calendar reformed by him, assisted by Sancroft, afterwards archbishop of Canterbury.—In 1673 he was presented by Sheldon, bishop of London, to the rectory of Laingdon in Essex; and, upon the promotion of that bishop to the see of Canterbury soon after, became one of his domestic chaplains. He was then doctor of divinity, and expected to be made a dean; but his improvement in the philosophical and mathematical sciences was so much the bent of his genius, that he did not much pursue his private advantage. The truth is, he was a helpless man, as to worldly affairs, and his tenants and relations imposed upon him, cozened him of the profits of his parsonage, and kept him so indigent, that he wanted necessaries, even ink and paper, to his dying day. He was for some time confined to the King's-bench prison for debt; but, in March 1682, was invited by Dr. Whitler to live in the college of physicians. Here he continued till June following; when he was obliged,<cb/> by his ill state of health, to remove to the house of a grandchild of his in St. Margaret's Church-yard, Westminster. But he died at the house of Mr. Cothorne, reader of the church of St. Giles's in the Fields, December the 12th, 1685, in the 74th year of his age, and was interred at the expence of Dr. Busby, master of Westminster school, and Mr. Sharp, rector of St. Giles's, in the rector's vault under that church.— Dr. Pell published some other things not yet mentioned, a list of which is as follows: viz,</p><p>1. An Exercitation concerning Easter; 1644, in 4to.</p><p>2. A Table of 10,000 square numbers, &c; 1672, folio.</p><p>3. An Inaugural Oration at his entering upon the Professorship at Breda.</p><p>4. He made great alterations and additions to Rhonius's Algebra, printed at London 1668, 4to, under the title of, An Introduction to Algebra; translated out of the High Dutch into English by Thomas Branker, much altered and augmented by D. P. (Dr. Pell). Also a Table of Odd Numbers, less than 100,000, shewing those that are incomposite, &c, supputated by the same Thomas Branker.</p><p>5. His Controversy with Longomontanus concerning the Quadrature of the Circle; Amsterdam, 1646, 4to.</p><p>He likewise wrote a Demonstration of the 2d and 10th books of Euclid; which piece was in MS. in the library of lord Brereton in Cheshire: as also Archimedes's Arenarius, and the greatest part of Diophantus's 6 books of Arithmetic; of which author he was preparing, Aug. 1644, a new edition, in which he intended to have corrected the translation, and made new illustrations. He designed likewise to publish an edition of Apollonius, but laid it aside, in May, 1645, at the desire of Golius, who was engaged in an edition of that author from an Arabic manuscript given him at Aleppo 18 years before. Letters of Dr. Pell to Sir Charles Cavendish, in the Royal Society.</p><p>Some of his manuscripts he left at Brereton in Cheshire, where he resided some years, being the seat of William lord Brereton, who had been his pupil at Breda. A great many others came into the hands of Dr. Busby; which Mr. Hook was desired to use his endeavours to obtain for the Society. But they continued buried under dust, and mixed with the papers and pamphlets of Dr. Busby, in four large boxes, till 1755; when Dr. Birch, secretary to the Royal Society, procured them for that body, from the trustees of Dr. Busby. The collection contains not only Pell's mathematical Papers, letters to him, and copies of those from him, &c, but also several manuscripts of Walter Warner, the mathematician and philosopher, who lived in the reigns of James the First and Charles the First.</p><p>Dr. Pell invented the method of ranging the several steps of an algebraical calculus, in a proper order, in so many distinct lines, with the number affixed to each step, and a short description of the operation or process in the line. He also invented the character ÷ for division, <*> for involution, and <*> for evolution.</p><p>PENCIL <hi rend="italics">of Rays,</hi> in Optics, is a double cone, or pyramid, of rays, joined together at the base; as<pb n="207"/><cb/> BGSC: the one cone having its vertex in some point of the object at B, and the crystalline humour, or the glass GLS for its base; and the other having its base on the same glass, or crystalline, but its vertex in the point of convergence, as at C. <figure/></p></div1><div1 part="N" n="PENDULUM" org="uniform" sample="complete" type="entry"><head>PENDULUM</head><p>, in Mechanics, any heavy body, so suspended as that it may swing backwards and forwards, about some fixed point, by the force of gravity.</p><p>These alternate ascents and descents of the Pendulum, are called its Oscillations, or Vibrations; each complete oscillation being the descent from the highest point on one side, down to the lowest point of the arch, and so on up to the highest point on the other side. The point round which the Pendulum moves, or vibrates, is called its Centre of Motion, or Point of Suspension; and a right line drawn through the centre of motion, parallel to the horizon, and perpendicular to the plane in which the Pendulum moves, is called the Axis of Oscillation. There is also a certain point within every Pendulum, into which, is all the matter that composes the Pendulum were collected, or condensed as into a point, the times in which the vibrations would be performed, would not be altered by such condensation; and this point is called Centre of Oscillation. The length of the Pendulum is always estimated by the distance of this point below the centre of motion; being usually near the bottom of the Pendulum; but in a cylinder, or any other uniform prism or rod, it is at the distance of one third from the bottom, or twothirds from and below the centre of motion.</p><p>The length of a Pendulum, so measured to its centre of oscillation, that it will perform each vibration in a second of time, thence called the second's Pendulum, has, in the latitude of London, been generally taken at 39 2/10 or 39 1/5 inches; but by some very ingenious and accurate experiments, the late celebrated Mr. George Graham found the true length to be 39 128/1000, inches, or 39 1/8 inches very nearly.</p><p>The length of the Pendulum vibrating seconds at Paris, was found by Varin, Des Hays, De Glos, and Godin, to be 440 5/<*> lines; by Picard 440 1/2 lines; and by Mairan 440 1<*>/30 lines.</p><p>Galileo was the first who made use of a heavy body annexed to a thread, and suspended by it, for measuring time, in his experiments and observations. But according to Sturmius, it was Riccioli who first observed the isochronism of Pendulums, and made use of them in measuring time. After him, Tycho, Langrene, Wendeline, Mersenne, Kircher, and others, observed the same thing; though, it is said, without any intimation of what had been done by Riccioli. But it was the celebrated Huygens who first demonstrated the principles and properties of Pendulums, and probably the first who applied them to clocks. He demon-<cb/> strated, that if the centre of motion were perfectly fixed and immoveable, and all manner of friction, and resistance of the air, &c, removed, then a Pendulum, once set in motion, would for ever continue to vibrate without any decrease of motion, and that all its vibrations would be perfectly isochronal, or performed in the same time. Hence the Pendulum has universally been considered as the best chronometer or measurer of time. And as all Pendulums of the same length perform their vibrations in the same time, without regard to their different weights, it has been suggested, by means of them, to establish an universal standard for all countries. On this principle Mouton, canon of Lyons, has a treatise, De Mensura posteris transmittenda; and several others since, as Whitehurst, &c. See <hi rend="italics">Universal</hi> <hi rend="smallcaps">Measure.</hi></p><p>Pendulums are either simple or compound, and each of these may be considered either in theory, or as in practical mechanics among artisans. <figure/></p><p><hi rend="italics">A Simple</hi> <hi rend="smallcaps">Pendulum</hi>, in Theory, consists of a single weight, as A, considered as a point, and an inflexible right line AC, supposed void of gravity or weight, and suspended from a fixed point or centre C, about which it moves.</p><p><hi rend="italics">A Compound</hi> <hi rend="smallcaps">Pendulum</hi>, in Theory, is a Pendulum consisting of several weights moveable about one common centre of motion, but connected together so as to retain the same distance both from one another, and from the centre about which they vibrate.</p><p><hi rend="italics">The Doctrine and Laws of</hi> <hi rend="smallcaps">Pendulums.</hi>—1. A Pendulum raised to B, through the arc of the circle AB, will fall, and rise again, through an equal arc, to a point equally high, as D; and thence will fall to A, and again rise to B; and thus continue rising and falling perpetually. For it is the same thing, whether the body fall down the inside of the curve BAD, by the force of gravity, or be retained in it by the action of the string; for they will both have the same effect; and it is otherwise known, from the oblique descents of bodies, that the body will descend and ascend along the curve in the manner above described.</p><p>Experience also consirms this theory, in any finite number of oscillations. But if they be supposed infinitely continued, a difference will arise. For the resistance of the air, and the friction and rigidity of the string about the centre C, will take off part of the force acquired in falling; whence it happens that it will not rise precisely to the same point from whence it fell.</p><p>Thus, the ascent continually diminishing the oscillation, this will be at last stopped, and the Pendulum will hang at rest in its natural direction, which is perpendicular to the horizon.</p><p>Now as to the real time of oscillation in a circular arc BAD: it is demonstrated by mathematicians, that if , denote the circumference of a circle whose diameter is 1; feet or 193 inches, the space a heavy body falls in the first second of time; and the length of the Pendulum; also the height of the arch of vibration; then the<pb n="208"/><cb/> time of each oscillation in the arc BAD will be equal to into the infinite series , where is the diameter of the arc described, or twice the length of the Pendulum.</p><p>And here, when the arc is a small one, as in the case of the vibrating Pendulum of a clock, all the terms of this series after the 2d may be omitted, on account of their smallness; and then the time of a whole vibration will be nearly equal to . So that the times of vibration of a Pendulum in different small arcs of the same circle, are as , or 8 times the radius, added to the versed sine of the semiarc.</p><p>And farther, if D denote the number of degrees in the semiarc AB, whose versed fine is <hi rend="italics">a,</hi> then the quantity last mentioned, for the time of a whole vibration, is changed to . And therefore the times of vibration in different small arcs, are as , or as the number 52524 added to the square of the number of degrees in the semiarc AB. See my Conic Sections and Select Exercises, p. 190. <figure/></p><p>2. Let CB be a semicycloid, having its base EC parallel to the hori<*>on, and its vertex B downwards; and let CD be the other half of the cycloid, in a similar position to the former. Suppose a Pendulum string, of the same length with the curve of each semicycloid BC, or CD, having its end sixed in C, and the thread applied all the way close to the cycloidal curve BC, and consequently the body or Pendulum weight coinciding with the point B. If now the body be let go from B, it will descend by its own gravity, and in descending it will unwind the string from off the arch BC, as at the position CGH; and the ball G will describe a semicycloid BHA, equal and similar to BGC, when it has arrived at the lowest point A; after which, it will continue its motion, and ascend, by another equal and similar semicycloid AKD, to the same height D, as it fell from at B, the string now wrapping itself upon the other arch CID. From D it will descend again, and pass along the whole cycloid DAB, to the point B; and thus perform continual successive oscillations between B and D, in the curve of a cycloid; as it before oscillated in the curve of a circle, in the former case.<cb/></p><p>This contrivance to make the Pendulum oscillate in the curve of a cycloid, is the invention of the celebrated Huygens, to make the Pendulum perform all its vibrations in equal times, whether the arch, or extent of the vibration be great or small; which is not the case in a circle, where the larger arcs take a longer time to run through them, than the smaller ones do, as is well known both from theory and practice.</p><p>The chief properties of the cycloidal Pendulum then, as demonstrated by Huygens, are the following. 1st, That the time of an oscillation in all arcs, whether larger or smaller, is always the same quantity, viz, whether the body begin to descend from the point B, and describe the semiarch BA; or that it begins at H, and describes the arch HA; or that it sets out from any other point; as it will still descend to the lowest point A in exactly the same time. And it is farther proved, that the time of a whole vibration through any double arc BAD, or HAK. &c, is in proportion to the time in which a heavy body will freely fall, by the force of gravity, through a space equal to (1/2)AC, half the length of the Pendulum, as the circumference of a circle is to its diameter. So that, if feet denote the space a heavy body falls in the first second of time, the circumference of a circle whose diameter is 1, and the length of the Pendulum; then, because, by the nature of descents by gravity, that is the time in which a body will fall through (1/2)<hi rend="italics">r,</hi> or half the length of the Pendulum; therefore, by the above proportion, as , which is the time of an entire oscillation in the cycloid.</p><p>And this conclusion is abundantly confirmed by experience. For example, if we consider the time of a vibration as 1 second, to find the length of the Pendulum that will so oscillate in 1 second; this will give the equation ; which reduced, gives inches = 39.11 or 39 1/9 inches, for the length of the second's Pendulum; which the best experiments shew to be about 39 1/8 inches.</p><p>3. Hence also, we have a method of determining, from the experimented length of a Pendulum, the space a heavy body will fall perpendicularly through in a given time: for, since , therefore, by reduction, is the space a body will fall through in the first second of time, when <hi rend="italics">r</hi> denotes the length of the second's Pendulum; and as constant experience shews that this length is nearly 39 1/8 inches, in the latitude of London, in this case <hi rend="italics">g</hi> or becomes inches = 16 1/12 feet, very nearly, for the space a body will fall in the first second of time, in the latitude of London: a fact which has been abundantly confirmed by experiments made there. And in the same manner, Mr. Huygens found the same space fallen through at Paris, to be 15 French feet.</p><p>The whole doctrine of Pendulums, oscillating between two semicycloids, both in theory and practice,<pb n="209"/><cb/> was delivered by that author, in his Horologium Oscillatorium, sive Demonstrationes de Motu Pendulorum. And every thing that regards the motion of Pendulums has since been demonstrated in different ways, and particularly by Newton, who has given an admirable theory on the subject, in his Principia, where he has extended to epicycloids the properties demonstrated by Huygens of the cycloids.</p><p>4. As the cycloid may be considered as coinciding, in A, with any small are of a circle described from the centre C, passing through A, where it is known the two curves have the same radius and curvature; therefore the time in the small arc of such a circle, will be nearly equal to the time in the cycloid; so that the times in very small circular arcs are equal, because these small arcs may be considered as portions of the cycloid, as well as of the circle. And this is one great reason why the Pendulums of clocks are made to oscillate in as small arcs as possible, viz, that their oscillations may be the nearer to a constant equality.</p><p>This may also be deduced from a comparison of the times of vibration in the circle, and in the cycloid, as laid down in the foregoing articles. It has there been shewn, that the times of vibration in the circle and cycloid are thus, viz, time in the circle nearly , time in the cycloidal arc ; where it is evident, that the former always exceeds the latter in the ratio of to 1; but this ratio always approaches nearer to an equality, as the arc, or as its versed sine <hi rend="italics">a,</hi> is smaller; till at length, when it is very small, the term <hi rend="italics">a</hi>/8<hi rend="italics">r</hi> may be omitted, and then the times of vibration become both the same quantity, viz .</p><p>Farther, by the same comparison, it appears, that the time lost in each second, or in each vibration of the second's Pendulum, by vibrating in a circle, instead of a cycloid, is <hi rend="italics">a</hi>/8<hi rend="italics">r,</hi> or ; and consequently the time lost in a whole day of 24 hours, is (5/3)D<hi rend="sup">2</hi> nearly. In like manner, the seconds lost per day by vibrating in the are of ▵ degrees, is (5/3)▵<hi rend="sup">2</hi>. Therefore if the Pendulum keep true time in one of these arcs, the seconds lost or gained per day, by vibrating in the other, will be (5/3)(D<hi rend="sup">2</hi> - ▵<hi rend="sup">2</hi>). So, for example, if a Pendulum measure true time in an arc of 3 degrees, on each side of the lowest point, it will lose 11 2/3 seconds a day by vibrating 4 degrees; and 26 2/3 seconds a day by vibrating 5 degrees; and so on.</p><p>5. The action of gravity is less in those parts of the earth where the oscillations of the same Pendulum are slower, and greater where these are swifter; for the time of oscillation is reciprocally proportional to √<hi rend="italics">g.</hi> And it being found by experiment, that the oscillations of the same Pendulum are slower near the equator, than in places farther from it; it follows that the force of<cb/> gravity is less there; and consequently the parts about the equator are higher or farther from the centre, than the other parts; and the shape of the earth is not a true sphere, but somewhat like an oblate spheroid, flatted at the poles, and raised gradually towards the equator. And hence also the times of the vibration of the same Pendulum, in different latitudes, afford a method of determining the true figure of the earth, and the proportion between its axis and the equatorial diameter.</p><p>Thus, M. Richer found by an experiment made in the island Cayenna, about 4 degrees from the equator, where a Pendulum 3 feet 8 2/5 lines long, which at Paris vibrated seconds, required to be shortened a line and a quarter to make it vibrate seconds. And many other observations have confirmed the same principle. See Newton's Principia, lib. 3, prop. 20. By comparing the different observations of the French astronomers, Newton apprehends that 2 lines may be considered as the length a seconds Pendulum ought to be decreased at the equator.</p><p>From some observations made by Mr. Campbell, in 1731, in Black<*>river, in Jamaica, 18° north latitude, it is collected, that if the length of a simple Pendulum that swings seconds in London, be 39.126 English inches, the length of one at the equator would be 39.00, and at the poles 39.206. Philos. Trans. numb. 432; or Abr. vol. 8, part 1, pa. 238.</p><p>And hence Mr. Emerson has computed the following Table, shewing the length of a Pendulum that swings seconds at every 5th degree of latitude, as also the length of the degree of latitude there, in English miles. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Degrees of Lat.</cell><cell cols="1" rows="1" role="data">Length of Pendulum.</cell><cell cols="1" rows="1" role="data">Length of the Degree.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">inches.</cell><cell cols="1" rows="1" role="data">miles.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">39.027</cell><cell cols="1" rows="1" rend="align=center" role="data">68.723</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=center" role="data">39.029</cell><cell cols="1" rows="1" rend="align=center" role="data">68.730</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">39.032</cell><cell cols="1" rows="1" rend="align=center" role="data">68.750</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=center" role="data">39.036</cell><cell cols="1" rows="1" rend="align=center" role="data">68.783</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">39.044</cell><cell cols="1" rows="1" rend="align=center" role="data">68.830</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" rend="align=center" role="data">39.057</cell><cell cols="1" rows="1" rend="align=center" role="data">68.882</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">39.070</cell><cell cols="1" rows="1" rend="align=center" role="data">68.950</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" rend="align=center" role="data">39.084</cell><cell cols="1" rows="1" rend="align=center" role="data">69.020</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=center" role="data">39.097</cell><cell cols="1" rows="1" rend="align=center" role="data">69.097</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=center" role="data">39.111</cell><cell cols="1" rows="1" rend="align=center" role="data">69.176</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" rend="align=center" role="data">39.126</cell><cell cols="1" rows="1" rend="align=center" role="data">69.256</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" rend="align=center" role="data">39.142</cell><cell cols="1" rows="1" rend="align=center" role="data">69.330</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=center" role="data">39.158</cell><cell cols="1" rows="1" rend="align=center" role="data">69.401</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" rend="align=center" role="data">39.168</cell><cell cols="1" rows="1" rend="align=center" role="data">69.467</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=center" role="data">39.177</cell><cell cols="1" rows="1" rend="align=center" role="data">69.522</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">75</cell><cell cols="1" rows="1" rend="align=center" role="data">39.185</cell><cell cols="1" rows="1" rend="align=center" role="data">69.568</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">80</cell><cell cols="1" rows="1" rend="align=center" role="data">39.191</cell><cell cols="1" rows="1" rend="align=center" role="data">69.601</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" rend="align=center" role="data">39.195</cell><cell cols="1" rows="1" rend="align=center" role="data">69.620</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" rend="align=center" role="data">39.197</cell><cell cols="1" rows="1" rend="align=center" role="data">69.628</cell></row></table></p><p>6. If two Pendulums vibrate in similar arcs, the times of vibration are in the sub-duplicate ratio of their lengths. And the lengths of Pendulums vibrating in similar arcs, are in the duplicate ratio of the times<pb n="210"/><cb/> of a vibration directly; or in the reciprocal duplicate ratio of the number of oscillations made in any one and the same time. For, the time of vibration <hi rend="italics">t</hi> being as , where <hi rend="italics">p</hi> and <hi rend="italics">g</hi> are constant or given, therefore <hi rend="italics">t</hi> is as √<hi rend="italics">r,</hi> and <hi rend="italics">r</hi> as <hi rend="italics">t</hi><hi rend="sup">2</hi>. Hence therefore the length of a half-second Pendulum will be 1/4<hi rend="italics">r</hi> or inches; and the length of the quarter-second Pendulum will be inches; and so of others.</p><p>7. The foregoing laws, &c, of the motion of Pendulums, cannot strictly hold good, unless the thread that sustains the ball be void of weight, and the gravity of the whole ball be collected into a point. In practice therefore, a very fine thread, and a small ball, but of a very heavy matter, are to be used. But a thick thread, and a bulky ball, disturb the motion very much; for in that case, the simple Pendulum becomes a compound one; it being much the same thing, as if s<*>ral weights were applied to the same inflexible rod in several places.</p><p>8. M. Krafft in the new Petersburgh Memoirs, vols 6 and 7, has given the result of many experiments upon Pendulums, made in different parts of Russia, with deductions from them, from whence he derives this theorem: If <hi rend="italics">x</hi> be the length of a Pendulum that swings seconds in any given latitude <hi rend="italics">l,</hi> and in a temperature of 10 degrees of Reaumur's thermometer, then will the length of that Pendulum, for that latitude, be thus expressed, viz, lines of a French foot. And this expression agrees very nearly, not only with all the experiments made on the Pendulum in Russia, but also, with those of Mr. Graham, and those of Mr. Lyons in 79° 50′ north latitude, where he found its length to be 441.38 lines. See <hi rend="smallcaps">Oblateness.</hi></p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Pendulum</hi>, in Mechanics, an expression commonly used among artists, to distinguish such Pendulums as have no provision for correcting the effects of heat and cold, from those that have such provision. Also Simple Pendulum, and Detached Pendulum, are terms sometimes used to denote such Pendulums as are not connected with any clock, or clock-work.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Pendulum</hi>, in Mechanics, is a Pendulum whose rod is composed of two or more wires or bars of metal. These, by undergoing different degrees of expansion and contraction, when exposed to the same heat or cold, have the difference of expansion or contraction made to act in such manner as to preserve constantly the same distance between the point of suspension, and centre of oscillation, although exposed to very different and various degrees of heat or cold. There are a great variety of constructions for this purpose; but they may be all reduced to the Gridiron, the Mercurial, and the Lever Pendulum.</p><p>It may be just observed by the way, that the vulgar method of remedying the inconvenience arising from the extension and contraction of the rods of common Pendulums, is by applying the bob, or small ball, with a screw, at the lower end; by which means the Pendulum is at any<cb/> time made longer or shorter, as the ball is screwed downwards or upwards, and thus the time of its vibration is kept continually the same.</p><p>The <hi rend="italics">Gridiron</hi> <hi rend="smallcaps">Pendulum</hi> was the invention of Mr. John Harrison, a very ingenious artist, and celebrated for his invention of the watch for finding the difference of longitude at sea, about the year 1725; and of several other time keepers and watches since that time; for all which he received the parliamentary reward of between 20 and 30 thousand pounds. It consists of 5 rods of steel, and 4 of brass, placed in an alternate order, the middle rod being of steel, by which the Pendulum ball is suspended; these rods of brass and steel, thus placed in an alternate order, and so connected with each other at their ends, that while the expansion of the steel rods has a tendency to lengthen the Pendulum, the expansion of the brass rods, acting upwards, tends to shorten it. And thus, when the lengths of the brass and steel rods are duly proportioned, their expansions and contractions will exactly balance and correct each other, and so preserve the Pendulum invariably of the same length. The simplicity of this ingenious contrivance is much in its favour; and the difficulty of adjustment seems the only objection to it.</p><p>Mr. Harrison in his first machine for measuring time at sea, applied this combination of wires of brass and steel, to prevent any alterations by heat or cold; and in the machines or clocks he has made for this purpose, a like method of guarding against the irregularities arising from this cause is used.</p><p>The <hi rend="italics">Mercurial</hi> <hi rend="smallcaps">Pendulum</hi> was the invention of the ingenious Mr. Graham, in consequence of several experiments relating to the materials of which Pendulums might be formed, in 1715. Its rod is made of brass, and branched towards its lower end, so as as to embrace a cylindric glass vessel 13 or 14 inches long, and about 2 inches diameter; which being filled about 12 inches deep with mercury, forms the weight or ball of the Pendulum. If upon trial the expansion of the rod be found too great for that of the mercury, more mercury must be poured into the vessel: if the expansion of the mercury exceeds that of the rod, so as to occasion the clock to go fast with heat, some mercury must be taken out of the vessel, so as to shorten the column. And thus may the expansion and contraction of the quicksilver in the glass be made exactly to balance the expansion and contraction of the Pendulum rod, so as to preserve the distance of the centre of oscillation from the point of suspension invariably the same.</p><p>Mr. Graham made a clock of this sort, and compared it with one of the best of the common sort, for 3 years together; when he found the errors of his but about one-eighth part of those of the latter. Philos. Trans. numb. 392.</p><p>The <hi rend="italics">Lever</hi> <hi rend="smallcaps">Pendulum.</hi> From all that appears concerning this construction of a Pendulum, we are inclined to believe that the idea of making the difference of the expansion of different metals operate by means of a lever, originated with Mr. Graham, who in the year 1737 constructed a Pendulum, having its rod composed of one bar of steel between two of brass, which acted upon the short end of a lever, to the other end of which, the ball or weight of the Pendulum was sus- pended.<pb n="211"/><cb/></p><p>This Pendulum however was, upon trial, found to move by jerks; and therefore laid aside by the inventor, to make way for the mercurial Pendulum, just mentioned.</p><p>Mr. Short informs us in the Philos. Trans. vol. 47, art. 88, that a Mr. Frotheringham, a quaker in Lincolnshire, caused a Pendulum of this kind to be made: it consisted of two bars, one of brass, and the other of steel, fastened together by screws, with levers to raise or let down the bulb; above which these levers were placed. M. Cassini too, in the History of the Royal Academy of Sciences at Paris, for 1741, describes two sorts of Pendulums for clocks, compounded of bars of brass and steel, and in which he applies a lever to raise or let down the bulb of the Pendulum, by the expansion or contraction of the bar of brass.</p><p>Mr. John Ellicott also, in the year 1738, constructed a Pendulum on the same principle, but differing from Mr. Graham's in many particulars. The rod of Mr. Ellicott's Pendulum was composed of two bars only; the one of brass, and the other of steel. It had two levers, each sustaining its half of the ball or weight; with a spring under the lower part of the ball to relieve the levers from a considerable part of its weight, and so to render their motion more smooth and easy. The one lever in Mr. Graham's construction was above the ball: whereas both the levers in Mr. Ellicott's were within the ball; and each lever had an adjusting screw, to lengthen or shorten the lever, so as to render the adjustment the more perfect. See the Philos. Trans. vol. 47, p. 479; where Mr. Ellicott's methods of construction are described, and illustrated by figures.</p><p>Notwithstanding the great ingenuity displayed by these very eminent artists on this construction, it must farther be observed, in the history of improvements of this nature, that Mr. Cumming, another eminent artist, has given, in his Essays on the Principles of Clock and Watch-work, Lond. 1766, an ample description, with plates, of a construction of a Pendulum with levers, in which it seems he has united the properties of Mr. Graham's and Mr. Ellicott's, without being liable to any of the defects of either. The rod of this Pendulum is composed of one flat bar of brass, and two of steel; he uses three levers within the ball of the Pendulum; and, among many other ingenious contrivances, for the more accurate adjusting of this Pendulum to mean time, it is provided with a small ball and screw below the principal ball or weight, one entire revolution of which on its screw will only alter the rate of the clock's going one second per day; and its circumference is divided into 30, one of which divisions will therefore alter its rate of going one second in a month.</p><p><hi rend="smallcaps">Pendulum</hi> <hi rend="italics">Clock,</hi> is a clock having its motion regulated by the vibration of a Pendulum.</p><p>It is controverted between Galileo and Huygens, which of the two first applied the Pendulum to a clock. For the pretensions of each, see <hi rend="smallcaps">Clock.</hi></p><p>After Huygens had discovered, that the vibration made in arcs of a cycloid, however unequal they might be in extent, were all equal in time; he soon perceived, that a Pendulum applied to a clock, so as to make it describe arcs of a cycloid, would rectify the otherwise <*>oidable irregularities of the motion of the clock;<cb/> since, though the several causes of those irrogularities should occasion the Pendulum to make greater or smaller vibrations, yet, by virtue of the cycloid, it would still make them perfectly equal in point of time; and the motion of the clock governed by it, would therefore be preserved perfectly equable. But the difficulty was, how to make the Pendulum describe arcs of a cycloid; for naturally the Pendulum, being tied to a fixed point, can only describe circular arcs about it.</p><p>Here M. Huygens contrived to fix the iron rod or wire, which bears the ball or weight, at the top to a silken thread, placed between two cycloidal cheeks, or two little arcs of a cycloid, made of metal. Hence the motion of vibration, applying successively from one of those arcs to the other, the thread, which is extremely flexible, easily assumes the figure of them, and by that means causes the ball or weight at the bottom to describe a just cycloidal arc.</p><p>This is doubtless one of the most ingenious and useful inventions many ages have produced: by means of which it has been asserted there have been clocks that would not vary a single second in several days: and the same invention also gave rise to the whole doctrine of involute and evolute curves, with the radius and degree of curvature, &c.</p><p>It is true, the Pendulum is still liable to its irregularities, how minute soever they may be. The silken thread by which it was suspended, shortens in moist weather, and lengthens in dry; by which means the length of the whole Pendulum, and consequently the times of the vibrations, are somewhat varied.</p><p>To obviate this inconvenience, M. De la Hire, instead of a silken thread, used a little fine spring; which was not indeed subject to shorten and lengthen, from those causes; yet he found it grew stiffer in cold weather, and then made its vibrations faster than in warm; to which also we may add its expansion and contraction by heat and cold. He therefore had recourse to a stiff wire or rod, firm from one end to the other. Indeed by this means he renounced the advantages of the cycloid; but he found, as he says, by experience, that the vibrations in circular arcs are performed in times as equal, provided they be not of too great extent, as those in cycloids. But the experiments of Sir Jonas Moore, and others, have demonstrated the contrary.</p><p>The ordinary causes of the irregularities of Pendulums Dr. Derham ascribes to the alterations in the gravity and temperature of the air, which increase and diminish the weight of the ball, and by that means make the vibrations greater and less; an accession of weight in the ball being found by experiment to accelerate the motion of the Pendulum; for a weight of 6 pounds added to the ball, Dr. Derham found made his clock gain 13 seconds every day.</p><p>A general remedy against the inconveniences of Pendulums, is to make them long, the ball heavy, and to vibrate but in small arcs. These are the usual means employed in England; the cycloidal checks being g<*>nerally neglected. See the foregoing article.</p><p>Pendulum clocks resting against the same rail have been found to influence each other's motion. See the Philos. Trans. numb. 453, sect. 5 and 6, where Mr. Ellicott has given a curious and exact account of this phenomenon.<pb n="212"/><cb/></p><p><hi rend="smallcaps">Pendulum</hi> <hi rend="italics">Royal,</hi> a name used among us for a clock, whose Pendulum swings seconds, and goes 8 days without winding up; shewing the hour, minute, and second. The numbers in such a piece are thus calculated. First cast up the seconds in 12 hours, which are the beats in one turn of the great wheel; and they will be found to be 43200 = 12 X 60 X 60. The swing wheel must be 30, to swing 60 seconds in one of its revolutions; now let the half of 43200, viz 21600, be divided by 30, and the quotient will be 720, which must be separated into quotients. The first of these must be 12, for the great wheel, which moves round once in 12 hours. Now 720 divided by 12, gives 60, which may also be conveniently broken into two quotients, as 10 and 6, or 12 and 5, or 8 and 7 1/2; which last is most convenient: and if the pinions be all taken 8, the work will stand thus: <table><row role="data"><cell cols="1" rows="1" role="data">8 )</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" rend="align=right" role="data">(     12</cell></row><row role="data"><cell cols="1" rows="1" role="data">8 )</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" rend="align=right" role="data">(       8</cell></row><row role="data"><cell cols="1" rows="1" role="data">8 )</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">( 7 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">   30</cell></row></table></p><p>According to this computation, the great wheel will go round once in 12 hours, to shew the hour; the next wheel once in an hour, to shew the minutes; and the swing-wheel once in a minute, to shew the seconds. See <hi rend="smallcaps">Clock-work.</hi></p><p><hi rend="italics">Ballistic</hi> <hi rend="smallcaps">Pendulum.</hi> See <hi rend="smallcaps">Ballistic</hi> <hi rend="italics">Pendulum.</hi></p><p><hi rend="italics">Level</hi> <hi rend="smallcaps">Pendulum.</hi> See <hi rend="smallcaps">Level.</hi></p><p><hi rend="smallcaps">Pendulum</hi> <hi rend="italics">Watch.</hi> See <hi rend="smallcaps">Watch.</hi></p></div1><div1 part="N" n="PENETRABILITY" org="uniform" sample="complete" type="entry"><head>PENETRABILITY</head><p>, capability of being penetrated. See <hi rend="smallcaps">Impenetrability.</hi></p></div1><div1 part="N" n="PENETRATION" org="uniform" sample="complete" type="entry"><head>PENETRATION</head><p>, the act by which one thing enters another, or takes up the place already possessed by another.</p><p>The schoolmen define Penetration the co-existence of two or more bodies, so that one is present, or has its extension in the same place as the other.</p><p>Most philosophers hold the penetration of bodies absurd, i. e. that two bodies should be at the same time in the same place; and accordingly impenetrability is laid down as one of the essential properties of matter.</p><p>What is popularly called Penetration, only amounts to the matter of one body's being admitted into the vacuity of another. Such is the Penetration of water through the substance of gold.</p></div1><div1 part="N" n="PENINSULA" org="uniform" sample="complete" type="entry"><head>PENINSULA</head><p>, is a portion or extent of land which is almost surrounded with water, being joined to the continent only by an isthmus, or narrow neck. Such is Africa, the greatest Peninsula in the world, which is joined to Asia, by the neck at the end of the Red Sea; fuch also is Peloponnesus, or the Morea, joined to Greece: and Jutland, &c. Peninsula is the same with what is otherwise called Chersonesus.</p></div1><div1 part="N" n="PENNY" org="uniform" sample="complete" type="entry"><head>PENNY</head><p>, formerly a piece of silver coin, but now an imaginary sum, equal to two copper coins called a halfpenny.</p><p>The Penny was the first silver coin struck in England by our Saxon ancestors, being the 240th part of their<cb/> pound, and its true weight was about 22 1/2 grains Troy.</p><p>In Etheldred's time, the Penny was the 20th part of the Troy ounce, and equal in weight to our three pence; which value it retained till the time of Edward the Third.</p><p>Till the time of King Edward the First, the Penny was struck with a cross so deeply sunk in it, that it might, on occasion, be easily broken, and parted into two halves, thence called Halfpennies; or into four, thence called Fourthings, or Farthings. But that Prince coined it without the cross; instead of which he struck round Halfpence and Farthings. Though there are said to be instances of such round Halfpence having been made in the reign of Henry the First, if not also in that of the two Williams.</p><p>Edward the First also reduced the weight of the Penny to a standard; ordering that it should weigh <*>2 grains of wheat, taken out of the middle of the ear. This Penny was called the Penny Sterling; and 20 of them were to weigh an ounce; whence the Penny became a weight as well as a coin.</p><p>By the 9th of Edward the Third, it was diminished to the 26th part of the Troy ounce; by the 2d of Henry the Sixth it was the 32d part; by the 5th of Edward the Fourth, it became the 40th, and also by the 36th of Henry the Eighth, and afterwards, the 45th; but by the 2d of Elizabeth, 60 Pence were coined out of the ounce, and during her reign 62, which last proportion is still observed in our times.</p><p>The Penny Sterling is now disused as a coin; and scarce subsists, but as a money of account, containing two copper Halfpence, or the 12th part of a shilling, or the 240th part of a pound.</p><p>The French Penny, or Denier, is of two kinds; the Paris Penny, called Denier Parisis; and the Penny of Tours, called Denier Tournois.</p><p>The Dutch Penny, called Pennink, or Pening, is a real money, worth about one-fifth more than the French Penny Tournois. The Pennink is also used as a money of account, in keeping books by pounds, florins, and patards; 12 Penninks make the patard, and 20 patards the florin.</p><p>At Hamburg, Nuremberg, &c, the Penny or Pfennig of account is equal to the French Penny Tournois. Of these, 8 make the krieuk; and 60 the florin of those cities; also 90 the French crown, or 4s 6d sterling.</p><p><hi rend="smallcaps">Penny</hi>-<hi rend="italics">Weight,</hi> a Troy weight, being the 20th part of an ounce, containing 24 grains; each grain weighing a grain of wheat gathered out of the middle of the ear, well dried. The name took its rise from its being actually the weight of one of our ancient silver Pennies. See the foregoing article.</p></div1><div1 part="N" n="PENTAGON" org="uniform" sample="complete" type="entry"><head>PENTAGON</head><p>, in Geometry, a plane figure consisting of five angles, and consequently sive sides also. If the angles be all equal, it is a regular Pentagon.</p><p>It is a remarkable property of the Pentagon, that its side is equal in power to the sides of a hexagon and a decagon inscribed in the same circle; that is, the square of the side of the Pentagon, is equal to both the squares taken together of the sides of the other two sigures; and consequently those three sides will consti-<pb n="213"/><cb/> tute a right-angled triangle. Euclid, book 13, prop. 10.</p><p>Pappus has also demonstrated, that 12 regular Pentagons contain more than 20 triangles inscribed in the same circle; lib. 5, prop. 45.</p><p>The dodecahedron, which is the fourth regular body or solid, is contained under 12 equal and regular Pentagons.</p><p><hi rend="italics">To find the Area of a Regular</hi> <hi rend="smallcaps">Pentagon.</hi> Multiply the square of its side by 1.7204774, or by 5/4 of the tangent of 54°, or by . Hence if <hi rend="italics">s</hi> de<*> note the side of the Pentagon, its area will be .</p></div1><div1 part="N" n="PENTAGRAPH" org="uniform" sample="complete" type="entry"><head>PENTAGRAPH</head><p>, otherwise called a Parallelogram, a mathematical instrument for copying designs, prints, plans, &c, in any proportion.</p><p>The common Pentagraph (Plate xix, fig. 2) consists of four rulers or bars, of metal or wood, two of them from 15 to 18 inches long, the other two half that length. At the ends, and in the middle, of the long rulers, as also at the ends of the shorter ones, are holes upon the exact fixing of which the perfection of the instrument chiefly depends. Those in the middle of the long rulers are to be at the same distance from those at the end of the long ones, and those of the short ones; so that, when put together, they may always make a parallelogram.</p><p>The instrument is fitted together for use, by several little pieces, particularly a little pillar, number 1, having at one end a nut and screw, joining the two long rulers together; and at the other end a small knot for the instrument to slide on. The piece numb. 2 is a rivet with a screw and nut by which each short ruler is fastened to the middle of each long one. The piece numb. 3 is a pillar, one end of which, being hollowed into a screw, has a nut fitted to it; and at the other end is a worm to screw into the table; when the instrument is to be used, it joins the ends of the two short rulers. The piece numb. 4 is a pen, or pencil, or portcrayon, screwed into a little pillar. Lastly, the piece numb. 5 is a brass point, moderately blunt, screwed likewise into a little pillar.</p><p><hi rend="italics">Use of the</hi> <hi rend="smallcaps">Pentagraph.</hi>—1. To copy a design in the same size or scale as the original. Screw the worm numb. 3 into the table; lay a paper under the pencil numb. 4, and the design under the point numb. 5. This done, conducting the point over the several lines and parts of the design, the pencil will draw or repeat the same on the paper.</p><p>2. When the design is to be reduced - ex. gr. to half the scale; the worm must be placed at the end of the long ruler numb. 4, and the paper and pencil in the middle. In this situation conduct the brass point over the several lines of the design, as before; and the pencil at the same time will draw its copy in the proportion required; the pencil here only moving half the lengths that the point moves.</p><p>3. On the contrary, when the design is to be enlarged to a double size; the brass point, with the design, must be placed in the middle at numb. 3, the pencil and paper at the end of the long ruler, and the worm at the other end.</p><p>4. To reduce or enlarge in other proportions, there<cb/> are holes drilled at equal distances on each ruler; viz, all along the short ones, and half way of the long ones, for placing the brafs point, pencil, and worm, in a right line in them; i. e. if the piece carrying the point be put in the third hole, the other two pieces must be put each in its third hole; &c.</p></div1><div1 part="N" n="PENTANGLE" org="uniform" sample="complete" type="entry"><head>PENTANGLE</head><p>, a plane figure of five angles, or the same as the <hi rend="smallcaps">Pentagon.</hi></p></div1><div1 part="N" n="PENUMBRA" org="uniform" sample="complete" type="entry"><head>PENUMBRA</head><p>, in Astronomy, a faint or partial shade, in an eclipse, observed between the perfect shadow, and the full light.</p><p>The Penumbra arises from the magnitude of the sun's body: were he only a luminous point, the shadow would be all perfect; but by reason of the diameter of the sun it happens, that a place which is not illuminated by the whole body of the sun, does yet receive rays from some part of it.</p><p>Thus, suppose S the sun, and T the moon, and the shadow of the latter projected on a plane, as GH (Plate xix, fig. 3). The true proper shadow of T, viz GH, will be encompassed with an imperfect shadow, or Penumbra, HL and GE, each portion of which is illuminated by an entire hemisphere of the sun.</p><p>The degree of light or shade of the Penumbra, will be more or less in different parts, as those parts lie open to the rays of a greater or less part of the sun's body; thus from L to H, and from E to G, the light continually diminishes; and in the consines of G and H, the Penumbra is darkest, and becomes lost and confounded with the total shade: as near E and L it is thin and confounded with the total light.</p><p>A Penumbra must be found in all eclipses, whether of the sun, the moon, or the other planets, primary or secondary; but it is most considerable with us in eclipses of the sun; which is the case here referred to.</p><p>The Penumbra extends infinitely in length, and grows still wider and wider; two rays drawn from the two extremities of the earth's diameter, and which proceed always diverging, form its two edges; all that infinite diverging space, included between lines passing through E and L, is the Penumbra, except the cone o<*> the shadow in the middle of it. <figure/></p><p>To determine how much of the surface of the earth can be involved in the Penumbra, let the apparent semidiameter of the sun be supposed the greatest, or about 16′ 20″, which is when the earth is in her perihelion; also let the moon be in her apogee, and therefore at her greatest distance from the earth, or about 64 of the earth's semidiameters. Let KNC be the earth, T the moon, and MKN the Penumbra, involving the part of the earth from K to N, which it is required to find. Here then are given the angle KMC = 16′ 20″, TC = 64, KC = 1, and OT = 11/40 of KC. Hence, in the right-angled triangle OTM, as fin. OMT: radius : : OT : TM = 210 1/2OT = 58KC nearly. Therefore semidiameters of the earth. Then, in the triangle KMC, there are given<pb n="214"/><cb/> KC = <*>, and MC = 122, also the angle KMC = 16′ 20″, to sind the angle C; thus, as KC:MC::sin. [angle] KMC:sin. [angle] MKP = 35° 25′ 35″; from this take the [angle] KMC - - 0 16 20, leaves the [angle] C - - - 35 9 11, the double of which is the arc KN 70 18 22, or nearly a space of 4866 miles in diameter.</p></div1><div1 part="N" n="PERAMBULATOR" org="uniform" sample="complete" type="entry"><head>PERAMBULATOR</head><p>, an instrument for measuring distances; called also Pedometer, Waywiser, and Surveying Wheel.</p><p>This wheel is contrived to measure out a pole, or 16 1/2 feet, in making two revolutions; consequently its circumference is 8 1/4 feet, and its diameter 2.626 feet, or 2 feet 5 1/2 inches and 12/1000 parts, very nearly. It is either driven sorward by two handles, by a person walking; or is drawn by a coach wheel, &c, to which it is attached by a pole. It contains various movements, by wheels, or clock-work, with indices on its face, which is like that of a clock, to point out the distance passed over, in miles, f<*>longs, poles, yards, &c.</p><p>Its advantages are its readiness and expedition; being very useful for measuring roads, and great distances on level ground. See the fig. Plate xvii, fig. 6.</p></div1><div1 part="N" n="PERCH" org="uniform" sample="complete" type="entry"><head>PERCH</head><p>, in Surveying, a square measure, being the 40th part of a rood, or the 160th part of an acre; that is, the square of a pole or rod, of the length of 5 1/2 yards, or 16 1/2 feet.</p><p><hi rend="smallcaps">Perch</hi> is by some also made to mean a measure of length; being the same as the rod or pole of 5 1/2 yards or 16 1/2 feet long. But it is better, for preventing confusion, to distinguish them.</p></div1><div1 part="N" n="PERCUSSION" org="uniform" sample="complete" type="entry"><head>PERCUSSION</head><p>, in Physics, the impression a body makes in falling or striking upon another; or the shock or collision of two bodies, which meeting alter each other's motion.</p><p>Percussion is either Direct or Oblique. It is also either of Elastic or Nonelastic bodies, which have each their different laws. It is true, we know of no bodies in nature that are either perfectly elastic or the contrary; but all partaking that property in different degrees; even the hardest and the softest being not entirely divested of it. But, for the sake of perspicuity, it is usual, and proper, to treat of these two separately and apart.</p><p><hi rend="italics">Direct</hi> <hi rend="smallcaps">Percussion</hi> is that in which the impulse is made in the direction of a line perpendicular at the place of impact, and which also passes through the common centre of gravity of the two striking bodies. As is the case in two spheres, when the line of the direction of the stroke passes through the centres of both spheres; for then the same line, joining their centres, passes perpendicularly through the point of impact. And</p><p><hi rend="italics">Oblique</hi> <hi rend="smallcaps">Percussion</hi>, is that in which the impulse is made in the direction of a line that does not pass through the common centre of gravity of the striking bodies; whether that line of direction is perpendicular to the place of impact, or not.</p><p>The force of Percussion is the same as the momentum, or quantity of motion, and is represented by the product arising from the mass or quantity of matter moved, multiplied by the velocity of its motion; and that without any regard to the time or duration of action; for its action is considered totally independent of time, or but as for an instant, or an infinitely small time.<cb/></p><p>This consideration will enable us to resolve a question that has been greatly canvassed among philosophers and mathematicians, viz, what is the relation between the force of Percussion and mere pressure or weight? For we hence infer, that the former force is insinitely, or incomparably, greater than the latter. For, let M denote any mass, body, or weight, having no motion or velocity, but simply its pressure; then will that pressure or force be denoted by M itself, if it be considered as acting for some certain finite assignable time; but, considered as a force of Percussion, that is, as acting but for an infinitely small time, its velocity being 0<*> or nothing, its percussive force will be 0 X M, that is 0, or nothing; and is therefore less than any the smallest percussive force whatever. Again, let us consider the two forces, viz, of Percussion and pressure, with respect to the effects they produce: Now the intensity of any force is very well measured and estimated by the effect it produces in a given time: But the effect of the pressure M, in 0 time, or an infinitely small time, is nothing at all; that is, it will not, in an infinitely small time, produce, for example, any motion, either in itself, or in any other body: its intensity therefore, as its effect, is insinitely less than any the smallest force of Percussion. It is true, indeed, that we see motion and other considerable effects produced by mere pressure, and to counteract which it will require the opposition of some considerable percussive force: but then it must be observed, that the former has been an infinitely longer time than the latter in producing its effect; and it is no wonder in mathematics that an infinite number of infinitely small quantities makes up a finite one. It has therefore only been for want of considering the circumstance of <hi rend="italics">time,</hi> that any question could have arisen on this head. Hence the two forces are related to each other, only as a surface is to a solid or body: by the motion of the surface through an infinite number of points, or through a finite right line, a solid or body is generated: and by the action of the pressure for an infinite number of moments, or for some finite time, a quantity equal to a given percussive force is generated: but the surface itself is infinitely less than any solid, and the pressure infinitely less than any percussive force. This point may be easily illustrated by some familiar instances, which prove at least the enormous disproportion between the two forces, if not also their absolute incomparability. And first, the blow of a small hammer, upon the head of a nail, will drive the nail into a board; when it is hard to conceive any weight so great as will produce a like effect, i. e. that will sink the nail as far into the board, at least unless it is left to act for a very considerable time: and even after the greatest weight has been laid as a pressure on the head of the nail, and has sunk it as far as it can as to sense, by remaining for a long time there without producing any farther sensible effect; let the weight be removed from the head of the nail, and instead of it, let it be struck a small blow with a hammer, and the nail will immediately sink farther into the wood. Again, it is also well known, that a ship-carpenter, with a blow of his mallet, will drive a wedge in below the greatest ship whatever, lying aground, and so overcome her weight, and lift her up. Lastly, let us consider a man with a club to strike a small ball, upwards or in<pb n="215"/><cb/> any other direction; it is evident that the ball will acquire a certain determinate velocity by the blow, suppose that of 10 feet per second, or minute, or any other time whatever: now it is a law, universally allowed in the communication of motion, that when different bodies are struck with equal forces, the velocities communicated are reciprocally as the weights of the bodies that are struck; that is, that a double body, or weight, will acquire half the velocity from an equal blow; a body 10 times as great, one 10th of the velocity; a body 100 times as great, the 100th part of the velocity; a body a million times as great, the millionth part of the velocity; and so on without end: from whence it follows, that there is no body or weight, how great soever, but will acquire some sinite degree of velocity, and be overcome, by any given small finite blow, or Percussion.</p><p>It appears that Des Cartes, first of any, had some ideas of the laws of Percussion; though it must be acknowledged, in some cases perhaps wide of the truth. The first who gave the true laws of motion in nonelastic bodies, was Doctor Wallis, in the Philos. Trans. numb. 43, where he also shews the true cause of reflections in other bodies, and proves that they proceed from their elasticity. Not long after, the celebrated Sir Christopher Wren and Mr. Huygens imparted to the Royal Society the laws that are observed by perfectly elastic bodies, and gave exactly the same construction, though each was ignorant of what the other had done. And all those laws, thus published in the Philos. Trans. without demonstration, were afterwards demonstrated by Dr. Keill, in his Philos. Lect. in 1700; and they have since been followed by a multitude of other authors.</p><p>In Percussion, we distinguish at least three several sorts of bodies; the perfectly hard, the perfectly soft, and the perfectly elastic. The two former are considered as utterly void of elasticity; having no force to separate them, or throw them off from each other again, after collision; and therefore either remaining at rest, or else proceeding uniformly forward together as one body or mass of matter.</p><p>The laws of Percussion therefore to be considered, are of two kinds: those for elastic, and those for nonelastic bodies.</p><p>The one only general principle, for determining the motions of bodies from Percussion, and which belongs equally to both the sorts of bodies, i. e. both the elastic and nonelastic, is this: viz, that there exists in the bodies the same momentum, or quantity of motion, estimated in any one and the same direction, both before the stroke and after it. And this principle is the immediate result of the third law of nature or motion, that reaction is equal to action, and in a contrary direction; from whence it happens, that whatever motion is communicated to one body by the action of another, exactly the same motion doth this latter lose in the same direction, or exactly the same does the former communicate to the latter in the contrary direction.</p><p>From this general principle too it results, that no alteration takes place in the common centre of gravity of bodies by their actions upon one another; but that the said common centre of gravity perseveres in the<cb/> same state, whether of rest or of uniform motion, both before and after the shock of the bodies.</p><p>Now, from either of these two laws, viz, that of the preservation of the same quantity of motion, in one and the same direction, and that of the preservation of the same state of the centre of gravity, both before and after the shock, all the circumstances of the motions of both the kinds of bodies after collision may be made out; in conjunction with their own peculiar and separate constitutions, namely, that of the one sort being elastic, and the other nonelastic.</p><p>The effects of these different constitutions, here alluded to, are these; that nonelastic bodies, on their shock, will adhere together, and either remain at rest, or else move together as one mass with a common velocity; or if elastic, they will separate after the shock with the very same relative velocity with which they met and shocked. The former of these consequences is evident, viz, that nonelastic bodies keep together as one mass after they meet; because there exists no power to separate them; and without a cause there can be no effect. And the latter consequence results immediately from the very definition and essence of elasticity itself, being a power always equal to the force of compression, or shock; and which restoring force therefore, acting the contrary way, will generate the same relative velocity between the bodies, or the same quantity of matter, as before the shock, and the same motion also of their common centre of gravity. <figure/></p><p>To apply now the general principle to the determination of the motions of bodies after their shock; let B and <hi rend="italics">b</hi> be any two bodies, and V and <hi rend="italics">v</hi> their respective velocities, estimated in the direction AD; which quantities V and <hi rend="italics">v</hi> will be both positive if the bodies both move towards D, but one of them as <hi rend="italics">v</hi> will be negative if the body <hi rend="italics">b</hi> move towards A, and <hi rend="italics">v</hi> will be = 0 if the body <hi rend="italics">b</hi> be at rest. Hence then BV is the momentum of B towards D, and <hi rend="italics">bv</hi> is the momentum of <hi rend="italics">b</hi> towards D, whose sum is BV + <hi rend="italics">bv,</hi> which is the whole quantity of motion in the direction AD, and which momentum must also be preserved after the shock.</p><p>Now if the bodies have no elasticity, they will move together as one mass B + <hi rend="italics">b</hi> after they meet, with some common velocity, which call <hi rend="italics">y,</hi> in the direction AD; therefore the momentum in that direction after the shock, being the product of the mass and velocity, will be . But the momenta, in the same direction, before and after the impact, are equal, that is ; from which equation any one of the quantities may be determined when the rest are given. So, if we would find the common velocity after the stroke, it will be , equal to the sum of the momenta divided by the sum of the bodies; which is also equal to the velocity of the common centre of gravity of the two bodies, both before and after the collision. The signs of the terms, in this value of <hi rend="italics">y,</hi> will be all positive, as<pb n="216"/><cb/> above, when the bodies move both the same way AD; but one term <hi rend="italics">bv</hi> must be made negative when the motion of <hi rend="italics">b</hi> is the contrary way; and that term will be absent or nothing, when <hi rend="italics">b</hi> is at rest, before the shock.</p><p>Again, for the case of elastic bodies, which will separate after the stroke, with certain velocities, <hi rend="italics">x</hi> and <hi rend="italics">z,</hi> viz, <hi rend="italics">x</hi> the velocity of B, and <hi rend="italics">z</hi> the velocity of <hi rend="italics">b</hi> after the collision, both estimated in the direction AD, which quantities will be either positive, or negative, or nothing, according to the circumstances of the masses B and <hi rend="italics">b,</hi> with those of their celerities before the stroke. Hence then B<hi rend="italics">x</hi> and <hi rend="italics">bz</hi> are the separate momenta after the shock, and B<hi rend="italics">x</hi> + <hi rend="italics">bz</hi> their sum, which must be equal to the sum BV + <hi rend="italics">bv</hi> in the same direction before the stroke: also <hi rend="italics">z</hi> - <hi rend="italics">x</hi> is the relative velocity with which the bodies separate after the blow, and which must be equal to V - <hi rend="italics">v</hi> the same with which they meet; or, which is the same thing, that ; that is, the sum of the two velocities of the one body, is equal to the sum of the velocities of the other, taken before and after the stroke; which is another notable theorem. Hencethen, for determining the two unknown quantities <hi rend="italics">x</hi> and <hi rend="italics">z,</hi> there are these two equations, viz, , and ; or ; the resolution of which equations gives those two velocities as below, viz, , and .</p><p>From these general values of the velocities, which are to be understood in the direction AD, any particular cases may easily be drawn. As, if the two bodies B and <hi rend="italics">b</hi> be equal, then B - <hi rend="italics">b</hi> = 0, and B + <hi rend="italics">b</hi> = 2B, and the two velocities in that case become, after impulse, <hi rend="italics">x</hi> = <hi rend="italics">v,</hi> and <hi rend="italics">z</hi> = V, the very same as they were before, but changed to the contrary bodies, i. e. the bodies have taken each other's velocity that it had before, and with the same sign also. So that, if the equal bodies were before both moving the same way, or towards D, they will do the same after, but with interchanged velocities. But if they before moved contrary ways, B towards D, and <hi rend="italics">b</hi> towards A, they will rebound contrary ways, B back towards A, and <hi rend="italics">b</hi> towards D, each with the other's velocity. And, lastly, if one body, as <hi rend="italics">b,</hi> were at rest before the stroke, then the other B will be at rest after it, and <hi rend="italics">b</hi> will go on with the motion that B had before. And thus may any other particular cases be deduced from the first general values of <hi rend="italics">x</hi> and <hi rend="italics">z.</hi></p><p>We may now conclude this article with some remarks on these motions, and the mistakes of some authors concerning them. And first, we observe this striking difference between the motions that are communicated by elastic and by nonelastic bodies, viz, that a nonelastic body, by striking, communicates to the body it strikes, exactly its whole momentum; as is evident. But the stroke of an elastic body may either communicate its whole motion to the body it strikes, or it may communicate only a part of it, or it may even communicate more than it had. For, if the striking body remain at rest after the stroke, it has<cb/> just lost all its motion, and therefore has communicated all it had; but if it still move forward in the same direction, it has still some motion left in that direction, and therefore has only communicated a part of what motion it had; and if the striking body rebound back, and move in the contrary direction, the other body has received not only the whole of the motion that the first had, but also as much more as the first has acquired in the contrary direction.</p><p>It has been denied by some authors, and in the Encyclopédie, that the same quantity of motion remains after the shock, as before it; and hence they seize an opportunity to reprehend the Cartesians for making that assertion, which they do, not only with respect to the case of two bodies, but also of all the bodies in the whole universe. And yet nothing is more true, if the motion be considered as estimated always in one and the same direction, esteeming that as negative, which is in the contrary or opposite direction. For it is a general law of nature, that no motion, nor force, can be generated, nor destroyed, nor changed, but by some cause which must produce an equal quantity in the opposite direction. And this being the case in one body, or two bodies, it must necessarily be the case in all bodies, and in the whole sola<*> system, since all bodies act upon one another. And hence also it is manifest, that the common centre of gravity of the whole solar system must always preserve its original condition, whether it be of rest or of uniform motion; since the state of that centre is not changed by the mutual actions of bodies upon one another, any more than their quantity of motion, in one and the same direction.</p><p>What may have led authors into the mistake above alluded to, which they bring no proof of, seems to be the discovery of M. Huygens, that the sums of the two products are equal, both before and after the shock, that are made by multiplying each body by the square of its velocity, viz, that , where V and <hi rend="italics">v</hi> are the velocities before the shock, and <hi rend="italics">x</hi> and <hi rend="italics">z</hi> the velocities after it. Such an expression, namely the product of the mass by the square of the velocity, is called the vis viva, or living force; and hence it has been inferred that the whole vis viva before the shock, or BV<hi rend="sup">2</hi> + <hi rend="italics">bv</hi><hi rend="sup">2</hi>, is equal to that after the stroke, or B<hi rend="italics">x</hi><hi rend="sup">2</hi> + <hi rend="italics">bz</hi><hi rend="sup">2</hi>; which is indeed very true, as will be shewn presently. But when they hence infer, both that therefore the forces of bodies in motion are as the squares of the velocities, and that there is not the same quantity of motion between the two striking bodies, both before and after the shock, they are grossly mistaken, and thereby shew that they are ignorant of the true derivation of the equation . For this equation is only a consequence of the very principle above laid down, and which is not acceded to by those authors, viz, that the quantity of motion is the same before and after the shock, or that , the truth of which last equation they deny, because they think the former one is true, never dreaming that they may be both true, and much less that the one is a consequence of the other, and derived from it; which however is now found to be the case, as is proved in this manner:</p><p>It has been shewn that the sum of the two momenta,<pb n="217"/><cb/> in the same direction, before and after the stroke, are equal, or that ; and also that the sum of the two velocities of the one body, is equal to the sum of those of the other, or that ; and it is now proposed to shew that from these two equations there results the third equation , or the equation of the living forces.</p><p>Now because , by transposition it is - ; which fhews that the difference between the two momenta of the one body, before and after the stroke, is equal to the difference between those of the other body; which is another notable theorem. But now, to derive the equation of the vis viva, set down the two foregoing equations, and multiply them together, so shall the products give the said equation required; thus Mult. , the equat. of the momenta, by , the equat. of the velocities, produc. , or , the very equation of the vis viva required. Which was to be proved.</p><p>When the elasticity of the bodies is not perfect, but only partially so, as is the case with all the bodies we know of, the determination of the motions after collision may be determined in a similar manner. See Keill's Lect. Philos. lect. 14, theor. 29, at the end. And for the geometrical determinations after impact, see the article <hi rend="smallcaps">Collision.</hi></p><p><hi rend="italics">Centre of</hi> <hi rend="smallcaps">Percussion</hi>, is the point in which the shock or impulse of a body which strikes another is the greatest that it can be. See <hi rend="smallcaps">Cenfre.</hi></p><p>The Centre of Percussion is the same as the centre of oscillation, when the striking body moves round a fixed axis. See <hi rend="smallcaps">Oscillation.</hi></p><p>But if all the parts of the striking body move with a parallel motion, and with the same velocity, then the Centre of Percussion is the same as the centre of gravity.</p><p>PERFECT <hi rend="smallcaps">Number</hi>, is one that is equal to the sum of all its aliquot parts, when added together. Eucl. lib. 7, def. 22. As the number 6, which is , the sum of all its aliquot parts; also 28, for , the sum of all its aliquot parts.</p><p>It is proved by Euclid, in the last prop. of book the 9th, that if the common geometrical series of numbers 1, 2, 4, 8, 16, 32, &c, be continued to such a number of terms, as that the sum of the said series of terms shall be a prime number, then the product of this sum by the last term of the series will be a perfect number.</p><p>This same rule may be otherwife expressed thus: If <hi rend="italics">n</hi> denote the number of terms in the given series 1, 2, 4, 8, &c.; then it is well known that the sum of all the terms of the series is , and it is evident that the last term is 2<hi rend="sup">n - 1</hi>: consequently the rule becomes thus, viz, = a perfect number, whenever is a prime number.</p><p>Now the sums of one, two, three, four, &c, terms of the series 1, 2, 4, 8, &c, form the series 1, 3, 7, 15, 31, &c; so that the number will be found perfect<cb/> whenever the corresponding term of this series is a prime, as 1, 3, 7, 31, &c. Whence the table of perfect numbers may be found and exhibited as follows; where the 1st column shews the number of terms, or the value of <hi rend="italics">n;</hi> the 2d column is the last term of the series 1, 2, 4, 8, &c, and is expressed by 2<hi rend="sup">n - 1</hi>; the 3d column contains the corresponding sums of the said series, or the values of the quantity ; which numbers in this 3d column are easily constructed by adding always the last number in this column to the next following number in the 2d column: and lastly, the 4th column shews the correspondent Persect Numbers, or the values of , the product of the numbers in the 2d and 3d columns, when , or the number in the 3d column, is a prime number; the products in the other cases being omitted, as not Perfect Numbers. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Values of <hi rend="italics">n</hi></cell><cell cols="1" rows="1" role="data">Values of 2<hi rend="sup">n - 1</hi></cell><cell cols="1" rows="1" role="data">Values of </cell><cell cols="1" rows="1" role="data">Perf. Numbers, or </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">28</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">496</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">8128</cell></row></table></p><p>Hence the first four Perfect Numbers are found to be 6, 28, 496, 8128; and thus the table might be continued to find others, but the trouble would be very great, for want of a general method to distinguish which numbers are primes, as the case requires. Several learned mathematicians have endeavoured to facilitate this business, but hitherto with only a small degree of perfection. After the foregoing four Perfect Numbers, there is a long interval before any more occur. The first eight are as follow, with the factors and products which produce them: <table><row role="data"><cell cols="1" rows="1" role="data">The first Perfect Numbers.</cell><cell cols="1" rows="1" role="data">Their values.</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">2</hi> - 1) 2</cell></row><row role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">3</hi> - 1) 2<hi rend="sup">2</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">496</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">5</hi> - 1) 2<hi rend="sup">4</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">8128</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">7</hi> - 1) 2<hi rend="sup">6</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">33550336</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">13</hi> - 1) 2<hi rend="sup">12</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">8589869056</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">17</hi> - 1) 2<hi rend="sup">16</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">137438691328</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">19</hi> - 1) 2<hi rend="sup">18</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">2305843008139952128</cell><cell cols="1" rows="1" role="data">= (2<hi rend="sup">31</hi> - 1) 2<hi rend="sup">30</hi></cell></row></table></p><p>See several considerable tracts on the subject of Perfect Numbers in the Memoirs of the Petersburgh Academy, vol. 2 of the new vols, and in several other volumes.</p><p>PERIÆCI. See <hi rend="smallcaps">Perioeci.</hi></p><p>PERIGÆUM, or <hi rend="smallcaps">Perigee</hi>, is that point of the orbit of the sun or moon, which is the nearest to the earth. In which sense it stands opposed to Apogee, which is the most distant point from the earth.<pb n="218"/><cb/></p><div2 part="N" n="Perigee" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Perigee</hi></head><p>, in the Ancient Astronomy, denotes a point in a planet's orbit, where the centre of its epicycle is at the least distance from the earth.</p></div2></div1><div1 part="N" n="PERIHELION" org="uniform" sample="complete" type="entry"><head>PERIHELION</head><p>, <hi rend="smallcaps">Perihelium</hi>, that point in the orbit of a planet or comet which is nearest to the sun. In which sense it stands opposed to Aphelion, or Aphelium, which is the highest or most distant point from the sun.</p><p>Instead of this term, the Ancients used Perigeum; because they placed the earth in the centre.</p></div1><div1 part="N" n="PERIMETER" org="uniform" sample="complete" type="entry"><head>PERIMETER</head><p>, in Geometry, the ambit, limit, or outer bounds of a figure; being the sum of all the lines by which it is inclosed or formed.</p><p>In circular figures, &c, instead of this term, the word circumference or periphery is used.</p></div1><div1 part="N" n="PERIOD" org="uniform" sample="complete" type="entry"><head>PERIOD</head><p>, in Astronomy, the time in which a star or planet makes one revolution, or returns again to the same point in the heavens.</p><p>The sun's, or properly the earth's tropical period, is 365 days 5 hours 48 minutes 45 seconds 30 thirds. That of the moon is 27 days 7 hours 43 minutes. That of the other planets as below.</p><p>There is a wonderful harmony between the distances of the planets from the sun, and their Periods round him; the great law of which is, that the squares of the Periodic times are always proportional to the cubes of their mean distances from the sun.</p><p>The Periods, both tropical and sydereal, with the proportions of the mean distances of the several planets are as follow: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Planets</cell><cell cols="1" rows="1" role="data">Tropical Periods</cell><cell cols="1" rows="1" role="data">Sydereal Periods</cell><cell cols="1" rows="1" role="data">Proport. Dists.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" rend="align=right" role="data">87<hi rend="sup">d</hi> 23<hi rend="sup">h</hi> 14′</cell><cell cols="1" rows="1" rend="align=right" role="data">87<hi rend="sup">d</hi> 23<hi rend="sup">h</hi> 16′</cell><cell cols="1" rows="1" rend="align=right" role="data">36710</cell></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" rend="align=right" role="data">224 16 42</cell><cell cols="1" rows="1" rend="align=right" role="data">224 16 49</cell><cell cols="1" rows="1" rend="align=right" role="data">72333</cell></row><row role="data"><cell cols="1" rows="1" role="data">Earth</cell><cell cols="1" rows="1" rend="align=right" role="data">365  5 49</cell><cell cols="1" rows="1" rend="align=right" role="data">365  6  9</cell><cell cols="1" rows="1" rend="align=right" role="data">100000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" rend="align=right" role="data">686 22 18</cell><cell cols="1" rows="1" rend="align=right" role="data">686 23 31</cell><cell cols="1" rows="1" rend="align=right" role="data">152369</cell></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" rend="align=right" role="data">4330  8 58</cell><cell cols="1" rows="1" rend="align=right" role="data">4332  8 51</cell><cell cols="1" rows="1" rend="align=right" role="data">520110</cell></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" rend="align=right" role="data">10749  7 22</cell><cell cols="1" rows="1" rend="align=right" role="data">10761 14 37</cell><cell cols="1" rows="1" rend="align=right" role="data">953800</cell></row><row role="data"><cell cols="1" rows="1" role="data">Georgian or  Herschel</cell><cell cols="1" rows="1" rend="align=right" role="data">30456  1 41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1908180</cell></row></table></p><p>As to the comets, the Periods of very few of them are known. There is one however of between 75 and 76 years, which appeared for the last time in 1759; another was supposed to have its Period of 129 years, which was expected to appear in 1789 or 1790, but it did not; and the comet which appeared in 1680 it is thought has its Period of 575 years.</p><div2 part="N" n="Period" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Period</hi></head><p>, in Chronology, denotes an epoch, or interval of time, by which the years are reckoned; or a series of years by which time is measured, in different nations. Such are the Calippic and Metonic Periods, two different corrections of the Greek calendar, the Julian Period, invented by Joseph Scaliger; the Victorian Period, &c.</p><p><hi rend="italics">Calippic</hi> <hi rend="smallcaps">Period.</hi> See <hi rend="smallcaps">Calippic</hi> <hi rend="italics">Period.</hi></p><p><hi rend="italics">Constantinopolitan</hi> <hi rend="smallcaps">Period</hi>, is that used by the Greeks, and is the same as the <hi rend="italics">Julian</hi> <hi rend="smallcaps">Period</hi>, which see.</p><p><hi rend="italics">Chaldaic</hi> <hi rend="smallcaps">Period.</hi> See <hi rend="smallcaps">Saros.</hi><cb/></p><p><hi rend="italics">Dionysian</hi> <hi rend="smallcaps">Period.</hi> See <hi rend="italics">Victorian</hi> <hi rend="smallcaps">Period.</hi></p><p><hi rend="italics">Hipparchus's</hi> <hi rend="smallcaps">Period</hi>, is a series or cycle of 304 solar years, returning in a constant round, and restoring the new and full moons to the same day of the sólar year; as Hipparchus thought.</p><p>This Period arises by multiplying the Calippic Period by 4. Hipparchus assumed the quantity of the solar year to be 365d. 5h. 55m. 12 sec. and hence he concluded, that in 304 years Calippus's Period would err a whole day. He therefore multiplied the Period by 4, and from the product cast away an entire day. But even this does not restore the new and full moons to the same day throughout the whole Period: but they are sometimes anticipated 1d. 8h. 23 m. 29 sec. 20 thirds.</p><p><hi rend="italics">Julian</hi> <hi rend="smallcaps">Period</hi>, so called as being adapted to the Julian year, is a series of 7980 Julian years; arising from the multiplications of the cycles of the sun, moon, and indiction together, or the numbers 28, 19, 15; commencing on the 1st. day of January in the 764th Julian year before the creation, and therefore is not yet completed. This comprehends all other cycles, Periods and epochs, with the times of all memorable actions and histories; and therefore it is not only the most general, but the most useful of all Periods in Chronology.</p><p>As every year of the Julian Period has its particular solar, lunar, and indiction cycles, and no two years in it can have all these three cyeles the same, every year of this Period becomes accurately distinguished from another.</p><p>This Period was invented by Joseph Scaliger, as containing all the other epochs, to facilitate the reduction of the years of one given epoch to those of another. It agrees with the Constantinopolitan Period, used by the Greeks, except in this, that the cycles of the sun, moon, and indiction, are reckoned differently; and also in that the first year of the Constantinopolitan Period differs from that of the Julian Period.</p><p>To find the year answering to any given year of the Julian Period, and vice versa; see <hi rend="smallcaps">Epoch.</hi></p><p><hi rend="italics">Metonic</hi> <hi rend="smallcaps">Period.</hi> See <hi rend="smallcaps">Cycle</hi> <hi rend="italics">of the Moon.</hi></p><p><hi rend="italics">Victorian</hi> <hi rend="smallcaps">Period</hi>, an interval of 532 Julian years; at the end of which, the new and full moons return again on the same day of the Julian year, according to the opinion of the inventor Victorinus, or Victorius, who lived in the time of pope Hilary.</p><p>Some ascribe this Period to Dionysius Exiguus, and hence they call it the Dionysian Period: others again call it the Great Paschal Cycle, because it was invented for computing the time of Easter.</p><p>The Victorian Period is produced by multiplying the solar cycle 28 by the lunar cycle 19, the product being 532. But neither does this restore the new and full moons to the same day throughout its whole duration, by 1d. 16h. 58m. 59s. 40 thirds.</p></div2><div2 part="N" n="Period" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Period</hi></head><p>, in Arithmetic, is a distinction made by a point, or a comma, after every 6th place, or figure; and is used in numeration, for the readier distinguishing and naming the several figures or places, which are thus distinguished into Periods of six figures each. See <hi rend="smallcaps">Numeration.</hi></p><p><hi rend="smallcaps">Period</hi> is also used in Arithmetic, in the extraction<pb n="219"/><cb/> of roots, to point off, or separate the figures of the given number into Periods, or parcels, of as many figures each as are expressed by the degree of the root to be extracted, viz, of two places each for the square root, three places for the cube root, and so on.</p></div2></div1><div1 part="N" n="PERIODIC" org="uniform" sample="complete" type="entry"><head>PERIODIC</head><p>, or <hi rend="smallcaps">Periodical</hi>, appertaining to Period, or going by periods. Thus, the Periodical motion of the moon, is that of her monthly period or course about the earth, called her Periodical month, containing 27 days 7 hours 45 minutes.</p><p><hi rend="smallcaps">Periodical</hi> <hi rend="italics">Month.</hi> See <hi rend="smallcaps">Month.</hi></p><p>PERIŒCI, or <hi rend="smallcaps">Perioecians</hi>, in Geography, are such as live in opposite points of the same parallel of latitude. Hence they have the same seasons at the same time, with the same phenomena of the heavenly bodies; but their times of the day are opposite, or differ by 12 hours, being noon with the one when it is midnight with the other.</p><p>PERIPATETIC <hi rend="italics">Philosophy,</hi> the system of philosophy taught and established by Aristotle, and maintained by his followers, the Peripatetics. See A<hi rend="smallcaps">RISTOTLE.</hi></p></div1><div1 part="N" n="PERIPATETICS" org="uniform" sample="complete" type="entry"><head>PERIPATETICS</head><p>, the followers of Aristotle. Though some derive their establishment from Plato himself, the master of both Xenocrates and Aristotle.</p></div1><div1 part="N" n="PERIPHERY" org="uniform" sample="complete" type="entry"><head>PERIPHERY</head><p>, in Geometry, is the circumference, or bounding line, of a circle, ellipse, or other regular curvilineal figure. See <hi rend="smallcaps">Circumference</hi>, and <hi rend="smallcaps">Circle.</hi></p></div1><div1 part="N" n="PERISCII" org="uniform" sample="complete" type="entry"><head>PERISCII</head><p>, or <hi rend="smallcaps">Periscians</hi>, those inhabitants of the earth, whose shadows do, in one and the same day, turn quite round to all the points of the compass, without disappearing.</p><p>Such are the inhabitants of the two frozen zones, or who live within the compass of the arctic and antarctic circles; for, as the sun never sets to them, after he is once up, but moves quite round about, so do their shadows also.</p></div1><div1 part="N" n="PERISTYLE" org="uniform" sample="complete" type="entry"><head>PERISTYLE</head><p>, in the ancient Architecture, a place or building encompassed with a row of columns on the inside; by which it is distinguished from the periptere, where the columns are disposed on the outside.</p><p><hi rend="smallcaps">Peristyle</hi> is also used, by modern writers, for a range of columns, either within or without a building.</p></div1><div1 part="N" n="PERITROCHIUM" org="uniform" sample="complete" type="entry"><head>PERITROCHIUM</head><p>, in Mechanics, is a wheel or circle, concentric with the base of a cylinder, and moveable together with it, about an axis. The axis, with the wheel, and levers sixed in it to move it, make that mechanical power, called <hi rend="smallcaps">Axis</hi> <hi rend="italics">in Peritrochio,</hi> which see.</p><p>PERMUTATIONS <hi rend="italics">of Quantities,</hi> in Algebra, the <hi rend="smallcaps">Alternations, Changes</hi>, or different C<hi rend="smallcaps">OMBINATIONS</hi> of any number of things. See those terms.</p></div1><div1 part="N" n="PERPENDICULAR" org="uniform" sample="complete" type="entry"><head>PERPENDICULAR</head><p>, in Geometry, or <hi rend="smallcaps">Normal.</hi> One line is Perpendicular to another, when the former meets the latter so as to make the angles on both sides of it equal to each other. And those angles are called right angles. And hence, to be Perpendicular to, or to make right-angles with, means one and the same<cb/> thing. So, when the angle ABC is equal to the angle ABD, the line AB is said to be Perpendicular, or normal, or at right angles to the line CD. <figure/></p><p>A line is Perpendicular to a curve, when it is perpendicular to the tangent of the curve at the point of contact.</p><p>A line is Perpendicular to a plane, when it is Perpendicular to every line drawn in the plane through the bottom of the Perpendicular. And one plane is Perpendicular to another, when a line in the one plane is Perpendicular to the other plane.</p><p>From the very principle and motion of a Perpendicular, it follows, 1. That the Perpendicularity is mutual, if the first AB is perpendicular to the second CD, then is the second Perpendicular to the first.—2. That only one Perpendicular can be drawn from one point in the same place.—3. That if a Perpendicular be continued through the line it was drawn Perpendicular to; the continuation BE will also be Perpendicular to the same. —4. That if there be two points, A and E, of a right line, each of which is at an equal distance from two points, C and D, of another right line; those lines are Perpendiculars.—5. That a line which is Perpendicular to another line, is also Perpendicular to all the parallels of the other.—6. That a Perpendicular is the shortest of all those lines which can be drawn from the same point to the same right line. Hence the distance of a point from a line or plane, is a line drawn from the point Perpendicular to the line or plane: and hence also the altitude of a figure is a Perpendicular let fall from the vertex to the base.</p><p><hi rend="italics">To Erect a Perpendicular</hi> from a given point in a line. —1. When the given point B is near the middle of the line; with any interval in the compasses take the two equal parts BC, BD: and from the two centres C and D, with any radius greater than BC or BD, strike two arcs intersecting in F; then draw BFA the Perpendicular required.</p><p>2. When the given point G is at or near the end of the line; with any centre I and radius IG describe an are HGK through G; then a ruler laid by H and I will cut the are in the point K, through which the Perpendicular GK must be drawn. <figure/></p><p><hi rend="italics">To let fall a Perpendicular</hi> upon a given line LM from a given point N. With the centre N, and a convenient radius, describe an arc cutting the given line in L and M; with these two centres, and any other convenient radius, strike<pb n="220"/><cb/> two other arcs intersecting in O, the point through which the Perpendicular NOP must be drawn.</p><p><hi rend="italics">Note,</hi> that Perpendiculars are best drawn, in practice, by means of a square, laying one side of it along the given line, and the other to pass through the given point.</p><div2 part="N" n="Perpendicular" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Perpendicular</hi></head><p>, in Gunnery, is a small instrument used for finding the centre line of a piece, in the operation of pointing it to a given object. See <hi rend="italics">Pointing of a</hi> <hi rend="smallcaps">Gun.</hi></p><p><hi rend="smallcaps">Perpetual</hi> <hi rend="italics">Motion.</hi> See <hi rend="smallcaps">Motion.</hi></p><p><hi rend="italics">Circle of</hi> <hi rend="smallcaps">Perpetual</hi> <hi rend="italics">Occultation and Apparition.</hi> See <hi rend="smallcaps">Circle.</hi></p></div2><div2 part="N" n="Perpetual" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Perpetual</hi></head><p>, or <hi rend="italics">Endless Screw.</hi> See <hi rend="smallcaps">Screw.</hi></p></div2></div1><div1 part="N" n="PERPETUITY" org="uniform" sample="complete" type="entry"><head>PERPETUITY</head><p>, in the Doctrine of Annuities, is the number of years in which the simple interest of any principal sum will amount to the same as the principal itself. Or it is the quotient arising by dividing 100, or any other principal, by its interest for one year. Thus, the Perpetuity, at the rate of 5 per cent. interest, is ; at 4 per cent. ; &c.</p></div1><div1 part="N" n="PERRY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PERRY</surname> (Captain <foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, was a celebrated English engineer. After acquiring great reputation for his skill in this country, he resided many years in Russia, having been recommended to the czar Peter while in England, as a person capable of serving him on a variety of occasions relating to his new design of establishing a fleet, making his rivers navigable, &c. His salary in this service was to be 300l. per annum, besides travelling expences and subsistence money on whatever service he should be employed, with a farther reward to his satisfaction at the conclusion of any work he should finish.</p><p>After some conversation with the czar himself, particularly respecting a communication between the rivers Volga and Don, he was employed on that work for three summers successively; but not being well supplied with men, partly on account of the ill success of Peter's arms against the Swedes at the battle of Narva, and partly by the discouragement of the governor of Astracan, he was ordered at the end of 1707 to stop, and next year was employed in refitting the ships at Veronise, and 1709 in making the river of that na ne navigable. But after repeated disappointments, and a variety of fruitless applications for his salary, he at length quitted the kingdom, under the protection of Mr. Whitworth, the English ambassador, in 1712. (See his Narrative in the Preface to <hi rend="italics">The State of Russia.</hi>)</p><p>In 1721 he was employed in stopping the breach at Dagenham, made in the bank of the river Thames, near the village of that name in Essex, and about 3 miles below Woolwich, in which he happily succeeded, after several other persons had failed in that undertaking. He was also employed, the same year, about the harbour at Dublin, and published at that time an Answer to the objections made against it.—Beside this piece, Captain Perry was author of, <hi rend="italics">The State of Russia,</hi> 1716, 8vo; and <hi rend="italics">An Account of the Stopping of Dagenham Breach,</hi> 1721, 8vo.—He died February the 11th 1733.</p></div1><div1 part="N" n="PERSEUS" org="uniform" sample="complete" type="entry"><head>PERSEUS</head><p>, a constellation of the northern hemisphere, being one of the 48 ancient asterisms.</p><p>The Greeks fabled that this is Perseus, whom they<cb/> make the son of Jupiter by Danae. The father of that lady had been told, that he should be killed by his grandchild, and having only Danae to take care of, he locked her up; but Jupiter found his way to her in a shower of gold, and Perseus verisied the oracle. He cut off also the head of the gorgon, and affixed it to his shield; and after many other great exploits he rescued Andromeda, the daughter of Cassiopeia, whom the sea nymphs, in revenge for that lady's boasting of superior beauty, had fastened to a rock to be devoured by a monster. Jupiter his father in honour of the exploit, they say, afterwards took up the hero, and the whole family with him, into the skies.</p><p>The number of stars in this constellation, in Ptolomy's catalogue, are 29; in Tycho's 29, in Hevelius's 46, and in the Britannic catalogue 59.</p><p>PERSIAN <hi rend="italics">Wheel,</hi> in Mechanics, a machine for raising a quantity of water, to serve for various purposes. Such a wheel is represented in plate xx, fig. 1; with which water may be raised by means of a stream AB turning a wheel CDE, according to the order of the letters, with buckets <hi rend="italics">a, a, a, a,</hi> &c, hung upon the wheel by strong pins <hi rend="italics">b, b, b, b,</hi> &c, fixed in the side of the rim; which must be made as high as the water is intended to be raised above the level of that part of the stream in which the wheel is placed. As the wheel turns, the buckets on the right hand go down into the water, where they are filled, and return up full on the left hand, till they come to the top at K; where they strike against the end <hi rend="italics">n</hi> of the fixed trough M, by which they are overset, and so empty the water into the trough; from whence it is to be conveyed in pipes to any place it is intended for: and as each bucket gets over the trough, it falls into a perpendicular position again, and so goes down empty till it comes to the water at A, where it is filled as before. On each bucket is a spring <hi rend="italics">r,</hi> which going over the top or crown of the bar <hi rend="italics">m</hi> (fixed to the trough M) raises the bottom of the bucket above the level of its mouth, and so causes it to empty all its water into the trough.</p><p>Sometimes this wheel is made to raise water no higher than its axis; and then instead of buckets hung upon it, its spokes C, <hi rend="italics">d, e, f, g, h,</hi> are made of a bent form, and hollow within; these hollows opening into the holes C, D, E, F, in the outside of the wheel, and also into those at O in the box N upon the axis. So that, as the holes C, D, &c, dip into the water, it runs into them; and as the wheel turns, the water rises in the hollow spokes, <hi rend="italics">c, d,</hi> &c, and runs out in a stream P from the holes at O, and falls into the trough Q, from whence it is conveyed by pipes.</p><div2 part="N" n="Persian" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Persian</hi></head><p>, or <hi rend="smallcaps">Persic</hi>, in Architecture, a name common to all statues of men; serving instead of columns to support entablatures.</p><p><hi rend="smallcaps">Persian</hi> <hi rend="italics">Era</hi> and <hi rend="italics">Year.</hi> See <hi rend="smallcaps">Epoch</hi> and <hi rend="smallcaps">Year.</hi></p></div2></div1><div1 part="N" n="PERSPECTIVE" org="uniform" sample="complete" type="entry"><head>PERSPECTIVE</head><p>, the art of delineating visible objects on a plane surface, such as they appear at a given distance, or height, upon a transparent plane, placed commonly perpendicular to the horizon, between the eye and the object. This is particularly called</p><p><hi rend="italics">Linear</hi> <hi rend="smallcaps">Perspective</hi>, as regarding the position, magnitude, form, &c, of the several lines, or contours of objects, and expressing their diminution.</p><p>Some make this a branch of Optics; others an art <pb/><pb/><pb n="221"/><cb/> and science derived from it: its operations however are all geometrical.</p><p><hi rend="italics">History of</hi> <hi rend="smallcaps">Perspective.</hi> This art derives its origin from painting, and particularly from that branch of it which was employed in the decorations of the theatre, where landscapes were chiefly introduced. Vitruvius, in the proem to his 7th book, says that Agatharchus, at Athens, was the first author who wrote upon this subject, on occasion of a play exhibited by Æschylus, for which he prepared a tragic scene; and that afterwards the principles of the art were more distinctly taught in the writings of Democritus and Anaxagoras, the disciples of A gatharchus, which are not now extant.</p><p>The Perspective of Euclid and of Heliodorus Larisseus contains only some general elements of optics, that are by no means adapted to any particular practice; though they furnish some materials that might be of service even in the linear Perspective of painters.</p><p>Geminus, of Rhodes, a celebrated mathematician, in Cicero's time, also wrote upon this science.</p><p>It is also evident that the Roman artists were acquainted with the rules of Perspective, from the account which Pliny (Nat. Hist. lib. 35, cap. 4) gives of the representation on the scene of those plays given by Claudius Pulcher; by the appearance of which the crows were so deceived, that they endeavoured to settle on the fictitious roofs. However, of the theory of this Art among the Ancients we know nothing; as none of their writings have escaped the general wreck of ancient literature in the dark ages of Europe. Doubtless this art must have been lost, when painting and sculpture no longer existed. However, there is reason to believe that it was practised much later in the Eastern empire.</p><p>John Tzetzes, in the 12th century, speaks of it as well acquainted with its importance in painting and statuary. And the Greek painters, who were employed by the Venetians and Florentines, in the 13th century, it seems brought some optical knowledge along with them into Italy: for the disciples of Giotto are commended for observing Perspective more regularly than any of their predecessors in the art had done; and he lived in the beginning of the 14th century.</p><p>The Arabians were not ignorant of this art; as may be presumed from the optical writings of Alhazen, about the year 1100. And Vitellus, a Pole, about the year 1270, wrote largely and learnedly on optics. And, of our own nation, friar Bacon, as well as John Peckham, archbishop of Canterbury, treated this subject with surprising accuracy, considering the times in which they lived.</p><p>The first authors who professedly laid down rules of Perspective, were Bartolomeo Bramantino, of Milan, whose book, Regole di Perspectiva, e Misure delle Antichita di Lombardia, is dated 1440; and Pietro del Borgo, likewise an Italian, who was the most ancient author met with by Ignatius Danti, and who it is supposed died in 1443. This last writer supposed objects placed beyond a transparent tablet, and so to trace the images, which rays of light, emitted from them, would make upon it. And Albert Durer constructed a machine upon the principles of Borgo, by which he could trace the Perspective appearance of ob- jects.<cb/></p><p>Leon Battista Alberti, in 1450, wrote his treatise De Pictura, in which he treats chiefly of Perspective.</p><p>Balthazar Per<*>zzi, of Siena, who died in 1536, had diligently studied the writings of Borgo; and his method of Perspective was published by Serlio in 1540. To him it is said we owe the discovery of points of distance, to which are drawn all lines that make an angle of 45° with the ground line.</p><p>Guido Ubaldi, another Italian, soon after discovered, that all lines that are parallel to one another, if they be inclined to the ground line, converge to some point in the horizontal line; and that through this point also will pass a line drawn from the eye parallel to them. His Perspective was printed at Pisaro in 1600, and contained the first principles of the method afterwards discovered by Dr. Brook Taylor.</p><p>In 1583 was published the work of Giacomo Barozzi, of Vignola, commonly called Vignola, intitled The two Rules of Perspective, with a learned commentary by Ignatius Danti. In 1615 Marolois' work was printed at the Hague, and engraved and published by Hondius. And in 1625, Sirigatti published his treatise of Perspective, which is little more than an abstract of Vignola's.</p><p>Since that time the art of Perspective has been gradually improved by subsequent geometricians, particularly by professor Gravesande, and still more by Dr. Brook Taylor, whose principles are in a great measure new, and far more general than those of any of his predecessors. He did not confine his rules, as they had done, to the horizontal plane only, but made them general, so as to affect every species of lines and planes, whether they were parallel to the horizon or not; and thus his principles were made universal. Besides, from the simplicity of his rules, the tedious progress of drawing out plans and elevations for any object, is rendered useless, and therefore avoided; for by this method, not only the fewest lines imaginable are required to produce any Perspective representation, but every sigure thus drawn will bear the nicest mathematical examination. Farther, his system is the only one calculated for answering every purpose of those who are practitioners in the art of design; for by it they may produce either the whole, or only so much of an object as is wanted; and by sixing it in its proper place, its apparent magnitude may be determined in an instant. It explains also the Perspective of shadows, the reflection of objects from polished planes, and the inverse practice of Perspective.</p><p>His Linear Perspective was first published in 1715; and his New Principles of Linear Perspective in 1719, which he intended as an explanation of his first treatise. And his method has been chiefly followed by all others since.</p><p>In 1738 Mr. Hamilton published his Stereography, in 2 vols folio, after the manner of Dr. Taylor. But the neatest system of Perspective, both as to theory and practice, on the samo principles, is that of Mr. Kirby. There are also good treatises on the subject, by Desargues, de Bosse, Albertus, Lamy, Niceron, Pozzo the Jesuit, Ware, Cowley, Priestley, Ferguson, Emerson, Malton, Henry Clarke, &c, &c.</p><p><hi rend="italics">Of the Principles of</hi> <hi rend="smallcaps">Perspective.</hi> To give an idea<pb n="222"/><cb/> of the first principles and nature of this art; suppose a transparent plane, as of glass &c, HI raised perpendicularly on a horizontal plane; and the spectator S directing his eye O to the triangle ABC: if now we conceive the rays AO, BO, CO, &c, in their passage through the plane, to leave their traces or vestiges in <hi rend="italics">a, b, c,</hi> &c, on the plane; there will appear the triangle <hi rend="italics">abc;</hi> which, as it strikes the eye by the same rays <hi rend="italics">a</hi>O, <hi rend="italics">b</hi>O, <hi rend="italics">c</hi>O, by which the reflected particles of light from the triangle are transmitted to the same, it will exhibit the true appearance of the triangle ABC, though the object should be removed, the same distance and height of the eye being preserved.</p><p>The business of Perspective then, is to shew by what certain rules the points <hi rend="italics">a, b, c,</hi> &c, may be found geometrically: and hence also we have a mechanical method of delineating any object very accurately. <figure/></p><p>Hence it appears that <hi rend="italics">abc</hi> is the section of the plane of the picture with the rays, which proceed from the original object to the eye: and therefore, when this is parallel to the picture, its representation will be both parallel to the original, and similar to it, though smaller in proportion as the original object is farther from the picture. When the original object is brought to coincide with the picture, the representation is equal to the original; but as the object is removed farther and farther from the picture, its image will become smaller and smaller, and also rise higher and higher in the picture, till at last, when the object is supposed to be at an infinite distance, its image will vanish in an imaginary point, exactly as high above the bottom of the picture as the eye is above the ground plane, upon which the spectator, the picture, and the original object are supposed to stand.</p><p>This may be familiarly illustrated in the following manner: Suppose a person at a window looks through an upright pane of glass at any object beyond; and, keeping his head steady, draws the figure of the object upon the glass, with a black-lead pencil, as if the point of the pencil touched the object itself; he would then have a true representation of the object in Perspective, as it appears to his eye. For properly drawing upon the glass, it is necessary to lay it over with strong gum water, which will be fit for drawing upon when dry, and will then retain the traces of the pencil. The person should also look through a small hole in a thin plate of metal, fixed about a foot from the glass, between it and his eye; keeping his eye close to the hole, other-<cb/> wise he might shift the position of his head, and so make a false delineation of the object.</p><p>Having traced out the figure of the object, he may go over it again, with pen and ink; and when that is dry, cover it with a sheet of paper, tracing the image upon this with a pencil; then taking away the paper, and laying it upon a table, he may finish the picture, by giving it the colours, lights, and shades, as he sees them in the object itself; and thus he will have a true resemblance of the object on the paper. <hi rend="center"><hi rend="italics">Of certain Definitions in</hi> <hi rend="smallcaps">Perspective.</hi></hi></p><p><hi rend="italics">The point of sight,</hi> in Perspective, is the point E, where the spectator's eye should be placed to view the <figure/> picture. And the <hi rend="italics">point of sight,</hi> in the picture, called also the <hi rend="italics">centre of the picture,</hi> is the point C directly opposite to the eye, where a perpendicular from the eye at E meets the picture. Also this perpendicular EC is the <hi rend="italics">distance of the picture:</hi> and if this distance be transferred to the horizontal line on each side of the point C, as is sometimes done, the extremes are called the points of distance.</p><p>The <hi rend="italics">original plane,</hi> or <hi rend="italics">geometrical plane,</hi> is the plane KL upon which the real or original object ABGD is situated. The line OI, where the ground plane cuts the bottom of the picture, is called the <hi rend="italics">section</hi> of the original plane, the <hi rend="italics">ground-line,</hi> the <hi rend="italics">line of the base,</hi> or the <hi rend="italics">fundamental line.</hi></p><p>If an original line AB be continued, so as to intersect the picture, the point of intersection R is called the intersection of that original line, or its <hi rend="italics">intersecting point.</hi> The <hi rend="italics">horizontal plane</hi> is the plane <hi rend="italics">abgd,</hi> which passes through the eye, parallel to the horizon, and cuts the Perspective plane or picture at right angles; and the <hi rend="italics">horizontal line bg</hi> is the common intersection of the horizontal plane with the picture.</p><p>The <hi rend="italics">vertical plane</hi> is that which passes through the eye at right angles both to the ground plane and to the picture, as ECSN. And the <hi rend="italics">vertical line</hi> is the common section of the vertical plane and the picture, as CN.</p><p>The <hi rend="italics">line of station</hi> SN is the common section of the vertical plane with the ground plane, and perpendicular to the ground line OI.<pb n="223"/><cb/></p><p>The <hi rend="italics">line of the height of the eye</hi> is a perpendicular, as ES, let fall from the eye upon the ground plane.</p><p>The <hi rend="italics">vanishing line</hi> of the original plane, is that line where a plane passing through the eye, parallel to the original plane, cuts the picture: thus <hi rend="italics">bg</hi> is the vanishing line of ABGD, being the greatest height to which the image can rise, when the original object is insinitely distant.</p><p>The <hi rend="italics">vanishing point</hi> of the original line, is that point where a line drawn from the eye, parallel to that original line, intersects the picture: thus C and <hi rend="italics">g</hi> are the vanishing points of the lines AB and <hi rend="italics">ki.</hi> All lines parallel to each other have the same vanishing point.</p><p>If from the point of sight a line be drawn perpendicular to any vanishing line, the point where that line intersects the vanishing line, is called the centre of that vanishing line: and the <hi rend="italics">distance of a vanishing line</hi> is the length of the line which is drawn from the eye, perpendicular to the said line.</p><p><hi rend="italics">Measuring points</hi> are points from which any lines in the Perspective plane are measured, by laying a ruler from them to the divisions laid down upon the ground line. The measuring point of all lines parallel to the ground line, is either of the points of distance on the horizontal line, or point of sight. The measuring point of any line perpendicular to the ground line, is in the point of distance on the horizontal line; and the measuring point of a line oblique to the ground line is found by extending the compasses from the vanishing point of that line to the point of distance on the perpendicular, and setting off on the horizontal line. <hi rend="center"><hi rend="italics">Some general Maxims or Theorems in</hi> <hi rend="smallcaps">Perspective.</hi></hi> <figure/></p><p>1. The representation <hi rend="italics">ab,</hi> of a line AB, is part of a line SC, which passes through the intersecting point S, and the vanishing point C, of the original line AB.</p><p>2. If the original plane be parallel to the picture, it can have no vanishing line upon it; consequently the representation will be parallel. When the original is perpendicular to the ground line, as AB, then its vanishing point is in C, the centre of the picture, or point of sight; because EC is perpendicular to the picture, and therefore parallel to AB.</p><p>3. The image of a line bears a certain proportion to its original. And the image may be determined by transferring the length or distance of the given line to<cb/> the intersecting line; and the distance of the vanishing point to the horizontal line; i. e. by bringing both into the plane of the picture.</p><p><hi rend="smallcaps">Prob.</hi> <hi rend="italics">To find the representation of an Objective point</hi> A. —Draw A1 and A2 at pleasure, intersecting the bot- <figure/> tom of the picture in 1 and 2; and from the eye E draw EH parallel to A1, and EL parallel to A2; then draw H1 and L2, which will intersect each other in <hi rend="italics">a,</hi> the representation of the point A.</p><p><hi rend="smallcaps">Otherwise.</hi> Let H be the given objective point. <figure/> From which draw HI perpendicular to the fundamental line DE. From the fundamental line DE cut off IK = IH : through the point of sight F draw a horizontal line FP, and make FP equal to the distance of the eye SK: lastly, join FI and PK, and their intersection <hi rend="italics">h</hi> will be the appearance of the given objective point H, as required.</p><p>And thus, by finding the representations of the two points, which are the extremes of a line, and connecting them together, there will be formed the representation of the line itself. In like manner, the representations of all the lines or sides of any figure or solid, determine those of the solid itself; which therefore are thus put into Perspective.</p><p><hi rend="italics">Aerial</hi> <hi rend="smallcaps">Perspective</hi>, is the art of giving a due diminution or gradation to the strength of light, shade, and colours of objects, according to their different distances, the quantity of light which falls upon them, and the medium through which they are seen.</p><p><hi rend="smallcaps">Perspective</hi> <hi rend="italics">Machine,</hi> is a machine for readily and easily making the Perspective drawing and appearance of any object, with little or no skill in the art. There have been invented various machines of this kind. One of which may even be seen in the works of Albert<pb n="224"/><cb/> Durer. A very convenient one was invented by Dr. Bevis, and is described by Mr. Ferguson, in his Perspective, pa. 113. And another is described in Kirby's Perspective, pa. 65.</p><p><hi rend="smallcaps">Perspective</hi> <hi rend="italics">Plan,</hi> or <hi rend="italics">Plane,</hi> is a glass or other transparent surface supposed to be placed between the eye and the object, and usually perpendicular to the horizon.</p><p><hi rend="italics">Scenographic</hi> <hi rend="smallcaps">Perspective.</hi> See <hi rend="smallcaps">Scenography.</hi></p><p><hi rend="smallcaps">Perspective</hi> <hi rend="italics">of Shadows.</hi> See <hi rend="smallcaps">Shadow.</hi></p><p><hi rend="italics">Specular</hi> <hi rend="smallcaps">Perspective</hi>, is that which <*>epresents the objects in cylindrical, conical, spherical, or other mirrors.</p></div1><div1 part="N" n="PERTICA" org="uniform" sample="complete" type="entry"><head>PERTICA</head><p>, a sort of comet, being the same with <hi rend="smallcaps">Veru.</hi></p></div1><div1 part="N" n="PETARD" org="uniform" sample="complete" type="entry"><head>PETARD</head><p>, a military engine, somewhat resembling in shape a high-crowned hat; serving sormerly to break down gates, barricades, draw-bridges, or the like works intended to be surprised. It is about 8 or 9 inches wide, and weighs from 55 to 70 pounds. Its use was chiefly in a clandestine or private attack, to break down the gates &c. It has also been used in countermines, to break through the enemies galleries, and give vent to their mines: but the use of Petards is now discontinued.——Their invention is ascribed to the French Hugonots in the year 1579. Their most signal exploit was the taking the city Cahors by means of them, as we are told by d'Aubigné.</p></div1><div1 part="N" n="PETIT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PETIT</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, a considerable mathematician and philosopher of France, was born at Montluçon in the diocese of Bourges, in the year 1589 according to some, but in 1600 according to others. — He first cultivated the mathematics and philosophy in the place of his nativity; but in 1633 be repaired to Paris, to which place his reputation had procured him an invitation. Here he became highly celebrated for his ingenious writings, and for his connections with Pascal, Des Cartes, Mersenne, and the other great men of that time. He was employed on several occasions by cardinal Richelieu; he was commissioned by this minister to visit the sea-ports, with the title of the king's engineer; and was also sent into Italy upon the king's business. He was at Tours in 1640, where he married; and was asterwards made intendant of the fortifications. Baillet, in his Life of Des Cartes, says, that Petit had a great genius for mathematics; that he excelled particularly in astronomy; and had a singular passion for experimental philosophy. About 1637 he returned to Paris from Italy, when the Dioptrics of Des Cartes were much spoken of. He read them, and communicated his objections to Mersenne, with whom he was intimately acquainted. And yet he soon after embraced the principles of Des Cartes, becoming not only his friend, but his partisan and defender also. He was intimately connected with Pascal, with whom he made at Rouen the same experiments concerning the vacuum, which Torricelli had before made in Italy; and was assured of their truth by frequent repetitions. This was in 1646 and 1647; and though there appears to be a long interval from this date to the time of his death, we meet with no other memoirs of his lise. He died August the 20th 1667 at Lagn<*>, near Paris, whither he had retired for some time before his decease.</p><p>Petit was the author of several works upon phy-<cb/> sical and astronomical subjects; the principal of which are,</p><p>1. Chronological Discourse, &c, 1636, 4to. In defence of Scaliger.</p><p>2. Treatise on the Proportional Compasses.</p><p>3. On the Weight and Magnitude of Metals.</p><p>4. Construction and Use of the Artillery Calipers.</p><p>5. On a Vacuum.</p><p>6. On Eclipses.</p><p>7. On Remedies against the Inundations of the Seine at Paris.</p><p>8. On the Junction of the Ocean with the Mediterranean sea, by means of the rivers Aude and Garonne.</p><p>9. On Comets.</p><p>10. On the proper Day for celebrating Easter.</p><p>11. On the Nature of Heat and Cold, &c.</p></div1><div1 part="N" n="PETTY" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PETTY</surname> (Sir <foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, a singular instance of a universal genius, was the elder son of Anthony Petty, a clothier at Rumsey in Hampshire, where he was born May the 16th, 1623. While a boy he took great delight in spending his time among the artificers there, whose trades he could work at when but 12 years of age. He then went to the grammar-school in that place, where at 15 he became master of the Latin, Greek, and French languages, with arithmetic and those parts of practical geometry and astronomy useful in navigation. Soon after, he went to the university of Caen in Normandy; and after some stay there he returned to England, where he was preferred in the king's navy. In 1643, when the civil war grew hot, and the times troublesome, he went into the Netherlands and France for three years; and having vigorously prosecuted his studies, especially in physic, at Utrecht, Leyden, Amsterdam, and Paris, he returned home to Rumsey. In 1647 he obtained a patent to teach the art of double writing for 17 years. In 1648 he published at London, “Advice to Mr. Samuel Hartlib, for the advancement of some particular parts of learning.” At this time he adhered to the prevailing party of the nation; and went to Oxford, where he taught anatomy and chemistry, and was created a doctor of physic, and grew into such repute that the philosophical meetings, which preceded and laid the foundation of the Royal Society, were first held at his house. In 1650 he was made professor of anatomy there; and soon after a member of the college of physicians in London, as also professor of music at Gresham college London. In 1652 he was appointed physician to the army in Ireland; as also to three lord lieutenants successively, Lambert, Fle<*>twood, and Henry Cromwell. In Ireland he acquired a great fortune, but not without suspicions and charges of unfair practices in his offices. After the rebellion was over in Ireland, he was appointed one of the commissioners for dividing the sorfeited lands to the army who suppressed it. When Henry Cromwell became lieutenant of that kingdom, in 1655, he appointed Dr. Petty his secretary, and clerk of the council: he likewise procured him to be elected a burgess for Westloo in Cornwall, in Richard Cromwell's parliament, which met in January 1658. But, in March following, Sir Hierom Sankey, member for Woodstock in Oxfordshire, impeached him of high crimes and misdemeanors in the execution of his office.<pb n="225"/><cb/> This gave the doctor a great deal of trouble, as he was summoned before the House of Commons; and notwithstanding the strenuous endeavours of his friends, in their recommendations of him to secretary Thurloe, and the defence he made before the house, his enemies procured his dismission from his public employments, in 1659. He then retired to Ireland, till the restoration of king Charles the Second; soon after which he came into England, where he was very graciously received by the king, resigned his professorship at Gresham college, and was appointed one of the commissioners of the Court of Claims. Likewise, April the 11th, 1661, he received the honour of knighthood, and the grant of a new patent, constituting him surveyor-general of Ireland, and was chosen a member of parliament there.</p><p>Upon the incorporating of the Royal Society, he was one of the first members, and of its first council. And though he had left off the practice of physic, his name was continued as an honorary member of the college of physicians in 1663.</p><p>About this time he invented his double bottomed ship, to sail against wind and tide, and afterwards presented a model of this ship to the Royal Society; to whom also, in 1665, he communicated “A Discourse about the Building of Ships,” containing some curious secrets in that art. But, upon trial, finding his ship failed in some respects, he at length gave up that project.</p><p>In 1666 Sir William drew up a treatise, called <hi rend="italics">Verbum Sapienti,</hi> containing an account of the wealth and expences of England, and the method of raising taxes in the most equal manner.—The same year, 1666, he suffered a considerable loss by the fire of London.— The year following he married Elizabeth, daughter of Sir Hardresse Waller; and afterwards set up iron works and pilchard fishing, opened lead mines and a timber trade in Kerry, which turned to very good account. But all these concerns did not hinder him from the pursuit of both political and philosophical speculations, which he thought of public utility, publishing them either separately or by communication to the Royal Society, particularly on finances, taxes, political arithmetic, land carriage, guns, pumps, &c.</p><p>Upon the first meeting of the Philosophical Society at Dublin, upon the plan of that at London, every thing was submitted to his direction; and when it was formed into a regular society, he was chosen president in Nov. 1684. Upon this occasion he drew up a “Catalogue of mean, vulgar, cheap, and simple Experiments,” proper for the infant state of the society, and presented it to them; as he did also his <hi rend="italics">Supellex Philosophica,</hi> consisting of 45 instruments requisite to carry on the design of their institution. In 1685 he made his will; in which he declares, that being then about 60, his views were fixed upon improving his lands in Ireland, and to promote the trade of iron, lead, marble, sish, and timber, which his estate was capable of. And as for studies and experiments, “I think now, says he, to confine the same to the anatomy of the people, and political arithmetic; as also the improvement of ships, land-carriages, guns, and pumps, as of most use to mankind, not blaming the study of other men.” But a few years after, all his pursuits were<cb/> determined by the effects of a gangrene in his foot, occasioned by the swelling of the gout, which put a period to his life, at his house in Piccadilly, Westminster, Dec. 16, 1687, in the 65th year of his age. His corpse was carried to Rumsey, and there interred, near those of his parents.</p><p>Sir William Petty died possessed of a very large fortune, as appears by his will; where he makes his real estate about 6,500l. per annum, his personal estate about 45,000l. his bad and desperate debts 30,000l. and the demonstrable improvements of his Irish estate, 4000l. per annum; in all, at 6 per cent. interest, 15,000l. per annum. This estate came to his family, which consisted of his widow and three children, Charles, Henry, and Anne: of whom Charles was created baron of Shelbourne, in the county of Waterford in Ireland, by king William the Third; but dying without issue, was succeeded by his younger brother Henry, who was created viscount Dunkeron, in the county of Kerry, and earl of Shelbourne Feb. 11, 1718. He married the lady Arabella Boyle, sister of Charles earl of Cork, who brought him several children. He was member of parliament for Great Marlow in Buckinghamshire, and a fellow of the Royal Society: he died April 17, 1751. Anne was married to Thomas Fitzmorris, baron of Kerry and Lixnaw, and died in Ireland in the year 1737.</p><p>The variety of pursuits, in which Sir William Petty was engaged, shews him to have had a genius capable of any thing to which he chose to apply it: and it is very extraordinary, that a man of so active and busy a spirit could find time to write so many things, as it appears he did, by the following catalogue.</p><p>1. Advice to Mr. S. Hartlib &c; 1648, 4to.—2. A Brief of Proceedings between Sir Hierom Sankey and the author &c; 1659, folio.—3. Reflections upon some persons and things in Ireland, &c; 1660, 8vo.—4. A Treatise of Taxes and Contribution, &c; 1662, 1667, 1685, 4to, all without the author's name. This last was re-published in 1690, with two other anonymous pieces, “The Privileges and Practice of Parliaments,” and “The Politician Discovered;” with a new titlepage, where it is said they were all written by Sir William, which, as to the first, is a mistake.—5. Apparatus to the History of the Common Practice of Dyeing;” printed in Sprat's History of the Royal Society, 1667, 4to.—6. A Discourse concerning the Use of Duplicate Proportion, together with a New Hypothesis of Springing or Elastic Motions; 1674, 12mo.—7. Colloquium Davidis cum Anima sua, &c; 1679, folio.— 8. The Politician Discovered, &c; 1681, 4to.— 9. An Essay in Political Arithmetic; 1682, 8vo.— 10. Observations upon the Dublin Bills of Mortality in 1681, &c; 1683, 8vo.—11. An Account of some Experiments relating to Land-carriage, Philos. Trans. numb. 161.—12. Some Queries for examining Mineral Waters, ibid. numb. 166.—13. A Catalogue of Mean, Vulgar, Cheap, and Simple Experiments, &c; ibid. numb. 167.—14. Maps of Ireland, being an Actual Survey of the whole Kingdom, &c; 1685, folio.— 15. An Essay concerning the Multiplicat<*>on of Mankind; 1686, 8vo.—16. A further Assertion concerning the magnitude of London, vindicating it, &c; Philos. Trans. numb. 185.—17. Two Essays in Politi-<pb n="226"/><cb/> cal Arithmetic; 1687, 8vo.—18. Five Essays in Political Arithmetic; 1687, 8vo.—19. Observations upon London and Rome; 1687, 8vo.</p><p>His posthumous pieces are, (1), Political Arithmetic; 1690, 8vo, and 1755, with his life prefixed.— (2), The Political Anatomy of Ireland, with Verbum Sapienti, 1691, 1719 —(3), A Treatise of Naval Philosophy; 1691, 12mo.—(4), What a complete Treatise of Navigation should contain; Philos. Trans. numb. 198.—(5), A Discourse of making Cloth with Sheep's Wool; in Birch's Hist. of the Roy. Soc.— (6), Supellex Philosophica; ibid.</p><p>PHÆNOMENON. See <hi rend="smallcaps">Phenomenon.</hi></p></div1><div1 part="N" n="PHARON" org="uniform" sample="complete" type="entry"><head>PHARON</head><p>, the name of a game of chance. See De Mo<*>vre's Doctrine of Chances, pa. 77 and 105.</p></div1><div1 part="N" n="PHASES" org="uniform" sample="complete" type="entry"><head>PHASES</head><p>, in Astronomy, the various appearances, or quantities of illumination of the moon, Venus, Mercury, and the other planets, by the sun. These Phases are very observable in the moon with the naked eye; by which she sometimes increases, sometimes wanes, is now bent into horns, and again appears a half circlé; at other times she is gibbous, and again a full circular face. And by help of the telescope, the like yariety of Phases is observed in Venus, Mars, &c.</p><p>Copernicus, a little before the use of telescopes, foretold, that after ages would find that Venus underwent all the changes of the moon; which prophecy was first fulsilled by Galileo, who, directing his telescope to Venus, observed her Phases to emulate those of the moon; being sometimes full, sometimes horned, and sometimes gibbous.</p><p><hi rend="smallcaps">Phases</hi> <hi rend="italics">of an Eclipse.</hi> To determine these for any time: Find the moon's place in her visible way for that moment; and from that point as a centre, with the interval of the moon's semidiameter, describe a circle: In like manner find the sun's place in the ecliptic, from which, with the semidiameter of the sun, describe another circle: The intersection of the two circles shews the Phases of the eclipse, the quantity of obscuration, and the position of the cusps or horns.</p></div1><div1 part="N" n="PHENOMENON" org="uniform" sample="complete" type="entry"><head>PHENOMENON</head><p>, or <hi rend="smallcaps">Phænomenon</hi>, an appearance in physics, an extraordinary appearance in the heavens, or on earth; either discovered by observation of the celestial bodies, or by physical experiments, the cause of which is not obvious. Such are meteors, comets, uncommon app<*>arance of stars and planets, earthquakes, &c. Such also are the effects of the magnet, phosphorus, &c.</p></div1><div1 part="N" n="PHILOLAUS" org="uniform" sample="complete" type="entry"><head>PHILOLAUS</head><p>, of Crotona, was a celebrated philosopher of the Ancients. He was of the school of Pythagoras, to whom that philosopher's Golden Verses have been ascribed. He made the heavens his chief object of contemplation; and has been said to be the author of that true system of the world which Copernicus afterwards revived; but erroneously, because there is undoubted evidence that Pythagoras learned that system in Egypt. On that erroneous supposition however it was, that Bulliald placed the name of Philolaus at the head of two works, written to illustrate and confirm that system.</p><p>“He was (says Dr. Enfield, in his History of Philosophy) a disciple of Archytas, and flourished in the time of Plato. It was from him that Plato purchased the written records of the Pythagorean system, contra-<cb/> ry to an express oath taken by the society of Pythagoreans, pledging themselves to keep secret the mysteries of their sect. It is probable that among these books were the writings of Timæus, upon which Plato formed the dialogue which bore his name. Plutarch relates, that Philolaus was one of the persons who escaped from the house which was burned by Cylon, during the life of Pythagoras; but this account cannot be correct. Philolaus was contemporary with Plato, and therefore certainly not with Pythagoras. Interfering in affairs of state, he fell a sacrisice to political jealonsy.</p><p>“Philolaus treated the doctrine of nature with great subtlety, but at the same time with great obscurity; referring every thing that exists to mathematical principles. He taught, that reason, improved by mathematical learning, is alone capable of judging concerning the nature of things: that the whole world consists of infinite and finite; that number subsists by itself, and is the chain by which its power sustains the eternal frame of things; that the Monad is not the sole principle of things, but that the Binary is necessary to furnish materials from which all subsequent numbers may be produced; that the world is one whole, which has a fiery centre, about which the ten celestial spheres revolve, heaven, the sun, the planets, the earth, and the moon; that the sun has a vitreous surface, whence the fire diffused through the world is reflected, rendering the mirror from which it is reflected visible; that all things are preserved in harmony by the law of necessity; and that the world is liable to destruction both by fire and by water. From this summary of the doctrine of Philolaus it appears probable that, following Timæus, whose writings he possessed, he so far departed from the Pythagorean system as to conceive two independent principles in nature, God and matter, and that it was from the same source that Plato derived his doctrine upon this subject.”</p></div1><div1 part="N" n="PHILOSOPHER" org="uniform" sample="complete" type="entry"><head>PHILOSOPHER</head><p>, a person well versed in philosophy; or who makes a profession of, or applies himself to, the study of nature or of morality.</p><p>PHILOSOPHICAL <hi rend="smallcaps">Transactions</hi>, those of the Royal Society. See <hi rend="smallcaps">Transactions.</hi></p></div1><div1 part="N" n="PHILOSOPHIZING" org="uniform" sample="complete" type="entry"><head>PHILOSOPHIZING</head><p>, the act of considering some object of our knowledge, examining its properties, and the phenomena it exhibits, and enquiring into their causes or effects, and the laws of them; the whole conducted according to the nature and reason of things, and directed to the improvement of knowledge.</p><p><hi rend="italics">The Rules of</hi> <hi rend="smallcaps">Philosophizing</hi>, as established by Sir Isaac Newton, are, 1. That no more causes of a natural effect be admitted than are true, and suffice to account for its phenomena. This agrees with the sentiments of most philosophers, who hold that nature does nothing in vain; and that it were vain to do that by many things, which might be done by fewer.</p><p>2. That natural effects of the same kind, proceed from the same causes. Thus, for instance, the cause of re<*>piration is one and the same in man and brute; the cause of the descent of a stone, the same in Europe as in America; the cause of light, the same in the sun and in culinary fire; and the cause of reflection, the same in the planets as the earth.</p><p>3. Those qualities of bodies which are not capable of being heightened, and remitted, and which are found<pb n="227"/><cb/> in all bodies on which experiments can be made, must be considered as universal qualities of all bodies. Thus, the extension of body is only perceived by our senses, nor is it perceivable in all bodies: but since it is found in all that we have perception of, it may be affirmed of all. So we find that several bodies are hard; and argue that the hardness of the whole only arises from the hardness of the parts: whence we infer that the particles, not only of those bodies which are sensible, but of all others, are likewise hard. Lastly, if all the bodies about the earth gravitate towards the earth, and this according to the quantity of matter in each; and if the moon gravitate towards the earth also, according to its quantity of matter; and the sea again gravitate towards the moon; and all the planets and comets gravitate towards each other: it may be affirmed universally, that all bodies in the creation gravitate towards each other. This rule is the foundation of all natural philosophy.</p></div1><div1 part="N" n="PHILOSOPHY" org="uniform" sample="complete" type="entry"><head>PHILOSOPHY</head><p>, the knowledge or study of nature or morality, founded on reason and experience. Literally and originally, the word signified a love of wisdom. But by Philosophy is now meant the knowledge of the nature and reasons of things; as distinguished from history, which is the bare knowledge of facts; and from mathematics, which is the knowledge of the quantity and measures of things.</p><p>These three kinds of knowledge ought to be joined as much as possible. History furnishes matter, principles, and practical examinations; and mathematics completes the evidence.</p><p>Philosophy being the knowledge of the reasons of things, all arts must have their peculiar Philosophy which constitutes their theory: not only law and physic, but the lowest and most abject arts are not without their reasons. It is to be observed that the bare intelligence and memory of philosophical propositions, without any ability to demonstrate them, is not Philosophy, but history only. However, where such propositions are determinate and true, they may be usefully applied in practice, even by those who are ignorant of their demonstrations. Of this we see daily instances in the rules of arithmetic, practical geometry, and navigation; the reasons of which are often not understood by those who practise them with success. And this success in the application produces a conviction of mind, which is a kind of medium between Philosophical or scientific knowledge, and that which is historical only.</p><p>If we consider the difference there is between natural philosophers, and other men, with regard to their knowledge of phenomena, we shall find it consists not in an exacter knowledge of the efficient cause that produces them, for that can be no other than the will of the Deity; but only in a greater and more enlarged comprehension, by which analogies, harmonies, and agreements are described in the works of nature, and the particular effects explained; that is, reduced to general rules, which rules grounded on the analogy and uniformness observed in the production of natural effects, are more agreeable, and sought after by the mind; for that they extend our prospect beyond what is present, and near to us, and enable us to make very probable conjectures, touching things that may have happened<cb/> at very great distances of time and place, as well as to predict things to come; which sort of endeavour towards omniscience is much affected by the mind. Berkley, Princip. of Hum. Knowledge, sect. 104, 105.</p><p>From the first broachers of new opinions, and the first founders of schools, Philosophy is become divided into several sects, some ancient, others modern; such are the Platonists, Peripatetics, Epicureans, Stoics, Pyrrhonians, and Academics; also the Cartesians, Newtonians, &c. See the particular articles for each.</p><p>Philosophy may be divided into two branches, or it may be considered under two circumstances, theoretical and practical.</p><p><hi rend="italics">Theoretical</hi> or <hi rend="italics">Speculative</hi> <hi rend="smallcaps">Philosophy</hi>, is employed in mere contemplation. Such is physics, which is a bare contemplation of nature, and natural things.</p><p>Theoretical Philosophy again is usually subdivided into three kinds, viz, pneumatics, physics or somatics, and metaphysics or ontology.</p><p>The first considers being, abstractedly from all matter: its objects are spirits, their natures, properties, and effects. The second considers matter, and material things: its objects are bodies, their properties, laws, &c.</p><p>The third extends to each indifferently: its objects are body or spirit.</p><p>In the order of our discovery, or arrival at the knowledge of them, physics is first, then metaphysics; the last arises from the two first considered together.</p><p>But in teaching, or laying down these several branches to others, we observe a contrary order; beginning with the most universal, and descending to the more particular. And hence we see why the Peripatetics call metaphysics, and the Cartesians pneumatics, the <hi rend="italics">prima philosophia.</hi></p><p>Others prefer the distribution of Philosophy into four parts, viz, 1. Pneumatics, which considers and treats of spirits. 2. Somatics, of bodies. 3. The third compounded of both, anthropology, which considers man, in whom both body and spirit are found. 4. Ontosophy, which treats of what is common to all the other three.</p><p>Again, Philosophy may be divided into three parts; intellectual, moral, and physical: the intellectual part comprises logic and metaphysics; the moral part contains the laws of nature and nations, ethics and politics; and lastly the physical part comprehends the doctrine of bodies, animate or inanimate: these, with their various subdivisions, will comprize the whole of Philosophy.</p><p><hi rend="italics">Practical</hi> <hi rend="smallcaps">Philosophy</hi>, is that which lays down the rules of a virtuous and happy life; and excites us to the practice of them. Most authors divide it into two kinds, answerable to the two sorts of human actions to be directed by it; viz, Logic, which governs the operations of the understanding; and Ethics, properly so called, which direct those of the will.</p><p>For the several particular sorts of Philosophy, see the articles, Arabian, Aristotelian, Atomical, Cartesian, Corpuscular, Epicurean, Experimental, Hermetical, Leibnitzian, Mechanical, Moral, Natural, Newtonian, Oriental, Platonic, Scholastic, Socratic, &c. &c.<pb n="228"/><cb/></p></div1><div1 part="N" n="PHOENIX" org="uniform" sample="complete" type="entry"><head>PHOENIX</head><p>, a constellation of the southern hemisphere. This is one of the new-added asterisms, unknown to the Ancients, and is not visible in our northern parts of the globe. There are 13 stars in this constellation.</p></div1><div1 part="N" n="PHONICS" org="uniform" sample="complete" type="entry"><head>PHONICS</head><p>, otherwise called <hi rend="smallcaps">Acoustics</hi>, is the doctrine or science of sounds.</p><p>Phonics may be considered as an art analogous to Optics; and may be divided, like that, into Direct, Refracted, and Reflected. These branches, the bishop of Ferns, in allusion to the parts of Optics, denominates Phonics, Diaphonics, and Cataphonics. See <hi rend="smallcaps">Acoustics.</hi></p></div1><div1 part="N" n="PHOSPHORUS" org="uniform" sample="complete" type="entry"><head>PHOSPHORUS</head><p>, a matter which shines, or even burns spontaneously, and without the application of any sensible fire.</p><p>Phosphori are either natural or artificial.</p><p><hi rend="italics">Natural</hi> <hi rend="smallcaps">Phosphori</hi>, are matters which become luminous at certain times, without the assistance of any art or preparation. Such are the glow-worms, frequent in our colder countries; lantern-flies, and other shining insects, in hot countries; rotten-wood; the eyes, blood, scales, flesh, sweat, feathers, &c, of several animals; diamonds, when rubbed after a certain manner, or after having been exposed to the sun or light; sugar and sulphur, when pounded in a dark place; sea water, and some mineral waters, when briskly agitated; a cat's or horse's back, duly rubbed with the hand, &c, in the dark; nay Dr. Croon tells us, that upon rubbing his own body briskly with a well-warmed shirt, he has frequently made both to shine; and Dr. Sloane adds, that he knew a gentleman of Bristol, and his son, both whose stockings would shine much after walking.</p><p>All natural Phosphori have this in common, that they do not shine always, and that they never give any heat.</p><p>Of all the natural Phosphori, that which has occasioned the greatest speculation, is the</p><p><hi rend="italics">Barometrical</hi> or <hi rend="italics">Mercurial</hi> <hi rend="smallcaps">Phosphorus.</hi> M. Picard first observed, that the mercury of his barometer, when shaken in a dark place, emitted light. And many fanciful explanations have been given of this phenomenon, which however is now found to be a mere electrical effect.</p><p>Mr. Hawksbee has several experiments on this appearance. Passing air forcibly through the body of quicksilver, placed in an exhausted receiver, the parts were violently driven against the side of the receiver, and gave all around the appearance of fire; continuing thus till the receiver was half full again of air.</p><p>From other experiments he found, that though the appearance of light was not producible by agitating the mercury in the same manner in the common air, yet that a very fine medium, nearly approaching to a va- <*>uum, was not at all necessary. And lastly, from other experiments he found that mercury inclosed in water, which communicated with the open air, by a violent shaking of the vessel in which it was inclosed, emitted particles of light in great plenty, like little <*>ars.</p><p>By including the vessel of mereury, &c, in a receiver, and exhausting the air, the phenomenon was changed; and upon shaking the vessel, instead of sparks of light,<cb/> the whole mass appeared one continued circle of light.</p><p>Farther, if mercury be inclosed in a glass tube, close stopped, that tube is found, on being rubbed, to give much more light, than when it had no mercury in it. When this tube has been rubbed, after raising successively its extremities, that the mercury might flow from one end to the other, a light is seen creeping in a serpentine manner all along the tube, the mercury being all luminous. By making the mercury run along the tube afterwards without rubbing it, it emitted some light, though much less than before; this proves that the friction of the mercury against the glass, in running along, does in some measure electrify the glass, as the rubbing it with the hand does, only in a much less degree. This is more plainly proved by laying some very light down near the tube, for this will be attracted by the electricity raised by the running of the mercury, and will rise to that part of the glass along which the mercury runs; from which it is plain, that what has been long known in the world under the name of the Phosphorus of the barometer, is not a Phosphorus, but merely a light raised by electricity, the mercury electrifying the tube. Philos. Trans. numb. 484.</p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Phosphori</hi>, are such as owe their luminous quality to some art or preparation. Some of these are made by the maceration of plants alone, and without any fire; such as thread, linen cloth, but above all paper: the luminous appearance of this last, which it is now known is an electrical phenomenon, is greatly increased by heat. Almost all bodies, by a proper treatment, have that power of shining in the dark, which at first was supposed to be the property of one, and afterwards only of a few. See Philos. Trans. numb. 478, in vol. 44, pa. 83.</p><p>Of Artificial Phosphori there are three principal kinds: the first <hi rend="italics">burning,</hi> which consumes every combustible it touches; the other two have no sensible heat, and are called the <hi rend="italics">Bononian</hi> and <hi rend="italics">Hermetic</hi> Phosphorus; to which class others of a similar kind may be referred.</p><p><hi rend="italics">The Burning</hi> <hi rend="smallcaps">Phosphorus</hi>, is a combination of phlogiston with a peculiar acid, and consequently a species of sulphur, tending to decompose itself, and so as to take fire on the access of air only. This may be made of urine, blood, hairs, and generally of any part of an animal that yields an oil by distillation, and most easily of urine. It is of a yellowish colour, and of the consistence of hard wax, in the condition it is left by the distillation; in which state it is called <hi rend="italics">phosphorus fulgurans,</hi> from its corruscations; and <hi rend="italics">phosphorus smaragdinus,</hi> because its light is often green or blue, especially in places that are not very dark; and from its consistence it is called solid Phosphorus. It dissolves in all kinds of distilled oils, in which state it is called liquid Phosphorus. And it may be ground in all kinds of fat pomatums, in which way it makes a luminous unguent.—So that these sorts are all the same preparation, under different circumstances.</p><p>The discovery of this Phosphorus was made in 1677, by one Brandt, a citizen of Hamburgh, in his researches for the philosopher's stone. And the method was afterwards found out both by Kunckel, and Mr. Boyle, from only learning that urine was the chief sub-<pb n="229"/><cb/> stance of it; since then it has been called Kunckell's Phosphorus. It is prepared by first evaporating the urine to a rob, or the consistence of honey, and afterwards distilling it in a very s<*>rong heat, &c. See Mem. Acad. Paris 1737; Philos. Trans. numb. 196, or Abr. vol. 3, pa. 346; Mem. Acad. Berlin 1743.</p><p>Many curious and amusing experiments are made with Phosphorus; as by writing with it, when the letters will appear like flame in the dark, though in the light nothing appears but a dim smoke; also a little bit of it rubbed between two papers, presently takes fire, and burns vehemently; &c. By washing the face, or hands, &c, with liquid Phosphorus, they will shine very considerably in the dark, and the lustre will be communicated to adjacent objects, yet, without hurting the skin: on bringing in the candle, the shining disappears, and no change is perceivable.</p><p><hi rend="italics">Bolognian</hi> or <hi rend="italics">Bononian</hi> <hi rend="smallcaps">Phosphorus</hi>, is a preparation of a stone called the Bononian stone, from Bologna, a city in Italy, near which it is found. This Phosphorus has no sensible heat, and only becomes luminous after being exposed to the sun or day light. For the method of preparing it, see the Mem. Acad. Berlin 1749 and 1750.</p><p>The <hi rend="italics">Hermetic</hi> <hi rend="smallcaps">Phosphorus</hi>, or third kind, is a preparation of English chalk, with aqua fortis, or spirit of nitre, by the fire. It makes a body considerably softer than the Bolognian stone, but having otherwise all the same qualities. It is also callėd Baldwin's Phosphorus, from its inventor, a German chemist, called also Hermes in the society of the Naturæ Curiosorum, whence its other name Hermetic: it was discovered a little before the year 1677. See Acad. Par. 1693, pa. 271; and Grew's Mus. Reg. Soc. p. 353.</p><p><hi rend="italics">Ammoniacal</hi> <hi rend="smallcaps">Phosphorus</hi>, first discovered by Homberg, is a combination of quick-lime with the acid of sal ammoniac, from which it receives its phlogiston. Mem. Acad. Par. 1693.</p><p><hi rend="italics">Antimonial</hi> <hi rend="smallcaps">Phosphorus</hi>, is a kind discovered by Mr. Geoffroy in his experiments on antimony. Mem. Acad. Par. 1736.</p><p><hi rend="smallcaps">Phosphorus</hi> <hi rend="italics">of the Berne-stone,</hi> a name given to a stone from Berne, in Switzerland, where it is found, and which becomes a kind of Phosphorus when heated. Mem. Acad. Paris 1724.</p><p><hi rend="italics">Canton's</hi> <hi rend="smallcaps">Phosphorus</hi>, a very good kind, prepared by Mr. Canton, an ingenious philosopher, from calcined oyster shells. Philos. Trans. vol. 58, pa. 337.</p><p><hi rend="smallcaps">Phopshorus</hi> <hi rend="italics">Fæcalis,</hi> a very good kind, exhibiting many wonderful phenomena, and prepared, by Mr. Homberg, from human dung mixed with alum. Mem. Acad. Par. 1711.</p><p><hi rend="smallcaps">Phosphorus</hi> <hi rend="italics">Metallorum,</hi> a name given by some chemists to a preparation of a certain mineral spar, found in the mines of Saxony, and other places where there is copper. Philos. Trans. numb. 244, p. 365.</p><p><hi rend="smallcaps">Phosphorus</hi> <hi rend="italics">of Sulphur,</hi> a new-discovered species, which readily takes fire on being exposed to the open air, and invented by M. Le Fevre. Mem. Acad. Par. 1728.</p><div2 part="N" n="Phosphorus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Phosphorus</hi></head><p>, in Astronomy, is the morning star, or the planet Venus, when she rises before the sun. The Latins call it Lucifer, the French Etoile de berger, and the Greeks Phosphorus.<cb/></p></div2></div1><div1 part="N" n="PHYSICAL" org="uniform" sample="complete" type="entry"><head>PHYSICAL</head><p>, something belonging to nature, or existing in it. Thus, we say a Physical point, in opposition to a mathematical one, which last only exists in the imagination. Or a Physical substance or body, in opposition to spirit, or metaphysical substance, &c.</p><div2 part="N" n="Physical" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Physical</hi></head><p>, or <hi rend="italics">Sensible Horizon.</hi> See <hi rend="smallcaps">Horizon.</hi></p><p><hi rend="smallcaps">Physico</hi>-<hi rend="italics">Mathematics,</hi> or <hi rend="italics">Mixed Mathematics,</hi> includes those branches of Physics which, uniting observation and experiment to mathematical calculation, explain mathematically the phenomena of nature.</p></div2></div1><div1 part="N" n="PHYSICS" org="uniform" sample="complete" type="entry"><head>PHYSICS</head><p>, called also <hi rend="italics">Physiology,</hi> and <hi rend="italics">Natural Philosophy,</hi> is the doctrine of natural bodies, their phenomena, causes, and effects, with their various affections, motions, operations, &c. So that the immediate and proper objects of Physics, are body, space, and motion.</p><p>The origin of Physics is referred, by the Greeks, to the Barbarians, viz, the brachmans, the magi, and the Hebrew and Egyptian priests. From these it passed to the Greek sages or sophi, particularly to Thales, who it is said first professed the study of nature in Greece. Hence it descended into the schools of the Pythagoreans, the Platonists, and the Peripatetics; from whence it passed into Italy, and thence through the rest of Europe. Though the druids, bards, &c, had a kind of system of Physics of their own.</p><p>Physics may be divided, with regard to the manner in which it has been handled, and the persons by whom, into</p><p><hi rend="italics">Symbolical</hi> <hi rend="smallcaps">Physics</hi>, or such as was couched under symbols: such was that of the old Egyptians, Pythagoreans, and Platonists; who delivered the properties of natural bodies under arithmetical and geometrical characters, and hieroglyphics.</p><p><hi rend="italics">Peripatetical</hi> <hi rend="smallcaps">Physics</hi>, or that of the Aristotelians, who explained the nature of things by matter, form, and privation, elementary and occult qualities, sympathies, antipathies, attractions, &c.</p><p><hi rend="italics">Experimental</hi> <hi rend="smallcaps">Physics</hi>, which enquires into the reasons and natures of things from experiments: such as those in chemistry, hydrostatics, pneumatics, optics, &c. And</p><p><hi rend="italics">M<*>chanical</hi> or <hi rend="italics">Corpuscular</hi> <hi rend="smallcaps">Physics</hi>, which explain<*> the appearances of nature from the matter, motion, structure, and figure of bodies and their parts: all according to the settled laws of nature and mechanics. See each of these articles under its own head.</p></div1><div1 part="N" n="PIASTER" org="uniform" sample="complete" type="entry"><head>PIASTER</head><p>, a Spanish money, more usually called Piece of Eight, about the value of 4s. 6d.</p></div1><div1 part="N" n="PIAZZA" org="uniform" sample="complete" type="entry"><head>PIAZZA</head><p>, popularly called Piache, an Italian name for a portico, or covered walk, supported by arches.</p></div1><div1 part="N" n="PICARD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PICARD</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an able mathematician of France, and one of the most learned astronomers of the 17th century, was born at Fleche, and became priest and prior of Rillie in Anjou. Coming afterwards to Paris, his superior talents for mathematics and astronomy soon made him known and respected. In 1666 he was appointed astronomer in the Academy of Sciences. And five years after, he was sent, by order of the king, to the castle of Uraniburgh, built by Tycho Brahe in Denmark, to make astronomical observations there; and from thence he brought the original manuscripts, written by Tycho Brahe; which are the more valuable, as they differ in many places from the printed copies, and<pb n="230"/><cb/> contain a book more than has yet appeared. These discoveries were followed by many others, particularly in astronomy: He was one of the first who applied the telescope to astronomical quadrants: he first executed the work called, <hi rend="italics">La Connoissance des Temps,</hi> which he calculated from 1679 to 1683 inclusively: he first observed the light in the vacuum of the barometer, or the mercurial phosphorus: he also first of any went through several parts of France, to measure the degrees of the French meridian, and first gave a chart of the country, which the Cassinis afterwards carried to a great degree of perfection. He died in 1682 or 1683, leaving a name dear to his friends, and respectable to his contemporaries and to posterity. His works are,</p><p>1. A treatise on Levelling.</p><p>2. Practical Dialling by calculation.</p><p>3. Fragments of Dioptrics.</p><p>4. Experiments on Running Water.</p><p>5. Of Measurements.</p><p>6. Mensuration of Fluids and Solids.</p><p>7. Abridgment of the Measure of the Earth.</p><p>8. Journey to Uraniburgh, or Astronomical Observations made in Denmark.</p><p>9. Astronomical Observations made in divers parts of France.</p><p>10. La Connoissance des Temps, from 1679 to 1683.</p><p>All these, and some other of his works, which are much esteemed, are given in the 6th and 7th volumes of the Memoris of the Academy of Sciences.</p></div1><div1 part="N" n="PICCOLOMINI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PICCOLOMINI</surname> (<foreName full="yes"><hi rend="smallcaps">Alexander</hi></foreName>)</persName></head><p>, archbishop of Patras, and a native of Sienna, where he was born about the year 1508. He was of an illustrious and ancient family, which came originally from Rome, but afterwards settled at Sienna. He composed with success for the theatre; but he was not more distinguished by his genius, than by the purity of his manners, and his regard to virtue. His charity was great; and was chiefly exerted in relieving the necessities of men of letters. He was the first who made use of the Italian language in writing upon philosophical subjects. He died at Sienna the 12th of March 1578, at 70 years of age, leaving behind him a number of works in Italian, on a variety of subjects. A particular catalogue of them may be seen in the Typographical Dictionary; the principal of which are the following:</p><p>1. Various Dramatical pieces.</p><p>2. A treatise on the Sphere.</p><p>3. A Theory of the Planets.</p><p>4. Translation of Aristotle's Art of Rhetoric and Poetry.</p><p>5. A System of Morality, published at Venice, 1575, in 4to; translated into French by Peter de Larivey, and printed at Paris, 1581, in 4to.</p><p>These, with a variety of other works, prove his extensive knowledge in natural philosophy, mathematics, and theology.</p><p><hi rend="smallcaps">Piccolomini</hi> (Francis), of the same family with the foregoing, was born in 1520, and taught philosophy with success, for the space of 22 years, in the most celebrated universities of Italy, and afterwards retired to Sienna, where he died, in 1604, at 84 years of age. He was so much and so generally respected, that the city went into mourning on his death.</p><p>Piccolomini laboured to revive the doctrine of Plato,<cb/> and endeavoured also to imitate the manners of that philosopher. He had for his rival the famous James Zabar Alla, whom he excelled in facility of expression and neatness of diction; but to whom he was much inferior in point of argument, because he did not examine matters to the bottom as the other did; but passed too rapidly from one proposition to another.</p></div1><div1 part="N" n="PICKET" org="uniform" sample="complete" type="entry"><head>PICKET</head><p>, <hi rend="italics">Picquet,</hi> or <hi rend="italics">Piquet,</hi> in Fortification &c. a stake sharp at one end, and usually shod with iron; used in laying out ground, to mark the several bounds and angles of it. There are also larger Pickets, driven into the earth, to hold together fascines or faggots, in works that are thrown up in haste. As also various forts of smaller Pickets for divers other uses.</p></div1><div1 part="N" n="PIECES" org="uniform" sample="complete" type="entry"><head>PIECES</head><p>, in Artillery, include all sorts of great guns and mortars; meaning Pieces of ordnance, or of artillery.</p></div1><div1 part="N" n="PIEDOUCHE" org="uniform" sample="complete" type="entry"><head>PIEDOUCHE</head><p>, in Architecture, a littlè stand, or pedestal, either oblong or square, enriched with mouldings; serving to support a bust, or other little figure; and is more usually called a bracket pedestal.</p></div1><div1 part="N" n="PIEDROIT" org="uniform" sample="complete" type="entry"><head>PIEDROIT</head><p>, in Architecture, a kind of square pillar, or pier, partly hid within a wall. Differing from the Pilaster by having no regular base nor capital.</p><p><hi rend="smallcaps">Piedroit</hi> is also used for a part of the solid wall annexed to a door or window; comprehending the doorpost, chambranle, tableau, leaf, &c.</p></div1><div1 part="N" n="PIER" org="uniform" sample="complete" type="entry"><head>PIER</head><p>, in Building, denotes a mass of stone, &c, opposed by way of fortress, against the force of the sea, or a great river, for the security of ships lying in any harbour or haven. Such are the Piers at Dover, or Ramsgate, or Yarmouth, &c.</p><p><hi rend="smallcaps">Piers</hi> are also used in Architecture for a kind of pilasters, or buttresses, raised for support, strength, and sometimes for ornament.</p><p><hi rend="italics">Circular</hi> <hi rend="smallcaps">Piers</hi>, are called Massive Columns, and are either with or without caps. These are often seen in Saracenic architecture.</p><div2 part="N" n="Piers" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Piers</hi></head><p>, of a Bridge, are the walls built to support the arches, and from which they spring as bases, to stand upon.</p><p>Piers should be built of large blocks of stone, solid throughout, and cramped together with iron, which will make the whole as one solid stone. Their extremities, or ends, from the bottom, or base, up to highwater mark, ought to project sharp out with a saliant angle, to divide the stream. Or perhaps the bottom part of the Pier should be built slat or square up to about half the height of low-water mark, to encourage a lodgment against it for the sand and mud, to cover the foundation; lest, being lest bare, the water should in time undermine and ruin it. The best form of the projection for dividing the stream, is the triangle; and the longer it is, or the more acute the saliant angle, the better it will divide it, and the less will the force of the water be against the Pier; but it may be sufficient to make that angle a right one, as it will render the masonry stronger, and in that case the perpendicular projection will be equal to half the breadth or thickness of the Pier. In rivers where large heavy craft navigate, and pass the arches, it may perhaps be better to make the ends semicircular; for though this figure does not divide the water so well as the triangle, it will better turn off, and bear the shock of the craft.<pb n="231"/><cb/></p><p>The thickness of the Piers ought to be such as will make them of weight, or strength, sufficient to support their interjacent arch, independent of the assistance of any other arches. And then, if the middle of the Pier be run up to its full height, the centring may be struck, to be used in another arch, before the hanches or spandrels are filled up. They ought also to be made with a broad bottom on the foundation, and gradually diminished in thickness by offsets up to low-water mark. <figure/></p><p><hi rend="italics">To find the thickness</hi> FG <hi rend="italics">of the Piers,</hi> necessary to support an arch ABM, this is a general rule. Let K be the centre of gravity of the half arch ADCB, A = its area; KL perpendicular to AM the span of the arch, OB its height, and BC its thickness at the crown: then is the thickness of the pier</p><p>Some authors pretend to give numbers, in tables, for this purpose; but they are very erroneous. See my treatise on the Principles of Bridges, sect. 3.</p></div2></div1><div1 part="N" n="PIKE" org="uniform" sample="complete" type="entry"><head>PIKE</head><p>, an offensive weapon, consisting of a shaft of wood, 12 or 14 feet long, headed with a flat-pointed steel, called the spear.</p><p>Pliny says the Lacedemonians were the inventors of the Pike. The Macedonian phalanx was evidently a battalion of Pikemen.</p><p>The Pike was long used by the infantry, to enable them to sustain the attack of the cavalry; but it is now taken from them, and the bayonet, fixed to the muzzle of the firelock, is given instead of it.—It is still used by some officers of infantry, under the name of spontoon.</p><p><hi rend="italics">Half</hi> <hi rend="smallcaps">Pike</hi> is the weapon carried by an officer of soot; being only 8 or 9 feet long.</p></div1><div1 part="N" n="PILASTER" org="uniform" sample="complete" type="entry"><head>PILASTER</head><p>, in Architecture, a square column, sometimes insulated, but more frequently let within a wall, and only projecting by a 4th or 5th part of its thickness.</p><p>The Pilaster is different in the different orders; borrowing the name of each order, and having the same proportions, and the same capitals, members, and ornaments, with the columns themselves.</p><p><hi rend="italics">Demi</hi> <hi rend="smallcaps">Pilaster</hi>, called also <hi rend="italics">Membretto,</hi> is a Pilaster that supports an arch; and it generally stands against a pier or column.<cb/></p></div1><div1 part="N" n="PILES" org="uniform" sample="complete" type="entry"><head>PILES</head><p>, in Building, are large stakes, or beams, sharpened at the end, and shod with iron, to be driven into the ground, for a foundation to build upon in marshy places.</p><p>Amsterdam, and some other cities, are wholly built upon Piles. The stoppage of Dagenham-breach was effected by dove-tail Piles, that is by Piles mortised into one another by a dovetail joint.</p><p>Piles are driven down by blows of a large iron weight, ram, or hammer, dropped continually upon them from a height, till the Pile is sunk deep enough into the ground.</p><p>Notwithstanding the momentum, or force of a body in motion, is as the weight multiplied by the velocity, or simply as its velocity, the weight being given, or constant; yet the effect of the blow will be nearly as the square of that velocity, the effect being the quantity the Pile sinks in the ground by the stroke. For the force of the blow, which is transferred to the Pile, being destroyed, in some certain definite time, by the friction of the part which is within the earth, which is nearly a constant quantity; and the spaces, in constant forces, being as the squares of the velocities; therefore the effects, which are those spaces sunk, are nearly as the square of the velocities; or, which is the same thing, nearly as the heights fallen by the ram or hammer, to the head of the Pile. See, upon this subject, Leopold Belidor, also Desaguliers's Exper. Philos. vol. 1, pa. 336, and vol. 2, pa. 417: and Philos. Trans. 1779, pa. 120.</p><p>There have been various contrivances for raising and dropping the hammer, for driving down the Piles; some simple and moved by strength of men, and some complex and by machinery; but the completest PileDriver is esteemed that which was employed in driving the Piles in the foundation of Westminster bridge. This machine was the invention of a Mr. Vauloue, and the description of it is as follows.</p><p><hi rend="italics">Description of Vauloue's</hi> <hi rend="smallcaps">Pile</hi>-<hi rend="italics">Driver.</hi> See fig. 2, pl. xx. A is the great upright shaft or axle, carrying the great wheel B and drum C, and turned by horses attached to the bars S, S. The wheel B turns the trundle X, having a fly O at the top, to regulate the motion, and to act against the horses, and keep them from falling when the heavy ram Q is disengaged to drive the Pile P down into the mud &c. in the bottom of the river. The drum C is loose upon the shaft A, but is locked to the wheel B by the bolt Y. On this drum the great rope HH is wound; one end of it being fixed to the drum, and the other to the follower G, passing over the pulleys I and K. In the follower G are contained the tongs F, which take hold of the ram Q, by the staple R for drawing it up. D is a spiral or fusee fixed to the drum, on which winds the small rope T which goes over the pulley U, under the pulley V, and is fastened to the top of the frame at 7. To the pulleyblock V is hung the counterpoise W, which hinders the follower from accelerating as it goes down to take hold of the ram: for, as the follower tends to acquire velocity in its descent, the line T winds downwards upon the fusee, on a larger and larger radius, by which means the counterpoise W acts stronger and stronger against it; and so allows it to come down with only a moderate and uniform velocity. The bolt Y locks the<pb n="232"/><cb/> drum to the great wheel, being pushed upward by the small lever 2, which goes through a mortise in the shaft A, turns upon a pin in the bar 3 fixed into the great wheel B, and has a weight 4, which always tends to push up the bolt Y through the wheel into the drum. L is the great lever turning on the axis <hi rend="italics">m,</hi> and resting upon the forcing bar 5, 5, which goes down through a hollow in the shaft A, and bears upon the little lever 2.</p><p>By the horses going round, the great rope H is wound about the drum C, and the ram Q is drawn up by the tongs F in the follower G, till they come between the inclined planes E; which, by shutting the tongs at the top, open them below, and so discharge the ram, which falls down between the guide posts <hi rend="italics">bb</hi> upon the Pile P, and drives it by a few strokes as far into the ground as it can go, or as is defired; after which, the top part is sawed off close to the mud, by an engine for that purpose. Immediately after the ram is discharged, the piece 6 upon the follower G takes hold of the ropes <hi rend="italics">aa,</hi> which raise the end of the lever L, and cause its end N to descend and press down the forcing bar 5 upon the little lever 2, which, by drawing down the bolt Y, unlocks the drum C from the great wheel B; and then the follower, being at liberty, comes down by its own weight to the ram; and the lower ends of the tongs slip over the staple R, and the weight of their heads causes them to fall outward, and shuts upon it. Then the weight 4 pushes up the bolt Y into the drum, which locks it to the great wheel, and so the ram is drawn up as before.</p><p>As the follower comes down, it causes the drum to turn backward, and unwinds the rope from it, while the horses, the great wheel, trundle, and fly, go on with an uninterrupted motion: and as the drum is turning backward, the counterpoise W is drawn up, and its rope T wound upon the spiral fusee D.</p><p>There are several holes in the under side of the drum, and the bolt Y always takes the first one that it finds when the drum stops by the falling of the follower upon the ram; till which stoppage, the bolt has not time to slip into any of the holes.</p><p>The peculiar advantages of this engine are, that the weight, called the ram, or hammer, may be raised with the least force; that, when it is raised to a proper height, it readily disengages itself and falls with the utmost freedom; that the forceps or tongs are lowered down speedily, and instantly of themselves again lay hold of the ram, and lift it up; on which account this machine will drive the greatest number of piles in the least time, and with the fewest labourers.</p><p>This engine was placed upon a barge on the water, and so was easily conveyed to any place desired. The ram was a ton weight; and the guides <hi rend="italics">b, b,</hi> by which it was let fall, were 30 feet high.</p><p>A new machine for driving piles has been invented lately by Mr. S. Bunce of Kirby-street, Hatton-street, London. This, it is said, will drive a greater number of Piles in a given time than any other; and that it can be constructed more simply to work by horses than Vauloue's engine above described.</p><p>Fig. 3 and 4, plate xx, represent a side and front section of the machine. The chief parts are, A, fig. 3, which are two endless ropes or chains, connected by cross pieces of iron B (fig. 4) corresponding with two<cb/> cross grooves cut diametrically opposite in the wheel C (fig. 3) into which they are received; and by which means the rope or chain A is carried round. FHK is a side-view of a strong wooden frame moveable on the axis H. D is a wheel, over which the chain passes and turns within at the top of the frame. It moves occasionally from F to G upon the centre H, and is kept in the position F by the weight I fixed to the end K. In fig. 5, L is the iron ram, which is connected with the cross pieces by the hook M. N is a cylindrical piece of wood suspended at the hook at O, which by sliding freely upon the bar that connects the hook to the ram, always brings the hook upright upon the chain when at the bottom of the machine, in the position of GP. See fig. 3.</p><p>When the man at S turns the usual crane-work, the ram being connected to the chain, and passing between the guides, is drawn up in a perpendicular direction; and when it is near the top of the machine, the projecting bar Q of the hook strikes against a cross piece of wood at R (fig. 3); and consequently discharges th<*> ram, while the weight I of the moveable frame instantly draws the upper wheel into the position shewn at F, and keeps the chain free of the ram in its descent. The hook, while descending, is prevented from catching the chain by the wooden piece N: for that piece being specisically lighter than the iron weight below, and moving with a less degree of velocity, cannot come into contact with the iron, till it is at the bottom, and the ram stops. It then falls, and again connects the hook with the chain, which draws up the ram, as before.</p><p>Mr. Bunce has made a model of this machine, which performs perfectly well; and he observes, that, as the motion of the wheel C is uninterrupted, there appears to be the least possible time lost in the operation.</p><p><hi rend="smallcaps">Pile</hi> is also used among Architects, for a mass or body of building.</p><div2 part="N" n="Pile" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Pile</hi></head><p>, in Artillery, denotes a collection or heap of shot or shells, piled up by horizontal courses into either a pyramidal or else a wedge-like form; the base being an equilateral triangle, a square, or a rectangle. In the triangle and square, the Pile terminates in a single ball or point, and forms a pyramid, as in plate xix, fig. 4 and 5, but with the rectangular base, it finishes at top in a row of balls, or an edge, forming a wedge, as in fig. 6.</p><p>In the triangular and square Piles, the number of horizontal rows, or courses, or the number counted on one of the angles from the bottom to the top, is always equal to the number counted on one side, in the bottom row. And in rectangular Piles, the number of rows, or courses, is equal to the number of balls in the breadth of the bottom row, or shorter side of the base<*> also, in this case, the number in the top row, or edge, is one more than the difference between the length and breadth of the base. All which is evident from the inspection of the figures, as above.</p><p>The courses in these Piles are figurate numbers.</p><p>In a triangular Pile, each horizontal course is a triangular number, produced by taking the successive sums of the ordinate numbers, viz,<pb n="233"/><cb/></p><p>And the number of shot in the triangular Pile, is the sum of all these triangular numbers, taken as far, or to as many terms, as the number in one side of the base. And therefore, to find this sum, or the number of all the shot in the Pile, multiply coutinually together, the number in one side of the base row, and that number increased by 1, and the same number increased by 2; then 1/6 of the last product will be the answer, or number of all the shot in the Pile.</p><p>That is, is the sum; where <hi rend="italics">n</hi> is the number in the bottom row.</p><p>Again, in Square Piles, each horizontal course is a square number, produced by taking the square of the number in its side, or the successive sums of the odd numbers, thus, .</p><p>And the number of shot in the square Pile is the sum of all these square numbers, continued so far, or to as many terms, as the number in one side of the base. And therefore, to find this sum, multiply continually together, the number in one side of the bottom course, and that number increased by 1, and double the same number increased by 1; then 1/6 of the last product will be the sum or answer.</p><p>That is, is the sum.</p><p>In a rectangular Pile, each horizontal course is a rectangle, whose two sides have always the same difference as those of the base course, and the breadth of the top row, or edge, being only 1: because each course in ascending has its length and breadth always less by 1 than the course next below it. And these rectangular courses are found by multiplying successively the terms or breadths 1, 2, 3, 4, &c, by the same terms added to the constant difference of the two sides <hi rend="italics">d;</hi> thus, .</p><p>And the number of shot in the rectangular Pile is the sum of all these rectangles, which, it is evident, consist of the sum of the squares, together with the sum of an arithmetical progression, continued till the number of terms be the difference between the length and breadth of the base, and 1 less than the edge or top row. And therefore, to find this sum, multiply continually together, the number in the breadth of the base row, the same number increased by 1, and double the same number increased by 1, and also increased by triple the difference between the length and breadth of the base; then 1/6 of the last product will be the answer.</p><p>That is, is the sum. where <hi rend="italics">b</hi> is the breadth of the base, and <hi rend="italics">d</hi> the difference between the length and breadth of the bottom course.</p><p>As to incomplete Piles, which are only frustums,<cb/> wanting a similar small Pile at the top; it is evident that the number in them will be found, by-first computing the number in the whole Pile, as if it were complete, and also the number in the small Pile wanting at top, both by their proper rule; and then subtracting the one number from the other.</p><p>In piling of shot, when room is an object, it may be observed that the square Pile is the least eligible, of any, as it takes up more room, in proportion to the number of shot contained in it, than either of the other two forms; and that the rectangular Pile is the most eligible, as taking up the least room in proportion to the number it contains.</p></div2></div1><div1 part="N" n="PILLAR" org="uniform" sample="complete" type="entry"><head>PILLAR</head><p>, a kind of irregular column, round, and insulated, or detached from the wall. Pillars are not restricted to any rules, their parts and proportions being arbitrary; such for example as those that support Saracenic vaults, and other buildings, &c.</p></div1><div1 part="N" n="PINION" org="uniform" sample="complete" type="entry"><head>PINION</head><p>, in Mechanics, is an arbor, or spindle, in the body of which are several notches, which are catched by the teeth of a wheel that serves to turn it round. Or a Pinion is any lesser wheel that plays in the teeth of a larger.</p><p>In a watch, &c, the notches of a Pinion are called leaves, and not teeth, as in other wheels; and their number is commonly 4, 5, 6, 8, &c.</p><p><hi rend="smallcaps">Pinion</hi> <hi rend="italics">of Report,</hi> is that Pinion, in a watch, commonly sixed on the arbor of a great wheel: and which used to have but four leaves in old watches; it drives the dial-wheel, and carries about the hand.</p><p>The number of turns to be laid upon the Pinion of report, is found by this proportion: as the beats in one turn of the great wheel, are to the beats in an h<*>ur, so are the hours on the face of the clock (viz 12 or 24), to the quotient of the hour-wheel or dial-wheel divided by the Pinion of report, that is, by the number of turns which the Pinion of report hath in one turn of the dialwheel. Which in numbers is . —Or thus; as the hours of the watch's going, are to the numbers of the turns of the fusee, so are the hours of the face, to the quotient of the Pinion of report. So, if the hours be 12, then as ; but if 24, then as .</p><p>This rule may serve to lay the Pinion of report on any other wheel, thus: as the beats in one turn of any wheel, are to the beats in an hour, so are the hours of the face, or dial-plate, of the watch, to the quotient of the dial-wheel divided by the Pinion of report, fixed on the spindle of the aforesaid wheel.</p></div1><div1 part="N" n="PINT" org="uniform" sample="complete" type="entry"><head>PINT</head><p>, a measure of capacity, being the 8th part of a <*>allon, both in ale and wine measure, &c. The wine Pint of pure spring water, weighs near 17 ounces avoirdupois, and the ale Pint a little above 20 ounces.</p><p>The Paris Pint contains about 2 pounds of common water. And the Scotch Pint contains 108 2/3 cubic inches, and therefore contains 3 English Pints.</p></div1><div1 part="N" n="PISCES" org="uniform" sample="complete" type="entry"><head>PISCES</head><p>, the 12th sign or constellation in the zodiac; in the form of two fishes tied together by the tails.</p><p>The Greeks, who have some fable to account for the origin of every constellation, tell us, that when Venus and Cupid were one time on the banks of the Euphrates, there appeared before them that terrible giant Typhon, who was so long a terror to all the Gods. These deities immediately, they say, threw themselves<pb n="234"/><cb/> into the water, and were there changed into these two si<*>hes, the Pisces, by which they escaped the danger. But the Egyptians used the signs of the zodiac as part of their hieroglyphic language, and by the 12 they conveyed an idea of the proper employment during the 12 months of the year. The Ram and the Boll had, at that time, taken to the increase of their flock, the young of those animals being then growing up; the maid Virgo, a reaper in the sield, spoke the approach of harvest; Sagittary declared autumn the time for hunting; and the Pisces, or fishes tied together, in token of their being taken, reminded men that the approach of spring was the time for sishing.</p><p>The Ancients, as they gave one of the 12 months of the year to the patronage of each of the 12 superior deities, so they also dedicated to, or put under the tutelage of each, one of the 12 signs of the zodiac. In this division, the fishes naturally fell to the share of Neptune; and hence arises that rule of the astrologers, which throws every thing that regards the fate of fleets and merchandize, under the more immediate patronage and protection of this constellation.</p><p>The stars in the sign Pisces are, in Ptolomy's catalogue 38, in Tycho's 36, in Hevelius's 39, and in the Britannic catalogue 113.</p><p>PISCIS <hi rend="italics">Australis,</hi> the Southern Fish, is a constellation of the southern hemisphere, being one of the old 48 constellations mentioned by the Ancients.</p><p>The Greeks have here again the fable of Venus and her son throwing themselves into the sea, to escape from the terrible Typhon. This fable is probably borrowed from the hieroglyphics of the Egyptians. With them, a fish represented the sea, its element; and Typhon was probably a land flood, perhaps represented by the sign Aquarius, or water pourer, whose stream or river is represented as swallowed up by this fish, as the land floods and rivers are by the sea. And Venus was some queen, perhaps Semiramis, otherwise called Hamamah, who took to the river or the sea with her son, in a vessel, to avoid the flood, &c.</p><p>The remarkable star Fomahaut, of the 1st magnitude, is just in the mouth of this fish. The stars of this constellation are, in Ptolomy's catalogue 18, and in Flamsteed's 24.</p><p><hi rend="smallcaps">Piscis</hi> <hi rend="italics">Volans,</hi> the Flying Fish, is a small constellation of the southern hemisphere, unknown to the Ancients, but added by the Moderns. It is not visible in our latitude, and contains only 8 stars.</p></div1><div1 part="N" n="PISTOLE" org="uniform" sample="complete" type="entry"><head>PISTOLE</head><p>, a gold coin in Spain, Italy, Switzerland, &c, of the value of about 16s. 6d.</p></div1><div1 part="N" n="PISTON" org="uniform" sample="complete" type="entry"><head>PISTON</head><p>, a part or member in several machines, particularly pumps, air-pumps, syringes, &c; called also the Embolus, and popularly the Sucker.</p><p>The Piston of a pump is a short cylinder of wood or metal, fitted exactly to the cavity of the barrel, or body; and which, being worked up and down alternately, raises the water; and when raised, presses it again, so as to make it force up a valve with which it is furnished, and so escape through the spout of the pump.</p><p>There are two sorts of Pistons used in pumps; the one with a valve, called a bucket; and the other without a valve, called a forcer.<cb/></p></div1><div1 part="N" n="PLACE" org="uniform" sample="complete" type="entry"><head>PLACE</head><p>, in Philosophy, that part of insinite which any body possesses.</p><p>Aristotle and his followers divide Place into External and Internal.</p><p><hi rend="italics">Internal</hi> <hi rend="smallcaps">Place</hi>, is that space or room which the body contains. And</p><p><hi rend="italics">External</hi> <hi rend="smallcaps">Place</hi>, is that which includes or contains the body; and is by Aristotle called the first or concave and immoveable surface of the ambient body.</p><p>Newton better, and more intelligibly, distinguishes Place into Absolute and Relative.</p><p><hi rend="italics">Absolute</hi> and <hi rend="italics">Primary</hi> <hi rend="smallcaps">Place</hi>, is that part of infinite and immoveable space which a body possesses. And</p><p><hi rend="italics">Relative,</hi> or <hi rend="italics">Secondary</hi> <hi rend="smallcaps">Place</hi>, is the space it possesses considered with regard to other adjacent objects.</p><p>Dr. Clark adds another kind of Relative Place, which he calls Rel<*>ively Common Place; and desines it, that part of any moveable or measurable space which a body possesses; which Place moves together with the body.</p><div2 part="N" n="Place" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Place</hi></head><p>, Mr. Locke observes, is sometimes likewise taken for that portion of insinite space possessed by the material world; though this, he adds, were more properly called extension. The proper idea of Place, according to him, is the relative position of any thing, with regard to its distance from certain fixed points; whence it is said a thing has or has not changed Place, when its distance is or is not altered with respect to those bodies.</p></div2><div2 part="N" n="Place" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Place</hi></head><p>, in Optics, or <hi rend="italics">Optical</hi> <hi rend="smallcaps">Place</hi>, is the point to which the eye refers an object.</p><p><hi rend="italics">Optic</hi> <hi rend="smallcaps">Place</hi> of a star, is a point in the surface of the mundane sphere in which a spect<*>tor sees the centre of the star, &c.—This is divided into True and Apparent.</p><p><hi rend="italics">True,</hi> or <hi rend="italics">Real Optic</hi> <hi rend="smallcaps">Place</hi>, is that point of the surface of the sphere, where a spectator at the centre of the earth would see the star, &c.</p><p><hi rend="italics">Apparent,</hi> or <hi rend="italics">Visible Optic</hi> <hi rend="smallcaps">Place</hi>, is that point of the surface of the sphere, where a spectator at the surface of the earth sees the star, &c.</p><p>The distance between these two optic Places makes what is called the Parallax.</p><p><hi rend="smallcaps">Place</hi> <hi rend="italics">of the Sun,</hi> or <hi rend="italics">Moon,</hi> or <hi rend="italics">Star,</hi> or <hi rend="italics">Planet,</hi> in Astronomy, simply denotes the sign and degree of the zodiac which the luminary is in; and is usually expressed either by its latitude and longitude, or by its right ascension and declination.</p><p><hi rend="smallcaps">Place</hi> <hi rend="italics">of Radiation,</hi> in Optics, is the interval or space in a medium, or transparent body, through which any visible object radiates.</p></div2><div2 part="N" n="Place" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Place</hi></head><p>, in Geometry, usually called <hi rend="italics">Locus,</hi> is a line used in the solution of problems, being that in which the determination of every case of the problem lies. See <hi rend="smallcaps">Locus</hi>, <hi rend="italics">Plane, Simple, Solid,</hi> &c.</p></div2><div2 part="N" n="Place" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Place</hi></head><p>, in War and Fortification, a general name for all kinds of fertresses, where a party may defend themselves.</p><p><hi rend="smallcaps">Place</hi> <hi rend="italics">of Arms,</hi> a strong part where the arms &c are deposited, and where usually the soldiers assemble and are drawn up.</p></div2></div1><div1 part="N" n="PLAFOND" org="uniform" sample="complete" type="entry"><head>PLAFOND</head><p>, or <hi rend="smallcaps">Platfond</hi>, in Architecture, the cieling of a room.</p><p>PLAIN &c. See <hi rend="smallcaps">Plane.</hi><pb n="235"/><cb/></p></div1><div1 part="N" n="PLAN" org="uniform" sample="complete" type="entry"><head>PLAN</head><p>, a representation of something, drawn on a plane. Such as maps, charts, and ichnogrophies.</p><div2 part="N" n="Plan" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Plan</hi></head><p>, in Architecture, is particularly used for a draught of a building; such as it appears, or is intended to appear, on the ground; shewing the extent, division, and distribution of its area into apartments, rooms, passages, &c. It is also called the Ground Plot, Platform, and Ichnography of the building; and is the first device or sketch the architect makes.</p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Plan</hi>, is that in which the solid and vacant parts are represented in their natural proportion.</p><p><hi rend="italics">Raised</hi> <hi rend="smallcaps">Plan</hi>, is that where the elevation, or upright, is shewn upon the geometrical Plan, so as to hide the distribution.</p><p><hi rend="italics">Perspective</hi> <hi rend="smallcaps">Plan</hi>, is that which is conducted and exhibited by degradations, or diminutions, according to the rules of Perspective.</p></div2></div1><div1 part="N" n="PLANE" org="uniform" sample="complete" type="entry"><head>PLANE</head><p>, or <hi rend="smallcaps">Plain</hi>, in Geometry, denotes a Plane figure, or a surface lying evenly between its bounding lines. Euclid.</p><p>Some desine a Plane, a surface, from every point of whose perimeter a right line may be drawn to every other point in the same, and always coinciding with it.</p><p>As the right line is the shortest extent from one point to another, so is a Plane the shortest extension between one line and another.</p><p><hi rend="smallcaps">Planes</hi> are much used in Astronomy, conic sections, spherics, &c, for imaginary surfaces, supposed to cut and pass through solid bodies.</p><p>When a Plane cuts a cone parallel to one side, it makes a parabola; when it cuts the cone obliquely, an cllipse or hyperbola; and when parallel to its base, a circle. Every section of a sphere is a circle.</p><p>The sphere is wholly explained by Planes, conceived to cut the celestial bodies, and to fill the areas or circumferences of the orbits. They are differently inclined to each other; and by us the inhabitants of the earth, the Plane of whose orbit is the Plane of the ecliptic, their inclination is estimated with regard to this Plane.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">of a Dial,</hi> is the surface on which a dial is supposed to be described.</p><div2 part="N" n="Plane" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Plane</hi></head><p>, in Mechanics. A <hi rend="italics">Horizontal</hi> <hi rend="smallcaps">Plane</hi>, is a Plane that is level, or parallel to the horizon.</p><p><hi rend="italics">Inclined</hi> <hi rend="smallcaps">Plane</hi>, is one that makes an oblique angle with a horizontal Plane.</p><p>The doctrine of the motion of bodies on Inclined Planes, makes a very considerable article in mechanics, and has been fully explained under the articles, M<hi rend="smallcaps">ECHANICAL</hi> <hi rend="italics">Powers,</hi> and <hi rend="smallcaps">Inclined</hi> <hi rend="italics">Plane.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">of Gravity,</hi> or <hi rend="italics">Gravitation,</hi> is a Plane supposed to pass through the centre of gravity of the body, and in the direction of its tendency; that is, perpendicular to the horizon.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">of Reflection,</hi> in Catoptrics, is a Plane which passes through the point of reslection; and is perpendicular to the Plane of the glass, or reflecting body.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">of Refraction,</hi> is a Plane passing through the incident and refracted ray.</p><p><hi rend="italics">Perspective</hi> <hi rend="smallcaps">Plane</hi>, is a Plane transparent surface, usually perpendicular to the horizon, and placed between the spectator's eye and the object he views; through which the optic rays, emitted from the several points of the object, are supposed to pass to the eye, and in their<cb/> passage to leave marks that represent them on the said Plane.—Some call this the Table, or Picture, because the draught or Perspective of the object is supposed to be upon it. Others call it the Section, from its cutting the visual rays; and others again the Glass, from its supposed transparency.</p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Plane</hi>, in Perspective, is a Plane parallel to the horizon, upon which the object is supposed to be placed that is to be drawn.</p><p><hi rend="italics">Horizontal</hi> <hi rend="smallcaps">Plane</hi>, in Perspective, is a Plane passing through the spectator's eye, parallel to the horizon.</p><p><hi rend="italics">Vertical</hi> <hi rend="smallcaps">Plane</hi>, in Perspective, is a Plane passing through the spectator's eye, perpendicular to the geometrical Plane, and usually at right angles to the perspective Plane.</p><p><hi rend="italics">Objective</hi> <hi rend="smallcaps">Plane</hi>, in Perspective, is any Plane situate in the horizontal Plane, of which the representation in perspective is required.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">of the H<*>ropter,</hi> in Optics, is a Plane passing through the horopter AB, and perpendicular to a Plane passing through the two optic axes CH and CI. See the fig. to the article <hi rend="smallcaps">Horopter.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">of the Projection,</hi> is the Plane upon which the sphere is projected.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Angle,</hi> is an angle contained under two lines or surfaces.—It is so called in contradistinction to a solid angle, which is formed by three or more Planes.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Triangle,</hi> is a triangle formed by three right lines; in opposition to a spherical and a mixt triangle.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Trigonometry</hi> is the doctrine of Plane triangles, their measures, proportions, &c. See T<hi rend="smallcaps">RIGONOMETRY.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Glass,</hi> or <hi rend="italics">Mirror,</hi> in Optics, is a glass or mirror having a slat or even surface.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Chart,</hi> in Navigation, is a sea-chart, having the meridians and parallels represented by parallel straight lines; and consequently having the degrees of longitude the same in every part. See <hi rend="smallcaps">Chart.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Number,</hi> is that which may be produced by the multiplication of two numbers the one by the other. Thus, 6 is a plane number, being produced by the multiplication of the two numbers 2 and 3; also 15 is a Plane number, being produced by the multiplication of the numbers 3 and 5. See <hi rend="smallcaps">Number.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Place, Locus Planus,</hi> or <hi rend="italics">Locus ad Planum,</hi> is a term used by the ancient geometricians, for a geometrical locus, when it was a right line or a <*>ircle, in opposition to a solid place, which was one of the conic sections.</p><p>These Plane Loci are distinguished by the Moderns into Loci ad Rectum, and Loci ad Circulum. See L<hi rend="smallcaps">OCUS.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Problem,</hi> is such a one as cannot be resolved geometrically, but by the intersection either of a right line and a circle, or of the circumferences of two circles. Such as this problem following: viz, Given the hypothenuse, and the sum of the other two sides, of a rightangled triangle; to find the triangle. Or this <*> Of four given lines to form a trapezium of a given area.</p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Sailing,</hi> in Navigation, is the art of working the several cases and varieties in a ship's motion on a Plane chart <*> or of navigating a ship upon principles<pb n="236"/><cb/> deduced from the notion of the earth's being an extended Plane.</p><p>This principle, though notoriously false, yet places being laid down accordingly, and a long voyage broken into many short ones, the voyage may be performed tolerably well by it, especially near the same meridian.</p><p>In P<*>ain Sailing it is supposed that these three, the rhumb line, the meridian, and parallel of latitude, will always form a right-angled triangle; and so posited, as that the perpendicular side will represent part of the meridian, or north and south line, containing the difference of latitude; the base of the triangle, the departure, or east-and west line; and the hypothenuse the distance sailed. The angle at the vertex is the course; and the angle at the base, the complement of the course; any two of which, besides the right angle, being given, the triangle may be protracted, and the other three parts found.</p><p>For the doctrine of Plane Sailing, see <hi rend="smallcaps">Sailing.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Scale,</hi> is a thin ruler, upon which are graduated the lines of chords, sines, tangents, secants, leagues, rhumbs, &c; being of great use in most parts of the mathematics, but especially in navigation. See its description and use under <hi rend="smallcaps">Scale.</hi></p><p><hi rend="smallcaps">Plane</hi> <hi rend="italics">Table,</hi> an instrument much used in landsurveying; by which the draught, or plan, is taken upon the spot, as the survey or measurement goes on, without any future protraction, or plotting.</p><p>This instrument consists of a Plane rectangular board, of any convenient size, the centre of which, when used, is fixed by means of screws to a three-legged stand, having a ball and socket, or universal joint, at the top, by means of which, when the legs are fixed on the ground, the table is inclined in any direction. To the table belongs,</p><p>1. A frame of wood, made to fit round its edges, for the purpose of fixing a sheet of paper upon the table. The one side of this frame is usually divided into equal parts, by which to draw lines across the table, parallel or perpendicular to the sides; and the other side of the frame is divided into 360 degrees, from a centre which is in the middle of the table; by means of which the table is to be used as a theodolite, &c.</p><p>2. A magnetic needle and compass screwed into the side of the table, to point out directions and be a check upon the sights.</p><p>3. An index, which is a brass two<*>foot scale, either with a small telescope, or open sights erected perpendicularly upon the ends. These sights and the fiducial edge of the index are parallel, or in the same Plane. <hi rend="center"><hi rend="italics">General Use of the</hi> <hi rend="smallcaps">Plane</hi> <hi rend="italics">Table.</hi></hi></p><p>To use this instrument properly, take a sheet of writing or drawing paper, and wet it to make it expand; then spread it flat upon the table, pressing down the frame upon the edges, to stretch it, and keep it fixed there; and when the paper is become dry, it will, by shrinking again, stretch itself smooth and flat from any cramps or unevenness. Upon this paper is to be drawn the plan or form of the thing measured.</p><p>The general use of this instrument, in land-surveying, is to begin by setting up the table at any part of the ground you think the most proper, and make a point upon a convenient part of the paper or table, to repre-<cb/> sent that point of the ground; then fix in that point of the paper one leg of the compasses, or a fine steel pin, and apply to it the fiducial edge of the index, moving it round the table, close by the pin, till through the sights you perceive some point desired, or remarkable object, as the corner of a field, or a picket set up, &c; and from the station point draw a dry or obscure line along the fiducial edge of the index. Then turn the index to another object, and draw a line on the paper towards it. Do the same by another; and so on till as many objects are set as may be thought necessary. Then measure from your station towards as many of the objects as may be necessary, and no more, taking the requisite offsets to corners or crooks in the hedges, &c; laying the measured distances, from a proper scale, down upon the respective lines on the paper. Then move the table to any of the proper places measured to, for a second station, fixing it there in the original position, turning it about its centre for that purpose, both till the magnetic needle point to the same degree of the compass as at first, and also by laying the fiducial edge of the index along the line between the two stations, and turning the table till through the index the former station can be seen; and then fix the table there: from this new station repeat the same operations as at the former; setting several objects, that is, drawing lines towards them, on the paper, by the edge of the index, measuring and laying osf the distances. And thus proceed from station to station; measuring only such lines as are necessary, and determining as many as you can by intersecting lines of direction drawn from different stations.</p><p><hi rend="italics">Of Shifting the Paper on the</hi> <hi rend="smallcaps">Plane</hi> <hi rend="italics">Table.</hi> When one paper is full of the lines &c measured, and the survey is not yet completed; draw a line in any manner through the farthest point of the last station line to which the work can be conveniently laid down; then take the sheet off the table, and fix another fair sheet in its place, drawing a line upon it, in a part of it the most convenient for the rest of the work, to represent the line drawn at the end of the work on the former paper. Then fold or cut the old sheet by the line drawn upon it; apply it so to the line on the new sheet, and, as they lie together in that position, continue or produce the last station line of the old sheet upon the new one; and place upon it the remainder of the measurement of that line, beginning at where the work left off on the old sheet. And so on, from one sheet to another, till the whole work is completed.</p><p>But it is to be noted, that if the said joining lines, upon the old and new sheet, have not the same inclination to the side of the table, the needle will not respect or point to the original degree of the compass, when the table is rectified. But if the needle be required to respect still the same degree of the compass, the easiest way then of drawing the lines in the same position, is to draw them both parallel to the same sides of the table, by means of the equal parallel divisions marked on the other two sides of the frame.</p><p>When the work of surveying is done, and you would fasten all the sheets together into one piece, or rough plan, the aforesaid lines are to be accurately joined together, in the same manner as when the lines were transferred from the old sheets to the new ones.</p><p>See more full directions for the use of the Plane<pb n="237"/><cb/> Table, illustrated with various examples, in my Treatise on Mensuration, 2d edit. pa. 509 &c.</p></div2></div1><div1 part="N" n="PLANET" org="uniform" sample="complete" type="entry"><head>PLANET</head><p>, literally a wanderer, or a wandering star, in opposition to a star, properly so called, which remains sixed. It is a celestial body, revolving around the sun, or some other planet, as a centre, or at least as a focus, and with a moderate degree of excentricity, so that it never is so much farther from the sun at one time than at another, but that it can be seen as well from one part of its orbit as another; as distinguished from the comets, which on the farthest part of their trajectory go osf to such vast distances, as to remain a long time invisible.</p><p>The Planets are usually distinguished into Primary and Secondary.</p><p><hi rend="italics">Primary</hi> <hi rend="smallcaps">Planets</hi>, called also simply Planets, are those which move round the sun, as their centre, or focus of their orbit. Such as Mercury, Venus, the Earth, Mars, Jupiter, Saturn, the Georgian or Herschel, and perhaps others. And the</p><p><hi rend="italics">Secondary</hi> <hi rend="smallcaps">Planets</hi>, are such as move round some primary one, as their centre, in the same manner as the primary ones do about the sun. Such as the moon, which moves round the earth, as a secondary; and the three, Jupiter, Saturn, and Georgian, have each several secondary Planets, or moons, moving round them.</p><p>Till very lately the number of the primary Planets was esteemed only six, which it was thought constituted the whole number of them in the solar system; viz, Mercury, Venus, the Earth, Mars, Jupiter, and Saturn; all of which it appears were known to the astronomers of all ages, who never dreamt of an increase to their number. But a seventh has been lately discovered, by Dr. Herschel, viz, on March the 13th, 1781, lying beyond all the rest, and now called the Georgian, or Herschel: and possibly others may still remain undiscovered to this day.</p><p>The primary Planets are again distinguished into Superior and Inferior.</p><p>The Superior Planets are those that are above the earth, or farther from the sun than the earth is; as, Mars, Jupiter, Saturn, and the Georgian or Herschel. And</p><p>The Inferior Planets are those that are below the earth, or that are nearer the sun than the earth is; which are Venus and Mercury.</p><p>The Planets were represented by the same characters as the chemists use to represent their metals by, on account of some supposed analogy between those celestial and the subterraneous bodies. Thus,</p><p>Mercury, the messenger of the Gods, represented by <*>, the same as that metal, imitating a man with wings on his head and feet, is a small bright planet, with a light tinct of blue, the sun's constant attendant, from whose side it never departs above 28°, and by that means is usually hid in his splendor. It performs its course around him in about 3 months.</p><p>Venus, the goddess of love, marked <*>, from the figure of a woman, the same as denotes copper, from a slight tinge of that colour, or verging to a light straw colour. She is a very bright Planet, revolving next above Mercury, and never appears above 48 degrees from the sun, finishing her course about him in about seven months. When this Planet goes before the sun,<cb/> or is a morning star, it has been called Phosphorus, and also Lucifer; and when following him, or when it shines in the evening as an evening star, it is called Hesperus.</p><p>Tellus, the Earth, next above Venus, is denoted by <*>, and performs its course about the sun in the space of a year.</p><p>Mars, the god of war, characterized <*>, a man holding out a spear, the same as iron, is a ruddy fierycoloured Planet, and finishes his course about the sun in about 2 years.</p><p>Jupiter, the chief god, or thunderer, marked <*>, to represent the thunderbolts, denoting the same as tin, from his pure white brightness. This Planet is next above Mars, and completes its course round the sun in about 12 years.</p><p>Saturn, the father of the Gods, is expressed by <*>, to imitate an old man supporting himself with a staff, and is the same as denotes lead, from his feeble light and dusky colour. He revolves next above Jupiter, and performs his course in about 30 years.</p><p>Lastly, the Georgian, or Herschel, is denoted by <*>, the initial of his name, with a cross for the christian Planet, or that discovered by the christians. This is the highest, or outermost, of the known Planets, and revolves around the sun in the space of about 90 years.</p><p>From these descriptions a person may easily distinguish all the Planets, except the last, which requires the aid of a telescope. For if after sun-set he sees a Planet nearer the east than the west, he may conclude it is neither Venus nor Mercury; and he may determine whether it is Saturn, Jupiter, or Mars, by the colour, light, and magnitude: by which also he may distinguish between Venus and Mercury.</p><p>It is probable that all the Planets are dark opake bodies, similar to the earth, and for the following reasons.</p><p>1. Because, in Mercury, Venus, and Mars, only that part of the disk is found to shine which is illuminated by the sun; and again, Venus and Mercury, when between the sun and the <*>arth, appear like maculæ or dark spots on the sun's face: from which it is evident, that those three Planets are opake bodies, illuminated by the borrowed light of the sun. And the same appears of Jupiter, from his being void of light in that part to which the shadow of his satellites reaches as well as in that part turned from the sun: and that his satellites are opake, and reslect the sun's light, like the moon, is abundantly shewn. Moreover, since Saturn, with his ring and satellites, and als<*> Herschel, with his satellites, only yield a faint light, considerably fainter than that of the rest of the Planets, and than that of the fixed stars, though these be vastly more remote; it is past a doubt that these Planets too, with their attendants, are opake bodies.</p><p>2. Since the sun's light is not transmitted through Mercury or Venus, when placed against him, it is plain they are dense opake bodies; which is likewise evident of Jupiter, from his hiding the satellites in his shadow; and therefore, by analogy, the same may be concluded of Saturn and Herschel.</p><p>3. From the variable spots of Venus, Mars, and Jupiter, it is evident that these Planets have a changeable atmosphere; which sort of atmosphere may, by a like argument, be inferred of the satellites of Jupiter; and<pb n="238"/><cb/> therefore, by similitude, the same may be concluded of the other Planets.</p><p>4. In like manner, from the mountains observed in the moon and Venus, the same may be supposed in the other Planets.</p><p>5. Lastly, since all these Planets are opake bodies, shining with the sun's borrowed light, are furnished with mountains, and are encompassed with a changeable atmosphere; they consequently have waters, seas &c, as well as dry land, and are bodies like the moon, and therefore like the earth. And heuce, it seems also probable, that the other Planets have their animal inhabitants, as well as our earth has. <hi rend="center"><hi rend="italics">Of the Orbits of the</hi> <hi rend="smallcaps">Planets.</hi></hi></p><p>Though all the primary Planets revolve about the sun, their orbits are not circles, but ellipses, having the sun in one of the foci. This circumstance was first found out by Kepler, from the observations of Tycho Brahe: before that, all astronomers took the planetary orbits for eccentric circles.</p><p>The Planes of these orbits do all intersect in the sun; and the line in which the plane of each orbit cuts that of the earth, is called the Line of the nodes; and the two points in which the orbits themselves touch that plane, are the Nodes; also the angle in which each plane cuts that of the ecliptic, is called the Inclination of the plane or orbit.—The distance between the centre of the sun, and the centre of each orbit, is called the excentricity of the Planet, or of its orbit. <hi rend="center"><hi rend="italics">The Motions of the</hi> <hi rend="smallcaps">Planets.</hi></hi></p><p>The motions of the primary Planets are very simple, and tolerably uniform, as being compounded only of a projectile motion, forward in a right line, which is a tangent to the orbit, and a gravitation towards the sun at the centre. Besides, being at such vast distances from each other, the effects of their mutual gravitation towards one another are in a considerable degree, though not altogether, insensible; for the action of Jupiter upon Saturn, for ex. is found to be 1/204 of the action of the sun upon Saturn, by comparing the matter of Jupiter with that of the sun, and the square of the distance of each from Saturn. So that the elliptic orbit of Saturn will be found more just, if its focus be supposed not in the centre of the sun, but in the common centre of gravity of the sun and Jupiter, or rather in the common centre of gravity of the sun and all the Planets below Saturn. And in like manner, the elliptic orbit of any other Planet will be found more accurate, by supposing its focus to be in the common centre of gravity of the sun and all the Planets that are below it. But the matter is far otherwise, in respect of the secondary Planets: for every one of these, though it chiefly gravitates towards its respective primary one, as its centre, yet at equal distances from the sun, it is also attracted towards him with an equally accelerated gravity, as the primary one is towards him; but at a greater distance with less, and at a nearer distance with greater: from which double tendency towards the sun, and towards their own primary Planets, it happens, that the motion of the satellites, or secondary Planets, comes to be very much compounded, and affected with various inequalities.<cb/></p><p>The motions even of the primary Planets, in their elliptic orbits, are not equable, because the sun is not in their centre, but their focus. Hence they move; sometimes faster, and sometimes slower, as they are nearer to or farther from the sun; but yet these irregularities are all certain, and follow according to an immutable <figure/> law. Thus, the ellipsis PEA &c representing the orbit of a Planet, and the focus S the sun's place: the axis of the ellipse AP, is the line of the apses; the point A, the higher apsis or aphelion; P the lower apsis or perihelion; CS the eccentricity; and ES the Planet's mean distance from the sun. Now the motion of the Planet in its perihelion P is swiftest, but in its aphelion A it is slowest; and at E the motion as well as the distance is a mean, being there such as would describe the whole orbit in the same time it is really described in. And the law by which the motion in every point is regulated, is this, that a line or radius drawn from the centre of the sun to the centre of the Planet, and thus carried along with an angular motion, does always describe an elliptic area proportional to the time; that is, the trilineal area ASB, is to the area ASG, as the time the Planet is in moving over AB, to the time it is in moving over AG. This law was first found out by Kepler, from observations; and has since been accounted for and demonstrated by Sir Isaac Newton, from the general laws of attraction and projectile motion.</p><p>As to the periods and velocities of the Planets, or the times in which they perform their courses, they are found to have a wonderful harmony with their distances from the sun, and with one another: the nearer each Planet being to the sun, the quicker still is its motion, and its period the shorter, according to this general and regular law; viz, that the squares of their periodical times are as the cubes of their mean distances from the sun or focus of their orbits. The knowledge of this law we owe also to the sagacity of Kepler, who found that it obtained in all the primary Planets; as astronomers have since found it also to hold good in the secondary ones. Kepler indeed deduced this law merely from observation, by a comparison of the several distances of the Planets with their periods or times: the glory of investigating it from physical principles is due to Sir Isaac Newton, who has demonstrated that, in the present state of nature, such a law was inevitable.</p><p>The phenomena of the Planets are, their Conjunctions, Oppositions, Elongations, Stations, Retrogradations, Phases, and Eclipses; for which see the respective articles.</p><p>For a view of the comparative magnitudes of the Planets; and for a view of their several distances, &c; see the articles <hi rend="smallcaps">Orbit</hi> and <hi rend="smallcaps">Solar System</hi>, as also Plate xxi, fig. 1.</p><p>The following Table contains a synopsis of the distances, magnitudes, periods, &c, of the several Planets, according to the latest observations and improve- ments.<pb n="239"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=8 align=center" role="data"><hi rend="smallcaps">Table</hi> <hi rend="italics">of the</hi> <hi rend="smallcaps">Planetary Motions, Distances</hi>, &c.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Anno 1784.</cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Mercury.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Venus.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Earth.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Mars.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Jupiter.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Saturn.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="smallcaps">Herschel</hi>, or <hi rend="smallcaps">Geor gian</hi>, 1782.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Greatest Elongation of Inferior, and Parallax of Superior Planets.</cell><cell cols="1" rows="1" rend="align=center" role="data">28° 20′</cell><cell cols="1" rows="1" rend="align=center" role="data">47° 48′</cell><cell cols="1" rows="1" rend="align=center" role="data">* *</cell><cell cols="1" rows="1" rend="align=center" role="data">47° 24′</cell><cell cols="1" rows="1" rend="align=center" role="data">11° 51′</cell><cell cols="1" rows="1" rend="align=center" role="data">6° 29′</cell><cell cols="1" rows="1" rend="align=center" role="data">3° 4′1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Periodical Revotions round the Sun.</cell><cell cols="1" rows="1" rend="align=center" role="data">87<hi rend="sup">d</hi> 23<hi rend="sup">h</hi> 15 1/2<hi rend="sup">m</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">22<hi rend="sub">d</hi>4 16<hi rend="sub">h</hi> 4<hi rend="sub">m</hi>9 1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">3<hi rend="sub">d</hi>65 6<hi rend="sub">h</hi> 9<hi rend="sub">m</hi> 1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">68<hi rend="sub">d</hi>6 23<hi rend="sub">h</hi> 30<hi rend="sub">m</hi> 3/4</cell><cell cols="1" rows="1" rend="align=center" role="data">433<hi rend="sub">d</hi>2 8<hi rend="sub">h</hi> 51<hi rend="sub">m</hi> 1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">107<hi rend="sub">d</hi>61 1<hi rend="sub">h</hi>4 36<hi rend="sub">m</hi> 3/4</cell><cell cols="1" rows="1" rend="align=center" role="data">304<hi rend="sub">d</hi>45 1<hi rend="sub">h</hi>8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diurnal Rotations upon their Axes.</cell><cell cols="1" rows="1" rend="align=center" role="data">* * *</cell><cell cols="1" rows="1" rend="align=center" role="data">23<hi rend="sup">h</hi> 22<hi rend="sup">m</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">23<hi rend="sup">h</hi> 56<hi rend="sup">m</hi> 4<hi rend="sup">s</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">24<hi rend="sup">h</hi> 39<hi rend="sup">m</hi> 22<hi rend="sup">s</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">9<hi rend="sup">h</hi> 56<hi rend="sup">m</hi></cell><cell cols="1" rows="1" rend="align=center" role="data">* *</cell><cell cols="1" rows="1" rend="align=center" role="data">* *</cell></row><row role="data"><cell cols="1" rows="1" role="data">Inclinations of their Orbits to the Ecliptic.</cell><cell cols="1" rows="1" rend="align=center" role="data">7° 0′</cell><cell cols="1" rows="1" rend="align=center" role="data">3° 23′1/3</cell><cell cols="1" rows="1" rend="align=center" role="data">* *</cell><cell cols="1" rows="1" rend="align=center" role="data">1° 51′</cell><cell cols="1" rows="1" rend="align=center" role="data">1° 19′1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">2° 30′1/3</cell><cell cols="1" rows="1" rend="align=center" role="data">48′ 0″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Place of the Ascending Node.</cell><cell cols="1" rows="1" rend="align=center" role="data">1<hi rend="sup">s</hi> 15° 46′3/4</cell><cell cols="1" rows="1" rend="align=center" role="data">2<hi rend="sup">s</hi> 14° 44′</cell><cell cols="1" rows="1" rend="align=center" role="data">* * *</cell><cell cols="1" rows="1" rend="align=center" role="data">1<hi rend="sup">s</hi> 17° 59′</cell><cell cols="1" rows="1" rend="align=center" role="data">3<hi rend="sup">s</hi> 8° 50′</cell><cell cols="1" rows="1" rend="align=center" role="data">3<hi rend="sup">s</hi> 21° 48′3/4</cell><cell cols="1" rows="1" rend="align=center" role="data">3<hi rend="sup">s</hi> 13° 1′</cell></row><row role="data"><cell cols="1" rows="1" role="data">Place of the Aphelion, or point farthest from the Sun.</cell><cell cols="1" rows="1" rend="align=center" role="data">8<hi rend="sup">s</hi> 14° 13′</cell><cell cols="1" rows="1" rend="align=center" role="data">10<hi rend="sup">s</hi> 9° 38′</cell><cell cols="1" rows="1" rend="align=center" role="data">9<hi rend="sup">s</hi> 9° 15′1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">5<hi rend="sup">s</hi> 2° 6′1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">6<hi rend="sup">s</hi> 10° 57 1/2′</cell><cell cols="1" rows="1" rend="align=center" role="data">9<hi rend="sup">s</hi> 0° 45′1/2</cell><cell cols="1" rows="1" rend="align=center" role="data">11<hi rend="sup">s</hi> 23°23</cell></row><row role="data"><cell cols="1" rows="1" role="data">Greatest Apparent Diameters, seen from the Earth.</cell><cell cols="1" rows="1" rend="align=center" role="data">11″</cell><cell cols="1" rows="1" rend="align=center" role="data">58″</cell><cell cols="1" rows="1" rend="align=center" role="data">*</cell><cell cols="1" rows="1" rend="align=center" role="data">25″</cell><cell cols="1" rows="1" rend="align=center" role="data">46″</cell><cell cols="1" rows="1" rend="align=center" role="data">20″</cell><cell cols="1" rows="1" rend="align=center" role="data">4″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diameters in English Miles; that of the Sun being 883217.</cell><cell cols="1" rows="1" rend="align=center" role="data">3222</cell><cell cols="1" rows="1" rend="align=center" role="data">7687</cell><cell cols="1" rows="1" rend="align=center" role="data">7964</cell><cell cols="1" rows="1" rend="align=center" role="data">4189</cell><cell cols="1" rows="1" rend="align=center" role="data">89170</cell><cell cols="1" rows="1" rend="align=center" role="data">79042</cell><cell cols="1" rows="1" rend="align=center" role="data">35109</cell></row><row role="data"><cell cols="1" rows="1" role="data">Proportional Mean Distances from the Sun.</cell><cell cols="1" rows="1" rend="align=center" role="data">38710</cell><cell cols="1" rows="1" rend="align=center" role="data">72333</cell><cell cols="1" rows="1" rend="align=center" role="data">100000</cell><cell cols="1" rows="1" rend="align=center" role="data">152369</cell><cell cols="1" rows="1" rend="align=center" role="data">520098</cell><cell cols="1" rows="1" rend="align=center" role="data">953937</cell><cell cols="1" rows="1" rend="align=center" role="data">1903421</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mean Distances from the Sun in Semidiameters of the Earth.</cell><cell cols="1" rows="1" rend="align=center" role="data">9210</cell><cell cols="1" rows="1" rend="align=center" role="data">17210</cell><cell cols="1" rows="1" rend="align=center" role="data">23799</cell><cell cols="1" rows="1" rend="align=center" role="data">36262</cell><cell cols="1" rows="1" rend="align=center" role="data">123778</cell><cell cols="1" rows="1" rend="align=center" role="data">227028</cell><cell cols="1" rows="1" rend="align=center" role="data">453000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mean Distances from the Sun in English Miles.</cell><cell cols="1" rows="1" rend="align=center" role="data">37 millions</cell><cell cols="1" rows="1" rend="align=center" role="data">68 millions</cell><cell cols="1" rows="1" rend="align=center" role="data">95 millions</cell><cell cols="1" rows="1" rend="align=center" role="data">144 millions</cell><cell cols="1" rows="1" rend="align=center" role="data">490 millions</cell><cell cols="1" rows="1" rend="align=center" role="data">900 millions</cell><cell cols="1" rows="1" rend="align=center" role="data">1800 millions</cell></row><row role="data"><cell cols="1" rows="1" role="data">Eccentricities or Distance of the Focus from the Centre.</cell><cell cols="1" rows="1" rend="align=center" role="data">7960</cell><cell cols="1" rows="1" rend="align=center" role="data">510</cell><cell cols="1" rows="1" rend="align=center" role="data">1680</cell><cell cols="1" rows="1" rend="align=center" role="data">14218</cell><cell cols="1" rows="1" rend="align=center" role="data">25277</cell><cell cols="1" rows="1" rend="align=center" role="data">53163</cell><cell cols="1" rows="1" rend="align=center" role="data">4759</cell></row><row role="data"><cell cols="1" rows="1" role="data">Proportion of Light and Heat; that of the Earth being 100.</cell><cell cols="1" rows="1" rend="align=center" role="data">668</cell><cell cols="1" rows="1" rend="align=center" role="data">191</cell><cell cols="1" rows="1" rend="align=center" role="data">100</cell><cell cols="1" rows="1" rend="align=center" role="data">43</cell><cell cols="1" rows="1" rend="align=center" role="data">3.7</cell><cell cols="1" rows="1" rend="align=center" role="data">1.1</cell><cell cols="1" rows="1" rend="align=center" role="data">0.276</cell></row><row role="data"><cell cols="1" rows="1" role="data">Proportion of Bulk; that of the Sun being 1380000.</cell><cell cols="1" rows="1" rend="align=center" role="data">1/15</cell><cell cols="1" rows="1" rend="align=center" role="data">8/9</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">7/24</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2/5</cell><cell cols="1" rows="1" rend="align=center" role="data">1000</cell><cell cols="1" rows="1" rend="align=center" role="data">90</cell></row><row role="data"><cell cols="1" rows="1" role="data">Proportion of Density; that of the Sun being 1/4.</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">1 1/4</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">.7</cell><cell cols="1" rows="1" rend="align=center" role="data">.23</cell><cell cols="1" rows="1" rend="align=center" role="data">.02</cell><cell cols="1" rows="1" rend="align=center" role="data">*</cell></row></table><pb n="240"/><cb/></p><p>A Planet's motion, or distance from its apogee, is called the mean anomaly of the Planet, and is measured by the<*> area it describes in the given time: when the Planet arrives at the middle of its orbit, or the point E, the area or time is called the true anomaly. When the Planet's motion is reckoned from the first point of Aries, it is called its motion in longitude; which is either mean or true; viz, mean, which is such as it would have were it to move uniformly in a circle; and true, which is that with which the Planet actually describes its orbit, and is measured by the are of the ecliptic it describes. And hence may be found the Planet's place in its orbit for any given time after it has left the aphelion: for suppose the area of the ellipsis be so divided by the line SG, that the whole elliptic area may have the same proportion to the part ASG, as the whole periodical time in which the Planet describes its whole orbit, has to the given time; then will G be the Planet's place in its orbit sought.</p></div1><div1 part="N" n="PLANETARIUM" org="uniform" sample="complete" type="entry"><head>PLANETARIUM</head><p>, an astronomical machine, contrived to represent the motions, orbits, &c, of the planets, as they really are in nature, or according to the Copernican system. The larger sort of them are called Orreries. See <hi rend="smallcaps">Orrery.</hi></p><p>A very remarkable machine of this sort was invented by Huygens, and described in his Opusc. Posth. tom. 2. p. 157, edit. Amst. 1728. And it is still preserved among the curiosities of the university at Leyden.</p><p>In this Planetarium, the five primary planets perform their revolutions about the sun, and the moon performs her revolution about the earth, in the same time that they are really performed in the heavens. Also the orbits of the moon and planets are represented with their true proportions, eccentricity, position, and declination from the ecliptic or orbit of the earth. So that by this machine the situation of the planets, with the conjunctions, oppositions, &c, may be known, not only for the present time, but for any other time either past or yet to come; as in a perpetual ephemeris.</p><p>There was exhibited in London, viz. in the year 1791, a still much more complete Planetarium of this sort; called “a Planetarium or astronomical machine, which exhibits the most remarkable phenomena, motions, and revolutions of the universe. Invented, and partly executed, by the celebrated M. Phil. Matthew Hahn, member of the academy of sciences at Erfurt. But finished and completed by Mr. Albert de Mylius.” This is a most stupendous and elaborate machine; consisting of the solar system in general, with all the orbits and planets in their due proportions and positions; as also the several particular planetary systems of such as have satellites, as of the earth, Jupiter, &c; the whole kept in continual motion by a chronometer, or grand eight-day-clock; by which all these systems are made perpetually to perform all their motions exactly as in nature, exhibiting at all times the true and real motions, positions, aspects, phenomena, &c, of all the celestial bodies, even to the very diurnal rotation of the planets, and the unequal motions in their elliptic orbits. A description was published of this most superb machine; and it was purchased and sent as one of the presents to the emperor of China, in the embassy of Lord Macartney, in the year 1793.</p><p>But the Planetariums or orreries now most commonly<cb/> used, do not represent the true times of the celestial motions, but only their proportions; and are not kept in continual motion by a clock, but are only turned round occasionally with the hand, to help to give young beginners an idea only of the planetary system; as also, if constructed with sufficient accuracy, to resolve problems, in a coarse way, relating to the motions of the planets, and of the earth and moon, &c.</p><p>Dr. Desaguliers (Exp. Philos. vol. 1, p. 430.) describes a Planetarium of his own contrivance, which is one of the best of the common sort. The machine is contrived to be rectified or set to any latitude; and then by turning the handle of the Planetarium, all the planets perform their revolutions round the sun in proportion to their periodical times, and they carry i<*>dices which shew the longitudes of the planets, by pointing to the divisions graduated on circles for that purpose.</p><p>The Planetarium represented in fig. 1, plate xxii. is an instrument contrived by Mr. Wm. Jones, of Holborn, London, mathematical instrument maker, who has paid considerable attention to such machines, to bring them to a great degree of simplicity and perfection. It represents in a general manner, by various parts of its machinery, all the motions and phenomena of the planetary system. This machine consists of, the Sun in the centre, with the Planets in the order of their distance from him, viz. Mercury, Venus, the Earth and Moon, Mars, Jupiter with his moons, and Saturn with his ring and moons; and to it is also occasionally applied an extra long arm for the Georgian Planet and his two moons. To the earth and moon is applied a frame CD, containing only four wheels and two pinions, which serve to preserve the earth's axis in its due parallelism in its motion round the sun, and to give the moon at the same time her due revolution about the earth. These wheels are connected with the wheelwork in the round box below, and the whole is set in motion by the winch H. The arm M that carries round the moon, points out on the plate C her age and phases for any situation in her orbit, upon which they are engraved. In like manner the arm points out her place in the ecliptic B, in signs and degrees, called her geocentric place, that is, as seen from the earth. The moon's orbit is represented by the flat rim A; the two joints of it, upon which it turns, denoting her nodes; and the orbit being made to incline to any required angle. The terrella, or little earth, of this machine, is usually made of a three inch globe papered, &c, for the purpose; and by means of the terminating wire that goes over it, points out the changes of the seasons, and the different lengths of days and nights more conspicuously. By this machine are seen at once all the Planets in motion about the Sun, with the same respective velocities and periods of revolution which they have in the heavens; the wheelwork being calculated to a minute of time, from the latest discoveries. See Mr. Jones's Description of his new portable Orrery.</p></div1><div1 part="N" n="PLANETARY" org="uniform" sample="complete" type="entry"><head>PLANETARY</head><p>, something that relates to the planets. Thus, we say Planetary worlds, Planetary inhabitants, Planetary motions, &c. Huygens and Fontenelle bring several probable arguments for the reality of Planetary worlds, animals, plants, men, &c.</p><p><hi rend="smallcaps">Planetary</hi> <hi rend="italics">System,</hi> is the system or assemblage of the Planets, primary and secondary, moving in their <pb/><pb/><pb/><pb/><pb n="241"/><cb/> respective orbits, round their common centre the sun. See <hi rend="italics">Solar</hi> <hi rend="smallcaps">System.</hi></p><p><hi rend="smallcaps">Planetary</hi> <hi rend="italics">Days.</hi> With the Ancients, the week was shared among the seven planets, each planet having its day. This we learn from Dion Cassius and Plutarch, Sympos. lib. 4. q. 7. Herodotus adds, that it was the Egyptians who first discovered what god, that is what planet, presides over each day; for that among this people the planets were directors. And hence it is, that in most European languages the days of the week are still denominated from the planets; as Sunday, Monday, &c.</p><p><hi rend="smallcaps">Planetary</hi> <hi rend="italics">Dials,</hi> are such as have the Planetary hours inscribed on them.</p><p><hi rend="smallcaps">Planetary</hi> <hi rend="italics">Hours,</hi> are the 12th parts of the artificial day and night. See <hi rend="italics">Planetary</hi> <hi rend="smallcaps">Hour.</hi></p><p><hi rend="smallcaps">Planetary</hi> <hi rend="italics">Squares,</hi> are the squares of the seven numbers from 3 to 9, disposed magically. Cornelius Agrippa, in his book of magic, has given the construction of the seven Planetary squares. And M. Poignard, canon of Brussels, in his treatise on sublime squares, gives new, general, and easy methods, for making the seven Planetary squares, and all others to infinity, by numbers in all sorts of progressions. See <hi rend="smallcaps">Magic</hi> <hi rend="italics">square.</hi></p><p><hi rend="smallcaps">Planetary</hi> <hi rend="italics">Years,</hi> the periods of time in which the several planets make their revolutions round the sun, or earth.—As from the proper revolution of the earth, or the apparent revolution of the sun, the solar year takes its original; so from the proper revolutions of the rest of the planets about the earth, as many sorts of years do arise; viz, the Saturnian year, which is defined by 29 Egyptian years 174 days 58 minutes, equivalent in a round number to 30 solar years. The Jovial year, containing 11 years 317 days 14 hours 59 minutes. The Martial year, containing 1 year 321 days 23 hours 31 minutes. For Venus and Mercury, as their years, when judged of with regard to the earth, are almost equal to the solar year; they are more usually estimated from the sun, the true centre of their motions: in which case the former is equal to 224 days 16 hours 49 minutes; and the latter to 87 days 23 hours 16 minutes.</p></div1><div1 part="N" n="PLANIMETRY" org="uniform" sample="complete" type="entry"><head>PLANIMETRY</head><p>, that part of geometry which considers lines and plane figures, without any regard to heights or depths.—Planimetry is particularly restricted to the mensuration of planes and other surfaces; as contradistinguished from Stereometry, or the mensuration of solids, or capacities of length, breadth and depth.</p><p>Planimetry is performed by means of the squares of long measures, as square inches, square feet, square yards, &c; that is, by squares whose side is an inch, a foot, a yard, &c. So that the area or content of any surface is said to be found, when it is known how many such square inches, feet, yards, &c, it contains. See <hi rend="smallcaps">Mensuration</hi> and <hi rend="smallcaps">Surveying.</hi></p></div1><div1 part="N" n="PLANISPHERE" org="uniform" sample="complete" type="entry"><head>PLANISPHERE</head><p>, a projection of the sphere, and its various circles, on a plane; as upon paper or the like. In this sense, maps of the heavens and the earth, exhibiting the meridians and other circles of the sphere, may be called Planispheres.</p><p>Planisphere is sometimes also considered as an astronomical instrument, used in observing the motions of the heavenly bodies; being a projection of the celestial sphere upon a plane, representing the stars, constellations,<cb/> &c, in their proper situations, distances, &c. As the Astrolabe, which is a common name for all such projections.</p><p>In all Planispheres, the eye is supposed to be in a point, viewing all the circles of the sphere, and referring them to a plane beyond them, against which the sphere is as it were flattened: and this plane is called the Plane of Projection, which is always some one of the circles of the sphere itself, or parallel to some one.</p><p>Among the infinite number of Planispheres which may be furnished by the different planes of projection, and the different positions of the eye, there are two or three that have been preferred to the rest. Such as that of Ptolomy, where the plane of projection is parallel to the equator: that of Gemma Frisius, where the plane of projection is the colure, or solstitial meridian, and the eye the pole of the meridian, being a stereographical projection: or that of John de Royas, a Spaniard, whose plane of projection is a meridian, and the eye placed in the axis of that meridian, at an infinite distance; being an orthographical projection, and called the Analemma.</p><p>PLANO-<hi rend="italics">Concave</hi> glass or lens, is that which is plane on one side, and concave on the other. And</p><p><hi rend="smallcaps">Plano</hi>-<hi rend="italics">Convex</hi> glass or lens, is that which is plane on one side, and convex on the other. See <hi rend="smallcaps">Lens.</hi></p><p>PLAT-<hi rend="smallcaps">Band</hi>, in Architecture, is any flat square moulding, whose height much exceeds its projecture. Such are the faces of an architrave, and the Platbands of the modillions of a cornice.</p></div1><div1 part="N" n="PLATFORM" org="uniform" sample="complete" type="entry"><head>PLATFORM</head><p>, in Artillery and Gunnery, a small elevation, or a floor of wood, stone, or the like, on which cannon, &c, are placed, for more conveniently working and firing them.</p><div2 part="N" n="Platform" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Platform</hi></head><p>, in Architecture, a row of beams that support the timber-work of a roof, lying on the top of the walls, where the entablature ought to be raised. Also a kind of flat walk, or plane floor, on the top of a building; from whence a fair view may be taken of the adjacent grounds. So, an edifice is said to be covered with a Platform, when it has no arched roof.</p></div2></div1><div1 part="N" n="PLATO" org="uniform" sample="complete" type="entry"><head>PLATO</head><p>, one of the most celebrated among the ancient philosophers, being the founder of the sect of the Academics, was the son of Aristo, and born at Athens, about 429 years before Christ. He was of a royal and illustrious family, being descended by his father from Codrus, and by his mother from Solon. The name given him by his parents was <hi rend="italics">Aristocles;</hi> but being of a robust make, and remarkably broad-shouldered, from this circumstance he was nick-named <hi rend="italics">Plato</hi> by his wrestling-master, which name he retained ever after.</p><p>From his infancy, Plato distinguished himself by his lively and brilliant imagination. He eagerly imbibed the principles of poetry, music, and painting. The charms of philosophy however prevailing, drew him from those of the fine arts; and at the age of twenty he attached himself to Socrates only, who called him the <hi rend="italics">Swan of the Academy.</hi> The disciple profited so well of his master's lessons, that at twenty-five years of age he had the reputation of a consummate sage. He lived with Socrates for eight years, in which time he committed to writing, according to the custom of the students, the purport of a great number of his master's excellent lectures, which he digested by way of philoso-<pb n="242"/><cb/> phical conversations; but made so many judicious additions and improvements of his own, that Socrates, hearing him one day recite his Lysis, cried out, O Hercules! how many fine sentiments does this young man ascribe to me, that I never thought of! And Laertius assures us, that he composed several discourses which Socrates had no manner of hand in. At the time when Socrates was first arraigned, Plato was a junior senator, and he assumed the orator's chair to plead his master's cause, but was interrupted in that design, and the judges passed sentence of condemnation upon Socrates. Upon this occasion Plato begged him to accept from him a sum of money sufficient to purchase his enlargement, but Socrates peremptorily refused the generous offer, and suffered himself to be put to death.</p><p>The philosophers who were at Athens were so alarmed at the death of Socrates, that most of them fled, to avoid the cruelty and injustice of the government. Plato retired to Megara, where he was kindly entertained by Euclid the philosopher, who had been one of the first scholars of Socrates, till the storm should be over. Afterwards he determined to travel in pursuit of knowledge; and from Megara he went to Italy, where he conferred with Eurytus, Philolaus, and Archytas, the most celebrated of the Pythagoreans, from whom he learned all his natural philosophy, diving into the most profound and mysterious secrets of the Pythagoric doctrines. But perceiving other knowledge to be connected with them, he went to Cyrene, where he studied geometry and other branches of mathematics under Theodorus, a celebrated master.</p><p>Hence he travelled into Egypt, to learn the theology of their priests, with the sciences of arithmetic, astronomy, and the nicer parts of geometry. Having taken also a survey of the country, with the course of the Nile and the canals, he settled some time in the province of Sais, learning of the sages there their opinions concerning the universe, whether it had a beginning, whether it moved wholly or in part, &c; also concerning the immortality and transmigration of souls: and here it is also thought he had some communication with the books of Moses.</p><p>Plato's curiosity was not yet satisfied. He travelled into Persia, to consult the magi as to the religion of that country. He designed also to have penetrated into India, to learn of the Brachmans their manners and customs; but was prevented by the wars in Asia.</p><p>Afterwards, returning to Athens, he applied himself to the teaching of philosophy, opening his school in the Academia, a place of exercise in the suburbs of the city; from whence it was that his followers took the name of Academics.</p><p>Yet, settled as he was, he made several excurfions abroad: one in particular to Sicily, to view the fiery ebullitions of Mount Etna. Dionysius the tyrant then reigned at Syracuse; a very bad man. Plato however went to visit him; but, instead of flattering him like a courtier, reproved him for the disorders of his court, and the injustice of his government. The tyrant, not used to disagreeable truths, grew enraged at Plato, and would have put him to death, if Dion and Aristomenes, formerly his scholars, and then favourites of that prince, had not powerfully interceded for him. Dionysius<cb/> however delivered him into the hands of an envoy of the Lacedemonians, who were then at war with the Athenians: and this envoy, touching upon the coast of Ægina, sold him for a slave to a merchant of Cyrene: who, as soon as he had bought him, liberated him, and sent him home to Athens.</p><p>Some time after, he made a second voyage into Sicily, in the reign of Dionysius the younger; who sent Dion, his minister and favourite, to invite him to court, that he might learn from him the art of governing his people well. Plato accepted the invitation, and went; but the intimacy between Dion and Plato raising jealousy in the tyrant, the former was disgraced, and the latter sent back to Athens. But Dion, being taken into favour again, persuaded Dionysius to recall Plato, who received him with all the marks of goodwill and friendship that a great prince could give. He sent out a fine galley to meet him, and went himself in a magnificent chariot, attended by all his court, to receive him. But this prince's uneven temper hurried him into new suspicions. It seems indeed that these apprehensions were not altogether groundless: for Ælian says, and Cicero was of the same opinion, that Plato taught Dion how to dispatch the tyrant, and to deliver the people from oppression. However this may be, Plato was offended and complained; and Dionysius, incensed at these complaints, resolved to put him to death: but Archytas, who had great interest with the tyrant, being informed of it by Dion, interceded for the philosopher, and obtained leave for him to retire.</p><p>The Athenians received him joyfully at his return, and offered him the administration of the government; but he declined that honour, choosing rather to live quietly in the Academy, in the peaceable contemplation and study of philosophy; being indeed so desirous of a private retirement that he never married. His fame drew disciples from all parts, when he would admit them, as well as invitations to come to reside in many of the other Grecian states; but the three that most distinguished themselves, were Spusippus his nephew, who continued the Academy after him, Xenocrates the Caledonian, and the celebrated Aristotle. It is said also that Theophrastus and Demosthenes were two of his disciples. He had it seems so great a respect for the science of geometry and the mathematics, that he had the following inscription painted in large letters over the door of his academy; L<hi rend="smallcaps">ET NO ONE ENTER HERE, UNLESS HE HAS A TASTE FOR</hi> G<hi rend="smallcaps">EOMETRY AND THE Mathematics!</hi></p><p>But as his great reputation gained him on the one hand many disciples and admirers, so on the other it raised him some emulators, especially among his fellowdisciples, the followers of Socrates. Xenophon and he were particularly disaffected to each other. Plato was of so quiet and even a temper of mind, even in his youth, that he never was known to express a pleasure with any greater emotion than that of a smile; and he had such a perfect command of his passions, that nothing could provoke his anger or resentment; from hence, and the subject and style of his writings, he acquired the appellation of the <hi rend="italics">Divine Plato.</hi> But although he was naturally of a reserved and very pensive disposition; yet, according to Aristotle, he was affable, courteous, and perfectly good-natured; and sometimes would conde-<pb n="243"/><cb/> scend to crack little innocent jokes on his intimate acquaintances. Of his affability there needs no greater proof than his civil manner of conversing with the philosophers of his own times, when pride and envy were at their height. His behaviour to Diogenes is always mentioned in his history. This Cynic was greatly offended, it seems, at the politeness and sine taste of Plato, and used to catch all opportunities of snarling at him. Dining one day at his table with other company, when trampling upon the tapestry with his dirty feet, he uttered this brutish sarcasm, “I trample upon the pride of Plato:” to which the latter wisely and calmly replied, “with a greater pride.”</p><p>This extraordinary man, being arrived at 81 years of age, died a very easy and peaceable death, in the midst of an entertainment, according to some; but, according to Cicero, as he was writing. Both the life and death of this philosopher were calm and undisturbed; and indeed he was finely composed for happiness. Beside the advantages of a noble birth, he had a large and comprehensive understanding, a vast fund of wit and good taste, great evenness and sweetness of temper, all cultivated and refined by education and travel; so that it is no wonder he was honoured by his countrymen, esteemed by strangers, and adored by his scholars. Tully perfectly adored him: he tells us that he was justly called by Panætius, the divine, the most wise, the most sacred, the Homer of philosophers; thinks, that if Jupiter had spoken Greek, he would have done it in Plato's style, &c. But, panegyric aside, Plato was certainly a very wonderful man, of a large and comprehensive mind, an imagination infinitely fertile, and of a most flowing and copious eloquence. However, the strength and heat of fancy prevailing over judgment in his composition, he was too apt to soar beyond the limits of earthly things, to range in the imaginary regions of general and abstracted ideas; on which account, though there is always a greatness and sublimity in his manner, he did not philosophize so much according to truth and nature as Aristotle, though Cicero did not scruple to give him the preference.</p><p>The writings of Plato are all in the way of dialogue, where he seems to deliver nothing from himself, but every thing as the sentiments and opinions of others, of Socrates chiefly, of Timæus, &c. His style, as Aristotle observed, is between prose and verse: on which account some have not scrupled to rank him among the poets: and indeed, beside the elevation and grandeur of his style, his matter is frequently the offspring of imagination, instead of doctrines or truths deduced from nature. The first edition of Plato's works in Greek, was printed by Aldus at Venice in 1513: but a Latin version of them by Marsilius Ficinus had been printed there in 1491. They were reprinted together at Lyons in 1588, and at Francfort in 1602. The famous printer Henry Stephens, in 1578, gave a beautiful and correct edition of Plato's works at Paris, with a new Latin version by Serranus, in three volumes folio.</p></div1><div1 part="N" n="PLATONIC" org="uniform" sample="complete" type="entry"><head>PLATONIC</head><p>, something that relates to Plato, his school, philosophy, opinions, or the like.</p><p>PLATONIC <hi rend="italics">Bodies,</hi> so called from Plato who treated of them, are what are otherwise called the regular bodies. They are five in number; the tetraedron, the hexaedron, the octaedron, the dodecaedron, and<cb/> the icosaedron. See each of these articles, as also <hi rend="smallcaps">Regular Bodies.</hi></p><p><hi rend="smallcaps">Platonic</hi> <hi rend="italics">Year,</hi> or the <hi rend="italics">Great Year,</hi> is a period of time determined by the revolution of the equinoxes, or the time in which the stars and constellations return to their former places, in respect of the equinoxes.</p><p>The Platonic year, according to Tycho Brahe, is 25816 solar years, according to Riccioli 25920, and according to Cassini 24800 years.</p><p>This period being once accomplished, it was an opinion among the ancients, that the world was to begin anew, and the same series of things to return over again.</p></div1><div1 part="N" n="PLATONISM" org="uniform" sample="complete" type="entry"><head>PLATONISM</head><p>, the doctrine and sentiments of Plato and his followers, with regard to philosophy, &c. His disciples were called Academics, from Academia, the name of a villa in the suburbs of Athens where he opened his school. Among these were Xenocrates, Aristotle, Lycurgus, Demosthenes, and Isocrates. In physics, he chiefly followed Heraclitus; in ethics and politics, Socrates; and in metaphysics, Pythagoras.</p><p>After his death, two of the principal of his disciples, Xenocrates and Aristotle, continuing his office, and teaching, the one in the Academy, the other in the Lycæum, formed two sects, under different names, though in other respects the same; the one retaining the denomination of <hi rend="smallcaps">Academics</hi>, the other assuming that of <hi rend="smallcaps">Peripatetics.</hi> See these two articles.</p><p>Afterwards, about the time of the first ages of Christianity, the followers of Plato quitted the title of <hi rend="italics">Academists,</hi> and took that of <hi rend="italics">Platonists.</hi> It is supposed to have been at Alexandria, in Egypt, that they first assumed this new title, after having restored the ancient academy, and re-established Plato's sentiments; which had many of them been gradually dropped and laid aside. Porphyry, Plotin, Iamblichus, Proclus, and Plutarch, are those who acquired the chief reputation among the Greek Platonists; Apuleius and Chalcidius, among the Latins; and Philo Judæus, among the Hebrews. The modern Platonists own Plotin the founder, or at least the reformer, of their sect.</p><p>The Platonic philosophy appears very consistent with the Mosaic; and many of the primitive fathers follow the opinions of that philosopher, as being favourable to Christianity. Justin is of opinion that there are many things in the works of Plato which this philosopher could not learn from mere natural reason; but thinks he must have learnt them from the books of Moses, which he might have read when in Egypt. Hence Numenius the Pythagorean expressly calls Plato the <hi rend="italics">Attic Moses,</hi> and upbraids him with plagiarism; because he stole his doctrine concerning God and the world from the books of Moses. Theodoret says expressly, that he has nothing good and commendable concerning the Deity and his worship, but what he took from the Hebrew theology; and Clemens Alexandrinus calls him the <hi rend="italics">Hebrew Philosopher.</hi> Gale is very particular in his proof of the point, that Plato borrowed his philosophy from the Scriptures, either immediately, or by means of tradition; and, beside the authority of the ancient writers, he brings some arguments from the thing itself. For example, Plato's confession, that the Greeks borrowed their knowledge of the one infinite God, from an ancient people, better and<pb n="244"/><cb/> nearer to God than they; by which people, our author makes no doubt, he meant the Jews, from his account of the state of innocence; as, that man was born of the earth, that he was naked, that he enjoyed a truly happy state, that he conversed with brutes, &c. In fact, from an examination of all the parts of Plato's philosophy, physical, metaphysical, and ethical, this author finds, in every one, evident marks of its sacred original.</p><p>As to the manner of the creation, Plato teaches, that the world was made according to a certain exemplar, or idea, in the divine architect's mind. And all things in the universe, in like manner, he shews, do depend on the efficacy of internal ideas. This ideal world is thus explained by Didymus: ‘Plato supposes certain patterns, or exemplars, of all sensible things, which he calls ideas; and as there may be various impressions taken off from the same seal, so he says are there a vast number of natures existing from each idea.’ This idea he supposes to be an eternal essence, and to occasion the several things in nature to be such as itself is. And that most beautiful and perfect idea, which comprehends all the rest, he maintains to be the world.</p><p>Farther, Plato teaches that the universe is an intelligent animal, consisting of a body and a soul, which he calls <hi rend="italics">the generated God,</hi> by way of distinction from what he calls the <hi rend="italics">immutable essence,</hi> who was the cause of the generated God, or the universe.</p><p>According to Plato, there were two sorts of inferior and derivative gods; the mundane gods, all of which had a temporary generation with the world; and the supramundane eternal gods, which were all of them, one excepted, produced from that one, and dependent on it as their cause. Dr. Cudworth says, that Plato asserted a plurality of gods, meaning animated or intellectual beings, or dæmons, superior to men, to whom honour and worship are due; and applying the appellation to the sun, moon, and stars, and also to the earth. He asserts however, at the same time, that there was one supreme God, the self originated being, the maker of the heaven and earth, and of all those other gods. He also maintains, that the Psyche, or universal mundane soul, which is a self-moving principle, and the immediate cause of all the motion in the world, was neither eternal nor self existent, but made or produced by God in time; and above this self-moving Psyche, but subordinate to the Supreme Being, and derived by emanation from him, he supposes an immoveable Nous or intellect, which was properly the Demiurgus, or framer of the world.</p><p>The first matter of which this body of the universe was formed, he observes, was a rude indigested heap, or chaos: Now, adds he, the creation was a mixed production; and the world is the result of a combination of necessity and understanding, that is, of matter, which he calls necessity, and the divine wisdom: yet so that mind rules over necessity; and to this necessity he ascribes the introduction and prevalence both of moral and natural evil.</p><p>The principles, or elements, which Plato lays down, are fire, air, water, and earth. He supposes two heavens, the Empyrean, which he takes to be of a fiery nature, and to be inhabited by angels, &c; and the Starry heaven, which he teaches is not adamantine, or solid, but liquid and spirable.<cb/></p><p>With regard to the human soul, Plato maintained its transmigration, and consequently its future immortality and pre-existence. He asserted, that human souls are here in a lapsed state, and that souls sinning should fall down into these earthly bodies. Eusebius expressly says, that Plato held the soul to be ungenerated, and to be derived by emanation from the first cause.</p><p>His physics, or doctrine <hi rend="italics">de corpore,</hi> is chiefly laid down in his Timæus, where he argues on the properties of body in a geometrical manner; which Aristotle takes occasion to reprehend in him. His doctrine <hi rend="italics">de mente</hi> is delivered in his 10th Book of Laws, and his Parmenides.</p><p>St. Augustine commends the Platonic philosophy; and even says, that the Platonists were not far from Christianity. It is also certain that most of the celebrated fathers were Platonists, and borrowed many of their explanations of scripture from the Platonic system. To account for this fact, it may be observed, that towards the end of the second century, a new sect of philosophers, called the modern, or later, Platonics, arose of a sudden, spread with amazing rapidity through the greatest part of the Roman empire, swallowed up almost all the other sects, and proved very detrimental to Christianity.</p><p>The school of Alexandria in Egypt, instituted by Ptolomy Philadelphus, renewed and reformed the Platonic philosophy. The votaries of this system distinguished themselves by the title of Platonics, because they thought that the sentiments of Plato concerning the Deity and invisible things, were much more rational and sublime than those of the other philosophers. This new species of Platonism was embraced by such of the Alexandrian Christians as were desirous to retain, with the profession of the gospel, the title, the dignity, and the habit of philosophers. Ammonius Saccas was its principal founder, who was succeeded by his disciple Plotinus, as this latter was by Porphyry, the chief of those formed in his school. From the time of Ammonius until the sixth century, this was almost the only system of philosophy publicly taught at Alexandria. It was brought into Greece by Plutarch, who renewed at Athens the celebrated Academy, from whence issued many illustrious philosophers. The general principle on which this sect was founded, was, that truth was to be pursued with the utmost liberty, and to be collected from all the different systems in which it lay dispersed. But none that were desirous of being ranked among these new Platonists, called in question the main doctrines; those, for example, which regarded the existence of one God, the fountain of all things; the eternity of the world; the dependance of matter upon the Supreme Being; the nature of souls; the plurality of gods, &c.</p><p>In the fourth century, under the reign of Valentinian, a dreadful storm of persecution arose against the Platonists; many of whom, being accused of magical practices, and other heinous crimes, were capitally convicted.</p><p>In the fifth century Proclus gave new life to the doctrine of Plato, and restored it to its former credit in Greece; with whom concurred many of the Christian doctors, who adopted the Platonic system. The<pb n="245"/><cb/> Platonic philosophers were generally opposers of Christianity; but in the sixth century. Chalcidius gave the Pagan system an evangelical aspect; and those who, before it became the religion of the state, ranged themselves under the standard of Plato, now repaired to that of Christ, without any great change of their system.</p><p>Under the emperor Justinian, who issued a particular edict, prohibiting the teaching of philosophy at Athens, which edict seems to have been levelled at modern Platonism, all the celebrated philosophers of this sect took refuge among the Persians, who were at that time the enemies of Rome; and though they returned from their voluntary exile, when the peace was concluded between the Persians and Romans, in 533, they could never recover their former credit, nor obtain the direction of the public schools.</p><p>Platonism however prevailed among the Greeks, and was by them, and particularly by Gemistius Pletho, introduced into Italy, and established, under the auspices of Cosmo de Medicis, about the year 1439, who ordered Marsilius Ficinus to translate into Latin the works of the most renowned Platonists.</p></div1><div1 part="N" n="PLATONISTS" org="uniform" sample="complete" type="entry"><head>PLATONISTS</head><p>, the followers of Plato; otherwise called Academics, from Academia, the name of the place that philosopher chose for his residence at Athens.</p></div1><div1 part="N" n="PLEIADES" org="uniform" sample="complete" type="entry"><head>PLEIADES</head><p>, an assemblage of seven stars in the neck of the constellation Taurus, the bull; although there are now only six of them visible to the naked eye. The largest of these is of the third magnitude, and called Lucido Pleiadum.</p><p>The Greeks fabled, that the name Pleiades was given to these stars from seven daughters of Atlas and Pleione one of the daughters of Oceanus, who having been the nurses of Bacchus, were for their services taken up to heaven and placed there as stars, where they still shine. The meaning of which fable may be, that Atlas first observed these stars, and called them by the names of the daughters of his wife Pleione.</p></div1><div1 part="N" n="PLENILUNIUM" org="uniform" sample="complete" type="entry"><head>PLENILUNIUM</head><p>, the full-moon.</p></div1><div1 part="N" n="PLENUM" org="uniform" sample="complete" type="entry"><head>PLENUM</head><p>, in Physics, signifies that state of things, in which every part of space, or extension, is supposed to be full of matter: in opposition to a Vacuum, which is a space devoid of all matter.</p><p>The Cartesians held the doctrine of an absolute Plenum; namely on this principle, that the essence of matter consists in extension; and consequently, there being every where extension or space, there is every where matter: which is little better than begging the question.</p></div1><div1 part="N" n="PLINTH" org="uniform" sample="complete" type="entry"><head>PLINTH</head><p>, in Architecture, a flat square member in form of a brick or tile; used as the foot or foundation of columns and pillars, &c.</p></div1><div1 part="N" n="PLOT" org="uniform" sample="complete" type="entry"><head>PLOT</head><p>, in Surveying, the plan or draught of any parcel of ground; as a field, farm, or manor, &c.</p></div1><div1 part="N" n="PLOTTING" org="uniform" sample="complete" type="entry"><head>PLOTTING</head><p>, in Surveying, the describing or laying down on paper, the several angles and lines, &c, of a tract of land, that has been surveyed and measured.</p><p>Plotting is usually performed by two instruments, the protractor and Plotting-scale; the former serving to lay off all the angles that have been measured and set down, and the latter all the measured lines. See these two instruments under their respective names.</p><p><hi rend="smallcaps">Plotting</hi> <hi rend="italics">Scale,</hi> a mathematical instrument chiefly<cb/> used for the plotting of grounds in surveying, or setting off the lengths of the lines. It is either 6, 9, or 12 inches in length, and about an inch and half broad; being made either of box-wood, brass, ivory, or silver; those of ivory are the neatest.</p><p>This instrument contains various scales or divided lines, on both sides of it. On the one side are a number of plane scales, or scales of equal divisions, each of a different number to the inch; as also scales of chords, for laying down angles; and sometimes even the degrees of a circle marked on one edge, answering to a centre marked on the opposite edge, by which means it serves also as a protractor. On the other side are several diagonal scales, of different sizes, or different divisions to the inch; serving to take off lines expressed by numbers to three dimensions, as units, tens, and hundreds; as also a scale of divisions which are the 100th parts of a foot. But the most useful of all the lines that can be laid upon this instrument, though not always done, is a line or plane scale upon the two opposite edges, made thin for that purpose. This is a very useful line in surveying; for by laying the instrument down upon the paper, with its divided edge along a line upon which are to be laid off several distances, for the places of off-sets, &c; these distances are all transferred at once from the instrument to the line on the paper, by making small marks or points against the respective divisions on the edge of the scale. See fig. 2 & 3, plates xxi and xxii.</p><p><hi rend="smallcaps">Plotting</hi>-<hi rend="italics">Table,</hi> in Surveying, is used for a plane table, as improved by Mr. Beighton, who has obviated a good many inconveniencies attending the use of the common plane table. See Philos. Trans. numb. 461, sect. 1.</p></div1><div1 part="N" n="PLOUGH" org="uniform" sample="complete" type="entry"><head>PLOUGH</head><p>, or <hi rend="smallcaps">Plow</hi>, in Navigation, an ancient mathematical instrument, made of box or pear-tree, and used to take the height of the sun or stars, in order to find the latitude. This instrument admits of the degrees to be very large, and has been much esteemed by many artists; though now quite out of use.</p><p>PLUMB-<hi rend="smallcaps">Line</hi>, a term among artificers for a line perpendicular to the horizon.</p></div1><div1 part="N" n="PLUMMET" org="uniform" sample="complete" type="entry"><head>PLUMMET</head><p>, <hi rend="smallcaps">Plumb-rule</hi>, or <hi rend="smallcaps">Plumb-line</hi>, an instrument used by masons, carpenters, &c, to draw perpendiculars; in order to judge whether walls, &c, be upright, or planes horizontal, and the like.</p></div1><div1 part="N" n="PLUNGER" org="uniform" sample="complete" type="entry"><head>PLUNGER</head><p>, in Mechanics, a solid brass cylinder, used as a forcer in forcing pumps.</p></div1><div1 part="N" n="PLUS" org="uniform" sample="complete" type="entry"><head>PLUS</head><p>, in Algebra, the affirmative or positive sign, +, signifying more or addition, or that the quantity following it is either to be considered as a positive or affirmative quantity, or that it is to be added to the other quantities; so , is read thus, 4 plus 6 is equal to 10. See <hi rend="smallcaps">Affirmative</hi> <hi rend="italics">Sign.</hi></p><p>The more early writers of Algebra, as Lucas de Burgo, Cardan, Tartaglia, &c, wrote the word mostly at full length. Afterwards the word was contracted or abbreviated, using one or two of its first letters; which initial was, by the Germans I think, corrupted to the present character +; which I find first used by Stifelius, printed in his Arithmetic.</p></div1><div1 part="N" n="PLUVIAMETER" org="uniform" sample="complete" type="entry"><head>PLUVIAMETER</head><p>, a machine for measuring the quantity of rain that falls. There is described in the Philos. Trans. (numb. 473, or Abridg. x. 456), by Robert Pickering, under the name of an Ombrameter,<pb n="246"/><cb/> an instrument of this kind. It consists of a tin funnel <hi rend="italics">d,</hi> whose surface is an inch square (fig. 6, plate xx); a flat board <hi rend="italics">aa;</hi> and a glass tube <hi rend="italics">bb,</hi> set into the middle of it in a groove; and an index with divisions <hi rend="italics">cc;</hi> the board and tube being of any length at pleasure. The bore of the tube is about half an inch, which Mr. Pickering says is the best size. The machine is sixed in some free and open place, as the top of the house, &c.</p><p>The Rain-gage employed at the house of the Royal Society is described by Mr. Cavendish, in the Philos. <figure/> Trans. for 1776, p. 384. The vessel which receives the rain is a conical funnel, strengthened at the top by a brass ring, 12 inches in diameter. The sides of the funnel and inner lip of the brass ring are inclined to the horizon, in an angle of above 65°; and the outer lip in an angle of above 50°; which are such degrees of steepness, that there seems no probability either that any rain which falls within the funnel, or on the inner lip of the ring, shall dash out, or that any which falls on the outer lip shall dash into the funnel. The annexed figure is a vertical section of the funnel, ABC and <hi rend="italics">abc</hi> being the brass ring, BA and <hi rend="italics">ba</hi> the inner lip, and BC and <hi rend="italics">bc</hi> the outer.</p><p>Note, that in fixing Pluviameters care should be taken that the rain may have free access to them, without being impeded or overshaded by buildings, &c; and therefore the tops of houses are mostly to be preferred. Also when the quantities of rain collected in them, at different places, are compared together, the instruments ought to be fixed at the same height above the ground at both places; because at different heights the quantities are always different, even in the same place. And hence also, any register or account of rain in the Pluviameter, ought to be accompanied with a note of the height above the ground the instrument is placed at. See <hi rend="italics">Quantity of</hi> <hi rend="smallcaps">Rain.</hi></p></div1><div1 part="N" n="PNEUMATICS" org="uniform" sample="complete" type="entry"><head>PNEUMATICS</head><p>, that branch of natural philosophy which treats of the weight, pressure, and elasticity of the air, or elastic fluids, with the effects arising from them. Wolfius, instead of Pneumatics, uses the term Aerometry.</p><p>This is a sister science to Hydrostatics; the one considering the air in the same manner as the other does water. And some consider Pneumatics as a branch of mechanics; because it considers the air in motion, with the consequent effects.</p><p>For the nature and properties of air, see the article <hi rend="smallcaps">Air</hi>, where they are pretty largely treated of. To which may be added the following, which respects more particularly the science of Pneumatics, as contained in a few propositions, and their corollaries.</p><p><hi rend="smallcaps">Prop.</hi> I. <hi rend="italics">The Air is a heavy fluid body, which surrounds and gravitates upon all parts of the surface of the earth.</hi></p><p>These properties of air are proved by experience. That it is a fluid, is evident from its easily yielding to<cb/> any the least force impressed upon it, with little or no sensible resistance.</p><p>But when it is moved briskly, by any means, as by a fan, or a pair of bellows; or when any body is moved swiftly through it; in these cases we become sensible of it as a body, by the resistance it makes in such motions, and also by its impelling or blowing away any light substances. So that, being capable of resisting, or moving other bodies by its impulse, it must itself be a body, and be heavy, like all other bodies, in proportion to the matter it contains; and therefore it will press upon all bodies that are placed under it.</p><p>And being a fluid, it will spread itself all over upon the earth; also like other fluids it will gravitate upon, and press every where upon the earth's surface.</p><p>The gravity and pressure of the air is also evident from many experiments. Thus, for instance, if water, or quick-silver, be poured into the tube ACE, and the <figure/> air be suffered to press upon it, in both ends of the tube; the fluid will rest at the same height in both the legs: but if the air be drawn out of one end as E, by any means; then the air pressing on the other end A, will press down the fluid in this leg at B, and raise it up in the other to D, as much higher than at B, as the pressure of the air is equal to. By which it appears, not only that the air does really press, but also what the quantity of that pressure is equal to. And this is the principle of the Barometer.</p><p><hi rend="smallcaps">Prop.</hi> II. <hi rend="italics">The air is also an elastic fluid, being condensible and expansible. And the law it observes in this respect is this, namely, that its density is always proportional to the force by which it is compressed.</hi></p><p>This property of the air is proved by many experiments. Thus, if the handle of a syringe be pushed inwards, it will condense the inclosed air into a less space; by which it is shewn to be condensible. But the included air, thus condensed, will be felt to act strongly against the hand, and to resist the force compressing it more and more; and on withdrawing the hand, the handle is pushed back again to where it was at first. Which shews that the air is elastic.</p><p>Again, fill a strong bottle half full with water, and then insert a pipe into it, putting its lower end down near to the bottom, and cementing it very close round the mouth of the bottle. Then if air be strongly injected through the pipe, as by blowing with the mouth or otherwise, it will pass through the water from the lower end, and ascend up into the part before occupied<pb n="247"/><cb/> by the air at G, and the whole mass of air become there condensed because the water is not easily compressed into a less space. But on removing the force which injected the air at F, the water will begin to rise from thence in a jet, being pushed up the pipe by the increased elasticity of the air G, by which it presses on the surface of the water, and forces it through the pipe, till as much be expelled as there was air forced in; when the air at G will be reduced to the same density as at first, and, the balance being restored, the jet will cease.</p><p>Likewise, if into a jar of water AB, be inverted an empty glass tumbler C, or such like; the water will <figure/> enter it, and partly fill it, but not near so high as the water in the jar, compressing and condensing the air into a less space in the upper part C, and causing the glass to make a sensible resistance to the hand in pushing it down. But on removing the hand, the elasticity of the internal condensed air throws the glass up again.—All these shewing that the air is condensible and elastic.</p><p>Again, to shew the rate or proportion of the elasticity to the condensation; take a long slender glass tube, open at the top A, bent near the bottom or close end B, and equally wide throughout, or at least in the part BD (2d fig. above). Pour in a little quicksilver at A, just to cover the bottom to the bend at CD, and to stop the communication between the external air and the air in BD. Then pour in more quicksilver, and observe to mark the corresponding heights at which it stands in the two legs: so, when it rises to H in the open leg AC, let it rise to E in the close one, reducing its included air from the natural bulk BD to the contracted space BE, by the pressure of the column H<hi rend="italics">e;</hi> and when the quicksilver stands at I and K, in the open leg, let it rise to F and G in the other, reducing the air to the respective spaces BF, BG, by the weights of the columns I<hi rend="italics">f,</hi> K<hi rend="italics">g.</hi> Then it is always found, that the condensations and elasticities are as the compressing weights, or columns of the quicksilver and the atmosphere together. So, if the natural bulk of the air BD be compressed into the spaces BE, BF, BG, or reduced by the spaces DE, DF, DG, which are 1/4, 1/2, 3/4 of BD, or as the numbers 1, 2, 3; then the atmosphere, together with the corresponding column H<hi rend="italics">e,</hi> I<hi rend="italics">f,</hi> K<hi rend="italics">g,</hi> will also be found to be in the same proportion, or as the numbers 1, 2, 3: and then the weights of the quicksilver are thus, viz, H<hi rend="italics">e</hi> = (1/3)A, I<hi rend="italics">f</hi> = A, and K<hi rend="italics">g</hi> = 3A; where A denotes the weight of the atmosphere. Which shews<cb/> that the condensations are directly as the compressing forces. And the elasticities are also in the same proportion, since the pressures in AC are sustained by the elasticities in BD.</p><p>From the foregoing principles may be deduced many useful remarks, as in the following corollaries, viz:</p><p><hi rend="italics">Corol.</hi> 1. The space that any quantity of air is confined in, is reciprocally as the force that compresses it. So, the forces which confine a quantity of air in the <figure/> cylindrical spaces AG, BG, CG, are reciprocally a<*> the same, or reciprocally as the heights AD, BD, CD. And therefore, if to the two perpendicular lines AD, DH, as asymptotes, the hyperbola IKL be described, and the ordinates AI, BK, CL be drawn; then the forces which confine the air in the spaces AG, BG, CG, will be as the corresponding ordinates AI, BK, CL, since these are reciprocally as the abscisses AD, BD, CD, by the nature of the hyperbola.</p><p><hi rend="italics">Corol.</hi> 2. All the air near the earth is in a state of compression, by the weight of the incumbent atmosphere.</p><p><hi rend="italics">Corol.</hi> 3. The air is denser near the earth, than in high places; or denser at the foot of a mountain, than at the top of it. And the higher above the earth, the rater it is.</p><p><hi rend="italics">Corol.</hi> 4. The spring or elasticity of the air, is equal to the weight of the atmosphere above it; and they will produce the same effects; since they are always sustained and balanced by each other.</p><p><hi rend="italics">Corol.</hi> 5. If the density of the air be increased, preserving the same heat or temperature; its spring or elasticity will likewise be increased, and in the same proportion.</p><p><hi rend="italics">Corol.</hi> 6. By the gravity and pressure of the atmosphere upon the surfaces of fluids, the fluids are made to rise in any pipes or vessels, when the spring or pressure within is diminished or taken off.</p><p><hi rend="smallcaps">Prop.</hi> III. <hi rend="italics">Heat increases the elasticity of the air, and cold diminisbes it. Or heat expands, and cold contracts and condenses the air.</hi></p><p>This property is also proved by experience.</p><p>Thus, tie a bladder very close, with some air in it; and lay it before the fire; then as it warms, it will more and more distend the bladder, and at last burst it, if the heat be continued and increased high enough. But if the bladder be removed from the fire; it will contract again to its former state by cooling.——It was upon this principle that the first air-balloons were made by Montgolfier: for by heating the air within them, by a fire underneath, the hot air distends them to a size which occupies a space in the atmosphere whose weight of common air exceeds that of the balloon.</p><p>Also, if a cup or glass, with a little air in it, be inverted into a vessel of water; and the whole be heated<pb n="248"/><cb/> over the fire, or otherwise: the air in the top will expand till it fill the glass, and expel the water out of it; and part of the air itself will follow, by continuing or increasing the heat.</p><p>Many other experiments to the same effect might be adduced, all proving the properties mentioned in the proposition.</p><p><hi rend="italics">Schol.</hi> Hence, when the force of the elasticity of the air is considered, regard must be had to its heat or temperature; the same quantity of air being more or less elastic, as its heat is more or less. And it has been found by experiment that its elasticity is increased at the following rate, viz, by the 435th part, by each degree of heat expressed by Fahrenheit's thermometer, of which there are 180 between the freezing and boiling heat of water. It has also been found (Philos. Trans. 1777, pa. 560 &c), that water expands the 6666th part, with each degree of heat; and mercury the 9600th part by each degree. Moreover, the relative or specific gravities of these three substances, are as follow<*> viz, <hi rend="brace"><note anchored="true" place="unspecified">when the barom. is at 30, and the thermom. at 55.</note> Air 1.232 Water 1000 Mercury 13600</hi> Also these numbers are the weights of a cubic foot of each, in the same circumstances of the barometer and thermometer.</p><p><hi rend="smallcaps">Prop.</hi> IV. <hi rend="italics">The weight or pressure of the atmosphere, upon any base at the surface of the earth, is equal to the weight of a column of quicksilver of the same base, and its height between</hi> 28 <hi rend="italics">and</hi> 31 <hi rend="italics">inches.</hi></p><p>This is proved by the barometer, an instrument which measures the pressure of the air; the description of which see under its proper article. For at some seasons, and in some places, the air sustains and balances a column of mercury of about 28 inches; but at others, it balances a column of 29, or 30, or near 31 inches high; seldom in the extremes 28 or 31, but commonly about the means 29 or 30, and indeed mostly near 30. A variation which depends partly on the different degrees of heat in the air near the surface of the earth, and partly on the commotions and changes in the atmosphere, from winds and other causes, by which it is accumulated in some places, and depressed in others, being thereby rendered denser and heavier, or rarer and lighter; which changes in its state are almost continually happening in any one place. But the medium state is from 29 1/2 to 30 inches.</p><p><hi rend="italics">Corol.</hi> 1. Hence the pressure of the atmosphere upon every square inch at the earth's surface, at a medium, is very near 15 pounds avoirdupois. For, a cubic foot of mercury weighing nearly 13600 ounces, a cubic inch of it will weigh the 1728th part of it, or almost 8 ounces, or half a pound, which is the weight of the atmosphere for every inch of the barometer upon a base of a square inch; and therefore 29 3/4 inches, the medium height of the barometer, weighs almost 15 pounds, or rather 14 (3/4)lb very nearly.</p><p><hi rend="italics">Corol.</hi> 2. Hence also the weight or pressure of the atmosphere, is equal to that of a column of water from 32 to 35 feet high, or on a medium 33 or 34 feet high. For water and quicksilver are in weight nearly as 1 to 13.6; so that the atmosphere will balance a<cb/> column of water 13.6 times higher than one of quicksilver; consequently inches or 34 feet, is near the medium height of water, or it is more nearly 33 3/4 feet. And hence it appears that a common sucking pump will not raise water higher than about 34 feet. And that a syphon will not run if the perpendicular height of the top of it be more than 33 or 34 feet.</p><p><hi rend="italics">Corol.</hi> 3. If the air were of the same uniform density, at every height, up to the top of the atmosphere, as at the surface of the earth; its height would be about 5 1/4 miles at a medium. For the weights of the same volume of air and water, are nearly as 1.232 to 1000; therefore as feet, or 5 1/4 miles very nearly. And so high the atmosphere would be, if it were all of uniform density, like water. But, instead of that, from its expansive and elastic quality, it becomes continually more and more rare the farther above the earth, in a certain proportion which will be treated of below.</p><p><hi rend="italics">Corol.</hi> 4. From this prop. and the last, it follows that the height is always the same, of an uniform atmosphere above any place, which shall be all of the uniform density with the air there, and of equal weight or pressure with the real height of the atmosphere above that place, whether it be at the same place at different times, or at any different places or heights above the earth; and that height is always about 27600 feet, or 5 1/4 miles, as found above in the 3d corollary. For, as the density varies in exact proportion to the weight of the column, it therefore requires a column of the same height in all cases, to make the respective weights or pressures. Thus, if W and <hi rend="italics">w</hi> be the weights of atmosphere above any places, D and <hi rend="italics">d</hi> their densities, and H and <hi rend="italics">h</hi> the heights of the uniform columns, of the same densities and weights: Then , and ; therefore W/D or H is equal to <hi rend="italics">w</hi>/<hi rend="italics">d</hi> or <hi rend="italics">h;</hi> the temperature being the same.</p><p><hi rend="smallcaps">Prop.</hi> V. <hi rend="italics">The density of the atmosphere, a<*> different heights above the earth, decreases in such sort, that when the heights increase in arithmetical progression, the densities decrease in geometrical progression.</hi></p><p>Let the perpendicular line AP, erected on the earth, be conceived to be divided into a great number of very <figure/> small parts A, B, C, D, &c, forming so many thin strata of air in the atmosphere, all of different density, gradually decreasing from the greatest at A: then the density of the several strata A, B, C, D, &c, will be in geometrical progression decreasing.</p><p>For, as the strata A, B, C, &c, are all of equal thickness, the quantity of matter in each of them, is as the density there; but the density in any one, being as the compressing force, is as the weight or quantity of matter from that place upward to the top of the atmosphere; therefore the quantity of matter in each stratum, is also as the whole quantity from that place upwards. Now if from the whole weight at any<pb n="249"/><cb/> place as B, the weight or quantity in the stratum B be subtracted, the remainder will be the weight at the next higher stratum C; that is, from each weight subtracting a part which is proportional to itself, leaves the next weight; or, which is the same thing, from each density subtracting a part which is always proportional to itself, leaves the next density. But when any quantities are continually diminished by parts which are proportional to themselves, the remainders then form a series of continued proportionals; and consequently these densities are in geometrical progression.</p><p>Thus, if the first density be D, and from each there be taken its <hi rend="italics">n</hi>th part; then there remains its (<hi rend="italics">n</hi> - 1)/<hi rend="italics">n</hi> part, or the <hi rend="italics">m</hi>/<hi rend="italics">n</hi> part, putting <hi rend="italics">m</hi> for <hi rend="italics">n</hi> - 1; and therefore the series of densities will be D, <hi rend="italics">m</hi>/<hi rend="italics">n</hi> D, <hi rend="italics">m</hi><hi rend="sup">2</hi>/<hi rend="italics">n</hi><hi rend="sup">2</hi> D, <hi rend="italics">m</hi><hi rend="sup">3</hi>/<hi rend="italics">n</hi><hi rend="sup">3</hi> D, &c, <hi rend="italics">m</hi>/<hi rend="italics">n</hi> being the common ratio of the series.</p><p><hi rend="italics">Schol.</hi> Because the terms of an arithmetical series, are proportional to the logarithms of the terms of a geometrical series; therefore different altitudes above the earth's surface, are as the logarithms of the densities, or weights of air, at those altitudes. So that, if D denote the density at the altitude A, and <hi rend="italics">d</hi> the density at the altitude <hi rend="italics">a;</hi> then A being as the logarithm of D, and <hi rend="italics">a</hi> as the logarithm of <hi rend="italics">d,</hi> the dif. of altitude <hi rend="italics">A</hi> - <hi rend="italics">a</hi> will be as the log. of <hi rend="italics">D</hi> - log. of <hi rend="italics">d,</hi> or as log. of D/<hi rend="italics">d</hi>.</p><p>And if A = 0, or D the density at the surface of the earth, then any altitude above the surface <hi rend="italics">a,</hi> is as the log. of D/<hi rend="italics">d</hi>. Or, in general, the log. of D/<hi rend="italics">d</hi> is as the altitude of the one place above the other, whether the lower place be at the surface of the earth, or any where else.</p><p>And from this property is derived the method of determining the heights of mountains, and other eminences, by the barometer, which is an instrument that measures the weight or density of the air at any place. For by taking with this instrument, the pressure or density at the foot of a hill for instance, and again at the top of it, the difference of the logarithms of these two presfures, or the logarithms of their quotient, will be as the difference of altitude, or as the height of the hill; supposing the temperatures of the air to be the same at both places, and the gravity of air not altered by the different distances from the earth's centre.</p><p>But as this formula expresses only the relations between different altitudes, with respect to their densities, recourse must be had to some experiment, to obtain the real altitude which corresponds to any given density, or the density which corresponds to a given altitude. Now there are various experiments by which this may be done. The first, and most natural, is that which results from the known specific gravity of air, with respect to the whole pressure of the atmosphere on the surface of the earth. Now, as the altitude <hi rend="italics">a</hi> is always as<cb/> log. D/<hi rend="italics">d</hi>, assume <hi rend="italics">h</hi> so that <hi rend="italics">a</hi> may be , where <hi rend="italics">h</hi> will be of one constant value for all altitudes; and to determine that value, let a case be taken in which we know the altitude <hi rend="italics">a</hi> corresponding to a known density <hi rend="italics">d:</hi> as for instance take <hi rend="italics">a</hi> = 1 foot, or 1 inch, or some such small altitude; and because the density D may be measured by the pressure of the atmosphere, or the uniform column of 27600 feet, when the temperature is 55°; therefore 27600 feet will denote the density D at the lower place, and 27599 the less density <hi rend="italics">d</hi> at one foot above it; consequently this equation arises, viz, of 27600/27599, which, by the nature of logarithms, is nearly nearly; and hence <hi rend="italics">h</hi> = 63451 feet; which gives for any altitude whatever, this general theorem, viz, , or feet, or fathoms; where M is the column of mercury which is equal to the pressure or weight of the atmosphere at the bottom, and <hi rend="italics">m</hi> that at the top of the altitude <hi rend="italics">a;</hi> and where M and <hi rend="italics">m</hi> may be taken in any measure, either feet, or inches, &c.</p><p><hi rend="italics">Note,</hi> that this formula is adapted to the mean temperature of the air 55°. But for every degree of temperature different from this, in the medium between the temperatures at the top and bottom of the altitude <hi rend="italics">a,</hi> that altitude will vary by its 435th part; which must be added when the medium exceeds 55°, otherwise subtracted.</p><p><hi rend="italics">Note</hi> also, that a column of 30 inches of mercury varies its length by about the 320th part of an inch for every degree of heat, or rather the 9600th part of the whole volume.</p><p>But the same formula may be rendered much more convenient for use, by reducing the factor 10592 to 10000, by changing the temperature proportionably from 55°: thus, as the difference 592 is the 18th part of the whole factor 10592; and as 18 is the 24th part of 435; therefore the corresponding change of temperature is 24°, which reduces the 55° to 31°. So that the formula becomes of M/<hi rend="italics">m</hi> fathoms when the temperature is 31 degrees; and for every degree above that, the result must be increased by so many times its 435th part.</p><p>See more on this head under the article B<hi rend="smallcaps">AROMETER</hi>, at the end.</p><p>By the weight and pressure of the atmosphere, the effect and operations of Pneumatic engines may be accounted for, and explained; such as syphons, pumps, barometers, &c. See each of these articles, also <hi rend="smallcaps">Air.</hi></p><p><hi rend="smallcaps">Pneumatic</hi> <hi rend="italics">Engine,</hi> the same as the <hi rend="smallcaps">Air-Pump.</hi></p><p>POCKET <hi rend="italics">Electrical Apparatus.</hi>—This is a contrivance of Mr. William Jones, in Holborn, the form of which is represented in plate xxiii, fig. 4.<pb n="250"/><cb/></p><p>This small machine is capable of a tolerably strong charge, or accumulation of electricity, and will give a small shock to one, two, three, or a greater number of persons.</p><p>A is the Leyden phial or jar that holds the charge. B is the discharger to discharge the jar when required without electrifying the person that holds it. C is a ribbon prepared in a peculiar manner so as to be excited, and communicate its electricity to the jar. D are two hair, &c, skin rubbers, which are to be placed on the first and middle fingers of the left hand. <hi rend="center"><hi rend="italics">To charge the Far.</hi></hi></p><p>Place the two finger-caps D on the first and middle finger of the left hand; hold the jar A at the same time, at the joining of the red and black on the outside between the thumb and first singer of the same hand; then take the ribbon in your right hand, and steadily and gently draw it upwards between the two rubbers D, on the two fingers; taking care at the same time, the brass ball of the jar is kept nearly close to the ribbon, while it is passing through the fingers. By repeating this operation twelve or fourteen times, the electrical fire will pass into the jar which will become charged, and by placing the discharger C against it, as in the plate, you will see a sensible spark pass from the ball of the jar to that of the discharger. If the apparatus is dry and in good order, you will hear the crackling of the fire when the ribbon is passing through the fingers, and the jar will discharge at the distance represented in the figure. <hi rend="center"><hi rend="italics">To electrify a Person.</hi></hi></p><p>You must desire him to take the jar in one hand, and with the other touch the nob of it: or, if diversion is intended, desire the person to smell at the nob of it, in expectation of smelling the scent of a rose or a pink; this last mode has occasioned it to be sometimes called the Magic Smelling Bottle.</p><p>POETICAL <hi rend="italics">Numbers.</hi> See <hi rend="smallcaps">Numbers.</hi></p><p><hi rend="smallcaps">Poetical</hi> <hi rend="italics">Rising</hi> and <hi rend="italics">Setting.</hi> See <hi rend="smallcaps">Rising</hi> and <hi rend="smallcaps">Setting.</hi></p><p>The ancient poets, referring the rising and setting of the stars to that of the sun, make three kinds of rising and setting, viz, Cosmical, Acronical, and Heliacal. See each of these words in its place.</p></div1><div1 part="N" n="POINT" org="uniform" sample="complete" type="entry"><head>POINT</head><p>, a term used in various arts and sciences.</p><div2 part="N" n="Point" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Point</hi></head><p>, in Architecture. Arches of the third Point, and Arches of the fourth Point. See <hi rend="smallcaps">Arches.</hi></p></div2><div2 part="N" n="Point" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Point</hi></head><p>, in Astronomy, is a term applied to certain parts or places marked in the heavens, and distinguished by proper epithets.</p><p>The four grand points or divisions of the horizon, viz, the east, west, north, and south, are called the Cardinal Points.—The zenith and nadir are the Vertical Points. —The Points where the orbits of the planets cut the plane of the ecliptic, are called the Nodes.—The Points where the ecliptic and equator intersect, are called the Equinoctial Points. In particular, that where the sun ascends towards the north pole is called the Vernal Point; and that where he descends towards the south, the Autumnal Point — The highest and lowest Points of the ecliptic are called the Solstitial Points. Particu larly, the former of them the Estival or Summer Point; the latter, the Brumal or Winter Point.<cb/></p></div2><div2 part="N" n="Points" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Points</hi></head><p>, in Electricity, are those acute terminations of bodies which facilitate the passage of the electrical fluid either <hi rend="italics">from</hi> or <hi rend="italics">to</hi> such bodies.</p><p>Mr. Jallabert was probably the first person who observed that a body pointed at one end, and round at the other, produced different appearances upon the same body, according as the pointed or round end was presented to it. But Dr. Franklin first observed and evinced the whole effect of pointed bodies, both in drawing and throwing off electricity at greater distances than other bodies could do it; though he candidly acknowledges, that the power of Points to throw off the electric sire was communicated to him by his sriend Mr. Thomas Hopkinson.</p><p>Dr. Franklin electrified an iron shot, 3 or 4 inches in diameter, and observed that it would not attract a thread when the Point of a needle, communicating with the earth, was presented to it; and he found it even impossible to electrify an iron shot when a sharp needle lay upon it. This remarkable property, possessed by pointed bodies, of gradually and silently receiving or throwing off the electric fluid, has been evinced by a variety of other familiar experiments.</p><p>Thus, if one hand be applied to the outside coating of a large jar fully charged, and the Point of a needle held in the other, be directed towards the knob of the jar, and moved gradually near it, till the Point of the needle touch the knob or ball, the jar will be entirely discharged, so as to give no shock at all, or one that is hardly sensible. In this case the Point of the needle has gradually and silently drawn away the superabundant electricity from the electrified jar.</p><p>Farther, if the knob of a brass rod be held at such a distance from the prime conductor, that sparks may easily escape from the latter to the former, whilst the machine is in motion; then if the Point of a needle be presented, though at twice the distance of the rod from the conductor, no more sparks will be seen passing to the rod. When the needle is removed, the sparks will be seen; but upon presenting it again, they will again disappear. So that the Point of the needle draws off silently almost all the fluid, which is thrown by the cylinder or globe of the machine upon the prime conductor. This experiment may be varied, by fixing the needle upon the prime conductor with the point upward; and then, though the knob of a discharging rod, or the knuckle of the singer, be brought very near the prime conductor, and the excitation be very strong, little or no spark will be perceived.</p><p>The influence of points is also evinced in the amusing experiment, commonly called the electrical horse-race, and many others. See <hi rend="smallcaps">Thunder</hi>-<hi rend="italics">house.</hi></p><p>The late Mr. Henly exhibited the efficacy of pointed bodies, by suspending a large bladder, well blown, and covered with gold, silver, or brass leaf, by means of gum-water, at the end of a silken thread 6 or 7 feet long, hanging from the cieling of a room, and electrifying the bladder by giving it a strong spark with the knob of a charged bottle: upon presenting to it the knob of a wire, it caused the bladder to move towards the knob, and when nearly in contact gave it a spark, thus discharging its electricity. By giving the bladder another charge, and presenting the Point of a needle to it, the bladder was not attracted by the Point, but<pb n="251"/><cb/> rather receded from it, especially when the needle was suddenly presented towards it.</p><p>But experiments evincing the efficacy of pointed bodies for silently receiving or throwing off the electric fluid, may be infinitely diversified, according to the fancy or convenience of the electrician.</p><p>It may be observed, that in the case of points throwing off or receiving electricity, a current of air is sensible at an electrified Point, which is always in the direction of the Point, whether the electricity be positive or negative. A fact which has been well ascertained by many electricians, and particularly by Dr. Priestley and Sig. Beccaria. The former contrived to exhibit the influence of this current on the flame of a candle, presented to a pointed wire, electrified negatively, as well as positively. The blaft was in both cases alike, and so strong as to lay bare the greatest part of the wick, the flame being driven from the Point; and the effect was the same whether the electric fluid issued out of the Point or entered into it. He farther evinced this phenomenon by means of thin light vanes; and he found, as Mr. Wilson had before observed, that the vanes would not turn in vacuo, nor in a close unexhausted receiver where the air had no sree circulation. And in much the same manner, Beccaria exhibited to sense the influence of the wind or current of air driven from points.</p><p>As to the <hi rend="italics">Theory</hi> of the phenomena of Points, these are accounted for in a variety of ways, by different authors, though perhaps by none with perfect satisfaction. See Franklin's writings on Electricity; Lord Mahon's Principles of Electricity, 1779; Beccaria's Artisicial Electricity, 1776, pa. 331; and Priestlcy's History of Electricity, vol. 2, pa. 191, edit. 1775.</p><p>As to the <hi rend="italics">Application</hi> of the doctrine of Points; it may be observed that there is not a more important fact in the history of electricity, than the use to which the discovery of the efficacy of pointed bodies has been applied.</p><p>Dr. Franklin, having ascertained the identity of electricity and lightning, was presently led to propose a cheap and easy method of securing buildings from the damage of lightning, by fixing a pointed metal rod higher than any part of the building, and communicating with the ground, or with the nearest water. And this contrivance was actually executed in a variety of cases; and has usually been thought an excellent preservative against the terrible effects of lightning.</p><p>Some few instances however having occurred, in which buildings have been struck and damaged, though provided with these conductors; a controversy arose with regard to their expediency and utility. In this controversy Mr. Benjamin Wilson took the lead, and Dr. Musgrave, and some few other electricians, the least acquainted with the subject, concurred with him in their opposition to pointed elevated conductors. These alledge, that every Point, as such, solicits the lightning, and thus contributes not only to increase the quantity of every actual discharge, but also frequently to occasion a discharge when it might not otherwise have happened: whereas, say they, if instead of pointed conductors, those with blunted terminations were used, they would as effectually answer the purpose of conveying away the lightning safely, without the same tendency to increase or invite it. Accordingly, Mr. Wilson, in a<cb/> letter to the marquis of Rockingham (Philos. Trans. vol. 54, art. 44), expresses his opinion, that, in order to prevent lightning from doing mischief to high buildings, large magazines, and the like, instead of the elevated external conductors, that, on the inside of the highest part of such building, and within a foot or two of the top, it may be proper to fix a rounded bar of metal, and to continue it down along the side of the wall to any kind of moisture in the ground.</p><p>On the other hand, it is urged by the advocates for pointed conductors, that Points, instead of increasing an actual discharge, really prevent a discharge where it would otherwise happen, and that blunted conductors tend to invite the clouds charged with lightning. And it seems to be a certain fact, that though a sharp Point will draw off a charge of electricity silently at a much greater distance than a knob, yet a knob will be struck with a full explosion or shock, the charge being the same in both cases, at a greater distance than a sharp Point.</p><p>The efficacy of pointed bodies for preventing a stroke of lightning, is ingeniously explained by Dr. Franklin in the following manner:—An eye, he says, so situated as to view horizontally the underside of a thunder-cloud, will see it very ragged, with a number of separate fragments or small clouds one under another; the lowest sometimes not far from the earth. These, as so many stepping-stones, assist in conducting a stroke between a cloud and a building. To represent these by a<*> experiment, he directs to take two or three locks of fine loose cotton, and connect one of them with the prime conductor by a fine thread of 2 inches, another to that, and a third to the second, by like threads, which may be spun out of the same cotton. He then directs to turn the globe, and says we shall see these locks extending themselves towards the table, as the lower small clouds do towards the earth; but that, on presenting a sharp Point, erect under the lowest, it will shrink up to the second, the second up to the sirst, and all together to the prime conductor, where they will continue as long as the Point continues under them. May not, he adds, in like manner, the small electrified clouds, whose equilibrium with the earth is soon restored by the Point, rise up to the main body, and by that means occasion so large a vacancy, as that the grand cloud cannot strike in that place? Letters, pa. 121.</p><p>Mr. Henly too, as well as several other persons, with a view of determining the question, whether Points or knobs are to be preferred for the terminations of conductors, made several experiments, shewing in a variety of instances, the efsicacy of Points in silently drawing off the electricity, and preventing strokes which would happen to knobs in the same situation. Philos. Trans. vol. 64, part 2, art. 18. See also <hi rend="smallcaps">Thunder</hi>- <hi rend="italics">Hous.</hi></p><p>Indeed it has been universally allowed, that in cases where the quantity of electricity, with which thunderclouds are charged, is small, or when they move slowly in their passage to and over a building, pointed conductors, which draw off the electrical fluid silently, within the distance at which rounded ends will explode, will gradually exhaust them, and thus contribute to prevent a stroke and preserve the buildings to which they are annexed.<pb n="252"/><cb/></p><p>But it has been said by those who are averse to the use of such conductors, that if clouds, of great extent, and highly electrified, should be driven directly over them with great velocity, or if a cloud hanging directly over buildings to which they are annexed, suddenly receives a charge by explosion from another cloud at a distance, so as to enable it instantly to strike into the earth, these pointed conductors must take the explosion; on account of their greater readiness to admit electricity at a much greater distance than those that are blunted, and in proportion to the difference of that striking distance, do mischief instead of good: and therefore, they add, that such pointed conductors, though they may be sometimes advantageous, are yet at other times prejudicial: and that, as the purpose for which conductors are fixed upon buildings, is not to protect them from one particular sort of clouds only, but if possible from all, it cannot be advisable to use that kind of conductors which, if they diminish danger on the one hand, will increase it on the other. Besides, it is alleged, that if pointed conductors are at<*> tended with any the slightest degree of danger, that danger must be considerably augmented by carrying them high up into the air, and by fixing them upon every angle of a building, and by making them project in every direction. Such is the reasoning of Dr. Musgrave: see his paper in the Philos. Trans. vol. 68, part 2, art. 36.</p><p>Mr. Wilson too, dissenting from the report of a committee of the Royal Society, appointed to inspect the damage done by lightning to the house of the Board of Ordnance, at Purfleet, in 1777, was led to justify his dissent, and to disparage the use of pointed and elevated conductors, by means of a magnificent apparatus he constructed, with which he might produce effects si<*>ilar to those that had happened in the case referred to the consideration and decision of the committee. With this view he procured a model of the Board-house at Purfleet, resembling it as nearly as possible in every essential appendage, and furnished with conductors of different lengths and terminations. And to construct a substitute for a cloud, he joined together the broad rims of 120 drums, forming together a cylinder of 155 feet in length, and above 16 inches in diameter; and this immense cylinder, of about 600 square feet of coated surface, was connected occasionally with one end of a wire 4800 feet long. As this bulky apparatus, representing the thunder-cloud, could not conveniently be put in motion, he contrived to accomplish the same end by moving the model of the building, with a velocity answering to that of the cloud, which he states, at a moderate computation, to be about 4 or 5 miles an hour. This apparatus was charged by a machine with one glass cylinder, about 10 or 11 feet from its nearest end; and the whole of the apparatus was disposed in the great room of the Pantheon, and applied to use in a variety of experiments. But it is impossible within the limits of this article to do justice to Mr. Wilson's experiments, or to the inferences which he deduces from them. Suffice it just to observe, that most of his experiments, in which the model of the house, which was passed swiftly under the artificial cloud, and having annexed to it either the pointed or<cb/> blunt conductors at the same or different heights, were intended to shew, that pointed conductors are struck at a greater distance, and with a higher elevation, than the blunted ones: and from all his experiments made with pointed and rounded conductors, provided the circumstances be the same in both, he infers, that the rounded ones are much the safer of the two; whether the lightning proceeds from one cloud or from several; that those are still safer which rise little or nothing above the highest part of the building; and that this safety arises from the greatest resistance exerted at the larger surface. See Philos. Trans. for 1778, pa. 232.</p><p>The committee of the Royal Society however, which was composed of nine of the most distinguished electricians in the kingdom, and to whom was referred the consideration of the most effectual method of securing the powder-magazines at Purfleet against the effects of lightning, express their united opinion, that elevated sharp rods, constructed and disposed in the manner which they direct, are preferable to low conductors terminated in rounded ends, knobs, or balls of metal; and that the experiments and reasonings, made and alleged to the contrary by Mr. Wilson, are inconclusive.</p><p>Mr. Nairne also, in order to obviate the objections of Mr. Wilson and others, and to vindicate the preference generally given to high and pointed conductors, constructed a much more simple apparatus than that of Mr. Wilson, with which he made a number of well-designed and well-conducted experiments, which seem to prove the point as far as it is capable of being proved by an artificial electrical apparatus. From these last experiments it appears, that though the point was struck by means of a swift motion of the artificial cloud, yet a small ball of 3 tenths of an inch diameter was struck farther off than the Point, and a larger ball at a much greater distance than either, even with the swiftest motion. Upon the whole, Mr. Nairne seems to be justi<*>ied in preferring elevated pointed conductors; next to them, those that are pointed, though they rise but little above the highest part of a building; and after them, those that are terminated in a ball, and placed even with the highest part of the building. See Philos. Trans. 1778, pa. 823.</p><p>On the other part, Dr. Musgrave, not yet satisfied, gave in another paper, being “Reasons for dissenting from the Report of the Committee appointed to consider of Mr. Wilson's Experiments; including Remarks on some Experiments exhibited by Mr. Nairne;” which is inserted, by mistake, before Mr. Nairne's paper, being at pa. 801 of the same volume.</p><p>And farther, Mr. Wilson has another paper, on the same subject, at pa. 999 of the same vol. of Philos. Trans. for 1778, entitled, “New Experiments upon the Leyden Phial, respecting the termination of conductors;” repeating and asserting his former objections and reasonings.</p><p>In the Philos. Trans. too for 1779, pa. 454, Mr. William Swift has a paper, farther prosecuting this subject; making various experiments with simple and ingenious machinery, with models of houses and clouds, and with various sorts of conductors. From the experiments he infers in general, that “the whole current<pb n="253"/><cb/> of these experiments tends to shew the preference of Points to balls, in order to diminish and draw off the electric matter when excited, or to prevent it from accumulating; and consequently the propriety or even necessity of terminating all conductors with Points, to make them useful to prevent damage to buildings from lightning. Nay the very construction of all electrical machines, in which it is necessary to round all the parts, and to avoid making edges and points which would hinder the matter from being excited, will, J imagine, on reflection, be another corroborating proof of the result of the experiments themselves.”</p><p>There were other communications made to the Royal Society upon the important subject of conductors, some of which were received, and others rejected. Upon the whole, this contest turned out one of the most extraordinary that ever was agitated in the Society; producing the most remarkable disputes, differences, and strange consequences, that ever the Society experienced since it had existence; consequences which manifested themselves in various instances for many years after, and which continue to this very day. All which, with the various secret springs and astonishing intrigues, may probably be given to the public on some other occasion.</p></div2><div2 part="N" n="Point" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Point</hi></head><p>, in Geometry, according to Euclid, is that which has no parts, or is indivisible; being void of all extension, both as to length, breadth, and depth.</p><p>This is what is otherwise called the Mathematical Point, being the intersection of two lines, and is only conceived by the imagination; yet it is in this that all magnitude begins and ends; the extremes of a line being Points; the extremes of a surface, Lines; and the extremes of a solid, Surfaces. And hence some define a Point, the inceptive of magnitude.</p><p><hi rend="italics">Proportion of Mathematical</hi> <hi rend="smallcaps">Points.</hi> It is a popular maxim, that all infinites are equal; yet is the maxim false, whether of quantities infinitely great, or infinitely little. Dr. Halley instances in several infinite quantities which are in a finite proportion to each other; and some that are infinitely greater than others. See I<hi rend="smallcaps">NFINITE</hi> <hi rend="italics">Quantity.</hi></p><p>And the same is shewn by Mr. Robarts, of infinitely small quantities, or mathematical Points. He demonstrates, for instance, that the Points of contact between circles and their tangents, are in the subduplicate ratio of the diameters of the circles; that the Point of contact between a sphere and a plane is insinitely greater than between a circle and a line; and that the Points of contact in spheres of different magnitudes, are to each other as the diameters of the spheres. Philos. Trans. vol. 27, pa. 470.</p><p><hi rend="italics">Conjugate</hi> <hi rend="smallcaps">Point</hi>, is used for that Point into which the conjugate oval, belonging to some kind of curves, vanishes. Maclaurin's Alg. pa. 308.</p><p><hi rend="smallcaps">Point</hi> <hi rend="italics">of Contrary Flexure,</hi> &c. See <hi rend="smallcaps">Inflexion, Retrogradation</hi> or <hi rend="smallcaps">Retrogression</hi>, &c, of curves.</p><p><hi rend="smallcaps">Points</hi> <hi rend="italics">of the Compass,</hi> or <hi rend="italics">Horizon,</hi> &c, in Geography and Navigation, are the Points of division when the whole circle, quite around, is divided into 32 equal parts. These Points are therefore at the distance of the 32d part of the circle, or 11° 15′, from each other; hence 5° 37′ 1/2 is the distance of the half points, and<cb/> 2° 48′ 3/4 is the distance of the quarter Points. See <hi rend="smallcaps">Compass.</hi> The principal of these are the four cardinal Points, east, west, north and south.</p><p>Point is also used for a cape or headland, jutting ont into the sea.——The seamen say two Points of land are one in another, when they are in a right line, the one behind the other.</p></div2><div2 part="N" n="Point" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Point</hi></head><p>, in Optics. As the</p><p><hi rend="smallcaps">Point</hi> <hi rend="italics">of Concourse</hi> or <hi rend="italics">Concurrence,</hi> is that in which converging rays meet; and is usually called focus.</p><p><hi rend="smallcaps">Point</hi> of <hi rend="italics">Dispersion, Incidence, Reflection, Refraction,</hi> and <hi rend="italics">Radiant</hi> <hi rend="smallcaps">Point.</hi> See these several articles.</p></div2><div2 part="N" n="Point" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Point</hi></head><p>, in Perspective, is a term used for various parts or places, with regard to the perspective plane. As, the</p><p><hi rend="smallcaps">Point</hi> <hi rend="italics">of Sight,</hi> or <hi rend="italics">of the eye,</hi> called also the Principal Point, is the Point on a plane where a perpendicular from the eye meets it. See <hi rend="smallcaps">Perspective.</hi></p><p>Some authors, however, by the Point of Sight, or Vision, mean the Point where the eye is actually placed, and where all the rays terminate. See <hi rend="smallcaps">Perspective.</hi></p><p><hi rend="smallcaps">Point</hi> <hi rend="italics">of Distance,</hi> is a Point in a horizontal line, at the same distance from the principal Point as the eye is from the same. See <hi rend="smallcaps">Perspective.</hi></p><p><hi rend="italics">Third</hi> <hi rend="smallcaps">Point</hi>, is a Point taken at discretion in the line of distance, where all the diagonals meet that are drawn from the divisions of the geometrical plane.</p><p><hi rend="italics">Objective</hi> <hi rend="smallcaps">Point</hi>, is a Point on a geometrical plane, whose representation on the perspective plane is required.</p><p><hi rend="italics">Accidental Point,</hi> and <hi rend="italics">Visual</hi> <hi rend="smallcaps">Point.</hi> See A<hi rend="smallcaps">CCIDENTAL</hi> and <hi rend="smallcaps">Visual.</hi></p><p><hi rend="smallcaps">Point</hi> <hi rend="italics">of View,</hi> with regard to Building, Painting, &c, is a Point at a certain distance from a building, or other object, where the eye has the most advantageous view or prospect of the same. And this Point is usually at a distance equal to the height of the building.</p></div2><div2 part="N" n="Point" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Point</hi></head><p>, in Physics, is the smallest or least sensible object of sight, marked with a pen, or point of a compass, or the like. This is popularly called a Physical Point, and of such does all physical magnitude consist.</p><p><hi rend="smallcaps">Point-Blanc</hi>, <hi rend="italics">Point-Blank,</hi> in Gunnery, denotes the horizontal or level position of a gun, or having its muzzle neither elevated nor depressed. And the Pointblanc range, is the distance the shot goes, before it strikes the level ground, when discharged in the horizontal or Point-blanc direction. Or sometimes this means the distance the ball goes horizontally in a straightlined direction.</p></div2></div1><div1 part="N" n="POINTING" org="uniform" sample="complete" type="entry"><head>POINTING</head><p>, in Artillery and Gunnery, is the laying a piece of ordnance in any proposed direction, either horizontal, or elevated, or depressed, to any angle. This is usually effected by means of the gunner's quadrant, which, being applied to, or in, the muzzle of the piece, shews by a plummet the degree of elevation or depression.</p><div2 part="N" n="Pointing" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Pointing</hi></head><p>, in Navigation, is the marking on the chart in what Point, or place, the vessel is.——This is done by means of the latitude and longitude, after these are known, or found by observation or computation. Thus, draw a line, with a pencil, across the chart according to the latitude; and another across the other way according to the longitude; then the inter-<pb n="254"/><cb/> section of these two lines, is the Point or place on the chart where the ship is; which is then marked black with a pen, and the pencil lines rubbed out. From the Point or place, thus found, the chart readily shews the direct distance and course run, as also yet to run to the intended port, &c.</p></div2></div1><div1 part="N" n="POLAR" org="uniform" sample="complete" type="entry"><head>POLAR</head><p>, something that relates to the poles of the world: as polar virtue, polar tendency.</p><p><hi rend="smallcaps">Polar</hi> <hi rend="italics">Circles,</hi> are two lesser circles of the sphere, or globe, one round each pole, and at the same distance from it as is equal to the sun's greatest declination or the obliquity of the ecliptic; that is, at present 23° 28′.—The space included within each polar circle, is the frigid zone; and to every part of this space, the sun never sets at some time of the year, and never rises at another time; each of these being a longer duration as the place is nearer the pole.</p><p><hi rend="smallcaps">Polar</hi> <hi rend="italics">Dials,</hi> are such as have their planes parallel to some great circle passing through the poles, or to some one of the hour-circles; so that the pole is neither elevated above the plane, nor depressed below it.—This dial, therefore, can have no centre; and consequently its style, substyle, and hour-lines, are parallel.—This will therefore be an horizontal dial to those who live at the equator.</p><p><hi rend="smallcaps">Polar</hi> <hi rend="italics">Projection,</hi> is a representation of the earth, or heavens, projected on the plane of one of the polar circles.</p><p><hi rend="smallcaps">Polar</hi> <hi rend="italics">Regions,</hi> are those parts of the earth which <*>ie near the north and south poles.</p></div1><div1 part="N" n="POLARITY" org="uniform" sample="complete" type="entry"><head>POLARITY</head><p>, the quality of a thing having poles, or pointing to, or respecting some pole: as the magnetic needle, &c.</p><p>By heating an iron bar, and letting it cool again in a vertical position, it acquires a polarity, or magnetic virtue: the lower end becoming the north pole, and the upper end the south pole. But iron bars acquire a polarity by barely continuing a long time in an erect position, even without heating them. Thus, the upright iron bars of some windows, &c, are often sound to have poles: Nay, an iron rod acquires a polarity, by the mere holding it erect; the lower end, in that case, attracting the south end of a magnetic needle; and the upper, the north end. But these poles are mutable, and shift with the situation of the rod.</p><p>Some modern writers, particularly Dr. Higgins, in his Philosophical Essay concerning Light, have maintained the polarity of the parts of matter, or that their simple attractions are more forcible in one direction, or axis of each atom, than in any other.</p></div1><div1 part="N" n="POLES" org="uniform" sample="complete" type="entry"><head>POLES</head><p>, in Astronomy, the extremities of the axis upon which the whole sphere of the world revolves; or the points on the surface of the sphere through which the axis passes. These are on every side at the distance of a quadrant, or 90°, from every point of the equinoctial, and are called, by way of eminence, the poles of the world. That which is visible to us in Europe, or raised above our horizon, is called the Arctic or North Pole; and its opposite one, the Antarctic or South Pole.</p><div2 part="N" n="Poles" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Poles</hi></head><p>, in Geography, are the extremities of the earth's axis; or the points on the surface of the earth through which the axis passes. Of which, that elevated<cb/> above our horizon is called the Arctic or North Pole; and the opposite one, the Antarctic or South Pole.</p><p>In consequence of the situation of the Poles, with the inclination of the earth's axis, and its parallelism during the annual motion of our globe round the sun, the Poles have only one day and one night throughout the year, each being half a year in length. And because of the obliquity with which the rays of the sun fall upon the polar regions, and the great length of the night in the winter season, it is commonly supposed the cold is so intense, that those parts of the globe which lie near the Poles have never been fully explored, though the attempt has been repeatedly made by the most celebrated navigators. And yet Dr. Halley was of opinion, that the solstitial day, at the Pole, is as hot as at the equator when the sun is in the zenith; because all the 24 hours of that day under the Pole the sun-beams are inclined to the horizon in an angle of 23° 28′; whereas at the equator, though the sun becomes vertical, yet he shines no more than 12 hours, being absent the other 12 hours: and besides, that during 3 hours 8 minutes of the 12 hours which he is above the horizon there, he is not so much elevated as at the Pole. Experience however seems to shew that this opinion and reasoning of Dr. Halley are not well founded: for in all the parts of the earth that we know, the middle of summer is always the less hot the farther the place is from the equator, or the nearer it is to the Pole.</p><p>The great object for which navigators have ventured themselves in the frozen seas about the north pole, was to find out a more quick and ready passage to the East Indies. And this has been attempted three several ways: one by coasting along the northern parts of Europe and Asia, called the north-east passage; another, by sailing round the northern part of the American continent, called the north-west passage; and the third, by sailing directly over the pole itself.</p><p>The possibility of succeeding in the north-east was for a long time believed; and in the last century many navigators, particularly the Hollanders, attempted it with great fortitude and perseverance. But it was always found impossible to surmount the obstacles which nature had thrown in the way; and subsequent attempts have in a manner demonstrated the impossibility of ever sailing eastward along the northern coast of Asia. The reason of this impossibility is, that in proportion to the extent of land, the cold is always greater in winter, and vice versa. This is the case even in temperate climates; but much more so in those frozen regions when the sun's influence, even in summer, is but small. Hence, as the continent of Asia extends a vast way from west to east, and has besides the continent of Europe joined to it on the west, it follows, that about the middle part of that tract of land the cold should be greater than any where else. Experience has determined this to be fact; and it now appears, that about the middle of the northern part of Asia, the ice never thaws; neither have even the hardy Russians and Siberians themselves been able to overcome the difficulties they meet with in that part of their voyages.</p><p>With regard to the north-west passage, the same difficulties occur as in the other. According to Captain Cook's voyage, it appears that if there is any strait<pb n="255"/><cb/> which divides the continent of America into two, it must lie in a higher latitude than 70°, and consequently be perpetually frozen up. And therefore if a northwest passage can be found, it must be by sailing round the whole American continent, instead of seeking a passage through it, which some have supposed to exist in the bottom of Baffin's Bay. But the extent of the American continent to the northward is yet unknown; and there is a possibility of its being joined to that part of Asia between the Piasida and Chatanga, which has never yet been circumnavigated. Indeed a rumour has lately gone abroad of some remarkable inlet being observed on the western coast of North America, which it is guessed may possibly lead to some communication with the eastern side, by the lakes, or a passage into Hudson's Bay: but there seems little or no probability of any success this way, in which many fruitless attempts have been made at various times. It remains therefore to consider, whether there is any probability of attaining the wished-for passage by sailing directly north, between the eastern and western continents.</p><p>The late celebrated mathematician, Mr. Maclaurin, was so fully persuaded of the practicability of passing by this way to the South and Indian seas, that he used to say, if his other avocations would permit, he would undertake the voyage of trial, even at his own expence.</p><p>The practicability of this method, which would lead directly to the Pole itself, has also been ingeniously supported by Mr. Daines Barrington, in some tracts published in the years 1775 and 1776, in consequence of the unsuccessful attempt made by captain Phipps in the year 1773, to reach a higher northern latitude than 81°. Mr. Barrington instances a great number of navigators who have reached very high northern latitudes; nay, some who have been at the Pole itself, or gone beyond it. From all which he concludes, that if the voyage be attempted at a proper time of the year, there would not be any great difficulty in reaching the Pole. Those vast pieces of ice which commonly obstruct the navigators, he thinks, proceed from the mouths of the great Asiatic rivers which run northward into the frozen ocean, and are driven eastward and westward by the currents. But, though we should suppose them to come directly from the Pole, still our author thinks that this affords an undeniable proof that the Pole itself is sree from ice; because, when the pieces leave it, and come to the southward, it is impossible that they can at the same time accumulate at the Pole.</p><p><hi rend="italics">The Altitude or Elevation of the</hi> <hi rend="smallcaps">Pole</hi>, is an arch of the meridian intercepted between the Pole and the horizon of any places, and is equal to the latitude of the place.</p><p><hi rend="italics">To observe the Altitude of the</hi> <hi rend="smallcaps">Pole.</hi> With a quadrant, observe both the greatest and least meridian altitude of the Pole star. Then half the sum of the two altitudes, will be the height of the Pole, or the latitude of the place; and half the difference of the same will be the distance of the star from the Pole. But, for accuracy, the observed altitudes should be corrected on account of refraction, before their sum or difference is taken. See <hi rend="smallcaps">Refraction.</hi></p></div2><div2 part="N" n="Pole" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Pole</hi></head><p>, in Spherics, or the Pole of a great circle, is<cb/> a point upon the sphere equally distant srom every part of the circumference of the great circle; or a point 90° distant from the circumference in any part of it.— The zenith and nadir are the Poles of the horizon; and the Poles of the equator are the same with those of the sphere or globe.</p></div2><div2 part="N" n="Poles" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Poles</hi></head><p>, in Magnetics, are two points in a loadstone, corresponding to the Poles of the world; one pointing to the north, and the other to the south.</p><p>If the stone be broken in ever so many pieces, every fragment will still have its two Poles. And if a magnet be bisected by a plane perpendicular to the axis; the two points before joined will become opposite Poles, one in each segment.</p><p>To touch a needle, &c, with a magnet, that part intended for the north end is touched with the south Pole of the magnet; and that intended for the south end, with the north Pole; for the Poles of the needle become contrary to those of the magnet.</p><p>A piece of iron acquires a polarity by only holding it upright; though its Poles are not fixed, but shift, and are inverted as the iron is. Fire destroys all fixed Poles; but it strengthens the mutable ones.</p><p>Dr. Gilbert says, the end of a rod being heated, and left to cool pointing northward, it becomes a sixed north Pole; if southward, a fixed south Pole. When the end is cooled, held downward, it acquires rather more magnetism than if cooled horizontally towards the north. But the best way is to cool it a little inclined to the north. Repeating the operations of heating and cooling does not increase the effect.</p><p>Dr. Power says, if a rod be held northwards, and the north end be hammered in that position, it will become a fixed north Pole; and contrarily if the south end be hammered. The heavier the blows are, cæteris paribus, the stronger will the magnetism be; and a few hard blows have as much effect as a great number. And what is said of hammering, is to be likewise understood of filing, grinding, sawing, &c; nay, a gentle rubbing, when long continued, will produce Poles.</p><p>Old punches and drills have all fixed north poles; because they are almost constantly used downwards. New drills have either mutable Poles, or weak north ones. Drilling with such a one southward horizontally, it is a chance if you produce a fixed south Pole; much less if you drill south downwards; but by drilling south upwards, you always make a fixed south Pole.</p><p>Mr. Ballard says, that in 6 or 7 drills, made in his presence, the bit of each became a north Pole, merely by hardening.</p><p>A weak fixed Pole may degenerate into a mutable one in a day, or even in a few minutes, by holding it in a position contrary to its pole. The loadstone itself will not make a fixed Pole in every piece of iron: if the iron be thick, it is necessary that it have some considerable length.</p><p><hi rend="smallcaps">Pole</hi> <hi rend="italics">of a Glass,</hi> in Optics, is the thickest part of a convex glass, or the thinnest part of a concave one; being the same as what is otherwise called the vertex of the glass; and which, when truly ground, is exactly in the middle of its surface.</p></div2><div2 part="N" n="Pole" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Pole</hi></head><p>, or <hi rend="italics">Rod,</hi> in Surveying, is a lineal measure containing 5 1/2 yards, or 16 1/2 feet.—The square of it is called a square Pole; but more usually a perch, or a rod.<pb n="256"/><cb/></p><p><hi rend="smallcaps">Pole-Star</hi>, is a star of the 2d magnitude near the north Pole, in the end of the tail of Ursa Minor, or the Little Bear. Its mean place in the heavens for the beginning of 1790, was as follows: viz, <table><row role="data"><cell cols="1" rows="1" role="data">Right Ascension</cell><cell cols="1" rows="1" rend="align=right" role="data">12°</cell><cell cols="1" rows="1" rend="align=right" role="data">31′</cell><cell cols="1" rows="1" role="data">47″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Annual variat. in ditto</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"> 4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Declination</cell><cell cols="1" rows="1" rend="align=right" role="data">88</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data"> 8</cell></row><row role="data"><cell cols="1" rows="1" role="data">Annual variat. in ditto</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">19 6/10</cell></row></table></p><p>The nearness of this star to the Pole, on which account it is always above the horizon in these northern latitudes, makes it very useful in Navigation, &c, for determining the meridian line, the elevation of Pole, and consequently the latitude of the place, &c.</p></div2></div1><div1 part="N" n="POLEMOSCOPE" org="uniform" sample="complete" type="entry"><head>POLEMOSCOPE</head><p>, in Optics, an oblique kind of prospective glass, contrived for the seeing of objects that do not lie directly before the eye. It was invented by Hevelius, in 1637, and is the same as <hi rend="smallcaps">Opera</hi> <hi rend="italics">Glass</hi>; which see.</p><p>POLITICAL <hi rend="italics">Arithmetic,</hi> the application of arithmetical calculations to political uses and subjects; such as the public revenues, the number of people, the extent and value of lands, taxes, trade, commerce, or whatever relates to the power, strength, riches, &c, of a nation or commonwealth. Or, as Davenant concisely defines it, the art of reasoning by figures, upon things relating to government.</p><p>The chief authors who have attempted calculations of this kind, are, Sir William Petty, Major Graunt, Dr. Halley, Dr. Davenant, Mr. King, and Dr. Price.</p><p>Sir William Petty, among many other articles, states that, in his time, the people in England were about six millions, and their annual expence about 7l. each; that the rent of the lands was about eight millions, and the interests and profits of the personal estates as much; that the rent of the houses in England was four millions, and the profits of the labour of all the people twenty-six millions yearly; that the corn used in England, at 5s. the bushel for wheat, and 2s. 6d. for barley, amounts to ten millions per annum; that the navy of England required 36,000 men to man it, and the trade and other shipping about 48,000; that the whole people in England, Scotland, and Ireland, together, were about nine millions and a half; and those in France about thirteen millions and a half; and in the whole world about 350 millions; also that the whole cash of England, in current money, was then about six millions sterling. See his Political Arith. p. 74, &c.</p><p>Mr. Davenant gives some good reasons why many of Sir W. Petty's numbers are not to be entirely depended on; and advances others of his own, founded on the observations of Mr. Greg. King. Some of the particulars are, that the land of England is thirty-nine millions of acres; that the number of people in London was about 530,000, and in all England five millions and 2 half, increasing 9000 annually, or about the 600th part; the yearly rent of the lands ten millions, and that of the houses two millions; the produce of all kinds of grain 9 millions. Davenant's Essay upon the probable methods &c, in his works, vol. 6.</p><p>Major Graunt, in his observations on the bills of mortality, computes, that there are 39,000 square miles of land in England, or 25 million acres in England and<cb/> Wales, and 4,600,000 persons, making about 5 acres and a half to each person; that the people of London were 640,000; and states the several numbers of persons living at the different ages.</p><p>Sir William Petty, in his discourse about duplicate proportion, farther states, that it is found by experience, that there are more persons living between 16 and 26 than of any other age; and from thence he infers, that the square roots of every number of men's ages under 16, whose root is 4, shew the proportion of the probability of such persons reaching the age of 70 years: thus, the probability of reaching that age by persons of the <table><row role="data"><cell cols="1" rows="1" role="data">ages of</cell><cell cols="1" rows="1" rend="align=right" role="data">16,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">and 1,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">are as</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" rend="align=right" role="data">1,</cell><cell cols="1" rows="1" role="data">respectively.</cell></row></table> Also that the probabilities of their order of dying, at ages above that, are as the square-roots of the ages: thus, the probabilities of the order of dying first, <table><row role="data"><cell cols="1" rows="1" role="data">of the ages</cell><cell cols="1" rows="1" rend="align=right" role="data">16,</cell><cell cols="1" rows="1" rend="align=right" role="data">25,</cell><cell cols="1" rows="1" rend="align=right" role="data">36,</cell><cell cols="1" rows="1" role="data">&c,</cell></row><row role="data"><cell cols="1" rows="1" role="data">are as the roots</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell><cell cols="1" rows="1" rend="align=right" role="data">5,</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> that is, the odds are 5 to 4 that a person of 25 dies before one of 16, and so on, declining up to 70 years of age.</p><p>Dr. Halley has made a very exact estimation of the degrees of mortality of mankind, from a curious table of the births and burials, at the city of Bre<*>lau, in Silesia; with an attempt to ascertain the price of annuities upon lives, and many other curious particulars. See the Philos. Trans. vol. 17, pa. 596. Another table of this kind is given by Mr. Simpson, for the city of London; and several by Dr. Price, for many different places.</p><p>Mr. Kerseboom, of Holland, has many and curious calculations and tables of the same kind. From his observations on the births of the people in England, it appears, that the number of males born, is in proportion to that of the females, as 18 to 17; and that the inhabitants living in Holland are in the same pro. portion.</p><p>Dr. Brackenridge has given an estimate of the number of people in England, formed both from the number of houses, and also from the quantity of bread consumed. Upon the former principle, he finds the number of houses in England and Wales to be about 900,000; and, allowing 6 persons to each house, the number of people near 5 millions and a half. And upon the latter principle, estimating the quantity of corn consumed at home at 2 millions of quarters, and 3 persons to every quarter of corn, makes the number of people 6 millions. See Philos. Trans. vol. 49, art. 45 and 113.</p><p>Dr. Derham, from a great number of registers of places, finds the proportions of the marriages to the births and burials; and Dr. Price has done the same for still more places; the mediums of all which are, <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Marriages to Births, as</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dr. Derham</cell><cell cols="1" rows="1" rend="align=center" role="data">1 to 4.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dr. Price</cell><cell cols="1" rows="1" rend="align=center" role="data">1 to 3.9</cell></row></table></p><p>See Philos. Trans. Abr. vol. 7, part 4, pa. 46; also Dr. Price's Observations on Reversionary Payments; and the articles of this Dictionary, <hi rend="smallcaps">Expectation</hi> <hi rend="italics">of</hi><pb n="257"/><cb/> <hi rend="italics">Life,</hi> <hi rend="smallcaps">Life</hi>-<hi rend="italics">Annuities,</hi> <hi rend="smallcaps">Mortality, Population</hi>, &c.</p></div1><div1 part="N" n="POLLUX" org="uniform" sample="complete" type="entry"><head>POLLUX</head><p>, in Astronomy, the hind twin, or the posterior part of the constellation Gemini.</p><p><hi rend="smallcaps">Pollux</hi> is also a sixed star of the second magnitude, in the constellation Gemini, or the Twins. See C<hi rend="smallcaps">ASTOR</hi> and <hi rend="italics">Poll<*>x,</hi> also <hi rend="smallcaps">Gemini.</hi></p></div1><div1 part="N" n="POLYACOUSTICS" org="uniform" sample="complete" type="entry"><head>POLYACOUSTICS</head><p>, instruments contrived to multiply sounds, as polyscopes or multiplying glasses do the images of objects.</p><p>POLYEDRON. See <hi rend="smallcaps">Polyhedron.</hi></p></div1><div1 part="N" n="POLYGON" org="uniform" sample="complete" type="entry"><head>POLYGON</head><p>, in Geometry, a figure of many angles; and consequently of many sides also; for every figure has as many sides as angles. If the angles be all equal among themselves, the polygon is said to be a regular one; otherwise, it is irregular. Polygons also take particular names according to the number of their sides; thus a Polygon of 3 sides is called a trigon, 4 sides " a tetragon, 5 sides " a pentagon, 6 sides " a hexagon, &c. and a circle may be considered as a Polygon of an infinite number of small sides, or as the limit of the Polygons.</p><p>Polygons have various properties, as below:</p><p>1. Every Polygon may be divided into as many triangles as it hath sides.</p><p>2. The angles of any Polygon taken together, make twice as many right angles, wanting 4, as the figure hath sides. Thus, if the Polygon has 5 sides; the double of that is 10, from which subtracting 4, leaves 6 right angles, or 540 degrees, which is the sum of the 5 angles of the pentagon. And this property, as well as the former, belongs to both regular and irregular Polygons.</p><p>3. Every regular Polygon may be either inscribed in a circle, or described about it. But not so of the irregular ones, except the triangle, and another particular case as in the following property.</p><p>An equilateral figure inscribed in a circle, is always equiangular.—But an equiangular figure inscribed in a circle is not always equilateral, but only when the number of sides is odd. For if the sides be of an even number, then they may either be all equal; or else half of them may be equal, and the other half equal to each other, but different from the former half, the equals being placed alternately.</p><p>4. Every Polygon, circumscribed about a circle, is equal to a right-angled triangle, of which one leg is the radius of the circle, and the other the perimeter or sum of all the sides of the Polygon. Or the Polygon is equal to half the rectangle under its perimeter and the radius of its inscribed circle, or the perpendicular from its centre upon one side of the Polygon.</p><p>Hence, the area of a circle being less than that of its circumscribing Polygon, and greater than that of its inscribed Polygon, the circle is the limit of the inscribed and circumscribed Polygons: in like manner the circumference of the circle is the limit between the perimeters of the said Polygons: consequently the circle is equal to a right-angled triangle, having one leg<cb/> equal to the radius, and the other leg equal to the circumference; and therefore its area is found by multiplying half the circumference by half the diameter. In like manner, the area of any Polygon is found by multiplying half its perimeter by the perpendicular demitted from the centre upon one side.</p><p>5. The following Table exhibits the most remarkable particulars in all the Polygons, up to the dodecagon of 12 sides; viz, the angle at the centre AOB, the angle of the Polygon C or CAB or double of OAB, and the area of the Polygon when each side AB is 1. (See the following figure.) <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">No. of sides.</cell><cell cols="1" rows="1" rend="align=center" role="data">Name of Polygon.</cell><cell cols="1" rows="1" rend="align=center" role="data">Ang. O at cent.</cell><cell cols="1" rows="1" rend="align=center" role="data">Ang. C. of Polyg.</cell><cell cols="1" rows="1" rend="align=center" role="data">Area.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">Trigon</cell><cell cols="1" rows="1" rend="align=right" role="data">120°</cell><cell cols="1" rows="1" rend="align=right" role="data">60°</cell><cell cols="1" rows="1" rend="align=right" role="data">0.4330127</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">Tetragon</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" rend="align=right" role="data">1.0000000</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">Pentagon</cell><cell cols="1" rows="1" rend="align=right" role="data">72</cell><cell cols="1" rows="1" rend="align=right" role="data">108</cell><cell cols="1" rows="1" rend="align=right" role="data">1.7204774</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">Hexagon</cell><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell><cell cols="1" rows="1" rend="align=right" role="data">2.5980762</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">Heptagon</cell><cell cols="1" rows="1" rend="align=right" role="data">51 3/7</cell><cell cols="1" rows="1" rend="align=right" role="data">128 4/7</cell><cell cols="1" rows="1" rend="align=right" role="data">3.6339124</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">Octagon</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">135</cell><cell cols="1" rows="1" rend="align=right" role="data">4.8284271</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">Nonagon</cell><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">140</cell><cell cols="1" rows="1" rend="align=right" role="data">6.1818242</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">Decagon</cell><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="align=right" role="data">144</cell><cell cols="1" rows="1" rend="align=right" role="data">7.6942088</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">Undecagon</cell><cell cols="1" rows="1" rend="align=right" role="data">32 8/11</cell><cell cols="1" rows="1" rend="align=right" role="data">147 3/11</cell><cell cols="1" rows="1" rend="align=right" role="data">9.3656399</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">Dodecagon</cell><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=right" role="data">150</cell><cell cols="1" rows="1" rend="align=right" role="data">11.1961524</cell></row></table></p><p>By means of the numbers in this Table, any Polygons may be constructed, or their areas found: thus, (1st) <hi rend="italics">To inscribe a Polygon in a given Circle.</hi> At the centre make the angle O equal to the angle at the centre of the proposed Polygon, found in the 3d colum<*> of the Table, the legs cutting the circle in A and B; and join A and B which will be one side of the Polygon. Then take AB between the compasses, and apply it continually round the circumference, to complete the Polygon.</p><p>(2d) <hi rend="italics">Upon the given Line</hi> AB <hi rend="italics">to describe a regular Polygon.</hi> From the extremities draw the two lines AO and BO, making the angles A and B each equal to half the angle of the Polygon, found in the 4th column of the Table, and their intersection O will be the centre of the circumscribed circle: then apply AB continually round the circumference as before.</p><p>(3d) <hi rend="italics">To describe a Polygon about a given Circle.</hi>— At the centre O make the angle <figure/> of the centre as in the 1st art. its legs cutting the circle in <hi rend="italics">a</hi> and <hi rend="italics">b:</hi> join <hi rend="italics">ab,</hi> and parallel to it draw AB to touch the circle: and meeting O<hi rend="italics">a</hi> and O<hi rend="italics">b</hi> produced in A and B: with the radius OA, or OB, describe a circle, and around its circumference apply continual AB, which will complete the P<*>lygon as before.</p><p>(4th) <hi rend="italics">To find the Area of any regular Polygon.</hi>— Multiply the square of its side by the tabular area, found on the line of its name in the last column of the Table, and the product will be the area. Thus, to<pb n="258"/><cb/> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">0.4330127</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">400</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">173.2050800</cell></row></table> find the area of the trigon, or equilateral triangle, whose side is 20. The square of 20 being 400, multiply the tabular area .4330127 by 400, as in the margin, and the product 173.20508 will be the area.</p><p>6. There are several curious algebraical theorems for inscribing Polygons in circles, or finding the chord of any proposed part of the circumference, which is the same as angular sections. These kinds of sections, or parts and multiples of arcs, were sirst treated of by Vieta, as shewn in the Introduction to my Log. pa. 9, and since pursued by several other mathematicians, in whose works they are usually to be found. Many other particulars relating to Polygons may also be seen in my Mensuration, 2d edit. pa. 20, 21, 22, 23, 113, &c.</p><div2 part="N" n="Polygon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Polygon</hi></head><p>, in Fortisication, denotes the figure or perimeter of a fortress, or fortified place. This is either Exterior or Interior.</p><p><hi rend="italics">Exterior</hi> <hi rend="smallcaps">Polygon</hi> is the perimeter or figure formed by lines connecting the points of the bastions to one another, quite round the work. And</p><p><hi rend="italics">Interior</hi> <hi rend="smallcaps">Polygon</hi>, is the perimeter or figure formed by lines connecting the centres of the bastions, quite around.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Polygons</hi>, is a line on some sectors, containing the homologous sides of the first nine regular Polygons inscribed in the same circle; viz, from an equilateral triangle to a dodecagon.</p><p>POLYGONAL <hi rend="italics">Numbers,</hi> are the continual or successive sums of a rank of any arithmeticals beginning at 1, and regularly increafing; and therefore are the first order of figurate numbers; they are called Polygonals, because the number of points in them may be arranged in the form of the several Polygonal figures in geometry, as is illustrated under the article <hi rend="smallcaps">Figurate</hi> <hi rend="italics">Numbers,</hi> which see.</p><p>The several sorts of Polygonal numbers, viz, the triangles, squares, pentagons, hexagons, &c, are formed from the addition of the terms of the arithmetical series, having respectively their common difference 1, 2, 3, 4, &c; viz, if the common difference of the arithmeticals be 1, the sums of their terms will form the triangles; if 2, the squares; if 3, the pentagons; if 4, the hexagons, &c. Thus: <table><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Arith. Progres.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">2 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">3 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">4 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">5 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">6 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">7 .</cell></row><row role="data"><cell cols="1" rows="1" role="data">Triang. Nos.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">3 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">6 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">10 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">15 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">21 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">28 .</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Arith. Progres.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">3 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">5 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">7 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">9 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">11 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">13 .</cell></row><row role="data"><cell cols="1" rows="1" role="data">Square Numbers</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">4 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">9 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">16 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">25 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">36 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">49 .</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Arith. Progres.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">4 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">7 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">10 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">13 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">16 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">19 .</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pentagonal Nos.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">5 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">12 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">22 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">35 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">51 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">70 .</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Arith. Progres.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">5 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">9 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">13 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">17 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">21 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">25 .</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hexagonal Nos.</cell><cell cols="1" rows="1" role="data">1 ,</cell><cell cols="1" rows="1" role="data">6 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">15 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">28 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">45 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">66 ,</cell><cell cols="1" rows="1" rend="align=right" role="data">91 .</cell></row></table></p><p>The <hi rend="italics">Side</hi> of a Polygonal number is the number of points in each side of the Polygonal figure when the points in the number are ranged in that form. And this is also the same as the number of terms of the arithmeticals that are added together in composing the Po-<cb/> lygonal number; or, in short, it is the number of the term from the beginning. So, in the 2d or squares, 1 2 3 4 <figure/> 1 4 9 16 the side of the first (1) is 1, that of the second (4) is 2, that of the third (9) is 3, that of the fourth (16) is 4, and so on. And</p><p>The <hi rend="italics">Angles,</hi> or Numbers of Angles, are the same as those of the figure from which the number takes its name. So the angles of the triangular numbers are 3, of the square ones 4, of the pentagonals 5, of the hexagonals 6, and so on. Hence, the angles are 2 more than the common difference of the arithmetical series from which any rank of Polygonals is formed: so the arithmetical series has for its common difference the number 1 or 2 or 3 &c as follows, viz, 1 in the triangles, 2 in the squares, 3 in the pentagons, &c; and, in general, if <hi rend="italics">a</hi> be the number of angles in the Polygon, then <hi rend="italics">a</hi> - 2 is = <hi rend="italics">d</hi> the common difference of the arithmetical series, or the number of angles.</p><p><hi rend="smallcaps">Prob.</hi> 1. <hi rend="italics">To find any Polygonal Number proposed;</hi> having given its side <hi rend="italics">n</hi> and angles <hi rend="italics">a.</hi> The Polygonal number being evidently the sum of the arithmetical progression whose number of terms is <hi rend="italics">n</hi> and common difference <hi rend="italics">a</hi> - 2, and the sum of an arithmetical progression being equal to half the product of the extremes by the number of terms, the extremes being 1 and ; therefore that number, or this sum, will be or , where <hi rend="italics">d</hi> is the common difference of the arithmeticals that form the Polygonal number, and is always 2 less than the number of angles <hi rend="italics">a.</hi></p><p>Hence, for the several sorts of Polygons, any particular number whose side is <hi rend="italics">n,</hi> will be found from either of these two formulæ, by using for <hi rend="italics">d</hi> its values 1, 2, 3, 4, &c; which gives these following formulæ for the Polygonal number in each sort, viz, the <table><row role="data"><cell cols="1" rows="1" role="data">Triangular</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Square</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pentagonal</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hexagonal</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Heptagonal</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">&c.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p><hi rend="smallcaps">Prob.</hi> 2. <hi rend="italics">To find the Sum of any Number of Polygonal Numbers of any order.</hi>—Let the angles of the Polygon<pb n="259"/><cb/> be <hi rend="italics">a,</hi> or the common difference of the arithmeticals that form the Polygonals, <hi rend="italics">d;</hi> and <hi rend="italics">n</hi> the number of terms in the Polygonal series, whose sum is sought: then is or the sum of the <hi rend="italics">n</hi> terms sought.</p><p>Hence, substituting successively the numbers 1, 2, 3, 4, &c, for <hi rend="italics">d,</hi> there is obtained the following particular cases, or formulæ, for the sums of <hi rend="italics">n</hi> terms of the several ranks of Polygonal numbers, viz, the sum of the <table><row role="data"><cell cols="1" rows="1" role="data">Triangulars</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Squares</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pentagonals</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hexagonals</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data">Heptagonals</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">&c</cell><cell cols="1" rows="1" role="data"/></row></table></p></div2></div1><div1 part="N" n="POLYGRAM" org="uniform" sample="complete" type="entry"><head>POLYGRAM</head><p>, in Geometry, a figure consisting of many lines.</p></div1><div1 part="N" n="POLYHEDRON" org="uniform" sample="complete" type="entry"><head>POLYHEDRON</head><p>, or <hi rend="smallcaps">Polyedron</hi>, a body or solid contained by many rectilinear planes or sides.</p><p>When the sides of the Polyhedron are regular polygons, all similar and equal, then the Polyhedron becomes a regular body, and may be inscribed in a sphere; that is, a sphere may be described about it, so that its surface shall touch all the angles or corners of the solid. There are but five of these regular bodies, viz, the tetraedron, the hexaedron or cube, the octaedron, the dodecaedron, and the icosaedron. See <hi rend="smallcaps">Regular</hi> <hi rend="italics">Body,</hi> and each of those five bodies severally.</p><p><hi rend="italics">Gnomonical</hi> <hi rend="smallcaps">Polyhedron</hi>, is a stone with several faces, on which are projected various kinds of dials. Of this sort, that in the Privy-garden, London, now gone to ruin, was esteemed the sinest in the world.</p><div2 part="N" n="Polyhedron" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Polyhedron</hi></head><p>, in Optics. See <hi rend="smallcaps">Polyscope.</hi></p><p>POLYHEDROUS <hi rend="italics">Figure,</hi> in Geometry, a solid contained under many sides or planes. See P<hi rend="smallcaps">OLYHEDRON.</hi></p></div2></div1><div1 part="N" n="POLYNOMIAL" org="uniform" sample="complete" type="entry"><head>POLYNOMIAL</head><p>, in Algebra, a quantity of many <*>ames or terms, and is otherwise called a Multinomial. As , &c. See <hi rend="smallcaps">Multinomial.</hi></p></div1><div1 part="N" n="POLYOPTRUM" org="uniform" sample="complete" type="entry"><head>POLYOPTRUM</head><p>, in Optics, a glass through which objects appear multiplied, but diminished. The Polyoptrum differs both in structure and phenomena from the common multiplying glasses called Polyhedra or Polyscopes.</p><p><hi rend="italics">To construct the Polyoptrum.</hi>—From a glass AB, <figure/> plane on both sides, and about 3 fingers thick, cut out spherical segments, scarce a 5th part of a digit in diameter.—If then the glass be removed to such a distance from the eye, that you can take in all the cavities at one view, you will see the same object, as if<cb/> through so many several concave glasses as there are cavities, and all exceeding small—Fit this, as an objectglass, in a tube ABCD, whose aperture AB is equal to the diameter of the glass, and the other CD is equal to that of an eye-glass, as for instance about a finger's breadth. The length of the tube AC is to be accommodated to the object-glass and eye-glass, by trial. In CD fit a convex eye-glass, or in its stead a meniscus having the diftance of its principal focus a little larger than the length of the tube; so that the point from which the rays diverge after refraction in the objectglass, may be in the focus. If then the eye be applied near the eye-glass, a single object will be seen repeated as often as there are cavities in the object-glass, but still diminished.</p></div1><div1 part="N" n="POLYSCOPE" org="uniform" sample="complete" type="entry"><head>POLYSCOPE</head><p>, or <hi rend="smallcaps">Polyhedron</hi>, in Optics, is a multiplying glass, being a glass or lens which represents a single object to the eye as if it were many. It consists of several plane surfaces, disposed into a convex form, through every one of which the object is seen.</p><p><hi rend="italics">Phenomena of the Polyscope.</hi>—1. If several rays, as EF, AB, CD, fall parallel on the surface of a Poly- <figure/> scope, they will continue parallel after refraction. If then the Polyscope be supposed regular, LH, HI, IM will be as tangents cutting the spherical convex lens in F, B, and D; and consequently, rays falling on the points of contact, intersect the axis. Wherefore, since the rest are parallel to these, they will also mutually intersect each other in G.</p><p>Hence, if the eye be placed where parallel rays decussate, rays of the same object will be propagated to it still parallel from the several sides of the glass. Wherefore, since the crystalline humour, by its convexity, unites parallel rays, the rays will be united in as many different points of the retina, <hi rend="italics">a, b, c,</hi> as the glass has sides. Consequently the eye, through a Polyscope, sees the object repeated as many times as there are sides. And hence, since rays coming from very remote ohjects are parallel, a remote object is seen through a Polyscope as often repeated as that has sides.</p><p>2. If rays AB, AC, AD, coming from a radiant <figure/> point A, fall on several sides of a regular Polyscope;<pb n="260"/><cb/> after refraction they will decussate in G, and proceed on a little diverging.</p><p>Hence, if the eye be placed where the rays decussate after coming from the several planes, the rays will be propagated to it from the several planes a little diverging, or as if they proceeded from different points. But since the crystalline humour, by its convexity, collects rays from several points into the same point; the rays will be united in as many different points of the retina, <hi rend="italics">a, b, c,</hi> as the glass has sides; and consequently the eye, being placed in the focus G, will see even a near object through the Polyscope as often repeated as that has sides.</p><p>Thus may the images of objects be multiplied in a camera obscura, by placing a Polyscope at its aperture, and adding a convex lens at a due distance from it. And it makes a very pleafant appearance, if a prism be applied so that the coloured rays of the sun refracted from it be received on the Polyscope: for by this means they will be thrown on a paper or wall near at hand in little lucid specks, much exceeding the brightness of any precious stone; and in the focus of the Polyscope, where the rays decussate (for in this experiment they are received on the convex side) will be a star of surprising lustre.</p><p>Farther, if images be painted in water-colours in the areolæ or little squares of a Polyscope, and the glass be applied to the aperture of a camera obscura; the sun's rays, passing through it, will carry with them the images, and project them on the opposite wall.—This artisice bears a resemblance to that other, by which an image on paper is projected on the camera; viz, by wetting the paper with oil, and straining it tight in a frame; then applying it to the aperture of the camera obscura, so that the rays of a candle may pass through it upon the Polyscope.</p><p><hi rend="italics">To make an Anamorphosis, or Deformed Image, which shall appear regular and beautiful through a Polyscope, or Multiplying Glass.</hi>—At one end of a horizontal table erect another perpendicularly, upon which a figure may be designed; and on the other end erect another, to ferve as a fulcrum or support, moveable on the horizontal one. To the fulcrum apply a plano-convex Polyfcope, consisting, for example, of 24 plane triangles; and let the Polyscope be fitted in a draw-tube, of which that end towards the eye may have only a very small aperture, and a little farther off than the focus. Remove the fulcrum from the other perpendicular table, till it be out of the distance of the focus; and the more so, as the image is to be greater. Before the little aperture place a lamp; and trace the luminous areolæ projected from the sides of the Polyscope, with a black lead pencil, on the vertical plane, or a paper applied upon it.</p><p>In the several areolæ, design the different parts of an image, in such a manner as that, when joined together, they may make one whole, looking every now and then through the tube to guide and correct the colours, and to see that the several parts match and fit well together. As to the intermediate space, it may be filled up with any figures or designs at pleasure, contriving it so, as that to the naked eye the whole may exhibit some appearance very different from that intended to appear through the Polyscope.<cb/></p><p>The eye, now looking through the small aperture of the tube, will see the several parts and members dispersed among the areolæ to exhibit one continued image, all the intermediate parts disappearing. See A<hi rend="smallcaps">NAMORPHOSIS.</hi></p></div1><div1 part="N" n="POLYSPASTON" org="uniform" sample="complete" type="entry"><head>POLYSPASTON</head><p>, in Mechanics, a machine so called by Vitruvius, consisting of an assemblage of several pullies, used for raising heavy weights.</p></div1><div1 part="N" n="PONTON" org="uniform" sample="complete" type="entry"><head>PONTON</head><p>, or <hi rend="smallcaps">Pontoon</hi>, a kind of flat-bottomed boat, whose carcass of wood is lined within and without with tin. Some nations line them on the outside only, and that with plates of copper, which is better. Our Pontoons are 21 feet long, nearly 5 feet broad, and 2 feet 1 1/2 inch deep within. They are carried along with an army upon carriages, to make temporary bridges, called Pontoon-bridges. See the next article.</p><p><hi rend="smallcaps">Pontoon</hi>-<hi rend="italics">Bridge,</hi> a bridge made of Pontoons slipped into the water, and moored by anchors and o<*>herwife fastened together by ropes, at small distances from one another; then covered by beams of timber passing over them; upon which is laid a flooring of boards. By this means, whole armies of infantry, cavalry, and artillery are quickly passed over rivers.—For want of Pontoons, &c, bridges are sometimes formed of empty powder casks, or powder barrels, which support the beams and flooring. Julius Cæsar and Aulus Gellius both mention Pontoons (pontones); but theirs were no more than a kind of square flat vessels, proper for carrying over horse &c.</p><p>PONT-<hi rend="smallcaps">Volant</hi>, or <hi rend="italics">Flying-bridge,</hi> is a kind of bridge used in sieges, for surprising a post or outwork that has but narrow moats. It is made of two small bridges laid over each other, and so contrived that, by means of cords and pullies placed along the sides of the under bridge, the upper may be pushed forwards, till it join the place where it is designed to be sixed. The whole length of both ought not to be above 5 fathoms, lest it should break with the weight of the men.</p></div1><div1 part="N" n="PORES" org="uniform" sample="complete" type="entry"><head>PORES</head><p>, are the small interstices between the particles of matter which compose bodie<*>; and are either empty, or filled with some insensible medium.</p><p>Condensation and rarefaction are only performed by closing and opening the Pores. Also the transparency of bodies is supposed to arise from their Pores being directly opposite to one another. And the matter of insensible perspiration is conveyed through the Pores of the cutis.</p><p>Mr. Boyle has a particular essay on the porosity of bodies, in which he proves that the most solid bodies have some kind of Pores: and indeed if they had not, all bodies would be alike specisically heavy.</p><p>Sir Isaac Newton shews, that bodies are much more rare and porous than is commonly believed. Water, for example, is 19 times lighter and rarer than gold; and gold itself is so rare, as very readily, and without the least opposition, to transmit magnetic effluvia, and easily to admit even quicksilver into its pores, and to let water pass through it: for a concave sphere of gold hath, when filled with water, and soldered up, upon pressing it with a great force, suffered the water to squeeze through it, and stand all over its outside, in multitudes of small drops like dew, without bursting or cracking the gold. Whence it may be concluded,<pb n="261"/><cb/> that gold has more pores than solid parts, and consequently that water has above 40 times more Pores than parts. Hence it is that the magnetic effluvia passes freely through all cold bodies that are not magnetic; and that the rays of light pass, in right lines, to the greatest distances through pellucid bodies.</p></div1><div1 part="N" n="PORIME" org="uniform" sample="complete" type="entry"><head>PORIME</head><p>, <hi rend="italics">Porima,</hi> in Geometry, a kind of easy lemma, or theorem so easily demonstrated, that it is almost self-evident: such, for example, as that a chord is wholly within the circle.—Porime stands opposed to Aporime, which denotes a proposition so difficult, as to be almost impossible to be demonstrated, or effected. Such as the quadrature of the circle, &c.</p></div1><div1 part="N" n="PORISM" org="uniform" sample="complete" type="entry"><head>PORISM</head><p>, <hi rend="italics">Porisma,</hi> in Geometry, has by some been desined a general theorem, or canon, deduced from a geometrical locus, and serving for the solution of other general and difficult problems. Proclus derives the word from the Greek <foreign xml:lang="greek">porizw</foreign>, <hi rend="italics">I establish,</hi> and conclude from something already done and demonstrated: and accordingly he defines Porisma a theorem drawn occasionally from some other theorem already proved: in which sense it agrees with what is otherwise called corollary.</p><p>Pappus says, a Porism is that in which something was proposed to be investigated.</p><p>Others derive it from <foreign xml:lang="greek">w=o/ros</foreign>, <hi rend="italics">a passage,</hi> and make it of the nature of a lemma, or a proposition necessary for passing to another more important one.</p><p>But Dr. Simson, rejecting the crroneous accounts that have been given of a Porism, desines it a proposition, either in the form of a problem or a theorem, in which it is proposed either to investigate, or demonstrate.</p><p>Euclid wrote three books of Porisms, being a curious collection of various things relating to the analysis of the more difficult and general problems. Those books however are lost; and nothing remains in the works of the ancient geometricians concerning this subject, besides what Pappus has preserved, in a very imperfect and obscure state, in his Mathematical Collections, viz, in the introduction to the 7th book.</p><p>Several attempts have been made to restore these writings in some degree, besides that which Pappus has left upon the subject. Thus, Fermat has given a few propositions of this kind; which are to be found in the collection of his works, in folio, 1679, pa. 116. The like was done by Bullialdus, in his Exercitationes Geometricæ, 4to, 1657. Dr. Robert Simson gave also a specimen, in two propositions, in the Philos. Trans. vol. 32, pa. 330; and besides left behind him a considerable treatise on the subject of Porisms, which has been printed in an edition of his works, at the expence of the earl of Stanhope, in 4to, 1776.</p><p>The whole three books of Euclid were also restored by that ingenious mathematician Albert Girard, as appears by two notices that he gave, first in his Trigonometry, printed in French, at the Hague, in 1629, and also in his edition of the works of Stevinus, printed at Leyden in 1634, pa. 459; but whether his intention of publishing them was ever carried into execution, I have not been able to learn.</p><p>A learned paper on the subject of Porisms, by the very ingenious Professor Playfair, has just been inserted in the 3d volume of the <hi rend="italics">Transactions</hi> of the Royal So-<cb/> ciety of Edinburgh. As this paper contains a number of curious observations on the geometry of the Ancients in general, as well as forms a complete treatise as it were on Porism in particular, a pretty considerable abstract of it cannot but be deemed in this place very useful and important.</p><p>“The restoration of the ancient books of geometry (says the learned professor) would have been impossible, without the coincidence of two circumstances, of which, though the one is purely accidental, the other is essentially connected with the nature of the mathematical sciences. The first of these circumstances is the preservation of a short abstract of those books, drawn up by Pappus Alexandrinus, together with a series of such lemmata, as he judged useful to facilitate the study of them. The second is, the necessary connection that takes place among the objects of every mathematical work, which, by excluding whatever is arbitrary, makes it possible to determine the whole course of an investigation, when only a few points in it are known. From the union of these circumstances, mathematics has enjoyed an advantage of which no other branch of knowledge can partake; and while the critic or the historian has only been able to lament the fate of those books of Livy and Tacitus which are lost, the geometer has had the high satisfaction to behold the works of Euclid and Apollonius reviving under his hands.</p><p>“The first restorers of the ancient books were not, however, aware of the full extent of the work which they had undertaken. They thought it sufficient to demonstrate the propositions, which they knew from Pappus, to have been contained in those books; but they did not follow the antient method of investigation, and few of them appear to have had any idea of the elegant and simple analysis by which these propositions were originally discovered, and by which the Greek Geometry was peculiarly distinguished.</p><p>“Among these few, Fermat and Halley are to be particularly remarked. The former, one of the greatest mathematicians of the last age, and a man in all respects of superior abilities, had very just notions of the geometrical analysis, and appears often abundantly skilful in the use of it; yet in his restoration of the Loci Plani, it is remarkable, that in the most difficult propositions, he lays aside the analytical method, and contents himself with giving the synthetical demonstration. The latter, among the great number and variety of his literary occupations, found time for a most attentive study of the ancient mathematicians, and was an instance of, what experience shews to be much rarer than might be expected, a man equally well acquainted with the ancient and the modern geometry, and equally disposed to do justice to the merit of both. He restored the books of Apollonius, on the problem De Sectione Spatii, according to the true principles of the ancient analysis.</p><p>“These books, however, are but short, so that the first restoration of considerable extent that can be reckoned complete, is that of the Loci Plani by Dr. Simson, published in 1749, which, if it differs at all from the work it is intended to replace, seems to do so only by its greater excellence. This much at least is certain, that the method of the ancient geometers does not appear to greater advantage in the most entire of their<pb n="262"/><cb/> writings, than in the restoration above mentioned; and that Dr. Simson has often sacrificed the elegance to which his own analysis would have led, in order to tread more exactly in what the lemmata of Pappus pointed out to him, as the track which Apollonius had pursued.</p><p>“There was another subject, that of Porisms, the most intricate and enigmatical of any thing in the ancient geometry, which was still reserved to exercise the genius of Dr. Simson, and to call forth that enthusiastic admiration of antiquity, and that unwearied perseverance in research, for which he was so peculiarly distinguished. A treatise in three books, which Euclid had composed on Porisms, was lost, and all that remained concerning them was an abstract of that treatise, inserted by Pappus Alexandrinus in his Mathematical Collections, in which, had it been entire, the geometers of later times would doubtless have found wherewithal to console themselves for the loss of the original work. But unfortunately it has suffered so much from the injuries of time, that all which we can immediately learn from it is, that the Ancients put a high value on the propositions which they called Porisms, and regarded them as a very important part of their analysis. The Porisms of Euclid are said to be, “Collectio artificio“sissima multarum rerum quæ spectant ad analysin dif“siciliorum et generalium problematum.” The curiosity, however, which is excited by this encomium is quickly disappointed; for when Pappus proceeds to explain what a Porism is, he lays down two desinitions of it, one of which is rejected by him as imperfect, while the other, which is stated as correct, is too vague and indefinite to convey any useful information.</p><p>“These defects might nevertheless have been supplied, if the enumeration which he next gives of Euclid's Propositions had been entire; but on account of the extreme brevity of his enunciations, and their reference to a diagram which is lost, and for the constructing of which no directions are given, they are all, except one, perfectly unintelligible. For these reasons, the fragment in question is so obscure, that even to the learning and penetration of Dr. Halley, it seemed impossible that it could ever be explained; and he therefore concluded, after giving the Greek text with all possible correctness, and adding the Latin translation, “Hactenus Porismatum descriptio nec mihi intellecta, “nec lectori profutura. Neque aliter sieri potuit, tam “ob defectum schematis cujus sit mentio, quam ob “omissa quædam et transposita, vel aliter vitiata in pro“positionis generalis expositione, unde quid sibi velit “Pappus haud mihi datum est conjicere. His adde “dictionis modum nimis contractum, ac in re difficili, “qualis hæc est, minime usurpandum.”</p><p>“It is true, however, that before this time, Fermat had attempted to explain the nature of Porisms, and not altogether without success. Guiding his conjectures by the desinition which Pappus censures as imperfect, because it defined Porisms only “ab accidente,” viz. “Porisma est quod desicit hypothesi a Theoremate Lo“cali,” he formed to himself a tolerably just notion of these propositions, and illustrated his general description by examples that are in effect Porisms. But he was able to proceed no farther; and he neither proved, that his notion of a Porism was the same with Euclid's, nor<cb/> attempted to restore, or explain any one of Euclid'<*> propositions; much less did he suppose, that they were to be investigated by an analysis peculiar to themselves. And so imperfect indeed was this attempt, that the complete restoration of the Porisms was necessary to prove, that Fermat had even approximated to the truth.</p><p>“All this did not, however, deter Dr. Simson from turning his thoughts to the same subject, which he appears to have done very early, and long before the publication of the Loci Plani in 1749.</p><p>“The account he gives of his progress, and of the obstacles he enconntered, will be always interesting to mathematicians. “Postquam vero apud Pappum le“geram, Porismata Euclidis collectionem fuisse artifi“ciosissimam multarum rerum, quæ spectant ad analysin “difficiliorum et generalium problematum, magno “desiderio tenebar, aliquid de iis cognoscendi; quare “sæpius et multis variisque viis tum Pappi propositio“nem generalem, mancam et imperfectam, tum pri“mum lib. i.</p><p>“Porisma, quod solum ex omnibus in tribus libris “integrum adhuc manet, intelligere et restituere “conabar; frustra tamen, nihil enim proficiebam. “Cumque cogitationes de hac re multum mihi tempo“ris consumpserint, atque molestæ admodum evaserint, “firmiter animum induxi hæc nunquam in posterum “investigare; præsertim cum optimus geometra Hal“leius spem omnem de iis intelligendis abjecisset. Un“de quoties menti occurrebant, toties eas arcebam. “Postea tamen accidit, ut improvidum et propositi im“memorem invaserint, meque detinuerint donec tan“dem lux quædam effulserit, quæ spem mihi faciebat “inveniendi saltem Pappi propositionem generalem, “quam quidem multa investigatione tandem restitui. “Hæc autem paulo post una cum Porismate primo “lib. i. impressa est inter Transactiones Phil. anni 1723, “num. 177.”</p><p>“The propositions mentioned, as inserted in the Philosophical Transactions for 1723, are all that Dr. Simson published on the subject of Porisms during his life, though he continued his investigations concerning them, and succeeded in restoring a great number of Euclid's propositions, together with their analysis. The propositions thus restored form a part of that valuable edition of the posthumous works of this geometer which the mathematical world owes to the munificence of the late earl Stanhope.</p><p>“The subject of Porisms is not, however, exhausted, nor is it yet placed in so clear a light as to need no farther illustration. It yet remains to enquire into the probable origin of these propositions, that is to say, into the steps by which the ancient geometers appear to have been led to the discovery of them.</p><p>“It remains also to point out the relations in which they stand to the other classes of geometrical truths; to consider the species of analysis, whether geometrical or algebraical, that belongs to them; and, if possible, to assign the reason why they have so long escaped the notice of modern mathematicians. It is to these points that the following observations are chiefly directed.</p><p>“I begin with describing the steps that appear to have led the ancient geometers to the discovery of Porisms; and must here supply the want of express testi-<pb n="263"/><cb/> mony by probable reasonings, such as are necessary, whenever we would trace remote discoveries to their sources, and which have more weight in mathematics than in any other of the sciences.</p><p>“It cannot be doubted, that it has been the solution of problems, which, in all states of the mathematical sciences, has led to the discovery of most geometrical truths. The first mathematical enquiries, in particular, must have occurred in the form of questions, where something was given, and something required to be done; and by the reasonings necessary to answer these questions, or to discover the relation between the things that were given, and those that were to be found, many truths were suggested, which came afterwards to be the subjects of separate demonstration. The number of these was the greater, that the ancient geometers always undertook the solution of problems with a scrupulous and minute attention, which would scarcely suffer any of the collateral truths to escape their observation. We know from the examples which they have left us, that they never considered a problem as resolved, till they had distinguished all its varieties, and evolved separately every different case that could occur, carefully remarking whatever change might arise in the construction, from any change that was supposed to take place among the magnitudes which were given.</p><p>“Now as this cautious method of proceeding was not better calculated to avoid error, than to lay hold of every truth that was connected with the main object of enquiry, these geometers soon observed, that there were many problems which, in certain circumstances, would admit of no solution whatever, and that the general construction by which they were resolved would fail, in consequence of a particular relation being supposed among the quantities which were given.</p><p>“Such problems were then said to become impossible; and it was readily perceived, that this always happened, when one of the conditions prescribed was inconsistent with the rest, so that the supposition of their being united in the same subject, involved a contradiction. Thus, when it was required to divide a given line, so that the rectangle under its segment, should be equal to a given space, it was evident, that if this space was greater than the square of half the given line, the thing required could not possibly be done; the two conditions, the one defining the magnitude of the line, and the other that of the rectangle under its segments, being then inconsistent with one another. Hence an infinity of beautiful propositions concerning the maxima and the minima of quantities, or the limits of the possible relations which quantities may stand in to one another.</p><p>“Such cases as these would occur even in the solution of the simplest problems; but when geometers proceeded to the analysis of such as were more complicated, they must have remarked, that their constructions would sometimes fail, for a reason directly contrary to that which has now been assigned. Instances would be found where the lines that, by their intersection, were to determine the thing sought, instead of intersecting one another, as they did in general, or of not meeting at all, as in the above-mentioned case of impossibility, would coincide with one another entirely, and leave the question of consequence unresolved. But<cb/> though this circumstance must have created considerable embarrassment to the geometers who first observed it, as being perhaps the only instance in which the language of their own science had yet appeared to them ambiguous or obscure, it would not probably be long till they found out the true interpretation to be put on it. After a little reflexion, they would conclude, that since, in the general problem, the magnitude required was determined by the intersection of the two lines above mentioned, that is to say, by the points common to them both; so, in the case of their coincidence, as all their points were in common, every one of these points must afford a solution; which solutions therefore must be infinite in number; and also, though infinite in number, they must all be related to one another, and to the things given, by certain laws, which the position of the two coinciding lines must necessarily determine.</p><p>“On enquiring farther into the peculiarity in the state of the data which had produced this unexpected result, it might likewise be remarked, that the whole proceeded from one of the conditions of the problem involving another, or necessarily including it; so that they both together made in fact but one, and did not leave a sufficient number of independent conditions, to confine the problem to a single solution, or to any determinate number of solutions. It was not difficult afterwards to perceive, that these cases of problems formed very curious propositions, of an intermediate nature between problems and theorems, and that they admitted of being enunciated separately, in a manner peculiarly elegant and concise. It was to such propositions, so enunciated, that the ancient geometers gave the name of Porisms.</p><p>“This deduction requires to be illustrated by examples.” Mr. Playfair then gives several problems by way of illuftration; one of which, which may here suffice to shew the method, is as follows:</p><p>“A triangle ABC being given, and also a point D, to draw through D a straight line DG, such, that, perpendiculars being drawn to it from the three angles of the triangle, viz, AE, BG, CF, the sum of the two perpendiculars on the same side of DG, shall be equal to the remaining perpendicular: or, that AE and BG together, may be equal to CF. <figure/></p><p>“Suppose it done: Bisect AB in H, join CH, and draw HK perpendicular to DG.</p><p>“Because AB is bisected in H, the two perpendioulars AE and BG are together double of HK; and as they are also equal to CF by hypothesis, CF must be double of HK; and CL of LH. Now, GH is given in position, and magnitude; therefore the point L is<pb n="264"/><cb/> given; and the point D being also given, the line DL is given in position, which was to be found.</p><p>“The construction was obvious. Bisect AB in H, join CH, and take HL equal to one third of CH; the straight line which joins the points D and L is the line required.</p><p>“Now, it is plain, that while the triangle ABC remains the same, the point L also remains the same, wherever the point D may be. The point D may therefore coincide with L; and when this happens, the position of the line to be drawn is left undetermined; that is to say, any line whatever drawn through L will satis<*> fy the conditions of the problem. Here therefore we have another indesinite case of a problem, and of consequence another Porism, which may be thus enunciated: “A triangle being given in position, a point in it may be found, such, that any straight line whatever being drawn through that point, the perpendiculars drawn to this straight line from the two angles of the triangle which are on one side of it, will be together equal to the perpendicular that is drawn to the same line from the angle on the other side of it.</p><p>“This Porisin may be made much more general; for if, instead of the angles of a triangle, we suppose ever so many points to be given in a plane, a point may be found such, that any straight line being drawn through it, the sum of all the perpendiculars that fall on that line from the given points on one side of it, is equal to the sum of the perpendiculars that fall on it from all the points on the other side of it.</p><p>“Or still more generally, any number of points being given not in the same plane, a point may be found, through which if any plane be supposed to pass, the sum of all the perpendiculars which fall on that plane from the points on one side of it, is equal to the sum of all the perpendiculars that fall on the same plane from the points on the other side of it. It is unnecessary to observe, that the point to be found in these propositions, is no other than the centre of gravity of the given points; and that therefore we have here an example of a Porism very well known to the modern geometers, though not distinguished by them from other theorems.”</p><p>After some examples of other Porisms, and remarks upon them, the author then adds,</p><p>“From this account of the origin of Porisms, it follows, that a Porism may be defined, <hi rend="italics">A proposition affirming the possibility of finding such conditions as will render a certain problem indeterminate, or capable of innumerable solutions.</hi></p><p>“To this definition, the different characters which Pappus has given will apply without difficulty. The propositions described in it like those which he mentions, are, strictly speaking, neither theorems nor problems, but of an intermediate nature between both; for they neither simply enunciate a truth to be demonstrated, nor propose a question to be solved: but are affirmations of a truth, in which the determination of an unknown quantity is involved. In as far therefore as they assert, that a certain problem may become indeterminate, they are of the nature of theorems; and in as far as they seek to discover the conditions by which that is brought about, they are of the nature of pro- blems.<cb/></p><p>“In the preceding definition also, and the instances from which it is deduced, we may trace that imperfect description of Porisms which Pappus ascribes to the later geometers, viz, “Porisma est quod desicit hypothesi a theoremate locali.” Now, to understand this, it must be observed, that if we take the converse of one of the propositions called <hi rend="italics">Loci,</hi> and make the construction of the figure a part of the hypothesis, we have what was called by the Ancients a Local Theorem. And again, if, in enunciating this theorem, that part of the hypothesis which contains the construction be suppressed, the proposition arising from thence will be a Porism; for it will enunciate a truth, and will also require, to the full understanding and investigation of that truth, that something should be found, viz, the circumstance in the construction, supposed to be omitted.</p><p>“Thus when we say; If from two given points E and D (2d fig. above), two lines EF and FD are inflected to a third point F, so as to be to one another in a given ratio, the point F is in the circumference of a circle given in position: we have a <hi rend="italics">Locus.</hi></p><p>“But when conversely it is said; If a circle ABC, of which the centre is O, be given in position, as also a point E, and if D be taken in the line EO, so that the rectangle EO OD be equal to the square of AO, the semidiameter of the circle; and if from E and D, the lines EF and DF be inflected to any point whatever in the circumference ABC; the ratio of EF to DF will be a given ratio, and the same with that of EA to AD: we have a local theorem.</p><p>“And, lastly, when it is said; If a circle ABC be given in position, and also a point E, a point D may be found, such, that if the two lines EF and FD be inflected from E and D to any point whatever F, in the circumference, these lines shall have a given ratio to one another: the proposition becomes a Porism.</p><p>“Here it is evident, that the local theorem is changed into a Porism, by leaving out what relates to the determination of the point D, and of the given ratio. But though all propositions formed in this way, from the conversion of <hi rend="italics">Loci,</hi> be Porisms, yet all Porisms are not formed from the conversion of <hi rend="italics">Loci.</hi> The first and second of the precéding, for instance, cannot by conversion be changed into <hi rend="italics">Loci;</hi> and therefore the desinition which describes all Porisms as being so convertible, is not sufficiently comprehensive. Fermat's idea of Porisms, as has been already observed, was founded wholly on this definition, and therefore could not fail to be imperfect.</p><p>“It appears, therefore, that the definition of Porisms given above agrees with Pappus's idea of these propositions, as far at least as can be collected from the imperfect fragments which contain his general description of them. It agrees also with Dr. Simson's definition, which is this: “Porisma est proposicio in qua proponitur demonstrare rem aliquam, vel plures datas esse, cui, vel quibus, ut et cuilibet ex rebus innumeris, non quidem datis, sed quæ ad ea quæ data sunt eandem habent relationem, convenire ostendendum est affectionem quandam communem in propositione descriptam.</p><p>“It cannot be denied, that there is a considerable degree of obscurity in this definition; notwithstanding of which it is certain, that every proposition to which it<pb n="265"/><cb/> applies must contain a <hi rend="italics">problematical</hi> part, viz “in qua proponitur demonstrare rem aliquam, vel plures datas esse,” and also a <hi rend="italics">theor<*>tical</hi> part, which contains the property, or <hi rend="italics">communis asfectio,</hi> affirmed of certain things which have been previously described.</p><p>“It is also evident, that the subject of every such proposition, is the relation between magnitudes of three different kinds; determinate magnitudes which are given; determinate magnitudes which are to be found; and indeterminate magnitudes which, though unlimited in number, are connected with the others by some common property. Now, these are exactly the conditions contained in the desinitions that have been given here.</p><p>“To confirm the truth of this theory of the origin of Porisms, or at least the justness of the notions founded on it, I must add a quotation from an Essay on the same subject, by a member of this society, the extent and correctness of whose views make every coincidence with his opinions peculiarly flattering. In a paper read several years ago before the Philosophical Society, Professor Dugald Stewart defined a Porism to be “A proposition affirming the possibility of finding one or more of the conditions of an indeterminate theorem.” Where, by an indeterminate theorem, as he had previously explained it, is meant one which expresses a relation between certain quantities that are indeterminate, both in magnitude and in number. The near agreement of this with the definition and explanations which have been given above, is too obvious to require to be pointed out; and I have only to observe, that it was not long after the publication of Simson's posthumous works, when, being both of us occupied in speculations concerning Porisms, we were led separately to the conclusions which I have now stated.</p><p>“In an enquiry into the origin of Porisms, the etymology of the term ought not to be forgotten. The question indeed is not about the derivation of the word <foreign xml:lang="greek">*porisma</foreign>, for concerning that there is no doubt; but about the reason why this term was applied to the class of propositions above described. Two opinions may be formed on this subject, and each of them with considerable probability: 1<hi rend="italics">mo.</hi> One of the significations of <foreign xml:lang="greek">porizw</foreign>, is <hi rend="italics">to acquire or obtain;</hi> and hence <foreign xml:lang="greek">*porisma</foreign>, <hi rend="italics">the thing obtained or gained.</hi></p><p>“Accordingly, Scapula says, <hi rend="italics">Es<*> vox a geometris desumpta qui theorema aliquid ex demonstrativo syllogismo necessario sequens inferentes, illud quasi lucrari dicuntur, quod non ex professo quidem theorematis bujus instituta sit demonstratio, sed tamen ex demonstratis recte sequatur.</hi> In this sense Euclid uses the word in his Elements of Geometry, where he calls the corollaries of his proposition, <hi rend="italics">Porismata.</hi> This circumstance creates a prefumption, that when the word was applied to a particular class of propositions, it was meant, in both cases, to convey nearly the same idea, as it is not at all probable, that so correct a writer as Euclid, and so scrupulous in his use of words, should employ the same term to express two ideas which are perfectly different. May we not therefore conjecture, that these propositions got the name of Porisms, entirely with a reference to their origin. According to the idea explained above, they would in general occur to mathematicians when engaged in the solution of the more difficult problems, and would arise<cb/> from those particular cases, where one of the conditions of the data involved in it some one of the rest. Thus a particular kind of theorem would be obtained, following as a corollary from the solution of the problem: and to this theorem the term <foreign xml:lang="greek">*porisma</foreign> might be very properly applied, since, in the words of Scapula, already quoted, <hi rend="italics">Non ex professo theorematis bujus instituta sit demonstratio, sed tamen ex demonstratis re<*>e sequatur.</hi></p><p>“2do. But-though this interpretation agrees so well with the supposed origin of Porisms, it is not free from difficulty. The verb <foreign xml:lang="greek">w=orizw</foreign> has another signification, <hi rend="italics">to find out, to discover, to devise;</hi> and is used in this sense by Pappus, when he says that the propositions called Porisms, asford great delight, <foreign xml:lang="greek">tois dunamenois oran kai porizma</foreign>, <hi rend="italics">to those who are able to understand and inv<*>stigate.</hi> Hence comes <foreign xml:lang="greek">porismos</foreign>, <hi rend="italics">the act of finding out or discovering,</hi> and from <foreign xml:lang="greek">porismos</foreign>, in this sense, the same author evidently considers <foreign xml:lang="greek">*porisma</foreign> as being derived. His words are, <foreign xml:lang="greek">*efasan de</foreign> (<foreign xml:lang="greek">o(i arxaioi</foreign>) <foreign xml:lang="greek">*porisma eina.to w=roteinomanon eis *porismon aut<*> w=rotei<*>oman<*></foreign>, <hi rend="italics">the Ancients said, that a Porism is something proposed for the</hi> sinding out, <hi rend="italics">or</hi> discovering <hi rend="italics">of the very thing proposed.</hi> It seems singular, however, that Porisms should have taken their name from a circumstance common to them with so many other geometrical truths; and if this was really the case, it must have been on account of the enigmatical form of their enunciations, which required, that in the analysis of these propositions, a sort of double discovery should be made, not only of the Truth, but also of the Meaning <hi rend="italics">of the very thing which was proposed.</hi> They may therefore have been called <hi rend="italics">Porismata,</hi> or <hi rend="italics">investigations,</hi> by way of eminence.</p><p>“We might next proceed to consider the particular Porisms which Dr. Simson has restored, and to shew, that every one of them is the indeterminate case of some problem. But of this it is so easy for any one, who has attended to the preceding remarks, to satisfy himself, by barely examining the enunciations of those propositions, that the detail unto which it would lead seems to be unnecessary. I shall therefore go on to make some observations on that kind of analysis which is particularly adapted to the investigation of Porisms.</p><p>“If the idea which we have given of these propositions be just, it follows, that they are always to be discovered by considering the cases in which the construction of a problem fails in consequence of the lines which, by their intersection, or the points which, by their position, were to determine the magnitude required, happening to coincide with one another—a Porism may therefore be deduced from the problem it belongs to, in the same manner that the propositions concerning the <hi rend="italics">ma<*>ima</hi> and <hi rend="italics">minima</hi> of quantities are deduced from the problems of which they form the limitations; and such no doubt is the most natural and most obvious analysis of which this class of propositions will admit.</p><p>“It is not, however, the only one that they will admit of; and there are good reasons for wishing to be provided with another, by means of which, a Porism that is any how suspected to exist, may be found out, independently of the general solution of the problem to which it belongs. Of these reasons, one is, that the Porism may perhaps admit of being investigated more easily than the general problem admits of being resolved:<pb n="266"/><cb/> and another is, that the former, in almost every case, helps to discover the simplest and most elegant solution that can be given of the latter.</p><p>“It is desirable to have a method of investigating Porisms, which does not require, that we should have previously resolved the problems they are connected with, and which may always serve to determine, whether to any given problem there be attached a Porism, or not. Dr. Simson's Analysis may be considered as answering to this description; for as that geometer did not regard these propositions at all in the light that is done here, nor in relation to their origin, an independent analysis os this kind, was the only one that could occur to him; and be has accordingly given one which is extremely ingenious, and by no means easy to be invented, but which he uses with great skilfulness and dexterity throughout the whole of his Restoration.</p><p>“It is not easy to ascertain whether this be the precise method used by the Ancients. Dr. Simson had here nothing to direct him but his genius, and has the full merit of the sirst inventor. It seems probable, however, that there is at least a great affinity between the methods, since the <hi rend="italics">lemmata</hi> given by Pappus as necessary to Euclid's demonstrations, are subservient also to those of our modern geometer.</p><p>“It is, as we have seen, a general principle that a problem is converted into a Porism, when one, or when two, of the conditions of it, necessarily involve in them some one of the rest. Suppose then that two of the conditions are exactly in that state which determines the third; then, while they remain fixed or given, should that third one be supposed to vary, or differ, ever so little, from the state required by the other two, a contradiction will ensue. Therefore if, in the hypothesis of a problem, the conditions be so related to one another as to render it indeterminate, a Porism is produced; but if, of the conditions thus related to one another, some one be supposed to vary, while the others continue the same, an absurdity follows, and the problem becomes impossible. <hi rend="italics">Wherever therefore any problem admits both of an indeterminate, and an impossible case, it is certain, that these cases are nearly related to one another, and that some of the conditions by which they are produced, are common to both.</hi></p><p>“It is supposed above, that <hi rend="italics">two</hi> of the conditions of a problem involve in them a third, and wherever that happens, the conclusion which has been deduced will invariably take place.</p><p>“But a Porism may sometimes be so simple, as to arise from the mere coincidence of <hi rend="italics">one</hi> condition of a problem with another, though in no case whatever, any inconsistency can take place between them. Thus, in the second of the foregoing propositions, the coincidence of the point given in the problem with another point, viz, the centre of gravity of the given triangle, renders the problem indeterminate; but as there is no relation of distance, or pofition, between these points, that may not exist, so the problem has no impossible case belonging to it. There are, however, comparatively but few Porisms so simple in their origin as this, or that arise from problems in which the conditions are so little complicated; for it usually happens, that a problem which can become indefinite, may also become<cb/> impossible; and if so, the connection between these cases, which has been already explained, never fails to take place.</p><p>“Another species of impossibility may frequently arise from the porifmatic case of a problem, which will very much affect the application of geometry to astronomy, or any of the sciences of experiment or observation. For when a problem is to be resolved by help of data furnished by experiment or observation, the first thing to be considered is, whether the data so obtained, be sufficient for determining the thing sought; and in this a very erroneous judgment may be formed, if we rest satisfied with a general view of the subject: For though the problem may in general be resolved from the data that we are provided with, yet these data may be so related to one another in the case before us, that the problem will become indeterminate, and instead of one solution, will admit of an insinite number.</p><p>“Suppose, for instance, that it were required to determine the position of a point F from knowing that it was situated in the circumference of a given circle ABC, and also from knowing the ratio of its distances from two given points E and D; it is certain that in general these data would be sufficient for determining the situation of F. But nevertheless, if E and D should be so situated, that they were in the same straight line with the centre of the given circle; and if the rectangle under their distances from that centre, were also equal to the square of the radius of the circle, then, the position of F could not be determined.</p><p>“This particular instance may not indeed occur in any of the practical applications of geometry; but there is one of the fame kind which has actually occurred in astronomy: And as the history of it is not a little singular, affording besides an excellent illustration of the nature of Porisms, I hope to be excused for entering into the following detail concerning it.</p><p>“Sir Isaac Newton having demonstrated, that the trajectory of a comet is a parabola, reduced the actual determination of the orbit of any particular comet to the solution of a geometrical problem, depending on the properties of the parabola, but of such considerable difficulty, that it is necessary to take the assistance of a more elementary problem, in order to find, at least nearly, the distance of the comet from the earth, at the times when it was observed. The expedient for this purpose, suggested by Newton himself, was to consider a small part of the comet's path as rectilineal, and described with an uniform motion, so that four observations of the comet being made at moderate intervals of time from one another, four straight lines would be determined, viz, the four lines joining the places of the earth and the comet, at the times of observation, acros<*> which if a straight line were drawn, so as to be cut by them in three parts, in the same ratios with the intervals of time abovementioned; the line so drawn would nearly represent the comet's path, and by its intersection with the given lines, would determine, at least nearly, the distances of the comet from the earth at the time of observation.</p><p>“The geometrical problem here employed, of drawing a line to be divided by four other lines given in position, into parts having given ratios to one another, had been already resolved by Dr. Wallis and Sir Chris-<pb n="267"/><cb/> topher Wren, and to their solutions Sir Isaac Newton added three others of his own, in different parts of his works. Yet none of all these geometers observed that peculiarity in the problem which rendered it inapplicable to astronomy. This was first done by M. Boscovich, but not till after many trials, when, on its application to the motion of comets, it had never led to any satisfactory result. The errors it produced in some instances were so considerable, that Zanotti, seeking to determine by it the orbit of the comet of 1739, found, that his construction threw the comet on the side of the sun opposite to that on which he had actually observed it. This gave occasion to Boscovich, some years afterwards, to examine the different cases of the problem, and to remark that, in one of them, it became indeterminate, and that, by a curious coincidence, this happened in the only case which could be supposed applicable to the astronomical problem abovementioned; in other words, he found, that in the state of the data, which must there always take place, innumerable lines might be drawn, that would be all cut in the same ratio, by the four lines given in position. This he demonstrated in a dissertation published at Rome in 1749, and since that time in the third volume of his <hi rend="italics">Opuscula.</hi> A demonstration of it, by the same author, is also inserted at the end of Castillon's Commentary on the <hi rend="italics">Arithmetica Universalis,</hi> where it is deduced from a construction of the general problem, given by Mr. Thomas Simpson, at the end of his Elements of Geometry. The proposition, in Boscovich's words, is this: Problema quo quæritur recta linea quæ quatuor rectas positione datas<*>ita secet, ut tria ejus segmenta <*>int invicem in ratione data, evadit aliquando indeterminatum, ita ut per quodvis punctum cujusvis ex iis quatuor rectis duci possit recta linea, quæ ei conditioni faciat satis.</p><p>“It is needless, I believe, to remark, that the proposition thus enunciated is a Porism, and that it was discovered by Bos<*>ovich, in the same way, in which I have supposed Porisms to have been first discovered by the geometers of antiquity.</p><p>“A question nearly connected with the origin of Porisms still remains to be solved, namely, from what cause has it arisen that propositions which are in themselves so important, and that actually occupied so considerable a place in the ancient geometry, have been so little remarked in the modern? It cannot indeed be said, that propositions of this kind were wholly unknown to the Moderns before the restoration of what Euclid had written concerning them; for besides M. Boscovich's proposition, of which so much has been already said, the theorem which asserts, that in every system of points there is a centre of gravity, has been shewn above to be a Porism; and we shall see hereafter, that many of the theorems in the higher geometry belong to the same class of propositions. We may add, that some of the elementary propositions of geometry want only the proper form of enunciation to be perfect Po<*>isms. It is not therefore strictly true, that none of the propositions called Porisms have been known to the Moderns; but it is certain, that they have not met, from them, with the attention they met with from the Ancients, and that they have not been distinguished as a separate class of propositions. The cause of this difference is undoubtedly to be sought for in a comparison<cb/> of the methods employed for the solution of geometrical problems in ancient and modern times.</p><p>“In the solution of such problems, the geometers of antiquity proceeded with the utmost caution, and were careful to remark every particular case, that is to say, every change in the construction, which any change in the state of the data could produce. The different conditions from which the solutions were derived, were supposed to vary one by one, while the others remained the same; and all their possible combinations being thus enumerated, a separate solution was given, whereever any considerable change was observed to have taken place.</p><p>“This was so much the case, that the <hi rend="italics">Sectio Rationis,</hi> a geometrical problem of no great difficulty, and one of which the solution would be dispatched, according to the methods of the modern geometry, in a single page, was made by Apollonius, the subject of a treatise con- <*>ting of two books. The first book has seven general divisions, and twenty-four cases; the second, fourteen general divisions, and seventy-three cases, each of which cases is separately considered. Nothing, it is evident, that was any way connected with the problem, could escape a geometer, who proceeded with such minuteness of investigation.</p><p>“The same scrupulous exactness may be remarked in all the other mathematical researches of the Ancients; and the reason doubtless is, that the geometers of those ages, however expert they were in the use f their analysis, had not sufficient experience in its powers, to trust to the more general applications of it. That principle which we call the <hi rend="italics">law of continuity,</hi> and which connects the whole system of mathematical truths by a chain of insensible gradations, was scarcely known to them, and has been unfolded to us, only by a more extensive knowledge of the mathematical sciences, and by that most perfect mode of expressing the relations of quantity, which forms the language of algebra; and it is this principle alone which has taught us, that though in the solution of a problem, it may be impossible to conduct the investigation without assuming the data in a <hi rend="italics">particular</hi> state, yet the result may be perfectly <hi rend="italics">general,</hi> and will accommodate itself to every case with such wonderful versatility, as is scarcely credible to the most experienced mathematician, and such as often forces h<*>m to stop, in the midst of his calculus, and look back, with a mixture of diffidence and admiration, on the unforeseen harmony of his conclusions. All this was unknown to the Ancients; and therefore they had no resource, but to apply their analysis separately to each particular case, with that extreme caution which has just been deseribed; and in doing so, they were likely to remark many peculiarities, which more extensive views, and more expeditious methods of investigation, might perhaps have induced them to overlook.</p><p>“To rest satissied, indeed, with too general results, and not to descend sufficiently into particular details, may be considered as a vice that naturally arises out of the excellence os the modern analysis. The esfect which thi<*> has had, in concealing from us the class of propositions we are now considering, cannot be better illustrated than by the example of the Porism discovered by Boscovich, in the manner related above. Though the problem from which that Porism is derived, was<pb n="268"/><cb/> resolved by several mathematicians of the sir<*> eminence, among whom also was Sir Isaac Newton, yet the Porism which, as it happens is the most important case of it, was not observed by any of them. This is the more remarkable, that Sir Isaac Newton takes notice of the two most simple cases, in which the problem obviously admits of innumerable solutions, viz, when the lines given in position are either all parallel, or all meeting in a point, and these two hypotheses he therefore expressly excepts. Yet he did not remark, that there are other circumstances which may render the solution of the problem indeterminate as well as these; so that the porismatic case considered above, escaped his observation: and if it escaped the observation of one who was accustomed to penetrate so far into matters infinitely more obscure, it was because he satissied himself with a general construction, without pursuing it into its particular cases. Had the solution been conducted after the manner of Euclid or Apollonius, the Porism in question must infallibly have been discovered.”</p><p>PORISTIC <hi rend="italics">Method,</hi> is that which determines when, by what means, and how many different ways, a problem may be resolved.</p></div1><div1 part="N" n="PORTA" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PORTA</surname> (<foreName full="yes"><hi rend="smallcaps">John Baptista</hi></foreName>)</persName></head><p>, called also in Italy <hi rend="italics">Giovan Batista de la Porta,</hi> of Naples, lived about the end of the 16th century, and was famous for his skill in philosophy, mathematics, medicine, natural history, &c, as well as for his indefatigable endeavours to improve and propagate the knowledge of those sciences. With this view, he not only established private schools for particular sciences, but to the ut<*>ost of his power promoted public academies. He had no small share in establishing the academy at <hi rend="italics">Gli Ozioni,</hi> at Naples, and had one in his own house, called <hi rend="italics">de Secreti,</hi> into which none were admitted members, but such as had made some new discoveries in nature. He died at Pisa, in the kingdom of Naples, in the year 1615.</p><p>Porta gave the fullest proof of an extensive genius, and wrote a great many works; the principal of which are as follow:</p><p>1. His <hi rend="italics">Natural Magic;</hi> a book abounding with curious experiments; but containing nothing of magic, the common acceptation of the word, as he pretends to nothing above the power of nature.</p><p>2. <hi rend="italics">Elements of Curve Lines.</hi></p><p>3. <hi rend="italics">A Treatise of Distillation.</hi></p><p>4. <hi rend="italics">A Treatise of Arithmetic.</hi></p><p>5. <hi rend="italics">Concerning Secret Letter-writing.</hi></p><p>6. <hi rend="italics">Of Optical Refractions.</hi></p><p>7. <hi rend="italics">A Treatise of Fortification.</hi></p><p>8. <hi rend="italics">A Treatise of Physiognomy.</hi></p><p>Beside some Plays and other pieces os less note.</p></div1><div1 part="N" n="PORTAIL" org="uniform" sample="complete" type="entry"><head>PORTAIL</head><p>, in Architecture, the face or frontispiece of a church, viewed on the side in which the great door is placed. It means also the great door or gate itself of a palace, castle, &c.</p></div1><div1 part="N" n="PORTAL" org="uniform" sample="complete" type="entry"><head>PORTAL</head><p>, in Architecture, a term used for a little square corner of a room, cut off from the rest of the room by the wainscot; frequent in the ancient buildings, but now disused.</p><p><hi rend="smallcaps">Portal</hi> is sometimes also used for a little gate, portella; where there are two gates, a large and a small one.<cb/></p><p><hi rend="smallcaps">Portal</hi> is sometimes also used for a kind of arch of joiner's work before a door.</p></div1><div1 part="N" n="PORTCULLICE" org="uniform" sample="complete" type="entry"><head>PORTCULLICE</head><p>, called also <hi rend="italics">Herse,</hi> and <hi rend="italics">Sarrasin,</hi> in Fortification, an assemblage of several large pieces of wood laid or joined across one another, like a harrow, and each pointed at the bottom with iron. These were formerly used to be hung over the gateways of fortified places, to be ready to let down in case of a surprize, when the enemy should come so quick, as not to allow time to shut the gates. But the orgues are now more generally used, being found to answer the purpose better.</p><p>PORT-<hi rend="smallcaps">Fire</hi>, in Gunnery, a paper tube, about 10 inches long, filled with a composition of meal-powder, sulphur, and nitre, rammed moderately hard; used to <*>ire guns and mortars, instead of match.</p></div1><div1 part="N" n="PORTICO" org="uniform" sample="complete" type="entry"><head>PORTICO</head><p>, in Architecture, is a kind of gallery, raised upon arches, under which people walk for shelter.</p></div1><div1 part="N" n="POSITION" org="uniform" sample="complete" type="entry"><head>POSITION</head><p>, or <hi rend="italics">Site,</hi> or <hi rend="italics">Situation,</hi> in Physics, is an affection of place, expressing the manner of a body's being in it.</p><div2 part="N" n="Position" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Position</hi></head><p>, in Architecture, denotes the situation of a building, with respect to the points of the horizon. The best it is thought is when the four sides point directly to the four winds.</p></div2><div2 part="N" n="Position" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Position</hi></head><p>, in Astronomy, relates to the sphere. The position of the sphere is either right, parallel, or oblique, whence arise the inequality of days, the difference of seasons, &c.</p><p><hi rend="italics">Circles of</hi> <hi rend="smallcaps">Position</hi>, are circles passing through the common intersections of the horizon and meridian, and through any degree of the ecliptic, or the centre of any star, or other point in the heavens; used for finding out the position or situation of any star. These are usually counted six in number, cutting the equator into twelve equal parts, which the astrologers call the celestial houses.</p></div2><div2 part="N" n="Position" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Position</hi></head><p>, in Arithmetic, called also False Position, or Supposition, or Rule of False, is a rule so called, because it consists in calculating by false numbers supposed or taken at random, according to the process described in any question or problem proposed, as if they were the true numbers, and then from the results, compared with that given in the question, the true numbers are found. It is sometimes also called Trialand-Error, because it proceeds by trials of false numbers, and thence finds out the true ones by a comparison of the errors.</p><p>Position is either Single or Double.</p><p><hi rend="italics">Single</hi> <hi rend="smallcaps">Position</hi> is when only one supposition is employed in the calculation. And</p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Position</hi> is that in which two suppositions are employed.</p><p>To the rule of Position properly belong such questions as cannot be resolved from a direct process by any of the other usual rules in arithmetic, and in which the required numbers do not ascend above the first power: such, for example, as most of the questions usually brought to exercise the reduction of simple equations in algebra. But it will not bring out true answers when the numbers sought ascend above the first power; for then the results are not proportional to the Positions,<pb n="269"/><cb/> or supposed numbers, as in the single rule; nor yet the errors to the difference of the true number and each Position, as in the double rule. Yet in all such cases, it is a very good approximation, and in exponential equations, as well as in many other things, it succeeds better than perhaps any other method whatever.</p><p>Those questions, in which the results are proportional to their suppositions, belong to Single Position: such are those which require the multiplication or division of the number sought by any n<*>mber; or in which it is to be increased or diminished by itself any number of times, or by any part or parts of it. But those in which the results are not proportional to their positions, belong to the double rule: such are those, in which the numbers sought, or their multiples or parts, are increased or diminished by some given absolute number, which is no known part of the number sought.</p><p>To work by <hi rend="italics">the Single Rule of</hi> <hi rend="smallcaps">Position.</hi> Suppose, take, or assume any number at pleasure, for the number sought, and proceed with it as if it were the true number, that is, perform the fame operations with it as, in the question, are described to be performed with the number required: then if the result of those operations be the same with that mentioned or given in the question, the supposed number is the same as the true one that was required; but if it be not, make this proportion, viz, as your result is to that in the question, so is your supposed false number, to the true one required.</p><p><hi rend="italics">Example.</hi> Suppose that a person, after spending 1/3 and 1/4 of his money, has yet remaining 60l.; what ium had he at first?</p><p>Suppose he had at first 120l. <table><row role="data"><cell cols="1" rows="1" role="data">Now 1/3 of 120 is</cell><cell cols="1" rows="1" role="data">  40</cell></row><row role="data"><cell cols="1" rows="1" role="data">and 1/4 of it is</cell><cell cols="1" rows="1" role="data">  30</cell></row><row role="data"><cell cols="1" rows="1" role="data">their sum is</cell><cell cols="1" rows="1" role="data">  70</cell></row><row role="data"><cell cols="1" rows="1" role="data">which taken from</cell><cell cols="1" rows="1" role="data">120</cell></row><row role="data"><cell cols="1" rows="1" role="data">leaves remaining</cell><cell cols="1" rows="1" role="data">  50, instead of 60.</cell></row></table></p><p>Therefore as the sum at first. <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Proof.</hi></cell><cell cols="1" rows="1" role="data">1/3 of 144 is</cell><cell cols="1" rows="1" role="data">  48</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1/4 of it is</cell><cell cols="1" rows="1" role="data">  36</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">their sum</cell><cell cols="1" rows="1" role="data">  84</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">taken from</cell><cell cols="1" rows="1" role="data">144</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">leaves just</cell><cell cols="1" rows="1" role="data">  60 as per quest.</cell></row></table> <hi rend="center"><hi rend="italics">To work by the Double Rule of</hi> <hi rend="smallcaps">Position.</hi></hi></p><p>In this rule, make two different suppositions, or assumptions, and work or perform the operations with each, described in the question, exactly as in the single rule: and if neither of the supposed numbers solve the question, that is, produce a result agreeing with that in the question; then observe the errors, or how much each of the false results differs from the true one, and also whether they are too great or too little; marking them with + when too great, and with - when too little. Next multiply, crosswise, each position by the error of the other; and if the errors be of the same affection, that is both +, or both -, subtract the one<cb/> product from the other, as also the one error from the other, and divide the former of these two remainders by the latter, for the answer, or number sought. But if the errors be unlike, that is, the one +, and the other -, add the two products together, and also the two errors together, and divide the former sum by the latter, for the answer.</p><p>And in this rule it is particularly useful to remember this part of the rule, viz. to subtract when the errors are alike, both + or both -, but to add when unlike, or the one + and the other -.</p><p><hi rend="italics">Example.</hi> A son asking his father how old he was, received this answer: Your age is now 1/4 of mine; but 5 years ago your age was only 1/5 of mine at that time. What then were their ages?</p><p>First, suppose the son 15; then the father's; also, 5 years ago the son was 10, and the father's must be 55, but ought to be 10 X 5 or 50, therefore the error is 5-. Again, suppose the son 22; then is the father's; also 5 years ago the son was 17, and the father's then 83, but ought to be 17 X 5, or 85, therefore the error is 2 +.</p><p>And the errors, being unlike, must be added, theirsum being 7. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Then 15</cell><cell cols="1" rows="1" rend="align=center" role="data"> 22</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">  5</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" rend="align=center" role="data">110</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"> 30</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7) 140 (20 the son's age, and consequently 80 the father's.</cell></row></table></p><p>This rule of Position, or trial-and-error, is a good general way of approximating to the roots of the higher equations, to which it may be applied even before the equation is reduced to a final or simple state, by which it often saves much trouble in such reductions. It is also eminently useful in resolving exponential equations, and equations involving arcs, or sines, &c, or logarithms, and in short in any equations that are very intricate and difficult. And even in the extraction of the higher roots of common numbers, it may be very usefully applied. As for instance, to extract the 3d or cubic root of the number 20.—Here it is evident that the root is greater than 2 and less than 3; making these two numbers therefore the suppositions, the process-will be thus: <table><row role="data"><cell cols="1" rows="1" role="data">1st sup.  2<hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2d sup.    3<hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">27</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">given number</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">given number</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1st error</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2d error</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">+</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">            12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">add</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">             7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">            19 )</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" rend="colspan=5 align=center" role="data">2.63 the first approximation.</cell></row></table><pb n="270"/><cb/></p><p>Again, as it thus appears the cube root of 20 is near 2.6 or 2.7, make supposition of these two, and repeat the proeess with them, thus: <table><row role="data"><cell cols="1" rows="1" role="data">Ist sup. 2.6<hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">17.576</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2d sup. 2.7<hi rend="sup">3</hi>=</cell><cell cols="1" rows="1" rend="align=right" role="data">19.683</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">given number</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">20.     </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">given number</cell><cell cols="1" rows="1" rend="align=right" role="data">20.     </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Ist error</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2.424</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">   2d error</cell><cell cols="1" rows="1" rend="align=right" role="data">0.317</cell><cell cols="1" rows="1" role="data">-</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2.7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2.6</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16968</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1902</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4848 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">634 </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">        2.424</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6.5448</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">subtr.</cell><cell cols="1" rows="1" rend="align=right" role="data">.8242</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">         .317</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">.8242</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">        2.107</cell><cell cols="1" rows="1" role="data">)</cell><cell cols="1" rows="1" rend="align=right" role="data">5.7206</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">(2.714 root sought.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>The rule of Position passed from the Moors into Europe, by Spain and Italy, along with their algebra, or method of equations, which was probably derived from the former.</p></div2><div2 part="N" n="Position" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Position</hi></head><p>, in Geometry, respects the situation, bearing, or direction of one thing, with regard to another. And Euclid says, “Points, lines, and angles, which have and keep always one and the same place and situation, are said to be given by Position or situation.” Data, def. 4.</p><p>POSITIVE <hi rend="italics">Quantities,</hi> in Algebra, such as are of a real, affirmative, or additive nature; and which either have, or are supposed to have, the affirmative or positive sign + before them; as <hi rend="italics">a</hi> or + <hi rend="italics">a,</hi> or <hi rend="italics">bc,</hi> &c. It is used in contradistinction from negative quantities, which are defective or subductive ones, and marked by the sign -; as - <hi rend="italics">a,</hi> or - <hi rend="italics">ab.</hi></p></div2></div1><div1 part="N" n="POSTERN" org="uniform" sample="complete" type="entry"><head>POSTERN</head><p>, or <hi rend="italics">Sally-port,</hi> in Fortification, a small gate, usually made in the angle of the flank of a bastion, or in that of the curtain, or near the orillon, defcending into the ditch; by which the garrison can march in and out, unperceived by the enemy, either to relieve the works, or to make private sallies, &c.—It means also any private or back door.</p></div1><div1 part="N" n="POSTICUM" org="uniform" sample="complete" type="entry"><head>POSTICUM</head><p>, in Architecture, the postern gate, or back-door of any fabric.</p></div1><div1 part="N" n="POSTULATE" org="uniform" sample="complete" type="entry"><head>POSTULATE</head><p>, a demand, petition, or a problem of so obvious a nature, as to need neither demonstration, nor explication, to render it either more plain or certain. This definition will nearly agree also to an axiom, which is a self-evident theorem, as a Postulate is a self-evident problem.</p><p>Euclid lays down these three Postulates in his Elements; viz, 1st, That from one point to another a line can be drawn. 2d, That a right line can be produced out at pleasure. 3d, That with any centre and radius a circle may be described.</p><p>As to axioms, he has a great number; as, That two things which are equal to one and the same thing, are equal to each other, &c.</p></div1><div1 part="N" n="POUND" org="uniform" sample="complete" type="entry"><head>POUND</head><p>, a certain weight; which is of two kinds, viz., the<*>pound troy, and the pound avoirdupois; the former consisting of 12 ounces troy, and the latter of 16 ounces avoirdupois.—The pound troy is to the pound avoirdupois as 5760 to 6999 1/2, or nearly 576 to 700</p><p><hi rend="smallcaps">Pound</hi> also is an imaginary money used in account-<cb/> ing, in several countries. Thus, in England there i<*> the Pound sterling, containing 20 shillings; in France the Pound or livre Tournois and Parisis; in Holland and Flanders, a Pound or livre de gros, &c.—The term arose from hence, that the ancient pound sterling, though it only contained 240 pence, as ours does; yet each penny being equal to five of ours, the pound of silver weighed a Pound troy.</p></div1><div1 part="N" n="POUNDER" org="uniform" sample="complete" type="entry"><head>POUNDER</head><p>, in Artillery, a term used to express a certain weight of shot or ball, or how many pounds weight the proper ball is for any cannon: as a 24 pounder, a 12 pounder, &c.</p></div1><div1 part="N" n="POWDER" org="uniform" sample="complete" type="entry"><head>POWDER</head><p>, <hi rend="italics">Gun.</hi> See <hi rend="smallcaps">Gunpowder.</hi></p><p><hi rend="smallcaps">Powder</hi>-<hi rend="italics">Triers.</hi> See <hi rend="smallcaps">Eprouvette.</hi></p></div1><div1 part="N" n="POWER" org="uniform" sample="complete" type="entry"><head>POWER</head><p>, in Mechanics, denotes some force which, being applied to a machine, tends to produce motion; whether it does actually produce it or not. In the former case, it is called a moving Power; in the latter, a sustaining power.</p><p><hi rend="smallcaps">Power</hi> is also used in Mechanics, for any of the six simple machines, viz. the lever, the balance, the screw, the wheel and axle, the wedge, and the pulley.</p><p><hi rend="smallcaps">Power</hi> <hi rend="italics">of a Glass,</hi> in Optics, is by some used for the distance between the convexity and the solar focus.</p><div2 part="N" n="Power" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Power</hi></head><p>, in Arithmetic, the produce of a number, or other quantity, arising by multiplying it by itself, any number of times.</p><p>Any number is called the first power of itself. If it be multiplied once by itself, the product is the second power, or square; if this be multiplied by the first power again, the product is the third power, or cube; if this be multiplied by the first power again, the product is the fourth power, or biquadratic; and so on; the Power being always denominated from the number which exceeds the multiplications by one or unity, which number is called the index or exponent of the Power, and is now set at the upper corner towards the right of the given quantity or root, to denote or express the Power. Thus, 3 or 3<hi rend="sup">1</hi> = 3 is the 1st power of 3, 3 X 3 or 3<hi rend="sup">2</hi> = 9 is the 2d power of 3, 3<hi rend="sup">2</hi> X 3 or 3<hi rend="sup">3</hi> = 27 is the 3d power of 3, 3<hi rend="sup">3</hi> X 3 or 3<hi rend="sup">4</hi> = 81 is the 4th power of 3, &c. &c.</p><p>Hence, to raise a quantity to a given Power or dignity, is the same as to find the product arising from its being multiplied by itself a certain number of times; for example, to raise 2 to the 3d power, is the same thing as to find the factum, or product . The operation of raising Powers, is called Involution.</p><p>Powers, of the same degree, are to one another in the ratio of the roots as manisold as their common exponent contains units: thus, squares are in a duplicate ratio of the roots; cubes in a triplicate ratio; 4th powers in a quadruplicate ratio.—And the Powers of proportional quantities are also proportional to one another: so, if , then, in any Powers also, .</p><p>The particular names of the several Powers, as introduced by the Arabians, were, square, cube, quadratoquadratum or biquadrate, sursolid, cube squared, second sursolid, quadrato-quadrato-quadratum, cube of the<pb n="271"/><cb/> cube, square of the sursolid, third sursolid, and so on, according to the <hi rend="italics">products</hi> of the indices.</p><p>And the names given by Diophantus, who is followed by Vieta and Oughtred, are, the side or root, square, cube, quadrato-quadratum, quadrato-cubus, cubo-cubus, quadrato-quadrato-cubus, quadrato-cubo-cubus, cubocubo-cubus, &c, according to the <hi rend="italics">sums</hi> of the indices.</p><p>But the moderns, after Des Cartes, are contented to distinguish most of the Powers by the exponents; as 1st, 2d, 3d, 4th, &c.</p><p>The characters by which the several Powers are denoted, both in the Arabic and Cartesian notation are as follow: <table><row role="data"><cell cols="1" rows="1" role="data">Arab.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"><hi rend="italics">R</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">q</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">bq</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">qc</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics"> Bs</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">tq</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">bc</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Cart.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">0</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">1</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">4</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">5</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">6</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">a</hi><hi rend="sup">7</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">a</hi><hi rend="sup">8</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><hi rend="sup">9</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">512</cell></row></table></p><p>Hence, 1st. The Powers of any quantity, form a series of geometrical proportionals, and their exponents a series of arithmetical proportionals, in such sort that the addition of the latter answers to the multiplication of the former, and the subtraction of the latter answers to the division of the former, &c; or in short, that the latter, or exponents, are as the logarithms of the former, or Powers. Thus, , and ; ; also , and ; .</p><p>2d. The 0 Power of any quantity, as <hi rend="italics">a</hi><hi rend="sup">0</hi>, is = 1.</p><p>3d. Powers of the same quantity are multiplied, by adding their exponents: Thus,</p><p>4th. Powers are divided by subtracting their exponents. Thus,</p><p>5th. Powers are also considered as negative ones, or having negative exponents, when they denote a divisor, or the denominator of a fraction. So , and , and , &c. And hence any quantity may be changed from the denominator to the numerator, or from a divisor to a multiplier, or vice versa, by changing the sign of its exponent; and the whole series of Powers proceeds indefinitely both ways from 1 or the 0 Power, positive on the one hand, and negative on the other. Thus, &c <hi rend="italics">a</hi><hi rend="sup">-4</hi> <hi rend="italics">a</hi><hi rend="sup">-3</hi> <hi rend="italics">a</hi><hi rend="sup">-2</hi> <hi rend="italics">a</hi><hi rend="sup">-1</hi> <hi rend="italics">a</hi><hi rend="sup">0</hi> <hi rend="italics">a</hi><hi rend="sup">1</hi> <hi rend="italics">a</hi><hi rend="sup">2</hi> <hi rend="italics">a</hi><hi rend="sup">3</hi> <hi rend="italics">a</hi><hi rend="sup">4</hi> &c, or &c 1/<hi rend="italics">a</hi><hi rend="sup">4</hi> 1/<hi rend="italics">a</hi><hi rend="sup">3</hi> 1/<hi rend="italics">a</hi><hi rend="sup">2</hi> 1/<hi rend="italics">a</hi> 1 <hi rend="italics">a</hi> <hi rend="italics">a</hi><hi rend="sup">2</hi> <hi rend="italics">a</hi><hi rend="sup">3</hi> <hi rend="italics">a</hi><hi rend="sup">4</hi> &c.</p><p>Powers are also denoted with fractional exponents, or<cb/> even with surd or irrational ones; and then the numerator denotes the Power raised to, and the denominator the exponent of some root to be extracted: Thus, , and , and , &c. And these are sometimes called imperfect powers, or surds.</p><p>When the quantity to be raised to any Power is positive, all its Powers must be positive. And when the radical quantity is negative, yet all its even Powers must be positive: because - X - gives +: the odd Powers only being negative, or when their exponents are odd numbers: Thus, the Powers of - <hi rend="italics">a,</hi> are + 1, - <hi rend="italics">a,</hi> + <hi rend="italics">a</hi><hi rend="sup">2</hi>, - <hi rend="italics">a</hi><hi rend="sup">3</hi>, + <hi rend="italics">a</hi><hi rend="sup">4</hi>, - <hi rend="italics">a</hi><hi rend="sup">5</hi>, + <hi rend="italics">a</hi><hi rend="sup">6</hi>, &c. where the even Powers <hi rend="italics">a</hi><hi rend="sup">2</hi>, <hi rend="italics">a</hi><hi rend="sup">4</hi>, <hi rend="italics">a</hi><hi rend="sup">6</hi> are positive, and the odd Powers <hi rend="italics">a,</hi> <hi rend="italics">a</hi><hi rend="sup">3</hi>, <hi rend="italics">a</hi><hi rend="sup">5</hi> are negative.</p><p>Hence, if a Power have a negative sign, no even root of it can be assigned; since no quantity multiplied by itself an even number of times, can give a negative product. Thus √- <hi rend="italics">a</hi><hi rend="sup">2</hi>, or the square or 2d root of - <hi rend="italics">a</hi><hi rend="sup">2</hi>, cannot be assigned; and is called an impossible root, or an imaginary quantity.—Every Power has as many roots, real and imaginary, as there are units in the exponent.</p><p>M. De la Hire gives a very odd property common to all Powers. M. Carre had observed with regard to the number 6, that all the natural cubic numbers, 8, 27, 64, 125, having their roots less than 6, being divided by 6, the remainder of the division is the root itself; and if we go farther, 216, the cube of 6, being divided by 6, leaves no remainder; but the divisor 6 is itself the root. Again, 343, the cube of 7, being divided by 6, leaves 1; which added to the divisor 6, makes the root 7, &c. M. De la Hire, on considering this, has found that all numbers, raised to any Power whatever, have divisors, which have the same effect with regard to them, that 6 has with regard to cubic numbers. For finding these divisors, he discovered the following general rule, viz, If the exponent of the Power of a number be even, i. e. if the number be raised to the 2d, 4th, 6th, &c Power, it must be divided by 2; the remainder of the division, when there <*>s any, added to 2, or to a multiple of 2, gives the root of this number, corresponding to its Power, i. e. the 2d, 6th, &c root.</p><p>But if the exponent of the power be an uneven number, i. e. if the number be raised to the 3d, 5th, 7th, &c Power; the double of this exponent will be the divisor, which has the property abovementioned. Thus is it found in 6, the double of 3, the exponent of the Power of the cubes: so also 10, the double of 5, is the divisor of all 5th Powers; &c.</p><p>Any Power of the natural numbers 1, 2, 3, 4, 5, 6, &c, as the <hi rend="italics">n</hi>th Power, has as many orders of differences as there are units in the common exponent of all the numbers; and the last of those differences is a constant quantity, and equal to the continual product , continued till the last factor, or the number of factors, be <hi rend="italics">n,</hi> the exponent of the Powers. Thus,<pb n="272"/><cb/> <*>he <*>st Powers 1, 2, 3, 4, 5, &c, have but one order of dif<*>erences 1 1 1 1 &c, and that difference is 1. The 2d <*>wrs. 1, 4, 9, 16, 25, &c, have two orders of differences 3 5 7 9 2 2 2 and the last of these is . The 3d Pwrs. 1, 8, 27, 64, 125, &c, have three orders of differences 7 19 37 61 12 18 24 6 6 and the last of these is .</p><p>In like manner, the 4th or last differences of the 4th Powers, are each ; and the 5th or last differences of the 5th Powers, are each . And so on. Which property was first noticed by Peletarius.</p><p>And the same is true of the Powers of any other arithmetical progression , &c, viz, , &c, the number of the orders of differences being still the same exponent <hi rend="italics">n,</hi> and the last of those orders each equal to , the same product of factors as before, multiplied by the same Power of the common difference <hi rend="italics">d</hi> of the series of roots: as was shewn by Briggs.</p><p>And hence arises a very eafy and general way of raising all the Powers of all the natural numbers, viz, by common addition only, beginning at the last differences, and adding them all continually, one after another, up to the Powers themselves. Thus, to generate the series of cubes, or 3d Powers, adding always 6, the common 3d difference gives the 2d differences 12, 18, 24, &c; and these added to the 1st of the 1st differences 7, gives the rest of the said 1st differences; and these again added to the 1st cube 1, gives the rest of the series of cubes, 8, 27, 64, &c, as below. <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3dD.</cell><cell cols="1" rows="1" role="data">2dD.</cell><cell cols="1" rows="1" role="data">1stD.</cell><cell cols="1" rows="1" role="data">Cubes.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 1</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 8</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 27</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 64</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">125</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">216</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">&c.</cell></row></table></p><p><hi rend="italics">Commensurable in</hi> <hi rend="smallcaps">Power</hi>, is said of quantities which, though not commensurable themselves<*> have their squares, or some other Power of them, commensurable. Euclid confines it to squares. Thus, the diagonal and side of a square are commensurable in Power, their squares being as 2 to 1, or commensurable; though they are not commensurable themselves, being as √2 to 1.</p><p><hi rend="smallcaps">Power</hi> <hi rend="italics">of an Hyperbola,</hi> is the square of the 4th part of the conjugate axis.</p><p>PRACTICAL Arithmetic, Geometry, Mathematics, &c, is the part that regards the practice, or ap-<cb/> plication, as contradistinguished from the theoretical part.</p></div2></div1><div1 part="N" n="PRACTICE" org="uniform" sample="complete" type="entry"><head>PRACTICE</head><p>, in Arithmetic, is a rule which expeditiously and compendiously answers questions in the golden rule, or rule-of-three, especially when the first term is 1. See rules for this purpose in all the books of practical arithmetic.</p><p>PRECESSION <hi rend="italics">of the Equinoxes,</hi> is a very slowmotion of them, by which they change their place, going from east to west, or backward, <hi rend="italics">in antecedentia,</hi> as astronomers call it, or contrary to the order of the signs.</p><p>From the late improvements in astronomy it appears, that the pole, the solstices, the equinoxes, and all the other points of the ecliptic, have a retrograde motion, and are constantly moving from east to west, or from Aries towards Pisces, &c; by means of which, the equinoctial points are carried farther and farther back, among the preceding signs or stars, at the rate of about 50″ 1/4 each year; which retrograde motion is called the Precession, Recession, or Retrocession of the Equinoxes.</p><p>Hence, as the stars remain immoveable, and the equinoxes go backward, the stars will seem to move more and more eastward with respect to them; for which reason the longitudes of all the stars, being reckoned from the first point of Aries, or the vernal equinox, are continually increasing.</p><p>From this <*>ause it is, that the constellations seem all to have changed the places assigned to them by the ancient astronomers. In the time of Hipparchus, and the oldest astronomers, the equinoctial points were <*>ixed to the first stars of Aries and Libra: but the signs do not now answer to the same points; and the stars which were then in conjunction with the sun when he was in the equinox, are now a whole sign, or 30 degrees, to the eastward of it: so, the first star of Aries is now in the portion of the ecliptic, called Taurus; and the stars of Taurus are now in Gemini; and those of Gemini in Cancer; and so on.</p><p>This seeming change of place in the stars was first observed by Hipparchus of Rhodes, who, 128 years before Christ, found that the longitudes of the stars in his time were greater than they had been before observed by Tymochares, and than they were in the sphere of Eudoxus, who wrote 380 years before Christ. Ptolomy also perceived the gradual change in the longitudes of the stars; but he stated the quantity at too little, making it but 1° in 100 years, which is at the rate of only 36″ per year. Y-hang, a Chinese, in the year 721, stated the quantity of this change at 1° in 83 years, which is at the rate of 43″ 1/2 per year. Other more modern astronomers have made this precession still more, but with some small differences from each other; and it is now usually taken at 50″ 1/4 per year. All these rates are deduced from a comparison of the longitude of certain stars as observed by more ancient astronomers, with the later observations of the same stars; viz. by subtracting the former from the latter, and dividing the remainder by the number of years in the interval between the dates of the observations. Thus, by a medium of a great number of comparisons, the quantity of the annual change has been fixed at 50″ 1/4, according to which rate it will require 25791 years for the equinoxes to make their revolution westward quite around the circle, and return to the same point again.<pb n="273"/><cb/></p><p>Thus, by taking the longitudes of the principal stars established by Tycho Brahe, in his book Astronomiæ Instauratæ Progymnasmata, pa. 208 and 232, for the beginning of 1586, and comparing them with the same as determined for the year 1750, by M. de la Caille, for that interval of 164 years, there will be obtained the following differences of longitude of several stars; viz, <table><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" role="data">Arietis</cell><cell cols="1" rows="1" role="data">2°</cell><cell cols="1" rows="1" role="data">17′</cell><cell cols="1" rows="1" role="data">37″</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Aldebaran</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">45</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">  1</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">26</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Regulus</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">32</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" role="data">Virginis</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">  1</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" role="data">Libræ</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Antares</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">28</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" role="data">Tauri</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">58</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">38</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" role="data">Cancri</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">12</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" role="data">Leonis</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">38</cell></row><row role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">10</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Medium of these 15 stars</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">35</cell></row></table> which divided by 164, the interval of years, gives 50″.336, or nearly 50″ 1/2, or after the rate of 1° 23′ 53″ 1/3 in 100 years, or 25,748 years for the whole revolution, or circle of 360 degrees. And nearly the same conclusion results from the longitudes of the stars in the Britannic catalogue, compared with those of the still later catalogues. See De la Lande's Astronomie, in several places.</p><p>The Ancients, and even some of the Moderns, have taken the equinoxes to be immoveable; and ascribed that change in the distance of the stars from it, to a real motion of the orb of the sixed stars, which they supposed had a slow revolution about the poles of the ecliptic; so as that all the stars perform their circuits in the ecliptic, or its parallels, in the space of 25,791 years; after which they should all return again to their former places.</p><p>This period the Ancients called the Platonic, or great year; and imagined that at its completion every thing would begin as at first, and all things come round in the same order as they have done before.</p><p>The phenomena of this retrograde motion of the equinoxes, or intersections of the equinoctial with the ecliptic, and consequently of the conical motion of the earth's axis, by which the pole of the equator describes a small circle in the same period of time, may be understood and illustrated by a scheme, as follows: Let NZSVL be the earth, SONA its axis produced to the starry heavens, and terminating in A, the present north pole of the heavens, which is vertical to N, the north pole of the earth. Let EOQ be the equator, T<*>Z the tropic of cancer, and VT<*> the tropic of capricorn; VOZ the ecliptic, and BO its axis, both of which are immoveable among the stars. But as the equinoctial points recede in the ecliptic, the earth's<cb/> axis SON is in motion upon the earth's centre O, in such a manner as to describe the double cone NO<hi rend="italics">n</hi> <figure/> and SO<hi rend="italics">s,</hi> round the axis of the ecliptic BO, in the time that the equinoctial points move round the ecliptic, which is 25,791 years; and in that length of time, the north pole of the earth's axis, produced, describes the circle ABCDA in the starry heavens, round the pole of the ecliptic, which keeps immoveable in the centre of that circle. The earth's axis being now 23° 28′ inclined to the axis of the ecliptic, the circle ABCDA, described by the north pole of the earth's axis produced to A, is 46° 56′ in diameter, or double the inclination of the earth's axis. In consequence of this, the point A, which is at present the north pole of the heavens, and near to a star of the 2d magnitude in the <*>nd of the Little Bear's tail, must be deserted by the earth's axis; which moving backwards 1 degree every 71 2/3 years nearly, will be directed towards the star or point B in 6447 3/4 years hence; and in double of that time, or 12,895 1/2 years, it will be directed towards the star or point C; which will then be the north pole of the heavens, although it is at present 8 1/2 degrees south of the zenith of London L. The present position of the equator EOQ will then be changed into <hi rend="italics">e</hi>O<hi rend="italics">q,</hi> the tropic of cancer T<*>Z into V<hi rend="italics">t</hi><*>, and the tropic of capricorn VT<*> into <hi rend="italics">t</hi><*>Z; as is evident by the figure. And the sun, in the same part of the heavens where he is now over the earthly tropic of capricorn, and makes the shortest days and longest nights in the northern hemisphere, will then be over the earthly tropic of cancer, and make the days longest and nights shortest. So that it will require 12,895 1/2 years yet more, or from that time, to bring the north pole N quite round, so as to be directed toward that point of the heavens which is vertical to it at present. And then, and not till then, the same stars which at present describe the equator, tropics, and polar circles, &c, by the earth's diurnal motion, will describe them over again.</p><p>From this shifting of the equinoctial points, and with them all the signs of the ecliptic, it follows, that those stars which in the infancy of astronomy were in Aries, are now found in Taurus; those of Taurus in Gemini, &c. Hence likewise it is, that the stars which rose or set at any particular season of the year, in the times of Hesiod, Eudoxus, Virgil, Pliny, &c,<pb n="274"/><cb/> by no means answer at this time to their descriptions.</p><p>As to the physical cause of the Precession of the equinoxes, Sir Isaac Newton demonstrates, that it arises from the broad or slat spheroidal sigure of the earth; which itself arises from the earth's rotation about its axis: for as more matter has thus been accumulated all round the equatorial parts than any where else on the earth, the sun and moon, when on either side of the equator, by attracting this redundant manner, bring the equator sooner under them, in every return towards it, than if there was no such accumulation.</p><p>Sir Isaac Newton, in determining the quantity of the annual Precession from the theory of gravity, on supposition that the equatorial diameter of the earth is to the polar diameter, as 230 to 229, finds the sun's action sufficient to produce a Precession of 9″ 1/8 only; and collecting from the tides the proportion between the sun's force and the moon's to be as 1 to 4 1/2, he settles the mean Precession resulting from their joint actions, at 50″; which, it must be owned, is nearly the same as it has since been found by the best observations; and yet several other mathematicians have since objected to the truth of Sir Isaac Newton's computation.</p><p>Indeed, to determine the quantity of the Precession arising from the action of the sun, is a problem that has been much agitated among modern mathematicians; and although they seem to agree as to Newton's mistake in the solution of it, they have yet generally disagreed from one another. M. D'Alembert, in 1749, printed a treatise on this subject, and claims the honour of having been the first who rightly determined the method of resolving problems of this kind. The subject has been also considered by Euler, Frisius, Silvabelle, Walmesley, Simpson, Emerson, La Place, La Grange, Landen, Milner, and Vince.</p><p>M. Silvabelle, stating the ratio of the earth's axis to be that of 178 to 177, makes the annual Precession caused by the sun 13″ 52‴, and that of the moon - - 34 17; making the ratio of the lunar force to the solar, to be that of 5 to 2; also the nutation of the earth's axis caused by the moon, during the time of a semirevolution of the pole of the moon's orbit, i. e. in 9 1/3 years, he makes 17″ 51‴.—M. Walmesley, on the supposition that the ratio of the earth's diameters is that of 230 to 229, and the obliquity of the ecliptic to the equator 23° 28′ 30″, makes the annual Precession, owing to the sun's force, equal to 10″.583; but supposing the ratio of the diameters to be that of 178 to 177, that Precession will be 13″.675.—Mr. Simpson, by a different method of calculation, determines the whole annual precession of the equinoxes caused by the sun, at 21″ 6‴; and he has pointed out the errors of the computations proposed by M. Silvabelle and M. Walmesley.—Mr. Milner's deduction agrees with that of Mr. Simpson, as well as Mr. Vince's; and their papers contain besides several curious particulars relative to this subject. But for the various principles and reasonings of these mathematicians, see Philos. Trans. vol. 48, pa 385; vol. 49, pa. 704; vol. 69, pa. 505; and vol. 77. pa. 363; as also the writings of Simpson, Emer-<cb/> son, Landen, &c; also De la Lande's Astronomie, and the Memoirs of the Acad. Sci. in several places.</p><p>As to the effect of the planets upon the equinoctial points, M. De la Place, in his new researches on this article, finds that their action causes those points to advance by 0″.2016 in a year, along the equator, or 0″.1849 along the ecliptic; from whence it follows that the quantity of the luni-solar Precession must be 50″.4349, since the total observed Precession is 50″ 1/4, or 50″.25.</p><p><hi rend="italics">To find the Precession in right ascension and declination.</hi> Put <hi rend="italics">d</hi> = the declination of a star, and <hi rend="italics">a</hi> = its right ascension; then their annual variations of Precessions will be nearly as follow: viz, 20″.084 X cos. <hi rend="italics">a</hi> = the annual preces. in declinat. and 46″.0619 + 20″.084 X sin. <hi rend="italics">a</hi> X tang. <hi rend="italics">d</hi> = that of right ascension. See the Connoissance des Temps for 1792, pa. 206, &c.</p></div1><div1 part="N" n="PRESS" org="uniform" sample="complete" type="entry"><head>PRESS</head><p>, in Mechanics, is a machine made of iron or wood, serving to compress or squeeze any body very close, by means of screws.</p><p>The common Presses consist of six members, or pieces; viz, two flat and smooth planks; between which the things to be pressed are laid; two screws, or worms, fastened to the lower plank, and passing through two holes in the upper; and two nuts, serving to drive the upper plank, which is moveable, against the lower, which is stable, and without motion.</p><p>PRESSION. See <hi rend="smallcaps">Pressure.</hi></p></div1><div1 part="N" n="PRESSURE" org="uniform" sample="complete" type="entry"><head>PRESSURE</head><p>, is properly the action of a body which makes a continual effort or endeavour to move another; such as the action of a heavy body supported by a horizontal table; in contradistinction from percussion, or a momentary force or action. Pressure equally respects both bodies, that which presses, and that which is pressed; from the mutual equality of action and reaction.</p><p>Pressure, in the Cartesian Philosophy, is an impulsive kind of motion, or rather an endeavour to move, impressed on a fluid medium, and propagated through it. In such a pressure the Cartesians suppose the action of light to consist. And in the various modifications of this Pressure, by the surfaces of bodies, on which that medium presses, they suppose the various colours to consist, &c. But Newton shews, that if light consisted only in a Pressure, propagated without actual motion, it could not agitate and warm such bodies as reflect and refract it, as we actually find it does; and if it consisted in an instantaneous motion, or one propagated to all distances in an instant, as such Pressure supposes, there would be required an infinite force to produce that motion every moment, in every lucid particle. Farther, if light consisted either in Pressure, or in motion propagated in a fluid medium, whether instantaneously, or in time, it must follow, that it would inflect itself <hi rend="italics">ad umbram;</hi> for Pressure, or motion, in a fluid medium, cannot be propagated in right lines, beyond any obstacle which shall hinder any part of the motion; but will inflect and diffuse itself, every way, into those parts of the quiescent medium which lie beyond the said obstacle.<pb n="275"/><cb/> Thus the force of gravity tends downward; but the Pressure which arises from that force of gravity, tends every way with an equable force; and, with equal ease and force, is propagated in crooked lines, as in straight ones. Waves on the surface of water, while they slide by the sides of any large obstacle, do inflect, dilate, and diffuse themselves gradually into the quiescent water lying beyond the obstacle. The waves, pulses, or vibrations of the air, in which sounds consist, do manifestly inflect themselves, though not so much as the waves of water; for the sound of a bell, or of a cannon, can be heard over a hill, which intercepts the sonorous object from our sight; and sounds are propagated as easily through crooked tubes, as through straight ones. But light is never observed to go in curved lines, nor to inflect itself <hi rend="italics">ad umbram;</hi> for the fixed stars do immediately disappear on the interposition of any of the planets; as well as some parts of the sun's body, by the interposition of the Moon, or Venus, or Mercury.</p><p><hi rend="smallcaps">Pressure</hi> <hi rend="italics">of Air, Water, &c.</hi> See <hi rend="smallcaps">Air, Water</hi>, &c.</p><p>The effects anciently ascribed to the fuga vacui, are now accounted for from the weight and Pressure of the air.</p><p>The Pressure of the air on the surface of the earth, is balanced by a column of water of the same base, and about 34 feet high; or of one of Mercury of near 30 inches high; and upon every square inch at the earth's surface, that Pressure amounts to about 14 3/4 pounds avoirdupois. The elasticity of the air is equal to that Pressure, and by means of that Pressure, or elasticity, the air would rush into a vacuum with a velocity of about 1370 feet per second. At different heights above the earth's surface, the Pressure of the air is as its density and elasticity, and each decreases in such sort, that when the heights above the surface increase in arithmetical progression, the Pressure &c decrease in geometri- <figure/> cal progression: and hence if the axis BC of a logarithmic curve AD be erected perpendicular to the horizon, and if the ordinate AB denote the Pressure, or elasticity, or density of the air, at the earth's sursace, then will any other absciss <table><row role="data"><cell cols="1" rows="1" role="data">EF</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(9)">}</hi>denote the Pressure &c at the altitude<hi rend="size(9)">{</hi></cell><cell cols="1" rows="1" role="data">BE,</cell></row><row role="data"><cell cols="1" rows="1" role="data">GH</cell><cell cols="1" rows="1" role="data">BG,</cell></row><row role="data"><cell cols="1" rows="1" role="data">IK</cell><cell cols="1" rows="1" role="data">BI,</cell></row></table></p><p>The Pressure of water, as this fluid is every where of the same density, is as its depth at any place, and in all directions the same; and upon a square foot of sursace, every foot in height presses with the force of a weight of 1000 ounces or 62 (1/2)lbs avoirdupois. And hence, if AB be the depth <figure/><cb/> of water in any vessel, and BE denote its Pressure at the depth B; by joining AE and drawing any other ordinates FC, HI; then shall these ordinates FG, HI, &c, denote the Pressure at the corresponding depths AG, AI, &c; also the area of the triangle ABE will denote the whole Pressure against the whole upright side AB, and which therefore is but half the Pressure on the bottom of the same area as the side. Moreover, if a hole were opened in the bottom or side of the vessel at B, the water, from the Pressure of the superincumbent fluid, would issue out with the velocity of 8√AB feet per second nearly; AB being estimated in feet.</p><p><hi rend="italics">Centre of</hi> <hi rend="smallcaps">Pressure</hi>, in Hydrostatics, is that point of any plane, to which, if the total Pressure were applied, its effect upon the plane would be the same as when it was distributed unequally over the whole; or it is that point in which the whole Pressure may be conceived to be united; or it is that point to which, if a force were applied equal to the total Pressure, but with an opposite direction, it would exactly balance, or restrain the effect of the Pressure, so that the body pressed on will not incline to either side. Thus, if ABCD (2d fig. above) be a vessel of water, and the side BC be pressed upon with a force equivalent to 20 pounds of water, this force is unequally distributed over BC, for the parts near B are less pressed than those near C, which are at a greater depth; and therefore the efforts of all the particular Pressures are united in some point E, which is nearer to C than to B; and that point E is called the centre of Pressure: if to that point a force equivalent to 20 pounds weight be applied, it will affect the plane BC in the same manner as by the Pressure of the water distributed unequally over the whole; and if to the same point the same force be applied in a contrary direction to that of the Pressure of the water, the force and the Pressure will balance each other, and by opposite endeavours destroy each other's effects. Supposing a cord EFG fixed at E, and passing over the pulley F, has a weight of 20 pounds annexed to it, and that the part of the cord FE is perpendicular to BC; then the effort of the weight G is equal, and its direction contrary, to that of the Pressure of the water. Now if E be the centre of Pressure, these two powers will be in equilibrio, and mutually defeat each other's endeavours.</p><p>This point E, or the centre of Pressure, is the same with the centre of percussion of the plane BC, the point of suspension being B, the surface of the water. And if the plane be oblique, the case is still the same, taking for the axis of suspension, the intersection of that plane and the surface of the sluid, both produced if necessary. See Cotes's Lectures, pa. 40, &c.</p><p>The centre of Pressure upon a plane parallel. to the horizon, or upon any plane where the Pressure is uniform, is the same as the centre of gravity of that plane. For the Pressure acts upon every part in the same manner as gravity does.</p><p>PRIMARY <hi rend="italics">Planets,</hi> are those which revolve round the sun as a centre. Such are the planets Mercury, Venus, Terra the Earth, Mars, Jupiter, Saturn, and Herfchel, and perhaps others. They are thus called, in contradistinction from the secondary planets, or satellites, which revolve about their respective Primaries. See <hi rend="smallcaps">Planet.</hi><pb n="276"/><cb/></p></div1><div1 part="N" n="PRIMES" org="uniform" sample="complete" type="entry"><head>PRIMES</head><p>, denote the first divisions into which some whole or integer is divided. As, a minute, or Prime minute, the 60th part of a degree; or the first place of decimals, being the 10th parts of units; or the first division of inches in duodecimals, being the 12th parts of inches; &c.</p><p><hi rend="smallcaps">Prime</hi> <hi rend="italics">Numbers,</hi> are those which can only be measured by unity, or exactly divided without a remainder, 1 being the only aliquot part: as 2, 3, 5, 7, 11, 13, 17,<cb/> &c. And they are otherwise called Simple, or Incomposite numbers. No even number is a Prime, because every even number is divisible by 2. No number that ends with 0 or 5 is a Prime, the former being divisible by 10, and the latter by 5. The following Table contains all the Prime numbers, and all the odd composite numbers, under 10,000, with the least Prime divisors of these; the description, nature, and use of which, see immediately following the Table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=36 align=center" role="data"><hi rend="italics">A Table of Prime and Composite Odd Numbers, under</hi> 10,000.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">33</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">01</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">01</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">03</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">03</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">07</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">07</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">09</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">09</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">31</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">1<*></cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell></row></table><pb n="277"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=36 align=center" role="data"><hi rend="italics">A Table of Prime and Composite Odd Numbers, under</hi> 10,000.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">67</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">01</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">01</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">03</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">03</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">07</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">07</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">09</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">09</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell 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cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell></row></table><pb n="278"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=34 align=center" role="data"><hi rend="italics">A Table of Prime and Composite Odd Numbers, under</hi> 10,000.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" role="data">99</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">01</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">01</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">03</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">03</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">07</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">07</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell 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role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">97</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell></row></table><pb n="279"/><cb/></p><p>Out of the foregoing Table, are omitted all the odd numbers that end with 5, because it is known that 5 is a divisor, or aliquot part of every such number.—— The disposition of the Prime and composite odd numbers in this Table, is along the top line, and down the first or left-hand column; while their least Prime divisors are placed in the angles of meeting in the body of the page. Thus, the figures along the top line, viz, 0, 1, 2, 3, 4, &c, to 99, are so many hundreds; and those down the first column, from 1 to 99 also, are units or ones; and the former of these set before the latter, make up the whole number, whether it be Prime or composite; just like the disposition of the natural numbers in a table of logarithms. So the 16 in the top line, joined with the 19 in the first column, makes the number 1619: the angle of their meeting, viz, of the column under 16, and of the line of 19, being blank, shews that the number 1619 has no aliquot part or divisor, or that it is a Prime number. In like manner, all the other numbers are Primes that have no figure in their angle of meeting, as the numbers 41, 401, 919, &c. But when the two parts of any number have some figure in their angle of meeting, that figure is the least divisor of the number, which is therefore not a Prime, but a composite number: so 301 has 7 for its least divisor, and 803 has 11 for its least divisor, and 1633 has 23 for its least divisor.</p><p>Hence, by the foregoing Table, are immediately known at sight all the Prime numbers up to 10,000; and hence also are readily found all the divisors or aliquot parts of the composite numbers, namely in this manner: Find the least divisor of the given number in the Table, as above; divide the given number by this divisor, and consider the quotient as another or new number, of which find the least divisor also in the Table, dividing the said quotient by this last divisor; and so on, dividing always the last quotient by its least divisor found in the Table, till a quotient be found that is a Prime number: then are the said divisors and the last or Prime quotient, all the simple or Prime divisors of the first given number; and if these simple divisors be multiplied together thus, viz, every two, and every three, and every four, &c, of them together, the several products will make up the compound divisors or aliquot parts of the first given number; noting, that if the given number be an even one, divide it by 2 till an odd number come out.</p><p>For example, to find all the divisors or component factors of the number 210. This being an even number, dividing it by 2, one of its divisors, gives 105; and this ending with 5, dividing it by 5, another of its factors, gives 21; and the least divisor of 21, by the <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">42</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">105</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">210</cell></row></table> Table is 3, the quotient from which is 7; therefore all the Prime or simple factors of the given number. are 2, 3, 5, 7. Set these therefore down in the first line as in the margin; then multiply the 2 by the 3, and set the product 6 below the 3; next multiply the 5 by all that precede it, viz, 2, 3, 6, and set the products below the 5; lastly multiply the 7 by all the seven factors preceding it, and set the products below the 7; so shall we have all the fac-<cb/> tors or divisors of the given number 210, which are these, viz, 2, 3, 5, 6, 7, 10, 14, 15, 21, 30, 35, 42, 70, 105.</p><p><hi rend="smallcaps">Prime</hi> <hi rend="italics">Vertical,</hi> is that vertical circle, or azimuth, which is perpendicular to the meridian, and passes through the east and west points of the horizon.</p><p><hi rend="smallcaps">Prime</hi> <hi rend="italics">Verticals,</hi> in Dialling, or <hi rend="smallcaps">Prime</hi>-<hi rend="italics">Vertical</hi> Dials, are those that are projected on the plane of the Prime vertical circle, or on a plane parallel to it. These are otherwise called direct, erect, north, or south dials.</p><p><hi rend="smallcaps">Prime</hi> <hi rend="italics">of the Moon,</hi> is the new moon at her first appearance, for about 3 days after her change. It means also the <hi rend="smallcaps">Golden</hi> <hi rend="italics">Number;</hi> which see.</p><p>PRIMUM <hi rend="italics">Mobile,</hi> in the Ptolomaic Astronomy, is supposed to be a vast sphere, whose centre is that of the world, and in comparison of which the earth is but a point. This they describe as including all other spheres within it, and giving motion to them, turning itself and all the rest quite round in 24 hours.</p></div1><div1 part="N" n="PRINCIPAL" org="uniform" sample="complete" type="entry"><head>PRINCIPAL</head><p>, in Arithmetic, or in Commerce, is the sum lent upon interest, either simple or compound.</p><p><hi rend="smallcaps">Principal</hi> <hi rend="italics">Point,</hi> in Perspective, is a point in the perspective plane, upon which falls the principal ray, or line from the eye perpendicular to the plane. This point is in the intersection of the horizontal and vertical planes; and is also called the <hi rend="italics">point of sight,</hi> and <hi rend="italics">point of the eye,</hi> or <hi rend="italics">centre of the picture,</hi> or again the <hi rend="italics">point of concurrence.</hi></p><p><hi rend="smallcaps">Principal</hi> <hi rend="italics">Ray,</hi> in Perspective, is that which passes from the spectator's eye perpendicular to the picture or perspective plane, and so meeting it in the principal point.</p></div1><div1 part="N" n="PRINGLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PRINGLE</surname> (Sir <foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, Baronet, the late worthy president of the Royal Society, was born at Stichelhouse, in the county of Roxburgh, North Britain, April 10, 1707. His father was Sir John Pringle, of Stichel, Bart. and his mother Magdalen Elliott, was sister to Sir Gilbert Elliott, of Stobs, Baronet. He was the youngest of several sons, three of whom, besides himself, arrived to years of maturity. After receiving his grammatical education at home, he was sent to the university of St. Andrews, where having staid some years, he removed to Edinburgh in 1727, to study physic, that being the profession which he now determined to follow. He staid however only one year at Edinburgh, being desirous of going to Leyden, which was then the most celebrated school for medicine in Europe. Dr. Boerhaave, who had brought that university into great reputation, was considerably advanced in years, and Mr. Pringle was desirous of benefiting by that great man's lectures. After having gone through his proper course of studies at Leyden, he was admitted, in 1730, to his doctor of physic's degree; upon which occasion his inaugural dissertation, <hi rend="italics">De Marcore Senili,</hi> was printed. On quitting Leyden, Dr. Pringle returned and settled at Edinburgh as a physician, where, in 1734, he was appointed, by the magistrates and council of the city, to be joint professor of pneumatics and moral philosophy with Mr. Scott, during this gentleman's life, and sole professor after his decease; being also admitted at the same time a member of the university. In discharging the duties of this new employ-<pb n="280"/><cb/> ment, his text-book was Puffendorff <hi rend="italics">de Officio Hominis et Civis;</hi> agreeably to the method he pursued through life, of making fact and experiment the basis of science.</p><p>Dr. Pringle continued in the practice of Physic at Edinburgh, and in duly performing the office of professor, till 1742, when he was appointed physician to the earl of Stair, who then commanded the Britifh army. By the interest of this nobleman, Dr. Pringle was constituted, the same year, physician to the military hospital in Flanders, with a salary of 20 shillings a-day, and the right to half pay for life. On this occasion he was permitted to retain his professorship of moral philosophy; two gentlemen, Messrs. Muirhead and Cleghorn teaching in his absence, as long as he requested it. The great attention which Dr. Pringle paid to his duty as an army physician, is evident from every page of his <hi rend="italics">Treatise on the Diseases of the Army,</hi> in the execution of which office he was sometimes exposed to very imminent dangers. He soon after also met with no small affliction in the retirement of his great friend the earl of Stair, from the army. He offered to resign with his noble patron, but was not permitted: he was therefore obliged to content himself with testifying his respect and gratitude to him, by accompanying the earl 40 miles on his return to England; after which he took leave of him with the utmost regret.</p><p>But though Dr. Pringle was thus deprived of the immediate protection of a nobleman who knew and esteemed his worth, his conduct in the duties of his station procured him effectual support. He attended the army in Flanders through the campaign of 1744, and so powerfully recommended himself to the duke of Cumberland, that in the spring following he had a commission, appointing him physician-general to the king's forces in the Low Countries, and parts beyond the seas; and on the next day he received a second commission from the duke, constituting him physician to the royal hospitals in those countries. In consequence of these promotions, he the same year resigned his professorship in the university of Edinburgh.</p><p>In 1745 he was also with the army in Flanders; but was recalled from that country in the latter end of the year, to attend the forces which were to be sent against the rebels in Scotland. At this time he had the honour of being chosen F. R. S. and the Society had good reason to be pleased with the addition of such a member. In the beginning of 1746, Dr. Pringle accompanied, in his official capacity, the duke of Cumberland in his expedition against the rebels; and remained with the forces, after the battle of Culloden, till their return to England the following summer. In 1747 and 1748, he again attended the army abroad; but in the autumn of 1748, he embarked with the forces for England, on the signing of the treaty of Aix-la-Chapelle.</p><p>From that time he mostly resided in London, where, from his known skill and experience, and the reputation he had acquired, he might reasonably expect to succeed as a physician. In 1749 he was appointed physician in ordinary to the duke of Cumberland. And in 1750 he published, in a letter to Dr. Mead, <hi rend="italics">Observations on the Gaol or Hospital Fever:</hi> this piece, with some alterations, was afterwards included in his grand work on the <hi rend="italics">Diseases of the Army.</hi><cb/></p><p>In this, and the two following years Dr. Pringle communicated to the Royal Society his celebrated <hi rend="italics">Experiments upon Septic and Antiseptic Substances, with Remarks relating to their Use in the Theory of Medicine;</hi> some of which were printed in the Philosophical Transactions, and the whole were subjoined, as an appendix, to his <hi rend="italics">Observations on the Diseases of the Army.</hi> Those experiments procured for the ingenious author the honour of Sir Godfrey Copley's gold medal; besides gaining him a high and just reputation as an experimental philosopher. He gave also many other curious papers to the Royal Society: thus, in 1753, he presented, <hi rend="italics">An Account of several Persons seized with the Gaol Fever by working in Newgate; and of the Manner by which the Infection was communicated to one entire Family;</hi> in the Philos. Trans. vol. 48. His next communication was, <hi rend="italics">A remarkable case of Fragility, Flexibility, and Dissolution of the Bones;</hi> in the same vol.—In the 49th volume, are accounts which he gave of an Earthquake felt at Brussels; of another at Glasgow and Dunbarton; and of the Agitation of the Waters, Nov. 1, 1756, in Scotland and at Hamburgh.—The 50th volume contains his Observations on the Case of lord Walpole, of Woollerton; and a Relation of the Virtues of Soap, in Dissolving the Stone.—The next volume is enriched with two of the doctor's articles, of considerable length, as well as value. In the first, he hath collected, digested, and related, the different accounts that had been given of a very extraordinary Fiery Meteor, which appeared the 26th of November 1758; and in the second he hath made a variety of remarks upon the whole, displaying a great degree of philosophical sagacity.—Besides his communications in the Philosophical Transactions, he gave, in the 5th volume of the Edinburgh Medical Essays, an Account of the Success of the <hi rend="italics">Vitrum ceratum Antimonii.</hi></p><p>In 1752, Dr. Pringle married Charlotte, the second daughter of Dr. Oliver, an eminent physician at Bath: a connection which however did not last long, the lady dying in the space of a few years. And nearly about the time of his marriage, he gave to the public the first edition of his <hi rend="italics">Observations on the Diseases of the Army;</hi> which afterwards went through many editions with improvements, was translated into the French, the German, and the Italian languages, and deservedly gained the author the highest credit and encomiums. The utility of this work however was of still greater importance than its reputation. From the time that the doctor was appointed a physician to the army, it seems to have been his grand object to lessen, as far as lay in his power, the calamities of war; nor was he without considerable success in his noble and benevolent design. The benefits which may be derived from our author's great work, are not solely confined to gentlemen of the medical profession. General Melville, a gentleman who unites with his military abilities the spirit of philosophy, and the feelings of humanity, was enabled, when governor of the Neutral Islands, to be singularly useful, in consequence of the instructions he had received from Dr. Pringle's book, and from personal conversation with him. By taking care to have his men always lodged in large, open, and airy apartments, and by never letting his forces remain long enough in swampy places to be injured by the noxious air of such places, the general was the<pb n="281"/><cb/> happy instrument of saving the lives of seven hundred soldiers.</p><p>Though Dr. Pringle had not for some years been called abroad, he still held his place of physician to the army; and in the war that began in 1755, he attended the camps in England during three seasons. In 1758, however, he entirely quitted the service of the army; and being now determined to fix wholly in London, he was the same year admitted a licentiate of the college of physicians.—After the accession of king George the 3d to the throne of Great Britain, Dr. Pringle was appointed, in 1761, physician to the queen's household; and this honour was succeeded, by his being constituted, in 1763, physician extraordinary to the queen. The same year he was chosen a member of the Academy of Sciences at Haarlem, and elected a fellow of the Royal College of Physicians in London.—In 1764, on the decease of Dr. Wollaston, he was made physician in ordinary to the queen. In 1766 he was elected a foreign member, in the physical line, of the Royal Society of Sciences at Gottingen, and the same year he was raised to the dignity of a baronet of Great Britain. In 1768 he was appointed physician in ordinary to the late princess dowager of Wales.</p><p>After having had the honour to be several times elected into the council of the Royal Society, Sir John Pringle was at length, viz, Nov. 30, 1772, in consequence of the death of James West Esq. elected president of that learned body. His election to this high station, though he had so respectable a character as the late Sir James Porter for his opponent, was carried by a very considerable majority. Sir John Pringle's conduct in this honourable station fully justified the choice the Society made of him as their president. By his equal, impartial, and encouraging behaviour, he secured the good will and best exertions of all for the general benefit of science, and true interests of the Society, which in his time was raised to the pinnacle of honour and credit. Instead of splitting the members into opposite parties, by cruel, unjust, and tyrannical conduct, as has sometimes been the case, to the ruin of the best interests of the Society, Sir John Pringle cherished and happily united the endeavours of all, collecting and directing the energy of every one to the common good of the whole. He happily also struck out a new way to distinction and usefulness, by the discourses which were delivered by him, on the annual assignment of Sir Godfrey Copley's medal. This gentleman had originally bequeathed five guineas, to be given at each anniversary meeting of the Royal Society, by the determination of the president and council, to the person who should be the author of the best paper of experimental observations for the year. In process of time, this pecuniary reward, which <*>ould never be an important consideration to a man of an enlarged and philosophical mind, however narrow his circumstances might be, was changed into the more liberal sorm of a gold medal; in which form it is become a truly honourable mark of distinction, and a just and laudable object of ambition. No doubt it was always usual for the president, on the delivery of the medal, to pay some compliment to the gentleman on whom it was bestowed; but the custom of making a set speech on the occasion, and of entering into the history of that part of philosophy to which the experiments, or the subject of the<cb/> paper related, was first introduced by Martin Folkes Esq. The discourses however which he and his successors delivered, were very short, and were only inserted in the minute-books of the Society. None of them had ever been printed before Sir John Pringle was raised to the chair. The first speech that was made by him being much more elaborate and extended than usual, the publication of it was desired; and with this request, it is said, he was the more ready to comply, as an absurd account of what he had delivered had appeared in a newspaper. Sir John was very happy in the subject of his first discourse. The discoveries in magnetism and electricity had been succeeded by the inquiries into the various species of air. In these enquiries, Dr. Priestley, who had already greatly distinguished himselt by his electrical experiments, and his other philosophical pursuits and labours, took the principal lead. A paper of his, intitled, <hi rend="italics">Observations on different Kinds of Air,</hi> having been read before the Society in March 1772, was adjudged to be deserving of the gold medal; and Sir John Pringle embraced with pleasure the occasion of celebrating the important communications of his friend, and of relating with accuracy and sidelity what had previously been discovered upon the subject.</p><p>It was not intended, we believe, when Sir John's first speech was printed, that the example should be followed: but the second discourse was so well received by the Society, that the publication of it was unanimously requested. Both the discourse itself, and the subject on which it was delivered, merited such a distinction. The composition of the second speech is evidently superior to that of the former one; Sir John having probably been animated by the favourable reception of his first effort. His account of the Torpedo, and of Mr. Walsh's ingenious and admirable experiments relative to the electrical properties of that extraordinary fish, is singularly curious. The whole discourse abounds with ancient and modern learning, and exhibits the worthy president's knowledge in natural history, as well as in medicine, to great advantage.</p><p>The third time that he was called upon to display his abilities at the delivery of the annual medal, was on a very beautiful and important occasion. This was no less than Mr. (now Dr.) Maskelyne's successful attempt completely to establish Newton's system of the universe, by his <hi rend="italics">Observations made on the Mountain Schehallien, for finding its attraction.</hi> Sir John laid hold of this opportunity to give a perspicuous and accurate relation of the several hypotheses of the Ancients, with regard to the revolutions of the heavenly bodies, and of the noble discoveries with which Copernicus enriched the astronomical world. He then traces the progress of the grand principle of gravitation down to Sir Isaac's illustrious confirmation of it; to which he adds a concise account of Messrs. Bouguer's and Condamine's experiment at Chimboraço, and of Mr. Maskelyne's at Schehallien. If any doubts still remained with respect to the truth of the Newtonian system, they were now completely removed.</p><p>Sir John Pringle had reason to be peculiarly satissied with the subject of his fourth discourse; that subject being perfectly congenial to his disposition and studies. His own life had been much employed in pointing out the means which tended not only to cure, but to pre-<pb n="282"/><cb/> vent the diseases of mankind; and it is probable, from his intimate friendship with captain Cook, that he might suggest to that sagacious commander some of the rules which he followed, in order to preserve the health of the crew of his ship, during his voyage round the world. Whether this was the case, or whether the method pursued by the captain to attain so salutary an end, was the result alone of his own reflections, the success of it was astonishing; and this celehrated voyager seemed well entitled to every honour which could be bestowed. To him the Society assigned their gold medal, but he was not present to receive the honour. He was gone out upon the voyage, from which he never returned. In this last voyage he continued equally successful in maintaining the health of his men.</p><p>The learned president, in his fifth annual dissertation, had an opportunity of displaying his knowledge in a way in which it had not hitherto appeared. The discourse took its rise from the adjudication of the prize medal to Mr. Mudge, then an eminent surgeon at Plymouth, on account of his valuable paper, containing <hi rend="italics">Directions for making the best Composition for the Metals of Reflecting Telescopes, together with a Description of the Process for Grinding, Polishing, and giving the Great Speculum the true Parabolic form.</hi> Sir John hath accurately related a variety of particulars, concerning the invention of reflecting telescopes, the subsequent improvements of these instruments, and the state in which Mr. Mudge found them, when he first set about working them to a greater perfection, till he had truly realized the expectation of Newton, who, above an hundred years ago, presaged that the public would one day possess a parabolic speculum, not accomplished by mathematical rules, but by mechanical devices.</p><p>Sir John Pringle's sixth and last discourse, to which he was led by the assignment of the gold medal to myself, on account of my paper intitled, <hi rend="italics">The Force of fired Gunpowder, and the Initial Velocity of Cannon Balls, determined by Experiments,</hi> was on the theory of gunnery. Though Sir John had so long attended the army, this was probably a subject to which he had heretofore paid very little attention. We cannot however help admiring with what perspicuity and judgment he hath stated the progress that was made, from time to time, in the knowledge of projectiles, and the scientific perfection to which it has been said to be carried in my paper. As Sir John Pringle was not one of those who delighted in war, and in the shedding of human blood, he was happy in being able to shew that even the study of artillery might be useful to mankind; and therefore this is a topic which he hath not forgotten to mention. Here ended our author's discourses upon the delivery of Sir Godfrey Copley's medal, and his presidency over the Royal Society at the same time, the delivering that medal into my hand being the last office he ever performed in that capacity; a ceremony which was attended by a greater number of the members, than had ever met together before upon any other occasion. Had he been permitted to preside longer in that chair, he would doubtless have found other occasions of displaying his acquaintance with the history of philosophy. But the opportunities which he had of signalizing himself in this respect were important in themselves, happily varied, and sufficient to gain him a solid and lasting reputation.<cb/></p><p>Several marks of literary distinction, as we have already seen, had been conferred upon Sir John Pringle, before he was raised to the president's chair. But after that event they were bestowed upon him in great abundance, having been elected a member of almost all the literary societies and institutions in Europe. He was also, in 1774, appointed physician extraordinary to the king.</p><p>It was at rather a late period of life when Sir John Pringle was chosen to be president of the Royal Society, being then 65 years of age. Considering therefore the great attention that was paid by him to the various and important duties of his office, and the great pains he took in the preparation of his discourses, it was natural to expect that the burthen of his honourable station should grow heavy upon him in a course of time. This burthen, though not increased by any great addition to his life, for he was only 6 years president, was somewhat augmented by the accident of a fall in the area in the back part of his house, from which he received some hurt. From these circumstances some persons have affected to account for his resigning the chair at the time when he did. But Sir John Pringle was naturally of a strong and robust frame and constitution, and had a fair prospect of being well able to discharge the duties of his situation for many years to come, had his spirits not been broken by the most cruel harassings and baitings in his office. His resolution to quit the chair arose from the disputes introduced into the Society, concerning the question, whether pointed or blunted electrical conductors are the most efficacious in preserving buildings from the pernicious effects of lightning, and from the cruel circumstances attending those disputes. These drove him from the chair. Such of those circumstances as were open and manifest to every one, were even of themselves perhaps quite sufficient to drive him to that resolution. But there were yet others of a more private nature, which operated still more powerfully and directly to produce that event; which may probably hereafter be laid before the public, when I shall give to them the history of the most material transactions of the Royal Society, especially those of the last 22 years, which I have from time to time composed and prepared with that view.</p><p>His intention of resigning however, was disagreeable to his friends, and the most distinguished members of the Society, who were many of them perhaps ignorant of the true motive for it. Accordingly, they earnestly solicited him to continue in the chair; but, his resolution being fixed, he resigned it at the anniversary meeting in 1778, immediately on delivering the medal, at the conclusion of his speech, as mentioned above.</p><p>Though Sir John Pringle thus quitted his particular relation to the Royal Society, and did not attend its meetings so constantly as he had formerly done, he still retained his literary connections in general. His house continued to be the resort of ingenious and philosophical men, whether of his own country, or from abroad; and he was frequent in his visits to his friends. He was held in particular esteem by eminent and learned foreigners, none of whom came to England without waiting upon him, and paying him the greatest respect. He treated them, in return, with distinguished civility and regard. When a number of gentlemen met at<pb n="283"/><cb/> his table, foreigners were usually a part of the company.</p><p>In 1780 Sir John spent the summer on a visit to Edinburgh; as he did also that of 1781; where he was treated with the greatest respect. In this last visit he presented to the Royal College of Physicians in that city, the result of many years labour, being ten folio volumes of <hi rend="italics">Medical and Physical Observations,</hi> in manuscript, on condition that they should neither be published, nor lent out of the library of the college on any account whatever. He was at the same time preparing two other volumes, to be given to the university, containing the formulas referred to in his annotations. He returned again to London, and continued for some time his usual course of life, receiving and paying visits to the most eminent literary men, but languishing and declining in his health and spirits, till the 18th of January 1782, when he died, in the 75th year of his age; the account of his death being every where received in a manner which shewed the high sense that was entertained of his merit.</p><p>Sir John Pringle's eminent character as a practical physician, as well as a medical author, is so well known, and so universally acknowledged, that an enlargement upon it cannot be necessary. In the exercise of his profession he was not rapacious; being ready, on various occasions, to give his advice without pecuniary views. The turn of his mind led him chiefly to the love of science, which he built on the firm basis of fact. With regard to philosophy in general, he was as averse to theory, unsupported by experiments, as he was with respect to medicine in particular. Lord Bacon was his favourite author; and to the method of investigation recommended by that great man, he steadily adhered. Such being his intellectual character, it will not be thought surprising that he had a dislike to Plato. And to metaphysical disquisitions he lost all regard in the latter part of his life.</p><p>Sir John had no great fondness for poetry. He had not even any distinguished relish for the immortal Shakespeare: at least he seemed too highly sensible of the defects of that illustrious bard, to give him the proper degree of estimation. Sir John had not in his youth been neglectful of philological enquiries, nor did he desert them in the last stages of his life, but cultivated even to the last a knowledge of the Greek language. He paid a great attention to the French language; and it is said that he was fond of Voltaire's critical writings. Among all his other pursuits, he never forgot the study of the English language. This he regarded as a matter of so much consequence, that he took uncommon pains with regard to the style of his compositions; and it cannot be denied, that he excelled in perspicuity, correctness, and propriety of expression. His six discourses in particular, delivered at the annual meetings of the Royal Society, on occasion of the prize medals, have been universally admired as elegant compositions, as well as critical and learned differtations. And this characteristic of them, seemed to increase and heighten, from year to year: a circumstance which argues rather an improvement of his faculties, than any decline of them, and that even after the accident which it was pretended occasioned his descent from the president's chair. So excellent indeed were these compositions esteem-<cb/> ed, that envy used to asperse his character with the imputation of borrowing the hand of another in those learned discourses. But how false such aspersion was, I, and I believe most of the other gentlemen who had the honour of receiving the annual medal from his hands, can fully testify. For myself in particular, I can witness for the last, and perhaps the best, that on the theory and improvements in gunnery, having been present or privy to his composition of every part of it.—Though our author was not fond of poetry, he had a great affection for the sister art, music. Of this art he was not merely an admirer, but became so far a practitioner in it, as to be a performer on the violoncello, at a weekly concert given by a society of gentlemen at Edinburgh. Besides a close application to medical and philosophical science, during the latter part of his life, he devoted much time to the study of divinity: this being with him a very favourite and intere<*>ting object.</p><p>If, from the intellectual, we pass on to the moral character of Sir John Pringle, we shall find that the ruling feature of it was integrity. By this principle he was uniformly actuated in the whole of his conduct and behaviour. He was equally distinguilhed for his sobriety. I and other persons have heard him declare, that he had never once in his life been intoxicated with liquor. In his friendships, he was ardent and steady. The intimacies which were formed by him, in the early part of his life, continued unbroken to the decease of the gentlemen with whom they were made; and were kept up by a regular correspondence, and by all the good offices that lay in his power.</p><p>With regard to Sir John's external manner of deportment, he paid a very respectful attention to those who were honoured with his friendship and esteem, and to such strangers as came to him well recommended. Foreigners in particular had good reason to be satisfied with the uncommon pains which he took to shew them every mark of civility and regard. He had however at times somewhat of a dryness and reserve in his behaviour, which had the appearance of coldness; and this was the case when he was not perfectly pleased with the persons who were introduced to him, or who happened to be in his company. His sense of integrity and dignity would not permit him to adopt that false and supersicial politeness, which treats all men alike, though ever so different in point of real estimation and merit, with the same shew of cordiality and kindness. He was above assuming the profession, without the reality of respect.</p></div1><div1 part="N" n="PRISM" org="uniform" sample="complete" type="entry"><head>PRISM</head><p>, in Geometry, is a body, or solid, whose two ends are any plane figures which are parallel, equal, and similar; and its sides, connecting those ends, are parallelograms.—Hence, every section parallel to the ends, is the same kind of equal and similar figure as the ends themselves are; and the Prism may be considered as generated by the parallel motion of this plane figure.</p><p>Prisms take their several particular names from the figure of their ends. Thus, when the end is a triangle, it is a Triangular Prism; when a square, a Square Prism; when a pentagon, a Pentagonal Prism; when a hexagon, a Hexagonal Prism; and so on. And hence the denomination Prism comprises also the cube and parallelopipedon, the former being a square Prism, and<pb n="284"/><cb/> the latter a rectangular one. And even a cylinder may be considered as a round Prism, or one that has an infinite number of sides. Also a Prism is said to be regular or irregular, according as the figure of its end is a regular or an irregular polygon.</p><p>The Axis of a Prism, is the line conceived to be drawn lengthways through the middle of it, connecting the centre of one end with that of the other end.</p><p>Prisms, again, are either right or oblique.</p><p>A <hi rend="italics">Right</hi> <hi rend="smallcaps">Prism</hi> is that whose sides, and its axis, are perpendicular to its ends; like an upright tower. And</p><p>An <hi rend="italics">Oblique</hi> <hi rend="smallcaps">Prism</hi>, is when the axis and sides are oblique to the ends; so that, when set upon one end, it inclines on one hand, like an inclined tower.</p><p>The principal properties of Prisms, are,</p><p>1. That all Prisms are to one another in the ratio compounded of their bases and heights.</p><p>2. Similar Prisms are to one another in the triplicate ratio of their like sides.</p><p>3. A Prism is triple of a pyramid of equal base and height; and the solid content of a Prism is found by multiplying the base by the perpendicular height.</p><p>4. The upright surface of a right Prism, is equal to a rectangle of the same height, and its breadth equal to the perimeter of the base or end. And therefore such upright surface of a right Prism, is found by multiplying the perimeter of the base by the perpendicular height. Also the upright surface of an oblique Prism is found by computing those of all its parallelogram sides separately, and adding them together.</p><p>And if to the upright surface be added the areas of the two ends, the sum will be the whole surface of the Prism.</p><div2 part="N" n="Prism" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Prism</hi></head><p>, in Dioptrics, is a piece of glass in form of a triangular Prism: which is much used in experiments concerning the nature of light and colours.</p><p>The use and phenomena of the Prism arise from its sides not being parallel to each other; from whence it separates the rays of light in their passage through it, by coming through two sides of one and the same angle.</p><p>The more general of these phenomena are enumerated and illustrated under the article Colour; which are sufficient to prove, that colours do not either consist in the contorsion of the globules of light, as Des Cartes imagined; nor in the obliquity of the pulses of the etherial matter, as Hook fancied; nor in the constipation of light, and its greater or less concitation, as Dr. Barrow conjectured; but that they are original and unchangeable properties of light itself.</p></div2></div1><div1 part="N" n="PRISMOID" org="uniform" sample="complete" type="entry"><head>PRISMOID</head><p>, is a solid, or body, somewhat resembling a prism, but that its ends are any dissimilar parallel plane figures of the same number of sides; the upright sides being trapezoids.—If the ends of the Prismoid be bounded by dissimilar curves, it is sometimes called a cylindroid.</p><p>PROBABILITY <hi rend="italics">of an Event,</hi> in the Doctrine of Chances, is the ratio of the number of chances by which the event may happen, to the number by which it may both happen and fail. So that, if there be constituted a fraction, of which the numerator is the number of chances for the events happening, and the denominator the number for both happening and failing, that fraction<cb/> will properly express the value of the Probability of the event's happening. Thus, if an event have 3 chances for happening, and 2 for failing, the sum of which being 5, the fraction 3/5 will fitly represent the Probability of its happening, and may be taken to be the measure of it. The same thing may be said of the Probability of failing, which will likewise be measured by a fraction, whose numerator is the number of chances by which it may fail, and its denominator the whole number of chances both for its happening and failing: so the Probability of the failing of the above event, which has 2 chances to fail, and 3 to happen, will be expressed or measured by the fraction 2/5.</p><p>Hence, if there be added together the fractions which express the Probability for both happening and failing, their sum will always be equal to unity or 1; since the sum of their numerators will be equal to their common denominator. And since it is a certainty that an event will either happen or fail, it follows that a certainty, which may be considered as an infinitely great degree of Probability, is fitly represented by unity. See Simpson's or Demoivre's Doctrine of Chances; also Bernoulli's Ars Conjectandi; Monmort's Analyse des Jeux de Hasard; or M. De Parcieu's Essais sur les Probabilites de la Vie humaine. See also <hi rend="smallcaps">Expectation</hi>, and <hi rend="smallcaps">Gaming.</hi></p><p><hi rend="smallcaps">Probability</hi> <hi rend="italics">of Life.</hi> See <hi rend="smallcaps">Expectation</hi> <hi rend="italics">of Life,</hi> and <hi rend="smallcaps">Life<*></hi> <hi rend="italics">Annuities.</hi></p></div1><div1 part="N" n="PROBLEM" org="uniform" sample="complete" type="entry"><head>PROBLEM</head><p>, in Geometry, is a proposition in which some operation or construction is required. As, to bisect a line, to make a triangle, to raise a perpendicular, to draw a circle through three points, &c.</p><p>A Problem, according to Wolsius, consists of three parts: The proposition, which expresses what is to be done; the resolution, or solution, in which are orderly rehearsed the several steps of the process or operation; and the demonstration, in which it is shewn, that by doing the several things prescribed in the resolution, the thing required is obtained.</p><div2 part="N" n="Problem" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Problem</hi></head><p>, in Algebra, is a question or proposition which requires some unknown truth to be investigated or discovered; and the truth of the discovery demonstrated.</p></div2><div2 part="N" n="Problem" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Problem</hi></head><p>, <hi rend="italics">Kepler's.</hi> See <hi rend="smallcaps">Kepler's</hi> <hi rend="italics">Problem.</hi></p></div2><div2 part="N" n="Problem" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Problem</hi></head><p>, <hi rend="italics">Determinate, Diophantine, Indeterminate, Limited, Linear, Losal, Plane, Solid, Sursolid,</hi> and <hi rend="italics">Unlimited.</hi> See the adjectives.</p><p><hi rend="italics">Deliacal</hi> <hi rend="smallcaps">Problem</hi>, in Geometry, is the doubling of a cube. This amounts to the same thing as the finding of two mean proportionals between two given lines: whence this also is called the Deliacal Problem. See <hi rend="smallcaps">Duplication.</hi></p></div2></div1><div1 part="N" n="PROCLUS" org="uniform" sample="complete" type="entry"><head>PROCLUS</head><p>, an eminent philosopher and mathematician among the later Platonists, was born at Constantinople in the year 410, of parents who were both able and willing to provide for his instruction in all the various branches of learning and knowledge. He was first sent to Xanthus, a city of Lycia, to learn grammar: from thence to Alexandria, where he was under the best masters in rhetoric, philosophy, and mathematics: and from Alexandria he removed to Athens, where he attended the younger Plutarch, and Syrian, both of them celebrated philosophers. He succeeded the latter in<pb n="285"/><cb/> the government of the Platonic school at Athens; where he died in 485, at 75 years of age.</p><p>Marinus of Naples, who was his successor in the school, wrote his life; the first perfect copy of which was published, with a Latin version and notes, by Fabricius at Hamburgh, 1700, in 4to; and afterwards subjoined to his <hi rend="italics">Bibliotheca Latina,</hi> 1703, in 8vo.</p><p>Proclus wrote a great number of pieces, and upon many different subjects; as, commentaries on philosophy, mathematics, and grammar; upon the whole works of Homer, Hesiod, and Plato's books of the republic: he wrote also on the construction of the Astrolabe. Many of his pieces are lost; some have been published; and a few remain still in manuscript only. Of the published, there are four very elegant hymns; one to the Sun, two to Venus, and one to the Muses. There are commentaries upon several pieces of Plato; upon the four books of Ptolomy's work <hi rend="italics">de Judiciis Astrorum;</hi> upon the first book of Euclid's Elements; and upon Hesiod's <hi rend="italics">Opera et Dies.</hi> There are also works of Proclus upon philosophical and astronomical subjects; particularly the piece <hi rend="italics">De Sph<*>ra,</hi> which was published, 1620, in 4to, by Bainbridge, the Savilian professor of astronomy at Oxford. He wrote also 18 arguments against the Christians, which are still extant, and in which he attacks them upon the question, whether the world be eternal? the affirmative of which he maintains.</p><p>The character of Proclus is the same as that of all the later Platonists, who it seems were not less enthusiasts and madmen, than the Christians their contemporaries, whom they represented in this light. Proclus was not reckoned quite orthodox by his own order: he did not adhere so rigorously, as Julian and Porphyry, to the doctrines and principles of his master: “He had, says Cudworth, some peculiar fancies and whimsies of his own, and was indeed a confounder of the Platonic theology, and a mingler of much unintelligible stuff with it.”</p></div1><div1 part="N" n="PROCYON" org="uniform" sample="complete" type="entry"><head>PROCYON</head><p>, in Astronomy, a fixed star, of the second magnitude, in Canis Minor, or the Little Dog.</p></div1><div1 part="N" n="PRODUCING" org="uniform" sample="complete" type="entry"><head>PRODUCING</head><p>, in Geometry, denotes the continuing a line, or drawing it farther out, till it have an assigned length.</p></div1><div1 part="N" n="PRODUCT" org="uniform" sample="complete" type="entry"><head>PRODUCT</head><p>, in Arithmetic, or Algebra, is the factum of two numbers, or quantities, or the quantity arising from, or produced by, the multiplication of, two or more numbers &c together. Thus, 48 is the product of 6 multiplied by 8.—In multiplication, unity is in proportion to one factor, as the other factor is to the product. So .</p><p>In Algebra, the product of simple quantities is expressed by joining the letters together like a word, and prefixing the product of the numeral coefficients: So the product of <hi rend="italics">a</hi> and <hi rend="italics">b</hi> is <hi rend="italics">ab,</hi> of 3<hi rend="italics">a</hi> and 4<hi rend="italics">bc</hi> is 12<hi rend="italics">abc.</hi> But the product of compound factors or quantities is expressed by setting the sign of multiplication between them, and binding each compound factor in a vinculum: so the product of and is , or .</p><p>In geometry, a rectangle answers to a product, its length and breadth being the two factors; because the numbers expressing the length and breadth being mul-<cb/> tiplied together, produce the content or area of the rectangle.</p></div1><div1 part="N" n="PROFILE" org="uniform" sample="complete" type="entry"><head>PROFILE</head><p>, in Architecture, the figure or draught of a building, fortification, or the like; in which are expressed the several heights, widths, and thicknesses, such as they would appear, were the building cut down perpendicularly from the roof to the foundation. Whence the Profile is also called the Section, and sometimes the Orthographical Section; and by Vitruvius the Sciography. In this sense, Profile amounts to the same thing with Elevation; and so stands opposed to a Plan or Ichnography.</p><p><hi rend="smallcaps">Profile</hi> is also used for the contour, or outline of a figure, building, member of architecture, or the like; as a base, a cornice, &c.</p></div1><div1 part="N" n="PROGRESSION" org="uniform" sample="complete" type="entry"><head>PROGRESSION</head><p>, an orderly advancing or proceeding in the same manner, course, tenor, proportion, &c.</p><p>Progression is either Arithmetical, or Geometrical.</p><p><hi rend="italics">Arithmetical</hi> <hi rend="smallcaps">Progression</hi>, is a series of quan<*>ities proceeding by continued equal differences, either increasing or decreasing. Thus, <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">increasing</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" role="data">&c,</cell><cell cols="1" rows="1" role="data">or</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">decreasing</cell><cell cols="1" rows="1" role="data">21,</cell><cell cols="1" rows="1" role="data">18,</cell><cell cols="1" rows="1" role="data">15,</cell><cell cols="1" rows="1" role="data">12,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" role="data">&c;</cell><cell cols="1" rows="1" role="data"/></row></table> where the former progression increases continually by the common difference 2, and the latter series or Progression decreases continually by the common difference 3.</p><p>1. And hence, to construct an arithmetical Progression, from any given first term, and with a given common difference; add the common difference to the first term, to give the 2d; to the 2d, to give the 3d; to the 3d, to give the 4th; and so on; when the series is ascending or increasing: but subtract the common difference continually, when the series is a descending one.</p><p>2. The chief property of an arithmetical Progression, and which arises immediately from the nature of its construction, is this; that the sum of its extremes, or first and last terms, is equal to the sum of every pair of intermediate terms that are equidistant from the extremes, or to the double of the middle term when there is an uneven number of the terms. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Thus,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" role="data">11,</cell><cell cols="1" rows="1" role="data">13,</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13,</cell><cell cols="1" rows="1" role="data">11,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" role="data">7,</cell><cell cols="1" rows="1" role="data">5,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" role="data">1,</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Sums</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14</cell></row></table> where the sum of every pair of terms is the same number 14. <table><row role="data"><cell cols="1" rows="1" role="data">Also,</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + 2<hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + 3<hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + 4<hi rend="italics">d,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + 4<hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + 3<hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + 2<hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">d,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">sums</cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 4<hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 4<hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 4<hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 4<hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> + 4<hi rend="italics">d</hi></cell></row></table></p><p>3. And hence it follows, that double the sum of all the terms in the series, is equal to the sum of the two extremes multiplied by the number of the terms; and consequently, that the single sum of all the terms of the series, is equal to half the said product. So the sum of the 7 terms<pb n="286"/><cb/> 1, 3, 5, 7, 9, 11, 13, is . And the sum of the five terms , is .</p><p>4. Hence also, if the first term of the Progression be 0, the sum of the series will be equal to half the product of the last term multiplied by the number of terms: i. e. the sum of , where <hi rend="italics">n</hi> is the number of terms, supposing 0 to be one of them. That is, in other words, the sum of an arithmetical Progression, whether finite or insinite, whose first term is 0, is to the sum of as many times the greatest term, in the ratio of 1 to 2.</p><p>5. In like manner, the sum of the squares of the terms of such a series, beginning at 0, is to the sum of as many terms each equal to the greatest, in the ratio of 1 to 3. And</p><p>6. The sum of the cubes of the terms of such a series, is to the sum of as many times the greatest term, in the ratio of 1 to 4.<cb/></p><p>7. And universally, if every term of such a Progression be raised to the <hi rend="italics">m</hi> power, then the sum of all those powers will be to the sum of as many terms equal to the greatest, in the ratio of <hi rend="italics">m</hi> + 1 to 1. That is, the sum , is to , in the ratio of 1 to .</p><p>8. A synopsis of all the theorems, or relations, in an arithmetical Progression, between the extremes or first and last term, the sum of the series, the number of terms, and the common difference, is as sollows: viz, if <hi rend="italics">a</hi> denote the least term, <hi rend="italics">z</hi> the greatest term, <hi rend="italics">d</hi> the common difference, <hi rend="italics">n</hi> the number of terms, <hi rend="italics">s</hi> the sum of the series; then will each of these five quantities be expressed in terms of the others, as below: .<cb/> And most of these expressions will become much simpler if the first term be 0 instead of <hi rend="italics">a.</hi></p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Progression</hi>, is a series of quantities proceeding in the same continual ratio or proportion, either increasing or decreasing; or it is a series of quantities that are continually proportional; or which increase by one common multiplier, or decreafe by one common divisor; which common multiplier or divisor is called the common ratio. As, <table><row role="data"><cell cols="1" rows="1" role="data">increasing,</cell><cell cols="1" rows="1" rend="align=right" role="data">1,</cell><cell cols="1" rows="1" rend="align=right" role="data">2,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" role="data">8,</cell><cell cols="1" rows="1" rend="align=right" role="data">16,</cell><cell cols="1" rows="1" role="data">&c,</cell></row><row role="data"><cell cols="1" rows="1" role="data">decreasing,</cell><cell cols="1" rows="1" rend="align=right" role="data">81,</cell><cell cols="1" rows="1" rend="align=right" role="data">27,</cell><cell cols="1" rows="1" role="data">9,</cell><cell cols="1" rows="1" role="data">3,</cell><cell cols="1" rows="1" rend="align=right" role="data">1,</cell><cell cols="1" rows="1" role="data">&c;</cell></row></table> where the former progression increases continually by the common multiplier 2, and the latter decreases by the common divisor 3. Or ascending, <hi rend="italics">a, ra, r</hi><hi rend="sup">2</hi><hi rend="italics">a, r</hi><hi rend="sup">3</hi><hi rend="italics">a,</hi> &c, or descending, <hi rend="italics">a, a/r, a/r</hi><hi rend="sup">2</hi>, <hi rend="italics">a/r</hi><hi rend="sup">3</hi>, &c; where the first term is <hi rend="italics">a,</hi> and common ratio <hi rend="italics">r.</hi></p><p>1. Hence, the same principal properties obtain in a geometrical Progression, as have been remarked of the arithmetical one, using only multiplication in the geometricals for addition in the arithmeticals, and division in the former for subtraction in the latter. So that, to construct a geometrical Progression, from any given first term, and with a given common ratio; multiply the 1st term continually by the common ratio, for the rest of the terms when the series is an ascending one;<cb/> or divide continually by the common ratio, when it is a descending Progression.</p><p>2. In every geometrical Progression, the product of the extreme terms, is equal to the product of every pair of the intermediate terms that are equidistant from the extremes, and also equal to the square of the middle term when there is a middle one, or an uneven number of the terms. <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Thus,</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">2,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" role="data">8,</cell><cell cols="1" rows="1" role="data">16,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">prod.</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">16</cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data">Also</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">a,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">ra,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">2</hi><hi rend="italics">a,</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">3</hi><hi rend="italics">a,</hi></cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a,</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">3</hi><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">2</hi><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">ra</hi></cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">prod.</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">r</hi><hi rend="sup">4</hi><hi rend="italics">a</hi><hi rend="sup">2</hi></cell></row></table></p><p>3. The last term of a geometrical Progression, is equal to the first term multiplied, or divided, by the ratio raised to the power whose exponent is less by 1 than the number of terms in the series; so <hi rend="italics">z</hi> = <hi rend="italics">ar</hi><hi rend="sup">n - 1</hi> when the series is an ascending one, or , when it is a descending Progression.</p><p>4. As the sum of all the antecedents, or all the terms except the least, is to the sum of all the consequents, or all the terms except the greatest, so is 1 to <hi rend="italics">r</hi> the ratio. For,<pb n="287"/><cb/> if be all except the last, then are all except the first; where it is evident that the former is to the latter as 1 to <hi rend="italics">r,</hi> or the former multiplied by <hi rend="italics">r</hi> gives the latter. So that, <hi rend="italics">z</hi> denoting the last term, <hi rend="italics">a</hi> the first term, and <hi rend="italics">r</hi> the ratio, also <hi rend="italics">s</hi> the sum of all the terms; then , or . And from this equation all the relations among the four quantities <hi rend="italics">a, z, r, s,</hi> are easily derived; such as, ; viz, multiply the greatest term by the ratio, subtract the least term from the product, then the remainder divided by 1 less than the ratio, will give the sum of the series. And if the least term <hi rend="italics">a</hi> be 0, which happens when the descending Progression is infinitely continued, then the sum is barely . As in the infinite Progression &c, where <hi rend="italics">z</hi> = 1, and <hi rend="italics">r</hi> = 2, it is <hi rend="italics">s</hi> or .</p><p>5. The first or least term of a geometrical Progression, is to the sum of all the terms, as the ratio minus 1, to the <hi rend="italics">n</hi> power of the ratio minus 1; that is .</p><p>Other relations among the five quantities <hi rend="italics">a, z, r, n, s,</hi> where <hi rend="italics">a</hi> denotes the least term, <hi rend="italics">z</hi> the greatest term, <hi rend="italics">r</hi> the common ratio, <hi rend="italics">n</hi> the number of terms, <hi rend="italics">s</hi> the sum of the Progression, are as below; viz, . And the other values of <hi rend="italics">a, z,</hi> and <hi rend="italics">r</hi> are to be found from these equations, viz, . <hi rend="center">For other sorts of Progressions, see <hi rend="smallcaps">Series.</hi></hi></p></div1><div1 part="N" n="PROJECTILE" org="uniform" sample="complete" type="entry"><head>PROJECTILE</head><p>, or <hi rend="smallcaps">Project</hi>, in Mechanics, is any<cb/> body which, being put into a violent motion by an external force impressed upon it, is dismissed from the agent, and left to pursue its course. Such as a stone thrown out of the hand o<*> a sling, an arrow from a bow, a ball from a gun, &c.</p></div1><div1 part="N" n="PROJECTILES" org="uniform" sample="complete" type="entry"><head>PROJECTILES</head><p>, the science of the motion, velocity, flight, range, &c, of a projectile put into violent motion by some external cause, as the force of gunpowder, &c. This is the foundation of gunnery, under which article may be found all that relates peculiarly to that branch.</p><p>All bodies, being indifferent as to motion or rest, will necessari<*>y continue the state they are put into, except so far as they are hindered, and forced to change it by some new cause. Hence, a Projectile, put in motion, must continue eternally to move on in the same right line, and with the same uniform or constant velocity, were it to meet with no resistance from the medium, nor had any force of gravity to encounter.</p><p>In the first case, the theory of Projectiles would be very simple indeed; for there would be nothing more to do, than to compute the space passed over in a given time by a given constant velocity; or either of these, from the other two being given.</p><p>But by the constant action of gravity, the Projectile is continually deflected more and more from its right-lined course, and that with an accelerated velocity; which, being combined with its Projectile impulse, causes the body to move in a curvilineal path, with a variable motion, which path is the curve of a parabola, as will be proved below; and the determination of the range, time of flight, angle of projection, and variable velocity, constitutes what is usually meant by the doctrine of Projectiles, in the common acceptation of the word.</p><p>What is said above however, is to be understood of Projectiles moving in a non-resisting medium; for when the resistance of the air is also considered, which is enormously great, and which very much impedes the first Projectile velocity, the path deviates greatly from the parabola, and the determination of the circumstances of its motion becomes one of the most complex and difficult problems in nature.</p><p>In the first place therefore it will be proper to consider the common doctrine of Projectiles, or that on the parabolic theory, or as depending only on the nature of gravity and the Projectile motion, as abstracted from the resistance of the medium.</p><p>Little more than 200 years ago, philosophers took the line described by a body projected horizontally, such as a bullet out of a cannon, while the force of the powder greatly exceeded the weight of the bullet, to be a right line, after which they allowed it became a curve. Nicholas Tartaglia was the first who perceived the mistake, maintaining that the path of the bullet was a curved line through the whole of its extent. But it was Galileo who first determined what particular curve it is that a Projectile describes; shewing that the path of a bullet projected horizontally from an eminence, was a parabola; the vertex of which is the point where the bullet quits the cannon. And the same is proved generally, in the 2d section following, when the projection is made in any direction whatever, viz, that the<pb n="288"/><cb/> curve is always a parabola, supposing the body moves in a non-resisting medium. <hi rend="center"><hi rend="italics">The Laws of the Motion of</hi> <hi rend="smallcaps">Projectiles.</hi></hi></p><p>I. If a heavy body be projected perpendicularly, it will continue to ascend or descend perpendicularly; because both the projecting and the gravitating force are found in the same line of direction.</p><p>II. If a body be projected in free space, either parallel to the horizon, or in any oblique direction; it will, by this motion, in conjunction with the action of gravity, describe the curve line of a parabola. <figure/></p><p>For let the body be projected from A, in the direction AD, with any uniform velocity; then in any equal portions of time it would, by that impulse alone, describe the equal spaces AB, BC, CD, &c, in the line AD, if it were not drawn continually down below that line by the action of gravity. Draw BE, CF, DG, &c, in the direction of gravity, or perpendicular to the horizon; and take BE, CF, DG, &c, equal to the spaces through which the body would descend by its gravity in the same times in which it would uniformly pass over the spaces AB, AC, AD, &c, by the Projectile motion. Then, since by these motions, the body is carried over the space AB in the same time as the space BE, and the space AC in the same time as the space CF, and the space AD in the same time as the space DG, &c; therefore, by the composition of motions, at the end of those times the body will be found respectively in the points E, F, G, &c, and consequently the real path of the Projectile will be the curve line AEFG &c. But the spaces AB, AC, AD, &c, being described by uniform motion, are as the times of description; and the spaces BE, CF, DG, &c, described in the same times by the accelerating force of gravity, are as the squares of the times; consequently the perpendicular descents are as the squares of the spaces in AD, that is BE, CF, DG, &c, are respectively proportional to AB<hi rend="sup">2</hi>, AC<hi rend="sup">2</hi>, AD<hi rend="sup">2</hi>, &c, which is the same as the property of the parabola. Therefore the path of the Projectile is the parabolic line AEFG &c, to which AD is a tangent at the point A.</p><p>Hence, 1. The horizontal velocity of a Projectile is always the same constant quantity, in every point of the curve; because the horizontal motion is in a con-<cb/> stant ratio to the motion in AD, which is the uniform Projectile motion; viz, the constant horizontal velocity being to the Projectile velocity, as radius to the cosine of the angle DAH, or angle of elevation or depression of the piece above or below the horizontal line AH.</p><p>2. The velocity of the Projectile in the direction of the curve, or of its tangent, at any point A, is as the secant of its angle BAI of direction above the horizon. For the motion in the horizontal direction AI being constant, and AI being to AB as radius to the secant of the angle A; therefore the motion at A, in AB, is as the secant of the angle A.</p><p>3. The velocity in the direction DG of gravity, or perpendicular to the horizon, at any point G of the curve, is to the first uniform Projectile velocity at A, as 2GD to AD. For the times of describing AD and DG being equal, and the velocity acquired by freely descending through DG being such as would carry the body uniformly over twice DG in an equal time, and the spaces described with uniform motions being as the velocities, it follows that the space AD is to the space 2DG, as the Projectile velocity at A is to the perpendicular velocity at G.</p><p>III. The velocity in the direction of the curve, at any point of it, as A, is equal to that which is generated by gravity in freely descending through a space which is equal to one-fourth of the parameter of the diameter to the parabola at that point. <figure/></p><p>Let PA or AB be the height due to the velocity of the Projectile at any point A, in the direction of the curve or tangent AC, or the velocity acquired by falling through that height; and complete the parallelogram ACDB. Then is CD = AB or AP the height due to the velocity in the curve at A; and CD is also the height due to the perpendicular velocity at D, which will therefore be equal to the former: but, by the last corollary, the velocity at A is to the perpendicular velocity at D, as AC to 2CD; and as these velocities are equal, therefore AC or BD is equal to 2CD or 2AB; and hence AB or AP is equal to 1/2BD or 1/4 of the parameter of the diameter AB by the nature of the parabola.</p><p>Hence, 1. If through the point P, the line PL be drawn perpendicular to AP; then the velocity in the curve at every point, will be equal to the velocity acquired by falling through the perpendicular distance<pb n="289"/><cb/> of the point from the said line PL; that is, a body falling freely through PA, acquires the velocity in the curve at A, EF, " at F, KD, " at D, LH, " at H. The reason of which is, that the line PL is what is called the Directrix of the parabola, the property of which is, that the perpendicular to it, from every point of the curve, is equal to one-fourth of the parameter of the diameter at that point, viz, PA = 1/4 the parameter of the diameter at A, EF = " at F, KD = " at D, LH = " at H.</p><p>2. If a body, after falling through the height PA, which is equal to AB, and when it arrives at A if its course be changed, by reflection from a firm plane AI, or otherwise, into any direction AC, without altering the velocity; and if AC be taken equal to 2AP or 2AB, and the parallelogram be completed; the body will describe the parabola passing through the point D.</p><p>3. Because AC = 2AB or 2CD or 2AP, therefore AC<hi rend="sup">2</hi> = 2AP . 2CD or AP . 4CD; and because all the perpendiculars EF, CD, GH are as AE<hi rend="sup">2</hi>, AC<hi rend="sup">2</hi>, AG<hi rend="sup">2</hi>; therefore also AP . 4EF = AE<hi rend="sup">2</hi>, and AP . 4GH = AG<hi rend="sup">2</hi>, &c; and because the rectangle of the extremes is equal to the rectangle of the means, of four proportionals, therefore it is always, , and , and , and so on.</p><p>IV. Having given the Direction of a Projectile, and the Impetus or Altitude due to the sirst velocity; to determine the Greatest Height to which it will rise, and the Random or Horizontal Range. <figure/></p><p>Let AP be the height due to the Projectile velocity at A, or the height which a body must fall to acquire the same velocity as the projectile has in the curve at A; also AG the direction, and AH the horizon. Upon AG let fall the perpendicular PQ, and on AP the perpendicular QR; so shall AR be equal to the greatest altitude CV, and 4RQ equal to the horizontal range AH. Or, having drawn PQ perpendicular to AG, take AG = 4AQ, and draw GH perpendicular to AH; then AH is the range.</p><p>For by the last cor. ... , and by sim.triangles, ... , or ;<cb/> therefore AG = 4AQ; and, by similar triangles, AH = 4RQ.</p><p>Also, if V be the vertex of the parabola, then AB or 1/2AG = 2AQ, or AQ = QB; consequently AR = BV which is = CV by the nature of the parabola.</p><p>Hence, 1. Because the angle Q is a right angle, which is the angle in a semicircle, therefore if upon AP as a diameter a semicircle be described, it will pass through the point Q. <figure/></p><p>2. If the Horizontal Range and the Projectile Velocity be given, the Direction of the piece so as to hit the object H will be thus easily found: Take AD = 1/4AH, and draw DQ perpendicular to AH, meeting the semicircle described on the diameter AP in Q and <hi rend="italics">q;</hi> then either AQ or A<hi rend="italics">q</hi> will be the direction of the piece. And hence it appears, that there are two directions AB and A<hi rend="italics">b</hi> which, with the same Projectile velocity, give the very same horizontal range AH; and these two directions make equal angles <hi rend="italics">q</hi>AD and QAP with AH and AP, because the arc PQ is equal to the arc A<hi rend="italics">q.</hi></p><p>3. Or if the Range AH and Direction AB be given; to find the Altitude and Velocity or Impetus: Take AD = 1/4AH, and erect the perpendicular DQ meeting AB in Q; so shall DQ be equal to the greatest altitude CV. Also erect AP perpendicular to AH, and QP to AQ; so shall AP be the height due to the velocity.</p><p>4. When the body is projected with the same velocity, but in different directions; the horizontal ranges AH will be as the sines of double the angles of elevation. Or, which is the same thing, as the rectangle of the sine and cosine of elevation. For AD or RQ, which is 1/4AH, is the sine of the arc AQ, which measures double the angle QAD of elevation.</p><p>And when the direction is the same, but the velocities different, the horizontal ranges are as the square of the velocities, or as the height AP which is as the square of the velocity; for the sine AD or RQ, or 1/4AH, is as the radius, or as the diameter AP</p><p>Therefore, when both are different, the ranges are in the compound ratio of the squares of the velocities, and the sines of double the angles of elevation.</p><p>5. The greatest range is when the angle of elevation is half a right angle, or 45°. For the double of 45 is 90°, which has the greatest sine. Or the radius OS, which is 1/4 of the range, is the greatest sine.</p><p>And hence the greatest range, or that at an elevation of 45°, is just double the altitude AP which is due to<pb n="290"/><cb/> the velocity. Or equal to 4VC. And consequently, in that case, C is the focus of the parabola, and AH its parameter.</p><p>And the ranges are equal at angles equally above and below 45°.</p><p>6. When the elevation is 15°, the double of which, or 30°, having its sine equal to half the radius, consequently its range will be equal to AP, or half the greatest range at the elevation of 45°; that is, the range at 15° is equal to the impetus or height due to the projectile velocity.</p><p>7. The greatest altitude CV, being equal to AR, is as the versed sine of double the angle of elevation, and also as AP or the square of the velocity. Or as the square of the sine of elevation, and the square of the velocity; for the square of the sine is as the versed sine of the double angle.</p><p>8. The time of flight of the projectile, which is equal to the time of a body falling freely through GH or 4CV, 4 times the altitude, is therefore as the square root of the altitude, or as the projectile velocity and sine of the elevation.</p><p>9. And hence may be deduced the following set of theorems, for finding all the circumstances relating to projectiles on horizontal planes, having any two of them given. Thus, let <hi rend="italics">s, c, t</hi> = sine, cosine, and tang. of elevation, S, <hi rend="italics">v</hi> = sine and vers. of double the elevation, R the horizontal range, T the time of flight, V the projectile velocity, H the greatest height of the projectile, <hi rend="italics">g</hi> = 16 1/12 feet, and <hi rend="italics">a</hi> = the impetus or the altitude due to the velocity V. Then, .</p><p>And from any of these, the angle of direction may be found.</p><p>V. To determine the Range on an oblique plane; having given the Impetus or the Velocity, and the Angle of Direction.</p><p>Let AE be the oblique plane, at a given angle above or below the horizontal plane AH; AG the direction of the piece; and AP the altitude due to the projectile velocity at A. <figure/><cb/></p><p>By the last prop. find the horizontal range AH to the given velocity and direction; draw HE perpendicular to AH meeting the oblique plane in E; draw EF parallel to the direction AG, and FI parallel to HE; so shall the projectile pass through I, and the range on the oblique plane will be AI. This is evident from prob. 17 of the Parabola in my treatise on Conic Sections, where it is proved, that if AH, AI be any two lines terminated at the curve, and IF, HE be parallel to the axis; then is EF parallel to the tangent AG. <figure/></p><p>Hence, 1. If AO be drawn perpendicular to the plane AI, and AP be bisected by the perpendicular STO; then with the centre O describing a circle through A and P, the same will also pass through <hi rend="italics">q,</hi> because the angle GAI, formed by the tangent AG and AI, is equal to the angle AP<hi rend="italics">q,</hi> which will therefore stand upon the same arc A<hi rend="italics">q.</hi></p><p>2. If there be given the Range and Velocity, or the Impetus, the Direction will then be easily found thus: Take A<hi rend="italics">k</hi> = 1/4AI, draw <hi rend="italics">kq</hi> perpendicular to AH, meeting the circle described with the radius AO in two points <hi rend="italics">q</hi> and <hi rend="italics">q;</hi> then A<hi rend="italics">q</hi> or A<hi rend="italics">q</hi> will be the direction of the piece. And hence it appears that there are two directions, which, with the same impetus, give the very same range AI, on the oblique plane. And these two directions make equal angles with AI and AP, the plane and the perpendicular, because the arc P<hi rend="italics">q</hi> = the arc A<hi rend="italics">q.</hi> They also make equal angles with a line drawn from A through S, because the arc S<hi rend="italics">q</hi> = the arc S<hi rend="italics">q.</hi></p><p>3. Or, if there be given the Range AI, and the Direction A<hi rend="italics">q;</hi> to find the Velocity or Impetus. Take A<hi rend="italics">k</hi> = 1/4AI, and erect <hi rend="italics">kq</hi> perpendicular to AH meeting the line of direction in <hi rend="italics">q;</hi> then draw <hi rend="italics">q</hi>P making the angle A<hi rend="italics">q</hi>P = the angle A<hi rend="italics">kq;</hi> so shall AP be the impetus, or altitude due to the projectile velocity.</p><p>4. The range on an oblique plane, with a given elevation, is directly as the rectangle of the cosine of the direction of the piece above the horizon and the sine of the direction above the oblique plane, and reciprocally as the square of the cosine of the angle of the plano above or below the horizon.<pb n="291"/><cb/></p><p>For put <hi rend="italics">s</hi> = sin. [angle] <hi rend="italics">q</hi>AI or AP<hi rend="italics">q,</hi> <hi rend="italics">c</hi> = cos. [angle] <hi rend="italics">q</hi>AH or sin. PA<hi rend="italics">q,</hi> C = cos. [angle] IAH or sin. A<hi rend="italics">kd</hi> or A<hi rend="italics">kq</hi> or A<hi rend="italics">q</hi>P. Then, in the tri. AP<hi rend="italics">q,</hi> ... , and in the tri. A<hi rend="italics">kq,</hi> ... , therefore by compos. ... .</p><p>So that the oblique range .</p><p>Hence the range is the greatest when A<hi rend="italics">k</hi> is the greatest, that is when <hi rend="italics">kq</hi> touches the circle in the middle point S, and then the line of direction passes through S, and bisects the angle formed by the oblique plane and the vertex. Also the ranges are equal at equal angles above and below this direction for the maximum.</p><p>5. The greatest height <hi rend="italics">tv</hi> or <hi rend="italics">kq</hi> of the projectile, above the plane, is equal to . And therefore it is as the impetus and square of the sine of direction above the plane directly, and square of the cosine of the plane's inclination reciprocally. For C (sin. A<hi rend="italics">q</hi>P) : <hi rend="italics">s</hi> (sin. AP<hi rend="italics">q</hi>) :: AP : A<hi rend="italics">q,</hi> and C (sin. A<hi rend="italics">kq</hi>) : <hi rend="italics">s</hi> (sin. <hi rend="italics">k</hi>A<hi rend="italics">q</hi>) :: A<hi rend="italics">q</hi> : <hi rend="italics">kq,</hi> thereforeby comp. C<hi rend="sup">2</hi> : <hi rend="italics">s</hi><hi rend="sup">2</hi> :: AP : <hi rend="italics">kq.</hi></p><p>6. The time of flight in the curve A<hi rend="italics">v</hi>I is = , where <hi rend="italics">g</hi> = 16 1/12 feet. And therefore it is as the velocity and sine of direction above the plane directly, and cosine of the plane's inclination reciprocally. For the time of describing the curve, is equal to the time of falling freely through GI or 4<hi rend="italics">kq</hi> or . Therefore, the time being as the square root of the distance, the time of flight.</p><p>7. From the foregoing corollaries may be collected the following set of theorems, relating to projects made on any given inclined planes, either above or b<*>low the horizontal plane. In which the letters denote as before, namely, <hi rend="italics">c</hi> = cos. of direction above the horizon, C = cos. of inclination of the plane, <hi rend="italics">s</hi> = sin. of direction above the plane, R the range on the oblique plane, T the time of flight, V the projectile velocity, H the greatest height above the plane, <hi rend="italics">a</hi> the impetus, or alt. due to the velocity V, <hi rend="italics">g</hi> = 16 1/12 feet. Then .<cb/> And from any of these, the angle of direction may be found. <hi rend="center"><hi rend="italics">Of the Path of</hi> <hi rend="smallcaps">Projectiles</hi> <hi rend="italics">as depending on the Resistance of the A.r.</hi></hi></p><p>For a long time after Galileo, philosophers seemed to be satisfied with the parabolic theory of Projectiles, deeming the effect of the air's resistance on the path as of no consequence. In process of time, however, <*>as the true philosophy began to dawn, they began to suspect that the resistance of the medium might have some effect upon the Projectile curve, and they set themselves to consider this subject with some attention.</p><p>Huygens, supposing that the resistance of the air was proportional to the velocity of the moving body, concluded that the line described by it would be a kind of logarithmic curve.</p><p>But Newton, having clearly proved, that the resistance to the body is not proportional to the velocity itself, but to the square of it, shews, in his Principia, that the line a Projectile describes, approaches nearer to an hyperbola than a parabola. Schol. prop. 10, lib. 2. Thus, if AGK be a curve of <figure/> the hyperbolic kind, one of whose asymptotes is NX, perpendicular to the horizon AK, and the other IX inclined to the same, where VG is reciprocally as DN<hi rend="sup">n</hi>, whose index is <hi rend="italics">n:</hi> this curve will nearer represent the path of a Projectile thrown in the direction AH in the air, than a parabola. Newton indeed says, that these hyperbolas are not accurately the curves that a Projectile makes in the air; for the true ones are curves which about the vertex are more distant from the asymptotes, and in the parts remote from the axis approach nearer to the asymptotes than these hyperbolas; but that in practice these hyperbolas may be used instead of those more compounded ones. And if a body be projected from A, in the right line AH, and AI be drawn parallel to the asymptote NX, and GT a tangent to the curve at the vertex: Then the density of the medium in A will be reciprocally as the tangent AH, and the body's velocity will be as , and the resistance of the medium will be to gravity, as AH to .</p><p>M. John Bernoulli constructed this curve by means of the quadrature of some transcendental curves, at the<pb n="292"/><cb/> request of Dr. Keil, who proposed this problem to him in 1718. It was also resolved by Dr. Taylor; and another solution of it may be found in Hermann's Phoronomia.</p><p>The commentators Le Sieur and Jacquier say, that the description of the curve in which a Projectile moves, is so very perplexed, that it can scarcely be expected any deduction should be made from it, either to philosophical or mechanical purposes: vol. 2. pa. 118.</p><p>Dan. Bernoulli too proved, that the resistance of the air has a very great esfect on swift motions, such as those of cannon shot. He concludes from experiment, that a ball which ascended only 7819 feet in the air, would have ascended 58750 feet in vacuo, being near eight times as high. Comment. Acad. Petr. tom. 2.</p><p>M. Euler has farther investigated the nature of this curve, and directed the calculation and use of a number of tables for the solution of all cases that occur in gunncry, which may be accomplished with nearly as much expedition as by the common parabolic principles. Memoirs of the Academy of Berlin, for the year 1753.</p><p>But how rash and erroneous the old opinion of the inconsiderable resistance of the air is, will easily appear from the experiments of Mr. Robins, who has shewn that, in some cases, this resistance to a cannon ball, amounts to more than 20 times the weight of the ball; and I myself, having prosecuted this subject far beyond any former example, have sometimes found this resistance amount to near 100 times the weight of the ball, viz, when it moved with a velocity of 2000 feet per second, which is a rate of almost 23 miles in a minute. What errors then may not be expected from an hypothesis which neglects this force, as inconsiderable! Indeed it is easy to shew, that the path of such Projectiles is neither a parabola nor nearly a parabola. For, by that theory, if the ball, in the instance last mentioned, flew in the curve of a parabola, its horizontal range, at 45° elevation, will be found to be almost 24 miles; whereas it often happens that the ball, with such a velocity, ranges far short of even one mile.</p><p>Indeed the falseness of this hypothesis almost appears at sight, even in Projectiles slow enough to have their motion traced by the eye; for they are seen to descend through a curve manifestly shorter and more inclined to the horizon than that in which they ascended, and the highest point of their flight, or the vertex of the curve, is much nearer to the place where they fall on the ground, than to that from whence they were at first discharged. These things cannot for a moment be doubted of by any one, who in a proper situation views the flight of stones, arrows, or fhells, thrown to any confiderable distance.</p><p>Mr. Robins has not only detected the errors of the parabolic theory of gunnery, which takes no account of the resistance of the air, but shews how to compute the real range of resisted bodies. But for the method which he proposes, and the tables he has computed for this purpose, see his Tracts of Gunnery,<cb/> pa. 183, &c, vol. 1; and also Euler's Commentary on the same, translated by Mr. Hugh Brown, in 1777.</p><p>There is an odd circumstance which often takes place in the motion of bodies projected with considerable force, which shews the great complication and difficulty of this subject; namely, that bullets in their flight are not only depressed beneath their original direction by the action of gravity, but are also frequently driven to the right or left of that direction by the action of some other force.</p><p>Now if it were true that bullets varied their direction by the action of gravity only, then it ought to happen that the errors in their flight to the right or left of the mark they were aimed at, fhould increase in the proportion of the distance of the mark from the piece only. But this is contrary to all experience; the same piece which will carry its bullet within an inch of the intended mark, at 10 yards distance, cannot be relied on to 10 inches in 100 yards, much less to 30 in 300 yards.</p><p>And this inequality can only arise from the track of the bullet being incurvated sideways as well as downwards; for by this means the distance between the incurvated line and the line of direction, will increase in a much greater ratio than that of the distance; these lines coinciding at the mouth of the piece, and afterwards separating in the manner of a curve from its tangent, if the mouth of the piece be considered as the point of contact.</p><p>This is put beyond a doubt from the experiments made by Mr. Robins; who found alfo that the direction of the shot in the perpendicular line was not less uncertain, falling sometimes 200 yards short of what it did at other times, although there was no visible cause of difference in making the experiment. And I myself have often experienced a difference of one-fifth or one-sixth of the whole range, both in the deflection to the right or left, and also in the extent of the range, of cannon shot.</p><p>If it be asked, what can be the cause of a motion so different from what has been hitherto supposed? It may be answered, that the deflection in question must be owing to some power acting obliquely to the progressive motion of the body, which power can be no other than the resistance of the air. And this resistance may perhaps act obliquely to the progressive motion of the body, from inequalities in the resisted surface; but its general cause is doubtless a whirling motion acquired by the bullet about an axis, by its friction against the sides of the piece; for by this motion of rotation, combined with the progressive motion, each part of the ball's surface will strike the air in a direction very different from what it would do if there was no such whirl; and the obliquity of the action of the air, arising from this cause, will be greater, according as the rotatory motion of the bullet is greater in proportion to its progressive motion. Tracts, vol. 1, p. 149, &c.</p><p>M. Euler, on the contrary, attributes this deflection of the ball to its sigure, and very little to its rotation: for if the ball was perfectly round, though its centre of gravity did not coincide, the deflection from the axis of the cylinder, or line of direction sideways, would be very inconsiderable. But when it is not round, it will<pb n="293"/><cb/> generally go to the right or left of its direction, and so much the more, as its range is greater. From his reasoning on this subject he infers, that cannon shot, which are made of iron, and rounder and less susceptible of a change of sigure in passing along the cylinder than those of lead, are more certain than musket shot. True Principles of Gunnery investigated, 1777, p. 304, &c.</p></div1><div1 part="N" n="PROJECTION" org="uniform" sample="complete" type="entry"><head>PROJECTION</head><p>, in Mechanics, the act of giving a projectile its motion.</p><p>If the direction of the force, by which the projectile is put in motion, be perpendicular to the horizon, the Projection is said to be perpendicular; if parallel to the apparent horizon, it is said to be an horizontal Projection; and if it make an oblique angle with the horizon, the Projection is oblique. In all cases the angle which the line of direction makes with the horizontal line, is called the angle of Elevation of the projectile, or of Depression when the line of direction points below the horizontal line.</p><div2 part="N" n="Projection" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Projection</hi></head><p>, in Perspective, denotes the appearance or representation of an object on the perspective plane. So, the Projection of a point, is a point, where the optic ray passes from the objective point through the plane to the eye; or it is the point where the plane cuts the optic ray. — And hence it is easy to conceive what is meant by the projection of a line, a plane, or a solid.</p><p><hi rend="smallcaps">Projection</hi> <hi rend="italics">of the Sphere in Plano,</hi> is a representation of the several points or places of the surface of the sphere, and of the circles described upon it, upon a transparent plane placed between the eye and the sphere, or such as they appear to the eye placed at a given distance. For the laws of this Projection, see P<hi rend="smallcaps">ERSPECTIVF;</hi> the Projection of the sphere being only a particular case of perspective.</p><p>The chief use of the Projection of the sphere, is in the construction of planispheres, maps, and charts; which are said to be of this or that Projection, according to the several situations of the eye, and the perspective plane, with regard to the meridians, parallels, and other points or places to be represented.</p><p>The most usual Projection of maps of the world, is that on the plane of the meridian, which exhibits a right sphere; the first meridian being the horizon. The next is that on the plane of the equator, which has the pole in the centre, and the meridians the radii of a circle, &c; and this represents a parallel sphere. See <hi rend="smallcaps">Map.</hi>—The primitive circle is that great circle.</p><p>The Projection of the sphere is usually divided into Orthographic and Stereographic; to which may be added Gnomonic.</p><p><hi rend="italics">Orthographic</hi> <hi rend="smallcaps">Projection</hi>, is that in which the surface of the sphere is drawn upon a plane, cutting it in the middle; the eye being placed at an infinite distance vertically to one of the hemispheres. And</p><p><hi rend="italics">Stereographic</hi> <hi rend="smallcaps">Projection</hi> of the sphere, is that in which the surface and circles of the sphere are drawn upon the plane of a great circle, the eye being in the pole of that circle.</p><p><hi rend="italics">Gnomonical</hi> <hi rend="smallcaps">Projection</hi> <hi rend="italics">of the Sphere,</hi> is that in which<cb/> the surface of the sphere is drawn upon a plane without side of it, commonly touching it, the eye being at the centre of the sphere. See <hi rend="smallcaps">Ggnomonical</hi> <hi rend="italics">Projection.</hi> <hi rend="center"><hi rend="italics">Laws of the Orthographic Projection.</hi></hi></p><p>1. The rays coming from the eye, being at an infinite distance, and making the Projection, are parallel to each other, and perpendicular to the plane of Projection.</p><p>2. A right line perpendicular to the plane of Projection, is projected into a point, where that line meets the said plane.</p><p>3. A right line, as AB, or CD, <figure/> not perpendicular, but either parallel or oblique to the plane of the Projection, is projected into a right line, as EF or GH, and is always comprehended between the extreme perpendiculars AE and BF, or CG and DH.</p><p>4. The Projection of the right line AB is the greatest, when AB is parallel to the plane of the Projection.</p><p>5. Hence it is evident, that a line parallel to the plane of the Projection, is projected into a right line equal to itself; but a line that is oblique to the plane of Projection, is projected into one that is less than itself.</p><p>6. A plane surface, as ACBD, perpendicular to the plane of the <figure/> Projection, is projected into the right line, as AB, in which it cuts that plane —Hence it is evident, that the circle ACBD perpendicular to the plane of Projection, passing through its centre, is pro jected into that diameter AB in which it cuts the plane of the Projection. Also any arch as C<hi rend="italics">c</hi> is projected into O<hi rend="italics">o,</hi> equal to <hi rend="italics">ca,</hi> the right sine of that arch; and the complemental arc <hi rend="italics">c</hi>B is projected into <hi rend="italics">o</hi>B, the versed sine of the same arc <hi rend="italics">c</hi>B.</p><p>7. A circle parallel to the plane of the Projection, is projected into a circle equal to itself, having its centre the same with the centre of the Projection, and its radius equal to the cosine of its distance from the plane. And a circle oblique to the plane of the Projection, is projected into an ellipsis, whose greater axis is equal to the diameter of the circle, and its less axis equal to double the cosine of the obliquity of the circle, to a radius equal to half the greater axis. <hi rend="center"><hi rend="italics">Properties of the Stereographic Projection.</hi></hi></p><p>1. In this Projection a right circle, or one perpendicular to the plane of Projection, and passing through the eye, is projected into a line of half tangents.</p><p>2. The Projection of all other circles, not passing through the projecting point, whether parallel or oblique, are projected into circles.<pb n="294"/><cb/> <figure/></p><p>Thus, let ACEDB represent a sphere, cut by a plane RS, passing through the centre I, perpendicular to the diameter EH, drawn from E the place of the eye; and let the section of the sphere by the plane RS be the circle CFDL, whose poles are H and E. Suppose now AGB is a circle on the sphere to be projected, whose pole most remote from the eye is P; and the visual rays from the circle AGB meeting in E, form the cone AGBE, of which the triangle AEB is a section through the vertex E, and diameter of the base AB: then will the figure <hi rend="italics">agbf,</hi> which is the Projection of the circle AGB, be itself a circle. Hence, the middle of the projected diameter is the centre of the projected circle, whether it be a great circle or a small one: Also the poles and centres of all circles, parallel to the plane of Projection, fall in the centre of the Projection: And all oblique great circles cut the primitive circle in two points diametrically opposite.</p><p>2. The projected diameter of any circle subtends an angle at the eye equal to the distance of that circle from its nearest pole, taken on the sphere; and that angle is bisected by a right line joining the eye and that pole. Thus, let the plane RS cut the sphere HFEG through <figure/> its centre I; and let ABC be any oblique great circle, whose diameter AC is projected into <hi rend="italics">ac;</hi> and KOL any small circle parallel to ABC, whose diameter KL is projected in <hi rend="italics">kl.</hi> The distances of those circles from their pole P, being the arcs AHP, KHP; and the angles <hi rend="italics">a</hi>E<hi rend="italics">c, k</hi>E<hi rend="italics">l,</hi> are the angles at the eye, subtended by their projected diameters, <hi rend="italics">ac</hi> and <hi rend="italics">kl.</hi> Then is the angle <hi rend="italics">a</hi>E<hi rend="italics">c</hi> measured by the arc AHP, and the angle <hi rend="italics">k</hi>E<hi rend="italics">l</hi> measured by the arc KHP; and those angles are bisected by EP.</p><p>3. Any point of a sphere is projected at such a distance from the centre of Projection, as is equal to the tangent of half the arc intercepted between that point and the pole opposite to the eye, the semidiameter of the sphere being radius. Thus, let C<hi rend="italics">b</hi>EB be a great circle of the sphere, whose centre is <hi rend="italics">c,</hi> GH the plane of Projection cutting the diameter of the sphere in <hi rend="italics">b</hi><cb/> and B; also E and C the poles of the section by that plane; and <hi rend="italics">a</hi> the projection of A. Then <hi rend="italics">ca</hi> is equal <figure/> the tangent of half the arc AC, as is evident by drawing CF = the tangent of half that arc, and joining <hi rend="italics">c</hi>F.</p><p>4. The angle made by two projected circles, is equal to the angle which these circles make on the sphere. For let IACE and ABL be two circles on a sphere <figure/> intersecting in A; E the projecting point; and RS the plane of Projection, in which the point A is projected in <hi rend="italics">a,</hi> in the line IC, the diameter of the circle ACE. Also let DH and FA be tangents to the circles ACE and ABL. Then will the projected angle <hi rend="italics">daf</hi> be equal to the spherical angle BAC.</p><p>5. The distance between the poles of the primitive circle and an oblique circle, is equal to the tangent of half the inclination of those circles; and the distance of their centres, is equal to the tangent of their inclination; the semidiameter of the primitive being radius. For let AC be the diameter of a circle, whose poles are P and Q, and inclined to the plane of Projection in the angle AIF; and let <hi rend="italics">a, c, p</hi> be the Projections of the points A, C, P; also let H<hi rend="italics">a</hi>E be the projected oblique circle, whose centre is <hi rend="italics">q.</hi> Now when the plane of Projection becomes the primitive circle, whose pole is I; then is I<hi rend="italics">p</hi> = tangent of half the angle AIF, or of half<pb n="295"/><cb/> the arch AF; and I<hi rend="italics">q</hi> = tangent of AF, or of the angle FH<hi rend="italics">a</hi> = AIF. <figure/></p><p>6. If through any given point in the primitive circle, an oblique circle be described; then the centres of all other oblique circles passing through that point, will be in a right line drawn through the centre of the first oblique circle, and perpendicular to a line passing through that centre, the given point, and the centre of the pri- <figure/> mitive circle. Thus, let GACE be the primitive circle, ADEI a great circle described through D, its centre being B. HK is a right line drawn through B perpendicular to a right line CI passing through D and B and the centre of the primitive circle. Then the centres of all other great circles, as FDG, passing through D, will fall in the line HK.</p><p>7. Equal arcs of any two great circles of the sphere will be intercepted between two other circles drawn on the sphere through the remotest poles of those great circles. For let PBEA be a sphere, on which AGB and <figure/> CFD are two great circles, whose remotest poles are E and P; and through these poles let the great circle PBEC and the small circle PGE be drawn, cutting the great circles AGB and CFD in the points B, G, D, F.<cb/> Then are the intercepted arcs BG and DF equal to one another.</p><p>8. If lines be drawn from the projected pole of any great circle, cutting the peripherics of the projected circle and plane of Projection; the intercepted arcs of those peripherics are equal; that is, the arc BG = <hi rend="italics">df.</hi></p><p>9. The radius of any lesser circle, whose plane is perpendicular to that of the primitive circle, is equal to the tangent of that lesser circle's distance from its pole; and the secant of that distance is equal to the distance of the centres of the primitive and lesser circle. For let P be the pole, and AB the diameter of a lesser circle, its plane being perpendicular to that of the primi- <figure/> tive circle, whose centre is C: then <hi rend="italics">d</hi> being the centre of the projected lesser circle, <hi rend="italics">da</hi> is equal to the tangent of the arc PA, and <hi rend="italics">d</hi>C = the secant of PA. See <hi rend="smallcaps">Stereographic</hi> <hi rend="italics">Projection.</hi></p><p><hi rend="italics">Mercator's</hi> <hi rend="smallcaps">Projection.</hi> See <hi rend="smallcaps">Mercator</hi> and <hi rend="smallcaps">Chart.</hi></p><p><hi rend="smallcaps">Projection</hi> of Globes, &c. See <hi rend="smallcaps">Globe</hi>, &c.</p><p><hi rend="italics">Polar</hi> <hi rend="smallcaps">Projection.</hi> See <hi rend="smallcaps">Polar.</hi></p><p><hi rend="smallcaps">Projection</hi> <hi rend="italics">of Sbadows.</hi> See <hi rend="smallcaps">Shadow.</hi></p></div2><div2 part="N" n="Projection" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Projection</hi></head><p>, or <hi rend="smallcaps">Projecture</hi>, in Building, the outjetting or prominency which the mouldings and members have, beyond the plane or naked of the wall, column, &c.</p><p><hi rend="italics">Monstrous</hi> <hi rend="smallcaps">Projection.</hi> See <hi rend="smallcaps">Anamorphosis.</hi></p><p>PROJECTIVE <hi rend="italics">Dialling,</hi> a manner of drawing the hour lines, the furniture &c of dials, by a method of projection on any kind of furface whatever, without regard to the situation of those surfaces, either as to declination, reclination, or inclination. See D<hi rend="smallcaps">IALLING.</hi></p></div2></div1><div1 part="N" n="PROLATE" org="uniform" sample="complete" type="entry"><head>PROLATE</head><p>, or <hi rend="smallcaps">Oblong</hi> <hi rend="italics">Spberoid,</hi> is a spheroid produced by the revolution of a semiellipsis about its longer diameter; being longest in the direction of that axis, and resembling an egg, or a lemon.</p><p>It is so called in opposition to the oblate or short spheroid, which is formed by the rotation of a semiellipsis about its shorter axis; being therefore shortest in the direction of its axis, or flatted at the poles, and so resembling an orange, or perhaps a turnip, according to the degree of flatness; and which is also the figure of the earth we inhabit, and perhaps of the planets also; having their equatorial diameter longer than the polar. See <hi rend="smallcaps">Spheroid.</hi></p></div1><div1 part="N" n="PROMONTORY" org="uniform" sample="complete" type="entry"><head>PROMONTORY</head><p>, in Geography, is a rock or high point of land projecting out into the sea. The extremity of which towards the sea is usually called a Cape, or Headland.</p></div1><div1 part="N" n="PROPORTION" org="uniform" sample="complete" type="entry"><head>PROPORTION</head><p>, in Arithmetic &c, the equality<pb n="296"/><cb/> or similitude of ratios. As the four numbers 4, 8, 15, 30 are proportionals, or in proportion, because the ratio of 4 to 8 is equal or similar to the ratio of 15 to 30, both of them being the same as the ratio of 1 to 2.</p><p>Euclid, in the 5th definition of the 5th book, gives a general definition of four proportionals, or when, of four terms, the first has the same ratio to the 2d, as the 3d has to the 4th, viz, when any equimultiples whatever of the first and third being taken, and any equimultiples whatever of the 2d and 4th; if the multiple of the sirst be less than that of the 2d, the multiple of the 3d is also less than that of the 4th; or if the multiple of the first be equal to that of the 2d, the multiple of the 3d is also equal to that of the 4th; or if the multiple of the first be greater than that of the 2d, the multiple of the 3d is also greater than that of the 4th. And this definition is general for all kinds of magnitudes or quantities whatever, though a very obscure one.</p><p>Also, in the 7th book, Euclid gives another definition of proportionals, viz, when the first is the same equimultiple of the 2d, as the 3d is of the 4th, or the same part or parts of it. But this definition appertains only to numbers and commensurable quantities.</p><p>Proportion is often confounded with ratio; but they are quite different things. For, ratio is properly the relation of two magnitudes or quantities of one and the same kind; as the ratio of 4 to 8, or of 15 to 30, or of 1 to 2; and so implies or respects only two terms or things. But Proportion respects four terms or things, or two ratios which have each two terms. Though the middle term may be common to both ratios, and then the Proportion is expressed by three terms only, as 4, 8, 64, where 4 is to 8 as 8 to 64.</p><p>Proportion is also sometimes confounded with progression. In fact, the two often coincide; the difference between them only consisting in this, that progression is a particular species of Proportion, being indeed a continued Proportion, or such as has all the terms in the same ratio, viz, the 1st to the 2d, the 2d to the 3d, the 3d to the 4th, &c; as the terms 2, 4, 8, 16, &c; so that progression is a series or continuation of Proportions.</p><p>Proportion is either continual, or discrete or interrupted.</p><p>The Proportion is continual when every two adjacent terms have the same ratio, or when the consequent of each ratio is the antecedent of the next following ratio, and so all the terms form a progression; as 2, 4, 8, 16, &c; where 2 is to 4 as 4 to 8, and as 8 to 16, &c.</p><p>Discrete or interrupted Proportion, is when the consequent of the first ratio is different from the antecedent of the 2d, &c; as 2, 4, and 3, 6.</p><p>Proportion is also either Direct or Inverse.</p><p><hi rend="italics">Direct</hi> <hi rend="smallcaps">Proportion</hi> is when more requires more, or less requires less. As it will require more men to perform more work, or fewer men for less work, in the same time.</p><p><hi rend="italics">Inversc</hi> or <hi rend="italics">Reciprocal</hi> <hi rend="smallcaps">Proportion</hi>, is when more requires less, or less requires more. As it will require more men to perform the same work in less time, or fewer men in more time. Ex. If 6 men can perform a piece of work in 15 days, how many men can do the same in 10 days. Then,<cb/> <hi rend="brace"><note anchored="true" place="unspecified">the answer.</note> reciprocally - as 1/15 to 1/10 so is 6 : 9 or inversely - as 10 to 15 so is 6 : 9</hi></p><p>Proportion, again, is distinguished into Arithmetical, Geometrical, and Harmonical.</p><p><hi rend="italics">Arithmetical</hi> <hi rend="smallcaps">Proportion</hi> is the equality of two arithmetical ratios, or differences. As in the numbers 12, 9, 6; where the difference between 12 and 9, is the same as the difference between 9 and 6, viz 3.</p><p>And here the sum of the extreme terms is equal to the sum of the means, or to double the single mean when there is but one. As .</p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Proportion</hi> is the equality between two geometrical ratios, or between the quotients of the terms. As in the three 9, 6, 4, where 9 is to 6 as 6 is to 4, thus denoted ; for 9/6 = 6/4, being each equal 3/2 or 1 1/2.</p><p>And in this Proportion, the rectangle or product of the extreme terms, is equal to that of the two means, or the square of the single mean when there is but one. For .</p><p><hi rend="italics">Harmonical</hi> <hi rend="smallcaps">Proportion</hi>, is when the first term is to the third, as the difference between the 1st and 2d is to the difference between the 2d and 3d; or in four terms when the 1st is to the 4th, as the difference between the 1st and 2d is to the difference between the 3d and 4th; or the reciprocals of an arithmetical Proportion are in harmonical Proportion. As 6, 4, 3; because ; or because 1/5, 1/4, 1/3 are in arithmetical Proportion, making . Also the four 24, 16, 12, 9 are in harmonical Proportion, because .</p><p>See <hi rend="smallcaps">Proportionals.</hi></p><p><hi rend="italics">Compass of</hi> <hi rend="smallcaps">Proportion</hi>, a name by which the French, and some English authors, call the Sector.</p><p><hi rend="italics">Rule of</hi> <hi rend="smallcaps">Proportion</hi>, in Arithmetic, a rule by which a 4th term is found in Proportion to three given terms. And is popularly called the Golden Rule, or Rule of Three.</p></div1><div1 part="N" n="PROPORTIONAL" org="uniform" sample="complete" type="entry"><head>PROPORTIONAL</head><p>, relating to Proportion. As, Proportional Compasses, Parts, Scales, Spirals, &c. See the several terms.</p><p><hi rend="smallcaps">Proportional</hi> <hi rend="italics">Compasses,</hi> are compasses with two pair of opposite legs, like a St. Andrew's cross, by which any space is enlarged or diminished in any proportion.</p><p><hi rend="smallcaps">Proportional</hi> <hi rend="italics">Part,</hi> is a part of some number that is analogous to some other part or number; such as the Proportional parts in the logarithms, and other tables.</p><p><hi rend="smallcaps">Proportional</hi> <hi rend="italics">Scales,</hi> called also Logarithmic Scales, are the logarithms, or artificial numbers, placed on lines, for the ease and advantage of multiplying and dividing &c, by means of compasses, or of sliding rulers. These are in effect so many lines of numbers, as they are called by Gunter, but made single, double, triple, or quadruple; beyond which they seldom go. See G<hi rend="smallcaps">UNTER'S</hi> <hi rend="italics">Scale,</hi> <hi rend="smallcaps">Scale</hi>, &c.</p><p><hi rend="smallcaps">Proportional</hi> <hi rend="italics">Spiral.</hi> See <hi rend="smallcaps">Spiral.</hi></p></div1><div1 part="N" n="PROPORTIONALITY" org="uniform" sample="complete" type="entry"><head>PROPORTIONALITY</head><p>, the quality of Proportionals. This term is used by Gregory St. Vincent, for the proportion that is between the exponents of four ratios.</p></div1><div1 part="N" n="PROPORTIONALS" org="uniform" sample="complete" type="entry"><head>PROPORTIONALS</head><p>, are the terms of a proportion; consisting of two extremes, which are the first<pb n="297"/><cb/> and last terms of the set, and the means, which are the rest of the terms. These Proportionals may be either arithmeticals, geometricals, or harmonicals, and in any number above two, and also either continued or discontinued.</p><p>Pappus gives this beautiful and simple comparison of the three kinds of Proportionals, arithmetical, geometrical, and harmonical, viz, <hi rend="italics">a, b, c</hi> being the first, second and third terms in any such proportion, then <hi rend="brace"><note anchored="true" place="unspecified">: : <hi rend="italics">a</hi> - <hi rend="italics">b</hi> : <hi rend="italics">b</hi> - <hi rend="italics">c.</hi></note> In the arithmeticals, <hi rend="italics">a a</hi> in the geometricals, <hi rend="italics">a</hi> : <hi rend="italics">b</hi> in the harmonicals, <hi rend="italics">a</hi> : <hi rend="italics">c</hi></hi></p><p>See <hi rend="smallcaps">Mean</hi> <hi rend="italics">Proportional.</hi></p><p>Continued Proportionals form what is called a progression; for the properties of which see P<hi rend="smallcaps">ROGRESSION.</hi> <hi rend="center">I. <hi rend="italics">Properties of Arithmetical</hi> <hi rend="smallcaps">Proportionals.</hi></hi></p><p>(For what respects Progressions and Mean Proportionals of all sorts, see <hi rend="smallcaps">Mean</hi>, and <hi rend="smallcaps">Progression.</hi>)</p><p>1. Four Arithmetical Proportionals, as 2, 3, 4, 5, are still Proportionals when inversely, 5, 4, 3, 2; or alternately, thus, 2, 4, 3, 5; or inversely and alternately, thus 5, 3, 4, 2.</p><p>2. If two Arithmeticals be added to the like terms of other two Arithmeticals, of the same difference or arithmetical ratio, the sums will have double the same difference or arithmetical ratio. So, to 3 and 5, whose difference is 2, add 7 and 9, whose difference is also 2, the sums 10 and 14 have a double diff. viz 4. And if to these sums be added two other numbers also in the same difference, the next sums will have a triple ratio or difference; and so on. Also, whatever be the ratios of the terms that are added, whether the same or different, the sums of the terms will have such arithmetical ratio as is composed of the sums of the others that are added. <table><row role="data"><cell cols="1" rows="1" role="data"> So</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">,</cell><cell cols="1" rows="1" rend="align=right" role="data">5,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" role="data"> and</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">,</cell><cell cols="1" rows="1" rend="align=right" role="data">10,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data"> and</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">make</cell><cell cols="1" rows="1" rend="align=right" role="data">22</cell><cell cols="1" rows="1" role="data">,</cell><cell cols="1" rows="1" rend="align=right" role="data">31,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">9.</cell></row></table></p><p>On the contrary, if from two Arithmeticals be subtracted others, the difference will have such arithmetical ratio as is equal to the differences of those. <table><row role="data"><cell cols="1" rows="1" role="data">So from</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">16,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">4</cell></row><row role="data"><cell cols="1" rows="1" role="data">take</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">10,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">3</cell></row><row role="data"><cell cols="1" rows="1" role="data">leaves</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">1</cell></row><row role="data"><cell cols="1" rows="1" role="data">Also from</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">9,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" role="data">take</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">5,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">2</cell></row><row role="data"><cell cols="1" rows="1" role="data">leaves</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">4,</cell><cell cols="1" rows="1" role="data">whose dif. is</cell><cell cols="1" rows="1" role="data">0</cell></row></table></p><p>3. Hence, if Arithmetical Proportionals be multitiplied or divided by the same number, their difference, or arithmetical ratio, is also multiplied or divided by the same number.<cb/> <hi rend="center">II. <hi rend="italics">Properties of Geometrical Proportionals.</hi></hi></p><p>The properties relating to mean Proportionals are given under the term <hi rend="smallcaps">Mean</hi> <hi rend="italics">Proportional;</hi> some are also given under the article Proportion; and some additional ones are as below:</p><p>1. To find a 3d Proportional to two given numbers, or a 4th Proportional to three: In the former case, multiply the 2d term by itself, and divide the product by the 1st: and in the latter case, multiply the 2d term by the 3d, and divide the product by the 1st. So , the 3d prop. to 2 and 6: and , the 4th prop. to 2, 6, and 5.</p><p>2. If the terms of any geometrical ratio be augmented or diminished by any others in the same ratio, or proportion, the sums or differences will still be in the same ratio or proportion. So if , then is . And if the terms of a ratio, or proportion, be multiplied or divided by any one and the same number, the products and quotients will still be in the same ratio, or proportion. Thus, .</p><p>3. If a set of continued Proportionals be either augmented or diminished by the same part or parts of themselves, the sums or differences will also be Proportionals. Thus if <hi rend="italics">a, b, c, d,</hi> &c be Propors. then are &c also Propors. where the common ratio is .</p><p>And if any single quantity be either augmented or diminished by some part of itself, and the result be also increased or diminished by the same part of itself, and this third quantity treated in the same manner, and so on; then shall all these quantities be continued Proportionals. So, beginning with the quantity <hi rend="italics">a,</hi> and taking always the <hi rend="italics">n</hi>th part, then shall , &c be Proportionals, or &c Propors. the common ratio being .</p><p>4. If one set of Proportionals be multiplied or divided by any other set of Proportionals, each term by each, the products or quotients will also be Proportionals. Thus, if , and ; then is , and .</p><p>5. If there be several continued Proportionals, then whatever ratio the 1st has to the 2d, the 1st to the 3d<pb n="298"/><cb/> shall have the duplicate of the ratio, the 1st to the 4th the triplicate of it, and so on. So in <hi rend="italics">a, na, n</hi><hi rend="sup">2</hi><hi rend="italics">a, n</hi><hi rend="sup">3</hi><hi rend="italics">a,</hi> &c, the ratio being <hi rend="italics">n;</hi> then <hi rend="italics">a</hi> : <hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">a,</hi> or 1 to <hi rend="italics">n</hi><hi rend="sup">2</hi>, the duplicate ratio, and <hi rend="italics">a</hi> : <hi rend="italics">n</hi><hi rend="sup">3</hi><hi rend="italics">a,</hi> or 1 to <hi rend="italics">n</hi><hi rend="sup">3</hi>, the triplicate ratio, and so on.</p><p>6. In three continued Proportionals, the difference between the 1st and 2d term, is a mean Proportional between the 1st term and the second difference of all the terms. Thus, in the three Propor. <hi rend="italics">a, na,</hi> <hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">a;</hi> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Terms</cell><cell cols="1" rows="1" role="data">1st difs.</cell><cell cols="1" rows="1" role="data">2d dif.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">a</hi> - <hi rend="italics">na</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">na</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">na</hi> - <hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">n</hi><hi rend="sup">2</hi><hi rend="italics">a</hi> - 2<hi rend="italics">na</hi> + <hi rend="italics">a,</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> then . Or in the numbers 2, 6, 18; <table><row role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8 the 2d difference;</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> then 2, 4, 8 are Proportionals.</p><p>7. When four quantities are in proportion, they are also in proportion by inversion, composition, division &c; thus, <hi rend="italics">a, na, b, nb</hi> being in proportion, viz, 1 ; then by 2. Inversion ; 3. Alternation ; 4. Composition ; 5. Conversion ; <hi rend="brace"><note anchored="true" place="unspecified">6. Division</note></hi>. <hi rend="center">III. <hi rend="italics">Properties of Harmonical Proportionals.</hi></hi></p><p>1. If three or four numbers in Harmonical Proportion, be either multiplied or divided by any number, the products or quotients will also be Harmonical Proportionals. Thus, 6, 3, 2 being harmon. Propor. then 12, 6, 4 are also harmon. Propor. and 6/2, 3/2, 2/2 are also harmon. Propor.</p><p>2. In the three Harmonical Proportionals <hi rend="italics">a, b, c,</hi> when any two of these are given, the 3d can be found from the definition of them, viz, that ; for hence the harmonical mean, and the 3d harmon. to <hi rend="italics">a</hi> and <hi rend="italics">b.</hi></p><p>3. And of the four Harmonicals, <hi rend="italics">a, b, c, d,</hi> any three being given, the fourth can be found from the definition of them, viz, that for thence the three <hi rend="italics">b, c, d,</hi> will be thus found, viz, .</p><p>4. If there be four numbers disposed in order, as 2, 3, 4, 6, of which one extreme and the two middle terms are in Arithmetical Proportion, and the other<cb/> extreme and the same middle terms are in Harmonical Proportion; then are the four terms in Geometrical Proportion: so here the three 2, 3, 4 are arithmeticals, and the three 3, 4, 6 are harmonicals, then the four 2, 3, 4, 6 are geometricals.</p><p>5. If between any two numbers, as 2 and 6, there be interposed an arithmetical mean 4, and also a harmonical mean 3, the four will then be geometricals, viz, .</p><p>6. Between the three kinds of proportion, there is this remarkable difference; viz, that from any given number there can be raised a continued arithmetical series increasing ad infinitum, but not decreasing; while the harmonical can be decreased ad insinitum, but not increased; and the geometrical admits of both.</p></div1><div1 part="N" n="PROPOSITION" org="uniform" sample="complete" type="entry"><head>PROPOSITION</head><p>, is either some truth advanced, and shewn to be such by demonstration; or some operation proposed, and its solution shewn. In short, it is something proposed either to be demonstrated, or to be done or performed. The former is a theorem, and the latter is a problem.</p></div1><div1 part="N" n="PROSTHAPHERESIS" org="uniform" sample="complete" type="entry"><head>PROSTHAPHERESIS</head><p>, in Astronomy, the difference between the true and mean motion, or between the true and mean place, of a planet, or between the true and equated anomaly; called also Equation of the Orbit, or Equation of the Centre, or simply the Equation; and it is equal to the angle formed at the planet, and subtended by the excentricity of its orbit.</p><p>Thus, if S be the sun, and P the place of a planet in its orbit APB, whose centre is C. <figure/> Then the mean anomaly is the [angle] ACP, and the true anomaly the [angle] ASP, the difference of which is the [angle] CPS, which is the Prosthapheresis; which is so called, because it is sometimes to be added to, and sometimes to be subtracted from the mean motion, to give the true one; as is evident from the sigure.</p></div1><div1 part="N" n="PROTRACTING" org="uniform" sample="complete" type="entry"><head>PROTRACTING</head><p>, or <hi rend="smallcaps">Protraction</hi>, in Surveying, the act of plotting or laying down the dimensions taken in the field, by means of a Protractor, &c: Protracting makes one part of surveying.</p><p><hi rend="smallcaps">Protracting</hi>-<hi rend="italics">Pin,</hi> a fine pointed pin, or needle, fitted into a handle, used to prick off degrees and minutes from the limb of the Protractor.</p></div1><div1 part="N" n="PROTRACTOR" org="uniform" sample="complete" type="entry"><head>PROTRACTOR</head><p>, a mathematical instrument, chiefly used in surveying, for laying down angles upon paper, &c.</p><p>The simplest, and most natural Protractor consists of a semicircular limb ADB (fig. 7, pl. xix) commonly of metal, divided into 180°, and subtended by a diameter AB; in the middle of which is a small notch C,<pb n="299"/><cb/> called the centre of the Protractor. And for the convenience of reckoning both ways, the degrees are numbered from the left hand towards the right, and from the right hand towards the left.</p><p>But this instrument is made much more commodious by transferring the divisions from the circumference to the edge of a ruler, whose side EF is parallel to AB, which is easily done by laying a ruler on the centre C, and over the several divisions on the semicircumference ADB, and marking the intersections of that ruler on the line EF: so that a ruler with these divisions marked on one of its sides as above, and returned down the two ends, and numbered both ways as in the circular Protractor, the fourth or blank side representing the diameter of the circle, is both a more useful form than the circular Protractor, and better adapted for putting into a case.</p><p><hi rend="italics">To make any Angle with the Protractor.</hi>—Lay the diameter of the Protractor along the given line which is to be one side of the angle, and its centre at the given angular point; then make a mark opposite the given degree of the angle found on the limb of the instrument, and, removing the Protractor, by a plane ruler laid over that point and the centre, draw a line, which will form the angle sought.</p><p>In the same way is any given angle measured, to sind the number of degrees it contains.</p><p>This Protractor is also very useful in drawing one line perpendicular to another; which is readily done by laying the Protractor across the given line, so that both its centre and the 90th degree on the opposite edge fall upon the line, also one of the edges passing over the given point, by which then let the perpendicular be drawn.</p><p><hi rend="italics">The Improved</hi> <hi rend="smallcaps">Protractor</hi> is an instrument much like the former, only furnished with a little more apparatus, by which an angle may be set off to a single minute.</p><p>The chief addition is an index attached to the centre, about which it is moveable, so as to play freely and steadily over the limb. Beyond the limb the index is divided, on both edges, into 60 equal parts of the portions of circles, intercepted by two other right lines drawn from the centre, so that each makes an angle of one degree with lines drawn to the assumed points from the centre.</p><p>To set off an angle of any number of degrees and minutes with this Protractor, move the index, so that one of the lines drawn on the limb, from one of the forementioned points, may fall upon the number of degrees given; and prick off as many of the equal parts on the proper edge of the index as there are minutes given; then drawing a line from the centre to that point so pricked off, the required angle is thus formed with the given line or diameter of the Protractor.</p><p>PROVING <hi rend="italics">of Gunpowder.</hi> See <hi rend="smallcaps">Eprouvette</hi>, and <hi rend="smallcaps">Gunpowd<*>r.</hi></p><p>PSEUDO <hi rend="smallcaps">Stella</hi>, any kind of meteor or phenomenon, appearing in the heavens, and resembling a star.</p></div1><div1 part="N" n="PTOLEMAIC" org="uniform" sample="complete" type="entry"><head>PTOLEMAIC</head><p>, or <hi rend="smallcaps">Ptolomaic</hi>, something relating to Ptolomy; as the Ptolomaic System, the Ptolomaic Sphere, &c. See <hi rend="smallcaps">System, Sphere</hi>, &c.<cb/></p></div1><div1 part="N" n="PTOLEMY" org="uniform" sample="complete" type="entry"><head>PTOLEMY</head><p>, or <hi rend="smallcaps">Ptolomy, (Claudius</hi>), a very celebrated geographer, astronomer, and mathematician, among the Ancients, was born at Pelusium in Egypt, about the 70th year of the Christian era; and died, it has been said, in the 78th year of his age, and in the year of Christ 147. He taught astronomy at Alexandria in Egypt, where he made many astronomical observations, and composed his other works. It is certain that he flourished in the reigns of Marcus Antoninus and Adrian: for it is noted in his Canon, that Antoninus Pius reigned 23 years, which shews that he himself survived him; he also tells us in one place, that he made a great many observations upon the fixed stars at Alexandria, in the second year of Antoninus Pius; and in another, that he observed an eclipse of the moon, in the ninth year of Adrian; from which it is reasonable to conclude that this astronomer's observations upon the heavens were many of them made between the year 125 and 140.</p><p>Ptolomy has always been reckoned the prince of astronomers among the Ancients, and in his works has left us an entire body of that science. He has preserved and transmitted to us the observations and principal discoveries of the Ancients, and at the same time augmented and enriched them with his own. He corrected Hipparchus's catalogue of the sixed stars; and formed tables, by which the motions of the sun, moon, and planets, might be calculated and regulated. He was indeed the first who collected the scattered and detached observations of the Ancients, and digested them into a system; which he set forth in his <foreign xml:lang="greek">*msgalh *suntacis</foreign>, sive <hi rend="italics">Magna Constructio,</hi> divided into 13 books. He adopts and exhibits here the ancient system of the world, which placed the earth in the centre of the universe; and this has been called from him, the Ptolomaic System, to distinguish it from those of Copernicus and Tycho Brahe.</p><p>About the year 827 this work was translated by the Arabians into their language, in which it was called <hi rend="italics">Almagestum,</hi> by order of one of their kings; and from Arabic into Latin, about 1230, by the encouragement of the emperor Frederic the 2d. There were also other versions from the Arabic into Latin; and a manuscript of one, done by Girardus Cremonensis, who flourished about the middle of the 14th century, Fabricius says, is still extant in the library of All Souls College in Oxford. The Greek text of this work began to be read in Europe in the 15th century; and was first publisned by Simon Grynæus at Basil, 1538, in folio, with the eleven books of commentaries by Theon, who flourished at Alexandria in the reign of the elder Theodosius. In 1541 it was reprinted at Basil, with a Latin version by George Trapezond; and again at the same place in 1551, with the addition of other works of Ptolomy, and Latin versions by Camerarius. We learn from Kepler, that this last edition was used by Tycho.</p><p>Of this principal work of the ancient astronomers, it may not be improper to give here a more particular account. In general, it may be observed, that the work is founded upon the hypothesis of the earth's being at rest in the centre of the universe, and that the heavenly bodies, the stars and planets, all move around it in solid orbs, whose motions are all directed by one, which Pto-<pb n="300"/><cb/> lumy cailed the <hi rend="italics">Primum Mobile,</hi> or First Mover, of which he discourses at large. But, to be more particular, this great work is divided into 13 books.</p><p>In the first book, Ptolomy shews, that the earth is in the centre of those orbs, and of the universe itself, as he underslood it: he represents the earth as of a spherical figure, and but as a point in comparison of the rest of the heavenly bodies: he treats concerning the several circles of the earth, and their distances from the equator; as also of the right and oblique ascension of the heavenly bodies in a right sphere.</p><p>In the 2d book, he treats of the habitable parts of the earth; of the elevation of the pole in an oblique sphere, and the various angles which the several circles make with the horizon, according to the different latitude of places; also of the phenomena of the heavenly bodies depending on the same.</p><p>In the 3d book, he treats of the quantity of the year, and of the unequal motion of the sun through the zodiac: he here gives the method os computing the mean motion of the sun, with tables of the same; and likewise treats of the inequality of days and nights.</p><p>In the 4th book, he treats of the lunar motions, and their various phenomena: he gives tables for sinding the moon's mean motions, with her latitude and longitude: he discourses largely concerning lunar epicycles; and by comparing the times of a great number of eclipses, mentioned by Hipparchus, Calippus, and others, he has computed the places of the sun and moon, according to their mean motions, from the first year of Nabonazar, king of Egypt, to his own time.</p><p>In the 5th book, he treats of the instrument called the Astrolabe: he treats also of the eccentricity of the lunar orbit, and the inequality of the moon's motion, according to her distance from the sun: he also gives tables, and an universal canon for the inequality of the lunar motions: he then treats of the different aspects or phases of the moon, and gives a computation of the diameter of the sun and moon, with the magnitude of the sun, moon and earth compared together; he states also the different measures of the distance of the sun and moon, according as they are determined by ancient mathematicians and philosophers.</p><p>In the 6th book, he treats of the conjunctions and oppositions of the sun and moon, with tables for computing the mean time when they happen; of the boundaries of solar and lunar eclipses; of the tables and methods of computing the eclipses of the sun and moon, with many other particulars.</p><p>In the 7th book, he treats of the sixed stars; and shews the methods of describing them, in their various constellations, on the surface of an artificial sphere or globe: he rectifies the places of the stars to his own time, and shews how different those places were then, from what they had been in the times of Timocharis, Hipparchus, Aristillus, Calippus, and others: he then lays down a catalogue of the stars in each of the northern constellations, with their latitude, longitude, and magnitudes.</p><p>In the 8th book, he gives a like catalogue of the stars in the constellations of the southern hemisphere, and in the 12 signs or constellations of the zodiac. This is the first catalogue of the stars now extant, and forms<cb/> the most valuable part of Ptolomy's works. He then treats of the galaxy, or milky-way; also of the planetary aspects, with the rising and setting of the sun, moon, and stars.</p><p>In the 9th book, he treats of the order of the sun, moon, and planets, with the periodical revolutions of the five planets; then he gives tables of the mean motions, beginning with the theory of Mercury, and shewing its various phenomena with respect to the earth.</p><p>The 10th book begins with the theory of the planet Venus, treating of its greatest distance from the sun; of its epicycle, eccentricity, and periodical motions: it then treats of the same particulars in the planet Mars.</p><p>The 11th book treats of the same circumstances in the theory of the planets Jupiter and Saturn. It also corrects all the planetary motions from observations made from the time of Nabonazar to his own.</p><p>The 12th book treats of the retrogressive motion of the several planets; giving also tables of their stations, and of the greatest distances of Venus and Mercury from the sun.</p><p>The 13th book treats of the several hypotheses of the latitude of the sive planets; of the greatest latitude, or inclination of the orbits of the five planets, which are computed and disposed in tables; of the rising and setting of the planets, with tables of them. Then follows a conclusion or winding up of the whole work.</p><p>This great work of Ptolomy will always be valuable on account of the observations he gives of the places of the stars and planets in former times, and according to ancient philosophers and astronomers that were then extant; but principally on account of the large and curious catalogue of the stars, which being compared with their places at present, we thence deduce the true quantity of their slow progressive motion according to the order of the signs, or of the precession of the equinoxes.</p><p>Another great and important work of Ptolomy was, his <hi rend="italics">Geography,</hi> in 7 books; in which, with his usual sagacity, he searches out and marks the situation of places according to their latitudes and longitudes; and he was the sirst that did so. Though this work must needs fall far short of perfection, through the want of necessary observations, yet it is of considerable merit, and has been very useful to modern geographers. Cellarius indeed suspects, and he was a very competent judge, that Ptolomy did not use all the care and application which the nature of his work required; and his reason is, that the author delivers himself with the same fluency and appearance of certainty, concerning things and places at the remotest distance, which it was impossible he could know any thing of, that he does concerning those which lay the nearest to him, and fall the most under his cognizance. Salmasius had before made some remarks to the same purpose upon this work of Ptolomy. The Greek text of this work was first published by itself at Basil in 1533, in 4to: afterward with a Latin version and notes by Gerard Mercator at Amsterdam, 1605; which last edition was reprinted at the same place, 1618, in folio, with neat geographical tables, by Bertius.<pb n="301"/><cb/></p><p>Other works of Ptolomy, though less considerable than these two, are still extant. As, <hi rend="italics">Libri quatuor de Judiciis Astrorum,</hi> upon the first two books of which Cardan wrote a commentary.—<hi rend="italics">Fructus Librorum suorum;</hi> a kind of supplement to the former work.—<hi rend="italics">Recensio Chronologica Regum:</hi> this, with another work of Ptolomy, <hi rend="italics">De Hypothesibus Planetarum,</hi> was published in 1620, 4to, by John Bainbridge, the Savilian professor of Astronomy at Oxford: And Scaliger, Petavius, Dodwell, and the other chronological writers, have made great use of it.—<hi rend="italics">Apparentiæ Stellarum Inerrantium:</hi> this was published at Paris by Petavius, with a Latin version, 1630, in folio; but from a mutilated copy, the defects of which have since been supplied from a perfect one, which Sir Henry Saville had communicated to archbishop Usher, by Fabricius, in the 3d volume of his <hi rend="italics">Bibliotheca Græca.—Elementorum Harmonicorum libri tres;</hi> published in Greek and Latin, with a commentary by Porphyry the philosopher, by Dr. Wallis at Oxford, 1682, in 4to; and afterwards reprinted there, and inserted in the 3d volume of Wallis's works, 1699, in folio.</p><p>Mabillon exhibits, in his <hi rend="italics">German Travels,</hi> an effigy of Ptolomy looking at the stars through an optical tube; which effigy, he says, he found in a manuscript of the 13th century, made by Conradus a monk. Hence some have fancied, that the use of the telescope was known to Conradus. But this is only matter of mere conjecture, there being no facts or testimonies, nor even probabilities, to support such an opinion.</p><p>It is rather likely that the tube was nothing more than a plain open one, employed to strengthen and defend the eye-sight, when looking at particular stars, by excluding adventitious rays from other stars and objects; a contrivance which no observer of the heavens can ever be supposed to have been without.</p></div1><div1 part="N" n="PULLEY" org="uniform" sample="complete" type="entry"><head>PULLEY</head><p>, one of the five mechanical powers; consisting of a little wheel, being a circular piece of wood or metal, turning on an axis, and having a channel around it, in its edge or circumference, in which a cord slides and so raises up weights. <figure/></p><p>The Latins call it Trochlea; and the seamen, when fitted with a rope, a Tackle. An assemblage of several <cb/> Pulleys is called a System of Pulleys, or Polyspaston: some of which are in a block or case, which is fixed; and others in a block which is moveable, and rises with the weight. The wheel or rundle is called the Sheave or Shiver; the axis on which it turns, the Gudgeon; and the fixed piece of wood or iron, into which it is put, the Block.</p><p><hi rend="italics">Doctrine of the</hi> <hi rend="smallcaps">Pulley.</hi>—1. If the equal weights P and W hang by the cord BB upon the pulley A, whose block <hi rend="italics">b</hi> is fixed to the beam HI, they will counterpoise each other, just in the same manner as if the cord were cut in the middle, and its two ends hung upon the hooks fixed in the Pulley at A and A, equally distant from the centre.</p><p>Hence, a single Pulley, if the lines of direction of the power and the weight be tangents to the periphery, neither assists nor impedes the power, but only changes its direction. The use of the Pulley therefore, is when the vertical direction of a power is to be changed into an horizontal one; or an ascending direction into a descending one; &c. This is found a good provision for the safety of the workmen employed in drawing with the Pulley. And this change of direction by means of a Pulley has this farther advantage; that if any power can exert more force in one direction than another, we are hence enabled to employ it with its greatest effect; as for the convenience of a horse to draw in a horizontal direction, or such like.</p><p>But the great use of the Pulley is in combining several of them together; thus forming what Vitruvius and others call Polyspasta; the advantages of which are, that the machine takes up but little room, is easily removed, and raises a very great weight with a moderate force.</p><p>2. When a weight W hangs at the lower end of the moveable block <hi rend="italics">p</hi> of the Pulley D, and the chord GF goes under the Pulley, it is plain that the part G of the cord bears one half of the weight W, and the part F the other half of it; for they bear the whole between them; therefore whatever holds the upper end of either rope, sustains one half of the weight; and thus the power P, which draws the cord F by means of the cord E, passing over the fixed pulley C, will sustain the weight W when its intensity is only equal to the half of W; that is, in the case of one moveable Pulley, the power gained is as 2 to 1, or as the number of ropes G and F to the one rope E.</p><p>In like manner, in the case of two moveable Pulleys P and L, each of these also doubles the power, and produces a gain of 4 to 1, or as the number of the ropes Q, M, S, K, sustaining the weight W, to the 1 rope O sustaining the power T; that is, W is to T as 4 to 1. And so on, for any number of moveable Pulleys, viz, 3 such Pulleys producing an increase of power as 6 to 1; 4 Pulleys, as 8 to 1; &c; each power adding 2 to the number. Also the effect is the same, when the Pulleys are disposed as in the fixed block X, and the other two as in the moveable block Y; these in the lower block giving the same advantage to the power, when they rise all together in one block with the weight.</p><p>But if the lower Pulleys do not rise all together in one block with the weight, but act upon one another, having the weight only fastened to the lowest of them, the <pb n="302"/><cb/> force of the power is still more increased, each power doubling the former numbers, the gain of power in this case proceeding in the geometrical progression, 1, 2, 4, 8, 16, &c, according to the powers of 2; whereas in the former case, the gain was only in arithmetical progression, increasing by the addition of 2. Thus, a power whose intensity is equal to 8lb applied at <hi rend="italics">a</hi> will, by means of the lower Pulley A, sustain 16lb; and a power equal to 4lb at <hi rend="italics">b,</hi> by means <figure/> of the Pulley, will sustain the power of 8lb acting at <hi rend="italics">a,</hi> and consequently the weight of 16lb at W; also a third power equal to 2lb at <hi rend="italics">c,</hi> by means of the Pulley C, will sustain the power of 4lb at <hi rend="italics">b;</hi> and a 4th power of 1lb at <hi rend="italics">d,</hi> by means of the Pulley D, will sustain the power 2 at <hi rend="italics">c,</hi> and consequently the power 4 at B, and the power 8 at A, and the weight 16 at W.</p><p>3. It is to be noted however, that, in whatever proportion the power is gained, in that very same proportion is the length of time increased to produce the same effect. For when a power moves a weight by means of several Pulleys, the space passed over by the power is to the space passed over by the weight, as the weight is to the power. Hence, the smaller a force is that sustains a weight by means of Pulleys, the slower is the weight raised; so that what is saved or gained in force, is always spent or lost in time: which is the general property of all the mechanical powers.</p><p>The usual methods of arranging Pulleys in their blocks, may be reduced to two. The first consists in placing them one by the side of another, upon the same pin; the other, in placing them directly under one another, upon separate pins. Each of these methods however is liable to inconvenience; and Mr. Smeaton, to avoid the impediments to which these combinations are subject, proposes to combine these two methods in one. See the Philos. Trans. vol. 47, pa. 494.</p><p>Some instances of such combinations of Pulleys are exhibited in the following figures; beside which, there are also other varieties of forms.</p><p>A very considerable improvement in the construction of Pulleys has been made by Mr. James White, who has obtained a patent for his invention, and of which he gives the following description. The last of the three following figures shews the machine, consisting of two Pulleys Q and R, one fixed and the other moveable. Each of these has six concentric grooves, capable of having a line put round them, and thus acting like as many different Pulleys, having diameters equal to those of the grooves. Supposing then each of the grooves to be a distinct Pulley, and that all their diameters were equal, it is evident that if the weight 144 were to be raised by pulling at S till the Pulleys touch each other, the first Pulley must receive the length of line as many times as there are parts of the line hanging <cb/> between it and the lower Pulley. In the present case, there are 12 lines, <hi rend="italics">b, d, f,</hi> &c, hanging between the <figure/> two pulleys, formed by its revolution about the six upper lower grooves. Hence as much line must pass over the uppermost Pulley as is equal to twelve times the distance of the two. But, from an inspection of the figure, it is plain, that the second Pulley cannot receive the full quantity of line by as much as is equal to the distance betwixt it and the first. In like manner, the third Pulley receives less than the first by as much as is the distance between the first and third; and so on to the last, which receives only one twelfth of the whole. For this receives its share of line <hi rend="italics">n</hi> from a fixed point in the upper frame, which gives it nothing; while all the others in the same frame receives the line partly by turning to meet it, and partly by the line coming to meet them.</p><p>Supposing now these Pulleys to be equal in size, and to move freely as the line determines them, it appears evident, from the nature of the system, that the number of their revolutions, and consequently their velocities, must be in proportion to the number of suspending parts that are between the fixed point above mentioned and each Pulley respectively. Thus the outermost Pulley would go twelve times round in the time that the Pulley under which the part <hi rend="italics">n</hi> of the line, if equal to it, would revolve only once; and the intermediate times and velocities would be a series of arithmetical proportionals, of which, if the first number were 1, the last would always be equal to the whole number of terms. Since then the revolutions of equal and distinct Pulleys are measured by their velocities, and that it is possible to find any proportion of velocity, on a single body running on a centre, viz, by finding proportionate distances from that centre; it follows, that if the diameters of certain grooves in the same substance be exactly adapted to the above series (the line itself being supposed inelastic, and of no magnitude) the necessity <pb/><pb/><pb n="303"/><cb/> of using several Pulleys in each frame will be obviated, and with that some of the inconveniencies to which the use of the Pulley is liable.</p><p>In the figure referred to, the coils of rope by which the weight is supported, are represented by the lines <hi rend="italics">a, b, c</hi> &c; <hi rend="italics">a</hi> is <hi rend="italics">the line of traction,</hi> commonly called the fall, which passes over and under the proper grooves, until it is fastened to the upper frame just above <hi rend="italics">n.</hi> In practice, however, the grooves are not arithmetical proportions, nor can they be so; for the diameter of the rope employed must in all cases be deducted from each term; without which the smaller grooves, to which the said diameter bears a larger proportion than to the larger ones, will tend to rise and fall faster than they, and thus introduce worse defects than those which they were intended to obviate.</p><p>The principal advantage of this kind of Pulley is, that it destroys lateral friction, and that kind of shaking motion which is so inconvenient in the common Pulley. And lest (says Mr. White) this circumstance should give the idea of weakness, I would observe, that to have pins for the pulleys to run on, is not the only nor perhaps the best method; but that I sometimes use centres fixed to the Pulleys, and revolving on a very short bearing in the side of the frame, by which strength is increased, and friction very much diminished; for to the last moment the motion of the Pulley is perfectly circular: and this very circumstance is the cause of its not wearing out in the centre as soon as it would, assisted by the ever increasing irregularities of a gullied bearing. These Pulleys, when well executed, apply to jacks and other machines of that nature with peculiar advantage, both as to the time of going and their own durability; and it is possible to produce a system of Pulleys of this kind of six or eight parts only, and adapted to the pockets, which, by means of a skain of sewing silk, or a clue of common thread, will raise upwards of an hundred weight.</p><p>As a system of Pulleys has no great weight, and lies in a small compass, it is easily carried about, and can be applied for raising weights in a great many cases, where other engines cannot be used. But they are subject to a great deal of friction, on the following accounts; viz, 1st, because the diameters of their axes bear a very considerable proportion to their own diameters; 2d, because in working they are apt to rub against one another, or against the sides of the block; 3dly, because of the stiffness of the rope that goes over and under them. See Ferguson's Mech. pa. 37, 4to.</p><p>But the friction of the Pulley is now reduced to nothing as it were, by the ingenious Mr. Garnett's patent friction rollers, which produce a great saving of labour and expence, as well as in the wear of the machine, both when applied to Pulleys and to the axles of wheel-carriages. His general principle is this; between the axle and nave, or centre pin and box, a hollow space is left, to be filled up by solid equal rollers nearly touching each other. These are furnished with axles inserted into a circular ring at each end, by which their relative distances are preserved; and they are kept parallel by means of wires fastened to the rings between the rollers, and which are rivetted to them.</p><p>The above contrivance is exhibited in the annexed figure; where ABCD represents a piece of metal to <cb/> be inserted into the bo<*> or nave, of which E is the centrepin or axle, and 1, 1, 1, &c, rollers of metal having <figure/> axes inserted in the brazen circle which passes through their centres; and both circles being rivetted together by means of bolts passing between the rollers from one side of the nave to the other; and thus they are always kept separate and parallel.</p></div1><div1 part="N" n="PUMP" org="uniform" sample="complete" type="entry"><head>PUMP</head><p>, in Hydraulics, a machine for raising water, and other fluids.</p><p>Pumps are probably of very ancient use. Vitruvius ascribes the invention to Ctesebes of Athens, some say of Alexandria, about 120 years before Christ. They are now of various kinds. As the Sucking Pump, the Lifting Pump, the Forcing Pump, Ship Pumps, Chain Pumps, &c. By means of the lifting and forcing Pumps, water may be raised to any height, with a sufficient power, and an adequate apparatus: but by the sucking Pump the water may, by the general pressure of the atmosphere on the surface of the well, be raised only about 33 or 34 feet; though in practice it is seldom applied to the raising it much above 28; because, from the variations observed in the barometer, it appears that the air may sometimes be lighter than 33 feet of water; and whenever that happens, for want of the due counterpoise, this Pump may fail in its performance.</p><p><hi rend="italics">The Common Sucking</hi> <hi rend="smallcaps">Pump.</hi>—This consists of a pipe, of wood or metal, open at both ends, having a fixed valve in the lower part of it opening upwards, and a moveable valve or bucket by which the water is drawn or lifted up. This bucket is just the size of the bore of the Pump-pipe, in that part where it works, and leathered round so as to sit it very close, that no air may pass by the sides of it; the valve hole being in the middle of the bucket. The bucket is commonly worked in the upper part of the barrel by a short rod, and another fixed valve placed just below the descent of the bucket. Thus, (fig. 1, pl. 23), AB is the Pump-pipe, C the lower fixed valve, opening upwards, and D is the bucket, or moving valve, also opening upwards.</p><p>In working the Pump; draw up the bucket D, by means of the Pump rod, having any sort of a handle fixed to it: this draws up the water that is above it, or if not, the air; in either case the water pushes up the valve C, and enters to supply the void left between C and D, being pushed up by the pressure of the atmosphere on the surface of the water in the well below. Next, the bucket D is pushed down, which shuts the <pb n="304"/><cb/> valve C, and prevents the return of the water downwards, which opens the valve D, by which the water ascends above it. And thus, by repeating the strokes of the Pump-rod handle, the valves alternately open and shut, and the water is drawn up at every stroke, and runs out at the nozle or spout near the top.</p><p><hi rend="italics">The Lifting</hi> <hi rend="smallcaps">Pump</hi> differs from the sucking Pump only in the disposition of its valves and the form of its piston frame. This kind of Pump is represented in fig. 2, pl. 23; where the lower valve D is moveable, being worked up and down with the Pump rod, which lifts the water up, and so opens the upper valve C, which is fixed, and permits the water to issue through it, and run out at top. Then as the piston D descends, the weight of the water above C shuts that valve C, and so prevents its return, till that valve be opened again by another lift of the piston D. And so alternately.</p><p><hi rend="italics">The Forcing</hi> <hi rend="smallcaps">Pump</hi> raises the water through the sucker, or lower valve C (fig. 3, pl. 23), in the same manner as the sucking Pump; but as the piston or plunger D has no valve in it, the water cannot get above it when this is pushed down again; instead of which, a side pipe is inserted between C and D, having a fixed valve at E opening upwards, through which the water is forced out of the Pump by pushing down the plunger D.</p><p><hi rend="italics">Observations on Pumps.</hi>—The force required to work a Pump, is equal to the weight of water raised at each stroke, or equal to the weight of water filling the cavity of the pipe, and its height equal to the length of the stroke made by the piston. Hence if <hi rend="italics">d</hi> denote the diameter of the pipe, and <hi rend="italics">l</hi> the length of the stroke, both in inches; then is .7854<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">l</hi> the content of the water raised at a stroke, in inches, or .0028<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">l</hi> in ale gallons; and the weight of it is (<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">l</hi>)/220 ounces or ((<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">l</hi>)/3520)lb. But if the handle of the pump be a lever which gains in the power of <hi rend="italics">p</hi> to 1, the force of the hand to work the Pump will be only ((<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">l</hi>)/(3520<hi rend="italics">p</hi>))lb, or, when <hi rend="italics">p</hi> is 5 for instance, it will be ((<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">l</hi>)/17600)lb. And all these over and above the friction of the moving parts of the Pump.</p><p>To the forcing Pump is sometimes adapted an air vessel, which, being compressed by the water, by its elasticity acts upon the water again, and forces it out to a great distance, and in a continued stream, instead of by jerks or jets. So, Mr. Newsham's water engine, for extinguishing fire, consists of two forcing Pumps, which alternately drive water into a close vessel of air, by which means the air in it is condensed, and compresses the water so strongly, that it rushes out with great impetuosity and force through a pipe that comes down into it, making a continued uniform stream.</p><p>By means of forcing Pumps, water may be raised to any height whatever above the level of a river or spring; and machines may be contrived to work these Pumps, either by a running stream, a fall of water, or by horses.</p><p><hi rend="italics">Ctesebes</hi>'s <hi rend="smallcaps">Pump</hi>, acts both by suction and by pression. Thus, a brass cylinder ABCD (fig. 5, pl. 23) furnished with a valve at L, is placed in the water. In this is fitted the piston KM, made of green wood, which will not swell in the water, and is adjusted to the <cb/> aperture of the cylinder with a covering of leather, but without any valve. Another tube NH is fitted on at H, with a valve I opening upwards.—Now the piston being raised, the water opens the valve L, and rises into the cavity of the cylinder. When the piston is depressed again, the valve I is opened, and the water is driven up through the tube HN.</p><p>This was the Pump used among the Ancients, and that from which both the others have been deduced. Sir Samuel Morland has endeavoured to increase its force by lessening the friction; which he has done to good effect, so as to make it work with very little.</p><p>There are various kinds of Pumps used in ships, for throwing the water out of the hold, and upon other occasions, as the Chain Pump, &c.</p><p><hi rend="italics">Air</hi>-<hi rend="smallcaps">Pump</hi>, in Pneumatics, is a machine, by means of which the air is emptied out of vessels, and a kind of vacuum produced in them. For the particulars of which, see <hi rend="smallcaps">Air</hi>-<hi rend="italics">Pump.</hi></p></div1><div1 part="N" n="PUNCHEON" org="uniform" sample="complete" type="entry"><head>PUNCHEON</head><p>, a measure for liquids, containing 1/3 of a tun, or a hogshead and 1/3, or 84 gallons.</p></div1><div1 part="N" n="PUNCHINS" org="uniform" sample="complete" type="entry"><head>PUNCHINS</head><p>, or <hi rend="smallcaps">Punchions</hi>, in Building, short pieces of timber placed to support some considerable weight.</p><p>PUNCTATED <hi rend="italics">Hyperbola,</hi> in the higher geometry, an hyperbola whose conjugate oval is infinitely small, that is, a point.</p><p>PUNCTUM <hi rend="italics">ex Comparatione,</hi> is either focus, in the ellipse or hyperbola; so called by Apollonius, because the rectangle under two abscisses made at the focus, is equal to one fourth part of what he calls the figure, which is the square of the conjugate axis, or the rectangle under the transverse and the parameter.</p></div1><div1 part="N" n="PURBACH" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">PURBACH</surname> (<foreName full="yes"><hi rend="smallcaps">George</hi></foreName>)</persName></head><p>, a very eminent mathematician and astronomer, was born at Purbach, a town upon the confines of Bavaria and Austria, in 1423, and educated at Vienna. He afterwards visited the most celebrated universities in Germany, France, and Italy; and found a particular friend and patron in cardinal Cusa at Rome. Returning to Vienna, he was appointed mathematical professor, in which office he continued till his death, which happened in 1461, in the 39th year of his age only, to the great loss of the learned world.</p><p>Purbach composed a great number of pieces, upon mathematical and astronomical subjects; and his fame brought many students to Vienna, and among them, the celebrated Regiomontanus, between whom and Purbach there subsisted the strictest friendship and union of studies till the death of the latter. These two laboured together to improve every branch of learning, by all the means in their power, though astronomy seems to have been the favourite of both; and had not the immature death of Purbach prevented his further pursuits, there is no doubt but that, by their joint industry, astronomy would have been carried to very great perfection. That this is not merely surmise, may be learnt from those improvements which Purbach actually did make, to render the study of it more easy and practicable. His first essay was, to amend the Latin translation of Ptolomy's Almagest, which had been made from the Arabic version: this he did, not by the help of the Greek text, for he was unacquainted with that language, but by drawing the most probable conjectures from a strict attention to the sense of the author. <pb n="305"/><cb/></p><p>He then proceeded to other works, and among them, he wrote a tract, which he entitled, <hi rend="italics">An Introduction to Arithmetic;</hi> then a treatise on <hi rend="italics">Gnomonics,</hi> or <hi rend="italics">Dialling,</hi> with tables suited to the difference of climates or latitudes; likewise a small tract concerning the <hi rend="italics">Altitudes of the Sun,</hi> with a table; also, <hi rend="italics">Astrolabie Canons,</hi> with a table of the parallels, proportioned to every degree of the equinoctial.</p><p>After this, he constructed Solid Spheres, or Celestial Globes, and composed a new table of fixed stars, adding the longitude by which every star, since the time of Ptolomy, had increased. He likewise invented various other instruments, among which was the Gnomon, or Geometrical Square, with canons and a table for the use of it.</p><p>He not only collected the various tables of the Primum Mobile, but added new ones. He made very great improvements in Trigonometry, and by introducing the table of Sines, by a decimal division of the radius, he quite changed the appearance of that science: he supposed the radius to be divided into 600000 equal parts, and computed the sines of the arcs, for every ten minutes, in such equal parts of the radius, by the decimal notation, instead of the duodecimal one delivered by the Greeks, and preserved even by the Arabians till our author's time; a project which was completed by his friend Regiomontanus, who computed the sines to every minute of the quadrant, in 1000000th parts of the radius.</p><p>Having prepared the tables of the fixed stars, he next undertook to reform those of the planets, and constructed some entirely new ones. Having finished his tables, he wrote a kind of Perpetual Almanac, but chiefly for the moon, answering to the periods of Meton and Calippus; also an Almanac for the Planets, or, as Regiomontanus afterwards called it, an Ephemeris, for many years. But observing there were some planets in the heavens at a great distance from the places where they were described to be in the tables, particularly the sun and moon (the eclipses of which were observed frequently to happen very different from the times predicted), he applied himself to construct new tables, particularly adapted to eclipses; which were long after famous for their exactness. To the same time may be referred his sinishing that celebrated work, entitled, <hi rend="italics">A New Theory of the Planets,</hi> which Regiomontanus afterwards published the first of all the works executed at his new printing-house.</p><p>PURE <hi rend="italics">Hyperbola,</hi> is an Hyperbola without any oval, node, cusp, or conjugate point; which happens through the impossibility of two of its roots.</p><p><hi rend="smallcaps">Pure</hi> <hi rend="italics">Mathematics, Proposition, Quadratics,</hi> &c. See the several articles.</p></div1><div1 part="N" n="PURLINES" org="uniform" sample="complete" type="entry"><head>PURLINES</head><p>, in Architecture, those pieces of timber that lie across the rafters on the inside, to keep them from sinking in the middle of their length.</p></div1><div1 part="N" n="PYRAMID" org="uniform" sample="complete" type="entry"><head>PYRAMID</head><p>, a solid having any plane figure for its base, and its sides triangles whose vertices all meet in a point at the top, called the vertex of the pyramid; the base of each triangle being the sides of the plane base of the Pyramid.—The number of triangles is equal to the number of the sides of the base; and a cone is a round Pyramid, or one having an infinite number of sides. —The Pyramid is also denominated from its base, <cb/> being triangular when the base is a triangle, quadrangular when a quadrangle, &c.</p><p>The <hi rend="italics">axis</hi> of the Pyramid, is the line drawn from the vertex to the centre of the base. When this axis is perpendicular to the base, the Pyramid is said to be a <hi rend="italics">right</hi> one; otherwise it is <hi rend="italics">oblique.</hi></p><p>1. A Pyramid may be conceived to be generated by a line moved about the vertex, and so carried round the perimeter of the base.</p><p>2. All Pyramids having equal bases and altitudes, are equal to one another: though the figures of their bases should even be different.</p><p>3. Every Pyramid is equal to one-third of the circumscribed prism, or a prism of the same base and altitude; and therefore the solid content of the Pyramid is found by multiplying the base by the perpendicular altitude, and taking 1/3 of the product.</p><p>4. The upright surface of a Pyramid, is found by adding together the areas of all the triangles which form that surface.</p><p>5. If a Pyramid be cut by a plane parallel to the base, the section will be a plane figure similar to the base; and these two figures will be in proportion to each other as the squares of their distances from the vertex of the Pyramid.</p><p>6. The centre of gravity of a Pyramid is distant from the vertex 3/4 of the axis.</p><p><hi rend="italics">Frustum of a</hi> <hi rend="smallcaps">Pyramid</hi>, is the part left at the bottom when the top is cut off by a plane parallel to the base.</p><p>The solid content of the Frustum of a Pyramid is found, by first adding into one sum the areas of the two ends and the mean proportional between them, the 3d part of which sum is a medium section, or is the base of an equal prism of the same altitude; and therefore this medium area or section multiplied by the altitude gives the solid content. So, if A denote the area of one end, <hi rend="italics">a</hi> the area of the other end, and <hi rend="italics">b</hi> the height; then 1/3 (A + <hi rend="italics">a</hi> + √(A<hi rend="italics">a</hi>)) is the medium area or section, and 1/3 (A + <hi rend="italics">a</hi> + √(A<hi rend="italics">a</hi>)) X <hi rend="italics">b</hi> is the solid con- tent.</p><p><hi rend="smallcaps">Pyramids</hi> <hi rend="italics">of Egypt,</hi> are very numerous, counting both great and small; but the most remarkable are the three Pyramids of Memphis, or, as they are now called, of Gheisa or Gize. They are square Pyramids, and the dimensions of the greatest of them, are 700 feet on each side of the base, and the oblique height or slant side the same; its base covers, or stands upon, nearly 11 acres of ground. It is thought by some that these Pyramids were designed and used as gnomons, for astronomical purposes; and it is remarkable that their four sides are accurately in the direction of the four cardinal points of the compass, east, west, north, and south.</p><p>PYRAMIDAL <hi rend="italics">Numbers,</hi> are the sums of polygonal numbers, collected after the same manner as the polygonal numbers themselves are found from arithmetical progressions.</p><p>These are particularly called First Pyramidals. The sums of First Pyramidals are called Second Pyramidals; and the sums of the 2d are 3d Pyramidals; and so on. Particularly, those arising from triangular numbers, are called Prime Triangular Pyramidals; those arising <pb n="306"/><cb/> from pentagonal numbers, are called Prime Pentagonal Pyramidals; and so on. <table><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">The numbers</cell><cell cols="1" rows="1" role="data">1,</cell><cell cols="1" rows="1" role="data">4,</cell><cell cols="1" rows="1" role="data">10,</cell><cell cols="1" rows="1" role="data">20,</cell><cell cols="1" rows="1" role="data">35,</cell><cell cols="1" rows="1" role="data">&c,</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">formed by adding the tri-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi> 1,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">3,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">6,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">10,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">15,</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">&c,</cell></row><row role="data"><cell cols="1" rows="1" role="data">  angulars</cell></row></table> are usually called simply by the name of Pyramidals; and the general formula for finding them is <hi rend="italics">n</hi> X (<hi rend="italics">n</hi> - 1)/2 X (<hi rend="italics">n</hi> - 2)/3; so the 4th Pyramidal is found by substituting 4 for <hi rend="italics">n;</hi> the 5th by substituting 5 for <hi rend="italics">n;</hi> &c. See <hi rend="smallcaps">Figurate</hi> <hi rend="italics">Numbers,</hi> and <hi rend="smallcaps">Polygonal</hi> <hi rend="italics">Numbers.</hi></p></div1><div1 part="N" n="PYRAMIDOID" org="uniform" sample="complete" type="entry"><head>PYRAMIDOID</head><p>, is sometimes used for the parabolic spindle, or the solid formed by the rotation of a semiparabola about its base or greatest ordinate. See <hi rend="smallcaps">Parabolic</hi> <hi rend="italics">Spindle.</hi></p></div1><div1 part="N" n="PYROMETER" org="uniform" sample="complete" type="entry"><head>PYROMETER</head><p>, or fire-measurer, a machine for measuring the expansion of solid bodies by heat.</p><p>Musschenbroek was the first inventor of this instrument; though it has since received several improvements by other philosophers. He has given a table of the expansions of the different metals, with various degrees of heat. Having prepared cylindric rods of iron, steel, copper, brass, tin, and lead, he exposed them first to a Pyrometer with one flame in the middle; then with two flames; then successively with three, four, and five flames. The effects were as in the following Table, where the degrees of expansion are marked in parts equal to the 12500th part of an inch. <table rend="border"><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Expansion of</cell><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" role="data">Steel</cell><cell cols="1" rows="1" role="data">Copp.</cell><cell cols="1" rows="1" role="data">Brass</cell><cell cols="1" rows="1" role="data">Tin</cell><cell cols="1" rows="1" role="data">Lead</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">By 1 flame</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" role="data">153</cell><cell cols="1" rows="1" role="data">155</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">By 2 flames</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(10)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=3" role="data">117</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">123</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">115</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">220</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="rowspan=3" role="data">274</cell></row><row role="data"><cell cols="1" rows="1" role="data">  placed close</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  together</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">By 2 flames at</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(10)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=3" role="data">109</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">94</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">92</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">141</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">219</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">263</cell></row><row role="data"><cell cols="1" rows="1" role="data">  2 1/2 inches dis-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  tant</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">By 3 flames close</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(10)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">142</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">168</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">193</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">275</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  together</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">By 4 flames close</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(10)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">211</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">270</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">270</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">361</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  together</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">By 5 flames</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">230</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">377</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>Tin easily melts when heated by two flames placed close together; and lead with three flames close together, when they burn long.</p><p>It hence appears that the expansions of any metal are in a less degree than the number of flames: so two flames give less than a double expansion, three flames less than a triple expansion, and so on, always more and more below the ratio of the number of flames. And the flames placed together cause a greater expansion, than with an interval between them.</p><p>For the construction of Musschenbroek's Pyrometer, with alterations and improvements upon it by Desaguliers, see Defag. Exper. Philos. vol. 1, pa. 421; see also Musschenbroek's translation of the Experiments of the Academy del Cimento, printed at Leyden in 1731; <cb/> and for a Pyrometer of a new construction, by which the expansions of metals in boiling fluids may be examined and compared with Fahrenheit's thermometer, see Musschenb. Introd. ad Philos. Nat. 4to, 1762, vol. 2, pa. 610.</p><p>But as it has been observed, that Musschenbroek's Pyrometer was liable to some objections, these have been removed in a good measure by Ellicott, who has given a description of his improved Pyrometer in the Philos. Trans. numb. 443; and the same may be seen in the Abridg. vol. 8, pa. 464. This instrument measures the expansions to the 7200th part of an inch; and by means of it, Mr. Ellicott sound, upon a medium, that the expansions of bars of different metals, as nearly of the same dimensions as possible, by the same degree of heat, were as below: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Gold,</cell><cell cols="1" rows="1" role="data">Silver,</cell><cell cols="1" rows="1" role="data">Brass,</cell><cell cols="1" rows="1" role="data">Copper,</cell><cell cols="1" rows="1" role="data">Iron,</cell><cell cols="1" rows="1" role="data">Steel,</cell><cell cols="1" rows="1" role="data">Lead.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">103</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">149</cell></row></table></p><p>The great difference between the expansions of iron and brass, has been applied with good success to remove the irregularities in pendulums arising from heat. Philos. Trans. vol. 47, pa. 485.</p><p>Mr. Graham used to measure the minute expansions of metal bars, by advancing the point of a micrometer screw, till it sensibly stopped against the end of the bar to be measured. This screw, being small and very lightly hung, was capable of agreement within the 3000 or 4000th part of an inch. And on this general principle Mr. Smeaton contrived his Pyrometer, in which the measures are determined by the contact of a piece of metal with the point of a micrometer screw. This instrument makes the expansions sensible to the 2345th part of an inch. And when it is used, both the instrument and the bar, to be measured, are immerged in a cistern of water, heated to any degree, up to boiling, by means of lamps placed under the cistern; and the water communicates the same degree of heat to the instrument and bar, and to a mercurial thermometer immerged in it, for ascertaining that degree.</p><p>With this Pyrometer Mr. Smeaton made several experiments, which are arranged in a table; and he remarks, that their result agrees very well with the proportions of expansions of several metals given by Mr. Ellicott. The following Table shews how much a foot in length of each metal expands by an increase of heat corresponding to 180 degrees of Fahrenheit's thermometer, or to the difference between the temperatures of freezing and boiling water, expressed in the 10000th part of an inch. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1.</cell><cell cols="1" rows="1" role="data">White glass barometer tube</cell><cell cols="1" rows="1" role="data">100</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2.</cell><cell cols="1" rows="1" role="data">Martial regulus of antimony</cell><cell cols="1" rows="1" role="data">130</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3.</cell><cell cols="1" rows="1" role="data">Blistered steel</cell><cell cols="1" rows="1" role="data">138</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4.</cell><cell cols="1" rows="1" role="data">Hard steel</cell><cell cols="1" rows="1" role="data">147</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5.</cell><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" role="data">151</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6.</cell><cell cols="1" rows="1" role="data">Bismuth</cell><cell cols="1" rows="1" role="data">167</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7.</cell><cell cols="1" rows="1" role="data">Copper, hammered</cell><cell cols="1" rows="1" role="data">204</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8.</cell><cell cols="1" rows="1" role="data">Copper 8 parts, mixed with 1 part tin</cell><cell cols="1" rows="1" role="data">218</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9.</cell><cell cols="1" rows="1" role="data">Cast brass</cell><cell cols="1" rows="1" role="data">225</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10.</cell><cell cols="1" rows="1" role="data">Brass 16 parts, with 1 of tin</cell><cell cols="1" rows="1" role="data">229</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">11.</cell><cell cols="1" rows="1" role="data">Brass wire</cell><cell cols="1" rows="1" role="data">232</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12.</cell><cell cols="1" rows="1" role="data">Speculum metal</cell><cell cols="1" rows="1" role="data">232</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">13.</cell><cell cols="1" rows="1" role="data">Spelter solder, viz 2 parts brass and 1 zinc</cell><cell cols="1" rows="1" role="data">247</cell></row></table> <pb n="307"/><cb/> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">14.</cell><cell cols="1" rows="1" role="data">Fine pewter</cell><cell cols="1" rows="1" role="data">274</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">15.</cell><cell cols="1" rows="1" role="data">Grain tin</cell><cell cols="1" rows="1" role="data">298</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16.</cell><cell cols="1" rows="1" role="data">Soft solder, viz lead 2 and tin 1</cell><cell cols="1" rows="1" role="data">301</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">17.</cell><cell cols="1" rows="1" role="data">Zinc 8 parts with tin 1, a little hammered</cell><cell cols="1" rows="1" role="data">323</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">18.</cell><cell cols="1" rows="1" role="data">Lead</cell><cell cols="1" rows="1" role="data">344</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">19.</cell><cell cols="1" rows="1" role="data">Zinc or spelter</cell><cell cols="1" rows="1" role="data">353</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20.</cell><cell cols="1" rows="1" role="data">Zinc hammered half an inch per foot</cell><cell cols="1" rows="1" role="data">373</cell></row></table></p><p>For a farther account of this instrument, with its use, see Philos. Trans. vol. 48, pa. 598.</p><p>Mr. Ferguson has constructed, and described a Pyrometer (Lect. on Mechanics, Suppl. pa. 7, 4to), which makes the expansion of metals by heat visible to the 45000th part of an inch. And another plan of a Pyrometer has lately been invented by M. De Luc, in consequence of a hint suggested to him by Mr. Ramsden: for an account of which, with the principle of its construction and use, both in the comparative measure of the expansions of bodies by heat, and the measure of their absolute expansion, as well as the experiments made with it, see M. De Luc's elaborate essay on Pyrometry &c, in the Philos. Trans. vol. 68, pa. 419— 546.</p><p>Other very nice and ingenious contrivances, for the measuring of expansions by heat, have been made by Mr. Ramsden; which he has successfully applied in the case of the measuring rods and chains lately employed, by General Roy and Col. Williams, in measuring the base on Hounslow Heath, &c; which determine the expansions, to great minuteness, for each degree of the thermometer. See Philos. Trans. 1785, &c.</p></div1><div1 part="N" n="PYROPHORUS" org="uniform" sample="complete" type="entry"><head>PYROPHORUS</head><p>, the name usually given to that substance by some called black phosphorus; being a chemical preparation possessing the singular property of kindling spontaneously when exposed to the air; which was accidentally discovered by M. Homberg, who prepared it of alum and human fæces. Though it has since been found, by the son of M. Lemeri, that the fæces are not necessary to it, but that honey, sugar, flour, and any animal or vegetable matter, may be used instead of the fæces; and M. De Suvigny has shewn that most vitriolic salts may be substituted for the alum. See Priestley's Observ. on Air, vol. 3, Append. p. 386, and vol. 4, Append. p. 479.</p></div1><div1 part="N" n="PYROTECHNY" org="uniform" sample="complete" type="entry"><head>PYROTECHNY</head><p>, the art of sire, or the science which teaches the application and management of fire in several operations.</p><p>Pyrotechny is of two kinds, military and chemical.</p><p><hi rend="italics">Military</hi> <hi rend="smallcaps">Pyrotechny</hi>, is the science of artificial fire-works, and fire-arms, teaching the structure and use both of those employed in war, as gunpowder, cannon, shells, carcasses, mines, fusees, &c; and of those made for amusement, as rockets, stars, serpents, &c.</p><p>Some call Pyrotechny by the name Artillery; though that word is usually confined to the instruments employed in war. Others choose to call it Pyrobology, or rather Pyroballogy, or the art of missile fires.</p><p>Wolfius has reduced Pyrotechny into a kind of mixt mathematical art. Indeed it will not allow of geometrical demonstrations; but he brings it to tolerable rules and reasons; whereas it had formerly been treated by authors at random, and without regard to any reasons at all. See the several articles <hi rend="smallcaps">Cannon, Gunpowder, Rocket, Shell</hi>, &c. <cb/></p><p><hi rend="italics">Chemical</hi> <hi rend="smallcaps">Pyrotechny</hi>, is the art of managing and applying fire in distillations, calcinations, and other operations of chemistry.</p><p>Some reckon a third kind of Pyrotechny, viz, the art of susing, refining, and preparing metals.</p></div1><div1 part="N" n="PYTHAGORAS" org="uniform" sample="complete" type="entry"><head>PYTHAGORAS</head><p>, one of the greatest philosophers of antiquity, was born about the 47th Olympiad, or 590 years before Christ. His father's principal residence was at Samos, but being a travelling merchant, his son Pythagoras was born at Sidon in Syria; but soon returning home again, our philosopher was brought up at Samos, where he was educated in a manner that was answerable to the great hopes that were conceived of him. He was called “the youth with a fine head of hair;” and from the great qualities that soon appeared in him, he was regarded as a good genius sent into the world for the benefit of mankind.</p><p>Samos however afforded no philosophers capable of satisfying his thirst for knowledge; and therefore, at 18 years of age, he resolved to travel in quest of them elsewhere. The fame of Pherecydes drew him first to the island of Syros: from hence he went to Miletus, where he conversed with Thales. He then travelled to Phœnicia, and stayed some time at Sidon, the place of his birth; and from hence he passed into Egypt, where Thales and Solon had been before him.</p><p>Having spent 25 years in Egypt, to acquire all the learning and knowledge he could procure in that country, with the same view he travelled through Chaldea, and visited Babylon. Returning after some time, he went to Crete; and from hence to Sparta, to be instructed in the laws of Minos and Lycurgus. He then returned to Samos; which, finding under the tyranny of Polycrates, he quitted again, and visited the several countries of Greece. Passing through Peloponnesus, he stopped at Phlius, where Leo then reigned; and in his conversation with that prince, he spoke with so much eloquence and wisdom, that Leo was at once ravished and surprised.</p><p>From Peloponnesus he went into Italy, and passed some time at Heraclea, and at Tarentum, but made his chief residence at Croton; where, after reforming the manners of the citizens by preaching, and establishing the city by wise and prudent counsels, he opened a school, to display the treasures of wisdom and learning he possessed. It is not to be wondered, that he was soon attended by a crowd of disciples, who repaired to him from different parts of Greece and Italy.</p><p>He gave his scholars the rules of the Egyptian priests, and made them pass through the austerities which he himself had endured. He at first enjoined them a five years silence in the school, during which they were only to hear; after which, leave was given them to start questions, and to propose doubts, under the caution however, to say, “not a little in many words, but much in a few.” Having gone through their probation, they were obliged, before they were admitted, to bring all their fortune into the common stock, which was managed by persons chosen on purpose, and called œconomists, and the whole community had all things in common.</p><p>The necessity of concealing their mysteries induced the Egyptians to make use of three sorts of styles, or ways of expressing their thoughts; the simple, the <pb n="308"/><cb/> hieroglyphical, and the symbolical. In the simple, they spoke plainly and intelligibly, as in common conversation; in the hieroglyphical, they concealed their thoughts under certain images and characters; and in the symbolical, they explained them by short expressions, which, under a sense plain and simple, included another wholly figurative. Pythagoras borrowed these three different ways from the Egyptians, in all the instructions he gave; but chiefly imitated the symbolical style, which he thought very proper to inculcate the greatest and most important truths: for a symbol, by its double sense, the proper and the figurative, teaches two things at once; and nothing pleases the mind more, than the double image it represents to our view.</p><p>In this manner Pythagoras delivered many excellent things concerning God and the human soul, and a great variety of precepts, relating to the conduct of life, political as well as civil; he made also some considerable discoveries and advances in the arts and sciences. Thus, among the works ascribed to him, there are not only books of physic, and books of morality, like that contained in what are called his <hi rend="italics">Golden Verses,</hi> but treatises on politics and theology. All these works are lost: but the vastness of his mind appears from the wonderful things he performed. He delivered, as antiquity relates, several cities of Italy and Sicily from the yoke of slavery; he appeased seditions in others; and he softened the manners, and brought to temper the most savage and unruly spirits, of several people and tyrants. Phalaris, the tyrant of Sicily, it is said, was the only one who could withstand the remonstrances of Pythagoras; and he it seems was so enraged at his discourses, that he ordered him to be put to death. But though the lectures of the philosopher could make no impression on the tyrant, yet they were sufficient to reanimate the Sicilians, and to put them upon a bold action. In short, Phalaris was killed the same day that he had fixed for the death of the philosopher.</p><p>Pythagoras had a great veneration for marriage; and therefore himself married at Croton, a daughter of one of the chief men of that city, by whom he had two sons and a daughter: one of the sons succeeded his father in the school, and became the master of Empedocles: the daughter, named Damo, was distinguished both by her learning and her virtues, and wrote an excellent commentary upon Homer. It is related, that Pythagoras had given her some of his writings, with express commands not to impart them to any but those of his own family; to which Damo was so scrupulously obedient, that even when she was reduced to extreme poverty, she refused a great sum of money for them.</p><p>From the country in which Pythagoras thus settled and gave his instructions, his society of disciples was called the Italic sect of philosophers, and their reputation continued for some ages afterwards, when the Academy and the Lycæum united to obscure and swallow up the Italic lect. Pythagoras's disciples regarded the words of their master as the oracles of a god; his authority alone, though unsupported by reason, passed with them for reason itself: they looked on him as the most perfect image of God among men. His house was called the temple of Ceres, and his court yard the temple of the Muses: and when he went into towns, it was said he <cb/> went thither, “not to teach men, but to heal them.”</p><p>Pythagoras however was persecuted by bad men in the last years of his life; and some say he was killed in a tumult raised by them against him; but according to others, he died a natural death, at 90 years of age, about 497 years before Christ.</p><p>Beside the high respect and veneration the world has always had for Pythagoras, on account of the excellence of his wisdom, his morality, his theology, and politics, he was renowned as learned in all the sciences, and a considerable inventor of many things in them; as arithmetic, geometry, astronomy, music, &c. In arithmetic, the common multiplication table is, to this day, still called Pythagoras's table. In geometry, it is said he invented many theorems, particularly these three; 1st, Only three polygons, or regular plane figures, can fill up the space about a point, viz, the equilateral triangle, the square, and the hexagon: 2d, The sum of the three angles of every triangle, is equal to two right angles: 3d, In any right-angled triangle, the square on the longest side, is equal to both the squares on the two shorter sides: for the discovery of this last theorem, some authors say he offered to the gods a hecatomb, or a sacrifice of a hundred oxen; Plutarch however says it was only one ox, and even that is questioned by Cicero, as inconsistent with his doctrine, which forbade bloody sacrisices: the more accurate therefore say, he sacrificed an ox made of flour, or of clay; and Plutarch even doubts whether such sacrifice, whatever it was, was made for the said theorem, or for the area of the parabola, which it was said Pythagoras also found out.</p><p>In astronomy his inventions were many and great. It is reported he discovered, or maintained the true system of the world, which places the sun in the centre, and makes all the planets revolve about him; from him it is to this day called the old or Pythagorean system; and is the same as that lately revived by Copernicus. He first discovered, that Lucifer and Hesperus were but one and the same, being the planet Venus, though formerly thought to be two different stars. The invention of the obliquity of the zodiac is likewise ascribed to him. He first gave to the world the name <foreign xml:lang="greek">*ko<*>m<*></foreign>, <hi rend="italics">Kosmos,</hi> from the order and beauty of all things comprehended in it; asserting that it was made according to musical proportion: for as he held that the sun, by him and his followers termed the fiery globe of unity, was seated in the midst of the universe, and the earth and planets moving around him, so he held that the seven planets had an harmonious motion, and their distances from the sun corresponded to the musical intervals or divisions of the monochord.</p><p>Pythagoras and his followers held the transmigration of souls, making them successively occupy one body after another: on which account they abstained from flesh, and lived chiefly on vegetables.</p><p><hi rend="smallcaps">Pythagoras</hi>'s <hi rend="italics">Table,</hi> the same as the multiplication-table; which see.</p></div1><div1 part="N" n="PYTHAGOREAN" org="uniform" sample="complete" type="entry"><head>PYTHAGOREAN</head><p>, or <hi rend="smallcaps">Pythagoric</hi> <hi rend="italics">System,</hi> among the Ancients, was the same as the Copernican system among the Moderns. In this system, the sun is supposed at rest in the centre, with the earth and all the planets revolving about him, each in their orbits. See <hi rend="smallcaps">System.</hi> <pb/><pb/><pb n="309"/><cb/></p><p>It was so called, as having been maintained and cultivated by Pythagoras, and his followers; not that it was invented by him, for it was much older.</p><p><hi rend="smallcaps">Pythagorean</hi> <hi rend="italics">Theorem,</hi> is that in the 47th proposition of the 1st book of Euclid's Elements; viz, that in a right-angled triangle, the square of the longest side, is equal to the sum of both the squares of the two shorter sides. It has been said that Pythagoras offered a hecatomb, or sacrifice of 100 oxen, to the <cb/> gods, for inspiring him with the discovery of so remarkable a property.</p></div1><div1 part="N" n="PYTHAGOREANS" org="uniform" sample="complete" type="entry"><head>PYTHAGOREANS</head><p>, a sect of ancient philosophers, who followed the doctrines of Pythagoras. They were otherwise called the Italic sect, from the circumstance of his having settled in Italy. Out of his school proceeded the greatest philosophers and legislators, Zaleucus, Charondas, Archytas, &c. See the article <hi rend="smallcaps">Pythagoras.</hi></p><p>PYXIS <hi rend="italics">Nautica,</hi> the seaman's compass. </p></div1></div0><div0 part="N" n="Q" org="uniform" sample="complete" type="alphabetic letter"><head>Q</head><cb/><div1 part="N" n="QUADRAGESIMA" org="uniform" sample="complete" type="entry"><head>QUADRAGESIMA</head><p>, a denomination given to the time of Lent, from its consisting of about 40 days; commencing on Ash Wednesday.</p><p><hi rend="smallcaps">Quadragesima</hi> <hi rend="italics">Sunday,</hi> is the first Sunday in Lent, or the 1st Sunday after Ash Wednesday.</p></div1><div1 part="N" n="QUADRANGLE" org="uniform" sample="complete" type="entry"><head>QUADRANGLE</head><p>, or <hi rend="smallcaps">Quadrangular</hi> <hi rend="italics">figure,</hi> in Geometry, is a plane figure having four angles; and consequently four fides also.</p><p>To the class of Quadrangles belong the square, parallelogram, trapezium, rhombus, and rhomboides.— A square is a regular Quadrangle; a trapezium an irregular one.</p></div1><div1 part="N" n="QUADRANT" org="uniform" sample="complete" type="entry"><head>QUADRANT</head><p>, in Geometry, is either the quarter or 4th part of a circle, or the 4th part of its circumference; the arch of which therefore contains 90 degrees.</p><p><hi rend="smallcaps">Quadrant</hi> also denotes a mathematical instrument, of great use in astronomy and navigation, for taking the altitudes of the sun and stars, as also taking angles in surveying, heights-and-distances, &c.</p><p>This instrument is variously contrived, and furnished with different apparatus, according to the various uses it is intended for; but they have all this in common, that they consist of the quarter of a circle, whose limb or arch is divided into 90° &c. Some have a plummet suspended from the centre, and are furnished either with plain sights, or a telescope, to look through.</p><p>The principal and most useful Quadrants, are the common Surveying Quadrant, the Astronomical Quadrant, Adams's Quadrant, Cole's Quadrant, Collins's or Sutton's Quadrant, Davis's Quadrant, Gunter's Quadrant, Hadley's Quadrant, the Horodictical Quadrant, and the Sinical Quadrant, &c. Of these in their order.</p><p>1. <hi rend="italics">The Common, or Surveying</hi> <hi rend="smallcaps">Quadrant.</hi>—This instrument ABC, fig. 1, pl. 24, is made of brass, or wood, &c; the limb or arch of which BC is divided into 90°, and each of these farther divided into as many equal parts as the space will allow, either diagonally or otherwise. On one of the radii AC, are fitted two <cb/> moveable sights; and to the centre is sometimes also annexed a label, or moveable index AD, bearing two other sights; but instead of these last sights, there is sometimes fitted a telescope. Also from the centre hangs a thread with a plummet; and on the under side or face of the instrument is fitted a ball and socket, by means of which it may be put into any position. The general use of it is for taking angles in a vertical plane, comprehended under right lines going from the centre of the instrument, one of which is horizontal, and the other is directed to some visible point. But besides the parts above described, there is often added on the face, near the centre, a kind of compartment EF, called a Quadrat, or Geometrical Square, which is a kind of separate instrument, and is particularly useful in Altimetry and Longimetry, or Heights-and-Distances.</p><p>This Quadrant may be used in different situations; in each of them, the plane of the instrument must be set parallel to that of the eye and the objects whose angular distance is to be taken. Thus, for observing heights or depths, its plane must be disposed vertically, or perpendicular to the horizon; but to take horizontal angles or distances, its plane must be disposed parallel to the horizon.</p><p>Again, heights and distances may be taken two ways, viz, by means of the fixed sights and plummet, or by the label; as also, either by the degrees on the limb, or by the Quadrat. Thus, fig. 2 pl. 24 shews the manner of taking an angle of elevation with this Quadrant; the eye is applied at C, and the instrument turned vertically about the centre A, till the object R be seen through the sights on the radius AC; then the angle of elevation RAH, made with the horizontal line KAH, is equal to the angle BAD, made by the plumb line and the other radius of the Quadrant, and the quantity of it is shewn by the degrees in the arch BD cut off by the plumb line AD.</p><p>See the use of the instrument in my Mensuration, under the section of Heights-and-Distances.</p><p>2. <hi rend="italics">The Astronomical</hi> <hi rend="smallcaps">Quadrant</hi>, is a large one, usu- <pb n="310"/><cb/> ally made of brass or iron bars; having its limb EF (fig. 3 pl. 24) nicely divided, either diagonally or otherwise, into degrees, minutes, and seconds, if room will permit, and furnished either with two pair of plain sights or two telescopes, one on the side of the Quadrant at AB, and the other CD moveable about the centre by means of the screw G. The dented wheels I and H serve to direct the instrument to any object or phenomenon.</p><p>The application of this useful instrument, in taking observations of the sun, planets, and fixed stars, is obvious; for being turned horizontally upon its axis, by means of the telescope AB, till the object is seen through the moveable telescope, then the degrees &c cut by the index, give the altitude &c required.</p><p>3. <hi rend="italics">Cole's</hi> <hi rend="smallcaps">Quadrant</hi>, is a very useful instrument, invented by Mr. Benjamin Cole. It consists of fix parts, viz, the staff AB (fig. 11 pl. 24); the quadrantal arch DE; three vanes A, B, C; and the vernier FG. The staff is a bar of wood about 2 feet long, an inch and a quarter broad, and of a sufficient thickness to prevent it from bending or warping. The quadrantal arch is also of wood; and is divided into degrees and 3d parts of degrees, to a radius of about 9 inches; and to its extremities are fitted two radii, which meet in the centre of the Quadrant by a pin, about which it easily moves. The sight-vane A is a thin piece of brass, near two inches in height, and one broad, set perpendicularly on the end of the staff A, by means of two screws passing through its foot. In the middle of this vane is drilled a small hole, through which the coincidence or meeting of the horizon and solar spot is to be viewed. The horizon-vane B is about an inch broad, and 2 inches and a half high, having a slit cut through it of near an inch long, and a quarter of an inch broad; this vane is fixed in the centre-pin of the instrument, in a perpendicular position, by means of two screws passing through its foot, by which its position with respect to the sight-vane is always the same, their angle of inclination being equal to 45 degrees. The shade-vane C is composed of two brass plates. The one, which serves as an arm, is about 4 1/2 inches long, and 3/4 of an inch broad, being pinned at one end to the upper limb of the Quadrant by a screw, about which it has a small motion; the other end lies in the arch, and the lower edge of the arm is directed to the middle of the centre-pin: the other plate, which is properly the vane, is about 2 inches long, being fixed perpendicularly to the other plate, at about half an inch distance from that end next the arch; this vane may be used either by its shade, or by the solar spot cast by a convex lens placed in it. And because the wood-work is often subject to warp or twist, therefore this vane may be rectified by means of a screw, so that the warping of the instrument may occasion no error in the observation, which is performed in the following manner: Set the line G on the vernier against a degree on the upper limb of the Quadrant, and turn the screw on the backside of the limb forward or backward, till the hole in the sight-vane, the centre of the glass, and the sunk spot in the horizon-vane, lie in a right line.</p><p><hi rend="italics">To find the Sun's Altitude by this instrument.</hi> Turn your back to the sun, holding the staff of the instru- <cb/> ment with the right hand, so that it be in a vertical plane passing through the sun; apply one eye to the sight-vane, looking through that and the horizonvane till the horizon be seen; with the left hand slide the quadrantal arch upwards, till the solar spot or shade, cast by the shade-vane, fall directly upon the spot or slit in the horizon-vane; then will that part of the quadrantal arch, which is raised above G or S (according as the observation respects either the solar spot or shade) shew the altitude of the sun at that time. But for the meridian altitude, the observation must be continued, and as the sun approaches the meridian, the sea will appear through the horizon-vane, which completes the observation; and the degrees and minutes, counted as before, will give the sun's meridian altitude: or the degrees counted from the lower limb upwards will give the zenith distance.</p><p>4. <hi rend="italics">Adams's</hi> <hi rend="smallcaps">Quadrant</hi>, differs only from Cole's Quadrant, just described, in having an horizontal vane, with the upper part of the limb lengthened; so that the glass, which casts the solar spot on the horizonvane, is at the same distance from the horizon-vane as the sight-vane at the end of the index.</p><p>5. <hi rend="italics">Collins's</hi> or <hi rend="italics">Sutton's</hi> <hi rend="smallcaps">Quadrant</hi>, (fig. 8 pl. 24) is a stereographic projection of one quarter of the sphere between the tropics, upon the plane of the ecliptic, the eye being in its north pole; and fitted to the latitude of London. The lines running from right to left, are parallels of altitude; and those crossing them are azimuths. The smaller of the two circles, bounding the projection, is one quarter of the tropic of Capricorn; and the greater is a quarter of the tropic of Cancer. The two ecliptics are drawn from a point on the left edge of the Quadrant, with the characters of the signs upon them; and the two horizons are drawn from the same point. The limb is divided both into degrees and time; and by having the sun's altitude, the hour of the day may here be found to a minute. The quadrantal arches next the certre contain the calendar of months; and under them, in another arch, is the sun's declination. On the projection are placed several of the most remarkable fixed stars between the tropics; and the next below the projection is the Quadrant and line of shadows.</p><p><hi rend="italics">To find the Time of the Sun's Rising or Setting, his Amplitude, his Azimuth, Hour of the Day, &c. by this Quadrant.</hi> Lay the thread on the day and the month, and bring the bead to the proper ecliptic, either of summer or winter, according to the season, which is called <hi rend="italics">rectifying;</hi> then by moving the thread bring the bead to the horizon, in which case the thread will cut the limb in the point of the time of the sun's rising or setting before or after 6: and at the same time the bead will cut the horizon in the degrees of the sun's amplitude.—Again, observing the sun's altitude with the Quadrant, and supposing it found to be 45° on the 5th of May, lay the thread over the 5th of May; then bring the bead to the summer ecliptic, and carry it to the parallel of altitude 45°; in which case the thread will cut the limb at 55° 15′, and the hour will be seen among the hour-lines to be either 41m. past 9 in the morning, or 19m. past 2 in the afternoon.—Lastly, the bead shews among the azimuths the sun's distance from the south 50° 41′. <pb n="311"/><cb/></p><p>But note, that if the sun's altitude be less than what it is at 6 o'clock, the operation must be performed among those parallels above the upper horizon; the bead being rectified to the winter ecliptic.</p><p>6. <hi rend="italics">Davis's</hi> <hi rend="smallcaps">Quadrant</hi>, the same as the B<hi rend="smallcaps">ACKSTAFF;</hi> which see.</p><p>7. <hi rend="italics">Gunner's</hi> <hi rend="smallcaps">Quadrant</hi>, (fig. 6 pl. 24), sometimes called the <hi rend="italics">Gunner's Square,</hi> is used for elevating and pointing cannon, mortars, &c, and consists of two branches either of wood or brass, between which is a quadrantal arch divided into 90°, and furnished with a thread and plummet.</p><p>The use of this instrument is very easy; for if the longer branch, or bar, be placed in the mouth of the piece, and it be elevated till the plummet cut the degree necessary to hit a proposed object, the thing is done.</p><p>Sometimes on the sides of the longer bar, are noted the division of diameters and weights of iron balls, as also the bores of pieces.</p><p>8. <hi rend="italics">Gunter's</hi> <hi rend="smallcaps">Quadrant</hi>, so called from its inventor Edmund Gunter, (fig. 4 pl. 24) beside the apparatus of other Quadrants, has a stereographic projection of the sphere on the plane of the equinoctial; and also a calendar of the months, next to the divisions of the limb; by which, beside the common purposes of other Quadrants, several useful questions in astronomy, &c, are easily resolved.</p><p><hi rend="italics">Use of Gunter's Quadrant.</hi> — 1. To find the sun's meridian altitude for any given day, or conversely the day of the year answering to any given meridian altitude. Lay the thread to the day of the month in the scale next the limb; then the degree it cuts in the limb is the sun's meridian altitude. And, contrariwise, the thread being set to the meridian altitude, it shews the day of the month.</p><p>2. To find the hour of the day. Having put the bead, which slides on the thread, to the sun's place in the ecliptic, observe the sun's altitude by the Quadrant; then if the bead be laid over the same in the limb, the bead will fall upon the hour required. On the contrary, laying the bead on a given hour, having first rectified or set it to the sun's place, the degree cut by the thread on the limb gives the altitude.</p><p>Note, the bead may be rectified otherwise, by bringing the thread to the day of the month, and the bead to the hour-line of 12.</p><p>3. To find the sun's declination from his place given; and the contrary. Bring the bead to the sun's place in the ecliptic, and move the thread to the line of declination ET, so shall the bead cut the degree of declination required. On the contrary, the bead being adjusted to a given declination, and the thread moved to the ecliptic, the bead will cut the sun's place.</p><p>4. The sun's place being given, to find the right ascension; or contrariwise. Lay the thread on the sun's place in the ecliptic, and the degree it cuts on the limb is the right ascension sought. And the converse.</p><p>5. The sun's altitude being given, to find his azimuth; and contrariwise. Rectify the bead for the time, as in the second article, and observe the sun's altitude; bring the thread to the complement of that <cb/> altitude; then the bead will give the azimuth sought, among the azimuth-lines.</p><p>9. <hi rend="italics">Hadley's</hi> <hi rend="smallcaps">Quadrant</hi>, (fig. 7 pl. 24) so called from its inventor John Hadley, Esq. is now universally used as the best of any for nautical and other observations.</p><p>It seems the first idea of this excellent instrument was suggested by Dr. Hooke; for Dr. Sprat, in his History of the Royal Society, pa. 246, mentions the invention of a new instrument for taking angles by reflection, by which means the eye at once sees the two objects both as touching the same point, though distant almost to a semicircle; which is of great use for making exact observations at sea. This instrument is described and illustrated by a figure in Hooke's Posthumous works, pa. 503. But as it admitted of only one reflection, it would not answer the purpose. The matter however was at last effected by Sir Isaac Newton, who communicated to Dr. Halley a paper of his own writing, containing the description of an instrument with two reflections, which soon after the doctor's death was sound among his papers by Mr. Jones, by whom it was communicated to the Royal Society, and it was published in the Philos Trans. for the year 1742. See also the Abridg. vol. 8, pa. 129. How it happened that Dr. Halley never mentioned this in his lifetime, is hard to say; but it is very extraordinary; more especially as Mr. Hadley had described, in the Transac. for 1731, his instrument, which is constructed on the same principles. See also Abr. vol. 6, pa. 139. Mr. Hadley, who was well acquainted with Sir Isaac Newton, might have heard him say, that Dr. Hooke's proposal could be effected by means of a double reflection; and perhaps in consequence of this hint, he might apply himself, without any previous knowledge of what Newton had actually done, to the construction of his instrument. Mr. Godfrey too, of Pennsylvania, had recourse to a similar expedient; for which reason some gentlemen of that colony have ascribed the invention of this excellent instrument to him. The truth may probably be, that each of these gentlemen discovered the method independent of one another. See Abr. Philos. Trans. vol. 8, pa. 366; also Trans. of the American Society, vol. 1, pa. 21 Appendix.</p><p>This instrument consists of the following particulars: 1. An octant, or the 8th part of a circle, ABC. 2. An index D. 3. The speculum E. 4. Two horizontal glasses, F, G. 5. Two screens, Kand K. 6. Two sight-vanes, H and I.</p><p>The octant consists of two radii, AB, AC, strengthened by the braces L, M, and the arch BC; which, though containing only 45°, is nevertheless divided into 90 primary divisions, each of which stands for degrees, and are numbered 0, 10, 20, 30, &c, to 90; beginning at each end of the arch for the convenience of numbering both ways, either for altitudes or zenith distances: also each degree is subdivided into minutes, by means of a vernier. But the number of these divisions varies with the size of the instrument.</p><p>The index D, is a flat bar, moveable about the centre of the instrument; and that part of it which slides over the graduated arch, BC, is open in the middle, with Vernier's scale on the lower part of it; <pb n="312"/><cb/> and underneath is a screw, serving to fasten the index against any division.</p><p>The speculum E is a piece of flat glass, quick silvered on one side, set in a brass box, and placed perpendicular to the plane of the instrument, the middle part of the former coinciding with the centre of the latter: and because the speculum is fixed to the index, the position of it will be altered by the moving of the index along the arch. The rays of an observed object are received on the speculum, and from thence reflected on one of the horizon glasses, F or G; which are two small pieces of looking glass placed on one of the limbs, their faces being turned obliquely to the speculum, from which they receive the reflected rays of objects. This glass F has only its lower part silvered, and set in brass-work; the upper part being left transparent to view the horizon. The glass G has in its middle a transparent slit, through which the horizon is to be seen. And because the warping of the materials, and other accidents, may distend them from their true situation, there are three screws passing through their feet, by which they may be easily replaced.</p><p>The screens are two pieces of coloured glass, set in two square brass frames K and K, which serve as screens to take off the glare of the sun's rays, which would otherwise be too strong for the eye; the one is tinged much deeper than the other; and as they both move on the same centre, they may be both or either of them used: in the situation they appear in the figure, they serve for the horizon-glass F; but when they are wanted for the horizon-glass G, they must be taken from their present situation, and placed on the Quadrant above G.</p><p>The sight-vanes are two pins, H and I, standing perpendicularly to the plane of the instrument: that at H has one hole in it, opposite to the transparent slit in the horizon-glass G; the other, at I, has two holes in it, the one opposite to the middle of the transparent part of the horizon-glass F, and the other rather lower than the quick-silvered part: this vane has a piece of brass on the back of it, which moves round a centre, and serves to cover either of the holes.</p><p><hi rend="italics">Of the Observations.</hi>—There are two sorts of observations to be made with this instrument: the one is when the back of the observer is turned towards the object, and therefore called the <hi rend="italics">back observation;</hi> the other when the face of the observer is turned towards the object, which is called the <hi rend="italics">fore-observation.</hi></p><p><hi rend="italics">To Rectify the Instrument for the Fore-observation:</hi> Slacken the screw in the middle of the handle behind the glass F; bring the index close to the button <hi rend="italics">h;</hi> hold the instrument in a vertical position, with the arch downwards; look through the right-hand hole in the vane I, and through the transparent part of the glass F, for the horizon; and if it lie in the same right line with the image of the horizon seen on the silvered part, the glass F is rightly adjusted; but if the two horizontal lines disagree, turn the screw which is at the end of the handle backward or forward, till those lines coincide; then fasten the middle screw of the handle, and the glass is rightly adjusted.</p><p><hi rend="italics">To take the Sun's Altitude by the Fore-observation.</hi> Having fixed the screens above the horizon-glass F, and suited them proportionally to the strength of the <cb/> sun's rays, turn your face towards the sun, holding the instrument with your right hand, by the braces L and M, in a vertical position, with the arch downward; put your eye close to the right-hand hole in the vane I, and view the horizon through the transparent part of the horizon-glass F, at the same time moving the index D with the left hand, till the reflex solar spot coincides with the line of the horizon; then the degrees counted from C, or that end next your body, will give the sun's altitude at that time, observing to add or subtract 16 minutes according as the upper or lower edge of the sun's reflex image is made use of.</p><p>But to get the sun's meridian altitude, which is the thing wanted for finding the latitude; the observations must be continued; and as the sun approaches the meridian the index D must be continually moved towards B, to maintain the coincidence between the reflex solar spot and the horizon; and consequently as long as this motion can maintain the same coincidence, the observation must be continued, till the sun has reached the meridian, and begins to descend, when the coincidence will require a retrograde motion of the index, or towards C; and then the observation is finished, and the degrees counted as before will give the sun's meridian altitude, or those from B will give the zenith distance; observing to add the semi-diameter, or 16′, when his lower edge is brought to the horizon; or to subtract 16′, when the horizon and upper edge coincide.</p><p><hi rend="italics">To take the Altitude of a Star by the Fore-observation.</hi> Through the vane H, and the transparent slit in the glass G, look directly to the star; and at the same time move the index, till the image of the horizon behind you, being reflected by the great speculum, be seen in the silvered part of G, and meet the star; then will the index shew the degrees of the star's altitude.</p><p><hi rend="italics">To Rectify the Instrument for the Back-observation.</hi> Slacken the screw in the middle of the handle, behind the glass G; turn the button <hi rend="italics">h</hi> on one side, and bring the index as many degrees before 0 as is equal to double the dip of the horizon at your height above the water; hold the instrument vertical, with the arch downward; look through the hole of the vane H; and if the horizon, seen through the transparent slit in the glass G, coincide with the image of the horizon seen in the silvered part of the same glass, then the glass G is in its proper position; but if not, set it by the handle, and fasten the screw as before.</p><p><hi rend="italics">To take the Sun's Altitude by the Back-observation.</hi> Put the stem of the screens K and K into the hole <hi rend="italics">r,</hi> and in proportion to the strength or faintness of the sun's rays, let either one or both or neither of the frames of those glasses be turned close to the face of the limb; hold the instrument in a vertical position, with the arch downward, by the braces L and M, with the left hand; turn your back to the sun, and put one eye close to the hole in the vane H, observing the horizon through the transparent slit in the horizon glass G; with the right hand move the index D, till the reflected image of the sun be seen in the silvered part of the glass G, and in a right line with the horizon; swing your body to and fro, and if the observation be well made, the sun's image will be observed to brush the horizon, and the degrees reckoned from C, or that part of the arch farthest from your <pb n="313"/><cb/> body, will give the sun's altitude at the time of observation; observing to add 16′ or the sun's semidiameter if the sun's upper edge be used, and subtract the same for the lower edge.</p><p>The directions just given, for taking altitudes at sea, would be sufficient, but for two corrections that are necessary to be made before the altitude can be accurately determined, viz, one on account of the observer's eye being raised above the level of the sea, and the other on account of the refraction of the atmosphere, especially in small altitudes.</p><p>The following tables therefore shew the corrections to be made on both these accounts. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">TABLE I.</cell><cell cols="1" rows="1" rend="colspan=6" role="data">TABLE II.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Dip of the Hori-</cell><cell cols="1" rows="1" rend="colspan=6" role="data">Refractions of the Stars &c in</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">zon of the Sea.</cell><cell cols="1" rows="1" rend="colspan=6" role="data">Altitude.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Height</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dip of</cell><cell cols="1" rows="1" role="data">Appar.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Refrac-</cell><cell cols="1" rows="1" role="data">Appar.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Refrac-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of the</cell><cell cols="1" rows="1" rend="colspan=2" role="data">the Ho-</cell><cell cols="1" rows="1" role="data">Altitude</cell><cell cols="1" rows="1" role="data">Altitude</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Eye.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">rizon.</cell><cell cols="1" rows="1" role="data">in Deg.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">tion.</cell><cell cols="1" rows="1" role="data">in Deg.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">tion.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">Feet.</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">47</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">1/4</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">23</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">1/2</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">57</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">48</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">33</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">10</cell></row></table> <hi rend="center"><hi rend="italics">General Rules for these Corrections.</hi></hi></p><p>1. In the fore-observations, add the sum of both corrections to the observed zenith distance, for the true zenith distance: or subtract the said sum from the observed altitude, for the true one. 2. In the backobservation, add the dip and subtract the refraction, for altitudes; and for zenith distances, do the contrary, viz, subtract the dip, and add the refraction.</p><p><hi rend="italics">Example.</hi> By a back observation, the altitude of the sun's lower edge was found by Hadley's Quadrant to be 25° 12′; the eye being 30 feet above the horizon. By the tables, the dip on 30 feet is 5′ 14″, and the refraction on 25° 12′ is 2′ 1″. Hence <table><row role="data"><cell cols="1" rows="1" role="data">Appar. alt. lower limb</cell><cell cols="1" rows="1" role="data">25° 12′  0″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sun's semidiameter, sub.</cell><cell cols="1" rows="1" role="data"> 0  16   0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Appar. alt. of centre</cell><cell cols="1" rows="1" role="data">24  56   0</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dip. of horizon, add</cell><cell cols="1" rows="1" role="data"> 0   5  14</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">25   1  14</cell></row><row role="data"><cell cols="1" rows="1" role="data">Refraction, subtract</cell><cell cols="1" rows="1" role="data"> 0   2   1</cell></row><row role="data"><cell cols="1" rows="1" role="data">True alt. of centre</cell><cell cols="1" rows="1" role="data">24  59  13</cell></row></table></p><p>In the case of the moon, besides the two corrections above, another is to be made for her parallaxes. But <cb/> for all these particulars, see the Requisite Tables for the Nautical Almanac, also Robertson's Navigation, vol. 2, pa. 340 &c, edit. 1780.</p><p>10. <hi rend="italics">Horodictical</hi> <hi rend="smallcaps">Quadrant</hi>, a pretty commodious instrument, and is so called from its use in telling the hour of the day. Its construction is as follows. From the centre of the Quadrant C, (fig. 5 pl. 24), whose limb AB is divided into 90°, describe seven concentric circles at any intervals; and to these add the signs of the zodiac, in the order represented in the figure. Then, applying a ruler to the centre C and the limb AB, mark upon the several parallels the degrees corresponding to the altitude of the sun, when in them, for the given hours; connect the points belonging to the same hour with a curve line, to which add the number of the hour. To the radius CA fit a couple of fights, and to the centre of the Quadrant C tie a thread with a plummet, and upon the thread a bead to slide.</p><p>Now if the bead be brought to the parallel in which the sun is, and the Quadrant be directed to the sun, till a visual ray pass through the sights, the bead will shew the hour. For the plummet, in this situation, cuts all the parallels in the degrees corresponding to the sun's altitude. And since the bead is in the parallel which the sun describes, and because hourlines pass through the degrees of altitude to which the sun is elevated every hour, therefore the bead must shew the present hour.</p><p>11. <hi rend="italics">Sinical</hi> <hi rend="smallcaps">Quadrant</hi>, is one of some use in Navigation. It consists of several concentric quadrantal arches, divided into 8 equal parts by means of radii, with parallel right lines crossing each other at right angles. Now any one of the arches, as BC, (fig. 10 pl. 24) may represent a Quadrant of any great circle of the sphere, but is chiefly used for the horizon or meridian. If then BC be taken for a Quadrant of the horizon, either of the sides, as AB, may represent the meridian; and the other side, AC, will represent a parallel, or line of east-and-west; all the other lines, parallel to AB, will be also meridians; and all those parallel to AC, east-and-west lines, or parallels. Again, the eight spaces into which the arches are divided by the radii, represent the eight points of the compass in a quarter of the horizon; each containing 11° 15′. The arch BC is likewise divided into 90°, and each degree subdivided into 12′, diagonalwise. To the centre is fixed a thread, which, being laid over any degree of the Quadrant, serves to divide the horizon.</p><p>If the sinical Quadrant be taken for a fourth part of the meridian, one side of it, AB, may be taken for the common radius of the meridian and equator; and then the other, AC, will be half the axis of the world. The degrees of the circumference, BC, will represent degrees of latitude; and the parallels to the side AB, assumed from every point of latitude to the axis AC, will be radii of the parallels of latitude, as likewise the cosine of those latitudes.</p><p>Hence, suppose it be required to find the degrees of longitude contained in 83 of the lesser leagues in the parallel of 48°: lay the thread over 48° of latitude on the circumference, and count thence the 83 leagues on AB, beginning at A; this will terminate in H, allow- <pb n="314"/><cb/> ing every small interval four leagues. Then tracing out the parallel HE, from the point H to the thread; the part AE of the thread shews that 125 greater or equinoctial leagues make 6° 15′; and therefore that the 83 lesser leagues AH, which make the difference of longitude of the course, and are equal to the radius of the parallel HE, make 6° 15′ of the said parallel.</p><p>When the ship sails upon an oblique course, such course, beside the north and south greater leagues, gives lesser leagues easterly and westerly, to be reduced to degrees of longitude of the equator. But these leagues being made neither on the parallel of departure, nor on that of arrival, but in all the intermediate ones, there must be found a mean proportional parallel between them. To find this, there is on the instrument a scale of cross latitudes. Suppose then it were required to find a mean parallel between the parallels of 40° and 60°; take with the compasses the middle between the 40th and 60th degree on the scale: this middle point will terminate against the 51st degree, which is the mean parallel sought.</p><p>The chief use of the sinical Quadrant, is to form upon it triangles similar to those made by a ship's way with the meridians and parallels; the sides of which triangles are measured by the equal intervals between the concentric Quadrants and the lines N and S, E and W: and every 5th line and arch is made deeper than the rest. Now suppose a ship has sailed 150 leagues northeast-by-north, or making an angle of 33° 45′ with the north part of the meridian: here are given the course and distance sailed, by which a triangle may be formed on the instrument similar to that made by the ship's course; and hence the unknown parts of the triangle may be found. Thus; supposing the centre A to represent the place of departure; count, by means of the concentric circles along the point the ship sailed on, viz, AD, 150 leagues: then in the triangle AED, similar to that of the ship's course, find AE = difference of latitude, and DE = difference of longitude, which must be reduced according to the parallel of latitude come to.</p><p><hi rend="italics">Sutton's</hi> <hi rend="smallcaps">Quadrant.</hi> See <hi rend="italics">Collins's</hi> <hi rend="smallcaps">Quadrant.</hi></p><p>12. <hi rend="smallcaps">Quadrant</hi> <hi rend="italics">of Altitude,</hi> (fig. 9 pl. 24) is an appendix to the artificial globe, consisting of a thin slip of brass, the length of a quarter part of one of the great circles of the globe, and graduated. At the end, where the division terminates, is a nut riveted on, and furnished with a screw, by means of which the instrument is fitted on the meridian, and moveable round upon the rivet to all points of the horizon, as represented in the figure referred to.—Its use is to serve as a scale in measuring of altitudes, amplitudes, azimuths, &c.</p><p>QUADRANTAL <hi rend="italics">Triangle,</hi> is a spherical triangle, which has one side equal to a quadrant or quarter part of a circle.</p></div1><div1 part="N" n="QUADRAT" org="uniform" sample="complete" type="entry"><head>QUADRAT</head><p>, called also <hi rend="italics">Geometrical Square,</hi> and <hi rend="italics">Line of Shadows:</hi> it is often an additional member on the face of Gunter's and Sutton's quadrants; and is chiefly useful in taking heights or depths. See my Mensuration, the chap. on Altimetry and Longimetry, or Heights and Distances.</p><div2 part="N" n="Quadrat" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quadrat</hi></head><p>, in Astrology, is the same as quartile, being an aspect of the heavenly bodies when they are <cb/> distant from each other a quadrant, or 90°, or 3 signs, and is thus marked □.</p><p>QUADRATIC <hi rend="italics">Equations,</hi> in Algebra, are those in which the unknown quantity is of two dimensions, or raised to the 2d power. See <hi rend="smallcaps">Equation.</hi></p><p>Quadratic equations are either simple, or affected, that is compound.</p><p><hi rend="italics">A Simple</hi> <hi rend="smallcaps">Quadratic</hi> <hi rend="italics">equation,</hi> is that which contains the 2d power only of the unknown quantity, without any other power of it: as <hi rend="italics">x</hi><hi rend="sup">2</hi> = 25, or <hi rend="italics">y</hi><hi rend="sup">2</hi> = <hi rend="italics">ab.</hi> And in this case, the value of the unknown quantity is found by barely extracting the square root on both sides of the equation: so in the equations above, it will be <hi rend="italics">x</hi> = ± 5, and <hi rend="italics">y</hi> = ± √<hi rend="italics">ab;</hi> where the sign of the root of the known quantity is to be taken either plus or minus, for either of these may be considered as the sign of the value of the root <hi rend="italics">x,</hi> since either of these, when squared, make the same square, , and also; and hence the root of every quadratic or square, has two values.</p><p><hi rend="italics">Compound</hi> or <hi rend="italics">Affected</hi> <hi rend="smallcaps">Quadratics</hi>, are those which contain both the 1st and 2d powers of the unknown quantity; as , or , where <hi rend="italics">n</hi> may be of any value, and then <hi rend="italics">x</hi><hi rend="sup">n</hi> is to be considered as the root or unknown quantity.</p><p>Affected quadratics are usually distinguished into three forms, according to the signs of the terms of the equation: Thus, . But the method of extracting the root, or finding the value of the unknown quantity <hi rend="italics">x,</hi> is the same in all of them. And that method is usually performed by what is called completing the square, which is done by taking half the coefficient of the 2d term or single power of the unknown quantity, then squaring it, and adding that square to both sides of the equation, which makes the unknown side a complete square. Thus, in the equation , the coefficient of the 2d term being <hi rend="italics">a,</hi> its half is (1/2)<hi rend="italics">a,</hi> the square of which is (1/4)<hi rend="italics">a</hi><hi rend="sup">2</hi>, and this added to both sides of the equation, it becomes , the former side of which is now a complete square. The square being thus completed, its root is next to be extracted; in order to which, it is to be observed that the root on the unknown side consists of two terms, the one of which is always <hi rend="italics">x</hi> the square root of the first term of the equation, and the other part is (1/2)<hi rend="italics">a</hi> or half the coefficient of the 2d term: thus then the root of <hi rend="italics">x</hi><hi rend="sup">2</hi> + <hi rend="italics">ax</hi> + (1/4)<hi rend="italics">a</hi><hi rend="sup">2</hi> the first side of the completed equation being <hi rend="italics">x</hi> + (1/2)<hi rend="italics">a,</hi> and the root of the other side (1/4)<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi> being ± √(<hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">b</hi>), it follows that , and hence, by transposing (1/2)<hi rend="italics">a,</hi> it is , the two values of <hi rend="italics">x,</hi> or roots of the given equation . And thus is found the root, or value of <hi rend="italics">x,</hi> in the three forms of equations above mentioned: thus, . <pb n="315"/><cb/> Where it is observable that, because of the double sign ±, every form has two roots: in the 1st and 2d forms those roots are the one positive and the other negative, the positive root being the less of the two in the 1st form, but the greater in the 2d form; and in the 3d form the roots are both positive. Again, the two roots of the 1st and 2d forms, are always both of them real; but in the 3d form, the two roots are either both real or both imaginary, viz, both real when (1/4)<hi rend="italics">a</hi><hi rend="sup">2</hi> is greater than <hi rend="italics">b,</hi> or both imaginary when (1/4)<hi rend="italics">a</hi><hi rend="sup">2</hi> is less than <hi rend="italics">b,</hi> because in this case (1/4)<hi rend="italics">a</hi><hi rend="sup">2</hi> - <hi rend="italics">b</hi> will be a negative quantity, the root of which is impossible, or an imaginary quantity.</p><p><hi rend="italics">Example</hi> of the 1st form, let . Here then <hi rend="italics">a</hi> = 6, and <hi rend="italics">b</hi> = 7; then .</p><p><hi rend="italics">Example</hi> of the 2d form, let . Here also <hi rend="italics">a</hi> = 6, and <hi rend="italics">b</hi> = 7; then ; the same two roots as before, with the signs changed.</p><p><hi rend="italics">Example</hi> of the 3d form, let . Here again <hi rend="italics">a</hi> = 6, and <hi rend="italics">b</hi> = 7; then , the two roots both real.</p><p>But if ; then <hi rend="italics">a</hi> = 6, and <hi rend="italics">b</hi> = 11, which gives the two roots both imaginary.</p><p>All equations whatever that have only two different powers of the unknown quantity, of which the index of the one is just double to that of the other, are resolved like Quadratics, by completing the square. Thus, the equation , by completing the square becomes <hi rend="italics">;</hi> whence, extracting the root on both sides, , where the root <hi rend="italics">x</hi> has four values, because the given equation rises to the 4th power. See <hi rend="smallcaps">Equation.</hi></p></div2></div1><div1 part="N" n="QUADRATRIX" org="uniform" sample="complete" type="entry"><head>QUADRATRIX</head><p>, or <hi rend="smallcaps">Quadratix</hi>, in Geometry, is a mechanical line, by means of which, right lines are found equal to the circumference of circles, or other curves, and of the parts of the same. Or, more accurately, the <hi rend="italics">Quadratrix of a curve,</hi> is a transcendental curve described on the same axis, the ordinates of which being given, the quadrature of the correspondent parts in the other curve is likewise given. See <hi rend="smallcaps">Curve.</hi>—Thus, for example, the curve AND may be <figure/> called the Quadratrix of the parabola AMC, when the area APMA bears some such relation as the following to the absciss AP or ordinate PN, viz, <cb/> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">when</cell><cell cols="1" rows="1" role="data">APM = PN<hi rend="sup">2</hi>,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">APM = AP</cell><cell cols="1" rows="1" role="data">X PN,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">or</cell><cell cols="1" rows="1" role="data">APM = <hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">X PN,</cell></row></table> where <hi rend="italics">a</hi> is some given constant quantity.</p><p>The most distinguished of these Quadratices are, those of Dinostrates and of Tschirnhausen for the circle, and that of Mr. Perks for the hyperbola.</p><p><hi rend="smallcaps">Quadratrix</hi> <hi rend="italics">of Dinostrates,</hi> is a curve AMD, by which the quadrature of the circle is effected, though not geometrically, but only mechanically. It is so called from its inventor Dinostrates; <figure/> and the genesis or description of it is as follows: Divide the quadrantal arc ANB into any number of equal parts, in the points N, <hi rend="italics">n, n,</hi> &c; and also the radius AC into the same number of parts at the points P, <hi rend="italics">p, p,</hi> &c. To the points of N, <hi rend="italics">n, n,</hi> &c, draw the radii CN, C<hi rend="italics">n,</hi> &c; and from the points P, <hi rend="italics">p,</hi> &c, the parallels to CB, as PM, P<hi rend="italics">m,</hi> &c: then through all the points of intersection draw the curve AM<hi rend="italics">m</hi>D, and it will be the Quadratrix of Dinostrates.</p><p>Or the same curve may be conceived as described by a continued motion, thus: Conceive a radius CN to revolve with a uniform motion about the centre C, from the position AC to the position BC; and at the same time a ruler PM always moving uniformly parallel towards CB; the two uniform motions being so regulated that the radius and the ruler shall arrive at the position BC at the same time. For thus the continual intersection M, <hi rend="italics">m,</hi> &c. of the revolving radius, and moving ruler, will describe the Quadratrix AM<hi rend="italics">m</hi> &c. Hence,</p><p>1. <hi rend="italics">For the Equation of the Quadratrix:</hi> Since, from the relation of the uniform motions, it is always, AB : AN :: AC : AP; therefore if AB = <hi rend="italics">a,</hi> AC = <hi rend="italics">r,</hi> AP = <hi rend="italics">x,</hi> and AN = <hi rend="italics">z,</hi> it will be <hi rend="italics">a : z :: r : x,</hi> or <hi rend="italics">ax</hi> = <hi rend="italics">rz,</hi> which is the equation of the curve.</p><p>Or, if <hi rend="italics">s</hi> denote the sine NE of the arc AN, and <hi rend="italics">y</hi> = PM the ordinate of the curve AM, its absciss AP being <hi rend="italics">x;</hi> then, by similar triangles, CE : CP :: EN : PM, that is, , and hence , the equation of the curve. And when the relation between AB and AN is given, in terms of that between AC and AP, hence will be expressed the relation between the sine EN and the radius CB, or <hi rend="italics">s</hi> will be expressed in terms of <hi rend="italics">r</hi> and <hi rend="italics">x;</hi> and consequently the equation of the curve will be expressed in terms of <hi rend="italics">r, x,</hi> and <hi rend="italics">y</hi> only.</p><p>2. The base of the Quadratrix CD is a third proportional to the quadrant AB and the radius AC or CB; i. e. CD : CB :: CB : AB. Hence the rectification and quadrature of the circle.</p><p>3. A quadrantal arc DF de- <figure/> scribed with the centre C and radius CD, will be equal in length to the radius CA or CB.</p><p>4. CDF being a quadrant inscribed in the Quadratrix AMD, if the base CD be = 1, and the circular arc DG = <hi rend="italics">x;</hi> then in the area . See <hi rend="smallcaps">Quadrature.</hi> <pb n="316"/><cb/></p><p><hi rend="smallcaps">Quadratrix</hi> <hi rend="italics">of Tscbirnhau- <figure/> sen,</hi> is a transcendental curve AM<hi rend="italics">m</hi>B by which the quadrature of the circle is likewise effected. This was invented by M. Tschirnhausen, and its genesis, in imitation of that of Dinostrates, is as follows: Divide the quadrant ANC, and the radius AC, each into equal parts, as before; and from the points P, <hi rend="italics">p,</hi> &c, draw the lines PM, <hi rend="italics">pm,</hi> &c, parallel to CB; also from the points N, <hi rend="italics">n,</hi> &c, the lines NM, <hi rend="italics">nm,</hi> &c, parallel to the other radius AC; so shall all the intersections M, <hi rend="italics">m,</hi> &c, be in the curve of the Quadratrix AM<hi rend="italics">m</hi>B.</p><p><hi rend="italics">Now for the Equation of this Quadratrix;</hi> it is, as before, .</p><p>Or, because here <hi rend="italics">y</hi> = PM = EN = <hi rend="italics">s;</hi> therefore <hi rend="italics">s,</hi> as before, expressed in terms of <hi rend="italics">r</hi> and <hi rend="italics">x,</hi> gives the equation of this Quadratrix in terms of <hi rend="italics">r, x,</hi> and <hi rend="italics">y,</hi> and that in a simpler form than the other. Thus, from the nature of the circle and the construction of the Quadratrix, it is , where A, B, C, &c, are the preceding terms; which is the equation of the curve or Quadratrix of Tschirnhausen.</p><p>By either Quadratix, it is evident that an arc or angle is easily divided into three, or any other number of equal parts; viz, by dividing the corresponding radius, or part of it, into the same number of equal parts: for AN is always the same part of AB, that AP is of AC.</p></div1><div1 part="N" n="QUADRATURE" org="uniform" sample="complete" type="entry"><head>QUADRATURE</head><p>, in Astronomy, that aspect or position of the moon when she is 90° distant from the sun. Or, the Quadratures or quarters are the two middle points of the moon's orbit between the points of conjunction and opposition, viz, the points of the 1st and 3d quarters; at which times the moon's face shews half full, being dichotomized or bisected.</p><p>The moon's orbit is more convex in the Quadratures than in the syzygies, and the greater axis of her orbit passes through the Quadratures, at which points also the is most distant from the earth.—In the Quadratures, and within 35° of them, the apses of the moon go backwards, or move in antecedentia; but in the syzygies the contrary.—When the nodes are in the Quadratures, the inclination of the moon's orbit is greatest, but least when they are in the syzygies.</p><p><hi rend="smallcaps">Quadrature</hi> <hi rend="italics">Lines,</hi> or <hi rend="italics">Lines of</hi> <hi rend="smallcaps">Quadrature</hi>, are two lines often placed on Gunter's sector. They are marked with the letter Q, and the figures 5, 6, 7, 8, 9, 10; of which Q denotes the side of a square, and the figures denote the sides of polygons of 5, 6, 7, &c sides. Also S denotes the semidiameter of a circle, and 90 a line equal to the quadrant or 90° in circumference.</p><div2 part="N" n="Quadrature" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quadrature</hi></head><p>, in Geometry, is the squaring of a figure, or reducing it to an equal square, or finding a square equal to the area of it.</p><p>The Quadrature of rectilineal figures falls under <cb/> common geometry, or mensuration; as amounting to no more than the finding their areas, or superficies; which are in effect their squares: which was fully effected by Euclid.</p><p>The <hi rend="smallcaps">Quadrature</hi> <hi rend="italics">of Curves,</hi> that is, the measuring of their areas, or the finding a rectilineal space equal to a proposed curvilineal one, is a matter of much deeper speculation; and makes a part of the sublime or higher geometry. The lunes of Hypocrates are the first curves that were squared, as far as we know of. The circle was attempted by Euclid and others before him: he shewed indeed the proportion of one circle to another, and gave a good method of approximating to the area of the circle, by describing a polygon between any two concentric circles, however near their circumferences might be to each other. At this time the conic sections were admitted in geometry, and Archimedes, perfectly, for the first time, squared the parabola, and he determined the relations of spheres, spheroids, and conoids, to cylinders and cones; and by pursuing the method of exhaustions, or by means of inscribed and circumscribed polygons, he approximated to the periphery and area of the circle; shewing that the diameter is to the circumference nearly as 7 to 22, and the area of the circle to the square of the diameter as 11 to 14 nearly. Archimedes likewise determined the relation between the circle and ellipse, as well as that of their similar parts: It is probable too that he attempted the hyperbola; but it is not likely that he met with any success, since approximations to its area are all that can be given by the various methods that have since been invented. Beside these figures, he left a treatise on a spiral curve; in which he determined the relation of its area to that of the circumscribed circle; as also the relation of their sectors.</p><p>Several other eminent men among the Ancients wrote upon this subject, both before and after Euclid and Archimedes; but their attempts were usually confined to particular parts of it, and made according to methods not essentially different from theirs. Among these are to be reckoned Thales, Anaxagoras, Pythagoras, Bryson, Antiphon, Hypocrates of Chios, Plato, Apollonius, Philo, and Ptolomy; most of whom wrote upon the Quadrature of the circle; and those after Archimedes, by his method, usually extended the approximation to a higher degree of accuracy.</p><p>Many of the Moderns have also prosecuted the same problem of the Quadrature of the circle, after the same methods, to still greater lengths; such are Vieta, and Metius; whose ratio between the diameter and the circumference, is that of 113 to 355, which is within about (3/10000000) of the true ratio; but above all, Ludolph van Collen, or a Ceulen, who, with an amazing degree of industry and patience, by the same methods, extended the ratio to 36 places of figures, making the ratio to be that of 1 to 3.14159,26535,89793,23846,26433,83279,50288 + or 9 -. And the same was repeated and confirmed by his editor Snellius. See <hi rend="smallcaps">Diameter</hi>, and <hi rend="smallcaps">Circle;</hi> also the Preface to my Mensuration.</p><p>Though the Quadrature, especially of the circle, be a thing which many of the principal mathematicians, among the Ancients, were very solicitous about; yet nothing of this kind has been done so considerable, as <pb n="317"/><cb/> about and since the middle of the last century; when, for example, in the year 1657, Sir Paul Neil, Lord Brouncker, and Sir Christopher Wren geometrically demonstrated the equality of some curvilineal spaces to rectilineal ones. Soon after this, other persons did the like in other curves; and not long afterwards the thing was brought under an analytical calculus, the first specimen of which ever published, was given by Mercator in 1688, in a demonstration of Lord Brouncker's Quadrature of the hyperbola, by Dr. Wallis's method of reducing an algebraical fraction into an infinite series by division.</p><p>Though, by the way, it appears that Sir Isaac Newton had discovered a method of attaining the area of all quadrable curves analytically, by his Method of Fluxions, before the year 1668. See his <hi rend="italics">Fluxions,</hi> also his <hi rend="italics">Analysis per Æquationes Numero Terminorum Infinitas,</hi> and his <hi rend="italics">Introductio ad Quadraturam Curvarum;</hi> where the Quadratures of Curves are given by general methods.</p><p>It is contested, between Mr. Huygens and Sir Christopher Wren, which of the two first found out the Quadrature of any determinate cycloidal space. Mr. Leibnitz afterwards discovered that of another space; and Mr. Bernoulli, in 1699, found out the Quadrature of an infinity of cycloidal spaces, both segments and sectors &c.</p><p>As to the Quadrature of the Circle in particular, or the finding a square equal to a given circle, it is a problem that has employed the mathematicians of all ages, but still without the desired success. This depends on the ratio of the diameter to the circumference, which has never yet been determined in precise numbers. Many persons have approached very near this ratio; for which see <hi rend="smallcaps">Circle.</hi></p><p>Strict geometry here failing, mathematicians have had recourse to other means, and particularly to a sort of curves called quadratices: but these being mechanical curves, instead of geometrical ones, or rather transcendental instead of algebraical ones, the problem cannot fairly be effected by them.</p><p>Hence recourse has been had to analytics. And the problem has been attempted by three kinds of algebraical calculations. The first of these gives a kind of transcendental Quadratures, by equations of indefinite degrees. The second by vulgar numbers, though irrationally such; or by the roots of common equations, which for the general Quadrature is impossible. The third by means of certain series, exhibiting the quantity of a circle by a progression of terms. See <hi rend="smallcaps">Series.</hi></p><p>Thus, for example, the diameter of a circle being 1, it has been found that the quadrant, or one-fourth of the circumference, is equal to (1/1) - (1/3) + (1/5) - (1/7) + (1/9) &c, making an infinite series of fractions, whose common numerator is 1, and denominators the natural series of odd numbers; and all these terms alternately will be too great, and too little. This series was discovered by Leibnitz and Gregory. And the same series is also the area of the circle.</p><p>If the sum of this series could be found, it would give the Quadrature of the circle: but this is not yet done; nor is it at all probable that it ever will be done; <cb/> though the impossibility has never yet been demonstrated.</p><p>To this it may be added, that as the same magnitude may be expressed by several different series, possibly the circumference of the circle may be expressed by some other series, whose sum may be found. And there are many other series, by which the quadrant, or area, to the diameter, has been expressed; though it has never been found that any one of them is actually summable. Such as this series, 1 - (1/6) - (1/40) - (1/112) &c, invented by Newton; with innumerable others.</p><p>But though a definite Quadrature of the whole circle was never yet given, nor of any aliquot part of it; yet certain other portions of it have been squared. The first partial Quadrature was given by Hippocrates of Chios; who squared a portion called, from its figure, the <hi rend="italics">lune,</hi> or <hi rend="italics">lunule;</hi> but this Quadrature has no dependence on that of the circle. And some modern geometricians have found out the Quadrature of any portion of the lune taken at pleasure, independently of the Quadrature of the circle; though still subject to a certain restriction, which prevents the Quadrature from being perfect, and what the geometricians call absolute and indefinite. See <hi rend="smallcaps">Lune.</hi> And for the Quadrature of the different kinds of curves, see their several particular names.</p><p><hi rend="smallcaps">Quadratures</hi> <hi rend="italics">by Fluxions.</hi>—The most general method of Quadratures yet discovered, is that of Newton, by means of Fluxions, and is as <figure/> follows. AC being any curve to be squared, AB an absciss, and BC an ordinate perpendicular to it, also <hi rend="italics">bc</hi> another ordinate indefinitely near to the former. Putting AB = <hi rend="italics">x,</hi> and BC = <hi rend="italics">y;</hi> then is B<hi rend="italics">b</hi> = <hi rend="italics">x</hi><hi rend="sup">.</hi> the fluxion of the absciss, and <hi rend="italics">yx</hi><hi rend="sup">.</hi> = C<hi rend="italics">b</hi> the fluxion of the area ABC sought. Now let the value of the ordinate <hi rend="italics">y</hi> be found in terms of the absciss <hi rend="italics">x,</hi> or in a function of the absciss, and let that function be called X, that is <hi rend="italics">y</hi> = X; then substituting X for <hi rend="italics">y</hi> in <hi rend="italics">yx</hi><hi rend="sup">.</hi>, gives X<hi rend="italics">x</hi><hi rend="sup">.</hi> the fluxion of the area; and the fluent of this, being taken, gives the area or Quadrature of ABC as required, for any curve, whatever its nature may be.</p><p><hi rend="italics">Ex.</hi> Suppose for example, AC to be a common parabola; then its equation is , where <hi rend="italics">p</hi> is the parameter; which gives , the value of <hi rend="italics">y</hi> in a function of <hi rend="italics">x,</hi> and is what is called X above; hence then is the fluxion of the area; and the fluent of this is of the circumscribing rectangle BD; which therefore is the Quadrature of the parabola.</p><p>Again, if AC be a circle whose diameter is <hi rend="italics">d;</hi> then its equation is , which gives , and the fluxion of the area . But as the fluent of this cannot be found in finite terms, the quantity √(<hi rend="italics">dx</hi> - <hi rend="italics">x</hi><hi rend="sup">2</hi>) is thrown into a series, and then the fluxion of the area <pb n="318"/><cb/> is ; and the fluent of this gives for the general expression of the area ABC. Now when the space becomes a semicircle, <hi rend="italics">x</hi> becomes = <hi rend="italics">d,</hi> and then the series above becomes for the area of the semicircle whose diameter is <hi rend="italics">d.</hi></p><p>In spirals CAR, or curves <figure/> referred to a centre C; put <hi rend="italics">y</hi> = any radius CR, <hi rend="italics">x</hi> = BN the arc of a circle described about the centre C, at any distance CB = <hi rend="italics">a,</hi> and C<hi rend="italics">nr</hi> another ray indefinitely near CNR: then , and by sim. fig. the fluxion of the area described by the revolving ray CR; then the fluent of this, for any particular case, will be the Quadrature of the spiral. So if, for instance, it be Archimedes's spiral, in which <hi rend="italics">x : y</hi> in a constant ratio suppose as <hi rend="italics">m : n,</hi> or <hi rend="italics">my</hi> = <hi rend="italics">nx,</hi> and ; hence then the fluxion of the area; the fluent of which is the general Quadrature of the spiral of Archimedes.</p></div2></div1><div1 part="N" n="QUADRILATERAL" org="uniform" sample="complete" type="entry"><head>QUADRILATERAL</head><p>, or <hi rend="smallcaps">Quadrilateral</hi> <hi rend="italics">Figure,</hi> is a figure comprehended by four right lines; and having consequently also four angles, for which reason it is otherwise called a quadrangle.</p><p>The general term Quadrilateral comprehends these several particular species or figures, viz, the square, parallelogram, rectangle, rhombus, rhomboides, and trapezium.</p><p>If the opposite sides be parallel, the Quadrilateral is a parallelogram. If the parallelogram have its angles right ones, it is a rectangle; if oblique, it is an oblique one. The rectangle having all its sides equal, becomes a square; and the oblique parallelogram having all its sides equal, is a rhombus, but if only the opposites be equal, it is a rhomboides. All other forms of the Quadrilateral, are trapeziums, including all the irregular shapes of it.</p><p>The sum of all the four angles of any Quadrilateral, is equal to 4 right angles. Also, the two opposite angles of a Quadrilateral inscribed in a circle taken together, are equal to two right angles. And in this case the rectangle of the two diagonals, is equal to the sum of the two rectangles of the opposite sides. For the properties of the particular species of Quadrilaterals, see their respective names, <hi rend="smallcaps">Square, Rectangle, Parallelogram, Rhombus, Rhomboides</hi>, and <hi rend="smallcaps">Trapezium.</hi></p></div1><div1 part="N" n="QUADRIPARTITION" org="uniform" sample="complete" type="entry"><head>QUADRIPARTITION</head><p>, is the dividing by 4, or <cb/> into four equal parts.—Hence <hi rend="italics">quadripartite,</hi> &c, the 4th part, or something parted into four.</p></div1><div1 part="N" n="QUADRUPLE" org="uniform" sample="complete" type="entry"><head>QUADRUPLE</head><p>, is four-fold, or something taken four times, or multiplied by 4; and so is the converse of Quadripartition.</p></div1><div1 part="N" n="QUALITY" org="uniform" sample="complete" type="entry"><head>QUALITY</head><p>, denotes generally the property or affection of some being, by which it affects our senses in a certain way, &c.</p><p><hi rend="italics">Sensible Qualities</hi> are such as are the more immediate object of the senses: as figure, taste, colour, smell, hardness, &c.</p><p><hi rend="italics">Occult Qualities,</hi> among the Ancients, were such as did not admit of a rational solution in their way.</p><p>Dr. Keil demonstrates, that every Quality which is propagated in orbem, such as light, heat, cold, odour, &c, has its efficacy or intensity either increased, or decreased, in a duplicate ratio of the distances from the centre of radiation inversely. So at double the distance from the earth's centre, or from a luminous or hot body, the weight or light or heat, is but a 4th part; and at 3 times the distance, they are 9 times less, or a 9th part, &c.</p><p>Sir Isaac Newton lays it down as one of the rules of philosophizing, that those Qualities of bodies that are incapable of being intended and remitted, and which are found to obtain in all bodies upon which the experiment could ever be tried, are to be esteemed universal Qualities of all bodies.</p><p><hi rend="smallcaps">Quality</hi> <hi rend="italics">of Curvature,</hi> in the higher geometry, is used to signify its form, as it is more or less inequable, or as it is varied more or less in its progress through different parts of the curve. Newton's Method of Fluxions, pa. 75; and Maclaurin's Fluxions, art. 369.</p></div1><div1 part="N" n="QUANTITY" org="uniform" sample="complete" type="entry"><head>QUANTITY</head><p>, denotes any thing capable of estimation, or mensuration; or which being compared with another thing of the same kind, may be said to be either greater or less, equal or unequal to it.</p><p>Mathematics is the doctrine or science of Quantity.</p><p><hi rend="italics">Physical</hi> or <hi rend="italics">Natural</hi> <hi rend="smallcaps">Quantity</hi>, is of two kinds: 1st, that which nature exhibits in matter, and its extension; and 2dly, in the powers and properties of natural bodies; as gravity, motion, light, heat, cold, density, &c.</p><p>Quantity is popularly distinguished into continued and discrete.</p><p><hi rend="italics">Continued</hi> <hi rend="smallcaps">Quantity</hi>, is when the parts are connected together, and is commonly called magnitude; which is the object of geometry.</p><p><hi rend="italics">Discrete</hi> <hi rend="smallcaps">Quantity</hi>, is when the parts, of which it consists, exist distinctly, and unconnected; which makes what is called multitude or number, the object of arithmetic.</p><p>The notion of continued Quantity, and its difference from discrete, appears to some without foundation. Mr. Machin considers all mathematical Quantity, or that for which any symbol is put, as nothing else but number, with regard to some measure, which is considered as 1; for that we know nothing precisely how much any thing is, but by means of number. The notion of continued Quantity, without regard to some measure, is indistinct and confused; and though some species of such Quantity, considered physically, may be described by motion, as lines by the motion of <pb n="319"/><cb/> points, and surfaces by the motion of lines; yet the magnitudes, or mathematical Quantities, are not made by the motion, but by numbering according to a measure. Philos. Trans. numb. 447, pa. 228.</p><p><hi rend="smallcaps">Quantity</hi> <hi rend="italics">of Action.</hi> See <hi rend="smallcaps">Action.</hi></p><p><hi rend="smallcaps">Quantity</hi> <hi rend="italics">of Curvature</hi> at any point of a curve is determined by the circle of curvature at that point, and is reciprocally proportional to the radius of curvature.</p><p><hi rend="smallcaps">Quantity</hi> <hi rend="italics">of Matter</hi> in any body, is its measure arising from the joint consideration of its magnitude and density, being expressed by, or proportional to the product of the two. So, if M and <hi rend="italics">m</hi> denote the magnitude of two bodies, and D and <hi rend="italics">d</hi> their densities; then DM and <hi rend="italics">dm</hi> will be as their Quantities of matter.</p><p>The Quantity of matter of a body is best discovered by its absolute weight, to which it is always proportional, and by which it is measured.</p><p><hi rend="smallcaps">Quantity</hi> <hi rend="italics">of Motion,</hi> or the <hi rend="italics">Momentum,</hi> of any body, is its measure arising from the joint consideration of its Quantity, and the velocity with which it moves. So, if <hi rend="italics">q</hi> denote the Quantity of matter, and <hi rend="italics">v</hi> the velocity of any body; then <hi rend="italics">qv</hi> will be its quantity of motion.</p><div2 part="N" n="Quantities" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quantities</hi></head><p>, in Algebra, are the expressions of indefinite numbers, that are usually represented by letters. Quantities are properly the subject of Algebra; which is wholly conversant in the computation of such Quantities.</p><p>Algebraic Quantities are either <hi rend="italics">given</hi> and <hi rend="italics">known,</hi> or else they are <hi rend="italics">unknown</hi> and <hi rend="italics">sought.</hi> The given or known Quantities are represented by the first letters of the alphabet, as <hi rend="italics">a, b, c, d, e,</hi> &c, and the unknown or required Quantities, by the last letters, as <hi rend="italics">z, y, x, w,</hi> &c.</p><p>Again, Algebraic Quantities are either positive or negative.</p><p>A positive or affirmative Quantity, is one that is to be added, and has the sign + or plus prefixed, or understood; as <hi rend="italics">ab</hi> or + <hi rend="italics">ab.</hi> And a negative or privative Quantity, is one that is to be subtracted, and has the sign - or minus prefixed; as - <hi rend="italics">ab.</hi></p></div2></div1><div1 part="N" n="QUART" org="uniform" sample="complete" type="entry"><head>QUART</head><p>, a measure of capacity, being the quarter or 4th part of some other measure. The English Quart is the 4th part of the gallon, and contains two pints. The Roman Quart, or quartarius, was the 4th part of their congius. The French, besides their Quart or pot of 2 pints, have various other Quarts, distinguished by the whole of which they are Quarters; as Quart de muid, and Quart de boisseau.</p></div1><div1 part="N" n="QUARTER" org="uniform" sample="complete" type="entry"><head>QUARTER</head><p>, the 4th part of a whole, or one part of an integer, which is divided into four equal portions.</p><div2 part="N" n="Quarter" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quarter</hi></head><p>, in weights, is the 4th part of the quintal, or hundred weight; and so contains 28 pounds.</p><p><hi rend="smallcaps">Quarter</hi> is also a dry measure, containing of corn 8 bushels striked; and of coals the 4th part of a chaldron.</p><p>Quarter, in Astronomy, the moon's period, or lunation, is divided into 4 stages or Quarters; each containing between 7 and 8 days. The first Quarter is from the new moon to the quadrature; the second is from thence to the full moon, and so on. <cb/></p></div2><div2 part="N" n="Quarter" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quarter</hi></head><p>, in Navigation, is the Quarter or 4th part of a point, wind, or rhumb; or of the distance between two points &c. The Quarter contains an arch of 2° 48′ 45″, being the 4th part of 11° 15′, which is one point.</p><p><hi rend="smallcaps">Quarter</hi> <hi rend="italics">Round,</hi> in Architecture, is a term used by the workmen for any projecting moulding, whose contour is a Quarter of a circle, or nearly so.</p></div2></div1><div1 part="N" n="QUARTILE" org="uniform" sample="complete" type="entry"><head>QUARTILE</head><p>, an aspect of the planets when they are at the distance of 3 signs or 90° from each other: and is denoted by the character □.</p><p>QUEUE <hi rend="smallcaps">D'ARONDE</hi>, or <hi rend="italics">Swallow's Tail,</hi> in Fortification, is a detached or outwork, whose sides spread or open towards the campaign, or draw narrower and closer towards the gorge. Of this kind are either single or double tenailles, and some horn-works, whose sides are not parallel, but are narrow at the gorge, and open at the head, like the figure of a swallow's tail.</p><p>On the contrary, when the sides are less than the gorge, the work is called <hi rend="italics">contre Queue d'aronde.</hi></p><p><hi rend="smallcaps">Queue</hi> <hi rend="italics">d'aronde,</hi> in Carpentry, a method of jointing, called also dove-tailing.</p></div1><div1 part="N" n="QUINCUNX" org="uniform" sample="complete" type="entry"><head>QUINCUNX</head><p>, denotes (5/12)ths of any thing. So 10 is quincunx of 24, being (5/12) of it.</p><div2 part="N" n="Quincunx" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quincunx</hi></head><p>, in Astronomy, is that position, or aspect, of the planets, when distant from each other by (5/12)ths. of the whole circle, or 5 signs out of the 12, that is 150 degrees. The Quincunx is marked Q, or V<hi rend="italics">c.</hi></p></div2></div1><div1 part="N" n="QUINDECAGON" org="uniform" sample="complete" type="entry"><head>QUINDECAGON</head><p>, is a plane figure of 15 angles, and consequently the same number of sides. When those are all equal, it is a regular Quindecagon, otherwise not.</p><p>Euclid shews how to inscribe this figure in a circle, prop. 16, lib. 4. And the side of a regular Quindecagon, so inscribed, is equal in power to the half difference between the side of the equilateral triangle, and the side of the pentagon; and also to the difference of the perpendiculars let fall on both sides, taken together.</p><p>QUINQUAGESIMA-<hi rend="italics">Sunday,</hi> is the same as Shrove-Sunday, and is so called as being about the 50th day before Easter, being indeed the 7th Sunday before it. Anciently the term Quinquagesima was used for Whitsunday, and for the 50 days between Easter and Whitsunday; but to distinguish this Quinquagesima from that before Easter, it was called the paschal Quinquagesima.</p></div1><div1 part="N" n="QUINQUEANGLED" org="uniform" sample="complete" type="entry"><head>QUINQUEANGLED</head><p>, or Quinqueangular, consisting of 5 angles.</p></div1><div1 part="N" n="QUINTAL" org="uniform" sample="complete" type="entry"><head>QUINTAL</head><p>, the weight of a hundred pounds, in most countries; but in England it is the hundred weight, or 112 pounds. Quintal was also formerly used for a weight of lead, iron, or other common metal, usually equal to a hundred pounds, at 6 score to the hundred.</p></div1><div1 part="N" n="QUINTILE" org="uniform" sample="complete" type="entry"><head>QUINTILE</head><p>, in Astronomy, an aspect of the planets when they are distant the 5th part of the zodiac, or 72 degrees; and is marked thus, C, or O.</p></div1><div1 part="N" n="QUINTUPLE" org="uniform" sample="complete" type="entry"><head>QUINTUPLE</head><p>, is five-fold, or five times as much as another thing.</p></div1><div1 part="N" n="QUOIN" org="uniform" sample="complete" type="entry"><head>QUOIN</head><p>, in Architecture, an angle or corner of stone or brick walls. When these stand out beyond the rest of the wall, their edges being chamferred off, they are called <hi rend="italics">rustic Quoins.</hi></p><div2 part="N" n="Quoin" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Quoin</hi></head><p>, in Artillery, is a loose wedge of wood, which <pb n="320"/><cb/> is put in below the breech of a cannon, to raise or depress it more or less.</p></div2></div1><div1 part="N" n="QUOTIENT" org="uniform" sample="complete" type="entry"><head>QUOTIENT</head><p>, in Arithmetic, is the result of the operation of division, or the number that arises by dividing the dividend by the divisor, shewing how often the latter is contained in the former. Thus the Quotient of 12 divided by 3 is 4; which is usually thus disposed, or expressed, <cb/> 3) 12 (4 the quotient, or thus the Quotient, or thus 12/3 like a vulgar fraction; all these meaning the same thing. —In division, as the divisor is to the dividend, so is unity or 1 to the Quotient; thus 3 : 12 :: 1 : 4 the Quotient. </p></div1></div0><div0 part="N" n="R" org="uniform" sample="complete" type="alphabetic letter"><head>R</head><cb/><p>RADIANT <hi rend="italics">Point,</hi> or <hi rend="smallcaps">Radiating</hi> <hi rend="italics">Point,</hi> is any point from whence rays proceed.</p><p>Every Radiant point diffuses innumerable rays all around: but those rays only are visible from which right lines can be drawn to the pupil of the eye; because the rays are all in right lines. All the rays proceeding from the same Radiant continually diverge; but the crystalline collects or reunites them again.</p><div1 part="N" n="RADIATION" org="uniform" sample="complete" type="entry"><head>RADIATION</head><p>, is the casting or shooting forth of rays of light as from a centre.—Every visible body is a radiating body; it being only by means of its rays that it affects the eye.—The surface of a radiating or visible body, may be conceived as consisting of radiant points.</p><p>RADICAL <hi rend="italics">Sign,</hi> in Algebra, the sign or character denoting the root of a quantity; and is this √. So √2 is the square root of 2, and √<hi rend="sup">3</hi>2 is the cube root of <*>, &c.</p></div1><div1 part="N" n="RADIOMETER" org="uniform" sample="complete" type="entry"><head>RADIOMETER</head><p>, a name which some writers give to the Radius Astronomicus, or Jacob's Staff. See <hi rend="smallcaps">Fore-Staff.</hi></p></div1><div1 part="N" n="RADIUS" org="uniform" sample="complete" type="entry"><head>RADIUS</head><p>, in Geometry, the semidiameter of a circle; or a right line drawn from the centre to the circumference.—It is implied in the definition of a circle, and it is apparent from its construction, that all the radii of the same circle are equal.—The Radius is sometimes called, in Trigonometry, the Sinus Totus, or whole sine.</p><div2 part="N" n="Radius" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Radius</hi></head><p>, in the Higher Geometry. <hi rend="smallcaps">Radius</hi> <hi rend="italics">of the Evoluta,</hi> <hi rend="smallcaps">Radius</hi> <hi rend="italics">Osculi,</hi> called also the <hi rend="italics">Radius of concavity,</hi> and the <hi rend="italics">Radius of curvature,</hi> is the right line CB, representing a thread, by whose evolution from off the curve <figure/> AC, upon which it was wound, the curve AB is formed. Or it is the Radius of a circle having the same curvature, in a given point of the curve at B, with that of the curve in that point. See C<hi rend="smallcaps">URVATURE</hi> and <hi rend="smallcaps">Evolute</hi>, where the method of finding this Radius may be seen. <cb/></p><p><hi rend="smallcaps">Radius</hi> <hi rend="italics">Astronomicus,</hi> an instrument usually called Jacob's Staff, the Cross-staff, or Fore-staff.</p></div2><div2 part="N" n="Radius" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Radius</hi></head><p>, in Mechanics, is applied to the spokes of a wheel; because issuing like rays from its centre.</p></div2><div2 part="N" n="Radius" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Radius</hi></head><p>, in Optics. See <hi rend="smallcaps">Ray.</hi></p><p><hi rend="smallcaps">Radius</hi> <hi rend="italics">Vector,</hi> is used for a right line drawn from the centre of force in any curve in which a body is supposed to move by a centripetal force, to that point of the curve where the body is supposed to be.</p></div2></div1><div1 part="N" n="RADIX" org="uniform" sample="complete" type="entry"><head>RADIX</head><p>, or <hi rend="italics">Root,</hi> is a certain finite expression or function, which, being evolved or expanded according to the rules proper to its form, shall produce a series. That finite expression, or Radix, is also the value of the infinite series. So 1/3 is the radix of .3333 &c, because 1/3 being evolved or expanded, by dividing 1 by 3, gives the infinite series .3333 &c. In like manner, the Radix .</p><p>See my Tracts, vol. 1, pa. 9, and 31, &c.</p></div1><div1 part="N" n="RAFTERS" org="uniform" sample="complete" type="entry"><head>RAFTERS</head><p>, in Architecture, are pieces of timber which stand by pairs on the raising-piece, or wall plate, and meet in an angle at the top, forming the roof of a building. <pb n="321"/><cb/></p></div1><div1 part="N" n="RAIN" org="uniform" sample="complete" type="entry"><head>RAIN</head><p>, water that descends from the atmosphere in the form of drops of a considerable size. Rain is apparently a precipitated cloud; as clouds are nothing but vapours raised from moisture, waters, &c. By this circumstance it is distinguished from dew and fog: in the former of which the drops are so small that they are quite invisible; and in the latter, though their size be larger, they seem to have very little more specific gravity than the atmosphere itself, and may therefore be reckoned hollow spherules rather than drops.</p><p>It is universally agreed, that Rain is produced by the water previously absorbed by the heat of the sun, or otherwise, from the terraqueous globe, into the atmosphere, as vapours, or vesiculæ. These vesiculæ, being specifically lighter than the atmosphere, are buoyed up by it, till they arrive at a region where the air is in a just balance with them; and there they float, till by some new agent they are converted into clouds, and thence either into Rain, snow, hail, mist, or the like.</p><p>But the agent in this formation of the clouds into Rain, and even of the vapours into clouds, has been much controverted. Most philosophers will have it, that the cold, which constantly occupies the superior regions of the air, chills and condenses the vesiculæ, at their arrival from a warmer quarter; congregates them together, and occasions several of them to coalesce into little masses: and thus their quantity of matter increasing in a higher proportion than their surface, they become an overload to the thin air, and so descend in Rain.</p><p>Dr. Derham accounts for the precipitation, hence; that the vesiculæ being full of air, when they meet with a colder air than that they contain, this is contracted into a less space: and consequently the watry shell or case becomes thicker, so as to become heavier than the air, &c.</p><p>But this separation cannot be ascribed to cold, since Rain often takes place in very warm weather. And though we should suppose the condensation owing to the cold of the higher regions, yet there is a remarkable fact which will not allow us to have recourse to this supposition: for it is certain that the drops of Rain increase in size considerably as they descend. On the top of a hill for instance, they will be small and inconsiderable, forming only a drizzling shower; but half way down the hill it is much more considerable; and at the bottom the drops will be very large, descending in an impetuous Rain. Which shews that the atmosphere condenses the vapours as well where it is warm as where it is cold.</p><p>Others allow the cold only a part in the action, and bring in the winds as sharers with it: alledging, that a wind blowing against a cloud will drive its vesiculæ upon one another, by which means several of them, coalescing as before, will be enabled to descend; and that the effect will be still more considerable, if two opposite winds blow together towards the same place: they add, that clouds already formed, happening to be aggregated by fresh accessions of vapour continually ascending, may thence be enabled to descend.</p><p>Yet the grand cause, according to Rohault, is still behind. That author conceives it to be the heat of the air, which, after continuing for some time near the earth, is at length carried up on high by a wind, and <cb/> there thawing the snowy villi or flocks of the half frozen vesiculæ, it reduces them into drops; which, coalescing, descend, and have their dissolution perfected in their progress through the lower and warmer stages of the atmosphere.</p><p>Others, as Dr. Clarke, &c, ascribe this descent of the clouds rather to an alteration of the atmosphere than of the vesiculæ; and suppose it to arise from a diminution of the spring or elastic force of the air. This elasticity, which depends chiefly or wholly on the dry terrene exhalations, being weakened, the atmosphere sinks under its burden; and the clouds fall, on the common principle of precipitation.</p><p>Now the small vesiculæ, by these or any other causes, being once upon the descent, will continue to descend notwithstanding the increase of resistance they every moment meet with in their progress through still denser and denser parts of the atmosphere. For as they all tend toward the same point, viz, the centre of the earth, the farther they fall, the more coalitions will they make; and the more coalitions, the more matter will there be under the same surface; the surface only increasing as the squares, but the solidity as the cubes of the diameters: and the more matter under the same surface, the less friction or resistance there will be to the same matter.</p><p>Thus then, if the causes of rain happen to act early enough to precipitate the ascending vesiculæ, before they are arrived at any considerable height, the coalitions being few in so short a descent, the drops will be proportionably small; thus forming what is called dew. If the vapours prove more copious, and rise a little higher, there is produced a mist or fog. A little higher still, and they produce a small rain, &c. If they neither meet with cold nor wind enough to condense or dissipate them; they form a heavy, thick, dark sky, which lasts sometimes several days, or even weeks.</p><p>But later writers on this part of philosophical science have, with greater shew of truth, considered Rain as an electrical phenomenon. Signior Beccaria reckons Rain, hail, and snow, among the effects of a moderate electricity in the atmosphere. Clouds that bring Rain, he thinks are produced in the same manner as thunder clouds, only by a moderate electricity. He describes them at large; and the resemblance which all their phenomena bear to those of thunder clouds, is very striking. He notes several circumstances attending Rain without lightning, which render it probable that it is produced by the same cause as when it is accompanied with lightning. Light has been seen among the clouds by night in rainy weather; and even by day rainy clouds are sometimes seen to have a brightness evidently independent of the sun. The uniformity with which the clouds are spread, and with which the Rain falls, he thinks are evidences of an uniform cause like that of electricity. The intensity also of electricity in his apparatus, usually corresponded very nearly to the quantity of Rain that fell in the same time. Sometimes all the phenomena of thunder, lightning, hail, Rain, snow, and wind, have been observed at one time; which shews the connection they all have with some common cause. Signior Beccaria therefore supposes that, previous to Rain, a quantity of electric <pb n="322"/><cb/> matter escapes out of the earth, in some place where there is a redundancy of it; and in its ascent to the higher regions of the air, collects and conducts into its path a great quantity of vapours. The same cause that collects, will condense them more and more; till, in the places of the nearest intervals, they come almost into contact, so as to form small drops; which, uniting with others as they fall, come down in the form of Rain. The Rain will be heavier in proportion as the electricity is more vigorous, and the cloud approaches more nearly to a thunder cloud: &c. See <hi rend="italics">Lettere dell Elettricismo;</hi> and Priestley's Hist. &c of Electricity, vol. 1, pa. 427, &c, 8vo. And for farther accounts of the phenomena of Rain &c, see <hi rend="smallcaps">Barometer</hi>, E<hi rend="smallcaps">VAPORATION, Ombrometer, Pluviameter, Vapour</hi>, &c. See also the Theory of Rain, by Dr. James Hutton, art. 2 vol. 1 of Transactions of the Royal Society of Edinburgh.</p><p><hi rend="italics">Quantity of</hi> <hi rend="smallcaps">Rain.</hi> As to the general quantity of Rain that falls, with its proportion in several places at the same time, and in the same place at different times, there are many observations, journals, &c, in the Philos. Trans. the Memoirs of the French Academy, &c. And upon measuring the rain that falls annually, its depth, on a medium, is found as in the following table: <hi rend="center"><hi rend="italics">Mean Annual Depth of Rain for several Places.</hi></hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">At</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Observed by</hi></cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">Inch.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Townley, in Lancashire</cell><cell cols="1" rows="1" role="data">Mr. Townley</cell><cell cols="1" rows="1" role="data">42 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Upminster, in Essex</cell><cell cols="1" rows="1" role="data">Dr. Derham</cell><cell cols="1" rows="1" role="data">19 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Zurich, Swisserland</cell><cell cols="1" rows="1" role="data">Dr. Scheuchzer</cell><cell cols="1" rows="1" role="data">32 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Pisa, in Italy</cell><cell cols="1" rows="1" role="data">Dr. Mich. Ang. Tilli</cell><cell cols="1" rows="1" role="data">43 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data">Paris, in France</cell><cell cols="1" rows="1" role="data">M. De la Hire</cell><cell cols="1" rows="1" role="data">19</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lisle, Flanders</cell><cell cols="1" rows="1" role="data">M. De Vauban</cell><cell cols="1" rows="1" role="data">24</cell></row></table> <hi rend="center"><hi rend="italics">Quantity of Rain fallen in several Years at Paris and Upminster.</hi></hi> <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">At Paris.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Years.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">At Upminster.</hi></cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Inches 21.37</cell><cell cols="1" rows="1" role="data">1700</cell><cell cols="1" rows="1" rend="align=center" role="data">19.03</cell><cell cols="1" rows="1" rend="align=left" role="data">Inches</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27.77</cell><cell cols="1" rows="1" role="data">1701</cell><cell cols="1" rows="1" rend="align=center" role="data">18.69</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17.45</cell><cell cols="1" rows="1" role="data">1702</cell><cell cols="1" rows="1" rend="align=center" role="data">20.38</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18.51</cell><cell cols="1" rows="1" role="data">1703</cell><cell cols="1" rows="1" rend="align=center" role="data">23.99</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21.20</cell><cell cols="1" rows="1" role="data">1704</cell><cell cols="1" rows="1" rend="align=center" role="data">15.80</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14.82</cell><cell cols="1" rows="1" role="data">1705</cell><cell cols="1" rows="1" rend="align=center" role="data">16.93</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20.19</cell><cell cols="1" rows="1" role="data">Mediums</cell><cell cols="1" rows="1" rend="align=center" role="data">19.14</cell><cell cols="1" rows="1" role="data"/></row></table> <hi rend="center"><hi rend="italics">Medium Quantity of Rain at London, for several Years, from the Philos. Trans.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data">Viz, in</cell><cell cols="1" rows="1" role="data">1774</cell><cell cols="1" rows="1" role="data">26.328</cell><cell cols="1" rows="1" role="data">inches.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1775</cell><cell cols="1" rows="1" role="data">24.083</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1776</cell><cell cols="1" rows="1" role="data">20.354</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1777</cell><cell cols="1" rows="1" role="data">25.371</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1778</cell><cell cols="1" rows="1" role="data">20.772</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1779</cell><cell cols="1" rows="1" role="data">26.785</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1780</cell><cell cols="1" rows="1" role="data">17.313</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data"> Medium of these 7 years</cell><cell cols="1" rows="1" role="data">23.001</cell><cell cols="1" rows="1" role="data"/></row></table> <cb/></p><p>See also Philos. Trans. Abr. vol. 4, pt. 2, pa. 81, &c<*> and vol. 10 in many places; also the Meteorological Journal of the Royal Society, published annually in the Philos. Trans. and the article <hi rend="smallcaps">Pluviameter</hi> or <hi rend="smallcaps">Ombrometer.</hi></p><p>It is reasonably to the expected, and all experience shews, that the most Rain falls in places near the sea coast, and less and less as the places are situated more inland. Some differences also arise from the circumstances of hills, valleys, &c. So when the quantity of Rain fallen in one year at London, is 20 inches, that on the western coast of England will often be twice as much, or 40 inches, or more. Those winds also bring most Rain, that blow from the quarter in which is the most and nearest sea; as our west and south-west winds.</p><p>It is also found, by the pluviameter or Rain-gage, that, in any one place, the more Rain is collected in the instrument, as it is placed nearer the ground; without any appearance of a difference, between two places, on account of their difference of level above the sea, provided the instrument is but as far from the ground at the one place, as it is from the ground at the other. These effects are remarked in the Philos. Trans. for 1769 and 1771, the former by Dr. Heberden, and the latter by Mr. Daines Barrington. Dr. Heberden says, “A comparison having been made between the quantity of Rain, which fell in two places in London, about a mile distant from one another, it was found, that the Rain in one of them constantly exceeded that in the other, not only every month, but almost every time that it rained. The apparatus used in each of them was very exact, and both made by the same artist; and upon examining every probable cause, this unexpected variation did not appear to be owing to any mistake, but to the constant effect of some circumstance, which not being supposed to be of any moment, had never been attended to. The Rain-gage in one of these places was fixed so high, as to rise above all the neighbouring chimnies; the other was considerably below them; and there appeared reason to believe, that the difference of the quantity of Rain in these two places was owing to this difference in the placing of the vessel in which it was received. A funnel was therefore placed above the highest chimnies, and another upon the ground of the garden belonging to the same house, and there was found the same difference between these two, though placed so near one another, which there had been between them, when placed at similar heights in different parts of the town. After this fact was sufficiently ascertained, it was thought proper to try whether the difference would be greater at a much greater height; and a Rain-gage was therefore placed upon the square part of the roof of Westminster Abbey. Here the quantity of Rain was observed for a twelvemonth, the Rain being measured at the end of every month, and care being taken that none should evaporate by passing a very long tube of the funnel into a bottle through a cork, to which it was exactly fitted. The tube went down very near to the bottom of the bottle, and therefore the Rain which fell into it would soon rise above the end of the tube, so that the water was no where open to the air except <pb n="323"/><cb/> for the small space of the area of the tube: and by trial it was found that there was no sensible evaporation through the tube thus fitted up.</p><p>The following table shews the result of these obsertions. <hi rend="center">From July the 7th 1766, to July the 7th 1767, there fell in a Rain-gage, fixed</hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Below the</cell><cell cols="1" rows="1" role="data">Upon the</cell><cell cols="1" rows="1" rend="align=left" role="data">Upon West-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">1766.</cell><cell cols="1" rows="1" role="data">top of a</cell><cell cols="1" rows="1" role="data">top of a</cell><cell cols="1" rows="1" rend="align=left" role="data">minster Ab-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">house.</cell><cell cols="1" rows="1" role="data">house.</cell><cell cols="1" rows="1" rend="align=left" role="data">bey.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">From the 7th to</cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Inches.</hi></cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">Inches.</hi></cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">Inches.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data"> the end of July</cell><cell cols="1" rows="1" rend="align=center" role="data">3.591</cell><cell cols="1" rows="1" rend="align=right" role="data">3.210</cell><cell cols="1" rows="1" rend="align=right" role="data">2.311</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">August</cell><cell cols="1" rows="1" rend="align=center" role="data">0.558</cell><cell cols="1" rows="1" rend="align=right" role="data">0.479</cell><cell cols="1" rows="1" rend="align=right rowspan=2" role="data"><hi rend="size(6)">}</hi>0.508</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">September</cell><cell cols="1" rows="1" rend="align=center" role="data">0.421</cell><cell cols="1" rows="1" rend="align=right" role="data">0.344</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">October</cell><cell cols="1" rows="1" rend="align=center" role="data">2.364</cell><cell cols="1" rows="1" rend="align=right" role="data">2.061</cell><cell cols="1" rows="1" rend="align=right" role="data">1.416</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">November</cell><cell cols="1" rows="1" rend="align=center" role="data">1.079</cell><cell cols="1" rows="1" rend="align=right" role="data">0.842</cell><cell cols="1" rows="1" rend="align=right" role="data">0.632</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">December</cell><cell cols="1" rows="1" rend="align=center" role="data">1.612</cell><cell cols="1" rows="1" rend="align=right" role="data">1.258</cell><cell cols="1" rows="1" rend="align=right" role="data">0.994</cell></row><row role="data"><cell cols="1" rows="1" role="data">1767,</cell><cell cols="1" rows="1" role="data">January</cell><cell cols="1" rows="1" rend="align=center" role="data">2.071</cell><cell cols="1" rows="1" rend="align=right" role="data">1.455</cell><cell cols="1" rows="1" rend="align=right" role="data">1.035</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">February</cell><cell cols="1" rows="1" rend="align=center" role="data">2.864</cell><cell cols="1" rows="1" rend="align=right" role="data">2.494</cell><cell cols="1" rows="1" rend="align=right" role="data">1.335</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" rend="align=center" role="data">1.807</cell><cell cols="1" rows="1" rend="align=right" role="data">1.303</cell><cell cols="1" rows="1" rend="align=right" role="data">0.587</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" rend="align=center" role="data">1.437</cell><cell cols="1" rows="1" rend="align=right" role="data">1.213</cell><cell cols="1" rows="1" rend="align=right" role="data">0.994</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" rend="align=center" role="data">2.432</cell><cell cols="1" rows="1" rend="align=right" role="data">1.745</cell><cell cols="1" rows="1" rend="align=right" role="data">1.142</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" rend="align=center" role="data">1.997</cell><cell cols="1" rows="1" rend="align=right" role="data">1.426</cell><cell cols="1" rows="1" rend="align=right rowspan=2" role="data"><hi rend="size(6)">}</hi>1.145</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">July 7</cell><cell cols="1" rows="1" rend="align=center" role="data">0.395</cell><cell cols="1" rows="1" rend="align=right" role="data">0.309</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">22.608</cell><cell cols="1" rows="1" rend="align=right" role="data">18.139</cell><cell cols="1" rows="1" rend="align=right" role="data">12.099</cell></row></table></p><p>By this table it appears, that there fell below the top of a house above a fifth part more Rain, than what fell in the same space above the top of the same house; and that there fell upon Westminster Abbey not much above one half of what was found to fall in the same space below the tops of the houses. This experiment has been repeated in other places with the same result. What may be the cause of this extraordinary difference, has not yet been discovered; but it may be useful to give notice of it, in order to prevent that error, which would frequently be committed in comparing the Rain of two places without attending to this circumstance.”</p><p>Such were the observations of Dr. Heberden on first announcing this circumstance, viz, of different quantities of Rain falling at different heights above the ground. Two years afterward, Daines Barrington Esq. made the following experiments and observations, to shew that this effect, with respect to different places, respected only the several heights of the instrument above the ground at those places, without regard to any real difference of level in the ground at those places.</p><p>Mr. Barrington caused two other Rain-gages, exactly like those of Dr. Heberden, to be placed, the one upon mount Rennig, in Wales, and the other on the plane below, at about half a mile's distance, the perpendicular height of the mountain being 450 yards, or 1350 feet; each gage being at the same height above the surface of the ground at the two stations. <cb/> <hi rend="center">The results of the Experiment are as below:</hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">1770.</cell><cell cols="1" rows="1" role="data">Bottom of the</cell><cell cols="1" rows="1" role="data">Top of the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">mountain.</cell><cell cols="1" rows="1" role="data">mountain.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">Inches.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Inches.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">From</cell><cell cols="1" rows="1" role="data">July 6 to 16</cell><cell cols="1" rows="1" rend="align=center" role="data">0.709</cell><cell cols="1" rows="1" rend="align=center" role="data">0.648</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">July 16 to 29</cell><cell cols="1" rows="1" rend="align=center" role="data">2.185</cell><cell cols="1" rows="1" rend="align=center" role="data">2.124</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">July 29 to Aug. 10.</cell><cell cols="1" rows="1" rend="align=center" role="data">0.610</cell><cell cols="1" rows="1" rend="align=center" role="data">0.656</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Sept. 9 both bottles had</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">run over.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Sept. 9 to 30</cell><cell cols="1" rows="1" rend="align=center" role="data">3.234</cell><cell cols="1" rows="1" rend="align=center" role="data">2.464</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Oct. 17. both bottles had</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data"> run over.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Oct. 17 to 22</cell><cell cols="1" rows="1" rend="align=center" role="data">0.747</cell><cell cols="1" rows="1" rend="align=center" role="data">0.885</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Oct. 22 to 29</cell><cell cols="1" rows="1" rend="align=center" role="data">1.281</cell><cell cols="1" rows="1" rend="align=center" role="data">1.388</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Nov. 20 both bottles were</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data"> broken by the frost</cell><cell cols="1" rows="1" rend="align=center" role="data">8.766</cell><cell cols="1" rows="1" rend="align=center" role="data">8.165</cell></row></table></p><p>“The inference to be drawn from these experiments, Mr. Barrington observes, seems to be, that the increase of the quantity of Rain depends upon its nearer approximation to the earth, and scarcely at all upon the height of places, provided the Rain-gages are fixed at about the same distance from the ground.</p><p>“Possibly also a much controverted point between the inhabitants of mountains and plains may receive a solution from these experiments; as in an <hi rend="italics">adjacent valley, at least,</hi> very nearly the same quantity of Rain appears to fall within the same period of time as upon the neighbouring mountains.”</p><p>Dr. Heberden also adds the following note. “It may not be improper to subjoin to the foregoing account, that, in places where it was first observed, a different quantity of Rain would be collected, according as the Rain-gages were placed above or below the tops of the neighbouring buildings; the Rain-gage below the top of the house, into which the greater quantity of Rain had for several years been found to fall, was above 15 feet above the level of the other Rain-gage, which in another part of London was placed above the top of the house, and into which the lesser quantity always fell. This difference therefore does not, as Mr. Barrington justly remarks, depend upon the greater quantity of atmosphere, through which the Rain descends: though this has been supposed by some, who have thence concluded that this appearance might readily be solved by the accumulation of more drops, in a descent through a great depth of atmosphere.”</p></div1><div1 part="N" n="RAINBOW" org="uniform" sample="complete" type="entry"><head>RAINBOW</head><p>, <hi rend="italics">Iris,</hi> or simply the <hi rend="italics">Bow,</hi> is a meteor in form of a party-coloured arch, or semicircle, exhibited in a rainy sky, opposite to the sun, by the refraction and reflection of his rays in the drops of falling rain. There is also a secondary, or fainter bow, usually seen investing the former at some distance. Among naturalists, we also read of lunar Rainbows, marine Rainbows, &c.</p><p>The Rainbow, Sir Isaac Newton observes, never appears but where it rains in the sunshine; and it may be represented artificially, by contriving water to fall <pb n="324"/><cb/> in small drops, like rain, through which the sun shining, exhibits a bow to a spectator placed between the sun and the drops, especially if there be disposed beyond the drops some dark body, as a black cloth, or such like.</p><p>Some of the ancients, as appears by Aristotle's tract on Meteors, knew that the Rainbow was caused by the refraction of the sun's light in drops of falling rain. Long afterwards, one Fletcher of Breslaw, in a treatise which he published in 1571, endeavoured more particularly to account for the colours of the Rainbow by means of a double refraction, and one reflection. But he imagined that a ray of light, after entering a drop of rain, and suffering a refraction, both at its entrance and exit, was afterwards reflected from another drop, before it reached the eye of the spectator. It seems he overlooked the reflection at the farther side of the drop, or else he imagined that all the bendings of the light within the drop would not make a sufficient curvature, to bring the ray of the sun to the eye of the spectator. But Antonio de Dominis, bishop of Spalato, about the year 1590, whose treatise <hi rend="italics">De Radiis Visûs et Lucis</hi> was published in 1611 by J. Bartolus, first advanced, that the double refraction of Fletcher, with an intervening reflection, was sufficient to produce the colours of the Rainbow, and also to bring the rays that formed them to the eye of the spectator, without any subsequent reflection. He distinctly describes the progress of a ray of light entering the upper part of the drop, where it suffers one refraction, and after being by that thrown upon the back part of the inner surface, is from thence reflected to the lower part of the drop; at which place undergoing a second refraction, it is thereby bent so as to come directly to the eye. To verify this hypothesis, he procured a small globe of solid glass, and viewing it when it was exposed to the rays of the sun, in the same manner in which he had supposed the drops of rain were situated with respect to them, he actually observed the same colours which he had seen in the true Rainbow, and in the same order. Thus this author shewed how the interior bow is formed in round drops of rain, viz, by two refractions of the sun's rays and one reflection between them; and he likewise shewed that the exterior bow is formed by two refractions and two sorts of reflections between them in each drop of water.</p><p>The theory of A. de Dominis was adopted, and in some degree improved with respect to the exterior bow, by Des Cartes, in his treatise on Meteors; and indeed he was the first who, by applying mathematics to the investigation of this surprising appearance, ever gave a tolerable theory of the Rainbow. Philosophers were however still at a loss when they endeavoured to assign reasons for all the particular colours, and for the order of them. Indeed nothing but the doctrine of the different refrangibility of the rays of light, a discovery which was reserved for the great Newton, could furnish a complete solution of this difficulty.</p><p>Dr. Barrow, in his Lectiones Opticæ, at Lect. 12, n. 14, says, that a friend of his (by whom we are to understand Mr. Newton) communicated to him a way of determining the angle of the Rainbow, which was hinted to Newton by Slusius, without making a table of the refractions, as Des Cartes did. The doctor shews <cb/> the method; as also several other matters, at n. 14, 15, 16, relating to the Rainbow, worthy the genius of those two eminent men. But the subject was given more perfectly by Newton afterwards, viz, in his Optics, prop. 9; where he makes the breadth of the interior bow to be nearly 2° 15′, that of the exterior 3° 40′, their distance 8° 25′, the greatest semidiameter of the interior bow 42° 17′, and the least of the exterior 50° 42′, when their colours appear strong and perfect.</p><p>The doctrine of the Rainbow may be illustrated and confirmed by experiment in several different ways. Thus, by hanging up a glass globe, full of water, in the sun-shine, and viewing it in such a posture that the rays which come from the globe to the eye, may include an angle either of 42° or 50° with the sun's rays; for ex. if the angle be about 42°, the spectator will see a full red colour in that side of the globe opposite to the sun. And by varying the position so as to make that angle gradually less, the other colours, yellow, green, and blue, will appear successively, in the same side of the globe, and that very bright. But if the angle be made about 50°, suppose by raising the globe, there will appear a red colour in that side of the globe toward the sun, though somewhat faint; and if the angle be made greater, as by raising the globe still higher, this red will change successively to the other colours, yellow, green, and blue. And the same changes are observed by raising or depressing the eye, while the globe is at rest. Newton's Optics, pt. 2, prop. 9, prob. 4.</p><p>Again, a similar bow is often observed among the waves of the sea (called the <hi rend="italics">marine Rainbow),</hi> the upper parts of the waves being blown about by the wind, and so falling in drops. This appearance is also seen by moon light (called the <hi rend="italics">lunar Rainbow),</hi> though seldom vivid enough to render the colours distinguishable. Also it is sometimes seen on the ground, when the sun shines on a very thick dew. Cascades and fountains too, whose waters are in their fall divided into drops, exhibit Rainbows to a spectator, if properly situated during the time of the sun's shining; and even water blown violently out of the mouth of an observer, standing with his back to the sun, never fails to produce the same phenomenon. The artificial Rainbow may even be produced by candle light on the water which is ejected by a small fountain or jet d'eau. All these are of the same nature, and they depend upon the same causes; some account of which is as follows. <figure/> <pb n="325"/><cb/></p><p>Let the circle WQGB represent a drop of water, or a globe, upon which a beam of parallel light falls, of which let TB represent a ray falling perpendicularly at B, and which consequently either passes through without refraction, or is reflected directly back from Q. Suppose another ray IK, incident at K, at a distance from B, and it will be refracted according to a certain ratio of the sines of incidence and refraction to each other, which in rain water is as 529 to 396, to a point L, whence it will be in part transmitted in the direction LZ, and in part reflected to M, where it will again in part be reflected, and in part transmitted in the direction MP, being inclined to the line described by the incident ray in the angle IOP. Another ray AN, still farther from B, and consequently incident under a greater angle, will be refracted to a point F, still farther from Q, whence it will be in part reflected to G, from which place it will in part emerge, forming an angle AXR with the incident AN, greater than that which was formed between the ray MP and its incident ray. And thus, while the angle of incidence, or distance of the point of incidence from B, increases, the distance between the point of reflection and Q, and the angle formed between the incident and emergent reflected rays, will also increase; that is, as far as it depends on the distance from B: but as the refraction of the ray tends to carry the point of reflection towards Q, and to diminish the angle formed between the incident and emergent reflected ray, and that the more the greater the distance of the point of incidence from B, there will be a certain point of incidence between B and W, with which the greatest possible distance between the point of reflection and Q, and the greatest possible angle between the incident and emergent reflected ray, will correspond. So that a ray incident nearer to B shall, at its emergence after reflection, form a less angle with the incident, by reason of its more direct reflection from a point nearer to Q; and a ray incident nearer to W, shall at its emergence form a less angle with the incident, by reason of the greater quantity of the angles of refraction at its incidence and emergence. The rays which fall for a considerable space in the vicinity of that point of incidence with which the greatest angle of emergence corresponds, will, after emerging, form an angle with the incident rays differing insensibly from that greatest angle, and consequently will proceed nearly parallel to each other; and those rays which fall at a distance from that point will emerge at various angles, and consequently will diverge. Now, to a spectator, whose back is turned towards the radiant body, and whose eye is at a considerable distance from the globe or drop, the divergent light will be scarcely, if at all, perceptible; but if the globe be so situated, that those rays that emerge parallel to each other, or at the greatest possible angle with the incident, may arrive at the eye of the spectator, he will, by means of those rays, behold it nearly with the same splendour at any distance.</p><p>In like manner, those rays which fall parallel on a globe, and are emitted after two reflections, suppose at the points F and G, will emerge at H parallel to each other, when the angle they make with the incident AN is the least possible; and the globe must be <cb/> seen very resplendent when its position is such, that those parallel rays fall on the eye of the spectator.</p><p>The quantities of these angles are determined by calculation, the proportion of the sines of incidence and refraction to each other being known. And this proportion being different in rays which produce different colours, the angles must vary in each. Thus it is found, that the greatest angle in rain water for the least refrangible, or red rays, emitted parallel after one reflection, is 42° 2′, and for the most refrangible or violet rays, emitted parallel after one reflection, 40° 17′; likewise, after two reflections, the least refrangible, or red rays, will be emitted nearly parallel under an angle of 50° 57′, and the most refrangible, or violet, under an angle of 54° 7′; and the intermediate colours will be emitted nearly parallel at intermediate angles.</p><p>Suppose now, that O is the spectator's eye, and OP a line drawn parallel to the sun's rays, SE, SF, SG, and SH; <figure/> and let POE, POF, POG, POH be angles of 40° 17′, 42° 2′, 50° 57′, and 54° 7′ respectively; then these angles turned about their common side OP, will with their other sides OE, OF, OG, OH describe the verges of the two Rainbows, as in the figure. For, if E, F, G, H be drops placed any where in the conical superficies described by OE, OF, OG, OH, and be illuminated by the sun's rays SE, SF, SG, SH; the angle SEO being equal to the angle POE, or 40° 17′, will be the greatest angle in which the most refrangible rays can, after one reflection, be refracted to the eye, and therefore all the drops in the line OE must send the most refrangible rays most copiously to the eye, and so strike the sense with the deepest violet colour in that region. In like manner, the angle SFO being equal to the angle POF, or 42° 2′, will be the greatest in which the least refrangible rays after one reflection can emerge out of the drops, and therefore those rays must come most copiously to the eye from the drops in the line OF, and strike the sense with the deepest red colour in that region. And, by the same argument, the rays which have the intermediate degrees of refrangibility will come most copiously from drops between E and F, and strike the senses with the intermediate colours in the order which their degrees of refrangibility require; that is, in the <pb n="326"/><cb/> progress from E to F, or from the inside of the bow to the outside, in this order, violet, indigo, blue, green, yellow, orange, red. But the violet, by the mixture of the white light of the clouds, will appear faint, and inclined to purple.</p><p>Again, the angle SGO being equal to the angle POG, or 50° 57′, will be the least angle in which the least refrangible rays can, after two reflections, emerge out of the drops, and therefore the least refrangible rays must come most copiously to the eye from the drops in the line OG, and strike the sense with the deepest red in that region. And the angle SHO being equal to the angle POH, or 54° 7′, will be the least angle in which the most refrangible rays, after two reflections, can emerge out of the drops, and therefore those rays must come most copiously to the eye from the drops in the line OH, and strike the sense with the deepest violet in that region. And, by the same argument, the drops in the regions between G and H will strike the sense with the intermediate colours in the order which their degrees of refrangibility require; that is, in the progress from G to H, or from the inside of the bow to the outside, in this order, red, orange, yellow, green, blue, indigo, and violet. And since the four lines OE, OF, OG, OH may be situated any where in the above-mentioned conical superficies, what is said of the drops and colours in these lines, is to be understood of the drops and colours every where in those superficies.</p><p>Thus there will be made two bows of colours, an interior and stronger, by one reflection in the drops, and an exterior and fainter by two; for the light becomes fainter by every reflection; and their colours will lie in a contrary order to each other, the red of both bows bordering upon the space GF, which is between the bows. The breadth of the interior bow, EOF, measured across the colours, will be 1° 15′, and the breadth of the exterior GOH, will be 3° 10′, also the distance between them GOF, will be 8° 55′, the greatest semidiameter of the innermost, that is, the angle POF, being 42° 2′, and the least semidiameter of the outermost POG being 50° 57′. These are the measures of the bows as they would be, were the sun but a point; but by the breadth of his body, the breadth of the bows will be increased by half a degree, and their distance diminished by as much; so that the breadth of the inner bow will be 2° 15′, that of the outer 3° 40′, their distance 8° 25′; the greatest semidiameter of the interior bow 42° 17′, and the least of the exterior 50° 42′. And such are the dimensions of the bows in the heavens found to be, very nearly, when their colours appear strong and perfect.</p><p>The light which comes through drops of rain by two refractions without any reflection, ought to appear strongest at the distance of about 26 degrees from the sun, and to decay gradually both ways as the distance from the sun increases and decreases. And the same is to be understood of light transmitted through spherical hailstones. If the hail be a little flatted, as it often is, the light transmitted may grow so strong at a little less distance than that of 26°, as to form a halo about the sun and moon; which halo, when the stones are duly figured, may be coloured, and then it must be <cb/> red within, by the least refrangible rays, and blue without, by the most refrangible ones.</p><p>The light which passes through a drop of rain after two refractions, and three or more reflections, is scarce strong enough to cause a sensible bow.</p><p>As to the dimension of the Rainbow, Des Cartes first determined its diameter by a tentative and indirect method; laying it down, that the magnitude of the bow depends on the degree of refraction of the fluid; and assuming the ratio of the sine of incidence to that of refraction, to be in water as 250 to 187. But Dr. Halley, in the Philos. Trans. number 267, gave a simple direct method of determining the diameter of the Rainbow from the ratio of the refraction of the fluid being given; or, vice versa, the diameter of the Rainbow being given, to determine the refractive power of the fluid. And Dr. Halley's principles and construction were farther explained by Dr. Morgan, bishop of Ely, in his Dissertation on the Rainbow, among the notes upon Rohault's System of Philosophy, part 3, chap. 17.</p><p>From the theory of the Rainbow, all the particular phenomena of it are easily deducible. Hence we see, 1st, Why the iris is always of the same breadth; because the intermediate degrees of refrangibility of the rays between red and violet, which are its extreme colours, are always the same.</p><p>2dly, Why the bow shifts its situation as the eye does; and, as the popular phrase has it, flies from those who follow it, and follows those that fly from it; the coloured drops being disposed under a certain angle, about the axis of vision, which is different in different places: whence also it follows, that every different spectator sees a different bow.</p><p>3dly, Why the bow is sometimes a larger portion of a circle, sometimes a less: its magnitude depending on the greater or less part of the surface of the cone, above the surface of the earth, at the time of its appearance; and the higher the sun, always the less the Rainbow.</p><p>4thly, Why the bow never appears when the sun is above a certain altitude; the surface of the cone, in which it should be seen, being lost in the ground at a little distance from the eye, when the sun is above 42° high.</p><p>5thly, Why the bow never appears greater than a semicircle, on a plane; since, be the sun never so low, and even in the horizon, the centre of the bow is still in the line of aspect; which in this case runs along the earth, and is not at all raised above the surface. Indeed if the spectator be placed on a very considerable eminence, and the sun in the horizon, the line of aspect, in which the centre of the bow is, will be considerably raised above the horizon. And if the eminence be very high, and the rain near, it is possible the bow may be an entire circle.</p><p>6thly, How the bow may chance to appear inverted, or the concave side turned upwards; viz, a cloud happening to intercept the rays, and prevent their shining on the upper part of the arch: in which case, only the lower part appearing, the bow will seem as if turned upside down; which has probably been the case in several prodigies of this kind, related by authors. <pb n="327"/><cb/></p><p><hi rend="italics">Lunar</hi> <hi rend="smallcaps">Rainbow.</hi> The moon sometimes also exhibits the phenomenon of an iris, by the refraction of her rays in the drops of rain in the night-time.</p><p>Aristotle says, he was the first that ever observed it; and adds, it is never seen but at the time of the full moon; her light at other times being too faint to affect the sight after two refractions and one reflection.</p><p>The lunar iris has all the colours of the solar, very distinct and pleasant; only fainter, both from the different intensity of the rays, and the different disposition of the medium.</p><p><hi rend="italics">Marine</hi> <hi rend="smallcaps">Rainbow.</hi> This is a phenomenon sometimes observed in a much agitated sea; when the wind, sweeping part of the tops of the waves, carries them aloft; so that the sun's rays, falling upon them, are refracted, &c, as in a common shower, and there paint the colours of the bow. These bows are less distinguishable and bright than the common bow: but then they exceed as to numbers, there being sometimes 20 or 30 seen together. They appear at noon day, and in a position opposite to that of the common bow, the concave side being turned upwards, as indeed it ought to be.</p><p>RAIN-<hi rend="smallcaps">Gage</hi>, an instrument for measuring the quantity of rain that falls. It is the same as O<hi rend="smallcaps">MBROMETER</hi>, or <hi rend="smallcaps">Pluviameter</hi>, which see.</p><p>RAKED <hi rend="italics">Table,</hi> or <hi rend="smallcaps">Raking</hi> <hi rend="italics">Table,</hi> in Architecture, a member hollowed in the square of a pedestal, or elsewhere.</p></div1><div1 part="N" n="RAM" org="uniform" sample="complete" type="entry"><head>RAM</head><p>, in Astronomy. See <hi rend="smallcaps">Aries.</hi></p><div2 part="N" n="Ram" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ram</hi></head><p>, <hi rend="italics">battering.</hi> See <hi rend="smallcaps">Battering</hi> <hi rend="italics">Ram.</hi></p><p><hi rend="smallcaps">Rams-Horns</hi>, in Fortification, a name given by Belidor to the Tenailles.</p></div2></div1><div1 part="N" n="RAMPART" org="uniform" sample="complete" type="entry"><head>RAMPART</head><p>, or <hi rend="smallcaps">Rampier</hi>, in Fortification, a massy bank or elevation of earth around a place, to cover it from the direct fire of an enemy, and of sufficient thickness to resist the efforts of their cannon for many days. It is formed into bastions, curtains, &c.</p><p>Upon the Rampart the soldiers continually keep guard, and the pieces of artillery are planted for defence. Also, to shelter the men from the enemy's shot, the outside of the Rampart is built higher than the rest, i. e. a parapet is raised upon it with a platform. It is encompassed with a moat or ditch, out of which is dug the earth that forms the Rampart, which is raised sloping, that the earth may not slip down, and having a berme at bottom, or is otherwise fortified, being lined with a facing of brick or stone.</p><p>The height of the Rampart need not be more than 3 fathoms, this being sufficient to cover the houses from the battery of the cannon; neither need its thickness be more than 10 or 12, unless more earth come out of the ditch than can otherwise be bestowed.</p><p>The Ramparts of halfmoons are the better for being low, that the small fire of the defendants may the better reach the bottom of the ditch; but yet they must be so high as not to be commanded by the covertway.</p><p><hi rend="smallcaps">Rampart</hi> is also used, in civil architecture, for the void space left between the wall of a city and the houses. This is what the Romans called Pomœrium, where it was forbidden to build, and where they planted <cb/> rows of trees for the people to walk and amuse themselves under.</p></div1><div1 part="N" n="RAMUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">RAMUS</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, a celebrated French mathematician and philosopher, was born in 1515, in a village of Vermandois in Picardy. He was descended of a good family, which had been reduced to extreme poverty by the wars and other misfortunes. His own life too, says Bayle, was the sport of fortune. In his infancy he was twice attacked by the plague. At 8 years of age, a thirst for learning urged him to go to Paris; but he was soon forced by poverty to leave that city. He returned to it again as soon as he could; but, being unable to support himself, he left it a second time: yet his passion for study was so violent, that notwithstanding his bad success in the two former visits, he ventured upon a third. He was maintained there some months by one of his uncles; after which he was obliged to become a servant in the college of Navarre. Here he spent the day in waiting upon his masters, and the greatest part of the night in study.</p><p>After having finished classical learning and rhetoric, he went through a course of philosophy, which took him up three years and a half in the schools. The thesis, which he made for his master of arts degree, offended every one; for he maintained in it, that all that Aristotle had advanced was false; and he gave very good answers to the objections of the professors. This success encouraged him to examine the doctrine of Aristotle more closely, and to combat it vigorously: but he confined himself chiefly to his logic. The two first books he published, the one entitled, <hi rend="italics">Institutiones Dialecticæ,</hi> the other <hi rend="italics">Aristotelicæ Animadversiones,</hi> occasioned great disturbances in the university of Paris. The professors there, who were adorers of Aristotle, ought to have refuted Ramus's books, if they could, by writings and lectures: but instead of confining themselves within the just bounds of academical wars, they prosecuted this anti-peripatetic before the civil magistrate, as a man who was going to sap the foundations of religion. They raised such clamours, that the cause was carried before the parliament of Paris: but, perceiving that it would be examined equably, his enemies by their intrigues took it from that tribunal, to bring it before the king's council, in 1543. The king ordered, that Ramus and Anthony Govea, who was his principal adversary, should choose two judges each, to pronounce on the controversy, after they should have ended their disputation; while he himself appointed a deputy. Ramus appeared before the five judges, though three of them were his declared enemies. The dispute lasted two days, and Govea had all the advantages he could desire; Ramus's books being prohibited in all parts of the kingdom, and their author sentenced not to teach philosophy any longer; upon which his enemies triumphed in the most indecent manner.</p><p>The year after, the plague made great havoc in Paris, and forced most of the students in the college of Presle to quit it; but Ramus, being prevailed upon to teach in it, soon drew together a great number of auditors. The Sorbonne attempted in vain to drive him from that college; for he held the headship of that <pb n="328"/><cb/> house by arrêt of parliament. Through the patronage and protection of the cardinal of Lorrain, he obtained from Henry the 2d, in 1547, the liberty of speaking and writing, and the regal professorship of philosophy and eloquence in 1551. The parliament of Paris had, before this, maintained him in the liberty of joining philosophical lectures to those of eloquence; and this arrêt or decree had put an end to several prosecutions, which Ramus and his pupils had suffered. As soon as he was made regius professor, he was fired with a new zeal for improving the sciences, notwithstanding the hatred of his enemies, who were never at rest.</p><p>Ramus bore at that time a part in a very singular affair. About the year 1550, the royal professors corrected among other abuses, that which had crept into the pronunciation of the Latin tongue. Some of the clergy followed this regulation; but the Sorbonnists were much offended at it as an innovation, and defended the old pronunciation with great zeal. Things at length were carried so far, that a minister, who had a good living, was very ill treated by them; and caused to be ejected from his benefice for having pronounced <hi rend="italics">quisquis, quanquam,</hi> according to the new way, instead of <hi rend="italics">kiskis, kankam,</hi> according to the old. The minister applied to the parliament; and the royal professors, with Ramus among them, fearing he would fall a victim to the credit and authority of the faculty of divines, for presuming to pronounce the Latin tongue according to their regulations, thought it incumbent on them to assist him. Accordingly, they went to the court of justice, and represented in such strong terms the indignity of the prosecution, that the minister was cleared, and every person had the liberty of pronouncing as he pleased.</p><p>Ramus was bred up in the Catholic religion, but afterwards deserted it. He began to discover his new principles by removing the images from the chapel of his college of Presle, in 1552. Hereupon such a persecution was raised against him by the Religionists, as well as Aristotelians, that he was driven out of his professorship, and obliged to conceal himself. For that purpose, with the king's leave he went to Fontainbleau; where, by the help of books in the king's library, he prosecuted geometrical and astronomical studies. As soon as his enemies found out his retreat, they renewed their persecutions; and he was forced to conceal himself in several other places. In the mean time, his curious and excellent collection of books in the college of Presle was plundered: but after a peace was concluded in 1563, between Charles the 9th and the Protestants, he again took possession of his employment, maintained himself in it with vigour, and was particularly zealous in promoting the study of the mathematics.</p><p>This continued till the second civil war in 1567, when he was forced to leave Paris, and shelter himself among the Hugonots, in whose army he was at the battle of St. Denys. Peace having been concluded some months after, he was restored to his professorship; but, foreseeing that the war would soon break out again, he did not care to venture himself in a fresh storm, and therefore obtained the king's leave to visit the universities of Germany. He accordingly undertook this journey in 1568, and received great honours <cb/> wherever he came. He returned to France, after the third war in 1571; and lost his life miserably, in the massacre of St. Bartholomew's day, 1572, at 57 years of age. It is said, that he was concealed in a granary during the tumult; but discovered and dragged out by some peripatetic doctors who hated him; these, after stripping him of all his money under pretence of preserving his life, gave him up to the assassins, who, after cutting his throat and giving him many wounds, threw him out of the window; and his bowels gushing out in the fall, some Aristotelian scholars, encouraged by their masters, spread them about the streets; then dragged his body in a most ignominious manner, and threw it into the river.</p><p>Ramus was a great orator, a man of universal learning, and endowed with very fine qualities. He was sober, temperate, and chaste. He ate but little, and that of boiled meat; and drank no wine till the latter part of his life, when it was prescribed by the physicians. He lay upon straw; rose early, and studied hard all day; and led a single life with the utmost purity. He was zealous for the protestant religion, but at the same time a little obstinate, and given to contradiction. The protestant ministers did not love him much, for he made himself a kind of head of a party, to change the discipline of the protestant churches: his design was to introduce a democratical government in the church, but this design was traversed, and defeated in a national synod. His sect flourished however for some time afterwards, spreading pretty much in Scotland and England, and still more in Germany.</p><p>He published a great many books; but mathematics was chiefly obliged to him. Of this kind, his writings were principally these following:</p><p>1. <hi rend="italics">Scholarum Mathematicarum libri</hi> 31.</p><p>2. <hi rend="italics">Arithmeticæ libri duo.—Algebræ libri duo.—Geometriæ libri</hi> 27.</p><p>These were greatly enlarged and explained by Schoner, and published in 2 volumes 4to. There were several editions of them; mine is that of 1627, at Frankfort.—The Geometry, which is chiefly practical, was translated into English by William Bedwell, and published in 4to, at London, 1636.</p><p>RANDOM-<hi rend="smallcaps">Shot</hi>, is a shot discharged with the axis of the gun elevated above the horizontal or point-blank direction.</p><div2 part="N" n="Random" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Random</hi></head><p>, of a shot, also sometimes means the range of it, or the distance to which it goes at the first graze, or where it strikes the ground. See <hi rend="smallcaps">Range.</hi></p></div2></div1><div1 part="N" n="RANGE" org="uniform" sample="complete" type="entry"><head>RANGE</head><p>, in Gunnery, sometimes means the path a shot flies in. But more usually,</p><p><hi rend="smallcaps">Range</hi> now means the distance to which the shot flies when it strikes the ground or other object, called also the amplitude of the shot. But Range is the term in present use.</p><p>Were it not for the resistance of the air, the greatest Range, on a horizontal plane, would be when the shot is discharged at an angle of 45° above the horizon; and all other Ranges would be the less, the more the angle of elevation is above or below 45°; but so as that at equal distances above and below 45°, the two Ranges are equal to each other. But, on account of the resistance of the air, the Ranges are altered, and that in different proportions, both for the different <pb n="329"/><cb/> sizes of the shot, and their different velocities: so that the greatest Range, in practice, always lies below the elevation of 45°, and the more below it as the shot is smaller, and as its velocity is greater; so as that the smallest balls, discharged with the greatest velocity in practice, ranges the farthest with an elevation of 30° or under, while the largest shot, with very small velocities, range farthest with nearly 45° elevation; and at all the intermediate degrees in the other cases. See <hi rend="smallcaps">Projectiles.</hi></p></div1><div1 part="N" n="RARE" org="uniform" sample="complete" type="entry"><head>RARE</head><p>, in Physics, is the quality of a body that is very porous, whose parts are at a great distance from one another, and which contains but little matter under a great magnitude. In which sense Rare stands opposed to dense.</p><p>The corpuscular philosophers, viz, the Epicureans, Gassendi<*>s, Newtonians, &c, assert that bodies are rarer, some than others, in virtue of a greater quantity of pores, or of vacuity lying between their parts or particles. The Cartesians hold, that a greater rarity only consists in a greater quantity of materia subtilis contained in the pores. And lastly, the Peripatetics contend, that rarity is a new quality superinduced upon a body, without any dependence on either vacuity or subtile matter.</p></div1><div1 part="N" n="RAREFACTION" org="uniform" sample="complete" type="entry"><head>RAREFACTION</head><p>, in Physics, the rendering a body rarer, that is bringing it to expand or occupy more room or space, without the accession of new matter: and it is opposed to condensation. The more accurate writers restrict the term Rarefaction to that kind of expansion which is effected by means of heat: and the expansion from other causes they term <hi rend="italics">dilatation;</hi> if indeed there be other causes; for though some philosophers have attributed it to the action of a repulsive principle in the matter itself; yet from the many discoveries concerning the nature and properties of the electric fluid and fire, there is great reason to believe that this repulsive principle is no other than elementary fire.</p><p>The Cartesians deny any such thing as absolute Rarefaction: extension, according to them, constituting the essence of matter, they are obliged to hold all extension equally full. Hence they make Rarefaction to be no other than an accession of fresh, subtile, and insensible matter, which, entering the parts of bodies, sensibly distends them.</p><p>It is by Rarefaction that gunpowder has its effect; and to the same principle also we owe eolipiles, thermometers, &c. As to the air, the degree to which it is rarefiable exceeds all imagination, experience having shewn it to be far above 10,000 times more than the usual state of the atmosphere; and as it is found to be about 1000 times denser in gunpowder than the atmosphere, it follows that experience has found it differ by about 10 millions of times. Perhaps indeed its degree of expansion is absolutely beyond all limits.</p><p>Such immense Rarefaction, Newton observes, is inconceivable on any other principle than that of a repelling force inherent in the air, by which its particles mutually fly from one another. This repelling force, he observes, is much more considerable in air than in other bodies, as being generated from the most fixed bodies, and that with much difficulty, and scarce without fermentation; those particles being always <cb/> found to fly from each other with the greatest force, which, when in contact, cohere the most firmly together. See <hi rend="smallcaps">Air.</hi></p><p>Upon the Rarefaction of the air is founded the useful method of measuring altitudes by the barometer, in all the cases of which, the rarity of the air is found to be inversely as the force that compresses it, or inversely as the weight of all the air above it at any place.</p></div1><div1 part="N" n="RARITY" org="uniform" sample="complete" type="entry"><head>RARITY</head><p>, thinness, subtlety, or the contrary to density.</p></div1><div1 part="N" n="RATCH" org="uniform" sample="complete" type="entry"><head>RATCH</head><p>, or <hi rend="smallcaps">Rash</hi>, in Clock-Work, a sort of wheel having 12 fangs, which serve to lift up the detents every hour, to make the clock strike.</p></div1><div1 part="N" n="RATCHETS" org="uniform" sample="complete" type="entry"><head>RATCHETS</head><p>, in a Watch, are the small teeth at the bottom of the fusee, or barrel, that stop it in winding up.</p></div1><div1 part="N" n="RATIO" org="uniform" sample="complete" type="entry"><head>RATIO</head><p>, according to Euclid, is the habitude or relation of two magnitudes of the same kind in respect of quantity. So the ratio of 2 to 1 is double, that of 3 to 1 triple, &c. Several mathematicians have found fault with Euclid's definition of a Ratio, and others have as much defended it, especially Dr. Barrow, in his Mathematical Lectures, with great skill and learning.</p><p>Ratio is sometimes confounded with proportion, but very improperly, as being quite different things; for proportion is the similitude or equality or identity of two Ratios. So the Ratio of 6 to 2 is the same as that of 3 to 1, and the Ratio of 15 to 5 is that of 3 to 1 also; and therefore the Ratio of 6 to 2 is similar or equal or the same with that of 15 to 5, which constitutes proportion, which is thus expressed, 6 is to 2 as 15 to 5, or thus 6 : 2 :: 15 : 5, which means the same thing. So that Ratio exists between two terms, but proportion between two Ratios or four terms.</p><p>The two quantities that are compared, are called the <hi rend="italics">terms</hi> of the Ratio, as 6 and 2; the first of these 6 being called the <hi rend="italics">antecedent,</hi> and the latter 2 the <hi rend="italics">consequent.</hi> Also the <hi rend="italics">index</hi> or <hi rend="italics">exponent</hi> of the Ratio, is the quotient of the two terms: so the index of the Ratio of 6 to 2 is 6/2 or 3, and which is therefore called a <hi rend="italics">triple Ratio.</hi></p><p>Wolfius distinguishes Ratios into <hi rend="italics">rational</hi> and <hi rend="italics">irrational.</hi></p><p><hi rend="italics">Rational</hi> <hi rend="smallcaps">Ratio</hi> is that which can be expressed between two rational numbers; as the Ratio of 6 to 2, or of 6√3 to 2√3, 3 to 1. And</p><p><hi rend="italics">Irrational</hi> <hi rend="smallcaps">Ratio</hi> is that which cannot be expressed by that of one rational number to another; as the Ratio of √6 to √2, or of √3 to root √1, that is √3 to 1, which cannot be expressed in rational numbers.</p><p>When the two terms of a Ratio are equal, the Ratio is said to be that of <hi rend="italics">equality;</hi> as of 3 to 3, whose index is 1, denoting the single or equal Ratio. But when the terms are not equal, as of 6 to 2, it is a <hi rend="italics">Ratio of inequality.</hi></p><p>Farther, when the antecedent is the greater term, as in 6 to 2, it is said to be the <hi rend="italics">Ratio of greater inequality:</hi> but when the antecedent is the less term, as in the Ratio of 2 to 6, it is said to be <hi rend="italics">the Ratio of</hi> <pb n="330"/><cb/> <hi rend="italics">less inequality.</hi> In the former case, if the less term be an aliquot part of the greater, the Ratio of greater inequality is said to be <hi rend="italics">multiplex</hi> or <hi rend="italics">multiple;</hi> and the Ratio of the less inequality, <hi rend="italics">sub-multiple.</hi> Particularly, in the first case, if the exponent of the Ratio be 2, as in 6 to 3, the Ratio is called <hi rend="italics">duple</hi> or <hi rend="italics">double;</hi> if 3, as in 6 to 2, it is <hi rend="italics">triple;</hi> and so on. In the second case, if the Ratio be 1/2, as in 3 to 6, the Ratio is called <hi rend="italics">subduple;</hi> if 1/3, as in 2 to 6, it is <hi rend="italics">subtriple;</hi> and so on.</p><p>If the greater term contain the less once, and one aliquot part of the same over; the Ratio of the greater inequality is called <hi rend="italics">superparticular,</hi> and the Ratio of the less <hi rend="italics">subsuperparticular.</hi> Particularly, in the first case, if the exponent be 3/2 or 1 1/2, it is called <hi rend="italics">sesquialterate;</hi> if 4/3 or 1 1/3, <hi rend="italics">sesquitertial;</hi> &c. In the other case, if the exponent be 7/3, the Ratio is called <hi rend="italics">subsesquialterate;</hi> if 3/4, it is <hi rend="italics">subsesquitertial.</hi></p><p>When the greater term contains the less once and several aliquot parts over, the Ratio of the greater inequality is called <hi rend="italics">superpartiens,</hi> and that of the less inequality is <hi rend="italics">subsuperpartiens.</hi> Particularly, in the former case, if the exponent be 5/3 or 1 2/3, the Ratio is called <hi rend="italics">superbipartiens tertias;</hi> if the exponent be 7/4 or 1 3/4, <hi rend="italics">supertripartiens quartas;</hi> if 11/7 or 1 4/7, <hi rend="italics">superquadripartiens septimas;</hi> &c. In the latter case, if the exponent be the reciprocals of the former, or 3/5, the Ratio is called <hi rend="italics">subsuperbipartiens tertias;</hi> if 4/7, <hi rend="italics">subsupertripartiens quartas;</hi> if 7/11, <hi rend="italics">subsuperquadripartiens septimas;</hi> &c.</p><p>When the greater term contains the less several times, and some one part over; the ratio of the greater inequality is called <hi rend="italics">multiplex superparticular;</hi> and the Ratio of the less inequality is called <hi rend="italics">submultiplex subsuperparticular.</hi> Particularly, in the former case, if the exponent be 5/2 or 2 1/2, the ratio is called <hi rend="italics">dupla sesquialtera;</hi> if 13/4 or 3 1/4, <hi rend="italics">tripla sesquiquarta,</hi> &c. In the latter case, if the exponent be 2/5, the Ratio is called <hi rend="italics">subdupla subsesquialtera;</hi> if 4/13, <hi rend="italics">subtripla subsesquiquarta,</hi> &c. Lastly, when the greater term contains the less several times, and several aliquot parts over; the Ratio of the greater inequality is called <hi rend="italics">multiplex superpartiens;</hi> that of the less inequality, <hi rend="italics">submultiplex subsuperpartiens.</hi> Particularly, in the former case, if the exponent be 8/3 or 2 2/3, the Ratio is called <hi rend="italics">dupla superbipartiens tertias;</hi> if 25/7 or 3 4/7, <hi rend="italics">tripla superbiquadripartiens septimas,</hi> &c. In the latter case, if the exponent be 3/8, the Ratio is called <hi rend="italics">subdupla subsuperbipartiens tertias;</hi> if 7/25, <hi rend="italics">subtripla subsuperquadripartiens septimas;</hi> &c.</p><p>These are the various denominations of rational Ratios, names which are very necessary to the reading of the ancient authors; though they occur but rarely among the modern writers, who use instead of them the smallest numeral terms of the Ratios; such 2 to 1 for duple, and 3 to 2 for sesquialterate, &c.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Ratio</hi>, is that which is made up of two or more other Ratios, viz, by multiplying the exponents together, and so producing the compound Ratio of the product of all the antecedents to the product of all the consequents. <table><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Thus the compound Ratio of 5</cell><cell cols="1" rows="1" role="data">to  3,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">and 7</cell><cell cols="1" rows="1" role="data">to  4,</cell></row><row role="data"><cell cols="1" rows="1" role="data">is the Ratio of</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">to 12.</cell></row></table> Particularly, if a Ratio be compounded of two equal Ratios, it is called the <hi rend="italics">duplicate Ratio;</hi> if of three equal <cb/> Ratios, the <hi rend="italics">triplicate Ratio;</hi> if of four equal Ratios, the <hi rend="italics">quadruplicate Ratio;</hi> and so on, according to the powers of the exponents, for all <hi rend="italics">multiplicate Ratios.</hi> So the several multiplicate Ratios of <table><row role="data"><cell cols="1" rows="1" role="data">the simple Ratio of</cell><cell cols="1" rows="1" role="data"> 3 to 2, are thus, viz.</cell></row><row role="data"><cell cols="1" rows="1" role="data">the duplicate Ratio</cell><cell cols="1" rows="1" role="data"> 9 :  4,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the triplicate Ratio</cell><cell cols="1" rows="1" role="data">27 :  8,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the quadruplicate Ratio</cell><cell cols="1" rows="1" role="data">81 : 16, &c.</cell></row></table></p><p><hi rend="italics">Properties of</hi> <hi rend="smallcaps">Ratios.</hi> Some of the more remarkable properties of Ratios are as follow:</p><p>1. The like multiples, or the like parts, of the terms of a Ratio, have the same Ratio as the terms themselves. So <hi rend="italics">a</hi> : <hi rend="italics">b,</hi> and <hi rend="italics">na</hi> : <hi rend="italics">nb,</hi> and <hi rend="italics">a</hi>/<hi rend="italics">n</hi> : <hi rend="italics">b</hi>/<hi rend="italics">n</hi> are all the same Ratio.</p><p>2. If to, or from, the terms of any Ratio, be added or subtracted either their like parts, or their like multiples, the sums or remainders will still have the same Ratio. So <hi rend="italics">a</hi> : <hi rend="italics">b,</hi> and <hi rend="italics">a</hi> ± <hi rend="italics">na</hi> : <hi rend="italics">b</hi> ± <hi rend="italics">nb,</hi> and <hi rend="italics">a</hi> ± <hi rend="italics">a</hi>/<hi rend="italics">n</hi> : <hi rend="italics">b</hi> ± <hi rend="italics">b</hi>/<hi rend="italics">n</hi> are all the same Ratio.</p><p>3. When there are several quantities in the same continued Ratio, <hi rend="italics">a, b, c, d, e,</hi> &c. whatever Ratio the first has to the 2d, the 1st to the 3d has the duplicate of that Ratio, the 1st to the 4th has the triplicate of that Ratio, the 1st to the 5th has the quadruplicate of it, and so on. Thus, the terms of the continued Ratio being 1, <hi rend="italics">r, r</hi><hi rend="sup">2</hi>, <hi rend="italics">r</hi><hi rend="sup">3</hi>, <hi rend="italics">r</hi><hi rend="sup">4</hi>, <hi rend="italics">r</hi><hi rend="sup">5</hi>, &c, where each term has to the following one the Ratio of 1 to <hi rend="italics">r,</hi> the Ratio of the 1st to the 2d; then 1 : <hi rend="italics">r</hi><hi rend="sup">2</hi> is the duplicate, 1 : <hi rend="italics">r</hi><hi rend="sup">3</hi> the triplicate, 1 : <hi rend="italics">r</hi><hi rend="sup">4</hi> the quadruplicate, and so on, according to the powers of 1 : <hi rend="italics">r.</hi></p><p>For other properties see <hi rend="smallcaps">Proportion.</hi></p><p><hi rend="italics">To approximate to a</hi> <hi rend="smallcaps">Ratio</hi> <hi rend="italics">in smaller Terms.</hi>—Dr. Wallis, in a small tract at the end of Horrox's works, treats of the nature and solution of this problem, but in a very tedious way; and he has prosecuted the same to a great length in his Algebra, chap. 10 and 11, where he particularly applies it to the Ratio of the diameter of a circle to its circumference. Mr. Huygens too has given a solution, with the reasons of it, in a much shorter and more natural way, in his Descrip. Autom. Planet. Opera Reliqua, vol. 1, pa. 174.</p><p>So also has Mr. Cotes, at the beginning of his Harmon. Mensurarum. And several other persons have done the same thing, by the same or similar methods. The problem is very useful, for expressing a Ratio in small numbers, that shall be near enough in practice, to any given Ratio in large numbers, such as that of the diameter of a circle to its circumference. The principle of all these methods, consists in reducing the terms of the proposed Ratio into a series of what are called continued fractions, by dividing the greater term by the less, and the less by the remainder, and so on, always the last divisor by the last remainder, after the manner of finding the greatest common measure of the two terms; then connecting all the quotients &c together in a series of continued fractions; and lastly collecting gradually these fractions together one after another. So if <hi rend="italics">b</hi>/<hi rend="italics">a</hi> be any fraction, or exponent of any Ratio; then dividing thus, <pb n="331"/><cb/> <figure/> gives <hi rend="italics">c, e, g, i,</hi> &c, for the several quotients, and these, formed in the usual way, give the approximate value of the given Ratio in a series of continued fractions; thus, . Then collecting the terms of this series, one after another, so many values of <hi rend="italics">b</hi>/<hi rend="italics">a</hi> are obtained, always nearer and nearer; the first value being <hi rend="italics">c</hi> or <hi rend="italics">c</hi>/1, the next , the 3d value ; in like manner, the 4th value is ; the 5th value is ; &c. From whence comes this general rule: Having found any two of these values, multiply the terms of the latter of them by the next quotient, and to the two products add the corresponding terms of the former value, and the sums will be the terms of the next value, &c.</p><p>For example, let it be required to find a series of Rations in lesser numbers, constantly approaching to the Ratio of 100000 to 314159, or nearly the Ratio of the diameter of a circle to its circumference. Here first dividing, thus, <figure/> there are obtained the quotients 3, 7, 15, 1, 25, 1, 7, 4. Hence 3 or 3/1 = <hi rend="italics">c,</hi> the 1st value; , the 2d value; , the 3d value; <cb/> , the 4th value; and so on; where the successive continual approximating values of the proposed Ratio are 3/1, 22/7, 333/106, 355/113, &c; the 2d of these, viz. 22/7, being the approximation of Archimedes; and the 4th, viz 355/133, is that of Metius, which is very near the truth, being equal <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">to 3.1415929,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the more accurate Ratio being</cell><cell cols="1" rows="1" rend="align=right" role="data">" 3.1415927.</cell></row></table></p><p><hi rend="italics">The doctrine of Ratios and Proportions,</hi> as delivered by Euclid, in the fifth book of his Elements, is considered by most persons as very obscure and objectionable, particularly the definition of proportionality; and several ingenious gentlemen have endeavoured to elucidate that subject. Among these, the Rev. Mr. Abram Robertson, of Christ Church College, Oxford, lecturer in geometry in that university, printed a neat little paper there in 1789, for the use of his classes, being a demonstration of that definition, in 7 propositions, the substance of which is as follows. He first premises this advertisement:</p><p>“As demonstrations depending upon proportionality pervade every branch of mathematical science, it is a matter of the highest importance to establish it upon clear and indisputable principles. Most mathematicians, both ancient and modern, have been of opinion that Euclid has fallen short of his usual perspicuity in this particular. Some have questioned the truth of the definition upon which he has founded it, and, almost all who have admitted its truth and validity have objected to it <hi rend="italics">as a definition.</hi> The author of the following propositions ranks himself amongst objectors of the last mentioned description. He thinks that Euclid must have founded the definition in question upon the reasoning contained in the first six demonstrations here given, or upon a similar train of thinking; and in his opinion <hi rend="italics">a definition</hi> ought to be as simple, or as free from a multiplicity of conditions, as the subject will admit.”</p><p>He then lays down these four definitions:</p><p>“1. Ratio is the relation which one magnitude has to another, of the same kind, with respect to quantity.”</p><p>“2. If the first of four magnitudes be exactly as great when compared to the second, as the third is when compared to the fourth, the first is said to have to the second the same Ratio that the third has to the fourth.”</p><p>“3. If the first of four magnitudes be greater, when compared to the second, than the third is when compared to the fourth, the first is said to have to the second a greater Ratio than the third has to the fourth.”</p><p>“4. If the first of four magnitudes be less, when compared to the second, than the third is when compared to the fourth, the first is said to have to the second a less Ratio than the third has to the fourth.”</p><p>Mr. Robertson then delivers the propositions, which are the following:</p><p>“<hi rend="italics">Prop.</hi> 1. If the first of four magnitudes have to the second, the same Ratio which the third has to the fourth; <pb n="332"/><cb/> then, if the first be equal to the second, the third is equal to the fourth; if greater, greater; if less, less.”</p><p>“<hi rend="italics">Prop.</hi> 2. If the first of four magnitudes be to the second as the third to the fourth, and if any equimultiples whatever of the first and third be taken, and also any equimultiples of the second and fourth; the multiple of the first will be to the multiple of the second as the multiple of the third to the multiple of the fourth.”</p><p>“<hi rend="italics">Prop.</hi> 3. If the first of four magnitudes be to the second as the third to the fourth, and if any like aliquot parts whatever be taken of the first and third, and any like aliquot parts whatever of the second and fourth, the part of the first will be to the part of the second as the part of the third to the part of the fourth.”</p><p>“<hi rend="italics">Prop.</hi> 4. If the first of four magnitudes be to the second as the third to the fourth, and if any equimultiples whatever be taken of the first and third, and any whatever of the second and fourth; if the multiple of the first be equal to the multiple of the second, the multiple of the third will be equal to the multiple of the fourth; if greater, greater; if less, less.”</p><p>“<hi rend="italics">Prop.</hi> 5. If the first of four magnitudes be to the second as the third is to a magnitude less than the fourth, then it is possible to take certain equimultiples of the first and third, and certain equimultiples of the second and fourth, such, that the multiple of the first shall be greater than the multiple of the second, but the multiple of the third not greater than the multiple of the fourth.”</p><p>“<hi rend="italics">Prop.</hi> 6. If the first of four magnitudes be to the second as the third is to a magnitude greater than the fourth, then certain equimultiples can be taken of the first and third, and certain equimultiples of the second and fourth, such, that the multiple of the first shall be less than the multiple of the second, but the multiple of the third not less than the multiple of the fourth.”</p><p>“<hi rend="italics">Prop.</hi> 7. If any equimultiples whatever be taken of the first and third of four magnitudes, and any equimultiples whatever of the second and fourth; and if when the multiple of the first is less than that of the second, the multiple of the third is also less than that of the fourth; or if when the multiple of the first is equal to that of the second, the multiple of the third is also equal to that of the fourth; or if when the multiple of the first is greater than that of the second, the multiple of the third is also greater than that of the fourth: then, the first of the four magnitudes shall be to the second as the third to the fourth.”</p><p>And all these propositions Mr. Robertson demonstrates by strict mathematical reasoning.</p></div1><div1 part="N" n="RATIONAL" org="uniform" sample="complete" type="entry"><head>RATIONAL</head><p>, in Arithmetic &c, the quality of numbers, fractions, quantities, &c, when they can be expressed by common numbers; in contradistinction to irrational or surd ones, which cannot be expressed in common numbers. Suppose any quantity to be 1; there are infinite other quantities, some of which are commensurable to it, either simply, or in power: these Euclid calls <hi rend="italics">Rational quantities.</hi> The rest, that are incommensurable to 1, he calls <hi rend="italics">irrational quantities,</hi> or <hi rend="italics">surds.</hi></p><p><hi rend="smallcaps">Rational</hi> <hi rend="italics">Horizon,</hi> or <hi rend="italics">True Horizon,</hi> is that whose plane is conceived to pass through the centre of the earth; and which therefore divides the globe into two equal portions or hemispheres. It is called the Rational horiz on, because only conceived by the understanding; <cb/> in opposition to the sensible or apparent horizon, or that which is visible to the eye.</p></div1><div1 part="N" n="RAVELIN" org="uniform" sample="complete" type="entry"><head>RAVELIN</head><p>, in Fortification, was anciently a flat bastion, placed in the middle of a curtain. But</p><p><hi rend="smallcaps">Ravelin</hi> is now a detached work, composed only of two faces, which form a salient angle usually without flanks. Being a triangular work resembling the point of a Bastion with the flanks cut off. It raised before the curtain, on the counterscarf of the place; and serving to cover it and the adjacent flanks from the direct fire of an enemy. It is also used to cover a bridge or a gate, and is always placed without the moat.</p><p>There are also double Ravelins, which serve to defend each other; being so called when they are joined by a curtain.</p><p>What the engineers call a Ravelin, the men usually call a demilune, or halfmoon.</p></div1><div1 part="N" n="RAY" org="uniform" sample="complete" type="entry"><head>RAY</head><p>, in Geometry, the same as <hi rend="smallcaps">Radius.</hi></p><div2 part="N" n="Ray" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ray</hi></head><p>, in Optics, a beam or line of light, propagated from a radiant point, through any medium.</p><p>If the parts of a Ray of light lie all in a straight line between the radiant point and the eye, the Ray is said to be <hi rend="italics">direct:</hi> the laws and properties of which make the subject of Optics.—If any of them be turned out of that direction, or bent in their passage, the Ray is said to be <hi rend="italics">refracted.</hi>—If it strike on the surface of any body, and be thrown off again, it is said to be <hi rend="italics">reflected.</hi>—In each case, the Ray, as it falls either directly on the eye, or on the point of reflection, or of refraction, is said to be <hi rend="italics">incident.</hi></p><p>Again, if several Rays be propagated from the radiant object equidistantly from one another, they are called <hi rend="italics">parallel</hi> Rays. If they come inclining towards each other, they are called <hi rend="italics">converging</hi> Rays. And if they go continually receding from each other, they are called <hi rend="italics">diverging</hi> Rays.</p><p>It is from the different circumstances of Rays, that the several kinds of bodies are distinguished in Optics. A body, for example, that diffuses its own light, or emits Rays of its own, is called a <hi rend="italics">radiating</hi> or <hi rend="italics">lucid</hi> or <hi rend="italics">luminous</hi> body. If it only reflect Rays which it receives from another, it is called an <hi rend="italics">illuminated</hi> body. If it only transmit Rays, it is called a <hi rend="italics">transparent</hi> or <hi rend="italics">translucent</hi> body. If it intercept the Rays, or refuse them passage, it is called an <hi rend="italics">opaque</hi> body.</p><p>It is by means of Rays reflected from the several points of illuminated objects to the eye, that they become visible, and that vision is performed; whence such Rays are called <hi rend="italics">visual</hi> Rays.</p><p>The Rays of light are not homogeneous, or similar, but differ in all the properties we know of; viz, refrangibility, reflexibility, and colour. It is probably from the different refrangibility that the other differences have their rise; at least it appears that those Rays which agree or differ in this, do so in all the rest. It is not however to be understood that the property or effect called colour, exists in the Rays of light themselves; but from the different sensations the differently disposed Rays excite in us, we call them <hi rend="italics">red Rays, yellow Rays,</hi> &c. Each beam of light however, as it comes from the sun, seems to be compounded of all the sorts of Rays mixed together; and it is only by splitting or separating the parts of it, that these different sorts become observable; and this is done by transmitting the <pb n="333"/><cb/> beam through a glass prism, which refracting it in the passage, and the parts that excite the different colours having different degrees of refrangibility, they are thus separated from one another, and exhibited each apart, and appearing of the different colours.</p><p>Beside refrangibility, and the other properties of the Rays of light already ascertained by observation and experiment, Sir I. Newton suspects they may have many more; particularly a power of being inflected or bent by the action of distant bodies; and those Rays which differ in refrangibility, he conceives likewise to differ in flexibility.</p><p>These Rays he suspects may be very small bodies emitted from shining substances. Such bodies may have all the conditions of light: and there is that action and reaction between transparent bodies and light, which very much resembles the attractive force between other bodies. Nothing more is required for the production of all the various colours, and all the degrees of refrangibility, but that the Rays of light be bodies of different sizes; the least of which may make violet the weakest and darkest of the colours, and be the most easily diverted by refracting surfaces from its rectilinear course; and the rest, as they are larger and larger, may make the stronger and more lucid colours, blue, green, yellow, and red. See <hi rend="smallcaps">Colour, Light</hi>, R<hi rend="smallcaps">EFRACTION, Reflection, Inflection, Converging, Diverging</hi>, &c, &c.</p><p><hi rend="italics">Reflected</hi> <hi rend="smallcaps">Rays</hi>, those Rays of light which are reflected, or thrown back again, from the surfaces of bodies upon which they strike. It is sound that, in all the Rays of light, the angle of reflection is equal to the angle of incidence.</p><p><hi rend="italics">Refracted</hi> <hi rend="smallcaps">Rays</hi>, are those Rays of light, which are bent or broken, in passing out of one medium into another.</p><p><hi rend="italics">Pencil of</hi> <hi rend="smallcaps">Rays</hi>, a number of Rays issued from a point of an object, and diverging in the form of a cone.</p><p><hi rend="italics">Principal</hi> <hi rend="smallcaps">Ray</hi>, in Perspective, is the perpendicular distance between the eye and the vertical plane or table, as some call it.</p><p><hi rend="smallcaps">Ray</hi> <hi rend="italics">of Curvature.</hi> See <hi rend="italics">Radius of</hi> <hi rend="smallcaps">Curvature.</hi></p></div2></div1><div1 part="N" n="REAUMUR" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">REAUMUR</surname> (<foreName full="yes"><hi rend="smallcaps">Rene - Antoine - Ferchault</hi></foreName>, Sieur de)</persName></head><p>, a respectable French philosopher, was born at Rochelle in 1683. After the usual course of school education, he was sent to Poitiers to study philosophy, and, in 1699, to Bourges to study the law, the profession for which he was intended. But philosophy and mathematics having very early been his favourite pursuits, he quitted the law, and repaired to Paris in 1703, to pursue those sciences to the best advantage; and here his character procured him a seat in the Academy in the year 1708; which he held till the time of his death, which happened the 18th of November 1757, at 74 years of age.</p><p>Reaumur soon justified the choice that was made of him by the Academy. He made innumerable observations, and wrote a great multitude of pieces upon the various branches of natural philosophy. His History of Insects, in 6 vols. quarto, at Paris, is his principal work. Another edition was printed in Holland, in <cb/> 12 vols. 12mo. He made also great and useful discoveries concerning iron; shewing how to change common wrought iron into steel, how to soften cast iron, and to make works in cast iron as fine as in wrought iron. His labours and discoveries concerning iron were rewarded by the duke of Orleans, regent of the kingdom, by a pension of 12 thousand livres, equal to about 500l. Sterling; which however he would not accept but on condition of its being put under the name of the Academy, who might enjoy it after his death. It was owing to Reaumur's endeavours that there were established in France manufactures of tin plates, of porcelain in imitation of china-ware, &c. They owe to him also a new thermometer, which bears his name, and is pretty generally used on the continent, while that of Fahrenheit is used in England, and some few other places. Reaumur's thermometer is a spirit one, having the freezing point at 0, and the boiling point at 80.</p><p>Reaumur is esteemed as an exact and clear writer; and there is an elegance in his style and manner, which is not commonly found among those who have made only the sciences their study. He is represented also as a man of a most amiable disposition, and with qualities to make him beloved as well as admired. He left a great variety of papers and natural curiosities, which he bequeathed to the Academy of Sciences.</p><p>The works published by him, are the following.</p><p>1. The Art of changing Forged Iron into Steel; of Softening Cast Iron; and of making works of Cast Iron, as fine as of Wrought Iron. Paris, 1722, 1 vol. in 4to.</p><p>2. Natural History of Insects, 6 vols. in 4to.</p><p>His memoirs printed in the volumes of the Academy of Sciences, are very numerous, amounting to upwards of a hundred, and on various subjects, from the year 1708 to 1763, several papers in almost every volume.</p></div1><div1 part="N" n="RECEIVER" org="uniform" sample="complete" type="entry"><head>RECEIVER</head><p>, <hi rend="italics">of an Air Pump,</hi> is part of its apparatus; being a glass vessel placed on the top of the plate, out of which the air is to be exhausted.</p></div1><div1 part="N" n="RECEPTION" org="uniform" sample="complete" type="entry"><head>RECEPTION</head><p>, in Astrology, is a dignity befalling two planets when they exchange houses: for example, when the sun arrives in Cancer, the house of the moon; and the moon, in her turn, arrives in the sun's house.— The same term is also used when two planets exchange exaltation.</p><p>RECESSION <hi rend="italics">of the Equinoxes.</hi> See <hi rend="smallcaps">Precession</hi> <hi rend="italics">of the Equinoxes.</hi></p></div1><div1 part="N" n="RECIPROCAL" org="uniform" sample="complete" type="entry"><head>RECIPROCAL</head><p>, in Arithmetic, &c, is the quotient arising by dividing 1 by any number or quantity. So, the Reciprocal of 2 is 1/2; of 3 is 1/3, and of <hi rend="italics">a</hi> is 1/<hi rend="italics">a,</hi> &c. Hence, the Reciprocal of a vulgar fraction is found, by barely making the numerator and the denominator mutually change places: so the Reciprocal of 1/2 is 2/1 or 2; of 2/3, is 3/2; of <hi rend="italics">a</hi>/<hi rend="italics">b,</hi> is <hi rend="italics">b</hi>/<hi rend="italics">a,</hi> &c. Hence also, any quantity being multiplied by its Reciprocal, the product is always equal to unity or 1 : so , and , and . <pb n="334"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="italics">Table of</hi> <hi rend="smallcaps">Reciprocals.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">61</cell><cell cols="1" rows="1" role="data">0163934</cell><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" role="data">0082645</cell><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" role="data">0055249</cell><cell cols="1" rows="1" role="data">241</cell><cell cols="1" rows="1" role="data">0041494</cell><cell cols="1" rows="1" rend="align=right" role="data">301</cell><cell cols="1" rows="1" role="data">0033223</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">62</cell><cell cols="1" rows="1" role="data">0161290</cell><cell cols="1" rows="1" role="data">122</cell><cell cols="1" rows="1" role="data">0081967</cell><cell cols="1" rows="1" role="data">182</cell><cell cols="1" rows="1" role="data">0054945</cell><cell cols="1" rows="1" role="data">242</cell><cell cols="1" rows="1" role="data">0041322</cell><cell cols="1" rows="1" rend="align=right" role="data">302</cell><cell cols="1" rows="1" role="data">0033113</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">3333333</cell><cell cols="1" rows="1" rend="align=right" role="data">63</cell><cell cols="1" rows="1" role="data">0158730</cell><cell cols="1" rows="1" role="data">123</cell><cell cols="1" rows="1" role="data">0081300</cell><cell cols="1" rows="1" role="data">183</cell><cell cols="1" rows="1" role="data">0054645</cell><cell cols="1" rows="1" role="data">243</cell><cell cols="1" rows="1" role="data">0041152</cell><cell cols="1" rows="1" rend="align=right" role="data">303</cell><cell cols="1" rows="1" role="data">0033003</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">64</cell><cell cols="1" rows="1" role="data">015625</cell><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" role="data">0080645</cell><cell cols="1" rows="1" role="data">184</cell><cell cols="1" rows="1" role="data">0054348</cell><cell cols="1" rows="1" role="data">244</cell><cell cols="1" rows="1" role="data">0040984</cell><cell cols="1" rows="1" rend="align=right" role="data">304</cell><cell cols="1" rows="1" role="data">0032895</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" role="data">0153846</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">008</cell><cell cols="1" rows="1" role="data">185</cell><cell cols="1" rows="1" role="data">0054054</cell><cell cols="1" rows="1" role="data">245</cell><cell cols="1" rows="1" role="data">0040816</cell><cell cols="1" rows="1" rend="align=right" role="data">305</cell><cell cols="1" rows="1" role="data">0032787</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">1666666</cell><cell cols="1" rows="1" rend="align=right" role="data">66</cell><cell cols="1" rows="1" role="data">0151515</cell><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" role="data">0079365</cell><cell cols="1" rows="1" role="data">186</cell><cell cols="1" rows="1" role="data">0053763</cell><cell cols="1" rows="1" role="data">246</cell><cell cols="1" rows="1" role="data">004065</cell><cell cols="1" rows="1" rend="align=right" role="data">306</cell><cell cols="1" rows="1" role="data">0032680</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">1428571</cell><cell cols="1" rows="1" rend="align=right" role="data">67</cell><cell cols="1" rows="1" role="data">0149254</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">0078740</cell><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" role="data">0053476</cell><cell cols="1" rows="1" role="data">247</cell><cell cols="1" rows="1" role="data">0040486</cell><cell cols="1" rows="1" rend="align=right" role="data">307</cell><cell cols="1" rows="1" role="data">0032573</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" rend="align=right" role="data">68</cell><cell cols="1" rows="1" role="data">0147059</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">0078125</cell><cell cols="1" rows="1" role="data">188</cell><cell cols="1" rows="1" role="data">0053191</cell><cell cols="1" rows="1" role="data">248</cell><cell cols="1" rows="1" role="data">0040323</cell><cell cols="1" rows="1" rend="align=right" role="data">308</cell><cell cols="1" rows="1" role="data">0032468</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">1111111</cell><cell cols="1" rows="1" rend="align=right" role="data">69</cell><cell cols="1" rows="1" role="data">0144928</cell><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" role="data">0077519</cell><cell cols="1" rows="1" role="data">189</cell><cell cols="1" rows="1" role="data">0052910</cell><cell cols="1" rows="1" role="data">249</cell><cell cols="1" rows="1" role="data">0040161</cell><cell cols="1" rows="1" rend="align=right" role="data">309</cell><cell cols="1" rows="1" role="data">0032362</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell 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cols="1" rows="1" role="data">0090090</cell><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" role="data">0058480</cell><cell cols="1" rows="1" role="data">231</cell><cell cols="1" rows="1" role="data">0043290</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">0034364</cell><cell cols="1" rows="1" rend="align=right" role="data">351</cell><cell cols="1" rows="1" role="data">0028490</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">52</cell><cell cols="1" rows="1" role="data">0192308</cell><cell cols="1" rows="1" rend="align=right" role="data">112</cell><cell cols="1" rows="1" role="data">0089286</cell><cell cols="1" rows="1" role="data">172</cell><cell cols="1" rows="1" role="data">0058141</cell><cell cols="1" rows="1" role="data">232</cell><cell cols="1" rows="1" role="data">0043103</cell><cell cols="1" rows="1" role="data">292</cell><cell cols="1" rows="1" role="data">0034246</cell><cell 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role="data">0087719</cell><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" role="data">0057471</cell><cell cols="1" rows="1" role="data">234</cell><cell cols="1" rows="1" role="data">0042735</cell><cell cols="1" rows="1" role="data">294</cell><cell cols="1" rows="1" role="data">0034014</cell><cell cols="1" rows="1" rend="align=right" role="data">354</cell><cell cols="1" rows="1" role="data">0028248</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">0181818</cell><cell cols="1" rows="1" rend="align=right" role="data">115</cell><cell cols="1" rows="1" role="data">0086957</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" role="data">0057143</cell><cell cols="1" rows="1" role="data">235</cell><cell cols="1" rows="1" role="data">0042553</cell><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" role="data">0033898</cell><cell cols="1" rows="1" rend="align=right" role="data">355</cell><cell cols="1" rows="1" role="data">0028169</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">56</cell><cell cols="1" rows="1" role="data">0178571</cell><cell cols="1" rows="1" rend="align=right" role="data">116</cell><cell cols="1" rows="1" role="data">0086207</cell><cell cols="1" rows="1" role="data">176</cell><cell cols="1" rows="1" role="data">0056818</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" role="data">0042373</cell><cell cols="1" rows="1" role="data">296</cell><cell cols="1" rows="1" role="data">0033783</cell><cell cols="1" rows="1" rend="align=right" role="data">356</cell><cell cols="1" rows="1" role="data">0028070</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" role="data">0175439</cell><cell cols="1" rows="1" rend="align=right" role="data">117</cell><cell cols="1" rows="1" role="data">0085470</cell><cell cols="1" rows="1" role="data">177</cell><cell cols="1" rows="1" role="data">0056497</cell><cell cols="1" rows="1" role="data">237</cell><cell cols="1" rows="1" role="data">0042194</cell><cell cols="1" rows="1" role="data">297</cell><cell cols="1" rows="1" role="data">0033670</cell><cell cols="1" rows="1" rend="align=right" role="data">357</cell><cell cols="1" rows="1" role="data">0028011</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">58</cell><cell cols="1" rows="1" role="data">0172414</cell><cell cols="1" rows="1" rend="align=right" role="data">118</cell><cell cols="1" rows="1" role="data">0084745</cell><cell cols="1" rows="1" role="data">178</cell><cell cols="1" rows="1" role="data">0056180</cell><cell cols="1" rows="1" role="data">238</cell><cell cols="1" rows="1" role="data">0042017</cell><cell cols="1" rows="1" role="data">298</cell><cell cols="1" rows="1" role="data">0033557</cell><cell cols="1" rows="1" rend="align=right" role="data">358</cell><cell cols="1" rows="1" role="data">0027933</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">59</cell><cell cols="1" rows="1" role="data">0169490</cell><cell cols="1" rows="1" rend="align=right" role="data">119</cell><cell cols="1" rows="1" role="data">0084034</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" role="data">0055866</cell><cell cols="1" rows="1" role="data">239</cell><cell cols="1" rows="1" role="data">0041841</cell><cell cols="1" rows="1" role="data">299</cell><cell cols="1" rows="1" role="data">0033445</cell><cell cols="1" rows="1" rend="align=right" role="data">359</cell><cell cols="1" rows="1" role="data">0027855</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" role="data">0166666</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell><cell cols="1" rows="1" role="data">0083333</cell><cell cols="1" rows="1" role="data">180</cell><cell cols="1" rows="1" role="data">0055555</cell><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" role="data">0041666</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">0033333</cell><cell cols="1" rows="1" rend="align=right" role="data">360</cell><cell cols="1" rows="1" role="data">0027777</cell></row></table><pb n="335"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="italics">Table of</hi> <hi rend="smallcaps">Reciprocals.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">361</cell><cell cols="1" rows="1" role="data">0027701</cell><cell cols="1" rows="1" rend="align=right" role="data">421</cell><cell cols="1" rows="1" role="data">0023753</cell><cell cols="1" rows="1" role="data">481</cell><cell cols="1" rows="1" role="data">0020790</cell><cell cols="1" rows="1" role="data">541</cell><cell cols="1" rows="1" role="data">0018484</cell><cell cols="1" rows="1" role="data">601</cell><cell cols="1" rows="1" role="data">0016639</cell><cell cols="1" rows="1" rend="align=right" role="data">661</cell><cell cols="1" rows="1" role="data">0015129</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">362</cell><cell cols="1" rows="1" role="data">0027624</cell><cell cols="1" rows="1" rend="align=right" role="data">422</cell><cell cols="1" rows="1" role="data">0023697</cell><cell cols="1" rows="1" role="data">482</cell><cell cols="1" rows="1" role="data">0020747</cell><cell cols="1" rows="1" role="data">542</cell><cell cols="1" rows="1" role="data">0018450</cell><cell cols="1" rows="1" role="data">602</cell><cell cols="1" rows="1" role="data">0016611</cell><cell cols="1" rows="1" rend="align=right" role="data">662</cell><cell cols="1" rows="1" role="data">0015106</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">363</cell><cell cols="1" rows="1" role="data">0027548</cell><cell cols="1" rows="1" rend="align=right" role="data">423</cell><cell cols="1" rows="1" role="data">0023641</cell><cell cols="1" rows="1" role="data">483</cell><cell cols="1" rows="1" role="data">0020704</cell><cell cols="1" rows="1" role="data">543</cell><cell cols="1" rows="1" role="data">0018416</cell><cell cols="1" rows="1" role="data">603</cell><cell cols="1" rows="1" role="data">0016584</cell><cell cols="1" rows="1" rend="align=right" role="data">663</cell><cell cols="1" rows="1" role="data">0015083</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">364</cell><cell cols="1" rows="1" role="data">0027473</cell><cell cols="1" rows="1" rend="align=right" role="data">424</cell><cell cols="1" rows="1" role="data">0023585</cell><cell cols="1" rows="1" role="data">484</cell><cell cols="1" rows="1" role="data">0020661</cell><cell cols="1" rows="1" role="data">544</cell><cell cols="1" rows="1" role="data">0018382</cell><cell cols="1" rows="1" role="data">604</cell><cell cols="1" rows="1" role="data">0016556</cell><cell cols="1" rows="1" rend="align=right" role="data">664</cell><cell cols="1" rows="1" role="data">0015060</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">365</cell><cell cols="1" rows="1" role="data">0027397</cell><cell cols="1" rows="1" rend="align=right" role="data">425</cell><cell cols="1" rows="1" role="data">0023529</cell><cell cols="1" rows="1" role="data">485</cell><cell cols="1" rows="1" role="data">0020619</cell><cell cols="1" rows="1" role="data">545</cell><cell cols="1" rows="1" role="data">0018349</cell><cell cols="1" rows="1" role="data">605</cell><cell cols="1" rows="1" role="data">0016529</cell><cell cols="1" rows="1" rend="align=right" role="data">665</cell><cell cols="1" rows="1" role="data">0015038</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">366</cell><cell cols="1" rows="1" role="data">0027322</cell><cell cols="1" rows="1" rend="align=right" role="data">426</cell><cell cols="1" rows="1" role="data">0023474</cell><cell cols="1" rows="1" role="data">486</cell><cell cols="1" rows="1" role="data">0020576</cell><cell cols="1" rows="1" role="data">546</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">642</cell><cell cols="1" rows="1" role="data">0015576</cell><cell cols="1" rows="1" rend="align=right" role="data">702</cell><cell cols="1" rows="1" role="data">0014245</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">403</cell><cell cols="1" rows="1" role="data">0024814</cell><cell cols="1" rows="1" rend="align=right" role="data">463</cell><cell cols="1" rows="1" role="data">0021598</cell><cell cols="1" rows="1" role="data">523</cell><cell cols="1" rows="1" role="data">0019120</cell><cell cols="1" rows="1" role="data">583</cell><cell cols="1" rows="1" role="data">0017153</cell><cell cols="1" rows="1" role="data">643</cell><cell cols="1" rows="1" role="data">0015552</cell><cell cols="1" rows="1" rend="align=right" role="data">703</cell><cell cols="1" rows="1" role="data">0014225</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">404</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">645</cell><cell cols="1" rows="1" role="data">0015504</cell><cell cols="1" rows="1" rend="align=right" role="data">705</cell><cell cols="1" rows="1" role="data">0014184</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">406</cell><cell cols="1" rows="1" role="data">0024631</cell><cell cols="1" rows="1" rend="align=right" role="data">466</cell><cell cols="1" rows="1" role="data">0021459</cell><cell cols="1" rows="1" role="data">326</cell><cell cols="1" rows="1" role="data">0019011</cell><cell cols="1" rows="1" role="data">586</cell><cell cols="1" rows="1" role="data">0017065</cell><cell cols="1" rows="1" role="data">646</cell><cell cols="1" rows="1" role="data">0015480</cell><cell cols="1" rows="1" rend="align=right" role="data">706</cell><cell cols="1" rows="1" role="data">0014164</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">407</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">648</cell><cell cols="1" rows="1" role="data">0015432</cell><cell cols="1" rows="1" rend="align=right" role="data">708</cell><cell cols="1" rows="1" role="data">0014124</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">409</cell><cell cols="1" rows="1" role="data">0024450</cell><cell cols="1" rows="1" rend="align=right" role="data">469</cell><cell cols="1" rows="1" role="data">0021322</cell><cell cols="1" rows="1" role="data">529</cell><cell cols="1" rows="1" role="data">0018904</cell><cell cols="1" rows="1" role="data">589</cell><cell cols="1" rows="1" role="data">0016978</cell><cell cols="1" rows="1" role="data">649</cell><cell cols="1" rows="1" role="data">0015408</cell><cell cols="1" rows="1" rend="align=right" role="data">709</cell><cell cols="1" rows="1" role="data">0014104</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">410</cell><cell cols="1" rows="1" 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role="data">0024213</cell><cell cols="1" rows="1" rend="align=right" role="data">473</cell><cell cols="1" rows="1" role="data">0021142</cell><cell cols="1" rows="1" role="data">533</cell><cell cols="1" rows="1" role="data">0018762</cell><cell cols="1" rows="1" role="data">593</cell><cell cols="1" rows="1" role="data">0016863</cell><cell cols="1" rows="1" role="data">653</cell><cell cols="1" rows="1" role="data">0015314</cell><cell cols="1" rows="1" rend="align=right" role="data">713</cell><cell cols="1" rows="1" role="data">0014025</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">414</cell><cell cols="1" rows="1" role="data">0024155</cell><cell cols="1" rows="1" rend="align=right" role="data">474</cell><cell cols="1" rows="1" role="data">0021097</cell><cell cols="1" rows="1" role="data">534</cell><cell cols="1" rows="1" role="data">0018727</cell><cell cols="1" rows="1" role="data">594</cell><cell cols="1" rows="1" role="data">0016835</cell><cell cols="1" rows="1" role="data">654</cell><cell cols="1" rows="1" role="data">0015291</cell><cell cols="1" rows="1" rend="align=right" role="data">714</cell><cell cols="1" rows="1" role="data">0014006</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">415</cell><cell cols="1" rows="1" role="data">0024096</cell><cell cols="1" rows="1" rend="align=right" role="data">475</cell><cell cols="1" rows="1" role="data">0021053</cell><cell cols="1" rows="1" role="data">535</cell><cell cols="1" rows="1" role="data">0018692</cell><cell cols="1" rows="1" role="data">595</cell><cell cols="1" rows="1" role="data">0016807</cell><cell cols="1" rows="1" role="data">655</cell><cell cols="1" rows="1" role="data">0015267</cell><cell cols="1" rows="1" rend="align=right" role="data">715</cell><cell cols="1" rows="1" role="data">0013986</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">416</cell><cell cols="1" rows="1" role="data">0024038</cell><cell cols="1" rows="1" rend="align=right" role="data">476</cell><cell cols="1" rows="1" role="data">0021008</cell><cell cols="1" rows="1" role="data">536</cell><cell cols="1" rows="1" role="data">0018657</cell><cell cols="1" rows="1" role="data">596</cell><cell cols="1" rows="1" role="data">0016779</cell><cell cols="1" rows="1" role="data">656</cell><cell cols="1" rows="1" role="data">0015244</cell><cell cols="1" rows="1" rend="align=right" role="data">716</cell><cell cols="1" rows="1" role="data">0013966</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">417</cell><cell cols="1" rows="1" role="data">0023981</cell><cell cols="1" rows="1" rend="align=right" role="data">477</cell><cell cols="1" rows="1" role="data">0020964</cell><cell cols="1" rows="1" role="data">537</cell><cell cols="1" rows="1" role="data">0018622</cell><cell cols="1" rows="1" role="data">597</cell><cell cols="1" rows="1" role="data">0016750</cell><cell cols="1" rows="1" role="data">657</cell><cell cols="1" rows="1" role="data">0015221</cell><cell cols="1" rows="1" rend="align=right" role="data">717</cell><cell cols="1" rows="1" role="data">0013947</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">418</cell><cell cols="1" rows="1" role="data">0023923</cell><cell cols="1" rows="1" rend="align=right" role="data">478</cell><cell cols="1" rows="1" role="data">0020921</cell><cell cols="1" rows="1" role="data">538</cell><cell cols="1" rows="1" role="data">0018587</cell><cell cols="1" rows="1" role="data">598</cell><cell cols="1" rows="1" role="data">0016722</cell><cell cols="1" rows="1" role="data">658</cell><cell cols="1" rows="1" role="data">0015198</cell><cell cols="1" rows="1" rend="align=right" role="data">718</cell><cell cols="1" rows="1" role="data">0013928</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">419</cell><cell cols="1" rows="1" role="data">0023866</cell><cell cols="1" rows="1" rend="align=right" role="data">479</cell><cell cols="1" rows="1" role="data">0020877</cell><cell cols="1" rows="1" role="data">539</cell><cell cols="1" rows="1" role="data">0018553</cell><cell cols="1" rows="1" role="data">599</cell><cell cols="1" rows="1" role="data">0016694</cell><cell cols="1" rows="1" role="data">659</cell><cell cols="1" rows="1" role="data">0015175</cell><cell cols="1" rows="1" rend="align=right" role="data">719</cell><cell cols="1" rows="1" role="data">0013908</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">420</cell><cell cols="1" rows="1" role="data">0023810</cell><cell cols="1" rows="1" rend="align=right" role="data">480</cell><cell cols="1" rows="1" role="data">0020833</cell><cell cols="1" rows="1" role="data">540</cell><cell cols="1" rows="1" role="data">0018518</cell><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">0016666</cell><cell cols="1" rows="1" role="data">660</cell><cell cols="1" rows="1" role="data">0015151</cell><cell cols="1" rows="1" rend="align=right" role="data">720</cell><cell cols="1" rows="1" role="data">0013888</cell></row></table><pb n="336"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="italics">Table of</hi> <hi rend="smallcaps">Reciprocals.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Recip.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">721</cell><cell cols="1" rows="1" role="data">0013870</cell><cell cols="1" rows="1" rend="align=right" role="data">768</cell><cell cols="1" rows="1" role="data">0013021</cell><cell cols="1" rows="1" role="data">815</cell><cell cols="1" rows="1" role="data">0012270</cell><cell cols="1" rows="1" role="data">862</cell><cell cols="1" rows="1" role="data">0011601</cell><cell cols="1" rows="1" role="data">909</cell><cell cols="1" rows="1" role="data">0011001</cell><cell cols="1" rows="1" rend="align=right" role="data">956</cell><cell cols="1" rows="1" role="data">0010460</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">722</cell><cell cols="1" rows="1" role="data">0013850</cell><cell cols="1" rows="1" rend="align=right" role="data">769</cell><cell cols="1" rows="1" role="data">0013004</cell><cell cols="1" rows="1" role="data">816</cell><cell 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role="data">0010121</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">754</cell><cell cols="1" rows="1" role="data">0013263</cell><cell cols="1" rows="1" rend="align=right" role="data">801</cell><cell cols="1" rows="1" role="data">0012484</cell><cell cols="1" rows="1" role="data">848</cell><cell cols="1" rows="1" role="data">0011792</cell><cell cols="1" rows="1" role="data">895</cell><cell cols="1" rows="1" role="data">0011173</cell><cell cols="1" rows="1" role="data">942</cell><cell cols="1" rows="1" role="data">0010616</cell><cell cols="1" rows="1" rend="align=right" role="data">989</cell><cell cols="1" rows="1" role="data">0010111</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">755</cell><cell cols="1" rows="1" role="data">0013245</cell><cell cols="1" rows="1" rend="align=right" role="data">802</cell><cell cols="1" rows="1" role="data">0012469</cell><cell cols="1" rows="1" role="data">849</cell><cell cols="1" rows="1" role="data">0011779</cell><cell cols="1" rows="1" role="data">896</cell><cell cols="1" rows="1" role="data">0011161</cell><cell cols="1" rows="1" role="data">943</cell><cell cols="1" rows="1" role="data">0010604</cell><cell cols="1" rows="1" rend="align=right" role="data">990</cell><cell cols="1" rows="1" role="data">0010101</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">756</cell><cell cols="1" rows="1" role="data">0013228</cell><cell cols="1" rows="1" rend="align=right" role="data">803</cell><cell cols="1" rows="1" role="data">0012453</cell><cell cols="1" rows="1" role="data">850</cell><cell cols="1" rows="1" role="data">0011765</cell><cell cols="1" rows="1" role="data">897</cell><cell cols="1" rows="1" role="data">0011148</cell><cell cols="1" rows="1" role="data">944</cell><cell cols="1" rows="1" role="data">0010593</cell><cell cols="1" rows="1" rend="align=right" role="data">991</cell><cell cols="1" rows="1" role="data">0010091</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">757</cell><cell cols="1" rows="1" role="data">0013210</cell><cell cols="1" rows="1" rend="align=right" role="data">804</cell><cell cols="1" rows="1" role="data">0012438</cell><cell cols="1" rows="1" role="data">851</cell><cell cols="1" rows="1" role="data">0011751</cell><cell cols="1" rows="1" role="data">898</cell><cell cols="1" rows="1" role="data">0011136</cell><cell cols="1" rows="1" role="data">945</cell><cell cols="1" rows="1" role="data">0010582</cell><cell cols="1" rows="1" rend="align=right" role="data">992</cell><cell cols="1" rows="1" role="data">0010081</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">758</cell><cell cols="1" rows="1" role="data">0013193</cell><cell cols="1" rows="1" rend="align=right" role="data">805</cell><cell cols="1" rows="1" role="data">0012422</cell><cell cols="1" rows="1" role="data">852</cell><cell cols="1" rows="1" role="data">0011737</cell><cell cols="1" rows="1" role="data">899</cell><cell cols="1" rows="1" role="data">0011123</cell><cell cols="1" rows="1" role="data">946</cell><cell cols="1" rows="1" role="data">0010571</cell><cell cols="1" rows="1" rend="align=right" role="data">993</cell><cell cols="1" rows="1" role="data">0010070</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">759</cell><cell cols="1" rows="1" role="data">0013175</cell><cell cols="1" rows="1" rend="align=right" role="data">806</cell><cell cols="1" rows="1" role="data">0012407</cell><cell cols="1" rows="1" role="data">853</cell><cell cols="1" rows="1" role="data">0011723</cell><cell cols="1" rows="1" role="data">900</cell><cell cols="1" rows="1" role="data">0011111</cell><cell cols="1" rows="1" role="data">947</cell><cell cols="1" rows="1" role="data">0010560</cell><cell cols="1" rows="1" rend="align=right" role="data">994</cell><cell cols="1" rows="1" role="data">0010060</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">760</cell><cell cols="1" rows="1" role="data">0013158</cell><cell cols="1" rows="1" rend="align=right" role="data">807</cell><cell cols="1" rows="1" role="data">0012392</cell><cell cols="1" rows="1" role="data">854</cell><cell cols="1" rows="1" role="data">0011710</cell><cell cols="1" rows="1" role="data">901</cell><cell cols="1" rows="1" role="data">0011099</cell><cell cols="1" rows="1" role="data">948</cell><cell cols="1" rows="1" role="data">0010549</cell><cell cols="1" rows="1" rend="align=right" role="data">995</cell><cell cols="1" rows="1" role="data">0010050</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">761</cell><cell cols="1" rows="1" role="data">0013141</cell><cell cols="1" rows="1" rend="align=right" role="data">808</cell><cell cols="1" rows="1" role="data">0012376</cell><cell cols="1" rows="1" role="data">855</cell><cell cols="1" rows="1" role="data">0011696</cell><cell cols="1" rows="1" role="data">902</cell><cell cols="1" rows="1" role="data">0011086</cell><cell cols="1" rows="1" role="data">949</cell><cell cols="1" rows="1" role="data">0010537</cell><cell cols="1" rows="1" rend="align=right" role="data">996</cell><cell cols="1" rows="1" role="data">0010040</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">762</cell><cell cols="1" rows="1" role="data">0013123</cell><cell cols="1" rows="1" rend="align=right" role="data">809</cell><cell cols="1" rows="1" role="data">0012361</cell><cell cols="1" rows="1" role="data">856</cell><cell cols="1" rows="1" role="data">0011682</cell><cell cols="1" rows="1" role="data">903</cell><cell cols="1" rows="1" role="data">0011074</cell><cell cols="1" rows="1" role="data">950</cell><cell cols="1" rows="1" role="data">0010526</cell><cell cols="1" rows="1" rend="align=right" role="data">997</cell><cell cols="1" rows="1" role="data">0010030</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">763</cell><cell cols="1" rows="1" role="data">0013106</cell><cell cols="1" rows="1" rend="align=right" role="data">810</cell><cell cols="1" rows="1" role="data">0012346</cell><cell cols="1" rows="1" role="data">857</cell><cell cols="1" rows="1" role="data">0011669</cell><cell cols="1" rows="1" role="data">904</cell><cell cols="1" rows="1" role="data">0011062</cell><cell cols="1" rows="1" role="data">951</cell><cell cols="1" rows="1" role="data">0010515</cell><cell cols="1" rows="1" rend="align=right" role="data">998</cell><cell cols="1" rows="1" role="data">0010020</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">764</cell><cell cols="1" rows="1" role="data">0013089</cell><cell cols="1" rows="1" rend="align=right" role="data">811</cell><cell cols="1" rows="1" role="data">0012330</cell><cell cols="1" rows="1" role="data">858</cell><cell cols="1" rows="1" role="data">0011655</cell><cell cols="1" rows="1" role="data">905</cell><cell cols="1" rows="1" role="data">0011050</cell><cell cols="1" rows="1" role="data">952</cell><cell cols="1" rows="1" role="data">0010504</cell><cell cols="1" rows="1" rend="align=right" role="data">999</cell><cell cols="1" rows="1" role="data">0010010</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">765</cell><cell cols="1" rows="1" role="data">0013072</cell><cell cols="1" rows="1" rend="align=right" role="data">812</cell><cell cols="1" rows="1" role="data">0012315</cell><cell cols="1" rows="1" role="data">859</cell><cell cols="1" rows="1" role="data">0011641</cell><cell cols="1" rows="1" role="data">906</cell><cell cols="1" rows="1" role="data">0011038</cell><cell cols="1" rows="1" role="data">953</cell><cell cols="1" rows="1" role="data">0010493</cell><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" role="data">001</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">766</cell><cell cols="1" rows="1" role="data">0013055</cell><cell cols="1" rows="1" rend="align=right" role="data">813</cell><cell cols="1" rows="1" role="data">0012300</cell><cell cols="1" rows="1" role="data">860</cell><cell cols="1" rows="1" role="data">0011628</cell><cell cols="1" rows="1" role="data">907</cell><cell cols="1" rows="1" role="data">0011025</cell><cell cols="1" rows="1" role="data">954</cell><cell cols="1" rows="1" role="data">0010482</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">767</cell><cell cols="1" rows="1" role="data">0013038</cell><cell cols="1" rows="1" rend="align=right" role="data">814</cell><cell cols="1" rows="1" role="data">0012285</cell><cell cols="1" rows="1" role="data">861</cell><cell cols="1" rows="1" role="data">0011614</cell><cell cols="1" rows="1" role="data">908</cell><cell cols="1" rows="1" role="data">0011013</cell><cell cols="1" rows="1" role="data">955</cell><cell cols="1" rows="1" role="data">0010471</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table><cb/></p><p>Of the preceding Table, the use is evidently to shorten arithmetical calculations, and will appear eminently great to those mathematicians and others who are frequently concerned in such kinds of computations. The structure of the Table is evident; the first column contains the natural series of numbers from 1 to 1000, the 2d the Reciprocals. These Reciprocals (which are no other than the decimal values of the quotients resulting from the division of unity or 1 by each of the several numbers from 1 to 1000) are not only useful in shewing by inspection the quotient when the dividend is unity, but are also applied with much advantage in turning many divi- <cb/> sions into multiplications, which are much easier performed, and are done by multiplying the Reciprocal of the divisor (as found in the Table) by the dividend, for the quotient; they will also apply to good purpose in summing the terms of many converging series.</p><p>The Reciprocals are carried on to 7 places of decimals (for the column of Reciprocals must be accounted all decimal figures, although they have not the decimal point placed before them, which is omitted to save room), each being set down to the nearest figure in the last place, that is, when the next figure beyond the last set down in the Table came out a 5 or more, the last <pb n="337"/><cb/> figure was increased by 1, otherwise not; excepting in the repetends which occurred among the Reciprocals, where the real last figure is always set down; the Reciprocals, which in the Table consist of less than seven figures, are those which terminate, and are complete within that number; such as .5 the Reciprocal of 2, .25 the Reciprocal of 4, &c.</p><p><hi rend="smallcaps">Reciprocal</hi> <hi rend="italics">Figures,</hi> in Geometry, are such as have the antecedents and conse- <figure/> quents of the same ratio in both figures. So, in the two rectangles BE and BD, if AB: DC :: BC : AE, then those rectangles are reciprocal figures; and are also equal.</p><p><hi rend="smallcaps">Reciprocal</hi> <hi rend="italics">Proportion,</hi> is when, in four quantities, the two latter terms have the Reciprocal ratio of the two former, or are proportional to the Reciprocals of them. Thus, 24, 15, 5, 8 form a Reciprocal proportion, because .</p><p><hi rend="smallcaps">Reciprocal</hi> <hi rend="italics">Ratio,</hi> of any quantity, is the ratio of the Reciprocal of the quantity.</p><p>RECIPROCALLY. One quantity is Reciprocally as another, when the one is greater in proportion as the other is less; or when the one is proportional to the Reciprocal of the other. So <hi rend="italics">a</hi> is Reciprocally as <hi rend="italics">b,</hi> when <hi rend="italics">a</hi> is always proportional to 1/<hi rend="italics">b.</hi> Like as in the mechanic powers, to perform any effect, the less the power is, the greater must be the time of performing it; or, as it is said, what is gained in power, is lost in time. So that, if <hi rend="italics">p</hi> denote any power or agent, and <hi rend="italics">t</hi> the time of its performing any given service; then <hi rend="italics">p</hi> is as 1/<hi rend="italics">t,</hi> and <hi rend="italics">t</hi> is as 1/<hi rend="italics">p;</hi> that is, <hi rend="italics">p</hi> and <hi rend="italics">t</hi> are Reciprocally proportionals to each other.</p></div1><div1 part="N" n="RECKONING" org="uniform" sample="complete" type="entry"><head>RECKONING</head><p>, in Navigation, is the estimating the quantity of a ship's way; or of the course and distance run. Or, more generally, a ship's Reckoning is that account, by which it may at any time be known where the ship is, and consequently on what course or courses she must steer to gain her intended port. The Reckoning is usually performed by keeping an account of the courses steered, and the distance run, with any accidental circumstances that occur. The courses steered are observed by the compass; and the distances run are estimated from the rate of running, and the time run upon each course. The rate of running is measured by the log, from time to time; which however is liable to great irregularities. Anciently Vitruvius, for measuring the rate of sailing, advised an axis to be passed through the sides of the ship, with two large heads protending out of the ship, including wheels touching the water, by the revolution of which the space passed over in a given time is measured. And the same has been since recommended by Snellius.</p><div2 part="N" n="Reckoning" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Reckoning</hi></head><p>, <hi rend="italics">Dead.</hi> See <hi rend="smallcaps">Dead</hi> <hi rend="italics">Reckoning.</hi></p><p>RECLINATION <hi rend="italics">of a Plane,</hi> in Dialling, is the angular quantity which a dial-plane leans backwards, from an exactly upright or vertical plane, or from the zenith. <cb/></p></div2></div1><div1 part="N" n="RECLINER" org="uniform" sample="complete" type="entry"><head>RECLINER</head><p>, or <hi rend="smallcaps">Reclining</hi> <hi rend="italics">Dial,</hi> is a dial whose plane reclines from the perpendicular, that is, leans backwards, or from you, when you stand before it.</p><div2 part="N" n="Recliner" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Recliner</hi></head><p>, <hi rend="italics">Declining,</hi> or <hi rend="italics">Declining</hi> <hi rend="smallcaps">Reclining</hi> <hi rend="italics">Dial,</hi> is one which neither stands perpendicularly, nor opposite to one of the cardinal points.</p></div2></div1><div1 part="N" n="RECOIL" org="uniform" sample="complete" type="entry"><head>RECOIL</head><p>, or <hi rend="smallcaps">Rebound</hi>, the resilition, or flying backward, of a body, especially a fire-arm. This is the motion by which, upon explosion, it starts or flies backwards; and the cause of it is the resistance of the ball and the impelling force of the powder, which acts equally on the gun and on the ball. It has been commonly said by authors, that the momentum of the ball is equal to that of the gun with its carriage together; but this is a mistake; for the latter momentum is nearly equal to that of the ball and half the weight of the powder together, moving with the velocity of the ball. So that, if the gun, and the ball with half the powder, were of equal weight, the piece would recoil with the same velocity as the ball is discharged. But the heavier any body is, the less will its velocity be, to have the same momentum, or force; and therefore so many times as the cannon and carriage is heavier than the ball and half the powder, just as many times will the velocity of the ball be greater than that of the gun; and in the same ratio nearly is the length of the barrel before the charge, to the quantity the gun Recoils in the time the ball is passing along the bore of the gun. So, if a 24 pounder of 10 feet long be 6400lb weight, and charged with 6lb of powder; then, when the ball quits the piece, the gun will have Recoiled (28/6400) X 10 = 7/160 of a foot, or nearly half an inch.</p></div1><div1 part="N" n="RECORDE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">RECORDE</surname> (<foreName full="yes"><hi rend="smallcaps">Robert</hi></foreName>)</persName></head><p>, a learned physician and mathematician, was born of a good family in Wales, and flourished in the reigns of Henry the 8th, Edward the 6th, and Mary. There is no account of the exact time of his birth, though it must have been early in the 16th century, as he was entered of the university of Oxford about the year 1525, where he was elected fellow of Allsouls college in 1531. Making physic his profession, he went to Cambridge, where he was honoured with the degree of doctor in that faculty, in 1545, and highly esteemed by all that knew him for his great knowledge in several arts and sciences. He afterwards returned to Oxford, where, as he had done before he went to Cambridge, he publicly taught arithmetic, and other branches of the mathematics, with great applause. It seems he afterwards repaired to London, and it has been said he was physician to Edward the 6th and Mary, to which princes he dedicates some of his books; and yet he ended his days in the King's-bench prison, Southwark, where he was confined for debt, in the year 1558, at a very immature age.</p><p>Recorde published several mathematical books, which are mostly in dialogue, between the master and scholar. They are as follow:</p><p>1. <hi rend="italics">The Pathway to Knowledge,</hi> containing the first Principles of Geometrie, as they may moste aptly be applied unto practise, bothe for use of Instrumentes Geometricall and Astronomicall, and also for Projection of Plattes much necessary for all sortes of men. Lond. 4to, 1551.</p><p>2. <hi rend="italics">The Ground of Arts,</hi> teaching the perfect worke and practice of Arithmeticke, both in whole numbers <pb n="338"/><cb/> and fractions, after a more easie and exact forme then in former time hath beene set forth, 8vo, 1552.—This work went through many editions, and was corrected and augmented by several other persons; as first by the famous Dr. John Dee; then by John Mellis, a schoolmaster, 1590; next by Robert Norton; then by Robert Hartwell, practitioner in mathematics, in London; and lastly by R. C. and printed in 8vo, 1623.</p><p>3. <hi rend="italics">The Castle of Knowledge,</hi> containing the Explication of the Sphere bothe Celestiall and Materiall, and divers other things incident thereto. With sundry pleasaunt proofes and certaine newe demonstrations not written before in any vulgare woorkes. Lond. folio, 1556.</p><p>4. <hi rend="italics">The Whetstone of Witte,</hi> which is the seconde part of Arithmetike: containing the Extraction of Rootes: the Cossike Practise, with the rules of Equation: and the woorkes of Surde Nombers. Lond. 4to, 1557.— For an analysis of this work on Algebra, with an account of what is new in it, see pa. 79 of vol. 1, under the article <hi rend="smallcaps">Algebra.</hi></p><p>Wood says he wrote also several pieces on physic, anatomy, politics, and divinity; but I know not whether they were ever published. And Sherburne says that he published <hi rend="italics">Cosmographiæ Isagogen;</hi> also that he wrote a book, <hi rend="italics">De Arte faciendi Horologium;</hi> and another, <hi rend="italics">De Usu Globorum, & de Statu Temporum;</hi> which I have never seen.</p></div1><div1 part="N" n="RECTANGLE" org="uniform" sample="complete" type="entry"><head>RECTANGLE</head><p>, in Geometry, is a right-angled parallelogram, or a right-angled quadrilateral figure.</p><p>If from any point O, lines be drawn to all the four <figure/> angles of a Rectangle; then the sum of the squares of the lines drawn to the opposite corners will be equal, in whatever part of the plane the point O is situated; viz, . For other properties of the Rectangle, see <hi rend="smallcaps">Parallelogram;</hi> for the Rectangle being a species of the parallelogram, whatever properties belong to the latter, must equally hold in the former.</p><p><hi rend="italics">For the Area of a</hi> <hi rend="smallcaps">Rectangle.</hi> Multiply the length by the breadth or <figure/> height.—<hi rend="italics">Otherwise;</hi> Multiply the product of the two diagonals by half the sine of their angle at the intersection.</p><p>That is, AB X AC, or AD X BC X 1/2 sin. [angle]P = area. A Rectangle, as of two lines AB and AC, is thus denoted, AB X AC, or AB . AC; or else thus expressed, the Rectangle of, or under, AB and AC.</p><div2 part="N" n="Rectangle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Rectangle</hi></head><p>, in Arithmetic, is the same with product or factum. So the Rectangle of 3 and 4, is 3 X 4 or 12; and of <hi rend="italics">a</hi> and <hi rend="italics">b</hi> is <hi rend="italics">a</hi> X <hi rend="italics">b</hi> or <hi rend="italics">ab.</hi></p></div2></div1><div1 part="N" n="RECTANGLED" org="uniform" sample="complete" type="entry"><head>RECTANGLED</head><p>, <hi rend="smallcaps">Right-angled</hi>, or R<hi rend="smallcaps">ECTANGULAR</hi>, is applied to figures and solids that have at least one right angle, if not more. So a Right-angled triangle, has one right angle: a Right-angled parallelogram <cb/> is a rectangle, and has four right angles. Such also are squares, cubes, parallelopipedons.</p><p>Solids are also said to be Rectangular with respect to their situation, viz, when their axis is perpendicular to their base; as right cones, pyramids, cylinders, &c.</p><p>The Ancients used the phrase <hi rend="italics">Rectangular section of a cone,</hi> to denote a parabola; that conic section, before Apollonius, being only considered in a cone having its vertex a right-angle. And hence it was, that Archimedes entitled his book of the quadrature of the parabola, by the name of <hi rend="italics">Rectanguli Coni Sectio.</hi></p></div1><div1 part="N" n="RECTIFICATION" org="uniform" sample="complete" type="entry"><head>RECTIFICATION</head><p>, in Geometry, is the finding of a right line equal to a curve. The Rectification of curves is a branch of the higher geometry, a branch in which the use of the inverse method of Fluxions is especially useful. This is a problem to which all mathematicians, both ancient and modern, have paid the greatest attention, and particularly as to the Rectification of the circle, or finding the length of the circumference, or a right line equal to it; but hitherto without the perfect effect: upon this also depends the quadrature of the circle, since it is demonstrated that the area of a circle is equal to a right angled triangle, of which one of the sides about the right angle is the radius, and the other equal to the circumference: but it is much to be feared that neither the one nor the other will ever be accomplished. Innumerable approximations however have been made, from Archimedes, down to the mathematicians of the present day. See <hi rend="smallcaps">Circle</hi> and <hi rend="smallcaps">Circumference.</hi></p><p>The first person who gave the Rectification of any curve, was Mr. Neal, son of Sir Paul Neal, as we find at the end of Dr. Wallis's treatise on the Cissoid; where he says, that Mr. Neal's Rectification of the curve of the semicubical parabola, was published in July or August, 1657. Two years after, viz in 1659, Van Haureat, in Holland, also gave the Rectification of the same curve; as may be seen in Schooten's Commentary on Des Cartes's Geometry.</p><p>The most comprehensive method of Rectification of curves, is by the inverse method of fluxions, which is thus: Let AC<hi rend="italics">c</hi> be any curve line, AB an absciss, and <figure/> BC a perpendicular ordinate; also <hi rend="italics">bc</hi> another ordinate indefinitely near to BC; and C<hi rend="italics">d</hi> drawn parallel to the absciss AB. Put the absciss AB = <hi rend="italics">x,</hi> the ordinate BC = <hi rend="italics">y,</hi> and the curve AC = <hi rend="italics">z:</hi> then is C<hi rend="italics">d</hi> = B<hi rend="italics">b</hi> = <hi rend="italics">x</hi><hi rend="sup">.</hi> the fluxion of the absciss AB, and <hi rend="italics">cd</hi> = <hi rend="italics">y</hi><hi rend="sup">.</hi> the fluxion of the ordinate BC, also C<hi rend="italics">c</hi> = <hi rend="italics">z</hi><hi rend="sup">.</hi> the fluxion of the curve AB. Hence because C<hi rend="italics">cd</hi> may be considered as a plane right-angled triangle, , or ; and therefore ; which is the fluxion of the length of any curve; and consequently, out of this equation expelling either <hi rend="italics">x</hi><hi rend="sup">.</hi> or <hi rend="italics">y</hi><hi rend="sup">.</hi>, by means of the particular equation expressing the nature of the curve in question, the fluents of the resulting equation, being then taken, will give the length of the curve, in finite terms when it is <pb n="339"/><cb/> rectifiable, otherwise in an infinite series, or in a logarithmic or exponential &c expression, or by means of some other curve, &c.</p><p>Ex. 1. <hi rend="italics">To rectify the common parabola.</hi>—In this case, the equation of the curve is , where <hi rend="italics">a</hi> is half the parameter. The fluxion of this equation is , and hence ; this being substituted in the general equation , it becomes ; the correct fluents of which give , which is the length of the curve AC, when it is a parabola.</p><p>And the same might be expressed by an infinite series, by expanding the quantity √(<hi rend="italics">aa</hi> + <hi rend="italics">yy</hi>). See my Mensuration, pa. 361, 2d edit.</p><p>Ex. 2. <hi rend="italics">To rectify the Circle.</hi>—The equation of the circle may be expressed either in terms of the sine, or versed sine, or tangent, or secant, &c, and the radius. Let therefore the radius of the circle be DA or DC = <hi rend="italics">r,</hi> the versed sine AB = <hi rend="italics">x,</hi> the right sine BC = <hi rend="italics">y,</hi> the tangent CE = <hi rend="italics">t,</hi> and the secant DE = <hi rend="italics">s;</hi> then, by the nature of the circle, we have these equations, ; and by means of the fluxions of these equations, with the general equation , are obtained the following fluxional forms for the fluxion of the curve, the fluent of any one of which will be the curve itself, viz, . Hence the value of the curve, from the fluent of each of these, gives the four following forms, in series, viz, the curve, putting <hi rend="italics">d</hi> = 2<hi rend="italics">r</hi> the diameter, is .</p><p>See my Mensur. 2d edit. pa. 118 &c, also most treatises on Fluxions.</p><p>It is evident that the simplest of these series is the third, or that which is expressed in terms of the tangent. It will therefore be the properest form to calculate an example by in numbers. And for this purpose it will be convenient to assume some arc whose tangent, or at least its square, is known to be some small finite number. Now the arc of 45° it is known has its tangent equal to the radius; and therefore, taking the radius <hi rend="italics">r</hi> = 1, and consequently the tangent of 45° or <hi rend="italics">t</hi> = 1 also, in this case the arc of 45° to the radius 1, <cb/> or the quadrant to the diameter 1, will be = 1 - 1/3 + 1/5 - 1/7 + 1/9 &c. But as this series converges very slowly, some smaller arch must be taken, that the series may converge faster; such as the arc of 30°, whose tangent is = √(1/3) = .5773502, or its square ; and hence, after the first term, the succeeding terms will be found by dividing always by 3, and these quotients divided by the absolute numbers 3, 5, 7, 9, &c; and lastly adding every other term together into two sums, the one the sum of the positive terms, and the other the sum of the negative ones, then lastly the one sum taken from the other leaves the length of the arc of 30°, which is the 12th part of the whole circumference when the radius is 1, or the 6th part when the diameter is 1, and consequently 6 times that arc will be the length of the whole circumference to the diameter 1; therefore multiply the 1st term √(1/3) by 6, and the product is √(36/3) or √12 = 3.4641016; hence the operation will be conveniently made as follows: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ Terms.</cell><cell cols="1" rows="1" role="data">- Terms.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1)</cell><cell cols="1" rows="1" role="data">3.4641016</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">3.4641016</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3)</cell><cell cols="1" rows="1" role="data">1.1547005</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.3849002</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5)</cell><cell cols="1" rows="1" role="data">3849002</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">769800</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7)</cell><cell cols="1" rows="1" role="data">1283001</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">183286</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9)</cell><cell cols="1" rows="1" role="data">427667</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">47519</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11)</cell><cell cols="1" rows="1" role="data">142556</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12960</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13)</cell><cell cols="1" rows="1" role="data">47519</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">3655</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15)</cell><cell cols="1" rows="1" role="data">15840</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1056</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17)</cell><cell cols="1" rows="1" role="data">5280</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">311</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19)</cell><cell cols="1" rows="1" role="data">1760</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">93</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21)</cell><cell cols="1" rows="1" role="data">587</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23)</cell><cell cols="1" rows="1" role="data">196</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25)</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27)</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">(</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><*></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right colspan=2" role="data">+ 3.5462332</cell><cell cols="1" rows="1" rend="align=right" role="data">-0.4046406</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right colspan=2" role="data">- 0.4046406</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3.1415926</cell><cell cols="1" rows="1" role="data">the circumference.</cell></row></table> Various other series for the Rectification of the circle may be seen in different parts of my Mensuration, as at pa. 121, 122, 137, 138, 422, &c. See also my paper on this subject in the Philos. Trans. vol. 66, pa. 476.</p></div1><div1 part="N" n="RECTIFIER" org="uniform" sample="complete" type="entry"><head>RECTIFIER</head><p>, in Navigation, is an instrument used for determining the variation of the compass, in order to rectify the ship's course. It consists of two circles, either laid upon, or let into one another, and so fastened together in their centres that they represent two compasses, the one fixed, and the other moveable. Each is divided into 32 points of the compass, and 360°, and numbered both ways, from the north and the south, ending at the east and west in 90°. The fixed compass represents the horizon, in which the north, and all the other points, are liable to variation. In the centre of <pb n="340"/><cb/> the moveable compass is fastened a silk thread, long enough to reach the outside of the fixed compass: but when the instrument is made of wood, an index is used instead of the thread.</p><p>RECTIFYING <hi rend="italics">of Curves.</hi> See R<hi rend="smallcaps">ECTIFICATION.</hi></p><p><hi rend="smallcaps">Rectifying</hi> <hi rend="italics">of the Globe or Sphere,</hi> is a previous adjustment of it, to prepare it for the solution of problems. This usually consists in placing it in the same position as the true sphere of the world has at some certain time proposed; which is done first by elevating the pole above the horizon as much as the latitude of the place is, then bringing the sun's place for the given day, found in the ecliptic, to the graduated side of the brass or general meridian, next move the hour-index to the upper hour of 12, so shall the globe be Rectified for noon of that day; and if the globe be turned about till the hour-index point at any proposed hour, then is the globe in the real position of the earth at that time, if the whole globe be set in the north and south position by means of the compass.</p></div1><div1 part="N" n="RECTILINEAL" org="uniform" sample="complete" type="entry"><head>RECTILINEAL</head><p>, <hi rend="smallcaps">Rectilinear</hi>, or <hi rend="italics">Right-lined,</hi> is the quality or nature of figures that are bounded by right lines, or formed by right lines.</p><p>RECURRING <hi rend="italics">Series,</hi> is a series constituted in such a manner, that having taken at pleasure any number of its terms, each following term shall be related to the same number of preceding terms according to a constant law of relation. See <hi rend="italics">Recurring</hi> <hi rend="smallcaps">Series.</hi></p></div1><div1 part="N" n="RED" org="uniform" sample="complete" type="entry"><head>RED</head><p>, in Physics, or Optics, one of the simple or primary colours of natural bodies, or rather of the rays of light.—The Red rays are the least refrangible of all the rays of light. And hence, as Newton supposes the different degrees of refrangibility to arise from the different magnitudes of the luminous particles of which the rays consist; therefore the Red rays, or Red light, is concluded to be that which consists of the largest particles. See <hi rend="smallcaps">Colour</hi> and <hi rend="smallcaps">Light.</hi></p><p>Authors distinguish three general kinds of Red: one bordering on the blue, as colombine, or dove-colour, purple, and crimson; another bordering on yellow, as flame-colour and orange; and between these extremes is a medium, which is that which is properly called Red.</p></div1><div1 part="N" n="REDANS" org="uniform" sample="complete" type="entry"><head>REDANS</head><p>, or <hi rend="smallcaps">Redant</hi>, or <hi rend="smallcaps">Redens</hi>, in Fortification, is a kind of work indented like the teeth of a saw, with salient and re-entering angles; to the end that one part may flank or defend another. It is called also <hi rend="italics">saw work,</hi> and <hi rend="italics">indented work.</hi></p><p>Redans are often used in fortifying of walls, where it is not necessary to be at the expence of building bastions; as when they stand on the side of a river, or a marsh, or the sea, &c. But the fault of such fortification is, that the besiegers from one battery may ruin both the sides of the tenaille or front of a place, and make an assault without fear of being ensiladed, since the defences are ruined.</p><p>The parapet of the corridor also is frequently Redented, or carried on by the way of Redans.</p></div1><div1 part="N" n="REDINTEGRATION" org="uniform" sample="complete" type="entry"><head>REDINTEGRATION</head><p>, is the taking or finding the integral or fluent again, from the fluxion. See <hi rend="smallcaps">Fluxion</hi> and <hi rend="smallcaps">Fluent.</hi></p></div1><div1 part="N" n="REDOUBT" org="uniform" sample="complete" type="entry"><head>REDOUBT</head><p>, or <hi rend="smallcaps">Redoute</hi>, in Fortification, a small fort, without any defence but in front, used in trenches, <cb/> lines of circumvallation, contravallation, and approach, as also for the lodging of corps de garde, and to defend passages.</p><p><hi rend="italics">A Detached</hi> <hi rend="smallcaps">Redoubt</hi>, is a kind of work resembling a ravelin, with flanks, placed beyond the glacis.—It is made to occupy some spot of ground which might be advantageous to the besiegers; and also to oblige the enemy to open his trenches farther off than he would otherwise do.</p><p>REDUCING <hi rend="italics">Scale,</hi> or <hi rend="smallcaps">Surveying</hi> <hi rend="italics">Scale,</hi> is a broad, thin slip of box, or ivory, having several lines and scales of equal parts upon it; used by surveyors for turning chains and links into roods and acres, by inspection. They use it also to reduce maps and draughts from one dimension to another.</p></div1><div1 part="N" n="REDUCTION" org="uniform" sample="complete" type="entry"><head>REDUCTION</head><p>, in general, is the bringing or changing some thing to a different form, state, or denomination.</p><div2 part="N" n="Reduction" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Reduction</hi>, in <hi rend="italics">Arithmetic</hi></head><p>, is commonly understood of the changing of money, weights, or measures, to other denominations, of the same value; and it is of two kinds, <hi rend="italics">Reduction Descending,</hi> which is the changing a number to its equivalent value in a lower denomination; as pounds into shillings or pence: and <hi rend="italics">Reduction Ascending,</hi> which is the changing numbers to higher denominations; as pence to shillings or pounds.</p><p><hi rend="smallcaps">Rule.</hi> <hi rend="italics">To perform Reduction;</hi> consider how many of the less denomination make one of the greater, as how many pence make a shilling, or how many shillings make a pound; and multiply by that number when the Reduction is descending, but divide by it when it is ascending. So to reduce 23l. into pence; and conversely those pence into pounds; multiply or divide by 12 and 20, as here below. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data"><hi rend="italics">l.</hi></cell><cell cols="1" rows="1" role="data">12 )</cell><cell cols="1" rows="1" rend="align=right" role="data">5520</cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">460</cell><cell cols="1" rows="1" role="data"><hi rend="italics">sh.</hi></cell><cell cols="1" rows="1" role="data">20 )</cell><cell cols="1" rows="1" rend="align=right" role="data">460</cell><cell cols="1" rows="1" role="data"><hi rend="italics">sh.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data"><hi rend="italics">l.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5520</cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p><hi rend="smallcaps">Reduction</hi> <hi rend="italics">of Fractions.</hi> See <hi rend="smallcaps">Fraction</hi>, and D<hi rend="smallcaps">ECIMAL.</hi></p><p><hi rend="smallcaps">Reduction</hi> <hi rend="italics">of Equations,</hi> in Algebra. See E<hi rend="smallcaps">QUATION.</hi></p><p><hi rend="smallcaps">Reduction</hi> <hi rend="italics">os Curves.</hi> See <hi rend="smallcaps">Curve.</hi></p><p><hi rend="smallcaps">Reduction</hi> <hi rend="italics">of a Figure, Design, or Draught,</hi> is the making a copy of it, either larger or smaller than the original, but still preserving the form and proportion.</p><p>Figures and plans are reduced, and copied, in various ways; as by the Pentagraph, and Proportional compasses. See <hi rend="smallcaps">Pentagraph</hi>, and <hi rend="italics">Proportional</hi> C<hi rend="smallcaps">OMPASSES.</hi> The best of the other methods of reducing are as below. <hi rend="center"><hi rend="italics">To reduce a Simple Rectilinear Figure by Lines.</hi></hi></p><p>Pitch upon a point P any where about the given figure ABCDE, either within it, or without it, or in one side or angle; but near the middle is best. From that point P draw lines through all the angles; upon one <pb n="341"/><cb/> of which take P<hi rend="italics">a</hi> to PA in the proposed proportion of the scales, or linear dimensions; then draw <hi rend="italics">ab</hi> parallel to AB, <hi rend="italics">bc</hi> to BC, &c; so shall <hi rend="italics">abcde</hi> be the reduced figure sought, either greater or smaller than the original. <figure/></p><p><hi rend="italics">To Reduce a Figure by a Scale.</hi>—Measure all the sides, and diagonals, of the figure, as ABCDE, by a scale; and lay down the same measures respectively, from another scale, in the proportion required.</p><p><hi rend="italics">To Reduce a Map, Design, or Figure, by Squares.</hi>— Divide the original into a number of little squares; and divide a fresh paper, of the dimensions required, into the same number of other squares, either greater or smaller as required. This done, in every square of the second figure, draw what is found in the corresponding square of the first or original figure.</p><p>The cross lines forming these squares, may be drawn with a pencil, and these rubbed out again after the work is finished. But a more ready and convenient way, especially when such Reductions are often wanted, would be to keep always at hand frames of squares ready made, of several sizes; for by only just laying them down upon the papers, the corresponding parts may be readily copied. These frames may be made of four stiff or inflexible bars, strung across with horse hairs, or fine catgut.</p><p><hi rend="smallcaps">Reduction</hi> <hi rend="italics">to the Ecliptic,</hi> in Astronomy, is the difference between the argument <figure/> of latitude, as NP, and an arc of the ecliptic NR, intercepted between the place of a planet, and the node.—To find this Reduction, or difference; in the right-angled spherical triangle NPR, are given the angle of inclination, and the argument of latitude NP; to find NR; then the difference between NP and NR is the Reduction sought.</p><p>REDUNDANT <hi rend="italics">Hyperbola,</hi> is a curve of the higher kind, so called because it exceeds the conical hyperbola in the number of legs; being a triple hyperbola, with 6 hyperbolic legs. See Newton's Enum. Lin. tertii Ordinis, nomina formarum, &c.</p><p>RE-ENTERING <hi rend="italics">Angle,</hi> in Fortification, is an angle whose point is turned inwards, or towards the place.</p><p>REFLECTED <hi rend="italics">Ray,</hi> or <hi rend="italics">Vision,</hi> is that which is made by the reflection of light, or by light first re- <cb/> ceived upon the surface of some body, and thence reflected again. See <hi rend="smallcaps">Ray, Vision</hi>, and R<hi rend="smallcaps">EFLECTION.</hi></p></div2></div1><div1 part="N" n="REFLECTING" org="uniform" sample="complete" type="entry"><head>REFLECTING</head><p>, or <hi rend="smallcaps">Reflexive</hi>, <hi rend="italics">Dial,</hi> is a kind of dial which shews the hour by means of a thin piece of looking-glass plate, duly placed to throw the sun's rays to the top of a cieling, on which the hour-lines are drawn.</p></div1><div1 part="N" n="REFLECTION" org="uniform" sample="complete" type="entry"><head>REFLECTION</head><p>, or <hi rend="smallcaps">Reflexion</hi>, in Mechanics, is the return, or regressive motion of a moveable body, occasioned by the resistance of another body, which hinders it from pursuing its former course of direction.</p><p>Reflection is conceived, by the latest and best authors, as a motion peculiar to elastic bodies, by which, after striking on others which they cannot remove, they recede, or turn back, or aside, by their elastic power.</p><p>On this principle it is asserted, that there may be, and is, a period of rest between the incidence and the reflection; since the reflected motion is not a continuation of the other, but a new motion, arising from a new cause or principle, viz, the power of elasticity.</p><p>It is one of the great laws of Reflection, that the angle of incidence is equal to the angle of Reflection; i. e. that the angle which the direction of motion of a striking body makes with the surface of the body struck, is equal to the angle made between the same surface and the direction of motion after the stroke. See <hi rend="smallcaps">Incidence</hi> and <hi rend="smallcaps">Percussion.</hi></p><p><hi rend="smallcaps">Reflection</hi> <hi rend="italics">of the Rays of Light,</hi> like that of other bodies, is their motion after being reflected from the surfaces of bodies.</p><p>The Reflection of the rays of light from the surfaces of bodies, is the means by which those bodies become visible. And the disposition of bodies to reflect this or that kind of rays most copiously, is the cause of their being of this or that colour. Also, the Reflection of light, from the surfaces of mirrors, makes the subject of catoptrics.</p><p>The Reflection of light, Newton has shewn, is not effected by the rays striking on the very parts of the bodies; but by some power of the body equally diffused throughout its whole surface, by which it acts upon the ray, attracting or repelling it without any real immediate contact. This power he also shews is the same by which, in other circumstances, the rays are refracted; and by which they are at first emitted from the lucid body.</p><p>Dr. Priestley says, it is not more probable, that the rays of light are transmitted from the sun, with an uniform disposition to be reflected or refracted, according to the circumstances of the bodies on which they impinge; and that the transmission of some of the rays, apparently under the same circumstances, with others that are reflected, is owing to the minute vibrations of the small parts of the surfaces of the mediums through which the rays pass; vibrations that are independent of action and reaction between the bodies and the particles of light at the time of their impinging, though probably excited by the action of preceding rays. Hist. of Light and Colours, pa. 309.</p><p>Newton concludes his account of the Reflection of light with observing, that if light be reflected not by impinging on the solid parts of bodies, but by some other principle, it is probable that as many of its <pb n="342"/><cb/> rays as impinge on the solid parts of bodies are not reflected, but stifled and lost in the bodies. Otherwise, he says, we must suppose two kinds of Reflection; for should all the rays be reflected which impinge on the internal parts of clear water or crystal, those substances would rather have a cloudy colour, than a clear transparency. To make bodies look black, it is necessary that many rays be stopped, retained and lost in them; and it does not seem probable that any rays can be stopped and stifled in them, which do not impinge on their parts: and hence, he says, we may understand, that bodies are much more rare and porous than is commonly believed. However, M. Bouguer disputes the fact of light being stifled or lost by impinging on the solid parts of bodies.</p><div2 part="N" n="Reflection" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Reflection</hi></head><p>, in Catoptrics, is the return of a ray of light from the polished surface of a speculum or mirror, as driven thence by some power residing in it.</p><p>The ray thus returned is called a <hi rend="italics">reflex</hi> or <hi rend="italics">reflected ray,</hi> or a <hi rend="italics">ray of Reflection;</hi> and the point of the speculum where the ray commences, is <figure/> called the <hi rend="italics">point of Reflection.</hi> Thus, the ray AB, proceeding from the radiant A, and striking on the point of the speculum B, being returned thence to C, BC represents the reflected ray, and B the point of Reflection; in respect of which, AB represents the incident ray, or ray of incidence, and B the point of incidence; also the angle CBE is the angle of Reflection, and ABD the angle of incidence; where DE is the reflecting surface, or at least a tangent to it at the point B. Though some count the angle of incidence and of Reflection from the perpendicular BF.</p><p><hi rend="italics">General Laws of</hi> <hi rend="smallcaps">Reflection.</hi>—1. <hi rend="italics">When a ray of light is reflected from a speculum of any form, the angle of incidence is always equal to the angle of Reflection.</hi> This law obtains in the percussions of all kinds of bodies; and consequently must do so in those of light; and the proof of it may be seen at the article I<hi rend="smallcaps">NCIDENCE.</hi></p><p>This law is confirmed also by experiments on all bodies; and on the rays of light in this manner: A ray from the sun falling on a mirror, in a dark room, through a small hole, you will have the pleasure to see it rebound, so as to make the angle of Reflection equal to the angle of incidence. And the same may be shewn in various other ways; thus ex. gr. placing a semicircle DFE on a mirror DE, its centre on B, and its limb or plane perpendicular to the speculum; and assuming equal arcs DG and EH; place an object in A, and the eye in C: then will the object be seen by a ray reflected from the point B. But by covering B, the object will cease to be seen.</p><p>II. <hi rend="italics">Every point of a speculum reflects rays falling on it, from every part of an object.</hi></p><p>III. <hi rend="italics">If the eye C and the radiant point A change places, the point will continue to radiate upon the eye, in the Jame course or path as before.</hi></p><p>IV. <hi rend="italics">The plane of Reflection is perpendicular to the surface of the speculum; and it passes through the centre in spherical specula.</hi></p><p><hi rend="smallcaps">Reflection</hi> <hi rend="italics">of the Moon,</hi> is a term used by some <cb/> authors for what is otherwise called <hi rend="italics">her variation;</hi> being the 3d inequality in her motion, by which her true place out of the quadratures differs from her place twice equated.</p><p><hi rend="smallcaps">Reflection</hi> is also used in the Copernican system, for the distance of the pole from the horizon of the disc; which is the same thing as the sun's declination in the Ptolomaic system.</p><p>REFLECTOIRE <hi rend="smallcaps">Curve.</hi> See <hi rend="italics">Reflectoire</hi> <hi rend="smallcaps">Curve.</hi></p><p>REFLEXIBILITY <hi rend="italics">of the rays of light,</hi> is that property by which they are disposed to be reflected. Or, it is their disposition to be turned back into the same medium, from any other medium on whose surface they fall. Hence those rays are said to be more or less reflexible, which are returned back more or less easily under the same incidence. Thus, if light pass out of glass into air, and by being inclined more and more to the common surface of the glass and air, begins at length to be totally reflected by that surface, those sorts of rays which at like incidences are reflected most copiously, or the rays which by being inclined begin soonest to be totally reflected, are the most reflexible rays.</p><p>That rays of light are of different colours, and endued with different degrees of reflexibility, was first discovered by Sir I. Newton; and it is shewn by the following experiment. Applying a prism DFE to <figure/> the aperture C of a darkened room in such manner that the light be reflected from the base in G; the violet rays are seen first reflected into HG; the other rays continuing still refracted to I and K. After the violet, the blue are all reflected; then the green, &c.—Hence it appears, that the differently coloured rays differ in degree of Reflexibility. And from other experiments it appears that those rays which are most reflexible, are also most refrangible.</p><p>REFLUX <hi rend="italics">of the Sea,</hi> is the ebbing of the water, or its return from the shore; being so called, because it is the opposite motion to the flood or flux. See <hi rend="smallcaps">Tide.</hi></p><p>REFRACTED <hi rend="italics">Angle,</hi> or <hi rend="italics">Angle of Refraction,</hi> in Optics, is the angle which the refracted ray makes with the refracting surface; or sometimes it denotes the complement of that, or the angle it makes with the perpendicular to the said surface.</p><p><hi rend="smallcaps">Refracted</hi> <hi rend="italics">Dials,</hi> or <hi rend="italics">Refracting Dials,</hi> are such as shew the hour by means of some refracting transparent fluid.</p><p><hi rend="smallcaps">Refracted</hi> <hi rend="italics">Ray,</hi> or <hi rend="italics">Ray of</hi> <hi rend="smallcaps">Refraction</hi>, is a <pb n="343"/><cb/> ray after it is broken or bent, at the common surface of two different mediums, where it passes from the one into the other. See <hi rend="smallcaps">Ray</hi>, and R<hi rend="smallcaps">EFRACTION.</hi></p></div2></div1><div1 part="N" n="REFRACTION" org="uniform" sample="complete" type="entry"><head>REFRACTION</head><p>, in Mechanics, is the deviation of a moving body from its direct course, by reason of the different density of the medium it moves in; or a flexion and change of determination, occasioned by a body's passing obliquely out of one medium into another of a different density. <figure/></p><p>Thus a ball A, moving in the air in the line AB, and falling obliquely on the surface of the water CD, does not proceed straight in the same direction, as to E, but deviates or is deflected to F. Again, if the ball move in water in the line AB, and fall obliquely on a surface of air CD; it will in this case also deviate from the same continued direction BE, but now the contrary way, and will go to G, on the other side of it. Now the deflection in either case is called the <hi rend="italics">Refraction,</hi> the Refraction being towards the denser surface BD in the former case, but from it in the latter.</p><p>These Refractions are supposed to arise from hence; that the ball arriving at B, in the first case finds more resistance or opposition on the one side O, or from the side of the water, than it did from the side P, or that of the air; and in the latter more resistance from the side P, which is now the side of the water, than the side O, which is that of the air. And so for any other different media: a visible instance of which is often perceived in the falling of shot or shells into the earth, as clay &c, when the perforation is found to rise a little upwards, toward the surface. However another reason is assigned for the Refraction of the rays of light, whose Refractions lie the contrary way to those above, as will be seen in what follows, viz, that water by its greater attraction accelerates the motion of the rays of light more than air does.</p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">of Light,</hi> in Optics, is an inflection or deviation of the rays from their rectilinear course on passing obliquely out of one medium into another, of a different density.</p><p>That a body may be refracted, it is necessary that it should fall obliquely on the second medium: in perpendicular incidence there is no Refraction. Yet Voscius and Snellius imagined they had observed a perpendicular ray of light undergo a Refraction; a perpendicular object appearing in the water nearer than it really was: but this was attributing that to a Refraction of the perpendicular rays, which was owing to <cb/> the divergency of the oblique rays after refraction, from a nearer point. Yet there is a manifest Refraction even of perpendicular rays found in island crystal.</p><p>Rohault adds, that though an oblique incidence be necessary in all other mediums we know of, yet the obliquity must not exceed a certain degree; if it do, the body will not penetrate the medium, but will be reflected instead of being refracted. Thus, cannon-balls, in sea engagements, falling very obliquely on the surface of the water, are observed to bound or rise from it, and to sweep the men from off the enemy's decks. And the same thing happens to the little stones with which children make their ducks and drakes along the surface of the water.</p><p>The ancients confounded Refraction with Reflection; and it was Newton who first taught the true difference between them. He shews however that there is a good deal of analogy between them, and particularly in the case of light.</p><p>The laws of Refraction of the rays of light in mediums differently terminated, i. e. whose surfaces are plane, concave, and convex, make the subject of Dioptrics.—By Refraction it is, that convex glasses, or lenses, collect the rays, magnify objects, burn, &c; and hence the foundation of microscopes, telescopes, &c.—And by Refraction it is, that all remote objects are seen out of their real places; particularly, that the heavenly bodies are apparently higher than they are in reality. The Refraction of the air has many times so uncertain an influence on the places of celestial objects, near the horizon, that wherever Refraction is concerned, the conclusions deduced from observations that are much affected by it, will always remain doubtful, and sometimes too precarious to be relied on. See Dr. Bradley in Philos. Trans. number 485.</p><p>As to the cause of Refraction, it does not appear that any person before Des Cartes attempted to explain it; this he undertook to do by the resolution of forces, on the principles of mechanics; in consequence of which, he was obliged to suppose that light passes with more ease through a dense medium than a rare one: thus, the ray AC falling obliquely on a denser medium at C is supposed to be acted on by two forces, one of them impelling it in the direction AL, and the other in AK, which alone can be affected by the change of medium: and since, after the ray has entered the denser medium, it approaches the perpendicular CI, it is plain that this force must have received an increase, whilst the other continued the same.</p><p>The first person who questioned the truth of this explanation of the cause of Refraction, was Fermat; he asserted, contrary to Des Cartes, that light suffers greater resistance in water than in air, and greater in glass than in water; and h<*> maintained that the resistance of different mediums, with respect to light, is in proportion to their densities. Leibnitz also adopted the same general idea; and they reasoned upon the subject in the following manner. Nature, say they, accomplishes her ends by the shortest methods; and therefore light ought to pass from one point to another, either by the shortest course, or by that in which the least time is required. But it is plain that the path in which light passes, when it falls obliquely upon a den- <pb n="344"/><cb/> ser medium, is not the most direct or the shortest; and therefore it must be that in which the least time is spent. And whereas it is demonstrable, that light falling obliquely upon a denser medium (in order to take up the least time possible, in passing from a point in one medium to a point in the other) must be refracted in such a manner, that the sine of the angles of incidence and Refraction must be to one another, as the different facilities with which light is transmitted in those mediums; it follows that, since light approaches the perpendicular when it passes obliquely from air into water, the facility with which water suffers light to pass through it, is less than that of the air; so that the light meets with greater resistance in water than in air.</p><p>This method of arguing from final causes could not satisfy philosophers. Dr. Smith observes, that it agrees only to the case of Refraction at a plane surface; and that the hypothesis is altogether arbitrary.</p><p>Dechales, in explaining the law of Refraction, supposes that every ray of light is composed of several smaller rays, which adhere to one another; and that they are refracted towards the perpendicular, in passing into a denser medium, because one part of the ray meets with more resistance than another part; so that the former traverses a smaller space than the latter; in consequence of which the ray must necessarily bend a little towards the perpendicular. This hypothesis was adopted by the celebrated Dr. Barrow, and indeed some say, he was the author of it. On this hypothesis it is plain, that mediums of a greater refractive power, must give a greater resistance to the passage of the rays of light, than mediums of a less refractive power; which is contrary to fact.</p><p>The Bernoullis, both father and son, have attempted to explain the cause of Refraction on mechanical principles; the former on the equilibrium of forces, and the latter on the same principles with the supposition of etherial vortices: but neither of these hypotheses have gained much credit.</p><p>M. Mairan supposes a subtle fluid, filling the pores of all bodies, and extending, like an atmosphere, to a small distance beyond their surfaces; and then he supposes that the Refraction of light is nothing more than a necessary and mechanical effect of the incidence of a small body in those circumstances. There is more, he says, of the refracting fluid, in water than in air, more in glass than in water, and in general more in a dense medium than in one that is rarer.</p><p>Maupertuis supposes that the course which every ray takes, in passing out of one medium into another, is that which requires the least quantity of action, which depends upon the velocity of the body and the space it passes over; so that it is in proportion to the sum of the products arising from the spaces multiplied by the velocities with which bodies pass over them. From this principle he deduces the necessity of the sine of the angle of incidence being in a constant proportion to that of Refraction; and also all the other laws relating to the propagation and reflection of light.</p><p>Dr. Smith (in his Optics, Remarks, p. 70) observes, that all other theories for explaining the reflection and Refraction of light, except that of Newton, suppose that it strikes upon bodies and is resisted by <cb/> them; which has never been proved by any deduction from experience. On the contrary, it appears by various considerations, and might be shewn by the observations of Mr. Molyneux and Dr. Bradley on the parallax of the sixed stars, that their rays are not at all impeded by the rapid motion of the earth's atmosphere, nor by the object glass of the telescope, through which they pass. And by Newton's theory of Refraction, which is grounded on experience only, it appears that light is so far from being resisted and retarded by Refraction into any dense medium, that it is swifter there than in vacuo in the ratio of the sine of incidence in vacuo to the sine of Refraction into the dense medium. Priestley's Hist. of Light, &c, p. 102 and 333.</p><p>Newton shews that the Refraction of light is not performed by the rays salling on the very surface of bodies; but that it is effected, without any contact, by the action of some power belonging to bodies, and extending to a certain distance beyond their surfaces; by which same power, acting in other circumstances, they are also emitted and reflected.</p><p>The manner in which Refraction is performed by mere attraction, without contact, may be thus accounted for: Suppose HI the boundary of two mediums, N and O; the first <figure/> the rarer, ex. gr. air; the second the denser, ex. gr. glass; the attraction of the mediums here will be as their densities. Suppose <hi rend="italics">p</hi> S to be the distance to which the attracting force of the denser medium exerts itself within the rarer. Now let a ray of light A<hi rend="italics">a</hi> fall obliquely on the surface which separates the mediums, or rather on the surface <hi rend="italics">p</hi>S, where the action of the second and more resisting medium commences: as the ray arrives at <hi rend="italics">a,</hi> it will begin to be turned out of its rectilinear course by a superior force, with which it is attracted by the medium O, more than by the medium N; hence the ray is bent out of its right line in every point of its passage between <hi rend="italics">p</hi>S and RT, within which distance the attraction acts; and therefore between these lines it describes a curve <hi rend="italics">a</hi>B<hi rend="italics">b;</hi> but beyond RT, being out of the sphere of attraction of the medium N, it will proceed uniformly in a right line, according to the direction of the curve in the point <hi rend="italics">b.</hi></p><p>Again, suppose N the denser and more attracting medium, O the rarer, and HI the boundary as before; and let RT be the distance to which the denser medium exerts its attractive force within the rarer: even when the ray has passed the point B, it will be within the sphere of the superior attraction of the denser medium; but that attraction acting in lines perpendicular to its surface, the ray will be continually drawn from its straight course BM perpendicularly towards HI: thus, having two forces or directions, it will have a compound motion, by which, instead of BM, it will describe B<hi rend="italics">m,</hi> which B<hi rend="italics">m</hi> will in strictness be a curve. Lastly, after it has arrived at <hi rend="italics">m,</hi> being out of the influence of the medium N, it will persist uniformly, in a right line, in the direction in which the extremity of <pb n="345"/><cb/> the curve leaves it.—Thus we see how Refraction is performed, both towards the perpendicular DE, and from it.</p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">in Dioptrics,</hi> is the inflexion or bending of the rays of light, in passing the surfaces of glasses, lenses, and other transparent bodies of different densities. Thus, a ray, as AB, falling obliquely from the radiant A, upon a point B, in a diaphanous surface HI, rarer or denser than the medium along which it was propagated from the radiant, has its direction there altered by the action of the new medium; and instead of proceeding to M, it deviates, as for ex. to C.</p><p>This deviation is called the <hi rend="italics">Refraction of the ray;</hi> BC the <hi rend="italics">refracted ray,</hi> or <hi rend="italics">line of Refraction;</hi> and B the <hi rend="italics">point of Refraction.</hi>—The line AB is also called the <hi rend="italics">line of incidence;</hi> and in respect of it, B is also called the <hi rend="italics">point of incidence.</hi> The plane in which both the incident and refracted ray are found, is called the <hi rend="italics">plane of Refraction;</hi> also a right line BE drawn in the refracting medium perpendicular to the refracting surface at the point of Refraction B, is called the <hi rend="italics">axis of Refraction;</hi> and its continuation DB along the medium through which the ray falls, is called the <hi rend="italics">axis of incidence.</hi>—Farther, the angle ABI, made by the incident ray and the refracting surface, is usually called the <hi rend="italics">angle of incidence;</hi> and the angle ABD, between the incident ray and the axis of incidence, is the <hi rend="italics">angle of inclination.</hi> Moreover, the angle MBC, between the refracted and incident rays, is called the <hi rend="italics">angle of Refraction;</hi> and the angle CBE, between the refracted ray and the axis of Refraction, is the <hi rend="italics">refracted angle.</hi> But it is also very common to call the angles ABD and CBE made by the perpendicular with the incident and refracted rays, the <hi rend="italics">angles of incidence and Refraction.</hi></p><p><hi rend="italics">General Laws of</hi> <hi rend="smallcaps">Refraction.</hi>—I. <hi rend="italics">A ray of light in its passage out of a rarer medium into a denser,</hi> ex. gr. <hi rend="italics">out of air into water or into glass, is refracted towards the perpendicular,</hi> i. e. <hi rend="italics">towards the axis of Refraction.</hi> Hence, the refracted angle is less than the angle of inclination; and the angle of Refraction less than that of incidence; as they would be equal were the ray to proceed straight from A to M.</p><p>II. <hi rend="italics">The ratio of the sines of the angles</hi> ABD, CBE, <hi rend="italics">made by the perpendicular with the incident and refracted rays, is a constant and fixed ratio;</hi> whatever be the obliquity of the-incident ray, the mediums remaining. Thus, the Refraction out of air, into water, is nearly as 4 to 3, and into glass it is nearly as 3 to 2. As to air in particular, it is shewn by Newton, that a ray of light, in traversing quite through the atmosphere, is refracted the same as it would be, were it to pass with the same obliquity out of a vacuum immediately into air of equal density with that in the lowest part of the atmosphere.</p><p>The true law of Refraction was first discovered by Willebrord Snell, professor of Mathematics at Leyden; who found by experiment that the cosecants of the angles of incidence and Refraction, are always in the same ratio. It was commonly attributed however to Des Cartes; who, having seen it in a MS. of Snell's, first published it in his Dioptrics, without naming Snellius, as Huygens asserts; Des Cartes having only <cb/> altered the form of the law, from the ratio of the cosecants, to that of the sines, which is the same thing.</p><p>It is to be observed however, that as the rays of light are not all of the same degree of refrangibility, this constant ratio must be different in different kinds: so that the ratio mentioned by authors, is to be understood of rays of the mean refrangibility, i. e. of green rays. The difference of Refraction between the least and most refrangible rays, that is, between violet and red rays, Newton shews, is about the 2/55 of the whole Refraction of the mean refrangible; which difference, he allows, is so small, that it seldom needs to be regarded.</p><p>Different transparent substances have indeed very different degrees of Refraction, and those not according to any regular law; as appears by many experiments of Newton, Euler, Hawksbee, &c. See Newton's Optics, 3d edit. pa. 247; Hawksbee's Experim. pa. 292; Act. Berlin. 1762, pa. 302; Priestley's Hist. of Light &c. pa. 479.</p><p>Whence the different refractive powers in different fluids arise, has not been determined. Newton shews, that in many bodies, as glass, crystal, selenites, pseudo-topaz, &c, the refractive power is indeed proportionable to their densities; whilst in sulphureous bodies, as camphor, linseed, and olive oil, amber, spirit of turpentine, &c, the power is two or three times greater than in other bodies of equal density; and yet even these have the refractive power with respect to each other, nearly as their densities. Water has a refractive power in a medium degree between those two kinds of substances; whilst salts and vitriols have refractive powers in a middle degree between those of earthy substances and water, and accordingly are composed of those two sorts of matter. Spirit of wine has a refractive power in a middle degree between those of water and oily substances; and accordingly it seems to be composed of both, united by fermentation. It appears therefore, that all bodies seem to have their refractive powers nearly proportional to their densities, excepting so far as they partake more or less of sulphureous oily particles, by which those powers are altered.</p><p>Newton suspected that different degrees of heat might have some effect on the refractive power of bodies; but his method of determining the general Refraction was not sufficiently accurate to ascertain this circumstance. Euler's method however was well adapted to this purpose: from his experiments he insers, that the focal distance of a single lens of glass diminishes with the heat communicated to it; which diminution is owing to a change in the refractive power of the glass itself, which is probably increased by heat, and diminished by cold, as well probably as that of all other translucent substances.</p><p>From the law above laid down it follows, that one angle of inclination, and its corresponding refracted angle, being found by observation, the refracted angles corresponding to the several other angles of inclination are thence easily computed. Now, Zahnius and Kircher have found, that if the angle of inclination be 70°, the refracted angle, out of air into glass, will be 38° 50′; on which principle Zahnius has constructed a table of those Refractions for the several degrees of the <pb n="346"/><cb/> angle of inclination; a specimen of which here follows: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Angle of In-</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Refracted</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Angle of Re-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">clination.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Angle.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">fraction.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">55</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">54</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">56</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">55</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">57</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">44</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">25</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">31</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">20</cell></row></table></p><p>Hence it appears, that if the angle of inclination be less than 20°, the angle of Refraction out of air into glass is almost 1/3 of the angle of inclination; and therefore a ray is refracted to the axis of Refraction by almost a third part of the quantity of its angle of inclination. And on this principle it is that Kepler, and most other dioptrical writers, demonstrate the Refractions in glasses; though in estimating the law of these Refractions he followed the example of Alhazen and Vitello, and sought to discover it in the proportion of the angles, and not in that of the sines, or cosecants, as discovered by Snellius, as mentioned above.</p><p>The refractive powers of several substances, as determined by different philosophers, may be seen in the following tables; in which the ray is supposed to pass out of air into each of the substances, and the annexed numbers shew the proportion to unity or 1, between the sines of the angles of incidence and Refraction. <table><row role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">1. By Sir Isaac Newton's Observations.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Air</cell><cell cols="1" rows="1" role="data">0.9997</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rain water</cell><cell cols="1" rows="1" role="data">1.3358</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of wine</cell><cell cols="1" rows="1" role="data">1.3698</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of vitriol</cell><cell cols="1" rows="1" role="data">1.4285</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alum</cell><cell cols="1" rows="1" role="data">1.4577</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil olive</cell><cell cols="1" rows="1" role="data">1.4666</cell></row><row role="data"><cell cols="1" rows="1" role="data">Borax</cell><cell cols="1" rows="1" role="data">1.4667</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gum Arabic</cell><cell cols="1" rows="1" role="data">1.4771</cell></row><row role="data"><cell cols="1" rows="1" role="data">Linseed oil</cell><cell cols="1" rows="1" role="data">1.4814</cell></row><row role="data"><cell cols="1" rows="1" role="data">Selenites</cell><cell cols="1" rows="1" role="data">1.4878</cell></row><row role="data"><cell cols="1" rows="1" role="data">Camphor</cell><cell cols="1" rows="1" role="data">1.5000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Dantzick vitriol</cell><cell cols="1" rows="1" role="data">1.5000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Nitre</cell><cell cols="1" rows="1" role="data">1.5238</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sal gem</cell><cell cols="1" rows="1" role="data">1.5455</cell></row><row role="data"><cell cols="1" rows="1" role="data">Glass</cell><cell cols="1" rows="1" role="data">1.5500</cell></row><row role="data"><cell cols="1" rows="1" role="data">Amber</cell><cell cols="1" rows="1" role="data">1.5556</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rock crystal</cell><cell cols="1" rows="1" role="data">1.5620</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of turpentine</cell><cell cols="1" rows="1" role="data">1.5625</cell></row><row role="data"><cell cols="1" rows="1" role="data">A yellow pseudo-topaz</cell><cell cols="1" rows="1" role="data">1.6429</cell></row><row role="data"><cell cols="1" rows="1" role="data">Island crystal</cell><cell cols="1" rows="1" role="data">1.6666</cell></row><row role="data"><cell cols="1" rows="1" role="data">Glass of antimony</cell><cell cols="1" rows="1" role="data">1.8889</cell></row><row role="data"><cell cols="1" rows="1" role="data">A Diamond</cell><cell cols="1" rows="1" role="data">2.4390</cell></row></table> <cb/> <table><row role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">2. By Mr. Hawksbee.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Water</cell><cell cols="1" rows="1" role="data">1.3359</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of honey</cell><cell cols="1" rows="1" role="data">1.3359</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of amber</cell><cell cols="1" rows="1" role="data">1.3377</cell></row><row role="data"><cell cols="1" rows="1" role="data">Human urine</cell><cell cols="1" rows="1" role="data">1.3419</cell></row><row role="data"><cell cols="1" rows="1" role="data">White of an egg</cell><cell cols="1" rows="1" role="data">1.3511</cell></row><row role="data"><cell cols="1" rows="1" role="data">French brandy</cell><cell cols="1" rows="1" role="data">1.3625</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of wine</cell><cell cols="1" rows="1" role="data">1.3721</cell></row><row role="data"><cell cols="1" rows="1" role="data">Distilled vinegar</cell><cell cols="1" rows="1" role="data">1.3721</cell></row><row role="data"><cell cols="1" rows="1" role="data">Gum ammoniac</cell><cell cols="1" rows="1" role="data">1.3723</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aqua regia</cell><cell cols="1" rows="1" role="data">1.3898</cell></row><row role="data"><cell cols="1" rows="1" role="data">Aqua fortis</cell><cell cols="1" rows="1" role="data">1.4044</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of nitre</cell><cell cols="1" rows="1" role="data">1.4076</cell></row><row role="data"><cell cols="1" rows="1" role="data">Crystalline humour of an ox's eye</cell><cell cols="1" rows="1" role="data">1.4635</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of vitriol</cell><cell cols="1" rows="1" role="data">1.4262</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of turpentine</cell><cell cols="1" rows="1" role="data">1.4833</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of amber</cell><cell cols="1" rows="1" role="data">1.5010</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of cloves</cell><cell cols="1" rows="1" role="data">1.5136</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of cinnamon</cell><cell cols="1" rows="1" role="data">1.5340</cell></row></table> <table><row role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">3. By Mr. Euler, junior.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Rain or distilled water</cell><cell cols="1" rows="1" role="data">1.3358</cell></row><row role="data"><cell cols="1" rows="1" role="data">Well water</cell><cell cols="1" rows="1" role="data">1.3362</cell></row><row role="data"><cell cols="1" rows="1" role="data">Distilled vinegar</cell><cell cols="1" rows="1" role="data">1.3442</cell></row><row role="data"><cell cols="1" rows="1" role="data">French wine</cell><cell cols="1" rows="1" role="data">1.3458</cell></row><row role="data"><cell cols="1" rows="1" role="data">A solution of gum arabic</cell><cell cols="1" rows="1" role="data">1.3467</cell></row><row role="data"><cell cols="1" rows="1" role="data">French brandy</cell><cell cols="1" rows="1" role="data">1.3600</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ditto a stronger kind</cell><cell cols="1" rows="1" role="data">1.3618</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of wine rectified</cell><cell cols="1" rows="1" role="data">1.3683</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ditto more highly rectified</cell><cell cols="1" rows="1" role="data">1.3706</cell></row><row role="data"><cell cols="1" rows="1" role="data">White of an egg</cell><cell cols="1" rows="1" role="data">1.3685</cell></row><row role="data"><cell cols="1" rows="1" role="data">Spirit of nitre</cell><cell cols="1" rows="1" role="data">1.4025</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of Provence</cell><cell cols="1" rows="1" role="data">1.4651</cell></row><row role="data"><cell cols="1" rows="1" role="data">Oil of turpentine</cell><cell cols="1" rows="1" role="data">1.4822</cell></row></table></p><p>III. <hi rend="italics">When a ray passes out of a denser medium into a rarer, it is refracted from the perpendicular, or from the axis of Refraction.</hi></p><p>This is exactly the reverse of the 2d law, and the quantity of Refraction is equal in both cases, or both forwards and backwards; so that a ray would take the same course back, by which another passed forward, viz, if a ray would pass from A by B to C, another would pass from C by B to A. Hence, in this case, the angle of Refraction is greater than the angle of inclination. Hence also, if the angle of inclination be less than 30°, MBC is nearly equal to 1/3 of MBE; therefore MBC is 1/2 of CBE; consequently, if the Refraction be out of glass into air, and the angle of inclination less than 30°, the ray is refracted from the axis of Refraction by almost the half of the angle of inclination. And this is the other dioptrical principle used by most authors after Kepler, to demonstrate the Refractions of glasses.</p><p>If the Refraction be out of air into glass, the ratio of the sines of inclination and Refraction is as 3 to 2, or more accurately as 17 to 11; if out of air into water as 4 to 3; therefore if the course be the contrary way, viz, out of glass or water into air, the ratio of the sines will be, in the former case as 2 to 3 or 11 to 17, and in the latter as 3 to 4. So that, if the Refraction be from water or glass into air, and the angle of inci- <pb n="347"/><cb/> dence or inclination be greater than about 48 1/2 degrees in water, or greater than about 40° in glass, the ray will not be refracted into air; but will be reflected into a line which makes the angle of reflection equal to the angle of incidence; because the sines of 48 1/2 and 40° are to the radius, as 3 to 4, and as 11 to 17 nearly; and therefore when the sine has a greater proportion to the radius than above, the ray will not be refracted.</p><p>IV. <hi rend="italics">A ray falling on a curve surface, whether concave or convex, is refracted after the same manner as if it fell on a plane which is a tangent to the curve in the point of incidence.</hi> Because the curve and its tangent have the point of contact common to both, where the ray is refracted. <hi rend="center"><hi rend="italics">Laws of</hi> <hi rend="smallcaps">Refraction</hi> <hi rend="italics">in Plane Surfaces.</hi></hi></p><p>1. If parallel rays, AB and CD, be refracted out of one transparent medium into another of a different density, they will continue parallel after Refraction, as BE and DF. Hence a glass that is plane on both sides, being turned either directly or obliquely to the sun, &c, the light passing through it will be propagated in the same manner as if the glass were away. <figure/></p><p>2. If two rays CD and CP, proceeding from the same radiant C, and falling on a plane surface of a different density, so that the points of Refraction D and P be equally distant from the perpendicular of incidence GK, the refracted rays DF and PQ have the same virtual focus, or the same point of dispersion G.— Hence, when refracted rays, falling on the eye placed out of the perpendicular of incidence, are either equally distant from the perpendicular, or very near each other, they will flow upon the eye as if they came to it from the point G; consequently the point C will be seen by the refracted rays as in G. And hence also, if the eye be placed in a dense medium, objects in a rarer will appear more remote than they are; and the place of the image, in any case, may be determined from the ratio of Refraction: Thus, to fishes swimming under water, objects out of the water must appear farther distant than in reality they are. But, on the contrary, if the eye at E be placed in a rarer medium, then an object G placed in a denser, appears, at C, nearer than it is; and the place of the image may be determined in any given case by the ratio of Refraction: and thus the bottom of a vessel full of water is raised by Refraction a third part of its depth, with respect to an eye placed perpendicularly over the refracting surface; and thus also fishes and other bodies, under water, appear nearer than they really are.</p><p>3. If the eye be placed in a rarer medium; then an object seen in a denser, by a ray refracted in a plane surface, will appear larger than it really is. But if the eye be in a denser medium, and the object in a rarer, <cb/> the object will appear less than it is. And, in each case, the apparent magnitude FQ is to the real one EH, as the rectangle CK . GL to GK . CL, or in the compound ratio of the distance CK of the point to which the rays tend before Refraction, from the refracting surface DP, to the distance GK of the eye from the same, and of the distance GL of the object EH from the eye, to its distance CL from the point to which the rays tend before Refraction.—Hence, if the object be very remote, CL will be physically equal to GL; and then the real magnitude EL is to the apparent maguitude FL, as GK to CK, or as the distance of the eye G from the refracting plane, to the distance of the point of convergence F from the same plane. And hence also, objects under water, to an eye in the air, appear larger than they are; and to fishes under water, objects in the air appear less than they are. <hi rend="center"><hi rend="italics">Laws of</hi> <hi rend="smallcaps">Refraction</hi> <hi rend="italics">in Spherical Surfaces, both concave and convex.</hi></hi></p><p>1. A ray of light DE, parallel <figure/> to the axis, after a single refraction at E, meets the axis in the point F, beyond the centre C.</p><p>2. Also in that case, the semidiameter CB or CE will be to the refracted ray EF, as the sine of the angle of refraction to the sine of the angle of inclination BCE. But the distance of the focus, or point of concurrence from the centre, CF, is to the refracted ray EF, as the sine of the refracted angle to the sine of the angle of inclination.</p><p>3. Hence also, in this case, the distance BF of the focus from the refracting surface, must be to CF its distance from the centre, in a ratio greater than that of the sine of the angle of inclination to the sine of the refracted angle. But those ratios will be nearly equal when the rays are very near the axis, and the angle of inclination BCE is only of a few degrees. And when the Refraction is out of air into glass, then <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">For rays near the axis,</cell><cell cols="1" rows="1" role="data">For more distant rays,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">BF:FC::3:2,</cell><cell cols="1" rows="1" role="data">BF:FC > 3:2,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">BC:BF::1:3.</cell><cell cols="1" rows="1" role="data">BC:BF < 1:3.</cell></row></table> But if the Refraction be out of air into water, then <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">For rays near the axis,</cell><cell cols="1" rows="1" role="data">For more distant rays,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">BF:FC::4:3,</cell><cell cols="1" rows="1" role="data">BF:FC > 4:3,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">BC:BF::1:4,</cell><cell cols="1" rows="1" role="data">BC:BF < 1:4.</cell></row></table></p><p>Hence, as the sun's rays are parallel as to sense, if they fall on the surface of a solid glass sphere, or of a sphere full of water, they will not meet the axis within the sphere: so that Vitello was mistaken when he imagined that the sun's rays, falling on the surface of a crystalline sphere, were refracted to the centre.</p><p>4. If a ray HE fall parallel to the axis FA, out of a rarer medium, on the concave spherical surface BE of a denser one; the refracted ray EN will diverge from the point of the axis F, so that FE will be to FC, in the ratio of the sine of the angle of inclination, to the sine of the refracted angle. Consequently FB to FC is in a greater ratio than that; unless when the rays are very near the axis, and the angle BCE is very small, <pb n="348"/><cb/> for then FB will be to FC nearly in that ratio. And hence, in the cases of Refraction out of air into water or glass, the ratios of BC, BF and CF, will be the same as specified in the last article.</p><p>5. If a ray DE, parallel to the <figure/> axis FC, pass out of a denser into a rarer spherical convex medium, it will diverge from the axis after Refraction; and the distance FC of the point of dispersion, or of the virtual focus F, from the centre of the sphere, will be to its semidiameter CE or CB, as the sine of the refracted angle is to the sine of the angle of Refraction; but to the portion of the refracted ray drawn back, FE, it will be in the ratio of the sine of the refracted angle to the sine of the angle of inclination. Consequently FC will be to FB, in a greater ratio than this last one: unless when the rays DE fall very near the axis FC, for then FC to FB will be very nearly in that ratio.</p><p>Hence, when the Refraction is out of glass into air; then, <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">For rays near the axis,</cell><cell cols="1" rows="1" role="data">For more distant rays,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">FC:FB::3:2,</cell><cell cols="1" rows="1" role="data">FC:FB > 3:2,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">BC:BF::1:2.</cell><cell cols="1" rows="1" role="data">BC:BF > 1:2,</cell></row></table> But when the Refraction is out of water into air; then <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">For rays near the axis,</cell><cell cols="1" rows="1" role="data">For more distant rays,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">FC:FB::4:3,</cell><cell cols="1" rows="1" role="data">FC:FB > 4:3,</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">BC:BF::1:3.</cell><cell cols="1" rows="1" role="data">BC:BF > 1:3.</cell></row></table></p><p>6. If the ray HE fall parallel to the axis CF, from a denser medium, upon the surface of a spherically concave rarer one; the refracted ray will meet with the axis in the point F, so that the distance CF from the centre, will be to the refracted ray FE, as the sine of the refracted angle, to the sine of the angle of inclination. Consequently FC will be to FB, in a greater ratio than that above mentioned: unless when the rays are very near the axis, for then FC is to FB very nearly in that ratio; and the three FB, FC, BC are, in the cases of air, water and glass, in the numeral ratios as specified at the end of the last article. See Wolsius, Elem. Mathes. tom. 3 p. 179 &c. <hi rend="center"><hi rend="smallcaps">Refraction</hi> <hi rend="italics">in a Glass Prism.</hi></hi></p><p>ABC being the transverse section of a prism; if a ray of light DE fall obliquely upon it out of the air; instead of proceeding straight on to F, being refracted <figure/> towards the perpendicular IE, it will decline to G. Again, since the ray EG, passing out of glass into air, falls obliquely on BC, it will be refracted to M, so as <cb/> to recede from the perpendicular GO. And hence arise the various phenomena of the prism. See <hi rend="smallcaps">Colour.</hi> <hi rend="center"><hi rend="smallcaps">Refraction</hi> <hi rend="italics">in a Convex Lens.</hi></hi></p><p>If parallel rays, AB, CD, EF, fall on the surface of a convex lens XBZ (the last fig. above); the perpendicular ray AB will pass unrefracted to K, where emerging, as before, perpendicularly, into air, it will proceed straight on to G. But the rays CD and EF, falling obliquely out of air into glass, at D and F, will be refracted towards the axis of Refraction, or towards the perpendiculars at D and F, and so decline to Q and P: where emerging again obliquely out of the glass into the surface of the air, they will be refracted from the perpendicular, and proceed in the directions QG and PG, meeting in G. And thus also will all the other rays be refracted so as to meet the rest near the place G. See <hi rend="smallcaps">Focus</hi> and <hi rend="smallcaps">Lens.</hi>—Hence the great property of convex glasses; viz, that they collect parallel rays, or make them converge into a point. <hi rend="center"><hi rend="smallcaps">Refraction</hi> <hi rend="italics">in a Concave Lens.</hi></hi></p><p>Parallel rays AB, CD, EF, falling on a concave lens GBHIMK, the ray AB falling perpendicularly on the glass at B, will pass unrefracted to M; where, being still perpendicular, it will pass into the air to L, with- <figure/> out Refraction. But the ray CD, falling obliquely on the surface of the glass, will be refracted towards the perpendicular at D, and proceed to Q; where again falling obliquely out of the glass upon the surface of air, it will be refracted from the perpendicular at Q, and proceed to V. After the same manner the ray EF is first refracted to Y, and thence to Z.—Hence the great property of concave glasses; viz, that they disperse parallel rays, or make them diverge. See <hi rend="smallcaps">Lens.</hi> <hi rend="center"><hi rend="smallcaps">Refraction</hi> <hi rend="italics">in a Plane Glass.</hi></hi></p><p>If parallel rays EF, GH, IK, (the last fig. above) fall obliquely on a plane glass ABCD; the obliquity being the same in all, by reason of their parallelism, they will be all equally refracted towards the perpendicular; and accordingly, being still parallel at M, O, and Q, they will pass out into the air equally refracted again from the perpendicular, and still parallel. Thus will the rays EF, GH, and IK, at their entering the glass, be inflected towards the right; and in their going out as much inflected to the left; so that the first Refraction is here undone by the second, thereby causing the rays on their emerging from the glass, to be parallel to their first direction before they entered it; though not so as that the object is seen in its true place; for the ray RQ, being produced back again, will not coincide with the ray IK, but will fall to the left of it; and this the more as the glass is thicker; however, as <pb n="349"/><cb/> to the colour, the second Refraction does really undo the first. See <hi rend="smallcaps">Colour.</hi></p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">in Astronomy,</hi> or <hi rend="smallcaps">Refraction</hi> <hi rend="italics">of the Stars,</hi> is an inflexion of the rays of those luminaries, in passing through our atmosphere; by which the apparent altitudes of the heavenly bodies are increased.</p><p>This Refraction arises from hence, that the atmosphere is unequally dense in different stages or regions; rarest of all at the top, and densest of all at the bottom; which inequality in the same medium, makes it equivalent to several unequal mediums, by which the course of the ray of light is continually bent into a continued curve line. See <hi rend="smallcaps">Atmosphere.</hi>—And Sir Isaac Newton has shewn, that a ray of light, in passing from the highest and rarest part of the atmosphere, down to the lowest and densest, undergoes the same quantity of Refraction that it would do in passing immediately, at the same obliquity, out of a vacuum into air of equal density with that in the lowest part of the atmosphere.</p><p>The effect of this Refraction may <figure/> be thus conceived. Suppose ZV a quadrant of a vertical circle described from the centre of the earth T, under which is AB a quadrant of a circle on the surface of the earth, and GH a quadrant of the surface of the atmosphere. Then suppose SE a ray of light emitted by a star at S, and falling on the atmosphere at E: this ray coming out of the ethereal medium, which is much rarer than our air, or perhaps out of a perfect vacuum, and falling on the surface of the atmosphere, will be refracted towards the perpendicular, or inclined down more towards the earth; and since the upper air is again rarer than that near the earth, and grows still denser as it approaches the earth's surface, the ray in its progress will be continually refracted, so as to arrive at the eye in the curve line EA. Then supposing the right line AF to be a tangent to the arch at A, the ray will enter the eye at A in the direction of AF; and therefore the star will appear in the heavens at Q, instead of S, higher or nearer the zenith than the star really is.</p><p>Hence arise the phenomena of the crepusculum or twilight; and hence also it is that the moon is sometimes seen eclipsed, when she is really below the horizon, and the sun above it.</p><p>That there is a real Refraction of the stars &c, is deduced not only from physical considerations, and from arguments a priori, and a similitudine, but also from precise astronomical observation: for there are numberless observations by which it appears that the sun, moon, and stars rise much sooner, and appear higher, than they should do according to astronomical calculations. Hence it is argued, that as light is propagated in right lines, no rays could reach the eye from a luminary below the horizon, unless they were deflected out of their course, at their entrance into the atmosphere; and therefore it appears that the rays are refracted in passing through the atmosphere.</p><p>Hence the stars appear higher by Refraction than they really are; so that to bring the observed or apparent altitudes to the true ones, the quantity of Refraction must be subtracted. And hence, the ancients, <cb/> as they were not acquainted with this Refraction, reckoned their altitudes too great, so that it is no wonder they sometimes committed considerable errors. Hence also, Refraction lengthens the day, and shortens the night, by making the sun appear above the horizon a little before his rising, and a little after his setting. Refraction also makes the moon and stars appear to rise sooner and set later than they really do. The apparent diameter of the sun or moon is about 32′; the horizontal refraction is about 33′; whence the sun and moon appear <hi rend="italics">wholly</hi> above the horizon when they are entirely below it. Also, from observations it appears that the Refractions are greater nearer the pole than at lesser latitudes, causing the sun to appear some days above the horizon, when he is really below it; doubtless from the greater density of the atmosphere, and the greater obliquity of the incidence.</p><p>Stars in the zenith are not subject to any Refraction: those in the horizon have the greatest of all: from the horizon, the Refraction continually decreases to the zenith. All which follows from hence, that in the first case, the rays are perpendicular to the medium; in the second, their obliquity is the greatest, and they pass through the largest space of the lower and denser part of the air, and through the thickest vapours; and in the third, the obliquity is continually decreasing.</p><p>The air is condensed, and consequently Refraction is increased, by cold; for which reason it is greater in cold countries than in hot ones. It is also greater in cold weather than in hot, in the same country; and the morning Refraction is greater than that of the evening, because the air is rarefied by the heat of the sun in the day, and condensed by the coldness of the night. Refraction is also subject to some small variation at the same time of the day in the finest weather.</p><p>At the same altitudes, the sun, moon, and stars all undergo the same Refraction: for at equal altitudes the incident rays have the same inclinations; and the sines of the refracted angles are as the sines of the angles of inclination, &c.</p><p>Indeed Tycho Brahe, who first deduced the Refractions of the sun, moon, and stars, from observation, and whose table of the Refraction of the stars is not much different from those of Flamsteed and Newton, except near the horizon, makes the solar Refractions about 4′ greater than those of the fixed stars; and the lunar Refractions also sometimes greater than those of the stars, and sometimes less. But the theory of Refractions, found out by Snellius, was not fully understood in his time.</p><p>The horizontal Refraction, being the greatest, is the cause that the sun and moon appear of an oval form at their rising and setting; for the lower edge of each being more refracted than the upper edge, the perpendicular diameter is shortened, and the under edge appears more flatted also.—Hence also, if we take with an instrument the distance of two stars when they are in the same vertical and near the horizon, we shall find it considerably less than if we measure it when they are both at such a height as to suffer little or no Refraction; because the lower star is more elevated than the higher. There is also another alteration made by Refraction in the apparent distance of stars: when two stars are in the same almicantar, or parallel of declination, their ap- <pb n="350"/><cb/> parent distance is less than the true; for since Refraction makes each of them higher in the azimuth or vertical in which they appear, it must bring them into parts of the vertical where they come nearer to each other; because all vertical circles converge and meet in the zenith. This contraction of distance, according to Dr. Halley (Philos. Trans. numb. 368) is at the rate of at least one second in a degree; so that, if the distance between two stars in a position parallel to the horizon measure 30°, it is at most to be reckoned only 29° 59′ 30″.</p><p>The quantity of the Refraction at every altitude, from the horizon, where it is greatest, to the zenith where it is nothing, has been determined by observation, by many astronomers; those of Dr. Bradley and Mr. Mayer are esteemed the most correct of any, being nearly alike, and are now used by most astronomers. Doctor Bradley, from his observations, deduced this very simple and general rule for the Refraction <hi rend="italics">r</hi> at any altitude <hi rend="italics">a</hi> whatever; viz, as rad. 1 : cotang. the Refraction in seconds.</p><p>This rule, of Dr. Bradley's, is adapted to these states of the barometer and thermometer, viz, <cb/> either 29.6 inc. barom. and 50° thermometer, or 30 — barom. and 55 thermometer, for both which states it answers equally the same. But for any other states of the barometer and thermometer, the Refraction above-found is to be corrected in this manner; viz, if <hi rend="italics">b</hi> denote any other height of the barometer in inches, and <hi rend="italics">t</hi> the degrees of the thermometer, <hi rend="italics">r</hi> being the Refraction uncorrected, as found in the manner above. Then as 29.6:<hi rend="italics">b</hi>::<hi rend="italics">r</hi>:R the Refraction corrected on account of the barometer, and 400:450 <hi rend="italics">t</hi>::R: the Refraction corrected both on account of the barometer and thermometer; which sinal corrected Refraction is therefore . Or, to correct the same Refraction <hi rend="italics">r</hi> by means of the latter state, viz, barom. 30 and therm. 55, it will be as , and the correct Refraction.</p><p>From Dr. Bradley's rule, , the following Table of the mean astronom. Refrac. is computed. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=18" role="data"><hi rend="italics">Mean Astronomical Refractions in Altitude.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Apparent</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Refraction.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Apparent</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Refraction.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Apparent</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Refraction.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Apparent</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Refraction.</cell><cell cols="1" rows="1" role="data">Apparent</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Refraction.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Altitude.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Altitude.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Altitude.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Altitude.</cell><cell cols="1" rows="1" role="data">Altitude.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0°</cell><cell cols="1" rows="1" role="data">0′</cell><cell cols="1" rows="1" role="data">33′</cell><cell cols="1" rows="1" role="data">0″</cell><cell cols="1" rows="1" role="data">3°</cell><cell cols="1" rows="1" role="data">0′</cell><cell cols="1" rows="1" role="data">14′</cell><cell cols="1" rows="1" role="data">36″</cell><cell cols="1" rows="1" role="data">8°</cell><cell cols="1" rows="1" role="data">30′</cell><cell cols="1" rows="1" role="data">6′</cell><cell cols="1" rows="1" role="data">8″</cell><cell cols="1" rows="1" role="data">20°</cell><cell cols="1" rows="1" role="data">0′</cell><cell cols="1" rows="1" role="data">2′</cell><cell cols="1" rows="1" role="data">35″</cell><cell cols="1" rows="1" rend="align=center" role="data">54°</cell><cell cols="1" rows="1" rend="align=center" role="data">41″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=center" role="data">55</cell><cell cols="1" rows="1" rend="align=center" role="data">40</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" rend="align=center" role="data">56</cell><cell cols="1" rows="1" rend="align=center" role="data">38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=center" role="data">57</cell><cell cols="1" rows="1" rend="align=center" role="data">37</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" rend="align=center" role="data">58</cell><cell cols="1" rows="1" rend="align=center" role="data">35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=center" role="data">59</cell><cell cols="1" rows="1" rend="align=center" role="data">34</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">60</cell><cell cols="1" rows="1" rend="align=center" role="data">33</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=center" role="data">61</cell><cell cols="1" rows="1" rend="align=center" role="data">31</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" rend="align=center" role="data">62</cell><cell cols="1" rows="1" rend="align=center" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" rend="align=center" role="data">63</cell><cell cols="1" rows="1" rend="align=center" role="data">29</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">47</cell><cell 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role="data">55</cell><cell cols="1" rows="1" rend="align=center" role="data">82</cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" rend="align=center" role="data">83</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" rend="align=center" role="data">84</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" rend="align=center" role="data">85</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" rend="align=center" role="data">86</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" rend="align=center" role="data">87</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" rend="align=center" role="data">88</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" rend="align=center" role="data">89</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row></table> <pb n="351"/><cb/></p><p>Mr. Mayer says his rule was deduced from theory, and, when reduced from French measure and Reaumur's thermometer, to English measure and Fahrenheit's thermometer, it is this, the Refraction in seconds, corrected for both barometer and thermometer: where the letters denote the same things as before, except <hi rend="italics">A,</hi> which denotes the angle whose tangent is .</p><p>Mr. Simpson too (Dissert. pa. 46 &c) has ingeniously determined by theory the astronomical Refractions, from which he brings out this rule, viz, As 1 to .9986 or as radius to sine of 86° 58′ 30″, so is the sine of any given zenith distance, to the sine of an arc; then 2/11 of the difference between this arc and the zenith distance, is the Refraction sought for that zenith distance. And by this rule Mr. Simpson computed a Table of the mean Refractions, which are not much different from those of Dr. Bradley and Mr. Mayer, and are as in the following Table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=9" role="data"><hi rend="italics">Mr. Simpson's Table of Mean Refractions.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Appa-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Appa-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Appa-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">rent</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Refrac-</cell><cell cols="1" rows="1" role="data">rent</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Refrac-</cell><cell cols="1" rows="1" role="data">rent</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Refrac-</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alti-</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">tion.</cell><cell cols="1" rows="1" role="data">Alti-</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">tion.</cell><cell cols="1" rows="1" role="data">Alti-</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">tion.</cell></row><row role="data"><cell cols="1" rows="1" role="data">tude.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">tude.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">tude.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">0°</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">0″</cell><cell cols="1" rows="1" role="data">17°</cell><cell cols="1" rows="1" role="data">2′</cell><cell cols="1" rows="1" role="data">50″</cell><cell cols="1" rows="1" role="data">38°</cell><cell cols="1" rows="1" role="data">1′</cell><cell cols="1" rows="1" role="data">7″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">58</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">54</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">50</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">47</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">44</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">24</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">19</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4 1/2</cell></row></table></p><p>It is evident that all observed altitudes of the heavenly bodies ought to be diminished by the numbers taken out of the foregoing Table. It is also evident that the Refraction diminishes the right and oblique ascensions of a star, and increases the descensions: it increases the northern declination and latitude, but decreases the southern: in the eastern part of the heavens it diminishes the longitude of a star, but in the western part of the heavens it increases the same. <cb/></p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">of Altitude,</hi> is an arc of a vertical circle, as AB, by which the altitude of a star AC is increased by the Refraction. <figure/></p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">of Ascension and Descension,</hi> is an arc DE of the equator, by which the ascension and descension of a star, whether right or oblique, is increased or diminished by the Refraction.</p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">of Declination,</hi> is an arc BF of a circle of declination, by which the declination of a star DA or EF is increased or diminished by Refraction.</p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">of Latitude</hi> is an arc AG of a circle of latitude, by which the latitude of a star AH is increased or diminished by the Refraction.</p><p><hi rend="smallcaps">Refraction</hi> <hi rend="italics">of Longitude</hi> is an arc 1H of the ecliptic, by which the longitude of a star is increased or diminished by means of the Refraction.</p><p><hi rend="italics">Terrestrial</hi> <hi rend="smallcaps">Refraction</hi>, is that by which terrestrial objects appear to be raised higher than they really are, in observing their altitudes. The quantity of this Refraction is estimated by Dr. Maskelyne at one-tenth of the distance of the object observed, expressed in degrees of a great circle. So, if the distance be 10000 fathoms, its 10th part 1000 fathoms, is the 60th part of a degree of a great circle on the earth, or 1′, which therefore is the Refraction in the altitude of the object at that distance. (Requisite Tables, 1766, pa. 134).</p><p>But M. Le Gendre is induced, he says, by several experiments, to allow only 1/14th part of the distance for the Refraction in altitude. So that, upon the distance of 10000 fathoms, the 14th part of which is 714 fathoms, he allows only 44″ of terrestrial Refraction, so many being contained in the 714 fathoms. See his Memoir concerning the Trigonometrical operations, &c.</p><p>Again, M. de Lambre, an ingenious French astronomer, makes the quantity of the Terrestrial Refraction to be the 11th part of the arch of distance. But the English measurers, Col. Edw. Williams, Capt. Mudge, and Mr. Dalby, from a multitude of exact observations made by them, determine the quantity of the medium Refraction to be the 12th part of the said distance.</p><p>The quantity of this Refraction, however, is found to vary considerably, with the different states of the weather and atmosphere, from the 15th part of the distance, to the 9th part of the same; the medium of which is the 12th part, as above mentioned.</p><p>Some whimsical effects of this Refraction are also related, arising from peculiar situations and circumstances. Thus, it is said, any person standing by the side of the river Thames at Greenwich, when it is high- <pb n="352"/><cb/> water there, he can see the cattle grazing on the Isle of Dogs, which is the marshy meadow on the other side of the river at that place; but when it is low-water there, he cannot see any thing of them, as they are hid from his view by the land wall or bank on the other side, which is raised higher than the marsh, to keep out the waters of the river. This curious effect is probably owing to the moist and dense vapours, just above and rising from the surface of the water, being raised higher or lifted up with the surface of the water at the time of high tide, through which the rays pass, and are the more refracted.</p><p>Again, a similar instance is related in a letter to me, from an ingenious friend, Mr. Abr. Crocker of Frome in Somersetshire, dated January 12, 1795. “My Devonshire friend,” says he, (whose seat is in the vicinity of the town of Modbury, 12 miles in a geographical line from Maker tower near Plymouth) “being on a pleasure spot in his garden, on the 4th of December 1793, with some friends, viewing the surrounding country, with an achromatic telescope, descried an object like a perpendicular pole standing up in the chasm of a hedge which bounded their view at about 9 miles distance; which, from its direction, was conjectured to be the flagstaff on Maker tower.—Directing the glass, on the morning of the next day, to the same part of the horizon, a flag was perceived on the pole; which corroborated the conjecture of the preceding day. This day's view also discovered the pinnacles and part of the shaft of the tower.—Viewing the same spot at 8 in the morning on the 9th of January 1794, the whole tower and part of the roof of the church, with other remote objects not before noticed, became visible.</p><p>“It is necessary to give you the state of the weather there, on those days. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1793.</cell><cell cols="1" rows="1" role="data">Barometer.</cell><cell cols="1" rows="1" role="data">Thermom.</cell><cell cols="1" rows="1" role="data">Wind</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Dec. 4</cell><cell cols="1" rows="1" role="data">29.93, rising</cell><cell cols="1" rows="1" rend="align=center" role="data">36.0</cell><cell cols="1" rows="1" rend="align=center" role="data">N.E.</cell><cell cols="1" rows="1" role="data">Frosty morning,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">a mist over the</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">land below.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">29.97, rising</cell><cell cols="1" rows="1" rend="align=center" role="data">35.2</cell><cell cols="1" rows="1" rend="align=center" role="data">W.</cell><cell cols="1" rows="1" role="data">Ditto.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1794.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Jan. 9</cell><cell cols="1" rows="1" role="data">30.01, falling</cell><cell cols="1" rows="1" rend="align=center" role="data">29.8</cell><cell cols="1" rows="1" rend="align=center" role="data">W.</cell><cell cols="1" rows="1" role="data">Hardwhitefrost,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">a fog over the</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">lowlands; clear</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">in the surround-</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">ing country.</cell></row></table></p><p>“The singularity of this phenomenon has occasioned repeated observations on it; from all which it appears that the summer season, and wet windy weather, are unfavourable to this refracted elevation; but that calm frosty weather, with the absence of the sun, are favourable to it.</p><p>“From hence a question arises; what is the principal or most general cause of atmospheric Refraction, which produces such extraordinary appearances?”</p><p>The following is also a copy of a letter to Mr. Crocker on this curious phenomenon, from his friend above mentioned, viz, Mr. John Andrews, of Traine, near Modbury, dated the 1st of February 1795.</p><p>“My good Friend,</p><p>“Finding, by your favour of last Sunday, the pro- <cb/> ceedings which are going on in respect to my observations on the phenomenon of <hi rend="italics">Looming,</hi> I have thought it necessary to bestow about half this day in preparing, what I am obliged to call, in my way, <hi rend="italics">drawings,</hi> illustrative of those observations.—I have endeavoured to distinguish, by different tints and shades, the grounds which lie nearer or more remote; but this will perhaps be better explained by the letters of reference, which I have inserted as they may be serviceable in future correspondence.—I believe the drawings, rough as they are, give a tolerably exact representation of the scenes: they may be properly copied to send to London by one of your ingenious sons.—I have been attentive in my observations, or rather in looking out for observations, during the late hard frosts, which you will be surprised to learn, have (except on one or two days) been very unpropitious to the phenomenon; but they have compensated for that disappointment, by a discovery, that a dry frost, though ever so intense, has no tendency to produce it. A hoar frost, or that kind of dewy vapour which, in a sufficient degree of cold, occasions a hoar frost, appears essentially necessary. This took place pretty favourably on the 6th of January, when the elevation was equal to that represented in the third drawing <hi rend="italics">(see plate</hi> 25, <hi rend="italics">fig.</hi> 3), much like what it was on the 9th of January 1794, and confirmed me as to the certainty of some peculiar appearances, hinted at in my letter of the 14th of that month, but not there described. What I allude to, was a fluctuating appearance of two horizons, one above the other, with a complete vacancy between them, exactly like what may be often observed looking through an uneven pane of glass. Divers instances of this were seen by my brother and myself on the 6th of last January; continually varying and intermitting, but not rapidly, so that they were capable of distinct observation.—Till that day I had formed, as I thought, a plausible theory, to account for, as well this latter, as all the other phenomena; but now, unless my imagination deceives me, I am left in impenetrable darkness. The vacant line of separation, you will take notice, would often increase so much in breadth, as to efface entirely the upper of the two horizons; forming then a kind of dent or gap in the remaining horizon, which horizon at the places contiguous to the extremities of the vacancy, seemed of the same height as the upper horizon was, before effaced. This vacancy was several times seen to approach and take in the tower, and immediately to admit a view of the whole or most part of its body (like that in the third drawing) which was not the case before: exactly, to all appearance, as if it had opened a gap for that purpose in the intercepted ground.—It remains therefore to be determined by future observations, whether the separation is effected by an elevation of the upper, or depression of the lower horizon; and if the latter, why the vacancy does not cause the tower to disappear, as well as the intervening ground?—As an opportunity for this purpose may not soon occur, I hope you will not wait for it, in your communications to him who is, Dear Sir, yours very truly, <hi rend="smallcaps">John Andrews.</hi>”</p><p>See the representations in plate 25, of the appearances, in three different states of the atmosphere, with the explanations of them. <pb n="353"/><cb/></p><p>REFRANGIBILITY <hi rend="italics">of Light,</hi> the disposition of the rays to be refracted. And a greater or less Refrangibility, is a disposition to be more or less refracted, in passing at equal angles of incidence into the same medium.</p><p>That the rays of light are differently refrangible, is the foundation of Newton's whole theory of light and colours; and the truth and circumstances of the principle he evinced from such experiments as the following. <figure/></p><p>Let EG represent the window-shutter of a dark room, and F a hole in it, through which the light passes, from the luminous object S, to the glass prism ABC within the room, which refracts it towards the opposite side, or a screen, at PT, where it appears of an oblong form; its length being about five times the breadth, and exhibiting the various colours of the rainbow; whereas without the interposition of the prism, the ray of light would have proceeded on in its first direction to D. Hence then it follows,</p><p>1. That the rays of light are refrangible. This appears by the ray being refracted from its original direction SHD, into another one, HP or HT, by passing through a different medium.</p><p>2. That the ray SFH is a compound one, which, by means of the prism, is decompounded or separated into its parts, HP, HT, &c, which it hence appears are all endued with different degrees of Refrangibility, as they are transmitted to all the intermediate points from T to P, and there painting all the different colours.</p><p>From this, and a great variety of other experiments, Newton proved, that the blue rays are more refracted than the red ones, and that there is likewise unequal refraction in the intermediate rays; and upon the whole it appears that the sun's rays have not all the same Refrangibility, and consequently are not of the same nature. It is also observed that those rays which are most refrangible, are also most reflexible. See R<hi rend="smallcaps">EFLEXIBILITY;</hi> also Newton's Optics, pa. 22 &c, 3d edit.</p><p>The difference between Refrangibility and reflexibility was first discovered by Sir Isaac Newton, in 1671-2, and communicated to the Royal Society, in a letter dated Feb. 6 of that year, which was published in the Philos. Trans. numb. 80, pa. 3075; and from that time it was vindicated by him, from the objections of several authors; particularly Pardies, Mariotte, Linus or Lin, and other gentlemen of the English college at Liege; and at length it was more fully laid down, illustrated, and confirmed, by a great variety of experiments, in his excellent treatise on Optics.</p><p>But farther, as not only these colours of light produced by refraction in a prism, but also those <cb/> reflected from opaque bodies, have their different degrees of Refrangibility and reflexibility; and as a white light arises from a mixture of the several coloured rays together, the same great author concluded that all homogeneous light has its proper colour, corresponding to its degree of Refrangibility, and not capable of being changed by any reflexions, or any refractions; that the sun's light is composed of all the primary colours; and that all compound colours arise from the mixture of the primary ones, &c.</p><p>The different degrees of Refrangibility, he conjectures to arise from the different magnitude of the particles composing the different rays. Thus, the most refrangible rays, that is the red ones, he supposes may consist of the largest particles; the least refrangible, i. e. the violet rays, of the smallest particles; and the intermediate rays, yellow, green, and blue, of particles of intermediate sizes. See <hi rend="smallcaps">Colour.</hi></p><p>For the method of correcting the effect of the different Refrangibility of the rays of light in glasse, see A<hi rend="smallcaps">BERRATION</hi> and <hi rend="smallcaps">Telescope.</hi></p></div1><div1 part="N" n="REGEL" org="uniform" sample="complete" type="entry"><head>REGEL</head><p>, or <hi rend="smallcaps">Rigel</hi>, a fixed star of the first magnitude, in the left foot of Orion.</p><p>REGIOMONTANUS. See <hi rend="italics">John</hi> <hi rend="smallcaps">Muller.</hi></p></div1><div1 part="N" n="REGION" org="uniform" sample="complete" type="entry"><head>REGION</head><p>, of the Air or Atmosphere. Authors divide the atmosphere into three stages, called the upper, middle, and lower Regions.—The lowest Region is that in which we breathe, and is bounded by the reflexion of the sun's rays, that is, by the height to which they rebound from the earth.—The middle Region is that in which the clouds reside, and where meteors are formed, &c; extending from the extremity of the lowest, to the tops of the highest mountains.— The upper Region commences from the tops of the mountains, and reaches to the utmost limits of the atmosphere. In this Region there probably reigns a perpetual equable calmness, clearness, and serenity.</p><p><hi rend="italics">Elementary</hi> <hi rend="smallcaps">Region</hi>, according to the Aristotelians, is a sphere terminated by the concavity of the moon's orb, comprehending the earth's atmosphere.</p><p><hi rend="italics">Ethereal</hi> <hi rend="smallcaps">Region</hi>, is the whole extent of the universe, comprising all the heavens with the orbs of the fixed stars and other celestial bodies.</p><div2 part="N" n="Region" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Region</hi></head><p>, in Geography, a country or particular division of the earth, or a tract of land inhabited by people of the same nation.</p><p><hi rend="smallcaps">Regions</hi> <hi rend="italics">of the Moon.</hi> Modern astronomers divide the moon into several Regions, or provinces, to each of which they give its proper name.</p><p><hi rend="smallcaps">Regions</hi> <hi rend="italics">of the Sea,</hi> are the two parts into which the whole depth of the sea is conceived to be divided. The upper of these extends from the surface of the water, down as low as the rays of the sun can pierce, and extend their influence; and the lower Region extends from thence to the bottom of the sea.</p><p><hi rend="italics">Subterranean</hi> <hi rend="smallcaps">Regions.</hi> These are three, into which the earth is divided, at different depths below the surface, according to different degrees of cold or warmth; and it is imagined that the 2d or middlemost of these Regions is the coldest of the three.</p></div2></div1><div1 part="N" n="REGIS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">REGIS</surname> (<foreName full="yes"><hi rend="smallcaps">Peter Sylvain</hi></foreName>)</persName></head><p>, a French philosopher, and great propagator of Cartesianism, was born in Agenois 1632.</p><p>He studied the languages and philosophy under the <pb n="354"/><cb/> Jesuits at Cahors, and afterwards divinity in the university of that town, being designed for the church. His progress in learning was so uncommon, that at the end of four years he was offered a doctor's degree without the usual charges; but he did not think it became him till he should study also in the Sorbonne at Paris. He accordingly repaired to the capital for that purpose; but he soon became disgusted with theology; and, as the philosophy of Des Cartes began at that time to make a noise through the lectures of Rohault, he conceived a taste for it, and gave himself up entirely to it.</p><p>Having, by attending those lectures, and by close study, become an adept in that philosophy, he went to Toulouse in 1665, where he set up lectures in it himself. Having a clear and fluent manner, and a happy way of explaining his subject, he drew all sorts of people to his discourses; the magistrates, the literati, the ecclesiastics, and the very women, who all now affected to renounce the ancient philosophy.</p><p>In 1671, he received at Montpellier the same applauses for his lectures as at Toulouse. Finally, in 1680 he returned to Paris; where the concourse about him was such, that the sticklers for Peripateticism began to be alarmed. These applying to the archbishop of Paris, he thought it expedient, in the name of the king, to put a stop to the lectures; which accordingly were discontinued for several months. Afterwards his whole life was spent in propagating the new philosophy, both by lectures, and by publishing books. In defence of his system, he had disputes with Huet, Du Hamel, Malbranche, and others. His works, though abounding with ingenuity and learning, have been neglected in consequence of the great discoveries and advancement in philosophic knowledge that has been since made.— He was chosen a member of the Academy of Sciences in 1699; and died in 1707, at 75 years of age.</p><p>His works, which he published, are,</p><p>1. <hi rend="italics">A System of Philosophy;</hi> containing Logic, Metaphysics, and Morals; in 1690, 3 vols in 4to. being a compilation of the different ideas of Des Cartes.—It was reprinted the year after at Amsterdam, with the addition of a Discourse upon Ancient and Modern Philosophy.</p><p>2. <hi rend="italics">The Use of Reason and of Faith.</hi></p><p>3. An Answer to Huet's Censures of the Cartesian Philosophy; and an Answer to Du Hamel's Critical Reflections.</p><p>4. Some pieces against Malbranche, to shew that the apparent magnitude of an object depends solely on the magnitude of its image, traced on the retina.</p><p>5. A small piece upon the question, Whether Pleasure makes our present happiness?</p></div1><div1 part="N" n="REGRESSION" org="uniform" sample="complete" type="entry"><head>REGRESSION</head><p>, or <hi rend="smallcaps">Retrogradation</hi> <hi rend="italics">of Curves,</hi> &c. See <hi rend="smallcaps">Retrogradation.</hi></p><p>REGULAR <hi rend="italics">Figure,</hi> in Geometry, is a figure that is both equilateral and equiangular, or having all its sides and angles equal to one another.</p><p>For the dimensions, properties, &c, of regular figures, see <hi rend="smallcaps">Polygon.</hi></p><p><hi rend="smallcaps">Regular</hi> <hi rend="italics">Body,</hi> called also <hi rend="italics">Platonic Body,</hi> is a body or solid comprehended by like, equal, and regular plane figures, and whose solid angles are all equal.</p><p>The plane figures by which the solid is contain- <cb/> ed, are the faces of the solid. And the sides of the plane figures are the edges, or linear sides of the solid.</p><p>There are only five Regular Solids, viz,</p><p>The tetraedron, or regular triangular pyramid, having 4 triangular faces;</p><p>The hexaedron, or cube, having 6 square faces;</p><p>The octaedron, having 8 triangular faces;</p><p>The dodecaedron, having 12 pentagonal faces;</p><p>The icosaedron, having 20 triangular faces.</p><p>Besides these five, there can be no other Regular Bodies in nature.</p><p><hi rend="smallcaps">Prob.</hi> 1. <hi rend="italics">To construct or form the Regular Solids.</hi>— See the method of describing these figures under the article <hi rend="smallcaps">Body.</hi></p><p>2. <hi rend="italics">To find either the Surface or the Solid Content of any of the Regular Bodies.</hi>—Multiply the proper tabular area or surface (taken from the following Table) by the square of the linear edge of the solid, for the superficies. And</p><p>Multiply the tabular solidity, in the last column of the Table, by the cube of the linear edge, for the solid content. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="italics">Surfaces and Solidities of Regular Bodies, the side being</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="italics">unity or</hi> 1.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Name.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Surface.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Solidity.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">sides.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" rend="align=left" role="data">Tetraedron</cell><cell cols="1" rows="1" role="data">1.7320508</cell><cell cols="1" rows="1" role="data">0.1178513</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" rend="align=left" role="data">Hexaedron</cell><cell cols="1" rows="1" role="data">6.0000000</cell><cell cols="1" rows="1" role="data">1.0000000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=left" role="data">Octaedron</cell><cell cols="1" rows="1" role="data">3.4641016</cell><cell cols="1" rows="1" role="data">0.4714045</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">12</cell><cell cols="1" rows="1" rend="align=left" role="data">Dodecaedron</cell><cell cols="1" rows="1" role="data">20.6457788</cell><cell cols="1" rows="1" role="data">7.6631189</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=center" role="data">20</cell><cell cols="1" rows="1" rend="align=left" role="data">Icosaedron</cell><cell cols="1" rows="1" role="data">8.6602540</cell><cell cols="1" rows="1" role="data">2.1816950</cell></row></table></p><p>3. The Diameter of a Sphere being given, to find the side of any of the Platonic bodies, that may be either inscribed in the sphere, or circumscribed about the sphere, or that is equal to the sphere.</p><p>Multiply the given diameter of the sphere by the proper or corresponding number, in the following Table, answering to the thing sought, and the product will be the side of the Platonic body required. <table rend="border"><row role="data"><cell cols="1" rows="1" role="data">The diam. of a</cell><cell cols="1" rows="1" role="data">That may be</cell><cell cols="1" rows="1" role="data">That may be cir-</cell><cell cols="1" rows="1" role="data">That is equal</cell></row><row role="data"><cell cols="1" rows="1" role="data">sphere being 1,</cell><cell cols="1" rows="1" role="data">inscribed in the</cell><cell cols="1" rows="1" role="data">cumscribed about</cell><cell cols="1" rows="1" role="data">to the sphere,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the side of a</cell><cell cols="1" rows="1" role="data">sphere, is</cell><cell cols="1" rows="1" role="data">the sphere, is</cell><cell cols="1" rows="1" role="data">is</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Tetraedron</cell><cell cols="1" rows="1" role="data">0.816497</cell><cell cols="1" rows="1" role="data">2.44948</cell><cell cols="1" rows="1" role="data">1.64417</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Hexaedron</cell><cell cols="1" rows="1" role="data">0.577350</cell><cell cols="1" rows="1" role="data">1.00000</cell><cell cols="1" rows="1" role="data">0.88610</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Octaedron</cell><cell cols="1" rows="1" role="data">0.707107</cell><cell cols="1" rows="1" role="data">1.22474</cell><cell cols="1" rows="1" role="data">1.03576</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Dodecaedron</cell><cell cols="1" rows="1" role="data">0.525731</cell><cell cols="1" rows="1" role="data">0.66158</cell><cell cols="1" rows="1" role="data">0.62153</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Icosaedron</cell><cell cols="1" rows="1" role="data">0.356822</cell><cell cols="1" rows="1" role="data">0.44903</cell><cell cols="1" rows="1" role="data">0.40883</cell></row></table></p><p>4. The side of any of the five Platonic bodies being given, to find the diameter of a sphere, that may either be inscribed in that body, or circumscribed about it, or that is equal to it.—As the respective number in the Table above, under the title, <hi rend="italics">inscribed, circumscribed,</hi> or <hi rend="italics">equal,</hi> is to 1, so is the side of the given Platonic <pb n="355"/><cb/> body, to the diameter of its inscribed, circumscribed, or equal sphere.</p><p>5. The side of any one of the five Platonic bodies being given; to find the side of any of the other four bodies, that may be equal in solidity to that of the given body.—As the number under the title <hi rend="italics">equal</hi> in the last column of the table above, against the given Platonic body, is to the number under the same title, against the body whose side is sought, so is the side of the given Platonic body, to the side of the body sought.</p><p>See demonstrations of many other properties of the Platonic bodies, in my Mensuration, part 3 sect. 2 pa. 249, &c, 2d edition.</p><p><hi rend="smallcaps">Regular</hi> <hi rend="italics">Curve.</hi> See <hi rend="smallcaps">Curve.</hi></p><p>REGULATOR <hi rend="italics">of a Watch,</hi> is a small spring belonging to the balance, serving to adjust the going, and to make it go either faster or slower.</p></div1><div1 part="N" n="REGULUS" org="uniform" sample="complete" type="entry"><head>REGULUS</head><p>, in Astronomy, a star of the first magnitude, in the constellation Leo; called also, from its situation, <hi rend="italics">Cor Leonis,</hi> or the <hi rend="italics">Lion's Heart;</hi> by the Arabs, <hi rend="italics">Alhabor;</hi> and by the Chaldeans, <hi rend="italics">Kalbeleced,</hi> or <hi rend="italics">Karbeleceid;</hi> from an opinion of its influencing the affairs of the heavens; as Theon observes.</p><p>The longitude of Regulus, as fixed by Flamsteed, is 25° 31′ 21″, and its latitude 0° 26′ 38″ north. See <hi rend="smallcaps">Leo.</hi></p></div1><div1 part="N" n="REINFORCE" org="uniform" sample="complete" type="entry"><head>REINFORCE</head><p>, in Gunnery, is that part of a gun next the breech, which is made stronger to resist the force of the powder. There are usually two Reinforces in each piece, called the first and second Reinforce. The second is somewhat smaller than the first, because the inflamed powder in that part is less strong.</p><p><hi rend="smallcaps">Reinforce</hi> <hi rend="italics">Rings</hi> of a cannon, are flat mouldings, like iron hoops, placed at the breech end of the first and second Reinforce, projecting beyond the rest of the metal about a quarter of an inch.</p></div1><div1 part="N" n="REINHOLD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">REINHOLD</surname> (<foreName full="yes"><hi rend="smallcaps">Erasmus</hi></foreName>)</persName></head><p>, an eminent astronomer and mathematician, was born at Salfeldt in Thuringia, a province in Upper Saxony, the 11th of October 1511. He studied mathematics under James Milichi at Wittemberg, in which university he afterwards became professor of those sciences, which he taught with great applause. After writing a number of useful and learned works, he died the 19th of February 1553, at 42 years of age only. His writings are chiefly the following:</p><p>1. <hi rend="italics">Thcoriæ novæ Planetarum G. Purbachii,</hi> augmented and illustrated with diagrams and Scholia in 8vo, 1542; and again in 1580.—In this work, among other things worthy of notice, he teaches (pa. 75 and 76) that the centre of the lunar epicycle describes an <hi rend="italics">oval figure</hi> in each monthly period, and that the orbit of Mercury is also of the same oval figure.</p><p>2. <hi rend="italics">Ptolomy's Almagest,</hi> the first book, in Greek, with a Latin version, and Scholia, explaining the more obscure passages; in 8vo, 1549.—At the end of pa. 123 he promises an edition of Theon's Commentaries, which are very useful for understanding Ptolomy's meaning; but his immature death prevented Reinhold from giving this and other works which he had projected.</p><p>3. <hi rend="italics">Prutenicæ Tabulæ Cœlestium Motuum,</hi> in 4to, <cb/> 1551; again in 1571; and also in 1585.—Reinhold spent seven years labour upon this work, in which he was assisted by the munisicence of Albert, duke of Prussia, from whence the tables had their name. Reinhold compared the observations of Copernicus with those of Ptolomy and Hipparchus, from whence he constructed these new tables, the uses of which he has fully explained in a great number of precepts and canons, forming a complete introduction to practical astronomy.</p><p>4. <hi rend="italics">Primus liber Tabularum Directionum;</hi> to which are added, the <hi rend="italics">Canon Fœcundus,</hi> or Table of Tangents, to every minute of the quadrant; and New Tables of Climates, Parallels and Shadows, with an Appendix containing the second Book of the Canon of Directions; in 4to, 1554.—Reinhold here supplies what was omitted by Regiomontanus in his Table of Directions, &c; shewing the finding of the sines, and the construction of the tangents, the sines being found to every minute of the quadrant, to the radius 10,000,000; and he produced the Oblique Ascensions from 60 degrees to the end of the quadrant. He teaches also the use of these tables in the solution of spherical problems.</p><p>Reinhold prepared likewise an edition of many other works, which are enumerated in the Emperor's Privilege, prefixed to the Prutenic Tables. Namely, Ephemerides for several years to come, computed from the new tables. Tables of the Rising and Setting of several Fixed Stars, for many different climates and times. The illustration and establishment of Chronology, by the eclipses of the luminaries, and the great conjunctions of the planets, and by the appearance of comets, &c. The Ecclesiastical Calendar. The History of Years, or Astronomical Calendar. <hi rend="italics">Isagoge Spherica,</hi> or Elements of the Doctrine of the Primum Mobile. <hi rend="italics">Hypotyposes Orbium Cœlestium,</hi> or the Theory of Planets. Construction of a New Quadrant. The Doctrine of Plane and Spherical Triangles. Commentaries on the work of Copernicus. Also Commentaries on the 15 books of Euclid, on Ptolomy's Geography, and on the Optics of Alhazen the Arabian.—Reinhold also made Astronomical Observations, but with a wooden quadrant, which observations were seen by Tycho Brahe when he passed through Wittemberg in the year 1575, who wondered that so great a cultivator of astronomy was not furnished with better instruments.</p><p>Reinhold left a son, named also Erasmus after himself, an eminent mathematician and physician at Salfeldt. He wrote a small work in the German language, on Subterranean Geometry, printed in 4to at Erfurt 1575.—He wrote also concerning the New Star which appeared in Cassiopeia in the year 1572; with an Astrological Prognostication, published in 1574, in the German language.</p></div1><div1 part="N" n="RELAIS" org="uniform" sample="complete" type="entry"><head>RELAIS</head><p>, in Fortification, a French term, the same with berme.</p></div1><div1 part="N" n="RELATION" org="uniform" sample="complete" type="entry"><head>RELATION</head><p>, in Mathematics, is the habitude or respect of quantities of the same kind to each other, with regard to their magnitude; more usually called <hi rend="italics">ratio.</hi>—And the equality, identity, or sameness of two such Relations, is called proportion.</p><div2 part="N" n="Relation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Relation</hi></head><p>, <hi rend="italics">Inharmonical,</hi> in Musical Composition, <pb n="356"/><cb/> is that whose extremes form a false or unnatural interval, incapable of being sung.—This is otherwise called a <hi rend="italics">false Relation,</hi> and stands opposed to a just or true one.</p><p>RELATIVE <hi rend="italics">Gravity, Levity, Motion, Necessity, Place, Space, Time, Velocity, &c.</hi> See the several substantives.</p></div2></div1><div1 part="N" n="RELIEVO" org="uniform" sample="complete" type="entry"><head>RELIEVO</head><p>, in Architecture, denotes the sally or projecture of any ornament.</p></div1><div1 part="N" n="REMAINDER" org="uniform" sample="complete" type="entry"><head>REMAINDER</head><p>, is the difference between two quantities, or that which is left after subtracting one from the other.</p></div1><div1 part="N" n="RENDERING" org="uniform" sample="complete" type="entry"><head>RENDERING</head><p>, in Building. See <hi rend="smallcaps">Pargeting.</hi></p><p>REPELLING <hi rend="italics">Power,</hi> in Physics, is a certain power or faculty, residing in the minute particles of natural bodies, by which, under certain circumstances, they mutually fly from each other. This is the reverse or opposite of the attractive power. Newton shews, from observation, that such a force does really exist; and he argues, that as in algebra, where positive quantities cease, there negative ones begin; so in physics, where the attractive force ceases, there a Repelling force must begin.</p><p>As the Repelling power seems to arise from the same principle as the attractive, only exercised under different circumstances, it is governed by the same laws. Now the attractive power we find is stronger in small bodies, than in great ones, in proportion to the masses; therefore the Repelling is so too. But the rays of light are the most minute bodies we know of; and therefore their Repelling force must be the greatest. It is computed by Newton, that the attractive force of the rays of light is above 1000000000000000, or one thousand million of millions of times stronger than the force of gravity on the surface of the earth: hence arises that inconceivable velocity with which light must move to reach from the sun to the earth in little more than 7 minutes of time. For the rays emitted from the body of the sun, by the vibrating motion of its parts, are no sooner got without the sphere of attraction of the sun, than they come within the action of the Repelling power.</p><p>The elasticity or springiness of bodies, or that property by which, after having their figure altered by an external force, they return to their former shape again, follows from the Repelling power. See R<hi rend="smallcaps">EPULSION.</hi></p><p>REPERCUSSION. See <hi rend="smallcaps">Reflection.</hi></p></div1><div1 part="N" n="REPETEND" org="uniform" sample="complete" type="entry"><head>REPETEND</head><p>, in Arithmetic, denotes that part of an infinite decimal fraction, which is continually repeated ad infinitum. Thus in the numbers 2.13 13 13 &c. the figures 13 are the Repetend, and marked thus 1<hi rend="sup">.</hi>3<hi rend="sup">.</hi>.</p><p>These Repetends chiefly arise in the reduction of vulgar fractions to decimals. Thus, 1/3 = 0.333 &c = 0.3<hi rend="sup">.</hi>; and 1/6 = 01666 &c = 0.16<hi rend="sup">.</hi>; and 1/7 = 0.142857 142857 &c = 0.1<hi rend="sup">.</hi>42857<hi rend="sup">.</hi>. Where it is to be observed, that a point is set over the figure of a single Repetend, and a point over the first and last figure when there are several that repeat.</p><p>Repetends are either <hi rend="italics">single</hi> or <hi rend="italics">compound.</hi></p><p>A <hi rend="italics">Single</hi> <hi rend="smallcaps">Repetend</hi> is that in which only one figure repeats; as 0.3<hi rend="sup">.</hi>, or 0.6<hi rend="sup">.</hi>, &c. <cb/></p><p>A <hi rend="italics">Compound</hi> <hi rend="smallcaps">Repetend</hi>, is that in which two or more figures are repeated; as .1<hi rend="sup">.</hi>3<hi rend="sup">.</hi>, or .2<hi rend="sup">.</hi>15<hi rend="sup">.</hi>, or .1<hi rend="sup">.</hi>42857<hi rend="sup">.</hi>, &c.</p><p><hi rend="italics">Similar</hi> <hi rend="smallcaps">Repetends</hi> are such as begin at the same place, and consist of the same number of figures: as .3<hi rend="sup">.</hi> and .6<hi rend="sup">.</hi>, or 1.3<hi rend="sup">.</hi>41<hi rend="sup">.</hi> and 2.1<hi rend="sup">.</hi>56<hi rend="sup">.</hi>.</p><p><hi rend="italics">Dissimilar</hi> <hi rend="smallcaps">Repetends</hi> begin at different places, and consist of an unequal number of figures.</p><p><hi rend="italics">To find the finite Value of any Repetend,</hi> or to reduce it to a Vulgar Fraction. Take the given repeating figure or figures for the numerator; and for the denominator, take as many 9's as there are recurring figures or places in the given Repetend. .</p><p>Hence it follows, that every such infinite Repetend has a certain determinate and finite value, or can be expressed by a terminate vulgar fraction. And consequently, that an infinite decimal which does not repeat or circulate, cannot be completely expressed by a finite vulgar fraction.</p><p>It may farther be observed, that if the numerator of a vulgar fraction be 1, and the denominator any prime number, except 2 and 5, the decimal which shall be equal to that vulgar fraction, will always be a Repetend, beginning at the first place of decimals; and this Repetend must necessarily be a submultiple, or an aliquot part of a number expressed by as many 9's as the Repetend has figures; that is, if the Repetend have six figures, it will be a submultiple of 999999; if four figures, a submultiple of 9999 &c. From whence it follows, that if any prime number be called <hi rend="italics">p,</hi> the series 9999 &c, produced as far as is necessary, will always be divisible by <hi rend="italics">p,</hi> and the quotient will be the Repetend of the decimal fraction = 1/<hi rend="italics">p.</hi></p><p>RESIDUAL <hi rend="italics">Figure,</hi> in Geometry, the figure remaining after subtracting a less from a greater.</p><p><hi rend="smallcaps">Residual</hi> <hi rend="italics">Root,</hi> is a root composed of two parts or members, only connected together with the sign — or minus. Thus, <hi rend="italics">a</hi> — <hi rend="italics">b,</hi> or 5 — 3, is a residual root; and is so called, because its true value is no more than the residue, or difference between the parts <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> or 5 and 3, which in this case is 2.</p><p>RESIDUUM <hi rend="italics">of a Charge,</hi> in Electricity, first discovered by Mr. Gralath, in Germany, in 1746, is that part of the charge that lay on the uncoated part of a Leyden phial, which does not part with all its electricity at once; so that it is afterwards gradually diffused to the coating.</p></div1><div1 part="N" n="RESISTANCE" org="uniform" sample="complete" type="entry"><head>RESISTANCE</head><p>, or <hi rend="smallcaps">Resisting</hi> <hi rend="italics">Force,</hi> in Physics, any power which acts in opposition to another, so as to destroy or diminish its effect. <pb n="357"/><cb/></p><p>There are various kinds of Resistance, arising from the various natures and properties of the resisting bodies, and governed by various laws: as, the Resistance of solids, the Resistance of fluids, the Resistance of the air, &c. Of each of these in their order, as below.</p><p><hi rend="smallcaps">Resistance</hi> <hi rend="italics">of Solids,</hi> in Mechanics, is the force with which the quiescent parts of solid bodies oppose the motion of others contiguous to them.</p><p>Of these, there are two kinds. The first where the resisting and the resisted parts, i. e. the moving and quiescent bodies, are only contiguous, and do not cohere; constituting separate bodies or masses. This Resistance is what Leibnitz calls <hi rend="italics">Resistance of the surface,</hi> but which is more properly called <hi rend="italics">friction:</hi> for the laws of which, see the article <hi rend="smallcaps">Friction.</hi></p><p>The second case of Resistance, is where the resisting and resisted parts are not only contiguous, but cohere, being parts of the same continued body or mass. This Resistance was first considered by Galileo, and may properly be called <hi rend="italics">renitency.</hi></p><p>As to what regards the Resistance of bodies when struck by others in motion, see <hi rend="smallcaps">Percussion</hi>, and <hi rend="smallcaps">Collision.</hi></p><p><hi rend="italics">Theory of the Resistance of the Fibres of Solid Bodies.</hi> —To conceive an idea of this Resistance, or renitency of the parts, suppose a cylindrical body suspended vertically by one end. Here all its parts, being heavy, tend downwards, and endeavour to separate the two contiguous planes or surfaces where the body is the weakest; but all the parts of them resist this separation by the force with which they cohere, or are bound together. Here then are two opposite powers; viz, the weight of the cylinder, which tends to break it; and the force of cohesion of the parts, which resists the fracture.</p><p>If now the base of the cylinder be increased, without increasing its length; it is evident that both the Resistance and the weight will be increased in the same ratio as the base; and hence it appears that all cylinders of the same matter and length, whatever their bases be, have an equal Resistance, when vertically suspended.</p><p>But if the length of the cylinder be increased, without increasing its base, its weight is increased, while the Resistance or strength continues unaltered; consequently the lengthening has the effect of weakening it, or increases its tendency to break.</p><p>Hence to find the greatest length a cylinder of any matter may have, when it just breaks with the addition of another given weight, we need only take any cylinder of the same matter, and fasten to it the least weight that is just sufficient to break it; and then consider how much it must be lengthened, so that the weight of the part added, together with the given weight, may be just equal to that weight, and the thing is done. Thus, let <hi rend="italics">l</hi> denote the first length of the cylinder, <hi rend="italics">c</hi> its weight, <hi rend="italics">g</hi> the given weight the lengthened cylinder is to bear, and <hi rend="italics">w</hi> the least weight that breaks the cylinder <hi rend="italics">l,</hi> also <hi rend="italics">x</hi> the length sought; then as <hi rend="italics">l</hi> : <hi rend="italics">x</hi> : : <hi rend="italics">c</hi> : (<hi rend="italics">cx</hi>)/<hi rend="italics">l</hi> = the weight of the longest cylinder sought; and this, together with the given <cb/> weight <hi rend="italics">g,</hi> must be equal to <hi rend="italics">c</hi> together with the weight <hi rend="italics">w;</hi> hence then the whole length of the cylinder sought. If the cylinder must just break with its own weight, then is <hi rend="italics">g</hi> = 0, and in that case is the whole length that just breaks by its own weight. By this means Galileo found that a copper wire, and of consequence any other cylinder of copper, might be extended to 4801 braccios or fathoms of 6 feet each.</p><p>If the cylinder be fixed by one end into a wall, with the axis horizontally; the force to break it, and its Resistance to fracture, will here be both different; as both the weight to cause the fracture, and the Resistance of the fibres to oppose it, are combined with the effects of the lever; for the weight to cause the fracture, whether of the weight of the beam alone, or combined with an additional weight hung to it, is to be supposed collected into the centre of gravity, where it is considered as acting by a lever equal to the distance of that centre beyond the face of the wall where the cylinder or other prism is fixed; and then the product of the said whole weight and distance, will be he momentum or force to break the prism. Again, the Resistance of the fibres may be supposed collected into the centre of the transverse section, and all acting there at the end of a lever equal to the vertical semidiameter of the section, the lowest point of that diameter being immoveable, and about which the whole diameter turns when the prism breaks; and hence the product of the adhesive force of the fibres multiplied by the said semidiameter, will be the momentum of Resistance, and must be equal to the former momentum when the prism just breaks.</p><p>Hence, to find the length a prism will bear, fixed so horizontally, before it breaks, either by its own weight, or by the addition of any adventitious weight; take any length of such a prism, and load it with weights till it just break. Then, put <hi rend="italics">l</hi> = the length of this prism, <hi rend="italics">c</hi> = its weight, <hi rend="italics">w</hi> = the weight that breaks it, <hi rend="italics">a</hi> = distance of weight <hi rend="italics">w,</hi> <hi rend="italics">g</hi> = any given weight to be borne, <hi rend="italics">d</hi> = its distance, <hi rend="italics">x</hi> = the length required to break.</p><p>Then <hi rend="italics">l</hi> : <hi rend="italics">x</hi> : : <hi rend="italics">c</hi> :(<hi rend="italics">cx</hi>)/<hi rend="italics">l</hi> the weight of the prism <hi rend="italics">x,</hi> and its momentum; also <hi rend="italics">dg</hi> = the momentum of the weight <hi rend="italics">g;</hi> therefore (<hi rend="italics">cx</hi><hi rend="sup">2</hi>)/(2<hi rend="italics">l</hi>) + <hi rend="italics">dg</hi> is the momentum of the prism <hi rend="italics">x</hi> and its added weight. In like manner 1/2<hi rend="italics">cl</hi> + <hi rend="italics">aw</hi> is that of the former or short prism and the weight that brake it; consequently is the length sought, that just <pb n="358"/><cb/> breaks with the weight <hi rend="italics">g</hi> at the distance <hi rend="italics">d.</hi> If this weight <hi rend="italics">g</hi> be nothing, then is the length of the prism that just breaks with its own weight.</p><p>If two prisins of the same matter, having their bases and lengths in the same proportion, be suspended horizontally; it is evident that the greater has more weight than the lesser, both on account of its length, and of its base; but it has less Resistance on account of its length, considered as a longer arm of a lever, and has only more Resistance on account of its base; therefore it exceeds the lesser in its momentum more than it does in its Resistance, and consequently it must break more easily.</p><p>Hence appears the reason why, in making small machines and models, people are apt to be mistaken as to the Resistance and strength of certain horizontal pieces, when they come to execute their designs in large, by observing the same proportions as in the small.</p><p>When the prism, fixed vertically, is just about to break, there is an equilibrium between its positive and relative weight; and consequently those two opposite powers are to each other reciprocally as the arms of the lever to which they are applied, that is, as half the diameter to half the axis of the prism. On the other hand, the Resistance of a body is always equal to the greatest weight which it will just sustain in a vertical position, that is, to its absolute weight. Therefore, substituting the absolute weight for the Resistance, it appears, that the absolute weight of a body, suspended horizontally, is to its relative weight, as the distance of its centre of gravity from the fixed point or axis of motion, is to the distance of the centre of gravity of its base from the same.</p><p>The discovery of this important truth, at least of an equivalent to it, and to which this is reducible, we owe to Galileo. On this system of Resistance of that author, Mariotte made an ingenious remark, which gave birth to a new system. Galileo supposes that where the body breaks, all the sibres break at once; so that the body always resists with its whole absolute force, or the whole force that all its fibres have in the place where it breaks. But Mariotte, finding that all bodies, even glass itself, bend before they break, shews that fibres are to be considered as so many little bent springs, which never exert their whole force, till stretched to a certain point, and never break till entirely unbent. Hence those nearest the fulcrum of the lever, or lowest point of the fracture, are stretched less than those farther off, and consequently employ a less part of their force, and break later.</p><p>This consideration only takes place in the horizontal situation of the body: in the vertical, the fibres of the base all break at once; so that the absolute weight of the body must exceed the united Resistance of all its fibres; a greater weight is therefore required here than in the horizontal situation, that is, a greater weight is required to overcome their united Resistance, than to overcome their several Resistances one after another.</p><p>Varignon has improved on the system of Mariotte, <cb/> and shewn that to Galileo's system, it adds the consideration of the centre of percussion. In each system, the section, where the body breaks, moves on the axis of equilibrium, or line at the lower extremity of the same section; but in the second, the fibres of this section are continually stretching more and more, and that in the same ratio, as they are situated farther and farther from the axis of equilibrium, and consequently are still exerting a greater and greater part of their whole force.</p><p>These unequal extensions, like all other forces, must have some common centre where they are united, making equal efforts on each side of it; and as they are precisely in the same proportion as the velocities which the several points of a rod moved circularly would have to one another, the centre of extension of the section where the body breaks, must be the same as its centre of percussion. Galileo's hypothesis, where fibres stretch equally, and break all at once, corresponds to the case of a rod moving parallel to itself, where the centre of extension or percussion does not appear, as being confounded with the centre of gravity.</p><p>Hence it follows, that the Resistance of bodies in Mariotte's system, is to that in Galileo's, as the distance of the centre of percussion, taken on the vertical diameter of the fracture, is to the whole of that diameter. Hence also, the Resistance being less than what Galileo imagined, the relative weight must also be less, and in the ratio just mentioned. So that, after conceiving the relative weight of a body, and its Resistance equal to its absolute weight, as two contrary powers applied to the two arms of a lever, in the hypothesis of Galileo, there needs nothing to change it into that of Mariotte, but to imagine that the Resistance, or the absolute weight, is become less, in the ratio above mentioned, every thing else remaining the same.</p><p>One of the most curious, and perhaps the most useful questions in this research, is to find what figure a body must have, that its Resistance may be equal or proportional in every part to the force tending to break it. Now to this end, it is necessary, some part of it being conceived as cut off by a plane parallel to the fracture, that the momentum of the part retrenched be to its Resistance, in the same ratio as the momentum of the whole is to its Resistance; these four powers acting by arms of levers peculiar to themselves, and are proportional in the whole, and in each part, of a solid of equal Resistance. From this proportion, Varignon easily deduces two solids, which shall resist equally in all their parts, or be no more liable to break in one part than in another: Galileo had found one before. That discovered by Varignon is in the form of a trumpet, and is to be fixed into a wall at its greater end; so that its magnitude or weight is always diminished in proportion as its length, or the arm of the lever by which its weight acts, is increased. It is remarkable that, howsoever different the two systems may be, the solids of equal Resistance are the same in both.</p><p>For the Resistance of a solid supported at each end, as of a beam between two walls, see <hi rend="smallcaps">Beam.</hi></p><p><hi rend="smallcaps">Resistance</hi> <hi rend="italics">of Fluids,</hi> is the force with which <pb n="359"/><cb/> bodies, moving in fluid mediums, are impeded and retarded in their motion.</p><p>A body moving in a fluid is resisted from two causes. The first of these is the cohesion of the parts of the fluid. For a body, in its motion, separating the parts of a fluid, must overcome the force with which those parts cohere. The second is the inertia, or inactivity of matter, by which a certain force is required to move the particles from their places, in order to let the body pass.</p><p>The retardation from the first cause is always the same in the same space, whatever the velocity be, the body remaining the same; that is, the Resistance is as the space run through, in the same time: but the velocity is also in the same ratio of the space run over in the same time: and therefore the Resistance, from this cause, is as the velocity itself.</p><p>The Resistance from the second cause, when a body moves through the same fluid with different velocities, is as the square of the velocity. For, first the Resistance increases according to the number of particles or quantity of the fluid struck in the same time; which number must be as the space run through in that time, that is, as the velocity: but the Resistance also increases in proportion to the force with which the body strikes against every part; which force is also as the velocity of the body, so as to be double with a double velocity, and triple with a triple one, &c: therefore, on both these accounts, the Resistance is as the velocity multiplied by the velocity, or as the square of the velocity. Upon the whole therefore, on account of both causes, viz, the tenacity and inertia of the fluid, the body is resisted partly as the velocity and partly as the square of the velocity.</p><p>But when the same body moves through different fluids with the same velocity, the Resistance from the second cause follows the proportion of the matter to be removed in the same time, which is as the density of the fluid.</p><p>Hence therefore, if <hi rend="italics">d</hi> denote the density of the fluid, <hi rend="italics">v</hi> the velocity of the body, and <hi rend="italics">a</hi> and <hi rend="italics">b</hi> constant coefficients: then <hi rend="italics">adv</hi><hi rend="sup">2</hi> + <hi rend="italics">bv</hi> will be proportional to the whole Resistance to the same body, moving with different velocities, in the same direction, through fluids of different densities, but of the same tenacity.</p><p>But, to take in the consideration of different tenacities of fluids; if <hi rend="italics">t</hi> denote the tenacity, or the cohesion of the parts of the fluid, then <hi rend="italics">adv</hi><hi rend="sup">2</hi> + <hi rend="italics">btv</hi> will be as the said whole Resistance.</p><p>Indeed the quantity of Resistance from the cohesion of the parts of fluids, except in glutinous ones, is very small in respect of the other Resistance; and it also increases in a much lower degree, being only as the velocity, while the other increases as the square of the velocity, and rather more. Hence then the term <hi rend="italics">btv</hi> is very small in respect of the other term <hi rend="italics">adv</hi><hi rend="sup">2</hi>; and consequently the Resistance is nearly as this latter term; or nearly as the square of the velocity. Which rule has been employed by most authors, and is very near the truth in slow motions; but in very rapid ones, it differs considerably from the truth, as we shall perceive below; not indeed from the omission of the small term <hi rend="italics">ctv,</hi> due to the cohesion, but from the want of the full <cb/> counter pressure on the hinder part of the body, a vacuum, either perfect or partial, being left behind the body in its motion; and also perhaps to some compression or accumulation of the fluid against the fore part of the body. Hence,</p><p>To conceive the Resistance of fluids to a body moving in them, we must distinguish between those fluids which, being greatly compressed by some incumbent weight, always close up the space behind the body in motion, without leaving any vacuity there; and those fluids which, not being much compressed, do not quickly fill up the space quitted by the body in motion, but leave a kind of vacuum behind it. These differences, in the resisting fluids, will occasion very remarkable varieties in the laws of their Resistance, and are absolutely necessary to be considered in the determination of the action of the air on shot and shells; for the air partakes of both these affections, according to the different velocities of the projected body.</p><p>In treating of these Resistances too, the fluids may be considered either as continued or discontinued, that is, having their particles contiguous or else as separated and unconnected; and also either as elastic or nonelastic. If a fluid were so constituted, that all the particles composing it were at some distance from each other, and having no action between them, then the Resistance of a body moving in it would be easily computed, from the quantity of motion communicated to those particles; for instance, if a cylinder moved in such a fluid in the direction of its axis, it would communicate to the particles it met with, a velocity equal to its own, and in its own direction, when neither the cylinder nor the parts of the fluid are elastic: whence, if the velocity and diameter of the cylinder be known, and also the density of the fluid, there would thence be determined the quantity of motion communicated to the fluid, which (as action and reaction are equal) is the same with the quantity lost by the cylinder, and consequently the Resistance would thus be ascertained.</p><p>In this kind of discontinued fluid, the particles being detached from each other, every one of them can pursue its own motion in any direction, at least for some time, independent of the neighbouring ones; so that, instead of a cylinder moving in the direction of its axis, if a body with a surface oblique to its direction be supposed to move in such a fluid, the motion which the parts of the fluid will hence acquire, will not be in the direction of the resisted body, but perpendicular to its oblique surface; whence the Resistance to such a body will not be estimated from the whole motion communicated to the particles of the fluid, but from that part of it only which is in the direction of the resisted body. In fluids then, where the parts are thus discontinued from each other, the different obliquities of that surface which goes foremost, will occasion considerable changes in the Resistance; although the transverse section of the solid should in all cases be the same: And Newton has particularly determined that, in a fluid thus constituted, the Resistance of a globe is but half the Resistance of a cylinder of the same diameter, moving, in the direction of its axis, with the same velocity.</p><p>But though the hypothesis of a fluid thus constituted <pb n="360"/><cb/> be of great use in explaining the nature of Resistances, yet we know of no such fluid existing in nature; all the fluids with which we are conversant being so formed, that their particles either lie contiguous to each other, or at least act on each other in the same manner as if they did: consequently, in these fluids, no one particle that is contiguous to the resisted body, can be moved, without moving at the same time a great number of others, some of which will be distant from it; and the motion thus communicated to a mass of the fluid, will not be in any one determined direction, but different in all the particles, according to the different positions in which they lie in contact with those from which they receive their impulse<hi rend="italics">;</hi> whence, great numbers of the particles being diverted into oblique directions, the Resistance of the moving body, which will depend on the quantity of motion communicated to the fluid in its own direction, will be different in quantity from what it would be in the foregoing supposition, and its estimation becomes much more complicated and oper<*>se.</p><p>If the fluid be compressed by the incumbent weight of its upper parts (as all fluids are with us, except at their very surface), and if the velocity of the moving body be much less than that with which the parts of the fluid would rush into a void space, in consequence of their compression; it is evident, that in this case the space left by the moving body will be instantaneously filled up by the fluid; and the parts of the fluid against which the foremost part of the body presses in its motion, will, instead of being impelled forwards in the direction of the body, in some measure circulate towards the hinder part of the body, in order to restore the equilibrium, which the constant influx of the fluid behind the body would otherwise destroy; whence the progressive motion of the fluid, and consequently the Resistance of the body, which depends upon it, would in this instance be much less, than in the hypothesis where each particle is supposed to acquire, from the stroke of the resisting body, a velocity equal to that with which the body moved, and in the same direction. Newton has determined, that the Resistance of a cylinder, moving in the direction of its axis, in such a compressed fluid as we have here treated of, is but one-fourth part of the Resistance to the same cylinder, if it moved with the same velocity in a fluid constituted in the manner described in the first hypothesis, each fluid being supposed of the same density.</p><p>But again, it is not only in the quantity of their Resistance that these fluids differ, but also in the different manner in which they act upon solids of different forms moving in them. In the discontinued fluid, first described, the obliquity of the foremost surface of the moving body would diminish the Resistance; but the same thing does not hold true in compressed fluids, at least not in any considerable degree; for the chief Resistance in compressed fluids arises from the greater or less facility with which the fluid, impelled by the fore part of the body, can circulate towards its hinder part; and this being little, if at all, affected by the form of the moving body, whether it be cylindrical, conical, or spherical, it follows, that while the transverse section of the body is the same, and consequently the quan- <cb/> tity of impelled fluid also, the change of figure in the body will scarcely affect the quantity of its Resistance.</p><p>And this case, viz, the Resistance of a compressed fluid to a solid, moving in it with a velocity much less than what the parts of the fluid would acquire from their compression, has been very fully considered by Newton, who has ascertained the quantity of such a Resistance, according to the different magnitudes of the moving body, and the density of the fluid. But he expressly informs us that the rules he has laid down, are not generally true, but only upon a supposition that the compression of the fluid be increased in the greater velocities of the moving body: however, some unskilful writers, who have followed him, overlooking this caution, have applied his determination to bodies moving with all sorts of velocities, without attending to the different compressions of the fluids they are resisted by; and by this means they have accounted the Resistance, for instance, of the air to musket and cannon shot, to be but about one-third part of what it is found to be by experience.</p><p>It is indeed evident that the resisting power of the medium must be increased, when the resisted body moves so fast that the fluid cannot instantaneously press in behind it, and fill the deserted space; for when this happens, the body will be deprived of the pressure of the fluid behind it; which in some measure balanced its Resistance, or at least the fore pressure, and must support on its fore part the whole weight of a column of the fluid, over and above the motion it gives to the parts of the same; and besides, the motion in the particles driven before the body, is less affected in this case by the compression of the fluid, and consequently they are less deflected from the direction in which they are impelled by the resisted surface; whence it happens that this species of Resistance approaches more and more to that described in the first hypothesis, where each particle of the fluid being unconnected with the neighbouring ones, pursued its own motion, in its own direction, without being interrupted or deflected by their contiguity; and therefore, as the Resistance of a discontinued fluid to a cylinder, moving in the direction of its axis, is 4 times greater than the Resistance of a fluid sufficiently compressed of the same density, it follows that the Resistance of a fluid, when a vacuity is left behind the moving body, may be near 4 times greater than that of the same fluid, when no such vacuity is formed; for when a void space is thus left, the Resistance approaches in its nature to that of a discontinued fluid.</p><p>This then may probably be the case in a cylinder moving in the same compressed fluid, according to the different degrees of its velocity; so that if it set out with a great velocity, and moves in the fluid till that velocity be much diminished, the resisting power of the medium may be near 4 times greater in the beginning of its motion than in the end.</p><p>In a globe, the difference will not be so great, because, on account of its oblique surface, its Resistance in a discontinued medium is but about twice as much as in one properly compressed; for its oblique surface diminishes its Resistance in one case, and not in the other: however, as the compression of the medium, <pb n="361"/><cb/> even when a vacuity is left behind the moving body, may yet confine the oblique motion of the parts of the fluid, which are driven before the body, and as in an elastic fluid, such as our air is, there will be some degree of condensation in those parts; it is highly probable that the Resistance of a globe, moving in a compressed fluid with a very great velocity, may greatly exceed the proportion of the Resistance to slow motions.</p><p>And as this increase of the resisting power of the medium will take place, when the velocity of the moving body is so great, that a perfect vacuum is left behind it, so some degree of augmentation will be sensible in velocities much short of this; for even when, by the compression of the fluid, the space left behind the body is instantaneously filled up; yet, if the velocity with which the parts of the fluid rush in behind, is not much greater than that with which the body moves, the same reasons that have been urged above, in the case of an absolute vacuity, will hold in a less degree in this instance; and therefore it is not to be supposed that, in the increased Resistance which has been hitherto treated of, it immediately vanishes when the compression of the fluid is just sufficient to prevent a vacuum behind the resisted body; but we must consider it as diminishing only according as the velocity, with which the parts of the fluid follow the body, exceeds that with which the body moves.</p><p>Hence then it may be concluded, that if a globe sets out in a resisting medium, with a velocity much exceeding that with which the particles of the medium would rush into a void space, in consequence of their compression, so that a vacuum is necessarily left behind the globe in its motion; the Resistance of this medium to the globe will be much greater, in proportion to its velocity, than what we are sure, from Sir I. Newton, would take place in a slower motion. We may farther conclude, that the resisting power of the medium will gradually diminish as the velocity of the globe decreases, till at last, when it moves with velocities which bear but a small proportion to that with which the particles of the medium follow it, the Resistance becomes the same with what is assigned by Newton in the case of a compressed fluid.</p><p>And from this determination may be seen, how false that position is, which asserts the Resistance of any medium to be always in the duplicate ratio of the velocity of the resisted body; for it plainly appears, by what has been said, that this can only be considered as nearly true in small variations of velocity, and can never be applied in comparing together the Resistances to all velocities whatever, without incurring the most enormous errors. See Robins's Gunnery, chap. 2 prop. 1, and my Select Exercises pa. 235 &c. See also the articles <hi rend="smallcaps">Resistance</hi> <hi rend="italics">of the Air,</hi> <hi rend="smallcaps">Projectile</hi>, and <hi rend="smallcaps">Gunnery.</hi></p><p>Resistance and retardation are used indifferently for each other, as being both in the same proportion, and the same Resistance always generating the same retardation. But with regard to different bodies, the same Resistance frequently generates different retardations; the Resistance being as the quantity of motion, and the retardation that of the celerity. For the difference and measure of the two, see <hi rend="smallcaps">Retardation.</hi></p><p>The retardations from this Resistance may be com- <cb/> pared together, by comparing the Resistance with the gravity or quantity of matter. It is demonstrated that the Resistance of a cylinder, which moves in the direction of its axis, is equal to the weight of a column of the fluid, whose base is equal to that of the cylinder, and its altitude equal to the height through which a body must fall in vacuo, by the force of gravity, to acquire the velocity of the moving body. So that, if <hi rend="italics">a</hi> denote the area of the face or end of the cylinder, or other prism, <hi rend="italics">v</hi> its velocity, and <hi rend="italics">n</hi> the specific gravity of the fluid; then, the altitude due to the velocity <hi rend="italics">v</hi> being (<hi rend="italics">v</hi><hi rend="sup">2</hi>)/(4<hi rend="italics">g</hi>), the whole Resistance, or motive force <hi rend="italics">m,</hi> will be ; the quantity <hi rend="italics">g</hi> being = 16 1/12 feet, or the space a body falls, in vacuo, in the first second of time. And the Resistance to a globe of the same diameter would be the half of this.—Let a ball, for instance, of 3 inches diameter, be moved in water with a celerity of 16 feet per second of time: now from experiments on pendulums, and on falling bodies, it has been found, that this is the celerity which a body acquires in falling from the height of 4 feet; therefore the weight of a cylinder of water of 3 inches diameter, and 4 feet high, that is a weight of about 12 lb 4 oz, is equal to the Resistance of the cylinder; and consequently the half of it, or 6 lb 2 oz is that of the ball. Or, the formula (<hi rend="italics">anv</hi><hi rend="sup">2</hi>)/(4<hi rend="italics">g</hi>) gives oz, or 12 lb 4 oz, for the Resistance of the cylinder, or 6 lb 2 oz for that of the ball, the same as before.</p><p>Let now the Resistance, so discovered, be divided by the weight of the body, and the quotient will shew the ratio of the retardation to the force of gravity. So if the said ball, of 3 inches diameter, be of cast iron, it will weigh nearly 61 ounces, or 3 4/5 lb; and the Resistance being 6 lb 2 oz, or 98 ounces; therefore, the Resistance being to the gravity as 98 to 61, the retardation, or retarding force, will be 98/61 or 1 3/5, the force of gravity being 1. Or thus; because <hi rend="italics">a</hi> the area of a great circle of the ball, is = <hi rend="italics">pd</hi><hi rend="sup">2</hi>, where <hi rend="italics">d</hi> is the diameter, and <hi rend="italics">p</hi> = .7854, therefore the Resistance to the ball is ; and because its solid content is , and its weight (2/3)N<hi rend="italics">pd</hi><hi rend="sup">3</hi>, where N denotes its specific gravity; therefore, dividing the Resistance or motive force <hi rend="italics">m</hi> by the weight <hi rend="italics">w,</hi> gives the retardation, or retarding force, that of gravity being 1; which is therefore as the square of the velocity directly, and as the diameter inversely; and this is the reason why a large ball overcomes the Resistance better than a small one, of the same density. See my Select Exercises, pa. 225 &c.</p><p><hi rend="smallcaps">Resistance</hi> <hi rend="italics">of Fluid Mediums to the Motion of Falling Bodies.</hi>—A body freely descending in a fluid, is accelerated by the relative gravity of the body, (that is, the difference between its own absolute gravity and that of a like bulk of the fluid), which continually acts upon it, yet not equably, as in a vacuum: the Resistance of the fluid occasions a retardation, or diminution <pb n="362"/><cb/> of acceleration, which diminution increases with the velocity of the body. Hence it happens, that there is a certain velocity, which is the greatest that a body can acquire by falling; for if its velocity be such, that the Resistance arising from it becomes equal to the relative weight of the body, its motion can be no longer accelerated; for the motion here continually generated by the relative gravity, will be destroyed by the Resistance, or the force of Resistance is equal to the relative gravity, and the body forced to go on equably: for after the velocity is arrived at such a degree, that the resisting force is equal to the weight that urges it, it will increase no longer, and the globe must afterward continue to descend with that velocity uniformly. A body continually comes nearer and nearer to this greatest celerity, but can never attain accurately to it. Now, N and <hi rend="italics">n</hi> being the specific gravities of the globe and fluid, N - <hi rend="italics">n</hi> will be the relative gravity of the globe in the fluid, and therefore is the weight by which it is urged downward; also is the Resistance, as above; therefore these two must be equal when the velocity can be no farther increased, or <hi rend="italics">m</hi> = <hi rend="italics">w,</hi> that is ; and hence is the said uniform or greatest velocity to which the body may attain; which is evidently the greater in the subduplicate proportion of <hi rend="italics">v</hi> the diameter of the ball. But <hi rend="italics">v</hi> is always = √4<hi rend="italics">gfs,</hi> the velocity generated by any accelerative force <hi rend="italics">f</hi> in describing the space <hi rend="italics">s;</hi> which being compared with the former, it gives <hi rend="italics">s</hi> = (4/3)<hi rend="italics">d,</hi> when <hi rend="italics">f</hi> is = (N - <hi rend="italics">n</hi>)/<hi rend="italics">n;</hi> that is, the greatest velocity is that which is generated by the accelerating force (N - <hi rend="italics">n</hi>)/<hi rend="italics">n</hi> in passing over the space (4/3)<hi rend="italics">d</hi> or 4/3 of the diameter of the ball, or it is equal to the velocity generated by gravity in describing the space . For ex. if the ball be of lead, which is about 11 1/4 times the density of water; then , and ; that is, the uniform or greatest velocity of a ball of lead, descending in water, is equal to that which a heavy body acquired by falling in vacuo through a space equal to 13 2/3 of the diameter of the ball, which velocity is nearly, or 8 times the root of the same space.</p><p>Hence it appears, how soon small bodies come to their greatest or uniform velocity in descending in a fluid, as water, and how very small that velocity is: which explains the reason of the slow precipitation of mud, and small particles, in water, as also why, in <cb/> precipitations, the larger and gross particles descend soonest, and the lowest.</p><p>Farther, where N = <hi rend="italics">n,</hi> or the density of the fluid is equal to that of the body, then N - <hi rend="italics">n</hi> = 0, consequently the velocity and distance descended are each nothing, and the body will just float in any part of the fluid.</p><p>Moreover, when the body is lighter than the fluid, then N is less than <hi rend="italics">n,</hi> and N - <hi rend="italics">n</hi> becomes a negative quantity, or the force and motion tend the contrary way, that is, the ball will ascend up towards the top of the fluid by a motive force which is as <hi rend="italics">n</hi> - N. In this case then, the body ascending by the action of the fluid, is moved exactly by the same laws as a heavier body falling in the fluid. Wherever the body is placed, it is sustained by the fluid, and carried up with a force equal to the difference of the weight of a quantity of the fluid of the same bulla as the body, from the weight of the body; there is therefore a force which continually acts equably upon the body; by which not only the action of gravity of the body is counteracted, so as that it is not to be considered in this case; but the body is also carried upwards by a motion equably accelerated, in the same manner as a body heavier than a fluid descends by its relative gravity: but the equability of acceleration is destroyed in the same manner by the Resistance, in the ascent of a body lighter than the fluid, as it is destroyed in the descent of a body that is heavier.</p><p>For the circumstances of the correspondent velocity, space, and time, &c, of a body moving in a fluid in which it is projected with a given velocity, or descending by its own weight, &c, see my Select Exercises, prop. 29, 30, 31, and 32, pag. 221 &c.</p><p><hi rend="smallcaps">Resistance</hi> <hi rend="italics">of the Air,</hi> in Pneumatics, is the force with which the motion of bodies, particularly of projectiles, is retarded by the opposition of the air or atmosphere. See <hi rend="smallcaps">Gunnery, Projectiles</hi>, &c.</p><p>The air being a fluid, the general laws of the Resistance of fluids obtain in it; subject only to some variations and irregularities from the different degrees of density in the different stations or regions of the atmosphere.</p><p>The Resistance of the air is chiefly of use in military projectiles, in order to allow for the differences caused in their flight and range by it. Before the time of Mr. Robins, it was thought that this Resistance to the motion of such heavy bodies as iron balls and shells, was too inconsiderable to be regarded, and that the rules and conclusions derived from the common parabolic theory, were sufficiently exact for the common practice of gunnery. But that gentleman shewed, in his New Principles of Gunnery, that, so far from being inconsiderable, it is in reality enormously great, and by no means to be rejected without incurring the grossest errors; so much so, that balls or shells which range, at the most, in the air, to the distance of two or three miles, would in a vacuum range to 20 or 30 miles, or more. To determine the quantity of this Resistance, in the case of different velocities, Mr. Robins discharged musket balls, with various degrees of known velocity, against his ballistic pendulums, placed at several different distances, and so discovered by experiment the quantity of velocity lost, when passing through those distances <pb n="363"/><cb/> or spaces of air, with the several known degrees of celerity. For having thus known, the velocity lost or destroyed, in passing over a certain space, in a certain time, (which time is very nearly equal to the quotient of the space divided by the medium velocity between the greatest and least, or between the velocity at the mouth of the gun and that at the pendulum); that is, knowing the velocity <hi rend="italics">v,</hi> the space <hi rend="italics">s,</hi> and time <hi rend="italics">t,</hi> the resisting force is thence easily known, being equal to (<hi rend="italics">vb</hi>)/(2<hi rend="italics">gt</hi>) or (<hi rend="italics">v</hi>V<hi rend="italics">b</hi>)/(2<hi rend="italics">gs</hi>), where <hi rend="italics">b</hi> denotes the weight of the ball, and V the medium velocity above mentioned. The balls employed upon this occasion by Mr. Robins, were leaden ones, of 1/12 of a pound weight, and 3/4 of an inch diameter; and to the medium velocity of <table><row role="data"><cell cols="1" rows="1" role="data">1600 feet the Resistance was</cell><cell cols="1" rows="1" rend="align=center" role="data">11 lb,</cell></row><row role="data"><cell cols="1" rows="1" role="data">1065 feet " it was</cell><cell cols="1" rows="1" rend="align=center" role="data">2 4/5;</cell></row></table> but by the theory of Newton, before laid down, the former of these should be only 4 1/2 lb, and the latter 2 lb: so that, in the former case the real Resistance is more than double of that by the theory, being increased as 9 to 22; and in the lesser velocity the increase is from 2 to 2 4/5, or as 5 to 7 only.</p><p>Mr. Robins also invented another machine, having a whirling or circular motion, by which he measured the Resistances to larger bodies, though with much smaller velocities: it is described, and a figure of it given, near the end of the 1st vol. of his works.</p><p>That this resisting power of the air to swift motions is very sensibly increased beyond what Newton's theory for slow motions makes it, seems hence to be evident. By other experiments it appears that the Resistance is very sensibly increased, even in the velocity of 400 feet. However, this increased power of Resistance diminishes as the velocity of the resisted body diminishes, till at length, when the motion is sufficiently abated, the actual Resistance coincides with that supposed in the theory nearly. For these varying Resistances Mr. Robins has given a rule, extending to 1670 feet velocity.</p><p>Mr. Euler has shewn, that the common doctrine of Resistance answers pretty well when the motion is not very swift, but in swift motions it gives the Resistance less than it ought to be, on two accounts. 1. Because in quick motions, the air does not fill up the space behind the body fast enough to press on the hinder parts, to counterbalance the weight of the atmosphere on the fore part. 2. The density of the air before the ball being increased by the quick motion, will press more strongly on the fore part, and so will resist more than lighter air in its natural state. He has shewn that Mr. Robins has restrained his rule to velocities not exceeding 1670 feet per second; whereas had he extended it to greater velocities, the result must have been erroneous; and he gives another formula himself, and deduces conclusions differing from those of Mr. Robins. See his Principles of Gunnery investigated, translated by Brown in 1777, pa. 224 &c.</p><p>Mr. Robins having proved that, in very great changes <cb/> of velocity, the Resistance does not accurately follow the duplicate ratio of the velocity, lays down two positions, which he thought might be of some service in the practice of artillery, till a more complete and accurate theory of Resistance, and the changes of its augmentation, may be obtained. The first of these is, that till the velocity of the projectile surpass 1100 or 1200 feet in a second, the Resistance may be esteemed to be in the duplicate ratio of the velocity; and the second is, that when the velocity exceeds 1100 or 1200 feet, then the absolute quantity of the Resistance will be near 3 times as great as it should be by a comparison with the smaller velocities. Upon these principles he proceeds in approximating to the actual ranges of pieces with small angles of elevation, viz, such as do not exceed 8° or 10°, which he sets down in a table, compared with their corresponding potential ranges. See his Mathematical Tracts, vol. 1 pa. 179 &c. But we shall see presently that these positions are both without foundation; that there is no such thing as a sudden or abrupt change in the law of Resistance, from the square of the velocity to one that gives a quantity three times as much; but that the change is slow and gradual, continually from the smallest to the highest velocities; and that the increased real Resistance no where rises higher than to about double of that which Newton's theory gives it.</p><p>Mr. Glenie, in his History of Gunnery, 1776, pa. 49, observes, in consequence of some experiments with a rifled piece, properly fitted for experimental purposes, that the Resistance of the air to a velocity somewhat less than that mentioned in the first of the above propositions, is considerably greater than in the duplicate ratio of the velocity; and that, to a celerity somewhat greater than that stated in the second, the Resistance is considerably less than that which is treble the Resistance in the said ratio. Some of Robins's own experiments seem necessarily to make it so; since, to a velocity no quicker than 400 feet in a second, he found the Resistance to be somewhat greater than in that ratio. But the true value of the ratio, and other circumstances of this Resistance, will more fully appear from what follows.</p><p>The subject of the Resistance of the air, as begun by Robins, has been prosecuted by myself, to a very great extent and variety, both with the whirling machine, and with cannon balls of all sizes, from 1 lb to 6 lb weight, as well as with figures of many other different shapes, both on the fore part and hind part of them, and with planes set at all varieties of angles of inclination to the path or motion of the same; from all which I have obtained the real Resistance to bodies for all velocities, from 1 up to 2000 feet per second; together with the law of the Resistance to the same body for all different velocities, and for different sizes with the same velocity, and also for all angles of inclination; a full account of which would make a book of itself, and must be reserved for some other occasion. In the mean time, some general tables of conclusions may be taken as below. <pb n="364"/><cb/> <table rend="border"><head><hi rend="smallcaps">Table</hi> I. <hi rend="italics">Resistances of different Bodies.</hi></head><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Smad</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Large Hemis.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Cone</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Resif.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Veloc.</cell><cell cols="1" rows="1" role="data">Hemis.</cell><cell cols="1" rows="1" role="data">Cylin-</cell><cell cols="1" rows="1" role="data">Whole</cell><cell cols="1" rows="1" role="data">as the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">per</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">der</cell><cell cols="1" rows="1" role="data">globe</cell><cell cols="1" rows="1" role="data">power</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">See</cell><cell cols="1" rows="1" role="data"><*>at</cell><cell cols="1" rows="1" role="data">flat</cell><cell cols="1" rows="1" role="data">round</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">vertex</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">base</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">of the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">side</cell><cell cols="1" rows="1" role="data">side</cell><cell cols="1" rows="1" role="data">side</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">vrloe.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">feet</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">.028</cell><cell cols="1" rows="1" role="data">.051</cell><cell cols="1" rows="1" role="data">.020</cell><cell cols="1" rows="1" role="data">.028</cell><cell cols="1" rows="1" role="data">.064</cell><cell cols="1" rows="1" role="data">.050</cell><cell cols="1" rows="1" role="data">.027</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">.048</cell><cell cols="1" rows="1" role="data">.096</cell><cell cols="1" rows="1" role="data">.039</cell><cell cols="1" rows="1" role="data">.048</cell><cell cols="1" rows="1" role="data">.109</cell><cell cols="1" rows="1" role="data">.090</cell><cell cols="1" rows="1" role="data">.047</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">.072</cell><cell cols="1" rows="1" role="data">.148</cell><cell cols="1" rows="1" role="data">.063</cell><cell cols="1" rows="1" role="data">.071</cell><cell cols="1" rows="1" role="data">.162</cell><cell cols="1" rows="1" role="data">.143</cell><cell cols="1" rows="1" role="data">.068</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">.103</cell><cell cols="1" rows="1" role="data">.211</cell><cell cols="1" rows="1" role="data">.092</cell><cell cols="1" rows="1" role="data">.098</cell><cell cols="1" rows="1" role="data">.225</cell><cell cols="1" rows="1" role="data">.205</cell><cell cols="1" rows="1" role="data">.094</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">.141</cell><cell cols="1" rows="1" role="data">.284</cell><cell cols="1" rows="1" role="data">.123</cell><cell cols="1" rows="1" role="data">.129</cell><cell cols="1" rows="1" role="data">.298</cell><cell cols="1" rows="1" role="data">.278</cell><cell cols="1" rows="1" role="data">.125</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">.184</cell><cell cols="1" rows="1" role="data">.368</cell><cell cols="1" rows="1" role="data">.160</cell><cell cols="1" rows="1" role="data">.168</cell><cell cols="1" rows="1" role="data">.382</cell><cell cols="1" rows="1" role="data">.360</cell><cell cols="1" rows="1" role="data">.162</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">.233</cell><cell cols="1" rows="1" role="data">.464</cell><cell cols="1" rows="1" role="data">.199</cell><cell cols="1" rows="1" role="data">.211</cell><cell cols="1" rows="1" role="data">.478</cell><cell cols="1" rows="1" role="data">.456</cell><cell cols="1" rows="1" role="data">.205</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.287</cell><cell cols="1" rows="1" role="data">.573</cell><cell cols="1" rows="1" role="data">.242</cell><cell cols="1" rows="1" role="data">.260</cell><cell cols="1" rows="1" role="data">.587</cell><cell cols="1" rows="1" role="data">.565</cell><cell cols="1" rows="1" role="data">.255</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">.349</cell><cell cols="1" rows="1" role="data">.698</cell><cell cols="1" rows="1" role="data">.292</cell><cell cols="1" rows="1" role="data">.315</cell><cell cols="1" rows="1" role="data">.712</cell><cell cols="1" rows="1" role="data">.688</cell><cell cols="1" rows="1" role="data">.310</cell><cell cols="1" rows="1" role="data">2.052</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">.418</cell><cell cols="1" rows="1" role="data">.836</cell><cell cols="1" rows="1" role="data">.347</cell><cell cols="1" rows="1" role="data">.376</cell><cell cols="1" rows="1" role="data">.850</cell><cell cols="1" rows="1" role="data">.826</cell><cell cols="1" rows="1" role="data">.370</cell><cell cols="1" rows="1" role="data">2.042</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">.492</cell><cell cols="1" rows="1" role="data">.988</cell><cell cols="1" rows="1" role="data">.409</cell><cell cols="1" rows="1" role="data">.440</cell><cell cols="1" rows="1" role="data">1.000</cell><cell cols="1" rows="1" role="data">.979</cell><cell cols="1" rows="1" role="data">.435</cell><cell cols="1" rows="1" role="data">2.036</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">.573</cell><cell cols="1" rows="1" role="data">1.154</cell><cell cols="1" rows="1" role="data">.478</cell><cell cols="1" rows="1" role="data">.512</cell><cell cols="1" rows="1" role="data">1.166</cell><cell cols="1" rows="1" role="data">1.145</cell><cell cols="1" rows="1" role="data">.595</cell><cell cols="1" rows="1" role="data">2.031</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">.661</cell><cell cols="1" rows="1" role="data">1.336</cell><cell cols="1" rows="1" role="data">.552</cell><cell cols="1" rows="1" role="data">.589</cell><cell cols="1" rows="1" role="data">1.346</cell><cell cols="1" rows="1" role="data">1.327</cell><cell cols="1" rows="1" role="data">.581</cell><cell cols="1" rows="1" role="data">2.031</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">.754</cell><cell cols="1" rows="1" role="data">1.538</cell><cell cols="1" rows="1" role="data">.634</cell><cell cols="1" rows="1" role="data">.673</cell><cell cols="1" rows="1" role="data">1.546</cell><cell cols="1" rows="1" role="data">1.526</cell><cell cols="1" rows="1" role="data">.663</cell><cell cols="1" rows="1" role="data">2.033</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">.853</cell><cell cols="1" rows="1" role="data">1.757</cell><cell cols="1" rows="1" role="data">.722</cell><cell cols="1" rows="1" role="data">.762</cell><cell cols="1" rows="1" role="data">1.763</cell><cell cols="1" rows="1" role="data">1.745</cell><cell cols="1" rows="1" role="data">.752</cell><cell cols="1" rows="1" role="data">2.038</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">.959</cell><cell cols="1" rows="1" role="data">1.998</cell><cell cols="1" rows="1" role="data">.818</cell><cell cols="1" rows="1" role="data">.858</cell><cell cols="1" rows="1" role="data">2.002</cell><cell cols="1" rows="1" role="data">1.986</cell><cell cols="1" rows="1" role="data">.848</cell><cell cols="1" rows="1" role="data">2.044</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1.073</cell><cell cols="1" rows="1" role="data">2.258</cell><cell cols="1" rows="1" role="data">.922</cell><cell cols="1" rows="1" role="data">.959</cell><cell cols="1" rows="1" role="data">2.260</cell><cell cols="1" rows="1" role="data">2.246</cell><cell cols="1" rows="1" role="data">.949</cell><cell cols="1" rows="1" role="data">2.047</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">1.196</cell><cell cols="1" rows="1" role="data">2.542</cell><cell cols="1" rows="1" role="data">1.033</cell><cell cols="1" rows="1" role="data">1.060</cell><cell cols="1" rows="1" role="data">2.540</cell><cell cols="1" rows="1" role="data">2.528</cell><cell cols="1" rows="1" role="data">1.057</cell><cell cols="1" rows="1" role="data">2.051</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Mean</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">pro<*>or.</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" role="data">119</cell><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">285</cell><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" role="data">2.040</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No<*>.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell></row></table></p><p>In this Table are contained the Resistances to several forms of bodies, when moved with several degrees of velocity, from 3 feet per second to 20. The names of the bodies are at the tops of the columns, as also which end went foremost through the air; the different velocities are in the first column, and the Resistances on the same line, in their several columns, in avoirdupois ounces and decimal parts. So on the first line are contained the Resistances when the bodies move with a velocity of 3 feet in a second, viz, in the 2d column for the small hemisphere, of 4 3/4 inches diameter, its Resistance .028 oz when the flat side went foremost; in the 3d and 4th columns the Resistances to a larger hemisphere, first with the flat side, and next the round side foremost, the diameter of this, as well as all the following figures being 6 5/8 inches, and therefore the area of the great circle = 32 sq. inches, or 2/9 of a sq. foot; then in the 5th and 6th columns are the Resistances to a cone, first its vertex and then its base foremost, the altitude of the cone being 6 5/8 inches, the same as the diameter of its base; in the 7th column the Resistance to the end of the cylinder, and in the 8th that against the whole globe or sphere. All the numbers show the real weights which are equal to the Resistances; and at the bottoms of the columns are placed proportional numbers, which shew the mean proportions of the Resistances of all the figures to one another, with any velocity. Lastly, in the 9th column are placed the exponents of the power of the velocity which the Resistances in the 8th column bear to each other, viz, which that of the 10 feet velocity bears to each of the following ones, the medium of all of them being as the 2.04 power of the velocity, that is, very little above the square or second power of the velocity, so far as the velocities in this Table extend. <cb/></p><p>From this Table the following inferences are easily deduced.</p><p>1. That the Resistance is nearly in the same proportion as the surfaces; a small increase only taking place in the greater surfaces, and for the greater velocities. Thus, by comparing together the numbers in the 2d and 3d columns, for the bases of the two hemispheres, the areas of which bases are in the proportion of 17 3/4 to 32, or 5 to 9 very nearly, it appears that the numbers in those two columns, expressing the Resistances, are nearly as 1 to 2 or 5 to 10, as far as the velocity of 12 feet; but after that, the Resistances on the greater surface increase gradually more and more above that proportion.</p><p>2. The Resistance to the same surface, with different velocities, is, in these slow motions, nearly as the square of the velocity; but gradually increases more and more above that proportion as the velocity increases. This is manifest from all the columns; and the index of the power of the velocity is set down in the 9th column, for the Resistances in the 8th, the medium being 2.04; by which it appears that the Resistance to the same body is, in these slow motions, as the 2.04 power of the velocity, or nearly as the square of it.</p><p>3. The round ends, and sharp ends, of solids, suffer less Resistance than the flat or plane ends, of the same diameter; but the sharper end has not always the less Resistance. Thus, the cylinder, and the flat ends of the hemisphere and cone, have more Resistance, than the round or sharp ends of the same; but the round side of the hemisphere has less Resistance than the sharper end of the cone.</p><p>4. The Resistance on the base of the hemisphere, is to that on the round, or whole sphere, as 2 1/3 to 1, instead of 2 to 1, as the theory gives that relation. Also the experimented Resistance, on each of these, is nearly 1/4 more than the quantity assigned by the theory.</p><p>5. The Resistance on the base of the cone, is to that on the vertex, nearly as 2 3/10 to 1; and in the same ratio is radius to the sine of the angle of inclination of the side of the cone to its path or axis. So that, in this instance, the Resistance is directly as the sine of the angle of incidence, the transverse section being the same.</p><p>6. When the hinder parts of bodies are of different forms, the Resistances are different, though the foreparts be exactly alike and equal; owing probably to the different pressures of the air on the hinder parts. Thus, the Resistance to the fore part of the cylinder, is less than on the equal flat surface of the cone, or of the hemisphere; because the hinder part of the cylinder is more pressed or pushed, by the following air than those of the other two figures; also, for the same reason, the base of the hemisphere suffers a less Resistance than that of the cone, and the round side of the hemisphere less than the whole sphere. <pb n="365"/><cb/> <table rend="width=70% border"><head><hi rend="smallcaps">Table</hi> II. <hi rend="italics">Resistances both by Experiment and Theory,</hi></head><head><hi rend="italics">to a Globe of 1965 Inches Diameter.</hi></head><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Veloc. per</cell><cell cols="1" rows="1" role="data">Resist. by</cell><cell cols="1" rows="1" role="data">Resist. by</cell><cell cols="1" rows="1" role="data">Ratio of</cell><cell cols="1" rows="1" role="data">Resist. as</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Exper.</cell><cell cols="1" rows="1" role="data">Theory.</cell><cell cols="1" rows="1" role="data">Exper. to</cell><cell cols="1" rows="1" role="data">the power</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">sec. in feet.</cell><cell cols="1" rows="1" role="data">oz.</cell><cell cols="1" rows="1" role="data">oz.</cell><cell cols="1" rows="1" role="data">Theory.</cell><cell cols="1" rows="1" role="data">of the veloc.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">0.006</cell><cell cols="1" rows="1" role="data">0.005</cell><cell cols="1" rows="1" role="data">1.20</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">0.024 1/2</cell><cell cols="1" rows="1" role="data">0.020</cell><cell cols="1" rows="1" role="data">1.23</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">0.055</cell><cell cols="1" rows="1" role="data">0.044</cell><cell cols="1" rows="1" role="data">1.25</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">0.100</cell><cell cols="1" rows="1" role="data">0.079</cell><cell cols="1" rows="1" role="data">1.27</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">0.157</cell><cell cols="1" rows="1" role="data">0.123</cell><cell cols="1" rows="1" role="data">1.28</cell><cell cols="1" rows="1" role="data">2.022</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">0.23</cell><cell cols="1" rows="1" role="data">0.177</cell><cell cols="1" rows="1" role="data">1.30</cell><cell cols="1" rows="1" role="data">2.055</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">0.42</cell><cell cols="1" rows="1" role="data">0.314</cell><cell cols="1" rows="1" role="data">1.33</cell><cell cols="1" rows="1" role="data">2.068</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">0.67</cell><cell cols="1" rows="1" role="data">0.491</cell><cell cols="1" rows="1" role="data">1.36</cell><cell cols="1" rows="1" role="data">2.075</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">100</cell><cell cols="1" rows="1" role="data">2.72</cell><cell cols="1" rows="1" role="data">1.964</cell><cell cols="1" rows="1" role="data">1.38</cell><cell cols="1" rows="1" role="data">2.059</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">200</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7.9</cell><cell cols="1" rows="1" role="data">1.40</cell><cell cols="1" rows="1" role="data">2.041</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">300</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">18.7</cell><cell cols="1" rows="1" role="data">1.41</cell><cell cols="1" rows="1" role="data">2.039</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">400</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">31.4</cell><cell cols="1" rows="1" role="data">1.43</cell><cell cols="1" rows="1" role="data">2.039</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">500</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">1.47</cell><cell cols="1" rows="1" role="data">2.044</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">600</cell><cell cols="1" rows="1" role="data">107</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">1.51</cell><cell cols="1" rows="1" role="data">2.051</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">700</cell><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" role="data">1.57</cell><cell cols="1" rows="1" role="data">2.059</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">800</cell><cell cols="1" rows="1" role="data">205</cell><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" role="data">1.63</cell><cell cols="1" rows="1" role="data">2.067</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">900</cell><cell cols="1" rows="1" role="data">271</cell><cell cols="1" rows="1" role="data">159</cell><cell cols="1" rows="1" role="data">1.70</cell><cell cols="1" rows="1" role="data">2.077</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1000</cell><cell cols="1" rows="1" role="data">350</cell><cell cols="1" rows="1" role="data">196</cell><cell cols="1" rows="1" role="data">1.78</cell><cell cols="1" rows="1" role="data">2.086</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1100</cell><cell cols="1" rows="1" role="data">442</cell><cell cols="1" rows="1" role="data">238</cell><cell cols="1" rows="1" role="data">1.86</cell><cell cols="1" rows="1" role="data">2.095</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1200</cell><cell cols="1" rows="1" role="data">546</cell><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" role="data">1.90</cell><cell cols="1" rows="1" role="data">2.102</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1300</cell><cell cols="1" rows="1" role="data">661</cell><cell cols="1" rows="1" role="data">332</cell><cell cols="1" rows="1" role="data">1.99</cell><cell cols="1" rows="1" role="data">2.107</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1400</cell><cell cols="1" rows="1" role="data">785</cell><cell cols="1" rows="1" role="data">385</cell><cell cols="1" rows="1" role="data">2.04</cell><cell cols="1" rows="1" role="data">2.111</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1500</cell><cell cols="1" rows="1" role="data">916</cell><cell cols="1" rows="1" role="data">442</cell><cell cols="1" rows="1" role="data">2.07</cell><cell cols="1" rows="1" role="data">2.113</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1600</cell><cell cols="1" rows="1" role="data">1051</cell><cell cols="1" rows="1" role="data">503</cell><cell cols="1" rows="1" role="data">2.09</cell><cell cols="1" rows="1" role="data">2.113</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1700</cell><cell cols="1" rows="1" role="data">1186</cell><cell cols="1" rows="1" role="data">568</cell><cell cols="1" rows="1" role="data">2.08</cell><cell cols="1" rows="1" role="data">2.111</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1800</cell><cell cols="1" rows="1" role="data">1319</cell><cell cols="1" rows="1" role="data">636</cell><cell cols="1" rows="1" role="data">2.07</cell><cell cols="1" rows="1" role="data">2.108</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1900</cell><cell cols="1" rows="1" role="data">1447</cell><cell cols="1" rows="1" role="data">709</cell><cell cols="1" rows="1" role="data">2.04</cell><cell cols="1" rows="1" role="data">2.104</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">2000</cell><cell cols="1" rows="1" role="data">1569</cell><cell cols="1" rows="1" role="data">786</cell><cell cols="1" rows="1" role="data">2.00</cell><cell cols="1" rows="1" role="data">2.098</cell></row></table></p><p>In the first column of this Table are contained the several velocities, gradually from o up to the great velocity of 2000 feet per second, with which a ball or globe moved. In the 2d column are the experimented Resistances, in averdupois ounces. In the 3d column are the correspondent Resistances, as computed by the foregoing theory. In the 4th column are the ratios of these two Resistances, or the quotients of the former divided by the latter. And in the 5th or last, the indexes of the power of the velocity which is proportional to the experimented Resistance; which are found by comparing the Resistance of 20 feet velocity with each of the following ones.</p><p>From the 2d, 3d and 4th columns it appears, that at the beginning of the motion, the experimented Resistance is nearly equal to that computed by theory; but that, as the velocity increases, the experimented Resistance gradually exceeds the other more and more, till at the velocity of 1300 feet the former becomes just double the latter; after which the difference increases a little farther, till at the velocity of 1600 or 1700, where that excess is the greatest, and is rather less than 2 1/10; after this, the difference decreases gradually as the velocity increases, and at the velocity of 2000, the former Resistance again becomes just double the latter.</p><p>From the last column it appears that, near the begin- <cb/> ning, or in slow motions, the Resistances are nearly as the square of the velocities; but that the ratio gradually increases, with some small variation, till at the velocity of 1500 or 1600 feet it becomes as the 2 1/9 power of the velocity nearly, which is its highest ascent; and after that it gradually decreases again, as the velocity goes higher. And similar conclusions have also been derived from experiments with larger balls or globes.</p><p>And hence we perceive that Mr. Robins's positions are erroneous on two accounts, viz, both in stating that the Resistance changes suddenly, or all at once, from being as the square of the velocity, so as then to become as some higher and constant power; and also when he states the Resistance as rising to the height of 3 times that which is given by the theory: since the ratio of the Resistance both increases gradually from the beginning, and yet never ascends higher than 2 9/100 of the theory. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="smallcaps">Table</hi> III. <hi rend="italics">Resistance to a Plane, set at various An-</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="italics">gles of Inclination to its Path.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Angle with the</cell><cell cols="1" rows="1" role="data">Experim. Re-</cell><cell cols="1" rows="1" role="data">Resist. by this</cell><cell cols="1" rows="1" rend="align=left" role="data">Sines of the An-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">sistances.</cell><cell cols="1" rows="1" role="data">Formula.</cell><cell cols="1" rows="1" rend="align=left" role="data">gles to Radius</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Path.</cell><cell cols="1" rows="1" role="data">oz.</cell><cell cols="1" rows="1" role="data">.84<hi rend="italics">s</hi><hi rend="sup">1.342c</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">.840.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">0°</cell><cell cols="1" rows="1" role="data">.000</cell><cell cols="1" rows="1" role="data">.000</cell><cell cols="1" rows="1" role="data">.000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">.015</cell><cell cols="1" rows="1" role="data">.009</cell><cell cols="1" rows="1" role="data">.073</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">.044</cell><cell cols="1" rows="1" role="data">.035</cell><cell cols="1" rows="1" role="data">.146</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">.082</cell><cell cols="1" rows="1" role="data">.076</cell><cell cols="1" rows="1" role="data">.217</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">.133</cell><cell cols="1" rows="1" role="data">.131</cell><cell cols="1" rows="1" role="data">.287</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">25</cell><cell cols="1" rows="1" role="data">.200</cell><cell cols="1" rows="1" role="data">.199</cell><cell cols="1" rows="1" role="data">.355</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">30</cell><cell cols="1" rows="1" role="data">.278</cell><cell cols="1" rows="1" role="data">.278</cell><cell cols="1" rows="1" role="data">.420</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">.362</cell><cell cols="1" rows="1" role="data">.363</cell><cell cols="1" rows="1" role="data">.482</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">.448</cell><cell cols="1" rows="1" role="data">.450</cell><cell cols="1" rows="1" role="data">.540</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">.534</cell><cell cols="1" rows="1" role="data">.535</cell><cell cols="1" rows="1" role="data">.594</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">.619</cell><cell cols="1" rows="1" role="data">.613</cell><cell cols="1" rows="1" role="data">.643</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">55</cell><cell cols="1" rows="1" role="data">.684</cell><cell cols="1" rows="1" role="data">.680</cell><cell cols="1" rows="1" role="data">.688</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">60</cell><cell cols="1" rows="1" role="data">.729</cell><cell cols="1" rows="1" role="data">.736</cell><cell cols="1" rows="1" role="data">.727</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">65</cell><cell cols="1" rows="1" role="data">.770</cell><cell cols="1" rows="1" role="data">.778</cell><cell cols="1" rows="1" role="data">.761</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" role="data">.803</cell><cell cols="1" rows="1" role="data">.808</cell><cell cols="1" rows="1" role="data">.789</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">75</cell><cell cols="1" rows="1" role="data">.823</cell><cell cols="1" rows="1" role="data">.826</cell><cell cols="1" rows="1" role="data">.811</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">80</cell><cell cols="1" rows="1" role="data">.835</cell><cell cols="1" rows="1" role="data">.836</cell><cell cols="1" rows="1" role="data">.827</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">85</cell><cell cols="1" rows="1" role="data">.839</cell><cell cols="1" rows="1" role="data">.839</cell><cell cols="1" rows="1" role="data">.838</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">90</cell><cell cols="1" rows="1" role="data">.840</cell><cell cols="1" rows="1" role="data">.840</cell><cell cols="1" rows="1" role="data">.840</cell></row></table></p><p>In the 2d column of this Table are contained the actual experimented Resistances, in ounces, to a plane of 32 square inches, or 2/9 of a square foot, moved through the air with a velocity of exactly 12 feet per second, when the plane was set so as to make, with the direction of its path, the corresponding angles in the first column.</p><p>And from these I have deduced this formula, or theorem, viz, .84<hi rend="italics">s</hi><hi rend="sup">1.842<hi rend="italics">c</hi></hi>, which brings out very nearly the same numbers, and is a general theorem for every angle, for the same plane of 2/9 of a foot, and moved with the same velocity of 12 feet in a second of time; where <hi rend="italics">s</hi> is the sine, and <hi rend="italics">c</hi> the cosine of the angles of inclination in the first column. <pb n="366"/><cb/></p><p>If a theorem be desired for any other velocity <hi rend="italics">v,</hi> and any other plane whose area is <hi rend="italics">a,</hi> it will be this: (1/38)<hi rend="italics">av</hi><hi rend="sup">2</hi><hi rend="italics">s</hi><hi rend="sup">1.842<hi rend="italics">c</hi></hi>, or more nearly (1/42)<hi rend="italics">av</hi><hi rend="sup">2.04</hi><hi rend="italics">s</hi><hi rend="sup">1.842<hi rend="italics">c</hi></hi>; which denotes the Resistance nearly to any plane surface whose area is <hi rend="italics">a,</hi> moved through the air with the velocity <hi rend="italics">v,</hi> in a direction making with that plane an angle, whose sine is <hi rend="italics">s,</hi> and cosine <hi rend="italics">c.</hi></p><p>If it be water or any other fluid, different from air, this formula will be varied in proportion to the density of it.</p><p>By this theorem were computed the numbers in the 3d column; which it is evident agree very nearly with the experiment Resistances in the 2d column, excepting in two or three of the small numbers near the beginning, which are of the least consequence. In all other cases, the theorem gives the true Resistance very nearly. In the 4th or last column are entered the sines of the angles of the first column, to the radius .84, in order to compare them with the Resistances in the other columns. From whence it appears, that those Resistances bear no sort of analogy to the sines of the angles, nor yet to the squares of the sines, nor to any other power of them whatever. In the beginning of the columns, the sines much exceed the Resistances all the way till the angle be between 55 and 60 degrees; after which the sines are less than the Resistances all the way to the end, or till the angle become of 90 degrees.</p><p>Mr. James Bernoulli gave some theorems for the Resistances of different figures, in the Acta Erud. Lips. for June 1693, pa. 252 &c. But as these are deduced from theory only, which we find to be so different from experiment, they cannot be of much use. Messieurs Euler, D'Alembert, Gravesande, and Simpson, have also written pretty largely on the theory of Resistances, besides what had been done by Newton.</p><p><hi rend="italics">Solid of Least</hi> <hi rend="smallcaps">Resistance.</hi> Sir Isaac Newton, from his general theory of Resistance, deduces the figure of a solid which shall have the least Resistance of the same base, height and content. <figure/></p><p>The figure is this. Suppose DNG to be a curve of such a nature, that if from any point N the ordinate NM be drawn perpendicular to the axis AB; and from a given point G there be drawn GR parallel to a tangent at N, and meeting the axis produced in R; then if MN be to GR, as GR<hi rend="sup">3</hi> to 4BR X BG<hi rend="sup">2</hi>, a solid described by the revolution of this figure about its axis AB, moving in a medium from A towards B, is less resisted than any other circular solid of the same base, &c.</p><p>This theorem, which Newton gave without a demonstration, has been demonstrated by several mathe- <cb/> maticians, as Facio, Bernoulli, Hospital, &c. See Maclaurin's Flux. sect. 606 and 607; also Horslev's edit. of Newton, vol. 2, pag. 390. See also Act. Erud. 1699, pa. 514; and Mem. de l'Acad. &c; also Robins's View of Newton's method for comparing the Resistance of Solids, 8vo, 1734; and Simpson's Fluxions, art. 413; or my Principles of Bridges, prop. 11 and 12.</p><p>M. Bouguer has resolved this problem in a very general manner; not in supposing the solid to be formed by a revolution, of any figure whatever. The problem, as enunciated and resolved by M. Bouguer, is this: Any base being given, to find what kind of solid must be formed upon it, so that the impulse upon it may be the least possible. Properly however it ought to be the retardive force, or the impulse divided by the weight or mass of matter in the body, that ought to be the minimum.</p></div1><div1 part="N" n="RESOLUTION" org="uniform" sample="complete" type="entry"><head>RESOLUTION</head><p>, in Physics, the reduction of a body into its original or natural state, by a dissolution or separation of its aggregated parts. Thus, snow and ice are said to be resolved into water; water resolves in vapour by heat; and vapour is again resolved into water by cold; also any compound is resolved into its ingredients, &c.</p><p>Some of the modern philosophers, particularly Boyle, Mariotte, Boerhaave, &c, maintain, that the natural state of water is to be congealed, or in ice; in as much as a certain degree of heat, which is a foreign and violent agent, is required to make it fluid: so that near the pole, where this foreign agent is wanting, it constantly retains its fixed or icy state.</p><div2 part="N" n="Resolution" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Resolution</hi></head><p>, or <hi rend="smallcaps">Solution</hi>, in Mathematics, is an orderly enumeration of several things to be done, to obtain what is required in a problem.</p><p>Wolfius makes a problem to consist of three parts: The <hi rend="italics">proposition</hi> (or what is properly called the <hi rend="italics">problem),</hi> the <hi rend="italics">Resolution,</hi> and the <hi rend="italics">demonstration.</hi></p><p>As soon as a problem is demonstrated, it is converted into a theorem; of which the Resolution is the hypothesis; and the proposition the thesis.</p><p>For the process of a mathematical Resolution, see the following article.</p><p><hi rend="smallcaps">Resolution</hi> in <hi rend="italics">Algebra,</hi> or <hi rend="italics">Algebraical</hi> R<hi rend="smallcaps">ESOLUTION</hi>, is of two kinds; the one practised in numerical problems, the other in geometrical ones.</p><p><hi rend="italics">In Resolving a Numerical Problem Algebraically,</hi> the method is this. First, the given quantities are distinguished from those that are sought; and the former denoted by the initial letters of the alphabet, but the latter by the last letters.—2. Then as many equations are formed as there are unknown quantities. If that cannot be done from the proposition or data, the problem is indeterminate; and certain arbitrary assumptions must be made, to supply the defect, and which can satisfy the question. When the equations are not contained in the problem itself, they are to be found by particular theorems concerning equations, ratios, proportions, &c.—Since, in an equation, the known and unknown quantities are mixed together, they must be separated in such a manner, that the unknown one remain alone on one side, and the known ones on the other. This reduction, or separation, is made by addition, subtraction, multiplication, division, extraction <pb n="367"/><cb/> of roots, and raising of powers; resolving every kind of combination of the quantities, by their counter or reverse ones, and performing the same operation on all the quantities or terms, on both sides of the equation, that the equality may still be preserved.</p><p><hi rend="italics">To Resolve a Geometrical Problem Algebraically.</hi>— The same sort of operations are to be performed, as in the former article; besides several others, that depend upon the nature of the diagram, and geometrical properties. As 1st, the thing required or proposed, must be supposed done, the diagram being drawn or constructed in all its parts, both known and unknown. 2. We must then examine the geometrical relations which the lines of the figure have among themselves, without regarding whether they are known or unknown, to find what equations arise from those relations, for finding the unknown quantities. 3. It is often necessary to form similar triangles and rectangles, sometimes by producing of lines, or drawing parallels and perpendiculars, and forming equal angles, &c; till equations can be formed, from them, including both the known and unknown quantities.</p><p>If we do not thus arrive at proper equations, the thing is to be tried in some other way. And sometimes the thing itself, that is required, is not to be sought directly, but some other thing, bearing certain relations to it, by means of which it may be found.</p><p>The final equation being at last arrived at, the geometrical construction is to be deduced from it, which is performed in various ways according to the different kinds of equations.</p><p><hi rend="smallcaps">Resolution</hi> <hi rend="italics">of Forces,</hi> or <hi rend="italics">of Motion,</hi> is the resolving or dividing of any one force or motion, into several others, in other directions, but which, taken together, shall have the same effect as the single one; and it is the reverse of the composition of forces or motions. See these articles.</p><p>Any single direct force AD, <figure/> may be resolved into two oblique forces, whose quantities and directions are AB, AC, having the same effect, by describing any parallelogram ABDC, whose diagonal is AD. And each of these may, in like manner, be resolved into two others; and so on, as far as we please. And all these new forces, or motions, so found, when acting together, will produce exactly the same effect as the single original one. See also <hi rend="smallcaps">Collision</hi>, P<hi rend="smallcaps">ERCUSSION, Motion</hi>, &c.</p></div2></div1><div1 part="N" n="REST" org="uniform" sample="complete" type="entry"><head>REST</head><p>, in Physics, the continuance of a body in the same place; or its continual application or contiguity to the same parts of the ambient and contiguous bodies.—See <hi rend="smallcaps">Space.</hi></p><p>Rest is either <hi rend="italics">absolute</hi> or <hi rend="italics">relative,</hi> as place is.</p><p>Some define Rest to be the state of a thing without motion; and hence again Rest becomes either absolute or relative, as motion is.</p><p>Newton defines true or absolute Rest to be the continuance of a body in the same part of absolute and immoveable space; and relative Rest to be the continuance of a body in the same part of relative space.</p><p>Thus, in a ship under sail, relative Rest is the continuance of a body in the same part of the ship. But true <cb/> or absolute Rest is its continuance in the same part of universal space in which the ship itself is contained.</p><p>Hence, if the earth be really and absolutely at Rest, the body relatively at Rest in the ship will really and absolutely move, and that with the same velocity as the ship itself. But if the earth do likewise move, there will then arise a real and absolute motion of the body at Rest; partly from the real motion of the earth in absolute space, and partly from the relative motion of the ship on the sea. Lastly, if the body be likewise relatively moved in the ship, its real motion will arise partly from the real motion of the earth in immoveable space, and partly from the relative motion of the ship on the sea, and of the body in the ship.</p><p>It is an axiom in philosophy, that matter is indifferent as to Rest or motion. Hence Newton lays it down, a<*> a law of nature, that every body perseveres in its state, either of Rest or uniform motion, except so far as it is disturbed by external causes.</p><p>The Cartesians assert, that firmness, hardness, or solidity of bodies, consists in this, that their parts are at Rest with regard to each other; and this Rest they establish as the great nexus, or principle of cohesion, by which the parts are connected together. On the other hand, they make fluidity to consist in a perpetual motion of the parts, &c. But the Newtonian philosophy furnishes us with much better solutions.</p><p>Maupertuis asserts, that when bodies are in equilibrio, and any small motion is impressed on them, the quantity of action resulting will be the least possible. This he calls the law of <hi rend="italics">Rest;</hi> and from this law he deduces the fundamental proposition of statics. See Berlin Mem. tom. 2, pa. 294. And from the same principle too he deduces the laws of percussion.</p></div1><div1 part="N" n="RESTITUTION" org="uniform" sample="complete" type="entry"><head>RESTITUTION</head><p>, in Physics, the returning of elastic bodies, forcibly bent, to their natural state; by some called the <hi rend="italics">motion of Restitution.</hi></p></div1><div1 part="N" n="RETARDATION" org="uniform" sample="complete" type="entry"><head>RETARDATION</head><p>, in Physics, the act of retarding, that is, of delaying the motion or progress of a body, or of diminishing its velocity.</p><p>The Retardation of moving bodies arises from two great causes, the resistance of the medium, and the force of gravity.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Retardation</hi> <hi rend="italics">from the Resistance</hi> is often confounded with the resistance itself; because, with respect to the same moving body, they are in the same proportion.</p><p>But with respect to different bodies, the same resistance often generates different Retardations. For if bodies of equal bulk, but different densities, be moved through the same fluid with equal velocity, the fluid will act equally on each; so that they will have equal resistances, but different Retardations; and the Retardations will be to each other, as the velocities which might be generated by the same forces in the bodies proposed; that is, they are inversely as the quantities of matter in the bodies, or inversely as the densities.</p><p>Suppose then bodies of equal density, but of unequal bulk, to move equally fast through the same fluid; then their resistances increase according to their superficies, that is as the squares of their diameters; but the quantities of matter are increased according to their mass or magnitude, that is as the cubes of their diameters: the resistances are the quantities of motion; <pb n="368"/><cb/> the Retardations are the celerities arising from them; and dividing the quantities of motion by the quantities of matter, we shall have the celerities; therefore the Retardations are directly as the squares of the diameters, and inversely as the cubes of the diameters, that is inversely as the diameters themselves.</p><p>If the bodies be of equal magnitude and density, and moved through different fluids, with equal celerity, their Retardations are as the densities of the fluids. And when equal bodies are carried through the same fluid with different velocities, the Retardations are as the squares of the velocities.</p><p>So that, if <hi rend="italics">s</hi> denote the superficies of a body, <hi rend="italics">w</hi> its weight, <hi rend="italics">d</hi> its diameter, <hi rend="italics">v</hi> the velocity, and <hi rend="italics">n</hi> the density of the fluid medium, and N that of the body; then, in similar bodies, the resistance is as <hi rend="italics">nsv</hi><hi rend="sup">2</hi> or as <hi rend="italics">nd</hi><hi rend="sup">2</hi><hi rend="italics">v</hi><hi rend="sup">2</hi>, and the Retardation, or retarding force, as (<hi rend="italics">nsv</hi><hi rend="sup">2</hi>)/<hi rend="italics">w,</hi> or as .</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Retardation</hi> <hi rend="italics">from Gravity</hi> is peculiar to bodies projected upwards. A body thrown upwards is retarded after the same manner as a falling body is accelerated; only in the one case the force of gravity conspires with the motion acquired, and in the other it acts contrary to it.</p><p>As the force of gravity is uniform, the Retardation from that cause will be equal in equal times. Hence, as it is the same force which generates motion in the falling body, and diminishes it in the rising one, a body rises till it lose all its motion; which it does in the same time in which a body falling would have acquired a velocity equal to that with which the body was thrown up.</p><p>Also, a body thrown up, will rise to the same height from which, in falling, it would acquire the same velocity with which it was thrown up: therefore the heights which bodies can rise to, when thrown up with different velocities, are to each other as the squares of the velocities.</p><p>Hence, the Retardations of motions may be compared together. For they are, first, as the squares of the velocities; 2dly, as the densities of the fluids through which the bodies are moved; 3dly, inversely as the diameters of those bodies; 4thly, inversely as the densities of the bodies themselves; as expressed by the theorem above, viz, (<hi rend="italics">nv</hi><hi rend="sup">2</hi>)/(N<hi rend="italics">d</hi>).</p><p><hi rend="italics">The Laws of</hi> <hi rend="smallcaps">Retardation</hi>, are the very same as those for acceleration; motion and velocity being destroyed in the one case, in the very same quantity and proportion as it is generated in the other.</p></div1><div1 part="N" n="RETICULA" org="uniform" sample="complete" type="entry"><head>RETICULA</head><p>, or <hi rend="smallcaps">Reticule</hi>, in Astronomy, a contrivance for measuring very nicely the quantity of eclipses, &c.</p><p>This instrument, introduced some years since by the Paris Acad. of Sciences, is a little frame, consisting of 13 sine silken threads, parallel to, and equidistant from each other; placed in the focus of object-glasses of telescopes; that is, in the place where the image of the luminary is painted in its full extent. Consequently the diameter of the sun or moon is thus seen divided into 12 equal parts or digits: so that, to find the quantity of <cb/> the eclipse, there is nothing to do but to number the parts that are dark, or that are luminous.</p><p>As a square Reticule is only proper for the diameter of the luminary, not for the circumference of it, it is sometimes made circular, by drawing 6 concentric equidistant circles; which represents the phases of the eclipse perfectly.</p><p>But it is evident that the Reticule, whether square or circular, ought to be perfectly equal to the diameter or circumference of the sun or star, such as it appears in the focus of the glass; otherwise the division cannot be just. Now this is no easy matter to effect, because the apparent diameter of the sun and moon differs in each eclipse; nay that of the moon differs from itself in the progress of the same eclipse.—Another imperfection in the Reticule is, that its magnitude is determined by that of the image in the focus; and of consequence it will only fit one certain magnitude.</p><p>But M. de la Hire has found a remedy for all these inconveniences, and contrived that the same Reticule shall serve for all telescopes, and all magnitudes of the luminary in the same eclipse. The principle upon which his invention is founded, is that two object-glasses applied against each other, having a common focus, and these forming an image of a certain magnitude, this image will increase in proportion as the distance between the two glasses is increased, as far as to a certain limit. If therefore a Reticule be taken of such a magnitude, as just to comprehend the greatest diameter the sun or moon can ever have in the common focus of two object-glasses applied to each other, there needs nothing but to remove them from each other, as the star comes to have a less diameter, to have the image still exactly comprehended in the same Reticule.</p><p>Farther, as the silken threads are subject to swerve from the parallelism, &c, by the different temperature of the air, another improvement is, to make the Reticule of a thin looking-glass, by drawing lines or circles upon it with the fine point of a diamond. See M<hi rend="smallcaps">ICROMETER.</hi></p><p>RETIRED <hi rend="smallcaps">Flank</hi>, in Fortification. See <hi rend="smallcaps">Flank.</hi></p><p>RETROCESSION <hi rend="italics">of Curves, &c.</hi> See R<hi rend="smallcaps">ETROGRADATION.</hi></p><p><hi rend="smallcaps">Retrocession</hi> <hi rend="italics">of the Equinox.</hi> See <hi rend="smallcaps">Precession.</hi></p></div1><div1 part="N" n="RETROGRADATION" org="uniform" sample="complete" type="entry"><head>RETROGRADATION</head><p>, or <hi rend="smallcaps">Retrogression</hi>, in Astronomy, is an apparent motion of the planets, by which they seem to go backwards in the ecliptic, and to move contrary to the order or succession of the signs.</p><p>When a planet moves in consequentia, or according to the order of the signs, as from Aries to Taurus, from Taurus to Gemini, &c, which is from west to east, it is said to be <hi rend="italics">direct.</hi>—When it appears for some days in the same place, or point of the heavens, it is said to be <hi rend="italics">stationary.</hi>—And when it goes in antecedentia, or backwards to the following signs, or contrary to the order of the signs, which is from east to west, it is said to be <hi rend="italics">retrograde.</hi> All these different affections or circumstances, may happen in all the planets, except the sun and moon, which are seen to go direct only. But the times of the superior and inferior planets being retrograde are different; the former appearing so about their opposition, and the latter about their conjunc- <pb n="369"/><cb/> tion. The intervals of time also between two Retrogradations of the several planets, are very unequal: <table><row role="data"><cell cols="1" rows="1" role="data">In Saturn it is</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">year</cell><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">days,</cell></row><row role="data"><cell cols="1" rows="1" role="data">In Jupiter "</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">In Mars "</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">In Venus "</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" rend="align=right" role="data">220</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">In Mercury "</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" rend="align=right" role="data">115</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>Again, Saturn continues retrograde 140 days, Jupiter 120, Mars 73, Venus 42, and Mercury 22; or nearly so; for the several Retrogradations of the same planet are not constantly equal.</p><p>These various circumstances however in the motions of the planets are not real, but only apparent; as the inequalities arise from the motion and position of the earth, from whence they are viewed; for when they are considered as seen from the sun, their motions appear always uniform and regular. These inequalities are thus explained:</p><p>Let S denote the sun; and ABCD &c the path or orbit of the earth, moving from west to east, and in that order; also GK &c the orbit of a superior planet, as Saturn for instance, moving the same way, or in the direction GKLG, but with a much less celerity than <figure/> the earth's motion. Now when the earth is at the point A of its orbit, let Saturn be at G, in conjunction with the sun, when it will be seen at P in the zodiac, or among the stars; and when the earth has moved from A to B, let Saturn have moved from G to H in its orbit, when it will be seen in the line BHQ, and will appear to have moved from P to Q in the zodiac; also when the earth has got to C, let Saturn be arrived at I, but found at R in the zodiac, where being seen in the line CIR, it appears stationary, or without motion in the zodiac at R. But after this, Saturn will appear for some time in Retrogradation, viz, moving backwards, or the contrary way: for when the earth has moved to D, Saturn will have got to K, and, being seen in the line DKQ, will appear to have moved retrograde in the zodiac from R to Q; about which place the planet, ceasing to recede any farther, again becomes stationary, and afterward proceeds forward again; for while the earth moves from D to E, and Saturn from K to L, this latter, being now seen in the line ELR, appears to <cb/> have moved forward in the zodiac from Q to R. And so on; the superior planets always becoming retrograde a little before they are in opposition to the sun, and continuing so till some time after the opposition: the retrograde motion being swiftest when the planet is in the very opposition itself; and the direct motion swiftest when in the conjunction. The arch RQ which the planet describes while thus retrograde, is called the arch of Retrogradation. These arches are unequal in all the planets, being greatest in the most distant. and gradually less in the nearer ones.</p><p>In like manner may be shewn the circumstances of the Retrogradations of the inferior planets; by which it will appear, they become stationary a little before their inferior conjunction, and go retrograde till a little time after it; moving the quickest retrograde just at that conjunction, and the quickest direct just at the superior or further conjunction.</p><p><hi rend="smallcaps">Retrogradation</hi> <hi rend="italics">of the Nodes of the Moon,</hi> is a motion of the line of the nodes of her orbit, by which it continually shifts its situation from east to west, contrary to the order of the signs, completing its retrograde circulation in the period of about 19 years: after which time, either of the nodes, having receded from any point of the ecliptic, returns to the same again.—Newton has demonstrated, in his Principia, that the Retrogradation of the moon's nodes is caused by the action of the sun, which continually drawing this planet from her orbit, deflects this orbit from a plane, and causes its intersection with the ecliptic continually to vary; and his determinations on this point have been confirmed by observation.</p><p><hi rend="smallcaps">Retrogradation</hi> <hi rend="italics">of the Sun,</hi> a motion by which in some situations, in the torrid zone, he seems to move retrograde or backwards.</p><p>When the sun is in the torrid zone, and has his declination AM greater than the latitude of the place AZ, but either northern or southern as that is (last fig. above), the sun will appear to go retrograde, or backwards, both before and after noon. For draw the vertical circle ZGN to be a tangent to the sun's diurnal circle MGO in G, and another ZON through the sun's rising, at O: then it is evident, that all the intermediate vertical circles cut the sun's diurnal circle twice: first in the arc GO, and the second time in the arc GI. So that, as the sun ascends through the arc GO, he continually arrives at farther and farther verticals. But as he continues his ascent through the arc GI, he returns to his former verticals; and therefore is seen retrograde for some time before noon. And in like manner it may be shewn that he does the same thing for some time after noon. Hence, as the shadow always tends opposite to the sun, the shadow will be retrograde twice every day in all places of the torrid zone, where the sun's declination exceeds the latitude.</p><p>But the same thing can never happen without the tropics, in a natural way.</p><div2 part="N" n="Retrogradation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Retrogradation</hi></head><p>, or <hi rend="smallcaps">Retrogression</hi>, in the Higher Geometry, is the same with what is otherwise called <hi rend="italics">contrary flexion</hi> or <hi rend="italics">flexure.</hi> See <hi rend="smallcaps">Flexure</hi>, and <hi rend="smallcaps">Inflexion.</hi> <pb n="370"/><cb/></p></div2></div1><div1 part="N" n="RETROGRADE" org="uniform" sample="complete" type="entry"><head>RETROGRADE</head><p>, denotes backward, or contrary to the forward or natural direction. See R<hi rend="smallcaps">ETROGRADATION.</hi></p></div1><div1 part="N" n="RETROGRESSION" org="uniform" sample="complete" type="entry"><head>RETROGRESSION</head><p>, or <hi rend="smallcaps">Retrocession.</hi> The same with <hi rend="smallcaps">Retrogradation.</hi></p><p>RETURNING <hi rend="italics">Stroke,</hi> in Electricity, is an expression used by lord Mahon (now earl Stanhope) to denote the effect produced by the return of the electric fire into a body from which, in certain circumstances, it has been expelled.</p><p>To understand properly the meaning of these terms, it must be premised that, according to the noble author's experiments, an insulated smooth body, immerged within the electrical atmosphere, but beyond the striking distance of another body, charged positively, is at the same time in a state of threefold electricity. The end next to the charged body acquires negative electricity, the farther end is positively electrified; while a certain part of the body, somewhere between its two extremes, is in a natural, unelectrified, or neutral state; so that the two contrary electricities balance each other. It may farther be added, that if the body be not insulated, but have a communication with the earth, the whole of it will be in a negative state. Suppose then a brass ball, which may be called A, to be constantly placed at the striking distance of a prime conductor; so that the conductor, the instant when it becomes fully charged, explodes into it. Let another large or second conductor be suspended, in a perfectly insulated state, farther from the prime conductor than the striking distance, but within its electrical atmosphere: let a person standing on an insulated stool touch this second conductor very lightly with a finger of his right hand; while, with a finger of his left hand, he communicates with the earth, by touching very lightly a second brass ball fixed at the top of a metallic stand, on the floor, which may be called B. Now while the prime conductor is receiving its electricity, sparks pass (at least if the distance between the two conductors is not too great) from the second conductor to the right hand of the insulated person; while similar and simultaneous sparks pass out from the finger of his left hand into the second metallic ball B, communicating with the earth. At length however the prime conductor, having acquired its full charge, suddenly strikes into the ball A, of the first metallic stand, placed for that purpose at the striking distance. The explosion being made, and the prime conductor suddenly robbed of its elastic atmosphere, its pressure or action on the second conductor, and on the insulated person, as suddenly ceases; and the latter instantly feels a smart Returning Stroke, though he has no direct or visible communication (except by the floor) with either of the two bodies, and is placed at the distance of 5 or 6 feet from both of them. This Returning Stroke is evidently occasioned by the sudden re-entrance of the electric fire naturally belonging to his body and to the second conductor, which had before been expelled from them by the action of the charged prime conductor upon them; and which returns to its former place in the instant when that action or elastic pressure ceases. When the second conductor and the insulated person are placed in the densest part of the electrical atmo- <cb/> sphere of the prime conductor, or just beyond the striking distance, the effects are still more considerable; the Returning Stroke being extremely severe and pungent, and appearing considerably sharper than even the main stroke itself, received directly from the prime conductor. Lord Mahon observes, that persons and animals may be destroyed, and particular parts of buildings may be much damaged, by an electrical Returning Stroke, occasioned even by some very distant explosion from a thunder cloud; possibly at the distance of a mile or more. It is certainly not difficult to conceive that a charged extensive thunder cloud must be productive of effects similar to those produced by the prime conductor; but perhaps the effects are not so great, nor the danger so terrible, as it seems have been apprehended. If the quantity of electric fluid naturally contained, for example, in the body of a man, were immense or indefinite, then the estimate between the effects producible by a cloud, and those caused by a prime conductor, might be admitted; but surely no electrical cloud can expel from a body more than the natural quantity of electricity which this contains. On the sudden removal therefore of the pressure by which this natural quantity had been expelled, in consequence of the explosion of the cloud into the earth, no more (at the utmost) than his whole natural stock of electricity can re-enter his body, provided it be so situated, that the returning fire of other bodies must necessarily pass through his body. But perhaps we have no reason to suppose that this quantity is so great, as that its sudden re-entrance into his body should destroy or injure him.</p><p>Allowing therefore the existence of the Returning Stroke, as sufficiently ascertained, and well illustrated, in a variety of circumstances, by the author's experiments, the magnitude and danger of it do not seem to be so alarming as he apprehends. See Lord Mahon's Principles of Electricity, &c. 4to. 1779, pa. 76, 113, and 131. Also Monthly Review, vol. 62, pa. 436.</p><p>REVERSION <hi rend="italics">of Series,</hi> in Algebra, is the finding the value of the root, or unknown quantity, whose powers enter the terms of an infinite series, by means of another infinite series in which it is not contained. As, in the infinite series &c; then if there be found &c, that series is inverted, or its root <hi rend="italics">x</hi> is found in an infinite series of other terms.</p><p>This was one of Newton's improvements in analysis, the first specimen of which was given in his Analysis per Æquationes Numero Terminorum Infinitas; and it is of great use in resolving many problems in various parts of the mathematics.</p><p>The most usual and general way of Reversion, is to assume a series, of a proper form, for the value of the required unknown quantity; then substitute the powers of this value, instead of those of that quantity into the given series; lastly compare the resulting terms with the said given series, and the values of the assumed coefficients will thus be obtained. So, to revert the series , &c, or to find the value of <hi rend="italics">x</hi> in terms of <hi rend="italics">z;</hi> assume it thus, <pb n="371"/><cb/> &c; then by involving this series, for the several powers of <hi rend="italics">x,</hi> and multiplying the corresponding powers by <hi rend="italics">a, b, c,</hi> &c, there results <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">z</hi> = <hi rend="italics">a</hi>A<hi rend="italics">z</hi></cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi>B<hi rend="italics">z</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi>C<hi rend="italics">z</hi><hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi>D<hi rend="italics">z</hi><hi rend="sup">4</hi>,</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi>A<hi rend="sup">2</hi><hi rend="italics">z</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data">2<hi rend="italics">b</hi>AB<hi rend="italics">z</hi><hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data">2<hi rend="italics">b</hi>AC<hi rend="italics">z</hi><hi rend="sup">4</hi></cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi>B<hi rend="sup">2</hi><hi rend="italics">z</hi><hi rend="sup">4</hi></cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi>A<hi rend="sup">3</hi><hi rend="italics">z</hi><hi rend="sup">3</hi></cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data">3<hi rend="italics">c</hi>A<hi rend="sup">2</hi>B<hi rend="italics">z</hi><hi rend="sup">4</hi></cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"><hi rend="italics">d</hi>A<hi rend="sup">4</hi><hi rend="italics">z</hi><hi rend="sup">4</hi></cell><cell cols="1" rows="1" role="data"/></row></table> Then by comparing the corresponding terms of this last series, or making their coefficients equal, there are obtained these equations, viz, , &c, which give these values of the assumed coefficients, viz, ; &c. and consequently &c; which is therefore a general formula or theorem for every series of the same kind, as to the powers of the quantity <hi rend="italics">x.</hi> Thus, for</p><p><hi rend="italics">Ex.</hi> Suppose it were required to revert the series , &c.</p><p>Here <hi rend="italics">a</hi> = 1, <hi rend="italics">b</hi> = -1, <hi rend="italics">c</hi> = 1, <hi rend="italics">a</hi> = -1, &c; which values of these letters being substituted in the theorem, there results , &c, which is that series reverted, or the value of <hi rend="italics">x</hi> in it.</p><p>In the same way it will be found that the theorem for reverting the series .</p><p>Various methods of Reversion may be seen as given by De Moivre in the Philos. Trans. number 240; or Maclaurin's Algebra pa. 263; or Stuart's Explanation of Newton's Analysis, &c. pa. 455; or Coulson's Comment on Newton's Flux. pa. 219; or Horsley's ed. of Newton's works vol. <*>1, pa. 291; or Simpson's Flux. vol. 2, pa. 302: or most authors on Algebra. <cb/></p></div1><div1 part="N" n="REVETEMENT" org="uniform" sample="complete" type="entry"><head>REVETEMENT</head><p>, in Fortification, a strong wall built on the outside of the rampart and parapet, to support the earth, and prevent its rolling into the ditch.</p></div1><div1 part="N" n="REVOLUTION" org="uniform" sample="complete" type="entry"><head>REVOLUTION</head><p>, in Geometry, the motion of rotation of a line about a fixed point or centre, or of any figure about a fixed axis, or upon any line or surface. Thus, the Revolution of a given line about a fixed centre, generates a circle; and that of a rightangled triangle about one side, as an axis, generates a cone; and that of a semicircle about its diameter, generates a sphere or globe, &c.</p><div2 part="N" n="Revolution" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Revolution</hi></head><p>, in Astronomy, is the period of a star, planet, or comet, &c; or its course from any point of its orbit, till it return to the same again.</p><p>The planets have a twofold Revolution. The one about their own axis, usually called their <hi rend="italics">diurnal rotation,</hi> which constitutes their day. The other about the sun, called their <hi rend="italics">annual Revolution,</hi> or <hi rend="italics">period,</hi> constituting their year.</p></div2></div1><div1 part="N" n="REYNEAU" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">REYNEAU</surname> (<foreName full="yes"><hi rend="smallcaps">Charles-Rene</hi></foreName>)</persName></head><p>, commonly called Father Reyneau, a noted French mathematician, was born at Brissac in the province of Anjou, in the year 1656. At 20 years of age he entered himself among the Oratorians, a kind of religious order, in which the members lived in community without making any vows, and applied themselves chiefly to the education of youth. He was soon after sent, by his superiors, to teach philosophy at Pezenas, and then at Toulon. This requiring some acquaintance with geometry, he contracted a great affection for this science, which he cultivated and improved to a great extent; in consequence he was called to Angers in 1683, to fill the mathematical chair; and the Academy of Angers elected him a member in 1694.</p><p>In this occupation Father Reyneau, not content with making himself master of every thing worth knowing, which the modern analysis, so fruitful in sublime speculations and ingenious discoveries, had already produced, undertook to reduce into one body, for the use of his scholars, the principal theories scattered here and there in Newton, Descartes, Leibnitz, Bernoulli, the Leipsic Acts, the Memoirs of the Paris Academy, and in other works; treasures which by being so widely dispersed, proved much less useful than they otherwise might have been. The fruit of this undertaking, was his <hi rend="italics">Analyse Demontrée,</hi> or Analysis Demonstrated, which he published in 2 volumes 4to, 1708.</p><p>Father Reyneau called this useful work, Analysis Demonstrated, because he demonstrates in it several methods which had not been demonstrated by the authors of them, or at least not with sufficient perspicuity and exactness; for it often happens that, in matters of this kind, a person is clear in a thing, without being able to demonstrate it. Some persons too have been so mistakingly fond of glory as to make a secret of their demonstrations, in order to perplex those, whom it would become them much better to instruct. This book of Reyneau's was so well approved, that it soon became a maxim, at least in France, that to follow him was the best, if not the only way, to make any extra- <pb n="372"/><cb/> ordinary progress in the mathematics. This was considering him as the first master, as the Euclid of the sublime geometry.</p><p>Reynean, aster thus giving lessons to those who undei stood something of geometry, thought proper to draw up some for such as were utterly unacquainted with that science. This was in some measure a condescension in him, but his passion to be useful made it easy and agreeable. In 1714 he published a volume in 4to on calculation, under the title of <hi rend="italics">Science du Calcul des Grandeurs,</hi> of which the then Censor Royal, a most intelligent and impartial judge, says, in his approbation of it, that “though several books had already appeared upon the same subject, such a treatise as that before him was still wanting, as in it every thing was handled in a manner sufficiently extensive, and at the same time with all possible exactness and perspicuity.” In fact, though most branches of the mathematics had been well treated of before that period, there were yet no good elements, even of practical geometry. Those who knew no more than what precisely such a book ought to contain, knew too little to complete a good one; and those who knew more, thought themselves probably above the task; whereas Reyneau possessed at once all the learning and modesty necessary to undertake and execute such a work.</p><p>As soon as the Royal Academy of Sciences at Paris, in consequence of a regulation made in the year 1716, opened its doors to other learned men, under the title of <hi rend="italics">Free Associates,</hi> Father Reynean was admitted of the number. The works however which we have already mentioned, besides a small piece upon <hi rend="italics">Logic,</hi> are the only ones he ever published, or probably ever composed, except most of the materials for a second volume of his <hi rend="italics">Science du Calcul,</hi> which he left behind him in manuscript. The last years of his life were attended with too much sickness to admit of any extraordinary application. He died in 1728, at 72 years of age, not more regretted on account of his great learning, than of his many virtues, which all conspired in an eminent degree to make that learning agreeable to those about him, and useful to the world. The first men in France deemed it an honour and a happiness to count him among their friends. Of this number were the chancellor of that kingdom, and Father Mallebranche, of whom Reyncau was a zealous and faithful disciple.</p></div1><div1 part="N" n="RHABDOLOGY" org="uniform" sample="complete" type="entry"><head>RHABDOLOGY</head><p>, or <hi rend="smallcaps">Rabdology</hi>, in Arithmetic, a name given by Napier to a method of performing some of the more difficult operations of numbers by means of certain square little rods. Upon these are inscribed the simple numbers; then by shifting them according to certain rules, those operations are performed by simply adding or subtracting of the numbers as they stand upon the rods. See Napier's Rabdologia, printed in 1617. See also the article <hi rend="smallcaps">Napier</hi>'s <hi rend="italics">Bones.</hi></p><p>RHEO-STATICS, is used by some for the statics, or the science of the equilibrium of fluids.</p></div1><div1 part="N" n="RHETICUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">RHETICUS</surname> (<foreName full="yes"><hi rend="smallcaps">George Joachim</hi></foreName>)</persName></head><p>, a noted German astronomer and mathematician, who was the colleague of Reinhold in the university of Wittemberg, being joint professors of mathematics there toge- <cb/> ther. He was born at Feldkirk in Tyrol the 15th of February 1514. After imbibing the elements of the mathematics at Tiguri with Oswald Mycone, he went to Wittemberg, where he diligently cultivated that science. Here he was made master of philosophy in 1535, and professor in 1537. He quitted this situation however two years after, and went to Fruenburg to put him under the assistance of the celebrated Copernicus, being induced to this step by his zeal for astronomical pursuits, and the great fame which Copernicus had then acquired. Rheticus assisted this astronomer for some years, and constantly exhorted him to perfect his work, <hi rend="italics">De Revolutionibus,</hi> which he published after the death of Copernicus, viz, in 1543, folio, at Norimberg, together with an illustration of the same in a narration, dedicated to Schoner. Here too, to render astronomical calculations more accurate, he began his very elaborate canon of sines, tangents and secants, to 15 places of sigures, and to every 10 seconds of the quadrant, a design which he did not live quite to complete. The canon of sines however to that radius, for every 10 seconds, and for every single second in the first and last degree of the quadrant, computed by him, was published in fulio at Francfort 1613 by Pitiscus, who himself added a few of the first sines computed to 22 places of figures. But the larger work, or canon of sines, tangents and secants, to every 10 seconds, was perfected and published after his death, viz, in 1596, by his disciple Valentine Otho, mathematician to the Electoral Prince Palatine; a particular account and analysis of which work may be seen in the Historical Introduction to my Logarithms, pa. 9.</p><p>After the death of Copernicus, Rheticus returned to Wittemberg, viz, in 1541 or 1542, and was again admitted to his office of professor of mathematics. The same year, by the recommendation of Melancthon, he went to Norimberg, where he found certain manuscripts of Werner and Regiomontanus. He afterwards taught mathematics at Leipsic. From Saxony he departed a second time, for what reason is not known, and went to Poland; and from thence to Cassovia in Hungary, where he died December the 4th, 1576, near 63 years of age.</p><p>His <hi rend="italics">Narratio de Libris Revolutionum Copernici,</hi> was first published at Gedunum in 4to, 1540; and afterwards added to the editions of Copernicus's work. He also composed and published <hi rend="italics">Ephemerides,</hi> according to the doctrine of Copernicus, till the year 1551.</p><p>Rheticus also projected other works, and partly executed them, though they were never published, of various kinds, astronomical, astrological, geographical, chemical, &c; as they are more particularly mentioned in his letter to Peter Ramus in the year 1568, which Adrian Romanus inserted in the preface to the first part of his Idea of Mathematics.</p><p>RHOMB <hi rend="smallcaps">Solid</hi>, consists of two equal and right cones joined together at their bases.</p></div1><div1 part="N" n="RHOMBOID" org="uniform" sample="complete" type="entry"><head>RHOMBOID</head><p>, or <hi rend="smallcaps">Rhomboides</hi>, in Geometry, a quadrilateral figure, whose opposite sides and angles are equal; but which is neither equilateral nor equiangular.</p></div1><div1 part="N" n="RHOMBUS" org="uniform" sample="complete" type="entry"><head>RHOMBUS</head><p>, is an oblique equilateral parallelogram; <pb n="373"/><cb/> on a quadrilateral figure, whose sides are equal and parallel, but the four angles not all equal, two of the opposite ones being obtuse, and the other two opposite ones acute.</p><p>The two diagonals of a Rhombus intersect at right angles; but not of a rhomboides.</p><p>As to the area of the Rhombus or rhomboides, it is found, like that of all other parallelograms, by multiplying the length or base by the perpendicular breadth.</p><p><hi rend="smallcaps">Rhombus</hi>-<hi rend="italics">Solid.</hi> See <hi rend="smallcaps">Rhomb</hi>-<hi rend="italics">Solid.</hi></p></div1><div1 part="N" n="RHUMB" org="uniform" sample="complete" type="entry"><head>RHUMB</head><p>, <hi rend="smallcaps">Rumb</hi>, or <hi rend="smallcaps">Rum</hi>, in Navigation, a vertical circle of any given place; or the intersection of a part of such a circle with the horizon. Rhumbs therefore coincide with points of the world, or of the horizon. And hence mariners distinguish the Rhumbs by the same names as the points and winds. But we may observe, that the Rhumbs are denominated from the points of the compass in a different manner from the winds: thus, at sea, the north-east wind is that which blows from the north-east point of the horizon towards the ship in which we are; but we are said to sail upon the north-east Rhumb, when we go towards the north-east.</p><p>They usually reckon 32 Rhumbs, which are represented by the 32 lines in the rose or card of the compass.</p><p>Aubin defines a Rhumb to be a line on the terrestrial globe, or sea-compass, or sea-chart, representing one of the 32 winds which serve to conduct a vessel. So that the Rhumb a vessel pursues is conceived as its route, or course.</p><p>Rhumbs are divided and subdivided like points of the compass. Thus, the whole Rhumb answers to the cardinal point. The half Rhumb to a collateral point, or makes an angle of 45 degrees with the former. And the quarter Rhumb makes an angle of 22° 30′ with it. Also the half-quarter Rhumb makes an angle of 11° 15′ with the same.</p><p>For a table of the Rhumbs, or points, and their distances from the meridian, see <hi rend="smallcaps">Wind.</hi></p><p><hi rend="smallcaps">Rhumb-Line</hi>, <hi rend="italics">Loxodromia,</hi> in Navigation, is a line prolonged from any point of the compass in a nautical chart, except the four cardinal points: or it is the line which a ship, keeping in the same collateral point, or rhumb, describes throughout its whole course.</p><p>The chief property of the Rhumb-line, or loxodromia, and that from which some authors define it, is, that it cuts all the meridians in the same angle.</p><p>This angle is called the <hi rend="italics">angle of the Rhumb,</hi> or the <hi rend="italics">loxodromic</hi> angle. And the angle which the Rhumbline makes with any parallel to the equator, is called the <hi rend="italics">complement of the Rhumb.</hi></p><p>An idea of the origin and properties of the Rhumbline, the great foundation of Navigation, may be conceived thus: a vessel beginning its course, the wind by which it is driven makes a certain angle with the meridian of the place; and as we shall suppose that the vessel runs exactly in the direction of the wind, it makes the same angle with the meridian which the wind makes. Supposing then the wind to continue the <cb/> same, as each point or instant of the progress may be esteemed the beginning, the vessel always makes the same angle with the meridian of the place where it is each moment, or in each point of its course which the wind makes.</p><p>Now a wind, for example, that is north-east, and which consequently makes an angle of 45 degrees with the meridian, is equally north-east wherever it blows, and makes the same angle of 45 degrees with all the meridians it meets. And therefore a vessel, driven by the same wind, always makes the same angle with all the meridians it meets with on the surface of the earth.</p><p>If the vessel sail north or south, it describes the great circle of a meridian. If it runs east or west, it cuts all the meridians at right angles, and describes either the circle of the equator, or else a circle parallel to it.</p><p>But if the vessel sails between the two, it does not then describe a circle; since a circle, drawn obliquely to a meridian, would cut all the meridians at unequal angles, which the vessel cannot do. It describes theresore another curve, the essential property of which is, that it cuts all the meridians in the same angle, and it is called the <hi rend="italics">loxodromy,</hi> or <hi rend="italics">loxodromic curve,</hi> or <hi rend="italics">Rhumbline.</hi></p><p>This curve, on the globe, is a kind of spiral, tending continually nearer and nearer to the pole, and making an infinite number of circumvolutions about it, but without ever arriving exactly at it. But the spiral Rhumbs on the globe become proportional spirals in the stereographic projection on the plane of the equator.</p><p>The length of a part of this Rhumb-line, or spiral, then, is the distance run by the ship while she keeps in the same course. But as such a spiral line would prove very perplexing in the calculation, it was necessary to have the ship's way in a right line; which right line however must have the essential properties of the curve line, viz, to cut all the meridians at right angles. The method of effecting which, see under the article <hi rend="smallcaps">Chart.</hi></p><p>The are of the Rhumb-line is not the shortest distance between any two places through which it passes; for the shortest distance, on the surface of the globe, is an arc of the great circle passing through those places; so that it would be a shorter course to sail on the arc of this great circle: but then the ship cannot be kept in the great circle, because the angle it makes with the meridians is continually varying, more or less.</p><p>Let P be the pole, RW the <figure/> equator, ABCDEP a spiral Rhumb, divided into an indefinite number of equal parts at the points B,C,D, &c; through which are drawn the meridians, PS, PT, PV, &c, and the parallels FB, KC, LD, &c, also draw the parallel AN. Then, as a ship sails along the Rhumbline towards the pole, or in the direction ABCD &c, from A to E, the distance sailed AE <pb n="374"/><cb/> is made up of all the small equal parts of the Rhumb AB + BC + CD + DE; and the sum of all the small differences of latitude AF + BG + CH + DI make up the whole difference of latitude AM or EN; and the sum of all the small parallels FB + GC + HD + IE is what is called the departure in plane sailing; and ME is the meridional distance, or distance between the first and last meridians, measured on the last parallel; also RW is the difference of longitude, measured on the equator. So that these last three are all different, viz, the departure, the meridional distance, and the difference of longitude.</p><p>If the ship sail towards the equator, from E to A; the departure, difference of latitude, and difference of longitude, will be all three the same as before; but the meridional distance will now be AN, instead of ME; the one of these AN being greater than the departure FB + GC + HD + IE, and the other ME is less than the same; and indeed that departure is nearly a mean proportional between the two meridional distances ME, AN. Other properties are as below.</p><p>1. All the small elementary triangles ABF, BCG, CDH, &c, are mutually similar and equal in all their parts. For all the angles at A, B, C, D, &c are equal, being the angles which the Rhumb makes with the meridians, or the angles of the course; also all the angles F, G, H, I, are equal, being right angles; therefore the third angles are equal, and the triangles all similar. Also the hypotenuses AB, BC, CD, &c, are all equal by the hypothesis; and consequently the triangles are both similar and equal.</p><p>2. As radius : distance run AE :: sine of course [angle]A : departure FB + GC &c, :: cosin. of course [angle]A : dif. of lat. AM. For in any one ABF of the equal elementary triangles, which may be considered as small right-angled plane triangles, it is, as rad. or sin. [angle]F : sin. course A :: AB : FB :: (by composition) the sum of all the distances AB + BC + CD &c : the sum of all the departures FB + GC + HD &c.</p><p>And, in like manner, as radius : cos. course A :: AB : AF :: AB + BC &c : AF + BG &c.</p><p>Hence, of these four things, the course, the difference of latitude, the departure, and the distance run, having any two given, the other two are found by the proportions above in this article.</p><p>By means of the departure, the length of the Rhumb, or distance run, may be connected with the longitude and latitude, by the following two theorems.</p><p>3. As radius : half the sum of the cosines of both the latitudes, of A and E :: dif. of long. RW : departure.</p><p>Because RS : FB :: radius : sine of PA or cos. RA, and VW : IE :: radius : sine of PE or cos. EW.</p><p>4. As radius : cos. middle latitude :: dif. of longitude : departure.—Because cosine of middle latitude is nearly equal to half the sum of the cosines of the two extreme latitudes.</p></div1><div1 part="N" n="RICCIOLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">RICCIOLI</surname> (<foreName full="yes"><hi rend="smallcaps">Joannes Baptista</hi></foreName>)</persName></head><p>, a learned Ita- <cb/> lian astronomer, philosopher, and mathematician, was born in 1598, at Ferrara, a city in Italy, in the dominions of the Pope. At 16 years of age he was admitted into the society of the Jesuits. He was endowed with uncommon talents, which he cultivated with extraordinary application; so that the progress he made in every branch of literature and science was surprising. He was first appointed to teach rhetoric, poetry, philosophy, and scholastic divinity, in the Jesuits' colleges at Parma and Bologna; yet applied himself in the mean time to making observations in geography, chronology, and astronomy. This was his natural bent, and at length he obtained leave from his superiors to quit all other employment, that he might devote himself entirely to those sciences.</p><p>He projected a large work, to be divided into three parts, and to contain as it were a complete system of philosophical, mathematical, and astronomical knowledge. The first of these parts, which regards astronomy, came out at Bologna in 1651, 2 vols. folio, with this title, <hi rend="italics">J. B. Riccioli Almagestum Novum, Astronomiam veterein novamque complectens, observationibus aliorum et propriis, novisque theorematibus, problematibus ac tabulis promotam.</hi> Riccioli imitated Ptolomy in this work, by collecting and digesting into proper order, with observations, every thing ancient and modern, which related to his subject; so that Gassendus very justly called his work, “Promptuarium et thesaurum ingentem Astronomiæ.”</p><p>In the sirst volume of this work, he treats of the sphere of the world, of the sun and moon, with their eclipses; of the fixed stars, of the planets, of the comets and new stars, of the several mundane systems, and six sections of general problems serving to astronomy, &c. —In the second volume, he treats of trigonometry, or the doctrine of plane and spherical triangles; proposes to give a treatise of astronomical instruments, and the optical part of astronomy (which part was never published); treats of geography, hydrography, with an epitome of chronology.—The third, comprehends observations of <*>e sun, moon, eclipses, fixed stars and planets, with precepts and tables of the primary and secondary motions, and other astronomical tables.</p><p>Riccioli printed also, two other works, in folio, at Bologna, viz,</p><p>2. <hi rend="italics">Astronomia Reformata,</hi> 1665; the design of which was, that of considering the various hypotheses of several astronomers, and the difficulty thence arising of concluding any thing certain, by comparing together all the best observations, and examining what is most certain in them, thence to reform the principles of astronomy.</p><p>3. <hi rend="italics">Chronologia Reformata,</hi> 1669.</p><p>Riccioli died in 1671, at 73 years of age.</p><p>RICOCHET Firing, in the Military Art, is a method of firing with small charges, and pieces elevated but in a small degree, as from 3 to 6 degrees. The word signifies duck-and drake, or rebounding, because the ball or shot, thus discharged, goes bounding and rolling along, and killing or destroying every thing in its way, like the bounding of a flat stone along the surface of water when thrown almost horizontally. <pb n="375"/><cb/></p></div1><div1 part="N" n="RIDEAU" org="uniform" sample="complete" type="entry"><head>RIDEAU</head><p>, in Fortification, a small elevation of earth, extending itself lengthways on a plain; serving to cover a camp, or give an advantage to a post.</p><p><hi rend="smallcaps">Rideau</hi> is sometimes also used for a trench, the earth of which is thrown up on its side, to serve as a parapet for covering the men.</p><p>RIFLE <hi rend="smallcaps">Guns</hi>, in the Military Art, are those whose barrels, instead of being smooth on the inside, are formed with a number of spiral channels, making each about a turn and a half in the length of the barrel. These carry their balls both farther and truer than the common pieces. For the nature and qualities of them, see Robins's Tracts, vol. 1 pa. 328 &c.</p></div1><div1 part="N" n="RIGEL" org="uniform" sample="complete" type="entry"><head>RIGEL</head><p>, in Astronomy. See <hi rend="smallcaps">Regel.</hi></p></div1><div1 part="N" n="RIGHT" org="uniform" sample="complete" type="entry"><head>RIGHT</head><p>, in Geometry, something that lies evenly or equally, without inclining or bending one way or another. Thus, a Right-line is that whose small parts all tend the same way. In this sense, Right means the same as straight, as opposed to curved or crooked.</p><p><hi rend="smallcaps">Right</hi>-<hi rend="italics">Angle,</hi> that which one line makes with another upon which it stands so as to incline neither to one side nor the other. And in this sense the word Right stands opposed to oblique.</p><p><hi rend="smallcaps">Right</hi>-<hi rend="italics">angled,</hi> is said of a figure when its sides are at Right angles or perpendicular to each other.—This sometimes holds in all the angles of the figure, as in squares and rectangles; sometimes only in part, as in right-angled triangles.</p><p><hi rend="smallcaps">Right</hi> <hi rend="italics">Cone,</hi> or <hi rend="italics">Cylinder,</hi> or prism, or pyramid, one whose axis is at right-angles to the base.</p><p><hi rend="smallcaps">Right</hi>-<hi rend="italics">lined Angle,</hi> one formed by Right lines.</p><p><hi rend="smallcaps">Right</hi> <hi rend="italics">Sine,</hi> one that stands at Right-angles to the diameter; as opposed to versed sine.</p><p><hi rend="smallcaps">Right</hi> <hi rend="italics">Sphere,</hi> is that where the equator cuts the horizon at Right angles; or that which has the poles in the horizon, and the equinoctial in the zenith.</p><p>Such is the position of the sphere with regard to those who live at the equator, or under the equinoctial. The consequences of which are; that they have no latitude, nor elevation of the pole; they see both poles of the world, and all the stars rise, culminate and set; also the sun always rises and descends at Right angles, and makes their days and nights equal. In a Right sphere, the horizon is a meridian; and if the sphere be supposed to revolve, all the meridians successively become horizons, one aster another.</p><p><hi rend="smallcaps">Right</hi> Ascension, Descension, Parallax, &c. See the respective Articles.</p><p><hi rend="smallcaps">Right</hi> <hi rend="italics">Circle,</hi> in the Stereographic Projection of the Sphere, is a circle at Right angles to the plane of projection, or that is projected into a Right line.</p><p><hi rend="smallcaps">Right</hi> <hi rend="italics">Sailing,</hi> is that in which a voyage is performed on some one of the four cardinal points, east, west, north, or south.</p><p>If the ship sail on a meridian, that is, north or south, she does not alter her longitude, but only changes the latitude, and that just as much as the number of degrees she has run.</p><p>But if she sail on the equator, directly east or west, <cb/> she varies not her latitude, but only changes the longitude, and that just as much as the number of degrees she has run.</p><p>And if she sail directly east or west upon any parallel, she again does not change her latitude, but only the longitude; yet not the same as the number of degrees of a great circle she hath sailed, as on the equator, but more, according as the parallel is remoter from the equinoctial towards the pole. For the less any parallel is, the greater is the difference of longitude answering to the distance run.</p></div1><div1 part="N" n="RIGIDITY" org="uniform" sample="complete" type="entry"><head>RIGIDITY</head><p>, a brittle hardness; or that kind of hardness which is supposed to arise from the mutual indentation of the component particles within one another. Rigidity is opposed to ductility, malleability, &c.</p></div1><div1 part="N" n="RING" org="uniform" sample="complete" type="entry"><head>RING</head><p>, in Astronomy and Navigation, an instrument used for taking the sun's altitude &c. It is usually of brass, about 9 inches diameter, suspended by a little swivel, at the distance of 45° from the point of which is a perforation, which is the centre of a quadrant of 90° divided in the inner concave surface.</p><p>To use it, let it be held up by the swivel, and turned round to the sun, till his rays, falling through the hole, make a spot among the degrees, which marks the altitude required.</p><p>This instrument is preferred before the astrolabe, because the divisions are here larger than on that instrument.</p><div2 part="N" n="Ring" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Ring</hi></head><p>, of Saturn, is a thin, broad, opaque circular arch, encompassing the body of that planet, like the wooden horizon of an artificial globe, without touching it, and appearing double, when seen through a good telescope.</p><p>This Ring was sirst discovered by Huygens, who, after frequent observation of the planet, perceived two lucid points, like ansæ or handles, arising out from the body in a right line. Hence as in subsequent observations he always found the fame appearance, he concluded that Saturn was encompassed with a permanent Ring; and accordingly produced his New System of Saturn, in 1659. However, Galileo sirst discovered that the figure of Saturn was not round.</p><p>Huygens makes the space between the globe of Saturn and the Ring equal to the breadth of the Ring, or rather more, being about 22000 miles broad; and the greatest diameter of the Ring, in proportion to that of the globe, as 9 to 4 But Mr. Pound, by an excellent micrometer applied to the Huygenian glass of 123 feet, determined this proportion, more exactly, to be as 7 to 3.</p><p>Observations have also determined, that the plane of the Ring is inclined to the plane of the ecliptic in an angle of 30 degrees; that the Ring probably turns, in the direction of its plane, round its axis, because when it is almost edgewise to us, it appears rather thicker on one side of the planet than on the other; and the thickest edge has been seen on different sides at different times: the sun shines almost 15 of our years together on one side of Saturn's Ring without setting, and as long on the other in its turn; so that the Ring is visible to the inhabitants of that planet for almost 15 <pb n="376"/><cb/> of our years, and as long invisible, by turns, if its axis has no inclination to its Ring; but if the axis of the planet be inclined to the Ring, ex. gr. about 30 degrees, the Ring will appear and disappear once every natural day to all the inhabitants within 30 degrees of the equator, on both sides, frequently eclipsing the sun in a Saturnian day. Moreover, if Saturn's axis be so inclined to his Ring, it is perpendicular to his orbit; by which the inconvenience of different seasons to that planet is avoided. <cb/></p><p>This Ring, seen from Saturn, appears like a large luminous arch in the heavens, as if it did not belong to the planet.</p><p>When we see the Ring most open, its shadow upon the planet is broadest; and from that time the shadow grows narrower, as the Ring appears to do to us; until, by Saturn's annual motion, the sun comes to the plane of the Ring, or even with its edge; which, being then directed towards us, becomes invisible, on account of its thinness. <figure/><cb/></p><p>The phenomena of Saturn's Ring are illustrated by a view of this figure. Let S be the sun, ABCDEFGH Saturn's orbit, and IKLMNO the earth's orbit. Both Saturn and the earth move according to the order of the letters; and when Saturn is at A, his Ring is turned edgewise to the sun S, and he is then seen from the earth as if he had lost his Ring, let the earth be in any part of its orbit whatever, except between N and O; for whilst it describes that space, Saturn is apparently so near the sun as to be hid in his beams. As Saturn goes from A to C, his Ring appears more and more open to the earth; at C the Ring appears most open of all; and seems to grow narrower and narrower as Saturn goes from C to E; and when he comes to E, the Ring is again turned edgewise both to the sun and earth; and as neither of its sides is illuminated, it is invisible to us, because its edge is too thin to be perceptible; and Saturn appears again as if he had lost his Ring. But as he goes from E to G, his Ring opens more and more to our view on the under side; and seems just as open at G as it was at C, and may be seen in the night time from the earth in any part of its orbit, except about M, when the sun hides the planet from our view.</p><p>As Saturn goes from G to A, his Ring turns more and more edgewise to us, and, therefore, it seems to grow narrower and narrower; and at A it disappears as before.</p><p>Hence, while Saturn goes from A to E, the sun shines on the upper side of his Ring, and the under side is dark; and whilst he goes from E to A, the sun shines on the under side of his Ring, and the upper side is dark. The Ring disappears twice in every annual revolution of Saturn, viz, when he is in the 19th degree of Pisces and of Virgo, and when Saturn is in the middle between these points, or in the 19th degree either of Gemini or of Sagittarius, his Ring appears most open to us; and then its longest diameter is to its shortest, as 9 to 4. Ferguson's Astr. sect. 204.</p><p>There are various hypotheses concerning this Ring. Kepler, in his Epitom. Astron. Copern. and after him <cb/> Dr. Halley, in his Enquiry into the Causes of the Variation of the Needle, Phil. Trans. No 195, suppose our earth may be composed of several crusts or shells, one within another, and concentric to each other. If this be the case, it is possible the Ring of Saturn may be the fragment or remaining ruin of his formerly exterior shell, the rest of which is broken or fallen down upon the body of the planet. And some have supposed that the Ring may be a congeries or series of moons revolving about the planet.</p><p>Later observations have thrown much more light upon this curious phenomenon, especially respecting its dimensions, and rotation, and division into two or more parts. De la Lande and De la Place say, that Cassini saw the breadth of the Ring divided into two separate parts that are equal, or nearly so. Mr. Short assured M. De la Lande, that he had seen many divisions upon the Ring, with his 12 feet telescope. And Mr. Hadley, with an excellent 5 1/2 feet reflector, saw the Ring divided into two parts. Several excellent theories have been given in the French Memoirs, particularly by De la Place, contending for the division of the Ring into many parts. But finally the observations of Dr. Herschel, in several volumes of the Philos. Trans. seem to confirm the division into two concentric parts only. The dimensions of these two Rings, and the space between them, he states in the following proportion to each other. <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">Miles.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">Inner diam. of smaller Ring</cell><cell cols="1" rows="1" rend="align=right" role="data">146345</cell></row><row role="data"><cell cols="1" rows="1" role="data">Outside diam. of ditto</cell><cell cols="1" rows="1" rend="align=right" role="data">184393</cell></row><row role="data"><cell cols="1" rows="1" role="data">Inner diam. of larger Ring</cell><cell cols="1" rows="1" rend="align=right" role="data">190248</cell></row><row role="data"><cell cols="1" rows="1" role="data">Outside diam. of ditto</cell><cell cols="1" rows="1" rend="align=right" role="data">204883</cell></row><row role="data"><cell cols="1" rows="1" role="data">Breadth of the inner Ring</cell><cell cols="1" rows="1" rend="align=right" role="data">20000</cell></row><row role="data"><cell cols="1" rows="1" role="data">Breadth of the outer Ring</cell><cell cols="1" rows="1" rend="align=right" role="data">7200</cell></row><row role="data"><cell cols="1" rows="1" role="data">Breadth of the vacant space</cell><cell cols="1" rows="1" rend="align=right" role="data">2839</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Ring revolves in its own plane, in 10<hi rend="sup">h</hi> 32′ 15″.4.</cell></row></table> So that the outside diameter of the larger Ring is almost 26 times the diameter of the earth.</p><p>Dr. Herschel adds, Some theories and observations, <pb n="377"/><cb/> of other persons, “lead us to consider the question, whether the construction of this Ring is of a nature so as permanently to remain in its present state? or whether it be liable to continual and frequent changes, in such a manner as in the course of not many years, to be seen subdivided into narrow slips, and then again as united into one or two circular planes only. Now, without entering into a discussion, the mind seems to revolt, even at first sight, against an idea of the chaotic state in which so large a mass as the Ring of Saturn must needs be, if phenomena like these can be admitted. Nor ought we to indulge a suspicion of this being a reality, unless repeated and well-confirmed observations had proved, beyond a doubt, that this Ring was actually in so fluctuating a condition.” But from his own observations he concludes, “It does not appear to me that there is a sufficient ground for admitting the Ring of Saturn to be of a very changeable nature, and I guess that its phenomena will hereafter be so fully explained, as to reconcile all observations. In the mean while, we must withhold a final judgment of its construction, till we can have more observations. Its division however into two very unequal parts, can admit of no doubt.” See Philos. Trans. vol. 80 pa. 4, 481 &c, and the vol. for 1792, pa. 1 &c. also Hist. de l'Acad. des Scienc. de Paris, 1787, pa. 249 &c.</p><p><hi rend="smallcaps">Rings</hi> <hi rend="italics">of Colours,</hi> in Optics, a phenomenon first observed in thin plates of various substances, by Boyle, and Hook, but afterwards more fully explained by Newton.</p><p>Mr. Boyle having exhibited a variety of colours in colourless liquors, by shaking them till they rose in bubbles, as well as in bubbles of soap and water, and also in turpentine, procured glass blown so thin as to exhibit similar colours; and he observes, that a feather of a proper shape and size, and also a black ribband, held at a proper distance between his eye and the sun, shewed a variety of little rainbows, as he calls them, with very vivid colours. Boyle's Works by Shaw, vol. 2, p. 70. Dr. Hook, about nine years after the publication of Mr. Boyle's Treatise on Colours, exhibited the coloured bubbles of soap and water, and observed, that though at first it appeared white and clear, yet as the film of water became thinner, there appeared upon it all the colours of the rainbow. He also described the beautiful colours that appear in thin plates of Muscovy glass; which appeared, through the microscope, to be ranged in Rings surrounding the white specks or flaws in them, and with the same order of colours as those of the rainbow, and which were often repeated ten times. He also took two thin pieces of glass, ground plane and polished, and putting them one upon another, pressed them till there began to appear a red coloured spot in the middle; and pressing them closer, he observed several Rings of colours encompassing the first place, till, at last, all the colours disappeared out of the middle of the circles, and the central spot appeared white. The first colour that appeared was red, then yellow, then green, then blue, then purple; then again red, yellow, green, blue, and purple; and again in the same order; so that he sometimes counted nine or ten of these circles, the red immediately next to the purple; and the last colour that <cb/> appeared before the white was blue; so that it began with red, and ended with purple. These Rings, he fays, would change their places, by changing the position of the eye, so that, the glasses remaining the same, that part which was red in one position of the eye, was blue in a second, green in the third, &c. Birch's Hist. of the Royal Society, vol. 3, pa. 54.</p><p>Newton, having demonstrated that every different colour consists of rays which have a different and specific degree of refrangibility, and that natural bodies appear of this or that colour, according to their disposition to reflect this or that species of rays (see <hi rend="smallcaps">Colour</hi>), pursued the hint suggested by the experiments of Dr. Hook, already recited, and casually noticed by himself, with regard to thin transparent substances. Upon compressing two prisms hard together, in order to make their sides touch one another, he observed, that in the place of contact they were perfectly transparent, which appeared like a dark spot, and when it was looked through, it seemed like a hole in that air, which was formed into a thin plate, by being impressed between the glasses. When this plate of air, by turning the prisms about their common axis, became so little inclined to the incident rays, that some of them began to be transmitted, there arose in it many slender arcs of colours, which increased, as the motion of the prisms was continued, and bended more and more about the transparent spot, till they were completed into circles, or Rings, surrounding it; and afterwards they became continually more and more contracted.</p><p>By another experiment, with two object glasses, he was enabled to observe distinctly the order and quality of the colours from the central spot, to a very considerable distance. Next to the pellucid central spot, made by the contact of the glasses, succeeded blue, white, yellow, and red. The next circuit immediately surrounding these, consisted of violet, blue, green, yellow, and red. The third circle of colours was purple, blue, green, yellow, and red. The fourth circle consisted of green and red. All the succeeding colours became more and more imperfect and dilute, till, after three or four revolutions, they ended in perfect whiteness.</p><p>When these Rings were examined in a darkened room, by the coloured light of a prism cast on a sheet of white paper, they became more distinct, and visible to a far greater number than in the open air. He sometimes saw more than twenty of them, whereas in the open air he could not discern above eight or nine.</p><p>From other curious observations on these Rings, made by different kinds of light thrown upon them, he inferred, that the thicknesses of the air between the glasses, where the Rings are successively made, by the limits of the seven colours, red, orange, yellow, green, blue, indigo, and violet, in order, are one to another as the cube roots of the squares of the eight lengths of a chord, which sound the notes in an octave, sol, la, fa, sol, la, mi, fa, sol; that is, as the cube roots of the squares of the numbers 1, 8/9, 5/6, 3/4, 2/3, 3/5, 9/6, 1/2. These Rings appeared of that prismatic colour, with which they were illuminated, and by projecting the prismatic colours immediately upon the glasses, he found that the light, which fell on the dark spaces between <pb n="378"/><cb/> the coloured Rings, was transmitted through the glasses without any change of colour. From this circumstance he thought that the origin of these Rings is manifest; because the air between the glasses is disposed according to its various thickness, in some places to reflect, and in others to transmit the light of any particular colour, and in the same place to reflect that of one colour, where it transmits that of another.</p><p>In examining the phenomena of colours made by a denser medium surrounded by a rarer, such as those which appear in plates of Muscovy glass, bubbles of soap and water, &c, the colours were found to be much more vivid than the others, which were made with a rarer medium surrounded by a denser.</p><p>From the preceding phenomena it is an obvious deduction, that the transparent parts of bodies, according to their several series, reflect rays of one colour and transmit those of another; on the same account that thin plates, or bubbles, reflect or transmit those rays, and this Newton supposed to be the reason of all their colours. Hence also he has inferred, that the size of those component parts of natural bodies that affect the light, may be conjectured by their colours. See C<hi rend="smallcaps">OLOUR</hi> and <hi rend="smallcaps">Reflection.</hi></p><p>Newton, pursuing his discoveries concerning the colours of thin substances, found that the same were also produced by plates of a considerable thickness, divisible into lesser thicknesses. The Rings formed in both cases have the same origin, with this difference, that those of the thin plates are made by the alternate reflexions and transmissions of the rays at the second surface of the plate, after one passage through it; but that, in the case of a glass speculum, concave on one side, and convex on the other, and quicksilvered over on the convex side, the rays go through the plate and return before they are alternately reflected and transmitted. Newton's Optics, p. 169, &c. or Newton's Opera, Horsley's edit. vol. 4, p. 121, &c. p. 184, &c.</p><p>The abbé Mazeas, in his experiments on the Rings of colours that appear in thin plates, has discovered several important circumstances attending them, which were overlooked by the sagacious Newton, and which tend to invalidate his theory for explaining them. In rubbing the flat side of an object-glass against another piece of flat and smooth glass, he found that they adhered very firmly together after this friction, and that the same colours were exhibited between these plane glasses, which Newton had observed between the convex object glass of a telescope, and another that was plane; and that the colours were in proportion to their adhesion. When the surfaces of pieces of glass, that are transparent and well polished, are equally pressed, a resistance will be perceived; and wherever this is felt, two or three very fine curve lines will be discovered, some of a pale red, and others of a faint green. If the friction be continued, the red and green lines increase in number at the place of contact; the colours being sometimes mixed without any order, and sometimes disposed in a regular manner; in which case the coloured lines are generally concentric circles, or ovals, more or less elongated, as the surfaces are more or less united.</p><p>When the colours are formed, the glassesadhere with <cb/> considerable force; but if the glasses be separated suddenly, the colours will appear immediately upon their being put together, without the least friction. Beginning with the slightest touch, and increasing the pressure by insensible degrees, there first appears an oval plate of a faint red, and in the centre of it a spot of light green, which enlarges by the pressure, and becomes a green oval, with a red spot in the centre; and this enlarging in its turn, discovers a green spot in its centre. Thus the red and green succeed one another in turns, assuming different shades, and having other colours mixed with them. The greatest difference between these colours exhibited between plane surfaces, and those by curve ones, is, that, in the former case, pressure alone will not produce them, except in the case above mentioned.</p><p>In rubbing together two prisms, with very small refracting angles, which were joined so as to form a parallelopiped, the colours appeared with a surprising lustre at the places of contact, and differently coloured ovals appeared.</p><p>In the centre there was a black spot, bordered by a deep purple; next to this appeared violet, blue, orange, red tinged with purple, light green, and faint purple.</p><p>The other Rings appeared to the naked eye to consist of nothing but faint reds and greens. When these coloured glasses were suspended over the flame of a candle, the colours disappeared suddenly, though they still adhered; but being suffered to cool, the colours returned to their former places, in the same order as before. At first the abbé Mazeas had no doubt but that these colours were owing to a thin plate of air between the glasses, to which Newton has aferibed them; but the remarkable difference in the circumstances attending those produced by the flat plates and those produced by the object glasses of Newton, convinced him that the air was not the cause of this appearance. The colours of the flat plates vanished at the approach of flame, but those of the object glasses did not. Nor was this difference owing to the plane glasses being less compressed than the convex ones; for though the former were compressed ever so much by a pair of forceps, it did not in the least hinder the effect of the flame. Afterwards he put both the plane glasses and the convex ones into the receiver of an air-pump, suspending the former by a thread, and keeping the latter compressed by two strings; but he observed no change in the colours of either of them, in the most perfect vacuum that he could make. Suspecting still that the air adhered to the surface of the glasses, so as not to be separated from them by the force of the pump, he had recourse to other experiments, which rendered it still more improbable that the air should be the cause of these colours. Having laid the coloured plates, after warming them gradually, on burning coals; and thus, when they were nearly red, rubbing them together, he observed the same coloured circles and ovals as before. When he ceased to press upon them, the colours seemed to vanish; but they returned, as he renewed the friction. In order to determine whether the colours were owing to the thickness of some matter interposed between the glasses, he rubbed them toge- <pb n="379"/><cb/> ther with suet and other soft substances between them; yet his endeavour to produce the colours had no effect. However by continuing the friction with some degree of violence, he observed, that a candle appeared through them encompassed with two or three concentric greens, and with a lively red inclining to yellow, and a green like that of an emerald, and at length the Rings assumed the colours of blue, yellow, and violet. The abbé was confirmed in his opinion that there must be some error in Newton's hypothesis, by considering that, according to his measures, the colours of the plates varied with the difference of a millionth part of an inch; whereas he was satisfied that there must have been much greater differences in the distance between his glasses, when the colours remained unchanged. From other experiments he concluded, that the plate of water introduced between the glasses was not the cause of their colours, as Newton apprehended; and that the coloured Rings could not be owing to the compression of the glasses. After all, he adds, that the theory of light, thus reflected from thin plates, is too delicate a subject to be completely ascertained by a small number of observations. Berlin Mem. for 1752, or Memoires Presentes, vol. 2, pa. 28—43. M. du Tour repeated the experiments of the abbé Mazeas, and added some observations of his own. See Mem. Pres. vol. 4, pa. 288.</p><p>Musschenbroeck is also of opinion, that the colours of thin plates do not depend upon the air; but as to the cause of them, he acknowledges that he could not satisfy himself about it. Introd. ad Phil. Nat. vol. 2, p. 738.</p><p>See on this subject Priestley's Hist. of Light, &c. per. 6, sect. 5, pa. 498, &c.</p><p>For an account of the Rings of colours produced by electrical explosions, see <hi rend="italics">Colours</hi> of <hi rend="italics">natural bodies,</hi> C<hi rend="smallcaps">IRCULAR</hi> <hi rend="italics">spots,</hi> and <hi rend="smallcaps">Fairy</hi> <hi rend="italics">circles.</hi></p></div2></div1><div1 part="N" n="RISING" org="uniform" sample="complete" type="entry"><head>RISING</head><p>, in Astronomy, the appearance of the sun, or a star, or other luminary, above the horizon, which before was hid beneath it.</p><p>By reason of the refraction of the at mosphere, the heavenly bodies always appear to rise before their time; that is, they are seen above the horizon, while they are really below it, by about 33′ of a degree.</p><p>There are three poetical kinds of Rising of the stars. See <hi rend="smallcaps">Acronical, Cosmical</hi>, and <hi rend="smallcaps">Heliacal.</hi></p></div1><div1 part="N" n="RIVER" org="uniform" sample="complete" type="entry"><head>RIVER</head><p>, in Geography, a stream or current of fresh water, flowing in a bed or channel, from a source or spring, into the sea.</p><p>When the stream is not large enough to bear boats, or small vessels, loaden, it is properly called by the diminutive, <hi rend="italics">rivulet</hi> or <hi rend="italics">brook;</hi> but when it is considerable enough to carry larger vessels, it is called by the general name River.</p><p>Rivulets have their rise sometimes from great rains, or great quantities of thawed snow, especially in mountainous places; but they more usually arise from springs.</p><p>Rivers themselves all arise either from the confluence of several rivulets, or from lakes.</p><div2 part="N" n="River" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">River</hi></head><p>, in Physics, denotes a stream of water running by its own gravity, from the more elevated parts of the earth towards the lower parts, in a natural bed or channel open above. <cb/></p><p>When the channel is artificial, or cut by art, it is called a canal; of which there are two kinds, viz, that whose channel is every where open, without sluices, called an artificial River, and that whose water is kept up and let off by means of sluices, which is properly a <hi rend="italics">canal.</hi></p><p>Modern philosophers end eavour to reduce the motion and flux of Rivers to precise laws; and with this view they have applied geometry and mechanics to this subject; so that the doctrine of Rivers is become a part of the new philosophy.</p><p>The authors who have most distinguished themselves in this branch, are the Italians, the French, and the Dutch, but especially the first, and among them more especially Gulielmini, and Ximenes.</p><p>Rivers, says Gulielmini, usually have their sources in mountains or elevated grounds; in the descent from which it is mostly that they acquire the velocity, or acceleration, which maintains their future current. In proportion as they advance farther, this velocity diminishes, on account of the continual friction of the water against the bottom and sides of the channel; as well as from the various obstacles they meet with in their progress, and from their arriving at length in plains where the descent is less, and consequently their inclination to the horizon greater. Thus the Reno, a River in Italy, which he says gave occasion, in some measure, to his speculations, is found to have near its mouth a declivity of scarce 52 seconds.</p><p>When the acquired velocity is quite spent, through the many obstacles, so that the current becomes horizontal, there will then nothing remain to propagate the motion, and continue the stream, but the depth, or the perpendicular pressure of the water, which is always proportional to the depth. And, happily for us, this resource increases, as the occasion for it increases; for in proportion as the water loses of the velocity acquired by the descent, it rises and increases in its depth.</p><p>It appears from the laws of motion pertaining to bodies moved on inclined planes, that when water flows freely upon an inclined bed, it acquires a velocity, which is always as the square root of the quantity of descent of the bed. But in an horizontal bed, opened by sluices or otherwise, at one or both ends, the water flows out by its gravity alone.</p><p>The upper parts of the water of a River, and those at a distance from the banks, may continue to flow, from the simple cause or principle of declivity, how small soever it be; for not being detained by any obstacle, the minutest difference of level will have its effect; but the lower parts, which roll along the bottom, will scarce be sensible of so small a declivity; and will only have what motion they receive from the pressure of the superincumbent waters.</p><p>The greatest velocity of a River is about the middle of its depth and breadth, or that point which is the farthest possible from the surface of the water, and from the bottom and sides of the bed or channel. Whereas, on the contrary, the least velocity of the water is at the bottom and sides of the bed, because there the desistance arising from friction is the greatest, which is communicated to the other parts of the section of the <pb n="380"/><cb/> River inversely as the distances from the bottom and sides.</p><p>To find whether the water of a River, almost horizontal, flows by means of the velocity acquired in its descent, or by the pressure of its depth; set up an obstacle perpendicular to it; then if the water rise and swell immediately against the obstacle, it runs by virtue of its fall; but if it first stop a little while, in virtue of its pressure.</p><p>Rivers, according to this author, almost always make their own beds. If the bottom have originally been a large declivity, the water, hence falling with a great force, will have swept away the most elevated parts of the soil, and carrying them lower down, will gradually render the bottom more nearly horizontal.</p><p>The water having made its bed horizontal, becomes so itself, and consequently rakes with the less force against the bottom, till at length that force becomes only equal to the resistance of the bottom, which is now arrived at a state of permanency, at least for a considerable time; and the longer according to the quality of the soil, clay and chalk resisting longer than sand or mud.</p><p>On the other hand, the water is continually wearing away the brims of its channel, and this with the more force, as, by the direction of its stream, it impinges more directly against them. By this means it has a continual tendency to render them parallel to its own course. At the same time that it has thus rectified its edges, it has widened its own bed, and thence becoming less deep, it loses part of its force and pressure: this it continues to do till there is an equilibrium between the force of the water and the resistance of its banks, and then they will remain without farther change. And it appears by experience that these equilibriums are all real, as we find that Rivers only dig and widen to a certain pitch.</p><p>The very reverse of all these things does also on some occasions happen. Rivers, whose waters are thick and muddy, raise their bed, by depositing part of the heterogeneous matters contained in them: they also contract their banks, by a continual opposition of the same matter, in brushing over them. This matter, being thrown aside far from the stream of water, might even serve, by reason of the dullness of the motion, to form new banks.</p><p>If these various causes of resistance to the motion of flowing waters did not exist, viz, the attraction and continual friction of the bottom and sides, the inequalities in both, the windings and angles that occur in their course, and the diminution of their declivity the farther they recede from their springs, the velocity of their currents would be accelerated to 10, 15, or even 20 times more than it is at present in the same Rivers, by which they would become absolutely unnavigable.</p><p>The union of two Rivers into one, makes the whole flow the swifter, because, instead of the friction of four shores, they have only two to overcome, and one bottom instead of two; also the stream, being farther distant from the banks, goes on with the less interruption, besides, that a greater quantity of water, moving with a greater velocity, digs deeper in the bed, and of <cb/> course retrenches of its former width. Hence also it is, that Rivers, by being united, take up less space on the surface of the earth, and are more advantageous to low grounds, which drain their superfluous moisture into them, and have also less occasion for dykes to prevent their overflowing.</p><p>A very good and simple method of measuring the velocity of the current of a River, or canal, is the following. Take a cylindrical piece of dry, light wood, and of a length something less than the depth of the water in the River; about one end of it let there be suspended as many small weights, as may keep the cylinder in a vertical or upright position, with its head just above water. To the centre of this end fix a small straight rod, precisely in the direction of the cylinder's axis; to the end that, when the instrument is suspended in the water, the deviations of the rod from a perpendicularity to the surface of it, may indicate which end of the cylinder goes foremost, by which may be discovered the different velocities of the water at different depths; for when the rod inclines forward, according to the direction of the current, it is a proof that the surface of the water has the greatest velocity; but when it reclines backward, it shews that the swiftest current is at the bottom; and when it remains perpendicular, it is a sign that the velocities at the top and bottom are equal.</p><p>This instrument, being placed in the current of a River or canal, receives all the percussions of the water throughout the whole depth, and will have an equal velocity with that of the whole current from the surface to the bottom at the place where it is put in, and by that means may be found, both with exactness and ease, the mean velocity of that part of the River for any determinate distance and time.</p><p>But to obtain the mean velocity of the whole section of the River, the instrument must be put successively both in the middle and towards the sides, because the velocities at those places are often very different from each other. Having by this means found the several velocities, from the spaces run over in certain times, the arithmetical mean proportional of all these trials, which is found by dividing the common sum of them all by the number of the trials, will be the mean velocity of the River or canal. And if this medium velocity be multiplied by the area of the transverse section of the waters at any place, the product will be the quantity running through that place in a second of time.</p><p>If it be required to find the velocity of the current only at the surface, or at the middle, or at the bottom, a sphere of wood loaded, or a common bottle corked with a little water in it, of such a weight as will remain suspended in equilibrium with the water at the surface or depth which we want to measure, will be better for the purpose than the cylinder, because it is only affected by the water of that sole part of the current where it remains suspended.</p><p>It follows from what has been said in the former part of this article, that the deeper the waters are in their bed in proportion to its breadth, the more their motion is accelerated; so that their velocity increases in the inverse ratio of the breadth of the bed, and also <pb n="381"/><cb/> of the magnitude of the section; whence, in order to augment the velocity of water in a River or canal, without augmenting the declivity of the bed, we must increase the depth of the channel, and diminish its breadth. And these principles are agreeable to observation; as it is well known, that the velocity of flowing waters depends much more on the quantity and depth of the water, and on the compression of the upper parts on the lower, than on the declivity of the bed; and therefore the declivity of a River must be made much greater in the begiuning than toward the end of its course; where it should be almost insensible. If the depth or volume of water in a River or canal be considerable, it will suffice, in the part next the mouth, to allow one foot of declivity through 6000, or 8000, or even (according to Dechales, De Fontibus et Fluviis, prop. 49) 10000 feet in horizontal extent; at most it need not be above 1 in 6 or 7 thousand. From hence the quantity of declivity in equal spaces must slowly and gradually increase as far as the current is to be made fit for navigation; but in such a manner, as that at this upper end there may not be above one foot of perpendicular declivity in 4000 feet of horizontal extent.</p><p>To conclude this article, M. de Buffon observes, that people accustomed to Rivers can easily foretell when there is going to be a sudden increase of water in the bed from floods produced by sudden falls of rain in the higher countries through which the Rivers pass. This they perceive by a particular motion in the water, which they express by saying, that the River's bottom moves, that is, the water at the bottom of the channel runs off faster than usual; and this increase of motion at the bottom of a River always announces a sudden increase of water coming down the stream. Nor, says he, is their opinion ill grounded; because the motion and weight of the waters coming down, though not yet arrived, must act upon the waters in the lower parts of the River, and communicate by impulsion part of their motion to them, within a certain distance.</p><p>On the subject of this article, see an elaborate treatise on Rivers and canals, in the Philos. Trans. vol. 69, pa. 555 &c, by Mr. Mann, who has availed himself of the observations of Gulielmini, and most other writers.</p></div2></div1><div1 part="N" n="RIXDOLLAR" org="uniform" sample="complete" type="entry"><head>RIXDOLLAR</head><p>, a silver coin, struck in several states and free cities in Germany, as also in Flanders, Poland, Denmark, Sweden, &c.</p><p>There is but little difference between the Rixdollar and the dollar, another silver coin struck in Germany, each being nearly equal to the French crown of three livres, or the Spanish piece of eight, or 4s. 6d, sterling.</p></div1><div1 part="N" n="ROBERVAL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROBERVAL</surname> (<foreName full="yes"><hi rend="smallcaps">Giles-Personne</hi></foreName>)</persName></head><p>, an eminent French mathematician, was born in 1602, at Roberval, a parish in the diocese of Beauvais. He was first professor of mathematics at the College of Maitre-Gervais, and afterwards at the College-royal. A similarity of taste connected him with Gassendi and Morin; the latter of whom he succeeded in the mathematical chair at the Royal College, without quitting however that of Ramus. <cb/></p><p>Roberval made experiments on the Torricellian vacuum: he invented two new kinds of balance, one of which was proper for weighing air; and made many other curious experiments. He was one of the first members of the ancient Academy of Sciences of 1666; but died in 1675, at 73 years of age. His principal works are,</p><p>I. A treatise on Mechanics.</p><p>II. A work entitled Aristarchus Samos.</p><p>He had several memoirs inserted in the volumes of the Academy of Sciences of 1666, viz,</p><p>1. Experiments concerning the Pressure of the Air.</p><p>2. Observations on the Composition of Motion, and on the Tangents of Curve Lines.</p><p>3. The Recognition of Equations.</p><p>4. The Geometrical Resolution of Plane and Cubic Equations.</p><p>5. Treatise on Indivisibles.</p><p>6. On the Trochoid, or Cycloid.</p><p>7. A Letter to Father Mersenne.</p><p>8. Two Letters from Torricelli.</p><p>9. A new kind of Balance.</p><p>ROBERVALLIAN <hi rend="italics">Lines,</hi> a name given to certain lines, used for the transformation of figures: thus called from their inventor Roberval.</p><p>These lines bound spaces that are infinitely extended in length, which are nevertheless equal to other spaces that are terminated on all fides.</p><p>The abbot Gallois, in the Memoirs of the Royal Academy, anno 1693, observes, that the method of transforming figures, explained at the latter end of Roberval's treatise of Indivisibles, was the same with that afterwards published by James Gregory, in his Geometria Universalis, and also by Barrow in his Lectiones Geometricæ; and that, by a letter of Torricelli, it appears, that Roberval was the inventor of this manner of transforming figures, by means of certain lines, which Torricelli therefore called Robervallian Lines.</p><p>He adds, that it is highly probable, that J. Gregory first learned the method in the journey he made to Padua in 1668, the method itself having been known in Italy from the year 1646, though the book was not published till the year 1692.</p><p>This account David Gregory has endeavoured to refute, in vindication of his uncle James. His answer is inserted in the Philos. Trans. of 1694, and the abbot rejoined in the French Memoirs of the Academy of 1703.</p></div1><div1 part="N" n="ROBINS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROBINS</surname> (<foreName full="yes"><hi rend="smallcaps">Benjamin</hi></foreName>)</persName></head><p>, an English mathematician and philosopher of great genius and eminence, was born at Bath in Somersetshire, 1707. His parents were of low condition, and Quakers; and consequently neither able from their circumstances, nor willing from their religious profession, to have him much instructed in that kind of learning which they are taught to despise as human. Nevertheless, he made an early and surprising progress in various branches of science and literature, particularly in the mathematics; and his friends being desirous that he might continue his pursuits, and that his merit might not be buried in obscurity, wished that he could be properly recommended <pb n="382"/><cb/> to teach that science in London. Accordingly, a specimen of his abilities in this way was sent up thither, and shewn to Dr. Pemberton, the author of the “View of Sir Isaac Newton's Philosophy;” who, thence conceiving a good opinion of the writer, for a farther trial of his skill sent him some problems, which Robins resolved very much to his satisfaction. He then came to London, where he confirmed the opinion which had been preconceived of his abilities and knowledge.</p><p>But though Robins was possessed of much more skill than is usually required in a common teacher; yet being very young, it was thought proper that he should employ some time in perusing the best writers upon the sublimer parts of the mathematics, before he should undertake publicly the instruction of others. In this interval, besides improving himself in the modern languages, he had opportunities of reading in particular the works of Archimedes, Apollonius, Fermat, Huygens, De Witt, Slusius, Gregory, Barrow, Newton, Taylor, and Cotes. These authors he readily understood without any assistance, of which he gave frequent proofs to his friends: one was, a demonstration of the last proposition of Newton's treatise on Quadratures, which was thought not undeserving a place in the Philosophical Transactions for 1727.</p><p>Not long after, an opportunity offered him of exhibiting to the public a specimen also of his knowledge in Natural Philosophy. The Royal Academy of Sciences at Paris had proposed, among their prize questions in 1724 and 1726, to demonstrate the laws of motion in bodies impinging on one another. John Bernoulli here condescended to be a candidate; and as his dissertation lost the reward, he appealed to the learned world by printing it in 1727. In this piece he endeavoured to establish Leibnitz's opinion of the force of bodies in motion from the effects of their striking against springy materials; as Poleni had before attempted to evince the same thing from experiments of bodies falling on soft and yielding substances. But as the insufficiency of Poleni's arguments had been demonstrated in the Philosophical Transactions, for 1722; so Robins published in the Present State of the Republic of Letters, for May 1728, a Coufutation of Bernoulli's performance, which was allowed to be unanswerable.</p><p>Robins now began to take scholars; and about this time he quitted the garb and profession of a Quaker; for, having neither enthusiasm nor superstition in his nature, as became a mathematician, he soon shook off the prejudices of such early habits. But though he professed to teach the mathematics only, he would frequently assist particular friends in other matters; for he was a man of universal knowledge: and the confinement of this way of life not suiting his disposition, which was active, he gradually declined it, and went into other courses, that required more exercise. Hence he tried many laborious experiments in gunnery; believing that the resistance of the air had a much greater effect on swist projectiles, than was generally supposed. And hence he was led to consider those mechanic arts that depend upon mathematical principles, in which he might employ his invention: as, the constructing of mills, the building of bridges, draining of fens, ren- <cb/> dering of rivers navigable, and making of harbours. Among other arts of this kind, fortification very much engaged his attention; in which he met with opportunities of perfecting himself, by a view of the principal strong places of Flanders, in some journeys he made abroad with persons of distinction.</p><p>On his return home from one of these excursions, he found the learned here amused with Dr. Berkeley's treatise, printed in 1734, entitled, “The Analyst;” in which an examination was made into the grounds of the doctrine of Fluxions, and occasion thence taken to explode that method. Robins was therefore advised to clear up this affair, by giving a full and distinct account of Newton's doctrines, in such a manner, as to obviate all the objections, without naming them, which had been advanced by Berkeley; and accordingly he published, in 1735, <hi rend="italics">A Discourse concerning the Nature and Certainty of Sir Isaac Newton's Method of Fluxions, and of Prime and Ultimate Ratios.</hi> This is a very clear, neat, and elegant performance: and yet some persons, even among those who had written against The Analyst, taking exception at Robins's manner of defending Newton's doctrine, he afterwards wrote two or three additional discourses.</p><p>In 1738, he defended Newton against an objection, contained in a note at the end of a Latin piece, called “Matho, five Cosmotheoria puerilis,” written by Baxter, author of the “Inquiry into the Nature of the Human Soul:” and the year after he printed <hi rend="italics">Remarks on Euler's Treatise of Motion,</hi> on <hi rend="italics">Smith's System of Optics,</hi> and on <hi rend="italics">Jurin's Discourse of Distinct and Indistinct Vision,</hi> annexed to Dr. Smith's work.</p><p>In the mean time Robins's performances were not confined to mathematical subjects: for, in 1739, there came out three pamphlets upon political affairs, which did him great honour. The first was entitled, <hi rend="italics">Observations on the present Convention with Spain:</hi> the second, <hi rend="italics">A Narrative of what passed in the Common Hall of the Citizens of London, assembled for the Election of a Lord Mayor:</hi> the third, <hi rend="italics">An Address to the Electors and other free Subjects of Great Britain, occasioned by the late Succession; in which is contained a Particular Account of all our Negociations with Spain, and their Treatment of us for above ten years past.</hi> These were all published without our author's name; and the first and last were so universally esteemed, that they were generally reputed to have been the production of the great man himself, who was at the head of the opposition to Sir Robert Walpole. They proved of such consequence to Mr. Robins, as to occasion his being employed in-a very honourable post; for, the patriots at length gaining ground against Sir Robert, and a committee of the House of Commons being appointed to examine into his past conduct, Robins was chosen their secretary. But after the committee had presented two reports of their proceedings, a sudden stop was put to their farther progress, by a compromise between the contending parties.</p><p>In 1742, being again at leisure, he published a small treatise, entitled, <hi rend="italics">New Principles of Gunnery;</hi> containing the result of many experiments he had made, by which are discovered the force of gunpowder, and the difference in the resisting power of the air to swist and <pb n="383"/><cb/> slow motions. To this treatise was prefixed a full and learned account of the progress which modern fortification had made from its first rise; as also of the invention of gunpowder, and of what had already been performed in the theory of gunnery. It seems that the occasion of this publication, was the disappointment of a situation at the Royal Military Academy at Woolwich. On the new modelling and establishing of that Academy, in 1741, our author and the late Mr. Muller were competitors for the place of professor of fortification and gunnery. Mr. Muller held then some post in the Tower of London, under the Board of Ordnance, so that, notwithstanding the great knowledge and abilities of our author, the interest which Mr. Muller had with the Board of Ordnance carried the election in his favour. Upon this disappointment Mr. Robins, indignant at the affront, determined to shew them, and the world, by his military publications, what sort of a man he was that they had rejected.</p><p>Upon a discourse containing certain experiments being published in the Philosophical Transactions, with a view to invalidate some of Robins's opinions, he thought proper, in an account he gave of his book in the same Transactions, to take notice of those experiments: and in consequence of this, several dissertations of his on the resistance of the air were read, and the experiments exhibited before the Royal Society, in 1746 and 1747; for which he was presented with the annual gold medal by that Society.</p><p>In 1748 came out Anson's Voyage round the World; which, though it bears Walter's name in the title-page, was in reality written by Robins. Of this voyage the public had for some time been in expectation of seeing an account, composed under that commander's own inspection: for which purpose the reverend Richard Walter was employed, as having been chaplain on board the Centurion the greatest part of the expedition. Walter had accordingly almost finished his task, having brought it down to his own departure from Macao for England; when he proposed to print his work by subscription. It was thought proper however that an able judge should first review and correct it, and Robins was appointed; when, upon examination, it was resolved, that the whole should be written entirely by Robins, and that what Walter had done, being mostly taken verbatim from the journals, should serve as materials only. Hence it was that the whole of the introduction, and many dissertations in the body of the work, were composed by Robins, without receiving the least hint from Walter's manuscript; and what he had transcribed from it regarded chiefly the wind and weather, the currents, courses, bearings, distances, offings, soundings, moorings, the qualities of the ground they anchored on, and such particulars as usually fill up a seaman's account. No production of this kind ever met with a more favourable reception, four large impressions having been sold off within a year: it was also translated into most of the European languages; and it still supports its reputation, having been repeatedly reprinted in various sizes. The fifth edition at London in 1749 was revised and corrected by Robins himself; and the 9th edition was printed there in 1761.</p><p>Thus becoming famous for his elegant talents in <cb/> writing, he was requested to compose an apology for the unfortunate affair at Prestonpans in Scotland. This was added as a preface to the Report of the Proceedings and Opinion of the Board of General Officers on their Examination into the Conduct of Lieutenant General Sir John Cope, &c, printed at London in 1749; and this preface was esteemed a master-piece in its kind.</p><p>Robins had afterwards, by the favour of lord Anson, opportunities of making farther experiments in Gunnery; which have been published since his death, in the edition of his works by his friend Dr. Wilson. He also not a little contributed to the improvements made in the Royal Observatory at Greenwich, by procuring for it, through the interest of the same noble person, a second mural quadrant, and other instruments; by which it became perhaps the completest of any observatory in the world.</p><p>His reputation being now arrived at its full height, he was offered the choice of two very considerable employments. The first was to go to Paris, as one of the commissaries for adjusting the limits in Acadia; the other, to be engineer general to the East India Company, whose forts, being in a most ruinous condition, wanted an able person to put them into a proper state of defence. He accepted the latter, as it was suitable to his genius, and as the Company's terms were both advantageous and honourable. He designed, if he had remained in England, to have written a second part of the Voyage round the World; as appears by a letter from lord Anson to him, dated Bath, Oct. 22, 1749, as follows.</p><p>“Dear Sir, when I last saw you in town, I forgot to ask you, whether you intended to publish the second volume of my Voyage before you leave us; which I confess I am very sorry for. If you should have laid aside all thoughts of favouring the world with more of your works, it will be much disappointed, and no one in it more than your very obliged humble servant, “<hi rend="smallcaps">Anson.</hi>”</p><p>Robins was also preparing an enlarged edition of his New Principles of Gunnery: but, having provided himself with a complete set of astronomical and other instruments, for making observations and experiments in the Indies, he departed hence at Christmas in 1749; and after a voyage, in which the ship was near being cast away, he arrived at India in July following. There he immediately set about his proper business with the greatest diligence, and formed complete plans for Fort St. David and Madras: but he did not live to put them into execution. For the great difference of the climate from that of England being beyond his constitution to support, he was attacked by a sever in September the same year; and though he recovered out of this, yet about eight months after he sell into a langnishing condition, in which he continued till his death, which happened the 29th of July 1751, at only 44 years of age.</p><p>By his last will, Mr. Robins lest the publishing of his Mathematical Works to his honoured and intimate friend Martin Folkes, Esq. president of the Royal Society, and to Dr. James Wilson; but the former of these <pb n="384"/><cb/> gentlemen being incapacitated by a paralytic disorder, for some time before his death, they were afterwards published by the latter, in 2 volumes 8vo, 1761. To this collection, which contains his mathematical and philosophical pieces only, Dr. Wilson has presixed an account of Mr. Robins, from which this memoir is chiefly extracted. He added also a large appendix at the end of the second volume, containing a great many curious and critical matters in various interesting parts of the mathematics. As to Mr. Robins's own papers in these two volumes, they are as follow: viz, in vol. I,</p><p>1. New Principles of Gunnery. First printed in 1742.</p><p>2. An Account of that book. Read before the Royal Society, April the 14th and 21st 1743.</p><p>3. Of the Resistance of the Air. Read the 12th of June 1746.</p><p>4. Of the Resistance of the Air; together with the Method of computing the Motions of Bodies projected in that Medium. Read June 19, 1746.</p><p>5. Account of Experiments relating to the Resistance of the Air. Read the 4th of June 1747.</p><p>6. Of the Force of Gunpowder, with the Computation of the Velocities thereby communicated to military projectiles. Read the 25th of June 1747.</p><p>7. A Comparison of the Experimental Ranges of Cannon and Mortars, with the Theory contained in the preceding papers. Read the 27th of June 1751.</p><p>8. Practical Maxims relating to the Effects and Management of Artillery, and the Flight of Shells and Shot.</p><p>9. A Proposal for increasing the Strength of the British Navy. Read the 2d of April 1747.</p><p>10. A Letter to Martin Folkes, Esq. President of the Royal Society. Read the 7th of January 1748.</p><p>11. A Letter to Lord Anson. Read the 26th of October 1749.</p><p>12. On Pointing, or Directing of Cannon to strike distant objects.</p><p>13. Observations on the Height to which Rockets ascend. Read the 4th of May 1749.</p><p>14. An Account of some Experiments on Rockets, by Mr. Ellicott.</p><p>15. Of the Nature and Advantage of Rifled Barrel Pieces, by Mr. Robins. Read the 2d of July 1747.</p><p>In volume II are,</p><p>16. A Discourse concerning the Nature and Certa nty of Sir Isaac Newton's Methods of Fluxions, and of Prime and Ultimate Ratios.</p><p>17. An Account of the preceding Discourse.</p><p>18. A Review of some of the principal Objections, that have been made to the Doctrine of Fluxions and Ultimate Proportions, with some Remarks on the different Methods, that have been taken to obviate them.</p><p>19. A Dissertation shewing, that the Account of the Doctrines of Fluxions and of Prime and Ultimate Ratios, delivered in Mr. Robins's Discourse, is agreeable to the real Meaning of their great Inventor.</p><p>20. A Demonstration of the Eleventh Proposition of Sir Isaac Newton's Treatise of Quadratures. <cb/></p><p>21. Remarks on Bernoulli's Discourse upon the Laws of the Communication of Motion.</p><p>22. An Examination of a Note concerning the Sun's Parallax, published at the end of Baxter's Matho.</p><p>23. Remarks on Euler's Treatise of Motion; Dr. Smith's System of Optics; and Dr. Jurin's Essay on Distinct and Indistinct Vision.</p><p>24. Appendix by the Publisher.</p><p>It is but justice to say, that Mr. Robins was one of the most accurate and elegant mathematical writers that our language can boast of; and that he made more real improvements in Artillery, the flight and the resistance of projectiles, than all the preceding writers on that subject. His New Principles of Gunnery were translated into several other languages, and commented upon by several eminent writers. The celebrated Euler translated the work into the German language, accompanied with a large and critical commentary; and this work of Euler's was again translated into English in 1714, by Mr. Hugh Brown, with Notes, in one volume 4to.</p></div1><div1 part="N" n="ROBINS" org="uniform" sample="complete" type="entry"><head>ROBINS</head><p>, or <hi rend="smallcaps">Robyns (John</hi>), an English mathematician, was born in Staffordshire about the close of the 15th century, as he was entered a student at Oxford in 1516, where he was educated for the church. But the bent of his genius lay to the sciences, and he soon made such a progress, says Wood, in “the pleasant studies of mathematics and astrology, that he became the ablest person in his time for those studies, not excepted his friend Record, whose learning was more general. At length, taking the degree of bachelor of divinity in 1531, he was the year following made by king Henry the VIIIth (to whom he was chaplain) one of the canons of his college in Oxon, and in December 1543 canon of Windsor, and in fine chaplain to Queen Mary, who had him in great veneration for his learning. Among several things that he hath written relating to astrology (or astronomy) I find these following: “<hi rend="italics">De Culminatione Fixarum Stellarum, &c.</hi><lb/> <hi rend="italics">De Ortu & O<*>casu Stellarum Fixarum, &c.</hi><lb/> <hi rend="italics">Annotationes Astrologicæ, &c. lib.</hi> 3.<lb/> <hi rend="italics">Annotationes Edwardo VI.</hi><lb/> <hi rend="italics">Tractatus de Prognosticatione per Eclipsin.</hi><lb/></p><p>“All which books, that are in MS, were some time in the choice library of Mr. Thomas Allen of Glocester Hall. After his death, coming into the hands of Sir Kenelm Digby, they were by him given to the Bodleian library, where they yet remain. It is also said, that he the said Robyns hath written a book intitled, <hi rend="italics">De Portentosis Cometis,</hi> but such a thing I have not yet seen, nor do I know any thing else of the author, only that paying his last debt to nature the 25th of August 1558, he was buried in the chappel of St. George at Windsore.”</p></div1><div1 part="N" n="ROCKET" org="uniform" sample="complete" type="entry"><head>ROCKET</head><p>, in Pyrotechny, an artificial firework, usually consisting of a cylindrical case of paper, filled with a composition of certain combustible ingredients; which being tied to a rod, mounts into the air to a considerable height, and there bursts. These are called <hi rend="italics">Sky Rockets.</hi> Beside which, there are others called <hi rend="italics">Water Rockets,</hi> from their acting in water. <pb n="385"/><cb/></p><p>The composition with which Rockets are filled, consists of the three following ingredients, viz, saltpetre, charcoal, and sulphur, all well ground; and in the smaller sizes, gunpowder dust is also added. But the proportions of all the ingredients vary with the weight of the Rocket, as in the following Table. <hi rend="center"><hi rend="italics">Compositions for Rockets of Various Sizes.</hi></hi> <table><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">The General Composition for Rockets is,</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Saltpetre</cell><cell cols="1" rows="1" role="data">4 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Sulphur</cell><cell cols="1" rows="1" role="data">1 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Charcoal</cell><cell cols="1" rows="1" role="data">1 lb.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">But for large Rockets,</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Saltpetre</cell><cell cols="1" rows="1" role="data">4 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Sulphur</cell><cell cols="1" rows="1" role="data">1 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Mealpowder</cell><cell cols="1" rows="1" role="data">1 lb.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2 align=center" role="data">For Rockets of a Middle Size,</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Saltpetre</cell><cell cols="1" rows="1" role="data">3 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Sulphur</cell><cell cols="1" rows="1" role="data">2 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Mealpowder</cell><cell cols="1" rows="1" role="data">1 lb.</cell></row><row role="data"><cell cols="1" rows="1" role="data">       Charcoal</cell><cell cols="1" rows="1" role="data">1 lb.</cell></row></table></p><p>When Rockers are intended to mount upwards, they have a long slender rod sixed to the lower end, to direct their motion.</p><p><hi rend="italics">Theory of the Flight of Rockets.</hi>—Mariotte takes the rise of Rockets to be owing to the impulse or resistance of the air against the flame. Desaguliers accounts for it thus.</p><p>Conceive the Rocket to have no vent at the choke, and to be set on fire in the conical bore; the consequence would be, either that the Rocket would burst in the weakest place, or that, if all parts were equally strong, and able to sustain the impulse of the flame, the Rocket would burn out immoveable. Now, as the force of the flame is equable, suppose its action downwards, or that upwards, sufficient to lift 40 pounds; as these forces are equal, but their directions contrary, they will destroy each other's action.</p><p>Imagine then the Rocket opened at the choke; by this means the action of the flame downwards is taken away, and there remains a force equal to 40 pounds acting upwards, to carry up the Rocket, and the stick or rod it is tied to. Accordingly we find that if the composition of the Rocket be very weak, so as not to give an impulse greater than the weight of the Rocket and stick, it does not rise at all; or if the composition be slow, so that a small part of it only kindles at first, the Rocket will not rise.</p><p>The stick serves to keep it perpendicular; for if the Rocket should begin to tumble, moving round a point in the choke, as being the common centre of gravity of Rocket and stick, there would be so much friction against the air, by the stick between the centre and the point, and the point would beat against the air with so much velocity, that the reaction of the medium would restore it to its perpendicularity. When the composition is burnt out, and the impulse upwards has ceased, the common centre of gravity is brought lower towards the middle of the stick; by which means the velocity of the point of the stick is decreased, and that of the <cb/> point of the Rocket is increased; so that the whole will tumble down, with the Rocket end foremost.</p><p>All the while the Rocket burns, the common centre of gravity is shifting and getting downwards, and still the faster and the lower as the stick is lighter; so that it sometimes begins to tumble before it is quite burnt out: but when the stick is too heavy, the common centre of gravity will not get so low, but that the Rocket will rise straight, though not so fast.</p><p>From the experiments of Mr. Robins, and other gentlemen, it appears that the Rockets of 2, 3, or 4 inches diameter, rise the highest; and they found them rise to all heights in the air, from 400 to 1254 yards, which is about three quarters of a mile. See Robins's Tracts, vol. 2, pa. 317, and the Philos. Trans. vol. 46, pa. 578.</p></div1><div1 part="N" n="ROD" org="uniform" sample="complete" type="entry"><head>ROD</head><p>, or <hi rend="italics">Pole,</hi> is a long measure, of 16 1/2 feet, or 5 1/2 yards, or the 4th part of a Gunter's chain, for landmeasuring.</p></div1><div1 part="N" n="ROEMER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROEMER</surname> (<foreName full="yes"><hi rend="smallcaps">Olaus</hi></foreName>)</persName></head><p>, a noted Danish astronomer and mathematician, was born at Arhusen in Jutland, 1644; and at 18 years of age was sent to the university of Copenhagen. He applied assiduously to the study of the mathematics and astronomy, and became so expert in those sciences, that when Picard was sent by Lewis the XIVth in 1671, to make observations in the north, he was greatly surprised and pleased with him. He engaged him to return with him to France, and had him presented to the king, who honoured him with the dauphin as a pupil in mathematics, and settled a pension upon him. He was joined with Picard and Cassini, in making astronomical observations; and in 1672 he was admitted a member of the academy of sciences.</p><p>During the ten years he resided at Paris, he gained great reputation by his discoveries; yet it is said he complained afterwards, that his coadjutors ran away with the honour of many things which belonged to him. Here it was that Roemer, first of any one, found out the velocity with which light moves, by means of the eclipses of Jupiter's satellites. He had observed for many years that, when Jupiter was at his greatest distance from the earth, where he could be observed, the emersions of his first satellite happened constantly 15 or 16 minutes later than the calculation gave them. Hence he concluded that the light reflected by Jupiter took up this time in running over the excess of distance, and consequently that it took up 16 or 18 minutes in running over the diameter of the earth's orbit, and 8 or 9 in coming from the sun to us, provided its velocity was nearly uniform. This discovery had at first many opposers; but it was afterwards confirmed by Dr. Bradley in the most ingenious and beautiful manner.</p><p>In 1681 Roemer was recalled back to his own country by Christian the Vth, king of Denmark, who made him professor of astronomy at Copenhagen. The king employed him also in reforming the coin and the architecture, in regulating the weights and measures, and in measuring and laying out the high roads throughout the kingdom; offices which he discharged with the greatest credit and satisfaction. In consequence he was honoured by the king with the appointment of chancellor of the exchequer and other dignities. Finally he became counsellor of state and burgomaster of, Copen- <pb n="386"/><cb/> hagen, under Frederic the IVth, the successor of Christian. Roemer was preparing to publish the result of his observations, when he died the 19th of September 1710, at 66 years of age: but this loss was supplied by Horrebow, his disciple, then professor of astronomy at Copenhagen, who published, in 4to, 1753, various observations of Roemer, with his method of observing, under the title of <hi rend="italics">Basis Astronomiæ.</hi>—He had also printed various astronomical observations and pieces, in several volumes of the Memoirs of the Royal Academy of Sciences at Paris, of the institution of 1666, particularly vol. 1 and 10 of that collection.</p></div1><div1 part="N" n="ROHAULT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROHAULT</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, a French philosopher, was the son of a rich merchant at Amiens, where he was born in 1620. He cultivated the languages and belles lettres in his own country, and then was sent to Paris to study philosophy. He seems to have been a great lover of truth, at least what he thought so, and to have sought it with much impartiality. He read the ancient and modern philosophers; but Des Cartes was the author who most engaged his notice. Accordingly he became a zealous follower of that great man, and drew up an abridgment and explanation of his philosophy with great clearness and method. In the preface to his <hi rend="italics">Physics,</hi> for so his work is called, he makes no scruple to say, that “the abilities and accomplishments of this philosopher must oblige the whole world to confess, that France is at least as capable of producing and raising men versed in all arts and branches of knowledge, as ancient Greece.” Clerselier, well known for his translation of many pieces of Des Cartes, conceived such an affection for Rohault, on account of his attachment to this philosopher, that he gave him his daughter in marriage against all the remonstrances of his family.</p><p>Rohault's Physics were written in French, but have been translated into Latin by Dr. Samuel Clarke, with notes, in which the Cartesian errors are corrected upon the Newtonian system. The fourth and best edition of <hi rend="italics">Rohault's Physica,</hi> by Clarke, is that of 1718, in 8vo. He wrote also, <hi rend="italics">Elemens de Mathematiques,</hi><lb/> <hi rend="italics">Traité de Mechanique,</hi> and<lb/> <hi rend="italics">Entretiens sur la Philosophie.</hi><lb/></p><p>But these dialogues are founded and carried on upon the principles of the Cartesian philosophy, which has now little other merit, than that of having corrected the errors of the Ancients. Rohault died in 1675, and left behind him the character of an amiable, as well as a learned and philosophic man.</p><p>His posthumous works were collected and printed in two neat little volumes, first at Paris, and then at the Hague in 1690. The contents of them are, 1. The first 6 books of Euclid. 2. Trigonometry. 3. Practical Geometry. 4. Fortification. 5. Mechanics. 6. Perspective. 7. Spherical Trigonometry. 8. Arithmetic.</p></div1><div1 part="N" n="ROLLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROLLE</surname> (<foreName full="yes"><hi rend="smallcaps">Michel</hi></foreName>)</persName></head><p>, a French mathematician, was born at Ambert, a small town in Auvergne, the 21st of April 1652. His first studies and employments were under notaries and attorneys; occupations but little suited to his genius. He went to Paris in 1675, with the only resource of fine penmanship, and subsisted by <cb/> giving lessons in writing. But as his inclination for the mathematics had drawn him to that city, he attended the masters in this science, and soon became one himself. Ozanam proposed a question in arithmetic to him, to which Rolle gave so clear and good a solution, that the minister Colbert made him a handsome gratuity, which at last grew into a sixed pension. He then abandoned penmanship, and gave himself up entirely to algebra and other branches of the mathematics. His conduct in life gained him many friends; in which his scientific merit, his peaceable and regular behaviour, with an exact and scrupulous probity of manners, were his only solicitors.</p><p>Rolle was chosen a member of the Ancient Academy of Sciences in 1685, and named second geometricalpensionary on its renewal in 1699; which he enjoyed till his death, which happened the 5th of July 1719, at 67 years of age.</p><p>The works published by Rolle, were,</p><p>I. A Treatise of Algebra; in 4to, 1690.</p><p>II. A method of resolving Indeterminate Questions in Algebra; in 1699. Besides a great many curious pieces inserted in the Memoirs of the Academy of Sciences, as follow:</p><p>1. A Rule for the Approximation of Irrational Cubes: an. 1666, vol. 10.</p><p>2. A Method of Resolving Equations of all Degrees which are expressed in General Terms: an. 1666, vol. 10.</p><p>3. Remarks upon Geometric Lines: 1702 and 1703.</p><p>4. On the New System of Infinity: 1703, pa. 312.</p><p>5. On the Inverse Method of Tangents: 1705, pa. 25, 171, 222.</p><p>6. Method of finding the Foci of Geometric Lines of all kinds: 1706, pa. 284.</p><p>7. On Curves, both Geometrical and Mechanical, with their Radii of Curvature: 1707, pa. 370.</p><p>8. On the Construction of Equations: 1708, and 1709.</p><p>9. On the Extermination of the Unknown Quantities in the Geometrical Analysis: 1709, pa. 419.</p><p>10. Rules and Remarks for the Construction of Equations: 1711, pa. 86.</p><p>11. On the Application of Diophantine Rules to Geometry: 1712.</p><p>12. On a Paradox in Geometric Effections: 1713, pa. 243.</p><p>13. On Geometric Constructions: 1713, pa. 261, and 1714, pa. 5.</p></div1><div1 part="N" n="ROLLING" org="uniform" sample="complete" type="entry"><head>ROLLING</head><p>, or <hi rend="italics">Rotation,</hi> in Mechanics, a kind of circular motion, by which the moveable body turns round its own axis, or centre, and continually applies new parts of its surface to the body it moves upon. Such is that of a wheel, a sphere, a garden roller, or the like.</p><p>The motion of Rolling is opposed to that of sliding; in which latter motion the same surface is continually applied to the plane it moves along.</p><p>In a wheel, it is only the circumference that properly and simply rolls; the rest of the wheel proceeds in a compound angular kind of motion, and partly <pb n="387"/><cb/> rolls, partly slides. The want of distinguishing between which two motions, occasioned the difficulty of that celebrated problem of Aristotle's Wheel.</p><p>The friction of a body in rolling, is much less than the friction in sliding. And hence arises the great use of wheels, rolls, &c, in machines; as much of the action as possible being laid upon it, to make the resistance the less.</p><p>ROMAN <hi rend="italics">Order,</hi> in Architecture, is the same as the composite. It was invented by the Romans, in the time of Augustus; and it is made up of the Ionic and Corinthian orders, being more ornamental than either.</p></div1><div1 part="N" n="RONDEL" org="uniform" sample="complete" type="entry"><head>RONDEL</head><p>, in Fortification, a round tower, sometimes erected at the foot of a bastion.</p></div1><div1 part="N" n="ROOD" org="uniform" sample="complete" type="entry"><head>ROOD</head><p>, a square measure, being a quantity of land just equal to the 4th part of an acre, or equal to 40 perches or square poles.</p></div1><div1 part="N" n="ROOF" org="uniform" sample="complete" type="entry"><head>ROOF</head><p>, in Architecture, the uppermost part of a building; being that which forms the covering of the whole. In this sense, the Roof comprises the timber work, together with its furniture, of slate, or tile, or lead, or whatever else serves for a covering: though the carpenters usually restrain Roof to the timberwork only.</p><p>The form of a Roof is various: viz, 1. <hi rend="italics">Pointed,</hi> when the ridge, or angle formed by the two sides, is an acute angle.—2. <hi rend="italics">Square,</hi> when the pitch or angle of the ridge is a right angle, called the true pitch.—3. <hi rend="italics">Flat</hi> or pediment Roof, being only pediment pitch, or the angle very obtuse. There are also various other forms, as hip Roofs, valley Roofs, hopper Roofs, double ridges, platforms, round, &c.—In the true pitch, when the sides form a square or right angle, the girt over both sides of the Roof, is accounted equal to the breadth of the building and the half of the same.</p></div1><div1 part="N" n="ROOKE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROOKE</surname> (<foreName full="yes"><hi rend="smallcaps">Lawrence</hi></foreName>)</persName></head><p>, an English astronomer and geometrician, was born at Deptford in Kent, 1623, and educated at Eton school. From hence he removed to King's College, Cambridge, in 1639. After taking the degree of master of arts in 1647, he retired into the country. But in the year 1650 he went to Oxford, and settled in Wadham College, that he might have the company of, and receive improvement from Dr. Wilkins, and Mr. Seth Ward the Astronomy Professor; and that he might also accompany Mr. Boyle in his chemical operations.</p><p>After the death of Mr. Foster, he was chosen Astronomy Professor in Gresham College, London, in the year 1652. He made some observations upon the comet at Oxford, which appeared in the month of December that year; which were printed by Mr. Seth Ward the year following. And, in 1655, Dr. Wallis publishing his treatise on Conic Sections, he dedicated that work to those two gentlemen.</p><p>In 1657, Mr. Rooke was permitted to exchange the astronomy professorship for that of geometry. This step might seem strange, as astronomy still continued to be his favourite study; but it was thought to have been from the convenience of the lodgings, which opened behind the reading hall, and therefore were proper for the reception of those gentlemen after the lec- <cb/> tures, who in the year 1660 formed the Royal Society there.</p><p>Mr. Rooke having thus successively enjoyed those two places some years before the restoration in 1658, most of those gentlemen who had been accustomed to assemble with him at Oxford, coming to London, joined with other philosophical gentlemen, and usually met at Gresham College to hear Mr. Rooke's lectures, and afterwards withdrew into his apartment; till their meetings were interrupted by the quartering of soldiers in the college that year. And after the Royal Society came to be formed and settled into a regular body, Mr. Rooke was very zealous and serviceable in promoting that great and useful institution; though he did not live till it received its establishment by the Royal charter.</p><p>The Marquis of Dorchester, who was not only a patron of learning, but learned himself, used to entertain Mr. Rooke at his seat at Highgate after the restoration, and bring him every Wednesday in his coach to the Royal Society, which then met on that day of the week at Gresham College. But the last time Mr. Rooke was at Highgate, he walked from thence; and it being in the summer, he overheated himself, and taking cold after it, he was thrown into a fever, which cost him his life. He died at his apartments at Gresham College the 27th of June 1662, in the 40th year of his age.</p><p>One other very unfortunate circumstance attended his death, which was, that it happened the very night that he had for some years expected to finish his accurate observations on the satellites of Jupiter. When he found his illness prevented him from making that observation, Dr. Pope says, he sent to the Society his request, that some other person, properly qualified, might be appointed for that purpose; so intent was he to the last on making those curious and useful discoveries, in which he had been so long engaged.</p><p>Mr. Rooke made a nuncupatory will, leaving what he had to Dr. Ward, then lately made bishop of Exeter: whom he permitted to receive what was due upon bond, if the debtors offered payment willingly, otherwise he would not have the bonds put in suit: “for, said he, as I never was in law, nor had any contention with any man, in my life-time; neither would I be so after my death.”</p><p>Few persons have left behind them a more agreeable character than Mr. Rooke, from every person that was acquainted with him, or with his qualifications; and in nothing more than for his veracity: for what he asserted positively, might be fully relied on: but if his opinion was asked concerning any thing that was dubious, his usual answer was, “I have no opinion.” Mr. Hook has given this copious, though concise character of him: “I never was acquainted with any person who knew more, and spoke less, being indeed eminent for the knowledge and improvement of astronomy.” Dr. Wren and Dr. Seth Ward describe him, as a man of profound judgment, a vast comprehension, prodigious memory, and solid experience. His skill in the mathematics was reverenced by all the lovers of those studies, and his perfection in many other sorts of learning deserves no less admiration; but above all, as another writer characterizes him, his extensive know- <pb n="388"/><cb/> ledge had a right influence on the temper of his mind, which had all the humility, goodness, calmness, strength, and sincerity, of a sound and unaffected philosopher. These accounts give us his picture only in miniature; but his successor, Dr. Isaac Barrow, has drawn it in full proportion, in his oration at Gresham College; which is too long to be inserted in this place.</p><p>His writings were chiefly;</p><p>1. <hi rend="italics">Observations on the Comet</hi> of Dec. 1652. This was printed by Dr. Seth Ward, in his Lectures on Comets, 4to, 1653.</p><p>2. <hi rend="italics">Directions for Seamen going to the East and West Indies.</hi> Published in the Philosophical Transactions for Jan. 1665.</p><p>3. <hi rend="italics">A. Method of Observing the Eclipses of the Moon &c.</hi> In the Philos. Trans. for Feb. 1666.</p><p>4. <hi rend="italics">A Discourse concerning the Observations of the Eclipses of the Satellites of Jupiter.</hi> In the History of the Royal Society, pa. 183.</p><p>5. <hi rend="italics">An Account of an Experiment made with Oil in a long Tube.</hi> Read to the Royal Soc. April 23, 1662 — By this experiment it was found, that the oil sunk when the sun shone out, and rose when he was clouded; the proportions of which are set down in the account.</p></div1><div1 part="N" n="ROOT" org="uniform" sample="complete" type="entry"><head>ROOT</head><p>, in Arithmetic and Algebra, denotes a quantity which being multiplied by itself produces some higher power; or a quantity considered as the basis or foundation of a higher power, out of which this arises and grows, like as a plant from its Root.</p><p>In the involution of powers, from a given Root, the Root is also called the first power; when this is once multiplied by itself, it produces the square or second power; this multiplied by the Root again, makes the cube or 3d power; and so on. And hence the Roots also come to be denominated the square-Root, or cube-Root, or 2d Root, or 3d Root, &c, according as the given power or quantity is considered as the square, or cube, or 2d power, or 3d power, &c. Thus, 2 is the square-Root or 2d Root of 4, and the cube-Root or 3d Root of 8, and the 4th Root of 16, &c.</p><p>Hence, the square-Root is the mean proportional between 1 and the square or given power; and the cube-Root is the first of two mean proportionals between 1 and the given cube; and so on.</p><p><hi rend="italics">To Extract the Root</hi> of a given number or power. This is the same thing as to find a number or quantity, which being multiplied the proper number of times, will produce the given number or power. So, to find the cube Root of 8, is finding the number 2, which multiplied twice by itself produces the given number 8.</p><p>For the usual methods of extracting the Roots of Numbers, see the common treatises on Arithmetic.</p><p>A Root, of any power, that consists of two parts, is called a binomial Root; as 12 or 10 + 2. If it consist of three parts, it is a trinomial Root; as 126 or 100 + 20 + 6. And so on.</p><p>The extraction of the Roots of algebraic quantities, is also performed after the same manner as that of numbers; as may be seen in any treatise on algebra. See also the article <hi rend="smallcaps">Extraction</hi> of Roots. <cb/></p><p>A general method for all Roots, is also by Newton's binomial theorem. See <hi rend="smallcaps">Binomial</hi> <hi rend="italics">Theorem.</hi></p><p>Finite approximating rules for the extraction of Roots have also been given by several authors, as Raphson, De Lagney, Halley, &c. See the articles A<hi rend="smallcaps">PPROXIMATION</hi> and <hi rend="smallcaps">Extraction.</hi> See also Newton's Universal Arith. the Appendix; Philos. Trans. numb. 210, or Abridg. vol. 1, pa. 81; Maclaurin's Alg. pa. 242; Simpson's Alg. pa. 155; or his Essays, pa. 82, or his Dissertations, pa. 102, or his Select Exerc. pa. 215: where various general theorems for approximating to the Roots of pure powers are given. See also <hi rend="smallcaps">Equation</hi> and <hi rend="smallcaps">Reduction</hi> of Equations, A<hi rend="smallcaps">PPROXIMATION</hi>, and <hi rend="smallcaps">Converging.</hi></p><p>But the most commodious and general rule of any, for such approximations, I believe, is that which has been invented by myself, and explained in my Tracts, vol. 1, pa. 49: which theorem is this; . That is, having to extract the <hi rend="italics">n</hi>th Root of the given number N; take <hi rend="italics">a</hi><hi rend="sup">n</hi> the nearest rational power to that given quantity N, whether greater or less, its Root of the same kind being <hi rend="italics">a;</hi> then the required Root, or √<hi rend="sup">n</hi>N, will be as is expressed in this formula above; or the same expressed in a proportion will be thus: the Root sought very nearly. Which rule includes all the particular rational formulas of De Lagney, and Halley, which were separately investigated by them; and yet this general formula is perfectly simple and easy to apply, and more easily kept in mind than any one of the said particular formulas.</p><p><hi rend="italics">Ex.</hi> Suppose it be required to double the cube, or to extract the cube Root of the number 2.</p><p>Here N = 2, <hi rend="italics">n</hi> = 3, the nearest Root <hi rend="italics">a</hi> = 1, also <hi rend="italics">a</hi><hi rend="sup">3</hi> = 1; hence, for the cube Root the formula becomes .</p><p>But ; therefore as 4 : 5 :: 1 : 5/4 = 1.25 = the Root nearly by a first approximation.</p><p>Again, for a second approximation, take <hi rend="italics">a</hi> = 5/4, and consequently ; therefore as 378 : 381, or as 126 : 127 :: 5/4 : 635/504 = 1.259921 &c, for the required cube Root of 2, which is true even in the last place of decimals.</p><p><hi rend="smallcaps">Root</hi> <hi rend="italics">of an Equation,</hi> denotes the value of the unknown quantity in an equation; which is such a <pb n="389"/><cb/> quantity, as being substituted instead of that unknown letter, into the equation, shall make all the terms to vanish, or both sides equal to each other. Thus, of the equation , the Root or value of <hi rend="italics">x</hi> is 3, because substituting 3 for <hi rend="italics">x,</hi> makes it become 9 + 5 = 14. And the Root of the equation is 4, because 2 X 4<hi rend="sup">2</hi> = 32. Also the Root of the equation .</p><p>For the Nature of Roots, and for extracting the several Roots of equations, see <hi rend="smallcaps">Equation.</hi></p><p>Every equation has as many Roots, or values of the unknown quantity, as are the dimensions or highest power in it. As a simple equation one Root, a quadratic two, a cubic three, and so on.</p><p>Roots are positive or negative, real or imaginary, rational or radical, &c. See <hi rend="smallcaps">Equation.</hi></p><p><hi rend="italics">Cubic</hi> <hi rend="smallcaps">Root.</hi> This is threefold, even for a simple cubic. So the cube Root of <hi rend="italics">a</hi><hi rend="sup">3</hi>, is either <cb/> . And even the cube Root of 1 itself is either .</p><p><hi rend="italics">Real and Imaginary</hi> <hi rend="smallcaps">Roots.</hi> The odd Roots, as the 3d, 5th, 7th, &c Roots, of all real quantities, whether positive or negative, are real, and are respectively positive or negative. So the cube Root of <hi rend="italics">a</hi><hi rend="sup">3</hi> is <hi rend="italics">a,</hi> and of - <hi rend="italics">a</hi><hi rend="sup">3</hi> is - <hi rend="italics">a.</hi></p><p>But the even Roots, as the 2d, 4th, 6th, &c, are only real when the quantity is positive; being imaginary or impossible when the quantity is negative. So the square Root of <hi rend="italics">a</hi><hi rend="sup">2</hi> is <hi rend="italics">a,</hi> which is real; but the square Root of - <hi rend="italics">a</hi><hi rend="sup">2</hi>, that is, √(- <hi rend="italics">a</hi><hi rend="sup">2</hi>), is imaginary or impossible; because there is no quantity, neither + <hi rend="italics">a</hi> nor - <hi rend="italics">a,</hi> which by squaring will make the given negative square - <hi rend="italics">a</hi><hi rend="sup">2</hi>. <hi rend="center"><hi rend="smallcaps">Table</hi> <hi rend="italics">of</hi> <hi rend="smallcaps">Roots</hi>, &c.</hi><cb/></p><p>THE following Table of Roots, Squares, and Cubes, is very useful in many calculations, and will serve to find square-Roots and cube Roots, as well as square and cubic powers. The Table consists of three columns: in the first column are the series of common numbers, or Roots, 1, 2, 3, 4, 5, 6, &c; in the second column are the squares, and in the third column the cubes of the same. For example, to find the square or the cube of the number or Root 49. Finding this number 49 in the first column; upon the same line with it, stands its square 2401 in the second column, and its cube 117649 in the third column.</p><p>Again, to find the square Root of the number 700. Near the beginning of the Table, it appears that the next less and greater tabular squares are 676 and 729, whose Roots are 26 and 27, and therefore the square Root of 700 is between 26 and 27. But a little further on, viz, among the hundreds, it appears that the required Root lies between 26.4 and 26.5, the tabular squares of these being 696.96 and 702.25, cutting off the proper part of the figures for <cb/> decimals. Take the difference between the less square 696.96 and the given number 700, which gives 3.04, and divide the half of it, viz 1.52, by the less given tabular Root, viz 26.4, and the quotient 575 gives as many more figures of the Root, to be joined to the first three, and thus making the Root equal to 26.4575, which is true in all its places.</p><p>Also to find the cube Root of the number 7000; near the beginning of the Table, among the tens, it appears that the cube Root of this number is between 19 and 20; but farther on, among the hundreds, it appears that it lies between 19.1 and 19.2, allowing for the proper number of integers. But if more figures are required; from the given number 7000 take the next less tabular one, or the cube of 19.1, viz 6967871, and there remains 32.129, the 3d part of which, or 10.730, divide by the square of 19.1, viz 364.81, found on the same line, and the quotient 293 is the next three figures of the Root, and therefore the whole cubic Root is 19.1293, which is true in all its figures.—The Table follows. <pb n="390"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="smallcaps">Table</hi> <hi rend="italics">of Square and Cubic</hi> <hi rend="smallcaps">Roots.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">4096</cell><cell cols="1" rows="1" role="data">262144</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">16129</cell><cell cols="1" rows="1" role="data">2048383</cell><cell cols="1" rows="1" role="data">190</cell><cell cols="1" rows="1" role="data">36100</cell><cell cols="1" rows="1" role="data">6859000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">4225</cell><cell cols="1" rows="1" role="data">274625</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">16384</cell><cell cols="1" rows="1" role="data">2097152</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data">36481</cell><cell cols="1" rows="1" role="data">6967871</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">4356</cell><cell cols="1" rows="1" role="data">287496</cell><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" role="data">16641</cell><cell cols="1" rows="1" role="data">2146689</cell><cell cols="1" rows="1" 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role="data">6751269</cell><cell cols="1" rows="1" role="data">252</cell><cell cols="1" rows="1" role="data">63504</cell><cell cols="1" rows="1" role="data">16003008</cell></row></table><pb n="391"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="italics">Table of Square and Cubic</hi> <hi rend="smallcaps">Roots.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">253</cell><cell cols="1" rows="1" role="data">64009</cell><cell cols="1" rows="1" role="data">16194277</cell><cell cols="1" rows="1" role="data">316</cell><cell cols="1" rows="1" role="data">99856</cell><cell cols="1" rows="1" role="data">31554496</cell><cell cols="1" rows="1" role="data">379</cell><cell cols="1" rows="1" role="data">143641</cell><cell cols="1" rows="1" role="data">54439939</cell><cell cols="1" rows="1" role="data">442</cell><cell cols="1" rows="1" role="data">195364</cell><cell cols="1" rows="1" role="data">86350888</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">254</cell><cell cols="1" rows="1" role="data">64516</cell><cell cols="1" rows="1" role="data">16387064</cell><cell cols="1" rows="1" role="data">317</cell><cell cols="1" rows="1" role="data">100489</cell><cell cols="1" rows="1" role="data">31855013</cell><cell cols="1" rows="1" role="data">380</cell><cell cols="1" rows="1" role="data">144400</cell><cell cols="1" rows="1" role="data">54872000</cell><cell cols="1" rows="1" role="data">443</cell><cell cols="1" rows="1" role="data">196249</cell><cell cols="1" rows="1" role="data">86938307</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">255</cell><cell cols="1" rows="1" role="data">65025</cell><cell cols="1" rows="1" role="data">16581375</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">101124</cell><cell cols="1" rows="1" role="data">32157432</cell><cell cols="1" rows="1" role="data">381</cell><cell cols="1" rows="1" role="data">145161</cell><cell cols="1" rows="1" role="data">55306341</cell><cell cols="1" rows="1" role="data">444</cell><cell cols="1" rows="1" role="data">197136</cell><cell cols="1" rows="1" role="data">87528384</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">65536</cell><cell cols="1" rows="1" role="data">16777216</cell><cell cols="1" rows="1" role="data">319</cell><cell cols="1" rows="1" role="data">101761</cell><cell cols="1" rows="1" role="data">32461759</cell><cell cols="1" rows="1" role="data">382</cell><cell cols="1" rows="1" role="data">145924</cell><cell cols="1" rows="1" role="data">55742968</cell><cell cols="1" rows="1" role="data">445</cell><cell cols="1" rows="1" role="data">198025</cell><cell cols="1" rows="1" role="data">88121125</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">257</cell><cell cols="1" rows="1" role="data">66049</cell><cell cols="1" rows="1" role="data">16974593</cell><cell cols="1" rows="1" role="data">320</cell><cell cols="1" rows="1" role="data">102400</cell><cell cols="1" rows="1" role="data">327680<*>0</cell><cell cols="1" rows="1" role="data">383</cell><cell cols="1" rows="1" role="data">146689</cell><cell cols="1" rows="1" role="data">5618<*>887</cell><cell cols="1" rows="1" role="data">446</cell><cell cols="1" rows="1" role="data">198916</cell><cell cols="1" rows="1" role="data">88716536</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">258</cell><cell cols="1" rows="1" role="data">66564</cell><cell cols="1" rows="1" role="data">17173512</cell><cell cols="1" rows="1" role="data">321</cell><cell cols="1" rows="1" role="data">103041</cell><cell cols="1" rows="1" role="data">33076161</cell><cell cols="1" rows="1" role="data">384</cell><cell cols="1" rows="1" role="data">147456</cell><cell cols="1" rows="1" role="data">56623104</cell><cell cols="1" rows="1" role="data">447</cell><cell cols="1" rows="1" role="data">199809</cell><cell cols="1" rows="1" role="data">89314623</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">259</cell><cell cols="1" rows="1" role="data">67081</cell><cell cols="1" rows="1" role="data">17373979</cell><cell cols="1" rows="1" role="data">322</cell><cell cols="1" rows="1" role="data">103684</cell><cell cols="1" rows="1" role="data">33386248</cell><cell cols="1" rows="1" role="data">385</cell><cell cols="1" rows="1" role="data">148225</cell><cell cols="1" rows="1" role="data">57066625</cell><cell cols="1" rows="1" role="data">448</cell><cell cols="1" rows="1" role="data">200704</cell><cell cols="1" rows="1" role="data">89915392</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">260</cell><cell cols="1" rows="1" role="data">67600</cell><cell cols="1" rows="1" role="data">17576000</cell><cell cols="1" rows="1" role="data">323</cell><cell cols="1" rows="1" role="data">104329</cell><cell cols="1" rows="1" role="data">33698267</cell><cell cols="1" rows="1" role="data">386</cell><cell cols="1" rows="1" role="data">148996</cell><cell cols="1" rows="1" role="data">57512456</cell><cell cols="1" rows="1" role="data">449</cell><cell cols="1" rows="1" role="data">201601</cell><cell cols="1" rows="1" role="data">90518849</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" role="data">68121</cell><cell cols="1" rows="1" role="data">17779581</cell><cell cols="1" rows="1" role="data">324</cell><cell cols="1" rows="1" role="data">104976</cell><cell cols="1" rows="1" role="data">34012224</cell><cell cols="1" rows="1" role="data">387</cell><cell cols="1" rows="1" role="data">149769</cell><cell cols="1" rows="1" role="data">57960603</cell><cell cols="1" rows="1" role="data">450</cell><cell cols="1" rows="1" role="data">202500</cell><cell cols="1" rows="1" role="data">91125000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">262</cell><cell cols="1" rows="1" role="data">68644</cell><cell cols="1" rows="1" role="data">17984728</cell><cell cols="1" rows="1" role="data">325</cell><cell cols="1" rows="1" role="data">105625</cell><cell cols="1" rows="1" role="data">34328125</cell><cell cols="1" rows="1" role="data">388</cell><cell cols="1" rows="1" role="data">150544</cell><cell cols="1" rows="1" role="data">58411072</cell><cell cols="1" rows="1" role="data">451</cell><cell cols="1" rows="1" role="data">203401</cell><cell cols="1" rows="1" role="data">91733851</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">263</cell><cell cols="1" rows="1" role="data">69169</cell><cell cols="1" rows="1" role="data">18191447</cell><cell cols="1" rows="1" role="data">326</cell><cell cols="1" rows="1" role="data">106276</cell><cell cols="1" rows="1" role="data">34645976</cell><cell cols="1" rows="1" role="data">389</cell><cell cols="1" rows="1" role="data">151321</cell><cell cols="1" rows="1" role="data">58863869</cell><cell cols="1" rows="1" role="data">452</cell><cell cols="1" rows="1" role="data">204304</cell><cell cols="1" rows="1" role="data">92345408</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">264</cell><cell cols="1" rows="1" role="data">69696</cell><cell cols="1" rows="1" role="data">18399744</cell><cell cols="1" rows="1" role="data">327</cell><cell cols="1" rows="1" role="data">106929</cell><cell cols="1" rows="1" role="data">34965783</cell><cell cols="1" rows="1" role="data">390</cell><cell cols="1" rows="1" role="data">152100</cell><cell cols="1" rows="1" role="data">59319000</cell><cell cols="1" rows="1" role="data">453</cell><cell cols="1" rows="1" role="data">205209</cell><cell cols="1" rows="1" role="data">92959677</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">265</cell><cell cols="1" rows="1" role="data">70225</cell><cell cols="1" rows="1" role="data">18609625</cell><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" role="data">107584</cell><cell cols="1" rows="1" role="data">35287552</cell><cell cols="1" rows="1" role="data">391</cell><cell cols="1" rows="1" role="data">152881</cell><cell cols="1" rows="1" role="data">59776471</cell><cell cols="1" rows="1" role="data">454</cell><cell cols="1" rows="1" role="data">206116</cell><cell cols="1" rows="1" role="data">93576664</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">266</cell><cell cols="1" rows="1" role="data">70756</cell><cell cols="1" rows="1" role="data">18821096</cell><cell cols="1" rows="1" role="data">329</cell><cell cols="1" rows="1" role="data">108241</cell><cell cols="1" rows="1" role="data">35611289</cell><cell cols="1" rows="1" role="data">392</cell><cell cols="1" rows="1" role="data">153664</cell><cell cols="1" rows="1" role="data">60236288</cell><cell cols="1" rows="1" role="data">455</cell><cell cols="1" rows="1" role="data">207025</cell><cell cols="1" rows="1" role="data">94196375</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">267</cell><cell cols="1" rows="1" role="data">71289</cell><cell cols="1" rows="1" role="data">19034163</cell><cell cols="1" rows="1" role="data">330</cell><cell cols="1" rows="1" role="data">108900</cell><cell cols="1" rows="1" role="data">35937000</cell><cell cols="1" rows="1" role="data">393</cell><cell cols="1" rows="1" role="data">154449</cell><cell cols="1" rows="1" role="data">60698457</cell><cell cols="1" rows="1" role="data">456</cell><cell cols="1" rows="1" role="data">207936</cell><cell cols="1" rows="1" role="data">94818816</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">268</cell><cell cols="1" rows="1" role="data">71824</cell><cell cols="1" rows="1" role="data">19248832</cell><cell cols="1" rows="1" role="data">331</cell><cell cols="1" rows="1" role="data">109561</cell><cell cols="1" rows="1" role="data">36264691</cell><cell cols="1" rows="1" role="data">394</cell><cell cols="1" rows="1" role="data">155236</cell><cell cols="1" rows="1" role="data">61162984</cell><cell cols="1" rows="1" role="data">457</cell><cell cols="1" rows="1" role="data">208849</cell><cell cols="1" rows="1" role="data">95443993</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">269</cell><cell cols="1" rows="1" role="data">72361</cell><cell cols="1" rows="1" role="data">19465109</cell><cell cols="1" rows="1" role="data">332</cell><cell cols="1" rows="1" role="data">110224</cell><cell cols="1" rows="1" role="data">36594368</cell><cell cols="1" rows="1" role="data">395</cell><cell cols="1" rows="1" role="data">156025</cell><cell cols="1" rows="1" role="data">61629875</cell><cell cols="1" rows="1" role="data">458</cell><cell cols="1" rows="1" role="data">209764</cell><cell cols="1" rows="1" role="data">96071912</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">270</cell><cell cols="1" rows="1" role="data">72900</cell><cell cols="1" rows="1" role="data">19683000</cell><cell cols="1" rows="1" role="data">333</cell><cell cols="1" rows="1" role="data">110889</cell><cell cols="1" rows="1" role="data">36926037</cell><cell cols="1" rows="1" role="data">396</cell><cell cols="1" rows="1" role="data">156816</cell><cell cols="1" rows="1" role="data">62099136</cell><cell cols="1" rows="1" role="data">459</cell><cell cols="1" rows="1" role="data">210681</cell><cell cols="1" rows="1" role="data">96702579</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">271</cell><cell cols="1" rows="1" role="data">73441</cell><cell cols="1" rows="1" role="data">19902511</cell><cell cols="1" rows="1" role="data">334</cell><cell cols="1" rows="1" role="data">111556</cell><cell cols="1" rows="1" role="data">37259704</cell><cell cols="1" rows="1" role="data">397</cell><cell cols="1" rows="1" role="data">157609</cell><cell cols="1" rows="1" role="data">62570773</cell><cell cols="1" rows="1" role="data">460</cell><cell cols="1" rows="1" role="data">211600</cell><cell cols="1" rows="1" role="data">97336000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">272</cell><cell cols="1" rows="1" role="data">73984</cell><cell cols="1" rows="1" role="data">20123648</cell><cell cols="1" rows="1" role="data">335</cell><cell cols="1" rows="1" role="data">112225</cell><cell cols="1" rows="1" role="data">37595375</cell><cell cols="1" rows="1" role="data">398</cell><cell cols="1" rows="1" role="data">158404</cell><cell cols="1" rows="1" role="data">63044792</cell><cell cols="1" rows="1" role="data">461</cell><cell cols="1" rows="1" role="data">212521</cell><cell cols="1" rows="1" role="data">97972181</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">273</cell><cell cols="1" rows="1" role="data">74529</cell><cell cols="1" rows="1" role="data">20346417</cell><cell cols="1" rows="1" role="data">336</cell><cell cols="1" rows="1" role="data">112896</cell><cell cols="1" rows="1" role="data">37933056</cell><cell cols="1" rows="1" role="data">399</cell><cell cols="1" rows="1" role="data">159201</cell><cell cols="1" rows="1" role="data">63521199</cell><cell cols="1" rows="1" role="data">462</cell><cell cols="1" rows="1" role="data">213444</cell><cell cols="1" rows="1" role="data">98611128</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">274</cell><cell cols="1" rows="1" role="data">75076</cell><cell cols="1" rows="1" role="data">20570824</cell><cell cols="1" rows="1" role="data">337</cell><cell cols="1" rows="1" role="data">113569</cell><cell cols="1" rows="1" role="data">38272753</cell><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">160000</cell><cell cols="1" rows="1" role="data">64000000</cell><cell cols="1" rows="1" role="data">463</cell><cell cols="1" rows="1" role="data">214369</cell><cell cols="1" rows="1" role="data">99252847</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">275</cell><cell cols="1" rows="1" role="data">75625</cell><cell cols="1" rows="1" role="data">20796875</cell><cell cols="1" rows="1" role="data">338</cell><cell cols="1" rows="1" role="data">114244</cell><cell cols="1" rows="1" role="data">386144<*>2</cell><cell cols="1" rows="1" role="data">401</cell><cell cols="1" rows="1" role="data">160801</cell><cell cols="1" rows="1" role="data">64481201</cell><cell cols="1" rows="1" role="data">464</cell><cell cols="1" rows="1" role="data">215296</cell><cell cols="1" rows="1" role="data">99897344</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">276</cell><cell cols="1" rows="1" role="data">76176</cell><cell cols="1" rows="1" role="data">21024576</cell><cell cols="1" rows="1" role="data">339</cell><cell cols="1" rows="1" role="data">114921</cell><cell cols="1" rows="1" role="data">38958219</cell><cell cols="1" rows="1" role="data">402</cell><cell cols="1" rows="1" role="data">161604</cell><cell cols="1" rows="1" 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role="data">139876</cell><cell cols="1" rows="1" role="data">52313624</cell><cell cols="1" rows="1" role="data">437</cell><cell cols="1" rows="1" role="data">190969</cell><cell cols="1" rows="1" role="data">83453453</cell><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">250000</cell><cell cols="1" rows="1" role="data">125000000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">312</cell><cell cols="1" rows="1" role="data">97344</cell><cell cols="1" rows="1" role="data">30371328</cell><cell cols="1" rows="1" role="data">375</cell><cell cols="1" rows="1" role="data">140625</cell><cell cols="1" rows="1" role="data">52734375</cell><cell cols="1" rows="1" role="data">438</cell><cell cols="1" rows="1" role="data">191844</cell><cell cols="1" rows="1" role="data">84027672</cell><cell cols="1" rows="1" role="data">501</cell><cell cols="1" rows="1" role="data">251001</cell><cell cols="1" rows="1" role="data">125751501</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">313</cell><cell cols="1" rows="1" role="data">97969</cell><cell cols="1" rows="1" role="data">30664297</cell><cell cols="1" rows="1" role="data">376</cell><cell cols="1" rows="1" role="data">141376</cell><cell cols="1" rows="1" role="data">53157376</cell><cell cols="1" rows="1" role="data">439</cell><cell cols="1" rows="1" role="data">192721</cell><cell cols="1" rows="1" role="data">84604519</cell><cell cols="1" rows="1" role="data">502</cell><cell cols="1" rows="1" role="data">252004</cell><cell cols="1" rows="1" role="data">126506008</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" role="data">98596</cell><cell cols="1" rows="1" role="data">30959144</cell><cell cols="1" rows="1" role="data">377</cell><cell cols="1" rows="1" role="data">142129</cell><cell cols="1" rows="1" role="data">53582633</cell><cell cols="1" rows="1" role="data">440</cell><cell cols="1" rows="1" role="data">193600</cell><cell cols="1" rows="1" role="data">85184000</cell><cell cols="1" rows="1" role="data">503</cell><cell cols="1" rows="1" role="data">253009</cell><cell cols="1" rows="1" role="data">127263527</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">315</cell><cell cols="1" rows="1" role="data">99225</cell><cell cols="1" rows="1" role="data">31255875</cell><cell cols="1" rows="1" role="data">378</cell><cell cols="1" rows="1" role="data">142884</cell><cell cols="1" rows="1" role="data">54010152</cell><cell cols="1" rows="1" role="data">441</cell><cell cols="1" rows="1" role="data">194481</cell><cell cols="1" rows="1" role="data">85766121</cell><cell cols="1" rows="1" role="data">504</cell><cell cols="1" rows="1" role="data">254016</cell><cell cols="1" rows="1" role="data">128024064</cell></row></table><pb n="392"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="italics">Table of Square and Cube</hi> <hi rend="smallcaps">Roots.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell><cell cols="1" rows="1" role="data">Root.</cell><cell cols="1" rows="1" role="data">Square.</cell><cell cols="1" rows="1" role="data">Cube.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">505</cell><cell cols="1" rows="1" role="data">255025</cell><cell cols="1" rows="1" role="data">128787625</cell><cell cols="1" rows="1" role="data">568</cell><cell cols="1" rows="1" role="data">322624</cell><cell cols="1" rows="1" role="data">183250432</cell><cell cols="1" rows="1" role="data">631</cell><cell cols="1" rows="1" role="data">398161</cell><cell cols="1" rows="1" role="data">251239591</cell><cell cols="1" rows="1" role="data">694</cell><cell cols="1" rows="1" role="data">481636</cell><cell cols="1" rows="1" role="data">334255384</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">506</cell><cell cols="1" rows="1" role="data">256036</cell><cell cols="1" rows="1" role="data">129554216</cell><cell cols="1" rows="1" role="data">569</cell><cell cols="1" rows="1" role="data">323761</cell><cell cols="1" rows="1" role="data">184220009</cell><cell cols="1" rows="1" role="data">632</cell><cell cols="1" rows="1" role="data">399424</cell><cell cols="1" rows="1" role="data">252435968</cell><cell cols="1" rows="1" role="data">695</cell><cell cols="1" rows="1" role="data">483025</cell><cell cols="1" rows="1" role="data">335702375</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">507</cell><cell cols="1" rows="1" role="data">257049</cell><cell cols="1" rows="1" role="data">130323843</cell><cell cols="1" rows="1" role="data">570</cell><cell cols="1" rows="1" role="data">324900</cell><cell cols="1" rows="1" role="data">185193000</cell><cell cols="1" rows="1" role="data">633</cell><cell cols="1" rows="1" role="data">400689</cell><cell cols="1" rows="1" role="data">253636137</cell><cell cols="1" rows="1" role="data">696</cell><cell cols="1" rows="1" role="data">484416</cell><cell cols="1" rows="1" role="data">337153536</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">508</cell><cell cols="1" rows="1" role="data">258064</cell><cell cols="1" rows="1" role="data">131096512</cell><cell cols="1" rows="1" role="data">571</cell><cell cols="1" rows="1" role="data">326041</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">510</cell><cell cols="1" rows="1" role="data">260100</cell><cell cols="1" rows="1" role="data">132651000</cell><cell cols="1" rows="1" role="data">573</cell><cell cols="1" rows="1" role="data">328329</cell><cell cols="1" rows="1" role="data">188132517</cell><cell cols="1" rows="1" role="data">636</cell><cell cols="1" rows="1" role="data">404496</cell><cell cols="1" rows="1" role="data">257259456</cell><cell cols="1" rows="1" role="data">699</cell><cell cols="1" rows="1" role="data">488601</cell><cell cols="1" rows="1" role="data">341532099</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">511</cell><cell cols="1" rows="1" role="data">261121</cell><cell cols="1" rows="1" role="data">133432831</cell><cell cols="1" rows="1" role="data">574</cell><cell cols="1" rows="1" role="data">329476</cell><cell cols="1" rows="1" role="data">189119224</cell><cell cols="1" rows="1" role="data">637</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">135005697</cell><cell cols="1" rows="1" role="data">576</cell><cell cols="1" rows="1" role="data">331776</cell><cell cols="1" rows="1" role="data">191102976</cell><cell cols="1" rows="1" role="data">639</cell><cell cols="1" rows="1" role="data">408321</cell><cell cols="1" rows="1" role="data">260917119</cell><cell cols="1" rows="1" role="data">702</cell><cell cols="1" rows="1" role="data">492804</cell><cell cols="1" rows="1" role="data">345948008</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">514</cell><cell cols="1" rows="1" role="data">264196</cell><cell cols="1" rows="1" role="data">135796744</cell><cell cols="1" rows="1" role="data">577</cell><cell cols="1" rows="1" role="data">332929</cell><cell cols="1" rows="1" role="data">192100033</cell><cell cols="1" rows="1" role="data">640</cell><cell cols="1" rows="1" role="data">409600</cell><cell cols="1" rows="1" role="data">262144000</cell><cell cols="1" rows="1" 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cols="1" rows="1" role="data">335241</cell><cell cols="1" rows="1" role="data">194104539</cell><cell cols="1" rows="1" role="data">642</cell><cell cols="1" rows="1" role="data">412164</cell><cell cols="1" rows="1" role="data">264609288</cell><cell cols="1" rows="1" role="data">705</cell><cell cols="1" rows="1" role="data">497025</cell><cell cols="1" rows="1" role="data">350402625</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">517</cell><cell cols="1" rows="1" role="data">267289</cell><cell cols="1" rows="1" role="data">138188413</cell><cell cols="1" rows="1" role="data">580</cell><cell cols="1" rows="1" role="data">336400</cell><cell cols="1" rows="1" role="data">195112000</cell><cell cols="1" rows="1" role="data">643</cell><cell cols="1" rows="1" role="data">413449</cell><cell cols="1" rows="1" role="data">265847707</cell><cell cols="1" rows="1" role="data">706</cell><cell cols="1" rows="1" role="data">498436</cell><cell cols="1" rows="1" 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role="data">197137368</cell><cell cols="1" rows="1" role="data">645</cell><cell cols="1" rows="1" role="data">416025</cell><cell cols="1" rows="1" role="data">268336125</cell><cell cols="1" rows="1" role="data">708</cell><cell cols="1" rows="1" role="data">501264</cell><cell cols="1" rows="1" role="data">354894912</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">520</cell><cell cols="1" rows="1" role="data">270400</cell><cell cols="1" rows="1" role="data">140608000</cell><cell cols="1" rows="1" role="data">583</cell><cell cols="1" rows="1" role="data">339889</cell><cell cols="1" rows="1" role="data">198155287</cell><cell cols="1" rows="1" role="data">646</cell><cell cols="1" rows="1" role="data">417316</cell><cell cols="1" rows="1" role="data">269586136</cell><cell cols="1" rows="1" role="data">709</cell><cell cols="1" rows="1" role="data">502681</cell><cell cols="1" rows="1" role="data">356400829</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">521</cell><cell cols="1" rows="1" role="data">271441</cell><cell cols="1" rows="1" role="data">141420761</cell><cell cols="1" rows="1" role="data">584</cell><cell cols="1" rows="1" role="data">341056</cell><cell cols="1" rows="1" role="data">199176704</cell><cell cols="1" rows="1" role="data">647</cell><cell cols="1" rows="1" role="data">418609</cell><cell cols="1" rows="1" role="data">270840023</cell><cell cols="1" rows="1" role="data">710</cell><cell cols="1" rows="1" role="data">504100</cell><cell cols="1" rows="1" role="data">357911000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">522</cell><cell cols="1" rows="1" role="data">272484</cell><cell cols="1" rows="1" role="data">142236648</cell><cell cols="1" rows="1" role="data">585</cell><cell cols="1" rows="1" role="data">342225</cell><cell cols="1" rows="1" role="data">200201625</cell><cell cols="1" rows="1" role="data">648</cell><cell cols="1" rows="1" 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role="data">747</cell><cell cols="1" rows="1" role="data">558009</cell><cell cols="1" rows="1" role="data">416832723</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">559</cell><cell cols="1" rows="1" role="data">312481</cell><cell cols="1" rows="1" role="data">174676879</cell><cell cols="1" rows="1" role="data">622</cell><cell cols="1" rows="1" role="data">386884</cell><cell cols="1" rows="1" role="data">240641848</cell><cell cols="1" rows="1" role="data">685</cell><cell cols="1" rows="1" role="data">469225</cell><cell cols="1" rows="1" role="data">321419125</cell><cell cols="1" rows="1" role="data">748</cell><cell cols="1" rows="1" role="data">559504</cell><cell cols="1" rows="1" role="data">418508992</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">560</cell><cell cols="1" rows="1" role="data">313600</cell><cell cols="1" rows="1" role="data">175616000</cell><cell cols="1" rows="1" role="data">623</cell><cell cols="1" rows="1" role="data">388129</cell><cell cols="1" rows="1" role="data">241804367</cell><cell cols="1" rows="1" role="data">686</cell><cell cols="1" rows="1" role="data">470596</cell><cell cols="1" rows="1" role="data">322828856</cell><cell cols="1" rows="1" role="data">749</cell><cell cols="1" rows="1" role="data">561001</cell><cell cols="1" rows="1" role="data">420189749</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">561</cell><cell cols="1" rows="1" role="data">314721</cell><cell cols="1" rows="1" role="data">176558481</cell><cell cols="1" rows="1" role="data">624</cell><cell cols="1" rows="1" role="data">389376</cell><cell cols="1" rows="1" role="data">242970624</cell><cell cols="1" rows="1" role="data">687</cell><cell cols="1" rows="1" role="data">471969</cell><cell cols="1" rows="1" role="data">324242703</cell><cell cols="1" rows="1" role="data">750</cell><cell cols="1" rows="1" role="data">562500</cell><cell cols="1" rows="1" role="data">421875000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">562</cell><cell cols="1" rows="1" role="data">315844</cell><cell cols="1" rows="1" role="data">177504328</cell><cell cols="1" rows="1" role="data">625</cell><cell cols="1" rows="1" role="data">390625</cell><cell cols="1" rows="1" role="data">244140625</cell><cell cols="1" rows="1" role="data">688</cell><cell cols="1" rows="1" role="data">473344</cell><cell cols="1" rows="1" role="data">325660672</cell><cell cols="1" rows="1" role="data">751</cell><cell cols="1" rows="1" role="data">564001</cell><cell cols="1" rows="1" role="data">423564751</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">563</cell><cell cols="1" rows="1" role="data">316969</cell><cell cols="1" rows="1" role="data">178453547</cell><cell cols="1" rows="1" role="data">626</cell><cell cols="1" rows="1" role="data">391876</cell><cell cols="1" rows="1" role="data">245314376</cell><cell cols="1" rows="1" role="data">689</cell><cell cols="1" rows="1" role="data">474721</cell><cell cols="1" rows="1" role="data">327082769</cell><cell cols="1" rows="1" role="data">752</cell><cell cols="1" rows="1" role="data">565504</cell><cell cols="1" rows="1" role="data">425259008</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">564</cell><cell cols="1" rows="1" role="data">318096</cell><cell cols="1" rows="1" role="data">179406144</cell><cell cols="1" rows="1" role="data">627</cell><cell cols="1" rows="1" role="data">393129</cell><cell cols="1" rows="1" role="data">246491883</cell><cell cols="1" rows="1" role="data">690</cell><cell cols="1" rows="1" role="data">476100</cell><cell cols="1" rows="1" role="data">328509000</cell><cell cols="1" rows="1" role="data">753</cell><cell cols="1" rows="1" role="data">567009</cell><cell cols="1" rows="1" role="data">426957777</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">565</cell><cell cols="1" rows="1" role="data">319225</cell><cell cols="1" rows="1" role="data">180362125</cell><cell cols="1" rows="1" role="data">628</cell><cell cols="1" rows="1" role="data">394384</cell><cell cols="1" rows="1" role="data">247673152</cell><cell cols="1" rows="1" role="data">691</cell><cell cols="1" rows="1" role="data">477481</cell><cell cols="1" rows="1" role="data">329939371</cell><cell cols="1" rows="1" role="data">754</cell><cell cols="1" rows="1" role="data">568516</cell><cell cols="1" rows="1" role="data">428661064</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">566</cell><cell cols="1" rows="1" role="data">320356</cell><cell cols="1" rows="1" role="data">181321496</cell><cell cols="1" rows="1" role="data">629</cell><cell cols="1" rows="1" role="data">395641</cell><cell cols="1" rows="1" role="data">248858189</cell><cell cols="1" rows="1" role="data">692</cell><cell cols="1" rows="1" role="data">478864</cell><cell cols="1" rows="1" role="data">331373888</cell><cell cols="1" rows="1" role="data">755</cell><cell cols="1" rows="1" role="data">570025</cell><cell cols="1" rows="1" role="data">430368875</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">567</cell><cell cols="1" rows="1" role="data">321489</cell><cell cols="1" rows="1" role="data">182284263</cell><cell cols="1" rows="1" role="data">630</cell><cell cols="1" rows="1" role="data">396900</cell><cell cols="1" rows="1" role="data">250047000</cell><cell cols="1" rows="1" role="data">693</cell><cell cols="1" rows="1" role="data">480249</cell><cell cols="1" rows="1" role="data">332812557</cell><cell cols="1" rows="1" role="data">756</cell><cell cols="1" rows="1" role="data">571536</cell><cell cols="1" rows="1" role="data">432081216</cell></row></table><pb n="393"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=12" role="data"><hi rend="italics">Table of Square and Cubic</hi> <hi rend="smallcaps">Roots.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Root</cell><cell cols="1" rows="1" role="data">Square</cell><cell cols="1" rows="1" role="data">Cube</cell><cell cols="1" rows="1" role="data">Root</cell><cell cols="1" rows="1" role="data">Square</cell><cell cols="1" rows="1" role="data">Cube</cell><cell cols="1" rows="1" role="data">Root</cell><cell cols="1" rows="1" role="data">Square</cell><cell cols="1" rows="1" role="data">Cube</cell><cell cols="1" rows="1" role="data">Root</cell><cell cols="1" rows="1" role="data">Square</cell><cell cols="1" rows="1" role="data">Cube</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">757</cell><cell cols="1" rows="1" role="data">573049</cell><cell cols="1" rows="1" role="data">433798093</cell><cell cols="1" rows="1" role="data">820</cell><cell cols="1" rows="1" role="data">672400</cell><cell cols="1" rows="1" role="data">551368000</cell><cell cols="1" rows="1" role="data">883</cell><cell cols="1" rows="1" role="data">779689</cell><cell cols="1" rows="1" role="data">688465387</cell><cell cols="1" rows="1" role="data">946</cell><cell cols="1" rows="1" role="data">894916</cell><cell cols="1" rows="1" role="data">846590536</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">758</cell><cell cols="1" rows="1" role="data">574564</cell><cell cols="1" rows="1" role="data">435519512</cell><cell cols="1" rows="1" role="data">821</cell><cell cols="1" rows="1" role="data">674041</cell><cell cols="1" rows="1" role="data">553387661</cell><cell cols="1" rows="1" role="data">884</cell><cell cols="1" rows="1" role="data">781456</cell><cell cols="1" rows="1" role="data">690807104</cell><cell cols="1" rows="1" role="data">947</cell><cell cols="1" rows="1" role="data">896809</cell><cell cols="1" rows="1" role="data">849378123</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">759</cell><cell cols="1" rows="1" role="data">576081</cell><cell cols="1" rows="1" role="data">437245479</cell><cell cols="1" rows="1" role="data">822</cell><cell cols="1" rows="1" role="data">675684</cell><cell cols="1" rows="1" role="data">555412248</cell><cell cols="1" rows="1" role="data">885</cell><cell cols="1" rows="1" role="data">783225</cell><cell cols="1" rows="1" role="data">693154125</cell><cell cols="1" rows="1" role="data">948</cell><cell cols="1" rows="1" role="data">898704</cell><cell cols="1" rows="1" role="data">851971392</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">760</cell><cell cols="1" rows="1" role="data">577600</cell><cell cols="1" rows="1" role="data">438976000</cell><cell cols="1" rows="1" role="data">823</cell><cell cols="1" rows="1" role="data">677329</cell><cell cols="1" rows="1" role="data">557441767</cell><cell cols="1" rows="1" role="data">886</cell><cell cols="1" rows="1" role="data">784996</cell><cell cols="1" rows="1" role="data">695506456</cell><cell cols="1" rows="1" role="data">949</cell><cell cols="1" rows="1" role="data">900601</cell><cell cols="1" rows="1" role="data">854670349</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">761</cell><cell cols="1" rows="1" role="data">579121</cell><cell cols="1" rows="1" role="data">440711081</cell><cell cols="1" rows="1" role="data">824</cell><cell cols="1" rows="1" role="data">678976</cell><cell cols="1" rows="1" role="data">559476224</cell><cell cols="1" rows="1" role="data">887</cell><cell cols="1" rows="1" role="data">786769</cell><cell cols="1" rows="1" role="data">697864103</cell><cell cols="1" rows="1" role="data">950</cell><cell cols="1" rows="1" role="data">902500</cell><cell cols="1" rows="1" role="data">857375000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">762</cell><cell cols="1" rows="1" role="data">580644</cell><cell cols="1" rows="1" role="data">442450728</cell><cell cols="1" rows="1" role="data">825</cell><cell cols="1" rows="1" role="data">680625</cell><cell cols="1" rows="1" role="data">561515625</cell><cell cols="1" rows="1" role="data">888</cell><cell cols="1" rows="1" role="data">788544</cell><cell cols="1" rows="1" role="data">700227072</cell><cell cols="1" rows="1" role="data">951</cell><cell cols="1" rows="1" role="data">904401</cell><cell cols="1" rows="1" role="data">860085351</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">763</cell><cell cols="1" rows="1" role="data">582169</cell><cell cols="1" rows="1" role="data">444194947</cell><cell cols="1" rows="1" role="data">826</cell><cell cols="1" rows="1" role="data">682276</cell><cell cols="1" rows="1" role="data">563559976</cell><cell cols="1" rows="1" role="data">889</cell><cell cols="1" rows="1" role="data">790321</cell><cell cols="1" rows="1" role="data">702595369</cell><cell cols="1" rows="1" role="data">952</cell><cell cols="1" rows="1" role="data">906304</cell><cell cols="1" rows="1" role="data">862801408</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">764</cell><cell cols="1" rows="1" role="data">583696</cell><cell cols="1" rows="1" role="data">445943744</cell><cell cols="1" rows="1" role="data">827</cell><cell cols="1" rows="1" role="data">683920</cell><cell cols="1" rows="1" role="data">565609283</cell><cell cols="1" rows="1" role="data">890</cell><cell cols="1" rows="1" role="data">792100</cell><cell cols="1" rows="1" role="data">704969000</cell><cell cols="1" rows="1" role="data">953</cell><cell cols="1" rows="1" role="data">908209</cell><cell cols="1" rows="1" role="data">865523177</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">765</cell><cell cols="1" rows="1" role="data">585225</cell><cell cols="1" rows="1" role="data">447697125</cell><cell cols="1" rows="1" role="data">828</cell><cell cols="1" rows="1" role="data">685584</cell><cell cols="1" rows="1" role="data">567663552</cell><cell cols="1" rows="1" role="data">891</cell><cell cols="1" rows="1" role="data">793881</cell><cell cols="1" rows="1" role="data">707347971</cell><cell cols="1" rows="1" role="data">954</cell><cell cols="1" rows="1" role="data">910116</cell><cell cols="1" rows="1" role="data">868250664</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">766</cell><cell cols="1" rows="1" role="data">586756</cell><cell cols="1" rows="1" role="data">449455096</cell><cell cols="1" rows="1" role="data">829</cell><cell cols="1" rows="1" role="data">687241</cell><cell cols="1" rows="1" role="data">569722789</cell><cell cols="1" rows="1" role="data">892</cell><cell cols="1" rows="1" role="data">795664</cell><cell cols="1" rows="1" role="data">709732288</cell><cell cols="1" rows="1" role="data">955</cell><cell cols="1" rows="1" role="data">912025</cell><cell cols="1" rows="1" role="data">870983875</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">767</cell><cell cols="1" rows="1" role="data">588289</cell><cell cols="1" rows="1" role="data">451217663</cell><cell cols="1" rows="1" role="data">830</cell><cell cols="1" rows="1" role="data">688900</cell><cell cols="1" rows="1" role="data">571787000</cell><cell cols="1" rows="1" role="data">893</cell><cell cols="1" rows="1" role="data">797449</cell><cell cols="1" rows="1" role="data">712121957</cell><cell cols="1" rows="1" role="data">956</cell><cell cols="1" rows="1" role="data">913936</cell><cell cols="1" rows="1" role="data">873722816</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">768</cell><cell cols="1" rows="1" role="data">589824</cell><cell cols="1" rows="1" role="data">452984832</cell><cell cols="1" rows="1" role="data">831</cell><cell cols="1" rows="1" role="data">690561</cell><cell cols="1" rows="1" role="data">573856191</cell><cell cols="1" rows="1" role="data">894</cell><cell cols="1" rows="1" role="data">799236</cell><cell cols="1" rows="1" role="data">714516984</cell><cell cols="1" rows="1" role="data">957</cell><cell cols="1" rows="1" role="data">915849</cell><cell cols="1" rows="1" role="data">876467493</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">769</cell><cell cols="1" rows="1" role="data">591361</cell><cell cols="1" rows="1" role="data">454756609</cell><cell cols="1" rows="1" role="data">832</cell><cell cols="1" rows="1" role="data">692224</cell><cell cols="1" rows="1" role="data">575930368</cell><cell cols="1" rows="1" role="data">895</cell><cell cols="1" rows="1" role="data">801025</cell><cell cols="1" rows="1" role="data">716917375</cell><cell cols="1" rows="1" role="data">958</cell><cell cols="1" rows="1" role="data">917764</cell><cell cols="1" rows="1" role="data">879217912</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">770</cell><cell cols="1" rows="1" role="data">592900</cell><cell cols="1" rows="1" role="data">456533000</cell><cell cols="1" rows="1" role="data">833</cell><cell cols="1" rows="1" role="data">693889</cell><cell cols="1" rows="1" role="data">578009537</cell><cell cols="1" rows="1" role="data">896</cell><cell cols="1" rows="1" role="data">802816</cell><cell cols="1" rows="1" role="data">719323136</cell><cell cols="1" rows="1" role="data">959</cell><cell cols="1" rows="1" role="data">919681</cell><cell cols="1" rows="1" role="data">881974079</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">771</cell><cell cols="1" rows="1" role="data">594441</cell><cell cols="1" rows="1" role="data">458314011</cell><cell cols="1" rows="1" role="data">834</cell><cell cols="1" rows="1" role="data">695556</cell><cell cols="1" rows="1" role="data">580093704</cell><cell cols="1" rows="1" role="data">897</cell><cell cols="1" rows="1" role="data">804609</cell><cell cols="1" rows="1" role="data">721734273</cell><cell cols="1" rows="1" role="data">960</cell><cell cols="1" rows="1" role="data">921600</cell><cell cols="1" rows="1" role="data">884736000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">772</cell><cell cols="1" rows="1" role="data">595984</cell><cell cols="1" rows="1" role="data">460099648</cell><cell cols="1" rows="1" role="data">835</cell><cell cols="1" rows="1" role="data">697225</cell><cell cols="1" rows="1" role="data">582182875</cell><cell cols="1" rows="1" role="data">898</cell><cell cols="1" rows="1" role="data">806404</cell><cell cols="1" rows="1" role="data">724150792</cell><cell cols="1" rows="1" role="data">961</cell><cell cols="1" rows="1" role="data">923521</cell><cell cols="1" rows="1" role="data">887503681</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">773</cell><cell cols="1" rows="1" role="data">597529</cell><cell cols="1" rows="1" 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role="data">864900</cell><cell cols="1" rows="1" role="data">804357000</cell><cell cols="1" rows="1" role="data">993</cell><cell cols="1" rows="1" role="data">986049</cell><cell cols="1" rows="1" role="data">979146657</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">805</cell><cell cols="1" rows="1" role="data">648025</cell><cell cols="1" rows="1" role="data">521660125</cell><cell cols="1" rows="1" role="data">868</cell><cell cols="1" rows="1" role="data">753424</cell><cell cols="1" rows="1" role="data">653972032</cell><cell cols="1" rows="1" role="data">931</cell><cell cols="1" rows="1" role="data">866761</cell><cell cols="1" rows="1" role="data">806954491</cell><cell cols="1" rows="1" role="data">994</cell><cell cols="1" rows="1" role="data">988036</cell><cell cols="1" rows="1" role="data">982107784</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">806</cell><cell cols="1" rows="1" role="data">649636</cell><cell cols="1" rows="1" role="data">523606616</cell><cell cols="1" rows="1" role="data">869</cell><cell cols="1" rows="1" role="data">755161</cell><cell cols="1" rows="1" role="data">656234909</cell><cell cols="1" rows="1" role="data">932</cell><cell cols="1" rows="1" role="data">868624</cell><cell cols="1" rows="1" role="data">809557568</cell><cell cols="1" rows="1" role="data">995</cell><cell cols="1" rows="1" role="data">990025</cell><cell cols="1" rows="1" role="data">985074875</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">807</cell><cell cols="1" rows="1" role="data">651249</cell><cell cols="1" rows="1" role="data">525557943</cell><cell cols="1" rows="1" role="data">870</cell><cell cols="1" rows="1" role="data">756900</cell><cell cols="1" rows="1" role="data">658503000</cell><cell cols="1" rows="1" role="data">933</cell><cell cols="1" rows="1" role="data">870489</cell><cell cols="1" rows="1" role="data">812166237</cell><cell cols="1" rows="1" role="data">996</cell><cell cols="1" rows="1" role="data">992016</cell><cell cols="1" rows="1" role="data">988047936</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">808</cell><cell cols="1" rows="1" role="data">652864</cell><cell cols="1" rows="1" role="data">527514112</cell><cell cols="1" rows="1" role="data">871</cell><cell cols="1" rows="1" role="data">758641</cell><cell cols="1" rows="1" role="data">660776311</cell><cell cols="1" rows="1" role="data">934</cell><cell cols="1" rows="1" role="data">872356</cell><cell cols="1" rows="1" role="data">814780504</cell><cell cols="1" rows="1" role="data">997</cell><cell cols="1" rows="1" role="data">994009</cell><cell cols="1" rows="1" role="data">991026973</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">809</cell><cell cols="1" rows="1" role="data">654481</cell><cell cols="1" rows="1" role="data">529475129</cell><cell cols="1" rows="1" role="data">872</cell><cell cols="1" rows="1" role="data">760384</cell><cell cols="1" rows="1" role="data">663054848</cell><cell cols="1" rows="1" role="data">935</cell><cell cols="1" rows="1" role="data">874225</cell><cell cols="1" rows="1" role="data">817400375</cell><cell cols="1" rows="1" role="data">998</cell><cell cols="1" rows="1" role="data">996004</cell><cell cols="1" rows="1" role="data">994011992</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">810</cell><cell cols="1" rows="1" role="data">656100</cell><cell cols="1" rows="1" role="data">531441000</cell><cell cols="1" rows="1" role="data">873</cell><cell cols="1" rows="1" role="data">762129</cell><cell cols="1" rows="1" role="data">665338617</cell><cell cols="1" rows="1" role="data">936</cell><cell cols="1" rows="1" role="data">876096</cell><cell cols="1" rows="1" role="data">820025856</cell><cell cols="1" rows="1" role="data">999</cell><cell cols="1" rows="1" role="data">993001</cell><cell cols="1" rows="1" role="data">997002999</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">811</cell><cell cols="1" rows="1" role="data">657721</cell><cell cols="1" rows="1" role="data">533411731</cell><cell cols="1" rows="1" role="data">874</cell><cell cols="1" rows="1" role="data">763876</cell><cell cols="1" rows="1" role="data">667627624</cell><cell cols="1" rows="1" role="data">937</cell><cell cols="1" rows="1" role="data">877969</cell><cell cols="1" rows="1" role="data">822656953</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">1000000</cell><cell cols="1" rows="1" role="data">1000000000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">812</cell><cell cols="1" rows="1" role="data">659344</cell><cell cols="1" rows="1" role="data">535387328</cell><cell cols="1" rows="1" role="data">875</cell><cell cols="1" rows="1" role="data">765625</cell><cell cols="1" rows="1" role="data">669921875</cell><cell cols="1" rows="1" role="data">938</cell><cell cols="1" rows="1" role="data">879844</cell><cell cols="1" rows="1" role="data">825293672</cell><cell cols="1" rows="1" role="data">1001</cell><cell cols="1" rows="1" role="data">1002001</cell><cell cols="1" rows="1" role="data">1003003001</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">813</cell><cell cols="1" rows="1" role="data">660969</cell><cell cols="1" rows="1" role="data">537366797</cell><cell cols="1" rows="1" role="data">876</cell><cell cols="1" rows="1" role="data">767376</cell><cell cols="1" rows="1" role="data">672221376</cell><cell cols="1" rows="1" role="data">939</cell><cell cols="1" rows="1" role="data">881721</cell><cell cols="1" rows="1" role="data">827936019</cell><cell cols="1" rows="1" role="data">1002</cell><cell cols="1" rows="1" role="data">1004004</cell><cell cols="1" rows="1" role="data">1006012008</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">814</cell><cell cols="1" rows="1" role="data">662596</cell><cell cols="1" rows="1" role="data">539353144</cell><cell cols="1" rows="1" role="data">877</cell><cell cols="1" rows="1" role="data">769129</cell><cell cols="1" rows="1" role="data">674526133</cell><cell cols="1" rows="1" role="data">940</cell><cell cols="1" rows="1" role="data">883600</cell><cell cols="1" rows="1" role="data">830584000</cell><cell cols="1" rows="1" role="data">1003</cell><cell cols="1" rows="1" role="data">1006009</cell><cell cols="1" rows="1" role="data">1009027027</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">815</cell><cell cols="1" rows="1" role="data">664225</cell><cell cols="1" rows="1" role="data">541343375</cell><cell cols="1" rows="1" role="data">878</cell><cell cols="1" rows="1" role="data">770884</cell><cell cols="1" rows="1" role="data">676836152</cell><cell cols="1" rows="1" role="data">941</cell><cell cols="1" rows="1" role="data">885481</cell><cell cols="1" rows="1" role="data">833237621</cell><cell cols="1" rows="1" role="data">1004</cell><cell cols="1" rows="1" role="data">1008016</cell><cell cols="1" rows="1" role="data">1012048064</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">816</cell><cell cols="1" rows="1" role="data">665856</cell><cell cols="1" rows="1" role="data">543338496</cell><cell cols="1" rows="1" role="data">879</cell><cell cols="1" rows="1" role="data">772641</cell><cell cols="1" rows="1" role="data">679151439</cell><cell cols="1" rows="1" role="data">942</cell><cell cols="1" rows="1" role="data">887364</cell><cell cols="1" rows="1" role="data">835896888</cell><cell cols="1" rows="1" role="data">1005</cell><cell cols="1" rows="1" role="data">1010025</cell><cell cols="1" rows="1" role="data">1015075125</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">817</cell><cell cols="1" rows="1" role="data">667489</cell><cell cols="1" rows="1" role="data">545338513</cell><cell cols="1" rows="1" role="data">880</cell><cell cols="1" rows="1" role="data">774400</cell><cell cols="1" rows="1" role="data">681472000</cell><cell cols="1" rows="1" role="data">943</cell><cell cols="1" rows="1" role="data">889249</cell><cell cols="1" rows="1" role="data">838561807</cell><cell cols="1" rows="1" role="data">1006</cell><cell cols="1" rows="1" role="data">1012036</cell><cell cols="1" rows="1" role="data">1018108216</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">818</cell><cell cols="1" rows="1" role="data">669124</cell><cell cols="1" rows="1" role="data">547343432</cell><cell cols="1" rows="1" role="data">881</cell><cell cols="1" rows="1" role="data">776161</cell><cell cols="1" rows="1" role="data">683797841</cell><cell cols="1" rows="1" role="data">944</cell><cell cols="1" rows="1" role="data">891136</cell><cell cols="1" rows="1" role="data">841232384</cell><cell cols="1" rows="1" role="data">1007</cell><cell cols="1" rows="1" role="data">1014049</cell><cell cols="1" rows="1" role="data">1021147343</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">819</cell><cell cols="1" rows="1" role="data">670761</cell><cell cols="1" rows="1" role="data">549353259</cell><cell cols="1" rows="1" role="data">882</cell><cell cols="1" rows="1" role="data">777924</cell><cell cols="1" rows="1" role="data">686128968</cell><cell cols="1" rows="1" role="data">945</cell><cell cols="1" rows="1" role="data">893025</cell><cell cols="1" rows="1" role="data">843908625</cell><cell cols="1" rows="1" role="data">1008</cell><cell cols="1" rows="1" role="data">1016064</cell><cell cols="1" rows="1" role="data">1024192512</cell></row></table><pb n="394"/></p><p>The following is another Table of the Square Roots of the first 1000 Numbers to 10 places of decimal figures beside the integers, which needs no farther explanation, as Numbers stand always in the first column, and their Square Roots in the next. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=8" role="data"><hi rend="italics">Table of Square</hi> <hi rend="smallcaps">Roots</hi> <hi rend="italics">to ten Decimal Places.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1.0000000000</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">8.0000000000</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">11.2694276696</cell><cell cols="1" rows="1" role="data">190</cell><cell cols="1" rows="1" role="data">13.7840487521</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1.4142135624</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">8.0622577483</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">11.3137084990</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data">13.8202749611</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1.7320508076</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">8.1240384046</cell><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" role="data">11.3578166916</cell><cell cols="1" rows="1" role="data">192</cell><cell cols="1" rows="1" role="data">13.8564064606</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2.0000000000</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">8.1853527719</cell><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" role="data">11.4017542510</cell><cell cols="1" rows="1" role="data">193</cell><cell cols="1" rows="1" role="data">13.8924439894</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2.2360679775</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">8.2462112512</cell><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data">11.4455231423</cell><cell cols="1" rows="1" role="data">194</cell><cell cols="1" rows="1" role="data">13.9283882772</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2.4494897428</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">8.3066238629</cell><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" role="data">11.4891252931</cell><cell cols="1" rows="1" role="data">195</cell><cell cols="1" rows="1" role="data">13.9642400438</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2.6457513111</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">8.3666002653</cell><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" role="data">11.5325625947</cell><cell cols="1" rows="1" role="data">196</cell><cell cols="1" rows="1" role="data">14.0000000000</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2.8284271247</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">8.4261497732</cell><cell cols="1" rows="1" role="data">134</cell><cell cols="1" rows="1" role="data">11.5758369028</cell><cell cols="1" rows="1" role="data">197</cell><cell cols="1" rows="1" role="data">14.0356688441</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3.0000000000</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">8.4852813742</cell><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" role="data">11.6189500386</cell><cell cols="1" rows="1" role="data">198</cell><cell cols="1" rows="1" role="data">14.0712472795</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3.1622776602</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">8.5440037453</cell><cell cols="1" rows="1" role="data">136</cell><cell cols="1" rows="1" role="data">11.6619037897</cell><cell cols="1" rows="1" role="data">199</cell><cell cols="1" rows="1" role="data">14.1067359797</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3.3166247904</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">8.6023252670</cell><cell cols="1" rows="1" role="data">137</cell><cell cols="1" rows="1" role="data">11.7046999111</cell><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">14.1421356237</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3.4641016151</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">8.6602540378</cell><cell cols="1" rows="1" role="data">138</cell><cell cols="1" rows="1" role="data">11.7473443808</cell><cell cols="1" rows="1" role="data">201</cell><cell cols="1" rows="1" role="data">14.1774468788</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3.6055512755</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">8.7177978871</cell><cell cols="1" rows="1" role="data">139</cell><cell cols="1" rows="1" role="data">11.7898261226</cell><cell cols="1" rows="1" role="data">202</cell><cell cols="1" rows="1" role="data">14.2126704036</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">3.7416573868</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">8.7749643874</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data">11.8321595662</cell><cell cols="1" rows="1" 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rows="1" role="data">17.7763888346</cell><cell cols="1" rows="1" role="data">379</cell><cell cols="1" rows="1" role="data">19.4679223339</cell><cell cols="1" rows="1" role="data">442</cell><cell cols="1" rows="1" role="data">21.0237960416</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">254</cell><cell cols="1" rows="1" role="data">15.9373774505</cell><cell cols="1" rows="1" role="data">317</cell><cell cols="1" rows="1" role="data">17.8044938148</cell><cell cols="1" rows="1" role="data">380</cell><cell cols="1" rows="1" role="data">19.4935886896</cell><cell cols="1" rows="1" role="data">443</cell><cell cols="1" rows="1" role="data">21.0475651798</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">255</cell><cell cols="1" rows="1" role="data">15.9687194227</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">17.8325545001</cell><cell cols="1" rows="1" role="data">381</cell><cell cols="1" rows="1" role="data">19.5192212959</cell><cell cols="1" rows="1" role="data">444</cell><cell cols="1" rows="1" role="data">21.0713075057</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">16.0000000000</cell><cell cols="1" rows="1" role="data">319</cell><cell cols="1" rows="1" role="data">17.8605710995</cell><cell cols="1" rows="1" role="data">382</cell><cell cols="1" rows="1" role="data">19.5448202857</cell><cell cols="1" rows="1" role="data">445</cell><cell cols="1" rows="1" role="data">21.0950231097</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">257</cell><cell cols="1" rows="1" role="data">16.0312195419</cell><cell cols="1" rows="1" role="data">320</cell><cell cols="1" rows="1" role="data">17.8885438200</cell><cell cols="1" rows="1" role="data">383</cell><cell cols="1" rows="1" role="data">19.5703857908</cell><cell cols="1" rows="1" 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role="data">501</cell><cell cols="1" rows="1" role="data">22.3830292856</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">313</cell><cell cols="1" rows="1" role="data">17.6918060130</cell><cell cols="1" rows="1" role="data">376</cell><cell cols="1" rows="1" role="data">19.3907194297</cell><cell cols="1" rows="1" role="data">439</cell><cell cols="1" rows="1" role="data">20.9523268398</cell><cell cols="1" rows="1" role="data">502</cell><cell cols="1" rows="1" role="data">22.4053565024</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" role="data">17.7200451467</cell><cell cols="1" rows="1" role="data">377</cell><cell cols="1" rows="1" role="data">19.4164878389</cell><cell cols="1" rows="1" role="data">440</cell><cell cols="1" rows="1" role="data">20.9761769<*>34</cell><cell cols="1" rows="1" role="data">503</cell><cell cols="1" rows="1" role="data">22.4276614920</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">315</cell><cell cols="1" rows="1" role="data">17.7482393493</cell><cell cols="1" rows="1" role="data">378</cell><cell cols="1" rows="1" role="data">19.4422220952</cell><cell cols="1" rows="1" role="data">441</cell><cell cols="1" rows="1" role="data">21.0000000000</cell><cell cols="1" rows="1" role="data">504</cell><cell cols="1" rows="1" role="data">22.4499441206</cell></row></table><pb n="396"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=8" role="data"><hi rend="italics">Table of Square</hi> <hi rend="smallcaps">Roots.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">505</cell><cell cols="1" rows="1" role="data">22.4722050542</cell><cell cols="1" rows="1" role="data">568</cell><cell cols="1" rows="1" role="data">23.8327505756</cell><cell cols="1" rows="1" role="data">631</cell><cell cols="1" rows="1" role="data">25.1197133742</cell><cell cols="1" rows="1" role="data">694</cell><cell cols="1" rows="1" role="data">26.3438797446</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">506</cell><cell cols="1" rows="1" role="data">22.4944437584</cell><cell cols="1" rows="1" role="data">569</cell><cell cols="1" rows="1" role="data">23.8537208838</cell><cell cols="1" rows="1" role="data">632</cell><cell cols="1" rows="1" role="data">25.1396101800</cell><cell cols="1" rows="1" role="data">695</cell><cell cols="1" rows="1" role="data">26.3628526529</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">507</cell><cell cols="1" rows="1" role="data">22.5166604984</cell><cell cols="1" rows="1" role="data">570</cell><cell cols="1" rows="1" role="data">23.8746727726</cell><cell cols="1" rows="1" role="data">633</cell><cell cols="1" rows="1" role="data">25.1594912508</cell><cell cols="1" rows="1" role="data">696</cell><cell cols="1" rows="1" role="data">26.3818119165</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">508</cell><cell cols="1" rows="1" role="data">22.5388553392</cell><cell cols="1" rows="1" role="data">571</cell><cell cols="1" rows="1" role="data">23.8956062907</cell><cell cols="1" rows="1" role="data">634</cell><cell cols="1" rows="1" role="data">25.1793566201</cell><cell cols="1" rows="1" role="data">697</cell><cell cols="1" rows="1" role="data">26.4007575649</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" 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rend="align=right" role="data"><cell cols="1" rows="1" role="data">564</cell><cell cols="1" rows="1" role="data">23.7486841741</cell><cell cols="1" rows="1" role="data">627</cell><cell cols="1" rows="1" role="data">25.0399680511</cell><cell cols="1" rows="1" role="data">690</cell><cell cols="1" rows="1" role="data">26.2678510731</cell><cell cols="1" rows="1" role="data">753</cell><cell cols="1" rows="1" role="data">27.4408454680</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">565</cell><cell cols="1" rows="1" role="data">23.7697286480</cell><cell cols="1" rows="1" role="data">628</cell><cell cols="1" rows="1" role="data">25.0599281723</cell><cell cols="1" rows="1" role="data">691</cell><cell cols="1" rows="1" role="data">26.2868788562</cell><cell cols="1" rows="1" role="data">754</cell><cell cols="1" rows="1" role="data">27.4590604355</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">566</cell><cell cols="1" rows="1" role="data">23.7907545067</cell><cell cols="1" rows="1" role="data">629</cell><cell cols="1" rows="1" role="data">25.0798724080</cell><cell cols="1" rows="1" role="data">692</cell><cell cols="1" rows="1" role="data">26.3058928759</cell><cell cols="1" rows="1" role="data">755</cell><cell cols="1" rows="1" role="data">27.4772633281</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">567</cell><cell cols="1" rows="1" role="data">23.8117617996</cell><cell cols="1" rows="1" role="data">630</cell><cell cols="1" rows="1" role="data">25.0998007960</cell><cell cols="1" rows="1" role="data">693</cell><cell cols="1" rows="1" role="data">26.3248931622</cell><cell cols="1" rows="1" role="data">756</cell><cell cols="1" rows="1" role="data">27.4954541697</cell></row></table><pb n="397"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=8" role="data"><hi rend="italics">Table of Square</hi> <hi rend="smallcaps">Roots.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Square Root.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">757</cell><cell cols="1" rows="1" role="data">27.5136329844</cell><cell cols="1" rows="1" role="data">818</cell><cell cols="1" rows="1" role="data">28.6006992922</cell><cell cols="1" rows="1" role="data">879</cell><cell cols="1" rows="1" role="data">29.6479324743</cell><cell cols="1" rows="1" role="data">940</cell><cell cols="1" rows="1" role="data">30.6594194335</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">758</cell><cell cols="1" rows="1" role="data">27.5317997959</cell><cell cols="1" rows="1" role="data">819</cell><cell cols="1" rows="1" role="data">28.6181760425</cell><cell cols="1" rows="1" role="data">880</cell><cell cols="1" rows="1" role="data">29.6647939484</cell><cell cols="1" rows="1" role="data">941</cell><cell cols="1" rows="1" role="data">30.6757233004</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">759</cell><cell cols="1" rows="1" role="data">27.5499546279</cell><cell cols="1" rows="1" role="data">820</cell><cell cols="1" rows="1" role="data">28.6356421266</cell><cell cols="1" rows="1" role="data">881</cell><cell cols="1" rows="1" role="data">29.6816441593</cell><cell cols="1" rows="1" role="data">942</cell><cell cols="1" rows="1" role="data">30.6920185064</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">760</cell><cell cols="1" rows="1" role="data">27.5680975042</cell><cell cols="1" rows="1" role="data">821</cell><cell cols="1" rows="1" role="data">28.6530975638</cell><cell cols="1" rows="1" role="data">882</cell><cell cols="1" rows="1" role="data">29.6984848098</cell><cell cols="1" rows="1" role="data">943</cell><cell cols="1" rows="1" role="data">30.7083050656</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">761</cell><cell cols="1" rows="1" role="data">27.5862284483</cell><cell cols="1" rows="1" role="data">822</cell><cell cols="1" rows="1" role="data">28.6705423737</cell><cell cols="1" rows="1" role="data">883</cell><cell cols="1" rows="1" role="data">29.7153159162</cell><cell cols="1" rows="1" role="data">944</cell><cell cols="1" rows="1" role="data">30.7245829915</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">762</cell><cell cols="1" rows="1" role="data">27.6043474837</cell><cell cols="1" rows="1" role="data">823</cell><cell cols="1" rows="1" role="data">28.6879765756</cell><cell cols="1" rows="1" role="data">884</cell><cell cols="1" rows="1" role="data">29.7321374946</cell><cell cols="1" rows="1" role="data">945</cell><cell cols="1" rows="1" role="data">30.7408522979</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">763</cell><cell cols="1" rows="1" role="data">27.6224546339</cell><cell cols="1" rows="1" role="data">824</cell><cell cols="1" rows="1" role="data">28.7054001888</cell><cell cols="1" rows="1" role="data">885</cell><cell cols="1" rows="1" role="data">29.7489495613</cell><cell cols="1" rows="1" role="data">946</cell><cell cols="1" rows="1" role="data">30.7571129985</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">764</cell><cell cols="1" rows="1" role="data">27.6405499222</cell><cell cols="1" rows="1" role="data">825</cell><cell cols="1" rows="1" role="data">28.7228132327</cell><cell cols="1" rows="1" role="data">886</cell><cell cols="1" rows="1" role="data">29.7657521323</cell><cell cols="1" rows="1" role="data">947</cell><cell cols="1" rows="1" role="data">30.7733651069</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">765</cell><cell cols="1" rows="1" role="data">27.6586333719</cell><cell cols="1" rows="1" role="data">826</cell><cell cols="1" rows="1" role="data">28.7402157264</cell><cell cols="1" rows="1" role="data">887</cell><cell cols="1" rows="1" role="data">29.7825452237</cell><cell cols="1" rows="1" role="data">948</cell><cell cols="1" rows="1" role="data">30.7896086367</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">766</cell><cell cols="1" rows="1" role="data">27.6767050062</cell><cell cols="1" rows="1" role="data">827</cell><cell cols="1" rows="1" role="data">28.7576076891</cell><cell cols="1" rows="1" role="data">888</cell><cell cols="1" rows="1" role="data">29.7993288515</cell><cell cols="1" rows="1" role="data">949</cell><cell cols="1" rows="1" role="data">30.8058436015</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">767</cell><cell cols="1" rows="1" role="data">27.6947648483</cell><cell cols="1" rows="1" role="data">828</cell><cell cols="1" rows="1" role="data">28.7749891399</cell><cell cols="1" rows="1" role="data">889</cell><cell cols="1" rows="1" role="data">29.8161030318</cell><cell cols="1" rows="1" role="data">950</cell><cell cols="1" rows="1" role="data">30.8220700148</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">768</cell><cell cols="1" rows="1" role="data">27.7128129211</cell><cell cols="1" rows="1" role="data">829</cell><cell cols="1" rows="1" role="data">28.7923600978</cell><cell cols="1" rows="1" role="data">890</cell><cell cols="1" rows="1" role="data">29.8328677804</cell><cell cols="1" rows="1" role="data">951</cell><cell cols="1" rows="1" role="data">30.8382878902</cell></row><row rend="align=right" role="data"><cell cols="1" 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cols="1" rows="1" role="data">832</cell><cell cols="1" rows="1" role="data">28.8444102037</cell><cell cols="1" rows="1" role="data">893</cell><cell cols="1" rows="1" role="data">29.8831055950</cell><cell cols="1" rows="1" role="data">954</cell><cell cols="1" rows="1" role="data">30.8868904230</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">772</cell><cell cols="1" rows="1" role="data">27.7848879789</cell><cell cols="1" rows="1" role="data">833</cell><cell cols="1" rows="1" role="data">28.8617393793</cell><cell cols="1" rows="1" role="data">894</cell><cell cols="1" rows="1" role="data">29.8998327755</cell><cell cols="1" rows="1" role="data">955</cell><cell cols="1" rows="1" role="data">30.9030742807</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">773</cell><cell cols="1" rows="1" role="data">27.8028775489</cell><cell cols="1" rows="1" role="data">834</cell><cell cols="1" rows="1" 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role="data">28.5482048472</cell><cell cols="1" rows="1" role="data">876</cell><cell cols="1" rows="1" role="data">29.5972971739</cell><cell cols="1" rows="1" role="data">937</cell><cell cols="1" rows="1" role="data">30.6104557300</cell><cell cols="1" rows="1" role="data">998</cell><cell cols="1" rows="1" role="data">31.5911379979</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">816</cell><cell cols="1" rows="1" role="data">28.5657137142</cell><cell cols="1" rows="1" role="data">877</cell><cell cols="1" rows="1" role="data">29.6141857899</cell><cell cols="1" rows="1" role="data">938</cell><cell cols="1" rows="1" role="data">30.6267856622</cell><cell cols="1" rows="1" role="data">999</cell><cell cols="1" rows="1" role="data">31.6069612586</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">817</cell><cell cols="1" rows="1" role="data">28.5832118559</cell><cell cols="1" rows="1" role="data">878</cell><cell cols="1" rows="1" role="data">29.6310647801</cell><cell cols="1" rows="1" role="data">939</cell><cell cols="1" rows="1" role="data">30.6431068921</cell><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">31.6227766017</cell></row></table><pb n="398"/><cb/></p></div1><div1 part="N" n="ROTA" org="uniform" sample="complete" type="entry"><head>ROTA</head><p>, in Mechanics. See <hi rend="smallcaps">Wheel.</hi></p><p><hi rend="smallcaps">Rota</hi> <hi rend="italics">Aristotelica,</hi> or <hi rend="italics">Aristotle's Wheel,</hi> denotes a celebrated problem in mechanics, concerning the motion or rotation of a wheel about its axis; so called because first noticed by Aristotle.</p><p>The difficulty is this. While a circle makes a revolution on its centre, advancing at the same time in a right line along a plane, it describes, on that plane, a right line which is equal to its circumference. Now if this circle, which may be called the deferent, carry with it another smaller circle, concentric with it, like the nave of a coach wheel; then this little circle, or nave, will describe a line in the time of the revolution, which shall be equal to that of the large wheel or circumference itself; because its centre advances in a right line as fast as that of the wheel does, being in reality the same with it.</p><p>The solution given by Aristotle, is no more than a good explication of the difficulty.</p><p>Galileo, who next attempted it, has recourse to an infinite number of infinitely little vacuities in the right line described by the two circles; and imagines that the little circle never applies its circumference to those vacuities; but in reality only applies it to a line equal to its own circumference; though it appears to have applied it to a much larger. But all this is nothing to the purpose.</p><p>Tacquet will have it, that the little circle, making its rotation more slowly than the great one, does on that account describe a line longer than its own circumference; yet without applying any point of its circumference to more than one point of its base. But this is no more satisfactory than the former.</p><p>After the fruitless attempts of so many great men, M. Dortous de Meyran, a French gentleman, had the good fortune to hit upon a solution, which he sent to the Academy of Sciences; where being examined by Mess. de Louville and Soulmon, appointed for that purpose, they made their report that it was satisfactory. The solution is to this effect:</p><p>The wheel of a coach is only acted on, or drawn in a right line; its rotation or circular motion arises purely from the resistance of the ground upon which it is applied. Now this resistance is equal to the force which draws the wheel in the right line, inasmuch as it defeats that direction; of consequence the causes of the two motions, the one right and the other circular, are equal. And hence the wheel describes a right line on the ground equal to its circumference.</p><p>As for the nave of the wheel, the case is otherwise. It is drawn in a right line by the same force as the wheel; but it only turns round because the wheel does so, and can only turn in the same time with it. Hence it follows, that its circular velocity is less than that of the wheel, in the ratio of the two circumferences; and therefore its circular motion is less than the rectilinear one. Since then it necessarily describes a right line equal to that of the wheel, it can only do it partly by sliding, and partly by revolving, the sliding part being more or less as the nave itself is smaller or larger. See <hi rend="smallcaps">Cycloid.</hi></p></div1><div1 part="N" n="ROTATION" org="uniform" sample="complete" type="entry"><head>ROTATION</head><p>, <hi rend="italics">Rolling,</hi> in Mechanics. See R<hi rend="smallcaps">OLLING.</hi></p><div2 part="N" n="Rotation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Rotation</hi></head><p>, in Geometry, the circumvolution of a <cb/> surface round an immoveable line, called the <hi rend="italics">axis of Rotation.</hi> By such Rotation of planes, the figures of certain regular solids are formed or generated. Such as, a cylinder by the Rotation of a rectangle, a cone by the Rotation of a triangle, a sphere or globe by the Rotation of a semicircle, &c.</p><p>The method of cubing solids that are generated by such Rotation, is laid down by Mr. Demoivre, in his specimen of the use of the doctrine of fluxions, Philos. Trans. numb. 216; and indeed by most of the writers on Fluxions. In every such solid, all the sections perpendicular to the axis are circles, and therefore the fluxion of the solid, at any section, is equal to that circle multiplied by the fluxion of the axis. So that, if <hi rend="italics">x</hi> denote an absciss of that axis, and <hi rend="italics">y</hi> an ordinate to it in the revolving plane, which will also be the radius of that circle; then, <hi rend="italics">n</hi> being put for 3.1416, the area of the circle is <hi rend="italics">ny</hi><hi rend="sup">2</hi>, and consequently the fluxion of the solid is <hi rend="italics">ny</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">.</hi>; the fluent of which will be the content of the solid.</p><p>Such solid may also be expressed in terms of the generating plane and its centre of gravity; for the solid is always equal to the product arising from the generating plane multiplied by the path of its centre of gravity, or by the line described by that centre in the Rotation of the plane. And this theorem is general, by whatever kind of motion the plane is moved, in describing a solid.</p></div2><div2 part="N" n="Rotation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Rotation</hi></head><p>, <hi rend="italics">Revolution,</hi> in Astronomy. See R<hi rend="smallcaps">EVOLUTION.</hi></p><p><hi rend="italics">Diurnal</hi> <hi rend="smallcaps">Rotation.</hi> See <hi rend="smallcaps">Diurnal</hi>, and <hi rend="smallcaps">Earth.</hi></p></div2></div1><div1 part="N" n="ROTONDO" org="uniform" sample="complete" type="entry"><head>ROTONDO</head><p>, or <hi rend="smallcaps">Rotundo</hi>, in Architecture, popular term for any building that is round both within and withoutside, whether it be a church, hall, a saloon, a vestibule, or the like.</p></div1><div1 part="N" n="ROUND" org="uniform" sample="complete" type="entry"><head>ROUND</head><p>, <hi rend="smallcaps">Roundness, Rotundity</hi>, the property of a circle and sphere or globe &c.</p></div1><div1 part="N" n="ROWNING" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">ROWNING</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an ingenious English mathematician and philosopher, was fellow of Magdalen College, Cambridge, and afterwards Rector of Anderby in Lincolnshire, in the gift of that society. He was a constant attendant at the meetings of the Spalding Society, and was a man of a great philosophical habit and turn of mind, though of a cheerful and companionable disposition. He had a good genius for mechanical contrivances in particular. In 1738 he printed at Cambridge, in 8vo, <hi rend="italics">A Compendious System of Natural Philosophy,</hi> in 2 vols 8vo; a very ingenious work, which has gone through several editions. He had also two pieces inserted in the Philosophical Transactions, viz, 1. <hi rend="italics">A Description of a Barometer wherein the Scale of Variation may be increased at pleasure;</hi> vol. 38, pa. 39. And 2. <hi rend="italics">Direction for making a Machine for finding the Roots of Equations universally, with the Manner of using it;</hi> vol. 60, pa. 240.—Mr. Rowning died at his lodgings in Carey-street near Lincoln's-Inn Fields, the latter end of November 1771, at 72 years of age.</p><p>Though a very ingenious and pleasant man, he had but an unpromising and forbidding appearance: he was tall, stooping in the shoulders, and of a sallow downlooking countenance.</p><p>ROYAL <hi rend="italics">Oak, Robur Carolinum,</hi> in Astronomy, one of the new southern constellations, the stars of <pb n="399"/><cb/> which, according to Sharp's catalogue, annexed to the Britannic, are 12.</p><p><hi rend="smallcaps">Royal</hi> <hi rend="italics">Society of England,</hi> is an academy or body of persons, supposed to be eminent for their learning, instituted by king Charles the IId, for promoting natural knowledge.</p><p>This once illustrious body originated from an assembly of ingenious men, residing in London, who, being inquisitive into natural knowledge, and the new and experimental philosophy, agreed, about the year 1645, to meet weekly on a certain day, to discourse upon such subjects. These meetings, it is said, were suggested by Mr. Theodore Haak, a native of the Palatinate in Germany; and they were held sometimes at Dr. Goddard's lodgings in Wood-street, sometimes at a convenient place in Cheapside, and sometimes in or near Gresham College. This assembly seems to be that mentioned under the title of the <hi rend="italics">Invisible, or Philosophical College,</hi> by Mr. Boyle, in some letters written in 1646 and 1647. About the years 1648 and 1649, the company which formed these meetings, began to be divided, some of the gentlemen removing to Oxford, as Dr. Wallis, and Dr. Goddard, where, in conjunction with other gentlemen, they held meetings also, and brought the study of natural and experimental philosophy into fashion there; meeting first in Dr. Petty's lodgings, afterwards at Dr. Wilkins's apartments in Wadham College, and, upon his removal, in the lodgings of Mr. Robert Boyle; while those gentlemen who remained in London continued their meetings as before. The greater part of the Oxford Society coming to London about the year 1659, they met once or twice a week in Term-time at Gresham College, till they were dispersed by the public distractions of that year, and the place of their meeting was made a quarter for soldiers. Upon the restoration, in 1660, their meetings were revived, and attended by many gentlemen, eminent for their character and learning.</p><p>They were at length noticed by the government, and the king granted them a charter, first the 15th of July 1662, then a more ample one the 22d of April 1663, and thirdly the 8th of April 1669; by which they were erected into a corporation, <hi rend="italics">consisting of a president, council, and fellows, for promoting natural knowledge,</hi> and endued with various privileges and authorities.</p><p>Their manner of electing members is by ballotting; and two-thirds of the members present are necessary to carry the election in favour of the candidate. The council consists of 21 members, including the president, vice-president, treasurer, and two secretaries; ten of which go out annually, and ten new members are elected instead of them, all chosen on St. Andrew's day. They had formerly also two curators, whose business it was to perform experiments before the society.</p><p>Each member, at his admission, subscribes an engagement, that he will endeavour to promote the good of the society; from which he may be freed at any time, by signifying to the president that he desires to withdraw.</p><p>The charges are five guineas paid to the treasurer at admission; and one shilling per week, or 52s. per year, <cb/> as long as the person continues a member; or, in lieu of the annual subscription, a composition of 25 guineas in one payment.</p><p>The ordinary meetings of the society, are once a week, from November till the end of Trinity term the next summer. At first, the meeting was from 3 o'clock till 6 afternoon. Afterwards, their meeting was from 6 till 7 in the evening, to allow more time for dinner, which continued for a long series of years, till the hour of meeting was removed, by the present president, to between 8 and 9 at night, that gentlemen of fashion, as was alleged, might have the opportunity of coming to attend the meetings after dinner.</p><p>Their design is to “make faithful records of all the “works of nature or art, which come within their “reach; so that the present, as well as after ages, “may be enabled to put a mark on errors which have “been strengthened by long prescription; to restore “truths that have been long neglected; to push those “already known to more various uses; to make the “way more passable to what remains unrevealed, “&c.”</p><p>To this purpose they have made a great number of experiments and observations on most of the works of nature; as eclipses, comets, planets, meteors, mines, plants, earthquakes, inundations, springs, damps, fires, tides, currents, the magnet, &e: their motto being <hi rend="italics">Nullius in Verba.</hi> They have registered experiments, histories, relations, observations, &c, and reduced them into one common stock. They have, from time to time, published some of the most useful of these, under the title of Philosophical Transactions, &c. usually one volume each year, which were, till lately, very respectable, both for the extent or magnitude of them, and for the excellent quality of their contents. The rest, that are not printed, they lay up in their registers.</p><p>They have a good library of books, which has been formed, and continually augmenting, by numerous donations. They had also a museum of curiosities in nature, kept in one of the rooms of their own house in Crane Court Fleet-street, where they held their meetings, with the greatest reputation, for many years, keeping registers of the weather, and making other experiments; for all which purposes those apartments were well adapted. But, disposing of these apartments, in order to remove into those allotted them in Somerset Place, where having neither room nor convenience for such purposes, the museum was obliged to be disposed of, and their useful meteorological registers discontinued for many years.</p><p>Sir Godfrey Copley, bart. left 5 guineas to be given annually to the person who should write the best paper in the year, under the head of experimental philosophy, this reward, which is now changed to a gold medal, is the highest honour the society can bestow; and it is conferred on St. Andrew's day: but the communications of late years have been thought of so little importance, that the prize medal remains sometimes for years undisposed of.</p><p>Indeed this once very respectable society, now consisting of a great proportion of honorary members, who do not usually communicate papers; and many scientific members being discouraged from making their <pb n="400"/><cb/> usual communications, by what is deemed the present arbitrary government of the society; the annual volumes have in consequence become of much less importance, both in respect of their bulk and the quality of their contents.</p><p><hi rend="smallcaps">Royal</hi> <hi rend="italics">Society of Scotland.</hi> See <hi rend="smallcaps">Society.</hi></p><p>RUDOLPHINE <hi rend="italics">Tables,</hi> a set of astronomical tables that were published by the celebrated Kepler, and so called from the emperor Rudolph or Rudolphus.</p></div1><div1 part="N" n="RULE" org="uniform" sample="complete" type="entry"><head>RULE</head><p>, <hi rend="italics">The Carpenters,</hi> a folding ruler generally used by carpenters and other artificers; and is otherwise called the sliding Rule.</p><p>This instrument consists of two equal pieces of boxwood, each one foot in length, connected together by a folding joint. One side or face, of the Rule, is divided into inches, and half quarters, or eighths. On the same face also are several plane scales, divided into 12th parts by diagonal lines; which are used in planning dimensions that are taken in feet and inches. The edge of the Rule is commonly divided decimally, or into 10ths; viz, each foot into 10 equal parts, and each of these into 10 parts again, or 100dth parts of the foot: so that by means of this last scale, dimensions are taken in feet and tenths and hundreds, and multiplied together as common decimal numbers, which is the best way.</p><p>On the one part of the other face are four lines, marked A, B, C, D, the two middle ones B and C being on a slider, which runs in a groove made in the stock. The same numbers serve for both these two middle lines, the one line being above the numbers, and the other below them.</p><p>These four lines are logarithmic ones, and the three A, B, C, which are all equal to one another, are double lines, as they proceed twice over from 1 to 10. The lowest line D is a single one, proceeding from 4 to 40. It is also called the girt line, from its use in casting up the contents of trees and timber: and upon it are marked WG at 17.15, and AG at 18.95, the wine and ale gauge points, to make this instrument serve the purpose of a gauging rule.</p><p>Upon the other part of this face is a table of the value of a load, or 50 cubic feet, of timber, at all prices, from 6 pence to 28. a foot.</p><p>When 1 at the beginning of any line is accounted only 1, then the 1 in the middle is 10, and the 10 at the end 100; and when the 1 at the beginning is accounted 10, then 1 in the middle is 100, and the 10 at the end 1000; and so on. All the smaller divisions being also altered proportionally.</p><p>By means of this Rule all the usual operations of arithmetic may be easily and quickly performed, as multiplication, division, involution, evolution, finding mean proportionals, 3d and 4th proportionals, or the Rule-of-three, &c. For all which, see my Mensuration, part 5, sect. 3, 2d edition.</p><p><hi rend="smallcaps">Rules</hi> <hi rend="italics">of Philosophizing.</hi> See <hi rend="smallcaps">Philosophizing.</hi></p><div2 part="N" n="Rule" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Rule</hi></head><p>, in Arithmetic, denotes a certain mode of operation with figures to find sums or numbers unknown, and to facilitate computations.</p><p>Each Rule in arithmetic has its particular name, according to the use for which it is intended. The first four, which serve as a foundation of the whole art, are <cb/> called <hi rend="italics">addition, subtraction, multiplication,</hi> and <hi rend="italics">division.</hi></p><p>From these arise numerous other Rules, which are indeed only applications of these to particular purposes and occasions; as the Rule-of-three, or Golden Rule, or Rule of Proportion; also the Rules of Fellowship, Interest, Exchanges, Position, Progressions, &c, &c. For which, see each article severally.</p><p><hi rend="smallcaps">Rule</hi>-<hi rend="italics">of-Three,</hi> or <hi rend="italics">Rule of Proportion,</hi> commonly called the <hi rend="italics">Golden Rule</hi> from its great use, is a Rule that teaches how to find a 4th proportional number to three others that are given.</p><p>As, if 3 degrees of the equator contain 208 miles, how many are contained in 360 degrees, or the whole circumference of the earth?</p><p>The Rule is this: State, or set the three given terms down in the form of the first three terms of a proportion, stating them proportionally, thus: <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">deg.</cell><cell cols="1" rows="1" role="data">mil.</cell><cell cols="1" rows="1" role="data">deg.</cell><cell cols="1" rows="1" role="data">miles.</cell></row><row role="data"><cell cols="1" rows="1" role="data">as</cell><cell cols="1" rows="1" role="data"> 3 :</cell><cell cols="1" rows="1" role="data">208 ::</cell><cell cols="1" rows="1" role="data">360 :</cell><cell cols="1" rows="1" role="data">24960</cell></row></table> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">360</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">12480</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">624    </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3)74880</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24960</cell></row></table> Then multiply the 2d and 3d terms together, and divide the product by the 1st term, so shall the quotient be the 4th term in proportion, or the answer to the question, which in this example is 24960 or nearly 25 thousand miles, for the circumference of the earth.</p><p>This rule is often considered as of two kinds, viz. <hi rend="italics">Direct,</hi> and <hi rend="italics">Inverse.</hi></p><p><hi rend="italics">Rule-of-Three Direct,</hi> is that in which more requires more, or less requires less. As in this; if 3 men mow 21 yards of grass in a certain time, how much will 6 men mow in the same time? Here more requires more, that is, 6 men, which are more than 3 men, will also perform more work, in the same time. Or if it were thus; if 6 men mow 42 yards, how much will 3 men mow in the same time? here then less requires less, or 3 men will perform proportionally less work, in the same time. In both these cases then, the Rule, or the proportion, is direct; and the stating must be thus, as 3 : 21 :: 6 : 42, or thus, as 6 : 42 :: 3 : 21.</p><p><hi rend="italics">Rule-of-Three Inverse,</hi> is when more requires less, or less requires more. As in this; if 3 men mow a certain quantity of grass in 14 hours, in how many hours will 6 men mow the like quantity? Here it is evident that 6 men, being more than 3, will perform the same work in less time, or fewer hours; hence then more requires less, and the Rule or question is inverse, and must be stated by making the number of men change places, thus, as 6 : 14 :: 3 : 7 hours, the time in which 6 men will perform the work; still multiplying the 2d and 3d terms together, and dividing by the 1st.</p><p>For various abbreviations, and other particulars re- <pb n="401"/><cb/> lating to these Rules, see any of the common books of arithmetic.</p><p><hi rend="smallcaps">Rule</hi>-<hi rend="italics">of-Five,</hi> or <hi rend="italics">Compound Rule-of-Three,</hi> is where two Rules-of-three are required to be wrought, or to be combined together, to find out the number sought.</p><p>This Rule may be performed, either by working the two statings or proportions separately, making the result or 4th term of the 1st operation to be the 2d term of the last proportion; or else by reducing the two statings into one, by multiplying the two first terms together, and the two third terms together, and using the products as the 1st and 3d terms of the compound stating. As, if the question be this: If 100l. in 2 years yield 9l. interest, how much will 500l. yield in 6 years. Here, the two statings are, <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>: 9 ::<hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">500</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell></row></table></p><p>Then, to work the two statings separately, <table><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">as</cell><cell cols="1" rows="1" role="data">100 :</cell><cell cols="1" rows="1" role="data">9 ::</cell><cell cols="1" rows="1" role="data">500 :</cell><cell cols="1" rows="1" role="data">45l.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">and</cell><cell cols="1" rows="1" role="data">2 :</cell><cell cols="1" rows="1" role="data">45 ::</cell><cell cols="1" rows="1" role="data">6 :</cell><cell cols="1" rows="1" role="data">135l.</cell></row></table> so that 135l. is the interest or answer sought. But to work by one stating, it will be thus, <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">200</cell><cell cols="1" rows="1" rend="align=right" role="data">: 9 :: 3000</cell><cell cols="1" rows="1" role="data">: 135l. the answer.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2.00) 270.00</cell><cell cols="1" rows="1" role="data">(135l.</cell></row></table></p><p>See the books of arithmetic for more particulars.</p><p><hi rend="italics">Central</hi> <hi rend="smallcaps">Rule.</hi> See <hi rend="smallcaps">Central</hi> <hi rend="italics">Rule.</hi></p><p><hi rend="italics">Parallel</hi> <hi rend="smallcaps">Ruler.</hi> See <hi rend="smallcaps">Parallel</hi> <hi rend="italics">Ruler.</hi></p></div2></div1><div1 part="N" n="RUMB" org="uniform" sample="complete" type="entry"><head>RUMB</head><p>, or <hi rend="smallcaps">Rum.</hi> See <hi rend="smallcaps">Rhumb.</hi></p><p><hi rend="smallcaps">Rumb</hi>-<hi rend="italics">Line,</hi> or <hi rend="italics">Loxodromic.</hi> See <hi rend="smallcaps">Rhumb</hi>-<hi rend="italics">Line.</hi></p></div1><div1 part="N" n="RUSTIC" org="uniform" sample="complete" type="entry"><head>RUSTIC</head><p>, in Architecture, denotes a manner of building in imitation of simple or rude nature, rather than according to the rules of art.</p><p><hi rend="smallcaps">Rustic</hi> <hi rend="italics">Quoins.</hi> See <hi rend="smallcaps">Quoin.</hi></p><p><hi rend="smallcaps">Rustic</hi> <hi rend="italics">Work</hi> is where the stones in the face &c of a building, instead of being smooth, are hatched or picked with the point of an instrument. <cb/></p><p><hi rend="italics">Regular</hi> <hi rend="smallcaps">Rustics</hi>, are those in which the stones are chamfered off at the edges, and form angular or square recesses of about an inch deep at their jointings, or beds, and ends.</p><p><hi rend="smallcaps">Rustic</hi> <hi rend="italics">Order,</hi> is an order decorated with rustic quoins, or rustic work, &c.</p></div1><div1 part="N" n="RUTHERFORD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">RUTHERFORD</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>, D. D.)</persName></head><p>, an ingenious English philosopher, was the son of the Rev. Thomas Rutherford, rector of Papworth Everard in the county of Cambridge, who had made large collections for the history of that county.</p><p>Our author was born the 13th of October 1712. He studied at Cambridge, and became fellow of St. John's college, and regius professor of divinity, in that university; afterwards rector of Shenfield in Essex, and of Barley in Hertfordshire, and archdeacon of Essex. He died the 5th of October 1771, at 59 years of age.</p><p>Dr. Rutherford, besides a number of theological writings, published, at Cambridge,</p><p>1. <hi rend="italics">Ordo Institutionum Physicarum,</hi> 1743, in 4to.</p><p>2. <hi rend="italics">A System of Natural Philosophy,</hi> in 2 vols, 4to, 1748. A work which has been much esteemed.</p><p>3. He communicated also to the Gentleman's Society at Spalding, a curious correction of Plutarch's description of the instrument used to renew the Vestal fire, as relating to the triangle with which the instrument was formed. It was nothing else, it seems, but a concave speculum, whose principal focus, which collected the rays, is not in the centre of concavity, but at the distance of half a diameter from its surface. But some of the Ancients thought otherwise, as appears from prop. 31 of Euclid's Catoptrics.</p><p>The writer of his epitaph says, “He was eminent no less for his piety and integrity, than his extensive learning; and filled every public station in which he was placed with general approbation. In private life, his behaviour was truly amiable. He was esteemed, beloved, and honoured by his family and friends; and his death was sincerely lamented by all who had ever heard of his well deserved character.” <pb n="402"/></p></div1></div0><div0 part="N" n="S" org="uniform" sample="complete" type="alphabetic letter"><head>S</head><cb/><p>S, IN books of Navigation, &c, denotes south. So also S. E. is south-east; S. W. south-west; and S. S. E. south-south-east, &c. See <hi rend="smallcaps">Compass.</hi></p><div1 part="N" n="SAGITTA" org="uniform" sample="complete" type="entry"><head>SAGITTA</head><p>, in Astronomy, the <hi rend="italics">Arrow</hi> or <hi rend="italics">Dart,</hi> a conftellation of the northern hemisphere near the eagle, and one of the 48 old asterisms. The Greeks say that this constellation owes its origin to one of the arrows of Hercules, with which he killed the eagle or vulture that gnawed the liver of Prometheus.</p><p>The stars in this constellation, in the catalogues of Ptolomy, Tycho, and Hevelius, are only 5, but in Flamsteed's they are extended to 18.</p><div2 part="N" n="Sagitta" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sagitta</hi></head><p>, in Geometry, is a term used by some writers for the absciss of a curve.</p></div2><div2 part="N" n="Sagitta" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sagitta</hi></head><p>, in Trigonometry &c, is the same as the versed sine of an arch; being so called because it is like a dart or arrow, standing on the chord of the arch.</p></div2></div1><div1 part="N" n="SAGITTARIUS" org="uniform" sample="complete" type="entry"><head>SAGITTARIUS</head><p>, <hi rend="smallcaps">Sagittary</hi>, the <hi rend="italics">Archer,</hi> one of the signs of the zodiac, being the 9th in order, and marked with the character <figure/> of a dart or arrow. This constellation is drawn in the figure of a Centaur, or an animal half man and half horse, in the act of shooting an arrow from a bow. This figure the Greeks feign to be Crotus, the son of Eupheme, the nurse of the muses. Among more ancient nations the figure was probably. meant for a hunter, to denote the hunting season, when the sun enters this sign.</p><p>The stars in this constellation are, in Ptolomy's catalogue 31, in Tycho's 14, in Hevelius's 22, and in the Britannic catalogue 69.</p></div1><div1 part="N" n="SAILING" org="uniform" sample="complete" type="entry"><head>SAILING</head><p>, in a general sense, denotes the movement by which a vessel is wafted along the surface of the water, by the action of the wind upon her sails.</p><p>Sailing is also used for the art or act of navigating; or of determining all the cases of a ship's motion, by means of sea charts &c. These charts are constructed either on the supposition that the earth is a large extended flat surface, whence we obtain those that are called plane charts; or on the supposition that the earth is a sphere, whence are derived globular charts. Accordingly, Sailing may be distinguished into two general kinds, viz, <hi rend="italics">plane Sailing,</hi> and <hi rend="italics">globular Sailing.</hi> Sometimes indeed a third sort is added, viz, <hi rend="italics">spheroidical Sailing,</hi> which proceeds upon the supposition of the spheroidical figure of the earth.</p><p><hi rend="italics">Plane</hi> <hi rend="smallcaps">Sailing</hi> is that which is performed by means of a plane chart; in which case the meridians are considered as parallel lines, the parallels of latitude are at right angles to the meridians, the lengths of the degrees on the meridians, equator, and parallels of latitude, are every where equal. <cb/></p><p>In Plane Sailing, the principal terms and circumstances made use of, are, course, distance, departure, difference of latitude, rhumb, &c; for as to longitude, that has no place in plane Sailing, but belongs properly to globular or spherical sailing. For the explanation of all which terms, see the respective articles.</p><p>If a ship sails either due north or south, she sails on a meridian, her distance and difference of latitude are the same, and she makes no departure; but where the ship sails either due east or west, she runs on a parallel of latitude, making no difference of latitude, and her departure and distance are the same. It may farther be observed, that the departure and difference of latitude always make the legs of a right-angled triangle, whose hypotenuse is the distance the ship has sailed; and the angles are the course, its complement, and the right angle; therefore among these four things, course, distance, difference of latitude, and departure, any two of them being given, the rest may be found by plane trigonometry. <figure/></p><p>Thus, in the annexed figure, suppose the circle FHFH to represent the horizon of the place A, from whence a ship sails; AC the rhumb she sails upon, and C the place arrived at: then HH represents the parallel of latitude she sailed from, and CC the parallel of the latitude arrived in: so that AD becomes the difference of latitude. DC the departure, AC the distance sailed, [angle]DAC is the course, and [angle]DCA the comp. of the course. And all these particulars will be alike represented, whether the ship sails in the NE, or NW, or SE, or SW quarter of the horizon.</p><p>From the same figure, in which AE or AF or AH represents the rad. of the tables, EB the sine of the course, AB the cosine of the course, we may easily deduce all the proportions or canons, as they are usually called by mariners, that can arise in Plane Sailing; because the triangles ADC and ABE and AFG are evidently similar These proportions are exhibited in the following Table, which consists of 6 cases, according to the varieties of the two parts that can be given. <pb n="403"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Case.</cell><cell cols="1" rows="1" role="data">Given.</cell><cell cols="1" rows="1" role="data">Required.</cell><cell cols="1" rows="1" role="data">Solutions.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">[angle]A and AC, i. e. course and distance.</cell><cell cols="1" rows="1" role="data">AD and DC, i. e. difference of latitude and departure.</cell><cell cols="1" rows="1" role="data">AE : AB :: AC : AD, i. e. rad. : s. course :: dist. : dif. lat. AE : EB :: AC : DC, i. e. rad. : cos. course :: dist. : depart.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">[angle]A and AD, i. e. course and difference of latitude.</cell><cell cols="1" rows="1" role="data">AC and DC, i. e. distance and departure.</cell><cell cols="1" rows="1" role="data">AB: AE :: AD : AC, i. e. cos. cour. : rad. :: dif. lat. : dist. AB : BE :: AD : DC, i. e. cos.cour.:s.cour.::dif.lat.:dep.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">[angle]A and DC, i. e. course and departure.</cell><cell cols="1" rows="1" role="data">AC and AD, i. e. distance and difference of latitude.</cell><cell cols="1" rows="1" role="data">BE : AE :: DC : AC, i. e. s. cour. : rad. :: depart. : dist. BE : AB :: DC : AD, i. e. s.cour.:cos.cour.::dep.:dif.lat.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">AC and AD, i. e. distance and difference of latitude.</cell><cell cols="1" rows="1" role="data">[angle]A and DC, i. e. course and departure.</cell><cell cols="1" rows="1" role="data">AC : AD :: AE : AB, i. e. dist. : dif. lat. :: rad. : cos. course. AE : EB :: AC : DC, i. e. rad. : s. course :: dist. : depart.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">AC and DC, i. e. distance and departure.</cell><cell cols="1" rows="1" role="data">[angle]A and AD, i. e. course and difference of latitude.</cell><cell cols="1" rows="1" role="data">AC : DC :: AE : EB, i. e. dist. : dep. :: rad. : s. course. AE : AB :: AC : AD, i. e. rad. : cos. cour. :: dist. : dif. lat.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">AD and DC, i. e. difference of latitude and departure.</cell><cell cols="1" rows="1" role="data">[angle]A and AC, i. e. course and distance.</cell><cell cols="1" rows="1" role="data">AD : DC :: AF : FG, i. e. dif. lat. : dep. :: rad. : tang. course. BE : AE :: DC : AC, i. e. s. cour. : rad. :: dep. : dist.</cell></row></table><cb/></p><p>For the ready working of any single course, there is a table, called a <hi rend="italics">Traverse Table,</hi> usually annexed to treatises of navigation; which is so contrived, that by finding the given course in it, and a distance not exceeding 100 or 120 miles, the usual extent of the table; then the difference of latitude and the departure are had by inspection. And the same table will serve for greater distances, by doubling, or trebling, or quadrupling, &c, or taking proportional parts. See <hi rend="smallcaps">Traverse</hi> <hi rend="italics">Table.</hi></p><p>An ex. to the first case may suffice to shew the method. Thus, A ship from the latitude 47° 30′ N, has sailed SW by S 98 miles; required the departure made, and the latitude arrived in.</p><p>1. <hi rend="italics">By the Traverse Table.</hi> In the column of the course, viz 3 points, against the distance 98, stands the number 54.45 miles for the departure, and 81.5 miles for the diff. of lat.; which is 1° 21′ 1/2; and this being taken from the given lat. 47° 30′, leaves 46° 8′1/2 for the lat. come to.</p><p>2. <hi rend="italics">By Construction.</hi> Draw the me- <figure/> ridian AD; and drawing an arc, with the chord of 60, make PQ or angle A equal to 3 points; through Q draw the distance AQE = 98 miles, and through E the departure ED perp. to AD. Then, by measuring, the diff. of lat. AD measures about 81 1/2 miles, and the departure DE about 54 1/2 miles. <cb/></p><p>3. <hi rend="italics">By Computation.</hi> <table><row role="data"><cell cols="1" rows="1" role="data">First, as radius</cell><cell cols="1" rows="1" rend="align=right" role="data">10.00000</cell></row><row role="data"><cell cols="1" rows="1" role="data">to sin. course 33° 45′</cell><cell cols="1" rows="1" rend="align=right" role="data">9.74474</cell></row><row role="data"><cell cols="1" rows="1" role="data">so dist. 98</cell><cell cols="1" rows="1" rend="align=right" role="data">1.99123</cell></row><row role="data"><cell cols="1" rows="1" role="data">to depart. 54.45</cell><cell cols="1" rows="1" rend="align=right" role="data">1.73597</cell></row><row role="data"><cell cols="1" rows="1" role="data">Again, as radius</cell><cell cols="1" rows="1" rend="align=right" role="data">10.00000</cell></row><row role="data"><cell cols="1" rows="1" role="data">to cos. course</cell><cell cols="1" rows="1" rend="align=right" role="data">9.91985</cell></row><row role="data"><cell cols="1" rows="1" role="data">so dist. 98</cell><cell cols="1" rows="1" rend="align=right" role="data">1.99123</cell></row><row role="data"><cell cols="1" rows="1" role="data">to diff. of lat. 81.48</cell><cell cols="1" rows="1" rend="align=right" role="data">1.91108</cell></row></table></p><p>4. <hi rend="italics">By Gunter's Scale.</hi> The extent from radius, or 8 points, to 3 points, on the line of sine rhumbs, applied to the line of numbers, will reach from 98 to 54 1/2 the departure. And the extent from 8 points to 5 points, of the rhumbs, reaches from 98 to 81 1/2 on the line of numbers, for the difference of latitude.</p><p>And in like manner for other cases.</p><p><hi rend="italics">Traverse</hi> <hi rend="smallcaps">Sailing</hi>, or <hi rend="italics">Compound Courses,</hi> is the uniting of several cases of plane sailing together into one; as when a ship sails in a zigzag manner, certain distances upon several different courses, to find the whole difference of latitude and departure made good on all of them. This is done by working all the cases separately, by means of the traverse table, and constructing the figure as in this example. <pb n="404"/><cb/></p><p><hi rend="italics">Ex.</hi> A ship sailing from a place in latitude 24° 32′ N, has run five different courses and distances, as set down in the 1st and 2d columns of the following traverse table; required her present latitude, with the departure, and the direct course and distance, between the place sailed from, and the place come to. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=6" role="data"><hi rend="italics">Traverse Table.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Courses.</cell><cell cols="1" rows="1" role="data">Dist.</cell><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data">E</cell><cell cols="1" rows="1" role="data">W</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">SW b S</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">25.0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">37.4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">ESE</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">19.1</cell><cell cols="1" rows="1" role="data">46.2</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">SW</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">21.2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">21.2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">SE b E</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">33.3</cell><cell cols="1" rows="1" role="data">49.9</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">SW b S 1/4 W</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">50.6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">37.5</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">149.2</cell><cell cols="1" rows="1" rend="align=center" role="data">96.1</cell><cell cols="1" rows="1" rend="align=center" role="data">96.1</cell></row></table> Here, by finding, in the general traverse table, the difference of latitude and departure answering to each course and distance, they are set down on the same lines with each course, and in their proper columns of northing, southing, easting, or westing, according to the quarter of the compass the ship sails in, at each course. As here, there is no northing, the differences of latitude are all southward, also two departures are eastward, and three are westward. Then, adding up the numbers in each column, the sum of the eastings appears to be exactly equal to the sum of the westings, consequently the ship is arrived in the same meridian, without making any departure; and the southings, or difference of latitude being 149.2 miles or minutes, <table><row role="data"><cell cols="1" rows="1" role="data">that is</cell><cell cols="1" rows="1" role="data">  2°</cell><cell cols="1" rows="1" role="data">29′,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">which taken from</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">32,</cell><cell cols="1" rows="1" role="data">the latitude dep. from,</cell></row><row role="data"><cell cols="1" rows="1" role="data">leaves</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">3 N,</cell><cell cols="1" rows="1" role="data">the latitude come to.</cell></row></table></p><p><hi rend="italics">To Construct this Traverse.</hi> <figure/> With the chord of 60 degrees describe the circle N 135 S &c, and quarter it by the two perpendicular diameters; then from S set upon it the several courses, to the points marked 1, 2, 3, 4, 5, through which points draw lines from the centre A, or conceive them to be drawn; lastly, upon the first line lay off the first distance 45 from A to B, also draw BC = 50 and parallel to A 2, and CD = 30 parallel to A 3, and DE = 60 parallel to A 4, and EF = 63, parallel to A 5; then it is found that the point F falls exactly upon the meridian NAF produced, thereby shewing that there is no departure; and by measuring AF, it gives 149 miles for the difference of latitude.</p><p><hi rend="italics">Oblique</hi> <hi rend="smallcaps">Sailing</hi>, is the resolution of certain cases and problems in Sailing by oblique triangles, or in which oblique triangles are concerned. <cb/></p><p>In this kind of Sailing, it may be observed, that <hi rend="italics">to set an object,</hi> means to observe what rhumb or point of the nautical compass is directed to it. And the <hi rend="italics">bearing</hi> of an object is the rhumb on which it is seen; also the bearing of one place from another, is reckoned by the name of the rhumb passing through those two places.</p><p>In every figure relating to any case of plane Sailing, the bearing of a line, not running from the centre of the circle or horizon, is found by drawing a line parallel to it, from the centre, and towards the same quarter.</p><p><hi rend="italics">Ex.</hi> A ship sailing at sea, observed a point of land to bear E by S; and then after sailing NE 12 miles, its bearing was found to be SE by E. Required the place of that point, and its distance from the ship at the last observation. <figure/></p><p><hi rend="italics">Construction.</hi> Draw the meridian line NAS, and, assuming A for the first place of the ship, draw AC the E by S rhumb, and AB the NE one, upon which lay off 12 miles from A to B; then draw the meridian BT parallel to NS, from which set off the SE by E point BC, and the point C will be the place of the land required; then the distance BC measures 26 miles.</p><p><hi rend="italics">By Computation.</hi> Here are given the side AB, and the two angles A and B, viz, the [angle]A = 5 points or 56° 15′, and the [angle]B = 9 points or 101° 15′; consequently the [angle]C = 2 points or 22° 30′. Then, by plane trigonometry, <table><row role="data"><cell cols="1" rows="1" role="data">As sin. [angle]C 22° 30′</cell><cell cols="1" rows="1" rend="align=right" role="data">9.58284</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. [angle]B 56  15</cell><cell cols="1" rows="1" rend="align=right" role="data">9.91985</cell></row><row role="data"><cell cols="1" rows="1" role="data">So is AB 12 miles</cell><cell cols="1" rows="1" rend="align=right" role="data">1.07918</cell></row><row role="data"><cell cols="1" rows="1" role="data">To BC 26.073 miles</cell><cell cols="1" rows="1" rend="align=right" role="data">1.41619</cell></row></table></p><p><hi rend="smallcaps">Sailing</hi> <hi rend="italics">to Windward,</hi> is working the ship towards that quarter of the compass from whence the wind blows.</p><p>For rightly understanding this part of navigation, it will be necessary to explain the terms that occur in it, though most of them may be seen in their proper places in this work.</p><p>When the wind is directly, or partly, against a ship's direct course for the place she is bound to, she reaches her port by a kind of zigzag or z like course; which is made by sailing with the wind first on one side of the ship, and then on the other side.</p><p>In a ship, when you look towards the head,<lb/> <hi rend="italics">Starboard</hi> denotes the right hand side.<lb/> <pb n="405"/><cb/> <hi rend="italics">Larboard</hi> the left hand side.<lb/> <hi rend="italics">Forwards,</hi> or <hi rend="italics">afore,</hi> is towards the head.<lb/> <hi rend="italics">Aft,</hi> or <hi rend="italics">abaft,</hi> is towards the stern.<lb/></p><p>The <hi rend="italics">beam</hi> signifies athwart or across the middle of the ship.</p><p>When a ship sails the same way that the wind blows, she is said to sail or run before the wind; and the wind is said to be <hi rend="italics">right aft,</hi> or <hi rend="italics">right astern;</hi> and her course is then 16 points, or the farthest possible, from the wind, that is from the point the wind blows from.—When the ship sails with the wind blowing directly across her, she is said to have the <hi rend="italics">wind on the beam;</hi> and her course is 8 points from the wind.—When the wind blows obliquely across the ship, the wind is said to be <hi rend="italics">abaft the beam</hi> when it pursues her, or blows more on the hinder part, but <hi rend="italics">before the beam</hi> when it meets or opposes her course, her course being more than 8 points from the wind in the former case, but less than 8 points in the latter case.—When a ship endeavours to sail towards that point of the compass from which the wind blows, she is said to <hi rend="italics">sail on a wind,</hi> or to <hi rend="italics">ply to windward.</hi>—And a vessel sailing as near as she can to the point from which the wind blows, she is said to be <hi rend="italics">close hauled.</hi> Most ships will lie within about 6 points of the wind; but sloops, and some other vessels, will lie much nearer. To know how near the wind a ship will lie; observe the course she goes on each tack, when she is close hauled; then half the number of points between the two courses, will shew how near the wind the ship will lie.</p><p>The <hi rend="italics">windward,</hi> or <hi rend="italics">weather side,</hi> is that side of the ship on which the wind blows; and the other side is called the <hi rend="italics">leeward,</hi> or <hi rend="italics">lee side.—Tacks</hi> and <hi rend="italics">sheets</hi> are large ropes fastened to the lower corners of the fore and main sails; by which either of these corners is hauled fore or aft.—When a ship sails on a wind, the windward tacks are always hauled forwards, and the leeward sheets aft.—The <hi rend="italics">starboard tacks are aboard,</hi> when the starboard side is to windward, and the larboard side to leeward. And the <hi rend="italics">larboard tacks are aboard,</hi> when the larboard side is to windward, and the starboard to leeward.</p><p>The most common cases in turning to windward may be constructed by the following precepts. Having drawn a circle with the chord of 60°, for the compass, or the horizon of the place, quarter it by drawing the meridian and parallel of latitude perpendicular to each other, and both through the centre; mark the place of the wind in the circumference; draw the rhumb passing through the place bound to, and lay on it, from the centre, the distance of that place. On each side of the wind lay off, in the circumference, the points or degrees shewing how near the wind the ship can lie; and draw these rhumbs.—Now the first course will be on one of these rhumbs, according to the tack the ship leads with. Draw a line through the place bound to, parallel to the other rhumb, and meeting the first; and this will shew the course and distance on the other tack.</p><p><hi rend="italics">Ex.</hi> The wind being at north, and a ship bound to a port 25 miles directly to windward; beginning with the starboard tacks, what must be the course and distance on each of two tacks to reach the port?</p><p><hi rend="italics">Construction.</hi> Having drawn the circle &c, as above described, where A is the port, AP and AQ the two rhumbs, each within 6 points of AN; in NA <cb/> produced take AB = 25 <figure/> miles, then B is the place of the ship; draw BC parallel to AP, and meeting QA produced in C; so shall BC and CA be the distances on the two tacks; the former being WNW, and the latter ENE.</p><p><hi rend="italics">Computation.</hi> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Here</cell><cell cols="1" rows="1" role="data">[angle]B = NAP = 6 points,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and</cell><cell cols="1" rows="1" role="data">[angle]A = NAQ = 6 points,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">theref.</cell><cell cols="1" rows="1" role="data">[angle]C = 4 points.</cell></row></table> So that all the angles are given, and the side AB, to find the other two sides AC and BC, which are equal to each other, because their opposite angles A and B are equal. Hence as sin. C : AB :: sin. A : BC, i. e. s. 45° : 25 :: s. 67° 30′ : 32 2/3 = BC or AC, the distance to be run on each tack.</p><p><hi rend="smallcaps">Sailing</hi> <hi rend="italics">in Currents,</hi> is the method of determining the true course and distance of a ship when her own motion is affected and combined with that of a current.</p><p>A <hi rend="italics">current</hi> or <hi rend="italics">tide</hi> is a progressive motion of the water, causing all floating bodies to move that way towards which the stream is directed.—The <hi rend="italics">setting</hi> of a tide, or current, is that point of the compass towards which the waters run; and the <hi rend="italics">drift</hi> of the current is the rate at which it runs per hour.</p><p>The drift and setting of the most remarkable tides and currents, are pretty well known; but for unknown currents, the usual way to find the drift and setting, is thus: Let three or four men take a boat a little way from the ship; and by a rope, fastened to the boat's stem, let down a heavy iron pot, or loaded kettle, into the sea, to the depth of 80 or 100 fathoms, when it can be done: by which means the boat will ride almost as steady as at anchor. Then heave the log, and the number of knots run out in half a minute will give the current's rate, or the miles which it runs per hour; and the bearing of the log shews the setting of the current.</p><p>A body moving in a current, may be considered in three cases: viz,</p><p>1. Moving with the current, or the same way it sets.</p><p>2. Moving against it, or the contrary way it sets.</p><p>3. Moving obliquely to the current's motion.</p><p>In the 1st case, or when a ship sails with a current, its velocity will be equal to the sum of its proper motion, and the current's drift. But in the 2d case, or when a ship sails against a current, its velocity will be equal to the difference of her own motion and the drift of the current: so that if the current drives stronger than the wind, the ship will drive astern, or lose way. In the 3d case, when the current sets oblique to the course of the ship, her real course, or that made good, will be somewhere between that in which the ship endeavours to go, and the track in which the current tries to drive her; and indeed it will always be along the diagonal of a parallelogram, of which one side represents the ship's course set, and the other adjoining side is the current's drift. <pb n="406"/><cb/></p><p>Thus, if ABDC be a parallelo- <figure/> gram. Now if the wind alone would drive the ship from A to B in the same time as the current alone would drive her from A to C: then, as the wind neither helps nor hinders the ship from coming towards the line CD, the current will bring her there in the same time as if the wind did not act. And as the current neither helps nor hinders the ship from coming towards the line BD, the wind will bring her there in the same time as if the current did not act. Therefore the ship must, at the end of that time, be sound in both those lines, that is, in their meeting D. Consequently the ship must have passed from A to D in the diagonal AD.</p><p>Hence, drawing the rhumbs for the proper course of the ship and of the current, and setting the distances off upon them, according to the quantity run by each in the given time; then forming a parallelogram of these two, and drawing its diagonal, this will be the real course and distance made good by the ship.</p><p><hi rend="italics">Ex.</hi> 1. A ship sails E. 5 miles an hour, in a tide setting the same way 4 miles an hour: required the ship's course, and the distance made good. <table><row role="data"><cell cols="1" rows="1" role="data">The ship's motion is</cell><cell cols="1" rows="1" role="data">5m. E.</cell></row><row role="data"><cell cols="1" rows="1" role="data">The current's motion is</cell><cell cols="1" rows="1" role="data">4m. E.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Theref. the ship's run is</cell><cell cols="1" rows="1" role="data">9m. E.</cell></row></table></p><p><hi rend="italics">Ex.</hi> 2. A ship sails SSW. with a brisk gale, at the rate of 9 miles an hour, in a current setting NNE. 2 miles an hour: required the ship's course, and the distance made good. <table><row role="data"><cell cols="1" rows="1" role="data">The ship's motion is</cell><cell cols="1" rows="1" role="data">SSW. 9m.</cell></row><row role="data"><cell cols="1" rows="1" role="data">The current's motion is</cell><cell cols="1" rows="1" role="data">NNE. 2m.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Theref. ship's true run is</cell><cell cols="1" rows="1" role="data">SSW. 7m.</cell></row></table></p><p><hi rend="italics">Ex.</hi> 3. A ship running south at the rate of 5 miles an hour, in 10 hours crosses a current, which all that time was setting east at the rate of 3 miles an hour: required the ship's true course and distance sailed.</p><p>Here the ship is first supposed to <figure/> be at A, her imaginary course is along the line AB, which is drawn south, and equal to 50 miles, the run in 10 hours; then draw BC east, and equal to 30 miles, the run of the ourrent in 10 hours. Then the ship is found at C, and her true path is in the line AC = 58.31 her distance, and her course is the angle at A = 30° 58′ from the south towards the east.</p><p><hi rend="italics">Globular</hi> <hi rend="smallcaps">Sailing</hi> is the estimating the ship's motion and run upon principles derived from the globular figure of the earth, viz, her course, distance, and difference of latitude and longitude.</p><p>The principles of this method are explained under the artioles <hi rend="smallcaps">Rhumb</hi>-<hi rend="italics">line, Mercator's</hi> <hi rend="smallcaps">Chart</hi>, and M<hi rend="smallcaps">ERIDIONAL</hi> <hi rend="italics">Parts;</hi> which see.</p><p>Globular Sailing, in the extensive sense here applied <cb/> to the term, comprehends <hi rend="italics">Parallel Sailing, Middle-latitude Sailing,</hi> and <hi rend="italics">Mercator's Sailing;</hi> to which may be added <hi rend="italics">Circular Sailing,</hi> or <hi rend="italics">Great-circle Sailing.</hi> Of each of which it may be proper here to give a brief account.</p><p><hi rend="italics">Parallel</hi> <hi rend="smallcaps">Sailing</hi> is the art of finding what distance a ship should run due east or west, in sailing from the meridian of one place to that of another place, in any parallel of latitude.</p><p>The computations in parallel sailing depend on the following rule: As radius,<lb/> To cosine of the lat. of any parallel;<lb/> So are the miles of long. between any two meridians,<lb/> To the dist. of these meridians in that parallel.<lb/> Also, for any two latitudes,<lb/> As the cosine of one latitude,<lb/> Is to the cosine of another latitude;<lb/> So is a given meridional dist. in the 1st parallel,<lb/> To the like meridional dist. in the 2d parallel.<lb/></p><p>Hence, counting 60 nautical miles to each degree of longitude, or on the equator; then, by the first rule the number of miles in each degree on the other parallels, will come out as in the following table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=6" role="data"><hi rend="italics">Table of Meridional Distances.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Miles.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Miles.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Miles.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">59.99</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">51.43</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">29.09</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">59.96</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">50.88</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">28.17</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">59.92</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">50.32</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">27.24</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">59.85</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">49.74</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">26.30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">59.77</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">49.15</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">25.36</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">59.67</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">48.54</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">24.41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">59.56</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">47.92</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">23.44</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">59.42</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">47.28</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">22.48</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">59.26</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">46.63</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">21.50</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">59.09</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">45.96</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">20.52</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">58.89</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">45.28</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">19.53</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">58.69</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">44.59</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">18.54</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">58.46</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">43.<*>8</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">17.54</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">58.22</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">43.16</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">16.54</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">57.95</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">42.43</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">15.53</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">57.67</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">41.68</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">14.51</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">57.38</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">40.92</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">13.50</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">57.06</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">40.15</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">12.48</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">56.73</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">39.36</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">11.45</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">56.38</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">38.57</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">10.42</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">56.01</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">37.76</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">9.38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">55.63</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">36.94</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">8.35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">55.23</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">36.11</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">7.32</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">54.81</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">35.27</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data">6.28</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">54.38</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">34.41</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">5.23</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">53.93</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">33.55</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">4.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">53.46</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">32.68</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">3.14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">52.97</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">31.79</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">2.09</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">52.47</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">30.90</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">1.05</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">51.96</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">30.00</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">0.00</cell></row></table> <pb n="407"/><cb/></p><p>See another table of this kind, allowing 69 1/<*> English miles to one degree, under the article D<hi rend="smallcaps">EGREE.</hi></p><p>To sind the meridional distance to any number of minutes between any of the whole degrees in the table, as for instance in the parallel of 48° 26′; take out the tabular distances for the two whole degrees between which the parallel or the odd minutes lie, as for 48° and 49°; subtract the one from the other, and take the proportional part of the remainder for the odd minutes, by multiplying it by those minutes, and dividing by 60; and lastly, subtract this proportional part from the greater tabular number. Thus, <table><row role="data"><cell cols="1" rows="1" role="data">Lat. 48°</cell><cell cols="1" rows="1" rend="align=right" role="data">40.15</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Lat. 49.</cell><cell cols="1" rows="1" rend="align=right" role="data">39.36</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As 60′ : 26′ ::</cell><cell cols="1" rows="1" rend="align=right" role="data">0.79</cell><cell cols="1" rows="1" role="data">rem. : 0.34</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">26</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">474</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">158  </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">60) 20.54</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0.34</cell><cell cols="1" rows="1" role="data">pro. part</cell></row><row role="data"><cell cols="1" rows="1" role="data">Taken from</cell><cell cols="1" rows="1" rend="align=right" role="data">40.15</cell><cell cols="1" rows="1" role="data">for lat. 48°</cell></row><row role="data"><cell cols="1" rows="1" role="data">Leaves merid. dist.</cell><cell cols="1" rows="1" rend="align=right" role="data">39.81</cell><cell cols="1" rows="1" role="data">for lat. 48° 26′</cell></row></table></p><p>And, in like manner, by the counter operation, to find what latitude answers to a given meridional distance. As, for ex. in what latitude 46.08 miles answer to a degree of longitude. <table><row role="data"><cell cols="1" rows="1" role="data">From</cell><cell cols="1" rows="1" rend="align=right" role="data">46.63</cell><cell cols="1" rows="1" role="data">for 39°</cell><cell cols="1" rows="1" rend="align=right" role="data">from 46.63</cell><cell cols="1" rows="1" role="data">for 39°</cell></row><row role="data"><cell cols="1" rows="1" role="data">Take</cell><cell cols="1" rows="1" rend="align=right" role="data">45.96</cell><cell cols="1" rows="1" role="data">for 40°</cell><cell cols="1" rows="1" rend="align=right" role="data">take 46.08</cell><cell cols="1" rows="1" role="data">given number.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Then as</cell><cell cols="1" rows="1" rend="align=right" role="data">0.67</cell><cell cols="1" rows="1" role="data"> : 60′ ::</cell><cell cols="1" rows="1" rend="align=right" role="data">0.55</cell><cell cols="1" rows="1" role="data">: 49′</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">67) 3300</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">49′</cell><cell cols="1" rows="1" role="data">pro. part.</cell></row></table> Therefore the latitude sought is 39° 49′.</p><p><hi rend="italics">Ex.</hi> 3. Given the latitude and meridional distance; to find the corresponding difference of longitude. As, if a ship, in latitude 53° 36′, and longitude 10° 18 east, sail due west 236 miles; required her present longitude.</p><p>Here, by the first rule, <table><row role="data"><cell cols="1" rows="1" role="data">As cos. lat.</cell><cell cols="1" rows="1" rend="align=right" role="data">53° 36′</cell><cell cols="1" rows="1" role="data">comp.</cell><cell cols="1" rows="1" rend="align=right" role="data">0.22664</cell></row><row role="data"><cell cols="1" rows="1" role="data">To radius</cell><cell cols="1" rows="1" rend="align=right" role="data">90 00 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10.00000</cell></row><row role="data"><cell cols="1" rows="1" role="data">So merid. dist.</cell><cell cols="1" rows="1" rend="align=right" role="data">236</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" rend="align=right" role="data">2.37291</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">To diff. long. 397.7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2.59955</cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data">Its 60th gives</cell><cell cols="1" rows="1" rend="align=right" role="data">6° 38′</cell><cell cols="1" rows="1" role="data">W. diff. long.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Taken from</cell><cell cols="1" rows="1" rend="align=right" role="data">10 18 </cell><cell cols="1" rows="1" role="data">E. long. from</cell></row><row role="data"><cell cols="1" rows="1" role="data">Leaves</cell><cell cols="1" rows="1" rend="align=right" role="data">3 40 </cell><cell cols="1" rows="1" role="data">E. long. come to.</cell></row></table></p><p><hi rend="italics">By the table;</hi> the length of a degree on the parallel of 53° 36′ is 35.6. <cb/> Then as 35.6 : 60 :: 236 : 397.7, the diff. of long. the same as before.</p><p><hi rend="italics">Middle-latitude</hi> <hi rend="smallcaps">Sailing</hi>, is a method of resolving the cases of globular Sailing by means of the middle latitude between the latitude departed from, and that come to. This method is not quite accurate, being only an approximation to the truth, and it makes use of the principles of plane Sailing and parallel Sailing conjointly.</p><p>The method is founded on the supposition that the departure is reckoned as a meridional distance in that latitude which is a middle parallel between the latitude sailed from, and the latitude come to. And the method is not quite accurate, because the arithmetical mean, or half sum of the cosines of two distant latitudes, is not exactly the cosine of the middle latitude, or half the sum of those latitudes; nor is the departure between two places, on an oblique rhumb, equal to the meridional distance in the middle latitude; as is presumed in this method. Yet when the parallels are near the equator, or near to each other, in any latitude, the error is not considerable.</p><p>This method seems to have been invented on account of the easy manner in which the several cases may be resolved by the traverse table, and when a table of meridional parts is wanting. The computations depend on the following rules:</p><p>1. Take half the sum, or the arithmetical mean, of the two given latitudes, for the middle latitude. Then,</p><p>2. As cosine of middle latitude, Is to the radius; So is the departure, To the diff. of longitude. And,</p><p>3. As cosine of middle latitude, Is to the tangent of the course; So is the difference of latitude, To the difference of longitude.</p><p><hi rend="italics">Mercator's</hi> <hi rend="smallcaps">Sailing</hi>, is the art of resolving the several cases of globular Sailing, by plane trigonometry, with the assistance of a table of meridional parts, or of logarithmic tangents. And the computations are performed by the following rules:</p><p>1. As meridional diff. lat. To diff. of longitude; So is the radius, To tangent of the course.</p><p>2. As the proper diff. lat. To the departure; So is merid. diff. lat. To diff. of longitude.</p><p>3. As diff. log. tang. half colatitudes, To tang. of 51° 38′ 09″; So is a given diff. longitude, To tangent of the course.</p><p>The manner of working with the meridional parts and logarithmic tangents, will appear from the two following cases.</p><p>1. Given the latitudes of two places; to sind their meridional difference of latitude.</p><p><hi rend="italics">By the Merid. Parts.</hi> When the places are both on <pb n="408"/><cb/> the same side of the equator, take the difference of the meridional parts answering to each latitude; but when the places are on opposite sides of the equator, take the sum of the same parts, for the meridional difference of latitude sought.</p><p><hi rend="italics">By the Log. Tangents.</hi> In the former case, take the difference of the long. tangents of the half colatitudes; but in the latter case, take the sum of the same; then the said difference or sum divided by 12 63, will give the meridional difference of latitude sought.</p><p>2. Given the latitude of one place, and the meridional difference of latitude between that and another place; to find the latitude of this latter place.</p><p><hi rend="italics">By the Merid. Parts.</hi> When the places have like names, take the sum of the merid. parts of the given lat. and the given diff.; but take the difference between the same when they have unlike names; then the result, being found in the table of meridional parts, will give the latitude sought.</p><p><hi rend="italics">By the Log. Tangents.</hi> Multiply the given meridional diff. of lat. by 12.63; then in the former case subtract the product from the log. tangent of the given half colatitude, but in the latter case add them; then seek the degrees and minutes answering to the result among the log. tangents, and these degrees, &c. doubled will be the colatitude sought.</p><p><hi rend="italics">Circular</hi> <hi rend="smallcaps">Sailing</hi>, or <hi rend="italics">Great-circle</hi> <hi rend="smallcaps">Sailing</hi>, is the art of finding what places a ship must go through, and what courses to steer, that her track may be in the arc of a great circle on the globe, or nearly so, passing through the place sailed from and the place bound to.</p><p>This method of Sailing has been proposed, because the shortest distance between two places on the sphere, is an arc of a great circle intercepted between them, and not the spiral rhumb passing through them, unless when that rhumb coincides with a great circle, which can only be on a meridian, or on the equator.</p><p>As the solutions of the cases in Mercator's Sailing are performed by plane triangles, in this method of Sailing they are resolved by means of spherical triangles. A great variety of cases might be here proposed, but those that are the most useful, and most commonly occur, pertain to the following problem.</p><p><hi rend="italics">Problem</hi> I. Given the latitudes and longitudes of two places on the earth; to find their nearest distance on the surface, together with the angles of position from either place to the other.</p><p>This problem comprehends 6 cases.</p><p><hi rend="italics">Case</hi> 1. When the two places lie under the same meridian; then their difference of latitude will give their distance, and the position of one from the other will be directly north and south.</p><p><hi rend="italics">Case</hi> 2. When the two places lie under the equator; their distance is equal to their difference of longitude, and the angle of position is a right angle, or the course from one to the other is due east or west.</p><p><hi rend="italics">Case</hi> 3. When both places are in the same parallel of latitude. Ex. gr. The places both in 37° north, but the longitude of the one 25° west, and of the other 76° 23′ west.</p><p>Let P denote the north pole, and A and B the two places on the same parallel BDA, also BIA their distance asunder, or the arc of a great circle <cb/> passing through them. Then <figure/> is the angle A or B that of position, and the angle BPA = 51° 23′ the difference of longitude, and the side PA or PB = 53° the colatitude.</p><p>Draw PI perp. to AB, or bisecting the angle at P. Then in the triangle API, right-angled at I, are given the hypotenuse AP = 53°, and the angle API = 25° 41′ 30″; to find the angle of position A or B = 73° 51′; and the half distance AI = 20° 15′1/2; this doubled gives 40° 31′ for the whole distance AB, or 2431 nautical miles, which is 31 miles less than the distance along ADB, or by parallel Sailing.</p><p><hi rend="italics">Case</hi> 4. When one place has latitude, and the other has none, or is under the equator. For example, suppose the Island of St. Thomas, lat. 0°, and long. 1° 0′ east, and Port St. Julian, in lat. 48° 51′ south, and long. 65° 10′ west. <table><row role="data"><cell cols="1" rows="1" role="data">Port St. Julian, lat.</cell><cell cols="1" rows="1" role="data">48° 51′ S.</cell><cell cols="1" rows="1" role="data">long.</cell><cell cols="1" rows="1" role="data">65° 10′W.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Isle St. Thomas</cell><cell cols="1" rows="1" role="data"> 0  00</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data"> 1  00 E</cell></row><row role="data"><cell cols="1" rows="1" role="data">Julian's colat.</cell><cell cols="1" rows="1" role="data">41  09 Diff.</cell><cell cols="1" rows="1" role="data">long.</cell><cell cols="1" rows="1" role="data"> 66 10</cell></row></table></p><p>Hence, if S denote the south pole, A the Isle St. Thomas at <figure/> the equator, and B St. Julian; then in the triangle are given SA a quadrant or 90°, BS = 41° 9′ the colat. of St. Julian, and the [angle]S = 66° 10′ the dif. of longitude; to find AB = 74° 35′ = 4475 miles, which is less by 57 miles than the distance found by Mercator's Sailing; also the angle of position at A = 51°22′, and the angle of position B = 108° 24′.</p><p><hi rend="italics">Case</hi> 5. When the two given places are both on the same side of the equator; for example the Lizard, and the island of Bermudas. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">The Lizard, lat.</cell><cell cols="1" rows="1" role="data">49° 57′N.</cell><cell cols="1" rows="1" role="data">long. 5°</cell><cell cols="1" rows="1" rend="align=left" role="data">21′W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Bermudas,</cell><cell cols="1" rows="1" role="data">32   35   N.</cell><cell cols="1" rows="1" role="data">63 </cell><cell cols="1" rows="1" rend="align=left" role="data">32   W.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">58 </cell><cell cols="1" rows="1" role="data">11</cell></row></table></p><p>Here, if P be the north pole, L the Lizard, and B Bermudas; there are given, <figure/> PL = 40° 03′ colat. of the Lizard, PB = 57 25 colat. of Bermudas, [angle]P = 58 11 diff. of longitude; to sind BL = 45° 44 = 2744 miles the distance, and [angle] of position B = 49° 27′, also [angle] of position L = 90° 31′.</p><p><hi rend="italics">Case</hi> 6. When the given places lie on different sides of the equator; as suppose St. Helena and Bermudas. Here <pb n="409"/><cb/></p><p>PB = 57° 25′ polar dist. Bermudas, <figure/></p><p>PH = 105 55 polar dist. St. Helena,</p><p>[angle]P = 57 43 diff. long.</p><p>To find BH = 73° 26′ = 4406 miles, the distance, also the angle of position H = 48° 0′, and the angle of position B = 121° 59′.</p><p>From the solutions of the foregoing cases it appears, that to sail on the arc of a great circle, the ship must continually alter her course; but as this is a difficulty too great to be admitted into the practice of navigation, it has been thought sufficiently exact to effect this business by a kind of approximation, that is, by a method which nearly approaches to the sailing on a great circle: namely, upon this principle, that in small arcs, the difference between the arc and its chord or tangent is so small, that they may be taken for one another in any nautical operations: and accordingly it is supposed that the great circles on the earth are made up of short right lines, each of which is a segment of a rhumb line. On this supposition the solution of the following problem is deduced.</p><p><hi rend="italics">Problem</hi> II. Having given the latitudes and longitudes of the places sailed from and bound to; to sind the successive latitudes on the arc of a great circle in those places where the alteration in longitude shall be a given quantity; together with the courses and distances between those places.</p><p>1. Find the angle of position at each place, and their distance, by one of the preceding cases.</p><p>2. Find the greatest latitude the great circle runs through, i. e. find the perpendicular from the pole to that circle; and also find the several angles at the pole, made by the given alterations of longitude between this perpendicular and the successive meridians come to.</p><p>3. With this perpendicular and the polar angles severally, find as many corresponding latitudes, by saying, as radius : tang. greatest lat. :: cos. 1st polar angle : tang. 1st lat. :: cos. 2d polar angle : tang. of 2d lat. &c.</p><p>4. Having now the several latitudes passed through, and the difference of longitude between each, then by Mercator's Sailing find the courses and distances between those latitudes. And these are the several courses and distances the ship must run, to keep nearly on the arc of a great circle.</p><p>The smaller the alterations in longitude are taken, the nearer will this method approach to the truth; but it is sufficient to compute to every 5 degrees of difference of longitude; as the length of an arc of 5 degrees differs from its chord, or tangent, only by 0.002.</p><p>The track of a ship, when thus directed nearly in the arc of a great circle, may be delineated on the Mercator's chart, by marking on it, by help of the latitudes and longitudes, the successive places where the ship is to alter her course; then those places or points, being joined by right lines, will shew the path along which the ship is to sail, under the proposed circumstances.</p><p>On the subject of these articles, see Robertson's Elements of Navigation, vol. 2. <cb/></p><p><hi rend="italics">Spheroidical</hi> <hi rend="smallcaps">Sailing</hi>, is computing the cases of navigation on the supposition or principles of the spheroidical figure of the earth. See Robertson's Navigation, vol. 2, b. 8. sect. 8.</p><div2 part="N" n="Sailing" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sailing</hi></head><p>, <hi rend="italics">in a more confined sense,</hi> is the art of conducting a ship from place to place, by the working or handling of her sails and rudder.</p><p>To bring Sailing 10 certain rules, M. Renau computes the force of the water, against the ship's rudder, stem, and side; and the force of the wind against her sails. In order to this, he first considers all fluid bodies, as the air, water, &c, as composed of little particles, which when they act upon, or move against any surface, do all move parallel to one another, or strike against the surface after the same manner. Secondly, that the motion of any body, with regard to the surface it strikes, must be either perpendicular, parallel, or oblique.</p><p>From these principles he computes, that the force of the air or water, striking perpendicularly upon a sail or rudder, is to the force of the same striking obliquely, in the duplicate ratio of radius to the sine of the angle of incidence: and consequently that all oblique forces of the wind against the sails, or of the water against the rudder, will be to one another in the duplicate ratio of the sines of the angles of incidence.</p><p>Such are the conclusions from theory; but it is very different in real practice, or experiments, as appears from the tables of experiments inserted at the article <hi rend="smallcaps">Resistance.</hi></p><p>Farther, when the different degrees of velocity are considered, it is also found that the forces are as the squares of the velocities of the moving air or water nearly; that is, a wind that blows twice as swift, as another, will have 4 times the force upon the sail; and when 3 times as swift, 9 times the force, &c. And it being also indifferent, whether we consider the motion of a solid in a fluid at rest, or of the fluid against the solid at rest; the reciprocal impressions being always the same; if a solid be moved with different velocities in the same fluid matter, as water, the different resistances which it will receive from that water, will be in the same proportion as the squares of the velocities of the moving body.</p><p>He then applies these principles to the motions of a ship, both forwards and sideways, through the water, when the wind, with certain velocities, strikes the sails in various positions. After this, the author proceeds to demonstrate, that the best position or situation of a ship, so as she may make the least lee-way, or side motion, but go to windward as much as possible, is this: that, let the sail have what situation it will, the ship be always in a line bisecting the complement of the wind's angle of incidence upon the sail. That is, supposing the sail in the position BC, and the wind blow- <figure/> ing from A to B, and consequently the angle of the wind's incidence on the sail is ABC, the complement of which is CBE: then must the ship be put in the position BK, or move in the line BL, bisecting the [angle] CBE. <pb n="410"/><cb/></p><p>He shews farther, that the angle which the sail ought to make with the wind, i. e. the angle ABC, ought to be but 24 degrees; that being the most advantageous situation to go to windward the most possible.</p><p>To this might be added many curious particulars from Borelli de Vi Percussionis, concerning the different directions given to a vessel by the rudder, when sailing with a wind, or floating without sails in a current: in the former case, the head of the ship always coming to the rudder, and in the latter always flying off from it; as also from Euler, Bouguer, and Juan, who have all written learnedly on this subject.</p></div2></div1><div1 part="N" n="SALIANT" org="uniform" sample="complete" type="entry"><head>SALIANT</head><p>, in Fortification, is said of an angle that projects its point outwards; in opposition to a reentering angle, which has its point turned inwards. Instances of both kinds of these we have in tenailles and star-works.</p></div1><div1 part="N" n="SALON" org="uniform" sample="complete" type="entry"><head>SALON</head><p>, or <hi rend="smallcaps">Saloon</hi>, in Architecture, a grand, lofty, spacious sort of hall, vaulted at top, and usually comprehending two stories, with two ranges of windows. It is sometimes built square, sometimes round or oval, sometimes octagonal, as at Marly, and sometimes in other forms.</p></div1><div1 part="N" n="SAP" org="uniform" sample="complete" type="entry"><head>SAP</head><p>, or <hi rend="smallcaps">Sapp</hi>, in Building, as to sap a wall, &c, is to dig out the ground from beneath it, so as to bring it down all at once for want of support.</p><div2 part="N" n="Sap" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sap</hi></head><p>, in the Military Art, denotes a work carried on under cover of gabions and fascines on the flank, and mantlets or stuffed gabions on the front, to gain the descent of a ditch, or the like.</p><p>It is performed by digging a deep trench, descending by steps from top to bottom, under a corridor, carrying it as far as the bottom of the ditch, when that is dry; or as far as the surface of the water, when wet.</p></div2></div1><div1 part="N" n="SAROS" org="uniform" sample="complete" type="entry"><head>SAROS</head><p>, in Chronology, a period of 223 lunar months. The etymology of the word is said to be Chaldean, signifying restitution, or return of eclipses; that is, conjunctions of the sun and moon in nearly the same place of the ecliptic. The Saros was a cycle like to that of Meto.</p></div1><div1 part="N" n="SARRASIN" org="uniform" sample="complete" type="entry"><head>SARRASIN</head><p>, or <hi rend="smallcaps">Sarrazin</hi>, in Fortification, a kind of port-cullis, otherwise called a herse, which is hung with ropes over the gate of a town or fortress, to be let fall in case of a surprise.</p></div1><div1 part="N" n="SATELLITES" org="uniform" sample="complete" type="entry"><head>SATELLITES</head><p>, in Astronomy, are certain secondary planets, moving round the other planets, as the moon does round the earth. They are so called because always found attending them, from rising to setting, and making the tour about the sun together with them.</p><p>The words moon and Satellite are sometimes used indifferently: thus we say, either Jupiter's moons, or Jupiter's Satellites; but usually we distinguish, restraining the term moon to the earth's attendant, and applying the term Satellite to the little moons more recently discovered about Jupiter, Saturn, and the Georgian planet, by the assistance of the telescope, which is necessary to render them visible. <cb/></p><p>The Satellites move round their primary planets, as their centres, by the same laws as those primary ones do round their centre the sun; viz, in such manner that, in the Satellites of the same planet, the squares of the periodic times are proportional to the cubes of their distances from the primary planet. For the physical cause of their motions, see <hi rend="smallcaps">Gravity.</hi> See also P<hi rend="smallcaps">LANETS.</hi></p><p>We know not of any Satellites beside those above mentioned, what other discoveries may be made by farther improvements in telescopes, time only can bring to light.</p><p><hi rend="smallcaps">Satellites</hi> <hi rend="italics">of Jupiter.</hi> There are are four little moons, or secondary planets now known performing their evolutions about Jupiter, as that planet does about the Sun.</p><p>Simon Marius, mathematician of the elector of Brandenburg, about the end of November 1609, observed three little stars moving round Jupiter's body, and proceeding along with him; and in January 1610, he found a 4th. In January 1610 Galileo also observed the same in Italy, and in the same year published his observations. These Satellites were also observed in the same month of January 1710, by Thomas Harriot, the celebrated author of a work upon algebra, and who made constant observations of these Satellites, from that time till the 26th of February 1612; as appears by his curious astronomical papers, lately discovered by Dr. Zach, at the seat of the earl of Egremont, at Petworth in Sussex.</p><p>One Antony Maria Schyrlæus di Reita, a capuchin of Cologne, imagined that, besides the four known Satellites of Jupiter, he had discovered five more, on December 29, 1642. But the observation being communicated to Gassendus, who had observed Jupiter on the same day, he soon perceived that the monk had mistaken five sixed stars, in the effusion of the water of Aquarius, marked in Tycho's catalogue 24, 25, 26, 27, 28, for Satellites of Jupiter.</p><p>When Jupiter comes into a line between any of his Satellites and the sun, the Satellite disappears, being then <hi rend="italics">eclipsed,</hi> or involved in his shadow.—When the Satellite goes behind the body of Jupiter, with respect to an observer on the earth, it is then said to be <hi rend="italics">occulted,</hi> being hid from our sight by his body, whether in his shadow or not.—And when the Satellite comes into a position between Jupiter and the Sun, it casts a shadow upon the face of that planet, which we see as an obscure round spot.—And lastly, when the Satellite comes into a line between Jupiter and us, it is said to <hi rend="italics">transit</hi> the disc of the planet, upon which it appears as a round black spot.</p><p>The periods or revolutions of Jupiter's Satellites, are found out from their conjunctions with that planet; after the same manner, as those of the primary planets are discovered from their oppositions to the sun. And their distances from the body of Jupiter, are measured by a micrometer, and est imated in semidiameters of that planet, and thence in miles.</p><p>By the latest and most exact observations, the periodical times and distances of these Satellites, and the angles under which their orbits are seen from the earth, at its mean distance from Jupiter, are as below: <pb n="411"/><cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=9" role="data"><hi rend="smallcaps">Satellites</hi> <hi rend="italics">of</hi> <hi rend="smallcaps">Jupiter.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Distances in</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Satel-</cell><cell cols="1" rows="1" rend="colspan=4 rowspan=2" role="data">Periodic Times.</cell><cell cols="1" rows="1" role="data">Semidia-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Miles.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Angles of</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">lites.</cell><cell cols="1" rows="1" role="data">meters.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Orbit.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data"> 1<hi rend="sup">d</hi></cell><cell cols="1" rows="1" role="data">18<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">34″</cell><cell cols="1" rows="1" rend="align=center" role="data">5 2/3</cell><cell cols="1" rows="1" rend="align=right" role="data">266,000</cell><cell cols="1" rows="1" role="data"> 3′</cell><cell cols="1" rows="1" role="data">55″</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" rend="align=center" role="data">9 1/59</cell><cell cols="1" rows="1" rend="align=right" role="data">423,000</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" rend="align=center" role="data">14 5/13</cell><cell cols="1" rows="1" rend="align=right" role="data">676,000</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">58</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" rend="align=center" role="data">25 3/10</cell><cell cols="1" rows="1" rend="align=right" role="data">1,189,000</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">30</cell></row></table></p><p>The eclipses of the Satellites, especially of those of Jupiter, are of very great use in astronomy. First, in determining pretty exactly the distance of Jupiter from the earth. A second advantage still more considerable, which is drawn from these eclipses, is the proof which they give of the progressive motion of light. It is demonstrated by these eclipses, that light does not come to us in an instant, as the Cartesians pretended, although its motion is extremely rapid. For if the motion of light were infinite, or came to us in an instant, it is evident that we should see the commencement of an eclipse of a Satellite at the same moment, at whatever distance we might be from it; but, on the contrary, if light move progressively, then it is as evident, that the farther we are from a planet, the later we shall be in seeing the moment of its eclipse, because the light will take up a longer time in arriving at us; and so it is found in fact to happen, the eclipses of these Satellites appearing always later and later than the true computed times, as the earth removes farther and farther from the planet. When Jupiter and the earth are at their nearest distance, being in conjunction both on the same side of the sun, then the eclipses are seen to happen the soonest; and when the sun is directly between Jupiter and the earth, they are at their greatest distance asunder, the distance being more than before by the whole diameter of the earth's annual orbit, or by double the earth's distance from the sun, then the eclipses are seen to happen the latest of any, and later than before by about a quarter of an hour. Hence therefore it follows, that light takes up a quarter of an hour in travelling across the orbit of the earth, or near 8 minutes in passing from the sun to the earth; which gives us about 12 millions of miles per minute, or 200,000 miles per second, for the velocity of light. A discovery that was first made by M. Roemer.</p><p>The third and greatest advantage derived from the eclipses of the Satellites, is the knowledge of the longitudes of places on the earth. Suppose two observers of an eclipse, the one, for example, at London, the other at the Canaries; it is certain that the eclipse will appear at the same moment to both observers; but as they are situated under different meridians, they count different hours, being perhaps 9 o'clock to the one, when it is only 8 to the other; by which observations of the true time of the eclipse, on communication, they find the difference of their longitudes to be one hour in time, which answers to 15 degrees of longitude.</p><p><hi rend="smallcaps">Satellites</hi> <hi rend="italics">of Saturn,</hi> are 7 little secondary planets revolving about him. <cb/></p><p>One of them, which till lately was reckoned the 4th in order from Saturn, was discovered by Huygens, the 25th of March 1655, by means of a telescope 12 feet long; and the 1st, 2d, 3d, and 5th, at different times, by Cassini; viz, the 5th in October 1671, by a telescope of 17 feet; the 3d in December 1672, by a telescope of Campani's, 35 feet long; and the first and second in March 1684, by help of Campani's glasses, of 100 and 136 feet. Finally, the 6th and 7th Satellites have lately been discovered by Dr. Herschel, with his 40 feet reflecting telescope, viz, the 6th on the 19th of August 1787, and the 7th on the 17th of September 1788. These two he has called the 6th and 7th Satellites, though they are nearer to the planet Saturn than any of the former five, that the names or numbers of these might not be mistaken or confounded, with regard to former observations of them.</p><p>Moreover, the great distance between the 4th and 5th Satellite, gave occasion to Huygens to suspect that there might be some intermediate one, or else that the 5th might have some other Satellite moving round it, as its centre. Dr. Halley, in the Philos. Trans. (numb. 145, or Abr. vol. 1. pa. 371) gives a correction of the theory of the motions of the 4th or Huygenian Satellite. Its true period he makes 11<hi rend="sup">d</hi> 22<hi rend="sup">h</hi> 41′ 6″.</p><p>The periodical revolutions, and distances of these Satellites from the body of Saturn, expressed in semidiameters of that planet, and in miles, are as follow. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=9" role="data"><hi rend="smallcaps">Satellites</hi> <hi rend="italics">of</hi> <hi rend="smallcaps">Saturn.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Distances in</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Satel-</cell><cell cols="1" rows="1" rend="colspan=4 rowspan=2" role="data">Periods.</cell><cell cols="1" rows="1" role="data">Semidi-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Miles.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Diam. of</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">lites.</cell><cell cols="1" rows="1" role="data">ameters.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Orbit.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data"> 1<hi rend="sup">d</hi></cell><cell cols="1" rows="1" role="data">21<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">18′</cell><cell cols="1" rows="1" role="data">27″</cell><cell cols="1" rows="1" rend="align=center" role="data">4 3/8</cell><cell cols="1" rows="1" rend="align=right" role="data">170,000</cell><cell cols="1" rows="1" role="data"> 1′</cell><cell cols="1" rows="1" role="data">27</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=center" role="data">5 1/2</cell><cell cols="1" rows="1" rend="align=right" role="data">217,000</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">52</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">8</cell><cell cols="1" rows="1" rend="align=right" role="data">303,000</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">36</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=center" role="data">18</cell><cell cols="1" rows="1" rend="align=right" role="data">704,000</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">18</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" rend="align=center" role="data">54</cell><cell cols="1" rows="1" rend="align=right" role="data">2,050,000</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> 4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" rend="align=center" role="data">3 5/9</cell><cell cols="1" rows="1" rend="align=right" role="data">135,000</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">14</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" rend="align=center" role="data">2 5/6</cell><cell cols="1" rows="1" rend="align=right" role="data">107,000</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">57</cell></row></table> The four first describe ellipses like to those of the ring, and are in the same plane. Their inclination to the ecliptic is from 30 to 31 degrees. The 5th describes an orbit inclined from 17 to 18 degrees with the orbit of Saturn; his plane lying between the ecliptic and those of the other Satellites, &c. Dr. Herschel observes that the 5th Satellite turns once round its axis exactly in the time in which it revolves about the planet Saturn; in which respect it resembles our moon, which does the same thing. And he makes the angle of its distance from Saturn, at his mean distance, 17′ 2″. Philos. Trans. 1792, pa. 22. See a long account of observations of these Satellites, with tables of their mean motions, by Dr. Herschel, Philos. Trans. 1790, pa. 427 &c.</p><p><hi rend="smallcaps">Satellites</hi> <hi rend="italics">of the Georgian Planet,</hi> or <hi rend="italics">Herschel,</hi> are two little moons that revolve about him, like those of <pb n="412"/><cb/> Jupiter and Saturn. These Satellites were discovered by Dr. Herschel, in the month of January 1787, who gave an account of them in the Philos. Trans. of that year, pa. 125 &c; and a still farther account of them in the vol. for 1788, pa. 364 &c; from which it appears that their synodical periods, and angular distances from their primary, are as follow: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Satellite.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Periods.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data"> 8<hi rend="sup">d</hi></cell><cell cols="1" rows="1" role="data">17<hi rend="sup">h</hi></cell><cell cols="1" rows="1" role="data">1′</cell><cell cols="1" rows="1" role="data">19″</cell><cell cols="1" rows="1" role="data">0′</cell><cell cols="1" rows="1" role="data">33″</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"> 1 1/2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">44 2/9</cell></row></table></p><p>The orbits of these Satellites are nearly perpendicular to the ecliptic; and in magnitude they are probably not less than those of Jupiter.</p><p><hi rend="smallcaps">Satellite</hi> <hi rend="italics">of Venus.</hi> Cassini thought he saw one, and Mr. Short and other astronomers have suspected the same thing. (Hist. de l'Acad. 1741, Philos. Trans. numb. 459). But the many fruitless searches that have been since made to discover it, leave room to suspect that it has been only an optical illusion, formed by the glasses of telescopes; as appears to be the opinion of F. Hell, at the end of his Ephemeris for 1766, and Boscovich, in his 5th Optical Dissertation.</p><p>Neither has it been discovered that either of the other planets Mars and Mercury have any Satellites revolving about them.</p></div1><div1 part="N" n="SATURDAY" org="uniform" sample="complete" type="entry"><head>SATURDAY</head><p>, the 7th or last day of the week, so called, as some have supposed, from the idol Seater, worshipped on this day by the ancient Saxons, and thought to be the same as the Saturn of the Latins. In astronomy, every day of the week is denoted by some one of the planets, and this day is marked with the planet <figure/> Saturn. Saturday answers to the Jewish sabbath.</p></div1><div1 part="N" n="SATURN" org="uniform" sample="complete" type="entry"><head>SATURN</head><p>, one of the primary planets, being the 6th in order of distance from the sun, and the outermost of all, except the Georgian planet, or Herschel, lately discovered; and is marked with the character <figure/>, denoting an old man supporting himself with a staff, representing the ancient god Saturn.</p><p>Saturn shines with but a feeble light, partly on account of his great distance, and partly from its dull red colour. This planet is perhaps one of the most engaging objects that astronomy offers to our view; it is surrounded with a double ring, one without the other, and beyond these by 7 Satellites, all in the plane of the rings; the rings and planets being all dark and dense bodies, like Saturn himself, these bodies casting their shadows mutually one upon another; though the reflected light of the rings is usually brighter than that of the planet itself.</p><p>Saturn has also certain obscure zones, or belts, appearing at times across his disc, like those of Jupiter, which are changeable, and are probably obscurations in his atmosphere. Dr. Herschel, Philos. Trans. 1790, shews that Saturn has a dense atmosphere; that he revolves about an axis, which is perpendicular to the plane of the rings; that his figure is, like the other planets, the oblate spheroid, being flatted at the poles, the polar diameter being to the equatorial one <cb/> as 10 to 11; that his ring has a motion of rotation in its own plane, its axis of motion being the same as that of Saturn himself, and its periodical time equal to 10<hi rend="sup">h</hi> 32′ 15″.4. See also <hi rend="smallcaps">Ring</hi>, and <hi rend="smallcaps">Satellite.</hi></p><p>Concerning the discovery of the ring and figure of Saturn; we find that Galileo first perceived that his figure is not round: but Huygens shewed, in his Systema Saturniana 1659, that this was owing to the positions of his ring; for his spheroidical form could only be seen by Herschel's telescope; though indeed Cassini, in an observation made June 19, 1692, saw the oval figure of Saturn's shadow upon his ring.</p><p>Mr. Bugge determines (Philos. Trans. 1787, pa. 42) the heliocentric longitude of Saturn's descending node to be 9<hi rend="sup">s</hi> 21° 5′ 8″1/2; and that the planet was in that node August 21, 1784, at 18<hi rend="sup">h</hi> 20′ 10″, time at Copenhagen.</p><p>The annual period of Saturn about the sun, is 10759 days 7 hours, or almost 30 years; and his diameter is about 67000 miles, or near 8 1/2 times the diameter of the earth; also his distance is about 9 1/2 times that of the earth. Hence some have concluded that his light and heat are entirely unfit for rational inhabitants. But that their light is not so weak as we imagine, is evident from their brightness in the night time. Besides, allowing the sun's light to be 45000 times as strong, with respect to us, as the light of the moon when full, the sun will afford 500 times as much light to Saturn as the full moon does to us, and 1600 times as much to Jupiter. So that these two planets, even without any moon, would be much more enlightened than we at first imagine; and by having so many, they may be very comfortable places of residence. Their heat, so far as it depends on the force of the sun's rays, is certainly much less than ours; to which no doubt the bodies of their inhabitants are as well adapted as ours are to the seasons we enjoy. And if it be considered that Jupiter never has any winter, even at his poles, which probably is also the case with Saturn, the cold cannot be so intense on these two planets as is generally imagined. To this may be added, that there may be something in the nature of their mould warmer than in that of our earth; and we find that all our heat does not depend on the rays of the sun; for if it did, we should always have the same months equally hot or cold at their annual return, which is very far from being the case.</p><p>See the articles <hi rend="smallcaps">Planet, Period, Ring</hi>, S<hi rend="smallcaps">ATELLITE.</hi></p></div1><div1 part="N" n="SAUCISSE" org="uniform" sample="complete" type="entry"><head>SAUCISSE</head><p>, in Artillery, a long train of powder inclosed in a roll or pipe of pitched cloth, and sometimes of leather, about 2 inches in diameter; serving to set fire to mines or caissons. It is usually placed in a wooden pipe, called an auget, to prevent its growing damp.</p><div2 part="N" n="Saucisson" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Saucisson</hi></head><p>, in Fortification, a kind of faggot, made of thick branches of trees, or of the trunks of shrubs, bound together; for the purpose of covering the men, and to serve as epaulements; and also to repair breaches, stop passages, make traverses over a wet ditch, &c.</p><p>The Saucisson differs from the fascine, which is only made of small branches; and by its being bound at both ends, and in the middle.</p></div2></div1><div1 part="N" n="SAVILLE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SAVILLE</surname> (Sir <foreName full="yes"><hi rend="smallcaps">Henry</hi></foreName>)</persName></head><p>, a very learned English- <pb n="413"/><cb/> man, the second son of Henry Saville, Esq. was born at Bradley, near Halifax, in Yorkshire, November the 30th, 1549. He was entered of Merton-college, Oxford, in 1561, where he took the degrees in arts, and was chosen fellow. When he proceeded master of arts in 1570, he read for that degree on the Almagest of Ptolomy, which procured him the reputation of a man eminently skilled in mathematics and the Greek language; in the former of which he voluntarily read a public lecture in the university for some time.</p><p>In 1578 he travelled into France and other countries; where, diligently improving himself in all useful learning, in languages, and the knowledge of the world, he became a most accomplished gentleman. At his return, he was made tutor in the Greek tongue to queen Elizabeth, who had a great esteem and liking for him.</p><p>In 1585 he was made warden of Merton-college, which he governed six-and-thirty years with great honour, and improved it by all the means in his power.— In 1596 he was chosen provost of Eton-college; which he filled with many learned men.—James the First, upon his accession to the crown of England, expressed a great regard for him, and would have preferred him either in church or state; but Saville declined it, and only accepted the ceremony of knighthood from the king at Windsor in 1604. His only son Henry dying about that time, he thenceforth devoted his fortune to the promoting of learning. Among other things, in 1619, he founded, in the university of Oxford, two lectures, or professorships, one in geometry, the other in astronomy; which he endowed with a salary of 1601. a year each, besides a legacy of 600l. to purchase more lands for the same use. He also furnished a library with mathematical books near the mathematical school, for the use of his professors; and gave 100l. to the mathematical chest of his own appointing: adding after wards a legacy of 40l. a year to the same chest, to the university, and to his professors jointly. He likewise gave 120l. towards the new-building of the schools, beside several rare manuscripts and printed books to the Bodleian library; and a good quantity of Greek types to the printing-press at Oxford.</p><p>After a life thus spent in the encouragement and promotion of science and literature in general, he died at Eton-college the 19th of February 1622, in the 73d year of his age, and was buried in the chapel there. On this occasion, the university of Oxford paid him the greatest honours, by having a public speech and verses made in his praise, which were published soon after in 4to, under the title of <hi rend="italics">Ultima Linea Savilii.</hi></p><p>As to the character of Saville, the highest encomiums are bestowed on him by all the learned of his time: by Casaubon, Mercerus, Meibomius, Joseph Scaliger, and especially the learned bishop Montague; who, in his <hi rend="italics">Diatribæ</hi> upon Selden's History of Tythes, styles him, “that magazine of learning, whose memory shall be honourable amongst not only the learned, but the righteous for ever.”</p><p>Several noble instances of his munificence to the republic of letters have already been mentioned: in the account of his publications many more, and even greater, will appear. These are,</p><p>1. <hi rend="italics">Four Books of the Histories of Cornelius Tacitus, and</hi> <cb/> <hi rend="italics">the Life of Agricola;</hi> with Notes upon them, in folio, dedicated to Queen Elizabeth, 1581.</p><p>2. <hi rend="italics">A View of certain Military Matters,</hi> or Commentaries concerning Roman Warfare, 1598.</p><p>3. <hi rend="italics">Rerum Anglicarum Scriptores post Bedam,</hi> &c. 1596. This is a collection of the best writers of our English history; to which he added chronological tables at the end, from Julius Cæsar to William the Conqueror.</p><p>4. <hi rend="italics">The Works of St. Chrysostom,</hi> in Greek, in 8 vols. folio, 1613. This is a very fine edition, and composed with great cost and labour. In the preface he says, “that having himself visited, about 12 years before, all the public and private libraries in Britain, and copied out thence whatever he thought useful to this design, he then sent some learned men into France, Germany, Italy, and the East, to transcribe such parts as he had not already, and to collate the others with the best manuscripts.” At the same time, he makes his acknowledgments to several eminent men for their assistance; as Thuanus, Velserus, Schottus, Casaubon, Ducæus, Gruter, Hoeschelius, &c. In the 8th volume are inserted Sir Henry Saville's own notes, with those of other learned men. The whole charge of this edition, including the several sums paid to learned men, at home and abroad, employed in finding out, transcribing, and collating the best manuscripts, is said to have amounted to no less than 80001. Several editions of this work were afterwards published at Paris.</p><p>5. In 1618 he published a Latin work, written by Thomas Bradwardin, abp. of Canterbury, against Pelagius, intitled, <hi rend="italics">De Causa Dei contra Pelagium, et de virtute causarum;</hi> to which he prefixed the life of Bradwardin.</p><p>6. In 1621 he published a collection of his own Mathematical Lectures on Euclid's Elements; in 4to.</p><p>7. <hi rend="italics">Oratio coram Elizabetha Regina Oxoniæ habita,</hi> anno 1592. Printed at Oxford in 1658, in 4to.</p><p>8. He translated into Latin king James's <hi rend="italics">Apology for the Oath of Allegiance.</hi> He also left several manuscripts behind him, written by order of king James; all which are in the Bodleian library. He wrote notes likewise upon the margin of many books in his library, particularly Eusebius's <hi rend="italics">Ecelesiastical History;</hi> which were afterwards used by Valesius, in his edition of that work in 1659.—Four of his letters to Camden are published by Smith, among <hi rend="italics">Camden's Letters,</hi> 1691, 4to.</p><p>Sir Henry Saville had a younger brother, <hi rend="italics">Thomas</hi> <hi rend="smallcaps">Saville</hi>, who was admitted probationer fellow of Merton-college, Oxford, in 1580. He afterwards travelled abroad into several countries. Upon his return he was chosen fellow of Eton-college; but he died at London in 1593. Thomas Saville was also a man of great learning, and an intimate friend of Camden; among whose letters, just mentioned, there are 15 of Mr. Saville's to him.</p></div1><div1 part="N" n="SAUNDERSON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SAUNDERSON</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">Nicholas</hi></foreName>)</persName></head><p>, an illustrious professor of mathematics in the university of Cambridge, and a fellow of the Royal Society, was born at Thurlston in Yorkshire in 1682. When he was but twelve months old, he lost not only his eye-fight, but his very eye-balls, by the small-pox; so that he could retain no more ideas of vision than if he had been born blind. At an early age, however, being of very pro- <pb n="414"/><cb/> mising parts, he was sent to the free-school at Penniston, and there laid the foundation of that knowledge of the Greek and Latin languages, which he afterwards improved so far, by his own application to the classic authors, as to hear the works of Euclid, Archimedes, and Diophantus read in their original Greek.</p><p>Having acquired a grammatical education, his father, who was in the excise, instructed him in the common rules of arithmetic. And here it was that his excellent mathematical genius first appeared: for he very soon became able to work the common questions, to make very long calculations by the strength of his memory, and to form new rules to himself for the better resolving of such questions as are often proposed to learners as trials of skill.</p><p>At the age of 18, our author was introduced to the acquaintance of Richard West, of Underbank, Esq. a lover of mathematics, who, observing Mr. Saunderson's uncommon capacity, took the pains to instruct him in the principles of algebra and geometry, and gave him every encouragement in his power to the prosecution of these studies. Soon after this he became acquainted also with Dr. Nettleton, who took the same pains with him. And it was to these two gentlemen that Mr. Saunderson owed his first institution in the mathematical sciences: they furnished him with books, and osten read and expounded them to him. But he soon surpassed his masters, and became sitter to teach, than to learn any thing from them.</p><p>His father, otherwise burdened with a numerous family, finding a difficulty in supporting him, his friends began to think of providing both for his education and maintenance. His own inclination led him strongly to Cambridge, and it was at length determined he should try his fortune there, not as a scholar, but as a master: or, if this design should not succeed, they promised themselves success in opening a school for him at London. Accordingly he went to Cambridge in 1707, being then 25 years of age, and his fame in a short time filled the university. Newton's Principia, Optics, and Universal Arithmetic, were the foundations of his lectures, and afforded him a noble field for the displaying of his genius; and great numbers came to hear a blind man give lectures on optics, discourse on the nature of light and colours, explain the theory of vision, the effect of glasses, the phenomenon of the rainbow, and other objects of sight.</p><p>As he instructed youth in the principles of the Newtonian philosophy, he soon became acquainted with its incomparable author, though he had several years before left the university; and frequently conversed with him on the most difficult parts of his works: he also held a friendly communication with the other eminent mathematicians of the age, as Halley, Cotes, Demoivre, &c.</p><p>Mr. Whiston was all this time in the mathematical professor's chair, and read lectures in the manner proposed by Mr. Saunderson on his settling at Cambridge; so that an attempt of this kind looked like an encroachment on the privilege of his office; but, as a goodnatured man, and an encourager of learning, he readily consented to the application of friends made in behalf of so uncommon a person.</p><p>Upon the removal of Mr. Whiston from his profes- <cb/> sorship, Mr. Saunderson's merit was thought so much superior to that of any other competitor, that an extraordinary step was taken in his favour, to qualify him with a degree, which the statute requires: in consequence he was chosen in 1711, Mr. Whiston's successor in the Lucasian professorship of mathematics, Sir Isaac Newton interesting himself greatly in his favour. His first performance, after he was seated in the chair, was an inaugural speech made in very elegant latin, and a style truly Ciceronian; for he was well versed in the writings of Tully, who was his favourite in prose, as Virgil and Horace were in verse. From this time he applied himself closely to the reading of lectures, and gave up his whole time to his pupils. He continued to reside among the gentlemen of Christ-college till the year 1723, when he took a house in Cambridge, and soon after married a daughter of Mr. Dickens, rector of Boxworth in Cambridgeshire, by whom he had a son and a daughter.</p><p>In the year 1728, when king George visited the university, he expressed a desire of seeing so remarkable a person; and accordingly our professor attended the king in the senate, and by his favour was there created doctor of laws.</p><p>Dr. Saunderson was naturally of a strong healthy constitution; but being too sedentary, and constantly confining himself to the house, he became a valetudinarian: and in the spring of the year 1739 he complained of a numbness in his limbs, which ended in a mortification in his foot, of which he died the 19th of April that year, in the 57th year of his age.</p><p>There was scarcely any part of the mathematics on which Dr. Saunderson had not composed something for the use of his pupils. But he discovered no intention of publishing any thing till, by the persuasion of his friends, he prepared his Elements of Algebra for the press, which after his death were published by subscription in 2 vols 4to, 1740.</p><p>He left many other writings, though none perhaps prepared for the press. Among these were some valuable comments on Newton's Principia, which not only explain the more difficult parts, but often improve upon the doctrines. These are published in Latin at the end of his posthumous Treatise on Fluxions, a valuable work, published in 8vo, 1756.—His manuscript lectures too, on most parts of natural philosophy, which I have seen, might make a considerable volume, and prove an acceptable present to the public if printed.</p><p>Dr. Saunderson, as to his character, was a man of much wit and vivacity in conversation, and esteemed an excellent companion. He was endued with a great regard to truth; and was such an enemy to disguise, that he thought it his duty to speak his thoughts at all times with unrestrained freedom. Hence his sentiments on men and opinions, his friendship or disregard, were expressed without reserve; a sincerity which raised him many enemies.</p><p>A blind man, moving in the sphere of a mathematician, seems a phenomenon difficult to be accounted for, and has excited the admiration of every age in which it has appeared. Tully mentions it as a thing scarce credible in his own master in philosophy, Diodotus; that he exercised himself in it with more assi- <pb n="415"/><cb/> duity after he became blind; and, what he thought next to impossible to be done without sight, that he professed geometry, describing his diagrams so exactly to his scholars, that they could draw every line in its proper direction. St. Jerome relates a still more remarkable instance in Didymus of Alexandria, who, though blind from his infancy, and therefore ignorant of the very letters, not only learned logic, but geometry also to very great perfection, which seems most of all to require sight. But, if we consider that the ideas of extended quantity, which are the chief objects of mathematics, may as well be acquired by the sense of feeling as that of sight, that a fixed and steady attention is the principal qualification for this study, and that the blind are by necessity more abstracted than others (for which reason it is said that Democritus put out his eyes, that he might think more intensely), we shall perhaps find reason to suppose that there is no branch of science so much adapted to their circumstances.</p><p>At first, Dr. Saunderson acquired most of his ideas by the sense of feeling; and this, as is commonly the case with the blind, he enjoyed in great perfection. Yet he could not, as some are said to have done, distinguish colours by that sense; for, after having made repeated trials, he used to say, it was pretending to impossibilities. But he could with great nicety and exactness observe the smallest degree of roughness or defect of polish in a surface. Thus, in a set of Roman medals, he distinguished the genuine from the false, though they had been counterfeited with such exactness as to deceive a connoisseur who had judged by the eye. By the sense of feeling also, he distinguished the least variation; and he has been seen in a garden, when observations have been making on the sun, to take notice of every cloud that interrupted the observation almost as justly as they who could see it. He could also tell when any thing was held near his face, or when he passed by a tree at no great distance, merely by the different impulse of the air on his face.</p><p>His ear was also equally exact. He could readily distinguish the 5th part of a note. By the quickness of this sense he could judge of the size of a room, and of his distance from the wall. And if ever he walked over a pavement, in courts or piazzas which reflected a sound, and was afterwards conducted thither again, he could tell in what part of the walk he stood, merely by the note it sounded.</p><p>Dr. Saunderson had a peculiar method of performing arithmetical calculations, by an ingenious machine and method which has been called his Palpable Arithmetic, and is particularly described in a piece prefixed to the first volume of his Algebra. That he was able to make long and intricate calculations, both arithmetical and algebraical, is a thing as certain as it is wonderful. He had contrived for his own use, a commodious notation for any large numbers, which he could express on his abacus, or calculating table, and with which he could readily perform any arithmetical operations, by the sense of feeling only, for which reason it was called his Palpable Arithmetic.</p><p>His calculating table was a smooth thin board, a little more than a foot square, raised upon a small frame so as to lie hollow; which board was divided into a <cb/> great number of little squares, by lines intersecting one another perpendicularly, and parallel to the sides of the table, and the parallel ones only one-tenth of an inch from each other; so that every square inch of the table was thus divided into 100 little squares. At every point of intersection the board was perforated by small holes, capable of receiving a pin; for it was by the help of pins, stuck up to the head through these holes, that he expressed his numbers. He used two sorts of pins, a larger and a smaller sort; at least their heads were different, and might easily be distinguished by feeling. Of these pins he had a large quantity in two boxes, with their points cut off, which always stood ready before him when he calculated. The writer of that account describes particularly the whole process of using the machine, and concludes, “He could place and displace his pins with incredible nimbleness and facility, much to the pleasure and surprize of all the beholders. He could even break off in the middle of a calculation, and resume it when he pleased, and could presently know the condition of it, by only drawing his fingers gently over the table.”</p></div1><div1 part="N" n="SAURIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SAURIN</surname> (<foreName full="yes"><hi rend="smallcaps">Joseph</hi></foreName>)</persName></head><p>, an ingenious French mathematician, was born in 1659, at Courtaison, in the principality of Orange. His father, minister at Grenoble, was a man of a very studious disposition, and was the first preceptor or instructor to our author; who made a rapid progress in his studies, and at a very early age was admitted a minister at Eure in Dauphiny. But preaching an offensive sermon, he was obliged to quit France in 1683. On this occasion he retired to Geneva; from whence he went into the State of Berne, and was appointed to a living at Yverdun. He was no sooner established in this his post, than certain theologians raised a storm against him. Saurin, disgusted with the controversy, and still more with the Swiss, where his talents were buried, passed into Holland, and from thence into France, where he put himself under the protection of the celebrated Bossu, to whom he made his abjuration in 1690, as it is suspected, that he might find protection, and have an opportunity of cultivating the sciences at Paris. And he was not disappointed: he met with many slattering encouragements; was even much noticed by the king, had a pension from the court, and was admitted of the Academy of Sciences in 1707, in the quality of geometrician. This science was now his chief study and delight; with many writings upon which he enriched the volumes of the Journal des Savans, and the Memoirs of the Academy of Sciences. These were the only works of this kind that he published: he was author of several other pieces of a controversial nature, against the celebrated Rousseau, and other antagonists, over whom with the assistance of government he was enabled to triumph. The latter part of his life was spent in more peace, and in cultivating the mathematical sciences; and he died the 29th of December 1737, of a lethargic fever, at 78 years of age.</p><p>The character of Saurin was lively and impetuous, endued with a considerable degree of that noble independence and loftiness of manner, which is apt to be mistaken for haughtiness or insolence; in consequence of which, his memory was attacked after his death, as his reputation had been during his life; and it was even <pb n="416"/><cb/> said he had been guilty of crimes, by his own confession, that ought to have been punished with death.</p><p>Saurin's mathematical and philosophical papers, printed in the Memoirs of the Academy of Sciences, which are pretty numerous, are to be found in the volumes for the years following; viz, 1709, 1710, 1713, 1716, 1718, 1720, 1722, 1723, 1725, 1727.</p></div1><div1 part="N" n="SAUVEUR" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SAUVEUR</surname> (<foreName full="yes"><hi rend="smallcaps">Joseph</hi></foreName>)</persName></head><p>, an eminent French mathematician, was born at La Fleche the 24th of March 1653. He was absolutely dumb till he was seven years of age; and then the organs of speech did not disengage so effectually, but that he was ever after obliged to speak very slowly and with difficulty. He very early discovered a great turn for mechanics, and was always inventing and constructing something or other in that way.</p><p>He was sent to the college of the Jesuits to learn polite literature, but made very little progress in poetry and eloquence. Virgil and Cicero had no charms for him; but he read with greediness books of arithmetic and geometry. However, he was prevailed on to go to Paris in 1670, and, being intended for the church, there he applied himself for a time to the study of philosophy and theology; but still succeeded no better. In short, mathematics was the only study he had any passion or relish for, and this he-cultivated with extraordinary success; for during his course of philosophy, he learned the first six books of Euclid in the space of a month, without the help of a master.</p><p>As he had an impediment in his voice, though otherwise endued with extraordinary abilities, he was advised by M. Bossuet, to give up all designs upon the church, and to apply himself to the study of physic: but this being utterly against the inclination of his uncle, from whom he drew his principal resources, Sauveur determined to devote himself to his favourite study, and to perfect himself in it, so as to teach it for his support; and in effect he soon became the fashionable preceptor in mathematics, so that at 23 years of age he had prince Eugene for his scholar.—He had not yet read the geometry of Des Cartes; but a foreigner of the first quality desiring to be taught it, he made himself master of it in an inconceivably small space of time. —Basset being a fashionable game at that time, the marquis of Dangeau asked him for some calculations relating to it, which gave such satisfaction, that Sauveur had the honour to explain them to the king and queen.</p><p>In 1681 he was sent with M. Mariotte to Chantilli, to make some experiments upon the waters there, which he did with much applause. The frequent visits he made to this place inspired him with the design of writing a treatise on fortification; and, in order to join practice with theory, he went to the siege of Mons in 1691, where he continued all the while in the trenches. With the same view also he visited all the towns of Flanders; and on his return he became the mathematician in ordinary at the court, with a pension for life.— In 1680 he had been chosen to teach mathematics to the pages of the Dauphiness. In 1686 he was appointed mathematical professor in the Royal College. And in 1696 admitted a member of the Academy of Sciences, where he was in high esteem with the members of that society.—He became also particularly acquainted with <cb/> the prince of Condé, from whom he received many marks of favour and affection. Finally, M. Vauban having been made marshal of France, in 1703, he proposed Sauveur to the king as his successor in the office of examiner of the engineers; to which the king agreed, and honoured him with a pension, which our author enjoyed till his death, which happened the 9th of July 1716, in the 64th year of his age</p><p>Sauveur, in his character, was of a kind obliging disposition, of a sweet, uniform, and unaffected temper; and although his fame was pretty generally spread abroad, it did not alter his humble deportment, and the simplicity of his manners. He used to say, that what one man could accomplish in mathematics, another might do also, if he chose it.</p><p>He was twice married. The first time he took a very singular precaution; for he would not meet the lady till he had been with a notary to have the conditions, he intended to insist on, reduced into a written form; for fear the sight of her should not leave him enough master of himself. This was acting very wisely, and like a true mathematician; who always proceeds by rule and line, and makes his calculations when his head is cool.—He had children by both his wives; and by the latter a son who, like himself, was dumb for the first seven years of his life.</p><p>An extraordinary part of Sauveur's character is, that though he had neither a musical voice nor ear, yet he studied no science more than music, of which he composed an entire new system. And though he was obliged to borrow other people's voice and ears, yet he amply repaid them with such demonstrations as were unknown to former musicians. He also introduced a new diction in music, more appropriate and extensive. He invented a new doctrine of sounds. And he was the first that discovered, by theory and experiment, the velocity of musical strings, and the spaces they describe in their vibrations, under all circumstances of tension and dimensions. It was he also who first invented for this purpose the monochord and the echometer. In short, he pursued his researches even to the music of the ancient Greeks and Romans, to the Arabs, and to the very Turks and Persians themselves; so jealous was he, lest any thing should escape him in the science of sounds.</p><p>Sauveur's writings, which consist of pieces rather than of set works, are all inserted in the volumes of the Memoirs of the Academy of Sciences, from the year 1700 to the year 1716, on various geometrical, mathematical, philosophical, and musical subjects.</p></div1><div1 part="N" n="SCALE" org="uniform" sample="complete" type="entry"><head>SCALE</head><p>, a mathematical instrument, consisting of certain lines drawn on wood, metal, or other matter, divided into various parts, either equal or unequal. It is of great use in laying down distances in proportion, or in measuring distances already laid down.</p><p>There are Scales of various kinds, accommodated to the several uses: the principal are the <hi rend="italics">plane Scale,</hi> the <hi rend="italics">diagonal Scale, Gunter's Scale,</hi> and the <hi rend="italics">plotting Scale.</hi></p><p><hi rend="italics">Plane or Plain</hi> <hi rend="smallcaps">Scale</hi>, a mathematical instrument of very extensive use and application; which is commonly made of 2 feet in length; and the lines usually drawn upon it are the following, viz, <pb n="417"/><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">1 Lines of</cell><cell cols="1" rows="1" role="data">Equal parts, and marked</cell><cell cols="1" rows="1" role="data">E. P.</cell></row><row role="data"><cell cols="1" rows="1" role="data">2 "</cell><cell cols="1" rows="1" role="data">Chords "</cell><cell cols="1" rows="1" role="data">Cho.</cell></row><row role="data"><cell cols="1" rows="1" role="data">3 "</cell><cell cols="1" rows="1" role="data">Rhumbs "</cell><cell cols="1" rows="1" role="data">Ru.</cell></row><row role="data"><cell cols="1" rows="1" role="data">4 "</cell><cell cols="1" rows="1" role="data">Sines "</cell><cell cols="1" rows="1" role="data">Sin.</cell></row><row role="data"><cell cols="1" rows="1" role="data">5 "</cell><cell cols="1" rows="1" role="data">Tangents "</cell><cell cols="1" rows="1" role="data">Tan.</cell></row><row role="data"><cell cols="1" rows="1" role="data">6 "</cell><cell cols="1" rows="1" role="data">Secants "</cell><cell cols="1" rows="1" role="data">Sec.</cell></row><row role="data"><cell cols="1" rows="1" role="data">7 "</cell><cell cols="1" rows="1" role="data">Semitangents "</cell><cell cols="1" rows="1" role="data">S. T.</cell></row><row role="data"><cell cols="1" rows="1" role="data">8 "</cell><cell cols="1" rows="1" role="data">Longitude "</cell><cell cols="1" rows="1" role="data">Long.</cell></row><row role="data"><cell cols="1" rows="1" role="data">9 "</cell><cell cols="1" rows="1" role="data">Latitude "</cell><cell cols="1" rows="1" role="data">Lat.</cell></row></table></p><p>1. The lines of equal parts are of two kinds, viz, simply divided, and diagonally divided. The first of these are formed by drawing three lines parallel to one another, and dividing them into any equal parts by short lines drawn across them, and in like manner subdividing the first division or part into 10 other equal small parts; by which numbers or dimensions of two figures may be taken off. Upon some rulers, several of these scales of equal parts are ranged parallel to each other, with figures set to them to shew into how many equal parts they divide the inch; as 20, 25, 30, 35, 40, 45, &c. The 2d or diagonal divisions are formed by drawing eleven long parallel and equidistant lines, which are divided into equal parts, and crossed <figure/> by other short lines, as the former; then the first of the equal parts have the two outermost of the eleven pa- <cb/> rallels divided into 10 equal parts, and the points of division being connected by lines drawn diagonally, the whole scale is thus divided into dimensions or numbers of three places of figures.</p><p>The other lines upon the scales are such as are commonly used in trigonometry, navigation, astronomy, dialling, projection of the sphere, &c, &c; and their constructions are mostly taken from the divisions of a circle, as follow:</p><p>Describe a circle with any convenient radius, and quarter it by drawing the diameters AB and DE at right angles to each other; continue the diameter AB out towards F, and draw the tangent line EG parallel to it; also draw the chords AD, DB, BE, EA. Then.</p><p>2. For the line of chords, divide a quadrant BE into 90 equal parts; on E as a centre, with the compasses transfer these divisions to the chord line EB, which mark with the corresponding numbers, and it will become a line of chords, to be transferred to the ruler.</p><p>3. For the line of rhumbs, divide the quadrant AD into 8 equal parts; then with the centre A transfer the divisions to the chord AD, for the line of rhumbs.</p><p>4. For the line of sines, through each of the divisions of the arc BE, draw right lines parallel to the radius BC, which will divide the radius CE into the sines, or versed sines, numbering it from C to E for the sines, and from E to C for the versed sines.</p><p>5. For the line of tangents, lay a ruler on C, and the several divisions of the arc BE, and it will intersect the line EG, which will become a line of tangents, and numbered from E to G with 10, 20, 30, 40, &c.</p><p>6. For the line of secants, transfer the distances between the centre C and the divisions on the line of tangents to the line BF, from the centre C, and these will give the divisions of the line of secants, which must be numbered from B towards F, with 10, 20, 30, &c.</p><p>7. For the line of semitangents, lay a ruler on D and the several divisions of the arc EB, which will intersect the radius CB in the divisions of the semitangents, which are to be marked with the corresponding figures of the arc EB.</p><p>The chief uses of the sines, tangents, secants, and semitangents, are to find the poles and centres of the several circles represented in the projections of the sphere.</p><p>8. For the line of longitude, divide the radius CD into 60 equal parts; through each of these, parallels to the radius BC will intersect the arc BD in as many points: from D as a centre the divisions of the arc BD being transferred to the chord BD, will give the divisions of the line of longitude.</p><p>If this line be laid upon the scale close to the line of chords, both inverted, so that 60° in the scale of longitude be against 0° in the chords, &c; and any degree of latitude be counted on the chords, there will stand opposite to it, in the line of longitude, the miles contained in one degree of longitude, in that latitude: the measure of 1 degree under the equator being 60 geographical miles. <pb n="418"/><cb/></p><p>9. For the line of latitude, lay a ruler on B, and the several divisions on the sines on CE, and it will intersect the arc AE in as many points; on A as a centre transfer the intersections of the arc AE to the chord AE, for the line of latitude.</p><p>See also Robertson's Description and use of Mathematical Instruments.</p><p><hi rend="italics">Decimal,</hi> or <hi rend="italics">Gunter's,</hi> or <hi rend="italics">Plotting,</hi> or <hi rend="italics">Proportional,</hi> or <hi rend="italics">Reducing</hi> <hi rend="smallcaps">Scale.</hi> See the several articles.</p><div2 part="N" n="Scale" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Scale</hi></head><p>, in Architecture and Geography, a line divided into equal parts, placed at the bottom of a map or draught, to serve as a common measure to all the parts of the building, or all the distances and places of the map.</p><p>In maps of large tracts, as kingdoms and provinces, &c, the Scale usually consists of miles; whence it is denominated a Scale of miles.—In more particular maps, as those of manors, &c, the Scale is usually of chains &c.—The Scales used in draughts of buildings mostly consist of modules, feet, inches, palms, fathoms, or the like.</p><p>To find the distance between two towns &c, in a map, the interval is taken in the compasses, and set off in the scale; and the number of divisions it includes gives the distance. The same method serves to find the height of a story, or other part in a design.</p><p><hi rend="italics">Front</hi> <hi rend="smallcaps">Scale</hi>, in Perspective, is a right line in the draught, parallel to the horizontal line; divided into equal parts, representing feet, inches, &c.</p><p><hi rend="italics">Flying</hi> <hi rend="smallcaps">Scale</hi>, is a right line in the draught, tending to the point of view, and divided into unequal parts, representing feet, inches, &c.</p><p><hi rend="italics">Differential</hi> <hi rend="smallcaps">Scale</hi>, is used for the scale of relation subtracted from unity. See <hi rend="smallcaps">Series.</hi></p><p><hi rend="smallcaps">Scale</hi> <hi rend="italics">of Relation,</hi> in Algebra, an expression denoting the relation of the terms of recurring series to each other. See <hi rend="smallcaps">Series.</hi></p><p><hi rend="italics">Hour</hi> <hi rend="smallcaps">Scale.</hi> See <hi rend="smallcaps">Hour.</hi></p></div2><div2 part="N" n="Scale" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Scale</hi></head><p>, in Music, is a denomination given to the arrangement of the six syllables, invented by Guido Aratino, <hi rend="italics">ut re mi fa sol la;</hi> called also gammut. It is called Scale, or ladder, because it represents a kind of ladder, by means of which the voice rises to acute, or sinks to grave; each of the six syllables being as it were one step of the ladder.</p><p><hi rend="smallcaps">Scale is</hi> also used for a series of sounds rising or falling towards acuteness or gravity, from any given pitch of tune, to the greatest distance that is sit or practicable, through such intermediate degrees as to make the succession most agreeable and perfect, and in which we have all the harmonical intervals most commodiously divided.</p><p>The scale is otherwise called an <hi rend="italics">universal system,</hi> as including all the particular systems belonging to music. See <hi rend="smallcaps">System.</hi></p><p>There were three different Scales in use among the Ancients, which had their denominations from the three several sorts of music, viz, the <hi rend="italics">diatonic, chromatic,</hi> and <hi rend="italics">inharmonic.</hi> Which see.</p></div2></div1><div1 part="N" n="SCALENE" org="uniform" sample="complete" type="entry"><head>SCALENE</head><p>, or <hi rend="smallcaps">Scalenous</hi> <hi rend="italics">triangle,</hi> is a triangle whose sides and angles are all unequal.—A cylinder or cone, whose axis is oblique or inclined to its base, is also said to be scalenous: though more frequently it is called oblique. <cb/></p></div1><div1 part="N" n="SCALIGER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SCALIGER</surname> (<foreName full="yes"><hi rend="smallcaps">Joseph Justus</hi></foreName>)</persName></head><p>, a celebrated French chronologer and critic, was the son of Julius Cæfar Scaliger, and born at Agen in France, in 1540. He studied in the college of Bourdeaux; after which his father took him under his own care, and employed him in transcribing his poems; by which means he obtained such a taste for poetry, that before he was 17 years old, he wrote a tragedy upon the subject of Oedipus, in which he introduced all the poetical ornaments of style and sentiment.</p><p>His father dying in 1558, he went to Paris the year following, with a design to apply himself to the Greek tongue. For this purpose he for two months attended the lectures of Turnebus; but finding that in the usual course he should be a long time in gaining his point, he shut himself up in his closet, and by constant application for two years gained a perfect knowledge of the Greek language. After which he applied himself to the Hebrew, which he learned by himself with great facility. And in like manner he ran through many other languages, till he could speak it is said no less than 13 ancient and modern ones. He made no less progress in the sciences; and his writings procured him the reputation of one of the greatest men of that or any other age. He embraced the reformed religion at 22 years of age. In 1563, he attached himself to Lewis Casteignier de la Roch Pazay, whom he attended in several journies. And, in 1593, the curators of the university of Leyden invited him to an honorary professorship in that university, where he lived 16 years, and where he died of a dropsy in 1609, at 69 years of age.</p><p>Scaliger was a man of great temperance; was never married; and was so close a student, that he often spent whole days in his study without eating: and though his circumstances were always very narrow, he constantly refused the presents that were offered him.</p><p>He was author of many ingenious works on various subjects. His elaborate work, <hi rend="italics">De Emendatione Temporum;</hi> his exquisite animadversions on Eusebius; with his <hi rend="italics">Canon Isagogicus Chronologiæ;</hi> and his accurate comment upon Manilius's <hi rend="italics">Astronom con,</hi> sufficiently evince his knowledge in the astronomy, and other branches of learning, among the Ancients, and who, according to the opinion of the celebrated Vieta, was far superior to any of that age. And he had no less a character given him by the learned Casaubon.—He wrote <hi rend="italics">Cyclometrica et Diatriba de Equinoctiorum Anticipatione.</hi> He wrote also notes upon Seneca, Varro, and Ausonius's Poems. But that which above all things renders the name of Scaliger memorable to posterity, is the invention of the Julian period, which consists of 7980 years, being the continued product of the three cycles, of the sun 28, the moon 19, and Roman indiction 15. This period had its beginning fixed to the 764th year before the creation, and is not yet completed, and comprehends all other cycles, periods, and epochas, with the times of all memorable actions and histories.—The collections intitled <hi rend="italics">Scaligeriana,</hi> were collected from his conversations by one of his friends; and being ranged in alphabetical order, were published by Isaac Vossius.</p></div1><div1 part="N" n="SCANTLING" org="uniform" sample="complete" type="entry"><head>SCANTLING</head><p>, a measure, size, or standard, by which the dimensions &c of things are to be determined. <pb n="419"/><cb/> The term is particularly applied to the dimensions of any piece of timber, with regard to its breadth and thickness.</p></div1><div1 part="N" n="SCAPEMENT" org="uniform" sample="complete" type="entry"><head>SCAPEMENT</head><p>, in Clock-work, a general term for the manner of communicating the impulse of the wheels to the pendulum. The ordinary Scapements consist of the swing-wheel and pallets only; but modern improvements have added other levers or detents, chiefly for the purposes of diminishing friction, or for detaching the pendulum from the pressure of the wheels during part of the time of its vibration. Notwithstanding the very great importance of the Scapement to the performance of clocks, no material improvement was made in it from the first application of the pendulum to clocks to the days of Mr. George Graham; nothing more was attempted before his time, than to apply the impulse of the swing-wheel. in such manner as was attended with the least friction, and would give the greatest motion to the pendulum. Dr. Halley discovered, by some experiments made at the Royal Observatory at Greenwich, that by adding more weight to the pendulum, it was made to vibrate larger arcs, and the clock went faster; by diminishing the weight of the pendulum, the vibrations became shorter, and the clock went slower; the result of these experiments being diametrically opposite to what ought to be expected from the theory of the pendulum, probably first roused the attention of Mr. Graham, and led him to such farther trials as convinced him, that this seeming paradox was occasioned by the retrograde motion, which was given to the swing-wheel by every construction of Scapement that was at that time in use; and his great sagacity soon produced a remedy for this defect, by constructing a Scapement which prevented all recoil of the wheels, and restored to the clock pendulum, wholly in theory, and nearly in practice, all its natural properties in its detached simple state; this Scapement was named by its celebrated inventor the <hi rend="italics">dead beat,</hi> and its great superiority was so universally acknowledged, that it was soon introduced into general use, and still continues in universal esteem. The importance of the Scapement to the accurate going of clocks, was by this improvement rendered so unquestionable, that artists of the first rate all over Europe, were forward in producing each his particular construction, as may be seen in the works of Thiout l'ainé, M. J. A. Lepante, M. le Roy, M. Ferdinand Bertoud, and Mr. Cummings' Elements of Clock and Watchwork, in which we have a minute description of several new and ingenious constructions of Scapements, with an investigation of the principles on which their claim to merit is founded; and a comparative view of the advantages or defects of the several constructions. Besides the Scapements described in the above works, many curious constructions have been produced by eminent artists, who have not published any account of them, nor of the motives which have induced each to prefer his favourite construction: Mr. Harrison, Mr. Hindley of York, Mr. Ellicot, Mr. Mudge, Mr. Arnold, Mr. Whitehurst, and many other ingenious artists of this country, have made Scapements of new and pecnliar constructions, of which we are unable, for the above reason, to give any farther account than that those of Mr. Harrison and Mr. Hindley had scarce <cb/> any friction, with a certain mode and quantity of recoil; those of all the other gentlemen, we believe, have been on the principle of the dead beat, with such other improvements as they severally judged most conducive to a good performance.</p><p>Count Bruhl has just published (in 1794) a small pamphlet, “On the Investigation of Astronomical Circles,” to which he has annexed, “a Description of the Scapement in Mr. Mudge's first Timekeeper, drawn up in August 1771.” Before entering upon the Description, the Count premises a few observations, in one of which he recognizes a hint concerning the nature of Mr. Mudge's Scapement, thrown out by this artist in a small tract printed by him in the year 1763, which is this: “The force derived from the mainspring should be made as equal as possible, by making the mainspring wind up another smaller spring at a less distance from the balance, at short intervals of time. <hi rend="italics">I think it would not be impracticable to make it wind up at every vibration, a small spring similar to the pendulum spring, that should immediately act on the balance, by which the whole force acting on the balance would be reduced to the greatest simplicity, with this advantage, that the force would increase in proportion to the arch.</hi>” From this hint, Count Bruhl is surprised that no other artists have taken up Mr. Mudge's invention. He then gives the Description of that invention as follows: “Mr. Mudge's Timekeeper has five wheels, with numbers high enough to admit pinions of twelve, and yet to go eight days. The Scapement consists of a wheel almost like that of a common crown wheel, and acts on pallets, each of which has a separate axis lying in the same line. To each pallet a spring is fixed in the shape of a pendulum spring; these springs are would up alternately by the action of the last wheel upon the pallets, which is performed in the following manner: —Whenever one of the pallets (for instance the upper one) is set in motion by a tooth of the wheel sliding upon it, and then resting against a hook, or, rather a bearing at its end, the balance is entirely detached from it, being then employed in carrying the other pallet the contrary way. When the balance returns from that vibration (partly by the force of the pendulum spring, and partly by that of one of the two small springs which it had bent by the motion of that pallet which it had carried along with itself) it lays hold of the upper pallet and carries it round in the same manner as it did before the lower one, and, of course, in the same direction which the upper pallet had received from the power of the mainspring at the time that it was quite unconnected with the balance. The communication of motion from the balance to the pallets, and vice versa, is effected by two pins fixed to a crank, which in following the balance, hit each its proper pallet alternately. By what has been said, it is evident that whatever inequality there may be in the power derived from the mainspring (provided the latter be sufficient to wind up those little pallet springs) it can never interfere with the regularity of the balance's motion, but at the instant of unlocking the pallets, which is so instantaneous an operation, and the resistance so exceedingly small, that it cannot possibly amount to any sensible error. The removal of this great obstacle was certainly never so effectually done by any other contri- <pb n="420"/><cb/> vance, and deserves the highest commendation, as a probable means to perfect a portable machine that will measure time correctly. But this is not the only, nor indeed the principal advantage which this timekeeper will possess over any other; for, as it is impossible to reduce friction to so small a quantity as not to affect the motion of a balance, the consequence of which is, that it describes sometimes greater and sometimes smaller arcs, it became necessary to think of some method by which the balance might be brought to describe those different arcs in the same time. If a balance could be made to vibrate without friction or resistance from the medium in which it moves, the mere expanding and contracting of the pendulum spring, would probably produce the so much wished-for effect, as its force is supposed to be proportional to the arcs described; but as there is no machine void of friction, and as from that cause, the velocity of every balance decreases more rapidly than the spaces gone through decrease, this inequality could only be removed by a force acting on the balance, which assuming different ratios in its different stages, could counterbalance that inequality. This very material and important remedy, Mr. Mudge has effected by the construction of his Scapement; for his pallet springs having a force capable of being increased almost at pleasure, at the commencement of every vibration, the proportion in their different degrees of tension may be altered till it answers the intended purpose. This shews how effectually Mr. Mudge's Scapement removes the two greatest difficulties that have hitherto baffled the attempts of every other artist, namely, the inequalities of the power derived from the main spring, and the irregularities arising from friction, and the variable resistance of the medium in which the balance moves. Although at the time I am writing this account of his invention, the machine is not yet finished; I am not the less confident that whenever it is, it will be found to be one of the most useful of any which has as yet appeared.”</p></div1><div1 part="N" n="SCARP" org="uniform" sample="complete" type="entry"><head>SCARP</head><p>, in Fortification, the interior slope of the ditch of a place; that is, the slope of that side of a ditch which is next to the place, or on the outside of the rampart at its foot, facing the champaign or open country. The slope on the outer side of the ditch is called the <hi rend="italics">counterscarp.</hi></p></div1><div1 part="N" n="SCENOGRAPHY" org="uniform" sample="complete" type="entry"><head>SCENOGRAPHY</head><p>, in Perspective, the perspective representation of a body on a plane; or a description and view of it in all its parts and dimensions, such as it appears to the eye in any oblique view.</p><p>This differs essentially from the ichnography and the orthography. The ichnography of a building, &c, represents the plan or ground work of the building, or section parallel to it; and the orthography the elevation, or front, or one side, also in its natural dimensions; but the Scenography exhibits the whole of the building that appears to the eye, front, sides, height, and all, not in their real dimensions or extent, but raised on the geometrical plan in perspective.</p><p>In architecture and fortification, Scenography is the manner of delineating the several parts of a building or fortress, as they are represented in perspective.</p><p><hi rend="italics">To exhibit the</hi> <hi rend="smallcaps">Scenography</hi> <hi rend="italics">of any body.</hi> 1. Lay down the basis, ground-plot, or plan, of the body, in the perspective ichnography, that is, draw the perspec- <cb/> tive appearance of the plan or basement, by the proper rules of perspective. 2. Upon the several points of the said perspective plan, raise the perspective heights, and connect the tops of them by the proper slope or oblique lines. So will the Scenography of the body be completed, when a proper shade is added. See <hi rend="smallcaps">Perspective.</hi></p></div1><div1 part="N" n="SCHEINER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SCHEINER</surname> (<foreName full="yes"><hi rend="smallcaps">Christopher</hi></foreName>)</persName></head><p>, a considerable German mathematician and astronomer, was born at Mundeilheim in Schwaben in 1575. He entered into the society of the Jesuits at 20 years of age; and afterwards taught the Hebrew tongue and the mathematics at Ingolstadt, Friburg, Brisac, and Rome. At length he became confessor to the archduke Charles, and rector of the college of the Jesuits at Neisse in Silesia, where he died in 1650, at 75 years of age.</p><p>Scheiner was chiefly remarkable for being one of the first who observed the spots in the sun with the telescope, though not the very first; for his observations of those spots were first made, at Ingolstadt, in the latter part of the year 1611, whereas Galileo and Harriot both observed them in the latter part of the year before, or 1610. Scheiner continued his observations on the solar phenomena for many years afterwards at Rome, with great assiduity and accuracy, constantly making drawings of them on paper, describing their places, figures, magnitude, revolutions and periods, so that Riccioli delivered it as his opinion that there was little reason to hope for any better observations of those spots. Des Cartes and Hevelius also say, that in their judgment, nothing can be expected of that kind more satisfactory. These observations were published in one volume folio, 1630, under the title of <hi rend="italics">Rosa Ursina,</hi> &c; almost every page of which is adorned with an image of the sun with the spots. He wrote also several smaller pieces relating to mathematics and philosophy, the principal of which are,</p><p>2. <hi rend="italics">Oculus, five Fundamentum Opticum, &c;</hi> which was reprinted at London, in 1652, in 4to.</p><p>3. <hi rend="italics">Sol Eclipticus, Disquisitiones Mathematicæ.</hi></p><p>4. <hi rend="italics">De Controversiis et Novitatibus Astronomicis.</hi></p></div1><div1 part="N" n="SCHEME" org="uniform" sample="complete" type="entry"><head>SCHEME</head><p>, a draught or representation of any geometrical or astronomical figure, or problem, by lines sensible to the eye; or of the celestial bodies in their proper places for any moment; otherwise called a diagram.</p><p><hi rend="smallcaps">Scheme</hi> <hi rend="italics">Arches.</hi> See <hi rend="smallcaps">Arch.</hi></p></div1><div1 part="N" n="SCHOLIUM" org="uniform" sample="complete" type="entry"><head>SCHOLIUM</head><p>, a note, remark, or annotation, occasionally made on some passage, proposition, or the like.</p><p>The term is much used in geometry, and other parts of the mathematics; where, after demonstrating a proposition, it is used to point out how it might be done some other way; or to give some advice or precaution, in order to prevent mistakes; or to add some particular use or application of it.</p><p>Wolfius has given abundance of curious and useful arts and methods, and a good part of the modern philosophy, with the description of mathematical instruments, &c; all by way of Scholia to the respective propositions in his Elementa Matheseos.</p></div1><div1 part="N" n="SCHONER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SCHONER</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a noted German philosopher and mathematician, was born at Carolostadt in the year 1477, and died in 1547, at 70 years of age.—His early propensity to those sciences may be deemed a just prognostication of the great progress <pb n="421"/><cb/> which he afterwards made in them. So that from his uncommon acquirements, he was chosen mathematical professor at Nuremburg when he was but a young man. He wrote a great many works, and was particularly famous for his astronomical tables, which he published after the manner of those of Regiomontanus, and to which he gave the title of <hi rend="italics">Resolutæ,</hi> on account of their clearness. But notwithstanding his great knowledge, he was, after the fashion of the times, much addicted to judicial astrology, which he took great pains to improve. The list of his writings is chiefly as follows:</p><p>1. Three Books of Judicial Astrology.</p><p>2. The Astronomical Tables named <hi rend="italics">Resolutæ.</hi></p><p>3. <hi rend="italics">De Usu Globi Stellif<*>ri; De Compositione Globi Cœlestis; De Usu Globi Terrestris, et de Compositione ejusdem.</hi></p><p>4. <hi rend="italics">Æquatorium Astronomicum.</hi></p><p>5. <hi rend="italics">Libellus de Distantiis Locorum per Instrumentum et Numeros Investigandis.</hi></p><p>6. <hi rend="italics">De Compositione Torqueti.</hi></p><p>7. <hi rend="italics">In Constructionem et Usum Rectanguli sive Radii Astronomici Annotationes.</hi></p><p>8. <hi rend="italics">Horarii Cylindri Canones.</hi></p><p>9. <hi rend="italics">Planisphærium, seu Meteoriscopium.</hi></p><p>10. <hi rend="italics">Organum Uranicum.</hi></p><p>11. <hi rend="italics">Instrumentum Impedimentorum Lunæ.</hi></p><p>All printed at Nuremburg, in folio, 1551.</p><p>Of these, the large treatise of dialling rendered him more known in the learned world than all his other works besides; in which he discovers a surprising genius and fund of learning of that kind.</p></div1><div1 part="N" n="SCHOOL" org="uniform" sample="complete" type="entry"><head>SCHOOL</head><p>, a place where the languages, humanities, or arts and sciences, &c, are taught.</p><p><hi rend="smallcaps">School</hi> is also used for a whole faculty, university, or sect; as Plato's school, the school of Epicurus, the school of Paris, &c.—The school of Tiberias was famous among the ancient Jews; and it is to this we owe the Massora, and Massoretes.</p><p><hi rend="smallcaps">School</hi> <hi rend="italics">Philosophy,</hi> &c. the same with <hi rend="italics">scholastic.</hi></p></div1><div1 part="N" n="SCIAGRAPHY" org="uniform" sample="complete" type="entry"><head>SCIAGRAPHY</head><p>, or <hi rend="smallcaps">Sciography</hi>, the profile or vertical section of a building; used to shew the inside of it.</p><div2 part="N" n="Sciagraphy" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sciagraphy</hi></head><p>, in Astronomy &c, is a term used by some authors for the art of finding the hour of the day or night, by the shadow of the sun, moon, stars, &c. See <hi rend="smallcaps">Dial.</hi></p></div2></div1><div1 part="N" n="SCIENCE" org="uniform" sample="complete" type="entry"><head>SCIENCE</head><p>, a clear and certain knowledge of any thing, founded on demonstration, or on self-evident principles. — In this sense, <hi rend="italics">doubting</hi> is opposed to science; and <hi rend="italics">opinion</hi> is the middle between the two.</p><p><hi rend="smallcaps">Science</hi> is more particularly used for a formed system of any branch of knowledge, comprehending the doctrine, reason, and theory of the thing, without any immediate application of it to any uses or offices of lise. And in this sense, the word is used in opposition to <hi rend="italics">art.</hi></p><p>Science may be divided into these three sorts: First, the knowledge of things, their constitutions, properties, and operations, whether material or immaterial. And this, in a little more enlarged sense of the word, may be called physics, or natural philosophy. Secondly, the skill of rightly applying our own powers and actions for the attainment of good and useful things, as <hi rend="italics">Ethics.</hi> Thirdly, the doctrine of signs; as words, logic, &c. <cb/></p></div1><div1 part="N" n="SCIENTIFIC" org="uniform" sample="complete" type="entry"><head>SCIENTIFIC</head><p>, or <hi rend="smallcaps">Scientifical</hi>, something relating to the pure and sublimer sciences; or that abounds in science, or knowledge.</p><p>A work, or method, &c, is said to be scientisical, when it is founded on the pure reason of things. or conducted wholly on the principles of them. In which sense the word stands opposed to narrative, arbitrary, opinionative, positive, tentative, &c.</p></div1><div1 part="N" n="SCIOPTIC" org="uniform" sample="complete" type="entry"><head>SCIOPTIC</head><p>, or <hi rend="smallcaps">Scioptric</hi> <hi rend="italics">Ball,</hi> a sphere or globe of wood, with a circular hole or perforation, where a lens is placed. It is so fitted that, like the eye of an animal, it may be turned round every way, to be used in making experiments of the darkened room.</p><p>SCIOPTRICS. See <hi rend="smallcaps">Camera Obscura.</hi></p><p>SCIOTHERICUM <hi rend="italics">Telescopium,</hi> is an horizontal dial, adapted with a telescope for observing the true time both by day and night, to regulate and adjust pendulum clocks, watches, and other time-keepers. It was invented by Mr. Molyneux, who published a book with this title, which contains an accurate description of this instrument, with all its uses and applications.</p></div1><div1 part="N" n="SCLEROTICA" org="uniform" sample="complete" type="entry"><head>SCLEROTICA</head><p>, one of the common membranes of the eye, on its hinder part. It is a large, thick, firm, hard, opaque membrane, extended from the external circumference of the cornea to the optic nerve, and forms much the greater part of the external globe of the eye. The Sclerotica and the cornea compose the case in which all the internal coats of the eye and its humours are contained.</p></div1><div1 part="N" n="SCONCES" org="uniform" sample="complete" type="entry"><head>SCONCES</head><p>, small forts, built for the defence of some pass, river, or other place. Some Sconces are made regular, of four, five, or six bastions; others are of smaller dimensions, fit for passes, or rivers; and others for the field.</p></div1><div1 part="N" n="SCORE" org="uniform" sample="complete" type="entry"><head>SCORE</head><p>, in Music, denotes partition, or the original draught of the whole composition, in which the several parts, viz the treble, second treble, bass, &c. are distinctly scored, and marked.</p></div1><div1 part="N" n="SCORPIO" org="uniform" sample="complete" type="entry"><head>SCORPIO</head><p>, the <hi rend="italics">Scorpion,</hi> the 8th sign of the zodiac, denoted by the character <figure/>, being a rude design of the animal of that name.</p><p>The Greeks, who would be supposed the inventors of astronomy, and who have, with that intent, fathered some story or other of their own upon every one of the constellations, give a very singular account of the origin of this one. They tell us that this is the creature which killed Orion. The story goes, that the famous hunter of that name boasted to Diana and Latona, that he would destroy every animal that was upon the earth; the earth, they say, enraged at this, sent forth the poisonous reptile the Scorpion, which insignificant creature stung him, that he died. Jupiter, they say, raised the Scorpion to the heavens, giving him this place among the constellations; and that afterwards Diana requested of him to do the same honour to Orion, which he at last consented to, but placed him in such a situation, that when the Scorpion rises, he sets.</p><p>But the Egyptians, or whatever early nation it was that framed the zodiac, probably placed this poisonous reptile in that part of the heavens to denote that when the sun arrived at it, fevers and sicknesses, the maladies of autumn, would begin to rage. This they represented by an animal whose sting was of power to occasion some <pb n="422"/><cb/> of them; and it was thus they formed all the constellations.</p><p>The ancients allotted one of the twelve principal among their deities to be the guardian for each of the 12 signs of the zodiac. The Scorpion, as their history of it made it a fierce and fatal animal that had killed the great Orion, fell naturally to the protection of the god of war; Mars is therefore its tutelary deity; and to this single circumstance is owing all that jargon of the astrologers, who tell us that there is a great analogy between the planet Mars and the constellation Scorpio. To this also is owing the doctrine of the alchymists, that iron, which they call Mars, is also under the dominion of the same constellation, and that the transmutation of that metal into gold can only be performed when the sun is in this sign.</p><p>The stars in Scorpio, in Ptolomy's catalogue, are 24; in that of Tycho 10, in that of Hevelius 20, but in that of Flamsteed and Sharp 44.</p><p><hi rend="smallcaps">Scorpion</hi> is also the name of an ancient military engine, used chiefly in the defence of walls, &c.</p><p>Marcellinus describes the Scorpion, as consisting of two beams bound together by ropes. From the middle of the two, rose a third beam, so disposed, as to be pulled up and let down at pleasure; and on the top of this were fastened iron hooks, where a sling was hung, either of iron or hemp; and under the third beam lay a piece of hair-cloth full of chaff, tied with cords. It had its name Scorpio, because when the long beam or tiller was erected, it had a sharp top in manner of a sting.</p><p>To use the engine, a round stone was put into the sling, and four persons on each side, loosening the beams bound by the ropes, drew back the erect beam to the hook; then the engineer, standing on an eminence, gave a stroke with a hammer on the chord to which the beam was fastened with its hook, which set it at liberty; so that hitting against the soft hair-cloth, it struck out the stone with a great force.</p></div1><div1 part="N" n="SCOTIA" org="uniform" sample="complete" type="entry"><head>SCOTIA</head><p>, in Architecture, a semicircular cavity or channel between the tores, in the bases of columns; and sometimes under the larmicr or drip, in the cornice of the Doric order. The workmen often call it the Casement, and it is also otherwise called the Trochilus.</p></div1><div1 part="N" n="SCREW" org="uniform" sample="complete" type="entry"><head>SCREW</head><p>, or <hi rend="smallcaps">Scrue</hi>, one of the six mechanical powers; chiefly used in pressing or squeezing bodies close, though sometimes also in raising weights.</p><p>The Screw is a spiral thread or groove cut round a cylinder, and everywhere making the same angle with the length of it. So that, if the surface of the cylinder, with this spiral thread upon it, were unfolded and stretched into a plane, the spiral thread would form a straight inclined plane, whose length would be to its height, as the circumference of the cylinder is to the distance between two threads of the Screw; as is evident by considering, that in making one round, the spiral rises along the cylinder the distance between the two threads.</p><p>Hence the threads of a Screw may be traced upon the smooth surface of a cylinder thus: Cut a sheet of paper into the form of a right-angled triangle, having its base to its height in the above proportion, viz, as the circumference of the cylinder of the Screw is to the intended distance between two threads; then wrap this <cb/> paper triangle about the cylinder, and the hypothenuse of it will trace out the line of the spiral thread.</p><p>When the spiral thread is upon the outside of a cylinder, the Screw is said to be a <hi rend="italics">male</hi> one. But if the thread be cut along the inner surface of a hollow cylinder, or a round perforation, it is said to be <hi rend="italics">female.</hi> And this latter is also sometimes called the <hi rend="italics">box</hi> or <hi rend="italics">nut.</hi></p><p>When motion is to be given to something, the male and female Screw are necessarily conjoined; that is, whenever the screw is to be used as a simple engine, or mechanical power. But when joined with an axis in peritrochio, there is no occasion for a female; but in that case it becomes part of a compound engine.</p><p>The Screw cannot properly be called a simple machine, because it is never used without the application of a lever, or winch, to assist in turning it. <hi rend="center"><hi rend="italics">Of the Force and Power of the Screw.</hi></hi></p><p>1. The force of a power applied to turn a Screw round, is to the force with which it presses upwards or downwards, setting aside the friction, as the distance between two threads is to the circumference where the power is applied.</p><p>For, the Screw being only an inclined plane, or half wedge, whose height is the distance between two threads, and its base the said circumference; and the force in the horizontal direction being to that in the vertical one as the lines perpendicular to them, viz, as the height of the plane, or distance of the two threads, is to the base of the plane, or circumference at the place where the power is applied; therefore the power is to the pressure, as the distance of two threads, is to that circumference.</p><p>2. Hence, when the Screw is put in motion; then the power is to the weight which would keep it in equilibrio, as the velocity of the latter is to that of the former. And hence their two momenta are equal, which are produced by multiplying each weight or power by its own velocity. Two different forms of Screw presses, are as below. <figure/></p><p>3. Hence we can easily compute the force of any machine turned by a Screw. Let the annexed figure represent a press driven by a Screw, whose threads are each a quarter of an inch asunder; and let the Screw be turned by a handle of 4 feet long from C to D; then if the natural force of a man, by which he can lift, or <pb n="423"/><cb/> pull, or draw, be 150 pounds; and it be required to determine with what force the Screw will press on the board, when the man turns the handle at C and D with his whole force. The diameter CD of the power being 4 feet, or 48 inches, its circumference is 48 X 3.1416 or 150<hi rend="sup">4</hi> nearly; and the distance of the threads being 1/4 of an inch; therefore the power is to the pressure, as 1/4 to 150 4/5 or as 1 to 603 1/5: but the power is equal to 150lb; therefore as 1 : 603 1/5 :: 150 : 90,480; and consequently the pressure at the bottom of the Screw, is equal to a weight of 90,480 pounds, independent of friction.</p><p>But the power has to overcome, not only the weight, or other resistance, but also the friction of the Screw, which in this machine is very great, in some cases equal to the weight itself, since it is sometimes sufficient to fustain the weight, when the power is taken off.</p><p>Mr. Hunter has described a new method of applying the Screw with advantage in particular cases, in the Philos. Trans. vol. 71, pa. 58 &c.</p><p><hi rend="italics">The Endless</hi> <hi rend="smallcaps">Screw</hi>, or <hi rend="italics">Perpetual</hi> <hi rend="smallcaps">Screw</hi>, is one which works in, and turns, a dented wheel DF, without a concave or female Screw; being so called because it may be turned for ever, without coming to an end. From the following schemes it is evident, that while the Screw turns once round, the wheel only advances the distance of one tooth. <figure/></p><p>1. If the power applied to the lever, or handle, of an endless Screw AB, be to the weight, in a ratio compounded of the periphery of the axis of the wheel EH, to the periphery described by the power in turning the handle, and of the revolutions of the wheel DF to the revolutions of the Screw CB, the power will balance the weight. Hence,</p><p>2. As the motion of the wheel is very slow, a small power may raise a very great weight, by means of an endless Screw. And therefore the chief use of such a Screw is, either where a great weight is to be raised through a little space; or where only a slow gentle motion is wanted. For which reason it is very useful in clocks and watches.</p><p>3. Having given the number of teeth, the distance of the power from the centre of the Screw B, the radius of the axis HE, and the power; to find the weight it will raise. Multiply the distance of the power from the centre of the Screw AB, by the number of the teeth, and the product will be the space passed through by the power, while the weight passes through a space equal to the periphery of the axis: then say, as the radius of <cb/> the axis is to the space of the power just found, so is the power to a 4th proportional, which will be the weight the power is able to sustain. Thus, if AB = 3, the radius of the axis HE = 1, the power 150 pounds, and the number of teeth of the wheel DF 48; then the weight will be found = 21,600 = 3 X 150 X 48. Whence it appears that the endless Screw exceeds all others in increasing the force of a power.</p><p>4. A machine for shewing the power of the Screw may be contrived in the following manner. Let the wheel C (last fig.) have a Screw <hi rend="italics">a b</hi> on its axis, working in the teeth of the wheel D, which suppose to be 48 in number. It is plain that for every revolution of the wheel C, and Screw <hi rend="italics">ab,</hi> by the winch A, the wheel D will be moved one tooth by the Screw; and therefore in 48 revolutions of the winch, the wheel D will be turned once round. Then if the circumference of a circle, described by the handle of the winch, be equal to the circumference of a groove <hi rend="italics">e</hi> round the wheel D, the velocity of the handle will be 48 times as great as the velocity of any given point in the groove. Consequently when a line G goes round the groove <hi rend="italics">e,</hi> and has a weight of 48lb hung to it below the pedestal EF, a power equal to one pound at the handle will balance and support the weight.</p><p><hi rend="italics">Archimedes's</hi> <hi rend="smallcaps">Screw</hi>, is a spiral pump, being a machine for raising water, first invented by Archimedes.</p><p>Its structure and use will be understood by the following description of it.</p><p>ABCD (Pl. xxiii, fig. 6) is a wheel, which is turned round, according to the order of those letters, by the fall of water EF, which need not be more than 3 feet. The axis G of the wheel is raised so as to make an angle of about 44° with the horizon; and on the top of that axle is a wheel H, which turns such another wheel I of the same number of teeth; the axle K of this last wheel being parallel to the axle G of the two former wheels. The axle G is cut into a double threaded Screw, as in the annexed figure (fig. 7), exactly resembling the Screw on the axis of the fly of a common jack, which must be what is called a right-handed Screw; if the first wheel turns in the direction ABCD; but a left-handed Screw, if the stream turns the wheel the contrary way; and the Screw on the axle G must be cut in a contrary way to that on the axle K, because these axles turn in contrary directions. These Screws must be covered close over with boards, like these of a cylindrical cask; and then they will be spiral tubes. Or they may be made of tubes or pipes of lead, and wrapt round the axles in shallow grooves cut in it, like the figure 8. The lower end of the axle G turns constantly in the stream that turns the wheel, and the lower ends of the spiral tubes are open into the water. So that, as the wheel and axle are turned round, the water rises in the spiral tubes, and runs out at L through the holes M, N, as they come about below the axle. These holes, of which there may be any number, as 4 or 6, are in a broad close ring on the top of the axle, into which ring the water is delivered from the upper open ends of the Screw tubes, and falls into the open box N. The lower end of the axle K turns on a gudgeon in the water in N; and the spiral tubes in that axle take up the water from N, and deliver it into another such box under the top of K; on which there may be such another <pb n="424"/><cb/> wheel as I, to turn a third axle by such a wheel upon it. And in this manner may water be raised to any proposed height, when there is a stream sufficient for that purpose to act on the broad float boards of the first wheel. Archimedes's Screw, or a still simpler form of it, is also represented in fig. 9.</p></div1><div1 part="N" n="SCROLLS" org="uniform" sample="complete" type="entry"><head>SCROLLS</head><p>, or <hi rend="smallcaps">Scrowls</hi>, or <hi rend="italics">Volutes,</hi> a term in Architecture. See <hi rend="smallcaps">Volutes.</hi></p><p>SCRUE. See <hi rend="smallcaps">Screw.</hi></p></div1><div1 part="N" n="SCRUPLE" org="uniform" sample="complete" type="entry"><head>SCRUPLE</head><p>, the least of the weights used by the ancients. Among the Romans it was the 24th part of an ounce, or the third part of a drachm.</p><p><hi rend="smallcaps">Scruple</hi> is still a small weight among us, equal to 20 grains, or the 3d part of a drachm. Among goldsmiths the scruple is 24 grains.</p><div2 part="N" n="Scruple" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Scruple</hi></head><p>, in Chronology, a small portion of time much used by the Chaldeans, Jews, Arabs, and other eastern people, in computations of time. It is the 1080th part of an hour, and by the Hebrews called <hi rend="italics">belakin.</hi></p></div2><div2 part="N" n="Scruples" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Scruples</hi></head><p>, in Astronomy. As</p><p><hi rend="smallcaps">Scruples</hi> <hi rend="italics">Eclipsed,</hi> denote that part of the moon's diameter which enters the shadow, expressed in the same measure in which the apparent diameter of the moon is expressed See <hi rend="smallcaps">Digit.</hi></p><p><hi rend="smallcaps">Scruples</hi> <hi rend="italics">of Half Duration,</hi> an arch of the moon's orbit, which the moon's centre describes from the beginning of an eclipse to its middle.</p><p><hi rend="smallcaps">Scruples</hi> <hi rend="italics">of Immersion,</hi> or <hi rend="italics">Incidence,</hi> an arch of the moon's orbit, which her centre describes from the beginning of the eclipse, to the time when the centre falls into the shadow. See <hi rend="smallcaps">Immersion.</hi></p><p><hi rend="smallcaps">Scruples</hi> <hi rend="italics">of Emersion,</hi> an arch of the moon's orbit, which her centre describes in the time from the first emersion of the moon's limb, to the end of the eclipse.</p></div2></div1><div1 part="N" n="SCYTALA" org="uniform" sample="complete" type="entry"><head>SCYTALA</head><p>, in Mechanics, a term which some writers use for a kind of radius, or spoke, standing out from the axis of a machine, as a lever or handle, to turn it round, and work it by.</p></div1><div1 part="N" n="SEA" org="uniform" sample="complete" type="entry"><head>SEA</head><p>, in Geography, is frequently used for that vast tract of water encompassing the whole earth, more properly called ocean. But</p><p><hi rend="smallcaps">Sea</hi> is more properly used for a particular part or division of the ocean, denominated from the countries it washes, or from other circumstances. Thus we say, the Irish sea, the Mediterranean sea, the Baltic sea, the Red sea, &c.</p><p><hi rend="smallcaps">Sea</hi> among sailors is variously applied, to a single wave, or to the agitation produced by a multitude of waves in a tempest, or to their particular progress and direction. Thus they say, a heavy sea broke over our quarter, or we shipped a heavy sea; there is a great sea in the offing; the sea sets to the southward. Hence a ship is said to head the sea, when her course is opposed to the setting or direction of the surges. A <hi rend="italics">Long Sea</hi> implies a steady and uniform motion of long and extensive waves. On the contrary, a <hi rend="italics">Short Sea</hi> is when they run irregularly, broken, and interrupted, so as frequently to burst over a vessel's side or quarter. <hi rend="center"><hi rend="italics">Properties and Affections of the</hi> <hi rend="smallcaps">Sea.</hi></hi></p><p>1. <hi rend="italics">General Motion of the Sea.</hi> M. Dassie of Paris, in a work long since published, has been at great pains <cb/> to prove that the Sea has a general motion, independent of winds and tides, and of more consequence in navigation than is usually supposed. He affirms that this motion is from east to west, inclining toward the north when the sun is on the north side of the equinoctial, but toward the south when he is on the south side of it. Philos. Trans. No. 135.</p><p>2. <hi rend="italics">Bason or Bottom of the</hi> <hi rend="smallcaps">Sea</hi>, or <hi rend="italics">Fundus Maris,</hi> a term used to express the bed or bottom of the sea in general. Mr. Boyle has published a treatise on this subject, in which he has given an account of its irregularities and various depths founded on the observa tions communicated to him by mariners.</p><p>Count Marsigli has, since his time, given a much fuller account of this part of the globe. The materials which compose the bottom of the Sea, may reasonably be supposed, in some degree, to influence the taste of its waters; and this anchor has made many experiments to prove that fossil coal, and other bituminous substances, which are found in plenty at the bottom of the Sea, may communicate in great part its bitterness to it.</p><p>It is a general rule among sailors, and is found to hold true in many instances, that the more the shores of any place are steep and high, forming perpendicular cliffs, the deeper the Sea is below; and that on the contrary, level shores denote shallow Seas. Thus the deepest part of the Mediterranean is generally allowed to be under the height of Malta. And the observation of the strata of earth and other fossils, on and near the shores, may serve to form a good judgment as to the materials to be found in its bottom. For the veins of salt and of bitumen doubtless run on the same, and in the same order, as we see them at land; and the strata of rocks that serve to support the earth of hills and elevated places on shore, serve also, in the same continued chain, to support the immense quantity of water in the bason of the Sea.</p><p>The coral fisheries have given occasion to observe that there are many, and those very large caverns or hollows in the bottom of the Sea, especially where it is rocky; and that the like caverns are sometimes found in the perpendicular rocks which form the steep sides of those fisheries. These caverns are often of great depth, as well as extent, and have sometimes wide mouths, and sometimes only narrow entrances into large and spacious hollows.</p><p>The bottom of the Sea is covered with a variety of matters, such as could not be imagined by any but those who have examined into it, especially in deep water, where the surface only is disturbed by tides and storms, the lower part, and consequently its bed at the bottom, remaining for ages perhaps undisturbed. The soundings, when the plummet first touches the ground on approaching the shores, give some idea of this. The bottom of the plummet is hollowed, and in that hollow there is placed a lump of tallow; which being the part that first touches the ground, the soft nature of the fat receives into it some part of those substances which it meets with at the bottom: this matter, thus brought up, is sometimes pure sand, sometimes a kind of sa d made of the fragment of shells, beaten to a sort of powder, sometimes it is made of a like powder of the several sorts of corals, and sometimes it is composed <pb n="425"/><cb/> of fragments of rocks; but beside these appearances, which are natural enough, and are what might well be expected, it brings up substances which are of the most beautiful colours. Marsigli Hist. Phys. de la Mer.</p><p>Dr. Donati, in an Italian work, containing an essay towards a natural history of the Adriatic Sea, printed at Venice in 1750, has related many curious observations on this subject, and which confirm the observations of Marsigli: having carefully examined the soil and productions of the various countries that surround the Adriatic Sea, and compared them with those which he took up from the bottom of the Sea, he found that there is very little difference between the former and the latter. At the bottom of the water there are mountains, plains, vallies, and caverns, similar to those upon land. The soil consists of different strata placed one upon another, and mostly parallel and correspondent to those of the rocks, islands, and neighbouring continents. They contain stones of different sorts, minerals, metals, various putrefied bodies, pumice stones, and lavas formed by volcanos.</p><p>One of the objects which most excited his attention, was a crust, which he discovered under the water, composed of crustaceous and testaceous bodies, and beds of polypes of different kinds, confusedly blended with earth, sand, and gravel; the different marine bodies which form this crust, are found at the depth of a foot or more, entirely petrified and reduced into marble; these he supposes are naturally placed under the Sea when it covers them, and not by means of volcanos and earthquakes, as some have conjectured. On this account he imagines that the bottom of the Sea is constantly rising higher and higher, with which other obvious causes of increase concur; and from this rising of the bottom of the Sea, that of its level or surface naturally results; in proof of which this writer recites a great number of facts. Philos. Trans. vol. 49, pa. 585.</p><p>3. <hi rend="italics">Luminousness of the</hi> <hi rend="smallcaps">Sea.</hi> This is a phenomenon that has been noticed by many nautical and philosophical writers. Mr. Boyle ascribes it to some cosmical law or custom of the terrestrial globe, or at least of the planetary vortex.</p><p>Father Bourzes, in his voyage to the Indies, in 1704, took particular notice of this phenomenon, and very minutely describes it, without assigning the true cause.</p><p>The Abbé Nollet was long of opinion, that the light of the Sea proceeded from electricity; and others have had recourse to the same principle, and shewn that the luminous points in the surface of the Sea are produced merely by friction.</p><p>There are however two other hypotheses, which have more generally divided between them the solution of this phenomenon; the one of these ascribes it to the shining of luminous insects or animalcules, and the other to the light proceeding from the putrefaction of animal substances. The Abbé Nollet, who at first considered this luminousness as an electrical phenomenon, having had an opportunity of observing the circumstances of it, when he was at Venice in 1749, relinquished his former opinion, and concluded that it was occasioned either by the luminous aspect, or by <cb/> some liquor or effluvia of an insect which he particularly describes, though he does not altogether exclude other causes, and especially the spawn or fry of fish.</p><p>The same hypothesis had also occurred to M. Vianelli; and both he and Grizellini, a physician in Venice, have given drawings of the insects from which they imagined this light to proceed.</p><p>A similar conjecture is proposed by a correspondent of Dr. Franklin, in a letter read at the Royal Society in 1756; the writer of which apprehends, that this appearance may be caused by a great number of little animals, floating on the surface of the Sea. And Mr. Forster, in his account of a voyage round the world with captain Cook, in the years 1772, 3, 4, and 5, describes this phenomenon as a kind of blaze of the Sea; and, having attentively examined some of the shining water, expresses his conviction that the appearance was occasioned by innumerable minute animals of a round shape, moving through the water in all directions, which show separately as so many luminous sparks when taken up on the hand: he imagines that these small gelatinous luminous specks may be the young fry of certain species of some medusæ, or blubber. And M. Dagelat and M. Rigaud observed several times, and in different parts of the ocean, such luminous appearances by vast masses of different animalcules; and a few days after the Sea was covered, near the coasts, with whole banks of small fish in innumerable multitudes, which they supposed had proceeded from the shining animalcules.</p><p>But M. le Roi, after giving much attention to this phenomenon, concludes that it is not occasioned by any shining insects, especially as, after carefully examining with a microscope some of the luminous points, he found them to have no appearance of an animal; and he also found that the mixture of a little spirit of wine with water just drawn from the Sea, would give the appearance of a great number of little sparks, which would continue visible longer than those in the ocean: the same effect was produced by all the acids, and various other liquors. M. le Roi is far from asserting that there are no luminous insects in the Sea; for he allows that several gentlemen have found them; but he is satisfied that the Sea is luminous chiefly on some other account, though he does not so much as offer a conjecture with respect to the true cause.</p><p>Other authors, equally dissatisfied with the hypothesis of luminous insects, for explaining the phenomenon which is the subject of this article, have ascribed it to some substance of the phosphoric kind, arising from putrefaction. The observations of F. Bourzes, above referred to, render it very probable, that the luminousness of the Sea arises from slimy and other putrescent matter, with which it abounds, though he does not mention the tendency to putrefaction, as a circumstance of any consequence to the appearance. But the experiments of Mr. Canton, which have the advantage of being easily made, seem to leave no room to doubt that the luminousness of the Sea is chiefly owing to putrefaction. And his experiments confirm an observation of Sir John Pringle's, that the quantity of salt contained in Sea-water hastens putrefaction; but since that precise quantity of salt which promotes putre- <pb n="426"/><cb/> faction the most, is less than that which is found in Sea-water, it is probable, Mr. Canton observes, that if the Sea were less salt, it would be more luminous. See Philos. Trans. vol. 59, pa. 446, and Franklin's Exper. and Observ. pa. 274. <hi rend="center"><hi rend="italics">Of the Depth of the Sea, its Surface, &c.</hi></hi></p><p>What proportion the superficies of the Sea bears to that of the land, is not accurately known, though it is said to be somewhat more than two to one. This proportion of the surface of the Sea to the land, has been found by experiment thus: taking the printed paper map or covering of a terrestrial globe, with a pair of scissors clip out the parts that are land, and those that are water; then weighing these parcels separately in a pair of fine scales, the land is found to be near 1/3, and the water rather more than 2/3 of the whole.</p><p>With regard to the profundity or depth of the Sea, Varenius affirms, that it is in some places unfathomable, and in others very various, being in certain places from 1/20th of a mile to 4 1/2 miles in depth, in other places deeper, but much less in bays than in oceans. In general, the depths of the Sea bear a great analogy to the height of mountains on the land, so far as is hitherto discovered.</p><p>There are two special reasons why the Sea does not increase by means of rivers, &c, running every where into it. The first is, because waters return from the Sea by subterranean cavities and aqueducts, through various parts of the earth. Secondly, because the quantity of vapours raised from the Sea, and falling in rain upon the land, only cause a circulation of the water, but no increase of it. It has been found by experiment and calculation, that in a summer's day, there may be raised in vapours from the surface of the Mediterranean Sea, 528 millions of tuns of water; and yet this Sea receiveth not, from all its nine great rivers, above 183 millions of tuns per day, which is but about a third part of what is exhausted in vapours; and this defect in the supply by the rivers, may serve to account for the continual influx of a current by the mouth or straits at Gibraltar. Indeed it is rather probable, that the waters of the Sea suffer a continual slow decrease as to their quantity, by sinking always deeper into the earth, by filtering through the fissures in the strata and component parts.</p></div1><div1 part="N" n="SEASONS" org="uniform" sample="complete" type="entry"><head>SEASONS</head><p>, certain portions or quarters of the year, distinguished by the signs which the sun then enters. Upon them depend the different temperatures of the air, different works in tillage, &c.</p><p>The year is divided into four Seasons, spring, summer, autumn, winter, which take their beginnings when the sun enters the first point of the signs Aries, Cancer, Libra, Capricorn.</p><p>The Seasons are very well illustrated by fig. 1, plate viii; where the candle at I represents the sun in the centre, about which the earth moves in the ecliptic ABCD, which cuts the equinoctial <hi rend="italics">abcd</hi> in the two equinoxes E and G. When the earth is in these two points, it is evident that the sun equally illuminates both the poles, and makes the days and nights equal all over the earth. But while the earth moves from G by C to <figure/>, the upper or north pole becomes more and more enlightened, the days become longer, and the nights shorter; so that when the earth is at <figure/>, or the sun at <figure/>, our <cb/> days are at the longest, as at midsummer. While the earth moves from <figure/> by D to E, our days continually decrease, by the north pole gradually declining from the sun, till at E or autumn they become equal to the nights, or 12 hours long. Again, while the earth moves from E by A to F, the north pole becomes always more and more involved in darkness, and the days grow always shorter, till at F or <figure/>, when it is midwinter to the inhabitants of the northern hemisphere. Lastly, while the earth moves from <figure/> by B to G, the north parts come more and more out of darkness, and the days grow continually longer, till at G the two poles are equally enlightened, and the days equal to the nights again. And so on continually year after year.</p></div1><div1 part="N" n="SECANT" org="uniform" sample="complete" type="entry"><head>SECANT</head><p>, in Geometry, a line that cuts another, whether right or curved; Thus <figure/> the line PA or PB, &c, is a Secant of the circle ABD, because cutting it in the point F, or G, &c. Properties of such Secants to the circle are as follow:</p><p>1. Of several Secants PA, PB, PD, &c, drawn from the same point P, that which passes through the centre C is the greatest; and from thence they decrease more and more as they recede farther from the centre; viz. PB less than PA, and PD less than PB, and so on, till they arrive at the tangent at E, which is the limit of all the Secants.</p><p>2. Of these Secants, the external parts PF, PG, PH, &c, are in the reverse order, increasing continually from F to E, the greater Secant having the less external part, and in such sort, that any Secant and its external part are in reciprocal proportion, or the whole is reciprocally as its external part, and consequently that the rectangle of every Secant and its external part is equal to a constant quantity, viz, the square of the tangent. That is, .</p><p>3. The tangent PE is a mean proportional between any Secant and its external part; as between PA and PF, or PB and PG, or PD and PH, &c.</p><p>4. The angle DPB, formed by two Secants, is measured by half the difference of its intercepted arcs DB and GH.</p><div2 part="N" n="Secant" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Secant</hi></head><p>, in Trigonometry, denotes a right line drawn from the centre of a circle, and, cutting the circumference, proceeds till it meets with a tangent to the same circle. <figure/> Thus, the line CD, drawn from the centre C, till it meets the tangent BD, is called a Secant; and particularly the Secant of the arc BE, to which BD is a tangent. In like manner, by producing DC to meet the tangent A<hi rend="italics">d</hi> in <hi rend="italics">d,</hi> then C<hi rend="italics">d,</hi> equal to CD, is the Secant of the arch AE which is the supplement of the arch BE. <pb n="427"/><cb/> So that an arch and its supplement have their Secants equal, only the latter one is negative to the former, being drawn the contrary way. And thus the Secants in the 2d and 3d quadrant are negative, while those in the 1st and 4th quadrants are positive.</p><p>The Secant CI of the arc EF, which is the complement of the former arch BE, is called the <hi rend="italics">cosecant</hi> of BE, or the Secant of its complement. The cosecants in the 1st and 2d quadrants are affirmative, but in the 3d and 4th negative.</p><p>The Secant of an arc is reciprocally as the cosine, and the cosecant reciprocally as the sine; or the rectangle of the Secant and cosine, and the rectangle of the cosecant and sine, are each equal to the square of the radius. For CD : CE :: CB : CH, or <hi rend="italics">s</hi> : <hi rend="italics">r</hi> :: <hi rend="italics">r</hi> : <hi rend="italics">c,</hi> and CI : CE :: CF : CK, or <foreign xml:lang="greek">s</foreign> : <hi rend="italics">r</hi> :: <hi rend="italics">r</hi> : <hi rend="italics">s;</hi> and consequently <hi rend="italics">r</hi><hi rend="sup">2</hi> = <hi rend="italics">cs</hi> = <hi rend="italics">s</hi><foreign xml:lang="greek">s</foreign>; where <hi rend="italics">r</hi> denotes the radius, <hi rend="italics">s</hi> the sine, <hi rend="italics">c</hi> the cosine, <hi rend="italics">s</hi> the Secant, and <foreign xml:lang="greek">s</foreign> the cosecant.</p><p>An arc <hi rend="italics">a,</hi> to the radius <hi rend="italics">r,</hi> being given, the Secant <hi rend="italics">s,</hi> and cosecant <foreign xml:lang="greek">s</foreign>, and their logarithms, or the logarithmic Secant and cosecant, may be expressed in infinite series, as follows, viz, where <hi rend="italics">m</hi> is the modulus of the system of logarithms.</p></div2><div2 part="N" n="Secants" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Secants</hi></head><p>, <hi rend="italics">Figure of.</hi> See <hi rend="smallcaps">Figure</hi> <hi rend="italics">of Secants.</hi></p></div2><div2 part="N" n="Secants" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Secants</hi></head><p>, <hi rend="italics">Line of.</hi> See <hi rend="smallcaps">Sector</hi>, and <hi rend="smallcaps">Scale.</hi></p></div2></div1><div1 part="N" n="SECOND" org="uniform" sample="complete" type="entry"><head>SECOND</head><p>, in Geometry, or Astronomy, &c, the 60th part of a prime or minute: either in the division of circles, or in the measure of time. A degree, or an hour, are each divided into 60 minutes, marked thus′; a minute is subdivided into 60 Seconds, marked thus ″; a Second into 60 thirds, marked thus ‴; &c.</p><p>We sometimes say a <hi rend="italics">Second minute,</hi> a <hi rend="italics">third minute,</hi> &c, but more usually only <hi rend="italics">Second, third,</hi> &c.</p><p>The Seconds pendulum, or pendulum that vibrates Seconds, in the latitude of London, is 39 1/8 inches long.</p><p>SECONDARY <hi rend="italics">Circles of the Ecliptic,</hi> are circles of longitude of the stars; or circles which, passing through the poles of the ecliptic, are at right angles to the ecliptic.</p><p>By means of these Secondary circles, all points in the heavens are referred to the ecliptic; that is, any star, planet, or other phenomenon, is understood to be in that point of the ecliptic, which is cut by the Secondary circle that passes through such star, &c.</p><p>If two stars be thus referred to the same point of the ecliptic, they are said to be in conjunction; if in opposite points, they are in opposition; if they are <cb/> referred to two points at a quadrant's distance, they are said to be in a quartile aspect, if the points differ a 6th part of the ecliptic, they are in sextile aspect, &c.</p><p>In general, all circles that intersect one of the six greater circles of the sphere at right angles, may be called Secondary circles. As the azimuth or vertical circles in respect of the horizon, &c; the meridian in respect of the equator, &c.</p><p><hi rend="smallcaps">Secondary</hi> <hi rend="italics">Planets,</hi> or <hi rend="italics">Satellites,</hi> are those moving round other planets as the centres of their motion, and along with them round the sun.</p></div1><div1 part="N" n="SECTION" org="uniform" sample="complete" type="entry"><head>SECTION</head><p>, in Geometry, denotes a side or surface appearing of a body, or figure, cut by another; or the place where lines, planes, &c, cut each other.</p><p>The common Section of two planes is always a right line; being the line supposed to be drawn by one plane in its cutting or entering the other. If a sphere be cut in any manner by a plane, the figure of the Section will be a circle; also the common intersection of the surfaces of two spheres, is the circumference of a circle; and the two common Sections of the surfaces of a right cone and a sphere, are the circumferences of circles if the axis of the cone pass through the centre of the sphere, otherwise not; moreover, of the two common Sections of a sphere and a cone, whether right or oblique, if the one be a circle the other will be a circle also, otherwise not. See my Tracts, tract 7, prop. 7, 8, 9.</p><p>The Sections of a cone by a plane, are five; viz, a triangle, circle, ellipse, hyperbola, and parabola. See each of these terms, as also <hi rend="smallcaps">Conic Section.</hi></p><p>Sections of Buildings and Bodies, &c, are either vertical, or horizontal, &c. The</p><p><hi rend="italics">Vertical</hi> <hi rend="smallcaps">Section</hi>, or simply the <hi rend="smallcaps">Section</hi>, of a building, denotes its profile, or a delineation of its heights and depths raised on the plan; as if the fabric had been cut asunder by a vertical plane, to discover the inside. And</p><p><hi rend="italics">Horizontal</hi> <hi rend="smallcaps">Section</hi> is the ichnography or ground plan, or a Section parallel to the horizon.</p></div1><div1 part="N" n="SECTOR" org="uniform" sample="complete" type="entry"><head>SECTOR</head><p>, of a Circle, is a portion of the circle comprehended between two radii and their included arc. Thus, <figure/> the mixt triangle ABC, contained between the two radii AC and BC, and the arc AB, is a Sector of the circle.</p><p>The Sector of a circle, as ABC, is equal to a triangle, whose base is the arc AB, and its altitude the radius AC or BC. And therefore the radius being drawn into the arc, half the product gives the area.</p><p><hi rend="italics">Similar</hi> <hi rend="smallcaps">Sectors</hi>, are those which have equal angles included between their radii. These are to each other as the squares of their bounding arcs, or as their whole circles.</p><p><hi rend="smallcaps">Sector</hi> also denotes a mathematical instrument, which is of great use in geometry, trigonometry, surveying, &c, in measuring and laying down and finding proportional quantities of the same kind: as between lines and lines, surfaces and surfaces, &c: whence the French call it the <hi rend="italics">compass of proportion.</hi> <pb n="428"/><cb/></p><p>The great advantage of the Sector above the common scales, &c, is, that it is contrived so as to suit all radii, and all scales. By the lines of chords, sines, &c, on the Sector, we have lines of chords, sines, &c, to any radius between the length and breadth of the Sector when open.</p><p>The Sector is founded on the 4th proposition of the 6th book of Euclid; where it is demonstrated, that similar triangles have their like sides proportional. An idea of the theory of its construction may be conceived thus. Let the lines AB, AC represent the legs of the Sector; <figure/> and AD, AE, two equal sections from the centre: then if the points BC and DE be connected, the lines BC and DE will be parallel; therefore the triangles ABC, ADE will be similar, and consequently the sides AB, BC, AD, DE proportional, that is, as AB : BC :: AD : DE; so that if AD be the half, 3d, or 4th part of AB, then DE will be a half, 3d, or 4th part of BC: and the same holds of all the rest. Hence, if DE be the chord, sine or tangent, of any arc, or of any number of degrees, to the radius AD, then BC will be the same to the radius AB.</p><p>The Sector, it is supposed, was the invention of Guido Baldo or Ubaldo, about the year 1568. The first printed account of it was in 1584, by Gaspar Mordente at Antwerp, who indeed says that his brother Fabricius Mordente invented it, in the year 1554. It was next treated of by Daniel Speckle, at Strasburgh, in 1589; after that by Dr. Thomas Hood, at London, in 1598: and afterwards by many other writers on practical geometry, in all the nations of Europe.</p><p><hi rend="italics">Description of the</hi> <hi rend="smallcaps">Sector.</hi> This instrument consists of two rules or legs, the longer the better, made of box, or ivory, or brass, &c, representing the radii, moveable round an axis or joint, the middle of which represents the centre; from whence several scales are drawn on the faces. See the fig. 1, plate xxvi.</p><p>The scales usually set upon Sectors, may be distinguished into single and double. The single scales are such as are set upon plane scales: the double scales are those which proceed from the centre; each of these being laid twice on the same face of the instrument, viz. once on each leg. From these scales, dimensions or distances are to be taken, when the legs of the instrument are set in an angular position.</p><p>The scales set upon the best Sectors are <table><row role="data"><cell cols="1" rows="1" rend="rowspan=14" role="data">Single<hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="rowspan=14" role="data"><hi rend="size(20)">}</hi>A line of<hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" rend="colspan=5" role="data">Inches, each divided into 8 and 10 parts,</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="colspan=5" role="data">Decimals, containing 100 parts.</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">Chords</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="rowspan=12" role="data"><hi rend="size(20)">}</hi>marked<hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">Cho.</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">Sines</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Sin.</cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">Tangents</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Tang.</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">Rhumbs</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Rhum.</cell></row><row role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">Latitude</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Lat.</cell></row><row role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">Hours</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Hou.</cell></row><row role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Longitude</cell><cell cols="1" rows="1" role="data">Lon.</cell></row><row role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Inclin. Merid.</cell><cell cols="1" rows="1" role="data">In. mer.</cell></row><row role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">  the</cell><cell cols="1" rows="1" rend="rowspan=4" role="data"><hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">Numbers</cell><cell cols="1" rows="1" role="data">Num.</cell></row><row role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">  loga-</cell><cell cols="1" rows="1" role="data">Sines</cell><cell cols="1" rows="1" role="data">Sin.</cell></row><row role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"> rithms</cell><cell cols="1" rows="1" role="data">Versed Sines</cell><cell cols="1" rows="1" role="data">V. Sin.</cell></row><row role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">    of</cell><cell cols="1" rows="1" role="data">Tangents</cell><cell cols="1" rows="1" role="data">Tan.</cell></row></table> <cb/> <table><row role="data"><cell cols="1" rows="1" rend="rowspan=7" role="data">Double<hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="rowspan=7" role="data"><hi rend="size(20)">}</hi>a line of<hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">Lines, or equal parts</cell><cell cols="1" rows="1" rend="rowspan=7" role="data"><hi rend="size(20)">}</hi>marked<hi rend="size(20)">{</hi></cell><cell cols="1" rows="1" role="data">Lin.</cell></row><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">Chords</cell><cell cols="1" rows="1" role="data">Cho.</cell></row><row role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">Sines</cell><cell cols="1" rows="1" role="data">Sin.</cell></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">Tangents to 45°</cell><cell cols="1" rows="1" role="data">Tan.</cell></row><row role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">Secants</cell><cell cols="1" rows="1" role="data">Sec.</cell></row><row role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">Tangents to above 45°</cell><cell cols="1" rows="1" role="data">Tan.</cell></row><row role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">Polygons</cell><cell cols="1" rows="1" role="data">Pol.</cell></row></table></p><p>The manner in which these scales are disposed on the Sector, is best seen in the figure.</p><p>The scales of lines, chords, sines, tangents, rhumbs, latitudes, hours, longitude, incl. merid. may be used, with the instrument either shut or open, each of these scales being contained on one of the legs only. The scales of inches, decimals, log. numbers, log. sines, log. versed sines, and log. tangents, are to be used with the Sector quite open, with the two rulers or legs stretched out in the same direction, part of each scale lying on both legs.</p><p>The double scales of lines, chords, sines, and lower tangents, or tangents under 45°, are all of the same radius or length; they begin at the centre of the instrument, and are terminated near the other extremity of each leg; viz, the lines at the division 10, the chords at 60, the sines at 90, and the tangents at 45; the remainder of the tangents, or those above 45°, are on other scales beginning at 1/4 of the length of the former, counted from the centre, where they are marked with 45, and run to about 76 degrees.</p><p>The secants also begin at the same distance from the centre, where they are marked with 10, and are from thence continued to as many degrees as the length of the Sector will allow, which is about 75°.</p><p>The angles made by the double scales of lines, or chords, of sines, and of tangents to 45 degrees, are always equal. And the angles made by the scales of upper tangents, and of secants, are also equal.</p><p>The scales of polygons are set near the inner edge of the legs; and where these scales begin, they are marked with 4, and from thence are figured backwards, or towards the centre, to 12.</p><p>From this disposition of the double scales, it is plain, that those angles that are equal to each other while the legs of the Sector were close, will still continue to be equal, although the Sector be opened to any distance.</p><p>The scale of inches is laid close to the edge of the Sector, and sometimes on the edge; it contains as many inches as the instrument will receive when opened; each inch being usually divided into 8, and also into 10 equal parts. The decimal scale lies next to this: it is of the length of the Sector when opened, and is divided into 10 equal parts, or primary divisions, and each of these into 10 other equal parts; so that the whole is divided into 100 equal parts: and by this decimal scale, all the other scales, that are taken from tables, may be laid down. The scales of chords, rhumbs, sines, tangents, hours, &c, are such as are described under Plane Scale.</p><p>The scale of logarithmic or artificial numbers, called Gunter's scale, or Gunter's line, is a scale expressing the logarithms of common numbers, taken in their natural order.</p><p>The construction of the double scale will be evident by inspecting the instrument. As to the scale of poly- <pb n="429"/><cb/> gons, it usually comprehends the sides of the polygons from 6 to 12 sides inclusive: the divisions are laid down by taking the lengths of the chords of the angles at the centre of each polygon, and laying them down from the centre of the instrument. When the polygons of 4 and 5 sides are also introduced, this line is constructed from a scale of chords, where the length of 90° is equal to that of 60° of the double scale of chords on the Sector.</p><p>In describing the use of the Sector, the terms <hi rend="italics">lateral distance</hi> and <hi rend="italics">transverse distance</hi> often occur. By the former is meant the distance taken with the compasses on one of the scales only, beginning at the centre of the Sector; and by the latter, the distance taken between any two corresponding divisions of the scales of the same name, the legs of the Sector being in an angular position. <hi rend="center"><hi rend="italics">Uses of the</hi> <hi rend="smallcaps">Sector.</hi></hi></p><p><hi rend="italics">Of the Line of Lines.</hi> This is useful, to divide a given line into any number of equal parts, or in any proportion, or to make scales of equal parts, or to find 3d and 4th proportionals, or mean proportionals, or to increase or decrease a given line in any proportion. Ex. 1. To divide a given line into any number of equal parts, as suppose 9: make the length of the given line a transverse distance to 9 and 9, the number of parts proposed; then will the transverse distance of 1 and 1 be one of the equal parts, or the 9th part of the whole; and the transverse distance of 2 and 2 will be 2 of the equal parts, or 2/9 of the whole line; and so on. 2. Again, to divide a given line into any number of parts that shall be in any assigned proportion, as suppose three parts, in the proportion of 2, 3, and 4. Make the given line a transverse distance to 9, the sum of the proposed numbers 2, 3, 4; then the transverse distances of these numbers severally will be the parts required.</p><p><hi rend="italics">Of the Scale of Chords.</hi> 1. To open the Sector to any angle, as suppose 50 degrees: Take the distance from the joint to 50 on the chords, the number of degrees proposed; then open the Sector till the transverse distance from 60 to 60, on each leg, be equal to the said lateral distance of 50; so shall the scale of chords make the proposed angle of 50 degrees.—By the converse of this operation, may be known the angle the Sector is opened to; viz, taking the transverse distance of 60, and applying it laterally from the joint.</p><p>2. To protract or lay down an angle of any given number of degrees. At any opening of the Sector, take the transverse distance of 60°, with which extent describe an arc; then take the transverse distance of the number of degrees proposed, and apply it to that arc; and through the extremities of this distance on the arc draw two lines from the centre, and they will form the angle as proposed. When the angle exceeds 60°, lay it off at twice or thrice.—By the converse operation any angle may be measured; viz, With any radius describe an arc from the angular point; set that radius transversely from 60 to 60; then take the distance of the intercepted arc and apply it transversely to the chords, which will shew the degrees in the given angle.</p><p><hi rend="italics">Of the Line of Polygons.</hi> 1. In a given circle to in- <cb/> scribe a regular polygon, for example an octagon. Open the legs of the Sector till the transverse distance from 6 to 6 be equal to the radius of the circle; then will the transverse distance of 8 and 8 be the side of the inscribed octagon. 2. Upon a line given to describe a regular polygon. Make the given line a transverse dis. to 5 and 5; and at that opening of the Sector take the transverse distance of 6 and 6; with which as a radius, from the extremities of the given line describe arcs to intersect each other, which intersection will be the centre of a circle in which the proposed polygon may be inscribed; then from that centre describe the said circle through the extremities of the given line, and apply this line continually round the circumference, for the several angular points of the polygon.—3. On a given right line as a base, to describe an isosceles triangle, having the angles at the base double the angle at the vertex. Open the Sector till the length of the given line fall transversely on 10 and 10 on each leg; then take the transverse distance to 6 and 6, and it will be the length of each of the equal sides of the triangle.</p><p><hi rend="italics">Of the Sines, Tangents, and Secants.</hi> By the several lines disposed on the sector, we have scales of several radii. So that, 1. Having a length or radius given, not exceeding the length of the Sector when opened, we can find the chord, sine, &c, to the same: for ex. suppose the chord, sine, or tangentof 20 degrees to a radius of 3 inches be required. Make 3 inches the opening or transverse distance to 60 and 60 on the chords; then will the same extent reach from 45 to 45 on the tangents, and from 90 to 90 on the sines; so that to whatever radius the line of chords is set, to the same are all the others set also. In this disposition therefore, if the transverse distance between 20 and 20 on the chords be taken with the compasses, it will give the chord of 20 degrees; and if the transverse of 20 and 20 be in like manner taken on the sines, it will be the sine of 20 degrees; and lastly, if the transverse distance of 20 and 20 be taken on the tangents, it will be the tangent of 20 degrees, to the same radius.—2. If the chord or tangent of 70 degrees were required. For the chord, the transverse distance of half the arc, viz 35, must be taken, as before; which distance taken twice gives the chord of 70 degrees. To find the tangent of 70 degrees, to the same radius, the scale of upper tangents must be used, the under one only reaching to 45: making therefore 3 inches the transverse distance to 45 and 45 at the beginning of that scale, the extent between 70 and 70 degrees on the same, will be the tangent of 70 degrees to 3 inches radius.— 3. To find the secant of an arc; make the given radius the transverse distance between 0 and 0 on the secants; then will the transverse distance of 20 and 20, or 70 and 70, give the secant of 20 or 70 degrees.— 4. If the radius, and any line representing a sine, tangent, or secant, be given, the degrees corresponding to that line may be found by setting the Sector to the given radius, according as a sine, tangent, or secant is concerned; then taking the given line between the compasses, and applying the two feet transversely to the proper scale, and sliding the feet along till they both rest on like divisions on both legs; then the divisions will shew the degrees and parts corresponding to the given line. <pb n="430"/><cb/> <hi rend="center"><hi rend="italics">Use of the Sector in Trigonometry, or in working any other proportions.</hi></hi></p><p>By means of the double scales, which are the parts more peculiar to the Sector, all proportions are worked by the property of similar triangles, making the sides proportional to the bases, that is, on the Sector, the lateral distances proportional to the transverse ones; thus, taking the distance of the first term, and applying it to the 2d, then the distance of the 3d term, properly applied, will give the 4th term: observing that the sides of triangles are taken off the line of numbers laterally, and the angles are taken transversely, off the sines or tangents or secants, according to the nature of the proportion. For example, in a plane triangle ABC, given two sides and an angle opposite to one of them, to find the rest; viz, given <figure/> AB = 56, AC = 64, and [angle]B = 46° 30′, to find BC and the angles A and C. In this case, the sides are proportional to the sines of their opposite angles; hence these proportions, as AC (64) : sin. [angle]B (46° 30′) :: AB (56) : sin. [angle]C, and as sin. B : AC :: sin. A : BC.</p><p>Therefore, to work these proportions by the Sector, take the lateral distance of 64 = AC from the lines, and open the Sector to make this a transverse distance of 46° 30′ = [angle]B, on the sines; then take the lateral distance of 56 = AB on the lines, and apply it transversely on the sines, which will give 39° 24′ = [angle]C. Hence, the sum of the angles B and C, which is 85° 54′, taken from 180°, leaves 94° 6′ = [angle]A. Then, to work the 2d proportion, the Sector being set at the same opening as before, take the transverse distance of 94° 6′ = [angle]A, on the sines, or, which is the same thing, the transverse distance of its supplement 85° 54′; then this applied laterally to the lines, gives 88 = the side BC sought.</p><p>For the complete history of the Sector, with its more ample and particular construction and uses, fee Robertson's <hi rend="italics">Treatise of such Mathematical Instruments, as are usually put into a Portable Case,</hi> the Introduction.</p><p><hi rend="smallcaps">Sector</hi> <hi rend="italics">of a Sphere,</hi> is the solid generated by the revolution of the Sector of a circle about one of its radii; the other radius describing the surface of a cone, and the circular arc a circular portion of the surface of the sphere of the same radius. So that the spherical Sector consists of a right cone, and of a segment of the sphere having the same common base with the cone. And hence the solid content of it will be found by multiplying the base or spherical surface by the radius of the sphere, and taking a 3d part of the product.</p><p><hi rend="smallcaps">Sector</hi> <hi rend="italics">of an ellipse,</hi> or <hi rend="italics">of an hyperbola,</hi> &c, is a part resembling the circular Sector, being contained by three lines, two of which are radii, or lines drawn from the centre of the figure to the curve, and the intercepted arc or part of that curve.</p><p><hi rend="italics">Astronomical</hi> <hi rend="smallcaps">Sector</hi>, an instrument invented by Mr. George Graham, for finding the difference in right ascension and declination between two objects, whose distance is too great to be observed through a sixed <cb/> telescope, by means of a micrometer. This instrument (fig. 2, pl. 26,) consists of a brass plate, called the Sector, formed like a T, having the shank CD, as a radius, about 2 1/2 feet long, and 2 inches broad at the end D, and an inch and a half at C; and the cross-piece AB, as an arch, about 6 inches long, and one and a half broad; upon which, with a radius of 30 inches, is described an arch of 10 degrees, each degree being divided in as many parts as are convenient. Round a small cylinder C, containing the centre of this arch, and fixed in the shank, moves a plate of brass, to which is fixed a telescope CE, having its line of collimation parallel to the plane of the Sector, and passing over the centre C of the arch AB, and the index of a Vernier's dividing plate, whose length, being equal to 16 quarters of a degree, is divided into 15 equal parts, fixed to the eye end of the telescope, and made to slide along the arch; which motion is performed by a long screw, G, at the back of the arch, communicating with the Vernier through a slit cut in the brass, parallel to the divided arch. Round the centre F of a circular brass plate <hi rend="italics">abc,</hi> of 5 inches diameter, moves a brass cross KLMN, having the opposite ends O and P of one bar turned up perpendicularly about 3 inches, to serve as supporters to the Sector, and screwed to the back of its radius; so that the plane of the Sector is parallel to the plane of the circular plate, and can revolve round the centre of that plate in this parallel position. A square iron axis HIF, 18 inches long, is screwed flat to the back of the circular plate along one of its diameters, so that the axis is parallel to the plane of the Sector. The whole instrument is supported on a proper pedestal, so that the said axis shall be parallel to the earth's axis, and proper contrivances are annexed to fix it in any position. The instrument, thus supported, can revolve round its axis HI, parallel to the earth's axis, with a motion like that of the stars, the plane of the Sector being always parallel to the plane of some hour circle, and consequently every point of the telescope describing a parallel of declination; and if the Sector be turned round the joint F of the circular plate, its graduated arch may be brought parallel to an hour-circle; and consequently any two stars, whose difference of declination does not exceed the degrees in that arch, will pass over it.</p><p>To observe their passage, direct the telescope to the preceding star, and fix the plane of the Sector a little to the westward of it; move the telescope by the screw G, and observe at the transit of each over the cross wires the time shewn by the clock, and also the division upon the arch AB, shewn by the index; then is the difference of the arches the difference of the declination; and that of the times shews the difference of the right ascension of those stars. For a more particular description of this instrument, see Smith's Optics, book iii, chap. 9.</p><p>SECULAR <hi rend="italics">Year,</hi> the same with Jubilee.</p></div1><div1 part="N" n="SECUNDANS" org="uniform" sample="complete" type="entry"><head>SECUNDANS</head><p>, an infinite series of numbers, beginning from nothing, and proceeding according to the squares of numbers in arithmetical progression, as 0, 1, 4, 9, 16, 25, 36, 49, 64, &c.</p></div1><div1 part="N" n="SEEING" org="uniform" sample="complete" type="entry"><head>SEEING</head><p>, the act of perceiving objects by the organ of sight; or the sense we have of external objects by means of the eye. <pb n="431"/><cb/></p><p>For the apparatus, or disposition of the parts necessary to Seeing, see <hi rend="smallcaps">Eye.</hi> And for the manner in which Seeing is performed, and the laws of it, see V<hi rend="smallcaps">ISION.</hi></p><p>Our best anatomists differ greatly as to the cause why we do not see double with the two eyes? Galen, and others after him, ascribe it to a coalition, or decussation, of the optic nerve, behind the os sphenoides. But whether they decussate or coalesce, or only barely touch one another, is not well agreed upon.</p><p>The Bartholines and Vesalius say expressly, they are united by a perfect confusion of their substance; Dr. Gibson allows them to be united by the closest conjunction, but not by a confusion of their fibres.</p><p>Alhazen, an Arabian philosopher of the 12th century, accounts for single vision by two eyes, by supposing that when two corresponding parts of the retina are affected, the mind perceives but one image.</p><p>Des Cartes and others account for the effect another way; viz, by supposing that the fibrillæ constituting the medullary part of those nerves, being spread in the retina of each eye, have each of them corresponding parts in the brain, so that when any of those fibrillæ are struck by any part of an image, the corresponding parts of the brain are affected by it. Somewhat like which is the opinion of Dr. Briggs, who takes the optic nerves of each eye to consist of homologous sibres, having their rise in the thalamus nervorum opticorum, and being thence continued to both the retinæ, which are composed of them; and farther, that those fibrillæ have the same parallelism, tension, &c, in both eyes; consequently when an image is painted on the same corresponding sympathizing parts of each retina, the same effects are produced, the same notice carried to the thalamus, and so imparted to the soul. Hence it is, that double vision ensues upon an interruption of the parallelism of the eyes; as when one eye is depressed by the finger, or their symphony is interrupted by disease: but Dr. Briggs maintains, that it is but in few subjects there is any decussation; and in none any conjunction more than mere contact; though his notion is by no means consonant to facts, fnd it is attended with many improbable circumstances.</p><p>It was the opinion of Sir Isaac Newton, and of many others, that objects appear single, because the two optic nerves unite before they reach the brain. But Dr. Porterfield shews, from the observation of several anatomists, that the optic nerves do not mix or confound their substance, being only united by a close cohesion; and objects have appeared single, where the optic nerves were found to be disjoined. To account for this phenomenon, this ingenious writer supposes, that, by an original law in our natures, we imagine an object to be situated somewhere in a right line drawn from the picture of it upon the retina, through the centre of the pupil; consequently the same object appearing to both eyes to be in the same place, we cannot distinguish it into two. In answer to an objection to this hypothesis, from objects appearing double when one eye is distorted, he says, the mind mistakes the position of the eye, imagining, that it had moved in a manner corresponding to the other, in which case the conclusion would have been just: in this he seems to have re- <cb/> course to the power of habit, though he disclaims that hypothesis. This principle however has been thought sufficient to account for this appearance.</p><p>Originally, every object making two pictures, one in each eye, is imagined to be double; but, by degrees, we find that when two corresponding parts of the retina are impressed, the object is but one; but if those corresponding parts be changed by the distortion of one of the eyes, the object must again appear double as at the first. This seems to be verified by Mr. Cheselden, who informs us, that a gentleman, who, from a blow on his head, had one eye distorted, found every object to appear double, but by degrees the most familiar ones came to appear single again, and in time all objects did so without amendment of the distortion. A similar case is mentioned by Dr. Smith.</p><p>On the other hand, Dr. Reid is of opinion, that the correspondence of the centres of two eyes, on which single vision depends, does not arise from custom, but from some natural constitution of the eye, and of the mind.</p><p>M. du Tour adopts an opinion, long before suggested by Gassendi, that the mind attends to no more than the image made in one eye at a time; in support of which, he produces several curious experiments; but as M. Buffon observes, it is a sufficient answer to this hypothesis, that we see more distinctly with two eyes than with one; and that when a round object is near us, we plainly see more of the surface in one case than in the other.</p><p>With respect to single vision with two eyes, Dr. Hartley observes, that it deserves particular attention, that the optic nerves of man, and such other animals as look the same way with both eyes, unite in the sella turrica in a ganglion, or little brain, as it may be called, peculiar to themselves, and that the associations between synchronous impressions on the two retinas, must be made sooner and cemented stronger on this account; also that they ought to have a much greater power over one another's image, than in any other part of the body. And thus an impression made on the right eye alone by a single object, propagates itself into the left, and there raises up an image almost equal in vividness to itself; and, consequently, when we see with one eye only, we may however have pictures in both eyes.</p><p>It is a common observation, says Dr. Smith, that objects seen with both eyes appear more vivid and stronger than they do to a single eye, especially when both of them are equally good. Porterfield on the Eye, vol. ii, pa. 285, 315. Smith's Optics, Remarks pa. 31. Reid's Inquiry, pa. 267. Mem. Présentes, pa. 514. Acad. Par. 1747. Mem. Pr. 334. Hartley on Man, vol. i, pa. 207. Priestley's Hist. of Light and Colours, pa. 663, .&c.</p><p>Whence it is that we see objects erect, when it is certain, that the images thereof are painted invertedly on the retina, is another difficulty in the theory of Seeing. Des Cartes accounts for it hence, that the notice which the soul takes of the object, does not depend on any image, nor any action coming from the object, but merely on the situation of the minute parts of the brain, whence the nerves arise. Ex. gr. the situation of a capillament brain, which occasions the <pb n="432"/><cb/> soul to see all those places lying in a right line with it.</p><p>But Mr. Molyneux gives another account of this matter. The eye, he observes, is only the organ, or instrument; it is the soul that sees. To enquire then, how the soul perceives the object erect by an inverted image, is to enquire into the soul's faculties. Again, imagine that the eye receives an impulse on its lower part, by a ray from the upper part of an object; must not the visive faculty be hereby directed to consider this stroke as coming from the top, rather than the bottom of the object, and consequently be determined to conclude it the representation of the top?</p><p>Upon these principles, we are to consider, that inverted is only a relative term, and that there is a very great difference between the real object, and the means or image by which we perceive it. When all the parts of a distant prospect are painted upon the retina (supposing that to be the seat of vision), they are all right with respect to one another, as well as the parts of the prospect itself; and we can only judge of an object being inverted, when it is turned reverse to its natural position with respect to other objects which we see and compare it with.</p><p>The eye or visive faculty (says Molyneux) takes no notice of the internal surface of its own parts, but uses them as an instrument only, contrived by nature for the exercise of such a faculty. If we lay hold of an upright stick in the dark, we can tell which is the upper or lower part of it, by moving our hand upward or downward; and very well know that we cannot feel the upper end by moving our hand downward. Just so, we find by experience and habit, that by directing our eyes towards a tall object, we cannot see its top by turning our eyes downward, nor its foot by turning our eyes upward; but must trace the object the same way by the eye to see it from head to foot, as we do by the hand to feel it; and as the judgement is informed by the motion of the hand in one case, so it is also by the motion of the eye in the other.</p><p>Molyneux's Dioptr. pa. 105, &c. Musschenbroek's Int. ad Phil. Nat. vol. ii, pa. 762. Ferguson's Lectures, pa. 132. See <hi rend="smallcaps">Sight, Visible</hi>, &c.</p></div1><div1 part="N" n="SEGMENT" org="uniform" sample="complete" type="entry"><head>SEGMENT</head><p>, in Geometry, is a part cut off the top of a figure by a line or plane; and the part remaining at the bottom, after the Segment is cut off, is called a <hi rend="italics">frustum,</hi> or a <hi rend="italics">zone.</hi> So, a</p><p><hi rend="smallcaps">Segment</hi> <hi rend="italics">of a Circle,</hi> is a part of the circle cut off by a chord, or a portion comprehended by an arch and its chord; and may be either greater or less than a semicircle. Thus, the portion ABCA is a Segment less than a semicircle; and ADCA a Segment greater.</p><p>The angle formed by lines <figure/> drawn from the extremities of a chord to meet in any point of the arc, is called an angle <hi rend="italics">in</hi> the Segment. So the angle ABC is an angle <hi rend="italics">in</hi> the Segment ABCA; and the angle ADC, an angle <hi rend="italics">in</hi> the Segment ADCA.</p><p>Also the angle B is said to be the angle <hi rend="italics">upon</hi> the <cb/> Segment ADC, and D the angle <hi rend="italics">on</hi> the Segment ABC.</p><p>The angle which the chord AC makes with a tangent EF, is called the angle <hi rend="italics">of</hi> a Segment; and it is equal to the angle in the alternate or supplemental Segment, or equal to the supplement of the angle in the same Segment. So the angle ACE is the angle <hi rend="italics">of</hi> the Segment ABC, and is equal to the angle ADC, or to the supplement of the angle B; also the angle ACF is the angle <hi rend="italics">of</hi> the Segment ADC, and is equal to the angle B, or to the supplement of the angle D.</p><p>The area of a Segment ABC, is evidently equal to the difference between the sector OABC of the same arc, and the triangle OAC on the same chord; the triangle being subtracted from the sector, to give the Segment, when less than a semicircle; but to be added when greater. See more rules for the Segment in my Mensuration, pa. 132 &c, 2d edition.</p><p><hi rend="italics">Similar</hi> <hi rend="smallcaps">Segments</hi>, are those that have their chords directly proportional to their radii or diameters, or that have similar arcs, or such as contain the same number of degrees.</p><p><hi rend="smallcaps">Segment</hi> <hi rend="italics">of a Sphere,</hi> is a part cut off by a plane.</p><p>The base of a Segment is always a circle. And the convex surfaces of different Segments, are to each other as their altitudes, or versed sines. And as the whole convex surface of the sphere is equal to 4 of its great circles, or 4 circles of the same diameter; so the surface of any Segment, is equal to 4 circles on a diameter equal to the chord of half the arc of the Segment. So that if <hi rend="italics">d</hi> denote the diameter of the sphere, or the chord of half the circumference, and <hi rend="italics">c</hi> the chord of half the arc of any other Segment, also <hi rend="italics">a</hi> the altitude or versed sine of the same; then, 3.1416<hi rend="italics">d</hi><hi rend="sup">2</hi> is the surface of the whole sphere, and 3.1416<hi rend="italics">c</hi><hi rend="sup">2</hi>, or 3.1416<hi rend="italics">ad,</hi> the surface of the Segment.</p><p>For the solid content of a Segment, there are two rules usually given; viz, 1. To 3 times the square of the radius of its base, add the square of its height; multiply the sum by the height, and the product by .5236. Or, 2dly, From 3 times the diameter of the sphere, subtract twice the height of the frustum; multiply the remainder by the square of the height, and the product by .5236. That is, in symbols, the solid content is either ; where <hi rend="italics">a</hi> is the altitude of the Segment, <hi rend="italics">r</hi> the radius of its base, and <hi rend="italics">d</hi> the diameter of the whole sphere.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Segments</hi>, are two particular lines, so called, on Gunter's sector. They lie between the lines of sines and superficies, and are numbered with 5, 6, 7, 8, 9, 10. They represent the diameter of a circle, so divided into 100 parts, as that a right line drawn through those parts, and perpendicular to the diameter, shall cut the circle into two Segments, the greater of which shall have the same proportion to the whole circle, as the parts cut off have to 100.</p></div1><div1 part="N" n="SELENOGRAPHY" org="uniform" sample="complete" type="entry"><head>SELENOGRAPHY</head><p>, the description and representation of the moon, with all the parts and appearances of her disc or face; like as geography does those of the earth. <pb/><pb/><pb n="433"/><cb/></p><p>Since the invention of the telescope, Selenography is very much improved. We have now distinct names for most of the regions, seas, lakes, mountains, &c, visible in the moon's body. Hevelius, a celebrated astronomer of Dantzic, and who published the first Selenography, named the several places of the moon from those of the earth. But Riccioli afterwards called them after the names of the most celebrated astronomers and philosophers. Thus, what the one calls <hi rend="italics">mons Porphyrites,</hi> the other calls <hi rend="italics">Aristarchus;</hi> what the one calls <hi rend="italics">Ætna, Sinai, Athos, Apenninus,</hi> &c, the other calls, <hi rend="italics">Copernicus, Posidonius, Tycho, Gassendus,</hi> &c.</p><p>M. Cassini has published a work called <hi rend="italics">Instructions Seleniques,</hi> and has published the best map of the moon.</p><p>SELEUOIDÆ, in Chronology, the era of the Seleucidæ, or the Syro-Macedonian era, which is a computation of time, commencing from the establishment of the Seleucidæ, a race of Greek kings, who reigned as successors of Alexander the Great, in Syria, as the Ptolomies did in Egypt. According to the best accounts, the first year of this era falls in the year 311 before Christ, which was 12 years after the death of Alexander.</p></div1><div1 part="N" n="SELL" org="uniform" sample="complete" type="entry"><head>SELL</head><p>, in Building, is of two kinds, viz, <hi rend="italics">GroundSell,</hi> which denotes the lowest piece of timber in a wooden building, and that upon which the whole superstructure is raised. And Sell of a window, or of a door, which is the bottom piece in the frame of them, upon which they rest.</p></div1><div1 part="N" n="SEMICIRCLE" org="uniform" sample="complete" type="entry"><head>SEMICIRCLE</head><p>, in Geometry, is half a circle, or a figure comprehended between the diameter of a circle, and half the circumference.</p><p><hi rend="smallcaps">Semicircle</hi> is also an instrument in Surveying, sometimes called the <hi rend="italics">graphometer.</hi></p><p>It consists of a semicircular limb or arch, as FIG (fig. 3, pl. 26) divided into 180 degrees, and sometimes subdivided diagonally or otherwise into minutes. This limb is subtended by a diameter FG, having two sights erected at its extremities. In the centre of the Semicircle, or the middle of the diameter, is fixed a box and needle; and on the same centre is fitted an alidade, or moveable index, carrying two other sights, as H, I: the whole being mounted on a staff, with a ball and socket &c.</p><p>Hence it appears, that the Semicircle is nothing but half a theodolite; with this only difference, that whereas the limb of the theodolite, being an entire circle, takes in all the 360° successively; while in the Semicircle the degrees only going from 1 to 180, it is usual to have the remaining 180°, or those from 180° to 360°, graduated in another line on the limb within the former.</p><p><hi rend="italics">To take an Angle with a Semicircle.</hi>—Place the instrument in such manner, as that the radius CG may hang over one leg of the angle to be measured, with the centre C over the vertex of the same. The first is done by looking through the sights F and G, at the extremities of the diameter, to a mark fixed up in one extremity of the leg; and the latter is had by letting fall a plummet from the centre of the instrument. This done, turn the moveable index HI on its centre towards the other leg of the angle, till through the sights fixed in it, you see a mark in the extremity of the leg. Then the degree which the index cuts on the limb, is the quantity or measure of the angle. <cb/></p><p>Other uses are the same as in the theodolite.</p><p>SEMICUBICAL <hi rend="smallcaps">Parabola</hi>, a curve of the 2d order, of such a nature that the cubes of the ordinates are proportional to the squares of the abscisses. Its equation is <hi rend="italics">ay</hi><hi rend="sup">2</hi> = <hi rend="italics">x</hi><hi rend="sup">3</hi>. This curve, AM<hi rend="italics">m,</hi> is one of <figure/> Newton's five diverging parabolas, being his 70th species; having a cusp at its vertex at A. It is otherwise named the Neilian parabola, from the name of the author who first treated of it.</p><p>The area of the space APM, is , or 4/15 of the circumscribing rectangle.</p><p>The content of the solid generated by the revolution of the space APM about the axis AP, is , or 1/4 of the circumscribing cylinder. And a circle equal to the surface of that solid may be found from the quadrature of an hyperbolic space.</p><p>Also the length of any arc AM of the curve may be easily obtained from the quadrature of a space contained under part of the curve of the common parabola, two semiordinates to the axis, and the part of the axis contained between them.</p><p>This curve may be described by a continued motion, viz, by fastening the angle of a square in the vertex of a common parabola; and then carrying the intersection of one side of this square and a long ruler (which ruler always moves perpendicularly to the axis of the parabola) along the curve of that parabola. For the intersection of the ruler, and the other side of the square will describe a Semicubical parabola. Maclaurin performs this without a common parabola, in his Geometria Organica.</p></div1><div1 part="N" n="SEMIDIAMETER" org="uniform" sample="complete" type="entry"><head>SEMIDIAMETER</head><p>, or <hi rend="italics">Radius,</hi> of a circle or sphere, is a line drawn from the centre to the circumference. And in any curve that has diameters and a centre, it is the radius, or half diameter, or a line drawn from the centre to some point in the curve.</p><p>The distances, diameters, &c, of the heavenly bodies, are usually estimated by astronomers in Semidiameters of the earth; the number of which terrestrial Semidiameters, contained in that of each of those planets, is as below. <table><row role="data"><cell cols="1" rows="1" role="data">The Earth</cell><cell cols="1" rows="1" rend="align=right" role="data">1   </cell><cell cols="1" rows="1" role="data">Semidiam.</cell></row><row role="data"><cell cols="1" rows="1" role="data">The Sun</cell><cell cols="1" rows="1" rend="align=right" role="data">111 1/4  </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">The Moon</cell><cell cols="1" rows="1" rend="align=right" role="data">0.27</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Mercury</cell><cell cols="1" rows="1" rend="align=right" role="data">0.38</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Venus</cell><cell cols="1" rows="1" rend="align=right" role="data">1.15</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Mars</cell><cell cols="1" rows="1" rend="align=right" role="data">0.65</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Jupiter</cell><cell cols="1" rows="1" rend="align=right" role="data">11.81</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Saturn</cell><cell cols="1" rows="1" rend="align=right" role="data">9.77</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Herschel</cell><cell cols="1" rows="1" rend="align=right" role="data">4.32</cell><cell cols="1" rows="1" role="data"/></row></table></p></div1><div1 part="N" n="SEMIDIAPENTE" org="uniform" sample="complete" type="entry"><head>SEMIDIAPENTE</head><p>, in Music, a defective or imperfect fifth, called usually by the Italians, <hi rend="italics">falsa quinta,</hi> and by us a false fifth. <pb n="434"/><cb/></p></div1><div1 part="N" n="SEMIDIAPASON" org="uniform" sample="complete" type="entry"><head>SEMIDIAPASON</head><p>, in Music, a defective or imperfect octave; or an octave diminished by a lesser semitone, or 4 commas.</p></div1><div1 part="N" n="SEMIDIATESSARON" org="uniform" sample="complete" type="entry"><head>SEMIDIATESSARON</head><p>, in Music, a defective fourth, called also a false fourth.</p></div1><div1 part="N" n="SEMIDITONE" org="uniform" sample="complete" type="entry"><head>SEMIDITONE</head><p>, in Music, is the lesser third, having its terms as 6 to 5.</p></div1><div1 part="N" n="SEMIORDINATES" org="uniform" sample="complete" type="entry"><head>SEMIORDINATES</head><p>, in Geometry, the halves of the ordinates or applicates, being the lines applied between the absciss and the curve.</p></div1><div1 part="N" n="SEMIPARABOLA" org="uniform" sample="complete" type="entry"><head>SEMIPARABOLA</head><p>, &c, in Geometry, the half of the whole parabola, &c.</p></div1><div1 part="N" n="SEMIQUADRATE" org="uniform" sample="complete" type="entry"><head>SEMIQUADRATE</head><p>, or <hi rend="smallcaps">Semiquartile</hi>, is an aspect of the planets, when distant from each other one sign and a half, or 45 degrees.</p></div1><div1 part="N" n="SEMIQUAVER" org="uniform" sample="complete" type="entry"><head>SEMIQUAVER</head><p>, in Music, the half of a quaver.</p></div1><div1 part="N" n="SEMIQUINTILE" org="uniform" sample="complete" type="entry"><head>SEMIQUINTILE</head><p>, is an aspect of the planets when distant from each other the half of a 5th of the circle, or by 36 degrees.</p></div1><div1 part="N" n="SEMISEXTILE" org="uniform" sample="complete" type="entry"><head>SEMISEXTILE</head><p>, an aspect of two planets, when they are distant from each other 30 degrees, or the half of a sextile, which is 2 signs or 60°. The Semisextile is marked <hi rend="italics">s. s.</hi></p></div1><div1 part="N" n="SEMITONE" org="uniform" sample="complete" type="entry"><head>SEMITONE</head><p>, in Music, a half tone or half note, one of the degrees or intervals of concords.</p><p>There are three degrees, or less intervals, by which a sound can move upwards and downwards, successively from one extreme of any concord to the other, and yet produce true melody. These degrees are the greater tone, the less tone, and the semitone. The ratios defining these intervals are these, viz, the greater tone 8 to 9, the less tone 9 to 10, and the Semitone 15 to 16. Its compass is 5 commas, and it has its name from being nearly half a whole, though it is really somewhat more.</p><p>There are several species of Semitones; but those that usually occur in practice are of two kinds, distinguished by the addition of greater and less. The first is expressed by the ratio of 16 to 15, or 16/15; and the second by 25 to 24, or 25/24. The octave contains 10 Semitones major, and 2 dieses, nearly, or 17 Semitones minor, nearly; for the measure of the octave <table><row role="data"><cell cols="1" rows="1" role="data">being expressed by the logarithm</cell><cell cols="1" rows="1" role="data">1,00000,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the Semitone major will be measured by</cell><cell cols="1" rows="1" role="data">0,09311,</cell></row><row role="data"><cell cols="1" rows="1" role="data">and the Semitone minor by</cell><cell cols="1" rows="1" role="data">0,05889.</cell></row></table> These two differ by a whole enharmonic diesis; which is an interval practicable by the voice. It was much in use among the Ancients, and is not unknown among modern practitioners. Euler Tent. Nov. Theor. Mus. pa. 107. See <hi rend="smallcaps">Interval.</hi></p><p>These Semitones are called <hi rend="italics">fictitious notes;</hi> and, with respect to the natural ones, they are expressed by characters called <hi rend="italics">slats</hi> and <hi rend="italics">sharps.</hi> The use of them is to remedy the defects of instruments, which, having their sounds fixed, cannot always be made to answer to the diatonic scale. By means of these, we have a new kind of scale, called the</p><p>SEMITONIC <hi rend="italics">Scale,</hi> or the <hi rend="italics">Scale of Semitones,</hi> <cb/> which is a scale or system of music, consisting of 12 degrees, or 13 notes, in the octave, being an improvement on the natural or diatonic scale, by inserting between each two notes of it, another note, which divides the interval or tone into two unequal parts, called Semitones.</p><p>The use of this scale is for instruments that have fixed sounds, as the organ, harpsichord, &c, which are exceedingly defective on the foot of the natural or diatonic scale. For the degrees of the scale being unequal, from every note to its octave there is a different order of degrees; so that from any note we cannot find every interval in a series of fixed sounds; which yet is necessary, that all the notes of a piece of music, carried through several keys, may be found in their just tune, or that the same song may be begun indifferently at any note, as may be necessary for accommodating some instrument to others, or to the voice, when they are to accompany each other in unison.</p><p>The diatonic scale, beginning at the lowest note, being first settled on an instrument, and the notes of it distinguished by their names <hi rend="italics">a, b, c, d, e, f, g;</hi> the inserted notes, or Semitones, are called fictitious notes, and take the name or letter below with a <figure/>, as <hi rend="italics">c</hi> <figure/> called <hi rend="italics">c</hi> sharp; signifying that it is a semitone higher than the sound of <hi rend="italics">c</hi> in the natural series; or this mark <figure/>, called a flat, with the name of the note above signifying it to be a Semitone lower.</p><p>Now 15/16 and 128/135 being the two Semitones the greater tone is divided into, and 15/16 and 24/25, the Semitones the less tone is divided into, the whole octave will stand as in the following scheme, where the ratios of each term to the next are written fraction-wise between them below. <hi rend="center"><hi rend="italics">Scale of Semitones.</hi></hi> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">c.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">e.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">f.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">f</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">g.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">g</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">b.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">cc.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">128/135</cell><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">24/25</cell><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">128/135</cell><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">24/25</cell><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">128/135</cell><cell cols="1" rows="1" role="data">15/16</cell></row></table> for the names of the intervals in this scale, it may be considered, that as the notes added to the natural scale are not designed to alter the species of melody, but leave it still diatonic, and only correct certain defects arising from something foreign to the office of the scale of music, viz, the sixing and limiting the sounds; we see the reason why the names of the natural scale are continued, only making a distinction of each into a greater and less. Thus an interval of one Semitone, is called a less second; of two Semitones, a greater second; of three Semitones, a less third; of four, a greater third, &c.</p><p>A second kind of Semitonie scale we have from another division of the octave into Semitones, which is performed by taking an harmonical mean between the extremes of the greater and less tone of the natural scale, which divides it into two Semitones nearly equal. Thus, the greater tone 8 to 9 is divided into two Semitones, which are 16 to 17, and 17 to 18; where 16, 17, 18, is an arithmetical division, the numbers representing the lengths of the chords; but if they represent the vibration, the lengths of the chords are reciprocal; viz as 1, 16/17, 8/9; which puts the greater Se- <pb n="435"/><cb/> mitone 16/17 next the lower part of the tone, and the lesser 17/18 next the upper, which is the property of the harmonical division. And after the same manner the less tone 9 to 10 is divided into two Semitones, 18 to 19, and 19 to 20; and the whole octave stands thus: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">c.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">e.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">f.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">f</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">g.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">g</hi><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a.</hi></cell><cell cols="1" rows="1" role="data"><figure/>.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">b.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">c.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">16/17</cell><cell cols="1" rows="1" role="data">17/18</cell><cell cols="1" rows="1" role="data">18/19</cell><cell cols="1" rows="1" role="data">19/20</cell><cell cols="1" rows="1" role="data">15/16</cell><cell cols="1" rows="1" role="data">16/17</cell><cell cols="1" rows="1" role="data">17/18</cell><cell cols="1" rows="1" role="data">18/19</cell><cell cols="1" rows="1" role="data">19/20</cell><cell cols="1" rows="1" role="data">16/17</cell><cell cols="1" rows="1" role="data">17/18</cell><cell cols="1" rows="1" role="data">15/16.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>This scale, Mr. Salmon tells us, in the Philosophical Transactions, he made an experiment of before the Royal Society, on chords, exactly in these proportions, which yielded a perfect concert with other instruments, touched by the best hands. Mr. Malcolm adds, that, having calculated the ratios of them, for his own satisfaction, he found more of them false than in the preceding scale, but then their errors were considerably less, which made amends. Malcolm's Music, chap. 10. § 2.</p><p>SENSIBLE <hi rend="italics">Horizon,</hi> or <hi rend="italics">Point,</hi> or <hi rend="italics">Quality,</hi> &c. See the substantives.</p></div1><div1 part="N" n="SEPTUAGESIMA" org="uniform" sample="complete" type="entry"><head>SEPTUAGESIMA</head><p>, in the Calendar, is the 9th Sunday before Easter, so called, as some have supposed, because it is near 70 days, though in reality it is only 63 days, before it.</p></div1><div1 part="N" n="SERIES" org="uniform" sample="complete" type="entry"><head>SERIES</head><p>, in Algebra, denotes a rank or progression of quantities or terms, which usually proceed according to some certain law. As the Series 1 + (1/2) + (1/4) + (1/8) + (1/16) &c, or the Series, 1 + (1/2) + (1/3) + (1/4) + (1/5) &c. where the former is a geometrical Series, proceeding by the constant division by 2, or the denominators multiplied by 2; and the latter is an harmonical Series, being the reciprocals of the arithmetical Series 1, 2, 3, 4, &c, or the denominators being continually increased by 1.</p><p>The doctrine and use of Series, one of the greatest improvements of the present age, we owe to Nicholas Mercator; though it seems he took the first hint of it from Dr. Wallis's Arithmetic of Infinites; but the genius of Newton first gave it a body and a form.</p><p>It is chiefly useful in the quadrature of curves; where, as we often meet with quantities which cannot be expressed by any precise definite numbers, such as is the ratio of the diameter of a circle to the circumference, we are glad to express them by a Series, which, infinitely continued, is the value of the quantity sought, and which is called an Infinite Series. <hi rend="center"><hi rend="italics">The Nature, Origin, &c, of</hi> <hi rend="smallcaps">Series.</hi></hi></p><p>Infinite Series commonly arise, either from a continued division, as was practised by Mercator, or the extraction of roots, as first performed by Newton, who also explained other general ways for the expanding of quantities into infinite Series, as by the binomial theorem. Thus, to divide 1 by 3, or to expand the fraction 1/3 into an infinite Series; by division in decimals in the ordinary way, the series is 0.3333 &c, or (3/10) + (3/100) + (3/1000) + (3/10000) &c, where the law of <cb/> continuation is manifest. Or, if the same fraction <*>/3 be set in this form 1/(2 + 1), and division be performed in the algebraic manner, the quotient will be &c. Or, if it be expressed in this form 1/3 = 1/(4 - 1), by a like division there will arise the Series, &c. And, thus, by dividing 1 by 5 - 2, or 6 - 3, or 7 - 4, &c, the Series answering to the fraction 1/3, may be found in an endless variety of infinite Series; and the finite quantity 1/3 is called the value or radix of the Series, or also its sum, being the number or sum to which the Series would amount, or the limit to which it would tend or approximate, by summing up its terms, or by collecting them together one after another.</p><p>In like manner, by dividing 1 by the algebraic sum <hi rend="italics">a</hi> + <hi rend="italics">c,</hi> or by <hi rend="italics">a</hi> - <hi rend="italics">c,</hi> the quotient will be in these two cases, as below, viz, &c. where the terms of each Series are the same, and they differ only in this, that the signs are alternately positive and negative in the former, but all positive in the latter.</p><p>And hence, by expounding <hi rend="italics">a</hi> and <hi rend="italics">c</hi> by any numbers whatever, we obtain an endless variety of infinite Series, whose sums or values are known. So, by taking <hi rend="italics">a</hi> or <hi rend="italics">c</hi> equal to 1 or 2 or 3 or 4, &c, we obtain these Series, and their values; &c.</p><p>And hence it appears, that the same quantity or radix may be expressed by a great variety of infinite Series, or that many different Series may have the same radix or sum.</p><p>Another way in which an infinite Series arises, is by the extraction of roots. Thus, by extracting the square root of the number 3 in the common way, we obtain its value in a series as follows, viz, &c; in which way of resolution the law of the progression <pb n="436"/><cb/> of the Series is not visible, as it is when sound by division. And the square root of the algebraic quantity <hi rend="italics">a</hi><hi rend="sup">2</hi> + <hi rend="italics">c</hi><hi rend="sup">2</hi> gives &c.</p><p>And a 3d way is by Newton's binomial theorem, which is a universal method, that serves for all sorts of quantities, whether fractional or radical ones: and by this means the same root of the last given quantity becomes &c. where the law of continuation is visible.</p><p>See <hi rend="smallcaps">Extraction</hi> <hi rend="italics">of Roots,</hi> and <hi rend="smallcaps">Binomial</hi> <hi rend="italics">Theorem.</hi></p><p>From the specimens above given, it appears that the signs of the terms may be either all plus, or alternately plus and minus. Though they may be varied in many other ways. It also appears that the terms may be either continually smaller and smaller, or larger and larger, or else all equal. In the first case therefore the Series is said to be a <hi rend="italics">decreasing</hi> one, in the 2d case an <hi rend="italics">increasing</hi> one, and in the 3d case an <hi rend="italics">equal</hi> one. Also the first Series is called a <hi rend="italics">converging</hi> one, because that by collecting its terms successively, taking in always one term more, the successive sums approximate or converge to the value or sum of the whole infinite Series. So, in the Series , &c, the first term 1/3 is too little, or below 1/2 which is the value or sum of the whole infinite Series proposed; the sum of the first two terms (1/3) + (1/9) is 4/9 = .4444 &c, is also too little, but nearer to 1/2 or .5 than the former; and the sum of three terms (1/3) + (1/9) + (1/27) is 13/27 = .481481 &c, is nearer than the last, but still too little; and the sum of four terms &c. which is again nearer than the former, but still too little; which is always the case when the terms are all positive. But when the converging Series has its terms alternately positive and negative, then the successive sums are alternately too great and too little, though still approaching nearer and nearer to the final sum or value. Thus in the Series &c, are too great, <cb/> four terms &c, are too great, and so on, alternately too great and too small, but every succeeding sum still nearer than the former, or converging.</p><p>In the second case, or when the terms grow larger and larger, the Series is called a <hi rend="italics">diverging</hi> one, because that by collecting the terms continually, the successive sums diverge, or go always farther and farther from the true value or radix of the Series; being all too great when the terms are all positive, but alternately too great and too little when they are alternately positive and negative. Thus, in the Series &c. the first term + 1 is too great, two terms 1 - 2 = - 1 are too little, three terms 1 - 2 + 4 = + 3 are too great, four terms 1 - 2 + 4 - 8 = - 5 are too little, and so on continually, after the 2d term, diverging more and more from the true value or radix 1/3, but alternately too great and too little, or positive and negative. But the alternate sums would be always more and more too great if the terms were all positive, and always too little if negative.</p><p>But in the third case, or when the terms are all equal, the Series of equals, with alternate signs, is called a <hi rend="italics">neutral</hi> one, because the successive sums, found by a continual collection of the terms, are always at the same distance from the true value or radix, but alternately positive and negative, or too great and too little. Thus, in the Series &c, the first term 1 is too great, two terms 1 - 1 = 0 are too little, three terms 1 - 1 + 1 = 1 too great, four terms 1 - 1 + 1 - 1 = 0 too little, and so on continually, the successive sums being alternately 1 and 0, which are equally different from the true value or radix 1/2, the one as much above it, as the other below it.</p><p>A Series may be terminated and rendered finite, and accurately equal to the sum or value, by assuming the supplement, after any particular term, and combining it with the foregoing terms. So, in the Series (1/2) - (1/4) + (1/8) - (1/16) &c, which is equal to 1/3, and found by dividing 1 by 2 + 1, after the first term, 1/2, of the quotient, the remainder is - (1/2), which divided by 2 + 1, or 3, gives - (1/6) for the supplement, which <pb n="437"/><cb/> combined with the first term 1/2, gives (1/2) - (1/6) = 1/3 the true sum of the Series. Again, after the first two terms (1/2) - (1/4), the remainder is + (1/4), which divided by the same divisor 3, gives 1/12 for the supplement, and this combined with those two terms (1/2) - (1/4), makes or 1/3 the same sum or value as before. And in general, by dividing 1 by <hi rend="italics">a</hi> + <hi rend="italics">c,</hi> there is obtained ; where, stopping the division at any term as <hi rend="italics">c<hi rend="sup">n</hi></hi>/<hi rend="italics">a</hi><hi rend="sup"><hi rend="italics">n</hi> + 1</hi>, the remainder after this term is <hi rend="italics">c</hi><hi rend="sup"><hi rend="italics">n</hi> + 1</hi>/<hi rend="italics">a</hi><hi rend="sup"><hi rend="italics">n</hi> + 1</hi>, which being divided by the same divisor <hi rend="italics">a</hi> + <hi rend="italics">c,</hi> gives <hi rend="italics">c</hi><hi rend="sup"><hi rend="italics">n</hi> + 1</hi>/(<hi rend="italics">a</hi><hi rend="sup"><hi rend="italics">n</hi> + 1</hi> (<hi rend="italics">a</hi> + <hi rend="italics">c</hi>)) for the supplement as above.</p><p><hi rend="italics">The Law of Continuation.</hi>—A Series being proposed, one of the chief questions concerning it, is to find the law of its continuation. Indeed, no universal rule can be given for this; but it often happens that the terms of the Series, taken two and two, or three and three, or in greater numbers, have an obvious and simple relation, by which the Series may be determined and produced indefinitely. Thus, if 1 be divided by 1 - <hi rend="italics">x,</hi> the quotient will be a geometrical progression, viz, 1 + <hi rend="italics">x</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi> + <hi rend="italics">x</hi><hi rend="sup">3</hi> &c, where the succeeding terms are produced by the continual multiplication by <hi rend="italics">x.</hi> In like manner, in other cases of division, other progressions are produced.</p><p>But in most cases the relation of the terms of a Series is not constant, as it is in those that arise by division. Yet their relation often varies according to a certain law, which is sometimes obvious on inspection, and sometimes it is found by dividing the successive terms one by another, &c. Thus, in the Series 1 + ((2/3)<hi rend="italics">x</hi>) + ((8/15)<hi rend="italics">x</hi><hi rend="sup">2</hi>) + ((16/35)<hi rend="italics">x</hi><hi rend="sup">3</hi>) + ((128/315)<hi rend="italics">x</hi><hi rend="sup">4</hi>)) &c, by dividing the 2d term by the 1st, the 3d by the 2d, the 4th by the 3d, and so on, the quotients will be (2/3)<hi rend="italics">x,</hi> (4/5)<hi rend="italics">x,</hi> (6/7)<hi rend="italics">x,</hi> (8/9)<hi rend="italics">x,</hi> &c; and therefore the terms may be continued indefinitely by the successive multiplication by these fractions. Also in the following Series 1 + ((1/6)<hi rend="italics">x</hi>)) + ((3/40)<hi rend="italics">x</hi><hi rend="sup">2</hi>) + ((5/128)<hi rend="italics">x</hi><hi rend="sup">3</hi>) + ((35/1152)<hi rend="italics">x</hi><hi rend="sup">4</hi>) &c, by dividing the adjacent terms successively by each other, the Series of quotients is <cb/> (1/6)<hi rend="italics">x,</hi> (9/20)<hi rend="italics">x,</hi> (25/42)<hi rend="italics">x,</hi> (49/72)<hi rend="italics">x,</hi> &c, or (1.1/2.3)<hi rend="italics">x,</hi> (3.3/4.5)<hi rend="italics">x,</hi> (5.5/6.7)<hi rend="italics">x,</hi> (7.7/8.9)<hi rend="italics">x,</hi> &c; and therefore the terms of the Series may be continued by the multiplication of these fractions.</p><p>Another method of expressing the law of a Series, is one that defines the Series itself, by its <hi rend="italics">general term,</hi> shewing the relation of the terms generally by their distances from the beginning, or by differential equations. To do this, Mr. Stirling conceives the terms of the Series to be placed as so many ordinates on a right line given by position, taking unity as the common interval between these ordinates. The terms of the Series he denotes by the initial letters of the alphabet, A, B, C, D, &c; A being the first, B the 2d, C the 3d, &c: and he denotes any term in general by the letter T, and the rest following it in order by T′, T″, T‴, T′′′′, &c; also the distance of the term T from any given term, or from any given intermediate point between two terms, he denotes by the indeterminate quantity <hi rend="italics">z:</hi> so that the distances of the terms T′, T″, T‴, &c, from the said term or point, will be <hi rend="italics">z</hi> + 1, <hi rend="italics">z</hi> + 2, <hi rend="italics">z</hi> + 3, &c; because the increment of the absciss is the common interval of the ordinates, or terms of the Series, applied to the absciss.</p><p>These things being premised, let this Series be proposed, viz, 1, (1/2)<hi rend="italics">x,</hi> (3/8)<hi rend="italics">x</hi><hi rend="sup">2</hi>, (5/16)<hi rend="italics">x</hi><hi rend="sup">3</hi>, (35/128)<hi rend="italics">x</hi><hi rend="sup">4</hi>, (63/256)<hi rend="italics">x</hi><hi rend="sup">5</hi>, &c; in which it is found, by dividing the terms by each other, that the relations of the terms are, , &c: then the relation in general will be defined by the equation , where <hi rend="italics">z</hi> denotes the distance of T from the first term of the SeriesFor by substituting 0, 1, 2, 3, 4, &c, successively instead of <hi rend="italics">z,</hi> the same relations will arise as in the proposed Series above. In like manner, if <hi rend="italics">z</hi> be the distance of T from the 2d term of the Series, the equation will be , as will appear by substituting the numbers - 1, 0, 1, 2, 3, &c, successively for <hi rend="italics">z.</hi> Or, if <hi rend="italics">z</hi> denote the place or number of the term T in the Series, its successive values will be 1, 2, 3, 4, &c, and the equation or general term will be .</p><p>It appears therefore, that innumerable differential equations may define one and the same Series, according to the different points from whence the origin of the absciss <hi rend="italics">z</hi> is taken. And, on the contrary, the same equation defines innumerable different Series, by taking different successive values of <hi rend="italics">z.</hi> For in the equation , which defines the foregoing Series <pb n="438"/><cb/> when 1, 2, 3, 4, &c are the successive values of the abscisses; if 1 1/2, 2 1/2, 3 1/2, 4 1/2, &c, be successively substituted for <hi rend="italics">z,</hi> the relations of the terms arising will be, , &c, from whence will arise the Series A, (2/3)A<hi rend="italics">x,</hi> (8/15)A<hi rend="italics">x</hi><hi rend="sup">2</hi>, (16/35)A<hi rend="italics">x</hi><hi rend="sup">3</hi>, (128/315)A<hi rend="italics">x</hi><hi rend="sup">4</hi>, &c, which is different from the former.</p><p>And thus the equation will always determine the Series from the given values of the absciss and of the first term, when the equation includes but two terms of the Series, as in the last example, where the first term being given, all the rest will be given.</p><p>But when the equation includes three terms, then two must be given; and three must be given, when it includes four; and so on. So, if there be proposed the Series <hi rend="italics">x,</hi> (1/6)<hi rend="italics">x</hi><hi rend="sup">3</hi>, (3/40)<hi rend="italics">x</hi><hi rend="sup">5</hi>, (5/128)<hi rend="italics">x</hi><hi rend="sup">7</hi>, (35/1152)<hi rend="italics">x</hi><hi rend="sup">9</hi>, &c, where the relations of the terms are, , &c. the equation defining this Series will be , where the successive values of <hi rend="italics">z</hi> are 1, 2, 3, 4, &c. See Stirling's Methodus Differentialis, in the introduction.</p><p>This may suffice to give a notion of these differential equations, defining the nature of Series. But as to the application of these equations in interpolations, and finding the sums of Series, it would require a treatise to explain it. We must therefore refer to that excellent one just quoted, as also to De Moivre's Miscellanea Analytica; and several curious papers by Euler in the Acta Petropolitana.</p><p>A Series often converges so slowly, as to be of no use in practice. Thus, if it were required to find the sum of the Series (1/(1.2)) + (1/(3.4)) + (1/(5.6)) + (1/(7.8)) + (1/(9.10)) &c, which Lord Brouncker found for the quadrature of the hyperbola, true to 9 figures, by the mere addition of the terms of the Series; Mr. Stirling computes that it would be necessary to add a thousand millions of terms for that purpose; for which the life of man would be too short. But by that gentleman's method, the sum of the Series may be found by a very moderate computation. See Method. Differ. pa. 26.</p><p>Series are of various kinds or descriptions. So,</p><p>An <hi rend="italics">Ascending</hi> <hi rend="smallcaps">Series</hi>, is one in which the powers of the indeterminate quantity increase; as 1 + <hi rend="italics">ax</hi> + <hi rend="italics">bx</hi><hi rend="sup">2</hi> + <hi rend="italics">cx</hi><hi rend="sup">3</hi> &c. And a</p><p><hi rend="italics">Descending</hi> <hi rend="smallcaps">Series</hi>, is one in which the powers decrease, or else increase in the denominators, which is the same thing; as 1 + <hi rend="italics">ax</hi><hi rend="sup">- 1</hi> + <hi rend="italics">bx</hi><hi rend="sup">- 2</hi> + <hi rend="italics">cx</hi><hi rend="sup">- 3</hi> &c, or 1 + (<hi rend="italics">a</hi>/<hi rend="italics">x</hi>) + (<hi rend="italics">b</hi>/<hi rend="italics">x</hi><hi rend="sup">2</hi>) + (<hi rend="italics">c</hi>/<hi rend="italics">x</hi><hi rend="sup">3</hi>) &c.</p><p><hi rend="italics">A Circular</hi> <hi rend="smallcaps">Series</hi>, which denotes a Series whose <cb/> sum depends on the quadrature of the circle. Such is the Series 1 + (1/3) + (1/5) - (1/7) + (1/9) &c: See Demoivre Miscel. Analyt. pa. 111, or my Mensur. pa. 119. Or the sum of the Series 1 + (1/4) + (1/9) + (1/16) + (1/25) &c, continued ad insinitum, according to Euler's discovery.</p><p><hi rend="italics">Continued Fraction</hi> or <hi rend="italics">Series,</hi> is a fraction of this kind, to infinity, . The first Series of this kind was given by Lord Brouncker, first president of the Royal Society, for the quadrature of the circle, as related by Dr. Wallis, in his Algebra, pa. 317. His series is , which denotes the ratio of the square of the diameter of a circle to its area. Mr. Euler has treated on this kind of Series, in the Petersburgh Commentaries, vol. 11, and in his Analys. Infinit. vol. 1, pa. 295, where he shews various uses of it, and how to transform ordinary fractions and common Series into continued fractions. A common fraction is transformed into a continued one, after the manner of seeking the greatest common measure of the numerator and denominator, by dividing the greater by the less, and the last divisor always by the last remainder. Thus to change 1461/59 to a continued fraction. <figure/> <pb n="439"/><cb/></p><p><hi rend="italics">Converging</hi> <hi rend="smallcaps">Series</hi>, is a Series whose terms continually decrease, or the successive sums of whose terms approximate or converge always nearer to the ultimate sum of the whole Series. And, on the contrary, a</p><p><hi rend="italics">Diverging</hi> <hi rend="smallcaps">Series</hi>, is one whose terms continually increase, or that has the successive sums of its terms diverging, or going off always the farther, from the sum or value of the Series.</p><p><hi rend="italics">Determinate</hi> <hi rend="smallcaps">Series</hi>, is a Series whose terms proceed by the powers of a determinate quantity; as 1 + (1/2) + (1/2<hi rend="sup">2</hi>) + (1/2<hi rend="sup">3</hi>) + &c. If that determinate quantity be unity, the Series is said to be determined by unity. De Moivre, Miscel. Analyt. pa. 111. And an</p><p><hi rend="italics">Indeterminate</hi> <hi rend="smallcaps">Series</hi> is one whose terms proceed by the powers of an indeterminate quantity <hi rend="italics">x;</hi> as <hi rend="italics">x</hi> + ((1/2)<hi rend="italics">x</hi><hi rend="sup">2</hi>) + ((1/3)<hi rend="italics">x</hi><hi rend="sup">3</hi>) + ((1/4)<hi rend="italics">x</hi><hi rend="sup">4</hi>) &c; or sometimes also with indeterminate exponents, or indeterminate coefficients.</p><p>The <hi rend="italics">Form of a</hi> <hi rend="smallcaps">Series</hi>, is used for that affection of an indeterminate Series, such as <hi rend="italics">ax<hi rend="sup">n</hi></hi> + <hi rend="italics">bx</hi><hi rend="sup"><hi rend="italics">n</hi> + <hi rend="italics">r</hi></hi> + <hi rend="italics">cx</hi><hi rend="sup"><hi rend="italics">n</hi> + 2<hi rend="italics">r</hi></hi> + <hi rend="italics">dx</hi><hi rend="sup"><hi rend="italics">n</hi> + 3<hi rend="italics">r</hi></hi> &c, which arises from the different values of the indices of <hi rend="italics">x.</hi> Thus, If <hi rend="italics">n</hi> = 1, and <hi rend="italics">r</hi> = 1, the Series will take the form <hi rend="italics">ax</hi> + <hi rend="italics">bx</hi><hi rend="sup">2</hi> + <hi rend="italics">cx</hi><hi rend="sup">3</hi> + <hi rend="italics">dx</hi><hi rend="sup">4</hi> &c. If <hi rend="italics">n</hi> = 1, and <hi rend="italics">r</hi> = 2, the form will be <hi rend="italics">ax</hi> + <hi rend="italics">bx</hi><hi rend="sup">3</hi> + <hi rend="italics">cx</hi><hi rend="sup">5</hi> + <hi rend="italics">dx</hi><hi rend="sup">7</hi> &c. If <hi rend="italics">n</hi> = 1/2, and <hi rend="italics">r</hi> = 1, the form is <hi rend="italics">ax</hi><hi rend="sup">1/2</hi> + <hi rend="italics">bx</hi><hi rend="sup">3/2</hi> + <hi rend="italics">cx</hi><hi rend="sup">3/2</hi> + <hi rend="italics">dx</hi><hi rend="sup">7/2</hi> &c. And If <hi rend="italics">n</hi> = 0, and <hi rend="italics">r</hi> = - 1, the form will be <hi rend="italics">a</hi> + <hi rend="italics">bx</hi><hi rend="sup">- 1</hi> + <hi rend="italics">cx</hi><hi rend="sup">-2</hi> + <hi rend="italics">dx</hi><hi rend="sup">- 3</hi> &c.</p><p>When the value of a quantity cannot be found exactly, it is of use in algebra, as well as in common arithmetic, to seek an approximate value of that quantity, which may be useful in practice. Thus, in arithmetic, as the true value of the square root of 2 cannot be assigned, a decimal fraction is found to a sufficient degree of exactness in any particular case; which decimal fraction is in reality, no more than an infinite series of fractions converging or approximating to the true value of the root sought. For the expression √2 = 1.414213 &c, is equivalent to this &c; or supposing <hi rend="italics">x</hi> = 10, to this &c, which last Series is a particular case of the more general indeterminate Series <hi rend="italics">ax<hi rend="sup">n</hi></hi> + <hi rend="italics">bx</hi><hi rend="sup"><hi rend="italics">n</hi> + <hi rend="italics">r</hi></hi> + <hi rend="italics">cx</hi><hi rend="sup"><hi rend="italics">n</hi> + 2<hi rend="italics">r</hi></hi> &c, viz, when <hi rend="italics">n</hi> = 0, <hi rend="italics">r</hi> = - 1, and the coefficients <hi rend="italics">a</hi> = 1, <hi rend="italics">b</hi> = 4, <hi rend="italics">c</hi> = 1, <hi rend="italics">d</hi> = 4, &c.</p><p>But the application of the notion of approximations in numbers, to species, or to algebra, is not so obvious. Newton, with his usual sagacity, took the hint, <cb/> and prosecuted it; by which were discovered general methods in the doctrine of infinite Series, which had before been treated only in a particular manner, though with great acuteness, by Wallis and a few others. See Newton's Method of Fluxions and Infinite Series, with Colson's Comment; as also the Analysis per Æquationes Numero Terminorum Infinitas, published by Jones in 1711, and since translated and explained by Stewart, together with Newton's Tract on Quadratures, in 1745. To these may be added Maclaurin's Algebra, part 2, chap. 10, pa. 244; and Cramer's Analyse des Lignes Courbes Algebraiques, chap. 7, pa. 148; and many other authors.</p><p>Among the various methods for determining the value of a quantity by a converging Series, that seems preferable to the rest, which consists in assuming an indeterminate Series as equal to the quantity whose value is sought, and afterwards determining the values of the terms of this assumed Series. For instance, suppose a logarithm were given, to find the natural number answering to it. Suppose the logarithm to be <hi rend="italics">z,</hi> and the corresponding number sought 1 + <hi rend="italics">x:</hi> then by the nature of logarithms and fluxions, . Now assume a Series for the value of the unknown quantity <hi rend="italics">x,</hi> and substitute it and its fluxion instead of <hi rend="italics">x</hi> and <hi rend="italics">x</hi><hi rend="sup">.</hi> in the last equation, then determine the assumed coefficients by comparing or equating the like terms of the equation. Thus, &c; hence, comparing the like terms of these two values of <hi rend="italics">x</hi><hi rend="sup">.</hi>, there arises <hi rend="italics">a</hi> = 1, <hi rend="italics">b</hi> = 1/2, <hi rend="italics">c</hi> = 1/6, <hi rend="italics">d</hi> = 1/24, &c; which values being substituted for <hi rend="italics">a, b, c,</hi> &c, in the assumed Series <hi rend="italics">ax</hi> + <hi rend="italics">bx</hi><hi rend="sup">2</hi> + <hi rend="italics">cx</hi><hi rend="sup">3</hi> &c, it gives &c; and consequently the number sought will be &c.</p><p>But the indeterminate Series <hi rend="italics">az</hi> + <hi rend="italics">bz</hi><hi rend="sup">2</hi> + <hi rend="italics">cz</hi><hi rend="sup">3</hi> &c, was here assumed arbitrarily, with regard to its exponents 1, 2, 3, &c, and will not succeed in all cases, because some quantities require other forms for the exponents. For instance, if from an arc given, it were required to find the tangent. Let <hi rend="italics">x</hi> = the tangent, and <hi rend="italics">z</hi> the arc, the radius being = 1. Then, from the nature of the circle we shall have . Now if, to find the value of <hi rend="italics">x,</hi> we suppose &c, and proceed as before, we shall find all the alternate coefficients <hi rend="italics">b, d, f,</hi> &c, or those of the even powers of <hi rend="italics">z,</hi> to be each = 0; and therefore the Series assumed is not of a proper form. <pb n="440"/><cb/> But supposing , &c, then we find <hi rend="italics">a</hi> = 1, <hi rend="italics">b</hi> = 1/3, <hi rend="italics">c</hi> = 2/15, <hi rend="italics">d</hi> = 17/315, &c, and consequently &c. And other quantities require other forms of Series.</p><p>Now to find a proper indeterminate Series in all cases, tentatively, would often be very laborious, and even impracticable. Mathematicians have therefore endeavoured to find out a general rule for this purpose; though till lately the method has been but imperfectly understood and delivered. Most authors indeed have explained the manner of finding the coefficients <hi rend="italics">a, b, c, d,</hi> &c, of the indeterminate Series <hi rend="italics">ax<hi rend="sup">n</hi></hi> + <hi rend="italics">bx</hi><hi rend="sup"><hi rend="italics">n</hi> + <hi rend="italics">r</hi></hi> + <hi rend="italics">cx</hi><hi rend="sup"><hi rend="italics">n</hi> + 2<hi rend="italics">r</hi></hi> &c, which is easy enough; but the values of <hi rend="italics">n</hi> and <hi rend="italics">r,</hi> in which the chief difficulty lies, have been assigned by many in a manner as if they were self-evident, or at least discoverable by an easy trial or two, as in the last example.</p><p>As to the number <hi rend="italics">n,</hi> Newton himself has shewn the method of determining it, by his rule for finding the first term of a converging Series, by the application of his parallelogram and ruler. For the particulars of this method, see the authors above cited; see also P<hi rend="smallcaps">ARALLELOGRAM.</hi></p><p>Taylor, in his Methodus Incrementorum, investigates the number <hi rend="italics">r;</hi> but Stirling observes that his rule sometimes fails. Lineæ Tert. Ordin. Newton. pa. 28. Mr. Stirling gives a correction of Taylor's rule, but says he cannot affirm it to be universal, having only found it by chance. And again</p><p>Gravesande observes, that though he thinks Stirling's rule never leads into an error, yet that it is not perfect. See Gravesande, De Determin. Form. Seriei Infinit. printed at the end of his Matheseos Universalis Elementa. This learned professor has endeavoured to rectify the rule. But Cramer has shewn that it is still defective in several respects; and he himself, to avoid the inconveniences to which the methods of former authors are subject, has had recourse to the first principles of the method of infinite Series, and has entered into a more exact and instructive detail of the whole method, than is to be met with elsewhere; for which reason, and many others, his treatise deserves to be particularly recommended to beginners.</p><p>But it is to be observed, that in determining the value of a quantity by a converging Series, it is not always necessary to have recourse to an indeterminate Series: for it is often better to find it by division, or by extraction of roots. See Newton's Meth. of Flux. and Inf. Series, above cited. Thus, if it were required to find the arc of a circle from its given tangent, that is, to find the value of <hi rend="italics">z</hi> in the given fluxional equation, , by an infinite Series: dividing <hi rend="italics">x</hi><hi rend="sup">.</hi> by 1 + <hi rend="italics">xx,</hi> the quotient will be the Series ; and taking the fluents of the terms, there results &c, which is the Series often used for the quadrature of the circle. If <hi rend="italics">x</hi> = 1, or the tangent of 45°, then <cb/> will &c = the length of an arc of 45°, or 1/8 of the circumference, to the radius 1, or 1/4 of the circumference to the diameter 1. Consequently, if 1 be the diameter, then 1 - (1/3) + (1/5) - (1/7) &c will be the area of the circle, because 1/4 of the circumference multiplied by the diameter, gives the area of the circle. And this Series was first given by Leibnitz and James Gregory.</p><p>See the form of the Series for the binomial theorem, determined, both as to the coefficients and exponents, in my Tracts, vol. 1, pa. 79.</p><p><hi rend="italics">Harmonical</hi> <hi rend="smallcaps">Series</hi>, the reciprocal of arithmeticals. See <hi rend="smallcaps">Harmonical.</hi></p><p><hi rend="italics">Hyperbolic</hi> <hi rend="smallcaps">Series</hi>, is used for a Series whose sum depends upon the quadrature of the hyperbola. Such is the Series (1/1) + (1/2) + (1/3) + (1/4) &c. De Moivre's Miscel. Analyt. pa. 111.</p><p><hi rend="italics">Interpolation of</hi> <hi rend="smallcaps">Series</hi>, the inserting of some terms between others, &c. See <hi rend="smallcaps">Interpolation.</hi></p><p><hi rend="italics">Interscendent</hi> <hi rend="smallcaps">Series.</hi> See <hi rend="smallcaps">Interscendent.</hi></p><p><hi rend="italics">Mixt</hi> <hi rend="smallcaps">Series</hi>, one whose sum depends partly on the quadrature of the circle, and partly on hit of the hyperbola. De Moivre, Miscel. Analyt. pa. 111.</p><p><hi rend="italics">Recurring</hi> <hi rend="smallcaps">Series</hi>, is used for a Series which is so constituted, that having taken at pleasure any number of its terms, each following term shall be related to the same number of preceding terms according to a constant law of relation. Thus, in the following Series, <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">e</hi></cell><cell cols="1" rows="1" role="data"> <hi rend="italics">f</hi></cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1 +</cell><cell cols="1" rows="1" role="data">2<hi rend="italics">x</hi> +</cell><cell cols="1" rows="1" role="data">3<hi rend="italics">x</hi><hi rend="sup">2</hi> +</cell><cell cols="1" rows="1" role="data">10<hi rend="italics">x</hi><hi rend="sup">3</hi> +</cell><cell cols="1" rows="1" role="data">34<hi rend="italics">x</hi><hi rend="sup">4</hi> +</cell><cell cols="1" rows="1" role="data">97<hi rend="italics">x</hi><hi rend="sup">5</hi></cell><cell cols="1" rows="1" role="data">&c,</cell></row></table> in which the terms being respectively represented by the letters <hi rend="italics">a, b, c,</hi> &c, set over them, we shall have , &c, &c, where it is evident that the law of relation between <hi rend="italics">d</hi> and <hi rend="italics">e,</hi> is the same as between <hi rend="italics">e</hi> and <hi rend="italics">f,</hi> each being formed in the same manner from the three terms which precede it in the Series.</p><p>The quantities 3<hi rend="italics">x</hi> - 2<hi rend="italics">x</hi><hi rend="sup">2</hi> + 5<hi rend="italics">x</hi><hi rend="sup">3</hi>, taken together and connected by their proper signs, form what De Moivre calls the <hi rend="italics">index,</hi> or the <hi rend="italics">scale of relation;</hi> though sometimes the bare coefficients 3 - 2 + 5 are called the scale of relation. And the scale of relation subtracted from unity, is called the <hi rend="italics">differential scale.</hi> On the subject of Recurring Series, see De Moivre's Miscel. Analyt. pa. 27 and 72, and his Doctrine of Chances, 3d edit. pa. 220; also Euler's Analys. Infinit. tom. 1, pa. 175.</p><p>Having given a recurring Series, with its scale of relation, the sum of the whole infinite Series will also be given. For instance, suppose a Series <pb n="441"/><cb/> <hi rend="italics">a</hi> + <hi rend="italics">bx</hi> + <hi rend="italics">cx</hi><hi rend="sup">2</hi> + <hi rend="italics">dx</hi><hi rend="sup">3</hi> &c, where the relation between the coefficient of any term and the coefficients of any two preceding terms may be expressed by <hi rend="italics">f</hi> - <hi rend="italics">g;</hi> that is, , &c; then will the sum of the Series, infinitely continued, be (<hi rend="italics">a</hi> + (<hi rend="italics">b</hi> - <hi rend="italics">fa</hi>) <hi rend="italics">x</hi>)/(1 - <hi rend="italics">fx</hi> + <hi rend="italics">gx</hi><hi rend="sup">2</hi>).</p><p>Thus, for example, assume 2 and 5 for the coefficients of the first two terms of a recurring Series; and suppose <hi rend="italics">f</hi> and <hi rend="italics">g</hi> to be respectively 2 and 1; then the recurring Series will be 2 + 5<hi rend="italics">x</hi> + 8<hi rend="italics">x</hi><hi rend="sup">2</hi> + 11<hi rend="italics">x</hi><hi rend="sup">3</hi> + 14<hi rend="italics">x</hi><hi rend="sup">4</hi> + 18<hi rend="italics">x</hi><hi rend="sup">5</hi> &c, and its sum . For the proof of which divide 2 + <hi rend="italics">x</hi> by (1 - <hi rend="italics">x</hi>)<hi rend="sup">2</hi>, and there arises the said Series 2 + 5<hi rend="italics">x</hi> + 8<hi rend="italics">x</hi><hi rend="sup">2</hi> + 11<hi rend="italics">x</hi><hi rend="sup">3</hi> &c. And similar rules might be derived for more complex cases.</p><p>De Moivre's general rule is this: 1. Take as many terms of the Series as there are parts in the scale of relation. 2. Subtract the scale of relation from unity, and the Remainder is the differential scale. 3. Multiply the terms taken in the Series by the differential scale, beginning at unity, and so proceeding orderly, remembering to leave out what would naturally be extended beyond the last of the terms taken. Then will the product be the numerator, and the differential scale will be the denominator of the fraction expressing the sum required.</p><p>But it must here be observed, that when the sum of a recurring Series extended to infinity, is found by De Moivre's rule, it ought to be supposed that the Series converges indefinitely, that is, that the terms may become less than any assigned quantity. For if the Series diverge, that is, if its terms continually increase, the rule does not give the true sum. For the sum in such case is infinite, or greater than any given quantity, whereas the sum exhibited by the rule, will often be finite. The rule therefore in this case only gives a fraction expressing the radix of the Series, by the expansion of which the Series is produced. Thus 1/((1 - <hi rend="italics">x</hi>)<hi rend="sup">2</hi>) by expansion becomes the recurring Series 1 + 2<hi rend="italics">x</hi> + 3<hi rend="italics">x</hi><hi rend="sup">2</hi> &c, whose scale of relation is 2 - 1, and its sum by the rule will be , the quantity from which the Series arose. But this quantity cannot in all cases be deemed equal to the infinite Series 1 + 2<hi rend="italics">x</hi> + 3<hi rend="italics">x</hi><hi rend="sup">2</hi> &c: for stop where you will, there will always want a supplement to make the product of the quotient by the divisor equal to the dividend. Indeed when the Series converges infinitely, the supplement, diminishing continually, becomes less than any assigned quantity, or equal to nothing; but in a diverging Series, this supplement becomes insinitely great, and the Series deviates indefinitely from the truth. See Colson's Comment on Newton's Method of Fluxions and Infinite Series, pa. 152; Stirling's Method. Differ. pa. 36; Bernoulli de Serieb. Insin. pa. 249; and Cramer's Analyse des Lignes Courbes, pa. 174.</p><p>A recurring Series being given, the sum of any <cb/> finite number of the terms of that Series may be found. This is prob. 3, pa. 73, De Moivre's Miscel. Analyt. and prob. 5, pa. 223 of his Doctrine of Chances. The solution is effected, by taking the difference between the sums of two infinite Series, differing by the terms answering to the given number; viz, from the sum of the whole infinite Series, commencing from the beginning, subtract the sum of another insinite number of terms of the same Series, commencing after so many of the first terms whose sum is required; and the difference will evidently be the sum of that number of terms of the Series. For example, to find the sum of <hi rend="italics">n</hi> terms of the infinite geometrical Series <hi rend="italics">a</hi> + <hi rend="italics">ax</hi> + <hi rend="italics">ax</hi><hi rend="sup">2</hi> + <hi rend="italics">ax</hi><hi rend="sup">3</hi> &c. Here are two insinite Series; the one beginning with <hi rend="italics">a,</hi> and the other with <hi rend="italics">ax<hi rend="sup">n</hi>,</hi> which is the next term after the first <hi rend="italics">n</hi> terms of the original Series. By the rule, the sum of the first infinite progression will be <hi rend="italics">a</hi>/(1 - <hi rend="italics">x</hi>), and the sum of the second <hi rend="italics">ax<hi rend="sup">n</hi></hi>/(1 - <hi rend="italics">x</hi>); the difference of which is (<hi rend="italics">a</hi> - <hi rend="italics">ax<hi rend="sup">n</hi></hi>)/(1 - <hi rend="italics">x</hi>), which is therefore the sum of the first <hi rend="italics">n</hi> terms of the Series. This quantity (<hi rend="italics">a</hi> - <hi rend="italics">ax<hi rend="sup">n</hi></hi>)/(1 - <hi rend="italics">x</hi>) is equal to (<hi rend="italics">ax<hi rend="sup">n</hi></hi> - <hi rend="italics">a</hi>)/(<hi rend="italics">x</hi> - 1) which last expression, putting , will be equivalent to this, (<hi rend="italics">lx</hi> - <hi rend="italics">a</hi>)/(<hi rend="italics">x</hi> - 1), which is the common rule for finding the sum of any geometric progression, having given the first term <hi rend="italics">a,</hi> the last term <hi rend="italics">l,</hi> and the ratio <hi rend="italics">x.</hi> See Miscel. Analyt. pa. 167, 168.</p><p>In a recurring Series, any term may be obtained whose place is assigned. For after having taken so many terms of the Series as there are terms in the scale of relation, the Series may be protracted till it reach the place assigned. But when that place is very distant from the beginning of the Series, the continuing the terms is very laborious; and therefore other methods have been contrived. See Miscel. Analyt. pa. 33; and Doctrine of Chances, pa. 224.</p><p>These questions have been resolved in many cases, besides those of recurring Series. But as there is no universal method for the quadrature of curves, neither is there one for the summation of Series; indeed there is a great analogy between these things, and similar difficulties arising in both. See the authors above cited.</p><p>The investigation of Daniel Bernoulli's method for finding the roots of algebraic equations, which is inserted in the Petersburgh Acts, tom. 3, pa. 92, depends upon the doctrine of recurring Series. See Euler's Analysis Infinitorum, tom 1, pa. 276.</p><p><hi rend="italics">Reversion of</hi> <hi rend="smallcaps">Series.</hi> See <hi rend="smallcaps">Reversion</hi> <hi rend="italics">of Series.</hi></p><p><hi rend="italics">Summable</hi> <hi rend="smallcaps">Series</hi>, is one whose sum can be accurately found. Such is the Series 1/2 + 1/4 + 1/8 &c, the sum of which is said to be unity, or, to speak more accurately, the limit of its sum is unity or 1.</p><p>An indefinite number of summable insinite Series <pb n="442"/><cb/> may be assigned: such are, for instance, all infinite recurring converging Series, and many others, for which, consult De Moivre, Bernoulli, Stirling, Euler, and Maclaurin; viz, Miscel. Analyt. pa. 110; De Serieb. Infinit. passim; Method. Different. pa. 34; Acta Petrop passim; Fluxions, art. 350.</p><p>The obtaining the sums of infinite Serieses of fractions has been one of the principal objects of the modern method of computation; and these sums may often be found, and often not. Thus the sums of the two following Series of geometrical progressionals are easily found to be 1 and 1/2, viz, &c.</p><p>But the Serieses of fractions that occur in the solution of problems, can seldom be reduced to geometric progressions; nor can any general rule, in cases so infinitely various, be given. The art here, as in most other cases, is only to be acquired by examples, and by a careful observation of the arts used by great authors in the investigation of such Series of fractions as they have considered. And the general methods of infinite Series, which have been carried so far by De Moivre, Stirling, Enler, &c, are often found necessary to determine the sum of a very simple Series of fractions. See the quotations above.</p><p>The sum of a Series of fractions, though decreasing continually, is not always sinite. This is the case of the Series 1/1 + 1/2 + 1/3 + 1/4 + 1/5 &c, which is the harmonic Series, consisting of the reciprocals of arithmeticals, the sum of which exceeds any given number whatever; and this is shewn from the analogy between this progression and the space comprehended by the common hyperbola and its asymptote; though the same may be shewn also from the nature of progressions. See James Bernoulli, de Seriebus Infin. But, what is curious, the square of it is finite, for if the same terms of the harmonic Series, 1/1 + 1/2 + 1/3 &c, be squared, forming the Series 1/1 + 1/4 + 1/9 &c, being the reciprocals of the squares of the natural Series of numbers; the sum of this Series of fractions will not only be limited, but it is remarkable that this sum will be precisely equal to the 6th part of the number which expresses the ratio of the square of the circumference of a circle to the square of its diameter. That is, if <hi rend="italics">c</hi> denote 3.14159 &c, the ratio of the circumference to the diameter, then is &c. This property was first discovered by Euler; and his investigation may be seen in the Acta Petrop. vol. 7. And Maclaurin has since observed, that this may easily be deduced from his Fluxions, art. 822. Philos. Trans. numb. 469. <cb/></p><p>It would require a whole treatise to enumerate the various kinds of Series of fractions which may or may not be summed. Sometimes the sum cannot be assigned, either because it is infinite, as in the harmonic Series 1/1 + 1/2 + 1/3 + 1/4 &c, or although its sum be finite (as in the Series 1/1 + 1/4 + 1/9 &c), yet its sum cannot be assigned in finite terms, or by the quadrature of the circle or hyperbola, which was the case of this Series before Euler's discovery; but yet the sum of any given number of the terms of the Series may be expeditiously found, and the whole sum may be assigned by approximation, independent of the circle. See Stirling's Method. Different. and De Moivre's Miscel. Analyt.</p><p>Besides the Serieses of fractions, the sums of which converge to a certain quantity, there sometimes occur others, which converge by a continued multiplication. Of this kind is the Series found by Wallis, for the quadrature of the circle, which he expresses thus, , where the character □ denotes the ratio of the square of the diameter to the area of the circle. Hence the denominator of this fraction, is to its numerator, both insinitely continued, as the circle is to the square of the diameter. It may farther be observed that this Series is equivalent to (9/8) X (25/24) X (49/48) X &c, or to (3<hi rend="sup">2</hi>/(3<hi rend="sup">2</hi> - 1)) X (5<hi rend="sup">2</hi>/(5<hi rend="sup">2</hi> - 1)) X (7<hi rend="sup">2</hi>/(7<hi rend="sup">2</hi> - 1)) X &c, that is, the product of the squares of all the odd numbers 3, 5, 7, 9, &c, is to the product of the same squares severally diminished by unity, as the square of the diameter is to the area of the circle. See Arithmet. Insinit. prop. 191, Oper. vol. 1, pa. 469. Id. Oper. vol. 2, pa. 819. And these products of fractions, and the like quantities arising from the continued multiplication of certain factors, have been particularly considered by Euler, in his Analysis Infinit. vol. 1, chap. 15, pa. 221.</p><p>For an easy and general method of summing all alternate Series, such as <hi rend="italics">a</hi> - <hi rend="italics">b</hi> + <hi rend="italics">c</hi> - <hi rend="italics">d</hi> &c, see my Tracts, vol. 1, pa. 11; and in the same vol. may be seen many other curious tracts on infinite Series.</p><p><hi rend="italics">Summation of Insinite</hi> <hi rend="smallcaps">Series</hi>, is the finding the value of them, or the radix from which they may be raised. For which, consult all the authors upon this science.</p><p>To find an infinite Series by extracting of roots; and to find an infinite Series by a presupposed Series; see <hi rend="smallcaps">Quadrature</hi> <hi rend="italics">of the Circle.</hi></p><p>To extract the roots of an infinite Series, see E<hi rend="smallcaps">XTRACTION</hi> <hi rend="italics">of Roots.</hi></p><p>To raise an infinite Series to any power, see I<hi rend="smallcaps">NVOLUTION</hi>, and <hi rend="smallcaps">Power.</hi></p><p><hi rend="italics">Transcendental</hi> <hi rend="smallcaps">Series.</hi> See <hi rend="smallcaps">Transcendental.</hi></p><p>There are many other important writings upon the subject of Infinite Series, besides those above quoted. A very good elementary tract on this science is that of James Bernoulli, intituled, <hi rend="italics">Tractatus de Seriebus Infini-</hi> <pb n="443"/><cb/> <hi rend="italics">tis,</hi> and annexed to his <hi rend="italics">Ars Conjectandi,</hi> published in 4to, 1713.</p></div1><div1 part="N" n="SERPENS" org="uniform" sample="complete" type="entry"><head>SERPENS</head><p>, in Astronomy, a constellation in the northern hemisphere, being one of the 48 old constellations mentioned by all the Ancients, and is called more particularly <hi rend="italics">Serpens Ophiuchi,</hi> being grasped in the hands of the constellation Ophiuchus. The Greeks, in their fables, have ascribed it sometimes to one of Triptolemus's dragons, killed by Carnabos; and sometimes to the serpent of the river Segaris, destroyed by Hercules. This is by some supposed to be the same as the author of the book of Job calls the <hi rend="italics">Crooked Serpent;</hi> but this expression more probably meant the constellation Draco, near the north pole.</p><p>The stars in the constellation Serpens, in Ptolomy's catalogue are 18, in Tycho's 13, in Hevelius's 22, and in the Britannic catalogue 64.</p></div1><div1 part="N" n="SERPENTARIUS" org="uniform" sample="complete" type="entry"><head>SERPENTARIUS</head><p>, a constellation of the northern hemisphere, being one of the 48 old constellations mentioned by all the Ancients. It is called also Ophiuchus, and anciently Æsculapius. It is in the figure of a man grasping the serpent.</p><p>The Greeks had different fables about this, and other constellations, because they were ignorant of the true meaning of them. Some of them say, it represents Carnabos, who killed one of the dragons of Triptolemus. Others say, it was Hercules, killing the serpent at the river Segaris. And others again say, it represents the celebrated physician Æsculapius, to denote his skill in medicine to cure the bite of the serpent.</p><p>The stars in the constellation Serpentarius, in Ptolomy's catalogue are 29, in Tycho's 15, in Hevelius's 40, and in the Britannic catalogue they are 74.</p><p>SERPENTINE <hi rend="italics">Line,</hi> the same with spiral.</p></div1><div1 part="N" n="SESQUI" org="uniform" sample="complete" type="entry"><head>SESQUI</head><p>, an expression of a certain ratio, viz, the second ratio of inequality, called also <hi rend="italics">superparticular</hi> ratio; being that in which the greater term contains the less once, and some certain part over; as 3 to 2, where the first term contains the second once, and unity over, which is a quota part of 2. Now if this part remaining be just half the less term, the ratio is called <hi rend="italics">sesquialiera;</hi> if the remaining part be a 3d part of the less term, as 4 to 3, the ratio is called <hi rend="italics">sesquitertia,</hi> or <hi rend="italics">sesquiterza;</hi> if a 4th part, as 5 to 4, the ratio is called <hi rend="italics">sesquiquarta;</hi> and so on continually, still adding to Sesqui the ordinal number of the smaller term.</p><p>In English we sometimes say, <hi rend="italics">sesquialteral,</hi> or <hi rend="italics">sesquialterate, sesquithird, sesquifourtk,</hi> &c.</p><p>As to the kinds of triples expressed by the particle <hi rend="italics">sesqui,</hi> they are these:</p></div1><div1 part="N" n="SESQUIALTERATE" org="uniform" sample="complete" type="entry"><head>SESQUIALTERATE</head><p>, <hi rend="italics">the greater perfect,</hi> which is a triple, where the breve is three measures, or semibreves.</p><div2 part="N" n="Sesquialterate" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sesquialterate</hi></head><p>, <hi rend="italics">greater imperfect,</hi> which is where the breve, when pointed, contains three measures, and without any point, two.</p></div2><div2 part="N" n="Sesquialterate" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sesquialterate</hi></head><p>, <hi rend="italics">less imperfect,</hi> a triple, where the semibreve with a point contains three measures, and two without.</p></div2><div2 part="N" n="Sesquialterate" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sesquialterate</hi></head><p>, in Arithmetic and Geometry, is a ratio between two numbers, or lines, &c, where the greater is equal to once and a half of the less. Thus 6 and 9 are in a Sesquialterate ratio, as also 20 and 30.</p></div2></div1><div1 part="N" n="SESQUIDITONE" org="uniform" sample="complete" type="entry"><head>SESQUIDITONE</head><p>, in Music, a concord resulting <cb/> from the sounds of two strings whose vibrations, in equal times, are to each other in the ratio of 5 to 6.</p><p>SESQUIDUPLICATE <hi rend="italics">Ratio,</hi> is that in which the greater term contains the less, twice and a half; as the ratio of 15 to 6, or 50 to 20.</p></div1><div1 part="N" n="SESQUIQUADRATE" org="uniform" sample="complete" type="entry"><head>SESQUIQUADRATE</head><p>, an aspect or position of the planets, when they are distant by 4 signs and a half, or 135 degrees.</p></div1><div1 part="N" n="SESQUIQUINTILE" org="uniform" sample="complete" type="entry"><head>SESQUIQUINTILE</head><p>, is an aspect of the planets when they are distant 1/5 of the circle and a half, or 108 degrees.</p><p>SESQUITERTIONAL <hi rend="italics">Proportion,</hi> is that in which the greater term contains the less once and one third; as 4 to 3, or 12 to 9.</p></div1><div1 part="N" n="SETTING" org="uniform" sample="complete" type="entry"><head>SETTING</head><p>, in Astronomy, the withdrawing of a star or planet, or its sinking below the horizon.</p><p>Astronomers and poets count three different kinds of Setting of the stars, viz, <hi rend="smallcaps">Achronical, Cosmical</hi>, and <hi rend="smallcaps">Heliacal.</hi> See these terms respectively.</p><div2 part="N" n="Setting" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Setting</hi></head><p>, in Navigation, Surveying, &c, denotes the observing the bearing or situation of any distant object by the compass, &c, to discover the angle it makes with the nearest meridian, or with some other line. See <hi rend="smallcaps">Bearing.</hi></p><p>Thus, to <hi rend="italics">set the land,</hi> or <hi rend="italics">the fun,</hi> by the compass, is to observe how the land bears on any point of the compass, or on what point of the compass the sun is. Also, when two ships come in fight of each other, to mark on what point the chace bears, is termed <hi rend="italics">Setting thechace by the compass.</hi></p><p><hi rend="smallcaps">Setting</hi> also denotes the direction of the wind, current, or sea, particularly of the two latter.</p><p>SEVEN <hi rend="smallcaps">Stars</hi>, a common denomination given to the cluster of stars in the neck of the sign Taurus, the bull, properly called the pleiades. They are so called from their number Seven which appear to the naked eye, though some eyes can discover only 6 of them; but by the help of telescopes there appears to be a great multitude of them.</p></div2></div1><div1 part="N" n="SEVENTH" org="uniform" sample="complete" type="entry"><head>SEVENTH</head><p>, <hi rend="italics">Septima,</hi> an interval in Music, called by the Greeks <hi rend="italics">heptachordon.</hi></p></div1><div1 part="N" n="SEXAGENARY" org="uniform" sample="complete" type="entry"><head>SEXAGENARY</head><p>, something relating to the number 60.</p><p><hi rend="smallcaps">Sexagenary</hi> <hi rend="italics">Arithmetic.</hi> See <hi rend="smallcaps">Sexagesimal.</hi></p><p><hi rend="smallcaps">Sexagenary</hi> <hi rend="italics">Tables,</hi> are tables of proportional parts, shewing the product of two Sexagenaries that are to be multiplied, or the quotient of two that are to be divided.</p></div1><div1 part="N" n="SEXAGESIMA" org="uniform" sample="complete" type="entry"><head>SEXAGESIMA</head><p>, the eighth Sunday before Easter; being so called because near 60 days before it.</p><p>SEXAGESIMAL or <hi rend="smallcaps">Sexagenary</hi> <hi rend="italics">Arithmetic,</hi> a method of computation proceeding by 60ths. Such is that used in the division of a degree into 60 minutes, of the minute into 60 seconds, of the second into 60 thirds, &c.</p></div1><div1 part="N" n="SEXAGESIMALS" org="uniform" sample="complete" type="entry"><head>SEXAGESIMALS</head><p>, or <hi rend="smallcaps">Sexagesimal</hi> <hi rend="italics">Fractions,</hi> are fractions whose denominators proceed in a sexagecuple ratio; that is, a prime, or the first minute = 1/60, a second = 1/3600, a third = 1/216000.</p><p>Anciently there were no other than Sexagesimals used in astronomical operations, for which reason they are sometimes called <hi rend="italics">astronomical fractions,</hi> and they are still retained in many cases, as in the divisions of time and of a circle; but decimal arithmetic is now much <pb n="444"/><cb/> used in the calculations. Sexagesimals were probably first used for the divisions of a circle, 360, or 6 times 60 making up the whole circumference, on account that 360 days made up the year of the Ancients, in which time the sun was supposed to complete his course in the circle of the ecliptic.</p><p>In these fractions, the denominator being always 60, or a multiple of it, it is usually omitted, and the numerator only written down: thus, 3° 45′ 24″ 40‴ &c, is to be read, 3 degrees, 45 minutes, 24 seconds, 40 thirds, &c.</p></div1><div1 part="N" n="SEXANGLE" org="uniform" sample="complete" type="entry"><head>SEXANGLE</head><p>, in Geometry, a figure having 6 angles, and consequently 6 sides also.</p></div1><div1 part="N" n="SEXTANS" org="uniform" sample="complete" type="entry"><head>SEXTANS</head><p>, a sixth part of certain things.</p><p>The Romans divided their <hi rend="italics">as,</hi> which was a pound of brass, into 12 ounces, called <hi rend="italics">uncia,</hi> from <hi rend="italics">unum;</hi> and the quantity of 2 ounces was called <hi rend="italics">sextans,</hi> as being the 6th part of the pound.</p><p><hi rend="smallcaps">Sextans</hi> was also a measure, which contained 2 ounces of liquor, or 2 cyathi.</p><div2 part="N" n="Sextans" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sextans</hi></head><p>, the Sextant, in Astronomy, a new constellation, placed across the equator, but on the south side of the ecliptic, and by Hevelius made up of some unformed stars, or such as were not included in any of the 48 old constellations. In Hevelius's catalogue it contains 11 stars, but in the Britannic catalogue 41.</p></div2></div1><div1 part="N" n="SEXTANT" org="uniform" sample="complete" type="entry"><head>SEXTANT</head><p>, denotes the 6th part of a circle, or an arch containing 60 degrees.</p><p><hi rend="smallcaps">Sextant</hi> is more particularly used for an astronomical instrument. It is made like a quadrant, excepting that its limb only contains 60 degrees. Its use and application are the same with those of the <hi rend="smallcaps">Quadrant;</hi> which see.</p></div1><div1 part="N" n="SEXTARIUS" org="uniform" sample="complete" type="entry"><head>SEXTARIUS</head><p>, an ancient Roman measure, containing 2 cotylæ, or 2 heminæ.</p></div1><div1 part="N" n="SEXTILE" org="uniform" sample="complete" type="entry"><head>SEXTILE</head><p>, the aspect or position of two planets, when they are distant the 6th part of the circle, viz, 2 signs or 60 degrees; and it is marked thus *.</p></div1><div1 part="N" n="SEXTUPLE" org="uniform" sample="complete" type="entry"><head>SEXTUPLE</head><p>, denotes 6 fold in general. But in music it denotes a mixed sort of triple time, which is beaten in double time.</p></div1><div1 part="N" n="SHADOW" org="uniform" sample="complete" type="entry"><head>SHADOW</head><p>, <hi rend="italics">Shade,</hi> in Optics, a certain space deprived of light, or where the light is weakened by the interposition of some opaque body before the luminary.</p><p>The doctrine of Shadows makes a considerable article in optics, astronomy, and geography; and is the general foundation of dialling.</p><p>As nothing is seen but by light, a mere shadow is invisible; and therefore when we say we see a shadow, we mean, partly that we see bodies placed in the Shadow, and illuminated by light reflected from collateral bo- i es, and partly that we see the confines of the light.</p><p>When the opaque body, that projects the Shadow, is perpendicular to the horizon, and the plane it is projected on is horizontal, the Shadow is called a <hi rend="italics">right</hi> one: such as the Shadows of men, trees, buildings, mountains, &c. But when the body is placed parallel to the horizon, it is called a <hi rend="italics">versed Shadow;</hi> as the arms of a man when stretched out, &c. <hi rend="center"><hi rend="italics">Laws of the Projection of Shadows.</hi></hi></p><p>1. Every opaque body projects a Shadow in the <cb/> same direction with the rays of light; that is, towards the part opposite to the light. Hence, as either the luminary or the body changes place, the Shadow likewise changes its place.</p><p>2. Every opaque body projects as many Shadows as there are luminaries to enlighten it.</p><p>3. As the light of the luminary is more intense, the shadow is the deeper. Hence, the intensity of the Shadow is measured by the degrees of light that space is deprived of In reality, the Shadow itself is not deeper; but it appears so, because the surrounding bodies are more intensely illuminated.</p><p>4. When the luminous body and opaque one are equal, the Shadow is always of the same breadth with the opaque body. But when the luminous body is the larger, the Shadow grows always less and less, the farther from the body. And when the luminous body is the smaller of the two, the Shadow increases always the wider, the farther from the body. Hence, the Shadow of an opaque globe is, in the first case a cylinder, in the second case it is a cone verging to a point, and in the third case a truncated cone that enlarges still the more the farther from the body Also, in all these cases, a transverse Section of the Shadow, by a plane, is a circle, respectively, in the three cases, equal, less, or greater than a great circle of the globe.</p><p>5. To find the length of the Shadow, or the axis of the shady cone, projected by a sphere, when it is illuminated by a larger one; the diameters and distance of the two spheres being known. Let C and D be the <figure/> centres of the two spheres, CA the semidiameter of the larger, and DB that of the smaller, both perpendicular to the side of the conical Shadow BEF, whose axis is DE, continued to C; and draw BG parallel to the same axis. Then, the two triangles AGB and BDE being similar, it will be AG : GB or CD :: BD : DE, that is, as the difference of the semidiameters is to the distance of the centres, so is the semidiameter of the opaque sphere to the axis of the Shadow, or the distance of its vertex from the said opaque sphere.</p><p>Ex. gr. If BD = 1 be the semidiameter of the earth, and AC = 101 the mean semidiameter of the sun, also their distance CD or GB = 24000; then as 100 : 24000 :: 1 : 240 = DE, which is the mean height of the earth's Shadow, in semidiameters of the base.</p><p>6. To find the length of the shadow AC projected by an opaque body AB; having given the altitude of the luminary, for ex. of the sun, above the horizon, viz, the angle C, and the height of the object AB. Here the proportion is, as tang. [angle] C : radius :: AB : AC.</p><p>Or, if the length of the Shadow AC be given, to find the height AB, it will be, as radius : tang. [angle] C :: AC : AB. <pb n="445"/><cb/></p><p>Or, if the length of the Shadow AC, and of the object AB, be given, to find the sun's altitude above t<*> horizon, or the angle at C. It will be, as AC : AB :: radius : tang. [angle] C sought. <figure/></p><p>7. To measure the height of any object, ex. gv. a tower AB, by means of its shadow projected on an horizontal plane.—At the extremity of the shadow, at C, erect a stick or pole CD, and measure the length of its shadow CE; also measure the length of the Shadow AC of the tower. Then, by similar triangles, it will be, as EC : CD :: CA : AB. So if EC = 10 feet, CD = 6 feet, and CA = 95 feet; then as 10 : 6 :: 95 : 57 feet = AB, the height of the tower sought.</p><div2 part="N" n="Shadow" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Shadow</hi></head><p>, in Geography. The inhabitants of the earth are divided, with respect to their shadows, into <hi rend="smallcaps">Ascii, Amrhiscii, Heteroscii</hi>, and <hi rend="smallcaps">Periscii.</hi> See these terms in their places.</p></div2><div2 part="N" n="Shadow" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Shadow</hi></head><p>, in Perspective, is of great use in this art. —Having given the appearance of an opaque body, and a luminous one, whose rays diverge, as a candle, or lamp, &c; to find the just appearance of the Shadow, according to the laws of perspective. The method is this: From the luminous body, which is here considered as a point, let fall a perpendicular to the perspective plane or table; and from the several angles, or raised points of the body, let fall perpendiculars to the same plane; then connect the points on which these latter perpendiculars fall, by right lines, with the point on which the first falls; continuing these lines beyond the side opposite to the luminary, till they meet with as many other lines drawn from the centre of the luminary through the said angles or raised points; so shall the points of intersection of these lines be the extremes on bounds of the Shadow. <figure/></p><p>For Example, to project the appearance of the Shadow of a prism ABCDEF, scenographically drli- <cb/> neated. Here M is the place of the perpendicular of the light L, and D, E, F those of the raised points A, B, C, of the prism; therefore, draw MEH, MDG, &c, and LBH, LAG, &c, which will give DEGH &c for the appearance of the Shadow.</p><p>As for those Shadows that are intercepted by other objects, it may be observed, that when the Shadow of a line falls upon any object, it must necessarily take the form of that object. If it fall upon another plane, it will be a right line; if upon a globe, it will be circular; and if upon a cylinder or cone, it will be circular, or oval, &c. If the body intercepting it be a plane, whatever be the situation of it, the shadow falling upon it might be found by producing that plane till it intercepted the perpendicular let fall upon it from the luminous body; for then a line drawn from that point would determine the Shadow, just as if no other plane had been concerned. But the appearance of all these Shadows may be drawn with less trouble, by first drawing it through these intercepted objects, as if they had not been in the way, and then making the Shadow to ascend perpendicularly up every perpendicular plane, and obliquely on those that are situated obliquely, in the manner described by Dr. Priestley, in his Perspective, pa. 73 &c.</p><p>Here we may observe in general, that since the Shadows of all objects which are cast upon the ground, will vanish into the horizontal line; so, for the same reason, the vanishing points of all Shadows, which are cast upon any inclined or other plane, will be somewhere in the vanishing line of that plane.</p><p>When objects are not supposed to be viewed by the light of the sun, or of a candle, &c, but only in the light of a cloudy day, or in a room into which the sun does not shine, there is no sensible Shadow of the upper part of the object, and the lower part only makes the neighbouring parts of the ground, on which it stands, a little darker than the vest. This imperfect obscure kind of Shadow is easily made, being nothing more than a shade on the ground, opposite to the side on which the light comes; and it may be continued to a greater or less distance, according to the supposed brightness of the light by which it is made. It is in this manner (in order to save trouble, and sometimes to prevent consusion) that the Shadows in most drawings are made. On this subject, see Priestley's Perspect. above quoted; also Kirby's Persp. book 2, ch. 4.</p><p>SHAFT <hi rend="italics">of a Column,</hi> in Building, is the body of it; thus called from its straightness: but by architects more commonly the Fust.</p><p><hi rend="smallcaps">Shaft</hi> is also used for the spire of a church steeple; and for the shank or tunnel of a chimney.</p></div2></div1><div1 part="N" n="SHARP" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SHARP</surname> (<foreName full="yes"><hi rend="smallcaps">Abraham</hi></foreName>)</persName></head><p>, an eminent mathematician, mechanist, and astronomer, was descended from an ancient family at Little-Horton, near Bradford, in the West Riding of Yorkshire, where he was born about the year 1651. At a proper age he was put apprentice to a merchant at Manchester; but his genius led him so strongly to the study of mathematics, both theoretical and practical, that he soon became uneasy in that situation of life. By the mutual consent therefore of his master and himself, though not altogether with that of his father, he quitted the business of a merchant. Upon this he removed to Liverpool, where he <pb n="446"/><cb/> gave himself up wholly to the study of mathematics, astronomy, &c; and where, for a subsistance, be opened a school, and taught writing and accounts, &c.</p><p>He had not been long at Liverpool when he accidentally fell in company with a merchant or tradesman vifiting that town from London, in whose house it seems the astronomer Mr. Flamsteed then lodged. With the view therefore of becoming acquainted with this eminent man, Mr. Sharp engaged himself with the merchant as a book-keeper. In consequence he soon contracted an intimate acquaintance and friendship with Mr. Flamsteed, by whose interest and recommendation he obtained a more prositable employment in the dockyard at Chatham; where he continued till his friend and patron, knowing his great merit in astronomy and mechanics, called him to his assistance, in contriving, adapting, and fitting up the astronomical apparatus in the Royal Observatory at Greenwich, which had been lately built, namely about the year 1676; Mr. Flamsteed being then 30 years of age, and Mr. Sharp 25.</p><p>In this situation he continued to assist Mr. Flamsteed in making observations (with the mural arch, of 80 inches radius, and 140 degrees on the limb, contrived and graduated by Mr. Sharp) on the meridional zenith distances of the fixed stars, sun, moon, and planets, with the times of their transits over the meridian; also the diameters of the sun and moon, and their eclipses, with those of Jupiter's satellites, the variation of the compass, &c. He assisted him also in making a catalogue of near 3000 fixed stars, as to their longitudes and magnitudes, their right ascensions and polar distances, with the variations of the same while they change their longitude by one degree.</p><p>But from the fatigue of continually observing the stars at night, in a cold thin air, joined to a weakly constitution, he was reduced to a bad state of health; for the recovery of which he desired leave to retire to his house at Horton; where, as soon as he found himself on the recovery, he began to fit up an observatory of his own; having first made an elegant and curious engine for turning all kinds of work in wood or brass, with a maundril for turning irregular figures, as ovals, roses, wreathed pillars, &c. Beside these, he made himself most of the tools used by joiners, clockmakers, opticians, mathematical instrument-makers, &c. The limbs or arcs of his large equatorial instrument, sextant, quadrant, &c, he graduated with the nicest accuracy, by diagonal divisions into degrees and minutes. The telescopes he made use of were all of his own making, and the lenses ground, figured, and adjusted with his own hands.</p><p>It was at this time that he assisted Mr. Flamsteed in calculating most of the tables in the second volume of his <hi rend="italics">Historia Cœlestis,</hi> as appears by their letters, to be seen in the hands of Mr. Sharp's friends at Horton. Likewise the curious drawings of the charts of all the constellations visible in our hemisphere, with the still more excellent drawings of the planispheres both of the northern and southern constellations. And though these drawings of the constellations were sent to be engraved at Amsterdam by a masterly hand, yet the originals far exceeded the engravings in point of beauty and elegance: these were published by Mr. Flamsteed, and both copies may be seen at Horton. <cb/></p><p>The mathematician meets with something extraordinary in Sharp's elaborate treatise of <hi rend="italics">Geometry Improved</hi> (in 4 to 1717, signed A. S. Philomath.), 1st, by a large and accurate table of segments of circles, its construction and various uses in the solution of several difficult problems, with compendious tables for finding a true proportional part; and their use in these or any other tables exemplified in making logarithms, or their natural numbers, to 60 places of figures; there being a table of them for all primes to 1100, true to 61 figures. 2d, His concise treatise of Polyedra, or solid bodies of many bases, both the regular ones and others: to which are added twelve new ones, with various methods of forming them, and their exact dimensions in surds, or species, and in numbers: illustrated with a variety of copperplates, neatly engraved by his own hands. Also the models of these polyedra he cut out in box wood with amazing neatness and accuracy. Indeed few or none of the mathematical instrument-makers could exceed him in exactly graduating or neatly engraving any mathematical or astronomical instrument, as may be seen in the equatorial instrument above mentioned, or in his sextant, quadrants and dials of various sorts; also in a curious armillary sphere, which, beside the common properties, has moveable circles &c, for exhibiting and resolving all spherical triangles; also his double sector, with many other instruments, all contrived, graduated and finished, in a most elegant manner, by himself. In short, he possessed at once a remarkably clear head for contriving, and an extraordinary hand for executing, any thing, not only in mechanics, but likewise in drawing, writing, and making the most exact and beautiful schemes or figures in all his calculations and geometrical constructions.</p><p>The quadrature of the circle was undertaken by him for his own private amusement in the year 1699, deduced from two different series, by which the truth of it was proved to 72 places of figures; as may be seen in the introduction to Sherwin's tables of logarithms; that is, if the diameter of a circle be 1, the circumference will be found equal to 3.1415926535897932 38462643383279502884197169399375105820974944 592307816405, &c. In the same book of Sherwin's may also be seen his ingenious improvements on the making of logarithms, and the constructing of the natural sines, tangents, and secants.</p><p>He also calculated the natural and logarithmic sines, tangents, and secants, to every second in the first minute of the quadrant: the laborious investigation of which may probably be seen in the archives of the Royal Society, as they were presented to Mr. Patrick Murdoch for that purpose; exhibiting his very neat and accurate manner of writing and arranging his figures, not to be equalled perhaps by the best penman now living.</p><p>The late ingenious Mr. Smeaton says (Philos. Trans. an. 1786, pa. 5, &c):</p><p>“In the year 1689, Mr. Flamsteed completed his mural arc at Greenwich; and, in the Prolegomena to his Historia Cœlestis, he makes an ample acknowledgment of the particular assistance, care, and industry of Mr. Abraham Sharp; whom, in the month of August 1688, he brought into the observatory, as his amanuensis and being as Mr. Flamsteed tells us, not <pb n="447"/><cb/> only a very skilful mathematician, but exceedingly expert in mechanical operations, he was principally employed in the construction of the mural arc; which in the compass of 14 months he finished, so greatly to the satisfaction of Mr. Flamsteed, that he speaks of him in the highest terms of praise.</p><p>“This celebrated instrument, of which he also gives the figure at the end of the Prolegomena, was of the radius of 6 feet 7 1/2 inches; and, in like manner as the sextant, was furnished both with screw and diagonal divisions, all performed by the accurate hand of Mr. Sharp. But yet, whoever compares the different parts of the table for conversion of the revolutions and parts of the screw belonging to the mural arc into degrees, minutes, and seconds, with each other, at the same distance from the zenith on different sides; and with their halves, quarters, &c, will find as notable a disagreement of the screw-work from the hand divisions, as had appeared before in the work of Mr. Tompion: and hence we may conclude, that the method of Dr. Hook, being executed by two such masterly hands as Tompion and Sharp, and found defective, is in reality not to be depended upon in nice matters.</p><p>“From the account of Mr. Flamsteed it appears also, that Mr. Sharp obtained the zenith point of the instrument, or line of collimation, by observation of the zenith stars, with the face of the instrument on the east and on the west side of the wall: and that having made the index stronger (to prevent flexure) than that of the sextant, and thereby heavier, he contrived, by means of pulleys and balancing weights, to relieve the hand that was to move it from a great part of its gravity. Mr. Sharp continued in strict correspondence with Mr. Flamsteed as long as he lived, as appeared by letters of Mr. Flamsteed's found after Mr. Sharp's death; many of which I have seen.</p><p>“I have been the more particular relating to Mr. Sharp, in the business of constructing this mural arc; not only because we may suppose it the first good and valid instrument of the kind, but because I look upon Mr. Sharp to have been the first person that cut accurate and delicate divisions upon astronomical instruments; of which, independent of Mr. Flamsteed's testimony, there still remain considerable proofs: for, after leaving Mr. Flamsteed, and quitting the department above-mentioned, he retired into Yorkshire, to the village of Little Horton, near Bradford, where he ended his days about the year 1743 (should be, in 1742); and where I have seen not only a large and very fine collection of mechanical tools, the principal ones being made with his own hands, but also a great variety of scales and instruments made with them, both in wood and brass, the divisions of which were so exquisite, as would not discredit the first artists of the present times: and I believe there is now remaining a quadrant, of 4 or 5 feet radius, framed of wood, but the limb covered with abrass plate; the subdivisions being done by diagonals, the lines of which are as finely cut as those upon the quadrants at Greenwich. The delicacy of Mr. Sharp's hand will indeed permanently appear from the copper-plates in a quarto book, published in the year 1718, intituled <hi rend="italics">Geometry Improved</hi> by A. Sharp, Philomath.” (or rather 1717, by A. S. Philomath.) “whereof not only the geometrical lines upon the plates, <cb/> but the whole of the engraving of letters and figures, were done by himself, as I was told by a person in the mathematical line, who very frequently attended Mr. Sharp in the latter part of his life. I therefore look upon Mr. Sharp as the first person that brought the affair of hand division to any degree of perfection.”</p><p>Mr. Sharp kept up a correspondence by letters with most of the eminent mathematicians and astronomers of his time, as Mr. Flamsteed, Sir Isaac Newton, Dr. Halley, Dr. Wallis, Mr. Hodgson, Mr. Sherwin, &c, the answers to which letters are all written upon the backs, or empty spaces, of the letters he received, in a short-hand of his own contrivance. From a great variety of letters (of which a large chest full remain with his friends) from these and many other celebrated mathematicians, it is evident that Mr. Sharp spared neither pains nor time to promote real science. Indeed, being one of the most accurate and indefatigable computers that ever existed, he was for many years the common resource for Mr. Flamsteed, Sir Jonas Moore, Dr. Halley, and others, in all sorts of troublesome and delicate calculations.</p><p>Mr. Sharp continued all his life a bachelor, and spent his time as recluse as a hermit. He was of a middle stature, but very thin, being of a weakly constitution; he was remarkably feeble the last three or four years before he died, which was on the 18th of July 1742, in the 91st year of his age.</p><p>In his retirement at Little Horton, he employed four or five rooms or apartments in his house for different purposes, into which none of his family could possibly enter at any time without his permission. He was seldom visited by any persons, except two gentlemen of Bradford, the one a mathematician, and the other an ingenious apothecary: these were admitted, when he chose to be seen by them, by the signal of rubbing a stone against a certain part of the outside wall of the house. He duly attended the dissenting chapel at Bradford, of which he was a member, every Sunday; at which time he took care to be provided with plenty of halfpence, which he very charitably suffered to be taken singly out of his hand, held behind him during his walk to the chapel, by a number of poor people who followed him, without his ever looking back, or asking a single question.</p><p>Mr. Sharp was very irregular as to his meals, and remarkably sparing in his diet, which he frequently took in the following manner. A little square hole, something like a window, made a communication between the room where he was usually employed in calculations, and another chamber or room in the house where a servant could enter; and before this hole he had contrived a sliding board: the servant always placed his victuals in this hole, without speaking or making any the least noise; and when he had a little leisure he visited his cupboard to see what it afforded to satisfy his hunger or thirst. But it often happened, that the breakfast, dinner, and supper have remained untouched by him, when the servant has gone to remove what was left—so deeply engaged had he been in calculations.—Cavities might easily be perceived in an old English oak table where he sat to write, by the frequent rubbing and wearing of his elbows.—<hi rend="italics">Gutta cavat lapidem, &c.</hi> <pb n="448"/><cb/></p><p>By Mr. Sharp's epitaph it appears that he was related to archbishop Sharp. And Mr. Sharp the eminent surgeon, who it seems has lately retired from business, is the nephew of our author. Another nephew was the father of Mr. Ramsden, the present celebrated instrument maker, who says that his grand uncle Abraham, our author, was some time in his younger days an exciseman; which occupation he quitted on coming to a patrimonial estate of about 200l. a year.</p><div2 part="N" n="Sharp" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sharp</hi></head><p>, in Music, a kind of artificial note, or character, thus formed <figure/>: this being presixed to any note, shews that it is to be sung or played a semitone or half note higher than the natural note is. When a Sharp is placed at the beginning of a stave or movement, it shews that all notes that are found on the same line, or space, throughout, are to be raised half a tone above their natural pitch, unless a natural intervene. When a Sharp occurs accidentally, it only affects as many notes as follow it on the same line or space, without a natural, in the compass of a bar.</p></div2></div1><div1 part="N" n="SHEAVE" org="uniform" sample="complete" type="entry"><head>SHEAVE</head><p>, in Mechanics, a solid cylindrical wheel, fixed in a channel, and moveable about an axis, as being used to raise or increase the mechanical powers applied to remove any body.</p></div1><div1 part="N" n="SHEERS" org="uniform" sample="complete" type="entry"><head>SHEERS</head><p>, aboard a ship, an engine used to hoist or displace the lower masts of a ship.</p></div1><div1 part="N" n="SHEKEL" org="uniform" sample="complete" type="entry"><head>SHEKEL</head><p>, or <hi rend="smallcaps">Shekle</hi>, an ancient Hebrew coin and weight, equal to 4 Attic drachmas, or 4 Roman denarii, or 28. 9 1/2d. sterling. According to father Mersenne, the Hebrew Shekel weighs 268 grains, and is composed of 20 oboli, each obolus weighing 16 grains of wheat.</p></div1><div1 part="N" n="SHILLING" org="uniform" sample="complete" type="entry"><head>SHILLING</head><p>, an English silver coin, equal to 12 pence, or the 20th part of a pound sterling.</p><p>This was a Saxon coin, being the 48th part of their pound weight. Its value at first was 5 pence; but it was reduced to 4 pence about a century before the conquest. After the conquest, the French solidus of 12 pence, which was in use among the Normans, was called by the English name of Shilling; and the Saxon Shilling of 4 pence took a Norman name, and was called the <hi rend="italics">groat,</hi> or <hi rend="italics">great</hi> coin, because it was the largest English coin then known in England. From this time, the Shilling underwent many alterations.</p><p>Many other nations have also their Shillings. The English Shilling is worth about 23 French sols; those of Holland and Germany about half as much, or 11 1/2 sols; those of Flanders about 9. The Dutch Shillings are also called <hi rend="italics">sols de gros,</hi> because equal to 12 gross. The Danes have copper Shillings, worth about one fourth of a farthing sterling.</p><p>In the time of Edward the 1st, the pound troy was the same as the pound sterling of silver, consisting of 20 Shillings; so that the Shilling weighed the 20th part of a pound, or more than half an ounce troy. But some are of opinion, there were no coins of this denomination, till Henry the 7th, in the year 1504, first coined silver pieces of 12 pence value, which we call Shillings. Since the reign of Elizabeth, a Shilling weighs the 62nd part of a pound troy, or 3 dwts 20 28/31 grs. the pound weight of silver making 62 Shillings. And hence the ounce of silver is worth 58. 2d. or 5 1/6 Shillings.</p></div1><div1 part="N" n="SHIVERS" org="uniform" sample="complete" type="entry"><head>SHIVERS</head><p>, in a ship, the seamen's term for those <cb/> little round wheels, in which the rope of a pully or block runs. They turn with the rope, and have pieces of brass in their ceatres, into which the pin of the block goes, and on which they turn.</p><p>SHORT SIGHTEDNESS. <hi rend="italics">myopia,</hi> a defect in the conformation of the eye, when the crystalline &c being too convex, the rays that enter the eye are refracted too much, and made to converge too fast, so as to unite before they reach the retina, by which means vision is rendered dim and confused.</p><p>It is commonly thought that Short-sightedness wears off in old age, on account of the eye becoming flatter; but Dr. Smith questions whether this be matter of fact, or only hypothesis. It is remarkable that Short-sighted persons commonly write a small hand, and love a small print, because they can see more of it at one view. That it is customary with them not to look at the person they converse with, because they cannot well see the motion of his eyes and features, and are therefore attentive to his words only. That they see more distinctly, and somewhat farther off, by a strong light, than by a weak one; because a strong light causes a contraction of the pupil, and consequently of the pencils, both here and at the retina, which lessens their mixture, and consequently the apparent consusion; and therefore, to see more distinctly, they almost close their eye-lids, for which reason they were anciently called <hi rend="italics">myopes.</hi> Smith's Optics, vol. 2, Rem. p. 10.</p><p>Dr. Jurin observes, that persons who are much and long accustomed to view objects at small distances, as students in general, watchmakers, engravers, painters in miniature, &c, see better at small distances, and worse at great distances, than other people. And he gives the reasons, from the mechanical effect of habit in the eye. Essay on Dist. and Indist. Vision.</p><p>The ordinary remedy for Short-sightedness is a concave lens, held before the eye; for this causing the rays to diverge, or at least diminishing much of their convergency, it makes a compensation for the too great convexity of the crystalline. Dr. Hook suggests another remedy; which is to employ a convex glass, in a position between the object and the eye, by means of which, the object may be made to appear at any distance from the eye, and so the eye be made to contemplate the picture in the same manner as if the object itself were in its place. But here unfortunately the image will appear inverted: for this however he has some whimsical expedients; viz, in reading to turn the book upside down, and to learn to write upside down. As to distant objects, the Doctor asserts, from his own experience, that with a little practice in contemplating inverted objects, one gets as good an idea of them as if seen in their natural posture.</p></div1><div1 part="N" n="SHOT" org="uniform" sample="complete" type="entry"><head>SHOT</head><p>, in the Military Art, includes all sorts of balls or bullets for fire arms, from the cannon to the pistol. As to those for mortars, they are usually called shells.</p><p>Shot are mostly of a round form, though there are other shapes. Those for cannon are of iron; but those for muskets and pistols are of lead.</p><p>Cannon shot and shells are usually set up in piles, or heaps, tapering from the base towards the top; the base being either a triangle, a square, or a rectangle; <pb n="449"/><cb/> from which the number in the pile is easily computed. See <hi rend="smallcaps">Pile.</hi></p><p>The weight and dimensions of balls may be found, the one from the other, whether they are of iron or of lead. Thus,</p><p>The weight of an iron ball of 4 inches diameter, is 9lb, and because the weight is as the cube of the diameter, therefore as 4<hi rend="sup">3</hi> : 9 :: <hi rend="italics">d</hi><hi rend="sup">3</hi> : (9/64) <hi rend="italics">d</hi><hi rend="sup">3</hi> = <hi rend="italics">w,</hi> the weight of the iron ball whose diameter is <hi rend="italics">d;</hi> that is, 9/64 of the cube of its diameter. And, conversely, if the weight be given, to find the diameter, it will be ; that is, take 64/9 or 7 1/9 of the weight, and the cube root of that will be the diameter of the iron ball.</p><p>For leaden balls; one of 4 1/4 inches diameter weighs 17 pounds; therefore as the cube of 4 1/4 is to 17, or nearly as 9 : 2 :: <hi rend="italics">d</hi><hi rend="sup">3</hi> : (2/9)<hi rend="italics">d</hi><hi rend="sup">3</hi> = <hi rend="italics">w,</hi> the weight of the leaden ball whose diameter is <hi rend="italics">d,</hi> that is, 2/9 of the cube of the diameter. On the contrary, if the weight be given, to find the diameter, it will be ; that is, 9/2 or 4 1/2 of the weight, and the cube root of the product. See my Conic Sections and Select Exercises, pa. 141.</p><p>SHOULDER <hi rend="italics">of a Bastion,</hi> in Fortification, is the angle where the face and the flank meet.</p></div1><div1 part="N" n="SHOULDERING" org="uniform" sample="complete" type="entry"><head>SHOULDERING</head><p>, in Fortisication. See <hi rend="italics">Epaulement.</hi></p><p>SHWAN-<hi rend="italics">pan,</hi> a Chinese instrument, composed of a number of wires, with beads upon them, which they move backwards and forwards, and which serves to assist them in their computations. See <hi rend="smallcaps">Abacus.</hi></p></div1><div1 part="N" n="SIDE" org="uniform" sample="complete" type="entry"><head>SIDE</head><p>, <hi rend="italics">latus,</hi> in Geometry. The side of a figure is a line making part of the periphery of any superficial figure, viz, a part between two successive angles.</p><p>In triangles, the sides are also called <hi rend="italics">legs.</hi> In a rightangled triangle, the two sides that include the right angle, are called <hi rend="italics">catheti,</hi> or sometimes the <hi rend="italics">base</hi> and <hi rend="italics">perpendicular;</hi> and the third side, the <hi rend="italics">hypothenuse.</hi></p><p><hi rend="smallcaps">Side</hi> <hi rend="italics">of a Polygonal Number,</hi> is the number of terms in the arithmetical progression that are summed up to form the number.</p><p><hi rend="smallcaps">Side</hi> <hi rend="italics">of a Power,</hi> is what is usually called the root or radix.</p><p><hi rend="smallcaps">Sides</hi> of <hi rend="italics">Horn-works, Crown-works, Double-tenailles,</hi> &c, are the ramparts and parapets which inclose them on the right and lest, from the gorge to the head.</p></div1><div1 part="N" n="SIDEREAL" org="uniform" sample="complete" type="entry"><head>SIDEREAL</head><p>, or <hi rend="smallcaps">Siderial</hi>, something relating to the stars. As Sidereal year, day, &c, being those marked out by the stars.</p><p><hi rend="smallcaps">Sidereal</hi> <hi rend="italics">Year.</hi> See <hi rend="smallcaps">Year.</hi></p><p><hi rend="smallcaps">Sidereal</hi> <hi rend="italics">Day,</hi> is the time in which any star appears to revolve from the meridian to the meridian again; which is 23 hours 56′ 4″ 6‴ of mean solar time; there being 366 Sidereal days in a year, or in the time of 365 diurnal revolutions of the sun; that is, exactly, if the equinoctial points were at rest in the heavens. But the equinoctial points go backward, with respect to the stars, at the rate of 50″ of a degree in a Julian year; which causeth the stars to have an apparent pro- <cb/> gressive motion eastward 50″ in that time. And as the sun's mean motion in the ecliptic is only 11 signs 29° 45′ 40″ 15‴ in 365 days, it follows, that at the end of that time he will be 14′ 19″ 45‴ short of that point of the ecliptic from which he set out at the beginning; and the stars will be advanced 50″ of a degree with respect to that point.</p><p>Consequently, if the sun's centre be on the meridian with any star on any given day of the year, that star will be 14′ 19″ 45‴ + 50″ or 15′ 9″ 45‴ east of the sun's centre, on the 365th day afterward, when the sun's centre is on the meridian; and therefore that star will not come to the meridian on that day till the sun's centre has passed it by 1′ 0″ 38‴ 57′′′′ of mean solar time; for the sun takes so much time to go through an arc of 15′ 9″ 45‴; and then, in 365<hi rend="sup">da</hi> 0<hi rend="sup">h</hi> 1′ 0″ 38‴ 57′′′′ the star will have just completed its 366th revolution to the meridian.</p><p>In the following table, of Sidereal revolutions, the first column contains the number of revolutions of the stars; the others next it shew the times in which these revolutions are made, as shewn by a well regulated clock; and those on the right hand shew the daily accelerations of the stars, that is, how much any star gains upon the time shewn by such a clock, in the corresponding revolutions. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Revol.</cell><cell cols="1" rows="1" rend="colspan=6" role="data">Times in which the re-</cell><cell cols="1" rows="1" rend="colspan=5" role="data">Accelerations of</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of the</cell><cell cols="1" rows="1" rend="colspan=6 rowspan=2" role="data">volutions are made.</cell><cell cols="1" rows="1" rend="colspan=5 rowspan=2" role="data">the stars.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">stars.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">da.</cell><cell cols="1" rows="1" role="data">ho.</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" role="data">sec.</cell><cell cols="1" rows="1" role="data">th.</cell><cell cols="1" rows="1" role="data">fo.</cell><cell cols="1" rows="1" role="data">ho.</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" role="data">sec.</cell><cell cols="1" rows="1" role="data">th.</cell><cell cols="1" rows="1" role="data">fo.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">59</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">59</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">58</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" 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role="data">43</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">36</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">199</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">23</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">299</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">360</cell><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data">364</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">366</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">3</cell></row></table> <pb n="450"/><cb/></p><p>This table will not differ the 279936000000th part of a second of time from the truth in a whole year. It was calculated by Mr. Ferguson; and it is the only table of the kind in which the recession of the equinoctial points has been taken into the calculation.</p><p>SIDUS <hi rend="italics">Georgium,</hi> a new primary planet, discovered by Dr. Herschel at Bath, in the night of March 13, 1781. It is sometimes also called the <hi rend="italics">Georgian Planet,</hi> and the <hi rend="italics">New Planet,</hi> from its having been newly or lately discovered, also <hi rend="italics">Herschel's Planet,</hi> from the name of its discoverer, and the <hi rend="italics">Planet Herschel,</hi> or simply <hi rend="italics">Herschel,</hi> by which name it is distinguished by the astronomers of almost all foreign nations. The planet is denoted by this character <figure/>, a Roman H as the initial of the name, the horizontal bar being crossed by a perpendicular line, forming a kind of cross, the emblem of Christianity, meaning thereby perhaps that its discovery was made by a Christian, or since the birth of Christ, as all the other planets were discovered long before that period.</p><p>This planet is the remotest of all those that are yet known, though not the largest, being in point of magnitude less than Saturn and Jupiter. Its light, says Dr. Herschel, is of a blueish-white colour, and its brilliancy between that of Venus and the moon. With a telescope that magnifies about 300 times, it appears to have a very well defined visible disk; but with instruments of a small power, it can hardly be distinguished from a fixed star of between the 6th and 7th magnitude. In a very fine clear night, when the moon is absent, a good eye will perceive it without a telescope.</p><p>From the observations and calculations of Dr. Herschel and other astronomers, the elements and dimenfions &c of this planet, have been collected as below. <table><row role="data"><cell cols="1" rows="1" role="data">Place of the node</cell><cell cols="1" rows="1" rend="align=right" role="data">2<hi rend="sup">s</hi> 11° 49′ 30″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Place of the aphelion in 1795</cell><cell cols="1" rows="1" rend="align=right" role="data">11  23  33  55 </cell></row><row role="data"><cell cols="1" rows="1" role="data">Inclination of the orbit</cell><cell cols="1" rows="1" rend="align=right" role="data">43  35 </cell></row></table> Time of the perihelion passage Sep. 7, 1799. <table><row role="data"><cell cols="1" rows="1" role="data">Eccentricity of the orbit</cell><cell cols="1" rows="1" rend="align=right" role="data">.8203</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Half the greater axis</cell><cell cols="1" rows="1" rend="align=right" role="data">19.0818</cell><cell cols="1" rows="1" role="data">of Earth's dist.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Revolution</cell><cell cols="1" rows="1" rend="align=center" role="data">83 1/3</cell><cell cols="1" rows="1" role="data">sidereal years</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diameter of the planet</cell><cell cols="1" rows="1" rend="align=right" role="data">34217</cell><cell cols="1" rows="1" role="data">miles</cell></row><row role="data"><cell cols="1" rows="1" role="data">Propor. of diam. to the earth's</cell><cell cols="1" rows="1" rend="align=right" role="data">4.3177</cell><cell cols="1" rows="1" role="data">to 1</cell></row><row role="data"><cell cols="1" rows="1" role="data">Its bulk to the earth's</cell><cell cols="1" rows="1" rend="align=right" role="data">80.4926</cell><cell cols="1" rows="1" role="data">to 1</cell></row><row role="data"><cell cols="1" rows="1" role="data">Its density as</cell><cell cols="1" rows="1" rend="align=right" role="data">.2204</cell><cell cols="1" rows="1" role="data">to 1</cell></row><row role="data"><cell cols="1" rows="1" role="data">Its quantity of matter</cell><cell cols="1" rows="1" rend="align=right" role="data">17.7406</cell><cell cols="1" rows="1" role="data">to 1</cell></row></table> <hi rend="center">And heavy bodies fall on its surface 18 feet 8 inches in one second of time.</hi></p></div1><div1 part="N" n="SIGN" org="uniform" sample="complete" type="entry"><head>SIGN</head><p>, in Algebra, a symbol or <hi rend="smallcaps">CHARACTER.</hi></p><div2 part="N" n="Signs" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Signs</hi></head><p>, <hi rend="italics">like, positive, negative, radical,</hi> &c. See the adjectives.</p></div2><div2 part="N" n="Sign" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sign</hi></head><p>, in Astronomy, a 12th part of the ecliptic, or zodiac; or a portion containing 30 degrees of the same.</p><p>The ancients divided the zodiac into 12 segments, called Signs; commencing at the point where the ecliptic and equinoctial intersect, and so counting forward from west to east, according to the course of the fun; these Signs they named from the 12 constellations which possessed those segments in the time of Hipparchus. But the constellations have since so changed their places, by the precession of the equinox, that Aries is now found in the sign called Taurus, and Taurus in that of Gemini, &c. <cb/></p><p>The names, and characters, of the 12 Signs, and their order, are as follow: Aries <figure/>, Taurus <figure/>, Gemini <foreign xml:lang="greek">*p</foreign>, Cancer <figure/>, Leo <figure/>, Virgo <figure/>, Libra <figure/>, Scorpio <figure/>, Sagittarius <figure/>, Capricornus <figure/>, Aquarius <figure/>, Pisces <figure/>; each of which, with the stars in them, see under its proper article, <hi rend="smallcaps">Aries, Taurus</hi>, &c.</p><p>The Signs are distinguished, with regard to the season of the year when the sun is in them, into vernal, æstival, autumnal, and brumal.</p><p><hi rend="italics">Vernal</hi> or <hi rend="italics">Spring</hi> <hi rend="smallcaps">Signs</hi>, are Aries, Taurus, Gemini.</p><p><hi rend="italics">Aestival</hi> or <hi rend="italics">Summer</hi> <hi rend="smallcaps">Signs</hi>, are Cancer, Leo, Virgo.</p><p><hi rend="italics">Autumnal</hi> <hi rend="smallcaps">Signs</hi>, are Libra, Scorpio, Sagittary.</p><p><hi rend="italics">Brumal</hi> or <hi rend="italics">Winter</hi> <hi rend="smallcaps">Signs</hi>, are Capricorn, Aquarius, Pisces.</p><p>The vernal and summer Signs are also called <hi rend="italics">northern</hi> Signs, because they are on the north side of the equinoctial; and the autumnal and winter Signs are called <hi rend="italics">southern</hi> ones, because they are on the south side of the same.</p><p>The Signs are also distinguished into <hi rend="italics">ascending</hi> and <hi rend="italics">descending,</hi> according as they are ascending toward the north, or descending toward the south. Thus, the</p><p><hi rend="italics">Ascending</hi> <hi rend="smallcaps">Signs</hi>, are the winter and spring signs, or those fix from the winter solstice to the summer solstice, viz, the Signs Capricorn, Aquarius, Pisces, Aries, Taurus, Gemini. And the</p><p><hi rend="italics">Descending</hi> <hi rend="smallcaps">Signs</hi> are the summer and autumn Signs, or the Signs Cancer, Leo, Virgo, Libra, Scorpio, Sagittary.</p></div2><div2 part="N" n="Signs" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Signs</hi></head><p>, <hi rend="italics">Fixed, Masculine,</hi> &c; see the adjectives.</p></div2></div1><div1 part="N" n="SILLON" org="uniform" sample="complete" type="entry"><head>SILLON</head><p>, in Fortification, an elevation of earth, made in the middle of the moat, to fortify it, when too broad. It is more usually called the Envelope.</p></div1><div1 part="N" n="SIMILAR" org="uniform" sample="complete" type="entry"><head>SIMILAR</head><p>, in Arithmetic and Geometry, the same with like. Similar things have the same disposition or conformation of parts, and differ in nothing but as to their quantity or magnitude; as two squares, or two circles, &c.</p><p>In Mathematics, Similar parts, as A, <hi rend="italics">a,</hi> have the same ratio to their wholes B, <hi rend="italics">b;</hi> and if the wholes have the same ratio to the parts, the parts are Similar.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">angles,</hi> are also equal angles.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">ares,</hi> of circles, are such as are like parts of their whole peripheries. And, in general, similar arcs of any like curves, are the like parts of the wholes.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">bodies,</hi> in Natural Philosophy, are such as have their particles of the same kind and nature one with another.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Curves.</hi> Two segments of two curves are said to be Similar when, any right-lined figure being inscribed within one of them, we can inscribe always a Similar rectilineal figure in the other.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Conic Sections,</hi> are such as are of the same kind, and have their principal axes and parameters proportional. So, two ellipses are figures of the same kind, but they are not Similar unless the axes of the one have the same ratio as the axes of the other. And the same of two hyperbolas, or two parabolas. And generally, those curves are Similar, that are of the same kind, and have their corresponding dimensions in the same ratio.—All circles are Similar figures.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Diameters of Conic Sections,</hi> are such as make equal angles with their ordinates.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Figures,</hi> or plane figures, are such as have all their angles equal respectively, each to each, and <pb n="451"/><cb/> their sides about the equal angles proportional. And the same of Similar polygons.—Similar plane figures have their areas or contents, in the duplicate ratio of their like sides, or as the squares of those sides.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Plane Numbers,</hi> are such as may be ranged into the form of Similar rectangles; that is, into rectangles whose sides are proportional. Such are 12 and 48; for the sides of 12 are 6 and 2, and the sides of 48 are 12 and 4, which are in the same proportion, viz, 6 : 2 :: 12 : 4.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Polygons,</hi> are polygons of the same number of angles, and the angles in the one equal severally to the angles in the other, also the sides about those angles proportional.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Rectangles,</hi> are those that have their sides about the like angles proportional.—All squares are Similar.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Segments of circles,</hi> are such as contain equal angles.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Solids,</hi> are such as are contained under the same number of Similar planes, alike situated.—Similar solids are to each other as the cubes of their like linear dimensions.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Solid Numbers,</hi> are those whose little cubes may be so ranged, as to form Similar parallelopipedons.</p><p><hi rend="smallcaps">Similar</hi> <hi rend="italics">Triangles,</hi> are such as are equiangular ones, or have all their three angles respectively equal in each triangle. For it is sufficient for triangles to be similar, that they be equiangular, because that being equiangular, they necessarily have their sides proportional, which is a condition of Similarity in all figures. As to other figures, having more sides than three, they may be equiangular, without having their sides proportional, and therefore without being similar.—Similar triangles are as the squares of their like sides.</p></div1><div1 part="N" n="SIMILITUDE" org="uniform" sample="complete" type="entry"><head>SIMILITUDE</head><p>, in Arithmetic and Geometry, denotes the relation of things that are similar to each other.</p><p>Euclid and, after him, most other authors, demonstrate every thing in geometry from the principle of congruity. Wolfius, instead of it, substitutes that of Similitude; which, he says, was communicated to him by Leibnitz, and which he finds of very considerable use in geometry, as serving to demonstrate many things directly, which are only demonstrable from the principle of congruity in a very tedious manner.</p></div1><div1 part="N" n="SIMPLE" org="uniform" sample="complete" type="entry"><head>SIMPLE</head><p>, something not mixed, or not compounded; in which sense it stands opposed to compound.</p><p>The elements are Simple bodies, from the composition of which there result all sorts of mixed bodies.</p><p><hi rend="smallcaps">Simple</hi> <hi rend="italics">Equation, Fraction,</hi> and <hi rend="italics">Surd.</hi> See the substantives.</p><p><hi rend="smallcaps">Simple</hi> <hi rend="italics">Quantities,</hi> in Algebra, are those that consist of one term only; as <hi rend="italics">a,</hi> or — <hi rend="italics">ab,</hi> or 3 <hi rend="italics">abc.</hi> In opposition to compound quantities, which consist of two or more terms; as <hi rend="italics">a</hi> + <hi rend="italics">b,</hi> or <hi rend="italics">a</hi> + 2<hi rend="italics">b</hi> - 3<hi rend="italics">ac.</hi></p><p><hi rend="smallcaps">Simple</hi> <hi rend="italics">Flank,</hi> and <hi rend="italics">Tenaille,</hi> in Fortification. See the substantives.</p><p><hi rend="smallcaps">Simple</hi> <hi rend="italics">Machine, Motion, Pendulum,</hi> and <hi rend="italics">Wheel,</hi> in Mechanics. See the substantives.</p><p>The simplest machines are always the most esteemed. And in geometry, the most simple demonstrations are the best.</p><p><hi rend="smallcaps">Simple</hi> <hi rend="italics">Problem,</hi> in Mathematics. See <hi rend="smallcaps">Linear</hi> <hi rend="italics">Problem.</hi></p><p><hi rend="smallcaps">Simple</hi> <hi rend="italics">Vision,</hi> in Optics. See <hi rend="smallcaps">Visic</hi></p></div1><div1 part="N" n="SIMPSON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SIMPSON</surname> (<foreName full="yes"><hi rend="smallcaps">Thomas</hi></foreName>)</persName></head><p>, F. R. S. a very eminent <cb/> mathematician, and professor of Mathematics in the Royal Military Academy at Woolwich, was born at Market Bosworth, in the county of Leicester, the 20th of August 1710. His father was a stuff-weaver in that town; and though in tolerable circumstances, yet, intending to bring up his son Thomas to his own business, he took so little care of his education, that he was only taught to read English. But nature had furnished him with talents and a genius for far other pursuits; which led him afterwards to the highest rank in the mathematical and philosophical sciences.</p><p>Young Simpson very soon gave indications of his turn for study in general, by eagerly reading all books he could meet with, teaching himself to write, and embracing every opportunity he could find of deriving knowledge from other persons. His father observing him thus to neglect his business, by spending his time in reading what he thought useless books, and following other such like pursuits, used all his endeavours to check such proceedings, and to induce him to follow his profession with steadiness and better effect. But after many struggles for this purpose, the differences thus produced between them at length rose to such a height, that our author quitted his father's house entirely.</p><p>Upon this occasion he repaired to Nuneaton, a town at a small distance from Bosworth, where he went to lodge at the house of a taylor's widow, of the name of Swinfield, who had been left with two children, a daughter and a son, by her husband, of whom the son, who was the younger, being but about two years older than Simpson, had become his intimate friend and companion. And here he continued some time, working at his trade, and improving his knowledge by reading such books as he could procure.</p><p>Among several other circumstances which, long before this, gave occasion to shew our author's early thirst for knowledge, as well as proving a fresh incitement to acquire it, was that of a large solar eclipse, which took place on the 11th day of May, 1724. This phenomenon, so awful to many who are ignorant of the cause of it, struck the mind of young Simpson with a strong curiosity to discover the reason of it, and to be able to predict the like surprising events. It was however several years before he could obtain his desire, which at length was gratified by the following accident. After he had been some time at Mrs. Swinfield's, at Nuneaton, a travelling pedlar came that way, and took a lodging at the same house, according to his usual custom. This man, to his profession of an itinerant merchant, had joined the more profitable one of a fortune-teller, which he performed by means of judicial astrology. Every one knows with what regard persons of such a cast are treated by the inhabitants of country villages; it cannot be surprising therefore that an untutored lad of nineteen should look upon this man as a prodigy, and, regarding him in this light, should endeavour to ingratiate himself into his favour; in which he succeeded so well, that the sage was no less taken with the quick natural parts and genius of his new acquaintance. The pedlar, intending a journey to Bristol fair, lest in the hands of young Simpson an old edition of Cocker's Arithmetic, to which was subjoined a short Appendix on Algebra, and a book upon Genitures, by Partridge the almanac maker. These books he had perused to so good purpose, during the absence of his <pb n="452"/><cb/> friend, as to excite his amazement upon his return; in consequence of which he set himself about erecting a genethliacal figure, in order to a presage of Thomas's future fortune.</p><p>This position of the heavens having been maturely considered <hi rend="italics">secundum artem,</hi> the wizard, with great confidence, pronounced, that, “within two years time Simpson would turn out a greater man than himself!”</p><p>In fact, our author profited so well by the encouragement and assistance of the pedlar, afforded him from time to time when he occasionally came to Nuneaton, that, by the advice of his friend, he at length made an open profession of casting nativities himself; from which, together with teaching an evening school, he derived a pretty pittance, so that he greatly neglected his weaving, to which indeed he had never manifested any great attachment, and soon became the oracle of Nuneaton, Bosworth, and the environs. Scarce a courtship advanced to a match, or a bargain to a sale, without previously consulting the infallible Simpson about the consequences. But as to helping people to stolen goods, he always declared that above his skill; and over life and death he declared he had no power: all those called <hi rend="italics">lawful questions</hi> he readily resolved, provided the persons were certain as to the horary <hi rend="italics">data</hi> of the horoscope: and, he has often declared, with such success, that if from very cogent reasons he had not been thoroughly convinced of the vain foundation and fallaciousness of his art, he never should have dropt it, as he afterwards found himself in conscience bound to do.</p><p>About this time he married the widow Swinfield, in whose house he lodged, though she was then almost old enough to be his grandmother, being upwards of fifty years of age. After this the family lived comfortably enough together for some short time, Simpson occasionally working at his business of a weaver in the daytime, and teaching an evening school or telling fortunes at night; the family being also farther assisted by the labours of young Swinfield, who had been brought up in the profession of his father.</p><p>But this tranquillity was soon interrupted, and our author driven at once from his home and the profession of astrology, by the following accident. A young woman in the neighbourhood had long wished to hear or know something of her lover, who had been gone to sea; but Simpson had put her off from time to time, till the girl grew at last so importunate, that he could deny her no longer. He asked her if she would be afraid if he should raise the devil, thinking to deter her; but she declared she feared neither ghost nor devil: so he was obliged to comply. The scene of action pitched upon was a barn, and young Swinfield was to act the devil or ghost; who being concealed under some straw in a corner of the barn, was, at a signal given, to rise slowly out from among the straw, with his face marked so that the girl might not know him. Every thing being in order, the girl came at the time appointed; when Simpson, after cautioning her not to be afraid, began muttering some mystical words, and chalking round about them, till, on the signal given, up rises the taylor slow and solemn, to the great terror of the poor girl, who, before she had seen half his shoulders, fell into violent fits, crying out it was the very image of her lover; and the effect upon her was so dreadful, <cb/> that it was thought either death or madness must be the consequence. So that poor Simpson was obliged immediately to abandon at once both his home and the profession of a conjuror.</p><p>Upon this occasion it would seem he fled to Derby, where he remained some two or three years, viz, from 1733 till 1735 or 1736; instructing pupils in an evening school, and working at his trade by day.</p><p>It would seem that Simpson had an early turn for versifying, both from the circumstance of a song written here in favour of the Cavendish family, on occasion of the parliamentary election at that place, in the year 1733; and from his first two mathematical questions that were published in the Ladies Diary, which were both in a set of verses, not ill written for the occasion. These were printed in the Diary for 1736, and therefore must at latest have been written in the year 1735. These two questions, being at that time pretty difficult ones, shew the great progress he had even then made in the mathematics; and from an expression in the first of them, viz, where he mentions his residence as being in latitude 52°, it appears he was not then come up to London, though he must have done so very soon after.</p><p>Together with his astrology, he had soon furnished himself with arithmetic, algebra, and geometry sufficient to be qualified for looking into the Ladies Diary (of which he had afterwards for several years the direction), by which he came to understand that there was a still higher branch of the mathematical knowledge than any he had yet been acquainted with; and this was the method of <hi rend="italics">Fluxions.</hi> But our young analyst was quite at a loss to discover any English author who had written on the subject, except Mr. Hayes; and his work being a folio, and then pretty scarce, exceeded his ability of purchasing: however an acquaintance lent him Mr. Stone's <hi rend="italics">Fluxions,</hi> which is a translation of the <hi rend="italics">Marquis de l'Hospital's Analyse des Infiniment Petits:</hi> by this one book, and his own penetrating talents, he was, as we shall see presently, enabled in a very few years to compose a much more accurate treatise on this subject than any that had before appeared in our language.</p><p>After he had quitted astrology and its emoluments, he was driven to hardships for the subsistence of his family, while at Derby, notwithstanding his other industrious endeavours in his own trade by day, and teaching pupils at evenings. This determined him to repair to London, which he did in 1735 or 1736.</p><p>On his first coming to London, Mr. Simpson wrought for some time at his business in Spitalfields, and taught mathematics at evenings, or any spare hours. His industry turned to so good account, that he returned down into the country, and brought up his wife and three children, she having produced her first child to him in his absence. The number of his scholars increasing, and his abilities becoming in some measure known to the public, he was encouraged to make proposals for publishing by subscription, A new Treatise of Fluxions: wherein the Direct and Inverse Methods are demonstrated after a new, clear, and concise Manner, with their Application to Physics and Astronomy: also the Doctrine of Infinite Series and Reverting Series universally, are amply explained, Fluxionary and Exponential Equations solved: together with a variety of new and curious Problems. <pb n="453"/><cb/></p><p>When Mr. Simpson first proposed his intentions of publishing such a work, he did not know of any English book, founded on the true principles of Fluxions, that contained any thing material, especially the practical part; and though there had been some very curious things done by several learned and ingenious gentlemen, the principles were nevertheless left obscure and defective, and all that had been done by any of them in <hi rend="italics">infinite series,</hi> very inconsiderable.</p><p>The book was published in 4to, in the year 1737, although the author had been frequently interrupted from furnishing the press so fast as he could have wished, through his unavoidable attention to his pupils for his immediate support. The principles of fluxions treated of in this work, are demonstrated in a method accurately true and genuine, not essentially different from that of their great inventor, being entirely expounded by finite quantities.</p><p>In 1740, Mr. Simpson published a Treatise on The Nature and Laws of Chance, in 4to. To which are annexed, Full and clear Investigations of two important Problems added in the 2d Edition of Mr. De Moivre's Book on Chances, as also two New Methods for the Summation of Series.</p><p>Our author's next publication was a 4to volume of Essays on several curious and interesting Subjects in Speculative and Mixed Mathhmatics; printed in the same year 1740: dedicated to Francis Blake, Esq. since Fellow of the Royal Society, and our author's good friend and patron.—Soon after the publication of this book, he was chosen a member of the Royal Academy at Stockholm.</p><p>Our author's next work was, The Doctrine of Annuities and Reversions, deduced from general and evident Principles: with useful Tables, shewing the Values of Single and Joint Lives, &c. in 8vo, 1742. This was followed in 1743, by an Appendix containing some Remarks on a late book on the same Subject (by Mr. Abr. De Moivre, F. R. S.) with Answers to some personal and malignant Representations in the Preface thereof. To this answer Mr. De Moivre never thought fit to reply. A new edition of this work has lately been published, augmented with the tract upon the same subject that was printed in our author's Select Exercises.</p><p>In 1743 also was published his Mathematical Dissertations on a variety of Physical and Analytical Subjects, in 4to; containing, among other particulars,</p><p>A Demonstration of the true Figure which the Earth, or any Planet, must acquire from its rotation about an Axis. A general Investigation of the Attraction at the Surfaces of Bodies nearly spherical. A Determination of the Meridional Parts, and the Lengths of the several Degrees of the Meridian, according to the true Figure of the Earth. An Investigation of the Height of the Tides in the Ocean. A new Theory of Astronomical Refractions, with exact Tables deduced from the same. A new and very exact Method for approximating the Roots of Equations in Numbers; which quintuples the number of Places at each Operation. Several new Methods for the Summation of Series. Some new and very useful Improvements in the Inverse Method of Fluxions. The work being dedicated to Martin Folkes, Esq. President of the Royal Society. <cb/></p><p>His next book was A Treatise of Algebra, wherein the fundamental Principles are demonstrated, and applied to the Solution of a variety of Problems. To which he added, The Construction of a great Number of Geometrical Problems, with the Method of resolving them numerically.</p><p>This work, which was designed for the use of young beginners, was inscribed to William Jones, Esq. F. R. S. and printed in 8vo, 1745. And a new edition appeared in 1755, with additions and improvements; among which was a new and general method of resolving all Biquadratic Equations, that are complete, or having all their terms. This edition was dedicated to James Earl of Morton, F. R. S. Mr. Jones being then dead. The work has gone through several other editions since that time: the 6th, or last, was in 1790.</p><p>His next work was, “Elements of Geometry, with their Application to the Mensuration of Superficies and Solids, to the Determination of Maxima and Minima, and to the Construction of a great Variety of geometrical Problems:” first published in 1747, in 8vo. And a second edition of the same came out in 1760, with great alterations and additions, being in a manner a new work, designed for young beginners, particularly for the gentlemen educated at the Royal Military Academy at Woolwich, and dedicated to Charles Frederick, Esq. Surveyor General of the Ordnance. And other editions have appeared since.</p><p>Mr. Simpson met with some trouble and vexation in consequence of the first edition of his Geometry. First, from some reflections made upon it, as to the accuracy of certain parts of it, by Dr. Robert Simson, the learned professor of mathematicks in the university of Glasgow, in the notes subjoined to his edition of Euclid's Elements. This brought an answer to those remarks from Mr. Simpson, in the notes added to the 2d edition as above; to some parts of which Dr. Simson again replied in his notes on the next edition of the said Elements of Euclid.</p><p>The second was by an illiberal charge of having stolen his Elements from Mr. Muller, the professor of fortification and artillery at the same academy at Woolwich, where our author was professor of geometry and mathematics. This charge was made at the end of the preface to Mr. Muller's Elements of Mathematics, in two volumes, printed in 1748; which was fully refuted by Mr. Simpson in the preface to the 2d edition of his Geometry.</p><p>In 1748 came out Mr. Simpson's Trigonometry, Plane and Spherical, with the Construction and Application of Logarithms, 8vo. This little book contains several things new and useful.</p><p>In 1750 came out, in two volumes, 8vo, The Doctrine and Application of Fluxions, containing, besides what is common on the Subject, a Number of new Improvements in the Theory, and the Solution of a Variety of new and very interesting Problems in different Branches of the Mathematics.—In the preface the author offers this to the world as a new book, rather than a second edition of that which was published in 1737, in which he acknowledges, that, besides errors of the press, there are several obscurities and defects, for want of experience, and the many disadvantages he then laboured under, in his first sally. <pb n="454"/><cb/></p><p>The idea and explanation here given of the first principles of Fluxions, are not essentially different from what they are in his former treatise, though expressed in other terms. The consideration of <hi rend="italics">time</hi> introduced into the general definition, will, he says, perhaps be disliked by those who would have fluxions to be <hi rend="italics">mere velocities:</hi> but the advantage of considering them otherwise, viz, not as the velocities themselves, but as magnitudes they would uniformly generate in a given time, appears to obviate any objection on that head. By taking fluxions as mere velocities, the imagination is confined as it were to a point, and without proper care insensibly involved in metaphysical difficulties. But according to this other mode of explaining the matter, less caution in the learner is necessary, and the higher orders of fluxions are rendered much more easy and intelligible. Besides, though Sir Isaac Newton defines fluxions to be the velocities of motions, yet he has recourse to the increments or moments generated in equal particles of time, in order to determine those velocities; which he afterwards teaches to expound by finite magnitudes of other kinds. This work was dedicated to George earl of Macclesfield.</p><p>In 1752 appeared, in 8vo, the <hi rend="italics">Select Exercises for young Preficients in the Mathematics.</hi> This neat volume contains, A great Variety of algebraical Problems, with their Solutions. A select Number of Geometrical Problems, with their Solutions, both algebraical and geometrical. The Theory of Gunnery, independent of the Conic Sections. A new and very comprehensive Method for finding the Roots of Equations in Numbers. A short Account of the first Principles of Fluxions. Also the Valuation of Annuities for single and joint Lives, with a Set of new Tables, far more extensive than any extant. This last part was designed as a supplement to his Doctrine of Annuities and Reversions; but being thought too small to be published alone, it was inserted here at the end of the Select Exercises; from whence however it has been removed in the last editions, and referred to its proper place, the end of the Annuities, as before mentioned. The examples that are given to each problem in this last piece, are according to the London bills of mortality; but the solutions are general, and may be applied with equal facility and advantage to any other table of observations. The volume is dedicated to John Bacon, Esq. F. R. S.</p><p>Mr. Simpson's Miscellaneous Tracts, printed in 4to, 1757, were his last legacy to the public: a most valuable bequest, whether we consider the dignity and importance of the subjects, or his sublime and accurate manner of treating them.</p><p>The first of these papers is concerned in determining the Precession of the Equinox, and the different Motions of the Earth's Axis, arising from the Attraction of the Sun and Moon. It was drawn up about the year 1752, in consequence of another on the same subject, by M. de Sylvabelle, a French gentleman. Though this gentleman had gone through one part of the subject with success and perspicuity, and his conclusions were perfectly conformable to Dr. Bradley's observations; it nevertheless appeared to Mr. Simpson, that he had greatly failed in a very material part, and that indeed the only very difficult one; that is, in the determination of the momentary alteration of the po- <cb/> sition of the earth's axis, caused by the forces of the sun and moon; of which forces, the quantities, but not the effects, are truly investigated. The second paper contains the Investigation of a very exact Method or Rule for finding the Place of a Planet in its Orbit, from a Correction of Bishop Ward's circular Hypothesis, by Means of certain Equations applied to the Motion about the upper Focus of the Ellipse. By this Method the Result, even in the Orbit of Mercury, may be found within a Second of the Truth, and that without repeating the Operation. The third shews the Manner of transferring the Motion of a Comet from a parabolic Orbit, to an elliptic one; being of great Use, when the observed Places of a (new) Comet are sound to differ sensibly from those computed on the Hypothesis of a parabolic Orbit. The fourth is an Attempt to shew, from mathematical Principles, the Advantage arising from taking the Mean of a Number of Observations, in practical Astronomy; wherein the Odds that the Result in this Way, is more exact than from one single Observation, is evinced, and the Utility of the Method in Practice clearly made appear. The fifth contains the Determination of certain Fluents, and the Resolution of some very useful Equations, in the higher Orders of Fluxions, by Means of the Measures of Angles and Ratios, and the right and versed Sines of circular Arcs. The 6th treats of the Resolution of algebraical Equations, by the Method of Surd-divisors; in which the Grounds of that Method, as laid down by Sir Isaac Newton, are investigated and explained. The 7th exhibits the Investigation of a general Rule for the Resolution of Isoperimetrical Problems of all Orders, with some Examples of the Use and Application of the said Rule. The 8th, or last part, comprehends the Resolution of some general and very important Problems in Mechanics and Physical Astronomy; in which, among other Things, the principal Parts of the 3d and 9th Sections of the first Book of Newton's Principia are demonstrated in a new and concise Manner. But what may perhaps best recommend this excellent tract, is the application of the general equations, thus derived, to the determination of the Lunar Orbit.</p><p>According to what Mr. Simpson had intimated at the conclusion of his Doctrine of Fluxions, the greatest part of this arduous undertaking was drawn up in the year 1750. About that time M. Clairaut, a very eminent mathematician of the French Academy, had started an objection against Newton's general law of gravitation. This was a motive to induce Mr. Simpson (among some others) to endeavour to discover whether the motion of the moon's apogee, on which that objection had its whole weight and foundation, could not be truly accounted for, without supposing a change in the received law of gravitation, from the inverse ratio of the squares of the distances. The success answered his hopes, and induced him to look farther into other parts of the theory of the moon's motion, than he had at first intended: but before he had completed his design, M. Clairaut arrived in England, and made Mr. Simpson a visit; from whom he learnt, that he had a little before printed a piece on that subject, a copy of which Mr. Simpson asterwards received as a present, and found in it the same things demonstrated, to which he himself had directed his enquiry, besides several others. <pb n="455"/><cb/></p><p>The facility of the method Mr. Simpson fell upon, and the extensiveness of it, will in some measure appear from this, that it not only determines the motion of the apogee, in the same manner, and with the same ease, as the other equations, but utterly excludes all that dangerous kind of terms that had embarrassed the greatest mathematicians, and would, after a great number of revolutions, entirely change the figure of the moon's orbit. From whence this important consequence is derived, that the moon's mean motion, and the greatest quantities of the several equations, will remain unchanged, unless disturbed by the intervention of some foreign or accidental cause. These tracts are inscribed to the Earl of Macclesfield, President of the Royal Society.</p><p>Besides the foregoing, which are the whole of the regular books or treatises that were published by Mr. Simpson, he wrote and composed several other papers and fugitive pieces, as follow:</p><p>Several papers of his were read at the meetings of the Royal Society, and printed in their Transactions: but as most, if not all of them, were afterwards inserted, with alterations or additions, in his printed volumes, it is needless to take any farther notice of them here.</p><p>He proposed, and resolved many questions in the Ladies Diaries, &c; sometimes under his own name, as in the years 1735 and 1736; and sometimes under feigned or fictitious names; such as, it is thought, Hurlothrumbo, Kubernetes, Patrick O'Cavenah, Marmaduke Hodgson, Anthony Shallow, Esq, and probably several others; see the Diaries for the years 1735, 1736, 42, 43, 53, 54, 55, 56, 57, 58, 59, and 60. Mr. Simpson was also the editor or compiler of the Diaries from the year 1754 till the year 1760, both inclusive, during which time he raised that work to the highest degree of respect. He was succeeded in the Editorship by Mr. Edw. Rollinson. See my Diarian Miscellany, vol. 3.</p><p>It has also been commonly supposed that he was the real editor of, or had a principal share in; two other periodical works of a miscellaneous mathematical nature; viz, the Mathematician, and Turner's Mathematical Exercises, two volumes, in 8vo, which came out in periodical numbers, in the years 1750 and 1751, &c. The latter of these seems especially to have been set on foot to afford a proper place for exposing the errors and absurdities of Mr. Robert Heath, the then conductor of the Ladies Diary and the Palladium; and which controversy between them ended in the disgrace of Mr. Heath, and expulsion from his office of editor to the Ladies Diary, and the substitution of Mr. Simpson in his stead, in the year 1753.</p><p>In the year 1760, when the plans proposed for erecting a new bridge at Blackfriars were in agitation, Mr. Simpson, among other gentlemen, was consulted upon the best form for the arches, by the New-bridge Committee. Upon this occasion he gave a preference to the semicircular form; and, besides his report to the Committee, some letters also appeared, by himself and others, on the same subject, in the public newspapers, particularly in the Daily Advertiser, and in Lloyd's Evening Post. The same were also collected in the Gentleman's Magazine for that year, page 143 and 144. <cb/></p><p>It is probable that this reference to him, gave occasion to the turning his thoughts more seriously to this subject, so as to form the design of composing a regular treatise upon it: for his family have often informed me, that he laboured hard upon this work for some time before his death, and was very anxious to have completed it, frequently remarking to them, that this work, when published, would procure him more credit than any of his former publications. But he lived not to put the finishing hand to it. Whatever he wrote upon this subject, probably fell, together with all his other remaining papers, into the hands of major Henry Watson, of the engineers, in the service of the India Company, being in all a large chest full of papers. This gentleman had been a pupil of Mr. Simpson's, and had lodged in his house. After Mr. Simpson's death, Mr. Watson prevailed upon the widow to let him have the papers, promising either to give her a sum of money for them, or else to print and publish them for her benefit. But neither of these was ever done; this gentleman always declaring, when urged on this point by myself and others, that no use could be made of any of the papers, owing to the very imperfect slate in which he said they were left. And yet he persisted in his refusal to give them up again.</p><p>From Mr. Simpson's writings, I now return to himself. Through the interest and solicitations of the beforementioned William Jones, Esq, he was, in 1743, appointed professor of mathematics, then vacant by the death of Mr. Derham, in the Royal Academy at Woolwich; his warrant bearing date August 25th. And in 1745 he was admitted a fellow of the Royal Society, having been proposed as a candidate by Martin Folkes, Esq. President, William Jones, Esq. Mr. George Graham, and Mr. John Machin, Secretary; all very eminent mathematicians. The president and council, in consideration of his very moderate circumstances, were pleased to excuse his admission fees, and likewise his giving bond for the settled future payments.</p><p>At the academy he exerted his faculties to the utmost, in instructing the pupils who were the immediate objects of his duty, as well as others, whom the superior officers of the ordnance permitted to be boarded and lodged in his house. In his manner of teaching, he had a peculiar and happy address; a certain dignity and perspicuity, tempered with such a degree of mildness, as engaged both the attention, esteem and friendship of his scholars; of which the good of the service, as well as of the community, was a necessary consequence.</p><p>It must be acknowledged however, that his mildness and easiness of temper, united with a more inactive state of mind, in the latter years of his life, rendered his services less useful; and the same very easy disposition, with an innocent, unsuspecting simplicity, and playfulness of mind, rendered him often the dupe of the little tricks of his pupils. Having discovered that he was fond of listening to little amusing stories, they took care to furnish themselves with a stock; so that, having neglected to learn their lessons perfect, they would get round him in a crowd, and, instead of demonstrating a proposition, would amuse him with some comical story, at which he would laugh and shake very heartily, especially if it were tinctured with somewhat of the <pb n="456"/><cb/> ludicrous or smutty; by which device they would contrive imperceptibly to wear out the hours allotted for instruction, and so avoid the trouble of learning and repeating their lesson. They tell also of various tricks that were practised upon him in consequence of the loss of his memory in a great degree, in the latter stage of his life.</p><p>It has been said that Mr. Simpson frequented low company, with whom he used to guzzle porter and gin: but it must be observed that the misconduct of his family put it out of his power to keep the company of of gentlemen, as well as to procure better liquor.</p><p>In the latter stage of his existence, when his life was in danger, exercise and a proper regimen were prescribed him, but to little purpose; for he sunk gradually into such a lowness of spirits, as often in a manner deprived him of his mental faculties, and at last rendered him incapable of performing his duty, or even of reading the letters of his friends; and so trifling an accident as the dropping of a tea-cup would flurry him as much as if a house had tumbled down.</p><p>The physicians advised his native air for his recovery; and in February, 1761, he set out, with much reluctance (believing he should never return) for Bosworth, along with some relations. The journey fatigued him to such a degree, that upon his arrival he betook himself to his chamber, where he grew continually worse and worse, to the day of his death, which happened the 14th of May, in the fiftyfirst year of his age.</p></div1><div1 part="N" n="SINE" org="uniform" sample="complete" type="entry"><head>SINE</head><p>, or <hi rend="italics">Right</hi> <hi rend="smallcaps">Sine</hi>, of an arc, in Trigonometry, a right line drawn from one extremity of the arc, perpendicular to the radius drawn to the other extremity of it: Or, it is half the chord <figure/> of double the arc. Thus the line DE is the sine of the arc BD; either because it is drawn from one end D of that arc, perpendicular to CB the radius drawn to the other end B of the arc; or also because it is half the chord DF of double the arc DBF. For the same reason also DE is the Sine of the arc AD, which is the supplement of BD to a semicircle or 180 degrees; that is, every Sine is common to two arcs, which are supplements to each other, or whose sum make up a semicircle, or 180 degrees.</p><p>Hence the Sines increase always from nothing at B till they become the radius CG, which is the greatest, being the Sine of the quadrant BG. From hence they decrease all the way along the second quadrant from G to A, till they quite vanish at the point A, thereby shewing that the Sine of the semicircle BGA, or 180 degrees, is nothing. After this they are negative all the way along the next semicircle, or 3d and 4th quadrants AFB, being drawn on the opposite side, or downwards from the diameter AB.</p><p><hi rend="italics">Whole</hi> <hi rend="smallcaps">Sine</hi>, or <hi rend="italics">Sinus Totus,</hi> is the Sine of the quadrant BG, or of 90 degrees; that is, the Whole Sine is the same with the radius CG.</p><p><hi rend="smallcaps">Sine</hi>-<hi rend="italics">Complement,</hi> or <hi rend="italics">Cosine,</hi> is the sine of an arc DG, which is the complement of another arc BD, to a quadrant. That is, the line DH is the Cosine of the arc BD; because it is the sine of DG which is the <cb/> complement of BD. And for the same reason DE is the Cosine of DG. Hence the sine and Cosine and radius, of any arc, form a right-angled triangle CDE or CDH, of which the radius CD is the hypotenuse; and therefore the square of the radius is equal to the sum of the squares of the sine and Cosine of any arc, that is, .</p><p>It is evident that the Cosine of o or nothing, is the whole radius CB. From B, where this Cosine is greatest, the Cosine decreases as the arc increases from B along the quadrant BDG, till it become o for the complete quadrant BG. After this, the Cosines, decreasing, become negative more and more all the way to the complete semicircle at A. Then the Cosines increase again all the way from A through I to B; at I the negation is destroyed, and the Cosine is equal to o or nothing; from I to B it is positive, and at B it is again become equal to the radius. So that, in general, the Cosines in the 1st and 4th quadrants are positive, but in the 2d and 3d negative.</p><p><hi rend="italics">Versed</hi> <hi rend="smallcaps">Sine</hi>, is the part of the diameter between the sine and the arc. So BE is the Versed Sine of the arc BD, and AE the Versed Sine of AD, also GH the Versed Sine of DG, &c. All Versed Sines are affirmative. The sum of the Versed Sine and cosine, of any arc or angle, is equal to the radius, that is, .—The sine, cosine, and Versed Sine, of an arc, are also the same of an angle, or the number of degrees &c, which it measures.</p><p>The Sines &c, of every degree and minute in a quadrant, are calculated to the radius 1, and ranged in tables for use. But because operations with these natural Sines require much labour in multiplying and dividing by them, the logarithms of them are taken, and ranged in tables also; and these logarithmic Sines are commonly used in practice, instead of the natural ones, as they require only additions and subtractions, instead of the multiplications and divisions. For the method of constructing the scales of Sines &c, see the article <hi rend="smallcaps">Scale.</hi></p><p>The Sines were introduced into trigonometry by the Arabians. And for the etymology of the word <hi rend="italics">Sine</hi> see Introduction to my Logarithms, pa. 17 &c. And the various ways of calculating tables of the Sines, may be seen in the same place, pa. 13 &c.</p><p><hi rend="italics">Theorems for the Sines, Cosines, &c,</hi> one from another. From the definitions of them, and the common property of right-angled triangles, with that of the circle, viz, that , are easily deduced these following values of the Sines, &c, viz, putting <hi rend="italics">s</hi> = the sine DE, <hi rend="italics">c</hi> = the cosine CE, <hi rend="italics">v</hi> = versed sine BE, v = suppl. versed sine AE, <hi rend="italics">r</hi> = radius AC or CB, <hi rend="italics">a</hi> = arc BD; then <pb n="457"/><cb/> . See many other curious expressions of this kind in Bougainville's Calcul Integral, and in Bertrand's Mathematics. <cb/></p><p>From some of the foregoing theorems the Sines of a great variety of angles, or number of degrees, may be computed. Ex. gr. as below. <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Angles.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Sines.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">90°</cell><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√(2 + √3) = <hi rend="italics">r</hi> X ((√6 + √2)/4)</cell></row><row role="data"><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√((5 + √5)/2)</cell></row><row role="data"><cell cols="1" rows="1" role="data">67 1/2</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√(2 + √2)</cell></row><row role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√3</cell></row><row role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√((3 + √5)/2) = <hi rend="italics">r</hi> X ((√5 + )1/4)</cell></row><row role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√2</cell></row><row role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√((5 - √5)/2)</cell></row><row role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">22 1/2</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√(2 - √2)</cell></row><row role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√((3 - √5)/2) = <hi rend="italics">r</hi> X ((√5 - 1)/4)</cell></row><row role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">(1/2)<hi rend="italics">r</hi>√(2 - √3) = <hi rend="italics">r</hi> X ((√6 - √2)/4)</cell></row></table></p><p>Radius being 1. Then for multiple arcs: the ;</p><p>That is, multiplying any Sine or cosine by 2<hi rend="italics">c,</hi> and the next preceding Sine or cosine subtracted from it, it gives the next following Sine or cosine. Hence <table><row role="data"><cell cols="1" rows="1" role="data">sin. 0<hi rend="italics">a</hi> = 0.</cell><cell cols="1" rows="1" role="data">cos. 0<hi rend="italics">a</hi> = 1 or radius.</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. <hi rend="italics">a</hi> = <hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data">cos. <hi rend="italics">a</hi> = <hi rend="italics">c.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 2<hi rend="italics">a</hi> = 2<hi rend="italics">sc.</hi></cell><cell cols="1" rows="1" role="data">cos. 2<hi rend="italics">a</hi> = <hi rend="italics">c</hi><hi rend="sup">2</hi> - <hi rend="italics">s</hi><hi rend="sup">2</hi>.</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 3<hi rend="italics">a</hi> = 3<hi rend="italics">sc</hi><hi rend="sup">2</hi> - <hi rend="italics">s</hi><hi rend="sup">3</hi>.</cell><cell cols="1" rows="1" role="data">cos. 3<hi rend="italics">a</hi> = <hi rend="italics">c</hi><hi rend="sup">3</hi> - 3 <hi rend="italics">cs</hi><hi rend="sup">2</hi>.</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 4<hi rend="italics">a</hi> = 4<hi rend="italics">sc</hi><hi rend="sup">3</hi> - 4<hi rend="italics">s</hi><hi rend="sup">3</hi><hi rend="italics">c.</hi></cell><cell cols="1" rows="1" role="data">cos. 4<hi rend="italics">a</hi> = <hi rend="italics">c</hi><hi rend="sup">4</hi> - 6<hi rend="italics">c</hi><hi rend="sup">2</hi><hi rend="italics">s</hi><hi rend="sup">2</hi> + <hi rend="italics">s</hi><hi rend="sup">4</hi>.</cell></row><row role="data"><cell cols="1" rows="1" role="data">sin. 5<hi rend="italics">a</hi> = 5<hi rend="italics">sc</hi><hi rend="sup">4</hi> - 10<hi rend="italics">s</hi><hi rend="sup">3</hi><hi rend="italics">c</hi><hi rend="sup">2</hi> + <hi rend="italics">s</hi><hi rend="sup">5</hi>.</cell><cell cols="1" rows="1" role="data">cos. 5<hi rend="italics">a</hi> = <hi rend="italics">c</hi><hi rend="sup">5</hi> - 10 <hi rend="italics">c</hi><hi rend="sup">3</hi><hi rend="italics">s</hi><hi rend="sup">2</hi> + 5<hi rend="italics">cs</hi><hi rend="sup">4</hi>.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">&c.</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> <cb/></p><p>And, in general, <hi rend="center"><hi rend="italics">Of the Tables of Sines, &c.</hi></hi></p><p>In estimating the quantity of the Sines &c, we assume radius for unity; and then compute the quantity <cb/> of the Sines, tangents, and secants, in fractions of it. From Ptolomy's Almagest we learn, that the ancients divided the radius into 60 parts, which they called degrees, and thence determined the chords in mi- <pb n="458"/><cb/> nutes, seconds, and thirds; that is, in sexagesimal fractions of the radius, which they likewise used in the resolution of triangles. As to the Sines, tangents and secants, they are modern inventions; the Sines being introduced by the Moors or Saracens, and the tangents and secants afterwards by the Europeans. See Introd. to my Logs. pa. 1 to 19.</p><p>Regiomontanus, at first, with the ancients, divided the radius into 60 degrees; and determined the Sines of the several degrees in decimal fractions of it. But he afterwards found it would be more convenient to assume 1 for radius, or 1 with any number of cyphers, and take the Sines in decimal parts of it; and thus he introduced the present method in trigonometry. In this way, different authors have divided the radius into more or fewer decimal parts; but in the common tables of Sines and tangents, the radius is conceived as divided into 10000000 parts; by which all the Sines are estimated.</p><p>An idea of some of the modes of constructing the tables of Sines, may be conceived from what here follows: First, by common geometry the sides of some of the regular polygons inscribed in the circle are computed, from the given radius, which will be the chords of certain portions of the circumference, denoted by the number of the sides; viz, the side of the triangle the chord of the 3d part, or 120 degrees; the side of the pentagon the chord of the 5th part, or 72 degrees; the side of the hexagon the chord of the 6th part, or 60 degrees; the side of the octagon the chord of the 8th part, or 45 degrees; and so on. By this means there are obtained the chords of several of such arcs; and the halves of these chords will be the Sines of the halves of the same arcs. Then the theorem will give the cosines of the same half arcs. Next, by bisecting these arcs continually, there will be found the Sines and cosines of a continued series as far as we please by these two theorems, . Then, by the theorems for the sums and differences of arcs, from the foregoing series, will be derived the Sines and cosines of various other arcs, till we arrive at length at the arc of 1′, or 1″, &c, whose Sine and cosine thus become known.</p><p>Or, rather, the sine of 1 minute will be much more easily found from the series , because the arc is equal to its Sine in small arcs; whence <hi rend="italics">s</hi> = <hi rend="italics">a</hi> only in such small arcs. But the length of the arc of 180° or 10800′ is known to be 3.14159265, &c; therefore, by proportion, as 10800′ : 1′ :: 3.14159265 : 0.0002908882 = <hi rend="italics">a</hi> the arc or <hi rend="italics">s</hi> the sine of 1′, which number is true to the last place of decimals. Then, for the cosine of 1′, it is the cosine of the same 1′.</p><p>Hence we shall readily obtain the Sines and cosines of all the multiples of 1′, as of 2′, 3′ 4′ 5′, &c, by the application of these two theorems, <cb/> ; for supposing <hi rend="italics">a</hi> = the arc of 1, then <hi rend="italics">c</hi> = 0.9999999577, and taking <hi rend="italics">n</hi> successively, equal to 1, 2, 3, 4, &c, the theorems for the Sines and cosines give severally the Sines and cosines of 1′, 2′, 3′, 4′, &c; viz, the Sines thus: .</p><p>In this manner then all the Sines and cosines are made, by only one constant multiplication and a subtraction, up to 30 degrees, forming thus the Sines of the first and last 30 degrees of the quadrant, or from 0 to 30° and from 60° to 90°; or, which will be much the same thing, the Sines only may be thus computed all the way up to 60°.</p><p>Then the Sines of the remaining 30°, from 60 to 90, will be found by one addition only for each of them, by means of this theorem, viz, ; that is, to the sine of any arc below 60°, add the Sine of its defect below 60, and the sum will be the Sine of another arc which is just as much above 60.</p><p>The Sines of all arcs being thus found, they give also very easily the versed sines, the tangents, and the secants. The versed sines are only the arithmetical complements to 1, that is, each cosine taken from the radius 1.</p><p>The tangents are found by these three theorems:</p><p>1. As cosine to sine, so is radius to tangent.</p><p>2. Radius is a mean proportional between the tangent and cotangent.</p><p>3. Half the difference between the tangent and cotangent, is equal to the tangent of the difference between the arc and its complement. Or, the sum arising from the addition of double the tangent of an arc with the tangent of half its complement, is equal to the tangent of the sum of that arc and the said half complement.</p><p>By the 1st and 2d of these theorems, the tangents are to be found for one half of the quadrant: then the other half of them will be found by one single addition, or subtraction, for each, by the 3d theorem.</p><p>This done, the secants will be all found by addition or subtraction only, by these two theorems: 1st. The secant of an arc, is equal to the sum of its tangent and the tangent of half its complement. 2nd. The secant <pb n="459"/><cb/> of an arc, is equal to the difference between the tangent of that arc and the tangent of the arc added to half its complement.</p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Sines</hi>, are the logarithmic Sines, or the logarithms of the Sines.</p><p><hi rend="italics">Curve</hi> or <hi rend="italics">Figure of the</hi> <hi rend="smallcaps">Sines.</hi> See <hi rend="smallcaps">Figure</hi> <hi rend="italics">of the Sines, &c.</hi> To what is there said of the figure of the Sines, may be here added as follows, from a property just given above, viz, if <hi rend="italics">x</hi> denote the absciss of this curve, or the corresponding circular arc, and <hi rend="italics">y</hi> its ordinate, or the Sine of that arc; then the equation of the curve will be this, ; where <hi rend="italics">h</hi> = 2.718281828, &c, the number whose hyp. log. is 1.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Sines</hi>, is a line on the sector, or Gunter's scale, &c, divided according to the Sines, or expressing the Sines. See those articles.</p><p><hi rend="smallcaps">Sine</hi> <hi rend="italics">of Incidence,</hi> or <hi rend="italics">of Refraction,</hi> is used for the Sine of the <hi rend="italics">angle</hi> of incidence, &c.</p><p>SINICAL <hi rend="italics">Quadrant,</hi> is a quadrant, made of wood or metal, with lines drawn from each side intersecting one another, with an index, divided by sines, also with 90 degrees on the limb, and two sights at the edge. Its use is to take the altitude of the sun. Instead of the sines, it is sometimes divided all into equal parts; and then it is used by seamen to resolve, by inspection, any problem of plane sailing.</p></div1><div1 part="N" n="SIPHON" org="uniform" sample="complete" type="entry"><head>SIPHON</head><p>, or <hi rend="smallcaps">Syphon</hi>, in Hydraulics, a crooked pipe or tube used in the raising of fluids, emptying of vessels, and in various hydrostatical experiments. It is otherwise called a crane.</p><p>Wolfius describes two vessels under the name of Siphons; the one cylindrical in the middle and conical at the two extremes; the other globular in the middle, with two narrow tubes fitted to it axis-wise; both serving to take up a quantity of liquid, and to retain it when up.</p><p>But the most usual Syphon is that which is here represented; where ABC is any <figure/> crooked tube, having two legs of unequal lengths; but such however that, in any position, the perpendicular altitude BD, of B above A, when AB is filled with any fluid, the weight of that fluid may not be more than about 15lb. upon every square inch of the base, or equal to the pressure of the atmosphere, because the pressure of the atmosphere will raise or suspend the fluid so high, when the tube is exhausted of air. This height is about 30 inches when the fluid is quicksilver, and about 34 feet when it is water; and so on for other fluids, according to the rarity of them.</p><p>To use the Siphon, in drawing off any fluid; immerse the shorter end A into the fluid, then suck or draw the air out by the other or lower end C, and the fluid will presently follow, and run out by the Siphon, from the vessel at A to the vessel at C; till such time as the surface of the fluid sink as low as the orifice at A, <cb/> <figure/> when the decanting will cease, and the Siphon will empty itself of the fluid, the whole of that which is in it running out at C. The principle upon which the Siphon acts, is this: when the tube is exhausted of air, the pressure of the atmosphere upon the surface of the fluid at D, forces it into the tube by the orifice at A, as in the barometer tube, and down the leg BC, if B is not above the surface at D more than 34 feet for water, or 30 inches for quicksilver, &c. Here, if the external leg of the Siphon terminate at E, on a horizontal level with the immersed end at A, or rather on a level with the water at D, the perpendicular pressures of the fluid in each leg, and of the external air, against each orisice, being alike in both, the fluid will be at rest in the Siphon, completely filling it, but without running or preponderating either way. But if the external end be the lower, terminating at C, then the fluid in this end being the heavier, or having more pressure, will preponderate and run out by the orifice at C; this would leave a vacuum at B but for the continual pressure of the atmosphere at D, which forces the fluid up by A to B, and so producing a continued motion of it through the tube, and a discharge or stream at C.</p><p>Instead of sucking out the air at C, another method is, first to fill the tube completely with the fluid, in an inverted position with the angle B downward; and, stopping the two orifices with the fingers, revert the tube again, and immerge the end A in the fluid; then take off the fingers, and immediately the stream commences from the end C.</p><p>Either of the two foregoing methods can be conveniently practised when the Siphon is small, and easily managed by the hand; as in decanting off liquors from casks, &c. But when the Siphon is very large, and many feet in height, as in exhausting water from a valley or pit, the following method is then recommended: Stop the orifice C, and, by means of an opening made in the top at B, fill the tube completely with water; then stop the opening at B with a plug, and open that at C; upon which the water will presently slow out at C, and <figure/> <pb n="460"/><cb/> so continue till that at A is exhausted. And this method of conveying water over a hill, from one valley to another, is described by Hero, the chief author of any consequence upon this subject among the Ancients. But in this experiment it must be noted, that the effect will not be produced when the hill at B is more than 33 or 34 feet above the surface of the water at A.</p><p>In an experiment of this kind, it is even said the water in the legs, unless it be purged of its air, will not rest at a height of quite 30 feet above the water in the vessels; because air will extricate itself out of the water, and getting above the water in the legs, press it downward, so that its height will be less to balance the pressure of the atmosphere. But with very sine, or capillary tubes, the experiment will succeed to a height somewhat greater; because the attraction of the matter of the very fine tube will attract the fluid, and support it at some certain height, independent of the pressure of the atmosphere. For which reason also it is, that the experiment succeeds for small heights in the exhausted receiver; as has been tried both with water and mercury, by Desaguliers and many other philosophers. Exper. Philos. vol. 2, pa. 168.</p><p>The figure of the vessel may be varied at pleasure, provided the orifice C be but below the level of the surface of the water to be drawn up, but still the farther it is below it, the quicker will the fluid run off. And if, in the course of the efflux, the orifice A be drawn out of the fluid; all the liquor in the Siphon will issue out at the lower orifice C; that in the leg BC dragging, as it were, that in the shorter leg AB after it.</p><p>But if a filled Siphon be so disposed, as that both orifices, A and C, be in the same horizontal line; the fluid will remain pendant in each leg, how unequal soever the length of the legs may be. So that fluids in Siphons seem, as it were, to form one continued body; the heavier part descending like a chain, and drawing the lighter after it.</p><p>The <hi rend="italics">Wirtemberg</hi> <hi rend="smallcaps">Siphon</hi>, is a very extraordinary <figure/> machine, performing several things which the common Siphon will not reach. This Siphon was projected by Jordan Pelletier, and executed at the expence of prince Frederic Charles, administrator of Wirtemberg, by his mathematician Shahackard, who made each branch 20 feet long, and set them 18 feet apart; and the description of it was published by Reiselius, the duke's physician. This gave occasion to Papin to invent another, which performed the same things, and is described in <cb/> the Philos. Trans. vol. 14, or Abr. vol. 1. Reiseline, in another paper in the same volume, ingenuously owns that this is the same with the Wirtemberg Siphon.</p><p>In this engine, though the legs be on the same level, yet the water rises up the one, and descends through the other: The water rises even through the aperture if the less leg be only half immerged in water: The Siphon has its effect after continuing dry a long time: Either of the apertures being opened, the other remaining shut for a whole day, and then opened, the water flows out as usual: Lastly, the water rises and falls indifferently through either leg.</p><p>Musschenbroek, in accounting for the operation of this Siphon, observes that no discharge could be made by it, unless the water applied to either leg cause the one to be shorter, and the other longer by its own weight. Introd. ad Phil. Nat. tom. 2, pa. 853, ed. 4to. 1762.</p></div1><div1 part="N" n="SIRIUS" org="uniform" sample="complete" type="entry"><head>SIRIUS</head><p>, the <hi rend="italics">Dog-star;</hi> a very bright star of the first magnitude, in the mouth of the constellation <hi rend="italics">Canis Major,</hi> or the Great Dog.</p><p>This is the brightest of all the stars in our firmanent, and therefore probably, says Dr. Maskelyne, the astronomer royal, the nearest to us of them all, in a paper recommending the discovery of its parallax, Philos. Trans. vol. 2, pa. 889. Some however suppose Arcturus to be the nearest.</p><p>The Arabs call it <hi rend="italics">Aschere, Elschecre, Scera;</hi> the Greeks, <hi rend="italics">Sirius;</hi> and the Latins, <hi rend="italics">Canicula,</hi> or <hi rend="italics">Canis candens.</hi> See <hi rend="smallcaps">Canicula.</hi></p><p>This is one of the earliest named stars in the whole heavens. Hesiod and Homer mention only four or five constellations, or stars, and this is one of them. Sirius and Orion, the Hyades, Pleiades, and Arcturus are almost the whole of the old poetical astronomy. The three last the Greeks formed of their own observation, as appears by the names; the two others were Egyptian. Sirius was so called from the Nile, one of the names of that river being Siris; and the Egyptians, seeing that river begin to swell at the time of a particular rising of this star, paid divine honours to the star, and called it by a name derived from that of the river, expressing the star of the Nile.</p></div1><div1 part="N" n="SITUS" org="uniform" sample="complete" type="entry"><head>SITUS</head><p>, in Algebra and Geometry, denotes the situation of lines, surfaces, &c. Wolfius delivers some things in geometry, which are not deduced from the common analysis, particularly matters depending on the <hi rend="italics">Situs</hi> of lines and figures. Leibnitz has even founded a particular kind of analysis upon it, called <hi rend="italics">Calculus Situs.</hi></p></div1><div1 part="N" n="SKY" org="uniform" sample="complete" type="entry"><head>SKY</head><p>, the blue expanse of the air or atmosphere.</p><p>The azure colour of the sky is attributed, by Newton, to vapours beginning to condense, having attained consistence enough to reflect the most reflexible rays, viz, the violet ones; but not enough to reflect any of the less reflexible ones.</p><p>De la Hire attributes it to our viewing a black object, viz the dark space beyond the regions of the atmosphere, through a white or lucid one, viz the air illuminated by the sun; a mixture of black and white always appearing blue. But this hypothesis is not originally his; being as old as Leonardo da Vinci.</p></div1><div1 part="N" n="SLIDING" org="uniform" sample="complete" type="entry"><head>SLIDING</head><p>, in Mechanics, is when the same point of a body, moving along a surface, describes a line on <pb n="461"/><cb/> that surface. Such is the motion of a parallelopipedon moved along a plane.</p><p>From Sliding arises friction.</p><p><hi rend="smallcaps">Sliding</hi> <hi rend="italics">Rule,</hi> a mathematical instrument serving to perform computations in gauging, measuring, &c, without the use of compasses; merely by the sliding of the parts of the instrument one by another, the lines and divisions of which give the answer or amount by inspection.</p><p>This instrument is variously contrived and applied by different authors, particularly Gunter, Partridge, Hunt, Everard, and Coggeshall; but the most usual and useful ones are those of the two latter.</p><p><hi rend="italics">Everard's</hi> <hi rend="smallcaps">Sliding</hi> <hi rend="italics">Rule</hi> is chiefly used in cask gauging. It is commonly made of box, 12 inches long, 1 inch broad, and (6/10) of an inch thick. It consists of three parts; viz, the stock just mentioned, and two thin slips, of the same length, sliding in small grooves in two opposite sides of the stock: consequently, when both these pieces are drawn out to their full extent, the instrument is 3 feet long.</p><p>On the first broad face of the instrument are four logarithmic lines of numbers; for the properties &c, of which, see <hi rend="smallcaps">Gunter</hi>'s <hi rend="italics">Line.</hi> The first, marked A, consisting of two radii numbered 1, 2, 3, 4, 5, 6, 7, 8, 9, 1; and then 2, 3, 4, 5, &c, to 10. On this line are four brass centre-pins, two in each radius; one in each of them being marked MB, for malt-bushel, is set at 2150.42 the number of cubic inches in a maltbushel; the other two are marked with A, for ale-gallon, at 282, the number of cubic inches in an ale gallon. The 2d and 3d lines of numbers are on the sliding pieces, and are exactly the same with the first; but they are distinguished by the letter B. In the first radius is a dot, marked <hi rend="italics">Si,</hi> at .707, the side of a square inscribed in a circle whose diameter is 1. Another dot, marked <hi rend="italics">Se.</hi> stands at .886, the side of a square equal to the area of the same circle. A third dot, marked W, is at 231, the cubic inches in a wine gallon. And a fourth, marked C, at 3.14, the circumference of the same circle whose diameter is 1. The fourth line of numbers, marked MD, to signify malt-depth, is a broken line of two radii, numbered 2, 10, 9, 8, 7, 6, 5, 4, 3, 2, 1, 9, 8, 7, &c; the number 1 being set directly against MB on the first radius.</p><p>On the second broad face, marked <hi rend="italics">cd,</hi> are several lines: as 1st, a line marked D, and numbered 1, 2, 3, &c, to 10. On this line are four centre pins: the first, marked WG, for wine-gauge, is at 17.15, the gaugepoint for wine gallons, being the diameter of a cylinder whose height is one inch, and content 231 cubic inches, or a wine gallon: the second centre-pin, marked AG, for ale-gauge, is at 18.95, the like diameter for an ale gallon: the 3d, marked MS, for malt square, is at 46.3, the square root of 2150.42, or the side of a square whose content is equal to the number of inches in a solid bushel: and the fourth, marked MR, for malt-round, is at 52.32, the diameter of a cylinder, or bushel, the area of whose base is the same 2150.42, the inches in a bushel. 2dly, Two lines of numbers on the sliding piece, on the other side, marked C. On these are two dots; the one, marked <hi rend="italics">c,</hi> at .0795, the area of a circle whose circumference is 1; and the other, marked <hi rend="italics">d,</hi> at .785, the area of the circle whose diame- <cb/> ter is 1. 3dly, Two lines of segments, each numbered 1, 2, 3, to 100; the first for finding the ullage of a cask, taken as the middle frustum of a spheroid, lying with its axis parallel to the horizon; and the other for finding the ullage of a cask standing.</p><p>Again, on one of the narrow sides, noted <hi rend="italics">c,</hi> are, 1st, a line of inches, numbered 1, 2, 3, &c to 12, each subdivided into 10 equal parts. 2dly, A line by which, with that of inches, we find a mean diameter for a cask, in the figure of the middle frustum of a spheroid: it is marked <hi rend="italics">Spheroid,</hi> and numbered 1, 2, 3, &c to 7. 3dly, A line for finding the mean diameter of a cask, in the form of the middle frustum of a parabolic spindle, which gaugers call the second variety of casks; it is therefore marked <hi rend="italics">Second Variety,</hi> and is numbered 1, 2, 3, &c.</p><p>4thly, A line by which is found the mean diameter of a cask of the third variety, consisting of the frustums of two parabolic conoids, abutting on a common base; it is therefore marked <hi rend="italics">Third Variety,</hi> and is numbered 1, 2, 3, &c.</p><p>On the other narrow face, marked <hi rend="italics">f,</hi> are 1st, a line of a foot divided into 100 equal parts, marked FM. 2dly, A line of inches, like that before mentioned, marked IM. 3dly, A line for finding the mean diameter of the fourth variety of casks, which is formed of the frustums of two cones, abutting on a common base. It is numbered 1, 2, 3, &c; and marked EC, for frustum of a cone.</p><p>On the backside of the two sliding pieces is a line of inches, from 12 to 36, for the whole extent of the 3 feet, when the pieces are put endwise, and against that, the correspondent gallons, and 100th parts, that any small tub, or the like open vessel, will contain at 1 inch deep.</p><p>For the various uses of this instrument, see the authors mentioned above, and most other writers on Gauging.</p><p><hi rend="italics">Coggeshall's</hi> <hi rend="smallcaps">Sliding</hi> <hi rend="italics">Rule</hi> is chiefly used in measuring the superficies and solidity of timber, masonry, brickwork, &c.</p><p>This consists of two rulers, each a foot long, which are united together in various ways. Sometimes they are made to slide by one another, like glaziers' rules: sometimes a groove is made in the sideof a common twofoot joint rule, and a thin sliding piece in one side, and Coggeshall's lines added on that side; thus forming the common or Carpenter's rule: and sometimes one of the two rulers is made to slide in a groove made in the side of the other.</p><p>On the Sliding side of the rule are four lines of numbers, three of which are double, that is, are lines to two radii, and the fourth is a single broken line of numbers. The first three, marked A, B, C, are sigured 1, 2, 3, &c to 9; then 1, 2, 3, &c to 10; the construction and use of them being the same as those on Everard's Sliding rule. The single line, called the <hi rend="italics">girt line,</hi> and marked D, whose radius is equal to the two radii of any of the other lines, is broken for the easier measuring of timber, and figured 4, 5, 6, 7, 8, 9, 10, 20, 30, &c. From 4 to 5 it is divided into 10 parts, and each 10th subdivided into 2; and so on from 5 to 10, &c.</p><p>On the backside of the rule are, 1st, a line of inch measure, from 1 to 12; each inch being divided and subdivided. 2dly, A line of foot measure, consisting <pb n="462"/><cb/> of one foot divided into 100 equal parts, and figured 10, 20, 30, &c.</p><p>The backside of the sliding piece is divided into inches, halves, &c, and figured from 12 to 24; so that when the slide is out, there may be a measure of 2 feet.</p><p>In the Carpenter's rule, the inch measure is on one side, continued all the way from 1 to 24, when the rule is unfolded, and subdivided into 8ths or half-quarters: on this side are also some diagonal scales of equal parts. And upon the edge, the whole length of 2 feet is divided into 200 equal parts, or 100ths of a foot.</p></div1><div1 part="N" n="SLING" org="uniform" sample="complete" type="entry"><head>SLING</head><p>, a string instrument, serving for the casting of stones &c with the greater violence.</p><p>Pliny, lib. 76, chap. 5, attributes the invention of the Sling to the Phœnicians; but Vegetius ascribes it to the inhabitants of the Balearic islands, who were celebrated in antiquity for the dextrous management of it. Florus and Strabo say, those people bore three kinds of Slings; some longer, others shorter, which they used according as their enemies were more remote or nearer hand. Diodorus adds, that the first served them for a head-band, the 2d for a girdle, and that the third they constantly carried with them in the hand. But it must be impossible to tell who were the first inventors of the Sling, as the instrument is so simple, and has been in general use by almost all nations. The instrument is much spoken of in the wars and history of the Israelites. David was so expert a slinger, that he ventured to go out, with one in his hand, against the giant and champion Goliath, and at a distance struck him on the forehead with the stone. And there were a number of left-handed men of one of the tribes of Israel, who it is said could Sling a stone at an hair's breadth.</p><p>The motion of a stone discharged from a Sling arises from its centrifugal force, when whirled round in a circle. The velocity with which it is discharged, is the same as that which it had in the circle, and is much greater than what can be given to it by the hand alone. And the direction in which it is discharged, is that of the tangent to the circle at the point of discharge. Whence its motion and effect may be computed as a projectile.</p></div1><div1 part="N" n="SLUSE" org="uniform" sample="complete" type="entry"><head>SLUSE</head><p>, or <hi rend="smallcaps">Slusius</hi> <hi rend="italics">(René Francis Walter</hi>) of Vise, a small town-in the county of Liege, where he enjoyed honours and preferment. He then became abbé of Amas, canon, councellor and chancellor of Liege, and made his name famous for his knowledge in theology, physics, and mathematics. The Royal Society of London elected him one of their members, and inserted several of his compositions in their Transactions. This very ingenious and learned man died at Liege in 1683, at 63 years of age.</p><p>Of Slusius's works there have been published, some learned letters, and a work intitled, <hi rend="italics">Mesolabium et Problemata solida;</hi> beside the following pieces in the Philosophical Transactions, viz,</p><p>1. Short and Easy Method of drawing Tangents to all Geometrical Curves; vol. 7, pa. 5143.</p><p>2. Demonstration of the same; vol. 8, pa. 6059, 6119.</p><p>3. On the Optic Angle of Alhazen; vol. 8, pa. 6139.</p></div1><div1 part="N" n="SMEATON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SMEATON</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, F. R. S. and a very cele- <cb/> brated civil engineer, was born the 28th of May 1724, at Austhorpe, near Leeds, in a house built by his grandfather, where the family have resided ever since, and where our author died the 28th of October 1792, in the 68th year of his age.</p><p>Mr. Smeaton seems to have been born an engineer. The originality of his genius and the strength of his understanding appeared at a very early age. His playthings were not those of children, but the tools men work with; and he had always more amusement in observing artificers work, and asking them questions, than in any thing else. Having watched some mill-wrights at work, he was one day, soon after, seen (to the distress of his family) on the top of his father's barn, fixing up something like a windmill. Another time, attending some men who were fixing a pump at a neighbouring village, and observing them cut off a piece of bored pipe, he contrived to procure it, of which he made a working pump that actually raised water. These anecdotes refer to circumstances that happened when he was hardly out of petticòats, and probably before he had reached the 6th year of his age. About his 14th or 15th year, he had made for himself an engine to turn rose-work; and he made several presents to his friends of boxes in ivory and wood, turned by him in that way.</p><p>His friend and partner in the Deptford Waterworks, Mr. John Holmes, an eminent clock and watch maker in the Strand, says, he visited Mr. Smeaton and spent a month with him at his father's house, in the year 1742, when consequently our author was about 18 years of age. Mr. Holmes could not but view young Smeaton's works with astonishment: he forged his own iron and steel, and melted his own metals; he had tools of every sort, for working in wood, ivory, and metals: he had made a lathe, by which he had cut a perpetual screw in brass, a thing very little known at that day.</p><p>Thus had Mr. Smeaton, by the strength of his genius, and indefatigable industry, acquired, at 18 years of age, an extensive set of tools, and the art of working in most of the mechanical trades, without the assistance of any master, and which he continued to do a part of every day when at the place where his tools were: and few men could work better.</p><p>Mr. Smeaton's father was an attorney, and was desirous of bringing him up to the same profession. He therefore came up to London in 1742, and for some time attended the courts in Westminster Hall. But finding that the profession of the law did not suit <hi rend="italics">the bent of his genius,</hi> as his usual expression was, he wrote a strong memorial to his father on the subject, whose good sense from that moment left Mr. Smeaton to pursue the bent of his genius in his own way.</p><p>Mr. Smeaton after this continued to reside in London, and about 1750 he commenced philosophical instrument maker, which he continued for some time, and became acquainted with most of the ingenious men of that time; and this same year he made his first communication to the Royal Society, being an account of Dr. Knight's improvements of the mariner's compass. Continuing his very useful labours, and making experiments, he communicated to that learned body, the two following years, a number of other ingenious improve- <pb n="463"/><cb/> ments, as will be enumerated in the list of his writings, at the end of this account of him.</p><p>In 1751 he began a course of experiments, to try a machine of his invention, for measuring a ship's way at fea; and also made two voyages in company with Dr. Knight to try it, as well as a compass of his own invention.</p><p>In 1753 he was elected a member of the Royal Society; and in 1759 he was honoured with their gold medal, for his paper concerning the natural powers of water and wind to turn mills, and other machines depending on a circular motion. This paper, he says, was the result of experiments made on working models in the years 1752 and 1753, but not communicated to the Society till 1759, having in the interval found opportunities of putting the result of these experiments into real practice, in a variety of cases, and for various purposes, so as to assure the Society he had found them to answer.</p><p>In 1754 his great thirst after experimental knowledge led him to undertake a voyage to Holland and the Low Countries, where he made himself acquainted with most of the curious works of art so frequent in those places.</p><p>In December 1755, the Edystone lighthouse was burnt down, and the proprietors, being desirous of rebuilding it in the most substantial manner, enquired of the earl of Macclesfield, then president of the Royal Society, who he thought might be the fittest person to rebuild it, when he immediately recommended our author. Mr. Smeaton accordingly undertook the work, which he completed with stone in the summer of 1759. Of this work he gives an ample description in a folio volume, with plates, published in 1791. A work which contains, in a great measure, the history of four years of his life, in which the originality of his genius is fully displayed, as well as his activity, industry, and perseverance.</p><p>Though Mr. Smeaton completed the building of the Edystone lighthouse in 1759, yet it seems he did not soon get into full business as a civil engineer; for in 1764, while in Yorkshire, he offered himself a candidate for one of the receivers of the Derwentwater estate; in which he succeeded, though two other persons, strongly recommended and powerfully supported, were candidates for the employment. In this appointment he was very happy, by the assistance and abilities of his partner Mr. Walton the younger, of Farnacres near Newcastle, one of the present receivers, who, taking upon himself the management and the accounts, left Mr. Smeaton leisure and opportunity to exert his abilities on public works, as well as to make many improvements in the mills, and in the estates of Greenwich hospital.</p><p>By the year 1775, he had so much business, as a civil engineer, that he was desirous of resigning the appointment for that hospital, and would have done it then, had not his friends prevailed upon him to continue in the office about two years longer.</p><p>Mr. Smeaton having thus got into full business as a civil engineer, it would be an endless task to enumerate all the variety of concerns he was engaged in. A very few of them however may be just mentioned in this place. —He made the river Calder navigable: a work that required great skill and judgment; owing to the very <cb/> impetuous floods in that river—He planned and attended the execution of the great canal in Scotland, for conveying the trade of the country, either to the Atlantic or German ocean; and having brought it to a conclusion, he declined a handsome yearly salary, that he might not be prevented from attending to the multiplicity of his other business.</p><p>On opening the great arch at London bridge, the excavation around and under the sterlings was so considerable, that it was thought the bridge was in greatdanger of falling; the apprehensions of the people on this head being so great, that few would pass over or under it. He was then in Yorkshire, where he was sent for by express, and he arrived in town with the greatest expedition. He applied himself immediately to examine it, and to sound about the sterlings as minutely as he could. The committee being called together, adopted his advice, which was, to repurchase the stones that had been taken from the middle pier, then lying in Moorfields, and to throw them into the river to guard the sterlings, a practice he had before adopted on other occasions. Nothing shews the apprehensions of the bridge falling, more than the alacrity with which his advice was pursued: the stones were repurchased that day; horses, carts, and barges were got ready, and the work instantly begun though it was Sunday morning. Thus Mr. Smeaton, in all human probability, saved London bridge from falling, and secured it till more effectual methods could be taken.</p><p>In 1771, he became, jointly with his friend Mr. Holmes above mentioned, proprietor of the works for supplying Deptford and Greenwich with water; which by their united endeavours they brought to be of general use to those they were made for, and moderately beneficial to themselves.</p><p>About the year 1785, Mr. Smeaton's health began to decline; in consequence he then took the resolution to endeavour to avoid any new undertakings in business as much as he could, that he might thereby also have the more leisure to publish some account of his inventions and works. Of this plan however he got no more executed than the account of the Edystone lighthouse, and some preparations for his intended treatise on mills; for he could not resist the solicitations of his friends in various works; and Mr. Aubert, whom he greatly loved and respected, being chosen chairman of Ramsgate harbour, prevailed upon him to accept the office of engineer to that harbour; and to their joint efforts the public are chiefly indebted for the improvements that have been made there within these few years; which fully appears in a report that Mr. Smeaton gave in to the board of trustees in 1791, which they immediately published.</p><p>It had for many years been the practice of Mr. Smeaton to spend part of the year in town, and the remainder in the country, at his house at Austhorpe; on one of these excursions in the country, while walking in his garden, on the 16th of September 1792, he was struck with the palsy, which put an end to his useful life the 28th of October following, to the great regret of a numerous set of friends and acquaintances.</p><p>The great variety of mills constructed by Mr. Smeaton, so much to the satisfaction and advantage of the owners, will shew the great use he made of his experi- <pb n="464"/><cb/> ments in 1752 and 1753. Indeed he scarcely trusted to theory in any case where he could have an opportunity to investigate it by experiment; and for this purpose he built a steam-engine at Austhorpe, that he might make experiments expressly to ascertain the power of Newcomen's steam-engine, which he improved and brought to a much greater degree of certainty, both in its construction and powers, than it was before.</p><p>During many years of his life, Mr Smeaton was a constant attendant on parliament, his opinion being continually called for. And here his natural strength of judgment and perspicuity of expression had their full display. It was his constant practice, when applied to, to plan or support any measure, to make himself fully acquainted with it, and be convinced of its merits, before he would be concerned in it. By this caution, joined to the clearness of his description, and the integrity of his heart, he seldom failed having the bill he supported carried into an act of parliament. No person was heard with more attention, nor had any one ever more confidence placed in his testimony. In the courts of law he had several compliments paid to him from the bench, by the late lord Mansfield and others, on account of the new light he threw upon difficult subjects.</p><p>As a civil engineer, he was perhaps unrivalled, certainly not excelled by any one, either of the present or former times. His building the Edystone lighthouse, were there no other monument of his fame, would establish his character. The Edystone rocks have obtained their name from the great variety of contrary <hi rend="italics">sets</hi> of the tide or current in their vicinity. They are situated nearly S. S. W. from the middle of Plymouth Sound. Their distance from the port of Plymouth is about 14 miles. They are almost in the line which joins the Start and the Lizard points; and as they lie nearly in the direction of vessels coasting up and down the channel, they were unavoidably, before the establishment of a light-house there, very dangerous, and often fatal to ships. Their situation with regard to the Bay of Biscay and the Atlantic is such, that they lie open to the swells of the bay and ocean, from all the south-western points of the compass; so that all the heavy seas from the south-west quarter come uncontroled upon the Edystone rocks, and break upon them with the utmost fury. Sometimes, when the sea is to all appearance smooth and even, and its surface unruffled by the slightest breeze, the <hi rend="italics">ground swell</hi> meeting the slope of the rocks, the sea beats upon them in a frightful manner, so as not only to obstruct any work being done on the rock, or even landing upon it, when, figuratively speaking, you might go to sea in a walnut-shell. That circumstances fraught with danger surrounding it should lead mariners to wish for a light-house, is not wonderful; but the danger attending the erection leads us to wonder that any one could be found hardy enough to undertake it. Such a man was first found in the person of Mr. H. Winstanley, who, in the year 1696, was furnished by the Trinity-house with the necessary powers. In 1700 it was finished; but in the great storm of November 1703, it was destroyed, and the projector perished in the ruins. In 1709 another, upon a different construction, was erected by a Mr. Rudyerd, which, in 1755, was unfortunately consumed by sire. The next <cb/> building was under the direction of Mr. Smeaton, who, having considered the errors of the former constructions, has judiciously guarded against them, and erected a building, the demolition of which seems little to be dreaded, unless the rock on which it is erected should perish with it.—Of his works, in constructing bridges, harbours, mills, engines, &c, &c, it were endless to speak. Of his inventions and improvements of philosophical instruments, as of the air-pump, the pyrometer, hygrometer, &c, &c, some idea may be formed from the list of his writings inserted below.</p><p>In his person, Mr. Smeaton was of a middle stature, but broad and strong made, and possessed of an excellent constitution. He had a great simplicity and plainness in his manners: he had a warmth of expression that might appear, to those who did not know him well, to border on harshness; but such as were more closely acquainted with him, knew it arose from the intense application of his mind, which was always in the pursuit of truth, or engaged in the investigation of difficult subjects. He would sometimes break out hastily, when any thing was said that was contrary to his ideas of the subject; and he would not give up any thing he argued for, till his mind was convinced by sound reasoning.</p><p>In all the social duties of life, Mr. Smeaton was exemplary; he was a most affectionate husband, a good father, a warm, zealous and sincere friend, always ready to assist those he respected, and often before it was pointed out to him in what way he could serve them. He was a lover and an encourager of merit wherever he found it; and many persons now living are in a great measure indebted for their present situation to his assistance and advice. As a companion, he was always entertaining and instructive, and none could spend their time in his company without improvement.</p><p>As to the list of his writings; beside the large work abovementioned, being the History of Edystone Lighthouse, and numbers of reports and memorials, many of which were printed, his communications to the Royal Society, and inserted in their Transactions, are as follow:</p><p>1. An Account of Dr. Knight's Improvements of the Mariner's Compass; an. 1750, pa. 513.</p><p>2. Some improvements in the Air-pump; an. 1752, pa. 413.</p><p>3. An Engine for raising Water by Fire; being an improvement on Savary's construction, to render it capable of working itself: invented by M. de Moura, of Portugal. Ib. pa. 436.</p><p>4. Description of a new Tackle, or Combination of Pulleys. Ib. 494.</p><p>5. Experiments upon a machine for measuring the Way of a Ship at Sea. An. 1754, pa. 532.</p><p>6. Description of a new Pyrometer. Ib. pa. 598.</p><p>7. Effects of Lightning on the Steeple and Church of Lestwithial in Cornwall. An. 1757, pa. 198.</p><p>8. Remarks on the different Temperature of the Air at Edystone Light-house, and at Plymouth. An. 1758, pa. 488.</p><p>9. Experimental enquiry concerning the natural powers of Water and Wind to turn mills and other machines depending on a circular motion. An. 1759, pa. 100. <pb n="465"/><cb/></p><p>10. On the Menstrual Parallax arising from the mutual gravitation of the earth and moon, its influence on the observation of the sun and planets, with a method of observing it. An. 1768, pa. 156.</p><p>11. Description of a new method of Observing the heavenly bodies out of the meridian. An. 1768, pa. 170.</p><p>12. Observations on a Solar Eclipse. An. 1769, pa. 286.</p><p>13. Description of a new Hygrometer. An. 1771, pa. 198.</p><p>14. An Experimental Examination of the quantity and proportion of Mechanical Power, necessary to be employed in giving different degrees of velocity to heavy bodies from a state of rest. An. 1776, pa. 450.</p></div1><div1 part="N" n="SMOKE" org="uniform" sample="complete" type="entry"><head>SMOKE</head><p>, or <hi rend="italics">Smoak,</hi> a humid matter exhaled in form of vapour by the action of heat, either external or internal; or Smoke consists of palpable particles, elevated by means of the rarefying heat, or by the force of the ascending current of air, from certain bodies exposed to heat; which particles vary much in their properties, according to the substances from which they are produced.</p><p>Sir Isaac Newton observes, that Smoke ascends in the chimney by the impulse of the air it floats in: for that air, being raresied by the heat of the fire underneath, has its specisic gravity diminished; and thus, being disposed to ascend itself, it carries up the Smoke along with it. The tail of a comet, the same author supposes, ascends from the nucleus after the same manner.</p><p>Smoke of fat unctuous woods, as fir, beech, &c, makes what is called lamp-black.</p><p>There are various inventions for preventing and curing smoky chimneys: as the æolipiles of Vitruvius, the ventiducts of Cardan, the windmills of Bernard, the capitals of Serlio, the little drums of Paduanus, and several artifices of De Lorme. See also the philosophical works of Dr. Franklin. Pans, resembling sugar pans, placed over the tops of chimneys, are useful to make them draw better; and the fire-grates called register-stoves, are always a sure remedy.</p><p>In the Philosophical Transactions is the description of an engine, invented by M. Dalesme, which consumes the Smoke of all sorts of wood so effectually, that the eye cannot discover it in the room, nor the nose distinguish the smell of it, though the fire be made in the middle of the room. It consists of several iron hoops, 4 or 5 inches in diameter, which shut into one another, and is placed on a trever.</p><p>The late invention called Argand's lamp, also consumes the Smoke, and gives a very strong light. Its principle is a thin broad cotton wick, rolled into the form of a hollow cylinder; the air passes up the hollow of it, and the Smoke is almost all consumed.</p><p><hi rend="smallcaps">Smoke</hi> <hi rend="italics">Jack,</hi> is a jack for turning a spit, turned by the Smoke of the kitchen fire, by means of thin iron sails set obliquely on an axis in the flue os the chimney. See <hi rend="smallcaps">Jack.</hi></p></div1><div1 part="N" n="SNELL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">SNELL</surname> (<foreName full="yes"><hi rend="smallcaps">Rodolph</hi></foreName>)</persName></head><p>, a respectable Dutch philosopher, was born at Oudenwater in 1546. He was some time professor of Hebrew and mathematics at Leyden, where he died in 1613, at 67 years of age. He was author of several works on geometry, and on all parts of <cb/> the philosophy of his time; but I have not obtained a particular list of them.</p><p><hi rend="smallcaps">Snell</hi> <hi rend="italics">(Willebrord),</hi> son of Rodolph above mentioned, an excellent mathematician, was born at Leyden in 1591, where he succeeded his father in the mathematical chair in 1613, and where he died in 1626, at only 35 years of age.</p><p>Willebrord Snell was author of several ingenious works and discoveries. Thus, it was he who first discovered the true law of the refraction of the rays of light; a discovery which he made before it was announced by Des Cartes, as Huygens assures us. Though the work which Snell prepared upon this subject, and upon optics in general, was never published, yet the discovery was very well known to belong to him, by several authors about his time, who had seen it in his manuscripts.—He undertook also to measure the earth. This he effected by measuring a space between Alcmaer and Bergen-op-zoom, the difference of latitude between these places being 1° 11′ 30″. He also measured another distance between the parallels of Alcmaer and Leyden; and from the mean of both these measurements, he made a degree to consist of 55021 French toises or fathoms. These measures were afterwards repeated and corrected by Musschenbroek, who found the degree to contain 57033 toises.—He was author of a great many learned mathematical works, the principal of which are,</p><p>1. <hi rend="italics">Apollonius Batavus;</hi> being the restoration of some lost pieces of Apollonius, concerning Determinate Section, with the Section of a Ratio and Space: in 4to, 1608, published in his 17th year.</p><p>2. <hi rend="italics">Eratosthenes Batavus;</hi> in 4to, 1617. Being the work in which he gives an account of his operations in measuring the earth.</p><p>3. A translation out of the Dutch language, into Latin, of Ludolph van Collen's book <hi rend="italics">De Circulo & Adscriptis,</hi> &c; in 4to, 1619.</p><p>4. <hi rend="italics">Cyclometricus, De Circuli Dimensione</hi> &c<hi rend="italics">;</hi> 4to, 1621. In this work, the author gives several ingenious approximations to the measure of the circle, both arithmetical and geometrical.</p><p>5. <hi rend="italics">Tiphis Batavus;</hi> being a treatise on Navigation and Naval Affairs; in 4to, 1624.</p><p>6. A posthumous treatise, being four books <hi rend="italics">Doctrinæ Triangulorum Canonicæ;</hi> in 8vo, 1627. In which are contained the canon of secants; and in which the construction of sines, tangents, and secants, with the dimension or calculation of triangles, both plane and spherical, are briefly and clearly treated.</p><p>7. Hessian and Bohemian Observations; with his own notes.</p><p>8. <hi rend="italics">Libra Astronomica & Philosophica;</hi> in which he undertakes the examination of the principles of Galileo concerning comets.</p><p>9. Concerning the Comet which appeared in 1618, &c.</p></div1><div1 part="N" n="SNOW" org="uniform" sample="complete" type="entry"><head>SNOW</head><p>, a well known meteor, formed by the freezing of the vapours in the atmosphere. It differs from hail and hoar-frost in being as it were crystallized, which they are not. This appears on examination of a flake of Snow by a magnifying glass; when the whole of it appears to be composed of fine shining spicula diverging like rays from a centre. As the flakes descend <pb n="466"/><cb/> through the atmosphere, they are continually joined by more of these radiated spicula, and thus increase in bulk like the drops of rain or hailstones; so that it seems as if the whole body of Snow were an infinite mass of icicles irregularly figured.</p><p>The lightness os Snow, although it is firm ice, is owing to the excess of its surface, in comparison to the matter contained under it; as even gold itself may be extended in surface, till it will float upon the least breath of air.</p><p>According to Beccaria, clouds of Snow differ in nothing from clouds of rain, but in the circumstance of cold that freezes them. Both the regular diffusion of the Snow, and the regularity of the structure of its parts, shew that clouds of Snow are acted upon by some uniform cause like electricity; and he endeavours to shew how electricity is capable of forming these figures. He was confirmed in his conjectures by observing, that his apparatus for shewing the electricity of the atmosphere, never failed to be electrified by Snow as well as by rain. Professor Wintrop sometimes found his apparatus electrified by Snow when driven about by the wind, though it had not been affected by it when the Snow itself was falling. A more intense electricity, according to Beccaria, unites the particles of hail more closely than the more moderate electricity does those of Snow, in the same manner as we see that the drops of rain which fall from the thunder-clouds, are larger than those which fall from others, though the former descend through a less space.</p><p>In the northern countries, the ground is covered with snow for several months; which proves exceedingly favourable for vegetation, by preserving the plants from those intense frosts which are common in such countries, and which would certainly destroy them. Bartholin ascribes great virtues to Snow-water, but experience does not seem to warrant his assertions. Snowwater, or ice-water, is always deprived of its fixed air: and those nations who live among the Alps, and use it for their constant drink, are subject to affections of the throat, which it is thought are occasioned by it.</p><p>From some late experiments on the quantity of water yielded by Snow, it appears that the latter gives only about one-tenth of its bulk in water.</p></div1><div1 part="N" n="SOCIETY" org="uniform" sample="complete" type="entry"><head>SOCIETY</head><p>, an assemblage or union of several learned persons, for their mutual assistance, improvement, or information, and for the promotion of philosophical or other knowledge. There are various philosophical Societies instituted in different parts of the world. See <hi rend="smallcaps">Royal</hi> <hi rend="italics">Society.</hi></p><p><hi rend="italics">American Philosophical</hi> <hi rend="smallcaps">Society</hi>, was established at Philadelphia in the year 1769, for promoting useful knowledge, under the direction of a patron, a president, three vice-presidents, a treasurer, four secretaries, and three curators. The first volume of their Transactions comprehends a period of two years, viz, from Jan. 1, 1769, to Jan. 1, 1771. Their labours seem to have been interrupted during the troubles in America, which commenced soon after; but since their termination, some more volumes have been published, containing a number of very ingenious and useful memoirs.</p><p><hi rend="italics">American Academy of Arts and Sciences,</hi> was established by a law of the Commonwealth of Massachusetts in North America, in the year 1780. <cb/></p><p><hi rend="italics">Boston Academy of Arts and Sciences.</hi> This is a Society similar to the former, which has lately been established at Boston in New England, under the title of the Academy of Arts and Sciences &c.</p><p><hi rend="italics">Berlin</hi> <hi rend="smallcaps">Society.</hi> The Society of Natural Historians at Berlin, was founded by Dr. Martini. There is also a Philosophical Society in the same place.</p><p><hi rend="italics">Brussels</hi> <hi rend="smallcaps">Society.</hi> The Imperial and Royal Academy of Sciences and Belles Lettres of Brussels was sounded in 1773. Several volumes of their Transactions have now been published.</p><p><hi rend="italics">Dublin</hi> <hi rend="smallcaps">Society.</hi> This is an Experimental Society, for promoting natural knowledge, which was instituted in 1777: the members meet once a week, and distribute three honorary gold medals annually for the most approved discovery, invention, or essay, on any mathematical or philosophical subject. The Society is under the direction of a president, two vice-presidents, and a secretary.</p><p><hi rend="italics">Edinburgh Philosophical</hi> <hi rend="smallcaps">Society</hi>, succeeded the Medical Society, and was formed upon the plan of including all the different branches of natural knowledge and the antiquities of Scotland. The meetings of this Society, interrupted in 1745, were revived in 1752; and in 1754 the first volume of their collection was published, under the title of Essays or Observations Physical and Literary, which has been succeeded by other volumes. This Society has been lately incorporated by royal charter, under the name of the Royal Society of Scotland, instituted for the advancement of learning and useful knowledge. The members are divided into two classes, physical and literary; and those who are near enough to Edinburgh to attend the meetings, pay a guinea on admission, and the same sum annually. The first meeting was held on the first Monday of August 1783; when there were chosen, a president, two vice-presidents, a secretary, treasurer, and a council of 12 persons. Three of the volumes of their Transactions have been published, which are very respectable both for their magnitude and contents.</p><p>In <hi rend="italics">France</hi> there have been several institutions of this kind for the improvement of science, besides those recounted under the word <hi rend="smallcaps">Academy:</hi> As, the Royal Academy at Soissons, founded in 1674; at Villefranche, Beaujolois, in 1679; at Nismes, in 1682; at Angers, in 1685; the Royal Society at Montpelier, in 1706, which is so intimately connected with the Royal Academy of Sciences of Paris, as to form with it, in some respects, one body; the literary productions of this Society are published in the memoirs of the academy: the Royal Academy of Sciences and Belles Lettres at Lyons, in 1700; at Bourdeaux, in 1703; at Marseilles, in 1726; at Rochelle, in 1734; at Dijon, in 1740; at Pau in Bern, in 1721; at Beziers, in 1723; at Montauban, in 1744; at Rouen, in 1744; at Amiens, in 1750; at Toulouse, in 1750; at Besançon, in 1752; at Metz, in 1760; at Arras, in 1773; and at Chalons sur Maine, in 1775. For other institutions of a similar nature, and their literary productions, see the articles <hi rend="smallcaps">Academy, Journal</hi>, and <hi rend="smallcaps">Transactions.</hi></p><p><hi rend="italics">Manchester Literary and Philosophical</hi> <hi rend="smallcaps">Society</hi>, is of considerable reputation, and has been lately established there, under the direction of two presidents, four vice-presidents, and two secretaries. The number <pb n="467"/><cb/> of members is limited to 50; besides these there are several honorary members, all of whom are elected by ballot; and the officers are chosen annually in April. Several valuable essays have been already read at the meetings of this Society.</p><p><hi rend="italics">Newcastle-upon-Tyne Literary and Philosophical</hi> S<hi rend="smallcaps">OCIETY.</hi> This Society was instituted the 7th of February 1793, under the direction of a president, four vice-presidents, two secretaries, a treasurer, which together with four of the ordinary members form a committee, all annually elected at a general meeting. The subjects proposed for the consideration and improvement of this Society, comprehend the mathematics, natural philosophy and history, chemistry, polite literature, antiquities, civil history, biography, questions of general law and policy, commerce, and the arts. From such ample scope in the objects of the Society, with the known respectability, zeal, and talents of the members, the greatest improvements and discoveries may be expected to be made in those important branches of useful knowledge.</p></div1><div1 part="N" n="SOCRATES" org="uniform" sample="complete" type="entry"><head>SOCRATES</head><p>, the chief of the ancient philosophers, was born at Alopece, a small village of Attica, in the 4th year of the 77th olympiad, or about 467 years before Christ. Sophroniscus, his father, being a statuary or carver of images in stone, our author followed the same profession for some time, for a subsistence. But being naturally averse to this profession, he only followed it when necessity compelled him; and upon getting a little before-hand, would for a while lay it aside. These intermissions of his trade were bestowed upon philosophy, to which he was naturally addicted; and this being observed by Crito, a rich philosopher of Athens, Socrates was at length taken from his shop, and put into a condition of philosophising at his ease and leisure.</p><p>He had various instructors in the sciences, as Anaxagoras, Archylaus, Damon, Prodicus, to whom may be added the two learned women Diotyma and Aspasia, of the last of whom he learned rhetoric: of Euenus he learned poetry; of Ichomachus, husbandry; and of Theodorus, geometry.</p><p>At length he began himself to teach; and was so eloquent, that he could lead the mind to approve or disapprove whatever he pleased; but never used this talent for any other purpose than to conduct his fellow citizens into the path of virtue. The academy of the Lycæum, and a pleasant meadow without the city on the side of the river Ilyssus, were places where he chiefly delivered his instructions, though it seems he was never out of his way in that respect, as he made use of all times and places for that purpose.</p><p>He is represented by Xenophon as excellent in all kinds of learning, and particularly instances arithmetic, geometry, and astrology or astronomy: Plato mentions natural philosophy; Idomeneus, rhetoric; Laertius, medicine. Cicero affirms, that by the testimony of all the learned, and the judgment of all Greece, he was, as well in wisdom, acuteness, politeness, and subtlety, as in eloquence, variety, and richness, in whatever he applied himself to, without exception, the prince of all.</p><p>It has been observed by many, that Socrates little <cb/> affected travel; his life being wholly spent at home, excepting when he went out upon military services. In the Peloponnesian war he was thrice personally engaged: upon which occasions it is said he outwent all the soldiers in hardiness: and if at any time, saith Alcibiades, as it often happens in war, the provisions failed, there were none who could bear the want of meat and drink like Socrates; yet, on the other hand, in times of feasting, he alone seemed to enjoy them; and though of himself he would not drink, yet being invited, he far outdrank every one, though he was never seen intoxicated.</p><p>To this great philosopher Greece was principally indebted for her glory and splendor. He formed the manners of the most celebrated persons of Greece, as Alcibiades, Xenophon, Plato, &c. But his great services and the excellent qualities of his mind could not secure him from envy, persecution, and calumny. The thirty tyrants forbad his instructing youth; and as he derided the plurality of the Pagan deities, he was accused of impiety. The day of trial being come, Socrates made his own defence, without procuring an advocate, as the custom was, to plead for him. He did not defend himself with the tone and language of a suppliant or guilty person, but, as if he were master of the judges themselves, with freedom, firmness, and some degree of contumacy. Many of his friends also spoke in his behalf; and lastly, Plato went up into the chair, and began a speech in these words: “Though I, Athenians, am the youngest of those that come up into this place”—but they stopped him, crying out, “of those that go down,” which he was thereupon constrained to do; and then proceeding to vote, they condemned Socrates to death, which was effected by means of poison, when he was 70 years of age. Plato gives an affecting account of his imprisonment and death, and concludes, “This was the end of the best, the wisest, and the justest of men.” And that account of it by Plato, Tully professes, he could never read without tears.</p><p>As to the person of Socrates, he is represented as very homely; he was bald, had a dark complexion, a flat nose, eyes sticking out, and a severe downcast look. But the defects of his person were amply compensated by the virtues and accomplishments of his mind. Socrates was indeed a man of all virtues; and so remarkably frugal, that how little soever he had, it was always enough. When he was amidst a great variety of rich and expensive objects, he would often say to himself, “How many things are there which I do not want!”</p><p>Socrates had two wives, one of which was the noted Xantippe; whom Aulus Gellius describes as an accursed froward woman, always chiding and scolding, by day and by night, and whom it was said he made choice of as a trial and exercise of his temper. Several instances are recorded of her impatience and his forbearance. One day, before some of his friends, she fell into the usual extravagances of her passion; when he, without answering a word, went abroad with them: but on his going out of the door, she ran up into the chamber, and threw down water upon his head; upon which, turning to his friends, “Did not I tell you <pb n="468"/><cb/> (says he), that after so much thunder we should have rain?” Another time she pulled his cloak from his shoulders in the open forum; and some of his friends advising him to beat her, “Yes (says he), that while we two fight, you may all stand by, and cry, Well done, Socrates; to him, Xantippe.”</p><p>They who affirm that Socrates wrote nothing, mean only in respect to his philosophy; for it is attested and allowed, that he assisted Euripides in composing tragedies, and was the author of some pieces of poetry. Dialogues also and epistles are ascribed to him: but his philosophical disputations were committed to writing only by his scholars; and that chiefly by Plato and Xenophon. The latter set the example to the rest in doing it first, and also with the greatest punctuality; as Plato did it with the most liberty, intermixing so much of his own, that it is hardly possible to know what part belongs to each. Hence Socrates, hearing him recite his Lysis, cried out, “How many things doth this young man feign of me!” Accordingly, the greatest part of his philosophy is to be found in the writings of Plato. To Socrates is ascribed the first introduction of moral philosophy. Man having a twofold relation to things divine and human, his doctrines were with regard to the former metaphysical, to the latter moral. His metaphysical opinions were chiefly, that, There are three principles of all things, God, matter, and idea. God is the universal intellect; matter the subject of generation and corruption; idea, an incorporeal substance, the intellect of God; God the intellect of the world. God is one, perfect in himself, giving the being and well-being of every creature.—That God, not chance, made the world and all creatures, is demonstrable from the reasonable disposition of their parts, as well for use as defence; from their care to preserve themselves, and continue their species.—That he particularly regards man in his body, appears from his noble upright form, and from the gift of speech; in his soul, from the excellency of it above others.—That God takes care of all creatures, is demonstrable from the benefit he gives them of light, water, fire, and fruits of the earth in due season. That he hath a particular regard of man, from the destination of all plants and creatures for his service; from their subjection to man, though they may exceed him ever so much in strength; from the variety of man's sense, accommodated to the variety of objects, for necessity, use, and pleasure; from reason, by which he discourseth through reminiscence from sensible objects; from speech, by which he communicates all he knows, gives laws, and governs states. Finally, that God, though invisible himself, at once sees all, hears all, is every where, and orders all.</p><p>As to the other great object of metaphysical research, the soul, Socrates taught, that it is pre-existent to the body, endued with the knowledge of eternal ideas, which in its union to the body it loseth, as stupefied, until awakened by discourse from sensible objects; on which account, all its learning is only reminiscence, a recovery of its first knowledge. That the body, being compounded, is dissolved by death; but that the soul, being simple, passeth into another life, incapable of corruption. That the souls of men are divine. That the souls of the good after death are in a happy state, united to God in a blessed inaccessible <cb/> place; that the bad in convenient places suffer condign punishment.</p><p>All the Grecian sects of philosophers refer their origin to the discipline of Socrates; particularly the Platonics, Peripatetics, Academics, Cyrenaics, Stoics, &c.</p></div1><div1 part="N" n="SOL" org="uniform" sample="complete" type="entry"><head>SOL</head><p>, in Astrology, &c, signifies the sun.</p></div1><div1 part="N" n="SOLAR" org="uniform" sample="complete" type="entry"><head>SOLAR</head><p>, something relating to the sun. Thus, we say Solar fire in contradistinction to culinary sire.</p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">Civil Month.</hi> See <hi rend="smallcaps">Month.</hi></p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">Cycle.</hi> See <hi rend="smallcaps">Cycle.</hi></p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">Comet.</hi> See <hi rend="smallcaps">Discus.</hi></p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">Eclipse,</hi> is a privation of the light of the sun, by the interposition of the opake body of the moon. See <hi rend="smallcaps">Eclipse.</hi></p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">Month, Rising, Spots.</hi> See the substantives.</p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">System,</hi> the order and disposition of the several heavenly bodies, which revolve round the sun as the centre of their motion; viz, the planets, primary and secondary, and the comets. See <hi rend="smallcaps">System.</hi></p><p><hi rend="smallcaps">Solar</hi> <hi rend="italics">Year.</hi> See <hi rend="smallcaps">Year.</hi></p></div1><div1 part="N" n="SOLID" org="uniform" sample="complete" type="entry"><head>SOLID</head><p>, in Physics, a body whose minute parts are connected together, so as not to give way, or slip from each other, on the smallest impression. The word is used in this sense, in contradistinction to fluid.</p><div2 part="N" n="Solid" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Solid</hi></head><p>, in Geometry, is a magnitude extended in every possible direction, quite around. Though it is commonly said to be endued with three dimensions only, length, breadth, and depth or thickness.</p><p>Hence, as all bodies have these three dimensions, and nothing but bodies, Solid and body are often used indiscriminately.</p><p>The extremes of Solids are surfaces. That is, Solids are terminated either by one surface, as a globe, or by several, either plane or curved. And from the circumstances of these, Solids are distinguished into regular and irregular.</p><p><hi rend="italics">Regular</hi> <hi rend="smallcaps">Solids</hi>, are those that are terminated by regular and equal planes. These are the tetraedron, hexaedron, or cube, octaedron, dodecaedron, and icosaedron; nor can there possibly be more than these five regular Solids or bodies, unless perhaps the sphere or globe be considered as one of an infinite number of sides. See these articles severally, also the article <hi rend="italics">Regular</hi> <hi rend="smallcaps">Body.</hi></p><p><hi rend="italics">Irregular</hi> <hi rend="smallcaps">Solids</hi>, are all such as do not come under the definition of regular ones: such as cylinder, cone, prism, pyramid, &c.</p><p>Similar Solids are to one another in the triplicate ratio of their like sides, or as the cubes of the same. And all sorts of prisms, as also pyramids, are to one another in the compound ratio of their bases and altitudes.</p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Angle,</hi> is that formed by three or more plane angles meeting in a point; like an angle of a die, or the point of a diamond well cut.</p><p>The sum of all the plane angles forming a Solid angle, is always less than 360°; otherwise they would constitute the plane of a circle, and not a Solid.</p><p><hi rend="italics">Atmosphere of</hi> <hi rend="smallcaps">Solids.</hi> See <hi rend="smallcaps">Atmosphere.</hi></p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Bastion.</hi> See <hi rend="smallcaps">Bastion.</hi></p><p><hi rend="italics">Cubature of</hi> <hi rend="smallcaps">Solids.</hi> See <hi rend="smallcaps">Cubature</hi> and S<hi rend="smallcaps">OLIDITY.</hi> <pb n="469"/><cb/></p><p><hi rend="italics">Measure of a</hi> <hi rend="smallcaps">Solid.</hi> See <hi rend="smallcaps">Measure.</hi></p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Foot.</hi> See <hi rend="smallcaps">Foot.</hi></p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Numbers,</hi> are those which arise from the multiplication of a plane number, by any other number whatever. Thus, 18 is a Solid number, produced from the plane number 6 and 3, or from 9 and 2.</p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Place.</hi> See <hi rend="smallcaps">Locus.</hi></p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Problem,</hi> is one which cannot be constructed geometrically; but by the intersection of a circle and a conic section, or by the intersection of two conic sections. Thus, to describe an isosceles triangle on a given base, so that either angle at the base shall be triple of that at the vertex, is a Solid problem, resolved by the intersection of a parabola and circle, and it serves to inscribe a regular heptagon in a given circle.</p><p>In like manner, to describe an isosceles triangle having its angles at the base each equal to 4 times that at the vertex, is a Solid problem, effected by the intersection of an hyperbola and a parabola, and serves to inscribe a regular nonagon in a given circle.</p><p>And such a problem as this has four solutions, and no more; because two conic sections can intersect but in 4 points.</p><p>How all such problems are constructed, is shewn by Dr. Halley, in the Philos. Trans. num. 188.</p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">of Least Resistance.</hi> See <hi rend="smallcaps">Resistance.</hi></p><p><hi rend="italics">Surfaces of</hi> <hi rend="smallcaps">Solids.</hi> See <hi rend="smallcaps">Area</hi> and S<hi rend="smallcaps">UPERFICIES.</hi></p><p><hi rend="smallcaps">Solid</hi> <hi rend="italics">Theorem.</hi> See <hi rend="smallcaps">Theorem.</hi></p></div2></div1><div1 part="N" n="SOLIDITY" org="uniform" sample="complete" type="entry"><head>SOLIDITY</head><p>, in Physics, a property of matter or body, by which it excludes every other body from that place which is possessed by itself.</p><p>Solidity in this sense is a property common to all bodies, whether solid or fluid. It is usually called <hi rend="italics">impenetrability;</hi> but Solidity expresses it better, as carrying with it somewhat more of positive than the other, which is a negative idea.</p><p>The idea of Solidity, Mr. Locke observes, arises from the resistance we find one body makes to the entrance of another into its own place. Solidity, he adds, seems the most extensive property of body, as being that by which we conceive it to fill space; it is distinguished from mere space, by this latter not being capable of resistance or motion.</p><p>It is distinguished from hardness, which is only a firm cohesion of the solid parts.</p><p>The difficulty of changing situation gives no more Solidity to the hardest body than to the softest; nor is the hardest diamond properly a jot more solid than water. By this we distinguish the idea of the extension of body, from that of the extension of space: that of body is the continuity or cohesion of solid, separable, moveable parts; that of space the continuity os unsolid, inseparable, immoveable parts.</p><p>The Cartesians however will, by all means, deduce Solidity, or as they call it impenetrability, from the nature of extension; they contend, that the idea of the former is contained in that of the latter; and hence they argue against a vacuum. Thus, say they, one cubic foot of extension cannot be added to another without having two cubic feet of extension; for each has in itself all that is required to constitute that magnitude. And hence they conclude, that every part of space is solid, or impenetrable, as of its own nature it <cb/> excludes all others. But the conclusion is false, and the instance they give follows from this, that the parts of space are immoveable, not from their being impenetrable or solid. See <hi rend="smallcaps">Matter.</hi></p><p><hi rend="smallcaps">Solidity</hi> is also used for hardness, or firmness; as opposed to fluidity; viz, when body is considered either as fluid or solid, or hard or firm.</p><div2 part="N" n="Solidity" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Solidity</hi></head><p>, in Geometry, denotes the quantity of space contained in a solid body, or occupied by it; called also the <hi rend="italics">solid content,</hi> or the <hi rend="italics">cubical content;</hi> for all solids are measured by cubes, whose sides are inches, or feet, or yards, &c; and hence the Solidity of a body is said to be so many cubic inches, feet, yards, &c, as will fill its capacity or space, or another of an equal magnitude.</p><p>The Solidity of a cube, parallelopipedon, cylinder, or any other prismatic body, i. e. one whose parallel sections are all equal and similar throughout, is found by multiplying the base by the height or perpendicular altitude. And of any cone or other pyramid, the Solidity is equal to one-third part of the same prism, because any pyramid is equal to the 3d part of its circumscribing prism. Also, because a sphere or globe may be considered as made up of an infinite number of pyramids, whose bases form the surface of the globe, and their vertices all meet in the centre, or having their common altitude equal to the radius of the globe; therefore the solid content of it is equal to onethird part of the product of its radius and surface. For the Solidity of other figures, see each figure separately.</p><p>The foregoing rules are such as are derived from common geometry. But there are in nature numberless other forms, which require the aid of other methods and principles, as follows.</p><p><hi rend="italics">Of the</hi> <hi rend="smallcaps">Solidity</hi> <hi rend="italics">of Bodies formed by a Plane revolving about any Axis, either within or without the Body.</hi>— Concerning such bodies, there is a remarkable property or relation between their Solidity and the path or line described by the centre of gravity of the revolving plane; viz, the Solidity of the body generated, whether by a whole revolution, or only a part of one, is always equal to the product arising from the generating plane drawn into the path or line described by its centre of gravity, during its motion in describing the body. And this rule holds true for figures generated by all sorts of motion whatever, whether rotatory, or direct or parallel, or irregularly zigzag, &c, provided the generating plane vary not, but continue the same throughout. And the same law holds true also for all surfaces any how generated by the motion of a right line. This is called the Centrobaric method. See my Mensuration, sect. 3, part 4, pa. 501, 2d edit.</p><p><hi rend="italics">Of the</hi> <hi rend="smallcaps">Solidity</hi> <hi rend="italics">of Bodies by the Method of Fluxions.</hi> —This method applies very advantageously in all cases also in which a body is conceived to be generated by the revolution of a plane figure about an axis, or, which is much the same thing, by the parallel motion of a circle, gradually expanding and contracting itself, according to the nature of the generating plane. And this method is particularly useful for the solids generated by any curvilineal plane figures. Thus, let the plane AED revolve about the axis AD; then it will generate the solid ABFEC. But as every ordinate DE, per- <pb n="470"/><cb/> pendicular to the axis AD, de- <figure/> scribes a circle BCEF in the revolution, therefore the same solid may be conceived as generated by a circle BCEF, gradually expanding itself larger and larger, and moving perpendicularly along the axis AD. Consequently the area of that circle being drawn into the fluxion of the axis, will produce the fluxion of the solid; and therefore the fluent, when taken, will give the Solidity of that body. That is, AD X circle BCF, (whose radius is DE, or diameter BE) is the fluxion of the Solidity.</p><p>Hence then, putting ; because <hi rend="italics">cy</hi><hi rend="sup">2</hi> is equal to the area of the circle BCF; therefore <hi rend="italics">cy</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">.</hi> is the fluxion of the solid. Consequently if the value of either <hi rend="italics">y</hi><hi rend="sup">2</hi> or <hi rend="italics">x</hi><hi rend="sup">.</hi> be found in terms of each other, from the given equation expressing the nature of the curve, and that value be substituted for it in the fluxional expression <hi rend="italics">cy</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">.</hi>, the fluent of the resulting quantity, being taken, will be the required Solidity of the body.</p><p>For Ex. Suppose the figure of a parabolic conoid, generated by the rotation of the common parabola ADE about its axis AD. In this case, the equation of the curve of the parabola is , where <hi rend="italics">p</hi> denotes the parameter of the axis. Substituting therefore <hi rend="italics">px</hi> instead of <hi rend="italics">y</hi><hi rend="sup">2</hi>, in the fluxion <hi rend="italics">cy</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">.</hi>, it becomes <hi rend="italics">cpxx</hi><hi rend="sup">.</hi>; and the fluent of this is for the Solidity; that is, half the product of the base of the solid drawn into its altitude; for <hi rend="italics">cy</hi><hi rend="sup">2</hi> is the area of the circular base BCF, and <hi rend="italics">x</hi> is the altitude. And so on for other such figures. See the content of each solid under its proper article.</p><p><hi rend="italics">For the</hi> <hi rend="smallcaps">Solidity</hi> <hi rend="italics">of Irregular Solids,</hi> or such as cannot be considered as generated by some regular motion or description; they must either be considered as cut or divided into several parts of known forms, as prisms, or pyramids, or wedges, &c, and the contents of these parts found separately. Or, in the case of the smaller bodies, of forms so irregular as not to be easily divided in that way, put them into some hollow regular vessel, as a hollow cylinder or parallelopipedon, &c; then pour in water or sand so as it may fill the vessel just up to the top of the inclosed irregular body, noting the height it rises to; then take out the body, and note the height the fluid again stands at; the difference of these two heights is to be considered as the altitude of a prism of the same base and form as the hollow vessel; and consequently the product of that altitude and base will be the accurate Solidity of the immerged body, be it ever so irregular.</p></div2></div1><div1 part="N" n="SOLSTICE" org="uniform" sample="complete" type="entry"><head>SOLSTICE</head><p>, in Astronomy, is the time when the sun is in one of the solstitial points, that is, when he is at the greatest distance from the equator, which is now nearly 23° 28′ on either side of it. It is so called, because the sun then seems to stand still, and not to change his place, as to declination, either way.</p><p>There are two Solstices, in each year, when the sun is at the greatest distance on the north and south sides of the ecliptic; viz, the <hi rend="italics">estival</hi> or <hi rend="italics">summer solstice,</hi> and the <hi rend="italics">hyemal</hi> or <hi rend="italics">winter solstice.</hi></p><p>The <hi rend="italics">Summer Solstice</hi> is when the sun is in the tropic of <cb/> Cancer; which is about the 21st of June, when he makes the longest day. And</p><p>The <hi rend="italics">Winter Solstice</hi> is when he enters the first degree of Capricorn; which is about the 22d day of December, when he makes the shortest day.</p><p>This is to be understood, as in our northern hemisphere; for in the southern, the sun's entrance into Capricorn makes their summer Solstice, and that into Cancer the winter one. So that it is more precise and determinate, to say the northern and southern Solstice.</p><p>SOLSTITIAL <hi rend="italics">Points,</hi> are those points of the ecliptic the sun is in at the times of the two Solstices, being the first points of Cancer and Capricorn, which are diametrically opposite to each other.</p><p><hi rend="smallcaps">Solstitial</hi> <hi rend="italics">Colure,</hi> is that which passes through the Solstitial points.</p></div1><div1 part="N" n="SOLUTION" org="uniform" sample="complete" type="entry"><head>SOLUTION</head><p>, in Mathematics, is the answering or resolving of a question or problem that is proposed. See <hi rend="smallcaps">Resolution</hi>, and <hi rend="smallcaps">Reduction</hi> <hi rend="italics">of Equations.</hi></p><div2 part="N" n="Solution" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Solution</hi></head><p>, in Physics, is the reduction of a solid or firm body, into a fluid state, by means of some menstruum.—Solution is often confounded with what is called dissolution, though there is a difference.</p></div2></div1><div1 part="N" n="SOSIGENES" org="uniform" sample="complete" type="entry"><head>SOSIGENES</head><p>, was an Egyptian mathematician, whose principal studies were chronology and the mathematics in general, and who flourished in the time of Julius Cæfar. He is represented as well versed in the mathematics and astronomy of the Ancients; particularly of those celebrated mathematicians, Thales, Archimedes, Hipparchus, Calippus, and many others, who had undertaken to determine the quantity of the solar year; which they had ascertained much nearer the truth than one can well imagine they should, with instruments so very imperfect; as may appear by reference to Ptolomy's Almagest.</p><p>It seems Sosigenes made great improvements, and gave proofs of his being able to demonstrate the certainty of his discoveries; by which means he became popular, and obtained repute with those who had a genius to understand and relish such enquiries. Hence he was sent for by Julius Cæsar, who being convinced of his capacity, employed him in reforming the calendar; and it was he who formed the Julian year which begins 45 years before the birth of Christ. His other works are lost since that period.</p></div1><div1 part="N" n="SOUND" org="uniform" sample="complete" type="entry"><head>SOUND</head><p>, in Geography, denotes a strait or inlet of the sea, between two capes or head-lands.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Sound</hi> is used, by way of eminence, for that celebrated strait which connects the German sea to the Baltic. It is situated between the island of Zealand and the coast of Schonen. It is about 16 leagues in length, and in general about 5 in breadth, except near the castle of Cronenberg, where it is but one; so that there is no passage for vessels but under the cannon of the fortress.</p><div2 part="N" n="Sound" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sound</hi></head><p>, in Physics, a perception of the mind, communicated by means of the ear; being an effect of the collision of bodies, and their consequent tremulous motion, communicated to the ambíent fluid, and so propagated through it to the organs of hearing.</p><p>To illustrate the cause of Sound, it is to be observed, 1st, That a motion is necessary in the sonorous body for the production of sound. 2dly, That this motion exists first in the small and insensible parts of the sonorous <pb n="471"/><cb/> bodies, and is excited in them by their mutual collision against each other, which produces the tremulous motion so observable in bodies that have a clear sound, as bells, musical chords, &c. 3dly, That this motion is communicated to, or produces a like motion in the air, or such parts of it as are fit to receive and propagate it. Lastly, That this motion must be communicated to those parts that are the proper and immediate instruments of hearing.</p><p>Now that motion of a sonorous body, which is the immediate cause of Sound, may be owing to two different causes; either the percussion between it and other hard bodies, as in drums, bells, chords, &c; or the beating and dashing of the sonorous body and the air immediately against each other, as in flutes, trumpets, &c.</p><p>But in both these cases, the motion, which is the consequence of the mutual action, as well as the immediate cause of the sonorous motion which the air conveys to the ear, is supposed to be an invisible, tremulous or undulating motion, in the small and insensible parts of the body. Perrault adds, that the visible motion of the grosser parts contributes no otherwise to Sound, than as it causes the invisible motion of the smaller parts, which he calls particles, to distinguish them from the sensible ones, which he calls parts, and from the smallest of all, which are called corpuscles.</p><p>The sonorous body having made its impression on the contiguous air, that impression is propagated from one particle to another, according to the laws of pneumatics.</p><p>A few particles, for instance, driven from the surface of the body, push or press their adjacent particles into a less space; and the medium, as it is thus rarefied in one place, becomes condensed in the other; but the air thus compressed in the second place, is, by its elasticity, returned back again, both to its former place and its former state; and the air contiguous to that is compressed; and the like obtains when the air less compressed, expanding itself, a new compression is generated. Therefore from each agitation of the air there arises a motion in it, analogous to the motion of a wave on the surface of the water; which is called a <hi rend="italics">wave</hi> or <hi rend="italics">undulation</hi> of air.</p><p>In each wave, the particles go and return back again, through very short equal spaces; the motion of each particle being analogous to the motion of a vibrating pendulum while it performs two oscillations; and most of the laws of the pendulum, with very little alteration, being applicable to the former.</p><p>Sounds are as various as are the means that concur in producing them. The chief varieties result from the figure, constitution, quantity, &c, of the sonorous body; the manner of percussion, with the velocity &c, of the consequent vibrations; the state and constitution of the medium; the disposition, distance, &c, of the organ; the obstacles between the organ and the sonorous object and the adjacent bodies. The most notable distinction of Sounds, arising from the various degrees and combinations of the conditions above mentioned, are into <hi rend="italics">loud</hi> and <hi rend="italics">low</hi> (or strong and weak); into <hi rend="italics">grave</hi> and <hi rend="italics">acute</hi> (or sharp and flat, or high and low); and into <hi rend="italics">long</hi> and <hi rend="italics">short.</hi> The management of which is the office of music. <cb/></p><p>Euler is of opinion, that no Sound making fewer vibrations than 30 in a second, or more than 7520, is distinguishable by the human ear. According to this doctrine, the limit of our hearing, as to acute and grave, is an interval of 8 octaves. Tentam. Nov. Theor. Mus. cap. 1, sect. 13.</p><p>The velocity of Sound is the same with that of the aerial waves, and does not vary much, whether it go with the wind or against it. By the wind indeed a certain quantity of air is carried from one place to another; and the Sound is accelerated while its waves move through that part of the air, if their direction be the same as that of the wind. But as Sound moves vastly swifter than the wind, the acceleration it will hereby receive is but inconsiderable; and the chief effect we can perceive from the wind is, that it increases and diminishes the space of the waves, so that by help of it the Sound may be heard to a greater distance than otherwise it would.</p><p>That the air is the usual medium of Sound, appears from various experiments in rarefied and condensed air. In an unexhausted receiver, a small bell may be heard to some distance; but when exhausted, it can scarce be heard at the smallest distance. When the air is condensed, the Sound is louder in proportion to the condensation, or quantity of air crowded in; of which there are many instances in Hauksbee's experiments, in Dr. Priestley's, and others.</p><p>Besides, sounding bodies communicate tremors to distant bodies; for example, the vibrating motion of a musical string puts others in motion, whose tension and quantity of matter dispose their vibrations to keep time with the pulses of air, propagated from the string that was struck. Galileo explains this phenomenon by observing, that a heavy pendulum may be put in motion by the least breath of the mouth, provided the blasts be often repeated, and keep time exactly with the vibrations of the pendulum; and also by the like art in raising a large bell.</p><p>It is not air alone that is capable of the impressions of Sound, but water also; as is manifest by striking a bell under water, the Sound of which may plainly enough be heard, only not so loud, and also a fourth deeper, according to good judges in musical notes. And Mersenne says, a Sound made under water is of the same tone or note, as if made in air, and heard under the water.</p><p>The velocity of Sound, or the space through which it is propagated in a given time, has been very differently estimated by authors who have written concerning this subject. Roberval states it at the rate of 560 feet in a second; Gassendus at 1473; Mersenne at 1474; Duhamel, in the History of the Academy of Sciences at Paris, at 1338; Newton at 968; Derham, in whose measure Flamsteed and Halley acquiesce, at 1142.</p><p>The reason of this variety is ascribed by Derham, partly to some of those gentlemen using strings and plummets instead of regular pendulums; and partly to the too small distance between the sonorous body and the place of observation; and partly to no regard being had to the winds.</p><p>But by the accounts since published by M. Cassini de Thury, in the Memoirs of the Royal Acad. of Scien- <pb n="472"/><cb/> ces at Paris, 1738, where cannon were fired at various as well as great distances, under many varieties of weather, wind, and other circumstances, and where the measures of the different places had been settled with the utmost exactness, it was found that Sound was propagated, on a medium, at the rate of 1038 French feet in a second of time. But the French foot is in proportion to the English as 15 to 16; and consequently 1038 French feet are equal to 1107 English feet. Therefore the difference of the measures of Derham and Cassini is 35 English feet, or 33 French feet, in a second. The medium velocity of Sound therefore is nearly at the rate of a mile, or 5280 feet, in 4 2/3 seconds, or a league in 14 seconds, or 13 miles in a minute. But sea miles are to land miles nearly as 7 to 6; and therefore Sound moves over a sea mile in 5 1/3 seconds nearly, or a sea league in 16 seconds.</p><p>Farther, it is a common observation, that persons in good health have about 75 pulsations, or beats of the artery at the wrist, in a minute; consequently in 75 pulsations, Sound flies about 13 land miles, or 11 1/7 sea miles, which is about 1 land mile in 6 pulses, or one sea mile in 7 pulses, or a league in 20 pulses.</p><p>And hence the distance of objects may be found, by knowing the time employed by Sound in moving from those objects to an observer. For Ex. On seeing the flash of a gun at sea, if 54 beats of the pulse at the wrist were counted before the report was heard; the distance of the gun will easily be found by dividing 54 by 20, which gives 2.7 leagues, or about 8 miles.</p><p>Upon the nature, production, and propagation of Sound, see the article <hi rend="smallcaps">Phonics</hi> and <hi rend="smallcaps">Echo;</hi> also the Memoirs of the Acad. and the Philos. Trans. in many places; Newton, Principia; Kircher, Mesurgia Universalis; Mersenne; Borelli, Del Suono; Priestley, Exper. and Observ. vol. 5; Hales, Sonorum Doctrina rationalis et experimentalis; 4to 1778. See also an ingenious treatise published 1790, by Mr. Geo. Saunders, on Theatres; in which he relates many experiments made by himself, on the nature and propagation of Sound. In this work, he shews the great effect of water, and some other bodies, in conducting of Sound, probably by rendering the air more dense near them. Some of his conclusions and observations are as follow:</p><p>Earth may be supposed to have a twofold property with respect to Sound. Being very porous, it absorbs Sound, which is counteracted by its property of conducting Sound, and occasions it to pass on a plane, in an equal proportion to its progress in air, unencumbered by any body.</p><p>If a Sound be sufficiently intense to impress the earth in its tremulous quality, it will be carried to a considerable distance, as when the earth is struck with any thing hard, as by the motion of a carriage, horses feet, &c.</p><p>Plaster is proportionally better than loose earth for conducting Sound, as it is more compact.</p><p>Clothes of every kind, particularly woollen cloths, are very prejudicial to Sound: their absorption of Sound, may be compared to that of water, which they greedily imbibe.</p><p>A number of people seated before others, as in the pit or gallery of a theatre, do considerably prevent the voice reaching those behind; and hence it is, that <cb/> we hear so much better in the front of the galleries, or of any situation, than behind others, though we may be nearer to the speaker. Our seats, rising so little above each other, occasion this defect, which would be remedied, could we have the seats to rise their whole height above each other, as in the ancient theatres.</p><p>Paint has generally been thought unfavourable to Sound, from its being so to musical instruments, whose effects it quite destroys.</p><p>Musical instruments mostly depend on the vibrative or tremulous property of the material, which a body of colour hardened in oil must very much alter; but we should distinguish that this regards the formation of Sound, which may not altogether be the case in the progress of it.</p><p>Water has been little noticed, with respect to its conducting Sound; but it will be found to be of the greatest consequence. I had often perceived in newly-finished houses, that while they were yet damp, they produced echoes; but that the echoing abated as they dried.</p><p>Exp. When I made the following experiment there was a gentle wind; consequently the water was proportionally agitated. I chose a quiet part of the river Thames, near Chelsea Hospital, and with two boats tried the distance the voice would reach. On the water we could distinctly hear a person read at the distance of 140 feet, on land at that of 76. It should be observed, that on land no noise intervened; but on the river some noise was occasioned by the flowing of the water against the boats; so that the difference on land and on water must be much more.</p><p>Watermen observe, that when the water is still, and the weather quite calm, if no noise intervene, a whisper may be heard across the river; and that with the current it will be carried to a much greater distance, and vice versa against the current.</p><p>Mariners well know the difference of Sound on sea and land.</p><p>When a canal of water was laid under the pit floor of the theatre of Argentino, at Rome, a surprising difference was observed; the voice has since been heard at the end very distinctly, where it was before scarce distinguishable. It is observable that, in this part, the canal is covered with a brick arch, over which there is a quantity of earth, and the timber floor over all.</p><p>The villa Simonetta near Milan, so remarkable for its echoes, is entirely over arcades of water.</p><p>Another villa near Rouen, remarkable for its echo, is built over subterraneous cavities of water.</p><p>A reservoir of water domed over, near Stanmore, has a strong echo.</p><p>I do not remember ever being under the arches of a stone bridge that did not echo; which is not always the case with similar structures on land.</p><p>A house in Lambeth Marsh, inhabited by Mr. Turtle, is very damp during winter, when it yields an echo which abates as the house becomes dry in summer.</p><p>Kircher observes, that echoes repeat more by night than during the day: he makes the difference to be double.</p><p>Dr. Plott says, the echo in Woodstock park repeated 17 times by day, and 20 by night. And Addison's <pb n="473"/><cb/> experiment at the Villa Simonetta was in a fog, when it produced 56 repetitions.</p><p>After all these instances, I think little doubt can remain of the influence water has on Sound; and I conclude that it conducts Sound more than any other body whatever.</p><p>After water, stone may be reckoned the best conductor of Sound. To what cause it may be attributed, I leave to future enquiries: I have confined myself to speak of facts only as they appear.</p><p>Stone is sonorous, but gives a harsh disagreeable tone, unfavourable to music.</p><p>Brick, in respect to Sound, has nearly the same properties as stone. Part of the garden wall of the late W. Pitt, Esq. of Kingston in Dorsetshire, conveys a whisper to the distance of near 200 feet.</p><p>Wood is sonorous, conductive, and vibrative; of all materials it produces a tone the most agreeable and melodious; and it is therefore the fittest for musical instruments, and for lining of rooms and theatres.</p><p>The common notion that whispering at one end of a long piece of timber would be heard at the other end, I found by experiment to be erroneous. A stick of timber 65 feet long being slightly struck at one end, a sound was heard at the other, and the tremor very perceptible: which is easily accounted for, when we consider the number or length of the fibres that compose it, each of which may be compared to a string of catgut.</p><p><hi rend="italics">For the Reflection, Refraction, &c, of</hi> <hi rend="smallcaps">Sound;</hi> see <hi rend="smallcaps">Echo</hi>, and <hi rend="smallcaps">Phonics.</hi></p><p><hi rend="italics">Articulate</hi> <hi rend="smallcaps">Sound.</hi> See <hi rend="smallcaps">Articulate.</hi></p></div2><div2 part="N" n="Sound" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sound</hi></head><p>, in Music, denotes a quality in the several agitations of the air, so as to make music or harmony.</p><p>Sound is the object of music; which is nothing but the art of applying Sounds, under such circumstances of tone and time, as to raise agreeable sensations. The principal affection of Sound, by which it becomes fitted to have this end, is that by which it is distinguished into acute and grave. This difference depends on the nature of the sonorous body; the particular figure and quantity of it; and even in some cases, on the part of the body where it is struck: and it is this that constitutes what are called <hi rend="italics">different tones.</hi></p><p>The cause of this difference appears to be no other than the different velocities of the vibrations of the sounding body. Indeed the tone of a Sound is found, by numerous experiments, to depend on the nature of those vibrations, whose differences we can conceive no otherwise than as having different velocities: and since it is proved that the small vibrations of the same chord are all performed in equal times, and that the tone of a Sound, which continues for some time after the stroke, is the same from first to last, it follows, that the tone is necessarily connected with a certain quantity of time in making each vibration, or each wave; or that a certain number of vibrations or waves, made in a given time, constitute a certain and determinate tone. From this principle are all the phænomena of tune deduced.</p><p>If the vibrations be isochronous, or performed in the <cb/> same time, the Sound is called musical, and is said to continue at the same pitch; and it is also accounted acuter, sharper, or higher than any other Sound, whose vibrations are slower, and therefore graver, flatter, or lower, than any other whose vibrations are quicker. See <hi rend="smallcaps">Unison.</hi></p><p>From the same principle arise what are called <hi rend="italics">concords,</hi> &c; which result from the frcquent unions and coincidences of the vibrations of two sonorous bodies, and consequently of the pulses or the waves of the air occasioned by them.</p><p>On the contrary, the result of less frequent coincidences of those vibrations, is what is called <hi rend="italics">discord.</hi></p><p>Another considerable distinction of musical Sounds, is that by which they are called <hi rend="italics">long</hi> and <hi rend="italics">short,</hi> owing to the continuation of the impulse of the efficient cause on the sonorous body for a longer or shorter time, as in the notes of a violin &c, which are made longer or shorter by strokes of different length or quickness. This continuity is properly a succession of several Sounds, or the effect of several distinct strokes, or repeated impulses, on the sonorous body, so quick, that we judge it one continued Sound, especially where it is continued in the same degree of strength; and hence arises the doctrine of <hi rend="italics">measure</hi> and <hi rend="italics">time.</hi></p><p>Musical Sounds are also divided into <hi rend="italics">simple</hi> and <hi rend="italics">compound;</hi> and that in two different ways. In the first, a Sound is said to be compound, when a number of successive vibrations of the sonorous body, and the air, come so fast upon the ear, that we judge them the same continued Sound; like as in the phenomenon of the circle of fire, caused by putting the fired end of a stick in a quick circular motion; where supposing the end of the stick in any point of the circle, the idea we receive of it there continues till the impression is renewed by a sudden return.</p><p>A <hi rend="italics">Simple</hi> <hi rend="smallcaps">Sound</hi> then, with regard to this composition, should be the effect of a single vibration, or of as many vibrations as are necessary to raise in us the idea of Sound.</p><p>In the second sense of composition, a simple Sound is the product of one voice, or one instrument, &c.</p><p>A <hi rend="italics">Compound</hi> <hi rend="smallcaps">Sound</hi> consists of the Sounds of several distinct voices or instruments all united in the same individual time, and measure of duration, that is, all striking the ear together, whatever their other differences may be. But in this sense again, there is a twofold composition; a natural and an artificial one.</p><p>The natural composition is that proceeding from the manifold reflections of the first Sound from adjacent bodies, where the reflections are not so sudden as to occasion echoes, but are all in the same tune with the first note.</p><p>The artificial composition, which alone comes under the musician's province, is that mixture of several Sounds, which being made by art, the ingredient Sounds are separable, and distinguishable from one another. In this sense the distinct Sounds of several voices or instruments, or several notes of the same instrument, are called simple Sounds, in contradistinction to the compound ones, in which, to answer the end <pb n="474"/><cb/> of music, the simples must have such an agreement in all relations, chiefly as to acuteness and gravity, as that the ear may receive the mixture with pleasure.</p><p>Another distinction of Sounds, with regard to music, is that by which they are said to be <hi rend="italics">smooth</hi> or <hi rend="italics">even,</hi> and <hi rend="italics">rough</hi> or <hi rend="italics">harsh,</hi> also <hi rend="italics">clear</hi> and <hi rend="italics">hoarse:</hi> the cause of which difference depends on the disposition and state of the sonorous body, or the circumstances of the place; but the ideas of the differences must be sought from observation.</p><p><hi rend="italics">Smooth</hi> and <hi rend="italics">Rough</hi> Sounds depend chiefly on the sounding body; of which we have a notable instance in strings that are uneven, and not of the same dimension and constitution throughout.</p><p>As to <hi rend="italics">clear</hi> and <hi rend="italics">hoarse</hi> Sounds, they depend on circumstances that are accidental to the sonorous body. Thus, a voice or instrument will be hollow and hoarse if sounded within an empty hogshead, that yet is clear and bright out of it: the effect is owing to the mixture of different Sounds, raised by reflections, which corrupt and change the species of the primitive Sound.</p><p>For Sounds to be fit to obtain the end of music, they ought to be smooth and clear, especially the first; since, without this, they cannot have one certain and discernible tone, capable of being compared to others, in a certain relation of acuteness, which the ear may judge of. So that, with Malcolm, we call that an harmonic or musical Sound which, being clear and even, is agreeable to the ear, and gives a certain and discernible tune (hence called tunable Sound), which is the subject of the whole theory of harmony.</p><p>Wood has a particular vibrating quality, owing to its elasticity; and all musical instruments made of this matter, are of a thickness proportioned to the superficies of the wood, and the tone they are to produce.</p><p>Metals are sonorous and vibrative, producing a harsh tone, very serviceable to some parts of music. Most wind instruments are made of metal, which is acted upon in its elastic and tremulous quality, being capable of being reduced very thin for that purpose. Instruments of this kind are such as horns, trumpets, &c. Some instruments however depend more on the form than the material; as flutes, for instance, which, if their lengths and bore be the same, have very little difference in their Sounds, whatever the matter of them may be. See <hi rend="smallcaps">Harmonical.</hi></p><p>SOUND-<hi rend="smallcaps">Board</hi>, the principal part of an organ, and that which makes the whole machine play. This Sound-board, or summer, is a reservoir into which the wind, drawn in by the bellows, is conducted by a portvent, and thence distributed into the pipes placed over the holes of its upper part. This wind enters them by valves, which open by pressing upon the stops or keys, after drawing the registers, which prevent the air from going into any of the other pipes beside those it is required in.</p><p><hi rend="smallcaps">Sound</hi>-<hi rend="italics">board</hi> denotes also a thin broad board placed over the head of a public speaker, to enlarge and extend or strengthen his voice.</p><p>Sound-boards, in theatres, are found by experience to be of no service; their distance from the speaker <cb/> being too great, to be impressed with sufficient force. But Sound-boards immediately over a pulpit have often a good effect, when the case is made of a just thickness, and according to certain principles.</p><p><hi rend="smallcaps">Sound</hi>-<hi rend="italics">Post,</hi> is a post placed withinside of a violin, &c, as a prop between the back and the belly of the instrument, and nearly under the bridge.</p></div2></div1><div1 part="N" n="SOUNDING" org="uniform" sample="complete" type="entry"><head>SOUNDING</head><p>, in Navigation, the act of trying the depth of the water, and the quality of the bottom, by a line and plummet, or other artifice.</p><p>At sea, there are two plummets used for this purpose, both shaped like the frustum of a cone or pyramid. One of these is called the hand-lead, weighing about 8 or 9lb; and the other the deep-sea-lead, weighing from 25 to 30lb. The former is used in shallow waters, and the latter at a great distance from the shore. The line of the hand-lead, is about 25 fathoms in length, and marked at every 2 or 3 fathoms, in this manner, viz, at 2 and 3 fathoms from the lead there are marks of black leather; at 5 fathoms a white rag, at 7 a red rag, at 10 and at 13 black leather, at 15 a white rag, and at 17 a red one.</p><p>Sounding with the hand-lead, which the seamen call heaving the lead, is generally performed by a man who stands in the main-chains to windward. Having the line all ready to run out, without interruption, he holds it nearly at the distance of a fathom from the plummet, and having swung the latter backwards and forwards three or four times, in order to acquire the greater velocity, he swings it round his head, and thence as far forward as is necessary; so that, by the lead's sinking whilst the ship advances, the line may be almost perpendicular when it reaches the bottom. The person sounding then proclaims the depth of the water in a kind of song resembling the cries of hawkers in a city; thus, if the mark of 5 be close to the surface of the water, he calls, ‘by the mark 5,’ and as there is no mark at 4, 6, 8, &c, he estimates those numbers, and calls, ‘by the dip four, &c.’ If he judges it to be a quarter or a half more than any particular number, he calls, ‘and a quarter 5,’ ‘and a half 4’ &c. If he conceives the depth to be three quarters more than a particular number, he calls it a quarter less than the next: thus, at 4 fathom 3/4, he calls, ‘a quarter less 5,’ and so on.</p><p>The deep-sea-lead line is marked with 2 knots at 20 fathom, 3 at 30, 4 at 40, &c to the end. It is also marked with a single knot at the middle of each interval, as at 25, 35, 45 fathoms, &c. To use this lead more effectually at sea, or in deep water on the sea-coast, it is usual previously to bring-to the ship, in order to retard her course: the lead is then thrown as far as possible from the ship on the line of her drift, so that, as it sinks, the ship drives more perpendicularly over it. The pilot feeling the lead strike the bottom, readily discovers the depth of the water by the mark on the line nearest its surface. The bottom of the lead, which is a little hollowed there for the purpose, being also well rubbed over with tallow, retains the distinguishing marks of the bottom, as shells, ooze, gravel, &c, which naturally adhere to it.</p><p>The depth of the water, and the nature of the ground, which are called the Soundings, are carefully marked in the log-book, as well to determine the distance of <pb n="475"/><cb/> the place from the shore, as to correct the observations of former pilots. Falconer.</p><p>For a machine to measure unfathomable depths of the sea, see <hi rend="smallcaps">Altitude.</hi></p><p><hi rend="smallcaps">Sounding</hi> <hi rend="italics">the pump,</hi> at sea, is done by letting fall a small line, with some weight at the end, down into the pump, to know what depth of water there is in it.</p></div1><div1 part="N" n="SOUTH" org="uniform" sample="complete" type="entry"><head>SOUTH</head><p>, one of the four cardinal points of the wind, or compass, being that which is directly opposite to the north.</p><p><hi rend="smallcaps">South</hi> <hi rend="italics">Direct Dials.</hi> See <hi rend="smallcaps">Prime</hi> <hi rend="italics">Verticals.</hi></p><p>SOUTHERN <hi rend="italics">Hemisphere, Signs, &c,</hi> those in the south side of the equator.</p></div1><div1 part="N" n="SOUTHING" org="uniform" sample="complete" type="entry"><head>SOUTHING</head><p>, in Navigation, the difference of latitude made by a ship in sailing to the southward.</p></div1><div1 part="N" n="SPACE" org="uniform" sample="complete" type="entry"><head>SPACE</head><p>, denotes room, place, distance, capacity, extension, duration, &c.</p><p>When Space is considered barely in length between any two bodies, it gives the same idea as that of distance. When it is considered in length, breadth, and thickness, it is properly called capacity. And when considered between the extremities of matter, which sills the capacity of Space with something solid, tangible, and moveable, it is then called extension.</p><p>So that extension is an idea belonging to body only; but Space may be considered without it. Therefore Space, in the general signification, is the same thing with distance considered every way, whether there be any matter in it or not.</p><p>Space is usually divided into <hi rend="italics">absolute</hi> and <hi rend="italics">relative.</hi></p><p><hi rend="italics">Absolute</hi> <hi rend="smallcaps">Space</hi> is that which is considered in its own nature, without regard to any thing external, which always remains the same, and is infinite and immoveable.</p><p><hi rend="italics">Relative</hi> <hi rend="smallcaps">Space</hi> is that moveable dimension, or measure of the former, which our senses define by its positions to bodies within it; and this the vulgar use for immoveable Space.</p><p>Relative Space, in magnitude and figure, is always the same with absolute; but it is not necessary it should be so numerically. Thus, when a ship is perfectly at rest, then the places of all things within her are the same both absolutely and relatively, and nothing changes its place: but, on the contrary, when the ship is under sail, or in motion, she continually passes through new parts of absolute Space; though all things on board, considered relatively, in respect to the ship, may yet be in the same places, or have the same situation and position, in regard to one another.</p><p>The Cartesians, who make extension the essence of matter, assert, that the Space any body takes up, is the same thing with the body itself; and that there is no such thing in the universe as mere Space, void of all matter; thus making Space or extension a substance. See this disproved under <hi rend="smallcaps">Vacuum.</hi></p><p>Among those too who admit a vacuum, and consequently an essential difference between Space and matter, there are some who assert that Space is a substance. Among these we find Gravesande, Introd. ad Philos. sect. 19.</p><p>Others again put Space into the same class of beings as time and number; thus making it to be no more than a notion of the mind. So that according to these authors, absolute Space, of which the Newtonians <cb/> speak, is a mere chimera. See the writings of the late bishop Berkley.</p><p>Space and time, according to Dr. Clarke, are attributes of the Deity; and the impossibility of annihilating these, even in idea, is the same with that of the necessary existence of the Deity.</p><div2 part="N" n="Space" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Space</hi></head><p>, in Geometry, denotes the area of any figure; or that which sills the interval or distance between the lines that terminate or bound it. Thus,</p><p>The Parabolic Space is that included in the whole parabola. The conchoidal Space, or the cissoidal Space, is what is included within the cavity of the conchoid or cissoid. And the asymptotic Space, is what is included between an hyperbolic curve and its asymptote. By the new methods now introduced, of applying algebra to geometry, it is demonstrated that the conchoidal and cissoidal Spaces, though infinitely extended in length, are yet only finite magnitudes or Spaces.</p></div2><div2 part="N" n="Space" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Space</hi></head><p>, in Mechanics, is the line a moveable body, considered as a point, is conceived to describe by its motion.</p></div2></div1><div1 part="N" n="SPANDREL" org="uniform" sample="complete" type="entry"><head>SPANDREL</head><p>, with Builders, <figure/> is the space included between the curve of an arch and the straight or right lines which inclose it; as the space <hi rend="italics">a,</hi> or <hi rend="italics">b.</hi></p><p>SPEAKING <hi rend="italics">Trumpet.</hi> See <hi rend="italics">Speaking</hi> <hi rend="smallcaps">Trumpet.</hi></p></div1><div1 part="N" n="SPECIES" org="uniform" sample="complete" type="entry"><head>SPECIES</head><p>, in Algebra, are the letters, symbols, marks, or characters, which represent the quantities in any operation or equation.</p><p>This short and advantageous way of notation was chiefly introduced by Vieta, about the year 1590; and by means of which he made many discoveries in algebra, not before taken notice of.</p><p>The reason why Vieta gave this name of Species to the letters of the alphabet used in algebra, and hence called Arithmetica Speciosa, seems to have been in imitation of the Civilians, who call cases in law that are put abstractedly, between John a Nokes and Tom a Stiles, between A and B; supposing those letters to stand for any persons indefinitely. Such cases they call Species: whence, as the letters of the alphabet will also as well represent quantities, as persons, and that also indefinitely, one quantity as well as another, they are properly enough called Species; that is general symbols, marks, or characters. From whence the literal algebra hath since been often called Specious Arithmetic, or Algebra in Species.</p><div2 part="N" n="Species" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Species</hi></head><p>, in Optics, the image painted on the retina by the rays of light reflected from the several points of the surface of an object, received in by the pupil, and collected in their passage through the crystalline, &c.</p><p>Philosophers have been in great doubt, whether the Species of objects, which give the soul an occasion of seeing, are an effusion of the substance of the body; or a mere impression which they make on all ambient bodies, and which these all reflect, when in a proper disposition and distance; or lastly, whether they are not some other more subtile body, as light, which receives all these impressions from bodies, and is continually sent and returning from one to another, with the different impressions and figures it has taken. But the moderns have decided this point by their invention of <pb n="476"/><cb/> artificial eyes, inwhich the Species of objects are received on a paper, in the same manner as they are received in the natural eye.</p></div2></div1><div1 part="N" n="SPECIFIC" org="uniform" sample="complete" type="entry"><head>SPECIFIC</head><p>, in Philosophy, that which is proper and peculiar to any thing; or that characterises it, and distinguishes it from every other thing. Thus, the attracting of iron is Specific to the loadstone, or is a Specific property of it.</p><p>A just definition should contain the Specific notion of the thing defined, or that which specifies and distinguishes it from every thing else.</p><p><hi rend="smallcaps">Specific</hi> <hi rend="italics">Gravily,</hi> in Hydrostatics, is the relative proportion of the weight of bodies of the same bulk. See <hi rend="italics">Specific</hi> <hi rend="smallcaps">Gravity.</hi></p><p><hi rend="smallcaps">Specific</hi> <hi rend="italics">Gravity of living men.</hi> Mr. John Robertson, late librarian to the Royal Society, in order to determine the Specific gravity of men, prepared a cistern 78 inches long, 30 inches wide, 30 inches deep; and having procured 10 men for his purpose, the height of each was taken and his weight; and afterwards they plunged successively into the cistern. A ruler or scale, graduated to inches and decimal parts, was fixed to one end of the cistern, and the height of the water shown by it was noted before each man went in, and to what height it rose when he immersed himself under its surface. The following table contains the several results of his experiments: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" role="data">Height.</cell><cell cols="1" rows="1" role="data">Weight.</cell><cell cols="1" rows="1" role="data">Water</cell><cell cols="1" rows="1" role="data">Solidity.</cell><cell cols="1" rows="1" role="data">Wt. of</cell><cell cols="1" rows="1" role="data">Specific</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">of</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">raised.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">water.</cell><cell cols="1" rows="1" role="data">gravity.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">men.</cell><cell cols="1" rows="1" role="data">Ft. In.</cell><cell cols="1" rows="1" role="data">lbs.</cell><cell cols="1" rows="1" role="data">Inches.</cell><cell cols="1" rows="1" role="data">Feet.</cell><cell cols="1" rows="1" role="data">lbs.</cell><cell cols="1" rows="1" role="data">(Wat. 1)</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">6  2</cell><cell cols="1" rows="1" role="data">161</cell><cell cols="1" rows="1" role="data">1.90</cell><cell cols="1" rows="1" role="data">2.573</cell><cell cols="1" rows="1" role="data">160.8</cell><cell cols="1" rows="1" role="data">1.001</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">5 10 3/8</cell><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" role="data">1.91</cell><cell cols="1" rows="1" role="data">2.586</cell><cell cols="1" rows="1" role="data">161.6</cell><cell cols="1" rows="1" role="data">0.901</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">5  9 1/2</cell><cell cols="1" rows="1" role="data">156</cell><cell cols="1" rows="1" role="data">1.85</cell><cell cols="1" rows="1" role="data">2.505</cell><cell cols="1" rows="1" role="data">156.6</cell><cell cols="1" rows="1" role="data">0.991</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">5  6 3/4</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data">2.04</cell><cell cols="1" rows="1" role="data">2.763</cell><cell cols="1" rows="1" role="data">172.6</cell><cell cols="1" rows="1" role="data">0.801</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">5  5 7/8</cell><cell cols="1" rows="1" role="data">158</cell><cell cols="1" rows="1" role="data">2.08</cell><cell cols="1" rows="1" role="data">2.817</cell><cell cols="1" rows="1" role="data">176.0</cell><cell cols="1" rows="1" role="data">0.900</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">5  5 1/2</cell><cell cols="1" rows="1" role="data">158</cell><cell cols="1" rows="1" role="data">2.17</cell><cell cols="1" rows="1" role="data">2.939</cell><cell cols="1" rows="1" role="data">183.7</cell><cell cols="1" rows="1" role="data">0.849</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">5  4 3/8</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data">2.01</cell><cell cols="1" rows="1" role="data">2.722</cell><cell cols="1" rows="1" role="data">170.1</cell><cell cols="1" rows="1" role="data">0.823</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">5  4 1/8</cell><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" role="data">1.79</cell><cell cols="1" rows="1" role="data">2.424</cell><cell cols="1" rows="1" role="data">151.5</cell><cell cols="1" rows="1" role="data">0.800</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">5  3 1/4</cell><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" role="data">1.73</cell><cell cols="1" rows="1" role="data">2.343</cell><cell cols="1" rows="1" role="data">146.4</cell><cell cols="1" rows="1" role="data">0.997</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">5  3 1/8</cell><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" role="data">1.85</cell><cell cols="1" rows="1" role="data">2.505</cell><cell cols="1" rows="1" role="data">156.6</cell><cell cols="1" rows="1" role="data">0.843</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=center" role="data">medium</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">5  6 2/3</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">146</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">1.933</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">2.618</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">163.6</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0.891</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=center" role="data">of all.</cell></row></table></p><p>One of the reasons, Mr. Robertson says, that induced him to make these experiments, was a desire of knowing what quantity of timber would be sufficient to keep a man afloat in water, thinking that most men were specifically heavier than river or common fresh water; but the contrary appears from the trials above recited; for, except the first, every man was lighter than an equal bulk of fresh water, and much more so than that of seawater. So that, if persons who fall into water had presence of mind enough to avoid the fright usual on such occasions, many might be preserved from drowning; and a piece of wood not larger than an oar, would buoy a man partly above water as long as he had strength or spirits to keep his hold. Philos. Trans. vol. 50, art. 5.</p><p>From the last line of the table appears the medium of all the circumstances of height, weight, &c; particu- <cb/> larly the mean Specific Gravity, 0.891, which is about 1/9 less than common water.</p></div1><div1 part="N" n="SPECTACLES" org="uniform" sample="complete" type="entry"><head>SPECTACLES</head><p>, an optical machine, consisting of two lenses set in a frame, and applied on the nose, to assist in defects of the organ of sight.</p><p>Old people, and all presbytæ, use Spectacles of convex lenses, to make amends for the flatness of the eye, which does not make the rays converge enough to have them meet in the retina.</p><p>Short-sighted people, or myopes, use concave lenses, to prevent the rays from converging so fast, on account of the greater roundness of the eye, or smallness of the sphere, which is such as to make them meet before they reach the retina.</p><p>F. Cherubin, a capuchin, describes a kind of Spectacle telescopes, for viewing remote objects with both eyes; and hence called <hi rend="italics">binoculi.</hi> Though F. Rheita had mentioned the same before him, in his Oculus Enoch et Eliæ. See <hi rend="smallcaps">Binocle.</hi> The same author invented a kind of Spectacles, with three or four glasses, which performed very well.</p><p>The invention of Spectacles has been much disputed. They were certainly not known to the ancients. Francisco Redi, in a learned treatise on Spectacles, contends that they were first invented between the years 1280 and 1311, probably about 1290; and adds, that Alexander de Spina, a monk of the order of Predicants of St. Catharine, at Pisa, first communicated the secret, which was of his own invention, upon learning that another person had it as well as himself.</p><p>The author tells us, that in an old manuscript still preserved in his library, composed in 1299, Spectacles are mentioned as a thing invented about that time: and that a celebrated Jacobin, one Jourdon de Rivalto, in a treatise composed in 1305, says expressly, that it was not yet 20 years since the invention of Spectacles. He likewise quotes Bernard Gordon in his Lilium Medicinæ, written the same year, where he speaks of a collyrium, good to enable an old man to read without Spectacles.</p><p>Musschenbroek observes, (Introd. vol. 2, pa. 786) that it is inscribed on the tomb of Salvinus Armatus, a nobleman of Florence, who died in 1317, that he was the inventor of Spectacles.</p><p>Du-Cange, however, carries the invention of Spectacles farther back; assuring us, that there is a Greek poem in manuscript in the French king's library, which shews that Spectacles were in use in the year 1150; however the dictionary of the Academy Della Crusca, under the word <hi rend="italics">occhiale,</hi> inclines to Redi's side; and quotes a passage from Jourdon's sermons, which says that Spectacles had not been 20 years in use; and Salvati has observed that those sermons were composed between the years 1330 and 1336.</p><p>It is probable that the first hint of the construction and use of Spectacles, was derived from the writings either of Alhazen, who lived in the 12th century, or of our own countryman Roger Bacon, who was born in 1214, and died in 1292, or 1294. The following remarkable passage occurs in Bacon's Opus Majus by Jebb, p. 352. Si vero homo aspiciat literas et alias res minutas per medium crystalli, vel vitri, vel alterius perspicui suppositi literis, et sit portio minor spheræ, cujus convexitas sit versus oculum et oculus sit in aëre, <pb n="477"/><cb/> longe melius videbit literas, et apparebunt ei majores.— Et ideo hoc instrumentum est utile senibus et habentibus oculos debiles: nam literam quantumcunque parvam possunt videre in sufficienti magnitudine. Hence, and from other passages in his writings, much to the same purpose, Molyneux, Plott, and others, have attributed to him the invention of reading-glasses. Dr. Smith indeed, observing that there are some mistakes in his reasoning on this subject, has disputed his claim. See Molyneux's Dioptr. p. 256. Smith's Optics, Rem. 86—89.</p><p>SPECULATIVE <hi rend="italics">Geometry, Mathematics, Music,</hi> and <hi rend="italics">Philosophy.</hi> See the <hi rend="smallcaps">Substantives.</hi></p></div1><div1 part="N" n="SPECULUM" org="uniform" sample="complete" type="entry"><head>SPECULUM</head><p>, or <hi rend="italics">Mirror,</hi> in Optics, any polished body, impervious to the rays of light: such as polished metals, and glasses lined with quicksilver, or any other opake matter, popularly called Looking-glasses; or even the surface of mercury or of water, &c.</p><p>For the several kinds and forms of Specula, plane, concave, and convex, with their theory and phenomena, see <hi rend="smallcaps">Mirror.</hi> And for their laws and effects, see <hi rend="smallcaps">Reflection</hi> and <hi rend="smallcaps">Burning</hi> <hi rend="italics">Glass.</hi></p><p>As for the Specula of reflecting telescopes, it may here be observed, that the perfection of the metal of which they should be made, consists in its hardness, whiteness, and compactness; for upon these properties the reflective powers and durability of the Specula depend. There are various compositions recommended for these Specula, in Smith's Optics, book 3, ch. 2, sect. 787; also by Mr. Mudge in the Philos. Trans. vol. 67; and in various other places, as by Mr. Edwards, in the Naut. Alm. for 1787, whose metal is the whitest and best of any that I have seen.—For the method of grinding, see <hi rend="smallcaps">Grinding.</hi></p><p>Mr. Hearne's method of cleaning a tarnished Speculum was this: Get a little of the strongest soap ley from the soap-makers, and having laid the Speculum on a table with its face upwards, put on as much of the ley as it will hold, and let it remain about an hour: then rub it softly with a silk or muslin, till the ley is all gone; then put on some spirit of wine, and rub it dry with another part of the silk or muslin. If the Speculum will not perform well after this, it must be new polished. A few faint spots of tarnish may be rubbed off with spirit of wine only, without the ley. Smith's Optics, Rem. p. 107.</p></div1><div1 part="N" n="SPHERE" org="uniform" sample="complete" type="entry"><head>SPHERE</head><p>, in Geometry, a solid body contained under one single uniform surface, every point of which is equally distant from a certain point in the middle called its centre.</p><p>The Sphere may be supposed <figure/> to be generated by the revolution of a semicircle ABD about its di meter AB, which is also called the <hi rend="italics">axis</hi> of the Sphere, and the extreme points of the axis, A and B, the <hi rend="italics">poles</hi> of the Sphere; also the middle of the axis C is the <hi rend="italics">centre,</hi> and half the axis, AC, the <hi rend="italics">radius.</hi> <hi rend="center"><hi rend="italics">Properties of the</hi> <hi rend="smallcaps">Sphere</hi>, are as follow.</hi></p><p>1. A Sphere may be considered as made up of an infinite number of pyramids, whose common altitude <cb/> is equal to the radius of the Sphere, and all their bases form the surface of the Sphere. And therefore the solid content of the Sphere is equal to that of a pyramid whose altitude is the radius, and its base is equal to the surface of the Sphere, that is, the solid content is equal to 1/3 of the product of its radius and surface.</p><p>2. A Sphere is equal to 2/3 of its circumscribing cylinder, or of the cylinder of the same height and diameter, and therefore equal to the cube of the diameter multiplied by .5236, or 2/3 of .7854; or equal to double a cone of the same base and height. Hence also different Spheres are to one another as the cubes of their diameters. And their surfaces as the squares of the same diameters.</p><p>3. The surface or superficies of any Sphere, is equal to 4 times the area of its great circle, or of a circle of the same diameter as the Sphere. Or</p><p>4. The surface of the whole Sphere is equal to the area of a circle whose radius is equal to the diameter of the Sphere. And, in like manner, the curve surface of any segment EDF, whether greater or less than a hemisphere, is equal to a circle whose radius is the chord line DE, drawn from the vertex D of the segment to the circumference of its base, or the chord of half its arc.</p><p>5. The curve surface of any segment or zone of a Sphere, is also equal to the curve surface of a cylinder of the same height with that portion, and of the same diameter with the Sphere. Also the surface of the whole Sphere, or of an hemisphere, is equal to the curve surface of its circumscribing cylinder. And the curve surfaces of their corresponding parts are equal, that are contained between any two places parallel to the base. And consequently the surface of any segment or zone of a Sphere, is as its height or altitude.</p><p>Most of these properties are contained in Archimedes's treatise on the Sphere and cylinder. And many other rules for the surfaces and solidities of Spheres, their segments, zones, frustums, &c, may be seen in my Mensuration, part 3, sect. 1, prob. 10, &c.</p><p>Hence, if <hi rend="italics">d</hi> denote the diameter or axis of a Sphere, <hi rend="italics">s</hi> its curve surface, <hi rend="italics">c</hi> its solid content, and <hi rend="italics">a</hi> = .7854 the area of a circle whose diam. is 1; then we shall, from the foregoing properties, have these following general values or equations, viz, .</p><p><hi rend="italics">Doctrine of the</hi> <hi rend="smallcaps">Sphere.</hi> See <hi rend="smallcaps">Spherics.</hi></p><p><hi rend="italics">Projection of the</hi> <hi rend="smallcaps">Sphere.</hi> See <hi rend="smallcaps">Projection.</hi></p><p><hi rend="smallcaps">Sphere</hi> <hi rend="italics">of Activity,</hi> of any body, is that determinate space or extent all around it, to which, and no farther, the effluvia or the virtue of that body reaches, and in which it operates according to the nature of the body. See <hi rend="smallcaps">Activity.</hi></p><div2 part="N" n="Sphere" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sphere</hi></head><p>, in Astronomy, that concave orb or expanse which invests our globe, and in which the hea- <pb n="478"/><cb/> venly bodies, the sun, moon, stars, planets, and comets, appear to be fixed at an equal distance from the eye. This is also called the Sphere of the world; and it is the subject of spherical astronomy.</p><p>This Sphere, as it includes the fixed stars, from whence it is sometimes called the <hi rend="italics">Sphere of the fixed stars,</hi> is immensely great. So much so, that the diameter of the earth's orbit is vastly small in respect of it; and consequently the centre of the Sphere is not sensibly changed by any alteration of the spectator's place in the several parts of the orbit: but still in all points of the earth's surface, and at all times, the inhabitants have the same appearance of the Sphere; that is, the fixed stars seem to possess the same points in the surface of the Sphere. For, our way of judging of the places &c of the stars, is to conceive right lines drawn from the eye, or from the centre of the earth, through the centres of the stars, and thence continued till they cut the Sphere; and the points where these lines so meet the Sphere, are the apparent places of those stars.</p><p>The better to determine the places of the heavenly bodies in the Sphere, several circles are conceived to be drawn in the surface of it, which are called circles of the Sphere.</p></div2><div2 part="N" n="Sphere" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Sphere</hi></head><p>, in Geography, &c, denotes a certain disposition of the circles on the surface of the earth, with regard to one another, which varies in the different parts of it.</p><p>The circles originally conceived on the surface of the Sphere of the world, are almost all transferred, by analogy, to the surface of the earth, where they are conceived to be drawn directly underneath those of the Sphere, or in the same positions with them; so that, if the planes of those of the earth were continued to the Sphere of the stars, they would coincide with the respective circles on it. Thus, we have an horizon, meridian, equator, &c, on the earth. And as the equinoctial, or equator, in the heavens, divides the Sphere into two equal parts, the one north and the other south, so does the equator on the surface of the earth divide its globe in the same manner. And as the meridians in the heavens pass through the poles of the equinoctial, so do those on the earth, &c. With regard then to the position of some of these circles in respect of others, we have a <hi rend="italics">right,</hi> an <hi rend="italics">oblique,</hi> and a <hi rend="italics">parallel</hi> Sphere.</p><p><hi rend="italics">A Right or Direct</hi> <hi rend="smallcaps">Sphere</hi>, (fig. 4, plate 26), is that which has the poles of the world PS in its horizon, and the equator EQ in the zenith and nadir. The inhabitants of this Sphere live exactly at the equator of the earth, or under the line. They have therefore no latitude, nor no elevation of the pole. They can see both poles of the world; all the stars do rise, culminate, and set to them; and the sun always rises at right-angles to their horizon, making their days and nights always of equal length, because the horizon bisects the circle of the diurnal revolution.</p><p><hi rend="italics">An Oblique</hi> <hi rend="smallcaps">Sphere</hi>, (fig. 5, plate 26), is that in which the equator EQ, as also the axis PS, cuts the horizon HO obliquely. In this Sphere, one pole P is above the horizon, and the other below it; and therefore the inhabitants of it see always the former pole, but never the latter; the sun and stars &c all rise and <cb/> set obliquely; and the days and nights are always varying, and growing alternately longer and shorter.</p><p><hi rend="italics">A Parallel</hi> <hi rend="smallcaps">Sphere</hi>, (fig. 6, plate 26), is that which has the equator in or parallel to the horizon, as well as all the sun's parallels of declination. Hence, the poles are in the zenith and nadir; the sun and stars move always quite around parallel to the horizon, the inhabitants, if any, being just at the two poles, having 6 months continual day, and 6 months night, in each year; and the greatest height to which the sun rises to them, is 23° 28′, or equal to his greatest declination.</p><p><hi rend="italics">Armillary or Artificial</hi> <hi rend="smallcaps">Sphere</hi>, is an astronomical instrument, representing the several circles of the Sphere in their natural order; serving to give an idea of the office and position of each of them, and to resolve various problems relating to them.</p><p>It is thus called, as consisting of a number of rings of brass, or other matter, called by the Latins <hi rend="italics">armillæ,</hi> from their resembling of bracelets or rings for the arm.</p><p>By this, it is distinguished from the globe, which, though it has all the circles of the Sphere on its surface; yet is not cut into armillæ or rings, to represent the circles simply and alone; but exhibits also the intermediate spaces between the circles.</p><p>Armillary Spheres are of different kinds, with regard to the position of the earth in them; whence they become distinguished into Ptolomaic and Copernican Spheres: in the first of which, the earth is in the centre, and in the latter near the circumference, according to the position which that planet obtains in those systems.</p><p><hi rend="italics">The Ptolomaic</hi> <hi rend="smallcaps">Sphere</hi>, is that commonly in use, and is represented in fig. 6, plate 2, vol. 1, with the names of the several circles, lines, &c of the Sphere inscribed upon it. In the middle, upon the axis of the Sphere, is a ball T, representing the earth, on the surface of which are the circles &c of the earth. The Sphere is made to revolve about the said axis, which remains at rest; by which means the sun's diurnal and annual courses about the earth are represented according to the Ptolomaic hypothesis: and even by means of this, all problems relating to the phenomena of the sun and earth are resolved, as upon the celestial globe, and after the same manner; which see described under <hi rend="smallcaps">Globe.</hi></p><p><hi rend="italics">Copernican</hi> <hi rend="smallcaps">Sphere</hi>, fig. 7, plate 26, is very different from the Ptolomaic, both in its constitution and use; and is more intricate in both. Indeed the instrument is in the hands of so few people, and its use so inconsiderable, except what we have in the other more common instruments, particularly the globe and the Ptolomaic Sphere, that any farther account of it is unnecessary.</p><p>Dr. Long had an Armillary Sphere of glass, of a very large size, which is described and represented in his Astronomy. And Mr. Ferguson constructed a similar one of brass, which is exhibited in his Lectures, p. 194 &c.</p></div2></div1><div1 part="N" n="SPHERICAL" org="uniform" sample="complete" type="entry"><head>SPHERICAL</head><p>, something relating to the sphere. As,</p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Angle,</hi> is the angle formed on the surface of a Sphere or globe by the circumferences of <pb n="479"/><cb/> two great circles. This angle, <figure/> formed by the circumferences, is equal to that formed by the planes of the same circles, or equal to the inclination of those two planes; or equal to the angle made by their tangents at the angular point. Thus, the inclination of the two planes CAF, CEF, forms the Spherical Angle ACE, equal to the tangential angle PCQ.</p><p>The measure of a Spherical Angle, ACE, is an arc of a great circle AE, described from the vertex C, as from a pole, and intercepted between the legs CA and CE.</p><p>Hence, 1st, Since the inclination of the plane CEF to the plane CAF, is every where the same, the angles in the opposite intersections, C and F, are equal.—2d, Hence the measure of a Spherical Angle ACE, is an arc described at the interval of a quadrant CA or CE, from the vertex C between the legs CA, CE.— 3d, If a circle of the sphere CEFG cut another AEBG, the adjacent angles AEC and BEC are together equal to two right angles; and the vertical angles AEC, BEF are equal to one another. Also all the angles formed at the same point, on the same side of a circle, are equal to two right angles, and all those quite around any point equal to four right angles.</p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Triangle,</hi> is a triangle formed upon the surface of a sphere, by the intersecting arcs of three great circles; as the triangle ACE.</p><p>Spherical Triangles are either <hi rend="italics">right-angled, oblique, equilateral, isosceles,</hi> or <hi rend="italics">scalene,</hi> in the same manner as plane triangles. They are also said to be <hi rend="italics">quadrantal,</hi> when they have one side a quadrant. Two sides or two angles are said to be of the <hi rend="italics">same affection,</hi> when they are at the same time either both greater, or both less than a quadrant or a right angle or 90°; and of <hi rend="italics">different offections,</hi> when one is greater and the other less than 90 degrees. <hi rend="center"><hi rend="italics">Properties of</hi> <hi rend="smallcaps">Spherical</hi> <hi rend="italics">Triangles.</hi></hi></p><p>1. Spherical Triangles have many properties in common with plane ones: Such as, That, in a triangle, equal sides subtend equal angles, and equal angles are subtended by equal sides: That the greater angles are subtended by the greater sides, and the less angles by the less sides.</p><p>2. In every Spherical Triangle, each side is less than a semicircle: any two sides taken together are greater than the third side: and all the three sides taken together are less than the whole circumference of a circle.</p><p>3. In every Spherical Triangle, any angle is less than 2 right angles; and the sum of all the three angles taken together, is greater than 2, but less than 6, right angles.</p><p>4. In an oblique Spherical Triangle, if the angles at the base be of the same affection, the perpendicular from the other angle falls within the triangle; but if they be of different affections, the perpendicular falls without the triangle.</p><p>Dr. Maskelyne's remarks on the properties of Spherical Triangles, are as follow: (See the Introd. to my Logs. pa. 160, 2d edition.) <cb/></p><p>5. “A Spherical Triangle is equilateral, isoscelar, or scalene, according as it has its three angles all equal, or two of them equal, or all three unequal; and vice versa.</p><p>6. The greatest side is always opposite the greatest angle, and the smallest side opposite the smallest angle.</p><p>7. Any two sides taken together are greater than the third.</p><p>8. If the three angles are all acute, or all right, or all obtuse; the three sides will be, accordingly, all less than 90°, or equal to 90°, or greater than 90°; and vice versa.</p><p>9. If from the three angles A, B, C, of a triangle ABC, as poles, there be described, upon the surface of the sphere, three arches of a great circle DE, DF, FE, forming by their intersections a new Spherical Triangle DEF; each side of the new triangle will be the supplement of the angle at its pole; and each angle of the same triangle, will be the supplement of the side opposite to it in the triangle ABC. <figure/></p><p>10. In any triangle GHI or G<hi rend="italics">h</hi>I, right angled in G, 1st, The angles at the hypotenuse are always of the same kind as their opposite sides; 2dly, The hypotenuse is less or greater than a quadrant, according as the sides including the right angle, are of the same or different kinds; that is to say, according as these same sides are either both acute, or both obtuse, or as one is acute and the other obtuse. And, vice versa, 1st, The sides including the right angle, are always of the same kind as their opposite angles; 2dly, The sides including the right angle will be of the same or different kinds, according as the hypotenuse is less or more than 90°; but one at least of them will be of 90°, if the hypotenuse is so.”</p><p><hi rend="italics">Of the Area of a</hi> <hi rend="smallcaps">Spherical</hi> <hi rend="italics">Triangle.</hi> The mensuration of Spherical Triangles and polygons was first found out by Albert Girard, about the year 1600, and is given at large in his <hi rend="italics">Invention Nouvelle en l'Algebre,</hi> pa. 50, &c; 4to, Amst. 1629. In any Spherical Triangle, the area, or surface inclosed by its three sides upon the surface of the globe, will be found by this proportion: As 8 right angles or 720°,<lb/> Is to the whole surface of the sphere;<lb/> Or, as 2 right angles or 180°,<lb/> To one great circle of the sphere;<lb/> So is the excess of the 3 angles above 2 right angles,<lb/> To the area of the Spherical Triangle.<lb/></p><p>Hence, if <hi rend="italics">a</hi> denote .7854, <hi rend="italics">d</hi> = diam. of the globe, and <hi rend="italics">s</hi> = sum of the 3 angles of the triangle; <pb n="480"/><cb/> then <hi rend="italics">add</hi> X (<hi rend="italics">s</hi> - 180)/180 = area of the Spherical Triangle.</p><p>Hence also, if <hi rend="italics">r</hi> denote the radius of the sphere, and <hi rend="italics">c</hi> its circumference; then the area of the triangle will thus be variously expressed; viz, , in square degrees, when the radius <hi rend="italics">r</hi> is estimated in degrees; for then the circumference <hi rend="italics">c</hi> is = 360°.</p><p>Farther, because the radius <hi rend="italics">r,</hi> of any circle, when estimated in degrees, is, =180/(3.14159 &c.) = 57.2957795, the last rule <hi rend="italics">r</hi> X ―(<hi rend="italics">s</hi> - 180), for expressing the area <hi rend="italics">A</hi> of the Spherical Triangle, in square degrees, will be barely very nearly.</p><p>Hence may be found the sums of the three angles in any Spherical Triangle, having its area <hi rend="italics">A</hi> known; for the last equation gives the sum .</p><p>So that, for a Triangle on the surface of the earth, whose three sides are known; if it be but small, as of a few miles extent, its area may be found from the known lengths of its sides, considering it as a plane Triangle, which gives the value of the quantity A; and then the last rule above will give the value of <hi rend="italics">s,</hi> the sum of the three angles; which will serve to prove whether those angles are nearly exact, that have been taken with a very nice instrument, as in large and extensive measurements on the surface of the earth.</p><p><hi rend="italics">Resolution of</hi> <hi rend="smallcaps">Spherical</hi> <hi rend="italics">Triangles.</hi> See <hi rend="smallcaps">Triangle</hi>, and <hi rend="smallcaps">Trigonometry.</hi></p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Polygon,</hi> is a figure of more than three sides, formed on the surface of a globe by the intersecting arcs of great circles.</p><p>The area of any Spherical Polygon will be found by the following proportion; viz, As 8 right-angles or 720°,<lb/> To the whole surface of the sphere;<lb/> Or, as 2 right angles or 180°,<lb/> To a great circle of the sphere;<lb/> So is the excess of all the angles above the product<lb/> of 180 and 2 less than the number of angles,<lb/> To the area of the spherical polygon.<lb/> <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">That is, putting <hi rend="italics">n</hi></cell><cell cols="1" rows="1" role="data">= the number of angles,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">s</hi></cell><cell cols="1" rows="1" role="data">= sum of all the angles,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">d</hi></cell><cell cols="1" rows="1" role="data">= diam. of the sphere,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">= .78539 &c;</cell></row></table> Then the area of the Spherical Polygon. <cb/></p><p>Hence other rules might be found, similar to those for the area of the Spherical Triangle.</p><p>Hence also, the sum <hi rend="italics">s</hi> of all the angles of any Spherical Polygon, is always less than 180<hi rend="italics">n,</hi> but greater than 180 (<hi rend="italics">n</hi> - 2), that is less than <hi rend="italics">n</hi> times 2 right angles, but greater than <hi rend="italics">n</hi> - 2 times 2 right angles.</p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Astronomy,</hi> that part of astronomy which considers the universe such as it appears to the eye. See <hi rend="smallcaps">Astronomy.</hi></p><p>Under Spherical Astronomy, then, come all the phenomena and appearances of the heavens and heavenly bodies, such as we perceive them, without any enquiry into the reason, the theory, or truth of them. By which it is distinguished from theorical astronomy, which considers the real structure of the universe, and the causes of those phenomena.</p><p>In the Spherical Astronomy, the world is conceived to be a concave Spherical surface, in whose centre is the earth, or rather the eye, about which the visible frame revolves, with stars and planets fixed in the circumference of it. And on this supposition all the other phenomena are determined.</p><p>The theorical astronomy teaches us, from the laws of optics, &c, to correct this Scheme and reduce the whole to a juster system.</p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Compasses.</hi> See <hi rend="smallcaps">Compasses.</hi></p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Geometry,</hi> the doctrine of the sphere; particularly of the circles described on its surface, with the method of projecting the same on a plane; and measuring their arches and angles when projected.</p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Numbers.</hi> See <hi rend="smallcaps">Circular</hi> <hi rend="italics">Numbers.</hi></p><p><hi rend="smallcaps">Spherical</hi> <hi rend="italics">Trigonometry.</hi> See <hi rend="italics">Spherical</hi> T<hi rend="smallcaps">RIGONOMETRY.</hi></p></div1><div1 part="N" n="SPHERICITY" org="uniform" sample="complete" type="entry"><head>SPHERICITY</head><p>, the quality of a sphere; or that by which a thing becomes spherical or round.</p></div1><div1 part="N" n="SPHERICS" org="uniform" sample="complete" type="entry"><head>SPHERICS</head><p>, the Doctrine of the sphere, particularly of the several circles described on its surface; with the method of projecting the same on a plane. See <hi rend="smallcaps">Projection</hi> <hi rend="italics">of the Sphere.</hi></p><p><hi rend="italics">A circle of the sphere</hi> is that which is made by a plane cutting it. If the plane pass through the centre, it is a <hi rend="italics">great</hi> circle: if not, it is a <hi rend="italics">little</hi> circle.</p><p>The <hi rend="italics">pole</hi> of a circle, is a point on the surface of the sphere equidistant from every point of the circumference of the circle. Hence every circle has two poles, which are diametrically opposite to each other; and all circles that are parallel to each other have the same poles. <hi rend="center"><hi rend="italics">Properties of the Circles of the Sphere.</hi></hi></p><p>1. If a sphere be cut in any manner by a plane, the section will be a circle. And a great circle when the section passes through the centre, otherwise it is a <hi rend="italics">little</hi> circle. Hence, all great circles are equal to each other: and the line of section of two great circles of the sphere, is a diameter of the sphere: and therefore two great circles intersect each other in points diametrically opposite; and make equal angles at those points; and divide each other into two equal parts; also any great circle divides the whole sphere into two equal parts.</p><p>2. If a great circle be perpendicular to any other circle, it passes through its poles. And if a great circle <pb n="481"/><cb/> pass through the pole of any other circle, it cuts it at right angles, and into two equal parts.</p><p>3. The distance between the poles of two circles, is equal to the angle of their inclination.</p><p>4. Two great circles passing through the poles of another great circle, cut all the parallels to this latter into similar arcs. Hence, an angle made by two great circles of the sphere, is equal to the angle of inclination of the planes of these great circles. And hence also the lengths of those parallels are to one another as the sines of their distances from their common pole, or as the cosines of their distances from their parallel great circle. Consequently, as radius is to the cosine of the latitude of any point on the globe, so is the length of a degree at the equator, to the length of a degree in that latitude.</p><p>5. If a great circle pass through the poles of another; this latter also passes through the poles of the former; and the two cut each other perpendicularly.</p><p>6. If two or more great circles intersect each other in the poles of another great circle; this latter will pass through the poles of all the former.</p><p>7. All circles of the sphere that are equally distant from the centre, are equal; and the farther they are distant from the centre, the less they are.</p><p>8. The shortest distance on the surface of a sphere, between any two points on that surface, is the arc of a great circle passing through those points. And the smaller the circle is that passes through the same points, the longer is the arc of distance between them. Hence the proper measure, or distance, of two places on the surface of the globe, is an arc of a great circle intercepted between the same. See Theodosius and other writers on Spherics.</p></div1><div1 part="N" n="SPHEROID" org="uniform" sample="complete" type="entry"><head>SPHEROID</head><p>, a solid body approaching to the figure of a sphere, though not exactly round, but having one of its diameters longer than the other.</p><p>This solid is usually considered as generated by the rotation of an oval plane figure about one of its axes. If that be the longer or transverse axis, the solid so generated is called an <hi rend="italics">oblong</hi> Spheroid, and sometimes <hi rend="italics">prolate,</hi> which resembles an egg, or a lemon; but if the oval revolve about its shorter axis, the solid will be an <hi rend="italics">oblate</hi> Spheroid, which resembles an orange, and in this shape also is the figure of the earth, and the other planets. <figure/></p><p>The axis about which the oval revolves, is called the <hi rend="italics">fixed</hi> axis, as AB; and the other CD is the <hi rend="italics">revolving</hi> axis: whichever of them happens to be the longer.</p><p>When the revolving oval is a perfect ellipse, the so- <cb/> lid generated by the revolution is properly called an <hi rend="italics">ellipsoid,</hi> as distinguished from the Spheroid, which is generated from the revolution of any oval whatever, whether it be an ellipse or not. But generally speaking, in common acceptation, the term Spheroid is used for an ellipsoid; and therefore, in what follows, they are considered as one and the same thing.</p><p>Any section of a Spheroid, by a plane, is an ellipse (except the sections perpendicular to the fixed axe, which are circles); and all parallel sections are similar ellipses, or having their transverse and conjugate axes in the same constant ratio; and the sections parallel to the fixed axe are similar to the ellipse from which the solid was generated. See my Mensuration pa. 267 &c, 2d edit.</p><p><hi rend="italics">For the Surface of a Spheroid,</hi> whether it be oblong or oblate. Let <hi rend="italics">f</hi> denote the fixed axe, <hi rend="italics">r</hi> the revolving axe, ; then will the surface <hi rend="italics">s</hi> be expressed by the following series, using the upper signs for the oblong spheroid, and the under signs for the oblate one; viz, &c; where the signs of the terms, after the first, are all negative for the oblong Spheroid, but alternately positive and negative for the oblate one.</p><p>Hence, because the factor 4<hi rend="italics">arf</hi> is equal to 4 times the area of the generating ellipse, it appears that the surface of the oblong Spheroid is less than 4 times the generating ellipse, but the surface of the oblate Spheroid is greater than 4 times the same: while the surface of the sphere falls in between the two, being just equal to 4 times its generating circle.</p><p>Huygens, in his Horolog. Oscillat. prop. 9, has given two elegant constructions for describing a circle equal to the superficies of an oblong and an oblate Spheroid, which he says he found out towards the latter end of the year 1657. As he gave no demonstrations of these, I have demonstrated them, and also rendered them more general, by extending and adapting them to the surface of any segment or zone of the Spheroid. See my Mensuration, pa. 308 &c, 2d ed. where also are several other rules and constructions for the surfaces of Spheroids, besides those of their segments, and frustums.</p><p><hi rend="italics">Of the Solidity of a Spheroid.</hi> Every Spheroid, whether oblong or oblate, is, like a sphere, exactly equal to two-thirds of its circumscribing cylinder. So that, if <hi rend="italics">f</hi> denote the fixed axe, <hi rend="italics">r</hi> the revolving axe, and ; then 2/3 <hi rend="italics">afr</hi><hi rend="sup">2</hi> denotes the solid content of either Spheroid. Or, which comes to the same thing, if <hi rend="italics">t</hi> denote the transverse, and <hi rend="italics">c</hi> the conjugate axe of the generating ellipse; then (2/3)<hi rend="italics">ac</hi><hi rend="sup">2</hi><hi rend="italics">t</hi> is the content of the oblong Spheroid, and (2/3)<hi rend="italics">act</hi><hi rend="sup">2</hi> is the content of the oblate Spheroid. Consequently, the proportion of the former solid to the latter, is as <hi rend="italics">c</hi> to <hi rend="italics">t,</hi> or as the less axis to the greater.</p><p>Farther, if about the two axes of an ellipse be ge- <pb n="482"/><cb/> nerated two spheres and two spheroids, the four solids will be continued proportionals, and the common ratio will be that of the two axes of the ellipse; that is, as the greater sphere, or the sphere upon the greater axe, is to the oblate Spheroid, so is the oblate Spheroid to the oblong Spheroid, and so is the oblong Spheroid to the less sphere, and so is the transverse axis to the conjugate. See my Mensuration, pa. 327 &c, 2d ed. where may be seen many other rules for the solid contents of Spheroids, and their various parts. See also Archimedes on Spheroids and Conoids.</p><p>Dr. Halley has demonstrated, that in a sphere, Mercator's nautical meridian line is a scale of logarithmic tangents of the half complements of the latitudes. But as it has been found that the shape of the earth is spheroidal, this figure will make some alteration in the numbers resulting from Dr. Halley's theorem. Maclaurin has therefore given a rule, by which the meridional parts to any Spheroid may be found with the same exactness as in a sphere. There is also an ingenious tract by Mr. Murdoch on the same subject. See Philos. Trans. No. 219. Mr. Cotes has also demonstrated the same proposition, Harm, Mens. pa. 20, 21. See <hi rend="smallcaps">Meridional</hi> <hi rend="italics">Parts.</hi></p><p><hi rend="italics">Universal</hi> <hi rend="smallcaps">Spheroid</hi>, a name given to the solid generated by the rotation of an ellipse about some other diameter, which is neither the transverse nor conjugate axis. This produces a figure resembling a heart. See my Mensuration, pa. 352, 2d ed.</p></div1><div1 part="N" n="SPINDLE" org="uniform" sample="complete" type="entry"><head>SPINDLE</head><p>, in Geometry, a solid body generated by the revolution of some curve line about its base or double ordinate AB; in op- <figure/> position to a conoid, which is generated by the rotation of the curve about its axis or absciss, perpendicular to its ordinate.</p><p>The Spindle is denominated circular, elliptic, hyperbolic, or parabolic, &c, according to the figure of its generating curve. See my Mensur. in several places.</p><div2 part="N" n="Spindle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Spindle</hi></head><p>, in Mechanics, sometimes denotes the axis of a wheel, or roller, &c; and its ends are the pivots.</p><p>See also <hi rend="italics">Double</hi> <hi rend="smallcaps">Cone.</hi></p></div2></div1><div1 part="N" n="SPIRAL" org="uniform" sample="complete" type="entry"><head>SPIRAL</head><p>, in Geometry, a curve line of the circular kind, which, in its progress, recedes always more and more from a point within, called its centre; as in winding from the vertex of a cone down to its base.</p><p>The first treatise on a Spiral is by Archimedes, who thus describes it: Divide the circumference of a circle A<hi rend="italics">pp</hi> &c into any number of equal parts, by a continual bisection at the points <hi rend="italics">pp</hi> &c. Divide also the radius AC into the same number of equal parts, and make C<hi rend="italics">m,</hi> C<hi rend="italics">m,</hi> C<hi rend="italics">m,</hi> &c, equal to 1, 2, 3, &c of these equal parts; then a line drawn, with a steady hand, drawn through all the points <hi rend="italics">m, m, m,</hi> &c, will trace out the Spiral.</p><p>This is more particularly called the <hi rend="italics">first</hi> Spiral, when it has made one complete revolution to the point A; and the space included between the Spiral and the radius CA, is the <hi rend="italics">Spiral space.</hi></p><p>The first Spiral may be continued to a <hi rend="italics">second,</hi> by describing another circle with double the radius of the <cb/> first; and the second may be continued to a <hi rend="italics">third,</hi> by a third circle; and so on. <figure/></p><p>Hence it follows, that the parts of the circumference A<hi rend="italics">p</hi> are as the parts of the radii C<hi rend="italics">m;</hi> or A<hi rend="italics">p</hi> is to the whole circumference, as C<hi rend="italics">m</hi> is to the whole radius. Consequently, if <hi rend="italics">c</hi> denote the circumference, <hi rend="italics">r</hi> the radius, then there arises this proportion <hi rend="italics">r</hi> : <hi rend="italics">c</hi> :: <hi rend="italics">x</hi> : <hi rend="italics">y,</hi> which gives for the equation of this Spiral; and which therefore it has in common with the quadratrix of Dinostrates, and that of Tschirnhausen: so that will serve for infinite Spirals and quadratrices. See Q<hi rend="smallcaps">UADRATRIX.</hi></p><p>The Spiral may also be conceived to be thus generated, by a continued uniform motion. If a right line, as AB (<hi rend="italics">last fig. above</hi>) having one end moveable about a fixed point at B, be uniformly turned round, so as the other end A may describe the circumference of a circle; and at the same time a point be conceived to move uniformly forward from B towards A, in the right line or radius AB, so that the point may describe that line, while the line generates the circle; then will the point, with its two motions, describe the curve B, 1, 2, 3, 4, 5, &c, of the same Spiral as before.</p><p>Again, if the point B be conceived to move twice as slow as the line AB, so that it shall get but half way along BA, when that line shall have formed the circle; and if then you imagine a new revolution to be made of the line carrying the point, so that they shall end their motion at last together, there will be formed a <hi rend="italics">double</hi> Spiral line, as in the last figure. From the manner of this description may easily be drawn these corollaries:</p><p>1. That the lines B12, B11, B10, &c, making equal angles with the first and second Spiral (as also B12, B10, B8), &c, are in arithmetical progression.</p><p>2. The lines B7, B10, &c, drawn any how to the first Spiral, are to one another as the arcs of the circle intercepted between BA and those lines; because whatever parts of the circumference the point A describes, as suppose 7, the point B will also have run over 7 parts of the line AB.</p><p>3. Any lines drawn from B to the second Spiral, as B18, B22, &c, are to each other as the aforesaid arcs, together with the whole circumference added on both sides: for at the same time that the point A runs over 12, or the whole circumference, or perhaps 7 parts more, shall the point B have run over 12, and 7 parts of the line AB, which is now supposed to the divided into 24 equal parts. <pb n="483"/><cb/></p><p>4. The first Spiral line is equal to half the circumference of the first circle; for the radii of the sectors, and consequently of the arcs, are in a simple arithmetic progression, while the circumference of the circle contains as many arcs equal to the greatest; therefore the circumference is in proportion to all those Spiral arcs, as 2 to 1.</p><p>5. The first Spiral space is equal to 1/3 of the first or circumscribing circle. That is, the area CABDE of the Spiral, is equal to 1/3 part of the circle described with the radius CE. In like manner, the whole Spiral area, generated by the ray drawn from the point C to the curve, when it makes two revolutions, is 2/3 of the circle described with the radius 2CE. <figure/></p><p>And, generally, the whole area generated by the ray from the beginning of the motion, till after any number <hi rend="italics">n</hi> of revolutions, is equal to n/3 of the circle whose radius is <*> X CE, that is equal to the 3d part of the space which is the same multiple of the circle described with the greatest ray, as the number of revolutions is of unity.</p><p>In like manner also, any sector or portion of the area of the Spiral, terminated by the curve C<hi rend="italics">m</hi>A and the right line CA, is equal to 1/3 of the circular sector CAG terminated by the right lines CA and CG, this latter being the situation of the revolving ray when the point that describes the curve sets out from C. See Maclaurin's Flux. Introd. pa. 30, 31. Se also Q<hi rend="smallcaps">UADRATURE</hi> of the Spiral of Archimedes.</p><div2 part="N" n="Spiral" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Spiral</hi></head><p>, <hi rend="italics">Logistic,</hi> or <hi rend="italics">Logarithmic.</hi> See <hi rend="smallcaps">Logistic</hi> and <hi rend="smallcaps">Quadrature.</hi></p><p><hi rend="smallcaps">Spiral</hi> <hi rend="italics">of Pappus,</hi> a Spiral formed on the surface of a sphere, by a motion similar to that by which the Spiral of Archimedes is described on a plane. This Spiral is so called from its inventor Pappus. Collect. Mathem. lib. 4 prop. 30. Thus, if C be the centre of the sphere, ARBA a great circle, P its pole; and while the quadrant PMA revolves about the pole P with an uniform motion, if a point proceeding from P move with a given velocity along the quadrant, it will trace upon the spherical surface the Spiral PZF<hi rend="italics">a.</hi></p><p>Now if we suppose the quadrant PMA to make a complete revolution in the same time that the point, which traces the Spiral on the surface of the sphere, describes the quadrant, which is the case considered by Pappus; then the portion of the spherical surface terminated by the whole Spiral, and the circle ARBA, and the quadrant PMA, will be equal to the square of the diameter AB. In any other case, the area PMA<hi rend="italics">a</hi> FZP is to the square of that diameter AB, as <cb/> the arc A<hi rend="italics">a</hi> is to the whole circumference ARBA. And this area is always to the spherical triangle PA<hi rend="italics">a,</hi> as a square is to its circumscribing circle, or as the diameter of a circle is to half its circumference, or as 2 is to 3.14159 &c. See Maclaurin's Fluxions, Introd. pa. 31—33.</p><p>The portion of the spherical surface, terminated by the quadrant PMA, with the arches AR, FR, and the spiral PZF, admits of a perfect quadrature, when the ratio of the arch A<hi rend="italics">a</hi> to the whole circumference can be assigned. See Maclaurin, ibid. pa. 33.</p><p><hi rend="italics">Parabolic</hi> <hi rend="smallcaps">Spiral.</hi> See <hi rend="smallcaps">Helicoid.</hi></p><p><hi rend="italics">Proportional</hi> <hi rend="smallcaps">Spiral</hi>, is generated by supposing the radius to revolve uniformly, and a point from the circumference to move towards the centre with a motion decreasing in geometrical progression. See L<hi rend="smallcaps">OGISTIC.</hi></p><p>From the nature of a decreasing geometrical progression, it is easy to conceive that the radius CA may be continually divided; and although each successive division becomes shorter than the next preceding one, yet there must be an infinite number of divisions or terms before the last of them become of no finite magnitude. Whence it follows, that this Spiral winds continually round the centre, without ever falling into it in any finite number of revolutions. <figure/></p><p>It is also evident that any Proportional Spiral cuts the intercepted radii at equal angles: for if the divisions A<hi rend="italics">d, de, ef, fg,</hi> &c, of the circumference be very small, the several radii will be so close to one another, that the intercepted parts AD, DE, EF, FG, &c, of the Spiral may be taken as right lines; and the triangles CAD, CDE, CEF, &c, will be similar, having equal angles at the point C, and the sides about those angles proportional; therefore the angles at A, D, E, F, &c, are equal, that is, the spiral cuts the radii at equal angles. Robertson's Elem. of Navig. book 2, pa. 87.</p><p>Proportional Spirals are such Spiral lines as the rhumb lines on the terraqueous globe; which, because they make equal angles with every meridian, must also make equal angles with the meridians in the stereographic projection on the plane of the equator, and therefore will be, as Dr. Halley observes, Proportional Spirals about the polar point. From whence he demonstrates, that the meridian line is a scale of log. tangents <pb n="484"/><cb/> of the half complements of the latitudes. See <hi rend="smallcaps">Rhumb, Loxodromy</hi>, and <hi rend="smallcaps">Meridional</hi> <hi rend="italics">Parts.</hi></p><p><hi rend="smallcaps">Spiral</hi> <hi rend="italics">Pump.</hi> See <hi rend="italics">Archimedes</hi>'s <hi rend="smallcaps">Screw.</hi></p></div2><div2 part="N" n="Spiral" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Spiral</hi></head><p>, in Architecture and Sculpture, denotes a curve that ascends, winding about a cone, or spire, so that all the points of it continually approach the axis.</p><p>By this it is distinguished from the Helix, which winds in the same manner about a cylinder.</p></div2></div1><div1 part="N" n="SPORADES" org="uniform" sample="complete" type="entry"><head>SPORADES</head><p>, in Astronomy, a name by which the ancients distinguished such stars as were not included in any constellation. These the moderns more usually call <hi rend="italics">unformed,</hi> or <hi rend="italics">extraconstellary</hi> stars.</p><p>Many of the Sporades of the ancients have been since formed into new constellations: thus, of those between Ursa Major and Leo, Hevelius has formed a constellation named Leo Minor; and of those between Ursa Minor and Auriga, he also formed the Lynx; and of those under the tail of Ursa Minor, another called Canis Venaticus; &c.</p></div1><div1 part="N" n="SPOTS" org="uniform" sample="complete" type="entry"><head>SPOTS</head><p>, in Astronomy, are dark places observed on the disks or faces of the sun, moon, and planets.</p><p>The Spots on the sun are seldom if ever visible, except through a telescope. I have indeed met with persons whose eyes were so good that they have declared they could distinguish the solar Spots; and it is mentioned in Josephus à Costa's Natural and Moral History of the West Indies, book 1, ch. 2, before the use of telescopes, that in Peru there are Spots to be seen in the sun, which are not to be seen in Europe. See a memoir by Dr. Zach, in the Astronomical Ephemeris of the Acad. of Berlin for 1788, relating to the discoveries and unpublished papers of Thomas Harriot the celebrated algebraist. In that memoir it is shewn, for the first time, that Harriot was also an excellent astronomer, both theoretical and practical; that he made innumerable observations with telescopes from the year 1610, and, amongst them, 199 observations of the solar Spots, with their drawings, calculations, and the determinations of the sun's revolution round his axis. These Spots were also discovered near about the same time by Galileo and Scheiner. See Joh. Fabricius Phrysius De Maculis in Sole observatis & apparente eorum cum sole conversione narratio, 1611; also Galileo's Istoria e Demonstrazioni intorne alle Machie Solare e loro accidenti, 1613.</p><p>Some distinguish the Spots into Maculæ, or dark Spots; and Faculæ, or bright Spots; but there seems but little foundation for any such division. They are very changeable as to number, form, &c; and are sometimes in a multitude, and sometimes none at all. Some imagine they may become so numerous, as to hide the whole face of the Sun, or at least the greater part of it; and to this they ascribe what Plutarch-mentions, viz, that in the first year of the reign of Augustus, the sun's light was so faint and obscure, that one might look steadily at it with the naked eye. To which Kepler adds, that in 1547, the Sun appeared reddish, as when viewed through a thick mist; and hence he conjectures that the Spots in the sun are a kind of dark smoke, or clouds, floating on his surface.</p><p>Some again will have them stars, or planets, passing over the body of the sun: but others, with more probability, think they are opake bodies, in manner <cb/> of crusts, formed like the scums on the surface of liquors.</p><p>Dr. Derham, from a variety of particulars, which he has recited, concerning the solar Spots, and their congruity to what we observe in our own globe, infers, that they are caused by the eruption of some new volcano in the sun, which pouring out at first a prodigious quantity of smoke and other opake matter, causeth the Spots: and as that fuliginous matter decays and spends itself, and the volcano at last becomes more torrid and flaming, so the Spots decay and become umbræ, and at last faculæ: which faculæ he supposes to be no other than more flaming lighter parts than any other parts of the sun. Philos. Trans. vol. 23, p. 1504, or Abr. vol. 4, p. 235.</p><p>Dr. Franklin (in his Exper. and Observ. p. 266.) suggests a conjecture, that the parts of the Sun's sulphur separated by fire, rise into the atmosphere, and there being freed from the immediate action of the fire, they collect into cloudy masses, and gradually becoming too heavy to be longer supported, they descend to the sun, and are burnt over again. Hence, he says, the Spots appearing on his face, which are observed to diminish daily in size, their consuming edges being of particular brightness.</p><p>For another solution of these phenomena, see M<hi rend="smallcaps">ACULÆ.</hi> Various other accounts and hypotheses of these Spots may be seen in many of the other volumes of the Philos. Trans. In one of these, viz, vol. 57, pa. 398, Dr. Horsley attempts to determine the height of the sun's atmosphere from the height of the solar Spots above his surface.</p><p>By means of the observations of these Spots, has been determined the period of the sun's rotation about his axis, viz, by observing their periodical return.</p><p>The lunar Spots are fixed: and astronomers reckon about 48 of them on the moon's face; to each of which they have given names. The 21st, called <hi rend="italics">Tycho,</hi> is one of the most considerable.</p><p><hi rend="italics">Circular</hi> <hi rend="smallcaps">Spots</hi>, in Electricity. See <hi rend="smallcaps">Circular</hi> Spots and <hi rend="smallcaps">Colours.</hi></p><p><hi rend="italics">Lucid</hi> <hi rend="smallcaps">Spots</hi>, <hi rend="italics">in the heavens,</hi> are several little whitish Spots, that appear magnified, and more luminous when seen through telescopes; and yet without any stars in them. One of these is in Andromeda's girdle, and was first observed in 1612, by Simon Marius: it has some whitish rays near its middle, is liable to several changes, and is sometimes invisible. Another is near the ecliptic, between the head and bow of Sagittarius; it is small, but very luminous. A third is in the back of the Centaur, which is too far south to be seen in Britain. A fourth, of a smaller size, is before Antinous's right foot, having a star in it, which makes it appear more bright. A fifth is in the constellation Hercules, between the stars <*> and <foreign xml:lang="greek">h</foreign>, which is visible to the naked eye, though it is but small, when the sky is clear and the moon absent. It is probable that with more powerful telescopes these lucid Spots will be found to be congeries of very minute fixed stars.</p><p><hi rend="italics">Planetary</hi> <hi rend="smallcaps">Spots</hi>, are those of the planets. Astronomers find that the planets are not without their spots. Jupiter, Mars, and Venus, when viewed through a telescope, shew several very remarkable ones: and it is <pb n="485"/><cb/> by the motion of these Spots, that the rotation of the planets about their axes is concluded, in the same manner as that of the sun is deduced from the apparent motion of his maculæ.</p></div1><div1 part="N" n="SPOUT" org="uniform" sample="complete" type="entry"><head>SPOUT</head><p>, or <hi rend="italics">Water</hi> <hi rend="smallcaps">Spout</hi>, an extraordinary meteor, or appearance, consisting of a moving column or pillar of water; called by the Latins <hi rend="italics">typho,</hi> and <hi rend="italics">sipho;</hi> and by the French <hi rend="italics">trompe,</hi> from its shape, which resembles a speaking trumpet, the widest end uppermost.</p><p>Its first appearance is in form of a deep cloud, the upper part of which is white, and the lower black. From the lower part of this cloud there hangs, or rather falls down, what is properly called the Spout, in manner of a conical tube, largest at top. Under this tube is always a great boiling and flying up of the water of the sea, as in a jet d'eau. For some yards above the surface of the sea, the water stands as a column, or pillar; from the extremity of which it spreads, and goes off, as in a kind of smoke. Frequently the cone descends so low as to the middle of this column, and continues for some time contiguous to it; though sometimes it only points to it at some distance, either in a perpendicular, or in an oblique line.</p><p>Frequently it can scarce be distinguished, whether the cone or the column appear the first, both appearing all of a sudden against each other. But sometimes the water boils up from the sea to a great height, without any appearance of a Spout pointing to it, either perpendicularly or obliquely. Indeed, generally, the boiling or flying up of the water has the priority, this always preceding its being formed into a column. For the most part the cone does not appear hollow till towards the end, when the sea water is violently thrown up along its middle, as smoke up a chimney: soon after this, the Spout or canal breaks and disappears; the boiling up of the water, and even the pillar, continuing to the last, and for some time afterwards; sometimes till the Spout form itself again, and appear anew, which it will do several times in a quarter of an hour. See a description of several Water-Spouts by Mr. Gordon, and by Dr. Stuart, in Phil. Trans. Abr. vol. iv, pa. 103 &c.</p><p>M. de la Pryme, from a near observation of two or three Spouts in Yorkshire, described in the Philosophical Transactions, num. 281, or Abr. vol. iv, pa. 106, concludes, that the Water Spout is nothing but a gyration of clouds by contrary winds meeting in a point, or centre; and there, where the greatest condensation and gravitation is, falling down into a pipe, or great tube, somewhat like Archimedes's spiral screw; and, in its working and whirling motion, absorbing and raising the water, in the same manner as the spiral screw does; and thus destroying ships &c.</p><p>Thus, June the 21st, he observed the clouds mightily agitated above, and driven together; upon which they became very black, and were hurried round; whence proceeded a most audible whirling noise like that usually heard in a mill. Soon after there issued a long tube, or Spout, from the centre of the congregated clouds, in which he observed a spiral motion, like that of a screw, by which the water was raised up.</p><p>Again, August 15, 1687, the wind blowing at the <cb/> same time out of the several quarters, created a great vortex and whirling among the clouds, the centre of which every now and then dropt down, in shape of a long thin black pipe, in which he could distinctly behold a motion like that of a screw, continually drawing upwards, and screwing up, as it were, wherever it touched.</p><p>In its progress it moved slowly over a grove of trees, which bent under it like wands, in a circular motion. Proceeding, it tore off the thatch from a barn, bent a huge oak tree, broke one of its greatest limbs, and threw it to a great distance. He adds, that whereas it is commonly said, the water works and rises in a column, before the tube comes to touch it, this is doubtless a mistake, owing to the fineness and transparency of the tubes, which do most certainly touch the surface of the sea, before any considerable motion can be raised in it; but which do not become opake and visible, till after they have imbibed a considerable quantity of water.</p><p>The dissolution of Water-Spouts he ascribes to the great quantity of water they have glutted: which, by its weight, impeding their motion, upon which their force, and even existence depends, they break, and let go their contents; which use to prove fatal to whatever is found underneath.</p><p>A notable instance of this may be seen in the Philosophical Transactions (num. 363, or Abr. vol iv. pa 108) related by Dr. Richardson. A Spout, in 1718, breaking on Emmotmoor, nigh Coln, in Lancashire, the country was immediately overflowed; a brook, in a few minutes, rose fix feet perpendicularly high; and the ground upon which the Spout fell, which was 66 feet over, was torn up to the very rock, which was no less than 7 feet deep; and a deep gulf was made for above half a mile, the earth being raised in vast heaps on each side. See a description and figure of a WaterSpout, with an attempt to account for it in Franklin's Exp. and Obs. pa. 226, &c.</p><p>Signor Beccaria has taken pains to show that Water-Spouts have an electrical origin. To make this more evident, he first describes the circumstances attending their appearance, which are the following.</p><p>They generally appear in calm weather. The sea seems to boil, and to send up a smoke under them, rising in a hill towards the Spout. At the same time, persons who have been near them have heard a rumbling noise. The form of a Water-Spout is that of a speaking trumpet, the wider end being in the clouds, and the narrower end towards the sea.</p><p>The size is various, even in the same Spout. The colour is sometimes inclining to white, and sometimes to black. Their position is sometimes perpendicular to the sea, sometimes oblique; and sometimes the Spout itself is in the form of a curve. Their continuance is very various, some disappearing as soon as formed, and some continuing a considerable time. One that he had heard of continued a whole hour. But they often vanish, and presently appear again in the same place. The very same things that Water-Spouts are at sea, are some kinds of whirlwinds and hurricanes by land. They have been known to tear up trees, to throw down buildings, and make caverns in the earth; and in all these cases, to scatter earth, bricks, stones, timber, &c, <pb n="486"/><cb/> to a great distance in every direction. Great quantities of water have been left, or raised by them, so as to make a kind of deluge; and they have always been attended by a prodigious rumbling noise.</p><p>That these phenomena depend upon electricity cannot but appear very probable from the nature of several of them; but the conjecture is made more probable from the following additional circumstances. They generally appear in months peculiarly subject to thunder-storms, and are commonly preceded, accompanied, or followed by lightning, rain, or hail, the previous state of the air being similar. Whitish or yellowish flashes of light have sometimes been seen moving with prodigious swiftness about them. And lastly, the manner in which they terminate exactly resembles what might be expected from the prolongation of one of the uniform protuberances of electrified clouds, mentioned before, towards the sea; the water and the cloud mutually attracting one another: for they suddenly contract themselves, and disperse almost at once; the cloud rising, and the water of the sea under it falling to its level. But the most remarkable circumstance, and the most favourable to the supposition of their depending on electricity, is, that they have been dispersed by presenting to them sharp pointed knives or swords. This, at least, is the constant practice of mariners, in many parts of the world, where these Water-Spouts abound, and he was assured by several of them, that the method has often been undoubtedly effectual.</p><p>The analogy between the phenomena of Water Spouts and electricity, he says, may be made visible, by hanging a drop of water to a wire communicating with the prime conductor, and placing a vessel of water under it. In these circumstances, the drop assumes all the various appearances of a Water Spout, both in its rise, form, and manner of disappearing. Nothing is wanting but the smoke, which may require a great force of electricity to become visible.</p><p>Mr. Wilcke also considers the Water-Spout as a kind of great electrical cone, raised between the cloud strongly electrified, and the sea or the earth, and he relates a very remarkable appearance which occurred to himself, and which strongly confirms his supposition. On the 20th of July 1758, at three o'clock in the afternoon, he observed a great quantity of dust rising from the ground, and covering a field, and part of the town in which he then was. There was no wind, and the dust moved gently towards the east, where appeared a great black cloud, which, when it was near its zenith, electrified his apparatus positively, and to as great a degree as ever he had observed it to be done by natural electricity. This cloud passed his zenith, and went gradually towards the west, the dust then following it, and continuing to rise higher and higher till it composed a thick pillar, in the form of a sugar-loaf, and at length seemed to be in contact with the cloud. At some distance from this, there came, in the same path, another great cloud, together with a long stream of smaller clouds, moving faster than the preceding. These clouds electrified his apparatus negatively, and when they came near the positive cloud, a flash of lightning was seen to dart through the cloud of dust, the positive cloud, the large negative cloud, and, as far as the eye could distinguish, the whole train of smaller negative clouds <cb/> which followed it. Upon this, the negative clouds spread very much, and dissolved in rain, and the air was presently clear of all the dust. The whole appearance lasted not above half an hour. See Priestley's Electr. vol. 1, pa. 438, &c.</p><p>This theory of Water-Spouts has been farther confirmed by the account which Mr. Forster gives of one of them, in his Voyage Round the World, vol. 1, pa. 191, &c. On the coast of New Zealand he had an opportunity of seeing several, one of which he has particularly described. The water, he says, in a space of fifty or sixty fathoms, moved towards the centre, and there rising into vapour, by the force of the whirling motion, ascended in a spiral form towards the clouds. Directly over the whirlpool, or agitated spot in the sea, a cloud gradually tapered into a long slender tube, which seemed to descend to meet the rising spiral, and soon united with it into a straight column of a cylindrical form. The water was whirled upwards with the greatest violence in a spiral, and appeared to leave a hollow space in the centre; so that the water seemed to form a hollow tube, instead of a solid column; and that this was the case, was rendered still more probable by the colour, which was exactly like that of a hollow glass tube. After some time, this last column was incurvated, and broke like the others; and the appearance of a flash of lightning which attended its disjunction, as well as the hail stones which fell at the time, seemed plainly to indicate, that WaterSpouts either owe their formation to the electric matter, or, at least, that they have some connection with it.</p><p>In Pliny's time, the seamen used to pour vinegar into the sea, to assuage and lay the Spout when it approached them: our modern seamen think to keep it off, by making a noise with filing and scratching violently on the deck; or by discharging great guns to disperse it.</p><p>See the figure of a Water-Spout, fig. 1, plate 27.</p></div1><div1 part="N" n="SPRING" org="uniform" sample="complete" type="entry"><head>SPRING</head><p>, in Natural History, a fountain or source of water, rising out of the ground.</p><p>The most general and probable opinion among philosophers, on the formation of Springs, is, that they are owing to rain. The rain-water penetrates the earth till such time as it meets a clayey soil, or stratum; which proving a bottom sufficiently solid to sustain and stop its descent, it glides along it that way to which the earth declines, till, meeting with a place or aperture on the surface, through which it may escape, it forms a Spring, and perhaps the head of a stream or brook.</p><p>Now, that the rain is sufficient for this effect, appears from hence, that upon calculating the quantity of rain and snow which falls yearly on the tract of ground that is to furnish, for instance, the water of the Seine, it is found that this river does not take up above onesixth part of it.</p><p>Springs commonly rise at the bottom of mountains; the reason is, that mountains collect the most waters, and give them the greatest descent the same way. And if we sometimes see Springs on high grounds, and even on the tops of mountains, they must come from other remoter places, considerably higher, along beds of clay, or clayey ground, as in their natural channels. So that if there happen to be a valley between a mountain on whose top is a Spring, and the mountain which is to <pb n="487"/><cb/> furnish it with water, the Spring must be considered as water conducted from a reservoir of a certain height, through a subterraneous channel, to make a jet of an almost equal height.</p><p>As to the manner in which this water is collected, so as to form reservoirs for the different kinds of Springs, it seems to be this: the tops of mountains usually abound with cavities and subterraneous caverns, formed by nature to serve as reservoirs; and their pointed summits, which seem to pierce the clouds, stop those vapours which float in the atmosphere; which being thus condensed, they precipitate in water, and by their gravity and fluidity easily penetrate through beds of sand and the lighter earth, till they become stopped in their descent by the denser strata, such as beds of clay, stone, &c, where they form a bason or cavern, and working a passage horizontally, or a little declining, they issue out at the sides of the mountains. Many of these Springs discharge water, which running down between the ridges of hills, unite their streams, and form rivulets or brooks, and many of these uniting again on the plain, become a river.</p><p>The perpetuity of divers Springs, always yielding the same quantity of water, equally when the least rain or vapour is afforded as when they are the greatest, furnish, in the opinion of some, considerable objections to the universality or sufficiency of the theory above. Dr. Derham mentions a Spring in his own parish of Upminster, which he could never perceive by his eye was diminished in the greatest droughts, even when all the ponds in the country, as well as an adjoining brook, had been dry for several months together; nor ever to be increased in the most rainy seasons, excepting perhaps for a few hours, or at most for a day, from sudden and violent rains. Had this Spring, he thought, derived its origin from rain or vapours, there would be found an increase and decrease of its water corresponding to those of its causes; as we actually find in such temporary Springs, as have undoubtedly their rise from rain and vapour.</p><p>Some naturalists therefore have recourse to the sea, and derive the origin of Springs immediately from thence. But how the sea-water should be raised up to the surface of the earth, and even to the tops of the mountains, is a difficulty, in the solution of which they cannot agree. Some fancy a kind of hollow subterranean rocks to receive the watery vapours raised from channels communicating with the sea, by means of an internal fire, and to act the part of alembics, in freeing them from their saline particles, as well as condensing and converting them into water. This kind of subterranean laboratory, serving for the distillation of seawater, was the invention of Des Cartes: see his Princip. part 4, § 64. Others, as De la Hire &c (Mem. de l'Acad. 1703) set aside the alembics, and think it enough that there be large subterranean reservoirs of water at the height of the sea, from whence the warmth of the bottom of the earth, &c, may raise vapours; which pervade not only the intervals and fissures of the strata, but the bodies of the strata themselves, and at length arrive near the surface; where, being condensed by the cold, they glide along on the first bed of clay they meet with, till they issue forth by some aperture in the ground. De la Hire adds, that the salts of stones and minerals may contribute to the de- <cb/> taining and fixing the vapours, and converting them into water. Farther, it is urged by some, that there is a still more natural and easy way of exhibiting the rise of the sea-water up into mountains &c, viz, by putting a little heap of sand, or ashes, or the like, into a bason of water; in which case the sand &c will represent the dry land, or an island; and the bason of water, the sea about it. Here, say they, the water in the bason will rise to the top of the heap, or nearly so, in the same manner, and from the same principle, as the waters of the sea, lakes, &c, rise in the hills. The principle of ascent in both is accordingly supposed to be the same with that of the ascent of liquids in capillary tubes, or between contiguous planes, or in a tube filled with ashes; all which are now generally accounted for by the doctrine of attraction.</p><p>Against this last theory, Perrault and others have urged several unanswerable objections. It supposes a variety of subterranean passages and caverns, communicating with the sea, and a complicated apparatus of alembics, with heat and cold, &c, of the existence of all which we have no sort of proof. Besides, the water that is supposed to ascend from the depths of the sea, or from subterranean canals proceeding from it, through the porous parts of the earth, as it rises in capillary tubes, ascends to no great height, and in much too small a quantity to furnish springs with water, as Perrault has sufficiently shewn. And though the sand and earth through which the water ascends may acquire some saline particles from it, they are nevertheless incapable of rendering it so fresh as the water of our fountains is generally found to be. Not to add, that in process of time the saline particles of which the water is deprived, either by subterranean distillation or filtration, must clog and obstruct those canals and alembics, by which it is supposed to be conveyed to our Springs, and the sea must likewise gradually lose a considerable quantity of its salt.</p><p><hi rend="italics">Different sorts of</hi> <hi rend="smallcaps">Springs.</hi> Springs are either such as run continually, called perennial; or such as run only for a time, and at certain seasons of the year, and therefore called <hi rend="italics">temporary</hi> Springs. Others again are called <hi rend="italics">intermitting</hi> Springs, because they flow and then stop, aud flow and stop again; and <hi rend="italics">reciprocating</hi> Springs, whose waters rise and fall, or flow and ebb, by regular intervals.</p><p>In order to account for these differences in Springs, let ABCDE (fig. 2, pl. 27) represent the declivity of a hill, along which the rain descends; passing through the fissures or channels BF, CG, DH, and LK, into the cavity or reservoir FGHKMI; from this cavity let there be a narrow drain or duct KE, which discharges the water at E. As the capacity of the reservoir is supposed to be large in proportion to that of the drain, it will furnish a constant supply of water to the spring at E. But if the reservoir FGHKMI be small, and the drain large, the water contained in the former, unless it is supplied by rain, will be wholly discharged by the latter, and the Spring will become dry: and so it will continue, even though it rains, till the water has had time to penetrate through the earth, or to pass through the channels into the reservoir; and the time necessary for furnishing a new supply to the drain KE will depend on the size of the fissures, the na- <pb n="488"/><cb/> ture of the soil, and the depth of the cavity with which it communicates. Hence it may happen, that the Spring at E may remain dry for a considerable time, and even while it rains; but when the water has found its way into the cavity of the hill, the Spring will begin to run. Springs of this kind, it is evident, may be dry in wet weather, especially if the duct KE be not exactly level with the bottom of the cavity in the hill, and discharge water in dry weather; and the intermissions of the Spring may continue several days. But if we suppose XOP to represent another cavity, supplied with water by the channel NO, as well as by fissures and clefts in the rock, and by the draining of the adjacent earth; and another channel STV, communicating with the bottom of it at S, ascending to T, and terminating on the surface at V, in the form of a siphon; this disposition of the internal cavities of the earth, which we may reasonably suppose that nature has formed in a variety of places, will serve to explain the principle of reciprocating Springs; for it is plain, that the cavity XOP must be supplied with water to the height QPT, before it can pass over the bend of the channel at T, and then it will flow through the longer leg of the siphon TV, and be discharged at the end V, which is lower than S. Now if the channel STV be considerably larger than NO, by which the water is principally conveyed into the reservoir XOP, the reservoir will be emptied of its water by the siphon; and when the water descends below its orifice S, the air will drive the remaining water out of the channel STV, and the Spring will cease to flow. But in time the water in the reservoir will again rise to the height QPT, and be discharged at V as before. It is easy to conceive, that the diameters of the channels NO and STV may be so proportioned to one ancther, as to afford an intermission and renewal of the Spring V at regular intervals. Thus, if NO communicates with a well supplied by the tide, during the time of flow, the quantity of water conveyed by it into the cavity XOP may be sufficient to fill it up to QPT; and STV may be of such a size as to empty it, during the time of ebb. It is easy to apply this reasoning to more complicated cases, where several reservoirs and siphons communicating with each other, may supply Springs with circumstances of greater variety. See Musschenbroek's Introd. ad Phil. Nat. tom. ii. pa. 1010. Desagu. Exp. Phil. vol. ii, pa. 173, &c.</p><p>We shall here observe, that Desaguliers calls those <hi rend="italics">reciprocating</hi> Springs which flow constantly, but with a stream subject to increase and decrease; and thus he distinguishes them from <hi rend="italics">intermitting</hi> Springs, which flow or stop alternately.</p><p>It is said that in the diocese of Paderborn, in Westphalia, there is a Spring which disappears after twentyfour hours, and always returns at the end of six hours with a great noise, and with so much force, as to turn three mills, not far from its source. It is called the Bolderborn, or boisterous Spring. Phil. Trans. num. 7, pa. 127.</p><p>There are many Springs of an extraordinary nature in our own country, which it is needless to recite, as they are explicable by the general principles already illustrated.</p><div2 part="N" n="Spring" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Spring</hi></head><p>, <hi rend="italics">Ver,</hi> in Astronomy and Cosmography, denotes one of the seasons of the year; commencing, in <cb/> the northern parts of the earth, on the day the sun enters the first degree of Aries, which is about the 21st day of March, and ending when the sun enters Cancer, at the summer solstice, about the 21st of June; Spring ending when the summer begins.</p><p>Or, more strictly and generally, for any part of the earth, or on either side of the equator, the Spring season begins when the meridian altitude of the sun, being on the increase, is at a medium between the greatest and least; and ends when the meridian altitude is at the greatest. Or the Spring is the season, or time, from the moment of the sun's crossing the equator till he rise to the greatest height above it.</p><p><hi rend="italics">Elater</hi> <hi rend="smallcaps">Spring</hi>, in Physics, denotes a natural faculty, or endeavour, of certain bodies, to return to their first state, after having been violently put out of the same by compressing, or bending them, or the like.</p><p>This faculty is usually called by philosophers, <hi rend="italics">elastic force,</hi> or <hi rend="italics">elasticity.</hi></p></div2><div2 part="N" n="Spring" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Spring</hi></head><p>, in Mechanics, is used to signify a body of any shape, perfectly elastic.</p><p><hi rend="italics">Elasticity of a</hi> <hi rend="smallcaps">Spring.</hi> See <hi rend="smallcaps">Elasticity.</hi></p><p><hi rend="italics">Length of a</hi> <hi rend="smallcaps">Spring</hi>, may, from its etymology, signify the length of any elastic body; but it is particularly used by Dr. Jurin to signify the greatest length to which a Spring can be forced inwards, or drawn outwards, without prejudice to its elasticity. He observes, this would be the whole length, were the Spring considered as a mathematical line; but in a material Spring, it is the difference between the whole length, when the Spring is in its natural situation, or the situation it will rest in when not disturbed by any external force, and the length or space it takes up when wholly compressed and closed, or when drawn out.</p><p><hi rend="italics">Strength or Force of a</hi> <hi rend="smallcaps">Spring</hi>, is used for the force or weight which, when the Spring is wholly compressed or closed, will just prevent it from unbending itself. Also the Force of a Spring partly bent or closed, is the force or weight which is just sufficient to keep the Spring in that state, by preventing it from unbending itself any farther.</p><p>The theory of Springs is founded on this principle, <hi rend="italics">ut intensio, sic vis;</hi> that is, the intensity is as the compressing force; or if a Spring be any way forced or put out of its natural situation, its resistance is proportional to the space by which it is removed from that situation. This principle has been verified by the experiments of Dr. Hook, and since him by those of others, particularly by the accurate hand of Mr. George Graham. Lectures De Potentia Restitutiva, 1678.</p><p>For elucidating this principle, on which the whole theory of Springs depends, suppose a Spring CL, resting at L against any immoveable support, but otherwise lying in its natural situation, and at full liberty. Then if this Spring be pressed inwards by any force <hi rend="italics">p,</hi> or from C towards L, through the space of one inch, and can be there detained by that force <hi rend="italics">p,</hi> the resistance of the Spring, and the force <hi rend="italics">p,</hi> exactly counterbalancing each other; then will the double force 2<hi rend="italics">p</hi> bend the Spring through the space of 2 inches, and the triple force 3<hi rend="italics">p</hi> through 3 inches, and the quadruple force 4<hi rend="italics">p</hi> through 4 inches, and so on. The space CL through which the Spring is bent, or by which its end C is removed from its natural situation, being al- <pb n="489"/><cb/> ways proportional to the force which will bend it so far, and will just detain it when so bent. On the other hand, if the end C be drawn outwards to any place <foreign xml:lang="greek">l</foreign>, and be there detained from returning back by any force <hi rend="italics">p,</hi> the space C<foreign xml:lang="greek">l</foreign>, through which it is so drawn outwards, will be also proportional to the force <hi rend="italics">p,</hi> which is just able to retain it in that situation. <figure/></p><p>It may here be observed, that the Spring of the air, or its elastic force, is a power of a different nature, and governed by different laws, from that of a palpable rigid Spring. For supposing the line LC to represent a cylindrical volume of air, which by compression is reduced to L<hi rend="italics">l,</hi> or by dilatation is extended to L<foreign xml:lang="greek">l</foreign>, its elastic force will be reciprocally as L<hi rend="italics">l</hi> or L<foreign xml:lang="greek">l</foreign>; whereas the force or resistance of a Spring is directly as C<hi rend="italics">l</hi> or C<foreign xml:lang="greek">l</foreign>.</p><p>This principle being premised, Dr. Jurin lays down a general theorem concerning the action of a body striking on one end of a Spring, while the other end is supposed to rest against an immoveable support.</p><p>Thus, if a Spring of the <figure/> strength P, and the length CL, lying at full liberty upon an horizontal plane, rest with one end L against an immoveable support; and a body of the weight M, moving with the velocity V, in the direction of the axis of the Spring, strike directly on the other end C, and so force the Spring inwards, or bend it through any space CB; and if a mean proportional CG be taken between (M/P) X CL and 2<hi rend="italics">a,</hi> where <hi rend="italics">a</hi> denotes the height to which a body would ascend in vacuo with the velocity V; and farther, if upon the radius R = CG be described the quadrant of a circle GFA: then,</p><p>1. When the Spring is bent through the right sine CB of any arc GF, the velocity <hi rend="italics">v</hi> of the body M is to the original velocity V, as the cosine BF is to the radius CG; that is <hi rend="italics">v</hi> : V :: BF : CG, or .</p><p>2. The time <hi rend="italics">t</hi> of bending the Spring through the same sine CB, is to T, the time of a heavy body's ascending in vacuo with the velocity V, as the corre- <cb/> sponding arc is to 2<hi rend="italics">a;</hi> that is <hi rend="italics">t</hi> : T :: GF : 2<hi rend="italics">a,</hi> or .</p><p>The doctor gives a demonstration of this theorem, and deduces a great many curious corollaries from it. These he divides into three classes. The first contains such corollaries as are of more particular use when the Spring is wholly closed before the motion of the body ceases: the second comprehends those relating to the case, when the motion of the body ceases before the Spring is wholly closed: and the third when the motion of the body ceases at the instant that the Spring is wholly closed.</p><p>We shall here mention some of the last class, as being the most simple; having first premised, that P = the strength of the Spring, L = its length, V = the initial velocity of the body closing the Spring, M = its mass, <hi rend="italics">t</hi> = time spent by the body in closing the Spring, A = height from which a heavy body will fall in vacuo in a second of time, <hi rend="italics">a</hi> = the height to which a body would ascend in vacuo with the velocity V, C = the velocity gained by the fall, <hi rend="italics">m</hi> = the circumference of a circle, whose diameter is 1. Then, the motion of the striking body ceasing when the Spring is wholly closed, it will be,</p><p>1. .</p><p>2. .</p><p>3. the first momentum.</p><p>4. If a quantity of motion MV bend a Spring through its whole length, and be destroyed by it; no other quantity of motion equal to the former, as <hi rend="italics">n</hi>M X (V/<hi rend="italics">n</hi>), will close the same Spring, and be wholly destroyed by it.</p><p>5. But a quantity of motion, greater or less than MV, in any given ratio, may close the same Spring, and be wholly destroyed in closing it; and the time spent in closing the Spring will be respectively greater or less, in the same given ratio.</p><p>6. The initial vis viva, or MV<hi rend="sup">2</hi> is = (C<hi rend="sup">2</hi>PL)/(2A); and 2<hi rend="italics">a</hi>M = PL; also the initial vis viva is as the rectangle under the length and strength of the Spring, that is, MV<hi rend="sup">2</hi> is as PL.</p><p>7. If the vis viva MV<hi rend="sup">2</hi> bend a Spring through its whole length, and be destroyed in closing it; any other vis viva, equal to the former, as <hi rend="italics">n</hi><hi rend="sup">2</hi>M X (V<hi rend="sup">2</hi>/<hi rend="italics">n</hi><hi rend="sup">2</hi>), will close the same Spring, and be destroyed by it.</p><p>8. But the time of closing the Spring by the vis viva <hi rend="italics">n</hi><hi rend="sup">2</hi>M X (V<hi rend="sup">2</hi>/<hi rend="italics">n</hi><hi rend="sup">2</hi>), will be to the time of closing it by the vis viva MV<hi rend="sup">2</hi>, as <hi rend="italics">n</hi> to 1.</p><p>9. If the vis viva MV<hi rend="sup">2</hi> be wholly consumed in closing a Spring, of the length L, and strength P; then the <pb n="490"/><cb/> vis viva <hi rend="italics">n</hi><hi rend="sup">2</hi>MV<hi rend="sup">2</hi> will be sufficient to close, 1st, Either a Spring of the length L and strength <hi rend="italics">n</hi><hi rend="sup">2</hi>P. 2d, Or a Spring of the length <hi rend="italics">n</hi>L and strength <hi rend="italics">n</hi>P. 3d, Or of the length <hi rend="italics">n</hi><hi rend="sup">2</hi>L and strength P. 4th, Or, if <hi rend="italics">n</hi> be a whole number, the number <hi rend="italics">n</hi><hi rend="sup">2</hi> of Springs, each of the length L and strength P.—It may be added, that it appears from hence, that the number of similar and equal Springs a given body in motion can wholly close, is always proportional to the squares of the velocity of that body. And it is from this principle that the chief argument, to prove that the force of a body in motion is as the square of its velocity, is deduced. See <hi rend="smallcaps">Force.</hi></p><p>The theorem given above, and its corollaries, will equally hold good, if the Spring be supposed to have been at first bent through a certain space, and by unbending itself to press upon a body at rest, and thus to drive that body before it, during the time of its expansion: only V, instead of being the initial velocity with which the body struck the Spring, will now be the final velocity with which the body parts from the Spring when totally expanded.</p><p>It may also be observed, that the theorem, &c, will equally hold good, if the Spring, instead of being pressed inward, be drawn outward by the action of the body. The like may be said, if the Spring be supposed to have been already drawn outward to a certain length, and in restoring itself draw the body after it. And lastly, the theorem extends to a Spring of any form whatever, provided L be the greatest length it can be extended to from its natural situation, and P the force which will confine it to that length. See Philos. Trans. num. 472, sect. 10, or vol. 43, art. 10.</p><p><hi rend="smallcaps">Spring</hi> is more particularly used, in the Mechanic Arts, for a piece of tempered steel, put into various machines to give them motion, by the endeavour it makes to unbend itself.</p><p>In watches, it is a fine piece of well-beaten steel, coiled up in a cylindrical case, or frame; which by stretching itself forth, gives motion to the wheels, &c.</p><p><hi rend="smallcaps">Spring</hi> <hi rend="italics">Arbor,</hi> in a Watch, is that part in the middle of the Spring-box, about which the Spring is wound or turned, and to which it is hooked at one end.</p><p><hi rend="smallcaps">Spring</hi> <hi rend="italics">Box,</hi> in a Watch, is the cylindrical case, or frame, containing within it the Spring of the watch.</p><p><hi rend="smallcaps">Spring</hi>-<hi rend="italics">Compasses.</hi> See <hi rend="smallcaps">Compasses.</hi></p><p><hi rend="smallcaps">Spring</hi> <hi rend="italics">of the Air,</hi> or its elastic force. See <hi rend="smallcaps">Air</hi>, and <hi rend="smallcaps">Elastioity.</hi></p><p><hi rend="smallcaps">Spring</hi>-<hi rend="italics">Tides,</hi> are the higher tides, about the times of the new and full moon. See <hi rend="smallcaps">Tide.</hi></p></div2><div2 part="N" n="Springy" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Springy</hi></head><p>, or <hi rend="italics">Elastic Body.</hi> See <hi rend="smallcaps">Elastic</hi> <hi rend="italics">Body.</hi></p></div2></div1><div1 part="N" n="SQUARE" org="uniform" sample="complete" type="entry"><head>SQUARE</head><p>, in Geometry, a quadrilateral figure, whose angles are right, and sides equal. Or it is an equilateral rectangle. Or an equilateral rectangular parallelogram.</p><p>A Square, and indeed any other parallelogram, is bisected by its diagonal. And the side of a Square is incommensurable to its diagonal, being in the ratio of 1 to √2. <cb/></p><p><hi rend="italics">To find the Area of a</hi> <hi rend="smallcaps">Square.</hi> Multiply the side by itself, and the product is the area. So, if the side be 10, the area is 100; and if the side be 12, the area is 144.</p><p><hi rend="smallcaps">Square</hi> <hi rend="italics">Foot,</hi> is a Square each side of which is equal to a foot, or 12 inches; and the area, or Square foot is equal to 144 square inches.</p><p><hi rend="italics">Geometrical</hi> <hi rend="smallcaps">Square</hi>, a compartment often added on the face of a quadrant, called also <hi rend="italics">Line of</hi> <hi rend="smallcaps">Shadows</hi>, and <hi rend="smallcaps">Quadrant.</hi></p><p><hi rend="italics">Gunner's</hi> <hi rend="smallcaps">Square.</hi> See <hi rend="smallcaps">Quadrant.</hi></p><p><hi rend="italics">Magic</hi> <hi rend="smallcaps">Square.</hi> See <hi rend="smallcaps">Magic</hi> <hi rend="italics">Square.</hi></p><p><hi rend="smallcaps">Square</hi> <hi rend="italics">Measures,</hi> the Squares of the lineal measures; as in the following Table of Square Measures: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Squa. Inches.</cell><cell cols="1" rows="1" role="data">Sq. Feet.</cell><cell cols="1" rows="1" role="data">Sq. Yards.</cell><cell cols="1" rows="1" role="data">Sq. Poles.</cell><cell cols="1" rows="1" role="data">S. Chs.</cell><cell cols="1" rows="1" role="data">Acres.</cell><cell cols="1" rows="1" role="data">S. Miles.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">144</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1296</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39204</cell><cell cols="1" rows="1" role="data">272 1/4</cell><cell cols="1" rows="1" role="data">30 1/4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">627264</cell><cell cols="1" rows="1" role="data">4356</cell><cell cols="1" rows="1" role="data">484</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6272640</cell><cell cols="1" rows="1" role="data">43560</cell><cell cols="1" rows="1" role="data">4840</cell><cell cols="1" rows="1" role="data">160</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4014489600</cell><cell cols="1" rows="1" role="data">27878400</cell><cell cols="1" rows="1" role="data">3097600.</cell><cell cols="1" rows="1" role="data">102400</cell><cell cols="1" rows="1" role="data">6400</cell><cell cols="1" rows="1" role="data">640</cell><cell cols="1" rows="1" role="data">1</cell></row></table></p><p><hi rend="italics">Normal</hi> <hi rend="smallcaps">Square</hi>, is an instrument, made of wood or metal, serving to describe and measure right angles; <figure/> such is ABC. It consists of two rulers or branches fastened together perpendicularly. When the two legs are moveable on a joint, it is called a bevel.</p><p>To examine whether the Square is exact or not. Describe a semicircle DBE, with any radius at pleasure; in the circumference of which apply the angle of the Square to any point as B, and the edge of one leg to one end of the diameter as D, then if the other leg pass just by the other extremity at E, the Square is true; otherwise not.</p><p><hi rend="smallcaps">Square</hi> <hi rend="italics">Number,</hi> is the product arising from a number multiplied by itself. Thus, 4 is the Square of 2, and 16 the Square of 4. The series of Square integers, is 1, 4, 9, 16, 25, 36, &c; which are the Squares of 1, 2, 3, 4, 5, 6, &c. Or the Square fractions 1/4, 4/9, 9/16, 16/25, 25/36, 36/49, &c, which are the Squares of 1/2, 2/3, 3/4, 4/5, 5/6, 6/7, &c.</p><p>A Square number is so called, either because it denotes the area of a Square, whose side is expressed by the root of the Square number; as in the annexed Square, <pb n="491"/><cb/> <figure/> which consists of 9 little squares, the side being equal to 3; or else, which is much the same thing, because the points in the number may be ranged <figure/> in the form of a Square, by making the root, or factor, the side of the Square.</p><p>Some properties of Squares are as follow: 1. Of the <table><row role="data"><cell cols="1" rows="1" role="data">Natural series of Squares,</cell><cell cols="1" rows="1" role="data">1<hi rend="sup">2</hi>, 2<hi rend="sup">2</hi>, 3<hi rend="sup">2</hi>,  4<hi rend="sup">2</hi>, &c,</cell></row><row role="data"><cell cols="1" rows="1" role="data">which are equal to</cell><cell cols="1" rows="1" role="data">1 , 4 , 9 , 16 , &c;</cell></row></table></p><p>The mean proportional <hi rend="italics">mn</hi> between any two of these Squares <hi rend="italics">m</hi><hi rend="sup">2</hi> and <hi rend="italics">n</hi><hi rend="sup">2</hi>, is equal to the less square <hi rend="italics">plus</hi> its root multiplied by the difference of the roots; or also equal to the greater square <hi rend="italics">minus</hi> its root multiplied by the said difference of the roots. That is, ; where is the difference of their roots.</p><p>2. An arithmetical mean between any two Squares <hi rend="italics">m</hi><hi rend="sup">2</hi> and <hi rend="italics">n</hi><hi rend="sup">2</hi>, exceeds their geometrical mean, by half the Square of the difference of their roots. That is .</p><p>3. Of three equidistant Squares in the Series, the geometrical mean between the extremes, is less than the middle Square by the Square of their common distance in the Series, or of the common difference of their roots. That is, ; where <hi rend="italics">m, n, p,</hi> are in arithmetical progression, the common difference being <hi rend="italics">d.</hi></p><p>4. The difference between the two adjacent <table><row role="data"><cell cols="1" rows="1" role="data">Squares <hi rend="italics">m</hi><hi rend="sup">2</hi>, and <hi rend="italics">n</hi><hi rend="sup">2</hi>, is</cell><cell cols="1" rows="1" role="data">;</cell></row><row role="data"><cell cols="1" rows="1" role="data">in like manner,</cell><cell cols="1" rows="1" role="data">, the differ-</cell></row></table> ence between the next two adjacent Squares <hi rend="italics">n</hi><hi rend="sup">2</hi> and <hi rend="italics">p</hi><hi rend="sup">2</hi>; and so on, for the next following Squares. Hence the difference of these differences, or the second difference of the Squares, is only, because ; that is, the second differences of the Squares are each the same constant number 2: therefore the first differences will be found by the continual addition of the number 2; and then the Squares themselves will be found by the continual addition of the first differences; and thus the whole series of Squares is constructed by addition only, as here below: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2d Diff.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1st Diff.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">&c.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Squares.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">&c.</cell></row></table> <cb/></p><p>And this method of constructing the table of Square numbers I sind first noticed by Peletarius, in his Algebra.</p><p>5. Another curious property, also noted by the same author, is, that the sum of any number of the cubes of the natural series 1, 2, 3, 4, &c, taken from the beginning, always makes a Square number; and that the series of Squares, so formed, have for their <table><row role="data"><cell cols="1" rows="1" role="data">roots the numbers</cell><cell cols="1" rows="1" role="data">1, 3, 6, 10, 15, 21, &c,</cell></row><row role="data"><cell cols="1" rows="1" role="data">the diffs. of which are</cell><cell cols="1" rows="1" role="data">1, 2, 3,  4,  5,  6, &c,</cell></row></table> viz, ; where <hi rend="italics">n</hi> is the number of the terms or cubes.</p><p><hi rend="smallcaps">Square</hi> <hi rend="italics">Root,</hi> a number considered as the root of a second power or Square number: or a number which multiplied by itself, produces the given number. See <hi rend="smallcaps">Extraction</hi> <hi rend="italics">of Roots,</hi> and also the article <hi rend="smallcaps">Root</hi>, where tables of Squares and roots are inserted.</p><p><hi rend="italics">T.</hi> <hi rend="smallcaps">Square</hi>, or <hi rend="italics">Tee</hi> <hi rend="smallcaps">Square</hi>, an instrument used in drawing, so called from its resemblance to the capital letter T.</p><p>This instrument consists of two straight rulers AB and <figure/> CD, fixed at right angles to each other. To which is sometimes added a third EF, moveable about the pin C, to set it to make any angle with CD.—It is very useful for drawing parallel and perpendicular lines, on the face of a smooth drawing-board.</p><p>SQUARED - <hi rend="italics">square,</hi> <hi rend="smallcaps">Squared</hi>-<hi rend="italics">cube,</hi> &c. See <hi rend="smallcaps">Power.</hi></p><p>SQUARING. See Q<hi rend="smallcaps">UADRATURE.</hi></p><p><hi rend="smallcaps">Squaring</hi> <hi rend="italics">the Circle,</hi> is the making or finding a Square whose area shall be equal to the area of a given circle.</p><p>The best mathematicians have not yet been able to resolve this problem accurately, and perhaps never will. But they can easily come to any proposed degree of approximation whatever; for instance, so near as not to err so much in the area, as a grain of sand would cover, in a circle whose diameter is equal to that of the orbit of Saturn. The following proportion is near enough the truth for any real use, viz, as 1 is to .88622692, so is the diameter of any circle, to the side of the square of an equal area. Therefore, if the diameter of the circle be called <hi rend="italics">d,</hi> and the side of the equal square <hi rend="italics">s;</hi> .</p><p>See <hi rend="smallcaps">Circle, Diameter</hi>, and <hi rend="smallcaps">Quadrature.</hi> <pb n="492"/><cb/></p></div1><div1 part="N" n="STADIUM" org="uniform" sample="complete" type="entry"><head>STADIUM</head><p>, an ancient Greek long measure, containing 125 geometrical paces, or 625 Roman feet; corresponding to our furlong.</p><p>Eight Stadia make a geometrical or Roman mile; and 20, according to Dacier, a French league: but according to others, 800 Stadia make 41 2/3 leagues.</p><p>Guilletiere observes, that the Stadium was only 600 Athenian feet, which amount to 625 Roman, or 566 French, or 604 English feet: so that the Stadium should have been only 113 geometrical paces. It must be observed however, that the Stadium was different at different times and places.</p></div1><div1 part="N" n="STAFF" org="uniform" sample="complete" type="entry"><head>STAFF</head><p>, <hi rend="italics">Almucantar's, Augural, Back, Cross, Fore, Offset, &c.</hi> See these several articles.</p></div1><div1 part="N" n="STAR" org="uniform" sample="complete" type="entry"><head>STAR</head><p>, <hi rend="smallcaps">Stella</hi>, in Astronomy, a general name for all the heavenly bodies.</p><p>The Stars are distinguished, from the phenomena, &c, into <hi rend="italics">sived</hi> and <hi rend="italics">erratic</hi> or <hi rend="italics">wandering.</hi></p><p><hi rend="italics">Erratic</hi> or <hi rend="italics">Wandering</hi> <hi rend="smallcaps">Stars</hi>, are those which are continually changing their places and distances, with regard to each other. These are what are properly called <hi rend="italics">planets.</hi> Though to the same class may likewise be referred comets or blazing Stars.</p><p><hi rend="italics">Fixed</hi> <hi rend="smallcaps">Stars</hi>, called also barely <hi rend="italics">Stars,</hi> by way of eminence, are those which have usually been observed to keep the same distance, with regard to each other.</p><p>The chief circumstances observable in the fixed Stars, are their <hi rend="italics">distance, magnitude, number, nature,</hi> and <hi rend="italics">motion.</hi> Of each of which in their order.</p><p><hi rend="italics">Distance of the Fixed</hi> <hi rend="smallcaps">Stars.</hi> The fixed Stars are so extremely remote from us, that we have no distances in the planetary system to compare to them. Their immense distance appears from hence, that they have no sensible parallax; that is, that the diameter of the earth's annual orbit, which is nearly 190 millions of miles, bears no sensible proportion to their distance.</p><p>Mr. Huygens (Cosmotheor. lib. 4) attempts to determine the distance of the Stars, by making the aperture of a telescope so small, as that the sun through it appears no larger than Sirius; which he found to be only as 1 to 27664 of his diameter, when seen with the naked eye. So that, were the sun's distance 27664 times as much as it is, it would then be seen of the same diameter with Sirius. And hence, supposing Sirius to be a sun of the same magnitude with our sun, the distance of Sirius will be found to be 27664 times the distance of the sun, or 345 million times the earth's diameter.</p><p>Dr. David Gregory investigated the distance of Sirius, by supposing it of the same magnitude with the sun, and of the same apparent diameter with Jupiter in opposition: as may be seen at large in his Astronomy, lib. 3, prop. 47.</p><p>Cassini (Mem. Acad. 1717), by comparing Jupiter and Sirius, when viewed through the same telescope, inferred, that the diameter of that planet was 10 times as great as that of the Star; and the diameter of Jupiter being 50″, he concluded that the diameter of Sirius was about 5″; supposing then that the real magnitude of Sirius is equal to that of the sun, and the distance of the sun from us 12000 diameters of the earth, and the apparent diameter of Sirius being to that of the sun as 1 to 384, the distance of Sirius becomes equal to 4608000 diameters of the earth. <cb/></p><p>These methods of Huygens, Gregory, and Cassini, are conjectural and precarious; both because the sun and Sirius are supposed of equal magnitude, and also because it is supposed the diameter of Sirius is determined with sufficient exactness.</p><p>Mr. Michell has proposed an enquiry into the probable parallax and magnitude of the fixed Stars, from the quantity of light which they afford us, and the peculiar circumstances of their situation. With this view he supposes, that they are, on a medium, equal in magnitude and natural brightness to the sun; and then proceeds to inquire, what would be the parallax of the sun, if he were to be removed so far from us, as to make the quantity of the light, which we should then receive from him, no more than equal to that of the sixed Stars. Accordingly, he assumes Saturn in opposition, as equal, or nearly equal in light to the brightest fixed Star. As the mean distance of Saturn from the sun is equal to about 2082 of the sun's semidiameters, the density of the sun's light at Saturn will consequently be less than at his own surface, in the ratio of the square of 2082 or 4334724 to 1: If Saturn therefore reflected all the light that falls upon him, he would be less luminous in that same proportion. And besides, his apparent diameter, in the opposition, being but about the 105th part of that of the sun, the quantity of light which we receive from him must be again diminished in the ratio of the square of 105 or 11025 to 1. Consequently, by multiplying these two numbers together, we shall have the whole of the light of the sun to that of Saturn, as the square nearly of 220,000 or 48,400,000,000 to 1. Hence, removing the sun to 220,000 times his present distance, he would still appear at least as bright as Saturn, and his whole parallax upon the diameter of the earth's orbit would be less than 2 seconds: and this must be assumed for the parallax of the brightest of the fixed Stars, upon the supposition that their light does not exceed that of Saturn.</p><p>By a like computation it may be found, that the distance, at which the sun would afford us as much light as we receive from Jupiter, is not less than 46,000 times his present distance, and his whole parallax in that case, upon the diameter of the earth's orbit, would not be more than 9 seconds; the light of Jupiter and Saturn, as seen from the earth, being in the ratio of about 22 to 1, when they are both in opposition, and supposing them to reflect equally in proportion to the whole of the light that falls upon them. But if Jupiter and Saturn, instead of reflecting the whole of the light that falls upon them, should really reflect only a part of it, as a 4th, or a 6th, which may be the case, the above distances must be increased in the ratio of 2 or 2 1/2 to 1, to make the sun's light no more than equal to theirs; and his parallax would be less in the same proportion. Supposing then that the fixed Stars are of the same magnitude and brightness with the sun, it is no wonder that their parallax should hitherto have escaped observation; since in this case it could hardly amount to 2 seconds, and probably not more than one in Sirius himself, though he had been placed in the pole of the ecliptic; and in those that appear much less luminous, as <foreign xml:lang="greek">g</foreign> Draconis, which is only of the 3d magnitude, it could hardly be expected to be sensible with such instruments as have hitherto been used. However, Mr. Mi- <pb n="493"/><cb/> chell suggests, that it is not impracticable to construct instruments capable of distinguishing even to the 20th part of a second, provided the air will admit of that degree of exactness. This ingenious writer apprehends that the quantity of light which we receive from Sirius, does not exceed the light we receive from the least fixed Star of the 6th magnitude, in a greater ratio than that of 1000 to 1, nor less than that of 400 to 1; and the smaller Stars of the 2d magnitude seem to be about a mean proportional between the other two. Hence the whole parallax of the least sixed Stars of the 6th magnitude, supposing them of the same size and native brightness with the sun, should be from about 2‴ to 3‴, and their distance from about 8 to 12 million times that of the sun: and the parallax of the smaller Stars of the 2d magnitude, upon the same supposition, should be about 12‴, and their distance about 2 million times that of the sun.</p><p>This author farther suggests, that, from the apparent situation of the Stars in the heavens, there is the greatest probability that the Stars are collected together in clusters in some places, where they form a kind of systems, whilst in others there are either few or none of them; whether this disposition be owing to their mutual gravitation, or to some other law or appointment of the Creator. Hence it may be inferred, that such double Stars, &c. as appear to consist of two or more Stars placed very near together, do really consist of Stars placed near together, and under the influence of some general law: and he proceeds to inquire whether, if the Stars be collected into systems, the sun does not likewise make one of some system, and which fixed Stars those are that belong to the same system with him.</p><p>Those Stars, he apprehends, which are found in clusters, and surrounded by many others at a small distance from them, belong probably to other systems, and not to ours. And those Stars, which are surrounded with nebulæ, are probably only very large Stars which, on account of their superior magnitude, are singly visible, while the others, which compose the remaining parts of the same system, are so small as to escape our sight. And those nebulæ in which we can discover either none or only a few Stars, even with the assistance of the best telescopes, are probably systems that are still more distant than the rest. For other particulars of this inquiry, see Philos. Trans. vol. 57, pa. 234 &c.</p><p>As the distance of the fixed Stars is best determined by their parallax, various methods have been pursued, though hitherto without success, for investigating it; the result of the most accurate observations having given us little more than a distant approximation; from which however we may conclude, that the nearest of the fixed Stars cannot be less than 40 thousand diameters of the whole annual orbit of the earth distant from us.</p><p>The method pointed out by Galileo, and attempted by Hook, Flamsteed, Molyneux, and Bradley, of taking the distances of such Stars from the zenith as pass very near it, has given us a much juster idea of the immense distance of the Stars, and furnished an approximation to their parallax, much nearer the truth, than any we had before. <cb/></p><p>Dr. Bradley assures us (Philos. Trans. num. 406, or Abr. vol. 6, pa. 162), that had the parallax amounted to a single second, or two at most, he should have perceived it in the great number of observations which he made, especially upon <foreign xml:lang="greek">g</foreign> Draconis; and that it seemed to him very probable, that the annual parallax of this Star does not amount to a single second, and consequently that it is above 400 thousand times farther from us than the sun.</p><p>But Dr. Herschel, to whose industry and ingenuity, in exploring the heavens, astronomy is already much indebted, remarks, that the instrument used on this occasion, being the same with the present zenith sectors, can hardly be allowed capable of shewing an angle of one or even two seconds, with accuracy: and besides, the Star on which the observations were made, is only a bright Star of the 3d magnitude, or a small Star of the 2d; and that therefore its parallax is probably much less than that of a Star of the first magnitude. So that we are not warrauted in inferring, that the parallax of the Stars in general does not exceed 1″, whereas those of the first magnitude may have, notwithstanding the result of Dr. Bradley's observations, a parallax of several seconds.</p><p>As to the method of zenith distances, it is liable to considerable errors, on account of refraction, the change of position of the earth's axis, arising from nutation, precession of the equinoxes, or other causes, and the aberration of light.</p><p>Dr. Herschel has proposed another method, by means of double Stars, which is free from these errors, and of such a nature, that the annual parallax, even if it should not exceed the 10th part of a second, may still become visible, and be ascertained at least much nearer than heretofore. This method, which was first proposed in an imperfect manner by Galileo, and has been also mentioned by other authors, is capable of every improvement which the telescope and mechanism of micrometers can furnish. To give a general idea of it, let O and E be two opposite points of the annual orbit, taken in the same <figure/> plane with two stars A, B, of unequal magnitudes. Let the angle AOB be observed when the earth is at O, and AEB be observed when the earth is at E. From the difference of these angles, when there is any, the parallax of the Stars may be computed, according to the theory subjoined. These two Stars ought to be as near as possible to each other, and also to differ as much in magnitude as we can find them.</p><p>This theory of the annual parallax of double Stars, with the method of computing from thence what is usually called the parallax of the fixed Stars, or of single Stars of the first magnitude, such as are nearest to us, supposes 1st, that the Stars are all about the size of the sun; and 2dly, that the difference in their apparent magnitudes, is owing to their different distances, so as that a Star of the 2d, 3d, or 4th magnitude, is 2, 3, or 4 times as far off as one of the first. These principles, which Dr. Herschel premises as po<*>lata, have so great a probability in their favour, that they will <pb n="494"/><cb/> scarcely be objected to by those who are in the least acquainted with the doctrine of chances. See Mr. Michell's Inquiry, &c. already cited. And Philos. Trans. vol. 57, pa. 234 . . . . 240. Also Dr. Halley, on the Number, Order, and Light of the fixed Stars, in the Philos. Trans. vol. 31, or Abr. vol. 6, pa. 148.</p><p>Therefore, let EO be the <figure/> whole diameter of the earth's annual orbit; and let A, B, C be three Stars situated in the ecliptic, in such a manner, that they may appear all in one line OABC when the earth is at O. Now if OA, AB, BC be equal to each other, A will be a Star of the first magnitude, B of the second, and C of the third. Let us next suppose the angle OAE, or parallax of the whole orbit of the earth, to be 1″ of a degree; then, because very small angles, having the same subtense EO, may be taken to be in the inverse ratio of the lines OA, OB, OC, &c, we shall have EBO = 1/2″, and ECO = 1/3″, &c, also because EA = AB nearly, the angle AEB = ABE = 1/2″; and because BC = 1/2 BO = 1/2 BE nearly, the angle BEC = 1/2 BCE = 1/6″, and hence AEC = 1/2 + 1/6 = 2/3″; from all which it follows that, when the earth is at E, the Stars A and B appear at 1/2″ distant from one another, the Stars A and C at 2/3″ distant, and the Stars B and C only 1/6″ distant. In like man<*> may be deduced a general expression for the parallax that will become visible in the change of distance between the two Stars, by the removal of the earth from one extreme of her orbit to the other. Let <hi rend="italics">P</hi> denote the total parallax of a fixed Star of the magnitude of the <hi rend="italics">M</hi> order, and <hi rend="italics">m</hi> the number of the order of a smaller Star, <hi rend="italics">p</hi> denoting the partial parallax to be observed by the change in the distance of a double Star; then is , which gives <hi rend="italics">P,</hi> when <hi rend="italics">p</hi> is found by observation.</p><p>For Ex. Suppose a Star of the 1st magnitude should have a small Star of the 12th magnitude near it; then will the partial parallax we are to expect to see be , or 11/12 of the total parallax of the larger Star; and if we should, by observation, find the partial parallax between two such Stars to amount to 1″, then will the total parallax . Again, if the Stars be of the 3d and 24th magnitude, the total parallax will be ; so that if by observation <hi rend="italics">p</hi> be found to be 1/10 of a second, the whole parallax <hi rend="italics">P</hi> will come out .</p><p>Farther, the Stars being still in the ecliptic, suppose <cb/> they should appear in one line, when the earth is in some other part of her orbit between E and O; then will the parallax be still expressed by the same algebraic formula, and one of the maxima will still lie at E, the other at O; but the whole effect will be divided into two parts, which will be in proportion to each other, as radius — sine to radius + sine of the Star's distance from the nearest conjunction or opposition.</p><p>When the Stars are any where out of the ecliptic, situated so as to appear in one line OABC perpendicular to EO, the maximum of parallax will still be expressed by ((<hi rend="italics">m</hi> - <hi rend="italics">M</hi>)/(<hi rend="italics">mM</hi>))<hi rend="italics">P;</hi> but there will arise another additional parallax in the conjunction and opposition, which will be to that which is found 90° before or after the sun, as the sine (<hi rend="italics">s</hi>) of the latitude of the Stars seen at O, is to radius (1); and the effect of this parallax will be divided into two parts; half of it lying on one side of the large Star, the other half on the other side of it. This latter parallax will also be compounded with the former, so that the distance of the Stars in the conjunction and opposition will then be represented by the diagonal of a parallelogram, whose sides are the two semiparallaxes; a general expression for which will be .</p><p>When the Stars are in the pole of the ecliptic, <hi rend="italics">s</hi> will be = 1, and the last formula becomes .</p><p>Again, let the Stars be at some distance, as 5″, from each other, and let them be both in the ecliptic. This case is resolvable into the first; for imagine the Star A to stand at I; then the angle AEI may be accounted equal to AOI; and as the foregoing formula, , gives us the angles AEB, AEC, we are to add AEI or 5″ to AEB, which will give IEB. In general, let the distance of the Stars be <hi rend="italics">d,</hi> and let the observed distance at E be <hi rend="italics">D;</hi> then will , and therefore the whole parallax of the annual orbit will be expressed by .</p><p>Suppose now the Stars to differ only in latitude, one being in the ecliptic, the other at some distance as 5″ north, when seen at O. This case may also be resolved by the former; for imagine the Stars B and C to be elevated at right angles above the plane of the figure, so that AOB, or AOC, may make an angle of 5″ at O; then instead of the lines OABC, EA, EB, EC, imagine them all to be planes at right angles to the figure; and it will appear that the parallax of the Stars in longitude, must be the same as if the small Star had been without latitude. And since the Stars B, C, by the motion of the earth from O to E, will not change their latitude, we shall have the following construction for finding the distance of the Stars AB and AC at E, and from thence the parallax <hi rend="italics">P.</hi> <pb n="495"/><cb/> Let the triangle <hi rend="italics">ab</hi><foreign xml:lang="greek">b</foreign> represent the <figure/> situation of the Stars; <hi rend="italics">ab</hi> is the subtense of 5″, the angle under which they are supposed to be seen at O. The quantity <hi rend="italics">b</hi><foreign xml:lang="greek">b</foreign> by the former theorem is found = ((<hi rend="italics">m</hi>-<hi rend="italics">M</hi>)/(<hi rend="italics">mM</hi>))<hi rend="italics">P,</hi> which is the partial parallax, that would have been seen by the earth's moving from O to E, if both Stars had been in the ecliptic; but, on account of the difference in latitude, it will now be represented by <hi rend="italics">a</hi><foreign xml:lang="greek">b</foreign>, the hypotenuse of the triangle <hi rend="italics">ab</hi><foreign xml:lang="greek">b</foreign>: therefore in general, putting <hi rend="italics">ab</hi> = <hi rend="italics">d,</hi> <hi rend="italics">a</hi><foreign xml:lang="greek">b</foreign> = <hi rend="italics">D,</hi> we have . Hence, <hi rend="italics">D</hi> being found by observation, and the three <hi rend="italics">d, m, M</hi> given, the total parallax is obtained.</p><p>When the Stars differ in longitude as well as latitude, this case may be resolved in the following manner. Let the triangle <hi rend="italics">ab</hi><foreign xml:lang="greek">b</foreign> represent the situation of the Stars, <hi rend="italics">ab</hi> = <hi rend="italics">d</hi> being their distance <figure/> seen at O, <hi rend="italics">a</hi><foreign xml:lang="greek">b</foreign> = <hi rend="italics">D</hi> their distance seen at E. That the change <hi rend="italics">b</hi><foreign xml:lang="greek">b</foreign>, which is produced by the earth's motion, will be truly expressed by ((<hi rend="italics">m</hi> - <hi rend="italics">M</hi>)/(<hi rend="italics">mM</hi>))<hi rend="italics">P,</hi> may be proved as before, by supposing the Star <hi rend="italics">a</hi> to have been placed at <foreign xml:lang="greek">a</foreign>. Now let the angle of position <hi rend="italics">ba</hi><foreign xml:lang="greek">a</foreign> be taken by a micrometer, or by any other method sufficiently exact; then, by resolving the triangle <hi rend="italics">ab</hi><foreign xml:lang="greek">a</foreign>, we obtain the longitudinal and latitudinal differences <hi rend="italics">a</hi><foreign xml:lang="greek">a</foreign> and <hi rend="italics">b</hi><foreign xml:lang="greek">a</foreign> of the two stars. Put <hi rend="italics">a</hi><foreign xml:lang="greek">a</foreign> = <hi rend="italics">x,</hi> <hi rend="italics">b</hi><foreign xml:lang="greek">a</foreign> = <hi rend="italics">y,</hi> and it will be <hi rend="italics">x</hi> + <hi rend="italics">b</hi><foreign xml:lang="greek">b</foreign> = <hi rend="italics">aq,</hi> whence .</p><p>If neither of the Stars should be in the ecliptic, nor have the same longitude or latitude, the last theorem will still serve to calculate the total parallax, whose maximum will lie in E. There will also arise another parallax, whose maximum will be in the conjunction and opposition, which will be divided, and lie on different sides of the large Star; but as the whole parallax is extremely small, it is not necessary to investigate every particular case of this kind; for by reason of the division of the parallax, which renders observations taken at any other time, except where it is greatest, very unfavourable, the formulæ would be of little use.</p><p>Dr. Herschel closes his account of this theory, with a general observation on the time and place where the maxima of parallax will happen. Thus, when two unequal Stars are both in the ecliptic, or, not being in the ecliptic, have equal latitudes, north or fouth, and the larger Star has most longitude, the maximum of the apparent distance will be when the sun's longitude is 90° more than the Star's, or when observed in the morning: and the minimum, when the longitude of the sun is 90° less than that of the Star, or when observed in the evening. But when the small Star has most <cb/> longitude, the maximum and minimum, as well as the time of observation, will be the reverse of the former. And when the Stars differ in latitude, this makes no alteration in the place of the maximum or minimum, nor in the time of observation; that is, it is immaterial which of the two Stars has the greater latitude. Philos. Trans. vol. 72, art. 11.</p><p>The distance of the Star <foreign xml:lang="greek">g</foreign> Draconis appears, by Bradley's observations, already recited, to be at least 400,000 times that of the sun, and the distance of the nearest fixed Star, not less than 40,000 diameters of the earth's annual orbit: that is, the distance from the earth, of the former at least 38,000,000,000,000 miles, and the latter not less than 7,600,000,000,000 miles. As these distances are immensely great, it may both be amusing, and help to a clearer and more familiar idea, to compare them with the velocity of some moving body, by which they may be measured.</p><p>The swiftest motion we know of, is that of light, which passes from the sun to the earth in about 8 minutes; and yet this would be above 6 years traversing the first space, and near a year and a quarter in passing from the nearest fixed Star to the earth. But a cannon ball, moving on a medium at the rate of about 20 miles in a minute, would be 3 million 8 hundred thousand years in passing from <foreign xml:lang="greek">g</foreign> Draconis to the earth, and 760 thousand years passing from the nearest fixed Star. Sound, which moves at the rate of about 13 miles in a minute, would be 5 million 600 thousand years traversing the former distance, and 1 million 128 thousand, in passing through the latter.</p><p>The celebrated Huygens pursued speculations of this kind so far, as to believe it not impossible, that there may be Stars at such inconceivable distances, that their light has not yet reached the earth since its creation.</p><p>Dr. Halley has also advanced, what he says seems to be a metaphysical paradox (Philos. Trans. number 364, or Abr. vol. 6, pa. 148), viz, that the number of fixed Stars must be more than finite, and some of them more than at a finite distance from others: and Addison has justly observed, that this thought is far from being extravagant, when we consider that the universe is the work of infinite power, prompted by infinite goodness, and having an infinite space to exert itself in; so that our imagination can set no bounds to it.</p><p><hi rend="italics">Magnitude of the fixed</hi> <hi rend="smallcaps">Stars.</hi> The magnitudes of the Stars appear to be very different from one another; which difference may probably arise, partly from a diversity in their real magnitude, but principally from their distances, which are different.</p><p>To the bare eye, the Stars appear of some sensible magnitude, owing to the glare of light arising from the numberless reflections from the aërial particles &c about the eye: this makes us imagine the Stars to be much larger than they would appear, if we saw them only by the few rays which come directly from them, so as to enter our eyes without being intermixed with others.</p><p>Any person may be sensible of this, by looking at a Star of the first magnitude through a long narrow tube; which, though it takes in as much of the sky as would hold a thousand such stars, scarce renders that one visible. <pb n="496"/><cb/></p><p>The Stars, on account of their apparently various sizes, have been distributed into several classes, called <hi rend="italics">magnitudes.</hi> The 1st class, or Stars of the first magnitude, are those that appear largest, and may probably be nearest to us. Next to these, are those of the 2d magnitude; and so on to the 6th, which comprehends the smallest Stars visible to the naked eye. All beyond these, that can be perceived by the help of telescopes, are called <hi rend="italics">telescopic</hi> stars. Not that all the Stars of each class appear justly of the same magnitude; there being great latitude in this respect; and those of the first magnitude appearing almost all different in lustre and size. There are also other Stars, of intermediate magnitudes, which astronomers cannot refer to one class rather than another, and therefore they place them between the two. Procyon, for instance, which Ptolomy makes of the first magnitude, and Tycho of the 2d, Flamsteed lays down as between the 1st and 2d. So that, instead of 6 magnitudes, we may say there are almost as many orders of Stars, as there are Stars; so great variations being observable in the magnitude, colour, and brightness of them.</p><p>There seems to be little chance of discovering with certainty the real size of any of the fixed Stars; we must therefore be content with an approximation, deduced from their parallax, if this should ever be found; and the quantity of light they afford us, compared with that of the sun. And to this purpose, Dr. Herschel informs us, that with a magnifying power of 6450, and by means of his new micrometer, he found the apparent diameter of <hi rend="italics">a</hi> Lyræ to be 0″.355.</p><p>The Stars are also distinguished, with regard to their situation, into <hi rend="italics">asterisms,</hi> or <hi rend="italics">constellations;</hi> which are nothing but assemblages of several neighbouring Stars, considered as constituting some determinate figure, as os an animal, &c, from which it is therefore denominated: a division as ancient as the book of Job, in which mention is made of Orion, the Pleiades, &c.</p><p>Besides the Stars thus distinguished into magnitudes and constellations, there are others not reduced to either. Those not reduced into constellations, are called <hi rend="italics">informes,</hi> or <hi rend="italics">unformed</hi> Stars; of which kind several, so left at large by the ancients, have since been formed into new constellations by the modern astronomers, and especially by Hevelius.</p><p>Those not reduced to classes or magnitudes, are called nebulous Stars; but such as only appear faintly in clusters, in form of little lucid spots, nebulæ, or clouds.</p><p>Ptolomy sets down five of such nebulæ, viz, one at the extremity of the right hand of Perseus, which appears through the telescope, thick set with Stars; one in the middle of the crab, called <hi rend="italics">Præsepe,</hi> or the Manger, in which Galileo counted above 40 Stars; one unformed near the sting of the Scorpion; another in the eye of Sagittarius, in which two Stars may be seen in a clear sky with the naked eye, and several more with the telescope; and the fisth in the head of Orion, in which Galileo counted 21 Stars.</p><p>Flamsteed observed a cloudy Star before the bow of Sagittarius, which consists of a great number of small Stars; and the Star <hi rend="italics">d</hi> above the right shoulder of this <cb/> constellation is encompassed with several more. Flamsteed and Cassini also discovered one between the great and little dog, which is very full of Stars, that are visible only by the telescope.</p><p>But the most remarkable of all the cloudy Stars, is that in the middle of Orion's sword, in which Huygens and Dr. Long observed 12 Stars, 7 of which (3 of them, now known to be 4, being very close together) seem to shine through a cloud, very lucid near the middle, but faint and ill defined about the edges. But the greatest discoveries of nebulæ and clusters of Stars, we owe to the powerful telescopes of Dr. Herschel, who has given accounts of some thousands of such nebulæ, in many of which the Stars seem to be innumerable, like grains of sand. See Philos. Trans. 1784, 1785, 1786, 1789. See <hi rend="smallcaps">Galaxy</hi>, and <hi rend="smallcaps">Magellanic</hi> <hi rend="italics">clouds,</hi> and <hi rend="italics">lucid</hi> <hi rend="smallcaps">Spots.</hi></p><p>Cassini is of opinion, that the brightness of these proceeds from Stars so minute, as not to be distinguished by the best glasses: and this opinion is fully confirmed by the observations of Dr. Herschel, whose powerful telescopes shew those lucid specks to be composed entirely of masses of small Stars, like heaps of sand.</p><p>There are also many Stars which, though they appear single to the naked eye, are yet discovered by the telescope to be double, triple, &c. Of these, several have been observed by Cassini, Hooke, Long, Maskelyne, Hornsby, Pigott, Mayer, &c; but Dr. Herschel has been much the most successful in observations of this kind; and his success has been chiefly owing to the very extraordinary magnifying powers of the Newtonian 7 feet reflector which he has used, and the advantage of an excellent micrometer of his own construction. The powers which he has used, have been 146, 227, 278, 460, 754, 932, 1159, 1536, 2010, 3168, and even 6450. He has already formed a catalogue, containing 269 double Stars, 227 of which, as far as he knows, have not been noticed by any other person. Among these there are also some Stars that are treble, double-double, quadruple, double-treble, and multiple. His catalogue comprehends the names of the Stars, and the number in Flamsteed's catalogue, or such a description of those that are not contained in it, as will be found sufficient to distinguish them; also the comparative size of the Stars; their colours as they appeared to his view; their distances determined in several different ways; their angle of position with regard to the parallel of declination; and the dates when he first perceived the Stars to be double, treble, &c. His observations appear to commence with the year 1776, but almost all of them were made in the years 1779, 1780, 1781.</p><p>Dr. Herschel has distributed the double Stars contained in his catalogue, into 6 different classes. In the first he has placed all those which require a very superior telescope, with the utmost clearness of air, and every other favourable circumstance, to be seen at all, or well enough to judge of them; and there are 24 of these. To the 2d class belong all those double Stars that are proper for estimations by the eye, and very delicate measures by the micrometer; the number being 38. The 3d class comprehends all those double Stars, that are between 5″ and 15″ asunder; the number of them being 46. The 4th, 5th, and 6th classes contain <pb n="497"/><cb/> double Stars that are from 15″ to 30″, and from 30″ to 1′, and from 1′ to 2′ or more asunder; of which there are 44 in the 4th class, 51 in the 5th class, and 66 in the 6th class: the last of this class is <foreign xml:lang="greek">a</foreign> Tauri, number 87 of Flamsteed, whose apparent diameter, upon the meridian measured with a power of 460 at a mean of two observations 1″ 46‴, and with a power of 932 at a mean of two observations 1″ 12‴. See the list at large, Philosoph. Trans. vol. 72, art. 12.</p><p>The Stars are also distinguished, in each constellation, by numbers, or by the letters of the alphabet. This sort of distinction was introduced by John Bayer, in his Uranometria, 1654; where he denotes the Stars, in each constellation, by the letters of the Greek alphabet, <foreign xml:lang="greek">a, b, g, d, e</foreign>, &c, viz, the most remarkable Star of each by <foreign xml:lang="greek">a</foreign>, the 2d by <foreign xml:lang="greek">b</foreign>, the 3d by <foreign xml:lang="greek">g</foreign>, &c; and when there are more Stars in a constellation than the characters in the Greek alphabet, he denotes the rest, in their order, by the Roman letters A, b, c, d, &c. But as the number of the Stars, that have been observed and registered in catalogues, since Bayer's time, is greatly increased, as by Flamsteed and others, the additional ones have been marked by the ordinal numbers 1, 2, 3, 4, 5, &c.</p><p>The <hi rend="italics">Number of</hi> <hi rend="smallcaps">Stars.</hi> The number of the Stars appears to be immensely great, almost infinite; yet have astronomers long since ascertained the number of such as are visible to the eye, which are much fewer than at first sight could be imagined. See <hi rend="smallcaps">Catalogue</hi> <hi rend="italics">of the Stars.</hi></p><p>Of the 3000 contained in Flamsteed's catalogue, there are many that are only visible through a telescope; and a good eye scarce ever sees more than a thousand at the same time in the clearest heaven; the appearance of innumerable more, that are frequent in clear winter nights, arising from our sight's being deceived by their twinkling, and from our viewing them confusedly, and not reducing them to any order. But nevertheless we cannot but think the Stars are almost, if not altogether, infinite. See Halley, on the number, order, and light of the fixed Stars, Philos. Trans. number 364, or Abr. vol. 6, pa. 148.</p><p>Riccioli, in his New Almagest, affirms, that a man who shall say there are above 20 thousand times 20 thousand, would say nothing improbable. For a good telescope, directed indifferently to almost any point of the heavens, discovers multitudes that are lost to the naked eye; particularly in the milky way, which some take to be an assemblage of Stars, too remote to be seen singly, but so closely disposed as to give a luminous appearance to that part of the heavens where they are. And this fact has been confirmed by Herschel's observations: though it is disputed by others, who contend that the milky way must be owing to some other cause.</p><p>In the single constellation of the Pleiades, instead of 6, 7, or 8 Stars seen by the best eye; Dr. Hook, with a telescope 12 feet long, told 78, and with larger glasses many more, of different magnitudes. And F. de Rheita affirms, that he has observed above 2000 Stars in the single constellation of Orion. The same author found above 188 in the Pleiades. And Huygens, looking at the Star in the middle of Orion's <cb/> sword, instead of one, found it to be 12. Galileo found 80 in the space of the belt of Orion's sword, 21 in the nebulous Star of his head, and above 500 in another part of him, within the compass of one or two degrees space, and more than 40 in the nebulous Star Præsepe.</p><p><hi rend="italics">The Changes that have happened in the</hi> <hi rend="smallcaps">Stars</hi> are very considerable. The first change that is upon record, was about 120 years before Christ; when Hipparchus, discovering a new Star to appear, was first induced to make a catalogue of the Stars, that posterity might perceive any future changes of the like nature.</p><p>In the year 1572, Cornelius Gemma and Tycho Brahe observed another new Star in the constellation Cassiopeia, which was likewise the occasion of Tycho's making a new catalogue. At first its magnitude and brightness exceeded the largest of the Stars, Sirius and Lyra; and even equalled the planet Venus when nearest the earth, and was seen in fair day-light. It continued 16 months; towards the latter end of which it began to dwindle, and at length, in March 1574, it totally disappeared, without any change of place in all that time.</p><p>Leovicius tells us of another Star appearing in the same constellation, about the year 945, which resembled that of 1572; and he quotes another ancient observation, by which it appears that a new Star was seen about the same place in 1264. Dr. Keil thinks these were all the same Star; and indeed the periodical intervals, or distance of time between these appearances, were nearly equal, being from 318 to 319 years; and if so, its next appearance may be expected about 1890.</p><p>Fabricius, in 1596, discovered another new Star, called the <hi rend="italics">stella mira,</hi> or <hi rend="italics">wonderful Star,</hi> in the neck of the whale, which has since been found to appear and disappear periodically, 7 times in 6 years, continuing in its greatest lustre for 15 days together; and is never quite extinguished. Its course and motion are described by Bulliald, in a treatise printed at Paris in 1667. Dr. Herschel has lately, viz, in the years 1777, 1778, 1779, and 1780, made several observations on this Star, an account of which may be seen in the Philos. Trans. vol. 70, art. 21.</p><p>In the year 1600, William Jansen discovered a changeable Star in the neck of the Swan, which gradually decreased till it became so small as to be thought to disappear entirely, till the years 1657, 1658, and 1659, when it regained its former lustre and magnitude; but soon decayed again, and is now of the smallest size.</p><p>In the year 1604, a new Star was seen by Kepler, and several of his friends, near the heel of the right foot of Serpentarius, which was particularly bright and sparkling; and it was observed to be every moment changing into some of the colours of the rainbow, except when it is near the horizon, at which time it was generally white. It surpassed Jupiter in magnitude, but was easily distinguished from him, by the steady light of the planet. It disappeared about the end of the year 1605, and has not been seen since that time.</p><p>Simon Marius discovered another in Andromeda's <pb n="498"/><cb/> girdle, in 1612 and 1613: though Bulliald says it had been seen before, in the 15th century.</p><p>In July 1670, Hevelius discovered a second changeable Star in the Swan, which was so diminished in October as to be scarce perceptible. In April following it regained its former lustre, but wholly disappeared in August. In March 1672 it was seen again, but appeared very small, and has not been visible since.</p><p>In 1686 a third changeable Star was discovered by Kirchius in the Swan, viz, the Star <hi rend="italics">X</hi> of that constellation, which returned periodically in about 405 days.</p><p>In 1672 Cassini saw a Star in the neck of the Bull, which he thought was not visible in Tycho's time, nor when Bayer made his figures.</p><p>It is certain, from the old catalogues, that many of the ancient Stars are not now visible. This has been particularly remarked with regard to the Pleiades.</p><p>M. Montanari, in his letter to the Royal Society in 1670, observes that there are now wanting in the heavens two Stars of the 2d magnitude, in the stern of the ship Argo, and its yard, which had been seen till the year 1664. When they first disappeared is not known; but he assures us there was not the least glimpse of them in 1668. He adds, he has observed many more changes in the fixed Stars, even to the number of a hundred. And many other changes of the Stars have been noticed by Cassini, Maraldi, and other observers. See Gregory's Astron. lib. 2, prop. 30.</p><p>But the greatest numbers of variable Stars have been observed of late years, and the most accurate observations made on their periods, &c, by Herschel, Goodricke, Pigott, &c, in the late volumes of the Philos. Trans. particularly in the vol. for 1786, where the last of these gentlemen has given a catalogue of all that have been hitherto observed, with accounts of the observations that have been made upon them.</p><p>Various hypotheses have been devised to aocount for such changes and appearances in the Stars. It is not probable they could be comets, as they had no parallax, even when largest and brightest. It has been supposed that the periodical Stars have vast dark spots, or dark sides, and very slow rotations on their axes, by which means they must disappear when the darker side is turned towards us. And as for those which break out suddenly with such lustre, these may perhaps be suns whose fuel is almost spent, and again supplied by some of their comets falling upon them, and occasioning an uncommon blaze and splendor for some time; which it is conjectured may be one use of the cometary part of our system.</p><p>Maupertuis, in his Dissertation on the figures of the Celestial Bodies (pa. 61—63), is of opinion that some Stars, by their prodigious swift rotation on their axes, may not only assume the figures of oblate spheroids, but that by the great centrifugal force arising from such rotations, they may become of the figures of mill-stones, or be reduced to flat circular planes, so thin as to be quite invisible when their edges are turned towards us, as Saturn's ring is in such position. But when very eccentric planets or comets go round any flat Star in orbits much inclined to its equator, the attraction of <cb/> the planets or comets in their perihelions must alter the inclination of the axis of that Star; on which account it will appear more or less large and luminous, as its broad side is more or less turned towards us. And thus he imagines we may account for the apparent changes of magnitude and lustre of those Stars, and also for their appearing and disappearing.</p><p>Hevelius apprehends (Cometograph. pa. 380), that the Sun and Stars are surrounded with atmospheres, and that by whirling round their axes with great rapidity, they throw off great quantities of matter into those atmospheres, and so cause great changes in them; and that thus it may come to pass that a Star, which, when its atmosphere is clear, shines out with great lustre, may at another time, when it is full of clouds and thick vapours, appear greatly diminished in brightness and magnitude, or even become quite invisible.</p><p><hi rend="italics">Nature of the fixed</hi> <hi rend="smallcaps">Stars.</hi> The immense distance of the Stars leaves us greatly at a loss about the nature of them. What we can gather for certain from their phenomena, is as follows:</p><p>1st, That the fixed Stars are greater than our earth: because if that was not the case, they could not be visible at such an immense distance.</p><p>2nd, The fixed Stars are farther distant from the earth than the farthest of the planets. For we frequently find the fixed Stars hid behind the body of the planets: and besides, they have no parallax, which the planets have.</p><p>3rd, The fixed Stars shine with their own light; for they are much farther from the Sun than Saturn, and appear much smaller than Saturn; but since, notwithstanding this, they are found to shine much brighter than that planet, it is evident they cannot borrow their light from the same source as Saturn does, viz, the Sun; but since we know of no other luminous body beside the Sun, whence they might derive their light, it follows that they shine with their own native light.</p><p>Besides, it is known, that the more a telescope magnifies, the less is the aperture through which the Star is seen; and consequently, the fewer rays it admits into the eye. Now since the Stars appear less in a telescope which magnifies two hundred times, than they do to the naked eye, insomuch that they seem to be only indivisible points, it proves at once that the Stars are at immense distances from us, and that they shine by their own proper light. If they shone by borrowed light, they would be as invisible without telescopes as the satellites of Jupiter are; for the satellites appear larger when viewed with a good telescope than the largest fixed Stars do.</p><p>Hence,</p><p>1. We deduce, that the fixed Stars are so many suns; for they have all the characters of suns.</p><p>2. That in all probability the Stars are not smaller than our sun.</p><p>3. That it is highly probable each Star is the centre of a system, and has planets or earths revolving round it, in the same manner as round our sun, i. e. it has opake bodies illuminated, warmed, and cherished by its light and heat. As we have incomparably more light from the moon than from all the Stars together, it is absurd to imagine that the Stars were made for no other purpose than to cast a faint light upon the earth; <pb n="499"/><cb/> especially since many more require the assistance of a good telescope to find them out, than are visible without that instrument. Our sun is surrounded by a system of planets and comets, all which would be invisible from the nearest fixed Star; and from what we already know of the immense distance of the Stars, it is easy to prove, that the sun, seen from such a distance, would appear no larger than a Star of the first magnitude.</p><p>From all this it is highly probable, that each Star is a sun to a system of worlds moving round it, though unseen by us; especially as the doctrine of a plurality of worlds is rational, and greatly manifests the power, the wisdom, and the goodness of the great creator.</p><p>How immense, then, does the universe appear! Indeed, it must either be infinite, or infinitely near it.</p><p>Kepler, it is true, denies that each Star can have its system of planets as ours has; and takes them all to be fixed in the same surface or sphere; urging, that were one twice or thrice as remote as another, it would be twice or thrice as small, supposing their real magnitudes equal; whereas there is no difference in their apparent magnitudes, justly observed, at all. But to this it is opposed, that Huygens has not only shewn, that fires and flames are visible at distances where other bodies, comprehended under equal angles, disappear; but it should likewise seem, that the optic theorem about the apparent diameters of objects, being reciprocally proportional to their distances from the eye, does only hold while the object has some sensible ratio to its distance.</p><p>As for periodical Stars, &c. see <hi rend="smallcaps">Changes</hi>, <hi rend="italics">&c of Stars,</hi> supra.</p><p><hi rend="italics">Motion of the</hi> <hi rend="smallcaps">Stars.</hi> The fixed Stars have two kinds of apparent motion; one called the <hi rend="italics">first, common,</hi> or <hi rend="italics">diurnal motion,</hi> arising from the earth's motion round its axis: by this they seem to be carried along with the sphere or firmament, in which they appear fixed, round the earth, from east to west, in the space of 24 hours.</p><p>The other, called the <hi rend="italics">second,</hi> or <hi rend="italics">proper motion,</hi> is that by which they appear to go backwards from west to east, round the poles of the ecliptic, with an exceeding slow motion, as describing a degree of their circle only in the space of 71 1/2 years, or 50 1/3 seconds in a year. This apparent motion is owing to the recession of the equinoctial points, which is 50 1/3 seconds of a degree in a year backward, or contrary to the order of the signs of the zodiac.</p><p>In consequence of this second motion, the longitude of the Stars will be always increasing. Thus, for example, the longitude of Cor Leonis was found at different periods, to be as follows: viz, <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">Year.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data"><hi rend="italics">Long.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">By Ptolomy, in</cell><cell cols="1" rows="1" role="data"> 138</cell><cell cols="1" rows="1" role="data">to be</cell><cell cols="1" rows="1" role="data"> 2°</cell><cell cols="1" rows="1" role="data">30′</cell></row><row role="data"><cell cols="1" rows="1" role="data">By the Persians, in</cell><cell cols="1" rows="1" role="data">1115</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">30</cell></row><row role="data"><cell cols="1" rows="1" role="data">By Alphonsus, in</cell><cell cols="1" rows="1" role="data">1364</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">40</cell></row><row role="data"><cell cols="1" rows="1" role="data">By Prince of Hesse, in</cell><cell cols="1" rows="1" role="data">1586</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">11</cell></row><row role="data"><cell cols="1" rows="1" role="data">By Tycho, in</cell><cell cols="1" rows="1" role="data">1601</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">17</cell></row><row role="data"><cell cols="1" rows="1" role="data">By Flamsteed, in</cell><cell cols="1" rows="1" role="data">1690</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">31 1/3</cell></row></table> Whence the proper motion of the Stars, according to the order of the signs, in circles parallel to the ecliptic, is easily inferred.</p><p>It was Hipparchus who first suspected this motion, upon comparing his own observations with those of Timocharis and Aristyllus. Ptolomy, who lived three <cb/> centuries after Hipparchus, demonstrated the same by undeniable arguments.</p><p>The increase of longitude in a century, as stated b<*> different astronomers, is as follows: <table><row role="data"><cell cols="1" rows="1" role="data">By</cell><cell cols="1" rows="1" role="data">Tycho Brahe</cell><cell cols="1" rows="1" role="data">1°</cell><cell cols="1" rows="1" role="data">25′</cell><cell cols="1" rows="1" role="data"> 0″</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Copernicus</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">40 1/5</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Flamsteed and Riccioli</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">20</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Bulliald</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">54</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Hevelius</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">46 5/<*></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Dr. Bradley, &c.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">55</cell></row></table> which is at the rate of 50 1/3 seconds per year.</p><p>From these data, the increase in the longitude of a Star for any given time, is easily had, and thence its longitude at any time: ex. gr. the longitude of Sirius, in Flamsteed's tables, for the year 1690, being 9° 49′ 1″, its longitude for the year 1800, is found by multiplying the interval of time, viz, 110 years, by 50 1/3, <table><row role="data"><cell cols="1" rows="1" role="data">the product 5537″, or</cell><cell cols="1" rows="1" role="data">  1° 32′ 17″, added to the</cell></row><row role="data"><cell cols="1" rows="1" role="data">given longitude</cell><cell cols="1" rows="1" role="data">  9   49      1</cell></row><row role="data"><cell cols="1" rows="1" role="data">gives the longitude</cell><cell cols="1" rows="1" role="data">11   21    18 for the year 1800.</cell></row></table></p><p>The chief phenomena of the fixed Stars, arising from their common and proper motion, besides their longitude, are their altitudes, right ascensions, declinations, occultations, culminations, risings, and settings.</p><p>Some have supposed that the latitudes of the Stars are invariable. But this supposition is founded on two assumptions, which are both controverted among astronomers. The one of these is, that the orbit of the earth continues unalterably in the same plane, and consequently that the ecliptic is invariable; the contrary of which is now very generally allowed.</p><p>The other assumption is, that the Stars are so fixed as to keep their places immoveably. Ptolomy, Tycho, and others, comparing their observations with those of the ancient astronomers, have adopted this opinion. But from the result of the comparison of our best modern observations, with such as were formerly made with any tolerable degree of exactness, there appears to have been a real change in the position of some of the fixed Stars, with respect to each other; and several Stars of the first magnitude have already been observed, and others suspected to have a proper motion of their own.</p><p>Dr. Halley (Philos. Trans. number 355, or Abr. vol. 4, p. 225) has observed, that the three following Stars, the Ball's eye, Sirius, and Arcturus, are now found to be above half a degree more southerly than the ancients reckoned them: that this difference cannot arise from the errors of the transcribers, because the declinations of the Stars, set down by Ptolomy, as observed by Timocharis, Hipparchus, and himself, shew their latitudes given by him are such as those authors intended: and it is scarce to be believed that those three observers could be deceived in so plain a matter. To this he adds, that the bright Star in the shoulder of Orion has, in Ptolomy, almost a whole degree more southerly latitude than at present: that an ancient observation, made at Athens in the year 509, as Bulliald supposes, of an appulse of the moon to the Bull's eye, shews that Star to have had less latitude at that time than it now has: that as to Sirius, it appears by Tycho's observations, that he found him 4 1/2 more northerly <pb n="500"/><cb/> than he is at this time. All these observations, compared together, seem to favour an opinion, that some of the Stars have a proper motion of their own, which changes their places in the sphere of heaven: this change of place, as Dr. Halley observes, may shew itself in so long a time as 1800 years, though it be entirely imperceptible in the space of one single century; and it is likely to be soonest discovered in such Stars as those just now mentioned; because they are all of the first magnitude, and may, therefore, probably be some of the nearest to our solar System. Arcturus, in particular, affords a strong proof of this: for if its present declination be compared with its place, as determined either by Tycho or Flamsteed, the difference will be found to be much greater than what can be suspected to arise from the uncertainty of their observations. See <hi rend="smallcaps">Arcturus</hi>, and Mr. Hornsby's enquiry into the quantity and direction of the proper motion of Arcturus, Phil. Trans. vol. 63, part 1, pa. 93, &c.</p><p>For an account of Dr. Bradley's observations, see the sequel of this article.</p><p>Dr. Herschel has also lately observed, that the diftance of the two Stars forming the double Star <foreign xml:lang="greek">g</foreign> Draconis, is 54″ 48‴, and their position 44° 19′ N. preceding. Whereas, from the right ascension and declination of these Stars in Flamsteed's catalogue, their distance, in his time, appears to have been 1′ 11″ .418, and their position 44° 23′ N. preceding. Hence he infers, that as the difference in the distance of these two Stars is so considerable, we can hardly account for it, otherwise than by admitting a proper motion in one or the other of the Stars, or in our solar system: most probably he says, neither of the three is at rest. He also suspects a proper motion in one of the double Stars, in Cauda Lyncis Media, and in <*> Ceti. Phil. Trans. vol. 72, part 1, p. 117, 143, 150.</p><p>It is reasonable to expect, that other instances of the like kind must also occur among the great number of visible Stars, because their relative positions may be altered by various means. For if our own solar system be conceived to change its place with respect to absolute space, this might, in process of time, occasion an apparent change in the angular distances of the fixed Stars; and in such a case, the places of the nearest Stars being more affected than of those that are very remote, their relative position might seem to alter, though the Stars themselves were really immoveable; and vice versa, we may surmise, from the observed motion of the Stars, that our sun, with all its planets and comets, may have a motion towards some particular part of the heavens, on account of a greater quantity of matter collected in a number of Stars and their surrounding planets there situated, which may perhaps occasion a gravitation of our whole solar system towards it. If this surmise should have any foundation, as Dr. Herschel observes, ubi supra, p. 103, it will shew itself in a series of some years; since from that motion there will arise another kind of hitherto unknown parallax (suggested by Mr. Michell, Philos. Trans. vol. 57, p. 252), the investigation of which may account for some part of the motions already observed in some of the principal Stars; and for the purpose of determining the direction and quantity of such a motion, accurate observations of the distance of Stars, that are near enough to be measured <cb/> with a micrometer, and a very high power of telescopes, may be of considerable use, as they will undoubtedly give us the relative places of those Stars to a much greater degree of accuracy than they can be had by instruments or sectors, and thereby much sooner enable us to discover any apparent change in their situation, occasioned by this new kind of secular or systematical parallax, if we may so express the change arising from the motion of the whole solar system.</p><p>And, on the other hand, if our system be at rest, and any of the Stars really in motion, this might likewise vary their apparent positions; and the more so, the nearer they are to us, or the swifter their motions are; or the more proper the direction of the motion is to be rendered perceptible by us. Since then the relative places of the Stars may be changed from such a variety of causes, considering the amazing distance at which it is certain some of them are placed, it may require the observations of many ages to determine the laws of the apparent changes, even of a single Star; much more difficult, therefore, must it be to settle the laws relating to all the most remarkable Stars.</p><p>When the causes which affect the places of all the Stars in general are known; such as the precession, aberration, and nutation, it may be of singular use to examine nicely the relative situations of particular Stars, and especially of those of the greatest lustre, which, it may be presumed, lie nearest to us, and may therefore be subject to more sensible changes, either from their own motion, or from that of our system. And if, at the same time the brighter Stars are compared with each other, we likewise determine the relative positions of some of the smallest that appear near them, whose places can be ascertained with sufficient exactness, we may perhaps be able to judge to what cause the change, if any be observable, is owing. The uncertainty that we are at present under, with respect to the degree of accuracy with which former astronomers could observe, makes us unable to determine several things relating to this subject; but the improvements, which have of late years been made in the methods of taking the places of the heavenly bodies, are so great, that a few years may hereafter be sufficient to settle some points, which cannot now be settled; by comparing even the earliest observations with those of the present age.</p><p>Dr. Hook communicated several observations on the apparent motions of the fixed Stars; and as this was a matter of great importance in astronomy, several of the learned were desirous of verifying and confirming his observations. An instrument was accordingly contrived by Mr. George Graham, and executed with surprising exactness.</p><p>With this instrument the Star <foreign xml:lang="greek">g</foreign>, in the constellation Draco, was frequently observed by Messrs. Molyneux, Bradley, and Graham, in the years 1725, 1726; and the observations were afterwards repeated by Dr. Bradley with an instrument contrived by the same ingenious person, Mr. Graham, and so exact, that it might be depended on to half a second. The result of these observations was, that the Star did not always appear in the same place, but that its distance from the zenith varied, and that the difference of the apparent places amounted to 21 or 22 seconds. Similar observations were made on other Stars, and a like apparent motion <pb n="501"/><cb/> was found in them, proportional to the latitude of the Star. This motion was by no means such as was to have been expected, as the effect of a parallax, and it was some time before any way could be found of accounting for this new phenomenon. At length Dr. Bradley resolved all its variety, in a satisfactory manner, by the motion of light and the motion of the earth compounded together. See <hi rend="smallcaps">Light</hi>, and Phil. Trans. No. 406, p. 364, or Abr. vol. vi, p. 149, &c.</p><p>Our excellent astronomer, Dr. Bradley, had no sooner discovered the cause, and settled the laws of aberration of the fixed Stars, than his attention was again excited by another new phenomenon, viz, an annual change of declination in some of the fixed Stars, which appeared to be sensibly greater than a precession of the equinoctial points of 50″ in a year, the mean quantity now usually allowed by astronomers, would have occasioned.</p><p>This apparent change of declination was observed in the Stars near the equinoctial colure, and there appearing at the same time an effect of a quite contrary nature, in some Stars near the solstitial colure, which seemed to alter their declination less than a precession of 50″ required, Dr. Bradley was thereby convinced, that all the phenomena in the different Stars could not be accounted for merely by supposing that he had assumed a wrong quantity for the precession of the equinoctial points. He had also, after many trials, sufficient reason to conclude, that these second unexpected deviations of the Stars were not owing to any imperfection of his instruments. At length, from repeated observations he began to guess at the real cause of these phenomena.</p><p>It appeared from the Doctor's observations, during his residence at Wansted, from the year 1727 to 1732, that some of the Stars near the solstitial colure had changed their declinations 9″ or 10″ less than a precession of 50″ would have produced; and, at the same time, that others near the equinoctial colure had altered theirs about the same quantity more than a like precession would have occasioned: the north pole of the equator seeming to have approached the Stars, which come to the meridian with the sun about the vernal equinox, and the winter solstice; and to have receded from those, which come to the meridian with the sun about the autumnal equinox and the summer solstice.</p><p>From the consideration of these circumstances, and the situation of the ascending node of the moon's orbit when he first began to make his observations, he suspected that the moon's action upon the equatorial parts of the earth might produce these effects.</p><p>For if the precession of the equinox be, according to Sir Isaac Newton's principles, caused by the actions of the sun and moon upon those parts; the plane of the moon's orbit being, at one time, above 10 degrees more inclined to the plane of the equator than at another, it was reasonable to conclude, that the part of the whole annual precession, which arises from her action, would, in different years, be varied in its quantity; whereas the plane of the ecliptic, in which the sun appears, keeping always nearly the same inclination to the equator, that part of the precession, which is owing to the sun's action, may be the same every year; and from hence it would follow, that although the mean annual precession, proceeding from the joint actions of <cb/> the sun and moon, were 50″; yet the apparent annual precession might sometimes exceed, and sometimes fall short of that mean quantity, according to the various situations of the nodes of the moon's orbit.</p><p>In the year 1727, the moon's ascending node was near the beginning of Aries, and consequently her orbit was as much inclined to the equator as it can at any time be; and then the apparent annual precession was found, by the Doctor's first year's observations, to be greater than the mean; which proved, that the Stars near the equinoctial colure, whose declinations are most of all affected by the precession, had changed theirs, above a tenth part more than a precession of 50″ would have caused. The succeeding year's observations proved the same thing; and, in three or four years' time, the difference became so considerable as to leave no room to suspect it was owing to any imperfection either of the instrument or observation.</p><p>But some of the Stars, that were near the solstitial colure, having appeared to move, during the same time, in a manner contrary to what they ought to have done, by an increase of the precession; and the deviations in them being as remarkable as in the others, it was evident that something more than a mere change in the quantity of the precession would be requisite to solve this part of the phenomenon. Upon comparing the observations of Stars near the solstitial colure, that were almost opposite to each other in right ascension, they were found to be equally affected by this cause. For whilst <foreign xml:lang="greek">g</foreign> Draconis appeared to have moved northward, the small Star, which is the 35th Camelopardali Hevelii, in the British catalogue, seemed to have gone as much towards the south; which shewed, that this apparent motion in both those Stars might proceed from a nutation of the earth's axis; whereas the comparison of the Doctor's observations of the same Stars formerly enabled him to draw a different conclusion, with respect to the cause of the annual aberrations arising from the motion of light. For the apparent alteration in <foreign xml:lang="greek">g</foreign> Draconis, from that cause, being as large again as in the other small Star, proved, that that did not proceed from a nutation of the earth's axis; as, on the contrary, this may.</p><p>Upon making the like comparison between the observations of other Stars, that lie nearly opposite in right ascension, whatever their situations were with respect to the cardinal points of the equator, it appeared, that their change of declination was nearly equal, but contrary; and such as a nutation or motion of the earth's axis would effect.</p><p>The moon's ascending node being got back towards the beginning of Capricorn in the year 1732, the Stars near the equinoctial colure appeared about that time to change their declinations no more than a precession of 50″ required; whilst some of those near the solstitial colure altered theirs above 2″ in a year less than they ought. Soon after the annual change of declination of the former was perceived to be diminished, so as to become less than 50″ of precession would cause, and it continued to diminish till the year 1736, when the moon's ascending node was about the beginning of Libra, and her orbit had the least inclination to the equator. But by this time, some of the Stars near the solstitial colure had altered their declinations 18″ <pb n="502"/><cb/> less since the year 1727, than they ought to have done from a precession of 50″. For <foreign xml:lang="greek">g</foreign> Draconis, which in those 9 years would have gone about 8″ more southerly, was observed, in 1736, to appear 10″ more northerly than it did in the year 1727.</p><p>As this appearance in <foreign xml:lang="greek">g</foreign> Draconis indicated a diminution of the inclination of the earth's axis to the plane of the ecliptic, and as several astronomers have supposed that inclination to diminish regularly; if this phenomenon depend upon such a cause and amounted to 18″ in 9 years, the obliquity of the ecliptic would, at that rate, alter a whole minute in 30 years; which is much faster than any observations before made would allow. The Doctor had therefore reason to think, that some part of this motion at least, if not the whole, was owing to the moon's action on the equatorial parts of the earth, which he conceived might cause a libratory motion of the earth's axis. But as he was unable to judge, from only 9 years observation, whether the axis would entirely recover the same position that it had in the year 1727, he found it necessary to continue his observations through a whole period of the moon's nodes; at the end of which he had the satisfaction to see, that the Stars returned into the same positions again, as if there had been no alteration at all in the inclination of the earth's axis; which fully convinced him, that he had guessed rightly as to the cause of the phenomenon. This circumstance proves likewise, that if there be a gradual diminution of the obliquity of the ecliptic, it does not arise only from an alteration in the position of the earth's axis, but rather from some change in the plane of the ecliptic itself; because the Stars, at the end of the period of the moon's nodes, appeared in the same places, with respect to the equator, as they ought to have done if the earth's axis had retained the same inclination to an invariable plane.</p><p>The Doctor having communicated these observations, and his suspicion of their cause, to the late Mr. Machin, that excellent geometrician soon after sent him a table, containing the quantity of the annual precession in the various positions of the moon's nodes, as also the corresponding nutations of the earth's axis; which was computed upon the supposition that the mean annual precession is 50″, and that the whole is governed by the pole of the moon's orbit only; and therefore Mr. Machin imagined, that the numbers in the table would be too large, as, in fact, they were found to be. But it appeared that the changes which Dr. Bradley had observed, both in the annual precession and nutation, kept the same law, as to increasing and decreasing, with the numbers of Mr. Machin's table. Those were calculated on the supposition, that the pole of the equator, during a period of the moon's nodes, moved round in the periphery of a little circle, whose centre was 23° 29′ distant from the pole of the ecliptic; having itself also an angular motion of 50″ in a year about the same pole. The north pole of the equator was conceived to be in that part of the small circle which is farthest from the north pole of the ecliptic at the same time when the moon's ascending node is in the beginning of Aries; and in the opposite point of it, when the same node is in Libra. <cb/></p><p>If the diameter of the little circle, in which the pole of the equator moves, be supposed equal to 18″, which is the whole quantity of the nutation, as collected from Dr. Bradley's observations of the Star <foreign xml:lang="greek">g</foreign> Draconis, then all the phenomena of the several Stars which he observed will be very nearly solved by this hypothesis. But for the particulars of his solution, and the application of his theory to the practice of astronomy, we must refer to the excellent author himself; our intention being only to give the history of the invention.</p><p>The corrections arising from the aberration of light, and from the nutation of the earth's axis, must not be neglected in astronomical observations; since such neglects might produce errors of near a minute in the polar distance of some Stars.</p><p>As to the allowance to be made for the aberration of light, Dr. Bradley assures us, that having again examined those of his own observations, which were most proper to determine the transverse axis of the ellipsis, which each Star seems to describe, he found it to be nearest to 40″; and this is the number he makes use of in his computations relating to the nutation.</p><p>Dr. Bradley says, in general, that experience has taught him, that the observations of such Stars as lie nearest the zenith, generally agree best with one another, and are therefore fittest to prove the truth of any hypothesis. Phil. Trans. N°. 485, vol. 45, p. 1, &c.</p><p>Monsieur d'Alembert has published a treatise, entitled, Recherches sur la Precession des Equinoxes, et sur la Nutation de la Terre dans le Systeme Newtonien, 4to. Paris, 1749. The calculations of this learned gentleman agree in general with Dr. Bradley's observations. But Monsieur d'Alembert finds, that the pole of the equator describes an ellipsis in the heavens, the ratio of whose axes is that of 4 to 3; whereas, according to Dr. Bradley, the curve described is either a circle or an ellipsis, the ratio of whose axes is as 9 to 8.</p><p>The several Stars in each constellation, as in Taurus, Bootes, Hercules, &c, see under the proper article of each constellation, <hi rend="smallcaps">Taurus, Bootes, Hercules</hi>, &c.</p><p>To learn to know the several fixed Stars by the globe, see <hi rend="smallcaps">Globe.</hi></p><p>The parallax and distance of the fixed Stars, see under <hi rend="smallcaps">Parallax</hi> and <hi rend="smallcaps">Distance.</hi></p><p><hi rend="italics">Circumpolar</hi> <hi rend="smallcaps">Stars.</hi> See <hi rend="smallcaps">Circumpolar.</hi></p><p><hi rend="italics">Morning</hi> <hi rend="smallcaps">Star.</hi> See <hi rend="smallcaps">Morning.</hi></p><p><hi rend="italics">Place of a</hi> <hi rend="smallcaps">Star.</hi> See <hi rend="smallcaps">Place.</hi></p><p><hi rend="italics">Pole</hi> <hi rend="smallcaps">Star.</hi> See <hi rend="smallcaps">Pole.</hi></p><p><hi rend="italics">Twinkling of the</hi> <hi rend="smallcaps">Stars.</hi> See <hi rend="smallcaps">Twinkling.</hi></p><p><hi rend="italics">Unformed</hi> <hi rend="smallcaps">Stars.</hi> See <hi rend="smallcaps">Informes.</hi></p><p>The following two catalogues of Stars are taken from Dr. Zach's Tabulæ Motuum Solis &c, and are adapted to the beginning of the year 1800. The former contains 381 Stars, shewing their names and Bayer's mark, their magnitude, declination, and right ascension, both in time and in arcs or degrees of a great circle, with the annual variations of the same. And the latter contains 162 principal Stars, shewing their declinations to seconds of a degree, with their annual variations. The explanations are sufficiently clear from the titles of the columns. <pb n="503"/> <table rend="border"><head>A <hi rend="smallcaps">Catalogue</hi> <hi rend="italics">of the most remarkable</hi> <hi rend="smallcaps">Fixed Stars</hi>, <hi rend="italics">with their Magnitudes, Right Ascensions, Declinations and Annual Variations, for the Beginning of the Year</hi> 1800.</head><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascension in degrees &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">56.79</cell><cell cols="1" rows="1" role="data">3.063</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">11.85</cell><cell cols="1" rows="1" role="data">45.95</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13.51</cell><cell cols="1" rows="1" role="data">3.059</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">22.66</cell><cell cols="1" rows="1" role="data">45.89</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">45.12</cell><cell cols="1" rows="1" role="data">3.301</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">16.75</cell><cell cols="1" rows="1" role="data">49.51</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">53.93</cell><cell cols="1" rows="1" role="data">3.262</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">29.01</cell><cell cols="1" rows="1" role="data">48.93</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Andromedæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">39.02</cell><cell cols="1" rows="1" role="data">3.161</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">45.31</cell><cell cols="1" rows="1" role="data">47.42</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">14.40</cell><cell cols="1" rows="1" role="data">3.311</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">35.95</cell><cell cols="1" rows="1" role="data">49.66</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">31.83</cell><cell cols="1" rows="1" role="data">3.001</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">57.40</cell><cell cols="1" rows="1" role="data">45.01</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">1.44</cell><cell cols="1" rows="1" role="data">3.389</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">21.64</cell><cell cols="1" rows="1" role="data">50.83</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">19.08</cell><cell cols="1" rows="1" role="data">3.093</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">46.15</cell><cell cols="1" rows="1" role="data">46.39</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">44.75</cell><cell cols="1" rows="1" role="data">3.505</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">11.29</cell><cell cols="1" rows="1" role="data">52.58</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">33.95</cell><cell cols="1" rows="1" role="data">3.103</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">29.20</cell><cell cols="1" rows="1" role="data">46.55</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Andromedæ</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">34.23</cell><cell cols="1" rows="1" role="data">3.297</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">33.38</cell><cell cols="1" rows="1" role="data">49.46</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">0.22</cell><cell cols="1" rows="1" role="data">3.531</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">3.33</cell><cell cols="1" rows="1" role="data">52.96</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16.99</cell><cell cols="1" rows="1" role="data">3.109</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">14.80</cell><cell cols="1" rows="1" role="data">46.63</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">50.58</cell><cell cols="1" rows="1" role="data">3.761</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">38.70</cell><cell cols="1" rows="1" role="data">56.42</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">42.07</cell><cell cols="1" rows="1" role="data">3.108</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">31.07</cell><cell cols="1" rows="1" role="data">46.62</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">30.70</cell><cell cols="1" rows="1" role="data">3.164</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">40.56</cell><cell cols="1" rows="1" role="data">47.46</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">1.77</cell><cell cols="1" rows="1" role="data">3.107</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">26.62</cell><cell cols="1" rows="1" role="data">46.61</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">50.72</cell><cell cols="1" rows="1" role="data">3.144</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">40.84</cell><cell cols="1" rows="1" role="data">47.16</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">10.01</cell><cell cols="1" rows="1" role="data">4.155</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30.13</cell><cell cols="1" rows="1" role="data">62.33</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">36.69</cell><cell cols="1" rows="1" role="data">2.953</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">10.33</cell><cell cols="1" rows="1" role="data">44.30</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Triang. Bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">42.74</cell><cell cols="1" rows="1" role="data">3.379</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">41.15</cell><cell cols="1" rows="1" role="data">50.68</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign>′</cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">34.52</cell><cell cols="1" rows="1" role="data">3.258</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">37.73</cell><cell cols="1" rows="1" role="data">48.87</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">36.77</cell><cell cols="1" rows="1" role="data">3.277</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">11.48</cell><cell cols="1" rows="1" role="data">49.15</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">48.86</cell><cell cols="1" rows="1" role="data">3.315</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">12.83</cell><cell cols="1" rows="1" role="data">49.73</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Andromedæ</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">41.05</cell><cell cols="1" rows="1" role="data">3.615</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">15.76</cell><cell cols="1" rows="1" role="data">54.23</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">preced. <foreign xml:lang="greek">a *y</foreign></cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">26.15</cell><cell cols="1" rows="1" role="data">.....</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">32.25</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">..</cell><cell cols="1" rows="1" role="data">..</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">55.27</cell><cell cols="1" rows="1" role="data">3.335</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">49.05</cell><cell cols="1" rows="1" role="data">50.02</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a *y</foreign></cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">38.13</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">31.95</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">..</cell><cell cols="1" rows="1" role="data">..</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">1.64</cell><cell cols="1" rows="1" role="data">3.308</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">24.55</cell><cell cols="1" rows="1" role="data">49.62</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti (Variab.)</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">14.70</cell><cell cols="1" rows="1" role="data">3.019</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">40.50</cell><cell cols="1" rows="1" role="data">45.29</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign>°</cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">9.29</cell><cell cols="1" rows="1" role="data">3.321</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">19.32</cell><cell cols="1" rows="1" role="data">49.81</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">s</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">28.09</cell><cell cols="1" rows="1" role="data">3.285</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">1.39</cell><cell cols="1" rows="1" role="data">49.28</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">14.17</cell><cell cols="1" rows="1" role="data">3.060</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">32.54</cell><cell cols="1" rows="1" role="data">45.90</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">53.42</cell><cell cols="1" rows="1" role="data">2.884</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">21.37</cell><cell cols="1" rows="1" role="data">43.27</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">57.18</cell><cell cols="1" rows="1" role="data">3.102</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">17.76</cell><cell cols="1" rows="1" role="data">46.53</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">36.02</cell><cell cols="1" rows="1" role="data">2.849</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">0.29</cell><cell cols="1" rows="1" role="data">42.74</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">*y</foreign>Y</cell><cell cols="1" rows="1" rend="align=left" role="data">Lilii Bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">57.70</cell><cell cols="1" rows="1" role="data">3.521</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">25.49</cell><cell cols="1" rows="1" role="data">52.81</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">*y</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lilii Aust.</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">14.43</cell><cell cols="1" rows="1" role="data">3.489</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">36.49</cell><cell cols="1" rows="1" role="data">52.34</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">r</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">35.65</cell><cell cols="1" rows="1" role="data">3.344</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">54.72</cell><cell cols="1" rows="1" role="data">50.16</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">r</foreign><hi rend="sup">3</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">9.35</cell><cell cols="1" rows="1" role="data">3.340</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">20.19</cell><cell cols="1" rows="1" role="data">50.10</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">39.72</cell><cell cols="1" rows="1" role="data">2.917</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">55.78</cell><cell cols="1" rows="1" role="data">43.75</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">48.01</cell><cell cols="1" rows="1" role="data">3.401</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">1.80</cell><cell cols="1" rows="1" role="data">51.01</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">24.42</cell><cell cols="1" rows="1" role="data">4.250</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">6.25</cell><cell cols="1" rows="1" role="data">63.75</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">50.07</cell><cell cols="1" rows="1" role="data">3.119</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">31.06</cell><cell cols="1" rows="1" role="data">46.66</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="504"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in degrees. &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Ceti</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">54.61</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">39.15</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">12.07</cell><cell cols="1" rows="1" role="data">3.846</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">1.04</cell><cell cols="1" rows="1" role="data">57.69</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">12.71</cell><cell cols="1" rows="1" role="data">3.393</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10.59</cell><cell cols="1" rows="1" role="data">50.89</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">25.98</cell><cell cols="1" rows="1" role="data">3.422</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">29.77</cell><cell cols="1" rows="1" role="data">51.33</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7.48</cell><cell cols="1" rows="1" role="data">2.904</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">52.20</cell><cell cols="1" rows="1" role="data">43.56</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">42.36</cell><cell cols="1" rows="1" role="data">3.433</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">35.39</cell><cell cols="1" rows="1" role="data">51.49</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6.85</cell><cell cols="1" rows="1" role="data">4.203</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">42.77</cell><cell cols="1" rows="1" role="data">63.05</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">16.37</cell><cell cols="1" rows="1" role="data">3.428</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">5.48</cell><cell cols="1" rows="1" role="data">51.42</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" rend="align=left" role="data">65</cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">55.52</cell><cell cols="1" rows="1" role="data">3.430</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">52.74</cell><cell cols="1" rows="1" role="data">51.45</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">f</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">54.65</cell><cell cols="1" rows="1" role="data">3.289</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">39.78</cell><cell cols="1" rows="1" role="data">49.33</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31.52</cell><cell cols="1" rows="1" role="data">2.883</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">52.84</cell><cell cols="1" rows="1" role="data">43.24</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">44.93</cell><cell cols="1" rows="1" role="data">4.203</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13.91</cell><cell cols="1" rows="1" role="data">63.05</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lucida Plei.</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">37.17</cell><cell cols="1" rows="1" role="data">3.535</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">17.56</cell><cell cols="1" rows="1" role="data">53.03</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">35.20</cell><cell cols="1" rows="1" role="data">3.734</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">48.00</cell><cell cols="1" rows="1" role="data">56.01</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">29.19</cell><cell cols="1" rows="1" role="data">3.977</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17.87</cell><cell cols="1" rows="1" role="data">59.66</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">42.25</cell><cell cols="1" rows="1" role="data">2.786</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">33.77</cell><cell cols="1" rows="1" role="data">41.79</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">A</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">53.46</cell><cell cols="1" rows="1" role="data">3.515</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">21.83</cell><cell cols="1" rows="1" role="data">52.72</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">25.23</cell><cell cols="1" rows="1" role="data">3.387</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">18.48</cell><cell cols="1" rows="1" role="data">50.80</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">24.68</cell><cell cols="1" rows="1" role="data">3.432</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">10.26</cell><cell cols="1" rows="1" role="data">51.48</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">34.93</cell><cell cols="1" rows="1" role="data">3.431</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">43.89</cell><cell cols="1" rows="1" role="data">51.46</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">27.66</cell><cell cols="1" rows="1" role="data">3.545</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">54.93</cell><cell cols="1" rows="1" role="data">53.17</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">31.13</cell><cell cols="1" rows="1" role="data">3.543</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">47.02</cell><cell cols="1" rows="1" role="data">53.14</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">56.98</cell><cell cols="1" rows="1" role="data">3.475</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">14.76</cell><cell cols="1" rows="1" role="data">52.12</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9.30</cell><cell cols="1" rows="1" role="data">3.401</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">19.53</cell><cell cols="1" rows="1" role="data">51.02</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">19.53</cell><cell cols="1" rows="1" role="data">3.399</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">52.91</cell><cell cols="1" rows="1" role="data">50.99</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præced. <foreign xml:lang="greek">a <*></foreign></cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">12.56</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">8.40</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Aldebaran</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">27.29</cell><cell cols="1" rows="1" role="data">3.421</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">49.38</cell><cell cols="1" rows="1" role="data">51.31</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">sequ. <foreign xml:lang="greek">a <*></foreign></cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">43.44</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">51.60</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">s</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">44.78</cell><cell cols="1" rows="1" role="data">3.406</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">11.77</cell><cell cols="1" rows="1" role="data">51.09</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">u</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">47.26</cell><cell cols="1" rows="1" role="data">2.329</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">48.83</cell><cell cols="1" rows="1" role="data">34.94</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">s</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">50.67</cell><cell cols="1" rows="1" role="data">3.409</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">40.87</cell><cell cols="1" rows="1" role="data">51.13</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">43.15</cell><cell cols="1" rows="1" role="data">2.615</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">47.21</cell><cell cols="1" rows="1" role="data">39.23</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">9.38</cell><cell cols="1" rows="1" role="data">3.565</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">20.75</cell><cell cols="1" rows="1" role="data">53.47</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">2.38</cell><cell cols="1" rows="1" role="data">2.948</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">35.74</cell><cell cols="1" rows="1" role="data">44.22</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">34.73</cell><cell cols="1" rows="1" role="data">2.863</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">40.95</cell><cell cols="1" rows="1" role="data">42.95</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Aurig.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">39.44</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">51.60</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Capella</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">56.16</cell><cell cols="1" rows="1" role="data">4.414</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">2.40</cell><cell cols="1" rows="1" role="data">66.21</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Aurig.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14.28</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">34.20</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">b</foreign> Orio.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">56.39</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">5.85</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Rigel</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">55.54</cell><cell cols="1" rows="1" role="data">2.867</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">53.10</cell><cell cols="1" rows="1" role="data">43.01</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">b</foreign> Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">24.57</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8.55</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" rend="align=center colspan=4" role="data">. . . .</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">39.38</cell><cell cols="1" rows="1" role="data">3.778</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">50.70</cell><cell cols="1" rows="1" role="data">56.67</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">24.54</cell><cell cols="1" rows="1" role="data">3.209</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">8.16</cell><cell cols="1" rows="1" role="data">48.13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leponis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">41.09</cell><cell cols="1" rows="1" role="data">2.565</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">16.40</cell><cell cols="1" rows="1" role="data">38.47</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">47.38</cell><cell cols="1" rows="1" role="data">3.057</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">53.69</cell><cell cols="1" rows="1" role="data">45.86</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="505"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Mag. nitude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascension in degrees, &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leporis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">54.94</cell><cell cols="1" rows="1" role="data">2.639</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">44.13</cell><cell cols="1" rows="1" role="data">39.59</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">42.41</cell><cell cols="1" rows="1" role="data">3.575</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">36.14</cell><cell cols="1" rows="1" role="data">53.62</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">4.15</cell><cell cols="1" rows="1" role="data">3.037</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">2.19</cell><cell cols="1" rows="1" role="data">45.55</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">40.57</cell><cell cols="1" rows="1" role="data">3.020</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">8.57</cell><cell cols="1" rows="1" role="data">45.30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Columbæ</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">25.03</cell><cell cols="1" rows="1" role="data">2.167</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">15.45</cell><cell cols="1" rows="1" role="data">32.50</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leporis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">9.02</cell><cell cols="1" rows="1" role="data">2.517</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15.34</cell><cell cols="1" rows="1" role="data">37.75</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">16.28</cell><cell cols="1" rows="1" role="data">2.839</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">4.21</cell><cell cols="1" rows="1" role="data">42.59</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Orio.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">27.74</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">56.10</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">20.57</cell><cell cols="1" rows="1" role="data">3.239</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">8.55</cell><cell cols="1" rows="1" role="data">48.59</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Orionis</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">35.59</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">23.85</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">101</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aurigæ</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">51.68</cell><cell cols="1" rows="1" role="data">4.398</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">55.22</cell><cell cols="1" rows="1" role="data">65.97</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">H</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Gemin. (prop.)</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">57.72</cell><cell cols="1" rows="1" role="data">3.642</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">25.77</cell><cell cols="1" rows="1" role="data">54.63</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">103</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">48.29</cell><cell cols="1" rows="1" role="data">3.623</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">4.34</cell><cell cols="1" rows="1" role="data">54.34</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">104</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">51.44</cell><cell cols="1" rows="1" role="data">3.624</cell><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">51.64</cell><cell cols="1" rows="1" role="data">54.36</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">39.08</cell><cell cols="1" rows="1" role="data">2.298</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">46.24</cell><cell cols="1" rows="1" role="data">34.47</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">53.76</cell><cell cols="1" rows="1" role="data">2.638</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">26.33</cell><cell cols="1" rows="1" role="data">39.57</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">107</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">5.48</cell><cell cols="1" rows="1" role="data">3.562</cell><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">22.22</cell><cell cols="1" rows="1" role="data">53.43</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">108</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">9.35</cell><cell cols="1" rows="1" role="data">3.463</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">20.32</cell><cell cols="1" rows="1" role="data">51.95</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">37.34</cell><cell cols="1" rows="1" role="data">3.695</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">20.05</cell><cell cols="1" rows="1" role="data">55.42</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Can. maj.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">41.80</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">27.00</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Sirius</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">19.91</cell><cell cols="1" rows="1" role="data">2.647</cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">58.65</cell><cell cols="1" rows="1" role="data">39.71</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">112</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Can. maj.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">26.68</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">40.20</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">113</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">46.21</cell><cell cols="1" rows="1" role="data">2.354</cell><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">33.20</cell><cell cols="1" rows="1" role="data">35.31</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">114</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">14.55</cell><cell cols="1" rows="1" role="data">3.567</cell><cell cols="1" rows="1" role="data">103</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">38.24</cell><cell cols="1" rows="1" role="data">53.47</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">115</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">15.39</cell><cell cols="1" rows="1" role="data">2.436</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">53.85</cell><cell cols="1" rows="1" role="data">36.54</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">116</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10.06</cell><cell cols="1" rows="1" role="data">3.594</cell><cell cols="1" rows="1" role="data">107</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30.94</cell><cell cols="1" rows="1" role="data">52.91</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">117</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis minoris</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">18.01</cell><cell cols="1" rows="1" role="data">3.261</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">30.21</cell><cell cols="1" rows="1" role="data">48.92</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">118</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Gemin.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">13.66</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">24.90</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">119</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Castor</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">48.81</cell><cell cols="1" rows="1" role="data">3.855</cell><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">12.15</cell><cell cols="1" rows="1" role="data">57.83</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Gemin.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">4.84</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">12.60</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">u</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">34.54</cell><cell cols="1" rows="1" role="data">3.715</cell><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">38.04</cell><cell cols="1" rows="1" role="data">55.72</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">122</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign>Can. min.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">40.27</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">4.05</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">123</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Procyon</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">49.10</cell><cell cols="1" rows="1" role="data">3.137</cell><cell cols="1" rows="1" role="data">112</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">16.50</cell><cell cols="1" rows="1" role="data">47.06</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Can. min.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">27.12</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">112</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">46.80</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Pollux</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">3.18</cell><cell cols="1" rows="1" role="data">3.687</cell><cell cols="1" rows="1" role="data">113</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">47.70</cell><cell cols="1" rows="1" role="data">55.31</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">b</foreign> Gemin.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">28.78</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">113</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">11.70</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">57.64</cell><cell cols="1" rows="1" role="data">3.545</cell><cell cols="1" rows="1" role="data">118</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">24.55</cell><cell cols="1" rows="1" role="data">53.18</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">y</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">23.25</cell><cell cols="1" rows="1" role="data">3.639</cell><cell cols="1" rows="1" role="data">119</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">48.73</cell><cell cols="1" rows="1" role="data">54.58</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">39.37</cell><cell cols="1" rows="1" role="data">3.266</cell><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">50.61</cell><cell cols="1" rows="1" role="data">48.99</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6′</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">10.35</cell><cell cols="1" rows="1" role="data">3.441</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">35.31</cell><cell cols="1" rows="1" role="data">51.61</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">6 7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">7.79</cell><cell cols="1" rows="1" role="data">3.491</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">56.87</cell><cell cols="1" rows="1" role="data">52.36</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">3.04</cell><cell cols="1" rows="1" role="data">3.189</cell><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">45.53</cell><cell cols="1" rows="1" role="data">47.83</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">41.86</cell><cell cols="1" rows="1" role="data">3.499</cell><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">27.85</cell><cell cols="1" rows="1" role="data">52.49</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">134</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">18.11</cell><cell cols="1" rows="1" role="data">3.428</cell><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">31.70</cell><cell cols="1" rows="1" role="data">51.42</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">10.14</cell><cell cols="1" rows="1" role="data">3.199</cell><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">32.03</cell><cell cols="1" rows="1" role="data">47.98</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="506"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascension in degrees, &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">136</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">48.86</cell><cell cols="1" rows="1" role="data">3.187</cell><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">12.84</cell><cell cols="1" rows="1" role="data">47.81</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">137</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">59.39</cell><cell cols="1" rows="1" role="data">3.290</cell><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">50.81</cell><cell cols="1" rows="1" role="data">49.35</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">138</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">31.82</cell><cell cols="1" rows="1" role="data">3.292</cell><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">57.26</cell><cell cols="1" rows="1" role="data">49.38</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">139</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">54.33</cell><cell cols="1" rows="1" role="data">3.263</cell><cell cols="1" rows="1" role="data">134</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">34.92</cell><cell cols="1" rows="1" role="data">48.95</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">c</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">30.37</cell><cell cols="1" rows="1" role="data">3.472</cell><cell cols="1" rows="1" role="data">134</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">35.48</cell><cell cols="1" rows="1" role="data">52.08</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">141</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">54.80</cell><cell cols="1" rows="1" role="data">3.120</cell><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">42.03</cell><cell cols="1" rows="1" role="data">46.80</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">58.32</cell><cell cols="1" rows="1" role="data">3.524</cell><cell cols="1" rows="1" role="data">138</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">34.77</cell><cell cols="1" rows="1" role="data">52.86</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">143</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Alphard</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">44.97</cell><cell cols="1" rows="1" role="data">2.935</cell><cell cols="1" rows="1" role="data">139</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">14.55</cell><cell cols="1" rows="1" role="data">44.03</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">144</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Hydræ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">9.19</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">17.85</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">145</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">c</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">9.26</cell><cell cols="1" rows="1" role="data">3.253</cell><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18.97</cell><cell cols="1" rows="1" role="data">48.80</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">27.65</cell><cell cols="1" rows="1" role="data">3.224</cell><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">54.82</cell><cell cols="1" rows="1" role="data">48.36</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">28.29</cell><cell cols="1" rows="1" role="data">3.434</cell><cell cols="1" rows="1" role="data">143</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">4.31</cell><cell cols="1" rows="1" role="data">51.51</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">148</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">21.84</cell><cell cols="1" rows="1" role="data">3.457</cell><cell cols="1" rows="1" role="data">145</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">27.54</cell><cell cols="1" rows="1" role="data">51.85</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">149</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">26.92</cell><cell cols="1" rows="1" role="data">3.243</cell><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">43.76</cell><cell cols="1" rows="1" role="data">48.65</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">150</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">37.99</cell><cell cols="1" rows="1" role="data">3.183</cell><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">29.89</cell><cell cols="1" rows="1" role="data">47.75</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">24.60</cell><cell cols="1" rows="1" role="data">3.289</cell><cell cols="1" rows="1" role="data">149</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9.04</cell><cell cols="1" rows="1" role="data">49.33</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Regulus</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">42.02</cell><cell cols="1" rows="1" role="data">3.204</cell><cell cols="1" rows="1" role="data">149</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">30.30</cell><cell cols="1" rows="1" role="data">48.06</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">153</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">28.58</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8.70</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">154</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">32.34</cell><cell cols="1" rows="1" role="data">3.361</cell><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">5.16</cell><cell cols="1" rows="1" role="data">50.42</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">155</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">55.22</cell><cell cols="1" rows="1" role="data">3.306</cell><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">48.23</cell><cell cols="1" rows="1" role="data">49.60</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">156</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">21.35</cell><cell cols="1" rows="1" role="data">3.635</cell><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">23.32</cell><cell cols="1" rows="1" role="data">54.52</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">157</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">r</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">15.77</cell><cell cols="1" rows="1" role="data">3.170</cell><cell cols="1" rows="1" role="data">155</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">56.49</cell><cell cols="1" rows="1" role="data">47.55</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">158</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">39.53</cell><cell cols="1" rows="1" role="data">3.709</cell><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">54.93</cell><cell cols="1" rows="1" role="data">55.63</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">159</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Crateris</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">4.55</cell><cell cols="1" rows="1" role="data">2.943</cell><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">8.25</cell><cell cols="1" rows="1" role="data">44.14</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">160</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">15.84</cell><cell cols="1" rows="1" role="data">3.847</cell><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">57.61</cell><cell cols="1" rows="1" role="data">57.70</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">161</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Crateris</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">50.06</cell><cell cols="1" rows="1" role="data">2.933</cell><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">30.97</cell><cell cols="1" rows="1" role="data">44.02</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">26.39</cell><cell cols="1" rows="1" role="data">3.199</cell><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">35.91</cell><cell cols="1" rows="1" role="data">47.98</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">163</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">44.23</cell><cell cols="1" rows="1" role="data">3.165</cell><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">3.49</cell><cell cols="1" rows="1" role="data">47.48</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">164</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Crateris</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">28.15</cell><cell cols="1" rows="1" role="data">2.981</cell><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">2.21</cell><cell cols="1" rows="1" role="data">44.72</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">i</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">28.32</cell><cell cols="1" rows="1" role="data">3.125</cell><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">4.85</cell><cell cols="1" rows="1" role="data">46.87</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">166</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">39.49</cell><cell cols="1" rows="1" role="data">3.085</cell><cell cols="1" rows="1" role="data">169</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">52.34</cell><cell cols="1" rows="1" role="data">46.28</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">167</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">u</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">42.82</cell><cell cols="1" rows="1" role="data">3.069</cell><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">42.29</cell><cell cols="1" rows="1" role="data">46.04</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">34.11</cell><cell cols="1" rows="1" role="data">3.087</cell><cell cols="1" rows="1" role="data">173</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">51.70</cell><cell cols="1" rows="1" role="data">46.31</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">169</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">b</foreign> Leonis</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">19.46</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">51.90</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">170</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Denebola</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">50.49</cell><cell cols="1" rows="1" role="data">3.062</cell><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">37.35</cell><cell cols="1" rows="1" role="data">45.93</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">16.38</cell><cell cols="1" rows="1" role="data">3.122</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5.70</cell><cell cols="1" rows="1" role="data">46.83</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">172</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">14.22</cell><cell cols="1" rows="1" role="data">3.212</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">33.33</cell><cell cols="1" rows="1" role="data">48.18</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">173</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">6.94</cell><cell cols="1" rows="1" role="data">3.062</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">44.10</cell><cell cols="1" rows="1" role="data">45.93</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">51.63</cell><cell cols="1" rows="1" role="data">3.067</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">54.47</cell><cell cols="1" rows="1" role="data">46.00</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">27.23</cell><cell cols="1" rows="1" role="data">3.021</cell><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">48.42</cell><cell cols="1" rows="1" role="data">45.32</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">176</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">32.31</cell><cell cols="1" rows="1" role="data">3.077</cell><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">4.58</cell><cell cols="1" rows="1" role="data">46.16</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">177</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">40.74</cell><cell cols="1" rows="1" role="data">3.067</cell><cell cols="1" rows="1" role="data">182</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">11.13</cell><cell cols="1" rows="1" role="data">46.01</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">178</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">54.39</cell><cell cols="1" rows="1" role="data">3.124</cell><cell cols="1" rows="1" role="data">185</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">35.92</cell><cell cols="1" rows="1" role="data">46.86</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">47.65</cell><cell cols="1" rows="1" role="data">2.661</cell><cell cols="1" rows="1" role="data">186</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">54.72</cell><cell cols="1" rows="1" role="data">39.91</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">180</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">33.85</cell><cell cols="1" rows="1" role="data">3.069</cell><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">27.72</cell><cell cols="1" rows="1" role="data">46.03</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="507"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Mag. nitude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in degrees. &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">12.58</cell><cell cols="1" rows="1" role="data">2.746</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">8.67</cell><cell cols="1" rows="1" role="data">41.19</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">182</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">33.66</cell><cell cols="1" rows="1" role="data">3.047</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">24.93</cell><cell cols="1" rows="1" role="data">45.71</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">183</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">13.33</cell><cell cols="1" rows="1" role="data">3.004</cell><cell cols="1" rows="1" role="data">193</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20.01</cell><cell cols="1" rows="1" role="data">45.06</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">184</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">36.46</cell><cell cols="1" rows="1" role="data">3.095</cell><cell cols="1" rows="1" role="data">194</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">6.97</cell><cell cols="1" rows="1" role="data">46.42</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">185</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4.31</cell><cell cols="1" rows="1" role="data">3.225</cell><cell cols="1" rows="1" role="data">197</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4.64</cell><cell cols="1" rows="1" role="data">48.38</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">186</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. a Virg.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">13.00</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">197</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">15.00</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Spica</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">40.11</cell><cell cols="1" rows="1" role="data">3.137</cell><cell cols="1" rows="1" role="data">198</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">1.66</cell><cell cols="1" rows="1" role="data">47.06</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">188</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">49.62</cell><cell cols="1" rows="1" role="data">2.425</cell><cell cols="1" rows="1" role="data">198</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">24.26</cell><cell cols="1" rows="1" role="data">36.37</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">189</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">i</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">10.79</cell><cell cols="1" rows="1" role="data">3.129</cell><cell cols="1" rows="1" role="data">199</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">41.83</cell><cell cols="1" rows="1" role="data">46.93</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">190</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">30.65</cell><cell cols="1" rows="1" role="data">3.064</cell><cell cols="1" rows="1" role="data">201</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">39.68</cell><cell cols="1" rows="1" role="data">45.96</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">46.36</cell><cell cols="1" rows="1" role="data">2.884</cell><cell cols="1" rows="1" role="data">204</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">35.35</cell><cell cols="1" rows="1" role="data">43.26</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">192</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">38.85</cell><cell cols="1" rows="1" role="data">2.355</cell><cell cols="1" rows="1" role="data">204</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">42.80</cell><cell cols="1" rows="1" role="data">35.88</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">193</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">9.21</cell><cell cols="1" rows="1" role="data">2.860</cell><cell cols="1" rows="1" role="data">206</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18.12</cell><cell cols="1" rows="1" role="data">42.90</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">194</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">58.88</cell><cell cols="1" rows="1" role="data">1.628</cell><cell cols="1" rows="1" role="data">209</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">43.24</cell><cell cols="1" rows="1" role="data">24.42</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">195</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">14.87</cell><cell cols="1" rows="1" role="data">3.179</cell><cell cols="1" rows="1" role="data">210</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">43.04</cell><cell cols="1" rows="1" role="data">47.68</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">196</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Arcturus</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">32.21</cell><cell cols="1" rows="1" role="data">2.722</cell><cell cols="1" rows="1" role="data">211</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">3.16</cell><cell cols="1" rows="1" role="data">40.83</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">197</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">36.46</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">211</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">6.90</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">198</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">19.14</cell><cell cols="1" rows="1" role="data">3.223</cell><cell cols="1" rows="1" role="data">212</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">47.16</cell><cell cols="1" rows="1" role="data">48.35</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">199</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">1.50</cell><cell cols="1" rows="1" role="data">2.428</cell><cell cols="1" rows="1" role="data">216</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">22.54</cell><cell cols="1" rows="1" role="data">36.42</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">35.56</cell><cell cols="1" rows="1" role="data">2.854</cell><cell cols="1" rows="1" role="data">217</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">53.42</cell><cell cols="1" rows="1" role="data">42.81</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">201</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">14.99</cell><cell cols="1" rows="1" role="data">2.612</cell><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">44.80</cell><cell cols="1" rows="1" role="data">39.33</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">202</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">22.95</cell><cell cols="1" rows="1" role="data">3.268</cell><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">44.22</cell><cell cols="1" rows="1" role="data">49.02</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">203</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">38.74</cell><cell cols="1" rows="1" role="data">3.299</cell><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">41.10</cell><cell cols="1" rows="1" role="data">49.49</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">204</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi> <figure/></cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">38.77</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">41.55</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">205</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">49.97</cell><cell cols="1" rows="1" role="data">3.289</cell><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">29.55</cell><cell cols="1" rows="1" role="data">49.34</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">206</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ minoris</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">27.55</cell><cell cols="1" rows="1" role="data">-0.329</cell><cell cols="1" rows="1" role="data">222</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">53.19</cell><cell cols="1" rows="1" role="data">-4.94</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">207</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">24.35</cell><cell cols="1" rows="1" role="data">3.482</cell><cell cols="1" rows="1" role="data">223</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">5.22</cell><cell cols="1" rows="1" role="data">52.23</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">208</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">24.99</cell><cell cols="1" rows="1" role="data">2.262</cell><cell cols="1" rows="1" role="data">223</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">14.85</cell><cell cols="1" rows="1" role="data">33.93</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">209</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">y</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">52.50</cell><cell cols="1" rows="1" role="data">2.580</cell><cell cols="1" rows="1" role="data">223</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">7.57</cell><cell cols="1" rows="1" role="data">38.70</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">210</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">15.61</cell><cell cols="1" rows="1" role="data">3.215</cell><cell cols="1" rows="1" role="data">226</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">54.21</cell><cell cols="1" rows="1" role="data">48.22</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">211</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">26.62</cell><cell cols="1" rows="1" role="data">2.409</cell><cell cols="1" rows="1" role="data">226</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">39.28</cell><cell cols="1" rows="1" role="data">36.13</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">212</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Coron. bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">51.97</cell><cell cols="1" rows="1" role="data">2.487</cell><cell cols="1" rows="1" role="data">227</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">59.55</cell><cell cols="1" rows="1" role="data">37.30</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">213</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Coron. bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">56.05</cell><cell cols="1" rows="1" role="data">2.465</cell><cell cols="1" rows="1" role="data">228</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">0.78</cell><cell cols="1" rows="1" role="data">36.97</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">214</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Coron. bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">34.88</cell><cell cols="1" rows="1" role="data">2.483</cell><cell cols="1" rows="1" role="data">229</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">43.13</cell><cell cols="1" rows="1" role="data">37.24</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">215</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ minoris</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">11.76</cell><cell cols="1" rows="1" role="data">-0.209</cell><cell cols="1" rows="1" role="data">230</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">56.39</cell><cell cols="1" rows="1" role="data">-3.14</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">216</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign><hi rend="sup">4</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">38.42</cell><cell cols="1" rows="1" role="data">3.365</cell><cell cols="1" rows="1" role="data">230</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">36.26</cell><cell cols="1" rows="1" role="data">50.48</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">217</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">21.22</cell><cell cols="1" rows="1" role="data">3.328</cell><cell cols="1" rows="1" role="data">231</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">18.34</cell><cell cols="1" rows="1" role="data">49.92</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">218</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">15.84</cell><cell cols="1" rows="1" role="data">2.861</cell><cell cols="1" rows="1" role="data">231</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">57.61</cell><cell cols="1" rows="1" role="data">42.91</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">219</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Gemma</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">13.29</cell><cell cols="1" rows="1" role="data">2.543</cell><cell cols="1" rows="1" role="data">231</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">19.35</cell><cell cols="1" rows="1" role="data">38.15</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">220</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">27.12</cell><cell cols="1" rows="1" role="data">3.433</cell><cell cols="1" rows="1" role="data">232</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">46.77</cell><cell cols="1" rows="1" role="data">51.49</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">221</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">25.21</cell><cell cols="1" rows="1" role="data">2.936</cell><cell cols="1" rows="1" role="data">223</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">18.00</cell><cell cols="1" rows="1" role="data">44.04</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">222</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">23.53</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">234</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">53.05</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">223</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">57.70</cell><cell cols="1" rows="1" role="data">2.756</cell><cell cols="1" rows="1" role="data">234</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">25.47</cell><cell cols="1" rows="1" role="data">41.34</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">224</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">10.30</cell><cell cols="1" rows="1" role="data">3.023</cell><cell cols="1" rows="1" role="data">234</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">34.55</cell><cell cols="1" rows="1" role="data">45.35</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">225</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">50.97</cell><cell cols="1" rows="1" role="data">2.969</cell><cell cols="1" rows="1" role="data">235</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">44.51</cell><cell cols="1" rows="1" role="data">44.54</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="508"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascension in degrees, &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">226</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Coron. bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">12.45</cell><cell cols="1" rows="1" role="data">2.515</cell><cell cols="1" rows="1" role="data">235</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6.71</cell><cell cols="1" rows="1" role="data">37.73</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">227</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">44.86</cell><cell cols="1" rows="1" role="data">3.457</cell><cell cols="1" rows="1" role="data">235</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">12.93</cell><cell cols="1" rows="1" role="data">51.86</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">228</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">33.34</cell><cell cols="1" rows="1" role="data">3.671</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">20.06</cell><cell cols="1" rows="1" role="data">55.06</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">229</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">46.43</cell><cell cols="1" rows="1" role="data">3.600</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">36.48</cell><cell cols="1" rows="1" role="data">54.00</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">230</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">y</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">0.91</cell><cell cols="1" rows="1" role="data">3.339</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">13.63</cell><cell cols="1" rows="1" role="data">50.09</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">231</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">12.93</cell><cell cols="1" rows="1" role="data">2.740</cell><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">13.98</cell><cell cols="1" rows="1" role="data">41.10</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">232</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">31.89</cell><cell cols="1" rows="1" role="data">3.521</cell><cell cols="1" rows="1" role="data">237</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">58.38</cell><cell cols="1" rows="1" role="data">52.82</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">233</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Coron. bor.</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">18.54</cell><cell cols="1" rows="1" role="data">2.483</cell><cell cols="1" rows="1" role="data">237</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">38.04</cell><cell cols="1" rows="1" role="data">37.24</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">234</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">41.18</cell><cell cols="1" rows="1" role="data">2.576</cell><cell cols="1" rows="1" role="data">238</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">17.72</cell><cell cols="1" rows="1" role="data">38.64</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">235</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">49.71</cell><cell cols="1" rows="1" role="data">3.465</cell><cell cols="1" rows="1" role="data">238</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">25.65</cell><cell cols="1" rows="1" role="data">51.97</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">236</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">8.28</cell><cell cols="1" rows="1" role="data">1.142</cell><cell cols="1" rows="1" role="data">239</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">4.27</cell><cell cols="1" rows="1" role="data">17.13</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">237</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">23.31</cell><cell cols="1" rows="1" role="data">3.465</cell><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">49.60</cell><cell cols="1" rows="1" role="data">51.96</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">238</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">52.80</cell><cell cols="1" rows="1" role="data">3.132</cell><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">11.95</cell><cell cols="1" rows="1" role="data">46.98</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">239</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">45.09</cell><cell cols="1" rows="1" role="data">3.154</cell><cell cols="1" rows="1" role="data">241</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">16.31</cell><cell cols="1" rows="1" role="data">47.30</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">5.83</cell><cell cols="1" rows="1" role="data">2.642</cell><cell cols="1" rows="1" role="data">243</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">27.41</cell><cell cols="1" rows="1" role="data">39.63</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">241</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Antares</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9.69</cell><cell cols="1" rows="1" role="data">3.645</cell><cell cols="1" rows="1" role="data">244</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">25.35</cell><cell cols="1" rows="1" role="data">54.68</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">242</cell><cell cols="1" rows="1" rend="align=left" role="data">* <foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">6.66</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">244</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">39.90</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">243</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">f</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">43.03</cell><cell cols="1" rows="1" role="data">3.418</cell><cell cols="1" rows="1" role="data">244</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">45.42</cell><cell cols="1" rows="1" role="data">51.27</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">244</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">18.33</cell><cell cols="1" rows="1" role="data">0.785</cell><cell cols="1" rows="1" role="data">245</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">34.92</cell><cell cols="1" rows="1" role="data">11.78</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">245</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">37.87</cell><cell cols="1" rows="1" role="data">2.579</cell><cell cols="1" rows="1" role="data">245</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">28.08</cell><cell cols="1" rows="1" role="data">38.68</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">246</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">26.96</cell><cell cols="1" rows="1" role="data">3.709</cell><cell cols="1" rows="1" role="data">245</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">44.36</cell><cell cols="1" rows="1" role="data">55.64</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">247</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">9.55</cell><cell cols="1" rows="1" role="data">3.287</cell><cell cols="1" rows="1" role="data">246</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">23.24</cell><cell cols="1" rows="1" role="data">49.30</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">248</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">45.64</cell><cell cols="1" rows="1" role="data">2.292</cell><cell cols="1" rows="1" role="data">248</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">24.67</cell><cell cols="1" rows="1" role="data">34.38</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">249</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">3.14</cell><cell cols="1" rows="1" role="data">2.046</cell><cell cols="1" rows="1" role="data">249</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">47.15</cell><cell cols="1" rows="1" role="data">30.69</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">250</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">38.68</cell><cell cols="1" rows="1" role="data">2.292</cell><cell cols="1" rows="1" role="data">253</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">40.16</cell><cell cols="1" rows="1" role="data">34.38</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">251</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">55.15</cell><cell cols="1" rows="1" role="data">3.424</cell><cell cols="1" rows="1" role="data">254</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">47.26</cell><cell cols="1" rows="1" role="data">51.36</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">252</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Herc.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">12.70</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">10.50</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">253</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">31.76</cell><cell cols="1" rows="1" role="data">2.726</cell><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">56.40</cell><cell cols="1" rows="1" role="data">40.89</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">254</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">49.41</cell><cell cols="1" rows="1" role="data">2.459</cell><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">21.17</cell><cell cols="1" rows="1" role="data">36.88</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">255</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">44.25</cell><cell cols="1" rows="1" role="data">3.669</cell><cell cols="1" rows="1" role="data">257</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">3.68</cell><cell cols="1" rows="1" role="data">55.04</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2.64</cell><cell cols="1" rows="1" role="data">4.057</cell><cell cols="1" rows="1" role="data">260</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">39.58</cell><cell cols="1" rows="1" role="data">60.85</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">257</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Ophi.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">45.90</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">28.50</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">258</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">38.97</cell><cell cols="1" rows="1" role="data">2.768</cell><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">44.55</cell><cell cols="1" rows="1" role="data">41.52</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">259</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Ophi.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">11.02</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">262</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">45.32</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">260</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">55.99</cell><cell cols="1" rows="1" role="data">1.348</cell><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">59.82</cell><cell cols="1" rows="1" role="data">20.22</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">261</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">35.77</cell><cell cols="1" rows="1" role="data">2.959</cell><cell cols="1" rows="1" role="data">263</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">56.54</cell><cell cols="1" rows="1" role="data">44.39</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">262</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">52.04</cell><cell cols="1" rows="1" role="data">3.003</cell><cell cols="1" rows="1" role="data">264</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">0.56</cell><cell cols="1" rows="1" role="data">45.05</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">263</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">54.59</cell><cell cols="1" rows="1" role="data">3.153</cell><cell cols="1" rows="1" role="data">267</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">38.87</cell><cell cols="1" rows="1" role="data">4<*>.30</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">264</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">37.47</cell><cell cols="1" rows="1" role="data">2.999</cell><cell cols="1" rows="1" role="data">267</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">21.99</cell><cell cols="1" rows="1" role="data">44.99</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">265</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">57.79</cell><cell cols="1" rows="1" role="data">1.389</cell><cell cols="1" rows="1" role="data">267</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">26.85</cell><cell cols="1" rows="1" role="data">20.83</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">266</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">58.05</cell><cell cols="1" rows="1" role="data">3.851</cell><cell cols="1" rows="1" role="data">268</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">30.76</cell><cell cols="1" rows="1" role="data">57.77</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">267</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Taur. Poniat.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">41.10</cell><cell cols="1" rows="1" role="data">2.993</cell><cell cols="1" rows="1" role="data">270</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">16.50</cell><cell cols="1" rows="1" role="data">44.90</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">268</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">48.37</cell><cell cols="1" rows="1" role="data">3.584</cell><cell cols="1" rows="1" role="data">270</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">5.63</cell><cell cols="1" rows="1" role="data">53.76</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">269</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 6</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16.90</cell><cell cols="1" rows="1" role="data">3.575</cell><cell cols="1" rows="1" role="data">270</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">13.57</cell><cell cols="1" rows="1" role="data">53.62</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">270</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">53.67</cell><cell cols="1" rows="1" role="data">3.984</cell><cell cols="1" rows="1" role="data">272</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">25.11</cell><cell cols="1" rows="1" role="data">59.76</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="509"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in degrees, &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">271</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">37.66</cell><cell cols="1" rows="1" role="data">3.705</cell><cell cols="1" rows="1" role="data">273</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">24.92</cell><cell cols="1" rows="1" role="data">55.57</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">272</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Lyræ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">40.12</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">277</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1.88</cell><cell cols="1" rows="1" role="data">. .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">273</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Wega</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">9.89</cell><cell cols="1" rows="1" role="data">1.994</cell><cell cols="1" rows="1" role="data">277</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">28.35</cell><cell cols="1" rows="1" role="data">29.91</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">274</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Lyræ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">40.00</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">277</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">50.00</cell><cell cols="1" rows="1" role="data">. .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">275</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">f</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">9.39</cell><cell cols="1" rows="1" role="data">3.747</cell><cell cols="1" rows="1" role="data">278</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">20.84</cell><cell cols="1" rows="1" role="data">56.21</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">276</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lyræ</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">42.87</cell><cell cols="1" rows="1" role="data">1.983</cell><cell cols="1" rows="1" role="data">279</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">43.03</cell><cell cols="1" rows="1" role="data">29.74</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">277</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">5.41</cell><cell cols="1" rows="1" role="data">3.625</cell><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">21.22</cell><cell cols="1" rows="1" role="data">54.38</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">278</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lyræ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">41.86</cell><cell cols="1" rows="1" role="data">2.211</cell><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">27.89</cell><cell cols="1" rows="1" role="data">33.16</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">279</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">s</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">51.40</cell><cell cols="1" rows="1" role="data">3.724</cell><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">50.99</cell><cell cols="1" rows="1" role="data">55.86</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">1.13</cell><cell cols="1" rows="1" role="data">3.623</cell><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">16.99</cell><cell cols="1" rows="1" role="data">54.35</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">281</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">Serpentis</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">3</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">18</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">46</cell><cell cols="1" rows="1" role="data">16.82</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">2.977</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">281</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">34</cell><cell cols="1" rows="1" role="data">12.35</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">44.66</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">3</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="rowspan=2" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18.32</cell><cell cols="1" rows="1" role="data">34.84</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">282</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lyræ</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">31.16</cell><cell cols="1" rows="1" role="data">2.095</cell><cell cols="1" rows="1" role="data">281</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">47.43</cell><cell cols="1" rows="1" role="data">31.42</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">14.15</cell><cell cols="1" rows="1" role="data">0.880</cell><cell cols="1" rows="1" role="data">282</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">32.20</cell><cell cols="1" rows="1" role="data">13.21</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">284</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lyræ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">27.04</cell><cell cols="1" rows="1" role="data">2.241</cell><cell cols="1" rows="1" role="data">282</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">45.55</cell><cell cols="1" rows="1" role="data">33.61</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">285</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">41.18</cell><cell cols="1" rows="1" role="data">3.595</cell><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">17.72</cell><cell cols="1" rows="1" role="data">53.92</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">286</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">26.45</cell><cell cols="1" rows="1" role="data">3.758</cell><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">36.82</cell><cell cols="1" rows="1" role="data">56.37</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">287</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Antinoi</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">38.10</cell><cell cols="1" rows="1" role="data">3.186</cell><cell cols="1" rows="1" role="data">283</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">31.55</cell><cell cols="1" rows="1" role="data">47.79</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">12.69</cell><cell cols="1" rows="1" role="data">2.755</cell><cell cols="1" rows="1" role="data">284</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10.37</cell><cell cols="1" rows="1" role="data">41.33</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">289</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">51.37</cell><cell cols="1" rows="1" role="data">3.574</cell><cell cols="1" rows="1" role="data">284</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">50.57</cell><cell cols="1" rows="1" role="data">53.61</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">290</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">y</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">15.27</cell><cell cols="1" rows="1" role="data">3.685</cell><cell cols="1" rows="1" role="data">285</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">49.04</cell><cell cols="1" rows="1" role="data">55.27</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">d</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 6</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">55.86</cell><cell cols="1" rows="1" role="data">3.517</cell><cell cols="1" rows="1" role="data">286</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">57.90</cell><cell cols="1" rows="1" role="data">52.76</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">292</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Draconis</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">27.95</cell><cell cols="1" rows="1" role="data">0.033</cell><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">59.21</cell><cell cols="1" rows="1" role="data">0.49</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">293</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">28.28</cell><cell cols="1" rows="1" role="data">1.383</cell><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">4.19</cell><cell cols="1" rows="1" role="data">20.73</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">294</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24.12</cell><cell cols="1" rows="1" role="data">3.008</cell><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">1.79</cell><cell cols="1" rows="1" role="data">45.12</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">38.53</cell><cell cols="1" rows="1" role="data">2.415</cell><cell cols="1" rows="1" role="data">290</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">37.97</cell><cell cols="1" rows="1" role="data">30.23</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">296</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">4 6</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">39.61</cell><cell cols="1" rows="1" role="data">1.511</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">54.19</cell><cell cols="1" rows="1" role="data">22.67</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">297</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Antinoi</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">22.16</cell><cell cols="1" rows="1" role="data">3.106</cell><cell cols="1" rows="1" role="data">291</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">32.37</cell><cell cols="1" rows="1" role="data">46.59</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">298</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">5.16</cell><cell cols="1" rows="1" role="data">1.645</cell><cell cols="1" rows="1" role="data">202</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">17.40</cell><cell cols="1" rows="1" role="data">24.68</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">299</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittæ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">9.25</cell><cell cols="1" rows="1" role="data">2.678</cell><cell cols="1" rows="1" role="data">292</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">18.74</cell><cell cols="1" rows="1" role="data">40.17</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">f</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">41.43</cell><cell cols="1" rows="1" role="data">3.520</cell><cell cols="1" rows="1" role="data">293</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">21.39</cell><cell cols="1" rows="1" role="data">52.80</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">301</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">g</foreign> Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">12.50</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">293</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">7.50</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">302</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">44.50</cell><cell cols="1" rows="1" role="data">2.837</cell><cell cols="1" rows="1" role="data">294</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">7.50</cell><cell cols="1" rows="1" role="data">42.59</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">303</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">g</foreign> Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">0.81</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">294</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">12.16</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">304</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">43.09</cell><cell cols="1" rows="1" role="data">1.869</cell><cell cols="1" rows="1" role="data">294</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">46.34</cell><cell cols="1" rows="1" role="data">28.02</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">305</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">35.87</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">294</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">58.05</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">306</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Atair</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">1.02</cell><cell cols="1" rows="1" role="data">2.918</cell><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">15.30</cell><cell cols="1" rows="1" role="data">43.78</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">307</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">52.37</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">5.55</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">308</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Antinoi</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">17.14</cell><cell cols="1" rows="1" role="data">3.058</cell><cell cols="1" rows="1" role="data">295</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">17.08</cell><cell cols="1" rows="1" role="data">45.87</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">39.56</cell><cell cols="1" rows="1" role="data">3.699</cell><cell cols="1" rows="1" role="data">296</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">53.46</cell><cell cols="1" rows="1" role="data">55.48</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">28.97</cell><cell cols="1" rows="1" role="data">2.939</cell><cell cols="1" rows="1" role="data">296</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">14.55</cell><cell cols="1" rows="1" role="data">44.08</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">311</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">58.52</cell><cell cols="1" rows="1" role="data">3.097</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">37.<*></cell><cell cols="1" rows="1" role="data">46.45</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">312</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">32.70</cell><cell cols="1" rows="1" role="data">3.330</cell><cell cols="1" rows="1" role="data">301</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">11.88</cell><cell cols="1" rows="1" role="data">49.95</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">313</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi> Capri.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">17.48</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">301</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">22.20</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">56.48</cell><cell cols="1" rows="1" role="data">3.331</cell><cell cols="1" rows="1" role="data">301</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">7.20</cell><cell cols="1" rows="1" role="data">49.96</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">315</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi> Capri.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">33.32</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">302</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">19.80</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="510"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in degrees, &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">316</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">31.35</cell><cell cols="1" rows="1" role="data">2.380</cell><cell cols="1" rows="1" role="data">302</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">50.19</cell><cell cols="1" rows="1" role="data">50.70</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">317</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">33.69</cell><cell cols="1" rows="1" role="data">3.337</cell><cell cols="1" rows="1" role="data">302</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">25.39</cell><cell cols="1" rows="1" role="data">50.06</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">45.50</cell><cell cols="1" rows="1" role="data">3.380</cell><cell cols="1" rows="1" role="data">302</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">22.54</cell><cell cols="1" rows="1" role="data">50.70</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">319</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">2.63</cell><cell cols="1" rows="1" role="data">2.148</cell><cell cols="1" rows="1" role="data">303</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">39.39</cell><cell cols="1" rows="1" role="data">32.22</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">320</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">26.45</cell><cell cols="1" rows="1" role="data">3.438</cell><cell cols="1" rows="1" role="data">304</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">36.72</cell><cell cols="1" rows="1" role="data">51.57</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">321</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">57.50</cell><cell cols="1" rows="1" role="data">2.801</cell><cell cols="1" rows="1" role="data">306</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">22.44</cell><cell cols="1" rows="1" role="data">42.01</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">322</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">10.32</cell><cell cols="1" rows="1" role="data">2.804</cell><cell cols="1" rows="1" role="data">307</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">34.74</cell><cell cols="1" rows="1" role="data">42.06</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">323</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">20.76</cell><cell cols="1" rows="1" role="data">2.780</cell><cell cols="1" rows="1" role="data">307</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">11.37</cell><cell cols="1" rows="1" role="data">41.70</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">324</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Deneb.</cell><cell cols="1" rows="1" rend="align=center" role="data">1 2</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">36.68</cell><cell cols="1" rows="1" role="data">2.034</cell><cell cols="1" rows="1" role="data">308</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">10.20</cell><cell cols="1" rows="1" role="data">30.51</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">325</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">28.55</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8.25</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">326</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">50.38</cell><cell cols="1" rows="1" role="data">3.255</cell><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">35.72</cell><cell cols="1" rows="1" role="data">48.83</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">327</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">g</foreign> Delphini</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">21.94</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">29.08</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">22.96</cell><cell cols="1" rows="1" role="data">2.783</cell><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">44.34</cell><cell cols="1" rows="1" role="data">41.75</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">329</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">6.70</cell><cell cols="1" rows="1" role="data">2.393</cell><cell cols="1" rows="1" role="data">309</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">40.56</cell><cell cols="1" rows="1" role="data">35.89</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">330</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">51.17</cell><cell cols="1" rows="1" role="data">3.233</cell><cell cols="1" rows="1" role="data">310</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">47.61</cell><cell cols="1" rows="1" role="data">48.65</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">331</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">4.59</cell><cell cols="1" rows="1" role="data">3.255</cell><cell cols="1" rows="1" role="data">311</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">8.80</cell><cell cols="1" rows="1" role="data">48.82</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">332</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">5 4</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">39.75</cell><cell cols="1" rows="1" role="data">3.384</cell><cell cols="1" rows="1" role="data">313</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">56.25</cell><cell cols="1" rows="1" role="data">50.76</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">333</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">41.24</cell><cell cols="1" rows="1" role="data">3.274</cell><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">18.64</cell><cell cols="1" rows="1" role="data">49.11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">334</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Equulei</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">48.78</cell><cell cols="1" rows="1" role="data">2.997</cell><cell cols="1" rows="1" role="data">316</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">11.73</cell><cell cols="1" rows="1" role="data">44.96</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">335</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">5.67</cell><cell cols="1" rows="1" role="data">3.355</cell><cell cols="1" rows="1" role="data">317</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">25.00</cell><cell cols="1" rows="1" role="data">50.33</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">336</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Equulei</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">57.71</cell><cell cols="1" rows="1" role="data">2.981</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">25.62</cell><cell cols="1" rows="1" role="data">44.72</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">337</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14.77</cell><cell cols="1" rows="1" role="data">3.286</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">41.53</cell><cell cols="1" rows="1" role="data">49.29</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">338</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cephei</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">46.85</cell><cell cols="1" rows="1" role="data">1.427</cell><cell cols="1" rows="1" role="data">318</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">42.69</cell><cell cols="1" rows="1" role="data">21.40</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">339</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">1.12</cell><cell cols="1" rows="1" role="data">3.165</cell><cell cols="1" rows="1" role="data">320</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">16.74</cell><cell cols="1" rows="1" role="data">47.48</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">340</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">52.29</cell><cell cols="1" rows="1" role="data">3.379</cell><cell cols="1" rows="1" role="data">321</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">4.34</cell><cell cols="1" rows="1" role="data">50.68</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">341</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cephei</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">1.18</cell><cell cols="1" rows="1" role="data">0.821</cell><cell cols="1" rows="1" role="data">321</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">17.76</cell><cell cols="1" rows="1" role="data">12.32</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">342</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">3 4</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">59.14</cell><cell cols="1" rows="1" role="data">3.329</cell><cell cols="1" rows="1" role="data">322</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">47.15</cell><cell cols="1" rows="1" role="data">49.93</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">343</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">27.98</cell><cell cols="1" rows="1" role="data">3.360</cell><cell cols="1" rows="1" role="data">322</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">59.74</cell><cell cols="1" rows="1" role="data">50.40</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">344</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">21.53</cell><cell cols="1" rows="1" role="data">2.943</cell><cell cols="1" rows="1" role="data">323</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">22.99</cell><cell cols="1" rows="1" role="data">44.15</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">345</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">59.63</cell><cell cols="1" rows="1" role="data">2.116</cell><cell cols="1" rows="1" role="data">323</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">54.38</cell><cell cols="1" rows="1" role="data">31.74</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">346</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">58.68</cell><cell cols="1" rows="1" role="data">3.310</cell><cell cols="1" rows="1" role="data">323</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">40.25</cell><cell cols="1" rows="1" role="data">49.65</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">347</cell><cell cols="1" rows="1" rend="align=left" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8.21</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">3.15</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">348</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">29.75</cell><cell cols="1" rows="1" role="data">3.067</cell><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">26.25</cell><cell cols="1" rows="1" role="data">46.00</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">349</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">18.89</cell><cell cols="1" rows="1" role="data">3.094</cell><cell cols="1" rows="1" role="data">332</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">43.39</cell><cell cols="1" rows="1" role="data">46.41</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">350</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">4.00</cell><cell cols="1" rows="1" role="data">3.065</cell><cell cols="1" rows="1" role="data">333</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">59.98</cell><cell cols="1" rows="1" role="data">45.97</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">351</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">31.68</cell><cell cols="1" rows="1" role="data">3.079</cell><cell cols="1" rows="1" role="data">334</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">55.26</cell><cell cols="1" rows="1" role="data">46.18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">352</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">s</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2.99</cell><cell cols="1" rows="1" role="data">3.186</cell><cell cols="1" rows="1" role="data">335</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">44.84</cell><cell cols="1" rows="1" role="data">47.79</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">353</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Lacertæ</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">7.61</cell><cell cols="1" rows="1" role="data">2.431</cell><cell cols="1" rows="1" role="data">335</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">54.14</cell><cell cols="1" rows="1" role="data">36.46</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">354</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">4.59</cell><cell cols="1" rows="1" role="data">3.079</cell><cell cols="1" rows="1" role="data">336</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">8.78</cell><cell cols="1" rows="1" role="data">46.19</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">355</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">k</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23.38</cell><cell cols="1" rows="1" role="data">3.117</cell><cell cols="1" rows="1" role="data">336</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">50.67</cell><cell cols="1" rows="1" role="data">46.76</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">356</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">29.06</cell><cell cols="1" rows="1" role="data">2.981</cell><cell cols="1" rows="1" role="data">337</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">15.89</cell><cell cols="1" rows="1" role="data">44.72</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">357</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">37.87</cell><cell cols="1" rows="1" role="data">3.792</cell><cell cols="1" rows="1" role="data">338</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">27.91</cell><cell cols="1" rows="1" role="data">41.88</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">358</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">4.70</cell><cell cols="1" rows="1" role="data">3.197</cell><cell cols="1" rows="1" role="data">339</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">10.57</cell><cell cols="1" rows="1" role="data">47.96</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">t</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">59.29</cell><cell cols="1" rows="1" role="data">3.190</cell><cell cols="1" rows="1" role="data">339</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">49.41</cell><cell cols="1" rows="1" role="data">47.85</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">360</cell><cell cols="1" rows="1" rend="align=left" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">10.52</cell><cell cols="1" rows="1" role="data">3.137</cell><cell cols="1" rows="1" role="data">340</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">37.87</cell><cell cols="1" rows="1" role="data">47.05</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="511"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=17" role="data"><hi rend="italics">Catalogue of the principal Fixed Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No. of Stars.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Names and Characters of the Stars.</cell><cell cols="1" rows="1" role="data">Magni- tude.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in time.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Right Ascens. in degrees, &c.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell><cell cols="1" rows="1" rend="colspan=4" role="data">Declination North and South.</cell><cell cols="1" rows="1" role="data">Annual Variat. in ditto.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"><hi rend="italics">h.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">m.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/100</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi> 1/1000</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">″ 1/100</cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data">+</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">361</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cephei</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">35.33</cell><cell cols="1" rows="1" role="data">2.109</cell><cell cols="1" rows="1" role="data">340</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">49.95</cell><cell cols="1" rows="1" role="data">31.63</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">362</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">1.80</cell><cell cols="1" rows="1" role="data">3.201</cell><cell cols="1" rows="1" role="data">341</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">27.06</cell><cell cols="1" rows="1" role="data">48.02</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">363</cell><cell cols="1" rows="1" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign>Pisc. aust.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">17.35</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">340</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">20.25</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">364</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Fomalhaut</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">33 60</cell><cell cols="1" rows="1" role="data">3.330</cell><cell cols="1" rows="1" role="data">341</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">24.00</cell><cell cols="1" rows="1" role="data">49.95</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Pisc. aust.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">38.04</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">342</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30.60</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">366</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">5.50</cell><cell cols="1" rows="1" role="data">2.874</cell><cell cols="1" rows="1" role="data">343</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">22.47</cell><cell cols="1" rows="1" role="data">43.11</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">367</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Markab</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">47.99</cell><cell cols="1" rows="1" role="data">2.964</cell><cell cols="1" rows="1" role="data">343</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">59.85</cell><cell cols="1" rows="1" role="data">44.46</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">368</cell><cell cols="1" rows="1" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign> Pegasi</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">35.64</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">343</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">54.60</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">369</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">f</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">4 5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">57.39</cell><cell cols="1" rows="1" role="data">3.109</cell><cell cols="1" rows="1" role="data">345</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">20.79</cell><cell cols="1" rows="1" role="data">46.64</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">370</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">y</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">23.05</cell><cell cols="1" rows="1" role="data">3.125</cell><cell cols="1" rows="1" role="data">346</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">45 68</cell><cell cols="1" rows="1" role="data">46.88</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">371</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">46.42</cell><cell cols="1" rows="1" role="data">3.057</cell><cell cols="1" rows="1" role="data">346</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">36.37</cell><cell cols="1" rows="1" role="data">45.85</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">372</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">y</foreign><hi rend="sup">3</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">32.63</cell><cell cols="1" rows="1" role="data">3.125</cell><cell cols="1" rows="1" role="data">347</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9.48</cell><cell cols="1" rows="1" role="data">46.88</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">S</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">373</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">11.40</cell><cell cols="1" rows="1" role="data">3.065</cell><cell cols="1" rows="1" role="data">351</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">50.96</cell><cell cols="1" rows="1" role="data">45.97</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">374</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">51.01</cell><cell cols="1" rows="1" role="data">3.066</cell><cell cols="1" rows="1" role="data">352</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">45.08</cell><cell cols="1" rows="1" role="data">45.99</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">375</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">5 6</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">10.88</cell><cell cols="1" rows="1" role="data">3.062</cell><cell cols="1" rows="1" role="data">354</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">43.18</cell><cell cols="1" rows="1" role="data">45.93</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">376</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">w</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">2.88</cell><cell cols="1" rows="1" role="data">3.061</cell><cell cols="1" rows="1" role="data">357</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">43.16</cell><cell cols="1" rows="1" role="data">45.92</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">377</cell><cell cols="1" rows="1" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign>Androm.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">45 47</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">358</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">22.05</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">378</cell><cell cols="1" rows="1" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">præc. <foreign xml:lang="greek">a</foreign> Androm.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">15.28</cell><cell cols="1" rows="1" role="data">3.060</cell><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">49.19</cell><cell cols="1" rows="1" role="data">45.90</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">379</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Andromedæ</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">4.32</cell><cell cols="1" rows="1" role="data">3.065</cell><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">4.95</cell><cell cols="1" rows="1" role="data">45.97</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">380</cell><cell cols="1" rows="1" role="data">*</cell><cell cols="1" rows="1" rend="align=left" role="data">seq. <foreign xml:lang="greek">a</foreign>Androm.</cell><cell cols="1" rows="1" rend="align=center" role="data">.</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">32.98</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">14.70</cell><cell cols="1" rows="1" role="data">. . .</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">381</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cassiopeiæ</cell><cell cols="1" rows="1" rend="align=center" role="data">2 3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">34.32</cell><cell cols="1" rows="1" role="data">3.051</cell><cell cols="1" rows="1" role="data">359</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">34.75</cell><cell cols="1" rows="1" role="data">45.76</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">N</cell><cell cols="1" rows="1" role="data"/></row></table> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=14" role="data"><hi rend="italics">Another</hi> <hi rend="smallcaps">Catalogue</hi> <hi rend="italics">of</hi> 162 <hi rend="smallcaps">Principal Stars</hi>, <hi rend="italics">shewing their Mean Declinations to Beginning</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=14" role="data"><hi rend="italics">of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Stars Names.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Mean Declin.</cell><cell cols="1" rows="1" role="data">Annual Va-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Stars Names.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Mean Declin.</cell><cell cols="1" rows="1" role="data">Annual Va-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">north.</cell><cell cols="1" rows="1" role="data">riation.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">north.</cell><cell cols="1" rows="1" role="data">riation.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Polaris</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 19.57</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lyræ</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 2.59</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Polaris</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Lyræ</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">- 18.20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">- 7.40</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">+ 13.59</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Castor</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 6.95</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ursæ majoris</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">- 13.21</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Castor</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Persei</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">+ 12.35</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Pollux</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">- 7.46</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Capella</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">+ 5.09</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 4.08</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 12.52</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">- 15.59</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">- 14.54</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Andromedæ</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">+ 20.25</cell></row></table><pb n="512"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=14" role="data"><hi rend="italics">The Mean Declinations of</hi> 162 <hi rend="italics">principal Stars for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Stars Names.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Mean Declin.</cell><cell cols="1" rows="1" role="data">Annual Va-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Stars Names.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Mean Declin.</cell><cell cols="1" rows="1" role="data">Annual Va-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">north.</cell><cell cols="1" rows="1" role="data">riation.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">north.</cell><cell cols="1" rows="1" role="data">riation.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Andromedæ</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">+ 20.25</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">- 2.22</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cygni</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">+ 7.04</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">- 11.01</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Gemma</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 12.50</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">- 11.75</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Gemma</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Aldebaran</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 8.16</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">- 16.46</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Aldebaran</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">45</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">+ 19.21</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 19.96</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 2.72</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">27</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">+ 12.68</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">- 4.56</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">+ 12.21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">- 16.10</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">+ 9.42</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">- 17.56</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">- 15.85</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Alcione</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">+ 11.88</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">- 4.75</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Electra</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">+ 12.04</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 19.22</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Atlas</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">+ 11.74</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">57</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Propus</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">+ 0.75</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">+ 20.04</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">t</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">+ 19.57</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">+ 20.04</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">- 0.89</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">+ 12.05</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">- 0.19</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">+ 4.83</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">- 0.19</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Regulus</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 17.24</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">+ 17.55</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Regulus</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">+ 17.55</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">- 13.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">- 5.83</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 3.05</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">- 12.28</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">- 9.67</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">- 19.54</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">- 8.38</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">- 12.57</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">- 19.43</cell><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">- 15.94</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">+ 3.05</cell><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Delphini</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">+ 11.73</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">- 17.72</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">- 18.24</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 4.48</cell><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 8.17</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">n</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">- 1.44</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">+ 16.10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Arcturus</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">- 19.10</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis minoris</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">- 6.51</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Arcturus</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">* - 19.10</cell><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 8.51</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Herculis</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">- 9.05</cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">48</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Bootis</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">- 18.00</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">+ 1.42</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Cancri</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">- 12.40</cell><cell cols="1" rows="1" role="data">101</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">+ 1.42</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Pegasi</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">+ 14.91</cell><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">- 12.60</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">+ 18.03</cell><cell cols="1" rows="1" role="data">103</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 11.94</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Arietis</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">+ 18.09</cell><cell cols="1" rows="1" role="data">104</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">50</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittæ</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">+ 7.73</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">- 11.97</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">- 17.18</cell><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">+ 8.86</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittæ</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">+ 7.73</cell><cell cols="1" rows="1" role="data">107</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">+ 8.86</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Tauri</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">+ 9.19</cell><cell cols="1" rows="1" role="data">108</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Procyon</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">- 7.51</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leonis</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">- 19.43</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">- 2.35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Geminorum</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">- 2.22</cell><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">- 19.66</cell></row></table><pb n="513"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=14" role="data"><hi rend="italics">The Mean Declinations of</hi> 162 <hi rend="smallcaps">Principal Stars</hi> <hi rend="italics">for the Beginning of the Year</hi> 1800.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Stars Names.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Mean Declin.</cell><cell cols="1" rows="1" role="data">Annual Va-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Stars Names.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Mean Declin.</cell><cell cols="1" rows="1" role="data">Annual Va-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">north.</cell><cell cols="1" rows="1" role="data">riation.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">south.</cell><cell cols="1" rows="1" role="data">riation.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">+ 3.97</cell><cell cols="1" rows="1" role="data">156</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">- 10.90</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">112</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 14.70</cell><cell cols="1" rows="1" role="data">157</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>+ 15.40</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">113</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">158</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">114</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 19.97</cell><cell cols="1" rows="1" role="data">159</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">+ 19.98</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">115</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">160</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">- 10.71</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">116</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">- 1.97</cell><cell cols="1" rows="1" role="data">161</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">+ 4.69</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">117</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquilæ</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">+ 6.44</cell><cell cols="1" rows="1" role="data">162</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">+ 5.33</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">118</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">+ 15.77</cell><cell cols="1" rows="1" role="data">163</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">i</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">- 17.14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">119</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Piscium</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">+ 17.73</cell><cell cols="1" rows="1" role="data">164</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">+ 20.04</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">h</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Antinoi</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">+ 8.63</cell><cell cols="1" rows="1" role="data">165</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Sirius</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">+ 4.43</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">121</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">- 3.38</cell><cell cols="1" rows="1" role="data">166</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Sirius</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">+ 4.33</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">122</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">- 18.72</cell><cell cols="1" rows="1" role="data">167</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">- 18.85</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center colspan=3" role="data">South Decl.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">123</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">i</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">- 15.86</cell><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">- 16.19</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">124</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">+ 19.86</cell><cell cols="1" rows="1" role="data">169</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Crateris</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">+ 19.11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">- 15.97</cell><cell cols="1" rows="1" role="data">170</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">i</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">- 15.82</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">126</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 17.15</cell><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">- 14.97</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">127</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">172</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">+ 1.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">128</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">- 3.02</cell><cell cols="1" rows="1" role="data">173</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leporis</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">- 3.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">129</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Antinoi</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">- 10.05</cell><cell cols="1" rows="1" role="data">174</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">- 13.81</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">130</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">- 2.60</cell><cell cols="1" rows="1" role="data">175</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">+ 10.03</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">131</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">- 17.81</cell><cell cols="1" rows="1" role="data">176</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">- 19.84</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">132</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">+ 9.77</cell><cell cols="1" rows="1" role="data">177</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">+ 10.52</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">133</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Serpentis</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">- 0.93</cell><cell cols="1" rows="1" role="data">178</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">- 0.09</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">134</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">+ 9.47</cell><cell cols="1" rows="1" role="data">179</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">- 4.95</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">+ 19.39</cell><cell cols="1" rows="1" role="data">180</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">+ 20.05</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">136</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">- 5.41</cell><cell cols="1" rows="1" role="data">181</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">+ 10.92</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">137</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">i</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Orionis</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">- 3.04</cell><cell cols="1" rows="1" role="data">182</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">o</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Sagittarii</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">- 4.51</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">138</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">- 15.39</cell><cell cols="1" rows="1" role="data">183</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">+ 19.94</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">139</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">f</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">- 19.44</cell><cell cols="1" rows="1" role="data">184</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Leporis</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">- 2.11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">+ 15.21</cell><cell cols="1" rows="1" role="data">185</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Corvi</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">+ 20.04</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">141</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Hydræ</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">+ 15.21</cell><cell cols="1" rows="1" role="data">186</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">+ 14.67</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Rigel</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">- 4.81</cell><cell cols="1" rows="1" role="data">187</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">+ 4.43</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">143</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">b</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">+ 13.82</cell><cell cols="1" rows="1" role="data">188</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">s</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">+ 9.38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">144</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Aquarii</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">- 18.89</cell><cell cols="1" rows="1" role="data">189</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">p</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Scorpii</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">+ 11.06</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">145</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Spica</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">+ 19.01</cell><cell cols="1" rows="1" role="data">190</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Antares</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">+ 8.75</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Spicæ</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">+ 19.01</cell><cell cols="1" rows="1" role="data">191</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Antares</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">+ 8.75</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">147</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ophiuchi</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">+ 8.02</cell><cell cols="1" rows="1" role="data">192</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">+ 5.18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">148</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">d</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Eridani</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">- 11.99</cell><cell cols="1" rows="1" role="data">193</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">e</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">+ 4.38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">149</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">m</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Ceti</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">+ 15.71</cell><cell cols="1" rows="1" role="data">194</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">z</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Canis majoris</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">+ 1.07</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">150</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">l</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Virginis</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">+ 17.01</cell><cell cols="1" rows="1" role="data">195</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Fomalhaut</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">- 19.01</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 10.47</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">1</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">153</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>- 10.50</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">154</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">a</foreign><hi rend="sup">2</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">Capricorni</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">155</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">g</foreign></cell><cell cols="1" rows="1" rend="align=left" role="data">Libræ</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">+ 12.63</cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell><cell cols="1" rows="1" role="data"> </cell></row></table><pb n="514"/><cb/></p><div2 part="N" n="Star" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Star</hi></head><p>, in Electricity, denotes the appearance of the electric matter on a point into which it enters. Beccaria supposes that the Star is occasioned by the difficulty with which the electric fluid is extricated from the air, which is an electric substance. See <hi rend="smallcaps">Brush.</hi></p></div2><div2 part="N" n="Star" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Star</hi></head><p>, in Fortification, denotes a small fort, having 5 or more points, or saliant and re-entering angles, flanking one another, and their faces 90 or 100 feet long.</p></div2><div2 part="N" n="Star" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Star</hi></head><p>, in Pyrotechny, a composition of combustible matters; which being borne, or thrown aloft into the air, exhibits the appearance of a real Star.—Stars are chiefly used as appendages to rockets, a number of them being usually inclosed in a conical cap, or cover, at the head of the rocket, and carried up with it to its utmost height, where the Stars, taking fire, are spread around, and exhibit an agreeable spectacle.</p><p><hi rend="italics">To make Stars.</hi>—Mix 3lbs of saltpetre, 11 ounces of sulphur, one of antimony, and 3 of gunpowder dust: or, 12 ounces of sulphur, 6 of saltpetre, 5 1/2 of gunpowder dust, 4 of olibanum, one of mastic, camphor, sublimate of mercury, and half an ounce of antimony and orpiment. Moisten the mass with gumwater, and make it into little balls, of the size of a chesnut; which dry either in the sun, or in the oven. These being set on fire in the air, will represent Stars.</p><p><hi rend="smallcaps">Star</hi>-<hi rend="italics">Board</hi> denotes the right hand side of a ship, when a person on board stands with the face looking forward towards the head or fore part of the ship. In contradistinction from <hi rend="italics">Larboard,</hi> which denotes the left hand side of the ship in the same circumstances.— They say, <hi rend="italics">Starboard the helm,</hi> or <hi rend="italics">helm a Starboard,</hi> when the man at the helm should put the helm to the right hand side of the ship.</p><p><hi rend="italics">Falling</hi> <hi rend="smallcaps">Star</hi>, or <hi rend="italics">Shooting</hi> <hi rend="smallcaps">Star</hi>, a luminous meteor darting rapidly through the air, and resembling a Star falling.—The explication of this phenomenon has puzzled all philosophers, till the modern discoveries in electricity have led to the most probable account of it. Signior Beccaria makes it pretty evident, that it is an electrical appearance, and recites the following fact in proof of it. About an hour after sunset, he and some friends that were with him, observed a falling Star directing its course towards them, and apparently growing larger and larger, but it disappeared not far from them; when it left their faces, hands, and clothes, with the earth, and all the neighbouring objects, suddenly illuminated with a diffused and lambent light, not attended with any noise at all. During their surprize at this appearance, a servant informed them that he had seen a light shine suddenly in the garden, and especially upon the streams which he was throwing to water it. All these appearances were evidently electrical; and Beccaria was confirmed in his conjecture, that electricity was the cause of them, by the quantity of electric matter which he had seen gradually advancing towards his kite, which had very much the appearance of a falling Star. Sometimes also he saw a kind of glory round the kite, which followed it when it changed its place, but left some light, for a small space of time, in the place it had quitted. Priestley's Elect. vol. 1, pa. 434, 8vo. See <hi rend="smallcaps">Ignis</hi> <hi rend="italics">Fatuus.</hi></p><p><hi rend="smallcaps">Star</hi>-<hi rend="italics">fort,</hi> or <hi rend="italics">Redoubt,</hi> in Fortification. See <hi rend="smallcaps">Star, Redoubt</hi>, and <hi rend="smallcaps">Fort.</hi> <cb/></p></div2></div1><div1 part="N" n="STARLINGS" org="uniform" sample="complete" type="entry"><head>STARLINGS</head><p>, or <hi rend="smallcaps">Sterlings</hi>, or <hi rend="italics">Fettecs,</hi> a kind of case made about a pier of stilts, &c, to secure it. See <hi rend="smallcaps">Stilts.</hi></p></div1><div1 part="N" n="STATICS" org="uniform" sample="complete" type="entry"><head>STATICS</head><p>, a branch of mathematics which considers weight or gravity, and the motion of bodies resulting from it.</p><p>Those who define mechanics, the science of motion, make Statics a part of it; viz, that part which considers the motion of bodies arising from gravity.</p><p>Others make them two distinct doctrines; restraining mechanics to the doctrine of motion and weight, as depending on, or connected with, the power of machines; and Statics to the doctrine of motion, considered merely as arising from the weight of bodies, without any immediate respect to machines. In this way, Statics should be the doctrine or theory of motion; and mechanics, the application of it to machines.</p><p>For the laws of Statics, see <hi rend="smallcaps">Gravity, Descent</hi>, &c.</p></div1><div1 part="N" n="STATION" org="uniform" sample="complete" type="entry"><head>STATION</head><p>, or <hi rend="smallcaps">Stationary</hi>, in Astronomy, the position or appearance of a planet in the same point of the zodiac, for several days. This happens from the observer being situated on the earth, which is far out of the centre of their orbits, by which they seem to proceed irregularly; being sometimes seen to go forwards, or from west to east, which is their natural <hi rend="italics">direction;</hi> sometimes to go backwards, or from east to west, which is their <hi rend="italics">retrogradation;</hi> and between these two states there must be an intermediate one, where the planet appears neither to go forwards nor backwards, but to stand still, and keep the same place in the heavens, which is called her <hi rend="italics">Station,</hi> and the planet is then said to be <hi rend="italics">Stationary.</hi></p><p>Apollonius Pergæus has shewn how to find the Stationary point of a planet, according to the old theory of the planets, which supposes them to move in epicycles; which was followed by Ptolomy in his Almag. lib. 12, cap. 1, and others, till the time of Copernicus. Concerning this, see Regiomontanus in Epitome Almagesti, lib. 12, prop. 1; Copernicus's Revolutiones Cœlest. lib. 5, cap. 35 and 36; Kepler in Tabulis Rudolphinis, cap. 24; Riccioli's Almag. lib. 7, sect. 5, cap. 2: Harman in Miscellan. Berolinens, pa. 197. Dr. Halley, Mr. Facio, Mr. De Moivre, Dr. Keil, and others have treated on this subject. See also the articles <hi rend="smallcaps">Retrograde</hi> and <hi rend="smallcaps">Stationary</hi> in this Dictionary.</p><div2 part="N" n="Station" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Station</hi></head><p>, in Practical Geometry &c, is a place pitched upon to make an observation, or take an angle, or such like, as in surveying, measuring heights-anddistances, levelling, &c.</p><p>An accessible height is taken from one Station; but an inaccessible height or distance is only to be taken by making two Stations, from two places whose distance asunder is known. In making maps of counties, provinces, &c, Stations are fixed upon certain eminencies &c of the country, and angles taken from thence to the several towns, villages, &c.—In surveying, the instrument is to be adjusted by the needle, or otherwise, to answer the points of the horizon at every Station; the distance from hence to the last Station is to be measured, and an angle is to be taken to the next Station; which process repeated includes the chief practice of <pb/><pb/><pb n="515"/><cb/> surveying.—In levelling, the instrument is rectified, or placed level at each Station, and observations made forwards and backwards.</p><p>There is a method of measuring distances at one Station, in the Philos. Trans. numb. 7, by means of a telescope. I have heard of another, by Mr. Ramsden; and have seen a third ingenious way by Mr. Green of Deptford, not yet published; this consists of a permanent scale of divisions, placed at any point whose distance is required; then the number of divisions seen through the telescope, gives the distance sought.</p><p><hi rend="smallcaps">Station</hi>-<hi rend="italics">Line,</hi> in Surveying, and <hi rend="italics">Line of Station,</hi> in Perspective. See <hi rend="smallcaps">Line.</hi></p></div2></div1><div1 part="N" n="STATIONARY" org="uniform" sample="complete" type="entry"><head>STATIONARY</head><p>, in Astronomy, the state of a planet when, to an observer on the earth, it appears for some time to stand still, or remain immoveable in the same place in the heavens. For as the planets, to such an observer, have sometimes a progressive motion, and sometimes a retrograde one, there must be some point between the two where they must appear Stationary. Now a planet will be seen Stationary, when the line that joins the centres of the earth and planet is constantly directed to the same point in the heavens, which is when it keeps parallel to itself. For all right lines drawn from any point of the earth's orbit, parallel to one another, do all point to the same star; the distance of these lines being insensible, in comparison of that of the fixed stars.</p><p>The planet Herschel is seen Stationary at the distance of from the sun; Saturn at somewhat more than 90°; Jupiter at the distance of 52°; and Mars at a much greater distance; Venus at 47°, and Mercury at 28°.</p><p>Herschel is Stationary days, Saturn 8, Jupiter 4, Mars 2, Venus 1 1/2, and Mercury 1/2 a day: though the several stations are not always equal; because the orbits of the planets are not circles which have the sun in their centre.</p></div1><div1 part="N" n="STEAM" org="uniform" sample="complete" type="entry"><head>STEAM</head><p>, the smoke or vapour arising from water, or any other liquid or moist body, when considerably heated. Subterranean Steams often affect the surface of the earth in a remarkable manner, and promote or prevent vegetation more than any thing else. It has been imagined that Steams may be the generative cause of both minerals and metals, and of all the peculiarities of springs. See Philos. Trans. vol. 5, pa. 1154, or Abr. vol. 2, pa. 833.—Of the use of the air to elevate the Steams of bodies, see pa. 2048 and 297 ib.— Concerning the warm and fertilizing temperature and Steams of the earth, see Phil. Trans. vol. 10, pa. 307 and 357. See also Dr. Hamilton “On the Ascent of Vapours.”</p><p>The Steam raised from hot water is an elastic fluid, which, like elastic air, has its elasticity proportional to its density when the heat is the same, or proportional to the heat when the density is the same. The Steam raised with the ordinary heat of boiling water, is almost 3000 times rarer than water, or about 3 1/2 times rarer than air, and has its elasticity about equal to that of the common air of the atmosphere. And by great heat it has been found that the Steam may be expanded into 14000 times the space of water, or may be made about 5 times stronger than the atmosphere. But from some accidents that have happened, <cb/> it appears that Steam, suddenly raised from water, or moist substances, by the immediate application of strong heat, is vastly stronger than the atmosphere, or even than gunpowder itself. Witness the accident that happened to a foundery of cannon at Moorfields, when upon the hot metal first running into the mould in the earth, some small quantity of water in the bottom of it was suddenly changed into Steam, which by its explosion, blew the foundery all to pieces. I remember another such accident at a foundery at Newcastle; the founder having purchased, among some old brass, a hollow brass ball that had been used for many years as a valve in a pump, withinside of which it would seem some water had got insinuated; and having put it into his fire to melt, when it had become very hot, it suddenly burst with a prodigious noise, and blew the adjacent parts of the furnace in pieces.</p><p>Steam may be applied to many purposes useful in life, but one of its chief uses is in the Steam-engine described in the following article.</p><p><hi rend="smallcaps">Steam</hi> <hi rend="italics">Engine,</hi> an engine for raising water by the force of Steam produced from boiling water; and often called the <hi rend="italics">Fire-engine,</hi> on account of the fire employed in boiling the water to produce the Steam. This is one of the most curious and useful machines, which modern art can boast, for raising water from ponds, wells, or pits, for draining mines, &c. Were it not for the use of this most important invention, it is probable we should not now have the benefit of coal fires in England; as our forefathers had, before the present century, excavated all the mines of coal as deep as it could be worked, without the benefit of this engine to draw the water from greater depths.</p><p>This engine is commonly a forcing pump, having its rod fixed to one end of a lever, which is worked by the weight or pressure of the atmosphere upon a piston, at the other end, a temporary vacuum being made below it, by suddenly condensing the Steam, that had been let into the cylinder in which this piston works, by a jet of cold water thrown into it. A partial vacuum being thus made, the weight of the atmosphere presses down the piston, and raises the other end of the straight lever with the water from the well &c. Then immediately a hole is uncovered in the bottom of the cylinder, by which a fresh fill of hot Steam rushes in from a boiler of water below it, which proves a counterbalance for the atmosphere above the piston, upon which the weight of the pump rods at the other end of the lever carries that end down, and raises the piston of the Steam cylinder. Immediately the Steam hole is shut, and the cock opened for injecting the cold water into the cylinder of Steam, which condenses it to water again, and thus making another vacuum below the piston, the atmosphere above it presses it down, and raises the pump rods with another lift of water; and so on continually. This is the common principle: but there are also other modes of applying the force of the Steam, as we shall see in the following short history of this invention and its various improvements.</p><p>The earliest account to be met with of the invention of this engine, is in the marquis of Worcester's small book intitled a Century of Inventions (being a description of 100 notable discoveries), published in the year 1663, where he proposed the raising of great quantities <pb n="516"/><cb/> of water by the sorce of Steam, raised from water by means of fire; and he mentions an engine of that kind, of his own contrivance, which could raise a continual stream like a fountain 40 feet high, by means of two cocks which were alternately and successively turned by a man to admit the Steam, and to re-sill the vessel with cold water, the fire being continually kept up.</p><p>However, this invention not meeting with encouragement, probably owing to the confused state of public affairs at that time, it was neglected, and lay dormant several years, until one Captain Thomas Savery, having read the marquis of Worcester's books, several years afterwards, tried many experiments upon the force and power of Steam; and at last hit upon a method of applying it to raise water. He then bought up and destroyed all the marquis's books that could be got, and claimed the honour of the invention to himself, and obtained a patent for it, pretending that he had discovered this secret of nature by accident. He contrived an engine which, after many experiments, he brought to some degree of perfection, so as to raise water in small quantities: but he could not succeed in raising it to any great height, or in large quantities, for the draining of mines; to effect which by his method, the Steam was required to be so strong as would have burst all his vessels; so that he was obliged to limit himself to raising the water only to a small height, or in small quantities. The largest engine he erected, was for the York-buildings Company in London, for supplying the inhabitants in the Strand and that neighbourhood with water.</p><p>The principle of this machine was as follows: H (fig. 3, pl. 27) represents a copper boiler placed on a furnace. E is a strong iron vessel, communicating with the boiler by means of a pipe at top, and with the main pipe AB by means of a pipe I at bottom; AB is the main pipe immersed in the water at B; D and C are two fixed valves, both opening upwards, one being placed above, and the other below the pipe of communication I. Lastly, at G is a cock that serves occasionally to wet and cool the vessel E, by water from the main pipe, and F is a cock in the pipe of communication between the vessel E and the boiler.</p><p>The engine is set to work, by filling the copper in part with water, and also the upper part of the main pipe above the valve C, the fire in the furnace being lighted at the same time. When the water boils strongly, the cock F is opened, the Steam rushes into the vessel E, and expels the air from thence through the valve C. The vessel E thus filled, and violently heated by the Steam, is suddenly cooled by the water which falls upon it by turning the cock C; the cock F being at the same time shut, to prevent any fresh accession of Steam from the boiler. Hence, the Steam in E becoming condensed, it leaves the cavity within almost intirely a vacuum; and therefore the pressure of the atmosphere at B forces the water through the valve D till the vessel E is nearly filled. The condensing cock G is then shut, and the Steam cock F again opened; hence the Steam, rushing into E, expels the water through the valve C, as it before did the air. Thus E becomes again filled with hot Steam, which is again cooled and condensed by the water from G, the supply of Steam being cut off by shutting F, as in the former <cb/> operation: the water consequently rushes through D, by the pressure of the atmosphere at B, and E is again silled. This water is forced up the main pipe through C, by opening F and shutting G, as before. And thus it is easy to conceive, that by this alternate opening and shutting the cocks, water will be continually raised, as long as the boiler continues to supply the Steam.</p><p>For the sake of perspicuity, the drawing is divested of the apparatus that serves to turn the two cocks at once, and of the contrivances for filling the copper to the proper quantity. But it may be found complete, with a full account of its uses and application, in Mr. Savery's book intituled the <hi rend="italics">Miner's Friend.</hi> The engines of this construction were usually made to work with two receivers or Steam vessels, one to receive the Steam, while the other was raising water by the condensation. This engine has been since improved, by admitting the end of the condensing pipe G into the vessel E, by which means the Steam is more suddenly and effectually condensed than by water on the outside of the vessel.</p><p>The advantages of this engine are, that it may be erected in almost any situation, that it requires but little room, and is subject to very little friction in its parts.—Its disadvantages are, that great part of the Steam is condensed and loses its force upon coming into contact with the water in the vessel E, and that the heat and elasticity of the Steam must be increased in proportion to the height that the water is required to be raised to. On both these accounts a large fire is required, and the copper must be very strong, when the height is considerable, otherwise there is danger of its bursting.</p><p>While captain Savery was employed in perfecting his engine, Dr. Papin of Marburg was contriving one on the same principles, which he describes in a small book published in 1707, intitled <hi rend="italics">Ars Nova ad Aquam Ignis adminiculo efficacissimè elevandam.</hi> Capt. Savery's engine however was much completer than that proposed by Dr. Papin.</p><p>About the same time also one Mons. Amontons of Paris was engaged in the same pursuit: but his method of applying the force of Steam was different from those before-mentioned; for he intended it to drive or turn a wheel, which he called a <hi rend="italics">fire-mill,</hi> which was to work pumps for raising water; but he never brought it to perfection. Each of these three gentlemen claimed the originality of the invention; but it is most probable they all took the hint from the book published by the marquis of Worcester, as before-mentioned.</p><p>In this imperfect state it continued, without farther improvements, till the year 1705, when Mr. Newcomen, an iron-monger, and Mr. John Cowley, a glazier, both of Dartmouth, contrived another way to raise water by Steam, bringing the engine to work with a beam and piston, and where the Steam, even at the greatest depths of mines, is not required to be greater than the pressure of the atmosphere: and this is the structure of the engine as it has since been chiefly used. These gentlemen obtained a patent for the sole use of this invention, for 14 years. The first proposal they made for draining of mines by this engine, was in the year 1711; but they were very coldly received by <pb n="517"/><cb/> many persons in the south of England, who did not understand the nature of it. In 1712 they came to an agreement with the owners of a colliery at Griff in Warwickshire, where they erected an engine with a cylinder of 22 inches diameter. At first they were under great difficulties in many things; but by the assistance of some good workmen they got all the parts put together in such a manner, as to answer their intention tolerably well: and this was the first engine of the kind erected in England. There was at first one man to attend the Steam-cock, and another to attend the injection cock; but they afterwards contrived a method of opening and shutting them by some small machinery connected with the working beam. The next engine erected by these patentees, was at a colliery in the county of Durham, about the year 1718, where was concerned, as an agent, Mr. Henry Beighton, F. R. S. and conductor of the Ladies' Diary from the year 1714 to the year 1744: this gentleman, not approving of the intricate manner of opening and shutting the cocks by strings and catches, as in the former engine, substituted the hanging beam for that purpose as at present used, and likewise made improvements in the pipes, valves, and some other parts of the engine.</p><p>In a few years afterwards, these engines came to be better understood than they had been; and their advantages, especially in draining of mines, became more apparent: and from the great number of them erected, they received additional improvements from different persons, till they arrived at their present degree of perfection: as will appear in the sequel, after we have a little considered the general principles of this engine, which are as follow. <hi rend="center"><hi rend="italics">Principles of the Steam Engine.</hi></hi></p><p>The principles on which this engine acts, are truly philosophical; and when all the parts of the machine are proportioned to each other according to these principles, it never fails to answer the intention of the engineer.</p><p>1. It has been proved in pneumatics, that the pressure of the atmosphere upon a square inch at the earth's surface, is about 14 3/4lb avoirdupois at a medium, or 11 1/2lb on a circular inch, that is on a circle of an inch diameter. And,</p><p>2. If a vacuum be made by any means in a cylinder, which has a moveable piston suspended at one end of a lever equally divided, the air will endeavour to rush in, and will press down the piston, with a force proportionable to the area of the surface, and will raise an equal weight at the other end of the lever.</p><p>3. Water may be rarefied near 14000 times by being reduced into Steam, and violently heated: the particles of it are so strongly repellent, as to drive away air of the common density, only by a heat sufficient to keep the water in a boiling state, when the Steam is almost 3000 times rarer than water, or 3 1/2 times rarer than air, as appears by an experiment of Mr. Beighton's: by increasing the heat, the Steam may be rendered much stronger; but this requires great strength in the vessels. This Steam may be again condensed into its former state by a jet of cold water dispersed through it; so that 14000 cubic inches of Steam admitted into a cy- <cb/> linder, may be reduced into the space of one cubic inch of water only, by which means a partial vacuum is obtained.</p><p>4. Though the pressure of the atmosphere be about 14 3/4 pounds upon every square inch, or 11 1/2 pounds upon a circular inch; yet, on account of the friction of the several parts, the resistance from some air which is unavoidably admitted with the jet of cold water, and from some remainder of Steam in the cylinder, the vacuum is very imperfect, and the piston does not descend with a force exceeding 8 or 9 pounds upon every square inch of its surface.</p><p>5. The gallon of water of 282 cubic inches weighs 10 1/5 pounds avoirdupois, or a cubic foot 62 1/2 pounds, or 1000 ounces. The piston being pressed by the atmosphere with a force proportional to its area in inches, multiplied by about 8 or 9 pounds, depresses that end of the lever, and raises a column of water in the pumps of equal weight at the other end, by means of the pump-rods suspended to it. When the Steam is again admitted, the pump-rods sink by their superior weight, and the piston rises; and when that Steam is condensed, the piston descends, and the pump-rods lift; and so on alternately as long as the piston works.</p><p>It has been observed above, that the piston does not descend with a force exceeding 8 or 9 pounds upon every square inch of its surface; but by reason of accidental frictions, and alterations in the density of the air, it will be safest, in calculating the power of the cylinder, to allow something less than 8 pounds for the pressure of the atmosphere, upon every square inch, viz 7lb. 10 oz, = 7.64lb, or just 6lb. upon every circular inch; and it being allowed that the gallon of water, of 282 cubic inches, weighs 10 1/5lb, from these premises the dimensions of the cylinder, pumps, &c, for any Steam-engine, may be deduced as follows:— Suppose <hi rend="italics">c</hi> = the cylinder's diameter in inches, <hi rend="italics">p</hi> = the pump's ditto, <hi rend="italics">f</hi> = the depth of the pit in fathoms, <hi rend="italics">g</hi> = gallons drawn by a stroke of 6 feet or a fathom, <hi rend="italics">h</hi> = the hogsheads drawn per hour, <hi rend="italics">s</hi> - the number of strokes per minute.</p><p>Then <hi rend="italics">c</hi><hi rend="sup">2</hi> is the area of the cylinder in circular inches, theref. 6<hi rend="italics">c</hi><hi rend="sup">2</hi> is the power of the cylinder in pounds.</p><p>And (<hi rend="italics">p</hi><hi rend="sup">2</hi> X .7854 X 72)/282 or (1/5)<hi rend="italics">p</hi><hi rend="sup">2</hi> is = <hi rend="italics">g</hi> the gallons contained in one fathom or 72 inches of any pump; which multiplied by <hi rend="italics">f</hi> fathoms, gives (1/5)<hi rend="italics">p</hi><hi rend="sup">2</hi><hi rend="italics">f</hi> for the gallons contained in <hi rend="italics">f</hi> fathoms of any pump whose diameter is <hi rend="italics">p.</hi></p><p>Hence (1/5)<hi rend="italics">p</hi><hi rend="sup">2</hi><hi rend="italics">f</hi> X 10 (1/5)lb. gives 2<hi rend="italics">p</hi><hi rend="sup">2</hi><hi rend="italics">f</hi> nearly, for the weight in pounds of the column of water which is to be equal to the power of the cylinder, which was before found equal to 6<hi rend="italics">c</hi><hi rend="sup">2</hi>. Hence then we have the 2d equation, viz, ; the first equation being . From which two equations, any particular circumstance may be determined.</p><p>Or if, instead of 6lb, for the pressure of the air on each circular inch of the cylinder, that force be sup- <pb n="518"/><cb/> posed any number as <hi rend="italics">a</hi> pounds; then will the power of the cyclinder be <hi rend="italics">ac</hi><hi rend="sup">2</hi>, and the 2d equation becomes , by substituting 5<hi rend="italics">g</hi> instead of <hi rend="italics">p</hi><hi rend="sup">2</hi>.</p><p>And farther, .</p><p>From a comparison of these equations, the following theorems are derived, which will determine the size of the cylinder and pumps of any Steam-engine capable of drawing a certain quantity of water from any assigned depth, with the pressure of the atmosphere on each circular inch of the cylinder's area.</p><p>These theorems are more particularly adapted to one pump in a pit. But it often happens in practice, that an engine has to draw several pumps of different diameters from different depths; and in this case, the square <cb/> of the diameter of each pump must be multiplied by its depth, and double the sum of all the products will be the weight of water drawn at each stroke, which is to be used instead of 2<hi rend="italics">p</hi><hi rend="sup">2</hi><hi rend="italics">f</hi> for the power of the cylinder.</p><p>The following is a Table, calculated from the foregoing theorems, of the powers of cylinders from 30 to 70 inches diameter; and the diameter and lengths of pumps which those cylinders are capable of working, from a 6 inch bore to that of 20 inches, together with the quantity of water drawn per stroke and per hour, allowing the engine to make 12 strokes of 6 feet per minute, and the pressure of the atmosphere at the rate of 7lb 10 oz per square inch, or 6lb per circular inch. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=8" role="data"><hi rend="italics">A</hi> <hi rend="smallcaps">Table</hi> <hi rend="italics">of</hi> <hi rend="smallcaps">Theorems</hi> <hi rend="italics">for the readier computing the</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=8" role="data"><hi rend="italics">Powers of a</hi> <hi rend="smallcaps">Steam-Engine.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(2<hi rend="italics">fp</hi><hi rend="sup">2</hi>)/<hi rend="italics">c</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(10<hi rend="italics">fg</hi>)/<hi rend="italics">c</hi><hi rend="sup">2</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(21<hi rend="italics">fh</hi>)/(2<hi rend="italics">c</hi><hi rend="sup">2</hi><hi rend="italics">s</hi>)</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">√((2<hi rend="italics">fp</hi><hi rend="sup">2</hi>)/<hi rend="italics">a</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">√((10<hi rend="italics">fg</hi>)/<hi rend="italics">a</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">√((21<hi rend="italics">fh</hi>)/(2<hi rend="italics">as</hi>))</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"><hi rend="italics">f</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(<hi rend="italics">ac</hi><hi rend="sup">2</hi>)/(2<hi rend="italics">p</hi><hi rend="sup">2</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(<hi rend="italics">ac</hi><hi rend="sup">2</hi>)/(10<hi rend="italics">g</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(2<hi rend="italics">ac</hi><hi rend="sup">2</hi><hi rend="italics">s</hi>)/(21<hi rend="italics">h</hi>)</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"><hi rend="italics">g</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi><hi rend="sup">2</hi>/5</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(<hi rend="italics">ac</hi><hi rend="sup">2</hi>)/(10<hi rend="italics">f</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(21<hi rend="italics">h</hi>)/(20<hi rend="italics">s</hi>)</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"><hi rend="italics">h</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(4<hi rend="italics">p</hi><hi rend="sup">2</hi><hi rend="italics">s</hi>)/21</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(20<hi rend="italics">gs</hi>)/21</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(2<hi rend="italics">ac</hi><hi rend="sup">2</hi><hi rend="italics">s</hi>)/(21<hi rend="italics">f</hi>)</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">√(5<hi rend="italics">g</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">√((<hi rend="italics">ac</hi><hi rend="sup">2</hi>)/(2<hi rend="italics">g</hi>))</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">√((21<hi rend="italics">h</hi>)/(4<hi rend="italics">s</hi>))</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data"><hi rend="italics">s</hi></cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(21<hi rend="italics">h</hi>)/(4<hi rend="italics">p</hi><hi rend="sup">2</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(21<hi rend="italics">h</hi>)/(20<hi rend="italics">g</hi>)</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">(21<hi rend="italics">fh</hi>)/(2<hi rend="italics">ac</hi><hi rend="sup">2</hi>)</cell></row></table><pb n="519"/> <table rend="border width=100%"><row role="data"><cell cols="1" rows="1" rend="colspan=18" role="data"><hi rend="smallcaps">Table</hi> <hi rend="italics">of the Power and Effects of</hi> <hi rend="smallcaps">Steam-Engines</hi>, <hi rend="italics">allowing</hi> 12 <hi rend="italics">Strokes, of</hi> 6 <hi rend="italics">Feet long each,</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=18" role="data"> <hi rend="italics">per Minute, and the pressure of the Air</hi> 7<hi rend="italics">lb</hi> 10<hi rend="italics">oz per Square Inch, or</hi> 6<hi rend="italics">lb per Circular</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=18" role="data"> <hi rend="italics">Inch.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center colspan=15" role="data">The Diameters of the Pumps in Inches.</cell><cell cols="1" rows="1" role="data">Power of the cylindersand weight of water in pounds.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=26 width=40%" role="data">The Diameters of the Cylinders in Inches.<hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">5400</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">5766</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">6144</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">3.</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">6534</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">693</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">7350</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">7776</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8214</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">8664</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">9126</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9600</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">108</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">10584</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">11616</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">12696</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">13824</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">15000</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">16224</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">17496</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">18816</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">101</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">20184</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">21600</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">23064</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">24546</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">26676</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">27744</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">29400</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left colspan=2" role="data">Quan. drawn at one stroke in gallons.</cell><cell cols="1" rows="1" role="data">7.2</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">16.2</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">24.2</cell><cell cols="1" rows="1" role="data">28.8</cell><cell cols="1" rows="1" role="data">33.8</cell><cell cols="1" rows="1" role="data">39.2</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">51.2</cell><cell cols="1" rows="1" role="data">57.8</cell><cell cols="1" rows="1" role="data">64.8</cell><cell cols="1" rows="1" role="data">72.2</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left colspan=2" role="data">Quan. drawn in one hour in hogsheads.</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">114</cell><cell cols="1" rows="1" role="data">148</cell><cell cols="1" rows="1" role="data">184</cell><cell cols="1" rows="1" role="data">228</cell><cell cols="1" rows="1" role="data">276</cell><cell cols="1" rows="1" role="data">328</cell><cell cols="1" rows="1" role="data">385</cell><cell cols="1" rows="1" role="data">447</cell><cell cols="1" rows="1" role="data">513</cell><cell cols="1" rows="1" role="data">583</cell><cell cols="1" rows="1" role="data">659</cell><cell cols="1" rows="1" role="data">738</cell><cell cols="1" rows="1" role="data">823</cell><cell cols="1" rows="1" role="data">912</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left colspan=2" role="data">Diameter of pumps.</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/></row></table><pb n="520"/><cb/></p><p>Let us now describe the several parts of an engine, and exemplify the application of the foregoing principies, in the construction of one of the completest of the modern engines. See fig. 4. pl. 27.</p><p>A represents the fire-place under the boiler, for the boiling of the water, and the ash-hole below it.</p><p>B, the boiler, filled with water about three feet above the bottom, made of iron plates.</p><p>C, the Steam pipe, through which the Steam passes from the boiler into the receiver.</p><p>D, the receiver, a close iron vessel, in which is the regulator or Steam-cock, which opens and shuts the hole of communication at each stroke.</p><p>E, the communication pipe between the receiver and the cylinder; it rises 5 or 6 inches up, in the inside of the cylinder bottom, to prevent the injected water from descending into the receiver.</p><p>F, the cylinder, of cast iron, about 10 feet long, bored smooth in the inside; it has a broad flanch in the middle on the outside, by which it is supported when hung in the cylinder-beams.</p><p>G, the piston, made to sit the cylinder exactly: it has a flanch rising 4 or 5 inches upon its upper surface, between which and the side of the cylinder a quantity of junk or oakum is stuffed, and kept down by weights, to prevent the entrance of air or water and the escaping of Steam.</p><p>H, the chain and piston shank, by which it is connected to the working beam.</p></div1><div1 part="N" n="II" org="uniform" sample="complete" type="entry"><head>II</head><p>, the working-beam or lever: it is made of two or more large logs of timber, bent together at each end, and kept at the distance of 8 or 9 inches from each other in the middle by the gudgeon, as represented in the Plate. The arch-heads, II, at the ends, are for giving a perpendicular direction to the chains of the piston and pump-rods.</p><p>K, the pump-rod which works in the sucking pump.</p><p>L, and draws the water from the bottom of the pit to the surface.</p><p>M, a cistern, into which the water drawn out of the pit is conducted by a trough, so as to keep it always full: and the superfluous water is carried off by another trough.</p><p>N, the jack-head pump, which is a sucking-pump wrought by a small lever or working-beam, by means of a chain connected to the great beam or lever near the arch <hi rend="italics">g</hi> at the inner end, and the pump-rod at the outer end. This pump commonly stands near the corner of the front of the house, and raises the column of water up to the cistern O, into which it is conducted by a trough.</p><p>O, the jack-head cistern for supplying the injection, which is always kept full by the pump N: it is sixed so high as to give the jet a sufficient velocity into the cylinder when the cock is opened. This cistern has a pipe on the opposite side for conveying away the supersluous water.</p></div1><div1 part="N" n="PP" org="uniform" sample="complete" type="entry"><head>PP</head><p>, the injection-pipe, of 3 or 4 inches diameter, which turns up in a curve at the lower end, and enters the cylinder bottom: it has a thin plate of iron upon the end <hi rend="italics">a,</hi> with 3 or 4 adjutage holes in it, to prevent the jet of cold water of the jack-head cistern <cb/> from flying up against the piston, and yet to condense the Steam each stroke, when the injection-cock is open.</p><p><hi rend="italics">c,</hi> a valve upon the upper end of the injection pipe within the cistern, which is shut when the engine is not working, to prevent any waste of the water.</p><p><hi rend="italics">f,</hi> a small pipe which branches off from the injection-pipe, and has a small cock to supply the piston with a little water to keep it air-tight.</p><p>Q, the working plug, suspended by a chain to the arch <hi rend="italics">g</hi> of the working beam. It is usually a heavy piece of timber, with a slit vertically down its middle, and holes bored horizontally through it, to receive pins for the purpose of opening and shutting the injection and Steam cocks, as it ascends and descends by the motion of the working beam.</p><p><hi rend="italics">h,</hi> the handle of the steam-cock or regulator. It is fixed to the regulator by a spindle which comes up through the top of the receiver. The regulator is a circular plate of brass or cast iron, which is moved horizontally by the handle <hi rend="italics">h,</hi> and opens or shuts the communication at the lower end of the pipe E within the receiver. It is represented in the plate by a circular dotted line.</p><p><hi rend="italics">ii,</hi> the spanner, which is a long rod or plate of iron for communicating motion to the handle of the regulator: to which it is fixed by means of a slit in the latter, and some pins put through to fasten it.</p><p><hi rend="italics">kl,</hi> the vibrating lever, called the Y, having the weight <hi rend="italics">k</hi> at one end and two legs at the other end. It is fixed to an horizontal axis, moveable about its centre-pins or pivots <hi rend="italics">mn,</hi> by means of the two shanks <hi rend="italics">op</hi> fixed to the same axis, which are alternately thrown backwards and forwards by means of two pins in the working plug; one pin on the outside depressing the shank <hi rend="italics">o,</hi> throws the loaded end <hi rend="italics">k</hi> of the Y from the cylinder into the position represented in the plate, and causes the leg <hi rend="italics">l</hi> to strike against the end of the spanner; which forcing back the handle of the regulator or steam cock, opens the communication, and permits the steam to fly into the cylinder. The piston immediately rising by the admission of the Steam, the working beam II rises; which also raises the working-plug, and another pin which goes through the slit raises the shank <hi rend="italics">p,</hi> which throws the end <hi rend="italics">k</hi> of the Z towards the cylinder, and, striking the end of the spanner, forces it forward, and shuts the regulator Steam-cock.</p><p><hi rend="italics">qr,</hi> the lever for opening and shutting the injection cock, called the F. It has two toes from its centre, which take between them the key of the injection cock. When the working-plug has ascended nearly to its greatest height, and shut the regulator, a pin catches the end <hi rend="italics">q</hi> of the F and raises it up, which opens the injection-cock, admits a jet of cold water to fly into the cylinder, and, condensing the Steam, makes a vacuum; then the pressure of the atmosphere bringing down the piston in the cylinder, and also the plugframe, another pin fixed in it catches the end of the lever in its descent, and, by pressing it down, shuts the injection-cock, at the same time the regulator is opened to admit Steam, and so on alternately; when the regulator is shut the injection is open, and when the former is open the latter is shut. <pb n="521"/><cb/></p><p>R, the hot-well, a small cistern made of planks, which receives all the waste water from the cylinder.</p><p>S, the sink-pit to convey away the water which is injected into the cylinder at each stroke. Its upper end is even with the inside of the cylinder bottom, its lower end has a lid or cover moveable on a hinge which serves as a valve to let out the injected water, and shuts close each stroke of the engine, to prevent the water being forced up again when the vacuum is made.</p><p>T, the feeding pipe, to supply the boiler with water from the hot-well. It has a cock to let in a large or small quantity of water as occasion requires, to make up for what is evaporated; it goes nearly down to the boiler bottom.</p><p>U, two gage cocks, the one larger than the other, to try when a proper quantity of water is in the boiler: upon opening the cocks, if one give Steam and the other water, it is right; if they both give Steam, there is too little water in the boiler; and if they both give water, there is too much.</p><p>W, a plate which is screwed on to a hole on the side of the boiler, to allow a passage into the boiler for the convenience of cleaning or repairing it.</p><p>X, the Steam-clack or puppet valve, which is a brass valve on the top of a pipe opening into the boiler, to let off the Steam when it is too strong. It is loaded with lead, at the rate of one pound to an inch square; and when the Steam is nearly strong enough to keep it open, it will do for the working of the engine.</p><p><hi rend="italics">f,</hi> the snifting valve, by which the air is discharged from the cylinder each stroke, which was admitted with the injection, and would otherwise obstruct the due operation of the engine.</p><p><hi rend="italics">tt,</hi> the cylinder-beams; which are strong joists going through the house for supporting the cylinder.</p><p><hi rend="italics">v,</hi> the cylinder cap of lead, soldered on the top of the cylinder, to prevent the water upon the piston from flashing over when it rises too high.</p><p><hi rend="italics">w,</hi> the waste-pipe, which conducts the superfluous water from the top of the cylinder to the hot-well.</p><p><hi rend="italics">xx,</hi> iron bars, called the catch-pins, fixed horizontally through each arch head, to prevent the beam descending too low in case the chain should break.</p><p><hi rend="italics">yy,</hi> two strong wooden springs, to weaken the blow given by the catch pins when the stroke is too long.</p><p><hi rend="italics">zz,</hi> two friction-wheels, on which the gudgeon or centre of the great beam is hung; they are the third or fourth part of a circle, and move a little each way as the beam vibrates. Their use is to diminish the friction of the axis, which, in so heavy a lever, would otherwise be very great.</p><p>When this engine is to be set to work, the boiler must be filled about three or four feet deep with water, and a large fire made under it; and when the Steam is found to be of a sufficient strength by the puppetclack, then by thrusting back the spanner, which opens the regulator or Steam-cock, the Steam is admitted into the cylinder, which raises the piston to the <cb/> top of the cylinder, and forces out all the air at the snifting valve; then by turning the key of the injectioncock, a jet of cold water is admitted into the cylinder, which condenses the Steam and makes a vacuum; and the atmosphere then pressing upon the piston, forces it down to the lower part of the cylinder, and makes a stroke by raising the column of water at the other end of the beam. After two or three strokes are made in this manner, by a man opening and shutting the cocks to try if they be right, then the pins may be put into the pin-holes in the working plug, and the engine left to turn the cocks of itself; which it will do with greater exactness than any man can do.</p><p>There are in some engines, methods of shutting and opening the cocks different from the one above described, but perhaps none better adapted to the purpose; and as the principles on which they all act are originally the same, any difference in the mechanical construction of the small machinery will have no influence of consequence upon the total effect of the grand machine.</p><p>The furnace or fire-place should not have the bars so close as to prevent the free admission of fresh air to the fire, nor so open as to permit the coals to fall through them; for which purpose two inches or thereabouts is sufficient for the distance betwixt the bars. The size of the furnace depends upon the size of the boiler; but in every case the ash-hole ought to be capacious to admit the air, and the greater its height the better. If the flame is conducted in a flue or chimney round the outside of the boiler, or in a pipe round the inside of it, it ought to be gradually diminished from the entrance at the furnace to its egress at the chimney; and the section of the chimney at that place should not exceed the section of the flue or pipe, and should also be somewhat less at the chimney-top.</p><p>The boiler or vessel in which the water is rarefied by the force of fire, may be made of iron plates, or cast iron, or such other materials as can withstand the effects of the fire, and the elastic force of the Steam. It may be considered as consisting of two parts; the upper part which is exposed to the Steam, and the under part which is exposed to the fire. The form of the latter should be such as to receive the full force of the fire in the most advantageous manner, so that a certain quantity of fuel may have the greatest possible effect in heating and evaporating the water; which is best done by making the sides cylindrical, and the bottom a little concave, and then conducting the flame by an iron flue or pipe round the inside of the boiler beneath the surface of the water, before it reach the chimney. For, by this means, after the fire in the furnace has heated the water by its effect on the bottom, the flame heats it again by the pipe being wholly included in the water, and having every part of its surface in contact with it; which is preferable to carrying it in a flue or chimney round the outside of the boiler, as a third or a half of the surface of the flame only could be in contact with the boiler, the other being spent upon the brick-work. This cylindric lower part may be less in its diameter than the upper part, and may contain from four to six feet perpendicular height of water in it. <pb n="522"/><cb/></p><p>The upper part of the boiler is best made hemispherical, for resisting the elasticity of the Steam; yet any other form may do, provided it be of sufficient strength for the purpose. The quick going of the engine depends much on the capaciousness of the boiler-top; for if it be too small, it requires the Steam to be heated to a great degree, to increase its elastic force so much as to work the engine. If the top is so capacious as to contain eight or ten times the quantity of Steam used each stroke, it will require no more fire to preserve its elasticity than is sufficient to keep the water in a proper state of boiling; this, therefore, is the best size for a boiler top. If the diameter of the cylinder be <hi rend="italics">c,</hi> and works a six-foot stroke, and the diameter of the boiler be supposed <hi rend="italics">b,</hi> then .</p><p>The effect of the injection in condensing the Steam in the cylinder, depends upon the height of the reservoir and the diameter of the adjutage. If the engine makes a 6 feet stroke, then the jackhead cistern should be 12 feet perpendicular above the bottom of the cylinder or the adjutage. The size of the adjutage may be from 1 to 2 inches in diameter; or if the cylinder be very large, it is proper to have three or four holes rather than one large one, in order that the jet may be dispersed the more effectually over the whole area of the cylinder. The injection pipe, or pipe of conduct, should be so large as to supply the injection freely with water; if the diameter of the injection pipe be called <hi rend="italics">p,</hi> and the diameter of the adjutage, <hi rend="italics">a,</hi> then .</p><p>For a further account of these engines, see Desaguliers's Exp. Philos. vol. 2, sect. 14, pa. 465, &c.; or for an abstract, Martin's Phil. Brit. number 461, or Nicholson's Nat. Philos. p. 83 &c. And for an account of the improvement made in the fire-engine by Mr. Payne, see Philos. Trans. number 461, or Martin's Philos. Brit. p. 87 &c.</p><p>Mr. Blakey communicated to the Royal Society, in 1752, remarks on the best proportions for Steam-engine cylinders of a given content: and Mr. Smeaton describes an engine of this kind, invented by Mr. De Moura of Portugal, being an improvement of Savery's construction, to render it capable of working itself: for both which accounts, see Philos. Trans. vol. 47 art. 29 and 72.</p><p>We are insormed in the new edit. of the Biograph. Brit. in the article Brindley, that in 1756 this gentleman, so well known for his concern in our inland navigations, undertook to erect a Steam-engine near Newcastle-under-Line, upon a new plan. The boiler of it was made with brick and stone, instead of iron plates, and the water was heated by iron flues of a peculiar construction; by which contrivances the consumption of fuel, necessary for working a Steam engine, was reduced one half. He introduced also in his engine, wooden cylinders, made in the manner of cooper's ware, instead of iron ones; the former being both cheaper and more easily managed in the shafts: and he likewise substituted wood for iron in the chains which worked at the end of the beam. He had formed designs of introducing other improvements into the con- <cb/> struction of this useful engine; but was discouraged by obstacles that were thrown in his way.</p><p>Mr. Blakey, some years ago, obtained a patent for his improvement of Savery's Steam-engine, by which it is excellently adapted for raising water out of ponds, rivers, wells, &c, and for forcing it up to any height wanted for supplying houses, gardens, and other places; though it has not power sufficient to drain off the water from a deep mine. The principles of his construction are explained by Mr. Ferguson, in the Supplement to his Lectures, pa. 19; and a more particular description of it, accompanied with a drawing, is given by the patentee himself in the Gentleman's Magazine for 1769, p. 392.</p><p>Mr. Blakey, it is said, is the first person who ever thought of making use of air as an intermediate body between Steam and water; by which means the Steam is always kept from touching the water, and consequently from being condensed by it: and on this new principle he has obtained a patent. The engine may be built at a trifling expence, in comparison of the common fire-engine now in use; it will seldom need repairs, and will not consume half so much fuel. And as it has no pumps with pistons, it is clear of all their friction; and the effect is equal to the whole strength or compressive force of the Steam; which the effect of the common fire-engine never is, on account of the great friction of the pistons in their pumps.</p><p>Ever since Mr. Newcomen's invention of the Steam fire engine, the great consumption of fuel with which it is attended, has been complained of as an immense drawback upon the profits of our mines. It is a known fact, that every fire-engine of considerable size consumes to the amount of three thousand pounds worth of coals in every year. Hence many of our engineers have endeavoured, in the construction of these engines, to save fuel. For this purpose, the fire-place has been diminished, the flame has been carried round from the bottom of the boiler in a spiral direction, and conveyed through the body of the water in a tube before its arrival at the chimney; some have used a double boiler, so that fire might act in every possible point of contact; and some have built a moor-stone boiler, heated by three tubes of flame passing through it. But the most important improvements which have been made in the Steam-engine for more than thirty years past, we owe to the skill of Mr. James Watt; of which we shall give some account: premising, that the internal structure of his new engines so much resembles that of the common ones, that those who are acquainted with them will not fail to understand the mechanism of his from the following description: he has contrived to observe an uniform heat in the cylinder of his engines, by suffering no cold water to touch it, and by protecting it from the air, or other cold bodies, by a surrounding case filled with Steam, or with hot air or water, and by coating it over with substances that transmit heat slowly. He makes his vacuum to approach nearly to that of the barometer, by condensing the Steam in a separate vessel, called the condenser, which may be cooled at pleasure without cooling the cylinder, either by an injection of cold water, or by surrounding <pb n="523"/><cb/> the condenser with it, and generally by both. He extracts the injection water, and detached air, from the cylinder or condenser by pumps, which are wrought by the engine itself, or blows them out by the Steam. As the entrance of air into the cylinder would stop the operation of the engines, and as it is hardly to be expected that such enormous pistons as those of Steamengines can move up and down, and yet be absolutely tight in the common engines; a stream of water is kept always running upon the piston, which prevents the entry of the air: but this mode of securing the piston, though not hurtful in the common ones, would be highly prejudicial to the new engines. Their pisston is therefore made more accurately; and the outer cylinder, having a lid, covers it, the Steam is introduced above the piston; and when a vacuum is produced under it, acts upon it by its elasticity, as the atmosphere does upon common engines by its gravity. This way of working effectually excludes the air from the inner cylinder, and gives the advantage of adding to the power, by increasing the elasticity of the Steam.</p><p>In Mr. Watt's engines, the cylinder, the great beams, the pumps, &c, stand in their usual positions. The cylinder is smaller than usual, in proportion to the load, and is very accurately bored.</p><p>In the most complete engines, it is surrounded at a small distance, with another cylinder, furnished with a bottom and a lid. The interstice between the cylinders communicates with the boilers by a large pipe, open at both ends: so that it is always filled with Steam, and thereby maintains the inner cylinder always of the same heat with the Steam, and prevents any condensation within it, which would be more detrimental than an equal condensation in the outer one. The inner cylinder has a bottom and piston as usual: and as it does not reach up quire to the lid of the outer cylinder, the Steam in the interstice has always free access to the upper side of the piston. The lid of the outer cylinder has a hole in its middle; and the piston rod, which is truly cylindrical, moves up and down through that hole, which is kept Steam-tight by a collar of oakum screwed down upon it. At the bottom of the inner cylinder, there are two regulating valves, one of which admits the Steam to pass from the interstice into the inner cylinder below the piston, or shuts it out at pleasure: the other opens or shuts the end of a pipe, which leads to the condenser. The condenser consists of one or more pumps furnished with clacks and buckets (nearly the same as in common pumps) which are wrought by chains fastened to the great working beam of the engine. The pipe, which comes from the cylinder, is joined to the bottom of these pumps, and the whole condenser stands immersed in a cistern of cold water supplied by the engine. The place of this cistern is either within the house or under the floor, between the cylinder and the lever wall; or without the house between that wall and the engine shaft, as conveniency may require. The condenser being exhausted of air by blowing, and both the cylinders being filled with Steam, the regulating valve which admits the Steam into the inner cylinder is shut, and the other regulator which communicates with the condenser is opened, and the Steam rushes into the vacuum of the condenser with <cb/> violence: but there it comes into contact with the cold sides of the pumps and pipes, and meets a jet of cold water, which was opened at the same time with the exhaustion regulator; these instantly deprive it of its heat, and reduce it to water; and the vacuum remaining perfect, more Steam continues to rush in, and be condensed until the inner cylinder be exhausted. Then the Steam which is above the piston, ceasing to be counteracted by that which was below it, acts upon the piston with its whole elasticity, and forces it to descend to the bottom of the cylinder, and so raises the buckets of the pumps which are hung to the other end of the beam. The exhaustion regulator is now shut, and the Steam one opened again, which, by letting in the Steam, allows the piston to be pulled up by the superior weight of the pump rods; and so the engine is ready for another stroke.</p><p>But the nature of Mr. Watt's improvement will be perhaps better understood from the following description of it as referred to a figure.—The cylinder or Steam vessel A, of this engine (fig. 5, pl. 27), is shut at bottom and opened at top as usual; and is included in an outer cylinder or case BB, of wood or metal, covered with materials which transmit heat slowly. This case is at a small distance from the cylinder, and close at both ends. The cover C has a hole in it, through which the piston rod E slides; and near the bottom is another hole F, by which the Steam from the boiler has always free entrance into this case or outer cylinder, and by the interstice GG between the two cylinders has access to the upper side of the piston HH. To the bottom of the inner cylinder A is joined a pipe I, with a cock or valve K, which is opened and shut when necessary, and forms a passage to another vessel L called a <hi rend="italics">Condenser,</hi> made of thin metal. This vessel is immersed in a cistern M full of cold water, and it is contrived so as to expose a very great surface externally to the water, and internally to the Steam. It is also made air-tight, and has pumps N wrought by the engine, which keep it always exhausted of air and water.</p><p>Both the cylinders A and BB being filled with Steam, the passage K is opened from the inner one to the condenser L, into which the Steam violently rushes by its elasticity, because that vessel is exhausted; but as soon as it enters it, coming into contact with the cold matter of the condenser, it is reduced to water, and, the vacuum still remaining, the Steam continues to rush in till the inner cylinder A below the piston is left empty. The Steam which is above the piston, ceasing to be counteracted by that which is below it, acts upon the piston HH, and forces it to descend to the bottom of the cylinder, and so raises the bucket of the pump by means of the lever. The passage K between the inner cylinder and the condenser is then shut, and another passage O is opened, which permits the Steam to pass from the outer cylinder, or from the boiler into the inner cylinder under the piston; and then the superior weight of the bucket and pump rods pulls down the outer end of the lever or great beam, and raises the piston, which is suspended to the inner end of the same beam.</p><p>The advantages that accrue from this construction are, first, that the cylinder being surrounded with the Steam from the boiler, it is kept always uniformly as hot as the Steam itself, and is therefore incapable of destroy- <pb n="524"/><cb/> ing any part of the Steam, which should fill it, as the common engines do. Secondly, the condenser being kept always as cold as water can be procured, and colder than the point at which it boils in vacuo, the Steam is perfectly condensed, and does not oppose the descent of the piston; which is therefore forced down by the full power of the Steam from the boiler, which is somewhat greater than that of the atmosphere.</p><p>In the common fire-engines, when they are loaded to 7 pounds upon the inch, and are of a middle size, the quantity of Steam which is condensed in restoring to the cylinder the heat which it had been deprived of by the former injection of cold water, is about one full of the cylinder, besides what it really required to fill that vessel; so that twice the full of the cylinder is employed to make it raise a column of water equal to about 7 pounds for each square inch of the piston: or, to take it more simply, a cubic foot of Steam raises a cubic foot of water about 8 feet high, besides overcoming the friction of the engine, and the resistance of the water to motion.</p><p>In the improved engine, about one full and a fourth of the cylinder is required to fill it, because the Steam is one-fourth more dense than in the common engine. This engine raises a load equal to 12 pounds and a half upon the square inch of the piston; and each cubic foot of Steam of the density of the atmosphere, raises one cultic foot of water 22 feet high.</p><p>The working of these engines is more regular and steady than the common ones, and from what has been said, their other advantages seem to be very considerable.</p><p>It is said, that the savings amount at least to two thirds of the fuel, which is an important object, especially where coals are dear. The new engines will raise from twenty thousand to twenty-four thousand cubic feet of water, to the height of twenty-four feet by one hundred weight of good pit coal: and Mr. Watt has proposed to produce engines upon the same principles, though somewhat differing in construction, which will require still much less fuel, and be more convenient for the purposes of mining, than any kind of engine yet used. Mr. Watt has also contrived a kind of mill wheel, which turns round by the power of Steam exerted within it.</p><p>The improvements above recited were invented by Mr. James Watt, at Glasgow, in Scotland, in 1764: he obtained the king's letters patent for the sole use of his invention in 1768; but meeting with difficulties in the execution of a large machine, and being otherwise employed, he laid aside the undertaking till the year 1774, when, in conjunction with Mr. Boulton near Birmingham, he completed both a reciprocating and rotative or wheel engine. He then applied to parliament for a prolongation of the term of his patent, which was granted by an act passed in 1775. Since that time, Mr. Watt and Mr. Boulton have erected several engines in Staffordshire, Shropshire, and Warwickshire, and a small one at Stratford near London. They have also lately finished another at Hawkesbury colliery near Coventry, which is justly supposed to be the most powerful engine in England. It has a cylinder 58 inches in diameter, which works a pump 14 inches in diameter, 65 fathoms high, and makes regularly twelve <cb/> strokes, each 8 feet long, in a minute. They have also erected several engines in Cornwall; one of which has a cylinder 30 inches in diameter, that works a pump 6 1/2 inches in diameter in two shafts, by flat rods with great friction, 300 feet distant from each other, 45 fathoms high in each shaft, equal in all to 90 fathoms, and can make 14 strokes, 8 feet long, in a minute, with a consumption of coals less than 20 bushels in 24 hours. The terms they offer to the public are, to take in lieu of all profits, one third part of the annual savings in fuel, which their engine makes when compared with a common engine of the same dimensions in the neighbourhood. The engines are built at the expence of those who use them, and Messrs. Boulton and Watt furnish such drawings, directions, and attendance, as may be necessary to enable a resident engineer to complete the machine. See the appendix to Pryce's Mineralogia, &c, 1778.</p><p>It has been said that some useful improvements have been made in the Steam engine by Mr. William Powel, who had lately the direction and care of an engine of this kind at a colliery near Swansea, in Glamorganshire.</p><p>It is hardly necessary to add, that Dr. Falck, in 1776, published an account and description of an improved Steam-engine, which, as he says, will, with the same quantity of fuel, and in an equal space of time, raise above double the quantity of water raised by any lever engine of the same dimensions; as he does not seem to have constructed even a working model of his proposed engine. The principal improvement, however, which he suggests, is to use two cylinders; into which the Steam is let alternately to ascend, by a common regulator, which always opens the communication of the Steam to one, whilst it shuts up the opening of the other: the piston rods are kept (by means of a wheel fixed to an arbour) in a continual ascending and descending motion, by which they move the common arbour, to which is affixed another wheel, moving the pump rods, in the same alternate direction as the piston rods, by which continual motion the pumps are kept in constant action.</p></div1><div1 part="N" n="STEELYARD" org="uniform" sample="complete" type="entry"><head>STEELYARD</head><p>, or <hi rend="smallcaps">Stilyard</hi>, in Mechanics, a kind of balance, called also, <hi rend="italics">Statera Romana,</hi> or the <hi rend="italics">Roman Balance,</hi> by means of which the weights of different bodies are discovered by using one single weight only. <figure/></p><p>The common Steelyard consists of an iron beam AB, <pb n="525"/><cb/> in which is assumed a point at pleasure, as C, on which is raised a perpendicular CD. On the shorter arm AC is hung a scale or bason to receive the bodies weighed: the moveable weight I is shifted backward and forward on the beam, till it be a counterbalance to 1, 2, 3, 4, &c pounds placed in the scale; and the points are noted where the constant weight I weighs, as 1, 2, 3, 4, &c pounds. From this construction of the Steelyard, the manner of using it is evident. But the instrument is very liable to deceit, and therefore is not much used in ordinary commerce.</p><p><hi rend="italics">Chinese</hi> <hi rend="smallcaps">Steelyard.</hi> The Chinese carry this Statera about them to weigh their gems, and other things of value. The beam or yard is a small rod of wood or ivory, about a foot in length: upon this are three rules of measure, made of a fine silver-studded work; they all begin from the end of the beam, whence the first is extended 8 inches, the second 6 1/2, the third 8 1/2. The first is the European measure, the other two seem to be Chinese measures. At the other end of the yard hangs a round scale, and at three several distances from this end are fastened so many slender strings, as different points of suspension. The first distance makes 1 3/5 or 8/5 of an inch, the second 3 1/5 or double the first, and the third 4 4/5 or triple of the first. When they weigh any thing, they hold up the yard by some one of these strings, and hang a sealed weight, of about 1 1/4<hi rend="smallcaps">OZ</hi> troy weight, upon the respective divisions of the rule, as the thing requires. Grew's Museum, pa. 369.</p><p><hi rend="italics">Spring</hi> <hi rend="smallcaps">Steelyard</hi>, is a kind of portable balance, serving to weigh any matter, from 1 to about 40 pounds.</p><p>It is composed of a brass or iron tube, into which goes a rod, and about that is wound a spring of tempered steel in a spiral form. On this rod are the divisions of pounds and parts of pounds, which are made by successively hanging on, to a hook fastened to the other end, 1, 2, 3, 4, &c, pounds.</p><p>Now the spring being fastened by a screw to the bottom of the rod; the greater the weight is that is hung upon the hook, the more will the spring be contracted, and consequently a greater part of the rod will come out of the tube; the proportions or quantities of which greater weights are indicated by the figures appearing against the extremity of the tube.</p><p><hi rend="smallcaps">Steelyard</hi>-<hi rend="italics">Swing.</hi> In the Philos. Trans. (no. 462, sect. 5) is given an account of a Steelyard swing, proposed as a mechanical method for assisting children labouring under deformities, owing to the contraction of the muscles on one side of the body. The crooked person is suspended with cords under his arm, and these are placed at equal distances from the centre of the beam. It is supposed that the gravity of the body will affect the contracted side, so as to put the muscles upon the stretch; and hence by degrees the defect may be remedied.</p></div1><div1 part="N" n="STEEPLE" org="uniform" sample="complete" type="entry"><head>STEEPLE</head><p>, an appendage usually raised on the western end of a church to contain the bells.—Steeples are denominated from their form, either <hi rend="italics">spires,</hi> or <hi rend="italics">towers.</hi> The first are such as rise continually diminishing like a cone or other pyramid. The latter are mere parallelopipedons, or some other prism, and are covered at top platform-like.—In each kind there is usually a <cb/> sort of windows, or loop-holes, to let out the sound, and so contrived as to throw it downward.</p><p>Masius, in his treatise on bells, treats likewise of Steeples. The most remarkable in the world, it is said, is that at Pisa, which leans so much to one side, that you fear every moment it will fall; yet is in no danger. This odd disposition, he observes, is not owing to a shock of an earthquake, as is generally imagined; but was contrived so at first by the architect; as is evident from the cielings, windows, doors, &c, which are all in the bevel.</p></div1><div1 part="N" n="STEERAGE" org="uniform" sample="complete" type="entry"><head>STEERAGE</head><p>, in a ship, that part next below the quarter-deck, before the bulk-head of the great cabin, where the steersman stands in most ships of war. In large ships of war it is used as a hall, through which it is necessary to pass to or from the great cabin. In merchant ships it is mostly the habitation of the lower officers and ship's crew.</p><div2 part="N" n="Steerage" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Steerage</hi></head><p>, in Sea-language, is also used to express the effort of the helm: and hence</p><p><hi rend="smallcaps">Steerage</hi>-<hi rend="italics">way</hi> is that degree of progressive motion communicated to a ship, by which she becomes susceptible of the effect of the helm to govern her course.</p></div2></div1><div1 part="N" n="STEERING" org="uniform" sample="complete" type="entry"><head>STEERING</head><p>, in Navigation, the art of directing the ship's way by the movements of the helm; or of applying its efforts to regulate her course when she advances.</p><p>The perfection of Steering consists in a vigilant attention to the motion of the ship's head, so as to check every deviation from the line of her course in the first inslant of its motion; and in applying as little of the power of the helm as possible. By this means she will run more uniformly in a straight path, as declining less to the right and left; whereas, if a greater effort of the helm be employed, it will produce a greater declination from the course, and not only increase the difficulty of Steering, but also make a crooked and irregular path through the water.</p><p>The helmsman, or steersman, should diligently watch the movements of the head by the land, clouds, moon, or stars; because, although the course is in general regulated by the compass, yet the vibrations of the needle are not so quickly perceived, as the sallies of the ship's head to the right or left, which, if not immediately restrained, will acquire additional velocity in every instant of their motion, and require a more powerful impulse of the helm to reduce them; the application of which will operate to turn her head as far on the contrary side of her course.</p><p>The phrases used in Steering a ship, vary according to the relation of the wind to her course. Thus, when the wind is large or fair, the phrases used by the pilot or officer who superintends the Steerage, are <hi rend="italics">port, starboard,</hi> and <hi rend="italics">steady:</hi> the first of which is intended to direct the ship's course farther to the right; the second to the left; and the last is designed to keep her exactly in the line on which she advances, according to the intended course. The excess of the first and second movement is called <hi rend="italics">hard-a-port,</hi> and <hi rend="italics">hard-a-starboard;</hi> the former of which gives her the greatest possible inclination to the right, and the latter an equal tendency to the left.—If, on the contrary, the wind be scant or foul, the phrases are <hi rend="italics">luff, thus,</hi> and <hi rend="italics">no nearer:</hi> the first of which is the order to keep her close to the wind; the second, to retain <pb n="526"/><cb/> her in her present situation; and the third, to keep her sails full.</p><p>STELLA. See <hi rend="smallcaps">Star.</hi></p><p>STENTOROPHONIC <hi rend="italics">Tube,</hi> a <hi rend="italics">Speaking Trumpet,</hi> or tube employed to speak to a person at a great distance. It has been so called from Stentor, a person mentioned in the 5th book of the Iliad, who, as Homer tells us, could call out louder than 50 men. The Stentorophonic horn of Alexander the Great is famous; with this it is said he could give orders to his army at the distance of 100 stadia, which is about 12 English miles.</p><p>The present speaking trumpet it is said was invented by Sir Samuel Moreland. But Derham, in his PhysicoTheology, lib. 4, ch. 3, says, that Kircher found out this instrument 20 years before Moreland, and published it in his Mesurgia; and it is farther said that Gaspar Schottus had seen one at the Jesuits' College at Rome. Also one Conyers, in the Philos. Trans. number 141, gives a description of an instrument of this kind, different from those commonly made. Gravesande, in his Philosophy, disapproves of the usual figures of these instruments; he would have them to be parabolic conoids, with the focus of one of its parabolic sections at the mouth.—Concerning this instrument, see Sturmy's Collegium Curiosum, Pt. 2, Tentam. 8; also Philos. Trans. vol. 6, pa. 3056, vol. 12, pa. 1027, or Abridg. vol. 1, pa. 505.</p><p>STEREOGRAPHIC <hi rend="italics">Projection of the Sphere,</hi> is that in which the eye is supposed to be placed in the surface of the sphere. Or it is the projection of the circles of the sphere on the plane of some one great circle, when the eye, or a luminous point, is placed in the pole of that circle.—For the fundamental principles and chief properties of this kind of projection, see <hi rend="smallcaps">Projection.</hi></p></div1><div1 part="N" n="STEREOGRAPHY" org="uniform" sample="complete" type="entry"><head>STEREOGRAPHY</head><p>, is the art of drawing the forms of solids upon a plane.</p></div1><div1 part="N" n="STEVIN" org="uniform" sample="complete" type="entry"><head>STEVIN</head><p>, <hi rend="smallcaps">Stevinus (Simon</hi>), a Flemish mathematician of Bruges, who died in 1633. He was master of mathematics to prince Maurice of Nassau, and inspector of the dykes in Holland. It is said he was the inventor of the sailing chariots, sometimes made use of in Holland. He was a good practical mathematician and mechanist, and was author of several useful works: as, treatises on Arithmetic, Algebra, Geometry, Statics, Optics, Trigonometry, Geography, Astronomy, Fortification, and many others, in the Dutch language, which were translated into Latin, by Snellius, and printed in 2 volumes folio. There are also two editions in the French language, in folio, both printed at Leyden, the one in 1608, and the other in 1634, with curious notes and additions, by Albert Girard.—For a particular account of Stevin's inventions and improvements in Algebra, which were many and ingenious, see our article Algebra, vol. 1, pa. 82 and 83.</p></div1><div1 part="N" n="STEWART" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">STEWART</surname> (the Rev. Dr. <foreName full="yes"><hi rend="smallcaps">Matthew</hi></foreName>)</persName></head><p>, late professor of mathematics in the university of Edinburgh, was the son of the reverend Mr. Dugald Stewart, minister of Rothsay in the Isle of Bute, and was born at that place in the year 1717. After having finished his course at the grammar school, being intended by his father for the church, he was sent to the university of Glasgow, and was entered there as a student in 1734. <cb/> His academical studies were prosecuted with diligence and success; and he was so happy as to be particularly distinguished by the friendship of Dr. Hutcheson, and Dr. Simson the celebrated geometrician, under whom he made great progress in that science.</p><p>Mr. Stewart's views made it necessary for him to attend the lectures in the university of Edinburgh in 1741; and that his mathematical studies might suffer no interruption, he was introduced by Dr. Simson to Mr. Maclaurin, who was then teaching with so much success, both the geometry and the philosophy of Newton, and under whom Mr. Stewart made that proficiency which was to be expected from the abilities of such a pupil, directed by those of so great a master. But the modern analysis, even when thus powerfully recommended, was not able to withdraw his attention from the relish of the ancient geometry, which he had imbibed under Dr. Simson. He still kept up a regular correspondence with this gentleman, giving him an account of his progress, and of his discoveries in geometry, which were now both numerous and important, and receiving in return many curious communications with respect to the <hi rend="italics">Loci Plani,</hi> and the Porisms of Euclid. Mr. Stewart pursued this latter subject in a different, and new direction. In doing so, he was led to the discovery of those curious and interesting propositions, which were published, under the title of <hi rend="italics">General Theorems,</hi> in 1746. They were given without the demonstrations; but they did not fail to place their discoverer at once among the geometricians of the first rank. They are, for the most part, Porisms, though Mr. Stewart, careful not to anticipate the discoveries of his friend, gave them only the name of Theorems. They are among the most beautiful, as well as most general propositions, known in the whole compass of geometry, and are perhaps only equalled by the remarkable locus to the circle in the second book of Apollonius, or by the celebrated theorem of Mr. Cotes.</p><p>Such is the history of the invention of these propositions; and the occasion of the publication of them was as follows. Mr. Stewart, while engaged in them, had entered into the church, and become minister of Roseneath. It was in that retired and romantic situation, that he discovered the greater part of those theorems. In the summer of 1746, the mathematical chair in the university of Edinburgh became vacant, by the death of Mr. Maclaurin. The General Theorems had not yet appeared; Mr. Stewart was known only to his friends; and the eyes of the public were naturally turned on Mr. Stirling, who then resided at Leadhills, and who was well known in the mathematical world. He however declined appearing as a candidate for the vacant chair; and several others were named, among whom was Mr. Stewart. Upon this occasion he printed the <hi rend="italics">General Theorems,</hi> which gave their author a decided superiority above all the other candidates. He was accordingly elected professor of mathematics in the university of Edinburgh, in September 1747.</p><p>The duties of this office gave a turn somewhat different to his mathematical pursuits, and led him to think of the most simple and elegant means of explaining those difficult propositions, which were hitherto only accessible to men deeply versed in the modern analysis. In doing this, he was pursuing the object which, <pb n="527"/><cb/> of all others, he most ardently wished to attain, viz, the application of geometry to such problems as the algebraic calculus alone had been thought able to resolve. His solution of Kepler's problem was the first specimen of this kind which he gave to the world; and it was perhaps impossible to have produced one more to the credit of the method he followed, or of the abilities with which he applied it. Among the excellent solutions hitherto given of this famous problem, there were none of them at once direct in its method, and simple in its principles. Mr. Stewart was so happy as to attain both these objects. He founds his solution on a general property of curves, which, though very simple, had perhaps never been observed; and by a most ingenious application of that property, he shows how the approximation may be continued to any degree of accuracy, in a series of results which converge with great rapidity.</p><p>This solution appeared in the second volume of the Essays of the Philosophical Society of Edinburgh, for the year 1756. In the first volume of the same collection, there are some other propositions of Mr. Stewart's, which are an extension of a curious theorem in the 4th book of Pappus. They have a relation to the subject of Porisms, and one of them forms the 91st of Dr. Simson's Restoration.</p><p>It has been already mentioned, that Mr. Stewart had formed the plan of introducing into the higher parts of mixed mathematics, the strict and simple form of ancient demonstration. The prosecution of this plan produced the <hi rend="italics">Tracts Physical and Mathematical,</hi> which were published in 1761. In the first of these, Mr. Stewart lays down the doctrine of centripetal forces, in a series of propositions, demonstrated (if we admit the quadrature of curves) with the utmost rigour, and requiring no previous knowledge of the mathematics, except the elements of plane Geometry, and of Conic Sections. The good order of these propositions, added to the clearness and simplicity of the demonstrations, renders this Tract perhaps the best elementary treatise of Physical Astronomy that is any where to be found.</p><p>In the three remaining Tracts, our author had it in view to determine, by the same rigorous method, the effect of those forces which disturb the motions of a secondary planet. From this he proposed to deduce, not only a theory of the moon, but a determination of the sun's distance from the earth. The former, it is well known, is the most difficult subject to which mathematics have been applied, and the resolution required and merited all the clearness and simplicity which our author possessed in so eminent a degree. It must be regretted therefore, that the decline of Dr. Stewart's health, which began soon after the publication of the Tracts, did not permit him to pursue this investigation.</p><p>The other object of the Tracts was, to determine the distance of the sun, from his effect in disturbing the motions of the moon; and his enquiries into the lunar irregularities had furnished him with the means of accomplishing it.</p><p>The theory of the composition and resolution of forces enables us to determine what part of the solar force is employed in disturbing the motions of the moon; and therefore, could we measure the instanta- <cb/> neous effect of that force, or the number of feet by which it accelerates or retards the moon's motion in a second, we should be able to determine how many feet the whole force of the fun would make a body, at the distance of the moon, or of the earth, descend in a second of time, and consequently how much the earth is, in every instant, turned out of its rectilineal course. Thus the curvature of the earth's orbit, or, which is the same thing, the radius of that orbit, that is, the distance of the sun from the earth, would be determined. But the fact is, that the instantaneous effects of the sun's disturbing force are too minute to be measured; and that it is only the effect of that force, continued for an entire revolution, or some considerable portion of a revolution, which astronomers are able to observe.</p><p>There is yet a greater difficulty which embarrasses the solution of this problem. For as it is only by the difference of the forces exerted by the sun on the earth and on the moon, that the motions of the latter are disturbed, the farther off the sun is supposed, the less will be the force by which he disturbs the moon's motions; yet that force will not diminish beyond a fixed limit, and a certain disturbance would obtain, even if the distance of the sun were infinite. Now the sun is actually placed at so great a distance, that all the disturbances, which he produces on the lunar motions, are very near to this limit, and therefore a small mistake in estimating their quantity, or in reasoning about them, may give the distance of the sun infinite, or even impossible. But all this did not deter Dr. Stewart from undertaking the solution of the problem, with no other assistance than that which geometry could afford. Indeed the idea of such a problem had first occurred to Mr. Machin, who, in his book on the laws of the moon's motion, has just mentioned it, and given the result of a rude calculation (the method of which he does not explain), which assigns 8″ for the parallax of the sun. He made use of the motion of the nodes; but Dr. Stewart considered the motion of the apogee, or of the longer axis of the moon's orbit, as the irregularity best adapted to his purpose. It is well known that the orbit of the moon is not immoveable; but that, in consequence of the disturbing force of the sun, the longer axis of that orbit has an angular motion, by which it goes back about 3 degrees in every lunation, and completes an entire revolution in 9 years nearly. This motion, though very remarkable and easily determined, has the same fault, in respect of the present problem, that was ascribed to the other irregularities of the moon: for a very small part of it only depends on the parallax of the sun; and of this Dr. Stewart seems not to have been perfectly aware.</p><p>The propositions however which defined the relation between the sun's distance and the mean motion of the apogee, were published among the Tracts, in 1761. The transit of Venus happened in that same year: the astronomers returned, who had viewed that curious phenomenon, from the most distant stations; and no very satisfactory result was obtained from a comparison of their observations. Dr. Stewart then resolved to apply the principles he had already laid down; and, in 1763, he published his essay on the Sun's Distance, where the computation being actually made, the parallax of the sun was found to be no more than 6″ 9, <pb n="528"/><cb/> and consequently his distance almost 29875 semidiameters of the earth, or nearly 119 millions of miles.</p><p>A determination of the sun's distance, that so far exceeded all former estimations of it, was received with surprise, and the reasoning on which it was founded was likely to undergo a severe examination. But, even among astronomers, it was not every one who could judge in a matter of such difficult discussion. Accordingly, it was not till about 5 years after the publication of the sun's distance, that there appeared a pamphlet, under the title of <hi rend="italics">Four Propositions,</hi> intended to point out certain errors in Dr. Stewart's investigation, which had given a result much greater than the truth. From his desire of simplifying, and of employing only the geometrical method of reasoning, he was reduced to the necessity of rejecting quantities, which were considerable enough to have a great effect on the last result. An error was thus introduced, which, had it not been for certain compensations, would have become immediately obvious, by giving the sun's distance near three times as great as that which has been mentioned.</p><p>The author of the pamphlet, referred to above, was the first who remarked the dangerous nature of these simplifications, and who attempted to estimate the error to which they had given rise. This author remarked what produced the compensation above mentioned, viz, the immense variation of the sun's distance, which corresponds to a very small variation of the motion of the moon's apogee. And it is but justice to acknowledge that, besides being just in the points already mentioned, they are very ingenious, and written with much modesty and good temper. The author, who at first concealed his name, but has now consented to its being made public, was Mr. Dawson, a surgeon at Sudbury in Yorkshire, and one of the most ingenious mathematicians and philosophers this country now possesses.</p><p>A second attack was soon after this made on the Sun's Distance, by Mr. Landen; but by no means with the same good temper which has been remarked in the former. He fancied to himself errors in Dr. Stewart's investigation, which have no existence; he exaggerated those that were real, and seemed to triumph in the discovery of them with unbecoming exultation. If there are any subjects on which men may be expected to reason dispassionately, they are certainly the properties of number and extension; and whatever pretexts moralists or divines may have for abusing one another, mathematicians can lay claim to no such indulgence. The asperity of Mr. Landen's animadversions ought not therefore to pass uncensured, though it be united with sound reasoning and accurate discussion. The error into which Dr. Stewart had fallen, though first taken notice of by Mr. Dawson, whose pamphlet was sent by me to Mr. Landen as soon as it was printed (for I had the care of the edition of it) yet this gentleman extended his remarks upon it to greater exactness. But Mr. Landen, in the zeal of correction, brings many other charges against Dr. Stewart, the greater part of which seem to have no good foundation. Such are his objections to the second part of the investigation, where Dr. Stewart finds the relation between the disturbing force of the sun, and the motion of the apses of the lunar orbit. For this part, instead of being liable to objection, is deserving of the greatest praise, <cb/> since it resolves, by geometry alone, a problem which had eluded the efforts of some of the ablest mathematicians, even when they availed themselves of the utmost resources of the integral calculus. Sir Isaac Newton, though he assumed the disturbing force very near the truth, computed the motion of the apses from thence only at one half of what it really amounts to; so that, had he been required, like Dr. Stewart, to invert the problem, he would have committed an error, not merely of a few thousandth parts, as the latter is alleged to have done, but would have brought out a result double of the truth. <hi rend="italics">(Princip. Math. lib.</hi> 3, <hi rend="italics">prop.</hi> 3.) Machin and Callendrini, when commenting on this part of the Principia, found a like inconsistency between their theory and observation. Three other celebrated mathematicians, Clairaut, D'Alembert, and Euler, severally experienced the same difficulties, and were led into an error of the same magnitude. It is true, that, on resuming their computations, they found that they had not carried their approximations to a sufficient length, which when they had at last accomplished, their results agreed exactly with observation. Mr. Walmsley and Dr. Stewart were, I think, the first mathematicians who, employing in the solution of this difficult problem, the one the algebraic calculus, and the other the geometrical method, were led immediately to the truth; a circumstance so much for the honour of both, that it ought not to be forgotten. It was the business of an impartial critic, while he examined our author's reasonings, to have remarked and to have weighed these considerations.</p><p>The <hi rend="italics">Sun's Distance</hi> was the last work which Dr. Stewart published; and though he lived to see the animadversions made on it, that have been taken notice of above, he declined entering into any controversy. His disposition was far from polemical; and he knew the value of that quiet, which a literary man should rarely suffer his antagonists to interrupt. He used to say, that the decision of the point in question was now before the public; that if his investigation was right, it would never be overturned, and that if it was wrong, it ought not to be defended.</p><p>A few months before he published the Essay just mentioned, he gave to the world another work, entitled, <hi rend="italics">Propositiones More Veterum Demonstratæ.</hi> It consists of a series of geometrical theorems, mostly new; investigated, first by an analysis, and afterwards synthetically demonstrated by the inversion of the same analysis. This method made an important part in the analysis of the ancient geometricians; but few examples of it have been preserved in their writings, and those in the <hi rend="italics">Propositiones Geometricæ</hi> are therefore the more valuable.</p><p>Doctor Stewart's constant use of the geometrical analysis had put him in possession of many valuable propositions, which did not enter into the plan of any of the works that have been enumerated. Of these, not a few have found a place in the writings of Dr. Simson, where they will for ever remain, to mark the friendship of these two mathematicians, and to evince the esteem which Dr. Simson entertained for the abilities of his pupil. Many of these are in the work upon the Porisms, and others in the Conic Sections, viz, marked with the letter <hi rend="italics">x;</hi> also a theorem in the edition of Euclid's Data. <pb n="529"/><cb/></p><p>Soon after the publication of the <hi rend="italics">Sun's Distance,</hi> Dr. Stewart's health began to decline, and the duties of his office became burdensome to him. In the year 1772, he retired to the country, where he afterwards spent the greater part of his life, and never resumed his labours in the university. He was however so fortunate as to have a son to whom, though very young, he could commit the care of them with the greatest confidence. Mr. Dugald Stewart, having begun to give lectures for his father from the period above mentioned, was elected joint professor with him in 1775, and gave an early specimen of those abilities, which have not been confined to a single science.</p><p>After mathematical studies (on account of the bad state of health into which Dr. Stewart was falling) had ceased to be his business, they continued to be his amusement. The analogy between the circle and hyperbola had been an early object of his admiration. The extensive views which that analogy is continually opening; the alternate appearance and disappearance of resemblance in the midst of so much dissimilitude, make it an object that astonishes the experienced, as well as the young geometrician. To the consideration of this analogy therefore the mind of Dr. Stewart very naturally returned, when disengaged from other speculations. His usual success still attended his investigations; and he has left among his papers some curious approximations to the areas, both of the circle and hyperbola. For some years toward the end of his life, his health scarcely allowed him to prosecute study even as an amusement. He died the 23d of January 1785, at 68 years of age.</p><p>The habits of study, in a man of original genius, are objects of curiosity, and deserve to be remembered. Concerning those of Dr. Stewart, his writings have made it unnecessary to remark, that from his youth he had been accustomed to the most intense and continued application. In consequence of this application, added to the natural vigour of his mind, he retained the memory of his discoveries in a manner that will hardly be believed. He seldom wrote down any of his investigations, till it became necessary to do so for the purpose of publication. When he discovered any proposition, he would set down the enunciation with great accuracy, and on the same piece of paper would construct very neatly the figure to which it referred. To these he trusted for recalling to his mind, at any future period, the demonstration, or the analysis, however complicated it might be. Experience had taught him that he might place this confidence in himself without any danger of disappointment; and for this singular power, he was probably more indebted to the activity of his invention, than to the mere tenaciousness of his memory.</p><p>Though Dr. Stewart was extremely studious, he read but few books, and thus verified the observation of D'Alembert, that, of all the men of letters, mathematicians read least of the writings of one another. Our author's own investigations occupied him sufficiently; and indeed the world would have had reason to regret the misapplication of his talents, had he employed, in the mere acquisition of knowledge, that time which he could dedicate to works of invention.</p><p>It was Dr. Stewart's custom to spend the summer at <cb/> a delightful retreat in Ayrshire, where, after the academical labours of the winter were ended, he found the leisure necessary for the prosecution of his researches. In his way thither, he often made a visit to Dr. Simson of Glasgow, with whom he had lived from his youth in the most cordial and uninterrupted friendship. It was pleasing to observe, in these two excellent mathematicians, the most perfect esteem and affection for each other, and the most entire absence of jealousy, though no two men ever trode more nearly in the same path. The similitude of their pursuits served only to endear them to each other, as it will ever do with men superior to envy. Their sentiments and views of the science they cultivated, were nearly the same; they were both profound geometricians; they equally admired the ancient mathematicians, and were equally versed in their methods of investigation; and they were both apprehensive that the beauty of their favourite science would be forgotten, for the less elegant methods of algebraic computation. This innovation they endeavoured to oppose; the one, by reviving those books of the ancient geometry which were lost; the other, by extending that geometry to the most difficult enquiries of the moderns. Dr. Stewart, in particular, had remarked the intricacies, in which many of the greatest of the modern mathematicians had involved themselves in the application of the calculus, which a little attention to the ancient geometry would certainly have enabled them to avoid. He had observed too the elegant synthetical demonstrations that, on many occasions, may be given of the most difficult propositions, investigated by the inverse method of fluxions. These circumstances had perhaps made a stronger impression than they ought, on a mind already filled with admiration of the ancient geometry, and produced too unsavourable an opinion of the modern analysis. But if it be confessed that Dr. Stewart rated, in any respect too high, the merit of the former of these sciences, this may well be excused in the man whom it had conducted to the discovery of the <hi rend="italics">General Theorems,</hi> to the <hi rend="italics">solution of Kepler's Problem,</hi> and to an <hi rend="italics">accurate</hi> determination of the <hi rend="italics">Sun's disturbing force.</hi> His great modesty made him ascribe to the method he used, that success which he owed to his own abilities.</p><p>The foregoing account of Dr. Stewart and his writings, is chiefly extracted from the learned history of them, by Mr. Playfair, in the 1st volume of the Edinburgh Philosophical Transactions, pa. 57, &c.</p></div1><div1 part="N" n="STIFELS" org="uniform" sample="complete" type="entry"><head>STIFELS</head><p>, <hi rend="smallcaps">Stifelius (Michael</hi>), a Protestant minister, and very skilful mathematician, was born at Eslingen a town in Germany; and died at Jena in Thuringia, in the year 1567, at 58 years of age according to Vossius, but some others say 80. Stifels was one of the best mathematicians of his time. He published, in the German language, a treatise on Algebra, and another on the Calendar or Ecclesiastical computation. But his chief work, is the <hi rend="italics">Arithmetices Integra,</hi> a complete and excellent treatise, in Latin, on Arithmetic and Algebra, printed in 4to at Norimberg 1544. In this work there are a number of ingenious inventions, both in common arithmetic and in algebra; of which, those relating to the latter are amply explained under the article <hi rend="italics">Algebra</hi> in this dictionary, vol. 1, pa. 77 &c; to which may be added some par- <pb n="530"/><cb/> ticulars concerning the arithmetic, from my volume of <hi rend="italics">Tracts</hi> printed in 1786, pa. 68. In this original work are contained many curious things, some of which have mistakenly been ascribed to a much later date. He here treats pretty fully and ably, of progressional and figurate numbers, and in particular of the triangular table, for constructing both them and the coefficients of the terms of all powers of a binomial; which has been so often used since his time for these and other purposes, and which more than a century after was, by Pascal, otherwise called the Arithmetical Triangle, and who only mentioned some additional properties of the table. Stifels shews, that the horizontal lines of the table furnish the coefficients of the terms of the corresponding powers of a binomial; and teaches how to make use of them in the extraction of roots of all powers whatever. Cardan seems to ascribe the invention of that table to Stiselius; but I apprehend that is only to be understood of its application to the extraction of roots.</p><p>It is remarkable too, how our author, at p. 35 &c of the same book, treats of the nature and use of logarithms; not under that name indeed, but under the idea of a series of arithmeticals, adapted to a series of geometricals. He there explains all their uses; such as, that the addition of them answers to the multiplication of their geometricals; subtraction to division; multiplication of exponents, to involution; and dividing of exponents to evolution. He also exemplifies the use of them in cases of the Rule-of-three, and in finding mean proportionals between given terms, and such like, exactly as is done in logarithms. So that he seems to have been in the full possession of the idea of logarithms, and wanted only the necessity of troublesome calculations to induce him to make a table of such numbers.</p><p>Stifels was a zealous, though weak disciple of Luther. He took it into his head to become a prophet, and he predicted that the end of the world would happen on a certain day in the year 1553, by which he terrified many people. When the proposed day arrived, he repaired early, with multitudes of his followers, to a particular place in the open air, spending the whole day in the most fervent prayers and praises, in vain looking for the coming of the Lord, and the universal conflagration of the elements, &c.</p><p>STILE. See <hi rend="smallcaps">Style.</hi></p><p>STILYARD. See <hi rend="smallcaps">Steelyard.</hi></p></div1><div1 part="N" n="STOFLER" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">STOFLER</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, a German mathematician, was born at Justingen in Suabia, in 1452, and died in 1531, at 79 years of age. He taught mathematics at Tubinga, where he acquired a great reputation, which however he in a great measure lost again, by intermeddling with the prediction of future events. He announced a great deluge, which he said would happen in the year 1524, a prediction with which he terrified all Germany, where many persons prepared vessels proper to escape with from the floods. But happily the prediction failing, it enraged the astrologer, though it served to convince him of the vanity of his prognostications.—He was author of several works in mathematics, and astrology, full of foolish and chimerical ideas; such as,</p><p>1. Elucidatio Fabric. Ususque Astrolabii; fol. 1513.</p><p>2. Procli Sphæram Comment. fol. 154. <cb/></p><p>3. Cosmographicæ aliquot Descriptiones; 4to, 1537.</p></div1><div1 part="N" n="STONE" org="uniform" sample="complete" type="entry"><head>STONE</head><p>, (<hi rend="smallcaps">Edmund</hi>), a good Scotch mathematician, who was author of several ingenious works. I know not the particular place or date of his birth, but it was probably in the shire of Argyle, and about the beginning of the present century, or conclusion of the last. Nor have we any memoirs of his life, except a letter from the Chevalier de Ramsay, author of the Travels of Cyrus, in a letter to father Castel, a Jesuit at Paris, and published in the Memoires de Trevoux, p. 109, as follows: “True genius overcomes all the disadvantages of birth, fortune, and education; of which Mr. Stone is a rare example. Born a son of a gardener of the duke of Argyle, he arrived at 8 years of age before he learnt to read.—By chance a servant having taught young Stone the letters of the alphabet, there needed nothing more to discover and expand his genius. He applied himself to study, and he arrived at the knowledge of the most sublime geometry and analysis, without a master, without a conductor, without any other guide but pure genius.”</p><p>“At 18 years of age he had made these considerable advances without being known, and without knowing himself the prodigies of his acquisitions. The duke of Argyle, who joined to his military talents, a general knowledge of every science that adorns the mind of a man of his rank, walking one day in his garden, saw lying on the grass a Latin copy of Sir Isaac Newton's celebrated Principia. He called some one to him to take and carry it back to his library. Our young gardener told him that the book belonged to him. <hi rend="italics">To you?</hi> replied the Duke. <hi rend="italics">Do you understand geometry, Latin, Newton?</hi> I know a little of them, replied the young man with an air of simplicity arising from a profound ignorance of his own knowledge and talents. The Duke was surprised; and having a taste for the sciences, he entered into conversation with the young mathematician: he asked him several questions, and was astonished at the force, the accuracy, and the candour of his answers. <hi rend="italics">But how,</hi> said the Duke, <hi rend="italics">came you by the knowledge of all these things?</hi> Stone replied, <hi rend="italics">A servant taught me, ten years since, to read: does one need to know any thing more than the</hi> 24 <hi rend="italics">letters in order to learn every thing else that one wishes?</hi> The Duke's curiosity redoubled—he sat down upon a bank, and requested a detail of all his proceedings in becoming so learned.”</p><p>“<hi rend="italics">I first learned to read,</hi> said Stone: <hi rend="italics">the masons were then at work upon your house: I went near them one day, and I saw that the architect used a rule, compasses, and that he made calculations. I enquired what might be the meaning of and use of these things; and I was informed that there was a science called Arithmetic; I purchased a book of arithmetic, and I learned it.—I was told there was another science called Geometry: I bought the books, and I learnt geometry. By reading I found that there were good books in these two sciences in Latin: I bought a dictionary, and I learned Latin. I understood also that there were good books of the same kind in French: I bought a dictionary, and I learned French. And this, my lord, is what I have done: it seems to me that we may learn every thing when we know the</hi> 24 <hi rend="italics">letters of the alphabet.</hi></p><p>This account charmed the Duke. He drew this wonderful genius out of his obscurity; and he provided him with an employment which left him plenty <pb n="531"/><cb/> of time to apply himself to the sciences. He discovered in him also the same genius for music, for painting, for architecture, for all the sciences which depend on calculations and proportions.”</p><p>“I have seen Mr. Stone. He is a man of great simplicity. He is at present sensible of his own knowledge: but he is not puffed up with it. He is possessed with a pure and distinterested love for the mathematics; though he is not solicitous to pass for a mathematician; vanity having no part in the great labour he sustains to excel in that science. He despises fortune also; and he has solicited me twenty times to request the duke to give him less employment, which may not be worth the half of that he now has, in order to be more retired, and less taken off from his favourite studies. He discovers sometimes, by methods of his own, truths which others have discovered before him. He is charmed to find on these occasions that he is not a first inventor, and that others have made a greater progress than he thought. Far from being a plagiary, he attributes ingenious solutions, which he gives to certain problems, to the hints he has found in others, although the connection is but very distant,” &c.</p><p>Mr. Stone was author and translator of several useful works; viz.</p><p>1. A New Mathematical Dictionary, in 1 vol. 8vo, first printed in 1726.</p><p>2. Fluxions, in 1 vol. 8vo, 1730. The Direct Method is a translation from the French, of Hospital's Analyse des Infiniments Petits; and the Inverse Method was supplied by Stone himself.</p><p>3. The Elements of Euclid, in 2 vols. 8vo, 1731. A neat and useful edition of those Elements, with an account of the life and writings of Euclid, and a defence of his elements against modern objectors.</p><p>Beside other smaller works.</p><p>Stone was a fellow of the Royal Society, and had inserted in the Philos. Transactions (vol. 41, pa. 218) an “Account of two species of lines of the 3d order, not mentioned by Sir Isaac Newton, or Mr. Stirling.”</p></div1><div1 part="N" n="STRABO" org="uniform" sample="complete" type="entry"><head>STRABO</head><p>, a celebrated Greek geographer, philosopher, and historian, was born at Amasia, and was descended from a family settled at Gnossus in Crete. He was the disciple of Xenarchus, a Peripatetic philosopher, but at length attached himself to the Stoics. He contracted a strict friendship with Cornelius Gallus, governor of Egypt; and travelled into several countries, to observe the situation of places, and the customs of nations.</p><p>Strabo flourished under Augustus; and died under Tiberius about the year 25, in a very advanced age.— He composed several works; all of which are lost, except his <hi rend="italics">Geography,</hi> in 17 books; which are justly esteemed very precious remains of antiquity. The first two books are employed in showing, that the study of geography is not only worthy of a philosopher, but even necessary to him; the 3d describes Spain; the 4th, Gaul and the Britannic isles; the 5th and 6th, Italy and the adjacent isles; the 7th, which is imperfect at the end, Germany, the countries of the Getæ and Illyrii, Taurica, Chersonesus, and Epirus; the 8th, 9th, and 10th, Greece with the neighbouring isles; the four following, Asia within Mount Taurus; the 15th and 16th, Asia without Taurus, India, Persia, <cb/> Syria, Arabia; and the 17th, Egypt, Ethiopia, Carthage, and other parts of Africa.</p><p>Strabo's work was published with a Latin version by Xylander, and notes by Isaac Casaubon, at Paris 1620, in folio; but the best edition is that of Amsterdam in 1707, in 2 volumes folio, by the learned Theodore Janson of Almelooveen, with the entire notes of Xylander, Casaubon, Meursius, Cluver, Holsten, Salmasius, Bochart, Ez. Spanheim, Cellar, and others. To this edition is subjoined the <hi rend="italics">Chrestomathiæ,</hi> or Epitome of Strabo; which, according to Mr. Dodswell, who has written a very elaborate and learned dissertation about it, was made by some unknown person, between the years of Christ 676 and 996. It has been found of some use, not only in helping to correct the original, but in supplying in some measure the defect in the 7th book. Mr. Dodswell's dissertation is prefixed to this edition.</p></div1><div1 part="N" n="STRAIT" org="uniform" sample="complete" type="entry"><head>STRAIT</head><p>, or <hi rend="smallcaps">Straight</hi>, or <hi rend="smallcaps">Streight</hi>, in Hydrography, is a narrow channel or arm of the sea, shut up between lands on either side, and usually affording a passage out of one great sea into another. As the Straits of Magellan, of Le Maire, of Gibraltar, &c.</p><p><hi rend="smallcaps">Strait</hi> is also sometimes used, in Geography, for an isthmus, or neck of land between two seas, preventing their communication.</p></div1><div1 part="N" n="STRENGTH" org="uniform" sample="complete" type="entry"><head>STRENGTH</head><p>, <hi rend="italics">vis,</hi> force, power.</p><p>Some authors make the Strength of animals, of the same kind, to depend on the quantity of blood; but most on the size of the bones, joints, and muscles; though we find by daily experience, that the animal spirits contribute greatly to Strength at different times.</p><p>Emerson has most particularly treated of the Strength of bodies depending on their dimensions and weight. In the General Scholium after his propositions on this subject, he adds; If a certain beam of timber be able to support a given weight; another beam, of the same timber, similar to the former, may be taken so great, as to be able but just to bear its own weight: while any larger beam cannot support itself, but must break by its own weight; but any less beam will bear something more. For the Strength being as the cube of the depth; and the stress, being as the length and quantity of matter, is as the 4th power of the depth; it is plain therefore, that the stress increases in a greater ratio than the Strength. Whence it follows, that a beam may be taken so large, that the stress may far exceed the Strength: and that, of all similar beams, there is but one that will just support itself, and nothing more. And the like holds true in all machines, and in all animal bodies. And hence there is a certain limit, in regard to magnitude, not only in all machines and artificial structures, but also in natural ones, which neither art nor nature can go beyond; supposing them made of the same matter, and in the same proportion of parts.</p><p>Hence it is impossible that mechanic engines can be increased to any magnitude at pleasure. For when they arrive at a particular size, their several parts will break and fall asunder by their weight. Neither can any buildings of vast magnitudes be made to stand, but must fall to pieces by their great weight, and go to ruin. <pb n="532"/><cb/></p><p>It is likewise impossible for nature to produce animals of any vast size at pleasure: except some sort of matter can be found, to make the bones of, which may be so much harder and stronger than any hitherto known: or else that the proportion of the parts be so much altered, and the bones and muscles made thicker in proportion; which will make the animal distorted, and of a monstrous figure, and not capable of performing any proper actions. And being made similar and of common matter, they will not be able to stand or move; but, being burthened with their own weight, must fall down. Thus, it is impossible that there can be any animal so large as to carry a castle upon his back; or any man so strong as to remove a mountain, or pull up a large oak by the roots: nature will not admit of these things; and it is impossible that there can be animals of any sort beyond a determinate size.</p><p>Fish may indeed be produced to a larger size than land animals; because their weight is supported by the water. But yet even these cannot be increased to immensity, because the internal parts will press upon one another by their weight, and destroy their fabric.</p><p>On the contrary, when the size of animals is diminished, their Strength is not diminished in the same proportion as the weight. For which reason a small auimal will carry far more than a weight equal to its own, whilst a great one cannot carry so much as its weight. And hence it is that small animals are more active, will run faster, jump farther, or perform any motion quicker, for their weight, than large animals: for the less the animal, the greater the proportion of the Strength to the stress. And nature seems to know no bounds as to the smallness of animals, at least in regard to their weight.</p><p>Neither can any two unequal and similar machines resist any violence alike, or in the same proportion; but the greater will be more hurt than the less. And the same is true of animals; for large animals by falling break their bones, while lesser ones, falling higher, receive no damage. Thus a cat may fall two or three yards high, and be no worse, and an ant from the top of a tower.</p><p>It is likewise impossible in the nature of things, that there can be any trees of immense size; if there were any such, their limbs, boughs, and branches, must break off and fall down by their own weight. Thus it is impossible there can be an oak a quarter of a mile high; such a tree cannot grow or stand, but its limbs will drop off by their weight. And hence also smaller plants can better sustain themselves than large ones.</p><p>As to the due proportion of Strength in several bodies, according to their particular positions, and the weights they are to bear; he farther observes that, If a piece of timber is to be pierced with a mortise-hole, the beam will be stronger when it is taken out of the middle, than when taken out of either side. And in a beam supported at both ends, it is stronger when the hole is made in the upper side than when made in the under, provided a piece of wood is driven hard in to fill up the hole.</p><p>If a piece is to be spliced upon the end of a beam, to be supported at both ends; it will be the stronger <cb/> when spliced on the under side of a beam: but if the piece is supported only at one end, to bear a weight on the other; it is stronger when spliced on the upper side.</p><p>When a small lever, &c, is nailed to a body, to move it or suspend it by; the strain is greater upon the nail nearest the hand, or point where the power is applied.</p><p>If a beam be supported at both ends; and the two ends reach over the props, and be sixed down immoveable; it will bear twice as much weight, as when the ends only lie loose or free upon the supporters.</p><p>When a slender cylinder is to be supported by two pieces; the distance of the pins ought to be nearly 3/5 of the length of the cylinder, and the pins equidistant from its ends; and then the cylinder will endure the least bending or strain by its weight.</p><p>A beam sixed at one end, <figure/> and bearing a weight at the other; if it be cut in the form of a wedge, and placed with its parallel sides parallel to the horizon; it will be equally strong every where; and no sooner break in one place than another.</p><p>When a beam has all its sides cut in form of a concave <figure/> parabola, having the vertex at the end, and its absciss perpendicular to the axis of the solid, and the base a square, or a circle, or any regular polygon; such a beam sixed horizontally, at one end, is equally strong throughout for supporting its own weight.</p><p>Also when a wall faces the wind, and if the vertical section of it be a right-angled triangle; or if the fore part next the wind &c be perpendicular to the horizon, and the back part a sloping plane; such a wall will be equally strong in all its parts to resist the wind, if the parts of the wall cohere strongly together; but when it is built of loose materials, it is better to be convex on the back part in form of a parabola.</p><p>When a wall is to support a bank of earth or any fluid body, it ought to be built concave in form of a semicubical parabola, whose vertex is at the top of the wall, provided the parts of the wall adhere firmly together. But if the parts be loose, then a right line or sloping plane ought to be its figure. Such walls will be equally strong throughout</p><p>All spires of churches in the form of cones or pyramids, are equally strong in all parts to resist the wind. But when the parts do not cohere together, then they ought to be parabolic conoids, to be equally strong throughout.</p><p>Likewise if there be a pillar erected in form of the logarithmic curve, the asymptote being the axis; it cannot be crushed to pieces in one part sooner than in another, by its own weight. And if such a pillar be turned upside down, and suspended by the thick end, it will not be more liable to separate in one part than another, by its own weight. <pb n="533"/><cb/></p><p>Moreover, if AE be a beam <figure/> in form of a triangular prism; and if AD = (1/9)AB, and AI = (1/9) AC, and the edge or small similar prism ADIF be cut away parallel to the base; the remaining beam DIBEF will bear a greater weight P, than the whole ABCEG, or the part will be stronger than the whole; which is a paradox in Mechanics.</p><p>As to the Strength of several sorts of wood, drawn from experiments, he says, On a medium, a piece of good oak, an inch square, and a yard long, supported at both ends, will bear in the middle, for a very short time, about 330lb averdupois, but will break with more than that weight. But such a piece of wood should not, in practice, be trusted for any length of time, with more than a third or a fourth part of that weight. And the proportion of the Strength of several sorts of wood, he found to be as follows: <table><row role="data"><cell cols="1" rows="1" role="data">Box, oak, plumbtree, yew</cell><cell cols="1" rows="1" rend="align=center" role="data">11</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ash, elm</cell><cell cols="1" rows="1" rend="align=center" role="data">8 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Thorn, walnut</cell><cell cols="1" rows="1" rend="align=center" role="data">7 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Apple tree, elder, red fir, holly, plane</cell><cell cols="1" rows="1" rend="align=center" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Beech, cherry, hazle</cell><cell cols="1" rows="1" rend="align=center" role="data">6 2/3</cell></row><row role="data"><cell cols="1" rows="1" role="data">Alder, asp, Birch, white-fir, willow</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell></row><row role="data"><cell cols="1" rows="1" role="data">Iron</cell><cell cols="1" rows="1" rend="align=center" role="data">107</cell></row><row role="data"><cell cols="1" rows="1" role="data">Brass</cell><cell cols="1" rows="1" rend="align=center" role="data">50</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bone</cell><cell cols="1" rows="1" rend="align=center" role="data">22</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lead</cell><cell cols="1" rows="1" rend="align=center" role="data">6 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data">Fine free stone</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row></table></p><p>As to the Strength of bodies in direction of the fibres, he observes, A cylindric rod of good clean fir, of an inch circumference, drawn in length, will bear at extremity 400lb; and a spear of fir 2 inches diameter, will bear about 7 ton.—A rod of good iron, of an inch circumference, will bear near 3 ton weight. And a good hempen rope of an inch circumference, will bear 1000lb. at extremity.</p><p>All this supposes these bodies to be sound and good throughout; but none of them should be put to bear more than a third or a fourth part of that weight, especially for any length of time. From what has been said; if a spear of fir, or a rope, or a spear of iron, of <hi rend="italics">d</hi> inches diameter, were to lift 1/4 the extreme weight; then</p><p>The fir would bear 8 (4/5)<hi rend="italics">dd</hi> hundred weight.</p><p>The rope would bear 22<hi rend="italics">dd</hi> hundred weight.</p><p>The iron would bear 6 (3/4)<hi rend="italics">dd</hi> ton weight.</p><p>As to Animals; Men may apply their Strength several ways, in working a machine. A man of ordinary Strength turning a roller by the handle, can act for a whole day against a resistance equal to 30lb. weight; and if he works 10 hours a day, he will raise a weight of 30lb. through 3 1/2 feet in a second of time; or if the weight be greater, he will raise it so much less in proportion. But a man may act, for a small time, against a resistance of 50lb. or more.</p><p>If two men work at a windlass, or roller, they can more easily draw up 70lb, than one man can 30lb, provided the elbow of one of the handles be at right angles <cb/> to that of the other. And with a fly, or heavy wheel, applied to it, a man may do 1/3 part more work; and for a little while he can act with a force, or overcome a continual resistance, of 80l; and work a whole day when the resistance is but 40lb.</p><p>Men used to bear loads, such as porters, will carry, some 150lb, others 200 or 250lb. according to their Strength.</p><p>A man can draw but about 70 or 80lb. horizontally; for he can but apply about half his weight.</p><p>If the weight of a man be 140lb, he can act with no greater a force in thrusting horizontally, at the height of his shoulders, than 27lb.</p><p>As to Horses: A horse is, generally speaking, as strong as 5 men. A horse will carry 240 or 270lb. A horse draws to greatest advantage, when the line of direction is a little elevated above the horizon, and the power acts against his breast: and he can draw 200lb. for 8 hours a day, at 2 1/2 miles an hour. If he draw 240lb, he can work but 6 hours, and not go quite so fast. And in both cases, if he carries some weight, he will draw the better for it. And this is the weight a horse is supposed to be able to draw over a pulley out of a well. But in a cart, a horse may draw 1000lb, or even double that weight, or a ton weight, or more.</p><p>As the most force a horse can exert, is when he draws a little above the horizontal position: so the worst way of applying the strength of a horse, is to make him carry or draw uphill: And three men in a steep hill, carrying each 100lb, will climb up faster than a horse with 300l. Also, though a horse may draw in a round walk of 18 feet diameter; yet such a walk should not be less than 25 or 30 feet diameter. Emerson's Mechan. pa. 111 and 177.</p></div1><div1 part="N" n="STRIKE" org="uniform" sample="complete" type="entry"><head>STRIKE</head><p>, or <hi rend="smallcaps">Stryke</hi>, a measure, containing 4 bushels, or half a quarter.</p><p>STRIKING-<hi rend="italics">wheel,</hi> in a clock, the same as that by some called the <hi rend="italics">pin-wheel,</hi> because of the pins which are placed on the round or rim, the number of which is the quotient of the pinion divided by the pinion of the detent-wheel. In sixteen-day clocks, the first or great wheel is usually the pin-wheel; but in such as go 8 days, the second wheel is the pin-wheel, or strikingwheel.</p></div1><div1 part="N" n="STRING" org="uniform" sample="complete" type="entry"><head>STRING</head><p>, in Music. See <hi rend="smallcaps">Chord.</hi></p><p>If two Strings or chords of a musical instrument only differ in length; their tones, or the number of vibrations they make in the same time, are in the inverse ratio of their lengths. If they differ only in thickness, their tones are in the inverse ratio of their diameters.</p><p>As to the tension of Strings, to measure it regularly, they must be conceived stretched or drawn by weights; and then, cæteris paribus, the tones of two Strings are in a direct ratio of the square roots of the weights that stretch them; that is, ex. gr. the tone of a String stretched by a weight 4, is an octave above the tone of a String stretched by the weight <*>.</p><p>It is an observation of very old standing, that if a viol or lute-string be touched with the bow, or the hand, another String on the same instrument, or even on another, not far from it, if in unison with it, or in octave, or the like, will at the same time tremble of <pb n="534"/><cb/> its own accord. But it is now found, that it is not the whole of that other String that thus trembles, but only the parts, severally, according as they are unisons to the whole, or the parts, of the String so struck. Thus, supposing AB to be <figure/> an upper octave to <hi rend="italics">ab,</hi> and therefore an unison to each half of it, stopped at <hi rend="italics">c;</hi> if while <hi rend="italics">ab</hi> is open, AB be struck, the two halves of this other, that is, <hi rend="italics">ac,</hi> and <hi rend="italics">cb,</hi> will both tremble; but the middle point will be at rest; as will be easily perceived, by wrapping a bit of paper lightly about the string <hi rend="italics">ab,</hi> and moving it successively from one end of the string to the other. In like manner, if AB were an upper 12th to <hi rend="italics">ab,</hi> and consequently an unison to its three parts <hi rend="italics">ad, de, eb;</hi> then, <hi rend="italics">ab</hi> being open, if AB be struck, the three parts of the other, <hi rend="italics">ad, de, eb</hi> will severally tremble; but the points <hi rend="italics">d</hi> and <hi rend="italics">e</hi> remain at rest.</p><p>This, Dr. Wallis tells us, was first discovered by Mr. William Noble of Merton college; and after him by Mr. T. Pigot of Wadham college, without knowing that Mr. Noble had observed it before. To which may be added, that M. Sauveur, long afterwards, proposed it to the Royal Academy at Paris, as his own discovery, which in reality it might be; but upon his being informed, by some of the members then present, that Dr. Wallis had published it before, he immediately resigned all the honour of it. Philos. Trans. Abridg. vol. I, pa. 606.</p></div1><div1 part="N" n="STURM" org="uniform" sample="complete" type="entry"><head>STURM</head><p>, <hi rend="smallcaps">Sturmius (John Christopher</hi>), a noted German mathematician and philospher, was born at Hippolstein in 1635. He became professor of philosophy and mathematics at Altdorf, where he died in 1703, at 68 years of age.</p><p>He was author of several useful works on mathematics and philosophy, the most esteemed of which are,</p><p>1. His <hi rend="italics">Mathesis enucleata,</hi> in 1 vol. 8vo.</p><p>2. <hi rend="italics">Mathesis Fuvenilis,</hi> in 2 large volumes 8vo.</p><p>3. <hi rend="italics">Collegium Experimentale, sive Curiosum, in quo primaria Seculi superioris Inventa & Experimenta PhysicoMathematica, Speciatim Campanæ Urinatoriæ, Cameræ obscuræ, Tubi Torricelliani, seu Baroscopii, Antliæ Pneumaticæ, Thermometrorum Phænomena & Effecta; partim ac aliis jampridem exhibita, partim noviter istis superaddita, &c.</hi> in one large vol. 4to, Norimberg, 1701.</p><p>This is a very curious work, containing a multitude of interesting experiments, neatly illustrated by copperplate figures printed upon almost every page, by the side of the letter-press. Of these, the 10th experiment is an improvement on father Lana's project for navigating a small vessel suspended in the atmosphere by several globes exhausted of air.</p></div1><div1 part="N" n="STYLE" org="uniform" sample="complete" type="entry"><head>STYLE</head><p>, in Chronology, a particular manner of <cb/> counting time; as the <hi rend="italics">Old Style,</hi> the <hi rend="italics">New Style.</hi> See <hi rend="smallcaps">Calendar.</hi></p><p><hi rend="italics">Old</hi> <hi rend="smallcaps">Syyle</hi>, is the Julian manner of computing, as instituted by Julius Cæsar, in which the mean year consists of 365 1/4 days.</p><p><hi rend="italics">New</hi> <hi rend="smallcaps">Style</hi>, is the Gregorian manner of computation, instituted by pope Gregory the 13th, in the year 1582, and is used by most catholic countries, and many other states of Europe.</p><p>The Gregorian, or new Style, agrees with the true solar year, which contains only 365 days 5 hours 49 minutes. In the year of Christ 200, there was no difference of Styles. In the year 1582, when the new Style was first introduced, there was a difference of 10 days. At present there is 11 days difference, and accordingly at the diet of Ratisbon, in the year 1700, it was decreed by the body of protestants of the empire, that 11 days should be retrenched from the old Style, to accommodate it for the future to the new. And the same regulation has since passed into Sweden, Denmark, and into England, where it was established in the year 1752, when it was enacted, that in all dominions belonging to the crown of Great Britain, the supputation, according to which the year of our lord begins on the 25th day of March, shall not be used from and after the last day of December 1751; and that from thenceforth, the 1st day of January every year shall be reckoned to be the first day of the year: and that the natural day next immediately following the 2d day of September 1752, shall be accounted the 14th day of September, omitting the 11 intermediate nominal days of the common calendar. It is farther enacted, that all kinds of writings, &c, shall bear date according to the new method of computation, and that all courts and meetings &c, feasts, fasts, &c, shall be held and observed accordingly. And for preserving the calendar in the same regular course for the future, it is enacted, that the several years of our lord 1800, 1900, 2100, 2200, 2300, &c, except only every 400th year, of which the year 2000 shall be the first, shall be common years of 365 days, and that the years 2000, 2400, 2800, &c, and every other 400th year from the year 2000 inclusive, shall be leap years, consisting of 366 days. See <hi rend="smallcaps">Bissextile</hi> and <hi rend="smallcaps">Calendar.</hi></p><p>The following table shews by what number of days the new style differs from the old, from 5900 years before the birth of Christ, to 5900 years after it. The days under the sign — (viz from 6000 years before to 200 years after Christ) are to be subtracted from the old Style, to reduce it to the new; and the days under the sign + (viz from 200 to 5900 years after Christ) are to be added to the old Style, to reduce it to the new.—N.B. All the years mentioned in the table are leap years in the old Style; but those only that are marked with an L are leap years in the new. <pb n="535"/><cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Years before</cell><cell cols="1" rows="1" role="data">Days</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Years after</cell><cell cols="1" rows="1" role="data">Days</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Christ.</cell><cell cols="1" rows="1" role="data">diff.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Christ.</cell><cell cols="1" rows="1" role="data">diff.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">New Style.</cell><cell cols="1" rows="1" role="data">-</cell><cell cols="1" rows="1" rend="colspan=2" role="data">New Style.</cell><cell cols="1" rows="1" role="data">∓</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5900</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">- 2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5800</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">- 1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5700</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">5600</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">+ 1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5500</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">1</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5400</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">2</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5300</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">5200</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">700</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5100</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">800</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5000</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">900</cell><cell cols="1" rows="1" role="data">5</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4900</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">4800</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1100</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4700</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">1200</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4600</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1300</cell><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4500</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1400</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">4400</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1500</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4300</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">1600</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4200</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1700</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4100</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1800</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">4000</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1900</cell><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3900</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">2000</cell><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3800</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2100</cell><cell cols="1" rows="1" role="data">14</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3700</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2200</cell><cell cols="1" rows="1" role="data">15</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">3600</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2300</cell><cell cols="1" rows="1" role="data">16</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3500</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">2400</cell><cell cols="1" rows="1" role="data">16</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3400</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2500</cell><cell cols="1" rows="1" role="data">17</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3300</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2600</cell><cell cols="1" rows="1" role="data">18</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">3200</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2700</cell><cell cols="1" rows="1" role="data">19</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3100</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">2800</cell><cell cols="1" rows="1" role="data">19</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3000</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2900</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2900</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3000</cell><cell cols="1" rows="1" role="data">21</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">2800</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3100</cell><cell cols="1" rows="1" role="data">22</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2700</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">3200</cell><cell cols="1" rows="1" role="data">22</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2600</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3300</cell><cell cols="1" rows="1" role="data">23</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2500</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3400</cell><cell cols="1" rows="1" role="data">24</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">2400</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3500</cell><cell cols="1" rows="1" role="data">25</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2300</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">3600</cell><cell cols="1" rows="1" role="data">25</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2200</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3700</cell><cell cols="1" rows="1" role="data">26</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2100</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3800</cell><cell cols="1" rows="1" role="data">27</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">2000</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">3900</cell><cell cols="1" rows="1" role="data">28</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1900</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">4000</cell><cell cols="1" rows="1" role="data">28</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1800</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4100</cell><cell cols="1" rows="1" role="data">29</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1700</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4200</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">1600</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4300</cell><cell cols="1" rows="1" role="data">31</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1500</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">4400</cell><cell cols="1" rows="1" role="data">31</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1400</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4500</cell><cell cols="1" rows="1" role="data">32</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1300</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4600</cell><cell cols="1" rows="1" role="data">33</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">1200</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4700</cell><cell cols="1" rows="1" role="data">34</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1100</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">4800</cell><cell cols="1" rows="1" role="data">34</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1000</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4900</cell><cell cols="1" rows="1" role="data">35</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">900</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5000</cell><cell cols="1" rows="1" role="data">36</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">800</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5100</cell><cell cols="1" rows="1" role="data">37</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">700</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">5200</cell><cell cols="1" rows="1" role="data">37</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5300</cell><cell cols="1" rows="1" role="data">38</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5400</cell><cell cols="1" rows="1" role="data">39</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5500</cell><cell cols="1" rows="1" role="data">40</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">5600</cell><cell cols="1" rows="1" role="data">40</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5700</cell><cell cols="1" rows="1" role="data">41</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5800</cell><cell cols="1" rows="1" role="data">42</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">L</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5900</cell><cell cols="1" rows="1" role="data">43</cell></row></table> <cb/></p><p>The French nation has lately commenced another new Style, or computation of time, viz, in the year 1792; according to which, the year commences usually on our 22d of September. The year is divided into 12 months of 30 days each; and each month into 3 decades of 10 days each. For the names and computations of which, see the article <hi rend="smallcaps">Calendar.</hi></p><div2 part="N" n="Style" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Style</hi></head><p>, in Dialling, denotes the cock or gnomon, raised above the plane of the dial, to project a Shadow. —The edge of the Style, which by its shadow marks the hours on the face of the dial, is to be set according to the latitude, always parallel to the axis of the world.</p></div2></div1><div1 part="N" n="STYLOBATA" org="uniform" sample="complete" type="entry"><head>STYLOBATA</head><p>, or <hi rend="smallcaps">Stylobaton</hi>, in Architecture, the same with the pedestal of a column. It is sometimes taken for the trunk of the pedestal, between the cornice and the base, and is then called <hi rend="italics">truncus.</hi> It is also otherwise named <hi rend="italics">abacus.</hi></p><p>SUBCONTRARY posi- <figure/> tion, in Geometry, is when two equiangular triangles, as VAB and VCD are so placed as to have one common angle V at the vertex, and yet their bases not parallel. Consequently the angles at the bases are equal, but on the contrary sides; viz, the [angle]A = [angle]C, and the [angle]B = [angle]D.</p><p>If the oblique cone VAB or V<hi rend="italics">ab,</hi> having the circular base AEB, or <hi rend="italics">aeb,</hi> be so cut by a plane DEC, that the angle D be = the [angle]B, or the [angle]C = [angle]A, then the cone is said to be cut, by this plane, in a Subcontrary position to the base AEB, or <hi rend="italics">aeb;</hi> and in this case the section DEC is always a circle, as well as the base AEB or <hi rend="italics">aeb.</hi></p></div1><div1 part="N" n="SUBDUCTION" org="uniform" sample="complete" type="entry"><head>SUBDUCTION</head><p>, in Arithmetic, the same as Subtraction.</p><p>SUBDUPLE <hi rend="italics">Ratio,</hi> is when any number or quantitity is the half of another, or contained twice in it. Thus, 3 is said to be subduple of 6, as 3 is the half of 6, or is twice contained in it.</p><p>SUBDUPLICATE <hi rend="italics">Ratio,</hi> of any two quantities, is the ratio of their square roots, being the opposite to duplicate ratio, which is the ratio of the squares. Thus, of the quantities, <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> the subduplicate ratio is that of √<hi rend="italics">a</hi> to √<hi rend="italics">b</hi> or <hi rend="italics">a</hi><hi rend="sup">1/2</hi> to <hi rend="italics">b</hi><hi rend="sup">1/2</hi>, as the duplicate ratio is that of <hi rend="italics">a</hi><hi rend="sup">2</hi> to <hi rend="italics">b</hi><hi rend="sup">2</hi>.</p><p>SUBLIME <hi rend="italics">Geometry,</hi> the higher geometry, or that of curve lines. See <hi rend="smallcaps">Geometry.</hi></p></div1><div1 part="N" n="SUBLUNARY" org="uniform" sample="complete" type="entry"><head>SUBLUNARY</head><p>, is said of all things below the moon; as all things on the earth, or in its atmosphere, &c.</p></div1><div1 part="N" n="SUBMULTIPLE" org="uniform" sample="complete" type="entry"><head>SUBMULTIPLE</head><p>, the contrary of a multiple, being a number or quantity which is contained exactly a certain number of times in another of the same kind; or it is the same as an aliquot part of it. Thus, 3 is a Submultiple of 21, or an aliquot part of it, because 21 is a multiple of 3.</p><p><hi rend="smallcaps">Submultiple</hi> <hi rend="italics">Ratio,</hi> is the ratio of a Submultiple or aliquot part, to its multiple; as the ratio of 3 to 21.</p></div1><div1 part="N" n="SUBNORMAL" org="uniform" sample="complete" type="entry"><head>SUBNORMAL</head><p>, in Geometry, is the subperpendicular AC, or line under the perpendicular to the curve BC, a term used in curve lines to denote the distance AC in the axis, between the ordinate AB, and the per- <pb n="536"/><cb/> pendicular BC to the curve or to <figure/> the tangent. And the said perpendicular BC is the normal.</p><p>In all curves, the Subnormal AC is a 3d proportional to the subtangent TA and the ordinate AB; and in the parabola, it is equal to half the parameter of the axis.</p></div1><div1 part="N" n="SUBSTITUTION" org="uniform" sample="complete" type="entry"><head>SUBSTITUTION</head><p>, in Algebra, is the putting and using, in an equation, one quantity instead of another which is equal to it, but expressed after another manner. See <hi rend="smallcaps">Reduction</hi> of Equations.</p></div1><div1 part="N" n="SUBSTRACTION" org="uniform" sample="complete" type="entry"><head>SUBSTRACTION</head><p>, or <hi rend="smallcaps">Subtraction</hi>, in Arithmetic, is the taking of one number or quantity from another, to find the remainder or difference between them; and is usually made the second rule in arithmetic.</p><p>The greater number or quantity is called the <hi rend="italics">minuend,</hi> the less is the <hi rend="italics">subtrahend,</hi> and the <hi rend="italics">remainder</hi> is the <hi rend="italics">difference.</hi> Also the sign of Subtraction is -, or minus.</p><p><hi rend="smallcaps">Subtraction</hi> <hi rend="italics">of Whole Numbers,</hi> is performed by setting the less number below the greater, as in addition, units under units, tens under tens, &c; and then, proceeding from the right hand towards the lest, subtract or take each lower figure from that just above, and set down the several remainders or differences underneath; and these will compose the whole remainder or difference of the two given numbers. But when any one of the figures of the under number is greater than that of the upper, from which it is to be taken, you must add 10 (in your mind) to that upper figure, then take the under one from this sum, and set the difference underneath, carrying or adding 1 to the next under figure to be subtracted. Thus, for example, to subtract 2904821 from 37409732 <table><row role="data"><cell cols="1" rows="1" role="data">Minuend</cell><cell cols="1" rows="1" rend="align=right" role="data">37409732</cell></row><row role="data"><cell cols="1" rows="1" role="data">Subtrahend</cell><cell cols="1" rows="1" rend="align=right" role="data">2904821</cell></row><row role="data"><cell cols="1" rows="1" role="data">Difference</cell><cell cols="1" rows="1" rend="align=right" role="data">34504911</cell></row><row role="data"><cell cols="1" rows="1" role="data">Proof</cell><cell cols="1" rows="1" rend="align=right" role="data">37409732</cell></row></table></p><p><hi rend="italics">To prove Subtraction:</hi> Add the remainder or difference to the less number, and the sum will be equal to the greater when the work is right.</p><p><hi rend="smallcaps">Subtraction</hi> <hi rend="italics">of Decimals,</hi> is performed in the same manner as in whole numbers, by observing only to set the figures or places of the same kind under each other. Thus: <table><row role="data"><cell cols="1" rows="1" role="data">From</cell><cell cols="1" rows="1" role="data">351.04</cell><cell cols="1" rows="1" role="data">.479</cell><cell cols="1" rows="1" role="data">27</cell></row><row role="data"><cell cols="1" rows="1" role="data">Take</cell><cell cols="1" rows="1" role="data"> 72.71</cell><cell cols="1" rows="1" role="data">.0573</cell><cell cols="1" rows="1" role="data"> 0.936</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diff.</cell><cell cols="1" rows="1" role="data">278.33</cell><cell cols="1" rows="1" role="data">.4217</cell><cell cols="1" rows="1" role="data">26.064</cell></row></table></p><p><hi rend="italics">To Subtract Vulgar Fractions.</hi> Reduce the two fractions to a common denominator, if they have different ones; then take the less numerator from the greater, and set the remainder over the common denominator, for the difference sought.—N. B. It is best to set the less fraction after the greater, with the sign (-) of subtraction between them, and the mark of equality (=) after them.</p><p>Thus, <cb/></p><div2 part="N" n="Subtraction" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Subtraction</hi></head><p>, <hi rend="italics">in Algebra,</hi> is performed by changing the signs of all the terms of the subtrahend, to their contrary signs, viz, + into -, and - into +; and then uniting the terms with those of the minuend after the manner of addition of Algebra. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Ex. From</cell><cell cols="1" rows="1" role="data">+ 6<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Take</cell><cell cols="1" rows="1" role="data">+ 2<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Rem.</cell><cell cols="1" rows="1" role="data">6<hi rend="italics">a</hi> - 2<hi rend="italics">a</hi> = 4<hi rend="italics">a.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">From</cell><cell cols="1" rows="1" role="data">+ 6<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Take</cell><cell cols="1" rows="1" role="data">- 2<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Rem.</cell><cell cols="1" rows="1" role="data">6<hi rend="italics">a</hi> + 2<hi rend="italics">a</hi> = 8<hi rend="italics">a.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">From</cell><cell cols="1" rows="1" role="data">- 6<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Take</cell><cell cols="1" rows="1" role="data">+ 2<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Rem.</cell><cell cols="1" rows="1" role="data">- 6<hi rend="italics">a</hi> - 2<hi rend="italics">a</hi> = - 8<hi rend="italics">a.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">From</cell><cell cols="1" rows="1" role="data">- 6<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Take</cell><cell cols="1" rows="1" role="data">- 4<hi rend="italics">a</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Rem.</cell><cell cols="1" rows="1" role="data">- 6<hi rend="italics">a</hi> + 4<hi rend="italics">a</hi> = - 2<hi rend="italics">a.</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">From</cell><cell cols="1" rows="1" role="data">2<hi rend="italics">a</hi> - 3<hi rend="italics">x</hi> + 5<hi rend="italics">z</hi> - 6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Take</cell><cell cols="1" rows="1" role="data">6<hi rend="italics">a</hi> + 4<hi rend="italics">x</hi> + 5<hi rend="italics">z</hi> + 4</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Rem.</cell><cell cols="1" rows="1" role="data">- 4<hi rend="italics">a</hi> - 7<hi rend="italics">x</hi> 0 - 10</cell></row></table></p></div2></div1><div1 part="N" n="SUBSTILE" org="uniform" sample="complete" type="entry"><head>SUBSTILE</head><p>, or <hi rend="smallcaps">Substilar</hi> <hi rend="italics">Line,</hi> in Dialling, a right line upon which the stile or gnomon of a dial is erected, being the common section of the face of the dial and a plane perpendicular to it passing through the stile.</p><p>The angle included between this line and the stile, is called the elevation or height of the stile.</p><p>In polar, horizontal, meridional, and northern dials, the Substilar line is the meridional line, or line of 12 o'clock; or the intersection of the plane of the dial with that of the meridian.—In all declining dials, the Substile makes an angle with the hour line of 12, and this angle is called the distance of the Substile from the meridian.—In easterly and westerly dials, the substilar line is the line of 6 o'clock, or the intersection of the dial plane with the prime vertical. <table><row role="data"><cell cols="1" rows="1" role="data">SUBSUPERPARTICULAR.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi>See <hi rend="smallcaps">Ratio.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">SUBSUPERPARTICUS.</cell></row></table></p><p>SUBTANGENT <hi rend="italics">of a curve,</hi> is the line TA in the axis below the tangent TB, or limited between the tangent and ordinate to the point of contact. (See the last figure above).</p><p>The tangent, subtangent, and ordinate, make a rightangled triangle.</p><p>In all paraboliform and hyperboliform figures, the Subtangent is equal to the absciss multiplied by the exponent of the power of the ordinate in the equation of the curve. Thus, in the common parabola, whose property or equation is <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">2</hi>, the Subtangent is equal to 2<hi rend="italics">x,</hi> double the absciss. And if <hi rend="italics">ax</hi><hi rend="sup">2</hi> = <hi rend="italics">y</hi><hi rend="sup">3</hi>, or <pb n="537"/><cb/> <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">3/2</hi>, then the Subtangent is = (3/2)<hi rend="italics">x.</hi> Also if <hi rend="italics">a</hi><hi rend="sup"><hi rend="italics">m</hi></hi> <hi rend="italics">x</hi><hi rend="sup"><hi rend="italics">n</hi></hi> = <hi rend="italics">y</hi><hi rend="sup"><hi rend="italics">m</hi> + <hi rend="italics">n</hi></hi>, or <hi rend="italics">px</hi> = <hi rend="italics">y</hi><hi rend="sup">(<hi rend="italics">m</hi> + <hi rend="italics">n</hi>)/<hi rend="italics">n</hi></hi>, the Subtangent is = ((<hi rend="italics">m</hi> + <hi rend="italics">n</hi>)/<hi rend="italics">n</hi>)<hi rend="italics">x.</hi> See <hi rend="italics">Method of</hi> <hi rend="smallcaps">Tangents.</hi></p></div1><div1 part="N" n="SUBTENSE" org="uniform" sample="complete" type="entry"><head>SUBTENSE</head><p>, in Geometry, of an arc, is the same as the chord of the arc; but of an angle, it is a line drawn across from the one leg of the angle to the other, or between the two extremes of the arc that measures the angle.</p><p>SUBTRACTION. See <hi rend="smallcaps">Substraction.</hi></p></div1><div1 part="N" n="SUBTRIPLE" org="uniform" sample="complete" type="entry"><head>SUBTRIPLE</head><p>, is when one quantity is the 3d part of another; as 2 is Subtriple of 6. And <hi rend="smallcaps">Subtriple</hi> <hi rend="italics">Ratio,</hi> is the ratio of 1 to 3.</p><p>SUBTRIPLICATE <hi rend="italics">Ratio,</hi> is the ratio of the cube roots. So the Subtriplicate ratio of <hi rend="italics">a</hi> to <hi rend="italics">b,</hi> is the ratio of √<hi rend="sup">3</hi><hi rend="italics">a</hi> to √<hi rend="sup">3</hi><hi rend="italics">b,</hi> or of <hi rend="italics">a</hi><hi rend="sup">1/3</hi> to <hi rend="italics">b</hi><hi rend="sup">1/3</hi>.</p><p>SUCCESSION <hi rend="italics">of Signs,</hi> in Astronomy, is the order in which they are reckoned, or follow one another, and according to which the sun enters them; called also <hi rend="italics">consequentia.</hi> As Aries, Taurus, Gemini, Cancer, &c.</p><p>When a planet goes according to the order and succession of the signs, or <hi rend="italics">in consequentia,</hi> it is said to be direct; but retrograde when contrary to the succession of the signs, or <hi rend="italics">in antecedentia,</hi> as from Gemini to Taurus, then to Aries, &c.</p></div1><div1 part="N" n="SUCCULA" org="uniform" sample="complete" type="entry"><head>SUCCULA</head><p>, in Mechanics, a bare axis or cylinder with staves in it to move it round; but without any tympanum, or peritrochium.</p></div1><div1 part="N" n="SUCKER" org="uniform" sample="complete" type="entry"><head>SUCKER</head><p>, in Mechanics, a name by which sometimes is called the piston or bucket, in a sucking pump; and sometimes the pump itself is so called.</p><p>SUCKING-<hi rend="italics">Pump,</hi> the common pump, working by two valves opening upwards. See <hi rend="smallcaps">Pump.</hi></p></div1><div1 part="N" n="SUMMER" org="uniform" sample="complete" type="entry"><head>SUMMER</head><p>, the name of one of the seasons of the year, being one of the quarters when the year is divided into 4 quarters, or one half when the year is divided only into two, Summer and winter. In the former case, Summer is the quarter during which, in northern climates, the sun is passing through the three signs Cancer, Leo, Virgo, or from the time of the greatest declination, till the sun come to the equinoctial again, or have no declination; which is from about the 21st of June, till about the 22d of September. In the latter case, Summer contains the 6 warmer months, while the sun is on one side of the equinoctial; and winter the other 6 months, when the sun is on the other side of it.</p><p>It is said, that a frosty winter produces a dry Summer; and a mild winter, a wet Summer. See Philos. Trans. no. 458, sect. 10.</p><p><hi rend="smallcaps">Summer</hi> <hi rend="italics">Solstice,</hi> the time or point when the sun comes to his greatest declination, and nearest the zenith of the place. See <hi rend="smallcaps">Solstice.</hi></p></div1><div1 part="N" n="SUM" org="uniform" sample="complete" type="entry"><head>SUM</head><p>, the quantity produced by addition, or by adding two or more numbers or quantities together. So the Sum of 6 and 4 is 10, and the Sum of <hi rend="italics">a</hi> and <hi rend="italics">b</hi> is <hi rend="italics">a</hi> + <hi rend="italics">b.</hi></p></div1><div1 part="N" n="SUN" org="uniform" sample="complete" type="entry"><head>SUN</head><p>, <hi rend="smallcaps">Sol</hi>, Θ, in Astronomy, the great luminary <cb/> which enlightens the world, and by his presence constitutes day.</p><p>The Sun, which was reckoned among the planets in the infancy of astronomy, should rather be counted among the fixed stars. He only appears brighter and larger than they do, because we keep constantly near the Sun; whereas we are immensely farther from the stars. But a spectator, placed as near to any star as we are to the Sun, would probably see that star a body as large and as bright as the Sun appears to us; and, on the other hand, a spectator as far distant from the Sun as we are from the stars, would see the Sun as small as we see a star, divested of all his circumvolving planets; and he would reckon it one of the stars in numbering them.</p><p>According to the Pythagorean and Copernican hypothesis, which is now generally received, and has been demonstrated to be the true system, the Sun is the common centre of all the planetary and cometary system; around which all the planets and comets, and our earth among the rest, revolve, in different periods, according to their different distances from the Sun.</p><p>But the Sun, though thus eased of that prodigious motion by which the Ancients imagined he revolved daily round our earth, yet is he not a perfectly quiescent body. For, from the phenomena of his maculæ or spots, it evidently appears, that he has a rotation round his axis, like that of the earth by which our natural day is measured, but only slower. For, some of these spots have made their first appearance near the edge or margin of the Sun, from thence they have seemed gradually to pass over the Sun's face to the opposite edge, then disappear; and hence, after an absence of about 14 days, they have reappeared in their first place, and have taken the same course over again; finishing their entire circuit in 27 days 12<hi rend="sup">h</hi> 20<hi rend="sup">m</hi>; which is hence inferred to be the period of the Sun's rotation round his axis: and therefore the periodical time of the Sun's revolution to a fixed star is 25<hi rend="sup">d</hi> 15<hi rend="sup">h</hi> 16<hi rend="sup">m</hi>; because in 27<hi rend="sup">d</hi> 12<hi rend="sup">h</hi> 20<hi rend="sup">m</hi> of the month of May, when the observations were made, the earth describes an angle about the Sun's centre of 26° 22′, and therefore as the angular motion. 360° + 26°22′ : 360° :: 27<hi rend="sup">d</hi> 12<hi rend="sup">h</hi> 20<hi rend="sup">m</hi> : 25<hi rend="sup">d</hi> 15<hi rend="sup">h</hi> 16<hi rend="sup">m</hi>. This motion of the spots is from west to east: whence we conclude the motion of the Sun, to which the other is owing, to be from east to west.</p><p>Beside this motion round his axis, the Sun, on account of the various attractions of the surrounding planets, is agitated by a small motion round the centre of gravity of the system.—Whether the Sun and stars have any proper motion of their own in the immensity of space, however small, is not absolutely certain. Though some very accurate observers have intimated conjectures of this kind, and have made such a general motion not improbable. See <hi rend="smallcaps">Stars.</hi></p><p><hi rend="italics">As for the apparent annual motion of the</hi> <hi rend="smallcaps">Sun</hi> <hi rend="italics">round the earth;</hi> it is easily shewn, by astronomers, that the real annual motion of the earth, about the Sun, will cause such an appearance. A spectator in the Sun would see the earth move from west to east, for the same reason as we see the Sun move from east to west: and all the phenomena resulting from this annual motion in whichsoever of the bodies it be, will appear the same <pb n="538"/><cb/> from either. And hence arises that apparent motion of the Sun, by which he is seen to advance insensibly towards the eastern stars; in so much that, if any star, near the ecliptic, rise at any time with the Sun; after a few days the Sun will be got more to the east of the star, and the star will rise and set before him. <hi rend="center"><hi rend="italics">Nature, Properties, Figure, &c, of the</hi> <hi rend="smallcaps">Sun.</hi></hi></p><p>Those who have maintained that the substance of the Sun is fire, argue in the following manner: The Sun shines, and his rays, collected hy concave mirrors, or convex lenses, do burn, consume, and melt the most solid bodies, or else convert them into ashes, or glass: therefore, as the force of the solar rays is diminished, by their divergency, in a duplicate ratio of the distances reciprocally taken; it is evident that their force and effect are the same, when collected by a burning lens, or mirror, as if we were at such distance from the sun, where they were equally dense. The Sun's rays therefore, in the neighbourhood of the Sun, produce the same effects, as might be expected from the most vehement fire: consequently the Sun is of a fiery substance.</p><p>Hence it follows, that its surface is probably every where fluid; that being the condition of flame. Indeed, whether the whole body of the Sun be fluid, as some think; or solid, as others; they do not presume to determine: but as there are no other marks, by which to distinguish fire from other bodies, but light, heat, a power of burning, consuming, melting, calcining, and vitrifying; they do not see what should hinder but that the Sun may be a globe of fire, like our fires, invested with flame: and, supposing that the maculæ are formed out of the solar exhalations, they infer that the Sun is not pure fire; but that there are heterogeneous parts mixed along with it.</p><p>Philosophers have been much divided in opinion with respect to the nature of fire, light, and heat, and the causes that produce them: and they have given very different accounts of the agency of the Sun, with which, whether we consider them as substances or qualities, they are intimately connected, and on which they seem primarily to depend. Some, among whom we may reckon Sir Isaac Newton, consider the rays of light as composed of small particles, which are emitted from shining bodies, and move with uniform velocities in uniform mediums, but with variable velocities in mediums of variable densities. These particles, say they, act upon the minute constituent parts of bodies, not by impact, but at some indefinitely small distance; they attract and are attracted; and in being reflected or refracted, they excite a vibratory motion in the component particles. This motion increases the distance between the particles, and thus occasions an augmentation of bulk, or an expansion in every dimension, which is the most certain characteristic of fire. This expansion, which is the beginning of a disunion of the parts, being increased by the increasing magnitude of the vibrations proceeding from the continued agency of light, it may easily be apprehended, that the particles will at length vibrate beyond their sphere of mutual attraction, and thus the texture of the body will be altered or destroyed; from solid it may become fluid, as in melted gold; or <cb/> from being fluid, it may be dispersed in vapour, as in boiling water.</p><p>Others, as Boerhaave, represent fire as a substance <hi rend="italics">sui generis,</hi> unalterable in its nature, and incapable of being produced or destroyed; naturally existing in equal quantities in all places, imperceptible to our senses, and only discoverable by its effects, when, by various causes, it is collected for a time into a less space than that which it would otherwise occupy. The matter of this fire is not in any wise supposed to be derived from the Sun: the solar rays, whether direct or reflected, are of use only as they impel the particles of fire in parallel directions: that parallelism being destroyed, by intercepting the solar rays, the fire instantly assumes its natural state of uniform diffusion. According to this explication, which attributes heat to the matter of fire, when driven in parallel directions, a much greater degree must be given it when the quantity, so collected, is amassed into a focus; and yet the focus of the largest speculum does not heat the air or medium in which it is is found, but only bodies of densities different from that medium.</p><p>M. de Luc (Lettres Physiques) is of opinion, that the solar rays are the principal cause of heat; but that they heat such bodies only as do not allow them a free passage. In this remark he agrees with Newton; but then he differs totally from him, as well as from Boerhaave, concerning the nature of the rays of the Sun. He does not admit the emanation of any luminous corpuscles from the Sun, or other self-shining substances, but supposes all space to be filled with an ether of great elasticity and small density, and that light consists in the vibrations of this ether, as sound consists in the vibrations of the air. “Upon Newton's supposition, says an excellent writer, the cause by which the particles of light, and the corpuscles constituting other bodies are mutually attracted and repelled, is uncertain. The reason of the uniform diffusion of fire, of its vibration, and repercussion, as stated in Boerhaave's opinion, is equally inexplicable. And in the last mentioned hypothesis, we may add to the other difficulties attending the supposition of an universal ether, the want of a first mover to make the Sun vibrate. Of these several opinions concerning elementary fire, it may be said, as Cicero remarked upon the opinions of philosophers concerning the nature of the soul: <hi rend="italics">Harum sententiarum quæ vera sit, Deus aliquis viderit; quæ verisimillima, magna questio est.</hi>” Watson's Chem. Ess. vol. 1, pa. 164.</p><p><hi rend="italics">As to the Figure of the</hi> <hi rend="smallcaps">Sun;</hi> this, like the planets, is not perfectly globular, but spheroidical, being higher about the equator than at the poles. The reason of which is this: the Sun has a motion about his own axis; and therefore the solar matter will have an endeavour to recede from the axis, and that with the greater force as their distances from it, or the circles they move in, are greater: but the equator is the greatest circle; and the rest, towards the poles, continually decrease; therefore the solar matter, though at first in a spherical form, will endeavour to recede from the centre of the equator farther than from the centres of the parallels. Consequently, since the gravity, by which it is retained in its place, is supposed to be uniform throughout the whole Sun, it will really recede from the centre more at <pb n="539"/><cb/> the equator, than at any of the parallels; and hence the Sun's diameter will be greater through the equator, than through the poles; that is, the Sun's figure is not perfectly spherical, but spheroidical.</p><p><hi rend="italics">Several particulars of the Sun,</hi> related by Newton, in his Principia, are as follow:</p><p>1. That the density of the Sun's heat, which is proportional to his light, is 7 times as great at Mercury as with us; and therefore our water there would be all carried off, and boil away: for he found by experiments of the thermometer, that a heat but 7 times greater than that of the Sun beams in summer, will serve to make water boil.</p><p>2. That the quantity of matter in the Sun is to that in Jupiter, nearly as 1100 to 1; and that the distance of that planet from the Sun, is in the same ratio to the Sun's semidiameter.</p><p>3. That the matter in the Sun is to that in Saturn, as 2360 to 1; and the distance of Saturn from the Sun is in a ratio but little less than that of the Sun's semidiameter. And hence, that the common centre of gravity of the Sun and Jupiter is nearly in the superficies of the Sun; of the Sun and Saturn, a little within it.</p><p>4. And by the same mode of calculation it will be found, that the common centre of gravity of all the planets, cannot be more than the length of the solar diameter distant from the centre of the Sun. This common centre of gravity he proves is at rest; and therefore though the Sun, by reason of the various positions of the planets, may be moved every way, yet it cannot recede far from the common centre of gravity, and this, he thinks, ought to be accounted the centre of our world. Book 3, prop. 12.</p><p>5. By means of the solar spots it hath been discovered, that the Sun revolves round his own axis, without moving considerably out of his place, in about 25 days, and that the axis of this motion is inclined to the ecliptic in an angle of 87° 30′ nearly. The Sun's apparent diameter being sensibly longer in December than in June, the Sun must be proportionably nearer to the earth in winter than in Summer; in the former of which seasons therefore will be the perihelion, in the latter the aphelion: and this is also confirmed by the earth's motion being quicker in December than in June, as it is by about 1/15 part. For since the earth always describes equal areas in equal times, whenever it moves swifter, it must needs be nearer to the Sun: and for this reason there are about 8 days more from the sun's vernal equinox to the autumnal, than from the autumnal to the vernal.</p><p>6. That the Sun's diameter is equal to 100 diameters of the earth; and therefore the body of the Sun must be 1000000 times greater than that of the earth.—Mr. Azout assures us, that he observed, by a very exact method, the Sun's diameter to be no less than 21′ 45″ in his apogee, and not greater than 32′ 45″ in his perigee.</p><p>7. According to Newton, in his theory of the moon, the mean apparent diameter of the Sun is 32′ 12″.— The Sun's horizontal parallax is now fixed at 8″5/10.</p><p>8. If you divide 360 degrees (the whole ecliptic) by the quantity of the solar year, it will give 59′ 8″ &c, which therefore is the medium quantity of the Sun's daily motion: and if this 59′ 8″ be divided by 24, you <cb/> have the Sun's horary motion equal to 2′ 28″: and if this last be divided by 60, it will give his motion in a minute, &c. And in this way are the tables of the Sun's mean motion constructed, as placed in books of Astronomical tables and calculations.</p></div1><div1 part="N" n="SUNDAY" org="uniform" sample="complete" type="entry"><head>SUNDAY</head><p>, the first day of the week; thus called by our idolatrous ancestors, because set apart for the worship of the sun.</p><p>It is sometimes called the <hi rend="italics">Lord's Day,</hi> because kept as a feast in memory of our Lord's resurrection on this day: and also <hi rend="italics">Sabbath-day,</hi> because substituted under the new law instead of the Sabbath in the old law.</p><p>It was Constantine the Great who first made a law for the proper observation of Sunday; and who, according to Eusebius, appointed that it should be regularly celebrated throughout the Roman empire.</p><p><hi rend="smallcaps">Sunday</hi> <hi rend="italics">Letter.</hi> See <hi rend="smallcaps">Dominical</hi> <hi rend="italics">Letter.</hi></p></div1><div1 part="N" n="SUPERFICIAL" org="uniform" sample="complete" type="entry"><head>SUPERFICIAL</head><p>, relating to Superficies.</p></div1><div1 part="N" n="SUPERFICIES" org="uniform" sample="complete" type="entry"><head>SUPERFICIES</head><p>, or <hi rend="smallcaps">Surface</hi>, in Geometry, the outside or exterior face of any body. This is considered as having the two dimensions of length and breadth only, but no thickness; and therefore it makes no part of the substance or solid content or matter of the body.</p><p>The terms or bounds or extremities of a Superficies, are lines; and Superficies may be considered as generated by the motions of lines.</p><p>Superficies are either rectilinear, curvilinear, plane, concave, or convex. A</p><p><hi rend="italics">Rectilinear</hi> <hi rend="smallcaps">Superficies</hi>, is that which is bounded by right lines.</p><p><hi rend="italics">Curvilinear</hi> <hi rend="smallcaps">Superficies</hi>, is bounded by curve lines.</p><p><hi rend="italics">Plane</hi> <hi rend="smallcaps">Superficies</hi> is that which has no inequality in it, nor risings, nor sinkings, but lies evenly and straight throughout, so that a right line may wholly coincide with it in all parts and directions.</p><p><hi rend="italics">Convex</hi> <hi rend="smallcaps">Superficies</hi>, is that which is curved and rises outwards.</p><p><hi rend="italics">Concave</hi> <hi rend="smallcaps">Superficies</hi>, is curved and sinks inward.</p><p>The measure or quantity of a Surface, is called the <hi rend="italics">area</hi> of it. And the finding of this measure or area, is sometimes called the <hi rend="italics">quadrature</hi> of it, meaning the reducing it to an equal square, or to a certain number of smaller squares. For all plane figures, and the Surfaces of all bodies, are measured by squares; as square inches, or square feet, or square yards, &c; that is, squares whose sides are inches, or feet, or yards, &c. Our least superficial measure is the square inch, and other squares are taken from it according to the proportion in the following Table of superficial or square measure. <table><head><hi rend="italics">Table of Superficial or Square Measure.</hi></head><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">144</cell><cell cols="1" rows="1" role="data">square inches</cell><cell cols="1" rows="1" role="data">= 1</cell><cell cols="1" rows="1" role="data">square foot</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">square feet</cell><cell cols="1" rows="1" role="data">= 1</cell><cell cols="1" rows="1" role="data">square yard</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30 1/4</cell><cell cols="1" rows="1" role="data">square yards</cell><cell cols="1" rows="1" role="data">= 1</cell><cell cols="1" rows="1" role="data">square pole</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">square poles</cell><cell cols="1" rows="1" role="data">= 1</cell><cell cols="1" rows="1" role="data">square chain</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">square chains</cell><cell cols="1" rows="1" role="data">= 1</cell><cell cols="1" rows="1" role="data">acre</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">640</cell><cell cols="1" rows="1" role="data">acres</cell><cell cols="1" rows="1" role="data">= 1</cell><cell cols="1" rows="1" role="data">square mile.</cell></row></table></p><p>The Superficial measure of all bodies and figures depends entirely on that of a rectangle; and this is found by drawing or multiplying the length by the breadth of <pb n="540"/><cb/> it; as is proved from plane geometry only, in my Mensuration, pt. 2, sect. 1, prob. 1. From the area of the rectangle we obtain that of any oblique parallelogram, which, by geometry, is equal to a rectangle of equal base and altitude; thence a triangle, which is the half of such a parallelogram or rectangle; and hence, by composition, we obtain the Superficies of all other figures whatever, as these may be considered as made up of triangles only.</p><p>Beside this way of deriving the Superficies of all figures, which is the most simple and natural, as proceeding on common geometry alone, there are certain other methods; such as the methods of exhaustions, of fluxions, &c. See these articles in their places, as also Q<hi rend="smallcaps">UADRATURES.</hi></p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Superficies</hi>, a line usually found on the sector, and Gunter's scale. The description and use of which, see under <hi rend="smallcaps">Sector</hi> and <hi rend="smallcaps">Gunter</hi>'s <hi rend="italics">Scale.</hi></p><p>SUPERPARTICULAR <hi rend="italics">Proportion,</hi> or <hi rend="italics">Ratio,</hi> is that in which the greater term exceeds the less by unit or 1. As the ratio of 1 to 2, or 2 to 3, or 3 to 4, &c.</p><p>SUPERPARTIENT <hi rend="italics">Proportion,</hi> or <hi rend="italics">Ratio,</hi> is when the greater term contains the less term, once, and leaves some number greater than 1 remaining. As the ratio of 3 to 5, which is equal to that of 1 to 1 2/3; of 7 to 10, which is equal to that of 1 to 1 3/7; &c.</p></div1><div1 part="N" n="SUPPLEMENT" org="uniform" sample="complete" type="entry"><head>SUPPLEMENT</head><p>, of an arch, or angle, in Geometry or Trigonometry, is what it wants of a semicircle, or of 180 degrees; as the <hi rend="italics">complement</hi> is what it wants of a quadrant, or of 90 degrees. So, the Supplement of 50° is 130°; as the complement of it is 40°.</p></div1><div1 part="N" n="SURD" org="uniform" sample="complete" type="entry"><head>SURD</head><p>, in Arithmetic, denotes a number or quantity that is incommensurate to unity; or that is inexpressible in rational numbers by any known way of notation, otherwise than by its radical sign or index.—This is otherwise called an <hi rend="italics">irrational</hi> or <hi rend="italics">incommensurable number,</hi> as also an <hi rend="italics">imperfect power.</hi></p><p>These Surds arise in this manner: when it is proposed to extract a certain root of some number or quantity, which is not a complete power or a true figurate number of that kind; as, if its square root be demanded, and it is not a true square; or if its cube root be required, and it is not a true cube, &c; then it is impossible to assign, either in whole numbers, or in fractions, the exact root of such proposed number. And whenever this happens, it is usual to denote the root by setting before it the proper mark of radicality, which is √, and placing above this radical sign the number that shews what kind of root is required. Thus, √<hi rend="sup">2</hi>2 or √2 signifies the square root of 2, and √<hi rend="sup">3</hi>10 signifies the cube root of 10; which roots, because it is impossible to express them in numbers exactly, are properly called <hi rend="italics">Surd roots.</hi></p><p>There is also another way of notation, now much in use, by which roots are expressed by fractional indices, without the radical sign: thus, like as <hi rend="italics">x</hi><hi rend="sup">2</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">4</hi>, &c, denote the square, cube, 4th power, &c, of <hi rend="italics">x;</hi> so <hi rend="italics">x</hi><hi rend="sup">1/2</hi>, <hi rend="italics">x</hi><hi rend="sup">1/3</hi>, <hi rend="italics">x</hi><hi rend="sup">1/4</hi>, &c, denote the square root, cube root, 4th root, &c, of the same quantity <hi rend="italics">x.</hi>—The reason of this is plain enough; for since √<hi rend="italics">x</hi> is a geometrical mean proportional between 1 and <hi rend="italics">x,</hi> so 1/2 is an arithmetical mean between 0 and 1; and therefore, as 2 is the index <cb/> of the square of <hi rend="italics">x,</hi> 1/2 will be the proper index of its square root, &c.</p><p>It may be observed that, for convenience, or the sake of brevity, quantities which are not naturally Surds, are often expressed in the form of Surd roots. Thus √4, √(9/4), √27, are the same as 2, 3/2, 3.</p><p>Surds are either <hi rend="italics">simple</hi> or <hi rend="italics">compound.</hi></p><p><hi rend="italics">Simple</hi> <hi rend="smallcaps">Surds</hi>, are such as are expressed by one single term; as √2, or √<hi rend="sup">3</hi><hi rend="italics">a,</hi> &c.</p><p><hi rend="italics">Compound</hi> <hi rend="smallcaps">Surds</hi>, are such as consist of two or more simple Surds connected together by the signs + or -; as √3 + √2, or √3 - √2, or √<hi rend="sup">3</hi>(5 + √2): which last is called an <hi rend="italics">universal</hi> root, and denotes the cubic root of the sum arising by adding 5 and the root of 2 together. <hi rend="center"><hi rend="italics">Of certain Operations by Surds.</hi></hi></p><p>1. Such Surds as √2, √3, √5, &c. though they are themselves incommensurable with unity, according to the definition, are commensurable in power with it, because their powers are integers, which are multiples of unity. They may also be sometimes commensurable with one another; as √8 and √2, which are to one another as 2 to 1, as is found by dividing them by their greatest common measure, which is √2, for then those two become √4 = 2, and 1 the ratio.</p><p>2. <hi rend="italics">To reduce Rational Quantities to the form of any proposed Surd Roots.</hi>—Involve the rational quantity according to the index of the power of the Surd, and then prefix before that power the proposed radical sign.</p><p>Thus, </p><p>And in this way may a simple Surd fraction, whose radical sign refers to only one of its terms, be changed into another, which shall include both numerator and denominator. Thus, √2/5 is reduced to √(2/25), and 5/√<hi rend="sup">3</hi>4 to √<hi rend="sup">3</hi>(125/4): thus also the quantity <hi rend="italics">a</hi> reduced to the form of <hi rend="italics">x</hi><hi rend="sup">1/n</hi> or √<hi rend="sup">n</hi><hi rend="italics">x,</hi> is (―<hi rend="italics">a</hi><hi rend="sup">n</hi>)<hi rend="sup">1/n</hi> or √<hi rend="sup">n</hi><hi rend="italics">a</hi><hi rend="sup">n</hi>. And thus may roots with rational coefficients be reduced so as to be wholly affected by the radical sign; as</p><p>3. <hi rend="italics">To reduce Simple Surds, having different radical signs (which are called heterogeneal Surds) to others that may have one common radical sign, or which are homogeneal: Or to reduce roots of different names to roots of the same name.</hi>—Involve the powers reciprocally, each according to the index of the other, for new powers; and multiply their indices together, for the common index. Otherwise, as Surds may be considered as powers with fractional exponents, reduce these fractional exponents to fractions having the same value and a common denominator.</p><p>Thus, by the 1st way, √<hi rend="sup">n</hi><hi rend="italics">a</hi> and √<hi rend="sup">m</hi><hi rend="italics">x</hi> become √<hi rend="sup">mn</hi><hi rend="italics">a</hi><hi rend="sup">m</hi> and √<hi rend="sup">mn</hi><hi rend="italics">x</hi><hi rend="sup">n</hi>; <pb n="541"/><cb/> and, by the 2d way, <hi rend="italics">a</hi><hi rend="sup">1/n</hi> and <hi rend="italics">x</hi><hi rend="sup">1/m</hi> become (―<hi rend="italics">a</hi><hi rend="sup">m</hi>))<hi rend="sup">1/(mn)</hi> and (―<hi rend="italics">x</hi><hi rend="sup">n</hi>))<hi rend="sup">1/(mn)</hi>.</p><p>Also √3 and √<hi rend="sup">3</hi>2 are reduced to √<hi rend="sup">6</hi>27 and √<hi rend="sup">6</hi>4, which are equal to them, and have a common radical sign.</p><p>4. <hi rend="italics">To reduce Surds to their most simple expressions, or to the lowest terms possible.</hi>—Divide the Surd by the greatest power, of the same name with that of the root, which you can discover is contained in it, and which will measure or divide it without a remainder; then extract the root of that power, and place it before the quotient or Surd so divided; this will produce a new Surd of the same value with the former, but in more simple terms. Thus, √(16<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">x</hi>), by dividing by 16<hi rend="italics">a</hi><hi rend="sup">2</hi>, and prefixing its root 4<hi rend="italics">a,</hi> before the quotient √<hi rend="italics">x,</hi> becomes 4<hi rend="italics">a</hi>√<hi rend="italics">x;</hi> in like manner, √12 or √(4 X 3), becomes 2√3; And .</p><p>5. <hi rend="italics">To Add and Subtract Surds.</hi>—When they are reduced to their lowest terms, if they have the same irrational part, add or subtract their rational coefficients, and to the sum or difference subjoin the common irrational part.</p><p>Thus, .</p><p>Or such Surds may be added and subtracted, by first squaring them (by uniting the square of each part with double their product), and then extracting the root universal of the whole. Thus, for the first example above, .</p><p>If the quantities cannot be reduced to the same irrational part, they may just be connected by the signs + or -.</p><p>6. <hi rend="italics">To Multiply and Divide Surds.</hi>—If the terms have the same radical, they will be multiplied and divided like powers, viz, by adding their indices for multiplication, and subtracting them for division.</p><p>Thus, .</p><p>If the quantities be different, but under the same radical sign; multiply or divide the quantities, and place the radical sign to the product or quotient.</p><p>Thus, . <cb/></p><p>But if the Surds have not the same radical sign, reduce them to such as shall have the same radical sign, and proceed as before.</p><p>Thus, .</p><p>If the Surds have any rational coefficients, their product or quotient must be prefixed.</p><p>Thus, .</p><p>7. <hi rend="italics">Involution and Evolution of Surds.</hi>—Surds are involved, or raised to any power, by multiplying their indices by the index of the power; and they are evolved or extracted, by dividing their indices by the index of the root.</p><p>Thus, the square of .</p><p>Or thus: involve or extract the quantity under the radical sign according to the power or root required, continuing the same radical sign.</p><p>So the square of √<hi rend="sup">3</hi>2 is √<hi rend="sup">3</hi>4; and the square root of √<hi rend="sup">3</hi>4, is √<hi rend="sup">3</hi>2.</p><p>Unless the index of the power is equal to the name of the Surd, or a multiple of it, for in that case the power of the Surd becomes rational. Thus, the square of √3 is 3, and the cube of √<hi rend="sup">3</hi><hi rend="italics">a</hi><hi rend="sup">2</hi> is <hi rend="italics">a</hi><hi rend="sup">2</hi>.</p><p>Simple Surds are commensurable in power, and by being multiplied by themselves give, at length, rational quantities: but compound Surds, multiplied by themselves, commonly give irrational products. Yet, in this case, when any compound Surd is proposed, there is another compound Surd, which, multiplied by it, gives a rational product.</p><p>Thus, √<hi rend="italics">a</hi> + √<hi rend="italics">b</hi> multiplied by √<hi rend="italics">a</hi> - √<hi rend="italics">b</hi> gives <hi rend="italics">a</hi> - <hi rend="italics">b;</hi> and √<hi rend="sup">3</hi><hi rend="italics">a</hi> - √<hi rend="sup">3</hi><hi rend="italics">b</hi> mult. by √<hi rend="sup">3</hi><hi rend="italics">a</hi><hi rend="sup">2</hi> + √<hi rend="sup">3</hi>(<hi rend="italics">ab</hi>) + √<hi rend="sup">3</hi><hi rend="italics">b</hi><hi rend="sup">2</hi> gives <hi rend="italics">a</hi> - <hi rend="italics">b.</hi> The finding of such a Surd as multiplying the proposed Surd gives a rational product, is made easy by three theorems, delivered by Maclaurin, in his Algebra, pa. 109 &c.</p><p>This operation is of use in reducing Surd expressions to more simple forms. Thus, suppose a binomial Surd divided by another, as √20 + √12 by √5 - √3, the quotient might be expressed by ; but this will be expressed in a more simple form, by multiplying both numerator and denominator by such a Surd as makes the product of the denominator become a rational quantity: thus, multiplying them by √5 + √3, the fraction or quotient becomes. . To do this generally, see Maclaurin's Alg. p. 113.</p><p>When the square root of a Surd is required, it may be found nearly, by extracting the root of a rational quantity that approximates to its value. Thus, to find the <pb n="542"/><cb/> square root of 3 + 2√2; first calculate √2 = 1.41421; hence 3 + 2√2 = 5.82842, the root of which is nearly 2.41421.</p><p>In like manner we may proceed with any other proposed root. And if the index of the root be very high, a table of logarithms may be used to advantage: thus, to extract the root √<hi rend="sup">7</hi>(5 + √<hi rend="sup">13</hi>17); take the logarithm of 17, divide it by 13, find the number answering to the quotient, add this number to 5, find the log. of the sum, and divide it by 7, and the number answering to this quotient will be nearly equal to √<hi rend="sup">7</hi>(5 + √<hi rend="sup">13</hi>17).</p><p>But it is sometimes requisite to express the roots of Surds exactly by other Surds. Thus, in the first example, the square root of 3 + 2√2 is 1 + √2, for . For the method of performing this, the curious may consult Maclaurin's Algeb. p. 115, where also rules for trinomials &c may be found. See also the article <hi rend="smallcaps">Binomial</hi> <hi rend="italics">Roots,</hi> in this Dictionary.</p><p>For extracting the higher roots of a binomial, whose two members when squared are commensurable numbers, we have a rule in Newton's Arith. pa. 59, but without demonstration. This is supplied by Maclaurin, in his Alg. p. 120: as also by Gravesande, in his Matheseos Univers. Elem. p. 211.</p><p>It sometimes happens, in the resolution of cubic equations, that binomials of this form <hi rend="italics">a</hi> ± <hi rend="italics">b</hi>√-1 occur, the cube roots of which must be found; and to these Newton's rule cannot always be applied, because of the impossible or imaginary factor √-1; yet if the root be expressible in rational numbers, the rule will often yield to it in a short way, not merely tentative, the trials being confined to known limits. See Maclaurin's Alg. p. 127. It may be farther observed, that such roots, whether expressible in rational numbers or not, may be found by evolving the quantity <hi rend="italics">a</hi> + <hi rend="italics">b</hi>√-1 by Newton's binomial theorem, and summing up the alternate terms. Maclaurin, p. 130.</p><p>Those who are desirous of a general and elegant solution of the problem, <hi rend="italics">to extract any root of an impossible binomial</hi> a + b√-1, <hi rend="italics">or of a possible binomial</hi> a + √b, may have recourse to the appendix to Saunderson's Algebra, and to the Philos. Trans. number 451, or Abridg. vol. 8, p. 1. On the management of Surds, see also the numerous authors upon Algebra.</p><p>SURDESOLID. See <hi rend="smallcaps">Sursolid.</hi></p></div1><div1 part="N" n="SURFACE" org="uniform" sample="complete" type="entry"><head>SURFACE</head><p>, in Geometry. See <hi rend="smallcaps">Superficies.</hi></p><p>A <hi rend="italics">mathematical</hi> <hi rend="smallcaps">Surface</hi> is the mere exterior face of a body, but is not any part of it, being of no thickness, but only the bare figure or termination of the body.</p><p>A <hi rend="italics">Physical</hi> <hi rend="smallcaps">Surface</hi> is considered as of some very small thickness.</p></div1><div1 part="N" n="SURSOLID" org="uniform" sample="complete" type="entry"><head>SURSOLID</head><p>, or <hi rend="smallcaps">Surdesolid</hi>, in Arithmetic, the 5th power of a number, considered as a root. The number 2, for instance, considered as a root, produces the powers thus: <table><row role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">the root or 1st power,</cell></row><row role="data"><cell cols="1" rows="1" role="data">2 X</cell><cell cols="1" rows="1" rend="align=right" role="data">2 =</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">the square or 2d power,</cell></row><row role="data"><cell cols="1" rows="1" role="data">2 X</cell><cell cols="1" rows="1" rend="align=right" role="data">4 =</cell><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">the cube or 3d power,</cell></row><row role="data"><cell cols="1" rows="1" role="data">2 X</cell><cell cols="1" rows="1" rend="align=right" role="data">8 =</cell><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">the biquadratic or 4th power,</cell></row><row role="data"><cell cols="1" rows="1" role="data">2 X</cell><cell cols="1" rows="1" rend="align=right" role="data">16 =</cell><cell cols="1" rows="1" rend="align=right" role="data">32</cell><cell cols="1" rows="1" role="data">the Sursolid or 5th power.</cell></row></table> <cb/></p><p><hi rend="smallcaps">Sursolid Problem</hi>, is that which cannot be resolved but by curves of a higher kind than the conic sections.</p></div1><div1 part="N" n="SURVEYING" org="uniform" sample="complete" type="entry"><head>SURVEYING</head><p>, the art, or act, of measuring land. This comprises the three following parts; viz, taking the dimensions of any tract or piece of ground; the delineating or laying the same down in a map or draught; and finding the superficial content or area of the same; beside the dividing and laying out of lands.</p><p>The first of these is what is properly called <hi rend="italics">Surveying;</hi> the second is called <hi rend="italics">plotting,</hi> or <hi rend="italics">protracting,</hi> or <hi rend="italics">mapping;</hi> and the third <hi rend="italics">casting up,</hi> or <hi rend="italics">computing the contents.</hi></p><p>The first again consists of two parts, the making of observations for the angles, and the taking of lineal measures for the distances.</p><p>The former of these is performed by some of the following instruments; the theodolite, circumferentor, semicircle, plain-table, or compass, or even by the chain itself: the latter is performed by means either of the chain, or the perambulator. The description and manner of using each of these, see under its respective article or name.</p><p>It is useful in Surveying, to take the angles which the bounding lines form with the magnetic needle, in order to check the angles of the figure, and to plot them conveniently afterwards. But, as the difference between the true and magnetic meridian perpetually varies in all places, and at all times; it is impossible to compare two surveys of the same place, taken at distant times, by magnetic instruments, without making due allowance for this variation. See observations on this subject, by Mr. Molineux, Philos. Trans. number 230, p. 625, or Abr. vol. 1, p. 125.</p><p>The second branch of Surveying is performed by means of the protractor, and plotting scale. The description of which, see under their proper names.</p><p>If the lands in the survey are hilly, and not in any one plane, the measured lines cannot be truly laid down on paper, till they are reduced to one plane, which must be the horizontal one, because angles are taken in that plane. And in this case, when observing distant objects, for their elevation or depression, the following table shews the links or parts to be subtracted from each chain in the hypothenusal line, when the angle is the corresponding number of degrees. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=6" role="data">A <hi rend="smallcaps">Table</hi> <hi rend="italics">of the links to be subtracted out of every</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=6" role="data"> <hi rend="italics">chain in hypothenusal lines, of several degrees of</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=6" role="data"> <hi rend="italics">altitude or depression, for reducing them to hori-</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=6" role="data"> <hi rend="italics">zontal.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">links</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">links</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4°</cell><cell cols="1" rows="1" role="data">3′</cell><cell cols="1" rows="1" rend="align=center" role="data">1/4</cell><cell cols="1" rows="1" role="data">19°</cell><cell cols="1" rows="1" role="data">57′</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" rend="align=center" role="data">1/2</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">7</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=center" role="data">3/4</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">13</cell></row></table> <pb/><pb/><pb n="543"/><cb/></p><p>For example, if a station line measure 1250 links, or 12 1/2 chains, on an ascent, or a descent, of 11°; here it is after the rate of almost two links per chain, and it will be exact enough to take only the 12 chains at that rate, which make 24 links in all, to be deducted from 1250, which leaves 1226 links, for the length to be laid down.</p><p>Practical surveyors say, it is best to make this deduction at the end of every chain-length while measuring, by drawing the chain forward every time as much as the deduction is; viz, in the present instance, drawing the chain on 2 links at each chain-length.</p><p>The third branch of Surveying, namely computing or casting-up, is performed by reducing the several inclosures and divisions into triangles, trapeziums, and parallelograms, but especially the two former; then finding the areas or contents of these several figures, and adding them all together. <hi rend="center"><hi rend="italics">The Practice of Surveying.</hi></hi></p><p>1. Land is measured with a chain, called Gunter's chain, of 4 poles or 22 yards in length, which consists of 100 equal links, each link being 22/100 of a yard, or 66/100 of a foot, or 7.92 inches long, that is nearly 8 inches or 2/3 of a foot.</p><p>An acre of land is equal to 10 square chains, that is, 10 chains in length and 1 chain in breadth. Or it is 40 X 4 or 160 square poles. Or it is 220 X 22 or 4840 square yards. Or it is 1000 X 100 or 100000 square links. These being all the same quantity.</p><p>Also, an acre is divided into 4 parts called roods, and a rood into 40 parts called perches, which are square poles, or the square of a pole of 5 1/2 yards long, or the square of 1/4 of a chain, or of 25 links, which is 625 square links. So that the divisions of land measure will be thus: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">625</cell><cell cols="1" rows="1" role="data">sq. links</cell><cell cols="1" rows="1" role="data">= 1 pole or perch</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data"> perches</cell><cell cols="1" rows="1" role="data">= 1 rood</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">roods</cell><cell cols="1" rows="1" role="data">= 1 acre</cell></row></table></p><p>The length of lines, measured with a chain, are set down in links as integers, every chain in length being 100 links; and not in chains and decimals. Therefore, after the content is found, it will be in square links; then cut off five of the figures on the right-hand for decimals, and the rest will be acres. Those decimals are then multiplied by 4 for roods, and the decimals of these again by 40 for perches.</p><p><hi rend="italics">Ex.</hi> Suppose the length of a rectangular piece of ground be 792 links, and its breadth 385: to find the area in acres, roods, and perches. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">792</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">385</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">3960</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">6336 </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2376  </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">ac. ro. p.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3.04920</cell><cell cols="1" rows="1" role="data">Ans. 3   0  7</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">.19680</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7.87200</cell><cell cols="1" rows="1" role="data"/></row></table> <cb/></p><p>2. Among the various instruments for surveying, the plain-table is the easiest and most generally useful, especially in crooked difficult places, as in a town among houses, &c. But although the plain-table be the most generally useful instrument, it is not <hi rend="italics">always</hi> so; there being many cases in which sometimes one instrument is the properest, and sometimes another; nor is that surveyor master of his business who cannot in any case distinguish which is the fittest instrument or method, and use it accordingly: nay, sometimes no instrument at all, but barely the chain itself, is the best method, particularly in regular open fields lying together; and even when you are using the plain-table, it is often of advantage to measure such large open parts with the chain only, and from those measures lay them down upon the table.</p><p>The perambulator is used for measuring roads, and other great distances on level ground, and by the sides of rivers. It has a wheel of 8 1/4 feet, or half a pole, in circumference, upon which the machine turns; and the distance measured is pointed out by an index, which is moved round by clock work.</p><p>Levels, with telescopic or other sights, are used to find the level between place and place, or how much one place is higher or lower than another.</p><p>An offset-staff is a very useful and necessary instrument, for measuring the offsets and other short distances. It is 10 links in length, being divided and marked at each of the 10 links.</p><p>Ten small arrows, or rods of iron or wood, are used to mark the end of every chain length, in measuring lines. And sometimes pickets, or staves with flags, are set up as marks or objects of direction.</p><p>Various scales are also used in protracting and measuring on the plan or paper; such as plane scales, line of chords, protractor, compasses, reducing scales, parallel and perpendicular rulers, &c. Of plane scales, there should be several sizes, as a chain in 1 inch, a chain in 3/4 of an inch, a chain in 1/2 of an inch, &c. And of these, the best for use are those that are laid on the very edges of the ivory scale, to prick off distances by, without compasses. <hi rend="center">3. <hi rend="italics">The Field Book.</hi></hi></p><p>In surveying with the plain-table, a field-book is not used, as every thing is drawn on the table immediately when it is measured. But in surveying with the theodolite, or any other instrument, some sort of a fieldbook must be used, to write down in it a register or account of all that is done and occurs relative to the survey in hand.</p><p>This book every one contrives and rules as he thinks fittest for himself. The following is a specimen of a form that has formerly been much used. It is ruled into 3 columns: the middle, or principal column, is for the stations, angles, bearings, distances measured, &c; and those on the right and left are for the offsets on the right and lest, which are set against their corresponding distances in the middle column; as also for such remarks as may occur, and be proper to note for drawing the plan, &c.</p><p>Here Θ 1 is the first station, where the angle or bearing is 105° 25′. On the left, at 73 links in the <pb n="544"/><cb/> distance or principal line, is an offset of 92; and at 610 an offset of 24 to a cross hedge. On the right, at 0, or the beginning, an offset 25 to the corner of the field; at 248 Brown's boundary hedge commences; at 610 an offset 35; and at 954, the end of the first line, the 0 denotes its terminating in the hedge. And so on for the other stations.</p><p>Draw a line under the work, at the end of every station line, to prevent confusion. <table rend="border"><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Stations,</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Offsets and Remarks</cell><cell cols="1" rows="1" role="data">Bearings,</cell><cell cols="1" rows="1" role="data">Offsets and Remarks</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">on the left.</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" role="data">on the right.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Distances.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Θ1</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">105°25′</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">00</cell><cell cols="1" rows="1" role="data">25 corner</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">92</cell><cell cols="1" rows="1" rend="align=center" role="data">73</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">248</cell><cell cols="1" rows="1" role="data">Brown's hedge</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">cross a hedge 24</cell><cell cols="1" rows="1" rend="align=center" role="data">610</cell><cell cols="1" rows="1" role="data">35</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">954</cell><cell cols="1" rows="1" role="data">00</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Θ2</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">53°10′</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">00</cell><cell cols="1" rows="1" role="data">00</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">house corner 51</cell><cell cols="1" rows="1" rend="align=center" role="data">25</cell><cell cols="1" rows="1" role="data">21</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">120</cell><cell cols="1" rows="1" role="data">29 a tree</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" rend="align=center" role="data">734</cell><cell cols="1" rows="1" role="data">40 a stile</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Θ3</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">67°20′</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">61</cell><cell cols="1" rows="1" role="data">35</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">a brook 30  </cell><cell cols="1" rows="1" rend="align=center" role="data">248</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">639</cell><cell cols="1" rows="1" role="data">16 a spring</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">foot path 16</cell><cell cols="1" rows="1" rend="align=center" role="data">810</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">cross hedge 18</cell><cell cols="1" rows="1" rend="align=center" role="data">973</cell><cell cols="1" rows="1" role="data">20 a pond</cell></row></table> But a few skilful surveyers now make use of a different method for the field book, namely, beginning at the bottom of the page and writing upwards; by which they sketch a neat boundary on either hand, as they pass it; an example of which will be given below, with the plan of the ground to accompany it.</p><p>In smaller surveys and measurements, a very good way of setting down the work, is, to draw, by the eye, on a piece of paper, a figure resembling that which is to be measured; and so write the dimensions, as they are found, against the corresponding parts of the figure. And this method may be practised to a considerable extent, even in the larger surveys.</p><p>4. <hi rend="italics">To measure a line on the ground with the chain:</hi> Having provided a chain, with 10 small arrows, or rods, to stick one into the ground, as a mark, at the end of every chain; two persons take hold of the chain, one at each end of it, and all the 10 arrows are taken by one of them who goes foremost, and is called the leader; the other being called the follower, for distinction's sake.</p><p>A picket, or station staff, being set up in the direction of the line to be measured, if there do not appear some marks naturally in that direction; the follower <cb/> stands at the beginning of the line, holding the ring at the end of the chain in his hand, while the leader drags forward the chain by the other end of it, till it is stretched straight, and laid or held level, and the leader directed, by the follower waving his hand, to the right or left, till the follower see him exactly in a line with the mark or direction to be measured to: then both of them stretching the chain straight, and stooping and holding it level, the leader having the head of one of his arrows in the same hand by which he holds the end of the chain, he there sticks one of them down with it, while he holds the chain stretched. This done, he leaves the arrow in the ground, as a mark for the follower to come to, and advances another chain forward, being directed in his position by the follower standing at the arrow, as before; as also by himself now, and at every succeeding chain's length, by moving himself from side to side, till he brings the follower and the back mark into a line. Having then stretched the chain, and stuck down an arrow, as before, the follower takes up his arrow, and they advance again in the same manner another chain length. And thus they proceed till all the 10 arrows are employed, and are in the hands of the follower; and the leader, without an arrow, is arrived at the end of the 11th chain length. The follower then sends or brings the 10 arrows to the leader, who puts one of them down at the end of his chain, and advances with the chain as before. And thus the arrows are changed from the one to the other at every 10 chains' length, till the whole line is finished; the number of changes of the arrows shews the number of tens, to which the follower adds the arrows he holds in his hand, and the number of links of another chain over to the mark or end of the line. So if there have been 3 changes of the arrows, and the follower hold 6 arrows, and the end of the line cut off 45 links more, the whole length of the line is set down in links thus, 3645. <hi rend="center">5. <hi rend="italics">To take Angles and Bearings.</hi></hi></p><p>Let <hi rend="smallcaps">B</hi> and <hi rend="smallcaps">C</hi> be two ob- <figure/> jects, or two pickets set up perpendicular; and let it be required to take their bearings, or the angle formed between them at any station <hi rend="smallcaps">A.</hi></p><p>1<hi rend="italics">st. With the Plain Table.</hi> The table being covered with a paper, and fixed on its stand; plant it at the station <hi rend="smallcaps">A</hi>, and fix a fine pin, or a point of the compasses, in a proper part of the paper, to represent the point <hi rend="smallcaps">A:</hi> Close by the side of this pin lay the fiducial edge of the index, and turn it about, still touching the pin, till one object <hi rend="smallcaps">B</hi> can be seen through the sights: then by the fiducial edge of the index draw a line. In the very same manner draw another line in the direction of the other object <hi rend="smallcaps">C.</hi> And it is done.</p><p>2<hi rend="italics">d. With the Theodolite, &e.</hi> Direct the fixed sights along one of the lines, as <hi rend="smallcaps">AB</hi>, by turning the instrument about till you see the mark <hi rend="smallcaps">B</hi> through these sights; and there screw the instrument fast. Then <pb n="545"/><cb/> turn the moveable index about till, through its sights, you see the other mark C. Then the degrees cut by the index, upon the graduated limb or ring of the instrument, shew the quantity of the angle.</p><p>3<hi rend="italics">d. With the Magnetic Needle and Compass.</hi> Turn the instrument, or compass, so, that the north end of the needle point to the flower-de-luce. Then direct the sights to one mark, as B, and note the degrees cut by the needle. Then direct the sights to the other mark C, and note again the degrees cut by the needle. Then their sum or difference, as the case is, will give the quantity of the angle BAC.</p><p>4<hi rend="italics">th. By Measurement with the Chain, &c.</hi> Measure one chain length, or any other length, along both directions, as to b and c. Then measure the distance b c, and it is done.—This is easily transferred to paper, by making a triangle A b c with these three lengths, and then measuring the angle A as in Practical Geometry. <hi rend="center">6. <hi rend="italics">To Measure the Offsets.</hi></hi></p><p>A h i k l m n being a crooked hedge, or river, &c. From A measure in a straight direction along the side of it to B. And in measuring along this line AB observe when you are directly opposite any bends or corners of the hedge, as at c d, e, &c; and from thence measure the perpendicular offsets, ch, di, &c, with the offset-staff, if they are not very large, otherwise with the chain itself; and the work is done. And the register, or field-book, may be as follows: <table rend="border"><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Offs. left.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Base line AB</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">Θ</cell><cell cols="1" rows="1" role="data">A</cell></row><row role="data"><cell cols="1" rows="1" role="data">ch</cell><cell cols="1" rows="1" rend="align=right" role="data">62</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">Ac</cell></row><row role="data"><cell cols="1" rows="1" role="data">di</cell><cell cols="1" rows="1" rend="align=right" role="data">84</cell><cell cols="1" rows="1" rend="align=right" role="data">220</cell><cell cols="1" rows="1" role="data">Ad</cell></row><row role="data"><cell cols="1" rows="1" role="data">ek</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" rend="align=right" role="data">340</cell><cell cols="1" rows="1" role="data">Ae</cell></row><row role="data"><cell cols="1" rows="1" role="data">fl</cell><cell cols="1" rows="1" rend="align=right" role="data">98</cell><cell cols="1" rows="1" rend="align=right" role="data">510</cell><cell cols="1" rows="1" role="data">Af</cell></row><row role="data"><cell cols="1" rows="1" role="data">gm</cell><cell cols="1" rows="1" rend="align=right" role="data">57</cell><cell cols="1" rows="1" rend="align=right" role="data">634</cell><cell cols="1" rows="1" role="data">Ag</cell></row><row role="data"><cell cols="1" rows="1" role="data">Bn</cell><cell cols="1" rows="1" rend="align=right" role="data">91</cell><cell cols="1" rows="1" rend="align=right" role="data">785</cell><cell cols="1" rows="1" role="data">AB</cell></row></table> <figure/> <hi rend="center">7. <hi rend="italics">To Survey a triangular Field</hi> ABC.</hi> <hi rend="center">1<hi rend="italics">st. By the Chain.</hi></hi> <figure/> <table><row role="data"><cell cols="1" rows="1" role="data">AP</cell><cell cols="1" rows="1" rend="align=right" role="data">794</cell></row><row role="data"><cell cols="1" rows="1" role="data">AB</cell><cell cols="1" rows="1" rend="align=right" role="data">1321</cell></row><row role="data"><cell cols="1" rows="1" role="data">PC</cell><cell cols="1" rows="1" rend="align=right" role="data">826</cell></row></table></p><p>Having set up marks at the corners, which is to be done in all cases where there are not marks naturally; measure with the chain from A to P, where a perpendicular would fall from the angle C, and there measure from P to C; then complete the distance AB by measuring from P to B; setting down each of these measured distances. And thus, having the base and perpendicular, the area from them is easily found. Or having the place P of the perpendicular, the triangle is easily constructed. <cb/></p><p>Or, measure all the three sides with the chain, and note them down. From which the content is easily found, or the figure constructed. <hi rend="center">2<hi rend="italics">d. By taking one or more of the Angles.</hi></hi></p><p>Measure two sides AB, AC, and the angle A between them. Or measure one side AB, and the two adjacent angles A and B. From either of these ways the figure is easily planned: then by measuring the perpendicular CP on the plan, and multiplying it by half AB, you have the content. <hi rend="center">8. <hi rend="italics">To measure a Four-sided Field.</hi></hi> <hi rend="center">1<hi rend="italics">st. By the Chain.</hi></hi> <figure/> <table><row role="data"><cell cols="1" rows="1" role="data">AE</cell><cell cols="1" rows="1" role="data">214</cell><cell cols="1" rows="1" role="data">210</cell><cell cols="1" rows="1" role="data">DE</cell></row><row role="data"><cell cols="1" rows="1" role="data">AF</cell><cell cols="1" rows="1" role="data">362</cell><cell cols="1" rows="1" role="data">306</cell><cell cols="1" rows="1" role="data">BF</cell></row><row role="data"><cell cols="1" rows="1" role="data">AC</cell><cell cols="1" rows="1" role="data">592</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>Measure along either of the diagonals, as AC; and either the two perpendiculars DE, BF, as in the last problem; or else the sides AB, BC, CD, DA. From either of which the figure may be planned and computed as before directed. <hi rend="center"><hi rend="italics">Otherwise by the Chain.</hi></hi> <figure/> <table><row role="data"><cell cols="1" rows="1" role="data">AP</cell><cell cols="1" rows="1" rend="align=right" role="data">110</cell><cell cols="1" rows="1" role="data">352</cell><cell cols="1" rows="1" role="data">PC</cell></row><row role="data"><cell cols="1" rows="1" role="data">AQ</cell><cell cols="1" rows="1" rend="align=right" role="data">74</cell><cell cols="1" rows="1" role="data">595</cell><cell cols="1" rows="1" role="data">QD</cell></row><row role="data"><cell cols="1" rows="1" role="data">AB</cell><cell cols="1" rows="1" rend="align=right" role="data">1110</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table></p><p>Measure on the longest side, the distances AP, AQ, AB; and the perpendiculars PC, QD. <hi rend="center">2<hi rend="italics">d. By taking one or more of the Angles.</hi></hi></p><p>Measure the diagonal AC (see the first fig. above), and the angles CAB, CAD, ACB, ACD.—Or measure the four sides, and any one of the angles as BAD. <table><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Thus</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Or thus</cell></row><row role="data"><cell cols="1" rows="1" role="data">AC</cell><cell cols="1" rows="1" role="data">59<*></cell><cell cols="1" rows="1" role="data">AB</cell><cell cols="1" rows="1" role="data">486</cell></row><row role="data"><cell cols="1" rows="1" role="data">CAB</cell><cell cols="1" rows="1" role="data">37°20′</cell><cell cols="1" rows="1" role="data">BC</cell><cell cols="1" rows="1" role="data">394</cell></row><row role="data"><cell cols="1" rows="1" role="data">CAD</cell><cell cols="1" rows="1" role="data">41 15</cell><cell cols="1" rows="1" role="data">CD</cell><cell cols="1" rows="1" role="data">410</cell></row><row role="data"><cell cols="1" rows="1" role="data">ACB</cell><cell cols="1" rows="1" role="data">72 25</cell><cell cols="1" rows="1" role="data">DA</cell><cell cols="1" rows="1" role="data">462</cell></row><row role="data"><cell cols="1" rows="1" role="data">ACD</cell><cell cols="1" rows="1" role="data">54 40</cell><cell cols="1" rows="1" role="data">BAD</cell><cell cols="1" rows="1" role="data">78°35′</cell></row></table> <hi rend="center">9. <hi rend="italics">To Survey any Field by the Chain only.</hi></hi></p><p>Having set up marks at the corners, where necessary, of the proposed field ABCDEFG. Walk over the ground, and consider how it can best be divided into triangles and trapeziums; and measure them separately as in the last two problems. And in this way it will be proper to divide it into as few separate triangles, and as many trapeziums as may be, by drawing diago- <pb n="546"/><cb/> nals from corner to corner: and so, as that all the perpendiculars may fall within the figure. Thus, the following figure is divided into the two trapeziums ABCG, GDEF, and the triangle GCD. Then, in the first, beginning at A, measure the diagonal AC, and the two perpendiculars Gm, Bn. Then the base GC and the perpendicular Dq. Lastly the diagonal DF, and the two perpendiculars pE, oG. All which measures write against the corresponding parts of a rough figure drawn to resemble the figure to be surveyed, or set them down in any other form you choose. <figure/> <table><row role="data"><cell cols="1" rows="1" role="data">Am</cell><cell cols="1" rows="1" role="data">135</cell><cell cols="1" rows="1" rend="align=right" role="data">130</cell><cell cols="1" rows="1" role="data">mG</cell></row><row role="data"><cell cols="1" rows="1" role="data">An</cell><cell cols="1" rows="1" role="data">410</cell><cell cols="1" rows="1" rend="align=right" role="data">180</cell><cell cols="1" rows="1" role="data">nB</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ac</cell><cell cols="1" rows="1" role="data">550</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">-------</cell><cell cols="1" rows="1" rend="colspan=2" role="data">-------</cell></row><row role="data"><cell cols="1" rows="1" role="data">Cq</cell><cell cols="1" rows="1" role="data">152</cell><cell cols="1" rows="1" rend="align=right" role="data">230</cell><cell cols="1" rows="1" role="data">qD</cell></row><row role="data"><cell cols="1" rows="1" role="data">CG</cell><cell cols="1" rows="1" role="data">440</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">-------</cell><cell cols="1" rows="1" rend="colspan=2" role="data">-------</cell></row><row role="data"><cell cols="1" rows="1" role="data">FO</cell><cell cols="1" rows="1" role="data">206</cell><cell cols="1" rows="1" rend="align=right" role="data">120</cell><cell cols="1" rows="1" role="data">oG</cell></row><row role="data"><cell cols="1" rows="1" role="data">FP</cell><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" rend="align=right" role="data">80</cell><cell cols="1" rows="1" role="data">pE</cell></row><row role="data"><cell cols="1" rows="1" role="data">FD</cell><cell cols="1" rows="1" role="data">520</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> <hi rend="center"><hi rend="italics">Or thus:</hi></hi></p><p>Measure all the sides AB, BC, CD, DE, EF, FG, and GA; and the diagonals AC, CG, GD, DF. <hi rend="center"><hi rend="italics">Otherwise,</hi></hi></p><p>Many pieces of land may be very well surveyed, by measuring any base line, either within or without them, together with the perpendiculars let fall upon it from every corner of them. For they are by these means divided into several triangles and trapezoids, all whose parallel sides are perpendicular to the base line; and the sum of these triangles and trapeziums will be equal to the figure proposed if the base line fall within it; if not, the sum of the parts which are without being taken from the sum of the whole which are both within and without, will leave the area of the figure proposed.</p><p>In pieces that are not very large, it will be sufficiently exact to find the points, in the base line, where the several perpendiculars will fall, by means of the <hi rend="italics">cross,</hi> and from thence measuring to the corners for the lengths of the perpendiculars.—And it will be most convenient to draw the line so as that all the perpendiculars may fall within the figure.</p><p>Thus, in the following figure, beginning at A, and measuring along the line AG, the distances and perpendiculars, on the right and left, are as below. <figure/> <table><row role="data"><cell cols="1" rows="1" role="data">Ab</cell><cell cols="1" rows="1" rend="align=right" role="data">315</cell><cell cols="1" rows="1" rend="align=right" role="data">350</cell><cell cols="1" rows="1" role="data">bB</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ac</cell><cell cols="1" rows="1" rend="align=right" role="data">440</cell><cell cols="1" rows="1" rend="align=right" role="data">70</cell><cell cols="1" rows="1" role="data">cC</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ad</cell><cell cols="1" rows="1" rend="align=right" role="data">585</cell><cell cols="1" rows="1" rend="align=right" role="data">320</cell><cell cols="1" rows="1" role="data">dD</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ae</cell><cell cols="1" rows="1" rend="align=right" role="data">610</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">eE</cell></row><row role="data"><cell cols="1" rows="1" role="data">Af</cell><cell cols="1" rows="1" rend="align=right" role="data">990</cell><cell cols="1" rows="1" rend="align=right" role="data">470</cell><cell cols="1" rows="1" role="data">fF</cell></row><row role="data"><cell cols="1" rows="1" role="data">AG</cell><cell cols="1" rows="1" rend="align=right" role="data">1020</cell><cell cols="1" rows="1" rend="align=right" role="data">0 </cell><cell cols="1" rows="1" role="data"/></row></table> <cb/> <hi rend="center">10. <hi rend="italics">To Survey any Field with the Plain Table.</hi></hi> <hi rend="center">1<hi rend="italics">st. From one Station.</hi></hi></p><p>Plant the table at any <figure/> angle, as C, from whence all the other angles, or marks set up, can be seen; and turn the table about till the needle point to the flower-de-luce: and there screw it fast. Make a point for C on the paper on the table, and lay the edge of the index to C, turning it about there till through the sights you see the mark D; and by the edge of the index draw a dry or obscure line: then measure the distance CD, and lay that distance down on the line CD. Then turn the index about the point C, till the mark E be seen through the sights, by which draw a line, and measure the distance to E, laying it on the line from C to E. In like manner determine the positions of CA and CB, by turning the sights successively to A and B; and lay the lengths of those lines down. Then connect the points with the boundaries of the field, by drawing the black lines CD, DE, EA, AB, BC. <hi rend="center">2<hi rend="italics">d. From a Station within the Field.</hi></hi></p><p>When all the other parts <figure/> cannot be seen from one angle, choose some place O within; or even without, if more convenient, from whence the other parts can be seen. Plant the table at O, then fix it with the needle north, and mark the point O upon it. Apply the index successively to O, turning it round with the sights to each angle A, B, C, D, E, drawing dry lines to them by the edge of the index, then measuring the distances OA, OB, &c, and laying them down upon those lines. Lastly draw the boundaries AB, BC, CD, DE, EA. <hi rend="center">3<hi rend="italics">d. By going round the Figure.</hi></hi></p><p>When the figure is a wood or water, or from some other obstruction you cannot measure lines across it; begin at any point A, and measure round it, either within or without the figure, and draw the directions of all the sides thus: Plant the table at A, turn it with the needle to the north or flower-de-luce, fix it and mark the point A. Apply the index to A, turning it till you can see the point E, there draw a line; and then the point B, and there draw a line: then measure these lines, and lay them down from A to E and B. Next move the table to B, lay the index along the line AB, and turn the table about till you can see the mark A, and screw fast the table; in which position also the needle will again point to the flower-de-luce, as it will <pb n="547"/><cb/> do indeed at every station when the table is in the right position. Here turn the index about B till through the sights you see the mark C; there draw a line, measure BC, and lay the distance upon that line after you have set down the table at C. Turn it then again into its proper position, and in like manner find the next line CD. And so on quite round by E to A again. Then the proof of the work will be the joining at A: for if the work is all right, the last direction EA on the ground, will pass exactly through the point A on the paper; and the measured distance will also reach exactly to A. If these do not coincide, or nearly so, some error has been committed, and the work must be examined over again. <hi rend="center">11. <hi rend="italics">To Survey a Field with the Theodolite, &c.</hi></hi> <hi rend="center">1<hi rend="italics">st. From one Point or Station.</hi></hi></p><p>When all the angles can be seen from one point, as the angle C (last fig. but one), place the instrument at C, and turn it about till, through the fixed sights, you see the mark B, and there fix it. Then turn the moveable index about till the mark A is seen through the sights, and note the degrees cut on the instrument. Next turn the index successively to E and D, noting the degrees cut off at each; which gives all the angles BCA, BCE, BCD. Lastly, measure the lines CB, CA, CE, CD; and enter the measures in a field-book, or rather against the corresponding parts of a rough figure drawn by guess to resemble the field. <hi rend="center">2<hi rend="italics">d. From a Point within or without.</hi></hi></p><p>Plant the instrument at O, (last fig.) and turn it about till the fixed sights point to any object, as A; and there screw it fast. Then turn the moveable index round till the sights point successively to the other points E, D, C, B, noting the degrees cut off at each of them; which gives all the angles round the point O. Lastly, measure the distances OA, OB, OC, OD, OE, noting them down as before, and the work is done. <hi rend="center">3<hi rend="italics">d. By going round the Field.</hi></hi></p><p>By measuring round, <figure/> either within or without the field, proceed thus. Having set up marks at B, C, &c. near the corners as usual, plant the instrument at any point A, and turn it till the fixed index be in the direction AB, and there screw it fast: then turn the moveable index to the direction AF; and the degrees cut off will be the angle A. Measure the line AB, and plant the instrument at B, and there in the same manner observe the angle A. Then measure BC, and observe the angle C. Then measure the distance CD, and take the angle D. Then measure DE, and take the angle E. Then measure EF, and take the angle F. And lastly measure the distance FA.</p><p>To prove the work; add all the inward angles, A, B, C, &c, together, and when the work is right, their sum will be equal to twice as many right angles as the figure has sides, wanting 4 right angles. And when there is an angle, as F, that bends inwards, and <cb/> you measure the external angle, which is less than two right angles, subtract it from 4 right angles, or 360 degrees, to give the internal angle greater than a semicircle or 180 degrees.</p><p><hi rend="italics">Otherwise.</hi> Instead of observing the internal angles, you may take the external angles, formed without the figure by producing the sides further out. And in this case, when the work is right, their sum altogether will be equal to 360 degrees. But when one of them, as F, runs inwards, subtract it from the sum of the rest, to leave 360 degrees. <hi rend="center">12. <hi rend="italics">To Survey a Field with crooked Hedges, &c.</hi></hi></p><p>With any of the instruments measure the lengths and positions of imaginary lines running as near the sides of the field as you can; and in going along them measure the offsets in the manner before taught; and you will have the plan on the paper in using the plain table, drawing the crooked hedges through the ends of the offsets; but in surveying with the theodolite, or other instrument, set down the measures properly in a field-book, or memorandum-book, and plan them after returning from the field, by laying down all the lines and angles. <figure/></p><p>So, in surveying the piece ABCDE, set up mark: a, b, c, d, dividing it into as few sides as may be. Then begin at any station a, and measure the lines ab, bc, cd, da, and take their positions, or the angles a, b, c, d; and in going along the lines measure all the offsets, as at m, n, o, p, &c, along every station line.</p><p>And this is done either within the field, or without, as may be most convenient. When there are obstructions within, as wood, water, hills, &c; then measure without, as in the figure here below. <figure/> <pb n="548"/><cb/> <hi rend="center">13. <hi rend="italics">To Survey a Field or any other Thing, by Two Stations.</hi></hi></p><p>This is performed by choosing two stations, from whence all the marks and objects can be seen, then measuring the distance between the stations, and at each station taking the angles formed by every object, from the station line or distance.</p><p>The two stations may be taken either within the bounds, or in one of the sides, or in the direction of two of the objects, or quite at a distance, and without the bounds of the objects, or part to be surveyed.</p><p>In this manner, not only grounds may be surveyed, without even entering them, but a map may be taken of the principal parts of a country, or the chief places of a town, or any part of a river or coast surveyed, or any other inaccessible objects; by taking two stations, on two towers, or two hills, or such like. <figure/></p><p>When the plain table is used; plant it at one station m, draw a line m n on it, along which lay the edge of the index, and turn the table about till the sights point directly to the other station; and there screw it fast. Then turn the sights round m successively to all the objects ABC, &c, drawing a dry line by the edge of the index at each, as mA, mB, mC, &c. Then measure the distance to the other station, there plant the table, and lay that distance down on the station line from m to n. Next lay the index by the line nm, and turn the table about till the sights point to the other station m, and there screw it fast. Then direct the sights successively to all the objects A, B, C, &c, as before, drawing lines each time, as nA, nB, nC, &c: and their intersection with the former lines will give the places of all the objects, or corners, A, B, C, &c.</p><p>When the theodolite, or any other instrument for taking angles, is used; proceed in the same way, measuring the station distance mn, planting the instrument first at one station, and then at another; then placing the fixed sights in the direction mn, and directing the moveable sights to every object, noting the degrees cut off at each time. Then, these observations being planned, the intersections of the lines will give the objects as before.</p><p>When all the objects, to be surveyed, cannot be seen from two stations; then three stations may be used, or four, or as many as is necessary; measuring always <cb/> the distance from one station to another; placing the instrument in the same position at every station, by means described before; and from each station observing or setting every object that can be seen from it, by taking its direction or angular position, till every object be determined by the intersection of two or more lines of direction, the more the better. And thus may very extensive surveys be taken, as of large commons, rivers, coasts, countries, hilly grounds, and such like. <hi rend="center">14. <hi rend="italics">To Survey a Large Estate.</hi></hi></p><p>If the estate be very large, and contain a great number of fields, it cannot well be done by surveying all the fields singly, and then putting them together; nor can it be done by taking all the angles and boundaries that inclose it. For in these cases, any small errors will be so multiplied, as to render it very much distorted.</p><p>1st. Walk over the estate two or three times, in order to get a perfect idea of it, and till you can carry the map of it tolerably in your head. And to help your memory, draw an eye draught of it on paper, or at least, of the principal parts of it, to guide you.</p><p>2d. Choose two or more eminent places in the estate, for your stations, from whence you can see all the principal parts of it: and let these stations be as far distant from one another as possible; as the fewer stations you have to command the whole, the more exact your work will be: and they will be sitter for your purpose, if these station lines be in or near the boundaries of the ground, and especially if two or more lines proceed from one station.</p><p>3d. Take angles, between the stations, such as you think necessary, and measure the distances from station to station, always in a right line: these things must be done, till you get as many angles and lines as are sufficient for determining all your points of station. And in measuring any of these station distances, mark accurately where these lines meet with any hedges, ditches, roads, lanes, paths, rivulets, &c, and where any remarkable object is placed, by measuring its distance from the station line, and where a perpendicular from it cuts that line; and always mind, in any of these observations, that you be in a right line, which you will know by taking back sight and foresight, along your station line. And thus as you go along any main station line, take offsets to the ends of all hedges, and to any pond, house, mill, bridge, &c, omitting nothing that is remarkable. And all these things must be noted down; for these are your data, by which the places of such objects are to be determined upon your plan. And be sure to set marks up at the intersections of all hedges with the station line, that you may know where to measure from, when you come to survey these particular fields, which must immediately be done, as soon as you have measured that station line, whilst they are fresh in memory. In this way all your station lines are to be measured, and the situation of all places adjoining to them determined, which is the first grand point to be obtained. It will be proper for you to lay down your work upon paper every night, when you go home, that you may see how you go on.</p><p>4th. As to the inner parts of the estate, they must be <pb n="549"/><cb/> determined in like manner, by new station lines: for, after the main stations are determined, and every thing adjoining to them, then the estate must be subdivided into two or three parts by new station lines; taking inner stations at proper places, where you can have the best view. Measure these station lines as you did the first, and all their intersections with hedges, and all offsets to such objects as appear. Then you may proceed to survey the adjoining fields, by taking the angles that the sides make with the station line, at the intersections, and measuring the distances to each corner, from the intersections. For every station line will be a basis to all the future operations; the situation of all parts being entirely dependant upon them; and therefore they should be taken of as great a length as possible; and it is best for them to run along some of the hedges or boundaries of one or more fields, or to pass through some of their angles. All things being determined for these stations, you must take more inner ones, and so continue to divide and subdivide, till at last you come to single fields; repeating the same work for the inner stations, as for the outer ones, till all be done: and close the work as often as you can, and in as few lines as possible. And that you may choose stations the most conveniently, so as to cause the least labour, let the station lines run as far as you can along some hedges, and through as many corners of the fields, and other remarkable points, as you can. And take notice how one field lies by another; that you may not misplace them in the draught.</p><p>5th. An estate may be so situated, that the whole cannot be surveyed together; because one part of the estate cannot be seen from another. In this case, you may divide it into three or four parts, and survey the parts separately, as if they were lands belonging to different persons; and at last join them together.</p><p>6th. As it is necessary to protract or lay down your work as you proceed in it, you must have a scale of a due length to do it by. To get such a scale, you must measure the whole length of the estate in chains; then you must consider how many inches in length the map is to be; and from these you will know how many chains you must have in an inch; then make your scale, or choose one already made, accordingly.</p><p>7th. The trees in every hedge row must be placed in their proper situation, which is soon done by the plain table; but may be done by the eye without an instrument; and being thus taken by guess, in a rough draught, they will be exact enough, being only to look at; except it be such as are at any remarkable places, as at the ends of hedges, at stiles, gates, &c, and these must be measured. But all this need not be done till the draught is finished. And observe in all the hedges, what side the gutter or ditch is on, and consequently to whom the fences belong.</p><p>8th. When you have long stations, you ought to have a good instrument to take angles with; and the plain table may very properly be made use of, to take the several small internal parts, and such as cannot be taken from the main stations, as it is a very quick and ready instrument.</p><p>15. Instead of the foregoing method, an ingenious friend (Mr. Abraham Crocker), after mentioning the new and improved method of keeping the field book by <cb/> writing from bottom to top of the pages, observes that “In the former method of measuring a large estate, the accuracy of it depends on the correctness of the instruments used in taking the angles. To avoid the errors incident to such a multitude of angles, other methods have of late years been used by some few skilful surveyors; the most practical, expeditious, and correct, seems to be the following.</p><p>“As was advised in the foregoing method, so in this, choose two or more eminences, as grand stations, and measure a principal base line from one station to the other, noting every hedge, brook, or other remarkable object as you pass by it; measuring also such short perpendicular lines to such bends of hedges as may be near at hand. From the extremities of this base line, or from any convenient parts of the same, go off with other lines to some remarkable object situated towards the sides of the estate, without regarding the angles they make with the base line or with one another; still remembering to note every hedge, brook or other object that you pass by. These lines, when laid down by intersections, will with the base line form a grand triangle upon the estate; several of which, if need be, being thus laid down, you may proceed to form other smaller triangles and trapezoids on the sides of the former: and so on, until you finish with the enclosures individually.</p><p>“To illustrate this excellent method, let us take AB (in the plan of an estate, fig. 1, pl. 28) for the principal base line. From B go off to the tree at C; noting down, in the field-book, every cross hedge, as you measure on; and from C measure back to the first station at A, noting down every thing as before directed.</p><p>“This grand triangle being completed, and laid down on the rough-plan paper, the parts, exterior as well as interior, are to be completed by smaller triangles and trapezoids.</p><p>“When the whole plan is laid down on paper, the contents of each field might be calculated by the methods laid down below, at article 20.</p><p>“In countries where the lands are enclosed with high hedges, and where many lanes pass through an estate, a theodolite may be used to advantage, in measuring the angles of such lands; by which means, a kind of skeleton of the estate may be obtained, and the lane-lines serve as the bases of such triangles and trapezoids as are necessary to fill up the interior parts.”</p><p>The method of measuring the other cross lines, offsets and interior parts and enclosures, appears in the plan, fig. 1, last referred to.</p><p>16. Another ingenious correspondent (Mr. John Rodham of Richmond, Yorkshire) has also communicated the following example of the new method of surveying, accompanied by the field-book, and its corresponding plan. His account of the method is as follows.</p><p>The field-book is ruled into three columns. In the middle one are set down the distances on the chain line at which any mark, offset, or other observation is made; and in the right and left hand columns are entered, the offsets and observations made on the right and left hand respectively of the chain line.</p><p>It is of great advantage, both for brevity and per- <pb n="550"/><cb/> spicuity, to begin at the bottom of the leaf and write upwards; denoting the crossing of fences, by lines drawn across the middle column, or only a part of such a line on the right and left opposite the figures, to avoid confusion, and the corners of fields, and other remarkable turns in the fences where offsets are taken to, by lines joining in the manner the fences do, as will be best seen by comparing the book with the plan annexed, fig. 2, pl. 28.</p><p>The marks called, <hi rend="italics">a, b, c,</hi> &c, are best made in the fields, by making a small hole with a spade, and a chip or small bit of wood, with the particular letter upon it, may be put in, to prevent one mark being taken for another, on any return to it. But in general, the name of a mark is very easily had by referring in the book to the line it was made in. After the small alphabet is gone through, the capitals may be next, the print letters afterwards, and so on, which answer the purpose of so many different letters; or the marks may be numbered.</p><p>The letter in the left hand corner at the beginning of every line, is the mark or place measured <hi rend="italics">from;</hi> and, that at the right hand corner at the end, is the mark measured <hi rend="italics">to:</hi> But when it is not convenient to go exactly from a mark, the place measured from, is described <hi rend="italics">such a distance</hi> from <hi rend="italics">one mark</hi> towards <hi rend="italics">another;</hi> and where a mark is not measured to, the exact place is ascertained by saying, turn to the right or left hand, <hi rend="italics">such a distance</hi> to <hi rend="italics">such a mark,</hi> it being always understood that those distances are taken in the chain line.</p><p>The characters used, are <figure/> for <hi rend="italics">turn to the right band,</hi> <figure/> for turn to the left hand, and <figure/> placed over an offset, to shew that it is not taken at right angles with the chain line, but in the line with some straight fence; being chiefly used when crossing their directions, and is a better way of obtaining their true places than by offsets at right angles.</p><p>When a line is measured whose position is determined, either by former work (as in the case of producing a given line or measuring from one known place or mark to another) or by itself (as in the third side of a triangle) it is called a <hi rend="italics">fast line,</hi> and a double line across the book is drawn at the conclusion of it; but if its position is not determined (as in the second side of a triangle) it is called a <hi rend="italics">loose line,</hi> and a single line is drawn across the book. When a line becomes determined in position, and is afterwards continued, a double line half through the book is drawn.</p><p>When a loose line is measured, it becomes absolutely necessary to measure some line that will determine its position. Thus, the first line <hi rend="italics">ab,</hi> being the base of a triangle, is always determined; but the position of the second side <hi rend="italics">hj,</hi> does not become determined, till the third side <hi rend="italics">jb</hi> is measured; then the triangle may be constructed, and the position of both is determined.</p><p>At the beginning of a line, to fix a loose line to the mark or place measured from, the sign of turning to the right or left hand must be added (as at <hi rend="italics">j</hi> in the third line); otherwise a stranger, when laying down the work may as easily construct the triangle <hi rend="italics">hjb</hi> on the wrong side of the line <hi rend="italics">ah,</hi> as on the right one: but this error cannot be fallen into, if the sign above named be carefully observed.</p><p>In choosing a line to fix a loose one, care must be <cb/> taken that it does not make a very acute or obtuse angle; as in the triangle <hi rend="italics">p</hi>B<hi rend="italics">r,</hi> by the angle at B being very obtuse, a small deviation from truth, even the breadth of a point at <hi rend="italics">p</hi> or <hi rend="italics">r,</hi> would make the error at B when constructed very considerable; but by constructing the triangle <hi rend="italics">p</hi>B<hi rend="italics">q,</hi> such a deviation is of no consequence.</p><p>Where the words <hi rend="italics">leave off</hi> are written in the fieldbook, it is to signify that the taking of offsets is from thence discontinued; and of course something is wanting between that and the next offset.</p><p>The field-book above referred to, is engraved on plate 29, in parts, representing so many pages, each of which is supposed to begin at the bottom, and end at top. And the map or plan belonging to it, in fig. 2, pl. 28. <hi rend="center">17. <hi rend="italics">To Survey a County, or Large Tract of Land.</hi></hi></p><p>1st. Choose two, three, or four eminent places for stations; such as the tops of high hills or mountains, towers, or church steeples, which may be seen from one another; and from which most of the towns, and other places of note, may also be seen. And let them be as far distant from one another as possible. Upon these place raise beacons, or long poles, with flags of different colours flying at them; so as to be visible from all the other stations.</p><p>2d. At all the places, which you would set down in the map, plant long poles with flags at them of several colours, to distinguish the places from one another; fixing them upon the tops of church steeples, or the tops of houses, or in the centres of lesser towns.</p><p>But you need not have these marks at many places at once, as suppose half a score at a time. For when the angles have been taken, at the two stations, to all these places, the marks may be moved to new ones; and so successively to all the places you want. These marks then being set up at a convenient number of places, and such as may be seen from both stations; go to one of these stations, and with an instrument to take angles, standing at that station, take all the angles between the other station, and each of these marks, observing which is blue, which red, &c, and which hand they lie on; and set all down with their colours. Then go to the other station, and take all the angles between the first station, and each of the former marks, and set them down with the others, each against his fellow with the same colour. You may, if you can, also take the angles at some third station, which may serve to prove the work, if the three lines intersect in that point, where any mark stands. The marks must stand till the observations are finished at both stations; and then they must be taken down, and set up at fresh places. And the same operations must be performed, at both stations, for these fresh places; and the like for others. Your instrument for taking angles must be an exceeding good one, made on purpose with telescopic sights; and of three, four, or five feet radius. A circumferentor is reckoned a good instrument for this purpose.</p><p>3d. And though it is not absolutely necessary to measure any distance, because a stationary line being laid down from any scale, all the other lines will be <pb n="551"/><cb/> proportional to it; yet it is better to measure some of the lines, to ascertain the distances of places in miles; and to know how many geometrical miles there are in any length; and from thence to make a scale to measure any distance in miles. In measuring any distance, it will not be exact enough to go along the high roads; by reason of their turnings and windings, and hardly ever lying in a right line between the stations, which would cause endless reductions, and create trouble to make it a right line; for which reason it can never be exact. But a better way is to measure in a right line with a chain, between station and station, over hills and dales or level fields, and all obstacles. Only in case of water, woods, towns, rocks, banks, &c, where one cannot pass, such parts of the line must be measured by the methods of inaccessible distances; and besides, allowing for ascents and descents, when we meet with them. And a good compass that shews the bearing of the two stations, will always direct you to go straight, when you do not see the two stations; and in your progress, if you can go straight, you may take offsets to any remarkable places, likewise noting the intersection of the stationary line with all roads, rivers, &c.</p><p>4th. And from all the stations, and in the whole progress, be very particular in observing sea coasts, river mouths, towns, castles, houses, churches, windmills, watermills, trees, rocks, sands, roads, bridges, fords, ferries, woods, hills, mountains, rills, brooks, parks, beacons, sluices, floodgates, locks, &c; and in general all things that are remarkable.</p><p>5th. After you have done with the first and main station lines, which command the whole county; you must then take inner stations, at some places already determined; which will divide the whole into several partitions: and from these stations you must determine the places of as many of the remaining towns as you can. And if any remain in that part, you must take more stations, at some places already determined; from which you may determine the rest. And thus proceed through all the parts of the country, taking station after station, till we have determined all we want. And in general the station distances must always pass through such remarkable points as have been determined before, by the former stations.</p><p>6th. Lastly, the position of the station line you measure, or the point of the compass it lies on, must be determined by astronomical observation. Hang up a thread and plummet in the sun, over some part of the station line, and observe when the shadow runs along that line, and at that moment take the sun's altitude; then having his declination, and the latitude, the azimuth will be found by spherical trigonometry. And the azimuth is the angle the station line makes with the meridian; and therefore a meridian may easily be drawn through the map: Or a meridian may be drawn through it by hanging up two threads in a line with the pole star, when he is just north, which may be known from astronomical tables. Or thus; observe the star Alioth, or that in the rump of the great bear, being that next the square; or else Cassiopeia's hip; I say, observe by a line and plummet when either of these stars and the pole star come into a perpendicular; and at that time they are due north. There- <cb/> fore two perpendicular lines being fixed at that moment, towards these two stars, will give the position of the meridian. <hi rend="center">18. <hi rend="italics">To Survey a Town or City.</hi></hi></p><p>This may be done with any of the instruments for taking angles, but best of all with the plain table, where every minute part is drawn while in sight. It is proper also to have a chain of 50 feet long, divided into 50 links, and an offset-staff of 10 feet long.</p><p>Legin at the meeting of two or more of the principal streets, through which you can have the longest prospects, to get the longest station lines. There having fixed the instrument, draw lines of direction along those streets, using two men as marks, or poles set in wooden pedestals, or perhaps some remarkable places in the houses at the farther ends, as windows, doors, corners, &c. Measure these lines with the chain, taking offsets with the staff, at all corners of streets, bendings, or windings, and to all remarkable things, as churches, markets, halls, colleges, eminent houses, &c. Then remove the instrument to another station along one of these lines; and there repeat the same process as before. And so on till the whole is finished. <figure/></p><p>Thus, fix the instrument at A, and draw lines in the direction of all the streets meeting there; and measure AB, noting the street on the left at m. At the second station B, draw the directions of the streets meeting there; measure from B to C, noting the places of the streets at n and o as you pass by them. At the 3d station C, take the direction of all the streets meeting there, and measure CD. At D do the same, and measure DE, noting the place of the cross streets at p. And in this manner go through all the principal streets. This done, proceed to the smaller and intermediate streets; and lastly to the lanes, alleys, courts, yards, and every part that it may be thought proper to represent. <hi rend="center"><hi rend="italics">Of Planning, Computing, and Dividing.</hi></hi> <hi rend="center">19. <hi rend="italics">To Lay down the Plan of any Survey.</hi></hi></p><p>If the survey was taken with a plain table, you have a rough plan of it already on the paper which covered <pb n="552"/><cb/> the table. But if the survey was with any other instrument, a plan of it is to be drawn from the measures that were taken in the survey, and first of all a rough plan upon paper.</p><p>To do this, you must have a set of proper instruments, for laying down both lines and angles, &c; as scales of various sizes, the more of them, and the more accurate, the better; scales of chords, protractors, perpendicular and parallel rulers, &c. Diagonal scales are best for the lines, because they extend to three figures, or chains and links, which are hundredth parts of chains. And in using the diagonal scale, a pair of compasses must be employed to take off the lengths of the principal lines very accurately. But a scale with a thin edge divided, is much readier for laying down the perpendicular offsets to crooked hedges, and for marking the places of those offsets upon the station line; which is done at only one application of the edge of the scale to that line, and then pricking off all at once the distances along it. Angles are to be laid down either with a good scale of chords, which is perhaps the most accurate way; or with a large protractor, which is much readier when many angles are to be laid down at one point, as they are pricked off all at once round the edge of the protractor.</p><p>Very particular directions for laying down all sorts of figures cannot be necessary in this place, to any person who has learned practical geometry, or the construction of figures, and the use of his instruments. It may therefore be sufficient to observe, that all lines and angles must be laid down on the plan in the same order in which they were measured in the field, and in which they are written in the field-book; laying down first the angles for the position of lines, then the lengths of the lines, with the places of the offsets, and then the lengths of the offsets themselves, all with dry or obscure lines; then a black line drawn through the extremities of all the offsets, will be the hedge or bounding line of the field, &c. After the principal bounds and lines are laid down, and made to fit or close properly, proceed next to the smaller objects, till you have entered every thing that ought to appear in the plan, as houses, brooks, trees, hills, gates, stiles, roads, lanes, mills, bridges, woodlands, &c, &c.</p><p>The north side of a map or plan is commonly placed uppermost, and a meridian somewhere drawn, with the compass or flower-de-luce pointing north. Also, in a vacant place, a scale of equal parts or chains is drawn, and the title of the map in conspicuous characters, and embellished with a compartment. All hills must be shadowed, to distinguish them in the map. Colour the hedges with different colours; represent hilly grounds by broken hills and valleys; draw single dotted lines for foot-paths, and double ones for horse or carriage roads. Write the name of each field and remarkable place within it, and, if you choose, its content in acres, roods, and perches.</p><p>In a very large estate, or a county, draw vertical and horizontal lines through the map, denoting the spaces between them by letters, placed at the top, and bottom, and sides, for readily finding any field or other object, mentioned in a table.</p><p>In mapping counties, and estates that have uneven <cb/> grounds of hills and valleys, reduce all oblique lines, measured up hill and down hill, to horizontal straight lines, if that was not done during the survey, before they were entered in the field-book, by making a proper allowance to shorten them. For which purpose, there is commonly a small table engraven on some of the instruments for Surveying. <hi rend="center">20. <hi rend="italics">To Compute the Contents of Fields.</hi></hi></p><p>1st. Compute the contents of the figures, whether triangles, or trapeziums, &c, by the proper rules for the several figures laid down in measuring; multiplying the lengths by the breadths, both in links; the product is acres after you have cut off five figures on the right, for decimals; then bring these decimals to roods and perches, by multiplying first by 4, and then by 40. An example of which was given in the description of the chain, art. 1.</p><p>2d. In small and separate pieces, it is usual to cast up their contents from the measures of the lines taken in surveying them, without making a correct plan of them.</p><p>Thus, in the triangle in art. 7, where we had <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">AP = 794, and AB = 1321</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">PC =  826</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7926</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2642 </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10568  </cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">2 ) 10.91146</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5.45573</cell><cell cols="1" rows="1" rend="align=center" role="data">ac r p</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=center" role="data">Ans. 32 1 33 nearly</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1.82292</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">32.91680</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>Or the first example to art. 8, thus: <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">AE 214  210 </cell><cell cols="1" rows="1" rend="align=left" role="data">ED</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">AF 362  306 </cell><cell cols="1" rows="1" rend="align=left" role="data">FB</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">AC 592</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">516 </cell><cell cols="1" rows="1" rend="align=left colspan=2" role="data">sum of perp.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">592 </cell><cell cols="1" rows="1" rend="align=left" role="data">AC</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1032</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4644 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2580  </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3.05472</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">ac r p</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Ans. 3 0 8</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">.21888</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8.75520</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> <pb n="553"/><cb/></p><p>Or the 2d example to the same article, thus: <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">AP 110</cell><cell cols="1" rows="1" role="data">352</cell><cell cols="1" rows="1" rend="align=left" role="data">PC</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">AQ 745</cell><cell cols="1" rows="1" role="data">595</cell><cell cols="1" rows="1" rend="align=left" role="data">QD</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">AB 1110</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">PC 352 </cell><cell cols="1" rows="1" role="data">PC 352</cell><cell cols="1" rows="1" role="data">QD 595</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">AP 110 </cell><cell cols="1" rows="1" role="data">QD 595</cell><cell cols="1" rows="1" role="data">QB 365</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">------- </cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2 APC 38720 </cell><cell cols="1" rows="1" role="data">sum 947</cell><cell cols="1" rows="1" role="data">2975</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">------- </cell><cell cols="1" rows="1" role="data">PQ 635</cell><cell cols="1" rows="1" role="data">3570 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">1785  </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4735</cell><cell cols="1" rows="1" role="data">-----</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2841 </cell><cell cols="1" rows="1" role="data">217175</cell><cell cols="1" rows="1" rend="align=left" role="data">= 2QDB</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5682  </cell><cell cols="1" rows="1" role="data">601345</cell><cell cols="1" rows="1" rend="align=left" role="data">= 2PCDQ</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-----</cell><cell cols="1" rows="1" role="data">38720</cell><cell cols="1" rows="1" rend="align=left" role="data">= 2APC</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2PCDQ</cell><cell cols="1" rows="1" role="data">601345</cell><cell cols="1" rows="1" role="data">-----</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-------</cell><cell cols="1" rows="1" role="data">2) 8.57240</cell><cell cols="1" rows="1" rend="align=left" role="data">= dou. the whole</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4.2862</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">ac r p</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-----</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Ans. 4 1 5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1.1448</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-----</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.7920</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">-----</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>3d. In pieces bounded by very crooked and winding hedges, measured by offsets, all the parts between the offsets are most accurately measured separately as small trapezoids. Thus, for the example to art. 6, where <table><row role="data"><cell cols="1" rows="1" role="data">Ac  45</cell><cell cols="1" rows="1" role="data">62 ch</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ad 220</cell><cell cols="1" rows="1" role="data">84 di</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ae 340</cell><cell cols="1" rows="1" role="data">70 ek</cell></row><row role="data"><cell cols="1" rows="1" role="data">Af 510</cell><cell cols="1" rows="1" role="data">98 fl</cell></row><row role="data"><cell cols="1" rows="1" role="data">Ag 634</cell><cell cols="1" rows="1" role="data">57 gm</cell></row><row role="data"><cell cols="1" rows="1" role="data">AB 785</cell><cell cols="1" rows="1" role="data">91 Bn</cell></row></table> <hi rend="center">Then</hi> <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Ac 45</cell><cell cols="1" rows="1" role="data">ch 62</cell><cell cols="1" rows="1" role="data">di 84</cell><cell cols="1" rows="1" role="data">ek 70</cell><cell cols="1" rows="1" role="data">fl 98</cell><cell cols="1" rows="1" role="data">gm 57</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">ch  62</cell><cell cols="1" rows="1" role="data">di 84</cell><cell cols="1" rows="1" role="data">ek 70</cell><cell cols="1" rows="1" role="data">fl 98</cell><cell cols="1" rows="1" role="data">gm 57</cell><cell cols="1" rows="1" role="data">Bn 91</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" role="data">154</cell><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" role="data">145</cell><cell cols="1" rows="1" role="data">148</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">270 </cell><cell cols="1" rows="1" role="data">cd 175</cell><cell cols="1" rows="1" role="data">de 120</cell><cell cols="1" rows="1" role="data">ef 170</cell><cell cols="1" rows="1" role="data">fg 124</cell><cell cols="1" rows="1" role="data">gB 151</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2790</cell><cell cols="1" rows="1" role="data">730</cell><cell cols="1" rows="1" role="data">18480</cell><cell cols="1" rows="1" role="data">11760</cell><cell cols="1" rows="1" role="data">580</cell><cell cols="1" rows="1" role="data">148</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1022 </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">168  </cell><cell cols="1" rows="1" role="data">290 </cell><cell cols="1" rows="1" role="data">740 </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">146  </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">145  </cell><cell cols="1" rows="1" role="data">148  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">28560</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">25550</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">17980</cell><cell cols="1" rows="1" role="data">22348</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2790</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">25550</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">18480</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">28560</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">17980</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">22348</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2 )</cell><cell cols="1" rows="1" role="data">1.15708</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">ac r p</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">.57854</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Content 0 2 12</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2.31416</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12.56640</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> <cb/></p><p>4th. Sometimes such pieces as that above, are computed by finding a mean breadth, by dividing the sum of the offsets by the number of them, accounting that for one of them where the boundary meets the station line, as at A; then multiply the length AB by that mean breadth.</p><p>Thus: <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">00</cell><cell cols="1" rows="1" role="data">785</cell><cell cols="1" rows="1" rend="align=left" role="data">AB</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" rend="align=left" role="data">mean breadth</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">4710</cell><cell cols="1" rows="1" rend="align=left" role="data">       ac r p</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" role="data">4710 </cell><cell cols="1" rows="1" rend="align=left" role="data">Content 0   2   2 by this method,</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">which is 10 perches too little.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data">.51810</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7) 462</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">2.07240</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2.89600</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>But this method is always erroneous, except when the offsets stand at equal distances from one another.</p><p>5th. But in larger pieces, and whole estates, consisting of many fields, it is the common practice to make a rough plan of the whole, and from it compute the contents quite independent of the measures of the lines and angles that were taken in Surveying. For then new lines are drawn in the fields in the plan, so as to divide them into trapeziums and triangles, the bases and perpendiculars of which are measured on the plan by means of the scale from which it was drawn, and so multiplied together for the contents. In this way the work is very expeditiously done, and sufficiently correct; for such dimensions are taken, as afford the most easy method of calculation; and, among a number of parts, thus taken and applied to a scale, it is likely that some of the parts will be taken a small matter too little, and others too great; so that they will, upon the whole, in all probability, very nearly balance one another. After all the fields, and particular parts, are thus computed separately, and added all together into one sum, calculate the whole estate independent of the fields, by dividing it into large and arbitrary triangles and trapeziums, and add these also together. Then if this sum be equal to the former, or nearly so, the work is right; but if the sums have any considerable difference, it is wrong, and they must be examined, and recomputed, till they nearly agree.</p><p>A specimen of dividing into one triangle, or one trapezium, which will do for most single fields, may be seen in the examples to the last article; and a specimen of dividing a large tract into several such trapeziums and triangles, in article 9, where a piece is so divided, and its dimensions taken and set down; and again in articles 15, 16.</p><p>6th. But the chief secret in casting up, consists in finding the contents of pieces bounded by curved, or very irregular lines, or in reducing such crooked sides of fields or boundaries to straight lines, that shall inclose the same or equal area with those crooked sides, and so obtain the area of the curved figure by means of the right-lined one, which will commonly be a trape- <pb n="554"/><cb/> zium. Now this reducing the crooked sides to straight ones, is very easily and accurately performed thus: Apply the straight edge of a thin, clear piece of lanthorn-horn to the crooked line, which is to be reduced, in such a manner, that the small parts cut off from the crooked figure by it, may be equal to those which are taken in: which equality of the parts included and excluded, you will presently be able to judge of very nicely by a little practice: then with a pencil draw a line by the straight edge of the horn. Do the same by the other sides of the field or figure. So shall you have a straight-sided figure equal to the curved one; the contents of which, being computed as before directed, will be the content of the curved figure proposed.</p><p>Or, instead of the straight edge of the horn, a horsehair may be applied across the crooked sides in the same manner; and the easiest way of using the hair, is to string a small slender bow with it, either of wire, or cane, or whale-bone, or such like slender springy matter; for, the bow keeping it always stretched, it can be easily and neatly applied with one hand, while the other is at liberty to make two marks by the side of it, to draw the straight line by.</p><p><hi rend="italics">Ex.</hi> Thus, let it be required to find the contents of the same figure as in art. 12, to a scale of 4 chains to an inch. <figure/></p><p>Draw the four dotted straight lines AB, BC, CD, DA, cutting off equal quantities on both sides of them, which they do as near as the eye can judge: so is the crooked figure reduced to an equivalent right-lined one of four sides ABCD. Then draw the diagonal BD, which, by applying a proper scale to it, measures 1256. Also the perpendicular, or nearest distance, from A to this diagonal, measures 456; and the distance of C from it, is 428. Then <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">456</cell><cell cols="1" rows="1" role="data">2 ) 11.10304</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">428</cell><cell cols="1" rows="1" role="data">5.55152</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">884</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1256</cell><cell cols="1" rows="1" role="data">2.20608</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5024</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10048 </cell><cell cols="1" rows="1" role="data">8.24320</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10048  </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1110304</cell><cell cols="1" rows="1" role="data"/></row></table> <cb/></p><p>And thus the content of the trapezium, and consequently of the irregular figure, to which it is equal, is easily found to be 5 acres, 2 roods, 8 perches. <hi rend="center">21. <hi rend="italics">To Transfer a Plan to another Paper, &c.</hi></hi></p><p>After the rough plan is completed, and a fair one is wanted; this may be done, either on paper or vellum, by any of the following Methods.</p><p><hi rend="italics">First Method.</hi>—Lay the rough plan upon the clean paper, and keep them always pressed flat and close together, by weights laid upon them. Then, with the point of a fine pin or pricker, prick through all the corners of the plan to be copied. Take them asunder, and connect the pricked points on the clean paper, with lines; and it is done. This method is only to be practised in plans of such figures as are small and tolerably regular, or bounded by right lines.</p><p><hi rend="italics">Second Method.</hi>—Rub the back of the rough plan over with black lead powder; and lay the said black part upon the clean paper, upon which the plan is to be copied, and in the proper position. Then, with the blunt point of some hard substance, as brass, or such like, trace over the lines of the whole plan; pressing the tracer so much as that the black lead under the lines may be transferred to the clean paper; after which take off the rough plan, and trace over the leaden marks with common ink, or with Indian ink, &c.—Or, instead of blacking the rough plan, you may keep constantly a blacked paper to lay between the plans.</p><p><hi rend="italics">Third Method.</hi>—Another way of copying plans, is by means of squares. This is performed by dividing both ends and sides of the plan, which is to be copied, into any convenient number of equal parts, and connecting the corresponding points of division with lines; which will divide the plan into a number of small squares. Then divide the paper, upon which the plan is to be copied, into the same number of squares, each equal to the former when the plan is to be copied of the same size, but greater or less than the others, in the proportion in which the plan is to be increased or diminished, when of a different size. Lastly, copy into the clean squares, the parts contained in the corresponding squares of the old plan; and you will have the copy either of the same size, or greater or less in any proportion.</p><p><hi rend="italics">Fourth Method.</hi>—A fourth way is by the instrument called a pentagraph, which also copies the plan in any size required.</p><p><hi rend="italics">Fifth Method.</hi>—But the neatest method of any is this. Procure a copying frame or glass, made in this manner; namely, a large square of the best window glass, set in a broad frame of wood, which can be raised up to any angle, when the lower side of it rests on a table. Set this frame up to any angle before you, facing a strong light; fix the old plan and clean paper together with several pins quite around, to keep them together, the clean paper being laid uppermost, and <pb n="555"/><cb/> upon the face of the plan to be copied. Lay them, with the back of the old plan, upon the glass, namely, that part which you intend to begin at to copy first; and, by means of the light shining through the papers, you will very distinctly perceive every line of the plan through the clean paper. In this state then trace all the lines on the paper with a pencil. Having drawn that part which covers the glass, slide another part over the glass, and copy it in the same manner. And then another part. And so on till the whole be copied.</p><p>Then, take them asunder, and trace all the pencillines over with a sine pen and Indian ink, or with common ink.</p><p>And thus you may copy the finest plan, without injuring it in the least.</p><p>When the lines, &c, are copied upon the clean paper or vellum, the next business is to write such names, remarks, or explanations as may be judged necessary; laying down the scale for taking the lengths of any parts, a flower-de-luce to point out the direction, and the proper title ornamented with a compartment; and illustrating or colouring every part in such manner as shall seem most natural, such as shading rivers or brooks with crooked lines, drawing the representations of trees, bushes, hills, woods, hedges, houses, gates, roads, &c, in their proper places; running a single dotted line for a foot path, and a double one for a carriage road; and either representing the bases or the elevations of buildings, &c. <hi rend="center">22. <hi rend="italics">Of the Division of Lands.</hi></hi></p><p>In the division of commons, after the whole is surveyed and cast up, and the proper quantities to be allowed for roads, &c, deducted, divide the net quantity remaining among the several proprietors, by the rule of Fellowship, in proportion to the real value of their estates, and you will thereby obtain their proportional quantities of the land. But as this division supposes the land, which is to be divided, to be all of an equal goodness, you must observe that if the part in which any one's share is to be marked off, be better or worse than the general mean quality of the land, then you must diminish or augment the quantity of his share in the same proportion.</p><p>Or, which comes to the same thing, divide the ground among the claimants in the direct ratio of the value of their claims, and the inverse ratio of the quality of the ground allotted to each; that is, in proportion to the quotients arising from the division of the value of each person's estate, by the number which expresses the quality of the ground in his share.</p><p>But these regular methods cannot always be put in practice; so that, in the division of commons, the usual way is, to measure separately all the land that is of different values, and add into two sums the contents and the values; then, the value of every claimant's share is found, by dividing the whole value among them in proportion to their estates; and, lastly, by the 24th <cb/> article, a quantity is laid out for each person, that shall be of the value of his share before found.</p><p>23. <hi rend="italics">It is required to divide any given Quantity of Ground, or its Value, into any given Number of Parts, and in Proportion as any given Numbers.</hi></p><p>Divide the given piece, or its value, as in the rule of Fellowship, by dividing the whole content or value by the sum of the numbers expressing the proportions of the several shares, and multiplying the quotient severally by the said proportional numbers for the respective shares required, when the land is all of the same quality. But if the shares be of different qualities, then divide the numbers expressing the proportions or values of the shares, by the numbers which express the qualities of the land in each share; and use the quotients instead of the former proportional numbers.</p><p><hi rend="italics">Ex.</hi> 1. If the total value of a common be 2500 pounds, it is required to determine the values of the shares of the three claimants A, B, C, whose estates are of these values, 10000, and 15000, and 25000 pounds.</p><p>The estates being in proportion as the numbers 2, 3, 5, whose sum is 10, we shall have 2500 ÷ 10 = 250; which being severally multiplied by 2, 3, 5, the products 500, 750, 1250, are the values of the shares required.</p><p><hi rend="italics">Ex.</hi> 2. It is required to divide 300 acres of land among A, B, C, D, E, F, G, and H, whose claims upon it are respectively in proportion as the numbers 1, 2, 3, 5, 8, 10, 15, 20.</p><p>The sum of these proportional numbers is 64, by which dividing 300, the quotient is 4 ac. 2 r. 30 p. which being multiplied by each of the numbers, 1, 2, 3, 5, &c, we obtain for the several shares as below: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Ac.</cell><cell cols="1" rows="1" role="data">R.</cell><cell cols="1" rows="1" role="data">P.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">A</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">C</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">D</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">E</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">00</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">F</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">G</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">00</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Sum</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">00</cell></row></table></p><p><hi rend="italics">Ex.</hi> 3. It is required to divide 780 acres among A, B, and C, whose estates are 1000, 3000, and 4000 pounds a year; the ground in their shares being worth 5, 8, and 10 shillings the acre respectively.</p><p>Here their claims are as 1, 3, 4; and the qualities of their land are as 5, 8, 10; therefore their quantities must be as 1/5, 3/8, 2/5, or, by reduction, as 8, 15, 16. Now the sum of these numbers is 39; by which dividing the 780 acres, the quotient is 20; which being multiplied severally by the three numbers 8, 15, 16, the three products are 160, 300, 320, for the shares of A, B, C, respectively. <pb n="556"/><cb/></p><p>24. <hi rend="italics">To Cut off from a Plan a Given Number of Acres, &c, by a Line drawn from any Point in the Side of it.</hi></p><p>Let A be the given point in the annexed plan, from which a line is to be drawn cutting off suppose 5 ac. 2 r. 14 p. <figure/></p><p>Draw AB cutting off the part ABC as near as can be judged equal to the quantity proposed; and let the true quantity of ABC, when calculated, be only 4 ac. 3 r. 20 p. which is less than 5 ac. 2 r. 14 p. the true quantity, by 0 ac. 2 r. 34 p. or 71250 square links. Then measure AB, which suppose = 1234 links, and divide 71250, by 617 the half of it, and the quotient 115 links will be the altitude of the triangle to be added, and whose base is AB. Therefore if upon the centre B, with the radius 115, an arc be described; and a line be drawn parallel to AB, touching the arc, and cutting BD in D; and if AD be drawn, it will be the line cutting off the required quantity ADCA.</p><p><hi rend="smallcaps">Note.</hi> If the first piece had been too much, then D must have been set below B.</p><p>In this manner the several shares of commons, to be divided, may be laid down upon the plan, and transferred from thence to the ground itself.</p><p>Also for the greater ease and perfection in this business, the following problems may be added.</p><p>25. <hi rend="italics">From an Angle in a Given Triangle, to draw Lines to the opposite Side, dividing the Triangle into any Number of Parts, which shall be in any assigned Proportion to each other.</hi></p><p>Divide the base into the same number of parts, and in the same proportion, by article 22; then from the several points of division draw lines to the proposed angle, and they will divide the triangle as required.— For, the several parts are triangles of the same altitude, and which therefore are as their bases, which bases are taken in the assigned proportion.</p><p><hi rend="italics">Ex.</hi> Let the triangle ABC, of 20 acres, be divided into five parts, which shall be in proportion to the numbers 1, 2, 3, 5, 9; the lines of division to be drawn from A to CB, whose length is 1600 links. <cb/> <figure/></p><p>Here 1 + 2 + 3 + 5 + 9 = 20, and 1600 ÷ 20 = 80; which being multiplied by each of the proportional numbers, we have 80, 160, 240, 400, and 720. Therefore make C<hi rend="italics">a</hi> = 80, <hi rend="italics">ab</hi> = 160, <hi rend="italics">bc</hi> = 240, <hi rend="italics">cd</hi> = 400, and <hi rend="italics">d</hi>B = 720; then by drawing the lines A<hi rend="italics">a,</hi> A<hi rend="italics">b,</hi> A<hi rend="italics">c,</hi> A<hi rend="italics">d,</hi> the triangle is divided as required.</p><p>26. <hi rend="italics">From any Point in one side of a Given Triangle, to draw Lines to the other two Sides, dividing the Triangle into any Number of Parts which shall be in any assigned Ratio.</hi></p><p>From the given point D, draw DB to the angle opposite the side AC in which the point is taken; then divide the same side AC into as many parts AE, EF, FG, GC, and in the same proportion with the required parts of the triangle, like as was done in the last problem; and from the points of division draw lines EK, FI, GH, parallel to the line BD, and meeting the other sides of the triangle in K, I, H; lastly, draw KD, ID, HD, so shall ADK, KDI, IDHB, HDC be the parts required.—The example to this will be done exactly as the last. <figure/></p><p>For, the triangles ADK, KDI, IDB, being of the same height, are as their bases AK, KI, IB; which, by means of the parallels EK, FI, DB, are as AE, EF, FD; in like manner, the triangles CDH, HDB, are to each other as CG, GD: but the two triangles JDB, BDH, having the same base BD, are to each other as the distances of I and H from BD, or as FD to DG; consequently the parts DAK, DKI, DIBH, DHC, are to each other as AE, EF, FG, GC. <hi rend="center"><hi rend="italics">Surveying of Harbours.</hi></hi></p><p>The method of Surveying harbours, and of forming maps of them, as also of the adjacent coasts, sands, &c, depends on the same principles, and is chiefly conducted like that of common Surveying. The operation is indeed more complicated and laborious; as it is necessary to erect a number of signals, and to mark a variety of objects along the coast, with different bearings from one another, and the several parts of the harbour; and likewise to measure a great number of angles <pb n="557"/><cb/> at different stations, whether on the land or the water. For this purpose, the best instrument is Hadley's quadrant, as all these operations may be performed by it, not only with greater ease, but also with much more precision, than can be hoped for by any other means, as it is the only instrument in use, in which neither the exactness of the observations, nor the ease with which they may be made, are sensibly affected by the motion of a vessel: and hence a single observer, in a boat, may generally determine the situation of any place at pleasure, with a sufficient degree of exactness, by taking the angles subtended by several pairs of objects properly chosen upon shores round about him; but it will be still better to have two observers, or the same observer at different stations, to take the like angles to the several objects, and also to the stations. By this means, two angles and one side are given, in every triangle, from whence the situation of every part of them will be known. By such observations, when carefully made with good instruments, the situation of places may be easily determined to 20 or 30 feet, or less, upon every 3 or 4 miles. See Philos. Trans. vol. 55, pa. 70; also Mackenzie's Maritime Surveying.</p><p><hi rend="smallcaps">Surveying</hi> <hi rend="italics">Cross.</hi> See <hi rend="smallcaps">Cross.</hi></p><p><hi rend="smallcaps">Surveying</hi> <hi rend="italics">Quadrant.</hi> See <hi rend="smallcaps">Quadrant.</hi></p><p><hi rend="smallcaps">Surveying</hi> <hi rend="italics">Scale,</hi> the same with Reducing Scale.</p><p><hi rend="smallcaps">Surveying</hi> <hi rend="italics">Wheel.</hi> See <hi rend="smallcaps">Perambulator.</hi></p></div1><div1 part="N" n="SURVIVORSHIP" org="uniform" sample="complete" type="entry"><head>SURVIVORSHIP</head><p>, the doctrine of reversionary payments that depend upon certain contingencies, or contingent circumstances.</p><p>Payments which are not to be made till some future period, are termed <hi rend="italics">reversions,</hi> to distinguish them from payments that are to be made immediately.</p><p>Reversions are either <hi rend="italics">certain</hi> or <hi rend="italics">contingent.</hi> Of the former sort, are all sums or annuities, payable certainly or absolutely at the expiration of any terms, or on the extinction of any lives. And of the latter sort, are all such reversions as depend on any contingency; and particularly the Survivorship of any lives beyond or after other lives. An account of the former may be found under the articles Assurance, Annuities, and Life-annuities. But the latter form the most intricate and difficult part of the doctrine of reversions and lifeannuities; and the books in which this subject is treated most at large, and at the same time with the most precision, are Mr. Simpson's Select Exercises; Dr. Price's Reversionary Payments; and Mr. Morgan's Annuities and Assurances on Lives and Survivorships. The whole likewise of the 3d volume of Dodson's Mathematical Repository is on this subject; but his investigations are founded on De Moivre's false hypothesis, viz of an equal decrement of life through all its stages, and which is explained under Life-annuities: but as this hypothesis does not agree near enough to fact and experience, the rules deduced from it cannot be sufficiently correct. For this reason, Dr. Price, and also the ingenious Mr. Maseres, cursitor baron of the exchequer (in two volumes lately published, entitled the Principles of the Doctrine of Life Annuities), have discarded the valuations of lives grounded upon it; and the former in particular, in order to obviate all occasion for using them, has substituted in their stead, a great variety of new tables of the probabilities and values of lives, at every age and in every situation; calculated, not upon any hypothesis, <cb/> but in strict conformity to the best observations. These tables, added to other new tables of the same kind, in Mr. Baron Maseres's work just mentioned, form a complete set of tables, by which all questions relating to annuities on lives and Survivorships, may be answered with as much correctness as the nature of the subject allows.</p><p>Rules for calculating correctly, in most cases, the values of reversions depending on Survivorships, may be found in the three treatises just mentioned. Mr. Morgan, in particular, has gone a good way towards exhausting this subject, as far as any questions can include in them any Survivorships between two or three lives, either for terms, or the whole duration of the lives.</p><p>There is, however, one circumstance necessary to be attended to in calculating such values, to which no regard could be paid till lately. This circumstance is the shorter duration of the lives of males than of females; and the consequent advantage in favour of females in all cases of Survivorship. In the 4th edition of Dr. Price's Treatise on Reversionary Payments, this fact is not only ascertained, but separate tables of the duration and values of lives are given for males and females.</p></div1><div1 part="N" n="SUSPENSION" org="uniform" sample="complete" type="entry"><head>SUSPENSION</head><p>, in Mechanics, as in a balance, are those points in the axis or beam where the weights are applied, or from which they are suspended.</p><p>SUTTON's <hi rend="italics">Quadrant.</hi> See <hi rend="smallcaps">Quadrant.</hi></p></div1><div1 part="N" n="SWAN" org="uniform" sample="complete" type="entry"><head>SWAN</head><p>, in Astronomy. See <hi rend="smallcaps">Cygnus.</hi></p><p>SWALLOW's-<hi rend="smallcaps">Tail</hi>, in Fortification, is a single Tenaille, which is narrower towards the place than towards the country.</p><p>SWING-<hi rend="italics">Wheel,</hi> in a royal pendulum, is that wheel which drives the pendulum. In a watch, or balance clock, it is called the <hi rend="italics">crown-wheel.</hi></p><p>SYDEREAL <hi rend="italics">Day,</hi> or <hi rend="italics">Year.</hi> See <hi rend="smallcaps">Sidereal.</hi></p></div1><div1 part="N" n="SYMMETRY" org="uniform" sample="complete" type="entry"><head>SYMMETRY</head><p>, the relation of parity, both in respect of length, breadth, and height, of the parts necessary to compose a beautiful whole.</p><p>Symmetry arises from that proportion which the Greeks call <hi rend="italics">analogy,</hi> which is the relation of conformity of all the parts of a building, and of the whole, to some certain measure; upon which depends the nature of Symmetry.</p><p>According to Vitruvius, Symmetry consists in the union and conformity of the several members of a work to their whole, and of the beauty of each of the separate parts to that of the intire work; regard being had to some certain measure: so the body, for instance, is framed with Symmetry, by the due relation which the arm, elbow, hand, singers, &c, have to each other, and to their whole.</p></div1><div1 part="N" n="SYMPHONY" org="uniform" sample="complete" type="entry"><head>SYMPHONY</head><p>, is a consonance or concert of several sounds agreeable to the ear; whether they be vocal or instrumental, or both; called also <hi rend="italics">harmony.</hi></p><p>The Symphony of the Ancients went no farther than to two or more voices or instruments set to unison: for they had no such thing as music in parts; as is very well proved by Perrault: at least, if ever they knew such a thing, it must have been early lost.</p><p>It is to Guido Aretine, about the year 1022, that most writers agree in ascribing the invention of composition: it was he, they say, who first joined in one harmony several distinct melodies; and brought it even to <pb n="558"/><cb/> the length of 4 parts, viz. bass, tenor, counter-tenor, and treble.</p><p>The term Symphony is now applied to instrumental music, both that of pieces designed only for instruments, as sonatas and concertos, and that in which the instruments are accompanied with the voice, as in operas, &c.</p><p>A piece is said to be in grand Symphony, when, besides the bass and treble, it has also two other instrumental parts, viz, tenor and 5th of the violin.</p></div1><div1 part="N" n="SYNCHRONISM" org="uniform" sample="complete" type="entry"><head>SYNCHRONISM</head><p>, the being or happening of several things together, at or in the same time.</p><p>The happening or performing of several things in equal times, as the vibrations of pendulums, &c, is more properly called <hi rend="italics">isochronism:</hi> though some authors confound the two.</p></div1><div1 part="N" n="SYNCOPATION" org="uniform" sample="complete" type="entry"><head>SYNCOPATION</head><p>, in Music, denotes a striking or breaking of the time; by which the distinctness of the several times or parts of the measure is interrupted.</p><div2 part="N" n="Syncopation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Syncopation</hi></head><p>, or <hi rend="smallcaps">Syncope</hi>, is more particularly used for the connecting the last note of one measure or bar with the first of the following measure; so as to make only one note of both.</p><p><hi rend="smallcaps">Syncopation</hi> is also used when a note of one part ends on the middle of a note of the other part. This is otherwise called <hi rend="italics">binding.</hi></p><p>SYNODICAL <hi rend="italics">Month,</hi> is the period or interval of time in which the moon passes from one conjunction with the sun to another. This period is also called a <hi rend="italics">Lunation,</hi> since in this period the moon puts on all her phases, or appearances, as to increase and decrease. —Kepler found the quantity of the mean Synodical month to be 29 days, 12 hrs, 44 min. 3 sec. 11 thirds.</p><p>SYNTHESIS denotes a method of composition, as opposed to analysis.</p><p>In the Synthesis, or synthetic method, we pursue the truth by reasons drawn from principles before established, or assumed, and propositions formerly proved; thus proceeding by a regular chain till we come to the conclusion; and hence called also the <hi rend="italics">direct</hi> method, and <hi rend="italics">composition,</hi> in opposition to analysis or resolution.</p><p>Such is the method in Euclid's Elements, and most demonstrations of the ancient mathematicians, which proceed from definitions and axioms, to prove theorems &c, and from those theorems proved, to demoustrate others. See <hi rend="smallcaps">Analysis.</hi></p><p>SYNTHETICAL <hi rend="italics">Method,</hi> the method by Synthesis, or composition, or the direct method. See <hi rend="smallcaps">Synthesis.</hi></p><p>SYPHON. See <hi rend="smallcaps">Siphon.</hi></p></div2></div1><div1 part="N" n="SYRINGE" org="uniform" sample="complete" type="entry"><head>SYRINGE</head><p>, in Hydraulics, a small simple machine, serving first to imbibe or suck in a quantity of water, or other fluid, and then to squirt or expel the same with violence in a small jet.</p><p>The Syringe is just a small single sucking pump, without a valve, the water ascending in it on the same principle. It consists, like the pump, of a small cylinder, with an embolus or sucker, moving up and down in it by means of a handle, and fitting it very close within. At the lower end is either a small hole, or a smaller tube fixed to it than the body of the instrument, through which the fluid or the water is drawn up, and squirted out again. <cb/></p><p>Thus, the embolus being first pushed close down, introduce the lower end of the pipe into the fluid, then draw up, by the handle, the sucker, and the fluid will immediately follow, so as to fill the whole tube of the Syringe, and will remain there, even when the pipe is taken out of the fluid; but by thrusting forward the embolus, it will drive the water before it; and, being partly impeded by the smallness of the hole, or pipe, it will hence be expelled in a smart jet or squirt, and to the greater distance, as the sucker is pushed down with the greater force, or the greater velocity.</p><p>This ascent of the water the Ancients, who supposed a plenum, attributed to Nature's abhorrence of a vacuum; but the Moderns, more reasonably, as well as more intelligibly, attribute it to the pressure of the atmosphere on the exterior surface of the fluid. For, by drawing up the embolus, the cavity of the cylinder would become a vacuum, or the air left there extremely rarefied; so that being no longer a counterbalance to the air incumbent on the surface of the fluid, this prevails, and forces the water through the little tube, or hole, up into the body of the Syringe.</p></div1><div1 part="N" n="SYSTEM" org="uniform" sample="complete" type="entry"><head>SYSTEM</head><p>, in a general Sense, denotes an assemblage or chain of principles and conclusions: or the whole of any doctrine, the several parts of which are bound together, and follow or depend on each other. As a System of astronomy, a System of planets, a System of philosophy, a System of motion, &c.</p><div2 part="N" n="System" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">System</hi></head><p>, in Astronomy, denotes an hypothesis or a supposition of a certain order and arrangement of the several parts of the universe; by which astronomers explain all the phenomena or appearances of the heavenly bodies, their motions, changes, &c.</p><p>This is more peculiarly called the <hi rend="italics">System of the world,</hi> and sometimes the <hi rend="italics">Solar System.</hi></p><p>System and hypothesis have much the same signification; unless perhaps hypothesis be a more particular System, and System a more general hypothesis.</p><p>Some late authors indeed make another distinction: an hypothesis, say they, is a mere supposition or fiction, founded rather on imagination than reason; while a System is built on the firmest ground, and raised by the severest rules; it is founded on astronomical observations, and physical causes, and confirmed by geometrical demonstrations.</p><p>The most celebrated Systems of the world, are the Ptolomaic, the Copernican or Pythagorean, and the Tychonic: the economy of each of which is as follows.</p><p><hi rend="italics">Ptolomaic</hi> <hi rend="smallcaps">System</hi> is so called from the celebrated astronomer Ptolomy. In this System, the earth is placed at rest, in the centre of the universe, while the heavens are considered as revolving about it, from east to west, and carrying along with them all the heavenly bodies, the stars and planets, in the space of 24 hours.</p><p>The principal assertors of this System, are Aristotle, Hipparchus, Ptolomy, and many of the old philosophers, followed by the whole world, for a great number of ages, and long adhered to in many universities, and other places. But the late improvements in philosophy and reasoning, have utterly exploded this erroneous System from the place it so long held in the minds of men.</p><p><hi rend="italics">Copernican</hi> <hi rend="smallcaps">System</hi>, is that System of the world <pb/><pb/><pb/><pb/><pb/><pb/><pb n="559"/><cb/> which places the Sun at rest, in the centre of the world, and the earth and planets all revolving round him, in their several orbits. See this more particularly explained under the article <hi rend="smallcaps">Copernican</hi> <hi rend="italics">System.</hi></p><p><hi rend="italics">Solar</hi> or <hi rend="italics">Planetary</hi> <hi rend="smallcaps">System</hi>, is usually confined to narrower bounds; the stars, by their immense distance, and the little relation they seem to bear to us, being accounted no part of it. It is highly probable that each fixed star is itself a Sun, and the centre of a particular System, surrounded with a company of planets &c, which, in different periods, and at different distances, perform their courses round their respective sun, which enlightens, warms, and cherishes them. Hence we have a very magnificent idea of the world, and the immensity of it. Hence also arises a kind of System of Systems.</p><p>The Planetary System, described under the article <hi rend="smallcaps">Copernican</hi>, is the most ancient in the world. It was first of all, as far as we know, introduced into Greece and Italy by Pythagoras; from whom it was called the Pythagorean System. It was followed by Philolaus, Plato, Archimedes, &c: but it was lost under the reign of the Peripatetic philosophy; till happily retrieved about the year 1500 by Nic. Copernicus.</p><p><hi rend="italics">Tychonic</hi> <hi rend="smallcaps">System</hi>, was taught by Tycho, a Dane; who was born An. Dom. 1546. It supposes that the earth is fixed in the centre of the universe or firmament of stars, and that all the stars and planets revolve round the earth in 24 hours; but it differs from the Ptolomaic System, as it not only allows a menstrual motion to the moon round the earth, and that of the satellites about Jupiter and Saturn, in their proper periods, but it makes the sun to be the centre of the orbits of the primary planets Mercury. Venus, Mars, Jupiter, &c, in which they are carried round the sun in their respective years, as the sun revolves round the earth in a solar year; and all these planets, together with the sun, are supposed to revolve round the earth in 24 hours. This hypothesis was so embarrassed and perplexed, that very few persons embraced it. It was afterwards altered by Longomontanus and others, who allowed the diurnal motion of the earth on its own axis, but denied its annual motion round the sun. This hypothesis, partly true and partly false, is called the <hi rend="italics">Semi-Tychonic System.</hi> See the figure and economy of these Systems, in plates 30, 31, 32, 33. <cb/></p></div2><div2 part="N" n="System" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">System</hi></head><p>, in Music, denotes a compound interval; or an interval composed, or conceived to be composed of several less intervals. Such is the octave, &c.</p></div2></div1><div1 part="N" n="SYSTYLE" org="uniform" sample="complete" type="entry"><head>SYSTYLE</head><p>, in Architecture, the manner of placing columns, where the space between the two fusts consists of 2 diameters, or 4 modules.</p></div1><div1 part="N" n="SYZYGY" org="uniform" sample="complete" type="entry"><head>SYZYGY</head><p>, a term equally used for the conjunction and opposition of a planet with the sun.</p><p>On the phenomena and circumstances of the Syzygies, a great part of the lunar theory depends. See <hi rend="smallcaps">Moon.</hi> For,</p><p>1. It is shewn in the physical astronomy, that the force which diminishes the gravity of the moon in the Syzygies, is double that which increases it in the quadratures; so that, in the Syzygies, the gravity of the moon is diminished by a part which is to the whole gravity, as 1 to 89.36; for in the quadratures, the addition of gravity is to the whole gravity, as 1 to 178.73.</p><p>2. In the Syzygies, the disturbing force is directly as the distance of the moon from the earth, and inversely as the cube of the distance of the earth from the sun. And at the Syzygies, the gravity of the moon towards the earth receding from its centre, is more diminished than according to the inverse ratio of the square of the distance from that centre.—Hence, in the moon's motion from the Syzygies to the quadratures, the gravity of the moon towards the earth is continually increased, and the moon is continually retarded in her motion; but in the moon's motion from the quadratures to the Syzygies, her gravity is continually diminished, and the motion in her orbit is accelerated.</p><p>3. Farther, in the Syzygies, the moon's orbit, or circuit round the earth, is more convex than in the quadratures; for which reason she is less distant from the earth at the former than the latter.—Also, when the moon is in the Syzygies, her apses go backward, or are retrograde.—Moreover, when the moon is in the Syzygies, the nodes move in antecedentia fastest; then slower and slower, till they become at rest when the moon is in the quadratures.—Lastly, when the nodes are come to the Syzygies, the inclination of the plane of the orbit is the least of all.</p><p>However, these several irregularities are not equal in each Syzygy, being all somewhat greater in the conjunction than in the opposition. </p></div1></div0><div0 part="N" n="T" org="uniform" sample="complete" type="alphabetic letter"><head>T</head><cb/><div1 part="N" n="TABLE" org="uniform" sample="complete" type="entry"><head>TABLE</head><p>, in Architecture, a smooth, simple member or ornament, of various forms, but most commonly in that of a parallelogram.</p><div2 part="N" n="Table" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Table</hi></head><p>, in Perspective, is sometimes used for the <cb/> perspective plane, or the transparent plane upon which the objects are formed in their respective appearance.</p><p><hi rend="smallcaps">Table</hi> <hi rend="italics">of Pythagoras,</hi> is the same as the <hi rend="smallcaps">Multipli-</hi> <pb n="560"/><cb/> <hi rend="smallcaps">CATION</hi> Table; which see; as also <hi rend="smallcaps">Pythagoras</hi>'s Table.</p><p><hi rend="smallcaps">Tables</hi> <hi rend="italics">of Houses,</hi> among astrologers, are certain Tables, ready drawn up, for the assistance of practitioners in that a<*>t, for the erecting or drawing of figures or schemes. See <hi rend="smallcaps">House.</hi></p></div2><div2 part="N" n="Tables" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tables</hi></head><p>, in Mathematics, are systems or series of numbers, calculated to be ready at hand for expediting any sort of calculations in the various branches of mathematics.</p><p><hi rend="italics">Astronomical</hi> <hi rend="smallcaps">Tables</hi>, are computations of the motions, places, and other phenomena of the planets, both primary and secondary.</p><p>The oldest astronomical Tables, now extant, are those of Ptolomy, found in his Almagest. These however are not now of much use, as they no longer agree with the motions of the heavens.</p><p>In 1252, Alphonso XI, king of Castile, undertook the correcting of them, chiefly by the assistance of Isaac Hazen, a learned Jew; and spent 400,000 crowns on the business. Thus arose the <hi rend="italics">Alphonsine Tables,</hi> to which that prince himself prefixed a preface. But the deficiency of these also was soon perceived by Purbach and Muller, or Regiomontanus; upon which the latter, and after him Walther Warner, applied themselves to celestial observations, for farther improving them; but death, or various difficulties, prevented the effect of these good designs.</p><p>Copernicus, in his books of the celestial revolutions, gives other Tables, calculated by himself, partly from his own observations, and partly from the Alphonsine Tables.</p><p>From Copernicus's observations and theorems, Erasmus Reinhold afterwards compiled the <hi rend="italics">Prutenic Tables,</hi> which have been printed several times, and in several places.</p><p>Tycho Brahe, even in his youth, became sensible of the deficiency of the Prutenic Tables: which determined him to apply himself with so much vigour to celestial observations. From these he adjusted the motions of the sun and moon; and Longomontanus, from the same observations, made out Tables of the motions of the planets, which he added to the Theories of the same, published in his Astronomia Danica; those being called the <hi rend="italics">Danish Tables.</hi> And Kepler also, from the same observations, published in 1627 his <hi rend="italics">Rudolphine Tables,</hi> which are much esteemed.</p><p>These were afterwards, viz in 1650, changed into another form, by Maria Cunitia, whose Astronomical Tables, comprehending the effect of Kepler's physical hypothesis, are very easy, satisfying all the phenomena without any mention of logarithms, and with little or no trouble of calculation. So that the Rudolphine calculus is here greatly improved.</p><p>Mercator made a like attempt in his Astronomical Institution, published in 1676. And the like did J. Bap. Morini, whose abridgment of the Rudolphine Tables was prefixed to a Latin version of Street's Astronomia Carolina, published in 1705.</p><p>Lansbergius indeed endeavoured to discredit the Rudolphine Tables, and framed <hi rend="italics">Perpetual Tables,</hi> as he calls them, of the heavenly motions. But his attempt was never much regarded by the astronomers; and our <cb/> countryman Horrox warmly attacked him, in his defence of the Keplerian astronomy.</p><p>Since the Rudolphine Tables, many others have been framed, and published: as the <hi rend="italics">Philolaic Tables</hi> of Bulliald; the <hi rend="italics">Britannic Tables</hi> of Vincent Wing, calculated on Bulliald's hypothesis; the <hi rend="italics">Britannic Tables</hi> of John Newton; the French ones of the Count Pagan; the <hi rend="italics">Caroline Tables</hi> of Street, all calculated on Ward's hypothesis; and the <hi rend="italics">Novalmajestic Tables</hi> of Riccioli. Among these, however, the Philolaic and Caroline Tables are esteemed the best; insomuch that Mr. Whiston, by the advice of Mr. Flamsteed, thought fit to subjoin the Caroline Tables to his astronomical lectures.</p><p>The <hi rend="italics">Ludovician Tables,</hi> published in 1702, by De la Hire, were constructed wholly from his own observations, and without the assistance of any hypothesis; which, before the invention of the micrometer telescope and the pendulum clock, was held impossible.</p><p>Dr. Halley also long laboured to perfect another set of Tables; which were printed in 1719, but not published till 1752.</p><p>M. Monnier, in 1746, published, in his Institutions Astronomiques, Tables of the motions of the sun and moon, with the satellites, as also of refractions, and the places of the fixed stars. La Hire also published Tables of the planets, and La Caille Tables of the sun: Gael Morris published Tables of the sun and moon, and Mayer constructed Tables of the moon, which were published by the Board of Longitude. Tables of the same have also been computed by Charles Mason, from the principles of the Newtonian philosophy, which are found to be very accurate, and are employed in computing the Nautical Ephemeris. Many other sets of astronomical Tables have also been published by various persons and academies; and divers sets of them may be found in the modern books of astronomy, navigation, &c, of which those are esteemed the best and most complete, that are printed in Lalande's Astronomy. For an account of several, and especially of those published annually under the direction of the Commissioners of Longitude, see <hi rend="smallcaps">Almanac, Ephemeris</hi>, and L<hi rend="smallcaps">ONGITUDE.</hi></p><p><hi rend="italics">For</hi> <hi rend="smallcaps">Tables</hi> <hi rend="italics">of the Stars,</hi> see <hi rend="smallcaps">Catalogue.</hi></p><p><hi rend="smallcaps">Tables</hi> of <hi rend="italics">Sines, Tangents,</hi> and <hi rend="italics">Secants,</hi> used in trigonometry, &c, are usually called <hi rend="smallcaps">Canons.</hi> See <hi rend="smallcaps">Sine.</hi></p><p><hi rend="smallcaps">Tables</hi> <hi rend="italics">of Logarithms, Rhumbs, &c,</hi> used in geometry, navigation, &c, see <hi rend="smallcaps">Logarithm</hi>, and <hi rend="smallcaps">Rhumb.</hi></p></div2><div2 part="N" n="Tables" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tables</hi></head><p>, <hi rend="italics">Loxodromic,</hi> and <hi rend="italics">of Difference of Latitude and Departure,</hi> are Tables used in computing the way and reckoning of a ship on a voyage, and are published in most books of navigation.</p></div2></div1><div1 part="N" n="TACQUET" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">TACQUET</surname> (<foreName full="yes"><hi rend="smallcaps">Andrew</hi></foreName>)</persName></head><p>, a Jesuit of Antwerp, who died in 1660. He was a most laborious and voluminous writer in mathematics. His works were collected, and printed at Antwerp in one large volume in folio, 1669.</p></div1><div1 part="N" n="TACTION" org="uniform" sample="complete" type="entry"><head>TACTION</head><p>, in Geometry, the same as tangency, or touching. See <hi rend="smallcaps">Tangent.</hi></p></div1><div1 part="N" n="TALUS" org="uniform" sample="complete" type="entry"><head>TALUS</head><p>, or <hi rend="smallcaps">Talud</hi>, in Architecture, the inclination or slope of a work; as of the outside of a wall, when its thickness is diminished by degrees, as it rises in height, to make it the firmer. <pb n="561"/><cb/></p><div2 part="N" n="Talus" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Talus</hi></head><p>, in Fortification, means also the slope of a work, whether of earth or masonry.</p><p>The <hi rend="italics">Exterior Talus</hi> of a work, is its slope on the side outwards or towards the country; which is always made as little as possible, to prevent the enemy's escalade, unless the earth be bad, for then it is necessary to allow a considerable Talus for its parapet, and sometimes to support the earth with a slight wall, called a revetement.</p><p>The <hi rend="italics">Interior Talus</hi> of a work, is its slope on the inside, towards the place. This is larger than the former, and it has, at the angles of the gorge, and sometimes in the middle of the curtains, ramps, or sloping roads for mounting upon the terreplain of the rampart.</p><p><hi rend="italics">Superior</hi> <hi rend="smallcaps">Talus</hi> <hi rend="italics">of the Parapet,</hi> is a slope on the top of the parapet, that allows of the soldiers defending the covert-way with small-shot, which they could not do if it were level.</p></div2></div1><div1 part="N" n="TAMBOUR" org="uniform" sample="complete" type="entry"><head>TAMBOUR</head><p>, in Architecture, a term applied to the Corinthian and Composite capitals, as bearing some resemblance to a tambour or drum.</p></div1><div1 part="N" n="TAMUZ" org="uniform" sample="complete" type="entry"><head>TAMUZ</head><p>, in Chronology, the 4th month of the Jewish ecclesiastical year, answering to part of our June and July. The 17th day of this month is observed by the Jews as a fast, in memory of the destruction of Jerusalem by Nebuchadnezzar, in the 11th year of Zedekiah, and the 588th before Christ.</p></div1><div1 part="N" n="TANGENT" org="uniform" sample="complete" type="entry"><head>TANGENT</head><p>, in Geometry, is a line that touches a curve, &c, that is, which meets it in a point without cutting it there, though it be produced both ways; as the Tangent AB of the circle <figure/> BD. The point B, where the Tangent touches the curve, is called the <hi rend="italics">point of contact.</hi></p><p>The direction of a curve at the point of contact, is the same as the direction of the Tangent.</p><p>It is demonstrated in Geometry;</p><p>1. That a Tangent to a circle, as AB, is perpendicular to the radius BC drawn to the point of contact.</p><p>2. The Tangent AB is a mean proportional between AF and AE, the whole secant and the external part of it; and the same for any other secant drawn from the same point A.</p><p>3. The two Tangents AB and AD, drawn from the same point A, are always equal to one another. And therefore also, if a number of Tangents be drawn to different points of the curve quite around, and an equal length BA be set off upon each of them from the points of contact, the locus of all the points A will be a circle having the same centre C.</p><p>4. The angle of contact ABE, formed at the point of contact, between the Tangent AB and the arc BE, is less than any rectilineal angle.</p><p>5. The Tangent of an arc is the right line that limits the position of all the secants that can pass through the point of contact; though strictly speaking it is not one of the secants, but only the limit of them.</p><p>6. As a right line is the Tangent of a circle, when it touches the circle so closely, that no right line can be drawn through the point of contact between it and <cb/> the arc, or within the angle of contact that is formed by them; so, in general, when any right line touches an arc of any curve, in such a manner, that no right line can be drawn through the point of contact, between the right line and the arc, or within the angle of contact that is formed by them, then is that line the Tangent of the curve at the said point; as AB. <figure/></p><p>7. In all the conic sections; if C be the centre of the figure, and BG an ordinate drawn from the point of contact and perpendicular to <figure/> the axis; then is CG : CE :: CE : CA, or the semiaxis CE is a mean proportional between CG and CA.</p><div2 part="N" n="Tangent" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tangent</hi></head><p>, in Trigonometry. A <hi rend="smallcaps">Tangent</hi> <hi rend="italics">of an arc,</hi> is a right line drawn touching one extremity of the arc, and limited by a secant or line drawn through the centre and the other extremity of the arc. So, AG is the Tangent of the arc AB, or of the arc ABD; and AH is the Tangent of the arc AI, or of the arc AIDK.</p><p>The same are also the Tangents of the angles that are subtended or measured by the arcs.</p><p>Hence, 1. The Tangents in the 1st and 3d quadrants are positive, in the 2d and 4th negative, or drawn the contrary way. But of 0 or 180° the semicircle, the Tangent is 0 or nothing; while those of 90° or a quadrant, and 270° or 3 quadrants, are both infinite; the former infinitely positive, and the latter infinitely negative. That is, Between 0 and 90°, or bet. 180° and 270°, the Tangents are positive. Bet. 90° and 180°, or bet. 270° and 360°, the Tangents are negative.</p><p>2. The Tangent of an arc and the Tangent of its supplement, are equal, but of contrary affections, the one being positive, and the other negative; as of <hi rend="italics">a</hi> and 180° - <hi rend="italics">a,</hi> where <hi rend="italics">a</hi> is any arc. <table><row role="data"><cell cols="1" rows="1" role="data">Also</cell><cell cols="1" rows="1" rend="align=right" role="data">180° + <hi rend="italics">a</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" role="data">have the same Tangent, and of the</cell></row><row role="data"><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">  same affection.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Or</cell><cell cols="1" rows="1" rend="align=right" role="data">180° + <hi rend="italics">a</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" role="data">have the same Tangent, but of</cell></row><row role="data"><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" rend="align=right" role="data">180° - <hi rend="italics">a</hi></cell><cell cols="1" rows="1" role="data">  different affections.</cell></row></table></p><p>3. The Tangent of an arc is a 4th proportional to the cosine the sine and the radius; that is, CN : NB :: CA : AG. Hence, a canon of sines being made or given, the canon of Tangents is easily constructed from them.</p><p><hi rend="italics">Co</hi>-<hi rend="smallcaps">Tangent</hi>, contracted from complement-tangent, is the Tangent of the complement of the arc or angle, or of what it wants of a quadrant or 90°. So LM is the Cotangent of the arc AB, being the Tangent of its complement BL.</p><p>The Tangent is reciprocally as the cotangent; or the <pb n="562"/><cb/> Tangent and cotangent are reciprocally proportional with the radius. That is Tang. is as 1/cotan., or Tang. : radius :: radius : cotan. And the rectangle of the Tangent and cotangent is equal to the square of the radius; that is, Tan. X cot. = radius<hi rend="sup">2</hi>.</p><p><hi rend="italics">Artificial</hi> <hi rend="smallcaps">Tangents</hi>, or <hi rend="italics">logarithmic</hi> <hi rend="smallcaps">Tangents</hi>, are the logarithms of the tangents of arcs; so called, in contradistinction from the natural Tangents, or the Tangents expressed by the natural numbers.</p><p><hi rend="italics">Line of</hi> <hi rend="smallcaps">Tangents</hi>, is a line usually placed on the sector, and Gunter's scale; the description and uses of which see under the article <hi rend="smallcaps">Sector.</hi></p><p><hi rend="italics">Sub</hi>-<hi rend="smallcaps">Tangent</hi>, a line lying beneath the Tangent, being the part of the axis intercepted by the Tangent and the ordinate to the point of contact; as the line AG in the 2d and 3d figures above.</p><p><hi rend="italics">Method of</hi> <hi rend="smallcaps">Tangents</hi>, is a method of determining the quantity of the Tangent and subtangent of any algebraic curve; the equation of the curve being given.</p><p>This method is one of the great results of the doctrine of fluxions. It is of great use in Geometry; because that in determining the Tangents of curves, we determine at the same time the quadrature of the curvilinear spaces: on which account it deserves to be here particularly treated on. <hi rend="center"><hi rend="italics">To Draw the Tangent, or to find the Subtangent,</hi> of a curve.</hi></p><p>If AE be any curve, and E <figure/> any point in it, to which it is required to draw a Tangent TE. Draw the ordinate DE: then if we can determine the subtangent TD, by joining the points T and E, the line TE will be the Tangent sought.</p><p>Let <hi rend="italics">dae</hi> be another ordinate indefinitely near to DE, meeting the curve, or Tangent produced, in <hi rend="italics">e;</hi> and let E<hi rend="italics">a</hi> be parallel to the axis AD. Then is the elementary triangle E<hi rend="italics">ae</hi> similar to the triangle TDE; <table><row role="data"><cell cols="1" rows="1" role="data">and therefore</cell><cell cols="1" rows="1" role="data"><hi rend="italics">ea</hi> : <hi rend="italics">a</hi>E :: ED : DS;</cell></row><row role="data"><cell cols="1" rows="1" role="data">but</cell><cell cols="1" rows="1" role="data"><hi rend="italics">ea</hi> : <hi rend="italics">a</hi>E :: flux. ED : flux. AD;</cell></row><row role="data"><cell cols="1" rows="1" role="data">therefore</cell><cell cols="1" rows="1" role="data">flux. ED : flux. AD :: DE : DT;</cell></row><row role="data"><cell cols="1" rows="1" role="data">that is</cell><cell cols="1" rows="1" role="data">,</cell></row></table> which is therefore the value of the subtangent sought; where <hi rend="italics">x</hi> is the absciss AD, and <hi rend="italics">y</hi> the ordinate DE.</p><p>Hence we have this general rule: By means of the given equation of the curve, find the value either of <hi rend="italics">x</hi><hi rend="sup">.</hi> or <hi rend="italics">y</hi><hi rend="sup">.</hi>, or of <hi rend="italics">x</hi><hi rend="sup">.</hi>/<hi rend="italics">y</hi><hi rend="sup">.</hi>, which value substitute for it in the expression , and, when reduced to its simplest terms, it will be the value of the subtangent sought. This we may illustrate in the following examples.</p><p><hi rend="italics">Ex.</hi> 1. The equation defining a circle is , where <hi rend="italics">a</hi> is the radius; and the fluxion of this is ; hence ; this multi- <cb/> plied by <hi rend="italics">y,</hi> gives , which is a property of the circle we also know from common geometry.</p><p><hi rend="italics">Ex.</hi> 2. The equation defining the common parabola is , <hi rend="italics">a</hi> being the parameter, and <hi rend="italics">x</hi> and <hi rend="italics">y</hi> the absciss and ordinate in all cases. The fluxion of this is ; hence ; conseq. ; that is, the subtangent TD is double the absciss AD, or TA is = AD, which is a well-known property of the parabola.</p><p><hi rend="italics">Ex.</hi> 3. The equation defining an ellipsis is , where <hi rend="italics">a</hi> and <hi rend="italics">c</hi> are the semiaxes. The fluxion of it is ; hence the subtangent; or by adding CD which is = <hi rend="italics">a</hi> - <hi rend="italics">x,</hi> it becomes , a well-known property of the ellipse.</p><p><hi rend="italics">Ex.</hi> 4. The equation defining the hyperbola is , which is similar to that for the ellipse, having only + <hi rend="italics">x</hi><hi rend="sup">2</hi> for - <hi rend="italics">x</hi><hi rend="sup">2</hi>; hence the conclusion is exactly similar also, viz, .</p><p>And so on, for the Tangents to other curves.</p><p><hi rend="italics">The Inverse Method of</hi> <hi rend="smallcaps">Tangents.</hi> This is the reverse of the foregoing, and consists in finding the nature of the curve that has a given subtangent. The method of solution is to put the given subtangent equal to the general expression (<hi rend="italics">yx</hi><hi rend="sup">.</hi>)/<hi rend="italics">y</hi><hi rend="sup">.</hi>, which serves for all sorts of curves; then the equation reduced, and the fluenta taken, will give the fluential equation of the curve sought.</p><p><hi rend="italics">Ex.</hi> 1. To find the curve line whose subtangent is = (2<hi rend="italics">y</hi><hi rend="sup">2</hi>)/<hi rend="italics">a.</hi> Here , and the fluents of this give , the equation to a parabola, which therefore is the curve sought.</p><p><hi rend="italics">Ex.</hi> 2. To find the curve whose subtangent is = <hi rend="italics">yy</hi>/(2<hi rend="italics">a</hi> - <hi rend="italics">x</hi>), or a third proportional to 2<hi rend="italics">a</hi> - <hi rend="italics">x</hi> and <hi rend="italics">y.</hi> Here , the fluents of which give , the equation to a circle, which therefore is the curve sought. <pb n="563"/><cb/></p><p>TANTALUS's <hi rend="italics">Cup,</hi> in Hydraulics, is a cup, as A, with a hole in the bottom, and <figure/> the longer leg of a syphon BCED cemented into the hole; so that the end D of the shorter leg DE may always touch the bottom of the cup within. Then, if water be poured into this cup, it will rise in the shorter leg by its upward pressure, extruding the air before it through the longer leg, and when the cup is filled above the bend of the syphon at E, the pressure of the water in the cup will force it over the bend; from whence it will descend in the longer leg EB, and through the bottom at G, till the cup be quite emptied. The legs of this syphon are almost close together, and it is sometimes concealed by a small hollow statue, or figure of a man placed over it; the bend E being within the neck of the figure as high as the chin. So that poor thirsty Tantalus stands up to the chin in water, according to the fable, imagining it will rise a little higher, as more water is poured in, and he may drink; but instead of that, when the water comes up to his chin, it immediately begins to descend, and therefore, as he cannot stoop to follow it, he is lest as much tormented with thirst as ever. Ferguson's Lect. p. 72, 4to.</p></div2></div1><div1 part="N" n="TARRANTIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">TARRANTIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Lucius</hi></foreName>)</persName></head><p>, surnamed <hi rend="italics">Firmanus,</hi> because he was a native of Firmum, a town in Italy, flourished at the same time with Cicero, and was one of his friends. He was a mathematical philosopher, and therefore was thought to have great skill in judicial astrology. He was particularly famous by two horoscopes which he drew, the one the horoscope of Romulus, and the other of Rome. Plutarch says, “Varro, who was the most learned of the Romans in history, had a particular friend named Tarrantius, who, out of curiosity, applied himself to draw horoscopes, by means of astronomical tables, and was esteemed the most eminent in his time.” Historians controvert some particular circumstances of his calculations; but all agree in conferring on him the honorary title <hi rend="italics">Prince of astrologers.</hi></p></div1><div1 part="N" n="TARTAGLIA" org="uniform" sample="complete" type="entry"><head>TARTAGLIA</head><p>, or <hi rend="smallcaps">Tartalea (Nicholas</hi>), a noted mathematician who was born at Brescia in Italy, probably towards the conclusion of the 15th century, as we find he was a considerable master or preceptor in mathematics in the year 1521, when the first of his collection of questions and answers was written, which he afterwards published in the year 1546, under the title of <hi rend="italics">Quesiti et Inventioni diverse,</hi> at Venice, where he then resided as a public lecturer on mathematics, he having removed to this place about the year 1534. This work consists of 9 chapters, containing answers to a number of questions on all the different branches of mathematics and philosophy then in vogue. The last or 9th of these, contains the questions in Algebra, among which are those celebrated letters and communications between Tartalea and Cardan, by which our author put the latter in possession of the rules for cubic equations, which he first discovered in the year 1530.</p><p>But the first work of Tartalea's that was published, was his <hi rend="italics">Nova Scientia inventa,</hi> in 4to, at Venice in <cb/> 1537. This is a treatise on the theory and practice of gunnery, and the first of the kind, he being the first writer on the flight and path of balls and shells. This work was translated into English, by Lucar, and printed at London in 1588, in folio, with many notes and additions by the translator.</p><p>Tartalea published at Venice, in folio, 1543, the whole books of Euclid, accompanied with many curious notes and commentaries.</p><p>But the last and chief work of Tartalea, was his <hi rend="italics">Trattato di Numeri et Misure,</hi> in folio, 1556 and 1560. This is an universal treatise on arithmetic, algebra, geometry, mensuration, &c. It contains many other curious particulars of the disputes between our author and Cardan, which ended only with the death of Tartalea, before the last part of this work was published, or about the year 1558.</p><p>For many other circumstances concerning Tartalea and his writings, see the article <hi rend="smallcaps">Algebra</hi>, vol. 1, pa. 73.</p></div1><div1 part="N" n="TATIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">TATIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Achilles</hi></foreName>)</persName></head><p>, an ancient Greek writer of Alexandria; but the age he lived in is uncertain. According to Suidas, who calls him Statius, he was at first a Heathen, then a Christian, and afterwards a bishop. He wrote a book upon the Sphere, which seems to have been nothing more than a commentary upon Aratus. Part of it is extant, and was translated into Latin by father Petavius, under the title of <hi rend="italics">Isagoges in Phænomena Arati.</hi> He wrote also, <hi rend="italics">Of the Loves of Cliptophon and Leucippe,</hi> in 8 books. He is well spoken of by Photius.</p></div1><div1 part="N" n="TAURUS" org="uniform" sample="complete" type="entry"><head>TAURUS</head><p>, <hi rend="italics">the Bull,</hi> in Astronomy, one of the 12 signs in the zodiac, and the second in order.</p><p>The Greeks fabled that this was the bull which carried Europa safe across the seas to Crete; and that Jupiter, in reward for so signal a service, placed the creature, whose form he had assumed on that occasion, among the stars, and that this is the constellation formed of it. But it is probable that the Egyptians, or Babylonians, or whoever invented the constellations of the zodiac, placed this figure in that part of it which the sun entered about the time of the bringing forth of calves; like as they placed the ram in the first part of spring, as the lambs appear before them, and the two kids (for that was the original figure of the sign Gemini), afterward, to denote the time of the goats bringing forth their young.</p><p>In the constellation Taurus there are some remarkable stars that have names; as Aldebaran in the south or right eye of the bull, the cluster called the Pleiades in the neck, and the cluster called Hyades in the face.</p><p>The stars in the constellation Taurus, in Ptolomy's catalogue are 44, in Tycho's catalogue 43, in Hevelius's catalogue 51, and in the Britannic catalogue 141.</p></div1><div1 part="N" n="TEBET" org="uniform" sample="complete" type="entry"><head>TEBET</head><p>, or <hi rend="smallcaps">Thevet</hi>, the 4th month of the civil year of the Hebrews, and the 10th of their ecclesiastical year. It answered to part of our December and January, and had only 29 days.</p></div1><div1 part="N" n="TEETH" org="uniform" sample="complete" type="entry"><head>TEETH</head><p>, of various sorts of machines, as of mill wheels, &c. These are often called cogs by the workmen; and by working in the pinions, rounds, or trundles, the wheels are made to turn one another.</p><p>Mr. Emerson (in his Mechanics, prop. 25), treats of the theory of Teeth, and shews that they ought to <pb n="564"/><cb/> have the figure of epicycloids, for properly working in one another. Camus too (in his Cours de Mathematique, tom. 2, p. 349, &c, Edit. 1767) treats more fully on the same subject; and demonstrates that the Teeth of the two wheels should have the figures of epicycloids, but that the generating circles of these epicycloids should have their diameters only the half of what Mr. Emerson makes them.</p><p>Mr. Emerson observes, that the Teeth ought not to act upon one another before they arrive at the line which joins their centres. And though the inner or under sides of the Teeth may be of any form; yet it is better to make them both sides alike, which will serve to make the wheels turn backwards. Also a part may be cut away on the back of every Tooth, to make way for those os the other wheel. And the more Teeth that work together, the better; at least one Tooth should always begin before the other hath done working. The Teeth ought to be disposed in such manner as not to trouble or hinder one another, before they begin to work; and there should be a convenient length, depth and thickness given to them, as well for strength, as that they may more easily disengage themselves.</p></div1><div1 part="N" n="TELEGRAPH" org="uniform" sample="complete" type="entry"><head>TELEGRAPH</head><p>, a machine brought into use by the French nation, in the year 1793, contrived to communicate words or signals from one person to another at a great distance, in a very small space of time.</p><p>The Telegraph it seems was originally the invention of William Amontons, an ingenious philosopher, born in Normandy in the year 1663. See his life in this Dictionary, vol. 1, pa. 105; where it is related that he pointed out a method to acquaint people at a great distance, and in a very little time, with whatever one pleased. This method was as follows: let persons be placed in several stations, at such distances from each other, that, by the help of a telescope, a man in one slation may see a signal made by the next before him: this person immediately repeats the same signal to the third man; and this again to a fourth, and so on through all the stations to the last.</p><p>This, with considerable improvements, it seems has lately been brought into use by the French, and called a Telegraph. It is said they have availed themselves of this contrivance to good purpose, in the present war; and from the utility of the invention, it has also just been brought into use in this country.</p><p>The following account of this curious instrument is copied from Barrere's report in the sitting of the French Convention of August 15, 1794.—“The new-invented telegraphic language of signals is an artful contrivance to transmit thoughts, in a peculiar language, from one distance to another, by means of machines, which are placed at different distances, of from 12 to 15 miles from one another, so that the expression reaches a very distant place in the space of a few minutes. Last year an experiment of this invention was tried in the presence of several Commissioners of the Convention. From the favourable report which the latter made of the efficacy of the contrivance, the Committee of Public Welfare tried every effort to establish, by this means, a correspondence between Paris and the frontier places, beginning with Lisle. Almost a whole twelvemonth has been spent in collecting the necessary instruments for the machines, and to teach the people employed how to use them. At present, <cb/> the telegraphic language of signals is prepared in such a manner, that a correspondence may be conducted with Lisle upon every subject, and that every thing, nay even proper names, may be expressed; an answer may be received, and the correspondence thus be renewed several times a day. The machines are the invention of Citizen Chappe, and were constructed under his own eye; he also directs their establishment at Paris. They have the advantage of resisting the changes in the atmosphere, and the inclemencies of the seasons. The only thing which can interrupt their effect is, if the weather is so very bad and turbid that the objects and signals cannot be distinguished. By this invention, remoteness and distance almost disappear; and all the communications of correspondence are effected with the rapidity of the twinkling of an eye. The operations of Government can be very much facilitated by this contrivance, and the unity of the Republic can be the more consolidated by the speedy communication with all its parts. The greatest advantage which can be derived from this correspondence is, that, if one chooses, its object shall only be known to certain individuals, or to one individual alone, or to the extremities of any distance; so that the Committee of Public Welfare may now correspond with the Representative of the People at Lisle without any other persons getting acquainted with the object of the correspondence. Hence it follows that, were Lisle even besieged, we should know every thing at Paris that might happen in that place, and could send thither the Decrees of the Convention without the enemy's being able to discover or to prevent it.”—The description and figure of the French machine, as given in some English prints, are as follow.</p><p><hi rend="italics">Explanation of the Machine (Telegraph) placed on the Mountain of Bellville, near Paris, for the purpose of communicating Intelligence.</hi></p><p>AA is a beam or mast of wood, placed upright on a rising ground (fig. 3, pl. 28) which is about 15 or 16 feet high. BB is a beam or balance, moving upon the centre AA. This balance-beam may be placed vertically, or horizontally, or any how inclined, by means of strong cords, which are fixed to the wheel D, on the edge of which is a double groove, to receive the two chords. This balance is about 11 or 12 feet long, and 9 inches broad, having at the ends two pieces of wood CC, which likewise turn upon angles by means of four other cords that pass through the axis of the main balance, otherwise the balance would derange the cords; the pieces C are each about 3 feet long, and may be placed either to the right or lest, straight or square with the balance-beam. By means of these three, the combination of movement is said to be very extensive, remarkably simple, and easy to perform. Below is a small wooden gouge or hut, in which a person is employed to observe the movements of the machine. In the mountain nearest to this, another person is to repeat these movements, and a third to write them down. The time taken up for each movement is 20 seconds; of which the motion alone is 4 seconds, the other 16 the machine is stationary. The stations of this machine are about 3 or 4 leagues distance; and there is an observatory near the Committee of Public <pb/><pb/><pb n="565"/><cb/> Safety to observe the motions of the last, which is at Bellville. The signs are sometimes made in words, and sometimes in letters; when in words, a small flag is hoisted, and, as the alphabet may be changed at pleasure, it is only the corresponding person who knows the meaning of the signs. In general, news are given every day, about 11 or 12 o'clock; but the people in the wooden gouge observe from time to time, and, as soon as a certain signal is given and answered, they begin, from one end to the other, to move the machine. It is painted of a dark brown colour.</p><p>Such is the account given of the French invention. Various improved contrivances have been since made in England, and a pamphlet has lately been published, giving an account of some of them, by the Rev. J. Gamble, under the title of, <hi rend="italics">Observations and Telegraphic Experiments,</hi> from whence the following remarks are extracted.</p><p>The object proposed is, to obtain an intelligible figurative language, which may be distinguished at a distance, and by which the obvious delay in the dispatch of orders or information by messenger may be avoided.</p><p>On first reflection we find the practical modes of such distant communication must be confined to Sound and Vision. Each of which is in a great degree subject to the state of the atmosphere: as, independent of the wind's direction, it is known that the air is sometimes so far deprived of its elasticity, or whatever other quality the conveyance of sound depends on, that the heaviest ordnance is scarce heard farther than the shot flies; it is also well known, that in thick hazy weather the largest objects become totally obscured at a short distance. No instrument therefore designed for the purpose can be perfect. We can only endeavour to diminish these irremediable defects as much as may be.</p><p>It seems the Romans had a method in their walled cities, either by a hollow formed in the masonry, or by tubes affixed to it, so to confine and augment sound as to convey information to any part they wished; and in lofty houses it is now sometimes the custom to have a pipe, by way of speaking trumpet, to give orders from the upper apartments to the lower: by this mode of confining sound its volume may be carried to a very great distance; but beyond a certain extent the sound, losing articulation, would only convey alarm, not give directions.</p><p>Every city among the antients had its watch-towers; and the castra stativa of the Romans, had always some spot, elevated either by nature or art, from whence signals were given to the troops cantoned or foraging in the neighbourhood. But I believe they had not arrived to greater refinement than that on seeing a certain signal they were immediately to repair to their appointed stations.</p><p>A beacon or bonfire made of the first inflammable materials that offered, as the most obvious, is perhaps the most antient mode of general alarm; and by being previously concerted, the number or point where the fires appeared might have its particular intelligence affixed. The same observations may be referred to the throwing up of rockets, whose number or point from whence thrown may have its affixed signification.</p><p>Flags or ensigns with their various devices are of earliest invention, especially at sea; where, from the first idea, which most probably was that of a vane to <cb/> shew the direction of the wind, they have been long adopted as the distinguishing mark of nations, and are now so neatly combined by the ingenuity of a great naval commander, that by his system every requisite order and question is received and answered by the most distant ships of a fleet.</p><p>To the adopting this or a similar mode in land service, the following are objections: That in the latter case, the variety of matter necessary to be conveyed, is so infinitely greater, that the combinations would become too complicated. And if the person for whom the information is intended should be in the direction of the wind, the flag would then present a straight line only, and at a little distance be scarce visible. The Romans were so well aware of this inconvenience of flags, that many of their standards were solid, and the name manipulus denotes the rudest of their modes, which was a truss of hay fixed on a pole.</p><p>The principle of water always keeping its own level has been suggested, as a mode of conveying intelligence, by Mr. Daniel Brent, of Rotherhithe, and put in practice on a small scale. As for example, suppose a pipe AB to reach from London to Dover, and to have a per- <figure/> pendicular tube connected to each extremity, as AC and BD. Then, if the pipe be constantly filled with water to a certain height, as AE, it will also rise to its level in the opposite perpendicular tube BF; and if one inch of water be added in the tube AC, it will almost instantly produce a similar elevation of the tube BD; so that by corresponding letters being adapted to the tubes AC and BD, at different heights, intelligence might be conveyed. But the method is liable to such objections, that it is not likely it can ever be adopted to facilitate the object of very distant communication.</p><p>Full as many, if not greater objections, will perhaps operate against every mode of electricity being used as the vehicle of information.—And the requisite magnitude of painted or illuminated letters offers an unsurmountable obstacle; besides, in them one object would be lost, that of the language being figurative.</p><p>As to the French machine, it is evident that to every angular change of the greater beam or of the lesser end arms, a different letter or figure may be annexed. But where the whole difference consists in the variation of the angle of the greater or lesser pieces, much error may be expected, from the inaccuracy either of the operator or the observer: besides other inconveniences arising from the great magnitude of the machinery.</p><p>Another idea is perfectly numerical; which is to raise and depress a flag or curtain a certain number of times for each letter, according to a previously concerted system: as, suppose one elevation to mean A, two to mean B, and so on through the alphabet. But in this case, the least inaccuracy in giving or noting the <pb n="566"/><cb/> number changes the letter; and besides, the last letters of the alphabet would be a tedious operation.</p><p>Another method that has been proposed, is an ingenious combination of the magnetical experiment of Comus, and the telescopic micrometer. But as this is only an imperfect idea of Mr. Garnet's very ingenious machine, described in the latter part of this article, no farther notice need be taken of it here.</p><p>Mr. Gamble then proposes one on a new idea of his own. The principle of it is simply that of a Venetian blind, or rather what are called the lever boards of a brewhouse, which, when horizontal, present so small a surface to the distant observer, as to be lost to his view, but are capable of being in an instant converted into a screen of a magnitude adapted to the required distance of vision.—Let AB and CD (fig. 4. pl. 28), two upright posts sixed in the ground, and joined by the braces BD and EF, be considered as the frame work for 9 lever boards working upon centres in EB and DF, and opening in three divisions by iron rods connected with each three of the lever boards. Let <hi rend="italics">abcd</hi> and <hi rend="italics">efgh</hi> be two lesser frames fixed to the great one, having also three lever boards in each, and moving by iron rods, in the same manner as the others. If all these rods be brought so near the ground as to be in the management of the operator, he will then have five, of what may be called, keys to play on. Now as each of the handles <hi rend="italics">iklmn</hi> commands three lever boards, by raising any one of them, and fixing it in its place by a catch or hook, it will give a different appearance to the machine; and by the proper variation of these sive movements, there will be more than 25 of what may be called mutations, in each of which the machine exhibits a different appearance, and to which any letter or figure may be annexed at pleasure.</p><p>Should it be required to give intelligence in more than one direction, the whose machine may be easily made to turn to different points on a strong centre, after the manner of a single-post windmill.—To use this machine by night, another frame must be connected with the back part of the Telegraph, for raising five lamps, of different colours, behind the openings of the lever boards; these lamps by night answer for the opening by day.</p><p>M. Gamble gives also particular directions for placing and using the machine, and for writing down the several figures or movements.</p><p>I shall now conclude this article with a short idea of Mr. John Garnet's most simple and ingenious contrivance. This is merely a bar or plank turning upon a centre, like the sail of a windmill, and being moved into any position, the distant observer turns the tube of a telescope into the same position, by bringing a fixed wire within it to coincide with or parallel to the bar, which is a thing extremely easy to do. The centre of motion of the bar has a small circle about it, with letters and figures around the circumference, and an index moving round with the bar, pointing to any letter or mark that the operator wishes to set the bar to, or to communicate to the observer. The eye end of the telescope without has a like index and circle, with the corresponding letters or other marks. The consequence is obvious: the telescope being turned round till its wire cover or become parallel to the bar, <cb/> the index of the former necessarily points out the same letter or mark in its circle, as that of the latter, and the communication of sentiment is immediate and perfect. The use of this machine is so easy, that I have seen it put into the hands of two common labouring men, who had never seen it before, and they have immediately held a quick and distant conversation together.</p><p>The more particular description and figure of this machine, take as follows. ABDE (fig. 5, pl. 28), is the Telegraph, on whose centre of gravity C, about which it revolves, is a fixed pin, which goes through a hole or socket in the firm upright post G, and on the opposite side of which is fixed an index CI. Concentric to C, on the same post, is fixed a wooden or brass circle, of 6 or 8 inches diameter, divided into 48 equal parts, 24 of which represent the letters of the alphabet, and between the letters, numbers. So that the index, by means of the arm AB, may be moved to any letter or number. The length of the arm should be 2 1/2 or 3 feet for every mile of distance. Two revolving lamps of different colours suspended occasionally at A and B, the ends of the arm, would serve equally at night.</p><p>Let <hi rend="italics">ss</hi> (fig. 6, pl. 28) represent the section of the outward tube of a telescope perpendicular to its axis, and <hi rend="italics">xx</hi> the like section of the sliding or adjusting tube, on which is fixed an index II. On the part of the outward tube next to the observer, there is fixed a circle of letters and numbers, similarly divided and situated to the circle in figure 3; then the index II, by means of the sliding or adjusting tube, may be turned to any letter or number.—Now there being a cross hair, or fine silver wire <hi rend="italics">fg,</hi> fixed in the focus of the eye glass, in the same direction as the index II; so that when the arm AB (fig. 5) of the Telegraph is viewed at a distance through the telescope, the cross hair may be turned, by means of the sliding tube, to the same direction of the arm AB; then the index II (fig. 6) will point to the same letter or number on its own circle, as the index 1 (fig. 5) points to on the Telegraphic circle.</p><p>If, instead of using the letters and numbers to form words at length, they be used as signals, three motions of the arm will give above a hundred thousand different signals.</p></div1><div1 part="N" n="TELESCOPE" org="uniform" sample="complete" type="entry"><head>TELESCOPE</head><p>, an optical instrument which serves for discovering and viewing distant objects, either directly by glasses, or by reflection, by means of specula, or mirrors. Accordingly,</p><p>Telescopes are either refracting or reflecting; the former consisting of different lenses, through which the objects are seen by rays refracted through them to the eye; and the latter of specula, from which the rays are reflected and passed to the eye. The lens or glass turned towards the object, is called the <hi rend="italics">object-glass;</hi> and that next the eye, the <hi rend="italics">eye-glass;</hi> and when the Telescope consists of more than two lenses, all but that next the object are called <hi rend="italics">eye-glasses.</hi></p><p>The invention of the Telescope is one of the noblest and most useful these ages have to boast of: by means of it, the wonders of the heavens are discovered to us, and astronomy is brought to a degree of perfection which former ages could have no idea of. <pb/><pb/><pb n="567"/><cb/></p><p>The discovery indeed was owing rather to chance than design; so that it is the good fortune of the discoverer, rather than his skill or ability, we are indebted to: on this account it concerns us the less to know, who it was that first hit upon this admirable invention. Be that as it may, it is certain it must have been casual, since the theory it depends upon was not then known.</p><p>John Baptista Porta, a Neapolitan, according to Wolfius, first made a Telescope, which he infers from this passage in the <hi rend="italics">Magia Naturalis</hi> of that author, printed in 1560: “If you do but know how to join “the two (viz, the concave and convex glasses) rightly “together, you will see both remote and near objects, “much larger than they otherwise appear, and withal “very distinct. In this we have been of good help “to many of our friends, who either saw remote “things dimly, or near ones confusedly; and have “made them see every thing perfectly.”</p><p>But it is certain, that Porta did not understand his own invention, and therefore neither troubled himself to bring it to a greater perfection, nor ever applied it to celestial observation. Besides, the account given by Porta of his concave and convex lenses, is so dark and indistinct, that Kepler, who examined it by desire of the emperor Rudolph, declared to that prince, that it was perfectly unintelligible.</p><p>Thirty years afterwards, or in 1590, a Telescope 16 inches long was made, and presented to prince Maurice of Nassau, by a spectacle maker of Middleburg: but authors are divided about his name. Sirturus, in a treatise on the Telescope, printed in 1618, will have it to be John Lippersheim: and Borelli, in a volume expressly on the inventor of the Telescope, published in 1655, shews that it was Zacharias Jansen, or, as Wolsius writes it, Hansen.</p><p>Now the invention of Lippersheim is fixed by some in the year 1609, and by others in 1605: Fontana, in his <hi rend="italics">Novæ Observationes Cælestium et Terrestrium Rerum,</hi> printed at Naples in 1646, claims the invention in the year 1608. But Borelli's account of the discovery of Telescopes is so circumstantial, and so well authenticated, as to render it very probable that Jansen was the original inventor.</p><p>In 1620, James Metius of Alcmaer, brother of Adrian Metius who was professor of mathematics at Franeker, came with Drebel to Middleburg, and there bought Telescopes of Jansen's children, who had made them public; and yet this Adr. Metius has given his brother the honour of the invention, in which too he is mistakenly followed by Descartes.</p><p>But none of these artificers made Telescopes of above a foot and a half: Simon Marius in Germany, and Galileo in Italy, it is said, first made long ones fit for celestial observations; though, from the recently discovered astronomical papers of the celebrated Harriot, author of the Algebra, it appears that he must have made use of Telescopes in viewing the solar maculæ, which he did quite as early as they were observed by Galileo. Whether Harriot made his own Telescopes, or whether he had them from Holland, does not appear: it seems however that Galileo's were made by himself; for Le Rossi relates, that Galileo, being then at Venice, was told of a sort of optic glass <cb/> made in Holland, which brought objects nearer: upon which, setting himself to think how it should be, he ground two pieces of glass into form as well as he could, and fitted them to the two ends of an organpipe; and with these he shewed at once all the wonders of the invention to the Venetians, on the top of the tower of St. Mark. The same author adds, that from this time Galileo devoted himself wholly to the improving and perfecting the Telescope; and that he hence almost deserved all the honour usually done him, of being reputed the inventor of the instrument, and of its being from him called <hi rend="italics">Galileo's tube.</hi> Galileo himself, in his <hi rend="italics">Nuncius Sid<*>us,</hi> published in 1610, acknowledges that he first heard of the instrument from a German; and that, being merely informed of its effects, first by common report, and a few days after by letter from a French gentleman, James Badovere, at Paris, he himself discovered the construction by considering the nature of refraction. He adds in his <hi rend="italics">Saggiatore,</hi> that he was at Venice when he heard of the effects of prince Maurice's instrument, but nothing of its construction; that the first night after his return to Padua, he solved the problem, and made his instrument the next day, and soon after presented it to the Doge of Venice, who, in honour of his grand invention, gave him the ducal letters, which settled him for life in his lectureship, at Padua, and doubled his salary, which then became treble of what any of his predecessors had enjoyed before. And thus Galileo may be considered as an inventor of the Telescope, though not the first inventor.</p><p>F. Mabillon indeed relates, in his travels through Italy, that in a monastery of his own order, he saw a manuscript copy of the works of Commestor, written by one Conradus, who lived in the 13th century; in the 3d page of which was seen a portrait of Ptolomy, viewing the stars through a tube of 4 joints or draws: but that father does not say that the tube had glasses in it. Indeed it is more than probable, that such tubes were then used for no other purpose but to defend and direct the sight, or to render it more distinct, by singling out the particular object looked at, and shutting out all the foreign rays reflected from others, whose proximity might have rendered the image less precise. And this conjecture is verified by experience; for we have often observed that without a tube, by only looking through the hand, or even the fingers, or a pinhole in a paper, the objects appear more clear and distinct than otherwise.</p><p>Be this as it may, it is certain that the optical principles, upon which Telescopes are founded, are contained in Euclid, and were well known to the ancient geometricians; and it has been for want of attention to them, that the world was so long without that admirable invention; as doubtless there are many others lying hid in the same principles, only waiting for reflection or accident to bring them forth.</p><p>To the foregoing abstract of the history of the invention of the Telescope, it may be proper to add some particulars relating to the claims of our own celebrated countryman, friar Bacon, who died in 1294. Mr. W. Molyneux, in his Dioptrica Nova, pa. 256, declares his opinion, that Bacon did perfectly well understand all sorts of optic glasses, and knew likewise the <pb n="568"/><cb/> way of combining them, so as to compose some such instrument as our Telescope: and his son, Samuel Molyneux, asserts more positively, that the invention of Telescopes, in its first original, was certainly put in practice by an Englishman, friar Bacon; although its first application to astronomical purposes may probably be ascribed to Galileo. The passages to which Mr. Molyneux refers, in support of Bacon's claims, occur in his Opus Majus, pa. 348 and 357 of Jebb's edit. 1773. The first is as follows: <hi rend="italics">Si vero non sint corpora plana, per quæ visus videt, sed sphæria, tunc est magna diversitas; nam vel concavitas corporis est versus oculum vel convexitas:</hi> whence it is inferred, that he knew what a concave and a convex glass was. The second is comprised in a whole chapter, where he says, <hi rend="italics">De visione fracta majora sunt; nam de facili patet per canones supra dictos, quod maxima possunt apparere minima, et e contra, et longe distantia videbuntur propinquissime, et e converso. Nam possumus sic figurare perspicua, et taliter ea ordinare respectu nostri visus et rerum, quod frangentur radii, et flectentur quorsumcunque voluerimus, ut sub quocunque angulo voluerimus, videbimus rem prope vel longe, &c. Sic etiam faceremus solem et lunam et stellas descendere secundum apparentiam hic inferius, &c:</hi> that is, Greater things than these may be performed by refracted vision; for it is easy to understand by the canons above mentioned, that the greatest things may appear exceeding small, and the contrary; also that the most remote objects may appear just at hand, and the converse; for we can give such figures to transparent bodies, and dispose them in such order with respect to the eye and the objects, that the rays shall be refracted and bent towards any place we please; so that we shall see the object near at hand or at a distance, under any angle we please, &c. So that thus the sun, moon, and stars may be made to descend hither in appearance, &c. Mr. Molyneux has also cited another passage out of Bacon's Epistle ad Parisiensem, of the Secrets of Art and Nature, cap. 5, to this purpose, <hi rend="italics">Possunt etiam sic figurari perspicua, ut longissime posita appareant propinqua, et è contrario; ita quod ex incredibili distantia legeremus literas minutissimas, et numeraremus res quantumquo parvas, et stellas faceremus apparere quo vellemus:</hi> that is, Glasses, or diaphanous bodies may be so formed, that the most remote objects may appear just at hand, and the contrary; so that we may read the smallest letters at an incredible distance, and may number things though never so small, and may make the stars appear as near as we please.</p><p>Moreover, Doctor Jebb, in the dedication of his edition of the Opus Majus, produces a passage from a manuscript, to shew that Bacon actually applied Telescopes to astronomical purposes: <hi rend="italics">Sed longe magis quam hæc,</hi> says he, <hi rend="italics">oporterel homines haberi, qui bene, immo optime, scirent perspectivam et instrumenta ejus—quia instrumenta astronomia non vadunt nisi per visionem secundum leges istius scientiæ.</hi></p><p>From these passages, it is not unreasonable to conclude, that Bacon had actually combined glasses so as to have produced the effects which he mentions, though he did not complete the construction of Telescopes. Dr. Smith, however, to whose judgment particular deference is due, is of opinion that the celebrated friar wrote hypothetically, without having made any actual <cb/> trial of the things he mentions: to which purpose he observes, that this author does not assert one fingle trial or observation upon the sun or moon, or any thing else, though he mentions them both: on the other hand, he imagines some effects of Telescopes that cannot possibly be performed by them. He adds, that persons unexperienced in looking through Telescopes expect, in viewing any object, as for instance the face of a man, at the distance of one hundred yards, through a Telescope that magnifies one hundred times, that it will appear much larger than when they are close to it: this he is satisfied was Bacon's notion of the matter; and hence he concludes that he had never looked through a Telescope.</p><p>It is remarkable that there is a passage in Thomas Digges's Stratioticos, pa. 359, where he affirms that his father, Leonard Digges, among other curious practices, had a method of discovering, by perspective glasses set at due angles, all objects pretty far distant that the sun shone upon, which lay in the country round about; and that this was by the help of a manuscript book of Roger Bacon of Oxford, who he conceived was the only man besides his father (since Archimedes) who knew it. This is the more remarkable, because the Stratioticos was first printed in 1579, more than 30 years before Metius or Galileo made their discovery of those glasses; and therefore it has hence been thought that Roger Bacon was the first inventor of Telescopes, and Leonard Digges the next reviver of them. But from what Thomas Digges says of this matter, it would seem that the instrument of Bacon, and of his father, was something of the nature of a camera obscura, or, if it were a Telescope, that it was of the reflecting kind; although the term <hi rend="italics">perspective</hi> glass seems to favour a contrary opinion.</p><p>There is also another passage to the same effect in the preface to the Pantometria of Leonard Digges, but published by his son Thomas Digges, some time before the Stratioticos, and a second time in the year 1591. The passage runs thus: <hi rend="italics">My father by his continuall painfull practises, assisted with demonstrations mathematical, was able, and sundrie times hath by Proportional Glasses duely situate in convenient angles, not only discovered things farre off, read letters, numbered peeces of money with the very coyne and superscription thereof, cast by some of his frecnds of purpose upon downes in open fields, but also seven myles off declared what hath beene doone at that instant in private places: He hath also sundrie times by the sunne beames fixed</hi> (should be <hi rend="italics">fired) powder, and dischargde ordinance halfe a mile and more distante,</hi> &c.</p><p>But to whomsoever we ascribe the honour of first inventing the Telescope, the rationale of this admirable instrument, depending on the refraction of light in passing through mediums of different forms, was first explained by the celebrated Kepler, who also pointed out methods of constructing others, of superior powers, and more commodious application, than that first used: though something of the same kind, it is said, was also done by Maurolycus, whose treatise De Lumine et Umbra was published in 1575.</p><p><hi rend="italics">The Principal Effects of</hi> <hi rend="smallcaps">Telescopes</hi>, depend upon this plain maxim, viz, that objects appear larger in proportion to the angles which they subtend at the <pb n="569"/><cb/> eye; and the effect is the same, whether the pencils of rays, by which objects are visible to us, come directly from the objects themselves, or from any place nearer to the eye, where they may have been united, so as to form an image of the object; because they issue again from those points in certain directions, in the same manner as they did from the corresponding points in the objects themselves. In fact therefore, all that is effected by a Telescope, is first to make such an image of a distant object, by means of a lens or mirror, and then to give the eye some assistance for viewing that image as near as possible; so that the angle, which it shall subtend at the eye, may be very large, compared with the angle which the object itself would subtend in the same situation. This is done by means of an eye-glass, which so refracts the pencils of rays, as that they may afterwards be brought to their several foci, by the natural humours of the eye. But if the eye had been so formed as to be able to see the image, with sufficient distinctness, at the same distance, without an eye-glass, it would appear to him as much magnified, as it does to another person who makes use of a glass for that purpose, though he would not in all cases have so large a field of view.</p><p>Although no image be actually formed by the foci of the pencil without the eye, yet if, by the help of an eye-glass, the pencils of rays shall enter the pupil, just as they would have done from any place without the eye, the visual angle will be the same as if an image had been actually formed in that place. Priestley's History of Light &c, pa. 69, &c.</p><p><hi rend="italics">As to the Grinding of Telescopic Glasses,</hi> the first persons who distinguished themselves in that way, were two Italians, Eustachio Divini at Rome, and Campani at Bologna, whose fame was much superior to that of Divini, or that of any other person of his time; though Divini himself pretended, that in all the trials that were made with their glasses, his of a great focal distance performed better than those of Campani, and that his rival was not willing to try them fairly, viz, with equal eye-glasses. It is however generally supposed, that Campani really excelled Divini, both in the goodness and the focal length of his object-glasses.</p><p>It was with Campani's Telescopes that Cassini discovered the nearest satellites of Saturn. They were made at the express desire of Lewis XIV, and were of 86, 100, and 136, Paris feet focal length.</p><p>Campani's laboratory was purchased, after his death, by pope Benedict XIV, who made a present of it to the academy at Bologna called the Institute; and by the account which Fougeroux has given, we learn that (except a machine which Campani constructed, to work the basons on which he ground his glasses) the goodness of his lenses depended upon the clearness of his glass, his Venetian tripoli, the paper with which he polished his glasses, and his great skill and address as a workman. It does not appear that he made many lenses of a very great focal distance. Accordingly Dr. Hook, who probably speaks with the partiality of an Englishman, says that some glasses, made by Divini and Campani, of 36 and 50 feet focal distance, did not excel Telescopes of 12 or 15 feet made in England. He adds, that Sir Paul Neilli made Telescopes of 36 feet, pretty good; and one of 50, but not of proportionable goodness. <cb/></p><p>Afterwards, Mr. Reive first, and then Mr. Cox, who were the most celebrated in England, as grinders of optic glasses, made some good Telescopes of 50 and 60 feet focal distance; and Mr. Cox made one of 100, but how good Dr. Hook could not assert. Borelli also in Italy, made object-glasses of a great focal length, one of which he presented to the Royal Society. But, with respect to the focal length of Telescopes, these and all others were far exceeded by those of Auzout, who made one object-glass of 600 feet focus; but he was never able to manage it, so as to make any use of it. And Hartsoeker, it is said, made some of a still greater focal length. Philos. Trans. Abr. vol. i, p. 193. Hook's Exper. by Derham, p. 261. Priestley as above, p. 211. See <hi rend="smallcaps">Grinding.</hi></p><p>Telescopes are of several kinds, distinguished by the number and form of their lenses, or glasses, and denominated from their particular uses &c: such are the <hi rend="italics">terrestrial</hi> or <hi rend="italics">land Telescope,</hi> the <hi rend="italics">celestial</hi> or <hi rend="italics">astronomical Telescope;</hi> to which may be added, the <hi rend="italics">Galilean</hi> or <hi rend="italics">Dutch Telescope,</hi> the <hi rend="italics">reflecting Telescope,</hi> the <hi rend="italics">refracting Telescope,</hi> the <hi rend="italics">aërial Telescope, achromatic Telescope, &c.</hi></p><p><hi rend="italics">Galileo's,</hi> or the <hi rend="italics">Dutch Telescope,</hi> is one consisting of a convex object-glass, and a concave eyeglass.</p><p>This is the most ancient form of any, being the only kind made by the inventors, Galileo, &c. or known, before Huygens. The first Telescope, constructed by Galileo, magnified only 3 times; but he soon made another, which magnified 18 times: and afterwards, with great trouble and expence, he constructed one that magnified 33 times; with which he discovered the satellites of Jupiter, and the spots of the sun. The construction, properties. &c, of it, are as follow:</p><p><hi rend="italics">Construction of Galileo's,</hi> or the <hi rend="italics">Dutch</hi> <hi rend="smallcaps">Telescope.</hi> In a tube prepared for the purpose, at one end is fitted a convex object lens, either a plain convex, or convex on both sides, but a segment of a very large sphere: at the other end is fitted an eye-glass, concave on both sides, and the segment of a less sphere, so disposed as to be at the distance of the virtual focus before the image of the convex lens.</p><p>Let AB (fig. 10, pl. 23) be a distant object, from every point of which pencils of rays issue, and falling upon the convex glass DE, tend to their foci at FSG. But a concave lens HI (the focus of which is at FG) being interposed, the converging rays of each pencil are made parallel when they reach the pupil; so that by the refractive humours of the eye, they can easily be brought to a focus on the retina at PRQ. Also the pencils themselves diverging, as if they came from X, MXO is the angle under which the image will appear, which is much larger than the angle under which the object itself would have appeared. Such then is the Telescope that was at first discovered and used by philosophers: the great inconvenience of which is, that the field of view, which depends, not on the breadth of the eye-glass, as in the astronomical Telescope, but upon the breadth of the pupil of the eye, is exceedingly small: for since the pencils of the rays enter the eye very much diverging from one another, but few of them can be intercepted by the pupil; and this inconvenience increases with the magnifying power of the Telescope, so that philosophers may now well wonder <pb n="570"/><cb/> at the patience and address with which Galileo and others, with such an instrument, made the discoveries they did. And yet no other Telescope was thought of for many years after the discovery. Descartes, who wrote 30 years after, mentions no other as actually constructed, though Kepler had suggested some. Hence,</p><p>1. In an instrument thus framed, all people, except myopes, or short-sighted persons, must see objects distinctly in an erect situation, and increased in the ratio of the distance of the virtual focus of the eyeglass, to the distance of the focus of the object glass.</p><p>2. But for myopes to see objects distinctly through such an instrument, the eye-glass must be set nearer the object-glass, so that the rays of each pencil may not emerge parallel, but may fall diverging upon the eye; in which case the apparent magnitude will be altered a little, though scarce sensibly.</p><p>3. Since the focus of a plano-convex object lens, and the vertical focus of a plano-concave eye-lens, are at the distance of the diameter; and the focus of an object-glass convex on both sides, and the vertical focus of an eye-glass concave on both sides, are at the distance of a semidiameter; if the object-glass be planoconvex, and the eye-glass plano-concave, the Telescope will increase the diameter of the object, in the ratio of the diameter of the concavity to that of the convexity: if the object-glass be convex on both sides, and the eye-glass concave on both sides, it will magnify in the ratio of the semidiameter of the concavity to that of the convexity: if the object-glass be plano-convex, and the eye-glass concave on both sides, the semidiameter of the object will be increased in the ratio of the diameter of the convexity to the semidiameter of the concavity: and lastly, if the object-glass be convex on both sides, and the eye-glass plano-concave, the increase will be in the ratio of the diameter of the concavity to the semidiameter of the convexity.</p><p>4. Since the ratio of the semidiameters is the same as that of the diameters, Telescopes magnify the object in the same manner, whether the object-glass be planoconvex, and the eye-glass plano-concave; or whether the one be convex on both sides, and the other concave on both.</p><p>5. Since the semidiameter of the concavity has a less ratio to the diameter of the convexity than its diameter has, a Telescope magnifies more if the objectglass be plano-convex, than if it be convex on both sides. The case is the same if the eye-glass be concave on both sides, and not plano-concave.</p><p>6. The greater the diameter of the object-glass, and the less that of the eye-glass, the less ratio has the diameter of the object, viewed with the naked eye, to its semidiameter when viewed with a Telescope, and consequently the more is the object magnified by it.</p><p>7. Since a Telescope exhibits so much a less part of the object, as it increases its diameter more, for this reason, mathematicians were determined to look out for another Telescope, after having clearly found the imperfection of the first, which was discovered by ehance. Nor were their endeavours vain, as appears from the astronomical Telescope described below.</p><p>If the semidiameter of the eye-glass have too small <cb/> a ratio to that of the object-glass, an object through the Telescope will not appear sufficiently clear, because the great divergency of the rays will occasion the several pencils representing the several points of the object on the retina, to consist of too few rays.</p><p>It is also found that equal object-lenses will not bear the same eye-lenses, if they be differently transparent, or if there be a difference in their polish; a less transparent object-glass, or one less accurately ground, requiring a more spherical eye-glass than another more transparent, &c.</p><p>Hevelius recommends an object-glass convex on both sides, whose diameter is 4 feet; and an eye-glass concave on both sides, whose diameter is 4 1/2 tenths of afoot. An object-glass, equally convex on both sides, whose diameter is 5 feet, he observes, will require an eye-glass of 5 1/2 tenths; and adds, that the same eye-glass will also serve an object-glass of 8 or 10 feet.</p><p>Hence, as the distance between the object-glass and eye-glass is the difference between the distance of the vertical focus of the eye-glass, and the distance of the focus of the object glass; the length of the telescope is had by subtracting that from this. That is, the length of the Telescope is the difference between the diameters of the object-glass and eye-glass, if the former be plano-convex, and the latter plano-concave; or the difference between the semidiameters of the object-glass and eye-glass, if the former be convex on both sides, and the latter concave on both; or the difference between the semidiameter of the object-glass and the diameter of the eye-glass, if the former be convex on both sides, and the latter planoconcave; or lastly the difference between the diameter of the object-glass and the semidiameter of the eyeglass, if the former be plano-convex, and the latter concave on both sides. Thus, for instance, if the diameter of an object-glass, convex on both sides, be 4 feet, and that of an eye-glass, concave on both sides, be 4 1/2 tenths of a foot; then the length of the Telescope will be 1 foot and 7 1/2 tenths.</p><p><hi rend="italics">Astronomicel</hi> <hi rend="smallcaps">Telescope;</hi> this is one that consists of an object-glass, and an eye-glass, both convex. It is so called from being wholly used in astronomical observations.</p><p>It was Kepler who first suggested the idea of this Telescope; having explained the rationale, and pointed out the advantages of it in his Catoptrics, in 1611. But the first person who actually made an instrument of this construction, was father Scheiner, who has given a description of it in his Rosa Ursina, published in 1630. To this purpose he says, If you insert two similar convex lenses in a tube, and place your eye at a convenient distance, you will see all terrestrial objects, inverted indeed, but magnified and very distinct, with a considerable extent of view. He afterwards subjoined an account of a Telescope of a different construction, with two convex eye-glasses, which again reverses the images, and makes them appear in their natural position. Father Reita however soon after proposed a better construction, using three eye-glasses instead of two.</p><p><hi rend="italics">Construction of the Astronomical</hi> <hi rend="smallcaps">Telescope.</hi> The tube being prepared, an object-glass, either plano-con- <pb n="571"/><cb/> vex, or convex on both sides, but a segment of a large sphere, is fitted in at one end; and an eye-glass, convex on both sides, which is the segment of a small sphere, is fitted into the other end; at the common distance of the foci.</p><p>Thus the rays of each pencil issuing from every point of the object ABC, (fig. 3 pl. 30) passing through the object-glass DEF, become converging, and meet in their foci at IHG, where an image of the object will be formed. If then another convex lens KM, of a shorter focal length, be so placed, as that its focus shall be in IHG, the rays of each pencil, after passing through it, will become nearly parallel, so as to meet upon the retina, and form an enlarged image of the object at RST. If the process of the rays be traced, it will presently be perceived that this image must be inverted. For the pencil that issues from A, has its focus in G, and again in R, on the same side with A. But as there is always one inversion in simple vision, this want of inversion produces just the reverse of the natural appearance. The field of view in this Telescope will be large, because all the pencils that can be received on the surface of the lens KM, being converging after passing through it, are thrown into the pupil of the eye, placed in the common intersection of the pencils at P.</p><p><hi rend="italics">Theory of the Astronomical</hi> <hi rend="smallcaps">Telescope.</hi>—An eye placed near the focus of the eye-glass, of such a Telescope, will see objects distinctly, but inverted, and magnified in the ratio of the distance of the focus of the eye-glass to the distance of the focus of the objectglass.</p><p>If the sphere of concavity in the eye-glass of the Galilean Telescope, be eqnal to the sphere of convexity in the eye-glass of another Telescope, their magnifying power will be the same. The concave glass however being placed between the object-glass and its focus, the Galilean Telescope will be shorter than the other, by twice the focal length of the eye-glass. Consequently, if the length of the Telescopes be the same; the Galilean will have the greater magnifying power. Vision is also more distinct in these Telescopes, owing in part perhaps to there being no intermediate image between the eye and the object. Besides, the eye-glass being very thin in the centre, the rays will be less liable to be distorted by irregularities in the substance of the glass. Whatever be the cause, we can sometimes see Jupiter's satellites very clearly in a Galilean Telescope, of 20 inches or 2 feet long, when one of 4 or 5 feet, of the common sort, will hardly make them visible.</p><p>As the astronomical Telescope exhibits objects inverted, it serves commodiously enough for observing the stars, as it is not material whether <hi rend="italics">they</hi> be seen erect or inverted; but for terrestrial objects it is much less proper, as the inverting often prevents them from being known. But if a plane well-polished metal speculum, of an oval figure, and about an inch long, and inclined to the axis in an angle of 45°, be placed bebehind the eye-glass; then the eye, conveniently placed, will see the image, hence reflected, in the same magnitude as before, but in an erect situation; and therefore, by the addition of such a speculum, the <cb/> astronomical Telescope is thus rendered sit to observe terrestrial objects.</p><p>Since the focus of the glass convex on both sides is distant from the glass itself a semidiameter, and that of a plano-convex glass, a diameter; if the objectglass be convex on both sides, the Telescope will magnify the semidiameter of the object, in the ratio of the diameter of the eye-glass to the diameter of the object-glass; but if the object-glass be a plano-convex, in the ratio of the semidiameter of the eye-glass to the diameter of the object-glass. And therefore a Telescope magnifies more if the object-glass be a planoconvex, than if convex on both sides. And for the same reason, a Telescope magnifies more when the eyeglass is convex on both sides, than when it is planoconvex.</p><p>A Telescope magnifies the more, as the object-glass is a segment of a great sphere, and the eye-glass of a less one. And yet the eye-glass must not be too small in respect of the object-glass; for if it be, it will not refract rays enough to the eye from each point of the object; nor will it separate sufficiently those that come from different points; by which means the vision will be rendered obscure and confused.—De Chales observes, that an object-lens of 2 1/4 feet will require an eye-glass of 1 1/2 tenth of a foot; and an object-glass of 8 or 10 feet, an eye-glass of 4 tenths; in which he is confirmed by Eustachio Divini.</p><p><hi rend="italics">To shorten the Astronomical</hi> <hi rend="smallcaps">Telescope;</hi> that is, to construct a Telescope so, as that, though shorter than the common one, it shall magnify as much.</p><p>Having provided a drawing tube, fit in it an objectlens EO which is a segment of a moderate sphere: <figure/> let the first eye-glass BD be concave on both sides, and so placed in the tube, as that the focus of the objectglass A may be behind it, but nearer to the centre of the concavity G: then will the image be thrown in Q, so as that GA : GI : : AB : QI. Lastly, sit in another object-glass, convex on both sides, and a segment of a smaller sphere, so as that its focus may be in Q.</p><p>This Telescope will magnify the diameter of the object more than if the object-glass were to represent its image at the same distance EQ; and consequently a shorter Telescope, constructed this way, is equivalent to a longer in the common way. See Wolfius Elem. Math. vol. 3, p. 245.</p><p>Sir Isaac Newton furnishes us with another method of constructing the Telescope, in his catoptrical or reflecting Telescope, the construction of which is given below. See <hi rend="italics">Achromatic</hi> <hi rend="smallcaps">Telescope.</hi></p><p><hi rend="italics">Aërial</hi> <hi rend="smallcaps">Telescope</hi>, a kind of astronomical Telescope, the lenses of which are used without a tube. In strictness however, the aërial Telescope is rather a particular manner of mounting and managing long <pb n="572"/><cb/> Telescopes for celestial observation in the nighttime, by which the trouble of long unwieldy tubes is saved, than a particular kind of Telescope; and the contrivance was one of Huygens's. This invention was successfully practised by the inventor himself and others, particularly with us by Dr. Pound and Dr. Bradley, with an object-glass of 123 feet focal distance, and an apparatus belonging to it, made and presented by Huygens to the Royal Society, and described in his Astroscopia Compendiaria Tubi Optici Molimine Liberata, printed at the Hague in 1684.</p><p>The principal parts of this Telescope may be comprehended from a view of fig. 4, pl. 30, where AB is a long pole, or a mast, or a high tree, &c, in a groove of which slides a piece that carries a small tube LK in which is fixed an object glass; which tube is connected by a fine line, with another small tube OQ, which contains the eye-glass, &c.</p><p>La Hire contrived a little machine for managing the object-glass which is described Mem. de l'Acad. 1715. See Smith's Optics, book 3, chap. 10.</p><p>Hartsoeker, who made Telescopes of a very considerable focal length, contrived a method of using them without a tube, by fixing them to the top of a tree, a high wall, or the roof of a house. Miscel. Berol. vol. 1, p. 261.</p><p>Huygens's great Telescope, with which Saturn's true face, and one of his satellites were first discovered, consists of an object-glass of 12 feet, and an eye-glass of a little more than 3 inches; though he frequently used a Telescope of 23 feet long, with two eye-glasses joined together, each 1 1/2 inch diameter; so that the two were equal to one of 3 inches.</p><p>The same author observes, that an object-glass of 30 feet requires an eye-glass of 3 3/10 inches; and has given a table of proportions for constructing astronomical Telescopes, an abridgment of which is as follows: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Dist. of</cell><cell cols="1" rows="1" role="data">Diameter</cell><cell cols="1" rows="1" role="data">Dist. of</cell><cell cols="1" rows="1" role="data">Power or</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Foc. of</cell><cell cols="1" rows="1" role="data">of</cell><cell cols="1" rows="1" role="data">Foc. of</cell><cell cols="1" rows="1" role="data">Magnitude</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Obj. Glass.</cell><cell cols="1" rows="1" role="data">Apert.</cell><cell cols="1" rows="1" role="data">Eye-glass.</cell><cell cols="1" rows="1" role="data">of Diam.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Feet.</cell><cell cols="1" rows="1" role="data">Inches</cell><cell cols="1" rows="1" role="data">Inches</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">and Decim.</cell><cell cols="1" rows="1" role="data">and Decim.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">0.55</cell><cell cols="1" rows="1" role="data">0.61</cell><cell cols="1" rows="1" role="data"> 20</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">0.77</cell><cell cols="1" rows="1" role="data">0.85</cell><cell cols="1" rows="1" role="data"> 28</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">0.95</cell><cell cols="1" rows="1" role="data">1.05</cell><cell cols="1" rows="1" role="data"> 34</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">1.09</cell><cell cols="1" rows="1" role="data">1.20</cell><cell cols="1" rows="1" role="data"> 40</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">1.23</cell><cell cols="1" rows="1" role="data">1.35</cell><cell cols="1" rows="1" role="data"> 44</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">1.34</cell><cell cols="1" rows="1" role="data">1.47</cell><cell cols="1" rows="1" role="data"> 49</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">1.45</cell><cell cols="1" rows="1" role="data">1.60</cell><cell cols="1" rows="1" role="data"> 53</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">1.55</cell><cell cols="1" rows="1" role="data">1.71</cell><cell cols="1" rows="1" role="data"> 56</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">1.64</cell><cell cols="1" rows="1" role="data">1.80</cell><cell cols="1" rows="1" role="data"> 60</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1.73</cell><cell cols="1" rows="1" role="data">1.90</cell><cell cols="1" rows="1" role="data"> 63</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">2.12</cell><cell cols="1" rows="1" role="data">2.33</cell><cell cols="1" rows="1" role="data"> 77</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2.45</cell><cell cols="1" rows="1" role="data">2.70</cell><cell cols="1" rows="1" role="data"> 89</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">2.74</cell><cell cols="1" rows="1" role="data">3.01</cell><cell cols="1" rows="1" role="data">100</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">3.00</cell><cell cols="1" rows="1" role="data">3.30</cell><cell cols="1" rows="1" role="data">109</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">3.46</cell><cell cols="1" rows="1" role="data">3.81</cell><cell cols="1" rows="1" role="data">120</cell></row></table> <cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Dist. of</cell><cell cols="1" rows="1" role="data">Diameter</cell><cell cols="1" rows="1" role="data">Dist. of</cell><cell cols="1" rows="1" role="data">Power or</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Foc. of</cell><cell cols="1" rows="1" role="data">of</cell><cell cols="1" rows="1" role="data">Foc. of</cell><cell cols="1" rows="1" role="data">Magnitude</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Object-glass.</cell><cell cols="1" rows="1" role="data">Apert.</cell><cell cols="1" rows="1" role="data">Eye-glass.</cell><cell cols="1" rows="1" role="data">of Diam.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Feet.</cell><cell cols="1" rows="1" role="data">Inches</cell><cell cols="1" rows="1" role="data">Inches</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">and Decim.</cell><cell cols="1" rows="1" role="data">and Decim.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 50</cell><cell cols="1" rows="1" role="data"> 3.87</cell><cell cols="1" rows="1" role="data"> 4.26</cell><cell cols="1" rows="1" role="data">141</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 60</cell><cell cols="1" rows="1" role="data"> 4.24</cell><cell cols="1" rows="1" role="data"> 4.66</cell><cell cols="1" rows="1" role="data">154</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 70</cell><cell cols="1" rows="1" role="data"> 4.58</cell><cell cols="1" rows="1" role="data"> 5.04</cell><cell cols="1" rows="1" role="data">166</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 80</cell><cell cols="1" rows="1" role="data"> 4.90</cell><cell cols="1" rows="1" role="data"> 5.39</cell><cell cols="1" rows="1" role="data">178</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 90</cell><cell cols="1" rows="1" role="data"> 5.20</cell><cell cols="1" rows="1" role="data"> 5.72</cell><cell cols="1" rows="1" role="data">189</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data"> 5.49</cell><cell cols="1" rows="1" role="data"> 6.03</cell><cell cols="1" rows="1" role="data">200</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data"> 6.00</cell><cell cols="1" rows="1" role="data"> 6.60</cell><cell cols="1" rows="1" role="data">218</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">140</cell><cell cols="1" rows="1" role="data"> 6.48</cell><cell cols="1" rows="1" role="data"> 7.12</cell><cell cols="1" rows="1" role="data">235</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">160</cell><cell cols="1" rows="1" role="data"> 6.93</cell><cell cols="1" rows="1" role="data"> 7.62</cell><cell cols="1" rows="1" role="data">252</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">180</cell><cell cols="1" rows="1" role="data"> 7.35</cell><cell cols="1" rows="1" role="data"> 8.09</cell><cell cols="1" rows="1" role="data">267</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">200</cell><cell cols="1" rows="1" role="data"> 7.75</cell><cell cols="1" rows="1" role="data"> 8.53</cell><cell cols="1" rows="1" role="data">281</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">220</cell><cell cols="1" rows="1" role="data"> 8.12</cell><cell cols="1" rows="1" role="data"> 8.93</cell><cell cols="1" rows="1" role="data">295</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">240</cell><cell cols="1" rows="1" role="data"> 8.48</cell><cell cols="1" rows="1" role="data"> 9.33</cell><cell cols="1" rows="1" role="data">308</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">260</cell><cell cols="1" rows="1" role="data"> 8.83</cell><cell cols="1" rows="1" role="data"> 9.71</cell><cell cols="1" rows="1" role="data">321</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">280</cell><cell cols="1" rows="1" role="data"> 9.16</cell><cell cols="1" rows="1" role="data">10.08</cell><cell cols="1" rows="1" role="data">333</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">300</cell><cell cols="1" rows="1" role="data"> 9.49</cell><cell cols="1" rows="1" role="data">10.44</cell><cell cols="1" rows="1" role="data">345</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">400</cell><cell cols="1" rows="1" role="data">10.95</cell><cell cols="1" rows="1" role="data">12.05</cell><cell cols="1" rows="1" role="data">400</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">500</cell><cell cols="1" rows="1" role="data">12.25</cell><cell cols="1" rows="1" role="data">13.47</cell><cell cols="1" rows="1" role="data">445</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">600</cell><cell cols="1" rows="1" role="data">13.42</cell><cell cols="1" rows="1" role="data">14.76</cell><cell cols="1" rows="1" role="data">488</cell></row></table></p><p>Dr. Smith (Rem. p. 78) observes, that the magnifying powers of this table are not so great as Huygens himself intended, or as the best object-glasses now made will admit of. For the author, in his Astroscopia Compendiaria, mentions an object-glass of 34 feet focal distance, which, in astronomical observations, bore an eye-glass of 2 1/2 inches focal distance, and consequently magnified 163 times. According to this standard, a Telescope of 35 feet ought to magnify 166 times, and of 1 foot 28 times; whereas the table allows but 118 times to the former, and but 20 to the latter. Now 166/118 or 28/20 = 1.4; by which if we multiply the numbers in the given column of magnifying powers, we shall gain a new column, shewing how much those object-glasses ought to magnify if wrought up to the perfection of this standard.</p><p>The new apertures and eye-glasses must also be taken in the same proportions to one another, as the old ones have in the table; or the eye-glasses may be found by dividing the length of each Telescope by its magnifying power. And thus a new table may be easily made for this or any other more perfect standard when offered.</p><p>The rule for computing this table depends on the following theorem, viz, that in refracting Telescopes of different lengths, a given object will appear equally bright and equally distinct, when their linear apertures and the focal distances of their eye-glasses are severally in a subduplicate ratio of their lengths, or focal distances of their object-glasses; and then also the breadth of their apertures will be in the subduplicate ratio of their lengths.</p><p>The rule is this: Multiply the number of feet in the focal distance of any proposed object-glass by 3000, and the square-root of the product will give the breadth of its aperture in centesms, or 100th parts of an inch <*> <pb n="573"/><cb/> that is, √(3000F) is the breadth of the aperture in centesms of an inch, where F is the focal distance of the object-glass in feet. Also, the same breadth of the aperture increased by the 10th part of itself, gives the focal distance of the eye-glass in centesms of an inch. And the magnifying powers are as the breadths of the apertures.</p><p>If, in different Telescopes, the ratio between the object-glass and eye-glass be the same, the object will be magnified the same in both. Hence some may conclude the making of large Telescopes a needless trouble. But it must be remembered, that an eye-glass may be in a less ratio to a greater object-glass than to a smaller: thus, for example, in Huygens's Telescope of 25 feet, the eye-glass is 3 inches: now, keeping this proportion in a Telescope of 50 feet, the eye-glass should be 6 inches; but the table shews that 4 1/2 are sufficient. Hence, from the same table it appears, that a Telescope of 50 feet magnifies in the ratio of 1 to 141; whereas that of 25 feet only magnifies in the ratio of 1 to 100.</p><p>Since the distance of the lens is equal to the aggregate of the distances of the foci of the object and eye-glasses; and since the focus of a glass convex on each side is a semidiameter's distance from the lens, and that of a plano-convex at a diameter's distance from the same; the length of a Telescope is equal to the aggregate of the semidiameters of the lenses, if the object-glass be convex on both sides; and to the sum of the semidiameter of the eye-glass and the whole diameter of the object glass, if the object-glass be a plano-convex.</p><p>But as the diameter of the eye-glass is very small in respect of that of the object-glass, the length of the Telescope is usually estimated from the distance of the object-glass; i. e. from its semidiameter if it be convex on both sides, or its whole diameter if plano-convex. Thus, a Telescope is said to be 12 feet, if the semidiameter of the object-glass, convex on both sides, be 12 feet, &c.</p><p>Since myopes see near objects best; for them, the eye-glass is to be removed nearer to the object-glass, that the rays refracted through it may be the more diverging.</p><p>To take in the larger field at one view, some make use of two eye-glasses, the foremost of which is a segment of a larger sphere than that behind; to this it must be added, that if two lenses be joined immediately together, so as the one may touch the other, the focus is removed to double the distance which that of one of them would be at.</p><p><hi rend="italics">Land</hi> <hi rend="smallcaps">Telescope</hi>, or <hi rend="italics">Day</hi> <hi rend="smallcaps">Telescope</hi>, is one adapted for viewing objects in the day-time, on or about the earth. This contains more than two lenses, usually it has a convex object-glass, and three convex eye-glasses; exhibiting objects erect, yet different from that of Galileo.</p><p>In this Telescope, after the rays have passed the first eye-glass HI (fig. 2, pl. 30), as in the former construction, instead of being there received by the eye, they pass on to another equally convex lens, situated at twice its focal distance from the other, so that the rays of each pencil, being parallel in that whole interval, those pencils cross one another in the common <cb/> focus, and the rays constituting them are transmitted parallel to the second eye-glass LM; after which the rays of each pencil converge to other foci at NO, where a second image of the object is formed, but inverted with respect to the former image in EF. This image then being viewed by a third eye-glass QR, is painted upon the retina at XYZ, exactly as before, only in a contrary position.</p><p>Father Reita was the author of this construction; which is effected by fitting in at one end of a tube an object-glass, which is either convex on both sides, or plano-convex, and a segment of a large sphere; to this add three eye-glasses, all convex on both sides, and segments of equal spheres; disposing them in such a manner as that the distance between any two may be the aggregate of the distances of their foci. Then will an eye applied to the last lens, at the distance of its focus, see objects very distinctly, erect, and magnified in the ratio of the distance of the focus of one eye-glass, to the distance of the focus of the objectglass.</p><p>Hence, 1. An astronomical Telescope is easily converted into a Land Telescope, by using three eyeglasses for one; and the Land Telescope, on the contrary, into an astronomical one, by taking away two eye-glasses, the faculty of magnifying still remaining the same.</p><p>2. Since the distance of the eye-glasses is very small, the length of the Telescope is much the same as if you only used one.</p><p>3. The length of the Telescope is found by adding five times the semidiamer of the eye-glasses, to the diameter of the object-glass when this is a planoconvex, or to its semidiameter when convex on both sides.</p><p>Huygens first observed, both in the astronomical and Land Telescope, that it contributes considerably to the perfection of the instrument, to have a ring of wood or metal, with an aperture, a little less than the breadth of the eye-glass, sixed in the place where the image is found to radiate upon the lens next the eye: by means of which, the colours, which are apt to disturb the clearness and distinctness of the object, are prevented, and the whole compass taken in at one view, perfectly defined.</p><p>Some make Land Telescopes of three lenses, which yet represent objects erect, and magnified as much as the former. But such Telescopes are subject to very great inconveniences, both as the objects in them are tinged with false colours, and as they are distorted about the margin.</p><p>Some again use five lenses, and even more; but as some parts of the rays are intercepted in passing every lens, objects are thus exhibited dim and feeble.</p><p>Telescopes of this kind, longer than 20 feet, will be of hardly any use in observing terrestrial objects, on account of the continual motion of the particles of the atmosphere, which these powerful Telescopes render visible, and give a tremulous motion to the objects themselves.</p><p>The great length of dioptric Telescopes, adapted to any important astronomical purpose, rendered them extremely inconvenient for use; as it was necessary to increase their length in no less a proportion than the <pb n="574"/><cb/> duplicate of the increase of their magnifying power: so that, in order to magnify twice as much as before, with the same light and distinctness, the Telescope required to be lengthened 4 times; and to magnify thrice as much, 9 times the length, and so on. This unwieldiness of refracting Telescopes, possessing any considerable magnifying power, was one cause, why the attention of astronomers, &c, was directed to the discovery and construction of reflecting Telescopes. And indeed a refracting Telescope, even of 1000 feet focus, supposing it possible to make use of such an instrument, could not be made to magnify with distinctness more than 1000 times; whereas a reflecting Telescope, of 9 or 10 feet, will magnify 12 hundred times. The perfection of refracting Telescopes, it is well known, is very much limited by the aberration of the rays of light from the geometrical focus: and this arises from two different causes, viz, from the different degrees of refrangibility of light, and from the figure of the sphere, which is not of a proper curvature for collecting the rays in a single point. Till the time of Newton, no optician had imagined that the object glasses of Telescopes were subject to any other error beside that which arose from their spherical figure, and therefore all their efforts were directed to the construction of them, with other kinds of curvature: but that author had no sooner demonstrated the different refraugibility of the rays of light, than he discovered in this circumstance a new and a much greater cause of error in Telescopes. Thus, since the pencils of each kind of light have their foci in different places, some nearer and some farther from the lens, it is evident that the whole beam cannot be brought into any one point, but that it will be drawn the nearest to a point in the middle place between the focus of the most and least refrangible rays; so that the focus will be a circular space of a considerable diameter. Newton shews that this space is about the 55th part of the aperture of the Telescope, and that the focus of the most refrangible rays is nearer to the object-glass than the focus of the least refrangible ones, by about the 27 1/2 part of the distance between the object-glass, and the focus of the mean refrangible rays. But he says, that if the rays flow from a lucid point, as far from the lens on one side as their foci are on the other, the focus of the most refrangible rays will be nearer to the lens than that of the least refrangible, by more than the 14th part of the whole distance. Hence, he concludes, that if all the rays of light were equally refrangible, the error in Telescopes, arising from the spherical figure of the glass, would be many hundred times less than it now is; because the error arising from the spherical figure of the glass, is to that arising from the different refrangibility of the rays of light, as 1 to 5449. See <hi rend="smallcaps">Aberration.</hi></p><p>Upon the whole he observes, that it is a wonder that Telescopes represent objects so distinctly as they do. The reason of which is, that the dispersed rays are not scattered uniformly over all the circular space above-mentioned, but are infinitely more dense in the centre than in any other part of the circle; and that in the way from the centre to the circumference they grow continually rarer and rarer, till at the circumference they become infinitely rare: for which reason, <cb/> these dispersed rays are not copious enough to be visible, except about the centre of the circle. He also mentions another argument to prove, that the different refrangibility of the rays of light is the true cause of the imperfection of Telescopes. For the dispersions of the rays arising from the spherical figures of objectglasses, are as the cubes of their apertures; and therefore, to cause Telescopes of different lengths to magnify with equal distinctness, the apertures of the objectglasses, and the charges or magnifying powers ought to be as the cubes of the square roots of their lengths, which does not answer to experience. But the errors of the rays, arising from the different refrangibility, are as the apertures of the object-glasses; and thence, to make Telescopes of different lengths to magnify with equal distinctness, their apertures and charges ought to be as the square roots of their lengths; and this answers to experience.</p><p>Were it not for this different refrangibility of the rays, Telescopes might be brought to a sufficient degree of perfection, by composing the object-glass of two glasses with water between them. For by this means, the refractions on the concave sides of the glasses will very much correct the errors of the refractions on the convex sides, so far as they arise from their spherical figure: but on account of the different refrangibility of different kinds of rays, Newton did not see any other means of improving Telescopes by refraction only, except by increasing their length. Newton's Optics, pa. 73, 83, 89, 3d edition.</p><p>This important desideratum in the construction of dioptric Telescopes, has been since discovered by the ingenious Mr. Dollond; an account of which is given below.</p><p><hi rend="italics">Achromatic</hi> <hi rend="smallcaps">Telescope</hi>, is a name given to the refracting Telescope, invented by Mr. John Dollond, and so contrived as to remedy the aberration arising from colours, or the different refrangibility of the rays of light. See <hi rend="smallcaps">Achromatic.</hi></p><p>The principles of Mr. Dollond's discovery and construction, have been already explained under the articles <hi rend="smallcaps">Aberration</hi>, and <hi rend="smallcaps">Achromatic.</hi> The improvement made by Mr. Dollond in his Telescopes, by making two object-glasses of crown-glass, and one of flint, which was tried with success when concave eye-glasses were used, was completed by his son Peter Dollond; who, conceiving that the same method might be practised with success with convex eye-glasses, found, after a few trials, that it might be done. Accordingly he finished an object-glass of 5 feet focal length, with an aperture of 3 3/4 inches, composed of two convex lenses of crown-glass, and one concave of white flint glass. But apprehending afterward that the apertures might be admitted still larger, he completed one of 3 1/2 feet focal length, with the same aperture of 3 3/4 inches. Philos. Trans. vol. 55, p. 56.</p><p>But beside the obligation we are under to Mr. Dollond, for correcting the aberration of the rays of light in the focus of object-glasses, arising from their different refrangibility, he made another considerable improvement in Telescopes, viz, by correcting, in a great measure, both this kind of aberration, and also that which arises from the spherical form of lenses, by an expedient of a very different nature, viz, increasing <pb n="575"/><cb/> the number of eye-glasses. If any person, says he, would have the visual angle of a Telescope to contain 20 degrees, the extreme pencils of the field must be bent or refracted in an angle of 10 degrees; which, if it be performed by one eye-glass, will cause an aberration from the figure, in proportion to the cube of that angle: but if two glasses be so proportioned and situated, as that the refraction may be equally divided between them, they will each of them produce a refraction equal to half the required angle; and therefore, the aberration being in this case proportional to double the cube of half the angle, will be but a 4th part of that which is in proportion to the cube of the whole angle; because twice the cube of 1 is but 1/4 of the cube of 2: so that the aberration from the figure, where two eye-glasses are rightly proportioned, is but a 4th part of what it must unavoidably be, where the whole is performed by a single eye-glass. By the same way of reasoning, when the refraction is divided among three glasses, the aberration will be found to be but the 9th part of what would be produced from a single glass; because 3 times the cube of 1 is but the 9th part of the cube of 3. Whence it appears, that by increasing the number of eye-glasses, the indistinctness, near the borders of the field of a Telescope, may be very much diminished, though not entirely taken away.</p><p>The method of correcting the errors arising from the different refrangibility of light, is of a different consideration from the former: for, whereas the errors from the figure can only be diminished in a certain proportion to the number of glasses, in this they may be entirely corrected, by the addition of only one glass; as we find in the astronomical Telescope, that two eye-glasses, rightly proportioned, will cause the edges of objects to appear free from colours quite to the borders of the field. Also, in the day telescope, where no more than two eye-glasses are absolutely necessary for erecting the object, we find, by the addition of a third rightly situated, that the colours, which would otherwise confuse the image, are entirely removed: but this must be understood with some limitation; for though the different colours, which the extreme pencils must necessarily be divided into by the edges of the eye-glasses, may in this manner be brought to the eye in a direction parallel to each other, so as, by its humours, to be converged to a point in the retina, yet if the glasses exceed a certain length, the colours may be spread too wide to be capable of being admitted through the pupil or aperture of the eye; which is the reason that, in long Telescopes, constructed in the common way, with three eye-glasses, the field is always very much contracted.</p><p>These considerations first set Mr. Dollond upon contriving how to enlarge the field, by increasing the number of eye-glasses, without any hindrance to the distinctness or brightness of the image: and though others had been about the same work before, yet observing that the five-glass Telescopes, sold in the shops, would admit of farther improvement, he endeavoured to construct one with the same number of glasses in a better manner; which so far answered his expectations, as to be allowed by the best judges to be a considerable improvement on the former. Encouraged by this success, he resolved to try if he could not make <cb/> some farther enlargement of the field, by the addition of another glass, and by placing and proportioning the glasses in such a manner, as to correct the aberrations as much as possible, without any detriment to the distinctness: and at last he obtained as large a field as is convenient or necessary, and that even in the longest Telescopes that can be made. These Telescopes, with 6 glasses, having been well received both at home and abroad, the author has settled the date of the invention in a letter addressed to Mr. Short, and read at the Royal Society, March 1, 1753. Philos. Trans. vol. 48, art. 14.</p><p>Of the Achromatic Telescopes, invented by Mr. Dollond, there are several different sizes, from one foot to 8 feet in length, made and sold by his sons P. and J. Dollond. In the 17 - inch improved Achromatic Telescope, the object glass is composed of three glasses, viz, two convex of crown-glass, and one concave of white flint-glass: the focal distance of this combined object-glass is about 17 inches, and the diameter of the aperture 2 inches. There are 4 eye-glasses contained in the tube, to be used for land objects; the magnifying power with these is near 50 times; and they are adjusted to different sights, and to different distances of the object, by turning a finger screw at the end of the outer tube. There is another tube, containing two eye-glasses that magnify about 70 times, for astronomical purposes. The Telescope may be directed to any object by turning two screws in the stand on which it is sixed, the one giving a vertical motion, and the other a horizontal oue. The stand may be inclosed in the inside of the brass tube.</p><p>The object-glass of the 2 1/2 and 3 1/2 feet Telescopes is composed of two glasses, one convex of crown glass, and the other concave of white flint glass; and the diameters of their apertures are 2 inches and 2 3/4 inches. Each of them is furnished with two tubes; one for land objects, containing four eye-glasses, and another with two eye-glasses for astronomical uses. They are adjusted by buttons on the outside of the wooden tube; and the vertical and horizontal motions are given by joints in the stands. The magnifying power of the least of these Telescopes, with the eye-glass for land objects, is near 50 times, and with those for astronomical purposes, 80 times; and that of the greatest for land objects is near 70 times, but for astronomical observations 80 and 130 times; for this has two tubes, either of which may be used as occasion requires. This Telescope is also moved by a screw and rackwork, and the screw is turned by means of a Hook's joint.</p><p>These opticians also construct an Achromatic pocket perspective glass, or Galilean Telescope; so contrived, that all the different parts are put together and contained in one piece 4 1/2 inches long. This small Telescope is furnished with 4 concave eye-glasses, the magnifying powers of which are 6, 12, 18, and 28 times. With the greatest power of this Telescope, the satellites of Jupiter and the ring of Saturn may be easily seen. They have also contrived an Achromatic Telescope, the sliding tubes of which are made of very thin brass, which pass through springs or tubes; the outside tube being either of mahogany or brass. These Telescopes, which from their convenience for gentlemen in the army are called military Telescopes, have 4 convex eye- <pb n="576"/><cb/> glasses, whose surfaces and focal lengths are so proportioned, as to render the field of view very large. They are of 4 different lengths and sizes, usually called one foot, 2, 3, and 4 feet: the first is 14 inches when in use, and 5 inches when shut up, having the aperture of the object-glass 1 1/10 inch, and magnifying 22 times: the second 28 inches for use, 9 inches shut up, the aperture 1 6/10 inch, and magnifying 35 times; the third 40 inches, and 10 inches shut, with the aperture 2 inches, and magnifying 45 times; and the sourth 52 inches, and 14 inches shut, with the aperture 2 3/4 inches, and magnifying 55 times.</p><p>Mr. Euler, who, in a memoir of the Academy of Berlin for the year 1757, p. 323, had calculated the effects of all possible combinations of lenses in Telescopes and microscopes, published another long memoir on the subject of these Telescopes, shewing with precision of what advantages they are naturally capable. See Miscel. Taurin. vol. 3, par. 2, pag. 92.</p><p>Mr. Caleb Smith, having paid much attention to the subject of shortening and improving Telescopes, thought he had found it possible to rectify the errors which arise from the different degrees of refrangibility, on the principle that the sines of refraction of rays differently refrangible, are to one another in a given proportion, when their sines of incidence are equal; and the method he proposed for this purpose, was to make the specula of glass, instead of metal, the two surfaces having different degrees of concavity. But it does not appear that this scheme was ever carried into practice. See Philos. Trans. number 456, pa. 326, or Abr. vol. 8, pa. 113.</p><p>The ingenious Mr. Ramsden has lately described a new construction of eye-glasses for such Telescopes as may be applied to mathematical instruments. The construction which he proposes, is that of two planoconvex lenses, both of them placed between the eye and the observed image formed by the object-glass of the instrument, and thus correcting not only the aberration arising from the spherical figure of the lenses, but also that arising from the different refrangibility of light. For a more particular account of this construction, its principle, and its effects, see Philos. Trans. vol. 73, art. 5.</p><p>A construction, similar at least in its principle to that above, is ascribed, in the Synopsis Optica Honorati Fabri, to Eustachio Divini, who placed two equal narrow plano-convex lenses, instead of one eye lens, to his Telescopes, which touched at their vertices; the focus of the object-glass coinciding with the centre of the plano-convex lens next it. And this, it is said, was done at once both to make the rays that come parallel from the object fall parallel upon the eye, to exclude the colours of the rainbow from it, to augment the angle of sight, the field of view, the brightness of the object, &c. This was also known to Huygens, who sometimes made use of the same construction, and gives the theory of it in his Dioptrics. See Hugenii Opera Varia, vol. 4, ed. 1728.</p><div2 part="N" n="Telescope" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Telescope</hi></head><p>, <hi rend="italics">Reflecting,</hi> or <hi rend="italics">Catoptric,</hi> or <hi rend="italics">Catadioptric,</hi> is a Telescope which, instead of lenses, consists chiefly of mirrors, and exhibits remote objects by reflection instead of refraction.</p><p>A brief account of the history of the invention of this <cb/> important and useful Telescope, is as follows. The ingenious Mr. James Gregory, of Aberdeen, has been commonly considered as the first inventor of this Telescope—But it seems the first thought of a reflector had been suggested by Mersenne, about 20 years before the date of Gregory's invention: a hint to this purpose occurs in the 7th proposition of his Catoptrics, which was printed in 1651: and it appears from the 3d and 29th letters of Descartes, in vol. 2 of his Letters, which it is said were written in 1639, though they were not published till the year 1666, that Mersenne proposed a Telescope with specula to Descartes in that correspondence; though indeed in a manner so very unsatisfactory, that Descartes, who had given particular attention to the improvement of the Telescope, was so far from approving the proposal, that he endeavoured to convince Mersenne of its fallacy. This point has been largely discussed by Le Roi in the Encyclopedia, art. Telescope, and by Montucla in his Hist. des Mathem. tom. 2, p. 643.</p><p>Whether Gregory had seen Mersenne's treatise on optics and catoptrics, and whether he availed himself of the hint there suggested, or not, perhaps cannot now be determined. He was led however to the inventi<*> by seeking to correct two imperfections in the common Telescope: the first of these was its too great length, which made it troublesome to manage; and the second was the incorrectness of the image. It had been already demonstrated, that a pencil of rays could not be collected in a single point by a spherical lens; and also, that the image transmitted by such a lens would be in some degree incurvated. These inconveniences he thought might be obviated by substituting for the object-glass a metallic speculum, of a parabolical figure, to receive the image, and to reflect it towards a small speculum of the same metal; this again was to return the image to an eye-glass placed behind the great speculum, which was, for that purpose, to be perforated in its centre. This construction he published in 1663, in his Optica Promota. But as Gregory, according to his own account, possessed no mechanical skill, and could not find a workman capable of realizing his invention, after some fruitless trials, he was obliged to give up the thoughts of bringing Telescopes of this kind into use.</p><p>Sir Isaac Newton however interposed, to save this excellent invention from perishing, and to bring it forward to maturity. Having applied himself to the improvement of the Telescope, and imagining that Gregory's specula were neither very necessary, nor likely to be executed, he began with prosecuting the views of Descartes; who aimed at making a more perfect image of an object, by grinding lenses, not to the figure of a sphere, but to that formed from one of the conic sections. But, in the year 1666, having discovered the different refrangibility of the rays of light, and finding that the errors of Telescopes, arising from that cause alone, were much more considerable than such as were occasioned by the spherical figure of lenses, he was constrained to turn his thoughts to reflectors. The plague however interrupted his progress in this business; so that it was towards the end of 1668, or in the beginning of 1669, when, despairing of perfecting Telescopes by means of refracted light, <pb n="577"/><cb/> and recurring to the construction of reflectors, he set about making his own specula, and early in the year 1672 completed two small reflecting Telescopes. In these he ground the large speculum into a spherical concave, being unable to accomplish the parabolic form proposed by Gregory; but though he then despaired of performing that work by geometrical rules, yet (as he writes in a letter that accompanied one of these instruments, which he presented to the Royal Society) he doubted not but that the thing might in some measure be accomplished by mechanical devices. With a perseverance equal to his ingenuity, he, in a great measure, overcame another difficulty, which was to find a metallic substance that would be of a proper hardness, have the fewest pores, and receive the smoothest polish: this difficulty he deemed almost insurmountable, when he considered that every irregularity in a reflecting surface would make the rays of light deviate 5 or 6 times more out of their due course, than the like irregularities in a refracting surface. After repeated trials, he at last found a composition that answered in some degree, leaving it to those who should come after him to find a better. These difficulties have accordingly been since obviated by other artists, particularly by Dr. Mudge, the rev. Mr. Edwards, and Dr. Herschel, &c. Newton having succeeded so far, he communicated to the Royal Society a full and satisfactory account of the construction and performance of his Telescope. The Society, by their secretary Mr. Oldenburgh, transmitted an account of the discovery to Mr. Huygens, celebrated as a distinguished improver of the refractor; who not only replied to the Society in terms expressing his high approbation of the invention, but drew up a favourable account of the new Telescope, which he caused to be published in the Journal des Sçavans of the year 1672, and by this mode of communication it was soon known over Europe. See Huygenii Opera Varia, tom. 4.</p><p>Notwithstanding the excellence and utility of this contrivance, and the honourable manner in which it was announced to the world, it seems to have been greatly neglected for nearly half a century. Indeed when Newton had published an account of his Telescopes in the Philos Trans. M. Cassegrain, a Frenchman, in the Journal des Sçavans of 1672, claimed the honour of a similar invention, and said, that, before he heard of Newton's improvement, he had hit upon a better construction, by using a small convex mirror instead of the reflecting prism. This Telescope, which was the Gregorian one disguised, the large mirror being perforated, and which it is said was never executed by the author, is much shorter than the Newtonian; and the convex mirror, by dispersing the rays, serves greatly to increase the image made by the large concave mirror.</p><p>Newton made many objections to Cassegrain's construction, but several of them equally affect that of Gregory, which has been found to answer remarkably well in the hands of good artists.</p><p>Dr. Smith took the pains to make many calculations of the magnifying power, both of Newton's and Cassegrain's Telescopes, in order to their farther improvement, which may be seen in his Optics, Rem. p. 97. <cb/></p><p>Mr. Short, it is also said, made several Telescopes on the plan of Cassegrain.</p><p>Dr. Hook constructed a Reflecting Telescope (mentioned by Dr. Birch in his Hist. of the Royal Soc. vol. 3, p. 122) in which the great mirror was perforated, so that the spectator looked directly towards the object, and it was produced before the Royal Society in 1674. On this occasion it was said that this construction was first proposed by Mersenne, and afterwards repeated by Gregory, but that it never had been actually executed before it was done by Hook. A description of this instrument may be seen in Hook's Experiments, by Derham, p. 269.</p><p>The Society also made an unsuccessful attempt, by employing an artisicer to imitate the Newtonian construction; however, about half a century after the invention of Newton, a Reflecting Telescope was produced to the world, of the Newtonian construction, which the venerable author, ere yet he had finished his very distinguished course, had the satisfaction to find executed in such a manner, as left no room to fear that the invention would longer continue in obscurity. This effectual service to science was accomplished by Mr. John Hadley, who, in the year 1723, presented to the Royal Society a Telescope, which he had constructed upon Newton's plan. The two Telescopes which Newton had made, were but 6 inches long, were held in the hand for viewing objects, and in power were compared to a 6-feet refractor: but the radius of the sphere, to which the principal speculum of Hadley's was ground, was 10 feet 5 1/4 inches, and consequently its focal length was 62 5/8 inches. In the Philos. Trans. Abr. vol. 6, p. 133, may be seen a drawing and description of this Telescope, and also of a very ingenious but complex apparatus, by which it was managed. One of these Telescopes, in which the focal length of the large mirror was not quite 5 1/4 feet, was compared with the celebrated Huygenian Telescope, which had the focal length of its object-glass 123 feet; and it was found that the former would bear such a charge, as to make it magnify the object as many times as the latter with its due charge; and that it represented objects as distinctly, though not altogether so clear and bright. With this Reflecting Telescope might be seen whatever had been hitherto discovered by the Huygenian, particularly the transits of Jupiter's satellites, and their shades over the disk of Jupiter, the black list in Saturn's ring, and the edge of the shade of Saturn cast upon his ring. Five satellites of Saturn were also observed with this Telescope, and it afforded other observations on Jupiter and Saturn, which confirmed the good opinion which had been conceived of it by Pound and Bradley.</p><p>Mr. Hadley, after finishing two Telescopes of the Newtonian construction, applied himself to make them in the Gregorian form, in which the large mirror is perforated. This scheme he completed in the year 1726.</p><p>Dr. Smith prefers the Newtonian construction to that of Gregory; but if long experience be admitted as a final judge in such matters, the superiority must be adjudged to the latter; as it is now, and has been for many years past, the only instrument in request. <pb n="578"/><cb/></p><p>Mr. Hadley spared no pains, after having completed his construction, to instruct Mr. Molyneux and Dr. Bradley; and when these gentlemen had made a good proficiency in the art, being desirous that these Telescopes should become more public, they liberally communicated to some of the chief instrument makers of London, the knowledge they had acquired from him: and thus, as it is reasonable to imagine, reflectors were completed by other and better methods than even those in which they had been instructed. Mr. James Short in particular signalized himself as early as the year 1734, by his work in this way. He at first made his specula of glass; but finding that the light reflected from the best glass specula was much less than the light reflected from metallic ones, and that glass was very liable to change its form by its own weight, he applied himself to improve metallic specula; and, by giving particular attention to the curvature of them, he was able to give them greater apertures than other workmen could do; and by a more accurate adjustment of the specula, &c, he greatly improved the whole instrument. By some which he made, in which the larger mirror was 15 inches focal distance, he and some other persons were able to read in the Philos. Trans. at the distance of 500 feet; and they several times saw the five satellites of Saturn together, which greatly surprised Mr. Maclaurin, who gave this account of it, till he found that Cassini had sometimes seen them all with a 17 feet refractor. Short's Telescopes were all of the Gregorian construction. It is supposed that he discovered a method of giving the parabolic figure to his great speculum; a degree of perfection which Gregory and Newton despaired of attaining, and which Hadley it seems had never attempted in either of his Telescopes. However, the secret of working that configuration, whatever it was, it seems died with that ingenious artist. Though lately in some degree discovered by Dr. Mudge and others.</p><p>On the History of Reflecting Telescopes, see Dr. David Gregory's Elem. of Catopt. and Dioptr. Appendix by Desaguliers: Smith's Optics, book 3, c. 2, Rem. on art. 489: and Sir John Pringle's excellent Discourse on the Invention &c of the Reflecting Telescope.</p><p><hi rend="italics">Construction of the Reflecting Telescope of the Newtonian form.</hi>—Let ABCD (fig. 2, pl. 32) be a large tube, open at AD, and closed at BC, and its length at least equal to the distance of the focus from the metallic spherical concave speculum GH placed at the end BC. The rays EG, FH, &c, proceeding from a remote object PR, intersect one another somewhere before they enter the tube, so that EG and <hi rend="italics">eg</hi> are those that come from the lower part of the object, and <hi rend="italics">fh</hi> FH from its upper part: these rays, after falling on the speculum GH, will be reflected so as to converge and meet in <hi rend="italics">mn,</hi> where they will form a perfect image of the object. But as this image cannot be seen by the spectator, they are intercepted by a small plane metallic speculum KK, intersecting the axis at an angle of 45°, by which the rays tending to <hi rend="italics">m, n,</hi> will be reflected towards a hole LL in the side of the tube, and the image of the object will be thus formed in <hi rend="italics">q</hi>S; which image will be less distinct, because some of the rays which would otherwise fall on the concave speculum <cb/> GH, are intercepted by the plane speculum: it will nevertheless appear pretty distinct, because the aperture AD of the tube, and the speculum GH, are large. In the lateral hole LL is fixed a convex lens, whose focus is at S<hi rend="italics">q;</hi> and therefore this lens will refract the rays that proceed from any point of the image, so as at their exit they will appear parallel, and those that proceed from the extreme points S, <hi rend="italics">q,</hi> will converge after refraction, and form an angle at O, where the eye is placed; which will see the image S<hi rend="italics">q,</hi> as if it were an object, through the lens LL: consequently the object will appear enlarged, inverted, bright, and distinct. In LL may be placed lenses of different convexities, which, by being moved nearer to the image and farther from it, will represent the object more or less magnified, if the surface of the speculum GH be of a figure truly spherical. If, instead of one lens LL, three lenses be disposed in the same manner with the three eye-glasses of the refracting Telescope, the object will appear erect, but less distinct than when it is observed with one lens. On account of the position of the eye in this Telescope, it is extremely difficult to direct the instrument towards any object: Huygens therefore first thought of adding to it a small refracting Telescope, having its axis parallel to that of the reflector: this is called a <hi rend="italics">finder</hi> or <hi rend="italics">director.</hi> The Newtonian Telescope is also furnished with a suitable apparatus for the commodious use of it.</p><p>To determine the magnifying power of this Telescope, it is to be considered that the plane speculum KK is of no use in this respect: let us then suppose that one ray proceeding from the object coincides with the axis GLIA of the lens and speculum; let <hi rend="italics">bb</hi> be <figure/> another ray proceeding from the lower extremity of the object, and passing through the focus I of the speculum KH; this will be reflected in the direction <hi rend="italics">bid,</hi> parallel to the axis GLA, and falling on the lens <hi rend="italics">d</hi>L<hi rend="italics">d,</hi> will be refracted to G, so that GL will be equal to LI, and <hi rend="italics">d</hi>G = <hi rend="italics">d</hi>I. To the naked eye the object would appear under the angle I<hi rend="italics">bi</hi> = <hi rend="italics">b</hi>IA; but by means of the Telescope it appears under the angle <hi rend="italics">d</hi>GL = <hi rend="italics">d</hi>IL = I<hi rend="italics">di:</hi> and the angle I<hi rend="italics">di</hi> is to the angle I<hi rend="italics">bi</hi> as <hi rend="italics">b</hi>I to I<hi rend="italics">d;</hi> consequently the apparent magnitude by the Telescope, is to that with the naked eye, as the distance of the focus of the speculum from the speculum, to the distance of the focus of the lens from the lens.</p><p><hi rend="italics">Construction of the Gregorian Reflecting Telescope.</hi>— Let TYYT (fig. 3, pl. 32) be a brass tube, in which E<hi rend="italics">ld</hi>D is a metallic concave speculum, perforated in the middle at X; and EF a less concave mirror, so sixed by the arm or strong wire RT, which is moveable by means of a long screw on the outside of the tube, as to be moved nearer to, or farther from the larger speculum L<hi rend="italics">ld</hi>D; its axis being kept in the same line with that of the great one. Let AB represent a very remote object, from each part of which issue pencils of rays, as <hi rend="italics">cd,</hi> CD, from A the upper extremity of the <pb n="579"/><cb/> object, and IL, <hi rend="italics">il,</hi> from the lower part B; the rays IL, CD, from the extremities, crossing one another before they enter the tube. These rays, falling upon the larger mirror LD, are reflected from it into the focus KH, where they form an inverted image of the object AB, as in the Newtonian Telescope. From this image the rays, issuing as from an object, fall upon the small mirror EF, the centre of which is at <hi rend="italics">e,</hi> so that after reflection they would meet in their foci at QQ, and there form an erect image. But since an eye at that place could see but a small part of an object, in order to bring rays from more distant parts of it into the pupil, they are intercepted by the plano-convex lens MN, by which means a smaller erect image is formed at PV, which is viewed through the meniscus SS, by an eye at O. This meniscus both makes the rays of each pencil parallel, and magnifies the image PV. At the place of this image all the foreign rays are intercepted by the perforated partition ZZ. For the same reason the hole near the eye O is very narrow. When nearer objects are viewed by this Telescope, the small speculum EF is removed to a greater distance from the larger LD, so that the second image may be always formed in PV: and this distance is to be adjusted (by means of the screw on the outside of the great tube) according to the form of the eye of the spectator. It is also necessary that the axis of the Telescope should pass through the middle of the speculum EF, and its centre, the centre of the speculum LL, and the middle of the hole X, the centres of the lenses MN, SS, and the hole near O. As the hole X in the speculum LL can reflect none of the rays issuing from the object, that part of the image which corresponds to the middle of the object, must appear to the observer more dark and confused than the extreme parts of it. Besides, the speculum EF will also intercept many rays proceeding from the object; and therefore, unless the aperture TT be large, the object must appear in some degree obscure.</p><p>The magnifying power of this Telescope is estimated in the following manner. Let LD be the larger mirror (fig. 3, pl. 31), having its focus at G, and aperture in A; and FF the small mirror with the focus of parallel rays in I, and the axis of both the specula and lenses MN, SS, be in the right line DIGAOK. Let <hi rend="italics">bb</hi> be a ray of light coming from the lower extremity of a very distant visible object, passing through the focus G, and falling upon the point <hi rend="italics">b</hi> of the speculum LD; which, after being reflected from <hi rend="italics">b</hi> to F in a direction parallel to the axis of the mirror DAK, is reflected by the speculum F so as to pass through the focus I in the direction FIN to N, at the extremity of the lens MN, by which it would have been refracted to K; but by the interposition of another lens SS is brought to O, so that the eye in O sees half the object under the angle TOS. The angle G<hi rend="italics">b</hi>F, or AG<hi rend="italics">b,</hi> under which the object is viewed by the naked eye, is to SOT under which it is viewed by the Telescope, in the ratio of G<hi rend="italics">b</hi>F to IF<hi rend="italics">i</hi> = <hi rend="italics">n</hi>IN, of <hi rend="italics">n</hi>IN to NK<hi rend="italics">n,</hi> and of NK<hi rend="italics">n</hi> to SOT. <hi rend="center">But G<hi rend="italics">b</hi>F : IF<hi rend="italics">i</hi> :: DI : GA,</hi> <hi rend="center">and <hi rend="italics">n</hi>IN : <hi rend="italics">n</hi>KN :: <hi rend="italics">n</hi>K : <hi rend="italics">n</hi>I,</hi> <hi rend="center">and <hi rend="italics">n</hi>KN : SOT :: TO : TK;</hi> <cb/> theref. G<hi rend="italics">b</hi>F : SOT :: DI X <hi rend="italics">n</hi>K X TO : GA X <hi rend="italics">n</hi>I X TK. Musschenbroek's Introd. vol. 2, p. 819.</p><p>In Reflecting Telescopes of different lengths, a given object will appear equally bright and equally distinct, when their linear apertures, and also their linear breadths, are as the 4th roots of the cubes of their lengths; and consequently when the focal distances of their eyeglasses are also as the 4th roots of their lengths. See the demonstration of this proposition in Smith's Optics, art. 361.</p><p>Hence he has deduced a rule, by which he has computed the following table for Telescopes of different lengths, taking, for a standard, the middle eye-glass and aperture of Hadley's Reflecting Telescope, described in Philos. Trans. number 376 and 378: the focal distances and linear apertures being given in 1000th parts of an inch. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="italics">Table for Telescopes of different Lengths.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Length of</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Focal dist.</cell><cell cols="1" rows="1" role="data">Linear am-</cell><cell cols="1" rows="1" role="data">Linear a-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">the Tel. or</cell><cell cols="1" rows="1" role="data">plifying or</cell><cell cols="1" rows="1" role="data">perture of</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">focal dist. of</cell><cell cols="1" rows="1" role="data">of the</cell><cell cols="1" rows="1" role="data">magnifying</cell><cell cols="1" rows="1" role="data">the concave</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">the cone.</cell><cell cols="1" rows="1" role="data">Eye-glass.</cell><cell cols="1" rows="1" role="data">power.</cell><cell cols="1" rows="1" role="data">metal.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">feet</cell><cell cols="1" rows="1" role="data">inches</cell><cell cols="1" rows="1" role="data">---</cell><cell cols="1" rows="1" role="data">inches</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 1/2</cell><cell cols="1" rows="1" role="data">0.167</cell><cell cols="1" rows="1" role="data"> 36</cell><cell cols="1" rows="1" role="data"> 0.864</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">0.199</cell><cell cols="1" rows="1" role="data"> 60</cell><cell cols="1" rows="1" role="data"> 1.440</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">0.236</cell><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" role="data"> 2.448</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">0.261</cell><cell cols="1" rows="1" role="data">138</cell><cell cols="1" rows="1" role="data"> 3.312</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">0.281</cell><cell cols="1" rows="1" role="data">171</cell><cell cols="1" rows="1" role="data"> 4.104</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">0.297</cell><cell cols="1" rows="1" role="data">202</cell><cell cols="1" rows="1" role="data"> 4.843</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">0.311</cell><cell cols="1" rows="1" role="data">232</cell><cell cols="1" rows="1" role="data"> 5.568</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">0.323</cell><cell cols="1" rows="1" role="data">260</cell><cell cols="1" rows="1" role="data"> 6.240</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">0.334</cell><cell cols="1" rows="1" role="data">287</cell><cell cols="1" rows="1" role="data"> 6.888</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">0.344</cell><cell cols="1" rows="1" role="data">314</cell><cell cols="1" rows="1" role="data"> 7.536</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.353</cell><cell cols="1" rows="1" role="data">340</cell><cell cols="1" rows="1" role="data"> 8.160</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0.362</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" role="data"> 8.760</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0.367</cell><cell cols="1" rows="1" role="data">390</cell><cell cols="1" rows="1" role="data"> 9.360</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0.377</cell><cell cols="1" rows="1" role="data">414</cell><cell cols="1" rows="1" role="data"> 9.936</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0.384</cell><cell cols="1" rows="1" role="data">437</cell><cell cols="1" rows="1" role="data">10.488</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">0.391</cell><cell cols="1" rows="1" role="data">460</cell><cell cols="1" rows="1" role="data">11.040</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0.397</cell><cell cols="1" rows="1" role="data">483</cell><cell cols="1" rows="1" role="data">11.592</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0.403</cell><cell cols="1" rows="1" role="data">506</cell><cell cols="1" rows="1" role="data">12.143</cell></row></table></p><p>Mr. Hadley's Telescope, above mentioned, magnified 228 or 230 times; but we are informed that an object-metal of 3 1/4 feet focal distance was wrought by Mr. Hauksbee to so great a perfection, as to magnify 226 times, and therefore it was scarcely inferior to Hadley's of 5 1/2 feet. If Hauksbee's Telescope be taken for a new standard, it follows that a speculum of one foot focal distance ought to magnify 93 times, whereas the above table allows it but 60. Now 93/60 = 1.55, and the given column of magnifying powers multiplied by this number, gives a new column, shewing how much the object-metals ought to magnify if wrought up to the perfection of Hauksbee's. And thus a new table may be easily made for this or any other more perfect standard, taking also the new eye-glasses and apertures in the same ratio to one another as the old ones have in this table. Smith's Optics, Rem. p. 79. <pb n="580"/><cb/></p><p>The magnifying power of any Telescope may be easily found by experiment, viz, by looking with one eye through the Telescope upon an object of known dimensions, and at a given distance, and throwing the image upon another object seen with the naked eye. Dr. Smith has given a particular account of the process, Rem. p. 79.</p><p>But the easiest method of all, is to measure the diameter of the aperture of the object-glass, and that of the little image of it, which is formed at the place of the eye. For the proportion between these gives the ratio of the magnifying power, provided no part of the original pencil be intercepted by the bad construction of the Telescope. For in all cases the magnifying power of Telescopes, or microscopes, is measured by the proportion of the diameter of the original pencil, to that of the pencil which enters the eye. Priestley's Hist. of Light, p. 747.</p><p>But the most considerable, and indeed truly astonishing magnifying powers, that have ever been used, are those of Dr. Herschel's Reflecting Telescopes. Some account of these, and of the discoveries made by them, has been already introduced under the article Star. For his method of ascertaining them, see Philos. Trans. vol. 72, pa. 173 &c. See also several of the other late volumes of the Philos. Trans.</p><p>Dr. Herschel observes, that though opticians have proved, that two eye-glasses will give a more correct image than one, he has always (from experience) persisted in refusing the assistance of a second glass, which is sure to introduce errors greater than those he would correct. “Let us resign, says he, the double eyeglass to those who view objects merely for entertainment, and who must have an exorbitant field of view. To a philosopher, this is an unpardonable indulgence. I have tried both the single and double eye-glass of equal powers, and always found that the single eye-glass had much the superiority in point of light and distinctness. With the double eye-glass I could not see the belts in Saturn, which I very plainly saw with the single one. I would however except all those cases where a large field is absolutely necessary, and where power joined to distinctness is not the sole object of our view.” Philos. Trans. vol. 72, p. 95.</p><p>Mr. Green of Deptford has lately added both to the reflecting and refracting Telescope an apparatus, which fits it for the purposes of surveying, levelling, measuring angles and distances, &c. See his Description and Use of the improved Reflecting and Refracting Telescopes, and Scale of Surveying &c, 1778.— Mr. Ramsden too has lately adapted Telescopes to the like purpose of measuring distances from one station, &c.</p><p><hi rend="italics">Meridian</hi> <hi rend="smallcaps">Telescope</hi>, is one that is fixed at right angles to an axis, and turned about it in the plane of the meridian; and is otherwise called a <hi rend="italics">transit instrument.</hi>—The common use of it is to correct the motion of a clock or watch, by daily observing the exact time when the sun or a star comes to the meridian. It serves also for a variety of other uses. The transverse axis is placed horizontal by a spirit level. For the farther description and method of fixing this instrument by means of its levels &c, see Smith's Optics, p. 321. See also <hi rend="smallcaps">Transit</hi> <hi rend="italics">Instrument.</hi> <cb/></p><p>TELESCOPICAL <hi rend="italics">Stars,</hi> are such as are not visible to the naked eye, being only discernible by means of a telescope. See <hi rend="smallcaps">Star.</hi></p><p>All stars less than those of the 6th magnitude, are Telescopic to an ordinary eye.</p></div2></div1><div1 part="N" n="TEMPERAMENT" org="uniform" sample="complete" type="entry"><head>TEMPERAMENT</head><p>, in Music, usually denotes a rectifying or amending the false or imperfect concords, by transferring to them part of the beauty of the perfect ones.</p></div1><div1 part="N" n="TENACITY" org="uniform" sample="complete" type="entry"><head>TENACITY</head><p>, in Natural Philosophy, is that quality of bodies by which they sustain a considerable pressure or force without breaking; and is the opposite quality to fragility or brittleness. Mem. Acad. Berlin. 1745, p. 47.</p></div1><div1 part="N" n="TENAILLE" org="uniform" sample="complete" type="entry"><head>TENAILLE</head><p>, in Fortification, a kind of outwork, consisting of two parallel sides, with a front, having a re-entering angle. In fact, that angle, and the faces which compose it, are the Tenaille.</p><p>The Tenaille is of two kinds, <hi rend="italics">simple</hi> and <hi rend="italics">double.</hi></p><p><hi rend="italics">Simple</hi> or <hi rend="italics">Single</hi> <hi rend="smallcaps">Tenaille</hi>, is a large outwork, consisting of two faces or sides, including a re-entering angle.</p><p><hi rend="italics">Double,</hi> or <hi rend="italics">Flanked</hi> <hi rend="smallcaps">Tenaille</hi>, is a large outwork, consisting of two simple Tenailles, or three saliant and two re-entering angles.</p><p>The great defects of Tenailles are, that they take up too much room, and on that account are advantageous to the enemy; that the re-entering angle is not defended; the height of the parapet preventing the seeing down into it, so that the enemy can lodge there under cover; and the sides are not sufficiently flanked. For these reasons, Tenailles are now mostly excluded out of fortification by the best engineers, and never made but where time does not serve to form a hornwork.</p><p><hi rend="smallcaps">Tenaille</hi> <hi rend="italics">of the Place,</hi> is the front of the place, comprehended between the points of two neighbouring bastions; including the curtain, the two flanks raised on the curtain, and the two sides of the bastions which face one another. So that the Tenaille, in this sense, is the same with what is otherwise called the <hi rend="italics">face of a fortress.</hi></p><p><hi rend="smallcaps">Tenaille</hi> <hi rend="italics">of the Ditch,</hi> is a low work raised before the curtain, in the middle of the foss or ditch; the parapet of which is only 2 or 3 feet higher than the level ground of the ravelin.</p><p>The use of Tenailles in general, is to defend the bottom of the ditch by a grazing fire, and likewise the level ground of the ravelin, which cannot be so conveniently defended from any other place. The first sort do not defend the ditch so well as the others, because they are too oblique a defence; but as they are not subject to be enfiladed, Vauban has generally preferred them in the fortifying of places. Those of the second sort defend the ditch much better than the first, and add a low flank to those of the bastions; but as these flanks are liable to be enfiladed, they have not been much used. This defect however might be remedied, by making them so as to be covered by the extremities of the parapets of the opposite ravelins, or by some other work. And the same thing may be said of the third sort as of the second.</p><p>The <hi rend="italics">Ram's-horn</hi> is a curved Tenaille, raised in the foss before the flanks, and presenting its convexity to <pb n="581"/><cb/> the covered way. This work seems preferable to either of the other Tenailles, both on account of its simplicity, and the defence for which it is constructed.</p></div1><div1 part="N" n="TENAILLONS" org="uniform" sample="complete" type="entry"><head>TENAILLONS</head><p>, in Fortification, are works constructed on each side of the ravelin, much like the lunettes. They differ, as one of the faces of a Tenaillon is in the direction of the ravelin, whereas that of the lunette is perpendicular to it.</p></div1><div1 part="N" n="TENOR" org="uniform" sample="complete" type="entry"><head>TENOR</head><p>, in Music, the first mean or middle part, or that which is the ordinary pitch, or Tenor, of the voice, when not either raised to the treble, or lowered to the bass.</p></div1><div1 part="N" n="TENSION" org="uniform" sample="complete" type="entry"><head>TENSION</head><p>, the state of a thing tight, or stretched. Thus, animals sustain and move themselves by the Tension of their muscles and nerves. A chord, or string, gives an acuter or deeper sound, as it is in a greater or less degree of Tension, that is, more or less stretched or tightened.</p></div1><div1 part="N" n="TERM" org="uniform" sample="complete" type="entry"><head>TERM</head><p>, in Geometry, is the extreme of any magnitude, or that which bounds and limits its extent. So the Terms of a line, are points; of a superficies, lines; of a solid, superficies.</p><div2 part="N" n="Terms" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Terms</hi></head><p>, of an equation, or of any quantity, in Algebra, are the several names or members of which it is composed, separated from one another by the signs + or -. So, the quantity <hi rend="italics">ax</hi> + 2<hi rend="italics">bc</hi> - 3<hi rend="italics">ax</hi><hi rend="sup">2</hi>, consists of the three Terms <hi rend="italics">ax</hi> and 2<hi rend="italics">bc</hi> and 3<hi rend="italics">ax</hi><hi rend="sup">2</hi>.</p><p>In an equation, the Terms are the parts which contain the several powers of the same unknown letter or quantity: for if the same unknown quantity be found in several members in the same degree or power, they shall pass but for one Term, which is called a compound one, in distinction from a simple or single Term. Thus, in the equation , the <cb/> four terms are <hi rend="italics">x</hi><hi rend="sup">3</hi> and (―(<hi rend="italics">a</hi> - 3<hi rend="italics">b</hi>)).<hi rend="italics">x</hi><hi rend="sup">2</hi> and <hi rend="italics">acx</hi> and <hi rend="italics">b</hi><hi rend="sup">3</hi>; of which the second Term (―(<hi rend="italics">a</hi> - 3<hi rend="italics">b</hi>)).<hi rend="italics">x</hi><hi rend="sup">2</hi> is compound, and the other three are simple Terms.</p></div2><div2 part="N" n="Terms" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Terms</hi></head><p>, of a Product, or of a Fraction, or of a Ratio, or of a Proportion, &c, are the several quantities employed in forming or composing them. Thus, the Terms of the product <hi rend="italics">ab,</hi> are <hi rend="italics">a</hi> and <hi rend="italics">b;</hi> of the fraction 5/8, are 5 and 8; of the ratio 6 to 7, are 6 and 7; of the proportion <hi rend="italics">a</hi> : <hi rend="italics">b</hi> :: 5 : 9, are <hi rend="italics">a, b,</hi> 5, 9.</p><p><hi rend="smallcaps">Terms</hi> are also used for the several times or seasons of the year in which the public colleges or universities, or courts of law, are open, or sit. Such are the Oxford and Cambridge Terms; also the Terms for the courts of King's-Bench, Common Pleas, and the Exchequer, which are the high courts of common law. But the high court of Parliament, the Chancery, and inferior courts, do not observe the Terms.—The rest of the year, out of Term-time, is called <hi rend="italics">vacation.</hi></p><p>There are four law Terms in the year; viz,</p><p><hi rend="italics">Hilary-Term,</hi> which, at London, begins the 23d day of January, and ends the 12th of February.</p><p><hi rend="italics">Easter-Term,</hi> which begins the 3d Wednesday after Easter-day, and ends on the Monday next after Ascension-day.</p><p><hi rend="italics">Trinity-Term,</hi> which begins the Friday next after Trinity-Sunday, and ends the 4th Wednesday after Trinity-Sunday.</p><p><hi rend="italics">Michaelmas-Term,</hi> which begins the 6th of November, and ends the 28th of November.</p><p>All these terms have also their returns, the days of which are expressed in the following table of synopsis. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=8" role="data"><hi rend="italics">Table of the Law Terms, and their Returns.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Term</cell><cell cols="1" rows="1" role="data">Begin.</cell><cell cols="1" rows="1" role="data">1st Return</cell><cell cols="1" rows="1" role="data">2d Return</cell><cell cols="1" rows="1" role="data">3d Return</cell><cell cols="1" rows="1" role="data">4th Return</cell><cell cols="1" rows="1" role="data">5th Return</cell><cell cols="1" rows="1" role="data">End.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Hilary</cell><cell cols="1" rows="1" role="data">January 23</cell><cell cols="1" rows="1" role="data">January 20</cell><cell cols="1" rows="1" role="data">January 27</cell><cell cols="1" rows="1" role="data">February 3</cell><cell cols="1" rows="1" role="data">February 9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">February 12</cell></row><row role="data"><cell cols="1" rows="1" role="data">Easter</cell><cell cols="1" rows="1" role="data">3 Wed. af. East.</cell><cell cols="1" rows="1" role="data">2 Wks. af. East.</cell><cell cols="1" rows="1" role="data">3 Wks. af. East.</cell><cell cols="1" rows="1" role="data">4 Wks. af. East.</cell><cell cols="1" rows="1" role="data">5 Wks. af. East.</cell><cell cols="1" rows="1" role="data">Ascens. day</cell><cell cols="1" rows="1" role="data">Mond. af. Ascens.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Trinity</cell><cell cols="1" rows="1" role="data">Frid. af. Trin. S.</cell><cell cols="1" rows="1" role="data">Trinity Mond.</cell><cell cols="1" rows="1" role="data">1 Wk. af. Trin.</cell><cell cols="1" rows="1" role="data">2 Wks. af. Trin.</cell><cell cols="1" rows="1" role="data">3 Wks. af. Trin.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4th Wed. af. Trin. S.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Mich.</cell><cell cols="1" rows="1" role="data">November 6</cell><cell cols="1" rows="1" role="data">November 3</cell><cell cols="1" rows="1" role="data">November 12</cell><cell cols="1" rows="1" role="data">November 18</cell><cell cols="1" rows="1" role="data">November 25</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">November 28</cell></row></table> <hi rend="center">N. B. When the beginning or ending of any of these Terms happens on a Sunday, it is held on the Monday after.</hi><cb/></p><p><hi rend="italics">Oxford</hi> <hi rend="smallcaps">Terms.</hi> These are four; which begin and end as below: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Terms</cell><cell cols="1" rows="1" role="data">Begin.</cell><cell cols="1" rows="1" role="data">End.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lent Term</cell><cell cols="1" rows="1" role="data">January 14</cell><cell cols="1" rows="1" role="data">Sat. bef. Palm-Sund.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Easter Term</cell><cell cols="1" rows="1" role="data">Wed. af. Low-Sund.</cell><cell cols="1" rows="1" role="data">Thurs. bef. Whitsun.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Trinity Term</cell><cell cols="1" rows="1" role="data">Wed. af. Trin. Sund.</cell><cell cols="1" rows="1" role="data">Sat. after the Act.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Michaelmas T.</cell><cell cols="1" rows="1" role="data">October 10</cell><cell cols="1" rows="1" role="data">December 17</cell></row></table></p><p>N. B. The <hi rend="italics">Act</hi> is 1st Monday after the 6th of July. —When the day of the beginning or ending happens on a Sunday, the Terms begin or end the day after. <cb/></p><p><hi rend="italics">Cambridge</hi>-<hi rend="smallcaps">Terms.</hi> These are three, as below: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Terms</cell><cell cols="1" rows="1" role="data">Begin.</cell><cell cols="1" rows="1" role="data">End.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lent Term</cell><cell cols="1" rows="1" role="data">January 13</cell><cell cols="1" rows="1" role="data">Frid. bef. Palm-Sund.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Easter Term</cell><cell cols="1" rows="1" role="data">Wed. aft. Low-Sund.</cell><cell cols="1" rows="1" role="data">Frid. aft. Commence.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Michaelmas T.</cell><cell cols="1" rows="1" role="data">October 10</cell><cell cols="1" rows="1" role="data">December 16</cell></row></table></p><p>N. B. The <hi rend="italics">Commencement</hi> is the 1st Tuesday in July. —There is no difference on account of the beginning or ending being Sunday. <pb n="582"/><cb/></p><p><hi rend="italics">Scottish</hi> <hi rend="smallcaps">Terms.</hi> In Scotland, <hi rend="italics">Candlemas Term</hi> begins January 23d, and ends February the 12th. <hi rend="italics">Whitsuntide-Term</hi> begins May 25th, and ends June 15th. <hi rend="italics">Lammas-Term</hi> begins July the 20th, and ends August the 8th. <hi rend="italics">Martin<*>as-Term</hi> begins November the 3d, and ends November the 29th.</p><p><hi rend="italics">Irish</hi> <hi rend="smallcaps">Terms.</hi> In Ireland the Terms are the same as at London, except <hi rend="italics">Michaelmas-Term,</hi> which begins October the 13th, and adjourns to November the 3d, and thence to the 6th.</p></div2></div1><div1 part="N" n="TERMINATOR" org="uniform" sample="complete" type="entry"><head>TERMINATOR</head><p>, in Astronomy, a name sometimes given to the circle of illumination, from its property of terminating the boundaries of light and darkness.</p></div1><div1 part="N" n="TERRA" org="uniform" sample="complete" type="entry"><head>TERRA</head><p>, in Geography. See <hi rend="smallcaps">Earth.</hi></p><p><hi rend="smallcaps">Terra</hi>-<hi rend="italics">firma,</hi> in Geography, is sometimes used for a continent, in contradistinction to islands. Thus, Asia, the Indies, and South America, are usually distinguished into Terra firmas and islands.</p></div1><div1 part="N" n="TERRAQUEOUS" org="uniform" sample="complete" type="entry"><head>TERRAQUEOUS</head><p>, in Geography, an epithet given to our globe or earth, considered as consisting of land and water, which together constitute one mass.</p><p>TERRE-<hi rend="smallcaps">PLEIN</hi>, or <hi rend="smallcaps">Terre-plain</hi>, in Fortification, the top, platform, or horizontal surface of the rampart, upon which the cannon are placed, and where the defenders perform their office. It is so called, because it lies level, having only a little slope outwardly to counteract the recoil of the cannon. Its breadth is from 24 to 30 feet; being terminated by the parapet on the outer side, and inwardly by the inner talus.</p></div1><div1 part="N" n="TERRELLA" org="uniform" sample="complete" type="entry"><head>TERRELLA</head><p>, or little earth, is a magnet turned of a spherical figure, and placed so as that its poles, equator, &c, do exáctly correspond with those of the world. It was so first called by Gilbert, as being a just representation of the great magnetic globe we inhabit. Such a Terrella, it was supposed, if nicely poised, and hung in a meridian like a globe, would be turned round like the earth in 24 hours by the magnetic particles pervading it; but experience has shewn that this is a mistake.</p></div1><div1 part="N" n="TERRESTRIAL" org="uniform" sample="complete" type="entry"><head>TERRESTRIAL</head><p>, something relating to the earth. As Terrestrial globe, Terrestrial line, &c.</p><p>TERTIAN; denotes an old measure, containing 84 gallons, so called because it is the 3d part of a tun.</p></div1><div1 part="N" n="TERTIATE" org="uniform" sample="complete" type="entry"><head>TERTIATE</head><p>, in Gunnery. To Tertiate a great gun, is to examine the thickness of the metal at the muzzle, by which to judge of the strength of the piece, and whether it be sufficiently fortified or not.</p></div1><div1 part="N" n="TETRACHORD" org="uniform" sample="complete" type="entry"><head>TETRACHORD</head><p>, in Music, called by the moderns a <hi rend="italics">fourth,</hi> is a concord or interval of four tones.—The Tetrachord of the ancients, was a rank of four strings, accounting the Tetrachord for one tone, as it is often taken in music.</p></div1><div1 part="N" n="TETRADIAPASON" org="uniform" sample="complete" type="entry"><head>TETRADIAPASON</head><p>, or <hi rend="italics">quadruple diapason,</hi> is a musical chord, otherwise called a quadruple eighth, or a nine and twentieth.</p></div1><div1 part="N" n="TETRAEDRON" org="uniform" sample="complete" type="entry"><head>TETRAEDRON</head><p>, or <hi rend="smallcaps">Tetrahedron</hi>, in Geometry, is one of the five Platonic or regular bodies or solids, comprehended under four equilateral and equal triangles. Or it is a triangular pyramid of four equal and equilateral faces.</p><p>It is demonstrated in geometry, that the side of a Tetraedron is to the diameter of its circumscribing sphere, as √2 to √3; consequently they are incommensurable. <cb/></p><p>If <hi rend="italics">a</hi> denote the linear edge or side of a Tetraedron, <hi rend="italics">b</hi> its whole superficies, <hi rend="italics">c</hi> its solidity, <hi rend="italics">r</hi> the radius of its inscribed sphere, and R the radius of its circumscribing sphere; then the general relation among all these is expressed by the following equations, viz, <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> = 2<hi rend="italics">r</hi>√6</cell><cell cols="1" rows="1" role="data">= (2/3)R√6</cell><cell cols="1" rows="1" role="data">= √((1/3)<hi rend="italics">b</hi>√3)</cell><cell cols="1" rows="1" role="data">= √<hi rend="sup">3</hi>(6<hi rend="italics">c</hi>√2).</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi> = 24<hi rend="italics">r</hi><hi rend="sup">2</hi>√3</cell><cell cols="1" rows="1" role="data">= (8/3)R<hi rend="sup">2</hi>√3</cell><cell cols="1" rows="1" role="data">= <hi rend="italics">a</hi><hi rend="sup">2</hi>√3</cell><cell cols="1" rows="1" role="data">= 6√<hi rend="sup">3</hi>(<hi rend="italics">c</hi><hi rend="sup">2</hi>√3).</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">c</hi> = 8<hi rend="italics">r</hi><hi rend="sup">3</hi>√3</cell><cell cols="1" rows="1" role="data">= (8/27)R<hi rend="sup">3</hi>√3</cell><cell cols="1" rows="1" role="data">= (1/12)<hi rend="italics">a</hi><hi rend="sup">3</hi>√2</cell><cell cols="1" rows="1" role="data">= (1/36)<hi rend="italics">b</hi>√(2<hi rend="italics">b</hi>√3).</cell></row><row role="data"><cell cols="1" rows="1" role="data">R = 3<hi rend="italics">r</hi></cell><cell cols="1" rows="1" role="data">= (1/4)<hi rend="italics">a</hi>√6</cell><cell cols="1" rows="1" role="data">= (1/4)√(2<hi rend="italics">b</hi>√3)</cell><cell cols="1" rows="1" role="data">= (3/2)√<hi rend="sup">3</hi>((1/3)<hi rend="italics">c</hi>√3).</cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">r</hi> = 1/3R</cell><cell cols="1" rows="1" role="data">= (1/12)<hi rend="italics">a</hi>√6</cell><cell cols="1" rows="1" role="data">= (1/12)√(2<hi rend="italics">b</hi>√3)</cell><cell cols="1" rows="1" role="data">= (1/2)√<hi rend="sup">3</hi>((1/3)<hi rend="italics">c</hi>√3).</cell></row></table></p><p>See my Mensuration, pa. 248 &c, 2d ed. See also the articles <hi rend="smallcaps">Regular</hi> and <hi rend="smallcaps">Bodies.</hi></p></div1><div1 part="N" n="TETRAGON" org="uniform" sample="complete" type="entry"><head>TETRAGON</head><p>, in Geometry, a quadrangle, or a figure having 4 angles. Such as a square, a parallelogram, a rhombus, and a trapezium. It sometimes also means peculiarly a square.</p><div2 part="N" n="Tetragon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tetragon</hi></head><p>, in Astrology, denotes an aspect of two planets with regard to the earth, when they are distant from each other a 4th part of a circle, or 90 degrees. The Tetragon is expressed by the character □, and is otherwise called a square or quartile aspect.</p></div2></div1><div1 part="N" n="TETRAGONIAS" org="uniform" sample="complete" type="entry"><head>TETRAGONIAS</head><p>, a meteor, whose head is of a quadrangular figure, and its tail or train is long, thick, and uniform. It does not differ much from the meteor called <hi rend="italics">Trabs</hi> or beam.</p></div1><div1 part="N" n="TETRAGONISM" org="uniform" sample="complete" type="entry"><head>TETRAGONISM</head><p>, a term which some authors use to express the quadrature of the circle, because the quadrature is the finding a square equal to it.</p></div1><div1 part="N" n="TETRASPASTON" org="uniform" sample="complete" type="entry"><head>TETRASPASTON</head><p>, in Mechanics, a machine in which are four pulleys.</p></div1><div1 part="N" n="TETRASTYLE" org="uniform" sample="complete" type="entry"><head>TETRASTYLE</head><p>, in the Ancient Architecture, a building, and particularly a temple, with four columns in front.</p></div1><div1 part="N" n="THALES" org="uniform" sample="complete" type="entry"><head>THALES</head><p>, a celebrated Greek philosopher, and the first of the seven wisemen of Greece, was born at Miletum, about 640 years before Christ. After acquiring the usual learning of his own country, he travelled into Egypt and several parts of Asia, to learn astronomy, geometry, mystical divinity, natural knowledge or philosophy, &c. In Egypt he met for some time great favour from the king, Amasis; but he lost it again, by the freedom of his remarks on the conduct of kings, which it is said occasioned his return to his own country, where he communicated the knowledge he had acquired to many disciples, among the principal of whom were Anaximander, Anaximenes, and Pythagoras, and was the author of the Ionian sect of philosophers. He always however lived very retired, and re fused the proffered favours of many great men. He was often visited by Solon; and it is said he took great pleasure in the conversation of Thrasybulus, whose excellent wit made him forget that he was Tyrant of Miletum.</p><p>Laertius, and several other writers, agree, that he was the father of the Greek philosophy; being the first that made any researches into natural knowledge, and mathematics. His doctrine was, that water was the principle of which all the bodies in the universe are composed; that the world was the work of God; and that God sees the most secret thoughts in the heart of man. He said, that in order to live well, we ought to abstain from what we find fault with in others: that <pb n="583"/><cb/> bodily felicity consists in health; and that of the mind in knowledge. That the most ancient of beings is God, because he is uncreated: that nothing is more beautiful than the world, because it is the work of God; nothing more extensive than space, quicker than spirit, stronger than necessity, wiser than time. He used to observe, that we ought never to say that to any one which may be turned to our prejudice; and that we should live with our friends as with persons that may become our en[etilde]mies.</p><p>In Geometry, it has been said, he was a considerable inventor, as well as an improver; particularly in triangles. And all the writers agree, that he was the first, even in Egypt, who took the height of the pyramids by the shadow.</p><p>His knowledge and improvements in astronomy were very considerable. He divided the celestial sphere into five circles or zones, the arctic and antarctic circles, the two tropical circles, and the equator. He observed the apparent diameter of the sun, which he made equal to half a degree; and formed the constellation of the Little Bear. He observed the nature and course of eclipses, and calculated them exactly; one in particular, memorably recorded by Herodotus, as it happened on a day of battle between the Medes and Lydians, which, Laertius says, he had foretold to the Ionians. And the same author informs us, that he divided the year into 365 days. Plutarch not only confirms his general knowledge of eclipses, but that his doctrine was, that an eclipse of the sun is occasioned by the intervention of the moon, and that an eclipse of the moon is caused by the intervention of the earth.</p><p>His morals were as just, as his mathematics well grounded, and his judgment in civil affairs equal to either. He was very averse to tyranny, and esteemed monarchy little better in any shape.—Diogenes Laertius relates, that walking to contemplate the stars, he fell into a ditch; on which a good old woman, that attended him, exclaimed, “How canst thou know what is doing in the heavens, when thou seest not what is at thy feet?”—He went to visit Crœsus, who was marching a powerful army into Cappadocia, and enabled him to pass the river Halys without making a bridge. Thales died soon after, at above 90 years of age, it is said, at the Olympic games, where, oppressed with heat, thirst, and a load of years, he, in public view, sunk into the arms of his friends.</p><p>Concerning his writings, it remains doubtful whether he left any behind him; at least none have come down to us. Augustine mentions some books of Natural Philosophy; Simplicius, some written on Nautic Astrology; Laertius, two treatises on the Tropics and Equinoxes; and Suidas, a treatise on Meteors, written in verse.</p></div1><div1 part="N" n="THAMMUZ" org="uniform" sample="complete" type="entry"><head>THAMMUZ</head><p>, in Chronology, the 10th month of the year of the Jews, containing 29 days, and answering to our June.</p></div1><div1 part="N" n="THEMIS" org="uniform" sample="complete" type="entry"><head>THEMIS</head><p>, in Astronomy, a name given by some to the 3d satellite of Jupiter.</p></div1><div1 part="N" n="THEODOLITE" org="uniform" sample="complete" type="entry"><head>THEODOLITE</head><p>, an instrument much used in surveying, for taking angles, distances, altitudes, &c.</p><p>This instrument is variously made; different persons having their several ways of contriving it, each attempting to make it more simple and portable, more accurate <cb/> and expeditious, than others. It usually consists of a brass circle, about a foot diameter, cut in form of fig. 5, pl. 31; having its limb divided into 360 degrees, and each degree subdivided either diagonally, or otherwise, into minutes. Underneath, at <hi rend="italics">cc,</hi> are fixed two little pillars <hi rend="italics">bb</hi> (fig. 6), which support an axis, bearing a telescope, for viewing remote objects.</p><p>On the centre of the circle moves the index C, which is a circular plate, having a compass in the middle, the meridian line of which answers to the fiducial line <hi rend="italics">aa;</hi> at <hi rend="italics">bb</hi> are fixed two pillars to support an axis, bearing a telescope like the former, whose line of collimation answers to the fiducial line <hi rend="italics">aa.</hi> At each end of either telescope is, or may be, fixed a plain sight, for the viewing of nearer objects.</p><p>The ends of the index <hi rend="italics">aa</hi> are cut circularly, to sit the divisions of the limb B; and when that limb is diagonally divided, the fiducial line at one end of the index shews the degrees and minutes upon the limb. It is also furnished with cross spirit levels, for setting the plane of the circle truly horizontal; and a vertical arch, divided into degrees, for taking angles of elevation and depression. The whole instrument is mounted with a ball and socket, upon a three-legged staff.</p><p>Many Theodolites however have no telescopes, but only four plain sights, two of them fastened on the limb, and two on the ends of the index. Two different ones, mounted on their stand, are represented in fig. 2 and 3, plate 33.</p><p>The use of the Theodolite is abundantly shewn in that of the semicircle, which is only half a Theodolite. And the index and compass of the Theodolite serve also for a circumferentor, and are used as such.</p><p>The ingenious Mr. Ramsden has lately made a most excellent Theodolite, for the use of the military survey now carrying on in England.</p></div1><div1 part="N" n="THEODOSIUS" org="uniform" sample="complete" type="entry"><head>THEODOSIUS</head><p>, a celebrated mathematician, who flourished in the times of Cicero and Pompey; but the time and place of his death are unknown. This Theodosius, the Tripolite, as mentioned by Suidas, is probably the same with Theodosius the philosopher of Bythinia, who Strabo says excelled in the mathematical sciences, as also his sons; for the same person might have travelled from the one of those places to the other, and spent part of his life in each of them; like as Hipparchus was called by Strabo the Bythinian; but by Ptolomy and others the Rhodian.</p><p>Theodosius chiefly cultivated that part of geometry which relates to the doctrine of the sphere, concerning which he published three books. The first of these contains 22 propositions; the second 23; and the third 14; all demonstrated in the pure geometrical manner of the Ancients. Ptolomy made great use of these propositions, as well as all succeeding writers. These books were translated by the Arabians, out of the original Greek, into their own language. From the Arabic, the work was again translated into Latin, and printed at Venice. But the Arabic version being very defective, a more complete edition was published in Greek and Latin, at Paris 1558, by John Pena, Regius Professor of Astronomy. And Vitello acquired reputation by translating Theodosius into Latin. This author's works were also commented on and illustrated by Clavius, Heleganius, and Guarinus, and lastly by <pb n="584"/><cb/> De Chales, in his Cursus Mathematicus. But that edition of Theodosius's Spherics which is now most in use, was translated, and published, by our countryman the learned Dr. Barrow, in the year 1675, illustrated and demonstrated in a new and concise method. By this author's account, Theodosius appears not only to be a great master in this more difficult part of geometry, but the first considerable author of antiquity who has written on that subject.</p><p>Theodosius too wrote concerning the Celestial Houses; also of Days and Nights; copies of which, in Greek, are in the king's library at Paris. Of which there was a Latin edition, published by Peter Dasypody, in the year 1572.</p></div1><div1 part="N" n="THEON" org="uniform" sample="complete" type="entry"><head>THEON</head><p>, of Alexandria, a celebrated Greek philosopher and mathematician, who flourished in the 4th century, about the year 380, in the time of Theodosius the Great; but the time and manner of his death are unknown. His genius and disposition for the study of philosophy were very early improved by a close application to study; so that he acquired such a proficiency in the sciences, as to render his name venerable in history; and to procure him the honour of being president of the famous Alexandrian school. One of his pupils was the admirable Hypatia, his daughter, who succeeded him in the presidency of the school; a trust, which, like himself, she discharged with the greatest honour and usefulness. [See her life in its place in the first volume of this Dictionary.]</p><p>The study of nature led Theon to many just conceptions concerning God, and to many useful reflections in the science of moral philosophy; hence, it is said, he wrote with great accuracy on divine providence. And he seems to have made it his standing rule, to judge the truth of certain principles, or sentiments, from their natural or necessary tendency. Thus, he says, that a full persuasion, that the Deity sees every thing we do, is the strongest incentive to virtue; for he insists, that the most profligate have power to refrain their hands, and hold their tongues. when they think they are observed, or overheard, by some person whom they fear or respect. With how much more reason then, says he, should the apprehension and belief, that God sees all things, restrain men from sin, and constantly excite them to their duty? He also represents this belief, concerning the Deity, as productive of the greatest pleasure imaginable, especially to the virtuous, who might depend with greater confidence on the favour and protection of Providence. For this reason, he recommends nothing so much as meditation on the presence of God: and he recommended it to the civil magistrate, as a restraint on such as were profane and wicked, to have the following inscription written, in large characters, at the corner of every street; G<hi rend="smallcaps">OD SEES THEE</hi>, O <hi rend="smallcaps">Sinner.</hi></p><p>Theon wrote notes and commentaries on some of the ancient mathematicians. He composed also a book, entitled <hi rend="italics">Progymnasinata,</hi> a rhetorical work, written with great judgment and elegance; in which he criticised on the writings of some illustrious orators and historians; pointing out, with great propriety and judgment, their beauties and imperfections; and laying down proper rules for propriety of style. He recommends conciseness of expression, and perspicuity, as the principal orna- <cb/> ments. This book was printed at Basle, in the year 1541; but the best edition is that of Leyden, in 1626, in 8vo.</p></div1><div1 part="N" n="THEOPHRASTUS" org="uniform" sample="complete" type="entry"><head>THEOPHRASTUS</head><p>, a celebrated Greek philosopher, was the son of Melanthus, and was born at Eretus in Bœotia. He was at first the disciple of Lucippus, then of Plato, and lastly of Aristotle; whom he succeeded in his school, about the 322d year before the Christian era, and taught philosophy at Athens with great applause.</p><p>He said of an orator without judgment “that he was a horse without a bridle.” He used also to say, “There is nothing so valuable as time, and those who lavish it are the most inexcusable of all prodigals.”— He died at about 100 years of age.</p><p>Theophrastus wrote many works, the principal of which are the following—1. An excellent moral treatise entitled, <hi rend="italics">Characters,</hi> which, he says in the preface, he composed at 99 years of age. Isaac Casaubon has written learned commentaries on this small treatise. It has been translated from the Greek into French, by Bruyere; and it has also been translated into English.— 2. A curious treatise on Plants.—3. A treatise on fossils or stones; of which Dr. Hill has given a good edition, with an English translation, and learned notes, in 8vo.</p></div1><div1 part="N" n="THEOREM" org="uniform" sample="complete" type="entry"><head>THEOREM</head><p>, a proposition which terminates in theory, and which considers the properties of things already made or done. Or, a Theorem is a speculative proposition, deduced from several definitions compared together. Thus, if a triangle be compared with a parallelogram standing on the same base, and of the same altitude, and partly from their immediate definitions, and partly from other of their properties already determined, it is inferred that the parallelogram is double the triangle; that proposition is a Theorem.</p><p>Theorem stands contradistinguished from problem, which denotes something to be done or constructed, as a Theorem proposes something to be proved or demonstrated.</p><p>There are two things to be chiefly regarded in every Theorem, viz, the proposition, and the demonstration. In the first is expressed what agrees to some certain thing, under certain conditions, and what does not. In the latter, the reasons are laid down by which the understanding comes to conceive that it does or does not agree to it.</p><p>Theorems are of various kinds: as,</p><p><hi rend="italics">Universal</hi> <hi rend="smallcaps">Theorem</hi>, is that which extends to any quantity without restriction, universally. As this, that the rectangle or product of the sum and difference of any two quantities, is equal to the difference of their squares.</p><p><hi rend="italics">Particular</hi> <hi rend="smallcaps">Theorem</hi>, is that which extends only to a particular quantity. As this, in an equilateral rectilinear triangle, each angle is equal to 60 degrees.</p><p><hi rend="italics">Negative</hi> <hi rend="smallcaps">Theorem</hi>, is that which expresses the impossibility of any assertion. As, that the sum of two biquadrate numbers cannot make a square number.</p><p><hi rend="italics">Local</hi> <hi rend="smallcaps">Theorem</hi> is that which relates to a surface. As, that triangles of the same base and altitude are equal.</p><p><hi rend="italics">Plane</hi> <hi rend="smallcaps">Theorem</hi>, is that which relates to a surface that is either rectilinear or bounded by the circumference of a circle. As, that all angles in the same segment of a circle are equal.</p><p><hi rend="italics">Solid</hi> <hi rend="smallcaps">Theorem</hi>, is that which considers a space ter- <pb/><pb/><pb n="585"/><cb/> minated by a solid line; that is, by any of the three conic sections. As this, that if a right line cut two asymptotic parabolas, its two parts terminated by them shall be equal.</p><p><hi rend="italics">Reciprocal</hi> <hi rend="smallcaps">Theorem</hi>, is one whose converse is true. As, that if a triangle have two sides equal, it has also two angles equal: the converse of which is likewise true, viz, that if the triangle have two angles equal, it has also two sides equal.</p></div1><div1 part="N" n="THEORY" org="uniform" sample="complete" type="entry"><head>THEORY</head><p>, a doctrine which terminates in the sole speculation or consideration of its object, without any view to the practice or application of it.</p><p>To be learned in an art, &c, the Theory is sufficient; to be a master of it, both the Theory and practice are requisite.—Machines often promise very well in Theory, but fail in the practice.</p><p>We say Theory of the moon, Theory of the rainbow, of the microscope, of the camera obscura, &c.</p><p><hi rend="smallcaps">Theories</hi> <hi rend="italics">of the Planets,</hi> &c, are systems or hypotheses, according to which the astronomers explain the reasons of the phenomena or appearances of them.</p></div1><div1 part="N" n="THERMOMETER" org="uniform" sample="complete" type="entry"><head>THERMOMETER</head><p>, an instrument for measuring the temperature of the air, &c, as to heat and cold.</p><p>The Thermometer and thermoscope are usually accounted the same thing. But Wolfius makes a difference; and he also shews that what we call Thermometers, are really no more than thermoscopes.</p><p>The invention of the Thermometer is attributed to several persons by different authors, viz, to Sanctorio, Galileo, father Paul, and to Drebbel. Thus, the invention is ascribed to Cornelius Drebbel of Alcmar, about the beginning of the 17th century, by his countrymen Boerhaave (Chem. 1, pa. 152, 156), and Musschenbroeck (Introd. ad Phil. Nat. vol. 2, pa. 625).— Fulgenzio, in his Life of father Paul, gives him the honour of the first discovery.—Vincenzio Viviani (Vit. de l'Galil. pa. 67; also Oper. di Galil. pref. pa. 47) speaks of Galileo as the inventor of Thermometers.— But Sanctorino (Com. in Galen. Art. Med. pa. 736, 842, Com. in Avicen. Can. Fen. 1, pa. 22, 78, 219) expressly assumes to himself this invention: and Borelli (De Mot. Animal. 2, prop. 175) and Malpighi (Oper. Posth. pa. 30) ascribe it to him without reserve. Upon which Dr. Martine remarks, that these Florentine academicians are not to be suspected of partiality in favour of one of the Patavinian school.</p><p>But whoever was the first inventor of this instrument, it was at first very rude and imperfect; and as the various degrees of heat were indicated by the different contraction or expansion of air, it was afterwards found to be an uncertain and sometimes a deceiving measure of heat, because the bulk of the air was affected, not only by the difference of heat, but also by the variable weight of the atmosphere.</p><p>There are various kinds of Thermometers, the construction, defects, theory, &c, of which, are as follow.</p><p><hi rend="italics">The Air</hi> <hi rend="smallcaps">Thermometer.</hi>—This instrument depends on the rarefaction of the air. It consists of a glass tube BE (fig. 1, pl. 34) connected at one end with a large glass ball A, and at the other end immersed in an open vessel, or terminating in a bait DE, with a narrow orifice at D; which vessel, or ball, <cb/> contains any coloured liquor that will not easily freeze. Aquafortis tinged of a fine blue colour with solution of vitriol or copper, or spirit of wine tinged with cochineal, will answer this purpose. But the ball A must be first moderately warmed, so that a part of the air contained in it may be expelled through the orifice D; and then the liquor pressed by the weight of the atmosphere, will enter the ball DE, and rise, for example, to the middle of the tube at C, at a mean temperature of the weather; and in this state the liquor by its weight, and the air included in the ball and tube ABC, by its elasticity, will counterbalance the weight of the atmosphere. As the surrounding air becomes warmer, the air in the ball and the upper part of the tube, expanding by heat, will drive the liquor into the lower ball, and consequently its surface will descend; on the contrary, as the ambient air becomes colder, that in the ball is condensed, and the liquor, pressed by the weight of the atmosphere, will ascend: so that the liquor in the tube will ascend or descend more or less, according to the state of the air contiguous to the instrument. To the tube is affixed a scale of the same length, divided upwards and downwards, from the middle C, into 100 equal parts, by means of which may be observed the ascent and descent of the liquor in the tube, and consequently the variations also in the temperature of the atmosphere.</p><p>A similar Thermometer may be constructed by putting a small quantity of mercury, not exceeding the bulk of a pea, into the tube BC (fig. 4, pl. 33), bent into wreaths, that taking up the less height, it may be the more manageable, and less liable to harm; divide this tube into any number of equal parts to serve for a scale. Here the approaches of the mercury towards the ball A will shew the increase of the degree of heat. The reason of which is the same as in the former.</p><p>The defect of both these instruments consists in this, that they are liable to be acted on by a double cause: for, not only a decrease of heat, but also an increase of weight of the atmosphere, will make the liquor rise in the one, and the mercury in the other; and, on the contrary, either an increase of heat, or decrease of the weight of the atmosphere, will cause them to descend.</p><p>For these, and other reasons, Thermometers of this kind have been long disused. However, M. Amontons, in 1702, with a view of perfecting the aërial Thermometer, contrived his <hi rend="italics">Universal Thermometer.</hi> Finding that the changes produced by heat and cold in the bulk of the air were subject to invincible irregularities, he substituted for these the variations produced by heat in the elastic force of this fluid. This Thermometer consisted of a long tube of glass (fig. 3, pl. 34) open at one end, and recurved at the other end, which terminated in a ball. A certain quantity of air was compressed into this ball by the weight of a column of mercury, and also by the weight of the atmosphere. The effect of heat on this included air was to make it sustain a greater or less weight; and this effect was measured by the variation of the column of mercury in the tube, corrected by that of the barometer, with respect to the changes of the weight of the external air. This instrument, though much more perfect than the former, is nevertheless subject to very considerable defects and <pb n="586"/><cb/> inconveniences. Its length of 4 feet renders it unfit for a variety of experiments, and its construction is difficult and complex: it is extremely inconvenient for carriage, as a very small inclination of the tube would suffer the included air to escape: and the friction of the mercury in the tube, and the compressibility of the air, contribute to render the indications of this instrument extremely uncertain. Besides, the dilatation of the air is not so regularly proportional to its heat, nor is its dilatation by a given heat nearly so uniform as he supposed. This depends much on its moisture; for dry air does not expand near so much by a given heat, as air stored with watery particles. For these, and other reasons, enumerated by De Luc (Recherches sur les Mod. de l'Atmo. tom. 1, pa. 278 &c), this instrument was imitated by very few, and never came into general use.</p><p><hi rend="italics">Of the Florentine</hi> <hi rend="smallcaps">Thermometer.</hi>—The academists del Cimento, about the middle of the 17th century, considering the inconveniencies of the air Thermometers above described, attempted another, that should measure heat and cold by the rarefaction and condensation of spirit of wine; though much less than those of air, and consequently the alterations in the degree of heat likely to be much less sensible.</p><p>The spirit of wine coloured, was included in a very fine and cylindrical glass tube (fig. 2, pl. 34), exhausted of its air, having a hollow ball at one end A, and hermetically sealed at the other end D. The ball and tube are filled with rectified spirit of wine to a convenient height, as to C, when the weather is of a mean temperature, which may be done by inverting the tube into a vessel of stagnant coloured spirit, under a receiver of the air-pump, or in any other way. When the thermometer is properly filled, the end D is heated red hot by a lamp, and then hermetically sealed, leaving the included air of about 1/3 of its natural density, to prevent the air which is in the spirit from dividing it in its expansion. To the tube is applied a scale, divided from the middle, into 100 equal parts, upwards and downwards.</p><p>Now spirit of wine rarefying and condensing very considerably; as the heat of the ambient atmosphere increases, the spirit will dilate, and so ascend in the tube; and as the heat decreases, the spirit will descend; and the degree or quantity of the motion will be shewn by the attached scale.</p><p>These Thermometers could not be subject to any inconvenience by an evaporation of the liquor, or a variable gravity of the incumbent atmosphere. Instruments of this kind were first introduced into England by Mr. Boyle, and they soon came into general use among philosophers in other countries. They are however subject to considerable inconveniences, from the weight of the liquor itself, and from the elasticity of the air above it in the tube, both which prevent the freedom of its ascent; besides, the rarefactions are not exactly proportional to the surrounding heat. Moreover spirit of wine is incapable of bearing very great heat or very great cold: it boils sooner than any other liquor; and therefore the degrees of heat of boiling fluids cannot be determined by this Thermometer. And though it retains its fluidity in pretty severe cold, yet it seems not to condense very regularly in them: and at <cb/> Torneao, near the polar circle, the winter cold was so severe, as Maupertuis informs us, that the spirits were frozen in all their Thermometers. So that the degrees of heat and cold, which spirit of wine is capable of indicating, is much too limited to be of very great or general use.</p><p>Another great defect of these, and other Thermometers, is, that their degrees cannot be compared with each other. It is true they mark the variations of heat and cold; but each marks for itself, and after its own manner; because they do not proceed from any point of temperature that is common to all of them.</p><p>From these and various other imperfections in these Thermometers, it happens, that the comparisons of them become so precarious and defective: and yet the most curious and interesting use of them, is what ought to arise from such comparison. It is by this we should know the heat or cold of another season, of another year, another climate, &c; and what is the greatest degree of heat or cold that men and other animals can subsist in.</p><p>Reaumur contrived a new Thermometer, in which the inconveniences of the former are proposed to be remedied. He took a large ball and tube, the content or dimensions of which are known in every part; he graduated the tube, so that the space from one division to another might contain a 1000th part of the liquor, which liquor would contain 1000 parts when it stood at the freezing point: then putting the ball of his Thermometer and part of the tube into boiling water, he observed whether it rose 80 divisions: if it exceeded these, he changed his liquor, and by adding water lowered it, till upon trial it should just rise 80 divisions; or if the liquor, being too low, fell short of 80 divisions, he raised it by adding rectified spirit to it. The liquor thus prepared suited his purpose, and served for making a Thermometer of any size, whose scale would agree with his standard. Such liquor, or spirits, being about the strength of common brandy, may easily be had any where, or made of a proper degree of density by raising or lowering it.</p><p>The abbé Nollet made many excellent Thermometers upon Reaumur's principle. Dr. Martine however expresses his apprehensions that Thermometers of this kind cannot admit of such accuracy as might be wished. The balls or bulbs, being large, as 3 or 4 inches in diameter, are neither heated nor cooled soon enough to shew the variations of heat. Small bulbs and small tubes, he says, are much more convenient, and may be constructed with sufficient accuracy. Though it must be allowed that Reaumur, by his excellent scale, and by depriving the spirit of its air, and expelling the air by means of heat from the ball and tube of his Thermometer, has brought it to as much perfection as may be; yet it is liable to some of the inconveniences of spirit Thermometers, and is much inferior to mercurial ones. These two kinds do not agree together in indicating the same degrees of intense cold; for when the mercury has stood at 22° below 0, the spirit indicated only 18°, and when the mercury stood at 28° or 37° below 0, the spirit rested at 25° or 29°. See the description of Reaumur's Thermometer at large in Mem. de l'Acad. dés Scienc. an. 1730, pa. 645, Hist. pa. 15. Ib. an. 1731, pa. 354, Hist. pa. 7. <pb n="587"/><cb/></p><p><hi rend="italics">Mercurial</hi> <hi rend="smallcaps">Thermometer.</hi>—It is a most important circumstance in the construction of Thermometers, to procure a fluid that measures equal variations of heat by corresponding equal variations in its own bulk: and the fluid which possesses this essential requisite in the most perfect degree, is mercury: the variations in its bulk approaching nearer to a proportion with the corresponding variations of its heat, than any other fluid. Besides, it is the most easy to purge of its air; and is also the most proper for measuring very considerable variations of heat and cold, as it will bear more cold before freezing, and more heat before boiling, than any other fluid. Mercury is also more sensible than any other fluid, air excepted, or conforms more speedily to the several variations of heat. Moreover, as mercury is an homogeneous fluid, it will in every Thermometer exhibit the same dilatation or condensation by the same variations of heat.</p><p>Dr. Halley, though apprized only of some of the remarkable properties of mercury above recited, seems to have been the first who suggested the application of this fluid to the construction of Thermometers. Philos. Trans. Abr. vol. 2, pa. 34.</p><p>Boerhaave (Chem. 1, pa. 720) says, these mercurial Thermometers were first contrived by Olaus Roemer; but the claims of Fahrenheit of Amsterdam, who gave an account of his invention to the Royal Society in 1724, (Philos. Trans. num. 381, or Abr. vol. 7, pa. 49) have been generally allowed. And though Prius and others, in England, Holland, France, and other countries, have made this instrument as well as Fahrenheit, most of the mercurial Thermometers are graduated according to his scale, and are called <hi rend="italics">Fahrenheit's Thermometers.</hi></p><p>The cone or cylinder, which these Thermometers are often made with, instead of the ball, is made of glass of a moderate thickness, lest, when the exhausted tube is hermetically sealed, its internal capacity should be diminished by the weight of the ambient atmosphere. When the mercury is thoroughly purged of its air and moisture by boiling, the Thermometer is filled with a sufficient quantity of it; and before the tube is hermetically sealed, the air is wholly expelled from it by heating the mercury, so that it may be rarefied and ascend to the top of the tube. To the side of the tube is annexed a scale (fig. 3, pl. 34), which Fahrenheit divided into 600 parts, beginning with that of the severe cold which he had observed in Iceland in 1709, or that produced by surrounding the bulb of the Thermometer with a mixture of snow or beaten ice and sal ammoniac or sea salt. This he apprehended to be the greatest degree of cold, and accordingly he marked this, as the beginning of his scale, with 0; the point at which mercury begins to boil, he conceived to shew the greatest degree of heat, and this he made the limit of his scale. The distance between these two points he divided into 600 equal parts or degrees; and by trials he found at the freezing point, when water just begins to freeze, or snow or ice just begins to thaw, that the mercury stood at 32 of these divisions, therefore called the degree of the freezing point; and when the tube was immersed in boiling water, the mercury rose to 212, which therefore is the boiling point, and is just 180 degrees above the former or freezing point. <cb/> But the present method of making the scale of these Thermometers, which is the sort in most common use, is first to immerge the bulb of the Thermometer in ice or snow just beginning to thaw, and mark the place where the mercury stands with a 32; then immerge it in boiling water, and again mark the place where the mercury stands in the tube, which mark with the num. 212, exceeding the former by 180; dividing therefore the intermediate space into 180 equal parts, will give the scale of the Thermometer, and which may afterwards be continued upwards and downwards at pleasure.</p><p>Other Thermometers of a similar construction have been accommodated to common use, having but a portion of the above scale. They have been made of a small size and portable form, and adapted with appendages to particular purposes; and the tube with its annexed scale has often been enclosed in another thicker glass tube, also hermetically sealed, to preserve the Thermometer from injury. And all these are called <hi rend="italics">Fahrenheit's Thermometers.</hi></p><p>In 1733, M. De l'Isle of Petersburgh constructed a mercurial Thermometer (see fig. 3, pl. 34), on the principles of Reaumur's spirit Thermometer. In his Thermometer, the whole bulk of quicksilver, when immerged in boiling water, is conceived to be divided into 100,000 parts; and from this one fixed point the various degrees of heat, either above or below it, are marked in these parts on the tube or scale, by the various expansion or contraction of the quicksilver in all imaginable varieties of heat.—Dr. Martine apprehends it would have been better if De l'Isle had made the integer 100,000 parts, or fixed point, at freezing water, and from thence computed the dilatations or condensations of the quicksilver in those parts; as all the common observations of the weather, &c, would have been expressed by numbers increasing as the heat increased, instead of decreasing, or counting the contrary way. However, in practice it will not be very easy to determine exactly all the divisions from the alteration of the bulk of the contained fluid. And besides, as glass itself is dilated by heat, though in a less proportion than quicksilver, it is only the excess of the dilatation of the contained fluid above that of the glass that is observed; and therefore if different kinds of glass be differently affected by a given degree of heat, this will make a seeming difference in the dilatations of the quicksilver in the Thermometers constructed in the Newtonian method, either by Reaumur's rules or De l'Isle's. Accordingly it has been found, that the quicksilver in De l'Isle's Thermometers, has stood at different degrees of the scale when immerged in thawing snow: having stood in some at 154°, while in others it has been at 156 or even 158°.</p><p><hi rend="italics">Metallic</hi> <hi rend="smallcaps">Thermometer.</hi>—This is a name given to a machine composed of two metals, which, whilst it indicates the variations of heat, serves to correct the errors hence resulting in the going of pendulum clocks and watches. Instruments of this kind have been contrived by Graham, Le Roy, Ellicot, Harrison, and other eminent artificers. See the Philos. Traus. vol. 44, pa. 689, and vol. 45, pa. 129, and vol. 51, pa. 823, where the particular descriptions &c may be seen.</p><p>M. De Luc has likewise described two Thermometers <pb n="588"/><cb/> of metal, which he uses for correcting the effects of heat upon a barometer, and an hygrometer of his construction connected with them. See Philos. Trans. vol. 68, p. 437.</p><p><hi rend="italics">Oil</hi> <hi rend="smallcaps">Thermometers.</hi>—To this class belongs Newton's Thermometer, constructed in 1701, with linseed oil, instead of spirit of wine. This fluid has the advantage of being sufficiently homogeneous, and capable of a considerable rarefaction, not less than 15 times greater than that of spirit of wine. It has not been observed to freeze even in very great colds; and it sustains a great heat, about 4 times that of water, before it boils. With these advantages it was made use of by Sir I. Newton, who discovered by it the comparative degree of heat for boiling water, melting wax, boiling spirit of wine, and melting tin; beyond which it does not appear that this Thermometer was applied. The method he used for adjusting the scale of this oil Thermometer, was as follows: supposing the bulb, when immerged in thawing snow, to contain 10,000 parts, he found the oil expanded by the heat of the human body so as to take up a 39th more space, or 10256 such parts; and by the heat of water boiling strongly 10725; and by the heat of melting tin 11516. So that, reckoning the freezing point as a common limit between heat and cold, he began his scale there, marking it 0, and the heat of the human body he made 12°; and consequently, the degrees of heat being proportional to the degrees of rarefaction, or 256 : 725 :: 12 : 34, this number 34 will express the heat of boiling water; and, by the same rule, 72 that of melting tin. Philos. Trans. number 270, or Abridg. vol. 4, par. 2, p. 3.</p><p>There is an insuperable inconvenience attending all Thermometers made with oil, or any other viscid fluid, viz, that such liquor adheres too much to the sides of the tube, and so inevitably disturbs the regularity and uniformity of the Thermometer.</p><p><hi rend="italics">Of the fixed points of</hi> <hi rend="smallcaps">Thermometers.</hi>—Various methods have been proposed by different authors, for finding a fixed point or degree of heat, from which to reckon the other degrees, and adjust the scale; so that different observations and instruments might be compared together. Mr. Boyle was very sensible of the inconveniences arising from the want of a universal scale and mode of graduation; and he proposed either the freezing of the essential oil of aniseeds, or of distilled water, as a term to begin the numbers at, and from thence to graduate them according to the proportional dilatations or contractions of the included spirits.</p><p>Dr. Halley (Philos. Trans. Abr. vol. 2, p. 36) seems 10 have been fully apprized of the bad effects of the indefinite method of constructing Thermometers, and wished to have them adjusted to some determined points. What he seems to prefer, for this purpose, is the degree of temperature found in subterranean places, where the heat in summer or cold in winter appears to have no influence. But this degree of temperature, Dr. Martine shews, is a term for the universal construction of Thermometers, both inconvenient and precarious, as it cannot be easily ascertained, and as the difference of soils and depths may occasion a considerable variation. Another term of heat, which he thought might be of use in a general graduation of Thermometers, is that of boiling spirit of wine that has been highly rectified. <cb/></p><p>The first trace that occurs of the method of actually applying fixed points or terms to the Thermometer, and of graduating it, so that the unequal divisions of it might correspond to equal degrees of heat, is the project of Renaldini, professor at Padua, in 1694: it is thus described in the Acta Erud. Lips. “Take a slender tube, about 4 palms long, with a ball fastened to the same; pour into it spirit of wine, enough just to fill the ball, when surrounded with ice, and not a drop over: in this state seal the orifice of the tube hermetically, and provide 12 vessels, each capable of containing a pound of water, and somewhat more; and into the first pour 11 ounces of cold water, into the second 10 ounces, into the third 9, &c; this done, immerge the Thermometer in the first vessel, and pour into it one ounce of hot water, observing how high the spirit rises in the tube, and noting the point with unity; then remove the Thermometer into the second vessel, into which are to be poured 2 ounces of hot water, and note the place the spirit rises to with 2: by thus proceeding till the whole pound of water is spent, the instrument will be found divided into 12 parts, denoting so many terms or degrees of heat; so that at 2 the heat is double to that at 1, at 3 triple, &c.”</p><p>But this method, though plausible, Wolsius shews, is deceitful, and built on false suppositions; for it takes for granted, that we have one degree of heat, by adding one ounce of hot water to 11 of cold; two degrees by adding 2 ounces to 10, &c: it supposes also, that a single degree of heat acts on the spirit of wine, in the ball, with a single force; a double with a double force, &c: lastly it supposes, that if the effect be produced in the Thermometer by the heat of the ambient air, which is here produced by the hot water, the air has the same degree of heat with the water.</p><p>Soon after this project of Renaldini, viz, in 1701, Newton constructed his oil Thermometer, and placed the base or lowest fixed point of his scale at the temperature of thawing snow, and 12 at that of the human body, &c, as above explained.—De Luc observes, that the 2d term of this scale should have been at a greater distance from the first, and that the heat of boiling water would have answered the purpose better than that of the human body.</p><p>In 1702, Amontons contrived his universal Thermometer, the scale of which was graduated in the foling manner. He chose for the first term, the weight that counterbalanced the air included in his Thermometer, when it was heated by boiling water: and in this state he so adjusted the quantity of mercury contained in it, till the sum of its height in the tube, and of its height in the barometer at the moment of observation, was equal to 73 inches. Fixing this number at the point to which the mercury in the tube rose by plunging it in boiling water, it is evident that if the barometer at this time was at 28 inches, the height of the column of mercury in the Thermometer, above the level of that in the ball, was 45 inches; but if the height of the barometer was less by a certain quantity, the column of the Thermometer ought to be greater by the same quantity, and reciprocally. He formed his scale on the supposition, that the weight of the atmosphere was always equal to that of a column of mercury of 28 inches, and he divided it into inches <pb n="589"/><cb/> from the point 73 downward, marking the divisions with 72, 71, 70, &c, and subdividing the inches into lines. But as the weight of the atmosphere is variable, the barometer must be observed at the same time with the Thermometer, that the number indicated by this last instrument may be properly corrected, by adding or subtracting the quantity which the mercury is below or above 28 inches in the barometer. In this scale then, the freezing point is at 51 1/2 inches, corresponding to 32 degrees of Fahrenheit, and the heat of boiling water at 73 inches, answering to 212 of Fahrenheit's; and thus they may be easily compared together.</p><p>The fixed points of Fahrenheit's Thermometer, as has been already observed, are the congelation produced by sal ammoniac and the heat of boiling water. The interval between these points is divided into 212 equal parts; the former of these points being marked 0, and the other 212.</p><p>Reaumur in his Thermometer, the construction of which he published in 1730, begins his scale at an artificial congelation of water in warm weather, which, as he uses large bulbs for his glasses, gives the freezing point much higher than it should be, and at boiling water he marks 80 degrees, which point Dr. Martine thinks is more vague and uncertain than his freezing point. In order to determine the correspondence of his scale with that of Fahrenheit, it is to be considered that his boiling water heat, is really only the boiling heat of weakened spirit of wine, coinciding nearly, as Dr. Martine apprehends, with Fahrenheit's 180 degrees. And as his 10 1/4 degrees is the constant heat of the cave of the observatory at Paris, or Fahrenheit's 53°, he thence finds his freezing point, instead of answering just to 32°, to be somewhat above 34°.</p><p>De l'Isle's Thermometer, an account of which he presented to the Petersburgh Academy in 1733, has only one fixed point, which is the heat of boiling water, and, contrary to the common order, the several degrees are marked from this point downward, according to the condensations of the contained quicksilver, and consequently by numbers increasing as the heat decreases. The freezing point of De l'Isle's scale, Dr. Martine makes near to his 150°, corresponding to Fahrenheit's 32, by means of which they may be compared; but Ducrest says, that this point ought to be marked at least at 154°.</p><p>Ducrest, in his spirit Thermometer, constructed in 1740, made use of two fixed points; the first, or 0, indicated the temperature of the earth, and was marked on his scale in the cave of the Paris Observatory; and the other was the heat of boiling water, which that spirit in his Thermometer was made to endure, by leaving the upper part of the tube full of air. He divided the interval between these points into 100 equal parts; calling the divisions upward, degrees of heat, and those below 0, degrees of cold.—It is said that he has since regulated his Thermometer by the degree of cold indicated by melting ice, which he found to be 10 2/5.</p><p>The Florentine Thermometers were of two sorts. In one sort the freezing point, determined by the <cb/> degree at which the spirit stood in the ordinary cold of ice or snow (probably in a thawing state) and coinciding with 32° of Fahrenheit, fell at 20°; and in the other sort at 13 1/2. And the natural heat of the viscera of cows and deer, &c, raised the spirit in the latter, or less sort, to about 40°, coinciding with their summer heat, and nearly with 102° in Fahrenheit's; and in their other or long Thermometer, the spirit, when exposed to the great midsummer heat in their country, rose to the point at which they marked 80°.</p><p>In the Thermometer of the Paris Observatory, made of spirit of wine by De la Hire, the spirit always stands at 48° in the cave of the observatory, corresponding to 53 degrees in Fahrenheit's; and his 28° corresponded with 51 inches 6 lines in Amontons' Thermometer, and consequently with the freezing point, or 32° of Fahrenheit's.</p><p>In Poleni's Thermometer, made after the manner of Amontons', but with less mercury, 47 inches corresponded, according to Dr. Martine, with 51 in that of Amontons, and 53 with 59 1/2.</p><p>In the standard Thermometer of the Royal Society of London, according to which Thermometers were for a long time constructed in England, Dr. Martine found that 34 1/2 degrees answered to 64° in Fahrenheit, and 0 to 89.</p><p>In the Thermometers graduated for adjusting the degrees of heat proper for exotic plants, &c, in stoves and greenhouses, the middle temperature of the air is marked at 0, and the degrees of heat and cold are numbered both above and below. Many of these are made on no regular and fixed principles. But in that formerly much used, called Fowler's regulator, the spirit fell, in melting snow, to about 34° under 0; and Dr. Martine found that his 16° above 0, nearly coincided with 64° of Fahrenheit.</p><p>Dr. Hales (Statical Essays, vol. 1, p. 58), in his Thermometer, made with spirit of wine, and used in experiments on vegetation, began his scale with the lowest degree of freezing, or 32° of Fahrenheit, and carried it up to 100°, which he marked where the spirit stood when the ball was heated in hot water, upon which some wax floating first began to coagulate, and this point Dr. Martine found to correspond with 142° of Fahrenheit. But by experience it is found that Hales's 100 falls considerably above our 142.</p><p>In the Edinburgh Thermometer, made with spirit of wine, and used in the meteorological observations published in the Medical Essays, the scale is divided into inches and tenths. In melting snow the spirit stood at 8 2/10, and the heat of the human skin raised it to 22 2/10. Dr. Martine found that the heat of the person who graduated it, was 97 of Fahrenheit.</p><p>As it is often of use to compare different Thermometers, in order to judge of the result of former observations, I have annexed from Dr. Martine's Essays, the table by which he compared 15 different thermometers. See Plate 34, fig. 3.</p><p>There is a Thermometer which has often been used in London, called the Thermometer of Lyons, because <pb n="590"/><cb/> M. Cristin brought it there into use, which is made of mercury: the freezing point is marked 0, and the interval from that point to the heat of boiling water is divided into 100 equal degrees.</p><p>From the abstract of the history of the construction of Thermometers it appears, that freezing and boiling water have furnished the distinguishing points that have been marked upon almost all Thermometers. The inferior fixed point is that of freezing, which some have determined by the freezing of water, and others by the melting of ice, plunging the ball of the Thermometer into the water and ice, while melting, which is the best way. The superior fixed point of almost all Thermometers, is the heat of boiling water. But this point cannot be considered as fixed and certain, unless the heat be produced by the same degree of boiling, and under the same weight of the atmosphere; for it is found that the higher the barometer, or the heavier the atmosphere, the greater is the heat when the water boils. It is now agreed therefore that the operation of plunging the ball of the Thermometer in the boiling water, or suspending it in the steam of the same in an inclosed vessel, be performed when the water boils violently, and when the barometer stands at 30 English inches, in a temperature of 55° of the atmosphere, marking the height of the Thermometer then for the degree of 212 of Fahrenheit; the point of melting ice being 32 of the same; thus having 180 degrees between those two fixed points, so determined. This was Mr. Bird's method, who it is apprehended first attended to the state of the barometer, in the making of Thermometers. But these instruments may be made equally true under any pressure of the atmosphere, by making a proper allowance for the difference in the height of the barometer from 30 inches. M. De Luc, in his Recherches sur les Mod. de l'Atmosphere, from a series of experiments, has given an equation for the allowance on account of this difference, in Paris measure, which has been verified by Sir George Schuckburgh, Philos. Trans. 1775 and 1778; also Dr. Horsley, Dr. Maskelyne, and Sir George Shuckburgh have adapted the equation and rules, to English measures, and have reduced the allowances into tables for the use of the artist. Dr. Horsley's rule, deduced from De Luc's, is this: , where <hi rend="italics">h</hi> denotes the height of a Thermometer plunged in boiling water, above the point of melting ice, in degrees of Bird's Fahrenheit, and <hi rend="italics">z</hi> the height of the barometer in 10ths of an inch. From this rule he has computed the following table, for finding the heights, to which a good Bird's Fahrenheit will rise, when plunged in boiling water, in all states of the barometer, from 27 to 31 English inches; which will serve, among other uses, to direct instrument makers in making a true allowance for the effect of the variation of the barometer, if they should be obliged to finish a Thermometer at a time when the barometer is above or below 30 inches; though it is best to fix the boiling point when the barometer is at that height. <cb/> <hi rend="center"><hi rend="italics">Equation of the Boiling Point.</hi></hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Barometer.</cell><cell cols="1" rows="1" role="data">Equation.</cell><cell cols="1" rows="1" role="data">Difference.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">31.0</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 1.57</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30.5</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 0.79</cell><cell cols="1" rows="1" role="data">0.78</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30.0</cell><cell cols="1" rows="1" rend="align=right" role="data">0.00</cell><cell cols="1" rows="1" role="data">0.79</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">29.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 0.80</cell><cell cols="1" rows="1" role="data">0.80</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">29.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 1.62</cell><cell cols="1" rows="1" role="data">9.82</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">28.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 2.45</cell><cell cols="1" rows="1" role="data">0.83</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">28.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 3.31</cell><cell cols="1" rows="1" role="data">0.85</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">27.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 4.16</cell><cell cols="1" rows="1" role="data">0.86</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">27.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 5.04</cell><cell cols="1" rows="1" role="data">0.88</cell></row></table></p><p>The numbers in the first column of this table express heights of the quicksilver in the barometer in English inches and decimal parts: the 2d column shews the equation to be applied, according to the sign prefixed, to 212° of Bird's Fahrenheit, to find the true boiling point for every such state of the barometer. The boiling point for all intermediate states of the barometer may be had with sufficient accuracy by taking proportional parts, by means of the 3d column of differences of the equations. See Philos. Trans. vol. 64, art. 30; also Dr. Maskelyne's paper, vol. 64, art. 20.</p><p>Sir Geo. Shuckburgh (Philos. Trans. vol. 69, pa. 362) has also given several tables and rules relating to the boiling point, both from his own observations and De Luc's, form whence is extracted the following table, for the use of artists in constructing the Thermometer. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Height of</cell><cell cols="1" rows="1" role="data">Corr. of</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Differ-</cell><cell cols="1" rows="1" role="data">Correct.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Differ-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">the Baro-</cell><cell cols="1" rows="1" role="data">the Boil.</cell><cell cols="1" rows="1" role="data">accord. to</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">meter.</cell><cell cols="1" rows="1" role="data">Point.</cell><cell cols="1" rows="1" role="data">ences.</cell><cell cols="1" rows="1" role="data">De Luc.</cell><cell cols="1" rows="1" role="data">ences.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">26.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 7.09</cell><cell cols="1" rows="1" role="data">0.91</cell><cell cols="1" rows="1" rend="align=right" role="data">- 6.83</cell><cell cols="1" rows="1" role="data">0.90</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">26.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 6.18</cell><cell cols="1" rows="1" role="data">0.91</cell><cell cols="1" rows="1" rend="align=right" role="data">- 5.93</cell><cell cols="1" rows="1" role="data">0.89</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">27.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 5.27</cell><cell cols="1" rows="1" role="data">0.90</cell><cell cols="1" rows="1" rend="align=right" role="data">- 5.04</cell><cell cols="1" rows="1" role="data">0.88</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">27.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 4.37</cell><cell cols="1" rows="1" role="data">0.89</cell><cell cols="1" rows="1" rend="align=right" role="data">- 4.16</cell><cell cols="1" rows="1" role="data">0.87</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">28.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 3.48</cell><cell cols="1" rows="1" role="data">0.89</cell><cell cols="1" rows="1" rend="align=right" role="data">- 3.31</cell><cell cols="1" rows="1" role="data">0.86</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">28.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 2.59</cell><cell cols="1" rows="1" role="data">0.87</cell><cell cols="1" rows="1" rend="align=right" role="data">- 2.45</cell><cell cols="1" rows="1" role="data">0.83</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">29.0</cell><cell cols="1" rows="1" rend="align=right" role="data">- 1.72</cell><cell cols="1" rows="1" role="data">0.87</cell><cell cols="1" rows="1" rend="align=right" role="data">- 1.62</cell><cell cols="1" rows="1" role="data">0.82</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">29.5</cell><cell cols="1" rows="1" rend="align=right" role="data">- 0.85</cell><cell cols="1" rows="1" role="data">0.85</cell><cell cols="1" rows="1" rend="align=right" role="data">- 0.80</cell><cell cols="1" rows="1" role="data">0.80</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30.0</cell><cell cols="1" rows="1" rend="align=right" role="data">0.00</cell><cell cols="1" rows="1" role="data">0.85</cell><cell cols="1" rows="1" rend="align=right" role="data">0.00</cell><cell cols="1" rows="1" role="data">0.79</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30.5</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 0.85</cell><cell cols="1" rows="1" role="data">0.84</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 0.79</cell><cell cols="1" rows="1" role="data">0.78</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">31.0</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 1.60</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">+ 1.57</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>The Royal Society too, fully sensible of the importance of adjusting the fixed points of Thermometers, appointed a committee of seven gentlemen to consider of the best method for this purpose; and their report may be seen in the Philos. Trans. vol. 67, art. 37.</p><p>They observe, that although the boiling point be placed so much higher on some of the Thermometers now made, than on others, yet this does not produce any considerable error in the observations of the weather, at least in this climate; for an error of 1 1/2 degree in the position of the boiling point, will make an error only of half a degree in the position of 92°, and of not more <pb n="591"/><cb/> than a quarter of a degree in the point of 62°. It is only in nice experiments, or in trying the heat of hot liquors, that this error in the boiling point can be of much signification.</p><p>In adjusting the freezing, as well as the boiling point, the quicksilver in the tube ought to be kept of the same heat as that in the ball. When the freezing point is placed at a considerable distance from the ball, the pounded ice should be piled up very near to it; if it be not so piled, then the observed point, to be very accurate, should be corrected, according to the following table. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Heat of the</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Correction.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Air.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">42°</cell><cell cols="1" rows="1" role="data">.00087</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">.00174</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">.00261</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">.00348</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">.00435</cell></row></table></p><p>The correction in this table is expressed in 1000th parts of the distance between the freezing point and the surface of the ice: ex. gr. if the freezing point stand 6 inches above the surface of the ice, and the heat of the room be 62, then the point of 32 should be placed 6 X .00261, or .01566 of an inch lower down than the observed point.</p><p>The committee farther observe, that in trying the heat of liquors, care should be taken that the quicksilver in the tube of the Thermometer be heated to the same degree as that in the ball; or if this cannot be done conveniently, the observed heat should be corrected on that account; for the manner of doing which, and a table calculated for that purpose, see Philos. Trans. vol. 67, art. 37.</p><p>It was for some time thought, especially from the experiments at Petersburgh, that quicksilver suffered a cold of several hundred degrees below o before it congealed and became fixed and malleable; but later experiments have shewn that this persuasion was merely owing to a deception in the experiments, and later ones have made it appear that its point of congelation is no lower than — 40°, or rather — 39°, of Fahrenheit's scale. But that it will bear however to be cooled a few degrees below that point, to which it leaps up again on beginning to congeal; and that its rapid descent in a Thermometer, through many hundred degrees, when it has once passed the above-mentioned limit, proceeds merely from its great contraction in the act of freezing. See Philos. Trans. vol. 73, art. *20, 20, 21. <hi rend="center"><hi rend="italics">Miscellaneous Observations.</hi></hi></p><p>It is absolutely necessary that those who would derive any advantage from these instruments, should agree in using the same liquor, and in determining, according to the same method, the two fundamental points. If they agree in these fixed points, it is of no great importance whether they divide the interval between them into a greater or a less number of equal parts. The scale of Fahrenheit, in which the fundamental interval between 212°, the point of boiling water, <cb/> and 32° that of melting ice, is divided into 180 parts, should be retained in the northern countries, where Fahrenheit's Thermometer is used: and the scale in which the fundamental interval is divided into 80 parts, will serve for those countries where Reaumur's Thermometer is adopted. But no inconvenience is to be apprehended from varying the scale for particular uses, provided care be taken to signify into what number of parts the fundamental interval is divided, and the point where o is placed.</p><p>With regard to the choice of tubes, it is best to have them exactly cylindrical through their whole length. The capillary tubes are preferable to others, because they require smaller bulbs, and they are also more sensible, and less brittle. The most convenient size for common experiments has the internal diameter about the 40th or 50th of an inch, about 9 inches long, and made of thin glass, that the rise and fall of the mercury may be better seen.</p><p>For the whole process of filling, marking, and graduating, see De Luc's Recherches &c, tom. 1, p. 393, &c. <hi rend="center"><hi rend="italics">Experiments with</hi> <hi rend="smallcaps">Thermometers.</hi></hi></p><p>The following is a table of some observations made with Fahrenheit's Thermometer, the barometer standing at 29 inches, or little higher. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">At 600°</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Mercury boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">546 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Oil of vitriol boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">242 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Spirit of nitre boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">240 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Lixivium of tartar boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">213 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Cow's milk boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">212 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Water boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">206 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Human urine boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">190 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Brandy boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">175 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Alcohol boils.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">156 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Serum of blood and white of eggs harden.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">146 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Kills animals in a few minutes.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">108 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">to 99, Hens hatch eggs.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">107 </cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Heat of skin in ducks, geese, hens, pi-</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">103 </cell><cell cols="1" rows="1" role="data"> geons, partridges, and swallows.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">106 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Heat of skin in a common ague and fever.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">103 </cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Heat of skin in dogs, cats, sheep, oxen,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">100 </cell><cell cols="1" rows="1" role="data"> swine, and most other quadrupeds.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">99 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">to 92, Heat of the human skin in health.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">97 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Heat of a swarm of bees.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">96 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">A perch died in 3 minutes in water so warm.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">80 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Heat of air in the shade, in very hot weather.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">74 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Butter begins to melt.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">64 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Heat of air in the shade, in warm weather.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">55 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Mean temperature of air in England.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">43 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Oil of olives begins to stiffen and grow opake.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right rowspan=2" role="data">32 </cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Water just freezing, or snow and ice just</cell></row><row role="data"><cell cols="1" rows="1" role="data"> melting.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">30 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Milk freezes.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">28 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Urine and common vinegar freezes.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">25 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Blood out of the body freezes.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">20 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Burgundy, Claret, and Madeira freeze.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right rowspan=2" role="data">5 </cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">Greatest cold in Pennsylvania in 1731-2,</cell></row><row role="data"><cell cols="1" rows="1" role="data"> lat. 40°.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Greatest cold at Utrecht in 1728-9.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right rowspan=2" role="data">0 </cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">A mixture of snow and salt, which can freeze</cell></row><row role="data"><cell cols="1" rows="1" role="data"> oil of tartar per deliquium, but not brandy.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">-39 </cell><cell cols="1" rows="1" rend="colspan=2" role="data">Mercury freezes.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">        Martine's Essays, p. 284, &c.</cell></row></table> <pb n="592"/><cb/></p><p>On the general subject of Thermometers also see Martine's Essays, Medical and Philosophical. Desaguliers's Exp. Phil. vol. 2, p. 289. Musschenbroeck's Int. ad Phil. Nat. vol. 2, p. 625, ed. 1762. De Luc's Recherches sur les Modif. &c, tom. 1, part 2, ch. 2. Nollet's Leçons de Physique, tom. 4, p. 375.</p><p><hi rend="smallcaps">Thermometers</hi> <hi rend="italics">for particular uses.</hi>—In 1757, lord Cavendish presented to the Royal Society an account of a curious construction of Thermometers, of two different forms; one contrived to shew the greatest degree of heat, and the other the greatest cold, that may happen at any time in a person's absence. Philos. Trans. vol. 50, p. 300.</p><p>Since the publication of Mr. Canton's discovery of the compressibility of spirits of wine and other fluids, there are two corrections necessary to be made in the result given by lord Cavendish's Thermometer. For in estimating, for instance, the temperature of the sea at any depth, the Thermometer will appear to have been colder than it really was: and besides, the expansion of spirits of wine by any given number of degrees of Fahrenheit's Thermometer, is greater in the higher degrees than in the lower. For the method of making these two corrections by Mr. Cavendish, see Phipps's Voyage to the North Pole, p. 145.</p><p>Instruments of this kind, for determining the degree of heat or cold in the absence of the observer, have been invented and described by others. Van Swinden (Diss. sur la Comparaison du Therm. p. 253 &c) describes one, which he says was the first of the kind, made on a plan communicated by Bernoulli to Leibnitz. Mr. Kraft, he also tells us, made one nearly like it. Mr. Six has lately, viz, in 1782, proposed another construction of a Thermometer of the same kind, described in the Philos. Trans. vol. 72, p. 72 &c.</p><p>M. De Luc has described the best method of constructing a Thermometer, fit for determining the temperature of the air, in the measuring of heights by the barometer. He has also shewn how to divide the scale of a Thermometer, so as to adapt it for astronomical purposes in the observation of refractions. See Recherches &c, tom. 2, p. 35 and 265.</p><p>Mr. Cavallo, in 1781, proposed the construction of a <hi rend="italics">Thermometrical Barometer,</hi> which, by means of boiling water, might indicate the various gravity of the atmosphere, or the height of the barometer. This Thermometer, he says, with its apparatus, might be packed up into a small portable box, and serve for determining the heights of mountains &c, with greater facility, than with the common portable barometer. The instrument, in its present state, consists of a cylindrical tin vessel, about 2 inches in diameter, and 5 inches high, in which vessel the water is contained, which may be made to boil by the flame of a large waxcandle. The Thermometer is fastened to the tin vessel in such a manner, as that its bulb may be about an inch above the bottom. The scale of this Thermometer, which is of brass, exhibits on one side of the glass tube a few degrees of Fahrenheit's scale, viz, from 200° to 216°. On the other side of the tube are marked the various barometrical heights, at which the boiling water shews those particular degrees of heat which are set down in Sir Geo. Shuckburgh's table. With this instrument the barometrical height is shewn within one <cb/> 10th of an inch. The degrees of this Thermometer are rather longer than one 9th of an inch, and therefore may be divided into many parts, especially by a Nonius. But a considerable imperfection arises from the smallness of the tin vessel, which does not admit a sufficient quantity of water; but when the quantity of water shall be sufficiently large, as for instance 10 or 12 ounces, and is kept boiling in a proper vessel, its degree of heat under the same pressure of the atmosphere is very settled; whereas when a Thermometer is kept in a small quantity of boiling water, the mercury in its stem does not stand very steady, sometimes rising or falling so much as half a degree. Mr. Cavallo proposes a farther improvement of this instrument, in the Philos. Trans. vol. 71, p. 524.</p><p>The ingenious Mr. Wedgwood, so well known for his various improvements in the different sorts of pottery ware, has contrived to make a Thermometer for measuring the higher degrees of heat, by means of a distinguishing property of argillaceous bodies, viz, the diminution of their bulk by fire. This diminution commences in a low red heat, and proceeds regularly, as the heat increases, till the clay becomes vitrified. The total contraction of some good clays which he has examined in the strongest of his own fires, is considerably more than one-fourth part in every dimension. By measuring the contraction of such substances then, Mr. Wedgwood contrived to measure the most intense heats of ovens, furnaces, &c. For the curious particulars of which, see Philos. Trans. vol. 72, p. 305 &c.</p></div1><div1 part="N" n="THERMOSCOPE" org="uniform" sample="complete" type="entry"><head>THERMOSCOPE</head><p>, an instrument shewing the changes happening in the air with respect to heat and cold.</p><p>The word Thermoscope is often used indifferently with that of thermometer. There is some difference however in the literal import of the two; the first signifying an instrument that shews or exhibits the changes of heat &c to the eye; and the latter an instrument that measures those changes; so that a thermometer should be a more accurate Thermoscope.</p></div1><div1 part="N" n="THIR" org="uniform" sample="complete" type="entry"><head>THIR</head><p>, in Chronology, the name of the 5th month of the Ethiopians, which corresponds, according to Ludolf, to the month of January.</p></div1><div1 part="N" n="THIRD" org="uniform" sample="complete" type="entry"><head>THIRD</head><p>, in Music, a concord resulting from a mixture of two sounds containing an interval of 2 degrees: being called a third, because containing 3 terms, or sounds, between the extremes.</p><p>There is a greater and a less Third. The former takes its form from the sesquiquarta ratio, 4 to 5. The logarithm or measure of the octave 2/1 being 1.00000, the measure of the greater Third 5/4 will be 0.32193.— The <hi rend="italics">greater Third</hi> is by practitioners often taken for the third part of an octave; which is an error, since three greater Thirds fall short of the octave by a diesis; for .</p><p>The <hi rend="italics">lesser Third</hi> takes its form from the sesquiquinta ratio 5 to 6; the measure or logarithm of this lesser Third 6/5, being 0.26303, that of the octave 2/1 being 1.00000.</p><p>Both these Thirds are of great use in melody; making as it were the foundation and life of harmony.</p><p><hi rend="smallcaps">Third</hi>-<hi rend="italics">Point,</hi> or <hi rend="italics">Tierce-point,</hi> in Architecture, the point of section in the vertex of an equilateral triangle. —Arches or vaults of the Third Point, are those con- <pb n="593"/><cb/> sisting of two arches of a circle, meeting in an angle at top.</p><p>THREE-<hi rend="italics">legged-staff,</hi> an instrument consisting of three wooden legs, made with joints, so as to shut all together, and to take off in the middle for the better carriage. It has usually a ball and socket on the top; and its use is to support and adjust instruments for astronomy, surveying, &c.</p></div1><div1 part="N" n="THUNDER" org="uniform" sample="complete" type="entry"><head>THUNDER</head><p>, a noise in the lower region of the air, excited by a sudden explosion of electrical clouds; which are therefore called Thunder-clouds.</p><p>The phenomenon of Thunder is variously accounted for. Seneca, Rohault, and some other authors, both ancient and modern, account for Thunder, by supposing two clouds impending over one another, the upper and rarer of which, becoming condensed by a fresh accession of air raised by warmth from the lower parts of the atmosphere, or driven upon it by the wind, immediately falls sorcibly down upon the lower and denser cloud; by which fall, the air interposed between the two being compressed, that next the extremities of the two clouds is squeezed out, and leaves room for the extremity of the upper cloud to close tight upon the under; thus a great quantity of the air is enclosed, which at length escaping through some winding irregular vent or passage, occasions the noise called Thunder.</p><p>But this lame device could only reach at most to the case of Thunder heard without lightning; and therefore recourse has been had to other modes of solution. Thus, it has been said that Thunder is not occasioned by the falling of clouds, but by the kindling of sulphurous exhalations, in the same manner as the noise of the aurum fulminans. “There are sulphurous exhalations, says Sir I. Newton, always ascending into the air when the earth is dry; there they ferment with the nitrous acids, and, sometimes taking fire, generate Thunder, lightning, &c.”</p><p>The effects of Thunder are so like those of sired gunpowder, that Dr. Wallis thinks we need not scruple to ascribe them to the same cause; and the principal ingredients in gunpowder, we know, are nitre and sulphur; charcoal only serving to keep the parts separate, for their better kindling. Hence, if we conceive in the air a convenient mixture of nitrous and sulphurous particles; and those, by any cause, to be set on sire, such explosion may well follow, and with such noise and light as attend the firing of gunpowder; and being once kindled, it will run from place to place, different ways, as the exhalations happen to lead it; much as is found in a train of gunpowder.</p><p>But a third, and most probable opinion is, that Thunder is the report or noise produced by an electrical explosion in the clouds. Ever since the year 1752, in which the identity of the matter of lightning and of the electrical fluid has been ascertained, philosophers have generally agreed in considering Thunder as a concussion produced in the air by an explosion of electricity. For the illustration and proof of this theory, see <hi rend="smallcaps">Electricity</hi>, and <hi rend="smallcaps">Lightning.</hi></p><p>It may here be observed, that Mr. Henry Eeles, in a letter written in 1751, and read before the Royal Society in 1752, considers the electrical fire as the cause of Thunder, and accounts for it on this hypothesis; and he tells us, that he did not know of any other <cb/> person's having made the same conjecture. Philos. Trans. vol. 47, p. 524 &c.</p><p>That rattling in the noise of Thunder, which makes it seem as if it passed through arches, or were variously broken, is probably owing to the sound being excited among clouds hanging over one another, and the agitated air passing irregularly between them.</p><p>The explosion, if high in the air, and remote from us, will do no mischief; but when near, it may destroy trees, animals, &c.</p><p>This proximity, or small distance, may be estimated nearly by the interval of time between seeing the flash of lightning, and hearing the report of the Thunder, estimating the distance, after the rate of 1142 feet per second of time, or 3 2/3 seconds to the mile. Dr. Wallis observes, that commonly the difference between the two is about 7 seconds, which, at the rate above mentioned, gives the distance almost 2 miles. But sometimes it comes in a second or two, which argues the explosion very near us, and even among us. And in such cases, the doctor assures us, he has sometimes foretold the mischiefs that happened.</p><p>The noise of Thunder, and the flame of lightning, are easily made by art. If a mixture of oil or spirit of vitriol be made with water, and some filings of steel added to it, there will immediately arise a thick smoke, or vapour, out of the mouth of the vessel; and if a lighted candle be applied to this, it will take fire, and the flame will immediately descend into the vessel, which will be burst to pieces with a noise like that of a cannon.</p><p>This is so far analogous to Thunder and lightning, that a great explosion and fire are occasioned by it; but in this they differ, that this matter when once fired is destroyed, and can give no more explosions; whereas, in the heavens, one clap of Thunder usually follows another, and there is a continued succession of them for a long time. Mr. Homberg explained this by the lightness of the air above us, in comparison of that near, which therefore would not suffer all the matter so kindled to be dissipated at once, but keeps it for several returns.</p><p>THUNDERBOLT. When lightning acts with extraordinary violence, and breaks or shatters any thing, it is called a <hi rend="italics">Thunderbolt,</hi> which the vulgar, to fit it for such effects, suppose to be a hard body, and even a stone.—But that we need not have recourse to a hard solid body to account for the effects commonly attributed to the Thunderbolt, will be evident to any one, who considers those of the pulvis fulminans, and of gunpowder; but more especially the astonishing powers of electricity, when only collected and employed by human art, and much more when directed and exercised in the course of nature.</p><p>When we consider the known effects of electrical explosions, and those produced by lightning, we shall be at no loss to account for the extraordinary operations vulgarly ascribed to Thunderbolts. As stones and bricks struck by lightning are often found in a vitrified state, we may reasonably suppose, with Beccaria, that some stones in the earth, having been struck in this manner, gave occasion to the vulgar opinion of the Thunderbolt.</p><p><hi rend="smallcaps">Thunder</hi>-<hi rend="italics">clouds,</hi> in Physiology, are those clouds <pb n="594"/><cb/> which are in a state fit for producing lightning and thunder.</p><p>From Beccaria's exact and circumstantial account of the external appearances of Thunder-clouds, the following particulars are extracted.</p><p>The first appearance of a Thunder storm, which usually happens when there is little or no wind, is one dense cloud, or more, increasing very fast in size, and rising into the higher regions of the air. The lower surface is black and nearly level; but the upper finely arched, and well defined. Many of these clouds often seem piled upon one another, all arched in the same manner; but they are continually uniting, swelling, and extending their arches.</p><p>At the time of the rising of this cloud, the atmosphere is commonly full of a great many separate clouds, that are motionless, and of odd whimsical shapes. All these, upon the appearance of the Thunder-cloud, draw towards it, and become more uniform in their shapes as they approach; till, coming very near the Thundercloud, their limbs mutually stretch toward one another, and they immediately coalesce into one uniform mass. These he calls adscititious clouds, from their coming in, to enlarge the size of the Thunder-cloud. But sometimes the Thunder-cloud will swell, and increase very fast, without the conjunction of any adscititious clouds; the vapours in the atmosphere forming themselves into clouds wherever it passes. Some of the adscititious clouds appear like white fringes, at the skirts of the Thunder-cloud, or under the body of it, but they keep continually growing darker and darker, as they approach to unite with it.</p><p>When the Thunder-cloud is grown to a great size, its lower surface is often ragged, particular parts being detached towards the earth, but still connected with the rest. Sometimes the lower surface swells into various large protuberances bending uniformly downward; and sometimes one whole side of the cloud will have an inclination to the earth, and the extremity of it nearly touch the ground. When the eye is under the Thundercloud, after it is grown larger, and well formed, it is seen to sink lower, and to darken prodigiously; at the same time that a number of small adscititious clouds (the origin of which can never be perceived) are seen in a rapid motion, driving about in very uncertain directions under it. While these clouds are agitated with the most rapid motions, the rain commonly falls in the greatest plenty, and if the agitation be exceedingly great, it commonly hails.</p><p>While the Thunder-cloud is swelling, and extending its branches over a large tract of country, the lightning is seen to dart from one part of it to another, and often to illuminate its whole mass. When the cloud has acquired a sufficient extent, the lightning strikes between the cloud and the earth, in two opposite places, the path of the lightning lying through the whole body of the cloud and its branches. The longer this lightning continues, the less dense does the cloud become, and the less dark its appearance; till at length it breaks in different places, and shews a clear sky.</p><p>These Thunder clouds were sometimes in a positive as well as a negative state of electricity. The electricity continued longer of the same kind, in proportion as the Thunder-cloud was simple, and uniform in its di- <cb/> rection: but when the lightning changed its place, there commonly happened a change in the electricity of the apparatus, over which the clouds passed. It would change suddenly after a very violent flash of lightning, but the change would be gradual when the lightning was moderate, and the progress of the Thunder-cloud slow. Beccar. Lettere dell 'Elettricismo pa. 107; or Priestley's Hist. Elec. vol. 1, p. 397. See also <hi rend="smallcaps">Lightning.</hi></p><p><hi rend="smallcaps">Thunder</hi>-<hi rend="italics">House,</hi> in Electricity, is an instrument invented by Dr. James Lind, for illustrating the manner in which buildings receive damage from lightning, and to evince the utility of metallic conductors in preserving them from it.</p><p>A (fig. 1, pl. 35), is a board about 3/4 of an inch thick, and shaped like the gable end of a house. This board is fixed perpendicularly upon the bottom board B, upon which the perpendicular glass pillar CD is also fixed in a hole about 8 inches distant from the basis of the board A. A square hole ILMK, about a quarter of an inch deep, and nearly one inch wide, is made in the board A, and is filled with a square piece of wood, nearly of the same dimensions. It is nearly of the same dimensions, because it must go so easily into the hole, that it may drop off, by the least shaking of the instrument. A wire LK is fastened diagonally to this square piece of wood. Another wire IH of the same thickness, having a brass ball H, screwed on its pointed extremity, is fastened upon the board A: so also is the wire MN, which is shaped in a ring at O. From the upper extremity of the glass pillar CD, a crooked wire proceeds, having a spring socket F, through which a double knobbed wire slips perpendicularly, the lower knob G of which falls just above the knob H. The glass pillar DC must not be made very fast into the bottom board; but it must be fixed so that it may be pretty easily moved round its own axis, by which means the brass ball G may be brought nearer to or farther from the ball H, without touching the part EFG. Now when the square piece of wood LMIK (which may represent the shutter of a window or the like) is fixed into the hole so that the wire LK stands in the dotted representation IM, then the metallic communication from H to O is complete, and the instrument represents a house furnished with a proper metallic conductor; but if the square piece of wood LMIK be fixed so that the wire LK stands in the direction LK, as represented in the figure, then the metallic conductor HO, from the top of the house to its bottom, is interrupted at IM, in which case the house is not properly secured.</p><p>Fix the piece of wood LMIK, so that its wire may be as represented in the figure, in which case the metallic conductor HO is discontinued. Let the ball G be fixed at about half an inch perpendicular distance from the ball H; then, by turning the glass pillar DC, remove the former ball from the latter; by a wire or chain connect the wire EF with the wire Q of the jar P; and let another wire or chain, fastened to the hook O, touch the outside coating of the jar. Connect the wire Q with the prime conductor, and charge the jar; then, by turning the glass pillar DC, let the ball G come gradually near the ball H, and when they are arrived sufficiently near one another, you will observe, that the jar explodes and the piece of wood LMIK is <pb/><pb/><pb n="595"/><cb/> pushed out of the hole to a considerable distance from the Thunder-house.</p><p>Now the ball G, in this experiment, represents an electrified cloud, which, when it is arrived sufficiently near the top of the house A, the electricity strikes it; and as this house is not secured with a proper conductor, the explosion breaks part of it, i. e. knocks off the piece of wood IM.</p><p>Repeat the experiment with only this variation, viz, that this piece of wood IM be situated so that the wire LK may stand in the situation IM; in which case the conductor HO is not discontinued; and you will observe that the explosion will have no effect upon the piece of wood LM; this remaining in the hole unmoved; which shews the usefulness of the metallic conductor.</p><p>Farther, unscrew the brass ball H from the wire HI, so that this may remain pointed, and with this difference only in the apparatus repeat both the above experiments, and you will find that the piece of wood IM is in neither case moved from its place, nor will any explosion be heard; which not only demonstrates the preference of conductors with pointed terminations to those with blunted ones, but also shews that a house, furnished with sharpterminations, although not furnished with a regular conductor, is almost sufficiently guarded against the effects of lightning.</p><p>Mr. Henley, having connected a jar containing 509 square inches of coated surface with his prime conductor, observed that if it was so charged as to raise the index of his electrometer to 60°, by bringing the ball on the wire of the Thunder-house, to the distance of half an inch from that connected with the prime conductor, the jar would be discharged, and the piece in the Thunder-house thrown out to a considerable distance. Using a pointed wire for a conductor to the Thunder-house, instead of the knob, the charge being the same as before, the jar was discharged silently, though suddenly; and the piece was not thrown out of the Thunderhouse. In another experiment, having made a double circuit to the Thunder-house, the first by the knob, the second by a sharp-pointed wire, at an inch and a quarter distance from each other, but of exactly the same height (as in fig. 2) the charge being the same; although the knob was brought first under that connected with the prime conductor, which was raised half an inch above it, and followed by the point, yet no explosion could fall upon the knob; the point drew off the whole charge silently, and the piece in the Thunderhouse remained unmoved.</p><p>Phil. Trans. vol. 64, p. 136. See <hi rend="smallcaps">Points</hi> in Electricity.</p></div1><div1 part="N" n="THURSDAY" org="uniform" sample="complete" type="entry"><head>THURSDAY</head><p>, the 5th day of the Christian's week, but the 6th of the Jews. The name is from Thor, one of the Saxon Gods.</p></div1><div1 part="N" n="THUS" org="uniform" sample="complete" type="entry"><head>THUS</head><p>, in Sea-Language, a word used by the pilot in directing the helmsman or steersman to keep the ship in her present situation when sailing with a scant wind, so that she may not approach too near the direction of the wind, which would shiver her sails, nor fall to leeward, and run farther out of her course.</p></div1><div1 part="N" n="TIDES" org="uniform" sample="complete" type="entry"><head>TIDES</head><p>, two periodical motions of the waters of the sea; called also the <hi rend="italics">flux</hi> and <hi rend="italics">reflux,</hi> or the <hi rend="italics">ebb</hi> and <hi rend="italics">flow.</hi> <cb/></p><p>The Tides are found to follow periodically the course of the sun and moon, both as to time and quantity And hence it has been suspected, in all ages, that the Tides were somehow produced by the influence of these luminaries. Thus, several of the ancients, and among others, Pliny, Ptolomy, and Macrobius, were acquainted with the influence of the sun and moon upon the Tides; and Pliny says expressly, that the cause of the ebb and flow is in the sun, which attracts the waters of the ocean; and adds, that the waters rise in proportion to the proximity of the moon to the earth. It is indeed now well known, from the discoveries of Sir Isaac Newton, that the Tides are caused by the gravitation of the earth towards the sun and moon. Indeed the sagacious Kepler, long ago, conjectured this to be the cause of the Tides: “If, says he, the earth ceased to attract its waters towards itself, all the water in the ocean would rise and flow into the moon: the sphere of the moon's attraction extends to our earth, and draws up the water.” Thus thought Kepler, in his Introd. ad Theor. Mart. This surmise, for it was then no more, is now abundantly verified in the theory laid down by Newton, and by Halley, from his principles.</p><p><hi rend="italics">As to the Phenomena of the</hi> <hi rend="smallcaps">Tides:</hi> 1. The sea is observed to flow, for about 6 hours, from south towards north; the sea gradually swelling; so that, entering the mouths of rivers, it drives back the river-waters towards their heads, or springs. After a continual flux of 6 hours, the sea seems to rest for about a quarter of an hour; after which it begins to ebb, or retire back again, from north to south, for 6 hours more; in which time, the water sinking, the rivers resume their natural course. Then, after a seeming pause of a quarter of an hour, the sea again begins to flow, as before: and so on alternately.</p><p>2. Hence, the sea ebbs and flows twice a day, but falling every day gradually later and later, by about 48 minutes, the period of a flux and reflux being on an average about 12 hours 24 minutes, and the double of each 24 hours 48 minutes; which is the period of a lunar day, or the time between the moon's passing a meridian, and coming to it again. So that the sea flows as often as the moon passes the meridian, both the arch above the horizon, and that below it; and ebbs as often as she passes the horizon, both on the eastern and western side.</p><p>Other phenomena of the Tides are as below; and the reasons of them will be noticed in the Theory of the Tides that follows.</p><p>3. The elevation towards the moon a little exceeds the opposite one. And the quantity of the ascent of the water is diminished from the equator towards the poles.</p><p>4. From the sun, every natural day, the sea is twice elevated, and twice depressed, the same as for the moon. But the solar Times are much less than the lunar ones, on account of the immense distance of the sun; yet they are both subject to the same laws.</p><p>5. The Tides which depend upon the actions of the sun and moon, are not distinguished, but compounded, and so forming as to sense one united Tide, increasing and decreasing, and thus making neap and spring Tides: for, by the action of the sun, the <pb n="596"/><cb/> lunar Tide is only changed; which change varies every day, by reason of the inequality between the natural and lunar day.</p><p>6. In the syzygies the elevations from the action of both luminaries concur, and the sea is more elevated. But the sea ascends less in the quadratures; for where the water is elevated by the action of the moon, it is depressed by the action of the sun; and vice versa. Therefore, while the moon passes from the syzygy to the quadrature, the daily elevations are continually diminished: on the contrary, they are increased while the moon moves from the quadrature to the syzygy. At a new moon also, <hi rend="italics">cæteris paribus,</hi> the elevations are greater; and those that follow one another the same day, are more different than at full moon.</p><p>7. The greatest elevations and depressions are not observed till the 2d or 3d day after the new or full moon. And if we consider the luminaries receding from the plane of the equator, we shall perceive that the agitation is diminished, and becomes less, according as the declination of the luminaries becomes greater.</p><p>8. In the syzygies, and near the equinoxes, the Tides are observed to be the greatest, both luminaries being in or near the equator.</p><p>9. The actions of the sun and moon are greater, the nearer those bodies are to the earth; and the less, as they are farther off. Also the greatest Tides happen near the equinoxes, or rather when the sun is a little to the south of the equator, that is, a little before the vernal, and after the autumnal equinox. But yet this does not happen regularly every year, because some variation may arise from the situation of the moon's orbit, and the distance of the syzygy from the equinox.</p><p>10. All these phenomena obtain, as described, in the open sea, where the ocean is extended enough to be subject to these motions. But the particular situations of places, as to shores, capes, straits, &c, disturb these general rules. Yet it is plain, from the most common and universal observations, that the Tides follow the laws above laid down.</p><p>11. The mean force of the moon to move the sea, is to that of the sun, nearly as 4 1/2 to 1. And therefore, if the action of the sun alone produce a Tide of 2 feet, which it has been stated to do, that of the moon will be 9 feet; from which it follows, that the spring Tides will be 11 feet, and the neap Tides 7 feet high. But as to such elevations as far exceed these, they happen from the motion of the waters against some obstacles, and from the sea violently entering into straits or gulphs where the force is not broken till the water rises higher. <hi rend="center"><hi rend="italics">Theory of the</hi> <hi rend="smallcaps">Tides.</hi></hi></p><p>1. If the earth were entirely fluid, and quiescent, it is evident that its particles, by their mutual gravity towards each other, would form the whole mass into the figure of an exact sphere. Then suppose some power to act on all the particles of this sphere with an equal force, and in parallel directions; by <cb/> such a power the whole mass will be moved together, but its figure will suffer no alteration by it, being still the same perfect sphere, whose centre will have the same motion as each particle.</p><p>Upon this supposition, if the motion of the earth round the common centre of gravity of the earth and moon were destroyed, and the earth left to the influence of its gravitation towards the moon, as the acting power above mentioned; then the earth would fall or move straight towards the moon, but still retaining its true spherical figure.</p><p>But the fact is, that the effects of the moon's action, as well as the action itself, on different parts of the earth, are not equal: those parts, by the general law of gravity, being most attracted that are nearest the moon, and those being least attracted that are farthest from her, while the parts that are at a middle distance are attracted by a mean degree of force: besides, all the parts are not acted on in parallel lines, but in lines directed towards the centre of the moon: on both which accounts the spherical figure of the fluid earth must suffer some change from the action of the moon. So that, in falling, as above, the nearer parts, being most attracted, would fall quickest; the farther parts, being least attracted, would fall slowest; and the fluid mass would be lengthened out, and take a kind of spheroidical form.</p><p>Hence it appears, and what must be carefully observed, that it is not the action of the moon itself, but the inequalities in that action, that cause any variation from the spherical figure; and that, if this action were the same in all the particles as in the central parts, and operating in the same direction, no such change would ensue.</p><p>Let us now admit the parts of the earth to gravitate toward its centre: then, as this gravitation far exceeds the action of the moon, and much more exceeds the differences of her actions on different parts of the earth, the effect that results from the inequalities of these actions of the moon, will be only a small diminution of the gravity of those parts of the earth which it endeavoured in the former supposition to separate from its centre; that is, those parts of the earth which are nearest to the moon, and those that are farthest from her, will have their gravity toward the earth somewhat abated; to say nothing of the lateral parts. So that supposing the earth fluid, the columns from the centre to the nearest, and to the farthest parts, must rise, till by their greater height they be able to balance the other columns, whose gravity is less altered by the inequalities of the moon's action. And thus the figure of the earth must still be an oblong spheroid.</p><p>Let us now consider the earth, instead of falling toward the moon by its gravity, as projected in any direction, so as to move round the centre of gravity of the earth and moon: it is evident that in this case, the several parts of the fluid earth will still preserve their relative positions; and the figure of the earth will remain the same as if it fell freely toward the moon; that is, the earth will still assume a spheroidal form, having its longest diameter directed toward the moon. <pb n="597"/><cb/> <figure/></p><p>From the above reasoning it appears, that the parts of the earth directly under the moon, as at H, and also the opposite parts at D, will have the flood or highwater at the same time; while the parts, at B and F, at 90° distance, or where the moon appears in the horizon, will have the ebbs or lowest waters at that time.</p><p>Hence, as the earth turns round its axis from the moon to the moon again in 24 hours 48 minutes, this oval of water must shift with it; and thus there will be two Tides of flood and two of ebb in that time.</p><p>But it is further evident that, by the motion of the earth on her axis, the most elevated part of the water is carried beyond the moon in the direction of the rotation. So that the water continues to rise after it has passed directly under the moon, though the immediate action of the moon there begins to decrease, and comes not to its greatest elevation till it has got about half a quadrant farther. It continues also to descend after it has passed at 90° distance from the point below the moon, to a like distance of about half a quadrant. The greatest elevation therefore is not in the line drawn through the centres of the earth and moon, nor the lowest points where the moon appears in the horizon, but all these about half a quadrant removed eastward from these points, in the direction of the motion of rotation. Thus in open seas, where the water flows freely, the moon M is generally past the north and south meridian, as at <hi rend="italics">p,</hi> when the high water is at Z and at <hi rend="italics">n:</hi> the reason of which is plain, because the moon acts with the same force after she has passed the meridian, and thus adds to the libratory or waving motion, which the water acquired when she was in the meridian; and therefore the time of high water is not precisely at the time of her coming to the meridian, but some time after, &c.</p><p>Besides, the Tides answer not always to the same distance of the moon, from the meridian, at the same places; but are variously affected by the action of the sun, which brings them on sooner when the moon is in her first and third quarters, and keeps them back later when she is in her 2d and 4th; because, in the former case the Tide raised by the sun alone would be earlier than the Tide raised by the moon, and in the latter case later.</p><p>2. We have hitherto adverted only to the action of the moon in producing Tides; but it is manifest that, for the same reasons, the inequality of the sun's action on different parts of the earth, would produce a like <cb/> effect, and a like variation from the exact spherical figure of a fluid earth. So that in reality there are two Tides every natural day from the action of the sun, as there are in the lunar day from that of the moon, subject to the same laws; and the lunar Tide, as we have observed, is somewhat changed by the action of the sun, and the change varies every day on account of the inequality between the natural and the lunar day. Indeed the effect of the sun in producing Tides, because of his immense distance, must be considerably less than that of the moon, though the gravity toward the sun be much greater: for it is not the action of the sun or moon itself, but the inequalities in that action, that have any effect: the sun's distance is so great, that the diameter of the earth is but as a point in comparison with it, and therefore the difference between the sun's actions on the nearest and farthest parts, becomes vastly less than it would be if the sun were as near as the moon. However the immense bulk of the sun makes the effect still sensible, even at so great a distance; and therefore, though the action of the moon has the greatest share in producing the Tides, the action of the sun adds sensibly to it when they conspire together, as in the full and change of the moon, when they are nearly in the same line with the centre of the earth, and therefore unite their forces: consequently, in the syzygies, or at new and full moon, the Tides are the greatest, being what are called the <hi rend="italics">Spring-Tides.</hi> But the action of the sun diminishes the effect of the moon's action in the quarters, because the one raises the water in that case where the other depresses it; therefore the Tides are the least in the quadratures, and are called <hi rend="italics">Neap-Tides.</hi></p><p>Newton has calculated the effects of the sun and moon respectively upon the Tides, from their attractive powers. The former he finds to be to the force of gravity, as 1 to 12868200, and to the centrifugal force at the equator as 1 to 44527. The elevation of the waters by this force is considered by Newton as an effect similar to the elevation of the equatorial parts above the polar parts of the earth, arising from the centrifugal force at the equator; and as it is 44527 times less, he finds it to be 24 1/2 inches, or 2 feet and 1/2 an inch.</p><p>To find the force of the moon upon the water, Newton compares the spring-tides at the mouth of the river Avon, below Bristol, with the neap-tides there, and finds the proportion as 9 to 5; whence, after several necessary corrections, he concludes that the force of the moon to that of the sun, in raising the waters of the ocean, is as 4.4815 to 1; so that the force of the moon is able of itself to produce an elevation of 9 feet 1 3/4 inch, and the sun and moon together may produce an elevation of about 11 feet 2 inches, when at their mean distances from the earth, or an elevation of about 12 3/4 feet, when the moon is nearest the earth. The height to which the water is found to rise, upon coasts of the open and deep ocean, is agreeable enough to this computation.</p><p>Dr. Horsley estimates the force of the moon to that of the sun, as 5.0469 to 1, in his edit. of Newton's Princip. See the Princip. lib. 3, sect. 3, pr. 36 and 37; also Maclaurin's Dissert. de Causa Physica Fluxus et Refluxus Maris apud Phil. Nat. Princ. Math. Com- <pb n="598"/><cb/> ment. le Seur & Jacquier, tom. 3, p. 272. And other calculators make the proportion still more different.</p><p>3. It must be observed, that the spring-tides do not happen precisely at new and full moon, nor the neaptides at the quarters, but a day or two after; because, as in other cases, so in this, the effect is not greatest or least when the immediate influence of the cause is greatest or least. As, for example, the greatest heat is not on the day of the solstice, when the immediate action of the sun is greatest, but some time after; so likewise, if the actions of the sun and moon should suddenly cease, yet the Tides would continue to have their course for some time; and like also as the waves of the sea continue aster a storm.</p><p>4. The different distances of the moon from the earth produce a sensible variation in the Tides. When the moon approaches toward the earth, her action on every part increases, and the differences of that action, on which the Tides depend, also increase; and as the moon approaches, her action on the nearest parts increases more quickly than that on the remote parts, so that the Tides increase in a higher proportion as the moon's distances decrease. In fact, it is shewn by Newton, that the Tides increase in proportion as the cubes of the distances decrease; so that the moon at half her distance would produce a Tide 8 times greater.</p><p>The moon describes an oval about the earth, and at her nearest distance produces a Tide sensibly greater than at her greatest distance from the earth: and hence it is that two great spring-tides never succeed each other immediately; for if the moon be at her least distance from the earth at the change, she must be at her greatest distance at the full, having made half a revolution in the intervening time, and therefore the spring-tide then will be much less than that at the last change was; and for the same reason, if a great spring-tide happen at the time of full moon, the Tide at the ensuing change will be less.</p><p>5. The spring-tides are highest, and the neap-tides lowest, about the time of the equinoxes, or the latter end of March and September; and, on the contrary, the spring-tides are the lowest, and the neap-tides the highest, at the solstices, or about the latter end of June and December: so that the difference between the spring and neap Tides, is much more considerable about the equinoctial than the solstitial seasons of the year. To illustrate and evince the truth of this observation, let us consider the effect of the luminaries upon the Tides, when in and out of the plane of the equator. Now it is manifest, that if either the sun or moon were in the pole, they could not have any effect on the Tides; for their action would raise all the water at the equator, or at any parallel, quite around, to a uniform height; and therefore any place of the earth, in describing its parallel to the equator, would not meet, in its course, with any part of the water more elevated than another; so that there could be no Tide in any place, that is, no alteration in the height of the waters.</p><p>On the other hand, the effect of the sun or moon is greatest when in the equinoctial; for then the axis of the spheroidal figure, arising from their action, moves in the greatest circle, and the water is put into <cb/> the greatest agitation; and hence it is that the spring-tides produced when the sun and moon are both in the equinoctial, are the greatest of any, and the neaptides the least of any about that time. And when the luminary is any where between the equinoctial and the pole, the Tides are the smaller.</p><p>6. The highest spring tides are after the autumnal and before the vernal equinox: the reason of which is, because the sun is nearer the earth in winter than in summer.</p><p>7. Since the greatest of the two Tides happening in every diurnal revolution of the moon, is that in which the moon is nearest the zenith, or nadyr: for this reason, while the sun is in the northern signs, the greater of the two diurnal Tides in our climates, is that arising from the moon above the horizon; when the sun is in the southern signs, the greatest is that arising from the moon below the horizon. Thus it is found by observation that the evening Tides in the summer exceed the morning Tides, and in winter the morning Tides exceed the evening Tides. The difference is found at Bristol to amount to 15 inches, and at Plymouth to 12. It would be still greater, but that a fluid always retains an impressed motion for some time; so that the preceding Tides affect always those that follow them. Upon the whole, while the moon has a north declination, the greatest Tides in the northern hemisphere are when she is above the horizon, and the reverse while her declination is south.</p><p>8. Such would the Tides regularly be, if the earth were all over covered with the sea very deep, so that the water might freely follow the influence of the sun and moon; but, by reason of the shoalness of some places, and the narrowness of the straits in others, through which the Tides are propagated, there arises a great diversity in the effect according to the various circumstances of the places. Thus, a very slow and imperceptible motion of the whole body of water, where it is very deep, as 2 miles for instance, will suffice to raise its surface 10 or 12 feet in a Tide's time: whereas, if the same quantity of water were to be conveyed through a channel of 40 fathoms deep, it would require a very rapid stream to effect it in so large inlets as are the English channel, and the German ocean; whence the Tide is found to set strongest in those places where the sea grows narrowest, the same quantity of water being in that case to pass through a smaller passage. This is particularly observable in the straits between Portland and Cape la Hogue in Normandy, where the Tide runs like a sluice: and would be yet more so between Dover and Calais, if the Tide coming round the island did not check it.</p><p>This force, when once impressed, continues to carry the water above the ordinary height in the ocean, especially where the water meets a direct obstacle, as it does in St. Maloes; and where it enters into a long channel which, running far into the land, grows very strait at its extremity, as it does into the Severn sea at Chepstow and Bristol.</p><p>This shoalness of the sea, and the intercurrent continents, are the reasons that in the open ocean the Tides rise but to very small heights in proportion to what they do in wide-mouthed rivers, opening in the direc- <pb n="599"/><cb/> tion of the stream of the Tide; and that high water is not soon aster the moon's appulse to the meridian, but some hours after it, as it is observed upon all the western coast of Europe and Africa, from Ireland to the Cape of Good Hope; in all which a south-west moon makes high water; and the same it is said is the case on the western side of America. So that Tides happen to different places at all distances of the moon from the meridian, and consequently at all hours of the day.</p><p>To allow the Tides their full motion, the ocean in which they are produced, ought to be extended from east to west 90 degrees at least; because that is the distance between the places where the water is most raised and depressed by the moon. Hence it appears that it is only in the great oceans that such Tides can be produced, and why in the larger Pacific ocean they exceed those in the Atlantic ocean. Hence also it is obvious, why the Tides are not so great in the torrid zone, between Africa and America, where the ocean is narrower, as in the temperate zones on either side; and hence we may also understand why the Tides are so small in islands that are very far distant from the shores. It is farther manifest that, in the Atlantic ocean, the water cannot rise on one shore but by descending on the other; so that at the intermediate islands it must continue at a mean height between its elevations on those two shores. But when Tides pass over shoals, and through straits into bays of the sea, their motion becomes more various, and their height depends on many circumstances.</p><p>To be more particular. The Tide that is produced on the western coasts of Europe, in the Atlantic, corresponds to the situation of the moon already described. Thus it is high water on the western coasts of Ireland, Portugal and Spain, about the 3d hour after the moon has passed the meridian: from thence it flows into the adjacent channels, as it finds the easiest passage. One current from it, for instance, runs up by the south of England, and another comes in by the north of Scotland; they take a considerable time to move all this way, making always high water sooner in the places to which they first come; and it begins to fall at these places while the currents are still going on to others that are farther distant in their course. As they return, they are not able to raise the Tide, because the water runs faster off than it returns, till, by a new Tide propagated from the open ocean, the return of the current is stopped, and the water begins to rise again. The Tide propagated by the moon in the German ocean, when she is 3 hours past the meridian, takes 12 hours to come from thence to London bridge; so that when it is high water there, a new Tide is already come to its height in the ocean; and in some intermediate place it must be low water at the same time. Consequently when the moon has north declination, and we should expect the Tide at London to be the greatest when the moon is above the horizon, we find it is least; and the contrary when she has south declination.</p><p>At several places it is high water 3 hours before the moon comes to the meridian; but that Tide, which the moon pushes as it were before her, is only <cb/> the Tide opposite to that which was raised by her when she was 9 hours past the opposite meridian.</p><p>It would be endless to recount all the particular solutions, which are easy consequences from this doctrine: as, why the lakes and seas, such as the Caspian sea and the Mediterranean sea, the Black sea and the Baltic, have little or no sensible Tides: for lakes are usually so small, that when the moon is vertical she attracts every part of them alike, so that no part of the water can be raised higher than another: and having no communication with the ocean, it can neither increase nor diminish their water, to make it rise and fall; and seas that communicate by such narrow inlets, and are of so immense an extent, cannot speedily receive and empty water enough to raise or sink their surface any thing sensibly.</p><p>In general; when the time of high water at any place is mentioned, it is to be understood on the days of new and full moons.—Among pilots, it is customary to reckon the time of flood, or high water, by the point of the compass the moon bears on, at that time, allowing 3/4 of an hour for each point. Thus, on the full and change days, in places where it is flood at noon, the Tide is said to flow north and south, or at 12 o'clock: in other places, on the same days, where the moon bears 1, 2, 3, 4, or more points to the east or west of the meridian, when it is high water, the Tide is said to flow on such point; thus, if the moon bears SE, at flood, it is said to flow SE and NW, or 3 hours before the meridian, that is, at 9 o'clock; if it bears SW, it flows SW and NE, or at 3 hours after the meridian; and in like manner for the other points of the moon's bearing.</p><p>The times of high water in any place fall about the same hours after a period of about 15 days, or between one spring Tide and another; but during that period, the times of high water fall each day later by about 48 minutes.</p><p>On the subject of this article, see Newton Princ. Math. lib. 3, prop. 24, and De System. Mundi sect. 38, &c. Apud Opera edit. Horsley, tom. 3, pa. 52 &c. p. 203 &c. Maclaurin's Account of Newton's Discoveries, book 4, ch. 7. Ferguson's Astron. ch. 17. Robertson's Navig. book 6, sect. 7, 8, 9. Lalande's Astron. vol. 4.</p><p><hi rend="smallcaps">Tide</hi> <hi rend="italics">Dial,</hi> an instrument contrived by Mr. Ferguson, for exhibiting and determining the state of the Tides. For the construction and use of which see his Astron. p. 297.</p><p><hi rend="smallcaps">Tide</hi> <hi rend="italics">Tables,</hi> are tables commonly exhibiting the times of high water at sundry places, as they fall on the days of the full and change of the moon, and sometimes the height of them also. These are common in most books on Navigation, particularly Robertson's, and the 2d ed. of Tables requisite to be used with the Nautical Almanac. See one at <hi rend="smallcaps">High</hi>- <hi rend="italics">water.</hi></p></div1><div1 part="N" n="TIERCE" org="uniform" sample="complete" type="entry"><head>TIERCE</head><p>, or <hi rend="smallcaps">Teirce</hi>, a liquid measure, as of wine, oil, &c, containing 42 gallons, or the 3d part of a pipe; whence its name.</p></div1><div1 part="N" n="TIME" org="uniform" sample="complete" type="entry"><head>TIME</head><p>, a succession of phenomena in the universe; or a mode of duration, marked by certain periods and measures; chiefly indeed by the motion <pb n="600"/><cb/> and revolution of the luminaries, and particularly of the sun.</p><p>The idea of Time in general, Locke observes, we acquire by considering any part of infinite duration, as set out by periodical measures: the idea of any particular Time, or length of duration, as a day, an hour, &c, we acquire first by observing certain appearances at regular and seemingly equidistant periods. Now, by being able to repeat these lengths or measures of Time as often as we will, we can imagine duration, where nothing really endures or exists; and thus we imagine tomorrow, or next year, &c.</p><p>Some of the later school-philosophers define Time to be the duration of a thing whose existence is neither without beginning nor end: by this, Time is distinguished from eternity.</p><p>Aristotle and the Peripatetics define it, <hi rend="italics">numerus motus secundum prius & posterius,</hi> or a multitude of transient parts of motion, succeeding each other, in a continual flux, in the relation of priority and posteriority. Hence it should follow that Time is motion itself, or at least the duration of motion, considered as having several parts, some of which are continually succeeding to others. But on this principle, Time or temporal duration would not agree to bodies at rest, which yet nobody will deny to exist in Time, or to endure for a Time.</p><p>To avoid this inconvenience, the Epicureans and Corpuscularians made Time to be a sort of flux different from motion, consisting of infinite parts, continually and immediately succeeding each other, and this from eternity to eternity. But others directly explode this notion, as establishing an eternal being, independent of God. For how should there be a flux before any thing existed to flow? and what should that flux be, a substance, or an accident? According to the philosophic poet, “Time of itself is nothing, but from thought<lb/> Receives its rise; by labouring fancy wrought<lb/> From things consider'd, whilst we think on some<lb/> As present, some as past, or yet to come.<lb/> No thought can think on Time, that's still confest,<lb/> But thinks on things in motion or at rest.”<lb/> And so on. Vide Lucretius, book i.</p><p>Time may be distinguished, like place, into <hi rend="italics">absolute</hi> and <hi rend="italics">relative.</hi></p><p><hi rend="italics">Absolute</hi> <hi rend="smallcaps">Time</hi>, is Time considered in itself, and without any relation to bodies, or their motions.</p><p><hi rend="italics">Relative</hi> or <hi rend="italics">Apparent</hi> <hi rend="smallcaps">Time</hi>, is the sensible measure of any duration by means of motion.</p><p>Some authors distinguish Time into <hi rend="italics">astronomical</hi> and <hi rend="italics">civil.</hi></p><p><hi rend="italics">Astronomical</hi> <hi rend="smallcaps">Time</hi>, is that which is taken purely from the motion of the heavenly bodies, without any other regard.</p><p><hi rend="italics">Civil</hi> <hi rend="smallcaps">Time</hi>, is the former Time accommodated to civil uses, and formed or distinguished into years, months, days, &c.</p><p>Time makes the subject of chronology.</p><div2 part="N" n="Time" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Time</hi></head><p>, in music, is an affection of sound, by which it is said to be long or short, with regard to its continuance in the same tone or degree of tune. <cb/></p><p>Musical Time is distinguished into <hi rend="italics">common</hi> or <hi rend="italics">duple</hi> Time, and <hi rend="italics">triple</hi> Time.</p><p><hi rend="italics">Double, duple,</hi> or <hi rend="italics">common Time,</hi> is when the notes are in a duple duration of each other, viz, a semibreve equal to 2 minims, a minim to 2 crotchets, a crotchet to 2 quavers, &c.</p><p>Common or double Time is of two kinds. The first when every bar or measure is equal to a semibreve, or its value in any combination of notes of a less quantity. The second is where every bar is equal to a minim, or its value in less notes. The movements of this kind of measure are various, but there are three common distinctions; the first <hi rend="italics">slow,</hi> denoted at the beginning of the line by the mark C; the 2d <hi rend="italics">brisk,</hi> marked thus <figure/>; and the 3d <hi rend="italics">very brisk,</hi> thus marked <figure/>.</p><p><hi rend="italics">Triple Time</hi> is when the durations of the notes are triple of each other, that is, when the semibreve is equal to 3 minims, the minim to 3 crotchets, &c. and it is marked T.</p><p><hi rend="smallcaps">Time</hi>-<hi rend="italics">keepers,</hi> in a general sense, denote instruments adapted for measuring time. See <hi rend="smallcaps">Chronometer.</hi></p><p>In a more peculiar and definite sense, Time-keeper is a term first applied by Mr. John Harrison to his watches, constructed and used for determining the longitude at sea, and for which he received, at different times, the parliamentary reward of 20 thousand pounds. And several other artists have since received also considerable sums for their improvements of Time-keepers; as Arnold, Mudge, &c. See <hi rend="smallcaps">Longitude.</hi></p><p>This appellation is now become common among artists, to distinguish such watches as are made with extraordinary care and accuracy for nautical or astronomical observations.</p><p>The principles of Mr. Harrison's Time-keeper, as they were communicated by himself, to the commissioners appointed to receive and publish the same in the year 1765, are as below:</p><p>“In this Time-keeper there is the greatest care taken to avoid friction, as much as can be, by the wheel moving on small pivots, and in ruby-holes, and high numbers in the wheels and pinions.</p><p>“The part which measures time goes but the eighth part of a minute without winding up; so that part is very simple, as this winding-up is performed at the wheel next to the balance-wheel; by which means there is always an equal force acting at that wheel, and all the rest of the work has no more to do in the measuring of time than the person that winds up once a day.</p><p>“There is a spring in the inside of the fusee, which I will call a secondary main spring. This spring is always kept stretched to a certain tension by the main spring; and during the time of winding-up the Time-keeper, at which time the main-spring is not suffered to act, this secondary-spring supplies its place.</p><p>“In common watches in general, the wheels have about one-third the dominion over the balance, that the balance-spring has; that is, if the power which the balance-spring has over the balance be called three, <pb n="601"/><cb/> that from the wheel is one: but in this my Time-keeper, the wheels have only about one-eightieth part of the power over the balance that the balance spring has; and it must be allowed, the less the wheels have to do with the balance, the better. The wheels in a common watch having this great dominion over the balance, they can, when the watch is wound up, and the balance at rest, set the watch a-going; but when my Timekeeper's balance is at rest, and the spring is wound up, the force of the wheels can no more set it a-going, than the wheels of a common regulator can, when the weight is wound-up, set the pendulum a-vibrating; nor will the force from the wheels move the balance when at rest, to a greater angle in proportion to the vibration that it is to fetch, than the force of the wheels of a common regulator can move the pendulum from the perpendicular, when it is at rest.</p><p>“My Time-keeper's balance is more than three times the weight of a large sized common watch balance, and three times its diameter; and a common watch balance goes through about six inches of space in a second, but mine goes through about twenty-four inches in that time: so that had my Time-keeper only these advantages over a common watch, a good performance might be expected from it. But my Timekeeper is not affected by the different degrees of heat and cold, nor agitation of the ship; and the force from the wheels is applied to the balance in such a manner, together with the shape of the balance-spring, and (if I may be allowed the term) an artificial cycloid, which acts at this spring; so that from these contrivances, let the balance vibrate more or less, all its vibrations are performed in the same time; and therefore if it go at all, it must go true. So that it is plain from this, that such a Time-keeper goes entirely from principle, and not from chance.”</p><p>We must refer those who may desire to see a minute account of the construction of Mr. Harrison's Timekeeper, to the publication by order of the commissioners of longitude.</p><p>We shall here subjoin a short view of the improvements in Mr. Harrison's watch, from the account presented to the board of longitude by Mr. Ludlam, one of the gentlemen to whom, by order of the commissioners, Mr. Harrison discovered and explained the principle upon which his Time-keeper is constructed. The defects in common watches which Mr. Harrison proposes to remedy, are chiefly these: 1. That the main spring acts not constantly with the same force upon the wheels, and through them upon the balance: 2. That the balance, either urged with an unequal force, or meeting with a different resistance from the air, or the oil, or the friction, vibrates through a greater or less arch: 3. That these unequal vibrations are not performed in equal times: and, 4. That the force of the balance-spring is altered by a change of heat.</p><p>To remedy the first defect, Mr. Harrison has contrived that his watch shall be moved by a very tender spring, which never unrolls itself more than one-eighth part of a turn, and acts upon the balance through one wheel only. But such a spring cannot keep the watch in motion a long time. He has, therefore, joined another, whose office is to wind up the first <cb/> spring eight times in every minute, and which is itself wound up but once a day. To remedy the second defect, he uses a much stronger balance spring than in a common watch. For if the force of this spring upon the balance remains the same, whilst the force of the other varies, the errors arising from that variation will be the less, as the fixed force is the greater. But a stronger spring will require either a heavier or a larger balance. A heavier balance would have a greater friction. Mr. Harrison, therefore, increases the diameter of it. In a common watch it is under an inch, but in Mr. Harrison's two inches and two tenths. However, the methods already described only lessening the errors, and not removing them, Mr. Harrison uses two ways to make the times of the vibrations equal, though the arches may be unequal: one is to place a pin, so that the balance-spring pressing against it, has its force increased, but increased less when the variations are larger: the other to give the pallets such a shape, that the wheels press them with less advantage, when the vibrations are larger. To remedy the last defect, Mr. Harrison uses a bar compounded of two thin plates of brass and steel, about two inches in length, riveted in several places together, fastened at one end and having two pins at the other, between which the balance spring passes. If this bar be straight in temperate weather (brass changing its length by heat more than steel) the brass side becomes convex when it is heated, and the steel side when it is cold: and thus the pins lay hold of a different part of the spring in different degrees of heat, and lengthen or shorten it as the regulator does in a common watch.</p><p>The principles, on which Mr. Arnold's Time-keeper is constructed, are these: The balance is unconnected with the wheel work, except at the time it receives the impulse to make it continue its motion, which is only whilst it vibrates 10° out of 380° which is the whole vibration; and during this small interval it has little or no friction, but what is on the pivots, which work in ruby holes on diamonds. It has but one pallet, which is a plane surface formed out of a ruby, and has no oil on it. Watches of this construction, says Mr. Lyons, go whilst they are wound up; they keep the same rate of going in every position, and are not affected by the different forces of the spring; and the compensation for heat and cold is absolutely adjustable. Phipps's Voyage to the North Pole, p. 230. See L<hi rend="smallcaps">ONGITUDE.</hi></p></div2></div1><div1 part="N" n="TISRI" org="uniform" sample="complete" type="entry"><head>TISRI</head><p>, or <hi rend="smallcaps">Tizri</hi>, in chronology, the first Hebrew month of the civil year, and the 7th of the ecclesiastical or sacred year. It answered to part of our September and October.</p><p>TOD <hi rend="italics">of wool,</hi> is mentioned in the statute 12 Carol. II. c. 32, as a weight containing 2 stone, or 28 pounds.</p></div1><div1 part="N" n="TOISE" org="uniform" sample="complete" type="entry"><head>TOISE</head><p>, a French measure, containing 6 of their feet, similar to our fathom.</p></div1><div1 part="N" n="TONDIN" org="uniform" sample="complete" type="entry"><head>TONDIN</head><p>, or <hi rend="smallcaps">Tandino</hi>, in Architecture. See <hi rend="smallcaps">Tore.</hi></p></div1><div1 part="N" n="TONE" org="uniform" sample="complete" type="entry"><head>TONE</head><p>, or <hi rend="smallcaps">Tune</hi>, in Music, a property of sound, by which it comes under the relation of grave and acute; or the degree of elevation any sound has, from the degree of swiftness of the vibrations of the parts of the sonorous body. <pb n="602"/><cb/></p><p>For the cause, measure, degree, difference, &c, of Tones, see <hi rend="smallcaps">Tune.</hi></p><p>The word Tone is taken in four different senses among the ancients. 1, For any sound, 2, For a certain interval; as when it is said the difference between the diapente and diatessaron is a Tone. 3, For a certain locus or compass of the voice; in which sense they used the Dorian, Phrygian, Lydian Tones. 4. For tension; as when they speak of an acute, a grave, or a middle Tone. Wallis's Append. Ptolom. Harm. p. 172.</p><p><hi rend="smallcaps">Tone</hi> is more particularly used, in music, for a certain degree or interval of tune, by which a sound may be either raised or lowered from one extreme of a concord to the other, so as still to produce true melody.</p><p>In tempered scales of music, the Tones are made equal, but in a true and accurate practice of singing they are not so. Pepusch, in Philos. Trans. No. 481, p. 274.</p><p>Beside the concords, or harmonical intervals, musicians admit three less kinds of intervals, which are the measures and component parts of the greater, and are called <hi rend="italics">degrees.</hi></p><p>Of these degrees, two are called Tones, and the third a semitone. Their ratios in numbers are 8 to 9, called a <hi rend="italics">greater Tone;</hi> 9 to 10, called a <hi rend="italics">lesser Tone;</hi> and 15 to 16, a <hi rend="italics">semitone.</hi></p><p>The Tones arise out of the simple concords, and are equal to their differences. Thus the greater Tone, 8 : 9, is the difference of a 5th and a 4th; the less Tone 9 : 10, the difference of a less 3d and a 4th, or of a 5th and a greater 6th; and the semitone 15 : 16, the difference of a greater 3d and a 4th.</p><p>Of these Tones and semitones every concord is compounded, and consequently every one is resolvable into a certain number of them. Thus the less 3d consists of one greater Tone and one semitone: the greater 3d, of one greater Tone and one less Tone: the 4th, of one greater Tone, one less Tone, and one semitone: and the 5th, of two greater Tones, one less Tone, and one semitone.</p></div1><div1 part="N" n="TONSTALL" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">TONSTALL</surname> (<foreName full="yes"><hi rend="smallcaps">Cuthbert</hi></foreName>)</persName></head><p>, a learned English divine and mathematician, was born in the year 1476. He entered a student at the university of Oxford about the year 1491; but afterwards, being driven from thence by the plague, he went to Cambridge, and shortly after to the university of Padua in Italy, which was then in a flourishing state of literature, where his genius and learning acquired him great respect from every one, particularly for his knowledge in mathematics, philosophy, and jurisprudence.</p><p>Upon his return home, he met with great favours from the government, obtaining several church preferments, and the office of secretary to the cabinet of the king, Henry the 8th. This prince, having also employed him on several foreign embassies, was so well satisfied with his conduct, that he first gave him the bishopric of London in 1522, and afterwards that of Durham in 1530.</p><p>Tonstall approved at first of the dissolution of the marriage of his benefactor with Catherine of Spain, and even wrote a book in favour of that dissolution; but he afterwards condemned that work, and experi- <cb/> enced a great reverse of fortune. He was ejected from the see of Durham for his religion in the time of Edward the 6th, to which however he was restored again by queen Mary in the beginning of her reign, but was again expelled in 1559 when queen Elizabeth was settled in her throne, and he died in a prison a few months after, in the 84th year of his age.</p><p>Tonstall was doubtless one of the most learned men of his time. “He was, says Wood, a very good Grecian and Ebritian, an eloquent rhetorician, a skilful mathematician, a noted civilian and canonist, and a profound divine. But that which maketh for his greatest commendation, is, that Erasmus was his friend, and he a fast friend to Erasmus, in an epistle to whom from Sir Thomas More, I find this character of Tonstall, that, “As there was no man more adorned with knowledge and good literature, no man more severe and of greater integrity for his life and manners; so there was no man a more sweet and pleasant companion, with whom a man would rather choose to converse.”</p><p>His writings that were published, were chiefly the following:</p><p>1. <hi rend="italics">In Laudem Matrimonii,</hi> Lond. 1518, 4to.—But that for which he is chiefly entitled to a place in this work, was his book upon arithmetic, viz,</p><p>2. <hi rend="italics">De Arte Supputandi,</hi> Lond. 1522, 4to, dedicated to Sir Thomas More. This was afterwards several times printed abroad.</p><p>3. A Sermon on Palm Sunday before king Henry the 8th, &c. Lond. 1539 and 1633, 4to.</p><p>4. <hi rend="italics">De Veritate Corporis & Sanguinis Domini in Eucharistia.</hi> Lutet. 1554, 4to.</p><p>5. <hi rend="italics">Compendium in decem Libros Ethicorum Aristotelis.</hi> Par. 1554, in 8vo.</p><p>6. <hi rend="italics">Contra impios. Blasphematores Dei prædestinationis opera.</hi> Antw. 1555, 4to.</p><p>7. Godly and devout Prayers in English and Latin. 1558, in 8vo.</p></div1><div1 part="N" n="TOPOGRAPHY" org="uniform" sample="complete" type="entry"><head>TOPOGRAPHY</head><p>, is a description or draught of some particular place, or small tract of land; as that of a city or town, manor or tenement, field, garden, house, castle, or the like; such as surveyors set out in their plots, or make draughts of, for the information and satisfaction of the proprietors.</p><p>Topography differs from Chorography, as a particular from a more general.</p></div1><div1 part="N" n="TORNADO" org="uniform" sample="complete" type="entry"><head>TORNADO</head><p>, a sudden and violent gust of wind arising suddenly from the shore, and afterwards veering round all points of the compass like a hurricane; very frequent on the coast of Guinea.</p></div1><div1 part="N" n="TORRENT" org="uniform" sample="complete" type="entry"><head>TORRENT</head><p>, in Hydrography, a temporary stream of water, falling suddenly from mountains, &c, where there have been great rains, or an extraordinary thaw of snow; sometimes making great ravages in the plains.</p></div1><div1 part="N" n="TORRICELLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">TORRICELLI</surname> (<foreName full="yes"><hi rend="smallcaps">Evangeliste</hi></foreName>)</persName></head><p>, an illustrious mathematician and philosopher of Italy, was born at Faenza in 1608, and trained up in Greek and Latin literature by an uncle, who was a monk. Natural inclination led him to cultivate mathematical knowledge, which he pursued some time without a master; but at about 20 years of age, he went to Rome, where he <pb n="603"/><cb/> continued the pursuit of it under father Benedict Castelli. Castelli had been a scholar of the great Galileo, and had been appointed by the pope professor of mathematics at Rome. Torricelli made such progress under this master, that having read Galileo's <hi rend="italics">Dialogues,</hi> he composed a <hi rend="italics">Treatise concerning motion</hi> upon his principles. Castelli, surprised at the performance, carried it and read it to Galileo, who heard it with great pleasure, and conceived a high esteem and friendship for the author. Upon this, Castelli proposed to Galileo, that Torricelli should come and live with him; recommending him as the most proper person he could have, since he was the most capable of comprehending those sublime speculations, which his own great age, infirmities, and want of sight, prevented him from giving to the world. Galileo accepted the proposal, and Torricelli the employment, as things of all others the most advantageous to both. Galileo was at Florence, at which place Torricelli arrived in 1641, and began to take down what Galileo dictated, to regulate his papers, and to act in every respect according to his directions. But he did not long enjoy the advantages of this situation, as Galileo died at the end of only three months.</p><p>Torricelli was then about returning to Rome; but the Grand Duke engaged him to continue at Florence, making him his own mathematician for the present, and promising him the professor's chair as soon as it should be vacant.</p><p>Here he applied himself intensely to the study of mathematics, physics, and astronomy, making many improvements and some discoveries. Among others, he greatly improved the art of making microscopes and telescopes; and it is generally acknowledged that he first found out the method of ascertaining the weight of the atmosphere by a proportionate column of quicksilver, the barometer being called from him the <hi rend="italics">Torricellian tube,</hi> and <hi rend="italics">Torricellian experiment.</hi> In short, great things were expected from him, and great things would probably have been farther performed by him, if he had lived: but he died, after a few days illness, in 1647, when he was but just entered the 40th year of his age.</p><p>Torricelli published at Florence in 1644, a volume of ingenious pieces, intitled, <hi rend="italics">Opera Geometrica,</hi> in 4to. There was also published at the same place, in 1715, <hi rend="italics">Lezzioni Accademiche,</hi> consisting of 96 pages in 4to. These are discourses that had been pronounced by him upon different occasions. The first of them was to the academy of La Crusca, by way of thanks for admitting him into their body. The rest are upon subjects of mathematics and physics. Prefixed to the whole is a long life of Torricelli by Thomas Buonaventuri, a Florentine gentleman.</p></div1><div1 part="N" n="TORRICELLIAN" org="uniform" sample="complete" type="entry"><head>TORRICELLIAN</head><p>, a term very frequent among physical writers, used in the phrases, <hi rend="italics">Torricellian tube,</hi> or <hi rend="italics">Torricellian experiment,</hi> on account of the inventor Torricelli, a disciple of the great Galileo.</p><p><hi rend="smallcaps">Torricellian</hi> <hi rend="italics">Tube,</hi> is the barometer tube, being a glass tube, open at one end, and hermetically sealed at the other, about 3 feet long, and 2/10 of an inch in diameter.</p><p><hi rend="smallcaps">Torricellian</hi> <hi rend="italics">Experiment,</hi> or the filling the baro- <cb/> meter tube, is performed by filling the Torricellian tube with mercury, then stopping the open orifice with the finger, inverting the tube, and plunging that orifice into a vessel of stagnant mercury. This done, the finger is removed, and the tube sustained perpendicular to the surface of the mercury in the vessel.</p><p>The consequence is, that part of the mercury falls out of the tube into the vessel, and there remains only enough in the tube to fill about 30 inches of its capacity, above the surface of the stagnant mercury in the vessel; these being sustained in the tube by the pressure of the atmosphere on the surface of the stagnant mercury; and according as the atmosphere is more or less heavy, or as the winds, blowing upward or downward, heave up or depress the air, and so increase or diminish its weight and spring, more or less mercury is sustained, from 28 to 31 inches.</p><p>The Torricellian Experiment constitutes what we now call the <hi rend="italics">Barometer.</hi></p><p><hi rend="smallcaps">Torricellian</hi> <hi rend="italics">Vacuum,</hi> is the vacuum produced by filling a tube with mercury, and when inverted allowing it to descend to such a height as is counterbalanced by the pressure of the atmosphere, as in the Torricellian Experiment and Barometer, the vacuum being that part of the tube above the surface of the mercury.</p><p>TORRID <hi rend="italics">Zone,</hi> is that round the middle of the earth, extending to 23 1/2 degrees on both sides of the equator.</p></div1><div1 part="N" n="TORUS" org="uniform" sample="complete" type="entry"><head>TORUS</head><p>, or <hi rend="smallcaps">Tore</hi>, in Architecture, is a large round moulding in the bases of the columns.</p></div1><div1 part="N" n="TOUCAN" org="uniform" sample="complete" type="entry"><head>TOUCAN</head><p>, or <hi rend="italics">American Goose,</hi> is one of the modern constellations of the southern hemisphere, consisting of 9 small stars.</p></div1><div1 part="N" n="TRACTION" org="uniform" sample="complete" type="entry"><head>TRACTION</head><p>, or <hi rend="italics">Drawing,</hi> is the act of a moving power, by which the moveable is brought nearer to the mover, called also attraction.</p></div1><div1 part="N" n="TRACTRIX" org="uniform" sample="complete" type="entry"><head>TRACTRIX</head><p>, in Geometry, a curve line called also Catenaria; which see.</p></div1><div1 part="N" n="TRAJECTORY" org="uniform" sample="complete" type="entry"><head>TRAJECTORY</head><p>, a term often used generally for the path of any body moving either in a void, or in a medium that resists its motion; or even for any curve passing through a given number of points. Thus Newton, Princip. lib. 1, prob. 22, proposes to describe a Trajectory that shall pass through five given points.</p><p><hi rend="smallcaps">Trajectory</hi> <hi rend="italics">of a Comet,</hi> is its path or orbit, or the line it describes in its motion. This path, Hevelius, in his Cometographia, will have to be very nearly a right line; but Dr. Halley concludes it to be, as it really is, a very excentric ellipsis; though its place may often be well computed on the supposition of its being a parabola.—Newton, in prop. 41 of his 3d book, shews how to determine the Trajectory of a comet from three observations; and in his last prop. how to correct a Trajectory graphically described.</p></div1><div1 part="N" n="TRAMMELS" org="uniform" sample="complete" type="entry"><head>TRAMMELS</head><p>, in Mechanics, an instrument used by artificers for drawing ovals upon boards, &c. One part of it consists of a cross with two grooves at right angles; the other is a beam carrying two pins which slide in those grooves, and also the describing pencil. All the engines for turning ovals are constructed on the same principles with the Trammels: the only difference is, that in the Trammels the board is at rest, and the pen- <pb n="604"/><cb/> cil moves upon it: in the turning engine, the tool, which supplies the place of the pencil, is at rest, and the board moves against it. See a demonstration of the chief properties of these instruments by Mr. Ludlam, in the Philos. Trans. vol. 70, pa. 378 &c.</p></div1><div1 part="N" n="TRANSACTIONS" org="uniform" sample="complete" type="entry"><head>TRANSACTIONS</head><p>, <hi rend="italics">Philosophical,</hi> are a collection of the principal papers and matters read before certain philosophical societies, as the Royal Society of London, and the Royal Society of Edinburgh. These Transactions contain the several discoveries and histories of nature and art, either made by the members of those societies, or communicated by them from their correspondents, with the various experiments, observations, &c, made by them, or transmitted to them, &c.</p><p>The Philos. Trans. of the Royal Society of London were set on foot in 1665, by Mr. Oldenburg, the then secretary of that Society, and were continued by him till the year 1677. They were then discontinued upon his death, till January 1678, when Dr. Grew resumed the publication of them, and continued it for the months of December 1678, and January and February 1679, after which they were intermitted till January 1683. During this last interval their want was in some measure supplied by Dr. Hook's Philosophical Collections. They were also interrupted for 3 years, from December 1687 to January 1691, beside other smaller interruptions amounting to near a year and a half more, before October 1695, since which time the Transactions have been carried on regularly to the present day, with various degrees of credit and merit.</p><p>Till the year 1752 these Transactions were published in numbers quarterly, and the printing of them was always the single act of the respective secretaries till that time; but then the society thought fit that a committee should be appointed to consider the papers read before them, and to select out of them such as they should judge most proper for publication in the future Transactions. For this purpose the members of the couneil for the time being, constitute a standing committee: they meet on the first Thursday of every month, and no less than seven of the members of the committee (of which number the president, or in his absence a vice president, is always to be one) are allowed to be a <hi rend="italics">quorum,</hi> capable of acting in relation to such papers; and the question with regard to the publication of any paper, is always decided by the majority of votes taken by ballot.</p><p>They are published annually in two parts, at the expence of the society; and each fellow, or member, is entitled to receive one copy <hi rend="italics">gratis</hi> of every part published after his admission into the society. For many years past, the collection, in two parts, has made one volume in each year; and in the year 1793 the number of the volumes was 83, being 10 less than the number of the year in the century. They were formerly much respected for the great number of excellent papers and discoveries contained in them; but within the last dozen years there has been a great falling off, and the volumes are now considered as of very inferior merit, as well as quantity.</p><p>There is also a very useful Abridgment, of those <cb/> volumes of the Transactions that were published before the year 1752, when the society began to publish the Transactions on their own account. Those to the end of the year 1700 were abridged, in 3 volumes, by Mr. John Lowthorp: those from the year 1700 to 1720 were abridged, in 2 volumes, by Mr. Henry Jones: and those from 1719 to 1733 were abridged, in 2 volumes, by Mr. John Eames and Mr. John Martyn; Mr. Martyn also continued the abridgment of those from 1732 to 1744 in 2 volumes, and of those from 1744 to 1750 in 2 volumes; making in all 11 volumes, of very curious and useful matters in all the arts and sciences.</p><p>The Royal Society of Edinburgh, instituted in 1783, have also published 3 volumes of their Philosophical Transactions; which are deservedly held in the highest respect for the importance of their contents.</p><p>TRANSCENDENTAL <hi rend="italics">Quantities,</hi> among Geometricians, are indeterminate ones; or such as cannot be expressed or fixed to any constant equation: such is a transcendental curve, or the like.</p><p>M. Leibnitz has a dissertation in the Acta Erud. Lips. in which he endeavours to shew the origin of such quantities; viz, why some problems are neither plain, solid, nor sursolid, nor of any certain degree, but do <hi rend="italics">transcend</hi> all algebraic equations.</p><p>He also shews how it may be demonstrated without calculus, that an algebraic quadratrix for the circle or hyperbola is impossible: for if such a quadratrix could be found, it would follow, that by means of it any angle, ratio, or logarithm, might be divided in a given proportion of one right line to another, and this by one universal construction: and consequently the problem of the section of an angle, or the invention of any number of mean proportionals, would be of a certain finite degree. Whereas the different degrees of algebraic equations, and therefore the problem understood in general of any number of parts of an angle, or mean proportionals, is of an indefinite degree, and <hi rend="italics">transcends</hi> all algebraical equations.</p><p>Others define Transcendental equations, to be such fluxional equations as do not admit of fluents in common finite algebraical equations, but as expressed by means of some curve, or by logarithms, or by infinite series; thus the expression is a Transcendental equation, because the fluents cannot both be expressed in finite terms. And the equation which expresses the relation between an arc of a circle and its sine is a Transcendental equation; for Newton has demonstrated that this relation cannot be expressed by any finite algebraic equation, and therefore it can only be by an infinite or a Transcendental equation.</p><p>It is also usual to rank exponential equations among Transcendental ones; because such equations, although expressed in finite terms, have variable exponents, which cannot be expunged but by putting the equation into fluxions, or logarithms, &c. Thus, the exponential <pb n="605"/><cb/> equation .</p><p><hi rend="smallcaps">Transcendental</hi> <hi rend="italics">Curve,</hi> in the Higher Geometry, is such a one as cannot be defined by an algebraic equation; or of which, when it is expressed by an equation, one of the terms is a variable quantity, or a curve line. And when such curve line is a geometrical one, or one of the first degree or kind, then the Transcendental curve is said to be of the second degree or kind, &c.</p><p>These curves are the same with what Des Cartes, and others after him, call mechanical curves, and which they would have excluded out of geometry; contrary however to the opinion of Newton and Leibnitz; for as much as, in the construction of geometrical problems, one curve is not to be preferred to another as it is defined by a more simple equation, but as it is more easily described than that other: besides, some of these Transcendental, or mechanical curves, are found of greater use than almost all the algebraical ones.</p><p>M. Leibnitz, in the Acta Erudit. Lips. has given a kind of Transcendental equations, by which these Transcendental curves are actually defined, and which are of an indefinite degree, or are not always the same in every point of the curve. Now whereas algebraists use to assume some general letters or numbers for the quantities sought, in these Transcendental problems Leibnitz assumes general or indefinite equations for the lines sought; thus, for example, putting <hi rend="italics">x</hi> and <hi rend="italics">y</hi> for the absciss and ordinate, the equation he uses for a line required, is : by the help of which indefinite equation, he seeks for the tangent; and comparing that which results with the given property of tangents, he finds the value of the assumed letters <hi rend="italics">a, b, c,</hi> &c, and thus defines the equation of the line sought.</p><p>If the comparison abovementioned do not succeed, he pronounces the line sought not to be an algebraical, but a Transcendental one.</p><p>This supposed, he proceeds to find the species of Transcendency: for some Transcendentals depend on the general division or section of a ratio, or upon logarithms, others upon circular arcs, &c.</p><p>Here then, beside the symbols <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> he assumes a third, as <hi rend="italics">v,</hi> to denote the Transcendental quantity; and of these three he forms a general equation of the line sought, from which he finds the tangent according to the differential method, which succeeds even in Transcendental quantities. This found, he compares it with the given properties of the tangents, and so discovers not only the values of <hi rend="italics">a, b, c,</hi> &c, but also the particular nature of the Transcendental quantity.</p><p>Transcendental problems are very well managed by the method of fluxions. Thus, for the relation of a circular arc and right line, let <hi rend="italics">a</hi> denote the arc, and <hi rend="italics">x</hi> the versed sine, to the radius 1, then is ; and if the ordinate of a cycloid be <hi rend="italics">y,</hi> then is . <cb/></p><p>Thus is the analytical calculus extended to those lines which have hitherto been excluded, for no other cause but that they were thought incapable of it.</p></div1><div1 part="N" n="TRANSFORMATION" org="uniform" sample="complete" type="entry"><head>TRANSFORMATION</head><p>, in Geometry, is the changing or reducing of a figure, or of a body, into another of the same area, or the same solidity, but of a different form. As, to Transform or reduce a triangle to a square, or a pyramid to a parallelopipedon.</p><p><hi rend="smallcaps">Transformation</hi> <hi rend="italics">of Equations,</hi> in Algebra, is the changing equations into others of a different form, but of equal value. This operation is often necessary, to prepare equations for a more easy solution, some of the principal cases of which are as follow.—1. The signs of the roots of an equation are changed, viz, the positive roots into negative, and the negative roots into positive ones, by only changing the signs of the 2d, 4th, and all the other even terms of the equation. Thus, the roots of the equation ; whereas the roots of the same equation having only the signs of the 2d and 4th terms changed, viz, of .</p><p>2. To Transform an equation into another that shall have its roots greater or less than the roots of the proposed equation by some given difference, proceed as follows. Let the proposed equation be the cubic ; and let it be required to Transform it into another, whose roots shall be less than the roots of this equation by some given difference <hi rend="italics">d;</hi> if the root <hi rend="italics">y</hi> of the new equation must be the less, take it , and hence ; then instead of <hi rend="italics">x</hi> and its powers substitute <hi rend="italics">y</hi> + <hi rend="italics">d</hi> and its powers, and there will arise this new equation <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">(A) <hi rend="italics">y</hi><hi rend="sup">3</hi> + 3<hi rend="italics">dy</hi><hi rend="sup">2</hi> + 3<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">y</hi> + <hi rend="italics">d</hi><hi rend="sup">3</hi></cell><cell cols="1" rows="1" rend="rowspan=4" role="data"><hi rend="size(6)">}</hi> = 0,</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">- <hi rend="italics">ay</hi><hi rend="sup">2</hi> - 2<hi rend="italics">ady</hi> - <hi rend="italics">ad</hi><hi rend="sup">2</hi></cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">+ <hi rend="italics">by</hi> + <hi rend="italics">bd</hi> </cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">- <hi rend="italics">c</hi> </cell></row></table> whose roots are less than the roots of the former equation by the difference <hi rend="italics">d.</hi> if the roots of the new equation had been required to be greater than those of the old one, we must then have substituted , or , &c.</p><p>3. To take away the 2d or any other particular term out of an equation; or to Transform an equation, so as the new equation may want its 2d, or 3d, or 4th, &c term of the given equation , which is transformed into the equation (A) in the last article Now to make any term of this equation (A) vanish, is only to make the coefficient of that term = 0, which will form an equation that will give the value of the assumed quantity <hi rend="italics">d,</hi> so as to produce the desired effect, viz, to make that term vanish. So, to take away the 2d term, make , which makes the assumed quantity . To take away the 3d term, we must put the sum of the coefficients of that term = 0, that is , or ; then by resolving this quadratic equation, there is found the assumed quantity , by the substitution of which for <hi rend="italics">d,</hi> the 3d term will be taken away out of the equation.</p><p>In like manner, to take away the 4th term, we must make the sum of its coefficients ; <pb n="606"/><cb/> and so on for any other term whatever. And in the same manner we must also proceed when the proposed equation is not a cubic, but of any height whatever, as : this is first, by substituting <hi rend="italics">y</hi> + <hi rend="italics">d</hi> for <hi rend="italics">x,</hi> to be Transformed to this new equation <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">y</hi><hi rend="sup">n</hi> + <hi rend="italics">ndy</hi><hi rend="sup">n-1</hi> + (1/2)<hi rend="italics">n</hi>.(―(<hi rend="italics">n</hi> - 1)).<hi rend="italics">d</hi><hi rend="sup">2</hi><hi rend="italics">y</hi><hi rend="sup">n-2</hi> &c</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(6)">}</hi> = 0;</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">- <hi rend="italics">ay</hi><hi rend="sup">n-1</hi> - <hi rend="italics">a</hi>.(―(<hi rend="italics">n</hi> - 1)).<hi rend="italics">dy</hi><hi rend="sup">n-2</hi> &c</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">+ <hi rend="italics">by</hi><hi rend="sup">n-2</hi> &c</cell></row></table> then, to take away the 2d term, we must make , or ; to take away the 3d term, we must make , or ; and so on.</p><p>From whence it appears that, to take away the 2d term of an equation, we must resolve a simple equation; for the 3d term, a quadratic equation; for the 4th term, a cubic equation, and so on.</p><p>4. To multiply or divide the roots of an equation by any quantity; or to Transform a given equation to another, that shall have its roots equal to any multiple or submultiple of those of the proposed equation. This is done by substituting, for <hi rend="italics">x</hi> and its powers, <hi rend="italics">y</hi>/<hi rend="italics">m</hi> or <hi rend="italics">py,</hi> and their powers, viz, <hi rend="italics">y</hi>/<hi rend="italics">m</hi> for <hi rend="italics">x,</hi> to multiply the roots by <hi rend="italics">m;</hi> and <hi rend="italics">py</hi> for <hi rend="italics">x,</hi> to divide the roots by <hi rend="italics">p.</hi></p><p>Thus, to multiply the roots by <hi rend="italics">m,</hi> substituting <hi rend="italics">y</hi>/<hi rend="italics">m</hi> for <hi rend="italics">x</hi> in the proposed equation ; or multiply all by <hi rend="italics">m</hi><hi rend="sup">n</hi>, then is , an equation that hath its roots equal to <hi rend="italics">m</hi> times the roots of the proposed equation.</p><p>In like manner, substituting <hi rend="italics">py</hi> for <hi rend="italics">x,</hi> in the proposed equation, &c, it becomes , an equation that hath its roots equal to those of the proposed equation divided by <hi rend="italics">p.</hi></p><p>From whence it appears, that to multiply the roots of an equation by any quantity <hi rend="italics">m,</hi> we must multiply its terms, beginning at the 2d term, respectively by the terms of the geometrical series, <hi rend="italics">m, m</hi><hi rend="sup">2</hi>, <hi rend="italics">m</hi><hi rend="sup">3</hi>, <hi rend="italics">m</hi><hi rend="sup">4</hi>, &c. And to divide the roots of an equation by any quantity <hi rend="italics">p,</hi> that we must divide its terms, beginning at the 2d, by the corresponding terms of this series <hi rend="italics">p, p</hi><hi rend="sup">2</hi>, <hi rend="italics">p</hi><hi rend="sup">3</hi>, <hi rend="italics">p</hi><hi rend="sup">4</hi>, &c.</p><p>5. And sometimes, by these Transformations, equations are cleared of fractions, or even of <*>urds. Thus the equation <cb/> , by putting , or multiplying the terms, from the 2d, by the geometricals √<hi rend="italics">p, p, p</hi>√<hi rend="italics">p,</hi> is Transformed to .</p><p>6. An equation, as , may be Transformed into another, whose roots shall be the reciprocals of the roots of the given equation, by substituting 1/<hi rend="italics">y</hi> for <hi rend="italics">x;</hi> by which it becomes , or, multiplying all by <hi rend="italics">y</hi><hi rend="sup">3</hi>, the same becomes .</p><p>On this subject, see Newton's Alg. on the Transmutation of Equations; Maclaurin's Algeb. pt. 2, chap. 3 and 4. Saunderson's Algebra, vol. 2, pa. 687, &c.</p></div1><div1 part="N" n="TRANSIT" org="uniform" sample="complete" type="entry"><head>TRANSIT</head><p>, in Astronomy, denotes the passage of any planet, just before or over another planet or star; or the passing of a star or planet over the meridian, or before an astronomical instrument.</p><p>Venus and Mercury, in their Transits over the sun, appear like dark specks.</p><p>Doctor Halley computed the times of a number of these visible Transits, for the last and present century, and published in the Philos. Trans. numb. 193. See also Abridg. vol. 1, pa. 427 &c. A Synopsis of these Transits is as follows, those of Mercury happening in the months of April and October, and those of Venus in May and November, both old-style; and if 11 days be added to the dates below, the sums will give the times for the new-style. First for Mercury, and then for Venus.</p><p><hi rend="italics">A Series of the Moments when Mercury is seen in Conjunction with the Sun, and within his Disc, with the Distances of the same Planet from the Sun's Centre.</hi> <table rend="border"><head><hi rend="italics">In April, Old-Style.</hi></head><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Years.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Times of Mercury's</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Distances from the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Conjunction.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">the Sun's Centre.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">h.</cell><cell cols="1" rows="1" role="data">min.</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1615</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">21</cell><cell cols="1" rows="1" role="data">38*</cell><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">N</cell></row><row role="data"><cell cols="1" rows="1" role="data">1628</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">15*</cell><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">S</cell></row><row role="data"><cell cols="1" rows="1" role="data">1661</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">52*</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">N</cell></row><row role="data"><cell cols="1" rows="1" role="data">1674</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">S</cell></row><row role="data"><cell cols="1" rows="1" role="data">1707</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">N</cell></row><row role="data"><cell cols="1" rows="1" role="data">1720</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">43*</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">S</cell></row><row role="data"><cell cols="1" rows="1" role="data">1740</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">N</cell></row><row role="data"><cell cols="1" rows="1" role="data">1758</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=right" role="data">19</cell><cell cols="1" rows="1" role="data">20*</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">S</cell></row><row role="data"><cell cols="1" rows="1" role="data">1786</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">57*</cell><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">N</cell></row><row role="data"><cell cols="1" rows="1" role="data">1799</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">34*</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">S</cell></row></table> <pb n="607"/><cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=7" role="data"><hi rend="italics">In October, Old-Style.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Years.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Times of Mercury's</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Distances from the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Conjunction.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Sun's Centre.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">h.</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1605</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" rend="align=left" role="data">29</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1618</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data"> 4*</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1631</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" rend="align=left" role="data">37*</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1644</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=left" role="data">11</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1651</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" rend="align=left" role="data">20</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1664</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" rend="align=left" role="data">54*</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1677</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=left" role="data">28**</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1690</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" rend="align=left" role="data"> 2*</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1697</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" rend="align=left" role="data">11*</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1710</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" rend="align=left" role="data">45</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1723</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=left" role="data">19*</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1730</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" rend="align=left" role="data">28</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1736</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" rend="align=left" role="data">53**</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1743</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" rend="align=left" role="data"> 2**</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1756</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" rend="align=left" role="data">36</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1769</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=left" role="data">10</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1776</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=left" role="data">19</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">S</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center colspan=3" role="data">November</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1782</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">44*</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">N</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center colspan=3" role="data">October</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1789</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">53*</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">S</cell></row></table></p><p>“Those Transits which have the mark *, are but partly visible at London; but those which are marked **, are totally visible.</p><p>“Now it is to be observed, that at the ascending node of Mercury in the month of October, the diameter of the sun takes up 32′ 34″, and therefore the greatest duration of a central Transit is 5<hi rend="sup">h</hi> 29<hi rend="sup">m</hi>. But in the month of April the diameter of the sun is 31′ 54″, whence by reason of the slower motion of the planet, there arises the greatest duration 8<hi rend="sup">h</hi> 1<hi rend="sup">m</hi>. Now if Mercury approaches obliquely, these durations become shorter on account of the distance from the centre of the sun. And that the calculation may be more perfect, I have added the following Tables, in which are exhibited the half durations of these eclipses, to every minute of the distance seen from the centre of the sun. These added to or subtracted from the moment of conjunction found by the foregoing Table, give the beginning and end of the whole phenomenon.” <cb/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data"><hi rend="italics">April.</hi></cell><cell cols="1" rows="1" rend="colspan=3" role="data"><hi rend="italics">October.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Distance in</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Half dura-</cell><cell cols="1" rows="1" role="data">Distance in</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Half dura-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Min.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">tion.</cell><cell cols="1" rows="1" role="data">Min.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">tion.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">h.</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">h.</cell><cell cols="1" rows="1" role="data">m.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=left" role="data"> 0 1/2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">44 1/2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=left" role="data"> 0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">44</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">58 1/2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">43</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">56</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">41 1/2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">53</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">39 1/2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">48 1/<*></cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">36 1/2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">43</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">33</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">36</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">28 1/2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">28</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">23</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">18 1/2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">17</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data"> 7</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">10</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">54</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data"> 1</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">38</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data">51</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" rend="align=left" role="data">19</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data">39</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data">55</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data">24</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data">21 1/2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data"> 4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15 1/2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=left" role="data">56</cell><cell cols="1" rows="1" role="data">15 1/2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=left" role="data">50</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=left" role="data">30</cell></row></table> <hi rend="center"><hi rend="italics">Of the Visible Conjunction of Venus with the Sun.</hi></hi></p><p>“Though Venus is the most beautiful of all the stars, yet (says Dr. Halley) like the rest of her sex, she does not care to appear in sight without her borrowed ornaments, and her assumed splendor. For the confined laws of motion envy this spectacle to the mortals of a whole age, like the secular games of the Ancients; though it be far the most noble among all those that astronomy can pretend to shew. Now it shall be declared hereafter, that by this one observation alone, the distance of the sun from the earth may be determined with the greatest certainty which hitherto has been included within wide limits, because of the parallax which is otherwise insensible. But as to the periods, they cannot be described so accurately as those of Mercury, since Venus has been observed within the sun's disk but once since the beginning of the world, and that by our Horrox.” Dr. Halley then exhibits the principles of calculating these Transits, from whence he infers that,</p><p>“After 18 years Venus returns to the sun, taking away 2<hi rend="sup">d</hi> 10<hi rend="sup">h</hi> (52 1/2)<hi rend="sup">m</hi>, from the moment of the foregoing Transit; and the planet proceeds in a path which is 24′ 41″ more to the south than the former.</p><p>“After 235 years adding 2<hi rend="sup">d</hi> 10<hi rend="sup">h</hi> 9<hi rend="sup">m</hi>, Venus may again enter the sun, but in a more northern path by 11′ 33″. But if the foregoing year is bissextile, 3<hi rend="sup">d</hi> 10<hi rend="sup">h</hi> 9<hi rend="sup">m</hi> must be added.</p><p>“After 243 years Venus may also pass the sun, only taking away 0<hi rend="sup">h</hi> 43<hi rend="sup">m</hi> from the time of the former; <pb n="608"/><cb/> but proceeds more southerly by 13′ 8″. Now if the foregoing year be bissextile, add 23<hi rend="sup">h</hi> 17<hi rend="sup">m</hi>.</p><p>“And in all these appulses of Venus to the sun, in the month of November, the angle of her path with the ecliptic is 9° 5′, and her horary motion within the sun is 4′ 7″. And since the semidiameter of the sun is 16′ 21″, the greatest duration of the Transit of the centre of Venus comes out 7<hi rend="sup">h</hi> 56<hi rend="sup">m</hi>.</p><p>“Then let the sun and Venus be in conjunction at the descending node in the month of May; and by the same numbers the same intervals may be computed. After 8 years let there be taken away 2<hi rend="sup">d</hi> 6<hi rend="sup">h</hi> 55′. And Venus will make her Transit in a more northern path by 19′ 58″.</p><p>“After 235 years add 2<hi rend="sup">d</hi> 8<hi rend="sup">h</hi> 18<hi rend="sup">m</hi>, or if the foregoing year be bissextile 3<hi rend="sup">d</hi> 8<hi rend="sup">h</hi> 18<hi rend="sup">m</hi>, and you will have Venus more to the South by 9′ 21″.</p><p>“Lastly, after 243 years add 0<hi rend="sup">d</hi> 1<hi rend="sup">h</hi> 23<hi rend="sup">m</hi>, or if the foregoing year be bissextile 1<hi rend="sup">d</hi> 1<hi rend="sup">h</hi> 23<hi rend="sup">m</hi>, and Venus will be found in conjunction with the sun, but in a more northerly path by 10′ 37″.</p><p>“In every Transit within the sun at this node, the angle of Venus's path with the ecliptic is 8° 28′, and her horary motion is 4′ 0″; and the semidiameter of the sun subtending 15′ 51″, the greatest duration of the central Transit comes out also 7<hi rend="sup">h</hi> 56<hi rend="sup">m</hi>, exactly the same as at the other node.</p><p>“As to the epochs, from that only ingress which Horrox observed, the sun being then just ready to set, it is concluded, that Venus was in conjunction with the sun at London in the year 1639, Nov. 24<hi rend="sup">d</hi> 6<hi rend="sup">h</hi> 37<hi rend="sup">m</hi>, and that she declined towards the south 8′ 30″. But in the month of May no mortal has seen her as yet within the sun. But from my numbers, which I judge to be not very different from the heavens, it appears that Venus for the next time will enter the sun in 1761, May 25<hi rend="sup">d</hi> 17<hi rend="sup">h</hi> 55<hi rend="sup">m</hi>, that being the middle of the eclipse, and then will be distant from his centre 4′ 15″, towards the south. Hence and from the foregoing revolutions all the phenomena of this kind will be easily exhibited for a whole millennium, as I have computed them in the following Table. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=7" role="data"><hi rend="italics">In November.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Years.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Times of Con-</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Distance from the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">junction.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Sun's Centre.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">h.</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">918</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1161</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1396</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1631</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1639</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1874</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2109</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2117</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">S</cell></row></table> <cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=7" role="data"><hi rend="italics">In May.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Years.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Times of Con-</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Distance from the</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">junction.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Sun's Centre.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">d.</cell><cell cols="1" rows="1" role="data">h.</cell><cell cols="1" rows="1" role="data">m.</cell><cell cols="1" rows="1" role="data">′</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1048</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1283</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1291</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1518</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1526</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1761</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1769</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">N</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1996</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">S</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2004</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">N</cell></row></table></p><p>“As for the durations of these eclipses of Venus, they may be computed after the same manner as those of Mercury in respect of the centre. But since Venus's diameter is pretty large, and since the parallaxes also may bring a very notable difference as to time, a particular calculation must necessarily be made for every place.</p><p>“Now the diameter of Venus is so great, that while she adheres to the sun's limb almost 20 minutes of time will be elapsed, that is, when she applies directly to the sun. But when she is incident obliquely, she continues longer in the limb. Now that diameter, according to Horrox's observation, takes up 1′ 18″, when she is in conjunction with the sun at the ascending node, and 1′ 12″ at the other node.</p><p>“Now the chief use of these conjunctions is accurately to determine the sun's distance from the earth, or his parallax, which astronomers have in vain attempted to find by various other methods; for the minuteness of the angles required easily eludes the nicest instruments. But in observing the ingress of Venus into the sun, and her egress from the same, the space of time between the moments of the internal contacts, observed o a second of time, that is, to 1/15 of a second or 4‴ of an arch, may be obtained by the assistance of a moderate telescope and a pendulum clock, that is consistent with itself exactly for 6 or 8 hours. Now from two such observations rightly made in proper places, the distance of the sun within a 500th part may be certainly concluded, &c.” See <hi rend="smallcaps">Parallax.</hi></p><p><hi rend="smallcaps">Transit</hi> <hi rend="italics">Instrument,</hi> in Astronomy, is a telescope fixed at right angles to a horizontal axis; this axis being so supported that the line of collimation may move in the plane of the meridian.</p><p>The axis, to the middle of which the telescope is fixed, should gradually taper toward its ends, and terminate in cylinders well turned and smoothed; and a proper weight or balance is put on the tube, so that it may stand at any elevation when the axis rests on the supporters. Two upright posts of wood or stone, firmly fixed at a proper distance, are to sustain the supporters to this instrument; these supporters are two thick brass <pb n="609"/><cb/> plates, having well smoothed angular notches in their upper ends to receive the cylindrical arms of the axis; each of the notched plates is contrived to be moveable by a screw, which slides them upon the surfaces of two other plates immoveably fixed to the two upright posts; one plate moving in a vertical direction, and the other horizontally, they adjust the telescope to the planes of the horizon and meridian; to the plane of the horizon, by a spirit level hung in a position parallel to the axis, and to the plane of the meridian in the following manner. Observe the times by the clock when a circumpolar star, seen through this instrument, Transits both above and below the pole; then if the times of describing the eastern and western parts of its circuit be equal, the telescope is then in the plane of the meridian; otherwise the notched plates must be gently moved till the time of the star's revolution is bisected by both the upper and lower Transits, taking care at the same time that the axis keeps its horizontal position.</p><p>When the telescope is thus adjusted, a mark must be set up, or made, at a considerable distance (the greater the better) in the horizontal direction of the intersection of the cross wires, and in a place where it can be illuminated in the night-time by a lanthorn near it, which mark, being on a fixed object, will serve at all times afterwards to examine the position of the telescope, by first adjusting the tranverse axis by the level.</p><p>To adjust a clock by the sun's Transit over the meridian, note the times by the clock, when the preceding and following edges of the sun's limb touch the cross wires: the difference between the middle time and 12 hours, shews how much the mean, or clock time, is faster and slower than the apparent or solar time, for that day; to which the equation of time being applied, it will shew the time of mean noon for that day, by which the clock may be adjusted.</p></div1><div1 part="N" n="TRANSMISSION" org="uniform" sample="complete" type="entry"><head>TRANSMISSION</head><p>, in Optics, &c, denotes the property of a transparent or translucent body, by which it admits the rays of light to pass through its substance; in which sense, the word stands opposed to reflection.</p><p>For the cause of Transmission, or the reason why some bodies Transmit the rays, and others reflect them, see <hi rend="smallcaps">Transparency</hi> and <hi rend="smallcaps">Opacity.</hi></p><p>The rays of light, Newton observes, are subject to fits of easy Transmission and reflection. See <hi rend="smallcaps">Light</hi>, and <hi rend="smallcaps">Reflection.</hi></p></div1><div1 part="N" n="TRANSMUTATION" org="uniform" sample="complete" type="entry"><head>TRANSMUTATION</head><p>, or <hi rend="smallcaps">Transformation</hi>, in Geometry, denotes the reduction or change of one figure or body into another of the same area or solidity; as a triangle into a square, a pyramid into a cube, &c.</p><div2 part="N" n="Transmutation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Transmutation</hi></head><p>, in the Higher Geometry, has been used for the converting of a figure into another of the same kind and order, whose respective parts rise to the same dimensions in an equation, and admit the same tangents, &c.</p><p>If a rectilineal figure be to be Transmuted into another, it is sufficient that the intersections of the lines which compose it be transferred, and lines drawn through the same in the new figure. But if the figure to be Transmuted be curvilinear, the points, tangents, and <cb/> other right lines, by means of which the curve line is to be defined, must be transferred.</p></div2></div1><div1 part="N" n="TRANSOM" org="uniform" sample="complete" type="entry"><head>TRANSOM</head><p>, among Builders, the piece that is framed across a double light window.</p><div2 part="N" n="Transom" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Transom</hi></head><p>, among Mathematicians, denotes the vane of a cross-staff; being a wooden member fixed across it, with a square upon which it slides, &c.</p></div2></div1><div1 part="N" n="TRANSPARENCY" org="uniform" sample="complete" type="entry"><head>TRANSPARENCY</head><p>, or <hi rend="smallcaps">Translucency</hi>, in Physics, a quality in certain bodies, by which they give passage to the rays of light.</p><p>The Transparency of natural bodies, as glass, water, air, &c, is ascribed by some, to the great number and size of the pores or interstices between the particles of those bodies. But this account is very defective; for the most solid and opaque body in nature, that we know of, contains a great deal more pores than it does matter; surely a great deal more than is necessary for the passage of so very fine and subtle a body as light.</p><p>Aristotle, Des Cartes, &c, make. Transparency to consist in straightness or rectilineal direction of the pores; by means of which, say they, the rays can make their way through, without striking against the solid parts, and so being reflected back again. But this account, Newton shews, is imperfect; the quantity of pores in all bodies being sufficient to transmit all the rays that fall upon them, however those pores be situated with respect to each other.</p><p>The reason then why all bodies are not Transparent, is not to be ascribed to their want of rectilineal pores; but either to the unequal density of the parts, or to the pores being filled with some foreign matters, or to their being quite empty, by means of which the rays, in passing through, undergoing a great variety of reflections and refractions, are perpetually diverted different ways, till at length falling on some of the solid parts of the body, they are extinguished and absorbed.</p><p>Thus cork, paper, wood, &c, are opake; while glass, diamonds, &c, are Transparent; and the reason is, that in the neighbourhood of parts equal in density with respect to each other, as these latter bodies, the attraction being equal on every side, no reflection or refraction ensues: but the rays which entered the first surface of the body proceed quite through it without interruption, those few only excepted that chance to meet with the solid parts: but in the neighbourhood of parts that differ much in density, such as the parts of wood and paper are, both in respect of themselves and of the air, or the empty space in their pores; as the attraction is very unequal, the reflections and refractions must be very great; and therefore the rays will not be able to make their way through such bodies, but will be variously deflected, and at length quite stopped. See <hi rend="smallcaps">Opacity.</hi></p></div1><div1 part="N" n="TRANSPOSITION" org="uniform" sample="complete" type="entry"><head>TRANSPOSITION</head><p>, in Algebra, is the bringing any term of an equation over to the other side of it. Thus, if , and you make , then <hi rend="italics">a</hi> is said to be Transposed.</p><p>This operation is to be performed in order to bring all the known terms to one side of the equation, and all those that are unknown to the other side of it; and every term thus Transposed must always have its sign <pb n="610"/><cb/> changed, from + to -, or from - to +; which in fact is no more than subtracting or adding such term on both sides of the equation. See <hi rend="smallcaps">Reduction</hi> of Equations.</p><p>TRANSVERSE-<hi rend="italics">Axis,</hi> or <hi rend="italics">Diameter,</hi> in the Conic Sections, is the first or principal diameter, or axis. See <hi rend="smallcaps">Axis, Diameter</hi>, and <hi rend="smallcaps">Latus</hi> T<hi rend="smallcaps">RANSVERSUM.</hi></p><p>In an ellipse the Transverse is the longest of all the diameters; but the shortest of all in the hyperbola; and in the parabola the diameters are all equal, or at least in a ratio of equality.</p></div1><div1 part="N" n="TRAPEZIUM" org="uniform" sample="complete" type="entry"><head>TRAPEZIUM</head><p>, in Geometry, a plane figure contained under four right lines, of which both the oppofite pairs are not parallel.—When this figure has two of its sides parallel to each other, it is sometimes called a <hi rend="italics">trapezoid.</hi>—The chief properties of the Trapezium are as follow:</p><p>1. Any three sides of a Trapezium taken together, are greater than the third side.</p><p>2. The two diagonals of any Trapezium divide it into four proportional triangles, <hi rend="italics">a, b, c, d.</hi> That is, the triangle <hi rend="italics">a</hi> : <hi rend="italics">b</hi> :: <hi rend="italics">c</hi> : <hi rend="italics">d.</hi></p><p>3. The sum of all the four inward angles, A, B, C, D, taken together, is equal to 4 right angles, or 360°. <figure/></p><p>4. In a Trapezium ABCD, if all the sides be bisected, in the points E, F, G, H, the figure EFGH formed by joining the points of bisection will be a parallelogram, having its opposite sides parallel to the corresponding diagonals of the Trapezium, and the area of the said inscribed parallelogram is just equal to half the area of the Trapezium.</p><p>5. The sum of the squares of the diagonals of the Trapezium, is equal to twice the sum of the squares of the diagonals of the parallelogram, or of the two lines drawn to bisect the opposite sides of the Trapezium. That is, .</p><p>6. In any Trapezium, the sum of the squares of all the four sides, is equal to the sum of the squares of the two diagonals together with 4 times the square of the line KI joining their middle points. That is, (first fig. below) . <figure/></p><p>7. In any Trapezium, the sum of the two diago- <cb/> nals, is less than the sum of any four lines that can be drawn, to the four angles, from any point within the figure, beside the intersection of the diagonals.</p><p>8. The area of any Trapezium, is equal to half the rectangle or product under either diagonal and the sum of the two perpendiculars drawn upon it from the two opposite angles.</p><p>9. The area of any Trapezium may also be found thus: Multiply the two diagonals together, then that product, multiplied by the sine of their angle of intersection, to the radius 1, will be the area. That is, .</p><p>10. The same area will be otherwise found thus: Square each side of a Trapezium, add the squares of each pair of opposite sides together, subtract the less sum from the greater, multiply the remainder by the tangent of the angle of intersection of the diagonals (to radius 1), and 1/4 of the product will be the area. That is, .</p><p>11. The area of a Trapezoid, or one that has two sides parallel, is equal to the rectangle or product under the sum of the two parallel sides and the perpendicular distance between them.</p><p>12. If a Trapezium be inscribed in a circle; the sum of any two opposite angles is equal to two right angles; and if the sum of two opposite angles be equal to two right angles, the sum of the other two will also be equal to two right angles, and a circle may be described about it; and farther, if one side, as DC, be produced out, the external angle will be equal to the interior opposite angle. That is, (last fig. above) .</p><p>13. If a Trapezium be inscribed in a circle; the rectangle of the two diagonals, is equal to the sum of the two rectangles contained under the opposite sides. That is, .</p><p>14. If a Trapezium be inscribed in a circle; its area may be found thus: Multiply any two adjacent sides together, and the other two sides together; then add these two products together, and multiply the sum by the sine of the angle included by either of the pairs of sides that are multiplied together, and half this last product will be the area. That is, the area is equal either .</p><p>15. Or, when the Trapezium can be inscribed in a circle, the area may be otherwise found thus: Add all the four sides together, and take half the sum; then from this half subtract each side severally; multiply the four remainders continually together, and the square root of the last product will be the area.</p><p>16. Lastly, the area of the Trapezium inscribed in a circle may be otherwise found thus: <pb n="611"/><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">Put</cell><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">,</cell></row></table> then (√<hi rend="italics">mnp</hi>)/4<hi rend="sup"><hi rend="italics">r</hi></hi> = the area of the Trapezium.</p></div1><div1 part="N" n="TRAPEZOID" org="uniform" sample="complete" type="entry"><head>TRAPEZOID</head><p>, sometimes denotes a trapezium that has two of its sides parallel to each other; and sometimes an irregular solid figure, having four sides not parallel to each other.</p></div1><div1 part="N" n="TRAVERSE" org="uniform" sample="complete" type="entry"><head>TRAVERSE</head><p>, in Gunnery, is the turning a piece of ordnance about, as upon a centre, to make it point in any particular direction.</p><div2 part="N" n="Traverse" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Traverse</hi></head><p>, in Fortification, denotes a trench with a little parapet, sometimes two, one on each side, to serve as a cover from the enemy that might come in flank.</p></div2><div2 part="N" n="Traverse" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Traverse</hi></head><p>, in a wet foss, is a sort of gallery, made by throwing saucissons, joists, fascines, stones, earth, &c, into the foss, opposite the place where the miner is to be put, in order to fill up the ditch, and make a passage over it.</p><p><hi rend="smallcaps">Traverse</hi> also denotes a wall of earth, or stone, raised across a work, to stop the shot from rolling along it.</p><p><hi rend="smallcaps">Traverse</hi> also sometimes signifies any retrenchment, or line fortified with fascines, barrels or bags of earth, or gabions.</p></div2><div2 part="N" n="Traverse" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Traverse</hi></head><p>, in Navigation, is the variation or alteration of a ship's course, occasioned by the shifting of the winds, or currents, &c; or a Traverse is a compound course, consisting of several different courses and distances.</p><p><hi rend="smallcaps">Traverse</hi> <hi rend="italics">Sailing,</hi> is the method of working, or calculating, Traverses or compound courses, so as to bring them into one, &c.</p><p>Traverse Sailing is used when a ship, having sailed from one port towards another, whose course and distance from the former is known, is by reason of contrary winds, or other accidents, forced to shift and sail upon several courses, which are to be brought into one course, to learn, after so many turnings and windings, the true course and distance made good, or the true point the ship is arrived at; and so to know what must be the new course and distance to the intended port.</p><p><hi rend="italics">To Construct a Traverse.</hi> Assume a convenient point or centre, to begin at, to represent the place sailed from. From that point as a centre, with the chord of 60°, describe a circle, which quarter with two perpendicular lines intersecting in the centre, one to represent the meridian, or north-and-south line, and the other the east-and-west line. From the intersections of these lines with the circle, set off upon the circumference, the arcs or degrees, taken from the chords, for the several courses that have been sailed upon, marking the points they reach to in the circumference with the figures for the order or number of the courses, 1, 2, 3, 4, &c; and from the centre draw lines to these several points in the circumference, or conceive them to be drawn. Upon the first of these lines lay off the first distance sailed; from the extremity of this distance draw a line parallel to the second radius, or line drawn in the circle, upon which lay off the 2d distance; through <cb/> the end of this 2d distance draw a line parallel to the 3d radius, for the direction of the 3d course, and upon it lay off the 3d distance; and so on, through all the courses and distances. This done, draw a line from the centre to the end of the last distance, which will be the whole distance made good, and it will cut the circle in a point shewing the course made good. Lastly, draw a line from the end of the last distance to the point representing the port bound to, and it will shew the distance and course yet to be sailed, to gain that port. <hi rend="center"><hi rend="italics">To work a Traverse, or to compute it by the Traverse Table of Difference of Latitude and Departure.</hi></hi></p><p>Make a little tablet with 6 columns; the 1st for the courses, the 2d for the distances, the 3d for the northing, the 4th for the southing, the 5th for the easting, and the 6th for the westing; first entering the several courses and distances, in so many lines, in the 1st and 2d columns. Then, from the Traverse table, take out the quantity of the northings or southings, and eastings or westings, answering to the several given courses and distances, entering them on their corresponding lines, and in the proper columns of easting, westing, northing, and southing. This done, add up into one sum the numbers in each of these last four columns, which will give four sums shewing the whole quantity of easting, westing, northing, and southing made good; then take the difference between the whole easting and westing, and also between the northing and southing, so shall these shew the spaces made good in these two directions, viz, east or west, and north or south; which being compared with the given difference of latitude and departure, will shew those yet to be made good in sailing to the desired port, and thence the course and distance to it.</p><p><hi rend="italics">Example.</hi> A ship from the latitude 28° 32′ north, bound to a port distant 100 miles, and bearing NE by N, has run the following courses and distances, viz, 1st, NW by N dist. 20 miles; 2d, SW 40 miles; 3d, NE by E 60 miles; 4th, SE 55 miles; 5th, W by S 41 miles; 6th, ENE 66 miles. Required her present latitude, with the direct course and distance made good, and those for the port bound to.</p><p>The numbers being taken out of the Traverse table, and entered opposite the several courses and distances, the tablet will be as here follows: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Courses.</cell><cell cols="1" rows="1" role="data">Dist.</cell><cell cols="1" rows="1" role="data">North.</cell><cell cols="1" rows="1" role="data">South.</cell><cell cols="1" rows="1" role="data">East.</cell><cell cols="1" rows="1" role="data">West.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">NW by N</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">16.6</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">11.1</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">SW</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">28.3</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">28.3</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">NE by E</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">33.3</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">49.9</cell><cell cols="1" rows="1" role="data">.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">SE</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">38.9</cell><cell cols="1" rows="1" role="data">38.9</cell><cell cols="1" rows="1" role="data">.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">W by S</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">8.0</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">40.2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=left" role="data">ENE</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">25.3</cell><cell cols="1" rows="1" role="data">.</cell><cell cols="1" rows="1" role="data">61.0</cell><cell cols="1" rows="1" role="data">.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">75.2</cell><cell cols="1" rows="1" role="data">75.2</cell><cell cols="1" rows="1" role="data">149.8</cell><cell cols="1" rows="1" role="data">79.6</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">75.2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">79.6</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70.2</cell><cell cols="1" rows="1" role="data">Dep.</cell></row></table> <pb n="612"/><cb/> where the sums of the northings and southings, being both alike, 75.2, shews that the ship is come to the same parallel of latitude she set out from. And the difference between the sums of the eastings and westings, shews that the ship is 70.2 miles more to the eastward, that being the greater. Consequently the course made good is due east, and the distance is 70.2 miles.</p><p>But, by the Traverse table, the northing and easting to the proposed course NE by N, and distance 100, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">are thus, viz, northing</cell><cell cols="1" rows="1" role="data">83.1</cell><cell cols="1" rows="1" role="data">and easting</cell><cell cols="1" rows="1" role="data">55.6</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">diff. from made good</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">and easting</cell><cell cols="1" rows="1" role="data">70.2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">give northing</cell><cell cols="1" rows="1" role="data">83.1</cell><cell cols="1" rows="1" role="data">and westing</cell><cell cols="1" rows="1" role="data">14.6</cell></row></table> yet to be made good to arrive at the intended port; and therefore, by finding these in the Traverse table, answering to them are the intended course and distance, viz, distance 85, and course N 10° W.</p><p>The geometrical construction, according to the method before described, gives the figure as below: where A is the port set out from, B is the port bound to, <figure/> C is the place come to, by sailing the several courses and distances A<hi rend="italics">a, ab, bc, cd, de,</hi> and <hi rend="italics">e</hi>C; then CB is the distance to be sailed to arrive at the port B, and its course, or direction with the meridian, is nearly 10°, or the angle ACB, made with the east-and-west line, nearly 80°—<hi rend="italics">Note,</hi> the radii from the centre to the several points in the circumference, are omitted, to prevent a confusion in the figure.</p><p><hi rend="smallcaps">Traverse</hi>-<hi rend="italics">Board,</hi> in a ship, a small round board, hanging up in the steerage, and pierced full of holes in lines shewing the points of the compass: upon which, by moving a small peg from hole to hole, the steersman keeps an account how many glasses, that is half hours, the ship steers upon any point.</p><p><hi rend="smallcaps">Traverse</hi>-<hi rend="italics">Table,</hi> in Navigation, is the same with a table of difference of latitude and departure; being the difference of latitude and departure ready calculated to every point, half point, quarter point, degree, &c, of the quadrant; and for every distance, up to 50 or 100 or 120, &c. Though it may serve for any greater distance whatever, by adding two or more together; or by taking their halves, thirds, fourths, &c, and then doubling, tripling, quadrupling, &c, the difference of latitude and departure found to those parts of the distance. <cb/></p><p>This table is one of the most necessary and useful things a navigator has occasion for; for by it he can readily reduce all his courses and distances, run in the space of 24 hours, into one course and distance; whence he finds the latitude he is in, and the departure from the meridian.</p><p>One of the best tables of this kind is in Robertson's Navigation, at the end of book 7, vol. 1. The distances are there carried to 120, for the sake of more easy subdivisions; and it is divided into two parts; the first containing the courses for every quarter point of the compass, and the 2d adapted to every 15′, or quarter of a degree, in the quadrant. See <hi rend="smallcaps">Traverse</hi> <hi rend="italics">Sailing.</hi></p><p>A specimen of such a Traverse Table is the following, otherwise called a Table of Difference of Latitude and Departure. The distances are placed at top and bottom of the columns, from 1 to 10; but may be extended to any quantity by multiplying the parts, and taking out at several times. The courses, or angles of a rightangled triangle, are in a column, on both sides, each in two parts, the one containing the even points and quarter points, and the other whole degrees, as far as to 45°, or half the quadrant, on the left-hand side, and the other half quadrant, from 45° to 90°, returned upwards from bottom to top on the right-hand side. The corresponding Difference of Latitude and Departure are in two columns below or above the distances, viz, below them when the course or angle is within 45°, or found on the left-hand side; but above them when between 45 and 90°, or found on the right-hand side.</p><p>The same table serves also to work all cases of rightangled triangles, for any other purposes. For example, Suppose a given course be 15°, and distance 35 miles, to find the corresponding difference of latitude and the departure: Or, in a right-angled triangle, given the hypotenuse 35, and one angle 15°, to find the two legs.</p><p>Here, the distance 3 in the table must be accounted 30, moving the decimal point proportionally or one place in the other numbers; and those numbers taken out at twice, viz, once from the columns under 3 for the 30, and the other from the columns under the distance 5. Thus, on the line of 15°, and under the <table><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data"><hi rend="italics">Dist.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">Lat.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">Dep.</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">28.978</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" role="data">7.765</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">4.830</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" role="data">1.294</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">theref. for 35</cell><cell cols="1" rows="1" role="data">are</cell><cell cols="1" rows="1" role="data">33.808</cell><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" role="data">9.059</cell></row></table> So that the other two legs of the triangle are 33.808 and 9.059.—If the course had been 75°, or the complement of the former, which is only the other angle of the same triangle, and which is found on the same line of the table, but on the right-hand side of it: then the numbers in the columns will be the same as before, and will give the same sums for the two legs of the triangle, only with the contrary names, as to Latitude and Departure, which change places. <pb n="613"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=14" role="data"><hi rend="italics">A</hi> <hi rend="smallcaps">Table</hi> <hi rend="italics">of the Difference of Latitude and Departure, for Degrees and Quarter Points.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Course</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 1</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 2</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 3.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 4</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 5</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Course</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Pts.</cell><cell cols="1" rows="1" role="data">D.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">D.</cell><cell cols="1" rows="1" role="data">Pts.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">0.9998</cell><cell cols="1" rows="1" role="data">0.0175</cell><cell cols="1" rows="1" role="data">1.9997</cell><cell cols="1" rows="1" role="data">0.0349</cell><cell cols="1" rows="1" role="data">2.9995</cell><cell cols="1" rows="1" role="data">0.0524</cell><cell cols="1" rows="1" role="data">3.9994</cell><cell cols="1" rows="1" role="data">0.0698</cell><cell cols="1" rows="1" role="data">4.9992</cell><cell cols="1" rows="1" role="data">0.0873</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">0.9994</cell><cell cols="1" rows="1" role="data">0.0349</cell><cell cols="1" rows="1" role="data">1.9988</cell><cell cols="1" rows="1" role="data">0.0698</cell><cell cols="1" rows="1" role="data">2.9982</cell><cell cols="1" rows="1" role="data">0.1047</cell><cell cols="1" rows="1" role="data">3.9976</cell><cell cols="1" rows="1" role="data">0.1396</cell><cell cols="1" rows="1" role="data">4.9970</cell><cell cols="1" rows="1" role="data">0.1745</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">0 1/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.9988</cell><cell cols="1" rows="1" role="data">0.0491</cell><cell cols="1" rows="1" role="data">1.9976</cell><cell cols="1" rows="1" role="data">0.0981</cell><cell cols="1" rows="1" role="data">2.9964</cell><cell cols="1" rows="1" role="data">0.1472</cell><cell cols="1" rows="1" role="data">3.9952</cell><cell cols="1" rows="1" role="data">0.1963</cell><cell cols="1" rows="1" role="data">4.9940</cell><cell cols="1" rows="1" role="data">0.2453</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7 3/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">0.9986</cell><cell cols="1" rows="1" role="data">0.0523</cell><cell cols="1" rows="1" role="data">1.9973</cell><cell cols="1" rows="1" role="data">0.1047</cell><cell cols="1" rows="1" role="data">2.9959</cell><cell cols="1" rows="1" role="data">0.1570</cell><cell cols="1" rows="1" role="data">3.9945</cell><cell cols="1" rows="1" role="data">0.2093</cell><cell cols="1" rows="1" role="data">4.9931</cell><cell cols="1" rows="1" role="data">0.2617</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">0.9976</cell><cell cols="1" rows="1" role="data">0.0698</cell><cell cols="1" rows="1" role="data">1.9951</cell><cell cols="1" rows="1" role="data">0.1395</cell><cell cols="1" rows="1" role="data">2.9927</cell><cell cols="1" rows="1" role="data">0.2093</cell><cell cols="1" rows="1" role="data">3.9903</cell><cell cols="1" rows="1" role="data">0.2790</cell><cell cols="1" rows="1" role="data">4.9878</cell><cell cols="1" rows="1" role="data">0.3488</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">0.9962</cell><cell cols="1" rows="1" role="data">0.0872</cell><cell cols="1" rows="1" role="data">1.9924</cell><cell cols="1" rows="1" role="data">0.1743</cell><cell cols="1" rows="1" role="data">2.9886</cell><cell cols="1" rows="1" role="data">0.2615</cell><cell cols="1" rows="1" role="data">3.9848</cell><cell cols="1" rows="1" role="data">0.3486</cell><cell cols="1" rows="1" role="data">4.9810</cell><cell cols="1" rows="1" role="data">0.4358</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">0 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.9952</cell><cell cols="1" rows="1" role="data">0.0980</cell><cell cols="1" rows="1" role="data">1.9904</cell><cell cols="1" rows="1" role="data">0.1960</cell><cell cols="1" rows="1" role="data">2.9856</cell><cell cols="1" rows="1" role="data">0.2940</cell><cell cols="1" rows="1" role="data">3.9807</cell><cell cols="1" rows="1" role="data">0.3921</cell><cell cols="1" rows="1" role="data">4.9759</cell><cell cols="1" rows="1" role="data">0.4901</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">0.9945</cell><cell cols="1" rows="1" role="data">0.1045</cell><cell cols="1" rows="1" role="data">1.9890</cell><cell cols="1" rows="1" role="data">0.2091</cell><cell cols="1" rows="1" role="data">2.9836</cell><cell cols="1" rows="1" role="data">0.3136</cell><cell cols="1" rows="1" role="data">3.9781</cell><cell cols="1" rows="1" role="data">0.4181</cell><cell cols="1" rows="1" role="data">4.9726</cell><cell cols="1" rows="1" role="data">0.5226</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">0.9925</cell><cell cols="1" rows="1" role="data">0.1219</cell><cell cols="1" rows="1" role="data">1.9851</cell><cell cols="1" rows="1" role="data">0.2437</cell><cell cols="1" rows="1" role="data">2.9776</cell><cell cols="1" rows="1" role="data">0.3656</cell><cell cols="1" rows="1" role="data">3.9702</cell><cell cols="1" rows="1" role="data">0.4875</cell><cell cols="1" rows="1" role="data">4.9627</cell><cell cols="1" rows="1" role="data">0.6093</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">8</cell><cell cols="1" rows="1" role="data">0.9903</cell><cell cols="1" rows="1" role="data">0.1392</cell><cell cols="1" rows="1" role="data">1.9805</cell><cell cols="1" rows="1" role="data">0.2783</cell><cell cols="1" rows="1" role="data">2.9708</cell><cell cols="1" rows="1" role="data">0.4175</cell><cell cols="1" rows="1" role="data">3.9611</cell><cell cols="1" rows="1" role="data">0.5567</cell><cell cols="1" rows="1" role="data">4.9513</cell><cell cols="1" rows="1" role="data">0.6959</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">0 3/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.9892</cell><cell cols="1" rows="1" role="data">0.1467</cell><cell cols="1" rows="1" role="data">1.9784</cell><cell cols="1" rows="1" role="data">0.2935</cell><cell cols="1" rows="1" role="data">2.9675</cell><cell cols="1" rows="1" role="data">0.4402</cell><cell cols="1" rows="1" role="data">3.9567</cell><cell cols="1" rows="1" role="data">0.5869</cell><cell cols="1" rows="1" role="data">4.9459</cell><cell cols="1" rows="1" role="data">0.7337</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9</cell><cell cols="1" rows="1" role="data">0.9877</cell><cell cols="1" rows="1" role="data">0.1564</cell><cell cols="1" rows="1" role="data">1.9754</cell><cell cols="1" rows="1" role="data">0.3129</cell><cell cols="1" rows="1" role="data">2.9631</cell><cell cols="1" rows="1" role="data">0.4693</cell><cell cols="1" rows="1" role="data">3.9508</cell><cell cols="1" rows="1" role="data">0.6257</cell><cell cols="1" rows="1" role="data">4.9384</cell><cell cols="1" rows="1" role="data">0.7822</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" role="data">0.9848</cell><cell cols="1" rows="1" role="data">0.1736</cell><cell cols="1" rows="1" role="data">1.9696</cell><cell cols="1" rows="1" role="data">0.3473</cell><cell cols="1" rows="1" role="data">2.9544</cell><cell cols="1" rows="1" role="data">0.5209</cell><cell cols="1" rows="1" role="data">3.9392</cell><cell cols="1" rows="1" role="data">0.6946</cell><cell cols="1" rows="1" role="data">4.9240</cell><cell cols="1" rows="1" role="data">0.8682</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">11</cell><cell cols="1" rows="1" role="data">0.9816</cell><cell cols="1" rows="1" role="data">0.1908</cell><cell cols="1" rows="1" role="data">1.9633</cell><cell cols="1" rows="1" role="data">0.3816</cell><cell cols="1" rows="1" role="data">2.9449</cell><cell cols="1" rows="1" role="data">0.5724</cell><cell cols="1" rows="1" role="data">3.9265</cell><cell cols="1" rows="1" role="data">0.7632</cell><cell cols="1" rows="1" role="data">4.9081</cell><cell cols="1" rows="1" role="data">0.9540</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.9808</cell><cell cols="1" rows="1" role="data">0.1951</cell><cell cols="1" rows="1" role="data">1.9616</cell><cell cols="1" rows="1" role="data">0.3902</cell><cell cols="1" rows="1" role="data">2.9424</cell><cell cols="1" rows="1" role="data">0.5853</cell><cell cols="1" rows="1" role="data">3.9231</cell><cell cols="1" rows="1" role="data">0.7804</cell><cell cols="1" rows="1" role="data">4.9039</cell><cell cols="1" rows="1" role="data">0.9754</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">12</cell><cell cols="1" rows="1" role="data">0.9781</cell><cell cols="1" rows="1" role="data">0.2079</cell><cell cols="1" rows="1" role="data">1.9563</cell><cell cols="1" rows="1" role="data">0.4158</cell><cell cols="1" rows="1" role="data">2.9344</cell><cell cols="1" rows="1" role="data">0.6237</cell><cell cols="1" rows="1" role="data">3.9126</cell><cell cols="1" rows="1" role="data">0.8316</cell><cell cols="1" rows="1" role="data">4.8907</cell><cell cols="1" rows="1" role="data">1.0396</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">13</cell><cell cols="1" rows="1" role="data">0.9744</cell><cell cols="1" rows="1" role="data">0.2250</cell><cell cols="1" rows="1" role="data">1.9487</cell><cell cols="1" rows="1" role="data">0.4499</cell><cell cols="1" rows="1" role="data">2.9231</cell><cell cols="1" rows="1" role="data">0.6749</cell><cell cols="1" rows="1" role="data">3.8975</cell><cell cols="1" rows="1" role="data">0.8998</cell><cell cols="1" rows="1" role="data">4.8718</cell><cell cols="1" rows="1" role="data">1.1248</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">14</cell><cell cols="1" rows="1" role="data">0.9703</cell><cell cols="1" rows="1" role="data">0.2419</cell><cell cols="1" rows="1" role="data">1.9406</cell><cell cols="1" rows="1" role="data">0.4838</cell><cell cols="1" rows="1" role="data">2.9108</cell><cell cols="1" rows="1" role="data">0.7258</cell><cell cols="1" rows="1" role="data">3.8812</cell><cell cols="1" rows="1" role="data">0.9677</cell><cell cols="1" rows="1" role="data">4.8515</cell><cell cols="1" rows="1" role="data">1.2096</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1 1/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.9700</cell><cell cols="1" rows="1" role="data">0.2430</cell><cell cols="1" rows="1" role="data">1.9401</cell><cell cols="1" rows="1" role="data">0.4860</cell><cell cols="1" rows="1" role="data">2.9101</cell><cell cols="1" rows="1" role="data">0.7289</cell><cell cols="1" rows="1" role="data">3.8801</cell><cell cols="1" rows="1" role="data">0.9719</cell><cell cols="1" rows="1" role="data">4.8502</cell><cell cols="1" rows="1" role="data">1.2149</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6 3/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">0.9659</cell><cell cols="1" rows="1" role="data">0.2588</cell><cell cols="1" rows="1" role="data">1.9319</cell><cell cols="1" rows="1" role="data">0.5176</cell><cell cols="1" rows="1" role="data">2.8978</cell><cell cols="1" rows="1" role="data">0.7765</cell><cell cols="1" rows="1" role="data">3.8637</cell><cell cols="1" rows="1" role="data">1.0353</cell><cell cols="1" rows="1" role="data">4.8296</cell><cell cols="1" rows="1" role="data">1.2941</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">16</cell><cell cols="1" rows="1" role="data">0.9613</cell><cell cols="1" rows="1" role="data">0.2756</cell><cell cols="1" rows="1" role="data">1.9225</cell><cell cols="1" rows="1" role="data">0.5513</cell><cell cols="1" rows="1" role="data">2.8838</cell><cell cols="1" rows="1" role="data">0.8269</cell><cell cols="1" rows="1" role="data">3.8450</cell><cell cols="1" rows="1" role="data">1.1025</cell><cell cols="1" rows="1" role="data">4.8063</cell><cell cols="1" rows="1" role="data">1.3782</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.9569</cell><cell cols="1" rows="1" role="data">0.2903</cell><cell cols="1" rows="1" role="data">1.9139</cell><cell cols="1" rows="1" role="data">0.5806</cell><cell cols="1" rows="1" role="data">2.8708</cell><cell cols="1" rows="1" role="data">0.8209</cell><cell cols="1" rows="1" role="data">3.8278</cell><cell cols="1" rows="1" role="data">1.1611</cell><cell cols="1" rows="1" role="data">4.7847</cell><cell cols="1" rows="1" role="data">1.4514</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">17</cell><cell cols="1" rows="1" role="data">0.9563</cell><cell cols="1" rows="1" role="data">0.2924</cell><cell cols="1" rows="1" role="data">1.9126</cell><cell cols="1" rows="1" role="data">0.5847</cell><cell cols="1" rows="1" role="data">2.8689</cell><cell cols="1" rows="1" 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role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.8315</cell><cell cols="1" rows="1" role="data">0.5556</cell><cell cols="1" rows="1" role="data">1.6629</cell><cell cols="1" rows="1" role="data">1.1111</cell><cell cols="1" rows="1" role="data">2.4944</cell><cell cols="1" rows="1" role="data">1.6667</cell><cell cols="1" rows="1" role="data">3.3259</cell><cell cols="1" rows="1" role="data">2.2223</cell><cell cols="1" rows="1" role="data">4.1573</cell><cell cols="1" rows="1" role="data">2.7778</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">34</cell><cell cols="1" rows="1" role="data">0.8290</cell><cell cols="1" rows="1" role="data">0.5592</cell><cell cols="1" rows="1" role="data">1.6581</cell><cell cols="1" rows="1" role="data">1.1184</cell><cell cols="1" rows="1" role="data">2.4871</cell><cell cols="1" rows="1" role="data">1.6776</cell><cell cols="1" rows="1" role="data">3.3162</cell><cell cols="1" rows="1" role="data">2.2368</cell><cell cols="1" rows="1" role="data">4.1452</cell><cell cols="1" rows="1" role="data">2.7960</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">0.8192</cell><cell cols="1" rows="1" role="data">0.5736</cell><cell cols="1" rows="1" role="data">1.6383</cell><cell cols="1" rows="1" role="data">1.1472</cell><cell cols="1" rows="1" role="data">2.4575</cell><cell cols="1" rows="1" role="data">1.7207</cell><cell cols="1" rows="1" role="data">3.2766</cell><cell cols="1" rows="1" role="data">2.2943</cell><cell cols="1" rows="1" role="data">4.0958</cell><cell cols="1" rows="1" role="data">2.8679</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">36</cell><cell cols="1" rows="1" role="data">0.8090</cell><cell cols="1" rows="1" role="data">0.5878</cell><cell cols="1" rows="1" role="data">1.6180</cell><cell cols="1" rows="1" role="data">1.1756</cell><cell cols="1" rows="1" role="data">2.4271</cell><cell cols="1" rows="1" role="data">1.7634</cell><cell cols="1" rows="1" role="data">3.2361</cell><cell cols="1" rows="1" role="data">2.3511</cell><cell cols="1" rows="1" role="data">4.0451</cell><cell cols="1" rows="1" role="data">2.9389</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">3 1/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.8032</cell><cell cols="1" rows="1" role="data">0.5957</cell><cell cols="1" rows="1" role="data">1.6064</cell><cell cols="1" rows="1" role="data">1.1914</cell><cell cols="1" rows="1" role="data">2.4096</cell><cell cols="1" rows="1" role="data">1.7871</cell><cell cols="1" rows="1" role="data">3.2128</cell><cell cols="1" rows="1" role="data">2.3828</cell><cell cols="1" rows="1" role="data">4.0160</cell><cell cols="1" rows="1" role="data">2.9785</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4 3/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">37</cell><cell cols="1" rows="1" role="data">0.7986</cell><cell cols="1" rows="1" role="data">0.6018</cell><cell cols="1" rows="1" role="data">1.5973</cell><cell cols="1" rows="1" role="data">1.2036</cell><cell cols="1" rows="1" role="data">2.3959</cell><cell cols="1" rows="1" role="data">1.8054</cell><cell cols="1" rows="1" role="data">3.1945</cell><cell cols="1" rows="1" role="data">2.4073</cell><cell cols="1" rows="1" role="data">3.9932</cell><cell cols="1" rows="1" role="data">3.0091</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">38</cell><cell cols="1" rows="1" role="data">0.7880</cell><cell cols="1" rows="1" role="data">0.6157</cell><cell cols="1" rows="1" role="data">1.5760</cell><cell cols="1" rows="1" role="data">1.2313</cell><cell cols="1" rows="1" role="data">2.3640</cell><cell cols="1" rows="1" role="data">1.8470</cell><cell cols="1" rows="1" role="data">3.1520</cell><cell cols="1" rows="1" role="data">2.4626</cell><cell cols="1" rows="1" role="data">3.9401</cell><cell cols="1" rows="1" role="data">3.0783</cell><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">39</cell><cell cols="1" rows="1" role="data">0.7771</cell><cell cols="1" rows="1" role="data">0.6293</cell><cell cols="1" rows="1" role="data">1.5543</cell><cell cols="1" rows="1" role="data">1.2586</cell><cell cols="1" rows="1" role="data">2.3314</cell><cell cols="1" rows="1" role="data">1.8880</cell><cell cols="1" rows="1" role="data">3.1086</cell><cell cols="1" rows="1" role="data">2.5173</cell><cell cols="1" rows="1" role="data">3.8857</cell><cell cols="1" rows="1" role="data">3.1466</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">3 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.7730</cell><cell cols="1" rows="1" role="data">0.6344</cell><cell cols="1" rows="1" role="data">1.5460</cell><cell cols="1" rows="1" role="data">1.2688</cell><cell cols="1" rows="1" role="data">2.3190</cell><cell cols="1" rows="1" role="data">1.9032</cell><cell cols="1" rows="1" role="data">3.0920</cell><cell cols="1" rows="1" role="data">2.5376</cell><cell cols="1" rows="1" role="data">3.8650</cell><cell cols="1" rows="1" role="data">3.1720</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">40</cell><cell cols="1" rows="1" role="data">0.7660</cell><cell cols="1" rows="1" role="data">0.6428</cell><cell cols="1" rows="1" role="data">1.5321</cell><cell cols="1" rows="1" role="data">1.2856</cell><cell cols="1" rows="1" role="data">2.2981</cell><cell cols="1" rows="1" role="data">1.9284</cell><cell cols="1" rows="1" role="data">3.0642</cell><cell cols="1" rows="1" role="data">2.5712</cell><cell cols="1" rows="1" role="data">3.8302</cell><cell cols="1" rows="1" role="data">3.2139</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">41</cell><cell cols="1" rows="1" role="data">0.7547</cell><cell cols="1" rows="1" role="data">0.6561</cell><cell cols="1" rows="1" role="data">1.5094</cell><cell cols="1" rows="1" role="data">1.3121</cell><cell cols="1" rows="1" role="data">2.2641</cell><cell cols="1" rows="1" role="data">1.9682</cell><cell cols="1" rows="1" role="data">3.0188</cell><cell cols="1" rows="1" role="data">2.6242</cell><cell cols="1" rows="1" role="data">3.7736</cell><cell cols="1" rows="1" role="data">3.2803</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">0.7431</cell><cell cols="1" rows="1" role="data">0.6691</cell><cell cols="1" rows="1" role="data">1.4803</cell><cell cols="1" rows="1" role="data">1.3383</cell><cell cols="1" rows="1" role="data">2.2294</cell><cell cols="1" rows="1" role="data">2.0074</cell><cell cols="1" rows="1" role="data">2.9726</cell><cell cols="1" rows="1" role="data">2.6765</cell><cell cols="1" rows="1" role="data">3.7157</cell><cell cols="1" rows="1" role="data">3.3457</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">3 3/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0.7410</cell><cell cols="1" rows="1" role="data">0.6716</cell><cell cols="1" rows="1" role="data">1.4819</cell><cell cols="1" rows="1" role="data">1.3431</cell><cell cols="1" rows="1" role="data">2.2229</cell><cell cols="1" rows="1" role="data">2.0147</cell><cell cols="1" rows="1" role="data">2.9638</cell><cell cols="1" rows="1" role="data">2.6862</cell><cell cols="1" rows="1" role="data">3.7048</cell><cell cols="1" rows="1" role="data">3.3578</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">0.7314</cell><cell cols="1" rows="1" role="data">0.6820</cell><cell cols="1" rows="1" role="data">1.4628</cell><cell cols="1" rows="1" role="data">1.3640</cell><cell cols="1" rows="1" role="data">2.1941</cell><cell cols="1" rows="1" role="data">2.0460</cell><cell cols="1" rows="1" role="data">2.9254</cell><cell cols="1" rows="1" role="data">2.7280</cell><cell cols="1" rows="1" role="data">3.6568</cell><cell cols="1" rows="1" role="data">3.4100</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">0.7193</cell><cell cols="1" rows="1" role="data">0.6947</cell><cell cols="1" rows="1" role="data">1.4387</cell><cell cols="1" rows="1" role="data">1.3894</cell><cell cols="1" rows="1" role="data">2.1580</cell><cell cols="1" rows="1" role="data">2.0840</cell><cell cols="1" rows="1" role="data">2.8774</cell><cell cols="1" rows="1" role="data">2.7786</cell><cell cols="1" rows="1" role="data">3.5967</cell><cell cols="1" rows="1" role="data">3.4733</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">0.7071</cell><cell cols="1" rows="1" role="data">0.7071</cell><cell cols="1" rows="1" role="data">1.4142</cell><cell cols="1" rows="1" role="data">1.4142</cell><cell cols="1" rows="1" role="data">2.1213</cell><cell cols="1" rows="1" role="data">2.1213</cell><cell cols="1" rows="1" role="data">2.8284</cell><cell cols="1" rows="1" role="data">2.8284</cell><cell cols="1" rows="1" role="data">3.5355</cell><cell cols="1" rows="1" role="data">3.5355</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Pts.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Deg.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Deg.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Pts.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 1</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 2</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 3</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 4</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 5</cell></row></table><pb n="614"/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=14" role="data"><hi rend="smallcaps">Table</hi> <hi rend="italics">of the Difference of Latitude and Departure, for Degrees and Quarter Points.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Course</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 6</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 7</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 8.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 9</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 10</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Course</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Pts.</cell><cell cols="1" rows="1" role="data">D.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">D.</cell><cell cols="1" rows="1" role="data">Pts.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">5.9991</cell><cell cols="1" rows="1" role="data">0.1047</cell><cell cols="1" rows="1" role="data">6.9989</cell><cell cols="1" rows="1" role="data">0.1222</cell><cell cols="1" rows="1" role="data">7.9988</cell><cell cols="1" rows="1" role="data">0.1396</cell><cell cols="1" rows="1" role="data">8.9986</cell><cell cols="1" rows="1" role="data">0.1571</cell><cell cols="1" rows="1" role="data">9.9985</cell><cell cols="1" rows="1" role="data">0.1745</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data">5.9963</cell><cell cols="1" rows="1" role="data">0.2094</cell><cell cols="1" rows="1" role="data">6.9957</cell><cell cols="1" rows="1" role="data">0.2443</cell><cell cols="1" rows="1" role="data">7.9951</cell><cell cols="1" rows="1" role="data">0.2792</cell><cell cols="1" rows="1" role="data">8.9945</cell><cell cols="1" rows="1" role="data">0.3141</cell><cell cols="1" rows="1" role="data">9.9939</cell><cell cols="1" rows="1" role="data">0.3490</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">0 3/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.9928</cell><cell cols="1" rows="1" role="data">0.2944</cell><cell cols="1" rows="1" role="data">6.9916</cell><cell cols="1" rows="1" role="data">0.3435</cell><cell cols="1" rows="1" role="data">7.9904</cell><cell cols="1" rows="1" role="data">0.3925</cell><cell cols="1" rows="1" role="data">8.9892</cell><cell cols="1" rows="1" role="data">0.4416</cell><cell cols="1" rows="1" role="data">9.9880</cell><cell cols="1" rows="1" role="data">0.4907</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7 3/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data">5.9918</cell><cell cols="1" rows="1" role="data">0.3140</cell><cell cols="1" rows="1" role="data">6.9904</cell><cell cols="1" rows="1" role="data">0.3664</cell><cell cols="1" rows="1" role="data">7.9890</cell><cell cols="1" rows="1" role="data">0.4187</cell><cell cols="1" rows="1" role="data">8.9877</cell><cell cols="1" rows="1" role="data">0.4710</cell><cell cols="1" rows="1" role="data">9.9863</cell><cell cols="1" rows="1" role="data">0.5234</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">5.9854</cell><cell cols="1" rows="1" role="data">0.4185</cell><cell cols="1" rows="1" role="data">6.9829</cell><cell cols="1" rows="1" role="data">0.4883</cell><cell cols="1" rows="1" role="data">7.9805</cell><cell cols="1" rows="1" role="data">0.5580</cell><cell cols="1" rows="1" role="data">8.9781</cell><cell cols="1" rows="1" role="data">0.6278</cell><cell cols="1" rows="1" role="data">9.9756</cell><cell cols="1" rows="1" role="data">0.6976</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" role="data">5.9772</cell><cell cols="1" rows="1" role="data">0.5229</cell><cell cols="1" rows="1" role="data">6.9734</cell><cell cols="1" rows="1" role="data">0.6101</cell><cell cols="1" rows="1" role="data">7.9696</cell><cell cols="1" rows="1" role="data">0.6972</cell><cell cols="1" rows="1" role="data">8.9658</cell><cell cols="1" rows="1" role="data">0.7844</cell><cell cols="1" rows="1" role="data">9.9619</cell><cell cols="1" rows="1" role="data">0.8716</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">0 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.9711</cell><cell cols="1" rows="1" role="data">0.5881</cell><cell cols="1" rows="1" role="data">6.9663</cell><cell cols="1" rows="1" role="data">0.6861</cell><cell cols="1" rows="1" role="data">7.9615</cell><cell cols="1" rows="1" role="data">0.7841</cell><cell cols="1" rows="1" role="data">8.9567</cell><cell cols="1" rows="1" role="data">0.8822</cell><cell cols="1" rows="1" role="data">9.9518</cell><cell cols="1" rows="1" role="data">0.9802</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">7 1/2</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data">5.9671</cell><cell cols="1" rows="1" role="data">0.6272</cell><cell cols="1" rows="1" role="data">6.9617</cell><cell cols="1" rows="1" role="data">0.7317</cell><cell cols="1" rows="1" role="data">7.9562</cell><cell cols="1" rows="1" role="data">0.8362</cell><cell cols="1" rows="1" role="data">8.9507</cell><cell cols="1" rows="1" role="data">0.9408</cell><cell cols="1" rows="1" role="data">9.9452</cell><cell cols="1" rows="1" role="data">1.0453</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">7</cell><cell cols="1" rows="1" role="data">5.9553</cell><cell cols="1" rows="1" role="data">0.7312</cell><cell cols="1" rows="1" role="data">6.9478</cell><cell cols="1" rows="1" role="data">0.8531</cell><cell cols="1" rows="1" role="data">7.9404</cell><cell cols="1" rows="1" role="data">0.9750</cell><cell cols="1" rows="1" role="data">8.9329</cell><cell cols="1" rows="1" role="data">1.0968</cell><cell cols="1" rows="1" role="data">9.9255</cell><cell cols="1" rows="1" 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rows="1" role="data">1.6538</cell><cell cols="1" rows="1" role="data">6.7288</cell><cell cols="1" rows="1" role="data">1.9295</cell><cell cols="1" rows="1" role="data">7.6901</cell><cell cols="1" rows="1" role="data">2.2051</cell><cell cols="1" rows="1" role="data">8.6513</cell><cell cols="1" rows="1" role="data">2.4807</cell><cell cols="1" rows="1" role="data">9.6120</cell><cell cols="1" rows="1" role="data">2.7562</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">1 1/2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.7416</cell><cell cols="1" rows="1" role="data">1.7417</cell><cell cols="1" rows="1" role="data">6.6986</cell><cell cols="1" rows="1" role="data">2.0320</cell><cell cols="1" rows="1" role="data">7.6555</cell><cell cols="1" rows="1" role="data">2.3223</cell><cell cols="1" rows="1" role="data">8.6125</cell><cell cols="1" rows="1" 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role="data">5.2485</cell><cell cols="1" rows="1" role="data">6.7924</cell><cell cols="1" rows="1" role="data">5.9045</cell><cell cols="1" rows="1" role="data">7.5471</cell><cell cols="1" rows="1" role="data">6.5606</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">42</cell><cell cols="1" rows="1" role="data">4.4589</cell><cell cols="1" rows="1" role="data">4.0148</cell><cell cols="1" rows="1" role="data">5.2020</cell><cell cols="1" rows="1" role="data">4.6839</cell><cell cols="1" rows="1" role="data">5.9452</cell><cell cols="1" rows="1" role="data">5.3530</cell><cell cols="1" rows="1" role="data">6.6883</cell><cell cols="1" rows="1" role="data">6.0222</cell><cell cols="1" rows="1" role="data">7.4314</cell><cell cols="1" rows="1" role="data">6.6913</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">3 3/4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4.4457</cell><cell cols="1" rows="1" role="data">4.0294</cell><cell cols="1" rows="1" role="data">5.1867</cell><cell cols="1" rows="1" role="data">4.7009</cell><cell cols="1" rows="1" role="data">5.9276</cell><cell cols="1" rows="1" role="data">5.3725</cell><cell cols="1" rows="1" role="data">6.6680</cell><cell cols="1" rows="1" role="data">6.0440</cell><cell cols="1" rows="1" role="data">7.4095</cell><cell cols="1" rows="1" role="data">6.7156</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4 1/4</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">43</cell><cell cols="1" rows="1" role="data">4.3881</cell><cell cols="1" rows="1" role="data">4.0920</cell><cell cols="1" rows="1" role="data">5.1195</cell><cell cols="1" rows="1" role="data">4.7740</cell><cell cols="1" rows="1" role="data">5.8508</cell><cell cols="1" rows="1" role="data">5.4560</cell><cell cols="1" rows="1" role="data">6.5822</cell><cell cols="1" rows="1" role="data">6.1380</cell><cell cols="1" rows="1" role="data">7.3135</cell><cell cols="1" rows="1" role="data">6.8200</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">44</cell><cell cols="1" rows="1" role="data">4.3160</cell><cell cols="1" rows="1" role="data">4.1679</cell><cell cols="1" rows="1" role="data">5.0354</cell><cell cols="1" rows="1" role="data">4.8626</cell><cell cols="1" rows="1" role="data">5.7547</cell><cell cols="1" rows="1" role="data">5.5573</cell><cell cols="1" rows="1" role="data">6.4741</cell><cell cols="1" rows="1" role="data">6.2519</cell><cell cols="1" rows="1" role="data">7.1934</cell><cell cols="1" rows="1" role="data">6.9466</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">45</cell><cell cols="1" rows="1" role="data">4.2426</cell><cell cols="1" rows="1" role="data">4.2426</cell><cell cols="1" rows="1" role="data">4.9497</cell><cell cols="1" rows="1" role="data">4.9497</cell><cell cols="1" rows="1" role="data">5.6509</cell><cell cols="1" rows="1" role="data">5.6569</cell><cell cols="1" rows="1" role="data">6.3640</cell><cell cols="1" rows="1" role="data">6.3640</cell><cell cols="1" rows="1" role="data">7.0711</cell><cell cols="1" rows="1" role="data">7.0711</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="rowspan=2" role="data">Pts.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Deg.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" role="data">Dep.</cell><cell cols="1" rows="1" role="data">Lat.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Deg.</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">Pts.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 6</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 7</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 8</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 9</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Dist. 10</cell></row></table><pb n="615"/><cb/></p></div2></div1><div1 part="N" n="TREBLE" org="uniform" sample="complete" type="entry"><head>TREBLE</head><p>, in Music, the highest or acutest of the four parts in symphony, or that which is heard the clearest and shrillest in a concert. In the like sense we say, a Treble violin, Treble hautboy, &c.</p><p>In vocal music, the Treble is usually committed to boys and girls; their proper part being the Treble.</p><p>The Treble is divided into first or highest Treble, and second or bass Treble. The half Treble is the same with the counter-tenor.</p></div1><div1 part="N" n="TRENCHES" org="uniform" sample="complete" type="entry"><head>TRENCHES</head><p>, in Fortification, are ditches which the besiegers cut to approach more securely to the place attacked; whence they are called <hi rend="italics">lines of approach.</hi> Their breadth is 8 or 10 feet, and depth 6 or 7.</p><p>They say, <hi rend="italics">mount the Trenches,</hi> that is, go upon duty in them. To <hi rend="italics">relieve the Trenches,</hi> is to relieve such as have been upon duty there. The enemy is said to have <hi rend="italics">cleared the Trenches,</hi> when he has driven away or killed the soldiers who guarded them.</p><p><hi rend="italics">Tail of the</hi> <hi rend="smallcaps">Trench</hi>, is the place where it was begun. And the <hi rend="italics">Head</hi> is the place where it ends.</p><p><hi rend="italics">Opening of the</hi> <hi rend="smallcaps">Trenches</hi>, is when the besiegers first begin to work upon them, or to make them; which is usually done in the night.</p></div1><div1 part="N" n="TREPIDATION" org="uniform" sample="complete" type="entry"><head>TREPIDATION</head><p>, in the Ancient Astronomy, denotes what they call a libration of the 8th sphere; or a motion which the Ptolomaic system attributed to the firmament, to account for certain almost insensible changes and motions observed in the axis of the world; by means of which the latitudes of the fixed stars come to be gradually changed, and the ecliptic seems to approach reciprocally, first towards one pole, then towards the other.</p><p>This motion is also called the <hi rend="italics">motion of the first libration.</hi></p></div1><div1 part="N" n="TRET" org="uniform" sample="complete" type="entry"><head>TRET</head><p>, in Commerce, is an allowance made for the waste, or the dust, that may be mixed with any commodity; which is always 4 pounds on every 104 pounds weight. See <hi rend="smallcaps">Tare.</hi></p></div1><div1 part="N" n="TRIANGLE" org="uniform" sample="complete" type="entry"><head>TRIANGLE</head><p>, in Geometry, a figure bounded or contained by three lines or sides, and which consequently has three angles, from whence the figure takes its name.</p><p>Triangles are either plane or spherical or curvilinear. Plane when the three sides of the Triangle are right lines; but spherical when some or all of them are arcs of great circles on the sphere.</p><p>Plane Triangles take several denominations, both from the relation of their angles, and of their sides, as below. And 1st with regard to the sides. <figure/></p><p>An <hi rend="italics">Equilateral Triangle,</hi> is that which has all its three sides equal to one another; as A.</p><p>An <hi rend="italics">Isosceles</hi> or <hi rend="italics">Equicrural Triangle,</hi> is that which has two sides equal; as B.</p><p>A <hi rend="italics">Scalene Triangle</hi> has all its sides unequal; as C. <cb/></p><p>Again, with respect to the Angles. <figure/></p><p>A <hi rend="italics">Rectangular</hi> or <hi rend="italics">Right-angled Triangle,</hi> is that which has one right angle; as D.</p><p>An <hi rend="italics">Oblique Triangle</hi> is that which has no right angle, but all oblique ones; as E or F.</p><p>An <hi rend="italics">Acutangular</hi> or <hi rend="italics">Oxygone Triangle,</hi> is that which has three acute angles; as E.</p><p>An <hi rend="italics">Obtusangular</hi> or <hi rend="italics">Amblygone Triangle,</hi> is that which has an obtuse angle; as F.</p><p>A <hi rend="italics">Curvilinear</hi> or <hi rend="italics">Curvilineal Triangle,</hi> is one that has all its three sides curve lines.</p><p>A <hi rend="italics">Mixtilinear Triangle</hi> is one that has its sides some of them curves, and some right lines.</p><p>A <hi rend="italics">Spherical Triangle</hi> is one that has its sides, or at least some of them, arcs of great circles of the sphere.</p><p><hi rend="italics">Similar Triangles</hi> are such as have the angles in the one equal to the angles in the other, each to each.</p><p>The <hi rend="italics">Base</hi> of a Triangle, is any side on which a perpendicular is drawn from the opposite angle, called the <hi rend="italics">vertex;</hi> and the two sides about the perpendicular, or the vertex, are called the <hi rend="italics">legs.</hi></p><p><hi rend="italics">The Chief Properties of Plane Triangles,</hi> are as follow, viz, In any plane Triangle,</p><p>1. The greatest side is opposite to the greatest angle, and the least side to the least angle, &c. Also, if two sides be equal, their opposite angles are equal; and if the Triangle be equilateral, or have all its sides equal, it will also be equiangular, or have all its angles equal to one another.</p><p>2. Any side of a Triangle is less than the sum, but greater than the difference, of the other two sides.</p><p>3. The sum of all the three angles, taken together, is equal to two right angles.</p><p>4. If one side of a Triangle be produced out, the external angle, made by it and the adjacent side, is equal to the sum of the two opposite internal angles.</p><p>5. A line drawn parallel to one side of a Triangle, cuts the other two sides proportionally, the corresponding segments being proportional, each to each, and to the whole sides; and the Triangle cut off is similar to the whole Triangle.</p><p>If a perpendicular be let fall from any angle of a Triangle, as a vertical angle, upon the opposite side as a base; then</p><p>6. The rectangle of the sum and difference of the sides, is equal to twice the rectangle of the base and the distance of the perpendicular from the middle of the base.—Or, which is the same thing in other words,</p><p>7. The difference of the squares of the sides, is equal to the difference of the squares of the segments of the base. Or, as the base is to the sum of the sides, so is the difference of the sides, to the difference of the segments of the base.</p><p>8. The rectangle of the legs or sides, is equal to the rectangle of the perpendicular and the diameter of the circumscribing circle.</p><p>If a line be drawn bisecting any angle, to the base or opposite side; then, <pb n="616"/><cb/></p><p>9. The segments of the base, made by the line bisecting the opposite angle, are proportional to the sides adjacent to them.</p><p>10. The square of the line bisecting the angle, is equal to the difference between the rectangle of the sides and the rectangle of the segments of the base.</p><p>If a line be drawn from any angle to the middle of the opposite side, or bisecting the base; then</p><p>11. The sum of the squares of the sides, is equal to twice the sum of the squares of half the base and the line bisecting the base.</p><p>12. The angle made by the perpendicular from any angle and the line drawn from the same angle to the middle of the base, is equal to half the difference of the angles at the base.</p><p>13. If through any point D, within a Triangle ABC, three lines EF, GH, IK, be drawn parallel to the three sides of the Triangle; the continual products or solids made by the alternate segments of these lines will be equal; viz, . <figure/></p><p>14. If three lines AL, BM, CN, be drawn from the three angles through any point D within a Triangle, to the opposite sides; the solid products of the alternate segments of the sides are equal; viz, , (2d fig. above).</p><p>15. Three lines drawn from the three angles of a Triangle to bisect the opposite sides, or to the middle of the opposite sides, do all intersect one another in the same point D, and that point is the centre of gravity of the Triangle, and the distance AD of that point from any angle as D, is equal to double the distance DL from the opposite side; or one segment of any of these lines is double the other segment: moreover the sum of the squares of the three bisecting lines, is 3/4 of the sum of the squares of the three sides of the Triangle.</p><p>16. Three perpendiculars bisecting the three sides of a Triangle, all intersect in one point, and that point is the centre of the circumscribing circle.</p><p>17. Three lines bisecting the three angles of a Triangle, all intersect in one point, and that point is the centre of the inscribed circle.</p><p>18. Three perpendiculars drawn from the three angles of a Triangle, upon the opposite sides, all intersect in one point.</p><p>19. If the three angles of a Triangle be bisected by the lines AD, BD, CD (3d fig. above), and any one as BD be continued to the opposite side at O, and DP be drawn perp. to that side; then is [angle]ADO = [angle]CDP, or [angle]ADP = [angle]CDO.</p><p>20. Any Triangle may have a circle circumscribed about it, or touching all its angles, and a circle inscribed within it, or touching all its sides. <cb/></p><p>21. The square of the side of an equilateral Triangles is equal to 3 times the square of the radius of its circumscribing circle.</p><p>22. If the three angles of one Triangle be equal to the three angles of another Triangle, each to each; then those two Triangles are similar, and their like sides are proportional to one another, and the areas of the two Triangles are to each other as the squares of their like sides.</p><p>23. If two Triangles have any three parts of the one (except the three angles), equal to three corresponding parts of the other, each to each; those two Triangles are not only similar, but also identical, or having all their six corresponding parts equal, and their areas equal.</p><p>24. Triangles standing upon the same base, and between the same parallels, are equal; and Triangles upon equal bases, and having equal altitudes, are equal.</p><p>25. Triangles on equal bases, are to one another as their altitudes: and Triangles of equal altitudes, are to one another as their bases; also equal Triangles have their bases and altitudes reciprocally proportional.</p><p>26. Any Triangle is equal to half its circumscribing parallelogram, or half the parallelogram on the same or an equal base, and of the same or equal altitude.</p><p>27. Therefore the area of any Triangle is found, by multiplying the base by the altitude, and taking half the product.</p><p>28. The area is also found thus: Multiply any two sides together, and multiply the product by the sine of their included angle, to radius 1, and divided by 2.</p><p>29. The area is also otherwise found thus, when the three sides are given: Add the three sides together, and take half their sum; then from this half sum subtract each side severally, and multiply the three remainders and the half sum continually together; then the square root of the last product will be the area of the Triangle.</p><p>30. In a right-angled Triangle, if a perpendicular be let fall from the right angle upon the hypothenuse, it will divide it into two other Triangles similar to one another, and to the whole Triangle.</p><p>31. In a right-angled Triangle, the square of the hypothenuse is equal to the sum of the squares of the two sides; and, in general, any figure described upon the hypothenuse, is equal to the sum of two similar figures described upon the two sides.</p><p>32. In an isosceles Triangle, if a line be drawn from the vertex to any point in the base; the square of that line together with the rectangle of the segments of the base, is equal to the square of the side.</p><p>33. If one angle of a Triangle be equal to 120°; the square of the base will be equal to the squares of both the sides, together with the rectangle of those sides; and if those sides be equal to each other, then the square of the base will be equal to three times the square of one side, or equal to 12 times the square of the perpendicular from the angle upon the base.</p><p>34. In the same Triangle, viz, having one angle equal to 120°; the difference of the cubes of the sides, about that angle, is equal to a solid contained by the difference of the sides and the square of the base; and the sum of the cubes of the sides, is equal to a solid contained by the sum of the sides and the difference <pb n="617"/><cb/> between the square of the base and twice the rectangle of the sides.</p><p>There are many other properties of Triangles to be found among the geometrical writers; so Gregory St. Vincent has written a folio volume upon Triangles; there are also several in his Quadrature of the circle. See also other properties under the article <hi rend="smallcaps">Trigonometry.</hi></p><p>For the properties of spherical Triangles, see S<hi rend="smallcaps">PHERICAL</hi> <hi rend="italics">Triangles.</hi></p><p><hi rend="italics">Solution of</hi> <hi rend="smallcaps">Triangles.</hi> See <hi rend="smallcaps">Trigonometry.</hi></p><div2 part="N" n="Triangle" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Triangle</hi></head><p>, in Astronomy, one of the 48 ancient constellations, situated in the northern hemisphere. There is also the <hi rend="italics">Southern Triangle</hi> in the southern hemisphere, which is a modern constellation. The stars in the Northern Triangle are, in Ptolomy's catalogue 4, in Tycho's 4, in Hevelius's 12, and in the British catalogue 16.</p><p>The stars in the Southern Triangle are, in Sharp's catalogue, 5.</p><p><hi rend="italics">Arithmetical</hi> <hi rend="smallcaps">Triangle</hi>, a kind of numeral Triangle, or Triangle of numbers, being a table of certain numbers disposed in form of a Triangle. It was so called by Pascal; but he was not the inventor of this table, as some writers have imagined, its properties having been treated of by other authors, some centuries before him, as is shewn in my Mathematical Tracts, vol. 1, pa. 69 &c.</p><p>The form of the Triangle is as follows: <figure/></p><p>And it is constructed by adding always the last two numbers of the next two preceding columns together, to give the next succeeding column of numbers.</p><p>The first vertical column consists of units; the 2d a series of the natural numbers 1, 2, 3, 4, 5, &c; the 3d a series of Triangular numbers 1, 3, 6, 10, &c; the 4th a series of pyramidal numbers, &c. The oblique diagonal rows, descending from left to right, are also the same as the vertical columns. And the numbers taken on the horizontal lines are the co-efficients of the different powers of a binomial. Many other properties and uses of these numbers have been delivered by various authors, as may be seen in the Introduction to my Mathematical Tables, pages 7, 8, 75, 76, 77, 89, 2d edition.</p><p>After these, Pascal wrote a treatise on the Arithmetical Triangle, which is contained in the 5th volume of his works, published at Paris and the Hague in 1779, in 5 volumes, 8vo.</p><p>In this publication is also a description, taken from the 1st volume of the French Encyclopedie, art. <hi rend="italics">Arithmetique Machine,</hi> of that admirable machine in- <cb/> vented by Pascal at the age of 19, furnishing an easy and expeditious method of making all sorts of arithmetical calculations without any other assistance than the eye and the hand.</p></div2></div1><div1 part="N" n="TRIANGULAR" org="uniform" sample="complete" type="entry"><head>TRIANGULAR</head><p>, relating to a triangle; as</p><p><hi rend="smallcaps">Triangular</hi> <hi rend="italics">Canon,</hi> tables relating to trigonometry; as of sines, tangents, secants, &c.</p><p><hi rend="smallcaps">Triangular</hi> <hi rend="italics">Compasses,</hi> are such as have three legs or feet, by which any triangle, or three points, may be taken off at once. These are very useful in the construction of maps, globes, &c.</p><p><hi rend="smallcaps">Triangular</hi> <hi rend="italics">Numbers,</hi> are a kind of polygonal numbers; being the sums of arithmetical progressions, which have 1 for the common difference of their terms. <table><row role="data"><cell cols="1" rows="1" role="data">Thus, from these arithmeticals</cell><cell cols="1" rows="1" role="data">1 2 3</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" rend="align=right" role="data">5</cell><cell cols="1" rows="1" rend="align=right" role="data">6,</cell></row><row role="data"><cell cols="1" rows="1" role="data">are formed the Triang. Numb.</cell><cell cols="1" rows="1" role="data">1 3 6</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" rend="align=right" role="data">21,</cell></row></table> or the 3d column of the arithmetical triangle abovementioned.</p><p>The sum of any number <hi rend="italics">n</hi> of the terms of the Triangular numbers, which is also equal to the number of shot in a trianguar pile of balls, the number of rows, or the number in each side of the base, being <hi rend="italics">n.</hi></p><p>The sum of the reciprocals of the Triangular series, infinitely continued, is equal to 2; viz, 1 + 1/3 + 1/6 + 1/10 + 1/15 &c = 2.</p><p>For the rationale and management of these numbers, see Malcolm's Arith. book 5, ch. 2; and Simpson's Algeb. sec. 15.</p><p><hi rend="smallcaps">Triangular</hi> <hi rend="italics">Quadrant,</hi> is a sector furnished with a loose piece, by which it forms an equilateral triangle. Upon it is graduated and marked the calendar, with the sun's place, and other useful lines; and by the help of a string and a plummet, with the divisions graduated on the loose piece, it may be made to serve for a quadrant.</p></div1><div1 part="N" n="TRIBOMETER" org="uniform" sample="complete" type="entry"><head>TRIBOMETER</head><p>, in Mechanics, a term applied by Musschenbroek to an instrument invented by him for measuring the friction of metals. It consists of an axis formed of hard steel, passing through a cylindrical piece of wood: the ends of the axis, which are highly polished, are made to rest on the polished semicircular cheeks of various metals, and the degree of friction is estimated by means of a weight suspended by a fine silken string or ribband over the wooden cylinder. For a farther description and the figure of this instrument, with the results of various experiments performed with it, see Musschenb. Introd. ad Phil. Nat. vol. 1, p. 151.</p></div1><div1 part="N" n="TRIDENT" org="uniform" sample="complete" type="entry"><head>TRIDENT</head><p>, is a particular kind of parabola, used by Descartes in constructing equations of 6 dimensions. See the article <hi rend="italics">Cartesian</hi> <hi rend="smallcaps">Parabola.</hi></p></div1><div1 part="N" n="TRIGLYPH" org="uniform" sample="complete" type="entry"><head>TRIGLYPH</head><p>, in Architecture, is a member of the Doric Frize, placed directly over each column, and at equal distances in the intercolumnation, having two entire glyphs or channels engraven in it, meeting in an angle, and separated by three legs from the two demichannels of the sides. <pb n="618"/><cb/></p></div1><div1 part="N" n="TRIGON" org="uniform" sample="complete" type="entry"><head>TRIGON</head><p>, a figure of three angles, or a triangle.</p><div2 part="N" n="Trigon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Trigon</hi></head><p>, in Astrology. See <hi rend="smallcaps">Triplicity.</hi></p></div2><div2 part="N" n="Trigon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Trigon</hi></head><p>, in Astronomy, denotes an aspect of two planets when they are 120 degrees distant from each other; called also a Trine, being the 3d part of 360 degrees.—The Trigons of Mars and Saturn are by astrologers held malific or malignant aspects.</p></div2><div2 part="N" n="Tbigon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tbigon</hi></head><p>, in Dialling, is an instrument of a triangular form.</p></div2><div2 part="N" n="Trigon" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Trigon</hi></head><p>, in Music, denoted a musical instrument, used among the ancients. It was a kind of triangular lyre, or harp, invented by Ibycus; and was used at feasts, being played on by women, who struck it either with a quill, or beat it with small rods of different lengths and weights, to occasion a diversity in the sounds.</p><p>TRIGONAL <hi rend="italics">Numbers.</hi> See <hi rend="smallcaps">Triangular</hi> <hi rend="italics">Numbers.</hi></p><p>TRIGONOMETER <hi rend="italics">Armillary.</hi> See <hi rend="smallcaps">Armillary</hi> <hi rend="italics">Trigonometer.</hi></p></div2></div1><div1 part="N" n="TRIGONOMETRY" org="uniform" sample="complete" type="entry"><head>TRIGONOMETRY</head><p>, the art of measuring the sides and angles of triangles, either plane or spherical, from whence it is accordingly called either Plane Trigonometry, or Spherical Trigonometry.</p><p>Every triangle has 6 parts, 3 sides, and 3 angles; and it is necessary that three of these parts be given, to find the other three. In spherical Trigonometry, the three parts that are given, may be of any kind, either all sides, or all angles, or part the one and part the other. But in plane Trigonometry, it is necessary that one of the three parts at least be a side, since from three angles can only be found the proportions of the sides, but not the real quantities of them.</p><p>Trigonometry is an art of the greatest use in the mathematical sciences, especially in astronomy, navigation, surveying, dialling, geography, &c, &c. By it, we come to know the magnitude of the earth, the planets and stars, their distances, motions, eclipses, and almost all other useful arts and sciences. Accordingly we find this art has been cultivated from the earliest ages of mathematical knowledge.</p><p>Trigonometry, or the resolution of triangles, is founded on the mutual proportions which subsist between the sides and angles of triangles; which proportions are known by finding the relations between the radius of a circle and certain other lines drawn in and about the circle, called <hi rend="italics">chords, sines, tangents,</hi> and <hi rend="italics">secants.</hi> The ancients Menelaus, Hipparchus, Ptolomy, &c, performed their Trigonometry, by means of the chords. As to the sines, and the common theorems relating to them, they were introduced into Trigonometry by the Moors or Arabians, from whom this art passed into Europe, with several other branches of science. The Europeans have introduced, since the 15th century, the tangents and secants, with the theorems relating to them. See the history and improvements at large, in the Introduction to my Mathematical Tables.</p><p>The proportion of the sines, tangents, &c, to their radius, is sometimes expressed in common or natural numbers, which constitute what we call the <hi rend="italics">tables of natural sines, tangents, and secants.</hi> Sometimes it is expressed in logarithms, being the logarithms of the <cb/> said natural sines, tangents, &c; and these constitute the table of <hi rend="italics">artificial sines,</hi> &c. Lastly, sometimes the proportion is not expressed in numbers; but the several sines, tangents, &c, are actually laid down upon lines of scales; whence the <hi rend="italics">line of sines,</hi> of <hi rend="italics">tangents,</hi> &c. See <hi rend="smallcaps">Scale.</hi></p><p>In Trigonometry, as angles are measured by arcs of a circle described about the angular point, so the whole circumference of the circle is divided into a great number of parts, as 360 degrees, and each degree into 60 minutes, and each minute into 60 seconds, &c; and then any angle is said to consist of so many degrees, minutes and seconds, as are contained in the arc that measures the angle, or that is intercepted between the legs or sides of the angle.</p><p>Now the sine, tangent, and secant, &c, of every degree and minute, &c, of a quadrant, are calculated to the radius 1, and ranged in tables for use; as also the logarithms of the same; forming the triangular canon. And these numbers, so arranged in tables, form every species of right-angled triangles, so that no such triangle can be proposed, but one similar to it may be there found, by comparison with which, the proposed one may be computed by analogy or proportion.</p><p>As to the scales of chords, sines, tangents, &c, usually placed on instruments, the method of constructing them is exhibited in the scheme annexed to the article <hi rend="smallcaps">Scale;</hi> which, having the names added to each, needs no farther explanation.</p><p>There are usually three methods of resolving triangles, or the cases of Trigonometry; viz, geometrical construction, arithmetical computation, and instrumental operation. In the 1st method, the triangle in question is constructed by drawing and laying down the several parts of their magnitudes given, viz, the sides from a scale of equal parts, and the angles from a scale of chords, or other instrument; then the unknown parts are measured by the same scales, and so they become known.</p><p>In the 2d method, having stated the terms of the proportion according to rule, which terms consist partly of the numbers of the given sides, and partly of the sines, &c, of angles taken from the tables, the proportion is then resolved like all other proportions, in which a 4th term is to be found from three given terms, by multiplying the 2d and 3d together, and dividing the product by the first. Or, in working with the logarithms, adding the log. of the 2d and 3d terms together, and from the sum subtracting the log. of the 1st term, then the number answering to the remainder is the 4th term sought.</p><p>To work a case instrumentally, as suppose by the log. lines on one side of the two-foot scales: Extend the compasses from the 1st term to the 2d, or 3d, which happens to be of the same kind with it; then that extent will reach from the other term to the 4th. In this operation, for the sides of triangles, is used the line of numbers (marked Num.); and for the angles, the line of sines or tangents (marked sin. and tan.) according as the proportion respects fines or tangents.</p><p>In every case of triangles, as has been hinted before, <pb n="619"/><cb/> there must be three parts, one at least of which must be a side. And then the different circumstances, as to the three parts that may be given, admit of three cases or varieties only; viz,</p><p>1st. When two of the three parts given, are a side and its opposite angle.—2d, When there are given two sides and their contained angle.—3d, And thirdly, when the three sides are given.</p><p>To each of these cases there is a particular rule, or proportion, adapted, for resolving it by.</p><p>1st. <hi rend="italics">The Rule for the</hi> 1<hi rend="italics">st Case,</hi> or that in which, of the three parts that are given, an angle and its opposite side are two of them, is this, viz, That the sides are proportional to the sines of their opposite angles,</p><p>That is, <table><row role="data"><cell cols="1" rows="1" role="data">As one side given</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To the sine of its opposite angle</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So is another side given</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To the sine of its opposite angle.</cell><cell cols="1" rows="1" role="data"/></row></table> Or, <table><row role="data"><cell cols="1" rows="1" role="data">As the sine of an angle given</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To its opposite side</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So is the sine of another angle given</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To its opposite side.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>So that, to find an angle, we must begin the proportion with a given side that is opposite to a given angle; and to find a side, we must begin with an angle opposite to a given side.</p><p><hi rend="italics">Ex.</hi> Suppose, in the triangle ABC, there be given <figure/> AB = 365 feet, AC = 154.33 f. [angle]C = 98° 3′ to find the other side, and the angles. <hi rend="center">1. <hi rend="italics">Geometrically, by Construction.</hi></hi></p><p>Draw AC = 154.33 from a scale of equal parts: Make the angle C = 98° 3′, producing CB indefinitely: With centre A, and radius 365 feet, cross CB in B: Then join AB, and the figure is constructed. Then, by measuring the unknown angles and side, the former by the line of cords or otherwise, and the side by the line of equal parts, they will be found, as near as they can be measured, as below, viz, BC = 310; the [angle]A = 57°1/4; and [angle]B = 24°3/4. <hi rend="center">2. <hi rend="italics">Arithmetically, by Tables of Logs.</hi></hi> <table><row role="data"><cell cols="1" rows="1" role="data">As AB</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" rend="align=right" role="data">log. 2.5622929</cell></row><row role="data"><cell cols="1" rows="1" role="data">To AC</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">154.33</cell><cell cols="1" rows="1" rend="align=right" role="data">2.1884504</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sin. [angle]C</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 98° 3′</cell><cell cols="1" rows="1" rend="align=right" role="data">or 81° 57′ 9.9950993</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. [angle] B</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 24° 45′</cell><cell cols="1" rows="1" rend="align=right" role="data">9.6218568</cell></row><row role="data"><cell cols="1" rows="1" role="data">the sum</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 122 48</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">taken from</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 180 00</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">leaves [angle]A</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 57 12</cell><cell cols="1" rows="1" role="data"/></row></table> <cb/> Then, again, <table><row role="data"><cell cols="1" rows="1" role="data">As sin. [angle]C</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 98° 3′</cell><cell cols="1" rows="1" rend="align=right" role="data">log. 9.9956993</cell></row><row role="data"><cell cols="1" rows="1" role="data">To AB</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" rend="align=right" role="data">2.5622929</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sin. [angle]A</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 57° 12′</cell><cell cols="1" rows="1" rend="align=right" role="data">9.9245721</cell></row><row role="data"><cell cols="1" rows="1" role="data">To BC</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">309.86</cell><cell cols="1" rows="1" rend="align=right" role="data">2.4911657</cell></row></table> <hi rend="center">3. <hi rend="italics">Instrumentally, by Gunter's Lines.</hi></hi></p><p>In the first proportion, Extend the compasses from 365 to 154 1/3 on the line of numbers; and that extent will reach, upon the line of sines, from 82° to 24 3/4, which gives the angle B. And, in the second proportion, Extend from 98° to 57 1/4 on the sines; and that extent will reach, upon the numbers, from 365 to 310, or the side BC nearly.</p><p><hi rend="italics">2d Case,</hi> when there are Given two Sides and their contained angle, to find the rest, the rule is this: As the sum of the two given sides: Is to the difference of the sides:: So is the tang. of half the sum of the two opposite angles, or cotangent of half the given angle: To tang. of half the diff. of those angles.</p><p>Then the half diff. added to the half sum, gives the greater of the two unknown angles; and subtracted, leaves the less of the two angles.</p><p>Hence, the angles being now all known, the remaining 3d side will be found by the former case.</p><p><hi rend="italics">Ex.</hi> Suppose, in the triangle ABC, there be given <table><row role="data"><cell cols="1" rows="1" role="data">the side</cell><cell cols="1" rows="1" role="data">AC = 154.33</cell></row><row role="data"><cell cols="1" rows="1" role="data">the side</cell><cell cols="1" rows="1" role="data">BC = 309.86</cell></row><row role="data"><cell cols="1" rows="1" role="data">the included</cell><cell cols="1" rows="1" role="data">[angle]C =  98° 3′</cell></row></table> to find the other side and the angles.</p><p>1. <hi rend="italics">Geometrically.</hi>—Draw two indefinite lines making the angle C = 98° 3′: upon these lines set off CA = 154 1/3, and CB = 310: Join the points A and B, and the figure is made. Then, by measurement, as before, we find the [angle]A = 57 1/4; [angle]B 24 3/4; and side AB = 365. <hi rend="center">2. <hi rend="italics">By Logarithms.</hi></hi> <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">As CB + CA  =</cell><cell cols="1" rows="1" rend="align=left" role="data">464.19</cell><cell cols="1" rows="1" role="data">log. 2.6666958</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">To CB - CA  =</cell><cell cols="1" rows="1" rend="align=left" role="data">155.53</cell><cell cols="1" rows="1" role="data">2.1918142</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">So tan. 1/2 A + 1/2 B =</cell><cell cols="1" rows="1" rend="align=left" role="data">40° 58 1/2′</cell><cell cols="1" rows="1" role="data">9.9387803</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">To tan. 1/2 A - 1/2 B =</cell><cell cols="1" rows="1" rend="align=left" role="data">16  13 1/2</cell><cell cols="1" rows="1" role="data">9.4638987</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">sum gives [angle]A</cell><cell cols="1" rows="1" rend="align=left" role="data">57  12</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">diff. gives [angle]B</cell><cell cols="1" rows="1" rend="align=left" role="data">24  45</cell><cell cols="1" rows="1" role="data"/></row></table> Then, <table><row role="data"><cell cols="1" rows="1" role="data">As sin. [angle]B =</cell><cell cols="1" rows="1" role="data">24° 45′</cell><cell cols="1" rows="1" rend="align=right" role="data">log. 9.6218612</cell></row><row role="data"><cell cols="1" rows="1" role="data">To side AC =</cell><cell cols="1" rows="1" role="data">154.33</cell><cell cols="1" rows="1" rend="align=right" role="data">2.1884504</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sin. [angle]C =</cell><cell cols="1" rows="1" role="data">98° 3′,</cell><cell cols="1" rows="1" rend="align=right" role="data">or 81° 57 9.9956993</cell></row><row role="data"><cell cols="1" rows="1" role="data">To side AB =</cell><cell cols="1" rows="1" role="data">365</cell><cell cols="1" rows="1" rend="align=right" role="data">2.5622885</cell></row></table></p><p>3. <hi rend="italics">Instrumentally.</hi>—Extend the compasses from 464 to 155 1/2 upon the line of numbers; then that extent will reach, upon the line of tangents, from 41° to 16° 1/4. Then, in the 2d proportion, extend the compasses from 24° 3/4 to 82° on the sines; and that extent <pb n="620"/><cb/> will reach, upon the numbers, from 154 1/3 to 365, which is the third side.</p><p>3d <hi rend="italics">Case,</hi> is when the three sides are given, to find the three angles; and the method of resolving this case is, to let a perpendicular fall from the greatest angle, upon the opposite side or base, dividing it into two segments, and the whole triangle into two smaller rightangled triangles: then it will be, <table><row role="data"><cell cols="1" rows="1" role="data">As the base, or sum of the two segments</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">Is to the sum of the other two sides</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So is the difference of those sides</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">To the difference of the segments of the base.</cell></row></table></p><p>Then half this difference of the two segments added to the half sum, or half the base, gives the greater segments, and subtracted, gives the less. Hence, in each of the two right-angled triangles, there are given the hypotenuse, and the base, besides the right angle, to find the other angles by the 1st case.</p><p><hi rend="italics">Ex.</hi> In the trangle ABC, suppose there are given the three sides, to sind the three angles, viz, <table><row role="data"><cell cols="1" rows="1" role="data">AB = 365</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(15)">}</hi> to find the angles.</cell></row><row role="data"><cell cols="1" rows="1" role="data">AC = 154.33</cell></row><row role="data"><cell cols="1" rows="1" role="data">BC = 309.86</cell></row></table></p><p>1. <hi rend="italics">Geometrically.</hi>—Draw the base AB = 365: with the radius 154 1/3 and centre A describe an arc; and with the radius 310 and centre B describe another arc, cutting the former in C; then join AC and BC, and the triangle is constructed. And by measuring the angles, they are found, viz. [angle]A = 57° 1/4; [angle]B = 24° 3/4; [angle]C = 98° nearly.</p><p>2. <hi rend="italics">Arithmetically.</hi>—Having let fall the perpendicular CP, dividing the base into the two segments AP, PB, and the given triangle ABC into the two rightangled triangles ACP, BCP. Then, <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">As AB      =</cell><cell cols="1" rows="1" role="data">365   </cell><cell cols="1" rows="1" role="data">log. 2.5622929</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">To CB + CA =</cell><cell cols="1" rows="1" role="data">464.19</cell><cell cols="1" rows="1" role="data">2.6666958</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">So CB - CA =</cell><cell cols="1" rows="1" role="data">155.53</cell><cell cols="1" rows="1" role="data">2.1918142</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">To BP - PA =</cell><cell cols="1" rows="1" role="data">197.80</cell><cell cols="1" rows="1" role="data">2.2962171</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">its half =</cell><cell cols="1" rows="1" role="data">98.90</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1/2 AB =</cell><cell cols="1" rows="1" role="data">182.50</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">sum BP =</cell><cell cols="1" rows="1" role="data">281.40</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">dif. AP =</cell><cell cols="1" rows="1" role="data">83.60</cell><cell cols="1" rows="1" role="data"/></row></table> Then, in the triangle APC, right-angled at P, <table><row role="data"><cell cols="1" rows="1" role="data">As AC</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">154.33</cell><cell cols="1" rows="1" rend="align=right" role="data">log. 2.1884504</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. [angle]P</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 90°</cell><cell cols="1" rows="1" rend="align=right" role="data">10.0000000</cell></row><row role="data"><cell cols="1" rows="1" role="data">So AP</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 83.6</cell><cell cols="1" rows="1" rend="align=right" role="data">1.9222063</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. [angle]ACP</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">32° 48′</cell><cell cols="1" rows="1" rend="align=right" role="data">9.7337559</cell></row><row role="data"><cell cols="1" rows="1" role="data">its comp. [angle]A</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">57° 12</cell><cell cols="1" rows="1" role="data"/></row></table> <cb/> And in the triangle BPC, right-angled at P, <table><row role="data"><cell cols="1" rows="1" role="data">As BC</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">309.86</cell><cell cols="1" rows="1" rend="align=right" role="data">log. 2.4911655</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. [angle]P</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">90°</cell><cell cols="1" rows="1" rend="align=right" role="data">10.0000000</cell></row><row role="data"><cell cols="1" rows="1" role="data">So BP</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">281.4</cell><cell cols="1" rows="1" rend="align=right" role="data">2.4493241</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. [angle]BCP</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">65° 15′</cell><cell cols="1" rows="1" rend="align=right" role="data">9.9581586</cell></row><row role="data"><cell cols="1" rows="1" role="data">its comp. [angle]B</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">24  45</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Also to [angle]ACP</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">32  48′</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">add [angle]BCP</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">65  15</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">makes [angle]ACB</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">98   3</cell><cell cols="1" rows="1" role="data"/></row></table></p><p>3. <hi rend="italics">Instrumentally.</hi>—In the 1st proportion, Extend the compasses from 365 to 464 on the line of numbers, and that extent will reach, on the same line, from 155 1/2 to 197.8 nearly.—In the 2d proportion, Extend the compasses from 154 1/3 to 83.6 on the line of numbers, and that extent will reach, on the sines, from 90° to 32° 3/4 nearly.—In the 3d proportion, Extend the compasses from 310 to 281 1/2 on the line of numbers; then that extent will reach, on the sines, from 90° to 65° 1/4.</p><p>The foregoing three cases include all the varieties of plane triangles that can happen, both of right and oblique-angled triangles. But beside these, there are some other theorems that are useful upon many occasions, or suited to some particular forms of triangles, which are often more expeditious in use than the foregoing general ones; one of which, for right-angled triangles, as the case for which it serves so often occurs, may be here inserted, and is as follows.</p><p><hi rend="italics">Case</hi> 4. When, in a right-angled triangle, there are given the angles and one leg, to find the other leg, or the hypothenuse. Then it will, <figure/> <table><row role="data"><cell cols="1" rows="1" role="data">As radius</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To given leg AB</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. adjacent [angle]A</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To the opp. leg BC, and</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sec. of same [angle]A</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To hypot. AC</cell><cell cols="1" rows="1" role="data">.</cell></row></table></p><p><hi rend="italics">Ex.</hi> In the triangle ABC, right-angled at B, <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">Given the leg AB = 162</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(9)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">to find BC</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">and the [angle]A = 53° 7′ 48″</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">conseq. [angle] C = 36 52 12 </cell><cell cols="1" rows="1" role="data">and AC.</cell></row></table></p><p>1. <hi rend="italics">Geometrically.</hi>—Draw the leg AB = 162: Erect the indefinite perpendicular BC: Make the angle A = 53° 1/8, and the side AC will cut BC in C, and form the triangle ABC. Then, by measuring, there will be found AC = 270, and BC = 216.</p><p>2. <hi rend="italics">Arithmetically.</hi> <table><row role="data"><cell cols="1" rows="1" role="data">As radius</cell><cell cols="1" rows="1" role="data">= 10</cell><cell cols="1" rows="1" rend="align=right" role="data">log. 10.0000000</cell></row><row role="data"><cell cols="1" rows="1" role="data">To AB</cell><cell cols="1" rows="1" role="data">= 162</cell><cell cols="1" rows="1" rend="align=right" role="data">2.2095150</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tan. [angle]A</cell><cell cols="1" rows="1" role="data">= 53° 7′ 48″</cell><cell cols="1" rows="1" rend="align=right" role="data">10.1249372</cell></row><row role="data"><cell cols="1" rows="1" role="data">To BC</cell><cell cols="1" rows="1" role="data">= 216</cell><cell cols="1" rows="1" rend="align=right" role="data">2.3344522</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sec. [angle]A</cell><cell cols="1" rows="1" role="data">= 53° 7′ 48″</cell><cell cols="1" rows="1" rend="align=right" role="data">10.2218477</cell></row><row role="data"><cell cols="1" rows="1" role="data">To AC</cell><cell cols="1" rows="1" role="data">= 270</cell><cell cols="1" rows="1" rend="align=right" role="data">2.4313627</cell></row></table> <pb n="621"/><cb/></p><p>3. <hi rend="italics">Instrumentally.</hi>—Extend the compasses from 45° at the end of the tangents (the radius) to the tangent of 53° 1/8; then that extent will reach, on the line of numbers, from 162 to 216, for BC. Again, extend the compasses from 36° 52′ to 90 on the sines; then that extent will reach, on the line of numbers, from 162 to 270 for AC.</p><p><hi rend="italics">Note,</hi> another method, by making every side radius, is often added by the authors <figure/> on Trigonometry, which is thus: The given right-angled triangle being ABC, make first the hypotenuse AC radius, that is, with the extent of AC as a radius, and each of the centres A and C, describe arcs CD and AE; then it is evident that each leg will represent the sine of its opposite angle, viz, the leg BC the sine of the arc CD or of the angle A, and the leg AB the sine of the arc AE or of the angle C. Again, making either leg radius, the other leg will represent the tangent of its opposite angle, and the hypotenuse the secant of the same angle; thus, with radius AB and centre A describing the arc BF, BC represents the tangent of that arc, or of the angle A, and the hypotenuse AC the secant of the same; or with the radius BC and centre C describing the arc BG, the other leg AB is the tangent of that arc BG, or of the angle C, and the hypotenuse CA the secant of the same.</p><p>And then the general rule for all these cases is this, viz, that the sides bear to each other the same proportions as the parts or things which they represent. And this is called making every side radius.</p><p><hi rend="italics">Spherical</hi> <hi rend="smallcaps">Trigonometry</hi>, is the resolution and calculation of the sides and angles of spherical triangles, which are made by three intersecting arcs of great circles on a sphere. Here, any three of the six parts being given, even the three angles, the rest can be found; and the sides are measured or estimated by degrees, minutes, and seconds, as well as the angles.</p><p>Spherical Trigonometry is divided into right-angled and oblique-angled, or the resolution of right and oblique-angled spherical triangles. When the spherical triangle has a right angle, it is called a right-angled triangle, as well as in plane triangles; and when a triangle has one of its sides equal to a quadrant of a circle, it is called a quadrantal triangle.</p><p>For the resolution of spherical Triangles, there are various theorems and proportions, which are similar to those in plane Trigonometry, by substituting the sines of sides instead of the sides themselves, when the proportion respects sines; or tangents of the sides for the sides, when the proportion respects tangents, &c; some of the principal of which theorems are as follow:</p><p><hi rend="italics">Theor.</hi> 1. In any spherical triangle, the sines of the sides are proportional to the sines of their opposite angles. <cb/> <table><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor</hi> 2. In any right-angled triangle,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As radius</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of one side</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. of the adjacent angle</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. of the opposite side.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">  <hi rend="italics">Theor.</hi> 3. If a perpendicular be let fall from any angl</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">upon the base or opposite side of a spherical triangle</cell></row><row role="data"><cell cols="1" rows="1" role="data">it will be,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As the sine of the sum of the two sides</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To the sine of their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So cotan. 1/2 sum angles at the vertex</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. of half their difference.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 4.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As tang. half sum of the sides</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. half their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. 1/2 sum [angle]s at the base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. half their difference.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 5.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As cotan. 1/2 sum of [angle]s at the base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. half their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. 1/2 sum of [angle]s at the vertex</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. half their difference.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 6.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As tang. 1/2 sum segments of base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. half sum of the sides</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. half difference of the sides</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. 1/2 diff. segments of base.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 7.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As sin. sum of [angle]s at the base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. 1/2 sum segments of base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. of half their difference.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 8.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As sin. sum of segments of base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sin. sum of angles at the vertex</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of their difference.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 9.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As sine of the base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of the vertical angle</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So sin. of diff. segments of the base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. diff. [angle]s at vertex, when the perp.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">    falls within</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">Or so sin. sum segments of base</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. sum vertical [angle]s, where the perp.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">    falls without.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 10.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As cosin. half sum of the two sides</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To cosine of half their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So cotang. of half the included angle</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. half sum of opposite angles.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 11.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As sin. of half sum of two sides</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of half their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So cotang. half the included angle</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. 1/2 diff. of the oppos. angles.</cell><cell cols="1" rows="1" role="data"/></row></table> <pb n="622"/><cb/> <table><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 12.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As cosin. half sum of two angles</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To cosine of half their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. of half the included side</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. 1/2 sum of the opposite sides.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 13.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As sin. half sum of two angles</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sine of half their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So tang. half the included side</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To tang. 1/2 diff. of the opposite sides.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">  <hi rend="italics">Theor.</hi> 14. In a right-angled triangle,</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">As sin. sum of hypot and one side</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To sin. of their difference</cell><cell cols="1" rows="1" role="data">::</cell></row><row role="data"><cell cols="1" rows="1" role="data">So radius squared</cell><cell cols="1" rows="1" role="data">:</cell></row><row role="data"><cell cols="1" rows="1" role="data">To square of tang. 1/2 contained angle.</cell><cell cols="1" rows="1" role="data"/></row></table></p><p><hi rend="italics">Theor.</hi> 15. In any spherical triangle; The product of the sines of two sides and of the cosine of the included angle, added to the product of the cosines of those sides, is equal to the cosine of the third side; the radius being 1.</p><p><hi rend="italics">Theor.</hi> 16. In any spherical triangle; The product of the sines of two angles and of the cosine of the included side, minus the product of the cosines of those angles, is equal to the cosine of the third angle; the radius being 1.</p><p>By some or other of these theorems may all the cases of spherical triangles be resolved, both right angled and oblique: viz, the cases of right-angled triangles by the 1st and 2d theorems, and the oblique triangles by some of the other theorems.</p><p>In treatises on Trigonometry are to be found many other theorems, as well as synopses or tables of all the cases, with the theorem that is peculiar or proper to each. See the Introduction to my Mathematical Tables, p. 155 &c; or Robertson's Navigation, vol. 1, p. 162. See also Napier's Catholic or Universal Rule, in this Dictionary.</p><p>To the foregoing Theorems may be added the following synopsis of rules for resolving all the cases of plane and spherical triangles, under the title of <hi rend="center"><hi rend="italics">Trigonometrical Rules.</hi></hi></p><p>1. In a right-lined triangle, whose sides are A, B, C, and their opposite angles <hi rend="italics">a, b, c;</hi> having given any three of these, of which one is a side; to find the rest. <figure/></p><p>Put s for the sine, s′ the cosine, t the tangent, and t′ the cotangent of an arch or angle, to the radius <hi rend="italics">r;</hi> also L for a logarithm, and L′ its arithmetical complement. Then <cb/> <hi rend="center"><hi rend="italics">Case</hi> 1. When three sides A, B, C, are given.</hi> . <hi rend="center"><hi rend="italics">Case</hi> 2. Given two sides A, B, and their included angle <hi rend="italics">c.</hi></hi> . <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">If A = B; we shall have</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi> C = ((s.(1/2)<hi rend="italics">c</hi>)/<hi rend="italics">r</hi>) X 2A.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> = <hi rend="italics">b</hi> = 90° - (1/2)<hi rend="italics">c,</hi> and   </cell></row></table> <hi rend="center"><hi rend="italics">Case</hi> 3. When a side and its opposite angle are among the terms given.</hi> ; from which equations any term wanted may be found.</p><p>When an angle, as <hi rend="italics">a,</hi> is 90°, and A and C are given, then .</p><p><hi rend="italics">Note,</hi> When two sides A, B, and an angle <hi rend="italics">a</hi> opposite to one of them, are given; if A be less than B, then <hi rend="italics">b, c,</hi> C have each two values; otherwise, only one value.</p><p>II. In a spherical triangle, whose three sides are A, B, C, and their opposite angles <hi rend="italics">a, b, c;</hi> any three of these six terms being given, to find the rest. <figure/> <pb n="623"/><cb/> <hi rend="center"><hi rend="italics">Case</hi> 1. Given the three sides A, B, C.</hi> .</p><p>And the same for the other angles. <hi rend="center"><hi rend="italics">Case</hi> 2. Given the three angles.</hi></p><p>And the same for the other sides.</p><p><hi rend="italics">Note.</hi> The sign > signifies greater than, and < less; also <01> the difference. <hi rend="center"><hi rend="italics">Case</hi> 3. Given A, B, and included angle <hi rend="italics">c.</hi></hi></p><p>To find an angle <hi rend="italics">a</hi> opposite the side A, .</p><p>Again let <hi rend="italics">r</hi> : s′<hi rend="italics">c</hi> :: t.A : t.M, like or unlike A as <hi rend="italics">c</hi> is > or < 90°; and N = B <01> M.</p><p>Then s′M : s′N :: s′A, s′ C, like or unlike N as <hi rend="italics">c</hi> is > or < 90°. Or, . <hi rend="center"><hi rend="italics">Case</hi> 4. Given <hi rend="italics">a, b,</hi> and included side C.</hi> ; <cb/> . <hi rend="center"><hi rend="italics">Case</hi> 5. Given A, B, and an opposite angle <hi rend="italics">a.</hi></hi> .</p><p>But if A be equal to B, or to its supplement, or between B and its supplement; then is <hi rend="italics">b</hi> like to B: also <hi rend="italics">c</hi> is = <hi rend="italics">m</hi> ∓ <hi rend="italics">n,</hi> and C = M ± N, as B is like or unlike <hi rend="italics">a.</hi> <hi rend="center"><hi rend="italics">Case</hi> 6. Given <hi rend="italics">a, b,</hi> and an opposite side A.</hi> .</p><p>But if A be equal to B, or to its supplement, or between B and its supplement; then B is unlike <hi rend="italics">b,</hi> and only the less values of N, <hi rend="italics">n,</hi> are possible.</p><p><hi rend="italics">Note,</hi> When two sides A, B, and their opposite angles <hi rend="italics">a, b,</hi> are known; the third side C, and its opposite angle <hi rend="italics">c,</hi> are readily found thus: . <figure/></p><p>III. In a right-angled spheric triangle, where H is the hypotenuse, or side opposite the right angle, B, P the other two sides, and <hi rend="italics">b, p</hi> their opposite angles; any two of these five terms being given, to find the rest; the cases, with their solutions, are as in the following Table. <pb n="624"/><cb/></p><p>The same Table will also serve for the quadrantal triangle, or that which has one side = 90°, H being the angle opposite that side, B, P the other two angles, <cb/> and <hi rend="italics">b, p</hi> their opposite sides: observing, instead of H to take its supplement: and mutually change the terms <hi rend="italics">like</hi> and <hi rend="italics">unlike</hi> for each other where H is concerned. <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Case</cell><cell cols="1" rows="1" role="data">Given</cell><cell cols="1" rows="1" role="data">Req<hi rend="sup">d</hi></cell><cell cols="1" rows="1" rend="colspan=2" role="data">SOLUTIONS.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">s. H. : <hi rend="italics">r</hi> :: s B : s<hi rend="italics">b,</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">and is like B</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : t′H :: t. B : s′<hi rend="italics">p</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi>, > or < 90° as H is like or unlike B</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" role="data">P</cell><cell cols="1" rows="1" rend="align=left" role="data">s′B : <hi rend="italics">r</hi> :: s′H : s′P</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s′H :: s.<hi rend="italics">b</hi> : s.B,</cell><cell cols="1" rows="1" rend="align=left" role="data">like <hi rend="italics">b</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">P</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s′<hi rend="italics">b</hi> :: t.H : t.P</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi>, > or < 90° as H is like or unlike <hi rend="italics">b</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s′H :: t.<hi rend="italics">b</hi> : t′<hi rend="italics">p</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" rend="align=left" role="data">s.<hi rend="italics">b</hi> : <hi rend="italics">r</hi> :: s.B : s.H</cell><cell cols="1" rows="1" rend="align=left rowspan=3" role="data"><hi rend="size(6)">}</hi>, each > or < 90°; both values true</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">P</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : t.B :: t′<hi rend="italics">b</hi> : s.P</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">s′B : <hi rend="italics">r</hi> :: s′<hi rend="italics">b</hi> : s.<hi rend="italics">p</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : t′B :: s′<hi rend="italics">p</hi> : t′H,</cell><cell cols="1" rows="1" rend="align=left" role="data"> > or < 90 as B is like or unlike <hi rend="italics">p</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s′B :: s.<hi rend="italics">p</hi> : s′<hi rend="italics">b,</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">like B</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s.B :: t.<hi rend="italics">p</hi> : t.P,</cell><cell cols="1" rows="1" rend="align=left" role="data">like <hi rend="italics">p</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s′B :: s′P : s′H,</cell><cell cols="1" rows="1" rend="align=left" role="data"> > or < 90° as B is like or unlike P</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s.P :: t′B : t′<hi rend="italics">b,</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">like B</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">P</cell><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : s.B :: t′P : t′<hi rend="italics">p,</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">like P</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">b</hi></cell><cell cols="1" rows="1" role="data">H</cell><cell cols="1" rows="1" rend="align=left" role="data"><hi rend="italics">r</hi> : t′<hi rend="italics">b</hi> :: t′<hi rend="italics">p</hi> : s′H,</cell><cell cols="1" rows="1" rend="align=left" role="data"> > or < 90° as <hi rend="italics">b</hi> is like or unlike <hi rend="italics">p</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">B</cell><cell cols="1" rows="1" rend="align=left" role="data">s.<hi rend="italics">p</hi> : <hi rend="italics">r</hi> :: s′<hi rend="italics">b</hi> : s′B</cell><cell cols="1" rows="1" rend="align=left" role="data">like <hi rend="italics">b</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">p</hi></cell><cell cols="1" rows="1" role="data">P</cell><cell cols="1" rows="1" rend="align=left" role="data">s.<hi rend="italics">b</hi> : <hi rend="italics">r</hi> :: s′<hi rend="italics">p</hi> : s′P</cell><cell cols="1" rows="1" rend="align=left" role="data">like <hi rend="italics">p</hi></cell></row></table><cb/></p><p>The following Propositions and Remarks, concerning Spherical Triangles, (selected and communicated by the reverend Nevil Maskelyne, D. D. Astronomer Royal, F. R. S.) will also render the calculation of them perspicuous, and free from ambiguity.</p><p>“1. A spherical triangle is equilateral, isoscelar, or scalene, according as it has its three angles all equal, or two of them equal, or all three unequal; and <hi rend="italics">vice versa.</hi></p><p>2. The greatest side is always opposite the greatest angle, and the smallest side opposite the smallest angle.</p><p>3. Any two sides taken together, are greater than the third.</p><p>4. If the three angles are all acute, or all right, or all obtuse; the three sides will be, accordingly, all less than 90°, or equal to 90°, or greater than 90°; and <hi rend="italics">vice versa.</hi></p><p>5. If from the three angles A, B, C, of a tri- <figure/> angle ABC, as poles, there be described, upon the surface of the sphere, three arches of a great circle DE, DF, FE, forming by their intersections a new spherical triangle DEF; each side of the new triangle will be the supplement of the angle at its pole; and each angle of the same triangle, will be the supplement of the side opposite to it in the triangle ABC. <cb/></p><p>6. In any triangle ABC, or <figure/> A<hi rend="italics">b</hi>C, right angled in A, 1st, The angles at the hypotenuse are always of the same kind as their opposite sides; 2dly, The hypotenuse is less or greater than a quadrant, according as the sides including the right angle are of the same or different kinds; that is to say, according as these same sides are either both acute or both obtuse, or as one is acute and the other obtuse. And, <hi rend="italics">vice versa,</hi> 1st, The sides including the right angle, are always of the same kind as their opposite angles: 2dly, The sides including the right angle will be of the same or different kinds, according as the hypotenuse is less or more than 90°; but one at least of them will be of 90°, if the hypotenuse is so.”</p></div1><div1 part="N" n="TRILATERAL" org="uniform" sample="complete" type="entry"><head>TRILATERAL</head><p>, three sided, a term applied to all figures of three sides, or triangles.</p></div1><div1 part="N" n="TRILLION" org="uniform" sample="complete" type="entry"><head>TRILLION</head><p>, in Arithmetic, the number of a million of billions, or a million of million of millions.</p></div1><div1 part="N" n="TRIMMERS" org="uniform" sample="complete" type="entry"><head>TRIMMERS</head><p>, in Architecture, pieces of timber framed at right-angles to the joists, against the ways for chimneys to support the hearths, and the well-holes for stairs.</p><p>TRINE <hi rend="italics">Dimension,</hi> or <hi rend="italics">threefold dimension,</hi> includes length, breadth, and thickness. The Trine dimension is peculiar to bodies or solids.</p><div2 part="N" n="Trine" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Trine</hi></head><p>, in Astrology, is the aspect or situation of one planet with respect to another, when they are dis- <pb n="625"/><cb/> tant 1/3 part of the circle, or 4 signs, or 120 degrees. It is also called trigon, and is denoted by the character ▵.</p><p>TRINITY <hi rend="italics">Sunday,</hi> is the next after Whitsunday; so called, because on that day was anciently held a festival (as it still continues to be in the Romish Church) in honour of the Holy Trinity.—The observance of this festival was first enjoined by the 6th canon of the council of Arles, in 1260; and John the 22d, who distinguished himself so much by his opinion concerning the beatific vision, it is said, fixed the office for this festival in 1334.</p></div2></div1><div1 part="N" n="TRINODA" org="uniform" sample="complete" type="entry"><head>TRINODA</head><p>, or <hi rend="smallcaps">Trinodia</hi> <hi rend="italics">Terræ,</hi> in some ancient writers, denotes the quantity of 3 perches of land.</p></div1><div1 part="N" n="TRINOMIAL" org="uniform" sample="complete" type="entry"><head>TRINOMIAL</head><p>, in Algebra, is a quantity, or a root, consisting of three parts or terms, connected together by the signs + or -: as <hi rend="italics">a</hi> + <hi rend="italics">b</hi> - <hi rend="italics">c,</hi> or <hi rend="italics">x</hi> + <hi rend="italics">y</hi> + <hi rend="italics">z.</hi></p></div1><div1 part="N" n="TRIO" org="uniform" sample="complete" type="entry"><head>TRIO</head><p>, in Music, a part of a concert in which three persons sing; or rather a musical composition consisting of 3 parts.—Trios are the finest kind of musical composition, and please most in concerts.</p></div1><div1 part="N" n="TRIOCTILE" org="uniform" sample="complete" type="entry"><head>TRIOCTILE</head><p>, in Astrology, an aspect or situation of two planets, with regard to the earth, when they are 3 octants, or 3/8 of a circle, which is 135°, distant from each other.—This aspect, which some call the <hi rend="italics">sesquiquadrans,</hi> is one of the new aspects added to the old ones by Kepler.</p></div1><div1 part="N" n="TRIONES" org="uniform" sample="complete" type="entry"><head>TRIONES</head><p>, in Astronomy, a sort of constellation, or assemblage of 7 stars in the Ursa Major, popularly called <hi rend="italics">Charles's Wain.</hi>—From the <hi rend="italics">Septem Triones</hi> the north pole takes the denomination <hi rend="italics">Septentrio.</hi></p></div1><div1 part="N" n="TRIPARTITION" org="uniform" sample="complete" type="entry"><head>TRIPARTITION</head><p>, is a division by 3, or the taking of the 3d part of any number or quantity.</p></div1><div1 part="N" n="TRIPLE" org="uniform" sample="complete" type="entry"><head>TRIPLE</head><p>, threefold. See <hi rend="smallcaps">Ratio</hi> and S<hi rend="smallcaps">UBTRIPLE.</hi></p><div2 part="N" n="Triple" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Triple</hi></head><p>, in Music, is one of the species of measure or time, and is taken from hence, that the whole, or half measure, is divisible into 3 equal parts, and is beaten accordingly.</p><p>TRIPLICATE <hi rend="italics">Ratio,</hi> is the ratio which cubes, or any similar solids, bear to each other; and is the cube of the simple ratio, or this twice multiplied by itself. Thus 1 to 8 is the Triplicate ratio of 1 to 2, and 1 to 27 Triplicate of 1 to 3.</p></div2></div1><div1 part="N" n="TRIPLICITY" org="uniform" sample="complete" type="entry"><head>TRIPLICITY</head><p>, or <hi rend="smallcaps">Trigon</hi>, with Astrologers, is a division of the 12 signs, according to the number of the 4 elements, earth, water, air, fire; each division consisting of 3 signs, making the earthly Triplicity, the watery Triphcity, the airy Triplicity, and the fiery Triplicity.</p><p>Triplicity is sometimes confounded with trine aspect; though they are, strictly speaking, very different things; as Triplicity is only used with regard to the signs, and trine with regard to the planets. The signs of Triplicity are those which are of the same nature, and not those that are in trine aspect: thus Aries, Leo, and Sagittary are signs of Triplicity, because those signs are, by these writers, all supposed fiery.</p><p>The signs in each of the four Triplicities, are as follow: <cb/> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">Earthly.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Watery.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Airy.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">Fiery.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><*>Taurus.</cell><cell cols="1" rows="1" role="data"><*>Cancer.</cell><cell cols="1" rows="1" role="data"><foreign xml:lang="greek">*p</foreign>Gemini.</cell><cell cols="1" rows="1" role="data">G Arics.</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/>Virgo.</cell><cell cols="1" rows="1" role="data"><figure/>Scorpio.</cell><cell cols="1" rows="1" role="data"><figure/>Libra.</cell><cell cols="1" rows="1" role="data">W Leo.</cell></row><row role="data"><cell cols="1" rows="1" role="data"><*>Capricorn.</cell><cell cols="1" rows="1" role="data"><figure/>Pisces.</cell><cell cols="1" rows="1" role="data"><figure/>Aquarius.</cell><cell cols="1" rows="1" role="data"><figure/>Sagittary.</cell></row></table></p><p>TRIS-<hi rend="smallcaps">Diapason</hi>, or <hi rend="italics">Triple Diapason Chord,</hi> in Music, is what is otherwise called a <hi rend="italics">triple eighth.</hi></p></div1><div1 part="N" n="TRISECTION" org="uniform" sample="complete" type="entry"><head>TRISECTION</head><p>, the dividing a thing into three equal parts. The term is chiefly used in Geometry, for the division of an angle into three equal parts. The <hi rend="italics">Trisection of an angle</hi> geometrically, is one of those great problems whose solution has been so much sought for by mathematicians, for 2000 years past; being, in this respect, on a footing with the famous quadrature of the circle, and the duplicature of the cube.</p><p>The Ancients Trisected an angle by means of the conic sections, and the book of Inclinations; and Pappus enumerates several ways of doing it, in the 4th book of his Mathematical Collections, prop. 31, 32, 33, 34, 35, &c. He farther observes, that the problem of Trisecting an angle, is a solid problem, or a problem of the 3d degree, being expressed by the resolution of a cubic equation, in which way it has been resolved by Vieta, and others of the Moderns. See his Angular Sections, with those of other authors, and the Trisection in particular by cubic equations, as in Guisne's Application of Algebra to Geometry, in l'Hospital's Conic Sections, and in Emerson's Trigonometry, book 1, sec. 4. The cubic equation by which the problem of Trisection is resolved, is as follows: Let <hi rend="italics">c</hi> denote the chord of a given arc, or angle, and <hi rend="italics">x</hi> the cord of the 3d part of the same, to the radius 1; then is , by the resolution of which cubic equation is found the value of <hi rend="italics">x,</hi> or the chord of the 3d part of the given arc or angle, whose chord is <hi rend="italics">c;</hi> and the resolution of this equation, by Cardan's rule, gives the chord</p></div1><div1 part="N" n="TRISPAST" org="uniform" sample="complete" type="entry"><head>TRISPAST</head><p>, or <hi rend="smallcaps">Trispaston</hi>, in Mechanics, a machine with 3 pulleys, or an assemblage of 3 pulleys, for raising great weights; being a lower species of the polyspaston.</p></div1><div1 part="N" n="TRITE" org="uniform" sample="complete" type="entry"><head>TRITE</head><p>, in Music, the 3d musical chord in the system of the Ancients.</p></div1><div1 part="N" n="TRITONE" org="uniform" sample="complete" type="entry"><head>TRITONE</head><p>, in Music, a false concord, consisting of three tones, or a greater third, and a greater tone. Its ratio or proportion in numbers, is that of 45 to 32.</p></div1><div1 part="N" n="TROCHILE" org="uniform" sample="complete" type="entry"><head>TROCHILE</head><p>, in Architecture, is that hollow ring, or cavity, which runs round a column next to the tore.</p></div1><div1 part="N" n="TROCHLEA" org="uniform" sample="complete" type="entry"><head>TROCHLEA</head><p>, in Mechanics, one of the mechanic powers, more usually called the pulley. <pb n="626"/><cb/></p></div1><div1 part="N" n="TROCHOID" org="uniform" sample="complete" type="entry"><head>TROCHOID</head><p>, in the Higher Geometry, a curve described by a point in any part of the radius of a wheel, during its rotatory and progressive motions. This is the same curve as what is more usually called the <hi rend="italics">Cycloid,</hi> where the construction and properties of it are shewn.</p><p>TRONE <hi rend="italics">Weight,</hi> was the same with what we now call <hi rend="italics">Troy Weight.</hi></p><p><hi rend="smallcaps">Trone</hi> <hi rend="italics">Pound,</hi> in Scotland, contains 20 Scotch ounces. Or because it is usual to allow one to the score, the Trone-pound is commonly 21 ounces.</p><p><hi rend="smallcaps">Trone</hi>-<hi rend="italics">Stone,</hi> in Scotland, according to Sir John Skene, contains 19 1/2 pounds.</p></div1><div1 part="N" n="TROPHY" org="uniform" sample="complete" type="entry"><head>TROPHY</head><p>, in Architecture, an ornament which represents the trunk of a tree, charged or encompassed all around with arms or military weapons, both offensive and defensive.</p></div1><div1 part="N" n="TROPICAL" org="uniform" sample="complete" type="entry"><head>TROPICAL</head><p>, something relating to the Tropics. As, <hi rend="smallcaps">Tropical</hi>-<hi rend="italics">Winds.</hi> See <hi rend="smallcaps">Wind</hi>, and <hi rend="smallcaps">Trade</hi>- <hi rend="italics">Winds.</hi></p><p><hi rend="smallcaps">Tropical</hi> <hi rend="italics">Year,</hi> the space of time during which the sun passes round from a tropic, till his return to it again. See <hi rend="smallcaps">Year.</hi></p></div1><div1 part="N" n="TROPICS" org="uniform" sample="complete" type="entry"><head>TROPICS</head><p>, in Astronomy, two fixed circles of the sphere, drawn parallel to the equator, through the solstitial points, or at such distance from the equator, as is equal to the sun's greatest recess or declination, or to the obliquity of the ecliptic.</p><p>Of the two Tropics, that on the north side of the equator, passes through the first point of Cancer, and is therefore called the <hi rend="italics">Tropic of Cancer.</hi> And the other on the south side, passing through the first point of Capricorn, is called the <hi rend="italics">Tropic of Capricorn.</hi></p><p>To determine the distance between the two Tropics, and thence the sun's greatest declination, or the obliquity of the ecliptic; observe the sun's meridian altitude, both in the summer and winter solstice, and subtract the latter from the former, so shall the remainder be the distance between the two Tropics; and the half of this will be the quantity of the greatest declination, or the obliquity of the ecliptic; the medium of which is now 23° 28′ nearly.</p><div2 part="N" n="Tropics" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tropics</hi></head><p>, in Geography, are two lesser circles of the globe, drawn parallel to the equator through the beginnings of Cancer and Capricorn, being in the planes of the celestial Tropics, and consequently at 23° 28′ distance either way from the equator.</p><p>TROY-<hi rend="italics">Weight,</hi> anciently called <hi rend="italics">Trone weight,</hi> is supposed to be taken from a weight of the same name in France, and that from the name of the town of Troyes there.</p><p>The original of all weights used in England, was a corn or grain of wheat gathered out of the middle of the ear, and, when well dried, 32 of them were to make one pennyweight, 20 pennyweights 1 ounce, and 12 ounces 1 pound Troy. Vide Statutes of 51 Hen. III; 31 Ed. I. and 12 Hen. VII.</p><p>But afterward it was thought sufficient to divide the said pennyweight into 24 equal parts, called grains, being the least weight now in common use; so that the divisions of Troy weight now are these: <cb/> <table><row role="data"><cell cols="1" rows="1" role="data">24 grains</cell><cell cols="1" rows="1" role="data">= 1 pennyweight</cell><cell cols="1" rows="1" role="data"><hi rend="italics">dwt.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">20 pennyweights</cell><cell cols="1" rows="1" role="data">= 1 ounce</cell><cell cols="1" rows="1" role="data"><hi rend="italics">oz.</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data">12 ounces</cell><cell cols="1" rows="1" role="data">= 1 pound</cell><cell cols="1" rows="1" role="data"><hi rend="italics">lb.</hi></cell></row></table></p><p>By Troy-weight are weighed jewels, gold, silver, and all liquors.</p></div2></div1><div1 part="N" n="TRUCKS" org="uniform" sample="complete" type="entry"><head>TRUCKS</head><p>, among Gunners, are the small wooden wheels fixed on the axletrees of gun carriages, especially those for ship service, to move them about by.</p><p>TRUE <hi rend="italics">Conjunction,</hi> in Astronomy. See <hi rend="italics">True</hi> C<hi rend="smallcaps">ONJUNCTION.</hi></p><p><hi rend="smallcaps">True</hi> <hi rend="italics">Place of a Planet or Star,</hi> is a point in the heavens shewn by a right line drawn from the centre of the earth, through the centre of the star or planet.</p></div1><div1 part="N" n="TRUMPET" org="uniform" sample="complete" type="entry"><head>TRUMPET</head><p>, <hi rend="italics">Listening or Hearing,</hi> is an instrument invented by Joseph Landini, to assist the hearing of persons dull of that faculty, or to assist us to hear persons who speak at a great distance.</p><p>Instruments of this kind are formed of tubes, with a wide mouth, and terminating in a small canal, which is applied to the ear. The form of these instruments evidently shews how they conduce to assist the hearing; for the greater quantity of the weak and languid pulses of the air being received and collected by the large end of the tube, are reflected to the small end, where they are collected and condensed; thence entering the ear in this condensed state, they strike the tympanum with a greater force than they could naturally have done from the ear alone.</p><p>Hence it appears, that a speaking Trumpet may be applied to the purpose of a hearing Trumpet, by turning the wide end towards the sound, and the narrow end to the ear.</p><p><hi rend="italics">Speaking</hi> <hi rend="smallcaps">Trumpet</hi>, is a tube of a considerable length, from 6 to 15 feet, used for speaking with to make the voice be heard to a greater distance.</p><p>This tube, which is made of tin, is straight throughout its length, but opening to a large aperture outwards, and the other end terminating in a proper shape and size to receive both the lips in the act of speaking, the speaker pushing his voice or the sound outwards, by which means it may be heard at the distance of a mile or more.</p><p>The invention of this Trumpet is held to be modern, and has been ascribed to Sir Samuel Moreland, who called it the <hi rend="italics">tuba stentorophonica,</hi> and in a work of the same name, published at London in 1671, that author gave an account of it, and of several experiments made with it. With one of these instruments, of 5 1/2 feet long, 21 inches diameter at the greater end, and 2 inches at the smaller, tried at Deal-Castle, the speaker was heard to the distance of 3 miles, the wind blowing from the shore.</p><p>But it seems that Kircher has a better title to the invention; for it is certain that he had such an instrument before ever Moreland thought of his. That author, in his Phonurgia Nova, published in 1673, says, that the tromba, published last year in England, he invented 24 years before, and published in his Mesurgia. He adds, that Jac. Albanus Ghibbisius and Fr. Eschinardus ascribe it to him; and that G. Schottus testifies of him, that he had such an instrument in his chamber in the <pb n="627"/><cb/> Roman college, with which he could call to, and receive answers from the porter.</p><p>But, considering how famous the tube or horn of Alexander the Great was, it is rather strange that the Moderns should pretend to the invention. With his stentorophonic horn or tube he used to speak to his army, and make himself be distinctly heard, it is said, 100 stadia or furlongs. A figure of this tube is preserved in the Vatican; and it is nearly the same as that now in use. See <hi rend="smallcaps">Stentorophonic.</hi></p><p>The principle of this instrument is obvious; for as sound is stronger in proportion to the density of the air, it follows that the voice in passing through a tube, or Trumpet, must be greatly augmented by the constant reflection and agitation of the air through the length of the tube, by which it is condensed, and its action on the external air greatly increased at its exit from the tube.</p><p>It has been found, that a man speaking through a tube of 4 feet long, may be understood at the distance of 500 geometrical paces; with a tube 16 2/3 feet, at the distance of 1800 paces; and with a tube 24 feet long, at more than 2500 paces.</p><p>Although some advantage in heightening the sound, both in speaking and hearing, be derived from the shape of the tube, and the width of the outer end, yet the effect depends chiefly upon its length. As to the form of it, some have asserted that the best figure is that which is formed by the revolution of a parabola about its axis; the mouth-piece being placed in the focus of the parabola, and consequently the sonorous rays reflected parallel to the axis of the tube. But Mr. Martin observes, that this parallel reflection is by no means essential to increasing the sound: on the contrary, it prevents the infinite number of reflections and reciprocations of sound, in which, according to Newton, its augmentation chiefly consists; the augmentation of the impetus of the pulses of air being proportional to the number of repercussions from the sides of the tube, and therefore to its length, and to such a figure as is most productive of them. Hence he infers, that the parabolic Trumpet is the most unfit of any for this purpose; and he endeavours to shew, that the logarithmic or logistic curve gives the best form, viz, by a revolution about its axis. Martin's Philos. Brit. vol. 2, pa. 248, 3d edit.</p><p>But Cassegrain is of opinion that an hyperbola, having the axis of the tube for an asymptote, is the best figure for this instrument. Musschenb. Intr. ad Phil. Nat. tom. 2, pa. 926, 4to.</p><p>For other constructions of Speaking Trumpets, by Mr. Conyers, see Philos. Trans. numb. 141, for 1678.</p><p>TRUNCATED <hi rend="italics">Pyramid</hi> or <hi rend="italics">Cone,</hi> is the frustum of one, being the part remaining at the bottom, after the top is cut off by a plane parallel to the base. See F<hi rend="smallcaps">RUSTUM.</hi></p></div1><div1 part="N" n="TRUNNIONS" org="uniform" sample="complete" type="entry"><head>TRUNNIONS</head><p>, of a piece of ordnance, are those knobs or short cylinders of metal on the sides, by which it rests on the cheeks of the carriage. <cb/></p><p><hi rend="smallcaps">Trunnion</hi>-<hi rend="italics">Ring,</hi> is the ring about a cannon, next before the Trunnions.</p></div1><div1 part="N" n="TSCHIRNHAUSEN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">TSCHIRNHAUSEN</surname> (<foreName full="yes"><hi rend="smallcaps">Ernfroy Walter</hi></foreName>)</persName></head><p>, an ingenious mathematician, lord of Killingswald and of Stolzenberg in Lusatia, where he was born in 1651. After having served as a volunteer in the army of Holland in 1672, he travelled into most parts of Europe, as England, Germany, Italy, France, &c. He went to Paris for the third time in 1682; where he communicated to the Academy of Sciences, the discovery of the curves called, from him, Tschirnhausen's Caustics; and the Academy in consequence elected the inventor one of its foreign members. On returning to Italy, he was desirous of perfecting the science of optics; for which purpose he established two glass-works, from whence resulted many new improvements in dioptrics and physics, particularly the noted burning-glass which he presented to the regent.—It was to him too that Saxony owed its porcelane manufactory.</p><p>Content with the enjoyment of literary same, Tschirnausen refused all other honours that were offered him. Learning was his sole delight. He searched out men of talents, and gave them encouragement. He was often at the expence of printing the useful works of other men, for the benefit of the public; and died, beloved and regretted, the 11th of September 1708.</p><p>Tschirnausen wrote, <hi rend="italics">De Medicina Mentis & Corporis,</hi> printed at Amsterdam in 1687. And the following memoirs were printed in the volumes of the Academy of Sciences.</p><p>1. Observations on Burning Glasses of 3 or 4 feet diameter: vol. 1699.</p><p>2. Observations on the Glass of a Telescope, convex on both sides, of 32 feet focal distance; 1700.</p><p>3. On the Radii of Curvature, with the finding the Tangents, Quadratures, and Rectisications of many curves; 1701.</p><p>4. On the Tangents of Mechanical Curves; 1702.</p><p>5. On a method of Quadratures; 1702.</p></div1><div1 part="N" n="TUBE" org="uniform" sample="complete" type="entry"><head>TUBE</head><p>, a pipe, conduit, or canal; being a hollow cylinder, either of metal, wood, glass, or other matter, for the conveyance of air, or water, &c.</p><p>The term is chiefly applied to those used in physics, astronomy, anatomy, &c. On other ordinary occasions, we more usually say <hi rend="italics">pipe.</hi></p><p>In the memoirs of the French Academy of Sciences, Varignon has given a treatise on the proportions for the diameters of tubes, to give any particular quantities of water. The result of his paper gives these two analogies, viz, that the diminutions of the velocity of water, occasioned by its friction against the sides of Tubes, are as the diameters; the Tubes being supposed equally long: and the quantities of water issuing out at the Tubes, are as the square roots of their diameters, deducting out of them the quantity that each is diminished.</p><div2 part="N" n="Tube" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tube</hi></head><p>, in Astronomy, is sometimes used for telescope; but more properly for that part of it into which the lenses are fitted, and by which they are directed and used. <pb n="628"/><cb/></p></div2></div1><div1 part="N" n="TUESDAY" org="uniform" sample="complete" type="entry"><head>TUESDAY</head><p>, the 3d day of the week, so called from Tuesco, one of the Saxon Gods, similar to Mars; for which reason the astronomical mark for this day of the week, is <figure/>.</p></div1><div1 part="N" n="TUMBREL" org="uniform" sample="complete" type="entry"><head>TUMBREL</head><p>, is a kind of carriage with two wheels, used either in Husbandry for dung, or in Artillery to carry the tools of the pioneers, &c, and sometimes likewise the money of an army.</p></div1><div1 part="N" n="TUN" org="uniform" sample="complete" type="entry"><head>TUN</head><p>, is a measure for liquids, as wine, oil, &c.</p><p>The English Tun contains 2 pipes, or 4 hogsheads, or 252 gallons.</p></div1><div1 part="N" n="TUNE" org="uniform" sample="complete" type="entry"><head>TUNE</head><p>, or <hi rend="smallcaps">Tone</hi>, in Music, is that property of sounds by which they come under the relation of acute and grave.</p><p>If two or more sounds be compared together in this relation, they are either equal or unequal in the degree of Tune: such as are equal, are called <hi rend="italics">unisons.</hi> The unequal constitute what are called <hi rend="italics">intervals,</hi> which are the differences of Tone between sounds.</p><p>Sonorous bodies are found to differ in Tone: 1st, According to the different kinds of matter; thus the sound of a piece of gold, is much graver than that of a piece of silver of the same shape and dimensions. 2d, According to the different quantities of the same matter in bodies of the same figure; as a solid sphere of brass of 1 foot diameter, sounds acuter than a sphere of brass of 2 feet diameter.</p><p>But the measures of Tone are only to be sought in the relations of the motions that are the cause of sound, which are most discernible in the vibration of chords. Now, in general, we find that in two chords, all things being equal, excepting the tension, the thickness, or the length, the Tones are different; which difference can only be in the velocity of their vibratory motions, by which they perform a different number of vibrations in the same time; as it is known that all the small vibrations of the same chord are performed in equal times. Now the frequenter or quicker those vibrations are, the more acute is the Tone; and the slower and fewer they are in the same space of time, by so much the more grave is the Tone. So that any given note of a Tune is made by one certain measure of velocity of vibrations, that is, such a certain number of vibrations of a chord or string, in such a certain space of time, constitutes a determinate Tone.</p><p>This theory is strongly supported by the best and latest writers on music, Holder, Malcolm, Smith, &c, both from reason and experience. Dr. Wallis, who owns it very reasonable, adds, that it is evident the degrees of acuteness are reciprocally as the lengths of the chords; though, he says, he will not positively affirm that the degrees of acuteness answer the number of vibrations, as their only true cause: but his diffidence arises from hence, that he doubts whether the thing has been sufficiently confirmed by experiment.</p><p>TUNNAGE. See <hi rend="smallcaps">Tonnage.</hi></p></div1><div1 part="N" n="TURN" org="uniform" sample="complete" type="entry"><head>TURN</head><p>, is used for a circular motion; in which sense it agrees with revolution.</p><div2 part="N" n="Turn" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Turn</hi></head><p>, in Clock or Watch-work, particularly denotes the revolution of a wheel or pinion.</p><p>In calculation, the number of Turns which the pi- <cb/> nion hath, is denoted in common arithmetic thus, 5) 60 (12, where the pinion 5, playing in a wheel of 60, moves round 12 times in one Turns of the wheel. Now by knowing the number of Turns which any pinion hath, in one Turn of the wheel it works in, you may easily find how many Turns a wheel or pinion has at a greater distance; as the contrat-wheel, crownwheel, &c, by multiplying together the quotients, and the number produced is the number of Turns, as in the example here annexed: the first of <table><row role="data"><cell cols="1" rows="1" role="data">5)</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" rend="align=right" role="data">(11</cell></row><row role="data"><cell cols="1" rows="1" role="data">5)</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" rend="align=right" role="data">( 9</cell></row><row role="data"><cell cols="1" rows="1" role="data">5)</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" rend="align=right" role="data">( 8</cell></row></table> these three numbers has 11 Turns, the next 9, and the last 8; if you multiply 11 by 9, it produces 99; that is, in one Turn of the wheel 55, there are 99 Turns of the second pinion 5, or the wheel 40, which runs concentrical or on the same arbor with the second pinion 5: and if you again multiply 99 by the last quotient 8, it produces 792, which is the number of Turns the third pinion 5 hath. See <hi rend="smallcaps">Clock</hi>-<hi rend="italics">work,</hi> and P<hi rend="smallcaps">INION.</hi></p><p>TURNING <hi rend="italics">to windward,</hi> in Sea Language, denotes that operation in sailing when a ship endeavours to make a progress against the direction of the wind, by a compound course, inclined to the place of her destination.—This method of navigation is otherwise called <hi rend="italics">plying to windward.</hi></p><p>TUSCAN <hi rend="italics">Order,</hi> in Architecture, is the first, the simplest, and the strongest or most massive of any. Its column has 7 diameters in height; and its capital, base, and entablement, have no ornaments, and but few mouldings.</p><p>TWELFTH-<hi rend="italics">Day,</hi> the festival of the Epiphany, or the manifestation of Christ to the Gentiles, so called, as being the Twelfth day, exclusive, from the nativity or Christmas-day; of course it falls always on the 6th day of January.</p></div2></div1><div1 part="N" n="TWILIGHT" org="uniform" sample="complete" type="entry"><head>TWILIGHT</head><p>, in Astronomy, is that faint light which is perceived before the sun-rising, and after sunsetting. The Twilight is occasioned by the earth's atmosphere refracting the rays of the sun, and reflecting them among its particles.</p><p>The depression of the sun below the horizon, at the beginning of the morning, and end of the evening Twilight, has been variously stated, at different seasons, and by different observers: by Alhazen it was observed to be 19°; by Tycho 17°; by Rothman 24°; by Stevinus 18°; by Cassini 15°; by Riccioli, at the time of the equinox in the morning 16°, in the evening 20° 1/2; in the summer solstice in the morning 21° 25′, and in the winter 17° 15′. Whence it appears that the cause of the Twilight is variable; but, on a medium, about 18° of the sun's depression will serve tolerably well for our latitude, for the beginning and end of Twilight, and according to which Dr. Long, (in his Astronomy, vol. 1, pa. 258) gives the following Table, of the duration of Twilight, in different latitudes, and for several different declinations of the sun. <pb n="629"/> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Latitude.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">0</cell><cell cols="1" rows="1" rend="colspan=2" role="data">10</cell><cell cols="1" rows="1" rend="colspan=2" role="data">20</cell><cell cols="1" rows="1" rend="colspan=2" role="data">30</cell><cell cols="1" rows="1" rend="colspan=2" role="data">40</cell><cell cols="1" rows="1" rend="colspan=2" role="data">45</cell><cell cols="1" rows="1" rend="colspan=2" role="data">50</cell><cell cols="1" rows="1" rend="colspan=2" role="data">52 1/2</cell><cell cols="1" rows="1" rend="colspan=2" role="data">55</cell><cell cols="1" rows="1" rend="colspan=2" role="data">60</cell><cell cols="1" rows="1" rend="colspan=2" role="data">65</cell><cell cols="1" rows="1" rend="colspan=2" role="data">70</cell><cell cols="1" rows="1" rend="colspan=2" role="data">75</cell><cell cols="1" rows="1" rend="colspan=2" role="data">80</cell><cell cols="1" rows="1" rend="colspan=2" role="data">85</cell><cell cols="1" rows="1" rend="colspan=2" role="data">90</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Θ</cell><cell cols="1" rows="1" role="data">En-</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" role="data">m</cell><cell cols="1" rows="1" role="data">h</cell><cell cols="1" rows="1" 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role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">d</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><foreign xml:lang="greek"><*></foreign></cell><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">w</cell><cell cols="1" rows="1" role="data">n</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">n</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"><*></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">n</cell><cell cols="1" rows="1" role="data">c</cell><cell cols="1" rows="1" role="data">n</cell></row></table><cb/> Where <hi rend="italics">c d</hi> signify that it is then continual day, <hi rend="italics">c n</hi> continual night, and <hi rend="italics">w n</hi> that the Twilight lasts the whole night. <hi rend="center"><hi rend="italics">Prob.—To find the Beginning or End of Twilight.</hi></hi></p><p>In this problem, there are given the sides of an oblique spherical trian- <figure/> gle, to find an angle; viz, given the side ZP the colatitude of the place; PΘ the codeclination, or polar distance; and ZΘ the zenith distance, which is always equal to 108°, viz, 90° from the zenith to the horizon, and 18° more for the sun's distance below the horizon. For example, suppose the place London in latitude 51° 32′, and the time the 1st of May, when the sun's declination is 15° 12′ north. Here then ZP = 38° 28′ the complement of 51° 32′, and PΘ = 74° 48′, the complement of 15° 12′. Then the calculation is as follows. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">PΘ =</cell><cell cols="1" rows="1" role="data">74° 48′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">PZ =</cell><cell cols="1" rows="1" role="data">38  28</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">PΘ - PZ =</cell><cell cols="1" rows="1" role="data">36  20</cell><cell cols="1" rows="1" rend="align=center" role="data">= D</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">ZΘ =</cell><cell cols="1" rows="1" role="data">108  00</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">ZΘ + D =</cell><cell cols="1" rows="1" role="data">144  20</cell><cell cols="1" rows="1" rend="align=center" role="data">72°</cell><cell cols="1" rows="1" role="data">10′ = (1/2) ―(ZΘ + D)</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2 )</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">ZΘ - D =</cell><cell cols="1" rows="1" role="data">71  40</cell><cell cols="1" rows="1" rend="align=center" role="data">35</cell><cell cols="1" rows="1" role="data">50 = (1/2) ―(Z○ - D)</cell></row></table> <table><row role="data"><cell cols="1" rows="1" role="data">Then,</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Co-ar. sin. polar dist.</cell><cell cols="1" rows="1" role="data">= 74°</cell><cell cols="1" rows="1" role="data">48′</cell><cell cols="1" rows="1" rend="align=right" role="data">0.01547</cell></row><row role="data"><cell cols="1" rows="1" role="data">Co-ar. sin. colat.</cell><cell cols="1" rows="1" role="data">= 38</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" rend="align=right" role="data">0.20617</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sine (1/2) ―(ZΘ + D)</cell><cell cols="1" rows="1" role="data">= 72</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">9.97861</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sine (1/2) ―(ZΘ - D)</cell><cell cols="1" rows="1" role="data">= 35</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" rend="align=right" role="data">9.76747</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sum of these four logs.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">19.96772</cell></row><row role="data"><cell cols="1" rows="1" role="data">Half sum gives</cell><cell cols="1" rows="1" role="data">74° (28 1/2)′</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">9.98386</cell></row></table></p><p>Which doubled gives 148 57 for the angle ZPO. <cb/></p><p>This 148° 57′ reduced to time, at the rate of 15° per hour, gives 9<hi rend="sup">h</hi> 55<hi rend="sup">m</hi> 48<hi rend="sup">s</hi>, either before or after noon; that is, the twilight begins at 2<hi rend="sup">h</hi> 4<hi rend="sup">m</hi> 12<hi rend="sup">s</hi> in the morning, and ends at 9<hi rend="sup">h</hi> 55<hi rend="sup">m</hi> 48<hi rend="sup">s</hi> in the evening on the given day at London.</p><p>TWINKLING <hi rend="italics">of the Stars,</hi> denotes that tremulous motion which is observed in the light proceeding from the sixed stars.</p><p>This Twinkling in the stars has been variously accounted for. Alhazen, a Moorish philosopher of the 12th century, considers refraction as the cause of this phenomenon.</p><p>Vitello, in his Optics, (composed before the year 1270) pa. 449, ascribes the Twinkling of the stars to the motion of the air, in which the light is refracted; and he observes, in confirmation of this hypothesis, that they Twinkle still more when they are viewed in water put into motion.</p><p>Dr. Hook (Microgr. pa. 231, &c) ascribes this phenomenon to the inconstant and unequal refraction of the rays of light, occasioned by the trembling motion of the air and interspersed vapours, in consequence of variable degrees of heat and cold in the air, producing corresponding variations in its density, and also of the action of the wind, which must cause the successive rays to fall upon the eye in different directions, and consequently upon different parts of the retina at different times, and also to hit and miss the pupil alternately; and this also is the reason, he says, why the limbs of the sun, moon, and planets appear to wave or dance.</p><p>These tremors of the air are manifest to the eye by the tremulous motion of shadows cast from high towers; and by looking at objects through the smoke of a chimney, or through steams of hot water, or at objects situated beyond hot sands, especially if the air be moved transversely over them. But when stars are seen through telescopes that have large apertures; they Twinkle but little, and sometimes not at all. For, as Newton has observed, (Opt. pa. 98) the rays of light which pass through different parts of the aperture, tremble each of them apart, and by means of their various, and contrary tremors, fall at one and the same <pb n="630"/><cb/> time upon different points in the bottom of the eye, and their trembling motions are too quick and confused to be separately perceived. And all these illuminated points constitute one broad lucid point, composed of those many trembling points confusedly and insensibly mixed with one another by very short and swift-tremors, and so cause the star to appear broader than it is, and without any trembling of the whole.</p><p>Dr. Jurin, in his Essay upon Distinct and Indistinct Vision, has recourse to Newton's hypothesis of fits of easy refraction and reflection for explaining the Twinkling of the stars: thus, he says, if the middle part of the image of a star be changed from light to dark, and the adjacent ring at the same time be changed from dark to light, as must happen from the least motion of the eye towards or from the star, this will occasion such an appearance as Twinkling.</p><p>Mr. Michell (Philos. Trans. vol. 57, pa. 262) supposes that the arrival of fewer or more rays at one time, especially from the smaller or more remote fixed stars, may make such an unequal impression on the eye, as may at least have some share in producing this effect: since it may be supposed that even a single particle of light is sufficient to make a sensible impression on the organs of sight; so that very few particles arriving at the eye in a second of time, perhaps not more than three or four, may be sufficient to make an object constantly visible. See <hi rend="smallcaps">Light.</hi></p><p>Hence, he says, it is not improbable that the number of the particles of light which enter the eye in a second of time, even from Syrius himself, may not exceed 3 or 4 thousand, and from stars of the 2d magnitude they may probably not exceed 100. Now the apparent increase and diminution of the light, which we observe in the Twinkling of the stars, seem to be repeated at intervals not very unequal, perhaps about 4 or 5 times in a second. He therefore thought it reasonable to suppose, that the inequalities which will naturally arise from the chance of the rays coming sometimes a little denser, and sometimes a little rarer, in so small a number of them, as must fall upon the eye in the 4th or 5th part of a second, may be sufficient to account for this appearance.</p><p>Since these observations were published however, Mr. Michell (as we are informed by Dr. Priestley in his Hist. of Light, pa. 495) has entertained some suspicion, that the unequal density of light does not contribute to this effect in so great a degree as he had imagined; especially as he has observed that even Venus does sometimes Twinkle. This he once observed her to do remarkably when she was about 6 degrees high, though Jupiter, which was then about 16 degrees high, and was sensibly less luminous, did not Twinkle at all. If, notwithstanding the great number of rays which doubtless come to the eye from such a surface as this planet presents, its appearance be liable to be affected in this manner, it must be owing to such undulations in the atmosphere, as will probably render the effect of every other cause altogether insensible.</p><p>Musschenbroek suspects (Introd. ad Phil. Nat. vol. 2, sect. 1741, pa. 707) that the Twinkling of the stars arises from some affection of the eye, as well as the <cb/> state of the atmosphere. For, says he, in Holland, when the weather is frosty, and the sky very clear, the stars Twinkle most manifestly to the naked eye, though not in telescopes; and since he does not suppose there is any great exhalation, or dancing of the vapour, at that time, he questions whether the vivacity of the light, affecting the eye, may not be concerned in the phenomenon.</p><p>But this philosopher might have satisfied himself with respect to this hypothesis, by looking at the stars near the zenith, when the light traverses but a small part of the atmosphere, and therefore might be expected to affect the eye most sensibly. For he would have found that they do not Twinkle near so much as they do near the horizon, when much more of their light is intercepted by the atmosphere.</p><p>Some astronomers have lately endeavoured to explain the Twinkling of the fixed stars, by the extreme minuteness of their apparent diameter; so that they suppose the sight of them is intercepted by every mote that floats in the air. To this purpose Dr. Long observes (Astron. vol. 1, pa. 170) that our air near the earth is so full of various kinds of particles, which are in continual motion, that some one or other of them is perpetually passing between us and any star we look at, which makes us every moment alternately see it and lose sight of it: and this Twinkling of the stars, he says, is greatest in those that are nearest the horizon, because they are viewed through a great quantity of thick air, where the intercepting particles are most numerous; whereas stars that are near the zenith do not Twinkle so much, because we do not look at them through so much thick air, and therefore the intercepting particles, being fewer, come less frequently before them. With respect to the planets, it is observed that, because they are much nearer to us than the stars, they have a sensible apparent magnitude, so that they are not covered by the small particles floating in the atmosphere, and therefore do not Twinkle, but shine with a steady light.</p><p>The fallacy of this hypothesis appears from the observation of Mr. Michell, that no object can hide a star from us that is not large enough to exceed the apparent diameter of the star, by the diameter of the pupil of the eye; so that if a star were even a mathematical point, or of no diameter, the interposing object must still-be equal in size to the pupil of the eye; and indeed it must be large enough to hide the star from both eyes at the same time.</p><p>The principal cause therefore of the Twinkling of the stars, is now acknowledged to be the unequal refraction of light, in consequence of inequalities and undulations in the atmosphere.</p><p>Besides a variation in the quantity of light, it may here be added, that a momentary change of colour has likewise been observed in some of the fixed stars. Mr. Melville (Edinb. Essays, vol. 2, pa. 81) says, that when one looks steadfastly at Sirius, or any bright star, not much elevated above the horizon, its colour seems not to be constantly white, but appears tinctured, at every Twinkling, with red and blue. Mr. Melville could not entirely satisfy himself as to the cause of this phe- <pb n="631"/><cb/> nomenon; observing that the separation of the colours by the refractive power of the atmosphere, is probably too small to be perceived. Mr. Michell's hypothesis above mentioned, though not adequate to the explication of the Twinkling of the stars, may pretty well account for this circumstance. For the red and blue rays being much fewer than those of the intermediate colours, and therefore much more liable to inequalities from the common effect of chance, a small excess or defect in either of them will make a very sensible difference in the colour of the stars.</p><p>TYCHONIC <hi rend="italics">System,</hi> or <hi rend="italics">Hypothesis,</hi> is an order or arrangement of the heavenly bodies, of an intermediate nature between the Copernican and Ptolomaic; and is so called from its inventor Tycho Brahe. See <hi rend="smallcaps">System.</hi></p></div1><div1 part="N" n="TYMPAN" org="uniform" sample="complete" type="entry"><head>TYMPAN</head><p>, or <hi rend="smallcaps">Tympanum</hi>, in Architecture, is the area of a pediment, being that part which is on a level with the naked of the frize. Or it is the space included between the three cornices of a triangular pediment, or the two cornices of a circular one.</p><p><hi rend="smallcaps">Tympan</hi> is also used for that part of a pedestal called the <hi rend="italics">trunk</hi> or <hi rend="italics">dye.</hi></p><div2 part="N" n="Tympan" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tympan</hi></head><p>, among Joiners, is also applied to the pannels of doors. <cb/></p><p><hi rend="smallcaps">Tympan</hi> <hi rend="italics">of an Arch,</hi> is a triangular space or table in the corners of sides of an arch, usually hollowed and enriched, sometimes with branches of laurel, olive-tree, or oak; or with trophies, &c; sometimes with flying figures, as fame, &c; or sitting figures, as the cardinal virtues.</p></div2><div2 part="N" n="Tympan" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Tympan</hi></head><p>, in Mechanics, is a kind of wheel placed round an axis, or cylindrical beam, on the top of which are two levers, or fixed staves, for more easily turning the axis about, in order to raise a weight. The Tympanum is much the same with the peritrochium; but that the cylinder of the axis of the peritrochium is much shorter and less than the cylinder of the Tympanum.</p><p><hi rend="smallcaps">Tympanum</hi> of a machine, is also used for a hollow wheel, in which people or animals walk, to turn it; such as that of some cranes, calenders, &c.</p></div2></div1><div1 part="N" n="TYR" org="uniform" sample="complete" type="entry"><head>TYR</head><p>, in the Ethiopian Calendar, the name of the 5th month of the Ethiopian year. It commences on the 25th of December of the Julian year.</p></div1><div1 part="N" n="TYSHAS" org="uniform" sample="complete" type="entry"><head>TYSHAS</head><p>, among the Ethiopians, the name of the 4th month of their year, commencing the 27th of November in the Julian year. <hi rend="center">U <hi rend="smallcaps">AND</hi> V.</hi><cb/></p><p>V <hi rend="smallcaps">Is</hi> a numeral letter, in the Roman numeration, denoting 5 or five. And with a dash over the top thus ―V, it denoted 5000.</p></div1><div1 part="N" n="VACUUM" org="uniform" sample="complete" type="entry"><head>VACUUM</head><p>, in Physics, a space empty or devoid of all matter.</p><p>Whether there be any such thing in nature as an absolute Vacuum; or whether the universe be completely full, and there be an absolute plenum; is a question that has been agitated by the philosophers of all ages.</p><p>The Ancients, in their controversies, distinguished two kinds; a <hi rend="italics">Vacuum coacervatum,</hi> and a <hi rend="italics">Vacuum interspersum,</hi> or <hi rend="italics">disseminatum.</hi></p><p><hi rend="smallcaps">Vacuum</hi> <hi rend="italics">Coacervatum,</hi> is conceived as a considerably large space destitute of matter; such, for instance, as there would be, should God annihilate all the air, and other bodies, within the walls of a chamber.</p><p>The existence of such a Vacuum is maintained by the Pythagoreans, Epicureans, and the Atomists or Corpuscularians; most of whom assert, that such a Vacuum actually exists without the limits of the sensible world. But the modern Corpuscularians, who hold a <hi rend="italics">Vacuum coacervatum,</hi> deny that appellation; as conceiving that <cb/> such a Vacuum must be infinite, eternal, and uncreated.</p><p>According then to the later philosophers, there is no Vacuum coacervatum without the bounds of the sensible world; nor would there be any other Vacuum, provided God should annihilate divers contiguous bodies, than what amounts to a mere privation, or nothing; the dimensions of such a space, which the Ancients held to be real, being by these held to be mere negations; that is, in such a place there is so much length, breadth, and depth wanting, as a body must have to fill it. To suppose then, that when all the matter in a chamber is annihilated, there should yet be real dimensions, is to suppose, say they, corporeal dimensions without body; which is absurd.</p><p>The Cartesians however deny any <hi rend="italics">Vacuum coacervatum</hi> at all, and assert that if God should immediately annihilate all the matter, for example in a chamber, and prevent the ingress of any other matter, the consequence would be, that the walls would become contiguous, and include no space at all. They add, that if there be no matter in a chamber, the walls cannot be conceived otherwise than as contiguous; those things <pb n="632"/><cb/> being said to be contiguous, between which there is not any thing intermediate: but if there be no body between, there is, say they, no extension between; extension and body being the same thing: and if there be no extension between, then the walls are contiguous; and where is the Vacuum?—But this reasoning, or rather quibbling, is founded on the mistake, that body and extension are the same thing.</p><p><hi rend="smallcaps">Vacuum</hi> <hi rend="italics">Disseminatum,</hi> or <hi rend="italics">Interspersum,</hi> is that supposed to be naturally interspersed in and among bodies, in the interstices between different bodies, and in the pores of the same body.</p><p>It is this kind of Vacuum which is chiefly contested among the modern philosophers; the Corpuscularians strenuously asserting it; and the Peripatetics and Cartesians as tenaciously denying it. See <hi rend="smallcaps">Cartesian</hi> and <hi rend="smallcaps">Leienitzian.</hi></p><p>The great argument urged by the Peripatetics against a Vacuum interspersum, is, that there are divers bodies frequently seen to move contrary to their own nature and inclination; and that for no other apparent reason, but to avoid a Vacuum: whence they conclude, that nature abhors a Vacuum; and give us a new class of motions ascribed to the <hi rend="italics">fuga vacui</hi> or nature's flying a Vacuum. Such, they say, is the rise of water in a syringe, upon the drawing up of the piston; and such is the ascent of water in pumps, and the swelling of the flesh in a cupping glass, &c.—But since the weight, elasticity, &c, of the air have been ascertained by sure experiments, those motions and effects are universally, and justly, ascribed to the gravity and pressure of the atmosphere.</p><p>The Cartesians deny, not only the actual existence, but even the possibility of a Vacuum; and that on this principle, that extension being the essence of matter, or body, wherever extension is, there is matter; but mere space, or vacuity, is supposed to be extended; therefore it is material. Whoever asserts an empty space, say they, conceives dimensions in that space, i. e. he conceives an extended substance in it; and therefore he denies a Vacuum, at the same time that he admits it.—But Descartes, if we may believe some accounts, rejected a Vacuum from a complaisance to the taste which prevailed in his time, against his own first sentiments; and among his familiar friends he used to call his system his philosophical romance.</p><p>On the other hand, the corpuscular authors prove, not only the possibility, but the actual existence, of a Vacuum, from divers considerations; particularly from the consideration of motion in general; and that of the planets, comets, &c, in particular; as also from the fall of bodies; from the vibration of pendulums; from rarefaction and condensation; from the different specific gravities of bodies; and from the divisibility of matter into parts.</p><p>1. First, there could be no linear or progressive motion without a Vacuum; for if all space were full of matter, no body could be moved out of its place, for want of another place unoccupied, to move into. And this argument was stated even by Lucretius.</p><p>2. The motions of the planets and comets also prove a Vacuum. Thus, Newton argues, “that there is no such fluid medium as æther,” (to fill up the porous parts of all sensible bodies, and so make a plenum), <cb/> seems probable; because the planets and comets proceed with so regular and lasting a motion, through the celestial spaces; for hence it appears that those celestial spaces are void of all sensible resistance, and consequently of all sensible matter. Consequently if the celestial regions were as dense as water, or as quicksilver, they would resist almost as much as water or quicksilver; but if they were perfectly dense, without any interspersed vacuity, though the matter were ever so fluid and subtle, they would resist more than quicksilver does: a perfectly solid globe, in such a medium, would lose above half its motion, in moving 3 lengths of its diameter; and a globe not perfectly solid, such as the bodies of the planets and comets are, would be stopped still sooner. Therefore, that the motion of the planets and comets may be regular, and lasting, it is necessary that the celestial spaces be void of all matter; except perhaps some few and much rarefied effluvia of the planets and comets, and the passing rays of light.”</p><p>3. The same great author also deduces a Vacuum from the consideration of the weights of bodies; thus: “All bodies about the earth gravitate towards it; and the weights of all bodies, equally distant from the earth's centre, are as the quantities of matter in those bodies. If the æther therefore, or any other subtile matter, were altogether destitute of gravity, or did gravitate less than in proportion to the quantity of its matter; because (as Aristotle, Descartes, and others, argue) it differs from other bodies only in the form of matter; the same body might, by the change of its form, gradually be converted into a body of the same constitution with those which gravitate most in proportion to the quantity of matter: and, on the other hand, the heaviest bodies might gradually lose their gravity, by gradually changing their form; and so the weights would depend upon the forms of bodies, and might be changed with them; which is contrary to all experiment.”</p><p>4. The descent of bodies proves, that all space is not equally full; for the same author goes on, “If all spaces were equally full, the specific gravity of that fluid with which the region of the air would, in that case, be filled, would not be less than the specific gravity of quicksilver or gold, or any other the most dense body; and therefore neither gold, nor any other body, could descend in it. For bodies do not descend in a fluid, unless that fluid be specifically lighter than the body. But by the air-pump we can exhaust a vessel. till even a feather shall fall with a velocity equal to that of gold in the open air; and therefore the medium through which this feather falls, must be much rarer than that through which the gold falls in the other case. The quantity of matter therefore in a given space may be diminished by rarefaction: and why may it not be diminished ad infinitum? Add, that we conceive the solid particles of all bodies to be of the same density; and that they are only rarefiable by means of their pores; and hence a Vacuum evidently follows.”</p><p>5. “That there is a Vacuum, is evident too from the vibrations of pendulums: for since those bodies, in places out of which the air is exhausted, meet with no resistance to retard their motion, or shorten their vibrations; it is evident that there is no sensible matter in those spaces, or in the occult pores of those bodies.” <pb n="633"/><cb/></p><p>6. That there are interspersed vacuities, appears from matter's being actually divided into parts, and from the figures of those parts; for, on supposition of an absolute plenum, we do not conceive how any part of matter could be actually divided from that next adjoining, any more than it is possible to divide actually the parts of absolute space from one another: for by the actual division of the parts of a continuum from one another, we conceive nothing else understood, but the placing of those parts at a distance from one another, which in the continuum were at no distance from one another: but such divisions between the parts of matter must imply vacuities between them.</p><p>7. As for the figures of the parts of bodies, upon the supposition of a plenum, they must either be all rectilinear, or all concavo-convex; otherwise they would not adequately sill space; which we do not find to be true in fact.</p><p>8. The denying a Vacuum supposes what it is impossible for any one to prove to be true, viz, that the material world has no limits.</p><p>However, we are told by some, that it is impossible to conceive a Vacuum. But this surely must proceed from their having imbibed Descartes's doctrine, that the essence of body is constituted by extension; as it would be contradictory to suppose space without extension. To suppose that there are fluids penetrating all bodies and replenishing space, which neither resist nor act upon bodies, merely in order to avoid admitting a Vacuum, is feigning two sorts of matter, without any necessity or foundation; or is tacitly giving up the question.</p><p>Since then the essence of matter does not consist in extension, but in solidity, or impenetrability, the universe may be said to consist of solid bodies moving in a Vacuum: nor need we at all fear, lest the phenomena of nature, most of which are plausibly accounted for from a plenum, should become inexplicable when the plenitude is set aside. The principal ones, such as the tides; the suspension of the mercury in the barometer; the motion of the heavenly bodies, and of light, &c, are more easily and satisfactorily accounted for from other principles.</p><p><hi rend="smallcaps">Vacuum</hi> <hi rend="italics">Boileanum,</hi> is used to express that approach to a real Vacuum, which we arrive at by means of the air-pump. Thus, any thing put in a receiver so exhausted, is said to be put <hi rend="italics">in vacuo:</hi> and thus most of the experiments with the air-pump are said to be performed <hi rend="italics">in vacuo,</hi> or <hi rend="italics">in vacuo Boileano.</hi></p><p>Some of the principal phenomena observed of bodies in vacuo, are; that the heaviest and lightest bodies, as a guinea and a feather, fall here with equal velocity: —that fruits, as grapes, cherries, peaches, apples, &c, kept for any time in vacuo, retain their nature, freshness, colour, &c, and those withered in the open air recover their plumpness in vacuo:—all light and fire become immediately extinct in vacuo:—little or no sound is heard from a bell rung in vacuo:—a bladder half full of air, will distend the bladder, and lift up 40 pound weight in vacuo:—most animals soon expire in vacuo.</p><p>By experiments made in 1704, Dr. Derham found that animals which have two ventricals, and no foramen ovale, as birds, dogs, cats, mice, &c, die in less than half a minute; counting from the first exsuction: a <cb/> mole died in one minute; a bat lived 7 or 8. Insects, as wasps, bees, grasshoppers, &c, seemed dead in two minutes; but after being lest in vacuo 24 hours, they came to life again in the open air: snails continued 24 hours in vacuo, without appearing much incommoded. —Seeds planted in vacuo do not grow: Small beer dies, and loses all its taste, in vacuo: And air rushing through mercury into a Vacuum, throws the mercury in a kind of shower upon the receiver, and produces a great light in a dark room.</p><p>The air-pump can never produce a perfect Vacuum; as is evident from its structure, and the manner of its working: in effect, every exsuction only takes away a part of the air; so that there is still some left after any finite number of exsuctions. For the air-pump has no longer any effect but while the spring of the air remaining in the receiver is able to lift up the valves; and when the rarefaction is come to that degree, you can come no nearer to a Vacuum; unless perhaps the air valves can be opened mechanically, independent of the spring of the air, as it is said they are in some new improved air-pumps.</p><p><hi rend="italics">Torricellian</hi> <hi rend="smallcaps">Vacuum</hi>, is that made in the barometer tube, between the upper end and the top of the mercury. This is perhaps never a perfect and entire Vacuum; as all fluids are found to yield or to rise in elastic vapours, on the removal of the pressure of the atmosphere. See <hi rend="smallcaps">Torricellian</hi>, and <hi rend="smallcaps">Barometer.</hi></p></div1><div1 part="N" n="VALVE" org="uniform" sample="complete" type="entry"><head>VALVE</head><p>, in Hydraulics, Pneumatics, &c, is a kind of lid or cover to a tube or vessel, contrived to open one way; but which, the more forcibly it is pressed the other way, the closer it shuts the aperture: so that it either admits the entrance of a fluid into the tube, or vessel, and prevents its return; or permits it to escape, and prevents its re-entrance.</p><p>Valves are of great use in the air-pump, and other wind machines; in which they are usually made of pieces of bladder. In hydraulic engines, as the emboli of pumps, they are mostly of strong leather, of a round figure, and fitted to shut the apertures of the barrels or pipes. Sometimes they are made of two round pieces of leather enclosed between two others of brass; having divers perforations, which are covered with another piece of brass, moveable upwards and downwards, on a kind of axis, which goes through the middle of them all. Sometimes they are made of brass, covered over with leather, and furnished with a fine spring, which gives way upon a force applied against it; but upon the ceasing of that, returns the Valve over the aperture. See <hi rend="smallcaps">Pump.</hi> See also Desaguliers' Exper. Philos. vol. 2, p. 156, and p. 180.</p></div1><div1 part="N" n="VANE" org="uniform" sample="complete" type="entry"><head>VANE</head><p>, in a ship, &c, a thin slip of some kind of matter, placed on high in the open air, turning easily round on an axis or spindle, and veered about by the wind, to shew its direction or course.</p><div2 part="N" n="Vanes" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Vanes</hi></head><p>, in Mathematical or Philosophical Instruments, are sights made to slide and move upon crossstaves, fore-staves, quadrants, &c.</p></div2></div1><div1 part="N" n="VAPOUR" org="uniform" sample="complete" type="entry"><head>VAPOUR</head><p>, in Meteorology, a watery exhalation raised up either by the heat of the sun, or any other heat, as fire, &c. Vapour is considered as a thin vesicle of water, or other humid matter, filled or inflated with air; which, being rarefied to a certain degree by the action of heat, ascends to some height in the <pb n="634"/><cb/> atmosphere, where it is suspended, till it returns in form of rain, snow, or the like. An assemblage of a number of particles or vesicles of vapour, constitutes what is called a cloud.</p><p>Some use the term Vapour indifferently, for all fumes emitted, either from moist bodies, as fluids of any kind; or from dry bodies, as sulphur, &c. But Newton, and other authors, better distinguish between humid and dry fumes, calling the latter <hi rend="italics">exhalations.</hi></p><p><hi rend="italics">For the manner in which Vapours are raised, and again precipitated,</hi> see <hi rend="smallcaps">Cloud, Dew, Rain, Barometer</hi>, and particularly <hi rend="smallcaps">Evaporation.</hi></p><p>It may here be added, with respect to the principles of solution adopted to account for evaporation, and largely illustrated under that article, that Dr. Halley, about the beginning of the present century, seems to have been acquainted with the solvent power of air on water; for he says, that supposing the earth to be covered with water, and the sun to move diurnally round it, the air would of itself imbibe a certain quantity of aqueous Vapours, and retain them like salts dissolved in water; and that the air warmed by the sun would sustain a greater proportion of Vapours, as warm water will hold more dissolved salts; which would be discharged in dews, similar to the precipitation of salts on the cooling of liquors. Philos. Trans. Abr. vol. 2, p. 127.</p><p>Mr. Eeles, in 1755, endeavoured to account for the ascent of Vapour and exhalation, and their suspension in the atmosphere, by means of the electric fire. The sun, he acknowledges, is the great agent in detaching Vapour and exhalations from their masses, whether he acts immediately by himself, or by his rendering the electric fire more active in its vibrations: but their subsequent ascent he attributes entirely to their being rendered specifically lighter than the lower air, by their conjunction with electrical fire: each particle of Vapour, with the electrical fluid that surrounds it, occupying a greater space than the same weight of air. Mr. Eeles also endeavours to shew, that the ascent and descent of Vapour, attended by this fire, are the cause of all the winds, and that they furnish a satisfactory solution of the general phenomena of the weather and barometer. Philos. Trans. vol. 49, pa. 124.</p><p>Dr. Darwin, in 1757, published remarks on the theory of Mr. Eeles, with a view of confuting it; and attempting to account for the ascent of Vapours, by considering the power of expansion which the constituent parts of some bodies acquire by heat, and also that some bodies have a greater affinity to heat, or acquire it sooner, and retain it longer, than others. On these principles, he thinks, it is easily understood how water, whose parts appear from the æolipile to be capable of immeasurable expansion, should by heat alone become specifically lighter than the common atmosphere. A small degree of heat is sufficient to detach or raise the Vapour of water from the mass to which it belongs; and the rays of the sun communicate heat only to those bodies by which they are refracted, reflected, or obstructed, whence, by their impulse, a motion or vibration is caused in the parts of such bodies. Hence he infers, that the sphericles of Vapour will, by refracting the solar rays, acquire a constant heat, <cb/> though the surrounding atmosphere remain cold. If it be asked, how clouds are supported in the absence of the sun? It must be remembered, that large masses of Vapour must for a considerable time retain much of the heat they have acquired in the day; at the same time reflecting how small a quantity of heat was necessary to raise them, and that doubtless even a less will be sufficient to support them; as from the diminished pressure of the atmosphere at a given height, a less power may be able to continue them in their present state of rarefaction; and lastly, that clouds of particular shapes will be sustained or elevated by the motion they acquire from winds. Philos. Trans. vol. 50, p. 246.</p><p><hi rend="italics">For the Effect of Vapour in the Eormation of Springs,</hi> &c, see <hi rend="smallcaps">Spring</hi>, and <hi rend="smallcaps">River.</hi></p><p>The quantity of Vapour raised from the sea by the warmth of the sun, must be far greater than is commonly imagined. Dr. Halley has attempted to estimate it. For the result of his calculations, see <hi rend="smallcaps">Evaporation.</hi></p></div1><div1 part="N" n="VARIABLE" org="uniform" sample="complete" type="entry"><head>VARIABLE</head><p>, in Geometry and Analytics, is a term applied by mathematicians, to such quantities as are considered in a Variable or changeable state, either increasing or decreasing. Thus, the abscisses and ordinates of an ellipsis, or other curve line, are Variable quantities; because these vary or change their magnitude together, the one at the same time with the other. But some quantities may be Variable by themselves alone, or while those connected with them are constant: as the abscisses of a parallelogram, whose ordinates may be considered as all equal, and therefore constant. Also the diameter of a circle, and the parameter of a conic section, are <hi rend="italics">constant,</hi> while their abscisses are <hi rend="italics">Variable.</hi></p><p>Variable quantities are usually denoted by the last letters of the alphabet, <hi rend="italics">z, y, x,</hi> &c; while the constant ones are denoted by the leading letters, <hi rend="italics">a, b, c,</hi> &c.</p><p>Some authors, instead of <hi rend="italics">Variable</hi> and <hi rend="italics">constant</hi> quantities, use the terms <hi rend="italics">fluent</hi> and <hi rend="italics">stable</hi> quantities.</p><p>The indefinitely small quantity by which a Variable quantity is continually increased or decreased, in very small portions of time, is called the <hi rend="italics">differential,</hi> or <hi rend="italics">increment</hi> or <hi rend="italics">decrement.</hi> And the rate of its increase or decrease at any point, is called its <hi rend="italics">fluxion;</hi> while the Variable quantity itself is called the <hi rend="italics">fluent.</hi> And the calculation of these, is the subject of the new <hi rend="italics">Methodus Differentialis,</hi> or <hi rend="italics">Doctrine of Fluxions.</hi></p></div1><div1 part="N" n="VARENIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">VARENIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Bernard</hi></foreName>)</persName></head><p>, a learned Dutch geographer and physician, of the last century, who was author of the best mathematical treatise on Geography, intitled, <hi rend="italics">Geographia Universalis,</hi> in qua affectiones generalis Telluris explicantur. This excellent work has been translated into all languages, and was honoured by an edition, with improvements, by Sir Isaac Newton, for the use of his academical students at Cambridge.</p></div1><div1 part="N" n="VARIATION" org="uniform" sample="complete" type="entry"><head>VARIATION</head><p>, <hi rend="italics">of Quantities,</hi> in Algebra. See <hi rend="smallcaps">Changes</hi>, and <hi rend="smallcaps">Combination.</hi></p><div2 part="N" n="Variation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Variation</hi></head><p>, in Astronomy.—<hi rend="italics">The Variation of the Moon,</hi> called by Bulliald, the <hi rend="italics">Reflection of her Light,</hi> is the third inequality observed in the moon's motion; by which, when out of the quadratures, her true place differs from her place twice equated. See <hi rend="smallcaps">Place, Equation</hi>, &c.</p><p>Newton makes the moon's variation to arise partly from the form of her orbit, which is an ellipsis; and <pb n="635"/><cb/> partly from the inequality of the spaces, which the moon describes in equal times, by a radius drawn to the earth.</p><p><hi rend="italics">To find the Greatest Variation.</hi> Observe the moon's longitude in the octants; and to the time of observation compute the moon's place twice equated; then the difference between the computed and observed place, is the greatest Variation.</p><p>Tycho makes the greatest Variation 40′ 30″; and Kepler makes it 51′ 49″.—But Newton makes the greatest Variation, at a mean distance between the sun and the earth, to be 35′ 10″: at the other distances, the greatest Variation is in a ratio compounded of the duplicate ratio of the times of the moon's synodical revolution directly, and the triplicate ratio of the distance of the sun from the earth inversely. And therefore in the sun's apogee, the greatest Variation is 33′ 14″, and in his perigee 37′ 11″; provided that the eccentricity of the sun be to the transverse semidiameter of the orbis magnus, as 16 15/16 to 1000. Or, taking the mean motions of the moon from the sun, as they are stated in Dr. Halley's tables, then the greatest Variation at the mean distance of the earth from the sun will be 35′ 7″, in the apogee of the sun 33′ 27″, and in his perigee 36′ 51″. Philos. Nat. Princ. pr. 29, lib. 3.</p></div2><div2 part="N" n="Variation" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Variation</hi></head><p>, in Geography, Navigation, &c, a term applied to the deviation of the magnetic needle, or compass, from the true north point, either towards the east or west; called also the <hi rend="italics">declination.</hi> Or the Variation of the compass is properly defined, the angle which a magnetic needle, suspended at liberty, makes with the meridian line on an horizontal plane; or an arch of the horizon, comprehended between the true and the magnetic meridians.</p><p>In the sea-language, the Variation is usually called <hi rend="italics">north-easting,</hi> or <hi rend="italics">north-westing.</hi></p><p>All magnetic bodies are found to range themselves, in some sort, according to the meridian; but they seldom agree precisely with it: in one place they decline, from the north toward the east, in another toward the west; and that too differently at different times.</p><p>The Variation of the compass could not long remain a secret, after the invention of the compass itself: accordingly Ferdinand, the son of Columbus, in his life written in Spanish, and printed in Italian at Venice in 1571, asserts, that his father observed it on the 14th of September 1492: though others seem to attribute the discovery of it to Sebastian Cabat, a Venetian, employed in the service of our king Henry VII, about the year 1500.—It now appears however, that this Variation or declination of the needle was known even some centuries earlier, though it does not appear that the use of the needle itself in navigation was then known. For it seems there is in the library of the university of Leyden, a small manuscript tract on the Magnet, in Latin, written by one Peter Adsiger, bearing date the 8th of August 1269; in which the declination of the needle is particularly mentioned. Mr. Cavallo has printed the chief part of this letter in the Supplement to his Treatise on Magnetism, with a translation; and I think it is to be wished he had printed the whole of so curious a paper. The curiosity of this letter, says Mr. Cavallo, consists in its containing almost all that <cb/> is at present known of the subject, at least the most remarkable parts of it, mixed however with a good deal of absurdity. The laws of magnetic attraction, and of the communication of that power to iron, the directive property of the natural magnet, as well as of the iron that has been touched by it, and even the declination of the magnetic needle, are clearly and unequivocally mentioned in it.</p><p>As this Variation differs in different places, Gonzales d'Oviedi found there was none at the Azores; from whence some geographers thought fit in their maps to make the first meridian pass through one of these islands: it not being then known that the Variation altered in time. See <hi rend="smallcaps">Magnet;</hi> also Gilbert De Magnete, Lond. 1600, p. 4 and 5; or Purchas's Pilgrims, Lond. 1625, book 2, sect. 1.</p><p>Various are the hypotheses that have been framed to account for this extraordinary phenomenon: we shall only notice some of the latter, and more probable: just premising, that Robert Norman, the inventor of the Dipping needle, disputes against Cortes's notion, that the Variation was caused by a point in the heavens; contending that it should be sought for in the earth, and proposes how to discover its place.</p><p>The first is that of Gilbert (De Magnete, lib. 4, p. 151 &c), which is followed by Cabeus, &c. This notion is, that it is the earth, or land, that draws the needle out of its meridian direction: and hence they argue, that the needle varied more or less, as it was more or less distant from any great continent; and consequently that if it were placed in the middle of an ocean, equally distant from equal tracts of land on each side, eastward and westward, it would not decline either to the one or the other, but point exactly north and south. Thus, say they, in the Azores islands, which are equally distant from Africa on the east, and America on the west, there is no Variation: but as you sail from thence towards Africa, the needle begins to decline toward the east, and that still more and more till you reach the shore. If you proceed still farther eastward, the declination gradually diminishes again, by reason of the land left behind on the west, which continues to draw the needle. The same holds till you arrive at a place where the tracts of land on each side are equal; and there again the Variation will be nothing. But the misfortune is, the law does not hold universally; for multitudes of observations of the Variation, in different parts, made and collected by Dr. Halley, overturn the whole theory.</p><p>Others therefore have recourse to the frame and compages of the earth, considered as interspersed with rocks and shelves, which being generally found to run towards the polar regions, the needle comes to have a general tendency that way; but it seldom happens that their direction is exactly in the meridian, and the needle has consequently, for the most part, some Variation.</p><p>Others hold that divers parts of the earth have different degrees of the magnetic virtue, as some are more intermixed with heterogeneous matters, which prevent the free action or effect of it, than others are.</p><p>Others again ascribe all to magnetic rocks and iron mines, which, affording more of the magnetic matter than other parts, draw the needle more. <pb n="636"/><cb/></p><p>Lastly, others imagine that earthquakes, or high tides, have disturbed and dislocated several considerable parts of the earth, and so changed the magnetic axis of the globe, which was originally the same with the axis of the earth itself.</p><p>But none of these theories can be the true one; for still that great phenomenon, the <hi rend="italics">Variation of the Variation,</hi> i. e. the continual change of the declination, in one and the same place, is not accountable for, on any of these foundations, nor is it even consistent with them.</p><p>Doctor Hook communicated to the Royal Society, in 1674, a theory of the Variation; the substance of which is, that the magnet has its peculiar pole, distant 10 degrees from the pole of the earth, about which it moves, so as to make a revolution in 370 years: whence the Variation, he says, has altered of late about 10 or 11 minutes every year, and will probably <cb/> so continue to do for some time, when it will begin to proceed slower and slower, till at length it become stationary and retrograde, and so return back again. Birch's Hist. of the Royal Society, vol. 3, p. 131.</p><p>Dr. Halley has given a new system, the result of numerous observations, and even of a number of voyages made at the public expence on this account. The light which this author has thrown upon this obscure part of natural history, is very great, and of important consequence in navigation, &c. In this system he has reduced the several Variations in divers places to a precise rule, or order, which before appeared all precarious and arbitrary.</p><p>His theory will therefore deserve a more ample detail. The observations it is built upon, as laid down in the Philos. Trans. number 148, or Abr. vol. 2, p. 610, are as follow: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=18" role="data"><hi rend="italics">Observed Variations of the Needle in divers places, and at divers times.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">Longitude</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Year of</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Variation</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">Longitude</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Year of</cell><cell cols="1" rows="1" rend="colspan=2 rowspan=2" role="data">Variation</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Places observed at.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">from</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Latitude</cell><cell cols="1" rows="1" role="data">Obser-</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Places observed at.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">from</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Latitude</cell><cell cols="1" rows="1" role="data">Obser-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">London.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">vation.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">observed.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">London.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">vation.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">observed.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′ </cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′ </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′ </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′ </cell><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′ </cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">°</cell><cell cols="1" rows="1" role="data">′ </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">London</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0  </cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">31 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1580</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">15 e</cell><cell cols="1" rows="1" rend="align=left" role="data">Baldivia</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1670</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10 e</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1622</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" rend="align=left" role="data">Cape Aguillas</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">30 e</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">50 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1622</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1634</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5 e</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1675</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1672</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">30 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1675</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1683</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1675</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">30 e</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Paris</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">25 e</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">51 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1640</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1675</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">30 e</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1666</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0  </cell><cell cols="1" rows="1" rend="align=left" role="data">St. Helena</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1677</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">40 e</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1681</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Isle Ascension</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">50 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1678</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0 e</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Uraniburg</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">54 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1672</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">35 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Johanna</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">15 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1675</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">30 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Copenhagen</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">53 e</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">41 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1649</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">53 e</cell><cell cols="1" rows="1" rend="align=left" role="data">Mombasa</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1675</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">1672</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">45 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Zocatra</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">30 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1674</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Dantzick</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">23 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1679</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Aden, Mouth</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">47</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">13</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 n</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1674</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">15</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Montpelier</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">37 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1674</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">10 w</cell><cell cols="1" rows="1" rend="align=left" role="data">of Red Sea</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Brest</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">25 w</cell><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">23 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1680</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">45 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Diego Roiz</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1676</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">30 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Rome</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">50 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1681</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">30 e</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0  </cell><cell cols="1" rows="1" rend="align=center" role="data">1676</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">30 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Bayonne</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20 w</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">30 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1680</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">20 w</cell><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1676</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Hudson's Bay</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">40 w</cell><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">0 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1668</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">15 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Bombay</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">30 e</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1676</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">In Hudson's</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">57</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 w</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">61</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 n</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1668</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">29</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Cape Comorin</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">15 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1680</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">48 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Straits</cell><cell cols="1" rows="1" rend="align=left" role="data">Ballasore</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">30 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1680</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Beffin's Bay,</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=3" role="data">80</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">0 w</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">78</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">0 n</cell><cell cols="1" rows="1" rend="align=center rowspan=3" role="data">1616</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">57</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">0 w</cell><cell cols="1" rows="1" rend="align=left" role="data">Fort St. George</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">15 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1680</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">10 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Sir T. Smith's</cell><cell cols="1" rows="1" rend="align=left" role="data">West Point of</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">104</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">6</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">40 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1676</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">3</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">10 w</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sound</cell><cell cols="1" rows="1" role="data">Java</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">40 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1682</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1677</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">30 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">50 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1682</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">30 w</cell><cell cols="1" rows="1" rend="align=left" role="data">I. St. Paul</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1677</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">30 w</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">At Sea</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">0 w</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">0 n</cell><cell cols="1" rows="1" rend="align=center" role="data">1678</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">40 e</cell><cell cols="1" rows="1" rend="align=left" role="data">At Van Diemen's</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">142</cell><cell cols="1" rows="1" role="data">0 e</cell><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">25 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1642</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">0  </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Cape St. Au-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">35</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 w</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">28</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1670</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">5</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 e</cell><cell cols="1" rows="1" rend="align=left" role="data">At New Zea-</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">170</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">40</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">50 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1642</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">9</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 e</cell></row><row role="data"><cell cols="1" rows="1" role="data">gustine</cell><cell cols="1" rows="1" role="data">land</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Off the mouth</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">53</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 w</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">39</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1670</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">20</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 e</cell><cell cols="1" rows="1" rend="align=left" role="data">Three - kings</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">169</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">34</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">35 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1642</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">8</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">40 e</cell></row><row role="data"><cell cols="1" rows="1" role="data">of River Plate</cell><cell cols="1" rows="1" role="data">Isle in ditto</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Cape Frio</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">10 w</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">40 s</cell><cell cols="1" rows="1" rend="align=center" role="data">1670</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">10 e</cell><cell cols="1" rows="1" rend="align=left" role="data">I. Rotterdam in</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">184</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">20</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">15 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1642</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">6</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">20 e</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Entrance of</cell><cell cols="1" rows="1" rend="rowspan=3" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=3" role="data">68</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">0 w</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">52</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">30 s</cell><cell cols="1" rows="1" rend="align=center rowspan=3" role="data">1670</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">17</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">0 e</cell><cell cols="1" rows="1" rend="align=left" role="data">the South Sea</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">Magellan's</cell><cell cols="1" rows="1" rend="align=left" role="data">Coast of New</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">149</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">4</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1643</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">8</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">45 e</cell></row><row role="data"><cell cols="1" rows="1" role="data">Straits</cell><cell cols="1" rows="1" role="data">Guinea</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="align=left" role="data">West Entrance</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">75</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 w</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">53</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1670</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">14</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">10 e</cell><cell cols="1" rows="1" rend="align=left" role="data">West Point of</cell><cell cols="1" rows="1" rend="rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="rowspan=2" role="data">126</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0 e</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">0</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">26 s</cell><cell cols="1" rows="1" rend="align=center rowspan=2" role="data">1643</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">5</cell><cell cols="1" rows="1" rend="rowspan=2" role="data">30 e</cell></row><row role="data"><cell cols="1" rows="1" role="data">of ditto</cell><cell cols="1" rows="1" role="data">ditto</cell></row></table><pb n="637"/><cb/></p><p>Upon these observed Variations Dr. Halley makes several remarks, as to the Variation in different parts of the world at the time of his writing, eastward and westward, and the situation and direction of the lines or places of no Variation: from the whole he deduces the following theory.</p><p>Dr. <hi rend="italics">Halley's Theory of the Variation of the Needle.</hi> That the whole globe of the earth is one great magnet, having four magnetical poles, or points of attraction; near each pole of the equator two; and that in those parts of the world which lie nearly adjacent to any one of these magnetic poles, the needle is governed by it; the nearest pole being always predominant over the more remote.</p><p>The pole which at present is nearest to us, he conjectures to lie in or near the meridian of the Land's-end of England, and not above 7° from the north pole: by this pole, the Variations in all Europe and Tartary, and the North Sea, are chiefly governed; though still with some regard to the other northern pole, whose situation is in the meridian passing about the middle of California, and about 15° from the north pole of the world, to which the needle has chiefly respect in all North America, and in the two oceans on either side of it, from the Azores westward to Japan, and farther.</p><p>The two southern magnetic poles, he imagines, are rather more distant from the south pole of the world; the one being about 16° from it, on a meridian 20° to the westward of the Magellanic Streights, or 95° west from London: this pole commands the needle in all South America, in the Pacific Ocean, and the greatest part of the Ethiopic Ocean. The other magnetic pole seems to have the greatest power, and the largest dominion of all, as it is the most remote from the pole of the world, being little less than 20° distant from it, in the meridian which passes through New Holland, and the island Celebes, about 120° east from London: this pole is predominant in the south part of Africa, in Arabia, and the Red Sea, in Persia, India, and its islands, and all over the Indian sea, from the Cape of Good Hope eastward, to the middle of the Great South Sea that divides Asia from America.</p><p>Such, he observes, seems to be the present disposition of the magnetic virtue thoughout the whole globe of the earth. It is then shewn how this hypothesis accounts for all the Variations that have been observed of late, and how it answers to the several remarks drawn from the table.</p><p>It is there inferred that from the whole it appears, that the direction of the needle, in the temperate and frigid zones, depends chiefly upon the counterpoise of the forces of two magnetic poles of the same nature: as also why, under the same meridian, the Variation should be in one place 29 1/2 degrees west, and in another 20 1/2 degrees east.</p><p>In the torrid zone, and particularly about the equator, respect must be had to all the four poles, and their positions must be well considered, otherwise it will not be easy to determine what the Variation should be, the nearest pole being always strongest; yet so however as to be sometimes counterbalanced by the united forces of two more remote ones. Thus, in sailing from St. Helena, by the isle of Ascension, to the <cb/> equator, on the north-west course, the Variation is very little easterly, and unalterable in that whole track; because the South-American pole (which is much the nearest in the aforesaid places), requiring a great easterly variation, is counterpoised by the contrary attraction of the North-American and the Asiatic south poles; each of which singly is, in these parts, weaker than the American south pole; and upon the north-west course the distance from this latter is very little varied; and as you recede from the Asiatic south pole, the balance is still preserved by an access towards the North-American pole. In this case no notice is taken of the European north pole; its meridian being a little removed from those of these places, and of itself requiring the same Variations which are here found.</p><p>After the same manner may the Variations in other places about the equator be accounted for, upon Dr. Halley's hypothesis.</p><p><hi rend="italics">To observe the Variation of the Needle.</hi> Draw a meridian line, as directed under <hi rend="smallcaps">Meridian;</hi> then a stile being erected in the middle of it, place a needle upon it, and draw the right line which it hangs over. Thus will the quantity of the Variation appear.</p><p>Or thus: As the former method of finding the Variation cannot be applied at sea, others have been devised, the principal of which are as follow. Suspend a thread and plummet over the compass, till the shadow pass through the centre of the card; observe the rhumb, or point of the compass which the shadow touches when it is the shortest. For the shadow is then a meridian line; and consequently the Variation is shewn.</p><p>Or thus: Observe the point of the compass upon which the sun, or some star, rises and sets; bisect the arch intercepted between the rising and setting, and the line of bisection will be the meridian line; consequently the Variation is had as before. The same may also be obtained from two equal altitudes of the same star, observed either by day or night. Or thus: Observe the rhumb upon which the sun or star rises and sets; and from the latitude of the place find the eastern or western amplitude: for the difference between the amplitude, and the distance of the rhumb observed, from the eastern rhumb of the card, is the Variation sought.</p><p>Or thus: Observe the altitude of the sun, or some star S, whose declination is <figure/> known; and note the rhumb in the compass to which it then corresponds. Then in the triangle ZPS, are known three sides, viz, PZ the colatitude, PS the codeclination, and ZS the coaltitude; the angle PZS is thence found by spherical trigonometry; the supplement to which, viz AZS, is the azimuth from the south. Then the difference between the azimuth and the observed distance of the rhumb from the south, is the Variation sought. See <hi rend="italics">Azimuth</hi> <hi rend="smallcaps">Compass.</hi></p><p>The use of the Variation is to correct the courses a ship has steered by the compass, which must always be done before they are worked, or calculated. <pb n="638"/><cb/></p><p><hi rend="smallcaps">Variation</hi> <hi rend="italics">of the Variation,</hi> is a gradual and continual change in the Variation, observed in any place, by which the quantity of the Variation is found to be different at different times.</p><p>This Variation, according to Henry Bond (in his <hi rend="italics">Longitude found,</hi> Lond. 1670, pa. 6) “was first found to decrease by Mr. John Mair; 2dly, by Mr. Edmund Gunter: 3dly, by Mr. Henry Gellibrand; 4thly, by myself (Henry Bond) in 1640; and lastly, by Dr. Robert Hook, and others, in 1665;” which they found out by comparing together observations made at the same place, at different times. The discovery was soon known abroad; for Kircher, in his treatise intitled Magnes, first printed at Rome in 1641, says that our countryman Mr. John Greaves had informed him of it, and then he gives a letter of Mersenne's, containing a distinct account of it.</p><p>This continual change in the Variation, is gradual and universal, as appears by numerous observations. Thus, the Variation was,</p><p>At Paris, according to Orontius Finæus, <table><row role="data"><cell cols="1" rows="1" role="data">in 1550</cell><cell cols="1" rows="1" role="data"> 8°</cell><cell cols="1" rows="1" role="data"> 0′E.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1640</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data"> 0 E.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1660</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data"> 0   </cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1681</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data"> 2 W.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1759</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">10 W.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1760</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">20 W.</cell></row></table> M. De la Lande (Exposition du Calcul Astronomique) observes, that the Variation has changed, at Paris, 26° 20′ in the space of 150 years, allowing that in 1610 the Variation was 8° E: and since 1740 the needle, which was always used by Maraldi, is more than 3° advanced toward the west, beyond what it was at that period; which is a change after the rate nearly of 9′1/2 per year.</p><p>At Cape d'Agulhas, in 1600, it had no Variation; (whence the Portuguese gave it that name); <table><row role="data"><cell cols="1" rows="1" role="data">in 1622</cell><cell cols="1" rows="1" role="data">it was</cell><cell cols="1" rows="1" role="data"> 2°W.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1675</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data"> 8 W.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in 1692</cell><cell cols="1" rows="1" role="data">"</cell><cell cols="1" rows="1" role="data">11 W.</cell></row></table> which is a change of nearly 8′ per year. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">At St. Helena, the Variation, in 1600</cell><cell cols="1" rows="1" role="data">was 8°</cell><cell cols="1" rows="1" role="data"> 0′E.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1623</cell><cell cols="1" rows="1" role="data">" 6 </cell><cell cols="1" rows="1" role="data"> 0 E.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1677</cell><cell cols="1" rows="1" role="data">" 0 </cell><cell cols="1" rows="1" role="data">40 E.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1692</cell><cell cols="1" rows="1" role="data">" 1 </cell><cell cols="1" rows="1" role="data"> 0 W.</cell></row></table> which is a change of nearly 5′1/2 per year.</p><p>At Cape Comorin, the Variation, <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1620</cell><cell cols="1" rows="1" role="data">was 14°</cell><cell cols="1" rows="1" role="data">20′W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1680</cell><cell cols="1" rows="1" role="data">" 8 </cell><cell cols="1" rows="1" role="data">44 W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1688</cell><cell cols="1" rows="1" role="data">" 7 </cell><cell cols="1" rows="1" role="data">30 W.</cell></row></table> which is a change of nearly 6′1/2 per year. <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">At London, the Variation, in 1580</cell><cell cols="1" rows="1" role="data">was 11°</cell><cell cols="1" rows="1" role="data">15′E.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1622</cell><cell cols="1" rows="1" role="data">" 6 </cell><cell cols="1" rows="1" role="data"> 0 E.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1634</cell><cell cols="1" rows="1" role="data">" 4 </cell><cell cols="1" rows="1" role="data"> 5 E.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1657</cell><cell cols="1" rows="1" role="data">" 0 </cell><cell cols="1" rows="1" role="data"> 0   </cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1672</cell><cell cols="1" rows="1" role="data">" 2 </cell><cell cols="1" rows="1" role="data">30 W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1692</cell><cell cols="1" rows="1" role="data">" 6 </cell><cell cols="1" rows="1" role="data"> 0 W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1723</cell><cell cols="1" rows="1" role="data">" 14 </cell><cell cols="1" rows="1" role="data">17 W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1747</cell><cell cols="1" rows="1" role="data">" 17 </cell><cell cols="1" rows="1" role="data">40 W.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">in 1780</cell><cell cols="1" rows="1" role="data">" 22 </cell><cell cols="1" rows="1" role="data">41 W.</cell></row></table> <cb/> which is a change after the rate of 10′ per year, upon a course of exactly 200 years. See Philos. Trans. No. 148 and No. 383, or Abr. vol. 2, p. 615, and vol. 7, p. 290; and Philos. Trans. vol. 45, p. 280, and vol. 66, p. 393. On the subject of the Variation, see also Norman's New Attractive 1614; Burrows's Discovery of the Variation 1581; Bond's Longitude found, 1676; &c.</p><p>Mr. Thomas Harding, in the Transactions of the Royal Irish Academy, vol. 4, has given observations on the Variation of the magnetic needle, at Dublin, which are rather extraordinary. He says the change in the Variation at that place is <hi rend="italics">uniform.</hi> That from the year 1657, in which the Variation was nothing (the same as at London in that year), it has been going on at the medium rate of 12′ 20″ annually, and was in May 1791, 27° 23′ west: exceeding that at London now by 3 or 4 degrees. He brings proof of his assertion of the uniformity of the Variation, from different authentic records, and states the operations by which it is calculated. He concludes with recommending accuracy in marking the existing Variation when maps are made, as not only conducing to the exact definition of boundaries, but as laying the best foundation for a discovery of the longitude by sea or land.</p><p><hi rend="italics">Theory of the Variation of the Variation.</hi> According to Dr. Halley's theory, this change in the Variation of the compass, is supposed owing to the difference of velocity in the motions of the internal and external parts of the globe. From the observations that have been cited, it seems to follow, that all the magnetical poles have a motion westward, but yet not exactly round the axis of the earth, for then the Variations would continue the same in the same parallel of latitude, contrary to experience.</p><p>From the disagreement of such a supposition with experiments therefore, the learned author of the theory invented the following hypothesis: The external parts of the globe he considers as the shell, and the internal as a nucleus, or inner globe; and between the two he conceives a fluid medium. That inner earth having the same common centre and axis of diurnal rotation, may revolve with our earth every 24 hours: Only the outer sphere having its turbinating motion somewhat swifter or slower than the internal ball; and a very minute difference in length of time, by many repetitions, becoming sensible; the internal parts will gradually recede from the external, and they will appear to move, either eastward or westward, by the difference of their motions.</p><p>Now, supposing such an internal sphere, having such a motion, the two great difficulties in the former hypothesis are easily solved; for if this exterior shell of earth be a magnet, having its pole at a distance from the poles of diurnal rotation; and if the internal nucleus be likewise a magnet, having its poles in two other places, distant also from the axis; and these latter, by a slow gradual motion, change their place in respect of the external, a reasonable account may then be given of the four magnetical poles before mentioned, and also of the changes of the needle's Variation.</p><p>The author thinks that two of these poles are fixed, and the other two moveable; viz, that the fixed poles are the poles of the external cortex or shell of the <pb n="639"/><cb/> earth; and the other the poles of the magnetical nucleus, included and moveable within the former. From the observations he infers, that the motion is westwards, and consequently that the nucleus has not precisely attained the same velocity with the exterior parts in their diurnal rotation; but so very nearly equals it, that in 365 revolutions the difference is scarcely sensible.</p><p>That there is any difference of this kind, arises from hence, that the impulse by which the diurnal motion was impressed on the earth, was given to the external parts, and from thence in time communicated to the internal; but so as not yet perfectly to equal the velocity of the first motion impressed on the superficial parts of the globe, and still preserved by them.</p><p>As to the precise period, observations are wanting to determine it, though the author thinks we may reasonably conjecture that the American pole has moved westward 46° in 90 years, and that its whole period is performed in about 700 years.</p><p>Mr. Whiston, in his New Laws of Magnetism, raises several objections against this theory. See M<hi rend="smallcaps">AGNETISM.</hi></p><p>M. Euler, too, the son of the celebrated mathematician of that name, has controverted and censured Dr. Halley's theory. He thinks, that two magnetic poles, placed on the surface of the earth, will sufficiently account for the Variation: and he then endeavours to shew, how we may determine the declination of the needle, at any time, and on every part of the globe, from this hypothesis. For the particulars of this reasoning, see the Histoire de l'Academie des Sciences & Belles Lettres of Berlin, for 1757; also Mr. Cavallo's Treatise on Magnetism, p. 117.</p><p><hi rend="italics">Variation of the Needle by Heat and Cold.</hi>—There is a small Variation of the Variation of the magnetic needle, amounting only to a few minutes of a degree in the same place, at different hours of the same day, which is only discoverable by nice observations. Mr. George Graham made several observations of this kind in the years 1722 and 1723, professing himself altogether ignorant of the cause of the phenomena he observed. Philos. Trans. No. 383, or Abr. vol. 7, p. 290.</p><p>About the year 1750, Mr. Wargentin, secretary of the Swedish Academy of Sciences, took notice both of the regular diurnal Variation of the needle, and also of its being disturbed at the time of the aurora borealis, as recorded in the Philos. Trans. vol. 47, p. 126.</p><p>About the year 1756, Mr. Canton commenced a series of observations, amounting to near 4000, with an excellent Variation-compass, of about 9 inches diameter. The number of days on which these observations were made, was 603, and the Diurnal Variation on 574 of them was regular, so as that the absolute Variation of the needle westward was increasing from about 8 or 9 o'clock in the morning, till about 1 or 2 in the afternoon, when the needle became stationary for some time; after that, the absolute Variation westward was decreasing, and the needle came back again to its former situation, or nearly so, in the night, or by the next morning. The Diurnal Variation is irregular when the needle moves slowly eastward in the latter part of the morning, or westward in the latter <cb/> part of the afternoon; also when it moves much either way after night, or suddenly both ways in a short time. These irregularities seldom happen more than once or twice in a month, and are always accompanied, as far as Mr. Canton observed, with an aurora borealis.</p><p>Mr. Canton lays down and evinces, by experiment, the following principle, viz, that the attractive power of the magnet (whether natural or artificial) will decrease while the magnet is heating, and increase while it is cooling. He then proceeds to account for both the regular and irregular Variation. It is evident, he says, that the magnetic parts of the earth in the north, on the east side, and on the west side of the magnetic meridian, equally attract the north end of the needle. If then the eastern magnetic parts be heated faster by the sun in the morning, than the western parts, the needle will move westward, and the absolute Variation will increase: when the attracting parts of the earth on each side of the magnetic meridian have their heat increasing equally, the needle will be stationary, and the absolute Variation will then be greatest; but when the western magnetic parts are either heating faster, or cooling slower, than the eastern, the needle will move eastward, or the absolute Variation will decrease; and when the eastern and western magnetic parts are cooling equally fast, the needle will again be stationary, and the absolute Variation will then be least.</p><p>By this theory, the Diurnal Variation in the summer ought to exceed that in winter; and accordingly it is found by observation, that the Diurnal Variation in the months of June and July is almost double of that in December and January.</p><p>The irregular Diurnal Variation must arise from some other cause than that of heat communicated by the sun; and here Mr. Canton has recourse to subterranean heat, which is generated without any regularity as to time, and which will, when it happens in the north, affect the attractive power of the magnetic parts of the earth on the north end of the needle. That the air nearest the earth will be most warmed by the heat of it, is obvious; and this has been often noticed in the morning, before day, by means of thermometers at different distances from the ground. Philos. Trans. vol. 48, pa. 526.</p><p>Mr. Canton has annexed to his paper on this subject, a complete year's observations; from which it appears, that the Diurnal Variation increases from January to June, and decreases from June to December. Philos. Trans. vol. 51, pa. 398.</p><p>It has also been observed, that different needles, especially if touched with different loadstones, will differ a few minutes in their Variation. See Poleni Epist. Phil. Trans. num. 421.</p><p>Dr. Lorimer (in the Supp. to Cavallo's Magnetism) adduces some ingenious observations on this subject. It must be allowed, says he, according to the observations of several ingenious gentlemen, that the collective magnetism of this earth arises from the magnetism of all the ferruginous bodies contained in it, and that the magnetic poles should therefore be considered as the centres of the powers of those magnetic substances. These poles must therefore change their places according as the magnetism of such substances is affected, and if <pb n="640"/><cb/> with Mr. Canton we allow, that the general cause of the Diurnal Variation arises from the sun's heat in the forenoon and afternoon of the same day, it will naturally occur, that the same cause, being continued, may be sufficient to produce the general Variation of the magnetic needle for any number of years. For we must consider, that ever since any attentive observations have been made on this subject, the natural direction of the magnetic needle in Europe has been constantly moving, from west to east, and that in other parts of the world it has continued its motion with equal constancy.</p><p>As we must therefore admit, says Dr. Lorimer, that the heat in the different seasons depends chiefly on the sun, and that the months of July and August are commonly the hottest, while January and February are the coldest months of the year; and that the temperature of the other months falls into the respective intermediate degrees; so we must consider the influence of heat upon magnetism to operate in the like manner, viz, that for a short time it scarcely manifests itself; yet in the course of a century, the constancy and regularity of it becomes sufficiently apparent. It would therefore be idle to suppose, that such an influence could be derived from an uncertain or fortuitous cause. But if it be allowed to depend upon the constancy of the sun's motion, and this appears to be a cause sufficient to explain the phenomena, we should (agreeably to Newton's first law of philosophizing) look no farther.</p><p>As we therefore consider, says he, the magnetic powers of the earth to be concentrated in the magnetic poles, and that there is a diurnal Variation of the magnetic needle, these poles must perform a small diurnal revolution proportional to such Variation, and return again to the same point nearly. Suppose then that the sun in his diurnal revolution passes along the northern tropic, or along any parallel of latitude between it and the equator, when he comes to that meridian in which the magnetic pole is situated, he will be much nearer to it, than in any other; and in the opposite meridian he will of course be the farthest from it. As the influence of the sun's heat will therefore act most powerfully at the least, and less forcibly at the greatest distance, the magnetic pole will consequently describe a figure something of the elliptical kind; and as it is well known that the greatest heat of the day is some time after the sun has passed the meridian, the longest axis of this elliptical figure will lie north easterly in the northern, and south-easterly in the southern hemisphere. Again, as the influence of the sun's heat will not from those quarters have so much power, the magnetic poles cannot be moved back to the very same point, from which they set out; but to one which will be a little more northerly and easterly, or more southerly and easterly, according to the hemispheres in which they are situated. The figures therefore which they describe, may more properly be termed elliptoidal spirals.</p><p>In this manner the Variation of the magnetic needle in the northern hemisphere may be accounted for. But with respect to the southern hemisphere we must recollect, that though the lines of declination in the northern hemisphere have constantly moved from west to east, yet in the southern hemisphere, it is equally certain that they have moved from east to west, ever since any observations have been made on the subject. Hence <cb/> then the lines of magnetic declination, or Halleyan curves, as they are now commonly called, appear to have a contrary motion in the southern hemisphere, to what they have in the northern; though both the magnetic poles of the earth move in the same direction, that is from west to east.</p><p>In the northern hemisphere there was a line of no Variation, which had east Variation on its eastern side, and west Variation on its western side. This line evidently moved from west to east during the two last centuries; the lines of east Variation moving before it, while the lines of west Variation followed it with a proportional pace. These lines first passed the Azores or Western Islands, then the meridian of London, and after a certain number of years still later, they passed the meridian of Paris. But in the southern hemisphere there was another line of no Variation, which had east Variation on its western side, and west Variation on its eastern; the lines of east Variation moving before it, while those of the west Variation followed it. This line of no Variation first passed the Cape des Aiguilles, and then the Cape of Good Hope; the lines of 5°, 10°, 15°, and 20° west Variation following it, the same as was the case in the northern hemisphere, but in the contrary direction.</p><p>We may just farther mention the idea of Dr. Gowin Knight, which was, that this earth had originally received its magnetism, or rather that its magnetical powers had been brought into action, by a shock, which entered near the southern tropic, and passed out at the northern one. His meaning appears to have been, that this was the course of the magnetic fluid, and that the magnetic poles were at first diametrically opposite to each other. Though, according to Mr. Canton's doctrine, they would not have long continued so; for from the intense heat of the sun in the torrid zone, according to the principles already explained, the north pole must have soon retired to the north-eastward, and the south pole to the south-eastward. It is also curious to observe, that on account of the southern hemisphere being colder upon the whole than the northern hemisphere, the magnetic poles would have moved with unequal pace: that is, the north magnetic pole would have moved farther in any given time to the north-east, than the south magnetic pole could have moved to the south-east. And, according to the opinions of the most ingenious authors on this subject, it is generally allowed, that at this time the north magnetic pole is considerably nearer to the north pole of the earth, than the south magnetic pole is to the south pole of the earth.</p><p>It may farther be added, that several ingenious sea officers are of opinion, that in the western parts of the English Channel the Variation of the magnetic needle has already begun to decrease; having in no part of it ever amounted to 25°. There are however other persons who assert that the Variation is still increasing in the Channel, and as far westward as the 15th degree of longitude and 51° of latitude, at which place they say that it amounts to about 30°.</p><p><hi rend="italics">Of the Variation Chart.</hi> Doctor Halley having collected a multitude of observations made on the Variation of the needle in many parts of the world, was hence enabled to draw, on a Mercator's chart, certain lines, shewing the Variation of the compass in all those <pb n="641"/><cb/> places over which they passed, in the year 1700, when he published the first chart of this kind, called the <hi rend="italics">Variation Chart.</hi></p><p>From the construction of this chart it appears, that the longitude of any of those places may be found by it, when the latitude and the Variation in that place are known. Thus, having found the Variation of the compass, draw a parallel of latitude on the chart through the latitude found by observation; and the point where it cuts the curved line, whose Variation is the same with that observed, will be the ship's place. A similar project of thus finding the longitude, from the known latitude and inclination or dip of the needle, was before proposed by Henry Bond, in his treatise intitled, The Longitude Found, printed in 1676.</p><p>This method however is attended with two considerable inconveniences: 1st, That wherever the Variation lines run east and west, or nearly so, this way of finding the longitude becomes imperfect, as their intersection with the parallel of latitude must be very indesinite: and among all the trading parts of the world, this imperfection is at present found chiefly on the western coasts of Europe, between the latitudes of 45° and 53°; and on the eastern shores of North America, with some parts of the Western Ocean and Hudson's Bay, lying between the said shores: but for the other parts of the world, a Variation Chart may be attended with considerable benefit. However, the Variation curves, when they run east and west, may sometimes be applied to good purpose in correcting the latitude, when meridian observations cannot be had, as it often happens on the northern coasts of America, in the Western Ocean, and about Newfoundland; for if the Variation can be obtained exactly, then the east and west curve, answering to the Variation in the chart, will shew the latitude.</p><p>2dly, As the deviation of the magnetical meridian, from the true one, is subject to continual alteration, therefore a chart to which the Variation lines are fitted for any year, must in time become useless, unless new lines, shewing the state of the Variation at that time, be drawn on the chart: but as the change in the Variation is very slow, therefore new Variation Charts published every 7 or 8 years, will answer the purpose tolerably well. And thus it has happened that Halley's Variation Chart has become useless, for want of encouragement to renew it from time to time.</p><p>However, in the year 1744, Mr. William Mountaine and Mr. James Dodson published a new Variation Chart, adapted for that year, which was well received; and several instances of its great utility having been communicated to them, they fitted the Variation lines anew for the year 1756, and in the following year published the 3d Variation Chart, and also presented to the Royal Society a curious paper concerning the Variation of the magnetic needle, with a set of tables annexed, containing the result of upwards of 50 thousand observations, in six periodical reviews, from the year 1700 to 1756 inclusive, and adapted to every 5 degrees of latitude and longitude in the more frequented oceans; which paper and tables were printed in the Transactions for the year 1757.</p><p>From these tables of observations, such extraordi- <cb/> nary and whimsical irregularities occur in the Variation, that we cannot think it wholly under the direction of one general and uniform law; but rather conclude, with Dr. Gowen, in the 87th prop. of his Treatise upon Attraction and Repulsion, that it is influenced by various and different magnetic attractions, perhaps occasioned by the heterogeneous compositions in the great magnet, the earth.</p><p>Many other observations on the Variation of the magnetic needle, are to be found in several volumes of the Philos. Trans. See particularly vol. 48, p. 875; vol. 50, p. 329; vol. 56, p. 220; and vol. 61, p. 422.</p><p><hi rend="smallcaps">Variation</hi> <hi rend="italics">Compass.</hi> See <hi rend="smallcaps">Compass.</hi></p><p><hi rend="smallcaps">Variation</hi> <hi rend="italics">of Curvature,</hi> in Geometry, is used for that inequality or change which takes place in the curvature of all curves except the circle, by which their curvature is more or less in different parts of them. And this Variation constitutes the quality of the curvature of any line.</p><p>Newton makes the index of the inequality, or Variation of Curvature, to be the ratio of the fluxion of the radius of curvature to the fluxion of the curve itself: and Maclaurin, to avoid the perplexity that different notions, connected with the same terms, occasion to learners, has adopted the same definition: but he suggests, that this ratio gives rather the Variation of the ray of curvature, and that it might have been proper to have measured the Variation of Curvature rather by the ratio of the fluxion of the curvature itself to the fluxion of the curve; so that, the curvature being inversely as the radius of curvature, and consequently its fluxion as the fluxion of the radius itself directly, and the square of the radius inversely, its Variation would have been directly as the measure of it according to Newton's definition, and inversely as the square of the radius of curvature.</p><p>According to this notion, it would have been measured by the angle of contact contained by the curve and circle of curvature, in the same manner as the curvature itself is measured by the angle of contact contained by the curve and tangent. The reason of this remark may appear from this example: The Variation of curvature, according to Newton's explication, is uniform in the logarithmic spiral, the fluxion of the radius of curvature in this figure being always in the same ratio to the fluxion of the curve; and yet, while the spiral is produced, though its curvature decreases, it never vanishes; which must appear a strange paradox to those who do not attend to the import of Newton's definition. Newton's Method of Fluxions and Inf. Series, pa. 76. Maclaurin's Flux. art. 386. Philos. Trans. num. 468, pa. 342.</p><p>The Variation of curvature at any point of a conic section, is always as the tangent of the angle contained by the diameter that passes through the point of contact, and the perpendicular to the curve at the same point, or to the angle formed by the diameter of the section, and of the circle of curvature. Hence the Variation of curvature vanishes at the extremities of either axis, and is greatest when the acute angle, contained by the diameter, passing through the point of contact and the tangent, is least.</p><p>When the conic section is a parabola, the Variation is <pb n="642"/><cb/> as the tangent of the angle, contained by the right line drawn from the point of contact to the focus, and the perpendicular to the curve. See <hi rend="smallcaps">Curvature.</hi></p><p>From Newton's definition may be derived practical rules for the Variation of curvature, as follows:</p><p>1. Find the radius of curvature, or rather its fluxion; then divide this fluxion by the fluxion of the curve, and the quotient will give the Variation of curvature; exterminating the fluxions when necessary, by the equation of the curve, or perhaps by expressing their ratio by help of the tangent, or ordinate, or subnormal, &c.</p><p>2. Since <hi rend="italics">z</hi><hi rend="sup">.</hi><hi rend="sup">3</hi>/-<hi rend="italics">x</hi><hi rend="sup">.</hi><hi rend="italics">y</hi><hi rend="sup">..</hi>, or <hi rend="italics">z</hi><hi rend="sup">.</hi><hi rend="sup">3</hi>/-<hi rend="italics">y</hi><hi rend="sup">..</hi> (putting <hi rend="italics">x</hi><hi rend="sup">.</hi> = 1) denotes the radius of curvature of any curve <hi rend="italics">z,</hi> whose absciss is <hi rend="italics">x,</hi> and ordinate <hi rend="italics">y;</hi> if the fluxion of this be divided by <hi rend="italics">z</hi><hi rend="sup">.</hi>, and <hi rend="italics">z</hi><hi rend="sup">.</hi> and <hi rend="italics">z</hi><hi rend="sup">..</hi> be exterminated, the general value of the Variation will come out (-3<hi rend="italics">y</hi><hi rend="sup">.</hi><hi rend="italics">y</hi><hi rend="sup">..</hi><hi rend="sup">2</hi> + <hi rend="italics">y</hi><hi rend="sup">∴</hi> (1 + <hi rend="italics">y</hi><hi rend="sup">.</hi><hi rend="sup">2</hi>))/<hi rend="italics">y</hi><hi rend="sup">..</hi><hi rend="sup">2</hi>; then substituting the values of <hi rend="italics">y</hi><hi rend="sup">.</hi>, <hi rend="italics">y</hi><hi rend="sup">..</hi>, <hi rend="italics">y</hi><hi rend="sup">∴</hi> (found from the equation of the curve) into this quantity, it will give the Variation sought.</p><p><hi rend="italics">Ex.</hi> Let the curve be the parabola, whose equation is , the Variation sought. Emerson's Flux. pa. 228.</p></div2></div1><div1 part="N" n="VARIGNON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">VARIGNON</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, a celebrated French mathematician and priest, was born at Caen in 1654, and died suddenly in 1722, at 68 years of age. He was the son of an architect in middling circumstances, but had a college education, being intended for the church. An accident threw a copy of Euclid's Elements in his way, which gave him a strong turn to that kind of learning. The study of geometry led him to the works of Des Cartes on the same science, and there he was struck with that new light which has, from thence, spread over the world.</p><p>He abridged himself of the necessaries of life to purchase books of this kind, or rather considered them of that number, as indeed they ought to be. What contributed to heighten this passion in him was, that he studied in private: for his relations observing that the books he studied were not such as were commonly used by others, strongly opposed his application to them. As there was a necessity for his being an ecclesiastic, he continued his theological studies, yet not entirely sacrificing his favourite subject to them.</p><p>At this time the Abbé St. Pierre, who studied philosophy in the same college, became acquainted with him. A taste in common for rational subjects, whether physics or metaphysics, and continual disputations, formed the bonds of their friendship. They were mutually serviceable to each other in their studies. The Abbé, to enjoy Varignon's company with greater ease, lodged him with himself; thus, growing still more <cb/> sensible of his merit, he resolved to give him a fortune, that he might fully pursue his genius, and improve his talents; and, out of only 18 hundred livres a year, which he had himself, he conferred 300 of them upon Varignon.</p><p>The Abbé, persuaded that he could not do better than go to Paris to study philosophy, settled there in 1686, with M. Varignon, in the suburbs of St. Jacques. There each studied in his own way; the Abbé applying himself to the study of men, manners, and the principles of government; whilst Varignon was wholly occupied with the mathematics.</p><p>I, says Fontenelle, who was their countryman, often went to see them, sometimes spending two or three days with them. They had also room for a couple of visitors, who came from the same province. We joined together with the greatest pleasure. We were young, full of the first ardour for knowledge, strongly united, and, what we were not then perhaps disposed to think so great a happiness, little known. Varignon, who had a strong constitution, at least in his youth, spent whole days in study, without any amusement or recreation, except walking sometimes in fine weather. I have heard him say, that in studying after supper, as he usually did, he was often surprised to hear the clock strike two in the morning; and was much pleased that four hours rest were sufficient to refresh him. He did not leave his studies with that heaviness which they usually create; nor with that weariness that a long application might occasion. He left off gay <*>d lively, filled with pleasure, and impatient to renew it. In speaking of mathematics, he would laugh so freely, that it seemed as if he had studied for diversion. No condition was so much to be envied as his; his life was a continual enjoyment, delighting in quietness.</p><p>In the solitary suburb of St. Jacques, he formed however a connection with many other learned men; as Du Hamel, Du Verney, De la Hire, &c. Du Verney often asked his assistance in those parts of anatomy connected with mechanics: they examined together the positions of the muscles, and their directions; hence Varignon learned a good deal of anatomy from Du Verney, which he repaid by the application of mathematical reasoning to that subject.</p><p>At length, in 1687, Varignon made himself known to the public by a Treatise on New Mechanics, dedicated to the Academy of Sciences. His thoughts on this subject were, in effect; quite new. He discovered truths, and laid open their sources. In this work, he demonstrated the necessity of an equilibrium, in such cases as it happens in, though the cause of it is not exactly known. This discovery Varignon made by the theory of compound motions, and is what this essay turns upon.</p><p>This new Treatise on Mechanics was greatly admired by the mathematicians, and procured the author two considerable places, the one of Geometrician in the Academy of Sciences, the other of Professor of Mathematics in the College of Mazarine, to which he was the first person raised.</p><p>Varignon catched eagerly at the Science of Infinitesimals as soon as it appeared in the world, and became one of its most early cultivators. When that sublime and beautiful method was attacked in the Academy it- <pb n="643"/><cb/> self (for it could not escape the fate of all innovations) he became one of its most zealous defenders, and in its favour he put a violence upon his natural character, which abhorred all contention. He sometimes lamented, that this dispute had interrupted him in his enquiries into the Integral Calculation so far, that it would be difficult for him to resume his disquisition where he had left it off. He sacrificed Infinitesimals to the interest of Infinitesimals, and gave up the pleasure and glory of making a farther progress in them when called upon by duty to undertake their defence.</p><p>All the printed volumes of the Academy bear witness to his application and industry. His works are never detached pieces, but complete theories of the laws of motion, central forces, and the resistance of mediums to motion. In these he mak<*> such use of his rules, that nothing escapes him that has any connection with the subject he treats.</p><p>Geometrical certainty is by no means incompatible with obscurity and confusion, and those are sometimes so great, that it is surprising a mathematician should not miss his way in so dark and perplexing a labyrinth. The works of M. Varignon never occasion this disagreeable surprise, he makes it his chief care to place every thing in the clearest light; he does not, as some great men do, consult his ease by declining to take the trouble of being methodical, a trouble much greater than that of composition itself; he does not endeavour to acquire a reputation for profoundness, by leaving a great deal to be guessed by the reader.</p><p>He was perfectly acquainted with the history of mathematics. He learned it not merely out of curiosity, but because he was desirous of acquiring knowledge from every quarter. This historical knowledge is doubtless an ornament in a mathematician, but it is an ornament which is by no means without its utility. Indeed it may be laid down as a maxim, the more different ways the mind is occupied in, upon a subject, the more it improves.</p><p>Though Varignon's constitution did not seem easy to be impaired, assiduity and constant application brought upon him a severe disease in 1705. Great abilities are generally dangerous to the possessors. He was six months in danger, and three years in a languid state, which proceeded from his spirits being almost entirely exhausted. He said that sometimes when delirious with a fever, he thought himself in the midst of a forest, where all the leaves of the trees were covered with algebraical calculations. Condemned by his physicians, his friends, and himself, to lay aside all study, he could not, when alone in his chamber, avoid taking up a book of mathematics, which he hid as soon as he heard any person coming. He again resumed the attitude and behaviour of a sick man, and seldom had occasion to counterfeit.</p><p>In regard to his character, Fontenelle observes, that it was at this time that a writing of his appeared, in which he censured Dr. Wallis for having advanced that there are certain spaces more than infinite, which that great geometrician ascribes to hyperbolas. He maintained, on the contrary, that they were finite. The criticism was softened with all the politeness and respect imaginable; but a criticism it was, though he had written it only for himself. He let M. Carré see it, <cb/> when he was in a state that rendered him indifferent about things of that kind; and that gentleman, influenced only by the interest of the sciences, caused it to be printed in the memoirs of the Academy of Sciences, unknown to the author, who thus made an attack against his inclination.</p><p>He recovered from his disease; but the remembrance of what he had suffered did not make him more prudent for the future. The whole impression of his <hi rend="italics">Project for a New System of Mechanics,</hi> having been sold off, he formed a design to publish a second edition of it, or rather a work entirely new, though upon the same plan, but more extended. It must be easy to perceive how much learning he must have acquired in the interval; but he often complained, that he wanted time, though he was by no means disposed to lose any. Frequent visits, either of French or of foreigners, some of whom went to see him that they might have it to say that they had seen him; and others to consult him and improve by his conversation: works of mathematics, which the authority of some, or the friendship he had for others, engaged him to examine, and which he thought himself obliged to give the most exact account of; a literary correspondence with all the chief mathematicians of Europe; all these obstructed the book he had undertaken to write. Thus a man acquires reputation by having a great deal of leisure time, and he loses this precious leisure as soon as he has acquired reputation. Add to this, that his best scholars, whether in the College of Mazarine or the Royal College (for he had a professor's chair in both), sometimes requested private lectures of him, which he could not refuse. He sighed for his two or three months of vacation, for that was all the leisure time he had in the year; no sooner were they come but he retired into the country, where his time was entirely his own, and the days seemed always quickly ended.</p><p>Notwithstanding his great desire of peace, in the latter part of his life he was involved in a dispute. An Italian monk, well versed in mathematics, attacked him upon the subject of tangents and the angle of contact in curves, such as they are conceived in the arithmetic of infinites; he answered by the last memoir he ever gave to the Academy, and the only one which turned upon a dispute.</p><p>In the last two years of his life he was attacked with an asthmatic complaint. This disorder increased every day, and all remedies were ineffectual. He did not however cease from any of his customary business; so that, after having finished his lecture at the College of Mazarine, on the 22d of December 1722, he died suddenly the following night.</p><p>His character, says Fontenelle, was as simple as his superior understanding could require. He was not apt to be jealous of the fame of others: indeed he was at the head of the French mathematicians, and one of the best in Europe. It must be owned however, that when a new idea was offered to him, he was too hasty to object. The sire of his genius, the various insights into every subject, made too impetuous an opposition to those that were offered; so that it was not easy to obtain from him a favourable attention.</p><p>His works that were published separately, were,</p><p>1. Projet d'une Nouvelle Mechanique; 4to, Paris 1687. <pb n="644"/><cb/></p><p>2. Des Nouvelles Conjectures sur la Pesanteur.</p><p>3. Nouvelle Mechanique ou Statique, 2 tom. 4to, 1725.</p><p>As to his memoirs in the volumes of the Academy of Sciences, they are far too numerous to be here particularized; they extend through almost all the volumes, down to his death in 1722.</p><p>VASA <hi rend="italics">Concordiæ,</hi> in Hydraulics, are two vessels, so constructed, as that one of them, though full of wine, will not run a drop, unless the other, being full of water, do run also. Their structure and apparatus may be seen in Wolsius, Element. Mathes. tom. 3, Hydraul.</p></div1><div1 part="N" n="VAULT" org="uniform" sample="complete" type="entry"><head>VAULT</head><p>, in Architecture, an arched roof, so contrived, as that the several stones of which it consists, by their disposition into the form of a curve, mutually sustain each other; as the arches of bridges, &c.</p><p>Vaults are to be preferred, on many occasions, to soffits, or flat ceilings, as they give a greater rise and elevation, and are also more firm and durable.</p><p>The Ancients, Salmasius observes, had only three kinds of vaults: the first the <hi rend="italics">fornix,</hi> made cradlewise; the 2d, the <hi rend="italics">testudo,</hi> tortoise-wise, or oven-wise; the 3d, the <hi rend="italics">concha,</hi> made shell-wise.</p><p>But the Moderns subdivide these three sorts into a great many more, to which they give different names, according to their figures and use: some are circular, others elliptical, &c.</p><p>Again, the sweeps of some are larger, and others less portions of a sphere: all above hemispheres are called <hi rend="italics">high,</hi> or <hi rend="italics">surmounted Vaults;</hi> all that are less than hemispheres, are <hi rend="italics">low,</hi> or <hi rend="italics">surbased Vaults,</hi> &c.</p><p>In some the height is greater than the diameter; in others it is less: there are others again quite flat, only made with haunses; others oven-like, and others growing wider as they lengthen, like a trumpet.</p><p>Of Vaults, some are <hi rend="italics">single,</hi> others <hi rend="italics">double, cross, diagonal, horizontal, ascending, descending, angular, oblique, pendent,</hi> &c, &c. There are also <hi rend="italics">Gothic</hi> Vaults, with <hi rend="italics">pendentives,</hi> &c.</p><p><hi rend="italics">Master</hi> <hi rend="smallcaps">Vaults</hi>, are those which cover the principal parts of buildings; in contradistinction from the <hi rend="italics">less,</hi> or subordinate Vaults, which only cover some small part; as a passage, a gate, &c.</p><p><hi rend="italics">Double</hi> <hi rend="smallcaps">Vault</hi>, is such a one as, being built over another, to make the exterior decoration range with the interior, leaves a space between the convexity of the one, and the concavity of the other: as in the dome of St. Paul's at London, and that of St. Peter's at Rome.</p><p><hi rend="smallcaps">Vaults</hi> <hi rend="italics">with Compartiments,</hi> are such whose sweep, or inner face, is enriched with pannels of sculpture, separated by platbands. These compartiments, which are of different figures, according to the Vaults, and are usually gilt on a white ground, are made with stucco, on brick Vaults; as in the church of St. Peter's at Rome; and with plaster, on timber Vaults.</p><p><hi rend="italics">Theory of</hi> <hi rend="smallcaps">Vaults.</hi>—In a semicircular Vault, or arch, being a hollow cylinder cut by a plane through its axis, standing on two imposts, and all the stones that compose it, being cut and placed in such a manner, as that their joints, or beds, being prolonged, do all meet in the centre of the vault; it is evident that all the stones must be cut wedge-wise, or wider at top and above, <cb/> than below; by virtue of which, they sustain each other, and mutually oppose the effort of their weight, which determines them to fall.</p><p>The stone in the middle of the Vault, which is perpendicular to the horizon, and is called the <hi rend="italics">key of the Vault,</hi> is sustained on each side by the two contiguous stones, as by two inclined planes.</p><p>The second stone, which is on the right or left of the key-stone, is sustained by a third; which, by virtue of the figure of the Vault, is necessarily more inclined to the second, than the second is to the first; and consequently the second, in the effort it makes to fall, employs a less part of its weight than the first.</p><p>For the same reason, all the stones, reckoning from the keystone, employ still a less and less part of their weight to the last; which, resting on the horizontal plane, employs no part of its weight, or makes no effort to fall, as being entirely supported by the impost.</p><p>Now a great point to be aimed at in Vaults, is, that all the several stones make an equal effort to fall: to effect this, it is evident that as each stone, reckoning from the key to the impost, employs a still less and less part of its whole weight; the first only employing, for example, one-half; the 2d, one-third; the 3d, onefourth; &c; there is no other way to make these different parts equal, but by a proportionable augmentation of the whole; that is, the second stone must be heavier than the first, the third heavier than the second, and so on to the last, which should be vastly heavier.</p><p>La Hire demonstrates what that proportion is, in which the weights of the stones of a semicircular arch must be increased, to be in equilibrio, or to tend with equal forces to fall; which gives the firmest disposition that a vault can have. Before him, the architects had no certain rule to conduct themselves by; but did all at random. Reckoning the degrees of the quadrant of the circle, from the keystone to the impost; the length or weight of each stone must be so much greater, as it is farther from the key. La Hire's rule is, to augment the weight of each stone above that of the key stone, as much as the tangent of the arch to the stone exceeds the tangent of the arch of half the key. Now the tangent of the last stone becomes infinite, and consequently the weight should be so too; but as infinity has no place in practice, the rule amounts to this, that the last stone be loaded as much as possible, and the others in proportion, that they may the better resist the effort which the Vault makes to separate them; which is called the <hi rend="italics">shoot</hi> or <hi rend="italics">drift</hi> of the Vault.</p><p>M. Parent, and other authors, have since determined the curve, or figure, which the extrados or outside of a Vault, whose intrados or inside is spherical, ought to have, that all the stones may be in equilibrio.</p><p>The above rule of La Hire's has since been found not accurate. See <hi rend="smallcaps">Arch</hi>, and <hi rend="smallcaps">Bridge.</hi> See also my Treatise on the Principles of Bridges, and Emerson's Construction of Arches.</p><p><hi rend="italics">Key of a</hi> <hi rend="smallcaps">Vault.</hi> See <hi rend="smallcaps">Key</hi>, and <hi rend="smallcaps">Voussoir.</hi></p><p><hi rend="italics">Reins</hi> or <hi rend="italics">fillings up of a</hi> <hi rend="smallcaps">Vault</hi>, are the sides which sustain it.</p><p><hi rend="italics">Pendentive of a</hi> <hi rend="smallcaps">Vault.</hi> See <hi rend="smallcaps">Pendentive.</hi></p><p><hi rend="italics">Impost of a</hi> <hi rend="smallcaps">Vault</hi>, is the stone upon which is laid the first voussoir, or arch-stone of the Vault. <pb n="645"/><cb/></p></div1><div1 part="N" n="VEADAR" org="uniform" sample="complete" type="entry"><head>VEADAR</head><p>, in Chronology, the 13th month of the Jewish ecclesiastical year, answering commonly to our March; this month is intercalated, to prevent the beginning of Nisan from being removed to the end of February.</p></div1><div1 part="N" n="VECTIS" org="uniform" sample="complete" type="entry"><head>VECTIS</head><p>, in Mechanics, one of the simple mechanical powers, more usually called the <hi rend="smallcaps">Lever.</hi></p></div1><div1 part="N" n="VECTOR" org="uniform" sample="complete" type="entry"><head>VECTOR</head><p>, or <hi rend="italics">Radius Vector,</hi> in Astronomy, is a line supposed to be drawn from any planet moving round a centre, or the focus of an ellipse, to that centre, or focus. It is so called, because it is that line by which the planet seems to be carried round its centre; and with which it describes areas proportional to the times.</p></div1><div1 part="N" n="VELOCITY" org="uniform" sample="complete" type="entry"><head>VELOCITY</head><p>, or <hi rend="italics">Swistness,</hi> in Mechanics, is that affection of motion, by which a moving body passes over a certain space in a certain time. It is also called celerity; and it is always proportional to the space moved over in a given time, when the Velocity is uniform, or always the same during that time.</p><p>Velocity is either <hi rend="italics">uniform</hi> or <hi rend="italics">variable. Uniform,</hi> or equal <hi rend="italics">Velocity,</hi> is that with which a body passes always over equal spaces in equal times. And it is <hi rend="italics">variable,</hi> or <hi rend="italics">unequal,</hi> when the spaces passed over in equal times are unequal; in which case it is either <hi rend="italics">accelerated</hi> or <hi rend="italics">retarded</hi> Velocity; and this acceleration, or retardation, may also be equal or unequal, i. e. uniform or variable, &c. See <hi rend="smallcaps">Acceleration</hi>, and <hi rend="smallcaps">Motion.</hi></p><p>Velocity is also either <hi rend="italics">absolute</hi> or <hi rend="italics">relative. Absolute Velocity</hi> is that we have hitherto been considering, in which the Velocity of a body is considered simply in itself, or as passing over a certain space in a certain time. But <hi rend="italics">relative</hi> or <hi rend="italics">respective Velocity,</hi> is that with which bodies approach to, or recede from one another, whether they both move, or one of them be at rest. Thus, if one body move with the absolute Velocity of 2 feet per second, and another with that of 6 feet per second; then if they move directly towards each other, the relative velocity with which they approach is that of 8 feet per second; but if they move both the same way, so that the latter overtake the former, then the relative Velocity with which that overtakes it, is only that of 4 feet per second, or only half of the former; and consequently it will take double the time of the former before they come in contact together.</p><p><hi rend="smallcaps">Velocity</hi> <hi rend="italics">in a Right Line.</hi>—When a body moves with a uniform Velocity, the spaces passed over by it, in different times, are proportional to the times; also the spaces described by two different uniform Velocities, in the same time, are proportional to the Velocities; and consequently, when both times and Velocities are unequal, the spaces described are in the compound ratio of the times and Velocities. That is, S <*> TV, and <hi rend="italics">s</hi> <*> <hi rend="italics">tv;</hi> or S : <hi rend="italics">s</hi> :: TV : <hi rend="italics">tv.</hi> Hence also, V : <hi rend="italics">v</hi> :: S/T : <hi rend="italics">s</hi>/<hi rend="italics">t,</hi> or the Velocity is as the space directly and the time reciprocally.</p><p>But in uniformly accelerated motions; the last degree of Velocity uniformly gained by a body in beginning from rest, is proportional to the time; and the space described from the beginning of the motion, is as the product of the time and Velocity, or as the square of the Velocity, or as the square of the time. That is, <cb/> in uniformly accelerated motions, <hi rend="italics">v</hi> <figure/> <hi rend="italics">t,</hi> and <hi rend="italics">s</hi> <figure/> <hi rend="italics">tv</hi> or <figure/> <hi rend="italics">v</hi><hi rend="sup">2</hi> or <figure/> <hi rend="italics">t</hi><hi rend="sup">2</hi>. And, in fluxions, <hi rend="italics">s</hi><hi rend="sup">.</hi> = <hi rend="italics">vt</hi><hi rend="sup">.</hi>.</p><p><hi rend="smallcaps">Velocity</hi> <hi rend="italics">of Bodies moving in Curves.</hi>—According to Galileo's system of the fall of heavy bodies, which is now universally admitted among philosophers, the Velocities of a body falling vertically are, at each moment of its fall, as the square roots of the heights from whence it has fallen; reckoning from the beginning of the descent. And hence he inferred, that if a body descend along an inclined plane, the Velocities it has, at the different times, will be in the same ratio: for since its Velocity is all owing to its fall, and it only falls as much as there is perpendicular height in the inclined plane, the Velocity should be still measured by that height, the same as if the fall were vertical.</p><p>The same principle led him also to conclude, that if a body fall through several contiguous inclined planes, making any angles with each other, much like a stick when broken, the Velocity would still be regulated after the same manner, by the vertical heights of the different planes taken together, considering the last Velocity as the same that the body would acquire by a fall through the same perpendicular height.</p><p>This conclusion it seems continued to be acquiesced in, till the year 1672, when it was demonstrated to be false, by James Gregory, in a small piece of his intitled <hi rend="italics">Tentamina quædam Geometrica de Motu Penduli & Projectorum.</hi> This piece has been very little known, because it was only added to the end of an obscure and pseudonymous piece of his, then published, to expose the errors and vanity of Mr. Sinclair, professor of natural philosophy at Glasgow. This little jeu d'esprit of Gregory is intitled, <hi rend="italics">The great and new Art of Weighing Vanity: or a discovery of the Ignorance and Arrogance of the great and new Artist, in his Pseudo-Philosophical writings: by M. Patrick Mathers, Arch-Bedal to the University of S. Andrews.</hi> In the <hi rend="italics">Tentamina,</hi> Gregory shews what the real Velocity is, which a body acquires by descending down two contiguous inclined planes, forming an obtuse angle, and that it is different from the Velocity a body acquires by descending perpendicularly through the same height; also that the Velocity in quitting the first plane, is to that with which it enters the second, and in this latter direction, as radius to the cosine of the angle of inclination between the two planes.</p><p>This conclusion however, Gregory observes, does not apply to the motions of descent down any curve lines, because the contiguous parts of curve lines do not form any angle between them, and consequently no part of the Velocity is lost by passing from one part of the curve to the other; and hence he infers, that the Velocities acquired in descending down a continued curve line, are the same as by falling perpendicularly through the same height. This principle is then applied, by the author, to the motion of pendulums and projectiles.</p><p>Varignon too, in the year 1693, followed in the same track, shewing that the Velocity lost in passing from one right lined direction to another, becomes indefinitely small in the course of a curve line; and that therefore the doctrine of Galileo holds good for the descent of bodies down a curve line, viz, that the Velocity <pb n="646"/><cb/> acquired at any point of the curve, is equal to that which would be acquired by a fall through the same perpendicular altitude.</p><p>The nature of every curve is abundantly determined by the ratio of the ordinates to the corresponding abscisses; and the essence of curves in general may be conceived as consisting in this ratio, which may be varied in a thousand different ways. But this same ratio will be also that of two simple Velocities, by whose joint effect a body may describe the curve in question; and consequently the essence of all curves, in general, is the same thing as the concourse or combination of all the forces which, taken two by two, may move the same body. Thus we have a most simple and general equation of all possible curves, and of all possible Velocities. By means of this equation, as soon as the two simple Velocities of a body are known, the curve resulting from them is immediately determined.</p><p>It may be observed, in particular, according to this equation, that an uniform Velocity, combined with a Velocity that always varies as the square roots of the heights, the two produce the particular curve of a parabola, independent of the angle made by the directions of the two forces that give the Velocities; and consequently a cannon ball, shot either horizontally or obliquely to the horizon, must always describe a parabola, were it not for the resistance of the air.</p><p><hi rend="italics">Circular</hi> <hi rend="smallcaps">Velocity.</hi> See <hi rend="smallcaps">Circular.</hi></p><p><hi rend="italics">Initial</hi> <hi rend="smallcaps">Velocity</hi>, in Gunnery, denotes the Velocity with which military projectiles issue from the mouth of the piece by which they are discharged. This, it is now known, is much more considerable than was formerly apprehended. For the method of estimating it; <cb/> and the result of a variety of experiments, by Mr. Robins, and myself, &c, see the articles <hi rend="smallcaps">Gun</hi>, G<hi rend="smallcaps">UNNERY, Projectile</hi>, and <hi rend="smallcaps">Resistance.</hi></p><p>Mr. Robins had hinted in his New Principles of of Gunnery, at another method of measuring the Initial Velocities of military projectiles, viz, from the arc of vibration of the gun itself, in the act of expulsion, when it is suspended by an axis like a pendulum. And Mr. Thompson, in his experiments (Philos. Trans. vol. 71, p. 229) has pursued the same idea at considerable length, in a number of experiments, from whence he deduces a rule for computing the Velocity, which is somewhat different from that of Mr. Robins, but which agrees very well with his own experiments.</p><p>This rule however being drawn only from the experiments with a musket barrel, and with a small charge of powder, and besides being different from that in the theory as proposed by Robins; it was suspected that it would not hold good when applied to cannon, or other large pieces of ordnance, of different and various lengths, and to larger charges of powder. For this reason, a great multitude of experiments, as related in my Tracts, vol. 1, were instituted with cannon of various lengths and charged with many different quantities of powder; and the Initial Velocities of the shot were computed both from the vibration of a ballistic pendulum, and from the vibration of the gun itself; but the consequence was, that these two hardly ever agreed together, and in many cases they differed by almost 400 feet per second in the Velocity. A brief abstract for a comparison between these two methods, is contained in the following tablet, viz. <hi rend="center"><hi rend="italics">Comparison of the Velocities by the Gun and Pendulum.</hi></hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Gun</cell><cell cols="1" rows="1" rend="colspan=3" role="data">2 Ounces.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">4 Ounces.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">8 Ounces.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">16 Ounces.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">No.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Velocity by</cell><cell cols="1" rows="1" role="data">Diff.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Velocity by</cell><cell cols="1" rows="1" role="data">Diff.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Velocity by</cell><cell cols="1" rows="1" role="data">Diff.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Velocity by</cell><cell cols="1" rows="1" role="data">Diff.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Gun</cell><cell cols="1" rows="1" role="data">Pend.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Gun</cell><cell cols="1" rows="1" role="data">Pend.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Gun</cell><cell cols="1" rows="1" role="data">Pend.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Gun</cell><cell cols="1" rows="1" role="data">Pend.</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">830</cell><cell cols="1" rows="1" role="data">780</cell><cell cols="1" rows="1" rend="align=right" role="data">50</cell><cell cols="1" rows="1" role="data">1135</cell><cell cols="1" rows="1" role="data">1100</cell><cell cols="1" rows="1" rend="align=right" role="data">35</cell><cell cols="1" rows="1" role="data">1445</cell><cell cols="1" rows="1" role="data">1430</cell><cell cols="1" rows="1" rend="align=right" role="data">15</cell><cell cols="1" rows="1" role="data">1345</cell><cell cols="1" rows="1" role="data">1377</cell><cell cols="1" rows="1" role="data">-32</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">863</cell><cell cols="1" rows="1" role="data">835</cell><cell cols="1" rows="1" rend="align=right" role="data">28</cell><cell cols="1" rows="1" role="data">1203</cell><cell cols="1" rows="1" role="data">1180</cell><cell cols="1" rows="1" rend="align=right" role="data">23</cell><cell cols="1" rows="1" role="data">1521</cell><cell cols="1" rows="1" role="data">1580</cell><cell cols="1" rows="1" rend="align=right" role="data">-59</cell><cell cols="1" rows="1" role="data">1485</cell><cell cols="1" rows="1" role="data">1656</cell><cell cols="1" rows="1" role="data">-171</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">919</cell><cell cols="1" rows="1" role="data">920</cell><cell cols="1" rows="1" rend="align=right" role="data">-1</cell><cell cols="1" rows="1" role="data">1294</cell><cell cols="1" rows="1" role="data">1300</cell><cell cols="1" rows="1" rend="align=right" role="data">-6</cell><cell cols="1" rows="1" role="data">1631</cell><cell cols="1" rows="1" role="data">1790</cell><cell cols="1" rows="1" rend="align=right" role="data">-159</cell><cell cols="1" rows="1" role="data">1680</cell><cell cols="1" rows="1" role="data">1998</cell><cell cols="1" rows="1" role="data">-318</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">929</cell><cell cols="1" rows="1" role="data">970</cell><cell cols="1" rows="1" rend="align=right" role="data">-41</cell><cell cols="1" rows="1" role="data">1317</cell><cell cols="1" rows="1" role="data">1370</cell><cell cols="1" rows="1" rend="align=right" role="data">-53</cell><cell cols="1" rows="1" role="data">1669</cell><cell cols="1" rows="1" role="data">1940</cell><cell cols="1" rows="1" rend="align=right" role="data">-271</cell><cell cols="1" rows="1" role="data">1730</cell><cell cols="1" rows="1" role="data">2106</cell><cell cols="1" rows="1" role="data">-376</cell></row></table><cb/></p><p>In this table, the first column shews the number of the gun, as they were of different lengths; viz, the length of number 1 was 30 1/3 inches, number 2 was 40 1/3 inches, number 3 was 60 inches, and number 4 was 83 inches, nearly. After the first column, the rest of the table is divided into four spaces, for the four charges, 2, 4, 8, 16 ounces of powder: and each of these is divided into three columns: in the first of the three is the Velocity of the ball as determined from the vibration of the gun; in the second is the Velocity as determined from the vibration of the pendulum; and in the third is the difference between the two, being so many feet per second, which is marked with the nega- <cb/> tive sign, or —, when the former Velocity is too little, otherwise it is positive.</p><p>From the comparison contained in this table, it appears, in general, that the Velocities, determined by the two different ways, do not agree together; and that therefore the method of determining the Velocity of the ball from the recoil of the gun, is not generally true, although Mr. Robins and Mr. Thompson had suspected it to be so: and consequently that the effect of the inflamed powder on the recoil of the gun, is not exactly the same when it is fired without a ball, as when it is fired with one. It also appears, that this difference is no ways regular, neither in the different <pb n="647"/><cb/> guns with the same charge of powder, nor in the same gun with different charges: That with very small charges, the Velocity by the gun is greater than that by the pendulum; but that the latter always gains upon the former, as the charge is increased, and soon becomes equal to it; and afterwards goes on to exceed it more and more: That the particular charge, at which the two Velocities become equal, is different in the different guns; and that this charge is less, or the equality sooner takes place, as the gun is longer. And all this, whether we use the actual Velocity with which the ball strikes the pendulum, or the same increased by the Velocity lost by the resistance of the air, in its flight from the gun to the pendulum.</p></div1><div1 part="N" n="VENTILATOR" org="uniform" sample="complete" type="entry"><head>VENTILATOR</head><p>, a machine by which the noxious air of any close place, as an hospital, gaol, ship, chamber, &c, may be discharged and changed for fresh air.</p><p>The noxious qualities of bad air have been long known; and Dr. Hales and others have taken great pains to point out the mischiefs arising from foul air, and to prevent o<*> remedy them. That philosopher proposed an easy and effectual one, by the use of his Ventilators; the account of which was read before the Royal Society in May 1741; and a farther account of it may be seen in his Description of Ventilators, printed at London in 8vo, 1743; and still farther in part 2, p. 32, printed in 1758; where the uses and applications of them are pointed out for ships, and prisons, &c. For what is said of the foul air of ships may be applied to that of gaols, mines, workhouses, hospitals, barraeks, &c. In mines, Ventilators may guard against the suffocations, and other terrible accidents arising from damps. The air of gaols has often proved infectious; and we had a fatal proof of this, by the accident that happened some years since at the Old Bailey sessions. After that, Ventilators were used in the prison, which were worked by a small windmill, placed on the top of Newgate; and the prison became more healthy.</p><p>Dr. Hales farther suggests, that Ventilators might be of use in making salt; for which purpose there should be a stream of water to work them; or they might be worked by a windmill, and the brine should be in long narrow canals, covered with boards of canvas, about a foot above the surface of the brine, to confine the stream of air, so as to make it act upon the surface of the brine, and carry off the water in vapours. Thus it might be reduced to a dry salt, with a saving of fuel, in winter and summer, or in rainy weather, or any state of the air whatever. Ventilators, he apprehends, might also serve for drying linen hung in low, long, narrow galleries, especially in damp or rainy weather, and also in drying woollen cloths, after they are fulled or dyed; and in this case, the Ventilators might be worked by the fulling water-mill. Ventilators might also be an useful appendage to malt and hop kilns; and the same author is farther of opinion, that a ventilation of warm dry air from the adjoining stove, with a cautious hand, might be of service to trees and plants in green-houses; where it is well known that air full of the rancid vapours which perspire from the plants, is very unkindly to them, as well as the vapours from human bodies are to men: for fresh air is as necessary <cb/> to the healthy state of vegetables, as of animals.—Ventilators are also of excellent use for drying corn, hopsand malt.—Gunpowder may be thoroughly dried, by blowing air up through it by means of Ventilators; which is of great advantage to the strength of it. These Ventilators, even the smaller ones, will also serve to purify most easily, and effectually, the bad air of a ship's well, before a person is sent down into it, by blowing air through a trunk, reaching near the bottom of it. And in a similar manner may stinking water, and ill tasted milk, &c, be sweetened, viz, by passing a current of air through them, from bottom to top, which will carry the offensive particles along with it.</p><p>For these and other uses to which they might be applied, as well as for a particular account of the construction and disposition of Ventilators in ships, hospitals, prisons, &c, and the benefits attending them, see Hales's Treatise on Ventilators, part 2 passim; and the Philos. Trans. vol. 49, p. 332.</p><p>The method of drawing off air from ships by means of sire-pipes, which some have preferred to Ventilators, was published by Sir Robert Moray in the Philos. Trans. for 1665. These are metal pipes, about 2 1/2 inches diameter, one of which reaches from the fireplace to the well of the ship, and other three branches go to other parts of the ship; the stove hole and ash hole being closed up, the fire is supplied with air through these pipes. The defects of these, compared with Ventilators, are particularly examined by Dr. Hales, ubi supra, p. 113.</p><p>In the latter part of the year 1741, M. Triewald, military architect to the king of Sweden, informed the secretary to the Royal Society, that he had in the preceding spring invented a machine for the use of ships of war, to draw out the foul air from under their decks, which exhausted 36172 cubic feet of air in an hour, or at the rate of 21732 tuns in 24 hours. In 1742 he sent one of these to France, which was approved of by the Academy of Sciences at Paris, and the navy of France was ordered to be furnished with the like Ventilators.</p><p>Mr. Erasmus King proposed to have Ventilators worked by the fire engines, in mines. And Mr. Fitzgerald has suggested an improved method of doing this, which he has also illustrated by figures. See Philos. Trans. vol. 50, p, 727.</p><p>There are various ways of Ventilation, or changing the air of rooms. Mr. Tidd contrived to admit fresh air into a room, by taking out the middle upper sash pane of glass, and fixing in its place a frame box, with a round hole in its middle, about 6 or 7 inches diameter; in which hole are fixed, behind each other, a set of sails of very thin broad copper-plates, which spread over and cover the circular hole, so as to make the air which enters the room, and turning round these sails, to spread round in thin sheets sideways; and so not to incommode persons, by blowing directly upon them, as it would do if it were not hindered by the sails.</p><p>This method however is very unseemly and disagreeable in good rooms: and therefore, instead of it, the late ingenious Mr. John Whitehurst substituted another; which was, to open a small square or rectanglar hole in the party wall of the room, in the upper part near the cieling, at a corner or part distant from the <pb n="648"/><cb/> fire; and before it he placed a thin piece of metal or pasteboard &c, attached to the wall in its lower part just below the hole, but declining from it upwards, so as to give the air, that enters by the hole, a direction upwards against the cieling, along which it sweeps and disperses itself through the room, without blowing in a current against any person. This method is very useful to cure smoky chimneys, by thus admitting conveniently fresh air. A picture placed before the hole prevents the sight of it from disfiguring the room. This, and many other methods of Ventilating, he meant to have published, and was occupied upon, when death put an end to his useful labours. These have since been published, viz in 1794, 4to, by Dr. Willan.</p></div1><div1 part="N" n="VENUS" org="uniform" sample="complete" type="entry"><head>VENUS</head><p>, in Astronomy, one of the inferior planets, but the brightest and to appearance the largest of all the planets; and is designed by the mark <figure/>, supposed to be a rude representation of a female figure, with her trailing robe.</p><p>Venus is easily distinguished from all the other planets, by her whiteness and brightness, in which she exceeds all the rest, even Jupiter himself, and which is so considerable, that in a dusky place she causes an object to project a sensible shadow, and she is often visible in the day-time. Her place in the system is the second from the sun, viz, between Mercury and the earth, and in magnitude is about equal to the earth, or rather a little larger according to Dr. Herschel's observations.</p><p>As Venus moves round the sun, in a circle beneath that of the earth, she is never seen in opposition to him, nor indeed very far from him; but seems to move backward and forward, passing him from side to side, to the distance of about 47 or 48 degrees, both ways, which is her greatest elongation.</p><p>When she appears west of the sun, which is from her inferior conjunction to her superior, she rises before him, or is a morning star, and is called <hi rend="italics">Phosphorus,</hi> or <hi rend="italics">Lucifer,</hi> or the <hi rend="italics">Morning Star;</hi> and when she is eastwards from the sun, which is from her superior conjunction to her inferior, she sets after him, or is an evening star, and is called <hi rend="italics">Hesperus,</hi> or <hi rend="italics">Vesper,</hi> or the <hi rend="italics">Evening star:</hi> being each of those in its turn for 290 days.</p><p>The real diameter of Venus is nearly equal to that of the earth, being about 7900 miles; her apparent mean diameter seen from the earth 59″, seen from the sun, or her horizontal parallax, 30″; but as seen from the earth 18″.79 according to Dr. Herschel: her distance from the sun 70 million of miles; her eccentricity 7/1000 ths of the same, or 490,000 miles; the inclination of her orbit to the plane of the ecliptic 3° 23′; the points of their intersection or nodes are 14° of <hi rend="smallcaps">II</hi> and <figure/>; the place of her aphelion <figure/> 4° 20′; her axis inclined to her orbit 75° 0′; her periodical course round the sun 224 days 17 hours; the diurnal rotation round her axis very uncertain, being according to Cassini only 23 hours, but according to the observations of Bianchini it is in 24 days 8 hours; though Dr. Herschel thinks it cannot be so much. See also <hi rend="smallcaps">Planets.</hi></p><p>Venus, when viewed through a telescope, is rarely seen to shine with a full face, but has phases and changes just like those of the moon, being increasing, decreasing, horned, gibbous, &c: her illuminated part <cb/> being constantly turned toward the sun, or directed toward the cast when she is a morning star, and toward the west when an evening star.</p><p>These different phases of Venus were first discovered by Galileo; who thus fulfilled the prediction of Copernicus: for when this excellent astronomer revived the ancient Pythagorean system, asserting that the earth and planets move round the sun, it was objected that in such a case the phases of Venus should resemble those of the moon; to which Copernicus replied, that some time or other that resemblance would be found out. Galileo sent an account of the first discovery of these phases in a letter, written from Florence in 1611, to William de Medici, the duke of Tuscany's ambassador at Prague; desiring him to communicate it to Kepler. The letter is extant in the preface to Kepler's Dioptrics, and a translation of it in Smith's Optics, p. 416. Having recited the observations he had made, he adds, “We have hence the most certain, sensible decision and demonstration of two grand questions, which to this day have been doubtful and disputed among the greatest masters of reason in the world. One is, that the planets in their own nature are opake bodies, attributing to Mercury what we have seen in Venus: and the other is, that Venus necessarily moves round the sun; as also Mercury and the other planets; a thing well believed indeed by Pythagoras, Copernicus, Kepler, and myself, but never yet proved, as now it is, by ocular inspection upon Venus.”</p><p>Cassini and Campani, in the years 1665 and 1666, discoyered spots in the face of Venus: from the appearances of which the former ascertained her motion round her axis; concluding that this revolution was performed in less than a day; or at least that the bright spot which he observed, finished its period either by revolution or libration in about 23 hours. And de la Hire, in 1700, through a telescope of 16 feet, discovered spots in Venus; which he found to be larger than those in the moon.</p><p>The next observations of the same kind that occur, are those of signior Binanchini at Rome, in 1726, 1727, 1728, who, with Campani's glasses, discovered several dark spots in the disc of Venus, of which he gave an account and a representation in his book entitled Hesperi et Phosphori Nova Phenomena, published at Rome in 1728. From several successive observations Bianchini concludes, that a rotation of Venus about her axis was not completed in 23 hours, as Cassini imagined, but in 24 1/3 days; that the north pole of this rotation faced the 20th degree of Aquarius, and was elevated 15° above the plane of the ecliptic, and that the axis kept parallel to itself, during the planet's revolution about the sun. Cassini the son, though he admits the accuracy of Bianchini's observations, disputes the conclusion drawn from them, and finally observes, that if we suppose the period of the rotation of Venus to be 23 h. 20 min. it agrees equally well with the observations both of his father and Bianchini; but if she revolve in 24 d. 8 h. then his father's observations must be rejected as of no consequence.</p><p>In the Philos. Trans. 1792, are published the results of a course of observations on the planet Venus, begun in the year 1780, by Mr. Schroeter, of Lilientha<*>, Bremen. From these observations, the author infers, <pb n="649"/><cb/> that Venus has an atmosphere in some respects similar to that of our earth, but far exceeding that of the moon in density, or power to weaken the rays of the sun: that the diurnal period of this planet is probably much longer than that of other planets: that the moon also has an atmosphere, though less dense and high than that of Venus: and that the mountains of this planet are 5 or 6 times as high as those on the earth.</p><p>Dr. Herschel too, between the years 1777 and 1793, has made a long series of observations on this planet, accounts of which are given in the Philos. Trans. for 1793. The results of these observations are: that the planet revolves about its axis, but the time of it is uncertain: that the position of its axis is also very uncertain: that the planet's atmosphere is very considerable: that the planet has probably hills and inequalities on its surface, but he has not been able to see much of them, owing perhaps to the great density of its atmosphere; as to the mountains of Venus, no eye, he says, which is not considerably better than his, or assisted by much better instruments, will ever get a sight of them: and that the apparent diameter of Venus, at the mean distance from the earth, is 18″.79; from whence it may be inferred, that this planet is somewhat larger than the earth, instead of being less, as former astronomers have imagined.</p><p>Sometimes Venus is seen in the disc of the sun, in form of a dark round spot. These appearances, called Transits, happen but seldom, viz, when the earth is about her nodes at the time of her inferior conjunction. One of these transits was seen in England in 1639 by Mr. Horrox and Mr. Crabtree; and two in the present century, viz, the one June 6, 1761, and the other in June 1769. There will not happen another of them till the year 1874. See <hi rend="smallcaps">Parallax.</hi></p><p>Except such transits as these, Venus exhibits the same appearances to us regularly every 8 years; her conjunctions, elongations, and times of rising and setting, being very nearly the same, on the same days, as before.</p><p>In 1672 and 1686, Cassini, with a telescope of 34 feet, thought he saw a satellite move round this planet, at the distance of about 3/4 of Venus's diameter. It had the same phases as Venus, but without any well defined form; and its diameter scarce exceeded 1/4 of the diameter of Venus. Dr. Gregory (Astron. lib. 6, prop. 3) thinks it more than probable that this was a satellite; and supposes that the reason why it is not more frequently seen, is the unfitness of its surface to reflect the rays of the sun's light; as is the case of the spots in the moon; for if the whole disc of the moon were composed of such, he thinks she could not be seen so far as to Venus.</p><p>Mr. Short, in 1740, with a reflecting telescope of 16 1/2 inches focus, perceived a small star near Venus: with another telescope of the same focus, magnifying 50 or 60 times, and fitted with a micrometer, he found its distance from Venus about 10′; and with a magnifying power of 240, he observed the star assume the same phases with Venus; its diameter seemed to be about 1/3, or somewhat less, of the diameter of Venus; its light not so bright and vivid, but exceeding sharp and well defined. He viewed it for the space of an hour; but never had the good fortune to see it after the <cb/> first morning. Philos. Trans. number 459, p. 646, or Abr. vol. 8, p. 208.</p><p>M. Montaign, of Limoges in France, preparing for observing the transit of 1761, discovered in the preceding month of May a small star, about the distance of 20′ from Venus, the diameter of it being about 1/4 of that of the planet. Others have also thought they saw a like appearance. And indeed it must be acknowledged, that Venus may have a satellite, though it is difficult for us to see it. Its enlightened side can never be fully turned towards us, but when Venus is beyond the sun; in which case Venus herself appears little bigger than an ordinary star, and therefore her satellite may be too small to be perceived at such a distance. When she is between us and the sun, her moon has its dark side turned towards us; and when Venus is at her greatest elongation, there is but half the enlightened side of the moon turned toward us, and even then it may be too far distant to be seen by us. But it was presumed, that the two transits of 1761, and 1769, would afford opportunity for determining this point; and yet we do not find, although many observers directed their attention to this object, that any satellite was then seen in the sun's disc; unless we except two persons, viz, an anonymous writer in the London Chronicle of May 18, who says that he saw the satellite of Venus on the sun the day of the transit, at St. Neot's in Huntingdonshire; that it moved in a track parallel to that of Venus, but nearer the ecliptic; that Venus quitted the sun's disc at 31 minutes after 8, and the satellite at 6 minutes after 9; and M. Montaign at Limoges, whose account of his observations is in the Memoirs of the Academy of Paris, from whence the following certificate is extracted:—<hi rend="smallcaps">Certificate.</hi> “We having examined, by order of the Academy, the remarks of M. Baudouin on a new observation of the satellite of Venus, made at Limoges the 11th of May by M. Montaign. This fourth observation, of great importance for the theory of the satellite, has shewn that its revolution must be longer than appeared by the first three observations. M. Baudouin believes it may be fixed at 12 days; as to its distance, it appears to him to be 50 semidiameters of Venus; whence he infers that the mass of Venus is equal to that of the earth. This mass of Venus is a very essential element to astronomy, as it enters into many computations, and produces different phenomena: &c. <table><row role="data"><cell cols="1" rows="1" role="data">Signed</cell><cell cols="1" rows="1" role="data">L'Abbé De La Caille,</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">De La Lande.”</cell></row></table></p></div1><div1 part="N" n="VERBERATION" org="uniform" sample="complete" type="entry"><head>VERBERATION</head><p>, in Physics, a term used to express the cause of sound, which arises from a Verberation of the air, when struck, in divers manners, by the several parts of the sonorous body first put into a vibratory motion.</p></div1><div1 part="N" n="VERNAL" org="uniform" sample="complete" type="entry"><head>VERNAL</head><p>, something belonging to the spring season: as vernal signs, vernal equinox, &c.</p></div1><div1 part="N" n="VERNIER" org="uniform" sample="complete" type="entry"><head>VERNIER</head><p>, is a scale, or a division, well adapted for the graduation of mathematical instruments, so called from its inventor Peter Vernier, a gentleman of Franche Comté, who communicated the discovery to the world in a small tract, entitled La Construction, l'Usage, et les Proprietez du Quadrant Nouveau de Mathen a<*>ique &c, printed at Brussels in 1631. This <pb n="650"/><cb/> was an improvement on the method of division proposed by Jacobus Curtius, printed by Tycho in Clavius's Astrolabe, in 1593. Vernier's method of division, or dividing plate, has been very commonly, though erroneously, called by the name of Nonius; the method of Nonius being very different from that of Vernier, and much less convenient.</p><p>When the relative unit of any line is so divided into many small equal parts, those parts may be too numerous to be introduced, or if introduced, they may be too close to one another to be readily counted or estimated; for which reason there have been various methods contrived for estimating the aliquot parts of the small divisions, into which the relative unit of a line may be commodiously divided; and among those methods, Vernier's has been most justly preferred to all others. For the history of this, and other inventions of a similar nature, see Robins's Math. Tracts, vol. 2, p. 265, &c.</p><p>Vernier's scale is a small moveable arch, or scale, sliding along the limb of a quadrant, or any other graduated scale, and divided into equal parts, that are one less in number than the divisions of the portion of the limb corresponding to it. So, if we want to subdivide the graduations on any scale into for ex. 10 equal parts; we must make the Vernier equal in length to 11 of those graduations of the scale, but dividing the same length of the Vernier itself only into 10 equal parts; for then it is evident that each division on the Vernier will be 1/10th part longer than the gradations on the instrument, or that the division of the former is equal to 11/10 of the degree on the latter, as that gains 1 in 10 upon this.</p><p>Thus let AB be a part of the <figure/> upper end of a barometer tube, the quicksilver standing at the point C; from 28 to 31 is a part of the scale of inches, viz, from 28 inches to 31 inches, divided into 10ths of inches; and the middle piece, from 1 to 10, is the Vernier, that slides up and down in a groove, and having 10 of its divisions equal to 11 tenths of the inches, for the purpose of subdividing every 10th of the inch into 10 parts, or, the inches into centesms or 100th parts. In practice, the method of counting is by observing (when the Vernier is set with its index at top pointing exactly against the upper surface of the mercury in the tube) which division of the Vernier it is that exactly, or nearest, coincides with a division in the scale of 10ths of inches, for that will shew the number of 100ths, over the 10ths of inches next below the index at top. So, in the annexed figure, the top of the Vernier is between 2 and 3 tenths above the 30 inches of the barometer; and because the 8th division of the Vernier is seen to coincide with a division of the scale, this shews that it is 8 centesms more: so that the height of the quicksilver altogether, is 30.28, that is, 30 <cb/> inches, and 28 hundredths, or 2 tenths and 8 hundredths.</p><p>If the scale were not inches and 10ths, but degrees of a quadrant, &c, then the 8 would be 8/10 of a degree, or 48′; or if every division on the scale be 10 minutes, then the Vernier will subdivide it into single minutes, and the 8 will then be 8 minutes. And so for any other case.</p><p>By altering the number of divisions, either in the degrees or in the Vernier, or in both, an angle can be observed to many different degrees of accuracy. Thus, if a degree on a quadrant be divided into 12 parts, each being 5 minutes, and the length of the Vernier be 21 such parts, or 1° 3/4, and divided into 20 parts, then 1/12 X 1/20 = 1°/240 = 1′/4 = 15″, is the smallest division the Vernier will measure to: Or, if the length of the Vernier he 2° 7/12, and divided into 30 parts, then 1/12 X 1/30 = 1°/360 = 1′/6 = 10″, is the smallest part in this case: Also 1/12 X 1/50 = 1°/600 = 1′/10 = 6″, is the smallest part when the Vernier extends 4° 1/4. See Robertson's Navigation, book 5, p. 279.</p><p>For the method of applying the Vernier to a quadrant, see <hi rend="italics">Hadley's</hi> <hi rend="smallcaps">Quadrant.</hi> And for the application of it to a telescope, and the principles of its construction, see Smith's Optics, book 3, sect. 861.</p><p>VERSED-<hi rend="italics">Sine,</hi> of an arch, is the part of the diameter intercepted between the sine and the commencement of the arc; and it is equal to the difference between the radius and the cosine. See <hi rend="italics">Versed</hi>-<hi rend="smallcaps">Sine.</hi> And for <hi rend="italics">coversed sine,</hi> see <hi rend="smallcaps">Coversed</hi>-<hi rend="italics">Sine.</hi></p><p>VERTEX <hi rend="italics">of an Angle,</hi> is the angular point, or the point where the legs or sides of the angle meet.</p><p><hi rend="smallcaps">Vertex</hi> <hi rend="italics">of a Figure,</hi> is the uppermost point, or the vertex of the angle opposite to the base.</p><p><hi rend="smallcaps">Vertex</hi> <hi rend="italics">of a Curve,</hi> is the extremity of the axis or diameter, or it is the point where the diameter meets the curve; which is also the vertex of the diameter.</p><p><hi rend="smallcaps">Vertex</hi> <hi rend="italics">of a Glass,</hi> in Optics, the same as its pole.</p><p><hi rend="smallcaps">Vertex</hi> is also used, in Astronomy, for the point of the heavens vertically or perpendicularly over our heads, also called the zenith.</p><div2 part="N" n="Vertex" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Vertex</hi></head><p>, <hi rend="italics">Path of the.</hi> See <hi rend="smallcaps">Path.</hi></p></div2></div1><div1 part="N" n="VERTICAL" org="uniform" sample="complete" type="entry"><head>VERTICAL</head><p>, something relating to the vertex or highest point. As,</p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Point,</hi> in Astronomy, is the same with vertex, or zenith.—Hence a star is said to be Vertical, when it happens to be in that point which is perpendicularly over any place.</p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Circle,</hi> is a great circle of the sphere, passing through the zenith and nadir of a place.—The Vertical circles are also called <hi rend="italics">azimuths.</hi> The meridian of any place is a Vertical circle, viz, that particular one which passes through the north or south point of the horizon.—All the Vertical circles intersect one another in the zenith and nadir. <pb n="651"/><cb/></p><p>The use of the Vertical circles is to estimate or measure the height of the stars &c, and their distances from the zenith, which is reckoned on these circles; and to find their eastern and western amplitude, by observing how many degrees the Vertical, in which the star rises or sets, is distant from the meridian.</p><p><hi rend="italics">Prime</hi> <hi rend="smallcaps">Vertical</hi>, is that Vertical circle, or azimuth, which passes through the poles of the meridian; or which is perpendicular to the meridian, and passes through the equinoctial points.</p><p><hi rend="italics">Prime</hi> <hi rend="smallcaps">Verticals</hi>, in Dialling. See <hi rend="smallcaps">Prime</hi> <hi rend="italics">Verticals.</hi></p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">of the Sun,</hi> is the Vertical which passes through the centre of the sun at any moment of time. —Its use is, in Dialling, to find the declination of the plane on which the dial is to be drawn, which is done by observing how many degrees that Vertical is distant from the meridian, after marking the point or line of the shadow upon the plane at any times.</p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Dial.</hi> See <hi rend="italics">Vertical</hi> <hi rend="smallcaps">Dial.</hi></p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Line,</hi> in Dialling, is a line in any plane perpendicular to the horizon.—This is best found and drawn on an erect and reclining plane, by steadily holding up a string and plummet, and then marking two points of the shadow of the thread on the plane, a good distance from one another: and drawing a line through these marks.</p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Line,</hi> in Conics, is a line drawn on the Vertical plane, and through the vertex of the cone.</p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Line,</hi> in Perspective. See <hi rend="italics">Vertical</hi> <hi rend="smallcaps">Line.</hi></p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Plane,</hi> in Conics, is a plane passing through the vertex of a cone, and parallel to any conic section.</p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Plane,</hi> in Perspective. See <hi rend="smallcaps">Plane</hi> and <hi rend="smallcaps">Perspective.</hi></p><p><hi rend="smallcaps">Vertical</hi> <hi rend="italics">Angles,</hi> or <hi rend="italics">Opposite Angles,</hi> <figure/> in Geometry, are such as have their legs or sides continuations of each other, and which consequently have the same vertex or angular point. So the angles <hi rend="italics">a</hi> and <hi rend="italics">b</hi> are Vertical angles; as also the angles <hi rend="italics">c</hi> and <hi rend="italics">d.</hi></p></div1><div1 part="N" n="VERTICITY" org="uniform" sample="complete" type="entry"><head>VERTICITY</head><p>, is that property of the magnet or loadstone, or of a needle &c touched with it, by which it turns or directs itself to some peculiar point, as to its pole.—The attraction of the magnet was known long before its Verticity.</p></div1><div1 part="N" n="VERU" org="uniform" sample="complete" type="entry"><head>VERU</head><p>, a comet, according to some writers, resembling a spit, being nearly the same as the lonchites, only its head is rounder, and its train longer and sharper pointed.</p></div1><div1 part="N" n="VESPER" org="uniform" sample="complete" type="entry"><head>VESPER</head><p>, in Astronomy, called also <hi rend="italics">Hesperus,</hi> and the <hi rend="italics">Evening Star,</hi> is the planet Venus, when she is eastward of the sun, and consequently sets after him, and shines as an evening star.</p></div1><div1 part="N" n="VESPERTINE" org="uniform" sample="complete" type="entry"><head>VESPERTINE</head><p>, in Astronomy, is when a planet is descending to the west after sun-set, or shines as an evening star.</p><p>VIA <hi rend="smallcaps">Lactea</hi>, in Astronomy, the milky way, or Galaxy. See <hi rend="smallcaps">Galaxy.</hi></p><p><hi rend="smallcaps">Via Solis</hi>, or <hi rend="italics">sun's way,</hi> is used among astronomers, for the ecliptic line, or path in which the sun seems always to move.</p></div1><div1 part="N" n="VIBRATION" org="uniform" sample="complete" type="entry"><head>VIBRATION</head><p>, in Mechanics, a regular reciprocal <cb/> motion of a body, as, for example, a pendulum, which being freely suspended, swings or vibrates from side to side.</p><p>Mechanical authors, instead of Vibration, often use the term <hi rend="italics">oscillation,</hi> especially when speaking of a body that thus swings by means of its own gravity or weight.</p><p>The Vibrations of the same pendulum are all isochronal; that is, they are performed in an equal time, at least in the same latitude; for in lower latitudes they are found to be slower than in higher ones. See P<hi rend="smallcaps">ENDULUM.</hi> In our latitude, a pendulum 39 1/<*> inches long, vibrates seconds, making 60 Vibrations in a minute.</p><p>The Vibrations of a longer pendulum take up more time than those of a shorter one, and that in the subduplicate ratio of the lengths, or the ratio of the square roots of the lengths. Thus, if one pendulum be 40 inches long, and another only 10 inches long, the former will be double the time of the latter in performing a Vibration; for √40 : √10 :: √4 : √1, that is as 2 to 1. And because the number of Vibrations, made in any given time, is reciprocally as the duration of one Vibration, therefore the number of such Vibrations is in the reciprocal subduplicate ratio of the lengths of the pendulums.</p><p>M. Mouton, a priest of Lyons, wrote a treatise, expressly to shew, that by means of the number of Vibrations of a given pendulum, in a certain time, may be established an universal measure throughout the whole world; and may fix the several measures that are in use among us, in such a manner, as that they might be recovered again, if at any time they should chance to be lost, as is the case of most of the ancient measures, which we now only know by conjecture.</p><p><hi rend="italics">The</hi> <hi rend="smallcaps">Vibrations</hi> <hi rend="italics">of a Stretched Chord,</hi> or <hi rend="italics">String,</hi> arise from its elasticity; which power being in this case similar to gravity, as acting uniformly, the Vibrations of a chord follow the same laws as those of pendulums. Consequently the Vibrations of the same chord equally stretched, though they be of unequal lengths, are isochronal, or are performed in equal times; and the squares of the times of Vibration are to one another inversely as their tensions, or powers by which they are stretched.</p><p>The Vibrations of a spring too are proportional to the powers by which it is bent. These follow the same laws as those of the chord and pendulum; and consequently are isochronal; which is the foundation of spring watches.</p><p><hi rend="smallcaps">Vibrations</hi> are also used in <hi rend="italics">Physics,</hi> &c, and for several other regular alternate motions. Sensation, for instance, is supposed to be performed by means of the vibratory motion of the contents of the nerves, begun by external objects, and propagated to the brain.</p><p>This doctrine has been particularly illustrated by Dr. Hartley, who has extended it farther than any other writer, in establishing a new theory of our mental operations.</p><p>The same ingenious author also applies the doctrine of Vibrations to the explanation of muscular motion, which he thinks is performed in the same general manner as sensation and the perception of ideas. For a particular account of his theory, and the arguments by which it is supported, see his Observations on Man. vol. 1. <pb n="652"/><cb/></p><p>The several sorts and rays of light, Newton conceives to make Vibrations of divers magnitudes; which, according to those magnitudes, excite sensations of several colours; much after the same manner as Vibrations of air, according to their several magnitudes, excite sensations of several sounds. See the article C<hi rend="smallcaps">OLOUR.</hi></p><p>Heat, according to the same author, is only an accident of light, occasioned by the rays putting a fine, subtile, ethereal medium, which pervades all bodies, into a vibrative motion, which gives us that sensation. See <hi rend="smallcaps">Heat.</hi></p><p>From the Vibrations or pulses of the same medium, he accounts for the alternate fits of easy reflexion and easy transmission of the rays.</p><p>In the Philosophical Transactions it is observed, that the butterfly, into which the silk-worm is transformed, makes 130 Vibrations or motions of its wings, in one coition.</p></div1><div1 part="N" n="VIETA" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">VIETA</surname> (<foreName full="yes"><hi rend="smallcaps">Francis</hi></foreName>)</persName></head><p>, a very celebrated French mathematician, was born in 1540 at Fontenai, or Fontenaile-Comté, in Lower Poitou, a province of France. He was Master of Requests at Paris, where he died in 1603, being the 63d year of his age. Among other branches of learning in which he excelled, he was one of the most respectable mathematicians of the 16th century, or indeed of any age. His writings abound with marks of great originality, and the finest genius, as well as intense application. His application was such, that he has sometimes remained in his study for three days together, without eating or sleeping. His inventions and improvements in all parts of the mathematics were very considerable. He was in a manner the inventor and introducer of Specious Algebra, in which letters are used instead of numbers, as well as of many beautiful theorems in that science, a full explanation of which may be seen under the article A<hi rend="smallcaps">LGEBRA.</hi> He made also considerable improvements in geometry and trigonometry. His angular sections are a very ingenious and masterly performance: by these he was enabled to resolve the problem of Adrian Roman, proposed to all mathematicians, amounting to an equation of the 45th degree. Romanus was so struck with his sagacity, that he immediately quitted his residence of Wirtzbourg in Franconia, and came to France to visit him, and solicit his friendship. His Apollonius Gallus, being a restoration of Apollonius's tract on Tangencies, and many other geometrical pieces to be found in his works, shew the finest taste and genius for true geometrical speculations.—He gave some masterly tracts on Trigonometry, both plane and spherical, which may be found in the collection of his works, published at Leyden in 1646, by Schooten, besides another large and separate volume in folio, published in the author's life-time at Paris in 1579, containing extensive trigonometrical tables, with the construction and use of the same, which are particularly described in the introduction to my Logarithms, pa. 4 &c. To this complete treatise on Trigonometry, plane and spherical, are subjoined several miscellaneous problems and observations, such as, the quadrature of the circle, the duplication of the cube, &c. Computations are here given of the ratio of the diameter of a circle to the circumference, and of the length of the sine of 1 minute, both <cb/> to a great many places of figures; by which he found that the sine of 1 minute is <table><row role="data"><cell cols="1" rows="1" role="data">between</cell><cell cols="1" rows="1" role="data">2908881959</cell></row><row role="data"><cell cols="1" rows="1" role="data">and</cell><cell cols="1" rows="1" role="data">2908882056;</cell></row></table> also the diameter of a circle being 1000 &c, that the perimeter of the inscribed and circumscribed polygon of 393216 sides, will be as follows, viz, the <table><row role="data"><cell cols="1" rows="1" role="data">perim. of the inscribed polygon</cell><cell cols="1" rows="1" role="data">31415926535</cell></row><row role="data"><cell cols="1" rows="1" role="data">perim. of the circumscribed polygon</cell><cell cols="1" rows="1" role="data">31415926537</cell></row></table> and that therefore the circumference of the circle lies between those two numbers.</p><p>Vieta having observed that there were many faults in the Gregorian Calendar, as it then existed, he composed a new form of it, to which he added perpetual canons, and an explication of it, with remarks and objections against Clavius, whom he accused of having deformed the true Lelian reformation, by not rightly understanding it.</p><p>Besides those, it seems a work greatly esteemed, and the loss of which cannot be sufficiently deplored, was his <hi rend="italics">Harmonicon Cœleste,</hi> which, being communicated to father Mersenne, was, by some perfidious acquaintance of that honest-minded person, surreptitiously taken from him, and irrecoverably lost, or suppressed, to the great detriment of the learned world. There were also, it is said, other works of an astronomical kind, that have been buried in the ruins of time.</p><p>Vieta was also a profound decipherer, an accomplishment that proved very useful to his country. As the different parts of the Spanish monarchy lay very distant from one another, when they had occasion to communicate any secret designs, they wrote them in ciphers and unknown characters, during the disorders of the league: the cipher was composed of more than 500 different characters, which yielded their hidden contents to the penetrating genius of Vieta alone. His skill so disconcerted the Spanish councils for two years, that they published it at Rome, and other parts of Europe, that the French king had only discovered their ciphers by means of magic.</p></div1><div1 part="N" n="VINCULUM" org="uniform" sample="complete" type="entry"><head>VINCULUM</head><p>, in Algebra, a mark or character, either drawn over, or including, or some other way accompanying, a factor, divisor, dividend, &c, when it is compounded of several letters, quantities, or terms, to connect them together as one quantity, and shew that they are to be multiplied, or divided, &c, together.</p><p>Vieta, I think, first used the bar or line over the quantities, for a Vinculum, thus ―(<hi rend="italics">a</hi> + <hi rend="italics">b</hi>); and Albert Girard the parenthesis thus <hi rend="italics">(a</hi> + <hi rend="italics">b);</hi> the former way being now chiefly used by the English, and the latter by most other Europeans. Thus ―(<hi rend="italics">a</hi> + <hi rend="italics">b</hi>) X <hi rend="italics">c,</hi> or <hi rend="italics">(a</hi> + <hi rend="italics">b)</hi> X <hi rend="italics">c,</hi> denotes the product of <hi rend="italics">c</hi> and the sum <hi rend="italics">a</hi> + <hi rend="italics">b</hi> considered as one quantity. Also √(<hi rend="italics">a</hi> + <hi rend="italics">b</hi>), or √<hi rend="italics">(a</hi> + <hi rend="italics">b),</hi> denotes the square root of the sum <hi rend="italics">a</hi> + <hi rend="italics">b.</hi> Sometimes the mark: is set before a compound factor, as a Vinculum, especially when it is very long, or an infinite series; thus 3<hi rend="italics">a</hi> X: 1 - 2<hi rend="italics">x</hi> + 3<hi rend="italics">x</hi><hi rend="sup">2</hi> - 4<hi rend="italics">x</hi><hi rend="sup">3</hi> + 5<hi rend="italics">x</hi><hi rend="sup">5</hi> &c.</p></div1><div1 part="N" n="VINDEMIATRIX" org="uniform" sample="complete" type="entry"><head>VINDEMIATRIX</head><p>, or <hi rend="smallcaps">Vindemiator</hi>, a fixed star of the third magnitude, in the northern wing of the constellation Virgo. <pb n="653"/><cb/></p></div1><div1 part="N" n="VIRGO" org="uniform" sample="complete" type="entry"><head>VIRGO</head><p>, in Astronomy, one of the signs or constellations of the zodiac, which the sun enters about the 21st or 22d of August; being one of the 48 old constellations, and is mentioned by the astronomers of all ages and nations, whose works have reached us. Anciently the figure was that of a girl, almost naked, with an ear of corn in her hand, evidently to denote the time of harvest among the people who invented this sign, whoever they were. But the Greeks much altered the figure, with clothes, wings, &c, and variously explained the origin of it by their own fables: thus, they tell us that the virgin, now exalted into the skies, was, while on earth, that Justitia, the daughter of Astræus and Ancora, who lived in the golden age, and taught mankind their duty; but who, when their crimes increased, was obliged to leave the earth, and take her place in the heavens. Again, Hesiod gives the celestial maid another origin, and says she was the daughter of Jupiter and Themis. There are also others who depart from both these accounts, and make her to have been Erigone, the daughter of Icarius: while others make her Parthene, the daughter of Apollo, who placed her there; and others, from the ear of corn, make it a representation of Ceres; and others, from the obscurity of her head, of Fortune.</p><p>The ancients, as they gave each of the 12 months of the year to the care of some one of the 12 principal deities, so they also threw into the protection of each of these one of the 12 signs of the zodiac. Hence Virgo, from the ear of corn in her hand, naturally fell to the lot of Ceres, and we accordingly find it called Signum Cereris.</p><p>The stars in the constellation Virgo, in Ptolomy's catalogue, are 32; in Tycho's 33; in Hevelius's 50; and in the Britannic 110.</p><p>VIRTUAL <hi rend="italics">Focus,</hi> in Optics, is a point in the axis of a glass, where the continuation of a refracted ray meets it. Thus, let D be the centre, and DBE <figure/> the axis of the glass AB; upon which falls the ray FA. Now this ray will not proceed straight forward, as AH, after passing the glass, but will take the course AK, being deflected from the perpendicular AD. If then the refracted ray KA be produced, by AE, to the axis at E, this point E Mr. Molineux calls the <hi rend="italics">Virtual focus,</hi> or <hi rend="italics">point of divergence.</hi></p></div1><div1 part="N" n="VIS" org="uniform" sample="complete" type="entry"><head>VIS</head><p>, a Latin word, signifying force or power; adopted by writers on physics, to express divers kinds of natural powers or faculties.</p><p>The term Vis is either active or passive: the <hi rend="italics">Vis activa</hi> is the power of producing motion; the <hi rend="italics">Vis passiva</hi> is that of receiving or losing it. The <hi rend="italics">Vis activa</hi> is again subdivided into <hi rend="italics">Vis viva</hi> and <hi rend="italics">Vis mortua.</hi></p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Absoluta,</hi> or <hi rend="italics">absolute force,</hi> is that kind of centripetal force which is measured by the motion that would be generated by it in a given body, at a given <cb/> distance, and depends on the efficacy of the cause producing it.</p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Acceleratrix,</hi> or <hi rend="italics">accelerating force,</hi> is that centripetal force which produces an accelerated motion, and is proportional to the velocity which it generates in a given time; or it is as the motive or absolute force directly, and as the quantity of matter moved inversely.</p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Impressa</hi> is defined by Newton to be the action exercised on any body to change its state, either of rest or moving uniformly in a right line.</p><p>This force consists altogether in the action; and has no place in the body after the action is ceased: for the body perseveres in every new state by the <hi rend="italics">Vis inertiæ</hi> alone.</p><p>This Vis impressa may arise from various causes; as from percussion; pression, and centripetal force.</p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Inertiæ,</hi> or power of inactivity, is defined by Newton to be a power implanted in all matter, by which it resists any change endeavoured to be made in its state, that is, by which it becomes difficult to alter its state, either of rest or motion.</p><p>This power then agrees with the <hi rend="italics">Vis resistendi,</hi> or power of resisting, by which every body endeavours, as much as it can, to persevere in its own state, whether of rest or uniform rectilinear motion; which power is still proportional to the body, or to the quantity of matter in it, the same as the weight or gravity of the body; and yet it is quite different from, and even independent of the force of gravity, and would be and act just the same if the body were devoid of gravity. Thus, a body by this force resists the same in all directions, upwards or downwards or obliquely; whereas gravity acts only downwards.</p><p>Bodies only exert this power in changes brought on their state by some <hi rend="italics">Vis impressa,</hi> force impressed on them. And the exercise of this power is, in different respects, both resistance and impetus; resistance, as the body opposes a force impressed on it to change its state; and impetus, as the same body endeavours to change the state of the resisting obstacle. Phil. Nat. Princ. Math. lib. 1.</p><p>The <hi rend="italics">Vis inertiæ,</hi> the same great author elsewhere observes, is a passive principle, by which bodies persist in their motion or rest, and receive motion, in proportion to the force impressing it, and resist as much as they are resisted. See <hi rend="smallcaps">Resistance.</hi></p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Insita,</hi> or <hi rend="italics">innate force</hi> of matter, is a power of resisting, by which every body, as much as in it lies, endeavours to persevere in its present state, whether of rest or of moving uniformly forward in a right line. This force is always proportional to the quantity of matter in the body, and differs in nothing from the Vis inertiæ, but in our manner of conceiving it.</p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Centripeta.</hi> See <hi rend="smallcaps">Centripetal</hi> <hi rend="italics">Force.</hi></p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Motrix,</hi> or <hi rend="italics">moving force</hi> of a centripetal body, is the tendency of the whole body towards the centre, resulting from the tendency of all the parts, and is proportional to the motion which it generates in a given time; so that the Vis motrix is to the Vis acceleratrix, as the motion is to the celerity: and as the quantity of motion in a body is estimated by the product of the velocity into the quantity of matter, so the Vis motrix <pb n="654"/><cb/> arises from the Vis acceleratrix multiplied by the quantity of matter.</p><p>The followers of Leibnitz use the term <hi rend="italics">Vis motrix</hi> for the force of a body in motion, in the same sense as the Newtonians use the term <hi rend="italics">Vis inertiæ;</hi> this latter they allow to be inherent in a body at rest; but the former, or Vis motrix, a force inherent in the same body only whilst in motion, which actually carries it from place to place, by acting upon it always with the same intensity in every physical part of the line which it describes.</p><p><hi rend="smallcaps">Vis</hi> <hi rend="italics">Mortua,</hi> and <hi rend="smallcaps">Vis</hi> <hi rend="italics">Viva,</hi> in Mechanics, are terms used by Leibnitz and his followers for force, which they distinguish into two kinds, <hi rend="italics">Vis mortua,</hi> and <hi rend="italics">Vis viva;</hi> understanding by the former any kind of pressure, or an endeavour to move, not sufficient to produce actual motion, unless its action on a body be continued for some time; and by the latter, that force or power of acting which resides in a body in motion.</p></div1><div1 part="N" n="VISIBLE" org="uniform" sample="complete" type="entry"><head>VISIBLE</head><p>, something that is an object of vision or right, or the property of a thing seen.</p><p>The Cartesians say that light alone is the proper object of vision. But according to Newton, colour alone is the proper object of sight; colour being that property of light by which the light itself is Visible, and by which the images of opake bodies are painted on the retina. <hi rend="center"><hi rend="italics">As to the Situation and Place of Visible Objects:</hi></hi></p><p>Philosophers in general had formerly taken for granted, that the place to which the eye refers any Visible object, seen by reflection or refraction, is that in which the visual ray meets a perpendicular from the object upon the reflecting or the refracting plane. That this is the case with respect to plane mirrors is universally acknowledged; and some experiments with mirrors of other forms seem to favour the same conclusion, and thus afford reason for extending the analogy to all cases of vision. If a right line be held perpendicularly over a convex or concave mirror, its image seems to make one line with it. The same is the case with a right line held perpendicularly within water; for the part which is within the water seems to be a continuation of that which is without. But Dr. Barrow called in question this method of judging of the place of an object, and so opened a new field of inquiry and debate in this branch of science. This, with other optical investigations, he published in his Optical Lectures, first printed in 1674. According to him, we refer every point of an object to the place from which the pencils of light issue, or from which they would have issued, if no reflecting or refracting substance intervened. Pursuing this principle, Dr. Barrow proceeded to investigate the place in which the rays issuing from each of the points of an object, and that reach the eye after one reflection or refraction, meet; and he found that when the refracting surface was plane, and the refraction was made from a denser medium into a rarer, those rays would always meet in a place between the eye and a perpendicular to the point of incidence. If a convex mirror be used, the case will be the same; but if the mirror be plane, the rays will meet in the perpendicular, and beyond it if it be concave. He also determined, ac- <cb/> cording to these principles, what form the image of a right line will take when it is presented in different manners to a spherical mirror, or when it is seen through a refracting medium.</p><p>Dr. Barrow however notices an objection against the maxim above mentioned, concerning the supposed place of visible objects, and candidly owns that he was not able to give a satisfactory solution of it. The objection is this: Let an object be placed beyond the focus of a convex lens, and if the eye be close to the lens, it will appear confused, but very near to its true place. If the eye be a little withdrawn, the confusion will increase, and the object will seem to come nearer; and when the eye is very near the focus, the confusion will be very great, and the object will seem to be close to the eye. But in this experiment the eye receives no rays but those that are converging; and the point from which they issue is so far from being nearer than the object, that it is beyond it; notwithstanding which the object is conceived to be much nearer than it is, though no very distinct idea can be formed of its precise distance.</p><p>The first person who took much notice of Dr. Barrow's hypothesis, and the difficulty attending it, was Dr. Berkeley, who (in his Essay on a New Theory of Vision, p. 30) observes, that the circle formed upon the retina, by the rays which do not come to a focus, produce the same confusion in the eye, whether they cross one another before they reach the retina, or tend to it afterwards; and therefore that the judgment concerning distance will be the same in both the cases, without any regard to the place from which the rays originally issued; so that in this case, by receding from the lens, as the confusion increases, which always accompanies the nearness of an object, the mind will judge that the object comes nearer. See <hi rend="italics">Apparent</hi> <hi rend="smallcaps">Distance.</hi></p><p>M. Bouguer (in his Traité d' Optique, p. 104) adopts Barrow's general maxim, in supposing that we refer objects to the place from which the pencils of rays seemingly converge at their entrance into the pupil. But when rays issue from below the surface of a vessel of water, or any other refracting medium, he finds that there are always two different places of this seeming convergence: one of them of the rays that issue from it in the same vertical circle, and therefore fall with different degrees of obliquity upon the surface of the refracting medium; and another of those that fall upon the surface with the same degree of obliquity, entering the eye laterally with respect to one another. He says, sometimes one of these images is attended to by the mind, and sometimes the other; and different images may be observed by different persons. And he adds, that an object plunged in water affords an example of this duplicity of images.</p><p>G. W. Krafft has ably supported Barrow's opinion, that the place of any point seen by reflection from the surface of any medium, is that in which rays issuing from it, infinitely near to one another, would meet; and considering the case of a distant object viewed in a concave mirror, by an eye very near it, when the image, according to Euclid and other writers, would be between the eye and the object, and Barrow's rule cannot be applied, he says that in this case the speculum may <pb n="655"/><cb/> be considered as a plane, the effect being the same, only that the image is more obscure. Com. Petrepol. vol. 12, p. 252, 256. See Priestley's Hist. of Light &c, p. 89, 688.</p><p>From the principle above illustrated several remarkable phenomena of vision may be accounted for: as— That if the distance between two Visible objects be an angle that is insensible, the distant bodies will appear as if contiguous: whence, a continuous body being the result of several contiguous ones, if the distances between several Visibles subtend insensible angles, they will appear one continuous body; which gives a pretty illustration of the notion of a continuum.—Hence also parallel lines, and long vistas, consisting of parallel rows of trees, seem to converge more and more the farther they are extended from the eye; and the roofs and floors of long extended alleys seen, the former to descend, and the latter to ascend, and approach each other; because the apparent magnitudes of their perpendicular intervals are perpetually diminishing, while at the same time we mistake their distance. <hi rend="center"><hi rend="italics">As to the Different Distances of Visible Objects:</hi></hi></p><p>The mind perceives the distance of Visible objects, 1st, From the different configurations of the eye, and the manner in which the rays strike the eye, and in which the image is impressed upon it. For the eye disposes itself differently, according to the different distances it is to see; viz, for remote objects the pupil is dilated, and the crystalline brought nearer the retina, and the whole eye is made more globous; on the contrary, for near objects, the pupil is contracted, the crystalline thrust forwards, and the eye lengthened. The mode of performing this however, has greatly divided the opinions of philosophers. See Priestley's Hist. of Light &c, p. 638—652, where the several opinions of Descartes, Kepler, La Hire, are Le Roi, Porterfield, Jurin, Musschenbroek, &c, stated and examined.</p><p>Again, the distance of Visible objects is judged of by the angle the object makes; from the distinct or confused representation of the objects; and from the briskness or feebleness, or the rarity or density of the rays.</p><p>To this it is owing, 1st, That objects which appear obscure or confused, are judged to be more remote; a principle which the painters make use of to cause some of their figures to appear farther distant than others on the same plane. 2d, To this it is likewise owing, that rooms whose walls are whitened, appear the smaller; that fields covered with snow, or white flowers, shew less than when clothed with grass; that mountains covered with snow, in the night time, appear the nearer, and that opake bodies appear the more remote in the twilight. <hi rend="center"><hi rend="italics">The Magnitude of Visible Objects.</hi></hi></p><p>The quantity or magnitude of Visible objects, is known chiefly by the angle contained between two rays drawn from the two extremes of the object to the centre of the eye. An object appears so large as is the angle it subtends; or bodies seen under a greater angle <cb/> appear greater; and those under a less angle less, &c. Hence the same thing appears greater or less as it is nearer the eye or farther off. And this is called the apparent magnitude.</p><p>But to judge of the real magnitude of an object, we must consider the distance; for since a near and a remote object may appear under equal angles, though the magnitudes be different, the distance must necessarily be estimated, because the magnitude is great or small according as the distance is. So that the real magnitude is in the compound ratio of the distance and the apparent magnitude; at least when the subtended angle, or apparent magnitude, is very small; otherwise, the real magnitude will be in a ratio compounded of the distance and the fine of the apparent magnitude, nearly, or nearer still its tangent.</p><p>Hence, objects seen under the same angle, have their magnitudes in the same ratio as their distances. The chord of an arc of a circle appears of equal magnitude from every point in the circumference, though one point be vastly nearer than another. Or if the eye be fixed in any point in the circumference, and a right line be moved round so as its extremes be always in the periphery, it will appear of the same magnitude in every position. And the reason is, because the angle it subtends is always of the same magnitude. And hence also, the eye being placed in any angle of a regular polygon, the sides of it will all appear of equal magnitude; being all equal chords of a circle described about it.</p><p>If the magnitude of an object directly opposite to the eye be equal to its distance from the eye, the whole object will be distinctly seen, or taken in by the eye, but nothing more. And the nearer you approach an object, the less part you see of it.</p><p>The least angle under which an ordinary object becomes visible, is about one minute of a degree. <hi rend="center"><hi rend="italics">Of the Figure of Visible Objects</hi></hi></p><p>This is estimated chiefly from our opinion of the situation of the several parts of the object. This opinion of the situation, &c, enables the mind to apprehend an external object under this or that figure, more justly than any similitude of the images in the retina, with the object can; the images being often elliptical, oblong, &c, when the objects they exhibit to the mind are circles, or squares, &c.</p><p>The laws of vision, with regard to the figures of Visible objects, are,</p><p>1. That if the centre of the eye be exactly in the direction of a right line, the line will appear only as a point.</p><p>2. If the eye be placed in the direction of a surface, it will appear only as a line.</p><p>3. If a body be opposed directly towards the eye, so as only one plane of the surface can radiate on it, the body will appear as a surface.</p><p>4. A remote arch, viewed by an eye in the same plane with it, will appear as a right line.</p><p>5. A sphere, viewed at a distance, appears a circle.</p><p>6. Angular figures, at a distance, appear round.</p><p>7. If the eye look obliquely on the centre of a regular figure, or a circle, the true figure will not be seen; but the circle will appear oval, &c. <pb n="656"/><cb/></p><p><hi rend="smallcaps">Visible</hi> <hi rend="italics">Horizon, Place, &c.</hi> See the substantives.</p></div1><div1 part="N" n="VISION" org="uniform" sample="complete" type="entry"><head>VISION</head><p>, is the act of seeing, or of perceiving external objects by the organ of sight.</p><p>When an object is so disposed, that the rays of light, coming from all parts of it, enter the pupil of the eye, and present its image on the retina, that object is then seen. This is proved by experiment; for if the eye of any animal be taken out, and the skin and fat be carefully stripped off from the back part of it, till only the thin membrane, which is called the retina, remains to terminate it behind, and any object be placed before the front of the eye, the picture of that object will be seen figured as with a pencil on that membrane. There are thousands of experiments which prove that this is the mechanical effect of Vision, or seeing, but none of them all appear so conveniently as this, which is made with the very eye itself of an animal; an eye of an ox newly killed shews this happily, and with very little trouble. It will indeed appear singular in this, that the object is inverted, in the picture thus drawn of it, in the eye; and the case is the same in the eye of a living person.</p><p>Various other opinions however have been held concerning the means of Vision among philosophers.</p><p>The Platonists and Stoics held Vision to be effected by the emission of rays out of the eyes; conceiving that there was a sort of light thus darted out; which, with the light of the external air, taking hold as it were of the objects, rendered them visible; and thus returning back again to the eye, altered and new modified by the contact of the object, made an impression on the pupil, which gave the sensation of the object.</p><p>Our own countryman, Roger Bacon, distinguished as he was in many respects, also assents to the opinion that visual rays proceed from the eye; giving this reason for it, that every thing in nature is qualified to discharge its proper functions by its own powers, in the same manner as the sun, and other celestial bodies. Opus Majus, pa. 289.</p><p>The Epicureans held, that Vision is performed by the emanation of corporeal species or images from objects; or a sort of atomical effluvia continually flying off from the intimate parts of objects, to the eye.</p><p>The Peripatetics hold, with Epicurus, that Vision is performed by the reception of species; but they differ from him in the circumstances; for they will have the species (which they call <hi rend="italics">intentionales</hi>) to be incorporeal. It is true, Aristotle's doctrine of Vision, delivered in his chapter De Aspectu, amounts to no more than this, that objects must have some intermediate body, that by this they may move the organ of sight. To which he adds, in another place, that when we perceive bodies, it is their species, not their matter, that we receive; as a seal makes an impression on wax, without the wax receiving any thing of the seal.</p><p>But this vague and obscure account the Peripatetics have thought sit to improve. Accordingly, what their master calls species, the disciples, understanding of real proper species, assert, that every visible object expresses a perfect image of itself in the air contiguous to it; and this image another, somewhat less, in the next air; and the third another; and so on till the last image ar- <cb/> rives at the crystalline, which they hold for the chief organ of sight, or that which immediately moves the soul. These images they call <hi rend="italics">intentional species.</hi></p><p>The modern philosophers, as the Cartesians and Newtonians, give a better account of Vision. They all agree, that it is performed by rays of light reflected from the several points of objects received in at the pupil, refracted and collected in their passage, through the coats and humours, to the retina; and this striking, or making an impression, on so many points of it; which impression is conveyed, by the correspondent capillaments of the optic nerve, to the brain, &c.</p><p>Baptista Porta's experiments with the camera obscura, about the middle of the 16th century, convinced him that vision is performed by the intermission of something into the eye, and not by visual rays proceeding from the eye, as had been the general opinion before his time; and he was the first who fully satisfied himself and others upon this subject; though several philosophers still adhered to the old opinion.</p><p>As for the Peripatetic series or chain of images, it is a mere chimera; and Aristotle's meaning is better understood without than with them. In fact, setting these aside, the Aristotelian, Cartesian, and Newtonian doctrines of Vision, are very consistent with one another; for Newton imagines that Vision is performed chiefly by the vibrations of a fine medium (which penetrates all bodies) excited in the bottom of the eye by the rays of light, and propagated through the capillaments of the optic nerves, to the sensorium. And Des Cartes maintains, that the sun pressing the materia subtilis, with which the whole universe is every where filled, the vibrations and pulses of that matter reflected from objects, are communicated to the eye, and thence to the sensory: so that the action or vibration of a medium is equally supposed in all.</p><p>It is generally concluded then, that the images of objects are represented on the retina; which is only an expansion of the fine capillaments of the optic nerve, and from whence the optic nerve is continued into the brain. Now any motion or vibration, impressed on one extremity of the nerve, will be propagated to the other: hence the impulse of the several rays, sent from the several points of the object, will be propagated as they are on the retina (that is, in their proper colours, &c, or in particular vibrations, or modes of pressure, corresponding to them) to the place where those capillaments are interwoven into the substance of the brain. And thus is Vision brought to the common case of sensation.</p><p>Experience teaches us that the eye is capable of viewing objects at a certain distance, without any mental exertion. Beyond this distance, no mental exertion can be of any avail: but, within it, the eye possesses a power of adapting itself to the various occasions that occur, the exercise of which depends on the volition of the mind. How this is effected, is a problem that has very much engaged the attention of optical writers: but it is doubted whether it has yet been satisfactorily explained. The first theory for the solution of this problem is that of Kepler. He supposes that the ciliary processes contract the diameter of the eye, and lengthen its axis by a muscular power. But Mr. Thomas Young (in some ingenious Observations on Vision in the Philos. Trans. 1793) ob- <pb n="657"/><cb/> serves, that these processes neither appear to contain any muscular fibres, nor have any attachment by which they can be capable of performing this action.</p><p>Des Cartes ascribed this contraction and elongation to a muscularity of the crystalline, of which he supposed the ciliary processes to be the tendons: but he neither demonstrated this muscularity, nor sufficiently considered the connection with the ciliary processes.</p><p>De la Hire allows of no change in the eye, except the contraction and dilatation of the pupil: this opinion he founds on an experiment which Dr. Smith has shewn to be sallacious. Haller adopted his hypothesis, notwithstanding its inconsistency with the principles of optics and constant experience.</p><p>Dr. Pemberton supposes that the crystalline contains muscular fibres, by which one of its surfaces is flattened, while the other is made convex: but he has not demonstrated the existence of these fibres; and Dr. Jurin has proved that such a change as this is inadequate to the effect.</p><p>Dr. Porterfield conceives that the ciliary processes draw the crystalline forward, and make the cornea more convex. But the ciliary processes are incapable of this action; and it appears from Dr. Jurin's calculations, that a sufficient motion of this kind requires a very visible increase in the length of the axis of the eye; an increase which has never yet been observed.</p><p>Dr. Jurin maintains that the uvea, at its attachment to the cornea, is muscular; and that the contraction of this ring makes the cornea more convex. But this hypothesis is not sufficiently confirmed by observation.</p><p>Musschenbroek conjectures that the relaxation of this ciliary zone, which is nothing but the capsule of the vitreous humour where it receives the impression of the ciliary processes, permits the coats of the eye to push forward the crystalline and cornea. Such a voluntary relaxation however, Mr. Young observes, is wholly without example in the animal economy: besides, if it actually occurred, the coats of the eye could not act as he conceives; nor could they act in this manner without being observed. He adds, that the contraction of the ciliary zone is equally inadequate and unnecessary.</p><p>Mr. Young, having examined these theories, and some others of less moment, proceeds to investigate a more probable solution of this optical difficulty.—Adverting to the observation of Dr. Porterfield, that those who have been couched have not the power of accommodating the eye to different distances; and to the reflections of other writers on this subject; he was led to conclude that the rays of light, emitted by objects at a small distance, could only be brought to foci on the retina by a nearer approach of the crystalline to a spherical form; and he imagined that no other power was capable of producing this change, beside a muscularity of part, or of the whole of its capsule:—but, on closely examining first with the naked eye and then with a magnifier, the crystalline of an ox's eye turned out of its capsule, he discovered a structure which seemed to remove the difficulties that have long embarrassed this branch of optics.</p><p>“The crystalline of the ox, says he, is composed of various similar coats, each of which consists of six muscles, intermixed with a gelatinous substance, and attached to six membranous tendons. Three of the tendons <cb/> are anterior, three posterior; their length is about two-thirds of the semidiameter of the coat; their arrangement is that of three equal and equidistant rays, meeting in the axis of the crystalline; one of the anterior is directed towards the outer angle of the eye, and one of the posterior towards the inner angle, so that the posterior are placed opposite to the middle of the interstices of the anterior: and planes passing through each of the six, and through the axis, would mark on either surface six regular equidistant rays. The muscular fibres arise from both sides of each tendon; they diverge till they reach the greatest circumference of the coat; and, having passed it, they again converge, till they are attached respectively to the sides of the nearest tendons of the opposite surface. The anterior or posterior portion of the six, viewed together, exhibits the appearance of three penniform-radiated muscles. The anterior tendons of all the coats are situated in the same planes, and the posterior ones in the continuations of these planes beyond the axis. Such an arrangement of fibres can be accounted for on no other supposition than that of muscularity. This mass is inclosed in a strong membranous capsule, to which it is loosely connected by minute vessels and nerves; and the connection is more observable near its greatest circumference. Between the mass and its capsule is found a considerable quantity of an aqueous fluid, the liquid of the crystalline.</p><p>“When the will is exerted to view an object at a small distance, the influence of the mind is conveyed through the lenticular ganglion, formed from branches of the third and fifth pair of nerves by the filaments perforating the sclerotica, to the orbiculus ciliaris, which may be considered as an annular plexus of nerves and vessels; and thence by the ciliary processes to the muscle of the crystalline, which, by the contraction of its fibres, becomes more convex, and collects the diverging rays to a focus on the retina. The disposition of fibres in each coat is admirably adapted to produce this change; for, since the least surface that can contain a given bulk is that of a sphere (Simpson's Fluxions, pa. 486) the contraction of any surface must bring its contents nearer to a spherical form. The liquid of the crystalline seems to serve as a synovia in facilitating the motion, and to admit a sufficient change of the muscular part, with a smaller motion of the capsule.</p><p>Mr. Young proceeds to enquire whether these fibres can produce an alteration in the form of the lens sufficiently great to account for the known effects; and he finds, by calculation, that, supposing the crystalline to assume a spherical form, its diameter will be 642 thousandths of an inch, and its focal distance in the eye .926. Then, disregarding the thickness of the cornea, we find (by Smith, art. 370) that such an eye will collect those rays on the retina, which diverge from a point at the distance of 12 inches and 8 tenths. This is a greater change than is necessary for an ox's eye; for if it be supposed capable of distinct Vision at a distance somewhat less than 12 inches, yet it is probably far short of being able to collect parallel rays. The human crystalline is susceptible of a much greater change of form. The ciliary zone may admit of as much extension as this diminution of the diameter of the crystal- <pb n="658"/><cb/> line will require; and its elasticity will assist the cellular texture of the vitreous humour, and perhaps the gelatinous part of the crystalline, in restoring the indolent form.—Mr. Young apprehends that the sole office of the optic nerve is to convey sensation to the brain; and that the retina does not contribute to supply the lens with nerves.—As the human crystalline resembles that of the ox, it may reasonably be presumed that the action of both organs depends on the same general principles.</p><p>This theory of Mr. Young's however is strongly opposed by Dr. Hosack, (Philos. Trans. 1794, part 2, pa. 196). He contests the existence of the muscles, which Mr. Young has described, for several reasons. First, from the transparency they must possess; otherwise there would be some irregularity in the refraction of those rays which pass through the several parts, differing both in shape and density. Another circumstance is the number of these muscles. Mr. Young describes 6 in each lamina; and as Leuwenhoek makes 2000 laminæ in all, therefore the number of muscles must amount to 12 thousand, the action of which, Dr. Hosack apprehends, must exceed comprehension. But the existence of these muscles is still more doubtful, if the accuracy of Dr. Hosack's observations be admitted. With the assistance of the best glasses, and with the greatest attention, he could not discover the structure of the crystalline described by Mr. Young, but found it to be perfectly transparent. He first observed the lens in its viscid state, and then exposed different lenses to a moderate degree of heat, so that they became opaque and dry; and it was easy to separate the distinct layers described by Mr. Young. These were so numerous as not to admit of having, each of them, 6 muscles. Another consideration, which seems to prove that these layers possess no distinct muscles, is that, in this opaque state, they are not visible, but consist of an almost infinite number of concentric fibres, not divided into particular bundles, but similar to as many of the finest hairs of equal thickness, arranged in similar order. This regular structure of layers, composed of concentric sibres, Dr. Hosack thinks is much better adapted to the transmission of the rays of light than the irregular structure of muscles. Besides, it ought to be considered that the crystalline lens is not the most essential organ in viewing objects at different distances; and if this be the case, the power of the eye cannot be owing to any changes in this lens. It is a fact, says Dr. Hosack, that we can, in a great degree, do without it; as is the case after couching or extraction, by which operation all its parts must be destroyed. Dr. Porterfield, however, and Mr. Young, on his authority, maintain that patients, after the operation of couching, have not the power of accommodating the eye to different distances of objects. On the whole, Dr. Hosack concludes that no such muscles, as Mr. Young has described, exist, and that he must have been deceived by some other appearances that resembled muscles: neither will he allow the effects ascribed to the ciliary processes in changing the shape or situation of the lens.</p><p>Dr. Hosack then proceeds to illustrate the structure and use of the external muscles of the eye; which are 6 in number, 4 called recti or straight, and 2 oblique, and by means of which he thinks the business is effect- <cb/> ed. The common purposes to which these muscles are subservient are well known: but beside these, Dr. Hosack suggests that it is not inconsistent with the general laws of nature, nor even with the animal œconomy, to imagine that, from their combination, they should have a different action and an additional use. In describing the precise action of these muscles, he supposes an object to be seen distinctly first at the distance of 6 feet; in which case the picture of it falls exactly on the retina. He then directs his attention to another object at the distance of 6 inches, as nearly as possible in the same line. While he is viewing this, he loses sight of the first object, though the rays proceeding from it still fall on the eye; and hence he infers that the eye must have undergone some change; so that the rays meet either before or behind the retina. But, as rays from a more distant object concur sooner than those from a nearer one, the picture of the more remote object must fall before the retina, while the others form a distinct image upon it. But yet the eye continued in the same place; and therefore the retina must, by some means, have been removed to a greater distance from the forepart of the eye, so as to receive the picture of the nearer object. This object, he contends, could not be seen distinctly, unless the retina were removed to a greater distance, or the refracting power of the media through which the rays passed were augmented:—but as the lens is the chief refracting medium, if we admit that this has no power of changing itself, we are under the necessity of adopting the first of these two suppositions.</p><p>The next object of inquiry is, how the external muscles are capable of producing these changes. The recti are strong, broad, and flat, and arise from the back part of the orbit of the eye; and, passing over the ball as over a pulley, they are inserted by broad flat tendons at the anterior part of the eye. The oblique are inserted towards the posterior part by similar tendons. When these different muscles act jointly, the eye being in the horizontal position, and every muscle in action contracting itself, the four recti by their combination must compress the various parts of the eye and lengthen its axis, while the oblique muscles serve to keep the eye in its proper direction and situation. The convexity of the cornea, by means of its great elasticity, is also increased in proportion to the degree of pressure, and thus the rays of light passing through it are necessarily more converged. The elongation of the eye serves also to lengthen the media, in the aqueous, crystalline, and vitreous humours through which the rays pass, so that their powers of refraction are proportionably increased. This is the general effect of the contraction of the external muscles, according to Dr. Hosack's statement of it: to which it may be added, that we possess the same power of relaxing them in proportion to the greater distance of the object, till we arrive at the utmost extent of indolent Vision. Dr. Hosack also illustrates this hypothesis by some experiments.</p><p>The misrepresentations of Vision often depend upon the distance of the object. Thus, if an opake globe be placed at a moderate distance from the eye, the picture of it upon the retina will be a circle properly diversified with light and shade, so that it will excite in the mind the sensation of a sphere or globe; but, if <pb n="659"/><cb/> the globe be placed at a great distance from the eye, the distance between those lights and shades, which form the picture of a globe, will be imperceptible, and the globe will appear no otherwise than as a circular plane. In a luminous globe, distance is not necessary in order to take off the representation of prominent and flat; an iron bullet, heated very red hot, and held but a few yards distance from the eye, appears a plane, not a prominent body; it has not the look of a globe, but of a circular plane. It is owing to this misrepresentation of Vision that we see the sun and moon flat by the naked eye, and the planets also, through telescopes<*> flat. It is in this light that astronomers, when they speak of the sun, moon, and planets, as they appear to our view, call them the discs of the sun, moon, and planets, which we see.</p><p>The nearer a globe is to the eye, the smaller segment of it is visible, the farther off the greater, and at a due distance the half; and, on the same principle, the nearer the globe is to the eye, the greater is its apparent diameter, that is, under the greater angle it will appear, the farther off the globe is placed, the less is its apparent diameter. This is a proposition of importance, for, on this principle, we know that the same globe, when it appears larger, is nearer to our eye, and, when smaller, is farther off from it. Therefore, as we know that the globes of the sun and moon continue always of the same size, yet appear sometimes larger, and sometimes smaller, to us, it is evident, that they are sometimes nearer, and sometimes farther off from the place whence we view them. Two globes, of different magnitude, may be made to appear of exactly the same diameter, if they be placed at different distances, and those distances be exactly proportioned to their diameters. To this it is owing, that we see the sun and moon nearly of the same diameter; they are, indeed, vastly different in real bulk, but, as the moon is placed greatly nearer to our eyes, the apparent magnitude of that little globe is nearly the same with that of the greater.</p><p>In this instance of the sun and moon (for there cannot be a more striking one) we see the misrepresentation of Vision in two or three several ways. The apparent diameters of these globes are so nearly equal, that, in their several changes of place, they do, at times, appear to us absolutely equal, or mutually greater than one another. This is often to be seen, but it is at no time so obvious, and so perfectly evinced, as in eclipses of the sun, which are total. In these we see the apparent magnitudes of the two globes vary so much according to their distances, that sometimes the moon is large enough exactly to cover the disc of the sun, sometimes it is larger, and a part of it every where extends beyond the disc of the sun; and, on the contrary, sometimes it is smaller, and, though the eclipse be absolutely central, yet it is annular, or a part of the sun's disc is seen in the middle of the eclipsed part, enlightened, and surrounding the opake body of the moon in form of a lucid ring.</p><p>When au object, which is seen above, without other objects of comparison, is of a known magnitude, we judge of its distance by its apparent magnitude; and custom teaches us to do this with tolerable accuracy. This is a practical use of the misrepresentation of Vision, and in the same manner, knowing that we see <cb/> things, which are near us, distinctly, and those which are distant, confusedly, we judge of the distance of an object by the clearness, or confusion, in which we see it. We also judge yet more easily and truly of the distance of an object by comparing it to another seen at the same time, the distance of which is better known, and yet more by comparing it with several others, the distances of which are more or less known, or more or less easily judged of. These are the circumstances which assist us, even by the misrepresentation of Vision, to judge of distance; but, without one or more of these, the eye does not, in reality, enable us to judge concerning the distance of objects.</p><p>This misrepresentation, although it serves us on some occasions, yet is very limited in its effects. Thus, though it helps us greatly in distinguishing the distance of objects that are about us, both with respect to ourselves and them, and with respect to themselves with one another, yet it can do nothing with the very remote. We see that immense concave circle, in which we suppose the fixed stars to be placed, at all this vast remove from us, and no change of place that we could make to get nearer to it, would be of any avail for determining the distance of the stars from one another. If we look at three or four churches from a distance of as many miles, we see them stand in a certain position with regard to one another. If we advance a great deal nearer to them, we see that position differ, but, if we move forward only eight or ten feet, the difference is not seen.</p><div2 part="N" n="Vision" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Vision</hi></head><p>, in Optics. The laws of Vision, brought under mathematical demonstrations, make the subject of Optics, taken in the greatest latitude of that word: for, among mathematical writers, optics is generally taken, in a more restricted signification, for the doctrine of <hi rend="italics">direct Vision;</hi> catoptrics, for the doctrine of <hi rend="italics">reflected Vision;</hi> and dioptrics, for that of <hi rend="italics">refracted Vision.</hi></p><p><hi rend="italics">Direct</hi> or <hi rend="italics">Simple</hi> <hi rend="smallcaps">Vision</hi>, is that which is performed by means of direct rays; that is, of rays passing directly, or in right lines, from the radiant point to the eye. Such is that explained in the preceding article <hi rend="smallcaps">Vision.</hi></p><p><hi rend="italics">Reflected</hi> <hi rend="smallcaps">Vision</hi>, is that which is performed by rays reflected from speculums, or mirrors. The laws of which, see under <hi rend="smallcaps">Reflection</hi>, and <hi rend="smallcaps">Mirror.</hi></p><p><hi rend="italics">Refracted</hi> <hi rend="smallcaps">Vision</hi>, is that which is performed by means of rays refracted, or turned out of their way, by passing through mediums of different density; chiefly through glasses and lenses. The laws of this, see under the article <hi rend="smallcaps">Refraction.</hi></p><p><hi rend="italics">Arch of</hi> <hi rend="smallcaps">Vision.</hi> See <hi rend="smallcaps">Arch.</hi></p><p><hi rend="italics">Distinct</hi> <hi rend="smallcaps">Vision</hi>, is that by which an object is seen distinctly. An object is said to be seen distinctly, when its outlines appear clear and well defined, and the several parts of it, if not too small, are plainly distinguishable, so that they can easily be compared one with another, in respect to their figure, size, and colour.</p><p>In order to such Distinct Vision, it had commonly been thought that all the rays of a pencil, flowing from a physical point of an object, must be exactly united in a physical, or at least in a sensible point of the retina. But Dr. Jurin has made it appear from experiments, that such an exact union of rays is not always necessary <pb n="660"/><cb/> to Distinct Vision. He shews that objects may be seen with sufficient distinctness, though the pencils of rays issuing from the points of them do not unite precisely in the same point on the retina; but that since, in this case, pencils from every point either meet before they reach the retina, or tend to meet beyond it, the light that comes from them must cover a circular spot upon it, and will therefore paint the image larger than perfect Vision would represent it. Whence it follows, that every object placed either too near or too remote for perfect Vision, will appear larger than it is by a penumbra of light, caused by the circular spaces, which are illuminated by pencils of rays proceeding from the extremities of the object.</p><p>The smallest distance of perfect Vision, or that in which the rays of a single pencil are collected into a physical point on the retina in the generality of eyes, Dr. Jurin, from a number of observations, states at 5, 6, or 7 inches. The greatest distance of distinct and perfect Vision he found was more difficult to determine; but by considering the proportion of all the parts of the eye, and the refractive power of each, with the interval that may be discerned between two stars, the distance of which is known, he fixes it, in some cases, at 14 feet 5 inches; though Dr. Porterfield had restricted it to 27 inches only, with respect to his own eye.</p><p>For other observations on this subject, see Jurin's Essay on Distinct and Indistinct Vision, at the end of Smith's Optics; and Robins's Remarks on the same, in his Math. Tracts, vol. 2, pa. 278 &c. See also an ingenious paper on Vision in the Philos. Trans. 1793, pa. 169, by Mr. Thomas Young.</p><p><hi rend="italics">Field of</hi> <hi rend="smallcaps">Vision.</hi> See <hi rend="smallcaps">Field.</hi></p></div2></div1><div1 part="N" n="VISUAL" org="uniform" sample="complete" type="entry"><head>VISUAL</head><p>, relating to sight, or seeing.</p><p><hi rend="smallcaps">Visual</hi> <hi rend="italics">Angle,</hi> is the angle under which an object is seen, or which it subtends. See <hi rend="smallcaps">Angle.</hi></p><p><hi rend="smallcaps">Visual</hi> <hi rend="italics">Line.</hi> See <hi rend="smallcaps">Line.</hi></p><p><hi rend="smallcaps">Visual</hi> <hi rend="italics">Point,</hi> in Perspective, is a point in the horizontal line, where all the ocular rays unite. Thus, a person standing in a straight long gallery, and looking forward; the sides, floor, and cieling seem to meet and touch one another in this point, or common centre.</p><p><hi rend="smallcaps">Visual</hi> <hi rend="italics">Rays,</hi> are lines of light, conceived to come from an object to the eye.</p></div1><div1 part="N" n="VITELLIO" org="uniform" sample="complete" type="entry"><head>VITELLIO</head><p>, or <hi rend="smallcaps">Vitello</hi>, a Polish mathematician of the 13th century, as he flourished about 1254. We have of his a large <hi rend="italics">Treatise on Optics,</hi> the best edition of which is that of 1572. Vitello was the first optical writer of any consequence among the modern Europeans. He collected all that was given by Euclid, Archimedes, Ptolomy, and Alhazen; though his work is of but little use now.</p><p>VITREOUS <hi rend="italics">Humour,</hi> or <hi rend="italics">Vitreus Humor,</hi> denotes the third or glassy humour of the eye; thus called from its resemblance to melted glass. It lies under the crystalline; by the impression of which, its forepart is rendered concave. It greatly exceeds in quantity both the aqueous and crystalline humours taken together, and consequently occupies much the greatest part of the cavity of the globe of the eye. Scheiner says, that the refractive power of this humour is a medium between those of the aqueous, which does not <cb/> differ much from water, and of the crystalline, which is nearly the same with glass. Hawksbee makes its refractive power the same with that of water; and, according to Robertson, its specific gravity agrees nearly with that of water.</p></div1><div1 part="N" n="VITRUVIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">VITRUVIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Marcus Vitruvius Pollio</hi></foreName>)</persName></head><p>, a celebrated Roman architect, of whom however nothing is known, but what is to be collected from his ten books <hi rend="italics">De Architectura,</hi> still extant. In the preface to the sixth book he writes, that he was carefully educated by his parents, and instructed in the whole circle of arts and sciences; a circumstance which he speaks of with much gratitude, laying it down as certain, that no man can be a complete architect, without some knowledge and skill in every one of them. And in the preface to the first book he informs us, that he was known to Julius Cæsar; that he was afterwards recommended by Octavia to her brother Augustus Cæsar; and that he was so favoured and provided for by this emperor, as to be out of all fear of poverty as long as he might live.</p><p>It is supposed that Vitruvius was born either at Rome or Verona; but it is not known which. His books of architecture are addressed to Augustus Cæsar, and not only shew consummate skill in that particular science, but also very uncommon genius and natural abilities. Cardan, in his 16th book <hi rend="italics">De Subtilitate,</hi> ranks Vitruvius as one of the 12 persons, whom he supposes to have excelled all men in the force of genius and invention; and would not have scrupled to have given him the first place, if it could be imagined that he had delivered nothing but his own discoveries. Those 12 persons were, Euclid, Archimedes, Apollonius Pergæus, Aristotle, Archytas of Tarentum, Vitruvius, Achindus, Mahomet Ibn Moses the inventor or improver of Algebra, Duns Scotus, John Suisset surnamed the Calculator, Galen, and Heber of Spain.</p><p>The architecture of Vitruvius has been often printed; but the best edition is that of Amsterdam in 1649. Perrault also, the noted French architect, gave an excellent French translation of the same, and added notes and figures: the first edition of which was published at Paris in 1673, and the second much improved, in 1684. —Mr. William Newton too, an ingenious architect, and late Surveyor to the works at Greenwich Hospital, published in 1780 &c, curious commentaries on Vitruvius, illustrated with figures; to which is added a description, with figures, of the Military Machines used by the Ancients.</p></div1><div1 part="N" n="VIVIANI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">VIVIANI</surname> (<foreName full="yes"><hi rend="smallcaps">Vincentio</hi></foreName>)</persName></head><p>, a celebrated Italian mathematician, was born at Florence in 1621, some say 1622. He was a disciple of the illustrious Galileo, and lived with him from the 17th to the 20th year of his age. After the death of his great master, he passed two or three years more in prosecuting geometrical studies without interruption, and in this time it was that he formed the design of his Restoration of Aristeus. This ancient geometrician, who was contemporary with Euclid, had composed five books of problems <hi rend="italics">De Locis Solidis,</hi> the bare propositions of which were collected by Pappus, but the books are entirely lost; which Viviani undertook to restore by the force of his genius.</p><p>He broke this work off before it was finished, in order to apply himself to another of the same kind, and <pb n="661"/><cb/> that was, to restore the 5th book of Apollonius's Conic Sections. While he was engaged in this, the famous Borelli found, in the library of the Grand Duke of Tuscany, an Arabic manuscript, with a Latin inscription, which imported, that it contained the eight books of Apollonius's Conic Sections: of which the 8th however was not found to be there. He carried this manuscript to Rome, in order to translate it, with the assistance of a professor of the Oriental languages. Viviani, very unwilling to lose the fruits of his labours, procured a certificate that he did not understand the Arabic language, and knew nothing of that manuscript: he was so jealous on this head, that he would not even suffer Borelli to send him an account of any thing relating to it. At length he finished his book, and published it, 1659, in folio, with this title, <hi rend="italics">De Maximis & Minimis Geometrica Divinatio in quintum Conicorum Apollonii Pergæi.</hi> It was found that he had more than divined; as he seemed superior to Apollonius himself.</p><p>After this he was obliged to interrupt his studies for the service of his prince, in an affair of great importance, which was, to prevent the inundations of the Tiber, in which Cassini and he were employed for some time, though nothing was entirely executed.</p><p>In 1664 he had the honour of a pension from Louis the 14th, a prince to whom he was not subject, nor could be useful. In consequence he resolved to finish his Divination upon Aristeus, with a view to dedicate it to that prince; but he was interrupted in this task again by public works, and some negotiations which his master entrusted to him.—In 1666 he was honoured by the Grand Duke with the title of his first mathematician.—He resolved three problems, which had been proposed to all the mathematicians of Europe, and dedicated the work to the memory of Mr. Chapelain, under the title of <hi rend="italics">Enodatio Problematum</hi> &c.—He proposed the problem of the quadrable arc, of which Leibnitz and l'Hospital gave solutions by the Calculus Differentialis.—In 1669, he was chosen to fill, in the Royal Academy of Sciences, a place among the eight foreign associates. This new favour reanimated his zeal; and he published three books of his Divination upon Aristeus, at Florence in 1701, which he dedicated to the King of France. It is a thin folio, intitled, <hi rend="italics">De Locis Solidis secunda Divinatio Geometrica,</hi> &c. This was a second edition enlarged; the first having been printed at Florence in 1673.—Viviani laid out the fortune, which he had raised by the bounties of his prince, in building a magnificent house at Florence; in which he placed a bust of Galileo, with several inscriptions in honour of that great man; and died in 1703, at 81 years of age.</p><p>Viviani had, says Fontenelle, that innocence and simplicity of manners which persons commonly preserve, who have less commerce with men than with books; without that roughness and a certain savage fierceness which those often acquire who have only to deal with books, not with men. He was affable, modest, a fast and faithful friend, and, what includes many virtues in one, he was grateful in the highest degree for favours.</p></div1><div1 part="N" n="ULLAGE" org="uniform" sample="complete" type="entry"><head>ULLAGE</head><p>, <hi rend="italics">of a Cask,</hi> in Gauging, is so much as it wants of being full. <cb/></p></div1><div1 part="N" n="ULTERIOR" org="uniform" sample="complete" type="entry"><head>ULTERIOR</head><p>, in Geography, is applied to some part of a country or province, which, with regard to the rest of that country, is situate on the farther side of a river, or mountain, or other boundary, which divides the country into two parts.</p></div1><div1 part="N" n="ULTRAMUNDANE" org="uniform" sample="complete" type="entry"><head>ULTRAMUNDANE</head><p>, beyond the world, is that part of the universe supposed to be without or beyond the limits of our world or system.</p></div1><div1 part="N" n="UMBILICUS" org="uniform" sample="complete" type="entry"><head>UMBILICUS</head><p>, and <hi rend="smallcaps">Umeilical</hi> <hi rend="italics">Point,</hi> in Geometry, the same with focus.</p></div1><div1 part="N" n="UMBRA" org="uniform" sample="complete" type="entry"><head>UMBRA</head><p>, a Shadow. See <hi rend="smallcaps">Light, Shadow</hi>, P<hi rend="smallcaps">ENUMBRA</hi>, &c.</p></div1><div1 part="N" n="UNCIA" org="uniform" sample="complete" type="entry"><head>UNCIA</head><p>, a term generally used for the 12th part of a thing. In which sense it occurs in Latin writers, both for a weight, called by us an <hi rend="italics">ounce,</hi> and a measure called an <hi rend="italics">inch.</hi></p><p>UNCIÆ, in Algebra, first used by Vieta, are the numbers prefixed to the letters in the terms of any power of a binomial; now more usually, and generally, called <hi rend="italics">coefficients.</hi> Thus, in the 4th power of <hi rend="italics">a</hi> + <hi rend="italics">b,</hi> viz, <hi rend="italics">a</hi><hi rend="sup">4</hi> + 4<hi rend="italics">a</hi><hi rend="sup">3</hi><hi rend="italics">b</hi> + 6<hi rend="italics">a</hi><hi rend="sup">2</hi><hi rend="italics">b</hi><hi rend="sup">2</hi> + 4<hi rend="italics">ab</hi><hi rend="sup">3</hi> + <hi rend="italics">b</hi><hi rend="sup">4</hi>, the Unciæ are 1, 4, 6, 4, 1.</p><p>Briggs first shewed how to find these Unciæ, one from another, in any power, independent of the foregoing powers. They are now usually found by what is called Newton's binomial theorem, which is the same rule as Briggs's in another form. See <hi rend="smallcaps">Binomial.</hi></p></div1><div1 part="N" n="UNDECAGON" org="uniform" sample="complete" type="entry"><head>UNDECAGON</head><p>, is a polygon of eleven sides.</p><p>If the side of a regular Undecagon be 1, its area will be 9.3656399 = 11/4 X tang. of 73 7/11 degrees; and therefore if this number be multiplied by the square of the side of any other regular Undecagon, the product will be the area of that Undecagon. See my Mensuration, pa. 114 &c, 2d edit.</p></div1><div1 part="N" n="UNDETERMINED" org="uniform" sample="complete" type="entry"><head>UNDETERMINED</head><p>, is sometimes used for I<hi rend="smallcaps">NDETERMINATE.</hi></p><p>UNDULATORY <hi rend="italics">Motion,</hi> is applied to a motion in the air, by which its parts are agitated like the waves of the sea; as is supposed to be the case when the string of a musical instrument is struck. This Undulatory motion of the air is supposed the matter or caufe of sound.—Instead of the Undulatory, some authors choose to call this a <hi rend="italics">vibratory</hi> motion.</p><p>UNEVEN <hi rend="italics">Number,</hi> the same as odd number, or such as cannot be divided by 2 without leaving 1 remaining. The series of Uneven Numbers are 1, 3, 5, 7, 9, &c. See <hi rend="smallcaps">Number</hi>, and <hi rend="smallcaps">Odd</hi> <hi rend="italics">Number.</hi></p></div1><div1 part="N" n="UNGULA" org="uniform" sample="complete" type="entry"><head>UNGULA</head><p>, in Geometry, is a part cut off a cylinder, cone, &c, by a plane passing obliquely through the base, and part of the curve surface; so called from its resemblance to the (ungula) hoof of a horse &c. For the contents &c of such Ungulas, see my Mensuration, pa. 218—246, 2d edition.</p></div1><div1 part="N" n="UNICORN" org="uniform" sample="complete" type="entry"><head>UNICORN</head><p>, in Astronomy. See <hi rend="smallcaps">Monoceros.</hi></p></div1><div1 part="N" n="UNIFORM" org="uniform" sample="complete" type="entry"><head>UNIFORM</head><p>, or <hi rend="italics">Equable Motion,</hi> is that by which a body passes always with the same celerity, or over equal spaces in equal times. See <hi rend="smallcaps">Motion.</hi></p><p>In Uniform motions, the spaces described or passed over, are in the compound ratio of the times and velocities; but the spaces are simply as the times, when the velocity is given; and as the velocities, when the time is given. <pb n="662"/><cb/></p><p><hi rend="smallcaps">Uniform</hi> <hi rend="italics">Matter,</hi> in Natural Philosophy, is that which is all of the same kind and texture.</p></div1><div1 part="N" n="UNISON" org="uniform" sample="complete" type="entry"><head>UNISON</head><p>, in Music, is when two sounds are exactly alike, or the same note, or tone.</p><p>What constitutes a Unison, is the equality of the number of vibrations, made in the same time, by the two sonorous bodies.</p><p>It is a noted phenomenon in music, that an intense sound being raised, either with the voice, or a sonorous body, another sonorous body near it, whose tone is either Unison, or octave to that tone, will sound its proper note Unison, or octave, to the given note. The experiment is easily tried with the strings of two instruments; or with a voice and harpsichord; or a bell, or even a drinking glass.</p><p>This phenomenon is thus accounted for: one string being struck, and the air put into a vibratory motion by it; every other string, within the reach of that motion, will receive some impression from it: but each string can only move with a determinate velocity of recourses or vibrations; and all Unisons proceed from equal vibrations; and other concords from other proportions of vibration. The Unison string then, keeping equal pace with the sounding string, as having the same measure of vibrations, must have its motion continued, and still improved, till at length its motion become sensible, and it give a distinct sound. Other concording strings have their motions propagated in different degrees, according to the frequency of the coincidence of their vibrations with those of the sounded string: the octave therefore most sensibly; then the 5th; after which, the crossing of the motions prevents any effect.</p><p>This is illustrated, as Galileo first suggested, by the pendulum, which being set a-moving, the motion may be continued and augmented, by making frequent, light, coincident impulses; as blowing on it when the vibration is just finished: but if it be touched by any cross or opposite motion, and that frequently too, the motion will be interrupted, and cease altogether. So, of two Unison strings, if the one be forcibly struck, it communicates motion, by the air, to the other; and both performing their vibrations together, the motion of that other will be improved and heightened by the frequent impulses received from the vibrations of the first, because given precisely when the other has finished its vibration, and is ready to return: but if the vibrations of the chords be unequal in duration, there will be a crossing of motions, more or less, according to the proportion of the inequality; by which the motion of the untouched string will be so checked, as never to be sensible. And this we find to be the case in all consonances, except Unison, octave, and the fifth.</p></div1><div1 part="N" n="UNIT" org="uniform" sample="complete" type="entry"><head>UNIT</head><p>, <hi rend="smallcaps">Unite</hi>, or <hi rend="smallcaps">Unity</hi>, in Arithmetic, the number one, or one single individual part of discrete quantity. See <hi rend="smallcaps">Number.</hi>—The place of units, is the first place on the right hand in integer numbers.</p><p>According to Euclid, Unity is not a number, for he defines number to be a multitude of Units.</p></div1><div1 part="N" n="UNITY" org="uniform" sample="complete" type="entry"><head>UNITY</head><p>, the abstract or quality which constitutes or denominates a thing <hi rend="italics">one.</hi></p></div1><div1 part="N" n="UNIVERSE" org="uniform" sample="complete" type="entry"><head>UNIVERSE</head><p>, a collective name, signifying the assemblage of heaven and earth, with all things in them.</p><p>The Aneients, and after them the Cartesians, ima- <cb/> gine the Universe to be infinite; and the reason they give is, that it implies a contradiction to suppose it finite or bounded; since it is impossible not to conceive space beyond any limits that can be assigned it; which space, according to the Cartesians, is body, and consequently part of the Universe.</p><p>UNLIKE <hi rend="italics">Quantities,</hi> in Algebra, are such as are expressed by different letters, or by different powers of the same letter. Thus, <hi rend="italics">a,</hi> and <hi rend="italics">b,</hi> and <hi rend="italics">a</hi><hi rend="sup">2</hi>, and <hi rend="italics">ab</hi> are all Unlike quantities.</p><p><hi rend="smallcaps">Unlike</hi> <hi rend="italics">Signs,</hi> are the different signs + and -.</p><p>UNLIMITED or <hi rend="italics">Indeterminate Problem,</hi> is such a one as admits of many, or even of infinite answers. As, to divide a given triangle into two equal parts; or to describe a circle through two given points. See D<hi rend="smallcaps">IOPHANTINE</hi>, and <hi rend="smallcaps">Indeterminate.</hi></p><p>VOID <hi rend="italics">Space,</hi> in Physics. See <hi rend="smallcaps">Vacuum.</hi></p></div1><div1 part="N" n="VOLUTE" org="uniform" sample="complete" type="entry"><head>VOLUTE</head><p>, in Architecture, a kind of spiral scroll, and used in the Ionic and Composite capitals; of which it makes the principal characteristic and ornament.</p></div1><div1 part="N" n="VORTEX" org="uniform" sample="complete" type="entry"><head>VORTEX</head><p>, or <hi rend="italics">Whirlwind,</hi> in Meteorology, a sudden, rapid, violent motion of the air, in circular whirling directions.</p><p><hi rend="smallcaps">Vortex</hi> is also used for an eddy or whirlpool, or a body of water, in certain seas and rivers, which runs rapidly round, forming a sort of cavity in the middle.</p><div2 part="N" n="Vortex" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Vortex</hi></head><p>, in the Cartesian Philosophy, is a system or collection of particles of matter moving the same way, and round the same axis.</p><p>Such Vortices are the grand machines by which these philosophers solve most of the motions and other phenomena of the heavenly bodies. And accordingly, the doctrine of these Vortices makes a great part of the Cartesian philosophy.</p><p>The matter of the world they hold to have been divided at the beginning into innumerable little equal particles, each endowed with an equal degree of motion, both about its own centre, and separately, so as to constitute a fluid.</p><p>Several systems, or collections of this matter, they farther hold to have been endowed with a common motion about certain points, as common centres, placed at equal distances, and that the matters, moving round these, composed so many Vortices.</p><p>Then, the primitive particles of the matter they suppose, by these intestine motions, to become, as it were, ground into spherical figures, and so to compose globules of divers magnitudes; which they call the matter of the second element: and the particles rubbed, or ground off them, to bring them to that form, they call the matter of the first element.</p><p>And since there would be more of the first element than would suffice to fill all the vacuities between the globules of the second, they suppose the remaining part to be driven towards the centre of the Vortex, by the circular motion of the globules; and that being there amassed into a sphere, it would produce a body like the sun.</p><p>This sun being thus formed, and moving about its own axis with the common matter of the Vortex, would necessarily throw out some parts of its matter, through the vacuities of the globules of the second element constituting the Vortex; and this especially at such places as are farthest from its poles; receiving, at the same time, <pb n="663"/><cb/> in, by these poles, as much as it loses in its equatorial parts. And, by this means, it would be able to carry round with it those globules that are nearest, with the greater velocity; and the remoter, with less. And by this means, those globules, which are nearest the centre of the sun, must be smallest; because, were they greater, or equal, they would, by reason of their velocity, have a greater centrifugal force, and recede from the centre. If it should happen, that any of these sunlike bodies, in the centres of the several Vortices, should be so incrustated, and weakened, as to be carried about in the Vortex of the true sun; if it were of less solidity, or had less motion, than the globules towards the extremity of the solar Vortex, it would descend towards the sun, till it met with globules of the same solidity, and susceptible of the same degree of motion with itself; and thus, being fixed there, it would be for ever after carried about by the motion of the Vortex, without either approaching any nearer to the sun, or receding from it; and so would become a planet.</p><p>Supposing then all this; we are next to imagine, that our system was at first divided into several Vortices, in the centre of each of which was a lucid spherical body; and that some of these, being gradually incrustated, were swallowed up by others which were larger, and more powerful, till at length they were all destroyed, and swallowed up by the largest solar Vortex; except some few which were thrown off in right lines from one Vortex to another, and so become comets.</p><p>But this doctrine of Vortices is, at best, merely hypothetical. It does not pretend to shew by what laws and means the celestial motions are effected, so much as by what means they possibly might, in case it should have so pleased the Creator. But we have another principle which accounts for the same phenomena as well, nay, better than that of Vortices; and which we plainly find has an actual existence in the nature of things: and this is gravity, or the weight of bodies.</p><p>The Vortices, then, should be thrown out of philosophy, were it only that two different adequate causes of the same phenomena are inconsistent.</p><p>But there are other objections against them. For, 1°, if the bodies of the planets and comets be carried round the sun in Vortices, the bodies with the parts of the Vortex immediately investing them, must move with the same velocity, and in the same direction; and besides, they must have the same density, or the same vis inertiæ. But it is evident, that the planets and comets move in the very same parts of the heavens with different velocity, and in different directions. It follows, therefore, that those parts of the Vortex must revolve at the same time, in different directions, and with different velocities; since one velocity, and direction, will be required for the passage of the planets, and another for that of the comets.</p><p>2°, If it were granted, that several Vortices are contained in the same space, and do penetrate each other, and revolve with divers motions; since these motions must be conformable to those of the bodies, which are perfectly regular, and performed in conic sections; it may be asked, How they should have been preserved entire so many ages, and not disturbed and confounded <cb/> by the adverse actions and shocks of so much matter as they must meet withal?</p><p>3°, The number of comets is very great, and their motions are perfectly regular, observing the same laws with the planets, and moving in orbits, that are exceedingly eccentric. Accordingly, they move every way, and towards all parts of the heavens, freely pervading the planetary regions, and going frequently contrary to the order of the signs; which would be impossible unless these Vortices were away.</p><p>4°, If the planets move round the sun in Vortices, those parts of the Vortices next the planets, we have already observed, would be equally dense with the planets themselves: consequently the vortical matter, contiguous to the perimeter of the earth's orbit, would be as dense as the earth itself: and that between the orbits of the earth and Saturn, must be as dense, or denser. For a Vortex cannot maintain itself, unless the more dense parts be in the centre, and the less dense towards the circumference: and since the periodical times of the planets are in sesquialterate ratio of their distances from the sun, the parts of the Vortex must be in the same ratio. Whence it follows, that the centrifugal forces of the parts will be reciprocally as the squares of the distances. Such, therefore, as are at a greater distance from the centre, will endeavour to recede from it with the less force. Accordingly, if they be less dense, they must give way to the greater force, by which the parts nearer the centre endeavour to rise. Thus, the more dense will rise, and the less dense descend; and thus there will be a change of places, till the whole fluid matter of the Vortex be so adjusted as that it may rest in equilibrio.</p><p>Thus will the greatest part of the Vortex without the earth's orbit, have a degree of density and inactivity, not less than that of the earth itself. Whence the comets must meet with a very great resistance, which is contrary to all appearances. Cotes, Præf. ad Newt. Princip. The doctrine of Vortices, Newton observes, labours under many difficulties: for a planet to describe areas proportional to the times, the periodical times of a Vortex should be in a duplicate ratio of their distances from the sun; and for the periodical time of the planets, to be in a sesquiplicate proportion of their distances from the sun, the periodical times of the parts of the Vortex should be in the same proportion of their distances: and, lastly, for the less Vortices about Jupiter, Saturn, and the other planets, to be preserved, and swim securely in the sun's Vortex, the periodical times of the sun's Vortex should be equal. None of which proportions are found to obtain in the revolutions of the sun and planets round their axes. Phil. Nat. Princ. Math. apud Schol. Gen. in Calce.</p><p>Besides, the planets, according to this hypothesis, being carried about the sun in ellipses, and having the sun in the focus of each figure, by lines drawn from themselves to the sun, they always describe areas proportionable to the times of their revolutions, which that author shews the parts of no Vortex can do. Schol. prop. ult. lib. ii. Princip.</p><p>Again, Dr. Keill proves, in his Examination of Burnet's Theory, that if the earth were carried in a Vortex, it would move faster in the proportion of three to <pb n="664"/><cb/> two, when it is in Virgo than when it is in Pisces; which all experience proves to be false.</p><p>There is, in the Philosophical Transactions, a Physico-mathematical demonstration of the impossibility and insufficiency of Vortices to account for the Celestial Phenomena; by Mons. de Sigorne. See Num. 457. Sect. vi. pa. 409 et seq.</p><p>This author endeavours to shew, that the mechanical generation of a Vortex is impossible; and that it has only an axifugal force, and not a centrifugal and centripetal one; that it is not sufficient for explaining gravity and its properties; that it destroys Kepler's astronomical laws; and therefore he concludes, with Newton, that the hypothesis of Vortices is fitter to disturb than explain the celestial motions. We must refer to the dissertation itself for the proof of these assertions. See <hi rend="smallcaps">Cartesian Philosophy.</hi></p></div2></div1><div1 part="N" n="VOSSIUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">VOSSIUS</surname> (<foreName full="yes"><hi rend="smallcaps">Gerard John</hi></foreName>)</persName></head><p>, one of the most learned and laborious writers of the 17th century, was of a considerable family in the Netherlands: and was born in 1577, in the Palatinate near Heidelberg, at a place where his father, John Vossius, was minister. He first learned Latin, Greek, and Philosophy at Dort, where his father had settled, and died. In 1595 he went to Leyden, where he farther pursued these studies, joining mathematics to them, in which science he made a considerable progress. He became Master of Arts and Doctor in Philosophy in 1598; and soon after, Director of the College at Dort; then, in 1614, Director of the Theological College just founded at Leyden; and, in 1618, Professor of Eloquence and Chronology in the Academy there, the same year in which appeared his History of the Pelagian Controversy. This history procured him much odium and disgrace on the continent, but an ample reward in England, where archbishop Laud obtained leave of king Charles the 1st for Vossius to hold a prebendary in the church of Canterbury, while he resided at Leyden: this was in 1629, when he came over to be installed, took a Doctor of Laws degree at Oxford, and then returned.—In 1633 he was called to Amsterdam to fill the chair of a Professor of History; where he died in 1649, at 72 years of age; after having written and published as many works as, when they came to be collected and printed at Amsterdam in 1695 &c, made 6 volumes folio, works which will long continue to be read with pleasure and profit. The principal of these are, —1. <hi rend="italics">Etymologicon Linguæ Latinæ.</hi>—2. <hi rend="italics">De Origine & Progressu Idololatriæ.</hi>—3. <hi rend="italics">De Historicis Græcis.</hi>—4. <hi rend="italics">De Historicis Latinis.</hi>—5. <hi rend="italics">De Arte Grammatica.</hi>—6. <hi rend="italics">De Vitiis Sermonis & Glossematis Latino-Barbaris.</hi>—7. <hi rend="italics">Institutiones Oratoriæ.</hi>—8. <hi rend="italics">Institutiones Poeticæ.</hi>—9. <hi rend="italics">Ars Historica.</hi>—10. <hi rend="italics">De quatuor Artibus popularibus, Grammatice, Gymnastice, Musice, & Graphice.</hi>—11. <hi rend="italics">De Philologia.</hi>—12. <hi rend="italics">De Universa Matheseos Natura & Constitutione.</hi>—13. <hi rend="italics">De Philosophia.</hi>—14. <hi rend="italics">De Philosophorum Sectis.</hi>—15. <hi rend="italics">De Veterum Poetarum Temporibus.</hi></p><p><hi rend="smallcaps">Vossius</hi> <hi rend="italics">(Denis),</hi> son of the foregoing Gerard John, died at 22 years of age, a prodigy of learning, whose incessant studies brought on him so immature a death. There are of his, among other smaller pieces, Notes upon Cæsar's Commentaries, and upon Maimonides on Idolatry. <cb/></p><p><hi rend="smallcaps">Vossius</hi> (<hi rend="italics">Francis</hi>), brother of Denis and son of Gerard John, died in 1645, after having published a Latin poem in 1640, on a naval victory gained by the celebrated Van Tromp.</p><p><hi rend="smallcaps">Vossius</hi> (<hi rend="italics">Gerard</hi>), brother of Denis and Francis, and son of Gerard John, wrote Notes upon Paterculus, which were printed in 1639. He was one of the most learned critics of the 17th century; but died in 1640, like his two brothers, at a very early age, and before their father.</p><p><hi rend="smallcaps">Vossius</hi> (<hi rend="italics">Isaac</hi>), was the youngest son of Gerard John, and the only one that survived him. He was born at Leyden in 1618, and was a man of great talents and learning. His father was his only preceptor, and his whole time was spent in studying. His merit recommended him to a correspondence with queen Christina of Sweden, who employed him in some literary commissions. At her request, he made several journeys into Sweden, where he had the honour to teach her the Greek language; though she afterwards discarded him on hearing that he intended to write against Salmasius, for whom she had a particular regard. In 1663 he received a handsome present of money from Louis the 14th of France, accompanied with a complimentary letter from the minister Colbert.—In 1670 he came over to England, when he was created Doctor of Laws at Oxford, and king Charles the 2d made him Canon of Windsor; though he knew his character well enough to say, there was nothing that Vossius refused to believe, excepting the Bible. He appears indeed, by his publications, which are neither so numerous nor so useful as his father's, to have been a most credulous man, while he afforded many circumstances to bring his religious faith in question. He died at his lodgings in Windsor Castle, in 1688; leaving behind him the best private library, as it was then supposed, in the world; which, to the shame and reproach of England, was suffered to be purchased and carried away by the university of Leyden. His publications chiefly were:—1. <hi rend="italics">Periplus Scylacis Caryandensis, &c,</hi> 1639.—2. Justin, with Notes, 1640.—3. <hi rend="italics">Ignatii Epistolæ, & Barnabæ Epistola,</hi> 1646.—4. <hi rend="italics">Pomponius Mela de Situ Orbis,</hi> 1648.—5. <hi rend="italics">Dissertatio de vera Ætate Mundi, &c,</hi> 1659.—6. <hi rend="italics">De Septuaginta Interpretibus, &c,</hi> 1661.—7. <hi rend="italics">De Luce,</hi> 1662.—8. <hi rend="italics">De Motu Marium & Ventorum.</hi>—9. <hi rend="italics">De Nili & aliorum Fluminum Origine.</hi>—10. <hi rend="italics">De Poematum Cantu & Viribus Rythmi,</hi> 1673.—11. <hi rend="italics">De Sybillinis aliisque, quæ Christi natalem præcessere,</hi> 1679.—12. <hi rend="italics">Catullus, & in eum Isaaci Vossii Observationes,</hi> 1684.—13. <hi rend="italics">Variarum Observationum liber,</hi> 1685, in which are contained the following pieces: viz, <hi rend="italics">De Antiquæ Romæ & aliarum quarundam Urbium Magnitudine; De Artibus & Scientiis Sinarum; De Origine & Progressu Pulveris Bellici apud Europæos; De Triremium & Liburnicarum Constructione; De Emendatione Longitudinum; De patefacienda per Septeutrionem ad Japonenses & Indos Navigatione; De apparentibus in Luna circulis; Diurna Telluris conversione omnia gravia ad medium tendere.</hi></p></div1><div1 part="N" n="VOUSSOIRS" org="uniform" sample="complete" type="entry"><head>VOUSSOIRS</head><p>, vault-stones, are the stones which immediately form the arch of a bridge, &c, being cut somewhat in the manner of a truncated pyramid, their under sides constituting the intrados, to which their <pb n="665"/><cb/> joints or ends should be every where in a perpendicular direction.</p><p>The length of the middle Voussoir, or key-stone, and which is the least of all, should be about 1/15th or 1/16th of the span of the arch; from hence these stones should be made larger and larger, all the way down to the impost; that they may the better sustain the great weight which rests upon them, without being crushed or broken, and that they may also bind the firmer together.</p><p>To find the just length of the Voussoirs, or the figure of the extrados, when that of the intrados is given; see my Principles of Bridges, or Emerson's Construction of Arches, in his volume of Miscellanies.</p></div1><div1 part="N" n="URANIBURGH" org="uniform" sample="complete" type="entry"><head>URANIBURGH</head><p>, or celestial town, the name of a celebrated observatory, in a castle in the little island Weenen, in the Sound; built by the celebrated Danish astronomer, Tycho Brahe, who furnished it with instruments for observing the course and motions of the heavenly bodies.</p><p>This observatory, which was finished about the year 1580, had not subsisted above 17 years when Tycho, who little thought to have erected an edifice of so short a duration, and who had even published the figure and position of the heavens, which he had chosen for the moment to lay the first stone in, was obliged to abandon his country.</p><p>Soon after this, the persons to whom the property of the island was given, demolished the building: part of the ruins was dispersed into divers places: the rest served to build Tycho a handsome seat upon his ancient estate, which to this day bears the name of Uraniburgh; and it was here that Tycho composed his catalogue of the stars. Its latitude is 55° 54′ north, and longitude 12° 47′ east of Greenwich.</p><p>M. Picart, making a voyage to Uraniburgh, found that Tycho's meridian line, there drawn, deviated from the meridian of the world; which seems to confirm the conjecture of some persons, that the position of the meridian line may vary.</p></div1><div1 part="N" n="URSA" org="uniform" sample="complete" type="entry"><head>URSA</head><p>, in Astronomy, the Bear, a name common to two constellations of the northern hemisphere, near the pole, distinguished by <hi rend="italics">Major</hi> and <hi rend="italics">Minor.</hi></p><p><hi rend="smallcaps">Ursa</hi> <hi rend="italics">Major,</hi> or the <hi rend="italics">Great Bear,</hi> one of the 48 old constellations, and perhaps more ancient than many of the others; being familiarly known and alluded to by the oldest writers, and is mentioned by Homer as observed by navigators. It is supposed that this constellation is that mentioned in the book of Job, under the name of <hi rend="italics">Chesil,</hi> which our translation has rendered Orion, where it is said, “Canst thou loose the bands of Chesil (Orion)?” It is farther said that the Ancients represented each of these two constellations under the form of a waggon drawn by a team of horses, and the Greeks originally called them waggons and two bears; they are to this day popularly called the wains, or waggons, and the greater of them Charles's Wain. Hence is remarked the propriety of the expression, “loose the bands &c,” the binding and loosing being terms very applicable to a harness, &c.</p><p>Perhaps the Egyptians, or whoever else were the people that invented the constellations, placed those stats, which are near the pole, in the figure of a bear, as being an animal inhabiting towards the north pole, and making neither long journeys, nor swist motions. <cb/> But the Greeks, in their usual way, have adapted some of their fables to it. They say this bear was Callisto, daughter of Lycaon, king of Arcadia; that being debauched by Jupiter, he afterwards placed her in the heavens, as well as her son Arcturus.</p><p>The Greeks called this constellation Arctos and Helice, from its turning round the pole. The Latins from the name of the nymph, as variously written, Callisto, Megisto, and Flemisto, and from the Arabians, sometimes Feretrum Majus, the Great Bier. And the Ursa Minor, they called Feretrum Minus, the Little Bier. The Italians have followed the same custom, and call them Cataletto. They spoke also of the Phenicians being guided by the Lesser Bear, but the Greeks by the Greater.</p><p>There are two remarkable stars in this constellation, viz, those in the middle of his body, considered as the two hindermost of the wain, and called the pointers, because they always point nearly in a direction towards the north pole star, and so are useful in finding this star out.</p><p>The stars in Ursa Major, are, according to Ptolomy's catalogue, 35; in Tycho's 56; in Hevelius's 73; but in the Britannic catalogue 87.</p><p><hi rend="smallcaps">Ursa</hi> <hi rend="italics">Minor,</hi> the <hi rend="italics">Little Bear,</hi> called also <hi rend="italics">Arctos Minor, Phœnice,</hi> and <hi rend="italics">Cynosura,</hi> one of the 48 old constellations, and near the north pole, the large star in the tip of its tale being very near to it, and thence called the pole-star.</p><p>The Phenicians guided their navigations by this constellation, for which reason it was called Phenice, or the Phenician constellation. It was also called Cynosura by the Greeks, because, according to some, that was one of the dogs of the huntress Callisto, or the Great Bear; but according to others Cynosura was one of the Idæan nymphs that nursed the infant Jupiter; and some say that Callisto was another of them, and that, for their care, they were taken up together to the skies.</p><p>Ptolomy places in this constellation 8 stars, Tycho 7, Hevelius 12, and Flamsteed 24.</p></div1><div1 part="N" n="URSUS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">URSUS</surname> (<foreName full="yes"><hi rend="smallcaps">Nicholas Ratmarus</hi></foreName>)</persName></head><p>, a very extraordinary person, and distinguished in the science of astronomy, was born at Henstedt in Dithmarsen, in the duchy of Holstein, about the year 1550 He was a swineherd in his youth, and did not begin to read till he was 18 years of age; but then he employed all the hours he could spare from his daily labour, in learning to read and write. He afterwards applied himself to learn the languages; and, having a strong genius, made a rapid progress in Greek and Latin. He quickly learned also the French language, the mathematics, astronomy, and philosophy; and most of them without the assistance of a master.</p><p>Having left his native country, he gained a maintenance by teaching; which he did in Denmark in 1584, and on the frontiers of Pomerania and Poland in 1585. It was in this place that he invented a new system of astronomy, very little different from that of Tycho Brahe. This he communicated, in 1586, to the landgrave of Hesse, which gave rise to a terrible dispute between him and Tycho. This celebrated astronomer charged him with being a plagiary: who, as he related, happening to come with his master into his study, saw there, drawn on a piece of paper, the figure of <pb n="666"/><cb/> his system; and afterwards insolently boasted that he himself was the inventor of it. Ursus, upon this accusation, wrote furiously against Tycho, called the honour of his invention in question, ascribing the system to Apollonius Pergæus; and in short abused him in so brutal a manner, that he was going to be prosecuted for it.</p><p>Ursus was afterwards invited by the emperor to teach the mathematics in Prague; from which city, to avoid the presence of Tycho, he withdrew silently in 1589, and died soon after. <cb/></p><p>He made some improvements in trigonometry, and wrote several books, which discover the marks of his hasty studies; his erudition being indigested, and his style incorrect, as is almost always to be observed of persons that are late-learned.</p><p>VULPECULA <hi rend="italics">et</hi> <hi rend="smallcaps">Anser</hi>, the <hi rend="italics">Fox and Goose,</hi> in Astronomy, one of the new constellations of the northern hemisphere, made out of the unformed stars by Hevelius, in which he reckons 27 stars; but Flamsteed counts 35. </p></div1></div0><div0 part="N" n="W" org="uniform" sample="complete" type="alphabetic letter"><head>W</head><cb/><div1 part="N" n="WAD" org="uniform" sample="complete" type="entry"><head>WAD</head><p>, or <hi rend="smallcaps">Wadding</hi>, in Gunnery, a stopple of paper, hay, straw, old rope-yarn, or tow, rolled sirmly up like a ball, or a short cylinder, and forced into a gun upon the powder, to keep it close in the chamber; or put up close to the shot, to keep it from rolling out, as well as, according to some, to prevent the inflamed powder from dilating round the sides of the ball, by its windage, as it passes along the chace, which it was thought would much diminish the effort of the powder. But, from the accurate experiments lately made at Woolwich, it has not been found to have any such effect.</p></div1><div1 part="N" n="WADHOOK" org="uniform" sample="complete" type="entry"><head>WADHOOK</head><p>, or <hi rend="smallcaps">Worm</hi>, a long pole with a screw at the end, to draw out the wad, or the charge, or paper &c from a gun.</p></div1><div1 part="N" n="WAGGONER" org="uniform" sample="complete" type="entry"><head>WAGGONER</head><p>, in Astronomy, is the constellation Ursa Major, or the Great Bear, called also vulgarly Charles's Wain.</p><p><hi rend="smallcaps">Waggoner</hi> is also used for a routier, or book of charts, describing the seas, their coasts, &c.</p></div1><div1 part="N" n="WALLIS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WALLIS</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an eminent English mathematician, was the son of a clergyman, and born at Ashford in Kent, Nov. 23. 1616. After being instructed, at different schools, in grammar learning, in Latin, Greek, and Hebrew, with the rudiments of logic, music, and the French language, he was placed in Emanuel college, Cambridge. About 1640 he entered into orders, and was chosen fellow of Queen's college. He kept his fellowship till it was vacated by his marriage, but quitted his college to be chaplain to Sir Richard Darley; after a year spent in this situation, he spent two more as chaplain to lady Vere. While he lived in this family, he cultivated the art of deciphering, which proved very useful to him on several occasions: he met with rewards and preferment from the government at home for deciphering letters for them; and it is said, that the elector of Brandenburg sent him a gold chain and medal, for explaining for him some letters written in ciphers. <cb/></p><p>In 1643 he published <hi rend="italics">Truth Tryed,</hi> or Animadversions on lord Brooke's treatise, called The Nature of Truth &c; styling himself “a minister in London,” probably of St. Gabriel Fenchurch, the sequestration of which had been granted to him.—In 1644 he was chosen one of the scribes or secretaries to the assembly of divines at Westminster.</p><p>Academical studies being much interrupted by the civil wars in both the universities, many learned men from them resorted to London, and formed assemblies there. Wallis belonged to one of these, the members of which met once a week, to discourse on philosophical matters; and this society was the rise and beginning of that which was afterwards incorporated by the name of the Royal Society, of which Wallis was one of the most early members.</p><p>The Savilian professor of geometry at Oxford being ejected by the parliamentary visitors, in 1649, Wallis was appointed to succeed him, and he opened his lec. tures there the same year. In 1650 he published some Animadversions on a book of Mr. Baxter's, intitled, “Aphorisms of Justisication and the Covenant.” And in 1653, in Latin, a Grammar of the English tongue, for the use of foreigners; to which was added, a tract <hi rend="italics">De Loquela seu Sonorum formatione, &c,</hi> in which he considers philosophically, the formation of all sounds used in articulate speech, and shews how the organs being put into certain positions, and the breath pushed out from the lungs, the person will thus be made to speak, whether he hear himself or not. Pursuing these reflections, he was led to think it possible, that a deaf person might be taught to speak, by being directed so to apply the organs of speech, as the sound of each letter required, which children learn by imitation and srequent attempts, rather than by art. He made a trial or two with success; and particularly upon one Popham, which involved him in a dispute with Dr. Holder, of which some account has already been given in the life of that gentleman. <pb n="667"/><cb/></p><p>In 1654 he took the degree of Doctor in Divinity; and the year after became engaged in a long controversy with Mr. Hobbes. This philosopher having, in 1655, printed his treatise <hi rend="italics">De Corpore Philosophico,</hi> Dr. Wallis the same year wrote a confutation of it in Latin, under the title of <hi rend="italics">Elenchus Geometriæ Hobbianæ;</hi> which so provoked Hobbes, that in 1656 he published it in English, with the addition of what he called, “Six Lessons to the Professors of Mathematics in Oxford.” Upon this Dr. Wallis wrote an answer in English, intitled, “Due Correction for Mr. Hobbes; or School discipline for not saying his Lessons right,” 1656: to which Mr. Hobbes replied in a pamphlet called “<foreign xml:lang="greek">*s*t*i*g*m*a*i</foreign>, &c, or Marks of the absurd Geometry, Rural Language, Scottish Church politics, and Barbarisms, of John Wallis, 1657.” This was immediately rejoined to by Dr. Wallis, in <hi rend="italics">Hobbiani Puncti Dispunctio,</hi> 1657. And here this controversy seems to have ended, at this time: but in 1661 Mr. Hobbes printed <hi rend="italics">Examinatio & Emendatio Mathematicorum Hodiernorum in sex Dialogis;</hi> which occasioned Dr. Wallis to publish the next year, <hi rend="italics">Hobbius Hcautontimorumenos,</hi> addressed to Mr. Boyle.</p><p>In 1657 he collected and published his mathematical works, in two parts, entitled, <hi rend="italics">Mathesis Universalis,</hi> in 4to; and in 1658, <hi rend="italics">Commercium Epistolicum de Quæstionibus quibusdam Mathematicis nuper habitum,</hi> in 4to; which was a collection of letters written by many learned men, as Lord Brounker, Sir Kenelm Digby, Fermat, Schooten, Wallis, and others.</p><p>He was this year chosen <hi rend="italics">Custos Archivorum</hi> of the university. Upon this occasion Mr. Stubbe, who, on account of his friend Mr. Hobbes, had before waged war against Wallis, published a pamphlet, intitled, “The Savilian Professor's Case Stated,” 1658. Dr. Wallis replied to this; and Mr. Stubbe republished his case, with enlargements, and a vindication against the exceptions of Dr. Wallis.</p><p>Upon the Restoration he met with great respect; the king thinking favourably of him on account of some services he had done both to himself and his father Charles the first. He was therefore confirmed in his places, also admitted one of the king's chaplains in ordinary, and appointed one of the divines empowered to revise the book of Common Prayer. He complied with the terms of the act of uniformity, and continued a steady conformist till his death. He was a very useful member of the Royal Society; and kept up a literary correspondence with many learned men. In 1670 he published his <hi rend="italics">Mechanica; sive de Motu,</hi> 4to. In 1676 he gave an edition of <hi rend="italics">Archimedis Syracusani Arenarius & Dimensio Circuli;</hi> and in 1682 he published from the manuscripts, <hi rend="italics">Claudii Ptolomæi Opus Harmonicum,</hi> in Greek, with a Latin version and notes; to which he afterwards added, <hi rend="italics">Appendix de veterum Harmonica ad hodiernam comparata, &c.</hi> In 1685 he published some theological pieces; and, about 1690, was engaged in a dispute with the Unitarians; also, in 1692, in another dispute about the Sabbath. Indeed his books upon subjects of divinity are very numerous, but nothing near so important as his mathematical works.</p><p>In 1685 he published his History and Practice of Algebra, in folio; a work that is full of learned and useful matter. Besides the works above mentioned, he <cb/> published many others, particularly his <hi rend="italics">Arithmetic of Infinites,</hi> a book of genius and good invention, and perhaps almost his only work that is so, for he was much more distinguished for his industry and judgment, than for his genius. Also a multitude of papers in the Philos. Trans. in almost every volume, from the 1st to the 25th volume. In 1697, the curators of the University press at Oxford thought it for the honour of the university to collect the doctor's mathematical works, which had been printed separately, some in Latin, some in English, and published them all together in the Latin tongue, in 3 vols folio, 1699.</p><p>Dr. Wallis died at Oxford the 28th of October 1703, in the 88th year of his age, leaving behind him one son and two daughters. We are told that he was of a vigorous constitution, and of a mind which was strong, calm, serene, and not easily ruffled or discomposed. He speaks of himself, in his letter to Mr. Smith, in a strain which shews him to have been a very cautious and prudent man, whatever his secret opinions and attachments might be: he concludes, “It hath been my endeavour all along to act by moderate principles, being willing, whatever side was uppermost, to promote any good design, for the true interest of religion, of learning, and of the public good.”</p></div1><div1 part="N" n="WARD" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WARD</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">Seth</hi></foreName>)</persName></head><p>, an English prelate, chiefly famous for his knowledge in mathematics and astronomy, was the son of an attorney, and born at Buntingford, Hertfordshire, in 1617 or 1618. From hence he was removed and placed a student in Sidney college, Cambridge, in 1632. Here he applied with great vigour to his studies, particularly to the mathematics, and was chosen fellow of his college. In 1640 he was pitched upon by the Vice chancellor to be Prævaricator, which at Oxford is called Terræ-filius; whose office it was to make a witty speech, and to laugh at any thing or any body: a privilege which he exercised so freely, that the Vice-chancellor actually suspended him from his degree; though he reversed the censure the day following.</p><p>The civil war breaking out, Ward was involved not a little in the consequences of it. He was ejected from his fellowship for refusing the Covenant; against which he soon after joined with several others, in drawing up that noted treatise, which was afterwards printed. Being now obliged to leave Cambridge, he resided for some time with certain friends about London, and at other times at Aldbury in Surry, with the noted mathematician Oughtred, where he prosecuted his mathematical studies. He afterwards lived for the most part, till 1649, with Mr. Ralph Freeman at Aspenden in Hertfordshire, whose sons he instructed as their preceptor; after which he resided some months with lord Wenman, of Thame Park, in Oxfordshire.</p><p>He had not been long in this family before the visitation of the university of Oxford began; the effect of which was, that many learned and eminent persons were turned out, and among them Mr. Greaves, the Savilian professor of Astronomy: this gentleman laboured to procure Ward for his successor, whose abilities in his way were universally known and acknowledged; and effected it; Dr. Wallis succeeding to the Geometry professorship at the same time. Mr. Ward then entered himself of Wadham college, for the sake of <pb n="668"/><cb/> Dr. Wilkins, who was the warden; and he presently applied himself to bring the astronomy lectures, which had long been neglected and disused, into repute again; and for this purpose he read them very constantly, never missing one reading day, all the while he held the lecture.</p><p>In 1654, both the Savilian prosessors did their exercises, in order to proceed doctors in divinity; and when they were to be presented, Wallis claimed precedency. This occasioned a dispute; which being decided in favour of Ward, who was really the senior, Wallis went out grand compounder, and so obtained the precedency. In 1659, Ward was chosen president of Trinity college; but was obliged at the Restoration to resign that place. He had amends made him, however, by being presented in 1660 to the rectory of St. Laurence Jewry. The same year he was also installed precentor of the church of Exeter. In 1661 he became fellow of the Royal Society, and dean of Exeter; and the year following he was advanced to the bishopric of the same church. In 1667 he was translated to the see of Salisbury; and in 1671 was made chancellor of the order of the garter; an honour which he procured to be permanently annexed to the see of Salisbury, after it had been held by laymen for above 150 years.</p><p>Dr. Ward was one of those unhappy persons who have the misfortune to survive their senses, which happened in consequence of a fever ill cured: he lived till the Revolution, but without knowing any thing of the matter; and died in January 1689, about 71 years of age. He was the author of several Latin works in astronomy and different parts of the mathematics, which were thought excellent in their day; but their use has been superseded by later improvements and the Newtonian philosophy. Some of these were,</p><p>1. A Philosophical Essay towards an Eviction of the Being and Attributes of God, &c. 1652.</p><p>2. De Cometis, &c; 4to, 1653.</p><p>3. In Ismaelis Bullialdi Astronomia Inquisitio; 4to, 1653.</p><p>4. Idea Trigonometriæ demonstratæ; 4to, 1654.</p><p>5. Astronomia Geometrica; 8vo, 1656. In this work, a method is proposed, by which the astronomy of the planets is geometrically resolved, either upon the Elliptical or Circular motion; it being in the third or last part of this work that he proposes and explains what is called Ward's Circular Hypothesis.</p><p>6. Exercitatio epistolica in Thomæ Hobbii Philosophiam, ad D. Joannem Wilkins; 1656, 8vo.</p><p>But that by which he hath chiefly signalized himself, as to astronomical invention, is his celebrated approximation to the true place of a planet, from a given mean anomaly, founded upon an hypothesis, that the motion of a planet, though it be really performed in an elliptic orbit, may yet be considered as equable as to angular velocity, or with an uniform circular motion round the upper focus of the ellipse, or that next the aphelion, as a centre. By this means he rendered the praxis of calculation much easier than any that could be used in resolving what has been commonly called Kepler's problem, in which the coequate anomaly was to be immediately investigated from the mean elliptic one. His hypothesis agrees very well with those orbits <cb/> which are elliptical but in a very small degree, as that of the Earth and Venus: but in others, that are more elliptical, as those of Mercury, Mars, &c, this approximation stood in need of a correction, which was made by Bulliald. Both the method, and the correction, are very well explained and demonstrated, by Keill, in his Astronomy, lecture 24.</p></div1><div1 part="N" n="WARGENTIN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WARGENTIN</surname> (<foreName full="yes"><hi rend="smallcaps">Peter</hi></foreName>)</persName></head><p>, an ingenious Swedish mathematician and astronomer, was born Sept. 22, 1717, and died Dec. 13, 1783. He became secretary to the Academy at Stockholm in 1749, when he was only 32 years of age; and he became successively a member of most of the literary academies in Europe, as London, Paris, Petersburg, Gottingen, Upsal, Copenhagen, Drontheim, &c. In this country he is probably most known on account of his tables for computing the eclipses of Jupiter's satellites, which are annexed to the Nautical Almanac of 1779. I know not that he has published any separate work; but his communications were very numerous to several of those Academies of which he was a member; as the Academy of Stockholm, in which are 52 of his memoirs; in the Philosophical Transactions, the Upsal Acts, the Paris Memoirs, &c.</p></div1><div1 part="N" n="WATCH" org="uniform" sample="complete" type="entry"><head>WATCH</head><p>, a small portable machine, or movement, for measuring time; having its motion commonly regulated by a spiral spring. Perhaps, strictly speaking, watches are all such movements as <hi rend="italics">shew</hi> the parts of time; as clocks are such as <hi rend="italics">publish</hi> them, by striking on a bell, &c. But commonly, the term Watch is appropriated to such as are carried in the pocket; and clock to the large movements, whether they strike the hour or not.</p><p><hi rend="italics">Spring</hi> or <hi rend="italics">Pendulum</hi> <hi rend="smallcaps">Watches</hi> stand pretty much on the same principle with pendulum clocks. For if a pendulum, describing small circular arcs, make vibrations of unequal lengths, in equal times, it is because it describes the greater arc with a greater velocity; so a spring put in motion, and making greater and less vibrations, as it is more or less stiff, and as it has a greater or less degree of motion given it, performs them nearly in equal times. Hence, as the vibrations of the pendulum had been applied to large clocks, to rectify the inequality of their motions; so, to correct the unequal motions of the balance in Watches, a spring is added, by the isochronism of whose vibrations the correction is to be affected. The spring is usually wound into a spiral; that, in the little compass allotted it, it may be as long as possible; and may have strength enough not to be mastered, and dragged about, by the inequalities of the balance it is to regulate. The vibrations of the two parts, viz, the spring and the balance, should be of the same length; but so adjusted, as that the spring, being more regular in the length of its vibrations than the balance, may occasionally communicate its regularity to the latter.</p><p><hi rend="italics">The Invention of Spring or Pocket Watches,</hi> is due to the last age. It is true, it is said, in the history of Charles the 5th, that a Watch was presented to that prince: but this was probably no more than a kind of clock to be set on a table: some resemblance of which we have still remaining in the ancient pieces made before the year 1670. Some accounts also say, the first Watches were made at Nuremberg in 1500, by Peter <pb n="669"/><cb/> Hell, and were called Nuremberg eggs, on account of their oval form. And farther, that the same year George Purbach, a mathematician of Vienna, employed a watch that pointed to seconds, for astronomical observations, which was probably a kind of clock. In effect, it is between Hook and Huygens that the glory of this excellent invention lies: but to which of them it properly belongs, has been greatly disputed; the English ascribing it to the former, and the French, Dutch, &c, to the latter. Derham, in his Artificial Clockmaker, says roundly, that Dr. Hook was the inventor; and adds, that he contrived various ways of regulation: one way was with a loadstone: another with a tender straight spring, one end of which played backward and forward with the balance; so that the balance was to the spring as the ball of a pendulum, and the spring as the rod of the same: a third method was with two balances, of which there were divers sorts; some having a spiral spring to the balance for a regulalator, and others without. But the way that prevailed, and which still continues in mode, was with one balance, and one spring running round the upper part of the verge of it: though this has a disadvantage, which those with two springs &c were free from; in that, a sudden jerk, or confused shake will alter its vibrations, and flurry it very much.</p><p>The time of these inventions was about the year 1658; as appears, among other evidences, from an inscription on one of the double-balance Watches presented to king Charles the second, viz, Rob. Hook inven. 1658. T. Tompion fecit, 1675. The invention soon came into repute both at home and abroad; and two of the machines were sent for by the Dauphin of France. Soon after this, M. Huygens's Watch with a spiral spring got abroad, and made a great noise in England, as if the longitude could be found by it. It is certain however, that this invention was later than the year 1673, when his book De Horol. Oscillat. was published; in which there is no mention of this, though he speaks of several other contrivances in the same way.</p><p>One of these the lord Brounker sent for out of France, where M. Huygens had got a patent for them. This Watch agreed with Dr. Hook's, in the application of the spring to the balance; only that of Huygens had a longer spiral spring and its pulses and beats were much slower; also the balance, instead of turning quite round, as Dr. Hook's, turned several times every vibration. Huygens also invented divers other kinds of Watches, some of them without any string or chain at all, which he called pendulum Watches.</p><p>Mr. Derham suggests that he suspects Huygens's fancy was first set to work by some intelligence he might have of Hook's invention from Mr. Oldenburg, or some other of his correspondents in England; though Mr. Oldenburg vindicates himself against that charge, in the Philos. Trans. numbers 118 and 129.</p><p>Watches, since their first invention, have gone on in a continued course of improvement, and they have lately been brought to great perfection, both in England and in France, but more especially the former, particularly owing to the great encouragement that has been given to them by the Board of Longitude. Some of the chief writers and improvers of Watches, are, <cb/> Le Roy, Cummins, Harrison, Mudge, Emery, and Arnold, whose Watches are now in very high repute, and in frequent use in the navy and India ships, for keeping the longitude. See Derham's Artificial Clockmaker; Cummins's Principles of Clock and Watch work; Mudge's Thoughts on the Means of improving Watches, &c.</p><p><hi rend="italics">Striking</hi> <hi rend="smallcaps">Watches</hi>, are such as, besides the proper Watch part, for measuring time, have a clock part, for striking the hours, &c. These are real clocks; only moved by a spring instead of a weight; and are properly called pocket-clocks.</p><p><hi rend="italics">Repeating</hi> <hi rend="smallcaps">Watches</hi>, are such as, by pulling a string, &c, repeat the hour, quarter, or minute, at any time of the day or night.—This repetition was the invention of Mr. Barlow, being first put in practice by him in larger movements or clocks, about the year 1676. The contrivance immediately set the other artists to work, who soon contrived divers ways of effecting the same. But its application to pocket Watches was not known before K. James the second's reign; when the ingenious inventor above mentioned was soliciting a patent for it. The talk of a patent engaged Mr. Quare to resume the thoughts of a like contrivance, which he had in view some years before: he now effected it; and being pressed to endeavour to prevent Mr. Barlow's patent, a Watch of each kind was produced before the king and council; upon trial of which, the preference was given to Mr. Quare's. The difference between them was, that Barlow's was made to repeat by pushing in two pieces on each side the Watch-box; one of which repeated the hour, and the other the quarter: whereas Quare's was made to repeat by a pin that stuck out near the pendant, which being thrust in (as now is done by thrusting in the pendant itself) repeated both the hour and quarter with the same thrust. <hi rend="center"><hi rend="italics">Of the Mechanism of a</hi> <hi rend="smallcaps">Watch.</hi></hi></p><p>Watches, as well as clocks, are composed of wheels and pinions, with a regulator to direct the quickness or slowness of the wheels, and of a spring which communicates motion to the whole machine. But the regulator and spring of a Watch are vastly inferior to the weight and pendulum of a clock, neither of which can be employed in Watches. Instead of a pendulum, therefore, they are obliged to use a balance (Pl. 34, fig. 4) to regulate the motion of a Watch; and of a spring (fig. 6), which serves instead of a weight, to give motion to the wheels and balance.</p><p>The wheels of a Watch, like those of a clock, are placed in a frame, formed of two plates and four pillars. Fig. 3 represents the inside of a Watch, after the plate (Fig. 5) is taken off. A is the barrel which contains the spring (fig. 6); the chain is rolled about the barrel, with one end of it fixed to the barrel A, and the other to the fusee B.</p><p>When a Watch is wound up, the chain which was upon the barrel winds about the susee, and by this means the spring is stretched; for the interior end of the spring is fixed by a spring to the immoveable axis, about which the barrel revolves; the exterior end of the spring is fixed to the inside of the barrel, which turns upon an axis. It is there easy to perceive how the spring extends itself, and how its elasticity forces <pb n="670"/><cb/> the barrel to turn round, and consequently obliges the chain which is upon the fusee to unfold and turn the fusee; the motion of the fusee is communicated to the wheel CC; then by means of the teeth, to the pinion <hi rend="italics">c,</hi> which carries the wheel D; then to the pinion <hi rend="italics">d,</hi> which carries the wheel E; then to the pinion <hi rend="italics">e,</hi> which carries the wheel F; then to the pinion <hi rend="italics">f,</hi> upon which is the balance-wheel G, whose pivot runs in the piece A, called the potance, and B called a follower, which are fixed on the plate fig. 5. This plate, of which only a part is represented, is applied to that of fig. 3, in such a manner, that the pivots of the wheels enter into holes made in the plate fig. 3. Thus the impressed force of the spring is communicated to the wheels: and the pinion <hi rend="italics">f</hi> being then connected to the wheel F, obliges it to turn (fig. 7). This wheel acts upon the pallats of the verge 1, 2. (fig. 4) the axis of which carries the balance HH (fig. 4). The pivot I, in the end of the verge, enters into the hole G in the potance A (fig. 5). In this figure the pallats are represented; but the balance is on the other side of the plate, as may be seen in fig. 11. The pivot 3 of the balance enters into a hole of the cock BC (fig. 10), a perspective view of which is represented in fig. 12. Thus the balance, turns between the cock and the potance <hi rend="italics">c</hi> (fig. 5), as in a kind of cage. The action of the balance-wheel upon the pallats 1, 2, (fig. 4) is the same with that of the same wheel in the clock; i. e. in a Watch the balance-wheel obliges the balance to vibrate backwards and forwards like a pendulum.</p><p>At each vibration of the balance a pallat allows a tooth of the balance-wheel to escape; so that the quickness of the motion of the wheels is entirely determined by the quickness of the vibrations of the balance, and these vibrations of the balance and motion of the wheels are produced by the action of the spring.</p><p>But the quickness or slowness of the vibrations of the balance depends not solely upon the action of the great spring, but chiefly upon the action of the spring <hi rend="italics">abc,</hi> called the spiral spring (fig. 13) situated under the balance H, and represented in perspective (fig. 11); the exterior end of the spiral is fixed to the pin <hi rend="italics">a</hi> (fig. 13). This pin is applied near the plate in <hi rend="italics">a</hi> (fig. 11); the interior end of the spiral is fixed by a peg to the centre of the balance. Hence if the balance be turned upon itself, the plates remaining immoveable, the spring will extend itself, and make the balance perform one revolution. Now, after the spiral is thus extended, if the balance be left to itself, the elasticity of the spiral will bring back the balance, and in this manner the alternate vibrations of the balance are produced.</p><p>In fig. 7 all the wheels above described are represented in such a manner, that we may easily perceive at first sight how the motion is communicated from the barrel to the balance.</p><p>In fig. 8 are represented the wheels under the dialplate, by which the hands are moved. The pinion <hi rend="italics">a</hi> is adjusted to the force of the prolonged pivot of the wheel D (fig. 7), and is called a cannon pinion. This wheel revolves in an hour. The end of the axis of the pinion <hi rend="italics">a,</hi> upon which the minute hand is fixed, is square; the pinion (fig. 8) is indented into the wheel <hi rend="italics">b,</hi> which is carried by the pinion <hi rend="italics">a.</hi> Fig. 9 is a wheel fixed upon a barrel, into the cavity of which the pinion <cb/> <hi rend="italics">a</hi> enters, and upon which it turns freely. This wheel <hi rend="italics">d</hi> revolves in 12 hours, and carries along with it the hour-hand.</p></div1><div1 part="N" n="WATER" org="uniform" sample="complete" type="entry"><head>WATER</head><p>, in Physiology, a clear, insipid, and colourless fluid, coagulable into a transparent solid substance, called ice, when placed in a temperature of 32° of Fahrenheit's thermometer, or lower, but volatile and fluid in every degree of heat above that; and when pure, or freed from heterogeneous particles, is reckoned one of the four elements.</p><p>By some late experiments of Messrs. Lavoisier, Watt, Cavendish, Priestley, Kirwan, &c, it appears, that Water consists of dephlogisticated air, and inflammable air or phlogiston intimately united; or, as Mr. Watt conceives, of those two principles deprived of part of their latent heat. And in some instances it appears that air and Water are mutually convertible into each other. Thus, Mr. Cavendish (Philos. Trans. vol. 74, p. 128) recites several experiments, in which he changed common air into pure Water, by decomposing it in conjunction with inflammable air. Dr. Priestley likewise, having decomposed dephlogisticated and inflammable air, by firing them together by the electric explosion, found a manifest decomposition of Water, which, as nearly as he could judge, was equal in weight to that of the decomposed air. He also made a number of other curious experiments, which seemed to favour the idea of a conversion of Water into air, without absolutely proving it. The difficulty which M. De Lue and others have found in expelling all air from Water, is best accounted for on the supposition of the generation of air from Water; and admitting that the conversion of Water into air is effected by the intimate union of what is called the principle of heat with the Water, it appears sufficiently analogous to other changes, or rather combinations, of substances. Is not, says Dr. Priestley, the acid of nitre, and also that of vitriol, a thing as unlike to air as Water is, their properties being as remarkably different? And yet it is demonstrable that the acid of nitre is convertible into the purest respirable air, and probably by the union of the same principle of heat. Philos. Trans. vol. 73, p. 414 &c.</p><p>Indeed there seems to be Water in all bodies, and particles of almost all kinds of matter in Water; so that it is hardly ever sufficiently pure to be considered as an element. Water, if it could be had alone, and pure, Boerhaave argnes, would have all the requisites of an element, and be as simple as fire; but there is no expedient hitherto discovered for procuring it so pure. Rain Water, which seems the purest of all those we know of, is replete with infinite exhalations of all kinds, which it imbibes from the air: so that if siltered and distilled a thousand times, there still remain fæces. Besides this, and the numberless impurities it acquires after it is raised, by mixing with all sorts of effluvia in the atmosphere, and by falling upon and running over the earth, houses, and other places. There is also fire contained in all Water; as appears from its fluidity, which is owing to fire alone. Nor can any kinds of siltering through sand, stone, &c, free it entirely from salts &c. Nor have all the experiments that have been invented by the philosophers, ever been able to derive Water perfectly pure. Hence Boerhaave says, that he is convinced nobody ever saw a drop of pure Water; <pb n="671"/><cb/> that the utmost of its purity known, only amounts to its being free from this or that sort of matter; and that it can never, for instance, be quite deprived of salt; since air will always accompany Water, and air always contains salt.</p><p>Water seems to be diffused everywhere, and to be present in all space wherever there is matter. There are hardly any bodies in nature but what will yield Water: it is even asserted that fire itself is not without it. A single grain of the fiery salt, which in a moment's time will penetrate through a man's hand, readily imbibes half its weight of Water, and melts even in the driest air imaginable. Among innumerable instances, hartshorn, kept 40 years, and turned as hard and dry as any metal, so that it will yield sparks of fire when struck against a flint, yet being put into a glass vessel, and distilled, will afford 1/8th part of its quantity of Water. Bones dead and dried 25 years, and thus become almost as hard as iron, yet by distillation have yielded half their weight of Water. And the hardest stones, ground and distilled, always discover a portion of it. But hitherto no experiment shews, that Water enters as a principle into the combination of metallic matters, or even into that of vitrescible stones.</p><p>From such considerations, philosophers have been led to hold the opinion, that all things were made of Water. Basil Valentine, Paracelsus, Van Helmont, and others have maintained, that Water is the elemental matter or stamen of all things, and suffices alone for the production of all the visible creation. Thus too Newton: “All birds, beasts, and fishes, insects, trees, and vegetables, with their several parts, do grow out of Water, and watery tinctures, and salts; and by putrefaction they all return again to watery substances.” And the same doctrine is held, and confirmed by experiments, by Van Helmont, Boyle, and others.</p><p>But Dr. Woodward endeavours to shew that the whole is a mistake.—Water containing extraneous corpuscles, some of which, according to him, are the proper matter of nutrition; the Water being still found to afford so much the less nourishment, the more it is purified by distillation. So that Water, as such, does not seem to be the proper nutriment of vegetables; but only the vehicle which contains the nutritious particles, and carries them along with it, through all the parts of the plant.</p><p>Helmont however carries his system still farther, and imagines that all bodies may be reconverted into Water. His alkahest, he affirms, adequately resolves plants, animals, and minerals, into one liquor, or more, according to their several internal differences of parts; and the alkahest, being abstracted again from these liquors, in the same weight, and with the same virtues, as when it dissolved them, the liquors may, by frequent cohobations from chalk, or some other proper matter, be totally deprived of their seminal endowments, and at last return to their first matter; which is insipid Water.</p><p>Spirit of wine, of all other spirits, seems freest from Water: yet Helmont affirms, it may be so united with Water, as to become Water itself. He adds, that it is material Water, only under a sulphureous disguise. And the same thing he observes of all salts, and of oils, which may be almost wholly changed into Water. <cb/></p><p><hi rend="italics">No standard for the Weight and Purity of</hi> <hi rend="smallcaps">Water.</hi>— Water scarce ever continues two moments exactly of the same weight; by reason of the air and fire contained in it. The expansion of Water in boiling shews what effect the different degrees of fire have on the gravity of Water. This makes it difficult to fix the specific gravity of Water, in order to settle its degree of purity. However, the purest Water we can obtain, according to the experiments of M. Hawskbee, is 850 times heavier than air: or according to the experiments of Mr. Cavendish, the thermometer being at 50° and the barometer at 29 3/4, about 800 times as heavy as air: and according to the experiments of Sir Geo. Shuckburgh, when the barometer is at 29.27 and the thermometer at 53°, Water is 836 times heavier than air; whence also may be deduced this general proportion, which may be accounted a standard, viz, that, when the barometer is at 30° and the thermometer at 55°, then Water is 820 times heavier than air; also that in such a state the cubic foot of Water weighs 1000 ounces avoirdupois, and that of air 1.222, or 1 2/9 nearly, also that of mercury 13600 ounces; and for other states of the thermometer and barometer, the allowance is after this rate, viz, that the column of mercury in the barometer varies its length by the 10 thousandth part of itself for a change of each single degree of temperature, and Water changes by 3/20000 part of its height or magnitude by each degree of the same. However, we have not any very exact standard in air; for Water being so much heavier than air, the more Water there is contained in the air, the heavier of course must the air be; as indeed a considerable part of the weight of the atmosphere seems to arise from the Water that is in it.</p><p><hi rend="italics">Properties and Effects of</hi> <hi rend="smallcaps">Water.</hi>—Water is a very volatile body. It is entirely reduced into vapours and dissipated, when exposed to the fire and unconfined.</p><p>Water heated in an open vessel, acquires no more than a certain determinate degree of heat, whatever be the intensity of the fire to which it is exposed; which greatest degree of heat is when it boils violently.</p><p>It has been found that the degree of heat necessary to make Water boil, is variable, according to the purity of the Water and the weight of the atmosphere. The following table shews the degree of heat at which Water boils, at various heights of the barometer, being a medium between those resulting from the experiments of Sir Geo. Shuckburgh and M. De Luc: <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Height of the</cell><cell cols="1" rows="1" role="data">Heat of Boiling</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Barometer.</cell><cell cols="1" rows="1" role="data">Water.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Inches.</cell><cell cols="1" rows="1" role="data">°</cell></row><row role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">205</cell></row><row role="data"><cell cols="1" rows="1" role="data">26 1/2</cell><cell cols="1" rows="1" role="data">206</cell></row><row role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">206.9</cell></row><row role="data"><cell cols="1" rows="1" role="data">27 1/2</cell><cell cols="1" rows="1" role="data">207.7</cell></row><row role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">208.5</cell></row><row role="data"><cell cols="1" rows="1" role="data">28 1/2</cell><cell cols="1" rows="1" role="data">209.4</cell></row><row role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">210.3</cell></row><row role="data"><cell cols="1" rows="1" role="data">29 1/2</cell><cell cols="1" rows="1" role="data">211.2</cell></row><row role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">212.0</cell></row><row role="data"><cell cols="1" rows="1" role="data">30 1/2</cell><cell cols="1" rows="1" role="data">212.8</cell></row><row role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">213.6</cell></row></table> <pb n="672"/><cb/></p><p>Water is found the most penetrative of all bodies, after fire, and the most difficult to confine; passing through leather, bladders, &c, which will confine air; making its way gradually through woods; and is only retainable in glass and metals; nay it was found by experiment at Florence, that when shut up in a spherical vessel of gold, which was pressed with a great force, it made its way through the pores even of the gold itself.</p><p>Water, by this penetrative quality alone, may be inferred to enter the composition of all bodies, both vegetable, animal, fossil, and even mineral; with this particular circumstance, that it is easily, and with a gentle heat, separable again from bodies it had united with.</p><p>And yet the same Water, as little cohesive as it is, and as easily separated from most bodies, will cohere firmly with some others, and bind them together in the most solid masses; as in the tempering of earth, or ashes, clay, or powdered bones, &c, with Water, and then dried and burnt, when the masses become hard as stones, though without the Water they would be mere dust or powder. Indeed it appears wonderful that Water, which is otherwise an almost universal dissolvent, should nevertheless be a great coagulator.</p><p>Some have imagined that Water is incompressible, and therefore nonelastic; founding their opinion on the celebrated Florentine experiment above mentioned, with the globe of gold; when the Water being, as they say, incapable of condensation, rather than yield, transuded through the pores of the metal, so that the ball was found wet all over the outside; till at length making a clest in the gold, it spun out with great vehemence. But the truth of the conclusions drawn from this Florentine experiment has been very justly questioned; Mr. Canton having proved by accurate experiments, that Water is actually compressed even by the weight of the atmosphere. See <hi rend="smallcaps">Compression.</hi></p><p>Besides, the diminution of size which Water suffers when it passes to a less degree of heat, sufficiently shews that the particles of this fluid are, like those of all other known substances, capable of approaching nearer together.</p><p><hi rend="italics">Ditch</hi> <hi rend="smallcaps">Water</hi>, is often used as an object for the microscope, and seldom fails to afford a great variety of animalcules; often appearing of a greenish, reddish, or yellowish colour, from the great multitudes of them. And to the same cause is to be ascribed the green skim on the surface of such Water. Dunghill Water is also full of an immense crowd of animalcules.</p><p><hi rend="italics">Fresh</hi> <hi rend="smallcaps">Water</hi>, is said of that which is insipid, or without salt, and inodorous; being the natural and pure state of the element.</p><p><hi rend="italics">Hard</hi> <hi rend="smallcaps">Water</hi>, or <hi rend="italics">Crude</hi> <hi rend="smallcaps">Water</hi>, is that in which soap does not dissolve completely or uniformly, but is curdled. The dissolving power of hard Water is less than that of soft; and hence its unfitness for washing, bleaching, dyeing, boiling kitchen vegetables, &c.</p><p>The hardness of Water may arise either from salts, or from gas. That which arises from salts, may be discovered and remedied by adding some drops of a solution of fixed alkali; but the latter by boiling, or exposure to the open air.</p><p>Spring Waters are often hard; but river Water soft. Hard Waters are remarkably indisposed to corrupt; <cb/> they even preserve putrescible substances for a considerable length of time: hence they seem to be best fitted for keeping at sea, especially as they are so easily softened by a little alkaline salt.</p><p><hi rend="italics">Putrid</hi> <hi rend="smallcaps">Water</hi>, is that which has acquired an offensive smell and taste by the putrescence of animal or vegetable substances contained in it. This kind of Water is in the highest degree pernicious to the human frame, and capable of bringing on mortal diseases even by its smell. Quicklime put into water is useful to preserve it longer sweet; or even exposure to the air in broad shallow vessels. And putrid Water may be in a great measure sweetened, by passing a current of fresh air through it, from bottom to top.</p><p><hi rend="italics">Rain</hi> <hi rend="smallcaps">Water</hi> may be considered as the purest distilled Water, but impregnated during its passage through the air with a considerable quantity of phlogistic and putrescent matter; whence it is superior to any other in fertilizing the earth. Hence also it is inferior for domestic purposes to spring or river Water, even if it could be readily procured: but such as is gotten from spouts placed below the roofs of houses, the common way of procuring it in this country, is evidently very impure, and becomes putrid in a short time.</p><p><hi rend="italics">River</hi> or <hi rend="italics">Running</hi> <hi rend="smallcaps">Water</hi>, is next in purity to snow or distilled water; and for domestic purposes superior to both, in having less putrescent matter, and more fixed air. That however is much the purest that runs over a clean rocky or stony bottom.</p><p>River Waters generally putrefy sooner than those of springs. During the putrefaction, they throw off a part of their heterogeneous matter, and at length become sweet again, and purer than at first; after which they commonly preserve a long time: this is remarkably the case with the Thames Water, taken up about London; which is commonly used by seamen, in their voyages.</p><p><hi rend="italics">Salt</hi> <hi rend="smallcaps">Water</hi>, such as has much salt in it, so as to be sensible to the taste.</p><p><hi rend="italics">Sea</hi> <hi rend="smallcaps">Water</hi>, or Water of the sea, is an assemblage of bodies, in which Water can scarce be said to have the principal part: it is an universal colluvies of all the bodies in nature, sustained and kept swimming in Water as a vehicle: being a solution of common salt, sal catharticus amarus, a selenitic substance, and a compound of muriatic acid with magnesia, mixed together in various proportions. It may be freshened by simple distillation without any addition, and thus it has sometimes been useful in long voyages at sea. Sea Water by itself has a purgative quality, owing to the salts it contains; and has been greatly recommended in scrophulous disorders.</p><p>Sea Water is about 3 parts in 100 heavier than common Water; and its temperature at great depths is from 34 to 40 degrees; but near the surface it follows more nearly the temperature of the air.</p><p><hi rend="italics">Snow</hi> <hi rend="smallcaps">Water</hi>, is the purest of all the common Waters, when the snow has been collected pure. Kept in a warm place, in clean glass vessels, not closely stopped, but covered from dust, &c, snow water becomes in time putrid; though in well-stopped bottles it remains unaltered for several years. But distilled Water suffers no alteration in either circumstance.</p><p><hi rend="italics">Spring</hi> <hi rend="smallcaps">Water</hi> is commonly impregnated with a <pb/><pb/><pb n="673"/><cb/> small portion of imperfect neutral salt, extracted from the different strata through which it percolates. Some contain a vast quantity of stony matter, which they deposit as they run along, and thus form masses of stone; sometimes incrustating various animal and vegetable matters, which they are therefore said to petrify. Spring-Water is much used for domestic purposes, and on account of its coolness is an agreeable drink; but on account of its being usually somewhat hard, is inferior to that which has run for a considerable way in a channel.</p><p>Spring-water arises from the rain, and from the mists and moisture in the atmosphere. These falling upon hills and other parts of the earth, soak into the ground, and pass along till they find a vent out again, in the form of a spring.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Bellows,</hi> in Mechanics, are bellows, for blowing air into furnaces, that are worked by the force of water.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Clock.</hi> See <hi rend="smallcaps">Clepsydra.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Engine,</hi> an engine for extinguishing fires; or any engine to raise water; or any engine moved by the force of Water. See <hi rend="smallcaps">Engine</hi>, and <hi rend="smallcaps">Steam</hi>-<hi rend="italics">Engine.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Gage,</hi> an instrument for measuring the depth or quantity of any water. See <hi rend="smallcaps">Gage.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Level,</hi> is the true level which the surface of still Water takes, and is the truest of any.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Logged,</hi> in Sea-Language, denotes the state of a ship when, by receiving a great quantity of Water into her hold, by leaking, &c, she has become heavy and inactive upon the sea, so as to yield without resistance to the effort of every wave rushing over her deck.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Machine.</hi> See <hi rend="smallcaps">Machine.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Measure.</hi> Salt, sea-coal, &c, while on board vessels in the pool, or river, are measured with the corn-bushel heaped up; or else 5 striked pecks are allowed to the bushel. This is called Water-measure; and it exceeds Winchester-measure by about 3 gallons in the bushel.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Microscope.</hi> See <hi rend="smallcaps">Microscope.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Mill.</hi> See <hi rend="smallcaps">Mill.</hi></p><p><hi rend="italics">Motion of</hi> <hi rend="smallcaps">Water</hi>, in Hydraulics. The theory of the motion of running Water is one of the principal objects of hydraulics, and to which many eminent mathematicians have paid their attention. But it were to be wished that their theories were more consistent with each other, and with experience. The inquisitive reader may consult Newton's Principia, lib. 2, pr. 36, with the comment. Dan. Bernoulli's Hydrodynamica. J. Bernoulli, Hydraulica, Oper. tom. 4, pa. 389. Dr. Jurin, in the Philos. Trans. num. 452, or Abridg. vol. 8, pa. 282. Gravesande, Physic. Elem. Mathem. lib. 3, par. 2. Maclaurin's Flux. art. 537. Poleni de Castellis, Ximenes, D'Alembert, Bossu, Buat, and many others.</p><p>But notwithstanding the labours of all these eminent authors, this intricate subject still remains in a great measure obscure and uncertain. Even the simple case of the motion of running water, when it issues from a hole in the bottom of a vessel, has never yet been determined, so as to give universal satisfaction to the learned. On this head, it is now pretty generally allowed, <cb/> that the velocity of the issuing stream, is equal to that which a heavy body acquires by falling through the height of the fluid above the hole, as may be demonstrated by theory: but in practice, the quantity of the effluent Water is much less than what is given by this theory; owing to the obstruction to the motion in the hole, partly from the sides of it, and partly from the different directions of the parts of the Water in entering it, which thence obstruct each other's motion. And this obstruction, and the diminution in the quantity of Water run out, is still the more in proportion as the hole is the smaller; in such sort, that when the hole is very small, the quantity is diminished in the ratio of √2 to 1 very nearly, which is the ratio of the greatest diminution; and for larger holes, the diminution is always less and less. This fact is ascertained, or admitted by Newton, and all the other philosophers abovementioned, with some small variations.</p><p>That the velocity of the Water in the hole, or at least some part of it, as that for example in the middle of the stream, is equal to that abovementioned, is even evinced by experiment, by directing the stream either sideways, or upwards: for in the former case, it is found to range upon an horizontal plane, a distance that just answers to that velocity, by the nature of projectiles; and in the latter case, the jet rises nearly to the height of the Water in the vessel; which it could not do, if its velocity were not equal to that acquired by the free descent of a body through that height. Hence it is evident then, that the particles of the Water, which are in the hole at the same moment of time, do not all burst out with the same velocity; and, in fact, the velocity is found to decrease all the way from the middle of the hole, where it is greatest, towards the side or edge, where it is the least.</p><p>At a small distance from the hole, the diameter of the vein of Water is much less than that of the hole. Thus, if the diameter of the hole be 1, the diameter of the vein of Water just without it, will be 21/25, or 0.84, according to Newton's measure, who first observed this phenomenon; and according to Poleni's measure 0.78 nearly.</p><p>By the experiments of Buat (Principes d'Hydraulique), the quantity by theory is to that by experiment, for a small hole made in the thin side of a reservoir, as 8 to 5. When a short pipe is added to the hole outwards, of the length of two or three times its diameter, that ratio is as 16 to 13. And when <figure/> the short pipe is all within side the vessel, as in the margin, the same ratio becomes that of 3 to 2. Poleni also found that the quantity of Water flowing through a pipe or tube, was much greater than that through a hole of the same diameter in the thin side or bottom of the vessel, the height of the head of Water above each being the same. See also many other curious circumstances in Buat's Principes above mentioned.</p><p>Some authors give this rule for finding the height due to the velocity in a flat orifice, or a medium among all the parts of it, such that this medium velocity being drawn into the area of the hole, shall give the quantity per second that runs through: viz, let <hi rend="italics">A</hi> denote the <pb n="674"/><cb/> area of the surface of the Water in the vessel, <hi rend="italics">a</hi> the area of the orifice by which the Water issues, and <hi rend="italics">H</hi> the height of the Water above the orifice; then, as 2<hi rend="italics">A</hi> - <hi rend="italics">a</hi> : <hi rend="italics">A</hi> :: <hi rend="italics">H</hi> : <hi rend="italics">b,</hi> the height due to the medium velocity, or the height from which a body must freely descend, by the force of gravity, to acquire that mean velocity.</p><p>Authors are not yet agreed as to the force with which a vein of Water, spouting from a round hole in the side of a vessel, presses upon a plane directly opposed to the motion of the vein. Most authors agree, that the pressure of this vein, flowing uniformly, ought to be equal to the weight of a cylinder of Water, whose base is equal to the hole through which the Water flows, and its height equal to the height of the Water in the vessel above the hole. The experiments made by Mariotte, and others, seem to countenance this opinion. But Dan. Bernoulli rejects it, and estimates this pressure by the weight of a column of the fluid, whose diameter is equal to the contracted vein (according to Newton's observation abovementioned), and the height of which is equal to double the altitude due to the real velocity of the spouting Water; and this pressure is also equal to the force of repulsion, arising from the reaction of the spouting Water upon the vessel. The ingenious author remarks that he speaks only of single veins of Water, the whole of which are received by the planes upon which they press; for as to the pressures exerted by fluids surrounding the bodies they press upon, as the wind, or a river, the case is different, though confounded with the former by writers on this subject. Hydrodynamica, pa. 289.</p><p>Another rule however had been adopted by the Academicians of Paris, who made a number of experiments to confirm or establish it. Hist. Acad. Paris, ann. 1679, sect. 3, cap. 5.</p><p>D. Bernoulli, on the other hand, thinks his own theory sufficiently established by the experiments he relates; for the particulars of which see the Acta Petropolitana, vol. 8, pa. 122.</p><p>This ingenious author is of opinion that his theory of the quantity of the force of repulsion, exerted by a vein of spouting Water, might be usefully applied to move ships by pumping; and he thinks the motion produced by this repulsive force would fall little, if at all, short of that produced by rowing. He has given his reasons and computations at length in his Hydrodynamica, pa. 293 &c.</p><p>This science of the pressures exerted by Water or other fluids in motion, is what Bernoulli calls <hi rend="italics">Hydraulico-statica.</hi> This science differs from hydrostatics, which considers only the pressure of Water and other fluids at rest; whereas hydraulico-statics considers the pressure of Water in motion. Thus the pressure exerted by Water moving through pipes, upon the sides of those pipes, is an hydraulico-statical consideration, and has been erroneously determined by many, who have given no other rules in these cases, but such as are applicable only to the pressure of fluids at rest. See Hydrodynam. pa. 256 &c.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Poise.</hi> See <hi rend="smallcaps">Hydrometer</hi>, and A<hi rend="smallcaps">REOMETER.</hi></p><p>Dr. Hook contrived a Water-poise, which may be of good service in examining the purity &c of Water. It <cb/> consists of a round glass ball, like a bolt head, about 3 inches diameter, with a narrow stem or neck, the 24th of an inch in diameter; which being poised with red lead, so as to make it but little heavier than pure sweet Water, and thus fitted to one end of a fine balance, with a counterpoise at the other end; upon the least addition of even the 2000th part of salt to a quantity of Water, half an inch of the neck will emerge above the water. Philos. Trans. num. 197.</p><p><hi rend="italics">Raising of</hi> <hi rend="smallcaps">Water</hi>, in Hydraulics. The great use of raising Water by engines for the various purposes of life, is well known. Machines have in all ages been contrived with this view; a detail of the best of which, with the theory of their construction, would be very curious and instructive. M. Belidor has executed this in part in his Architecture Hydraulique. Dr. Desaguliers has also given a description of several engines to raise Water, in his Course of Experimental Philosophy, vol. 2, and there are several other fmaller works of the same kind.</p><p>Engines for raising Water are either such as throw it up with a great velocity, as in jets; or such as raise it from one place to another by a gentle motion. For the general theory of these engines, see Bernoulli's Hydrodynamica.</p><p>Desaguliers has settled the maximum of engines for raising water, thus: a man with the best Water engine cannot raise above one hogshead of Water in a minute, 10 feet high, to hold it all day; but he can do almost twice as much for a minute or two.</p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Spout.</hi> See <hi rend="smallcaps">Spout.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Wheel,</hi> an engine for raising Water in great quantity out of a deep well, &c. See <hi rend="smallcaps">Persian</hi>- <hi rend="italics">Wheel.</hi></p><p><hi rend="smallcaps">Water</hi>-<hi rend="italics">Works.</hi> See <hi rend="italics">Raising of</hi> <hi rend="smallcaps">Water.</hi></p></div1><div1 part="N" n="WAVE" org="uniform" sample="complete" type="entry"><head>WAVE</head><p>, in Physics, a volume of water elevated by the action of the wind &c, upon its surface, into a state of fluctuation, and accompanied by a cavity. The extent from the bottom or lowest point of one cavity, and across the elevation, to the bottom of the next cavity, is the breadth of the Wave.</p><p>Waves are considered as of two kinds, which may be distinguished from one another by the names of natural and accidental Waves. The natural Waves are those which are regularly proportioned in size to the strength of the wind which produces them. The accidental Waves are those occasioned by the wind's reacting upon itself by repercussion from hills or high shores, and by the dashing of the Waves themselves, otherwise of the natural kind, against rocks and shoals; by which means these Waves acquire an elevation much above what they can have in their natural state.</p><p>Mr. Boyle proved, by numerous experiments, that the most violent wind never penetrates deeper than 6 feet into the water; and it seems a natural consequence of this, that the water moved by it can only be elevated to the same height of 6 feet from the level of the surface in a calm; and these 6 feet of elevation being added to the 6 of excavation, in the part from whence that water so elevated was raised, should give 12 feet for the utmost elevation of a Wave. This is a calculation that does great honour to its author; as many experiments and <pb n="675"/><cb/> observations have proved that it is very nearly true in deep seas, where the Waves are purely natural, and have no accidental causes to render them larger than their just proportion.</p><p>It is not to be understood however, that no Wave of the sea can rise more than 6 feet above its natural level in open and deep water; for Waves vastly higher than these are formed in violent tempests in the great seas. These however are not to be accounted Waves in their natural state, but as compound Waves formed by the union of many others; for in these wide plains of water, when one Wave is raised by the wind, and would elevate itself up to the exact height of 6 feet, and no more, the motion of the water is so great, and the succession of Waves so quick, that while this is rising, it receives into it several other Waves, each of which would have been at the same height with itself; these run into the first Wave one after another, as it is rising; by which means its rise is continued much longer than it naturally would have been, and it becomes accumulated to an enormous size. A number of these complicated Waves rising together, and being continued in a long succession by the continuation of the storm, make the Waves so dangerous to ships, which the sailors in their phrase call mountains high.</p><p>Different Waves do not disturb one another when they move in different directions. The reason is, that whatever figure the surface of the water has acquired by the motion of the Waves, there may in that be an elevation and depression; as also such a motion as is required in the motion of a Wave.</p><p>Waves are often produced by the motion of a tremulous body, which also expand themselves circularly, though the body goes and returns in a right line; for the water which is raised by the agitation, descending, forms a cavity, which is every where surrounded with a rising.</p><p><hi rend="italics">The Motion of the</hi> <hi rend="smallcaps">Waves</hi>, makes an article in the Newtonian philosophy; that author having explained their motions, and calculated their velocity from mathematical principles, similar to the motion of a pendulum, and to the reciprocation of water in the two legs of a bent and inverted syphon or tube.</p><p>His proposition concerning such canal or tube is the 44th of the 2d book of his Principia, and is this: “If water ascend and descend alternately in the erected legs of a canal or pipe; and a pendulum be constructed, whose length between the point of suspension and the centre of oscillation, is equal to half the length of the water in the canal; then the water will ascend and descend in the same times in which the pendulum oscillates.” The author hence infers, in prop. 45, that the velocity of Waves is in the subduplicate ratio of their breadths; and in prop. 46, he proceeds “To find the velocity of Waves,” as follows: “Let a pendulum be constructed, whose length between the point of suspension and the centre of oscillation is equal to the breadth of the Waves; and in the time that the pendulum will perform one single oscillation, the Waves will advance forward nearly a space equal to their breadth. That which I call the breadth of the Waves, is the transverse measure lying between the deepest part of the hollows, or between the tops of the ridges. <cb/> Let ABCDEF represent the surface of stagnant water ascending and descending in successive Waves; also let <figure/> A, C, E, &c, be the tops of the Waves; and B, D, F, &c, the intermediate hollows. Because the motion of the Waves is carried on by the successive ascent and descent of the water, so that the parts of it, as A, C, E, &c, which are highest at one time, become lowest immediately after; and because the motive force, by which the highest parts descend and the lowest ascend, is the weight of the elevated water, that alternate ascent and descent will be analogous to the reciprocal motion of the water in the canal, and observe the same laws as to the times of its ascent and descent; and therefore (by prob. 44, above mentioned) if the distances between the highest places of the Waves A, C, E, and the lowest B, D, F, be equal to twice the length of any pendulum, the highest parts A, C, E, will become the lowest in the time of one oscillation, and in the time of another oscillation will ascend again. Therefore between the passage of each Wave, the time of two oscillations will intervene; that is, the Wave will describe its breadth in the time that the pendulum will oscillate twice; but a pendulum of 4 times that length, and which therefore is equal to the breadth of the Waves, will just oscillate once in that time. <hi rend="italics">Q. E. I.</hi></p><p>“<hi rend="italics">Corol.</hi> 1. Therefore Waves, whose breadth is equal to 39 1/8 inches, or 3 25/96 feet, will advance through a space equal to their breadth in one second of time; and therefore in one minute they will go over a space of 195 5/8 feet; and in an hour a space of 11737 feet, nearly, or 2 miles and almost a quarter.</p><p>“<hi rend="italics">Corol.</hi> 2. And the velocity of greater or less Waves, will be augmented or diminished in the subduplicate ratio of their breadth.</p><p>“These things (Newton adds) are true upon the supposition, that the parts of water ascend or descend in a right line; but in fact, that ascent and descent is rather performed in a circle; and therefore I propose the time defined by this proposition as only near the truth.”</p><p><hi rend="italics">Stilling</hi> <hi rend="smallcaps">Waves</hi> <hi rend="italics">by means of Oil.</hi> This wonderful property, though well known to the Ancients, as appears from the writings of Pliny, was for many ages either quite unnoticed, or treated as fabulous by succeeding philosophers. Of late it has, by means of Dr. Franklin, again attracted the attention of the learned; though it appears, from some anecdotes, that seafaring people have always been acquainted with it. In Martin's description of the Western Islands of Scotland, we have the following passage: “The steward of Kilda, who lives in Pabbay, is accustomed, in time of a storm, to tie a bundle of puddings, made of the fat of seafowl, to the end of his cable, and lets it fall into the sea behind his rudder. This, he says, hinders the Waves from breaking, and calms the sea.” Mr. Pennant, in his British Zoology, vol. iv, under the article <pb n="676"/><cb/> Seal, takes notice, that when these animals are devouring a very oily fish, which they always do under water, the Waves above are remarkably smooth; and by this mark the fishermen know where to find them. Sir Gilbert Lawson, who served long in the army at Gibraltar, assured Dr. Franklin, that the fishermen in that place are accustomed to pour a little oil on the sea, in order to still its motion, that they may be enabled to see the oysters lying at its bottom, which are there very large, and which they take up with a proper instrument. A similar practice obtains among fishermen in various other parts, and Dr. Franklin was informed by an old sea-captain, that the fishermen of Lisbon, when about to return into the river, if they saw too great a surf upon the bar, would empty a bottle or two of oil into the sea, which would suppress the breakers, and allow them to pass freely.</p><p>The Doctor having revolved in his mind all these pieces of information, became impatient to try the experiment himself. At last having an opportunity of observing a large pond very rough with the wind, he dropped a small quantity of oil upon it. But having at first applied it on the lee side, the oil was driven back again upon the shore. He then went to the windward side, and poured on about a tea-spoon full of oil; this produced an instant calm over a space several yards square, which spread amazingly, and extended itself gradually till it came to the lee-side; making all that quarter of the pond, perhaps half an acre, as smooth as glass. This experiment was often repeated in different places, and always with success. Our author accounts for it in the following manner:</p><p>“There seems to be no natural repulsion between water and air, to keep them from coming into contact with each other. Hence we find a quantity of air in water; and if we extract it by means of the air pump, the same water again exposed to the air will soon imbibe an equal quantity.—Therefore air in motion, which is wind, in passing over the smooth surface of water, may rub as it were upon that surface, and raise it into wrinkles; which, if the wind continues, are the elements of future Waves. The smallest Wave once raised does not immediately subside and leave the neighbouring water quiet; but in subsiding raises nearly as much of the water next to it, the friction of the parts making little difference. Thus a stone dropped into a pool raises first a single Wave round itself, and leaves it, by sinking to the bottom; but that first Wave subsiding raises a second, the second a third, and so on in circles to a great extent.</p><p>“A small power continually operating, will produce a great action. A finger applied to a weighty suspended bell, can at first move it but little; if repeatedly applied, though with no greater strength, the motion increases till the bell swings to its utmost height, and with a force that cannot be resisted by the whole strength of the arm and body. Thus the small first raised Waves being continually acted upon by the wind, are, though the wind does not increase in strength, continually increased in magnitude, rising higher and extending their bases, so as to include a vast mass of water in each Wave, which in its motion acts with great violence. But if there be a mutual repulsion between the particles <cb/> of oil, and no attraction between oil and water, oil dropped on water will not be held together by adhesion to the spot whereon it falls; it will not be imbibed by the water; it will be at liberty to expand itself; and it will spread on a surface that, besides being smooth to the most perfect degree of polish, prevents, perhaps by repelling the oil, all immediate contact, keeping it at a minute distance from itself; and the expansion will continue, till the mutual repulsion between the particles of the oil is weakened and reduced to nothing by their distance.</p><p>“Now I imagine that the wind blowing over water thus covered with a film of oil cannot easily catch upon it, so as to raise the first wrinkles, but slides over it, and leaves it smooth as it finds it. It moves the oil a little indeed, which being between it and the water, serves it to slide with, and prevents friction, as oil does between those parts of a machine that would otherwise rub hard together. Hence the oil dropped on the windward side of a pond proceeds gradually to leeward, as may be seen by the smoothness it carries with it quite to the opposite side. For the wind being thus prevented from raising the first wrinkles that I call the elements of Waves, cannot produce Waves, which are to be made by continually acting upon and enlarging those elements; and thus the whole pond is calmed.</p><p>“Totally therefore we might suppress the Waves in any required place, if we could come at the windward place where they take their rise. This in the ocean can seldom if ever be done. But perhaps something may be done on particular occasions to moderate the violence of the Waves when we are in the midst of them, and prevent their breaking when that would be inconvenient. For when the wind blows fresh, there are continually rising on the back of every great Wave a number of small ones, which roughen its surface, and give the wind hold, as it were, to push it with greater force. This hold is diminished by preventing the generation of those small ones. And possibly too, when a Wave's surface is oiled, the wind, in passing over it, may rather in some degree press it down, and contribute to prevent its rising again, instead of promoting it.</p><p>“This, as mere conjecture, would have little weight, if the apparent effects of pouring oil into the midst of Waves were not considerable, and as yet not otherwise accounted for.</p><p>“When the wind blows so fresh, as that the Waves are not sufficiently quick in obeying its impulse, their tops being thinner and lighter, are pushed forward, broken, and turned over in a white foam. Common Waves lift a vessel without entering it; but these, when large, sometimes break above and pour over it, doing great damage.</p><p>“That this effect might in any degree be prevented, or the height and violence of Waves in the sea moderated, we had no certain account; Pliny's authority for the practice of seamen in his time being slighted. But discoursing lately on this subject with his excellency Count Bentinck of Holland, his son the honourable Captain Bentinck, and the learned professor Allemand (to all whom I showed the experiment of smoothing in a windy day the large piece of water at the head of the <pb n="677"/><cb/> green park), a letter was mentioned which had been received by the Count from Batavia, relative to the saving of a Dutch ship in a storm by pouring oil into the sea.”</p><p>WAY <hi rend="italics">of a Ship,</hi> is sometimes used for her wake or track. But more commonly the term is understood of the course or progress which she makes on the water under sail: thus, when she begins her motion, she is said to be <hi rend="italics">under Way;</hi> when that motion increases, she is said to have <hi rend="italics">fresh Way</hi> through the water; when she goes apace, they say <hi rend="italics">she has a good Way;</hi> and the account of her rate of sailing by the log, they call, <hi rend="italics">keeping an account of her Way.</hi> And because most ships are apt to fall a little to the leeward of their true course; it is customary, in casting up the log-board, to allow something for her <hi rend="italics">leeward Way,</hi> or <hi rend="italics">leeway.</hi> Hence also a ship is said to have <hi rend="italics">head-Way,</hi> and <hi rend="italics">stern-Way.</hi></p></div1><div1 part="N" n="WAYWISER" org="uniform" sample="complete" type="entry"><head>WAYWISER</head><p>, an instrument for measuring the road, or distance travelled; called also P<hi rend="smallcaps">ERAMBULATOR</hi>, and <hi rend="smallcaps">Pedometer.</hi> See these two articles.</p><p>Mr. Lovell Edgworth communicated to the Society of Arts, &c, an account of a Way-wiser of his invention; for which he obtained a silver medal. This machine consists of a nave, formed of two round flat pieces of wood, 1 inch thick and 8 inches in diameter. In each of the pieces there are cut eleven grooves, 5/8 of an inch wide, and 3/8 deep; and when the two pieces are screwed together, they enclose eleven spokes, forming a wheel of spokes, without a rim: the circumference of the wheel is exactly one pole; and the instrument may be easily taken to pieces, and put up in a small compass. On each of the spokes there is driven a ferril, to prevent them from wearing out; and in the centre of the nave, there is a square hole to receive an axle. Into this hole is inserted an iron or brass rod, which has the thread of a very fine screw worked upon it from one end to the other; upon this screw hangs a nut which, as the rod turns round with the wheel, advances towards the nave of the wheel or recedes from it. The nut does this, because it is prevented from turning round with the axle, by having its centre of gravity placed at some distance below the rod, so as always to hang perpendicularly like a plummet. Two sides of this screw are filed away flat, and have figures engraved upon them, to shew by the progressive motion of the nut, how many circumvolutions of the wheel and its axle have been made: on one side the divisions of miles, furlongs, and poles are in a direct order, and on the other side the same divisions are placed in a retrograde order.</p><p>If the person who uses this machine places it at his right hand side, holding the axle loosely in his hands, and walks forward, the wheel will revolve, and the nut advance from the extremity of the rod towards the nave of the wheel. When two miles have been measured, it will have come close to the wheel. But to continue this measurement, nothing more is necessary than to place the wheel at the left hand of the operator; and the nut will, as he continues the course, recede from the axletree, till another space of two miles is measured.</p><p>It appears from the construction of this machine, that it operates like circular compasses; and does not, like the common wheel Way-wiser, measure the surface of every stone and molehill, &c, but passes over most of <cb/> the obstacles it meets with, and measures the chords only, instead of the arcs of any curved surfaces upon which it rolls.</p></div1><div1 part="N" n="WEATHER" org="uniform" sample="complete" type="entry"><head>WEATHER</head><p>, denotes the state or disposition of the atmosphere, with regard to heat and cold, drought and moisture, fair or foul, wind, rain, hail, srost, snow, fog, &c. See <hi rend="smallcaps">Atmosphere, Hail, Heat, Frost, Rain</hi>, &c.</p><p>There does not seem in all philosophy any thing of more immediate concernment to us, than the state of the Weather; as it is in, and by means of the atmosphere, that all plants are nourished, and all animals live and breathe; and as any alterations in the density, heat, purity, &c, of that, must necessarily be attended with proportionable ones in the state of these.</p><p>The great, but regular alterations, a little change of Weather makes in many parts of inanimate matter, every person knows, in the common instance of barometers, thermometers, hygrometers, &c; and it is owing partly to our inattention, and partly to our unequal and intemperate course of life, that we also, like many other animals, do not feel as great and as regular ones in the tubes, chords, and fibres of our own bodies.</p><p>To establish a proper theory of the Weather, it would be necessary to have registers carefully kept in divers parts of the globe, for a long series of years; from whence we might be enabled to determine the directions, breadth, and bounds of the winds, and of the weather they bring with them; with the correspondence between the Weather of divers places, and the difference between one sort and another at the same place. We might thus in time learn to foretell many great emergencies; as, extraordinary heats, rains, frosts, droughts, dearths, and even plagues, and other epidemical diseases, &c.</p><p>It is however but very few, and partial registers or accounts of the Weather, that have been kept. The Royal Society, the French Academy, and a few particular philosophers, have at times kept such registers as their fancies have dictated, but at no time a regular and correspondent series in many different places, at the same time, followed with particular comparisons and deductions from the whole, &c. The most of what has been done in this way, is as follows: The volumes of the Philosophical Transactions from year to year; the same, for instructions and examples pertaining to the subject, vol. 65, part 2, art. 16; Eras. Bartholin has observations of the Weather for every day in the year 1671: Mr. W. Merle made the like at Oxford, for 7 years: Dr. Plot did the same at the same place, for the year 1684: Mr. Hillier, at Cape Corse, for the years 1686 and 1687: Mr. Hunt and others at Gresham College, for the years 1695 and 1696: Dr. Derham at Upminster in Essex, for the years 1691, 1692, 1697, 1698, 1699, 1703, 1704, 1705: Mr. Townley, in Lancashire, in 1697, 1698: Mr. Cunningham, at Emin in China, for the years 1698, 1699, 1700, 1701: Mr. Locke, at Oats in Essex, 1692: Dr. Scheuchzer, at Zurich, 1708; and Dr. Tilly, at Pisa, the same year: Professor Toaldo, at Padua, for many years: Mr. T. Barker, at Lyndon, in Rutland, for many years in the Philos. Trans.: Mr. Dalton for Kendal, and Mr. Crosthwaite for Keswick, in the years <pb n="678"/><cb/> 1788, 1789, 1790, 1791, 1792, &c; and several others. The register now kept, for many years, in the Philos. Trans. contains an account, two times every day, of the thermometer, barometer, hygrometer, quantity of rain, direction and strength of the wind, and appearance of the atmosphere, as to fair, cloudy, foggy, rainy, &c. And if similar registers were kept in many other parts of the globe, and printed in such-like public Transactions, they might readily be consulted, and a proper use made of them, for establishing this science on the true basis of experiment.</p><p>From many experiments, some general observations have been made, as follow: That barometers generally rise and fall together, even at very distant places, and a consequent conformity and similarity of Weather; but this is the more uniformly so, as the places are nearer together, as might be expected. That the variations of the barometer are greater, as the places are nearer the pole; thus, for instance, the mercury at London has a greater range by 2 or 3 lines than at Paris; and at Paris, a greater than at Zurich; and at some places near the equator, there is scarce any variation at all. That the rain in Switzerland and Italy is much greater in quantity, for the whole year, than in Essex; and yet the rains are more frequent, or there are more rainy days, in Essex, than at either of those places. That cold contributes greatly to rain; and this apparently by condensing the suspended vapours, and so making them descend: thus, very cold months, or seasons, are commonly followed immediately by very rainy ones; and cold summers are always wet ones. That high ridges of mountains, as the Alps, and the snows with which they are covered, not only affect the neighbouring places by the colds, rains, vapours, &c, which they produce; but even distant countries, as England, often partake of their effects. See a collection of ingenious and meteorological observations and conjectures, by Dr. Franklin, in his Experiments, &c, pa. 182, &c. Also a Meteorological Register kept at Mansfield Woodhouse, from 1784 to 1794, Nottingham 1795, 8vo; and Kirwin's ingenious papers on this subject in the Transactions of the Irish Academy, vol. 5. See also the articles <hi rend="smallcaps">Evaporation, Rain</hi>, and <hi rend="smallcaps">Wind.</hi> <hi rend="center"><hi rend="italics">Other Prognostics and Observations,</hi> are as follow:</hi></p><p>That a thick dark sky, lasting for some time, without either sun or rain, always becomes first fair, and then foul, i. e. it changes to a fair clear sky, before it turns to rain. And the reason is obvious: the atmosphere is replete with vapours which, though sufficient to reflect and intercept the sun's rays from us, yet want density to descend; and while the vapours continue in the same state, the Weather will do so too: accordingly, such Weather is commonly attended with moderate warmth, and with little or no wind to disturb the vapours, and a heavy atmosphere to sustain them; the barometer being commonly high: but when the cold approaches, and by condensing the vapours drives them into clouds or drops, then way is made for the sun beams; till the fame vapours, by farther condensation, be formed into rain, and fall down in drops.</p><p>That a change in the warmth of the Weather is <cb/> followed by a change in the wind. Thus, the northerly and southerly winds, though commonly accounted the <hi rend="italics">causes</hi> of cold and warm Weather, are really the <hi rend="italics">effects</hi> of the cold or warmth of the atmosphere; of which Dr. Derham assures us he had so many confirmations, that he makes no doubt of it. Thus, it is common to see a warm southerly wind suddenly changed to the north, by the fall of snow or hail; or to see the wind, in a cold frosty morning, north, when the sun has well warmed the air, wheel towards the south; and again turn northerly or easterly in the cold evening.</p><p>That most vegetables expand their flowers and down in sunshiny Weather: and towards the evening, and against rain, close them again; especially at the beginning of their flowering, when their seeds are tender and sensible. This is visible enough in the down of Dandelion, and other downs: and eminently so in the flowers of pimpernel; the opening and shutting of which make what is called the countryman's <hi rend="italics">Weatherwiser,</hi> by which he foretels the Weather of the following day. The rule is, when the flowers are close shut up, it betokens rain, and foul Weather; but when they are spread abroad, fair Weather.</p><p>The stalk of trefoil, lord Bacon observes, swells against rain, and grows more upright: and the like may be observed, though less sensibly, in the stalks of most other plants. He adds, that in the stubble fields there is found a small red flower, called by the country people pimpernel, which opening in a morning, is a sure indication of a fine day.</p><p>It is very conceivable that vegetables should be affected by the same causes as the Weather, as they may be considered as so many hygrometers and thermometers, consisting of an infinite number of tracheæ, or air-vessels; by which they have an immediate communication with the air, and partake of its moisture, heat, &c.</p><p>Hence it is, that all wood, even the hardest and most solid, swells in moist Weather; the vapours easily insinuating into the pores, especially of the lighter and drier kinds. And hence is derived a very extraordinary use of wood, viz, for breaking rocks or milstones. The method at the quarries is this: Having cut a rock into the form of a cylinder, the workmen divide it into several thinner cylinders, of horizontal courses, by making holes at proper distances round the great one; into these holes they drive pieces of sallow wood, dried in an oven; these in moist Weather, imbibing the humidity from the air, swell, and acting like wedges they break or cleave the rock into several flat stones. And, in like manner, to separate large blocks of stone in the quarry, they wedge such pieces of wood into holes, forming the block into the intended shape, and then pour water upon the wedges, to produce the effect more immediately.</p><p><hi rend="smallcaps">Weather</hi>-<hi rend="italics">Glasses,</hi> are instruments contrived to shew the state of the atmosphere, as to heat, cold, moisture, weight, &c; and so to measure the changes that take place in those respects; by which means we are enabled to predict the alteration of Weather, as to rain, wind, frost, &c.</p><p>Under the class of Weather-glasses, are comprehended barometers, thermometers, hygrometers, manometers, and anemometers. <pb n="679"/><cb/></p></div1><div1 part="N" n="WEDGE" org="uniform" sample="complete" type="entry"><head>WEDGE</head><p>, in Geometry, is a solid having a rectangular base, and two of its oppo- <figure/> site sides ending in an acies or edge. Thus, AB is the rectangular base; and DC the edge; a perpendicular CE, from the edge to the base, is the height of the Wedge. When the length of the edge DC is equal to the length of the base BF, which is the most common form of it, the Wedge is equal to half a rectangular prism of the same base AB and height EC; or it is then a whole triangular prism, having the triangle BCG for its base, and AG or DC for its height. If the edge be more or less than AG, its solid content will be more or less. But, in all cases of the Wedge, the following is a general rule for finding the content of it, viz,</p><p>To twice the length of the base add the length of the edge, multiply the sum by the breadth of the base, and the product by the height of the Wedge; then 1/6 of the last product will be the solid content.</p><p>That is, the content. See this rule demonstrated, and illustrated with examples, in my Mensuration, p. 191, 2d edition.</p><div2 part="N" n="Wedge" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Wedge</hi></head><p>, in Mechanics, one of the five mechanical powers, or simple engines; being a geometrical Wedge, or very acute triangular prism, applied to the splitting of wood, or rocks, or raising great weights.</p><p>The Wedge is made of iron, or some other hard matter, and applied to the raising of vast weights, or separating large or very firm blocks of wood or stone, by introducing the thin edge of the Wedge, and driving it in by blows struck upon the back by hammers or mallets.</p><p>The Wedge is the most powerful of all the simple machines, having an almost unlimited and double advantage over all the other simple mechanical powers; both as it may be made vastly thin, in proportion to its height; in which consists its own natural power; and as it is urged by the force of percussion, or of smart blows, which is a force incomparably greater than any mere dead weight or pressure, such as is employed upon other machines. And accordingly we find it produces effects vastly superior to those of any other power whatever; such as the splitting and raising the largest and hardest rocks; or even the raising and lifting the largest ship, by driving a Wedge below it; which a man can do by the blow of a mallet: and thus the small blow of a hammer, on the back of a Wedge, appears to be incomparably greater than any mere pressure, and will overcome it.</p><p>To the Wedge may be referred all edge-tools, and tools that have a sharp point, in order to cut, cleave, slit, split, chop, pierce, bore, or the like; as knives, hatchets, swords, bodkins, &c.</p><p>In the Wedge, the friction against the sides is very great, at least equal to the force to be overcome; because the Wedge retains any position to which it is driven; and therefore the resistance is at least doubled by the friction.</p><p>Authors have been of various opinions concerning <cb/> the principle from whence the Wedge derives its power. Aristotle considers it as two levers of the first kind, inclined towards each other, and acting opposite ways. Guido Ubaldi, Mersenne, &c, will have them to be levers of the second kind. But De Lanis shews, that the Wedge cannot be reduced to any lever at all. Others refer the Wedge to the inclined plane. And others again, with De Stair, will hardly allow the Wedge to have any force at all in itself; ascribing much the greatest part to the mallet which drives it.</p><p>The doctrine of the force of the Wedge, according to some writers, is contained in this proposition: “If a power directly applied to the head of a Wedge, be to the resistance to be overcome, as the breadth of the back GB, is to the height EC; then the power will be equal to the resistance; and if increased, it will overcome it.”</p><p>But Desaguliers has proved that, when the resistance acts perpendicularly against the sides of the Wedge, the power is to the whole resistance, as the thickness of the back is to the length of both the sides taken together. And the same proportion is adopted by Wallis (Op. Math. vol. 1, p. 1016), Keill (Intr. ad Ver. Phys.), Gravesande (Elem. Math. Lib. 1, cap. 14), and by almost all the modern mathematicians. Gravesande indeed distinguishes the mode in which the Wedge acts, into two cases, one in which the parts of a block of wood, &c, are separated farther than the edge has penetrated to, and the other in which they have not separated farther: In his Scholium de Ligno findendo (ubi supra), he observes, that when the parts of the wood are separated before the Wedge, the equilibrium will be when the force by which it is pushed in, is to the resistance of the wood, as the line DE drawn from <figure/> the middle of the base to the side of the Wedge but perpendicular to the feparated side of the wood continued FG, is to the height of the Wedge DC; but when the parts of the wood are separated no farther than the Wedge is driven in, the equilibrium will be, when the power is to the resistance, as the half base AD, is to its side AC.</p><p>Mr. Ferguson, in estimating the proportion of equilibrium in the two cases last mentioned by Gravesande, agrees with this author, and other modern philosophers, in the latter case; but in the former he contends, that when the wood cleaves to any distance before the Wedge, as it generally does, then the power impelling the Wedge, will be to the resistance of the wood, as half its thickness, is to the length of either side of the cleft, estimated from the top or acting part of the Wedge: for, supposing the Wedge to be lengthened down to the bottom of the cleft, the power will be to the resistance, as half the thickness of the Wedge is to the length of either of its sides. See Ferguson's Lect. p. 40, &c, 4to. See also Desagu. Exp. Phil. vol. 1, p. 107; and Ludlam's Essay on the Power of the Wedge, printed in 1770; &c.</p><p>The generally acknowledged property of the Wedge, and the simplest way of demonstrating it, seem to be the following: When a Wedge is kept in equilibrio, the power acting against the back, is to the force acting <pb n="680"/><cb/> Perpendicularly against either side, as the breadth of the back AB, is to the length of the side AC or BC. —<hi rend="italics">Demonstra.</hi> For any three forces which sustain one another in equilibrio, are as the corresponding sides of a triangle that are drawn perpendicular to the directions in which the forces act. But AB is perpendicular to the force acting on the back, to drive the Wedge forward; and the sides AC and BC are perpendicular to the forces acting upon them; therefore the three forces are as the said lines AB, AC, BC.</p><p>Hence, the thinner a Wedge is, the greater is its effect, in splitting any body, or in overcoming any resistance against the side of the Wedge.</p></div2></div1><div1 part="N" n="WEDNESDAY" org="uniform" sample="complete" type="entry"><head>WEDNESDAY</head><p>, the 4th day of the week, formerly consecrated by the inhabitants of the northern nations to Woden or Oden; who, being reputed the author of magic and inventor of all the arts, was thought to answer to the Mercury of the Greeks and Romans, in honour of whom the same day was by them called <hi rend="italics">dies Mercurii;</hi> and hence it is denoted by astronomers by the character of Mercury <figure/>.</p></div1><div1 part="N" n="WEEK" org="uniform" sample="complete" type="entry"><head>WEEK</head><p>, a division of time that comprises seven days.</p><p>The origin of this division of Weeks, or of computing time by sevenths, is much controverted. It has often been thought to have taken its rise from the four quarters or intervals of the moon, between her changes of phases, which, being about 7 days distant, gave occasion to the division: but others more probably from the seven planets.</p><p>Be this however as it may, the division is certainly very ancient. The Syrians, Egyptians, and most of the oriental nations, appear to have used it from the earliest ages: though it did not get footing in the west till brought in by christianity. The Romans reckoned their days not by sevenths, but by ninths; and the ancient Greeks by decads, or tenths; in imitation of which the new French calendar seems to have been framed.</p><p>The Jews divided their time by Weeks, of 7 days each, as prescribed by the law of Moses; in which they were appointed to work 6 days, and to rest the 7th, in commemoration of the creation, which being effected in 6 days, God rested on the 7th.</p><p>Some authors will even have the use of Weeks, among the other eastern nations, to have proceeded from the Jews; but with little appearance of probability. It is with better reason that others suppose the use of Weeks, among the eastern nations, to be a remnant of the tradition of the creation, which they had still retained with divers others; or else from the number of the planets.</p><p>The Jews denominated the days of the Week, the first, second, third, fourth, and fifth; and the sixth day they named the preparation of the sabbath, or 7th day, which answered to our Saturday. And the like method is still kept up by the christian Arabs, Persians, Ethiopians, &c.</p><p>The ancient heathens denominated the days of the Week from the seven planets; which names are still mostly retained among the christians of the west: thus, the first day was called <hi rend="italics">dies solis, sun-day;</hi> the 2d <hi rend="italics">dies lunæ, moon-day;</hi> &c; a practice the more natural on Dion's principle, that the Egyptians took the division of the Week itself from the seven planets. <cb/></p><p>In fact, the true reason for these denominations seems to be founded in astrology. For the astrologers distributing the government and direction of all the hours in the Week among the seven planets, <figure/> Θ <figure/>, so as that the government of the first hour of the first day fell to Saturn, that of the second day to Jupiter, &c, they gave each day the name of the planet which, according to their doctrine, presided over the first hour of it, and that according to the order above stated. So that the order of the planets in the Week, bears little relation to that in which they follow in the heavens: the former being founded on an imaginary power each planet has, in its turn, on the first hour of each day.</p><p>Dion Cassius gives another reason for the denomination, drawn from the celestial harmony. For it being observed, that the harmony of the diatessaron, which consists in the ratio of 4 to 3, is of great force and effect in music; it was judged meet to proceed directly from Saturn to the Sun; because, according to the old system, there are three planets between Saturn and the Sun, and 4 from the Sun to the Moon</p><p>Our Saxon ancestors, before their conversion to Christianity, named the seven days of the Week from the Sun and Moon and some of their deified heroes, to whom they were peculiarly consecrated, and representing the ancient gods or planets; which names we received and still retain: Thus, Sunday was devoted to the Sun; Monday to the Moon; Tuesday to Tuisco; Wednesday to Woden; Thursday to Thor, the thunderer; Friday to Friga or Friya or Fræa, the wife of Thor; and Saturday to Seater. And nearly according to this order, the modern astronomers express the days of the Week by the seven planets as below: <table><row role="data"><cell cols="1" rows="1" role="data">Θ</cell><cell cols="1" rows="1" role="data">Sunday</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">Monday</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">Tuesday</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">Wednesday</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">Thursday</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">Friday</cell></row><row role="data"><cell cols="1" rows="1" role="data"><figure/></cell><cell cols="1" rows="1" role="data">Saturday.</cell></row></table></p><p>In the same order and number also do these obtain in the Hindoo days of the Week. See Kindersley's Specimens of Hindoo Literature, just published, 8vo.</p></div1><div1 part="N" n="WEIGH" org="uniform" sample="complete" type="entry"><head>WEIGH</head><p>, <hi rend="smallcaps">Way</hi>, or <hi rend="smallcaps">Wey</hi>, a weight of cheese, wool, &c, containing 256 pounds avoirdupois. Of corn, the Weigh contains 40 bushels; of barley or malt, 6 quarters.</p></div1><div1 part="N" n="WEIGHT" org="uniform" sample="complete" type="entry"><head>WEIGHT</head><p>, or <hi rend="italics">Gravity,</hi> in Physics, a quality in natural bodies, by which they tend downwards toward the centre of the earth. See <hi rend="smallcaps">Gravity.</hi></p><p>Weight, like gravity, may be distinguished into <hi rend="italics">absolute, specific,</hi> and <hi rend="italics">relative.</hi></p><p>Newton demonstrates, 1. That the Weights of all bodies, at equal distances from the centre of the earth, are directly proportional to the quantities of matter that each contains: Whence it follows, that the Weights of bodies have no dependence on their shapes or textures; and that all spaces are not equally full of matter.</p><p>2. On different parts of the earth's surface, the Weight of the same body is different; owing to the spheroidal figure of the earth, which causes the body on the surface to be nearer the centre in going from the equator toward the poles; and the increase in the Weight is <pb n="681"/><cb/> nearly in proportion to the versed sine of double the latitude; or, which is the same thing, to the square of the right sine of the latitude: the Weight at the equator to that at the pole, being as 229 to 230; or the whole increase of Weight from the equator to the pole, is the 229th part of the former.</p><p>3. That the Weights of the same body, at different distances above the earth, are inversely as the squares of the distances from the centre. So that, a body at the distance of the moon, which is 60 semidiameters from the earth's centre, would weigh only the 3600th part of what it weighs at the earth's surface.</p><p>4. That at different distances within the earth, or below the surface, the weights of the same body are directly as the distances from the earth's centre: so that, at half way toward the centre, a body would weigh but half as much, and at the very centre it would be no Weight at all.</p><p>5. A body immersed in a fluid, which is specifically lighter than itself, loses so much of its Weight, as is equal to the Weight of a quantity of the fluid of the same bulk with itself. Hence, a body loses more of its weight in a heavier fluid than in a lighter one; and therefore it weighs more in a lighter fluid than in a heavier one.</p><p>The Weight of a cubic foot of pure water, is 1000 ounces, or 62 1/2 pounds, avoirdupois. And the Weights of the cubic foot of other bodies, are as set down under the article <hi rend="italics">Specific</hi> <hi rend="smallcaps">Gravity.</hi></p><p>In the Philos. Trans. (number 458, p. 457 &c) is contained some account of the analogy between English Weights and measures, by Mr. Barlow. He states, that anciently the cubic foot of water was assumed as a general standard for liquids. This cubic foot, of 62 1/2 lb, multiplied by 32, gives 2000, the weight of a ton: and hence 8 cubic feet of water made a hogshead, and 4 hogsheads a tun, or ton, in capacity and denomination, as well as Weight.</p><p>Dry measures were raised on the same model. A bushel of wheat, assumed as a general standard for all sorts of grain, also weighed 62 1/2lb. Eight of these bushels make a quarter, and 4 quarters, or 32 bushels, a ton Weight. Coals were sold by the chaldron, supposed to weigh a ton, or 2000 pounds; though in reality it weighs perhaps upwards of 3000 pounds.</p><p>Hence a ton in Weight is the common standard for liquids, wheat, and coals. Had this analogy been adhered to, the confusion now complained of would have been avoided.—It may reasonably be supposed that corn and other commodities, both dry and liquid, were first sold by Weight; and that measures, for convenience, were afterwards introduced, as bearing some analogy to the Weights before used.</p><div2 part="N" n="Weight" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Weight</hi></head><p>, <hi rend="italics">Pondus,</hi> in Mechanics, denotes any thing to be raised, sustained, or moved by a machine; or any thing that in any manner resists the motion to be produced.</p><p>In all machines, there is a natural and sixed ratio between the Weight and the moving power: and if they be such as to balance each other in equilibria, and then the machine be put in motion by any other force; the Weight and power will always be reciprocally as the velocities of them, or of their centres of gravity; or their momentums will be equal, that is, the pro- <cb/> duct of the Weight multiplied by its velocity, will be equal to the product of the power multiplied by its velocity.</p></div2><div2 part="N" n="Weight" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Weight</hi></head><p>, in Commerce, denotes a body of a known Weight, appointed to be put into a balance against other bodies, whose Weight is required to be known. These Weights are usually of lead, iron, or brass; though in several parts of the East Indies common flints are used; and in some places a sort of little beans.</p><p>The diversity of Weights, in all nations, and at all times, makes one of the most perplexing circumstances in commerce, &c. And it would be a very great convenience if all nations could agree upon a universal standard, and system, both of Weights and measures.</p><p>Weights may be distinguished into <hi rend="italics">ancient</hi> and <hi rend="italics">modern, foreign</hi> and <hi rend="italics">domestic.</hi> <hi rend="center"><hi rend="italics">Modern</hi> <hi rend="smallcaps">Weights</hi>, <hi rend="italics">used in the several parts of Europe, and the Levant.</hi></hi></p><p><hi rend="italics">English</hi> <hi rend="smallcaps">Weights.</hi> By the 27th chapter of Magna Charta, the Weights are to be the same all over England: but for different commodities there are two different sorts, viz, <hi rend="italics">troy Weight,</hi> and <hi rend="italics">averdupois Weight.</hi></p><p>The origin from which both of these are raised, is the grain of wheat, gathered in the middle of the ear: 32 of these, well dried, made one pennyweight, 20 pennyweights " one ounce, and 12 ounces " one pound troy; by Stat. 51 Hen. III; 31 Edw. I; 12 Henry VII.</p><p>A learned writer has shewn that, by the laws of assize, from William the Conqueror to the reign of Henry VII, the legal pound Weight contained a pound of 12 ounces, raised from 32 grains of wheat; and the legal gallon measure contained 8 of those pounds of wheat, 8 gallons making the bushel, and 8 bushels the quarter.</p><p>Henry VII. altered the old English Weight, and introduced the troy pound in its stead, being 3 quarters of an ounce only heavier than the old Saxon pound, or 1-16th heavier. The first statute that directs the use of the averdupois Weight, is that of 24 Henry VIII; and the particular use to which this Weight is thus directed, is simply for weighing butcher's meat in the market; though it is now used for weighing all sorts of coarse and large articles. This pound contains 7000 troy grains; while the troy pound itself contains only 5760 grains, and the old Saxon pound Weight but 5400 grains. Philos. Trans. vol. 65, art. 3.</p><p>Hence there are now in common use in England, two different Weights, viz, troy Weight, and averdupois Weight, the former being employed in weighing such fine articles as jewels, gold, silver, silk, liquors, &c; and the latter for coarse and heavy articles, as bread, corn, flesh, butter, cheese, tallow, pitch, tar, iron, copper, tin, &c. and all grocery wares. And Mr. Ward supposes that it was brought into use from this circumstance, viz, as it was customary to allow larger Weight, of such coarse articles, than the law had expressly enjoined, and this he observes happened to be a 6th part more. Apothecaries buy their drugs by averdupois Weight, but they compound them by troy Weight, though under some little variation of name and divisions. <pb n="682"/><cb/></p><p>The troy or trone pound Weight in Scotland, which by statute is to be the same as the French pound, is commonly supposed equal to 15 3/4 English troy ounces, or 7560 grains; but by a mean of the standards kept by the dean of gild of Edinburgh, it weighs 7599 1/16 or 7600 grains nearly.</p><p>The following tables shew the divisions of the troy and averdupois Weights. <hi rend="center"><hi rend="italics">Table of Troy Weight, as used,</hi></hi> <hi rend="center">1. By the Goldsmiths, &c.</hi> <table><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Grains</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Pennywt.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">24</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">dwt.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Ounce</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">480</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">20</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 1 oz.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">Pound</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">5760</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=right" role="data">240</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data"> 12 = 1 lb.</cell></row></table> <hi rend="center">2. By the Apothecaries.</hi> <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Grains.</cell><cell cols="1" rows="1" rend="colspan=2" role="data">Scruples</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20 =</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" rend="align=left" role="data">[scruple]</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">Drams</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60 =</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" rend="align=left" role="data">=</cell><cell cols="1" rows="1" role="data">1 [dram]</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Ounces</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">480 =</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" rend="align=left" role="data">=</cell><cell cols="1" rows="1" role="data">8 =</cell><cell cols="1" rows="1" rend="align=left" role="data">  1 [ounce]</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">Pound</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5760 =</cell><cell cols="1" rows="1" role="data">288</cell><cell cols="1" rows="1" rend="align=left" role="data">=</cell><cell cols="1" rows="1" role="data">96 =</cell><cell cols="1" rows="1" rend="align=left" role="data"> 12 = 1 lb.</cell></row></table> <hi rend="center"><hi rend="italics">Table of Averdupois Weight.</hi></hi> <table><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">Drams</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Ounces</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">Pounds</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">256</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Quarters</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7168</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">448</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Hund. wt.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28672</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">1792</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">112</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data">Ton</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">573440</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">35840</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" role="data">2240</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=center" role="data">80</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=center" role="data">20</cell><cell cols="1" rows="1" role="data">=</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell></row></table></p><p>Mr. Ferguson (Lect. on Mech. p. 100, 4to) gives the following comparison between troy and averdupois Weight. <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">175</cell><cell cols="1" rows="1" role="data">troy pounds are equal to 144 averdup. pounds.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">175</cell><cell cols="1" rows="1" role="data">troy ounces are equal to 192 averdup. ounces.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">troy pound contains 5760 grains.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">averdupois pound contains 7000 grains.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">averdupois ounce contains 437 1/2 grains.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">averdupois dram contains 27.34375 grains.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">troy pound contains 13 oz. 2.651428576 drams</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data"/><cell cols="1" rows="1" role="data">  averdupois</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1</cell><cell cols="1" rows="1" role="data">averdup. lb. contains 1 lb 2 oz 11 dwts 16 gr troy</cell></row></table></p><p>The moneyers, jewellers, &c, have a particular class of Weights, for gold and precious stones, viz, <hi rend="italics">carat</hi> and <hi rend="italics">grain;</hi> and for silver, the <hi rend="italics">pennyweight</hi> and <hi rend="italics">grain.</hi> The moneyers have also a peculiar subdivision of the troy grain: thus, dividing <hi rend="center">the grain into 20 mites</hi> <hi rend="center">the mite into 24 droits</hi> <hi rend="center">the droit into 20 periots</hi> <hi rend="center">the periot into 24 blanks.</hi> <cb/></p><p>The dealers in wool have likewise a particular set of Weights; viz, the <hi rend="italics">sack, weigh, tod, stone,</hi> and <hi rend="italics">clove,</hi> the proportions of which are as below: viz, <table><row role="data"><cell cols="1" rows="1" role="data">the sack containing</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">weighs</cell></row><row role="data"><cell cols="1" rows="1" role="data">the weigh "</cell><cell cols="1" rows="1" role="data">6 1/2</cell><cell cols="1" rows="1" role="data">tods</cell></row><row role="data"><cell cols="1" rows="1" role="data">the tod "</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">stones</cell></row><row role="data"><cell cols="1" rows="1" role="data">the stone "</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">cloves</cell></row><row role="data"><cell cols="1" rows="1" role="data">the clove "</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">pounds.</cell></row></table></p><p>Also 12 sacks make a last or 4368 pounds. Farther, 56 lb of old hay, or 60 lb new hay, make a truss. 40 lb of straw make a truss. 36 trusses make a load, of hay or straw. 14 lb make a stone. 5 lb of glass a stone.</p><p><hi rend="italics">French</hi> <hi rend="smallcaps">Weights.</hi> The common or Paris pound Weight, is to the English troy pound, as 21 to 16, and to the averdupois pound as 27 to 25; it therefore contains 7560 troy grains; and it is divided into 16 ounces like the pound averdupois, but more particularly thus: the pound into 2 <hi rend="italics">marcs;</hi> the marc into 8 <hi rend="italics">ounces;</hi> the ounce into 8 <hi rend="italics">gros,</hi> or <hi rend="italics">drams;</hi> the gross or dram into 3 deniers, Paris scruples or pennyweights; and the pennyweight into 24 <hi rend="italics">grains;</hi> the grain being an equivalent to a grain of wheat. So that the Paris ounce contains 472 1/2 troy grains, and therefore it is to the English troy ounce as 63 to 64. But in several of the French provinces, the pound is of other different Weights. A <hi rend="italics">quintal</hi> is equal to 100 pounds.</p><p>The Weights above enumerated under the two articles of English and French Weights, are the same as are used throughout the greatest part of Europe; only under somewhat different names, divisions, and proportions. And besides, particular nations have also certain Weights peculiar to themselves, of too little consequence here to be enumerated. But to shew the proportion of these several Weights to one another, there may be here added a reduction of the divers pounds in use throughout Europe, by which the other Weights are estimated, to one standard pound, viz, the pound of Amsterdam, Paris, and Bourdeaux; as they were accurately calculated by M. Ricard, and published in the new edition of his Traité de Commerce, in 1722. <hi rend="center"><hi rend="italics">Proportion of the</hi> <hi rend="smallcaps">Weights</hi> <hi rend="italics">of the chief Cities in Europe to that of Amsterdam.</hi></hi> <table><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data">100 pounds of Amsterdam are equal to</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">108lbs of Alicant</cell><cell cols="1" rows="1" rend="colspan=2" role="data">100lbs of Bilboa</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Antwerp</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Bois le Duc</cell></row><row role="data"><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data">Archangel, or</cell><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" role="data">Bologna</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> 3 poedes</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Bourdeaux</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Arschot</cell><cell cols="1" rows="1" role="data">104</cell><cell cols="1" rows="1" role="data">Bourg en Bresse</cell></row><row role="data"><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data">Avignon</cell><cell cols="1" rows="1" role="data">103</cell><cell cols="1" rows="1" role="data">Bremen</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 98</cell><cell cols="1" rows="1" role="data">Basil</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">Breslaw</cell></row><row role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Bayonne</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Bruges</cell></row><row role="data"><cell cols="1" rows="1" role="data">166</cell><cell cols="1" rows="1" role="data">Bergamo</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Brussels</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 97</cell><cell cols="1" rows="1" role="data">Berg. op Zoom</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Cadiz</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 95 1/4</cell><cell cols="1" rows="1" role="data">Bergen, Norw.</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Cologne</cell></row><row role="data"><cell cols="1" rows="1" role="data">111</cell><cell cols="1" rows="1" role="data">Bern</cell><cell cols="1" rows="1" role="data">107 1/2</cell><cell cols="1" rows="1" role="data">Copenhagen</cell></row><row role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Besançon</cell><cell cols="1" rows="1" role="data"> 87</cell><cell cols="1" rows="1" role="data">Constantinople</cell></row></table> <pb n="683"/><cb/> <hi rend="center"><hi rend="smallcaps">Weights</hi> <hi rend="italics">continued.</hi></hi> <table><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data">100 pounds of Amsterdam are equal to</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">113 1/2lbs of Dantzic</cell><cell cols="1" rows="1" rend="colspan=2" role="data">154lbs of Messina</cell></row><row role="data"><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Dort</cell><cell cols="1" rows="1" role="data">168</cell><cell cols="1" rows="1" role="data">Milan</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 97</cell><cell cols="1" rows="1" role="data">Dublin</cell><cell cols="1" rows="1" role="data">120</cell><cell cols="1" rows="1" role="data">Montpelier</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 97</cell><cell cols="1" rows="1" role="data">Edinburgh</cell><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">Muscovy</cell></row><row role="data"><cell cols="1" rows="1" role="data">143</cell><cell cols="1" rows="1" role="data">Florence</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Nantes</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 98</cell><cell cols="1" rows="1" role="data">Franckfort, sur</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Nancy</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> Maine</cell><cell cols="1" rows="1" role="data">169</cell><cell cols="1" rows="1" role="data">Naples</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Gaunt</cell><cell cols="1" rows="1" role="data"> 98</cell><cell cols="1" rows="1" role="data">Nuremberg</cell></row><row role="data"><cell cols="1" rows="1" role="data"> 89</cell><cell cols="1" rows="1" role="data">Geneva</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Paris</cell></row><row role="data"><cell cols="1" rows="1" role="data">163</cell><cell cols="1" rows="1" role="data">Genoa</cell><cell cols="1" rows="1" role="data">112 1/2</cell><cell cols="1" rows="1" role="data">Revel</cell></row><row role="data"><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" role="data">Hamburgh</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">Riga</cell></row><row role="data"><cell cols="1" rows="1" role="data">125</cell><cell cols="1" rows="1" role="data">Koningsberg</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Rochel</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Leipsic</cell><cell cols="1" rows="1" role="data">146</cell><cell cols="1" rows="1" role="data">Rome</cell></row><row role="data"><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" role="data">Leyden</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Rotterdam</cell></row><row role="data"><cell cols="1" rows="1" role="data">143</cell><cell cols="1" rows="1" role="data">Leghorn</cell><cell cols="1" rows="1" role="data"> 96</cell><cell cols="1" rows="1" role="data">Rouen</cell></row><row role="data"><cell cols="1" rows="1" role="data">105 1/2</cell><cell cols="1" rows="1" role="data">Liege</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">S. Malo</cell></row><row role="data"><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" role="data">Lisbon</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">S. Sebastian</cell></row><row role="data"><cell cols="1" rows="1" role="data">114</cell><cell cols="1" rows="1" role="data">Lisle</cell><cell cols="1" rows="1" role="data">158 1/7</cell><cell cols="1" rows="1" role="data">Saragosa</cell></row><row role="data"><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">London, aver-</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">Seville</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"> dupois</cell><cell cols="1" rows="1" role="data">114</cell><cell cols="1" rows="1" role="data">Smyrna</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Louvain</cell><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" role="data">Stetin</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Lubeck</cell><cell cols="1" rows="1" role="data"> 81</cell><cell cols="1" rows="1" role="data">Stockholm</cell></row><row role="data"><cell cols="1" rows="1" role="data">141 1/2</cell><cell cols="1" rows="1" role="data">Lucca</cell><cell cols="1" rows="1" role="data">118</cell><cell cols="1" rows="1" role="data">Tholouse</cell></row><row role="data"><cell cols="1" rows="1" role="data">116</cell><cell cols="1" rows="1" role="data">Lyons</cell><cell cols="1" rows="1" role="data">151</cell><cell cols="1" rows="1" role="data">Turin</cell></row><row role="data"><cell cols="1" rows="1" role="data">114</cell><cell cols="1" rows="1" role="data">Madrid</cell><cell cols="1" rows="1" role="data">158 1/2</cell><cell cols="1" rows="1" role="data">Valencia</cell></row><row role="data"><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">Malines</cell><cell cols="1" rows="1" role="data">182</cell><cell cols="1" rows="1" role="data">Veniçe.</cell></row><row role="data"><cell cols="1" rows="1" role="data">123 1/2</cell><cell cols="1" rows="1" role="data">Marseilles</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table> <hi rend="center"><hi rend="italics">Ancient</hi> <hi rend="smallcaps">Weights.</hi></hi></p><p>1. The Weights of the ancient Jews, reduced to the English troy Weights, will stand as below: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">lb</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">dwt</cell><cell cols="1" rows="1" role="data">gr</cell></row><row role="data"><cell cols="1" rows="1" role="data">Shekel</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=center" role="data">9</cell><cell cols="1" rows="1" role="data"> 2 4/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Manch</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">10 2/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Talent</cell><cell cols="1" rows="1" rend="align=right" role="data">113</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">10 2/7</cell></row></table></p><p>2. Grecian and Roman Weights, reduced to English troy Weight, are as in the following table: <table><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">lb</cell><cell cols="1" rows="1" role="data">oz</cell><cell cols="1" rows="1" role="data">dwt</cell><cell cols="1" rows="1" role="data">gr</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lentes</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data"> 0 85/112</cell></row><row role="data"><cell cols="1" rows="1" role="data">Siliquæ</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data"> 3 1/28</cell></row><row role="data"><cell cols="1" rows="1" role="data">Obolus</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data"> 9 3/28</cell></row><row role="data"><cell cols="1" rows="1" role="data">Scriptulum</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" role="data">18 3/14</cell></row><row role="data"><cell cols="1" rows="1" role="data">Drachma</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">2</cell><cell cols="1" rows="1" role="data"> 6 9/14</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sextula</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">3</cell><cell cols="1" rows="1" role="data"> 0 6/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sicilicus</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">4</cell><cell cols="1" rows="1" role="data">13 2/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Duella</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">6</cell><cell cols="1" rows="1" role="data"> 1 5/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Uncia</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data"> 5 1/7</cell></row><row role="data"><cell cols="1" rows="1" role="data">Libra</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=right" role="data">10</cell><cell cols="1" rows="1" rend="align=right" role="data">18</cell><cell cols="1" rows="1" role="data">13 5/7</cell></row></table></p><p>The Roman ounce is the English averdupois ounce, which they divide into 7 denarii, as well as 8 drachms: and as they reckoned their denarius equal to the Attic drachm, this will make the Attic Weights one-eighth heavier than the correspondent Roman Weights. Arbuth. <cb/></p><p><hi rend="italics">Regulation of</hi> <hi rend="smallcaps">Weights</hi> <hi rend="italics">and Measures.</hi> This is a branch of the king's prerogative. For the public convenience, these ought to be universally the same throughout the nation, the better to reduce the prices of articles to equivalent values. But as Weight and measure are things in their nature arbitrary and uncertain, it is necessary that they be reduced to some sixed rule or standard. It is however impossible to fix such a standard by any written law or oral proclamation; as no person can, by words only, give to another an adequate idea of a pound Weight, or foot-rule. It is therefore expedient to have recourse to some visible, palpable, material standard; by forming a comparison with which, all Weights and measures may be reduced to one uniform size. Such a standard was anciently kept at Winchester: and we find in the laws of king Edgar, near a century before the conquest, an injunction that that measure should be observed throughout the realm.</p><p>Most nations have regulated the standard of measures of length from some parts of the human body; as the palm, the hand, the span, the foot, the cubit, the ell (<hi rend="italics">ulna</hi> or arm), the pace, and the fathom. But as these are of different dimensions in men of different proportions, ancient historians inform us, that a new standard of length was fixed by our king Henry the first; who commanded that the <hi rend="italics">ulna</hi> or ancient ell, which answers to the modern yard, should be made of the exact length of his own arm.</p><p>A standard of long measure being once gained, all others are easily derived from it; those of greater length by multiplying that original standard, those of less by dividing it. Thus, by the statute called <hi rend="italics">compositio ulnarum et perticarum,</hi> 5 1/2 yards make a perch; and the yard is subdivided into 3 seet, and each foot into 12 inches; which inches will be each of the length of 3 barley corns. But some, on the contrary, derive all measures, by composition, from the barley corn.</p><p>Superficial measures are derived by squaring those of length; and measures of capacity by cubing them.</p><p>The standard of Weights was originally taken from grains or corns of wheat, whence our lowest denomination of Weights is still called a <hi rend="italics">grain;</hi> 32 of which are directed, by the statute called <hi rend="italics">compositio mensurarum,</hi> to compose a pennyweight. 20 of which make an ounce, and 12 ounces a pound, &c.</p><p>Under king Richard the first it was ordained, that there should be only one Weight and one measure throughout the nation, and that the custody of the assize or standard of Weights and measures, should be committed to certain persons in every city and borough; from whence the ancient office of the king's ulnager seems to have been derived. These original standards were called <hi rend="italics">pondus regis,</hi> and <hi rend="italics">mensura domini regis,</hi> and are directed by a variety of subsequent statutes to be kept in the exchequer chamber, by an officer called the <hi rend="italics">clerk of the market,</hi> except the wine gallon, which is committed to the city of London, and kept in Guildhall.</p><p>The Scottish standards are distributed among the oldest boroughs. The elwand is kept at Edinburgh, the pint at Stirling, the pound at Lanark, and the firlot at Linlithgow.</p><p>The two principal Weights established in Great Britain, are troy Weight, and avoirdupois Weight, <pb n="684"/><cb/> as before mentioned. Under the head of the former it may farther be added, that</p><p>A carat is a Weight of 4 grains; but when the term is applied to gold, it denotes the degree of fineness. Any quantity of gold is supposed divided into 24 parts. If the whole mass be pure gold, it is said to be 24 carats fine; if there be 23 parts of pure gold, and one part of alloy or base metal, it is said to be 23 carats fine, and so on.</p><p>Pure gold is too soft to be used for coin. The standard coin of this kingdom is 23 carats fine. A pound of standard gold is coined into 44 1/2 guineas, and therefore every guinea mould weigh 5 dwts 9 39/<*> grains.</p><p>A pound of silver for coin contains 11 oz 2 dwts pure silver, and 18 dwts alloy: and standard silverplate, 11 ounces pure silver, with 1 ounce alloy. A pound of standard silver is coined into 62 shillings; and therefore the Weight of a shilling should be 3 dwts 20 28/31 grains. <hi rend="center"><hi rend="italics">Universal Standard for</hi> <hi rend="smallcaps">Weights</hi> <hi rend="italics">and Measures.</hi></hi></p><p>Philosophers, from their habits of generalizing, have often made speculations for forming a general standard for Weights and measures through the whole world. These have been devised chiefly of a philosophical nature, as best adapted to universality. After the invention of pendulum clocks, it first occurred that the length of a pendulum which should vibrate seconds, would be proper to be made a universal standard for lengths; whether it should be called a yard, or any thing else. But it was found, that it would be difficult in practice, to measure and determine the true length of such a pendulum, that is the distance between the point of suspension and the point of oscillation. Another cause of inaccuracy was afterwards discovered, when it was found that the seconds pendulum was of different lengths in all the different latitudes, owing to the spheroidal figure of the earth, which causes that all places in different latitudes are at different distances from the centre, and consequently the pendulums are acted upon by different forces of gravity, and therefore require to be of different lengths. In the latitude of London this is found to be 39 1/8 inches.</p><p>The Society of Arts in London, among their many laudable and patriotic endeavours, offered a handsome premium for the discovery of a proper standard for Weights and measures. This brought them many frivolous expedients, as well as one which was an improvement on the method of the pendulum, by one Hatton. This consisted in measuring the difference of the lengths of two pendulums of different times of vibration; which could be performed more easily and accurately than that of the length of one single pendulum. This method was put in practice, and fully explained and illustrated, by the late Mr. Whitehurst, in his attempt to ascertain an Universal Standard of Weights and Measures. But still the same kind of inaccuracy of measurement &c, obtains in this way, as in the single pendulum, though in a smaller degree.</p><p>Another method that has been proposed for this purpose, is the space that a heavy body falls freely <cb/> through in 1 second of time. But this is an experiment more difficult than the former to be made with accuracy; on which account, different persons will all make the space fallen to be of different quantities, which would give as many different standards of length. Add to this, that the spheroidal form of the earth here again introduces a diversity in the space, owing to the different distances from the centre, and the consequent diversity in the force of gravity by which the body falls. This space has been found to be 193 inches, or 16 1/12 feet, in the latitude of London; but it will be a different quantity in other latitudes.</p><p>Many other inferior expedients have also been proposed for the purpose of universal measures, and Weights, but there is another which now has the best prospect of success, and is at present under particular experiments, by the philosophers both of this and the French nation. This method is by the measure of the degrees of latitude; which would give a large quantity, and admit of more accurate measures, by subdivision, than what could be obtained by beginning from a small quantity, or measure, and thence to proceed increasing by multiples. This measure might be taken either from the extent of the whole compass of the earth, or of all the 360 degrees, or a medium degree among them all, or from the measure of a degree in the medium latitude of 45 degrees. It will also be most convenient to make the subdivisions. of this measure, when found, to proceed decimally, or continually by 10ths.</p><p>The universal standard for lengths being once established, those of Weights, &c, would easily follow. For instance, a vessel, of certain dimensions, being filled with distilled water, or some other homogeneous matter, the Weight of that may be considered as a standard for Weights.</p><p><hi rend="smallcaps">Weight</hi> of the Air, Water, &c. See those articles severally. See also <hi rend="smallcaps">Specific Gravity.</hi></p></div2></div1><div1 part="N" n="WERST" org="uniform" sample="complete" type="entry"><head>WERST</head><p>, a Russian measure of length, equal to 3500 English feet.</p></div1><div1 part="N" n="WEST" org="uniform" sample="complete" type="entry"><head>WEST</head><p>, one of the cardinal points of the horizon, or of the compass, diametrically opposite to the east, or lying on the left hand when we face the north. Or West is strictly the intersection of the prime vertical with the horizon, on that side where the sun sets.</p><p><hi rend="smallcaps">West</hi> <hi rend="italics">Wind,</hi> is also called <hi rend="italics">Zephyrus,</hi> and <hi rend="italics">Favonius.</hi></p><p><hi rend="smallcaps">West</hi> <hi rend="italics">Dial.</hi> See <hi rend="smallcaps">Dial.</hi></p><p>WESTERN <hi rend="italics">Amplitude, Horizon, Ocean.</hi> See the several articles.</p></div1><div1 part="N" n="WESTING" org="uniform" sample="complete" type="entry"><head>WESTING</head><p>, in Navigation, is the quantity of departure made good to the westward from the meridian.</p><p>WEY. See <hi rend="smallcaps">Weigh.</hi></p></div1><div1 part="N" n="WHALE" org="uniform" sample="complete" type="entry"><head>WHALE</head><p>, in Astronomy, one of the constellations. See <hi rend="smallcaps">Cetus.</hi></p></div1><div1 part="N" n="WHEEL" org="uniform" sample="complete" type="entry"><head>WHEEL</head><p>, in Mechanics, a simple machine, consisting of a circular piece of wood, metal, or other matter, that revolves on an axis. This is otherwise called <hi rend="italics">Wheel and Axle,</hi> or <hi rend="smallcaps">Axis</hi> <hi rend="italics">in Peritrochio,</hi> as a mechanical power, being one of the most frequent and useful of any. In this capacity of it, the Wheel is a kind of perpetual lever, and the axis another lesser one; or the radius of the Wheel and that of its axis may be considered as the longer and shorter arms of a lever, the centre of the Wheel being the fulcrum or point of <pb n="685"/><cb/> suspension. Whence it is, that the power of this machine is estimated by this rule, as the radius of the axis is to the radius of the Wheel or of the circumference, so is any given power, to the weight it will sustain.</p><p>Wheels, as well as their axes, are frequently dented, or cut into teeth, and are then of use upon innumerable occasions; as in jacks, clocks, mill-work, &c; by which means they are capable of moving and acting on one another, and of being combined together to any extent; the teeth either of the axis or circumference working in those of other Wheels or axles; and thus, by multiplying the power to any extent, an amazing great effect is produced.</p><p><hi rend="italics">To compute the power of a combination of Wheels;</hi> the teeth of the axis of every Wheel acting on those in the circumference of the next following. Multiply continually together the radii of all the axes, as also the radii of all the Wheels; then it will be, as the former product is to the latter product, so is a given power applied to the circumference, to the weight it can sustain. Thus, for example, in a combination of five Wheels and axles, to find the weight a man can sustain, or raise, whose force is equal to 150 pounds, the radii of the Wheels being 30 inches, and those of the axes 3 inches. Here 3 X 3 X 3 X 3 X 3 = 243, and 30 X 30 X 30 X 30 X 30 = 24300000, therefore as 243 : 24300000 :: 150 : 15000000 lb, the weight he can sustain, which is more than 6696 tons weight. So prodigious is the increase of power in a combination of Wheels!</p><p>But it is to be observed, that in this, as well as every other mechanical engine, whatever is gained in power, is lost in time; that is, the weight will move as much flower than the power, as the force is increased or multiplied, which in the example above is 100000 times flower.</p><p>Hence, having given any power, and the weight to be raised, with the proportion between the Wheels and axles necessary to that effect; to find the number of the Wheels and axles. Or, having the number of the Wheels and axles given, to find the ratio of the radii of the Wheels and axles. Here, putting <hi rend="italics">p</hi> = the power acting on the last wheel, <hi rend="italics">w</hi> = the weight to be raised, <hi rend="italics">r</hi> = the radius of the axles, R = the radius of the wheels, <hi rend="italics">n</hi> = the number of the wheels and axles; then, by the general proportion, as <hi rend="italics">r</hi><hi rend="sup">n</hi> : R<hi rend="sup">n</hi> :: <hi rend="italics">p</hi> : <hi rend="italics">w;</hi> therefore is a general theorem, from whence may be found any one of these five letters or quantities, when the other four are given. Thus, to find <hi rend="italics">n</hi> the number of Wheels: we have first . And to sind R/<hi rend="italics">r,</hi> the ratio of the Wheel to the axle; it is .</p><p><hi rend="smallcaps">Wheels</hi> <hi rend="italics">of a Clock, &c,</hi> are, the crown wheel, contrat wheel, great wheel, second wheel, third wheel, striking wheel, detent wheel, &c. <cb/></p><p><hi rend="smallcaps">Wheels</hi> <hi rend="italics">of Coaches, Carts, Waggons, &c.</hi> With respect to Wheels of carriages, the following particulars are collected from the experiments and observations of Desaguliers, Beighton, Camus, Ferguson, Jacob, &c.</p><p>1. The use of Wheels, in carriages, is twofold; viz, that of diminishing or more easily overcoming the resistance or friction from the carriage; and that of more easily overcoming obstacles in the road. In the first case the friction on the ground is transferred in some degree from the outer surface of the Wheel to its nave and axle; and in the latter, they serve easily to raise the carriage over obstacles and asperities met with on the roads. In both these cases, the height of the Wheel is of material consideration, as the spokes act as levers, the top of an obstacle being the fulcrum, their length enables the carriage more easily to surmount them; and the greater proportion of the Wheel to the axle serves more easily to diminish or to overcome the friction of the axle. See Jacob's Observations on Wheel Carriages, p. 23 &c.</p><p>2. The Wheels should be exactly round; and the fellies at right angles to the naves, according to the inclination of the spokes.</p><p>3. It is the most general opinion, that the spokes be somewhat inclined to the naves, so that the Wheels may be dishing or concave. Indeed if the Wheels were always to roll upon smooth and level ground, it would be best to make the spokes perpendicular to the naves, or to the axles; because they would then bear the weight of the load perpendicularly. But because the ground is commonly uneven, one Wheel often falls into a cavity or rut, when the other does not, and then it bears much more of the weight than the other does; in which case it is best for the Wheels to be dished, because the spokes become perpendicular in the rut, and therefore have the greatest strength when the obliquity of the road throws most of the weight upon them; whilst those on the high ground have less weight to bear, and therefore need not be at their full strength.</p><p>4. The axles of the Wheels should be quite straight, and perpendicular to the shafts, or to the pole. When the axles are straight, the rims of the Wheels will be parallel to each other, in which case they will move the easiest, because they will be at liberty to proceed straight forwards. But in the usual way of practice, the ends of the axles are bent downwards; which always keeps the sides of the Wheels that are next the ground nearer to one another than their upper sides are; and this not only makes the Wheels drag sideways as they go along, and gives the load a much greater power of crushing them than when they are parallel to each other, but also endangers the overturning the carriage when a Wheel falls into a hole or rut, or when the carriage goes on a road that has one side lower than the other, as along the side of a hill. Mr. Beighton however has offered several reasons to prove that the axles of Wheels ought not to be straight; tor which see Desaguliers's Exp. Phil. vol. 2, Appendix.</p><p>5. Large Wheels are found more advantageous for rolling than small ones, both with regard to their power as a longer lever, and to the degree of friction, and to the advantage in getting over holes, rubs, and <pb n="686"/><cb/> stones, &c. If we consider Wheels with regard to the friction upon their axles, it is evident that small Wheels, by turning oftener round, and swifter about the axles, than large ones, must have much more friction. Again, if we consider Wheels as they sink into holes or soft earth, the large Wheels, by sinking less, must be much easier drawn out of them, as well as more easily over stones and obstacles, from their greater length of lever or spokes. Desaguliers has brought this matter to a mathematical calculation, in his Experim. Philos. vol. 1, p. 171, &c. See also Jacob's Observ. p. 63.</p><p>From hence it appears then, that Wheels are the more advantageous as they are larger, provided they are not more than 5 or 6 feet diameter; for when they exceed these dimensions, they become too heavy; or if they are made light, their strength is proportionably diminished, and the length of the spokes renders them more liable to break: besides, horses applied to such Wheels would not be capable of exerting their utmost strength, by having the axles higher than their breasts, so that they would draw downwards; which is even a greater disadvantage than small Wheels have in occasioning the horses to draw upwards.</p><p>6. Carriages with 4 Wheels, as waggons or coaches, are much more advantageous than carriages with 2 Wheels, as carts and chaises; for with 2 wheels it is plain the tiller horse carries part of the weight, in one way or other: in going down hill, the weight bears upon the horse; and in going up hill, the weight falls the other way, and lifts the horse, which is still worse. Besides, as the Wheels sink into the holes in the roads, sometimes on one side, sometimes on the other, the shafts strike against the tiller's sides, which destroys many horses: moreover, when one of the Wheels sinks into a hole or rut, half the weight falls that way, which endangers the overturning of the carriage.</p><p>7. It would be much more advantageous to make the 4 Wheels of a coach or waggon large, and nearly of a height, than to make the fore Wheels of only half the diameter of the hind Wheels, as is usual in many places. The fore Wheels have commonly been made of a less size than the hind ones, both on account of turning short, and to avoid cutting the braces. Crane-necks have also been invented for turning yet shorter, and the fore Wheels have been lowered, so as to go quite under the bend of the crane-neck.</p><p>It is held, that it is a great disadvantage in small Wheels, that as their axle is below the bow of the horses breasts, the horses not only have the loaded carriage to draw along, but also part of its weight to bear, which tires them soon, and makes them grow much stiffer in their hams, than they would be if they drew on a level with the fore axle.</p><p>But Mr. Beighton disputes the propriety of fixing the line of traction on a level with the breast of a horse, and says it is contrary to reason and experience. Horses, he says, have little or no power to draw but what they derive from their weight; without which they could not take hold of the ground, and then they must slip, and draw nothing. Common experience also teaches, that a horse must have a certain weight on his back or shoulders, that he may draw the better. And <cb/> when a horse draws hard, it is observed that he bends forward, and brings his breast near the ground; and then if the Wheels are high, he is pulling the carriage against the ground. A horse tackled in a waggon will draw two or three ton, because the point or line of traction is below his breast, by the lowness of the Wheels. It is also common to see, when one horse is drawing a heavy load, especially up hill, his fore feet will rise from the ground; in which case it is usual to add a weight on his back, to keep his fore part down, by a person mounting on his back or shoulders, which will enable him to draw that load, which he could not move before. The greatest stress, or main business of drawing, says this ingenious writer, is to overcome obstacles; for on level plains the drawing is but little, and then the horse's back need be pressed but with a small weight.</p><p>8. The utility of broad Wheels, in amending and preserving the roads, has been so long and generally acknowledged, as to have occasioned the legislature to enforce their use. At the same time, the proprietors and drivers of carriages seem to be convinced by experience, that a narrow-wheeled carriage is more easily and speedily drawn by the same number of horses, than a broad-wheeled one of the same burthen: probably because they are much lighter, and have less friction on the axle.</p><p>On the subject of this article, see Jacob's Observ. &c. on Wheel-Carriages, 1773, p. 81. Desagul. Exper. Phil. vol. 1, p. 201. Ferguson's Lect. 4to, p. 56. Martin's Phil. Brit. vol. 1, p. 229.</p><p><hi rend="italics">Blowing</hi> <hi rend="smallcaps">Wheel</hi>, is a machine contrived by Desaguliers, for drawing the foul air out of any place, or for forcing in fresh, or doing both successively, without opening doors or windows. See Philos. Trans. number 437. The intention of this machine is the same as that of Hales's ventilator, but not so effectual, nor so convenient. See Desag. Exper. Philos. vol. 2, p. 563, 568.—This Wheel is also called a <hi rend="italics">centrifugal Wheel,</hi> because it drives the air with a centrifugal force.</p><p><hi rend="italics">Water</hi> <hi rend="smallcaps">Wheel</hi>, of a Mill, that which receives the impulse of the stream by means of ladle-boards or floatboards. M. Parent, of the Academy of Sciences, has determined that the greatest effect of an undershot Wheel, is when its velocity is equal to the 3d part of the velocity of the water that drives it; but it ought to be the half of that velocity, as is fully shewn in the article Mill, pa. 111. In fixing an undershot Wheel, it ought to be considered whether the water can run clear off, so as to cause no back-water to stop its motion. Concerning this article, see Desagul. Exp. Philos. vol. 2, p. 422. Also a variety of experiments and observations relating to undershot and overshot Wheels, by Mr. Smeaton, in the Philos. Trans. vol. 51, p. 100.</p><p><hi rend="italics">Aristotle's</hi> <hi rend="smallcaps">Wheel.</hi> See <hi rend="smallcaps">Rota</hi> <hi rend="italics">Aristotelica.</hi></p><p><hi rend="italics">Measuring</hi> <hi rend="smallcaps">Wheel.</hi> See <hi rend="smallcaps">Perambulator.</hi></p><p><hi rend="italics">Orffyreus's</hi> <hi rend="smallcaps">Wheel.</hi> See <hi rend="smallcaps">Orffyreus.</hi></p><p><hi rend="italics">Persian</hi> <hi rend="smallcaps">Wheel.</hi> See <hi rend="smallcaps">Persian.</hi></p><p><hi rend="smallcaps">Wheel</hi>-<hi rend="italics">Barometer.</hi> See <hi rend="smallcaps">Barometer.</hi></p><p>WHIRL-POOL, an eddy, vortex, or gulph, where the water is continually turning round.</p><p>WHIRLING-TABLE, a machine contrived for <pb/><pb/><pb n="687"/><cb/> representing several phenomena in philosophy, and nature; as, the principal laws of gravitation, and of the planetary motions in curvilinear orbits.</p><p>The figure of this instrument is exhibited fig. 1, pl. 35: where AA is a strong frame of wood; B a winch fixed on the axis C of the wheel D, round which is the catgut string F, which also goes round the small wheels G and K, crossing between them and the great wheel D. On the upper end of the axis of the wheel G, above the frame, is fixed the round board <hi rend="italics">d,</hi> to which may be occasionally fixed the bearer MSX. On the axis of the wheel H is fixed the bearer NTZ, and when the winch B is turned, the wheels and bearers are put into a Whirling motion. Each bearer has two wires W, X, and Y, Z, fixed and screwed tight into them at the ends by nuts on the outside; and when the nuts are unscrewed, the wires may be drawn out in order to change the balls U, V, which slide upon the wires by means of brass loops fixed into the balls, and preventing their touching the wood below them. Through each ball there passes a silk line, which is fixed to it at any length from the centre of the bearer to its end, by a nut-screw at the top of the ball; the shank of the screw going into the centre of the ball, and pressing the line against the under side of the whole which it goes through. The line goes from the ball, and under a small pulley sixed in the middle of the bearer; then up through a socket in the round plate (S and T) in the middle of each bearer; then through a slit in the middle of the square top (O and P) of each tower, and going over a small pulley on the top comes down again the same way, and is at last fastened to the upper end of the socket fixed in the middle of the round plate above mentioned. Each of these plates S and T has four round holes near their edges, by which they slide up and down upon the wires which make the corner of each lower. The balls and plates being thus connected, each by its particular line, it is plain that if the balls be drawn outward, or towards the end M and N of their respective bearers, the round plates S and T will be drawn up to the top of their respective towers O and P.</p><p>There are several brass weights, some of two, some of three, and others of four ounces, to be occasionally put within the towers O and P, upon the round plates S and T: each weight having a round hole in the middle of it, for going upon the sockets or axes of the plates, and being slit from the edge to the hole, that it may slip over the line which comes from each ball to its respective plate.</p><p>For a specimen of the experiments which may be made with this machine, may be subjoined the following.</p><p>1. Removing the bearer MX, put the loop of the line <hi rend="italics">b</hi> to which the ivory ball <hi rend="italics">a</hi> is fastened over a pin in the centre of the board <hi rend="italics">d,</hi> and turn the winch B; and the ball will not immediately begin to move with the board, but, on account of its inactivity, endeavour to remain in its state of rest. But when the ball has acquired the same velocity with the board, it will remain upon the same part of the board, having no relative motion upon it. However, if the board be suddenly stopped, the ball will continue to revolve upon <cb/> it, until the friction thereof stops its motion: so that matter resists every change of state, from that of rest to that of motion, and <hi rend="italics">vice versa.</hi></p><p>2. Put a longer cord to this ball; let it down through the hollow axis of the bearer MX and wheel G, and fix a weight to the end of the cord below the machine; and this weight, if left at liberty, will draw the ball from the edge of the Whirling board to its centre. Draw off the ball a little from the centre, and turn the winch; then the ball will go round and round with the board, and gradually fly farther from the centre, raising up the weight below the machine. And thus it appears that all bodies, revolving in circles, have a tendency to fly off from those circles, and must be retained in them by some power proceeding from or tending to the centre of motion. Stop the machine, and the ball will continue to revolve for some time upon the board; but as the friction gradually stops its motion, the weight acting upon it will bring it nearer and nearer to the centre in every revolution, till it brings it quite thither. Hence it appears, that if the planets met with any resistance in going round the sun, its attractive power would bring them nearer and nearer to it in every revolution, till they would fall into it.</p><p>3. Take hold of the cord below the machine with one hand, and with the other throw the ball upon the round board as it were at right angles to the cord, and it will revolve upon the board. Then, observing the velocity of its motion, pull the cord below the machine, and thus bring the ball nearer the centre of the board, and the ball will be seen to revolve with an increasing velocity, as it approaches the centre: and thus the planets which are nearest the sun perform quicker revolutions than those which are more remote, and move with greater velocity in every part of their respective circles.</p><p>4. Remove the ball <hi rend="italics">a,</hi> and apply the bearer MX, whose centre of motion is in its middle at <hi rend="italics">w,</hi> directly over the centre of the Whirling board <hi rend="italics">d.</hi> Then put two balls (V and U) of equal weight upon their bearing wires, and having fixed them at equal distances from their respective centres of motion. <hi rend="italics">w</hi> and <hi rend="italics">x</hi> upon their silk cords, by the screw nuts, put equal weights in the towers O and P. Lastly, put the catgut strings E and F upon the grooves G and H of the small wheels, which, being of equal diameters, will give equal velocities to the bearers above, when the winch B is turned; and the balls U and V will fly off toward M and N, and raise the weights in the towers at the same instant. This shews, that when bodies of equal quantities of matter revolve in equal circles with equal velocities, their centrifugal forces are equal.</p><p>5. Take away these equal balls, and put a ball of 6 ounces into the bearer MX, at a 6th part of the distance <hi rend="italics">wz</hi> from the centre, and put a ball of one ounce into the opposite bearer, at the whole distance <hi rend="italics">xy</hi> = <hi rend="italics">wz;</hi> and six the balls at these distances on their cords, by the screw nuts at the top: then the ball U, which is 6 times as heavy as the ball V, will be at only a 6th part of the distance from its centre of motion; and consequently will revolve in a circle of only a 6th part of the circumference of the circle in which V revolves. Let equal weights be put into the towers, and the winch be turned; which (as the catgut-string <pb n="688"/><cb/> is on equal wheels below, will cause the balls to revolve in equal times: but V will move 6 times as fast as U, because it revolves in a circle of 6 times its radius, and both the weights in the towers will rise at once. Hence it appears, that the centrifugal forces of revolving bodies are in direct proportion to their quantities of matter multiplied into their respective velocities, or into their distance from the centres of their respective circles.</p><p>If these two balls be fixed at equal distances from their respective centres of motion, they will move with equal velocities; and if the tower O has 6 times as much weight put into it as the tower P has, the balls will raise their weights exactly at the same moment: i. e. the ball U, being 6 times as heavy as the ball V, has 6 times as much centrifugal force in describing an equal circle with an equal velocity.</p><p>6. Let two balls, U and V, of equal weights, be sixed on their cords at equal distances from their respective centres of motion <hi rend="italics">w</hi> and <hi rend="italics">x;</hi> and let the catgut string E be put round the wheel K (whose circumference is only half that of the wheel H or G) and over the pulley <hi rend="italics">s</hi> to keep it tight, and let 4 times as much weight be put into the tower P as in the tower O. Then turn the winch B, and the ball V will revolve twice as fast as the ball U in a circle of the same diameter, because they are equidistant from the centres of the circles in which they revolve; and the weights in the towers will both rise at the same instant; which shews that a double velocity in the same circle will exactly balance a quadruple power of attraction in the centre of the circle: for the weights in the towers may be considered as the attractive forces in the centres, acting upon the revolving balls; which moving in equal circles, are as if they both moved in the same circle. Whence it appears that, if bodies of equal weights revolve in equal circles with unequal velocities, their centrifugal forces are as the squares of the velocities.</p><p>7. The catgut string remaining as before, let the distance of the ball V from the centre <hi rend="italics">x</hi> be equal to 2 of the divisions on its bearer; and the distance of the ball U from the centre <hi rend="italics">w</hi> be 3 and a 6th part; the balls themselves being equally heavy, and V making two revolutions by turning the winch, whilst U makes one; so that if we suppose the ball V to revolve in one moment, the ball U will revolve in 2 moments, the squares of which are 1 and 4: therefore, the square of the period of V is contained 4 times in the square of the period of U. But the distance of V is 2, the cube of which is 8, and the distance of U is 3 1/6, the cube of which is 32 very nearly, in which 8 is contained 4 times: and therefore, the squares of the periods V and U are to one another as the cubes of their distances from <hi rend="italics">x</hi> and <hi rend="italics">w,</hi> the centres of their respective circles. And if the weight in the tower O be 4 ounces, or equal to the square of 2, which is the distance of V from the centre <hi rend="italics">x;</hi> and the weight in the tower P be 10 ounces, nearly equal to the square of 3 1/6, the distance of U from <hi rend="italics">w;</hi> it will be found upon turning the machine by the winch, that the balls U and V will raise their respective weights at very nearly the same instant of time. This experiment confirms the famous proposition of Kepler, viz, that the squares of the periodical times of the planets round the sun are in propor- <cb/> tion as the cubes of their distances from him; and that the sun's attraction is inversely as the square of the distance from his centre.</p><p>8. Take off the string E from the wheels D and H, and let the string F remain upon the wheels D and G; take away also the bearer MX from the Whirlingboard <hi rend="italics">d,</hi> and instead of it put on the machine AB (fig. 2), fixing it to the centre of the board by the pins <hi rend="italics">c</hi> and <hi rend="italics">d,</hi> so that the end <hi rend="italics">ef</hi> may rise above the board to an angle of 30 or 40 degrees. On the upper part of this machine, there are two glass tubes <hi rend="italics">a</hi> and <hi rend="italics">b,</hi> close stopped at both ends, each tube being about three quarters full of water. In the tube <hi rend="italics">a</hi> is a little quicksilver, which naturally falls down to the end <hi rend="italics">a</hi> in the water; and in the tube <hi rend="italics">b</hi> is a small cork, floating on the top of the water, and small enough to rise or fall in the tube. While the board <hi rend="italics">b</hi> with this machine upon it continues at rest, the quicksilver lies at the bottom of the tube <hi rend="italics">a,</hi> and the cork floats on the water near the top of the tube <hi rend="italics">b.</hi> But, upon turning the winch and moving the machine, the contents of each tube fly off towards the uppermost ends, which are farthest from the centre of motion; the heaviest with the greatest force. Consequently, the quicksilver in the tube <hi rend="italics">a</hi> will fly off quite to the end <hi rend="italics">f,</hi> occupying its bulk of space, and excluding the water, which is lighter than itself: but the water in the tube <hi rend="italics">b,</hi> flying off to its higher end <hi rend="italics">c,</hi> will exclude the cork from that place, and cause it to descend toward the lowest end of the tube; for the heavier body, having the greater centrifugal force, will possess the upper part of the tube, and the lighter body will keep between the heavier and the lower part.</p><p>This experiment demonstrates the absurdity of the Cartesian doctrine of vortices; for, if a planet be more dense or heavy than its bulk of the vortex, it will fly off in it farther and farther from the sun; if less dense, it will come down to the lowest part of the vortex, at the sun: and the whole vortex itself, unless prevented by some obstacle, would fly quite off, together with the planets.</p><p>9. If a body be so placed upon the Whirling-board of the machine (fig. 1.) that the centre of gravity of the body be directly over the centre of the board, and the board be moved ever so rapidly by the winch B, the body will turn round with the board, without removing from its middle; for, as all parts of the body are in equilibrio round its centre of gravity, and the centre of gravity is at rest in the centre of motion, the centrifugal force of all parts of the body will be equal at equal distances from its centre of motion, and therefore the body will remain in its place. But if the centre of gravity be placed ever so little out of the centre of motion, and the machine be turned swiftly round, the body will fly off towards that side of the board on which its centre of gravity lies. Then if the wire C (fig. 3) with its little ball B be taken away from the semi-globe A, and the flat side <hi rend="italics">f</hi> of the semiglobe be laid upon the Whirling-board, so that their centres may coincide; if then the board be turned ever so quickly by the winch, the semi-globe will remain where it was placed: but if the wire C be screwed into the semi-globe at <hi rend="italics">d,</hi> the whole becomes one body, whose centre of gravity is at or near <hi rend="italics">d.</hi> Fix the pin <hi rend="italics">c</hi> <pb n="689"/><cb/> in the centre of the Whirling-board, and let the deep groove <hi rend="italics">b</hi> cut in the flat side of the semi-globe be put upon the pin, so that the pin may be in the centre of A (see fig. 4) where the groove is to be represented at <hi rend="italics">b,</hi> and let the board be turned by the winch, which will carry the little ball B (fig. 3) with its wire C, and the semi-globe A, round the centre-pin <hi rend="italics">c i;</hi> and then, the centrifugal force of the little ball B, weighing one ounce, will be so great as to draw off the semi-globe A, weighing two pounds, until the end of the groove at <hi rend="italics">c</hi> strikes against the pin <hi rend="italics">c,</hi> and so prevents A from going any farther: otherwise, the centrifugal force of B would have been great enough to have carried A quite off the whirling-board. Hence we see that, if the sun were placed in the centre of the orbits of the planets, it could not possibly remain there; for the centrifugal forces of the planets would carry them quite off, and the sun with them; especially when several of them happened to be in one quarter of the heavens. For the sun and planets are as much connected by the mutual attraction subsisting between them, as the bodies A and B are by the wire C fixed into them both. And even if there were but one planet in the whole heavens to go round ever so large a sun in the centre of its orbit, its centrifugal force would soon carry off both itself and the sun; for the greatest body placed in any part of free space could be easily moved; because, if there were no other body to attract it, it would have no weight or gravity of itself, and consequently, though it could have no tendency of itself to remove from that part of space, yet it might be very easily moved by any other substance.</p><p>10. As the centrifugal force of the light body B will not allow the heavy body A to remain in the centre of motion, even though it be 24 times as heavy as B; let the ball A (fig. 5) weighing 6 ounces be connected by the wire C with the ball B, weighing one ounce, and let the fork E be fixed into the centre of the Whirling-board; then, hang the balls upon the fork by the wire C in such a manner that they may exactly balance each other, which will be when the centre of gravity between them, in the wire at <hi rend="italics">d,</hi> is supported by the fork. And this centre of gravity is as much nearer to the centre of the ball A than to the centre B, as A is heavier than B; allowing for the weight of the wire on each side of the fork. Then, let the machine be moved, and the balls A and B will go round their common centre of gravity <hi rend="italics">d,</hi> keeping their balance, because either will not allow the other to fly off with it. For, supposing the ball B to be only one ounce in weight, and the ball A to be six ounces; then, if the wire C were equally heavy on each side of the fork, the centre of gravity <hi rend="italics">d</hi> would be 6 times as far from the centre of B as from the centre of A, and consequently B will revolve with a velocity 6 times as great as A does; which will give B 6 times as much centrifugal force as any single ounce of A has; but then as B is only one ounce, and A six ounces, the whole centrifugal force of A will exactly balance that of B; and therefore, each body will detain the other, so as to make it keep in its circle.</p><p>Hence it appears, that the sun and planets must all move round the common centre of gravity of the whole <cb/> system, in order to preserve that just balance which takes place among them.</p><p>11. Take away the forks and balls from the Whirling-board, and place the trough AB (fig. 6) thereon, fixing its centre to that of the board by the pin H. In this trough are two balls D and E of unequal weights, connected by a wire <hi rend="italics">f,</hi> and made to slide easily upon the wire stretched from end to end of the trough, and made fast by nut screws on the outside of the ends. Place these balls on the wire <hi rend="italics">c,</hi> so that their common centre of gravity <hi rend="italics">g,</hi> may be directly over the centre of the Whirling-board. Then turn the machine by the winch ever so swiftly, and the trough and balls will go round their centre of gravity, so as neither of them will fly off; because, on account of the equilibrium, each ball detains the other with an equal force acting against it. But if the ball E be drawn a little more towards the end of the trough at A, it will remove the centre of gravity towards that end from the centre of motion; and then, upon turning the machine, the little ball E will fly off, and strike with a considerable force against the end A, and draw the great ball B into the middle of the trough. Or, if the great ball D be drawn towards the end B of the trough, so that the centre of gravity may be a little towards that end from the centre of motion; and the machine be turned by the winch, the great ball D will fly off, and strike violently against the end B of the trough, and will bring the little ball E into the middle of it. If the trough be not made very strong, the ball D will break through it.</p><p>12. Mr. Ferguson has explained the reason why the tides rise at the same time on opposite sides of the earth, and consequently in opposite directions, by the following new experiment on the Whirling-table. For this purpose, let <hi rend="italics">a b c d</hi> (fig. 7) represent the earth, with its side <hi rend="italics">c</hi> turned toward the moon, which will then attract the water so as to raise them from <hi rend="italics">c</hi> to <hi rend="italics">g:</hi> and in order to shew that they will rise as high at the same time on the opposite side from <hi rend="italics">a</hi> to <hi rend="italics">e;</hi> let a plate AB (fig. 8) be fixed upon one end of the flat bar DC, with such a circle drawn upon it as <hi rend="italics">a b c d</hi> (fig. 7) to represent the round figure of the earth and sea; and an ellipse as <hi rend="italics">e f g h</hi> to represent the swelling of the tide at <hi rend="italics">e</hi> and <hi rend="italics">g,</hi> occasioned by the influence of the moon. Over this plate AB suspend the three ivory balls <hi rend="italics">e, f, g,</hi> by the silk lines <hi rend="italics">h, i, k,</hi> fastened to the tops of the wires H, I, K, so that the ball at <hi rend="italics">e</hi> may hang freely over the side of the circle <hi rend="italics">e,</hi> which is farthest from the moon M at the other end of the bar; the ball at <hi rend="italics">f</hi> over the centre, and the ball at <hi rend="italics">g</hi> over the side of the circle <hi rend="italics">g,</hi> which is nearest the moon. The ball <hi rend="italics">f</hi> may represent the centre of the earth, the ball <hi rend="italics">g</hi> water on the side next the moon, and the ball <hi rend="italics">e</hi> water on the opposite side. On the back of the moon M is fixed a short bar N parallel to the horizon, and there are three holes in it above the little weights <hi rend="italics">p, q, r.</hi> A silken thread <hi rend="italics">o</hi> is tied to the line <hi rend="italics">k</hi> close above the ball <hi rend="italics">g,</hi> and passing by one side of the moon M goes through a hole in the bar N, and has the weight <hi rend="italics">p</hi> hung to it. Such another thread <hi rend="italics">m</hi> is tied to the line <hi rend="italics">i,</hi> close above the ball <hi rend="italics">f,</hi> and, passing through the centre of the moon M and middle of the bar N, has the weight <hi rend="italics">q</hi> hung to it which is lighter than the weight <hi rend="italics">p.</hi> A third thread <hi rend="italics">m</hi> is tied to the line <hi rend="italics">h,</hi> close <pb n="690"/><cb/> above the ball <hi rend="italics">e,</hi> and, paffing by the other side, of the moon M through the bar N, has the weight <hi rend="italics">r</hi> hung to it, which is lighter than the weight <hi rend="italics">q.</hi> The use of these three unequal weights is to represent the moon's unequal attraction at different distances from her; so that if they are left at liberty, they will draw all the three balls towards the moon with different degrees of force, and cause them to appear as in fig. 9, in which case they are evidently farther from each other than if they hung freely by the perpendicular lines <hi rend="italics">h, i, k.</hi> Hence it appears, that as the moon attracts the side of the earth which is nearest her with a greater degree of force than she does the centre of the earth, she will draw the water on that side more than the centre, and cause it to rise on that side: and as she draws the centre more than the opposite side, the centre will recede farther from the surface of the water on that opposite side, and leave it as high there as she raised it on the side next her. For, as the centre will be in the middle between the tops of the opposite elevations, they must of course be equally high on both sides at the same time.</p><p>However, upon this supposition, the earth and moon would soon come together; and this would be the case if they had not a motion round their common centre of gravity, to produce a degree of centrifugal force, sufficient to balance their mutual attraction. Such motion they have; for as the moon revolves in her orbit every month, at the distance of 240000 miles from the earth's centre, and of 234000 miles from the centre of gravity of the earth and moon, the earth also goes round the same centre of gravity every month at the distance of 6000 miles from it, i. e. from it to the centre of the earth. But the diameter of the earth being, in round numbers, 8000 miles, its side next the moon is only 2000 miles from the common centre of gravity of the earth and moon, its centre 6000 miles from it, and its farthest side from the moon 10000 miles. Consequently the centrifugal forces of these parts are as 2000, 6000, and 10000; i. e. the centrifugal force of any side of the earth, when it is turned from the moon, is five times as great as when it is turned toward the moon. And as the moon's attraction, expressed by the number 6000 at the earth's centre, keeps the earth from flying out of this monthly circle, it must be greater than the centrifugal force of the waters on the side next her; and consequently, her greater degree of attraction on that side is sufficient to raise them; but as her attraction on the opposite side is less than the centrifugal force of the water there, the excess of this force is sufficient to raise the water just as high on the opposite side.</p><p>To prove this experimentally, let the bar DC with its furniture be fixed on the Whirling-board of the machine (fig. 1.) by pushing the pin P into the centre of the board; which pin is in the centre of gravity of the whole bar with its three balls, <hi rend="italics">e, f, g,</hi> and moon M. Now if the Whirling-board and bar be turned slowly round by the winch, till the ball <hi rend="italics">f</hi> hangs over the centre of the circle, as in fig. 10, the ball <hi rend="italics">g</hi> will be kept towards the moon by the heaviest weight <hi rend="italics">p</hi> (fig. 8), and the ball <hi rend="italics">e,</hi> on account of its greater centrifugal force, and the less weight <hi rend="italics">r,</hi> will fly off as far to the other side, as in fig. 10. And thus, whilst the <cb/> machine is kept turning, the balls <hi rend="italics">e</hi> and <hi rend="italics">g</hi> will hang over the ends of the ellipse <hi rend="italics">l f k.</hi> So that the centrifugal force of the ball <hi rend="italics">e</hi> will exceed the moon's attraction just as much as her attraction exceeds the centrifugal force of the ball <hi rend="italics">g,</hi> whilst her attraction just balances the centrifugal force of the ball <hi rend="italics">f,</hi> and makes it keep in its circle. Hence it is evident, that the tides must rise to equal heights at the same time on opposite sides of the earth. See Ferguson's Lectures on Mechanics, lect. 2, and Desag. Ex. Phil. vol. 1, lect. 5.</p></div1><div1 part="N" n="WHIRLWIND" org="uniform" sample="complete" type="entry"><head>WHIRLWIND</head><p>, a wind that rises suddenly, is exceedingly rapid and impetuous, in a Whirling direction, and often progressively also; but it is commonly soon spent.</p><p>Dr. Franklin, in his Physical and Meteorological Observations, read to the Royal Society in 1756, supposes a Whirlwind and a waterspout to proceed from the same cause: their only difference being, that the latter passes over the water, and the former over the land. This opinion is corroborated by the observations of M. de la Pryme, and many others, who have remarked the appearances and effects of both to be the same. They have both a progressive as well as a circular motion; they usually rise after calms and great heats, and mostly happen in the warmer latitudes: the wind blows every way from a large surrounding space, both to the waterspout and whirlwind; and a waterspout has, by its progressive motion, passed from the sea to the land, and produced all the phenomena and effects of a Whirlwind: so that there is no reason to doubt that they are meteors arising from the same general cause, and explicable upon the same principles, furnished by electrical experiments and discoveries. See <hi rend="smallcaps">Hurricane</hi>, and <hi rend="smallcaps">Waterspout.</hi> For Dr. Franklin's ingenious method of accounting for both these phenomena, see his Letters and Papers, &c, vol. 1, p. 191, 216, &c.</p><p>WHISPERING-<hi rend="italics">Places,</hi> are places where a Whisper, or other small noise, may be heard from one part to another, to a great distance. They depend on a principle, that the voice, &c, being applied to one end of an arch, easily passes by repeated reflections to the other. Thus, <figure/> let ABC represent the segment of a sphere; and suppose a low voice uttered at A, the vibrations extending themselves every way, some of them will impinge upon the points E, E, &c; and thence be reflected to the points F, F, &c; thence to G, G, &c; till at last they meet in C; where by their union they cause a much stronger sound than in any part of the segment <pb n="691"/><cb/> whatever, even louder than at the point from whence they set out. Accordingly, all the contrivance in a Whispering-place is, that near the person who Whispers, there be a smooth wall, arched either cylindrically, or elliptically, &c. A circular arch will do, but not so well.</p><p>Some of the most remarkable places for Whispering, are the following: viz, The prison of Dionysius at Syracuse, which increased a soft Whisper to a loud noise; or a clap of the hand to the report of a cannon, &c. The aqueducts of Claudius, which carried a voice 16 miles: beside divers others mentioned by Kircher in his Phonurgia. In England, the most considerable Whispering places are, the dome of St. Paul's church, London, where the ticking of a watch may be heard from side to side, and a very soft Whisper may be sent all round the dome: this Dr. Derham found to hold not only in the gallery below, but above upon the scaffold, where a Whisper would be carried over a person's head round the top of the arch, though there be a large opening in the middle of it into the upper part of the dome. And the celebrated Whispering-place in Gloucester cathedral, which is only a gallery above the east end of the choir, leading from one side of it to the other. See Birch's Hist. of the Royal Soc. vol. 1, pa. 120.</p></div1><div1 part="N" n="WHISTON" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WHISTON</surname> (<foreName full="yes"><hi rend="smallcaps">William</hi></foreName>)</persName></head><p>, an English divine, philosopher, and mathematician, of uncommon parts, learning, and extraordinary character, was born the 9th of December 1667, at Norton in the county of Leicester; where his father was rector. He was educated under his father till he was 17 years of age, when he was sent to Tamworth school, and two years after admitted of Clare-hall, Cambridge, where he pursued his studies, and particularly the mathematics, with great diligence. During this time he became afflicted with a great weakness of sight, owing to close study in a whitened room; which was in a good measure relieved by a little relaxation from study, and taking off the strong glare of light by hanging the place opposite his seat with green.</p><p>In 1693 he was made master of arts and fellow of the college, and soon after commenced one of the tutors; but his ill state of health soon after obliged him to relinquish this profession. Having entered into orders, in 1694 he became chaplain to Dr. More, bishop of Norwich; and while in this station he published his first work, intitled, <hi rend="italics">A New Theory of the Earth &c;</hi> in which he undertook to prove that the Mosaic doctrine of the earth was perfectly agreeable to reason and philosophy: which work, having much ingenuity, though it was written against by Mr. John Keill, brought considerable reputation to the author.</p><p>In the year 1698, bishop More gave him the living of Lowestoff in Suffolk, where he immediately went to reside, and devoted himself with great diligence to the discharge of that trust.—In the beginning of this century he was made Sir Isaac Newton's deputy, and afterwards his successor in the Lucasian professorship of mathematics; when he resigned his living at Lowestoff, and went to reside at Cambridge. From this time his publications became very frequent, both in theology and mathematics. Thus, in 1702 he published, A Short View of the Chronology of the Old Testament, <cb/> and of the Harmony of the four Evangelists.—In 1707, <hi rend="italics">Prælectiones Astronomicæ;</hi> beside eight Sermons on the Accomplishment of the Scripture Prophecies, preached at Boyle's lecture; and Newton's Arithmetica Universalis.—In 1708, Tacquet's Euclid, with select Theorems of Archimedes; the former of which had accidentally been his first introduction to the study of the mathematics.—In the same year he drew up an Essay upon the Apostolical Constitutions, which the Vice chancellor refused his licence for printing. The author tells us, he had read over the two first centuries of the church, and found that the Eusebian or Arian doctrine was chiefly the doctrine of those ages, which, though deemed heterodox, he thought it his duty to discover. —In 1709, he published a volume of Sermons and Essays on various subjects.—In 1710, Prælectiones Physico-Mathematicæ, which with the Prælectiones Astronomicæ, were translated and published in English. And it may be said, with no small honour to the memory of Mr. Whiston, that he was one of the first who explained the Newtonian philosophy in a popular way, so as to be intelligible to the generality of readers.— Among other things also, he translated the Apostolical Constitutions into English, which favoured the doctrine of the supremacy of the father and subordination of the son, vulgarly called the Arian heresy: Upon which his friends began to be alarmed for him; and the consequence shewed it was not groundless; for, Oct. 30, 1710, he was deprived of his professorship, and expelled the university of Cambridge, after he had been formally convened and interrogated for some days together.—At the conclusion of this year, he wrote his Historical preface, afterwards prefixed to his Primitive Christianity Revived, containing the reasons for his dissent from the commonly received notions of the Trinity, which work he published the next year, in 4 volumes 8vo, for which the Convocation fell upon him most vehemently.</p><p>In 1713, he and Mr. Ditton composed their scheme for finding the longitude, which they published the year following, a method which consisted in measuring distances by means of the velocity of sound; some more particulars of which are related in the life of Mr. Ditton.—In 1719, he published an ironical Letter of Thanks to doctor Robinson, bishop of London, for his late Letter to his clergy against the use of New Forms of Doxology. And, the same year, a Letter to the earl of Nottingham, Concerning the Eternity of the Son of God, and his Holy Spirit.—In 1720, he was proposed by Sir Hans Sloane and Dr. Halley to the Royal Society as a member; but was refused admittance by Sir Isaac Newton the president.</p><p>On Mr. Whiston's expulsion from Cambridge, he went to London, where he conferred with Doctors Clarke, Hoadly, and other learned men, who endeavoured to moderate his zeal, which however he would not suffer to be tainted or corrupted, and many were not much satisfied with the authority of these constitutions, but approved his integrity. Mr. Whiston now settled in London with his family; where, without suffering his zeal to be intimidated, he continued to write, and to propagate his Primitive Christianity with as much ardour as if he had been in the most flourishing circumstances; which however were so bad, that, in <pb n="692"/><cb/> 1721, a subscription was made for the support of his family, which amounted to 470l. For though he drew some profits from reading astronomical and philosophical lectures, and also from his publications, which were very numerous, yet these of themselves were very insufficient: nor, when joined with the benevolence and charity of those who loved and esteemed him for his learning, integrity, and piety, did they prevent his being frequently in great distress.—In 1722 he published an Essay towards restoring the true text of the Old Testament.—In 1724, The Literal Accomplishment of Scripture Prophecies.—Also, The Calculation of Solar Eclipses without Parallaxes.—In 1726, Of the Thundering Legion &c.—In 1727, A Collection of Authentic Records belonging to the Old and New Testament.—In 1730, Memoirs of the Life of Dr. Samuel Clarke.—In 1732, A Vindication of the Testimony of Phlegon, or an Account of the Great Darkness and Earthquake at our Saviour's Passion, described by Phlegon.—In 1736, Athanasian Forgeries, &c. And the Primitive Eucharist revived.—In 1737, The Astronomical Year, particularly of the Comet foretold by Sir Isaac Newton.—Also the Genuine Works of Flavius Josephus.—In 1739, Mr. Whiston put in his claim to the mathematical professorship at Cambridge, then vacant by the death of Dr. Saunderson, in a letter to Dr. Ashton, the master of Jesus-college; but no regard was paid to it.—In 1745, he published his Primitive New Testament in English.—In 1748, his Sacred History of the Old and New Testament. Also, Memoirs of his own Life and Writings, which are very curious.</p><p>Whiston continued many years a member of the established church; but at length forsook it, on account of the reading of the Athanasian Creed, and went over to the Baptists; which happened while he was at the house of Samuel Barker, Esq. at Lindon in Rutlandshire, who had married his daughter; where he died, after a week's illness, the 22d of August 1752, at upwards of 84 years of age.—We have mentioned the principal of his writings in the foregoing memoir; to which may be added, Chronological Tables, published in 1750.</p><p>The character of this conscientious and worthy man has been attempted by two very able personages, who were well acquainted with him, namely, bishop Hare and Mr. Collins, who unite in giving him the highest applauses, for his integrity, piety, &c.—Mr. Whiston left some children behind him; among them, Mr. John Whiston, who was for many years a very considerable bookseller in London.</p></div1><div1 part="N" n="WHITE" org="uniform" sample="complete" type="entry"><head>WHITE</head><p>, one of the colours of bodies. Though White cannot properly be said to be one colour, but rather a composition of all the colours together: for Newton has demonstrated that bodies only appear White by reflecting all the kinds of coloured rays alike; and that even the light of the sun is only White, because it consists of all colours mixed together.</p><p>This may be shewn mechanically in the following manner: Take seven parcels of coloured fine powders, the same as the primary colours of the rainbow, taking such quantities of these as shall be proportional to the respective breadths of these colours in the rainbow, which are of red 45 parts, orange 27, yellow 48, green <cb/> 60, blue 60, indigo 40, and of violet 80; then mix intimately together these seven parcels of powders, and the mixture will be a pretty White colour: and this is only similar to the uniting the prismatic colours together again, to form a White ray or pencil of light of the whole of them. The same thing is done conveniently thus: Let the flat upper surface of a top be divided into 360 equal parts, all around its edge; then divide the same surface into seven sectors in the proportion of the numbers above, by seven radii or lines drawn from the centre; next let the respective colours be painted in a lively manner on these spaces, but so as the edge of each colour may be made nearly like the colour next adjoining, that the separation may not be well distinguished by the eye; then if the top be made to spin, the colours will thus seem to be mixed all together, and the whole surface will appear of a uniform whiteness: and if a large round black spot be painted in the middle, so as there may be only a broad flat ring of colours around it, the experiment will succeed the better. See Newton's Optics, prop. 6, book 1; and Ferguson's Tracts, pa. 296.</p><p>White bodies are found to take heat slower than black ones; because the latter absorb or imbibe rays of all kinds and colours, and the former reflect them. Hence it is that black paper is sooner put in flame, by a burning-glass, than White; and hence also black clothes, hung up in the sun by the dyers, dry sooner than white ones.</p></div1><div1 part="N" n="WHITEHURST" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WHITEHURST</surname> (<foreName full="yes"><hi rend="smallcaps">John</hi></foreName>)</persName></head><p>, an ingenious English philosopher, was born at Congleton in the county of Cheshire, the 10th of April 1713, being the son of a clock and watch-maker there. Of the early part of his life but little is known; he who dies at an advanced age, leaving few behind him to communicate anecdotes of his youth. On his quitting school, where it seems the education he received was very defective, he was bred by his father to his own profession, in which he soon gave hopes of his future eminence.</p><p>It was very early in life that, from his vicinity to the many stupendous phenomena in Derbyshire, which were constantly presented to his observation, his attention was excited to enquire into the various causes of them.</p><p>At about the age of 21, his eagerness after new ideas carried him to Dublin, having heard of an ingenious piece of mechanism in that city, being a clock with certain curious appendages, which he was very desirous of seeing, and no less so of conversing with the maker. On his arrival however, he could neither procure a sight of the former, nor draw the least hint from the latter concerning it. Thus disappointed, he fell upon an expedient for accomplishing his design; and accordingly took up his residence in the house of the mechanic, paying the more liberally for his board, as he had hopes from thence of more readily obtaining the indulgence wished for. He was accommodated with a room directly over that in which the favourite piece was kept carefully locked up: and he had not long to wait for his gratification: for the artist, while one day employed in examining his machine, was suddenly called down stairs; which the young enquirer happening to overhear, softly slipped into the room, inspected the machine, and, presently satisfying himself as to the secret, escaped undis- <pb n="693"/><cb/> covered to his own apartment. His end thus compassed, he shortly after bid the artist farewell, and returned to his father in England.</p><p>About two or three years after his return from Ireland, he left Congleton, and entered into business for himself at Derby, where he soon got into great employment, and distinguished himself very much by several ingenious pieces of mechanism, both in his own regular line of bufiness, and in various other respects, as in the construction of curious thermometers, barometers, and other philosophical instruments, as well as in ingenious contrivances for water-works, and the erection of various larger machines: being consulted in almost all the undertakings in Derbyshire, and in the neighbouring counties, where the aid of superior skill, in mechanics, pneumatics, and hydraulics, was requisite.</p><p>In this manner his time was fully and usefully employed in the country, till, in 1775, when the act passed for the better regulation of the gold coin, he was appointed stamper of the money-weights; an office conferred upon him, altogether unexpectedly, and without solicitation. Upon this occasion he removed to London, where he spent the remainder of his days, in the constant habits of cultivating some useful parts of philosophy and mechanism. And here too his house became the constant resort of the ingenious and scientific at large, of whatever nation or rank, and this to such a degree, as very often to impede him in the regular prosecution of his own speculations.</p><p>In 1778, Mr. Whitehurst published his Inquiry into the Original State and Formation of the Earth; of which a second edition appeared in 1786, considerably enlarged and improved; and a third in 1792. This was the labour of many years; and the numerous investigations necessary to its completion, were in themselves also of so untoward a nature, as at times, though he was naturally of a strong constitution, not a little to prejudice his health. When he first entered upon this species of research, it was not altogether with a view to investigate the formation of the earth, but in part to obtain such a competent knowledge of subterraneous geography as might become subservient to the purposes of human life, by leading mankind to the discovery of many valuable substances which lie concealed in the lower regions of the earth.</p><p>May the 13th, 1779, he was elected and admitted a Fellow of the Royal Society. He was also a member of some other philosophical societies, which admitted him of their respective bodies, without his previous knowledge; but so remote was he from any thing that might favour of ostentation, that this circumstance was known only to a very few of his most confidential friends. Before he was admitted a member of the Royal Society, three several papers of his had been inserted in the Philosophical Transactions, viz, Thermometrical Observations at Derby, in vol. 57; An Account of a Machine for raising Water, at Oulton, in Cheshire, in vol. 65; and Experiments on Ignited Substances, vol. 66: which three papers were printed afterwards in the collection of his works in 1792.</p><p>In 1783 he made a second visit to Ireland, with a view to examine the Giant's Causeway, and other northern parts of that island, which he found to be chiefly composed of volcanic matter: an account and representa- <cb/> tions of which are inserted in the latter editions of his Inquiry. During this excursion, he erected an engine, for raising water from a well, to the summit of a hill, in a bleaching ground, at Tullidoi, in the county of Tyrone: it is worked by a current of water, and for its utility is perhaps unequalled in any country.</p><p>In 1787 he published, An Attempt toward obtaining Invariable Measures of Length, Capacity, and Weight, from the Mensuration of Time. His plan is, to obtain a measure of the greatest length that conveniency will permit, from two pendulums whose vibrations are in the ratio of 2 to 1, and whose lengths coincide nearly with the English standard in whole numbers. The numbers which he has chosen shew much ingenuity. On a supposition that the length of a seconds pendulum, in the latitude of London, is 39 1/5 inches, the length of one vibrating 42 times in a minute, must be 80 inches; and of another vibrating 84 times in a minute must be 20 inches; and their difference, 60 inches, or 5 feet, is his standard measure. By the experiments however, the difference between the lengths of the two pendulum rods, was found to be only 59.892 inches, instead of 60, owing to the error in the assumed length of the seconds pendulum, 39 1/5 inches being greater than the truth, which ought to be 39 1/8 very nearly. By this experiment, Mr. Whitehurst obtained a fact, as accurately as may be in a thing of this nature, viz, the difference between the lengths of two pendulum rods whose vibrations are known: a datum from whence may be obtained, by calculation, the true lengths of pendulums, the spaces through which heavy bodies fall in a given time, and many other particulars relating to the doctrine of gravitation, the figure of the earth, &c, &c.</p><p>Mr. Whitehurst had been at times subject to slight attacks of the gout, and he had for several years felt himself gradually declining. By an attack of that disease in his stomach, after a struggle of two or three months, it put an end to his laborious and useful life, on the 18th of February 1788, in the 75th year of his age, at his house in Bolt-court, Fleet-street, being the same house where another eminent self-taught philosopher, Mr. James Ferguson, had immediately before him lived and died.</p><p>For several years before his death, Mr. Whitehurst had been at times occupied in arranging and completing some papers, for a treatise on Chimneys, Ventilation, and Garden-stoves; which have since been collected and given to the public, by Dr. Willan, in 1794.</p><p>However respectable Mr. Whitehurst may have been in mechanics, and those parts of natural science which he more immediately cultivated, he was of still higher account with his acquaintance and friends on the score of his moral qualities. To say nothing of the uprightness and punctuality of his dealings in all transactions relative to business; few men have been known to possess more benevolent affections than he, or, being possessed of such, to direct them more judiciously to their proper ends. As to his person, he was above the middle stature, rather thin than otherwise, and of a countenance expressive at once of penetration and mildness. His fine gray locks, unpolluted by art, gave a venerable air to his whole appearance. In dress he was plain, in diet temperate, in his general intercourse with <pb n="694"/><cb/> mankind easy and obliging. In company he was cheerful or grave alike, according to the dictate of the occasion; with now and then a peculiar species of humour about him, delivered with such gravity of manner and utterance, that those who knew him but slightly were apt to understand him as serious, when he was merely playful. But where any desire of information on subjects in which he was conversant was expressed, he omitted no opportunity of imparting it.</p></div1><div1 part="N" n="WHITSUNDAY" org="uniform" sample="complete" type="entry"><head>WHITSUNDAY</head><p>, the 50th day or seventh sunday from Easter.—The season properly called Pentecost, is popularly called <hi rend="italics">Whitsuntide;</hi> because, it is said, in the primitive church, the newly baptized persons came to church between Easter and Pentecost in <hi rend="italics">white</hi> garments.</p></div1><div1 part="N" n="WILKINS" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WILKINS</surname> (Dr. <foreName full="yes">John</foreName>)</persName></head><p>, a very ingenious and learned English bishop and mathematician, was the son of a goldsmith at Oxford, and born in 1614. After being educated in Greek and Latin, in which he made a very quick progress, he was entered a student of New-Inn in that university, when he was but 13 years of age; but after a short stay there, he was removed to Magdalen Hall; where he took his degrees. Having entered into holy orders, he first became chaplain to William Lord Say, and afterwards to Charles Count Palatine of the Rhine, with whom he continued some time. Adhering to the Parliament during the civil wars, they made him warden of Wadham college about the year 1648. In 1656 he married the sister of Oliver Cromwell, then lord protector of England, who granted him a dispensation to hold his wardenship, notwithstanding his marriage. In 1659, he was by Richard Cromwell made master of Trinity college in Cambridge; but ejected the year following, upon the restoration. He was then chosen preacher to the society of Gray's Inn, and rector of St. Lawrence Jewry, London, upon the promotion of Dr. Seth Ward to the bishoprick of Exeter. About this time he became a member of the Royal Society, was chosen of their council, and proved one of their most eminent members. He was afterwards made dean of Rippon, and in 1668 bishop of Chester; but died of the stone in 1672, at 58 years of age.</p><p>Bishop Wilkins was a man who thought it prudent to submit to the powers in being; he therefore subscribed to the solemn league and covenant, while it was enforced; and was equally ready to swear allegiance to king Charles when he was restored: this, with his moderate spirit towards dissenters, rendered him not very agreeable to the churchmen; and yet several of them could not but give him one of the best of characters. Burnet writes that “he was a man of as great a mind, as true a judgment, as eminent virtues, and of as good a soul, as any he ever knew: that though he married Cromwell's sister, yet he made no other use of that alliance, but to do good offices, and to cover the university of Oxford from the sourness of Owen and Goodwin. At Cambridge, he joined with those who studied to propagate better thoughts, to take men off from being in parties, or from narrow notions, from superstitions conceits, and fierceness about opinions. He was also a great observer and promoter of experimental philosophy, which was then a new thing, and much looked after. He was naturally ambitious, but was the wisest clergyman I ever knew. He was a lover of <cb/> mankind, and had a delight in doing good.” The same historian mentions afterwards another quality which Wilkins possessed in a supreme degree, and which it was well for him he did, since he had great occasion for the use of it; and that was, says he, “a courage, which could stand against a current, and against all the reproaches with which ill-natured clergymen studied to load him.”</p><p>Of his publications, which are all of them very ingenious and learned, and many of them particularly curious and entertaining, the first was in 1638, when he was only 24 years of age, viz, The Discovery of a New World; or, A Discourse to prove, that it is probable there may be another Habitable World in the Moon; with a Discourse concerning the Possibility of a Passage thither.—In 1640, A Discourse concerning a New Planet, tending to prove that it is probable our earth is one of the Planets.—In 1641, Mercury; or, the Secret and Swift Messenger; shewing, how a man may with Privacy and Speed communicate his Thoughts to a Friend at any Distance, 8vo.—In 1648, Mathematical Magic; or, the Wonders that may be performed by Mathematical Geometry, 8vo. All these pieces were published entire in one volume 8vo, in 1708, under the title of, The Mathematical and Philosophical Works of the right rev. John Wilkins, &c; with a print of the author and general title page handsomely engraven, and an account of his life and writings. To this collection is also subjoined an abstract of a larger work, printed in 1668, folio, intitled, An Essay towards a Real Character and a Philosophical Language. These were all his mathematical and philosophical works; beside which, he wrote several tracts in theology, natural religion, and civil polity, which were much esteemed for their piety and moderation, and went through several editions.</p></div1><div1 part="N" n="WINCH" org="uniform" sample="complete" type="entry"><head>WINCH</head><p>, a popular term for a windlass. Also the bent handle for turning round wheels, grind-stones, &c.</p></div1><div1 part="N" n="WIND" org="uniform" sample="complete" type="entry"><head>WIND</head><p>, a current, or stream of air, especially when it is moved by some natural cause.</p><p>Winds are denominated from the point of the compass or horizon they blow from; as the east Wind, north Wind, south Wind, &c.</p><p>Winds are also divided into several kinds; as <hi rend="italics">general, particular, perennial, stated, variable, &c.</hi></p><p><hi rend="italics">Constant</hi> or <hi rend="italics">Perennial</hi> <hi rend="smallcaps">Winds</hi>, are those that always blow the same way; such as the remarkable one between the two tropics, blowing constantly from east to west, called also the <hi rend="italics">general trade-Wind.</hi></p><p><hi rend="italics">Stated</hi> or <hi rend="italics">Periodical</hi> <hi rend="smallcaps">Winds</hi>, are those that constantly return at certain times. Such are the sea and land breezes, blowing from land to sea in the morning, and from sea to land in the evening. Such also are the shifting or particular trade Winds, which blow one way during certain months of the year, and the contrary way the rest of the year.</p><p><hi rend="italics">Variable</hi> or <hi rend="italics">Erratic</hi> <hi rend="smallcaps">Winds</hi>, are such as blow without any regularity either as to time, place, or direction. Such as the Winds that blow in the interior parts of England, &c: though some of these claim their certain times of the day; as, the north-Wind is most frequent in the morning, the west-Wind about noon, and the south-Wind in the night. <pb n="695"/><cb/></p><p><hi rend="italics">General</hi> <hi rend="smallcaps">Wind</hi>, is such as blows at the same time the same way, over a very large tract of ground, most part of the year; as the general trade-Wind.</p><p><hi rend="italics">Particular</hi> <hi rend="smallcaps">Winds</hi>, include all others, excepting the general trade Winds.</p><p>Those peculiar to one little canton or province, are called <hi rend="italics">topical</hi> or <hi rend="italics">provincial Winds.</hi> The Winds are also divided, with respect to the points of the compass or of the horizon, into <hi rend="italics">cardinal</hi> and <hi rend="italics">collateral.</hi></p><p><hi rend="italics">Cardinal</hi> <hi rend="smallcaps">Winds</hi>, are those blowing from the four cardinal points, east, west, north, and south.</p><p><hi rend="italics">Collateral</hi> <hi rend="smallcaps">Winds</hi>, are the intermediate Winds between any two cardinal Winds, and take their names from the point of the compass or horizon they blow from.</p><p>In Navigation, when the Wind blows gently, it is called a <hi rend="italics">breeze;</hi> when it blows harder, it is called a <hi rend="italics">gale,</hi> or a <hi rend="italics">stiff gale;</hi> and when it blows very hard, a <hi rend="italics">storm.</hi></p><p>For a particular account of the trade-Winds, monsoons, &c, see Philos. Trans. number 183, or Abridg. vol. 2, p. 133. Also Robertson's Navigation book 5, sect. 6.</p><p>A Wind blowing from the sea, is always moist; as bringing with it the copious evaporation and exhalations from the waters: also, in summer, it is cool; and in winter warm. On the contrary, a Wind from the continent, is always dry; warm in summer, and cold in winter. Our northerly and southerly Winds however, which are usually accounted the causes of cold and warm weather, Dr. Derham observes, are really rather the effect of the cold or warmth of the atmosphere. Hence it is that we often find a warm southerly Wind suddenly change to the north, by the fall of snow or hail; and in a cold frosty morning, we find the Wind north, which afterward wheels about to the southerly quarter, when the sun has well warmed the air; and again in the cold evening, turns northerly, or easterly.</p><p><hi rend="italics">Physical Cause of</hi> <hi rend="smallcaps">Winds.</hi> Some philosophers, as Descartes, Rohault, &c, account for the general Wind, from the diurnal rotation of the earth; and from this general Wind they derive all the particular ones. Thus, as the earth turns eastward, the particles of the air near the equator, being very light, are left behind; so that, in respect of the earth's surface, they move westwards, and become a constant easterly wind, as they are found between the tropics, in those parallels of latitude where the diurnal motion is swiftest. But yet, against this hypothesis, it is urged, that the air, being kept close to the earth by the principle of gravity, would in time acquire the same degree of velocity that the earth's surface moves with, as well in respect of the diurnal rotation, as of the annual revolution about the sun, which is about 30 times swifter.</p><p>Dr. Halley therefore substitutes another cause, capable of producing a like constant effect, not liable to the same objections, but more agreeable to the known properties of the elements of air and water, and the laws of the motion of fluid bodies. And that is the action of the sun's beams, as he passes every day over the air, earth, and water, combined with the situation of the adjoining continents. Thus, the air which is less rarefied or expanded by heat, must have a motion towards those <cb/> parts which are more rarefied, and less ponderous, to bring the whole to an equilibrium; and as the sun keeps continually shifting to the westward, the tendency of the whole body of the lower air is that way. Thus a general easterly Wind is formed, which being impressed upon the air of a vast ocean, the parts impel one another, and so keep moving till the next return of the sun, by which so much of the motion as was lost, is again restored; and thus the easterly Wind is made perpetual. But as the air towards the north and south is less rarefied than in the middle, it follows that from both sides it ought to tend towards the equator.</p><p>This motion, compounded with the former easterly Wind, accounts for all the phenomena of the general trade-Winds, which, if the whole surface of the globe were sea, would blow quite round the world, as they are found to do in the Atlantic and the Ethiopic oceans. But the large continents of land in this middle tract, being excessively heated, communicate their heat to the air above them, by which it is exceedingly rarefied, which makes it necessary that the cooler and denser air should rush in towards it, to restore the equilibrium. This is supposed to be the cause why, near the coast of Guinea, the wind always sets in upon the land, blowing westerly instead of easterly.</p><p>From the same cause it happens, that there are such constant calms in that part of the ocean called the <hi rend="italics">rains;</hi> for this tract being placed in the middle, between the westerly Winds blowing on the coast of Guinea, and the easterly trade-Winds blowing to the westward of it; the tendency of the air here is indifferent to either, and so stands in equilibrio between both; and the weight of the incumbent atmosphere being diminished by the continual contrary Winds blowing from hence, is the reason that the air here retains not the copious vapour it receives, but lets it fall in so frequent rains.</p><p>It is also to be considered, that to the northward of the Indian ocean there is every where land, within the usual limits of the latitude of 30°, viz, Arabia, Persia, India, &c, which are subject to excessive heats when the sun is to the north, passing nearly vertical; but which are temperate enough when the sun is removed towards the other tropic, because of a ridge of mountains at some distance within the land, said to be often in winter covered with snow, over which the air as it passes must needs be much chilled. Hence it happens that the air coming, according to the general rule, out of the north-east, to the Indian sea, is sometimes hotter, sometimes colder, than that which, by a circulation of one current over another, is returned out of the south-west; and consequently sometimes the under current, or Wind, is from the north-east, sometimes from the south-west.</p><p>That this has no other cause, appears from the times when these Winds set, viz, in April: when the sun begins to warm these countries to the north, the southwest monsoons begin, and blow during the heats till October, when the sun being retired, and all things growing cooler northward, but the heat increasing to the south, the north-east Winds enter, and blow all the winter, till April again. And it is doubtless from the same principle, that to the southward of the equator, in part of the Indian ocean, the north-west Winds succeed the south-east, when the sun draws near the <pb n="696"/><cb/> tropic of Capricorn. Philos. Trans. num. 183; or Abridg. vol. 2, pa. 193.</p><p>But some philosophers, not satisfied with Dr. Halley's theory above recited, or thinking it not sufficient for explaining the various phenomena of the Wind, have had recourse to another cause, viz, the gravitation of the earth and its atmosphere towards the sun and moon, to which the tides are confessedly owing. They allege that, though we cannot discover aërial tides, of ebb or flow, by means of the barometer, because columns of air of unequal height, but different density, may have the same pressure or weight; yet the protuberance in the atmosphere, which is continually following the moon, must, say they, occasion a motion in all parts, and so produce a Wind more or less to every place, which conspiring with, or being counteracted by the Winds arising from other causes, makes them greater or less. Several dissertations to this purpose were published, on occasion of the subject proposed by the Academy of Sciences at Berlin, for the year 1746. But Musschenbroek will not allow that the attraction of the moon is the cause of the general Wind; because the east Wind does not follow the motion of the moon about the earth; for in that case there would be more than 24 changes, to which it would be subject in the course of a year, instead of two. Introd. ad Phil. Nat. vol. 2, pa. 1102.</p><p>And Mr. Henry Eeles, conceiving that the rarefaction of the air by the sun cannot simply be the cause of all the regular and irregular motions which we find in the atmosphere, ascribes them to another cause, viz, the ascent and descent of vapour and exhalation, attended by the electrical fire or fluid; and on this principle he has endeavoured to explain at large the general phenomena of the weather and barometer. Philos. Trans. vol. 49, pa. 124. <hi rend="center"><hi rend="italics">Laws of the Production of</hi> <hi rend="smallcaps">Wind.</hi></hi></p><p>The chief laws concerning the production of Wind, may be collected under the following heads.</p><p>1. If the spring of the air be weakened in any place more than in the adjoining places, a Wind will blow through the place where the diminution is; because the less elastic or forcible will give way to that which is more so, and thence induce a current of air into that place, or a Wind. Hence, because the spring of the air increases, as the compressing weight increases, and compressed air is denser than that which is less compressed; all Winds blow into rarer air, out of a place filled with a denser.</p><p>2. Therefore, because a denser air is specifically heavier than a rarer; an extraordinary lightness of the air in any place must be attended with extraordinary Winds, or storms. Now, an extraordinary fall of the mercury in the barometer shewing an extraordinary lightness of the atmosphere, it is no wonder if that foretels storms of Wind and rain.</p><p>3. If the air be suddenly condensed in any place, its spring will be suddenly diminished: and hence, if this diminution be great enough to affect the barometer, a Wind will blow through the condensed air. But since the air cannot be suddenly condensed, unless it has before been much rarefied, a Wind will blow through the air, as it cools, after having been violently heated. <cb/></p><p>4. In like manner, if air be suddenly rarefied, its spring is suddenly increased; and it will therefore flow through the air not acted on by the rarefying force. Hence a Wind will blow out of a place, in which the air is suddenly rarefied; and on this principle probably it is, that the sun, by rarefying the air, must have a great influence on the production of Winds.</p><p>5. Most caves are found to emit Wind, either more or less. Musschenbroek has enumerated a variety of causes that produce Winds, existing in the bowels of the earth, on its surface, in the atmosphere, and above it. See Introd. ad Phil. Nat. vol. 2, pa. 1116.</p><p>6. The rising and changing of the Winds are determined by weathercocks, placed on the tops of high buildings, &c. But these only indicate what passes about their own height, or near the surface of the earth. And Wolfius assures us, from observations of several years, that the higher Winds, which drive the clouds, are different from the lower ones, which move the weathercocks. Indeed it is no uncommon thing to see one tier of clouds driven one way by a Wind, and another tier just over the former driven the contrary way, by another current of air, and that often with very different velocities. And the late experiments with air balloons have proved the frequent existence of counter Winds, or currents of air, even when it was not otherwise visible, nor at all expected; by which they have been found to take very different and unexpected courses, as they have ascended higher and higher in the atmosphere. <hi rend="center"><hi rend="italics">Laws of the Force and Velocity of the</hi> <hi rend="smallcaps">Wind.</hi></hi></p><p>Wind being only air in motion, and the motion of a fluid against a body at rest, creating the same resistance as when the body moves with the same velocity through the fluid at rest; it follows, that the force of the Wind, and the laws of its action upon bodies, may be referred to those of their resistance when moved through it; and as these circumstances have been treated pretty fully under the article <hi rend="smallcaps">Resistance</hi> <hi rend="italics">of the Air,</hi> there is no occasion here to make a repetition of them. We there laid down both the quantity and laws of such a force, upon bodies of different shapes and sizes, moving with all degrees of velocity up to 2000 feet per second, and also for planes set at all degrees of obliquity, or inclination to the direction of motion; all which circumstances having, for the first time, been determined by real experiments.</p><p><hi rend="italics">As to the Velocity of the Wind:</hi> philosophers have made use of various methods for determining it. The method employed by Dr. Derham, was by letting light downy feathers fly in the air, and nicely observing the distance to which they were carried in any number of half seconds. He says that he thus measured the velocity of the Wind in the great storm of August 1705, which he found moved at the rate of 33 feet in half a second, or 45 miles per hour: whence he concludes, that the most vehement Wind does not fly at the rate of above 50 or 60 miles an hour; and that at a medium the velocity of Wind is at the rate of 12 or 15 miles per hour. Philos. Trans. number 313, or Abridg. vol. 4, p. 411.</p><p>Mr. Brice observes however, that experiments with feathers are liable to much uncertainty; as they hardly <pb n="697"/><cb/> ever go forward in a straight direction, but spirally, or else irregularly from side to side, or up and down.</p><p>He therefore considers the motion of a cloud, by means of its shadow over the surface of the earth, as a much more accurate measure of the velocity of the Wind. In this way he found that the Wind, in a considerable storm, moved at the rate of near 63 miles un hour; and when it blew a fresh gale, at the rate of 21 miles per hour; and in a small breeze it was near 10 miles an hour. Philos. Trans. vol. 56, p. 226.</p><p>The velocity and force of the Wind are also determined experimentally by various machines, called <hi rend="italics">anemometers, wind-measurers,</hi> or <hi rend="italics">wind-gages;</hi> the description of which see under these articles.</p><p>In the Philos. Trans. for 1759, p. 165, Mr. Smeaton has given a table, communicated to him by a Mr. Rouse, for shewing the force of the Wind, with several different velocities, which I shall insert below, as I find the numbers nearly agree with my own experiments made on the resistance of the air, when the resisting surfaces are reduced to the same size, by a due proportion for the resistance, which is in a higher degree than that of the surfaces.</p><p>N. B. The table of my results is printed in pa. 111, vol. 1, under the article <hi rend="smallcaps">Anemometer;</hi> where it is to be noted, that the numbers in the third column of that table, for the velocity of the Wind per hour, are all erroneously printed, only the 4th part of what each of them ought to be; so that those numbers must be all multiplied by 4. <hi rend="center">A Table of the different Velocities and Forces of the Wind, according to their common appellations.</hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Velocity of the</cell><cell cols="1" rows="1" role="data">Perpendi-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Wind</cell><cell cols="1" rows="1" role="data">cular force</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">on one sq.</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Common appellations of the</cell></row><row role="data"><cell cols="1" rows="1" role="data">Miles</cell><cell cols="1" rows="1" role="data">= feet</cell><cell cols="1" rows="1" role="data">foot, in a-</cell><cell cols="1" rows="1" rend="align=center colspan=2" role="data">Winds.</cell></row><row role="data"><cell cols="1" rows="1" role="data">in one</cell><cell cols="1" rows="1" role="data">in one</cell><cell cols="1" rows="1" role="data">verdupois</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">hour.</cell><cell cols="1" rows="1" role="data">second.</cell><cell cols="1" rows="1" role="data">pounds.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">1.47</cell><cell cols="1" rows="1" role="data">.005</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">Hardly perceptible.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2.93</cell><cell cols="1" rows="1" role="data">.020</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">Just perceptible.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4.40</cell><cell cols="1" rows="1" role="data">.044</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">5.87</cell><cell cols="1" rows="1" role="data">.079</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">Gentle pleasant wind.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">7.33</cell><cell cols="1" rows="1" role="data">.123</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">14.67</cell><cell cols="1" rows="1" role="data">.492</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">Pleasant brisk gale.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">22.00</cell><cell cols="1" rows="1" role="data">1.107</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">29.34</cell><cell cols="1" rows="1" role="data">1.968</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">Very brisk.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">36.67</cell><cell cols="1" rows="1" role="data">3.075</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">44.01</cell><cell cols="1" rows="1" role="data">4.429</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">High Winds.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">51.34</cell><cell cols="1" rows="1" role="data">6.027</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">58.68</cell><cell cols="1" rows="1" role="data">7.873</cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data"><hi rend="size(6)">}</hi></cell><cell cols="1" rows="1" rend="align=left rowspan=2" role="data">Very high.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">66.01</cell><cell cols="1" rows="1" role="data">9.963</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">73.35</cell><cell cols="1" rows="1" role="data">12.300</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">A storm or tempest.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">88.02</cell><cell cols="1" rows="1" role="data">17.715</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">A great storm.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">117.36</cell><cell cols="1" rows="1" role="data">31.490</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left" role="data">A hurricane.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" rend="rowspan=3" role="data">100</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">146.70</cell><cell cols="1" rows="1" rend="rowspan=3" role="data">49.200</cell><cell cols="1" rows="1" rend="align=left rowspan=3" role="data"><hi rend="size(6)">{</hi></cell><cell cols="1" rows="1" role="data">A hurricane that tears</cell></row><row role="data"><cell cols="1" rows="1" role="data">up trees, and carries</cell></row><row role="data"><cell cols="1" rows="1" role="data">buildings &c before it.</cell></row></table></p><p>The force of the Wind is nearly as the square of the velocity, or but little above it, in these velocities. But the force is much more than in the simple ratio of the <cb/> surfaces, with the same velocity, and this increase of the ratio is the more, as the velocity is the more. By accurate experiments with two planes, the one of 17 3/2 square inches, the other of 32, which are nearly in the ratio of 5 to 9, I found their resistances, with a velocity of 20 feet per second, to be, the one 1.196 ounces, and the other 2.542 ounces; which are in the ratio of 8 to 17, being an increase of between 1/5 and 1/6 part more than the ratio of the surfaces.</p><p><hi rend="smallcaps">Wind</hi>-<hi rend="italics">Gage,</hi> in Pneumatics, an instrument serving to determine the velocity and force of the Wind. See <hi rend="smallcaps">Anemometer, Anemoscope</hi>, and the article just above concerning the Force and Velocity of the Wind.</p><p>Dr. Hales had various contrivances for this purpose. He found (Statical Essays, vol. 2, p. 326) that the air rushed out of a smith's bellows, at the rate of 68 3/4 feet in a second of time, when compressed with a force of half a pound upon every square inch lying on the whole upper surface of the bellows. The velocity of the air, as it passed out of the trunk of his ventilators, was found to be at the rate of 3000 feet in a minute, which is at the rate of 34 miles an hour. The same author says, that the velocity with which impelled air passes out at any orifice, may be determined by hanging a light valve over the nose of a bellows, by pliant leathern hinges, which will be much agitated and lifted up from a perpendicular to a more than horizontal position by the force of the rushing air. There is also another more accurate way, he says, of estimating the velocity of air, viz, by holding the orisice of an inverted glass siphon full of water, opposite to the stream of air, by which the water will be depressed in one leg, and raised in the other, in proportion to the force with which the water is impelled by the air. Descrip. of Ventilators, 1743, p. 12. And this perhaps gave Dr. Lind the idea of his Wind-gage, described below.</p><p>M. Bougner contrived a simple instrument, by which may be immediately discovered the force which the Wind exerts on a given surface. This is a hollow tube AABB (fig. 14, pl. 30), in which a spiral spring CD is fixed, that may be more or less compressed by a rod FSD, passing through a hole within the tube at AA. Then having observed to what degree different forces or given weights are capable of compressing the spiral, mark divisions on the rod in such a manner, that the mark at S may indicate the weight requisite to force the spring into the situation CD: afterwards join at right angles to this rod at F, a plane surface CFE of any given area at pleasure; then let this instrument be opposed to the Wind, so that it may strike the surface perpendicularly, or parallel to the rod; then will the mark at S shew the weight to which the force of the Wind is equivalent.</p><p>Dr. Lind has also contrived a simple and easy apparatus of this kind, nearly upon the last idea of Dr. Hales mentioned above. This instrument is fully explained at the article <hi rend="smallcaps">Anemometer</hi>, vol. 1, pa. 111, and a figure of it given, pl. 3, fig. 4.</p><p>Mr. Benjamin Martin, from a hint first suggested by Dr. Burton, contrived an anemoscope, or Wind-gage, of a construction like a Wind-mill, with four sails; but the axis which the sails turn, is not cylindrical, but conical, like the fusee of a watch; about this susee winds a cord, having a weight at the end, which is <pb n="698"/><cb/> wound always, by the force of the Wind, upon the sails, till the weight just balances that force, which will be at a thicker part of the fusee when the Wind is strong, and at a smaller part of it when it is weaker. But although this instrument shews when a Wind is stronger or weaker, it will neither shew what is the actual velocity of the Wind, nor yet its force upon a square foot of direct surface; because the sails are set at an uncertain oblique angle to the Wind, and this acts at different distances from the axis or centre of motion. Martin's Phil. Brit. vol. 2, p. 211. See the fig. 5, plate 3, vol. 1.</p><p><hi rend="smallcaps">Wind</hi>-<hi rend="italics">Gun,</hi> the same as <hi rend="smallcaps">Air</hi>-<hi rend="italics">Gun;</hi> which see.</p><p><hi rend="smallcaps">Wind</hi>-<hi rend="italics">Mill,</hi> a kind of mill which receives its motion from the impulse of the Wind.</p><p>The internal structure of the Windmill is much the same with that of watermills: the difference between them lying chiefly in an external apparatus, for the application of the power. This apparatus consists of an axis EF (fig. 11, pl. 36), through which pass perpendicular to it, and to each other, two arms or yards, AB and CD, usually about 32 feet long: on these yards are formed a kind of sails, vanes, or flights, in a trapezoid form, with parallel ends; the greater of which HI is about 6 feet, and the less FG are determined by radii drawn from the centre E, to I and H.</p><p>These sails are to be capable of being always turned to the wind, to receive its impulse: for which purpose there are two different contrivances, which constitute the two different kinds of Windmills in common use.</p><p>In the one, the whole machine is supported upon a moveable arbor, or axis, fixed upright on a stand or foot; and turned round occasionally to suit the wind, by means of a lever.</p><p>In the other, only the cover or roof of the machine, with the axis and sails, in like manner turns round with a parallel or horizontal motion. For this purpose, the cover is built turret-wise, and encompassed with a wooden ring, having a groove, at the bottom of which are placed, at certain distances, a number of brass truckles; and within the groove is another ring, upon which the whole turret stands. To the moveable ring are connected beams <hi rend="italics">ab</hi> and <hi rend="italics">se;</hi> and to the beam <hi rend="italics">ab</hi> is fastened a rope at <hi rend="italics">b,</hi> having its other end fitted to a windlass, or axisin peritrochio: this rope being drawn through the iron hook G, and the windlass turned, the sails are moved round, and set fronting the wind, or with the axis pointing straight against the wind.</p><p>The internal mechanism of a Windmill is exhibited in fig. 12; where AHO is the upper room, and H<hi rend="italics">o</hi>Z the lower one; AB the axle-tree passing through the mill; STVW the sails covered with canvas, set obliquely to the wind, and turning round in the order of the letters; CD the cogwheel, having about 48 cogs or teeth, <hi rend="italics">a, a, a,</hi> &c, which carry round the lantern EF, having 8 or 9 trundles or rounds <hi rend="italics">c, c, c,</hi> &c, together with its upright axis GN; IK is the upper millstone, and LM the lower one; QR is the bridge, supporting the axis or spindle GN; this bridge is supported by the beams <hi rend="italics">cd,</hi> XY, wedged up at <hi rend="italics">c, d</hi> and X; ZY is the lifting tree, which stands upright; <hi rend="italics">ab</hi> and <hi rend="italics">ef</hi> are levers, whose centres of motion are Z and <hi rend="italics">e; fghi</hi> is a cord, with a stone <hi rend="italics">i,</hi> going about the pins <hi rend="italics">g</hi> and <hi rend="italics">h,</hi> and serving as a balance or counterpoise. The spindle <hi rend="italics">t</hi>N <cb/> is fixed to the upper millstone IK, by a piece of iron called the rynd, and fixed in the lower side of the stone, which is the only one that turns about, and its whole weight rests upon a hard stone, fixed in the bridge QR at N. The trundle EF, and its axis G<hi rend="italics">t,</hi> may be taken away; for it rests by its lower part at <hi rend="italics">t</hi> by a square socket, and the top runs in the edge of the beam <hi rend="italics">w.</hi> By bearing down the end <hi rend="italics">f</hi> of the lever <hi rend="italics">fe, b</hi> is raised, which raises ZY, and this raises YX, which lifts up the bridge QR, with the axis NG, and the upper stone IK; and thus the stones are fet at any distance. The lower or immoveable stone is fixed upon strong beams, and is broader than the upper one: the flour is conveyed through the tunnel <hi rend="italics">no</hi> into a chest; P is the hopper, into which is put the corn, which runs through the spout <hi rend="italics">r</hi> into the hole <hi rend="italics">t,</hi> and so falls between the stones, where it is ground to meal. The axis G<hi rend="italics">t</hi> is square, which shaking the spout <hi rend="italics">r,</hi> as it goes round, makes the corn run out; <hi rend="italics">rs</hi> is a string going about the pin <hi rend="italics">s,</hi> and serving to move the spout nearer to the axis or farther from it, so as to make the corn run faster or slower, according to the velocity and force of the wind. And when the wind is strong, the sails are only covered in part, or on one side, or perhaps only one half of two opposite sails. Toward the end B of the axletree is placed another cogwheel, trundle, and millstones, with an apparatus like that just described; so that the same axis moves two stones at once; and when only one pair is to grind, one of the trundles and its spindle are taken out: <hi rend="italics">xyl</hi> is a girth of pliable wood, fixed at the end <hi rend="italics">x;</hi> the other end <hi rend="italics">l</hi> being tied to the lever <hi rend="italics">km,</hi> moveable about <hi rend="italics">k;</hi> and the end <hi rend="italics">m</hi> being put down, draws the girth <hi rend="italics">xyl</hi> close to the cogwheel, which gently and gradually stops the motion of the mill, when required: <hi rend="italics">pq</hi> is a ladder for ascending to the higher part of the mill; and the corn is drawn up by means of a rope, rolled about the axis AB, when the mill is at work. See <hi rend="smallcaps">Mill.</hi> <hi rend="center"><hi rend="italics">Theory of the</hi> <hi rend="smallcaps">Windmill</hi>, <hi rend="italics">Position of the Sails, &c.</hi></hi></p><p>Were the sails set square upon their arms or yards, and perpendicular to the axletree, or to the wind, no motion would ensue, because the direct wind would keep them in an exact balance. But by setting them obliquely to the common axis, like the sails of a smokejack, or inclined like the rudder of a ship, the wind, by striking the surface of them obliquely, turns them about. Now this angle which the sails are to make with their common axis, or the degree of <hi rend="italics">weathering,</hi> as the mill-wrights call it, so as that the wind may have the greatest effect, is a matter of nice enquiry, and has much occupied the thoughts of the mathematician and the artist.</p><p>In examining the compound motions of the rudder of a ship, we find that the more it approaches to the direction of the keel, or to the course of the water, the more weakly this strikes it; but, on the other hand, the greater is the power of the lever to turn the vessel about. The obliquity of the rudder therefore has, at the same time, both an advantage and a disadvantage. It has been a point of inquiry therefore to find the position of the rudder when the ratio of the advantage over the disadvantage is the greatest. And M. Renau, in <pb n="699"/><cb/> his theory of the working of ships, has found, that the best situation of the rudder is when it makes an angle of about 55 degrees with the keel.</p><p>The obliquity of the sails, with regard to their axis, has precisely the same advantage, and disadvantage, with the obliquity of the rudder to the keel. And M. Parent, seeking by the new analysis the most advantageous situation of the sails on the axis, finds it the same angle of about 55 degrees. This obliquity has been determined by many other mathematicians, and found to be more accurately 54° 44′. See Maclaurin's Fluxions, p. 733; Simpson's Fluxions, prob. 17, p. 521; Martin's Philos. Britan. vol. 1, p. 220, vol. 2, p. 212; &c.</p><p>This angle, however, is only that which gives the wind the greatest force to put the sail in motion, but not the angle which gives the force of the wind a maximum upon the sail when in motion: for when the sail has a certain degree of velocity, it yields to the wind; and then that angle must be increased, to give the wind its full effect. Maclaurin, in his Fluxions, p. 734, has shewn how to determine this angle.</p><p>It may be observed, that the increase of this angle should be different according to the different velocities from the axletree to the further extremity of the sail. At the beginning, or axis, it should be 54° 44′; and thence continually increasing, giving the vane a twist, and so causing all the ribs of the vane to lie in different planes.</p><p>It is farther observed, that the ribs of the vane or sail ought to decrease in length from the axis to the extremity, giving the vane a curvilinear form; so that no part of the force of any one rib be spent upon the rest, but all move on independent of each other. The twist above mentioned, and the diminution of the ribs, are exemplified in the wings of birds.</p><p>As the ends of the sail nearest the axis cannot move with the same velocity which the tips or farthest ends have, although the wind acts equally strong upon them both, Mr. Ferguson (Lect. on Mech. pa. 52) suggests, that perhaps a better position than that of stretching them along the arms directly from the centre of motion, might be, to have them set perpendicularly across the farther ends of the arms, and there adjusted lengthwise to the proper angle: for in that case both ends of the sails would move with the same velocity; and being farther from the centre of motion they would have so much the more power, and then there would be no occasion for having them so large as they are generally made; which would render them lighter, and consequently there would be so much the less friction on the thick neck of the axle, when it turns in the wall.</p><p>Mr. Smeaton (Philos. Trans. 1759), from his experiments with Windmill sails, deduces several practical maxims: as,</p><p>1. That when the wind falls upon a concave surface, it is an advantage to the power of the whole, though every part, taken separately, should not be disposed to the best advantage. By several trials he has found that the curved form and position of the sails will be best regulated by the numbers in the following table. <cb/> <table><row role="data"><cell cols="1" rows="1" role="data">6th Parts of</cell><cell cols="1" rows="1" role="data"> Angle</cell><cell cols="1" rows="1" role="data">Angle with</cell></row><row role="data"><cell cols="1" rows="1" role="data">the radius or</cell><cell cols="1" rows="1" role="data">with the</cell><cell cols="1" rows="1" role="data">the plane of</cell></row><row role="data"><cell cols="1" rows="1" role="data">sail.</cell><cell cols="1" rows="1" role="data">axis.</cell><cell cols="1" rows="1" role="data">motion.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">1</cell><cell cols="1" rows="1" role="data">72°</cell><cell cols="1" rows="1" role="data">18°</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">2</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">19</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">3</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">18 middle.</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">4</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">16</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">5</cell><cell cols="1" rows="1" role="data">77 1/2</cell><cell cols="1" rows="1" role="data">12 1/2</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center" role="data">6</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data"> 7 end.</cell></row></table></p><p>2. That a broader sail requires a greater angle; and that when the sail is broader at the extremity, than near the centre, this shape is more advantageous than that of a parallelogram.</p><p>3. When the sails, made like sectors of circles, joining at the centre or axis, filled up about 7-8ths of the whole circular space, the effect was the greatest.</p><p>4. The velocity of Windmill sails, whether unloaded, or loaded so as to produce a maximum of effect, is nearly as the velocity of the Wind; their shape and position being the same.</p><p>5. The load at the maximum is nearly, but somewhat less than, as the square of the velocity of the wind.</p><p>6. The effects of the same sails at a maximum, are nearly, but somewhat less than, as the cubes of the velocity of the wind.</p><p>7. In sails of a similar figure and position, the number of turns in a given time, are reciprocally as the radius or length of the sail.</p><p>8. The effects of sails of similar figure and position, are as the square of their length.</p><p>9. The velocity of the extremities of Dutch mills, as well as of the enlarged sails, in all their usual positions, is considerably greater than the velocity of the wind.</p><p>M. Parent, in considering what figure the sails of a Windmill should have, to receive the greatest impulse from the wind, finds it to be a sector of an ellipsis, whose centre is that of the axletree of the mill; and the less semiaxis the height of 32 feet; as for the greater, it follows necessarily from the rule that directs the sail to be inclined to the axis in the angle of 55 degrees.</p><p>On this foundation he assumes four such sails, each being a quarter of an ellipse; which he shews will receive all the wind, and lose none, as the common ones do. These 4 surfaces, multiplied by the lever, with which the wind acts on one of them, express the whole power the wind has to move the machine, or the whole power the machine has when in motion.</p><p>A Windmill with 6 elliptical sails, he shews, would still have more power than one with only four. It would only have the same surface with the four; since the 4 contain the whole space of the ellipsis, as well as the 6. But the force of the 6 would be greater than that of the 4, in the ratio of 245 to 231. If it were desired to have only two sails, each being a semiellipsis, the surface would be still the same; but the power would be diminished by near 1-3d of that with 6 fails; because the greatness of the sectors would much shorten the lever with which the wind acts.</p><p>The same author has also considered which form, among the rectangular sails, will be most advantageous; <pb n="700"/><cb/> i. e. that which shall have the product of the surface by the lever of the wind, the greatest. The result of this enquiry is, that the width of the rectangular sail should be nearly double its length; whereas usually the length is made almost 5 times the width.</p><p>The power of the mill, with four of these new rectangular sails, M. Parent shews, will be to the power of four elliptic sails, nearly as 13 to 23; which leaves a considerable advantage on the side of the elliptic ones; and yet the force of the new rectangular sails will still be considerably greater than that of the common ones.</p><p>M. Parent also considers what number of the new sails will be most advantageous; and finds that the fewer the sails, the more surface there will be, but the power the less. Farther, the power of a Windmill with 6 sails is denoted by 14, that of another with 4 will be as 13, and another with 2 sails will be denoted by 9. That as to the common Windmill, its power still diminishes as the breadth of the sails is smaller, in proportion to the length: and therefore the usual proportion of 5 to 1 is exceedingly disadvantageous.</p></div1><div1 part="N" n="WINDWARD" org="uniform" sample="complete" type="entry"><head>WINDWARD</head><p>, in Sea Language, denotes any thing towards that point from whence the wind blows, in respect of a ship.</p><p><hi rend="italics">Sailing to</hi> <hi rend="smallcaps">Windward.</hi> See <hi rend="smallcaps">Sailing.</hi></p><p><hi rend="smallcaps">Windward</hi> <hi rend="italics">Tide,</hi> denotes a tide that runs against the wind.</p><p>WINDAGE <hi rend="italics">of a Gun,</hi> is the difference between the diameter of the bore of the gun and the diameter of the ball.</p><p>Heretofore the Windage appointed in the English service, viz, 1-20th of the diameter of the ball, which has been used almost from the beginning, has been far too much, owing perhaps to the first want of roundness in the ball, or to rust, foulness, or irregularities in the bore of the gun. But lately a beginning has been made to diminish the Windage, which cannot fail to be of very great advantage; as the shot will both go much truer, and have less room to bounce about from side to lide, to the great damage of the gun; and besides much less powder will serve for the same effect, as in some cases 1/3 or 1/2 the inflamed powder escapes by the Windage. The French allowance of Windage is 1-25th of the diameter of the ball.</p></div1><div1 part="N" n="WINDLASS" org="uniform" sample="complete" type="entry"><head>WINDLASS</head><p>, or <hi rend="smallcaps">Windlace</hi>, a particular machine used for raising heavy weights, as guns, stones, anchors, &c.</p><p>This is a very simple machine, consisting only of an axis or roller, supported horizontally at the two ends by two pieces of wood and a pulley: the two pieces of wood meet at top, being placed diagonally so as to prop each other; and the axis or roller goes through the two pieces, and turns in them. The pulley is fastened at top, where the pieces join. Lastly, there are two staves or hand spikes which go through the roller, to turn it by; and the rope, which comes over the pulley, is wound off and on the same.</p><div2 part="N" n="Windlass" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Windlass</hi></head><p>, in a Ship, is an instrument in small ships, placed upon the deck, just abaft the foremast. It is made of a long and thick piece of timber, either cylindrical, or octagonal, &c, in form of an axletree, placed horizontally across the ship, a foot or more above the deck; and it is turned about by the help of handspikes put into holes made for that purpose. <cb/></p><p>This machine will purchase or raise much more than a capstan, and that without any danger to those that heave; for if in heaving the Windlass about, any of the handspikes should happen to slip or break, the Windlass will stop of itself, as it does at the end of every pull or heave of the men, being prevented from returning by means of a catch that falls into notches. See fig. 15, pl. 35.</p></div2></div1><div1 part="N" n="WINDOW" org="uniform" sample="complete" type="entry"><head>WINDOW</head><p>, q. d. <hi rend="italics">wind-door,</hi> an aperture or opening in the wall of a house, to admit the air and light.</p><p>Before the use of glass became general, which was not till towards the end of the 12th century, the Windows in England seem generally to have been composed of paper, oiled, both to defend it against the weather, and to make it more transparent; as now is sometimes used in workshops and unfinished buildings. Some of the better sort were surnished with lattices of wood or sheets of linen. These it seems were fixed in frames, called <hi rend="italics">capsamenta,</hi> and hence our <hi rend="italics">casements</hi> still so common in some of the counties.</p><p>The chief rules with regard to Windows are, 1. That they be as few in number, and as moderate in dimensions, as may be consistent with other respects; inasmuch as all openings are weakenings.</p><p>2. That they be placed at a convenient distance from the angles or corners of the buildings: both for strength and beauty.</p><p>3. That they be made all equal one with another, in their rank and order; so that those on the right hand may answer to those on the left, and those above be right over those below: both for strength and beanty.</p><p>As to their dimensions, care is to be taken, to give them neither more nor less than is needful; regard being had to the size of the rooms, and of the building. The apertures of Windows in middle-sized houses, may be from 4 to 5 feet; in the smaller ones less; and in large buildings more. And the height may be double their width at the least: but in lofty rooms, or large buildings, the height may be a 4th, or 3d, or half their breadth more than the double.</p><p>Such are the proportions for Windows of the first story; and the breadth must be the same in the upper stories; but as to the height, the second story may be a 3d part lower than the first, and the third story a 4th part lower than the second.</p></div1><div1 part="N" n="WINTER" org="uniform" sample="complete" type="entry"><head>WINTER</head><p>, one of the four seasons or quarters of the year.</p><p>Winter properly commences on the day when the sun's distance from the zenith of the place is the greatest, or when his declination is the greatest on the contrary side of the equator; and it ends on the day when that distance is a mean between the greatest and least, or when he next crosses the equinoctial.</p><p>At and near the equator, the Winter, as well as the other seasons, return twice every year; but all other places have only one Winter in the year; which in the northern hemisphere begins when the sun is in the tropic of Capricorn, and in the southern hemisphere when he is in the tropic of Cancer: so that all places in the same hemisphere have their Winter at the same time.</p><p>Notwithstanding the coldness of this season it is proved in astronomy, that the sun is really nearer to the earth in our Winter than in summer: the reason of the <pb n="701"/><cb/> defect of heat being owing to the lowness of the sun, or to the obliquity of his rays.</p></div1><div1 part="N" n="WOLFF" org="uniform" sample="complete" type="entry"><head>WOLFF</head><p>, <hi rend="smallcaps">Wolfius, (Christian</hi>), baron of the Roman empire, privy counsellor to the king of Prussia, and chancellor to the university of Halle in Saxony, as well as member of many of the literary academies in Europe, was born at Breslau in 1679. After studying philosophy and mathematics at Breslau and Jena, he obtained permission to give lectures at Leipsic; which, in 1703, he opened with a dissertation called <hi rend="italics">Philosophia Practica Universalis, Methodo Mathematica conscripta,</hi> which served greatly to enhance the reputation of his talents. He published two other dissertations the same year; the first <hi rend="italics">De Rotis Dentatis,</hi> the other <hi rend="italics">De Algorithmo Insinitesimali Differentiali;</hi> which obtained him the honourable appellation of Assistant to the Faculty of Philosophy at Leipsic.</p><p>He now accepted the professorship of mathematics at Halle, and was elected into the society at Leipsic, at that time engaged in publishing the <hi rend="italics">Acta Eruditorum.</hi> After having inserted in this work many important pieces relating to mathematics and physics, he undertook, in 1709, to teach all the various branches of philosophy, beginning with a small Logical treatise in Latin, being Thoughts on the Powers of the Human Understanding. He carried himself through these great pursuits with amazing assiduity and ardour: the king of Prussia rewarded him with the office of counsellor to the court in 1721, and augmented the profits of that post by very considerable appointments: he was also chosen a member of the Royal Society of London and of Prussia.</p><p>In the midst of all this prosperity however, Wolff raised an ecclesiastical storm against himself, by a Latin oration he delivered in praise of the Chinese philosophy: every pulpit immediately resounded against his tenets; and the faculty of theology, who entered into a strict examination of his productions, resolving that the doctrine he taught was dangerous to the last degree, an order was obtained in 1723 for displacing him, and commanding him to leave Halle in 24 hours.</p><p>Wolff now retired to Cassel, where he obtained the professorship of mathematics and philosophy in the university of Marbourg, with the title of Counsellor to the Landgrave of Hesse; to which a prositable pension was annexed. Here he renewed his labours with redoubled ardour; and it was in this retreat that he published the greatest part of his numerous works.</p><p>In 1725, he was declared an honorary professor of the academy of sciences at Petersburg, and in 1733 was admitted into that of Paris. The king of Sweden also declared him one of the council of regency; but the pleasing situation of his new abode, and the multitude of honours which he had received, were too alluring to permit him to accept of many advantageous offers; among which was the office of president of the academy at Petersburg.</p><p>The king of Prussia too, who was now recovered from the prejudices he had been made to conceive against Wolff, wanted to re-establish him in the university of Halle in 1733, and made another attempt to effect it in 1739; which Wolff for a time thought fit to decline, but at last submitted: he returned therefore in 1741, invested with the characters of privy counsellor, vice <cb/> chancellor, and professor of the law of nature and of nations. The king afterwards, upon a vacancy, raised him to the dignity of chancellor of the university; and the elector of Bavaria created him a baron of the empire. He died at Halle in Saxony, of the gout in his stomach, in 1754, in the 76th year of his age, after a life silled up with a train of actions as wise and systematical as his writings, of which he composed in Latin and German more than 60 distinct pieces. The chief of his mathematical compositions, is his <hi rend="italics">Elementa Matheseos Universæ,</hi> the best edition of which is that of 1732 at Geneva, in 5 vols 4to; which does not however comprise his Mathematical Dictionary in the German language, in 1 vol. 8vo, nor many other distinct works on different branches of the mathematics, nor his System of Philosophy, in 23 vols. in 4to.</p><p>WORKING <hi rend="italics">to Windward,</hi> in Sea Language, is the operation by which a ship endeavours to make a progress against the wind.</p></div1><div1 part="N" n="WREN" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WREN</surname> (Sir <foreName full="yes"><hi rend="smallcaps">Christopher</hi></foreName>)</persName></head><p>, a great philosopher and mathematician, and one of the most learned and eminent architects of his age, was the son of the rev. Christopher Wren, dean of Windsor, and was born at Knoyle in Wiltshire in 1632. He studied at Wadham college, Oxford; where he took the degree of master of arts in 1653, and was chosen fellow of Allsouls college there. Soon after, he became one of that ingenious and learned society, who then met at Oxford for the improvement of natural and experimental philosophy, and which at length produced the Royal Society.</p><p>When very young, he discovered a surprising genius for the mathematics, in which science he made great advances before he was 16 years of age.—In 1657 he was made professor of astronomy in Gresham college, London; and his lectures, which were much frequented, tended greatly to the promotion of real knowledge: in his inaugural oration, among other things, he proposed several methods by which to account for the shadows returning backward 10 degrees on the dial of king Ahaz, by the laws of nature. One subject of his lectures was upon telescopes, to the improvement of which he had greatly contributed: another was on certain properties of the air, and the barometer. In the year 1658 he read a description of the body and different phases of the planet Saturn; which subject he proposed to investigate while his colleague, Mr. Rooke, then professor of geometry, was carrying on his observations upon the satellites of Jupiter. The same year he communicated some demonstrations concerning cycloids to Dr. Wallis, which were afterwards published by the doctor at the end of his treatise upon that subject. About that time also, he resolved the problem proposed by Pascal, under the feigned name of John de Montford, to all the English mathematicians; and returned another to the mathematicians in France, formerly proposed by Kepler, and then resolved likewise by himself, to which they never gave any solution.— In 1660, he invented a method for the construction of solar eclipses: and in the latter part of the same year, he with ten other gentlemen formed themselves into a society, to meet weekly, for the improvement of natural and experimental philosophy; being the foundation of the Royal Society.—In the beginning of 1661, he was chosen Savilian professor of astronomy at Oxford, <pb n="702"/><cb/> in the room of Dr. Seth Ward; where he was the same year created Doctor of Laws.</p><p>Among his other accomplishments, Dr. Wren had gained so considerable a skill in architecture, that he was sent for the same year, from Oxford, by order of king Charles the 2d, to assist Sir John Denham, surveyor general of the works.—In 1663, he was chosen fellow of the Royal Society; being one of those who were first appointed by the Council aster the grant of their charter. Not long after, it being expected that the king would make the society a visit, the lord Brounker, then president, by a letter requested the advice of Dr. Wren, concerning the experiments which might be mo<*>proper on that occasion: to whom the doctor recommended principally the Torricellian experiment, and the weather needle, as being not mere amusements, but useful, and also neat in their operation. Indeed upon many occasions Dr. Wren did great honour to that illustrious body, by many curious and useful discoveries, in astronomy, natural philosophy, and other sciences, related in the History of the Royal Society, where Dr. Sprat has inserted them from the registers and other books of the society to 1665. Among others of his productions there enumerated, is a lunar globe; representing the spots and various degrees of whiteness upon the moon's surface, with the h<*>s, eminences and cavities: the whole contrived fo, that by turning it round to the light, it shews all the lunar phases, with the various appearances that happen from the shadows of the mountains and valleys, &c: this lunar model was placed in the king's cabinet. Another of these productions, is a tract on the Doctrine of Motion that arises from the impact between two bodies, illustrated by experiments. And a third is, The History of the Seasons, as to the temperature, weather, productions, diseases, &c, &c. For which purpose he contrived many curious machines, several of which kept their own registers, tracing out the lines of variations, so that a person might know what changes the weather had undergone in his absence: as windgages, thermometers, barometers, hygrometers, raingages, &c.—He made also great additions to the new discoveries on pendulums; and among other things shewed, that there may be produced a natural standard for measure from the pendulum for common use.—He invented many ways to make astronomical observations more easy and accurate: He sitted and hung quadrants, sextants, and radii more commodiously than formerly: he made two telescopes to open with a joint like a sector, by which observers may infallibly take a distance to half minutes, &c. He made many sorts of retes, screws, and other devices, for improving telescopes to take small distances, and apparent diameters, to seconds: He made apertures for taking in more or less light, as the observer pleases, by opening and shutting, the better to sit glasses for crepusculine observations.—He added much to the theory of dioptrics; much to the manufacture of grinding good glasses: He attempted, and not without success, the making of glasses of other forms than spherical. He exactly measured and delineated the spheres of the humours of the eye, the proportions of which to one another were only guessed at before: a discussion shewing the reasons why we see objects erect, and that reflection conduces <cb/> as much to vision as refraction. He displayed a natural and easy theory of refractions, which exactly answered every experiment. He fully demonstrated all dioptrics in a few propositions, shewing not only, as in Kepler's Dioptrics, the common properties of glasses, but the proportions by which the individual rays cut the axis, and each other, upon which the charges of the telescopes, or the proportion of the eye-glasses and apertures, are demonstrably discovered.—He made constant observations on Saturn, and a true theory of that planet, before the printed discourse by Huygens, on that subject, appeared.—He made maps of the Pleiades and other telescopic stars: and proposed methods to determine the great question as to the earth's motion or rest, by the small stars about the pole to be seen in large telescopes—In navigation he made many improvements. He sramed a magnetical terella, which he placed in the midst of a plane board with a hole, into which the terella is half immersed, till it be like a globe with the poles in the horizon: the plane is then dusted over with steel filings from a sieve: the dust, by the magnetical virtue, becomes immediately figured into furrows that bend like a sort of helix, proceeding as it were out at one pole, and returning in by the other; the whole plane becoming figured like the circles of a planisphere.—It being a question in his time among the problems of navigation, to what mechanical powers sailing against the wind was reducible; he shewed it to be a wedge: and he demonstrated, how a transient sorce upon an oblique plane would cause the motion of the plane against the first mover: and he made an instrument mechanically producing the same effect, and shewed the reason of sailing on all winds. The geometrical mechanism of rowing, he shewed to be a lever on a moving or cedent fulcrum: for this end, he made instruments and experiments, to find the resistance to motion in a liquid medium; with other things that are the necessary elements for laying down the geometry of sailing, swimming, rowing, flying, and constructing of ships—He invented a very speedy and curious way of etching. He started many things towards the emendation of water-works. He likewise made some instruments for respiration, and for straining the breath from fuliginous vapours, to try whether the same breath, so purisied, will serve again.—He was the first inventor of drawing pictures by microscopical glasses. He found out perpetual, or at least longlived lamps, for keeping a perpetual regular heat, in order to various uses, as hatching of eggs and insects, production of plants, chemical preparations, imitating nature in producing fossils and minerals, keeping the motion of watches equal, for the longitude and astronomical uses.—He was the first author of the anatomical experiment of injecting liquor into the veins of animals. By this operation, divers creatures were immediately purged, vomited, intoxicated, killed, or revived, according to the quality of the liquor injected. Hence arose many other new experiments, particularly that of transfusing blood, which has been prosecuted in sundry curious instances. This is a short account of the principal discoveries which Dr. Wren presented, or suggested, to the Royal Society, or were improved by him.</p><p>As to his architectural works: It has before been <pb n="703"/><cb/> observed that he had been sent for to assist Sir John Denham. In 1665 he travelled into France, to examine the most beautiful edifices and curious mechanical works there, when he made many useful observations. Upon his return home, he was appointed architect, and one of the commissioners for repairing St. Paul's cathedral. Within a few days after the fire of London, 1666, he drew a plan for a new city, and presented it to the king; but it was not approved of by the parliament. In this model, the chief streets were to cross each other at right angles, with lesser streets between them; the churches, public buildings, &c, so disposed as not to interfere with the streets, and four piazzas placed at proper distances.—Upon the death of Sir John Denham, in 1668, he succeeded him in the office of surveyorgeneral of the king's works; and from this time he had the direction of a great many public edifices, by which he acquired the highest reputation. He built the magnificent theatre at Oxford, St. Paul's cathedral, the Monument, the modern part of Hampton Court, Chelsea-college, one of the wings of Greenwich hospital, the churches of St. Stephen Walbrook, and St. Mary-le-bow, with upwards of 60 other churches and public works, which that dreadful fire made necessary. In the management of which business, he was assisted in the measurements, and laying out of private property, by the ingenious Dr. Robert Hook. The variety of business in which he was by this means engaged, requiring his constant attendance and concern, he resigned his Savilian professorship at Oxford in 1673; and the year following he received from the king the honour of knighthood.—He was one of the commissioners who, on the motion of Sir Jonas Moore, surveyor-general of the ordnance, had been appointed to find out a proper place for erecting an observatory; and he proposed Greenwich, which was approved of; the foundation stone of which was laid the 10th of August 1675, and the building was presently finished under the direction of Sir Jonas, with the advice and assistance of Sir Christopher.</p><p>In 1680 he was chosen president of the Royal Society; afterwards appointed architect and commissioner of Chelsea-college; and in 1684, principal officer or comptroller of the works in Windsor-castle. Sir Christopher sat twice in Parliament, as a representative for two different boroughs. While he continued surveyor-general, his residence was in Scotland-yard; but after his removal from that office, in 1718, he lived in St. James's-street, Westminster. He died the 25th of February 1723, at 91 years of age; and he was interred with great solemnity in St. Paul's cathedral, in the vault under the south wing of the choir, near the <*>ast end.</p><p>As to his person, Sir Christopher Wren was of a low stature, and thin frame of body; but by temperance and skilful management he enjoyed a good state of health, to a very unusual length of life. He was modest, devout, strictly virtuous, and very communicative of his knowledge. Besides his peculiar eminence as an architect, his learning and knowledge were very extensive in all the arts and sciences, and especially in the mathematics.</p><p>Sir Christopher never printed any thing himself, but <cb/> several of his works have been published by others: some in the Philosophical Transactions, and some by Dr. Wallis and other friends.—His posthumous works and draughts were published by his son.</p></div1><div1 part="N" n="WRIGHT" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">WRIGHT</surname> (<foreName full="yes"><hi rend="smallcaps">Edward</hi></foreName>)</persName></head><p>, a noted English mathematician, who flourished in the latter part of the 16th century, and beginning of the 17th; dying in the year 1615. He was contemporary with Mr. Briggs, and much concerned with him in the business of the logarithms, the short time they were published before his death. He also contributed greatly to the improvement of navigation and astronomy. The following memoirs of him are translated from a Latin paper in the annals of Gonvile and Caius college in Cambridge, viz, “This year (1615) died at London, Edward Wright of Garveston in Norfolk, formerly a fellow of this college; a man respected by all for the integrity and simplicity of his manners, and also famous for his skill in the mathematical sciences: so that he was not undeservedly styled a most excellent mathematician by Richard Hackluyt, the author of an original treatise of our English navigations. What knowledge he had acquired in the science of mechanics; and how usefully he employed that knowledge to the public as well as private advantage, abundantly appear both from the writings he published, and from the many mechanical operations still extant, which are standing monuments of his great industry and ingenuity. He was the first undertaker of that difficult but useful work, by which a little river is brought from the town of Ware in a new canal, to supply the city of London with water; but by the tricks of others he was hindered from completing the work he had begun. He was excellent both in contrivance and execution, nor was he inserior to the most ingenious mechanic in the making of instruments, either of brass or any other matter. To his invention is owing whatever advantage Hondius's geographical charts have above others; for it was Wright who taught Jodocus Hondius the method of constructing them, which was till then unknown; but the ungrateful Hondius concealed the name of the true author, and arrogated the glory of the invention to himself. Of this fraudulent practice the good man could not help complaining, and justly enough, in the preface to his treatise of the Correction of Errors in the Art of Navigation; which he composed with excellent judgment, and after long experience, to the great advancement of naval affairs. For the improvement of this art he was appointed mathematical lecturer by the East-India Company, and read lectures in the house of that worthy knight Sir Thomas Smith, for which he had a yearly salary of 50 pounds. This office he discharged with great reputation, and much to the satisfaction of his hearers. He published in English a book on the doctrine of the sphere, and another concerning the construction of sun dials. He also prefixed an ingenious preface to the learned Gilbert's book on the loadstone. By these and other his writings, he has transmitted his same to latest posterity. While he was yet a fellow of this college, he could not be concealed in his private study, but was called forth to the public business of the nation, by the queen, about the year 1593. [Other accounts say 1589.] He was ordered to attend the earl of Cumberland in <pb n="704"/><cb/> some maritime expeditions. One of these he has given a faithful account of, in the manner of a journal or ephemeris, to which he has prefixed an elegant hydrographical chart of his own contrivance. A little before his death he employed himself about an English translation of the book of logarithms, then lately discovered by lord Napier, a Scotchman, who had a great affection for him. This posthumous work of his was published soon after, by his only son Samuel Wright, who was also a scholar of this college. He had formed many other useful designs, but was hindered by death from bringing them to perfection. Of him it may truly be said, that he studied more to serve the public than himself; and though he was rich in fame, and in the promises of the great, yet he died poor, to the scandal of an ungrateful age.” So far the memoir; other particulars concerning him, are as follow.</p><p>Mr. Wright first discovered the true way of dividing the meridian line, according to which the Mercator's charts are constructed, and upon which Mercator's sailing is founded. An account of this he sent from Caius college, Cambridge, where he was then a fellow, to his friend Mr. Blondeville, containing a short table for that purpose, with a specimen of a chart so divided, together with the manner of dividing it. All which Blondeville published, in 1594, among his Exercises. And, in 1597, the reverend Mr. William Barlowe, in his Navigator's Supply, gave a demonstration of this division as communicated by a friend.</p><p>At length, in 1599, Mr. Wright himself printed his celebrated treatise, intitled, <hi rend="italics">The Correction of certain Errors in Navigation,</hi> which had been written many years before; where he shews the reason of this division of the meridian, the manner of constructing his table, and its uses in navigation, with other improvements. In 1610 a second edition of Mr. Wright's book was published, and dedicated to his royal pupil, prince Henry; in which the author inserted farther improvements; particularly he proposed an excellent way of determining the magnitude of the earth; at the same time recommending very judiciously, the making our common measures in some certain proportion to that of a degree on its surface, that they might <cb/> not depend on the uncertain length of a barley-corn. Some of his other improvements were; The Table of Latitudes for dividing the meridian, computed as far as to minutes: An instrument, he calls the Sea rings, by which the variation of the compass, the altitude of the sun, and the time of the day, may be readily determined at once in any place, provided the latitude be known: The correcting of the errors arising from the eccentricity of the eye in observing by the cross-staff. A total amendment in the Tables of the declinations and places of the sun and stars, from his own observations, made with a six-foot quadrant, in the years 1594, 95, 96, 97: A sea-quadrant, to take altitudes by a forward or backward observation; having also a contrivance for the ready finding the latitude by the height of the pole-star, when not upon the meridian. And that this book might be the better understood by beginners, to this edition is subjoined a translation of Zamorano's Compendium; and added a large table of the variation of the compass as observed in very different parts of the world, to shew it is not occasioned by any magnetical pole. The work has gone through several other editions since. And, beside the books above mentioned, he wrote another on navigation, intitled, <hi rend="italics">The Haven-finding Art.</hi> Other accounts of him say also, that it was in the year 1589 that he first began to attend the earl of Cumberland in his voyages. It is also said that he made, for his pupil, prince Henry, a large sphere with curious movements, which, by the help of spring-work, not only represented the motions of the whole celestial sphere, but shewed likewise the particular systems of the sun and moon, and their circular motions, together with their places and possibilities of eclipsing each other: there is in it a work for a motion of 17100 years, if it should not be stopt, or the materials fail. This sphere, though thus made at a great expence of money and ingenious industry, was afterwards in the time of the civil wars cast aside, among dust and rubbish, where it was found, in the year 1646, by Sir Jonas Moore, who at his own expence restored it to its first state of perfection, and deposited it at his own house in the Tower, among his other mathematical instruments and curiosities. </p></div1></div0><div0 part="N" n="X" org="uniform" sample="complete" type="alphabetic letter"><head>X</head><cb/><div1 part="N" n="XENOCRATES" org="uniform" sample="complete" type="entry"><head>XENOCRATES</head><p>, an eminent philosopher among the ancient Greeks, was born at Chalcedon, and died 314 years before Christ, at about 90 years of age. He became early a disciple of Plato, studying under this great master at the same time with Aristotle, though he was not possessed of equal talents; the for- <cb/> mer wanting a spur, and the latter a bridle. He was fond of the mathematics; and permitted none of his scholars to be ignorant of them. There was something slovenly in the behaviour of Xenocrates; for which reason Plato frequently exhorted him to sacrifice to the graces. Seriousness and severity were always seen in his <pb n="705"/><cb/> deportment: yet notwithstanding this severe cast of mind, he was very compassionate. There was something extraordinary in the rectitude of his morals: he was absolute master of his passions; and was not fond of pleasure, riches, or applause. Indeed, so great was his reputation for sincerity and probity, that he was the only person whom the magistrates of Athens dispensed from confirming his testimony with an oath. And yet he was so ill treated by them, as to be sold because he could not pay the poll-tax laid upon foreigners. Demetrius Phalereus bought Xenocrates, paid the debt to the Athenians, and immediately gave him his liberty. At Alexander's request, he composed a treatise on the Art of Reigning; 6 books on Nature; 6 books on Philosophy; one on Riches, &c; but none of them have come down to these times:—His theology it seems was but poor stuff: Cicero refutes him in the first book of the Nature of the Gods.</p></div1><div1 part="N" n="XENOPHANES" org="uniform" sample="complete" type="entry"><head>XENOPHANES</head><p>, a Greek philosopher, born in Colophon, was, according to some authors, the disciple of Archelaus; in which case he must have been contemporary with Socrates. Others relate, that he taught himself all he knew, and that he lived at the same time with Anaximander: according to which account he must have flourished before Socrates, and about the 60th Olympiad, as Diogenes Laertius affirms. He founded the Eleatic sect; and wrote several poems on philosophical subjects; as also a great many on the foundation of Colophon, and on that of the colony of Elea. He wrote also against Homer and Hesiod. He was banished from his country, withdrew to Sicily, and lived in Zanche and Catana. His opinion with regard to the nature of God differs not much from that of Spinoza.—When he saw the Egyptians pour forth lamentations during their festivals, he thus advised them: “If the objects of your worship are Gods, do not <cb/> weep: if they are men, offer not sacrifices to them.” The answer he made to a man with whom he refused to play at dice, is highly worthy of a philosopher: This man calling him a coward, “Yes, replied he, I am excessively so with regard to all shameful actions.”</p></div1><div1 part="N" n="XENOPHON" org="uniform" sample="complete" type="entry"><head>XENOPHON</head><p>, a celebrated Greek general, philosopher, and historian, was born at Athens, and became early a disciple of Socrates, who, says Strabo, saved his life in battle. About the 50th year of his age he engaged in the expedition of Cyrus, and accomplished his immortal retreat in the space of 15 months. The jealousy of the Athenians banished him from his native city, for engaging in the service of Sparta and Cyrus. On his return therefore he retired to Scillus, a town of Elis, where he built a temple to Diana, which he mentions in his epistles, and devoted his leisure to philosophy and rural sports. But commotions arising in that country, he removed to Corinth, where it seems he wrote his Grecian History, and died at the age of 90, in the year 360 before Christ.</p><p>By his wife Philesia he had two sons, Diodorus and Gryllus. The latter rendered himself immortal by killing Epaminondas in the famous battle of Mantinea, but perished in that exploit, which his father lived to record.</p><p>The best editions of his works are those of Franckfort in 1674, and of Oxford, in Greek and Latin, in 1703, 5 vols. 8vo. Separately have been pubhshed his <hi rend="italics">Cyropœdia,</hi> Oxon. 1727, 4to, and 1736, 8vo. <hi rend="italics">Cyri Anabasis,</hi> Oxon. 1735, 4to, and 1747, 8vo. <hi rend="italics">Memorabilia Socratis,</hi> Oxon. 1741, 8vo.—His <hi rend="italics">Cyropœdia</hi> has been admirably translated into English by Spelman.</p></div1><div1 part="N" n="XIPHIAS" org="uniform" sample="complete" type="entry"><head>XIPHIAS</head><p>, in Astronomy, is the Dorado or Swordfish, a constellation of the southern hemisphere; being one of the new constellations added by modern astronomers; and consisting of 6 stars only. See <hi rend="smallcaps">Dorado.</hi> </p></div1></div0><div0 part="N" n="Y" org="uniform" sample="complete" type="alphabetic letter"><head>Y</head><cb/><div1 part="N" n="YARD" org="uniform" sample="complete" type="entry"><head>YARD</head><p>, a lineal measure, or measure of length, used in England and Spain chiefly to measure cloth, stuffs, &c. The Yard was settled by Henry the 1st, from the length of his own arm.</p><p>The English Yard contains 3 feet; and it is equal to 4-5ths of the English ell, to 7-9ths of the Paris ell, to 4-3ds of the Flemish ell, to 56-51sts of the Spanish vara or Yard.</p><div2 part="N" n="Yard" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Yard</hi></head><p>, or <hi rend="italics">Golden</hi> <hi rend="smallcaps">Yard</hi>, is also a popular name given to the 3 stars which compose the belt of Orion.</p></div2></div1><div1 part="N" n="YEAR" org="uniform" sample="complete" type="entry"><head>YEAR</head><p>, in the full extent of the word, is a system or cycle of several months, usually 12. Others define Year, in the general, a period or space of time, measured out by the revolution of some celestial body in <cb/> its orbit. Thus, the time in which the sixed stars make a revolution, is called the <hi rend="italics">great Year;</hi> and the times in which Jupiter, Saturn, the Sun, Moon, &c, complete their courses, and return to the same point of the zodiac, are respectively called the Years of Jupiter, and Saturn, and the Solar, and Lunar Years, &c.</p><p>As Year denoted originally a revolution, and was not limited to that of the sun; accordingly we find by the oldest accounts, that people have, at different times, expressed other revolutions by it, particularly that of the moon: and consequently that the Years of some accounts, are to be reckoned only months, and sometimes periods of 2, or 3, or 4 months. This will help us greatly in understanding the accounts that certain nations give of their own antiquity, and per- <pb n="706"/><cb/> haps of the age of men. We read expressly, in several of the old Greek writers, that the Egyptian Year, at one period, was only a month; and we are farther told that at other periods it was 3 months, or 4 months: and it is probable that the children of Israel followed the Egyptian account of their Years. The Egyptians talked, almost 2000 years ago, of having accounts of events 48 thousand Years distance. A great deal must be allowed to fallacy, on the above account; but beside this, the Egyptians had, in the time of the Greeks, the same ambition which the Chinese have at present, and wanted to pass themselves upon that people, as these others do upon us, for the oldest inhabitants of the earth. They had recourse also to the same means, and both the present and the early impostors have pretended to ancient observations of the heavenly bodies, and recounted eclipses in particular, to vouch for the truth of their accounts. Since the time in which the solar Year, or period of the earth's revolution round the sun, has been received, we may account with certainty; but for those remote ages, in which we do not know of a certainty what is meant by the term Year, it is impossible to form any conjecture of the duration of time in the accounts. The Babylonians pretend to an antiquity of the same romantic kind; they talk of 47 thousand Years in which they had kept observations; but we may judge of these as of the others, and of the observations as of the Years. The Egyptians speak of the stars having four times altered their courses in that period which they claim for their history, and that the sun set twice in the east. They were not such perfect astronomers, but, after a round-about voyage, they might perhaps mistake the east for the west when they came in again.</p><div2 part="N" n="Year" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Year</hi></head><p>, or <hi rend="smallcaps">Solar Year</hi>, properly, and by way of eminence so called, is the space of time in which the sun moves through the 12 signs of the ecliptic. This, by the observations of the best modern astronomers, contains 365 days, 5 hours, 48 min. 48 seconds: the quantity assumed by the authors of the Gregorian calendar is 365 days, 5 hours, 49 min. But in the civil or popular account, this Year only contains 365 days; except every 4th Year, which contains 366.</p><p>The vicissitude of seasons seems to have given occasion to the first institution of the Year. Man, naturally curious to know the cause of that diversity, soon found it was the proximity and distance of the sun; and therefore gave the name Year to the space of time in which that luminary performed his whole course, by returning to the same point of his orbit. According to the accuracy in their observations, the Year of some nations was more perfect than that of others, but none of them quite exact, nor whose parts did not shift with regard to the parts of the sun's course.</p><p>According to Herodotus, it was the Egyptians who first formed the Year, making it to contain 360 days, which they subdivided into 12 months, of 30 days each. Mercury Trismegistus added 5 days more to the account. And on this footing it is said that Thales instituted the Year among the Greeks; though that form of the Year did not hold throughout all Greece. Also, the Jewish, Syrian, Roman, Persian, Ethiopic, Arabic, &c Years, were all different. In fact, considering the imperfect state of astronomy in those ages, <cb/> it is no wonder that different people should disagree in the calculation of the sun's course. We are even assured by Diod. Siculus, lib. 1. Plutarch, in Numa, and Pliny, lib. 7, cap. 48, that the Egyptian Year itself was at first very different from that now represented.</p><p>The solar Year is either <hi rend="italics">astronomical</hi> or <hi rend="italics">civil.</hi></p><p>The <hi rend="italics">Astronomical Solar</hi> <hi rend="smallcaps">Year</hi>, is that which is determined precisely by astronomical observations; and is of two kinds, <hi rend="italics">tropical,</hi> and <hi rend="italics">sidereal</hi> or <hi rend="italics">astral.</hi></p><p><hi rend="italics">Tropical,</hi> or <hi rend="italics">Natural</hi> <hi rend="smallcaps">Year</hi>, is the time the sun takes in passing through the zodiac; which, as before observed, is 365d. 5h. 48m. 48sec.; or 365d. 5h. 49min. This is the only proper or natural Year, because it always keeps the same seasons to the same months.</p><p><hi rend="italics">Sidereal</hi> or <hi rend="italics">Astral</hi> <hi rend="smallcaps">Year</hi>, is the space of time the sun takes in passing from any fixed star, till his return to it again. This consists of 365d. 6h. 9m. 17 sec.; being 20m. 29 sec. longer than the true solar year.</p><p><hi rend="italics">Lunar</hi> <hi rend="smallcaps">Year</hi>, is the space of 12 lunar months. Hence, from the two kinds of synodical lunar months, there arise two kinds of lunar Years; the one <hi rend="italics">astronomical,</hi> the other <hi rend="italics">civil.</hi></p><p><hi rend="italics">Lunar Astronomical</hi> <hi rend="smallcaps">Year</hi>, consists of 12 lunar synodical months; and therefore contains 354d. 8h. 48m. 38sec. and is therefore 10d. 21h. om. 10 s. shorter than the solar Year. A difference which is the foundation of the Epact.</p><p><hi rend="italics">Lunar Civil</hi> <hi rend="smallcaps">Year</hi>, is either common or embolismic.</p><p>The <hi rend="italics">Common Lunar</hi> <hi rend="smallcaps">Year</hi> consists of 12 lunar civil months; and therefore contains 354 days. And</p><p>The <hi rend="italics">Embolismic</hi> or <hi rend="italics">Intercalary Lunar</hi> <hi rend="smallcaps">Year</hi>, consists of 13 lunar civil months, and therefore contains 384 days.</p><p>Thus far we have considered Years and months, with regard to astronomical principles, upon which the division is founded. By this, the various forms of civil Years that have formerly obtained, or that do still obtain, in divers nations, are to be examined.</p><p><hi rend="italics">Civil</hi> <hi rend="smallcaps">Year</hi>, is that form of Year which every nation has contrived or adopted, for computing their time by. Or the civil is the tropical Year, considered as only consisting of a certain number of whole days: the odd hours and minutes being set aside, to render the computation of time, in the common occasions of life, more easy. As the tropical Year is 365d. 5h. 49m. or almost 365d. 6h. which is 365 days and a quarter; therefore if the civil Year be made 365 days, every 4th year it must be 366 days, to keep nearly to the course of the sun. And hence the civil Year is either <hi rend="italics">common</hi> or <hi rend="italics">bissextile.</hi> The</p><p><hi rend="italics">Common Civil</hi> <hi rend="smallcaps">Year</hi>, is that consisting of 365 days; having seven months of 31 days each, four of 30 days, and one of 28 days; as indicated by the following well known memorial verses:</p><p>Thirty days hath September,<lb/> April, June, and November;<lb/> February twenty-eight alone,<lb/> And all the rest have thirty one.<lb/></p><p><hi rend="italics">Bissextile</hi> or <hi rend="italics">Leap</hi> <hi rend="smallcaps">Year</hi>, consists of 366 days; having one day extraordinary; called the intercalary, or bissextile day; and takes place every 4th Year. This additional day to every 4th Year, was first introduced <pb n="707"/><cb/> by Julius Cæsar; who, to make the civil Years keep pace with the tropical ones, contrived that the 6 hours which the latter exceeded the former, should make one day in 4 years, and be added between the 24th and 23d of February, which was their 6th of the calends of March; and as they then counted this day twice over, or had <hi rend="italics">bis sexto calendas,</hi> hence the Year itself came to be called <hi rend="italics">bis sixtus,</hi> and <hi rend="italics">bissextile.</hi></p><p>However, among us, the intercalary day is not introduced by counting the 23d of February twice over, but by adding a day at the end of that month, which therefore in that Year contains 29 days.</p><p>A farther reformation was made in this year by Pope Gregory. See <hi rend="italics">Gregorian</hi> <hi rend="smallcaps">Year, Calendar</hi>, B<hi rend="smallcaps">ISSEXTILE</hi>, and <hi rend="smallcaps">Leap</hi>-<hi rend="italics">Year.</hi></p><p>The Civil or Legal Year, in England, formerly commenced on the day of the Annunciation, or 25th of March; though the historical Year began on the day of the Circumcision, or 1st of January; on which day the German and Italian Year also begins. The part of the Year between these two terms was usually expressed both ways: as 1745.6, or 174 5/6. But by the act for altering the stile, the civil Year now commences with the 1st of January.</p><p><hi rend="italics">Ancient Roman</hi> <hi rend="smallcaps">Year.</hi> This was the lunar Year, which, as first settled by Romulus, contained only ten months, of unequal numbers of days in the following order: viz,</p><p>March 31; April 30; May 31; June 30; Quintilis 31; Sextilis 30; September 30; October 31; November 30; December 30; in all 304 days; which came short of the true lunar Year by 50 days; and of the solar by 61 days. Hence, the beginning of Romulus's Year was vague, and unfixed to any precise season; to remove which inconvenience, that prince ordered so many days to be added yearly as would make the state of the heavens correspond to the first month, without calling them by the name of any month.</p><p>Numa Pompilius corrected this irregular constitution of the Year, composing two new months, January and February, of the days that were used to be added to the former Year. Thus Numa's year consisted of 12 months, of different days, as follow; viz, <table><row role="data"><cell cols="1" rows="1" role="data">January</cell><cell cols="1" rows="1" role="data">29;</cell><cell cols="1" rows="1" role="data">February</cell><cell cols="1" rows="1" role="data">28;</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" role="data">31;</cell></row><row role="data"><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" role="data">29;</cell><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" role="data">29;</cell></row><row role="data"><cell cols="1" rows="1" role="data">Quintilis</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">Sextilis</cell><cell cols="1" rows="1" role="data">29;</cell><cell cols="1" rows="1" role="data">September</cell><cell cols="1" rows="1" role="data">29;</cell></row><row role="data"><cell cols="1" rows="1" role="data">October</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">November</cell><cell cols="1" rows="1" role="data">29;</cell><cell cols="1" rows="1" role="data">December</cell><cell cols="1" rows="1" role="data">29;</cell></row></table> in all 355 days; therefore exceeding the quantity of a lunar civil Year by one day; that of a lunar astronomical Year by 15<hi rend="sup">h</hi> 11<hi rend="sup">m</hi> 22<hi rend="sup">s</hi>; but falling short of the common solar Year by 10 days; so that its beginning was still vague and unfixed.</p><p>Numa, however, desiring to have it begin at the winter solstice, ordered 22 days to be intercalated in February every 2d Year, 23 every 4th, 22 every 6th, and 23 every 8th Year.</p><p>But this rule failing to keep matters even, recourse was had to a new way of intercalating; and instead of 23 days every 8th Year, only 15 were to be added. The care of the whole was committed to the pontifex maximus; who however, neglecting the trust, let things run to great confusion. And thus the Roman Year stood till Julius Cæsar reformed it. See <hi rend="smallcaps">Calen-</hi> <cb/> <hi rend="smallcaps">DAR.</hi> And for the manner of reckoning the days of the Roman months, see <hi rend="smallcaps">Calends, Nones</hi>, and <hi rend="smallcaps">Ides.</hi></p><p><hi rend="italics">Julian</hi> <hi rend="smallcaps">Year.</hi> This is in effect a solar Year, commonly containing 365 days; though every 4th Year, called Bissextile, it contains 366. The months of the Julian Year, with the number of their days, stood thus: <table><row role="data"><cell cols="1" rows="1" role="data">January</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">February</cell><cell cols="1" rows="1" role="data">28;</cell><cell cols="1" rows="1" role="data">March</cell><cell cols="1" rows="1" role="data">31;</cell></row><row role="data"><cell cols="1" rows="1" role="data">April</cell><cell cols="1" rows="1" role="data">30;</cell><cell cols="1" rows="1" role="data">May</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">June</cell><cell cols="1" rows="1" role="data">30;</cell></row><row role="data"><cell cols="1" rows="1" role="data">July</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">August</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">September</cell><cell cols="1" rows="1" role="data">30;</cell></row><row role="data"><cell cols="1" rows="1" role="data">October</cell><cell cols="1" rows="1" role="data">31;</cell><cell cols="1" rows="1" role="data">November</cell><cell cols="1" rows="1" role="data">30;</cell><cell cols="1" rows="1" role="data">December</cell><cell cols="1" rows="1" role="data">31.</cell></row></table> But every Bissextile Year had a day added in February, making it then to contain 29 days.</p><p>The mean quantity therefore of the Julian Year is 365 1/4 days, or 365<hi rend="sup">d</hi> 6<hi rend="sup">h</hi>; exceeding the true solar Year by somewhat more than 11 minutes; an excess which amounts to a whole day in almost 131 years. Hence the times of the equinoxes go backward, and fall earlier by one day in about 130 or 131 Years. And thus the Roman Year stood, till it was farther corrected by pope Gregory.</p><p>For settling this Year, Julius Cæsar brought over from Egypt, Sosigenes, a celebrated mathematician; who, to supply the defect of 67 days, which had been lost through the neglect of the priests, and to bring the beginning of the Year to the winter solstice, made one Year to consist of 15 months, or 445 days; on which account that Year was used to be called <hi rend="italics">annus confusionis,</hi> the <hi rend="italics">Year of confusion.</hi> See <hi rend="italics">Julian</hi> <hi rend="smallcaps">Calendar.</hi></p><p><hi rend="italics">Gregorian</hi> <hi rend="smallcaps">Year.</hi> This is the Julian Year corrected by this rule, viz, that instead of every secular or 100th Year being a bissextile, as it would be in the former way, in the new way three of them are common Years, and only the 4th is bissextile.</p><p>The error of 11 minutes in the Julian Year, by continual repetition, had accumulated to an error of 13 days from the time when Cæsar made his correction; by which means the equinoxes were greatly disturbed. In the Year 1582, the equinoxes were fallen back 10 days, and the full moons 4 days, more backward than they were in the time of the Nicene council, which was in the Year 325; viz, the former from the 20th of March to the 10th, and the latter from the 5th to the 1st of April. To remedy this increasing irregularity, pope Gregory the 13th, in the year 1582, called together the chief astronomers of his time, and concerted this correction, throwing out the 10 days above mentioned. He exchanged the lunar cycle for that of the epacts, and made the 4th of October of that Year to be the 15th; by that means restoring the vernal equinox to the 21st of March. It was also provided, by the omission of 3 intercalary days in 400 Years, to make the civil Year keep pace nearly with the solar Year, for the time to come. See <hi rend="smallcaps">Calendar.</hi></p><p>In the Year 1700, the error of 10 days was grown to 11; upon which, the protestant states of Germany, to prevent farther confusion, adopted the Gregorian correction. And the same was accepted also in England in the year 1752, when 11 days were thrown out after the 2d of September that Year, by accounting the 3d to be the 14th day of the month: calling this the new stile, and the former the old stile. And the Gregorian, or <pb n="708"/><cb/> new stile, is now in like manner used in most countries of Europe.</p><p>Yet this last correction is still not quite perfect; for as it has been shewn that in 4 centuries, the Julian Year gains 3<hi rend="sup">d</hi> 2<hi rend="sup">h</hi> 40<hi rend="sup">m</hi>; and as it is only the 3 days that are kept out in the Gregorian Year; there is still an excess of 2<hi rend="sup">h</hi> 40<hi rend="sup">m</hi> in 4 centuries, which amounts to a whole day in 36 centuries, or in 3600 Years. See C<hi rend="smallcaps">ALENDAR</hi>, <hi rend="italics">New</hi> or <hi rend="italics">Gregorian</hi> <hi rend="smallcaps">Stile</hi>, &c.</p><p><hi rend="italics">Egyptian</hi> <hi rend="smallcaps">Year</hi>, called also the <hi rend="italics">Year of Nabonassar,</hi> on account of the epoch of Nabonassar, is the solar Year of 365 days, divided into 12 months, of 30 days each, beside 5 intercalary days, added at the end. The order and names of these months are as follow: <table><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">1.</cell><cell cols="1" rows="1" role="data">Thoth;</cell><cell cols="1" rows="1" rend="align=right" role="data">2.</cell><cell cols="1" rows="1" role="data">Paophi;</cell><cell cols="1" rows="1" rend="align=right" role="data">3.</cell><cell cols="1" rows="1" role="data">Athyr;</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">4.</cell><cell cols="1" rows="1" role="data">Chojac;</cell><cell cols="1" rows="1" rend="align=right" role="data">5.</cell><cell cols="1" rows="1" role="data">Tybi;</cell><cell cols="1" rows="1" rend="align=right" role="data">6.</cell><cell cols="1" rows="1" role="data">Mecheir;</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">7.</cell><cell cols="1" rows="1" role="data">Phamenoth;</cell><cell cols="1" rows="1" rend="align=right" role="data">8.</cell><cell cols="1" rows="1" role="data">Pharmuthi;</cell><cell cols="1" rows="1" rend="align=right" role="data">9.</cell><cell cols="1" rows="1" role="data">Pachon;</cell></row><row role="data"><cell cols="1" rows="1" rend="align=right" role="data">10.</cell><cell cols="1" rows="1" role="data">Pauni;</cell><cell cols="1" rows="1" rend="align=right" role="data">11.</cell><cell cols="1" rows="1" role="data">Epiphi;</cell><cell cols="1" rows="1" rend="align=right" role="data">12.</cell><cell cols="1" rows="1" role="data">Mesori.</cell></row></table></p><p>As the Egyptian Year, by neglecting the 6 hours, in every 4 Years loses a whole day of the Julian Year, its beginning runs through every part of the Julian Year in the space of 1460 Years; after which, they meet again; for which reason it is called the <hi rend="italics">erratic</hi> Year. And because this return to the same day of the Julian Year, is performed in the space of 1460 Julian Years, this circle is called the Sothic period.</p><p>This Year was applied by the Egyptians to civil uses, till Anthony and Cleopatra were defeated; but the mathematicians and astronomers used it till the time of Ptolomy, who made use of it in his Almagest; so that the knowledge of it is of great use in asstronomy, for comparing the ancient observations with the modern.</p><p>The ancient Egyptians, we are told by Diodorus Siculus, (Plutarch, lib. 1, in the life of Numa, and Pliny, lib. 7, cap. 48) measured their Years by the course of the moon. At first they were only one month, then 3, then 4, like that of the Arcadians; and then 6, like that of the people of Acarnania. Those authors add, that it is on this account that they reckon such a vast number of Years from the beginning of the world; and that in the history of their kings, we meet with some who lived 1000, or 1200 Years. The same thing is maintained by Kircher; Oedip. Egypt. tom. 2, pa. 252. And a late author observes, that Varro has affirmed the same of all nations, that has been quoted of the Egyptians. By which means many account for the great ages of the more ancient patriarchs; expounding the gradual decrease in their ages, by the successive increase of the number of months in their years.</p><p>Upon the Egyptians being subdued by the Romans, they received the Julian Year, though with some alteration; for they still retained their ancient months, with the five additional days, and every 4th Year they intercalated another day, for the 6 hours, at the end of the Year, or between the 28th and 29th of August. Also, the beginning of their Year, or the first day of the month Thoth, answered to the 29th of August of the Julian Year, or to the 30th if it happened to be leap Year.</p><p><hi rend="italics">The Ancient Greek</hi> <hi rend="smallcaps">Year.</hi>—This was a lunar Year, <cb/> consisting of 12 months, which at first had each 30 days, then alternately 29 and 30 days, computed from the first appearance of the new moon; with the addition of an embolismic month of 30 days, every 3d, 5th, 8th, 11th, 14th, 16th, and 19th Year of a cycle of 19 Years; in order to keep the new and full moons to the same terms or seasons of the Year.</p><p>Their Year commenced with that new moon which was nearest to the summer solstice. And the order of the months, with the number of their days, were as follow: 1. <foreign xml:lang="greek">*ekatom<*>aiwn</foreign>, of 29 days; 2. <foreign xml:lang="greek">*mhtageitniwn</foreign> 30; 3. <foreign xml:lang="greek">*bohdromiwn</foreign> 29; 4. <foreign xml:lang="greek">*maimakth<*>iwn</foreign> 30; 5. <foreign xml:lang="greek">*puaneyiwn</foreign> 29; 6. <foreign xml:lang="greek">*poseidewn</foreign> 30; 7. <foreign xml:lang="greek">*gamhliwn</foreign> 29; 8. <foreign xml:lang="greek">*anqesh<*>iwn</foreign> 30; 9. <foreign xml:lang="greek">*ela<*>h<*>oliwn</foreign> 29; 10. <foreign xml:lang="greek">*m<*>nuxiwn</foreign> 30; 11. <foreign xml:lang="greek">*oarghliwn</foreign> 29; 12. <foreign xml:lang="greek">*suiro<*>oriwn</foreign> 30.— But many of the Greek nations had other names for their months.</p><p><hi rend="italics">The Ancient Jewish</hi> <hi rend="smallcaps">Year.</hi>—This is a lunar Year, usually consisting of 11 months, containing alternately 30 and 29 days. And it was made to agree with the solar Year, by adding 11, and sometimes 12 days, at the end of the Year, or by an embolismic month. The order and quantities of the months were as follow: 1. Nisan or Abib 30 days; 2. Jiar or Zius 29; 3. Siban or Sievan 30; 4. Thamuz or Tamuz 29; 5. Ab 30; 6. Elul 29; 7. Tisri or Ethanim 30; 8. Marchesvam or Bul 29; 9. Cisleu 30; 10. Tebeth 29; 11. Sabat or Schebeth 30; 12. Adar 30 in the embolismic year, but 29 in the common year. —Note, in the defective Year, Cisleu was only 29 days; and in the redundant Year, Marchesvam was 30.</p><p><hi rend="italics">The Modern Jewish</hi> <hi rend="smallcaps">Year</hi> is likewise lunar, consisting of 12 months in common Years, but of 13 in embolismic Years; which, in a cycle of 19 Years, are the 3d, 6th, 8th, 11th, 14th, 17th, and 19th. Its beginning is fixed to the new moon next after the autumnal equinox. The names and order of the months, with the number of the days, are as follow: 1. Tisri 30 days; 2. Marchesvan 29; 3. Cisleu 30; 4. Tebeth 29; 5. Schebeth 30; 6. Adar 29; 7. Veadar, in the embolismic year, 30; 8. Nisan 30; 9. Ilar 29; 10. Sivan 30; 11. Thamuz 29; 12. Ab 30; 13. Elul 29.</p><p><hi rend="italics">The Syrian</hi> <hi rend="smallcaps">Year</hi>, is a solar one, having its beginning fixed to the beginning of October in the Julian Year; from which it only differs in the names of the months, the quantities being the same; as follow: 1. Tishrin, answering to our October, and containing 31 days; 2. Latter Tishrin, containing, like November, 30 days; 3. Canun 31; 4. Latter Canun 31; 5. Shabat 28, or 29 in a leap-year; 6. Adar 31; 7. Nisan 30; 8. Aiyar 31; 9. Haziram 30; 10. Thamuz 31; 11. Ab 31; 12. Elul 30.</p><p><hi rend="italics">The Persian</hi> <hi rend="smallcaps">Year</hi>, is a solar one, of 365 days, consisting of 12 months of 30 days each, with 5 intercalary days added at the end. The months are as follow: 1. Asrudia meh; 2. Ardihascht meh; 3. Cardi meh; 4. Thir meh; 5. Merded meh; 6. Schabarir meh; 7. Mehar meh; 8. Aben meh; 9. Adar meh; 10. Di meh; 11. Behen meh; 12, Assirer meh. This Year is the same as the Egyptian Nabonassarean, and is called the <hi rend="italics">yezdegerdic Year,</hi> to distinguish it from the fixed solar Year, called the Gelalean Year, which the Persians began to use in the Year 1079, and which was <pb n="709"/><cb/> formed by an intercalation, made six or seven times in four Years, and then once every 5th Year.</p><p><hi rend="italics">The Arabic, Mahometan, and Turkish</hi> <hi rend="smallcaps">Year</hi>, called also the Year of the <hi rend="italics">Hegira,</hi> is a lunar Year, equal to 354<hi rend="sup">d</hi> 8<hi rend="sup">h</hi> 48<hi rend="sup">m</hi>, and consists of 12 months, containing alternately 30 and 29 days. Though sometimes it contains 13 months; the names &c being as follow: 1. Muharram of 30 days; 2. Saphar 29; 3. Rabia 30; 4. Latter Rabia 29; 5. Jomada 30; 6. Latter Jomada 29; 7. Rajab 30; 8. Shaaban 29; 9. Ramadan 30; 10. Shawal 29; 11. Dulkaadah 30; 12. Dulheggia 29, but in the embolismic year 30. An intercalary day is added every 2d, 5th, 7th, 10th, 13th, 15th, 18th, 21st, <cb/> 24th, 26th, 29th, in a cycle of 29 Years. The months commence with the first appearance of the new moons after the conjunctions.</p><p><hi rend="italics">Ethiopic</hi> <hi rend="smallcaps">Year</hi>, is a solar Year perfectly agreeing with the Actiac, except in the names of the months, which are; 1. Mascaram; 2. Tykympt; 3. Hydar; 4. Tyshas; 5. Tyr; 6. Jacatil; 7. Magabit; 8. Mijazia; 9. Ginbat; 10. Syne; 11. Hamel; 12. Hahase. Intercalary days 5. It commences with the Egyptian Year, on the 29th of August of the Julian Year.</p><p>YESDEGERDIC <hi rend="smallcaps">Year.</hi> See <hi rend="italics">Persian</hi> <hi rend="smallcaps">Year.</hi> </p></div2></div1></div0><div0 part="N" n="Z" org="uniform" sample="complete" type="alphabetic letter"><head>Z</head><cb/><div1 part="N" n="ZENITH" org="uniform" sample="complete" type="entry"><head>ZENITH</head><p>, in Astronomy, the vertical point, or point in the heavens directly overhead. Or, the Zenith is a point in the surface of the sphere, from which a right line drawn through the place of any spectator, passes through the centre of the earth.</p><p>The Zenith of any place, is also the pole of the horizon, being 90 degrees distant from every point of it. And through the Zenith pass all the azimuths, or vertical circles.</p><p>The point diametrically opposite to the Zenith, is called the <hi rend="italics">nadir,</hi> being the point in the sphere directly under our feet: and it is the Zenith to our antipodes, as our Zenith is their nadir.</p><p><hi rend="smallcaps">Zenith</hi>-<hi rend="italics">Distance,</hi> is the distance of the sun or star from our Zenith; and is the complement of the altitude, or what it wants of 90 degrees.</p></div1><div1 part="N" n="ZENO" org="uniform" sample="complete" type="entry"><head>ZENO</head><p>, <hi rend="smallcaps">Eleates</hi>, or of <hi rend="italics">Elea,</hi> one of the greatest philosophers among the Ancients, flourished about 500 years before the Christian æra. He was the disciple of Parmenides, and even, according to some writers, his adopted son. Aristotle asserts that he was the inventor of logic: but his logic seems to have been calculated and employed to perplex all things, and not to clear up any thing. For Zeno employed it only to dispute against all comers, and to silence his opponents, whether they argued right or wrong. Among many other subtleties and embarrassing arguments, he proposed some with regard to motion, denying that ther was any such thing in nature; and Aristotle, in the 6th book of his physics, has preserved some of them, which are extremely subtile, especially the famous argument named Achilles; which was to prove this proposition, that the swiftest animal could never overtake the slowest, as a greyhound a tortoise, if the latter set out a little fore the former: for suppose the tortoise to be 100 yards before the dog, and that this runs 100 times as fast as the other; then while the dog runs the first 100 yards, the tortoise runs <*>, and is therefore 1 yard <cb/> before the dog; again, while the dog runs over this yard, the tortoise will run the 100th part of a yard, and will be so much before the dog; and again, while the dog runs over this 100th part of a yard, the tortoise will have got the 100th part of that 100th part before him; and so on continually, says he, the dog will always be some small part behind the tortoise. But the fallacy will soon be detected, by considering where the tortoise will be when the dog has run over 200 yards; for as the former can have run only two yards in the same time, and therefore must then be 98 yards behind the dog, he consequently must have overtaken and passed the tortoise. It has been said that, to prove to him, or some disciple of his, that there is such a thing as motion, Diogenes the Cynic rose up and walked over the floor.—Zeno shewed great courage in suffering pain; for having joined with others to endeavour to restore liberty to his country, which groaned under the oppression of a tyrant, and the enterprize being discovered, he supported with extraordinary firmness the sharpest tortures. It is even said that he had the courage to bite off his tongue, and spit it in the tyrant's face, for fear of being forced, by the violence of his torments, to discover his accomplices. Some say that he was pounded to death in a mortar.</p><div2 part="N" n="Zeno" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Zeno</hi></head><p>, a celebrated Greek philosopher, was born at Citium, in the Isle of Cyprus, and was the founder of the Stoics; a sect which had its name from that of a portico at Athens, where this philosopher chose to hold his discourses. He was cast upon that coast by shipwreck; and he ever after regarded this as a great happiness, praising the winds for having so happily driven him into the port of Piræum.—Zeno was the disciple of Crates, and had a great number of followers. He made the sovereign good to consist in dying in conformity to nature, guided by the dictates of right reason. He acknowledged but one God; and admitted an inevitable destiny over all events. His <pb n="710"/><cb/> servant taking advantage of this last opinion, cried, while he was beating him for dishonesty, “I was destined to steal;” to which Zeno replied, “Yes, and to be beaten too.” This philosopher used to say, “That if a wise man ought not to be in love, as some pretended, none would be more miserable than beautiful and virtuous women, since they would have none for their admirers but fools.” He also said, “That a part of knowledge consists in being ignorant of such things as ought not to be known: that a friend is another self: that a little matter gives perfection to a work, though perfection is not a little matter.” He compared those who spoke well and lived ill, to the money of Alexandria, which was beautiful, but composed of bad metal.—It is said that being hurt by a fall, he took that as a sign he was then to quit this life, and laid violent hands on himself, about 264 years before Christ.</p><p>Cleanthes, Crysippus, and the other successors of Zeno maintained, that with virtue we might be happy in the midst even of disgrace and the most dreadful torments. They admitted the existence of only one God, the soul of the world, which they considered as his body, and both together forming a perfect being. It is remarked that, of all the sects of the ancient philosophers, this was one of those which produced the greatest men.</p><p>We ought not to confound the two Zenos above mentioned, with</p></div2><div2 part="N" n="Zeno" org="uniform" sample="complete" type="subentry"><head><hi rend="smallcaps">Zeno</hi></head><p>, a celebrated Epicurean philosopher, born at Sidon, who had Cicero and Pomponius Atticus for his disciples, and who wrote a book against the mathematics, which, as well as that of Possidonius's refutation of it, is lost; nor with several other Zenos mentioned in history.</p></div2></div1><div1 part="N" n="ZENSUS" org="uniform" sample="complete" type="entry"><head>ZENSUS</head><p>, or <hi rend="smallcaps">Zenzus</hi>, in Arithmetic and Algebra, a name used by some of the older authors, especially in Germany, for a square number, or the 2d power: being a corruption from the Italic <hi rend="italics">censi,</hi> of Pacioli, Tartalea, &c, or the Latin <hi rend="italics">census,</hi> which signified the same thing.</p></div1><div1 part="N" n="ZETETICE" org="uniform" sample="complete" type="entry"><head>ZETETICE</head><p>, or <hi rend="smallcaps">Zetetic</hi> <hi rend="italics">Method,</hi> in Mathematics, was the method made use of to investigate, or find out the solution of a problem; and was much the same thing as analytics, or the analytic method.</p><p>Vieta has an ingenious work of this kind in 5 books; <hi rend="italics">Zeteticorum libri quinque.</hi></p></div1><div1 part="N" n="ZOCCO" org="uniform" sample="complete" type="entry"><head>ZOCCO</head><p>, <hi rend="smallcaps">Zoccolo, Zocle</hi>, or <hi rend="smallcaps">Socle</hi>, in Architecture, a square body, less in height than breadth, placed under the bases of pedestals, statues, vases, &c. See <hi rend="smallcaps">Socle</hi> and <hi rend="smallcaps">Plinth.</hi></p></div1><div1 part="N" n="ZODIAC" org="uniform" sample="complete" type="entry"><head>ZODIAC</head><p>, in Astronomy, an imaginary ring or broad circle, in the heavens, in form of a belt or girdle, within which the planets all make their excursions. In the very middle of it runs the ecliptic, or path of the sun in his annual course; and its breadth, comprehending the deviations or latitudes of the planets, is by some authors accounted 16°, some 18, and others 20 degrees.</p><p>The Zodiac, cutting the equator obliquely, makes with it the same angle as the ecliptic, which is its middle line, which angle, continually varying, is now nearly equal to 23° 28′; which is called the obliquity of the <cb/> Zodiac or ecliptic, and is also the sun's greatest declination.</p><p>The Zodiac is divided into 12 equal parts, of 30 degrees each, called the signs of the Zodiac, being so named from the constellations which anciently passed them. But, the stars having a motion from west to east, those constellations do not now correspond to their proper signs; from whence arises what is called the <hi rend="italics">precession of the equinoxes.</hi> And therefore when a star is said to be in such a sign of the Zodiac, it is not to be understood of that constellation, but only of that dodecatemory or 12th part of it.</p><p>Cassini has also observed a tract in the heavens, within whose bounds most of the comets, though not all of them, are observed to keep, and which he therefore calls the <hi rend="italics">Zodiac of the comets.</hi> This he makes as broad as the other Zodiac, and marks it with signs or constellations, like that; as Antinous, Pegasus, Andromeda, Taurus, Orion, the Lesser Dog, Hydra, the Centaur, Scorpion, and Sagittary.</p><p>ZODIACAL <hi rend="italics">Light,</hi> a brightness sometimes observed in the zodiac, resembling that of the galaxy or milky way. It appears at certain seasons, viz, towards the end of winter and in spring, after sunset, or before his rising, in autumn and beginning of winter, resembling the form of a pyramid, lying lengthways with its axis along the zodiac, its base being placed obliquely with respect to the horizon. This phenomenon was first described and named by the elder Cassini, in 1683. It was afterwards observed by Fatio, in 1684, 1685, and 1686; also by Kirch and Eimmart, in 1688, 1689, 1691, 1693, and 1694. See Mairan, Suite des Mem. de l'Acad. Royale des Sciences 1731, pa. 3.</p><p>The Zodiacal light, according to Mairan, is the solar atmosphere, a rare and subtile fluid, either luminous by itself, or made so by the rays of the sun surrounding its globe; but in a greater quantity, and more extensively, about his equator, than any other part.</p><p>Mairan says, it may be proved from many observations, that the sun's atmosphere sometimes reaches as far as the earth's orbit, and there meeting with our atmosphere, produces the appearance of an Aurora borealis.</p><p>The length of the Zodiacal light varies sometimes in reality, and sometimes in appearance only, from various causes.</p><p>Cassini often mentions the great resemblance between the Zodiacal light and the tails of comets. The same observation has been made by Fatio: and Euler endeavoured to prove that they were owing to similar causes. See Decouverte de la Lumiere Celeste que paroit dans le Zodiaque, art. 41. Lettre à M. Cassini, printed at Amsterdam in 1686. Euler, in Mem. de l'Acad. de Berlin, tom. 2.</p><p>This light seems to have no other motion than that of the sun itself: and its extent from the sun to its point, is seldom less than 50 or 60 degrees in length, and more than 20 degrees in breadth: but it has been known to extend to 100 or 103°, and from 8 to 9° broad.</p><p>It is now generally acknowledged, that the electric fluid is the cause of the aurora borealis, ascribed by <pb n="711"/><cb/> Mairan to the solar atmosphere, which produces the Zodiacal light, and which is thrown off chiefly and to the greatest distance from the equatorial parts of the sun, by means of the rotation on his axis, and extending visibly as far as the orbit of the earth, where it falls into the upper regions of our atmosphere, and is collected chiefly towards the polar parts of the earth, in consequence of the diurnal revolution, where it forms the aurora borealis. And hence it has been suggested, as a probable conjecture, that the sun may be the fountain of the electrical fluid, and that the Zodiacal light, and the tails of comets, as well as the aurora borealis, the lightning, and artificial electricity, are its various and not very dissimilar modifications.</p></div1><div1 part="N" n="ZONE" org="uniform" sample="complete" type="entry"><head>ZONE</head><p>, in Geography and Astronomy, a division of the earth's surface, by means of parallel circles, chiefly with respect to the degree of heat in the different parts of that surface.</p><p>The ancient astronomers used the term Zone, to explain the different appearances of the sun and other heavenly bodies, with the length of the days and nights; and the geographers, as they used the climates, to mark the situation of places; using the term climate when they were able to be more exact, and the term Zone when less so.</p><p>The Zones were commonly accounted five in number; one a broad belt round the middle of the earth, having the equator in the very middle of it, and bounded, towards the north and south by parallel circles passing through the tropics of Cancer and Capricorn. This they called the <hi rend="italics">torrid Zone,</hi> which they supposed not habitable, on account of its extreme heat. Though sometimes they divided this into two equal torrid Zones, by the equator, one to the north, and the other south; and then the whole number of Zones was accounted 6.</p><p>Next, from the tropics of Cancer and Capricorn, to the two polar circles, were two other spaces called <hi rend="italics">temperate Zones,</hi> as being moderately warm; and these they supposed to be the only habitable parts of the carth. <cb/></p><p>Lastly, the two spaces beyond the temperate Zones, about either pole, bounded within the polar circles, and having the poles in the middle of them, are the two <hi rend="italics">frigid</hi> or <hi rend="italics">frozen Zones,</hi> and which they supposed not habitable, on account of the extreme cold there.</p><p>Hence, the breadth of the torrid Zone, is equal to twice the greatest declination of the sun, or obliquity of the ecliptic, equal to 46° 56′, or twice 23° 28′. Each frigid Zone is also of the same breadth, the distance from the pole to the polar circle being equal to the same obliquity 23° 28′. And the breadth of each temperate Zone is equal to 43° 4′, the complement of twice the same obliquity. See these Zones exhibited in plate 35, fig. 16.</p><p>The difference of Zones is attended with a great diversity of phenomena. 1. In the torrid Zone, the sun passes through the zenith of every place in it twice a year; making as it were two summers in the year; and the inhabitants of this Zone are called <hi rend="italics">amphiscians,</hi> because they have their noon-day shadows projected different ways in different times of the year, northward at one season, and southward at the other.</p><p>2. In the temperate and frigid Zones, the sun rises and sets every natural day of 24 hours. Yet every where, but under the equator, the artificial days are of unequal lengths, and the inequality is the greater, as the place is farther from the equator. The inhabitants of the temperate Zones are called <hi rend="italics">heteroscians,</hi> because their noon-day shadow is cast the same way all the year round, viz, those in the north Zone toward the north pole, and those in the south Zone toward the south pole.</p><p>3. Within the frigid Zones, the inhabitants have their artificial days and nights extended out to a great length; the sun sometimes skirting round a little above the horizon for many days together: and at another season never rising above the horizon at all, but making continual night for a considerable space of time. The inhabitants of these Zones are called <hi rend="italics">periscians,</hi> because sometimes they have their shadows going quite round them in the space of 24 hours. <pb/><pb/> <hi rend="center">ADDENDA <hi rend="smallcaps">ET</hi> CORRIGENDA.</hi></p></div1></div0><div0 part="N" n="A" org="uniform" sample="complete" type="alphabetic letter"><head>A</head><cb/><p>ACCELERATED <hi rend="italics">Motion,</hi> pa. 18, col. 1, line 17 from the bottom, <hi rend="italics">after</hi> second instant, <hi rend="italics">add,</hi> or small part of time.—l. 6 from the bottom, <hi rend="italics">for</hi> in every instant, <hi rend="italics">read</hi> at every moment.—l. 2 from bottom, <hi rend="italics">for</hi> 16 1/12, <hi rend="italics">read</hi> 32 1/6.—col. 2, l. 1 and 2, <hi rend="italics">for</hi> 32 1/6, 48 1/4, 64 1/3, <hi rend="italics">read</hi> 64 1/3, 96 1/2, 128 2/3.</p><p>ACCELERATING <hi rend="italics">Force,</hi> pa. 21, col. 2, l. 27, <hi rend="italics">for</hi> requires, <hi rend="italics">read</hi> acquires.</p><p>Pa. 22, col. 2, l. 16 from the bottom, for <hi rend="italics">t</hi> = <hi rend="italics">vt</hi><hi rend="sup">.</hi> read <hi rend="italics">s</hi><hi rend="sup">.</hi> = <hi rend="italics">vt</hi><hi rend="sup">.</hi>.—next line, for <hi rend="italics">t</hi> and <hi rend="italics">s,</hi> read <hi rend="italics">t</hi><hi rend="sup">.</hi> and <hi rend="italics">s</hi><hi rend="sup">.</hi>.</p><div1 part="N" n="ACHROMATIC" org="uniform" sample="complete" type="entry"><head>ACHROMATIC</head><p>, pa. 25, col. 2, l. 1<hi rend="sup">.</hi>4, <hi rend="italics">for</hi> fractions, <hi rend="italics">read</hi> refractions.</p><p>Pa. 26, col. 1, l. 12, <hi rend="italics">for</hi> Veritus, <hi rend="italics">read</hi> Veritas.</p><p>After l. 9, <hi rend="italics">add,</hi> Since this article was printed, I observe, in the 3d volume of the Edinburgh Philosophical Transactions, an account of a curious set of experiments, on the unequal refrangibility of light, with observations on Achromatic telescopes, by Dr. Robert Blair. This ingenious gentleman sets out with observing, “If the theory of the Achromatic telescope is so complete as it has been represented, may it not reasonably be demanded, whence it proceeds, that Hugenius and others could execute telescopes with single object glasses 8 inches and upwards in diameter, while a compound object glass of half these dimensions, is hardly to be met with? or how it can arise from any defect in the execution, that reflectors can be made so much shorter than Achromatic refractors of equal apertures, when it is well known that the latter are much less affected by any imperfections in the execution of the lenses composing the object glass, than reflectors are by equal defects in the figure of the great speculum?—The general answer made by artists to enquiries of this kind, is, that the fault lies in the imperfection of glass, and particularly in that kind of glass of which the concave lens of the compound object glass is formed, called flint glass.— It was in order to satisfy myself concerning the reality of this difficulty, and to attempt to remove it, that I engaged in the following course of experiments.”</p><p>Dr. Blair describes the apparatus and manner of making the experiments. He employed various prisins of different kinds of glass; also lenses of glass, and of <cb/> a great variety of fluid mediums, having different degrees of refraction. Having detailed the whole at considerable length, for which a reference must be made to the work itself, and it is very deserving of attentive perusal, he concludes with the following recapitulation of the contents and scope of the whole discourse.</p><p>“The unequal refrangibility of light, as discovered and fully explained by Sir Isaac Newton, so far stands its ground uncontroverted, that when the refraction is made in the confine of any medium whatever, and a vacuum, the rays of different colours are unequally refracted, the red-making rays being the least refrangible, and the violet-making rays the most refrangible.</p><p>“The discovery of what has been called a different dispersive power in different refractive mediums, proves those theorems of Sir Isaac Newton not to be universal, in which he concludes that the difference of refraction of the most and least refrangible rays, is always in a given proportion to the refraction of the mean refrangible ray. There can be no doubt that this position is true with respect to the mediums on which he made his experiments; but there are many exceptions to it.</p><p>“For the experiments of Mr. Dollond prove, that the difference of refraction between the red and violet rays, in proportion to the refraction of the whole pencil, is greater in some kinds of glass than in water, and greater in flint-glass than in crown-glass.</p><p>“The first set of experiments above recited, prove, that the quality of dispersing the rays in a greater degree than crown-glass, is not confined to a few mediums, but is possessed by a great variety of fluids, and by some of these in a most extraordinary degree. Solutions of metals, essential oils, and mineral acids, with the exception of the vitriolic, are most remarkable in this respect.</p><p>“Some consequences of the combinations of mediums of different dispersive powers, which have not been sufficiently attended to, are then explained. Although the greater refrangibility of the violet rays than of the red rays, when light passes from any medium whatever into a vacuum, may be considered as a law of nature, yet in the passage of light from one medium into another, it depends entirely on the qualities of the mediums, which of these rays shall be the most refrangible, or whether there shall be any difference in their refrangibility. <pb n="714"/><cb/></p><p>“The application of the demonstrations of Hugenius to the correction of the aberration from the spherical figures of lenses, whether solid or fluid, is then taken notice of, as being the next step towards perfecting the theory of telescopes.</p><p>“Next it appears from trials made with object-glasses of very large apertures, in which both aberrations are corrected as far as the principles will admit, that the correction of colour which is obtained by the common combination of two mediums which differ in dispersive power, is not complete. The homogeneal green rays emerge most refracted, next to these the united blue and yellow, then the indigo and orange united, and lastly the united violet and red, which are least refracted.</p><p>“If this production of colour were constant, and the length of the secondary spectrum were the same in all combinations of mediums when the whole refraction of the pencil is equal, the perfect correction of the aberration from difference of refrangibility would be impossible, and would remain an insurmountable obstacle to the improvement of dioptrical instruments.</p><p>“The object of the next experiment is, therefore, to search, whether nature affords mediums which differ in the degree in which they disperse the rays composing the prismatic spectrum, and at the same time separate the several orders of rays in the same proportion. For if such could be found, the above-mentioned secondary spectrum would vanish, and the aberration from difference of refrangibility might be removed. The result of this investigation was unsuccessful with respect to its principal object. In every combination that was tried, the same kind of uncorrected colour was observed, and it was thence concluded, that there was no direct method of removing the aberration.</p><p>“But it appeared in the course of the experiments, that the breadth of the secondary spectrum was less in some combinations than in others, and thence an indirect way opened, leading to the correction sought after; namely by forming a compound concave lens of the materials which produce most colour, and combining it with a compound convex lens formed of the materials which produce least colour; and it was observed in what manner this might be effected by means of three mediums, though apparently four are required.</p><p>“In searching for mediums best adapted for the above purpose, a very singular and important quality was detected in the muriatic acid. In all the dispersive mediums hitherto examined, the green rays, which are the mean refrangible in crown-glass, were found among the less refrangible, and thence occasion the uncorrected colour which has been described. In the muriatic acid, on the contrary, these same rays make a part of the more refrangible; and in consequence of this, the order of the colours in the secondary spectrum, formed by a combination of crown glass with this fluid, is inverted, the homogeneal green being now the least refrangible, and the united red and violet the most refrangible.</p><p>“This remarkable quality found in the marine acid led to complete success in removing the great defect of optical instruments, that dissipation or aberration of the rays, arising from their unequal refrangibility, which has rendered it impossible hitherto to converge all of them to one point either by single or opposite refractions. A fluid in which the particles of marine acid and metal- <cb/> line particles hold a due proportion, at the same time that it separates the extreme rays of the spectrum much more than crown-glass, refracts all the orders of rays exactly in the same proportion as the glass does; and hence rays of all colours, made to diverge by the refraction of the glass, may either be rendered parallel by a subsequent refraction made in the confine of the glass and this fluid, or by weakening the refractive density of the fluid, the refraction which takes place in the confine of it and glass, may be rendered as regular as reflexion, while the errors arising from unavoidable imperfections of workmanship, are far less hurtful than in reflexion, and the quantity of light transmitted by equal apertures of the telescopes much greater.</p><p>“Such are the advantages which the theory presents. In reducing this theory to practice, difficulties must be expected in the first attempts. Many of these it was necessary to surmount before the experiments could be completed. For the delicacy of the observations is such as to require a considerable degree of perfection in the execution of the object-glasses, in order to admit of the phenomena being rendered more apparent by means of high magnifying powers. Great pains seem to have been taken by mathematicians to little purpose, in calculating the radii of the spheres requisite for Achromatic telescopes, from their not considering that the object-glass itself is a much nicer test of the optical properties of refracting mediums than the gross experiments made by prisms, and that the results of their demonstrations cannot exceed the accuracy of the data, however much they may fall short of it.</p><p>“I shall conclude this paper, which has now greatly exceeded its intended bounds, by enumerating the several cases of unequal refrangibility of light, that their varieties may at once be clearly apprehended.</p><p>“In the refraction which takes place in the confine of every known medium and a vacuum, rays of different colours are unequally refrangible, and the red-making rays are least refrangible, and the violet-making rays are most refrangible.</p><p>“This difference of refrangibility of the red and violet rays is not the same in all mediums. Those mediums in which the difference is greatest, and which, by consequence, separate or disperse the rays of different colours most, have been distinguished by the term dispersive, and those mediums which separate the rays least have been called indispersive. Dispersive mediums differ from indispersive, and still more from each other, in another very essential circumstance.</p><p>“It appears from the experiments which have been made on indispersive mediums, that the mean refrangible light is always the same, and of a green colour.</p><p>“Now, in by far the largest class of dispersive mediums, including flint glass, metallic solutions, essential oils, the green light is not the mean refrangible order, but forms one of the less refrangible orders of light, being found in the prismatic spectrum nearer to the deep red than the extreme violet.</p><p>“In another class of dispersive mediums, which includes the muriatic and nitrous acids, this same green light becomes one of the more refrangible orders, being now found nearer to the extreme violet than the deep red. <pb n="715"/><cb/></p><p>“These are the varieties in the refrangibility of light, when the refraction takes place in the confine of a vacuum; and the phenomena will scarce differ sensibly in refractions made in the confine of dense mediums and air.</p><p>“But when light passes from one dense medium into another, the cases of unequal refrangibility are more complicated.</p><p>“In refractions made in the confine of mediums which differ only in strength, not in quality, as in the confine of water and crown-glass, or in the confine of the different kinds of dispersive fluids more or less diluted, the difference of refrangibility will be the same as above stated in the confine of dense mediums and air, only the whole refiaction will be less.</p><p>“In the confine of an indispersive medium, and a rarer medium belonging to either class of the dispersive, the red and violet rays may be rendered equally refrangible. If the dispersive power of the rare medium be then increased, the violet rays will become the least refrangible, and the red rays the most refrangible. If the mean refractive density of the two mediums be rendered equal, the red and violet rays will be refracted in opposite directions, the one towards, the other from the perpendicular.</p><p>“Thus it happens to the red and violet rays, whichsoever class of dispersive mediums be employed. But the refrangibility of the intermediate orders of rays, and especially of the green rays, will be different when the class of dispersive mediums is changed.</p><p>“Thus, in the first case, where the red and violet rays are rendered equally refrangible, the green rays will emerge most refrangible if the first class of dispersive mediums is used, and least refrangible if the second class is used. And in the other two cases, where the violet becomes least refrangible, and the red most refrangible, and where these two kinds of rays are refracted in opposite directions, the green rays will join the red if the first class of dispersive mediums be employed, and will arrange themselves with the violet if the second class be made use of.</p><p>“Only one case more of unequal refrangibility remains to be stated; and that is, when light is refracted in the confine of mediums belonging to the two different classes of dispersive fluids. In its transition, for example, from an essential oil, or a metallic solution, into the muriatic acid, the refractive density of these fluids may be so adjusted, that the red and violet rays shall suffer no refraction in passing from the one into the other, how oblique soever their incidence be. But the green rays will then suffer a considerable refraction, and this refraction will be from the perpendicular, when light passes from the muriatic acid into the essential oil, and towards the perpendicular, when it passes from the essential oil into the muriatic acid. The other orders of rays will suffer similar refractions, which will be greatest in those adjoining the green, and will diminish as they approach the deep red on the one hand, and the extreme violet on the other, where the refraction ceases entirely.</p><p>“The manner of the production of these effects, by the attraction of the several mediums, may be thus explained. We shall suppose the attractive forces, which <cb/> produce the refractions of the red, green and violet light, to be represented by the numbers, 8, 12, and 16, in glass; 6, 9, 14, in the metallic solution; 6, 11, 14, in the muriatic acid; and 6, 10, 14, in a mixture of these two fluids. The excess of attraction of glass for the red and violet light is equal to 2, whichsoever of the three fluids be employed. The refraction of these two orders of rays will therefore be the same in all the three cases. But the excess of attraction for the green light is equal to 3, when the metallic solution is used, and therefore the green light will be more refracted than the red and violet, in this case. When the muriatic acid is used, the excess of attraction of glass for the green light is only 1, and therefore the green light will now be less refracted than the red and violet.</p><p>“We shall next suppose the metallic solution and the acid to adjoin each other. The attractions of both these mediums, for the red light being 6, and for the violet light 14, these two orders of rays will suffer no refraction in the confine of the two fluids, the difference of their attractions being equal to nothing.</p><p>“But the attractive force of the metallic solution for the green ray being only 9, and that of the muriatic acid for the same ray being 11, the green light will be attracted towards the muriatic acid with the force 2; and therefore the difference between the refraction of the green light and the unrefracted red and violet light, which takes place in the confine of these fluids, will greatly exceed the difference of refraction of the green light, and equally refracted red and violet light, which is produced in the confine of glass and either of the fluids.</p><p>“Lastly, in a mixture of the two kinds of fluids, the attraction for the red, green and violet rays, being 6, 10 and 14, and that of the glass, 8, 12 and 16, the excess of the attraction of the glass for the green rays, is the same which it is for the red and violet rays. These three orders of rays will therefore suffer an equal refraction, being each of them attracted towards the glass with the force 2; and when this is the case, it appears, from the observations, that the indefinite variety of rays of intermediate colours and shades of colours, which altogether compose solar light, will also be regularly bent from their rectilinear course, constituting what has been termed a planatic refraction.”</p><p>In short, Dr. Blair says, that he “uses more transparent mediums than the common ones; avoids or greatly diminishes the reflections at the surfaces of the mediums; applies fluid mediums more homogeneous than thick flint or crown glass, which at the same time disperse the different coloured rays of light in the same proportion, by which means an image is produced perfectly Achromatic, which is but imperfectly so in Dollond's object glasses made of flint and crown glass combined.</p></div1><div1 part="N" n="ACOUSTICS" org="uniform" sample="complete" type="entry"><head>ACOUSTICS</head><p>, at the end, <hi rend="italics">add,</hi> But this statute was repealed by the 15th of Geo. the 3d, cap. 32.</p></div1><div1 part="N" n="AEROSTATION" org="uniform" sample="complete" type="entry"><head>AEROSTATION</head><p>, pa. 45, col. 2, l. 40, <hi rend="italics">for</hi> 800, <hi rend="italics">read</hi> 680.—l. 46, <hi rend="italics">for</hi> 28 1/3 <hi rend="italics">read</hi> 26.—l. 48, <hi rend="italics">for</hi> balloon <hi rend="italics">read</hi> parachute.—l. 51 and 52, <hi rend="italics">for</hi> 28 1/3 <hi rend="italics">read</hi> 26, and <hi rend="italics">for</hi> 13 <hi rend="italics">read</hi> 12.—l. 55, <hi rend="italics">read</hi> 2 feet 3 inches. <pb n="716"/><cb/></p><p>Pa. 46, col. 2, at the end of the article on <hi rend="italics">Aerostation, add,</hi> See an ingenious and learned treatise on the mathematical and physical principles of Airballoons, by the late Dr. Damen, professor of philosophy and mathematics in the University of Leyden, entitled, Physical and Mathematical Contemplations on Aerostatic Balloons, &c; in 8vo, at Utrecht, 1784.</p><p>Pa. 70, col. 1, l. 4, <hi rend="italics">dele</hi> -√(3 - 1 = 2).-l. 5, at the end <hi rend="italics">add</hi> -√(3 - 1) = 2.</p><p>Pa. 71, col. 1, l. 9, <hi rend="italics">for y</hi><hi rend="sup">2</hi> + 2<hi rend="italics">y</hi> - 7 <hi rend="italics">read</hi> ―(<hi rend="italics">y</hi><hi rend="sup">2</hi> + 2<hi rend="italics">y</hi>-7).</p><p>AFFECTED <hi rend="italics">Equations, add</hi> (from Francis Maseres, Esq.)—“This expression of <hi rend="italics">Affected Equations</hi> seems to require some further explanation. It was introduced by the celebrated Vieta, the great father and restorer of Algebra. He has many expressions peculiar to himself, and which have not been adopted by subsequent Algebraists. Amongst these are the following ones. He calls a set of quantities in continual geometrical proportion, (such as the quantities 1, <hi rend="italics">x, x</hi><hi rend="sup">2</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">4</hi>, <hi rend="italics">x</hi><hi rend="sup">5</hi>, <hi rend="italics">x</hi><hi rend="sup">6</hi> <hi rend="italics">x</hi><hi rend="sup">7</hi>, &c,) a set of <hi rend="italics">scalar</hi> quantities, or <hi rend="italics">magnitudines scalares;</hi> and, when there are several of these <hi rend="italics">scalar</hi> quantities mentioned together, (as in the compound quantity <hi rend="italics">x</hi><hi rend="sup">5</hi> + <hi rend="italics">ax</hi><hi rend="sup">4</hi>-<hi rend="italics">b</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">3</hi>,) he calls the highest quantity, or that which is farthest in the scale of quantities 1, <hi rend="italics">x, x</hi><hi rend="sup">2</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">4</hi>, <hi rend="italics">x</hi><hi rend="sup">5</hi>, <hi rend="italics">x</hi><hi rend="sup">6</hi>, <hi rend="italics">x</hi><hi rend="sup">7</hi>, &c. (to wit, the quantity <hi rend="italics">x</hi><hi rend="sup">5</hi> in the said compound quantity <hi rend="italics">x</hi><hi rend="sup">5</hi> + <hi rend="italics">ax</hi><hi rend="sup">4</hi>-<hi rend="italics">b</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">3</hi>,) <hi rend="italics">the power</hi> of the fundamental quantity <hi rend="italics">x,</hi> or of the second term in the said scale; and he calls the lower scalar quantities which are involved in the second and third terms of the said compound quantity <hi rend="italics">x</hi><hi rend="sup">5</hi>+<hi rend="italics">ax</hi><hi rend="sup">4</hi> -<hi rend="italics">b</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">3</hi>, to wit, the quantities <hi rend="italics">x</hi><hi rend="sup">4</hi> and <hi rend="italics">x</hi><hi rend="sup">3</hi>, (or, in our present language, the inferior powers of <hi rend="italics">x,</hi>) scalar quantities of a <hi rend="italics">parodic</hi> degree to <hi rend="italics">x</hi><hi rend="sup">5</hi>, or the power of the fundamental quantity <hi rend="italics">x.</hi> This word <hi rend="italics">parodic</hi> I take to be derived (though Vieta does not tell us so) from the Greek words <foreign xml:lang="greek">pxrx\</foreign> and <foreign xml:lang="greek">o(do\s</foreign>, which signify <hi rend="italics">near</hi> and <hi rend="italics">a way</hi> or <hi rend="italics">road,</hi> because these inferior scalar quantities <hi rend="italics">x</hi><hi rend="sup">3</hi> and <hi rend="italics">x</hi><hi rend="sup">4</hi> lie <hi rend="italics">in the way</hi> as you pass along in the scale of the aforesaid quantities 1, <hi rend="italics">x, x</hi><hi rend="sup">2</hi>, <hi rend="italics">x</hi><hi rend="sup">3</hi>, <hi rend="italics">x</hi><hi rend="sup">4</hi>, <hi rend="italics">x</hi><hi rend="sup">5</hi>, <hi rend="italics">x</hi><hi rend="sup">6</hi> <hi rend="italics">x</hi><hi rend="sup">7</hi>, &c, from 1 to <hi rend="italics">x</hi><hi rend="sup">5</hi>, which he calls the power of <hi rend="italics">x</hi> in the said compound quantity <hi rend="italics">x</hi><hi rend="sup">5</hi> + <hi rend="italics">ax</hi><hi rend="sup">4</hi>-<hi rend="italics">b</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">3</hi>. These inferiour scalar quantities <hi rend="italics">x</hi><hi rend="sup">3</hi> and <hi rend="italics">x</hi><hi rend="sup">4</hi> are therefore <hi rend="italics">parodic,</hi> or <hi rend="italics">situated in the way to,</hi> or are <hi rend="italics">leading to,</hi> the higher scalar quantity <hi rend="italics">x</hi><hi rend="sup">5</hi>. He then proceeds to define <hi rend="italics">a pure power</hi> and <hi rend="italics">an affected power,</hi> and tells us that <hi rend="italics">a pure power</hi> is a scalar quantity that is not affected with any <hi rend="italics">parodic,</hi> or <hi rend="italics">inferiour</hi> scalar quantity, and that <hi rend="italics">an affected power</hi> is a scalar quantity that is connected by addition, or subtraction with one, or more, <hi rend="italics">inferiour,</hi> or <hi rend="italics">parodic,</hi> scalar quantities, combined with co-efficients that raise them to the same dimension as the power itself, or make them <hi rend="italics">homogeneous</hi> to it, and consequently capable of being added to it, or subtracted from it. Thus <hi rend="italics">x</hi><hi rend="sup">5</hi> alone is a <hi rend="italics">pure power</hi> of <hi rend="italics">x,</hi> namely, its fifth power; and <hi rend="italics">x</hi><hi rend="sup">5</hi> + <hi rend="italics">ax</hi><hi rend="sup">4</hi>-<hi rend="italics">b</hi><hi rend="sup">2</hi><hi rend="italics">x</hi><hi rend="sup">3</hi> is <hi rend="italics">an affected power</hi> of <hi rend="italics">x,</hi> namely, its fifth power <hi rend="italics">affected by,</hi> or <hi rend="italics">connected with,</hi> the two <hi rend="italics">parodic,</hi> or <hi rend="italics">inferiour,</hi> scalar quantities <hi rend="italics">x</hi><hi rend="sup">3</hi> and <hi rend="italics">x</hi><hi rend="sup">4</hi>, which are multiplied into <hi rend="italics">bb</hi> and <hi rend="italics">a,</hi> in order to make <cb/> them <hi rend="italics">homogeneous</hi> to, or <hi rend="italics">of the same dimension with, x</hi> itself, and capable of being added to it or subtracted from it. See Schooten's Edition of Vieta's works, published at Leyden in Holland in the year 1646, pages 3 and 4.</p><p>“This, then, being the meaning of the expression, <hi rend="italics">a pure power</hi> and <hi rend="italics">an affected power,</hi> the meaning of the corresponding expressions of <hi rend="italics">a pure equation</hi> and <hi rend="italics">an affected equation</hi> follows from it of course: <hi rend="italics">a pure equation</hi> signifying an equation in which a pure power of an unknown quantity is declared to be equal to some known quantity; such as the equation ; and <hi rend="italics">an affected equation</hi> signifying an equation in which a power of an unknown quantity affected by, or connected, either by addition or subtraction, with, some inferiour powers of the same unknown quantity, (multiplied into proper co-efficients in order to make them <hi rend="italics">homogeneous</hi> to the said highest power of the said unknown quantity,) is declared to be equal to some known quantity; such as the equation . This I take to be the original meaning of the expression <hi rend="italics">an affected equation.</hi> But, as the language of <hi rend="italics">Vieta</hi> has not been adopted by subsequent writers of Algebra, I should think it would be more convenient to call them by some other name. And, perhaps those of <hi rend="italics">binomial, trinomial, quadrinomial, quinquinomial,</hi> and, in general, that of <hi rend="italics">multinomial</hi> equations, would be as convenient as any. Thus, , might all be called <hi rend="italics">binomial</hi> equations, because they would be equations in which a <hi rend="italics">binomial</hi> quantity, or quantity consisting of two terms that involved the unknown quantity <hi rend="italics">x,</hi> is declared to be equal to a known quantity; and, for a like reason, the equations , might be called <hi rend="italics">trinomial</hi> equations. And the like names might be given to equations of a greater number of terms. Dr. Hutton, I observe, in his excellent new Mathematical and Philosophical Dictionary, just now published, (Feb. 2. 1795,) calls them <hi rend="italics">compound</hi> equations; which is likewise a very proper name for them, and less obscure than that of <hi rend="italics">affected</hi> equations.”</p><p>Pa. 76, col. 1, l. 25, <hi rend="italics">for</hi> √(3 + 1) - √(3 - 1), <hi rend="italics">read</hi> √(3 + 1) - √(3 - 1).</p><p>Pa. 94, col. 2, l. 34, <hi rend="italics">for</hi> Spaniard, <hi rend="italics">read</hi> Portuguese.</p><p>Pa. 95, col. 2, after l. 21, or the end of the paragraph relating to Dr. Barrow, add as follows:—Of these lectures, the 13th deserves the most special notice, being entirely employed upon Equations, delivered in a very curious way. He there treats of the nature and number of their roots, and the limits of their magnitudes, from the description of lines accommodated to each, viz, treating the subject as a branch of the doctrine of maxima and minima, which, in the opinion of some persons, is the right way of considering them, and far preferable to the so much boasted invention of the generation of Equations from each other discovered by Harriot and Descartes.</p><p>Pa. 97, col. 2, after l. 3, <hi rend="italics">add</hi>—Dr. Waring and the Rev. M. Vince, of Cambridge, have both given many <pb n="717"/><cb/> improvements and discoveries in series and in other branches of analysis. Those of Mr. Vince are chiefly contained in the latter volumes of the Philosophical Transactions; where also are several of Dr. Waring's; but the bulk of this gentleman's improvements are contained in his separate publications, particularly the <hi rend="italics">Meditationes Algebraicæ,</hi> published in 1770; the <hi rend="italics">Proprietates Algebraicarum Curvarum,</hi> 1772; and the <hi rend="italics">Meditationes Analyticæ,</hi> 1776; an account of the chief contents of which, a friend has favoured me with, as follows. <hi rend="center"><hi rend="italics">Of Dr. Waring's Meditationes Algebraicæ.</hi></hi></p><p>The first chapter treats of the transformation of algebraical equations into others, of which the roots have given algebraical relation to the roots of the given equations.</p><p>The general resolution of this problem requires the finding the aggregates of each of the values of algebraical functions of the roots of the given equation: for this purpose the author begins with finding the sum of the <hi rend="italics">m</hi><hi rend="sup">th</hi> power of each of the roots of the equation by a series proceeding according to the dimensions of <hi rend="italics">p</hi> the sum of the roots: this series (when continued in infinitum and converges) finds also the sum of any root of the above-mentioned quantities. From this series is deduced the law of the reversion of the series , which finds <hi rend="italics">x</hi> in terms of <hi rend="italics">y;</hi> and also the law of a series, which expresses the greatest or least roots, and their powers or roots of a given algebraical equation, and which may be applied whether that root is possible or impossible, if the root be much greater or less than each of the remaining ones. All the powers and roots of this series, when continued in infinitum, observe the same law.</p><p>On this subject are further added some elegant theorems; of which, one finds the sum of all quantities of this kind <foreign xml:lang="greek">a</foreign><hi rend="sup">a</hi><foreign xml:lang="greek">b</foreign><hi rend="sup">b</hi><foreign xml:lang="greek">g</foreign><hi rend="sup">c</hi>, &c; where <foreign xml:lang="greek">a, b, g</foreign>, &c, denote the roots of the given equation. This has been since published by the celebrated mathematician Mr. le Grange in the Academy of Sciences at Paris.</p><p>There is also added a method of considerable utility in these matters; viz, the assuming equations whose roots are known, and thence deducing the coefficients of the equations sought: and also from the terms of an inferior equation deducing the terms of a superior.</p><p>The second chapter principally treats of the limits and number of impossible and affirmative and negative roots of algebraical equations.</p><p>Some new properties are added, of the limiting equations resulting from multiplying the successive terms of the given equation into an arithmetical series; and a method of finding limits between each of the roots of a given equation, since published in the Berlin Acts, and also some new methods of finding equations whose roots are limits between the roots of other equations. In theor. 4 and 5 are contained quantities which are always greater than certain others, when they are all possible; from whence may be deduced Newton's and several other rules for finding the number of impossible roots: these rules may be rendered somewhat more general by multiplying the given equations into others, whose roots are all possible, and finding whether im- <cb/> possible roots may be deduced by the rule in the resulting equation, which cannot from it be discovered in the given one. A rule is given, deduced from each successive four terms of the given equation, and consequently much more general than rules deduced from each successive three terms. The former always discovers the true number of impossible roots contained in quadratic and cubic equations, the latter in quadratic only. There is also a rule given for finding the number of impossible roots from an equation, of which the roots are the squares, &c, of the roots of a given equation; and a second from an equation of which the roots are the squares of the differences of the roots of a given equation; and a third rule for finding an equation, of which the root is ; if be the given equation, &c, these latter resolutions always discover the true number of impossible roots contained in cubic, biquadratic and sursolid equations; and also whether or not any impossible roots are contained in any given equation; and also from the last term whether the number of impossible roots contained be 2, 6, 10, &c, or 0, 4, 8, &c. The principle of a 4th rule is given by finding when two roots once, twice, thrice, &c, or four, &c, roots become equal. From a method given of finding the number of impossible roots contained in an equation involving only one unknown quantity, is deduced a method of discovering limits between which are contained any number of impossible roots in an equation involving two or more unknown quantities. From the number of impossible, affirmative and negative roots contained in a given equation, is delivered a method of finding the number of impossible, &c roots contained in an equation of which the roots have a given algebraical relation to the roots of the given equation.</p><p>The principles are subjoined of finding the number of affirmative and negative roots contained in an algebraical equation: but this necessarily supposes a method of finding the number of its impossible roots known. It is demonstrated, that if the equation be multiplied by <hi rend="italics">x</hi> - <hi rend="italics">a,</hi> then every change of signs in the given, will have one, or three, or five, &c in the resulting equation; and if it be multiplied by <hi rend="italics">x</hi> + <hi rend="italics">a,</hi> then every continuation from + to + or - <*>o -, will produce one, or three, or five, &c such continuations in the resulting, whence every equation will contain at least so many changes of signs in its successive terms as there are affirmative roots, and so many continued progresses from + to + and - to -, as there are negative. In a biquadratic , of which two roots are impossible, and <hi rend="italics">s</hi> an affirmative quantity, then it is demonstrated that the two possible ones will be both negative or both affirmative, according as <hi rend="italics">p</hi><hi rend="sup">3</hi> - 4<hi rend="italics">pq</hi> + 8<hi rend="italics">r</hi> is an affirmative or negative quantity, if the signs of the coefficients, <hi rend="italics">p, q, r, s</hi> are neither all affirmative, nor alternately - and +. The number of impossible and affirmative and negative roots contained in the equation is likewise given, &c. If , then the content of all the values <pb n="718"/><cb/> of the quantity <hi rend="italics">w</hi> will be to the content of all the values of the quantity <hi rend="italics">v</hi> :: ± <hi rend="italics">l</hi><hi rend="sup">n</hi> : <hi rend="italics">h</hi><hi rend="sup">m</hi>, from whence are deduced some properties of parabolic curves. <hi rend="italics">Ex. gr.</hi> Let the equation expressing the relation between the absciss <hi rend="italics">x</hi> and ordinate <hi rend="italics">y</hi> be . Then will the content under the (<hi rend="italics">n</hi> - 1) greatest ordinates be to the square of the content of all the distances between any two points in which the absciss cuts the curve :: <hi rend="italics">a</hi><hi rend="sup">n-1</hi> : <hi rend="italics">n</hi><hi rend="sup">n</hi>-2. The quotient of the content of all the sines divided by the content of all the cosines to the points in which the absciss cuts the curve, will be to the content of all the abovementioned greatest ordinates :: <hi rend="italics">n</hi><hi rend="sup">n</hi><hi rend="italics">a</hi> : 1. Similar propositions are deduced concerning the ordinates to the points of contrary flexure, &c.</p><p>The third chapter is versant, concerning, 1st finding the roots of equations or irrational quantities, which have given relations to each other: this is performed by substitution or division and finding the common divisors of the quantities resulting; and 2d concerning more <hi rend="italics">(n)</hi> equations containing a less number <hi rend="italics">(m)</hi> of supposed unknown quantities, which consequently require <hi rend="italics">n</hi>-<hi rend="italics">m</hi> equations, since named equations of condition; these are likewise deduced from the method of finding common divisors. 3dly, Concerning the resolution of equations; in this case is given, 1. The reduction or resolution of some recurring equations. 2. Some properties of the roots of the equation . 3. Resolution of a biquadratic , by reducing it to an equation . 4. A resolution of the biquadratic by adding (<hi rend="italics">p</hi><hi rend="sup">2</hi> + 2<hi rend="italics">n</hi>) <hi rend="italics">x</hi><hi rend="sup">2</hi> + 2<hi rend="italics">pnx</hi> + <hi rend="italics">n</hi><hi rend="sup">2</hi> to both sides of the equation, so as to complete the square; and the deducing that the values of <hi rend="italics">n</hi> are (<foreign xml:lang="greek">ab</foreign> + <foreign xml:lang="greek">gd</foreign>)/2, (<foreign xml:lang="greek">ag</foreign> + <foreign xml:lang="greek">bd</foreign>)/2, (<foreign xml:lang="greek">ad</foreign> + <foreign xml:lang="greek">bg</foreign>)/2; the values of √ (<hi rend="italics">q</hi> + <hi rend="italics">p</hi><hi rend="sup">2</hi> + 2<hi rend="italics">n</hi>) are (<foreign xml:lang="greek">a</foreign> + <foreign xml:lang="greek">b</foreign> - <foreign xml:lang="greek">g</foreign> - <foreign xml:lang="greek">d</foreign>)/2, (<foreign xml:lang="greek">a</foreign> + <foreign xml:lang="greek">g</foreign> - <foreign xml:lang="greek">b</foreign> - <foreign xml:lang="greek">d</foreign>)/2, &c, and the values of √ (<hi rend="italics">s</hi> + <hi rend="italics">n</hi><hi rend="sup">2</hi>) are (<foreign xml:lang="greek">ab</foreign> - <foreign xml:lang="greek">gd</foreign>)/2, (<foreign xml:lang="greek">ag</foreign> - <foreign xml:lang="greek">bd</foreign>)/2, &c; if <foreign xml:lang="greek">a, b, g, d</foreign>, are the roots of the given equation. 5. A resolution of equations as general as any yet discovered, viz, the assuming ; and exterminating the irrational quantities, viz, from assuming are deduced different resolutions of cubic; from different resolutions of biquadric; from the equations , are deduced De Moivre's equation, and several others of new formula not before delivered. 6. The resolution , first given by Euler, shewn to be a very particular; but this is rendered here much more general by assuming a more general resolution. 7. The resolution and reduction of equations from exterminating irrational quantities. 8. Reduction of some equations, when they are deduced from others by reducing them to the <cb/> original equations. 9. The finding a quantity, which multiplied into a given irrational will produce a rational quantity, and thence deducing from a given equation involving irrational quantities the dimensions to which the equation freed from them will ascend. 10. Let P = a series either ascending or descending according to the dimensions of <hi rend="italics">x,</hi> from thence is deduced the sum of a series consisting of its alternate terms, or terms at <hi rend="italics">(n)</hi> distance from each other. 11. It is proved, that Cardan's resolution of a cubic, is a resolution of an equation of 9 dimensions or three different cubics: similar principles are applied to some other equations. 12. General principles are given for the deducing the function of the roots of the given, which constitute the coefficients or roots of the transformed equation. E. g. Let a cubic equation , thence is shewn the function of the roots of <hi rend="italics">x,</hi> which constitute <hi rend="italics">z,</hi> and further the cases of the cubic, which are resolvable by the transformed equation, whose root is <hi rend="italics">z:</hi> the same principles are applied to biquadratics. 13. The correspondent impossible roots of a given irrational quantity are deduced; and also the different roots of a given resolution. 14. The biquadratic of the formula is distinguished into two quadratic equations involving only possible quantities, and thence every algebraic equation is proved to consist of simple and quadratic divisors involving only possible quantities. 15. A method is delivered of transforming irrational quantities into others; but it is cautioned, that in reduction and transformation correspondent roots should be used, otherwise it is probable that we shall fall into errors, of which examples are given. 16. The convergency of a root found by the common method of approximations is given; and it is discovered that the convergency principally depends on the quantity assumed for the root being much more near to one root than to any other; and independent of it, not on how near it is to a root.</p><p>The fourth chapter is principally conversant concerning more algebraical equations and their reductions to one. 1. It gives the law of the resolution of any number of simple equations; and the reduction of <hi rend="italics">n</hi> simple equations to <hi rend="italics">n</hi> - 1 by means of others. 2. The method of reducing more (<hi rend="italics">n</hi>) equations into one so as to exterminate <hi rend="italics">n</hi> - 1 unknown quantities by the method of common divisors, and further delivers the principles of investigating the roots or values of the unknown quantities, which result from this, or, which is much the same, from the common method of Erasmus Bartholinus, and which are not contained in the given equations. 3. If two algebraical equations of <hi rend="italics">n</hi> and <hi rend="italics">m</hi> dimensions of the unknown quantities <hi rend="italics">x</hi> and <hi rend="italics">y</hi> are reduced to one so as to exterminate one of the unknown quantities, the principles are given of finding the dimensions to which the other will ascend: if it ascends to <hi rend="italics">n</hi> X <hi rend="italics">m</hi> dimensions; then the sum of the roots depends on the terms of <hi rend="italics">n</hi> and <hi rend="italics">n</hi> - 1 dimensions in the one, and <hi rend="italics">m</hi> and <hi rend="italics">m</hi> - 1 in the other, and similarly of the products of every two; &c. From this principle are deduced several properties of algebraical curves. <pb n="719"/><cb/> The same principles are applied to more equations involving more unknown quantities. 4. Some two equations of given formulæ are reduced to one so as to exterminate one unknown quantity. 5. Two equations are likewise reduced to one so as to exterminate unknown quantities by means of insinite series. 6. A method of finding whether some equations contain the same roots of the unknown quantities as others. 7. From the correspondent roots of the unknown quantities in given equations are found the constitution of their coefficients; and from thence the aggregates of the functions of the roots of two or more equations. 8. Some things are given concerning the transformations of more equations than one, of their impossible roots, of their roots which have a given relation to each other. 9. Some reductions and resolutions of more equations involving more unknown quantities. 10. If two equations similarly involve two unknown quantities <hi rend="italics">x</hi> and <hi rend="italics">y;</hi> then the equation of which the root is <hi rend="italics">x</hi> or <hi rend="italics">y</hi> is demonstrated to have twice the dimensions of the equation whose root is any rational function of <hi rend="italics">x</hi> + <hi rend="italics">y</hi> or <hi rend="italics">x</hi><hi rend="sup">2</hi> + <hi rend="italics">y</hi><hi rend="sup">2</hi> or any rational recurring function of <hi rend="italics">x</hi> and <hi rend="italics">y;</hi> and if for <hi rend="italics">y</hi> be substituted - <hi rend="italics">y;</hi> then in the equation whose root is the resulting quantity the dimensions will be the same as in the equations whose root is <hi rend="italics">x</hi> or <hi rend="italics">y,</hi> but its formula will be of half the number of dimensions. The same principles are applied to more equations similarly involving more unknown quantities. 11. If there are two equations involving two unknown quantities, one deduced from the other, by some substitutions investigated from equations similarly involving two unknown quantities; then the equation whose root is one of the unknown quantities will be recurring. 12. Let A and B be functions of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> a method is given of finding, whether A is a function of B. 13. Methods of approximations to the roots of equations when they are unequal, or two or more nearly equal, possible or impossible; and also some remarks on the increments or decrements of the roots, in passing from one equation to others of the same number of dimensions are given.</p><p>The fifth chapter treats of rational and integral values of the unknown quantities of given equations.</p><p>1. It finds the rational and integral simple, quadratic, &c divisors (by a method different to Waessaner's) of a given equation, which involves one or more unknown quantities. 2. If two equations involve two unknown quantities <hi rend="italics">x</hi> and <hi rend="italics">y;</hi> the same irrationality, which is contained in <hi rend="italics">x</hi> will likewise be contained in its correspondent value of <hi rend="italics">y,</hi> unless two or more values of the quantity (<hi rend="italics">x</hi> or <hi rend="italics">y</hi>) are equal, &c. 3. A method is given of finding integral correspondent values of the unknown quantities of two or more equations involving as many unknown quantities. 4. A method is also delivered of deducing when a given equation can be resolved by means of square, cube, &c roots; and when by similar methods it can be reduced to equations of 1/2, 1/4, &c, its dimensions. 5. A method is given of finding a quantity or number, in which are contained all the divisors of any given rational or integral quantities. 6. A method different from Schooten's, Newton's, and Euler's, of extracting the root of a binomial surd <hi rend="italics">a</hi> + √<hi rend="italics">b</hi> is given, and the principle demonstrated on <cb/> which all the rules are founded given by Schooten, viz, the multiplying the binomial surd so that the <hi rend="italics">n</hi><hi rend="sup">th</hi> root of A<hi rend="sup">2</hi> - B can be extracted, where A + √B is the resulting surd; and it is further proved that multiplying the given surd <hi rend="italics">a</hi> + √<hi rend="italics">b</hi> into 2<hi rend="sup">n</hi> will render Newton's resolution as general as the others; and lastly the extraction of the (<hi rend="italics">m</hi><hi rend="sup">th</hi>) root of the quantity A + B√<hi rend="sup">n</hi><hi rend="italics">p</hi> + C√<hi rend="sup">n</hi>(<hi rend="italics">p</hi><hi rend="sup">2</hi>) + ... + √<hi rend="sup">n</hi>(<hi rend="italics">p</hi><hi rend="sup">n-1</hi>) is given. 7. The law of Dr. Wallis's approximations in terms of the successive quotients, as also of continual fractions is deduced. 8. A method of deducing the integral values of each of the unknown quantities <hi rend="italics">x, y, z, v,</hi> &c, contained in the equation in terms of quantities, for which may be assumed any whole numbers. 9. Two or more equations are reduced to one, so as to exterminate unknown quantities; and if the unknown quantities of the resulting equations be integral or fractional, then the unknown quantities of the given equations will also be integral or fractional. 10. Principles are delivered of deducing equations of which the unknown quantities admit of correspondent and known integral or rational values. 11. Correspondent integral or rational values of the unknown quantities in several equations are given, and from some values of the abovementioned kind given, are deduced others. 12. A method of denoting any numbers either by fours, fives, fixes, &c, and their powers; and similar properties deduced as in decimal arithmetic. 13. It is demonstrated that the sum of the divisors of the number 1. 2. 3 ... <hi rend="italics">x</hi> = N has to N a greater ratio than the sum of the divisors of any number L less than N has to L; and some other similar properties. 14. In the Philosophical Transactions are given properties similar to Mr. Euler's of the sum of divisors of the natural numbers, and some others. 15. Let , where <hi rend="italics">a, b, r, p</hi> and <hi rend="italics">q</hi> are whole numbers, then N2<hi rend="italics">m</hi> + 1 and N2<hi rend="italics">m</hi> + 2 can be compounded by (<hi rend="italics">m</hi> + 1) different ways of the quantities <hi rend="italics">p</hi><hi rend="sup">2</hi> + <hi rend="italics">rq</hi><hi rend="sup">2</hi>; the different ways were first given in the Medit. 16. Every number consists of 1, 2, 3 or 4 squares, and of 1, 2, 3, 4, .. 9 cubes, and therefore if a number N is equal to 3 squares or 8 cubes, the problem may not be possible. 17. Let <hi rend="italics">x</hi> and <hi rend="italics">z</hi> be any whole numbers, and <hi rend="italics">a</hi> and <hi rend="italics">b</hi> numbers prime to each other, then <hi rend="italics">ax</hi> + <hi rend="italics">bz</hi> can constitute any number, which exceeds <hi rend="italics">a</hi> X <hi rend="italics">b</hi> - <hi rend="italics">a</hi> - <hi rend="italics">b.</hi> 18. Let <hi rend="italics">r</hi> the greatest common divisor of <hi rend="italics">m</hi> and <hi rend="italics">n</hi> - 1, where <hi rend="italics">n</hi> is a prime number; the number of remainders from the division of the number 1<hi rend="sup">m</hi>, 2<hi rend="sup">m</hi>, 3<hi rend="sup">m</hi>, &c, in infinitum by <hi rend="italics">n</hi> will be (<hi rend="italics">n</hi> - 1)/<hi rend="italics">r</hi> + 1: from which are deduced several propositions. 19. Sir John Wilson's property delivered and demonstrated, viz, 1. 2. 3 ... <hi rend="italics">n</hi> - 1 + 1 will be divisible by <hi rend="italics">n,</hi> if <hi rend="italics">n</hi> be a prime number. 20. The sum of the powers 1<hi rend="sup">r</hi> + 2<hi rend="sup">r</hi> + 3<hi rend="sup">r</hi> + ... <hi rend="italics">x</hi><hi rend="sup">r</hi> are found divisible by <hi rend="italics">x.</hi> ―(<hi rend="italics">x</hi> + 1), if <hi rend="italics">r</hi> be a whole number; from whence is deduced an elegant property of all parabolas correspondent to the property of Archimedes of the inscribed triangles in a conical parabola. 21. Some properties of exponential equations; several other new properties of algebraical quantities and equations are given in these Meditations. They were sent to the Royal Socicty in 1757, and since published in the years 1760, 62, and 69, <pb n="720"/><cb/> <hi rend="center"><hi rend="italics">Properties of Algebraical Carves.</hi></hi></p><p>The equation expressing the relation between the absciss and its correspondent ordinates of a curve is transformed into another which expresses the relation between different abscissæ and their ordinates, from which is deduced, that there may be <hi rend="italics">n</hi> and not more different diameters in a curve of <hi rend="italics">n</hi> - 1 order, which cuts its ordinates in a given angle; and likewise that a diameter can have no more than <hi rend="italics">n</hi> - 1 different inclinations of its ordinates, unless the diameter be a general one. 2. The formula of the equations to curves, all whose diameters are parallel, or cut each other in a given point, or which have a general diameter to which the lines any how inclined are ordinates. 3. It is proved that there cannot be more than <hi rend="italics">n</hi>/<hi rend="italics">m</hi> different inclinations of parallel ordinates, which cut the curve in <hi rend="italics">n</hi> - <hi rend="italics">m</hi> points only, possible or impossible. 4. Something is added concerning diameters, which cut their ordinates on both sides into equal parts. 5. It is demonstrated that there are curves of any number of odd orders, that cut a right line in 2, 4, 6, &c, points only; and of any number of even orders that cut a right line in 3, 5, 7, &c points; and consequently that the order of the curve cannot be denounced from the number of points, in which it cuts a right line. 6. The principles are delivered of finding the asymptotes, parabolical legs, ovals, points, &c, of a curve, of which the equation marking the relation between the absciss and its ordinates is given; and also given the number of asymptotes, parabolical legs of different kinds, ovals, points of different kinds, the least order of a curve, which receives them, is deduced. 7. An equation expressing the relation between an absciss and its ordinates, is transformed into an equation expressing the relation between the distances from two or more points, the latter may be varied an infinite number of ways; and thence are deduced some properties. Many resolutions of this kind are only resolutions of a particular case contained in it; and consequently can never be deduced from any general reasoning; they are often deduced from some particular cases, which are known to answer several conditions of the problem. Transformations of a given curve into others by substitutions, and properties of the loci of some points are deduced, from which Mr. Cotes's property of algebraical curves, and others of a similar and somewhat different nature are derived. 8. Let a curve of <hi rend="italics">n</hi> dimensions have <hi rend="italics">n</hi> asymptotes, then the content of the <hi rend="italics">n</hi> abscissæ will be to the content of the <hi rend="italics">n</hi> ordinates, in the same ratio in the curve and asymptotes, the sum of their <hi rend="italics">(n)</hi> subnormals to ordinates perpendicular to their abscissæ will be equal to the curve and the asymptotes; and they will have the same central and diametrial curves. 9. Some propositions are added concerning the construction of equations, and some equations are constructed from the principles of Slusius.—If two curves of <hi rend="italics">n</hi> and <hi rend="italics">m</hi> dimensions have a common asymptote; or the terms of the equations to the curves of the greatest dimensions have a common divisor, then the curves cannot intersect each other in <hi rend="italics">n</hi> X <hi rend="italics">m</hi> points, possible or impossible. If the two curves have a common general centre, and intersect each other in <hi rend="italics">n</hi> X <hi rend="italics">m</hi> points, then the sum of the <cb/> affirmative abscissæ &c to those points will be equal to the sum of the negative; and the sum of the <hi rend="italics">n</hi> subnormals to a curve which has a general centre will be proportional to the distance from that centre. 10. Something is added on the description of curves. 11. No curve which has an hyperbolical leg of the conical kind can in general be squared. 12. It is demonstrated that no oval sigure, which does not intersect itself in a given point, can in general be expressed in finite algebraical terms. 13. Given an algebraical equation, and similarly equations expressing a relation between <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> &c; and also a fluxional quantity which is an algebraical function <hi rend="italics">(z)</hi> of <hi rend="italics">x</hi> and <hi rend="italics">y</hi> and their fluxions; a method is given of deducing an equation whose root is <hi rend="italics">z;</hi> and thence some properties of curves. 14. Properties similar to the subsequent of conic sections, are extended to curves of superior orders, viz, if lines be drawn from given points in them in given angles to four lines inscribed in the conic section, then will the rectangle under two of those lines be to the rectangle under the other two in a given ratio. Several properties are added, which follow from the application of algebraical propositions invented in the Medit. Algebr. to curve lines.</p><p>The second chapter treats of curvoids and epicurvoids, or curves generated by the rotation of given curves on right lines or curves, and gives a method of rectifying and squaring them; and from the radii of curvature of the generating curves being given, it deduces the length and radius of curvature of the curve generated at the correspondent point; it also asserts that from them may be deduced the construction of the fluxional equations of the different orders.</p><p>The third chapter treats of algebraical solids. 1. It deduces the equation to every section of a solid generated by the rotation of a curve round its axis; and from thence the different sections generated by the rotation of conic sections round their axis. 2. The equation to solids contains the relation between the two abscissæ and their ordinates, and the order of the solid may be distinguished according to the dimensions of the equation; or the solid may be defined by two equations expressing the relation between the three abovementioned quantities, and a fourth which may be the axis of the section: there is further given a method of deducing the equation to any section of these solids, and from it the equation to the curve projected on a plane by a given curve. 3. A method of deducing the projection of a curve or solid on each other. 4. If the equation be <hi rend="italics">x</hi> - <hi rend="italics">a</hi> = 0, (<hi rend="italics">x</hi> being the distance from a given point) then it may denote the periphery of a circle if one plane, or the surface of a globe if it refers to a solid. 5. Let <hi rend="italics">x</hi> and <hi rend="italics">y</hi> denote the distances from two respective points, then an equation expressing the relation between <hi rend="italics">x</hi> and <hi rend="italics">y</hi> designs the periphery of a curve, if contained in the same plane, or the surface of a solid generated by the rotation of a curve round its axis, passing through the two given points, if a solid. 6. An equation expressing the relation between lines drawn from three or more points may denote an equation to a solid. 7. If <hi rend="italics">x, z</hi> and <hi rend="italics">y</hi> denote the two abscisses and correspondent ordinates to a solid, and the terms of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> or <hi rend="italics">x</hi> and <hi rend="italics">z,</hi> or <hi rend="italics">y</hi> and <hi rend="italics">z;</hi> or <hi rend="italics">x, z</hi> and <hi rend="italics">y</hi> be similarly involved; then may the solid be divided into two <pb n="721"/><cb/> or six similar and equal parts; and if no unequal power of <hi rend="italics">x</hi> or <hi rend="italics">y</hi> or <hi rend="italics">z;</hi> or <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> &c; or <hi rend="italics">x, y</hi> and <hi rend="italics">z</hi> be contained in the equation, then the curve may further be divided in general into twice, four or six times the preceding number of equal parts. 7. Curves of double curvature are designed by two equations expressing the relation between two abscissæ and correspondent ordinates, or between lines drawn from three or more points; similar properties may be deduced from these as from the equations to curves.</p><p>Chapter the 4th treats of the maxims and minims of polygons inscribed and circumscribed about curves, and thence deduces certain quantities equal to each other, when maxims and minims are contained at every point of the curve: it further contains several properties of conic sections. 1. If any rectilinear figure circumscribes an ellipse, the content under the alternate segments of the line made by the points in which the line touches the ellipse will be equal. 2. If a right line cuts a conic section, and the parts of the line without the conic section on both sides are equal; and any rectilinear figure, which begins and ends at the bounds of the abovementioned line, be described round the conic section, then the contents under the alternate segments of the circumscribing lines as divided in the points of contact will be equal. 3. If two polygons be circumscribed about an ellipse, and the sides are cut by the points of contacts in the same ratios in the one as in the other; then will the areas of the two polygons be equal. 4. If two lines cut a conic section proportionally, i. e. they are divided by the conic section in the same ratio in the one as in the other, and if polygons be described round the conic section, terminated at the ends of those lines, of which the sides are divided by the points of contact in the same ratio in the one as in the other, then will the area of the two polygons be equal, as likewise the curvilinear area. 5. If all the sides of two polygons inscribed in an ellipse make the two angles at the same point equal, and two polygons of this kind be inscribed in the curve, then will the sum of the sides of the one polygon be equal to the sum of the sides of the other. Several other similar properties are added, as also properties of solids generated by the rotation of a conic section round its axis; to which I shall mention the three or four following. 1. The diagonals of a parallelogram circumscribing an ellipse or hyperbola will be conjugate diameters. 2. The sections of a solid generated by the rotation of a conic section round its axis, which pass through its focus, will have that point for the focus of all the sections. 3. If 4 perpendiculars be drawn from any point in an hyperbola to its periphery; and two lines from the same point to the asymptotes and the ordinates from the 4 points of the curve and the 2 of the asymptotes be drawn to the absciss; then will the sum of the resulting abscissæ to the former be double to the sum of the abscissæ to the latter. 4. If an arc of the periphery of a circle be divided into <hi rend="italics">n</hi> equal parts, <hi rend="italics">a,</hi> 2<hi rend="italics">a,</hi> 3<hi rend="italics">a,</hi> &c, and <hi rend="italics">p</hi> = chord of the arc 180 - <hi rend="italics">na,</hi> and <foreign xml:lang="greek">a</foreign> and <foreign xml:lang="greek">b</foreign> be the roots of the quadratic and radius 1: then will <foreign xml:lang="greek">a</foreign><hi rend="sup">n</hi>+<foreign xml:lang="greek">b</foreign><hi rend="sup">n</hi>=chord of the arc 180 - <hi rend="italics">na,</hi> from whence may be deduced the divisors of the quantity <hi rend="italics">x</hi><hi rend="sup">2n</hi> - <hi rend="italics">Ax</hi><hi rend="sup">n</hi> + 1; and also the equation whose roots are the distances of a point in the circle from those points of equal division, and further may be deduced <cb/> the sum of all the values of any algebraical function of those lines.</p><p>Most of the properties of circles given by Archimedes are extended to conic sections, and some of the algebraical and geometrical properties of Pappus are rendered more general; and the principles invented applied to many other cases. In the first edition of this book published in 1762 were nearly enumerated the lines of the fourth order on the same principles as Newton's enumeration of lines of the third order; but this has since been rejected by the author as not sufficiently distinguishing the curve, and as being of no great utility. <hi rend="center"><hi rend="italics">Meditationes Analyticæ.</hi></hi></p><p>The first <hi rend="italics">chapter</hi> treats of finding the fluxion of a fluent, when the quantity or fluent is considered as generated by motion; or the parts from the whole when the whole or quantity is considered as consisting of innumerable parts. It further gives the law of a series, which expresses the fluxion of an exponential of any order.</p><p><hi rend="italics">Chapter</hi> 2, is versant about the fluents of fluxions. 1. It finds the general fluent of a fluxion P<hi rend="italics">x</hi><hi rend="sup">.</hi>, when P is any algebraical function of <hi rend="italics">x</hi> however irrational but not exponential; for which intent it investigates the common divisors of any two quantities contained under the different vincula; and thence the common divisors of the resulting divisors, and so on; and likewise all the equal divisors contained in any of the abovementioned quantities; whence it so reduces the quantity P, that no equal nor common divisors may be contained in any of the resulting quantities under the different vincula; and from the common method deduces the terms of a series to the number, which the series is shewn to consist of, when it does not proceed in infinitum. 2. It demonstrates, that if the dimensions of <hi rend="italics">x</hi> in the denominator of P exceed its dimensions in the numerator by 1, then the fluent cannot be expressed in finite terms; and also if one factor of P be (A ± (A<hi rend="sup">2</hi> + <hi rend="italics">a</hi>)<hi rend="sup">1/2</hi>)<hi rend="sup"><foreign xml:lang="greek">l</foreign></hi>, where <hi rend="italics">a</hi> is an invariable quantity, and in some other cases the substitution required must be somewhat different. 3. The fluents of some fluential and exponential fluxions, or fluxions involving fluents and exponential quantities, are given. 4. A general method of discovering whether the fluent of any fluxion of any order involving one, two or more variable quantities, and their fluxions, can be expressed in terms of the variable quantities and their fluxions. 5. The correction of fluents of all orders, and thence the fluent contained between any values of the variable quantities and their fluxions, is given; in these corrections the same roots of the irrational quantities are to be used in the correction as in the fluent. 6. From the transformation of equations and the principles before delivered, are deduced fluents equal to each other. 7. Some exponential quantities given which continually change from possibility to impossibility, and from impossibility to possibility. 8. Is a method of finding whether the fluent of any fluxion contained between any limits are finite or not. 9. The sum of the fluents of a fluxion which is. an algebraical function of the letter <hi rend="italics">x</hi> multiplied into <hi rend="italics">x</hi><hi rend="sup">.</hi> can always be expressed by finite terms, circular arcs and logarithms, the extraction of the roots of equations being granted. <pb n="722"/><cb/> 10. Some fluxions involving irrational quantities are reduced to others, in which no irrationality is contained. 11. The general principles of deducing whether the fluent of a given fluxion can generally be expressed by finite algebraical terms, their circular arcs and logarithms. 12. Some equal correspondent fluents are found by substitutions deduced from equations in which two variable quantities are similarly involved. 13. Some necessary corrections are given of finding the fluents of all the fluxions of the formula <hi rend="italics">x</hi><hi rend="sup"><hi rend="italics">pn</hi> ± <foreign xml:lang="greek">s</foreign><hi rend="italics">n</hi> - 1</hi> <hi rend="italics">x</hi><hi rend="sup">.</hi> X R<hi rend="sup"><hi rend="italics">m</hi> ± <foreign xml:lang="greek">l</foreign></hi> X S<hi rend="sup"><hi rend="italics">o</hi> ± <foreign xml:lang="greek">m</foreign></hi> X T<hi rend="sup"><hi rend="italics">t</hi> X <foreign xml:lang="greek">n</foreign></hi> X &c, (where <foreign xml:lang="greek">s, l, m, n</foreign>, &c denote any whole numbers, and from <foreign xml:lang="greek">a</foreign> + <foreign xml:lang="greek">b</foreign> + <foreign xml:lang="greek">g</foreign> + &c, independent fluents; but perhaps not from <foreign xml:lang="greek">a</foreign> + <foreign xml:lang="greek">b</foreign> + <foreign xml:lang="greek">g</foreign> + &c fluents, which have different values of the quantities, <foreign xml:lang="greek">s, l, m, n</foreign>, &c. 14. The number of independent fluents of the formulæ <hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">q</foreign> + <foreign xml:lang="greek">a</foreign><hi rend="italics">n</hi> + <foreign xml:lang="greek">b</foreign><hi rend="italics">m</hi></hi> X (<hi rend="italics">a</hi> + <hi rend="italics">bx<hi rend="sup">n</hi></hi> + <hi rend="italics">cx<hi rend="sup">m</hi></hi>)<hi rend="sup"><foreign xml:lang="greek">l</foreign> + <foreign xml:lang="greek">p</foreign></hi> X <hi rend="italics">x</hi><hi rend="sup">.</hi>, where <foreign xml:lang="greek">a, b</foreign> and <foreign xml:lang="greek">p</foreign> denote whole affirmative numbers, &c; and the number of independent fluents of the formulæ X<hi rend="sup">.</hi>∫Y<hi rend="italics">x</hi><hi rend="sup">.</hi>, where X<hi rend="sup">.</hi> is a fluxion of which the fluent can be sound, from which can be deduced all of the same formula, is immediately known from the number of independent fluents of the formula Y<hi rend="italics">x</hi><hi rend="sup">.</hi> and XY<hi rend="italics">x</hi><hi rend="sup">.</hi> which determine all of those formulæ. 15. Let , and from some fluents of the fluxions of the formulæ <hi rend="italics">p</hi> X <hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">m</foreign><hi rend="italics">n</hi> - 1</hi> <hi rend="italics">x</hi><hi rend="sup">.</hi>, where <foreign xml:lang="greek">m</foreign> is a whole affirmative number, are determined the remaining ones of the same formula. 16. Something is added concerning finding the value of a fraction, when both the numerator and denominator vanish; and lastly from the fluents of some fluxions being given, the method of deducing the fluents of others.</p><p><hi rend="italics">Chapter</hi> 3, principally treats of algebraical and fluxional equations. 1. It gives the method of transforming two or more fluxional equations into one so as to exterminate one or more variable quantities and their fluxions, and finds the order of the resulting equation. 2. It reduces some fluxional equations into more. 3. A method of reducing fluxional equations involving fluents so as to exterminate the fluents. 3. Some cases are given, in which the two variable quantities contained in a given equation are expressed in terms of a third. 4. Given an algebraical equation expressing the relation between <hi rend="italics">x</hi> and <hi rend="italics">y;</hi> a method is given of finding the fluent of <hi rend="italics">yx</hi><hi rend="sup">.</hi><hi rend="sup">n</hi> or other fluxions in finite terms of <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> if they can be expressed by such; or else by infinite series; this was first taught in the Philosophical Transactions in the year 1764. 5. Something is added concerning the correction of fluxional equations. 6. A method of investigating, whether a given equation is the general fluent of a given fluxional equation. 7. The method of deducing, whether a given equation is a particular or general fluent of a given fluxional equation. In both by substituting for the fluxions their values deduced from the fluential equation their values &c in the <cb/> fluxional, the fluxional must result = 0; and in the general fluent there must be contained so many invariable quantities to be assumed at will independently as is the order of the fluent; and in both all the variable quantities must necessarily be variable, and no function of them vanish out of the fluxional equation from the substitution; for then all the conditions of the fluxional equation are answered by the fluential. 8. An investigation, when fluxional equations are integrable. 9. From some fluents are deduced others, <hi rend="italics">e. g.</hi> if the area between any two ordinates to one abscissa can in general be found, then the area between any two ordinates of any other abscissa can be found &c. 10. From given fluxional equations and the fluents of some fluxions are deduced the fluents of many others. 11. The fluent of the first order of a fluxional equation of the <hi rend="italics">n</hi>th order will have (<hi rend="italics">n</hi>) different values and <hi rend="italics">n</hi> different multipliers; and the fluent of the second order <hi rend="italics">n</hi>.(<hi rend="italics">n</hi> - 1)/2 different values, &c. 12. Let <foreign xml:lang="greek">a</foreign> = 0, <foreign xml:lang="greek">b</foreign> = 0, <foreign xml:lang="greek">g</foreign> = 0, &c, (<hi rend="italics">n</hi>) general fluents of the fluxional equation, <foreign xml:lang="greek">l</foreign> = 0, then will any function of the fluents <foreign xml:lang="greek">a, b, g</foreign>, &c be a fluent of the same fluxional equation <foreign xml:lang="greek">l</foreign> = 0. 13. From assuming equations, which contain only simple powers of the invariable quantities to be assumed at will, may easily be deduced fluxional equations, of which the general resolutions are known: 2. From assuming the values of any variable quantities and substituting then their fluxions for the variable quantities, &c. in any functions <foreign xml:lang="greek">p, r</foreign>, &c of the variables assumed, let the quantities resulting be A, B, &c; then generally will <foreign xml:lang="greek">p</foreign> = A, <foreign xml:lang="greek">r</foreign> = B, &c. be fluxional equations, of which the particular fluentials are known. It may be observed in this place as before, that from no general reasoning can particular fluents be deduced. 14. In the resolution of fluxional equations it is observed, that from the logarithmic and exponential quantities contained in the fluxional, may be deduced by chapter 1 the exponentials &c contained in the fluential: 2, and in a similar manner from the irrational quantities and denominators contained in it, the correspondent irrational quantities and denominators contained in the fluential: 3, the greatest dimensions of <hi rend="italics">y</hi> multiplied into <hi rend="italics">x</hi><hi rend="sup">.</hi> must be greater than those of <hi rend="italics">y</hi> into <hi rend="italics">y</hi><hi rend="sup">.</hi> by unity; when there are two of this kind &c, (<foreign xml:lang="greek">d</foreign><hi rend="italics">x</hi><hi rend="sup">.</hi> + <foreign xml:lang="greek">e</foreign><hi rend="italics">y</hi><hi rend="sup">.</hi>) the refolution is given; and so of more. 15. In the given equation, if the fluxion of the greatest order does not ascend to one dimension only; then by extraction &c so reduce the equation, that it may ascend to one dimension only; and thence find the fluent of any fluxion P<hi rend="sup">n</hi><hi rend="italics">y</hi><hi rend="sup">.</hi> + Q<hi rend="sup">n - 1</hi><hi rend="italics">y</hi><hi rend="sup">.</hi> + &c, + R′<hi rend="sup">n</hi><hi rend="italics">z</hi><hi rend="sup">.</hi> + &c. 16. Let a fluxional equation be given involving <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> in which <hi rend="italics">x</hi> flows uniformly, a method is given of finding whether it admits of a multiplier, which is a function of <hi rend="italics">x</hi> ∴ and similarly of multipliers of other formulæ. 17. The method of deducing the multipliers of fluxional equations by infinite series. 18. Some fluxional equations are reduced by substitutions, which substitutions are commonly easily deducible from the fluxional equation given. 19. Somewhat concerning the reduction of some fluxional equations to homogeneous, and concerning homogeneous equations of different orders; and of reducing an homogeneous fluxional equation of <hi rend="italics">n</hi> order to a fluxional equation of <hi rend="italics">n</hi> - 1 order: and <pb n="723"/><cb/> also of reducing <hi rend="italics">m</hi> fluxional equations of <hi rend="italics">n</hi> order to one of <hi rend="italics">mn</hi> - 1 orders, and so of all others to one degree less than the order generally occurring if they had not been homogeneous. 20. The substitution of an exponential for a variable quantity in equations which contain no exponential quantity; for sometimes <hi rend="italics">n</hi> has been substituted for a quantity which flows uniformly, and then <hi rend="italics">w</hi> supposed to flow uniformly, which leads to a false resolution. 21. A caution is given not to substitute homogeneous functions of no dimensions for variable quantities; and in the general resolution to observe, that there is contained an invariable quantity to be assumed at will, which is not contained in the fluxional equation. 22. Something more added concerning the fluents of , where <hi rend="italics">p, q, r,</hi> &c. are functions of <hi rend="italics">x,</hi> and so of some other fluxional equations. 23. Fluxional equations are deduced, of which the variable quantities cannot be expressed in terms of each other, but both may be expressed in terms of a third. 24. Every fluxion or fluent which is a function <hi rend="italics">x, y, z,</hi> and <hi rend="italics">x, y.</hi> &c. is expressed in terms of partial differences. 25. The resolution of some equations expressing the relation between partial differences &c is given. 26. Some observations on finding the fluents of fluxions, when the variable quantities become infinite.</p><p>The second book treats of increments and their integrals. 1. Some new laws of the increments are given. 2. The fluxion of the increment of P will be equal to the increment of the fluxion; where P is any function of <hi rend="italics">x,</hi> if only the fluxion of the increment of <hi rend="italics">x</hi> be equal to the increment of the fluxion. 3. Increments are reduced to others of given formulæ <hi rend="italics">e. g.</hi> <foreign xml:lang="greek">a</foreign> + <foreign xml:lang="greek">b</foreign>/<hi rend="italics">x</hi> + <foreign xml:lang="greek">g</foreign>/(<hi rend="italics">x</hi>(<hi rend="italics">x</hi>+<hi rend="italics">x<hi rend="sub">.</hi></hi>)) +, &c, and it is observed that if <foreign xml:lang="greek">b</foreign> be not = 0, then the integral cannot be found in finite terms of the variable quantity, &c. It may be observed, that Taylor, Monmort, &c, first found the integral of the two increments <hi rend="italics">x</hi>.―(<hi rend="italics">x</hi>-<hi rend="italics">x<hi rend="sub">.</hi></hi>).―(<hi rend="italics">x</hi>-2<hi rend="italics">x<hi rend="sub">.</hi></hi>) ... ―(<hi rend="italics">x</hi>-―(<hi rend="italics">n</hi>-1) <hi rend="italics">x<hi rend="sub">.</hi></hi>) and 1/(<hi rend="italics">x.</hi>―(<hi rend="italics">x</hi>-<hi rend="italics">x<hi rend="sub">.</hi></hi>) ...―(<hi rend="italics">x</hi>-―(<hi rend="italics">n</hi>-1)<hi rend="italics">x<hi rend="sub">.</hi></hi>)) but did not proceed much further (correspondent to the finding the fluxion of the fluent <hi rend="italics">x</hi><hi rend="sup">n</hi>); the increments of fluents have been since deduced, &c. In this book are discovered propositions correspondent to most of the inventions in fluxions, <hi rend="italics">e. g.</hi> a method of finding the integral of any increment expressed in algebraical or exponential terms of the variable quantity or quantities, and when the fluent cannot be expressed: it is observed that they cannot be expressed in finite terms of the variable <hi rend="italics">x,</hi> &c, if the dimensions of <hi rend="italics">x,</hi> &c, in the denominator exceed its dimensions in the numerator by 1; or if any factor in the denominator of the fraction reduced to its lower terms have not another contained likewise in the denominator, distant by a whole number, multiplied into the increment of <hi rend="italics">x.</hi> —The increments of some integrals are deduced from the integrals of other increments; the integrals of some incremental equations from different methods; their general integrals, and particular corrections, &c, &c; but here it is to be observed, that the general problem of increments cannot be extended beyond the particular of fluxions, <cb/> but somewhat more may be added, when both are joined together. The third book is versant concerning infinite series. 1. It gives the ratio of the apparent and real convergency. 2. A method of finding limits between which the sum of the series consists; and also whether the sum of the series is sinite or not from the terms being given or equation between the terms. 3. The convergency of the whole series is judged from the ratio of convergency of the terms at an insinite distance. 4. The series from the fluent converges, if the series from the fluxion does, there are several propositions on insinite series deducible from the common algebra. 5. Let an equation ; and <hi rend="italics">b</hi>/<hi rend="italics">a</hi> much greater than <hi rend="italics">c</hi>/<hi rend="italics">b,</hi> <hi rend="italics">c</hi>/<hi rend="italics">b</hi> than <hi rend="italics">d</hi>/<hi rend="italics">c;</hi> &c. then will all the roots be possible, and <hi rend="italics">a</hi>/<hi rend="italics">b</hi> an approximation to the least root, <hi rend="italics">b</hi>/<hi rend="italics">c</hi> to the next, &c: if an equation , and if one root be much less than any <hi rend="italics">m</hi> root, but much greater than the remaining; or if the equation be , then will the approximation to the above root be <hi rend="italics">i</hi>/<hi rend="italics">h</hi> - (<hi rend="italics">k</hi>/<hi rend="italics">i</hi> - (<hi rend="italics">gi</hi><hi rend="sup">2</hi>)/<hi rend="italics">h</hi><hi rend="sup">3</hi>) + &c. 6. Somewhat on the approximations when the approximation given is much more near to one, two, or more roots than to any other, and on the degree of convergency of the subsequent approximations deduced; and their ultimate approximations. 7. Given approximations to <hi rend="italics">m</hi> roots of a given equation are deduced more near approximations to them. 8. The incremental equation given and applied to approximations. 9. From given approximations to two or more unknown quantities contained in two or more equations are deduced more near approximations to them, either when the approximations given are more near to one, or to two, or more roots of one or more of the unknown quantities than to any others, and so of infinite equations. 10. New series are given for the fluents of different fluxions. 1. . The sine of the arc A±<hi rend="italics">e</hi> is S ± C<hi rend="italics">e</hi> - 1/2 S<hi rend="italics">e</hi><hi rend="sup">2</hi>, &c, and cosine of the same arc = C± S<hi rend="italics">e</hi> - 1/(2.3) C<hi rend="italics">e</hi><hi rend="sup">2</hi> ± &c. S and C being the sine and cosine of A, the fluent of the fluxion of an elliptical arc √((1-<hi rend="italics">cx</hi><hi rend="sup">2</hi>)<hi rend="italics">x</hi><hi rend="sup">.</hi>)/√(1 - <hi rend="italics">x</hi><hi rend="sup">3</hi>) which differs little from the arc of a circle when <hi rend="italics">e</hi> is a very small quantity = A′ - <hi rend="italics">c</hi>/2 X (1.A - <hi rend="italics">x</hi>P)/2 - &c, where , <pb n="724"/><cb/> , and A = arc of a circle of which the sine is <hi rend="italics">x.</hi></p><p>A similar series may be applied from the arc of an hyperbola or ellipse, to find a correspondent arc of an hyperbola or ellipse not much different from the preceding. In this method the series proceeds according to the dimensions of some small quantities, and the first term of the series is generally a near value of the quantity sought. These series properly instituted will generally converge the swiftest. 11. Something new is added concerning the fluent of the fluxional equation ; E and F being any quantities to be assumed at will; and of correspondent equations to logarithms, and finding their values when <hi rend="italics">z</hi> is increased by <hi rend="italics">e.</hi> 12. A series for the increase of the arc from a small increase of the tangent, fine, &c. 12. When the terms <hi rend="italics">a</hi> and <hi rend="italics">x</hi> of the binomial <hi rend="italics">a</hi>±<hi rend="italics">x</hi> are equal, the cases are given in which the series or the series <hi rend="italics">a</hi><hi rend="sup">m</hi><hi rend="italics">x</hi> = (<hi rend="italics">m</hi>/2)<hi rend="italics">a</hi><hi rend="sup">m-1</hi> <hi rend="italics">x</hi> + &c, &c. will ultimately converge. 13. If any algebraical quantity V a function of <hi rend="italics">x</hi> be reduced into a series proceeding according to the dimensions of <hi rend="italics">x,</hi> a general method of finding what are the limits between which it converges; or the series from ∫ V<hi rend="italics">x</hi><hi rend="sup">.</hi>, &c; and the method of interpolations so as to render them converging. 14. The convergency of different series are compared together. <hi rend="italics">e. g.</hi> is given : there is an erratum contained in this example, for <hi rend="italics">a</hi> - is sometimes printed instead of <hi rend="italics">a</hi> +: this series is easily deduced from Bernouilli's method of deducing infinite series, and has been since printed in the Philosophical Transactions. 15. Given algebraical or fluxional equations, and a fluxional quantity, a method is given of finding a series, which expresses the fluent of the fluxional quantity, from which principles are deduced new series for the area of a segment of a circle, the periphery of the ellipse, hyperbola, &c. 16. It is shewn, that serieses proceeding according to the dimensions of a quantity <hi rend="italics">x</hi> always diverge, when serieses for the same purpose proceeding according to the reciprocal of its dimensions converge; unless sometimes in the case when they both become the same. 17. As series proceeding in infinitum according to the dimensions of the quantity <hi rend="italics">x</hi> were first invented or used for the finding the fluents of fluxions, it being reduced into terms, whose fluents were known: so in finding integrals of increments it may be necessary to reduce the quantity into an infinite series of terms, whose integrals are known, and which converges. Examples of formulæ of serieses of this kind are given. 18. Methods are given of finding the value of one unknown quantity contained in one or more equations involving more unknown quantities, and the law of their convergencies <cb/> and the interpolations necessary to render serieses for finding fluents converging, similar principles may be applied to incremental and fluxional equations. 19. It is observed, that in finding the value of any variable quantity in a series proceeding according to the dimensions of another, there will occur in a fluxional or incremental equation of (<hi rend="italics">n</hi>) order in the series <hi rend="italics">n</hi> invariable quantities to be assumed at will; and also the fluxional equations, &c. from whence they will arise. 20. The finding the integral of <hi rend="italics">ż</hi>/<hi rend="italics">z,</hi> &c. 21. From the correspondent relation between the sums of two series resulting, which are functions of a variable quantity <hi rend="italics">y,</hi> when the relation between <hi rend="italics">x</hi> and <hi rend="italics">z</hi> two values of <hi rend="italics">y</hi> are given, is given a method of finding the coefficients of the series. 22. The rule generally called the reductio ad absurdum extended to more substitutions.</p><p>The fourth book treats of the summation of series, a method of correspondent values and several other problems. 1. Of finding the sum of a series expressed by a rational function of <hi rend="italics">z</hi> into <hi rend="italics">x</hi><hi rend="sup">n2</hi>; where <hi rend="italics">z</hi> denotes successively the numbers 1, 2, 3, &c, in infinitum. 2. Given an equation expressing the relation between the successive sums, the relation between the successive terms is known, and the <hi rend="italics">vice versa,</hi> &c. 3. It is found from an equation expressing the relation between the successive sums, terms and <hi rend="italics">z</hi> the distance from the first term of the series, whether the sum of the series is finite or not. 4. The difference between <hi rend="italics">z</hi><hi rend="sup">-0</hi> and ―(<hi rend="italics">z</hi>+1)<hi rend="sub">-0</hi>, where <hi rend="italics">z</hi> denotes the distance from the first term of the series, will be — 0 X <hi rend="italics">z</hi><hi rend="sup">-0-1</hi>, which is greater than the simple ratio let 0 be as small as possible, and consequently the sum of the series finite. 5. If a series <hi rend="italics">a</hi> + <hi rend="italics">bx</hi> + <hi rend="italics">cx</hi><hi rend="sup">2</hi> + <hi rend="italics">x</hi><hi rend="sup">3</hi>, of which at an infinite distance the preceding coefficients have to the subsequent the ratio of <hi rend="italics">r</hi>:1, be multiplied into a function = 0, when <hi rend="italics">x</hi>=<foreign xml:lang="greek">d</foreign>, then if <foreign xml:lang="greek">a</foreign> be greater than <hi rend="italics">r</hi> the series will diverge; if less converge. 6. From adding several terms of one or more series together may be formed a series, of which the sum from the sums of the preceding series is known. 6. Serieses are formed, of which the sums are known from varying the divisors, &c. 7. From given series are deduced others, of which the sums are known, and the sum of many series are deduced from finding the fluxions of fluents and fluents of fluxions. 8. From the relation between the different terms given is deduced the correspondent fluxional equation. 9. The finding the terms of any series, which can be deduced from given series; and thence deducing many series of which the sums can be found from the sum of the given series. 10. Series are given of which the sums can be found from finite terms, circular arcs, logarithms, elliptical and hyperbolical arcs. 11. From a general expression, when algebraical, fluxional, incremental, &c, for the sum of a series can be deduced a similar expression for the sum of every second, third, &c, terms. 12. An infinite series may be a particular resolution of infinite fluxional equations. 13. The terms of some series may be infinite and their sums known. 14. The general fluent of is given by a series of the same kind, and the same of some other fluxional equations. 15. A quantity is found which multiplied into a series <pb n="725"/><cb/> more swiftly converging gives a given series. 16. The first differences of the terms of some series are given; if the terms are in geometrical ratio to each other the abovementioned differences will also be in geometrical ratio to each other: whence it appears, that the series from this method of differences will converge least when the given series converges swiftest, &c, but not always the contrary. Several other propositions are added concerning the method of differences applied to series. 17. A parabolico-hyperbolical curve is drawn through any number of points, as also an algebraical solid —. 18. Something is given concerning the convergency &c. of series deduced from the differences of the numerators of a given series, of which the denominators constitute a geometrical progression. 19. A rule is given for rendering series converging, in which it is observed that the sum of so many terms should be found that <hi rend="italics">z</hi> the distance from the first term of the series may exceed the greatest root of the equation resulting from the quantity which expresses the term made = 0. 20. An equation expressing the relation between the sums and terms is reduced to an infinite fluxional equation expressing the relation between the sum or term, its fluxions, and <hi rend="italics">z</hi> the distance from the first term of the series. 21. From a method being known of finding the sum of a series, which involves one variable only, is given a method of finding the sum of series which involve more variable quantities: and from assuming sums of serieses of this kind are deduced their terms. 22. The sums of series are found consisting of irrational terms. 23. The principle of the convergency of the approximations found in drawing parabolical curves through given points. 24. Something new is given concerning the interpolations of quantities. 25. . if <foreign xml:lang="greek">a, b, g</foreign>, &c, are the roots of , &c. 26. Something is added concerning series from . 27. Nandens's Problems are somewhat extended. 28. Something is added on changing continual fractions into others. 29. A method of transforming series into continual factors. 30. A rule for finding the sine and cosine of <hi rend="italics">n</hi>/<hi rend="italics">m</hi> the arc; and transforming an algebraical equation into an equation expressed in terms of sines and cosines, and thence from an approximation to the sine is found one more near; the same might have been performed by tangents, cotangents, secants, cosecants, &c. 31. From some-fluents given have been found others, and consequently by reducing the fluents to infinite series from some infinite series given may others be deduced. 32. The fluent of (<hi rend="italics">x<foreign xml:lang="greek">a</foreign> x</hi><hi rend="sup">.</hi>)/(1 ± <hi rend="italics">x</hi><hi rend="sup">n</hi>) is found by approximation, where <foreign xml:lang="greek">a</foreign> is an irrational quantity, which method of finding approximations to the indices may be applied to other cases. 33. The sum of the fractions are found when the denominators = 0, and consequently each particular in- <cb/> finite. 34. It is asserted, that the sum of certain fractions given become = 0, when the terms are expressed by a fraction of which the denominator is a rational function of the distance from the first term of the series. 35. ∫<hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">a</foreign> - <foreign xml:lang="greek">b</foreign> - 1</hi><hi rend="italics">x</hi><hi rend="sup">.</hi>∫<hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">b</foreign> - <foreign xml:lang="greek">g</foreign> - 1</hi><hi rend="italics">x</hi><hi rend="sup">.</hi> ∫<hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">g</foreign> - <foreign xml:lang="greek">d</foreign> - 1</hi><hi rend="italics">x</hi><hi rend="sup">.</hi> X P, where , &c, will be to ∫<hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">b</foreign> - <foreign xml:lang="greek">a</foreign> - 1</hi><hi rend="italics">x</hi><hi rend="sup">.</hi> ∫<hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">a</foreign> - <foreign xml:lang="greek">g</foreign> - 1</hi><hi rend="italics">x</hi>∫<hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">g</foreign> - <foreign xml:lang="greek">d</foreign> - 1</hi><hi rend="italics">x</hi><hi rend="sup">.</hi>∫&c. X P :: <hi rend="italics">x</hi><hi rend="sup"><hi rend="italics">z</hi></hi> : <hi rend="italics">x</hi><hi rend="sup"><foreign xml:lang="greek">b</foreign></hi> if the fluents are contained between the same values of <hi rend="italics">x.</hi> 36. Are given some series consisting of two, of which the one converges, when the other diverges, and consequently the sum of both diverges; &c. 37. From the law of a series being given, the law of the series which expresses the square, or some function of the given series, is found.</p><p>1. A method of differences, which deduces from the sums given any successive sums, e. g. Let S<hi rend="sup">1</hi>, S<hi rend="sup">2</hi>, S<hi rend="sup">3</hi>, S<hi rend="sup">4</hi>, be the logarithms of the ratios <hi rend="italics">r</hi> : <hi rend="italics">r</hi> + <hi rend="italics">p,</hi> <hi rend="italics">r</hi> : <hi rend="italics">r</hi> + 2<hi rend="italics">p,</hi> <hi rend="italics">r</hi> : <hi rend="italics">r</hi> + 3<hi rend="italics">p,</hi> <hi rend="italics">r</hi> : <hi rend="italics">r</hi> + 4<hi rend="italics">p,</hi> then will the logarithm of <hi rend="italics">r</hi> : <hi rend="italics">r</hi> + 5<hi rend="italics">p</hi> be 5 X (S<hi rend="sup">4</hi> - S<hi rend="sup">2</hi>) + 10(S<hi rend="sup">2</hi> - S<hi rend="sup">3</hi>) nearly: then rules are given in general, and likewise their errors from the true values.</p><p>2. A method of correspondent values is given, e. g. Let <hi rend="italics">a, b, c, d,</hi> &c, be values of <hi rend="italics">x;</hi> and S<hi rend="sup">a</hi>, S<hi rend="sup">b</hi>, S<hi rend="sup">c</hi>, S<hi rend="sup">d</hi>, &c, correspondent values of <hi rend="italics">y;</hi> then may .</p><p>3. If the formula of the series be ; if the formula of the series be , which answers to Briggs's or Newton's method of interpolations; or the series will be <hi rend="italics">x</hi><hi rend="sup">h</hi>/<hi rend="italics">a</hi><hi rend="sup">h</hi> X ((<hi rend="italics">x</hi><hi rend="sup">k</hi> - <hi rend="italics">b</hi><hi rend="sup">k</hi>) (<hi rend="italics">x</hi><hi rend="sup">k</hi> - <hi rend="italics">c</hi><hi rend="sup">k</hi>) (<hi rend="italics">x</hi><hi rend="sup">k</hi> - <hi rend="italics">d</hi><hi rend="sup">k</hi>) &c)/((<hi rend="italics">a</hi><hi rend="sup">k</hi> - <hi rend="italics">b</hi><hi rend="sup">k</hi>) (<hi rend="italics">a</hi><hi rend="sup">k</hi> - <hi rend="italics">c</hi><hi rend="sup">k</hi>) (<hi rend="italics">a</hi><hi rend="sup">k</hi> - <hi rend="italics">d</hi><hi rend="sup">k</hi>) &c) X S<hi rend="sup">a</hi> + <hi rend="italics">x</hi><hi rend="sup">h</hi>/<hi rend="italics">b</hi><hi rend="sup">h</hi> X ((<hi rend="italics">x</hi><hi rend="sup">k</hi> - <hi rend="italics">a</hi><hi rend="sup">k</hi>) (<hi rend="italics">x</hi><hi rend="sup">k</hi> - <hi rend="italics">c</hi><hi rend="sup">k</hi>) (<hi rend="italics">x</hi><hi rend="sup">k</hi> - <hi rend="italics">d</hi><hi rend="sup">k</hi>) &c)/((<hi rend="italics">b</hi><hi rend="sup">k</hi> - <hi rend="italics">a</hi><hi rend="sup">k</hi>) (<hi rend="italics">b</hi><hi rend="sup">k</hi> - <hi rend="italics">c</hi><hi rend="sup">k</hi>) (<hi rend="italics">b</hi><hi rend="sup">k</hi> - <hi rend="italics">d</hi><hi rend="sup">k</hi>) &c) X S<hi rend="sup">b</hi> + &c; if the formula of the series be a general formula, which includes the preceding.</p><p>5. The series is given for deducing others when the number of correspondent values given are either even or odd, and the values of <hi rend="italics">x</hi> are equidistant from each other. 6. And also from correspondent values of <hi rend="italics">x</hi> and <hi rend="italics">y</hi> to a number of equidistant values of <hi rend="italics">x</hi> is deduced the value of <hi rend="italics">y</hi> to the next successive or any successive value of <hi rend="italics">x.</hi> 7. Some arithmetical theorems are deduced from the preceding propositions. 8. Another method is given of resolving the preceding problem. 9. A method of correcting the solution from a solution <pb n="726"/><cb/> given which finds (<hi rend="italics">n</hi>) values of <hi rend="italics">y</hi> to (<hi rend="italics">n</hi>) given values of <hi rend="italics">x</hi> true, and <hi rend="italics">m</hi> false to (<hi rend="italics">m</hi>) other values. 10. A similar resolution is added from correspondent values of <hi rend="italics">x, y, z,</hi> &c given; and more general resolutions. 11. Given the resolution of some cases, and formula in which the general is contained, a method is given in some cases of deducing it. 12. The principles of a method of deductions and reductions are added.</p><p><hi rend="italics">In a Pamphlet published at Cambridge,</hi> algebraical quantities are translated into probable relations, and some theorems on probabilities thence deduced; to which are adjoined,</p><p>1. The theorem ; this becomes the binomial theorem when <hi rend="italics">l</hi> = 0; and it will afford answers to similar cases when the whole number of chances are increased or diminished constantly by <hi rend="italics">l,</hi> as the binomial does when they remain the same, a similar multinomial theorem is given. In the same pamphlet are further added some new propositions on chances, on the values of lives, survivorships, &c. In these books are also contained the inventions of others on similar subjects, which in the prefaces are ascribed to their respective authors.</p><p><hi rend="italics">In the Philosophical Transactions</hi> are given some properties of numbers, &c, of which some have been published in the books above mentioned; to which may be subjoined something in mixed mathematics, viz, a paper on central forces, which extends not only to central forces, but also to forces applied in any other direction, as in the direction of the tangent, and consequently includes resistances, &c. It gives a rule for finding the forces tending to two or more given points when the curve described and velocity of the body in every point of it is given, <hi rend="italics">e. g.</hi> Let the curve be an ellipse, and the velocity the same at every point, and the two centres of force be the foci of the ellipse; then will the forces tending to the two foci be equal, and vary as the square of the sine of the angle contained between the distance from the centre of force to the point in which the body is situated, and the tangent to the curve at that point.</p><p>The method of deducing the fluxional equations which express the curve described by a body acted on by any forces tending to given points, or applied in any given directions: some other propositions are contained on similar subjects. 2. A paper on the fluxions of the attractions of lines, surfaces, and solids, and from the different methods of deducing them are found different fluents equal to each other: a third paper gives a solution of Kepler's problem of cutting the area of a circle described round a point by approximations, which also is applied to other cases; this like- <cb/> wise contains some other problems. Many of these discoveries have since been published, some in the London, and other foreign transactions.</p><p>Let , then will <hi rend="italics">l</hi> denote the log. of N to the modulus <hi rend="italics">e.</hi> If <hi rend="italics">e</hi> the modulus = 10, then will the system be the common or Briggs's system of logarithms. Logarithms, and the sums of some other serieses, of the formulæ <hi rend="italics">ax</hi><hi rend="sup">h</hi> + <hi rend="italics">bx</hi><hi rend="sup">h + k</hi> + &c may be deduced in a manner similar to that which was used by the Ancients for finding the sines of the arcs of circles.</p><p>To particularise the numerous propositions contained in these works, would exceed the limits of our design. Besides those already mentioned, others are interspersed through the whole works.</p></div1><div1 part="N" n="ANEMOMETER" org="uniform" sample="complete" type="entry"><head>ANEMOMETER</head><p>, p. 111, col. 2, l. 1, <hi rend="italics">after</hi> 12 ounces, <hi rend="italics">add</hi> or 3/4 of a pound. Owing to an oversight in the succeeding lines, of considering this 12 ounces as 12 pounds, in the calculations, several errors have been incurred, and the 3d column of the table of numbers, in that page, or the column for the velocity, has the numbers only 1/4 of what they ought to be, or they require to be all multiplied by 4, the square-root of 16, the number of ounces in a pound. Hence, in line 6, <hi rend="italics">for</hi> √12 <hi rend="italics">r.</hi> √3/4; l. 7 and 8, <hi rend="italics">for</hi> 22 4/5 <hi rend="italics">r.</hi> 91 1/5; l. 8, <hi rend="italics">for</hi> 15 1/2 <hi rend="italics">r.</hi> 62. And the whole succeeding table corrected will be as follows: <hi rend="center"><hi rend="italics">Table of the corresponding Height of Water, Force on a Square Foot, and Velocity of Wind,</hi></hi> <table rend="border"><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Height of</cell><cell cols="1" rows="1" role="data">Force of</cell><cell cols="1" rows="1" role="data">Velocity of</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Water.</cell><cell cols="1" rows="1" role="data">Wind.</cell><cell cols="1" rows="1" role="data">Wind per Hour</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Inches.</cell><cell cols="1" rows="1" role="data">Pounds.</cell><cell cols="1" rows="1" rend="align=right" role="data">Miles.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">0 1/4</cell><cell cols="1" rows="1" role="data">1.3</cell><cell cols="1" rows="1" rend="align=right" role="data">18.0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">0 1/2</cell><cell cols="1" rows="1" role="data">2.6</cell><cell cols="1" rows="1" rend="align=right" role="data">25.6</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">5.2</cell><cell cols="1" rows="1" rend="align=right" role="data">36.0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">10.4</cell><cell cols="1" rows="1" rend="align=right" role="data">50.8</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">15.6</cell><cell cols="1" rows="1" rend="align=right" role="data">62.0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">20.8</cell><cell cols="1" rows="1" rend="align=right" role="data">76.0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">26.0</cell><cell cols="1" rows="1" rend="align=right" role="data">80.4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">31.25</cell><cell cols="1" rows="1" rend="align=right" role="data">88.0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">36.5</cell><cell cols="1" rows="1" rend="align=right" role="data">95.2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">41.7</cell><cell cols="1" rows="1" rend="align=right" role="data">101.6</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">46.9</cell><cell cols="1" rows="1" rend="align=right" role="data">108.0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">52.1</cell><cell cols="1" rows="1" rend="align=right" role="data">113.6</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">57.3</cell><cell cols="1" rows="1" rend="align=right" role="data">119.2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">62.5</cell><cell cols="1" rows="1" rend="align=right" role="data">124.0</cell></row></table></p><p>In one instance Dr. Lind found that the force of the wind was such as to be equal 34 9/10 pounds, on a square foot; and this by proportion, in the foregoing table, will be found to answer to a velocity of 93 miles per hour.</p></div1><div1 part="N" n="ARCH" org="uniform" sample="complete" type="entry"><head>ARCH</head><p>, p. 137, col. 1, l. 29, <hi rend="italics">for</hi> such cases as they, <hi rend="italics">read</hi> such cases they. Line 30, <hi rend="italics">after</hi> hanches, <hi rend="italics">add,</hi> See <hi rend="smallcaps">Bridge.</hi></p></div1><div1 part="N" n="ARCHIMEDES" org="uniform" sample="complete" type="entry"><head>ARCHIMEDES</head><p>, p. 139, col. 1, l. 52 and 53, <hi rend="italics">for</hi> preface, a commentary, <hi rend="italics">read</hi> preface. We find here also Eutocius's commentary. Pa. 59, <hi rend="italics">after</hi> college, <hi rend="italics">add,</hi> who had the sole care of this edition. <pb n="727"/><cb/></p><p>ASSURANCE <hi rend="italics">on Lives.</hi> Pa. 150, col. 2, in the 3d paragraph, for want of sufficient information concerning the London and Royal Exchange Assurance Offices, that paragraph gives an imperfect and, in some respect, erroneous account of them: it refers to their state 30 years ago, but the Companies have since that, altered their method of proceeding. Instead of that paragraph therefore, take the following account of their present constitution; viz,</p><p><hi rend="italics">The London</hi> <hi rend="smallcaps">Assurance</hi>, is a corporation established by a charter of king George the 1st, viz, in 1720; under power of which, Assurances are made from the risk of sea-voyages, and from the danger of fire to houses and goods; the prices of which are regulated by the apparent risk to be assured. They also make Assurances on lives; the prices of which are formed on an estimation of the probable duration of life at different ages, on the consideration of the apparent health of the persons to be assured, and of their avocations in life.</p><p>This corporation, and the Royal Exchange corporation, gave each the sum of 150,000 pounds to government, for an <hi rend="italics">exclusive right</hi> of making Assurances as <hi rend="italics">corporate bodies.</hi> They are known to possess a large and undeniable fund to answer losses. And the prudent management of these corporations has enabled them, of late years, to increase gradually their dividends to the proprietors of their stock. This <hi rend="italics">exclusive privilege</hi> to make Assurances as corporate bodies, is of great advantage and convenience to the public; and as they act under a common seal, the assured may have a speedy and easy mode of recovering losses, and cannot be subject to any calls or deductions whatever. When their <cb/> charters were granted to them, it was enacted, that if a proprietor of the stock of one corporation should at the same time, directly or indirectly, be a proprietor of stock in the other corporation, the respective stock so held is to be forfeited, one moiety to the king, the other to the informer. This was evidently settled, to prevent their interest from becoming a joint one; so that they should be made to act in competition to each other, for the greater benefit of the public.</p><p><hi rend="italics">The Royal Exchange</hi> <hi rend="smallcaps">Assurance</hi>, is a corporation established by charter, as above, under the power of which, Assurances are made from the risk of sea voyages, and from the danger of fire to houses and goods; the prices of which are regulated by the greater or less risk supposed to be assured. They also make Assurances on lives, the prices of which are formed on estimation of the probable duration of life at different ages, and under different circumstances. The present rates of Assurances on lives are as in the table below. And though a duty on these Assurances should take place on the plan lately proposed to the House of Commons, there is no great probability that these prices will be increased.</p><p>This corporation has also, like the former, been empowered to grant life annuities by an act of parliament, which requires that the prices of the annuities should be expressed in tables, hung up in some conspicuous place in their offices, for public inspection; and no agreement for any price is valid, but such as shall be expressed in the tables last made and published by the corporation. <pb n="728"/> <hi rend="center"><hi rend="italics">From the Office of the</hi> <hi rend="smallcaps">Corporation</hi> <hi rend="italics">of the</hi> ROYAL EXCHANGE ASSURANCE, <hi rend="italics">on the</hi> ROYAL EXCHANGE, <hi rend="smallcaps">London.</hi></hi> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=25" role="data">RATES OF ASSURANCES ON LIVES.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=10" role="data">SINGLE LIVES.</cell><cell cols="1" rows="1" rend="colspan=15" role="data">JOINT LIVES.</cell></row><row role="data"><cell cols="1" rows="1" rend="rowspan=7" role="data">Age.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3" role="data">Premium per</cell><cell cols="1" rows="1" rend="colspan=5" role="data">For the Assurance of a Gross</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Premium per</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Premium per</cell><cell cols="1" rows="1" rend="colspan=3" role="data">ct. per. ann.</cell><cell cols="1" rows="1" rend="colspan=5" role="data">Sum, payable when One</cell><cell cols="1" rows="1" rend="colspan=10" role="data">For the Assurance of a Gross Sum, payable</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">cent. for an</cell><cell cols="1" rows="1" rend="colspan=3" role="data">cent. per an-</cell><cell cols="1" rows="1" rend="colspan=3" role="data">for an assur-</cell><cell cols="1" rows="1" rend="colspan=5" role="data">of Two Joint Lives that</cell><cell cols="1" rows="1" rend="colspan=10" role="data">when either of Two Joint Lives shall drop.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">assurance for</cell><cell cols="1" rows="1" rend="colspan=3" role="data">num, for an</cell><cell cols="1" rows="1" rend="colspan=3" role="data">ance for the</cell><cell cols="1" rows="1" rend="colspan=5" role="data">shall be named shall drop.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">one year.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">assurance for</cell><cell cols="1" rows="1" rend="colspan=3" role="data">whole con-</cell><cell cols="1" rows="1" rend="colspan=5" role="data">-------------------------</cell><cell cols="1" rows="1" rend="colspan=10" role="data">----------------------------------------</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3" role="data">seven years.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">tinuance of</cell><cell cols="1" rows="1" role="data">Age</cell><cell cols="1" rows="1" role="data">Age of the</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3" role="data">life.</cell><cell cols="1" rows="1" role="data">of the</cell><cell cols="1" rows="1" role="data">life against</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Premium</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3" role="data">Premium</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=3" role="data">Premium</cell></row><row role="data"><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data">life</cell><cell cols="1" rows="1" role="data">which the</cell><cell cols="1" rows="1" rend="colspan=3" role="data">per cent.</cell><cell cols="1" rows="1" role="data">Age.</cell><cell cols="1" rows="1" role="data">Age.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">per ct. per</cell><cell cols="1" rows="1" role="data">Age.</cell><cell cols="1" rows="1" role="data">Age.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">per cent.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">£.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">£.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">£.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" rend="align=left" role="data">to be</cell><cell cols="1" rows="1" rend="align=left" role="data">assuranceis</cell><cell cols="1" rows="1" rend="align=left colspan=3" role="data">per ann.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left colspan=3" role="data">annum.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=left colspan=3" role="data">per ann.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8 to 14</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" rend="align=left" role="data">assured.</cell><cell cols="1" rows="1" rend="align=left" role="data">to be made.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">£.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">£.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">d.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">£.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">s.</hi></cell><cell cols="1" rows="1" rend="align=center" role="data"><hi rend="italics">d.</hi></cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">------</cell><cell cols="1" rows="1" role="data">--------</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">----</cell><cell cols="1" rows="1" rend="colspan=3" role="data">--------</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row></table><pb n="729"/> <table rend="border"><row role="data"><cell cols="1" rows="1" role="data">By whom the As-</cell><cell cols="1" rows="1" role="data">Name, age, and</cell><cell cols="1" rows="1" role="data">Time for which</cell><cell cols="1" rows="1" role="data"><hi rend="italics">Conditions of Assurance made</hi></cell><cell cols="1" rows="1" role="data">Sum assured.</cell><cell cols="1" rows="1" role="data">Rate <hi rend="italics">per</hi> cent.</cell></row><row role="data"><cell cols="1" rows="1" role="data">surance is made.</cell><cell cols="1" rows="1" role="data">description of the</cell><cell cols="1" rows="1" role="data">the Assurance is</cell><cell cols="1" rows="1" role="data"><hi rend="italics">by Persons on their own</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">per</hi> annum.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">life to be As-</cell><cell cols="1" rows="1" role="data">made.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">Lives.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">sured.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">The Assurance to be void if</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">the person whose life is As-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">sured shall depart beyond</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">the limits of Europe, shall</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">die upon the seas, or enter</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">into or engage in any mili-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">tary or naval service what-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">ever, without the previous</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">consent of the company; or</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">shall come by death by sui-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">cide, duelling, or the hand</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">of Justice; or shall not be,</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">at the time the Assurance is</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">made, in good health.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">By whom the As-</cell><cell cols="1" rows="1" role="data">Name, age and</cell><cell cols="1" rows="1" role="data">Time for which</cell><cell cols="1" rows="1" role="data"><hi rend="italics">Conditions of Assurance made</hi></cell><cell cols="1" rows="1" role="data">Sum assured.</cell><cell cols="1" rows="1" role="data">Rate <hi rend="italics">per</hi> cent.</cell></row><row role="data"><cell cols="1" rows="1" role="data">surance is made.</cell><cell cols="1" rows="1" role="data">description of the</cell><cell cols="1" rows="1" role="data">the Assurance is</cell><cell cols="1" rows="1" role="data"><hi rend="italics">by Persons on the Lives of</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"><hi rend="italics">per</hi> annum.</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">life to be As-</cell><cell cols="1" rows="1" role="data">made.</cell><cell cols="1" rows="1" role="data"><hi rend="italics">others.</hi></cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">sured.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">The Assurance to be void</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">if the person whose life is</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Assured shall depart beyond</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">the limits of Europe, shall</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">die upon the seas, or enter</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">into or engage in any mili-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">tary or naval service what-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">ever, without the previous</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">consent of the company; or</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">shall not be at the time the</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">Assurance is made in good</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">health.</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Place and date of birth.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Reference to be made to two persons of repute to ascertain his or her identity.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">If had the small-pox.</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">Whether in the army or navy.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">Attendance daily from ten to half past two o'clock and from five to seven, Saturday in the afternoon excepted.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">The life Assured to appear at the office, or pay</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">10<hi rend="italics">s. per cent.</hi> on Assurances for one year.</cell><cell cols="1" rows="1" rend="colspan=3" role="data">The lives of persons engaged in the army or navy may be Assured by special agreement.</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=3" role="data">15<hi rend="italics">s. per cent.</hi> for more than one year, and not exceeding seven years. 20<hi rend="italics">s. per cent.</hi> if for the whole continuance of life.<hi rend="brace"><label>In the first payment only.</label></hi></cell><cell cols="1" rows="1" rend="colspan=3" role="data"><hi rend="italics">N. B.</hi> THE CORPORATION ALSO GRANT ANNUITIES ON LIVES.</cell></row></table><pb n="730"/><cb/></p><p>AUTOMATON. To the end of this article, in pa. 176, col. 2, may be added the following curious particulars, extracted from a letter of an ingenious gentleman since that article was published, viz, Thomas Collinfon, Esq. nephew of the late ingenious Peter Collinson, Esq. F. R. S. “Turning over the leaves of your late valuable publication (says my worthy correspondent), part 1. of the Mathematical and Philosophical Dictionary, I observed under the article <hi rend="italics">Automaton,</hi> the following:” ‘But all these seem to be inferior to M. Kempell's chess-player, which may truly be considered as the greatest master-piece in mechanics that ever appeared in the world;’ (upon which Mr. Collinson observes) “So it certainly would have been, had its scientific movements depended merely on mechanism. Being slightly acquainted with M. Kempell when he exhibited his chess-playing figure in London, I called on him about five years since at his house at Vienna; another gentleman and myself being then on a tour on the continent. The baron (for I think he is such) shewed me some working models which he had lately made—among them, an improvement on Arkwright's cotton-mill, and also one which he thought an improvement on Boulton and Watt's last steam-engine. I asked him after a piece of speaking mechanism, which he had shewn me when in London. It spoke as before, and I gave the same word as I gave when I first saw it, <hi rend="italics">Exploitation,</hi> which it distinctly pronounced with the French accent. But I particularly noticed, that not a word passed about the chess-player; and of course I did not ask to see it.—In the progress of the tour I came to Dresden, where becoming acquainted with Mr. Eden, our envoy there, by means of a letter given me by his brother lord Auckland, who was ambassador when I was at Madrid, he obligingly accompanied me in seeing several things worthy of attention. And he introduced my companion and myself to a gentleman of rank and talents, named Joseph Freidrick Freyhere, who seems completely to have discovered the <hi rend="italics">Vitality</hi> and soul of the chess-playing figure. This gentleman courteously presented me with the treatise he had published, dated at Dresden, Sept. 30, 1789, explaining its principles, accompanied with curious plates neatly coloured. This treatise is in the German language; and I hope soon to get a translation of it. A welltaught boy, very thin and small of his age (sufficiently so that he could be concealed in a drawer almost immediately under the chess-board), agitated the whole. Even after this abatement of its being strictly an automaton, much ingenuity remains to the contriver.— This discovery at Dresden accounts for the silence about it at Vienna; for I understand, by Mr. Eden, that Mr. Freyhere had sent a copy to baron Kempell: though he seems unwilling to acknowledge that Mr. F. has completely analysed the whole.</p><p>“I know that long and uninteresting letters are formidable things to men who know the value of time <cb/> and science: but as this happens to be upon the subject, forgive me for adding one very admirable piece of mechanism to those you have touched upon. When at Geneva, I called upon Droz, son of the original Droz of la Chaux de Fonds (where I also was). He shewed me an oval gold snuff box, about (if I recollect right) 4 inches and a half long, by 3 inches broad, and about an inch and a half thick. It was double, having an horizontal partition; so that it may be considered as one box placed on another, with a lid of course to each box—One contained snuff—In the other, as soon as the lid was opened, there rose up a very small bird, of green enamelled gold, sitting on a gold stand. Immediately this minute curiosity wagged its rail, shook its wings, opened its bill of white enamelled gold, and poured forth, minute as it was (being only three quarters of an inch from the beak to the extremity of the tail) such a clear melodious song, as would have filled a room of 20 or 30 feet square with its harmony.—Droz agreed to meet me at Florence; and we visited the Abbé Fontana together. He afterwards joined me at Rome, and exhibited his bird to the pope and the cardinals in the Vatican palace, to the admiration, I may say to the astonishment of all who saw and heard it.”</p><p>Another extract from a second letter upon the same subject, by Mr. Collinson, is as follows: “Permit me to speak of another Automaton of Droz's, which several years since he exhibited in England; and which, from my personal acquaintance, I had a commodious opportunity of particularly examining. It was a figure of a man, I think the size of life. It held in its hand a metal style; a card of Dutch vellum being laid under it. A spring was touched, which released the internal clockwork from its stop, when the figure immediately began to draw. Mr. Droz happening once to be sent for in a great hurry to wait upon some considerable personage at the west end of the town, left me in possession of the keys, which opened the recesses of all his machinery. He opened the drawing-master himself; wound it up; explained its leading parts; and taught me how to make it obey my requirings, as it had obeyed his own. Mr. Droz then went away. After the first card was finished, the figure rested. I put a second; and so on, to five separate cards, all different subjects: but five or six was the extent of its delineating powers. The first card contained, I may truly say, elegant portraits and likenesses of the king and queen, facing each other: and it was curious to observe with what precision the figure lifted up his pencil, in the transition of it from one point of the draft to another, without making the least slur whatever: for instance, in passing from the forehead to the eye, nose, and chin; or from the waving curls of the hair to the ear, &c. I have the cards now by me, &c, &c.”</p><p>Pa. 177, col. 1, l. 2, <hi rend="italics">for</hi> August <hi rend="italics">read</hi> September. <pb n="731"/></p></div1></div0><div0 part="N" n="B" org="uniform" sample="complete" type="alphabetic letter"><head>B</head><cb/><p>PAGE 195, col. 1, at the end of the article on Barometrical Measurements of Altitudes, <hi rend="italics">add,</hi> See a learned paper in vol. 1. of the Transactions of the R. Soc. of Edinburgh, “On the Causes which affect the Accuracy of Barometrical Measurements; by John Playfair, A. M. F. R. S. Edin. and Professor of Mathematics in the University of Edinburgh.” Also another by Dr. Damen, late Professor of Mathematics and Philosophy in the University of Leyden, intitled, “Dissertatio Physica & Mathematica de Montium Altitudine Barometro Metienda: Accedit Refractionis Astronomicæ Theoria; in 8vo, at the Hague, 1783.</p><p>Pa. 205, col. 1, after the life of Dan. Bernoulli, <hi rend="italics">add</hi> the following life of James.</p><div1 part="N" n="BERNOULLI" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">BERNOULLI</surname> (<foreName full="yes"><hi rend="smallcaps">James</hi></foreName>)</persName></head><p>, another mathematical branch of the foregoing celebrated family. He was born at Basil in October 1759; being the son of John Bernoulli, and grandson of the first John Bernoulli, before mentioned, and the nephew of Daniel Bernoulli last noticed above. Our author's elder brother John, who still lives at Berlin, is also well known in the republic of science, particularly for his astronomical labours.</p><p>The gentleman to whom this article relates, was educated, as most of his relations had been, for the profession of law: but his genius led him very early into the study of mathematics; and at 20 years of age he read public lectures on experimental philosophy in the university of Basil, for his uncle Daniel Bernoulli, whom he hoped to have succeeded as professor. Being disappointed in this view, he resolved to leave his native place, and to seek his fortune elsewhere; hence he accepted the office of secretary to Count Breuner, the emperor's envoy to the republic of Venice; and in this city he remained till the year 1786, when, on the recommendation of his countryman, M. Fuss, he was invited to Petersburgh to succeed M. Lexell in the academy there, where he continued till his death, which happened the 3d of July 1789, at not quite 30 years of age, and when he had been married only two months, to the youngest daughter of John Albert Euler, the son of the so celebrated Leonard Euler.</p><p><hi rend="italics">Impossible or Imaginary</hi> BINOMIAL. After this article, in pa. 208, the middle of col. 1, <hi rend="italics">add</hi> what here follows.</p><p>In the foregoing article are given several rules for the roots of Binomials. Dr. Maskelyne, the Astronomer Royal, has also given a method of finding any power of an Impossible Binomial, by another like Binomial. This rule is given in his Introduction prefixed to Taylor's Tables of Logarithms, pa. 56; and is as follows.</p><p>The logarithms of <hi rend="italics">a</hi> and <hi rend="italics">b</hi> being given, it is required to find the power of the Impossible Binomial <cb/> <hi rend="italics">a</hi> ± √-<hi rend="italics">b</hi><hi rend="sup">2</hi> whose index is <hi rend="italics">m</hi>/<hi rend="italics">n,</hi> that is, to find (<hi rend="italics">a</hi> ± √-<hi rend="italics">b</hi><hi rend="sup">2</hi>)<hi rend="sup"><hi rend="italics">m</hi>/<hi rend="italics">n</hi></hi> by another Impossible Binomial; and thence the value of (<hi rend="italics">a</hi> + √-<hi rend="italics">b</hi><hi rend="sup">2</hi>)<hi rend="sup"><hi rend="italics">m</hi>/<hi rend="italics">n</hi></hi> + (<hi rend="italics">a</hi> - √-<hi rend="italics">b</hi><hi rend="sup">2</hi>)<hi rend="sup"><hi rend="italics">m</hi>/<hi rend="italics">n</hi></hi>, which is always possible, whether <hi rend="italics">a</hi> or <hi rend="italics">b</hi> be the greater of the two.</p><p><hi rend="italics">Solution.</hi> Put <hi rend="italics">b</hi>/<hi rend="italics">a</hi> = tang. <hi rend="italics">z.</hi> Then , where the first or second of these two last expressions is to be used, according as <hi rend="italics">z</hi> is an extreme or mean arc; or rather, because <hi rend="italics">b</hi>/<hi rend="italics">a</hi> is not only the tangent of <hi rend="italics">z,</hi> but also of <hi rend="italics">z</hi> + 360°, <hi rend="italics">z</hi> + 720°, &c; therefore the factor in the answer will have several values, viz, 2 cos.(<hi rend="italics">m</hi>/<hi rend="italics">n</hi>)<hi rend="italics">z;</hi> 2 cos.(<hi rend="italics">m</hi>/<hi rend="italics">n</hi>)(<hi rend="italics">z</hi> + 360°); 2 cos.(<hi rend="italics">m</hi>/<hi rend="italics">n</hi>)(<hi rend="italics">z</hi> + 720°); &c; the number of which, if <hi rend="italics">m</hi> and <hi rend="italics">n</hi> be whole numbers, and the fraction <hi rend="italics">m</hi>/<hi rend="italics">n</hi> be in its least terms, will be equal to the denominator <hi rend="italics">n;</hi> otherwise infinite.</p><p><hi rend="italics">By Logarithms.</hi> Put log. <hi rend="italics">b</hi> + 10-log. <hi rend="italics">a</hi> = log. tan. <hi rend="italics">z.</hi> Then log. ; where the first or second expression is to be used, according as <hi rend="italics">z</hi> is an extreme or mean arc. Moreover by taking successively, l. cos.(<hi rend="italics">m</hi>/<hi rend="italics">n</hi>)<hi rend="italics">z;</hi> l. cos.(<hi rend="italics">m</hi>/<hi rend="italics">n</hi>)(<hi rend="italics">z</hi> + 360°); l.cos.<hi rend="italics">m</hi>/<hi rend="italics">n</hi> (<hi rend="italics">z</hi> + 720°); &c, there will arise several distinct answers to the question, agreeably to the remark above.</p><p>BINOMIAL <hi rend="italics">Theorem.</hi> Francis Maseres, Esq. (Cursitor Baron of the Exchequer) has communicated <pb n="732"/><cb/> the following observations on the Binomial theorem, and its demonstration; viz, About the year 1666 the celebrated Sir Isaac Newton discovered that, if <hi rend="italics">m</hi> were put for any whole number whatsoever, the coefficients of the terms of the <hi rend="italics">m</hi>th power of 1 + <hi rend="italics">x</hi> would be 1, <hi rend="italics">m</hi>/1, <hi rend="italics">m</hi>/1.(<hi rend="italics">m</hi> - 1)/2, <hi rend="italics">m</hi>/1.(<hi rend="italics">m</hi> - 1)/2.(<hi rend="italics">m</hi> - 2)/3, &c, till we come to the term (<hi rend="italics">m</hi> - (<hi rend="italics">m</hi> -1))/<hi rend="italics">m,</hi> which will be the last term. But how he discovered this proposition, he has not told us, nor has he even attempted to give a demonstration of it. Dr. John Wallis, of Oxford, informs us (in his Algebra, chap. 85, pa. 319) that he had endeavoured to find this manner of generating these coefficients one from another, but without success; and he was greatly delighted with the discovery, when he found that Mr. Newton had made it. But he likewise has omitted to give a demonstration of it, as well as Sir Isaac Newton; and probably he did not know how to demonstrate it.</p><p>Sir Isaac Newton, after he had discovered this rule for generating the coefficients of the powers of 1 + <hi rend="italics">x</hi> when the indexes of those powers were whole numbers, conjectured that it might possibly be true likewise when they were fractions. He therefore resolved to try whether it was or not, by applying it to such indexes in a few easy instances, and particularly to the indexes 1/2 and 1/3, which, if the rule held good in the case of fractional indexes, would enable him to find serieses equal to the values of ―(1 + <hi rend="italics">x</hi>))<hi rend="sup">1/2</hi> and ―(1 + <hi rend="italics">x</hi>))<hi rend="sup">1/3</hi>, or the square-root and the cube-root of the Binomial quantity 1 + <hi rend="italics">x.</hi> And, when he had in this manner obtained a series for ―(1 + <hi rend="italics">x</hi>))<hi rend="sup">1/2</hi>, which he suspected to be equal to ―(1 + <hi rend="italics">x</hi>))<hi rend="sup">1/2</hi>, or the square root of 1 + <hi rend="italics">x,</hi> he multiplied the said series into itself, and found that the product was 1 + <hi rend="italics">x;</hi> and when he had obtained a series for ―(1 + <hi rend="italics">x</hi>))<hi rend="sup">1/3</hi> he multiplied the said series twice into itself, and found that the product was 1 + <hi rend="italics">x;</hi> and thence he concluded that the former series was really equal to the square-root of 1 + <hi rend="italics">x,</hi> and that the latter series was really equal to its cube-root. And from these and a few more such trials, in which he found the rule to answer, he concluded universally that the rule was always true, whether the index <hi rend="italics">m</hi> stood for a whole number or a fraction of any kind, as 1/2, 1/3, 2/3, 3/2, 5/9, 9/5, or, in general <hi rend="italics">p</hi>/<hi rend="italics">q.</hi></p><p>After the discovery of this rule by Sir Isaac Newton, and the publication of it by Dr. Wallis, in his Algebra, chap. 85, in the year 1685, (which l believe was the first time it was published to the world at large, though it was inserted in Sir Isaac Newton's first letter to Mr. Oldenburgh, the secretary to the Royal Society, dated June 13, 1676, and the said letter was shewn to Mr. Leibnitz, and probably to some other of the learned mathematicians of that time) it remained for some years without a demonstration, either in the case of integral powers or of roots. At last however it was demon- <cb/> strated in the case of integral powers by means of the properties of the figurate numbers, by that learned, sagacious, and accurate mathematician Mr. James Bernoulli, in the 3d chapter of the 2d part of his excellent treatise <hi rend="italics">De Arte Conjectandi,</hi> or, <hi rend="italics">On the Art of forming reasonable Conjectures concerning Events that depend on Chance;</hi> which appears to me to be by much the best written treatise on the doctrine of Chances that has yet been published, though Mr. Demoivre's book on the same subject may have carried the doctrine something further. This treatise of Mr. James Bernoulli's was not published till the year 1713, which was some years after his death, which happened in August 1705; but there is reason to think that it was composed in the latter years of the preceding century, about the years 1696, 1697, 1698, 1699, and 1700, and even that some parts of it, or some of the propositions inserted in it, had been found out by the author in the years 1689, 1690, 1691, and 1692. For the first part of his very curious tract, intitled, <hi rend="italics">Positiones Arithmeticæ de Seriebus Infinitis</hi> was published at Basil or Basle in Switzerland (which was his native place, and in which he was at that time professor of mathematics) in the year 1689; and the second part of the said <hi rend="italics">Positiones</hi> (in the 19th Position of which those properties of the figurate numbers from which the Binomial theorem may be deduced, are set down) was published at the same place in the year 1692. But the demonstrations of those properties of the figurate numbers, and of the Binomial theorem, which depends upon them, were never as I believe communicated to the public till the year 1713, when the author's posthumous treatise <hi rend="italics">De Arte Conjectandi</hi> made its appearance. These demonstrations are founded on clear and simple principles, and afford as much satisfaction as can well be expected on the subject. But the full display and explanation of these principles, and the deduction of the said properties of the figurate numbers, and ultimately of the Binomial theorem, from them, is a matter of considerable length. It will not therefore be amiss to give a shorter proof of the truth of this important theorem, that shall not require a previous knowledge of the properties of the figurate numbers, but yet shall be equally conclusive with that which is derived from those properties. Now this may be done in the manner following.</p><p>Let us suppose that the coefficients of the terms of the first six powers of the Binomial quantity 1 + <hi rend="italics">x</hi> have been found, upon trial, to be such as would be produced by the general expressions 1, <hi rend="italics">m</hi>/1, <hi rend="italics">m</hi>/1.(<hi rend="italics">m</hi> - 1)/2, <hi rend="italics">m</hi>/1.(<hi rend="italics">m</hi> - 1)/2.(<hi rend="italics">m</hi> - 2)/3, &c, by substituting in them first 1, then 2, then 3, then 4, then 5, and lastly 6, instead of <hi rend="italics">m.</hi> This may easily be tried by raising the said first six powers of 1 + <hi rend="italics">x</hi> by repeated multiplications by 1 + <hi rend="italics">x</hi> in the common way, and afterwards finding the terms of the same powers by means of the said general expressions above; which will be found to produce the very same terms as arose from the multiplications. After these trials we shall be sure that those general expressions are the true values of the coefficients of the powers of 1 + <hi rend="italics">x</hi> at least in the said first six powers. And it will therefore only <pb n="733"/><cb/> remain to be proved that, since the rule is true in the said first six powers, it will also be true in the next following, or the 7th power, and consequently in the 8th, 9th and 10th powers, and in all higher powers whatsoever.</p><p>Now, if the coefficients of the 1st, 2d, 3d, 4th, and other following terms of (―(1 + <hi rend="italics">x</hi>)))<hi rend="sup"><hi rend="italics">m</hi></hi> be denoted by the letters <hi rend="italics">a, b, c, d,</hi> &c, respectively, it is evident from the nature of multiplication, that the coefficients of the 1st, 2d, 3d, 4th, and other following terms of the next higher power of 1 + <hi rend="italics">x,</hi> to wit, (―(1 + <hi rend="italics">x</hi>)))<hi rend="sup"><hi rend="italics">m</hi> + 1</hi> will be equal to <hi rend="italics">a, a</hi> + <hi rend="italics">b, b</hi> + <hi rend="italics">c, c</hi> + <hi rend="italics">d,</hi> &c, respectively, or to the sums of every two contiguous coefficients of the terms of the preceding series which is = (―(1 + <hi rend="italics">x</hi>)))<hi rend="sup"><hi rend="italics">m</hi></hi>. This will appear from the operation of multiplication, which is as follows. <table><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">bx</hi> + <hi rend="italics">cx</hi><hi rend="sup">2</hi> + <hi rend="italics">dx</hi><hi rend="sup">3</hi> + <hi rend="italics">ex</hi><hi rend="sup">4</hi> + &c</cell></row><row role="data"><cell cols="1" rows="1" role="data">1 + <hi rend="italics">x</hi></cell></row><row role="data"><cell cols="1" rows="1" role="data"><hi rend="italics">a</hi> + <hi rend="italics">bx</hi> + <hi rend="italics">cx</hi><hi rend="sup">2</hi> + <hi rend="italics">dx</hi><hi rend="sup">3</hi> + <hi rend="italics">ex</hi><hi rend="sup">4</hi> + &c</cell></row><row role="data"><cell cols="1" rows="1" role="data">  + <hi rend="italics">ax</hi> + <hi rend="italics">bx</hi><hi rend="sup">2</hi> + <hi rend="italics">cx</hi><hi rend="sup">3</hi> + <hi rend="italics">dx</hi><hi rend="sup">4</hi> + &c.</cell></row></table> Therefore, if (―(1 + <hi rend="italics">x</hi>)))<hi rend="sup"><hi rend="italics">m</hi></hi> is equal to the series , then (―(1 + <hi rend="italics">x</hi>)))<hi rend="sup"><hi rend="italics">m</hi> + 1</hi> will be equal to the series .</p><p>Now let <hi rend="italics">n</hi> be = <hi rend="italics">m</hi> + 1. We shall then have to prove that, if the coefficients <hi rend="italics">a, b, c, d,</hi> &c, be respectively equal to 1, <hi rend="italics">m</hi>/1, (<hi rend="italics">m</hi>/1).((<hi rend="italics">m</hi> - 1)/2), (<hi rend="italics">m</hi>/1).((<hi rend="italics">m</hi> - 1)/2).((<hi rend="italics">m</hi> - 2)/3), &c, the coefficients <hi rend="italics">a, a</hi> + <hi rend="italics">b, b</hi> + <hi rend="italics">c,</hi> &c, will be respectively equal to 1, <hi rend="italics">n</hi>/1, (<hi rend="italics">n</hi>/1).((<hi rend="italics">n</hi> - 1)/2), (<hi rend="italics">n</hi>/1).((<hi rend="italics">n</hi> - 1)/2).((<hi rend="italics">n</hi> - 2)/3), &c.</p><p>In order to prove this, there is nothing more to do than to collect together every two terms of the former of these two series, and then substitute into these sums, <hi rend="italics">n</hi> instead of <hi rend="italics">m</hi> + 1, when there will immediately come out the terms of the latter series as above, viz, . <hi rend="smallcaps">Q. E. D.</hi></p><p><hi rend="smallcaps">Binomial</hi> <hi rend="italics">Theorem, Improvement of.</hi> Mr Bonnycastle, of the Royal Mil. Acad. has lately discovered the following ingenious improvement of this theorem, which is now published for the first time.</p><p>This celebrated theorem has been given under various forms, since the time of its first invention; but the following property of it is conceived to be new, and capable of an application of which the original series is not susceptible.</p><p>The Newtonian theorem, in one of its most commodious forms, is <cb/> &c; and the new theorem here alluded to, is &c. Of which the investigation is as follows: &c. Then by connecting the several powers of <hi rend="italics">p</hi> with all the like powers of <hi rend="italics">n,</hi> the latter series will become ; which by abbreviation, &c, becomes . In which last series, the literal parts of the coefficients of the 3d, 4th, 5th, &c terms, are the square, cube, biquadrate, &c, of the coefficient of the 2d term, as will appear either from the actual involution of &c, or by comparing its several powers with the multinomial theorem of Demoivre.</p><p>From hence it follows that, <pb n="734"/><cb/> &c. And if &c be put = <hi rend="italics">s,</hi> we shall have &c, as was to be shewn. <cb/></p><p>By a similar mode of deduction, it may also be proved that ; where in this case .</p><p>In each of which formulæ, the index <hi rend="italics">n,</hi> may be considered either as a whole number, a fraction, a surd, a given or an unknown quantity, as the circumstance may require.</p><p>For the application of these theorems, see L<hi rend="smallcaps">OGARITHMS</hi>, and <hi rend="smallcaps">Exponential</hi> <hi rend="italics">Equations,</hi> following. </p></div1></div0><div0 part="N" n="C" org="uniform" sample="complete" type="alphabetic letter"><head>C</head><cb/><div1 part="N" n="CANAL" org="uniform" sample="complete" type="entry"><head>CANAL</head><p>, in general, denotes a long, round, hollow instrument, through which a fluid matter may be conveyed. In which sense, it amounts to the same as what is otherwise called a pipe, tube, channel, &c. Thus the Canal of an aqueduct, is the part through which the water passes; which, in the ancient works of this kind, is lined with a coat of mastic of a peculiar composition.</p><p><hi rend="smallcaps">Canal</hi> more particularly denotes a kind of artificial river, often furnished with locks and sluices, and sustained by banks or mounds. They are contrived for divers purposes; some for forming a communication between one place and another; as the Canals between Bruges and Ghent, or between Brussels and Antwerp: Others for the decoration of a garden, or house of pleasure; as the Canals of Versailles, Fontainbleau, St. James's Park, &c: And others are made for draining wet and marshy lands; which last however are more properly called water-gangs, drains, ditches, &c.</p><p>It is needless to enumerate the many advantages arising from Canals and artificial navigations. Their utility is now so apparent, that most nations in Europe give the highest encouragement to undertakings of this kind wherever they are practicable. Nor did their advantages escape the observation of the Ancients. From the earliest accounts of society we read of attempts to cut through large isthmuses, to make communications by water, either between one sea and another, or between different nations, or distant parts of the same nation, where land-carriage was long and expensive.</p><p>Egypt is full of Canals, dug to receive and distribute the waters of the Nile, at the time of its inundation. They are dry the rest of the year, except the Canal of Joseph, and four or five others, which may be ranked as considerable rivers. There were also subterraneous Canals, or tunnels, dug by an ancient king of Egypt, by which those lakes, formed by the inundations of the Nile, were conveyed into the Mediterranean sea. <cb/></p><p>Herodotus relates, that the Cnidians, a people of Coria, in Asia Minor, designed to cut through the isthmus which joins that peninsula to the continent; but were superstitious enough to give up the undertaking, because it was interdicted by an oracle.</p><p>Several kings of Egypt attempted to join the RedSea to the Mediterranean; a project which Cleopatra was very fond of. This Canal was begun, according to Herodotus, by Necus son of Psammeticus, who desisted from the attempt on an answer from the oracle, after having lost 120 thousand men in the enterprise. It was resumed and completed by Darius son of Hystaspes, or, according to Diodorus and Strabo, by Ptolomy Philadelphus; who relate that Darius relinquished the work on a representation made to him by unskilful engineers, that the Red-Sea, being higher than the land of Egypt, would overflow and drown the whole country. It was wide enough for two galleys to pass abreast, and its length was four days sailing. Diodorus adds, that it was also called Ptolomy's river; that this prince built a city at its mouth on the Red-Sea, which he called Arsinoë, from the name of his favourite sister; and that the Canal might be either opened or shut, as occasion required. Diod. Sic. lib. 1; Strabo, Geog. lib. 17; Herod. lib. 2. Soliman the 2d, emperor of the Turks, employed 50 thousand men in this great work; which was completed under the caliphate of Omar, about the year 635; but was afterward allowed to fall into neglect and disrepair; so that it is now difficult to discover any traces of it. Hist. Acad. Scienc. ann. 1703, pa. 110.</p><p>Both the Greeks and Romans intended to make a Canal across the Isthmus of Corinth, which joins the Morea and Achaia, for a navigable passage by the Ionian sea into the Archipelago. Demetrius, Julius Cæsar, Caligula, and Nero, made several unsuccessful efforts to open this passage. But as the Ancients were entirely ignorant of the use of water-locks, their whole <pb n="735"/><cb/> attention was employed in making level cuts, which is probably the chief reason why they so often failed in their attempts. Charlemagne formed a design of joining the Rhine and the Danube, to make a communication between the Ocean and the Black-Sea, by a Canal from the river Almutz which discharges itself into the Danube, to the Reditz, which falls into the Maine, which last falls into the Rhine near Mayence or Mentz: for this purpose he employed a prodigious number of workmen; but he met with so many obstacles from different quarters, that he was obliged to give up the attempt.</p><p>A new Canal for conveying the waters of the Nile from Ethiopia into the Red-Sea without passing into Egypt, was projected by Albuquerque, viceroy of India for the Portuguese, to render Egypt barren and unprofitable to the Turks.—M. Gaildereau attributes the frequency of the plague in Egypt, of late days, to the decay, or stopping up of these Canals; which happened upon the Turks becoming masters of the country.</p><p>In China, there is scarce a town or village without the advantage either of an arm of the sea, a navigable river, or a Canal, by which means navigation is rendered so common, that there are almost as many people on the water as the land. The great Canal of China, is one of the wonders of art, extending from north to south quite across the empire, from Pekin to Canton, a distance of 825 miles, and was made upwards of 800 years ago. Its breadth and depth are sufficient to carry barks of considerable burden, which are managed by sails and masts, as well as towed by hand. On this Canal it seems the emperor employs near ten thousand ships. It passes through, or by, 41 large cities; there are in it 75 vast locks and sluices, to keep up the water, and pass the ships where the ground will not admit of sufficient depth of channel, beside several thousand draw and other bridges. Indeed, F. Magaillane assures us, there are passages from one end of China to the other, the space of 600 French leagues, either by Canals or rivers, except a single day's journey by land, necessary to cross a mountain.</p><p>The French at present have many fine Canals. That of Briere, otherwise called the Canal of Burgundy, was begun under Henry IV, and finished under the direction of cardinal Richelieu in the reign of Louis XIII. This Canal makes a communication between the Loire and the Seine, and so to Paris. It extends 11 French great leagues from Briere to Montargis, and has 42 locks upon it.</p><p>The Canal of Orleans was begun in 1675, for establishing a communication also between the Seine and the Loire. It is considerably shorter than that of Briere, and has only 20 sluices.</p><p>The Canal of Bourbon was but lately undertaken: its design is to make a communication from the river Oise to Paris.</p><p>But the greatest and most useful work of this kind, is the junction of the Ocean with the Mediterranean by the Canal of Languedoc, called also the Canal of the two seas. It was proposed in the reigns of Francis I and Henry IV, and was begun and finished under Louis XIV; having been planned by Francis Riquet in the year 1666, and finished before his <cb/> death, which happened in 1680. It begins with a large reservoir 4000 paces in circumference, and 24 feet deep, which receives many springs from the mountain Noire. The Canal is about 200 miles in length, extending from Narbonne to Tholouse, being supplied by a number of rivulets in the way, and furnished with 104 locks or sluices, of about 8 feet rise each. In some places it is carried over bridges and aqueducts of vast height, which give passage underneath to other rivers; and in some places it is cut through solid rocks for a mile together.</p><p>The new Canal of the lake Ladoga, cut from Volhowa to the Neva, by which a communication is made between the Baltic, or rather Ocean, and the Caspian sea, was begun by the czar Peter the 1st in 1719: by means of which the English and Dutch merchandize is easily conveyed into Persia, without being obliged to double the Cape of Good Hope.—There was a former Canal of communication between the Ladoga lake and the river Wolga, by which timber and other goods had been brought from Persia to Petersburg; but the navigation of it was so dangerous, that a new one was undertaken.</p><p>The Spaniards have several times had in view the digging a Canal through the Isthmus of Darien, between North and South America, from Panama to Nombre de Dios, to make a ready communication between the Atlantic and the South Sea, and thus afford a straight passage to China and the East Indies.</p><p>In the Dutch, Austrian, and French Netherlands, there is a great number of Canals: that from Bruges to Ostend carries vessels of 200 tons. But it would be an endless task to describe the numberless Canals in Holland, Germany, Russia, &c. We may therefore only take a view of those in our own country.</p><p>In England, that ancient Canal from the river Nyne, a little below Peterborough, to the river Witham, three miles below Lincoln; called by the modern inhabitants Caerdike; may be ranked among the monuments of the Roman grandeur, though it is now most of it filled up. Morton will have it made under the emperor Domitian. Urns and medals have been discovered on the banks of this Canal, which seem to confirm that opinion. Yet some authors take it to be a Danish work. It was 40 miles in length; and, so far as appears from the ruins, must have been very broad and deep. Notwithstanding that early beginning, it is not long since Canals have been revived in this country. They are now however become very numerous, particularly in the counties of York, Lincoln, and Cheshire. Most of the counties between the mouth of the Thames and the Bristol channel are connected together either by natural or artificial navigations; those upon the Thames and Isis reaching within about 20 miles of those upon the Severn.</p><p>The Canal for supplying London with water by means of the New River, was projected and begun by Mr. Edward Wright, author of the celebrated treatise on Navigation, about the year 1608; but finished by Mr. (afterwards Sir Hugh) Middleton, five years after. This Canal commences near Ware, in Hertfordshire, and takes a course of 60 miles before it reaches the cistern at Islington, which supplies the several water pipes that convey it to the city and parts adjacent. In some <pb n="736"/><cb/> places it is 30 feet deep, and in others it is conveyed over a valley between two hills, by means of a trough supported on wooden arches, and rising above 23 feet in height.</p><p>The Duke of Bridgwater's Canal, projected and executed under the direction of Mr. Brindley, was begun about the year 1759. It was first designed only for conveying coals to Manchester, from a mine in the duke's estate; but has since been applied to many other useful purposes of inland navigation. This Canal begins at a place called Worsley-mill, about 7 miles from Manchester, where a bason is made capable of holding all the boats, and a great body of water which serves as a reservoir or head to the navigation. The Canal runs through a hill by a subterraneous passage, large enough for admitting long flat-bottomed boats, which are towed by a rail on each hand, near three quarters of a mile, to the coal-works. There the passage divides into two channels, one of which goes off 300 yards to the right, and the other as many to the left; and both may be continued at pleasure. The passage is in some places cut through the solid rock, and in others arched over with brick; and air-funnels, some of which are near 37 yards perpendicular, are cut, at certain distances, through the rock to the top of the hill. The arch at its entrance is about 6 feet wide, and about 5 feet high from the surface of the water; but widens within, so that in some places the boats may pass one another, and at the pits it is 10 feet wide. When the boats are loaded and brought out of the bason, five or six of them are linked together, and drawn along the Canal by a single horse, and thus reaching Manchester in a course of nine miles. It is broad enough for two barges to pass or go abreast; and on one side there is a good road for the passage of the people, and the horses or mules employed in the work. The Canal is raised over public roads by means of arches; and it passes over the navigable river Irwell near 50 feet above it; so that large vessels in full sail pass under the Canal, while the duke's barges are at the same time passing over them. This Canal joins that which passes from the river Mersey towards the Trent, taking in the whole a course of 34 miles.</p><p>The Lancaster Canal begins near Kendal, and terminates near Eccleston, comprehending the distance of 72 1/2 miles.</p><p>The Canal from Liverpool to Leeds is 108 1/3 miles: that from Leeds to Selby, 23 1/4 miles; from Chichester to Middlewich, 26 3/4 miles; from the Trent to the Mersey, 88 miles; from the Trent to the Severn, 46 1/2 miles. The Birmingham Canal joins this near Wolverhampton, and is 24 1/4 miles: the Droitwich Canal is 5 1/2 miles: the Covehtry Canal, commencing near Lichfield, and joining that of the Trent, is 36 1/4 miles: the Oxford Canal breaks off from this, and is 82 miles: the Chesterfield Canal joins the Trent near Gainsborough, and is 44 miles.</p><p>A communication is now formed, by means of this inland navigation, between Kendal and London, by way of Oxford; between Liverpool and Hull, by the way of Leeds; and between the Bristol channel and the Humber, by the junction formed between the Trent and the Severn. Other schemes have been projected, which the present spirit of improvement will probably soon carry into execution, of opening a communication <cb/> between the German and Irish seas, <*> reduce a <*> zardous navigation of more than 800 miles by sea, into a little more than 150 miles by land, or inland navigation; and also of joining the Isis with the Severn.</p><p>In Scotland, a navigable Canal between the Forth and Clyde, which divides that country into two parts, was thought of more than a century since, for transports and small ships of war. It was again projected in the year 1722, and a survey made; but nothing more was done till 1761, when the then lord Napier, at his own expence, had a survey, plan, and estimate made on a small scale. In 1764, the trustees for fisheries, &c, in Scotland, procured another survey, plan, and estimate of a Canal 5 feet deep, which was to cost 79,000 pounds. In 1766, a subscription was obtained by a number of the most respectable merchants in Glasgow, for making a Canal 4 feet deep and 24 feet in breadth; but when the bill was nearly obtained in parliament, it was given up on account of the smallness of the scale, and a new subscription set on foot for a Canal 7 feet deep, estimated at 150,000 pounds. This obtained the sanction of parliament; and the work was begun in 1768, by Mr. Smeaton the engineer. The extreme length of the Canal from the Forth to the Clyde is 35 miles, beginning at the mouth of the Carron, and ending at Dalmure Burnfoot on the Clyde, 6 miles below Glasgow, rising and falling 160 feet by means of 39 locks, 20 on the east side of the summit, and 19 on the west, as the tide does not ebb so low in the Clyde as in the Forth by 9 feet; and it was deepened to upwards of 8 feet. This Canal was finished a few years since, after having experienced some interruptions and delays, for want of resources, and is esteemed the greatest work of the kind in this island. Vessels drawing 8 feet water, with 19 feet in the beam and 73 feet in length, pass with ease; and the whole enterprise displays the art of man in a high degree. To supply the Canal with water was of itself a very great work. There is one reservoir of 50 acres 24 feet deep, and another of 70 acres 22 feet deep, in which many rivers and springs terminate, which it is expected will afford sufficient supply of water at all times. <hi rend="center"><hi rend="italics">The Practice of Canal Digging and Inland Navigations.</hi></hi></p><p>The particular operations necessary for making artificial navigations, depend upon a number of circumstances. The situation of the ground; the vicinity or connection with rivers; the ease or difficulty with which a proper quantity of water can be obtained: these and many other circumstances necessarily produce great variety in the structure of artificial navigations, and augment or diminish the labour and expence of executing them. When the ground is naturally level, and unconnected with rivers, the execution is easy, and the navigation is not liable to be disturbed by floods: but when the ground rises and falls, and cannot be reduced to a level, artificial methods of raising and lowering vessels must be employed; which likewise vary according to circumstances.</p><p>Sometimes a kind of temporary sluices are employed, to raise boats over falls or shoals in rivers, by a very simple operation. Two pillars of mason-work, with grooves, are fixed, one on each bank of the river, at <pb n="737"/><cb/> some distance below the shoal. The boat having passed these pillars, strong planks are let down across the river by pulleys into the grooves; by which means the water is dammed up to a proper height for allowing the boat to pass up the river over the shoal.</p><p>The Dutch and Flemings at this day sometimes, when obstructed by cascades, form an inclined plane or rolling-bridge upon dry land, along which their vessels are drawn from the river below the cascade, into the river above it. This it is said was the only method employed by the Ancients, and still sometimes used by the Chinese. These rolling-bridges consist of a number of cylindrical rollers which turn easily on pivots. And a mill is commonly built near; so that the same machinery may serve the double purpose of working the mill and drawing up vessels.</p><p>But in the present improved state of inland navigation, these falls and shoals are commonly surmounted by means of what are called locks or sluices. A lock is a bason placed lengthwise in a river or Canal, lined with walls of masonry on each side, and terminated by two gates placed across the Canal, where there is a cascade or natural fall of the country; and so construct. ed, that the bason being filled with water by an upper sluice to the level of the waters above, a vessel may ascend through the upper gate; or the water in the lock being reduced to the level of the water at the bottom of the cascade, the vessel may descend through the lower gate: for when the waters are brought to a level on either side, the gate on that side may be easily opened.</p><p>But as the lower gate is strained in proportion to the depth of water it supports, when the perpendicular height of the water exceeds 12 or 13 feet, it becomes necessary to have more locks than one. Thus, if the fall be 16 feet, two locks are required, each of 8 feet fall; and if the fall be 25 feet, three locks are necessary, each having 8 feet 4 inches fall.—It is evident that the sidewalls of locks should be made very strong: and where the natural foundation is bad, they should be founded on piles and platforms of wood. They should likewise slope outwards, in order to resist the pressure of the earth from behind.</p><p>To illustrate this by representations: Plate 37, fig. 1, is a perspective view of part of a Canal, with several locks &c; the vessel L being within the lock AC.— Fig. 2 is an elevation or upright section along the Canal; the vessel L about to enter.—Fig. 3, a like section of a lock full of water; the vessel L being raised to a level with the water in the superior Canal.—And fig. 4 is the plan or ground section of a lock: where L is a vessel in the inferior Canal; C, the under gate; A, the upper gate; GH, a subterraneous passage for letting water from the superior Canal run into the lock; and KF, a subterraneous passage for water from the lock to the inferior Canal.</p><p>X and Y (fig. 1) are the two flood-gates, each of which consists of two leaves, resting upon one another, so as to form an obtuse angle, the better to resist the pressure of the water. The first (X) prevents the water of the superior Canal from falling into the lock; and the second (Y) dams up and sustains the water in the lock. These flood-gates ought to be very strong, and to turn freely upon their hinges. They should also be <cb/> made very tight and close, that as little water as possible may be lost. And, to make them open and shut with ease, each leaf is furnished with a long lever A<hi rend="italics">b,</hi> A<hi rend="italics">b;</hi> C<hi rend="italics">b,</hi> C<hi rend="italics">b.</hi></p><p>By the subterraneous passage GH (fig. 2, 3, 4) which descends obliquely, by opening the sluice G, the water is let down from the superior Canal D into the lock, where it is stopped and retained by the gate C when shut, till the water in the lock comes to be on a level with the water in the superior Canal D; as represented in fig. 3. When, on the other hand, the water contained by the lock is to be let out, the passage GH must be shut, by letting down the sluice G; the gate A must also be shut, and the passage KF opened by raising the sluice K. A free passage being thus given to the water, it descends through KF, into the inferior Canal, until the water in the lock be on a level with the water in the inferior Canal B; as represented in fig. 2.</p><p>Now suppose it be required to raise the vessel L (fig. 2) from the inferior Canal B to the superior one D. If the lock be full of water, the sluice G must be shut, as also the gate A, and the sluice K opened, so that the water in the lock may run out till it become to a level with the water in the inferior Canal B. When the water in the lock comes to be on a level with the water at B, the leaves of the gate C are opened by the levers C<hi rend="italics">b,</hi> which is easily performed, the water on each side of the gate being in equilibrio; the vessel then sails into the lock. After this, the gate C and the sluice K are shut, and the sluice G opened, in order to fill the lock, till the water in the lock, and consequently the vessel, be upon a level with the water in the superior Canal D; as is represented in fig. 3. The gate A is then opened, and the vessel passes into the Canal D.</p><p>Again let it be required to make a vessel descend from the Canal D into the inferior Canal B. If the lock be empty, as in fig. 2, the gate C and sluice K must be shut, and the upper sluice G opened, so that the water in the lock may rise to a level with the water in the upper Canal D. Then, opening the gate A, the vessel will pass through into the lock. This done, shut the gate A and the sluice G; then open the sluice K, till the water in the lock be on a level with the water in the inferior Canal; this done, the gate C is opened, and the vessel passes along into the Canal B, as was required.</p><p>CATENARY. Line 4, <hi rend="italics">for</hi> ACB <hi rend="italics">read</hi> BAC.— 1. 6, <hi rend="italics">for</hi> A and B <hi rend="italics">read</hi> C and B. <hi rend="italics">After which add,</hi> It is otherwise called the <hi rend="italics">Elastic Curve.</hi></p><p>CHALDRON. Line 4, <hi rend="italics">for</hi> 2000 pounds, <hi rend="italics">read</hi> 28 cwt. or 3136 pounds. <hi rend="italics">At the end add,</hi> By act of parliament, a Newcastle Chaldron is to weigh 52 1/2 cwt, or 3 waggons of 17 1/2 cwt, or 6 carts of 8 3/4 cwt each, making 52 1/2 cwt to the Chaldron. The statute London Cha<*>n is to consist of 36 bushels heaped up, each bushel to contain a Winchester bushel and one quart, and to be 19 1/2 inches diameter externally. Now it has been found by repeated trials, that 15 London Chaldrons are equal to 8 Newcastle Chaldrons, which, reckoning 52 1/2 cwt to the latter, gives 28 cwt to the former, or 3136 lbs to the London Chaldron.</p><p>This I find nearly confirmed by experiment. I <pb n="738"/><cb/> weighed one peck of coals, which amounted to 21 3/4 lb. Then 4 times this gives 87 lb for the weight of the bushel; and 36 times the bushel gives 3132 lb for the Chaldron; to which if the weight of the odd quart be added, or 3 lb nearly, it gives 3135 lb for the weight of the Chaldron, which is only one pound less than by statute.</p><p>Pa. 287, col. 2, l. 20, <hi rend="italics">for</hi> YX - <hi rend="italics">a</hi> - <hi rend="italics">x, read</hi> YX = <hi rend="italics">a</hi> - <hi rend="italics">x.</hi></p><p>CIRCLE <hi rend="italics">of Curvature.</hi> To what is said of this article in the Dictionary, may be added what follows.</p><p>A circular are is the only curve line that is equally curved in every point. In all other curve lines, such as the are of an ellipse, or a parabola, or an hyperbola, or a cycloid, the curvature is different in different points, and the degree of curvature in any point is estimated by the curvature of a Circle which is said to have the same curvature as the proposed curve line in that point; by which is understood the Circle which, having the tangent of the proposed curve in the said point for its tangent, approaches so nearly to the proposed curve that no other Circle whatever can be drawn between it and that curve.</p><p>This Circle is also said to <hi rend="italics">osculate</hi> the curve in the said point, and is therefore often called the <hi rend="italics">osculating Circle,</hi> as well as the <hi rend="italics">Circle of equal curvature</hi> with the curve in the said point. And the radius of this Circle is called the <hi rend="italics">radius of curvature</hi> of the proposed curve in the said point; also its centre is called the <hi rend="italics">centre of curvature.</hi></p><p>Now there are some curve lines so very highly curved in some particular points, that every Circle, of how small a radius soever, having the tangent to the curve in one of those points for its tangent, will pass without the curve, or between the curve and its tangent. This, for example, is the case with the curve of a cycloid in the two points contiguous to its base, as also with the cissoid at its vertex. And in such points the curvature of these curves is said to be <hi rend="italics">infinite,</hi> because it is greater than the curvature of any Circle, how small soever. Also the radius of the Circle of curvature in such points is nothing; the length of that radius being always inversely or reciprocally as the degree of curvature at any point.</p><p>The theory of these Circles of equal curvature with curves in particular points was first cultivated by Apollonius in his Conic Sections: and it has since been carried much farther by several great mathematicians of modern times; particularly by Mr. Huygens in his doctrine of Evolute Curves and Curves of Evolution, and by the great Sir Isaac Newton. See <hi rend="smallcaps">Curvature.</hi></p></div1><div1 part="N" n="CLARKE" org="uniform" sample="complete" type="entry"><head><persName><surname full="yes">CLARKE</surname> (Dr. <foreName full="yes"><hi rend="smallcaps">Samuel</hi></foreName>)</persName></head><p>, a celebrated English divine, philosopher, and metaphysician, was the son of Edward Clarke, Esq. alderman of Norwich, and for several years one of its representatives in parliament; and was born there the 11th of October 1675. He was instructed in classical learning at the free-school of that town; and in 1691 removed thence to Caius college in Cambridge; where his uncommon abilities soon began to display themselves. Though the philosophy of Des Cartes was at that time the established philosophy of the <cb/> university, yet Clarke easily mastered the new system of Newton; and in order to his first degree of arts, performed a public exercise in the schools upon a question taken from it. He greatly contributed to the establishment of the Newtonian philosophy by an excellent translation of Rohault's Physics, with notes, which he finished before he was 22 years of age: a book which had been for some time the system used in the university, and founded upon Cartesian principles. This was first published in the year 1697, and it soon after went through several other editions, all with improvements.</p><p>Mr. Whiston relates that, in that year, 1697, while he was chaplain to Dr. Moore bishop of Norwich, he met with young Clarke, then wholly unknown to him, at a coffee-house in that city; where they entered into a conversation about the Cartesian philosophy, particularly Rohault's Physics, which Clarke's tutor, as he tells us, had put him upon translating. “The result of this conversation was, says Whiston, that I was greatly surprised that so young man as Clarke then was, should know so much of those sublime discoveries, which were then almost a secret to all, but to a few particular mathematicians. Nor did I remember (continues he) above one or two at the most, whom I had then met with, that seemed to know so much of that philosophy as Mr. Clarke.”</p><p>He afterwards turned his thoughts to divinity; and having taken holy orders, in 1698 he succeeded Mr. Whiston as chaplain to Dr. Moore bishop of Norwich, who was ever after his constant friend and patron. In 1699 he published two treatises: the one on Baptism, Confirmation, and Repentance; the other, Reflections on that part of a book called Amyntor, or a Defence of Milton's Life, which relates to the Writings of the Primitive Fathers, and the Canon of the New Testament. In 1701 he published A Paraphrase upon the Gospel of St. Matthew; which was followed in 1702 by the Paraphrases upon the Gospels of St. Mark and St. Luke, and soon after by a third volume upon St. John.</p><p>Mean while bishop Moore gave him the rectory of Drayton near Norwich, with a lectureship in that city. In 1704 he was appointed to preach Boyle's lecture; and the subject he chose was, The Being and Attributes of God. He succeeded so well in this, and gave so much satisfaction, that he was appointed to preach the same lecture the next year, when he chose for his subject, The Evidences of Natural and Revealed Religion. These sermons were first printed in two volumes, in 1705 and 1706; and contained some remarks on such objections as had been made by Hobbes and Spinoza, and other opposers of natural and revealed religion. In the 6th edition was added, A Discourse concerning the Connection of the Prophecies of the Old Testament, and the application of them to Christ.</p><p>About this time, Mr. Whiston informs us, he discovered that Mr. Clarke (having read much of the primitive writers) began to suspect that the Athanasian doctrine of the Trinity was not the doctrine of those early ages; and it was particularly remarked of him, that he never read the Athanasian Creed at his parish church.</p><p>In 1706 he published A Letter to Mr. Dodwell; answering all the arguments in his epistolary discourse against the immortality of the soul. Bishop Hoadley <pb n="739"/><cb/> observes, that in this letter he answered Mr. Dodwell in so excellent a manner, both with regard to the philosophical part, and to the opinions of some of the primitive writers, upon whom these doctrines were fixed, that it gave universal satisfaction. But this controversy did not stop here; for the celebrated Mr. Collins, coming in as a second to Dodwell, went much farther into the philosophy of the dispute, and indeed seemed to produce all that could be said against the immateriality of the soul, as well as the liberty of human actions. This enlarged the scene of the dispute; into which our author entered, and wrote with such a spirit of clearness and demonstration, as at once shewed him greatly superior to his adversaries in metaphysical and physical knowledge; making every intelligent reader rejoice that such an incident had happened to provoke and extort from him such excellent reasoning and perspicuity of expression.</p><p>In the midst of these labours, Mr. Clarke found time to shew his regard to mathematical and philosophical studies, with his exact knowledge and skill in them. And his natural affection and capacity for these studies were not a little improved by the friendship of Sir Isaac Newton; at whose request he translated his Optics into Latin in 1706. With this version Sir Isaac was so highly pleased, that he presented him with the sum of 500l. or 100l. to each of his five children.</p><p>The same year also, bishop Moore procured for him the rectory of St. Bennett's, Paul's Wharf, in London: and soon after carried him to court, and recommended him to the favour of queen Anne. She appointed him one of her chaplains in ordinary; and also presented him to the rectory of St. James's, Westminster, when it became vacant in 1709. Upon this occasion he took the degree of D. D. when the public exercise which he performed for it at Cambridge was highly admired.</p><p>The same year 1709, Dr. Clarke revised and corrected Whiston's translation of the Apostolical Constitutions into English, at his earnest request. In 1712 he published a most beautiful and pompous edition of Cæsar's Commentaries. And the same year, his celebrated book called, The Scripture Doctrine of the Trinity. Whiston informs us, that some time before the publication of this book, there was a message sent to the author by lord Godolphin, and others of queen Anne's ministers, importing, “That the affairs of the public were with difficulty then kept in the hands of those that were for liberty; that it was therefore an unseasonable time for the publication of a book that would make a great noise and disturbance; and that therefore they desired him to forbear till a fitter opportunity should offer itself:” which message, says he, the doctor paid no regard to, but went on according to the dictates of his own conscience with the publication of his book. The ministers however were very right in their conjectures; for the work made noise and disturbance enough, and occasioned a great many books and pamphlets, written by himself and others. Nor were these the whole that his work occasioned: it rendered the author obnoxious to the ecclesiastical power, and his book was complained of by the lower house of convention. The doctor drew up a preface, and afterwards gave in seve- <cb/> ral explanations, which seemed to satisfy the upper house; at least the affair was not brought to any issue, the members appearing desirous to prevent dissensions and divisions.</p><p>In 1715 and 1716 he had a dispute with the celebrated Leibnitz, concerning the principles of natural philosophy and religion; and a collection of the papers which passed between them, was published in 1717. This work was addressed to queen Caroline, then princess of Wales, who was pleased to have the controversy pass through her hands. It related chiefly to the subjects of liberty and necessity.</p><p>About the year 1718 he was presented by the lord Lechmere, to the master ship of Wigston's hospital in Leicestershire. In 1724 and 1725 he published 18 sermons, preached on several occasions. In 1727, on the death of Sir Isaac Newton, he had the offer of succeeding him as Master of the Mint, a place worth from 12 to 15 hundred a year: but to this secular preferment he could not reconcile himself; and therefore absolutely refused it.—In 1728 was published, a Letter from Dr. Clarke to Mr. Benjamin Hoadley, occasioned by the Controversy relating to the Proportion of Velocity and Force of Bodies in Motion; and printed in the Philosophical Transactions, num. 401.—In the beginning of 1729 he published the first 12 books of Homer's Iliad: a work which bishop Hoadley calls an accurate performance; and his notes, a treasury of grammatical and critical knowledge. And the same year came out, his Exposition of the Church Catechism, and 10 volumes of Sermons: books so well known and so generally approved, that they need no recommendation. But the same year, on Sunday the 11th of May, going to preach before the Judges at Serjeant's Inn, he was seized with a pain in his side, which made it impossible for him to perform his office. He was carried home and continued under his disorder till the 17th of the same month, when he died, in the 54th year of his age, after long enjoying a vigorous state of health, having scarce ever known sickness.</p><p>Three years after the doctor's death, appeared the other 12 books of the Iliad, published in 4to by his son, Mr. Samuel Clarke, who says in the preface, that his father had finished the annotations to the first three of those books, and as far as the 359th verse of the 4th; and had revised the text and version as far as verse 510 of the same book.</p><p>Dr. Clarke married Catherine, the only daughter of the Rev. Mr. Lockwood, rector of Little Missingham in the county of Norfolk, by whom he had seven children, four of whom survived him.</p><p>Queen Caroline took great pleasure in the doctor's conversation and friendship, seldom missing a week in which she did not receive some proof of the greatness of his genius, and the force of his understanding.</p><p>As to the character of Dr. Clarke, he is represented as possessing one of the best dispositions in the world, remarkably humane and tender, free and easy in his conversation, cheerful and even playful in his manner. Bishop Hare says of him, “He was a man who had all the good qualities that could meet together to recommend him. He was possessed of all the parts of learning that are valuable in a clergyman, in a degree that <pb n="740"/><cb/> few possess any single one. He has joined to a good skill in the three learned languages, a great compass of the best philosophy and mathematics, as appears by his Latin works; and his English ones are such a proof of his own piety, and of his knowledge in divinity, and have done so much service to religion, as would make any other man, that was not under a suspicion of heresy, secure of the friendship of all good churchmen, especially the clergy. And to all this piety and learning was joined, a temper happy beyond expression; a sweet, easy, modest, obliging behaviour adorned all his actions; and neither passion, vanity, insolence, or ostentation appeared either in what he said or wrote. This is the learning, this the temper of the man, whose study of the Scriptures has betrayed him into a suspicion of some heretical opinions. Bishop Hoadley too having remarked how great the doctor was in all branches of learning, adds, If in any one of these he had excelled only so much as he did in all, he would have been justly entitled to the character of a great man: but there is something so very extraordinary, that the same person should excel not only in those parts of knowledge which require the strongest judgment, but in those which require the greatest memory too. So that, in a very high degree, divinity and mathematics, experimental philosophy and classical learning, metaphysics and critical skill, were united in Dr. Clarke.—Much more may be seen, said in his praise by bishop Hoadley, Dr. Sykes, and Mr. Whiston, in their Memoirs of his life.</p></div1><div1 part="N" n="CLEF" org="uniform" sample="complete" type="entry"><head>CLEF</head><p>, or <hi rend="smallcaps">Cliff</hi>, in Music, a mark at the beginning of the lines of a song, which shews the tone or key in which the piece is to begin. Or, it is a letter marked on any line, which explains and gives the name to all the rest.</p><p>Anciently, every line had a letter marked for a Clef; but now a letter on one line suffices; since by this all the rest are known; reckoning up and down, in the order of the letters.</p><p>It is called the Clef, or key, because that by it are known the names of all the other lines and spaces; and consequently the quantity of every degree, or interval. But because every note in the octave is called a key, though in another sense, this letter marked is called peculiarly the <hi rend="italics">signed Clef;</hi> because, being written on any line, it not only signs and marks that one, but it also explains all the rest. By Clef, therefore, for distinction sake, is meant that letter, signed on a line, which explains the rest; and by key, the principal note of a song, in which the melody closes.</p><p>There are three of these signed Clefs, <hi rend="italics">c, f, g.</hi> The Clef of the highest part in a song, called <hi rend="italics">treble,</hi> or <hi rend="italics">alt,</hi> is <hi rend="italics">g,</hi> set on the second line counting upwards. The Clef of the bass, or the lowest part, is <hi rend="italics">f</hi> on the 4th line upwards. For all the other mean parts, the Clef is <hi rend="italics">c,</hi> sometimes on one, sometimes on another line. Indeed, some that are really mean parts, are sometimes set with the <hi rend="italics">g</hi> clef. It must however be observed, that the ordinary signatures of Clefs bear little resemblance to those letters. Mr. Malcolm thinks it would be well if the letters themselves were used. Kepler takes great pains to shew, that the common signatures are only cor- <cb/> ruptions of the letters they represent. The figures of these now are as follow:</p><p><figure/> Character of the treble Clef.</p><p><figure/> The mean Clef.</p><p><figure/> The bass Clef.</p><p>The Clefs are always taken fifths to one another. So the Clef <hi rend="italics">f</hi> being lowest, <hi rend="italics">c</hi> is a fifth above it, and <hi rend="italics">g</hi> a fifth above <hi rend="italics">c.</hi></p><p>When the place of the Clef is changed, which is not frequent in the mean Clef, it is with a design to make the system comprehend as many notes of the song as possible, and so to have the fewer notes above or below it. So that, if there be many lines above the Clef, and few below it, this purpose is answered by placing the Clef in the first or second line: but if there be many notes below the Clef, it is placed lower in the system. In effect, according to the relation of the other notes to the Clef note, the particular system is taken differently in the scale, the Clef line making one in all the variety.</p><p>But still, in whatever line of the particular system any Clef is found, it must be understood to belong to the same of the general system, and to be the same individual note or sound in the scale. By this constant relation of Clefs, we learn how to compare the several particular systems of the several parts, and to know how they communicate in the scale, that is, which lines are unison, and which not: for it is not to be supposed, that each part has certain bounds, within which another must never come. Some notes of the treble, for example, may be lower than some of the mean parts, or even of the bass. Therefore to put together into one system all the parts of a composition written separately, the notes of each part must be placed at the same distances above and below the proper Clef, as they stand in the separate system: and because all the notes that are consonant, or heard together, must stand directly over each other, that the notes belonging to each part may be distinctly known, they may be made with such differences as shall not confound, or alter their significations with respect to time, but only shew that they belong to this or that part. Thus we shall see how the parts change and pass through one another; and which, in every note, is highest, lowest, or unison.</p><p>It must here be observed, that for the performance of any single piece, the Clef only serves for explaining the intervals in the lines and spaces: so that it need not be regarded what part of any greater system it is; but the first note may be taken as high or low as we please. For as the proper use of the scale is not to limit the absolute degree of tone; so the proper use of the <hi rend="italics">signed</hi> Clef is not to limit the pitch, at which the first note of any part is to be taken; but to determine the tune of the rest, with respect to the first; and considering all the parts together, to determine the relation of their several notes by the relations of their Clefs in the scale: thus, their pitch of tune being determined in a certain note of one part, the other notes of that part are determined by the constant relations of the <pb n="741"/><cb/> letters of the scale, and the notes of the other parts by the relations of their Clefs.</p><p>In effect, for performing any single part, the Clef note may be taken in an octave, that is, at any note of the same name; provided we do not go too high, or too low, for finding the rest of the notes of a song. But in a concert of several parts, all the Clefs must be taken, not only in the relations, but also in the places of the system abovementioned; that every part may be comprehended in it.</p><p>The natural and artificial note expressed by the same letter, as <hi rend="italics">c</hi> and <hi rend="italics">c</hi><figure/>, are both set on the same line or space. When there is no character of flat or sharp, at the beginning with the Clef, all the notes are natural: and if in any particular place the artificial note be required, it is denoted by the sign of a flat or sharp, set on the line a space before that note.</p><p>If a sharp or flat be set at the beginning in any line or space with the Clef, all the notes on that line or space are artificial ones; that is, are to be taken a semitone higher or lower than they would be without such sign. And the same affects all their octaves above and below, though they be not marked so. In the course of the song, if the natural note be sometimes required, it is signified by the character <figure/>.</p><p>COMPASS. Pa. 314, col. 1, after l. 6 from the bottom, add, See also a new one in the Supplement to Cavallo's Treatise on Magnetism.</p><p>CONDORCET (<hi rend="smallcaps">John-Anthony-Nicholas</hi> <hi rend="italics">de</hi> <hi rend="smallcaps">Caritat</hi>, <hi rend="italics">Marquis of),</hi> member of the Institute of Bologna, of the Academies of Turin, Berlin, Stockholm, Upsal, Philadelphia, Petersbourg, Padua, &c, and secretary of the Paris Academy of Sciences, was born at Ribemont in Picardie, the 17th of September 1743. His early attachment to the sciences, and progress in them, soon rendered him a conspicuous character in the commonwealth of letters. He was received as a member of the Academy of Sciences at 25 years of age, namely, in March 1769, as Adjunct-Mecanician; afterwards, he became Associate in 1710, AdjunctSecretary in 1773, and sole Secretary soon after, which he enjoyed till his death, or till the dissolution of the Academy by the Convention.</p><p>Condorcet soon became an author, and that in the most sublime branches of science. He published his <hi rend="italics">Essais d'Analyse</hi> in several parts; the first part in 1765 (at 22 years of age); the second, in 1767; and the third, in 1768. These works are chiefly on the Integral Calculus, or the finding of Fluents, and make one volume in 4to.</p><p>He published the Eloges of the Academicians or members of the Academy of Sciences, from the year 1666 till 1700, in several volumes. He wrote also similar Eloges of the Academicians who died during the time that he discharged the important office of Secretary to the Academy; as well as the very useful histories of the different branches of science commonly prefixed to the volumes of Memoirs, till the volume for the year 1783, when it is to be lamented that so useful a part of the plan of the Academy was discontinued. <cb/></p><p>His other memoirs contained in the volumes of the Academy, are the following.</p><p>1. Tract on the Integral Calculus; 1765.</p><p>2. On the problem of Three Bodies; 1767.</p><p>3. Observations on the Integral Calculus; 1767.</p><p>4. On the Nature of Infinite Series; on the Extent of the Solutions which they give; and on a new method of Approximation for Differential Equations of all Orders; 1769.</p><p>5. On Equations for Finite Differences; 1770.</p><p>6. On Equations for Partial Differences; 1770.</p><p>7. On Differential Equations; 1770.</p><p>8. Additions to the foregoing Tracts; 1770.</p><p>9. On the Determination of Arbitrary Functions which enter the Integrals of Equations to Partial Differences; 1771.</p><p>10. Reflexions on the Methods of Approximation hitherto known for Differential Equations; 1771.</p><p>11. Theorem concerning Quadratures; 1771.</p><p>12. Inquiry concerning the Integral Calculus; 1772.</p><p>13. On the Calculation of Probabilities, part 1 and 2; 1781.</p><p>14. Continuation of the same, part 3; 1782.</p><p>15. Ditto, part 4; 1783.</p><p>16. Ditto, part 5; 1784.</p><p>Condorcet had the character of being a very worthy honest man, and a respectable author, though perhaps not a first-rate one, and produced an excellent set of Eloges of the deceased Academicians, during the time of his secretaryship. A late French political writer has observed of him, that he laboured to succeed to the literary throne of d'Alembert, but that he cannot be ranked among illustrious authors; that his works have neither animation nor depth, and that his style is dull and dry; that some bold attacks on religion and declamations against despotism have chiefly given a degree of fame to his writings.</p><p>On the breaking out of the troubles in France, Condorcet took a decided part on the side of the people, and steadily maintained the cause he had espoused amid all the shocks and intrigues of contending parties; till, under the tyranny of Robespierre, he was driven from the convention, being one of those members proscribed on the 31st of May 1793, and he died about April 1794. The manner of his death is thus described by the public prints of that time. He was obliged to conceal himself with the greatest care for the purpose of avoiding the fate of Brissot and the other deputies who where executed. He did not, however, attempt to quit Paris, but concealed himself in the house of a female, who, though she knew him only by name, did not hesitate to risk her own life for the purpose of preserving that of Condorcet. In her house he remained till the month of April 1794, when it was rumoured that a domiciliary visit was to be made, which obliged him to leave Paris. Although he had neither passport nor civic card, he escaped through the Barrier, and arrived at the Plain of Mont rouge, where he expected to find an asylum in the country-house of an intimate friend. Unfortunately this friend had set out for Paris, where he was to remain for three days.—During all this period, Condorcet wandered about the fields and in the woods, <pb n="742"/><cb/> not daring to enter an inn on account of not having a civic card. Half dead with hunger, fatigue, and fear, and scarcely able to walk on account of a wound in his foot, he passed the night under a tree.</p><p>At length his friend returned, and received him with great cordiality; but as it was deemed imprudent that he should enter the house in the day-time, he returned to the woods till night. In this short interval between morning and night his caution forsook him, and he resolved to go to an inn for the purpose of procuring food. He went to an inn at Clamars, and ordered an omlette. His torn clothes, his dirty cap, his meagre and pale countenance, and the greediness with which he devoured the omlette, fixed the attention of the persons in the inn, among whom was a member of the Revolutionary Committee of Clamars. This man conceiving him to be Condorcet, who had effected his escape from the Bicetre, asked him whence he came, whither he was going, and whether he had a passport? The confused manner in which he replied to these questions, induced the member to order him to be conveyed before the Committee, who, after an examination, sent him to the district of Boury la Reine. He was there interrogated again, and the unsatisfactory answers which he gave, determined the directors of the district to send him to prison on the succeeding day.—During <cb/> the night he was confined in a kind of dungeon. On the next morning, when his keeper entered with some bread and water for him, he found him stretched on the ground without any signs of life.</p><p>On inspecting the body, the immediate cause of his death could not be discovered, but it was conjectured that he had poisoned himself. Condorcet indeed always carried a dose of poison in his pocket, and he said to the friend who was to have received him into his house, that he had been often tempted to make use of it, but that the idea of a wife and daughter, whom he loved tenderly, restrained him. During the time that he was concealed at Paris, he wrote a history of the Progress of the Human Mind, in two volumes.</p><p>CUBICS. The method of resolving all the cases of Cubic equations by the tables of sines, tangents and secants, are thus given by Dr. Maskelyne, p. 57, Taylor's Logarithms.</p><p>“The following method is adapted to a Cubic equation, wanting the second term; therefore, if the equation has the second term, it must be first taken away in the usual manner. There are four forms of Cubic equations wanting the second term, whose roots, according to known rules equivalent to Cardan's, are as follow: <cb/></p><p>The roots of the first and second forms are negatives of each other; and those of the third and fourth are also negatives of each other. The first and second forms have only one root each. The third and fourth forms have also only one root each, when the quadratic surd √(<hi rend="italics">q</hi><hi rend="sup">2</hi>/4 - (<hi rend="italics">p</hi><hi rend="sup">3</hi>/27)) is possible; but have three roots each when that surd is impossible.</p><p>The roots of all the four forms may, in all cases, be easily computed as follows:</p><p><hi rend="italics">Forms</hi> 1<hi rend="italics">st and</hi> 2<hi rend="italics">d.</hi> Put ; where the upper sign belongs to the first form, and the lower sign to the second form.</p><p><hi rend="italics">Forms</hi> 3<hi rend="italics">d and</hi> 4<hi rend="italics">th.</hi> Put 2/<hi rend="italics">q</hi> X ―(<hi rend="italics">p</hi>/3))<hi rend="sup">3/2</hi> if less than unity, <cb/> else its reciprocal . Then,</p><p><hi rend="italics">Case</hi> 1<hi rend="italics">st.</hi> 2/<hi rend="italics">q</hi> X ―(<hi rend="italics">p</hi>/3))<hi rend="sup">3/2</hi> < unity. Put ; where the upper sign belongs to the third form, and the lower sign to the fourth form.</p><p><hi rend="italics">Case</hi> 2<hi rend="italics">d.</hi> 2/<hi rend="italics">q</hi> X ―(<hi rend="italics">p</hi>/3))<hi rend="sup">3/2</hi> > unity. Then <hi rend="italics">x</hi> has three values in each form, viz, ; where the upper signs belong to the third form, and the lower signs to the fourth form. <pb n="743"/><cb/></p><p><hi rend="italics">By Logarithms.</hi></p><p><hi rend="italics">Eorms</hi> 1<hi rend="italics">st and</hi> 2<hi rend="italics">d.</hi> ; and <hi rend="italics">x</hi> will be affirmative in the first form, and negative in the second form.</p><p><hi rend="italics">Forms</hi> 3<hi rend="italics">d and</hi> 4<hi rend="italics">th.</hi> (3/2) X log.<hi rend="italics">p</hi>/3 + 10 - log.<hi rend="italics">q</hi>/2 being less than 10 (which is case first) or log.<hi rend="italics">q</hi>/2 + 10 - (3/2) X log.<hi rend="italics">p</hi>/3 being less than 10 (which is case 2d) = log. cos. <hi rend="italics">z.</hi></p><p><hi rend="italics">Case</hi> 1<hi rend="italics">st.</hi> ; and <hi rend="italics">x</hi> will be affirmative in the third form, and negative in the 4th form.</p><p><hi rend="italics">Case</hi> 2<hi rend="italics">d.</hi> Here <hi rend="italics">x</hi> has three values. ; where the upper signs belong to the third form, and the lower signs to the fourth form; that is, the first value of <hi rend="italics">x</hi> in the third form is positive, and its second and third values negative; and the first value of <hi rend="italics">x</hi> in the fourth form is negative, and its second and third values affirmative.”</p><p>See also <hi rend="smallcaps">Irreducible</hi> <hi rend="italics">Case.</hi></p><p>CURVE. Pa. 350, col. 2, l. 35, for <hi rend="italics">dx</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi> r. ―(<hi rend="italics">dx</hi> + <hi rend="italics">x</hi><hi rend="sup">2</hi>).</p></div1></div0><div0 part="N" n="D" org="uniform" sample="complete" type="alphabetic letter"><head>D</head><p>DIPPING <hi rend="italics">Needle.</hi> Pa. 383, col. 2, after line 38 <hi rend="italics">add,</hi> See a new Dipping-needle by Dr. Lorimer, in the Philos. Trans. 1775, also in the Supplement to Cavallo's Treatise on Magnetism.</p><p>DOME. In plate 33 is represented the plan and elevation of a Dome constructed without centring, by Mr. S. Bunce; viz, Fig. 1 the plan, and Fig. 2 the elevation. The first course consists of the stones marked 1, 1, 1, &c, of different sizes, the large ones exactly twice the height of the small ones, placed alternately, and forming intervals to receive the stones marked 2, 2, 2. The other courses are continued in the same manner, according to the order of the figures to the top.</p><p>It is evident, from the converging or wedgelike form of the intervals, that the stones they receive can only be inserted from the outside, and cannot fall through: therefore the whole Dome may be built without centring or temporary support. To break the upright joints, the stones may be cut of the form marked in <cb/> Fig. 3; and those marked 16, 17, &c, near the keystones, may be enlarged as at Fig. 4.</p><p>Pa. 399, col. 2, line 10 from the bottom, <hi rend="italics">for</hi> D<hi rend="smallcaps">YMANICS</hi> <hi rend="italics">read</hi> <hi rend="smallcaps">Dynamics.</hi></p></div0><div0 part="N" n="E" org="uniform" sample="complete" type="alphabetic letter"><head>E</head><div1 part="N" n="EUTOCIUS" org="uniform" sample="complete" type="entry"><head>EUTOCIUS</head><p>, a respectable Greek mathematician, lived at Ascalon in Palestine about the year of Christ 550. He was one of the most considerable mathematicians that flourished about the decline of the sciences among the Greeks, and had for his preceptor Isidorus the principal architect of the church of St. Sophia at Constantinople. He is chiefly known however by his commentaries on the works of the two ancient authors, Archimedes and Apollonius. Those two commentaries are both excellent compositions, to which we owe many useful circumstances in the history of the mathematics.</p><p>His commentaries on Apollonius are published in Halley's edition of the works of that author; and those on Archimedes, first in the Basle edition, in Greek and Latin, in 1543, and since in some others, as the late Oxford edition. Of these commentaries, those rank the highest, which illustrate Archimedes's work on the Sphere and Cylinder; in one of which we have a recital of the various methods practised by the ancients in the solution of the Delian problem, or that of doubling the cube. The others are of less value; though it cannot but be regretted that Eutocius did not pursue his plan of commenting on all the works of Archimedes, with the same attention and diligence which he employed in his remarks on the sphere and cylinder.</p><p>Pa. 507, line 5 from the bottom, <hi rend="italics">for</hi> 3 + 1 <hi rend="italics">read</hi> 3 + 1/7.</p><p>Pa. 551, line 22 from the bottom, <hi rend="italics">for</hi> 7/50 <hi rend="italics">read</hi> 7/30.</p></div1></div0><div0 part="N" n="G" org="uniform" sample="complete" type="alphabetic letter"><head>G</head><div1 part="N" n="GROIN" org="uniform" sample="complete" type="entry"><head>GROIN</head><p>, with Builders, is the angle made by the intersection of two arches. It is of two kinds, regular and irregular; viz, Regular when both the arches have the same diameter, but an Irregular Groin when one arch is a semicircle and the other a semiellipsis. Groins are chiefly used in forming arched roofs, where one hollow arched vault intersects with another; as in the roofs of most churches, and some cellars in large houses.</p></div1></div0><div0 part="N" n="I" org="uniform" sample="complete" type="alphabetic letter"><head>I</head><p>IMPOSSIBLE <hi rend="italics">Binomial.</hi> See <hi rend="smallcaps">Binomial.</hi></p><p>IRREDUCIBLE <hi rend="italics">Case,</hi> in Algebra. Mr. Bonnycastle has communicated the following additional observations on this case, and, an improved solution by a table of sines. The</p><p><hi rend="smallcaps">Irreducible</hi> <hi rend="italics">Case, in Algebra,</hi> is a cubic equation of the form , having (1/27)<hi rend="italics">a</hi><hi rend="sup">3</hi> greater than (1/4)<hi rend="italics">b</hi><hi rend="sup">2</hi>, or 4<hi rend="italics">a</hi><hi rend="sup">3</hi> greater than 27<hi rend="italics">b</hi><hi rend="sup">2</hi>; in which case, it is well known, that the solution cannot be generally obtained, either by Cardan's rule, or any other which has yet been devised.</p><p>One of the most convenient methods of determining the roots of equations of this kind, is by means of a Table of Natural Sines, &c, for which purpose the following formulæ will be found extremely commodious, the arc, in each case, being always less than a quadrant, and therefore attended with no ambiguity. <pb n="744"/><cb/></p><p>If the equation be ; let A be put = arc whose cos. is ((3<hi rend="italics">b</hi>)/(2<hi rend="italics">a</hi>))√(3/<hi rend="italics">a</hi>) to rad. 1, then the three roots, or values of <hi rend="italics">x,</hi> will be as follows:</p><p>And, if the equation be ; let A be put = arc whose sine is ((3<hi rend="italics">b</hi>)/(2<hi rend="italics">a</hi>))√(3/<hi rend="italics">a</hi>) to rad. 1; then the three roots, or values of <hi rend="italics">x,</hi> will be as follows.</p><p>Ex. 1. Let , to find the 3 roots of the equation. Here .</p><p>Ex. 2d. Let , to find the 3 roots of the equation. Here ,</p><p>The investigation of this method is as follows:</p><p>It is shewn, by the writers on Trigonometry, that if <hi rend="italics">c</hi> be the cosine of any arc to rad. 1, 4<hi rend="italics">c</hi><hi rend="sup">3</hi> - 3<hi rend="italics">c</hi> will be the cosine of 3 times that arc; and consequently <hi rend="italics">c</hi> is the cosine of 1/3 of the arc whose cosine is 4<hi rend="italics">c</hi><hi rend="sup">3</hi> - 3<hi rend="italics">c,</hi> or any other equal quantity.</p><p>In order, therefore, to reduce the equation to this form, let ; then ; whence if 4<hi rend="italics">az</hi><hi rend="sup">2</hi> be put = 3, we shall have <cb/> , and consequently .</p><p>From which last equation, it appears that <hi rend="italics">y</hi> = cos. 1/3arc whose cos. is((3<hi rend="italics">b</hi>)/(2<hi rend="italics">a</hi>))√(3/<hi rend="italics">a</hi>); and therefore . or, if A be put = arc whose cos. is ((3<hi rend="italics">b</hi>)/(2<hi rend="italics">a</hi>))√(3/<hi rend="italics">a</hi>), <hi rend="italics">x</hi> is =2√(<hi rend="italics">a</hi>/3) X cos.A/3.</p><p>But the arc of which ((3<hi rend="italics">b</hi>)/(2<hi rend="italics">a</hi>))√(3/<hi rend="italics">a</hi>) is the cosine, is either A, A+360° or A+720°; whence ; the two latter of which being converted into sines, will give the same formulæ as in the rule.</p><p>In like manner, if <hi rend="italics">s</hi> be the sine of any arc to rad. 1, 3<hi rend="italics">s</hi> - 4<hi rend="italics">s</hi><hi rend="sup">3</hi> is well known to be the sine of 3 times that arc; and consequently <hi rend="italics">s</hi> is the fine of 1/3 of the arc whose sine is 3<hi rend="italics">s</hi> - 4<hi rend="italics">s</hi><hi rend="sup">3</hi>. Whence, to reduce the equation , to this form, let , as before; then ; where, if 4<hi rend="italics">az</hi><hi rend="sup">2</hi> be put = 3, we shall have , and consequently .</p><p>From which last equation it appears that <hi rend="italics">y</hi> = sine 1/3<hi rend="italics">arc</hi> whose fine is((3<hi rend="italics">b</hi>)/(2<hi rend="italics">a</hi>))√(3/<hi rend="italics">a</hi>), and therefore , which is the same as the rule, the other two roots being found as in the former case.</p></div0><div0 part="N" n="L" org="uniform" sample="complete" type="alphabetic letter"><head>L</head><div1 part="N" n="LOCK" org="uniform" sample="complete" type="entry"><head>LOCK</head><p>, for Canals, in Inland Navigations. See <hi rend="smallcaps">Canal.</hi></p><p>LOGARITHMS. Mr. Bonnycastle has communicated the following new method of making these useful numbers:</p><p><hi rend="smallcaps">Logarithms.</hi> The series now chiefly used in the computation of Logarithms were originally derived from the hyperbola, by means of which, and the logistic curve, the nature and properties of these numbers are clearly and elegantly explained.</p><p>The doctrine, however, being purely arithmetical, this mode of demonstrating it, by the intervention of certain curves, was considered, by Dr. Halley, as not conformable to the nature of the subject. <pb n="745"/><cb/></p><p>He has, accordingly, investigated the same series from the abstract principles of numbers; but his method, which is a kind of disguised fluxions, is, in many places, so extremely abstruse and obscure, that few have been able to comprehend his reasoning.</p><p>An easy and perspicuous demonstration, of this kind, was therefore still wanting; which may be obtained from the pure principles of Algebra, independently of the doctrine of Curves, as follows:</p><p>The Logarithm of any number, is the index of that power of some other number, which is equal to the given number.</p><p>Thus, if , the logarithm of <hi rend="italics">a</hi> is <hi rend="italics">x,</hi> which may be either positive or negative, and <hi rend="italics">r</hi> any number whatever, according to the different systems of Logarithms.</p><p>When <hi rend="italics">a</hi> = 1, it is plain that <hi rend="italics">x</hi> must be = 0, whatever be the value of <hi rend="italics">r;</hi> and consequently the Logarithm of 1 is always 0 in every system.</p><p>If <hi rend="italics">x</hi> = 1, it is also plain that <hi rend="italics">a</hi> must be = <hi rend="italics">r;</hi> and therefore <hi rend="italics">r</hi> is always the number in every system, whose Logarithm in that system is 1.</p><p>To find the Logarithm of any number, in any system, it is only necessary, from the equation , to find the value of <hi rend="italics">x</hi> in terms of <hi rend="italics">r</hi> and <hi rend="italics">a.</hi></p><p>This may be strictly effected, by means of a new property of the binomial theorem of Newton; which is given under its proper article in this Appendix. The general Logarithmic equation being , let , &c. See <hi rend="italics">Binomial</hi> <hi rend="smallcaps">Theorem</hi>, Appendix.</p><p>And if <hi rend="italics">p</hi> - <hi rend="italics">p</hi><hi rend="sup">2</hi>/2 + <hi rend="italics">p</hi><hi rend="sup">3</hi>/3 - <hi rend="italics">p</hi><hi rend="sup">4</hi>/4 + <hi rend="italics">p</hi><hi rend="sup">5</hi>/5 &c be put = <hi rend="italics">s,</hi> we shall have , which let be put = <hi rend="italics">q;</hi> then, by reverting the series <hi rend="italics">z</hi> or 1/<hi rend="italics">x</hi> will be found =(<hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c)/<hi rend="italics">s</hi>=(<hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> +(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c)/(<hi rend="italics">p</hi>-(1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c) and consequently .</p><p>The Logarithm of <hi rend="italics">a,</hi> or 1 + <hi rend="italics">p,</hi> is therefore =(<hi rend="italics">p</hi>-(1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c)/(<hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c); or, since , and , the Logarithm of <hi rend="italics">a</hi> is <cb/> =((<hi rend="italics">a</hi>-1)-(1/2)(<hi rend="italics">a</hi>-1)<hi rend="sup">2</hi>+(1/3)(<hi rend="italics">a</hi>-1)<hi rend="sup">3</hi>-(1/4)(<hi rend="italics">a</hi>-1)<hi rend="sup">4</hi>+(1/5)(<hi rend="italics">a</hi>-1)<hi rend="sup">5</hi>)/ ((<hi rend="italics">r</hi>-1)-(1/2)(<hi rend="italics">r</hi>-1)<hi rend="sup">2</hi>+(1/3)(<hi rend="italics">r</hi>-1)<hi rend="sup">3</hi>-(1/4)(<hi rend="italics">r</hi>-1)<hi rend="sup">4</hi>+(1/5)(<hi rend="italics">r</hi>-1)<hi rend="sup">5</hi>) &c; Which is a general expression for the Logarithm of any number, in any system of Logarithms, the radix <hi rend="italics">r</hi> being taken of any value, greater or less than 1.</p><p>But as <hi rend="italics">r</hi> in every system, is a constant quantity, being always the number whose Logarithm in the system to which it belongs is 1, the above expression may be simplified, either by assuming <hi rend="italics">r</hi> = to some particular number, and from thence finding the value of the series constituting the denominator; or by assuming this whole series = to some particular number, and from thence finding the value which must be given to the radix <hi rend="italics">r.</hi></p><p>By the latter of these methods, the denominator may be made to vanish, by assuming the value of the series of which it consists = 1, in which case, the Logarithm of 1 + <hi rend="italics">p</hi> becomes = <hi rend="italics">p</hi>-<hi rend="italics">p</hi><hi rend="sup">2</hi>/2+<hi rend="italics">p</hi><hi rend="sup">3</hi>/3-<hi rend="italics">p</hi><hi rend="sup">4</hi>/4+<hi rend="italics">p</hi><hi rend="sup">5</hi>/5 &c, or the Logarithm of , and <hi rend="italics">r,</hi> by reversion of series is found = 2.7182818 &c.</p><p>The system arising from this mode of determining the value of the radix <hi rend="italics">r,</hi> is that which furnishes what have been usually called hyperbolic Logarithms; and appears to be the simplest form the general expression admits of.</p><p>If, on the contrary, the radix <hi rend="italics">r</hi> be assumed = to some particular number, as for instance 10, the value of the series <hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c, or its equal (<hi rend="italics">r</hi>-1)-(1/2)(<hi rend="italics">r</hi>-1)<hi rend="sup">2</hi>+(1/3)(<hi rend="italics">r</hi>-1)<hi rend="sup">3</hi>-(1/4)(<hi rend="italics">r</hi>-1)<hi rend="sup">4</hi>+(1/5)(<hi rend="italics">r</hi>-1)<hi rend="sup">5</hi> &c will become = 2.30258509 &c, and the or the , which gives the system that furnishes Briggs's or the common Logarithms.</p><p>And, in like manner, by assuming any particular value for <hi rend="italics">r,</hi> and thence determining the value of the series <hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c, or its equal (<hi rend="italics">r</hi>-1)-(1/2)(<hi rend="italics">r</hi>-1)<hi rend="sup">2</hi>+(1/3)(<hi rend="italics">r</hi>-1)<hi rend="sup">3</hi>-(1/4)(<hi rend="italics">r</hi>-1)<hi rend="sup">4</hi>+(1/5)(<hi rend="italics">r</hi>-1)<hi rend="sup">5</hi> &c; or by assuming the same series of some particular value, and thence determining the value of <hi rend="italics">r,</hi> any system of Logarithms may be derived.</p><p>The series <hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c, or its equal (<hi rend="italics">r</hi>-1)-(1/2)(<hi rend="italics">r</hi>-1)<hi rend="sup">2</hi>+(1/3)(<hi rend="italics">r</hi>-1)<hi rend="sup">3</hi>-(1/4)(<hi rend="italics">r</hi>-1)<hi rend="sup">4</hi>+(1/5)(<hi rend="italics">r</hi>-1)<hi rend="sup">5</hi> &c, which forms the denominator of the above compound expression, exhibiting the Logarithms of numbers according to any system, is what was first called, by Cotes, the Modulus of the system, being always a constant quantity, depending only on the assumed value of <hi rend="italics">r.</hi></p><p>And, as the form of this series is exactly the same as that which constitutes the numerator, and which has been shewn to be the hyperbolic Logarithm of <hi rend="italics">a,</hi> it follows that the Modulus of any system of Logarithms is equal to the hyperbolic Logarithm of the radix of that <pb n="746"/><cb/> system, or of the number whose proper Logarithm in the system to which it belongs is 1.</p><p>The form of the series here obtained for the hyperbolic Logarithm of <hi rend="italics">a,</hi> is the same as that which was first discovered by Mercator; and if the series of Wallis be required, it may be investigated in a similar manner as follows:</p><p>The general Logarithmic equation being , as before, let and ; then , and .</p><p>And if <hi rend="italics">p</hi>+<hi rend="italics">p</hi><hi rend="sup">2</hi>/2+<hi rend="italics">p</hi><hi rend="sup">3</hi>/3+<hi rend="italics">p</hi><hi rend="sup">4</hi>/4+<hi rend="italics">p</hi><hi rend="sup">5</hi>/5 &c be put = <hi rend="italics">s,</hi> we shall have , which let be put = <hi rend="italics">q;</hi> then, by conversion of series, <hi rend="italics">z</hi> or 1/<hi rend="italics">x</hi> will be found =(<hi rend="italics">q</hi>+(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>+(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c)/<hi rend="italics">s</hi>=(<hi rend="italics">q</hi>+(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>+(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi>)/ (<hi rend="italics">p</hi>+(1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi>+(1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi>+(1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi>) and consequently .</p><p>The Logarithm of <hi rend="italics">a</hi> or 1/(1-<hi rend="italics">p</hi>) is, therefore, = (<hi rend="italics">p</hi> + (1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> + (1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c)/ (<hi rend="italics">q</hi> + (1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi> + (1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c); or since and , the Logarithm of <hi rend="italics">a</hi> is = ((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi> + (1/2)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">2</hi> + (1/3)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">3</hi> + (1/4)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">4</hi> + (1/5)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">5</hi> &c)/ ((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi> + (1/2)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">2</hi> + (1/3)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">3</hi> + (1/4)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">4</hi> + (1/5)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">5</hi> &c) Which is another general expression for the Logarithm of any number <hi rend="italics">a,</hi> in any system of Logarithms, that may be simplified in the same manner as the former, the denominator being still equal to the hyperbolic Logarithm of the radix <hi rend="italics">r;</hi> or, which is the same thing, to the Modulus of the system.</p><p>For if the series <hi rend="italics">q</hi> + (1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi> + (1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c, or its equal (<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi> + (1/2)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">2</hi> + (1/3)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">3</hi> + (1/4)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">4</hi> + (1/5)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">5</hi> &c, be assumed = 1, the hyperbolic Logarithm of 1/(1-<hi rend="italics">p</hi>) <cb/> will be = <hi rend="italics">p</hi> + (1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> + (1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c, or the hyperbolic Logarithm of ; and <hi rend="italics">r,</hi> by reversion of series will be found = 2.7182818, as before. And if, on the contrary, the radix <hi rend="italics">r</hi> be assumed = 10, the value of the series <hi rend="italics">q</hi> + (1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi> + (1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c, or its equal (<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi> + (1/2)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">2</hi> + (1/3)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">3</hi> + (1/4)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">4</hi> + (1/5)((<hi rend="italics">r</hi>-1)/<hi rend="italics">r</hi>)<hi rend="sup">5</hi> &c, will become = 2.30258509, as before; and the common Logarithm of , or the common Logarithm of .</p><p>Or the latter formula, for the Logarithm of 1/(1 - <hi rend="italics">p</hi>), or its equal <hi rend="italics">a,</hi> may be more concisely derived from the first, as follows:</p><p>The Logarithm of 1 + <hi rend="italics">p</hi> has been shewn to be = (<hi rend="italics">p</hi> - (1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> - (1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c)/ (<hi rend="italics">q</hi> - (1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi> - (1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c), and if - <hi rend="italics">p</hi> be substituted in the place of + <hi rend="italics">p,</hi> the logarithm of 1 - <hi rend="italics">p</hi> will become = (- <hi rend="italics">p</hi> - (1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi> - (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> - (1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi> - (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c)/ (<hi rend="italics">q</hi> - (1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi> - (1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">q</hi><hi rend="sup">5</hi> &c), whence the Logarithm of where the denominator is the same as in the first formula, <hi rend="italics">q</hi> being here = <hi rend="italics">r</hi> - 1.</p><p>If the denominator, in either of these general formulæ, be put = <hi rend="italics">m,</hi> the Logarithm of 1 + <hi rend="italics">p</hi> will be denoted by 1/<hi rend="italics">m</hi> X (<hi rend="italics">p</hi> - (1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> - (1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c, or the Logarithm of <hi rend="italics">a</hi> by (1/<hi rend="italics">m</hi>) X : (<hi rend="italics">a</hi>-1) - (1/2)(<hi rend="italics">a</hi>-1)<hi rend="sup">2</hi> + (1/3)(<hi rend="italics">a</hi>-1)<hi rend="sup">3</hi> - (1/4)(<hi rend="italics">a</hi>-1)<hi rend="sup">4</hi> + (1/5)(<hi rend="italics">a</hi>-1)<hi rend="sup">5</hi> &c. And the Logarithm of 1/(1 - <hi rend="italics">p</hi>) will be denoted by 1/<hi rend="italics">m</hi> X (<hi rend="italics">p</hi> + (1/2)<hi rend="italics">p</hi><hi rend="sup">2</hi> + (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> + (1/4)<hi rend="italics">p</hi><hi rend="sup">4</hi> + (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> &c, or the Logarithm of <hi rend="italics">a</hi> by 1/<hi rend="italics">m</hi> X : (<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi> + (1/2)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">2</hi> + (1/3)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">3</hi> + (1/4)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">4</hi> + (1/5)((<hi rend="italics">a</hi>-1)/<hi rend="italics">a</hi>)<hi rend="sup">5</hi> &c.</p><p>And since the sum of the Logarithms of any two numbers is equal to the Logarithm of their product, the Logarithm of (1 + <hi rend="italics">p</hi>)/(1 - <hi rend="italics">p</hi>) will become <pb n="747"/><cb/> = (2/<hi rend="italics">m</hi>) X (<hi rend="italics">p</hi> + (1/3)<hi rend="italics">p</hi><hi rend="sup">3</hi> + (1/5)<hi rend="italics">p</hi><hi rend="sup">5</hi> + (1/7)<hi rend="italics">p</hi><hi rend="sup">7</hi> &c), or the Logarithm of Which is a third general formula, that converges faster than either of the former.</p><p>The Logarithm of any number may, therefore, be exhibited universally, or according to any system of Logarithms, in the three following forms: . <hi rend="center">Or</hi> . And if <hi rend="italics">a</hi>+<hi rend="italics">b</hi> be put = <hi rend="italics">s,</hi> and <hi rend="italics">a</hi> <01> <hi rend="italics">b</hi> = <hi rend="italics">d,</hi> these general formulæ may be easily converted into the following: .</p><p>From which last expressions, if <hi rend="italics">d</hi> or its equal <hi rend="italics">a</hi> <01> <hi rend="italics">b</hi> be put = 1, we shall have, by proper substitution, and the nature of Logarithms: .</p><p>And from the addition and subtraction of these series, several others may be derived; but in the actual computation of Logarithms they will be found to possess little or no advantage above those here given. The same general formula may be derived from the original Logarithmic equation in a different way, thus: <cb/> Let , then ; or if <hi rend="italics">r</hi> be put =1/(1-<hi rend="italics">q</hi>), we shall have .</p><p>And by denoting <hi rend="italics">q</hi>-(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>-(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi> &c in the first case, or its equal <hi rend="italics">q</hi>+(1/2)<hi rend="italics">q</hi><hi rend="sup">2</hi>+(1/3)<hi rend="italics">q</hi><hi rend="sup">3</hi>+(1/4)<hi rend="italics">q</hi><hi rend="sup">4</hi>, in the latter case, by <hi rend="italics">m,</hi> these expressions will become ; and ; which are the two anti-Logarithmic series of Halley: from whence, by reversion of series, may be found the Logarithm of any number <hi rend="italics">a,</hi> as before.</p></div1></div0><div0 part="N" n="M" org="uniform" sample="complete" type="alphabetic letter"><head>M</head><p>MICROSCOPE. The following directions are given for using the New Universal Pocket Microscope, made and sold by W. and S. Jones, opticians, No. 135, Holborn, London. See fig. 4, pl. 33.</p><p>“This Microscope is adapted to the viewing of all sorts of objects, whether <hi rend="italics">transparent,</hi> or <hi rend="italics">opake;</hi> and for <hi rend="italics">insects, flowers, animalcules,</hi> and the infinite variety of the <hi rend="italics">minutiæ</hi> of Nature and Art, will be found the most complete and portable for the price, of any hitherto contrived.</p><p>Place the square pillar of the Microscope in the square socket at the foot D, and fasten it by the pin, as shewn in the figure. Place also in the foot, the reflecting mirror C. There are three lenses at the top shewn at A, which serve to magnify the objects. By using these lenses separately or combined, you make seven different powers. When transparent objects, such as are in the ivory sliders, number 4, are-to be viewed, you place the sliders over the spring, at the underside of the stage B; then looking through the lens or magnifier, at A, at the same time reflect up the light, by moving the mirrour C below, and move gently upwards or downwards as may be necessary, the stage B, upon its square pillar, till you see the object illuminated and distinctly magnified; and in this manner for the other objects.</p><p>For animalcules, you unscrew the brass box that is fitted at the stage B, containing two glasses, and leave the undermost glass upon the stage, to receive the fluids. If you wish to view thereon any moving insect, &c, it may be confined by screwing on the cover: of the two glasses, the concave is best for fluids. Should the objects be opake, such as seeds, &c; they are to be placed upon the black and white ivory round piece, number 3, which is fitted also to the stage B. If the objects are of a dark colour, you place them contrastedly on the white side of the ivory. If they are of a white, or a light colour, upon the blackened side. Some objects <pb n="748"/><cb/> will be more conveniently viewed, by sticking them on the point of number 2; or between the nippers at the other end, which open by pressing the two little brass pins. This apparatus is also fitted to a small hole in the stage, made to receive the support of the wire.</p><p>The brass forceps, number 1, serve to take up any small object by, in order to place them on the stage for view. The instrument may be readily converted into an hand Microscope, to view objects against the common light; and which, for some transparent ones, is better so. It is done by only taking out the pillar from its foot in D, turning it half round, and fixing it in again; the foot then becomes a useful handle, and the reflector C is laid aside.</p><p>The whole apparatus packs into a fish-skin case, 4 1/4 inches long, 2 1/4 inches broad, and 1 1/2 inches deep.</p><p>For persons more curious and nice in these sort of instruments, there is contrived a useful adjusting screw to the stage, represented at <hi rend="italics">e.</hi> It is first moved up and down like the other, to the focus nearly, and made fast by the small screw. The utmost distinctness of the object is then obtained, by gently turning the long fine threaded screw, at the same time you are looking through the magnifiers A. In this case, there may be also added an extraordinary deep magnifier, and a concave silver speculum, with a magnisier to screw on at A, which will serve for viewing the very small, and opake objects, in the completest manner, and render the instrument as comprehensive in its uses and powers, as those formerly sold under the name of <hi rend="italics">Wilson's Microscope.</hi>”</p><div1 part="N" n="MODULUS" org="uniform" sample="complete" type="entry"><head>MODULUS</head><p>, and MODULAR <hi rend="italics">Ratio.</hi> See p. 49 at the bottom.</p></div1></div0><div0 part="N" n="N" org="uniform" sample="complete" type="alphabetic letter"><head>N</head><div1 part="N" n="NUTATION" org="uniform" sample="complete" type="entry"><head>NUTATION</head><p>, in Astronomy, a kind of libratory motion of the earth's axis; by which its inclination to the plane of the ecliptic is continually varying, by a certain number of seconds, backwards and forwards. The whole extent of this change in the inclination of the earth's axis, or, which is the same thing, in the apparent declination of the stars, is about 19″, and the period of that change is little more than 9 years, or the space of time from its setting out from any point and returning to the same point again, about 18 years and 7 months, being the same as the period of the moon's motions, upon which it chiefly depends; being indeed the joint effect of the inequalities of the action of the sun and moon upon the spheroidal figure of the earth, by which its axis is made to revolve with a conical motion, so that the extremity of it describes a small circle, or rather an ellipse, of 19.1 seconds diameter, and 14″.2 conjugate, each revolution being made in the space of 18 years 7 months, according to the revolution of the moon's nodes.</p><p>This is a natural consequence of the Newtonian system of universal attraction; the first principle of which is, that all bodies mutually attract each other in the direct ratio of their masses, and in the inverse ratio of the squares of their distances. From this mutual attraction, combined with motion in a right line, Newton deduces the figure of the orbits of the planets, and particularly that of the earth. If this orbit were a circle, and if the earth's form were that of a perfect sphere, the attraction of the sun would have no other <cb/> effect than to keep the earth in its orbit, without causing any irregularity in the position of its axis. But neither is the earth's orbit a circle, nor its body a sphere; for the earth is sensibly protuberant towards the equator, and its orbit is an ellipsis, which has the sun in its focus. Now when the position of the earth is such, that the plane of the equator passes through the centre of the sun, the attractive power of the sun acts only so as to draw the earth towards it, still parallel to itself, and without changing the position of its axis; a circumstance which happens only at the time of the equinoxes. In proportion as the earth recedes from those points, the sun also goes out of the plane of the equator, and approaches that of the one or other of the tropics; the semidiameter of the earth, then exposed to the sun, being unequal to what it was in the former case, the equator is more powerfully attracted than the rest of the globe, which causes some alteration in its position, and its inclination to the plane of the ecliptic: and as that part of the orbit, which is comprised between the autumnal and vernal equinox, is less than that which is comprised between the vernal and autumnal, it follows, that the irregularity caused by the sun, during his passage through the northern signs, is not entirely compensated by that which he causes during his passage through the southern signs; and that the parallelism of the terrestrial axis, and its inclination to the ecliptic, is thence a little altered.</p><p>The like effect which the sun produces upon the earth, by his attraction, is also produced by the moon, which acts with greater force, in proportion as she is more distant from the equator. Now, at the time when her nodes agree with the equinoxial points, her greatest latitude is added to the greatest obliquity of the ecliptic. At this time therefore, the power which causes the irregularity in the position of the terrestrial axis, acts with the greatest force; and the revolution of the nodes of the moon being performed in 18 years 7 months, hence it happens that in this time the nodes will twice agree with the equinoxial points; and consequently, twice in that period, or once every 9 years, the earth's axis will be more influenced than at any other time.</p><p>That the moon has also a like motion, is shewn by Newton, in the first book of the Principia; but he observes indeed that this motion must be very small, and scarcely sensible.</p><p>As to the history of the Nutation, it seems there have been hints and suspicions of the existence of such a circumstance, ever since Newton's discovery of the system of the universal and mutual attraction of matter; some traces of which are found in his Principia, as above mentioned.</p><p>We find too, that Flamsteed had hoped, about the year 1690, by means of the stars near his zenith, to determine the quantity of the Nutation which ought to follow from the theory of Newton; but he gave up that project, because, says he, if this effect exists, it must remain insensible till we have instruments much longer than 7 feet, and more solid and better fixed than mine. Hist. Cælest. vol. 3, pa. 113.</p><p>And Horrebow gives the following passage, extracted from the manuscripts of his master Roemer, who died in 1710, whose observations he published in 1753, un- <pb n="749"/><cb/> der the title of <hi rend="italics">Basis Astronomiæ.</hi> By this paragraph it appears that Roemer suspected also a Nutation in the earth's axis, and had some hopes to give the theory of it: it runs thus; “Sed de altitudinibus non perinde certus reddebar, tàm ob refractionum varietatem quàm ob aliam nondum liquido perspectam causam; scilicet per hos duos annos, quemadmodum & alias, expertus sum esse quandam in declinationibus varietatem, quæ nec resractionibus nec parallaxibus tribui potest, sine dubio ad vacillationem aliquam poli terrestris referendam, cujus me verisimilem dare posse theoriam, observationibus munitam, spero.” Basis Astronomiæ, 1735, pa. 66.</p><p>These ideas of a Nutation would naturally present them selves to those who might perceive certain changes in the declinations of the stars; and we have seen that the first suspicions of Bradley in 1727, were that there was some Nutation of the earth's axis which caused the star <foreign xml:lang="greek">g</foreign> Draconis to appear at times more or less near the pole; but farther observations obliged him to search another cause for the annual variations (art. A<hi rend="smallcaps">BERRATION</hi>): it was not till some years after that he discovered the second motion which we now treat of, properly called the Nutation. See the art. <hi rend="smallcaps">Star</hi>, pa. 500 &c, where Bradley's discovery of it is given at length; to which may be farther added the following summary.</p><p>For the better explaining the discovery of the Nutation by Bradley, we must recur to the time when he observed the stars in discovering the aberration. He perceived in 1728, that the annual change of declination in the stars near the equinoxial colure, was greater than what ought to result from the annual precession of the equinoxes being supposed 50″, and calculated in the usual way; the star <foreign xml:lang="greek">h</foreign> Ursæ Majoris was in the month of September 1728, 20″ more south than the preceding year, which ought to have been only 18″; from whence it would follow that the precession of the equinoxes should be 55″1/2 instead of 50″, without ascribing the difference between the 18 and 20″ to the instrument, because the stars about the solstitial colure did not give a like difference. Philos. Trans. vol. 35, pa. 659.</p><p>In general, the stars situated near the equinoctial colure had changed their declination about 2″ more than they ought by the mean precession of the equinoxes, the quantity of which is very well known, and the stars near the solstitial colure the same quantity less than they ought; but, Bradley adds, whether these small variations arise from some regular cause, or are occasioned by some change in the sector, I am not yet able to determine. Bradley therefore ardently continued his observations for determining the period and the law of these variations; for which purpose he resided almost continually at Wansted till 1732, when he was obliged to repair to Oxford to succeed Dr. Halley; he still continued to observe with the same exactness all the circumstances of the changes of declination in a great number of stars. Each year he saw the periods of the aberration confirmed according to the rules he had lately discovered; but from year to year he found also other differences; the stars situated between the vernal equinox and the winter solstice approached nearer to the north pole, while the opposite ones receded farther from it: he began therefore to suspect that the action of the moon upon the elevated equatorial parts of <cb/> the earth might cause a variation or libration in the earth's axis: his sector having been left fixed at Wansted, he often went there to make observations for many years, till the year 1747, when he was fully satisfied of the cause and effects, and account of which he then communicated to the world. Philos. Trans. vol. 45, an. 1748.</p><p>“On account of the inclination of the moon's orbit to the ecliptic, says Dr. Maskelyne (Astronomical Observations 1776, pa. 2), and the revolution of the nodes in antecedentia, which is performed in 18 years and 7 months, the part of the precession of the equinoxes, owing to her action, is not uniform: but subject to an equation, whose maximum is 18″: and the obliquity of the ecliptic is also subject to a periodical equation of 9″.55; being greater by 19.1″ when the moon's ascending node is in Aries, than when it is in Libra. Both these effects are represented together, by supposing the pole of the earth to describe the periphery of an ellipsis, in a retrograde manner, during each period of the moon's nodes, the greater axis, lying in the solstitial colure, being 19.1″, and the lesser axis, lying in the equinoctial colure, 14.2″; being to the greater, as the cosine of double the obliquity of the ecliptic to the cosine of the obliquity itself. This motion of the pole of the earth is called the Nutation of the earth's axis, and was discovered by Dr. Bradley, by a series of observations of several stars made in the course of 20 years, from 1727 to 1747, being a continuation of those by which he had discovered the aberration of light. But the exact law of the motion of the earth's axis has been settled by the learned mathematicians d'Alembert, Euler, and Simpson, from the principles of gravity. The equation hence arising in the place of a fixed star, whether in longitude, right-ascension, or declination (for the latitudes are not affected by it) has been sometimes called Nutation, and sometimes Deviation.” And again (says the Doctor, pa. 8), the above “quantity 19.1″, of the greatest Nutation of the earth's axis in the solstitial colure, is what I found from a scrupulous calculation of all Dr. Bradley's observations of <foreign xml:lang="greek">g</foreign> Draconis, which he was pleased to communicate to me for that purpose. From a like examination of his observation of <foreign xml:lang="greek">h</foreign> Ursæ majoris, I found the lesser axis of the ellipsis of Nutation to be 14.1″, or only (1/10)th of a second less than what it should be from the observations of <foreign xml:lang="greek">g</foreign> Draconis. But the result from the observations of <foreign xml:lang="greek">g</foreign> Draconis is most to be depended upon.”</p><p>Mr. Machin, secretary of the Royal Society, to whom Bradley communicated his conjectures, soon perceived that it would be sufficient to explain, both the Nutation and the change of the precession, to suppose that the pole of the earth described a small circle. He stated the diameter of this circle at 18″, and he supposed that it was described by the pole in the space of one revolution of the moon's nodes. But later calculations and theory, have shewn that the pole describes a small ellipsis, whose axes are 19.1″ and 14.2″, as above mentioned.</p><p>To shew the agreement between the theory and observations, Bradley gives a great multitude of observations of a number of stars, taken in different positions; and out of more than 300 observations which he made, he found but 11 which were different from the mean by <pb n="750"/><cb/> so much as 2″. And by the supposition of the elliptic rotation, the agreement of the theory with observation comes out still nearer.</p><p>By the observations of 1740 and 1741, the star <foreign xml:lang="greek">h</foreign> Ursæ majoris appeared to be 3″ farther from the pole than it ought to be according to the observations of other years. Bradley thought this difference arose from some particular cause; which however was chiefly the fault of the circular hypothesis. He suspected also that the situation of the apogee of the moon might have some influence on the Nutation. He invited therefore the mathematicians to calculate all these effects of attraction, which has been ably done by d'Alembert, Euler, Walmesley, Simpson, and others; and the astronomers to continue to observe the positions of the smallest stars, as well as the largest, to discover the physical derangements which they may suffer, and which had been observed in some of them.</p><p>Several effects arise from the Nutation. The first of these, and that which is the most easily perceived, is the change in the obliquity of the ecliptic; the quantity of which ought to be varied from that cause by 18″ in about 9 years. Accordingly, the obliquity of the ecliptic was observed in 1764 to be 23° 28′ 15″, and in 1755 only 23° 28′ 5″: not only therefore had it not diminished by 8″, as it ought to have done according to the regular mean diminution of that obliquity; but it had even augmented by 10″; making together 18″, for the effect of the Nutation in the 9 years.</p><p>The Nutation changes equally the longitudes, the right-ascensions, and the declinations of the stars, as before observed; it is the latitudes only which it does not affect, because the ecliptic is immoveable in the theory of the Nutation.</p><p>Dr. Bradley illustrates the foregoing theory of Nutation in the following manner. Let P represent the mean place of the pole of the equator, about which point, as a centre, suppose the <figure/> true pole to move in the small circle ABCD, whose diameter is 18″. Let E be the pole of the ecliptic, and EP be equal to the mean distance between the poles of the equator and ecliptic; and suppose the true pole of the equator to be at A, when the moon's ascending node is in the beginning of Aries; and at B, when the node gets back to Capricorn; and at C, when the same node is in Libra: at which time the north pole of the equator being nearer the north pole of the ecliptic, by the whole diameter of the little circle AC, equal to 18″; the obliquity of the ecliptic will then be so much less than it was, when the moon's ascending node was in Aries. The point P is supposed to move round E, with an equal retrograde motion, answerable to the mean precession arising from the joint actions of the sun and moon: while the true pole of the equator moves round P, in the circumference ABCD, with a retrograde motion likewise, in a period of the moon's nodes, or of 18 years and 7 months. By this means, when the moon's ascending node is in Aries, and the true pole of the equator, at A, is moving from A towards B; it will approach the stars that come to the <cb/> meridian with the sun about the vernal equinox, and recede from those that come with the sun near the autumnal equinox, faster than the mean pole P does. So that, while the moon's node goes back from Aries to Capricorn, the apparent precession will seem so much greater than the mean, as to cause the stars that lie in the equinoctial colure to have altered their declination 9″, in about 4 years and 8 months, more than the mean precession would do; and in the same time, the north pole of the equator will seem to have approached the stars that come to the meridian with the sun of our winter solstice about 9″, and to have receded as much from those that come with the sun at the summer solstice.</p><p>Thus the phenomena before recited are in general conformable to this hypothesis. But to be more particular; let S be the place of a star, PS the circle of declination passing through it, representing its distance from the mean pole, and <*> PS its mean right-ascension. Thus if O and R be the points where the circle of declination cuts the little circle ABCD, the true pole will be nearest that star at O, and farthest from it at R; the whole difference amounting to 18″, or to the diameter of the little circle. As the true pole of the equator is supposed to be at A, when the moon's ascending node is in Aries; and at B, when that node gets back to Capricorn; and the angular motion of the true pole about P, is likewise supposed equal to that of the moon's node about E, or the pole of the ecliptic; since in these cases the true pole of the equator is 90 degrees before the moon's ascending node, it must be so in all others.</p><p>When the true pole is at A, it will be at the same distance from the stars that lie in the equinoctial colure, as the mean pole P is; and as the true pole recedes back from A towards B, it will approach the stars which lie in that part of the colure represented by P<*>, and recede from those that lie in P<figure/>; not indeed with an equable motion, but in the ratio of the sine of the distance of the moon's node from the beginning of Aries. For if the node be supposed to have gone backwards from Aries 30°, or to the beginning of Pisces, the point which represents the place of the true pole will, in the mean time, have moved in the little circle through an arc, as AO, of 30° likewise; and would therefore in effect have approached the stars that lie in the equinoctial colure P<*>, and have receded from those that lie in P <figure/> by 4 1/<*> seconds, which is the sine of 30° to the radius AP. For if a perpendicular fall from O upon AP, it may be conceived as part of a great circle, passing through the true pole and any star lying in the equinoctial colure. Now the same proportion that holds in these stars, will obtain likewise in all others; and from hence we may collect a general rule for finding how much nearer, or farther, any star is to, or from, the mean pole, in any given position of the moon's node.</p><p>For, If <hi rend="italics">from the right-ascension of the star, we subtract the distance of the moon's ascending node from Aries; then radius will be to the sine of the remainder, as 9″ is to the number of seconds that the star is nearer to, or farther from, the true, than the mean pole.</hi></p><p>This motion of the true pole, about the mean at P, will also produce a change in the right-ascension of the <pb n="751"/><cb/> stars, and in the places of the equinoctial points, as well as in the obliquity of the ecliptic; and the quantity of the equations, in either of these cases, may be easily computed for any given position of the moon's nodes.</p><p>Dr. Bradley then proceeds to find the exact quantity of the mean precession of the equinoctial points, by comparing his own observations made at Greenwich, with those of Tycho Brahe and others; the mean of all which he states at 1 degree in 71 1/2 years, or 50 1/3″ per year; in order to shew the agreement of the foregoing hypothesis with the phenomena themselves, of the alterations in the polar distances of the stars; the conclusions from which approach as near to a coincidence as could be expected on the foregoing circular hypothesis, the diameter of which is 18″; instead of the more accurate quantity 19.1″, as deduced by Dr. Maskelyne, and the elliptic theory as determined by the mathematicians, in which the greater axis (19.1″) is to the less axis (14.2″), as the cosine of the greatest declination is to the cosine of double the same.</p><p>To give an idea now of the Nutation of the stars, in longitude, right-ascension, and declination; suppose the pole of the equator to be at any time in the point O, also S the place of any star, and OH perpendicular to AE: then, like as AE is the solstitial colure when the pole of the equator was at A, and the longitude of the star S equal to the angle AES; so OE is the solstitial colure when that pole is at O, and the longitude is then only the angle OES; less than before by the angle AEO, which therefore is the Nutation in longitude: counting the longitudes from the solstitial instead of the equinoctial colure, from which they differ equally by 90 degrees, and therefore have the same difference AEO. Now the angle AEO will be as the line HO = sin. AO to radius PB = sin. AO X PB = sin. AO X 9″; therefore as , since AO is equal to longitude of the moon's node. This expression therefore gives the Nutation in longitude, supposing the maximum of Nutation, with Bradley, to be 18″; and it is negative, or must be subtracted from the mean longitude of the stars, when the moon's node is in the first 6 signs of its longitude, but additive in the latter 6, to give the true apparent longitude.</p><p>This equation of the Nutation in longitude is the same for all the stars; but that for the declination and right ascension is various for the different stars. In the foregoing figure, PS is the mean polar distance, or mean codeclination, of the star S, when the true place of the pole is O; and SO the apparent codeclination; also, the angle SPE is the mean right-ascension, and SOE the apparent one, counted from the solstitial colure; consequently OPS or OPF the difference between the right-ascension of the star and that of the pole, which is equal to the longitude of the node increased by 3 signs or 90 degrees; supposing OF to be a small arc perpendicular to the circle of declination PFS; then is SF = SO, and PF the Nutation in declination, or the quantity the declination of the star has increased; but radius 1 : 9″ :: cosin. OPF : PF = 9″ X cos. OPF; so that the equation of decli- <cb/> nation will be found by multiplying 9″ by the sine of the star's right-ascension diminished by the longitude of the node; for that angle is the complement of the angle SPO. This Nutation in declination is to be added to the mean declination to give the apparent, when its argument does not exceed 6 signs; and to be subtracted in the latter 6 signs. But the contrary for the stars having south declination.</p><p>To calculate the Nutation in right-ascension, we must find the difference between the angle SOE the apparent, and SPE the mean right-ascension, counted from the solstitial colure EO. Now the true rightascension SOE is equal to the difference between the two variable angles GOE and GOS; the former of which arises from the change of one of the variable circles EO, and depends only on the situation of the node or of that of the pole O; the latter GOS depends on the angle GPS which is the difference between the right-ascension of the star and the place of the pole O. Now in the spherical triangle GPE, which changes into GOE, the side GE and the angle G remain constant, and the other parts are variable; hence therefore the small variation PO of the side next the constant angle G, is to the small variation of the angle opposite to the constant side GE, as the tangent of the side PE opposite to the constant angle, is to the sine of the angle GPE opposite to the constant side; that is, as , the difference between the angles GOE and GPE. This is the change which the Nutation PO produces in the angle GPE, being the first part of the Nutation sought, and is common to all the stars and planets. It is to be subtracted from the mean right-ascension in the first 6 signs of the longitude of the node, and added in the other six.</p><p>In like manner is found the change which the Nutation produces in the other part of the right-ascension SPE, that is, in the angle SPG, which becomes SOG by the effect of the Nutation. This small variation will be calculated from the same analogy, by means of the triangle SOG, in which the angle G is constant, as well as the side SG, whilst SP changes into SO. Hence therefore, tang. SP : sin. SPG :: 9″ : variation of SPG, that is, the cotangent of the declination is to the cosine of the distance between the star and the node, as 9″ are to the quantity the angle SPG varies in becoming the angle SOG, being the second part of the Nutation in right-ascension; and if there be taken for the argument, the right-ascension of the star minus the longitude of the node, the equation will be subtractive in the first and last quadrant of the argument, and additive in the 2d and 3d, or from 3 to 9 signs. But the contrary for stars having south declination.</p><p>This second part of the Nutation in right-ascension affects the return of the sun to the meridian, and therefore it must be taken into the account in computing the equation of time. But the former part of the Nutation does not enter into that computation; because it only changes the place of the equinox, without changing the point of the equator to which a star corresponds, and consequently without altering the duration of the returns to the meridian. <pb n="752"/><cb/></p><p>All these calculations of the Nutation, above explained, are upon Machin's hypothesis, that the pole describes a circle; however Bradley himself remarked that some of his observations differed too much from that theory, and that such observations were found to agree better with theory, by supposing that the pole, instead of the circle, describes an ellipse, having its less axis DB = 16″ in the equinoctial colure, and the greater axis AC = 18″, lying in the solstitial colure. But as even this correction was not sufficient to cause all the inequalities to disappear entirely, Dr. Bradley referred the determination of the point to theoretical and physical investigation. Accordingly several mathematicians undertook the task, and particularly d'Alembert, in his Recherches fur la précession des equinoxes, where he determines that the pole really describes an ellipse, and that narrower than the one assumed above by Bradley, the greater axis being to the less, as the <cb/> cosine of 23° 28′ to the cosine of double the same. And as Dr. Maskelyne found, from a more accurate reduction of Bradley's observations, that the maximum of the Nutation gives 19.1″ for the greater axis, therefore the above proportion gives 14.2″ for the less axis of it; and according to these data, the theory and observations are now found to agree very near together.</p><p>See La Lande's Astron. vol. 3, art. 2874 &c, where he makes the corrections for the ellipse. He observes however that by the circular hypothesis alone, the computations may be performed as accurately as the observations can be made; and he concludes with some corrections and rules for computing the Nutation in the elliptic theory.</p><p>The following set of general tables very readily give the effect of Nutation on the elliptical hypothesis; they were calculated by the late M. Lambert, and are taken from the Connoissance des Temps for the year 1788. <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=15" role="data"><hi rend="italics">General Tables for Nutation in the Ellipse.</hi></cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="colspan=5" role="data"><hi rend="smallcaps">Table</hi> 1.</cell><cell cols="1" rows="1" rend="colspan=5" role="data"><hi rend="smallcaps">Table</hi> 2.</cell><cell cols="1" rows="1" rend="colspan=5" role="data"><hi rend="smallcaps">Table</hi> 3.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">De-</cell><cell cols="1" rows="1" role="data">0.6</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">2.8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">De-</cell><cell cols="1" rows="1" role="data">0.6</cell><cell cols="1" rows="1" role="data">1.7</cell><cell cols="1" rows="1" role="data">2.8</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">De-</cell><cell cols="1" rows="1" role="data">0.6</cell><cell cols="1" rows="1" role="data">1.7</cell><cell cols="1" rows="1" role="data">2.8</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" rend="align=right" role="data">grees</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">grees</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">grees</cell><cell cols="1" rows="1" role="data">- +</cell><cell cols="1" rows="1" role="data">- +</cell><cell cols="1" rows="1" role="data">- +</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=left" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data">″</cell><cell cols="1" rows="1" role="data"/></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">0.00</cell><cell cols="1" rows="1" role="data">3.93</cell><cell cols="1" rows="1" role="data">6.80</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">0.00</cell><cell cols="1" rows="1" role="data">0.58</cell><cell cols="1" rows="1" role="data">1.00</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">0.00</cell><cell cols="1" rows="1" role="data"> 7.71</cell><cell cols="1" rows="1" role="data">13.36</cell><cell cols="1" rows="1" role="data">30</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">0.14</cell><cell cols="1" rows="1" role="data">4.04</cell><cell cols="1" rows="1" role="data">6.86</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">0.02</cell><cell cols="1" rows="1" role="data">0.59</cell><cell cols="1" rows="1" role="data">1.01</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">0.27</cell><cell cols="1" rows="1" role="data"> 7.95</cell><cell cols="1" rows="1" role="data">13.50</cell><cell cols="1" rows="1" role="data">29</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">0.27</cell><cell cols="1" rows="1" role="data">4.16</cell><cell cols="1" rows="1" role="data">6.93</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">0.04</cell><cell cols="1" rows="1" role="data">0.61</cell><cell cols="1" rows="1" role="data">1.02</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">0.54</cell><cell cols="1" rows="1" role="data"> 8.18</cell><cell cols="1" rows="1" role="data">13.62</cell><cell cols="1" rows="1" role="data">28</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">0.41</cell><cell cols="1" rows="1" role="data">4.28</cell><cell cols="1" rows="1" role="data">6.99</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">0.06</cell><cell cols="1" rows="1" role="data">0.63</cell><cell cols="1" rows="1" role="data">1.02</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">0.81</cell><cell cols="1" rows="1" role="data"> 8.40</cell><cell cols="1" rows="1" role="data">13.75</cell><cell cols="1" rows="1" role="data">27</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">0.55</cell><cell cols="1" rows="1" role="data">4.39</cell><cell cols="1" rows="1" role="data">7.06</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">0.08</cell><cell cols="1" rows="1" role="data">0.64</cell><cell cols="1" rows="1" role="data">1.03</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">1.08</cell><cell cols="1" rows="1" role="data"> 8.63</cell><cell cols="1" rows="1" role="data">13.87</cell><cell cols="1" rows="1" role="data">26</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">0.68</cell><cell cols="1" rows="1" role="data">4.50</cell><cell cols="1" rows="1" role="data">7.11</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">0.10</cell><cell cols="1" rows="1" role="data">0.66</cell><cell cols="1" rows="1" role="data">1.04</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">1.35</cell><cell cols="1" rows="1" role="data"> 8.85</cell><cell cols="1" rows="1" role="data">13.98</cell><cell cols="1" rows="1" role="data">25</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">0.82</cell><cell cols="1" rows="1" role="data">4.61</cell><cell cols="1" rows="1" role="data">7.17</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">0.12</cell><cell cols="1" rows="1" role="data">0.68</cell><cell cols="1" rows="1" role="data">1.05</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">1.61</cell><cell cols="1" rows="1" role="data"> 9.07</cell><cell cols="1" rows="1" role="data">14.10</cell><cell cols="1" rows="1" role="data">24</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">0.95</cell><cell cols="1" rows="1" role="data">4.72</cell><cell cols="1" rows="1" role="data">7.23</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">0.14</cell><cell cols="1" rows="1" role="data">0.69</cell><cell cols="1" rows="1" role="data">1.06</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">1.88</cell><cell cols="1" rows="1" role="data"> 9.29</cell><cell cols="1" rows="1" role="data">14.20</cell><cell cols="1" rows="1" role="data">23</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">1.11</cell><cell cols="1" rows="1" role="data">4.83</cell><cell cols="1" rows="1" role="data">7.28</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">0.16</cell><cell cols="1" rows="1" role="data">0.71</cell><cell cols="1" rows="1" role="data">1.07</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">2.15</cell><cell cols="1" rows="1" role="data"> 9.50</cell><cell cols="1" rows="1" role="data">14.31</cell><cell cols="1" rows="1" role="data">22</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">1.23</cell><cell cols="1" rows="1" role="data">4.94</cell><cell cols="1" rows="1" role="data">7.33</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">0.18</cell><cell cols="1" rows="1" role="data">0.72</cell><cell cols="1" rows="1" role="data">1.07</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">2.41</cell><cell cols="1" rows="1" role="data"> 9.71</cell><cell cols="1" rows="1" role="data">14.41</cell><cell cols="1" rows="1" role="data">21</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">1.36</cell><cell cols="1" rows="1" role="data">5.05</cell><cell cols="1" rows="1" role="data">7.38</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.20</cell><cell cols="1" rows="1" role="data">0.74</cell><cell cols="1" rows="1" role="data">1.08</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">2.68</cell><cell cols="1" rows="1" role="data"> 9.92</cell><cell cols="1" rows="1" role="data">14.50</cell><cell cols="1" rows="1" role="data">20</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">1.50</cell><cell cols="1" rows="1" role="data">5.15</cell><cell cols="1" rows="1" role="data">7.42</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0.22</cell><cell cols="1" rows="1" role="data">0.75</cell><cell cols="1" rows="1" role="data">1.09</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">2.94</cell><cell cols="1" rows="1" role="data">10.12</cell><cell cols="1" rows="1" role="data">14.59</cell><cell cols="1" rows="1" role="data">19</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">1.63</cell><cell cols="1" rows="1" role="data">5.25</cell><cell cols="1" rows="1" role="data">7.47</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0.24</cell><cell cols="1" rows="1" role="data">0.77</cell><cell cols="1" rows="1" role="data">1.09</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">3.21</cell><cell cols="1" rows="1" role="data">10.32</cell><cell cols="1" rows="1" role="data">14.67</cell><cell cols="1" rows="1" role="data">18</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">1.77</cell><cell cols="1" rows="1" role="data">5.35</cell><cell cols="1" rows="1" role="data">7.51</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0.26</cell><cell cols="1" rows="1" role="data">0.78</cell><cell cols="1" rows="1" role="data">1.10</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">3.47</cell><cell cols="1" rows="1" role="data">10.52</cell><cell cols="1" rows="1" role="data">14.76</cell><cell cols="1" rows="1" role="data">17</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">1.90</cell><cell cols="1" rows="1" role="data">5.45</cell><cell cols="1" rows="1" role="data">7.55</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0.28</cell><cell cols="1" rows="1" role="data">0.80</cell><cell cols="1" rows="1" role="data">1.11</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">3.73</cell><cell cols="1" rows="1" role="data">10.72</cell><cell cols="1" rows="1" role="data">14.83</cell><cell cols="1" rows="1" role="data">16</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">2.03</cell><cell cols="1" rows="1" role="data">5.55</cell><cell cols="1" rows="1" role="data">7.58</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">0.30</cell><cell cols="1" rows="1" role="data">0.81</cell><cell cols="1" rows="1" role="data">1.11</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">3.99</cell><cell cols="1" rows="1" role="data">10.91</cell><cell cols="1" rows="1" role="data">14.90</cell><cell cols="1" rows="1" role="data">15</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">2.16</cell><cell cols="1" rows="1" role="data">5.65</cell><cell cols="1" rows="1" role="data">7.62</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0.32</cell><cell cols="1" rows="1" role="data">0.83</cell><cell cols="1" rows="1" role="data">1.12</cell><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">4.25</cell><cell cols="1" rows="1" role="data">11.10</cell><cell cols="1" rows="1" role="data">14.97</cell><cell cols="1" rows="1" role="data">14</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">2.30</cell><cell cols="1" rows="1" role="data">5.74</cell><cell cols="1" rows="1" role="data">7.65</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0.34</cell><cell cols="1" rows="1" role="data">0.84</cell><cell cols="1" rows="1" role="data">1.12</cell><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">4.51</cell><cell cols="1" rows="1" role="data">11.28</cell><cell cols="1" rows="1" role="data">15.03</cell><cell cols="1" rows="1" role="data">13</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">2.43</cell><cell cols="1" rows="1" role="data">5.83</cell><cell cols="1" rows="1" role="data">7.68</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">0.35</cell><cell cols="1" rows="1" role="data">0.85</cell><cell cols="1" rows="1" role="data">1.13</cell><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">4.77</cell><cell cols="1" rows="1" role="data">11.47</cell><cell cols="1" rows="1" role="data">15.09</cell><cell cols="1" rows="1" role="data">12</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">2.56</cell><cell cols="1" rows="1" role="data">5.92</cell><cell cols="1" rows="1" role="data">7.71</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0.37</cell><cell cols="1" rows="1" role="data">0.87</cell><cell cols="1" rows="1" role="data">1.13</cell><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">5.02</cell><cell cols="1" rows="1" role="data">11.65</cell><cell cols="1" rows="1" role="data">15.15</cell><cell cols="1" rows="1" role="data">11</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">2.68</cell><cell cols="1" rows="1" role="data">6.01</cell><cell cols="1" rows="1" role="data">7.73</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0.39</cell><cell cols="1" rows="1" role="data">0.88</cell><cell cols="1" rows="1" role="data">1.13</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">5.28</cell><cell cols="1" rows="1" role="data">11.82</cell><cell cols="1" rows="1" role="data">15.20</cell><cell cols="1" rows="1" role="data">10</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">2.81</cell><cell cols="1" rows="1" role="data">6.10</cell><cell cols="1" rows="1" role="data">7.75</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">0.41</cell><cell cols="1" rows="1" role="data">0.89</cell><cell cols="1" rows="1" role="data">1.14</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">5.53</cell><cell cols="1" rows="1" role="data">11.99</cell><cell cols="1" rows="1" role="data">15.24</cell><cell cols="1" rows="1" role="data"> 9</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">2.94</cell><cell cols="1" rows="1" role="data">6.19</cell><cell cols="1" rows="1" role="data">7.76</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">0.43</cell><cell cols="1" rows="1" role="data">0.91</cell><cell cols="1" rows="1" role="data">1.14</cell><cell cols="1" rows="1" role="data"> 8</cell><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">5.78</cell><cell cols="1" rows="1" role="data">12.16</cell><cell cols="1" rows="1" role="data">15.28</cell><cell cols="1" rows="1" role="data"> 8</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">3.07</cell><cell cols="1" rows="1" role="data">6.27</cell><cell cols="1" rows="1" role="data">7.77</cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">0.45</cell><cell cols="1" rows="1" role="data">0.92</cell><cell cols="1" rows="1" role="data">1.14</cell><cell cols="1" rows="1" role="data"> 7</cell><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">6.03</cell><cell cols="1" rows="1" role="data">12.32</cell><cell cols="1" rows="1" role="data">15.32</cell><cell cols="1" rows="1" role="data"> 7</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">3.19</cell><cell cols="1" rows="1" role="data">6.35</cell><cell cols="1" rows="1" role="data">7.79</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">0.47</cell><cell cols="1" rows="1" role="data">0.93</cell><cell cols="1" rows="1" role="data">1.14</cell><cell cols="1" rows="1" role="data"> 6</cell><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">6.28</cell><cell cols="1" rows="1" role="data">12.48</cell><cell cols="1" rows="1" role="data">15.35</cell><cell cols="1" rows="1" role="data"> 6</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">3.32</cell><cell cols="1" rows="1" role="data">6.43</cell><cell cols="1" rows="1" role="data">7.80</cell><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">0.49</cell><cell cols="1" rows="1" role="data">0.94</cell><cell cols="1" rows="1" role="data">1.15</cell><cell cols="1" rows="1" role="data"> 5</cell><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">6.52</cell><cell cols="1" rows="1" role="data">12.64</cell><cell cols="1" rows="1" role="data">15.37</cell><cell cols="1" rows="1" role="data"> 5</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">3.44</cell><cell cols="1" rows="1" role="data">6.51</cell><cell cols="1" rows="1" role="data">7.82</cell><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">0.50</cell><cell cols="1" rows="1" role="data">0.95</cell><cell cols="1" rows="1" role="data">1.15</cell><cell cols="1" rows="1" role="data"> 4</cell><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">6.76</cell><cell cols="1" rows="1" role="data">12.79</cell><cell cols="1" rows="1" role="data">15.39</cell><cell cols="1" rows="1" role="data"> 4</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">3.56</cell><cell cols="1" rows="1" role="data">6.58</cell><cell cols="1" rows="1" role="data">7.83</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">0.52</cell><cell cols="1" rows="1" role="data">0.96</cell><cell cols="1" rows="1" role="data">1.15</cell><cell cols="1" rows="1" role="data"> 3</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">7.01</cell><cell cols="1" rows="1" role="data">12.94</cell><cell cols="1" rows="1" role="data">15.41</cell><cell cols="1" rows="1" role="data"> 3</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">3.69</cell><cell cols="1" rows="1" role="data">6.66</cell><cell cols="1" rows="1" role="data">7.84</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">0.54</cell><cell cols="1" rows="1" role="data">0.97</cell><cell cols="1" rows="1" role="data">1.15</cell><cell cols="1" rows="1" role="data"> 2</cell><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">7.25</cell><cell cols="1" rows="1" role="data">13.09</cell><cell cols="1" rows="1" role="data">15.42</cell><cell cols="1" rows="1" role="data"> 2</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">3.81</cell><cell cols="1" rows="1" role="data">6.73</cell><cell cols="1" rows="1" role="data">7.85</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">0.56</cell><cell cols="1" rows="1" role="data">0.99</cell><cell cols="1" rows="1" role="data">1.15</cell><cell cols="1" rows="1" role="data"> 1</cell><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">7.48</cell><cell cols="1" rows="1" role="data">13.23</cell><cell cols="1" rows="1" role="data">15.43</cell><cell cols="1" rows="1" role="data"> 1</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">3.93</cell><cell cols="1" rows="1" role="data">6.80</cell><cell cols="1" rows="1" role="data">7.85</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">0.58</cell><cell cols="1" rows="1" role="data">1.00</cell><cell cols="1" rows="1" role="data">1.15</cell><cell cols="1" rows="1" role="data"> 0</cell><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">7.71</cell><cell cols="1" rows="1" role="data">13.36</cell><cell cols="1" rows="1" role="data">15.43</cell><cell cols="1" rows="1" role="data"> 0</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" rend="align=right" role="data">De-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" role="data">+ -</cell><cell cols="1" rows="1" rend="align=right" role="data">De-</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">- +</cell><cell cols="1" rows="1" role="data">- +</cell><cell cols="1" rows="1" role="data">- +</cell><cell cols="1" rows="1" rend="align=right" role="data">De-</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.11</cell><cell cols="1" rows="1" role="data">4.10</cell><cell cols="1" rows="1" role="data">3.9</cell><cell cols="1" rows="1" rend="align=right" role="data">grees</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.11</cell><cell cols="1" rows="1" role="data">4.10</cell><cell cols="1" rows="1" role="data">3.9</cell><cell cols="1" rows="1" rend="align=right" role="data">grees</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">5.11</cell><cell cols="1" rows="1" role="data">4.10</cell><cell cols="1" rows="1" role="data">3.9</cell><cell cols="1" rows="1" rend="align=right" role="data">grees</cell></row></table><pb n="753"/><cb/> <hi rend="center"><hi rend="italics">The Use of the Tables.</hi></hi></p><p>The right-ascension of a star minus the moon's mean longitude, gives the argument of the first of these three tables. The sum of the same two quantities gives the argument of the 2d table. Then the sum or the difference of the quantities found with these two arguments, will give the correction to be applied to the mean declination of the star, if it is north declination; but if it is southern, the signs + or - are to be changed into - and +.</p><p>From each of those two arguments for the declination subtracting 3 signs, or 90°, gives the arguments for correcting the right-ascension; the sum or difference of the quantities found, with these two arguments, in tables 1 and 2, is to be multiplied by the tangent of the star's declination, and to the product is to be added the quantity taken out of table 3, the argument of which is the mean longitude of the moon's ascending node: when the declination of the star is south, the tangent will be negative.</p><p><hi rend="italics">Example.</hi> To find the Nutation in right-ascension and declination for the star <foreign xml:lang="greek">a</foreign> Aquilæ, the 1st of July 1788. <table><row role="data"><cell cols="1" rows="1" role="data">Right-ascension of the star</cell><cell cols="1" rows="1" role="data">9<hi rend="sup">s</hi></cell><cell cols="1" rows="1" role="data">25°</cell><cell cols="1" rows="1" role="data"> 7′</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data">Long. of the moon's node</cell><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data"/></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">  ″</cell></row><row role="data"><cell cols="1" rows="1" role="data">Diff. being argument 1,</cell><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data"> 9</cell><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">+ 4.99</cell></row><row role="data"><cell cols="1" rows="1" role="data">Sum, argument 2,</cell><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">- 0.22</cell></row><row role="data"><cell cols="1" rows="1" rend="colspan=2" role="data">Correction of the declination</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">+ 4.77</cell></row></table> The above two arguments being each diminished by 3 signs, give, <table><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">s ° ′  ″  </cell></row><row role="data"><cell cols="1" rows="1" role="data">Argument 1</cell><cell cols="1" rows="1" rend="align=right" role="data">10 9 27 - 6.06</cell></row><row role="data"><cell cols="1" rows="1" role="data">Argument 2</cell><cell cols="1" rows="1" rend="align=right" role="data">3 10 47 + 1.13</cell></row><row role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="align=right" role="data">- 4.93</cell></row><row role="data"><cell cols="1" rows="1" role="data">Declin. of star north, its tangent</cell><cell cols="1" rows="1" rend="align=right" role="data">0.146</cell></row><row role="data"><cell cols="1" rows="1" role="data">The product is</cell><cell cols="1" rows="1" rend="align=right" role="data">- 0.72</cell></row><row role="data"><cell cols="1" rows="1" role="data">Long. of the <figure/>'s node, argum. 3</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 14.94</cell></row><row role="data"><cell cols="1" rows="1" role="data">Correction of right-ascension</cell><cell cols="1" rows="1" rend="align=right" role="data">+ 14.22</cell></row></table></p><p>In general, let <figure/> denote the longitude of the moon's ascending node; <hi rend="italics">r</hi> the right-ascension of a star or planet; <hi rend="italics">d</hi> its declination; the Nutation in declination and rightascension will be expressed by the two following formulæ; viz, the Nutation in declination = 7″.85 X sin. (<hi rend="italics">r</hi> - <figure/>) + 1″.15 X sin. (<hi rend="italics">r</hi> + <figure/>); and the Nutation in right-ascension = [7″.85 X sin. (<hi rend="italics">r</hi> - <figure/> - 90°) + 1″.15 X sin. (<hi rend="italics">r</hi> + <figure/> - 90°)] X tang. <hi rend="italics">d</hi> - 15″.43 X sin. <figure/>.</p><p>For the mathematical investigation of the effects of universal attraction, in producing the Nutation, &c, see d'Alembert's Recherches fur la Precession des Equinoxes; Silvabelle's Treatise on the Precession of the Equinoxes &c, in the Philos. Trans. an. 1754, p. 385; Walmesley's treatise De Præcessione Equinoctiorum et Axis Terræ Nutatione, in the Philos. Trans. an. 1756, <cb/> pa. 700; Simpson's Miscellaneous Tracts, pa. 1; and other authors. <hi rend="center">S</hi></p><p>STEAM. The observations on the different degrees of temperature acquired by water in boiling, under different pressures of the atmosphere, and the formation of the vapour from water under the receiver of an airpump, when, with the common temperatures, the pressure is diminished to a certain degree, have taught us that the expansive force of vapour or Steam is different in the different temperatures, and that in general it increases in a variable ratio as the temperature is raised.</p><p>But there was wanting, on this important subject, a series of exact and direct experiments, by means of which, having given the degree of temperature in boiling water, we may know the expansive force of the Steam rising from it; and vice versa. There was wanting also an analytical theorem, expressing the relation between the temperature of boiling water, and the pressure with which the force of its Steam is in equilibrium. These circumstances then have lately been accomplished by M. Betancourt, an ingenious Spanish philosopher, the particulars of which are described in a memoir communicated to the French Academy of Sciences in 1<*>90, and ordered to be printed in their collection of the Works of Strangers.</p><p>The apparatus which M. Betancourt makes use of, is a copper vessel or boiler, with its cover firmly soldered on. The cover has three holes, which close up with screws: the first is to put the water in and out; through the second passes the stem of a thermometer, which has the whole of its scale or graduations above the vessel, and its ball within, where it is immersed either in the water or the Steam according to the different circumstances; through the third hole passes a tube making a communication between the cavity of the boiler and one branch of an inverted syphon, which, containing mercury, acts as a barometer for measuring the pressure of the elastic vapour within the boiler. There is a fourth hole, in the side of the vessel, into which is inserted a tube, with a turn-cock, making a communication with the receiver of an air-pump, for extracting the air from the boiler, and to prevent its return.</p><p>The apparatus being prepared in good order, and distilled water introduced into the boiler by the first hole, and then stopped, as well as the end of the inverted syphon or barometer, M. Betancourt surrounded the boiler with ice, to lower the temperature of the water to the freezing point, and then extracting all the air from the boiler by means of the air-pump, the difference between the columns of mercury in the two branches of the barometer is the measure of the spring of the vapour arising from the water in that temperature. Then, lighting the fire below the boiler, he raised gradually the temperature of the water from 0 to 110 degrees of Reaumur's thermometer; being the same as from 32 to 212 degrees of Fahrenheit's; and for each degree of elevation in the temperature, he observed the height of the column of mercury which measured the elasticity or pressure of the vapour.</p><p>The results of M. Betancourt's experiments are con- <pb n="754"/><cb/> tained in a table of four columns, which are but little different, according to the different quantities of water in the vessel. It is here observable, that the increase in the expansive force of the vapour, is at first very slow; but gradually increasing faster and faster, till at last it becomes very rapid. Thus, the strength of the vapour, at 80 degrees, is only equal to 28 French inches of mercury; but at 110 degrees it is equal to no less than 98 inches, that is 3 times and a half more for the increase of only 30 degrees of heat.</p><p>To express analytically the relation between the degrees of temperature of the vapour, and its expansive force, this author employs a method devised by M. Prony. This method consists in conceiving the heights of the columns of mercury, measuring the expansive force, to represent the ordinates of a curve, and the degrees of heat as the abscisses of the same; making the ordinates equal to the sum of several logarithmic ones, which contain two indeterminates, and determining these quantities so that the curve may agree with a good number of observations taken throughout the whole extent of them. Then constructing the curve which results immediately from the experiments, and that given by the formula, these two curves are found to coincide almost perfectly together; the small differences being doubtless owing to the little irregularities in the experiments and in dividing the scale; so that the phenomena may be considered as truly represented by the formula.</p><p>M. Betancourt made also experiments with the vapour from spirit of wine, similar to those made with water; constructing the curve, and giving the formula proper to the same. From which is derived this remarkable result, that, for any one and the same degree of heat, the strength of the vapour of spirit of wine, is to that of water, always in the same constant ratio, viz, that of 7 to 3 very nearly; the strength of the former being always 2 1/3 times the strength of the latter, with the same degree of heat in the liquid. <hi rend="center"><hi rend="italics">Of the Formula, or Equation to the Curve.</hi></hi></p><p>The equation to the curve of temperature and pressure, denoting the relation between the abscisses and ordinates, or between the temperature of the vapour and its strength, is, for water, . Where <hi rend="italics">x</hi> denotes the abscisses of the curve, or the degrees of Reaumur's thermometer; and <hi rend="italics">y</hi> the corresponding ordinates, or the heights of the column of mercury in Paris inches, representing the strength or elasticity of the vapour answering to the number <hi rend="italics">x</hi> of degrees of the thermometer. Then, by comparing this formula with a proper number of the experiments, the values of the constant quantities come out as below: <hi rend="center"><hi rend="italics">b</hi> = 10.</hi> <hi rend="center"><hi rend="italics">a</hi> = 0.068831</hi> <hi rend="center"><hi rend="italics">c</hi> = 0.019438</hi> <hi rend="center"><*> = 0.013490</hi> <cb/> <hi rend="center"><hi rend="italics">e</hi> = - 4.689760</hi> <hi rend="center"><hi rend="italics">c″</hi> = 0.058622</hi> <hi rend="center"><hi rend="italics">e′</hi> = - 3.937600</hi> <hi rend="center"><hi rend="italics">c‴</hi> = 0.049220</hi></p><p>Hence it is evident by inspection, that the terms of the equation are very easy to calculate. For, <hi rend="italics">b</hi> being the radix or root of the common system of logarithms, and all the terms on the second side of the equation being the powers of <hi rend="italics">b,</hi> these terms are consequently the tabular natural numbers having the variable exponents for their logarithms. Now as <hi rend="italics">x</hi> rises only to the first power, and is multiplied by a constant number, and another constant number being added to the product, gives the variable exponent, or logarithm; to which then is immediately found the corresponding natural number in the table of logarithms.</p><p>In the above formula, the two last terms may be entirely omitted, as very small, as far as to the 90th degree of the thermometer; and even above that temperature those two terms make but a small part of the whole formula.</p><p>And for the spirit of wine the formula is . Where <hi rend="italics">x</hi> and <hi rend="italics">y,</hi> as before, denote the absciss and ordinate of the curve, or the temperature and expansive force of the vapour from the spirit of wine; also the values of the constant quantities are as below: <hi rend="center"><hi rend="italics">b</hi> = 10.</hi> <hi rend="center"><hi rend="italics">a</hi> = - 0.04853</hi> <hi rend="center"><hi rend="italics">c</hi> = 0.02393</hi> <hi rend="center"><hi rend="italics">a′</hi> = - 0.63414</hi> <hi rend="center"><hi rend="italics">c′</hi> = - 0.096532</hi> <hi rend="center"><hi rend="italics">e</hi> = - 2.509542</hi> <hi rend="center"><hi rend="italics">c″</hi> = 0.046473</hi> <hi rend="center"><hi rend="italics">e′</hi> = - 1.790192</hi> <hi rend="center"><hi rend="italics">c‴</hi> = 0.029448</hi> <hi rend="center"><hi rend="italics">A</hi> = 1.12647</hi></p><p>This formula is of the same nature as the former, having also the like ease and convenience of calculation; and perhaps more so; as the second term <hi rend="italics">b</hi><hi rend="sup"><hi rend="italics">a′</hi> + <hi rend="italics">c′x</hi></hi>, having its exponent wholly negative, soon diminishes to no value, so as to be omitted from the 10th degree of temperature; also the difference between the last two terms - <hi rend="italics">b</hi><hi rend="sup"><hi rend="italics">e</hi> + <hi rend="italics">c″x</hi></hi> + <hi rend="italics">b</hi><hi rend="sup"><hi rend="italics">e′</hi> + <hi rend="italics">c‴x</hi></hi> may be omitted till the 70th degree, for the same reason. So that, to the 10th degree of temperature the theorem is only ; and from the 10th to the 70th degree it is barely ; after which, for the last 15 or 20 degrees, for great accuracy, the last two terms may be taken in.</p><p>A compendium of the table of the experiments here follows, for the vapour of both water and spirit of wine, the temperature by Reaumur's thermometer, and the barometer in French inches. <pb n="755"/> <table rend="border"><head><hi rend="italics">Table of the Temperature and Strength of the Vapour of Water and Spirit of Wine, by Reaumur's Thermometer, and French Inches.</hi></head><row rend="align=center" role="data"><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">Height of the Barometer for</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" rend="colspan=2" role="data">Height of the Barometer for</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Degr. of</cell><cell cols="1" rows="1" role="data">Vapour of</cell><cell cols="1" rows="1" role="data">Vapour of</cell><cell cols="1" rows="1" role="data">Deg. of</cell><cell cols="1" rows="1" role="data">Vapour</cell><cell cols="1" rows="1" role="data">Vapour of</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Reau. Ther.</cell><cell cols="1" rows="1" role="data">Water.</cell><cell cols="1" rows="1" role="data">Spirit of Wine.</cell><cell cols="1" rows="1" role="data">Reau. Ther.</cell><cell cols="1" rows="1" role="data">of Water.</cell><cell cols="1" rows="1" role="data">Spirit of Wine.</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">1</cell><cell cols="1" rows="1" role="data">0.0176</cell><cell cols="1" rows="1" role="data">0.0043</cell><cell cols="1" rows="1" role="data">56</cell><cell cols="1" rows="1" role="data">7.6948</cell><cell cols="1" rows="1" role="data">18.4420</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">2</cell><cell cols="1" rows="1" role="data">0.0346</cell><cell cols="1" rows="1" role="data">0.0208</cell><cell cols="1" rows="1" role="data">57</cell><cell cols="1" rows="1" role="data">8.1412</cell><cell cols="1" rows="1" role="data">19.5081</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">3</cell><cell cols="1" rows="1" role="data">0.0538</cell><cell cols="1" rows="1" role="data">0.0478</cell><cell cols="1" rows="1" role="data">58</cell><cell cols="1" rows="1" role="data">8.6221</cell><cell cols="1" rows="1" role="data">20.6286</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">4</cell><cell cols="1" rows="1" role="data">0.0747</cell><cell cols="1" rows="1" role="data">0.0837</cell><cell cols="1" rows="1" role="data">59</cell><cell cols="1" rows="1" role="data">9.1071</cell><cell cols="1" rows="1" role="data">21.6071</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">5</cell><cell cols="1" rows="1" role="data">0.1038</cell><cell cols="1" rows="1" role="data">0.1279</cell><cell cols="1" rows="1" role="data">60</cell><cell cols="1" rows="1" role="data">9.6280</cell><cell cols="1" rows="1" role="data">23.0544</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">6</cell><cell cols="1" rows="1" role="data">0.1211</cell><cell cols="1" rows="1" role="data">0.1794</cell><cell cols="1" rows="1" role="data">61</cell><cell cols="1" rows="1" role="data">10.1767</cell><cell cols="1" rows="1" role="data">24.3451</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">7</cell><cell cols="1" rows="1" role="data">0.1508</cell><cell cols="1" rows="1" role="data">0.2377</cell><cell cols="1" rows="1" role="data">62</cell><cell cols="1" rows="1" role="data">10.7098</cell><cell cols="1" rows="1" role="data">25.6107</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">8</cell><cell cols="1" rows="1" role="data">0.1741</cell><cell cols="1" rows="1" role="data">0.3024</cell><cell cols="1" rows="1" role="data">63</cell><cell cols="1" rows="1" role="data">11.3602</cell><cell cols="1" rows="1" role="data">27.1444</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">9</cell><cell cols="1" rows="1" role="data">0.2073</cell><cell cols="1" rows="1" role="data">0.3733</cell><cell cols="1" rows="1" role="data">64</cell><cell cols="1" rows="1" role="data">11.9976</cell><cell cols="1" rows="1" role="data">28.6483</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">10</cell><cell cols="1" rows="1" role="data">0.2304</cell><cell cols="1" rows="1" role="data">0.4502</cell><cell cols="1" rows="1" role="data">65</cell><cell cols="1" rows="1" role="data">12.6687</cell><cell cols="1" rows="1" role="data">30.2262</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">11</cell><cell cols="1" rows="1" role="data">0.2681</cell><cell cols="1" rows="1" role="data">0.5130</cell><cell cols="1" rows="1" role="data">66</cell><cell cols="1" rows="1" role="data">13.3743</cell><cell cols="1" rows="1" role="data">31.8795</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">12</cell><cell cols="1" rows="1" role="data">0.3039</cell><cell cols="1" rows="1" role="data">0.6058</cell><cell cols="1" rows="1" role="data">67</cell><cell cols="1" rows="1" role="data">14.1161</cell><cell cols="1" rows="1" role="data">33.6114</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">13</cell><cell cols="1" rows="1" role="data">0.3419</cell><cell cols="1" rows="1" role="data">0.7040</cell><cell cols="1" rows="1" role="data">68</cell><cell cols="1" rows="1" role="data">14.8958</cell><cell cols="1" rows="1" role="data">35.4258</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">14</cell><cell cols="1" rows="1" role="data">0.3877</cell><cell cols="1" rows="1" role="data">0.8077</cell><cell cols="1" rows="1" role="data">69</cell><cell cols="1" rows="1" role="data">15.7153</cell><cell cols="1" rows="1" role="data">37.3232</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">0.4258</cell><cell cols="1" rows="1" role="data">0.9172</cell><cell cols="1" rows="1" role="data">70</cell><cell cols="1" rows="1" role="data">16.577 </cell><cell cols="1" rows="1" role="data">39.3076</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">16</cell><cell cols="1" rows="1" role="data">0.4778</cell><cell cols="1" rows="1" role="data">1.0330</cell><cell cols="1" rows="1" role="data">71</cell><cell cols="1" rows="1" role="data">17.482 </cell><cell cols="1" rows="1" role="data">41.3807</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">17</cell><cell cols="1" rows="1" role="data">0.5208</cell><cell cols="1" rows="1" role="data">1.1553</cell><cell cols="1" rows="1" role="data">72</cell><cell cols="1" rows="1" role="data">18.433 </cell><cell cols="1" rows="1" role="data">43.5465</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">18</cell><cell cols="1" rows="1" role="data">0.5730</cell><cell cols="1" rows="1" role="data">1.2846</cell><cell cols="1" rows="1" role="data">73</cell><cell cols="1" rows="1" role="data">19.433 </cell><cell cols="1" rows="1" role="data">45.8042</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">19</cell><cell cols="1" rows="1" role="data">0.6283</cell><cell cols="1" rows="1" role="data">1.4212</cell><cell cols="1" rows="1" role="data">74</cell><cell cols="1" rows="1" role="data">20.485 </cell><cell cols="1" rows="1" role="data">48.1589</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">20</cell><cell cols="1" rows="1" role="data">0.6872</cell><cell cols="1" rows="1" role="data">1.5655</cell><cell cols="1" rows="1" role="data">75</cell><cell cols="1" rows="1" role="data">21.587 </cell><cell cols="1" rows="1" role="data">50.6096</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">21</cell><cell cols="1" rows="1" role="data">0.7497</cell><cell cols="1" rows="1" role="data">1.7180</cell><cell cols="1" rows="1" role="data">76</cell><cell cols="1" rows="1" role="data">22.746 </cell><cell cols="1" rows="1" role="data">53.1593</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">22</cell><cell cols="1" rows="1" role="data">0.8159</cell><cell cols="1" rows="1" role="data">1.8791</cell><cell cols="1" rows="1" role="data">77</cell><cell cols="1" rows="1" role="data">23.965 </cell><cell cols="1" rows="1" role="data">55.8095</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">23</cell><cell cols="1" rows="1" role="data">0.8863</cell><cell cols="1" rows="1" role="data">2.0494</cell><cell cols="1" rows="1" role="data">78</cell><cell cols="1" rows="1" role="data">25.260 </cell><cell cols="1" rows="1" role="data">58.3968</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">24</cell><cell cols="1" rows="1" role="data">0.9610</cell><cell cols="1" rows="1" role="data">2.2293</cell><cell cols="1" rows="1" role="data">79</cell><cell cols="1" rows="1" role="data">26.588 </cell><cell cols="1" rows="1" role="data">61.3057</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">25</cell><cell cols="1" rows="1" role="data">1.0402</cell><cell cols="1" rows="1" role="data">2.4194</cell><cell cols="1" rows="1" role="data">80</cell><cell cols="1" rows="1" role="data">28.006 </cell><cell cols="1" rows="1" role="data">64.3524</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">26</cell><cell cols="1" rows="1" role="data">1.1239</cell><cell cols="1" rows="1" role="data">2.6202</cell><cell cols="1" rows="1" role="data">81</cell><cell cols="1" rows="1" role="data">29.455 </cell><cell cols="1" rows="1" role="data">67.4095</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">27</cell><cell cols="1" rows="1" role="data">1.2127</cell><cell cols="1" rows="1" role="data">2.8325</cell><cell cols="1" rows="1" role="data">82</cell><cell cols="1" rows="1" role="data">30.980 </cell><cell cols="1" rows="1" role="data">70.4967</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">28</cell><cell cols="1" rows="1" role="data">1.3068</cell><cell cols="1" rows="1" role="data">3.0568</cell><cell cols="1" rows="1" role="data">83</cell><cell cols="1" rows="1" role="data">32.575 </cell><cell cols="1" rows="1" role="data">73.7647</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">29</cell><cell cols="1" rows="1" role="data">1.4065</cell><cell cols="1" rows="1" role="data">3.2937</cell><cell cols="1" rows="1" role="data">84</cell><cell cols="1" rows="1" role="data">34.251 </cell><cell cols="1" rows="1" role="data">77.0764</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">30</cell><cell cols="1" rows="1" role="data">1.5019</cell><cell cols="1" rows="1" role="data">3.5441</cell><cell cols="1" rows="1" role="data">85</cell><cell cols="1" rows="1" role="data">35.984 </cell><cell cols="1" rows="1" role="data">80.4708</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">31</cell><cell cols="1" rows="1" role="data">1.6333</cell><cell cols="1" rows="1" role="data">3.8087</cell><cell cols="1" rows="1" role="data">86</cell><cell cols="1" rows="1" role="data">37.800 </cell><cell cols="1" rows="1" role="data">83.9351</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">32</cell><cell cols="1" rows="1" role="data">1.7413</cell><cell cols="1" rows="1" role="data">4.0883</cell><cell cols="1" rows="1" role="data">87</cell><cell cols="1" rows="1" role="data">39.697 </cell><cell cols="1" rows="1" role="data">87.4625</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">33</cell><cell cols="1" rows="1" role="data">1.8671</cell><cell cols="1" rows="1" role="data">4.3837</cell><cell cols="1" rows="1" role="data">88</cell><cell cols="1" rows="1" role="data">41.642 </cell><cell cols="1" rows="1" role="data">91.1366</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">34</cell><cell cols="1" rows="1" role="data">1.9980</cell><cell cols="1" rows="1" role="data">4.6958</cell><cell cols="1" rows="1" role="data">89</cell><cell cols="1" rows="1" role="data">43.730 </cell><cell cols="1" rows="1" role="data">94.6580</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">35</cell><cell cols="1" rows="1" role="data">2.1374</cell><cell cols="1" rows="1" role="data">5.0256</cell><cell cols="1" rows="1" role="data">90</cell><cell cols="1" rows="1" role="data">45.870 </cell><cell cols="1" rows="1" role="data">98.2764</cell></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">36</cell><cell cols="1" rows="1" role="data">2.2846</cell><cell cols="1" rows="1" role="data">5.3741</cell><cell cols="1" rows="1" role="data">91</cell><cell cols="1" rows="1" role="data">48.092 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">37</cell><cell cols="1" rows="1" role="data">2.4401</cell><cell cols="1" rows="1" role="data">5.6423</cell><cell cols="1" rows="1" role="data">92</cell><cell cols="1" rows="1" role="data">50.408 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">38</cell><cell cols="1" rows="1" role="data">2.6045</cell><cell cols="1" rows="1" role="data">6.1315</cell><cell cols="1" rows="1" role="data">93</cell><cell cols="1" rows="1" role="data">52.785 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">39</cell><cell cols="1" rows="1" role="data">2.7780</cell><cell cols="1" rows="1" role="data">6.5426</cell><cell cols="1" rows="1" role="data">94</cell><cell cols="1" rows="1" role="data">55.253 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">40</cell><cell cols="1" rows="1" role="data">2.9711</cell><cell cols="1" rows="1" role="data">6.9770</cell><cell cols="1" rows="1" role="data">95</cell><cell cols="1" rows="1" role="data">57.801 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">41</cell><cell cols="1" rows="1" role="data">3.1544</cell><cell cols="1" rows="1" role="data">7.4360</cell><cell cols="1" rows="1" role="data">96</cell><cell cols="1" rows="1" role="data">60.423 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">42</cell><cell cols="1" rows="1" role="data">3.3583</cell><cell cols="1" rows="1" role="data">7.9211</cell><cell cols="1" rows="1" role="data">97</cell><cell cols="1" rows="1" role="data">63.108 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">43</cell><cell cols="1" rows="1" role="data">3.5735</cell><cell cols="1" rows="1" role="data">8.4336</cell><cell cols="1" rows="1" role="data">98</cell><cell cols="1" rows="1" role="data">65.877 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">44</cell><cell cols="1" rows="1" role="data">3.8005</cell><cell cols="1" rows="1" role="data">8.9751</cell><cell cols="1" rows="1" role="data">99</cell><cell cols="1" rows="1" role="data">68.692 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">45</cell><cell cols="1" rows="1" role="data">4.0399</cell><cell cols="1" rows="1" role="data">9.5476</cell><cell cols="1" rows="1" role="data">100</cell><cell cols="1" rows="1" role="data">71.552 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">46</cell><cell cols="1" rows="1" role="data">4.2922</cell><cell cols="1" rows="1" role="data">10.1516</cell><cell cols="1" rows="1" role="data">101</cell><cell cols="1" rows="1" role="data">74.444 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">47</cell><cell cols="1" rows="1" role="data">4.5582</cell><cell cols="1" rows="1" role="data">10.7906</cell><cell cols="1" rows="1" role="data">102</cell><cell cols="1" rows="1" role="data">77.359 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">48</cell><cell cols="1" rows="1" role="data">4.8386</cell><cell cols="1" rows="1" role="data">11.4606</cell><cell cols="1" rows="1" role="data">103</cell><cell cols="1" rows="1" role="data">80.268 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">49</cell><cell cols="1" rows="1" role="data">5.1346</cell><cell cols="1" rows="1" role="data">12.1800</cell><cell cols="1" rows="1" role="data">104</cell><cell cols="1" rows="1" role="data">83.259 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">50</cell><cell cols="1" rows="1" role="data">5.4453</cell><cell cols="1" rows="1" role="data">12.9340</cell><cell cols="1" rows="1" role="data">105</cell><cell cols="1" rows="1" role="data">85.992 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">51</cell><cell cols="1" rows="1" role="data">5.7706</cell><cell cols="1" rows="1" role="data">13.7300</cell><cell cols="1" rows="1" role="data">106</cell><cell cols="1" rows="1" role="data">88.735 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">52</cell><cell cols="1" rows="1" role="data">6.1194</cell><cell cols="1" rows="1" role="data">14.5720</cell><cell cols="1" rows="1" role="data">107</cell><cell cols="1" rows="1" role="data">91.367 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">53</cell><cell cols="1" rows="1" role="data">6.4834</cell><cell cols="1" rows="1" role="data">15.4610</cell><cell cols="1" rows="1" role="data">108</cell><cell cols="1" rows="1" role="data">93.815 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">54</cell><cell cols="1" rows="1" role="data">6.8667</cell><cell cols="1" rows="1" role="data">16.4000</cell><cell cols="1" rows="1" role="data">109</cell><cell cols="1" rows="1" role="data">96.039 </cell><cell cols="1" rows="1" role="data"/></row><row rend="align=right" role="data"><cell cols="1" rows="1" role="data">55</cell><cell cols="1" rows="1" role="data">7.2798</cell><cell cols="1" rows="1" role="data">17.3930</cell><cell cols="1" rows="1" role="data">110</cell><cell cols="1" rows="1" role="data">98.356 </cell><cell cols="1" rows="1" role="data"/></row></table><pb n="756"/><cb/></p><p>M. Betancourt deduces several useful and ingenious consequences and applications from this course of experiments. He shews, for instance, that the effect of Steam engines must, in general, be greater in winter than in summer; owing to the different degrees of temperature in the water of injection. And from the very superior strength of the vapour of spirit of wine, over that of water, he argues that, by trying other fluids, some may be found, not very expensive, whose vapour may be so much stronger than that of water, with the same degree of heat, that it may be substituted instead of water in the boilers of Steam-engines, to the great saving in the very heavy expence of fuel: nay, he even declares, that spirit of wine itself might thus be employed in a machine of a particular construction, which, with the same quantity of fuel, and without any increase of expence in other things, shall produce an effect greatly superior to what is obtained from the steam of water. He makes several other observations on the working and improvement of Steamengines.</p><p>Another use of these experiments, deduced by M. Betancourt, is, to measure the height of mountains, by means of a thermometer, immersed in boiling water, which he thinks may be done with a precision equal, if not superior, to that of the barometer. As soon as I had obtained exact results of my experiments, says he, and was convinced that the degree of heat received by water depends absolutely on the pressure upon its surface, I endeavoured to compare my observations with such as have been made on mountains of different heights, to know what is the degree of heat which water can receive when the barometer stands at a determinate height; but from so few observations having been made of this kind, and the different ways employed in graduating instruments, it is difficult to draw any certain consequences from them.</p><p>The first observation which M. Betancourt compared with his experiments, is one mentioned in the Memoirs of the Academy of Sciences, anno 1740, page 92. It is there said, that M. Monnier having made water boil upon the mountain of Canigou, where the barometer stood at 20.18 inches, the thermometer immersed in this water stood at a point answering to 71 degrees of Reaumur: whereas in M. Betancourt's table of experiments, at an equal pressure upon the surface of the water, the thermometer stood at 73.7 degrees. This difference he thinks is owing partly to the want of precision in the observation, and partly to the different method of graduating the thermometer, and the neglect of purging the barometer tube of air.</p><p>M. Betancourt next compared his experiments with some observations made by M. De Luc on the tops of several mountains; in which, after reducing the scales of this gentleman to the same measures as his own, he finds a very near degree of coinoidence indeed. The following table contains a specimen of these comparisons, the instances being taken at random from De Luc's treatise on the Modifications of the Atmosphere. <cb/> <table rend="border"><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="italics">Degrees of Heat in Boiling Water upon the</hi></cell><cell cols="1" rows="1" role="data">Heat of the</cell></row><row role="data"><cell cols="1" rows="1" rend="align=center colspan=4" role="data"><hi rend="italics">Tops of Mountains, observed by De Luc.</hi></cell><cell cols="1" rows="1" role="data">Water in M.</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Places of</cell><cell cols="1" rows="1" role="data">Heat of</cell><cell cols="1" rows="1" role="data">Height of</cell><cell cols="1" rows="1" role="data">Heat of the</cell><cell cols="1" rows="1" rend="align=left" role="data">Betancourt's</cell></row><row rend="align=center" role="data"><cell cols="1" rows="1" role="data">Observation.</cell><cell cols="1" rows="1" role="data">the air.</cell><cell cols="1" rows="1" role="data">the Bar</cell><cell cols="1" rows="1" role="data">Wa. by Th.</cell><cell cols="1" rows="1" rend="align=left" role="data">Experim.</cell></row><row role="data"><cell cols="1" rows="1" role="data">Beaucaire</cell><cell cols="1" rows="1" role="data">14 1/4</cell><cell cols="1" rows="1" role="data">28.248</cell><cell cols="1" rows="1" role="data">80.37</cell><cell cols="1" rows="1" role="data">80.29</cell></row><row role="data"><cell cols="1" rows="1" role="data">Geneva</cell><cell cols="1" rows="1" role="data">12 1/2</cell><cell cols="1" rows="1" role="data">27.056</cell><cell cols="1" rows="1" role="data">79.33</cell><cell cols="1" rows="1" role="data">79.33</cell></row><row role="data"><cell cols="1" rows="1" role="data">Grange Town</cell><cell cols="1" rows="1" role="data">16 1/4</cell><cell cols="1" rows="1" role="data">24.510</cell><cell cols="1" rows="1" role="data">77.11</cell><cell cols="1" rows="1" role="data">77.42</cell></row><row role="data"><cell cols="1" rows="1" role="data">Lans le Bourg</cell><cell cols="1" rows="1" role="data"/><cell cols="1" rows="1" role="data">24-145</cell><cell cols="1" rows="1" role="data">77.18</cell><cell cols="1" rows="1" role="data">77.14</cell></row><row role="data"><cell cols="1" rows="1" role="data">Grange le F.</cell><cell cols="1" rows="1" role="data">15</cell><cell cols="1" rows="1" role="data">24.089</cell><cell cols="1" rows="1" role="data">76.76</cell><cell cols="1" rows="1" role="data">77.09</cell></row><row role="data"><cell cols="1" rows="1" role="data">Grenairon</cell><cell cols="1" rows="1" role="data">10 1/4</cell><cell cols="1" rows="1" role="data">20.427</cell><cell cols="1" rows="1" role="data">73.26</cell><cell cols="1" rows="1" role="data">73.89</cell></row><row role="data"><cell cols="1" rows="1" role="data">Glaciere de B.</cell><cell cols="1" rows="1" role="data"> 6 1/2</cell><cell cols="1" rows="1" role="data">19.677</cell><cell cols="1" rows="1" role="data">72.56</cell><cell cols="1" rows="1" role="data">73.24</cell></row></table> Where it is remarkable, that the difference between the two is of no consequence in such matters.</p><p>Many other advantages might be deduced from the exact knowledge of the effect which the pressure of the atmosphere has upon the heat which water can receive; one of which, M. Betancourt observes, is of too great importance in physics not to be mentioned. As soon as the thermometer became known to philosophers, almost every one endeavoured to find out two fixed points to direct them in dividing the scale of the instrument; having found that those of the freezing and boiling of water were nearly constant in different places, they gave these the preference over all others: but having discovered that water is capable of receiving a greater or less quantity of heat, according to the pressure of the atmosphere upon its surface, they felt the necessity of fixing a certain constant value to that pressure, which it was almost generally agreed should be equal to a column of 28 French inches of mercury. This agreement however did not remove all the difficulties. For instance, if it were required to construct at Madrid a thermometer that might be comparable with another made at Paris, the thing would be found impossible by the means hitherto known, because the barometer never rises so high as 27 inches at Madrid; and it was not certainly known how much the scale of the thermometer ought to be increased to have the point of boiling water in a place where the barometer is at 28 inches. But by making use of the foregoing observations, the thing appears very easy, and it is to be hoped that by the general knowledge of them, thermometers may be brought to great perfection, the accurate use of which is of the greatest importance in physics.</p><p>Besides, without being confined to the height of the barometer in the open air, in a given place, we may regulate a thermometer according to any one assigned heat of water, by means of such an apparatus as M. Betancourt's. For, in order to graduate a thermometer, having a barometer ready divided; it is evident that by knowing, from the foregoing table of experiments, the degree of heat answering to any one expansive force, we can thence assign the degree of the thermometer corresponding to a certain height of the barometer. A determination admitting of great precision, especially in the higher temperatures, where the motion of the barometer is so considerable in respect to that of the thermometer. </p></div1></div0></body> </text> </TEI>